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IISTORY  OF  MATHEMATICS 


VOLUME  II 

SPECIAL  TOPICS  OF 
ELEMENTARY  MATHEMATICS 


BY 
DAVID  EUGENE  SMITH 


,  1?25,  BY  DAVID   EUGENE  SMITH 
ALL  RIGHTS   RESERVED 

825-2 


PREFACE 

As  stated  in  Volume  I,  this  work  has  been  written  chiefly  for  the 
purpose  of  supplying  teachers  and  students  with  a  usable  textbook  on 
the  history  of  elementary  mathematics,  that  is,  of  mathematics 
through  the  first  steps  in  the  calculus.  The  subject  has  come  to  be 
recognized  as  an  important  one  in  the  preparation  of  teachers  and  in 
the  liberal  education  of  students  in  colleges  and  high  schools,  showing, 
as  it  does,  mathematics  as  constantly  progressing  instead  of  being  a 
static  mass  of  knowledge.  Through  a  consideration  of  the  history  of 
the  science  the  student  comes  to  appreciate  the  fact  that  mathematics 
has  continually  adjusted  itself  to  human  needs,  both  material  and 
intellectual ;  and  thus  he  comes  into  sympathy  with  the  effort  to  im- 
prove its  status,  either  adding  to  its  store  through  his  own  discoveries 
or  bettering  the  methods  of  presenting  it  to  those  to  whom  it  is  taught 
in  our  schools. 

In  Volume  I  the  reader  found  a  general  survey  of  the  progress  of 
elementary  mathematics  arranged  by  chronological  periods  with  ref- 
erence to  racial  and  geographical  conditions.  In  this  volume  he  will 
find  the  subject  treated  by  topics.  The  teacher  of  arithmetic  will 
now  see,  in  three  or  four  chapters,  a  kind  of  moving  picture  of  the 
growth  of  his  subject, — how  the  world  has  counted,  how  it  has  per- 
formed the  numerical  operations,  and  what  have  been  the  leading 
lines  of  applications  in  which  it  has  been  interested.  In  geometry  he 
will  see  how  the  subject  arose,  what  intellectual  needs  established  it 
so  firmly,  what  influences  led  to  its  growth  in  various  directions,  and 
what  human  interest  there  is  in  certain  of  the  great  basal  propositions. 
In  algebra  he  will  see,  partly  by  means  of  facsimiles,  how  the  symbol- 
ism has  grown,  how  the  equation  looked  three  thousand  years  ago,  the 
way  its  method  of  expression  has  changed  from  age  to  age,  and  how 
the  science  has  so  adjusted  itself  to  world  needs  as  now  to  be  a  neces- 
sity for  the  average  citizen  instead  of  a  mental  luxury  for  the  selected 

iii 


iv  PREFACE 

few.  He  will  learn  how  the  number  concept  has  enlarged  as  new  needs 
have  manifested  themselves,  and  how  the  world  struggled  with  frac- 
tions and  with  the  mysteries  of  such  artificial  forms  as  the  negative 
and  the  imaginary  number,  and  will  thus  have  a  still  clearer  vision  of 
mathematics  as  a  growing  science.  The  terminology  of  the  subject 
will  arouse  interest ;  the  common  units  of  measure  will  mean  some- 
thing more  than  mere  names ;  the  minutes  and  seconds  of  time  and  of 
angles  will  take  on  a  kind  of  human  aspect ;  and  the  calendar  will 
cease  to  be  the  mystery  that  it  is  to  the  youth.  Trigonometry  will 
have  a  new  interest  to  the  teacher  who  reads  what  Plutarch  tells  of 
the  shadow-reckoning  of  Thales,  and  of  the  independent  origin  of  the 
trigonometry  of  the  sphere ;  and  the  calculus,  which  the  freshman  or 
sophomore  burns  in  accordance  with  time-honored  tradition,  will  be 
seen  to  have  a  history  that  is  both  interesting  and  illuminating.  To 
see  in  its  genetic  aspect  the  subject  that  one  is  teaching  or  studying, 
and  to  see  how  the  race  has  developed  it,  is  oftentimes  to  see  how  it 
should  be  presented  to  the  constantly  arriving  new  generations,  and 
how  it  can  be  made  to  satisfy  their  intellectual  hunger. 

While  the  footnote  is  frequently  condemned  as  being  merely  an 
apology  for  obscurity  or  as  an  exhibition  of  pedantry,  it  would  be 
difficult,  in  a  work  of  this  kind,  to  dispense  with  its  aid.  There  are 
two  principal  justifications  for  such  a  device :  first,  it  enables  an 
author  to  place  the  responsibility  for  a  statement  that  may  be  open  to 
question ;  and,  second,  it  encourages  many  students  to  undertake 
further  study,  either  from  secondary  sources  or,  what  is  more  im- 
portant, from  the  original  writings  of  the  men  who  rank  among  the 
creators  of  mathematics.  With  these  two  points  in  mind,  footnotes 
have  been  introduced  in  such  a  way  as  to  be  used  by  readers  who 
wish  further  aid,  and  to  be  neglected  by  those  who  wish  merely  a  sum- 
mary of  historical  facts.  For  the  student  who  seeks  an  opportunity 
to  study  original  sources  a  slight  introduction  has  been  made  to  this 
field.  The  text  of  the  book  contains  almost  no  quotations  in  foreign 
languages,  the  result  being  that  the  reader  will  not  meet  with  linguistic 
difficulties  in  the  general  narrative.  In  the  notes,  however,  it  is  fre- 
quently desirable  to  quote  the  precise  words  of  an  author,  and  this 
has  been  done  witli  reference  to  such  European  languages  as  are  more 
or  less  familiar.  It  is  not  necessary  to  translate  literally  all  these 


PREFACE  V 

extracts,  since  the  text  itself  sets  forth  the  general  meaning.  Students 
who  have  some  general  knowledge  of  Latin,  French,  or  German  will 
have  little  difficulty,  and  in  many  cases  will  have  much  interest, 
in  seeing  various  statements  in  their  original  form.  For  special  reasons 
a  few  notes  have  been  given  in  Greek,  but  in  every  case  the  meaning 
is  evident  from  the  text. 

In  these  footnotes  and  occasionally  in  the  text  there  have,  in  this 
volume,  been  inserted  a  few  names  of  minor  importance  which  were 
purposely  omitted  in  Volume  I.  These  names  refer  to  certain  arith- 
meticians who  contributed  nothing  to  the  advance  of  mathematics, 
but  who,  through  popular  textbooks,  helped  to  establish  the  symbols 
and  terms  that  are  used  in  elementary  instruction.  In  such  cases 
all  that  has  seemed  necessary  in  the  way  of  personal  information  is  to 
give  the  approximate  dates.  In  the  case  of  names  of  particular  im- 
portance further  information  may  be  found  by  referring  to  the  Index. 

The  difficult  question  of  the  spelling  and  transliteration  of  proper 
names  is  always  an  annoying  one  for  a  writer  of  history.  There  is 
no  precise  rule  that  can  be  followed  to  the  satisfaction  of  all  readers. 
In  general  it  may  be  said  that  in  this  work  a  man's  name  is  given  as 
he  ordinarily  spelled  it,  if  this  spelling  has  been  ascertained.  To  this 
rule  there  is  the  exception  that  where  a  name  has  been  definitely 
Anglicized,  the  English  form  has  been  adopted.  For  example,  it  would 
be  mere  pedantry  to  use,  in  a  work  in  English,  such  forms  as  Platon 
and  Strabon,  although  it  is  proper  to  speak  of  Antiphon  and  Bryson 
instead  of  Antipho  and  Bryso.  When  in  doubt,  as  in  the  case  of 
Heron,  the  preference  has  been  given  to  the  transliteration  which 
most  clearly  represents  the  spelling  used  by  the  man  himself. 

In  many  cases  this  rule  becomes  a  matter  of  compromise,  and  then 
the  custom  of  a  writer's  modern  compatriots  is  followed.  An  exampU 
is  seen  in  the  case  of  Leibniz.   This  spelling  seems  to  be  gainin 
ground  in  our  language,  and  it  has  therefore  been  adopted  instear' 
Leibnitz,  even  though  the  latter  shows  the  English  pronuno* 
better  than  the  former.  Leibniz  himself  wrote  in  Latin,  and  the 
spelled  the  name  variously  in  the  vernacular.  There  seems,  thei 
to  be  no  better  plan  than  to  conform  to  the  spelling  of  those  t 
German  writers  who  appear  to  be  setting  the  standard  that  is  ) 
to  be  followed. 


vi  PREFACE 

In  connection  with  dates  of  events  before  the  Christian  era  the 
letters  B.C.  are  used;  in  connection  with  dates  after  the  beginning  of 
this  era  no  distinguishing  letters  are  added  except  in  a  few  cases  near 
the  beginning  of  the  period,  in  which  the  conventional  letters  A.D. 
have  occasionally  been  inserted  to  avoid  ambiguity.  With  some  hesi- 
tation, but  for  a  purpose  which  seems  valid,  dates  are  frequently 
given  in  parentheses  after  proper  names.  It  is  well  recognized  that 
a  precise  date,  like  1202  after  the  name  Fibonacci,  is  of  no  particular 
value  in  itself.  It  makes  no  difference,  in  ordinary  cases,  whether 
Fibonacci  wrote  his  Liber  Abaci  in  1202,  or  in  1180,  or  in  1220,  or 
whether  abacus  is  spelled  abbacus,  as  in  some  manuscripts,  or  in  the 
more  correct  Latin  form.  On  the  other  hand,  two  things  are  accom- 
plished by  a  free  use  of  such  dates.  In  the  first  place,  a  reader  is 
furnished  with  a  convenient  measuring  instrument ;  he  does  not  have 
to  look  in  the  Index  or  in  a  chronological  table  in  order  to  see  approx- 
imately where  the  particular  writer  belongs  in  the  world's  progress. 
The  casual  reader  may  well  be  pardoned  if  he  does  not  recall  where 
Bede,  Alcuin,  Gerbert,  Jordanus,  Fibonacci,  and  Roger  Bacon  stood 
chronologically  with  respect  to  one  another,  and  in  reading  a  technical 
history  of  this  kind  there  is  no  reason  why  he  should  not  be  relieved 
of  the  trouble  of  consulting  an  index  whenever  he  meets  with  such 
names  as  these.  In  the  second  place,  it  needs  no  psychologist  to  con- 
firm the  familiar  principle  that  the  mind  comes,  without  conscious 
effort,  to  associate  in  memory  those  things  which  the  eye  has  fre- 
quently associated  in  reading.  At  the  risk,  therefore,  of  disturbing 
the  minds  of  those  who  are  chiefly  interested  in  the  literary  aspect  of 
a  general  statement  of  the  progress  of  mathematics,  many  important 
dates  have  been  repeated,  especially  where  they  have  not  appeared  in 
he  pages  immediately  preceding. 

The  extent  of  a  bibliography  in  a  popular  work  of  this  kind  is 
Batter  of  judgment.    It  can  easily  run  to  great  length  if  the  writer 
ibliophile,  or  it  may  receive  but  little  attention.  The  purpose  of 
lists  of  books  for  further  study  is  that  the  student  may  have 
,  to  information  which  the  author  has  himself  used  and  which 
•esves  will  be  of  service  to  the  reader.   For  this  reason  the  sec- 
it  sources  mentioned  in  this  work  are  such  as  may  be  available, 
iany  cases  are  sure  to  be  so,  in  the  libraries  connected  with 


PREFACE  vii 

our  universities,  while  the  original  sources  are  those  which  are  of 
importance  in  the  development  of  elementary  mathematics  or  which 
may  be  of  assistance  in  showing  certain  tendencies. 

The  first  time  a  book  was  mentioned  in  Volume  I,  the  title,  date, 
and  place  of  publication  were  given,  together,  whenever  it  seemed 
necessary,  with  the  abbreviated  title  thereafter  used.  In  general  this 
plan  has  been  followed  in  Volume  II,  at  least  in  the  case  of  important 
works.  To  find  the  complete  title  at  any  time,  the  reader  has  only  to 
turn  to  the  Index  to  find  the  first  reference  to  the  book.  The  abbrevia- 
tion loc.  cit.  (for  loco  citato,  in  the  place  cited)  is  used  only  where 
the  work  has  been  cited  a  little  distance  back,  since  any  more  general 
use  of  the  term  would  be  confusing.  The  symbolism  "I,  7"  has  been 
used  for  "Vol.  I,  p.  7,"  in  order  to  conserve  space,  although  exceptions 
have  been  made  in  certain  ambiguous  cases,  as  in  the  references  to 
Heath's  Euclid,  references  to  Euclid  being  commonly  by  book  and 
proposition,  as  in  the  case  of  Euclid,  I,  47. 

The  standard  works  are  referred  to  as  given  on  pages  xiv-xvi  of 
Volume  I. 

In  the  selection  of  illustrations  the  general  plan  followed  has 
been  to  include  only  such  as  will  be  helpful  to  the  reader  or  likely  to 
stimulate  his  interest;  it  would  be  undesirable  to  attempt  to  give, 
even  if  this  were  possible,  illustrations  from  all  the  important  sources, 
for  this  would  tend  to  weary  him.  On  the  other  hand,  where  the 
reader  has  no  access  to  a  classic  that  is  being  described,  or  even  to  a 
work  which  is  mentioned  as  having  contributed  to  the  world's  prog- 
ress in  some  humbler  manner,  a  page  in  facsimile  is  often  of  value. 
It  is  evident  that  space  does  not  permit  of  the  use  of  such  biblio- 
graphical illustrations  as  those  which  comprise  a  large  part  of  the 
facsimiles  in  the  author's  Kara  Arithmetica. 

In  general  the  illustrations  have  been  made  from  the  original  books 
or  manuscripts  in  the  well-known  and  extensive  library  of  George  A. 
Plimpton,  Esq.,  who  has  been  very  generous  in  allowing  this  material 
to  be  used  for  this  purpose,  or  from  the  author's  collection  of  books, 
manuscripts,  mathematical  portraits  and  medals,  and  early  mathe- 
matical instruments. 

The  scheme  of  transliteration  and  pronunciation  of  proper  names 
is  set  forth  fully  on  pages  xvii-xxii  of  Volume  I.  Since  Arabic,  Persian, 


viii  PREFACE 

Hindu,  Chinese,  and  Japanese  names  are  used  less  frequently  in  this 
volume,  it  will  sufficiently  meet  the  needs  of  the  reader  if  he  refers  to 
the  scheme  there  given. 

As  in  Volume  I,  a  fe^v  topics  for  discussion  or  for  the  personal 
consideration  of  student^  are  suggested  at  the  close  of  each  chapter. 
Specific  questions  have  been  avoided,  the  purpose  being  not  so  much 
to  examine  the  reader  on  the  facts  set  forth  as  to  encourage  him  to 
pursue  his  reading  in  other  works  upon  the  subject.  In  most  cases 
this  reading  will  be  done  in  such  encyclopedias  as  may  be  available, 
and,  preferably,  in  other  histories  also;  but  in  any  case  the  reader 
will  have  his  attention  called  to  a  number  of  general  lines  for  further 
study,  and  he  will  have  the  consciousness  that  the  present  work  is 
merely  an  introduction  to  the  general  subject,  in  which,  it  is  hoped, 
his  interest  has  increased. 

On  account  of  the  extent  of  the  index  to  Volume  I  it  has  not  been 
combined  with  that  of  Volume  II.  It  should  therefore  be  consulted 
in  connection  with  the  index  to  this  volume,  particularly  with  respect 
to  biographical  and  bibliographical  references.  Since,  in  many  cases, 
textbooks  are  mentioned  so  frequently  as  to  render  a  complete  list  so 
long  as  to  be  burdensome  to  the  reader,  thus  defeating  its  purpose, 
such  works  are  included  only  when  the  author  is  not  mentioned  in 
Volume  I  and  when  the  work  is  of  such  importance  as  to  make  the 
reference  valuable. 

The  author  wishes  to  express  his  appreciation  of  the  aid  rendered 
by  various  friends  in  reading  the  proofsheets  of  both  volumes,  and 
especially  by  the  late  Herr  Gustaf  Enestrom  of  Stockholm,  by  Pro- 
fessor R.  C.  Archibald,  by  Mr.  Jekuthial  Ginsburg  of  New  York,  and 

by  Captain  E.  L.  Morss  of  Boston. 

DAVID  EUGENE  SMITH 


CONTENTS 

CHAPTER  PAGE 

I.  DEVELOPMENT  OF  THE  ARITHMETICA 

1.  GENERAL  SURVEY i 

2.  THE  SEVEN  LIBERAL  ARTS 2 

3.  EARLY  WRITERS  ON  NUMBER  THEORY 4 

4.  NAMES  FOR  ARITHMETIC 7 

5.  ELEMENTARY  CLASSIFICATIONS  OF  NUMBER       ...  n 

6.  UNITY 26 

7.  LATER  DEVELOPMENTS 29 

TOPICS  FOR  DISCUSSION 31 

II.  LOGISTIC  OF  NATURAL  NUMBERS 

1.  FUNDAMENTAL  OPERATIONS 32 

2.  READING  AND  WRITING  NUMBERS 36 

3.  ADDITION 88 

4.  SUBTRACTION 94 

5.  MULTIPLICATION 101 

6.  DIVISION 128 

7.  ROOTS 144 

8.  CHECKS  ON  OPERATIONS  . 151 

TOPICS  FOR  DISCUSSION 155 

III.  MECHANICAL  AIDS  TO  CALCULATION 

1.  THE  ABACUS 156 

2.  FINGER  RECKONING 196 

3.  MODERN  CALCULATING  MACHINES 202 

TOPICS  FOR  DISCUSSION 2< 

ix 


X  CONTENTS 

CHAPTER  PACK 

IV.  ARTIFICIAL  NUMBERS 

1.  COMMON  FRACTIONS 208 

2.  SEXAGESIMAL  FRACTIONS 228 

3.  DECIMAL  FRACTIONS    . 235 

4.  SURD  NUMBERS 251 

5.  NEGATIVE  NUMBERS 257 

6.  COMPLEX  NUMBERS 261 

7.  TRANSCENDENTAL  NUMBERS 268 

TOPICS  FOR  DISCUSSION 269 

V.  GEOMETRY 

1.  GENERAL  PROGRESS  OF  ELEMENTARY  GEOMETRY   .      .  270 

2.  NAME  FOR  GEOMETRY 273 

3.  TECHNICAL  TERMS  OF  EUCLIDEAN  GEOMETRY       .      .  274 

4.  AXIOMS  AND  POSTULATES 280 

5.  TYPICAL  PROPOSITIONS  OF  PLANE  GEOMETRY  .      .      .  284 

6.  TYPICAL  PROPOSITIONS  OF  SOLID  GEOMETRY     .      .      .  291 

7.  THE  THREE  FAMOUS  PROBLEMS 297 

8.  ANALYTIC  GEOMETRY 316 

9.  MODERN  GEOMETRY 331 

10.  PERSPECTIVE  AND  OPTICS 338 

11.  INSTRUMENTS  IN  GEOMETRY 344 

12.  THE  PROBLEM  OF  EARTH  MEASURE 368 

TOPICS  FOR  DISCUSSION 377 

VI.  ALGEBRA 

1.  GENERAL  PROGRESS  OF  ALGEBRA 378 

2.  NAME  FOR  ALGEBRA 386 

3.  TECHNICAL  TERMS 393 

4.  SYMBOLS  OF  ALGEBRA 395 

5.  FUNDAMENTAL  OPERATIONS 


CONTENTS  xi 

CHAPTER  PAGF, 

6.  CONTINUED  FRACTIONS 418 

7.  THE  WRITING  OF  EQUATIONS 421 

8.  THE  SOLUTION  OF  EQUATIONS 435 

9.  DETERMINANTS            .    * 475 

10.  RATIO,  PROPORTION,  AND  THE  RULE  OF  THREE    .      .  477 

11.  SERIES 494 

12.  LOGARITHMS 513 

13.  PERMUTATIONS,  COMBINATIONS,  PROBABILITY   .      .      .  524 

TOPICS  FOR  DISCUSSION 531 

VII.  ELEMENTARY  PROBLEMS 

1.  MATHEMATICAL  RECREATIONS 532 

2.  TYPICAL  PROBLEMS 536 

3.  COMMERCIAL  PROBLEMS 552 

4.  APPLICATIONS  OF  ALGEBRA 582 

5.  MAGIC  SQUARES 591 

TOPICS  FOR  DISCUSSION 599 

VIII.  TRIGONOMETRY 

1.  GENERAL  DEVELOPMENT  OF  TRIGONOMETRY    .     .      .  600 

2.  TRIGONOMETRIC  FUNCTIONS 614 

3.  TRIGONOMETRIC  TABLES         623 

4.  TYPICAL  THEOREMS 628 

TOPICS  FOR  DISCUSSION 633 

IX.  MEASURES 

1.  WEIGHT 634 

2.  LENGTH 640 

3.  AREAS 644 

4.  CAPACITY 644 


xii  CONTENTS 

CHAPTKR  PAGE 

5.  VALUE 645 

6.  METRIC  SYSTEM 648 

7.  TIME 651 

TOPICS  FOR  DISCUSSION 675 

X.  THE  CALCULUS 

1.  GREEK  IDEAS  OF  A  CALCULUS     .......  676 

2.  MEDIEVAL  IDEAS  OF  THE  CALCULUS 684 

3.  MODERN   FORERUNNERS  OF  THE  CALCULUS       .      .      .  685 

4.  NEWTON  AND  LEIBNIZ 692 

5.  JAPAN 701 

TOPICS  FOR  DISCUSSION 703 

INDEX 705 


HISTORY  OF  MATHEMATICS 

SPECIAL  TOPICS  OF  ELEMENTARY 
MATHEMATICS 

CHAPTER  I 

DEVELOPMENT  OF  THE  ARITHMETICA 

i.  GENERAL  SURVEY 

yr  .  \  . 

Nature  of  Arithmetica.  (As  stated  in  the  Preface)  it  is  the 

purpose  of  this  volume  to  set  forth  in  considerable  detail  the 
important  steps  in  the  historical  development  of  the  several 
branches  of  elementary  mathematics.  One  of  these  branches  is 
now  known  as  arithmetic,  a  name  which,  as  commonly  under- 
stood in  the  English-speaking  world,  has  little  or  no  relation  to 
the  arithmetic  of  the  ancients.  In  recent  times  the  word  has 
acquired  the  meaning  given  by  the  Greeks  and  Romans  to  logis- 
tic, or  the  art  of  computation,  a  much  more  humble  discipline 
than  that  which  they  called  arithmetic." 

"  In  order  tq^make  the  distinction  clear,  the  present  chapter 
will  set  forth  ^  sufficient  number  of  simple  details  of  the  ancient 
arithmetic  to  enable  the  student  to  form  an  idea  of  its  general 
nature,  and  the  second  chapter  will  consider  the  development  of 
that  elementary  art  which  now  bears  the  ancient  name.  It  will 
be  seen  that  the  science  which  formerly  appropriated  the  title 
was  not  related  to  ordinary  calculation  but  was  a  philosophical 
study  dealing  with  such  properties  as  might  now  find  place  in  a 
course  in  the  theory  of  numbers  if  the  latter  had  not  outgrown 
most  of  these  simple  number  relations  and  become  a  subject  for 
the  university  student. 


2  THE  SEVEN  LIBERAL  ARTS 

Modern  Theory.  The  modern  theory  of  numbers  has  so  little 
direct  relation  to  elementary  mathematics  that  its  history  need 
only  be  referred  to  briefly  in  this  volume.1  Certain  features  like 
prime  and  composite  numbers,  polygonal  numbers  (such  as 
squares),  and  solid  numbers  (such  as  cubes)  are  still  found  in 
elementary  mathematics,  however,  and  these  features  render 
essential  a  brief  statement  concerning  the  ancient  arithmetic. 
In  order  to  explain  the  position  of  this  science  in  the  ancient 
scheme  of  learning,  it  is  desirable  to  speak  first  of  the  general 
range  of  knowledge  according  to  the  Greek  schools  of  philoso- 
phy, and  to  distinguish  between  arithmetica,  the  classical  theory 
of  numbers,  and  arithmetic,  the  modern  art  of  computation. 

2.  THE  SEVEN  LIBERAL  ARTS 

The  Sevenfold  Division.  As  stated  in  Volume  I,  three  and 
seven  have  been  the  chief  among  mystic  numbers  in  all  times 
and  among  all  peoples.  Many  reasons  have  been  assigned  for 
this  universal  habit  of  the  race,  most  of  them  manifestly  fanci- 
ful, and  possibly  no  reason  can  be  adduced  that  will  command 
the  general  approval  of  scholars.  If,  however,  we  omit  the  num- 
ber five,  which  was  often  used  as  a  primitive  radix  and  thus  lost 
its  element  of  mystery,  a  fairly  satisfactory  explanation  is  found 
in  the  fact  that  three  and  seven  are  the  first  prime  numbers, — 
odd,  unfactorable,  unconnected  with  any  common  radix,  pos- 
sessed of  various  peculiar  properties,  and  thus  of  a  nature  to 
attract  attention  in  the  period  of  superstition  and  mysticism. 

One  of  the  many  results  of  this  veneration  for  these  numbers 
is  seen  in  the  fact  that  the  ancients  numbered  seven  great 
branches  of  learning,  just  as  they  numbered  the  Seven  Wonders 
of  the  World  and  the  Seven  Wise  Men  of  Greece.  They  sepa- 
rated these  branches  into  two  groups,  four  studies  making  up 
the  domain  of  science  as  recognized  by  the  Pythagoreans,  and 
three  constituting  the  nonscientific  domain.  Plato2  spoke  of 

1The  student  will  find  it  elaborately  treated  in  L.  E.  Dickson,  History  of 
the  Theory  of  Numbers,  3  vote.,  Washington,  1919-1923;  hereafter  referred  to 
as  Dickson,  Hist.  Th.  Numb. 

2  Republic,  IX.   See  also  Aristotle's  Politics,  VIII,  i. 


MEANING  OF  LIBERAL  ARTS  3 

the  liberal  arts  and  separated  them  into  two  groups,  but  he  did 
not  limit  them  to  any  definite  number.  The  scientific  group, 
consisting  of  arithmetic,  geometry,  spherics,  and  music,  con- 
stituted the  ancient  domain  of  mathematics. 

The  Seven  Liberal  Arts.  It  was  probably  in  the  work  of 
Capella  (c.  460),  that  the  seven  liberal  arts  were  first  distinctly 
specified.1  These  seven  arts  were  thenceforth  looked  upon  as 
necessary  to  the  education  of  free  men  (liberi).  They  were 
then  separated  into  the  quadrivium,2  constituting  the  Pythag- 
orean group,  and  the  trivium,3  made  up  of  grammar,  dialectics, 
and  rhetoric.4 

The  names  of  the  seven  arts  are  fairly  descriptive  of  the  sub- 
jects represented,  with  the  exception  of  spherics,  which  related 
to  mathematical  astronomy;  music,  which  related  only  to  the 
theory  of  harmony;5  and  arithmetic,  which  had  little  in  com- 
mon with  the  subject  known  in  English  by  this  name. 

^Varro  (ist  century  B.C.)  wrote  a  treatise  on  the  "nine  liberal  disciplines," 
but  the  work  is  not  extant.  Capella  introduced  the  liberal  arts  as  the  brides- 
maids at  the  marriage  of  Philology  and  Mercury.  Cassiodorus  (c.  47o-c.  564) 
placed  the  limit  definitely  at  seven  because  of  the  seven  pillars  in  the  Temple 
of  Wisdom  (Proverbs,  ix,  i). 

2  In  medieval  Latin  also  written  quadruvium,  the  quadruplex  via,  as  some 
writers  have  it.   The  term  in  its  literal  meaning  is  found  as  early  as  Juvenal. 
In  its  technical  educational  meaning  it  is  used  by  Cassiodorus. 

3  Also  written  truvium. 

4  As   "Hugnitio    natione    tuscus,    civis   pisanus,    episcopus    ferrariensis,"    to 
quote  a  medieval  record,  has  it:   "Et  uero  quia  gramatica  dialecta  rethorica 
dicuntur   triuuium   quadam   similitudine   quasi   triplex   uia   ad   idem    idest   ad 
eloquentiam  arismethica.  musica.  geometria.  astronomia.  quadam  simili  simili- 
tudine dicuntur  quadriuuium  quasi  quadruplex  uia  ad  idem  idest  ad  sapientiam." 
See  also  the  well-known  verse  quoted  in  Volume  I,  page  180. 

5 As  an  old  Latin  MS.  has  it: 

Musicorum  et  cantorum  magna  est  distantia: 
Isti  dicunt  illi  sciunt  quae  componit  musica. 

The  distinction  is  well  set  forth  in  B.  Veratti,  De'  Matematici  Italiani  anteriori 
all'  invenzione  della  stampa,  p.  4  (Modena,  1860).  See  also  P.  Tannery,  "Du 
role  de  la  musique  grecque  dans  le  developpement  de  la  mathematique  pure," 
Bibl.  Math.,  Ill  (3),  161  ;  E.  Narducci,  "Di  un  codice  .  .  .  dell' opera  di  Giorgio 
Pachimere :  wepl  rQv  T€<T<r6,pwv  fj.a0rjfji<LT<jjv"  Rendiconti  della  R.  Accad.  del  Lincei, 
Rome,  VII  (1891),  191. 


4  EARLY  WRITERS  ON  NUMBER  THEORY 

3.  EARLY  WRITERS  ON  NUMBER  THEORY 

Origin  of  the  Theory.  There  is  no  definite  trace  of  the  study 
of  the  theory  of  numbers  before  the  time  of  Thales  (c.  600  B.C.). 
Tradition  says  that  this  philosopher,  filled  with  the  lore  of  the 
Egyptians  and  probably  well  informed  concerning  the  mysticism 
of  the  Babylonians,  taught  certain  of  the  elementary  properties 
of  numbers  in  the  Ionic  School,  of  which  he  was  the  founder. 
Such  meager  knowledge  as  he  had  he  imparted  to  his  bril- 
liant disciple,  Pythagoras  (c.  540  B.C.),  who  thereupon  resorted 
to  the  priests  of  Egypt  and  probably  of  Babylon  for  further 
light.  In  the  school  which  he  established  at  Crotona,  in  south- 
ern Italy  (Magna  Grsecia),  he  elaborated  the  doctrines  of  his 
teachers,  including  ideas  which  are  distinctly  Oriental,  and  made 
the  first  noteworthy  beginning  in  the  theory  of  arithmetica. 

Little  by  little,  first  among  the  Pythagoreans  and  then  in 
other  schools  of  philosophy,  the  subject  grew,  a  little  being 
added  here  and  a  little  there,  until  the  time  finally  became  ripe 
for  the  appearance  of  treatises  in  which  the  accumulated 
knowledge  could  be  systematically  arranged. 

Books  on  the  Theory.  The  first  successful  effort  in  the 
preparation  of  an  expository  treatise  on  the  subject  was  made 
by  Euclid  (c.  300  B.C.),  who  is  often  known  only  as  a  geometer 
but  who  showed  great  genius  in  systematizing  mathematical 
knowledge  in  other  important  lines  as  well.  In  his  Elements  he 
devotes  Books  II,  V,  VII,  VIII,  IX,  X  (in  whole  or  in  part)  to 
the  theory  of  numbers  or  to  geometric  propositions  closely  re- 
lated thereto,  and  includes  such  propositions  as  the  following : 

If  four  numbers  are  proportional,  they  are  also  proportional 
alternately  (VII,  13). 

If  two  numbers  are  prime  to  two  numbers,  both  to  each,  their 
products  also  will  be  prime  to  one  another  (VII,  26). 

If  a  square  number  does  not  measure  a  square  number,  neither 
will  the  side  measure  the  side ;  and  if  the  side  does  not  measure  the 
side,  neither  will  the  square  measure  the  square  (VIII,  16). 

If  an  odd  number  measures  an  even  number,  it  will  also  measure 
the  half  of  it  (IX,  30). 


NICOMACHUS  AND  THEON  5 

The  next  worker  in  this  field  was  that  interesting  dilet- 
tante in  matters  mathematical,  Eratosthenes  (c.  230  B.C.),  who 
worked  on  a  method  of  finding  prime  numbers1  by  sifting  out 
the  composite  numbers  in  the  natural  series,  leaving  only  primes. 
This  he  did  by  canceling  the  even  numbers  except  2,  every  third 
odd  number  after  3,  every  fifth  odd  number  after  5,  and  so  on, 
the  result  being  what  the  ancient  writers  called  the  sieve.2 

His  friend  and  sometime  companion  Archimedes  (c.  225  B.C.) 
did  little  with  the  theory  of  arithmetica,  but  made  an  effort  to 
improve  upon  the  Greek  system  of  numbers,3  his  plan  involving 
the  counting  by  octads  (io8),  in  which  he  proceeded  as  far  as 
io52,  and  making  use  of  a  law  which  would  now  be  expressed  by 
such  a  symbolism  as  a"'an=.  a9"*",  although  he  made  no  specific 
mention  of  this  important  theorem. 

It  was  to  the  commentary  on  the  Timceus  of  Plato,  written  by 
Poseidonius  (c.  77  B.C.),  that  the  Greeks  invariably  went  for 
their  knowledge  of  the  number  theories  of  the  Pythagoreans.4 
This  is  seen  in  the  fact  that  the  phraseology  used  by  such  writers 
as  Theon  of  Smyrna  (c.  125)  and  Anatolius  (c.  280),  in  speak- 
ing of  this  subject,  is  simply  a  paraphrase  of  that  used  by 
Poseidonius. 

Nicomachus,  and  Theon  of  Smyrna.  The  first  noteworthy 
textbook  devoted  to  arithmetica  was  written  by  Nicomachus 
(c.  100),  a  Greek  resident  of  Gerasa  (probably  the  modern 
Jerash,  a  town  situated  about  fifty-six  miles  northeast  of  Jeru- 
salem). He  was  not  an  original  mathematician,  but  he  did 
for  the  theory  of  numbers  what  Euclid  had  done  for  elementary 
geometry  and  Apollonius  (c.  225  B.C.)  for  conic  sections,  —  he 
summarized  the  accumulated  knowledge  in  his  subject.  In  his 
work  are  found  such  statements  as  the  following:  "Now  fur- 


.  I,  p.  109.  2K6ffKivov  (kos'kinon),  Latin,  cribrum. 

3  Vol.  I,  p.  113.  See  also  his  ^a/u^-njs  (psammi'tes,  Latin  arenarius,  "sand 
reckoner"),  Archimedis  Opera  Omnia,  ed.  Heiberg  (Leipzig,  1880-1915),  with 
revisions. 

4F.  E.  Robbins,  "Posidonius  and  the  Sources  of  Pythagorean  Arithmology," 
Classical  Philology,  XV  (1920),  309.  On  Plato's  appreciation  of  the  value  of 
this  kind  of  work  see  F.  Cajori,  "  Greek  Philosophers  on  the  Disciplinary  Value 
of  Mathematics,"  The  Mathematics  Teacher  (December,  1920),  p.  57. 

ii 


6  EARLY  WRITERS  ON  NUMBER  THEORY 

thermore  every  square  upon  receiving  its  own  side  becomes 
heteromecic  ;  or,  by  Zeus,  on  being  deprived  of  its  own  side."1 
The  next  writer  of  note  was  Theon  of  Smyrna  (c.  125).  He 
added  several  new  propositions  to  the  theory,  two  of  them  being 
of  special  interest  :  (  i  )  If  n  be  any  number,  n2  or  n2  —  i  is  divisi- 
ble by  3,  by  4,  or  by  both  3  and  4  ;  and  if  n2  is  divisible  by  3  and 
not  by  4,  then  rr  —  i  is  divisible  by  4.  (2)  If  we  arrange  two 
groups  of  numbers  as  follows  : 

n^  =  i  -f  o  ^=1+0=1 

;/a=  i  +  i  d^    2+1=    3 

«8=2  +  3  rfg=    4  +  3-    7 

»4=5  +  7  </4=io  +  7  =  17 

nr  ~  nr  -  1  +  dr  „!  <^  =  2  ;zr  _  1  +  dr  _  l 

then  d2  is  of  the  form  2  «2  ±  i  ;  for  example,  d%=  i  =  2  «*  —  i, 
^/2a  =  9  =  2  ;/22  +!,•••.  The  numbers  ^4  were  called  by  Theon 
diameters.  It  is  interesting  to  observe  a  fact  unknown  to  him, 
namely,  that  the  ratios  rft  1^=1,  */2  :  ;/2=  f,  rf3  :  «8=  |,  •  •  •  are 
the  successive  convergents  of  the  continued  fraction 


and  hence  approach  nearer  and  nearer  the  square  root  of  2. 

Boethius.  Boethius  (c.  510)  appropriated  the  knowledge  of 
such  writers  as  Euclid,  Nicomachus,  and  Theon,  incorporating 
it  in  his  work  De  institutione  arithmetica  libri  duo  and  produc- 
ing a  textbook  that  was  used  in  all  the  important  schools  in  the 
Middle  Ages.  It  is  the  source  with  which  a  student  may  advan- 
tageously begin  his  study  of  this  subject. 

Later  Writers.  The  most  noteworthy  writer  on  the  subject 
in  the  medieval  period  is  Fibonacci  (1202),  and  with  respect  to 
him  and  subsequent  writers,  all  of  whom  have  been  considered 
in  Volume  I,  we  shall  later  speak  in  detail  as  necessity  arises. 

1  Introduction,  XX.  See  G.  Johnson,  The  Arithmetical  Philosophy  of  Nicom- 
achus of  Gerasa,  Lancaster,  1916,  hereafter  referred  to  as  Johnson,  Nicomachus. 
The  meaning  is  that  x2  ±  x  is  not  a  square  but  a  heteromecic  or  oblong  number. 


ARITHMETIC  AND  LOGISTIC  7 

4.  NAMES  FOR  ARITHMETIC 

Arithmetic  and  Logistic.  The  ancient  Greeks  distinguished 
between  arithmetic,1  which  was  the  theory  of  numbers  and  was 
therefore  even  more  abstract  than  geometry,2  and  logistic,3 
which  was  the  art  of  calculating.  These  two  branches  of  the 
study  of  numbers  continued  as  generally  separate  subjects  until 
the  time  of  printing,  although  often  with  variations  in  their 
names;  but  about  the  beginning  of  the  i6th  century  the  more 
aristocratic  name  of  "arithmetic"  came  to  be  applied  to  both 
disciplines.  This  use  of  the  term  was  not  universal,  however, 
and  even  today  the  Germans  reserve  the  word  Arithmetik  for  the 
theoretical  part  of  the  science  as  seen  in  the  operations  in  alge- 
bra, using  the  word  Rechnung  for  the  ancient  logistic.4  Various 
writers,5  preserved  the  word  "logistic"  in  the  i6th  century,  but 
in  the  older  sense  it  generally  dropped  out  of  use  thereafter. 

From  the  fact  that  computations  were  commonly  performed 
on  the  abacus,  the  name  of  this  instrument  was  used  in  the  early 
Middle  Ages  as  a  synonym  of  logistic.  Finally,  however,  the 
word  "abacus"  came  to  mean  any  kind  of  elementary  arith- 
metic,6 and  this  usage  obtained  long  after  printing  was  invented.7 

In  the  Middle  Ages  the  name  "arithmetic"  was  apparently 
not  in  full  favor,  perhaps  because  it  was  not  of  Latin  origin. 
Thus,  in  a  manuscript  attributed  to  Gerbert  the  word  is  spoken 
of  as  Greek,  the  Latin  being  "numerorum  scientia."* 

i'Api0MTtKri  (arithmetike'} ,  from  &pte/*6s  (arithmos1),  number.  It  passed  over 
into  Latin  as  arithmetica. 

2"Est  enim  Arithmetices  subjectum  purius  quiddam  &  magis  abstractum,  quam 
subjectum  Geometriae"  (J.  Wallis,  Opera  Mathematica,  I,  18  (Oxford,  1695)). 

3  Aoyio-TiK-/)  (logistike'},  which  passed  over  into  Latin  as  logistica. 

4  Compare  also  the  French  calcul. 

nj.  Noviomagus,  De  Numeris  libri  duo  (1539);  Buteo  (Lyons,  i559)> 
Schoner  edition  of  Ramus  (1586);  "Logistica  quam  uulgo  uocant  algoristicam 
et  algorismum"  (MS.  notes  in  the  1558  edition  of  Gemma  Frisius,  in  Mr.  Plimp- 
ton's library).  For  biographical  information  relating  to  such  writers  as  are  of 
particular  importance,  see  the  Index  of  Volume  I. 

6 As  in  Fibonacci's  Liber  Abaci  (1202). 

7  See  Ram  Arithmetica  for  many  works  bearing  such  titles  as  Libro  d'  abacho. 

8"Graece  Arithmetica,  latine  dicitur  numerorum  scientia,"  from  the  colo- 
phon of  the  "Liber  subtilissimus  de  arithmetica."  See  C.  F.  Hock,  Gerberto  o 
sia  Silvestro  U  Papa  .  .  .  trad,  del  .  .  .  Stelzi,  p.  206  (Milan,  1846). 


8  NAMES  FOR  ARITHMETIC 

xX".'. 

v  Vicissitudes  of  the  Term.  'The  word  " arithmetic,"  like  most 
other  words,  has  undergone  many  vicissitudes.  In  the  Middle 
Ages,  through  a  mistaken  idea  of  its  etymology,  it  took  an  extra 
r,  as  if  it  had  to  do  with  "metric."1  So  we  find  Plato  of  Tivoli, 
in  his  translation  (1116)  of  Abraham  Savasorda,  speaking  of 
"Boetius  in  arismetricis."2  The  title  of  the  work  of  Johannes 
Hispalensis,  a  few  yearsf  later  (c.  1140),  is  given  as  "Arismet- 
rica,"  and  fifty  years  later  than  this  we  find  Fibonacci  dropping 
the  initial  and  using  the  form  "Rismetrica."3  The  extra  r  is 
generally  found  in  the  Italian  literature  until  the  time  of  print- 
ing.4 From  Italy  it  passed  over  to  Germany,  where  it  is  not 
uncommonly  found  in  the  books  of  the  i6th  century,5  and  to 
France,  where  it  is  found  less  frequently.6  The  ordinary  varia- 
tions in  spelling  have  less  significance,  merely  illustrating,  as  is 
the  case  with  many  other  mathematical  terms,  the  vagaries  of 
pronunciation  in  the  uncritical  periods  of  the  world's  literatures/ 

i/'v 

1  Greek  /jLtrpov,  a  measure,  as  in  "metre"  and  "metrology." 

2  Abhandlungen,  XII,  16.   For  such  abridged  forms  see  the  Index  of  Volume  I. 

3  This  is  in  one  of  the  MSS.  formerly  owned  by  Boncompagni.   See  the  sale 
catalogue  of  his  library,  p.  104.   Fibonacci  (1202)  commonly  used  "abacus." 

4 -E.g.,  see  the  "Brani  degli  Annali  Decemvirali  posseduti  dall'  archive  De- 
cemvirale  di  Perugia,"  in  Boncompagni's  Bullettino,  XII,  432 ;  E.  Narducci, 
Catalogo  di  Manoscritti,  2d  ed.  (Rome,  1892),  No.  56,  p.  26;  hereafter  referred 
to  as  Narducci,  Catalogo  Manosc. 

5 E.g.,  "Die  Kunst  Arismetrica  die  aller  edelst  vnder  den  sybe  freyen 
klinsten,"  Kobel,  1514.  A  MS.  in  ScheubePs  (c.  1550)  handwriting  in  the 
Columbia  University  Library  has  "de  Arrismetris."  There  is  also  a  MS.  copy 
made  c.  1515  in  Rome,  by  a  Swedish  savant,  Peder  Mansson,  from  the  Mar- 
garita phylosophica  of  Gregorius  Reisch  (1503),  in  which  the  form  "Aris- 
metrice"  is  given.  See  Bibl.  Math.,  II  (2),  17. 

6  So  in  a  MS.  written  by  Rollandus  c.  1424  (see  Rara  Arithmetic^  p.  446) 
the  form  "arismetica"  is  usually  given,  but  the  form  "  arismetrica "  also  appears. 
In  an  unpublished  MS.  entitled  "traicte  d'Arismetricque  .  .  .  faite  et  compill6  A 
paris  en  Ian  mil  475"  (for  1475)  there  is  this  curious  etymology:  "Arismeticve 
est  vne  des  sept  ars  liberaulx  &  la  premiere  des  quatre  ars  Mathematique  En  la 
quelle  est  la  vertus  de  nombrer.   Et  est  dicte  de  ares  Nom  grec  qui  est  en  latin 
Virtus  Et  de  menos  aussi  nom  grec  qui  est  en  Latin  numerus  parquoy  est  dicte 
Vertus  de  Nombre."    E.  Narducci,  Catalogo  Manosc.,  No.  603,  p.  395. 

7  Thus,  we  have  "arimmetica"  throughout  Zuchetta's  work  of  1600  (see  Rara 
Arithmeticat  p.  425) ;  " eritmeticha "  in  a  i7th  century  MS.  (see  Narducci's  Cata- 
logo Manosc.,  No.  446,  i,  p.  267);  " aristmeticque "  in  an  anonymous  French 
work,  Paris,  1540;  "Alchorismi  de  pratica  Aricmetica,"  in  a  MS.  of  Sacrobosco, 
Ppncompagni  sale  catalogue,  No,  645, 


ALGORISM  g 

Origin  of  Algorism.  From  the  fact  that  the  arithmetic  of 
al-Khowarizmi  (c.  825)  was  translated  into  Latin  as  liber 
Algorismi  (the  book  of  al-Khowdrizmi),  arithmetic  based  on 
the  Hindu-Arabic  numerals,  more  especially  those  that  made 
use  of  the  zero,  came  to  be  called  algorism  as  distinct  from  the 
theoretical  work  with  numbers  which  was  still  called  arithmetic.1 
Since  al  often  changes  to  an  in  French,  we  have  "augrisme" 
and  "augrime," — forms  which  were  carried  over  to  England  as 
"augrim,"2  later  reverting  to  "algorism"3  or  the  less  satisfactory 
form  of  "algorithm."4 

The  prefix  al  was  dropped  from  this  word  by  most  Spanish 
writers,  giving  such  forms  as  "guarisma"5  and  "guarismo,"' 
and  in  other  countries  there  were  many  variations  that  were 
quite  as  curious.7 

The  word  troubled  many  of  the  early  Latin  writers,  and 
various  fanciful  etymologies  were  suggested,  the  best  conjec- 
ture being  that  of  Sacrobosco  that  it  came  from  Algus  or  Argus, 

1Thus,  that  part  of  the  Rollandus  MS.  (c.  1424)  relating  to  the  theory  is 
referred  to  in  the  phrase  "  Arismetrice  pars  primo  tractanda  est  speculatiua," 
while  the  other  part  is  called  "algorismus."  See  also  M.  Chaslcs,  Comptes 
rendus,  XVI,  162. 

2"U  ouer  the  wiche  degrees  ther  ben  nowmbres  of  augrym ;  .  .  .  &  the  nombres 
of  the  degres  of  tho  signes  ben  writen  in  Augrim."  Chaucer's  Astrolabe,  ed. 
Skeat,  p.  5. 

"Although  a  sypher  in  augrim  have  no  might  in  significacion  of  it-selve, 
yet  he  yeveth  power  in  significacion  to  other."  Chaucer,  The  Testament  of 
Love,  ed.  Skeat,  Bk.  II,  chap.  vii. 

3Thus,  Recorde  (c.  1542)  in  his  Grovnd  of  Arts  (as  spelled  in  the  1646 
edition)  :  "Some  call  it  Arsemetrick,  and  some  Augrime.  .  .  .  Both  names  are 
corruptly  written :  Arsemetrick  for  Arithmetick,  as  the  Greeks  call  it,  and 
Augrime  for  Algorisme,  as  the  Arabians  found  it."  1646  ed.,  p.  8. 

4One  eccentric  English  writer,  Daniel  Penning  (1750),  attempted  to  dis- 
tinguish algorithm,  as  first  principles,  from  algorism,  as  the  practice  of  these 
principles. 

5  As  in  the  Spanish  Suma  de  Arithmetica  of  Gaspard  de  Texeda,  Valladolid, 
1546.  The  separate  word  al  or  el  (the)  was  prefixed,  however,  and  the  form 
algoritmo  is  still  preserved. 

6".  .  .  de  vn  Filosofo  llamado  Algo,  y  por  aquesta  causa  fue  llamada  el 
Guarismo"  (Santa-Cruz,  a  Spanish  writer,  1594);  but  see  Kara  Arithmetica, 
p.  407. 

7 " Arismethique  qui  vulgayrement  est  appellee  argorisme"  (E.  de  la  Roche, 
a  French  writer,  1520).  We  also  find  such  forms  as  alkauresmus  and  alcho- 
charithmus  in  various  MSS.  of  the  same  period. 


io  NAMES  FOR  ARITHMETIC 

a  certain  philosopher,  this  being  merely  a  corruption  of  al- 
Khowarizmi.1  It  was  not  until  1849  that  the  true  etymology 
was  again  discovered.2 

The  Etymology  early  Recognized  and  Forgotten.  Very  likely 
the  etymology  of  the  term  "algorism"  was  known  to  such  early 
translators  or  writers  as  Johannes  Hispalensis3  (c.  1140)  and 
Adelard  of  Bath4  (c.  1120).  By  the  following  century,  how- 
ever, al-Khowarizmi  was  quite  forgotten  by  such  Latin  writers 
as  Sacrobosco5  (c.  1250)  and  Bacon  (c.  1250).  From  that  time 
on  we  have  the  word  loosely  used  to  represent  any  work  related 
to  computation  by  modern  numerals6  and  also  as  synonymous 

i-So  we  have  Chaucer's  expression, 

Thogh  Argus  the  noble  covnter 
Sete  to  rekene  in  hys  counter. 

Dethe  Blaunche  (c.  1369) 

This  derivation  was  followed  by  various  writers,  such  as  Santa-Cruz  (1594), 
Cataldi  (1602),  and  Tartaglia  (1556  ed.,  I,  fol.  3,  r.).  Of  the  other  fanciful 
etymologies  the  following  may  be  of  interest:  argris  (Greek)  4-  mos  (custom)  ; 
algos  (Greek  for  "white  sand")  +  ritmos  (calculation);  algos  (art)  +  rado 
(number) ;  Algorus,  the  name  of  a  Hindu  scholar;  Algor,  a  king  of  Castile.  See 
A.  Favaro,  Boncompagni's  Bullettino,  XII,  115;  M.  Cantor,  Mathematische 
Beitrdge  zum  Kulturleben  der  Volker,  p.  267  (Halle,  1863);  C.  I.  Gerhardt, 
Ueber  die  Entstehung  .  .  .  des  dekadhchen  Zahlensy stems.  Prog.,  p.  26,  n. 
(Salzwedel,  1853);  K.  Hunrath,  "Zum  Verstanclniss  des  Wortes  Algorismus," 
Bibl.  Math.,  I  (2),  70;  and  see  VIII  (2),  74.  P.  Ramus  (Scholarum  Mathe- 
maticarum  Libri  XXXI,  p.  112  (1569))  derived  it  from  al  (Arabic  for  "the") 
4-  dpifyuSs  (arithmos'y  number),  and  J.  Schoner  (1534  edition  of  the  Algorithmvs 
Demonstrates,  fol.  A  [iij],  v.)  did  the  same. 

2By  the  orientalist  J.  Reinaud  (1795-1867).  See  Mem.  de  I'lnstitut  na- 
tional de  Prance  des  inscriptions  et  belles-lettres,  XVIII,  303 ;  Boncompagni's 
Bullettino,  XII,  116.  Even  as  late  as  1861,  however,  L.  N.  Bescherelle's  well- 
known  French  dictionary  (Paris,  1861)  gave  al  (the)  +  ghor  (parchment),  and 
the  variants  algarthme,  algarisme.  See  also  Boncompagni's  Bullettino,  XIII,  557. 
"Incipit  prologus  in  libro  alghoarismi  de  pratica  arismetrice.  Qui  editus  est 
a  magistro  Johanne  yspalensi."  See  F.  Woepcke,  Journal  Asiatique,  I  (6),  519. . 

4  Who  uses  such  forms  as  algoritmi  and  algorizmi. 

6"Hanc  igitur  scientiam  numerandi  compendiosam  edidit  philosophus  nomine 
Algus,  unde  algorismus  nuncupatur,  vel  ars  numerandi,  vel  introductio  in 
numerum."  Halliwell  ed.,  p.  i. 

6"Ceste  signifiance  est  appellee  algorisme"  (MS.  of  c.  1275);  see  C.  Henry, 
Boncompagni's  Bullettino,  XV,  53.  "Secondo  Lalgorismo"  (Ghaligai,  1521) ; 
"...  calculandi  artem,  quam  uulgus  Algorithmum  uocat"  (Schoner,  1534)- 
So  the  MS.  of  Scheubel  (c.  1550),  already  mentioned  as  in  the  Columbia  Uni- 


CLASSIFICATIONS  OF  NUMBER  11 

with  the  fundamental  operations  themselves1  and  even  with  that 
form  of  arithmetic  which  makes  use  of  the  abacus.2 

Names  for  Logistic.  There  have  been  various  other  names 
for  logistic.  The  early  Italian  writers  often  spoke  of  a  practical 
arithmetic  as  a  practical  pratica,  or  pratiche*  Many  of  the 
Latin  writers  of  the  Renaissance,  particularly  in  the  i6th  cen- 
tury, spoke  of  it  as  the  art  of  computing  (ars  supputandi)  .5 
The  Dutch  writers  used  the  term  "ciphering/76  particularly  in 
the  1 6th  and  iyth  centuries,  and  from  this  source,  through  New 
Amsterdam,  came  the  common  use  of  the  word  in  the  early 
schools  of  America. 

In  Italy,  in  the  i5th  century,  logistic  occasionally  went  by 
the  name  of  the  minor  art,7  and  arithmetic  and  algebra  by  the 
name  of  the  major  art.8 


^    5.  ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 

-  Abstract  and  Concrete. v  The  distinction  between  abstract  and 
concrete  numbers  is  modern.  The  Greek  arithmeticians  were 
concerned  only  with  the  former,  while  the  writers  on  logistic 
naturally  paid  no  attention  to  such  fine  distinctions.  It  was  not 

versity  Library,  has  such  phrases  as  "Algebrae  fundamenta  seu  algorismus," 
"Algorismus  de  surdis,"  and  "  Algorithmus  quantitatis,"  showing  the  broader 
use  of  the  term,  Stifel  (1544)  used  the  term  in  the  same  way. 

iThus,  Thierfelder  (1587)  uses  "Der  Algorithmus"  and  "Die  Species" 
(p.  51)  as  synonymous.  Similarly,  "ALgorithmus  ist  ein  lehr  aus  der  man 
lernet  Addiren/Subtrahiren/Multipliciren  vnd  Diuidiren"  (Stifel,  1545). 

2As  in  the  Algoritmus  of  Klos  (1538),  the  first  Polish  arithmetic,  which  is 
purely  a  treatise  on  abacus  reckoning.  See  S.  Dickstein,  Bibl.  Math.,  IV  (2),  57. 
Similarly,  there  were  several  books  entitled  Algorithmus  linealis  published  in 
Germany  early  in  the  i6th  century,  all  dealing  with  the  abacus. 

8  As  in  the  Treviso  arithmetic  (1478). 

4  As  in  Cataneo's  arithmetic  (1546). 

5  Thus,  Tonstall   (1522)   calls  his  work  De  arte  supputandi,  a  title  already 
used  by  Clichtoveus  (1503)  in  the  abridged  form  of  Ars  supputadi.   Glareanus 
(1538)    speaks   of   the   "supputandi    ars,"    and   "  supputation  "    (for   computa- 
tion) was  a  term  in  common  use  in  England  until  the  iQth  century.    For  ex- 
ample, see  W.  Butler,  Arithmetical  Questions,  London,  2di  ed.,  1795. 

6Cyffering,  cyffer-konst,  cyffer-boeck,  and  the  like. 

7  L 'arte  minore.  BL'arte  maggiore\  or,  in  Latin,  Ars  magna. 


12      ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 

until  the  two  streams  of  ancient  number  joined  to  form  our  mod- 
ern elementary  arithmetic  that  it  was  thought  worth  while  to 
make  this  classification,  and  then  only  in  the  elementary  school. 

The  terms  "abstract"  and  "concrete"  were  slow  in  estab- 
lishing themselves.  The  mathematicians  did  not  need  them, 
and  the  elementary  teachers  had  not  enough  authority  to  stand- 
ardize them.  In  the  i6th  century  the  textbook  writers  began  to 
make  the  distinction  between  pure  number  and  number  to  which 
some  denomination  attached,  and  so  we  find  Trenchant  (1566), 
for  example,  speaking  of  absolute  and  denominate  number,  the 
latter  including  not  only  3  feet  but  also  3  fourths.1 

From  that  time  on  the  distinction  is  found  with  increasing 
frequency  in  elementary  works.  Such  refinements,  however,  as 
required  the  product  to  be  of  the  same  denomination  as  the 
multiplicand  are,  in  general,  igth  century  creations  of  the 
schools.  Thus  Hodder2  asserts  that  "Pounds  multiplied  by  20, 
are  shillings,"  and  every  scientist  today  recognizes  such  forms 
as  "20  Ib.  x  10  ft.  =  200  foot-pounds."< 

•  Digits,  Articles,  and  Composites/  One  of  the  oldest  classifica- 
tions of  numbers  is  based  upon  finger  symbolism.3  The  late 
Roman  writers  seem  to  have  divided  the  numbers  below  a  hun- 
dred into  fingers  (digiti),  joints  (articuli],  and  composites 

1"L'absolu  est  ccluy  qui  n'a  aucune  denomination:  comme  2,  7,  5,  tel 
nombre  est  abstret,  &  de  forme  nue  se  referant  a  la  Theorique.  Le  denomme: 
est  celuy  qui  si  prononce  auec  quelque  denomination  ...  &  se  refere  a  la 
Pratique."  The  latter  included  "le  vulgairement  denomme,  comme  8  aun," 
and  also  "le  rompu,  comme  |,"  although  he  says  that  in  practice  j  is  consid- 
ered as  abstract  unless  some  denomination  is  given  to  it:  "lequel  en  prati- 
quant  est  entendu  absolu  s'il  n'a  quelque  denomination  de  suget,  comme  disant 
|  d'aun"  (1578  ed.,  p.  16). 

Similarly,  Stifel :  "Numeri  abstract*  proprie  dicuntur,  iq  nulla  prorsus 
denomination^  habet"  (Arithmetica  Integra,  1544,  fol.  7,  v.).  Xylander  (1577) 
used  ledige  and  benannte  Zahlen.  21672  ed.,  p.  56. 

3 See  page  196.  Th.  Martin,  "Les  signes  numeraux,"  Annali  di  mat.  pura 
ed  applic.,  V,  257,  337,  and  reprint  (Rome,  1864);  hereafter  referred  to  as 
Martin,  Les  Signes  Num.  Suevus  (Arithmetica  Historica,  1593,  p.  3)  speaks 
of  the  finger  origin:  "Digitus  heist  ein  Finger  zal/die  unter  zehen  bedeut"; 
M.  Wilkens,  a  Dutch  arithmetician  (Arithmetica,  Groningen,  1630;  1669  ed., 
p.  i),  says:  "Dese  zijn  Digiti,  dat's  Enckel  ofte  vingergetalen " ;  and  many  other 
early  writers  have  similar  statements. 


DIGITS,  ARTICLES,  COMPOSITES  13 

(compositi)  of  fingers  and  joints,  the  joints  being  the  tens,  and 
the  composites  being  numbers  like  15,  27,  and  so  on.  In  a  pas- 
sage attributed,  but  doubtfully,  to  Boethius  it  is  said  that  this 
threefold  division  is  due  to  the  ancients.1  While  the  terms  were 
probably  known  in  early  times,,  they  were  not  used  commonly 
enough  to  appear  in  the  places  where  finger  symbolism  is  men- 
tioned.2 So  far  as  extant  works  are  concerned,  the  classification 
is  medieval. 

Meaning  of  "  Digit."  Since  there  are  ten  fingers,  it  is  probable 
that  the  digits  were  originally  the  numbers  from  one  to  ten 
inclusive ;  but  so  far  as  appears  from  treatises  now  extant  they 
were  the  numbers  from  one  to  nine  inclusive,  not  the  figures 
representing  these  numbers ;  that  is,  they  were  the  numbers  be- 
low the  first  "limit."  The  division  of  numbers  into  limits  or 
differences  (in  which  10,  20,  •  •  •,  90  were  of  the  first  order; 
100,  200,  •  •  •,  900,  of  the  second  order,  and  so  on)  is  found  in 
the  works  of  such  writers  as  Alcuin  (c.  780),  Jordahus  Nemora- 
rius  (c.  1225),  O'Creat  (c.  1150),  and  Sacrobosco  (c.  1250), 
and  was  evidently  common.3  Since  unity  was  not  considered  a 
number  until  modern  times,  it  was  sometimes  definitely  omitted, 
leaving  only  eight  digits. V  " 

1  Since  this  is  the  first  time  the  division  appears,  so  far  as  known,  the  pas- 
sage is  important  enough  to  be  quoted  in  the  original :  "  Digitos  vero,  quos- 
cunque  infra  primum  limitem,  id  est  omnes,  quos  ab  unitate  usque  ad  denariam 
summam  numeramus,  veteres  appellare  consueverunt.  Articuli  autem  omnes  a 
deceno  in  ordine  positi  et  in  infinitum  progressi  nuncupantur.  Compositi 
quippe  numeri  sunt  omnes  a  primo  limite  id  est  a  decem  usque  ad  secundum 
limitem  id  est  viginti  ceterique  sese  in  ordine  sequentes  exceptis  limitibus. 
Incompositi  autem  sunt  digiti  omnes  annumeratis  etiam  omnibus  limitibus." 
Boethius,  ed.  Friedlein,  p.  395.  See  also  G.  Enestrom,  Bibl.  Math.,  XI  (2),  116. 

2Pliny,  Hist.  Nat.j  34,  7;  2,  23;  Martin,  Les  Signes  Num.,  51. 

3G.  Enestrom,  "Sur  les  neuf  Mimites'  mentionnes  dans  1' '  Algorismus J  de 
Sacrobosco,"  Bibl.  Math.,  XI  (2),  97.  See  also  the  i2th  century  MS.  described 
by  M.  Chasles  in  the  Comptes  rendus,  XVI  (1843),  237;  the  Compotus  Rein- 
heri,  p.  28;  Boncompagni's  Bullettino,  X,  626;  S.  Gtinther,  Geschichte  des 
math.  Unterrichts,  p.  99  (Berlin,  1887)  (for  Bernelinus),  hereafter  referred  to 
as  Gunther,  Math.  Unterrichts. 

4E.g.,  by  Peletier  (1549):  "Le  Nombre  Entier  se  diuise  en  Simple,  Article, 
&  Compos6.  Le  Simple  est  le  Nombre  plus  bas  que  10 :  ce  sont  les  huict 
figures,  2,  3,  4,  5,  6,  7,  8,  9."  He  uses  numbers  and  figures  as  synonymous,  and 
uses  "simple"  for  "digit." 


14       ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 

v/ 
"  Meaning  of  "Article"  and  "Composite."   The  articles  were 

sometimes  limited  to  nine  in  number  (10,  20,  •  •  •,  90),  but  it 
was  more  common  to  take  any  multiple  of  ten. x  In  the  early 
printed  books  they  were  occasionally  called  decimal  numbers,1 
and  as  such  they  finally  disappeared. 

<  The  term  "composite,"  originally  referring  to  a  number  like 
17,  56,  or  237,  ceased  to  be  recognized  by  arithmeticians  in  this 
sense  because  Euclid  had  used  it  to  mean  a  nonprime  number.2 
This  double  meaning  of  the  word  led  to  the  use  of  such  terms  as 
" mixed "  and  "compound"  to  signify  numbers  like  16  and  345 

The  oldest  known  French  algorism  (c.  1275)  has  the  three- 
fold division4  above  mentioned,  as  does  also  the  oldest  one  in 
the  English  language  (c.  1300),  already  cited.  The  latter  work 
is  so  important  in  the  history  of  mathematics  in  this  language 
as  to  justify  a  further  brief  quotation : 

Some  numbur  is  called  digitus  latine,  a  digit  in  englys.  Somme 
nombur  is  called  articulus  latine.  An  Articul  in  englys.  Some  nombur 
is  called  a  composyt  in  englys.  .  .  . 

flSunt  digiti  numeri  qui  citra  denarium  sunt.5 

irThus,  Pellos  (1492,  fol.  4)  speaks  of  "numbre  simple,"  "nubre  desenal," 
and  "nubre  plus  que  desenal";  and  Ortega  (1512;  1515  ed.,  fols.  4,  5)  has 
"lo  numero  simplice,"  "lo  numero  decenale,"  and  "lo  numero  composto." 

z  Elements,  II,  def.  13.  For  other  Greek  usage  see  Heath's  Euclid,  Vol.  II, 
p.  286. 

Lazesio  (1526),  among  others,  pointed  out  this  twofold  usage  "sccudo  sacro 
busco  I  suo  algorismo"  and  "secodo  el  senso  di  Euclide"  (1545  cd.,  fol.  2). 
See  also  Pacioli,  Suma  (1494  ed.,  fols.  9,  19)  ;  Tartaglia,  General  Trattato  (1556, 
II,  fol.  i,  v.}  ;  Santa-Cruz  (1594;  1643  ed.,  fol.  2). 

Trenchant  (1566;  1578  eel.,  p.  223)  speaks  of  "Nombre  premier,  ou  incom- 
pose,"  and  "Nombre  second,  ou  compose","  a  natural  use  of  "second"  as  related  to 
"premier"  (prime),  and  the  same  usage  was  doubtless  common  at  that  time. 

3"Alius  aut  mixt'  siue  ppositus,"  in  the  Questio  hand  indigna  eiusqj  solutio 
ex  anrelio  Augustino,  c.  1507.  So  Hylles  (1592;  1600  cd.,  fol.  7)  says:  "The 
third  sort  are  numbers  MIXT  or  compound";  Digges  (1572;  1570  ed.,  p.  2)  uses 
"compound"  alone;  and  Hodder  (roth  cd.,  1672,  p.  5)  has  "A  Mixt,  or 
Compound."  Dutch  arithmeticians  avoided  the  difficulty  by  using  terms  in  the 
vernacular;  thus,  Mots  (1640)  gives  "De  enckel  getallen"  (digits),  "Punct 
ofte  leden-getallen "  (articles),  " t'samen-gevoeghde  getallen  (composites). 

4"Tu  dois  savoir  ki  sont  .3.  manieres  de  nombres  car  li  .1.  sont  degit  li  autre 
article,  li  autre  compost."  See  Ch.  Henry  in  Boncompagni's  Bullettino,  XV,  53. 

5  The  anonymous  writer  here  quotes  from  the  Carmen  de  Algorismo  of 
Alexandre  de  Villa  Dei  (c,  1240).  The  translation  follows. 


ARTICLE  AND  COMPOSITE  15 

fiHere  he  telles  qwat  is  a  digit,  Expone  versus  sic.  Nomburs  digitus 
bene  alle  nomburs  )>  at  ben  with-inne  ten,  as  nyne,  8.7.6.5.4.3.2.1.  .  .  . 
Articulis  ben  ben  alle  ]?at  may  be  deuidyt  into  nomburs  of  ten 
&  nothynge  leue  ouer,  as  twenty,  thretty,  fourty,  a  hundryth,  a 
thousand,  &  such  ofer.  .  .  .  Compositys  ben)  nomburs  |>at  bene  com- 
ponyt  of  a  digyt  &  of  an  articulle  as  fouretene,  fyftene,  sextene,  & 
such  oj>er. 

Recorde  (c.  1542)  sums  the  matter  up  by  saying: 

A  diget  is  any  numb  re  vnder  10.  .  .  .  And  10  with  all  other  that 
may  bee  diuided  into  x.  partes  iuste,  and  nothyng  remayne,  are 
called  articles,  suche  are  10,  20,  30,  40,  50,  &c.  100, 200,  &c.  1000.  &c. 
And  that  numbre  is  called  myxt,  that  contayneth  articles,  or  at  the 
least  one  article  and  a  digette :  as  12} 

At  best  such  a  classification  is  unwieldy,  and  many  of  the 
more  thoughtful  writers,  like  Fibonacci  (1202),  abandoned 
it  entirely.  Others,  like  Sacrobosco  (c.  1250),  struggled 
with  it  but  were  obscure  in  their  statements ; 2  while  Ramus 
very  wisely  (1555)  dismissed  the  whole  thing  as  " puerile 
and  fruitless."3 

v  All  that  is  left  of  the  ancient  discussion  is  now  represented 
by  the  word  "digit,"  which  is  variously  used  to  represent  the 
numbers  from  one  to  nine,  the  common  figures  for  these  num- 
bers, the  ten  figures  o,  i,  .  .  .,  9,  or  the  first  ten  numbers 
corresponding  to  the  fingers.* 

Significant  Figures.  "After  the  advent  of  the  Hindu- Arabic 
figures  into  Europe  (say  in  the  icth  century)  the  difference 
between  the  zero  and  the  other  characters  became  a  subject  of 
comment.  The  result  was  the  coining  of  the  name  "significant 
figures"  for  i,  2,  3,  •  •  •,  9.  At  the  present  time  the  meaning 

1i558  ed.  of  the  Grovnd  of  Artes,  fol.  Ciij.  Similar  classifications  are  found 
in  most  of  the  early  printed  books  of  a  theoretical  nature,  but  less  frequently  in 
the  commercial  books. 

2  Thus  Petrus  de  Dacia  (1291)  confessed  that  he  could  not  quite  understand 
Sacrobosco,  saying,  "ita  credo  auctorcm  esse  intelligendum." 

3"Puerilis  et  sine  ullo  fructu."  See  also  Boncompagni,  Trattati  d'Aritmetica, 
II,  27  (Rome,  1857);  J.  Havet,  Lettres  de  Gerbert,  p.  238  (Paris,  1889); 
Boncompagni's  Bullet  tino,  XIV,  91;  Abhandlungen,  III,  136. 


1 6      ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 

has  been  changed,  so  that  o  is  a  significant  figure  in  certain 
cases^  For  example,  if  we  are  told  to  give  log  20  to  four  signifi- 
cant figures,  we  write  1.301.  Similarly,  we  write  0.3010  for 
log  2,  and  7.550  for  V57-  The  term  is  doubtless  to  be  found  in 
medieval  manuscripts ;  at  any  rate  it  appears  in  the  early  printed 
arithmetics1  and  has  proved  useful  enough  to  be  retained  to  the 
present  time  in  spite  of  the  uncertainty  of  its  meaning. 

\  Odd  and  Even  Numbers.  The  distinction  between  odd  and 
even  numbers  is  one  of  the  most  ancient  features  in  the  science 
of  arithmetic.  The  Pythagoreans  knew  it,  and 
their  founder  may  well  have  learned  it  in  Egypt 
or  in  Babylon.  It  must  have  been  common  to 
a  considerable  part  of  the  race,  for  the  game  of 

1      "even  and  odd7'  has  been  played  in  one  form 

or  another  almost  from  time  immemorial,2  being 
ancient  even  in  Plato's  time.3  The  game  consisted  simply  in 
guessing  odd  or  even  with  respect  to  the  number  of  coins  or 
other  objects  held  in  the  hand. 

The  odd  number  was  also  called  by  the  geometric  name  of 

" gnomon,"  the  primitive  form  of  the  sundial.    If  such  a  figure 

<*-*•/ 

"LE.g.,  Licht  (1500);  Grammateus  (1518),  "neun  bedeutlich  figuren"  ;  Riese 
(1522),  "Die  ersten  neun  sind  bedeutlich";  Gemma  Frisius  (1540)  ;  Stifel  (1544), 
"Et  nouem  quidem  priores,  significatiuae  uocantur";  Peletier  (1549),  "Chacune 
des  neufs  premieres  (qui  sont  appellees  significatiues) "  .  .  .  ;  Recorde  (c.  1542), 
"The  other  nyne  are  called  Signifying  figures";  Trenchant  (1566). 

2  This  is  seen  in  such  expressions  as  d/9T* cur^uSs,  Apria  rj  irepirrd,  iralfriv,  fvyb  9? 
&£u*ya  iralfav.  This  £VJCL  y  d^vya, "  yokes  or  not-yokes,"  is  similar  to  the  Sanskrit 
"yuj"  and  "ayug"  for  even  and  odd.  Horace  couples  it  with  riding  a  hobby 
horse  as  a  childish  diversion : 

Ludere  par  impar,  equitare  in  harundine  longa. 

Satires,  II,  3,  248 

See  also  E.  B.  Tylor,  "History  of  Games,"  in  the  Fortnightly  Review,  May, 

1879,  P-  735- 

8  In  addition  to  the  references  to  the  Greek  theory  of  numbers  given  in 
Volume  I  and  in  this  chapter,  consult  Dickson,  Hist.  Th.  Numb.,  F.  von  Drie- 
berg,  Die  Arithmetik  der  Griechen,  Leipzig,  1819;  G.  Friedlein,  Die  Zahlzeichen 
und  das  elementare  Rechnen  der  Griechen  und  Ro'mer,  Erlangen,  1869;  Heath, 
History,  I,  67-117.  Heath  mentions  a  fragment  of  Philolaus  (c.  425  B.C.)  which 
says  that  "numbers  are  of  two  special  kinds,  odd  and  even,  with  a  third,  even- 
odd,  arising  from  a  mixture  of  the  two." 


ODD  AND  EVEN  NUMBERS 


is  turned  to  the  east  in  the  morning  and  to  the  west  in  the  after- 
noon, the  hours  can  be  read  on  the  horizontal  arm  as  in  the 
Egyptian  sun  clock  mentioned  in  Volume  I,  page  50.  Thus  we 
have  the  origin  of  the  right  shadow,  the  umbra  recta,  used  in 
early  trigonometry.  By  such  an  instrument  we  come  to  "  know  " x 
the  time,  and  by  facing  it  to  the  south  we  also  come  to  know  the 
seasons,  the  solstices,  and  the  length  of  the  year.2 

It  is  apparent  that  the  gnomon  here  shown  in  the  shaded  part 
of  the  figure  is  of  the  form  2  n  -f  i  and  hence,  as  stated  above,  is 

an  odd  number.3    It  is  also  apparent  that  ^(2^  +  1)  is  a 

0 

square,  that  is,  that  the  sum  of  the  first  n  odd  numbers, 
including  i,  is  a  square, — a  fact  well  known 
to  the  Greeks,  as  is  shown  by  the  works  of 
Theon  of  Smyrna4  (c.  125). 

That  there  is  luck  in  odd  numbers  is  one 
of  the  oldest  superstitions  of  the  race,  with 
such  occasional  exceptions  as  the  case  of 
the  general  fear  of  thirteen, — a  fear  that, 
seems  to  have  long  preceded  the  explanation 
that  it  arose  from  the  number  present  at  the  Last  Supper.5 

The  general  feeling  that  odd  numbers  are  fortunate  and  even 
numbers  unfortunate  comes  from  the  ancient  belief  that  odd 
numbers  were  masculine  and  even  numbers,  always  containing 
other  numbers,  were  feminine.  This  led  to  the  belief  that  odd 
numbers  were  divine  and  heavenly,  while  even  numbers  were 
human  and  earthly.  The  superstition  was  quite  general  among 
ancient  peoples.  Plato  says:  " The  gods  below  .  .  .  should  re- 
ceive everything  in  even  numbers,  and  of  the  second  choice,  and 

1  Greek  yv&nwv  (gno'mon),  one  who  knows,  from  yiyv&<rKciv,  yv&vai,  know. 

2  Heath,  History,  I,  78. 

3"  Gnomon  .  .  .  quod  Latini  amussim  seu  normam  vocant."  J.  C.  Heilbron- 
ner,  Historia  Matheseos  Universae  (Leipzig,  1742),  p.  173;  see  also  page  193; 
hereafter  referred  to  as  Heilbronner,  Historia. 

4  Theonis  Smyrnaei  .    .    .  expositio,  ed.   Hiller,  p.  31  (Leipzig,  1878).    On  this 
entire  discussion  see  also  Johnson,  Nicomachus,  and  especially  Heath,  History, 

1,77. 

5  Ernst  Boklen,  Die  Ungluckszahl  Dreizehn  und  ihre  mythische  Bedeutung, 
Leipzig,  1913,  with  extensive  bibliography. 


i8      ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 

ill  omen ;  while  the  odd  numbers,  and  of  the  first  choice,  and 
the  things  of  lucky  omen,  are  given  to  the  gods  above,"1  and 
the  phrase  "Deus  imparibus  numeris  gaudet"  ("God  delights  in 
odd  numbers")  probably  goes  back  to  the  time  of  Pythagoras.2 
The  superstition  runs  through  a  wide  range  -of  literature, 
Thus,  Shakespeare,  in  the  Merry  Wives  of  Windsor,  remarks 
that  "  there  is  divinity  in  odd  numbers,  either  in  nativity,  chance, 
or  death."  Such  beliefs  naturally  persist  among  the  less  ad- 
vanced peoples  and  are  common  even  today.) "For  example,  on 
the  island  of  Nicobar,  India,  an  odd  number  of  vessels  of  water 
are  dashed  against  the  hut  where  a  corpse  is  being  laid  out, 
and  the  stretcher  that  bears  it  must  always  contain  an  odd 

number  of  pegs.3 

/ 

-  Further  Classification.  The  Greeks  not  only  recognized  odd 
and  even  numbers,4  but  they  carried  the  classification  much  far- 
ther, including  what  Euclid  calls  "even-times-even  numbers," 
"even-times-odd  numbers, "and  "odd-times-odd  numbers."  His 
definitions  of  the  first  two  differ  from  those  given  by  Nicom- 
achus  (c.  100)  and  other  writers,5  with  whom  an  "even-times- 
even  number"  is  of  the  form  2";  an  "even-times-odd  number" 
is  of  the  form  2(2/74-1);  and  an  "odd-times-odd  number"  is  of 
the  form  (2/2  -f  i)  (2w  4- 1).  How  far  back  these  ideas  go  in 
Greek  arithmetic  is  unknown,  for  they  were  doubtless  trans- 
mitted orally  long  before  they  were  committed  to  writing.- 

Since  the  product  of  two  equal  numbers  represents  the  numer- 
ical area  of  a  square,  this  product  was  itself  called  a  square, — 
a  word  thus  borrowed  from  geometry.  The  product  of  two 
unequal  numbers  was  called  a  heteromecic  (different-sided) 
number.  Square  and  heteromecic  numbers  were  called  plane 

i-Laws,  Jowett  translation,  V,  100. 

2  On  the  general  number  theory  of  Pythagoras,  see  Heath,  History,  I,  65. 

3E.  H.  Man,  "Notes  on  the  Nicobarese,"  in  the  Indian  Antiquary,  1899,  p.  253. 

4 In  Euclid's  Elements,  VII,  6,  7,  &prtoi  and  irepicraol. 

5 For  particulars  see  Heath's  Euclid,  Vol.  II,  pp.  277,  281  seq.  For  the 
"odd-times-even  number,"  which  Euclid  seems  to  have  taken  as  synonymous 
with  an  "even-times-odd  number,"  see  ibid.,  p.  283;  on  the  general  "classifica- 
tions by  the  Greeks,  see  K.  G.  Hunger,  Die  arithmetische  Terminologie  der 
Griechen,  Prog.,  Hildburghausen,  1874. 


GREEK  CLASSIFICATIONS 


numbers,   while   the  product  of   three   numbers   was   called 
a  solid  number,  the  cube  being  a  special  case.     These  are 


pariter  tni* 

far  » 


O 


ARITHMETIC  AE 

N  I  C  O  L  A  V  S. 

Vt*  eft  alter  numem  far*  I  VST.  £J(J 
paritcr  impar  ,uel  A  parilM 
' 


o?  .  Eft  <utf  cw,  cum  primum  diuiditurjnox  fa 
indiuipbili*  ,  ut  14.   1  8  .  iz  ,    N  J  C  O  L  „ 
own  flnit  borum  nuMerorwn  txquifi* 
I  VST.  S 


rwi  diuidcnt  par  eftjfd  diuifortM  >rox  imp^r  ex«r 
gcf»  N  I  C  O  L.  Cwr  j  J  nominfc  illi  inditum  eft  ? 
1  V  S  T  .  ldeo,quod  qiulibct  eiitt  ordinis  numeri 
pares,fafltfitnt  per  impawn  midtiplicationcml  i& 

pariteritnpar  fa  ter,  fenariunt)  bis  quincj;  deuaritiM  conficiunt. 
Vemm  fi  cui  altiM  contetr.plffi  libtt,  eundeni  KO* 
cabit  imparcm  infita  qnantitate  >fed  pdrcm  in  deno 
ininaticne&Ib  exempli  gratia,denariM,  culm  al* 
ten  pun  eft  tptinariM9<]ui  quantitatejioc  c37  mo* 
uadum  congrcgatione  eft  imparted  quia  i  binayio 
denominator  ypar  iitdicabitttr.  QU£  ratio  nomini* 
ex  Boct  bio  colligitur  :  Alia  autcw  Euclidi  cjfi  uidc 
tur  -  N  I  C  O  L  .  Stint  ne  hide  de  Mo  aliquot  tbeo 

Symbol*  ex  remata  ?  I  V  S  T.  Quidni  *  Vnum  eft  ,  Sinu* 
tnerM  dimidium  impar  habuerit  ,  pariter  impaf 
eft  tantwn  .  Ham  hie  dmtaxat  txtrcmuin  •>  quoS, 

maxi* 

THEORY  OF  ODD  AND  EVEN   NUMBERS 


vi. 


From  the  arithmetic  of  Willichius  (1540).    The  page  also  illustrates  the  use  of 
the  catechism  method  in  the  i6th  century 

particular    types   of   the   figurate   numbers   mentioned   later. 
The  Boethian  arithmetic  made  much  of  this  classification  ;  x 

lBoetii  de  institutione  arithmetica  libri  duo,  ed.  Friedlein,  p.  17  (Leipzig, 
1867)  J  hereafter  referred  to  as  Boethius,  ed.  Friedlein. 

For  a  full  discussion  see  R.  Bombelli,  L'antica  numerazione  Italica,  cap.  x 
(Rome,  1876)  ;  hereafter  referred  to  as  Bombelli,  Antica  numer.  For  the 
status  of  the  classification  in  the  early  printed  books,  see  Pacioli,  Suma,  1494 
ed.,  fol.  [i],  v.(  =  A  [i],  v.). 


20      ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 

the  medieval  writers,  both  Arab1  and  Latin/  did  the  same;  and 
early  writers  in  the  vernaculars  simply  followed  the  custom.3 

Prime  Number.  Aristotle,  Euclid,  and  Theon  of  Smyrna 
defined  a  prime  number  as  a  number  "measured  by  no  number 
but  by  an  unit  alone,"  with  slight  variations  of  wording.  Since 
unity  was  not  considered  as  a  number,  it  was  frequently  not 
mentioned.  lamblichus  says  that  a  prime  number  is  also  called 
"odd  times  o<id,"  which  of  course  is  not  our  idea  of  such  a  num- 
ber. Other  names  were  used,  such  as  "eu  thy  me  trie"  and  "recti- 
linear," but  they  made  little  impression  upon  standard  writers.4 

The  name  "prime  number"  contested  for  supremacy  with 
"incomposite  number"  in  the  Middle  Ages,  Fibonacci  (1202) 
using  the  latter  but  saying  that  others  preferred  the  former.* 

Perfect  Numbers.  Conventionally  we  speak  of  the  aliquot 
parts  of  an  integral  number  as  the  integral  and  exact  divisors 
of  the  number,  including  unity  but  not  including  the  number 
itself.  A  number  is  said  to  be  deficient,  perfect,  or  abundant 
according  as  it  is  greater  than,  equal  to,  or  less  than  the  sum  of 
its  aliquot  parts.6 

!On  Savasorda  (c.  noo),  or  Abraham  bar  Chiia,  and  his  classification,  see 
Abhandlungen,  XII,  16.  On  al-Hassar  (c.  i2th  century),  see  Bibl.  Math,, 
II  (3),  17. 

2Thus  Jordanus  Nemorarius  (c.  1225):  "Par  numerus  est  qui  in  duo 
equalia  diuidi  potest.  Impar  est  in  quo  aliqua  prima  pars  est  absq}  pari: 
additq3  supra  parem  vnitate.  Parium  numeroru  alius  pariter  par :  alius  pariter 
ipar:  et  alius  impariter  par.  Pariter  par  est  que  nullus  impar  numerat. 
Pariter  ipar  est  que  quicunq}  pares  numerat.  Imparit^  par  est  que  quida  par 
scdm  pare  1  quida  scdm  impare  numerat."  1496  ed.,  fol.  b  (3). 

3  E.g.,  Chuquet,  La  Triparty  (1484) ;  see  Boncompagni's  Bullcttino,  XV,  619. 
Curtze  found  an  early  German  MS.  at  Munich  (No.  14,908,  Cod.  lat.  Monac.) 
with  such  terms  as  "gelich  oder  ungelich,"  "glich  unglich,"  and  the  like.    See 
Bibl.  Math.,  IX  (2),  39. 

4  Heath,  Euclid,  Vol.  I,  p.  146;  Vol.  II,  pp.  284,  285. 
6aNvmerorumquidam  sunt  incompositi,  et  sunt  illi  qui  in  arismetrica  et  in  geo- 

metria  primi  appellantur.  .  .  .  Arabes  ipsos  hasam  appellant.   Greci  coris  canon, 
nos  autem  sine  regulis  eos  appellamus."   Liber  Abaci,  I,  30. 

6  E.g.,  8  is  a  deficient  number,  since  8>i+2  +  4;6isa  perfect  number, 
since  6=1+2  +  3;  12  is  an  abundant  number,  since  i2<i  +  2  +  3+4  +  6. 
Various  other  names  are  given  to  abundant  and  deficient  numbers,  such  as 
redundant  or  overperfect  (farepreX^s,  vireprfreios)  and  defective  (AXonfc).  Heath, 
History,  I,  10,  74. 


PRIME  AND  PERFECT  NUMBERS  21 

This  classification  may  have  been  known  to  the  early  Pythag- 
oreans, but  we  have  no  direct  evidence  of  the  fact;  indeed, 
their  use  of  "perfect"  was  in  another  sense,  10  apparently  being 
considered  by  them  as  a  perfect  number.  &•' 

n 

Euclid  proved1  that  if  p  =  ^  2"  and  is  prime,  then  2np  is  per- 
fect.2 Nicomachus3  separated  even  numbers  into  the  classes 
above  mentioned,  and  gave  6,  28,  496,  and  8128  as  perfect 
numbers,  noting  the  fact  that  they  ended  in  6  or  8.  Theon 
of  Smyrna  (c.  125)  followed  the  classification  of  Nicom- 
achus, but  gave  only  two  perfect  numbers,  6  and  28.  lambli- 
chus4  (c.  325)  did  the  same,  but  asserted  that  there  was 
one  and  only  one  perfect  number  in  each  of  the  intervals 
i  .  •  •  10,  10  •  •  •  100,  100  •  •  •  1000,  1000  •  •  •  10,000,  and  so  on, 
and  that  the  perfect  numbers  end  alternately  in  6  and  8, — 
statements  which  are  untrue  but  which  are  found  repeated  in 
the  arithmetic5  of  Boethius  (c.  510).  Subsequent  writers  in 
the  Middle  Ages  and  the  Renaissance  frequently  followed 
Nicomachus  or  lamblichus. 

Fibonacci  (1202)  gave  |  •  22(22  —  i)  =  6,  ^  •  23(23—  i)  =  28, 
1 .  25(2*  —  i)  =  496  as  perfect  numbers,  and  so  in  general  for 
^  •  2P(2P  —  i)  where  2^—1  is  prime, — a  rule  which  holds  for  the 
first  eight  perfect  numbers  but  is  not  universal.6  Chuquet 
(1484)  gave  Euclid's  rule  and  repeated  the  ancient  error  that 
perfect  numbers  end  alternately  in  6  and  8. 

The  fifth  perfect  number,  33,550,336,  is  first  given,  so  far  as 
known,  in  an  anonymous  manuscript7  of  1456-1461.  Pacioli 
(1494)  incorrectly  gave  9,007,199,187,632,128  as  a  perfect 
number.8 

i  Elements,  IX,  36. 

2 On  all  this  work  see  Dickson,  Hist .  Th.  Numb.,  with  bibliography,  I,  i;  R.  C. 
Archibald,  Amer.  Math.  Month.,  XXVIII,  140,  with  valuable  references  to 
American  contributions. 

8 Arithmetica^  I,  14,  15.  4i668  ed.,  p.  43. 

5  Arithmetica,  I,  cap.  20,,  "De  generatione  numeri  perfecti." 

« Dickson,  Hist.  Th.  Numb.,  I,  13. 

7Codex  lat.  Monac.  14,908.   Dickson,  Hist.  Th.  Numb.,  i,  6. 

8"Sia  el  nuero  a  noi  pposto.  9007199187632128.  qle  como  e  ditto:  c  ^ 
Fol.  7,  v.  • 


22       ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 

Charles  de  Bouelles  (1509)  wrote  on  perfect  numbers1  and 
asserted,  without  proof,  that  every  perfect  number  is  even. 
He  stated  that  2"~'(2"~~I)  &  a  perfect  number  when  n  is 
odd,  which  is  substantially  the  incorrect  rule  of  Fibonacci. 
This  was  also  given  by  various  other  writers  of  the  i6th  cen- 
tury, including  as  good  mathematicians  as  Stifel2  (1544)  and 
Tartaglia3  (1556). 

Robert  Recorde4  (1557)  attempted  to  give  the  first  eight 
perfect  numbers,  but  three  in  his  list  were  incorrect.  Cataldi5 
showed  that  Pacioli's  pretended  fourteenth  perfect  number  is 
in  fact  abundant,  that  the  ancient  belief  that  all  perfect  numbers 
end  in  6  or  8  is  unfounded,  and  that  perfect  numbers  of  the  type 
given  by  Euclid's  rule  do  actually  end  in  6  or  8. 

Descartes  thought  that  Euclid's  rule  covered  all  even  perfect 
numbers  and  that  the  odd  perfect  numbers  were  all  of  the  type 
ps2,  where  p  is  a  prime.6 

Fermat  (1636)  and  Mersenne  (1634)  paid  much  attention  to 
the  subject,  and  their  investigations  contributed  to  the  theory 
of  prime  numbers.7 

Euler  at  first  (1739)  asserted  his  belief  that  2"~I(2"~I) 
is  a  perfect  number  for  n  =  i,  2,  3,  5,  7,  13,  17,  19,  31,  41,  and 
47,  but  afterward  (1750)  showed  that  he  was  in  error  with 
respect  to  41  and  47.  He  proved  that  every  even  perfect  num- 
ber is  of  Euclid's  type,  2"  ]P  2",  and  that  every  odd  perfect  num- 

o 

berisof  the  form r4A4"1^2, where  r  is  a  prime  of  the  form  4«+i.8 

There  are  many  references  to  perfect  numbers  in  general 

literature,9   in   Hebrew  and   Christian  writings   on   religious 

1MDe  Numeris  Perfectis,"  in  his  general  work  published  at  Paris  in  1509- 
1510.  See  Kara  Arithmetic^  p.  89. 

2  Arithmetica  Integra,  fols.  10,  n  (Niirnberg,  1544);  also  Die  Coss  Chris- 
toffs  Rudolffs,  fols.  10,  ii  (Konigsberg,  1553). 

3 La  seconda  Parte  del  General  Trattato,  fol.  146,  v.   Venice,  1556. 

4The  whetstone  of  witte,  fol.  [4,  v.].   London,  1557. 

^Trattato  dey  nvmeri  perfetti.    Bologna,  1603. 

*(Euvres,  II,  429.    Paris,  1898.  7Dickson,  Hist.  Th.  Numb.,  I,  11-13. 

8Ibid.t  p.  18.   For  the  later  theory,  consult  this  work,  I,  i. 

9  Thus  Macrobius,  in  his  Saturnalia,  says  that  six  "plenus  perfectus  atque 
diuinus  est."  Satvrnaliorvm  Liber  Vll,  cap.  xiii,  ed.  Eyssenhardt,  1868,  p.  446. 


PERFECT  AND  AMICABLE  NUMBERS  23 

doctrines,1  including  Isidorus  of  Seville  and  Rabbi  ben  Ezra, 
and  in  the  works  of  medieval  and  Renaissance  mystics.2 

Amicable  Numbers.3  Two  integral  numbers  are  said  to  be 
amicable4  if  each,  as  in  the  case  of  220  and  284,  is  equal  to  the 
sum  of  the  aliquot  parts  of  the  other.  These  two  numbers, 
probably  known  to  the  early  Pythagoreans,  are  mentioned  by 
lamblichus.  They  occupied  the  attention  of  the  Arabs,  as  in 
the  works  of  Tabit  ibn  Qorra  (c.  870).  It  was  asserted  by 
certain  Arab  writers  that  talismans  with  the  numbers  220  and 
284  had  the  property  of  establishing  a  union  or  close  friend- 
ship between  the  possessors,  and  this  statement  was  repeated 
by  later  European  writers,  including  Chuquet  (1484)  and 
Mersenne  (1634).% 

For  a  long  time  the  only  amicable  numbers  known  were  the 
two  given  above,  220  and  284,  but  in  1636  Fermat5  discovered 
a  second  pair,  17,296  =  24  •  23  •  47  and  18,416  =  24  •  1151, 
and  also  found  a  rule  for  determining  such  numbers.6  A 
third  pair  was  discovered  by  Descartes7  (1638),  namely, 
9,363, 584  =  27  •  191  •  383  and  9,437,056  =  27- 73,727.  Des- 
cartes gave  a  rule  which  he  asserted8  to  be  essentially  the 
same  as  Fermat's,  but  which  various  later  writers,  apparently 
ignorant  of  this  assertion,  assigned  to  Descartes  himself. 

Euler9  (1750)  made  a  greater  advance  in  this  field  than  any 
of  his  predecessors,  adding  fifty-nine  pairs  of  amicable  numbers 
of  the  type  am,  an,  in  which  a  is  relatively  prime  to  m  and  n, 
and  contributing  extensively  to  the  general  theory.  Dickson 


1£.g.,  J.  J.  Schmidt,  Biblischer  Mathematicus,  p.  20  (Ziillichau,  1736). 

2  See  Curtze's  mention  of  a  Munich  MS.  (No.  14,908,  Codex  lat.  Monac.)  in 
the  Bibl.  Math.,  IX  (2),  39,  with  five  perfect  numbers. 

Thierfelder  (1587,  fol.  A4,  r.)  says:  "Den  in  sechsz  tagen  hat  Gott  Himmel 
vnd  Erden/  vnd  alles  was  daririen  ist/  gemacht/  das  ist  ein  Trigonal  oder 
dreyeckichte  Zahl/  welche  Zahlen  fur  die  heiligen  Zahlen  gehalten  werden/  vnd 
ist  darzu  die  erste  perfect  Zahl."  3 Dickson,  Hist.  Th.  Numb.,  I,  36. 

4 The  terms  "amiable"  and  "agreeable"  are  also  used. 

6CEuvresy  1894,  II,  72,  208. 

6 The  rule  is  given  in  Dickson,  loc.  cit.,  p.  37. 

t(Euvres,  1898,  II,  93.  *GEuvres,  1898,  II,  148. 

9Opuscula  varii  argumenti,  3  vols.,  II,  23  (Berlin,  1746-1751).  See  also  Bibl. 
Math.,  IX  (3),  263;  X  (3),  80;  XIV  (3),  351;  Cantor,  Geschichte,  III,  616. 


24      ELEMENTARY  CLASSIFICATIONS  OF  NUMBER 


(1911)  has  obtained  two  new  pairs  of  amicable  numbers  and 
has  also  added  to  the  general  theory  of  the  subject.1 

Figurate  Numbers.    The  Greeks  were  deeply  interested  in 
numbers  which  are  connected  with  geometric  forms  and  which 
therefore  received  the  name  of  figurate 
numbers.2    These  are  triangular  if  capable 
of  being  pictured  thus : 


and  are  therefore  of  the  form 


FIGURATE    NUMBERS 

From  the  first  printed  They  are  square  if  they  can  be  represented 
by  squares,  such  as  JJ,  and  are  then 
of  the  form  n2.  They  are  pentagonal 

if  in  the   form  of  a  square  with  a  triangle  on  top,   thus: 


a      a 


so    that    the    form    is    n2  +  \  n  (  n  —  i  )  . 

Similarly,  there  are  hexagonal  numbers 

and  other  types  of  polygonal  numbers.3 

In  the  Greek  manuscripts  they  appeared  in  such  forms  as  those 

here  shown,  the  #'s  standing  for  I's  or  possibly  for  aptd^ 

(arithmos1  ',  number).4 

Related  to  figurate  numbers  there  are  the  linear  numbers. 
Under  this  name  Nicomachus  (c.  100)  included  the  natural 
numbers,  beginning  with  2  ;  side  and  diagonal  numbers5;  area, 


also  his  Hist.  Th.  Numb,  and  Amer.  Math.  Month.,  XXVIII,  195. 

2Boethius  defined  them  as  numbers  "qui  circa  figuras  geometricas  et  earum 
spatia  demensionesque  versantur."  Ed.  Friedlein,  p.  86,  1.  12.  See  also  Heath, 
History,  I,  76.  3Boethius,  ed.  Friedlein,  p.  98  seq. 

4  These  two  forms  are  from  a  loth  century  MS.  of  Nicomachus  in  Gottingen. 

6  The  irXevpiKol  ical  dtafjLcrpLKol  Api0/j.ot  of  Theon  of  Smyrna  (c.  125).  See  also 
Boethius,  ed.  Friedlein,  p.  90, 


FIGURATE  NUMBERS  25 

or  polygonal,  numbers1;  and  solid  numbers,2  including  cubic, 
pyramidal,  and  spherical  numbers.3    A  relic  of  such  numbers  is 

Pyrtmidm  numm  hoc  patio  digewitur. 


10 


ooo  •<»    •••9 

PYRAMIDAL  NUMBERS 

From  Joachim  Fortius  Ringelbergius,  Opera,  1531.   The  four  layers  of  the  two 
pyramidal  numbers  35  and  30  are  shown 

seen  in  problems  relating  to  the  piling  of  round  shot,  still  to  be 
found  in  algebras.  Indeed,  it  is  not  impossible  that  they  may 
have  been  suggested  to  the  ancients  by  the  piling  of  spheres  in 

1  See  Nicomachus,  Introd.,  II,  capp.  8-n. 
22r€p€ol.    Nicomachus,  Introd.,  II,  14. 
3Boethius,  he.  cit.,  pp.  104,  121. 


26  UNITY 

such  games  as  the  Castellum  nucum  to  which  Ovid  refers  in  his 
poem  De  Nuce,  where  the  pyramidal  number  is  mentioned.1 

,  Continuous  and  Discrete.  The  distinction  between  continuous 
and  discrete  magnitude  is  commonly  referred  to  the  Pythag- 
oreans or  even  to  Pythagoras  (c.  540  B.C.)  himself,2  the 
continuous  magnitude  being  geometric  and  the  discrete  being 
arithmetic.  The  distinction  was  recognized  by  various  Greek 
and  Latin  writers/3  appearing  in  the  works  of  such  medieval 
authors  as  Fibonacci  (1202)  and  Roger  Bacon  (c.  1250).? 

Cardinals  and  Ordinals.    The  distinction  between  cardinal 

and  ordinal  numbers  is  ancient,  but  the  names  are  relatively 

modern.    A  cardinal  number  is  a  number  on  which  arithmetic 

turns5  or  depends,  and  hence  is  a  number  of  importance/"  while 

ordinal  number  is  one  which  denotes  order.7 

6.  UNITY 

Unity.  Not  until  modern  times  was  unity  considered  a  num- 
ber. Euclid  defined  number  as  a  quantity  made  up  of  units,8 
and  in  this  he  is  followed  by  Nicomachus.9  Unity  was  defined 
by  Euclid  as  that  by  which  anything  is  called  "one."10  It  was 
generally  defined,  however,  as  the  source  of  number,  as  in  the 

1  Quattuor  in  nucibus,  non  amplius,  alea  tola  est, 

Cum  sibi  suppositis  additur  una  tribus. 

See  also  F.  Lindemann,  "Zur  Geschichte  der  Polyeder  und  der  Zahlzeichen," 
Sitzungsberichte  der  math.-physik.  Classe  der  K.  Bayerischen  Akad.  der  Wis- 
sensch.  zu  Miinchen,  XXVI,  625-757  (Munich,  1897). 

2"Ogni  quantita  .  .  .  secondo  Pythagora,  e  o  continua,  ouer  Discreta,  la 
continua  e  detta  Magnitudine,  ...  &  la  discreta  moltitudine."  Tartaglia,  Gene- 
ral Trattato,  I,  fol.  i,  r.  (Venice,  1556). 

3Boethius,  ed.  Friedlein,  pp.  8,  16;  Heath,  Euclid,  Vol.  I,  p.  234. 

4 E.g.,  in  the  Sloane  MS.  fol.  94  of  the  Communia.         5 Latin,  cardo,  a  hinge. 

6  Compare  cardinal,  a  prince  of  the  Church.    Glareanus  recognized  this  meta- 
phor: "Sunt  enim  quaedam,  quae  Cardinalia  appellant,  a  cardine  sumpta,  ut 
opinor,  metaphora,  quod  ut  in  cardine  ianua  uertitur,  ita  huius  artis  primum  ac 
praecipuum  negocium  in  hisce  consistat"  (1538;  1543  ed.,  fol.  3,  r.). 

7  On  the  history  of  these  terms  see  E.  Bortolotti,  "Definizioni  di  Numero," 
Esercitazioni  Matematiche,  II,  253,  and  Periodico  di  Matematiche,  II  (4),  413. 

8  'Api^s  8t  7-6  £K  ij.ovddwv  <rvyK<-tfj.evov  Tr\ij6os.  Elements,  VII,  def.  2.   See  also 
Heath,  History,  I,  69.  9Introd.,  I,  7,  i. 

i®MoJ>ds  fonv  tcaO*   yv  cKaffrov  r&v  OPTUV  ev  X^yercu.    Elements,  VII,  def.  i; 
ed.  Heath,  Vol.  II,  p.  279,  with  references  to  other  Greek  writers. 


EARLY  IDEAS  OF  UNITY  27 

anonymous  Theologumena?  a  Greek  work  of  the  early  Middle 
Ages.  The  dispute  goes  back  at  least  to  the  time  of  Plato,  for 
the  question  is  asked  in  the  Republic,  "To  what  class  do 
unity  and  number  belong?" — the  two  being  thus  put  into 
separate  categories. 

It  is  not  probable  that  Nicomachus  (c.  100)  intended  to  ex- 
clude unity  from  the  number  field  in  general,  but  only  from 
the  domain  of  polygonal  numbers.2  It  may  have  been  a  misin- 
terpretation of  the  passage  from  Nicomachus  that  led  Boethius3 
to  add  the  great  authority  of  his  name  to  the  view  that  one  is  not 
a  number.  Even  before  his  time  the  belief  seems  to  have  pre- 
vailed, as  in  the  case  of  Victorius  (457)  and  Capella  (c.  460), 
although  neither  of  these  writers  makes  the  direct  assertion.4 
Following  the  lead  of  Boethius,  the  medieval  writers  in  gen- 
eral, suchasal-Khowarizmi5  (c.  825),  Psellus0  (c.  1075),  Sava- 
sorda7  (c.  uoo),  Johannes  Hispalensis8  (c.  1140),  and  Rol- 
landus9  (c.  1424),  excluded  unity  from  the  number  fiefd.10  One 
writer,  Rabbi  ben  Ezra  (c.  1140),  seems,  however,  to  have 


i<mv  dpx^}  apt0fju>v,  Otcriv  IULTJ  €xov<ra.      Theologumena,  I,   I. 

2'H  (itv  novas  (nwelov  rbirov  ^7r^x°v^a  Ka<-  Tpbwov.  See  also  Johnson,  Nicomachiis, 
p.  7. 

3  "Numerus  est  unitatum  collectio."  Ed.  Friedlein,  p.  13,  1.  10.  In  the 
Latin  version  of  the  so-called  Boethian  geometry  it  is  asserted :  "  Primum 
autem  numerum  id  est  binariiim,  unitas  enim,  ut  in  arithmeticis  est  dictum, 
numerus  non  est,  sed  fons  et  origo  numerorum.  .  .  ."  Ed.  Friedlein,  p.  397, 
1.  19.  See  also  H.  Weissenborn,  Gerbert,  p.  219  (Berlin,  1888). 

4 "Unitas  ilia,  unde  omnis  numerorum  multitude  procedit."  From  the  Cal- 
culus of  Victorius;  see  Boncompagni's  Bullettino,  IV,  443. 

"Nee  dissimulandum  est  ex  eo  quod  monas  retractantibus  unum  solum  ipsam 
esse.  ab  eaque  cetera  procreari.  Omniumque  numerorum  solam  seminarium 
esse.  solamque  mensuram  et  incrementorum.  causam.  statumque  detrimentorum." 
From  a  fragment  of  Capella ;  see  E.  Narducci  in  Boncompagni's  Bullettino,XV,  566. 

5UQuia  unum  est  radix  uniuersi  numeri,  et  est  extra  numerum."  From  the 
supposed  translation  of  Adelard  of  Bath. 

6  "  Principium  itaque  omnis  numeri  est  Monas,  non-numerus  fons  numero- 
rum." See  the  1532  edition,  p.  13. 

7 "Numerus  est  ex  unitatibus  profusa  collectio"  (Plato  of  Tivoli's  translation, 
1145).  See  C.  H.  Haskins,  Bibl.  Math.,  XI  (3),  332. 

8 "Unitas  est  origo  et  prima  pars  numeri  .  .  .  sed  ipsa  extra  omnem  numerum 
intelligitur."  See  B.  Boncompagni,  Trattati  d'  Aritmetica,'!!,  25  (Rome,  1857); 
hereafter  referred  to  as  Boncompagni,  Trattati. 

9 See  Volume  I,  page  261.  "  Vnitas  non  est  numerus  sed  principia  numerorum" 
(Plimpton  MS.,  Pt.  I,  cap.  i).  10See  also  Boncompagni's  Bullettino,  XV,  126. 


28  UNITY 

approached  the  modern  idea.  In  his  Sefer  ha-Echad  (Book  on 
Unity}  there  are  several  passages  in  which  he  argues  that  one 
should  be. looked  upon  as  a  number. 

Most  of  the  authors  of  the  early  printed  books  excluded  unity, 
as  is  seen  in  the  works  of  Pacioli1  (1494),  Kobel2  (1514), 
Tzwivel3  (1505),  and  many  others.  Thus  the  English  writer 
Baker  (1568)  remarks  that  "an  vnitie  is  no  number  but  the 
beginning  and  original  of  number."4  In  the  i6th  century,  how- 
ever, the  more  thoughtful  writers  began  to  raise  the  question  as 
to  whether  this  exclusion  of  unity  from  the  number  field  was  not 
like  the  trivial  disputes  of  the  schoolmen,5  and  by  the  end  of  the 
century  it  was  recognized  that  the  ancient  definition  was  too 
narrow.  Thus  Hylles  (1592),  speaking  of  "an  vnit  or  an  in- 
teger (which  sometimes  I  also  cal  an  Ace),"  is  rather  afraid  to 
take  a  definite  stand  in  the  matter,  but  says  that  "the  latter 
writers,  as  namely  Ramus,  and  such  as  have  written  since  his 
time,  affirme  not  only  that  an  vnite  or  one,  is  a  number,  but 
also  that  euery  fraction  or  parte  of  an  vnite,  is  a  number.  .  .  . 
I  do  accompt  it  after  a  sorte  for  the  first  or  least  number  .  .  . 
euen  as  an  egg,  with0  in  power  possibilitie  containeth  a  bird 
though  really  and  actually  it  is  none."  Stevin  (1585),  a  much 
greater  man,  used  the  argument  that  a  part  is  of  the  same 
nature  as  the  whole,  and  hence  that  unity,  which  is  part  of  a 
collection  of  units,  is>  a  number.7  To  this  Antoine  Arnauld,  "le 

luEt  essa  vnita  no  e  numero :  ma  ben  principle  di  ciascun  numero"  (1494 
ed.,  fol.  9). 

2  "Daraus3  verstehstu  das  I.  kein  zal  ist/  sender  es  ist  ein  gebererin/  anfang/ 
vnnd  fundament  aller  anderer  zalen"  (Zwey  rechenbuchlin,  Frankfort  ed.,  1537, 
fol.  26).  It  is  also  in  his  Rechenbuchlin,  1531  ed.  dedication,  and  1549  ed.,  fol.  26. 

8"Unitas  em  numerus  non  est.  sed  fons  et  origo  numerorum"  (fol.  2). 

4 1580  ed.,  fol.  i. 

5 So  Gemma  Frisius  (1540)  makes  it  a  matter  of  authority:  "Nvmerum 
authorcs  vocant  multitudinem  ex  Vnitatibus  conflatum.  Itaque  Vnitas  ipsa  licet 
subinde  pro  numero  habeatur,  proprie  tamen  numerus  non  erit"  (1563  ed.,  fol.  5) . 

Also  Trenchant  (1566):  "...  Pvnit6  n'est  pas  nombre  .  .  .  Mais  en  la 
pratique,  ou  le  nombre  est  tousiours  adapte  a  quelque  suget  .  .  .  1'vnite*  est 
prinse  pour  nombre"  (1578  ed.,  p.  9). 

6S£c,  for  "  which."   From  the  1600  edition. 

7 "La  partie  est  de  mesme  nature  que  le  tout.  Unit6  est  partie  d'une  mul- 
titude d'  unitez  .  .  .  et  par  consequent  nombre."  See  also  the  Girard  edition, 
of  1634,  P-  !>  with  slight  change  in  wording. 


THEORY  OF  NUMBERS  29 

grand  Arnauld"  (1612-1694),  replied  that  the  argument  was 
worthless,  for  a  semicircle  is  not  a  circle.  Stevin  also  used  the 
argument  that  if  from  a  number  there  is  subtracted  no  number, 
the  given  number  remains;  but  if  from  3  we  take  i,  3  does  not 
remain;  hence  i  is  not  no  number.1  Tjie  school  arithmetics 
kept  the  Boethian  limitation  until  the  'close  of  the  i8th  century.2 
Another  common  notion  was  that  unity  is,  like  a  point,  in- 
capable of  division, — an  idea  also  due  to  the  Greeks.0 

7.  LATER  DEVELOPMENTS 

Higher  Domain  of  the  Arithmetica.  The  later  developments 
in  the  arithmetica  do  not  belong  to  the  domain  of  elementary 
mathematics.  Their  history  has  been  treated  with  great  erudi- 
tion by  Professor  Dickson  in  his  History  of  the  Theory  of  Num- 
bers.4 As  a  matter  of  general  information,  however,  a  few  of  the 
theorems  which  have  attracted  wide  attention  will  be  stated. 

Typical  Theorems.  In  1640  Fermat,  in  a  letter  to  Bernard 
Frenicle  de  Bessy  (c.  1602-1675),  set  forth  the  theorem  that 
if  p  is  any  prime  number  and  x  is  any  integer  not  divisible  by  p, 
then  a/"1--  i  is  divisible  by  p.  The  special  case  of  2  —  2  being 
divisible  by  the  prime  p  had  long  been  known  to  Chinese 
scholars,  but  the  general  theorem  is  due  to  Fermat.  Leibniz 
proved  the  proposition  some  time  before  1683. 

Euler  stated  Fermat's  theorem  in  a  communication  to  the 
Petrograd  Academy5  in  the  form:  If  n  +  i  is  a  prime  dividing 
neither  a  nor  6,  then  an—  bn  is  divisible  by  n  +i. 

As  stated  in  Volume  I,  page  459,  Wilson  discovered  (c.  1760) 
that  if  p  is  prime,  then  i  +  (p  ^-  i) !  is  a  multiple  of  p.  The 
manuscripts  of  Leibniz  now  preserved  at  Hannover  show  that 
he  knew  the  theorem  before  1683,  but  he  published  nothing  upon 

1  Abhandlungen,  XIV,  227. 

2 E.g.,  Ward's  Young  Mathematician's  Guide,  p.  4  (London,  1771). 

8  "II  punto  nella  Geometria,  &  IVnita  nell'  Arimmetica  non  e  capace  di  parti- 
mento.  Proclo  sopra  Euclide  lib.  2.c.xi."  Ciacchi,  Regole  Generali  d'  Abbaco, 
p.  352  (Florence,  1675). 

4 See  also  A.  Natucci,  //  Concetto  di  Numero,  Turin,  1923. 

5 Presented  in  1732,  published  in  1738. 


30  LATER  DEVELOPMENTS 

the  subject.  Lagrange  published  a  proof  of  the  theorem  in  1771, 
deduced  it  from  Fermat's  Theorem,  and  proved  its  converse. 
Fermat  gave  as  his  opinion  that  2*"+  i  is  always  prime,  but 
asserted  that  he  was  unable  to  prove  it.  Euler  (I732)1  showed 
that  Fermat's  opinion  was  not  warranted,  since 

225  +  i  =  641  •  6,700,417. 

Fermat's  connection  with  numbers  of  this  form  led  to  their 
being  called  "Fermat's  Numbers."2 

With  respect  to  the  sum  and  the  number  of  divisors  of  a 
number  there  is  an  extensive  literature.3  For  example,  Cardan 
(i537)  stated  that  a  product  P  of  k  distinct  primes  has 
i-f  2  +  22  +  •  •  •  +  2*~1aliquot  parts;  for  example,  that  3-5-7 
has  14-2  +  4  aliquot  parts.  This  rule  was  proved  by  Stifel  in 
his  Arithmetica  Integra  (1544).  Frans  van  Schooten  (1657) 
proved  that  a  product  of  k  distinct  primes  has  2k—  i  aliquot 
parts,  which  is  only  another  expression  for  Cardan's  rule. 

Descartes  (probably  in  1638)  showed  that  if  p  is  a  prime 
the  sum  of  the  aliquot  parts  of  pn  is  (pn—  i)/(p  —  i),  a  law 
simply  illustrated  by  the  cases  of  2*  and  f. 

Fermat  proposed  (1657)  two  problems:  (i)  Find  a  cube 
which,  when  increased  by  the  sum  of  its  aliquot  parts,  becomes 
a  square,  one  example  being  f  +  (i  +  7  +  f)  =  20" ;  (2)  find 
a  square  which,  when  increased  by  the  sum  of  its  aliquot  parts, 
becomes  a  cube.  Problems  of  this  general  nature  attracted  the 
attention  of  men  like  Frenicle  de  Bessy,  Lord  Brouncker, 
Wallis,  Frans  van  Schooten,  Ozanam,  and  various  later  scholars. 

Other  Subjects  of  Investigation.  Among  other  subjects  in- 
vestigated is  that  of  the  factors  of  numbers  that  can  be  ex- 
pressed in  the  form  of  an±b"\  for  example,  to  find  all  the 
prime  factors  of  245  —  i .  There  are  also  such  questions  as  the 
infinitude  of  primes  in  general ;  the  tests  for  primality ;  the  num- 
ber of  primes  between  assigned  limits;  the  curious  properties 
connected  with  the  digits  of  numbers ;  periodic  fractions ;  and 
the  general  theory  of  congruent  numbers. 

i  Published  in  1738.  2R.  C.  Archibald,  Amer.  Math.  Month.,  XXI,  247. 

3Dickson,  Hist.  Th.  Numb.,  I,  51. 


DISCUSSION  31 

TOPICS  FOR  DISCUSSION 

1.  The  numbers  three  and  seven  in  folklore  and  in  literature. 

2.  The  history  of  the  seven  liberal  arts. 

3.  Distinction  between  arithmetic  and  logistic  in  ancient  and 
medieval  times. 

4.  History  of  the  word  " algorism"  in  various  languages,  particu- 
larly with  reference  to  its  forms  and  significance. 

5.  Various  names  given  to  what  is  now  called  arithmetic  in  the 
period  known  as  the  Renaissance. 

6.  History   of   the   distinction   between   concrete   and   abstract 
numbers.    The  present  status  of  the  question,  including  that  of  opera- 
tions with  concrete  numbers. 

7.  History  of  the  finger  names  assigned  to  numbers,  and  the 
probable  reason  why  such  names  attracted  more  attention  in  early 
times  than  at  present. 

8.  Rise  of  the  idea  of  significant  figures  and  the  present  use  of 
the  term. 

9.  Probable  reasons  for  the  superstitions  in  regard  to  odd  and 
even  numbers  and  for  the  properties  assigned  to  them  at  various 
times  and  by  various  peoples. 

10.  The  gnomon  and  its  relation  to  numbers  and  to  other  branches 
of  mathematics. 

11.  Nature  of  and  probable  reason  for  certain  other  classifications 
of  number  in  ancient  times. 

12.  Probable  cause  for  the  special  interest  in  prime  numbers  ex- 
pressed by  the  ancients. 

13.  The  historical  development  of  the  interest  in  amicable  numbers 
and  the  present  status  of  the  theory. 

14.  The  historical  development  of  the  theory  of  perfect  numbers 
and  the  present  status  of  the  theory. 

15.  The  interest  in  figurate  numbers  among  the  Greeks  and  the 
traces  of  such  numbers  in  modern  times. 

1 6.  The  history  of  the  concept  of  unity  and  of  the  controversy 
with  respect  to  its  being  a  number. 

17.  Traces  of  ancient  arithmetic  and  logistic  in  modern  textbooks 
in  arithmetic  and  algebra. 

1 8.  Questions  relating  to  the  theory  of  numbers  and  attracting  the 
attention  of  mathematicians  during  the  igth  and  2oth  centuries. 


CHAPTER  II 

LOGISTIC  OF  NATURAL  NUMBERS 
i.  FUNDAMENTAL  OPERATIONS 

Number  of  Operations.  In  America  at  the  present  time  it  is 
the  custom  to  speak  of  four  fundamental  operations  in  arith- 
metic, that  is,  in  what  the  ancients  called  logistic.  This  number 
is,  however,  purely  arbitrary,  and  it  is  quite  possible  to  argue 
that  it  should  be  increased  to  nine  or  more,1  or  even  that  it 
should  be  decreased  to  one. 

The  Crafte  of  Nombrynge  (c.  1300)  enumerates  seven: 

jf  Here  tells  |?at  ]?er  ben  .7.  spices  or  partes  of  pis  craft.  The  first 
is  called  addition,  )>e  secunde  is  called  subtraction.  The  thryd  is 
called  duplacion.  The  4.  is  called  dimydicion.  The  5.  is  called  mul- 
tiplication. The  6.  is  called  diuision.  The  7.  is  called  extraccioh  of 
j>e  Rote.2 

Sacrobosco  (c.  1250)  had  already  spoken  of  nine  of  these 
operations,  —  numeration,  addition,  subtraction,  duplation,  me- 
diation, multiplication,  division,  progression,  and  the  extraction 
of  roots,3  —  and  Michael  Scott  had  done  the  same.4  This  was 
a  common  number  among  medieval  writers  and,  indeed,  in  the 


the  general  question  of  the  operations  see  J.  Tropfke,  Geschickte  der 
Element  ar-Mathematik,  I  (2)  (Leipzig,  1921),  hereafter  referred  to  as  Tropfke, 
Geschichte,  Suzan  R.  Benedict,  A  Comparative  Study  of  the  Early  Treatises  in- 
troducing into  Europe  the  Hindu  Art  of  Reckoning,  Dissertation,  Univ.  of 
Michigan,  1914. 

2R.  Steele,  The  Earliest  Arithmetics  in  English,  Oxford,  1923.  The  old  letter 
J>  is  our  t  h.  Since  it  slightly  resembles  our  letter  y,  the  old  word  J?e  (the)  is  often 
ignorantly  written  as  ye,  as  in  "ye  editor." 

3  In  his  Algorismus.    See  Volume  I,  page  222. 

4Santa-Cruz  (1594)  refers  to  this,  saying:  ".  .  .  las  especies  del  qual,  segu 
lua  de  sacrovosco,  y  Michael  Scoto,  son  nueue"  (1643  ed.,  fol.  9). 

32 


COMMON  OPERATIONS  33 

early  printed  books.1  Pacioli  (1494),  however,  claimed  credit 
to  himself  for  reducing  the  number  to  seven.2  In  due  time  a 
further  reduction  was  made  to  six,3  then  to  five,  as  with  most 
16th-century  writers,  and  then  to  four.  When  five  operations 
were  taken,  numeration  was  usually  the  first,  the  topic  properly 
including  notation.4  One  of  the  first  of  the  writers  of  any  note 
to  reduce  the  number  to  four  was  Gemma  Frisius  (1540),*  and 
such  was  his  influence  that  this  number  soon  became  common. 
There  have  been  those,  indeed,  who  gave  only  three  funda- 
mental operations,  multiplication  being  included  in  addition 
as  a  special  case.  This  number  was  given,  for  example,  by 
Elia  Misrachi  (c.  1500).° 

Duplation  and  Mediation.  The  four  operations  generally 
recognized  at  present  will  be  considered  later  in  this  chapter ; 
the  two  operations  of  duplation  (doubling)  and  mediation 
(halving),  with  the  reasons  for  their  use,  will  be  explained 
briefly  at  this  time. 

1£.g.,  Widman  (1489),  Peurbach  (c.  1460;  ist  ed.,  1492),  Huswirt  (1501), 
Tartaglia  (1556),  and  Santa-Cruz  (1594). 

2  He  says  that  nine  were  given  by  "Gioua  de  sacro  busco  e  Prodocimo  de 
beldemandis  da  padua  dignissimo  astronomo  e  molti  altri  in  loro  algorismi. 
Ma  noi  le  ditte  noue  reduremo  a  septe"  (fol.  19,  r.) . 

Since  the  names  have  some  interest,  the  list  is  reproduced :  "  La  prima  sira 
ditta  numeratioe  ouer  representatioe :  cioe  sapere  cognoscere  e  releuare  le  figure 
e  caratteri  del  nuero.  La  secoda  sira  ditta  additioe  ouer  recoglicre :  agiognere; 
sftmare  e  acozare.  La  terza  sira  ditta  subtractide  ouer  abattere :  sotrare :  cauare 
e  trare.  La  qrta  fia  ditta  multiplicatioe.  La  quinta  sira  ditta  diuisioe  ouer 
partire.  La  sexta  sira  ditta  ^gressioe.  La  septima  sira  ditta  delle  radici 
extractione."  Ibid. 

3 E.g.,  Glareanus  (1538):  "Eius  sex,  ut  in  epitome,  prosequcmur  species,  nu- 
merationem,  additionem,  subtractionem,  multiplicationem,  diuisionem,  ac  pro- 
gressionem"  (1543  ed.,  fol.  9). 

4Numeration  (Latin  numcratio,  from  numerus,  a  number)  has  lately  been 
used  to  mean  the  reading  of  numbers.  Since  medieval  writers  often  called  the 
characters  i,  2,  3,  ...  notae  (compare  the  "notes"  in  music),  the  writing  of 
numbers  has  been  called  notation.  The  distinction  is  one  chiefly  of  the  school- 
room. Ramus  (1569;  1586  ed.,  p.  i)  was  one  of  the  first  prominent  writers  to 
make  it:  "In  numero  spectatur  primum  notatio,  deinde  numeratio." 

5"Qvatuor  omnino  sunt  Arithmetices  species"  (1563  ed.,  fol.  6). 

6G.  Wertheim,  Die  Arithmetik  des  Elia  Misrachi,  Prog.,  Frankfort  a.  M., 
1893 ;  hereafter  referred  to  as  Wertheim,  Elia  Misrachi.  The  first  edition  of 
Misrachi  appeared  at  Constantinople,  1532  or  1533;  the  second,  at  Basel,  i546- 


34  FUNDAMENTAL  OPERATIONS 

The  Egyptians  often  multiplied  by  continued  doubling,1  thus 
saving  the  trouble  of  learning  a  multiplication  table.  This  was 
particularly  convenient  in  working  on  the  abacus.  On  this  ac- 
count duplation  was  generally  recognized  as  a  separate  topic 
until  the  i6th  century.  Moreover,  the  Egyptian  tables  of  meas- 
ure were  commonly  arranged  so  as  to  make  doubling  and  halving 
operations  of  great  importance.2  This  method  continued  as 
long  as  the  abacus  was  in  use,  and  persisted  for  some  time  after 
that  instrument  was  generally  abandoned.  An  interesting  illus- 
tration of  mediation  is  seen,  for  example,  in  official  papers  of 
Russia  prior  to  the  time  of  Peter  the  Great  (1672-1725),  the 
word  "half "  being  repeated  as  many  as  ten  times  to  indicate  a 
certain  division.3 

The  use  of  duplation  and  mediation  is  seen  in  many  of  the 
Arab  works,4  and  this  fact  influenced  such  medieval  transla- 
tors as  Johannes  Hispalensis5  (c.  1140)  and  Adelard  of  Bath6 
(c.  1120).  The  processes  were  common  in  the  theoretical  works 
of  the  1 5th  century7  but  not  in.  the  commercial  arithmetics,  at 
least  in  Italy.  The  early  printed  books  of  Germany  were  less 
progressive  in  this  respect  than  those  of  other  countries,  partly 


iThus,  7  x  15  =  2  x  2  x  15  +  2  x  15  +  15. 

2  See  the  common  use  of  the  fractions  £,  \,  |,  T\,  in  the  Edfu  Survey,  in 
H.  Brugsch,  Thesaurus  Inscriptionum  Mgyptiacarum,  Leipzig,  1883-1891, 
Vol.  Ill;  H.  Brugsch,  Numerorum  apud  Veteres  JEgyptios  Demoticorum  Doc- 
trina,  Berlin,  1849;  T.  E.  Peet,  The  Rhind  Mathematical  Papyrus,  London,  1923; 
hereafter  referred  to  as  Peet,  Rhind  Papyrus.  For  the  survival  of  doubling  and 
halving  until  the  present  time,  see  the  tables  in  Mahmoud  Bey,  "Le  systeme 
metrique  actuel  d'figypte,"  Journal  Asiatique,  I  (7),  69,  82. 

3V.  V.  Bobynin,  "Esquisse  de  1'histoire  du  calcul  fractionnaire,"  Bibl.  Math., 

X   (2),  97- 

4  E.g.,  al-Nasavi  (c.  1025),  on  whose  work  see  F.  Woepcke,  Journal  Asiatique, 
I  (6),  496;  al-Iiassar  (i2th  century),  on  whose  work  see  M.  Steinschneider, 
Abhandlungen,  III,  10;  al-Khowarizmi  (c.  825),  on  whose  work  and  al-Hassar's 
see  Suter,  Bibl.  Math.,  II  (3),  12. 

5 Who,  however,  speaks  of  them  merely  as  special  cases  of  multiplication  and 
division.  See  Boncompagni,  Trattati,  II,  38.  Similarly,  as  to  Gernardus  (i3th 
century?),  see  G.  Enestrom,  Bibl.  Math.,  XIII  (3),  289,  292. 

6  Boncompagni,  Trattati,  I,  10. 

7E.g.,  in  the  Rollandus  MS.  (1424)  ;  see  Rara  Arithmetica,  p.446.  Rollandus 
gives:  "addere.  sbfhere.  mediare.  duplare.  diuidere.  mltiplicare.  et  radices 
invenir6"  (fol.  2), 


NAMES  OF  THE  OPERATIONS  35 

because  of  the  continued  use  of  the  abacus  in  that  part  of 
Europe,  and  so  these  two  processes  are  found  in  the  works  of 
Tzwivel  (1505),  Kobel  (1514),  Grammateus  (1518),  Riese 
(1522),  Rudolff  (1526),  and  various  other  writers  of  that 
period,  often  with  a  statement  that  they  are  special  forms  of 
multiplication  and  division.1  Stifel  (1545)  uses  them  only 
apologetically,  and  Scheubel  (1545)  omits  them  entirely.2  They 
are  rarely  found  in  any  of  the  printed  arithmetics  of  Italy,3 
Spain,4  France,  or  England.  It  is  a  curious  fact,  however,  that 
Recorde  (c.  1542)  omits  them  with  integers  but  includes  them 
with  fractions, — a  vagary  that  endured  at  least  as  late  as  the 
1668  edition  of  his  Ground  of  Aries.  His  example  was  followed 
by  Baker  (1568),  who  had  the  notion  that  only  fractions  should 
be  used  with  fractions,  saying :  "If  you  will  double  anye  broken 
number  you  shall  divide  ye  same  by  £,"  and  giving  triplation 
and  quadruplation  in  the  same  way.  Gemma  Frisius  (1540) 
did  as  much  as  any  other  Continental  writer  to  show  the  absurd- 
ity of  following  those  "  stupid  people "  who  would  include 
these  operations.5 

Names  of  the  Operations.  The  awkward  expression  "the  four 
fundamental  operations"  is  modern.  Several  others  used  in 
the  past  possess  the  merit  of  greater  brevity,  and  some  of 
these  are  still  found  in  various  languages.  A  common  name 
is  "species,"  a  term  of  the  i3th  century  and  made  popular 
in  the  i6th  century  by  the  works  of  Riese  (1522)  and  Gemma 
Frisius  (1540). 6  Ramus  (1569)  used  both  "parts"  and 

iaDupliren  heist  zwifeltigen/  ist  nichts  anders  dafi  ein  zal  mit  2  multiplicirn. 
Medijren  heist  halb  machen  od'  halbiren/  ist  nichts  anders/  dan  ein  zal  in  2 
abteilen."  Kobel  (1514). 

2"De  duplatione  porro  &  Mediatione,  cum  ilia  multiplicationis,  haec  uero 
diuisionis  pars  sit,  scribere  quicq},  necesse  non  fuit."  "Tractatus  secundus"  of 
the  1545  edition  of  his  De  Numeris. 

3 E.g.,  Pacioli  (1494):  "Ma  noi  le  ditte  noue  reduremo  a  septe.  Peroche  la 
duplatioe  Iplicita  in  la  multiplicatioe :  ela  mediatioe  nella  diuisioe"  (fol.  19,  O. 

*E.g.,  Santa-Cruz  (1594):  "Y  porq  el  doblar  no  se  distingue  del  multiplicar, 
ni  el  mediar  del  partir"  (1643  ed.,  fol.  9). 

5 "Quid  vero  mouerit  stupidos  illos  nescio"  (1563  ed.,  fol.  12). 

6 In  the  Latin  editions:  "De  speciebus  Arithmetices";  "Vocamus  autem 
species  certas  operandi  .  .  ."  In  the  Italian  translation:  "Delle  Specie  dell'  Arit- 
metica." 


36  READING  AND  WRITING  NUMBERS 

"species,"1  while  most  of  the  Spanish2  and  Dutch3  arithmeti- 
cians of  the  1 6th  and  iyth  centuries  used  the  latter  only. 

A  common  Italian  name  in  the  i6th  century  was  "acts,"4 
although  "passions"  was  also  used.5  When  Clavius  wrote  his 
algebra  (1608),  he  used  the  word  operationes,  and  it  is  probable 
that  this  word  worked  down  from  algebra  to  arithmetic. 

Sequence  of  Operations.  Our  present  traditional  sequence 
has  by  no  means  been  generally  recognized,  particularly  in  rela- 
tion to  fractions.  Although  all  writers  place  notation  and  a 
certain  amount  of  addition  first,  there  has  been  little  further 
uniformity.  Abraham  bar  Chiia  (c.  1120),  Rabbi  ben  Ezra 
(£.1140),  and  Fibonacci  (1202),  for  example,  use  this  se- 
quence: multiplication,  division,  addition,  subtraction,  frac- 
tions, proportion,  and  roots.0  Gramma teus  (1518)  used  the 
order:  addition,  multiplication,  subtraction,  division,7 — an 
order  which  has  much  to  commend  it. 


2.  READING  AND  WRITING  NUMBERS 

Babylonian  Numerals.  Since  the  early  Babylonians  were 
without  papyrus  or  parchment,  they  doubtless  followed  the  cus- 
tom of  most  other  early  peoples  and  wrote  upon  leather.  Living 
on  an  alluvial  plain,  they  had  no  convenient  access  to  stone  for 
the  purpose  of  permanent  inscriptions,  except  in  the  northern 
region,  and  so  they  also  resorted  to  the  use  of  clay.  They 
wrote  by  pressing  into  the  clay  with  a  stylus,  the  result  being 
wedge-shaped  (cuneiform)  characters.  These  tablets  were  then 
baked  in  the  sun  or  in  a  kiln  of  some  kind,  and  thus  they 

^"Alii  faciunt  arithmeticae  partes  vel  species  .  .  . ,"  Arithmeticae  libri  dvo, 

p.  in. 

2E.g.,  Santa-Cruz  (1594):  ".  .  .  las  especies  del  qual  .  .  .  son  nueue." 
3Thus  Stockmans   (1589),  Houck   (1676),  and  others  speak  of  the  "vier 

specien." 

4So  Sfortunati  (1534)  speaks  of  the  "Cinque  atti  dell'  arithmetica." 
5Tartaglia  (1556)  prefers  "atti"  but  says  that  "altri  gli  dicone  Passioni  del 

numero"  (1592  ed.,  fol.  5).   The  word  comes  from  the  Latin  passio,  used  by 

late  Latin  writers  to  mean  "phenomenon";  originally,  something  endured. 
°M.  Steinschneider,  Abhandlungen,  III,  107.  7iS35  ed.,  fol.  Aiii. 


BABYLONIAN  NUMERALS  37 

became  fairly  permanent  records.  For  relatively  small  num- 
bers the  numerical  system  was  simple,  consisting  of  the  following 
characters:  v  , 


These  symbols  had  different  numerical  meanings,  however. 
The  Y  stood  not  only  for  i  but  also  for  60,  3600,  12,960,000, 
and  in  general  for  6on.  The  «<  stood  for  10  •  6ow,  and  hence  for 
10,  600,  36,000,  •  •  •.  In  every  case  the  context  was  depended 
upon  to  determine  which  value  was  to  be  taken.  Furthermore, 
we  often  find  the  units  represented  by  horizontal  strokes,  10 
represented  by  a  vertical  crossed  by  a  horizontal  stroke  (like 
a  plus  sign),  20  represented  by  a  vertical  crossed  by  two 
horizontals,  and  so  on.  In  certain  tablets  71  is  represented 
by  i  (for  60),  the  above  symbol  for  10,  and  a  horizontal  stroke 
for  the  unit. 

Y       YY       YYY       T      VYY      W      ^      V?       555 

BABYLONIAN  NUMERALS  FROM    I    TO   9 

The  forms  vary  in  shape,  but  this  gives  an  idea  of  the 

simpler  numerals  in  common  use.   For  the  correct  forms  as 

seen  in  the  clay  tablets,  see  page  39 

In  writing  their  numerals  the  Babylonians  made  a  slight  use 
of  the  subtractive  principle  with  which  we  are  familiar  in  con- 
nection with  the  Roman  notation.  For  example,  the  XIX  of 
the  Romans  is  equivalent  to  XX  —  I,  a  device  that  was  antici- 
pated some  two  thousand  years  by  the  Babylonians,  who  wrote 
«Y>v  for  19,  the  symbol  Y*~  (lal  or  Id}  meaning  minus.1  In 
this  case,  then,  we  have  20  —  i,  or  19.  It  has  been  suggested 
that  one  reason  for  writing  19  as  20  —  i  instead  of  10  +  9  is  that 

1  There  are  numerous  forms  for  this  symbol,  some  of  them  very  complex. 
See  H.  V.  Hilprecht,  Mathematical,  Metrological,  and  Chronological  Tablets 
from  the  Temple  Library  of  Nippur,  p.  23  (Philadelphia,  1906)  ;  hereafter  re- 
ferred to  as  Hilprecht,  Tablets.  See  also  G.  Reisner,  "  Altbabylonische  Maasse 
und  Gewichte,"  Sitzungsberichte  der  k.  Preussischen  Akad.  der  Wissensch., 
p.  417  (Berlin,  1896);  G.  Contenau,  "Contribution  a  1'histoire  economique 
d'Umma,"  Bibliotheque  de  Vfccole  des  hautes  etudes,  fascicule  219  (Paris,  I9IS)> 
with  excellent  facsimiles  of  various  numeral  forms.  See  especially  Plate  XIV  for 
the  representation  of  71  referred  to  in  the  text.  The  tablets  date  from  c.  2300 
to  c.  2200  B.C. 

ii 


38  READING  AND  WRITING  NUMBERS 

it  was  an  unlucky  number.  The  nineteenth  day  of  a  lunar  month 
was  the  forty-ninth  day  from  the  beginning  of  the  preceding 
month,  and  this  forty-ninth  day  was  one  to  be  specially  avoided. 
To  avoid  writing  19,  therefore,  the  Babylonians  resorted  to  writ- 
ing 20  —  i.  This  does  not,  however,  account  for  such  common 
forms  as  60  —  |  for  59!  and,  as  we  shall  presently  see,  the  ex- 
istence of  the  subtractive  principle  is  easily  explained  on  other 
and  more  rational  grounds. 

Since  the  larger  numbers  were  used  by  relatively  few  scholars, 
there  was  no  compelling  force  of  custom  to  standardize  them. 
The  variants  in  these  cases  are  not  of  importance  for  our  pur- 
poses, and  simply  a  few  of  the  numerals  will  serve  to  show  their 
nature.  These  illustrations1  date  from  c.  2400  B.C. 

O  3600 

3s>  36,000,     i.e.,  3600  x  10 

f$  72,000,     i.e.,  3600  x  (10  -f  10) 

OJY-  216,000 

Y<Y<Y<Y<  2400 

YYV<"Y*  171^  i.e.,  2x60+50  +  1+! 

9  10 

i9  m        36 

19,  i.e.y  20  —  i 

18,  i.e.,  20  —  2 

17,  i.e.,  20  —  3 

130-^  i.e.,  2x60  +  10  +  ! 

W"  S3,  *.e.,  50+3 

As  mentioned  later,,  the  Babylonians  also  used  a  circle  for 
zero,  at  least  to  the  extent  that  they  employed  it  to  represent 
the  absence  of  number,  but  it  played  little  part  in  their  system 

1G.  A.  Barton,  Haver  ford  College  Library  Collection  of  Cuneiform  Tablets, 
Part  I,  Philadelphia  [1905] ;  Allotte  de  la  Fuye,  "En-e-tar-zi  pat6si  de  Lagas," 
in  the  Hilprecht  Anniversary  Volume,  p.  121  (Chicago,  1909). 


BABYLONIAN  AND  CHINESE  NUMERALS 


39 


of  notation.    More  commonly  a  circle  simply  stood  for  10, 
particularly  in  the  early  inscriptions,  as  shown  on  page  38. 


TABLET  FROM  NIPPUR 

Contains  divisors  of  6o4,  the  quotients  being  in  geometric  progression.  Date 
c.  2400  B.C.  The  top  line  reads  2(  •  60),  s(  •  60),  and  12 (  •  60).  The  left-hand 
figure  is  the  original ;  the  right-hand  one  is  a  drawing.  Courtesy  of  the  University 

of  Pennsylvania 

Chinese  Numerals.    The  present  forms  of  the  Chinese  nu- 
merals from  i  to  10  are  as  follows: 


The  number  789  may  be  written  either  from  the  top  downwards 
or  from  left  to  right,  as  follows : 


40  READING  AND  WRITING  NUMBERS 

The  second  character  in  the  number  as  written  on  the  preced- 
ing line  means  hundred,  and  the  forth  character  ten. 

It  will  be  observed  that  the  figure  for  4,  probably  four  vertical 
marks  in  its  original  form,  resembles  the  figure  for  8  inclosed 
in  a  rectangle.  On  account  of  this  the  Chinese  have  given  it 
the  fanciful  name  of  "eight  in  the  mouth."1 

Chinese  merchants  also  use  the  following  forms  for  figures 
from  i  to  10:  ,  „  (|J  x  ^  ^  ±  ±  ^  + 

and  they  have  special  symbols  for  100,  1000,  and  io?ooo, 
besides  those  in  which  a  circle  is  used  for  zero. 

These  symbols  are  not  the  same  as  the  ancient  forms,  but  our 
knowledge  of  the  latter  is  imperfect.2  There  are  many  variants 
of  each  of  the  characters  given  above,  as  when  Ch'in  Kiu-shao 
(1247)  used  i  for  5,  and  both  x  and  *  for  p.3  The  numerals 
on  the  early  coins  also  show  the  variations  that  are  found 
from  time  to  time.  In  the  second  century  B.C.,  for  example, 
we  find  the  5  given  in  the  so-called  seal  characters  in  the  form 
H , — a  form  which  was  used  for  hundreds  of  years.4 

Rod  Numerals.  There  were  also  numerals  represented  by 
rods  placed  on  the  counting  board, — a  device  which  will  be 
described  in  Chapter  III.  These  numerals  appear  in  the 
Wu-ts'ao  Suan-king,  which  may  have  been  written  about  the 
beginning  of  our  era,  or  possibly  much  earlier,  and  are  found 

!L.  Vanh6e,  in  Toung-Pao,  reprint.  On  the  general  subject  of  the  Chinese 
numerals  in  their  historical  development  the  standard  work  is  that  of  F.  H.  Chal- 
fant  in  the  Memoirs  oj  the  Carnegie  Museum,  Vol.  IV,  No.  i,  and  Plate  XXIX. 

2 We  have,  however,  various  records  going  back  to  the  early  part  of  the 
Christian  Era.  For  example,  the  Metropolitan  Museum,  New  York,  has  a  land 
grant  of  403  in  which  the  forms  of  the  numerals  are  almost  the  same  as  those 
now  in  use. 

3Chalfant,  loc.  cit.\  Y.  Mikami,  The  Development  oj  Mathematics  in  China 
and  Japan,  p.  73  (Leipzig,  1913) ;  hereafter  referred  to  as  Mikami,  China.  On 
the  general  topic  see  S.  W.  Williams,  The  Middle  Kingdom,  New  York,  1882; 
1895  ed.,  I,  619;  hereafter  referred  to  as  Williams,  Middle  Kingdom.  J.  Hager, 
An  Explanation  of  the  Elementary  Characters  oj  the  Chinese,  London,  1801 ; 
J.  Legge,  The  Chinese  Classics,  2d  ed.,  I,  449  (Oxford,  1893). 

4H.  B.  Morse,  "Currency  in  China,"  Journal  of  the  North  China  Branch 
of  the  R.  Asiat.  Soc^  Shanghai,  reprint  (n.d.).  Valuable  on  the  history  of 
Chinese  money  and  weights. 


MONOGRAM    FORMS    OF    CHINESE    NUMERALS 

The  Japanese  sangi  were  sticks  used  for  representing  numbers  and  were  de- 
scended from  the  "bamboo  rods"  of  the  ancient  Chinese.  They  gave  rise  to  a 
sangi  method  of  writing  numbers.  From  a  work  by  the  Japanese  mathematician 
Fujita  Sadasuke  (1779)  in  Chinese  characters.  The  number  in  the  first  line  at  the 

top  is  46,431 


42  READING  AND  WRITING  NUMBERS 

even  as  late  as  the  ipth  century.  The  oldest  forms  for  the  units 
are  commonly  arranged  as  follows  : 

I  I!   Ill   Illl   Hill  TTT  TIT  TITT 
In  the  tens'  column  the  symbols  usually  appear  as  follows  : 

_   ___    ^      ===    J_  JL  =!=    =b 

the  arrangement  thereafter  alternating,  the  hundreds  being  like 
the  units,  and  so  on.  Sometimes  the  =f  was  used  instead  of 
TT  for  7  hundreds,  and  so  on,  and  similarly  for  8  and  9.  By  the 
common  plan  the  number  7436  would  appear  as  J=  IIII==T. 
In  this  system  the  zero  takes  the  form  of  a  circle  in  the  Sung 
Dynasty  (950-1280),  as  is  seen  in  a  work  of  I247,1  where  the 
subtraction  1,470,000  —  64,464  =  1,405,536  appears  as 

I^TOOOO 
lsO  =  ll!llsT  TXIIH-LX 

with  two  forms  for  4. 

These  numerals  were  frequently  written  in  the  monogram 
form;2  for  example,  123,456,789  appears  as  HIHIIIHdf- 

Hindu  Numerals.  The  history  of  those  Hindu-Arabic  nu- 
merals which  may  have  developed  into  our  modern  European 
forms  is  considered  later.  It  should  be  said,  however,  that 
there  are  various  other  systems  in  use  in  India  and  neighboring 
countries.  Of  these  the  most  interesting  is  the  modern  Sanskrit, 
the  numerals  being  as  follows  : 


These  characters  are  evidently  related  to  the  early  Brahmi 
forms  which  are  mentioned  later. 

aThe  Su-shu  Kiu-ch'ang  of  Ch'in  Kiu-shao.  See  A.  Wylie,  Chinese  Researches, 
Pt.  Ill,  p.  159  seq.  (Shanghai,  1897)  ;  L.  Vanhee,  in  Toung-Pao,  reprint,  thinks 
that  the  zero  reached  China  from  India  somewhat  earlier. 

2  A.  Vissiere,  Recherches  sur  I'origine  de  I'abaque  chinois  (Paris,  1892),  re- 
print from  the  Bulletin  de  Geographic]  hereafter  referred  to  as  Vissiere,  Abaque. 


HINDU  NUMERALS  43 

Of  the  numerals  of  the  same  general  character  and  in  use  in 
parts  of  Asia  adjacent  to  India,  the  following  are  types: 

or         23456789       10 

Siam  O  < 

Burma          Q      9       J 

Malabar        ^  ^     ft 

Thibet  °       I*     £    ^^   ^^ft  f   £     7 

Ceylon  C 

Malayalam  f 


Their  history  has  no  particular  significance,  however,  in  a  work 
of  this  nature,  since  the  forms  are  local  and  are  relatively 
modern.1 

An  American*  Place  Value.  When  Francisco  de  Cordoba 
landed  the  first  Spanish  expedition  on  the  coast  of  Yucatan,  in 
1517,  he  found  the  relics  of  a  highly  developed  civilization,  that 
of  the  Maya,  which  had  received  its  deathblow  in  the  wars  of 
the  preceding  century.2  Within  a  few  years  after  the  European 
invasion  the  independence  of  the  Maya  was  completely  lost. 
In  1565  Diego  de  Landa,  bishop  of  Merida,  in  northern  Yuca- 
tan, wrote  a  history  of  these  people,3  so  that  our  knowledge  of 
their  achievements  goes  back  to  about  the  beginning  of  the 
period  of  European  influence.  They  had  an  elaborate  calendar 
before  the  Spaniards  arrived,  and  capable  investigators  have 
asserted  that  the  Maya  cycle  began  as  early  as  3373  B.C. 

aOne  of  the  best  general  works  on  Eastern  notation  is  that  of  A.  P.  Pihan, 
Expose  des  Signes  de  Numeration  usites  chez  les  Peuples  Orientaux  Anciens  et 
Modernes,  Paris,  1860,  with  many  tables. 

2S,  G.  Morley,  An  Introduction  to  the  Study  of  the  Maya  Hieroglyphs, 
p.  6  (Washington,  1915).  This  work  should  be  consulted  for  details  respecting 
this  entire  topic.  Authorities  vary  as  to  the  plural  form  of  Maya,  some  giving 
Mayas  and  others  Maya.  See  also  C.  Thomas,  "Numeral  Systems  of  Mexico 
and  Central  America,"  Annual  Report  of  the  Bureau  of  American  Ethnology, 
XIX  (1897^1898),  853  (Washington,  1900). 

3  In  his  Relacion  de  las  Cosas  de  Yucatan,  a  work  not  printed  until  1864. 


44 


READING  AND  WRITING  NUMBERS 


The  Maya  counted  essentially  on  a  scale  of  20,  using  for  their 
basal  numerals  two  elements,  a  dot  (•)  and  a  dash  ( — ),  the 
former  representing  one  and  the  latter  five.  The  first  nine- 
teen numerals  were  as  follows,  reading  from  left  to  right : 


There  were  numerous  variants  of  these  forms,1  but  these  offer 
no  special  peculiarities  which  we  need  consider. 

The  most  important  feature  of  their  system  was  their  zero, 
the  character  <^£^>,  which  also  had  numerous  variants.  Since 
their  scale  was  vigesimal,  they  wrote  20  as  we  write  10,  using 
their  characters  for  i  and  zero.2  The  following  table  shows  the 
general  plan  that  was  used  when  they  wrote  on  flexible  material : 


Numerals 

« 

EEEE 

3? 

*=2~5^ 
^^^^^ 

=== 

Our  forms 

i 

i 

15.20 

360 
o 

19.360 
13.20 

0 

I  *7 

0 

o 

13 

Values 

20 

37 

300 

360 

7^3 

We  see  here  a  fairly  well  developed  place  value,  the  lowest 
order  being  units  from  i  to  19,  the  next  being  2o's  from  1-20 


Morley,  loc.  tit.,  p.  89,  which  should  be  consulted  for  a  description  of 
the  system. 

2  Their   special   hieroglyphic  for   20,  used  for   certain   purposes,   need   not 
concern  us.   On  the  word  "hieroglyphic'*  see  page  45,  note  2. 


MAYA  AND  EGYPTIAN  NUMERALS 


45 


to  17  •  20;  the  next  being  360*5  from  i  •  360  to  19  •  360;  the 
next  being  7200%  and  so  on/  representing  a  very  satisfactory 
system.  There  is  no  evidence  in  any  extant  record  that  it  was 
used  for  purposes  of  computation,  its  use  in  the  texts  being  merely 
to  express  the  time  elapsing  between  dates.  The  fact,  however, 
that  the  pebble  and  rod  are 
apparently  the  basal  elements 
in  the  writing  of  numbers 
leads  us  to  feel  that  we  have 
in  these  numerals  clear  evi- 
dence of  the  early  use  of  an 
abacus.  If,  as  many  ethnol- 
ogists believe,  there  is  a  con- 
nection between  the  Japanese 
and  certain  of  our  primitive 
Americans,  the  use  of  the  rods 
may  be  traced  back  to  Asia. 

Egyptian  Numerals.  The 
Egyptians  had  four  materials 
upon  which  they  could  con- 
veniently record  events.  One 
of  these  was  stone,  a  medium 
supplied  by  the  quarries  along 
certain  parts  of  the  Nile.  An- 
other medium  was  papyrus,  a 

kind  of  paper  made  from  strips  of  the  pulp  of  a  water  reed 
which  was  apparently  more  common  at  one  time  than  it  is  at 
present.  The  other  two  common  materials  were  wood  and  pieces 
of  pottery.  Leather  does  not  seem  to  have  been  so  commonly 
used  as  in  other  countries. 

In  writing  on  stone  the  Egyptians  took  time  for  the  work  and 
made  their  characters  with  great  care.  These  characters  are 
called  hieroglyphics.2  The  hieroglyphic  characters  were  com- 
monly written  from  right  to  left,  but  also  from  left  to  right. 

1For  complete  description  see  Morley,  loc.  cit.,  pp.  129-133. 
2 The  sacred   inscriptions;   from   the   Greek  Up6s  (hieros1),  sacred  +  y\t<j>€tv 
(gly'phein),  to  carve. 


EARLY    FORMS    OF    COMMON 
EGYPTIAN    NUMERALS 

From  a  piece  of  pottery   of  the  First 

Dynasty,  c.  3400  B.C.    The  symbols  for 

10  and  100  are  repeated  several  times 


46  READING  AND  WRITING  NUMBERS 

In  the  earlier  inscriptions  they  are  often  written  from  the  top 
down.  This  accounts  for  the  various  ways  of  writing  the  simple 
numerals,  a  character  often  being  found  facing  in  different 
directions.  For  our  present  purposes  it  suffices  to  give  the 
ordinary  form  of  hieroglyphic  numerals,1  as  follows: 


ii 


I          II          Iti  Illl  '"        "!  ""  !!!!  ft 

I          (I          III  Illl  ii         in  in  mi  in         I  I 

123  4  5        6  7  8  9        I0 

in    nn    nn    RR  nnnnnnn   9    99  i  f 


12     2O    40    70   IOO   2OO   IOOO   IO,OOO 


EGYPTIAN  NUMERALS 
Numerals  reading  from  left  to  right.   From  the  walls  of  a  temple  at  Luxor 

1  These  are  as  given  in  A.  Eisenlohr,  Ein  mathematisches  Handbuch  der  alien 
Aegypter,  2d.  ed.,  table  following  p.  8  (Leipzig,  1877) ;  hereafter  referred  to  as 
Eisenlohr,  Ahmes  Papyrus.  See  also  Peet,  Rhind  Papyrus;  J.  De  Morgan,  L'Hu- 
manitt  PrMstorique,  p.  115  (Paris,  1921). 


EGYPTIAN  AND  GREEK  NUMERALS 


47 


There    were    higher    numerals,    but    the    above    will    serve 
to   show   the   general    nature   of    the    characters    employed. 

While  the  hieroglyphic  forms  were 
used  in  writing  inscriptions  on  stone 
and  in  elaborate  treatises  on  papyrus, 
other  forms  were  early  developed  for 
rapid  writing  on  papyrus,  wood,  and 
pieces  of  pottery.  There  were  two  forms 
of  this  writing,  the  hieratic  (religious) 
and  the  demotic  (popular).  The  for- 
mer was  a  cursive  script  derived  from 
the  hieroglyphic,  and  the  latter  was  a 
somewhat  later  form  of  the  hieratic, 
beginning  in  the  yth  century  B.C.  After 
the  demotic  forms  came  into  general 
use,  the  hieratic  was  reserved  for 
religious  purposes. 

The   hieratic   writing   usually   pro- 
ceeded from  right  to  left,  although  in 
early  times  it  is  occasionally   found 
running  from  the  top  down.    The  numerals  to  10  were  of  the 
following  forms : x 

/     (\     111     MM 

123  4          5  6  7         8  9         10 

The  demotic  forms  offer  no  peculiarities  of  special  interest.2 

Greek  Numerals.  The  first  numeral  forms  of  the  Greeks  seem 
to  have  been  such  upright  strokes  as  were  used  in  all  Mediter- 
ranean countries,  and  perhaps  represented  the  fingers.  These 
strokes  were  repeated  as  far  as  the  needs  of  the  primitive  in- 
habitants required.  For  example,  in  a  stele  from  Corinth,  of 
about  the  sth  century  B.C.,  there  is  the  numeral  Mil  I  III,  referring 

1  These  are  taken  from  the  Ebers  Papyrus  as  copied  by  Eisenlohr.   Naturally 
the  forms  varied  with  different  scribes. 

2  For  a  careful  study  of  these  forms,  with  numerous  facsimiles,  see  H.  Brugsch, 
Numerorum  apud  Veteres  Mgyptios  Demoticorum  Doctrina,  Berlin,  1849. 


HIEROGLYPHIC  FOR  6000 
(ABOUT  500  B.C.) 

The  meaning  is, "  The  Falcon 
King  led  captive  6000  men 
of  the  Land  of  the  Harpoon 
Lake,"  there  being  a  harpoon 
just  below  this  in  the  original 
inscription 


EARLY  CYPRIOTE  NUMERALS 

From  a  fragment  of  a  temple  record  found  on  the  island  of  Cyprus.   In  the  last 
two  lines  the  numeral  for  6  (III    III)  appears  twice.    Courtesy  of  the  Metro- 
politan Museum  of  Art,  New  York 


EARLY  CYPRIOTE  NUMERALS 

The  lower  part  of  the  fragment  shown  above.  The  numerals  are  the  same  as 
those  on  the  tablets  found  at  Knossos,  Crete,  where  O  is  used  for  1000,  O  for  100, 
-  for  10,  and  I  for  i.  The  number  4  (Mil)  is  in  the  first  line  and  the  number  14 
(MM — )  in  the  line  next  to  the  last.  The  Phoenicians  also  used  these  symbols  for 
ten  and  one.  Courtesy  of  the  Metropolitan  Museum  of  Art,  New  York 


GREEK  NUMERALS 


49 


EARLY  CYPRIOTE  NUMERALS 

From  a  fragment  of  a  receptacle  in  a  sanc- 
tuary.  The  inscription  reads,  "Zeus's  por- 
tion of  wine  is  three  measures."    Courtesy 
of  the  Metropolitan  Museum  of  Art 


to  a  fine  of  eight  obols  for 
intruding  on  certain  prop- 
erty.1 Inscriptions  illus- 
trating this  usage  are  found 
not  only  in  Greece  but  in 
various  islands  of  the  east- 
ern Mediterranean  Sea,  as 
shown  in  the  illustrations 
of  monumental  records 
from  Cyprus.  By  the  time 
Greece  had  reached  the 
period  of  her  intellectual 
ascendancy  there  had  de- 
veloped a  system  of  nu- 
merals formed  from  initial 

letters  of  number  names.    These  forms  appear  in  records  of 

the  third  century  B.C.,  and  were  probably  in  use  much  earlier, 

although  the  custom  of  writ- 
ing large  numbers  in  words 

seems  to  have  been  general. 

Many  generations  later  the 

system  was  described  so  fully 

by  Herodianus,  a  prominent 

grammarian  of  the  latter  part 

of  the  second  century,  that 

the  symbols  were  thereafter 

known  as  Herodianic  numer- 
als, although  this  name  has  no 

worthy  sanction.    In  recent 

times  they  have  been  known 

as  Attic  numerals,  since  they 

are   the   only   pre-Christian 


number  forms  found  in  Attic 
inscriptions.  The  system  is 
also  known  as  the  acrophonic 
(initial)  system,  the  initials 


GREEK  NUMERALS  OF  THE 
PTOLEMAIC  PERIOD 

On  an  icosahedral  die  of  the  Ptolemaic  Pe- 
riod in  Alexandria,  just  before  the  Christian 
Era.  Such  dice  are  occasionally  found, 
usually  made  of  basalt  or  quartz.  This  one 
is  basalt, whitened  for  the  purpose  of  pho- 
tographing. From  the  author's  collection 


1  American  Journal  of  Archeology  (1919),  p.  353. 


SO  READING  AND  WRITING  NUMBERS 

of  the  several  number  names,  as  of  TreVre  (pen'  fie),  five,  being 
used,  singly  or  in  combination,  in  the  following  manner  : 

P,  an  old  form  for  TT,  the  letter  pi,  initial  of  TTENTE 
(perite),  five,  used  as  a  numeral  for  5; 

A,  the  capital  delta,  initial  of  AEKA  (dek'a),  ten,  used 
as  a  numeral  for  10;  it  is  often  written  like  O  in 
the  Greek  papyri,  and  an  inscription  at  Argos  has  O  ; 

H,  the  old  Attic  breathing,  like  our  h,  later  represented 
by  ',  initial  of  HEKATON  (hekatori),  hundred; 

X,  the  capital  chi,  initial  of  X\MO\(chil'ioi),  thousand; 

M,  the  capital  mu,  initial  of  MTPIOI  (myr'ioi),  ten 
thousand. 

These  numerals  were  frequently  combined,  thus  : 

F1  or  F,  pente-deka,  was  used  for  50  ; 
P,  pente-hekaton,  was  used  for  500  ; 

and  so  on  for  other  numbers. 

The  forms  of  the  letters  varied  in  different  cities  and  states  of 
Greece,  but  the  variants  need  not  concern  us  in  this  description.1 

The  following  will  show  how  the  characters  were  used  : 


Anil!  -19  MM  MM  -=40,000 

PAAAA  ^90  r  =  50,000 

but  in  the  manuscripts  the  forms  vary  so  much  as  often  to  be 
exceedingly  difficult  to  decipher. 

1G.  Friedlein,  Die  Zahlzeichen  und  das  elementare  Rechnen  der  Griechen  und 
Romer,  Erlangen,  1869;  F.  G.  Kenyon,  Paleography  of  Greek  Papyri,  Oxford, 
1899;  E.  S.  Roberts,  Greek  Epigraphy,  p.  96  (Cambridge,  1887);  J.  Gow, 
"The  Greek  Numeral  Alphabet,"  Journal  of  Philology  (1884),  p.  278;  S.  Rei- 
nach,  Trait6  d'fipigraphie  Grecque,  pp.  216,  218  (Paris,  1885)  ;  J.  P.  Mahaffy, 
"On  the  Numerical  Symbols  used  by  the  Greek  Historians,"  Trans,  of  the  Royal 
Soc.  of  Literature,  XXVII  (2),  160;  Heath,  History,  I,  29.  The  best  modern  treat- 
ment is  that  of  M.  N.  Tod,  "Three  Greek  Numeral  Systems,"  Journal  of  Hellenic 
Studies,  XXXIII,  27,  and  "The  Greek  Numeral  Notation,"  Annual  of  the  British 
School  at  Athens,  XVIII,  98.  On.  the  numerals  of  Crete  see  Sir  A.  J.  Evans,  The 
Palace  of  Minos,  p.  279>(London,  1921),  and  Scripta  Minoa,  p.  258  (Oxford,  1909)  . 


GREEK  NUMERALS  51 

To  these  may  be  added  the  following  characters  related  to 
numerical  work : 

T~  talent  and  also  \  obol 
h  =  drachma 

I  =  obol,  with  D  or  C  for  |  obol 
3  ^stater,  so  that  3333333  =  7  staters 
and  HHAAAAr  333  =  248  staters 
F1  =  5  talents,  ^  =  10  talents,  H  =  100  talents 

Contemporary  with  the  development  of  the  Ionic  alphabet 
we  find  numerical  values  assigned  to  the  letters,  somewhat  as 
we  use  letters  to  number  the  rows  of  seats  in  an  assembly  room. 
The  oldest  forms  that  we  have  are  substantially  as  follows : 

A=  i 

B  =  2 


H=    7 

N  =  13 

T=  19 

e=  8 

I  =  14 

Y=  20 

1=    9 

0=i5 

4»=2I 

K  =  10 

n  =  16 

X=22 

A=n 

P=i7 

Y=23 

1  =  6  M  — 12  3^i8  ft  ^24* 

These  were  used  very  early,  but  the  system  was  manifestly 
of  no  value  for  computation.  A  more  refined  alphabetic  system 
appeared  at  least  as  early  as  the  third  century  B.C.,  running 
parallel  with  the  more  primitive  systems. 

As  seen  above,  the  Greeks  had  twenty-four  letters  in  their 
common  Ionic  alphabet,  but  for  a  more  satisfactory  system  of 
numerals  they  needed  twenty-seven  letters.  They  therefore 
added  the  three  forms  F  or  C  (the  old  digamma),  S  or  some- 
times 9  (the  Phoenician  koph),  and  ^2  (perhaps  the  Phoenician 

1S.  Reinach,  loc.  cit.j  p.  220. 

2 A  modern  name  for  the  character  is  sampi  (<rav+ TTI,  san'pt] ,  suggested  because 
of  its  resemblance  to  TT  in  its  i$th  century  form.  The  form  in  the  2d  century 
was  ^,  and  it  may  go  back  to  the  T  (s),  which  was  used  from  the  5th  to  the 
2d  century  B.C.  See  Roberts,  loc.  cit,,  p.  10. 


A 

B 

r 

A 

E 

F 

I 

H 

e 

I 

2 

3 

4 

5 

6 

7 

8 

9 

1 

K 

A 

M 

N 

,3. 

0 

n 

s 

10 

20 

30 

40 

5o 

60 

70 

so 

9o 

p 

S 

T 

Y 

4> 

X 

V 

Q 

* 

IOO 

2OO 

300 

400 

500 

600 

700 

800 

900 

52  READING  AND  WRITING  NUMBERS 

shin  or  tsade\  after  which  they  arranged  their  system  as 
follows : 

Units 
Tens 

Hundreds 

To  distinguish  the  numerals  from  letters,  a  bar  was  commonly 
written  over  each  number,  as  in  the  case  of  A,  although  in  the 
Middle  Ages  the  letter  was  occasionally  written  as  if  lying  on  its 
side,  as  in  the  case  of  <.1 

The  capital  forms  were  used,  the  small  letters  being  an  in- 
vention of  a  much  later  period.  In  a  manuscript  of  the  loth 
century  in  Gottingen  the  small  letters  are  found,  and  there  are 
no  accents  when  these  numerals  appear  in  tables.  When,  how- 
ever, they  appear  in  the  text,  there  are  bars  superscribed  to  dis- 
tinguish the  numerals  from  words,  thus:  a,  e,  0,  iff,  etc.  In 
modern  books  the  forms  usually  appear  as  a! ,  /3',  7',  8',  and  so 
on,  the  accents  being  used  to  distinguish  the  numerals  from 
letters.  The  thousands  were  often  indicated  by  placing  a  bar 
to  the  left,  thus : 

/A,     /B,     /l~,     •••     for     1000,     2000,     3000,    •  •  •, 

these  appearing  in  modern  Greek  type  as  fa,  ,/8,  ,7,  •  •  • . 

The  myriads,  or  ten  thousands  (pvpioi,  myr'ioi},  were  rep- 
resented by  such  forms  as  the  following : 

Y  B  r 

M    or    M,    10,000;     M,    20,000;     M,    30,000,   and  so  on. 

In  late  Greek  manuscripts  the  symbol  °  was  used  for  myriad, 
as  in  the  case  of  JA  for  14  myriads  (140,000).  We  also  find 
such  forms  as  A  for  5  myriads  ( 50,000). 2 

1V.  Gardthauscn,  Die  Schrift,  Unterschriften  und  Chronologic  im  Byzanti- 
nischen  Mittelalter,  2d  ed.,  p.  360  (Leipzig,  1913) ;  hereafter  referred  to  as 
Gardthausen,  Die  Schrijt.  See  also  F.  E.  Robbins,  "A  Greco-Egyptian  Mathe- 
matical Papyrus,"  Classical  Philology  >  XVIII,  328. 

2  Gardthausen,  Die  Schrift,  p.  371. 


GREEK  AND  HEBREW  NUMERALS  53 

In  the  early  Christian  period  the  three  lines  of  letters  rep- 
resenting units,  tens,  and  hundreds  respectively  were  called 
verses  or  rows,  and  the  rectangular  arrangement  of  the  figures 
in  these  verses  was  probably  of  some  value  in  computation.1 

Hebrew  Numerals.  The  Jewish  scholars  used  the  letters  of 
their  alphabet  for  numeral  symbols  in  the  same  way  as  the 
Greeks  did.  We  find  this  usage  well  established  in  the  Macca- 
bean  period  (2d  century  B.C.),  but  it  is  probably  of  an  earlier 
date.  In  the  Talmud  the  numbers  above  400  are  formed  by 
composition,  500  being  formed  of  the  symbols  for  400  and  ioo,2 
and  900  being  a  combination  of  the  symbols  for  400,  400,  and 
ioo.3  Later  writers,  however,  followed  a  plan  introduced  by  the 
Massoretes,4  in  which  certain  final  forms  of  letters  were  used  for 
the  hundreds  above  400.  These  numeral  forms  as  now  recog- 
nized are  as  follows  : 

K         2         a         i         n         i         7         n         to 
i          23456789 


30  40  50  60          70          80  90 


IOO      20O     300     4OO      500     600     700     8OO     900 

The  thousands  were  represented  by  the  same  letters  as  the 
units.  Since  the  number  15  would  naturally  be  represented  by 
10  and  5,  read  from  right  to  left,  that  is,  by  rv,  and  since  these 
are  the  first  two  letters  of  the  word  nirvQhvh,  Jahveh,  Jehovah), 
the  Hebrews  wrote  9  +  6  (ito)  instead. 

^'Primus  igitur  versus  est  a  monade  usque  ad  enneadem,"  etc.  (Capella,  VII, 
745).  Favonius  Eulogius  (c.  400)  remarks:  "Primi  versus  absolutio  novenario 
numero  continetur."  See  J.  G.  Smyly,  in  "Melange  Nicole,"  Recueil  de  Me- 
moires  de  Philologie  Class,  et  d'Archeol.,  p.  514  (Geneva,  1905). 

2pn.  8pnn. 

4  The  scholars  engaged  in  the  work  of  Massorah,  the  establishing  of  the 
traditional  pronunciation  and  accents  of  the  Hebrew  scriptures.  The  work  ex- 
tended over  a  long  period,  closing  in  the  loth  century.  See  Jewish  Encyclo- 
pedia, IX,  348  (New  York,  1905).  For  the  zero,  see  Smith-Karpinski,  p.  60. 

II 


54  READING  AND  WRITING  NUMBERS 

Gematria.  The  fact  that  the  letters  of  various  ancient  alpha- 
bets had  numerical  values,  and  hence  were  used  in  computation, 
led  to  the  formation  of  a  mystic  pseudo-science  known  as 
gematria,  which  was  very  popular  among  the  Hebrews  as  well 
as  among  other  peoples. 

Although  it  had  many  modifications,  its  general  nature  may 
be  explained  by  saying  that  the  numerical  value  of  a  name 
could  be  considered  instead  of  the  name  itself.  If  two  names 
had  the  same  numerical  value,  this  fact  showed  some  relation 
between  the  individuals.  It  is  probable  that  666,  "the  number 
of  the  beast"  in  Revelations,  was  the  numerical  value  of  some 
name,  this  name  being  known  to  those  who  were  in  the  secret, 
but  being  now  lost.  It  is  not  improbable  that  it  referred  to 
"Nero  Caesar,"  which  name  has  this  value  when  written  in 
Hebrew.  For  nearly  two  thousand  years  attempts  have  been 
made  to  relate  the  number  to  different  individuals,  particularly 
to  those  of  a  religious  faith  differing  from  that  of  the  one  sug- 
gesting the  relationship.  Thus,  it  has  been  assigned  to  various 
popes,  to  Luther,  and  to  Mohammed ;  but  it  has  also  been  re- 
lated to  statesmen,  to  the  Latin  Church,  and  to  various  other 
classes  and  organizations.  In  some  cases  a  man's  name  and  its 
gematria  number  have  both  appeared  upon  his  tombstone.  An 
interesting  illustration  of  gematria  is  also  found  in  our  word 
"amen."  Written  in  Greek,  the  numerical  values  of  the  letters 
are  as  follows:  A(a)  =  i,  M (//)  =  40,  H(i/)  =  8,  N(i>)=  50,  the 
total  being  99.  On  this  account  we  find  in  certain  Christian 
manuscripts  the  number  99  written  at  the  end  of  a  prayer  to 
signify  "amen."1 

Roman  Numerals.  The  theories  of  the  origin  of  the  Roman 
numerals  are  for  the  most  part  untenable.  Priscian  (6th  cen- 
tury) believed  that  "I"  was  used  for  i  because  it  was  the 
initial  of  the  Greek  la,  a  dialectic  Greek  word  for  unity,2  al- 
though long  before  the  Greeks  had  any  written  language  it 
was  used  for  this  purpose  in  Egypt,  Babylon,  and  various 

1Gardthausen,  Die  Schrift,  p.  309. 

2See  the  1527  (Venice)  edition  of  Priscian,  fol.  271,  r.  For  the  feminine  of 
th(heis)  the  /Eolic  Greeks  used  fa;  the  other  Greeks,  pla.  Homer  used  both  forms. 


ROMAN  NUMERALS  55 

other  parts  of  the  ancient  world.  His  other  theories  were 
equally  unscientific  except  in  the  cases  of  C  and  M.  These 
symbols  he  took  to  be  the  initials  of  centum  (hundred)  and 
mille  (thousand),  and  there  was  enough  historical  evidence  for 
the  late  adoption  of  these  letters  as  symbols  for  100  and  1000 
to  justify  him  in  making  this  statement.  There  are  also  various 
theories  connected  with  stick-laying,  but  for  these  there  is  no 
historic  sanction. 

In  the  1 6th  century  Mattheus  Hostus1  asserted  that  the 
theory  of  the  early  grammarians  frivolum  est ;  and  while  his  own 
theories  were  generally  about  as  frivolous,  he  made  the  plausible 
suggestion  that  the  V  was  derived  from  the  open  hand,  the 
fingers  with  the  exception  of  the  thumb  being  held  together. 
This  led  naturally  to  taking  the  X  as  a  double  V, — a  view  held 
by  various  later  writers  and  receiving  powerful  support  from 
Mommsen  (1850),  the  great  German  authority  on  Latin  history 
and  epigraphy.  The  theory  is  not  inconsistent  with  the  fact 
that  the  V  is  occasionally  inverted  (A),  since  this  form,  al- 
though an  early  one,  may  have  developed  relatively  late  with 
respect  to  X  and  may  thus  have  represented  half  of  that  numeral. 

Mommsen's  most  important  suggestion,  however,  was  that 
C  and  M  are  not  primitive  forms  but  are  late  modifications  of 
such  forms,  influenced  by  the  initials  of  centum  and  mille.  The 
primitive  forms  for  50,  100,  and  1000  he  stated  to  be  the  Greek 
aspirates  X  (chi),  from  which  L  was  derived;  ©  (theta),  from 
which  comes  the  C ;  and  4>  (phi},  which  is  the  origin  of  the  M. 
As  to  this  theory  there  is  positive  evidence  that  one  of  the 
earliest  forms  for  X  (chi)  was  4^,  and  this,  with  the  later  forms 
vi,  1,  and  L,  was  used  for  50  in  the  inscriptions  of  about  the 
beginning  of  our  era. 

As  to  the  use  of  Q  for  100,  we  have  also  the  early  forms 
®,  ©,  O,  and  0.  If  the  last  of  these  were  written  rapidly  with 
a  stylus  or  a  reed  pen,  the  result  might  easily  resemble  C. 
We  have  not,  however,  any  of  these  transition  forms  extant, 
although  by  analogy  with  L  and  M  we  might  well  accept 
this  theory. 

lDe  numeratione  emendata  veteribus  Latinis  el  Graecis  usitata,  Antwerp,  1582. 


56  READING  AND  WRITING  NUMBERS 

The  4>  was  also  written  CD,  and  the  symbol  for  1000  is  very 
commonly  given  on  the  ancient  monuments  as  CID,  /h,  cU 
and  the  like,  so  that  this  part  of  the  theory  is  reasonable.  The 
M  as  a  numeral  is  unusual  on  the  older  monuments,  although 
an  expression  like  II M  for  2000,  where  M  evidently  stands  for 
the  word  mille,  is  not  uncommon.  Generally  the  Romans  used 
one  of  the  modifications  of  $  as  stated  above,  or  the  symbol  oo, 
which  is  probably  a  cursive  form  of  CID,  with  numerous  variants 
such  as  t><3  and  ^^  . 

As  to  the  X  for  10,  there  is  the  further  theory  that  it  may 
have  come  from  the  crossing  off  of  ten  single  strokes  for  i  by  a 
decussare  line,  either  as  JW44fflI  or  as  LJJ4HtTTT?  which  was  ab- 
breviated as  x?  This  is  analogous  to  the  possible  Egyptian 
plan  of  grouping  ten  strokes  by  an  arc  and  thus  obtaining  their 
symbol  fl .  There  is  much  to  commend  this  decussare  theory,  for 
20  was  commonly  written  "K  or fj,  and  similarly  for  30  and  40. 
If  this  is  the  origin  of  the  X  for  10,  then  the  V  and  A  were 
naturally  taken  as  halves  of  X.  On  the  whole,  this  seems  quite 
as  probable  as  the  hand  theory.  It  has  also  been  thought  that  X 
represents  the  crossed  hands,  thus  giving  two  fives. 

In  1887  Karl  Zangemeister2  advanced  the  theory  that  the 
entire  system  was  based  on  the  single  decussare  principle. 
Briefly,  a  crossing  line  multiplies  any  number  by  ten.  Hence 
we  have  I  and  X  for  i  and  10  respectively;  X  and  K  for  10 
and  100,  from  the  latter  of  which  the  X  finally  dropped  out, 
leaving  (,  which  became  our  C  under  the  influence  of  the  initial 
letter  for  centum;  and  $C  for  1000,  which  finally  became  the 
common  oo.  Although  the  theory  is  interesting,  it  has  never 
been  generally  accepted  by  Latin  epigraphists,  and  so  we  at 
present  fall  back  on  the  Mommsen  theory  as  the  most  prob- 
able of  any  thus  far  suggested.  It  is  quite  as  reasonable,  how- 
ever, to  believe  that  the  symbols  were  arbitrary  inventions  of 
the  priests. 

^Decussare  is  the  verb  form.  The  word  also  appears  as  decussatio,  decus- 
satim,  and  decussis,  according  to  the  sentence  construction. 

2"Entstehung  der  romischen  Zahlzeichen,"  Sitzungsberichte  der  Konigl.  Preuss. 
Akad.  der  Wissensch.,  XLIX,  ion,  with  a  bibliography  on  page  1013. 


ROMAN  NUMERALS  57 

An  examination  of  the  many  thousand  inscriptions  collected 
in  the  Corpus  Inscriptionum  Latinarum1  fails  to  solve  the  prob- 
lem of  origin,  but  it  shows  the  change  in  forms  from  century  to 
century.  This  change  is  even  more  marked  in  the  medieval 
manuscripts.  The  following  brief  notes  will  serve  to  show  how 
these  numerals  have  varied. 

The  I  is  always  a  vertical  stroke,  or  substantially  so.  Hori- 
zontal strokes  are  used  in  writing  certain  fractions.  In  late  me- 
dieval manuscripts  the  stroke  appears  as  i  or,  as  a  final  letter,  j. 

The  V  also  appears  on  the  early  monuments  as  U  or  A,  and 
is  frequently  found  in  such  contracted  forms  as  X,  for  15.  In 
the  medieval  manuscripts  it  varies  with  the  style  of  writing, 
appearing  as  V,  v,  U,  and  u.  In  the  late  Roman  times  the  char- 
acter <y;  with  numerous  variants,  was  used  for  6,  possibly  from 
the  Greek  numeral.  To  represent  eight,  for  example,  this  char- 
acter was  combined  with  II.2 

The  X  also  appears  on  the  monuments  as  X  or  1^ .  It  is  fre- 
quently combined  with  other  letters  in  such  forms  as  L**  for  70. 
In  the  medieval  manuscripts  it  is  often  written  as  a  small  letter. 

The  L  very  frequently  appears  on  the  monuments  of  about 
the  beginning  of  our  era  in  the  older  forms  of  4, ,  vL ,  and  JL.  In 
the  Middle  Ages  it  often  appears  as  a  small  letter,  as  in  a  case 
like  Clxviij  for  168. 

The  C  has  changed  less  than  the  other  forms,  appearing  on 
the  ancient  monuments  as  a  capital  and  frequently  in  the  later 
manuscripts  as  a  small  letter. 

The  D  is  generally  thought  to  be  merely  half  of  the  CIO 
which  stood  for  thousands.  It  is  occasionally  written  Cl  and 
appears  very  commonly  as  13  even  after  the  beginning  of 
printing.  In  the  Middle  Ages  it  appears  both  as  a  capital 
and  as  a  small  letter.  There  is  a  possibility  that  the  use 
of  D  to  represent  500  is  due  to  the  fact  that  the  Etruscans 
had  no  such  letter  in  their  early  alphabet,  and  consequently 
took  the  A  (delta)  for  this  purpose,  just  as  they  took  other 

1  Berlin,  1863  seq. 

2L.  A.  Chassant,  Dictionnaire  des  Abrtviations  .  .  .  du  Moyen  Age,  p.  114 
(Paris,  1884). 


58  READING  AND  WRITING  NUMBERS 

Greek  letters  for  numerical  purposes.  The  delta  was  then 
changed,  in  the  course  of  time,  to  the  form  with  which  we 
are  now  familiar.1 

When  the  Romans  used  the  M  in  representing  numbers,  it 
was  commonly  as  the  initial  of  mille,  thousand.  When  writ- 
ten with  other  numerals,  the  thousand  symbol  was  usually 
CIO,  A,  cb,  *b,  <*>>  E^,  *~^>9  or  some  similar  form,  as  in  the 
case  of  ooCIII  for  1103.  In  the  medieval  manuscripts  the  M, 
usually  a  capital,  replaced  the  earlier  forms,  as  in  the  number 
Mcccclxxxxiiij  for  1494. 

The  subtractive  principle  is  found  in  certain  cases  like  that  of 
IV  for  4,  that  is,  5  —  1.  This  principle  was,  as  we  have  seen, 
used  by  the  Babylonians  in  the  3d  millennium  B.C.  It  was  also 
used  by  the  Hebrews,  at  least  in  word  forms,  but  apparently  not 
before  the  Etruscans  and  Romans  used  it.  The  Etruscans2  pre- 
ceded the  Romans  in  recognizing  the  principle  and  made  a  more 
extensive  use  of  it.  They  commonly  wrote  their  numerals  from 
right  to  left,  and  so  we  have  such  forms  as  the  following : 3 

XIIIXX,  for  20  +  (io-3),  or  27; 

XIIXXX,  for  30+  (10-2), or  38; 

till,  for  50-3,  or  47; 

MTX,  for  (50  -  io)+  2,  or  42. 

"  The  Etruscans  also  used  ^  for  X,  and  so  we  find  such  forms 
as  ^^  for  XL;  when  read  from  right  to  left  this  means  our 
LX  (60),  but  when  written  ^<l>  it  means  our  XL  (40).*  Such 
forms  as  XII f)  for  the  Roman  LIX  are  also  found. 

Subtractive  Principle  Widespread.  The  subtractive  principle 
was  probably  used  by  various  other  early  peoples,  for  an 
immature  mind  finds  it  easier  to  count  backwards  by  one  or  two 
from  some  fixed  standard,  like  5,  10,  15,  20,  and  so  on,  than  to 
count  forwards  by  three  or  four.  Thus,  the  Romans  found  it 
easier  to  think  of  "two  from  twenty "  (duo  de  viginti)  than  of 

!B.  Lefebvre,  Notes  d'Histoire  des  Mathematiques,  p.  30  (Louvain,  1920). 
2R.  Brown,  "The  Etruscan  Numerals,"  Archaeological  Rev.,  July,  1889. 
s\V.  Corssen,  Ueber  die  Sprache  der  Etrusker,  2  vols.,  Leipzig,  1874,  1875. 

Corpus  Inscriptionum  Etruscarum,  I,  Nos.  23,  27,  32,  38,  et  passim  (Leipzig, 
1893-        ).  BIbid.,  4615. 


ROMAN  NUMERALS  59 

"eight  and  ten"  (octodecim) ,  and  of  "one  from  twenty"  than 
of  "nine  and  ten."  This  is  especially  the  case  with  numbers 
above  five,  since  the  difficulty  is  hardly  experienced  until  nine 
or  fourteen  is  reached. 

As  an  indication  of  the  tendency  of  primitive  peoples  to  use 
the  subtractive  principle  the  fact  may  be  mentioned  that  the 
Zufii  Indians,  whose  number  names  refer  to  the  fingers,  speak 
of  four  as  "all  the  fingers  almost  complete,"  and  of  nine  as 
"almost  all  are  held  up  with  the  rest,"  each  containing  the 
idea  of  subtraction.1  They  had  a  system  of  knot  numerals 
which  involved  the  same  principle.  A  medium  knot  indicated 
5,  and  this  with  a  small  knot  before  it  indicated  5  —  i,  whereas 
if  the  small  knot  came  after  the  medium  one  the  number  was 
5  +  1.  Similarly,  a  large  knot  indicated  10,  and  a  small 
knot  was  used  either  before  or  after  it  so  as  to  indicate  9  or  1 1 
respectively. 

Further  Cases  of  the  Subtractive  Principle.  It  is  because  of 
the  fact  that  the  difficulty  is  not  evident  with  so  simple  a 
number  as  4  that  the  Romans  did  not  commonly  use  the 
subtractive  principle  in  this  case,  preferring  the  form  INI  to 
the  form  IV.  They  used  the  principle  more  frequently  in  the 
case  of  9,  but  even  here  they  wrote  VI I II  oftener  than  IX.  In 
the  case  of  400  they  usually  wrote  CCCC,  but  occasionally  they 
used  CD.  Even  as  late  as  the  i6th  century  we  often  find  a 
number  like  1549  written  in  some  such  form  as  Mcccccxxxxviiij. 

Relics  of  the  subtractive  principle  are  seen  in  our  tendency  to 
say  "ten  minutes  of  (or  "to")  six"  instead  of  "fifty  minutes 
past  five,"  and  to  say  "a  quarter  of  (or  "to")  six"  rather  than 
"three  quarters  of  an  hour  past  five." 

There  is  a  possibility  that  the  Romans  avoided  IV,  the  initials 
of  IVPITER,  just  as  the  Hebrews  avoided  m  in  writing  15,  as 
the  Babylonians  avoided  their  natural  form  for  19,  and  as 
similar  instances  of  reverence  for  or  fear  of  deity  occur  in  other 
languages. 

XF.  H.  Gushing,  "Manual  Concepts,"  American  Anthropologist  (1892), p.  289; 
Th.  W.  Danzel,  Die  Anfdnge  der  Schrijt,  p.  55  (Leipzig,  1912) ;  L.  L.  Conant, 
Number  Concept,  p.  48  (New  York,  1896). 


60  READING  AND  WRITING  NUMBERS 

Even  when  the  subtractive  principle  was  used,  no  fixed 
standard  was  recognized.  The  number  19  was  commonly  writ- 
ten XIX,  but  not  infrequently  I XX.1  We  also  find  NX  for  8  and 
1 1 XX  for  1 8,  but  these  were  not  so  common.  It  is  quite  rare  to 
find  CD  for  400  or  CM  for  900,  and  forms  like  MCM  and  DCD 
were  never  used  in  ancient  or  medieval  times.  In  general,  there- 
fore, it  may  be  said  that  the  Romans  recognized  the  subtractive 
principle  but  did  not  make  much  use  of  it. 

Occasionally  this  principle  was  used  with  the  fraction  ^,  for 
which  the  Romans  wrote  the  letter  S,  initial  of  semis  (half). 
Thus  we  find  SXC  for  89 J  and  SXXC  for  79*. 

Large  Numbers.  The  Romans  had  relatively  little  need  for 
large  numbers,  and  so  they  developed  no  general  system  for 
writing  them.  The  current  belief  that  they  commonly  used  a 
bar,  or  vinculum,  over  a  number  to  multiply  it  by  1000  is  erro- 
neous. What  they  ordinarily  did,  if  they  used  numerical  sym- 
bols at  all,  was  to  take  some  such  forms  as  the  following : 

For  100,000:  CCCIOOO  171  4-  ^  @  ® 
For  10,000:  CCI3D  ^  v^  cd^  A  nln 
For  5,000:  103  h\  I/  1.3  fcs  IM 

To  represent  larger  numbers,  these  forms  were  repeated. 
Thus,  the  symbol  £3^,  used  for  100,000,  is  repeated  twenty- 
three  times  on  the  columna  rostrata?  making  2,300,000. 

In  the  Middle  Ages,  however,  we  find  such  forms  as  fY[  or  IX I 
for  million  and  [M]  for  hundred  million,  that  is,  for  ten  hundred 
thousand  and  one  thousand  hundred  thousand. 

Use  of  the  Bar.  The  Romans  commonly  placed  a  bar  over  a 
number  to  distinguish  it  from  a  word,  as  in  the  case  of  fjVIR 
for  duumviri  (two  men)  and  TTlVIR  for  the  triumvirate.  The 

1In  early  inscriptions  this  form  was  sometimes  used  for  21,  since  the  Romans 
occasionally  wrote  numbers  from  right  to  left,  like  the  early  Greeks. 

2  A  Roman  monument  set  up  in  the  Forum  to  commemorate  the  victory  of 
260  B.C.  over  the  Carthaginians.  This  is  the  earliest  noteworthy  example  of  the 
use  of  large  numbers  in  a  Roman  inscription. 


ROMAN  NUMERALS  61 

oldest  example  that  we  have  of  the  bar  to  indicate  thousands 
dates  from  about  50  B.C.  Cicero  (106-43  B.C.),  or,  more  prob- 
ably, some  late  copyist  of  his  works,  also  used  >(X.CD  and 
CCIOO  CCI33  CCCC  as  equivalent.  The  vinculum  is  found  fre- 
quently in  the  works  of  Pliny  (ist  century),  but  it  is  not  used 

Qugnam  fuerunt  noes  Roma* 

no  rum? 
J.      u 
V-     1* 
X.     to. 
I*.     *o. 
C.      100. 

^).  o»  io.   *oo.     Qjrmgento 

cxo.   CD.  ci3.  1000.     XiVf*.  MiKif. 


CMo,         -  CCiDD.ioooo. 

^  .  IDDD-  ^oooo.    Qjfffl^K<*gWlf4 


<  oo  o  o  o.  tyiwgenta  milltdt 


Romrfm  iwnfim  won  progrcdiuntur  ultra  dtchs  c<nun<t 
igm^^ 

GO.   xooo. 


CIO,  ID» 

ROMAN   NUMERALS 
From  the  work  of  Freigius,  a  Swiss  writer,  published  in  1582 

with  much  uniformity  and  we  are  not  sure  how  many  of  his 
numeral  forms  are  due  to  later  scribes.  In  the  Middle  Ages  the 
vinculum  was  called  a  titulus*  but  even  then  it  was  more  com- 
monly used  to  distinguish  numerals  from  words  than  to  indicate 
thousands. 


Bernelinus:  "Nam  sicut  prima  unitas  notatur  per  elementum  I,  ita 
millenarius  primus  per  idem  I,  superaddito  tantum  titulo."  A.  Olleris,  CEuvres  de 
Gerbert,  p.  360  (Paris,  1867). 


62 


READING  AND  WRITING  NUMBERS 


CD 

00  CC1M 


OCIOO 

C-C-1OO 

X 
X 

CC-I-CC 

DMC 

OMD 

IMI 


9000. 


10000. 


The    Romans   did^  not   use   the   double   bar    to    indicate 
1000  x  1000,  as  in  V  for  5,000,000,  but  it  is  said  to  be  oc- 
casionally seen  in  the  late 
Numcratio^ Middle  Ages.1 

Late  Coefficient  Method. 
In  the  later  Roman  times 
there  arose  a  kind  of  coeffi- 
cient method  of  represent- 
ing large  numbers.  Thus, 
Pliny  used  XI I M  for  12,000, 
and  we  have  a  relic  of  this 
method  in  our  modern  use 
of  10  M.  In  such  cases, 
however,  M  was  looked 
upon  as  abbreviation  for 
mille  rather  than  as  a  sym- 
bol for  1000,  although  the 
distinction  is,  of  course,  not 
noticeable.  We  find  the 
same  thing  in  the  Middle 
Ages,  as  when  O'Creat  (c. 
1150)  writes  XesM.  milia 
for  ten  thousand  thousand. 
A  somewhat  similar  usage 
appears  in  the  Compotus 
Reinheri  (i3th  century), 
where  IIIIor  milia.  ccca.l.vi 
appears  for  4356.  Even  as 
late  as  the  i6th  century 
the  same  plan  was  followed, 

as  when  Noviomagus  (1539)  wrote  HIM  for  3000  and  MM  for 
1,000,000,  and  when  Robert  Recorde  (c.  1542)  used  vj.C  for 
600,  ixM  for  9000,  CCC.M  for  300,000,  and  230  M,  MM,  M 
for  230  •  io12.  The  coefficients  were  often  written  above  the  M, 

Ixxx  C 

as  in  MM  for  80,000,000  and  MM  for  100,000,000,  in  a  manu- 


CCD3  « 

COIOD  OQ 

CCD3  ™  CD 

COIOp  H    «o 

CCD3CDCI3ci:> 

CCIDD  v>  *o  <* 

CCDD  CD  i* 

CCIDD   00  m 

CCDO '» 


ROMAN   NUMERALS 

From  Bongo's  work  on  the  mystery  of 
numbers,  Bergamo,  1584-1585 


//coo. 


12000. 


IjOOt. 


/4.000. 


XA.  Cappelli,  Dizionario  di  Abbreviature,  2d  ed.,  p.  Hi  (Milan,  1912). 


ROMAN  NUMERALS  63 

script  of  c.  I442.1  They  were  also  written  below,  as  inCxxiij  for 
123  and  C  MM  C  M  C  for  123,456,789.  in  the  arithmetic 

I      xxiij      iiij    Ivj    vij    Ixxxix 

of  Bartjens.2 

Epigraphical  Difficulties.  The  Romans  varied  their  numerals, 
often  according  to  the  pleasure  of  the  writers,  and  it  takes  a 
skilled  epigraphist  to  decipher  many  of  those  that  appear  upon 
the  amphorae  stating  the  amount  or  the  price  of  wine.  For  ex- 
ample, the  following  numbers,  which  were  taken  from  wine  jugs 
of  about  the  ist  century,  and  which  are  by  no  means  among 
the  most  difficult,  would  certainly  not  be  understood  by  the 
casual  observer : 


for  oo 

for  IXCS     =  89*- 


Such  forms  concern  chiefly  the  student  of  epigraphy,  how- 
ever. The  medieval  numerals  are  more  interesting,  since  they 
involve  new  methods,  and  hence  a  few  types  will  here  be  given : 


c  •  Ixiiij jj:cc  •  1  •  i,  for  164,351,  Adelard  of  Bath  (c.  1120). 

vi  •  dclxvi,  for  6666,  Radulph  of  Laon  (c.  1125). 
II.DCCC.XIIII,  for  2814,  Jordanus  Nemorarius  (c.  1225). 
MQCLVI,  for  1656,  a  monument  in  San  Marco,  Venice, 
do.  Io.  ic?  for  1599,  edition  of  Capella,  Leyden,  1599. 

xxyiii,  for  28,  edition  of  Horace,  Venice,  1520. 
IIIIxx   et  huit,  for  88,  a  Paris  treaty  of  1388." 

©DCXL,  for  1640,  edition  of  Petrus  Servius,  Rome, 

1640. 

four  Cli.M,  two  Cxxxiiii,  millions,  sixe  ClxxviiiM.  fiue  Clxvii, 
for  451,234,678,567,  Baker,  1568. 

1A  copy  of  Sacrobosco's  arithmetic  made  c.  1442.  See  Kara  Arithmetica, 
p.  450. 

2A  Dutch  work  of  the  i8th  century,  1792  ed.,  p.  8. 

3This  is  simply  the  French  quatre  mngt  (4  X  20)  and  is  common  in  medieval 
French  MSS. 


64  READING  AND  WRITING  NUMBERS 

To  the  many  other  peculiarities  of  this  system  it  is  not  pos- 
sible to  allow  further  space.  The  Roman  forms  persisted  in 
use,  especially  outside  of  Italy,  until  printed  arithmetics  made 
our  common  numerals  widely  known.  Even  at  the  present  time 
the  fishermen  of  Chioggia,  near  Venice,  use  forms  that  closely 
resemble  those  of  the  early  Etruscans,  so  persistent  is  custom 
in  the  humbler  occupations  of  man.1 

Our  Common  Notation.  When  we  come  to  consider  the  origin 
of  our  common  numerals,  we  are  confronted  by  various  theories, 
and  the  uncertainty  is  quite  as  marked  as  in  the  case  of  the 
Roman  system.  These  symbols  are  generally  believed  to  have 
originated  in  India,  to  have  been  carried  to  Bagdad  in  the  8th 
century,  and  thence  to  have  found  their  way  to  Europe.2  This 
is  not  certain,  for  various  authors  of  scientific  standing  have  at- 
tempted to  show  that  these  numerals  did  not  originate  in  India 
at  all,3  but  the  evidence  still  seems  much  more  favorable  to  the 
Hindu  origin  than  to  any  other  that  has  been  suggested.  The 
controversy  has  recently  centered  about  the  meaning  of  the 
word  hindasi,  which  is  often  used  by  the  Arabs  in  speaking  of 
the  numerals.  It  is  asserted  that  the  word  does  not  mean  Hindu, 
some  claiming  that  it  refers  to  Persia  and  others  that  it  means 
that  which  is  related  to  calculation.  It  is  difficult,  however,  to 
explain  away  the  following  words  of  Severus  Sebokht  (c.  650), 
written  in  the  yth  century  and  already  quoted  in  Volume  I : 

1  will  omit  all  discussion  of  the  science;  of  the  Hindus,  a  people 
not  the  same  as  the  Syrians;  their  subtle  discoveries  in  this  science 

aA.  P.  Ninni,  "Sui  segni  prealfabetici  usati  .  .  .  nella  numerazione  scritta  dai 
pescatori  Clodiensi,"  Atti  del  R.  Istituto  Veneto  delle  sci.  lett.  ed  arti,  VI  (6),  679. 

2  Smith  and  Karpinski,  The  Hindu-Arabic  Numerals,  with  bibliography,  Bos- 
ton, 1911  (hereafter  referred  to  as  Smith-Karpinski) ;  G.  F.  Hill,  The  Develop- 
ment of  Arabic  Numerals  in  Europe,  Oxford,   1915;   J.  A.  Decourdemanche, 
"Sur  la  filiation  des  chiffres  europeens  modernes  et  des  chiffres  modernes  des 
Arabes,"  Revue  d' Ethnographic  et  de  Sociologie,  Paris,  1912;  G.  Oppert,  "Ueber 
d.  Ursprung  der  Null,"  Zeitschrift  fur  Ethnographie,  XXXII,  122  (Berlin,  1900) ; 
G.  N.  Banerjee,  Hellenism  in  Ancient  India,  p.  202  (Calcutta,  1919). 

3JE.g.,  see  Carra  de  Vaux,  "Sur  Torigine  des  chiffres,"  Scientia,  XXI  (1917), 
273;  but  see  F.  Cajori,  "The  Controversy  on  the  Origin  of  our  Numerals," 
The  Scientific  Monthly,  IX,  458. 


OUR  COMMON  NOTATION  65 

of  astronomy,  discoveries  that  are  more  ingenious  than  those  of  the 
Greeks  and  the  Babylonians  ;  their  valuable  methods  of  calculation  ; 
and  their  computing  that  surpasses  description.  I  wish  only  to  say 
that  this  computation  is  done  by  means  of  nine  signs. 

Types  of  Early  Hindu  Numerals.  The  early  numerals  of  India 
were  of  various  types.1  The  earliest  known  forms  are  found  in 
the  inscriptions  of  King  Asoka,  the  great  patron  of  Buddhism, 
who  reigned  over  most  of  India  in  the  3d  century  B.C.  These 
symbols  are  not  uniform,  the  characters  varying  to  meet 
the  linguistic  conditions  in  different  parts  of  the  country. 
The  Karosthi  forms,  for  example,  are  merely  vertical  marks, 
I  ||  [HI]  Mil  Mill,  and  are  not  particularly  significant.  The 
Brahmi  characters  found  in  some  of  these  inscriptions  are  of 
greater  interest.  The  only  numerals  thus  far  found  in  the  Asoka 
edicts  are  as  follows  : 


l\\-+<6f  634 


5°    5°    2O°       20° 


The  Nana  Ghat  Inscriptions.  About  a  century  after  the  Asoka 
edicts  certain  records  were  inscribed  on  the  walls  of  a  cave  on 
the  top  of  the  Nana  Ghat  hill,  about  seventy-five  miles  from  the 
city  of  Poona.  A  portion  of  the  inscriptions  is  as  follows  : 


!For  discussion  and  bibliography  see  Smith-Karpinski,  p.  19. 


66  READING  AND  WRITING  NUMBERS 

The  probable  number  forms  contained  in  these  inscriptions 
are  as  follows  : 


1    ?OCtecf 

12  4  6         7        9       10     10     10 


o     *> 

2O   60     80     100   TOO   IOO     2OO          4OO 


T  7    TT 

700    1000     4000   6000   10,000   20,000 

The  next  important  trace  of  the  numerals  is  found  in  the 
caves  at  Nasik,  India.  These  are  of  the  ist  or  2d  century  and 
are  as  follows  : 


i  23          4  56789 

cc 


IO          IO  2O  40  70  IOO        2OO        500 

1   1   r  r   v  y- 

1000   2000   3000    4000    8000   70,000 

The  significant  feature  of  these  numerals  is  that  they  clearly 
resemble  the  Nana  Ghat  forms,  and  that  in  both  we  seem  to 
have  the  progenitors  of  our  present  numerals. 

It  should  be  understood,  however,  that  the  interpretations  of 
the  inscriptions  at  Nana  Ghat  and  Nasik  are  not  universally  ac- 
cepted. All  that  we  can  say,  in  our  present  state  of  knowledge, 
is  that  these  are  the  probable  number  forms  as  stated,  that  they 
resemble  some  of  the  numerals  that  were  transmitted  to  Europe 
as  of  Hindu  origin,  that  no  zero  appears  in  these  early  inscrip- 
tions, and  hence  that  the  place  value,  as  we  know  it  at  present, 
had  not  yet  been  developed. 


HINDU  NUMERALS  67 

Variants  of  Hindu  Forms.  The  variants  of  the  Hindu  forms 
preceding  the  invention  of  the  zero  may  be  seen  in  the  table 
shown  below. 

NUMERALS  12     3    45    67    8  9    10  20  30  40   w    60  70  80  90  100  200  1000 
I      *  Asoka        (  // 

:      2Saka  I  II  III  X  IXIIX     XX      ?  ?  73333?  Mill 

_      3Asoka         I  II       +      /£  G  /f 

-=      *       <*  7       PcfO  ^ 

-=5*lip7  <7?<*e     X 

'Ksatrapa  ~.~H^  A  ?333aCesrx 
-Kusana     -  ^H^F  M  Sp  <*  Q  V  X  0      X  CO© 
«Gupta       -,:™3^,?  A  (53^°^  S^W&yH 

°Valhab!     N  =  £       3f  J  ^  f  9  «.  Q  V  ^  $  J$&&      CT 
Nepal         ^^       ^TA        ?^Q^X  °       ^ 

Kalinga         ^JJiy?^^^^  ^0^ 

Vakataka  f^          ^    \\t»   6 

As  to  the  original  significance  of  these  forms  we  are  wholly 
ignorant  except  in  the  cases  of  the  first  three.  As  to  I,  II,  III, 
or  — ,  n,  E,  there  is,  of  course,  no  question.  The  vertical 
forms  may  have  represented  fingers  used  in  counting  or  they 
may  have  been  the  marks  that  one  naturally  makes  with  a 
stylus  or  brush  in  keeping  a  numerical  record.  The  horizontal 
forms  may  be  pictures  of  computing  sticks,  like  those  which  the 
Chinese  used  in  remote  times,  and  which  appear  in  the  Chinese 
numerals.  Such  sticks  are  naturally  laid  horizontally  with  re- 
spect to  the  eye.  The  earliest  Sumerian  forms  of  the  numerals 

iRarosthl  numerals,  Asoka  inscriptions,  c.  250  B.C.    For  sources  of  infor- 
mation with  respect  to  this  table  see  Smith-Karpinski,  p.  25. 
2 Same,  Saka  inscriptions,  probably  of  the  first  century  B.C. 
3BrahmI  numerals,  Asoka  inscriptions,  c.  250  B.C. 
4 Same,  Nana  Ghat  inscriptions,  c.  150  B.C. 
5  Same,  Nasik  inscription,  c.  100. 
6Ksatrapa  coins,  c.  200. 
7Kusana  inscriptions,  c.  150. 
8  Gupta  inscriptions,  c.  300  to  450.  9Valhabi,  c.  600. 


68  READING  AND  WRITING  NUMBERS 

were  horizontal,1  and  so  the  computing  rod  may  have  had  its 
origin  in  Sumeria.  The  later  Babylonian  forms  were  vertical, 
and  so  the  finger  computation  may  have  been  in  favor  at  that 
time.  The  Mediterranean  lands  adopted  the  vertical  forms,  and 
the  Far  East  preferred  the  horizontal.  From  the  vertical  II 
came  the  following  Egyptian  forms : 

Hieroglyphic   II 
Hieratic  l| 

Demotic          ty 

From  the  symbol  ||  came  also  the  Arabic  P,  which  is  our  2  if 
turned  on  its  side.  Indeed,  our  2  is  merely  a  cursive  form  of 
=  and  our  3  is  similarly  derived  from  E,  as  is  seen  from  a 
study  of  inscriptions  and  manuscripts.2  These  horizontal  forms 
were  early  used  by  the  Chinese  and  probably  found  their  way 
from  China  to  India. 

Fanciful  Theories.  Numerous  conjectures  have  been  made  as 
to  the  origin  of  the  characters  from  four  to  nine,  but  no  one  of 
them  has  had  any  wide  acceptance.  We  may  dismiss  at  once  all 
speculations  as  to  their  derivation  from  such  combinations 
as  12  and  0  and  from  the  number  of  sticks  that  might  be  laid 
down  to  make  the  figure  8.  Such  ideas  are  trivial  and  have  no 
sanction  from  the  study  of  paleography.  There  remain,  how- 
ever, various  scientific  theories,  as  that  the  forms  are  ancient 
initial  letters  of  number  words.3  None  of  these  theories,  how- 
ever, has  stood  the  test  of  scholarly  criticism,  and  today  we 
have  to  confess  that  we  are  entirely  ignorant  as  to  the  origin 
of  the  forms  which  began  possibly  in  India  in  Asoka's  time  and 
appear  as  the  common  numerals  which  we  use. 

iSir  H.  H.  Howard,  "On  the  Earliest  Inscriptions  from  Chaldea,"  Proceed- 
ings of  the  Society  of  Biblical  Archaeology,  XXI,  301. 

2  An  interesting  fact  in  relation  to  the  figure  2  is  that  the  Romans  often 
wrote  -f^  as  two  lines,  n ,  the  twelfth  being  understood  as  we  understand  tenths 
when  we  write  0.2.   They  also  wrote  this  character  cursively,  z,  which  is  the 
character  used  for  our  2  in  several  early  printed  books. 

3  For  details  of  this  theory  consult  the  bibliography  given  in  Smith-Karpinski, 
P- 30- 


ORIGIN  OF  THE  ZERO  69 

Origin  of  the  Zero.  The  origin  of 
tain  as  the  origia.af  our  other  numerals.  Without  it  the  Hindu 
numerals  would  be  no  better  than  many  others,  since  the  dis- 
tinguishing feature  of  our  present  system  is  its  place  value. 
The  earliest  undoubted  occurrence  of  a  zero  in  India  is  seen  in 
an  inscription  of  876  at  Gwalior.  In  this  inscription  50  and  270 
are  both  written  with  zeros.1  We  have  evidence,  however,  that 
a  place  value  was  recognized  at  an  earlier  period,  so  that  the 
zero  had  probably  been  known  for  a  long  time.  The  Baby- 
lonians, indeed,  had  used  a  character  for  the  absence  of  number, 
and  they  made  use  of  a  primitive  kind  of  place  value ; 2  but  they 
did  not  create  a  system  of  numeration  in  which  the  zero  played 
any  such  part  as  it  does  in  the  one  which  we  now  use.  There 
is  also  a  slight  approach  to  a  place  value  in  some  of  the  late 
Greek  works.  For  example,  Diophantus  seems,  judging  by 
certain  manuscripts,  to  have  used  -B-A^TTZ  for  23,587,  the  four 
points  about  the  B  being  a  late  Greek  symbol  for  myriads,  and 
the  position  of  A  determining  its  value  as  30  hundreds.3 

The  form  of  the  zero  may  have  been  suggested  by  an  empty 
circle,  by  the  Greek  use  of  omicron  (0)  to  indicate  a  lacuna,4 
by  the  horned  circle  used  in  the  Brahml  symbols  for  ten,  by  the 
Hindu  use  of  a  small  circle  (o),  as  well  as  a  dot,  to  indicate  a 
negative,  or  in  some  other  way  long  since  forgotten.  There  is 
no  probability  that  the  origin  will  ever  be  known,  and  there  is 
no  particular  reason  why  it  should  be.  We  simply  know  that 
the  world  felt  the  need  of  a  better  number  system,  and  that  the 
zero  appeared  in  India  as  early  as  the  gth  century,  and  probably 
some  time  before  that,  and  was  very  likely  a  Hindu  invention, 

The  Arabs  represented  5  by  a  character  that  looked  some- 
what like  the  Hindu  zero.  In  a  manuscript  of  1575  the  numer- 
als appear  as  ^  A  Vc;  &  Y*  /^  •  In  other  manuscripts  we  find 
such  forms  as  T,  CD,  and  o.  Because  of  the  resemblance  oi 
their  five  to  the  circle  the  Arabs  adopted  a  dot  for  their  zero 

1For  facsimiles  see  Smith-Karpinski,  p.  52. 
2  For  particulars  see  ibid.,  p.  51. 
8For  other  cases  see  Gardthausen,  Die  Schrift,  p.  372. 

4 Being  the  initial  of  ovdtv  (ouden'),  nothing.  Thus,  Archimedes  might  have 
used  0  to  indicate  the  absence  of  degrees  or  minutes.  See  Heath,  Archimedes,  Ixxi 


70  READING  AND  WRITING  NUMBERS 

For  purposes  of  comparison  the  Sanskrit  forms  are  here  re- 
peated and  the  modern  Arabic  forms  are  given : 

Sanskrit, 


Arabic, 


\rrioi\Ai. 


The  various  forms  of  the  numerals  used  in  India  after  the 
zero  appeared  may  be  judged  from  the  table  here  shown. 

126  80 


MS.   See  Volume  I,  page  164;  Smith-Karpinski,  pp.  40,  50. 

2  The  3,  4,  6,  from  H.  H.  Dhruva,  "The  Land-Grants  from  Sankheda,"  Epi- 
graphia  Indica,  Vol.  II,  pp.  19-24  with  plates;  date  595.  The  7,  i,  5,  from  Bhan- 
darkar,  "Daulatabad  Plates,"  Epigraphia  Indica,  Vot.*lX,  Part  V;  date  c,  798. 

8The  8,  7,  2,  from  "Buckhala  Inscription  of  Nagabhatta,"  Bhandarkar,  Epi- 
graphia Indica,  Vol.  IX,  Part  V;  date  815.  The  5  from  "The  Morbi  Copper- 
Plate,"  Bhandarkar,  Indian  Antiquary,  Vol.  II,  pp.  257-258,  with  plate;  date  804. 

4  The  8  from  the  above  Morbi  Copper-  Plate.  The  4,  5,  7,  9,  and  o  from  "  Asni 
Inscription  of  Mahipala,"  Indian  Antiquary,  Vol.  XVI,  pp.  174-175;  date  9^7. 

6  The  8,  9,  4,  from  "Rashtrakuta  Grant  of  Amoghavarsha,"  J.  F.  Fleet,  Indian 
Antiquary,  Vol.  XII,  pp.  263-272;  date  c.  972.  See  Biihler.  The  7,  3,  5,  from 
"Torkhede  Copper-Plate  Grant,"  Fleet,  Epigraphia  Indica,  Vol.  Ill,  pp.  53-58. 

6  From  "A  Copper-Plate  Grant  of  King  Tritochanapala  Chanlukya  of  Lata- 
desa,"  H.  H.  Dhruva,  Indian  Antiquary,  Vol.  XII,  pp.  196-205  ;  date  1050. 

7  A.  C.  Burnell,  South  Indian  Palaeography,  Plate  XXIII,  TelugXi-Caharese 
numerals  of  the  nth  century. 


NAME  FOR  ZERO  71 

The  following  are  later  European  and  Oriental  forms  : 
1234567         890 

1  /.  e  3  4  ,*  ^  7  v 

*  *     *   ^   /  *   £>    9 

3/    z  3  i-  f  <£  y 
4  I   T  T 


The  Name  for  Zero.  The  name  for  zero  is  not  settled  even  yet. 
Modern  usage  allows  it  to  be  called  by  the  name  of  the  letter  O, 
an  interesting  return  to  the  Greek  name  omicron  used  by  Buteo 
in  1559.  The  older  names  are  zero,  cipher,  and  naught.  The 
Hindus  called  it  sunya,  "void,"  and  this  term  passed  over  into 
Arabic  as  as-sifr  or  si  jr.  When  Fibonacci  (1202)  wrote  his 
Liber  Abaci,  he  spoke  of  the  character  as  zephirum.8  Maximus 
Planudes  (c.  1340)  called  it  tziphra9  and  this  form  was  used  by 
Fine  (1530)  in  the  i6th  century.  It  passed  over  into  Italian 
.as  zeuero,w  ceuero,^  and  zepiro,12  and  in  the  medieval  perjod  it 

1  and  2From  a  manuscript  of  the  second  half  of  the  i3th  century,  reproduced 
in  "Delia  vita  e  delle  opere  di  Leonardo  Pisano,"  Baldassare  Boncompagni,  Rome, 
1852,  in  Atti  dell'  Accademia  Pontificia  del  Nuovi  Lincei,  anno  V. 

3  and  4  From  a  i4th  century  manuscript. 

5  From  a  Thibetan  MS.  in  the  library  of  the  author. 

6  From  a  specimen  of  Thibetan  block  printing  in  the  library  of  the  author. 
7SSrada,  numerals  from  The  KashmirianAtharva-Veda,  reproduced  by  chromo- 

photography   from   the    manuscript   in    the    University    Library    at    Tubingen, 
M.  Bloomfield  and  R.  Garbe,  Baltimore,  1901. 

8".  .  .  quod  arabice  zephirum  appellatur." 

9  From  the  Greek  form  r£l<ppa  (tzi'phra)  ,  used  also  by  another  writer,  Neophy- 
tos,  about  the  same  time. 

10Thus  Jacopo  da  Firenze  (1307),  or  Magister  Jacobus  de  Florentia. 

11  As  in  the  arithmetic  of  Giovanni  de  Danti  of  Arezzo  (1370). 

12  As  in  a  translation  into  Latin  of  the  works  of  Avicenna. 


7.2  READING  AND  WRITING  NUMBERS 

had  various  other  forms,  including  sipos,  tsiphron,  zeron,  cifra, 
and  zero.  It  was  also  known  by  such  names  as  rota,  circulus, 
galgalj  omicron,  theca,  null,  and  figura  nihili? 

Numerals  outside  of  India.  The  first  definite  trace  that  we 
have  of  the  Hindu  numerals  outside  of  India  is  in  the  passage 
already  quoted  from  Severus  Sebokht  (c.  650).  From  this  it 
seems  clear  that  they  had  reached  the  monastic  schools  of 
Mesopotamia  as  early  as  650. 

The  next  fairly  definite  information  as  to  their  presence  in 
this  part  of  the  world,  and  with  a  zero,  is  connected  with  the 
assertion  that  a  set  of  astronomical  tables  was  taken  to  Bagdad 
in  773  and  translated  from  the  Sanskrit  into  Arabic  by  the 
caliph's  command.  There  is  ground  for  doubt  as  to  the  asser- 
tion, but  the  translation  is  said  to  have  been  made  by  al-Fazari 
(c.  773).  It  is  probable  that  the  numerals  were  made  known  in 
Bagdad  at  this  time,  and  they  were  certainly  known  by  the 
year  825.  About  that  year  al-Khowarizmi  recognized  their 
value  and  wrote  a  small  book  explaining  their  use.  This  book 
was  translated  into  Latin,  possibly  by  Adelard  of  Bath  (c.  1120) , 
under  the  title  Liber  Algorismi  de  numero  Indorum.2 

The  Hindu  forms  described  by  al-Khowarizmi  were  not  used 
by  the  Arabs,  however.  The  Bagdad  scholars  evidently  derived 
their  forms  from  some  other  source,  possibly  from  Kabul3  in 
Afghanistan,  where  they  may  have  been  modified  in  transit  from 
India.  These  numerals  have  been  still  further  modified  in  some 
respects,  and  at  present  are  often  seen  in  the  forms  given  on 
pages  70  and  71. 

The  Numerals  move  Westward.  Owing  to  the  fact  that  almost 
no  records  of  a  commercial  nature  have  been  preserved  from 

*For  a  full  discussion  see  Smith-Karpinski,  chap.  iv. 

2  The  Book  of  al-Khowarizmi  on  Hindu  number.   On  this  work  see  Smith- 
Karpinski,  pp.  5  seq.,  92  seq. 

3  It  is  curious  that  the  old  Biblical  name  of  Cabul  (i  Kings,  ix,  13;  Joshua, 
xix,  27)  should  be  found  in  Afghanistan.   Could  it  have  been  taken  there  by  the 
ruling  clan,  the  Duranis,  who  call  themselves  Beni  Israel  and  who  claim  descent 
from  the  Israelites  who  fled  to  the  Far  East  after  the  Assyrians  devastated 
Samaria?    If  so,  could  these  people,  who  also  claim  descent  from  Kish  (i  Samuel, 
ix,  i),  have  taken  the  numerals  from  Egypt  to  Afghanistan? 


GOBAR  NUMERALS 


73 


the  so-called  Dark  Ages  of  Europe,  and  that  the  number  of 
scientific  works  that  have  come  down  to  us  is  also  very  limited, 
we  cannot  say  when  the  Hindu- Arabic  numerals  first  found  their 
way  to  the  West.  There  are  good  reasons  for  believing  that 
they  reached  Alexandria  along  the  great  pathway  of  trade  from 
the  East  even  before  they  reached  Bagdad,  possibly  in  the 
5th  century,  but  without  the  zero.1  It  would  have  been  strange 
if  the  Alexandrian  merchants  of  that  time  and  later  had  not 
known  the  numeral  marks  on  goods  from  India,  China,  and 
Persia.  No  system  that  did  not  contain  a  zero,  however,  would 
have  attracted  much  attention,  and  so  this  one,  if  it  was  known 
at  all,  was  probably  looked  upon  only  as  a  part  of  the  necessary 
equipment  of  a  trader  with  the  East. 

The  Gobar  Numerals.  At  any  rate,  numerals  are  found  in 
Spain  as  early  as  the  loth  century 7  and  some  of  these  numerals 
differ  so  much  from  the  rest  that  they  evidently  came  through 
a  different  channel,  although  from  the  same  source.  These  were 
called  the  dust  numerals,2  possibly  because  they  were  written 
on  the  dust  abacus  instead  of  being  represented  by  counters. 
It  is  worthy  of  note  that  Alberuni  (c.  1000)  states  that  the 
Hindus  often  performed  numerical  computations  in  the  sand. 

If  these  numerals  reached  Alexandria  in  the  5th  century,  they 
probably  spread  along  the  coasts  of  the  Mediterranean  Sea,  be- 
coming known  in  all  the  leading  ports.  In  this  case  they  would 
have  been  familiar  to  the  merchants  for  purposes  of  trade  and  to 
the  inquisitive  for  reasons  of  curiosity.  The  soothsayer  and 
astrologer  would  have  adopted  them  as  part  of  the  mysticism  of 
their  profession,  and  the  scholar  would  have  investigated  them 
as  possibilities  for  the  advancement  of  science.  In  that  case  a 
man  like  Boethius  (c.  510)  would  have  been  apt  to  know  of 
them  and  perhaps  to  mention  them  in  his  writings. 

All  tM^^  The  gobar  numerals  exist  as  a  fact, 

and  this  is  their  possible  origin.  In  certain  manuscripts  of 

xFor  bibliography  and  discussion  see  Smith-Karpinski. 

2I3uHif  al-gobdr.  The  name  appears  in  Tunis  as  early  as  the  middle  of  the 
loth  century.  There  were  also  the  huruf  al-jumal,  or  alphabetic  numerals,  used 
by  the  Jews  and  probably  also  by  the  Arabs. 


74  READING  AND  WRITING  NUMBERS 

Boethius  there  appear  similar  forms,  but  these  manuscripts  are 
not  earlier  than  the  loth  century  and  were  written  at  a  time 
when  it  was  not  considered  improper  to  modernize  a  text.  They 
do  not  appear  in  the  arithmetic  of  Boethius,  where  we  might 
expect  to  find  them,  if  at  all,  but  in  his  geometry,  and  their 
introduction  breaks  the  continuity  of  the  text.  It  therefore 
seems  very  doubtful  that  they  were  part  of  the  original  work  of 
Boethius.  Since  any  forms  that  reached  Alexandria  would  prob- 
ably have  lacked  the  zero,  and  since  a  zero  appears  in  the  late 
Boethian  manuscripts,  there  is  the  added  reason  for  feeling  that 
at  least  part  and  very  likely  all  of  the  symbols  were  inserted 
by  copyists. 

These  gobar  numerals  varied  considerably  but  were  substan- 
tially as  shown  in  the  following  table  : 


-7  6  i 


'  2  3  2  ^ 

7  A  v  ^  0  rr 

JS)   6  ?  f  *  11* 

Gerbert  and  the  Numerals.  The  first  European  scholar  who  is 
definitely  known  to  have  taught  the  new  numerals  is  Gerbert 
(c.  980),  who  later  became  Pope  Sylvester  II  (999).  He  went 
to  Spain  in  967  and  may  have  learned  about  them  in  Barce- 
lona.0 He  probably  did  not  know  of  the  zero,  and  at  any  rate  he 

1  For  sources  of  information  with  respect  to  these  numerals  see  Smith-Karpinski, 
p.  69. 

2Al~Hassar's  forms,  H.  Suter,  Bibl.  Math.,  II  (3),  15. 

3  The  manuscript  from  which  these  are  taken  is  the  oldest  (970  A.D.)  Arabic 
document  known  to  contain  all  the  numerals. 

4  and  s  Woepcke,  "Introduction  au  calcul  Gobari  et  Hawal,"  Atti  dell'  Ac- 
cademia  Pontificia  dei  Nuovi  Lincei,  Vol.  XIX. 

6  On  this  question  see  Smith-Karpinski,  p.  no. 


GERBERT  AND  THE  NUMERALS  75 

did  not  know  its  real  significance.  He  placed  upon  counters  the 
nine  caracteres,  as  they  were  called  by  his  pupils  Bernelinus 
(c.  1020)  and  Richer,  and  used  these  counters  on  the  abacus. 
Such  counters,  probably  in  the  form  of  flattened  cones,  were 
called  apices,  a  term  also  used  in  connection  with  the  numerals 
themselves.  These  numerals  were  severally  called  by  the  names 
igin,  andras,  ormis,  arbasj  quimas,  calctis,  zenis,  temenias, 
celentis,  and  sipos?  The  origin  and  meaning  of  these  terms 
have  never  been  satisfactorily  explained,  but  the  words  seem  to 
be  Semitic.2 

The  oldest  definitely  elated  European  manuscript  that  con- 
tains these  numerals  was  written  in  Spain  in  976.    A  Spanish 


EARLY  EUROPEAN  NUMERALS 

Oldest  example  of  our  numerals  known  in  any  European  manuscript.   This 
manuscript  was  written  in  Spain  in  976 

copy  of  the  Origines  of  Isidorus,  dated  992,  contains  the  nu- 
merals with  the  exception  of  zero.  Dated  manuscripts  of  the 
Arabs  have  been  found  which  give  some  of  these  numerals 
a  century  earlier,  that  is,  in  874  and  888.  They  also  appear  in 
a  Shiraz  manuscript  of  970  and  in  an  Arabic  inscription  in  Egypt 
dated  961.  The  earliest  occurrence  of  these  numerals  in  a  date 
on  a  coin  is  found  on  a  piece  struck  in  Sicily  in  1138. 

There  is  good  reason  for  believing  that  Gerbert  obtained  his 
knowledge  of  the  numerals  from  studying  in  the  convent  of 
Santa  Maria  de  Ripoll,  a  well-known  center  of  learning  near 
Barcelona;3  indeed,  it  is  not  improbable  that  he  saw  the  very 

1  There  were  variants  of  these  forms.  The  sipos  does  not  appear  in  the 
works  of  the  pupils  of  Gerbert,  but  is  found  in  a  MS.  of  Radulph  of  Laon 
(c.  1125).  2Smith-Karpinski,  p.  118. 

3J.  M.  Burnam,  "A  Group  of  Spanish  Manuscripts,"  Bulletin  Hispanique, 
XXII,  229  (Bordeaux,  1920).  With  respect  to  the  library  in  this  convent  sec  R. 
Beer,  Die  Handschrijten  des  Klosters  Santa  Maria  de  Ripoll,  Vienna,  1907. 


76 


READING  AND  WRITING  NUMBERS 


manuscript  of  976  above  mentioned.  There  is  considerable 
evidence  to  support  the  belief  that  the  monks  in  this  cloister 
obtained  their  knowledge  of  these  numerals  through  mercantile 


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sources  which  were  in  communication  with  the  East,  rather  than 
through  any  Moorish  channels  in  Mohammedan  Spain. 

The  changes  in  the  forms  of  the  numerals  may  be  seen  in  the 
table  above.1  The  forms  as  they  appeared  just  before  the 

1  This  is  from  a  table  prepared  by  Mr.  G.  F.  Hill  of  the  British  Museum,  and 
is  reproduced  by  his  permission.  His  noteworthy  article  on  the  subject  appeared 
in  Archaologia,  LXII  (1910).  This  was  elaborated  in  book  form  under  the 
title,  The  Development  of  Arabic  Numerals  in  Europe,  Oxford,  1915. 


DEVELOPMENT  OF  THE  NUMERALS  77 

invention  of  European  printing  may  be 
seen  in  the  annexed  facsimile  from  a 
Latin  manuscript  written  by  Rollandus 
at  Paris,  c.  1424.  After  Europe  began 
to  print  books,  the  forms  varied  but 
little,  most  of  the  changes  being  due 
simply  to  the  fashions  set  by  designers 
of  type.  For  example,  the  figures  4  and 
5  were  changed  to  their  present  forms 
in  the  isth  century  and  have  since  then 
remained  fairly  well  standardized. 

Not  only  did  the  forms  of  the  nu- 
merals change  considerably  during  the 
Middle  Ages,  but  the  method  of  writing 
the  ordinary  numbers  also  varied  from 
century  to  century.  Some  scribes  al- 
ways placed  a  dot  before  and  after  each 
figure,  as  in  the  case  of  a  number 
like  .2.  Others  adopted  a  somewhat 
similar. plan  in  the  case  of  numbers 
having  several  figures.  For  example, 
one  writer  of  c.  I4OO1  gives  5.  7.  8.  2. 
for  5782,  and  one  of  1384  gives  1000. 
300.  80.  4  for  1384,  as  shown  in  the 
following  illustration  from  an  anony-  FROM  THE  ROLLANDUS 
mous  computus  written  in  Italy :  MANUSCRIPT  OF  c.  im 


£<*.    fl*  **-, 

NUMERALS  FROM  A  COMPUTUS  OF   1384 

The  method  of  writing  the  date,  1000.  300.  80.  4,  illustrates  the  difficulties  in 
using  the  numerals.   From  Mr.  Plimpton's  library 

1F.  J.  Studnicka,  Algorismus  prosaycus  magistri  Christani,  p.  9  (Prag,  1893). 
This  Magister  Christanus  was  Christanus  Prachaticensis,  or  Christian  of  Prag 
(born  1368;  died  1439)- 


78  READING  AND  WRITING  NUMBERS 

Early  English  Algorism.  An  interesting  illustration  of  the 
early  use  of  the  word  "algorism"  (algorym,  augrim)  in  the 
English  language  may  be  seen  in  a  manuscript  now  in  the  Brit- 
ish Museum,  dating  from  c.  isoo.1  The  first  page,  which  is 
here  shown  in  facsimile,  reads  as  follows : 

Hec  algorisms  ars  psens  dicitr  in  qua 
Talibs  indoi|  fruimr  bis  quiq  figuris.2 

This  boke  is  called  pe  boke  of  algorym  or  Augrym  after  lewder 
use.  And  pis  boke  tretys  pe  Craft  of  Nombryng,  pe  quych  crafte  is 
called  also  Algorym.  Ther  was  a  kyng  of  Inde  )>e  quich  heyth  Algor, 
&  he  made  pis  craft.  And  aft  his  name  he  called  hit  algory.  Or  els 
anoth  cause  is  quy  it  is  called  Algorym,  for  pe  latyn  word  of  hit  s. 
Algorismus  corns  of  Algos  grece  q  e  ars,  latine,  craft  on  englis,  and  rides 
q  e  nms,  latine,  A  nombr  on  englys.  inde  dr  algorismus  p  addicone 
huis  sillabe  ms  &  subtracconem  d  &  E,  qsi  ars  numandi.3 
|f  fforthermor4  ye  most  undrstonde  ft  in  ]>is  craft  ben  usid  teen 
figurys.  as  her  ben  writen  for  ensampul.  $.9.8.7.6.5.4.3.2.1. 
|f  Expone  pe  too  vsus  a  for;5  pis  psent  craft  is  called  Algorisms, 
in  pe  quych  we  use  teen  figurys  of  Inde.  Questio.  f[  Why  ten  fyguris 

1  It  was  first  privately  printed  by  the  Early  English  Text  Society  (transcription 
by  Robert  Steele),  London,  1894.  It  has  already  been  referred  to  in  Volume  I, 
page  238,  and  in  this  volume,  page  32. 

2 These  are  the  two  opening  lines  of  the  Carmen  de  Algorismo,  of  Alexandre 
de  Villedieu  (c.  1240).  They  should  read  as  follows  : 

Haec  algorismus  ars  praesens  dicitur;  in  qua 
Talibus  Indorum  fruimur  bis  quinque  figuris. 

It  is  translated  a  few  lines  later :  "  This  present  craft  is  called  Algorismus,  in 
the  which  we  use  ten  figures  of  India." 

3 "Inde  dicitur  Algorismus  per  addicionem  huius  sillabe  mus  &  subtraccionem 
d  &  e,  quasi  ars  numerandi  (Whence  it  is  called  Algorismus  by  the  addition  of 
this  syllable  mus,  and  the.  taking  away  of  d  and  e,  as  if  the  art  of  numbering)." 
This  idea  had  considerable  acceptance  in  the  i3th  century. 

4 "Furthermore,"  the  /  being  doubled  for  a  capital.  "Furthermore  you  must 
understand  that  in  this  craft  there  are  used  ten  figures."  The  forms  of  the  nu- 
merals given  in  the  original  were  the  common  ones  of  the  i2th  and  i3th  cen- 
turies. The  zero  was  not  usually  our  form,  but  frequently  looked  more  like 
the  Greek  phi.  The  7,  5,  and  4  changed  materially  in  the  latter  part  of  the 
i5th  century,  about  the  time  of  the  first  printed  books.  The  sequence  here  shown 
is  found  in  most  of  the  very  early  manuscripts,  the  zero  or  nine  being  at  the  left. 

5"  Explain  the  two  verses  afore." 


i<v 


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Folio  ir  of  the  manuscript 


FIRST  PAGE  OF  THE  CRAFT  OF  NOMBRYNG 

Egerton  MS.  2622  in  the  British  Museum,  one  of  the  earliest  manuscripts  in 
English  which  treat  of  any  phase  of  mathematics 


8o  READING  AND  WRITING  NUMBERS 

of  Inde.  Solucio.1  for  as  I  have  sayd  a  fore  )>>ai  wer  fonde  fyrst  in 
Inde  of  a  kyng  of  fat  Cuntre  j>t  was  called  Algor.  fl  Pma  sigt  uno 
duo  vo  scda 2  fl  Tercia  sigt  tria  sic  pcede  sinistre.  jf  Done  ad  extma 
venias  que  cifra  vocar.  jf  Capm  pmu  de  significacoe  figurarm  |f  In 
]>is  verse  is  notifide  f e  significacon  of  fese  figuris.  And  pus  expose 
)>e  verse  the  first  signifiyth  on.  J>e  secude  signi[fiyth  tweyn].3 

Reading  and  Writing  Large  Numbers.  One  of  the  most  strik- 
ing features  of  ancient  arithmetic  is  the  rarity  of  large  numbers. 
There  are  exceptions,  as  in  some  of  the  Hindu  traditions  of 
Buddha's  skill  with  numbers,4  in  the  records  on  some  of  the 
Babylonian  tablets,5  and  in  the  Sand  Reckoner6  of  Archimedes, 
with  its  number  system  extending  to  io03,  but  these  are  all 
cases  in  which  the  elite  of  the  mathematical  world  were  con- 
cerned ;  the  people,  and  indeed  the  substantial  mathematicians 
in  most  cases,  had  little  need  for  or  interest  in  numbers  of  any 
considerable  size. 

The  Million.  The  word  "million,"  for  example,  is  not  found 
before  the  i3th  century,  and  seems  to  have  come  into  use  in 
England  even  later.  William  Langland  (c.  1334-^;.  1400),  in 
Piers  Plowman,  says, 

Coueyte  not  his  goodes 
For  millions  of  moneye/ 
i"  Answer." 

2 "The  first  means  one,  the  second  two,  the  third  means  three,  and  thus  pro- 
ceed to  the  left  until  you  reach  the  last,  which  is  called  cifra."  The  author  is 
quoting  from  the  Carmen  of  Alexandre  de  Villedieu : 

Prima  significat  unum;  duo  vero  secunda; 
Tertia  significat  tria;  sic  precede  sinistre 
Donee  ad  extremam  venias,  quae  cifra  vocatur. 

3"Gapitulum  primum  de  significacione  figurarum  (Chapter  I,  On  the  meaning 
of  the  figures) ." 

"And  thus  explain  the  (Latin)  verse:  the  first  signineth  one." 

"The  second  (secunde)  signifieth  twain." 

4  Sir  Edwin  Arnold  speaks  of  this  in  The  Light  of  Asia.  See  Smith-Karpinski, 
p.  16.  5Hilprecht,  Tablets. 

6  ^afjLjjLlTrjs  (Psammi'tes) ,  translated  into  Latin  as  Arenarius.  For  the  text, 
see  Archimedis  opera  omnia,  ed.  Heiberg,  with  revisions,  II,  242  (Leipzig,  1880- 
1913)  ;  Heath,  Archimedes,  p.  221.  See  also  M.  Chasles,  in  the  Comptes  rendus, 
April  u,  1842;  the  preface  to  the  English  translation  of  the  work  by  Archimedes 
made  by  G.  Anderson,  London,  1784;  Heath,  History,  II,  81. 


THE  MILLION  81 

but  Maximus  Planudes  (c.  1340)  seems  to  have  been  among  the 
first  of  the  mathematicians  to  use  the  word.1  By  the  isth  cen- 
tury it  was  known  to  the  Italian  arithmeticians,  for  Ghaligai 


£  million  adoncba  fcDic  fotmor  per  litre  figure 
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tnttlemfara  .fanovno  million  ictcfleiKJomqud 
Iiiogo  laftgura  cberipjeiceiira  vtio  pero  bcne 
editovno  imlf0n.3&a.fc)iieftpmodo.  1  100000 
afria'vno  milfoil  ecento  roflifa  :ercbe  oltra  cl 


imli'a.3&a.fif        .  , 

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dmAioK  e  cento  nit'Im  :ttt  luogooete  ocjccucdc^ 
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Havnmflioncctitocvndc]cemi1fa  per  cbe  oltraclmilionccnco 
ediejce  miltaan  luo^o&e  numertoemfarfonolaftgura  cbe  tip 
Vei)tavno:ftd>ebedeedirovi!mt{iO]}cenroevndejcemi(ia.7£>a        1  1  it  i  oo 
inqueftomodo.  1  1  1  1  loo.tJinavnmiliancciuoe  vrtdccem^ 
lid  e  ccneorpercbc  oltra  cfnnlion  cento  e  vndece  mil/a:  to  luoaa 
oc  effimplt  ce  centenarfono  lafigura  cbe  tipzaenta  yno:fi  cbe 
bene  editovnmilion  cento  e  vndcice  mi'Ka  e  cenco.^a  mquefto 
modo,  j  1  1  1  1  lo.Ciriavnmi'Won,  cento  e  vudocemiUa  cento       1  1  1  1  1  1  o 
cdiejtfpcrcbeoitra  elmidbn  cento  evndcre  mffiaecento:! 
luo^oDcIcfimpIice&ejccnc'.rono  fefigoracbe  np:ejcentavt)o. 
^ainqneftomodo.  1  1  1  1  1  1  i.DtriavnmiUoncentoevnde       1  1  1  1  1  1  1 
JDC  mWacentoe  vndere.pcr  cbe  ancbe  m  (uogo  ode  Hmpllcc 
vnita  .fono  lafi^ura  cberip:ejccnta  vno/tcbe  bene  cdirovn- 
mflioncentoe  vndcjremilca  ccntoe  vndejce.ctcboft  pjocede'do  ***** 

perfma  .^^^^^.poncndo  fc«ip:e  aifuo  luogtqndefigurc  999y"9* 
rep:erentante  qneh'nmncK  oncroocjcenc  ocentenara.cbefi- 
nommaetcetera.eqiicftobnfta  cercba  loamai  (Iram  euro  DC! 
nntnerar.bencbeminfinftumnpo^ia  p:cccdcr.ma  cbonivna 
general  figura  mi'fo:cero  m'cbiarfrquanto  jwtciTeacbadcr.ct: 
farano  queflo  fottopott* 

THE  WRITING  OF  LARGE  NUMBERS  IN  1484 

From  Pietro  Borghi's  De  Arte  Mathematiche,  Venice,  1484.    This  illustration  is 
from  the  1488  edition 

(1521  ;  1552  ed.,  fol.  3)  relates  that  "Maestro  Paulo  da  Pisa"2 
read  the  seventh  order  as  millions.   It  first  appeared  in  a  printed 

1H.Waschke  translation,  p.  4n.  (Halle,  1878)  ;  hereafter  referred  to  asWaschke, 
Planudes.  The  word  simply  means  "great  thousand"  (from  mille  4-  on),  just 
as  salon  means  "great  hall"  (from  salle  4-  on)  and  balloon  means  "great  ball." 

2  "La  settima  dice  numero  di  milione."  This  Paul  of  Pisa  may  have  been 
the  Paolo  dell'  Abbaco  (Dagomari,  c.  1340)  mentioned  in  Volume  I,  page  232. 


82  READING  AND  WRITING  NUMBERS 

work  in  the  Treviso  arithmetic  of  1478.  Thereafter  it  found 
place  in  the  works  of  most  of  the  important  popular  Italian 
writers,  such  as  Borghi1  (1484),  Pellos2  (1492),  and  Pacioli3 
(1494),  but  outside  of  Italy  and  France  it  was  for  a  long  time 
used  only  sparingly.  Thus,  Gemma  Frisius  (1540)  used  "  thou- 
sand thousand"4  in  his  Latin  editions,  which  were  published  in 
the  North,  while  in  the  Italian  translation  (1567)  the  word 
millioni  appears.  Similarly,  Clavius  carried  his  German  ideas 
along  with  him  when  he  went  to  Rome,  and  when  (1583)  he 
wished  to  speak  of  a  thousand  thousand  he  almost  apologized 
for  using  "  million,"  referring  to  it  as  an  Italian  form  which 
needed  some  explanation.5 

In  Spain  the  word  cuento*  was  early  used  for  iofi,  the  word 
millon  being  reserved  for  io12.  When  the  latter  word  was 
adopted  by  mathematicians,  it  was  slow  in  coming  into  gen- 
eral use.7 

iall  miar  de  milliara  6  vuol  dir  il  million"    (1540  ed.,  fol.  5). 

2  His  names  beyond  units  are  desena,  centenal,  millier,  xa  de  m{ %  ca  de  ma, 
and  million  (fol.  2). 

3He  uses  milioni  (fol.  9)  but  no  higher  special  names,  although  he  repeats 
this  word,  as  in  "Migliara  de  milio  de  milio"  (fol.  19,  v.)y  adding:  "Et  sic  I 
sequetib9.  ^seqre." 

The  spelling  varies  in  the  early  books,  sometimes  appearing  as  miglioni 
(Pagani,  1591).  4  Millena  millia. 

BIn  the  Latin  edition  (1583):  "lam  vero  si  more  Italorum  millena  millia 
appellare  velimus  Milliones,  paucioribus  verbis  &  fortasse  significantius " 
(Epitome,  cap.  i). 

In  the  Italian  edition  (1586)  :  "Hora  se  secodo  il  costume  d'  Italia  vorremo 
vn  migliaio  di  migliaia  chiamare  millione,  con  manco  parole,  &  forse  piu 
significantemente "  (p.  14). 

In  De  Cosmographia  Libri  IV  by  Francesco  Barozzi  (c.  i538-c.  1587),  a 
work  published  in  Venice  in  1585,  it  is  stated  that  "septima  (nota)  pro  Mil- 
lenario  Millenarii  quern  vulgus  quidem  Millionem  appellant,  Latini  vero  Milleno 
Millio." 

GFrom  contar,  to  count  or  reckon.  Ciruelo,  whose  work  was  published  in 
Paris  in  1495,  says:  "Millies  millena:  quod  vulgariter  dicitur  cuento :  decies 
cuento/centies  cuento/millies  cuento/decies  millies  cuento/centies  millies  cuento 
[/millies  millies  cuento]/quod  vulgariter  dicitur  millon."  See  1513  ed.,  fol.  a2 
and  fol.  A8.  He  is  not,  however,  uniform  in  the  matter,  using  "millon"  as 
synonymous  with  "cuento"  in  other  places. 

7Aslate  as  the  1643  edition  of  Santa-Cruz  (1594)  it  was  necessary  to  explain 
the  word  thus:  "Millon  que  significa  mil  millares"  (fol.  13,  r.),  the  latter  being 
the  common  form, 


THE  MILLION  83 

France  early  took  the  word  "  million "  from  Italy,  as  when 
Chuquet  (1484)  used  it,  being  followed  by  De  la  Roche  (1520), 
after  which  it  became  fairly  common. 

The  conservative  Latin  writers  of  the  i6th  century  were 
very  slow  in  adopting  the  word.  Even  Tonstall  (1522),  who 
followed  such  eminent  Italian  writers  as  Pacioli,  did  not  com- 
monly use  it.  He  seems  to  have  been  influenced  by  the  fact 
that  the  Romans  had  no  use  for  large  numbers;1  or  by  the 
fact  that,  for  common  purposes,  it  sufficed  to  say  "thousand 
thousand,"  as  had  been  done  for  many  generations.2  He 
simply  mentions  the  word  as  a  piece  of  foreign  slang  to  be 
avoided.3  Other  Latin  writers  were  content  to  say  "  thousand 
thousand.  "4 

The  German  writers  were  equally  slow  in  abandoning  "thou- 
sand thousand "  for  "million,"  most  of  the  writers  of  the  i6th 
century  preferring  the  older  form.5  The  Dutch  were  even  more 
conservative,  continuing  the  old  form  later  than  the  writers  in 
the  neighboring  countries.6  Indeed,  for  the  ordinary  needs  of 
business  in  the  i6th  century,  the  word  "million"  was  a  luxury 
rather  than  a  necessity. 

!"Non  me  latet  Romanes  ueteres  prisco  more,  suos  numos  Sestertios  com- 
putates,  numerum  trascendentem  centum  millia  .  .  .  Latinc  n5  enunciasse  .  .  .  ." 

2Even  as  late  as  1501,  Huswirt,  a  German  scholar,  writes  "quadraginta  quat- 
tuor  mille  millia.  quingSta  millia  quinquaginta  noue  millia.  octingenta.  octoginta 
sex"  for  44,559,886. 

3"  Septimus  millena  millia:  uulgus  millione  barbare  uocat." 

4 So  Stifel  uses  "millia,  millies"  (Arithmetica  Integra,  1544  ed.,  fol.  i); 
Ramus  uses  "millena  millia"  (Libri  II,  1569,  p.  i) ;  Glarcanus  has  "mille  millia" 
(1538;  1543  ed.,  fol.  9). 

Thus,  "tausant  mal  tausant"  is  used  by  such  writers  as  Kobel  (Zwey 
Reckenbuchlin,  1514;  1537  ed.,  fol.  14),  Grammateus  (1518;  1535  ed.,  p.  5), 
Riese  (1522;  1529  ed.,  p.  3),  and  Rudolff  (1526;  1534  ed.,  fol.  3).  Rudolff, 
however,  uses  it  together  with  the  older  form  in  his  Rechenbuch  (1526),  and 
in  his  Exempelbuchlin  (1530;  1540  ed.,  exs.  62  and  137)  he  says:  "Vnd  wirt 
ein  million  mit  ziffcrn  geschriben  1000000,"  and  "ist  zehenmal  hundert  tausent." 

6 Thus,  "duysent  mael  duysent"  is  used  by  such  writers  as  Petri  (1567, 
fol.  i),  Raets  (1580,  fol.  A3,  with  "duysentich  duysent"),  Mots  (1640,  fol.  B2), 
Cardinael  (1659  fid.,  fol.  A8),  Willemsz  (1708  ed.,  p.  5),  and  Bartjens  (1792  ed., 
p.  8).  There  were  exceptions,  as  when  Wentsel  (Wenceslaus,  1599)  used  both 
"millioenen"  and  "millions"  (p.  2),  Stockmans  (1589;  1679  ed.,  p.  8)  occa- 
sionally used  "millioen,"  and  Starcken  (1714  ed.,  p.  2)  used  "million"  rather 
apologetically. 


84  READING  AND  WRITING  NUMBERS 

England  adopted  the  Italian  word  more  readily  than  the 
other  countries,  probably  owing  to  the  influence  of  Recorde1 
(c.  1542).  It  is  interesting  to  see  that  Poland  was  also  among 
the  first  to  recognize  its  value,  the  word  appearing  in  the 
arithmetic  of  Klos  in  1538. 

The  Billion.  Until  the  World  War  of  1914-1918  taught  the 
world  to  think  in  billions  there  was  not  much  need  for  number 
names  beyond  millions.  Numbers  could  be  expressed  in  figures, 
and  an  astronomer  could  write  a  number  like  9.1 5 -lo7,  or 
2.5  •  io20,  without  caring  anything  about  the  name.  Because  of 
this  fact  there  was  no  uniformity  in  the  use  of  the  word  "bil- 
lion." It  meant  a  thousand  million  (io9)  in  the  United  States 
and  a  million  million  (io12)  in  England,  while  France  commonly 
used  milliard  for  io9,  with  billion  as  an  alternative  term. 

Historically  the  billion  first  appears  as  io12,  as  the  English 
use  the  term.  It  is  found  in  this  sense  in  Chuquet's  number 
scheme2  (1484),  and  this  scheme  was  used  by  De  la  Roche 
(1520),  who  simply  copied  parts  of  Chuquet's  unpublished 
manuscript,  but  it  was  not  common  in  France  at  this  time,  and 
it  was  not  until  the  latter  part  of  the  i7th  century  that  it  found 
place  in  Germany.3  Although  Italy  had  been  the  first  country 
to  make  use  of  the  word  "  million,"  it  was  slow  in  adopting  the 
word  " billion."  Even  in  the  1592  edition  of  Tartaglia's  arith- 
metic the  word  does  not  appear.  Cataldi  (1602)  was  the  first 
Italian  writer  of  any  prominence  to  use  the  term,  but  he  sug- 

*"  203000000,  that  is,  CCiii  millios,"  "M.  of  millions,"  and  "x.M.  of  mil- 
lions" (1558  ed.,  fol.  C8). 

2  This  plan  is  historically  so  important  as  to  deserve  being  given  in  full. 
Chuquet  gives  the  6-figure  periods,  thus:  7453 24'8o430oo'7ooo23 '6543 21  (in 
which  8043000  should  be  804300) ,  and  then  says :  "  Ou  qui  veult  le  pmier 
point  peult  sigmffier  million  Le  second  point  byllion  Le  tiers  poit  tryllion  Le 
quart  quadrillion  Le  cinqe  quyllion  Le  sixe  sixlion  Le  sept?  septyllion  Le  huyte 
ottyllion  Le  neuf e  nonyllion  et  ansi  des  ault98  se  plus  oultre  on  vouloit  pceder 
fl  Item  Ion  doit  sauoir  que  ung  million  vault  mille  milliers  de  unitez.  et  ung 
byllion  vault  mille  milliers  de  millions,  et  tryllion  vault  mille  milliers  de  byl- 
lions."  From  A.  Marre's  autograph  copy  of  Chuquet.  See  also  Boncompagni's 
Bullettino,  Vol.  XIII,  p.  594. 

3F.  Unger,  Die  Methodik  der  praktischen  Arithmetik  in  historischer  Entwicke- 
lung,  p.  71  (Leipzig,  1888),  with  the  date  of  use  of  the  word  as  1681;  hereafter 
referred  to  as  Unger,  Die  Methodik. 


THE  BILLION  85 

gested  it  as  a  curiosity  rather  than  a  word  of  practical  value.1 
About  the  same  time  the  term  appeared  in  Holland,2  but  it  was 
not  often  recognized  by  writers  there  or  elsewhere  until  the 
1 8th  century,  and  even  then  it  was  not  used  outside  the  schools. 
Even  as  good  an  arithmetician  as  Guido  Grandi  (1671-1742) 
preferred  to  speak  of  a  million  million  rather  than  use  the 
shorter  term.3 

The  French  use  of  milliard,  for  io9,  with  billion  as  an  alterna- 
tive, is  relatively  late.  The  word  appears  at  least  as  early  as 
the  beginning  of  the  i6th  century  as  the  equivalent  both  of  io9 
and  of  io12,  the  latter  being  the  billion  of  England  today.4  By 
the  1 7th  century,  however,  it  was  used  in  Holland5  to  mean  io9, 
and  no  doubt  it  was  about  this  time  that  the  usage  began  to 
change  in  France. 

As  to  the  American  psage,  taking  a  billion  to  mean  a  thousand 
million  and  running  the  subsequent  names  by  thousands,  it  should 

aHe  generally  used  millions,  thousand  millions,  million  millions,  and  so  on 
(p.  2)  ;  but  he  sometimes  used  bilioni  for  io9,  although  even  then  he  preferred 
duilioni.  His  scheme  of  names  is  millioni,  bilioni  (or  duilioni),  trilioni,  quadri- 
lioni  (or  quattrilioni} ,  quintilioni,  for  io6,  io9,  io12,  io15,  and  io18.  Practica 
Aritmetica,  p.  5  (Bologna,  1602). 

2 Van  der  Schuere  (1600)  uses  millioen  (IOG),  bimillioen  (io12),  trlmillioen 
(io18),  and  quadrimillioen  (io24),  but  in  a  later  edition  (1634)  °f  his  arith- 
metic he  gives  bimillion  and  billion  (io9),  trimillion  and  trillion  (io12),  and  so 
on  to  nonemillion  and  nonilion.  Even  as  late  as  1710  Leonhard  Christoph 
Sturm  (Kurzer  Begriff  der  gesamten  Mathesis,  Frankfort  a.  d.  Oder,  1710)  used 
the  words  trimillionen  and  bimttlionen. 

3 In  his  Jstituzioni  di  Aritmetica  Practica,  p.  3  (Florence,  1740),  he  says: 
"millioni  de'  millioni  (che  possono  dirsi  Billioni)  e  li  millioni  di  millioni  di 
millioni  (che  si  chiamano  ancora  Trillioni).  .  .  .  E  cosi  se  fosse  piu  lungo  il 
numero,  vi  sarebbero  ancora  Quintillioni  .  .  .  Novillioni,  ec.  crescendosi  ciascuno 
da  ogni  sci  note." 

4Thus,  Trenchant  (1566)  uses  Miliars  (1578  ed.,  p.  14),  and  Peletier  (1549) 
says:  "Les  Frangois  ont  deux  mots  numeraux  significatifs :  Tun  au  septieme 
lieu,  qui  est  Million,  &  1'autre  au  treizieme,  qui  est  Milliart :  c'est  a  dire,  Million 
de  Millions"  (1607  ed.,  p.  15).  Peletier  states  that  the  word  was  used  by 
Budaeus,  and  in  the  latter's  De  Asse  et  partibus  eius  Libri  quinq$  (1514;  Paris 
edition  of  1532,  fol.  95,  v.}  the  following  appears:  "hoc  est  denas  myriadu 
myriadas,  quod  vno  verbo  nostrates  abaci  studiosi  Milliartu  appellat,  quasi 
millionu  millione."  In  Boissiere's  arithmetic  of  1554  there  is  a  statement  similar 
to  the  one  in  Peletier's  work.  In  E.  Develey,  Arithmetique  d'fimile,  2d  ed., 
Paris,  1802,  only  "billion"  is  used  for  1000  millions. 

5"milliart/ofte  duysent  millioenen,"  as  Houck's  arithmetic (1676,  p.  2)  has  it. 

n 


86  READING  AND  WRITING  NUMBERS 

be  said  that  this  is  due  in  part  to  French  influence  after  the  Revo- 
lutionary War,  although  our  earliest  native  American  arithmetic, 
the  Greenwood  book  of  1729,'  gave  the  billion  as  iofj,  the  trillion 
as  io127  and  so  on.  Names  for  large  numbers  were  the  fashion 
in  early  days,  Pike's  well-known  arithmetic  ( 1 788) ,  for  example, 
proceeding  to  duodecillions  before  taking  up  addition. 

Writing  Large  Numbers.  Although  it  is  nearly  a  thousand 
years  since  our  common  numerals  appeared  in  any  European 
manuscripts  now  extant,  we  have  not  even  yet  decided  on  the 
method  of  writing  large  numbers. 

Influenced  by  the  crosses  placed  on  the  thousands'  and  mil- 
lions' lines  of  the  abacus  (see  page  181)  to  aid  the  eye,  the  me- 
dieval writers  often  placed  a  dot  above  the  thousands  and  above 
every  third  place  beyond,  but  sometimes  they  placed  one  or 
more  dots  below,  and  these  customs  also  appear  in  the  early 
printed  books.  Thus,  we  have  such  a  form  as  6854973^  with 
the  occasional  variant  of  a  dot  over  the  units'  figure  also.3 
Recorde  (c.  1542)  gives  the  rule  as  follows: 

Fyrst  put  a  pricke  ouer  the  fourth  fygure,  and  so  ouer  the  vij. 
And  if  you  had  so  many  ouer  the  x,  xiij,  xvj,  and  so  forth,  still 
leauing  two  fygurs  betwene  eche  two  pricks.  And  those  roomes  be- 
twene  the  prickes  are  called  Ternaries.4 

Recorde  also  uses  a  bar  (virgula)  for  separating  the  figures, 
saying : 

*It  was  published  anonymously,  but,  as  is  stated  in  the  Weekly  News  Letter 
(Boston)  of  May  29,  1729,  was  written  by  Isaac  Greenwood,  sometime  professor 
of  mathematics  at  Harvard.  As  stated  in  Volume  I,  the  first  arithmetic  printed 
in  the  New  World  appeared  in  Mexico  in  1556 ;  the  first  in  what  is  now  the  United 
States  was  a  reprint  of  Hodder's  English  arithmetic,  Boston,  1719. 

2Tonstall  (1522,  fol.  C,),  Riese  (1522;  1529  ed.,  p.  3),  Rudolff  (1526, 
fol.  3),  Grammateus  (1518;  1535  ed.,  p.  5),  and  many  others.  Widman  (1489) 
recommends  but  does  not  use  this  plan:  "Vnd  setz  vff  ytlich  tausent  ain  punct 
da  by  man  mercken  mag  wie  vil  die  letst  figur  mer  tausent  bedeut  dann  die 
vor  ir"  (1519  ed.,  fol.  5,  v.). 

3Thus  Clavius  (Italian  edition  of  the  Epitome,  1586,  p.  14;  Latin  ed.,  1583, 
p.  10).  He  recommends,  however,  the  following:  42329089562800. 

4 Ground  of  Artes,  1558  ed.,  fol.  B8.  Similarly  in  Digges  (1572;  1579  ed., 
p.  2),  Baker  (1568;  1580  ed.,  fol.  4),  and  Hodder  (1672  ed.). 


LARGE  NUMBERS  87 

And  some  doo  parte  the  nubres  with  lynes  after  this  forme 
23o|864Jo89|oi5|34o,  where  you  see  as  many  lines  as  you  made 
pricks.1 

Some  writers  used  this  symbolism  in  grouping  by  sixes.2  Be- 
sides placing  one  dot  above  a  figure,  the  medieval  writers  often 

used  such  forms  as  243^56^93842 13  and  2437562938421,  and 

these  occasionally  appear  in  the  printed  works.  Fibonacci 
(1202)  used  the  arc,  as  in  67893^7'84io5296,4  but  this  was  not 
a  common  form.  A  few  of  the  other  variations  are  given  below : 

7.538.275.136  Pellos  (1492,  fol.  4) 

4.5.9-3.6.2.9.0.2.2  or  j  Reisch5    (1503,    Lib.    IIII, 

4593629022                J  Tract.  II,  Cap.  4) 

25783916627512346894352  Barozzi0  (1585) 

23.456.067.840.000.365.321  Santa-Cruz  (1594,  fol.  12,  r.) 

1,234,567  or  1.234.567  Greenwood  (1729) 

68|76s|43'2|i89|7i6[789|i32  Blassiere  (1769) 

The  groups  have  been  called  by  various  names,  such  as 
periods/  regions,8  and  ternaries,9  and  occasionally,  as  with 
Trenchant  (1566),  there  were  four  figures  in  the  right-hand 
group.10 

Spanish  Method  of  Writing  Large  Numbers.  One  of  the  most 
interesting  examples  of  the  writing  of  large  numbers  found  in 
the  books  of  the  i6th  century  is  seen  in  the  work  of  Texeda, 

1  Similarly  in  Gemma  Frisius  (1540),  Trenchant  (1566),  and  various  others. 
2 E.g.,  the  Dutch  arithmetic  of  Wilkens,  1669,  p.  8. 

3  As  in  a  i4th  century  algorismus  in  the  Columbia  University  Library.    This 
plan  is  also  followed  by  Tartaglia  (1556,  I,  fol.  7,  r.}. 
4 Liber  Abaci,  p.  i. 

5  These  cases  contain  errors  in  printing  in  the  first   (1503)   edition  of  the 
Margarita  phylosophica,  but  they  are  corrected  in  the  later  editions. 

6  Francesco  Barozzi,  De  Cosmographia  Libri  IV,  Venice,  1585. 

7"Haec  prima  est  periodus,"  etc.,  Ramus  (1569;  ed.  Schoner,  1586,  p.  2). 

8Santa-Cruz  (1594),  fol.  12. 

9Recorde,  as  quoted  above.  10As  in  10,500,340,8020  on  page  16. 


88  ADDITION 

a  Spanish  writer  of  1546.  In  seeking  to  explain  algorism  he 
writes  numbers  in  the  Spanish  style  (en  Castellano]  and  also  in 
algorism  (en  guarismo).  The  following  cases  are  typical: 

c.  Ix.  U  462  qs  .  .  ix  U  62 11  160  U  462  qs  009  U  621 

c.  iij  U.  75  qs    c.  ij  U  300  103  U  075  qs  102  U  300 

Dcccxcj  Uccxxxiiij  qs  Dlx  U.  891  U  234  qs  560/000 

vij  U  .  .  .  qs  Dxlv  Ucccclxijm  7  U  ooo  qs  545  U  462 

It  will  be  seen  that  in  the  Spanish  forms,  doubtless  owing  to 
the  Arab  influence,  there  is  a  tendency  (not  uniformly  carried 
out)  to  use  the  dot  for  zero.  Texeda  also  mixes  his  algoristic 
numerals  with  the  Roman, — a  custom  not  uncommon  after  the 
1 2th  century.  The  U  stands  for  thousands,  appearing  in  earlier 
times  as  U  with  several  variants,  and  being  of  uncertain 
origin.  The  qs  stands  for  quentos  (cuentos,  millions).  In  the 
1 6th  century  the  Greek  6  was  also  used  instead  of  U,  as  in 
XXXV0CCCXXVI  for  35,326,  and  637^500  for  637,500,  and 
in  the  i8th  century  it  often  degenerated  into  a  kind  of  in- 
verted C.2  In  Portugal  a  symbol  $  (cifrao}  was  used  as  early  as 
the  1 6th  century  for  the  same  purpose. 

3.  ADDITION 

Terminology  of  Addition.  The  name  of  the  operation  which 
we  call  addition  has  had  its  vicissitudes.  One  writer  of  the 
i3th  century,  for  example,  used  "aggregation"  instead.3  Writ- 

xln  the  original  the  ix  is  misprinted  x. 

The  numbers  at  the  left  are  en  castellano ;  those  at  the  right,  en  guarismo. 
The  illustrations  are  from  fols.  iij,  v.,  to  iiij,  v. 

The  number  is  160,462,009,621. 

See  F.  Cajori, "  Spanish  and  Portuguese  symbols  for  *  Thousands,' "  Amer.  Math. 
Month.,  XXIX,  201,  who  had  not  seen  Texeda;  he  suggests  that  the  U  came  from 
some  variant  of  the  Roman  symbol  for  thousand. 

2  See  the  "  Fragmentos  del  Archive  Particular  de  Antonio  Perez,  Secretario  de 
Felipe  II,"  Revista  de  Archives,  Bibliotecas  y  Museos,  XXV  (1920),  140 
(Madrid,  1920).  In  the  author's  library  are  several  Spanish  manuscripts  of 
c.  1725-1750  with  the  degenerate  form  of  0  referred  to  in  the  text. 

8<<Agregare  est  quoslibet  duos  numeros  uel  plures  in  unum  colligere"  (B.  Bon- 
compagni,  Trattati,  II,  30).  We  still  preserve  the  phrase  "in  the  aggregate." 
The  word  is  merely  the  Latin  for  T/xwm0^eu  (prostithen'ai) ,  used  by  Euclid 
and  Diophantus,  or  (rvvnOtvat  (syntithen'ai) ,  as  used  by  Heron  and  Pappus. 


TERMINOLOGY  89 

ing  about  the  year  1200,  Fibonacci  used  " composition"  and 
"collection"  as  well  as  "addition."1  Nearly  a  century  after 
Fibonacci  the  earliest  French  algorism  (c.  1275)  used  "as- 
semble"2 for  "add,"  and  two  centuries  later  the  first  printed 
arithmetic  used  "join."3  In  the  early  printed  books  the  word 
"summation"  was  a  rival  of  "addition,"4  and  we  still  speak  of 
summing  up,  and  of  summing  certain  numbers.  Addition  being 
the  operation  most  frequently  used,  the  operation  probably 
gave  rise  to  the  expression  "to  do  a  sum,"  meaning  to  solve  a 
problem.  Various  other  names  for  the  process  have  been  used,5 
but  they  have  no  special  significance.  With  the  English  tend- 
ency to  brevity,  there  is  little  prospect  of  change  in  this  lan- 
guage in  the  words  "add"  and  "addition." 

In  such  of  the  early  printed  arithmetics  as  were  intended  for 
popular  use  there  was  ordinarily  no  word  corresponding  to  our 
term  "addend."6  On  the  other  hand,  the  theoretical  books, 
generally  printed  in  Latin,  spoke  of  the  numeri  addendi,  that  is, 
the  "numbers  to  be  added,"7  and  from  this  came  the  word 
addendi  alone,  as  used  by  Fine  (iS3o)?8  Gemma  Frisius  (1540), 
and  later  writers.  From  this  we  have  our  English  "addends." 
Those  who  seek  for  a  change  have  occasionally  used  the  less 
familiar  "summands." 


aln  the  Latin,  compositio,  collectio,  and  additio. 

2"Se  tu  veus  assambler  .1.  nombre  a  autre"  (Boncompagni's  Bidlettino 
XV,  53).  3 /.<?.,  jongere.  Treviso  arithmetic,  1478 

4"Addirn  oder  Summirn,"  in  Rudolff's  arithmetic  of  1526  (1534  ed.,  fol.  3) 
Stifel's  Deutsche  Arithmetica  (1545,  fol.  i),  Albert's  arithmetic  of  1534,  anc 
many  others.  Grammateus  (1518)  has  "Additio  oder  Summierung,"  and  Adarr 
Riese  (1522;  1550  ed.)  has  a  chapter  on  "Addirn/Summirn/Zusamen  legen/ 
the  last  term  derived  from  counter  reckoning. 

5 E.g.,  the  German  Zusammenthmmg,  the  French  aiouster,  and  the  Italiar 
recogliere,  summare,  and  acozzare. 

6£.g.,  Recorde  (c.  1542),  Sfortunati  (1534),  Baker  (1568),  Digges  (1572) 
Peletier  (1549),  Trenchant  (1566),  Pagani  (1591),  and  Pacioli's  Suma  (1494) 
The  early  Dutch  arithmeticians  rarely  had  such  a  word,  and  even  the  Americar 
Greenwood  (1729)  does  hot  give  one. 

7  As  in  Scheubel   (1545,  p.   13),  Clavius   (1583),  Licht   (1500,  fol.  2),  anc 
many  others,  but  curiously  not  in  Tonstall  (1522).   There  were  also  such  ternu 
as  termini  addendi^  numeri  colligendi,  numeri  summandi,  and  the  like. 

8  Thus,  he  speaks  of  the  addendorum  summa  as  well  as  the  numeri  addend 
(iSSS  ed.,  fol.  3). 


90  ADDITION 

The  word  " addend"  was  frequently  used  to  refer  only  to  the 
lower  of  two  numbers  to  be  added,  as  in  the  following  case  from 
the  Margarita  phylosophica  (1503  ed.)  : 

4'6'7V  numerus  cui  debet  fieri  additio 
3 '2 '3 '2'  numerus  addendus 
7  9  i  i   numerus  jsductus 

It  was  also  used  by  many  writers  to  refer  to  all  the  numbers  to 
be  added  except  the  top  one.1 

The  result  obtained  in  addition  has  had  a  variety  of  names, 
although  "sum"  has  been  the  favorite.2  Next  in  order  of  popu- 
larity is  "product,"  a  term  used  for  the  result  of  any  operation, 
but  particularly  in  addition  and  multiplication.  It  was  popular 
in  Germany,3  especially  in  early  times,4  and  was  also  used  in  the 
Latin  countries:' 

Some  of  the  Latin  books  of  the  1 6th  century  also  used  numerus 
collectus,  based  upon  the  use  of  collectio  for  addition,  and  pos- 
sibly we  might  now  be  using  "collect"  for  "sum"  if  the  Church 
had  not  appropriated  the  term. 

The  Operation  of  Addition.  The  operation  of  addition  has 
not  changed  much  since  the  Hindu-Arabic  numerals  began  to 
be  used.)  Even  with  the  Roman  numerals  it  was  not  a  difficult 
process,  and  it  is  not  probable  that  a  Roman  banker  was  com- 


1  George  of  Hungary   (1499)   calls  only  the  lower  of  the  two  numbers  the 
numerus  addendus:  "et  numerus  addendus,  qui  debet  scribi  in  inferior!  ordine" 
(Budapest  reprint  of  1894,  p.  4).   The  same  usage  is  found  in  an  unpublished 
algorism  of  c.  1400  in  the  British  Museum  (SI.  3281,  fol.  4,  v.). 

2  Thus,  Chuquet  (1484)  uses  some,  and  similar  forms  appear  in  many  early 
printed  books,  including  those  of  Pacioli  (1494),  Fine  (1530),  Tonstall  (1522), 
Klos  (1538),  Sfortunati  (1534),  and  Riese  (1522). 

3  Giinther,  Math.  Unterrichts,  p.  316. 

4  Joannes    de    Muris    (c.    1350)    says    in   his    Quadripartttum  \     "Propositis 
namque  numeris  addicionis,  supra  figuras  cuiuslibet  numeri  calculis  situatis  adde 
singulam  singulis,  arcubus  obseruatis,  et  productum  signa  per  calculos  atque 
lege."    Abhandlungen,  V,  144.    This  is  also  interesting  because  it  describes  the 
use  of  counters  on  a  Gerbert  abacus. 

°An  interesting  case  sometimes  occurs,  as  in  Savonne's  work  of  1563,  where 
"sum"  is  used  for  addend,  and  "product"  for  the  result:  "Adiouster  est  mettre 
plusieurs  nombres  ou  sommes  ensemble  pour  en  sqauoir  le  produit." 


THE  OPERATION  91 

pelled  to  resort  to  the  abacus  in  ordinary  addition.    This  will 
easily  be  seen  by  considering  a  case  like  the  following : 

DCCLXXVII 
CC  X    VI 


DCCCCLXXXXIII 

We  might  write  this  result  CMXCIII,  but  a  Roman  would 
rarely  if  ever  have  done  so.  Even  in  the  i6th  century  we  find 
forms  analogous  to  this,  as  in  the  work  of  Texeda  (i546)/ 
where  we  have  the  following  parallel  arrangement : 

xx  j  Ucxx  vi  j  2 1 U 1 2  7 

x  vllccxviij  1511218 

ijUccccliiij  21/454 

jU.x.  lUoio 

xxxjUclxxxij  3iUi82 

IxxUDccccxcj  701199 1 

By  using  their  alphabetic  numerals  the  Greeks  were  able  to 
perform  various  operations  without  recourse  to  an  abacus,  al- 
though the  work  was  somewhat  more  complicated  than  it  is  with 
our  numerals.2 

v  Hindu  Method.  Bhaskara  (c.  1150)  gives  as  the  first  prob- 
lem in  the  Lilavati  the  following:3  "Dear  intelligent  Lilavati, 
if  thou  be  skilled  in  addition  .  .  . ,  tell  me  the  sum  of  two,  five, 
thirty-two,  a  hundred  and  ninety-three,  eighteen,  ten,  and  a 
hundred,  added  together."  In  a  commentary  on  this  work,  of 
unknown  date,  the  following  method  is  given : 

Sum  of  the  units,  2,  5,  2,  3,  8,  o,  o        20 

Sum  of  the  tens,  3,  9,  i,  i,  o  14 

Sum  of  the  hundreds,          i,  o,  o,  i  2 

Sum  of  the  sums, 360 

1Fol.  v,  v.         2For  details  as  to  the  Greek  methods  see  Heath,  History,  I,  52. 

3H.  T.  Colebrooke,  Algebra  with  Arithmetic  and  Mensuration  from  the  San- 
scrit, p.  5  (London,  1817)  ;  hereafter  referred  to  as  Colebrooke,  loc.  cit.,  or  to 
.special  topics  under  the  heads  Aryabhafa,  Brahmagupta,  Bhaskara,  Vija  Ganita, 
with  these  spellings. 


ADDITION 


3279 
10420 


909 


The  Hindus  seem  generally  to  have  written  the  sum  below  the 
addends,  beginning  with  units'  columns  as  we  do.  They  had  at 
one  time  another  method ,  however,  which  they 
designated  as  inverse  or  retrograde,  the  operator 
beginning  at  the  left  and  blotting  out  the  numbers 
as  they  were  corrected.1 

Arab  Method  and  its  Influence.  The  Arabs, 
on  the  other  hand,  often  wrote  the  sum  at  the 
top,  putting  the  figures  of  the  check  of  casting 
out  g's  at  the  side.2  This  plan  was 

adopted  by  Maximus  Planudes  (c.  1340),  the 

form  used  by  him  being  here  shown.3  ^ 
How  the  traces  of  the  Oriental  sand  table, 

with  its  easily  erased  figures,  and  the  traces  of 

the  old  counter-reckoning,  showed  themselves 

in  early  English  works  is  seen  in  the  following 

passage  in  The  Crajte  of  Nombrynge*  (c.  1300): 

lo  an  Ensampull  of  all 
326 
216 

Cast  6  to  6,  &  fere-of 5  wil  arise  twelue.  do  away  pe  hyer  6  &  write 
pere  2,  fat  is  pe  digit  of  pis  composit.  And  pen  write  pe  articulle 
fat  is  ten  ouer  pe  figuris  bed  of  twene  as  pus 

i 

322 
216 

^•This  method  is  here  indicated  by  canceling.  The  plan  is  one  naturally 
adapted  to  the  sand  abacus.  On  the  dispute  as  to  whether  the  Hindus  used  this 
abacus,  see  Chapter  III.  See  also  C.  I.  Gerhardt,  Etudes  historiques  sur  Varith- 
metique  de  position,  Prog.,  p.  4  (Berlin,  1856)  (hereafter  referred  to  as  Gerhardt, 
Etudes) ;  J.  Taylor,  Lilawati,  Introd.,  p.  7  (Bombay,  1816)  (hereafter  referred  to 
as  Taylor,  Lilawati) . 

2H.  Suter,  "Das  Rechenbuch  des  Abu  Zakarija  al-Hassar,"  Bibl.  Math., 
II  (3),  IS- 

3Waschke,  Planudes,  p.  6;  Gerhardt,  Etudes,  p.  20.  On  such  general  early 
methods  in  the  various  operations  see  F.  Woepcke,  Sur  I'introduction  de  I'Arith- 
mitique  Indienne  en  Occident  (Rome,  1859). 

4  See  pages  32  and  78. 

6  As  stated  on  page  32,  the  old  letter  ]>  is  our  th. 


ARABIC  INFLUENCE  93 

Now  cast  )?e  articulle  pat  standus  vpon  )>e  figuris  of  twene  bed  to  ]> e 
same  figure,  reken  fat  articul  bot  for  one,  and  pan  fere  will  arise  thre. 
fan  cast  fat  thre  to  fe  neper  figure,  fat  is  one,  &  fat  wul  be  foure.  do 
away  fe  figure  of  3,  and  write  fere  a  figure  of  foure.  and  let  fe  nefer 
figure  stonde  stil,  &  fan  worch  forth. 

This  is  the  oldest  known  satisfactory  explanation  of  an  example 
in  addition  in  our  language.1 

Special  Devices.  In  the  way  of  special  devices,  Gemma 
Frisius  (1540)  gives  one  that  is  still  used  in  adding  long 
columns.  It  consists  in  adding  each  column 
separately,  writing  the  several  results,  and  then 
adding  the  partial  sums  as  here  shown.  It  will 
be  observed  that  Gemma  writes  the  largest 
number  at  the  top,  the  object  being  to  more 


9279 
389 
479 

27 


22 

9 


9 


10147 


easily  place  the  various  orders  in  their  proper 
columns.2 

In  manuscripts  of  this  period  dots  are  some- 
times used,  as  is  the  case  today,  to  indicate  the 
figure  to  be  carried. 

,  Carrying  Process.  The  expression  "to  carry," 
as  used  in  addition,  is  an  old  one  and,  although  occasion- 
ally objected  to  by  teachers,  is  likely  to  remain  in  use.  It 
probably  dates  from  the  time  when  a  counter  was  actually 
carried  on  the  line  abacus  to  the  space  or  line  above,3  but 
it  was  not  common  in  English  works  until  the  i7th  century. 
Thus,  we  have  Recorde  (c.  1542)  using  "keepe  in  mynde," 
Baker  (1568)  saying  "keepe  the  other  in  your  minde,"  and 
Digges  (1572)  employing  the  same  phraseology  and  also  say- 
ing "keeping  in  memorie,"  and  "keeping  reposed  in  memorie." 
The  later  popularity  of  the  word  "carry"  in  English  is 
largely  due  to  Hodder  (3d  ed.,  1664).  In  the  i7th  cen- 

1On  similar  methods  in  the  medieval  manuscripts,  not  merely  in  addition  but 
in  the  other  operations,  see  L.  C.  Karpinski,  "Two  Twelfth  Century  Algorisms," 
Isis>  III,  396. 

2 "  Obseruandum  igitur  primo,  vti  maior  numerus  superiori  loco  scribatur, 
minores  huic  subscribantur "  (1575  ed.,  fol.  Ay).  8See  Chapter  III. 


94  SUBTRACTION 

tury  the  expression  "to  carry"  was  often  used  in  Italy.1  Ex- 
pressions like  "retain,",  "keep  in  mind,"  and  "hold"  have, 
however,  been  quite  as  common.2 

:    4.  SUBTRACTION 

-  Terminology  of  Subtraction.  As  with  addition,  so  with  sub- 
traction, the  name  of  the  process  and  the  names  of  the  numbers 
used  have  varied  greatly  and  are  not  settled  even  now.  Out- 
side the  school  the  technical  terms  of  arithmetic  are  seldom 
heard.  When  we  hear  a  statement  like  "Deduct  what  I  owe 
and  pay  me  the  rest,"  we  hear  two  old  and  long-used  terms  in- 
stead of  the  less  satisfactory  words  "subtract"  and  "difference." 

Terms  Meaning  Subtract.  While  the  word  "subtract,"  mean- 
ing to  draw  away  from  under,3  has  been  the  favorite  term  by 
which  to  indicate  the  operation,  it  has  by  no  means  enjoyed  a 
monopoly.  When  Fibonacci  (1202),  for  example,  wishes  to  say 
"I  subtract,"  he  uses  some  of  the  various  words  meaning  "I 
take."4  Instead  of  saying  "to  subtract"  he  says  "to  extract,"5 
and  hence  he  speaks  of  "extraction."0  These  terms,  as  also 
"detract,"7  which  Cardan8  (1539)  used,  are  etymologically 
rather  better  than  ours.  "Subduction"9  has  also  been  used  for 
"subtraction,"  both  in  Latin10  and  in  English.  Digges  (1572), 

^•As  in  "summa  senza  portare,"  "portare  decine,"  and  the  like.  See,  for  ex- 
ample, the  arithmetic  of  G.  M.  Figatelli  Centese,  fol.  21  (Bologna,  1664). 

2  E.g.,  "die  ander  behalt"  (Riese,  1522;  1533  cd.),  "behalt  die  ander  in  sinn, 
welche  ist  zu  geben  der  nechsten"  (Grammateus,  1518),  "et  altera  mente  reconda" 
(Clichtoveus,  1503;  c.  1507  ed.,  fol.  Da),  "&  secunda  reservanda"  (Ramus,  ed. 
Schoner,  1586,  p.  6),  "ie  .  .  .  retien  le  nombre  de  diszeines"  (Trenchant,  1566; 
1578  ed.,  p.  24). 

3Sub  (under)  +  trahere  (whence  tractum)  (to  draw). 

4Tollo,  aufero,  or  accipio.         5Extrahere  (to  draw  out  or  take  away  from). 

*  Extra[c]tio .  sPractica,  1539,  capp.  7-14. 

7Detrahere  (to  draw  or  take  from).  *Sub  (under)  +  ducere  (to  lead). 

10 E.g.,  Tonstall  (1522)  devotes  fifteen  pages  to  Subductio.  He  also  says: 
"Hanc  autem  eandem,  uel  deductionem  uel  subtractionem  appellare  Latine  licet" 
(1538  ed.,  p.  23;  1522  ed.,  fol.  £2,  r.}.  See  also  Ramus,  Libri  duo,  1569,  1580 
ed.,  p.  3;  Schol.  Math.,  1569  ed.,  p.  115.  Schoner,  in  his  notes  on  Ramus  (1586 
ed.,  p.  8),  uses  both  subduco  and  tollo  for  "I  subtract."  Gemma  Frisius  (1540) 
has  a  chapter  De  Subductione  sine  Subtractione,  and  Clavius  (1585  ed.,  p.  26) 
says:  "Subtractio  est  .  .  .  subductio."  In  his  arithmetic  Boethius  uses  sub- 
trahere,  but  in  the  geometry  attributed  to  him  he  prefers  subducere. 


TERMINOLOGY  95 

for  example,  says :  "To  subduce  or  subtray  any  sume,  is  wittily 
to  pull  a  lesse  fr5  a  bigger  nuber."  Our  common  expressions 
"to  diminish"  and  "to  deduct"  have  also  had  place  in  standard 
works,  as  in  the  translation  of  the  Liber  algorismi1  and  in  the 
work  of  Hylles  (i5Q2).2  Recorde  (c.  1542)  used  "rebate"  as 
a  synonym  for  "subtract,"  and  the  word  is  used  today  in  com- 
mercial matters  in  a  somewhat  similar  sense. 

In  a  manuscript  written  by  Christian  of  Prag3  (c.  1400) 
the  word  "subtraction"  is  at  first  limited  to  cases  in  which 
there  is  no  "borrowing."  Cases  in  which  "borrowing"  occurs 
he  puts  under  the  title  cautela  (caution),  and  gives  this  caption 
the  same  prominence  as  subtraction 

The  word  "subtract"  has  itself  had  an  interesting  history. 
The  Latin  sub  appears  in  French  as  sub,  soub,  sou,  and  sous, 
subtrahere  becoming  soustraire  and  subtractio  becoming  sous- 
traction^  Partly  because  of  this  French  usage,  and  partly  no 
doubt  for  euphony,  as  in  the  case  of  "abstract,"  there  crept  into 
the  Latin  works  of  the  Middle  Ages,  and  particularly  into  the 
books  printed  in  Paris  early  in  the  i6th  century,  the  form  subs- 
tractio*  From  France  the  usage  spread  to  Holland7  and  Eng- 
land, and  from  each  of  these  countries  it  came  to  America.  Until 
the  beginning  of  the  igth  century  "substract"  was  a  common 
form  in  England  and  America,8  and  among  those  brought  up  in 
somewhat  illiterate  surroundings  it  is  still  to  be  found. 

1  Which  uses  both  diminuere  and  subtrahere.    See  Boncompagni,  Trattati,  II,  32. 

2He  uses  "abate,"  "subtract,"  "deduct,"  and  "take  away."         3See  page  77. 

4 The  passage  begins:  "Cautela  ...  si  figura  inferioris  ordinis  non  poterit 
subtrahi  a  sibi  supraposita." 

GWith  such  variants  as  soubstraction,  soubstraire,  and  the  like. 

6  It  appears  in  the  Geometria  of  Gerbcrt,  but  the  MSS.  used  are  of  c.  1200  ; 
ed,  Olleris,  p.  430.  As  to  the  early  printed  books,  Clichtoveus  (1503),  for  ex- 
ample, generally  uses  substractio,  although  subtractio  is  occasionally  found.  See 
also  his  edition  of  Boethius,  and  see  the  1510  edition  of  Sacrobosco.  The  word 
also  appears  in  the  work  of  George  of  Hungary  (1499),  along  with  subtractio, 
so  that  the  usage  was  unsettled. 

7Thus  Wentsel  (1599),  Van  der  Schuere  (1600),  Mots  (1640),  and,  indeed, 
nearly  all  Dutch  writers  before  1800.  Petri  (1567;  1635  ed.),  however,  uses 
subtractio  in  the  Latin  form  and  subtraheert  in  the  Dutch,  and  Adriaen  Metius 
(1633;  1635  ed.)  also  omits  the  s. 

8Our  American  Greenwood  (1729),  for  example,  always  used  "substract"  and 
"substraction,"  but  dropped  the  s  in  "subtrahend." 


96  SUBTRACTION 

The  incorrect  form  was  never  common  in  Germany,1  prob- 
ably because  of  the  Teutonic  exclusion  of  international  terms." 

/Minuend  and  Subtrahend.  The  terms  "  minuend"  and  "  sub- 
trahend," still  in  use  in  elementary  schools,  are  abbreviations 
of  the  Latin  numerus  minuendus  (number  to  be  diminished) 
and  numerus  subtrahendus  (number  to  be  subtracted).3 

The  early  manuscripts  and  printed  books  made  no  use  of  our 
abridged  terms.  The  minuend  and  subtrahend  were  called  the 
higher  and  the  lower  numbers  respectively,  as  in  The  Crajte  of 
Nombrynge(c.  1300),  the  upper  and  under  numbers,1  the  num- 
ber from  which  we  subtract  and  the  subduced,5  the  total  and 
less,6  the  total  and  abatement,  and  the  total  and  deduction.7 
Among  the  most  popular  terms  have  been  "debt"  and  "pay- 
ment,"8 but  better  still  are  the  terms  "greater"  and  "less."9 

aAs  witness  Kobel  (1514;  if>49  ed.,  fol.  no),  Stifel  (Arithmetica  Integra, 
1544,  fol.  2),  Albert  (1534;  1561  ed.),  Thierfelder  (1587,  p.  n),  and  many  others. 

2  Their  early  writers  used  such  forms  as  abzihung  and  abzyhung,  instead  of 
"  subtraction,"  just  as  the  Dutch  used  such  terms  as  Af-trekkinge  (Van  der  Schuere, 
1600;  1624  ed.,  fol.  10).  While  the  Italians  used  abattere  and  cavare,  they  also 
used  sottrare  and  trarre  (as  in  Cataneo,  1546;  1567  ed.,  fol.  5). 

3 See  Boncompagni,  Trattati,  II,  33,  on  Johannes  Hispalensis  (c.  1140)  and 
his  use  of  numerus  minuendus. 

4 "Die  vnder  zal  sol  nit  ubertreffen  die  obern"  (Grammateus,  1518).  Tonstall 
(1522)  and  other  Latin  writers  have  numerus  superior  and  numerus  inferior;  the 
Italian  edition  of  Clavius  (1586)  has  numero  superiore  and  numero  inferiore. 

e" Numerus  ex  quo  subducitur"  and  "subducendus"  (Gemma  Frisius,  1540; 
1563  ed.,  fol.  9). 

6  Totalis,  minor,  used  by  Tzwivel  (1505),  Clichtoveus  (1503),  and  others. 

7Hylles,  1600,  fol.  19. 

8Thus,  the  Dutch-French  work  of  Wentsel  (1599,  p.  4)  has: 

Schult/Debte.                                  £.  15846 
Betaelt/paye, £.     5424 

Reste  £.  10422 

The  Dutch  names  in  the  i6th  and  i7th  centuries  were  generally  de  Schult  and 
de  Betaelinghe. 

Similarly,  we  have  the  Italian  debito,  pagato,  and  residuo  (as  in  the  1515 
edition  of  Ortega),  the  French  dette  and  paye,  as  well  as  la  superieure  &  infe- 
rieure  (Trenchant,  1566;  1578  ed.,  p.  30),  and  the  Spanish  recibo  and  gasto 
(Santa-Cruz,  1594;  1643  ed.,  fol.  20). 

9Sfortunati  (1534;  1544  ed.,  fol.  8),  il  numero  maggiore  and  II  numero  minore; 
G.  B.  di  S.  Francesco  (1689),  quantitd  maggiore  and  quantita  minore;  Raets 
(1580),  Het  meeste  ghetal  and  Het  minste  ghetal,  with  similar  forms  in  other 
languages.  See  also  Tartaglia,  1592  ed.,  fol.  9. 


TERMINOLOGY  97 

Name  for  Difference.  The  words  "difference"  and  " remain- 
der" have  never  been  popular,  in  spite  of  the  fact  that  they  are 
commonly  found  in  the  textbooks  of  today.  The  popular  term 
has  been  "rest,"  and  in  common  parlance  this  is  still  the  case, 
as  when  we  say  "Give  me  the  rest,"  "Take  the  rest."  It  appears 
in  the  first  printed  arithmetic1  and  is  found  generally  in  the 
works  of  the  Latin  countries.  Indeed,  the  verb  "to  rest"  was 
not  infrequently  used  to  mean  subtract.2  In  England,  Tonstall 
(1522),  writing  in  Latin,  used  sometimes  reliqua*  and  some- 
times an  expression  like  "the  number  sought."  Recorde 
(c.  1542)  introduced  "remayner"  or  "remainer,"  a  term  which 
Hylles  (1592)  also  used,  together  with  "remaynder,"  "re- 
maynes,"  and  "rest."  The  Latin  writers  commonly  used  nu- 
merus  residuus,4  differentia,  excessusf  and  reliqua.  Of  these 
terms  we  have  relics  in  our  language  in  the  forms  of  "dif- 
ference" and  "excess,"  and  another  term  commonly  used  by 
us  is  "balance."  \ 

An  interesting  illustration  of  the  use  of  expressions  which 
later  resulted  in  technical  terms  is  seen  in  the  following  from 
the  Margarita  phylosophica  (1503  ed.) : 

9001386  numerus  a  quo  debet  fieri  subtractio 
7532436  numerus  subtrahendus 
1468950  numerus  relictus 

":  The  Operation  of  Subtraction.  The  process  of  subtraction, 
unlike  the  processes  of  addition  and  multiplication,  has  never 
been  standardized.  There  are  four  or  five  methods  in  common 
use  today,  the  relative  advantage  of  any  one  over  the  others  not 
being  decided  enough  to  give  it  the  precedence.  A  brief  history 
of  a  few  of  the  more  prominent  methods  will  be  given.  ^ 

iTreviso,  1478,  p.  18. 

2Thus,  the  Spanish  writer  Santa-Cruz  (1594)  uses  restar;  and  Ortega  (1512; 
1515  ed.)  begins  a  chapter  per  sapere  restare  o  /  subtrahere. 

3 Various  other  writers  did  the  same.  Thus,  Glareanus  (1538)  has  relictum 
and  reliquum.  Fibonacci  (1202)  used  residuum  and  reliquus. 

4 E.g.,  Fine  (1530)  and  occasionally  Clavius  (1583).  An  unpublished  algorism 
of  c.  1400,  now  in  the  British  Museum  (SI.  3281,  fol.  4,  v.),  uses  a  q"  sbtrahi, 
subtrahed? ,  and  residuu  for  the  three  terms. 

5 Clavius  speaks  of  differentia  siue  excessus.    1585  ed.,  p.  133. 


98  SUBTRACTION 

v     i .  The  complementary  plan  is  based  upon  the  identity 
a  —b  =  a  4-  ( 10  —  b)  —10. 

In  particular,  to  find  13  —  8  we  may  substitute  the  simpler  proc- 
ess 13  +  2  and  then  subtract  10.  This  plan  is  today  used  in  the 
case  of  cologarithms  and  on  certain  types  of  calculating  ma- 
chines. It  is  not  a  modern  device,  however.  Bhaskara  (c.  1150) 
used  it  in  the  Lildvati?  and  no  doubt  it  was  even  then  an  old  one*. 
It  appears  in  The  Crajte  of  Nombrynge  (c.  1300),  and  the  dif- 
ficulty of  the  operation  is  apparent  from  the  following  extract : 

lo  an  Ensampul. 


take  4  out  of  2.  it  wyl  not  be,  perfore  borro  one  of  pe  next  figure,  pat  is 
2.  and  set  pat  ouer  pe  bed  of  pe  fyrst  2.  &  releue  it  for  ten.  and  pere2  pe 
secunde  stondes  write  i.  for  pou  tokest  on3  out  of  hym.  pan  take  pe 
neper  figure,  pat  is  4,  out  of  ten.  And  pen  leues  6.  cast 4  to  6  pe  figure  of 
pat  2  pat  stode  vnder  pe  hedde  of  i.  pat  was  borwed  &  rekened  for  10, 
and  pat  wylle  be  8.  do  away  pat  6  &  pat  2,  &  sette  pere  8,  &  lette  pe  neper 
figure  stonde  stille, 

and  so  on  with  equal  prolixity.  The  expression  to  "borro,"  used 
in  this  work,  was  already  old.  It  was  afterwards  used  by  Maxi- 
mus  Planudes  (c.  1340),  acquired  good  standing  in  the  works 
of  Recorde  (c.  1542)  and  Baker  (1568),  and  has  never  lost  its 
popularity. 

The  same  method  appears  in  the  Treviso  arithmetic5  (1478), 

1  Taylor,  Lilawati,  Introd.,  p.  7.  2For  "where."  3One. 

4/.e.,  add;  a  relic  of  the  abacus.    Compare  our  expression  "cast  accounts." 

5  The  author  adds  2  to  2,  the  result  being  4, 
and  then  adds  i  to  the  next  figure  of  the  subtra- 
hend, saying: 

"al  .4.  tu  die  iongere  i.  e  levera  .5.  poi  dira 
.5.  da  .5.  che  equale  da  equale :  resta  .o."  (Treviso 
arithmetic,  p.  [19]).  The  i  is  used  for  i,  as  on  page  97 
and  as  is  the  ;  in  the  following  problem  from  Huswirt. 


452 
348 


Lo  resto 


VARIOUS  METHODS 


99 


and  Huswirt  (1501)  solved  his  first  problem  in  subtraction  by 
this  means,  saying  : 

5  from  4  I  cannot.  I  take  the  distance1  of  the  lower  number,  that 
is,  5  from  10,  or  5,  and  this  I  add  to  the  upper  number,  4,  and  obtain 
9,  which  I  write  directly  under  the  bar  and  below 
the  5.  I  carry  the  j  in  mind  or  on  the  tablet,2 
first  canceling  the  4  and  5,  and  add  it  to  the 
next  number,  that  is,  to  9.  .  .  . 


59jojojoj4 
400 j 999 j 95 

J9o8joj8J9 


Among  other  authors  of  early  printed 
books  who  favored  the  plan  there  were  such 
writers  as  Petzensteiner  (1483),  Pellos  (1492),  Ortega  (1512), 
Fine  (1530),  Gemma  Frisius  (1540),  Ramus  (1555),  Albert 
(1534),  Baker  (1568),  and  Digges  (1572). 
Savonne  (1563)  also  used  it  and  indicated  the 
borrowing  of  ten  by  means  of  a  dot,  as  shown 
in  the  annexed  example  from  his  arithmetic. 

The   early   American   arithmeticians   looked 
with  some  favor  on  the  plan.   Thus,  Pikea  says: 

If  the  lower  figure  be  greater  than  the  upper,  borrow  ten  and  sub- 
tract the  lower  figure  therefrom :  To  this  difference  add  the  upper  figure. 

2.  The  borrowing  and  repaying  plan,  in  which  the  i  that  is 
borrowed  is  added  to  the  next  figure  of  the  lower  number,  is  one 
of  the  most  rapid  of  the  methods  in  use  today  and  has  for  a  long 
time  been  one  of  the  most  popular.  It  appears  in 
BorghPs  (1484)  well-known  work,  the  first  great 
commercial  arithmetic  to  be  printed.  Borghi  takes 
the  annexed  example  and  says,  in  substance:  "8 
from  14,  6;  8  from  15,  7;  10  from  13,  3;  3  from 
6,  3."  The  plan  was  already  old  in  Europe,  how- 
ever. Fibonacci4  (1202)  used  it,  and  so  did  Maximus  Planudes 
(c.  1340).  These  writers  seem  to  have  inherited  it  from  the 
Eastern  Arabs,  as  did  the  Western  Arab  writer  al-Qalasadi 

1  Distantly  for  the  complement. 

2  Very   likely   the   wax   tablet,   still   used   in    Germany   at   that   time.    See 
Chapter  III.        8i788;  1816  ed.,  p.  12.       4  Liber  Abaci,  Boncompagni  cd.,.I,  22. 


ipo  SUBTRACTION 

(c.  1475).  The  arrangement  of  figures  used  by  Maximus 
Planudes  in  the  subtraction  of  35843  from  54612  is  here  shown, 
the  remainder  being  placed  above  the  larger  num- 
6  ber,  after  the  Arab  and  Hindu1  custom.  The  top 

«  6  line  was  used  only  in  the  checking  process.    This 

— £~         method  of  borrowing  and   repaying  was  justly 
g  looked  upon  as  one  of  the  best  plans  by  most  of 

^  the  1 5th  and  i6th  century  writers,  and  we  have 

none  that  is  distinctly  superior  to  it  even  at  the 


present  time. 


3.  The  plan  of  simple  borrowing  is  the  one  in  which  the 
computer  says:  "7  from  12,  5;  2  from  3  (instead  of  3 
from  4),  i."  This  method  is  also  very  old.  It  appears  in  the 
writings  of  Rabbi  ben  Ezra2  (c.  1140),  the  computer  being 
advised  to  begin  at  the  left  and  to  look  ahead  to 
take  care  of  the  borrowing.  This  left-to-right  fea- 
ture is  Oriental3  and  was  in  use  in  India  a  century  42 
ago.4  It  was  the  better  plan  when  the  sand  table 
allowed  for  the  easy  erasure  of  figures,  but  it  had 
few  advocates  in  Europe.5 

When  the  computation  began  at  the  right,  the  borrowing 
plan  was  also  advocated  by  such  writers  as  Gernardus6  (i3th 
century?),  Sacrobosco7  (c.  1250),  and  Maximus  Planudes8 
(c.  1340).  The  writers  of  the  early  printed  arithmetics9  were 

1  Taylor,  Lttawati,  In  trod.,  p.  7. 

2Sefer  ha-Mispar,  ed.  Silberberg,  p.  29  (Frankfort  a.  M.,  1895) ;  hereafter 
referred  to  as  Silberberg,  Sefer  ha-Mispar. 

3It  is  found  in  the  works  of  al-Khowarizmi  (0.825),  Beha  Eddin  (c.  1600), 
Albanna  (c.  1300),  and  others.  See  H.  Suter,  Bibl.  Math.,  II  (3),  15. 

4  See  Taylor,  Lilawati,  Introduction. 

5 One  of  these  was  Ramus,  who  advocates  "subductio  fit  a  sinestra  dex- 
trorsum"  (Arith.  Libri  duo,  1569;  1580  ed.,  p.  4;  1586  ed.,  p.  8). 

^  Algorithmus  demonstratus,  I,  cap.  ix.  Formerly  attributed  to  Jordanus 
Nemorarius  (c.  1225).  See  G.  Enestrom,  Bibl.  Math.,  XIII  (3),  289,  292,  331. 

7See  J.  O.  Halliwell,  Rara  Mathematica  (London,  1838-1839),  2d  ed.,  1841, 
p.  7 ;  hereafter  referred  to  as  Halliwell,  Rara  Math. 

8  With  one  or  two  other  methods. 

9£.g.,  such  writers  as  Tzwivel  (1505),  Clichtoveus  (1510  edition  of  his  Boe- 
thius,  fol.  39),  Kobel  (1514;  1549  ed.,  fol.  120),  Stifel  (1544),  Ghaligai  (1521), 
Raets  (1580),  and  Clavius  (1583).  Some  of  the  more  pretentious  writers,  like 
Pacioli  (1494)  and  Tartaglia  (1556),  gave  all  three  methods. 


MULTIPLICATION  101 

not  unfavorable  to  it,  although  they  in  general  preferred  the 

borrowing  and  repaying  method. 

.4.  The  addition  method,  familiar  in  "making  change,"  is 

possibly  the  most  rapid  method  if  taught  from  the  first.    To 

subtract  87  from  243  the  computer  says:  "7  and  6 

are  13;  9  and  5  are  14;  i  and  i  are  2";  or  else 

he  says :  "7  and  6  are  13 ;  8  and  5  are  13 ;  o  and  i 

are  i,"  the  former  being  the  better.    The  method 

was  suggested  by  Buteo  (1559)  and  probably  by 

various  other  early  writers,  but  it  never  found  much 

favor  among  arithmeticians  until  the  igth  century.     It  has 

been  called  the  Austrian  Method,  because  it  was  brought 

to  the  attention  of  German  writers  by  Kuckuck  (1874),  who 

learned  of  it  through  the  Austrian  arithmetics  of  Mocnik  (1848) 

and  Josef  Salomon  (1849). 

v  5.  MULTIPLICATION 

General  Idea  of  Multiplication.  The  development  of  the  idea 
of  multiplication  and  of  the  process  itself  is  naturally  more  in- 
teresting than  the  evolution  of  the  more  primitive  and  less 
intellectual  processes  already  described.  Just  as  addition  is  a 
device  for  obtaining  results  that  could  be  reached  by  the  more 
laborious  method  of  counting,  so  multiplication  was  developed 
as  an  abridgment  of  addition.1  It  was  simply  a  folding  together 
of  many  equal  addends.  This  is  expressed  not  merely  in  the 
Latin  name2  but  in  the  corresponding  names  in  various  other 

1  Attention  was  called  to  this  fact  by  various  i6th  century  writers.  Thus 
Ramus  (1569)  remarks:  " Multiplicatio  est  qua  multiplicandus  toties  additur, 
quoties  unitas  in  multiplicante  continetur,  &  habetur  factus."  Schoner,  in  his 
commentary,  adds:  "Ideoq^  multiplicatio  est  additio,  sed  ejusdem  numeri 
secum,  no  diuersorunV  (1586  edition  of  the  Libri  duo,  p.  12).  Even  as  early 
as  c,  1341  Rhabdas  mentioned  the  same  fact.  See  P.  Tannery,  Notices  et  ex- 
traits  des  manuscrits  de  la  Bibl.  nat.,  XXXII,  155. 

2 From  multus  (many)  +  plicare  (to  fold) ;  compare  also  our  word  "mani- 
fold." The  term  is  simply  the  Latin  form  of  the  Greek  iroXvirXaffidfav  (poly- 
plasia'zein))  as  used  by  Euclid,  Pappus,  and  Diophantus,  or  Tro\\air\aatA^€iv 
(pollaplasia'zein) ,  as  used  by  Heron  and  Pappus,  the  latter  using  both 
forms.  Such  words  as  "three-ply"  and  "four-ply"  illustrate  this  use  of 
plicare. 

n 


102  MULTIPLICATION 

languages.1  The  Latin  writers  of  the  Middle  Ages  and  the 
Renaissance  speak  of  leading  a  number  into  this  multiplicity,2 
which  explains  our  use  of  the  expression  "a  into  b"  still  retained 
in  algebra  but  discarded  in  arithmetic.  ^ 

"-  Definition  of  Multiplication.  The  definition  of  multiplication 
has  often  disturbed  teachers  of  arithmetic  because  of  their 
failure  to  recognize  the  evolution  of  such  terms.  It  gave  no 
trouble  in  theancientarithmetica,  for  the  numbers  there  involved, 
in  speaking  of  such  a  process,  were  positive  integers ;  whereas  in 
the  ancient  logistica  no  attention,  so  far  as  we  know,  was  paid 
to  any  definitions  whatever.  When,  however,  the  notion  of  the 
necessity  of  exact  definition  entered  the  elementary  school, 
teachers  were  naturally  at  a  loss  in  adjusting  the  ancient  limita- 
tions to  the  multiplication  by  a  fraction  or  an  irrational  number, 
and  by  such  later  forms  as  a  negative  or  a  complex  number. 
One  of  the  best  of  the  elementary  definitions  referring  to 
integers,  and  at  the  same  time  one  of  the  oldest  in  our  language, 
is  found  in  The  Crafte  of  Nombrynge  (c.  1300) :  "multiplicacion 
is  a  bryngynge  to-geder  of  2  thynges  in  on  nombur,  fe  quych 
on  nombur  contynes  so  mony  tymes  on,  howe  mony  tymes  pere 
ben  vnytees  in  )>e  nowmbre  of  J>at  2."3  The  same  definition  is 
found  in  the  arithmetic  of  Maximus  Planudes  (c.  1340)^  in  the 
first  printed  arithmetic  (i478),5  in  the  first  noteworthy  com- 

1Compare  the  German  works  of  the  i$th  and  i6th  centuries,  with  their 
mannigfaltigen  and  vervieljachen.  Grammateus  (1518)  speaks  of  " Multiplicatio 
oder  Merung." 

2"fl  Si  aliquis  numerus  .  .  .  ducatur,"  as  Jordanus  Nemorarius  (c.  1225) 
says  (1406  edition  of  the  arithmetic,  fol.  €3,  et  passim}.  Similarly  Clichtoveus 
(1503),  "Duco  .4.  in  .3.  et  fit  .jz.";  Gemma  Frisius  (1540),  " Mvltiplicare,  est 
ex  ductu  vnius  numeri  in  alterum  numerum  producere,  qui  toties  habeat  in  se 
multiplicatum,  quoties  multipliers  vnitatem"  ;  and  many  others.  In  the  Latin 
edition  of  his  arithmetic  (1583  ;  1585  ed.,  p.  36)  Clavius  has  "Multiplicatio  est 
ductus  vnius  numeri  in  alium  .  .  .  Vt  numerus  6.  in  luimerum  5.  ...  duci 
dicitur  .  .  . ,"  but  in  the  Italian  edition  (1586,  p.  35)  he  uses  per  for  in,  thus: 
"  Moltiplicare  vn  numero  per  vn'  altro." 

3R.  Steele's  proof-sheet  edition,  p.  21  (London,  1894). 

4Waschke,  Planudes,  p.  13. 

8 "  Che  moltiplicare  vno  nuero  per  si  ouero  per  vno  altro  :  non  e  altro :  che  de 
do  nnmeri  ppositi :  trouere  vno  terzo  numero :  el  quale  tante  volte  contien  vno 
de  quelli  numeri :  quante  vnitade  sono  nel  altro.  Exempio  .2.  fia  .4.  fa  .8.  ecco  che 
.8.  cotie  in  se  tante  .4.  quante  vnitade  sono  nel  .2."  Treviso  arithmetic,  p.  [27], 


DEFINITION  103 

mercial  arithmetic  (I484),1  and  in  numerous  other  works.2 
Recorde  (c.  1542)  set  the  English  standard  by  saying,  "Multi- 
plication is  such  an  operacio  that  by  ij  sumes  producyth  the 
thyrde,  whiche  thyrde  sume  so  manye  times  shall  cotaine  the 
fyrst,  as  there  are  vnites  in  the  second."3 

A  somewhat  more  refined  definition,  including  the  notion  of 
ratio,  was  necessary  for  fractional  multipliers,  and  this  appeared 
occasionally  in  the  early  printed  books,  as  in  Huswirt  (1501).* 
Its  use  in  English  is  largely  due  to  the  influence  of  Cocker's 
popular  arithmetic  (1677),  where  it  appears  in  these  words: 
"  Multiplication  is  performed  by  two  numbers  of  like  kind,  for 
the  production  of  a  third,  which  shall  have  such  reason  [ratio] 
to  the  one,  as  the  other  hath  to  unite."  The  idea  is  Oriental,5 
appearing  in  various  Arab  and  Russian  works.6 

The  elementary  teacher  generally  objects  to  such  a  form  as 
2  ft.  x  3  ft.  =  6  sq.  ft.,  and  on  the  ground  of  pedagogical  theory 
there  is  some  reason  for  so  doing,  but  not  on  logical  or  historical 
grounds.  With  respect  to  logic,  it  all  depends  on  how  multipli- 
cation is  defined ;  while  with  respect  to  history  there  is  abundant 
sanction  for  the  form  in  the  works  of  early  and  contemporary 
writers.  For  example,  Savasorda7  (c.  1120)  and  Plato  of  Tivoli 
(in6)8  broaden  the  definition  in  such  a  way  as  to  allow  a  line 
to  serve  as  a  multiplier,  and  Baker  (1568)  remarks,  "If  you  wil 
multiply  any  number  by  shillinges  and  pence,"  an  expression 
commonly  paralleled  by  children  today.  Few  of  our  contem- 
porary physicists  would  see  anything  to  criticize  in  such  an  ex- 
pression as  6  ft.  x  10  Ib.  =  60  foot-pounds,  and  in  due  time  such 
forms  will  receive  more  recognition  in  elementary  arithmetics 

1Borghi,  1540  ed.,  fol.  6. 

2 E.g.,  Tonstall,  1522,  fol.  Gi;  Stifel,  Arithmetica  Integra,  1544,  fol.  2;  Sfor- 
tunati,  1534,  1544/5  ed.,  fol.  n;  Tartaglia,  1556;  Trenchant,  1566. 

3i558  ed.,  fol.  Gi.    Digges  (1572;  1579  ed.,  p.  4)  gives  the  same  form. 

4"MuItipIicatio  est  numeri  procreatio,  proportionabiliter  se  habentis  ad  mul- 
tiplicandu  sicut  multiplicans  ad  vnitatem  se  habet"  (fol.  3). 

5 See  Taylor,  Lilawati,  Introd.,  p.  15. 

6 E.g.,  Beha  Eddin  (c.  1600).  It  is  still  used  in  Russian  textbooks  on  arithmetic. 

7Abraham  bar  Chiia. 

8  In  his  translation  of  Savasorda :  "  Et  primum  quidem  exponemus,  quid  sig- 
nificare  velimus,  cum  dicimus :  multiplicatio  lineae  in  se  ipsam." 


104  MULTIPLICATION 

Terminology  of  Multiplication.  Of  the  terms  employed,  "mul- 
tiplicand" is  merely  a  contraction  of  numerus  multiplicands. 
In  The  Crajte  of  Nombrynge  (c.  1300)  it  is  explained  as  "Nu- 
merus multiplicands,  Anglice  f>e  nombur  ]>e  quych  to  be  mul- 
tiplied." In  most  of  the  early  printed  Latin  books  it  appears 
in  the  full  form,1  but  occasionally  the  numerus  was  dropped, 
leaving  only  multiplicandus?  and  this  led  the  non-Latin  writers 
to  use  the  single  term.3  A  few  of  the  Latin  writers  suggested 
multiplicatus*  so  that  we  had  at  one  time  a  fair  chance  of  adopt- 
ing "multiplicate."  In  their  vernacular,  however,  many  writers 
tended  to  use  no  technical  term  at  all,  simply  speaking  of  the 
number  multiplied,  as  the  Latin  writers  had  done,5  and  to  this 
custom  we  might  profitably  return.  It  is  hardly  probable  that 
such  terms  as  subtrahend,  minuend,  and  multiplicand,  signify- 
ing little  to  the  youthful  intelligence,  can  endure  much  longer. 

The  word  "multiplier"  has  had  a  more  varied  career.  The 
Crajte  of  Nombrynge  (c.  1300)  speaks  of  "numerus  multipli- 
cans.  Anglice,  fe  nombur  multipliynge,"  the  former  being  the 
Latin  name  for  "multiplying  number."  Since  the  word  nu- 
merus was  frequently  dropped6  by  Latin  writers,  in  the  trans- 
lations the  technical  term  appeared  as  a  single  word,  with  such 

1£.g.,  Clichtoveus  (1510  edition  of  Boethius,  fol.  35),  Tonstall  (1522), 
Grammateus  (1518),  Scheubel  (1545,  I,  cap.  4). 

2  E.g.,  Pacioli  (1494,  fol.  26),  Licht  (1500,  fol.  6),  Huswirt  (1501),  Ciruelo 
(1495),  Glareanus  (1538),  Fine  (1530). 

3Thus  Trenchant  used  multiplicands  (1566;  1578  ed.,  p.  35).  On  the  Italian 
writers  see  B.  Boncompagni,  Atti  d.  Accademia  Pontificia  di  Nuovi  Lincei,  XVI, 
520 ;  hereafter  referred  to  as  Atti  Pontif. 

4 E.g.,  Tzwivel  (1505)  used  "nuerus  multiplicadus  siue  multiplicatus,"  and 
Gemma  Frisius  (1540)  used  multiplicandus  and  multiplicatus  interchangeably. 
So  the  Treviso  arithmetic  (1478)  says:  "Intendi  bene.  che  ne  la  moltiplicatione 
sono  pricipalmente  do  numeri  necessarii.  zoe  el  nuero  moltiplicatore :  et  el  micro 
de  fir  moltiplicato "  (p.  [27]),  which  is  not  quite  the  same  usage. 

BThus  Chuquet  (1484)  simply  speaks  of  "le  nombre  multiplie,"  and  simi- 
larly with  Borghi  (1484),  Riese  (1522;  1529  ed.,  p.  8),  Sfortunati  (1534;  *544/5 
ed.,  fol.  n),  and  others.  Digges  (1572)  speaks  of  it  as  "the  other  summe,  or 
number  to  be  multiplied." 

6Thus  Johannes  Hispalensis  (c.  1140) ;  see  Boncompagni,  Trattati,  II,  41. 
Clichtoveus  (1510  edition  of  Boethius)  uses  multiplicand  both  as  an  adjective 
and  as  a  noun.  So  also  Huswirt  (1501,  fol.  3),  Ciruelo  (1495;  1513  ed.,  fol. 
A.  6),  Grammateus  (1518;  1523  ed.,  fol.  A  4),  Gemma  Frisius  (1540),  and 
many  others. 


TERMINOLOGY  105 

variants  as  "multiplicans,"1  "moltiplicante,"2  "multiplicator,"8 
"multipliant,"4  and  "multiplier.''5 

The  word  " product"  might  with  almost  equal  propriety  be 
applied  to  the  result  of  any  other  arithmetic  operation  as  well  as 
to  multiplication.  It  means  simply  a  result,6  but  it  has  some 
slightly  stronger  connection  with  multiplication  on  account  of 
the  use  of  the  verb  ducere  in  the  late  Latin  texts.7  It  has,  how- 
ever, been  used  in  the  other  operations  by  many  writers,  and  its 
special  application  to  the  result  of  multiplication  is  compara- 
tively recent.  The  tendency  to  simplify  the  language  of  the  ele- 
mentary school  will  naturally  lead  to  employing  some  such  term 
as  "result"  for  the  various  operations. 

The  authors  of  the  early  printed  books  often  took  the  sensible 
plan  of  having  no  special  name  for  the  result  in  multiplication. 
Certain  of  them  used  "sum"8  or  "sum  produced,"9  while  factus , 
a  natural  term  where  factor  is  employed,  had  its  advocates.10 
Finally,  however,  the  numerus  was  dropped11  from  numerus 
productus  j  and  "product"  remained.12  ^"' 

1Pacioli,  1494,  fol.  26.  2Cataldi,  1602,  p.  21. 

3 Multiplicador  (Pellos,  1492,  fol.  8),  multiplicatore  (Ortega,  1512;  1515  ed., 
fol.  16),  multiplicatour  (Baker,  1568;  1580  ed.,  fol.  16).  The  word  was  com- 
mon in  English.  Greenwood  used  it,  with  "multiplier,"  in  the  1729  American 
arithmetic. 

4Peletier,  1549;  1607  ed.,  p.  34. 

5 "The  lesse  is  named  the  Multiplicator  or  Multiplyer."  Digges,  1572;  1579 
ed.,  p.  5. 

6 Latin  producere  (to  lead  forth) ;  whence  productum  (that  which  is  led  forth). 

7  See  pages  101,  102. 

8  E.g.,  Pacioli  (1494,  fol.  26),  Ortega  (1512  ;  1515  ed.,  fol.  18),  and  Recorde 
(c.  1542  ;  1558  ed.,  fol.  G2).   Fine  (1530)  uses  it  as  well  as  numerus  productus. 

9 Thus  Hodder  (loth  ed.,  1672,  p.  25)  speaks  of  "The  Product,  or  sum 
produced."  Similarly,  Clichtoveus  (1503)  uses  both  numerus  productus  and 
tola  summa,  and  Glareanus  (1538)  uses  summa  producta. 

10  Thus,  Fibonacci  (1202)  uses  "factus  ex  multiplicatione."   He  also  speaks  of 
the  "contemptum  sub  duobus  numeris."   Ramus  (1569)  speaks  of  the  factus  in 
multiplication,  and  in  his  treatment  of  proportion  he  says:  "Factus  a  medio 
aequat  factum  ab  extremis." 

11  As  in  Licht  (1500),  Huswirt  (1501),  Gemma  Frisius  (1540),  and  Scheubel 

(iS45). 

l2Produit,  Trenchant  (1566) ;  produtto,  Sfortunati  (1534)  andCataldi  (1602), 
or  prodotto  by  later  Italian  writers.  Unlike  most  of  the  Latin  terms  it  found 
place  in  the  early  Teutonic  vocabulary,  as  seen  in  Werner  (1561)  and  such 
Dutch  writers  as  Petri  (1567),  Raets  (1580),  and  Coutereels  (1599), 


io6 


MULTIPLICATION 


4 

8 

i6 


The  Process  of  Multiplication.  We  know  but  little  about  the 

methods  of  multiplication  used  by  the  ancients.  The  Egyptians 
probably  made  some  use  of  the  duplation  plan,  1 7 
being  multiplied  by  1 5  as  shown  in  the  annexed 
scheme.1  It  is  also  probable  that  this  plan  was 
followed  by  other  ancient  peoples  and  by  their 
successors  for  many  generations,  which  accounts 
for  the  presence  of  the  chapter  on  duplation  in 
so  many  books  of  the  Renaissance  period.  In- 
deed, even  as  good  a  mathematician  as  Stifel 
multiplied  42  by  31  by  successive  duplation,  sub- 
stantially as  here  shown.2 
There  is  also  a  contemporary  example 

of  the  use  of  duplation  and  mediation, 

found  among  the  Russian  peasants  today. 

To  multiply  49  by  28  they  proceed  to 

double  28  and  to  halve  49,  thus: 


34 

68 

136 

272 

25S 


49 

28 


24 
56 


12 
112 


6 

224 


3 
448 


i 
896 


i 

42  = 

42 

2 

•42  = 

84 

4 

•  42  = 

168 

8 

•  42  = 

336 

16 

•  42  = 

672 

31 

-  42  = 

1302 

The  fractions  are  neglected  each  time,  and  finally  the  figures  in 
the  lower  row  which  stand  under  odd  numbers  are  added,  thus: 

28  +  448  +  896  =  1372, 
and  this  is  the  product  of  28  x  49. 3 

Greek  and  Roman  Methods.  The  abacus4  was  probably  used 
so  generally  in  ancient  times  that  we  need  hardly  speculate  on 
the  methods  of  multiplication  used  by  the  Greeks  and  Romans. 
It  is  quite  possible,  however,  that  the  Greeks  multiplied  upon 

1P.  Tannery,  Notices  et  extraits  des  manuscrits  de  la  Bibl.  nat.,  XXXII,  125; 
Pour  I'histoire  de  la  science  Hellene,  p.  82  (Paris,  1887) ;  Heath,  History,  I,  52. 
2 See  his  Rechenbuch  (1546),  p.  12.    He  uses  a  similar  process  for  division. 
3  In  the  above  case  we  have 


-Y- 


49 


24 

2-  28 


4.!3 

6 


*§  H 

12  6  3  i 

4-28         8-28         16-28         32-28 
Here  49  •  28  =  (32  -f  16  -f  i)  •  28  =  896  -f  448  +  28  =  1372. 
4  See  Chapter  III. 


PACIOLI'S  METHODS 


107 


their  wax  tablets  about  as  we  multiply,  but  beginning  with  their 
highest  order;1  and  there  is  no  good  reason  why  the  Romans 
should  not  have  done  very  nearly  the  same.  Indeed,  in  Texeda's 
arithmetic  (1546)  the  "Spanish  method"  with  Roman  numerals 
is  given  side  by  side  with  the  new  method  of  "Guarism,"  thus: 


-7U506 


22SIU800 

5251:420 

37U530 


ccclxxv vi  j  UD .  v  j 

ijqsccljUDccc. . 
DxxvUccccxx. 
xxxvijUDxxx. 
ijqsDcccxiiijUDccL 


It  should  be  observed,  however,  that  the  small  tradesman  has 
never  had  much  need  for  this  kind  of  work.2 

Pacioli's  Eight  Plans.  Our  first  real  interest  in  the  methods 
of  multiplication  starts  with  Bhaskara's  Lildvati  (c.  1150),  al- 
though we  have  a  few  earlier  sources.  Bhaskara  gives  five  plans 
and  his  commentators  add  two  more.  These  plans  had  increased 
to  eight  when  Pacioli  published  his  Suma  (1494),  and  these  will 
now  be  considered. 

L  Our  Common  Method  of  Multiplying.  Our  common  form  was 
called  by  Pacioli  "Multiplicatio  bericocoli  vel  scachierij,"  and 
appears  in  his  treatise  (fol.  26,  r.}  in  the  following  form : 


IV  JL  Ul  Llpll^aiUJl  UO  . 

Producentes. 
Multiplicans. 

9 
6 

8 

7 

7 
8 

6 
9 

schachieri 
Bericuocolo 

» 

8 

8 

8 

4 

7 

9 

1° 

o 

8 

["6" 

9 

i 

3 

2 

|5|9 

2 

5 

6| 

summa 


67048    164 


XP.  Tannery,  Notices  et  extraits  des  manuscrits  de  la  Bibl.  nat.,  XXXII,  126; 
Heath,  History,  I,  54. 

2  An  interesting  witness  to  this  fact  is  the  first  Bulgarian  arithmetic  (1833), 
described  in  L'Enseignement  Mathimatique  (1905),  p.  257. 


io8 


MULTIPLICATION 


12 


He  says  that  the  Venetians  called  this  the  method  "per 
scachieri"  because  of  its  resemblance  to  a  chessboard,1  while 
the  Florentines  called  it  "per  bericuocolo"  be- 
cause it  looked  like  the  cakes  called  by  this  name 
and  sold  in  the  fairs  of  Tuscany.2  In  Verona  it 
was  called  "per  organetto,"3  because  of  the  re- 
semblance of  the  lines  to  those  of  a  pipe  organ, 
and  "a  scaletta"  was  sometimes  used 
because  of  the  "little  stairs"  in  the 
figure,  as  seen  on  page  107.* 

This  method  is  not  found  directly 
in  the  Lildvati,  but  two  somewhat  similar  ones 
are  given.  In  the  first  of  these5  the  multiplier 
is  treated  as  a  one-figure  number  and  the  work 
begins  at  the  left,  as  shown  above ;  in  the  second,0 
which  is  shown  in  the  above  computation  at  the  right,  the  mul- 
tiplier is  separated  as  with  us,  but  the  work  begins  at  the  left 
as  in  the  preceding  case/  s  , 


36 

60 
l62O 


12 


270 
1620 


^Scacchero,  the  modern  scacchiere.  Our  word  "exchequer"  comes  from  the 
same  root.  See  page  188.  The  spelling  varies  often  in  the  same  book. 

2" .  .  .el  primo  e  detto  multiplicare  yP  Scachieri  in  vinegia  ouer  per  altro  nome 
per  bericuocolo  in  firenqa  ...  el  primo  modo  di  multiplicare  chiamano.  Beri- 
cuocolo :  perch'  pare  la  figura  de  qsti  bricuocoli :  o  cofortini  che  se  vendano  ale 
fcste"  (fols.  26,  r.,  28,  v.).  A  MS.  in  Dresden,  dated  1346,  has  "lo  modo  di  mol- 
tiplicare  per  ischachiere."  B.  Boncompagni,  Atti  Ponti}.,  XVI,  436,  439.  An  un- 
dated MS.  in  the  Biblioteca  Magliabechiana  (Florence,  C.  7.  No.  2645)  gives 
the  name  as  iscacherio,  scacherio,  and  ischacherio.  Cataneo  (1546;  1567  ed., 
fol.  10),  although  printing  his  work  in  Venice,  calls  the  method  biricvocolo. 
Those  who  do  not  have  access  to  Pacioli  may  find  the  methods  in  facsimile  in 
Boncompagni,  Scritti  inediti  del  P.  D.  Pietro  Cossali,  p.  116  (Rome,  1857), 
a  work  more  likely  to  be  found  in  university  libraries. 

3So  Feliciano  da  Lazesio  (1526)  says:  "Del  multiplicar per scachier vocabulo 
Venitiano,  ouer  baricocolo  uocabolo  Fiorentino,  ouer  multiplicar  per  organetto 
uocabolo  Veronese"  (1545  ed.,  fol.  12).  Similarly,  Tartaglia  (1556)  says: 
"Del  secondo  modo  di  multiplicare  detto  per  Scachero,  ouer  per  Baricocolo, 
ouer  per  Organetto"  (General  Trattato,  I,  23  (Venice,  1556)). 

4 "Multiplication  a  scaletta  .&.  aggregatione  a  bericocolo,"  in  a  MS.  at  Paris, 
described  by  Boncompagni,  Atti  Pontif.,  XVI,  331. 

5  The  Swarupa  gunanam,  "the  multiplier  as  a  factor."  It  is  Bhaskara's  first 
method.  For  the  method  of  Mahavlra  (c.  850)  see  his  Ganita-Sdra-Sangraha, 
Madras,  1912,  p.  9  of  the  translation  (hereafter  referred  to  as  Mahavlra}. 

6 The  St'hana  gunanam,  "multiplication  by  places."   It  is  his  fourth  method. 


CHESSBOARD  METHOD 


109 


CM 

XM 

M 

c 

X 

I 

4 

6 

i 

8 

I 

2 

i 

2 

8 

i 

5 

8 

2 

3 

Since  multiplication  on  the  abacus  required  no  symbol  for 
zero,  the  earlier  attempts  with  the  Hindu- Arabic  numerals  occa- 
sionally show  the  influence  of 
the  calculi.  This  is  seen  in  a 
Paris  MS.  in  which  the  mul- 
tiplication of  4600  by  23  is 
described  in  a  manner  leading 
to  the  form  here  shown.1  It  is 
possibly  in  forms  like  this  that 
the  chessboard  method  had  its 
origin. 

The  name  scachiero  was  used 
for  a  century  after  the  chess- 
board form  had  entirely  dis- 
appeared.   The  Treviso  arithmetic  (1478)  does  not  attempt 

to  mark   off   the   squares,   but  the   author 

uses  the  name,2  as  did  various  other  Italian 
writers.3  It  was  also  used  occasionally  in 
Germany,4  England,5  and  Spain/5  but  less  in 
other  countries. 

Even  after  this  method  was  generally 
adopted,  the  relative  position  of  the  figures 
was  for  a  long  time  unsettled.  In  the  oldest 
known  German  algorism7  the  multiplier 
appears  above  the  multiplicand.  In  the 
Rollandus  MS.  (Paris,  c.  1424)*  the  ar- 
rangement is  as  here  shown.  In  the  Treviso  arithmetic  the 
multiplier  is  sometimes  placed  at  the  right,  as  seen  in  the 


4 

5 

3 

4 

>j 

0 

i 

6 

i 

5 

i 

2 

1    5    3    ° 

XM.  Chasles,  Comptes  rendus,  XVI,  234  (1843). 

2"  .  .  .  attedi  al  terzo  modo.  zoe  al  moltiplicare  per  scachiero"  (fol.  19). 
3Borghi  (1484)  gives  only  this  method,  designating  it  "per  scachier." 
4 E.g.,  Petzensteiner  (1483)  says:  "Also  ich  wil  multipliciren  in  Scachir." 
5Recorde  (c.  1542)  speaks  of  "one  way  that  is  wrought  by  a  checker  table" 

(1558  ed.,  fol.  G8). 

6ThusTexeda  (1546) describes  multiplication  "escaqr  o  berricolo." 

7 A.  Nagl,  "Ueber  eine  Algorismus-Schrift  des  XII.  Jahrh.,"  Zeitschrift  fur 

Mathematik  und  Physik,  HI.  Abt,  XXXIV,   129.    This   journal  is  hereafter 

referred  to  as  Zeitschrift  (HI.  Abt.). 
8Rara  Arithmetica,  p.  446. 


no  MULTIPLICATION 

annexed  facsimile;  Widman  (1489)  gives  the  same  arrange- 
ment in  his  second  method;  and  this  lateral  position  of  the 

multiplier  is  preserved  in  our  syn- 
934*  thetic  multiplication  in  algebra. 

The  placing  of  the  multiplier  above 
the  multiplicand1  is  possibly  due 
to  the  fact  that  the  writers  of 
that  period  did  not  greatly  concern 

SCACCHEEO    MULTIPLICATION        themselves     as     to     whkh     of     the 

two  numbers  was  the  operator,  al- 

Frorn  the  Treviso  arithmetic,  1478.        .          ,      ,  ,, 

in  this  case  the  multiplier  is  placed     though  the  smaller  one  was  more 
at  the  right  often   chosen.    The   difficulty   of 

settling  down  to  a  definite  arrange- 
ment of  figures  is  seen  by  a  study  of  the  various  editions  of  the 
Taglientes'  popular  Libro  dabaco?  eight  of  which3  give  the 
following  examples : 

(1515)  (1520)  (1541)  (1547) 

456  456  456                       4S6 

23  _  23  23          23 

1368  1368"  136  8  1368 

912  912  912  912 


10488  10488  1048  8  10488 

(iSSo)  (1561)  (1564)  (1567) 

456  456        456         456 

23  23  ^23          23 

1368  1368  1368         1368 

912  912         912         912 


10488  10488  10488  10488 

The  various  other  editions  give  arrangements  similar  to  the 
above.  Some  of  these  forms  are  doubtless  due  to  printers' 
errors,  but  as  a  whole  they  go  to  show  that  a  definite  plan  had 

lE.g.,  Sfortunati,  1534;  1545  ed.,  fol.  u. 

2  1515,  the  work  of  two  authors.  3The  dates  appear  in  parentheses. 


THE  CASTLE  METHOD  in 

not  been  agreed  upon  in  the  i6th  century,  although  the  general 
chessboard  method  was  given  the  preference  by  most  writers,1 
other  methods  being  looked  upon  as  mere  curiosities.  Thus 
Hylles  (1600)  says: 

Also  you  shall  vnderstand,  that  there  are  besides  these  sundrie 
other  waies  of  Multiplication,  asvvell  with  squars  as  without  which  if 
you  list  to  learne  I  referre  you  to  M.  Records  ground  of  artes,  where 
you  may  finde  plentie  of  varietie. 

The  Castle  Method.  The  second  plan  of  multiplying  laid  down 
by  Pacioli  was,  on  account  of  the  form  of  the  work,  known  as 
"the  castle  "  or,  in  Florence,  "the  little  castle."2  The  signifi- 
cance of  the  name  is  best  understood  from  the  first  example 
given  by  Pacioli. 

9876  6  Per  .7 

6 
6789  i  ___  Proua 

61101000 
Castelucio  5431200 

476230     [sic] 
40734 
Suma  67048164  .1. 

It  will  be  observed  that  the  figures  are  arranged  somewhat 
like  the  wall  and  turret  of  a  castle.3  The  scheme  was  merely 
a  copy-book  invention  of  the  Italian  schoolmasters  and,  al- 
though enduring  until  the  close  of  the  i?th  century  or  later,4 
was  always  looked  upon  as  a  puerile  method.5 


Tartaglia  (1556)  calls  it  "vn  modo  generalissimo  da  nostri  antichi 
pratici  ritrouato,  &  pill  di  alcun'  altro  vsitato."  Pagani  (1591),  although  giving 
a  list  of  methods  like  Pacioli,  prefers  this  one,  calling  it  "molto  vago"  (very 
pretty)  and  "molto  sicuro"  (very  certain). 

2  "Del    multiplicare    per    castello    ouero    castelluccio    vocabulo    Florentine" 
(Feliciano,  1526;   1545  ed.,  fol.   12).    Pacioli   (1494)   says  of  it:   "El  secondo 
modo  di  multiplicare  e  detto  castellucio"  (fol.  26,  r.),  and  a  MS.  of  Benedetto  da 
Firenze  of  c.  1460  calls  it  "elchasteluccio."   In  Spain  (Texeda,  1546)  it  appears 
as  "El  .2.  modo  le  dize  castellucio." 

3  The  figures  at  the  right  are  the  proof  by  casting  out  y's. 

4  It  appears  in  Ciacchi's  Regole  Generali  d'  Abbaco,  fol.  83  (Florence,  1675). 
r'Thus  Pagani  (1591)  says:  "ma  piu  tosto  capriciose  ch'  vsitato,  &  vtile." 


112 


MULTIPLICATION 


The  Column  Method.  The  third  method  given  by  Pacioli  is 
known  as  the  column  or  tablet  plan.1  By  this  method  the  com- 
puter refers  to  the  elaborate  tables,  always  in  columns,  like 
those  used  by  the  Babylonians,  which  are  found  in  many  of  the 
1 5th  century  manuscripts.  It  is  essentially  nothing  but  a  step 
in  the  development  of  such  elaborate  and  convenient  multipli- 
cation tables  as  those  of  Crelle  and  others,  which  appeared  in 
the  i  gth  century. 

Cross  Multiplication.  Pacioli's  fourth  method  was  that  of 
cross  multiplication,  still  preserved  in  our  algebras  and  used,  in 
simple  cases,  by  many  computers.  To  this  he  gave  the  name 
"crocetta"  or  "casella,"2  adding  that  it  is  more  fantastic  and 
ingenious  than  the  others.3  His  most  elaborate  illustration  is 
that  of  78  x  9876: 


770328 

!"El  terqo  e  detto  multiplicare  p  colona  ouer  atauoletta"  (1494  ed.,  fol.  26). 
Cataneo  (1546;  1567  ed.,  fol.  8)  gives  it  another  name,  "Del  mvltiplicar  a  la 
memoria  detto  uulgarmente  Caselle  o  Librettine,"  and  the  Taglientes  (1515) 
speak  of  it  as  ";p  cholonella."  Texeda  (1546),  who  follows  Pacioli  very  closely, 
speaks  of  it  as  "colona  o  taboleta."  As  "per  colona"  it  is  the  first  method 
given  in  the  Treviso  (1478)  arithmetic,  and  this  name  is  also  used  by  Borghi 
(1484).  Tartaglia  (1556)  calls  it  the  oral  or  mental  plan  ("per  discorso,  ouer 
di  testa")  as  well  as  "per  colona"  and  "per  colonella." 

2"De  .4°.  mo  multiplicand!  dicto  crocetta  siue  casella."  The  Treviso  arith- 
metic calls  it  the  method  of  the  simple  little  cross :  "  Attendi  diligetamente  a  lo 
segondo  modo :  zoe  moltiplicare  per  croxetta  simplice."  The  name  crocetta  was 
the  more  common  one,  for  Feliciano  (1526)  speaks  of  it  as  "per  crocetta  o  voi 
dire  per  casela";  Cardan  (1539)  gives  it  only  the  name  "modus  multiplicadi 
P  cruceta";  and  Tartaglia  (1556)  and  Cataneo  (1546)  call  it  merely  "per 
crosetta."  CroceMa  means  a  little  cross,  and  Casella  (a  little  house)  is  often  used 
for  "pigeonhole." 

3  "...  piu  fantasia  e  ceruello  che  alcflo  d'  glialtri."  He  admires  it,  however, 
as  "bella  e  sotil  e  fo  bel  trouato"  (fol.  28,  r.). 


CROSS  MULTIPLICATION 


The  filling  of  the  vacant  places  by  zeros,  in  0078,  was  not 
unusual  among  the  Arabs.  Thus  in  a  manuscript  of  one  of  the 
works  of  Qosta  ibn  Luqa1  (c.  900)  the  multiplication  of  21,600 
by  4  appears  in  this  form : 


The  plan  is  ancient,  appearing  in  the  Lildvati  (c.  1150)  as 
the  tatst'ha  method,  or  method  of  the  stationary  multiplier,  in 
distinction  from  the  advancing 
multiplier,  where  the  multiplying 
figures  were  advanced  one  place 
to  the  right  after  each  partial 
product  was  found.  This  method 
is  shown  on  page  118.  The 
method  given  in  the  Lildvati  is 
fully  explained  by  Gane^a  (c. 
1535)  in  his  commentary  on 
Bhaskara.2  It  also  appears  in  the 
arithmetic  of  Planudes(c.  1340), 
but  in  the  forms  here  shown : 3 


840 
24 

35 


114048    76842 


432 
264 


1423 

0054 


While  the  first  of  these  cases  is 
simple,  it  is  doubtful  if  the  other 
two  were  practically  used. 

Some  of  the  work  of  the  Ta- 
glientes  (1515)  is  related  to  cross 
multiplication,  as  may  be  seen 
from  the  illustration  here  given. 

1The  MS.  is  dated  1106  A.H.  (1695  A.D.). 
2Colebrooke,  Lilavati,  p.  6  n. 


AN  ITALIAN  METHOD  OF 
MULTIPLICATION 

From  the  1541  edition  of  the 
Taglientes'  Opera  of  1515 


3Waschke,  Planudes,  p.  14. 


H4  MULTIPLICATION 

On  account  of  the  difficulty  of  setting  the  crossed  lines,  printers 
often  used  the  letter  X  between  the  multiplicand  and  the  multi- 
plier, and  this  may  have  suggested  to  Wright  (1618)  the  multi- 
plication sign  (  x  )  used  by  him  and  his  contemporaries.  In 
Pagani's  arithmetic  of  1591,  for  example,  the  work  in  cross  mul- 
tiplication appears  as  follows: 

3        2  4         3         2 

X        IXIXI 

2 5.  ;L_JL_4 

800  157248 

He  recognized  that  the  method  is  not  very  practical  with  num- 
bers of  more  than  two  figures.1 

The  Method  of  the  Quadrilateral.  The  fifth  method  given  by 
Pacioli  is  that  of  the  quadrilateral.2    It  was  really  nothing  but 
the   chessboard   plan   with   the   partial 
9    3    ^  products  slightly  shifted,  as  is  seen  in 

the  illustration  from  the  Treviso  arith- 
metic  (1478)  here  shown. 

Gelosia  Method.    There  seems  to  be 
no  good  reason  why  Pacioli  should  have 
MULTIPLICATION  PER      postponed  to  the  sixth  in  order  the  so- 
QUADRILATERO  called  gelosia,  or  grating,  method,  also 

From  the  Treviso  arith-     known  by  the  name  of  the  quadrilateral, 
metic,  1478  the  square,3  or  the  method  of  the  cells,4 

x"Il  moltiplicare  a  crocetta  di  tre  figure,  e  assai  piu  dificile  de  primo.  .  .  . 
II  moltiplicare  per  crocetta  di  4.  figure  e  piu  dificile  delli  sopra  nominati,  & 
quanto  piu  sono  figure,  tanto  piu  sono  dificile"  (p.  17). 

2"E1  quinto  mo  e  detto  ;>  qdrilatero"  (fol.  26,  r.) .  It  appears  in  Pagani's 
work  (1591)  as  "per  quadrato."  It  is  often  merged  with  the  gelosia  method 
next  mentioned,  as  when  Tartaglia  (1556)  calls  it  "Per  Quadrilatero,  ouero 
per  gelosia." 

3 "El  sexto  modo  e  detto  p  gelosia  ouer  graticola."  Tartaglia  (1556)  says: 
"II  quinto  modo  di  multiplicare  e  detto  Quadrilatero  qual  e  assai  bello,  perche 
in  quello  no  vi  occorre  a  tener  a  mente  le  decene."  Various  other  names  are 
used,  such  as  "modo  de  quadrato"  (Feliciano,  1526),  ",p  quadro"  (i6th  century 
MS.  of  Gio.  Dom.  Marchesi),  "per  squadrado"  (undated  Bologna  MS.). 

4".  .  .per  le  figure  de  le  camerete,"  "dala  fugura  dela  camerella,"  the  cells 
also  being  called  "camere  triangulate."  This  is  in  an  undated  Turin  MS.  See 
B.  Boncompagni,  Atti  Pontif.,  XVI,  448. 


GELOSIA  METHOD 


and  to  the  Arabs  after  the  i2th  century  by  such  names  as 
the  method  of  the  sieve1  or  method  of  the  net.2 

The  method  is  well  illustrated  by  two  examples  from  the 
Treviso  (1478)  book  here  shown.  It  will  be  observed  that  the 
diagonals  separate  the  tens  and  units  and  render  unnecessary 


/*>  / 


*/ 

0 


' 


3 

i 

4 


Somm<u   x 


FIRST  PRINTED  CASE  OF  GELOSIA  MULTIPLICATION 

From  the  Treviso  arithmetic  (1478),  showing  the  form  from  which  the  Napier 

rods  were  developed 

the  carrying  process  except  in  adding  the  partial  products. 
These  diagonals  sometimes  slant  one  way  and  sometimes  an- 
other, but  in  general  the  direction  from  the  upper  right-hand 
corner  to  the  lower  left-hand  corner  was  the  favorite.3 

The  method  is  very  old  and  might  have  remained  the  popular 
one  if  it  had  not  been  difficult  to  print  or  even  to  write  the  net. 
It  was  very  likely  developed  first  in  India,  for  it  appears  in 
Ganesa's  commentary  on  the  Lildvatl  and  in  other  Hindu 
works.4  From  India  it  seems  to  have  moved  northward  to 


TIn  the  writings  of  Albanna  (c.  1300),  possibly  due  to  his  commentator, 
al-Qalasadi  (c.  1475).  See  F.  Woepcke,  Journal  Asiatique,  I  (6),  512. 

2H,  Suter,  "Das  Rechenbuch  des  Abu  Zakarija  al-Hassar,"  Bibl.  Math., 
II  (3),  17- 

3  See  the  Marre  translation  of  Beha  Eddin's  (c.  1600)  Kholdfat  al  Hissab, 
Paris,  1846  ;  Rome,  1864,  p.  13  ;  hereafter  referred  to  as  Beha  Eddin,  Choldsat. 
Also  see  books  as  late  as  Giuseppe  Cortese's  Aritmetica  (Naples,  1716). 

The  contrary  direction  is  seen  in  MSS.  of  Ibn  al-Ha'im  (c.  1400),  dated 
1132  A.H.  (1720  A.D.),  Albanna  (c.  1300),  and  various  other  Arabic  writers. 

4Colebrooke,  Lildvati,  p.  7.  See  also  the  introduction  to  Taylor,  Lilawatij 
pp.  20,  33.  It  was  called  by  the  Hindus  Shabakh. 


n6 


MULTIPLICATION 


China,  appearing  there  in  an  arithmetic  of  IS93.1   It  also  found 
its  way  into  the  Arab  and  Persian  works,  where  it  was  the 

favorite  method  for  many 
generations. 

y  From  the  Arabs  it  passed 
over  to  Italy  and  is  found 
in  many  manuscripts  of  the 
*4th  and  isth  centuries.  In 
the  printed  books  it  ap- 
peared as  late  as  the  begin- 
ning of  the  1  8th  century,2 
but  more  as  a  curiosity  than 
as  a  practical  method. 


GELOSIA  METHOD  OF   MULTIPLYING 

From  an  anonymous  manuscript  written  in 

Florence  c.  1430.   The  author  describes  it 

as  Multiplicha  p  modo  de  Quadrato 


As  to  the  name  g 
Pacioli's  statement  is  more 
complete  than  that  of  any  of 
his  contemporaries  : 


The  sixth  method  of  multi- 
plying is  called  gelosia  or  gra- 

ticola  .  .  .    because  the  arrangement  of  the  work  resembles  a  lattice 
or  gelosia.    By  gelosia  we  understand  the  grating  which  it  is  the  cus- 


974  »  6  9 

GELOSIA  MULTIPLICATION  AS  GIVEN  BY  PACIOLI,  1494 
Showing  the  same  double  arrangement  of  diagonals  as  in  the  Treviso  book  of  1478 

1  Libri,  Histoire,  I,  386,  389. 

2 E.g.,  in  the  1690  edition  of  Coutereels's  Cyffer-Boeck;  in  Padre  Alessandro's 
Arimmetica  (Rome,  1714);  and  in  Giuseppe  Cortese's  Aritmetica  (Naples,  1716). 


REPIEGO  AND  SCAPEZZO  METHODS  117 

torn  to  place  at  the  windows  of  houses  where  ladies  or  nuns  reside,  sc 
they  cannot  easily  be  seen.  Many  such  abound  in  the  noble  city  oi 
Venice.1 

The  Repiego  Method.  Another  method  that  was  populai 
enough  to  survive  and  to  have  a  place  in  some  of  our  modern 
textbooks  was  called  by  the  early  Italians  the  "modo  per  re- 
piego,"  that  is,  the  method  by  composition,  or,  more  exactly,  by 
decomposition,  of  factors.  For  example,  to  multiply  by  72, 
multiply  by  9  and  then  by  8,  thus  saving  the  addition  of  partial 
products.2  It  is  one  of  several  methods  inherited  by  the  Italians 
through  the  Arabs,  from  Hindu  sources.3 

The  Scapezzo  Method.  Pacioli's  eighth  method  was  commonly 
known  among  the  Italians  as  a  scapezzo,  or  multiplying  by  the 
parts,  not  the  factors,  of  the  multiplier.4  Tartaglia  (1556)  gives 
as  an  illustration 

26  x  67  =  (3  +  4  +  5  +  6  +  8)  x  67 

=  201  +  268  +  335  +  402  +  536  =  1742  ; 

but  he  could  not  have  considered  it  as  other  than  a  curiosity 
although  it  was  recognized  by  such  writers  as  Ramus  anc 
Schoner.5  It  goes  back  to  Bhaskara  at  least.6 

1"Gelosia  intendiamo  quelle  graticelle  ch  si  costumono  mettere  ale  finestr 
de  le  case  doue  habitano  done  acio  no  si  possino  facilme  e  vedere  o  altri  religiosi 
Diche  molto  abonda  la  excelsa  cita  de  uinegia"  (fol.  28,  r.).  The  word  founi 
its  way  into  French  as  jalousie,  meaning  a  blind,  and  thence  passed  into  Ger 
man  and  was  carried  even  to  the  Far  East,  where  it  is  met  with  today. 

2  Pacioli  explains  the  term  thus :  "  Repiego  de  vn  numero  se  intende  el  pro 
ducto  de  doi  altri  numeri  che  multiplicati  vno  nel  laltro  fanno  quel  tal  numen 
aponto:  del  quale  essi  sonno  ditti  repieghi"  (fol.  28,  v.).  It  is  Tartaglia' 
third  method.  See  also  Terquem's  Bulletin,  Vol.  VI,  and  B.  Boncompagni,  Att 
Pontif.,  XVI,  404. 

/  3  It  appears  in  Taylor's  notes  on  the  Lildvati  under  the  name  vibhaga  gunanav 
(submultiple  multiplication).  See  translation,  p.  8n. 

4"  De  octauo  modo  multiplicand!  dicto  aschapec,c.o."  Pacioli,  1494,  fol.  29,  t 
Tartaglia  gives  the  name  as  "spezzato,  ouer  spezzatamente."  General  Trattatc 
1556,  I,  26.  In  Texeda's  Spanish  work  of  1546  it  appears  as  escape$o. 

Oi$86  ed.,  p.  16. 

6  See  Taylor's  translation,  p.  8.  The  name  in  his  notes  is  khanda  gunamn 
(parts  multiplication) . 


n8  MULTIPLICATION 

Minor  Methods  of  Multiplying.  Besides  the  leading  methods 
given  by  Pacioli  there  are  many  variations  to  be  found  in  other 
early  works.  One  of  the  most  valuable  is  the  left-to-right 
method,  the  "allo  adietro"  plan  of  Tartaglia,1  still  used  to  ad- 
vantage in  some  cases  but  hardly  worth  teaching.  The  Arabs 
occasionally  used  it,2  and  the  Hindus  varied  it  by  beginning 
with  the  lowest  order  of  the  multiplier  and  the  highest  order  of 
the  multiplicand.3 

Another  variant  is  seen  in  a  cancellation  method 
which  went  by  various  names.  The  Arabs  called 
it  the  Hindu  plan,4  and  Taylor5  found  the  Hindus 
using  it  early  in  the  igth  century.  y  Al-Nasavi 
(c.  102  5 )6  and  other  Arab  writers  thought  highly 
enough  of  the  method  to  give  it  a  place  in  their 
works.  It  may  be  illustrated  by  the  case  of 
76  x  43.  The  figures  were  first  written  as  shown 
in  the  upper  rectangle.  The  multiplication  began 
with  7x4=28.  As  soon  as  a  figure  had  served 
its  purpose  it  was  erased  on  the  abacus  and  its 
place  was  taken  by  another.  This  procedure  was 
modified  in  India,  probably  long  after  200  B.C.,  when  ink  came 
into  use,7  the  figures  being  canceled  as  here  shown.8 

In  the  1 5th  and  i6th  centuries  there  were  numerous  vagaries 
of  the  copy-book  makers,  the  extensive  discussion  of  which  is 


1  General  Trattato,  1556, 1,  25,  v.  Calandri  (1491)  gives  a  page  (fol.  8)  to  the 
method. 

2 E.g.,  al-Karkhi,  c.  1020.  See  H.  Hankel,  Geschichte  der  Mathematik,  Leip- 
zig, 1874,  pp.  56,  188;  hereafter  referred  to  as  Hankel,  Geschichte, 

3  An  elaborate  example  of  the  late  use  of  the  method  is  in  Marten  Jellen's 
Rekenkundige  Byzonderheden  (1779),  p.  13. 

4  This  translation  of  the  term  Hindasi,  used  by  many  Arab  writers,  is,  how- 
ever, disputed.   It  also  means  numerical,  a  translation  that  would  have  little 
significance  in  a  case  like  this.   See  page  64.  5Lilawati,  Introd.,  p.  9. 

6F.  \Y°epcke,  Journal  Asiattque,  I  (6),  497. 

7See  G.  Blihler,  Indische  Palaeographie,  pp.  5,  91  (Strasburg,  1896). 

8 This  example  is  from  an  Arab  arithmetician,  Mohammed  ibn  Abdallah  ibn 
'Aiyash,  Abu  Zakariya,  commonly  known  as  al-Hassar  (c.  1175?).  See 
H.  Suter,  Bibl.  Math.,  II  (3),  16.  Substantially  the  same  plan  is  used  by  al- 
Qalasad!  (c.  1475),  ibid.,  p.  17.  For  various  arrangements  followed  by  the 
Hindus  see  ColebrookeJ  LUdvati,  p.  7  n. 


SHORT  METHODS 


119 


not  worth  while.  Suffice  it  to  say  that  certain  teachers  had  their 
pupils  arrange  the  partial  products  in  the  form  of  a  rhombus, 
and  even  as  good  mathematicians  as  Tar- 
taglia  (1556)  and  Cataldi  (1602)  mul- 
tiplied "per  Rumbo"  and  "a  Rombo." 
Others  arranged  the  figures  so  that  the 
outline  of  the  work  looked  like  a  cup, 
chalice,  or  beaker.1  The  better  writers, 
however,  recognized  that  such  work  was 
time-consuming.2 

Labor-Saving  Short  Methods.  Besides 
the  general  methods  already  described 
there  were  many  special  devices  for  the 
saving  of  labor.  Even  when  the  multipli- 
cation table  was  learned,  the  medieval 
computers  did  not  require  it  beyond 
5  x  10,  and  various  plans  were  developed 
for  operating  under  this  limitation  when 
an  abacus  was  not  conveniently  at  hand. 
Of  these  methods,  one  of  the  best 
known  is  a  complementary  plan  that  is 
found  in  many  of  the  i6th 
century  books.  To  multiply 
7  by  8,  write  the  numbers 
with  their  complements  to  10,  as  here  shown. 
Then  either  8  —  3  or  7  —  2  is  5,  and  2X3=6, 
so  that  56  is  the  product,  the  operation  not  re- 
quiring the  multiplication  table  beyond  5  x  10  in  any  case.  It 
is  given  by  such  writers  as  Huswirt  (1501),  Fine  (1530), 
*Riese(i522),  Rudolff  (1526),  Stifel(iS44),  Recorde(c.  1542), 

^hus.Tartagliac^lls  the  plan  "Per  Coppa,  ouer  per  Calice,"  and  Cataldi 
says :  "  dalla  forma  lorb\si  possono  chiamare  a  Calice,  Coppa,  Tazza,  6  Bic- 
chiere."  In  Spanish  (TexecTa,  1546)  it  appears  as  "per  copa"  and  is  incorrectly 
given  as  "a  la  fracesa"  (a  French  method).  It  is  essentially  the  method  used 
by  Juan  Diez,  Mexico,  1556,  as  shown  in  the  illustration. 

2  So  Tartaglia :  "  trouate  piu  per  mostrar  vn  piu  sapere,  che  per  alcuna  vtilita" 
(1592  ed.,  fol.  40).  Other  curious  forms  are  given  by  Coutereels  (1690  ed.,  p.  8) 
under  the  title  "Vernakelijke  Multiplicatie"  (interesting  multiplication). 


726  400 

1*  9* 
*>)  54 
4  S 
4V 

S5S,  75Q_ 

THE  FIRST  EXAMPLE 

IN    MULTIPLICATION 

PRINTED       IN       THE 

NEW    WORLD 

From  the  Sumario  cope- 
dioso  of  Juan  Diez, 
published  in  Mexico  in 
1556.  The  problem  is 
to  multiply  978  by  875 
and  the  method  is  es- 
sentially per  copa,  that 
is,  the  method  of  the 
cup,  so  called  because 
the  figure  resembles  a 
drinking  cup 


7\/3 
&/\2 


120  MULTIPLICATION 

Peletier  (1549),  and  Baker  (1568),  while  Peurbach  (c.  1450; 
ist  ed.,  1492 )  speaks  of  it  as  an  ancient  rule.1  It  was  commonly 
used  in  connection  with  finger  reckoning.2  For  example,  to  find 
6x9,  raise  one  finger  on  one  hand  and  four  fingers  on  the  other 
hand,  these  representing  the  respective  complements  of  9  and  6. 
Then  multiply  the  standing  fingers  for  the  units  and  add  the 
closed  fingers  for  the  tens.  The  plan  is  still  in  use  among  the 
peasants  in  certain  parts  of  Russia  and  Poland.  As  a  variant 
of  this  method  the  following  plan  was  until  recently  in  use  in 
certain  towns  in  Russia :  Number  the  fingers  on  each  hand  from 
6  to  io?  and  give  to  each  the  value  10.  Then  the  products  of 
8x8  and  7x9  are  found  as  indicated  below: 

8x8  7x9 

10 — 10 

9 9 

IO  10 

J J 

*        I0  I0         ^ 

6 —   6 

8x8=6x104-2x2=64 

A  few  of  the  most  common  of  the  complementary  methods 
will  now  be  briefly  indicated. 

Various  Arab  and  Persian  writers3  multiplied  8  by  7  by  the 
relation  ab  =  lob  —  (10  —  a}b,  as  here  shown.4 

In  this  case  we  have 


9- 

10 

Q 

Q 

V 

8 

10 

10 

7 

6         I0 

/ 

-  I0     -   6 

7X9=4Xio4-2X 

104-1X3=63 

7X  8  =  10  x  8  ~-(io  — 7)x  8 
=  10  x  8  —  3x8 
=  80  —  24 
=  56. 


10  X  8  =  80 
3  x  8  =  24 

56 


The  method  was  considered  valuable  by  writers  like  Widman 
(1489),  Riese  (1522),  Rudolff  (1526),  and  Scheubel  (1545). 

i".  .  .  regulam  illam  antiquam."         3£.g.,  Beha  Eddin  (c.  1600).. 

2 See  pages  196  and  201.  4G.  Enestrom,  Bibl.  Math.,  VII  (3),  95. 


SPECIAL  METHODS  121 

Widman,  Rudolff,  and  Grammateus  (1518)  used  the  relation 

ad  =  10  d-  10(10  -a)  +  (io  —  a)(io  —  6>) 
=  io[*-(io-  *)]  +  (10  -a)  (10  -6), 

£rartatu0 

fQemultipUcatione  Caprmquftrttmt 
ipticatio  eft  numm  p:ocrcatio.p:opo2tionabi 


ntftn  ft  f?a  tef-eferbpli  gratia  j  ad  4  inulnplicarc  eft  nume 
rum  jz  piocreare.  qiu  fie  muttiplicando  videlicet  4  p:<tf 
procionantur  quemadmodumtniilripkcan0>fcittcct  j  vnirari  cotrcf^on) 
Dcr.quia  vrnc^  eft  piopojno  rripla.^rem  multtpltcacio  pTcrequinc  «r  <5» 
kne  mulnplicarionetn  Mgitomm  tnrcr  fe  fcur.Cuiue  tahe  Darur 


. 

>:rrjm  ponao.()ua0  (nrcr  fe  tnuInpijco-crproducTum  mfenue  fcnhr-PonJ 
t>coiffcrcnriam  vniusaD^foalteriua  rubrrafecerpzwipzodiKro  poftf 
oe.npjouenict  fttmma.vf  paret  in  figure  ^emplum,  fcpta  * 
8  z  DilTcrenru.  quocfunr.mulnplu:« 


4  . 

48  terfe^tcrunrS.quc 


.  . 

earidn«ouoiumnumerfl:um  infra  10  quowm  qudibet  DuabuB  figuri* 
fcnprueeft.pjoporiris  tfatpDuobuft  nummBptimajinftriMiB  cumpma 
fuperiojia  mulnpltc>ttf  p:o<reabirur  numeruovnavdoualtmtfuna  Krt 
hndu0.rivna.frnbarur.fi  ouabj.prtrnam  t'arumffnbe.fccnn^n1""10"* 
Do  in  menrc.  BetndJ  irmim  eafdem  fad  fe  oddac^r  pwdurro  Pi«wcn9 
tetnnguramtnmcnrt  refcruaram  a<Jtling6er  piouenicr  nttmemo  »MJ 


mauninumerozumrm  wtcwaccipi  oct   . 

crit  fumma.StautemDuabJ.pJtroam  l?aromfmbe.fe(udamvnttanflp« 

flenohbj  figun0aecipicndcadde.qua8fimulfmbf.cteritfumma 


COMPLEMENTARY   MULTIPLICATION 

From  Huswirt's  Enchiridion  nouus  Algorismi  summopere  visits  De  integris, 
Cologne,  1501.   Much  reduced 

illustrated  in  the  following  multiplication  of  8  by  7. 
Here  we  have 

7X  8  =  iox  8  —  10  x(io  — 7) 


=  80—30  +  1 
-56. 


10  x  8  = 

80 

—  10  x  3  =- 

-30 

+    3X2  = 

6 

56 

1 2  2  MULTIPLICATION 

Further  Algebraic  Relations-.  The  following  relations  also  had 
their  advocates : 

ab  =  10  (a  +  b  —  10)  +  ( 10- a)  (10  -  />) ; l 


ab  =  io(a~  b  +  2 •  b~  s)  +  (io—  a)(io  —  b}\ 
ab  =  (10  —  a)(io  —  b}  +  io(a  +  b}—  ioo;2 
(10  a  +  a) (10  6  +  6)  =  [(ioa  +  a) b  +  ab\  10  +  ab  ;3 
(ioa  +  b)(ioa  +  c)~(ioa  +  b  +  c}a  X  10  + fa',4 
(loa  +  b)(ioa  —  c)  =  iood*+  ioab  —  (ioac  +  bc)\* 
ab  =  (a  +  b—  10)  X  10  — (#—  io)(io—  /;),  a>  10,  b<  10  ;6 

(10  +  rt)  (10  ^  +  <;:)  =  (ab  +  10  £  +  c)  X  10  +  ^c; ; 
(3  0)2=  10  «2-  a2]  (3  a  +  i)2-  (3  af+  [(3  ^  +  i)  +  3  «] ; 

(3  a  -  i)2=  (3  *)2-  [(3  ^  -  i)  +  3  «]  ;7 
^2==  io<7  —  (io-~rt)  x  «; 
^2=  (1  of  X  10  —  (1  ^)2,  where  ^  =  3  m  ; 
az=(a  —  i)*+[a+(a  —  i)],  where  ^  =  3^  +  1; 
^2=(^  -fi)2— [0  + (fl-f-i)],  where  ^  =  3  m  +  2  ;8 

(a  +  *)(«-*)=rt2-*V. 
(5  0)2=io02x  2^; 

1£.g.,  Beha  Eddin  and  Riese. 

2  The  Petzensteiner  arithmetic,  Bamberg,  1483. 

3Al-Karkhi  (c.  1020),  as  in 

22  x  44  =  (22  x  4  -f  8)  10  +  2  x  4  =  968. 
4Al-Karkhi,  Beha  Eddin,  Tartaglia.   E.g., 

23  x  27  =  (23  -f  7)  x  2  x  10  4-  3  x  7  =  600  4-  21  =  621. 

5Al-Karkhi  and  Tartaglia.  6An  Arab  writer,  al-Kashi,  c.  1430. 

7Elia  Misrachi  (c.  1500)  and  Rabbi  ben  Ezra  (c.  1140). 

8  Rabbi  ben  Ezra,  who  recognized  that  the  limitations  on  a  were  unnecessary. 

9  Well  known  to  the  Greeks  and  given  by  Euclid. 


THE  MULTIPLICATION  TABLE  123 

(5«  +  02=(5*)M-[(5*  +  i)+5*];1 
(10  -f  a)  (10  +  S)=  100  +  io(a  +  b)  +  ab  ;2 


The  method  ,  of  aliquot  parts  was  also  well  known  in  the 
Middle  Ages,  both  in  Europe  and  among  the  Arabs,  and  the 
1  6th  century  writers  frequently  gave  our  common  rules  of  mul- 
tiplying by  numbers  like  1  1  and  15.  Beginning  at  least  as  early 
as  the  1  4th  century,  multiplication  by  numbers  ending  in  one 
or  more  zeros  was  commonly  effected  as  at  present.4 

Contracted  multiplication,  the  work  being  correct  to  a  given 
number  of  significant  figures,  is  a  development  intended  to  meet 
the  needs  of  modern  science.  It  began  to  assume  some  impor- 
tance in  the  i8th  century,5  although  a  beginning  had  already 
been  made  by  Burgi  (c.  1592)  and  Praetorius  (c.  I599).6 

The  Multiplication  Table.  The  oldest  known  arrangement  of 
the  multiplication  table  is  by  columns.  This  is  the  one  always 
found  on  the  Babylonian  cylinders  and  the  one  commonly  used 
by  the  Italian  writers  on  mercantile  arithmetic  in  the  formative 
period  of  the  subject.  In  general,  no  product  appeared  more 
than  once  ;  that  is,  after  2X3  =  6  was  given,  3x2  was  thought 

xElia  Misrachi  gives  various  rules  of  this  kind. 

2Huswirt  (1501). 

3  This  is  the  rule  of  quarter  squares,  which  still  has  its  advocates.  It  is  prob- 
ably due  to  the  Hindus.  See  A.  Hochheim,  Kafi  jil  Hisdb,  p.  7  (Halle  a.  S., 
1878)  (hereafter  referred  to  as  Hochheim,  Kdfi  jil  Hisdb)  ;  H.  Weissenborn, 
Gerbert,  p.  201  (Berlin,  1888).  It  is  found  in  the  Talkhys  of  Albanna  (c.  1300), 
the  work  of  al-Karkhi  (c.  1020)  mentioned  above,  and  the  works  of  Beha  Eddm 
(c.  1600)  and  other  Oriental  writers.  The  preferred  transliteration  of  the  name 
of  al-Karkhi's  work  is  al-Kdft  ft'l-Pfisdb,  but  the  more  familiar  title  as  given  in 
the  European  editions  has  been  adopted  in  this  work.  See  Volume  I,  page  283. 

4£.g.,  Maestro  Paolo  dell'  Abbaco  (c.  1340);  see  G.  Frizzo's  edition,  p.  42 
(Verona,  1883).  Bianchini's  correspondence  with  Regiomontanus  (1462)  con- 
tains it;  see  M.  Curtze,  Abhandlungen,  XII,  197,  270.  It  is  also  in  the  Treviso 
arithmetic  (1478),  Pellos  (1492),  and  other  early  works.  There  are,  however, 
various  cases  in  which  it  was  not  recognized  in  the  i6th  century. 

5  Greenwood's  American  arithmetic  (1729)  gives  the  reversed  multiplier. 

6M.  Curtze,  Zeitschrift  (HI.  Abt),  XL,  7. 


1  2  4  MULTIPLICATION 

to  be  unnecessary,  a  view  still  taken  by  Japanese  arithmeticians1 
and  having  much  to  commend  it.  The  early  Italian  mercantile 
arithmetics  gave,  for  purpose  of  easy  reference,  tables  with 
the  products  of  all  primes  to  47  x  47,  or  often  to  97  x  97. 
Computers  turned  to  these  columns  for  the  simpler  products 
needed  in  multiplication  per  colonna.  The  Italians  obtained 
the  idea  from  the  East,  Rhabdas  (1341)  giving  the  column 
tables  "  which  the  very  wise  Palamedes  taught  me.772 


MEDIEVAL  MULTIPLICATION  TABLE 

Part  of  a  table  from  an  anonymous  Italian  MS.  of  c.  1456,  but  apparently  a 
copy  of  an  earlier  work  of  c.  1420 

The  second  arrangement  was  the  square  form  generally  used 
by  nonmercantile  writers  and  known  as  the  Pythagorean  Table,3 
whereof,  as  Hylles  (1600)  remarks,  "Some  affirme  Pythagoras 
to  be  the  first  author."  This  mistaken  idea  was  held  by  various 
early  writers,4  although  the  better  ones  seem  to  have  recognized 

1  Smith   and  Mikami,  History   of  Japanese   Mathematics,  p.  37    (Chicago, 
1914)  ;  hereafter  referred  to  as  Smith-Mikami. 

2  P.  Tannery's  translation  in  Notices  et  extraits  des  manuscrits  de  la  Bibl. 
nat.y  XXXII,  167.   For  their  use  by  Benedetto  da  Firenze,  Luca  dell'  Abacho, 
and  others,  see  Rara  Arithmetica,  p.  464  and  elsewhere.   They  are  also  found  in 
Pacioli  (1494),  Pellos  (1492),  Borghi  (1484),  and  the  Treviso  book  (1478),  and 
in  many  other  works.   See  also  D.  E.  Smith,  "A  Greek  Multiplication  Table," 
Bibl.  Math.,  IX  (3),  193. 

3  Table   de   Pythagore,   Tabula   Pythagorica,    Mensa    Pythagorae,    Mensula 
Pythagorae,  Tavola  Pitagorica,  Mensa  Pythagorica,  and  other  similar  names 
are  common. 

4Thus  Kobel  (1514)  speaks  of  "Der  Pythagorisch  Tisch  oder  Tafel"  as 
"von  dem  Fiirste  Pythagora  geordnet"  (1518  ed.,  fol.  17). 


n 


t      «f  lY'Vani 
fnr  -run  - 


-'  rf"  /rt/tcm  ^\*<i/rtt   otf-nifTtt?~c{/t<fitt/'  nTi 

^^ 
fri/hH  «i/i.r  rrni 

/Tmif»-  «C  ("/  -nui   c]'it 


^  pf«M 


,     .  L^JffffSf,\ 

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MULTIPLICATION  TABLE  (c.  1500) 

The  table  as  it  appeared  in  an  anonymous  Latin  MS.  of  c.  1500,  being  the  same 
form  as  the  one  found  in  various  MSS.  of  Boethius 


1 2  6  MULTIPLICATION 

that  the  later  Pythagoreans  were  the  inventors.1  It  is  found  in 
the  arithmetic  of  Boethius2  and  in  a  work  attributed  to  Bede 
(c.  yio),3  but  the  fact  that  Rhabdas  (c.  1341)  does  not  give  it4 
suggests  that  the  Greeks  did  not  use  it.  It  was  common  in  the 
medieval  works5  and  in  the  early  printed  books.6  Some  writers 
carelessly  attributed  it  to  Boethius,7  while  others  arranged  tables 
of  addition,  subtraction,  and  division  on  the  same  plan  and  gave 
to  them  the  name  of  Pythagoras.8 

The  third  standard  form  was  the  triangular  array.  It  appears 
in  a  Prag  manuscript9  in  the  form  here  shown,  but  there 
are  several  variants.  It  is  given  in  The  Crajte  of  Nombrynge 

(c.  1300)  as  "a  tabul  of  figures,  where-by 
]>ou  schalt  se  a-nonn)  ryght  what  is  pe 
nounbre  fat  comes  of  pe  multiplicacion) 
of  2  digittes."  Widman  (1489)  speaks 
of  it  as  a  Hebrew  device,10  and  at  any 
rate  it  is  quite  likely  to  be  Arabic.11  It 
was  not  so  popular  in  the  early  textbooks  as  the  columnar  and 
square  arrangements,  although  it  was  used  by  such  writers  as 
Widman  (1489),  Gemma  Frisius  (1540),  Recorde  (c.  1542), 
Baker  (1568),  and  Trenchant  (1566). 

i'Thus  Boethius  says:  "Pythagorici  .  .  .  quam  ob  honorem  sui  praeceptoris 
mensarh  Pythagoream  nominabant"  (Friedlein  ed.,  p.  396).  See  also  A.  Favaro 
in  Boncompagni's  Bullettino,  XII,  148.  Clavius  (1583)  says:  "quod  Pythagoras 
earn  vel  primus  excogitauerit,  vel  certe  discipulos  suos  in  ea  mirifice  exercuerit." 

2  Friedlein  ed.,  p.  53.    On  the  text  see  Boncompagni's  Bullettino,  XV,  139. 

3De  arithmetic^  numeris,  of  doubtful  authorship,  where  the  "Pythagorica 
Mensa  sive  abacus  numerandi"  is  given  in  full  to  20  X  20,  with  the  more 
important  products  as  far  as  iooo2. 

4P.  Tannery,  Notices  et  extraits  des  manuscrits  de  la  Bibl.  nat.,  XXXII,  121. 

5 E.g.,  Jordanus  Nemorarius  (c.  1225),  Rollandus  (1424),  and  al-Kashf 
(c.  1430). 

6 E.g.,  Tzwivel  (1505)  and  such  commentators  as  Faber  Stapulensis  and 
Clichtoveus. 

7Thus  Stifel  (1545)  says:  "Disc,  tafel  hat  Boetius  gesetzt." 

8Possibly  Ramus  (1569)  began  this,  for  he  gives  these  tables  and  says:  "Hie 
Pythagoraeus  additionis  abacus  est,"  and  so  for  subtraction  and  division. 

9  See  S.  Giinther,  Boncompagni's  Bullettino,  XII,  p.  149;  very  likely  the  MS. 
of  Christian  of  Prag,  already  referred  to  on  pages  77,  95. 

10  "Das    erst    ist    eynn    taffel    geformiret    auff    den    triangel    geczogen    aus} 
hefrraischer  zungen   oder  iudischer." 
X1:lBeha  Eddin  (c.  1600)  gives  it  in  his  Kholdsat  al-hisdb. 


KINDS  OF  TABLES 


127 


The  extent  to  which 
ably  from  time  to  time, 
use  Crelle's  tables 
today,  go  back  to 
ancient  times,  one 
of  the  5th  century 
giving  the  impor- 
tant products  to 
50  x  looo.1  The 
medieval  writers 
were  usually  con- 
tent to  stop  with 
20  x  2O,2  however. 

For  tables  to 
be  committed  to 
memory  it  was  suf- 
ficient, in  the  days 
of  the  medieval 
abacus,  to  go  only 
to  5  x  10 ;  even 
4x9  was  far 
enough  for  prac- 
tical purposes.3 
Many  of  the  i6th 
century  writers 
outside  of  Italy 
found  it  necessary 
to  urge  their  pupils 


the  tables  were  carried  varied  consider- 
Tables used  for  reference,  as  we  might 


z  |4  P  JB   |'Q|«2li4l 
5  |6  \9 .  M'5|i'8zi 


eig  bA*emm*leift  Qowitt 
Arumggemet'n 


§19" 


TRIANGULAR  AND  SQUARE  FORMS  OF  THE 
MULTIPLICATION  TABLE 

From  Widman's  arithmetic  (Leipzig,  1489),  the 

edition  of  1500 


1"Victorii  Calculus  ex  codice  Vaticano  editus  a  Godofredo  Friedlein,"  in 
Boncompagni's  Bullettino,  IV,  443. 

2As  in  a  MS.  written  before  1284  and  copied  in  1385,  described  by  Stein- 
schneider  in  the  Bibl.  Math.,  XIII  (2),  40.  See  also  Beldamandi's  work  (1410), 
printed  in  1483,  where  the  products  extend  to  22  x  22. 

3  See  the  devices  for  finding  such  products  as  7  x  8,  page  119.  Thus  Rudolff 
(1526;  1534  ed.,  fol.  D  8)  says:  "Das  ein  mal  eins  .  .  .  musten  zum  ersten 
wol  in  kopff  fassen  /  doch  nit  weiter  dan  bis  auff  4  mal  9."  Clavius,  while 
recommending  the  learning  of  the  table  to  TO  x  10,  says:  "Qvod  si  huiusmodi 
tabula  in  promptu  no  sit,  vtendum  erit  hac  regula,"  namely,  the  one  given  on 
page  119. 


128  DIVISION 

very  strongly  to  learn  the  table,  showing  that  the  custom 
was  relatively  recent  in  countries  where  the  abacus  had  only 
just  been  abandoned  or  where  its  use  was  diminishing.1  One 
Spanish  writer  says  that  it  should  be  known  as  thoroughly  as 
the  Ave  Maria,2  and  Digges  (1572)  encourages  his  pupils  by 
saying:  "This  Table  therefore  first  printe  liuely  in  thy  remem- 
brance, and  then  boldly  proceede  farther,  all  difficultie  I  assure 
thee  is  past." 

It  may  interest  those  teachers  who  feel  that  they  must  insist 
upon  "two  threes  are  six"  instead  of  "two  times  three  are  six" 
to  know  that  the  former  has  at  least  some  kind  of  remote  sanc- 
tion in  a  terse  Latin  form,3  although  in  most  languages  the  use 
of  "times "  has  been  general. 

6.  DIVISION 

Definition  of  Division.  Division  has  generally  been  considered 
as  the  fourth  of  the  fundamental  operations,4  the  fifth  when 
numeration  is  included,  or  the  seventh  when  duplation  and 
mediation  are  considered  separately.  In  general  the  operation 

3 So  Chuquet  (1484)  says:  "(0tem  plus  est  necesze  de  sauoir  tout  de  cueur 
la  multiplication  dune  chascune  des  .10.  figures  par  soy  mesmes  et  aussi  par 
une  chascune  des  aultres  La  quelle  chose  est  appelle  le  petit  liuret  de  algorisme." 
(From  A.  Marre's  MS.  copy  in  the  author's  library.)  This  "livret  de  algo- 
risme" was  a  common  name  for  the  small  multiplication  table,  "gli  libretti 
minor!"  of  the  Italians,  "gli  libretti  maggiori"  referring  to  the  table  beyond 
10  x  10.  (Spelling  as  in  the  Dagomari  MS.  described  in  Rara  Arithmetica, 
p.  435.)  The  couplet  often  found  in  i6th  century  books, 

fl  Lern  wol  mit  fleisz  das  ein  mal  ein 
So  wirt  dir  alle  rechnung  gmein, 

appeared  first  in  print,  so  far  as  I  have  found,  in  Widman's  work  of  1489; 
1508  ed.,  fol.  ii. 

2".  .  .  laquale  tabula  bisogna  sapere  ad  memoria  como  la  Aue  Maria" 
(Ortega,  1512;  1515  ed.,  fol.  16).  Thierfelder  (1587)  says:  "Aber  wer  das  ein 
mal  eins  nicht  fertig  lernet  .  .  .  wird  nimermehr  keinen  fertigen  Rechner 
geben"  (p.  16).  Metius  is  equally  urgent:  "Tabula  Pitagorica,  dieman  wel  vast 
in  sijn  memorie  moet  hebben"  (1635  ed.,  p.  5). 

3Thus  Scheubel  (1545, 1,  cap.  4)  says:  "Sixies  septem  sunt  42.  septies  quinq3 
sunt  35,"  and  so  on,  in  which,  however,  the  word  "times"  is  concealed. 

*The  "quarto  atto"  of  the  Treviso  arithmetic 


DIVISION  DEFINED  129 

has  been  known  either  as  division1  or  as  partition,2  but  many 
writers  use  both  terms.3  Thus  Baker  (1568)  speaks  of  "Deui- 
sion  or  partition,"  and  Digges  (1572)  says"Todeuideorparte."4 
As  in  the  case  of  multiplication,  no  satisfactory  definition, 
adapted  to  the  understanding  of  beginners,  is  possible,  since 
the  concept  is  constantly  extended  as  the  pupil  proceeds.  To 
say  that  "diuision  sheweth  onlely  howe  often  the  lesse  summe 
is  conteyned  in  the  bigger,"5  or  that 

Diuision  doth  search  how  oft  the  diuisor 
In  Diuidend  may  be  quoted  or  found 
Whereof  the  quotient  is  the  decider,6 

is  to  exclude  cases  like  6  ft.  -*-  2  or  3  -s-  4,  although  the  latter  was 
intentionally  barred  out  by  many  writers7  for  the  reason  that 
a  result  like  f  could  not  be  " times"  in  the  primitive  use  of  the 
word.  The  early  idea  was  manifestly  that  of  an  integral  divisor 
and  an  integral  quotient.8 

A  second  definition  which  has  had  some  sanction  is  that  of 
finding  a  number  which  is  contained  as  many  times  in  the  divi- 
dend as  unity  is  contained  in  the  divisor.  It  has  long  been  used, 
being  found  in  Maximus  Planudes  (c.  1340)°  and  the  Treviso 
arithmetic.10  An  improvement  upon  this  definition,  and  quite 

*E.g.,  with  such  medieval  writers  as  Fibonacci  (1202),  Liber  Abaci,  p.  27. 
This  is  the  idea  of  measuring,  and  so  Euclid  used  the  term  per p£v  (metrtiri}  to 
mean  both  to  measure  and  to  divide. 

2 E.g.,  the  Treviso  arithmetic,  Huswirt  (1501),  Ghaligai  (1521),  Stifel 
(1544),  Scheubel  (1545),  Cataldi  (1602),  Ortega  (1512),  Savonne  (1563),  and 
Santa-Cruz  (1594).  This  form  was  preferred  by  Heron,  Pappus,  and  Diophan- 
tus,  all  of  whom  used  peplfrw  (meri'zein,  to  part). 

3  E.g.,  Pacioli  (1494),  Tartaglia  (1556),  Trenchant  (1566),  Clavius  (1583). 

4  So  with  some  of  the  Dutch  writers.   Thus  in  the  Dutch-French  work  of 
Wentsel  (1599) :  "Deuisio:  dat  is  deelinge,"  "Diuisio :  e'  esta  dire,  partir." 

5Digges,  1572;  IS79  ed.,  p.  8. 

6Hylles  (1600),  the  word  "quoted"  being  interesting  as  related  to  "quotient." 

7 Thus  Tzwivel  (1505)  says:  "Officiu}  diuisionis  est  cognoscere  quotiens 
minor  nuerus  in  maiore  re^piat,"  and  Peletier  (1549)  says:  'Vest  sqauoir  com- 
bien  de  fois  vn  moindre  nombre  est  contenu  en  vn  plus  grand." 

8 Thus  Clavius:  "Divisio  est  distributio  propositi  numeri  in  partes  ab  altcro 
numero  dato  denominatas"  (1583;  1585  ed.,  p.  48). 

9Waschke,  Planudes,  p.  23. 

10"Trouare  vno  terzo  nuero:  el  quale  se  troua  tante  volte  nel  mazore;  quate 
vnitade  sono  nel  menore." 


130  DIVISION 

sufficient  for  pure  number,  is  one  that  is  based  upon  ratio, — the 
finding  of  a  number  which  has  to  unity  the  same  ratio  as  the 
dividend  has  to  the  divisor.  It  is  often  found  in  the  i6th 
century  books.1 

It  was  natural  in  the  Middle  Ages,  when  division  as  per- 
formed on  the  abacus  was  often  based  upon  subtraction,  to  base 
the  definition  also  upon  the  latter  operation.2  This  plan  was 
followed  by  such  writers  as  Ramus,3  Schoner,4  and  Peletier5  and 
has  not  wholly  died  out  even  yet. 

Of  all  the  elementary  definitions  the  one  most  generally  ap- 
proved describes  the  operation  as  seeking  a  number  which,  mul- 
tiplied by  the  divisor,  is  equal  to  the  dividend,  and  it  serves  the 
purpose  fairly  well.  It  is  perhaps  the  oldest  definition  extant6 
and  it  has  the  sanction  of  many  scholarly  writers.7 

Two-fold  Nature  of  Division.  The  above  definitions  do  not,  in 
general,  distinguish  between  the  two  notions  of  division  illus- 
trated by  the  cases  6  ft.  -^  3  ft.  =  2  and  6  ft.  -*-  2  =  3  ft.,  al- 
though the  last  definition  includes  both  cases.  Rudolff  (1526) 
seems  to  have  been  the  first  to  make  this  distinction  perfectly 
clear,8  and  Stifel  (1545)  to  have  been  the  second.9  Tartaglia10 
also  gave  it,  and  thereafter  it  was  mentioned  by  various  writers 
of  the  1 6th  and  iyth  centuries. 

^'Diuisio  est  numeri  pcreatio  ^portionabiliter  se  ad  vnitatem  habetis  vt 
diuidedus  ad  diuisore"  (Huswirt,  1501,  fol.  5). 

2"Numerum  per  numerum  diuidcre  est  maiorem  secundum  quantitatem 
minoris  partiri,  uidelicet  minorem  de  maiore  tociens  subtrahi,  quociens  in  eo 
potent  inueniri."  Johannes  Hispalensis,  Liber  Algorismi  (c,  1140),  in  Bon- 
compagni's  Trattati,  II,  41. 

'AArith.  libri  duo,  1569.  ^Tabulae  Astronomicae,  1536,  fol.  A 30. 

5  1540;  1607  ed.,  p.  48,  as  a  secondary  definition. 

6J.  P.  A.  Erman,  Life  in  Ancient  Egypt ,  p.  364,  English  translation  by 
Tirard,  New  York,  1894  (hereafter  referred  to  as  Erman,  Egypt},  attributes  it 
to  the  Egyptians. 

7 E.g.,  Cataldi:  "II  partire  e  modo  di  trouare  vna  quantita,  quale  moltipli- 
cata  per  vna  quantita  proposta  (ouero  con  la  quale  moltiplicando  vna  quantita 
proposta)  produca  vna  quantita  data"  (1602,  p.  32). 

8"Diuidirn  heisst  abteilen.  Lernet  ein  zal  in  die  ander  teilen/auff  das  man 
sehe/wie  offt  eine  in  d'andern  beschlossen  werde/oder  wieuil  auff  einen  teil 
kome"  (1534  ed.,  fol.  8). 

^Deutsche  Arithmetica,  1545,  fol.  i,  where  it  is  more  clearly  stated. 
10 General  Trattato,  1556, 1,  fol.  27,  r. 


TERMINOLOGY  131 

Terminology  of  Division.  Early  writers  commonly  gave 
names  to  only  two  of  the  numbers  used  in  division,  the  numerus 
dividendus  (number  to  be  divided)  and  the  numerus  divisor.1 
These  are,  of  course,  not  technical  terms,  and  they  appear  as 
mere  colloquial  expressions  in  various  medieval  works.  Gradu- 
ally, however,  the  numerus  was  dropped  and  dividendus  and 
divisor  came  to  be  used  as  technical  nouns,  as  at  present.2  Such 
names  as  "answer"  or  " result"  were  commonly  used  for  quo- 
tient3 and  were  quite  as  satisfactory. 

The  names  of  the  terms  have  undergone  various  changes. 
The  divisor  has  frequently  been  called  the  "parter"4  or  the 
"dividens,"5  but  our  present  term  has  been  the  one  most  com- 
monly used.  The  dividend  has  generally  been  called  by  this 
name,  although  there  have  been  terms  equivalent  to  "partend," 
with  the  usual  linguistic  variants.0  The  quotient  has  frequently 
been  called  the  product,7  the  part,8  the  exiens,9  and  the  outcome,10 
but  the  term  used  by  English  writers  has  been  the  favorite  in 
most  of  the  leading  European  languages.11 

1Thus  Clichtoveus,  in  his  commentary  on  Boethius  (1503;  1510  ed.,  fol.  36), 
says:  "In  divisione  tres  requiruntur  numeri.  Primus  est  numerus  diuided^ 
&  maior/ex  hypothesi  dandus.  Secundus/numerus  diuisor  siue  diuidens :  etiam 
assignandus  ex  hypothesi.  Tertius  est  numerus  ex  diuisione  proueniens:  & 
hie  est  querendus,"  no  name  being  given  for  this  quotient  and  no  mention  being 
made  of  a  remainder. 

2  E.g.,  in  the  Rollandus  MS.  (1424),  where  quotiens  is  also  used.  Joannes  de 
Muris  (c.  1350)  used  dividendus  and  numerus  quociens,  but  not  divisor.  See 
Abhandlungen,  V,  145.  3From  quoties,  how  much. 

4"L'autre  qui  le  diuise,  s'apele  parteur,  partisseur,  ou  diuiseur"  (Trenchant, 
1566;  1578  ed.,  p,  51).  Chuquet  (1484)  calls  it  the  partiteur,  and  Cataldi 
(1602)  uses  il  partitore,  following  the  Treviso  book  and  other  Italian  works  of 
the  time.  Pellos  (1492),  writing  in  a  dialect  mixture  of  French,  Italian,  and 
Spanish,  called  it  the  partidor.  In  the  Teutonic  languages  it  appeared  in  the 
i6th  century  as  Theiler,  Deyler,  Teyler,  Deeler,  and  deylder. 

n".  .  .  nuer^  diuisor  siue  diuides,"  Tzwivel,  1505,  fol.  6,  and  various  other 
Latin  works. 

6Ortega  (1512;  1515  ed.)  calls  it  "la  partitione,"  as  he  calls  the  multiplicand 
"la  multiplicatione."  Santa-Cruz  (1594)  calls  it  "suma  partidera."  Digges' 
(1572)  writes  it  "diuident." 

7  E.g.,  Gemma  Frisius  (1540)  and  numerous  other  Latin  writers. 

8 E.g.,  the  Treviso  book  (1478)   gives  "la  parte." 

°£.g.,  Scheubel  (1545).  10In  Dutch,  the  UHkomst. 

11  Of  course  with  such  variants  as  quotiens  in  the  Latin  books,  cocienie  in 
the  Spanish  (Santa-Cruz,  1594),  and  so  on. 


132  DIVISION 

For  obvious  reasons  the  name  for  the  remainder  has  varied 
more  than  the  others.  The  medieval  Latin  writers  used  numerus 
residuus,  residuus,  and  residua,  and  various  other  related  terms, 
and  certain  later  authors  employed  the  same  word  for  the  re- 
mainder as  for  the  fraction  in  the  quotient.1  ^ 

The  Process  of  Division.  The  operation  of  division  was 
one  of  the  most  difficult  in  the  ancient  logistica,  and  even 
in  the  isth  century  it  was  commonly  looked  upon  in  the  com- 
mercial training  of  the  Italian  boy  as  a  hard  matter.2  Pacioli 
(1494)  remarked  that  "if  a  man  can  divide  well,  everything 
else  is  easy,  for  all  the  rest  is  involved  therein."  He  consoles 
the  learner,  however,  by  a  homily  on  the  benefits  of  hard  work.3 
So  impressed  was  Gerbert  (c.  980)  by  the  difficulties  to  be  over- 
come that  he  gave  no  less  than  ten  cases  in  division,  beginning 
with  units  by  units,  treated  by  continued  subtraction.4  Even 
as  late  as  1424  Rollandus  gave  only  the  simplest  cases  with 
small  numbers,  and  nearly  two  centuries  later  Hylles  (1600) 
recognized  the  difficulties  when  he  said,  "Diuision  is  esteemed 
•one  of  the  busiest  operations  of  Arithmetick,  and  such  as  re- 
quireth  a  mynde  not  wandering,  or  setled  vppon  other  matters."5 

Early  Form  of  Division.  Probably  the  oldest 
form  of  division  is  the  one  used  by  the  Egyptians. 
This  was  based  upon  the  processes  of  duplation 
and  mediation.  Thus,  to  divide  1 9  by  8  we  may  ar- 
range the  work  as  here  shown.  We  take  2x8 
=  1 6,  ^  of  8  =  4,  and  so  on,  and  select  the  num- 
bers in  the  right-hand  column  which  have  19  for 
their  sum;  for  example,  16  +  2  +  i  =  19.  The 

quotient  is  therefore  2 -f  |+|,  the  multipliers  being  marked 

here  by  asterisks.6 

*E.g.,  G.  B.  di  S.  Francesco  (1689).  In  the  case  of  7-*- 3  =2,  and  i  re- 
mainder, or  2^,  he  uses  auanzo  for  the  i  and  also  for  the  ^.  In  a  MS.  of  1736 
in  the  Woolwich  Academy,  England,  "remainer"  is  used  exclusively  for 
"remainder."  2"Dura  cosa  e  la  partita"  is  a  phrase  often  met. 

8"Peroche  nulla  virtus  est  sine  labore.   E  questo  aferma  el  phylosopho  q$ 
virtus  cftsistit  circa  difficile"  (fol.  32,  v.). 
•,'  *M.  Chasles,  Comptes  rendus,  XVI,  284. 

5  Fol.  37.  «Erman,  Egypt,  p.  365. 


EARLY  METHODS  133 

We  are  quite  ignorant  as  to  the  way  in  which  the  Greeks 
and  Romans  performed  the  operation  of  division  before  the 
Christian  Era.  We  have,  however,  a  case  described  in  the  4th 
century  by  Theon  of  Alexandria  (c.  390),  in  which  the  literal 
numeral  system  of  the  Greeks  is  used  and  the  work  is  not  un- 
like our  own,  except  that  sexagesimal  fractions  are  employed.1 
Since  we  know  so  little  of  the  development  of  the  operation 
among  the  ancients,  we  shall  proceed  at  once  to  the  history 
of  the  subject,  showing  particularly  how  long  division  was  per- 
formed after  the  introduction  of  our  modern  numerals,  say  from 
about  the  year  1000.  -^ 

Short  Division.  The  simplest  method,  however,  was  the  one 
which  we  call  in  English  short  division,  which  is  based  upon 
the  recognition  of  the  products  in  the  columns  of  the  multipli- 
cation table,  and  which  has  therefore  been  known  as  division 
by  the  column,2  by  rule,  or  by  the  table,  as  oral  division,  or  as 
division  in  the  head.3  The  method  is  illustrated  in  the  Treviso 
book  as  follows: 


Lo  partitore  .2.  7624 

La  parte  38i2 


o  lauanzo, 


which  means  that  7624-^2=3812,  with  o  remainder.  The 
arrangement  used  by  Sfortunati  (1534)  for  a  similar  case  is  seen 
in  the  following  example  taken  from  his  arithmetic  : 

Pi4 


74098ft 


irThe  details  are  given  in  Heath,  History,  I,  58. 

2Per  colona  (Treviso,  1478);  per  cholona  (Borghi,  1484). 

3"Partire  a  regolo:  ouer  a  tauoletta"  (Pacioli,  1494;  1523  ed.,  fol.  32). 
Pacioli  advises:  "E  comenza  a  partire  sempre  da  lultima  (more  arabG),"  that  is, 
to  begin  at  the  left  as  the  Arabs  do.  If  anyone  claims  that  the  method  is  diffi- 
cult, says  Pacioli,  "  Bonum  est  difficile  :  malum  autem  facile  .  .  .  Stultonim  infi- 
nitus  est  numerus"  (fol.  32,  i».).  "Del  primo  modo  de  partire  detto  per  colona, 
ouer  di  testa,  ouer  per  discorso,  ouer  per  toletta  .  .  .  aregolo,  ouer  alia  dritta, 
ouer  tauoletta."  Tartaglia,  General  Trattato,  1556,  I,  fol.  29;  1592  edition  of 
Arithmetica^  fol.  43. 

ii 


134  DIVISION 

meaning  that  1,037,382  -s-  14  =  74,098  {J.1  As  with  us,  the 
method  was  generally  used  only  with  a  divisor  of  one  figure2 
and  until  recently  has  not  been  very  popular  with  teachers,3 
requiring  as  it  did  some  attention  to  a  division  table.4 

Gerbert's  Method.  Of  the  methods  which  make  use  of  our 
common  numerals  in  long  division,  one  of  the  oldest  is  often 
attributed  to  Gerbert  (c.  980),  although  it  is  uncertain  whether 
he  originated  it  and  although  he  did  not  use  the  zero.5  It  may 
be  illustrated  by  the  simple  case  of  900  -*-  8.  The  process  con- 
sists of  dividing  900  by  10  —  2,  2  being  the  complement  of  the 
divisor,  and  was  essentially  as  follows  : 

10-  2)900(90  +  18  +  3+1  +  -!-.  =  112^ 
900  —  1  80 


I  8O 


36 

30™  6 
6  +  6  =  12 

IO  —  2 


The  form  actually  used  by  certain  of  the  successors  of  Gerbert 
may  be  seen  from  an  example  in  an  anonymous  manuscript  of 
the  1 2th  century  now  in  Paris,6  no  zero  appearing  in  the  compu- 
tation. The  combination  of  Roman  and  Hindu  numerals  is 


1 1544/5  ed.,  fol.  15,  under  "Partire  per  testa." 

2  "Si  chiama  Partire  a  Colonna,  quando  il  Partitore  sara  d'vn  Numero  solo." 
Gio.  Batt.  di  S.  Francesco,  1689,  p.  29. 

3 Pike's  very  widely  used  arithmetic  employs  long  division  in  the  cases  of 
175,817-^-3  and  293-^-8.  See  the  8th  edition,  New  York,  1816,  pp.  18,  60. 

4 Some  i6th  and  i7th  century  writers  in  Italy  gave  a  division  table,  and 
Onofrio  (1670)  speaks  of  his  as  "di  grandissima  vtilta."  The  Japanese  learn 
a  peculiar  division  table  for  their  soroban  and  the  Chinese  for  their  suan-pan. 
See  Smith-Mikami,  p.  40. 

5H.  Weissenborn,  Zur  Geschichte  der  Einfuhrung  der  jetzigen  Ziffern,  p.  14 
(Berlin,  1892);  Gerbert,  p.  169  (Berlin,  1888). 

6M.  Chasles,  Comptes  rendus,  XVI,  235,  243. 


GERBERT'S  METHOD 


135 


frequently  seen  in  this  period.   The  long  explanation  in  the 
manuscript  may  be  summarized  in  the  following  solution: 


[10-8] 


[2  x  90] 

[2  X   10] 

[80  +  20] 

[2  X   10] 

[2X2] 


[Quotient]  1 


c 

X 

I 

2 

8 

* 
1 

cS 

2 

i 

i 

4 

i 

i 

9 

2 

I 

r 

2 

Differentia 
Divisor 

I)ivclus 


lr  Dcnominaciones 


This  same  method  is  one  of  three  given  by  Adelard  of  Bath 
(Regulae  abaci ,  c.  1120),  who  attributes  it  to  Gerbert.  These 
three  methods  are  the  divisio  ferrea,  as  above ;  the  divisio  aurea, 
somewhat  like  our  long  division ;  and  the  divisio  permixta.2 

Division  by  Factors.  A  third  method  of  division  that  was 
common  in  the  late  Middle  Ages  consisted  in  using  the  factors 
of  the  divisor,  and  was  known  as  "per  repiego."3  By  this 
method  216  -*-  24  reduces  to  216  -*-  8  -*-  3,  the  object  being  to 

irThe  fraction  ^  was  neglected.   The  bracketed  matter  is  not  in  the  original. 

2 Of  the  "iron  division"  he  says:  "dLDe  ferreis  quidem  diuisorib}  [for  "divi- 
sionibus,"  as  in  two  MSS.]  hec  paucis  dicta  sufficiant.  Tamen  quia  super  his 
tractauit  gibertus  philosoph?  vir  subtilis  ingenij  diligenter  et  compendiose  qui- 
dam  eciam  quern  discipulum  eius  predicant  que  guichardum  nominant/diligenter 
et  prolixe."  See  Boncompagni's  BulleMino,  XIV,  67. 

3 Repiego  means  "refolding."  It  appears  with  various  spellings,  often 
ripiego.  In  Texeda's  Spanish  arithmetic  (1546)  it  appears  as  repriego.  It  was 
occasionally  called  "division  by  rule,"  a  name  also  given  to  short  division. 
Thus  in  a  i4th  century  MS.  in  Mr.  Plimpton's  library:  "Questo  e  partire  per 
regola  =  cioe.  Parti  9859  p  48  cioe  ,p  .6.  &  ^  .8.  sua  reghola.  .  .  .  fattiamo  fino 
alpartmeo  p  Regbolo."  See  also  the  repiego  method  of  multiplication,  page  117. 


136 


DIVISION 


secure  one-figure  divisors  that  could  be  handled  "per  tavoletta." 
The  illustration  given  by  Pacioli  (1494)  is  that  of  9876  •*-  48. 
He  first  divides  9876  by  6,  the  result  being  1646.  He  then  di- 
vides 1646  by  "the  other  number  of  the  repiego"1  and  obtains 
205!  or  205! .  ^  *s  still  used,  although  not  commonly  taught  in 
school. 

Division  by  Parts.  If  the  divisor  was  a  multiple  of  ten,  the 
1 6th  century  writers  frequently  resorted  to  "Partire  per  il 
scapezo,"  that  is,  "division  by  cutting  up"  the  dividend.  Thus, 
to  divide  84,789  by  20,  the  dividend  was  cut  by  a  bar,  8478(9, 
the  first  part  being  divided  by  2  and  the  9  being  divided  by  20, 
— a  plan  that  is  found  essentially  in  our  modern  books." 

The  Galley  Method.  By  far  the  most  common  plan  in  use 
before  1600  is  known  as  the  galley,  batello,  or  scratch,  method 
and  seems  to  be  of  Hindu  origin.  It  may  be  illustrated  by  the 
case  of  65,284  -^-594,  as  given  in  the  Treviso  arithmetic  (1478). 
To  make  the  work  clear,  the  first  six  steps  are  given  separately 
as  follows : 


(0 
65284 

594 

(4) 
5 


(2) 


/94 

(5) 

5 

If* 


(3) 


(284 


(6) 
5 


10 


10 


59 


5 


x"  .  .  .  dico  che  parta  p  laltro  numero  del  repiego:  cioe.  p  .8.  neuen  .205. 
sani :  e  auaza  .6."  (1494  ed.,  fol.  33,  r.). 

2  It  is  given  in  Le  Regoluzze  di  Maestro  Paolo  dell1  Abbaco  (i4th  century), 
ed.  Frizzo,  p.  43  (Verona,  1883).  The  relation  of  this  to  the  decimal  fraction 
is  discussed  on  page  238.  The  plan  is  given  by  many  writers,  including  Borghi 
(1484),  Sfortunati  (1534),  Cataneo  (1546),  Baker  (1568),  Digges  (1572),  and 
Pagani  (1591). 


THE  GALLEY  METHOD  137 

The  completed  work,  the  explanation  for  which  occupies  two 
and  one-half  pages,  is  as  follows  : 

/5 


109 


That  is,  65,284  -*-  594  =  109,  with  a  remainder  538. 

The  method  is  by  no  means  as  difficult  as  it  seems  at  first 
sight,  and  in  general  it  uses  fewer  figures  than  our  common  plan. 
Maximus  Planudes  (c.  1340)  throws  some  light  upon  its  early 
history,  saying  that  it  is  "very  difficult  to  perform  on  paper, 
with  ink,  but  it  naturally  lends  itself  to  the  sand 
abacus.  The  necessity  for  erasing  certain  num- 
bers and  writing  others  in  their  places  gives  rise 
to  much  confusion  where  ink  is  used,  but  on  the 
sand  table  it  is  easy  to  erase  numbers  with  the 
fingers  and  to  write  others  in  their  places."1  It 
thus  appears  that  this  method,  which  at  first 
seems  cumbersome,  is  a  natural  development  of 
a  satisfactory  method  used  on  the  sand  abacus. 
It  was  adopted  by  Fibonacci  (1202),  as  here  shown  for  the 
case  of  18,456  -*-  i;.2 

The  names  galea  and  batello  referred  to  a  boat  which  the 
outline  of  the  work  was  thought  to  resemble.3  An  interesting 

1From  the  French  translation  in  the  Journal  Asiatique,  I  (6),  240.  On  the 
Hindu  method,  see  Gerhardt,  Etudes,  p.  7. 

2 The  Boncompagni  edition  (I,  32)  gives  no  cancellation  marks,  and  very 
likely  Fibonacci  made  no  use  of  them. 

8 As  Pacioli  says:  "E  Qsto  vocabulo  li  aduene  a  tale  opare  jp  certa  simili- 
tudine  materiale  che  li  respode  del  offitio  e  acto  de  la  galea  materiale  qle  e 
legno  marittimo  acto  al  nauigare"  (1494  ed.,  fol.  34).  Tartaglia  remarks:  "fe 
detto  in  Vinetia  per  batello,  ouer  per  galea  per  certe  similitudini  di  figure" 
(1592  ed.,  fol.  48).  The  spelling  varied,  as  usual,  giving  such  forms  as  battello, 
vatelo,  galera,  and  galia.  There  was  occasionally  a  distinction  between  the 
galea  and  batello  forms,  as  in  Forestani,  Pratica  d*  Arithmetica,  Venice,  1603. 


DIVISION 


illustration  of  this  resemblance  is  seen  in  a  manuscript  of 
c.  1575,  as  here  shown.  Tartaglia1  tells  us  that  it  was  the  cus- 
tom of  Venetian  teachers  to  require  such  illustrations  from  their 
pupils  when  they  had  finished  the  work. 


GALLEY   DIVISION,    16TH    CENTURY 

From  an  unpublished  manuscript  of  a  Venetian  monk.   The  title  of  the  work 

is  "Opus  Arithmetica  D.  Honorati  veneti  monachj  coenobij  S.  Lauretij."    From 

Mr.  Plimpton's  library 

This  method  of  dividing  was  used  by  the  Arab  writers 
from  the  time  of  al-Khowarizmi  (c.  825),  of  course  with  va- 
riations. For  example,  al-Nasavi  (c.  1025),  in  finding  that 
2852-^-12  =  237 T82-,  used  the  form  on  page  139.  The  advancing 

1i5Q2  ed.,  fol.  53. 


THE  GALLEY  METHOD 


139 


of  the  divisor  one  place  to  the  right  each  time  is  here  seen  more 
clearly  than  in  the  usual  Italian  forms.  The  medieval  Latin 
writers  sometimes  called  this  feature  anteri- 
oratio.1  This  advancing  of  the  divisor  was 
not  universal,  however,  Rudolff  (1526)  tell- 
ing us  that  the  French  and  other  computers 
often  set  the  divisor  down  but  once.2 

As   already    stated,    the   galley   method 
was  the  favorite 
one    with    arith- 
meticians   before 


I  2 

493 

237 

2852 

12 
12 

I  2 


237 

8 


13 

O3OO 
1  1  4406)4400 

25666 

221 


FIRST    EXAMPLE    IN    LONG 

DIVISION  PRINTED  IN  THE 

NEW   WORLD 

From  the  Sumario  Compen- 
dioso  of  Juan  Diez,  Mexico, 
1556.  It  illustrates  the  galley 
method,  without  canceled  fig- 
ures, as  applied  to  the  case  of 
114,400  •*•  26  =  4400 


1600,  and  it  had  many  strong  advo- 
cates up  to  the  close  of  the  i8th 
century.3  It  is  found  occasionally 
without  cancel  marks,  probably 
owing  in  most  cases  to  the  lack  of 
the  necessary  canceled  types.4  With 
or  without  this  canceling,  the  method 
was  preferred  not  merely  by  com- 
mercial computers  but  also  by  such 
scientists  as  Regiomontanus.5  Even 
as  good  a  mathematician  as  Heil- 
bronner,  in  the  middle  of  the  i8th 
century,  preferred  it  in  all  long  ex- 
amples.0 One  reason  for  this  preference  was,  no  doubt,  that 
fewer  figures  were  used ;  but  even  more  important  was  the  fact 
that  the  work  was  more  compact, — an  important  item  before 

3  So  Sacrobosco  (c.  1250)  uses  this  word  and  also  the  verb  anteriorare.  From 
this,  no  doubt,  Chuquet  (1484)  was  led  to  use  anteriorer.  See  G.  Enestrom, 
Bibl.  Math.,  XIII  (2),  54;  Halliwell,  Kara  Math.,  p.  17. 

2"Frantzosen  vnd  etlich  ander  Nacion/welche  den  teyler  nit  mehr  dann  ein 
mal  setzcn/.  .  ."  (1534  ed.,  fol.  IT)- 

3 Among  those  who  preferred  it  to  any  other  are  Chuquet  (1484),  Widman 
(1489),  Riese  (1522),  Tonstall  (1522),  Kb'bcl  (1514),  Gemma  Frisius  (1540), 
Recorde  (c.  1542),  Baker  (1568),  Oughtred  (1631),  and  certain  Dutch  writers 
even  as  late  as  Bartjens  (1792). 

4 E.g.,  Pellos  (1492),  Grammateus  (1518),  Albert  (1534),  and  the  Mexican 
work  of  1556  as  shown  in  the  facsimile. 

°See  his  correspondence  with  Bianchini  in  the  Abhandlungen,  XII,  197. 

6Historia,  pp.  776  et  passim. 


140 


DIVISION 


the  days  of  cheap  paper.    Hodder,  late  in  the  ryth  century, 
says  that  he  "will  leave  it  to  the  censure  of  the  most  experienced 

to  judge,  whether  this  manner 
of  dividing  be  not  plain,  lineal, 
and  to  be  wrought  with  fewer 
Figures  than  any  which  is  com- 
monly taught/'1  and  in  this  he 
follows  the  testimony  of  many 
of  the  best  Italian  writers  for 
two  centuries  preceding.2  The 
method  is  still  taught  in  the 
Moorish  schools  of  North  Africa, 
and  doubtless  in  other  parts  of 
the  Mohammedan  world. 

Our  Long  Division.  It  is  im- 
possible to  fix  an  exact  date 
for  the  origin  of  our  present 
arrangement  of  figures  in  long 
division,  partly  because  it  de- 


i       4      4 

(Quotient) 
(Dividend) 

(Remainder) 
I  (Divisor) 

i 
i 

7 

2 

2 

9 

o 

5 
4 

8 

i 

4 
4 

o 

8 

i 

i 

2 

i 

2 

T 

2 

2  5)  62  5(25 

4 


22 
TO 


veloped  gradually.  We  find  in  various  Arab 
and  Persian  works  arrangements  substan- 
tially like  the  one  shown  above  for  the  case 
of  1729  -f- 12  =  144,  and  i  remainder.3  This 
resembles  our  method,  although  it  has  several 
points  in  common  with  the  galley  plan. 

In  the  1 4th  century  Maximus  Planudes 
gave  what  is  called  an  Arab  device.  This  is 
a  step  in  advance  of  the  one  given  above  and 
yet  is  quite  distinct  from  our  method.4  It 
appears  in  a  form  somewhat  like  the  one  here  shown  for  the 
case  of  625  -s-  25. 

^1672  ed.,  p.  54. 

2  Thus  Pagani  (1591)  :  "H  partire  a  Galera  e  molto  sicuro  &  legiadro  ch'  ogn' 
altro  partire,"  and  Pacioli  is  even  more  pronounced  in  his  opinion. 

3 This  is  a  composite  of  solutions  in  various  MSS.  examined,  including  several 
of  the  i6th  century.  See  also  the  work  of  al-Kashf  (c.  1430)  as  referred  to  in 
Taylor,  Lilawati,  Introd.,  p.  22;  Gerhardt,  fitudes,  p.  14. 

4Gerhardt,  Etudes,  p.  22. 


THE  "A  BAND  A"  METHOD 


141 


The  isth  century  saw  the  method  brought  into  its  present 
form  under  the  name  a  danda  ("by  giving").  This  name  came 
from  the  fact  that  when  a  partial  product 
is  subtracted  we  bring  down  the  next 
figure  arid  "give"  it  to  the  remainder.1 
An  excellent  illustration  from  a  manu- 
script of  c.  1460  is  here  shown,  but  it  will 
be  noticed  that  the  remainder  is  repeated 
each  time  before  the  "giving."  The  name 
danda,  or  dande  in  parts  of  Tuscany,  is 
still  used  to  designate  this  method  of 
dividing.'2  It  has,  however,  been  applied 
to  forms  quite  different  from  the  one 
shown  above.  For  example,  the  case  of 
49,289  -5-  23  =  2143  appears  in  the  form 
shown  below  in  a  i4th  century  Italian 
manuscript,3  and  the  author  speaks  of 
an  analogous  solution  as  a  danda.4  The 
earliest  printed  book  to  give  the  method 
is  CalandrPs  work  of  1491,  and  the  first 
example  of  the  kind 
is  shown  on  page  142. 
It  next  appeared  as 
the  third  method  of 
Pacioli,5  and  was 
given  with  increasing 
frequency  in  the  following  century,  but  rather  as  an  interesting 
than  as  a  particularly  valuable  device/5  With  the  opening  of 
the  i  yth  century  it  began  more  effectively  to  replace  the  galley 

1So  Cataneo  (1546)  says:  "£  chiamato  a  danda  il  detto  modo,  perche  a  ogni 
sottration  fatta  nel  operare  se  li  da  vna  o  piu  figure  dal  lato  destro"  (1567  ed., 
foh  15).  2 Boncompagni's  Buttettino,  XIII,  252  n. 

tRara  Arithmetica,  p  437.  4"Questo  sie'  ilpartire  adanda." 

5"De  tertio  modo  diuidendi  dicto  danda"  (1494  ed.,  fol.  33). 

6Pagani  (1591)  speaks  of  it  thus:  "Partire  a  danda  e  assai  bello,  &  vago." 
Cognet  (1573)  mentions  the  advantage  of  not  canceling:  "Les  Marchands 
Italiens,  pour  ne  trencher  aucune  figure,  divisent  en  la  sorte  qui  s'ensuit";  and 
Trenchant  (1566)  remarks  :  "II  y  a  vne  autre  belle  forme  de  partir,  sans  trencher 
aucune  figure,"  or  "sans  rien  couper." 


EARLY  EXAMPLE  OF 
LONG   DIVISION 

One    of   the   earliest   ex- 
amples    of    the    present 
method.  From  an  Italian 
MS.  of  c.  1460 


142 


DIVISION 


method.    Cataldi  (1602)  gives  it  as  his  first  method,  but  with 
the  quotient  below  the  dividend,  the  first  part  of  his  work 


;Parri 


Uicnnc 


'Parti  |  g   C  o         -parti  n>i  g 

i  —  Co  n>^  —  ix 

o  i/£-  i3>i/-& 

Uicnnc  i  I     ii 

-parti  >  g  | 


uicnnc  T^T 

*Paiti  to 

CP  — 

480 
uicnnc     i  Co 


1 


Uicnne    o 


JL>  ! 


FIRST  PRINTED  EXAMPLE  OF  MODERN  LONG  DIVISION 

From  Calandri's  arithmetic,  Florence,  1491.    The  problem  is  the  division  of 

53,497  by  83 

being  as  shown  on  page  143.   In  another  example  he  places 
the  quotient  at  the  right,  saying  that  this  is  the  custom  in 


THE  "A  BAND  A"  METHOD 


143 


37)46201 
1248- 

46 

37 
92 
74 


Milan.1    In  the  galley  method  the  most  convenient  place  for  the 
quotient  was  at  the  right ;  Cataldi's  attempt  at  placing  it  below 

was  awkward;  the  modern  custom  of 
placing  it  above  the  dividend  in  long 
division  is  the  best  of  all,  since  it  auto- 
matically locates  the  decimal  point. 

At  the  close  of  the  iyth  century  the 
modern  form  of  division  was  fairly  well 
established,  the  galley  method  being 
looked  upon  more  as  a  curiosity.2 

There  have  been  many  variants  of 
the  a  dand a  method,  but  the  only  one 
of  any  importance  is  that  which  omits  the  partial  products  as 
shown  below.  Cataldi  (1602)  calls  it  the  abbreviated  a  danda.3 
It  has  had  more  or  less  vogue  for  three  centuries,  but  it  requires 
too  much  mental  effort  to  become  common.  It  was  brought  to 
the  attention  of  American  teachers  by  Green- 
wood (1729),  who,  speaking  of  the  various 
methods,  remarked  that,  as  "most  of  the  rest 
are  at  best  an  unnecessary  Curiosity ;  I  shall 
confine  myself  wholly  to  the  Two  ITALIAN 
Methods;  which  are  the  most  usual,"  these 
two  being  a  danda  and  the  contracted  form. 
Of  the  various  methods  suggested  in  the 
1 6th  century  one  of  the  most  interesting  is  that  of  Apianus 
(1527),  particularly  as  it  suggested  the  scheme  of  decimal 
fractions.  To  divide  11,664  by  48,  Apianus  first  writes  the 
aliquot  parts  of  48,  with  a  corresponding  series  of  numbers 
based  on  48  as  a  unit,  substantially  as  follows: 


48 

•|  of  48  =  24, 
\  of  48=12, 
-|  of  48=  6, 
TV  of  48=  3, 


corresponds  to 


05 
025 
0125 
00625 


^'Partire  a  Danda  vsato  in  Milano." 

2Thus  Onofrio's  Aritmetica  (1670)  gives  it  as  "di  poco  6  nullo  profitto." 

8"  ...  a  Danda  abbreuiato"  (p.  88). 


144 


ROOTS 


He  then  observes  that  n  -s-  48  >  i,  n  -^-24  >  i,  n  -s-i2>i, 
but  ii  -5-  6  > i.    But  6  —  |  of  48,  and  hence  the  first  part  of  the 
quotient  is  0125.    The  rest  of  the  work  is 
substantially  as  follows: 
ii  —6  =  5 


1664 


0125 
OO62 

05 

°5 
o 


Facit  2  43 


5  -y  24  >  i 

S  -5-  3  >  i 

hence  we  write  y1^,  or  00625,  and  so  on. 
It  is  evident  that  Apianus  had  some  idea 
of  decimal  fractions  in  his  mind,  although 


it  was  not  developed  in  his  treatise. 

Clichtoveus  (1503)  gave  a  rule  based  upon  the  identity 


10  a 


Thus,  to  find  29 


4,  take  a  —  -  =  2  —  7=0; 


then  subtract  4 


(or  c)  as  often  as  possible  from  9  (or  b),  thus  finding  that 
9  -5-  4  =  2  J-.    The  final  quotient1  is  then  2\  -+  5,  or  71  . 

Whatever  method  of  dividing  was  used,  a  table  of  multiples 
of  the  divisor  was  early  recognized  as  desirable.  Such  tables 
are  found  in  many  works,  including  those  of  Recorde  (c.  1542), 
Fine  (1530),  Ramus  (1569),  Hylles  (1592),  and  Greenwood 
(1729). 

7.  ROOTS 

Finding  Square  and  Cube  Roots  of  Numbers.  The  Greeks 
found  the  square  root  of  a  number  by  a  method  similar  to  the 
one  commonly  set  forth  in  the  elementary  algebras  and  arith- 
metics of  the  present  time.  It  was  shown  geometrically  by 
Euclid2  that  (a  +  b)2  =  a2  +  2db  +  b2  (a  fact  that  was  prob- 

1  Edition  of  c.  1507,  fol.  D  4.  He  also  gives  a  rule  for  the  case  of  a<  %  c. 
For  a  few  further  notes  on  the  history  of  division  see  E.  Mathieu,  "M6thodes 
de  division  en  usage  a  la  fin  du  siecle  dernier,"  in  Journal  de  math,  iliment., 
V  (4),  97.  2  Elements,  II,  4. 


SQUARE  ROOT  145 

ably  known  long  before  his  time),  and  by  means  of  this  relation 
Theon  of  Alexandria  (c.  390),  using  sexagesimals,  found  the 
square  root  of  a  number  by  the  following  rule : 

When  we  seek  a  square  root,  we  take  first  the  root  of  the  nearest 
square  number.  We  then  double  this  and  divide  with  it  the  remainder 
reduced  to  minutes,  and  subtract  the  square  of  the  quotient ;  then  we 
reduce  the  remainder  to  seconds  and  divide  by  twice  the  degrees  and 
minutes  [of  the  whole  quotient  j.  \Ve  thus  obtain  nearly  the  root  of 
the  quadratic.1 

By  this  rule  he  finds  that  \/45oo  =  67  4'  55"  approximately. 

From  Greece  the  method  passed  over  to  the  Arabs  and  Hin- 
dus, with  no  particular  improvement.  Thus  Bhaskara  (c.  1150) 
writes  his  number  as  follows : 

l  —   l  — -  I 
88209 

and  then  proceeds  much  as  Theon  had  done.   He  says : 

Having  deducted  from  the  last  of  the  odd  digits2  the  square  number, 
double  its  root;  and  by  that  dividing  the  subsequent  even  digit3  and 
subtracting  the  square  of  the  quotient  from 
the  uneven  place,4  note  in  a  line  the  double 
of  the  quotient.5 


3°         5 

5 


One  of  the  most  interesting  medi- 
eval   examples   of   the    finding   of    a 
square  root  is  given  by  Maximus  Planudes   (c.  1340).     To 
find  vTj5  he  arranges  the  work  as  here  shown.     This  is  quite 


further  details  of  the  process  see  J.  Gow,  History  of  Greek  Mathe- 
matics, pp  54-57  (Cambridge,  1884)  (hereafter  referred  to  as  Gow,  Greek 
Math.)  ;  K.  Hunrath,  Ueber  das  Ausziehen  der  Quadratwurzel  bei  Griechen  und 
Indern,  Prog.,  Hadersleben,  1883. 

2  That  is,  from  8,  the  third  and  last  of  the  odd  places  denoted  by  a  vertical 
line,  counting  from  the  right. 

3  Really,  (882  —  400)  -*•  40  =  9+. 

4  Apparently  meaning  that  2g2  is  subtracted  from  882. 

5  For  the  rest  of  the  rule  see  Colebrooke's  LUdvati,  p.  9.  For  the  work  of  the 
Arabs  and  Persians,  see  the  Taylor  translation,  p.  23. 


146  ROOTS 

unintelligible  without  the  accompanying  explanation,  which 
may  be  condensed  as  follows  :  1 

235(15 

i 

2  13 

2  X  5  =     £0 

30  —  twice  the  root. 

Hence      15  =  the  root. 

But  is'2  ^225. 

2  ?  t  _  2  ^  ^         I 

Hence    -  -—  —  —  ~  —  -,  which  must  be  added. 
30  3 

Hence  the  root  is  15^. 

The  early  printed  arithmetics  generally  used  an  arrangement 
of  figures  similar  to  the  one  found  in  the  galley  method  of  di- 
vision. Thus  Pacioli  (1494)  gives  the  following:2 

Extractio  radicu 


08^8080 
11999 
I 


that  is,  \/99,98o,ooi  =  9999. 

Gradually,  in  the  i6th  century,  the  galley  method  gave  way 
to  our  modern  arrangement,  although  it  was  occasionally  used 
until  the  i8th  century/5  Among  the  early  writers  to  take  an 

*For  examples  of  a  more  elaborate  nature  see  Waschke,  Planudes. 

2Fol.  45,  r.  For  those  who  do  not  have  access  to  original  works  a  good  illustra- 
tion of  this  method  may  be  seen  in  the  Abhandktngen,  XII,  201,  269.  A  problem 
of  Chuquet's  (1484)  may  also  be  seen  in  Boncompagni's  Bullettino,  XIII,  695. 

3  Among  the  better  arithmeticians  that  used  it  in  the  iyth  century  was 
Wilkens,  a  Dutch  writer  (1630). 


SQUARE  ROOT  147 

important  step  toward  our  present  method  was  Cataneo  (1546), 
who  arranged  the  work  substantially  as  follows:1 


54756(234 

4  primo  duplata    4 

14  secondo  46 

12 
27 
9 


185 
184 

16 

16 
o 

Among  the  first  of  the  well-known  writers  to  use  our  method 
in  its  entirety  was  Cataldi,  in  his  Trattato  of  i6i3.2  Most 
early  writers  gave  directions  for  " pointing  off"  in  periods  of 
two  figures  each,  some  placing  dots  above,  as  in  824464s; 
some  placing  dots  below,  as  in  1 19925 4  or  as  in  21  17  84  O45; 
some  using  lines,  as  in  2  6  006  006  obo G ;  some  using  colons, 
as  in  13  :  01  :  76  :  64  : 7;  and  some  using  vertical  bars,  as  in 
94  2i|8o  73  55.8  Many  writers,  however,  did  not  separate  the 
figures  into  groups.9 

^Le  Pratiche,  Venice,  1567  ed.,  fol.  72. 

2  For  various  forms  used  by  other  writers  see  P.  Treutlein,  Abhandlungen, 
I,  64,  71- 

3 -E.g.,  Grammateus  (1518),  Scheubcl  (1545),  Hartwell  (1646  edition  of  Rec- 
orde's  Ground  of  Aries),  Wilkens  (1630),  and  the  American  Greenwood  (1729). 

4 E.g.,  Gemma  Frisius  (1540),  L.  Schemer  (1586),  Peletier  (1549),  Santa-Cruz 
(1594),  and  Metius  (1625).  Cardan  sometimes  placed  them  above  and  some- 
times below. 

5  This  from  the  Epitome  of  Clavius  (1583  ;  1585  ed.,  p.  309),  although  gen- 
erally (as  on  page  310)  he  places  the  dots  immediately  below  the  figures. 

°This  in  cube  root,  from  the  Rollandus  MS.  (1424). 

7 From  Ortega  (1512  ;  1515  ed.,  fol.  99).  He  also  writes  3:6:0:8: 
for  the  square  root. 

8  This  was  very  common  and  has  much  to  commend  it.  It  was  given  by 
Chuquet  (1484),  Pellos  (1492),  Fine  (153°),  Trenchant  (1566),  and  many 
others. 

9 E.g.,  the  Arab  al-Hassar  (c.  1175),  Cataneo  (see  the  example  above),  and 
Feliciano  da  Lazesio  (1526). 


148  ROOTS 

In  finding  the  square  root  and  the  cube  root  most  of  the 
early  writers1  gave  the  rules  without  any  explanation,  or  at  the 
most  with  merely  a  reference  to  the  fact  that  (a  +  b)2  = 
a2  4-  zab  +62.  Thus  Buteo  (1559)  proceeded  no  farther  with 
cube  root  than  to  find  the  first  figure,  saying  that  it  is  better  to 
use  a  table  of  cubes  ;2  and  more  than  a  century  later  de  Lagny3 
asserted  that  it  would  take  most  computers  more  than  a  month 
to  find  the  cube  root  of  696,536,483,318,640,035,073,641,037. 
Although  the  ponderous  work  of  Tonstall  (1522)  naturally 
included  roots,  Recorde  (c.  1542)  did  not  think  the  subject 
worthy  of  a  place  in  his  Ground  of  Artes* 

A  conviction  of  the  value  of  the  reasoning  involved  in  the 
subject  led  various  writers  in  the  i6th  century  to  give  clear 
explanations  based  on  the  geometric  diagram.5  The  use  of  the 
blocks  for  explaining  cube  root  was  found  somewhat  later,  and 
became  fairly  common  in  the  i7th  century.0  In  the  i7th  and 
1 8th  centuries  the  blocks  are  even  used  in  finding  the  fourth 
root,  x  cubes  being  taken,  each  composed  of  x*  cubes.7 

*E.g.,  Brahmagupta  (c.  628)  and  Bhaskara  (c.  1150),  ed.  Colebrooke,  pp.  10, 
279;  al-Karkhi  (c.  1020),  ed.  Hochheim,  II,  13.  Fibonacci  (1202)  described  cube 
root,  and  it  also  appears  in  Sacrobosco's  Algorismus  (c.  1250)  and  in  the  Carmen 
de  Algorismo  of  Alexandre  de  Villedieu  (c.  1240). 

2  He  gives  such  a  table  up  to  4O3,  and  a  rule  which  we  may  express  by  the 
approximation  formula  __ 


-f  r  =  a  +  • 


sNouveaux  Element  d'Arithmelique  et  d'Algebre,  Paris,  1697;  A.  De  Mor- 
gan, Arithmetical  Books,  p.  55  (London,  1847)  (hereafter  referred  to  as  De 
Morgan,  Arith.  Books). 

4 In  Hartwell's  edition,  however,  there  is  "An  Appendix  concerning  the 
Resolution  of  the  Square  and  Cube  in  Numbers,  to  the  finding  of  their  side," 
in  which  he  speaks  of  the  u  Quadrat  root,  or  the  side  of  any  Quadrat  number," 
and  gives  the  geometric  diagram  (1646  ed.,  p.  573). 

r'E.g.,  Tonstall  (1522,  fol.  TV8),  Trenchant  (1566),  L.  Schoner  (1586,  p.  255), 
and  Gemma  Frisius  (1540). 

6A  good  illustration  is  found  in  Hartwell's  edition  of  Recorde's  Ground  of 
Artes  (1646  ed.,  p.  587). 

7 -E.g.,  Cardinael  (1644;  1659  ed.,  fol.  E8).  When  Bartjens  (1633;  1752  ed., 
pp.  242,  243)  wishes  to  know  "Hoe  veel  is  de  V xx  van  576"  or  "de  Radix  xx 
uit  3136  is,"  such  being  his  two  symbols  for  square  root,  he  uses  the  diagram. 
He  then  uses  the  blocks  in  "kubicq-wortel"  when  he  "trekt  de  V#3  uit  5832" 
(p.  251),  and  in  fourth  root  when  he  "trekt  de\/#4  van  81450625."  See  also  the 
1676  edition,  p.  242. 


HIGHER  ROOTS  149 

Higher  Roots.  The  conviction  of  the  value  of  the  subject  as  a 
mental  exercise  led  various  writers  to  include  some  work  in 
higher  roots,  the  work  being  based  upon  a  knowledge  of  the 
binomial  coefficients.  These  coefficients  were  occasionally  ar- 
ranged in  the  triangular  form  subsequently  known  as  Pascal's 
Triangle.1  This  arrangement  was  known  to  the  Chinese2  as 
early  as  1303,  and  also  to  the  Arabs/3  and  in  Europe  it  appeared 
in  print  on  the  title-page  of  a  work  by  Apianus  published  in 
1527  and  in  a  work4  by  Scheubel  that  appeared  in  1545. 

This  arrangement  of  the  binomial  coefficients  was  first  seri- 
ously considered  in  a  printed  book,  in  connection  with  higher 
roots,  simultaneously  by  Stifel  (i544)5  and  Scheubel  (i545).6 
The  latter  finds  the  tenth  root  of  1,152,921,504,606,846,976,  for 
example,  to  be  64,  and  he  carries  the  work  as  far  as  to  the  find- 
ing of  a  24th  root.  A  little  later  it  was  used  in  France  by  such 
writers  as  Trenchant  (1566) 7  and  Peletier  (i54g),8  and  it  ap- 
peared also  in  the  works  of  various  Dutch  writers.9 

^Traite  du  triangle  arithmelique,  published  posthumously  in  1665.  The  form 
used  by  Pascal  is  given  later  (p.  510). 

2  It  appears  in  the  Szu-yuen  Yii-kien  of  Chu  Shi'-kie  (1303),  but  as  some- 
thing already  known.  See  Mikami,  China,  p.  90. 

3Cantor,  Geschichte,  I  (2),  645.  4Rara  Arithmetica,  pp.  156,  236. 

GIn  the  Arithmetica  Integra,  fol.  44.  As  to  their  use  in  his  Coss  (1554),  see 
Abhandlimgen,  I,  77  ;  II,  43. 

GDe  Numeris,  in  the  tractalus  quintus.    See  Kara  Arithmetica,  p.  236. 

7 "Doctrine  generate  pour  extrere  toutes  racines."  He  also  says:  "Pour 
fondement  de  la  quelle,  ray  forme  ce  trigone  seme  de  nombres,  s'imbolisans  & 
s'engendrans  les  vns  les  autres  par  vn  ordre  de  grandis- 
sime  consideration"  (1578  ed.,  p.  249).  It  will  be  ob- 
served that,  by  placing  i  at  each  end  of  each  row,  the 
successive  rows  give  the  coefficients  in  the  expansion  of 


3  •  3 
4-6-4 

•  10  •  10  • 
etc 


(a  +  b)n  for  n  equal  to  2,  3,  4,  ....  This  serves  as  a 
basis  for  the  general  rule  for  finding  the  wth  root  of 
any  number.  For  example,  to  find  the  fifth  root  we 
observe  that  the  arithmetic  triangle  gives  the  trial 
divisor  as  5 a4  and  the  complete  divisor  as  5  a4  +  ioa36 
-f  ioa'2b~  +  5<z&3  -f  b4,  a  principle  well  known  to 
writers  of  the  i6th  century. 

8  He  speaks  of  it  as  a  "Nouuelle  manierc  d'extraire  les  Racines,  generate 
pour  toutes  extractions,  jusques  a  infinite"  (1607  ed.,  pp.  107,  178,  252). 

9Thus  Van  der  Schuere  (1600)  speaks  of  the  "Drie-hoecks  wijze"  (triangle- 
like)  arrangement.  It  is  also  used  by  Bartjcns  (1633),  Cardinael  (1644),  an(i 
others. 

n 


ROOTS 

Abbreviated  Methods.  Attempts  at  abbreviating  the  process 
are  relatively  late.  One  of  the  most  popular  rules  for  the  abridg- 
ment of  square  root  is  attributed  to  Newton,  and  Greenwood 
(1729)  gives  it  as  follows : 

SIR  Isaac  Newton  takes  notice  of  a  very  useful  Contraction,  in 
these  Cases,  viz.  That  when  a  Root  is  carried  on  half  way  or  above, 
the  Number  oj  Figures  you  intend  it  shall  consist  oj ;  the  remaining 
Figures  may  be  obtained  by  Dividing  the  remainder  by  the  double 
oj  the  Radical  Figures.1 

The  Meaning  of  the  Term.  It  should  be  stated  in  this  connec- 
tion that  the  use  of  "root"  to  mean  the  square  root,  common 
in  Europe  today,  has  historic  sanction.  Indeed,  all  the  world 
still  recognizes  it  by  taking  the  symbol  V0  instead  of  -\fa  to 
indicate  the  positive  square  root  of  a.  The  usage,  however,  was 
not  entirely  general,  many  early  writers  specifying  the  square 
root  as  carefully  as  the  cube  root.2 

The  Arab  writers  conceived  a  square  number  to  grow  out  of 
a  root,  while  the  Latin  writers  thought  of  the  side  of  a  geo- 
metric square.  Hence  the  works  translated  from  the  Arabic 
have  radix  for  a  common  term,  while  those  inherited  from 
the  Roman  civilization  have  latus:1  Hence  the  Latin  writers 
"found7'  the  latus  and  the  Arab  writers  "extracted,"  or  pulled 
out,  the  root.  Our  arithmetics,  based  largely  upon  Arab  sources, 
still  use  "extract,"  although  the  older  usage  of  "find"  is  bet- 
ter. The  fact  that  from  radix  we  have  both  "radical"  and 

XP.  77.  See  Newton's  Arithmetica  Universalis,  p.  33  (Cambridge,  1707): 
"Ubi  vero  radix  ad  medietatem  aut  ultra  extracta  est,  caeterae  figurae  per  di- 
visionem  solam  obtineri  possunt." 

2 Thus  Suevus  (1593),  under  his  "Regvla  qvadrata,"  gives  "Extractio  Radi- 
cis  Quadratae,"  and  Digges  (1572)  speaks  of  "the  square  Radix"  "quadrat 
roote,"  and  "quadrate  root." 

Among  the  early  writers  who  used  "root"  for  "square  root"  were  al-Nasavt 
(c.  1025),  L.  Schoner  (1586),  Rollandus  (1424),  and  probably  Bhaskara  (c.  1150; 
Taylor,  Lilawati,  introduction,  p.  6). 

3  Schoner  speaks  of  this  in  De  numeris  figuratis  liber,  appended  to  his  1586 
edition  of  Ramus:  "Sic  9  est  aequilatcrus,  &  latus  ejus  est  3.  Hoc  latus  aequila- 
teri  ab  Arabibus  etiam  dicitur  Radix"  (p.  3).  Fibonacci  (Liber  Abaci,  p.  353) 
uses  "find"  instead  of  "extract"  with  the  word  "root,"  having  used  "extract" 
for  "subtract." 


CHECKS  ON  OPERATIONS  151 

"radish"  makes  the  use  of  "extract"  more  easily  understood.3 
This  use  is  found  in  various  modern  languages,2  but  is  by  no 
means  universal.  Thus  Digges  (1572)  says,  "To  find  the 
square  Radix,  or  Roote  of  any  number"  (p.  13),  although  he 
also  says,  "to  search  or  pull  out  the  Radix,  or  roote  cubical" 
(p.  i6).3 

8.  CHECKS  ON  OPERATIONS 

Need  for  Checks.  The  fact  that  the  intermediate  steps  in  a 
long  operation  were  erased  on  the  various  forms  of  the  abacus 
rendered  it  impossible  to  review  the  work  as  may  be  done  with 
our  present  methods.  It  was  therefore  necessary  that  some 
simple  check  should  be  used  to  determine  the  probable  accu- 
racy of  a  result.  The  inverse  operation  was  generally  too  long 
to  serve  the  purposes,  and  hence  other  methods  were  developed 
rather  early. 

Check  of  Nines.  Of  all  these  methods  the  check  of  nines  is 
probably  the  best  known.  It  is  simple  of  application  and  serves 
to  detect  most  of  the  errors  that  are  likely  to  occur.  The  origin 
of  the  method  is  obscure.  It  is fQundJn  the  works  of  various 
Arab  writers,  including  al-Khowarizmi  (c.  825),  al-Karkhi 
(c.  1020),  Beha  Eddin  (c.  1600),  and  others.  Avicenna  (c. 
1020),  however,  in  discussing  the  subject  of  roots,  speaks  of  it 
as  a  Hindu  method.4  On  the  contrary,  no  Hindu  writer  is 

lOn  the  use  of  "root"  see  Wertheim's  edition  of  Elia  Misrachi  (c.  1500), 
Sejer-IIamispary  p.  20  (Frankfort  a.  M.,  1893),  and  Tartaglia's  General  Trat- 
tato,  II,  fol.  53,  v.  (1556). 

2 E.g.,  " Uyttreckinge  der  wortelen"  (Cardinael,  1659  ed.,  p.  2),  and  "cavere 
la  radice  qvadra"  (Ciacchi,  1675  ed.,  p.  335). 

3 Of  other  forms  of  expression  the  following  are  types:  "7097,  cuius  tetra- 
gonicu  latus  inquirens  .  .  . ,"  Buteo  (1559;  1560  ed.,  p.  71)  ;  "...  sacar  rayz 
quadrada  .  .  . ,"  luan  Perez  de  Moya  (1562;  1615  ed.,  fol.  223),  sacar  mean- 
ing to  extract;  "Del  trare  la  radice  de  numeri  quadrat!"  (fol.  15),  but  "Del 
trouare  la  radice  Cubica"  (fol.  18,  v.)>  in  the  Italian  translation  of  Fine 
(Venice,  1587),  showing  both  ft extract"  and  "find";  "Del  modo  di  trar  la  radice 
quadra  .  .  ."  (fol.  182,  v.},  but  "La  estrattione  delle  radici  cube"  (fol.  187,  v.), 
Forestani,  Pratica  d'  Arithmetics  .  .  . ,  Venice,  1603. 

/  4"Fa'  1-tharik  al-hindaci,"  an  expression  that  has  been  variously  interpreted. 
See  F.  Woepcke,  Journal  Asiatique,  I  (6),  500;  Carra  de  Vaux,  "Sur  1'histoire 
de  l'arithme"tique  arabe,"  Bibl.  Math.,  XIII  (2),  33. 


1 52  CHECKS  ON  OPERATIONS 

known  to  have  used  it  before  the  i2th  century,1  while  the  Arabs 
certainly  used  it  early  in  the  9th  century.  Nevertheless,  as 
careful  a  writer  as  Paul  Tannery  is  convinced  that  the  evidence 
at  present  points  to  its  invention  in  India  but  to  its  first  con- 
siderable use  in  the  School  of  Bagdad.2 

There  is  some  interesting  evidence  of  the  recognition  of  the 
excess  of  nines  in  the  number  mysticism  of  one  of  the  late  Greco- 
Roman  writers,  Hippolytus,  who  seems  to  have  lived  in  the 
3d  century  and  who  wrote  several  theological  treatises  as  well  as 
a  canon  paschalis.  He  made  no  use  of  the  principle,  however, 
in  the  verification  of  computations,  and  so  far  as  we  know  he 
was  ignorant  of  this  application  of  the  theory/3  What  he  did 
was  to  make  use  of  gematria,  as  in  estimating  the  relative  ability 
of  individuals  by  means  of  the  numerical  values  of  the  letters 
of  their  names.  Instead,  however,  of  simply  stating  this  value 
in  the  usual  way,  he  stated  it  with  respect  to  the  modulus  nine. 
For  example,  the  numerical  value  of  Hector  ("Etcrwp)  is  1225, 
but  Hippolytus  gave  it  as  i,  which  is  the  excess  of  nines  in 
this  number.  He  spoke  of  this  plan  as  due  to  the  Pythago- 
reans, meaning,  no  doubt,  the  Neo-Pythagoreans  of  a  period 
much  later  than  that  of  Pythagoras  himself. 

The  check  of  nines  seems  to  have  come  into  general  use  in 
the  nth  century,  largely  due  to  the  influence  of  Avicenna 
(c.  1020)  and  his  contemporary,  al-Karkhi,  and  thereafter  it  is 
found  in  most  of  the  other  arithmetics  of  any  importance  for  a 
period  of  about  eight  hundred  years.  Albanna  (c.  1300)  speaks 
of  the  Arab  arithmeticians  as  giving  proofs  of  their  computa- 
tions by  the  checks  of  7,  8,  9,  and  n,  and  as  knowing  of  the 
checks  by  other  numbers  as  well. 

From  the  Arabs  this  method  of  checking  passed  over  to  the 
West,  appearing  in  the  works  of  the  Hebrew-Arabic  writer 
Kushyar  ibn  Lebban  (c.  1000),  the  Hebrew  Rabbi  ben  Ezra 

:LG.  R.  Kaye,  Indian  Mathematics,  p.  34  (Calcutta,  1915),  hereafter  referred 
to  as  Kaye,  Indian  Math.;  Taylor's  Lilawati,  p.  7. 

2P.  Tannery,  Mtmoires  Scientifiques,  I,  185  (Paris,  1912).  On  the  Arab 
writers  see  Boncompagni,  Trattati,  I,  12;  Bibl.  Math.,  II  (3),  17;  XIII  (2),  33; 
Hochheim,  Kaji  fit  Hisab,  II,  ion. 

3P.  Tannery,  Mtmoires  Scientifiques,  I,  185;  Tropfke,  I  (2),  58. 


CHECK  OF  NINES  153 

(c.  H4O),1  the  •  Hebrew-Christian  Johannes  Hispalensis  (c. 
1140),  and  the  Christian  writers  Fibonacci  (1202),  Maximus 
Planudes  (c.  1340),  and  their  successors. 

Fibonacci  called  the  excess  of  nines  the  pensa  or  portio2  of 
the  number,  and  used  it  as  a  check  in  multiplication  and  divi- 
sion. Maximus  Planudes  arranged  his  work  in  multiplication 
as  here  shown,  using  9  instead  of  o  in  the  case  of  a  zero  excess, 
and  apparently  believing  that  the  check  was 
a  complete  one.  Johannes  Hispalensis  and 
Fibonacci,  however,  recognized  its  limitations. 

In  the'  early  printed  arithmetics  the  check 
is  found  quite  generally.  Pacioli  (1494)  speaks 
of  it  as  "corrente  mercatoria  e  presta,"3  and  Widman  (1489) 
always  concludes  his  operations  by  the  query,  "  Wiltu  probirn?" 
Scheubel  (1545)  considered  the  matter  so  important  that  he  gave 
a  table  of  multiples  of  nine  for  the  convenience  of  computers.4 

The  failure  of  the  check  was  considered  at  some  length  by 
Pacioli,  but  Clavius5  was  especially  clear  in  his  treatment  of 
the  case.  So  important  was  the  whole  matter  considered  that 
Santa-Cruz  (1594)  devoted  twenty-two  pages  to  the  theory.6 
In  the  i  yth  century,  owing  to  the  general  acceptance  of  the 
modern  forms  of  computing,  the  revision  of  the  operations  be- 
came more  simple,  and  hence  some  of  the  leading  commercial 
arithmetics7  discarded  the  check  of  nines.  In  England,  how- 
ever, the  influence  of  Cocker8  served  to  make  it  very  popu- 
lar, and  such  influence  as  Greenwood  (1729)  had  in  America 
was  in  the  same  direction.  In  the  igth  century  it  dropped  out 
of  American  arithmetics  for  the  most  part,  but  after  1900  it 
began  to  appear  again. 

So  important  did  Tartaglia  (1556)  consider  the  check  of 
nines,  even  in  addition,  that  he  gave  a  table  of  the  excess  of 

1Silberberg,  Sefer  ha-Mispar,  p.  94.  2  Liber  Abaci,  I,  8. 

3Fol.  20  [numbered  10],  r. 

*De  Nvmeris,  I,  chap.  2,  p.  12  (1545).  He  did  the  same  for  7,  n,  13,  and 
19,  using  these  numbers  also  for  checking.  ^Epitome,  p.  22  (1583). 

°i643  ed.,  fol.  171.  See  also  Sfortunati  (1534;  *545  ed.,  fol.  8)  ;  Cataneo 
(1546  ;  1567  ed.,  fol.  18)  ;  and  Pagani,  p.  6  (1591). 

7 E.g.,  Eversdyck's  edition  of  Coutereels,  p.  33  (1658) ;  Mots  (1640). 

8Arithmetick,  London,  1677,  witn  later  editions. 


154  CHECKS  ON  OPERATIONS 

nines  for  each  number  from  o  to  go,1  a  waste  of  space  that  argues 
for  the  lack  of  appreciation  of  the  ease  with  which  one  casts  out 
the  nines  in  any  number,  however  large. 

Checks  with  Other  Numbers.  Any  other  number  besides  nine 
may  be  used  for  checking,  although  nine  is  the  most  convenient. 
The  use  of  other  numbers  is  found  in  the  works  of  various  Arab 
writers,  and  Fibonacci2  gives  the  checks  for  7,  9,  n.  Other 
medieval  and  Renaissance  writers3  also  give  such  numbers  as  2, 
3,  5,  6,  13,  and  19.  Several  of  the  early  printed  books  show  a 
preference  for  7  on  account  of  the  diminished  chance  of  error.4 
In  general,  however,  they  naturally  give  the  proof  by  nines 
the  preference.5 

Inverse  Operation.  Although  the  check  by  the  inverse  opera- 
tion took  more  time,  it  was  more  certain,  and  hence  it  found 
many  advocates.  It  is  so  simple  that  its  origin  is  probably 
remote,  although  it  is  not  until  the  Middle  Ages  that  we  find 
it  first  stated  definitely.6  It  appears  frequently  in  the  early 
printed  books, — for  example,  in  the  works  of  Clichtoveus 
(1503),  Albert  (1534),  and  Thierfelder  (1587).  Tartaglia 
(1556)  asserted  that  the  method  was  illogical,  since  subtrac- 
tion could  not  be  used  in  checking  addition,  for  the  reason  that 
it  was  taught  after  that  subject,7 — an  objection  that  is  of  no 
practical  significance. 

1  General  Trattato,  I,  fols.  8,  v.,  and  9,  r. 

2  Liber  Abaci,  pp.  8,  39,  45. 

:iB.  Boncompagni,  Atti  Poniij.,  XVI,  519.  Rudolff  (1526),  Apianus  (1527), 
Fischer  (1549),  Albert  (1534),  and  Scheubel  (1545)  are  particularly  worth  con- 
sulting. 

4 Thus  Pellos  (1492),  comparing  7  with  2,  says:  "<H.Item  sapias  che  ,pba  de 
.7.  es  la  plu  segura  ,pba  che  pusca  esser  air  la  ^ba  de  .2."  (fol.  18).  See  also 
Borghi  (1484). 

5 Thus  Clavius  (1583)  prefers  the  proof  "per  abiectionem  nouenarij "  or,  in 
the  Italian  edition,  "col  gettar  via  tutti  li  9,"  to  that  "per  abiectionem  sep- 
tenarij "  or  "  col  gettar  via  li  7 " ;  and  so  with  Chuquet,  Pacioli,  Buteo,  Tar- 
taglia, Cardan,  and  many  others.  The  proof  by  other  numbers  than  9  and  n  is 
not  often  found  after  about  1600. 

6  For  example,  in  the  Algorismus  prosaycus  magistri  Christani  (c.  1400) :  "Et 
nota,  quod  subtraccio  probat  addicionem  et  addicio  subtraccionem."   Studnicka 
ed.,  p.  9  (Prag,  1893). 

7  See  the  General  Trattato,  I,  8,  r. 


DISCUSSION  155 

TOPICS  FOR  DISCUSSION 

1.  The  number  and  the  nature  of  the  fundamental  operations, 
and  the  reasons  for  the  various  classifications. 

2.  Significance  of  duplation  and  mediation  in  the  development  of 
logistic,  particularly  in  early  times. 

3.  Difficulties  in  adequately  defining  the  fundamental  operations 
as  their  nature  expanded  from  time  to  time. 

4.  The  leading  principles  determining  systems  of  notation,  with 
illustrations  of  each  principle. 

5.  The  leading  systems  of  notation,  with  a  study  of  their  respec- 
tive merits. 

6.  The  significance  and  growth  of  the  concept  of  place  value  in 
the  writing  of  numbers. 

7.  The  history  of  the  Roman  numerals,  with  a  study  of  the  vari- 
ants from  century  to  century. 

8.  The  nature,  history,  and  significance  of  the  subtractive  prin- 
ciple in  the  writing  of  numbers. 

9.  The  history  of  our  common  numerals,  with  a  study  of  the 
variants  from  century  to  century. 

10.  The  reading  and  writing  of  large  numbers  at  various  periods 
and  in  various  systems. 

11.  The  terminology  used  from  time  to  time  in  connection  with 
the  common  operations. 

12.  Significant  features  of  the  work  in  addition  and  subtraction  at 
different  stages  of  the  development  of  these  operations  and  a  study  of 
the  relative  merits  of  the  various  methods. 

13.  A  study  of  the  different  methods  of  multiplying,  with  a  con- 
sideration of  the  relative  merits  of  each  and  of  the  probable  reason 
for  the  survival  of  the  present  common  method. 

14.  A  study  of  the  different  methods  of  division,  with  particular 
reference  to  the  contest  between  our  present  plan  (a  modification  of 
the  a  danda  arrangement)  and  the  galley  method. 

15.  Traces  of  early  methods  of  computations  in  our  present  oper- 
ations with  algebraic  polynomials. 

1 6.  The  historical  development  of  the  process  of  finding  roots  of 
numbers. 

17.  The  historical  development  of  the  various  methods  of  checking 
operations  with  integers. 


CHAPTER  III 

MECHANICAL  AIDS  TO  CALCULATION 
i.    THE  ABACUS 

Necessity  for  the  Abacus.  Since  the  numerals  of  the  ancients 
were  rather  unsuited  to  the  purposes  of  calculation,  it  is  prob- 
able that  some  form  of  mechanical  computation  was  every- 
where necessary  before  the  perfecting  of  the  modern  system. 
This  probability  becomes  the  stronger  when  we  consider  that 
all  convenient  writing  materials  were  late  developments  in  the 
history  of  civilization.  Papyrus  was  unknown  in  Greece  before 
the  7th  century  B.C.,  parchment  was  an  invention  of  the  sth 
century  B.C.,1  and  paper  is  a  comparatively  recent  product,2 
while  tablets  of  clay  or  wax  were  not  suitable  for  calculation. 

Meaning  of  the  Term.  In  earliest  times  the  word  "abacus"3 
seems  to  have  referred  to  a  table  covered  with  sand  or  with  fine 

1  Pliny  says,  of  the  2d  century. 

2 It  may  have  been  brought  into  Europe  in  the  i2th  century  by  the  Moors  of 
Spain,  but  specimens  dating  from  about  the  beginning  of  our  era  have  been 
found  on  the  eastern  borders  of  China. 

3 The  word  comes  from  the  Greek  a/3a£  (a'bax},  probably  from  the  Semitic 
p2X  (abq),  dust.  Numerous  other  etymologies  have  been  suggested.  Among 
the  most  interesting  is  one  given  by  Th.  Martin  (Les  Signes  Num.,  p.  34)  on 
the  authority  of  Orion  of  Thebes,  a  lexicographer  of  the  5th  century,  and  on 
that  of  several  other  scholars,- — namely,  that  the  word  comes  from  a  +  /3ct<m 
(a  +  ba'sis,  without  base),  referring  to  the  fact  that  the  computing  tablet  had 
no  feet.  A  recent  article  by  R.  Soreau  gives  the  improbable  suggestion  that 
a/3a£  simply  meant  a  numerical  table,  and  came  from  a',  0',  +  a£ta(a,  fc,  ax'ia, 
relating  to  value),  meaning  i,  2,  +  a£  ( indicating  numerical  values).  See  R. 
Soreau,  "Sur  1'origine  et  le  sens  du  mot  'abaque,'"  Comptes  rendus,  CLXVI, 
67.  The  question  was  debated  even  in  Pacioli's  time,  for  he  says  (Suma,  fol. 
19,  r.  (1494)):  "e  modo  arabico  e  chiamase  Abaco :  ouer  secodo  altri  e  dicta 
Abaco  dal  greco  vocabulo."  Of  the  various  guesses,  that  of  Joannes  de  Muris 
(c.  1350)  is  the  most  curious,  that  "abacus"  is  the  name  of  the  inventor:  "Non 
est  sub  silencio  transeundum  de  tabula  numerorum,  quam  abacus  adinuenit" 
(Quadripartitum,  chap,  xiv;  in  the  Abhandlungen,  V,  144). 

156 


THE  ABACUS 

dust,  the  figures  being  drawn  with  a  stylus  and  the  marks  being 
erased  with  the  finger  when  necessary.  This  at  any  rate  is  the 
testimony  of  etymology,  and  the  dust  tablet  seems  to  have 
been  the  earliest  form  of  the  instrument.1 

While  all  definite  knowledge  of  the  origin  of  the  abacus  is 
lost,  there  is  some  reason  for  attributing  it  to  Semitic  rather 
than  to  Aryan  sources.2 

The  dust  abacus  finally  gave  place  to  a  ruled  table  upon 
which  small  disks  or  counters  were  arranged  on  lines  to  in- 
dicate numbers.  This  form  was  in  common  use  in  Europe 
until  the  opening  of  the  xyth  century,  and  persisted  in  various 
localities  until  a  much  later  date. 

Meanwhile,  and  in  rather  remote  times,  a  third  form  of 
abacus  appeared  in  certain  parts  of  the  world.  Instead  of  lines 
on  which  loose  counters  were  placed  there  were  grooves  or  rods 
for  movable  balls  or  disks,  a  form  still  found  in  Russia,  China, 
Japan,  and  parts  of  Arabia. 

We  have,  then,  three  standard  types, — the  ancient  dust  board, 
which  probably  gave  the  name  to  the  abacus,  the  table  with 
loose  counters,  and  the  table  with  counters  fastened  to  the 
lines.  These  three,  with  their  characteristic  variants,  will  now 
be  explained.3 

The  Dust  Abacus.  The  dust  abacus  was  merely  a  kind  of 
writing  medium  of  little  greater  significance  in  computation 

1C.  G.  Knott,  "The  abacus  in  its  historic  and  scientific  aspects,"  Transac- 
tions of  the  Asiatic  Society  of  Japan,  Yokohama,  XIV,  18  ;  hereafter  referred 
to  as  Knott,  Abacus.  2 Knott,  Abacus,  pp.  33,  44. 

3 The  literature  of  the  subject  is  extensive.  The  following  are  some  of  the 
general  authorities  consulted:  Knott,  Abacus;  M.  Chasles,  Comptes  rendus, 
XVI,  1409;  F.  Woepcke,  Journal  Asiatique,  I  (6),  516;  Sir  E.  Clive  Bayley, 
Journal  of  the  Royal  Asiatic  Society,  XV  (N.  S.) ;  M.  Hiibner,  "Die  charakte- 
ristischen  Formen  des  Rechenbretts,"  Zeitschrift  fiir  Lehrmittelwesen  und  pdda- 
gogische  Literatury  II,  47  ;  D.  Martines,  Origine  e  progres.si  dell'  aritmetica,p.  19 
(Messina,  1865)  (hereafter  referred  to  as  Martines,  Origine  aritmet.)  ;  A.  Ter- 
rien  de  Lacouperie,  "The  Old  Numerals,  the  Counting  Rods  and  the  Swan-pan 
in  China,"  Numismatic  Chronicle,  III  (3),  297-340,  reprinted  in  London  in 
1888  (hereafter  referred  to  as  Lacouperie,  The  Old  Numerals).  The  most 
elaborate  and  scholarly  work  on  the  subject  is  F.  P.  Barnard,  The  Casting- 
Counter  and  the  Counting-Board,  Oxford,  1916  (hereafter  referred  to  as  Bar- 
nard, Counters'). 


158  THE  ABACUS 

than  the  clay  tablet  of  the  Babylonians,  the  wax  tablet  of  the 
Romans,  the  slate  of  the  Renaissance  period,  or  the  sheet  of 
paper  of  today.  In  its  use,  however,  is  to  be  found  the  explana- 
tion of  certain  steps  in  the  operations  with  numbers,  and  on 
this  account  it  deserves  mention. 

The  Hindus  seem  to  have  known  this  type  in  remote  times 
but  to  have  generally  discontinued  its  use.  Even  in  recent 
times,  however,  children  have  been  instructed  to  write  letters 
and  figures  in  the  dust  or  sand  on  the  floor  of  the  native  school 
before  being  allowed  to  use  the  common  materials  for  writing.1 
That  the  dust  abacus  was  common  a  century  ago  is  asserted  by 
Taylor  in  the  preface  to  his  edition  of  the  Lildvati. 

In  the  Greek  and  Roman  civilizations  the  dust  abacus  was 
also  well  known.  Figures  were  drawn  upon  it  with  a  stylus, 
called  by  the  Latin  writers  a  radius?  much  as  they  were  drawn 
on  the  slate  in  recent  times.  The  wax  tablet,  described  later, 
was  even  more  extensively  used. 

Nature  of  the  Counter  Abacus.  As  in  the  case  of  all  such  prim- 
itive instruments,  the  origin  of  the  counter  abacus  is  obscure.3 
We  only  know  that  in  very  early  times  there  seems  to  have 
been  a  widespread  knowledge  of  some  kind  of  instrument  in 
which  objects  (beads,  disks,  or  counters)  on  one  line  indicated 
units,  on  the  next  line  tens,  on  the  next  hundreds,  and  so  on. 
Some  general  idea  of  this  instrument  may  be  obtained  from  the 
illustrations  given  on  page  159.  The  first  one  shows  the  succes- 
sive steps  taken  in  the  addition  of  numbers.  The  second  illus- 
tration shows  the  use  of  the  abacus  in  multiplication.  Several 
variants  of  this  type  are  given  later. 


*Sir  E.  Give  Baylcy,  Journal  of  the  Royal  Asiatic  Society,  XV  (N.S.),  911., 
15,  and  XIV  (N.  S.),  Part  3  (in  the  reprint  of  the  article  "On  the  genealogy  of 
ancient  numerals"  it  appears  in  Part  II,  p.  71 ;  see  also  Part  I,  p.  19,  and  Part  II, 
PP-  5°>  54)  J  G.  R.  Kaye,  "The  use  of  the  abacus  in  Ancient  India,"  Journ.  and 
Proc.  of  Asiatic  Soc.  of  Bengal,  IV  (2),  293. 

2 "Ex  eadem  urbe  humilem  homunculum  a  pulvere  et  radio  excitabo,  qui 
multis  annis  post  fuit  Archimedes,"  Cicero,  Tusculan  Disputations,  V,  23,  64; 
"Descripsit  radio,"  Vergil,  Eclogues,  III,  41. 

3  On  the  history  in  general  see  A.  Nagl,  Die  Rechenpfennige  und  die  opera- 
the  Arithmetik,  Vienna,  1888. 


COUNTER  ABACUS 


159 


There  is  some  reason  for  believing  that  this  form  of  the 
abacus  originated  in  India,  Mesopotamia,  or  Egypt.  The  whole 

Tens  Units    Tens    Units    Tens   Units     Tens    Units     Tens  Units   Hundreds  Tens  Units 


First  Step  .Second  Step    Third   Step    Fourth  Step        Fifth    Step 
ADDITION    ON    THE    ABACUS 


Sixth  Step 


An  early  computer,  wishing  to  add  22  and  139,  might  have  proceeded  as  follows: 
Place  2  pebbles  on  the  units'  line,  as  shown  in  the  First  Step.  Then  place  9  more, 
as  shown  in  the  Second  Step.  Then  take  away  10  of  these  pebbles  and  add  one 
pebble  to  the  tens'  line,  as  shown  in  the  Third  Step.  Then  add  2  pebbles  to  the 
tens'  line  because  of  the  20  in  22,  as  shown  in  the  Fourth  Step.  Then  add  3  more 
because  of  the  30  in  139,  as  shown  in  the  Fifth  Step.  Finally  draw  a  line  for  hun- 
dreds, and  on  this  place  one  pebble  because  of  the  100  in  139.  The  answer  is  161 

matter  is,  however,  purely  speculative  at  the  present  time  and 
it  seems  improbable  that  it  will  ever  be  definitely  settled. 


4132 


2  X  4132  =  8264 


MULTIPLICATION  ON  THE  ABACUS 

Above  the  horizontal  line  in  the  middle  it  is  easily  seen  that  the  number  4132 
is  represented.  If  we  wish  to  multiply  this  by  2,  we  may  simply  double  the  ob- 
jects (in  this  case  the  black  dots)  below  the  line,  and  the  result  is  evidently  8264 


160  THE  ABACUS 

The  Abacus  in  Egypt.  That  the  Egyptians  used  an  abacus  is 
known  on  the  testimony  of  Herodotus,  who  says  that  they 
"  write  their  characters  and  reckon  with  pebbles,  bringing  the 
hand  from  right  to  left,  while  the  Greeks  go  from  left  to  right." 
This  right-to-left  order  was  that  of  the  Hieratic  script,  the 
writing  of  the  priestly  caste,  and  in  this  respect  there  is  prob- 
ably some  relation  between  this  script  and  the  abacus.1  No 
wall  pictures  thus  far  discovered  give  any  evidence  of  the  use 
of  the  abacus,  but  in  any  collection  of  Egyptian  antiquities 
there  may  be  found  disks  of  various  sizes  which  may  have  been 
used  as  counters.2 

The  Abacus  in  Babylonia.  We  have  as  yet  no  direct  evidence 
of  a  Babylonian  abacus.  The  probabilities  are,  however,  that 
the  Babylonians,  like  their  neighbors,  made  use  of  it.  Methods 
of  computing  were  never  chiefly  confined  to  the  learned  class 
whose  written  records  have  survived.  It  was  the  trader  first 
of  all  who  used  the  abacus,  and  it  was  he  who  carried  the  cus- 
toms and  manners  from  country  to  country.  Tradition  not  in- 
frequently assigns  the  origin  of  the  abacus  to  the  Middle  East, 
as  in  the  writings  of  lamblichus  (c.  325),  who  not  only  states 
that  Pythagoras  introduced  the  instrument  into  Greece,  but 
hints  that  he  may  have  brought  knowledge  of  this  kind  from 
Babylon.3  The  tradition  that  the  primitive  home  of  the  abacus 
was  in  or  near  Babylon  is  also  recorded  by  Radulph  of  Laon 
(c.  ii25)4  and  other  writers  who  had  no  special  knowledge  of 
the  subject. 

1On  the  Egyptian  abacus  see  M.  Cantor,  Geschichte,  I,  chap,  i;  J.  P.  Mahaffy, 
Old  Greek  Education,  p.  56  (New  York,  1882),  derives  the  Greek  abacus  from 
Egypt. 

2 In  a  papyrus  of  the  time  of  Menephtah  I  (1341-1321  B.C.,  Lepsius)  is  a 
drawing  which  looks  at  first  sight  like  an  abacus  (Cantor,  Geschichte,  I  (i),5i), 
but  which  is  more  likely  a  record  of  the  delivery  of  grain.  Numerous  similar 
illustrations  are  to  be  found  in  collections  of  Egyptian  antiquities,  as  in  the 
Archeological  Museum  at  Florence  (Egyptian  coll.,  2631  and  2652). 

*De  Vita  Pythagorae,  cap.  v,  §  22.  "Primo  itaque  ilium  in  arithmeticam  et 
geometriam  introduxit,  demonstrationibus  in  abaco  propositis.  .  .  ."  For  further 
evidence  as  to  the  Babylonian  origin  see  Volume  I,  page  40. 

4"Et  quum  instrumenti  hujus  Assirii  inventores  fuisse  perhibeantur."  From 
a  MS.  in  Paris,  transcribed  by  F.  Woepcke,  Journal  Asiatique,  I  (6),  48  n. 


EGYPT,  BABYLONIA,  AND  GREECE 


161 


The  Abacus  in  Greece.  The  abacus1  and  the  counters2  are 
mentioned  several  times  in  Greek  literature.  It  is  possible  that 
one  of  the  pictures  on  the  so-called  Darius  vase  in  the  Museum 
at  Naples  is  intended  to 
represent  such  an  instru- 
ment, although  what  vari- 
ous writers  have  stated  to 
be  an  abacus  may  be 
merely  the  table  of  the 
receiver  of  tribute.  In  the 
lowest  line  of  figures  in 
the  illustration  the  king's 
treasurer  may  be  seen  as 
the  figure  next  to  the  last 
one  on  the  left.  The 
other  figures  represent  the 
bearers  of  tribute.  On  the 
table  itself  are  the  letters 
MY H APO<T,  which  are  the 
ordinary  numerals  repre- 
senting ten  thousands, 
thousands,  hundreds,  tens, 
and  fives,  together  with 
the  symbols  for  the  obol, 
half  obol,  and  quarter  obol. 
These  symbols  resemble 
those  on  the  Salamis 
abacus  mentioned  below. 
The  receiver  of  tribute 
holds  a  diptych,  or  two- 
leaved  wax  tablet,  in  his  hand.  Upon  this  tablet  are  the  letters 
TAAATA:H,  which  seem  to  stand  for  TaXa(z>)ra  e(/earoV) 
(tal'anta  hekatori,  hundred  talents).  The  receiver  of  tribute 
seems  to  be  casting  something  on  the  table,  the  picture  refers 
to  the  Persian  wars  of  the  time  of  Darius,  these  wars  took  place 
about  500  B.C.,  and  coins  were  then  known ;  hence  he  may  have 


THE   DARIUS   VASE 

The  collector  of  tribute  mentioned  in  the 
text  is  the  figure  next  to  the  left-hand  one 
in  the  lowest  row.  He  has  a  tablet  in  one 
hand,  and  there  is  a  table  in  front.  From 
the  Museum  at  Naples 


L*A|3a£,  dpdiciov  (a'bax,  aba'kion). 


2*7/001  (pse'foi). 


1 62  THE  ABACUS 

been  casting  either  coins  or  counters.  The  one  thing  that  leads 
to  the  belief  that  the  table  is  an  abacus  is  the  numerals,  but  there 
are  no  lines  such  as  are  found  on  the  Salamis  specimen.  The 
date  of  the  vase  itself  is  unknown,  but  the  style  shows  it  to  be 
of  the  best  Greek  period.  It  was  found  in  iSsi.1 

Salamis  Abacus.  While  there  is  some  question  as  to  the  figure 
on  the  Darius  vase,  there  seems  to  be  little  respecting  an 
abacus  found  on  the  island  of  Salamis.  It  is  of  white  marble, 
1.49  m.  long  and  0.75  m.  wide,  and  is  broken  into  two  unequal 
parts,  but  is  otherwise  well  preserved  and  is  now  in  the  Epi- 
graphical  Museum  at  Athens.2  Of  the  history  of  this  specimen 
but  little  is  known.  It  was  found  before  the  days  of  the  careful 
keeping  of  records,  and  we  are  ignorant  of  its  date  and  of  the 
exact  place  in  which  it  was  discovered.  It  may  have  been  the 
computing  table  in  the  counting  house  of  some  dealer  in  ex- 
change, and  in  some  of  its  features  it  is  not  unlike  the  tables 
used  by  bankers  in  the  Middle  Ages ;  or,  as  Kubitschek  thinks, 
it  may  have  been  used  in  some  school.  The  theory  that  it  may 
have  been  used  in  scoring  games  of  some  kind  seems  to  have 
no  substantial  foundation.  In  any  case  it  was  apparently  used 
for  the  mechanical  representation  of  numbers  by  means  of 
counters.  It  should  be  observed  that,  although  the  crosses  are 
at  intervals  of  three  spaces,  the  first  is  not  on  the  fourth  line  as 
in  the  medieval  European  abacus. 

1The  vase  is  unusually  large,  being  1.3  meters  high.  For  a  good  description 
see  A.  Baumeister,  Denkmdler  des  klassischen  Altertums,  I,  408  (Munich,  1885). 
I  have  slightly  changed  the  inscription  from  a  personal  examination  of  the  vase. 
See  Heath,  History,  I,  48;  M.  N.  Tod,  "Greek  Numeral  Notation,"  Annual  of  the 
British  School  at  Athens,  XVIII,  124. 

2 A  description  was  first  published  by  Rangabe  in  the  Revue  Archeologique, 
III,  295  seq.,  with  a  comment  by  A.  J.  H.  Vincent,  p.  401.  Until  1899  all  repro- 
ductions of  the  stone  seem  to  have  been  derived  from  the  drawing  in  Rangabe's 
article.  In  that  year  Dr.  Nagl  (Zeitschrijt  (HI.  Abt),  IX,  337-357,  and  plate) 
published  an  illustration  of  the  abacus  under  the  mistaken  impression  that  it  was 
different  from  Rangabe's  specimen.  In  the  same  year  W.  Kubitschek  set  forth  the 
facts  and  gave  a  satisfactory  photograph  in  the  Wiener  Numismatische  Zeitschrift, 
XXXI,  393-398,  Plate  XXIV.  The  author  has  had  a  cast  taken  from  the  original, 
and  from  this  the  above  description  is  made.  See  also  Harper's  Dictionary  of 
Classical  Literature  and  Antiquities,  p.  2  (New  York,  1897)  ;  hereafter  referred  to 
as  Harper's  Diet.  Class.  Lit.\  Heath,  History,  I,  49-51;  Tod,  loc.  cit.,  p.  116. 


THE   SALAMIS   ABACUS 


1 64  THE  ABACUS 

It  will  be  seen  that  the  marble  slab  is  ruled  as  usual,  so  that 
counters  could  be  placed  on  the  lines.  On  three  sides  are  Greek 
characters  substantially  as  follows : 

H  i,  drachma,  a  mutilated  form  of  E,  for  ev 

P  5,  old  form  of  TT,  for  vreVre 

A  10,  for  Se/ca 

pn  50,  for  TT  and  A,  five  tens 

H         100,  for  HEKATON,  old  form  for  e/cardv 

H  500,  for  TT  and  H,  five  hundreds 

X  1000,  for  %t\fc(u 

I  the  obol 

C  the  half  obol 

T  the  quarter  obol 

X  for  xaX/covs,  the  eighth  of  an  obol 

I*1  5000,  for  TT  and  X,  five  thousands 

T  the  talent  of  6000  drachmas. 

The  lines  at  the  top  were  for  fractions.  In  the  illustration  the 
lines  and  symbols  have  been  accentuated  for  the  sake  of  clearness. 
As  to  whether  the  Greeks  commonly  used  loose  counters  or 
not  we  can  only  infer  from  this  single  extant  specimen  of  an 
abacus,  and  possibly  from  the  Darius  vase.  The  former  and 
possibly  the  latter  lead  us  to  believe  that  the  loose  counters 
were  preferred  to  those  sliding  on  wires  or  rods.  We  do  not 
know  any  details  as  to  the  actual  methods  of  computing,  and 
in  spite  of  the  effort  of  Herodotus  to  be  clear  on  the  subject1 
we  are  uncertain  whether  the  rows  were  horizontal  or  vertical 
with  respect  to  the  computer.2  It  seems  probable  that  the  Greeks 
made  less  use  of  the  abacus  than  the  Romans,  the  Greek  nu- 
merals being  better  adapted  to  the  purposes  of  computation, 
particularly  of  multiplication  and  division.3 

1  Liber  II,  cap.  36. 

2H.  Weissenborn,  Zur  Geschichte  der  Einfiihrung  der  jetzigen  Ziffern,  p.  2 
(Berlin,  1892),  and  authorities  cited. 

3J.  G.  Smyly,  "The  employment  of  the  alphabet  in  Greek  logistic,"  in  the 
Melanges  Jules  Nicole,  p.  521  (Geneva,  1905)  ;  H.  Suter,  Geschichte  der  math. 
Wissenschaften,  2d  ed.,  I,  n  (Zurich,  1873) ;  Heath,  History,  I,  52. 


GREECE  AND  ROME  165 

The  Abacus  in  Rome.  There  were  at  least  three  forms  of  aba- 
cus used  by  the  Romans, — a  grooved  table  with  beads,  a 
marked  table  for  counters,  and  the  primitive  dust  board.1  In 
respect  to  each  of  these  forms  Latin  writers  give  us  consider- 
able information.  Horace,  for  example,  speaks  of  the  school- 
boy with  his  bag  and  table  hung  upon  his  left  arm,  the  table 
referring  to  the  abacus  or  the  wax  tablet;2  and  Juvenal  men- 
tions both  the  table  and  the  counters.3  Cicero  refers  to  counters 
when  he  speaks  of  the  aera  (bronzes),  the  computing  pieces  be- 
ing then  made  of  bronze,4  and  Lucilius  the  satirist,  who  lived  a 
generation  earlier,  does  the  same.5  The  common  name  for  these 
counters  was,  however,  calculi  or  abaculi,  and  the  material  from 
which  they  were  made  was  originally  stone  and  later  ivory  and 
colored  glass.0  The  word  calculus  means  "pebble"  and  is  the 

1  S.  Hotiel,  in  his  review  of  Friedlein's  "  Die  Zahlzeichen,"  in  Boncompagni's 
Bullettino,  III,  78;  Glinther,  Math.  Unterrichts,  p.  95  n.;  A.  J.  H.  Vincent,  Re- 
vue Archeologique,  III,  401 ;  A.  Kuckuck,  Die  Rechenkunst  im  sechzehnten 
Jahrhundert,  p.  6  (Berlin,  1874).  Although  we  have  numerous  references  to  the 
use  of  loose  counters,  it  is  curious  that  no  ancient  writer  speaks  definitely  of  the 
ruled  table  on  which  they  are  used  ;  see  Gerhardt,  Etudes,  p.  16.  On  the  abacus 
as  a  gaming  table,  particularly  for  dice,  sec  G.  Oppert,  On  the  original  inhabit- 
ants of  Bharatavarsa  or  India,  p.  329  (London,  1893)  >  W.  Ramsay  and  R.  Lan- 
ciani,  Manual  of  Roman  Antiquities,  i7th  ed.,  p.  497  (London,  1901)  (hereafter 
referred  to  as  Ramsay  and  Lanciani) .  For  a  bibliography  and  description  of  the 
Kvfioi  (ku'boi)  and  tesserae  see  J.  Marquardt,  La  vie  privee  des  Remains, 
French  translation,  II,  522  (Paris,  1893). 

2Laevo  suspensi  loculos  tabulamque  lacerto. 

Sat.,  I,  6,  74 

3Computat  .  .  .  ponatur  calculus,  adsint 
Cum  tabula  pueri ;  numera  sestertia  quinque 
Omnibus  in  rebus,  numerentur  deinde  labores. 

Satire  IX,  40 

4"Si  aera  singula  probasti."   Philosoph.  Fragmenta,  V,  59. 
6  Hoc  est  ratio  ?  perversa  aera,  summa  est  subducta  improbe  ! 

L.  886,  ed.  Marx  ;  1.  740,  ed.  Lachmann 
GAdeo  nulla  uncia  nobis 
Est  eboris,  nee  tessellae  nee  calculus  ex  hac 
Materia.  Juvenal,  XI,  131 

Fragmenta  teporata  .  .  .  fundi  non  queunt  praeterquam  abrupta  sibimet  in 
guttas,  veluti  cum  calculi  fiunt,  quos  quidem  abaculos  appellant  aliquos  et  pluri- 
bus  modis  versicolores. —  Pliny,  Hist.  Nat.,  XXXVl,  26,  67 

Capitolinus  (Pertinax,  I,  4),  speaking  of  the  boyhood  of  Pertinax  (126-193), 
says :  "  Puer  litteris  elementariis  et  calculo  imbutus." 

Martial  (II,  48)  includes  among  his  modest  wants  "tabulamque  calculosque." 
II 


1 66  THE  ABACUS 

diminutive  of  calx,  a  piece  of  limestone  (often  referring  to  the 
special  form  of  chalk,  the  name  of  which  comes  from  the  same 
root).  It  is  therefore  our  word  "marble"  as  applied  to  the 
small  spheres  with  which  children  play  games.  From  it  came 
the  late  Latin  calculare?  to  calculate.  Teachers  of  calculation 
were  known  as  calculones  if  slaves,  but  calculator es  or  numeraril 
if  of  good  family.2  To  calculate  means  literally,  therefore,  to 
pebble,  and  a  calculator  is  a  pebbler.  The  word  calculi  was 
transmitted  by  the  Romans  to  medieval  Europe  and  was  in 
common  use  until  the  i6th  century.3 

We  are  not  sure  whether  the  small  disks  found  in  Roman  re- 
mains were  counters  for  purposes  of  calculation,  counters  for 
games  (like  American  poker  chips),  or  draughts.  The  games 
of  backgammon  and  draughts  are  both  very  old,4  and  the  for- 
mer is  our  nearest  approach,  aside  from  such  abaci  as  we  still 
use,  to  the  Roman  and  medieval  abacus.5 

The  abacus  in  which  the  beads  were  allowed  to  slide  in 
grooves  or  on  rods  is  not  mentioned  by  any  early  writer  and 
seems  to  have  been  of  relatively  late  invention.  Indeed,  in  the 
1 5th  and  i6th  centuries  it  was  commonly  asserted  that  Ap- 

irrhe  Romans  used  calculos  subducere  instead  of  calculare.  This  word,  in  the 
sense  of  "to  calculate,"  is  first  found  in  the  works  of  the  poet  Aurelius  Clemens 
Prudentius,  who  lived  in  Spain  £.400;  see  Nouvelles  Annales  de  Math.,  XVII, 
supplementary  bulletin,  p.  33. 

-Tertullian,  evidently  with  reference  to  the  dust  abacus,  calls  them  "primi 
numerorum  arenarii." 

3Thus  Clichtoveus,  in  his  arithmetic  of  1503  (1507  ed.,  fol.  b,  iij,  v.),  says: 
"Numeratio  calcularis  est  cuiusq^  numeri  suo  loco  et  limite  apta  per  calculos 
dispositio";  and  Noviomagus  (1539,  fol.  9,  r.)  says:  "  Ut  detur  autem  hac  forma 
in  calculis  seu  ut  nunc  fit  nummis." 

4 They  appear  in  various  Egyptian,  Greek,  and  Roman  remains.  For  example, 
in  the  British  Museum  is  an  ancient  model  of  an  Egyptian  barge  on  which  a 
game  of  draughts  is  in  progress,  and  A.  Baumeister  (Denkmaler  des  klassischen 
Altertums,  I,  354  (Munich,  1885))  has  reproduced  an  illustration  of  a  similar 
game  from  an  old  Greek  terracotta.  There  were  two  Roman  games,  the  ludus 
latrunculorum  and  Indus  duodecim  scriptorum,  on  which  pieces  called  calculi 
were  used,  but  their  exact  nature  is  unknown.  See  Ramsay  and  Lanciani,  p.  498  ; 
J.  Marquardt,  La  vie  privee  des  Romains,  French  translation,  II,  530  (Paris, 
1893);  Harper's  Diet.  Class*  Lit.,  p.  562;  J.  Bowring,  The  Decimal  System, 
p.  198  (London,  1854). 

5  It  was  probably  the  ludus  duodecim  scriptorum  already  mentioned,  or  the 
s  (diagr  animism  os')t  the  late  rd/SXa  (ta'bla) ,  of  the  Greeks. 


ROMAN  TYPES 


167 


puleius  invented  this  form  of  the  instrument  in  the  2d  cen- 
tury,1— a  statement  for  which  there  is  no  standard  authority. 
Our  knowledge  of  the  grooved  abacus  is  derived  from  a  few 
specimens  of  uncertain  date  which  have  come  down  to  modern 
times.  One  of  these,  formerly  owned  by  Marcus  Welser  of 
Augsburg,  was  made  of  metal,  is  said  to  have  been  4.2  cm.  long 


ROMAN   ABACUS 
Ancient  bronze  abacus  of  uncertain  date,  now  in  the  British  Museum 

and  3.5  cm.  wide,  and  had  nineteen  grooves  and  forty-five 
counters  or  buttons  (calculi)2.  Another  was  once  owned  by  the 
reformer  Ursinus  (c.  1575),  but  is  now  lost.  A  third  specimen, 
of  bronze,  is  now  in  the  Kircherian  Museum  at  Rome.  The  gen- 
eral plan  of  the  Roman  abacus  may  be  seen  from  the  illustration 
here  given,  representing  a  specimen  in  the  British  Museum. 

The  symbols  found  on  such  specimens  as  are  extant  are  usu- 
ally the  common  Roman  numerals  from  1,000,000  down  to  i, 

iUnger,  Die  Methodik,  p.  69. 

2  This  was  twice  described  before  it  was  lost,  once  in  Amsterdam  in  1674  and 
once  in  Niirnberg  in  1682.  The  measurements  are  questionable.  See  G.  A.  Saal- 
feld,  "Der  griechische  Einfluss  auf  Erziehung  und  Unterricht  in  Rom,"  Neue 
Jahrbiicher  fitr  Philologie,  CXXVI,  371- 


1 68  THE  ABACUS 

together  with  o  (or  ff)  for  uncia,  or  ^  of  the  as ;  S  for  semiuncia ; 
D  for  the  sicilicuSj  or  |  uncia]  and  Z  for  the  duella,  or  ^  uncia. 

The  Abacus  in  China.  At  the  present  time  the  use  of  the  aba- 
cus is  universal  in  China.  In  banks,  shops,  and  counting  houses 
of  all  kinds  the  computations  are  performed  on  the  suan-pan.1 
The  computer  works  very  rapidly,  like  an  expert  typist  or 
pianist,  and  secures  his  results  much  more  quickly  than  can  be 
done  by  our  common  Western  methods.*  He  learns  its  use  by 
practical  experience  in  business,  probably  as  the  Romans  and 
Greeks  learned  it,  and  not  in  the  village  schools.3 

The  suan-pan  is,  however,  a  relatively  late  development  of 
the  abacus  in  China,  appearing  first,  so  far  as  we  know  at  the 
present  time,  in  the  i2th  century.4  It  is  true  that  many  writers5 
have  placed  its  introduction  much  earlier,  but  there  is  no  defi- 
nite description  of  the  instrument  in  Chinese  before  about  1 175.° 

1The  term  means  computing  plate  or  computing  tray,  often  incorrectly  trans- 
lated as  computing  board.  It  is  also  called  the  su-pan,  and  there  are  other  variants. 
It  is  called  suinbon  in  Calcutta,  where  it  is  used  by  all  the  Chinese  shroffs  (com- 
puters, accountants,  cashiers)  in  the  counting  houses.  The  common  spelling  is 
suan-pan,  swan  p'an,  or  swan  pan.  The  instrument  is  also  in  common  use  in 
Siam  and  wherever  Chinese  merchants  have  determined  business  customs. 

2 See  Knott,  Abacus,  p.  44;  J.  D.  Bell,  Things  Chinese,  p.  i  (New  York, 
1904)  ;  J.  Goschkewitsch,  "Ueber  das  chinesische  Rechnenbrett,"  Arbeiten  der 
kaiserlich  Russischen  Gesandtschajt  zu  Peking,  I,  293  (Berlin,  1858)  ;  Smith- 
Mikami;  R.  van  Name,  "On  the  Abacus  of  China  and  Japan,"  Journal  of  the 
Amer.  Orient.  Soc.,  X  (Proceedings),  p.  ex;  J.  Bowring,  The  Decimal  System, 
p.  193  (London,  1854). 

3 A.  H.  Smith,  Village  Life  in  China,  p.  105  (New  York,  1899). 

4  One  of  the  most  scholarly  articles  on  the  history  of  the  suan-pan  is  the  one 
already  cited,  by  Lacouperie,  The  Old  Numerals,  pp.  297-340.    It  contains  an 
excellent  bibliography  of  the  subject  up  to  1883. 

5  J.  Hager,  An  Explanation  of  the  Elementary  Characters  of  the  Chinese,  p.  x 
(London,  1801) ;  H.  Cordier,  Bibliotheca  Sinica,  col.  509   (Paris,  1881-1895)  ; 
L.  Rodet,  "Le  Souan-pan  et  la  Banque  des  Argentiers,"  Bulletin  de  la  Societe 
Mathematique  de  France,  Vol.  VIII  (Paris,  1880).    Chinese  writers  record  that 
a  work,  Su-shuh  ki-i,  "Anecdotes  of  mathematics,"  written  about  200,  mentions 
various  methods  of  computing,  including  "bead  computation"  and  "hand  com- 
putation," but  the  work  gives  no  description  of  any  process.   See  also  an  inter- 
esting early  essay,  Smethurst,  "Account  of  the  shwan  pan"  Phil.  Trans.,  XLVI 
(1749),  22. 

6  This  occurs  in  two  works,  the  Pan  chu  tsih  and  the  Tseu  pan  tsih,  which 
appeared  in  the  Shun-hi  dynasty,  1174-1190.   They  describe  the  pan,  or  tray,  the 
word  suan-pan  not  being  then  in  use.    Indeed,  as  late  as  the  i6th  century  the 
name  pan  shih  (board  to  measure)  was  used.   See  Lacouperie,  p.  38. 


CHINESE  SUAN-PAN 


169 


As  to  the  origin  of  the  Chinese  abacus,  the  evidence  seems  to 
point  to  Central  or  Western  Asia.  At  the  time  of  its  appear- 
ance, China  was  largely  under  the  domination  of  the  Tangut  or 
Ho-si  state  and  of  the  Liao  and  Kin  Tartars.  The  Tangutans 
were  a  mercantile  race,  and  the  Tartars  were  favorable  to 
learning.  Moreover,  Arab  and  Persian  traders  are  known  to 


MODERN  CHINESE  ABACUS 
The  man-pan,  known  to  have  been  used  as  early  as  the  i2th  century 

have  been  in  Canton  in  the  8th  century,  and  the  Nestorians 
were  in  contact  with  the  northwest,  so  there  was  plenty  of  op- 
portunity for  such  a  simple  device  to  make  its  way  into  China 
from  Khorasan  or  some  neighboring  province.  The  fact  that 
it  seems  to  have  reached  Russia  from  Central  Asia1  adds  to  the 
belief  that  China  may  have  received  it  from  the  same  source.2 
Before  the  time  of  the  man-pan  the  counting  rods,  often 
called  the  bamboo  rods,  had  been  used  for  more  than  a  thou- 
sand years.  They  were  known  c.  542  B.C.8  and  are  referred  to 
as  counting  stalks  in  a  statement  of  Hiao-tze,  the  ruler  of  Ts'in 
from  361  to  337  B.C.  They  are  mentioned  again  about  215  B.C., 
and  some  specimens  of  this  period  were  displayed  in  a  museum 

iLarousse,  Grand  Dictionnaire  Universel,  I,  636;  Vissiere,  Abacque;  A.  Wylie, 
Notes  on  Chinese  Literature,  p.  91  (Shanghai,  1902). 

2Lacouperie,  The  Old  Numerals,  p.  41.  3See  Volume  I,  page  96. 


170  THE  ABACUS 

of  the  Emperor  Ngan  (397-419).  These  were  about  18  inches 
long,  some  made  of  bone  and  others  of  horn.  In  the  reign  of 
Wu-ti  (140-87  B.C.)  of  the  Han  dynasty,  it  is  related  that  an 
astronomer  Sang  Hung  (about  118  B.C.)  was  very  skillful  in 
his  use  of  the  rods.  In  the  third  century  of  our  era  it  is  re- 
corded that  Wang  Jung,  a  minister  of  state,  spent  his  nights 
in  reckoning  his  income  with  ivory  calculating  rods,  and  the 
expression  "to  reckon  with  ivory  rods"  is  still  used  as  an  allu- 
sion to  wealth.  In  the  time  of  the  Emperor  Ch'eng  (326-343) 
the  counting  rods  were  made  of  wood,  ivory,  or  iron,  and  two 
centuries  later  the  Emperor  Siuen  Wu  (500-516)  had  counting 
rods  cast  in  iron  for  the  use  of  his  people.1 

The  Chinese  historian  Mei  Wen-ting  (1633-1721),  in  his 

work  on  ancient  calculating  instruments,2  states  that  about  the 

beginning  of  the  Christian  era  271  rods  constituted  a  set,  or 

handful,  and  that  they  formed  a  hexagon  that  had  nine  rods 

on  a  side.    This  means  that  they  were  arranged  in  six  groups 

of  which  the  ends  of  each  formed  a  triangular  number  of 

i  -h  2  -f  •  •  •  4-  9  units,  or  45  in  all.      Six  of  these  make  6  x  45, 

or  270,  and  these  six  were  grouped  about 

•  •  one  central  rod,  making  271,  thus  afford- 

•  •  •  ing  an  illustration  of  the  use  of  figurate 

*  *  *  *  numbers  in  the  East. 

•  •••••  It  seems  from  Mei  Wen-ting's  work 

•  ••••••         that  the  rods  were  in  general  use  until 

«\\\\%\\\     the  i3th  century.    With  respect  to  the 
suan-pan,  he  places  the  date  somewhat 

later  than  other  writers,  saying,  "If  in  my  ignorance  I  may  be 
allowed  to  hazard  a  guess,  I  should  say  that  it  began  with  the 
first  years  of  the  Ming  dynasty,"  which  would  make  the  date 
about  1368.  Subsequent  writers  are  probably  correct,  however, 
in  placing  it  a  century  or  two  earlier. 

The  Abacus  in  Japan.  The  primitive  method  of  computing  in 
Japan  is  quite  unknown,  but  from  the  time  of  the  Empress 
Suiko  (593-628)  the  bamboo  rods  (chikusaku)  were  used.3 

1  Lacouperie,  The  Old  Numerals,  pp.  34-36  of  reprint. 

2Ku-suan-k'i-k'ao.  3  Smith-Mikami,  chap,  iii,  with  bibliography. 


JAPAN  AND  KOREA 


171 


These  were  round  sticks  about  2  mm.  in  diameter  and  12  cm. 
in  length,  but  because  of  their  liability  to  roll  they  were  in  due 
time  replaced  by  the  sanchu  or  sangi,  rectangular  prisms  about 
7  mm.  thick  and  5  cm.  long.  The  soroban,  the  name  being 
probably  the  Japanese  rendering  of  the  Chinese  word  suan-pan, 
was  developed  but  not  generally  adopted  in  the  i6th  century.1 


I !__.___ 


1- 


ft 


1 


THE  SANGI  BOARD  IN  JAPAN 

Intended  for  computation  with  the  sangi  (rods).    From  Sato  Shigeharu's  Tengen 

Shin  an  t  1698 

What  may  prove  to  be  a  relic  of  a  very  early  Japanese  sys- 
tem is  seen  in  the  tally  sticks  used  in  the  Luchu  (Liu  Kiu,  Riu 
Kiu)  Islands,  near  Formosa,  and  known  as  Sho-Chu-Ma? 

The  Abacus  in  Korea.  The  bamboo  rods  of  China  passed  over 
to  Japan  by  way  of  Korea,  and  in  the  latter  country  they  re- 

^mith-Mikami,  p.  IQ. 

2B.  H.  Chamberlain,  Journal  of  the  Anthropological  Institute  of  Great  Britain 
and  Ireland,  XXVII,  383.  For  a  brief  mention  of  these  tallies  see  the  Geographi- 
cal Journal,  June,  1895. 


172 


THE  ABACUS 


mained  in  use  long  after  they  were  abandoned  elsewhere.    The 
commercial  class  was  acquainted  with  the  suan-pan  for  a  long 


i 
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— 

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II 

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—    ^ 


SANGI  BOARD  WITH  NUMBERS  INDICATED 


From  Nishiwaki  RichyU's  Sampd  Tengen  Roku,  1714.    The  sangi  board  was  a 
board  ruled  as  shown,  the  sangi  being  placed  in  the  rectangles 


time  before  the  Japanese  conquest,  and  now  the  soroban  is 
common  among  the  officials.    But  most  of  those  who  were  edu- 


JAPANESE  SOROBAN 


173 


cated  in  the  native  schools  used  the  counting  sticks  until  recent 
times,  while  those  with  but  little  education  performed  their 


^•^'l^^M  iV  i I  'Nv^/t/^Fl 
;<^^^lr  ^u!j3|  #K^ 


THE  SANGI  BOARD  IN  USE 
From  Miyake  Kenryu's  Shojutsu  Sangaku  Zuye,  1716  (1795  ed.) 

simple  computations  mentally  or  on  their  fingers.  The  count- 
ing sticks  (Ka-tji  san)  were  of  bone,  as  in  the  illustration  on 
page  174,  or  of  bamboo  split  into  long  prisms.  About  a  hun- 


JAPANESE  ABACUS 

The  soroban,  known  to  have  been  used  in  Japan  as  early  as  the  i6th  century, 
and  in  universal  use  there  at  present 

dred  fifty  were  used  in  ordinary  calculation,  and  these  were 
kept  in  a  bamboo  case  on  the  computer's  desk.  The  sticks  were 
laid  as  follows  to  represent  the  first  twelve  numbers : 

I      II      III      Illl      X     X\      XII      XIII      Xllll     -     T     TT 
123456         7          8  9         10     ii      12 

In  computing,  the  Koreans  used  the  rods  in  substantially  the 
same  way  as  the  Chinese  and  Japanese  had  used  theirs.    The 


THE  ABACUS 


process  was  so  cumbersome  that  it  has  recently  given  way  to 
the  Chinese  and  Japanese  methods  with  the  suan-pan  and  the 

soroban}  The  Koreans  also 
used  pebbles  and  coins  for 
the  same  purpose.2 

The  Abacus  among  the 
Mohammedans.  The  Arabs, 
Persians,  Armenians,  and 
Turks  have  a  form  of  abacus 
which  differs  from  that  of 
the  Far  East  and  from  the 
one  used  by  the  Romans, 
having  ten  beads  on  each 
line.  Its  early  history  is 
unknown,  but  since  it  re- 
sembles neither  the  abacus 
of  China  nor  that  of  Western 
Europe,  it  probably  origi- 
nated among  the  Arab  or 
Persian  computers.  The 
Turks  call  it  the  coulba  and 
the  Armenians  the  choreb* 

This  form  of  the  abacus 
does  not  seem  to  have  been 
generally  used  by  the  Sara- 


KOREAN   COMPUTING   RODS 


cens  in  the  Middle  Ages.    In 


Computing  rods  made  of  bone.    Until 

quite   recently   these   were   used   in   the 

schools  of   Korea.    The  numbers  were 

represented  as  shown  on  page  173 

50n  the  mathematics  of  Korea  in  general,  see  P.  Lowell,  The  Land  of  the 
Morning  Calm,  p.  250  (Boston,  1886).  For  the  Song  yang  hoei  soan  fa  or  Song 
yang  hold  san  pep  (Treatise  on  Arithmetic  of  Yang  Hoei  of  the  Song  Dynasty), 
which  was  for  a  long  time  a  classic,  see  M.  Courant,  Bibliographic  Coreennet 
III,  i  (Paris,  1896).  See  also  the  Grammaire  Coreenne,  p.  44  (Yokohama,  1881), 
in  which  the  description  of  the  laying  of  the  sticks  recalls  the  Japanese  method 
and  differs  from  the  one  shown  on  page  173,  which  was  given  to  the  author  by  an 
educated  Korean  in  Peking.  '2  Grammaire  Coreenne,  loc.  tit. 

3Pacioli  speaks  of  this  form  of  the  abacus  when  he  says  that  the  orders  of 
numbers  increase  from  right  to  left  "more  arabu  de  simil  arte  pratica  primi 
inuetori  secodo  alcuni  vnde  p  ignoratia  et  vulgo  a  corropto  el  vocabulo  dicedo 
la  Abaco:  cioe  modo  arabico.  Che  loperare  suo  e  modo  arabico  e  chiamase 
Abaco:  ouer  secodo  altri  e  dicta  Abaco  dal  greco  vocabulo."  Suma,  fol.  19,  r. 
(Venice,  1494). 


MOHAMMEDAN  AND  RUSSIAN  TYPES  175 

that  period  the  dust  board  was  common  and  the  numeral  forms 
derived  from  being  written  on  such  a  tablet  were  therefore,  as 
already  stated,  called  in  the  schools  of  the  western  Arabs  the 
gobdr  (dust)  numerals.1  Thus  the  Moorish  writer  al-Qalasadi 
(c.  1475),  m  his  commentary  on  the  Talchis  of  Albanna  (c. 
1300),  speaks  of  "a  man  of  the  Indian  nation  who  took  fine 
powder  and  sprinkled  it  on  a  table  and  marked  on  it  the  multipli- 
cations, divisions,  or  other  operations,  and  this  is  the  origin  of 
the  term  gobdr"  (dust).2  Further  evidence  of  the  rarity  of  any* 
other  form  of  the  abacus  among  the  Saracens  in  the  Middle  Ages 
is  to  be  found  in  the  silence  of  Maximus  Planudes  (c.  1340) 
upon  the  subject ;  for  the  contact  with  the  East  of  one  writing 
upon  arithmetic  in  Constantinople  would  almost  certainly  have 
led  him  to  speak  of  the  bead  abacus  if  it  had  been  in  common 
use  among  the  Arabs  of  his  time.  It  may  be,  however,  that  the 
dust  abacus  was  used  in  some  parts  of  the  Mohammedan 
domain,  and  the  bead  abacus  in  other  parts,  the  latter  giving  to 
Christian  nations  the  line  abacus.  Some  reason  for  this  belief 
is  found  in  the  fact  that  certain  medieval  writers  derived  the 
word  " abacus"  from  the  Arabic,3  while  William  of  Malmes- 
bury,  although  by  no  means  a  reliable  chronicler,  writing  in  the 
1 2th  century,  says  that  Gerbert  (c.  1000)  obtained  his  idea  of 
the  instrument  from  the  Saracens.4  There  is  also  a  possible 
reference  to  the  line  abacus  by  Alchindi5  (c.  860). 

The  Abacus  in  Russia.  From  the  Mohammedan  countries  the 
bead  abacus  worked  its  way  northward,  and  in  comparatively 

JH.  Wcissenborn,  Gerbert,  p.  235  (Berlin,  1888) ;  Zur  Geschichte  der  Einfuh- 
rung  der  jetzigen  Ziffern,  p.  7  (Berlin,  1892)  ;  Smith-Karpinski,  p.  65. 

2  From  an  Arabic  MS.  in  Paris,  described  by  F.  Woepcke,  Journal  Asiatique, 
I  (6),  60. 

3 Thus  a  1 2th  century  MS.,  Regulae  abaci,  published  by  M.  Chasles  in  the 
Comptes  rendus  for  1843  (XVI,  218),  asserts,  "Ars  ista  vocatur  abacus:  hoc 
nomen  vero  arabicum  est  et  sonat  mensa." 

4  "Abacam  certe  primus  a  Sarazenis  rapiens,  regulas  dedit,  quae  a  sudantibus 
abacistis  vix  intclligentur."  On  the  unreliability  of  this  chronicler,  see  H.  Weissen- 
born,  Gerbert,  p.  236  (Berlin,  1888). 

5  The  reference  is  in  the  chapter  "De  numeris  per  lineas  &  grana  hordeacea 
multiplicandis  Liber  I"  of  the  Latin  translation  of  his  arithmetic.    See  H.  Weis- 
senborn,  Zur  Geschichte  der  Einfuhrung  der  jetzigen  Ziffern,  p.  7. 


176 


THE  ABACUS 


100,000 


10,000 


1,000 


recent  times  was  adopted  in  Russia.  It  is  still  found  in  every 
school,  shop,  and  bank  of  Russia  proper,  although  in  the  for- 
mer provinces  of  Finland  and  Poland  it  is  seldom  used.  The 

computers  handle  it 
with  much  the  same 
ease  as  the  Chinese 
show  in  their  use  of 
the  suan-pan,  and 
there  seems  to  be 
no  reason  why  they 
should  not  continue 
to  use  it  until  it  is 
replaced  by  more 
elaborate  calculating 
machines.  The  Rus- 
sians call  their  aba- 
cus the  s'choty,1  and 
the  form  is  the  same 
as  that  of  the  Arme- 
nian choreh  or  the 
Turkish coulba.  They 
occasionally  speak  of 
it  as  the  Chinese  aba- 
cus, so  that  there  is 
this  ground  for  the 

The  s'choty  of  the  Russians.   It  is  of  the  same  form        i    •       f|1qt   -f  :n 

as  the  Armenian  chorcb  and  the  Turkish  coulba        Claim  mal  ll  WdS  .m" 

troduced  from  China 

by  way  of  Siberia,  although  the  form  of  the  instrument  would 
go  to  show  that  it  came  from  the  South. 

In  the  1 6th  century  the  German  form  of  line  reckoning  was 
used  in  Poland,2  and  when  this  disappeared  it  was  not  replaced 
by  the  Russian  abacus  but  by  the  algorism  of  Western  Europe. 

1  Variously  transliterated,  but  this  form  gives  the  pronunciation  more  nearly 
than  the  others. 

2  The  third  arithmetic  printed  in  Poland,  but  the  first  in  the  Polish  language, 
is  that  of  Klos  (1538).   It  is  devoted  almost  entirely  to  this  kind  of  computation. 
See  the  Baraniecki  reprint  (Cracow,  1889)  and  Dickstein's  article  in  Bibl.  Math., 
IV  (2),  57. 


i  ruble 


ruble 


10  kopeks 
i  kopek 
\  kopek 


RUSSIAN  ABACUS 


THE  DUST  TABLE  177 

The  Abacus  in  Western  Europe.  In  medieval  times  in  Western 
Europe  the  abacus  had  various  names  and  forms.  The  fol- 
lowers of  Boethius  (c.  510)  called  it  the  Pythagorean  table 
(mensa  Pythagorica],  a  name  also  given  to  the  square  array  of 
the  multiplication  table.1  It  was  also  known  as  the  geometric 
table  (tabula  geometricalis,  mensa  geometrically},  table  of  the 
abacus  (tabula  abaci},  and  Pythagorean  arc  (arcus  Pythago- 
reus),  although  abax  or  abacus  was  the  common  medieval 
name."  So  common  was  this  name  that  the  verb  "to  abacus" 
became  recognized/"'  and  the  arithmeticians  of  about  the  nth 
century  and  later  were  occasionally  called  abacists. 

The  Dust  Table.  Of  the  various  forms  of  abacus  used  in 
Europe,  the  dust  table,  already  described  as  known  in  the 
Orient  and  in  classical  times,  was  one.  We  have  evidence  of  its 
use  at  the  close  of  the  9th  century,  when  Remigius  of  Auxerre 
(c.  900),  in  his  commentary  on  Capella's  arithmetic,  speaks  of 
the  table  as  being  sprinkled  with  blue  or  green  sand  and  the 

1  Cantor  called  attention  to  this  distinction  in  his  Mathematische  Beitrage 
zum  Kulturleben  der  Volker,  Halle,  1863,  p.  204  (hereafter  referred  to  as  Cantor, 
Beitrdge) ;  and  Enestrom  did  the  same  in  the  Bibl.  Math.,  I  (2),  90.  Adelard  of 
Bath  (c.  1 1 20),  in  his  Regulae  Abaci,  says:  ".  .  .  quidem  mcnsam  pithagoream 
ob  magistri  sui  rcuerentiam,  sed  post!  tame  abacum  dixerunt"  (Boncompagni's 
Bullettino,  XIV,  68).  In  this  he  apparently  had  in  mind  a  passage  in  the 
Ars  Geometria  of  Boethius  (ed.  Fricdlcin,  p.  396):  "Pythagorici  vero  .  .  .  de- 
scripserunt  sibi  quandam  formulam,  quam  ob  honorcm  sui  pracceptoris  mensam 
Pythagoream  nominabant  ...  a  postcrioribus  appcllabatur  abacus."  Adelard 
even  goes  so  far  as  to  assert,  with  no  foundation  except  tradition,  that  the 
abacus  is  due  to  Pythagoras  himself:  "Pythagorici  vero  hoc  opus  [abacum] 
composuerunt/ut  ea  que  magistro  suo  pitagora  doccnte  audierant.  ocul'  subiecta 
retinerent :  et  firmius  custodirent." 

2TurchilIus,  writing  on  the  abacus  about  1200,  says:  "Ab  antiquis  mensa 
pytagorica,  a  modernis  autem  uel  abax  vel  abacus  nuncupatur"  (Boncompagni's 
Bullettino,  XV,  135).  An  anonymous  MS.  of  the  i2th  century,  in  the  Vatican, 
says:  "(T)abula  abaci  qu£  pytagorea  msa  uocatur"  (ibid.,  pp.  132,  154).  See 
also  M.  Chasles,  "  Developpcments  et  details  historiques  sur  divers  points  du  sys- 
teme  de  I'abacus,"  Comptes  rendus,  XVI,  1393,  with  references  to  other  MSS. 

3 In  a  MS.  in  the  Bibliotheque  nationale  in  Paris  there  is  some  correspondence 
between  one  Radulph  of  Liege  and  Rogimbold  of  Cologne  in  the  early  part  of 
the  nth  century  in  which  the  writer  says:  "Hoc  si  abacizando  probaveris." 
In  the  same  MS.  there  is  a  letter  addressed  to  Hermannus  Contractus  (c.  1050) 
in  which  the  statement  is  made:  "Ut  meam  abicizandi  notem  inscitiam."  See 
Chasles,  loc.  cit.,  p.  1417. 


178  THE  ABACUS 

figures  as  being  drawn  with  a  radius.1  A  certain  Papias,  who 
wrote  a  Vocabularium  in  1053,  and  who  may  be  considered  as 
representing  the  knowledge  of  his  time;  also  speaks  of  the 
abacus  as  a  table  covered  with  green  sand.2 

The  Wax  Tablet.  Allied  to  the  dust  table  is  the  old  wax  tablet 
of  classical  times.  This  consisted  of  a  tablet  of  wood  or  bone 
on  which  a  thin  coat  of  black  wax  was  smeared,  the  figures  be- 
ing written  with  an  iron  stylus  of  which  one  end  was  pointed 
and  the  other  was  somewhat  spoon-shaped,  the  latter  being 
used  for  erasing  by  smoothing  the  wax  down  again.:i  This  tablet 
passed  into  the  medieval  schools  and  counting  houses,  and  spec- 
imens are  extant  which  were  in  use  as  late  as  the  i6th  century.1 

i"  Abacus  tabula  est  geometricalis  super  quam  spargebatur  puluis  uitreus  siuc 
glaucus.  Ibique  cum  radio  uirg$  formabantur  figure^  geometric/'  See  Boncom- 
pagni's  Bullettino,  XV,  572,  and  III,  84.  In  the  same  journal  (X,  625)  there 
is  a  description  of  a  medieval  MS.  of  unknown  date  in  which  the  following 
passage  appears:  "Abacus  vocatur  mensa  geometricalis  que  et  in  numcris  et 
formis  numerorum  diuisa  .  .  .  ." 

2 "Abacus  vel  abax  tabula:  in  qua  uiridi  pulue  formae  depinguntur."  From 
the  first  printed  edition  of  the  Vocabularium,  Milan,  1476,  fol.  2,  v.  See  also 
Boncompagni's  Bullettino,  XIV,  69. 

It  would  seem  that  Adelard  of  Bath  (c.  1120)  referred  to  this  form  of  abacus 
when  he  wrote:  "Vocatur  (Abacus)  ctiam  radius  geometricus,  quia  cum  ad 
multa  pertineat,  maxime  per  hoc  geometricae  subtilitatcs  nobis  illuminantur." 
Radulph  of  Laon  (c.  1125)  had  the  same  form  in  mind  in  writing  the  follow- 
ing: ".  .  .ad  arithmeticae  speculationis  investigandas  rationes,  et  ad  eos  qui 
musices  modulationibus  deserviunt  numeros,  necnon  et  ad  ea  quae  astrologorum 
sollerti  industria  de  variis  errantium  siderum  cursibus  .  .  .  Abacus  valde  neces- 
sarius  inveniatur."  See  Chasles,  loc.  cit.,  p.  1414. 

3  One  of  the  best  specimens  of  this  kind  seen  by  the  author  is  a  6th  century 
Roman  piece  in  the  Rylands  Library  at  Manchester,  England.  This  is  made 
of  bone  and  is  a  diptych  bound  with  iron  and  having  an  iron  hinge.  It  has 
three  iron  styli,  one  end  being  pointed  for  writing  and  the  other  end  being 
spoon-shaped  for  erasing. 

4There  is  an  elaborate  set  of  Comptes  de  1 'hotel  Saint  Louis,  written  in 
1256-1257,  in  the  Musee  des  Archives,  Paris.  It  consists  of  between  ten  and 
twenty  placques.  There  is  a  isth  century  specimen  in  the  Germanic  Museum 
at  Niirnberg,  from  southern  Germany;  a  piece  apparently  of  the  i6th  century, 
from  a  church  in  Switzerland,  may  be  seen  in  the  British  Museum;  but  the 
most  interesting  one  of  the  medieval  period  that  has  come  to  the  author's 
attention  is  in  the  Rathhaus  at  Goslar  a.  Harz, — a  book  of  eight  tablets  bound 
together  and  making  sixteen  pages,  two  compartments  to  a  page.  This  Goslar 
specimen  is  a  Burgerrolle  of  the  i4th  or  i5th  century,  the  numerals  being  Roman 
and  the  original  stylus  being  annexed. 


THE  WAX  TABLET  AND  THE  SLATE  179 

The  Slate.  In  the  later  Middle  Ages  the  slate  replaced  the 
wax  tablet  and  sand  table,  and  continued  in  use  until  the  manu- 
facture of  cheap  paper  rendered  it  nearly  obsolete  at  the  close 
of  the  i  gth  century.  The  earliest  printed  reference  to  it  is 


O" 

nfmt  tracfetae  pcrutilio  i  aeceftmud 
foeticiter  incipit.qui  degcneribue  caF 
coUttononi  fpcdc  p:ctcn  t  nnltaj.q  faltc 
iieceflaria  ad  b*  art?  sgiutoj  fiierat 

Tlncm  foqs  phmtae  libra  algo:ifmf  nuctipa 
ti0.m6e  area  numcroe  ogandi  fade  wane  e: 
atq;  dftierfo&c)  licet  bom  ejtiflcrctatq}  vcri 
crattfi  faflidicfi:tu  ,ppf  ipap  regulap  nwl' 


ipap  operationo^>bafoee:^Xbonc  fucrint  ud  we.  it  rat  a  tti 
am  tfti  modi  intm  faflidiofi:  cp  ft  in  aUq?  calculo  adrolotco  cnct 
ot!^'iTj:ca!culato!C  ogatoj  foam  a  capireincipcre  oponcbat:  da 
to  q?  erro:  fuas  adbuc  fatid  ^pfqaa0  cjrtftcrer.  i  boc  jppi  figu 
rao  in  fua  ogatoe  deleta0»3ndtgebat  col  calcolato:  feme  auq? 
lapidc  ud  fibi  ^fo:mi.fujj  quo  fcnbere  atqj  fadlitar  delete  pcJTj 
ftgurod  cu  gbufogabat  in  calailo  fuo^ltt  $a  bee  oia  food  fa;  I 


FIRST  PRINTED  REFERENCE   TO  A   SLATE 
From  Beldamandi's  work,  1483.   See  the  word  lapide  in  next  to  the  last  line 

probably  in  the  Algorismus  of  Prosdocimo  de'  Beldamandi,  a 
work  written  in  1410  and  first  printed  in  I4.83,1  in  which  the 
author  speaks  of  the  necessity  which  a  computer  has  for  a  slate 
from  which  he  can  easily  erase  what  he  has  written.2  The  gain 
to  the  art  of  computation  which  resulted  from  this  invention 
can  hardly  be  realized  at  the  present  time. 

!"Anno  domini  .1410.  die  .10.  lunij  compilata."   Sec  Rara  Arithmetica,  p.  13. 

2"Indigebat  etia  calculator  sem.p  aliq?  lapide  uel  sibi  Pformi,  su,p  quo 
scribere  atq^  faciliter  delere  poss}  figuras  cu  cjbus  o^pabat'  in  calculo  suo"  (1540 
ed.,  fol.  2,  v.).  Compare  also  J.  T.  Freigius,  Pcedagogvs,  Basel,  1582  :  "Numeri 
in  abaco  scribendi." 


i8o 


THE  ABACUS 


In  this  period,  but  we  do  not  know  precisely  when,  there 
came  into  general  use  the  blackboard,  arranged  for  hanging  on 
a  wall.  This  is  frequently  shown  in  the  illustrations  in  the  early 
printed  books,  as  in  the  case  of  Boschensteyn's  work  of  1514. 


EARLY  ILLUSTRATION   OF  A  BLACKBOARD 
From  Johann  Boschensteyn's  Rechenbiechlin,  Augsburg,  1514 

Gerberfs  Abacus.  To  Gerbert  (c.  1000)  there  is  attributed 
the  arc  or  column  abacus.1  If  we  could  put  one  counter  marked 
"4"  on  a  line  instead  of  putting  four  counters  upon  it,  there 
would  seem  at  first  thought  to  be  some  gain.  This  apparent  gain 
is  offset,  however,  by  the  loss  of  time  in  selecting  the  counters 
and  still  more  by  the  necessity  for  learning  certain  tables.  The 
plan  was  followed  by  Gerbert,  and  possibly  some  of  his  succes- 

1Arcus  Pythagoreiis,  tableau  a  colonnes. 


ARCUS  PYTHAGOREUS 


181 


sors,  the  counters  being  the  apices  already  mentioned  on  page  7  5 . 
They  represented  the  number  2 ,056,708,  for  example,  as  follows : 


Over  each  triad  of  columns  an  arc  was  drawn  to  aid  the  eye, 
whence  the  name  "arc  abacus ";  and  in  each  column  in  which 
a  number  was  to  be  represented  a  counter  bearing  that  number 
was  placed.  As  already  stated,  however,  the  gain  over  the  older 
form  was  more  apparent 
than  real,  for  the  computer 
was  under  the  necessity 
of  picking  out  the  right 
counter  each  time.  If  Ger- 
bert  had  understood  the 
significance  of  the  zero,  he 
would  not  have  used  this 
device. 

The  Line  Abacus.  The 
most  popular  abacus  of 
Western  Europe  consisted 
of  a  table  ruled  horizontally 
to  represent  different  deci- 


""  Ten  thousands 

Five  thousands 
^^  Thousands 

Five  hundreds 

Hundreds 

Fifties 

Tens 

Fives 
"  Units 

GENERAL  PLAN  OF  A  MEDIEVAL 
COMPUTING  TABLE 


This  plan  shows  the  arrangement  of  lines 
on  the  kind  of  computing  table  used  in 
most  parts  of  Europe  in  the  Middle  Ages 

mal  orders,  counters  being 

placed  upon  the  lines  and  in  the  spaces.  It  was  often  called  a 
calculating  table  or  simply  a  table/  and  in  England  it  received 
the  name  "counting  table"  or  "counter." 

The  illustration  of  the  line  abacus,  from  KobePs  work  of 
1514,  shows  the  form  which  was  common  in  all  Western  Europe 

1So  Hudal rich  Regius,  in  his  Epitome  (1536),  says,  " Abacus  vulgo  mensa 
dicitur  calculatoria  quibusdam  distincta  lineis";  and  Radulph  of  Laon  (c.  1125) 
asserts  that  "Gr^ci  enim  Mensam  abacum  dicunt."  See  Abhandlungen,  V,  96. 


182 


THE  ABACUS 


for  several  hundred  years.  The  line  nearest  the  computer  rep- 
resents units,  the  space  above  it,  fives;  the  second  line,  tens; 
the  second  space,  fifties;  and  so  on.1 

Representation  of  Numbers.  On  the  lines  and  in  the  spaces, 
counters  were  then  placed  as  shown  in  this  illustration  in  the 


COUNTER  RECKONING  IN  1514 
From  the  title-page  of  Kobel's  Rechenbiechlin,  Augsburg,  1514 

two  columns  on  the  table.  In  the  right-hand  column,  which  is 
the  left-hand  column  of  the  computer  who  sits  by  the  window, 
the  number  26  is  represented,  2  being  on  the  tens  line,  i  in  the 

1  One  of  the  best  of  the  older  authorities  on  this  type  is  the  work  of  T.  Snell- 
ing,  A  view  of  the  origin,  nature,  and  use  of  Jettons  or  Counters,  London,  1769. 
See  also  D.  E.  Smith,  Computing  Jetons,  New  York,  1921.  The  standard  author- 
ity, however,  is  Barnard's  work  already  mentioned. 


RECKONING  ON  THE  LINES  183 

fives  space,  and  i  on  the  units  line.  There  should  never  be 
more  than  one  counter  left  in  a  space  or  more  than  four  counters 
on  a  line;  for  if  there  are  five  on  a  line,  one  is  "carried"  to  the 
space  above,  and  if  there  are  two  in  a  space,  one  is  "  carried " 
to  the  line  above;  whence  our  expression  "to  carry"  in  addi- 
tion.1 The  thousands  and  millions  lines  were  each  marked  by  a 
small  cross.  This  aided  the  eye  in  reading  the  numbers  and  is 
the  origin  of  our  system  of  separating  the  figures  in  groups  of 
three  by  means  of  a  comma.2 

The  intervals  between  the  horizontal  lines  were  commonly 
called  "spaces"  (spacia),  and  the  divisions  made  by  the  vertical 
lines  were  called  cambien,  from  the  Italian  cambio  (exchange).3 

Reckoning  on  the  Lines.  Computation  on  this  form  of  abacus 
was  called  reckoning  "on  the  lines/'  and  many  of  the  early  Ger- 
man arithmetics  include  the  expression  "auf  den  Linien."4  As 
a  result,  a  boy  who  knew  his  abacus  was  said  to  "know  the 
lines."5  When  he  represented  a  number  by  means  of  counters 
on  the  lines,  he  was  said  to  "lay"  the  sum ; G  and  when  he  also 
knew  the  modern  form  of  computing  which  had  developed  in 
Italy,  he  was  said  to  be  able  to  reckon  "on  the  lines  and  with 
the  pen."7  He  was  often  advised  to  "lay  and  seize"  correctly, 
meaning  that  he  must  be  careful  to  place  the  counter  prop- 


1  Other  illustrations  from  early  printed  books  will  be  found  in  Kara  Arith- 
metica.  A  good  description  of  the  common  German  abacus  is  given  by  A.Kuckuck, 
Die  Rechenkunst  im  sechzehnten  J ahrhundert ,  p.  7  (Berlin,  1874). 

2"Diesclbe  verzeichne  mit  einem  Creutzlin,"  as  J.  Albert  says  in  his  arithmetic 
of  1534,  mentioned  below  in  note  4. 

3 Also  "Cambien  odcr  Bankir"  by  various  other  writers;  e.g.,  Kobel,  Zwey 
rechenbuchlin  (1514;  title  of  edition  of  1517).  Hudalrich  Regius  (1536)  used 
the  term  viculi,  and  J.  Albert  (1534)  speaks  of  the  divisions  as  Feldungen  as 
well  as  Cambien  and  Cambiere. 

4 For  example,  J.  Albert,  Rechenbuchlein  Auff  der  Federn,  Wittemberg,  1534 
(title  from  1561  ed.)  ;  A.  Riese,  Rechenung  auff  der  linihen  vnd  federn,  Erfurt, 
1522. 

5 "Die  Linien  zu  erkennen,  ist  zu  mercken,  das  die  underste  Linic  (welche 
die  erste  genent  wird)  bedeut  uns,  die  ander  hinauff  zehen,  die  dritte  hundert," 
etc.  J.  Albert,  loc.  cit. 

6 "Leg  zum  ersten  die  fl."  J.  Albert,  loc.  cit.  This  may  be  connected  with 
our  expression  "  to  lay  a  wager." 

7 As  in  Riese's  work  of  1522. 


184 


THE  ABACUS 


erly  and  pick  it  up  with  the  same  care.    Thus  Albert  (1534) 
tells  him : 

Write  right,  lay  right,  seize  right,  speak  right, 
And  you  will  always  get  the  answer  right.1 

Addition  and  Subtraction.  The  operation  of  addition  can  be 
understood  by  studying  the  illustration  on  page  185  from  Re- 
corded work  and  by  considering  the  following  figure  suggested 
by  Albert's  arithmetic : 


•,                          s 

• 

• 

•>•>•> 

• 

^    •  • 

__•  

(213  +  1450  +  2378  =  4041) 

Subtraction  was  merely  the  reverse  of  the  above  operation, 
and  the  word  " borrowing"  had  a  more  definite  meaning  than 
with  us.  The  following  figure  is  also  suggested  by  Albert's  work : 


\ 

• 

/ 

-  ^   «fc  ,  *    m  — 

• 

,.-.*,  *_A_  — 

(1534-186-1348) 

Multiplication  and  division  were  more  complicated  and  are 
not  of  enough  importance  to  warrant  a  description  in  this  work. 

Extent  of  Use  of  the  Line  Abacus.  During  the  isth  century 
the  line  abacus  furnished  almost  the  only  means  of  commercial 
computation  throughout  most  of  Western  Europe  north  of  the 

1 "  Schreib  recht/leg  recht/greiff  recht/sprich  recht/ 
So  kompt  allzeit  dein  Facit  recht." 


THE  OPERATIONS  185 

Alps.  In  the  i6th  century  we  find  it  given  prominently  in  the 
printed  arithmetics  of  Germany,  Holland,  Poland,  and  Austria, 
somewhat  less  prominently  in  England,  and  still  less  in  France. 
We  may  say  that  those  countries  which  were  chiefly  influenced 

ADDITION. 

Matter* 

be  eaficft  toap  m  tbts  artels  to  atste 
but  ttoo  fimtmcs  at  ones  togFtbct: 
tioti)  be  it,  pou  marc  a&oc  mcjMS  3!  tort  tci 
l>ou  anone  *  tljeccfoic  toljcnnc  pou  topttc 
abbe  ttoo  fumtnes,pou  (ball  fpifte  Cet  Oottme 
one  of  ttjenut  foicttt)  not  totitcfjc,  anb  tbcn 
bp  ttbiate  alpne  croflfe  the  otfycc  Ipnts.3n9 
aftcttoatuc  fettcbounctDcotticrfuinmc » Co 
ttjattDat  ipncmapc 
i>cb«tbcuetDcm;a0 
it  pou  tooulbe  aoot 

*^5P  tO  8  341  ,  pOU 

muttfctpoucffimcfl 


as  pou  fee  tore, 


tf  pou 

Ipft,  pou  mapc  aoar 
tDc  one  to  tl)c  ottjer  in  tlje  fame  place,  o:  cl« 
TOU  map  aCUf  tl)  :m  bottj:  toOul)cr-tu  a  ucto 
place  :  toljirt  \iiap,  br(^ul"c  ic  is  luoftplpucft 

3 

PAGE  FROM  ROBERT  RECORDE'S  GROUND  OF  ARTKS,  C.  1542 
This  page  shows  the  treatment  of  addition.    It  is  from  the  1558  edition 

by  the  customs  of  the  Italian  merchants  tended  to  abandon 
the  abacus,  while  those  which  were  in  closer  contact  with  the 
German  counting  houses  continued  to  use  it. 

The  popularity  of  the  method  may  be  seen  in  the  fact  that 
abacus-reckoning  was  a  favorite  subject  of  illustration  in  the 
title-pages  of  the  arithmetics  of  Adam  Riese,  Gemma  Frisius, 
and  Robert  Recorde,  which  were  among  the  most  widely  circu- 
lated textbooks  of  the  i6th  century. 


1 86  THE  ABACUS 

Origin  of  this  Form  of  Abacus.  It  is  not  known  when  this 
form  of  the  abacus  first  appeared.  Indeed,  there  is  a  break  of 
several  centuries  in  the  use  of  counters  in  any  manner.  We  are 
ignorant  as  to  how  the  Western  world  computed  at  the  begin- 
ning of  the  Middle  Ages,  or  what  method  Bede  (c.  710)  and 
Alcuin  (c.  775)  used  in  their  calculations.  In  the  i3th  century 
counters  were  used  for  practical  business  computation,  as  they 
had  been  in  the  Roman  days,  but  in  the  long  interval  the 
ancient  scheme  had  changed,  the  vertical  lines  giving  place 
to  the  horizontal.  When  or  where  this  change  took  place 
there  is  at  present  no  means  of  knowing.1  Fibonacci  (1202) 
names  three  methods  of  computing  in  use  in  his  day, — finger 
reckoning,  algorism  (Hindu  numerals),  and  the  Gerbert  aba- 
cus.2 Of  the  use  of  ordinary  counters  he  has  nothing  to  say. 
Certain  it  is  that  counters  were  generally  unknown  in  the  :5th 
century  in  Italy,  for  we  have  the  positive  assertion  of  the  Vene- 
tian patrician  Ermolao  Barbaro  (d.  1495)  that  they  were  used 
only  in  foreign  countries.3 

The  Counters  in  England.  The  most  common  of  the  English 
names  for  the  small  disks  used  on  the  line  abacus  was  " counter/' 
a  word  derived  from  the  Latin  computare  through  the  French 
forms  conteor  and  compteur  and  the  Middle  English  countere7 
cowntere,  and  countour*  So  in  a  work  entitled  Know  Thyself, 
written  about  1310,  we  are  told  to  "sitte  doun  and  take  coun- 
tures  rouncle. . . .  And  for  vche  a  synne  lay  thou  doun  on  Til  thou 
thi  synnes  haue  sought  vp  and  founde,"  and  in  a  work  of  1496 
mention  is  made  of  "A  nest  of  cowntouris  to  the  King."  In  the 
laws  of  Henry  VIII  (Act  32,  cap.  14,  1540)  we  read,  "Item  for 
euery  nest  of  compters  .xviii.s,"  so  that  the  expression  was  a 
common  one  and  referred  to  the  box  or  bag  full  of  computing 

XA.  Nagl,  Die  Rechenpfennige  und  die  operative  Arithmetik,  p.  8  (Vienna, 
1888) ;  A.  Kuckuck,  loc.  cit.,  p.  15. 

2"Computatio  manibus,  algorismus,  arcus  pictagore." 

8"Calculos  sive  abaculos  .  .  .  eos  essc  intclligo  .  .  .  qui  mos  hodie  apud 
barbaros  fere  omnes  servatur."  Nagl,  loc.  cit.,  p.  40. 

4 By  a  false  etymology  we  have  "comptroller,"  although  this  word  is  properly 
"  controller,"  one  who  controls  an  account,  from  the  Middle  Latin  contrarotulum 
(contra  +  rotulus)y  a  counter  roll,  a  check  list. 


COUNTERS  IN  ENGLAND 


187 


THE 


O  F 


G  R  O  V  N  D 

A  R  T  B  S: 

3Tce?ci)(nstt)Ctt)ooz&eflti&  pzacttfc  of 

0vftbmettUe>kotl)  in  \1?l)olc  numb  ies 

flirt)  junctions  ,  aftrr  a  moie  cafpct? 

anD  craetet  fozce,  then  anye  lyfte 

Ijattjtjpttwtobeene 


ucvs  uetb  f,D* 


J.R  O  B  E  R  T  B 
R  E  C  O  R  D  E 

SDocfo;  of  jd 


pieces.  Such  a  nest  is  probably  referred  to  by  Alexander  Barclay 
(c.  1475-1552),  in  his  Egloges,  when  he  speaks  of  "The  kitchin 
clarke  .  .  .  Jangling 
his  counters/7 

In  the  Middle  Ages 
in  England  it  seems 
to  have  been  the  cus- 
tom of  merchants,  ac- 
countants, and  judges 
who  had  to  consider 
financial  questions  to 
si  t  on  benches  ( banks ) 
with  checkered  boards 
and  counters  placed 
before  them.  Hence 
the  checkered  board 
came  to  represent  a 
money  changer's  of- 
fice, finally  becoming 
a  symbol  for  an  inn, 
probably  because  inn- 
keepers followed  the 
trade  of  the  money 
changer. 

In  Shakespeare's 
time  abacus  compu- 
tation was  in  low  re- 
pute, for  the  poet 
speaks  contemptu- 
ously of  a  shopkeeper 
as  a  " counter  caster."  Counters  apparently  lost  their  standing 
only  in  the  last  half  of  the  i6th  century,  for  Robert  Recorde, 
writing  c.  1542,  says :  "Nowe  that  you  haue  learned  the  common 
kyndes  of  Arithmetike  with  the  penne,  you  shall  see  the  same 
arte  in  counters/71  and  an  anonymous  arithmetic  of  1546  has 

1From  the  1558  edition  of  the  Ground  of  Aries,  in  which  Recorde  devotes 
forty  pages  to  this  phase  of  the  subject. 


COUNTER  RECKONING 

From  the  1558  edition  of  Recorde's  Ground 
of  Aries 


1 88  THE  ABACUS 

"An  introduction  for  to  lerne  to  reken  with  the  pen,  or  with 
the  counters  accordying  to  the  trewe  cast  of  Algorisme."  A  cen- 
tury later  Hartwell,  in  his  appendix  to  the  Ground  of  Artes 
(1646  ed.)7  speaks  of  ignorant  people  as  "any  that  can  but  cast 
with  Counters."  Even  in  the  first  half  of  the  i6th  century 
people  had  begun  to  doubt  the  value  of  line  reckoning,  for 
Palegrave  writes  in  1530:  "I  shall  reken  it  syxe  tymes  by 
aulgorisme  or  you  can  caste  it  ones  by  counters."  That  the 
abacus  died  out  here  before  it  did  in  Germany  is  also  evident 
from  the  fact  that  German  counters  of  the  isth  and  i6th  cen- 
turies are  very  common  in  numismatical  collections,  while  most 
of  those  used  in  England  at  this  time  were  imported.1 

From  the  use  of  "counter"  in  the  sense  described,  the  word 
came  to  mean  an  arithmetician.  Thus  we  find  in  one  of  the 
manuscripts  in  the  Cotton  library  the  statement,  "Ther  is  no 
countere  nor  clerke  Con  hem  recken  alle,"  and  Hoccleve  (1420) 
writes:  "In  my  purs  so  grete  sommes  be,  That  there  nys 
counter  in  alle  cristente  Whiche  that  kan  at  ony  nombre 
sette."2  The  word  also  came  to  mean  the  abacus  itself.  Thus, 
in  his  Dethe  Blaunche  (c.  1369)  Chaucer  says:  "Thogh  Argus3 
the  noble  covnter  Sete  to  rekene  in  hys  counter." 

Court  of  the  Exchequer.  Aside  from  the  mere  history  of  com- 
putation an  interest  attaches  to  the  abacus  in  England  because 
of  its  relation  to  the  Court  of  the  Exchequer,  the  Chambre 
de  Vechiquier  of  the  French.4  In  the  Dialogus  de  Scaccario 


1  Barnard,  Counters,  p.  63. 

2  See  Murray's  New  English  Dictionary ,  II,  1057. 

3  The  passage  comes  from  the  Roman  de  la  Rose,  in  which  this  name,  with 
also  the  spelling  Algus,  is  given  for  al-Khowarizmi.    Chaucer  also  speaks  of  the 
counters  as  "augrim  (i.e.,  algorism,  from  al-Khowarizmi)  stones."   On  this  sub- 
ject see  L.  C.  Karpinski,  "Augrim  Stones,"  Modern  Language  Notes,  November, 
1912  (Baltimore). 

4  The  best  original  source  of  information  as  to  the  exchequer  is  the  Dialogus  de 
Scaccario,  a  work  written  by  one  Fitz-Neal  in  1178-1179   (1181  according  to 
Stubbs)  and  first  edited  by  Madox  in  1711.  See  also  F.  Liebermann,  Einleitung  in 
den  Dialogus  de  Scaccario,  Gottingen,  1875,  and  the  Oxford  edition  of  1902.   It  is 
published  in  E.  F.  Henderson,  Select  Historical  Documents  of  the  Middle  Ages, 
p.  20  (London,  1892).    Consult  also  H.  Hall,  The  Antiquities  and  Curiosities  of 
the  Exchequer  y  London,  1891  (reviewed  somewhat  adversely  in  The  Nation,  New 


COURT  OF  THE  EXCHEQUER  189 

a  disciple  and  his  master  discuss  the  nature  of  the  exchequer 
as  follows: 

Disciple.   What  is  the  exchequer? 

Master.  The  exchequer1  is  a  quadrangular  surface  about  ten  feet 
in  length,  five  in  breadth,  placed  before  those  who  sit  around  it  in  the 
manner  of  a  table,  and  all  around  it  has  an  edge  about  the  height  of 
one's  four  fingers,  lest  anything  placed  upon  it  should  fall  off.  There 
is  placed  over  the  top  of  the  exchequer,  moreover,  a  cloth2  bought  at 
the  Easter  term,  not  an  ordinary  one  but  a  black  one  marked  with 
stripes,  the  stripes  being  distant  from  each  other  the  space  of  a  foot 
or  the  breadth  of  a  hand.  In  the  spaces  moreover  are  counters  placed 
according  to  their  values. 

The  rest  of  the  description  is  too  long  to  be  given,  but  it 
shows  that  a  kind  of  abacus,  although  not  the  one  above 
described,  characterized  this  ancient  court. 

The  counters  finally  came  to  be  used  to  keep  the  scores  in 
games,3  as  in  the  American  game  of  poker  and  in  the  use  of 
markers  in  billiards.  They  also  remained  in  the  schools  for  the 
purpose  of  teaching  the  pupils  the  significance  of  our  number 
system,  sometimes  in  the  form  of  an  abacus  with  ten  beads  on  a 

York,  February  25,  1892,  p.  157)  ;  R.  L.  Poole's  Ford  Lectures  at  Oxford  in  1911, 
published  under  the  title  The  Exchequer  in  the  Twelfth  Century,  J.  H.  Ramsay, 
The  Foundations  of  England,  II,  323  (London,  1898) ;  Martin,  Les  Signes  Num., 
p.  32 ;  J.  H.  Round,  The  Commune  of  London,  and  other  Studies,  p.  62  (London, 
1899) ;  C.  H.  Haskins,  "The  abacus  and  the  king's  curia,"  English  Historical  Re- 
view (1912),  p.  101.  On  the  Dialogus  as  the  earliest  work  on  English  government, 
consult  J.  R.  Green,  Short  History  of  the  English  People. 

!The  word  is  a  corrupt  form  of  the  Old  French  eschequier  and  Middle 
English  escheker,  based  on  the  mistaken  idea  that  the  Latin  ex-  is  taken  with 
scaccarium.  The  term  scaccarium  for  exchequer  first  appears  under  Henry  I, 
about  noo.  Before  him,  under  William  the  Conqueror  and  William  Rufus,  we 
find  the  terms  fiscus  and  thesaurus.  "Exchequer"  was  later  used  to  mean  a 
chessboard,  as  in  a  work  of  1300:  "And  bidde  the  pleie  at  the  escheker";  and 
in  Caxton's  work  on  chess  (c.  1475),  p.  I3S,  where  it  appears  as  eschequer. 

2  There  is  in  the  National  Museum  at  Munich  a  green  baize  cloth  embroid- 
ered in  yellow  with  the  ordinary  arrangement  of  the  medieval  German  abacus, 
intended  to  be  laid  on  the  computing  table  in  the  manner  here  described.  For 
illustrations  of  such  pieces  see  Barnard,  loc.  cit.,  plates. 

3 "They  were  marking  their  game  with  Counters."  Steele,  in  The  Tatler, 
No.  15  (1709). 


190 


THE  ABACUS 


line,  as  seen  in  primary  classes  today,  and  sometimes  for  the 
purpose  of  teaching  fractional  parts.    A  specimen  possibly  in- 

tended chiefly  for  this  purpose 
is  seen  in  an  abacus  formerly  used 
in  the  Blue  Coat  School  in  Lon- 
don, and  here  shown. 

Counters  in  Germany.  In  no 
country  was  the  line  abacus  more 
highly  esteemed  in  the  isth  and 
1  6th  centuries  than  in  Germany. 
Its  use  had  died  out  in  Italy,  the 
great  commercial  center  of  the 
world,  but  in  the  counting  houses 
of  Germany  it  was  almost  univer- 
sal until  the  era  of  printed  arith- 
metics. Indeed,  even  in  the  i6th 
century  its  superiority  was  stoutly 
maintained  by  various  German 
Rechenmeisters,1  and  as  late  as 
the  middle  of  the  i8th  century  its 
use  had  not  died  out  in  a  number 
of  the  towns.2  Even  after  the  lead- 
ing merchants  had  learned  the 
method  of  algorism  the  common 
,e  continued  to  do  their  sim- 

i  u       *i_          «_j       £ 

pie  sums  by  the  aid  of  counters 


ABACUS  FOR  TEACHING 
FRACTIONS 


ton's  collection 


dler,8  and  the  most  popular  of  the 


1Thus  Apianus  (Eyn  Newe  .  .  .  Kanffmansz  Rechnung,  Ingolstadt,  1527) 
asserted:  "...  die  Summirung  der  Register  in  gewicht  mass  vnd  miintz  durch 
die  rechenpfenning  auf  der  linie  brauchsamer  ist  vnd  vil  schneller  vnd  fiiglicher 
geschicht  dann  durch  die  federn  oder  kreide." 

2Heilbronner,  in  his  Historia,  p.  890,  says  that  in  his  time  the  counters  were 
still  used  "in  pluribus  Germaniae  atque  Galliae  provinciis  a  mercatoribus,"  and 
speaks  of  computing  on  the  line  as  "arithmetica  calculatoria  sive  linearis  est 
Scientia  numerandi  per  calculos  vel  nummos  metallicos." 

3  The  priest  Geiler  of  Kaiserberg  (1445-1510)  tells  of  peddlers'  selling  them 
in  his  day,  and  the  custom  doubtless  continued.  See  Bibl.  Math.,  IV  (3),  284. 


GERMANY  AND  FRANCE  191 

early  German  arithmetics  were  based  on  "  Rechnung  auf  Linien." l 
As  late  as  1587  Thierf elder  testified  to  the  fact  that  the  use  of 
counters  was  still  common  in  Germany,2  and  in  1591  two  arith- 
metics based  on  line  computation  were  published.3  Even  as  late 
as  162 1  a  textbook4  on  the  subject  appeared  at  Hildesheim.  One 
of  the  last  writers  to  describe  the  process  fully  was  Leonhard 
Christoph  Sturm,  whose  work  was  published  in  1701." 

The  common  German  name  for  the  counter  was  Rechen- 
pfennig"  although  Zahlpfennig7  and  Raitpfennig  were  also  used. 

Counters  in  France.  In  the  later  Middle  Ages  France,  like  all 
the  Latin  countries,  made  less  of  counter  reckoning  than  the 
Teutonic  lands.  The  first  printed  description  of  counter  reck- 
oning in  that  country  dates  from  about  isoo,8  and  although 

1So  Kobel's  well-known  arithmetics,  which  went  through  various  editions 
beginning  with  1514,  gave  only  the  counter  reckoning;  Adam  Riesc's  famous 
textbooks,  beginning  in  1518,  favored  it;  and  various  other  popular  textbooks 
gave  it  prominent  place.  Even  as  good  a  mathematician  as  Rudolff  introduced 
it  immediately  after  the  treatment  of  algorism  in  his  Kunstliche  rechnung  of  1526. 

The  first  arithmetics  printed  in  Germany  appeared  at  Bamberg  in  1482  and 
1483.  Neither  seems  to  have  had  anything  to  say  about  counters,  although  we 
have  only  a  fragment  of  the  earlier  one,  and  perhaps  this  failure  explains  the  lack 
of  popularity  of  these  books.  The  subject  was  so  popular  that  Stifel  (1544)  calls 
it  Haussrechnung.  On  the  general  subject,  see  H.  Schubert,  "Die  Rechenkunst 
im  16.  Jahrhundert,"  Deutsche  Blatter  jur  Erziehenden  Unterricht,  III,  69,  105. 

2 On  the  first  page  of  his  Rechenbuch  he  says:  "Wie  vil  sind  Arten  oder 
Weisen/die  im  Rechnen  am  meysten  gcbraucht  werden?  Furnehmlich  zwo/ 
Die  erst  mit  der  Feder/oder  Kreyden/durch  die  Ziffern.  Die  ander  mit  den 
Zahlpfenning  auff  den  Linien." 

3  One  (Swiss)  by  Mewrer  in  Zurich  and  the  other  by  Kauder  in  Regensburg. 

4 The  work  was  anonymous.  The  title  is  Ein  new  Rechenbuchlein  auf  Linien 
und  Ziffern.  See  Nagl,  Die  Rechenpfennige,  p.  27,  for  other  cases. 

^Kurtzer  Be  griff  .  .  .  Mathesis. 

6 Reckoning  penny.  The  spelling  varies.  Stifel,  for  example,  in  his  Deutsche 
Arithmetica,  1545,  fol.  i,  calls  them  Rechenpfenning  (both  singular  and  plural). 
Rechenpfennig  was  merely  a  translation  of  the  medieval  Latin  name,  as  is  seen 
in  the  arithmetic  of  Clichtovcus  (1503):  "quos  denarios  supputarios  vocant," 
and  in  his  commentary  on  Boethius  (1510,  fol.  33). 

7  Number   penny.    Rudolff,   in   his   edition    of    1534,    uses   this   form,   while 
Thierfelder,  in  his  Rechenbuch  of  1587,  uses  the  form  Zahlpfenning. 

8  In  this  anonymous  and  undated  work,  De  arte  numerandi  sine  arismetice 
(perfections)   summa  quadripartita,  the  author  treats  of  the  operations  "per 
proiectiles,"  and  says:  "hec  licet  breuiter  de  proiectilibus  sint  dicta,  negotiant! 
tamen  atque  se  exercenti  per  eos  frequenter,  abundantissime  hec  pauca  suf- 
ficient."   See  Treutlein  in  the  Abhandlungen,  I,  24. 


192  THE  ABACUS 

arithmetics  on  the  subject  are  not  so  common  as  in  Germany, 
they  are  sufficiently  numerous  to  show  that  the  system  was  well 
known.1  The  subject  dropped  out  of  the  business  textbooks  in 
the  last  quarter  of  the  i6th  century,  although  the  counter  was 
used  by  women  long  after  men  had  come  to  use  algorism,  writ- 
ing not  being  so  common  among  the  former  as  among  the  latter.2 

Because  the  counters  are  thrown  upon  the  table  the  medieval 
Latin  writers  often  called  them  projectiles?  The  French  trans- 
lated this  word,  omitting  the  prefix,  as  jetons*  a  word  which 
still  survives  in  France  to  mean  a  game  counter,  a  small  medal, 
or  a  token. 

The  older  French  jetons  frequently  bear  such  inscriptions  as 
"pour  les  Comtes"  and  "pour  les  Finances,"5  showing  their 
use.  Sometimes  the  legends  are  admonitory,  thus:  "Gectez, 
Entendez  au  Compte,"  "Gardez  vous  de  Mescomptes,"  and 
"Jettez  bien,  que  vous  ne  perdre  Rien."° 

The  Tally  and  Related  Forms.  The  subject  of  the  abacus 
should  not  be  dismissed  without  mention  of  the  tally  and  cer- 
tain other  related  forms.  The  tally  was  originally  a  piece  of 
wood  on  which  notches  or  scores7  were  cut  to  designate  num- 

1  Among  these  books  may  be  mentioned  the  following:  Clichtoveus,  Ars 
supputddi  tarn  per  calculos  q$  notas  arithmetics,  Paris,  1507;  Clichtoveus, 
De  Mystica  numerorum  (Paris,  1513,  fol.  33,  r.),  subdividing  supputatio  into 
calcularis  and  figuralis,  giving  five  common  operations  under  the  former  and 
eight  under  the  latter;  Blasius,  Liber  Arithmetics  Practice  (1513);  an  anony- 
mous Le  livre  des  Gctz  (about  1500),  in  which  is  taught  "la  pratique  de  bien 
scjavoir  center  aux  getz  comme  a  la  plume";  Cathalan,  Arithmetiqve,  Lyons, 
1555,  in  which  the  author  explains  "a  Chiffrer  &  compter  par  la  plume  &  par  les 
gestz";  an  anonymous  Arithmetique  par  les  jects,  Paris,  1559;  Trenchant, 
Arithmetiqve,  1566,  the  1578  edition  of  which  gives  thirteen  pages  to  "L'art 
et  moyen  de  calcvler  avec  les  Getons,"  the  1602  edition  dropping  the  subject. 

2 Thus  F.  Legendre,  in  his  arithmetic  of  1729,  says:  "Cette  maniere  de  cal- 
culer  est  plus  pratiquee  par  les  f  emmes  que  par  les  hommes.  Cependant  plusieurs 
personnes  qui  sont  employees  dans  les  finances  ct  dans  toutes  les  jurisdictions 
s'en  servent  avec  beaucoup  de  succes."  3Pro  (forward)  +  jacere  (to  throw). 

4 Also  found  in  the  following  forms:  jettons,  gects,  gectz,  getoers,  getoirs, 
gettoirs,  getteurs,  jectoers,  jectoirs,  jetoirs,  giets,  gietons,  and  gitones.  Consult 
Snelling,  loc.  cit.,  p.  2. 

5 Also  "Getoirs  de  la  chambre  des  comptes,  Le  Roi,"  "Ce  sont  les  getoirs 
des  ?tes  [Comptes].  La  Reinne." 

6See  also  Snelling,  loc.  cit.,  p.  3;  Nagl,  loc.  cit.,  p.  u;  and  Barnard,  where 
many  photographic  plates  may  be  consulted.  7 Scars;  related  to  "shear." 


TALLY  STICKS 


193 


bers.  The  word  comes  from  the  French  tattler  (to  cut),  whence 
our  word  "tailor."  The  root  is  also  seen  in  the  Italian  word 
intaglio  and  is  from  the  Latin  talea  (a  slender  stick)  .*  The  word 
has  even  been  connected  with  the  German  Zahl  (number) 
through  the  primitive  root  tal.2 

The  idea  of  keeping  numerical  records  on  a  stick  is  very 
ancient,  and  in  a  bas-relief  on  the  temple  of  Seti  I  (c.  1350  B.C.), 
at  Abydos,  Thot  is 
represented  as  indi- 
cating by  means  of 
notches  on  a  long 
frond  of  palm  the 
duration  of  the  reign 
of  Pharaoh  as  de- 
creed by  the  gods.3 

In  the  Middle  Ages 
the  tally  formed  the 
standard  means  of 
keeping  accounts.  It 

was  commonly  split  TALLY  STICKS  OF  1295 

so  as  to  allow  each 
party  to  have  a  rec- 
ord, whence  the  ex- 
pression "our  accounts  tally."4  The  root  also  appears  in  "tail," 
as  when  Piers  Plowman,  speaking  of  his  gold,  says  that  he  "toke 
it  by  taille,"  meaning  by  count ; 5  and  in  "tailage"  (or  "tallage") 
for  toll  or  tax,6— a  relic  of  the  days  when  "our  forefathers  had 
no  other  books  but  the  score  and  the  tally."7 

/ 

1  Consult  also  Greenough  and  Kittredge,  Words  and  their  Ways,  pp.  45,  266 
(New  York,  1901). 

2G.  Rosenhagen,  "Was  bedeutet  Zahl  ursprunglich ?  "  Zeitschrift  fur  deutschen 
Altertum  und  deutsche  Literatur,  LVII,  189. 

3 It  is  reproduced  in  G.  Maspero,  Dawn  of  Civilization,  3d  ed.  by  Sayce, 
p.  221  (New  York,  1897). 

4 Other  expressions,  like  "keeping  tally "  at  a  game,  "to  tally  up,"  and 
"stocks,"  are  traced  to  this  device. 

6  See  also  Chaucer,  General  Prologue  to  the  Canterbury  Tales,  1.  570. 

6 The  first  poll  tax  is  said  to  have  been  the  "tailage  of  groats"  levied  by 
Parliament  in  1377.  7  Shakespeare,  2  Henry  VI,  IV,  vii,  38. 


Fragments  found  at  Westminster  in  1904.   In  the 
author's  collection 


194  THE  ABACUS 

Usually  a  hazel  stick  was  prepared  by  the  "  tally  cutter,"  and 
the  notches  were  cut  before  it  was  split,  a  large  notch  meaning 
M  ( £1000),  a  smaller  one  C  (fioo),  a  still  smaller  one  X  (£10), 
and  so  on  down  to  pence.  In  earlier  times  the  twentieth  mark 
was  a  larger  scar  than  the  others,  and  the  number  was  there- 
fore called  a  score.1  The  system  was  used  in  the  English  Ex- 
chequer as  late  as  1812. 

It  is  interesting  also  to  note  the  relation  of  the  tally  stick  to 
modern  forms  of  investment.  Formerly,  if  a  man  lent  money 
to  the  Bank  of  England,  the  amount  was  cut  on  a  tally  stick.2 
This  was  then  split,  the  bank  keeping  the  "foil"  (folium,  leaf) 
and  the  lender  receiving  the  "stock"  (stipes),  thereby  becom- 
ing a  "stock"  holder  and  owning  "bank  stock."3 

The  tally  was  used  in  Germany  for  keeping  accounts  in  the 
i3th  and  i4th  centuries.  Even  at  the  beginning  of  the  isth 
century,  and  in  as  progressive  a  city  as  Frankfort  am  Main, 
the  so-called  Kerbenrechnung  was  common,4  nor  did  the  cus- 
tom die  out  in  Germany  and  Austria  until  the  igth  century.5 

1  Teutonic  Stiege,  a  word  often  used  for  twenty. 

2 The  British  Museum  has  a  number  of  tally  sticks,  from  1348  to  the  "hop 
tally"  still  used  in  Kent  and  Worcestershire.  There  are  six  very  perfect  but 
not  very  old  specimens  in  the  Museum  of  Folklore  at  Antwerp.  In  cleaning 
the  Chapel  of  the  Pyx  at  Westminster  in  1904  several  specimens  were  found. 
These  dated,  as  the  inscriptions  show,  from  1296,  and  some  of  them  came 
into  the  author's  possession  and  are  shown  on  page  193.  Scotch  "nick  sticks"  and 
Scandinavian  calendar  sticks  belong  to  the  same  general  class. 

On  the  method  of  cutting  and  using  tallies  in  England  see  the  Publications 
of  the  Pipe  Roll  Society,  Vol.  Ill  (London,  1884).  On  tally  charts  used  by 
sailors,  see  Zeitschrift  fur  Ethnologie,  XXXV,  672  (Berlin,  1903).  On  their 
use  as  a  means  of  communication  see  W.  von  Schulenberg,  Verhandlungen  d. 
Berliner  Gesellsch.  jur  Anthropologie,  Ethnol.,  und  Urgeschichte,  XVIII,  384; 
Zeitschrift  fur  Ethnologie,  XIV,  370.  On  their  other  uses  see  Gyula  v.  Sebestyen, 
"Ursprung  der  Bustrophedonschrift,"  Zeitschrift  fur  Ethnologie,  XXXV,  755- 

3PooIe,  loc.  cit.  See  also  C.  H.  Jenkinson,  "Early  wooden  tallies,"  Surrey 
Archaeological  Collection,  XXIII,  203.  4Giinther,  Math.  Unterrichts,  p.  287. 

5F.  Villicus,  Geschichte  der  Rechenkunst,  3d  ed.,  p.  15  (Vienna,  1897)  (here- 
after referred  to  as  Villicus,  Geschichte}  ;  R.  Andree,  Braunschweiger  Volkskunde, 
p.  247  (Braunschweig,  1901),  with  several  bibliographical  notes  of  value;  Ver- 
handlungen  d.  Berliner  Gesellsch.  fur  Anthropologie,  Ethnol.,  und  Urgeschichte, 
XI,  763.  With  our  "keeping  tally"  and  baseball  "scores"  compare  the  Ger- 
man "Er  hat  viel  auf  dem  Kerbholze."  For  interesting  accounts  of  the  use  of 
the  tally  in  Bohemia,  see  W.  Wattenbach,  Das  Schriftwesen  im  Mittelalterf 
3d  ed.,  p.  95  (Leipzig,  1896). 


THE  KNOTTED  CORDS  195 

In  Italy  the  tally  was  evidently  common  in  the  i6th  century, 
for  Tartaglia  (1556)  gives  a  picture  of  one  in  his  arithmetic, 
saying  that  one  of  the  two  parts  was  called  by  the  Latin  name 
tessera,  a  word  often  used  to  mean  a  counter.1 

Knotted  Cords.  Related  to  the  tally,  in  that  they  were  used 
for  recording  numbers  but  not  for  purposes  of  calculation,  are 
the  knotted  cords.  These  are  used  in  various  parts  of  the  world 
and  have  such  an  extended  history  that  only  a  passing  reference 
can  be  given  to  them.  Lao-tze,  "the  old  philosopher,"  as  the 
Chinese  call  him,  in  his  Tao-teh-king  of  the  6th  century  B.C., 
referring  to  the  earlier  use  of  this  device,  says,  "Let  the  people 
return  to  knotted  cords  (chieh  shing)  and  use  them."2  Herod- 
otus (IV,  98)  tells  us  that  the  king  of  Persia  handed  the  lonians 
a  thong  with  sixty  knots  as  a  calendar  for  two  months,  and  a 
similar  device  of  modern  India  may  be  seen  in  the  museum  at 
Madras.  Indeed,  in  taking  the  census  in  India  in  1872,  the  San- 
tals  in  the  wilder  parts  of  Santal  Parganas  used  knots  on  four 
colors  of  cords,  the  black  signifying  an  adult  man,  the  red  an 
adult  woman,  the  white  a  boy,  and  the  yellow  a  girl.  The  census 
was  taken  by  the  headmen,  who,  being  unable  to  write,  simply 
followed  the  popular  method  of  keeping  a  numerical  record.3 

In  the  New  World  the  knotted  cord  is  best  illustrated  in  the 
Peruvian  quipu.4  In  each  city  of  Peru  there  was,  at  the  time 


T"E  laltro  di  quest!  dui  pezzi  lo  chiamauano  Tessera."  General  Trattato, 
I,  fol.  3,  v.  (Venice,  1556). 

2 See  Carus's  English  edition  of  the  Tao-teh-king,  pp.  137,  272,  323  (Chicago, 
1898). 

8 Proceedings  of  the  Asiatic  Society  of  Bengal,  p.  192  (Calcutta,  1872). 

4  The  leading  work  on  the  general  question  of  the  quipu,  with  analyses  of  about 
fifty  specimens,  and  with  an  extensive  bibliography,  is  that  of  L.  L.  Locke,  The 
Quipu,  New  York,  1923,  published  by  the  American  Museum  of  Natural  History. 
There  is  an  article  by  the  same  author,  entitled  "  The  Ancient  Quipu,"  in  the 
American  Anthropologist  for  1912,  p.  325.  See  also  E.  Clodd,  Storia  dell'  Alfabeto, 
trad,  del  Nobili,  cap.  iii  (Turin,  1903),  or  the  English  original;  E.  B.  Tylor,  Early 
History  of  Mankind,  p.  160;  Westminster  Review,  London,  XI,  246;  A.  Treichel, 
Verhandlungen  d. Berliner  Gesellsch.  jur  Anthropologie,  Ethnol.,und  Urgeschichte, 
XVIII,  251;  W.  von  Schulenberg,  "Die  Knotenzeichen  der  Miiller,"  Zeitschrift 
fur  Ethnologie,  XXIX,  491 ;  H.  G.  Fegencz,  "  Kinderkunst  und  Kinderspiele," 
Anzeiger  d.  Ethnolog.  Abteilung  d.  Ungarischen  National-Museums,  Budapest, 
XI,  103. 


FINGER  RECKONING 


of  the  European  invasion,  a  quipucamayocuna  (a official  of  the 
knots")  who  may  have  performed  duties  not  unlike  those  of  a 

city  treasurer  today.  At  any 
rate,  we  have  no  evidence 
that  the  knots  were  used  for 
any  other  purpose  than  the 
recording  of  numerical  re- 
sults, just  as  the  Peruvian 
shepherd  today  uses  them  for 
keeping  account  of  his  herds.1 
The  knotted  cords  found 
in  various  forms  of  religious 
regalia  may  originally  have 
recorded  the  number  of 
prayers,  pilgrimages,  or  sac- 
rifices of  the  devotee.  Ex- 
amples of  these  are  seen  in 
the  Lama  rosary  (prenba) 
and  the  rosaries  of  the  Mo- 
hammedans, the  Buddhists 
of  Burma,  and  the  Catholic 
Christians.  Somewhat  simi- 
lar in  use  is  the  notched  pray- 
ing stick  of  the  pilgrim,  such  as  may  be  seen  at  the  shrine  of 
St.  Fin  Barr  at  Gouganebarra,  Ireland. 


SPECIMEN    OF    QUIPU 

The  knotted  cords  of  the  ancient 
Peruvians 


2.  FINGER  RECKONING 

Finger  Notation.  The  absence  or  rarity  of  suitable  writing 
material  led  most  early  peoples  to  represent  numbers  by  posi- 
tions of  the  fingers, — a  system  not  unlike  the  digital  language 
of  the  deaf  mutes  of  today.  While  this  is  manual  rather  than 
mechanical,  it  may  properly  be  explained  in  this  chapter.  It  is 
not  improbable  that  the  idea  developed  from  the  primitive 
method  of  counting  on  the  fingers,  usually  beginning  by  point- 


*They  are  also  related  to  the  wampum  of  the  American  Indian  and  possibly 
to  the  lo-shu  and  ho-t'u  symbols  of  the  ancient  Chinese  I -king. 


DIGITAL  NOTATION 


197 


ing  at  the  little  finger  of  the  left  hand 
with  the  second  finger  of  the  right,  this 
being  the  result  of  holding  the  hands 
in  a  natural  position  for  such  a  pur- 
pose. The  person  counting  would  thus 
proceed  from  right  to  left,  and  this 
may  have  influenced  some  of  the  early 
systems  of  writing  numbers.1 

The  general  purposes  of  digital  no- 
tation were  to  aid  in  bargaining  at 
the  great  international  fairs  with  one 
whose  language  was  not  understood, 
to  remember  numbers  in  computing 
on  an  abacus,  and  to  perform  simple 
calculations.2 

For  the  mere  representing  of  the 
small  numbers  of  everyday  life  the  left 
hand  sufficed.  In  this  way  it  became 
the  custom  to  represent  numbers  be- 
low 100  on  the  left  hand  and  the 
hundreds  on  the  right  hand.  Juvenal 
refers  to  this  custom  in  his  tenth 
satire,  saying:  " Happy  is  he  indeed 
who  has  postponed  the  hour  of  his 
death  so  long  and  finally  numbers  his 
years  upon  his  right  hand."3 


Pftpg 

rnV  :'\  r«j' 


FINGER  SYMBOLISM  ABOUT 
THE  YEAR   1140 

One  of  a  large  number  of 
drawings  in  a  manuscript 
copy  of  Bede's  works  in  the 
Biblioteca  Nacional  at  Mad- 
rid, dating  from  c.  1140.  The 
number  2000  is  indicated. 
From  a  photograph  by  Pro- 
fessor J.  M.  Burnam 


xOn  the  general  relation  of  the  finger  numbers  to  systems  of  counting  and 
writing  there  is  an  extensive  literature.  See,  for  example,  F.  W.  Eastlake,  The 
China  Review,  Hongkong,  IX,  251,  319,  with  a  statement  that  the  Chinese 
place  the  system  before  the  time  of  Confucius;  S.  W.  Koelle,  "Etymology  of  the 
Turkish  Numerals,"  Journal  of  the  Royal  Asiat.  Soc.,  London,  XVI  (N.  S.), 
141 ;  Sir  E.  Clive  Bayley,  ibid.,  XIV  (reprint,  part  2,  p.  45  n.) ;  M.  Barbieri, 
Notizie  istoriche  del  Mat.  e  Filosofi  .  .  .  di  Napoli,  p.  10  (Naples,  1778) ; 
Villicus,  Geschichte,  p.  6;  Bombelli,  Antica  Numer.,  I,  108,  115  n.,  with  bibliog- 
raphy and  plates. 

2P.  Treutlein,  Abhandlungen,  I,  21;  H.  Stoy,  Zur  Geschichte  des  Rechen- 
unterrichtSy  I.  Theil,  Diss.,  p.  31  (Jena,  1876) . 

3Felix  nimirum,  qui  tot  per  saecula  mortem 
Distulit  atque  suos  iam  dextra  computat  annos. 

ii 


198 


FINGER  RECKONING 


That  the  system  was  familiar  to  the  people  is  evident  from 
a  remark  of  Pliny1  to  the  effect  that  King  Numa  dedicated  a 
statue  of  two-faced  Janus,  the  fingers  being  put  in  a  position  to 
indicate  the  number  of  days  in  a  common  year,  and  Macrobius 
testifies  that  the  hundreds  were  indicated  on  the  right  hand. 


FINGER  SYMBOLISM   IN   THE   13TH   CENTURY 

From  the  Codex  Alcobatiensis  in  the  Biblioteca  Nacional  at  Madrid,  dating 
from  c.  1 200.    From  a  photograph  by  Professor  J.  M.  Burnam 

The  system  was  in  use  among  the  Greeks  in  the  5th  cen- 
tury B.C.,  for  Herodotus  tells  us  that  his  countrymen  knew  of 
it.  Among  the  Latin  writers  it  is  mentioned  by  Plautus,  Seneca, 
Ovid,  and  various  others.2 

1See  I.  Sillig,  edition  of  Pliny's  works,  V,  140  (Hamburg  and  Gotha,  1851), 
and  the  Hist.  Nat.,  XXXIV,  vii,  16,  33. 

2L.  J.  Richardson,  "Digital  Reckoning  among  the  Ancients,"  Amer.  Math. 
Month.,  XXIII,  7;  Bombelli,  Antica  Numer.,  p.  102;  A.  Dragoni,  Sul  Metodo 
aritmetico  degli  antichi  Romani,  p.  10  (Cremona,  1811)  ;  Giinther,  Math.  Un- 
terrichts,  p.  12.  Possibly  Juan  Perez  de  Moya  (1573)  was  correct  in  saying  that 
the  Egyptians  used  the  system  because  they  were  "friends  of  few  words," — "los 
Egipcianos  eran  amigos  de  pocas  palabras  .  .  .  destos  deuio  salir." 


fciftinctio fcnmda.Zracfata0<luartiig 


PACIOLI  ON  FINGER  SYMBOLISM 

From  the  Suma  of  Pacioli,  Venice,  1404.    The  two  columns  at  the  left  represent 
the  left  hand,  the  other  two  representing  the  right  hand 


200  FINGER  RECKONING 

Finger  symbolism  was  evidently  widely  spread  during  many 
centuries,  for  there  are  also  numerous  references  to  it  in  both 
the  Hebrew  and  the  Arabic  literature.  Our  precise  knowledge 
of  the  subject  is  due  chiefly,  however,  to  a  few  writers, — to  the 
Venerable  Bede  (c.  710),  Nicholas  Rhabdas  (c.  1341),  and  a 
Bavarian  writer,  Aventinus  (I522).1  In  the  works  of  these 
writers  the  system  is  fully  described,  but  brief  summaries,  often 
with  illustrations,  may  be  found  in  various  books  of  the  i6th 
century,  including  those  of  Andres2  (1515),  Recorde3  (c.  1542), 
Moya4  (1562),  Valerianus5  (1556),  and  Noviomagus6  (1539). 
The  later  literature  of  the  subject  is  also  extensive.7 

The  general  scheme  of  number  representations  may  be  suf- 
ficiently understood  from  the  illustrations  given  from  the  works 
of  Pacioli  and  Aventinus.  Bede  gives  a  description  of  upwards 
of  fifty  finger  symbols,  the  numbers  extending  through  one 
million.  No  other  such  extended  description  has  been  given 
except  the  one  of  Rhabdas,  but  the  works  of  Pacioli  and 
Aventinus  contain  what  are  probably  the  best-known  pictorial 
illustrations  of  the  process. 

1Bede,  "De  loquela  per  gestum  digitorum,"  in  his  Opera  Omnia,  I,  686  (Paris, 
1850);  Nicholas  Rhabdas,  "E/c0pa<m  roO  datcrvXiKov  ^rpov  ;  J.  Aventinus,  Ab acvs 
atqve  vetvstissima,  vetervm  latinorum  per  digitos  manusqj  numerandi  .  .  .  cosue- 
twd0,Nurnberg,  1522  (title  from  Regensburg  edition,  1532).  See  also  St.  Augustine, 
Enarrationes  in  Psalmos,  xlix,  9,  i;  Sermones,  ccxlviii,  ccxlix,  cclii;  and  Contra 
lulianum,  iii,  n,  22;  and  M.  Capella,  De  Nuptiis  Pkilologiae  et  Mercurii,  ii,  102, 
and  vii,  729  and  746.  On  the  Rhabdas  symbolism  see  Heath,  History,  II,  551. 

2Mosseru  Juan  Andres,  Sumario  breve  d'  la  prdtica  de  la  Arithmetica,  Va- 
lencia, 1515.  3  Ground  of  Artes,  London,  c.  1542. 

*  Arithmetics  Practica,  p.  627  (Salamanca,  1562). 

5  Joannes  Pierius  Valerianus  Belluncnsis,  Hieroglyphica,  p.  454   (Frankfort 
a.  M.,  1556;  1614  ed.). 

6  Cap.  XIII  of  the  1544  edition  of  his  De  nvmeris  llbri  II,  Cologne. 

7 -E.g.,  see  Abhandlungen,  V,  91,  100;  the  Basel  edition  of  St.  Jerome's  works, 
IX,  8  (1516) ;  L.  A.  Muratori,  Anecdota,  Naples,  1776,  with  the  "Liber  de  com- 
pute S.  Cyrilli  Alexandrini " ;  V.  Requeno,  Scoperta  delta  Chironomia,  Parma, 
1797,  with  illustrations;  M. 'Steinschneidcr,  BibL  Math.,  X  (2),  81 ;  A.  Marre, 
"Maniere  de  compter  des  anciens  avec  les  doigts  .  .  . ,"  Boncompagni's  Bul- 
lettino,  I,  309;  Bombelli,  Antica  Numer.,  cap.  xiv,  especially  p.  109  n.;  A.  F.  Pott, 
£)ie  quinare  und  vigesimale  Zdhlmethode  bei  Volkern  alter  Welttheile,  Halle, 
1847,  with  an  "Anhang  liber  Fingernamen,"  p.  225;  F.  T.  Elworthy,  The  Evil 
Eye,  p.  237  (London,  1895),  with  illustrations;  E.  A.  Bechtel,  "Finger-Counting 
among  the  Romans  in  the  Fourth  Century,"  Classical  Philology,  IV,  25. 


THE  SYMBOLS 


201 


The  representation  of  numbers  below  100  was  naturally 
more  uniform,  since  they  were  in  international  use  by  the 
masses,  while  the  representation  of  the  higher  numbers  was 
not  so  well  standardized. 

Finger  Computation.  From  finger  notation  there  developed  an 
extensive  use  of  finger  computation.  This  began,  of  course, 
with  simple  counting  on  the  fingers,  but  it  was  extended  to  in- 
clude particularly  the  simpler  cases  of  multiplication  needed 


AVENTINUS  ON  FINGER  SYMBOLS 

From  the  Abacvs  of  Johannes  Avcntinus,  Ntirnberg,  1522  (Regensburg 
edition  of  1532) 

by  the  illiterate.  For  example,  to  multiply  7  by  8,  raise  two 
fingers  on  one  hand  and  three  on  the  other,  since  5+2  =  7  and 
5  +  3  =  8.  Then  add  the  numbers  denoted  by  the  raised  fin- 
gers, 2  +  3  =  5,  and  multiply  those  denoted  by  the  others, 
3.2  =  6,  and  the  former  result  is  the  tens,  50,  and  the  latter  is 
the  units,  the  product  being  56.  This  depends,  of  course,  upon 
the  fact  that  (10  -  a)  (10  -  b)  =  10  (5  -  a  +  5  -  6)  +  aft.1 
The  same  principle  is  frequently  seen  in  written  work  in  arith- 
metic in  the  Middle  Ages,  since  by  its  use  it  was  unnecessary 
to  learn  the  multiplication  table  above  5-5.  In  a  somewhat 

1See  also  pages  119  and  120. 


202  MODERN  CALCULATING  MACHINES 

similar  way  we  may  find  the  product  of  numbers  from  10  to  15. 
For  example,  to  find  the  product  of  14  and  13,  raise  four  fingers 
on  one  hand  and  three  on  the  other,  since  14  =  10  +  4  and 
13  =  10  +  3.  Then  to  100  add  ten  times  the  sum  of  the  num- 
ber of  fingers  raised,  and  the  product  of  the  same  numbers,  the 
result  being  100  +  10  (4 -f  3)  +  4  •  3  =  182.  The  method  is 
evidently  general,  since 

(10  +  0)  (10  +  b)  =  100  +  io(a  +  b)  +  ab. 

Such  work  is  still  to  be  seen  in  various  parts  of  the  world. 

In  the  time  of  Fibonacci  (1202)  finger  symbols  were  still 
used,1  especially  in  remembering  certain  numbers  in  division.2 

3.  MODERN  CALCULATING  MACHINES 

Napier's  Rods.  The  first  important  improvement  on  the 
ancient  counter  computation  was  made  by  Napier  (1617).  In 
his  Rabdologia*  he  explains  a  system  of  rods  arranged  to  rep- 
resent the  gelosia  method  of  multiplication  as  seen  in  the  illus- 
tration on  page  203.  The  plan  shows  how  crude  were  the 
methods  of  calculating  even  as  late  as  the  iyth  century,  al- 
though it  would  have  had  some  value  in  connection  with  trigo- 
nometric functions  if  logarithms  had  not  been  invented.  These 
rods  were  commonly  known  as  Napier's  Bones,  as  in  Leybourn's 
The  Art  of  Numbring  By  Speaking-Rods:  Vulgarly  termed 
Nepeir's  Bones*  London,  1667.  They  attracted  considerable 
attention,  not  merely  in  Europe  but  also  in  China  and  Japan. 

lt{.  .  .  opportet  eos  qui  arte  abbaci  uti  uoluerint,  ut  subtiliores  et  ingeniores 
appareant  scire  computum  per  figuram  manuum,  secundum  magistrorum  abbaci 
usum  antiquitus  sapientissime  inuentam."  Liber  Abaci,  I,  5. 

2  So  Fibonacci  has  a  chapter,  "De  diuisione  numerorum  cordetenus  in  mam- 
bus  per  eosdem  numeros,"  with  such  expressions  as  "ponens  semper  in  manibus 
numeros  ex  diuisione  exeuntes."  Ibid.,  I,  30. 

3Rabdologiae,  Sev  Nvmerationis  Per  Virgulas  Libri  Dvo,  Edinburgh,  1617; 
Leyden,  1626.  Translations,  Verona,  1623;  Berlin,  1623.  Rabdologia  =  late 
Greek  pa(38o\oyla  (rhabdologi'a),  a  collection  of  rods,  from  pdpdos  (rhab'dos, 
rod)  +  \oyla  (logi'a,  collection).  Probably  Napier  took  the  word  from  the 
Glossaria  H.  Stephani,  Paris,  1573,  where  the  above  meaning  is  given. 

4W.  Leybourn  (c.  1670)  derived  Rabdologia  from  pd^dos (rhab'dos,  r 
(log' os,  speech),  and  this  etymology  is  still  accepted  by  some  writers. 


NAPIER'S  RODS 


203 


Modern  Machines.  The  essential  superiority  of  the  modern 
calculating  machine  over  an  instrument  like  the  man-pan  is 
that  the  carrying  of  the  tens 
is  done  mechanically  instead 
of  being  done  by  the  opera- 
tor. For  this  purpose  a  disk 
is  used  which  engages  a  sec- 
ond disk,  turning  the  latter 
one  unit  after  nine  units  have 
been  turned  on  the  former.1 

The  first  of  these  instru- 
ments seems  to  have  been 
suggested  by  a  Jesuit  named 
Johann  Ciermans,  in  1640^ 
but  apparently  nothing  was 
done  by  him  in  the  way  of 
actually  constructing  such  a 
machine. 

The  real  invention  may 
properly  be  attributed  to  Pas- 
cal (1642),  who,  at  the  age 
of  nineteen  and  after  many 
attempts,  made  an  instru- 
ment of  this  kind,  receiving 


NAPIER  S   RODS   IN   JAPAN 


The  Napier  Rods  found  their  way  into 
China  at  least  as  early  as  the  beginning 
of  the  1 8th  century,  and  into  Japan  in 
the  century  following.  This  illustration 
is  from  Hanai  Kenkichi's  Seisan  Sokuchi, 
of  the  middle  of  the  ipth  century 


(1649)  a  royal  privilege  for 
its    manufacture,3    and    one 
particularly  interesting  speci- 
men is  still  preserved  in  the  Conservatoire  National  des  Arts  et 
Metiers  at  Paris.    It  is  an  adding  machine  adapted  to  numbers 

i-For  a  succinct  description  of  modern  machines  see  F.  J.  W.  Whipple,  "Cal- 
culating Machines,"  in  E.  M.  Horsburgh,  Handbook  of  the  Exhibition  at  the 
Napier  Tercentenary  Celebration,  p.  69  (Edinburgh,  1914),  hereafter  referred  to 
as  Horsburgh,  Handbook.  For  slide  rules,  ibid.,  pp.  155  and  163. 

2Kiistner,  GeschicMe,  III,  438;  Cantor,  Geschichte,  II  (i),  657. 

3 Cantor,  Geschichte,  II  (i),  661 ;  M.  D'Ocagne,  Le  Calcul  simplifie  par  les 
procedes  m&caniques  et  graphiques,  Paris,  1894;  2d  cd.,  1905;  "Histoire  des  ma- 
chines a  calculer,"  Bulletin  de  la  Societe  d' Encouragement  pour  I'Industrie 
Nationale,  tome  132,  p.  554,  and  other  articles  in  the  same  number,  with  an 
extensive  bibliography  on  pages  739-759. 


204  MODERN  CALCULATING  MACHINES 

of  six  figures  and  is  one  of  the  later  attempts  of  Pascal.    On  the 
inside  of  the  box  is  this  inscription : 

Esto  probati  instrument!  symbolum  hoc ;  Blasius  Pascal ;  arver- 
nus,  inventor.  20  mai  I652.1 

In  the  same  museum  there  are  two  other  machines,  apparently 
also  of  Pascal's  make,  one  of  which  was  verified  and  presented 
by  a  collateral  descendant. 

In  1673  Sir  Samuel  Morland  (1625-1695),  an  English  diplo- 
mat, mathematician,  and  inventor,  made  a  machine  for  mul- 
tiplying, and  about  the  same  time  (1671)  Leibniz  constructed 
one  in  Germany.  In  1709  the  Marchese  Giovanni  Poleni 
(1683-1761),  then  professor  of  astronomy  at  Padua,  made  a 
similar  attempt  in  Italy;  and  in  1727  there  was  described 
in  Germany  a  machine  constructed  just  before  his  death  by 
Jacob  Leupold  (1674-1727),  a  Leipzig  mechanic.  These  vari- 
ous attempts  were  recorded  in  I7352  by  Christian  Ludwig  Ger- 
sten  (1701-1762),  then  professor  of  mathematics  at  Giessen,  in 
connection  with  a  description  of  a  machine  invented  by  him- 
self. It  was  not,  however,  until  the  igth  century  that  any 
great  advance  was  made.  In  1820  Charles  Babbage  began  the 
construction  of  a  machine  for  calculating  mathematical  tables, 
and  in  1823  the  Royal  Society  secured  aid  from  the  British 
government  to  enable  him  to  continue  his  work.  Babbage's 
progress  not  being  satisfactory,  this  aid  was  soon  withdrawn, 
but  the  work  continued  until  1856,  when  it  was  abandoned.8 
From  the  time  when  Babbage  began  to  the  present,  however, 
the  modern  calculating  machine  has  been  constantly  improved, 
first  by  Thomas  de  Colmar  (1820),  and  various  types  are  now 
in  extensive  use.4 

^'Let  this  signature  be  the  sign  of  an  approved  instrument.  Blaise  Pascal, 
of  Auvergne,  inventor.  May  20,  1652." 

2  Phil.  Trans.,  Abridgment,  1747,  VIII,  16. 

3 One  of  the  best  descriptions  of  this  machine  is  given  in  Babbage's  Calculat- 
ing Machine;  or  Difference  Engine,  printed  by  the  Victoria  and  Albert  Museum, 
London,  1872;  reprinted  in  1907. 

4  There  is  a  large  collection  in  the  Conservatoire  National  des  Arts  et  Metiers 
at  Paris. 


THE  SLIDE  RULE  205 

Slide  Rule.  In  1620  Edmund  Gunter  designed  the  logarithmic 
"line  of  numbers,"  on  which  the  distances  were  proportional  to 
the  logarithms  of  the  numbers  indicated.  This  was  known  as 
Gunter 's  Scale,  and  by  adding  or  subtracting  distances  by  the 
aid  of  compasses  it  was  possible  to  perform  multiplications  and 
divisions.  Thus  the  inventor  worked  out  the  principle  of  the 
slide  rule,  but  instead  of  having  the  sliding  attachment  he  used 
a  pair  of  compasses.1  This  instrument  was  subsequently  used 
in  navigation. 

In  1628  Edmund  Wingate  published  at  London  his  Con- 
struction and  Use  of  the  Line  of  Proportion,  but  this,  like 
Gunter's  Scale,  was  merely  a  rule  in  which  the  spaces  on  one 
side  indicated  numbers,  while  those  on  the  other  indicated  the 
mantissas  of  these  numbers. 

About  1622  William  Oughtred  invented  the  slide  rule,2  but 
descriptions  of  his  instrument  did  not  appear  in  print  until 
1632.  A  pupil  of  Oughtred's,  Richard  Delamain,  published  at 
London  in  1630  a  small  pamphlet  entitled  Grammelogia;  or  the 
Mat  hematic  all  Ring,  in  which  he  described  a  circular  slide  rule, 
apparently  of  his  own  invention.  Oughtred,  however,  seems 
unquestionably  to  have  invented  the  rectilinear  logarithmic  slide 
rule,  and  also,  independently  of  Delamain,  to  have  invented  a 
circular  one. 

In  the  year  1654  a  slide  rule  was  made  in  which  the  slide 
moved  between  parts  of  a  rigid  stock,  and  a  specimen  of  this 
type,  now  in  the  Science  Museum  at  South  Kensington,  is  in- 
scribed "Made  by  Robert  Bissaker,  1654,  for  T.W."3  Who 
this  T.  W.  was  we  do  not  know,  but  the  invention  was  a  notable 
step  in  the  development  of  the  modern  type. 

From  that  time  on  there  were  numerous  inventors  who  im- 
proved upon  the  instrument.  Among  them  were  various  obscure 
artisans,  but  there  was  also  Newton,  who  devised  a  system  of 


1On  this  entire  topic  see  F.  Cajori,  A  History  of  the  Logarithmic  Slide  Rule, 
New  York,  1909;  hereafter  referred  to  as  Cajori,  Slide  Rule. 

2F.  Cajori,  William  Oughtred,  p.  47  (Chicago,  1916) ;  hereafter  referred  to 
as  Cajori,  Oughtred. 

3Horsburgh,  Handbook,  p.  163. 


206  MODERN  CALCULATING  MACHINES 

concentric  circles  for  the  solution  of  equations.  In  the  i7th 
century  the  slide  rule  of  the  type  now  used  attracted  little 
attention,  either  in  England  or  on  the  continent.  In  the  fol- 
lowing century,  however,  its  value  began  to  be  recognized,  and 
the  instruments  in  use  at  that  time  resemble  in  several  particu- 
lars those  with  which  we  are  familiar.  About  1748  George 
Adams  made  spiral  slide  rules  that  were  carefully  engraved 
and  probably  were  of  a  higher  degree  of  accuracy  than  those 
of  his  predecessors. 

The  first  one  to  make  a  decided  step  in  advance,  however, 
was  William  Nicholson  (1753-1815).  He  described  (1787) 
the  various  types  of  rules  then  known,  and  suggested  note- 
worthy improvements,  particularly  in  the  way  of  a  rule  which, 
through  the  device  of  a  system  of  parallels,  gave  the  effect  of 
an  instrument  more  than  20  feet  in  length.  He  also  designed 
a  spiral  slide  rule,  apparently  ignorant  of  the  work  done  in  this 
field  by  various  predecessors.  At  about  the  same  time  various 
French  and  German  writers  contributed  to  the  perfecting  of 
the  instrument,  notably  Jean  Baptiste  Clairaut  (1720),  who 
designed  a  new  circular  slide  rule. 

The  most  marked  advance  in  the  middle  of  the  iQth  century 
was  made  by  Amedee  Mannheim1  (1831-1906),  who  (c.  1850) 
designed  the  Mannheim  Slide  Rule,  which  is  still  a  standard, 
although  modified  in  various  particulars.2  These  modifications 
related  (i)  to  increasing  the  length  of  the  scales  without  in- 
creasing the  size  of  the  instrument;  (2)  to  adapting  the  rule  to 
specialized  branches  of  science;  and  (3)  to  increasing  the 
mechanical  efficiency  of  the  device.3  Few  such  instruments 
have  gained  so  much  popularity  in  such  a  short  time. 

"t-L'Enseignement  Mathematique,  IX  (1907),  i6q. 

2 For  details  as  to  other  inventors  see  Cajori,  Slide  Rule.  On  the  history  of 
the  planimeter,  which  may  be  classified  among  instruments  relating  to  the  cal- 
culus or  among  those  having  to  do  with  calculation,  see  A.  Favaro,  "Beitrage 
zur  Geschichte  der  Planimeter,"  Separat-Abdruck  aus  der  Allgemeinen  Bauzei- 
tungj  Vienna,  1873.  The  instrument  seems  to  have  been  first  designed  c.  1814  by 
J.  M.  Hermann,  but  it  attracted  little  attention.  The  first  published  description 
was  that  of  an  Italian  inventor,  Gonella;  it  appeared  in  1825. 

3Horsburgh,  Handbook,  p.  156. 


DISCUSSION  207 

TOPICS  FOR  DISCUSSION 

1.  Consider  the  difficulties  of  multiplying  a  number  like  4275 
by  a  number  like  876,  using  only  the  Roman,  Greek,  Egyptian,  or 
Babylonian  numerals. 

2.  Reason  for  the  persistence  of  the  abacus  in  business  calcula- 
tions until  the  i7th  century. 

3.  Reason   for  abandoning   the  abacus  in   Italy   before  it  was 
abandoned  in  northern  Europe. 

4.  The  various  etymologies  of  the  term  " abacus"  as  given  in  such 
dictionaries  as  are  accessible. 

5.  Reason  for  believing  that  the  origin  of  the  line  abacus  may  be 
Semitic. 

6.  Etymologies  of  the  terms  used  in  connection  with  the  abacus, 
and  the  relation  of  the  word  "calculus"  to  other  words  in  our  language. 

7.  Words  used  in  various  languages  to  mean  computing  disks, 
with  their  etymologies. 

8.  Comparison  of  the  various  types  of  line  abacus,  with  a  discus- 
sion of  their  respective  merits. 

9.  The  various  forms  of  the  abacus  used  in  the  Far  East,  with  a 
comparison  of  their  merits. 

10.  A  study  of  the  evolution  of  the  paper  tablet  used  for  computa- 
tion by  pupils  at  the  present  time,  beginning  possibly  with  the  dust 
table  or  the  wax  tablet. 

11.  Gerbert's  abacus  and  its  chief  defects. 

12.  History  of  the  British  Court  of  the  Exchequer. 

13.  General  use  of  counters  in  the  countries  of  Western  Europe. 

14.  History  of  the  tally. 

15.  The  history  of  finger  symbols  and  finger  computation,  with 
special    reference   to   the   international   character   of    the   symbols 
themselves. 

1 6.  The  general  character  of  the  quipu  and  of  similar  knot-tying 
devices  in  various  parts  of  the  world. 

17.  Relation  of  the  "cat's  cradle"  to  the  knotted  cords  and  pos- 
sibly to  the  tying  together  of  the  stars  to  make  the  constellations  of 
ancient  astronomy. 

1 8.  Rise  of  the  modern  calculating  machine. 

19.  Types  and  history  of  modern  calculating  machines  and  pla- 
nimeters  as  described  in  current  encyclopedias. 


CHAPTER  IV 

ARTIFICIAL  NUMBERS 

i.  COMMON  FRACTIONS 

Origin  of  Artificial  Numbers.  The  natural  numbers  seem  to 
have  served  the  purposes  of  the  world  until  about  the  beginning 
of  the  historic  period.  Men  broke  articles  and  spoke  of  the 
broken  parts,  but  even  after  weights  came  into  use  it  was  not 
the  custom  to  speak  of  such  a  fraction  as  *J  of  a  pound.  The 
world  avoided  difficulties  of  this  kind  by  creating  such  smaller 
units  as  the  ounce  and  then  speaking  of  the  particular  number 
of  ounces.  For  example,  the  commercial  fractions  of  Rome 
were  referred  to  the  as,1  16  asses  making  a  denarius?  A  twelfth 
part  of  the  as  was  the  uncia,  whence  the  modern  "ounce"  and 
"inch."  Hence  the  Romans  used  this  scheme  : 

Multiples  of  the  as 

1  as  =  -^Q  denarius  =  2\  -f-  -^  =  Denarii  semuncia  sicilicus 

2  asses  =  \  denarius  =  ^  4-  o\  =  Denarii  uncia  semuncia 

3  asses  —  -^\  denarius  =  £  +  ^  =  Denarii  sextans  sicilicus 

15  asses  =  ¥$  denarius  ^  12  +  iV  ~  Denarii  deunx  sicilicus 

Submultiples  of  the  as 

\\,  deunx,  i.e.,  i  —  ^V*  de  uncia,  -^  taken  away.    The  symbol 

is  S  —  =  ~ ,  meaning  semis  +  fV 
{f,  dextans,  i.e.,  i  —  ±,  de  sextans,  J  taken  away.    The  symbol 

is  S  =  =,  semis  -f  ^$- 

1  Originally  a  pound  of  copper,  but  reduced  by  successive  depreciations  of 
coin  until  (191  B.C.)  it  weighed  half  an  ounce. 

2  Originally  a  coin  of  10  asses,  but  later  of  16  asses,  about  16  American  cents. 

208 


ROMAN  FRACTIONS  209 

g-,  dodrans,  i.e.,  i—  ^  de  quadrans,  |  taken  away.    The 

symbol  is  S  =—,  semis  +T\. 
3-,  bes,  i.e.,  bi  as  for  duae  partes,  |.    The  symbol  is  $=, 


iV?  septunx,  i.e.,  septem  unciae.  The  symbol  is  $— ,  semis-}- T12. 

-^2-,  semis,  half.    The  symbol  is  $,  2,  or  (. 

1527  quincunx,  i.e.,  quinque  unciae.    The  symbol  is  =  —  — . 

^,triens,  one  third.    The  symbol  is  ==. 

-^,  quadrans,  one  fourth.    The  symbol  is  =— . 

T22,  sextans,  one  sixth.    The  symbol  is  =. 

^2,  uncia,  ounce,  inch.    The  symbol  is  — . 

There  were  similar  special  names  and  symbols  for  oV  (setn- 
uncia,  ^,(),^(sicilicus,  j),  y2,  ^| 4,  2 -\  $  (scriptulum,scripulum, 
scrupulum,  9,  surviving  in  our  " scruple"),  and  other  fractions.1 

It  will  be  seen  that  the  Roman  merchant  could  speak  of  f 
of  a  denarius  as  6  asses,  of  ^  of  the  as  as  a  semuncia,  and  so 
on,  without  considering  fractions  at  all,  and  this  was  the  case 
with  all  ancient  peoples.  In  fact,  the  origin  of  such  compound 
numbers  as  3  yd.  2  ft.  8  in.  is  to  be  sought  in  the  effort  of  the 
world  to  avoid  the  use  of  fractions. 

Gradually,  however,  the  notion  of  a  unit  fraction  developed ; 
then  came  the  idea  of  a  general  fraction;  then  the  surd  ap- 
peared ;  and  so  on  through  various  types  of  fractions,  irrational 
numbers,  transcendental  numbers,  complex  numbers,  and  other 
kinds  of  artificial  numbers.  Each  was  created  to  satisfy  an 
intellectual  need,  and  in  due  time  each,  excepting  the  latest 
creations,  has  satisfied  important  practical  needs  as  well. 

First  Steps  in  Fractions.  The  first  satisfactory  treatment  of 
fractions  as  such  is  found  in  the  Ahmes  Papyrus  (c.  1550  B.C.).2 

1For  a  brief  discussion  of  Roman  fractions,  with  bibliography,  see  Pauly- 
Wissowa,  Real-Encyclopadie,  II,  1114  (Stuttgart,  1896);  hereafter  referred  to 
as  Pauly-Wissowa.  See  also  Ch.  Daremberg  and  E.  Saglio,  Dictionnaire  des 
Antiquitds  Grecques  et  Romaines,  Paris,  1877.  Of  the  early  printed  works  on 
the  subject  the  classical  one  is  G.  Bud6  (or  Budaeus) ,  De  asse  et  partibus  ejus, 
Libri  V,  Paris,  1516,  with  several  later  editions. 

2F.  Hultsch,  Die  Elemente  der  dgyptischen  Theilungsrechnung,  reprint  from 
the  Abhandl.  d.  k.  Sachs.  Gesellsch.  d.  Wissensch.,  Bd.  XXXIX;  Eisenlohr,  Ahmes 
Papyrus ;  Peet,  Rhind  Papyrus,  where  the  date  is  put  somewhat  earlier. 


210  COMMON  FRACTIONS 

Artificial  numbers  of  this  kind  had  already  been  used  by  the 
Babylonians,1  but  we  have  no  noteworthy  treatment  of  frac- 
tions prior  to  the  work  of  Ahmes.  The  notion  of  the  unit  frac- 
tion was  already  old  in  Egypt,  however,  for  the  tables  given  by 
Ahmes  bear  evidence  of  a  development  through  a  long  period. 
The  essential  feature  of  the  early  Egyptian  treatment  is  the 
unit  fraction.  The  arithmeticians  had  long  been  able  to  con- 
ceive of  TV ,2  but  they  had  no  plural  for  it  either  verbally*  or 
mentally.  By  the  time  of  Ahmes,  however,  an  idea  akin  to  that 
of  ratio  had  developed.  The  number  2  was  divided,  say  into 
43  equal  parts,  and  what  is  essentially  the  ratio  of  2  to  43,  or 
twice  ^,  was  expressed,  using  modern  symbols,  as 

2  :  43  =  -S3  +  sV  +  1^9  +  31)  i- 

Indeed,  most  of  the  ancient  theory  of  fractions  centered  about 
the  concept  of  ratio,  and  in  such  theoretical  works  as  that  of 
Boethius  it  lasted  until  the  i6th  century. 

In  the  Ahmes  Papyrus  the  fraction  ^  ?  for  example,  is  writ- 
ten il~,  where  the  dot  is  the  unit-fraction  symbol,  — »  is  the 
Ahmes  hieratic  symbol  for  40,  and  il  is  used  to  denote  2.3  It  is 
a  curious  fact  that  the  dot  is  occasionally  found  in  modern 
times  as  a  fraction  symbol,  as  in  the  case  of  -•-  and  |  for 
\  and  {  in  English  copy-books  of  the  i8th  century. 

How  these  unit  fractions  were  derived  we  do  not  know.  It  is 
evident  that  more  than  one  solution  is  possible,  but  it  is  not 
always  evident  why  any  given  one  should  be  preferred  to  any 
other.  For  example, 


—     1_  _L     1      _L-        1 
~~  30     '    "8"6  "•     G"4~7V 

"  36   +  86   +  6  4T  +  TV  2  +  T¥ 

==  40"  +  '86TT  +  TY20 

~  4  2~  +  86  +  T2  9  +  .TOT' 


ID.  E.  Smith,  "The  Mathematical  Tablets  of  Nippur,"  in  the  Bulletin  of  the 
Amer.  Math.  Soc.,  XIII  (2),  392;  H.  F.  Lutz,  "A  Mathematical  Cuneiform 
Tablet,"  American  Journal  of  Semitic  Languages,  XXXVI,  240. 

2 Re-met,  "mouth  of  ten."   Erman,  Egypt,  p.  365.   Compare  the  Hebrew  pe-esr. 

3For  the  complete  work  in  facsimile,  see  the  British  Museum  edition. 

4Eisenlohr,  Ahmes  Papyrus,  p.  12 ;  Peet,  Rhind  Papyrus,  p.  42. 


EGYPTIAN  FRACTIONS  211 

and  so  on,  to  which,  of  course,  may  be  added  ^  +  ^ .  Of  all 
these  possibilities  Ahmes  and  his  predecessors  took  the  form 

although  214  +  2G8+T?jV2  has  the  advantage  that  the  first  frac- 
tion is  nearer  the  value  of  ^  than  it  is  in  the  others.  Although 
there  are  numerous  rules  for  forming  the  unit  fractions,  no  one 
of  them  applies  to  all  the  cases.  This  shows  that  the  treatise 
combined  the  results  of  earlier  computers,  each  working  by  a 
secret  rule  of  his  own,  or  else  that  each  solution  was  worked  out 
laboriously  by  repeated  trials.1 

The  Egyptians  indicated  a  unit  fraction  by  a  fraction  symbol 
with  the  denomination  underneath.  In  hieroglyphics  this  sym- 
bol was  <==> ,  but  in  the  cursive  hieratic  writing  it  was  merely  a 
dot.  Thus,  1-,  -^0,  and  ^appear  respectively  as  iTrrf^n™,  and 
^  appears  as  either  Tr^  or  £fn  in  hieroglyphic, but  in  the  hieratic 
it  appears  as  shown  on  page  210.  For  i-  i  ?  i  ?  and  |  there  were 
special  symbols,  this  having  been  rendered  necessary  by  the 
frequent  use  of  these  fractions.  Thus  the  symbol  forf  was  <£K 

The  symbol  <o  was  also  used  with  a  different  meaning.  F'or 
example,  in  the  Archeological  Museum  at  Florence  there  is  a 
marble  cubit  divided  into  parts  marked  with  such  characters  as 
fr?  nli  Sf?.  -  m^  representing  3,  4,  5,  ...  1 6  fractional  parts, 
not  3,  I*  i>  •  •  •  TO •  ^n  ^e  Louvre  there  is  a  similar  measure 
made  of  wood  with  the  symbols  \\\<=>  •••  f)01. •••!!,'  O0-  ^ 
should  be  said,  however,  that  the  first  of  these  symbols  may 
be  looked  upon  as  meaning  J  if  we  consider  it  as  applying  to  ^ 
of  the  subdivision  of  the  cubit,  say  to  |  of  an  inch,  and 
similarly  for  the  other  fractions. 

1 E.  g.,  when  b  +  c  =  ka,  we  have 

a  i  i 


be       ,     b  -f  c  b  +  c 

b c  - 

a  a 

2           2 
and  this  gives  the  Ahmes  result  in  certain  cases  but  not  in  others.    Thus,  —  = » 

and  3  +  5—4.2.  This  fraction,  therefore,  is  equal  to  ^  -f  J(1,  but  Ahmes  gives 
\o  +  nV  See  Eisenlohr,  Ahmes  Papyrus,  p.  28;  G.  Lofia,  Bibl.  Math.,  VI  (2), 
97;  VH  (2),  84;  Peet,  Rhind  Papyrus,  p.  34. 


2i2  COMMON  FRACTIONS 

Later  Development  of  Unit  Fractions.  The  separation  of  a 
fraction  into  partial  fractions  is  an  illustration  of  the  force  of 
tradition.  The  predecessors  of  Ahmes  decomposed  their  quo- 
tients in  this  way,  and  so  Ahmes  did  the  same.  Although  the 
Greeks  had  meanwhile  developed  a  fairly  good  system  of  frac- 
tions, Heron  (c.  50?)  followed  the  Egyptian  tradition,  adopting 
the  standard  set  by  Ahmes  nearly  two  thousand  years  earlier.1 
Some  six  or  seven  centuries  later,  so  the  Akhmim  Papyrus 
(c.  8th  century)  informs  us,  the  identical  method  of  Ahmes  was 
still  in  vogue  in  the  temple  schools  of  Egypt.  Even  as  late  as  the 
loth  century  Rabbi  Sa'adia  ben  Joseph  al-Fayyumi2  (died  941 ) , 
a  Hebrew  writer  living  in  Egypt,  made  much  use  of  unit  frac- 
tions in  his  computations  relating  to  the  division  of  inheritances. 

Not  all  tables  of  fractions  made  by  the  Egyptians  followed 
precisely  the  Ahmes  type,  as  may  be  seen  in  one  dating  from 
about  the  4th  century  and  recently  acquired  by  the  University 
of  Michigan.3  This  table  gives  the  unit  fractional  parts  up  to 
tenths  of  the  units  from  i  to  9,  of  the  tens  from  10  to  90,  of  the 
hundreds  to  900,  and  of  the  thousands  to  9000.  It  then  gives 
the  elevenths,  twelfths,  and  so  on  to  the  seventeenths  of  the 
units  up  to  ii,  12,  and  so  on  to  17  respectively.  For  example, 
£  of  50  is  given  as  s|  1V,  and  ^T  of  9  as  £  £  A  4\. 

Upwards  of  two  centuries  after  Rabbi  Sa'adia,  Fibonacci 
gave  a  rule  for  separating  fractions  into  partial  fractions,4  of 
which  the  separation  into  unit  fractions  is  a  special  case.  Until 
recently  our  textbooks  in  algebra  have  given  similar  directions, 
although  the  subject  had  no  immediate  application  that  the 
pupil  could  then  understand. 

In  the  Middle  Ages  unit  fractions  were  sometimes  called 
"simple  fractions,"  the  more  general  form  being  known  as 
"composite  fractions."5  These  "simple  fractions"  were  not 

1  Professor  Loria  has  called  attention  to  the  fact  that  Heron  was  not  very  accu- 
rate about  it,  for  he  gives  ij-f  I  as  ^  -f  J  +  jV,  while  the  ^3  should  be  i :  (73  +  f ). 
Bibl.  Math.,  VII  (2),  88.  *~Traitt  des  successions,  ed.  Joel  Muller.  Paris,  1897. 

3L.  C.  Karpinski,  "Michigan  Mathematical  Papyrus,  No.  621,"  in  Isis,  vol.  iv. 

4 He  called  it  a  "regula  uniuersalis  in  disgregatione  partium  numerorum."  See 
Liber  Abaci,  p.  82. 

5 So  the  Rollandus  MS.  (c.  1424)  says:  "sicut  £••£••£  que  simplices  fractoes 
dicute.  sic  §  .  £ .  $  .que  coposite  siue  pregnates  dicuntV 


UNIT  AND  GENERAL  FRACTIONS  213 

infrequently  favored  even  by  Renaissance  mathematicians  of 
some  prominence,  Buteo,  for  example,  giving  1350534!  ^  as 
the  square  of  1162 1,1  even  though  he  knew  the  other  forms. 
As  late  as  the  1 7th  century  Russian  manuscripts  on  surveying 
speak  of  a  ''half-half-half-half-half-third"  of  a  certain  measure 
instead  of  -J$  of  the  measure,2  and  even  today  the  unit  fraction 
is  used  to  some  extent  in  the  diamond  trade  in  speaking  of 
parts  of  a  carat. 

The  difficulty  met  in  early  times  in  solving  problems  involv- 
ing fractions  is  illustrated  by  an  example  from  Ahmes:  "A 
number  together  with  its  fifth  makes  21  [;  find  the  number]." 
Our  solution  would  be  f  x  =  21,  whence  #  =  {j-X2i  =  i7£; 
but  Ahmes  went  through  substantially  this  process :  Multiply- 
ing i  and  \  by  5,  we  have  5  and  i,  which  make  6,  and  this  is 
too  small.  To  find  how  many  times  too  small  we  divide  21 
by  6,  the  result  being  |  and  3.  Multiplying  5  by  this  result, 
the  answer  is  17 1-3 

Development  of  the  General  Fraction.  It  seems  probable  that, 
except  in  very  simple  cases,  the  idea  of  a  fraction  with  numera- 
tor greater  than  unity  arose  in  Babylon.  Although  the  unit 
fraction  and  possibly  some  idea  of  the  sexagesimal  fraction 
appear  in  the  cuneiform  records  of  c.  2000  B.C.,  the  fraction 
forms  also  include  special  symbols  for  f,  -*$ ,  -j%,  f ,  and  other 
cases  of  a  like  degree  of  difficulty.  No  such  elaborate  treat- 
ment of  the  subject  as  that  given  by  Ahmes,  however,  has  been 
found  as  yet  among  the  Babylonian  remains.4  In  spite  of  this 
early  use  of  the  general  fraction,  our  present  forms  are  not  due 
to  Babylonian  influences,  at  least  not  directly,  but  apparently 
to  the  Hindu  arithmeticians.5 

i-Ioan  Bvteonis  De  Qvadratvra  circuit  Libri  duo,  p.  39.   Lyons,  1559. 

2V.  V.  Bobynin,  "Quelques  mots  sur  1'histoire  des  connaissances  mathema- 
tiques,"  Bibl.  Math.,  Ill  (2),  104. 

3Peet,  Rhind  Papyrus,  p.  62,  No.  27. 

4For  examples  of  the  non-unit  fractions  among  the  Babylonians  see  Contenau, 
loc.  cit.  (cited  on  page  37),  plates  3,  35,  and  100;  Peet,  Rhind  Papyrus,  p.  28. 

5R.  C.  Dutt,  History  of  Civilization  m  Ancient  India,  I,  273  (London,  1893) ; 
hereafter  referred  to  as  Dutt,  History  Civ.  in  Anc.  India.  See  also  V.  V.  Bobynin, 
"Esquisse  de  Fhistoire  du  calcul  fractionnaire,"  Bibl.  Math.,  X  (2),  100. 

ii 


214  COMMON  FRACTIONS 

Greek  Fractions.  The  Greeks  followed  the  ancient  plan  of 
avoiding,  by  the  use  of  submultiples,  the  difficulty  of  comput- 
ing with  fractions;  but  in  due  time  the  need  for  a  fraction 
symbolism  became  so  apparent  that  they  developed  a  sys- 
tem that  served  their  purposes  fairly  well.  They  designated  | 

(rpiTov,  tri'ton)  by  the  symbol  F,  the  F  being  the  symbol  for 
three.  This  was  further  abbreviated  to  1  .  Similarly,  for  \  they 
used  A  (for  four)  with  two  accent  marks,  thus:  "A.  In  the 
same  way  they  accented  their  other  numerals,  a  method  rep- 
resented in  modern  typography  by  7",  S",  e", 

The  more  common  fractional  unit,  one  half  (^iorv^he'misy)^ 
had  a  special  symbol,  (,  which  was  often  written  in  a  form 
resembling  the  Greek  S  or  the  Latin  S.  Two  thirds  (Sipoipov, 
di'moiron)  had  various  abbreviations,  such  as  (<T/y ',  that  is,|+^.1 

Aristarchus  (c.  260  B.C.)  wrote  the  word  for  the  numerator 
and  the  numeral  for  the  denominator,  as  we  might  write  "ten 
7ists."2  Various  methods  were  afterwards  used,  such  as  writ- 
ing the  numeral  for  each  term  but  doubling  it  for  the  denom- 
inator, as  in  the  case  of  2/5//5"(j8W/)  for  |;  or  writing  the 
numerator,  then  the  words  "in  part,"3  and  finally  the  de- 
nominator, as  in  the  case  of  "3,069,000  in  part  331,776,"  for 
^WiWe0"-4  Heron  .(c.  50?)  and  Diophantus  (c.  275)  used  a 
symbol  that  naturally  seems  strange  to  the  modern  reader, 
namely,  our  common  fraction  reversed ;  that  is,  they  wrote  the 
equivalent  of  \9  or  -\9-  for  four  nineteenths,5  and  similarly  in 
other  cases.  Ordinarily,  however,  the  unit  fraction  was  pre- 
ferred, }f  being  written  as  i  +  \  +  \  +  ^  .6 

Roman  Fractions.  As  already  stated,  the  Romans,  like  their 
predecessors,  avoided  fractions  to  a  great  extent  by  the  device 

1For  bibliographical  references  and  for  this  general  topic  see  Pauly-Wissowa, 
II,  1077;  Heath,  History,  I,  42. 

2A^a  oa".   All  the  Greek  symbols  used  hereafter  in  this  section  are  modern. 

8'Ev  v-optv,  from  phpiov,  a  piece,  portion,  or  section,  much  as  we  should 
say  "divided  by."  

4  In  modern  Greek  symbols,  re.  ,0  pop.  \y.  /x^oc. 

5Le->  '/(V)'  or  P°ssibly  y  ( VJ),  for  TV 

6/.e.,  (5'V'ic".    For  a  further  discussion,  see  Pauly-Wissowa  and  Heath. 


METHODS  OF  WRITING  215 

of  compound  numbers,  although  using  a  few  convenient  sym- 
bols. Even  such  names  as  semuncia  (half  -twelfth)  were  not 
numerous.  Marcus  Terentius  Varro1  (116-28  B.C.)  mentions 
twelve  such  fractions,  and  Volusius  Maecianus2  (2d  century) 
gives  only  two  more.  Of  the  later  Latin  writers,  Isidorus 
(c.  610)  mentions  only  eight  and  Papias  (nth  century)  has 
eighteen.3  Adelard  of  Bath  (c.  1120)  mentions  twenty-four. 

Chinese  Fractions.  The  Chinese  seem  to  have  made  use  of 
fractions  of  considerable  difficulty  at  a  very  early  date.4  The 
Chou-pe'i,  probably  of  about  1105  B.C.  but  possibly  much  ear- 
lier, has  various  problems  involving  such  numbers  as  247-^^, 
not  stated,  however,  in  numerical  symbols  but  given  in  words. 
The  work  includes  such  divisions  as  that  of  119,000  by  iSaf, 
both  of  these  expressions  being  multiplied  by  8  before  dividing. 
The  unit  fraction  also  entered  into  their  work,  as  it  did  in  all 
earlier  civilizations.  For  example,  in  the  Nine  Sections,  a  work 
of  very  uncertain  date  but  probably  of  the  second  millennium 
B.C.,5  there  is  given  the  problem  : 

There  is  a  field  whose  length  is  one  pu  and  a  half,  one-third  pu, 
one-fourth  pit,  and  one-fifth  pu.  If  the  area  is  240  square  pu,  what  is 
its  breadth  ? 

Present  Writing  of  Common  Fractions.  It  is  probable  that  our 
method  of  writing  common  fractions  is  due  essentially  to  the 
Hindus,  although  they  did  not  use  the  bar.  Brahmagupta 
(c.  628)  and  Bhaskara  (c.  1150),  for  example,  wrote  f  for  |.6 
The  Arabs  introduced  the  bar,  but  it  was  not  used  by  all  their 
writers,  and  when  Rabbi  ben  Ezra  (c.  1140)  adopted  the 
Moorish  forms  he  generally  omitted  it.7  It  is  ordinarily  found 


Lingua  Latina,  ist  ed.  s.l.a.,  but  Rome.  Hain  mentions  six  editions  s.l.a. 
before  1501,  and  one  dated  1474  and  another  1498. 

2Assis  Distributio^  ist  ed.,  Paris,  1565. 

*Vocabularium,  ist  ed.,  Milan,  1476.  For  a  full  discussion  of  these  fractions 
see  Boncompagni's  Bullettino,  XIV,  71,  IOQ.  Not  the  Papias  of  the  2d  century. 

4Y.  Mikami,  "Arithmetic  with  Fractions  in  Old  China,"  Archiv  for  Mathe- 
matik  og  Naturvidenskabj  Christiania,  XXXII,  No.  3. 

5  See  Volume  I,  page  32. 

6Taylor,  Lilawati,  Introd.,  p.  12;  text,  p.  24  n.;  Villicus,  Geschichte,  p.  54. 

7Silberberg,  Sefer  ha-Mispar,  p.  104. 


216  COMMON  FRACTIONS 

in  the  Latin  manuscripts  of  the  late  Middle  Ages,  but  when 
printing  was  introduced  it  was  frequently  omitted,  doubtless 
owing  to  typographical  difficulties.  This  inference  is  confirmed 
by  such  books  as  Rudolffs  Kunstliche  rechnung  (1526),  where 
the  bar  is  omitted  in  all  ordinary  fractions  like  |  and  a82  but  is 
inserted  in  all  fractions  printed  in  larger  type  and  in  those  hav- 
ing large  numbers.1  The  same  inference  is  drawn  from  his 
Exempel-Biichlin  (1530),  ^  having  the  bar  because  that  frac- 
tion was  in  the  font,  and  the  other  fractions  not  having  it 
because  of  the  necessity  for  piecing  them  up.  One  of  the  inter- 
esting evidences  of  the  troubles  of  early  printers  is  seen  in 
Ciacchi's  Regole  generali  d'  abbaco  (Florence,  1675),  where, 
in  order  to  secure  better  alignment,  every  fraction  in  the  book 
is  set  up  like  ^,  for  |.  The  difficulties  of  the  early  printers 
probably  account  also  for  such  forms  as  "Z3&  septe  octaui" 
for  2  3 1,  and  "Z3&V  octaui"  for  23|,inChiarino's  work  of  1481. 
The  omission  of  the  bar  was  not,  however,  entirely  a  matter 
of  typography.  Hylles  (1592),  for  example,  omitted  it  after 
the  first  fraction  in  a  case  like  |  of  f  of  |,  writing  this  expres- 
sion ^  •  |  •  I  and  saying : 

And  here  you  see  the  first  fractions  to  wit  ^  being  a  true  fraction, 
written  with  his  lyne  as  it  ought  to  be.  and  the  other  two  that  is  to 
say  |  and  I  to  be  written  without  any  lyne  as  their  vse  and  order  is.2 

Recorde  (c.  1542)  tells  us  that  "some  .  .  .  expresse  them  thus 

3 
* 

in  slope  forme,"3  as  here  shown :          f 

i 

2 

The  common  use  of  2/3  for  f  is  the  result  of  a  desire  to 
simplify  written  and  printed  forms.4 

1  Edition  of  1534  examined. 

2Fol.  n,  v.  Even  as  good  a  writer  as  Paolo  Casati,  Fabrica  et  vso  del  com- 
passo  di  proportione^  Bologna,  2d  ed.,  1685,  however,  omits  the  bar  entirely. 

3  Ground  of  Artes,  1558  ed.,  fol.  Riij,  v. 

4The  questionable  statement  that  2/3  comes  from  2  f  3,  the  f  meaning  fratto 
(fraction),  is  made  by  G.  Frizzo,  Le  Regoluzze  del  M.  Paolo  deW  Abbaco , 
Bologna,  1857;  2d  ed.,  enlarged,  Verona,  1883,  P-  45 ;  but  the  manuscript  was 
first  published  in  G.  Libri,  Histoire  des  Mathtmatiques,  III,  295. 


METHODS  OF  WRITING  2  1  7 

Since  the  bar  is  an  Oriental  device,  it  was  never  used  by  the 
Greeks  or  Romans  to  indicate  a  fraction,  at  least  in  the  way  that 
we  use  it  today.  In  Renaissance  times,  however,  when  Arab 
devices  mingled  with  classical  forms,  we  find  the  Roman  numer- 

IX 
als  occasionally  used1  in  cases  like  -rr^»  and  the  Greek  numerals 

employed2  in  a  similar  manner. 


I       ZH'efle  fi'gur  ift  t>n  befcefft  am  fi  artel 
flTI    garfscn/rtlfb  mag  limit  and)  aitt  ffinfftttl/ftw 
fecfcffau/atrt  ftbeittail  ODcrjtraf  fec^flau2c;jJrt&  rtlle 


VI       fciS  few  Beet  6  acfydt'l/tae  fern  fed^tml  to 
VlIT 


IX   &i$Si$w  bc^aigt  ann  nnc 
>Cl   IX  nau/fcer  XI»a  w  gans  madjcn 

XX 

XXXf    figt  taif/frae  ftt»  Qwcrt^tgt  tail  *^cr  aiite* 
am  gatt^mac^en  4 

3D$  fein 

w 


KOBEI/S  USE  OF  COMMON  FRACTIONS 

From  KobePs  Rechen  biechlin  (1514;  1518  ed.),  showing  the  attempt  to  use 
Roman  numerals  with  common  fractions 

The  Name  "Fraction."  The  word  "fraction"  is  from  the 
Latin  fr  anger  e  (to  break)  .  It  is  a  broken  number  and  was  often 
so  called.  Baker  (1568),  for  example,  speaks  of  "fractions  or 
broken  numbers/7  calling  a  fraction  of  a  fraction  a  "broken  of 
broken/'  and  various  other  English  writers  did  the  same.  The 
word  "fragment"  is  from  the  same  root  and  was  not  infre- 
quently used  for  "fraction."3 

1K6bel,  Ain  New  geordnet  Rechen  biechlin  (1514;  1518  ed.),  fol.  xxxiii,  r. 
2V.  Strigelius  (Strigel),  Arithmeticus  Libellvs,  Leipzig,  1563. 
3  Thus,  in  the  Italian  edition  (1586)   of  Clavius  (p.  75)   the  word  appears  as 
Jragmeto.   The  idea  goes  back  to  the  Egyptians.   See  Peet,  Rhind  Papyrus,  p.  15. 


2i8  COMMON  FRACTIONS 

The  use  of  this  root  has  not,  however,  been  universal. 
Boethius  (c.  510)  does  not  speak  of  fractions  as  such  in  his 
arithmetic,  introducing  instead  an  elaborate  system  of  ratios; 
but  in  the  geometry  attributed  to  him  there  is  a  chapter  De 
Minutiis?  so  that  if  he  spoke  of  fractions  at  all,  other  than  as 
ratios,  he  called  them  minutes,  and  in  this  he  was  followed  by 
various  medieval  writers.2  In  the  i2th  century,  for  example, 
Adelard  of  Bath  used  minutiae*  while  about  the  same  time 
Johannes  Hispalensis  preferred  jractiones*  In  the  translation  of 
al-Khowarizmi  attributed  to  Adelard,  however,  jraciones  is  used.5 
There  are  many  instances  in  the  early  printed  books  of  the 
use  of  the  two  terms  interchangeably,  each  signifying  a  common 
fraction.6  Several  reputable  writers  used  "parts"  as  a  synonym 
of  " fractions."7  In  English  the  word  "fraction"  appeared 
early,8  however,  and  has  been  the  general  favorite. 

Since  ruptus,  like  jractus,  means  broken,  this  has  been  the  root 
of  a  name  for  fraction.  In  Italian  it  appears  as  rotto  (plural, 
rotti}?  in  Spanish  as  rocto™  and  in  French  in  various  forms.11 


1Friedlein  ed.,  pp.  v,  425. 

2  We  shall  see  that  the  term  was  also  applied  specifically  to  sexagesimals, 
although  by  no  means  generally. 

3 In  his  Regular,  abaci.    See  Boncompagni's  Bullettino,  XIV,  109. 

4  In  his  Liber  Algorismi  de  pratica  arismetrice.  See  Boncompagni,  Trattati, 
II,  49;  Abhandlungen,  III,  in.  Fibonacci  (1202)  generally  used  fractio. 

r'For  jractiones.  Minuta  is  used  (for  minutae)  to  mean  sixtieths.  See  Bon- 
compagni, Trattati,  I,  17. 

"Thus  Huswirt  (1501):  "Minutia  siue  fractio  nihil  aliud  est  qj  pars  integri" 
(fol.  n,  r.)  ;  and  Clavius  (1583)  expresses  certain  of  his  quantities  "in  numeris 
fractis,  qui  alio  nomine  Mintutiae,  fractionesve  dici  solent  vulgares"  (p.  81). 

7 Thus  Fine  (1530):  "De  minutis,  siue  quotis  eorundem  integroru  partibus 
(quas  uulgares  appellant  fractiones)";  and  Gemma  Frisius  (1540):  "Fractiones 
minutias  aut  partes."  Gosselin,  in  his  translation  (Paris,  1578)  of  Tartaglia's 
arithmetic,  uses  parties  more  commonly  than  any  other  term  for  fractions. 
Hylles  (1592)  has  the  expression:  "fractions  of  fractions  (or  as  some  men  call 
them  particles,  that  is  as  you  would  say  parcels  of  parts) ."  Ramus  (1555)  speaks 
of  "fractio  sive  pars." 

8  Thus  Chaucer,  in  his  Astrolabe  (c.  1391),  uses  fraction. 

9Pacioli  (1494)   ordinarily  speaks  of  rotti,  although  he  also  uses  fractioni 
and  fracti  (fol.  48).   Most  of  the  i6th  century  Italian  writers  use  rotti. 
10Ortega  (1512). 

11Chuquet  (1484),  "nombres  routz";  Savonne  (1563),  "roupt";  and  later 
writers,  "nombre  rompu." 


VARIOUS  NAMES  FOR  FRACTION  219 

In  the  Teutonic  languages  the  custom  was  followed  of  using 
vernacular  expressions,  and  so  the  Latin  jractlo  appeared  as 
"broken  number."1 

Common  Fraction.  The  expression  "common  fraction77  was 
originally  used  to  distinguish  the  fractions  employed  in  trade 
from  the  sexagesimal  fractions  found  in  astronomy.  It  refers 
merely  to  the  form  of  writing  a  fraction,  -^  being  a  common 
fraction,  0.5  being  a  decimal,  and  30'  being  a  sexagesimal,  al- 
though the  values  of  the  three  are  the  same.  In  Latin  the  ex- 
pression was  fractiones vulgar es,  whence  the  "vulgar  fractions'7 
of  the  English.  The  adjective  "common77  is  used  at  present  in 
America,  although  this  has  not  always  been  the  case,2  nor  have 
the  English  uniformly  followed  their  present  usage.3 

Definition  of  Fraction.  In  general  a  fraction  has  been  defined 
as  one  or  more  parts,  or  equal  parts,  of  a  unit,4  sometimes  with 
the  limitation  that  the  numerator  must  be  less  than  the  denom- 
inator.5 Occasionally  the  more  scientific  writers  based  the  defi- 
nition upon  division,  usually  of  a  smaller  number  by  a  larger.0 
The  idea  of  an  improper  fraction,  like  | ,  is  a  late  development. 
Occasionally  a  i6th  century  writer  like  Recorde7  (c.  1542), 
Gemma  Frisius8  (1540),  or  Tartaglia9  (1556)  mentioned  this 
type  of  fraction  as  an  expression  of  division,  but  little  was  done 
with  it.  Complex  fractions,  those  in  which  a  fraction  appears 

1Thus  Riese  (1522)  speaks  of  "Ein  gebrochene  zal"  (1550  ed.,  fol.  I4»  v.), 
and  Grammateus  (1518)  has  a  chapter  "Von  Priichen,"  speaking  of  a  fraction 
as  "ein  iglicher  pruch  (welchen  man  in  latein  fraction  nennet)."  So  in  Dutch 
we  find  Raets  (1580)  speaking  of  "Die  ghebroken  ghetalen,"  and  Mots  (1640) 
and  others  speaking  of  "Ghebroken." 

2 Similarly  in  France,  instead  of  fraction  ordinaire  Trenchant  (1566)  used 
fraction  vulgaire.  Our  colonial  arithmeticians  usually  followed  the  English  use 
of  "vulgar." 

a  Thus  Digges  (1572)  speaks  of  "the  vulgare  or  common  Fractions." 

4 E.g.,  Pacioli  (1494):  "Rotto  e  vno  o  vero  piu  parti  de  vno  Itegro"  (fol. 
48,  f.) ;  Santa-Cruz  (1594) :  "Quebrados  es  vna  parte,  6  partes  dela  cosa  entera." 

5E.  g.,  Pagani's  arithmetic  (1591). 

6£.g.,  Ramus  (1555).  The  Dutch  arithmetic  of  Raets  (1580)  defines  a  frac- 
tion as  "een  ghetal  diuideert  met  een  grooter."  On  the  fusion  of  the  notions  oi 
fraction  and  quotient,  see  V.  V.  Bobynin,  Bibl.  Math.,  XIII  (2),  81. 

7  See  the  1558  edition  of  the  Ground  of  Artes,  fol.  S  vi,  v. 

8"Fractiones  quae  plus  Integro  valent."  9iS56  ed.,  I,  fol.  107,  r, 


220  COMMON  FRACTIONS 

in  either  numerator  or  denominator,  or  in  both,  are  older  than 
might  be  expected.  Rabbi  ben  Ezra,  for  example,  has  a  prob- 
lem involving  the  product  of  two  such  forms.1 

Terms  of  a  Fraction.  The  medieval  Latin  writers  found  it 
convenient  to  devise  names  for  the  terms  of  a  fraction  written 
after  the  Arab  manner,  and  so  they  called  the  upper  number 
by  such  names  as  numerator  (numberer)  and  numerus  (num- 
ber),2 while  the  lower  number  was  called  the  denominator 
(namer).  These  terms  are  hardly  destined  to  endure,  but  no 
others  have  been  generally  accepted.  Among  the  medieval  and 
Renaissance  writers  the  numerator  was  often  designated  by 
such  words  as  nominator*  "topterme,"  "top,"4  superior-,*  and 
denominato^  and  the  denominator  by  such  names  as  base,  in- 
ferior, and  denominante.  Both  the  numerator  and  denominator 
took  on  vernacular  forms  with  later  Teutonic  writers.7  In  the 
Latin  languages,  however,  the  favorite  names  were  numerator 
and  denominator,  the  former  of  which  Tartaglia  (1556)  speaks 
of  as  being  written  above  a  virgoletta  (little  bar),  and  the  latter 
as  being  written  below  it.8 


ha-Mispar,  39. 

2  So  the  Rollandus  MS.  (c.  1424)  speaks  of  the  "nuator  et  denomtor."    See 
also  the  correspondence  of  Regiomontanus  and  Bianchini,  Abhandlungen,  XII, 
287.    In  the  i6th  century  Ramus  (1555)  speaks  of  the  superior  terminus  as  the 
numerus  sive  numerator. 

3  .E.g.,  Digges  (1572),  although  he  also  used  numerator.   See  pages  20,  24,  27 
of  the  1579  edition. 

4Thus  Hylles  (1592):  "Numerator  which  also  for  more  shortnesse  is  some- 
times called  the  Topterme  or  top  onely:  and  that  the  lower  term  is  vsually 
called  the  Denominator  or  Base." 

5  As  in  Gemma  Frisius  (1540),  although  he  also  uses  numerator. 

6  Paolo  delP  Abaco  (c.  1340)  :  "Sappi  che  ogni  rotto  si  scrive  con  due  numeri  : 
il  minore  sta  sopra  la  verga  e  chiamasi  denominato  ;  e  il  maggiore  sotto  la  verga 
e  chiamasi  denominante"  (ed.  Frizzo,  1883,  p.  45).   The  name  was  used  by  vari- 
ous 1  6th  century  writers,  such  as  Sfortunati,  Nuovo  Lume  (1534). 

7  E.g.,  Widman  (1489)  has  the  Latin  forms,  but  a  little  later  the  words  Zeler 
and  Nenner,  with  variants,  came  into  general  use.   Occasionally  a  Dutch  writer 
like  Wentsel  (1599)   used  the  Latin  forms,  but  most  arithmeticians  preferred 
"teller"  and  "noemer,"  with  such  variants  as  "telder"  and  "nommer,"  and 
similarly  with  the  Scandinavian  writers. 

8"  .  .  .1'  uno  di  quali  e  detto  numeratore  (&  questo  si  scriue  sempre  sopra  vna 
virgoletta)  P  altro  e  chiamato  denominator,  e  questo  si  scriue  sempre  sotto  a 
quella  tal  virgoletta."  General  Trattato,  I,  106,  v.\  107,  r. 


REDUCTION  OF  FRACTIONS  221 

Reduction  of  Fractions.  Until  recently  the  reduction  of  a 
fraction  to  lower  or  lowest  terms  was  commonly  known  as 
abbreviation.  Thus  Digges  (1572)  says: 

To  abbreviate  any  Fragment,  is  to  bring  a  Fraction  to  his  lest  de- 
nomination. To  make  this  abbreuiation,  yee  must  diuide  the  Numer- 
ator of  the  Fraction,  and  so  in  the  like  maner  the  Denominator  by  the 
biggest  number,  that  is  some  common  part  of  them  both.1 

The  word  " depression"  was  also  used,  and,  like  " abbreviation," 
is  more  suggestive  than  " reduction,"2  which  sometimes  had  the 
special  meaning  of  bringing  fractions  to  a  common  denominator.3 
Before  the  invention  of  decimals  such  fractions  as  f  f  IMH 
were  not  uncommon,1  and  it  was  necessary  to  reduce  them  to 
lowest  terms  in  order  to  operate  with  them.  In  general  the 
cancellation  of  all  common  factors  was  not  convenient,  and 
hence  the  long  form  of  greatest  common  divisor  was  essential. 
First,  however,  factors  were  canceled.  A  factor  thus  elimi- 
nated was  called  by  the  Italians  a  schisatore.5  On  account  of 
the  necessity  for  recognizing  common  factors,  many  of  the 
early  manuscripts  and  printed  works  gave  the  ordinary  tests 
for  divisibility  by  2,  3,  and  5,  and  even  some  kind  of  test  for 

1i579  ed.,  p.  24.  So  Hodder  (1672  ed.)  says:  "I  would  abreviate  rs4o>"  and 
J.  Ward  (1771  ed.)  has  a  caption  "To  Abbreviate  or  Reduce  Fractions  into 
their  Lowest  or  Least  Denomination."  The  expression  is  much  older  than  this, 
however,  for  Chuquet  (1484)  says:  "Abreuier  est  poser  ou  escripre  vng  nombre 
rout  par  moins  de  figures  .  .  ."  (fol.  12,  r.y  of  his  MS.).  Early  Spanish  writers 
used  the  same  expression,  as  in  Santa-Cruz  (1594),  "De  abreuiar  quebrados." 
The  Dutch  writers  of  the  same  period  used  various  terms,  including  abbreviation 
verminderinge,  and  vercontinghe,  and  Van  der  Schuere  (1600)  says:  "om  ghe- 
broken  ghetallen  te  vercorten  ofte  minderen." 

2 Thus  Pacioli  (1494):  "De  vltima  depssione  fractorum  siue  modo  schisandi 
dicto,"  adding  "Che  I  fra^ese  si  chiama  Abreuier"  (fol.  48,  v.). 

3 E.g.,  Pellos  (1492,  fol.  21,  r.),  Chuquet  (1484,  fol.  10,  y.),  and  others.  This 
special  meaning  was  not  general,  for  Tartaglia  uses  it  in  the  broader  sense  (1592 
ed.,  I,  fol.  169,  r.). 

4  This  and  similar  fractions  are  in  the  Treviso  arithmetic  (1478).  Fractions 
like  T^W^Ar  and  48«WJ&$  are  ^iven  in  the  ^P^me  of  Clavius  (1583; 
1585  ed.,  pp.  77,  124). 

6 A  word  suggesting  canceling  "across"  (schisa),  whence  schisare,  to  reduce 
a  fraction.  So  Pacioli  (1494)  speaks  "De  diuersis  modis  in  ueniendi  schisa- 
torem"  (fol.  49,  r.),  and  Cataneo  (1546)  tells  "Come  si  schisino  i  rotti"  and 
speaks  of  "Lo  schisamento "  and  "di  schisare." 


222  COMMON  FRACTIONS 

divisibility  by  y.1  When  common  factors  were  not  readily  seen, 
the  greatest  common  divisor  was  resorted  to  at  once,  being 
found  by  the  Euclidean  method.2  This  is  given  in  al-Karkhi's 
Kdfi  fU  Hisab  (c.  iois),3  and  in  various  other  Oriental  works, 
manifestly  all  derived  from  Greek  sources. 

Greatest  Common  Divisor.  The  greatest  common  divisor  went 
by  various  names  in  the  early  printed  books.4  The  theoretical 
works  usually  gave  a  rule  for  finding  it,  although  the  mercan- 
tile works  often  omitted  the  subject  entirely,  the  former  mak- 
ing use  of  long  fractions  and  the  latter  ignoring  them.  One  of 
the  earliest  printed  rules  is  stated  by  Pacioli  (1494)  and  is 
credited  to  Boethius  (c.  Sio).5  In  this  the  smaller  term  is 
continually  subtracted  from  the  larger,  a  smaller  remainder 
from  that,  and  so  on,  an  evident  modification  of  the  Euclidean 
method.  Several  early  writers  used  the  latter  method  for  " ab- 
breviating," without  mentioning  the  greatest  common  divisor 
as  such.0 

Sequence  of  Operations.  By  analogy  to  the  sequence  of  opera- 
tions in  the  case  of  integers,  the  sequence  in  fractions  has 
generally  begun  with  addition.  Medieval7  and  Renaissance 
writers,  however,  often  took  the  more  sensible  course  of  begin- 
ning with  multiplication, — a  course  to  which  the  primary 
schools  have  now  returned.  Recorde  (c.  1542)  was  earnest  in 
his  advocacy  of  this  method,  saying : 

JA  good  illustration  of  the  use  of  these  tests  in  the  later  works  of  the  i6th 
century  may  be  found  in  Van  der  Schuere's  Arithmetica  (1600). 

2 That  is,  the  one  given  in  the  Elements.  See  Heath's  Euclid,  Vol.  II,  pp.  118,  299. 

3Hochheim  ed.,  p.  10. 

4  In  the  Latin  books  it  usually  appears  as  maximus  communis  divisor,  and  in 
the  Italian  works  as  il  maggior  comune  ripiego  (Cataneo's  spelling,  1546)  or  mas- 
sima  comune  misura  (Cataldi's  spelling,  1606). 

5"Vn  altro  modo  se  elice  da  Boetio  nel  secondo  della  sua  Arithmetica  per  tro- 
uare  ditto  schisatore"  (fol.  49,  v.).  See  Friedlein's  Boethius,  p.  77. 

6E.g.y  Chuquet  (1484),  under  "Aultre  stile  de  abreuir,"  and  the  Dutch 
writer  Petri  (1567).  Somewhat  similar  treatments  are  given  by  Baker  (1568), 
Raets  (1580),  Rudolff  (1526),  and  others.  The  phraseology  used  by  Gram- 
mateus  (1518)  is  interesting  ("ffPrikh  kleyner  zumachen"),  and  that  of 
Rudolff  is  analogous  to  it  ("Wie  man  gewiszlich  erkennen  mag/ob  ein  bruch 
mug  noch  kleiner  gemacht  werde  od'  nit"). 

7 For  example,  Abraham  ben  Ezra  (c.  1140). 


SEQUENCE  OF  OPERATIONS  223 

There  is  an  other  ordre  to  be  folowed  in  fractions  then  there  was 
in  whole  numbres.  for  in  whole  numbres  this  was  the  ordre,  Nu- 
meration, Addition,  Subtraction,  Multiplyplication,  Diuision  and 
Reduction,  but  in  fractions  (to  folowe  the  same  aptnesse  in  pro- 
cedyng  from  the  easyest  woorkes  to  the  harder)  we  muste  vse  this 
ordre  of  the  woorkes,  Numeration,  Multiplication,  Diuision,  Reduc- 
tion, Addition,  and  Subtractio. 

The  book  is  in  the  form  of  a  dialogue,  and  upon  the  pupil's 
saying,  "I  desyre  to  vnderstond  ye  reason,"  the  master  says: 

As  in  the  arte  of  whole  numbres  ordre  woulde  reasonablye  begyn 
with  the  easiest,  and  so  go  forwarde  by  degrees  to  the  hardest,  even 
so  reason  teacheth  in  Fractions  the  lyke  ordre.1 

Addition  and  Subtraction.  In  adding  or  subtracting,  early 
writers  usually  took  for  a  new  denominator  the  product  of  the 
given  denominators,  reducing  the  final  result  to  lowest  terms.2 
Because  of  the  size  of  the  common  denominator  thus  found;5 
the  early  Rechenmeisters  in  Germany  ordinarily  added  but  two 
fractions  at  a  time.  Although  the  plan  of  reducing  to  the  least 
common  denominator  before  adding  or  subtracting  was  occa- 
sionally used  by  isth  and  i6th  century  arithmeticians,4  it  was 
not  until  the  i  yth  century  that  it  began  to  be  generally  recog- 
nized,5 and  even  then  the  name  was  slow  of  acceptance.6 

1For  further  discussion  see  the  1558  edition  of  the  Ground  of  Artes,  fol. 
Riiii,  v.  The  same  order  is  followed  by  Pacioli  (1494,  fol.  51),  Pagani  (1591, 
pp.  34,  41),  and  others.  Giovanni  Battista  di  San  Francesco,  Elementi  Aritmetici 
(Rome,  1689),  even  begins  with  division  "that  it  may  be  better  understood." 

2 Thus,  i+£  =  if +1HJ  =  $¥=M2  =  ITV  The  method  is  given  by  Bhaskara 
(c.  1150);  see  Taylor's  translation,  p.  24.  It  appears  in  many  medieval  MSS. 
and  in  such  early  arithmetics  as  those  of  Petzensteiner  (1483),  Pellos  (1492), 
Riese  (1522),  Recorde  (c.  1542),  and  Baker  (1568), 

:t£.g.,  the  Dutch  arithmetician  Wilkens  (1630)  reduces  4,  •$,  g,  and  J  to 
96oths  before  adding. 

4Chuquet  (1484)  gives  it  (fols.  13  and  14),  and  it  is  found  in  such  works 
of  higher  class  as  those  of  Tartaglia  (1556)  and  Clavius  (1583). 

5 So  Cataldi  (1606)  reduces  to  the  "minor  commune  denominator";  the  well- 
known  Coutereels  (1599),  to  "het  minste  ghetal";  and  Wilkens  (1630),  to  the 
"Kleynste  gemeyne  Noemer." 

6 The  shorter  name  of  "general  denominator"  was  used  by  some  writers.  See 
Starcken's  Dutch  work  of  1714,  with  "General  Nenner."  Ramus  (1569)  sug- 
gested "cognomen"  for  common  denominator, — not  a  bad  term. 


224  COMMON  FRACTIONS 

The  arrangement  of  an  example  in  addition  was  somewhat 
uniform  before  the  iyth  century,  and  it  may  be  understood 
from  the  following  case  of  £  -f  f  as  given  by  Pacioli1  : 

40     18  10 


Multiplication  of  Fractions.  Although  our  present  interest  in 
the  multiplication  of  fractions  relates  to  such  simple  cases  as 
§  x  |  ,  it  is  desirable  to  set  forth  some  of  the  difficulties  met  by 
ancient  writers.  These  difficulties  appear  in  the  works  of  the 
Egyptians  and  Greeks,  but  they  are  sufficiently  evident  in  a 
single  example  given  by  Rabbi  Sa'adia  ben  Joseph  al-Fayyumi, 
a  Hebrew  scholar  of  the  loth  century  already  mentioned  on 
page  212.  The  problem,  which  shows  the  difficulties  met  with 
in  the  use  of  unit  fractions,  is  to  find  the  product  of  6i|  |  by 
6i|  i.  The  solution  is  substantially  as  follows:  61x10  —  610, 
61x20  —  1220,  61  x  40  =  2440,  61  x  60  =  3660,  the  last  three 
being  found  by  doubling  or  by  adding.  Then  61  x  61  =  3721, 
evidently  found  by  adding  61  to  3660.  Then  \  x  61  =  20^, 
and  20J  +  -J-  of  61  =  26|  (sic),  the  double  of  which  is  531. 
Adding  this  to  3721,  he  obtains  3774.3-,  and  this  increased  by  | 
gives  the  result,  3775.  What  he  tries  to  do  is  to  square  6iJ  § 
by  taking  61  2  +  2  x  61  x  (^  -f-  -9)  +  (£  -f  \-)2  (a  rule  which  was, 
of  course,  well  known),  but  he  fails  in  his  computation. 

With  respect  to  the  ordinary  operation  with  simple  fractions, 
the  process  of  multiplication  has  not  changed  materially  during 
the  last  few  centuries  except  that  cancellation  was  not  generally 
used  by  early  writers,2  although  a  few  of  the  better  arithme- 
ticians saw  its  advantages.3 


ed.,  fol.  51,  with  an  error  in  the  quotient  as  printed. 
2£.  g.,  Calandri  (1491)  multiplied    $  by  J  thus: 
Multiplica  J  uie  f 

3  _  .  _  4 

4  i# 
Fanno  % 

Even  in  the  Greenwood  American  arithmetic  (1729)  this  method  is  followed. 

3  Thus  Rudolff  (Kunstliche  rechnung,  1526)  says  "das  man  ein  ober  vnd  ein 
vnter  gegen  einander  mag  auffheben  oder  kleyner  machen"  during  the  operation. 


MULTIPLICATION  2  2  5 

In  the  matter  of  language,  the  schools  have  usually  protested 
against  the  broadening  of  the  meaning  of  any  technical  term. 
A  teacher  will  object  to  saying  "  f  times  4"  but  will  say 
"i|  times  4."  The  contest  is  an  old  one;  thus  Ortega  (1512) 
would  not  write  "3-|  ducats/'  preferring  the  awkward  expres- 
sion "3  ducats  and  one  fourth  of  a  ducat";1  but  Rudolff 
(1526),  at  about  the  same  time,  did  not  hesitate  to  speak  of 
"!  times"  a  number.2 

Of  the  various  special  rules,  most  of  which  came  from  the 
Arabs,  a  single  one  may  serve  as  a  type.  Expressed  in  modern 

symbols, 

J  3  ac      ac 

a     c       b       d 


J  0 

that  is,  £.«.  =  :*- 

thus  reducing  the  work  to  dividing  a  fraction  by  an  integer  and 
suggesting  cancellation  more  strongly.3 

Many  of  the  early  writers  expressed  concern  over  the  fact 
that  the  product  of  a  number  by  a  proper  fraction  was  less 
than  the  multiplicand.  Borghi  (1484)  seems  to  have  been  the 
first  author  of  a  printed  book  to  discuss  the  matter,  and  various 
1  6th  century  writers  had  much  to  say  about  it.4 

Few  writers  before  the  i?th  century  made  any  attempt  at 
explaining  the  process,  although  Trenchant  (1566)  devoted 
some  attention  to  it,  using  the  illustration  of  a  square  cut  into 
smaller  squares.5 

J"f[Se  3  ducati  e  vn  quarto  de  ducato  .  .  .  guadagnano  5  fiorini  e  vn  terzo," 
much  as  we  say  "a  dollar  and  a  half." 

2"  .  .  .dan  ich  hab  die  sechs  nur  ein  halbs  mal  haben  wollen"  (1534  ed., 
fol.  Ciiij,  v.}. 

3  This  is  given  by  al-Karkhi  (c.  1020)  . 

4  Among  them  was  Ramus  (1569).   In  the  1586  edition  (p.  73)  his  commen- 
tator, Schoner,  gives  a  whole  page  to  it.    Cataneo  (1546)  also  devotes  a  page  to 
it,  seeking  particularly  to  combat  Borghi  and  Pacioli.     (See  1567  edition,  fol.  21, 
v.)    Even  Tartaglia  did  not  see  the  point  of  the  controversy  (see  1592  ed.,  I, 
fol.  187,  r.). 

5Cardinael's  School  Boecken  (1650;  1674  ed.)  goes  into  the  jnatter  more 
fully,  using  several  diagrams. 


226  COMMON  FRACTIONS 

Division  of  Fractions.  Naturally  the  most  difficult  operation 
was  division.  Multiplication  by  the  inverted  divisor  is  so  simple 
that  we  hardly  realize  that  it  has  come  into  general  use  only 
recently,  although  it  was  known  in  the  early  Middle  Ages  by 
both  the  Hindus  and  the  Arabs.1  Influenced  by  the  notion  that 
only  fractions  could  deal  with  fractions,  medieval  writers  often 
substituted  for  the  division  of  a  fraction  by  an  integer  the 
process  of  multiplying  by  the  reciprocal  of  the  integer  ;  2  that  is, 

2  _i.  A  —  2    v   1 

3  r  •  4  —  3  x  ?• 

The  early  printed  books  gave  two  leading  methods.  The 
first  of  these  reduced  the  fractions  to  a  common  denominator 
and  took  the  quotient  of  the  numerators,3  as  in  the  case  of 

2  ^_  3  —  _8    _•_  _9    —  8 

3  '    1  ~~  12     *    12  ~~  9* 

The  second  method  is  one  of  cross  multiplication.  Thus,  in 
the  case  of  |  -f-  1  we  have 


\4     9 

which  involves  the  same  operations  that  enter  with  the  inverted 
divisor.  This  was  the  favorite  method  in  the  early  printed 
books,4  and  the  name  "cross  multiplication"  or  its  equivalent5 
was  common,  the  divisor  being  usually  placed  on  the  left,  but 
sometimes  on  the  right.6  One  writer  expresses  the  opinion 
that  the  divisor  was  placed  at  the  left  because  the  process 
may  have  come  from  the  Hebrews,  who  write  toward  the  left.7 

1Brahmagupta  (c.  628)  and  Bhaskara  (c.  1150)  both  gave  it  (Colebrooke 
translation,  pp.  17,  278),  and  al-Hassar  (c.  1175?)  recognized  it,  at  least  with 
integral  dividends  (Bibl.  Math.,  II  (3),  p.  36). 

2Thus  Rollandus  (c.  1424). 

3E.g.y  Chuquet  (1484,  fol.  16,  whose  manuscript  was  so  extensively  appro- 
priated by  De  la  Roche  and  in  part  printed  in  1520),  Trenchant  (1566),  and 
Ramus  (1555). 

4  E.g.,  Widman  (1489):  "Nu  wiltu  teile"  ^3  in  |  sprich  6  mal  9  ist  54  dy 
setz  fiir  den  zeler  vnd  sprich  darnach  5  mal  13  ist  65  die  setz  fiir  den  nener 
also  |f"  (1508  ed.,  fol.  30,  v.). 

5Thus  Hodder  (1672  ed.)  says  "multiply  cross  wise";  Riese  (1522),  "so  mul- 
tiplicir  im  creutz";  Peletier  (1549),  "multiplier  en  croix";  Pagani  (1591),  "molti- 
plica  in  croce." 

6  Thus  Hudalrich  Regius  (1536),  Pagani  (1591),  Mots  (1640),  and  others. 

7WentseI,  1599,  p.  88. 


DIVISION 


227 


The  idea  would  have  been  more  reasonable,  so  far  as  imme- 
diate origin  is  concerned,  if  he  had  spoken  of  the  Arabs. 


EARLY  DIVISION  OF  FRACTIONS 
From  an  anonymous  Italian  MS.  of  1545  in  Mr.  Plimpton's  library 

The  Inverted  Divisor.  As  already  said,  the  method  of  mul- 
tiplying by  the  inverted  divisor  was  known  to  certain  Hindu 
and  Arab  writers.  It  seems,  however,  to  have  dropped  out  of 


228  SEXAGESIMAL  FRACTIONS 

sight  for  three  or  four  hundred  years,  reappearing  in  StifePs 
works  in  IS44.1  It  was  not  at  once  accepted,  only  a  few  of  the 
1 6th  century  writers  making  any  use  of  it,2  but  in  the  iyth 
century  it  became  fairly  common. 

Before  the  inverted  divisor  came  into  general  use  there  were 
several  special  rules  that  met  with  some  favor.  One  of  these, 
given  by  Gemma  Frisius  (1540),  may  be  expressed  in  modern 

symbols  thus :  7 

J  a      ka  __  c 

~b^~c^Tb' 

as  in  -  -f-  —  =  -  -f-  — —  = =  ---  • 

5      13      5         13        4x5      20 

2.  SEXAGESIMAL  FRACTIONS 

Nature  of  Sexagesimals.  For  scientific  purposes  the  medieval 
writers  usually  followed  the  late  Alexandrian  astronomers  in 
the  use  of  fractions  written  on  the  scale  of  sixty.3  This  cus- 
tom has  continued  until  now  in  the  measures  of  time,  angles, 
and  arcs,  as  when  we  write  2  hr.  20  min.  45  sec.,  that  is, 
(2  +  ITF  +  TTGO'TF)  hours,  instead  of  (2  +  -$  +  jfa)  hours.  The 
measure  of  time  meets  a  popular  need,  and  so  the  sexagesimal 
fraction  gives  no  present  evidence  of  being  abandoned  for  this 
purpose,  but  for  circular  measure  it  is  losing  its  hold  as  decimals 
become  better  known,  and  seems  destined  soon  to  disappear. 

In  the  Middle  Ages  the  scientific  workers  carried  the  sexa- 
gesimal divisions  still  farther  than  the  Greeks,  as  if  we  were 
to  write  2  10'  30"  45'"  5iv  yv,  meaning  thereby 

2  +  i2  +  ^+45+JL+JL. 

60     6o2     6<y     6o4     6o& 

1<lEgo  Diuisionis  regulam  reduco  ad  regulam  Multiplicationis  Minutiarum,  hoc 
modo:  Diuisoris  terminos  commuto,"  etc.  Arithmetica  Integra,  1544,  fol.  6,  r. 

"Thu  im  also.  Den  Teyler  .  .  .  kere  vmb/also  ausz  dem  Zeler  werde  der 
nenner/vnd  ausz  dem  nenner  der  Zeler.  So  steht  derm  das  exemplum  mit  vmbge- 
kereten  Teyler  also"  (Deutsche  Arithmetica,  1545,  fol.  13,  v.). 

2 Among  them  were  Thierf elder  (1587)  and  Clavius  (1583).  The  latter  says: 
"ac  si  termini  diuisoris  commutentur,  &  regula  multiplications  seruetur"  (1585 
ed.,  p.  118,  and  similarly  the  Italian  edition  of  1586,  p.  106). 

3  Latin  sexagesimus  or  sexagensumus,  sixtieth,  from  sexaginta,  sixty. 


ORIGIN  OF  SEXAGESIMALS  229 

Thus  Sibt  al-Mariclini,1  an  astronomer  at  the  mosque  of  al- 
Azhar  in  Cairo  in  the  middle  of  the  isth  century,2  gave 
45°  So'-s- 1°  25'=.  33°  45'  $2"  $&"  28iv  14*  f  T  3iviii  45ix  52s'-  -  •, 
and  similar  cases  occur  in  many  medieval  works. 

Names  of  Sexagesimals.  Sexagesimals  were  usually  known  as 
physical  fractions  in  the  Middle  Ages.3  The  name  may  pos- 
sibly have  come  from  their  use  in  physics,  this  word  (more 
frequently  "physic"),  as  applied  to  natural  philosophy,  not 
being  so  recent  as  is  sometimes  thought.  On  the  other  hand, 
it  may  come  from  the  fact  that  the  denominators  were  under- 
stood to  proceed  in  the  natural4  order  of  the  powers  of  60, 
somewhat  as  we  speak  of  "natural  numbers"  at  present,  this 
being  an  opinion  expressed  in  the  i6th  century.5 

They  were  also  called  astronomical  fractions,6  the  reason 
being  quite  apparent.7 

Origin  of  Sexagesimals.  There  is  a  common  idea  that  sex- 
agesimal fractions  came  from  Babylon, — an  idea  which  arose 
from  the  fact  that  60  plays  an  important  part  in  the  number 

1Mohammed  ibn  Mohammed  ibn  Ahmed,  Abu  'Abdallah,  Bedr  ed-din  al- 
Misri,  born  in  1423,  died  in  1494/95.  He  wrote  a  number  of  works  on  arith- 
metic and  astronomy. 

2He  gives  the  result  only  to  31 VI",  the  fraction  then  repeating, —  an  interesting 
case  of  a  circulating  sexagesimal.  See  Carra  de  Vaux,  "Sur  Fhistoire  de  rarithme- 
tique  arabe,"  Bibl.  Math.,  XIII  (2),  33.  The  above  symbols  are,  of  course,  mod- 
ern. The  problem  is  substantially  that  of  45 £  -+•  iy\  —  334  +• 

3  Thus  we  find  in  the  MSS.  such  expressions  as  "Modum  representationis 
minuciarum  vulgarium  et  physicarum"  (anonymous  MS.  of  1466)  and  "Minucie 
duplices  sunt  scilicet  phisice  et  vulgares"  (anonymous  MS.  of  isth  century). 
In  the  early  printed  books  they  are  called  "fractiones  phisice"  (Ciruelo,  1495), 
"fraciones  fisicas"  (Texeda,  1546),  "  Minucciamenti  Fisici"  (Italian  edition  of 
Gemma  Frisius,  1567),  and  by  other  similar  names.  4<Mrts  (phy'sis,  nature). 

5Thus  Trenchant  (1566):  "S'apele  phisic,  c'est  a  dire,  naturel :  pour  ce  que 
ses  denominateurs,  &  caracteres,  sont  selon  1'ordre  naturel  du  nombre  com- 
menqant  a  1'vnite  (1578  ed.,  p.  19). 

6"De  Fractionibus  Astronomicis,  siue  de  minutiis  Physicis,"  as  Gemma 
Frisius  (1540)  says  in  his  Latin  editions,  the  Italian  having  "Rotti  Astrono- 
mici."  Trenchant  has  "Du  nombre  phisic,  ou  fractions  astronomiques."  Peletier 
(Pelctarius)  in  his  notes  on  Gemma  Frisius  (1563  ed.)  speaks  of  "Fractiones 
Astronomicae,  quas  vulgo  Physicis  vocat."  The  name  was  used  by  Abraham  ben 
Ezra  (c.  1140)  and  probably  by  the  late  Greek  writers. 

7  As  Peletier  (1549)  says,  because  they  "seruent  aux  supputations  des  mouue- 
ments  celestes." 

ii 


230  SEXAGESIMAL  FRACTIONS 

system  of  that  country.  The  assertion  of  this  origin  was  first 
made,  so  far  as  we  know,  by  Achilles  Tatius,  an  Alexandrian 
rhetorician  of  the  5th  or  6th  century.  It  has  also  been  assumed 
that  the  Babylonians  divided  the  circle  into  360  equal  parts, 
because  of  the  early  notion  that  a  year  consisted  of  360  clays, 
and  because  their  scientists  knew  that  the  radius  employed  in 
stepping  around  a  circle  divided  it  into  six  equal  arcs,  thus 
making  60  a  mystic  number.  This  reason  may  possibly  be  valid, 
but  there  is  no  authority  for  asserting  that  it  is  historical.  The 
Babylonians  divided  the  circle  into  8,  12,  120,  240,  and  480 
equal  parts,  but  not  into  360  such  parts.1  Thus  in  a  tablet 
from  the  palace  of  Sennacherib  (c.  700  B.C.)  now  in  the  British 
Museum  the  division  into  480  parts  is  given.  It  is  true  that 
six-spoked  wheels  are  found  represented  on  the  Babylonian 
monuments,  but  no  more  frequently  than  the  eight-spoked 
wheels,  and  the  six-spoked  type  is  more  common  in  Egypt 
where  the  number  60  was  not  used  to  any  great  extent.  It  would 
seem,  therefore,  that  the  number  60  was  not  derived  from  the 
division  of  the  circle  into  six  equal  arcs. 

It  is  true,  however,  that  the  Babylonians  wrote  the  equiva- 
lent of  ii  for  60  +  i,  in  for  6o2-f  6o-f~i,  and  44  26  for 
44  x  60  +  26,  although  there  is  no  reason  for  believing  that  this 
is  a  proof  of  their  use  of  sexagesimal  fractions.  In  a  certain 
tablet  of  c.  2000  B.C.,  for  example,  the  equivalent  of  the  square 
of  44  26  40  is  given  as325Si83i64.  This  may  be  inter- 
preted to  mean  the  square  of  either  44  x  6o2  4-  2  6  x  60  -h  40  or 

44  +  —  +  ^  v  In  the  latter  case  we  have  sexagesimal  fractions ; 

60     co- 
in the  former,  numbers  written  on  the  scale  of  sixty,  an  inter- 
pretation more  in  harmony  with  the  system  of  compound  num- 


1  On  this  entire  discussion  see  A.  H.  Sayce  and  R.  H.  M.  Bosanquet,  "  The 
Babylonian  Astronomy,"  in  Monthly  Notices  of  the  Royal  Astron.  Society,  XL, 
108;  E.  Hoppe,  Archiv  der  Math.,  XV  (3),  304;  E.  Loffler,  ibid.,  XVII  (3),  135; 
and  Hochheim,  Kafi  fit  Hisdb,  p.  23.  The  claim  that  the  Chinese  used  a  sexa- 
gesimal system  in  the  third  millennium  B.C.  (Vol.  I,  p.  24)  is  not  supported  by 
sufficient  evidence  to  be  considered  at  present.  It  is  very  improbable  that  it  in- 
volved anything  more  than  a  recognition  of  60  as  a  convenient  unit  for  sub- 
division. On  the  Greek  development  of  sexagesimals  see  Heath,  History,  I,  44. 


GREEK  USE  OF  SEXAGESIMALS  231 

bers  used  by  all  ancient  peoples.  Similarly,  we  find  the  case  of 
i  -*-8i,  but  whether  this  is  to  be  interpreted  as  having  the 
dividend  60  or  some  power  of  60  is  uncertain.  In  any  case  we 
have  no  evidence  of  any  such  general  use  of  sexagesimal  frac- 
tions as  is  found  among  the  Greek  astronomers.1 

The  division  of  the  circle  into  360  parts  as  practiced  by  such 
Greek  astronomers  as  Ptolemy  (c.  150)  was  probably  the  out- 
growth rather  than  the  origin  of  the  sexagesimal  system.  The 
Babylonians  counted  decimally  by  preference,  although  the 
base  of  60  played  a  considerable  part  in  their  system.  They 
counted  decimally  to  60,  that  is,  to  a  soss ;  then  by  sosses  and 
the  number  over  to  the  ner,  which  was  10  sosses,  or  600;  then 
by  ners,  sosses,  and  the  number  over  to  the  saru,  which  was 
6  ners,  or  3600;  but  they  never  counted  60,  360,  3600,  so  that 
360  was  not  a  natural  step  in  their  sexagesimal  system.2 

Greek  Use  of  Sexagesimals.  We  do  not  know  why  the  Greek 
astronomers  should  have  developed  a  scale  of  60  in  such  a  com- 
plete form,  although  we  can  readily  surmise  the  cause.  There 
seems  to  be  no  reason  to  doubt  that  the  number  60  was  sug- 
gested to  them  from  Babylon,  but  the  system  of  sexagesimal 
fractions,  as  we  know  and  use  it,  was,  so  far  as  now  appears, 
their  own  invention.  Ptolemy  used  these  fractions  to  represent 
his  chords  in  terms  of  a  radius  60  ;3  that  is,  the  chord  of  24° 
would  then  be  24.9494,  or,  in  sexagesimals,  24  56'  58".  It  seems 
clear,  however,  that  the  Greeks  needed  for  their  astronomical 
work  a  better  type  of  fraction  than  the  unit  type  of  the  Egyp- 
tians ;  that  their  habit  of  using  such  submultiples,  as  in  feet  and 
inches,  naturally  led  them  to  a  similar  usage  in  fractions,  as 
would  be  the  case  with  degrees  and  minutes ;  and  that  the  60 
of  Babylon  was  a  convenient  radix,  since  it  has  as  factors  2,3, 
4,  5,  6,  10,  12,  15,  20,  and  30,  and  so  permits  of  the  ready  use 
of  halves,  thirds,  fourths,  fifths,  sixths,  tenths,  twelfths,  and  so 

aFor  arguments  in  favor  of  the  fraction  interpretation  see  F.  Cajori,  "Sexa- 
gesimal Fractions  among  the  Babylonians,"  Amer.  Math.  Month.,  XXIX,  8.  See 
also  Heath,  History,  I,  29.  2Hilprecht,  Tablets. 

3 For  a  discussion  of  this  point  see  A.  Schiilke,  "Zur  Dezimalteilung  des  Win- 
kels,"  Zeitsch.  fur  math,  und  naturw.  Utiterr.,  XXVII,  339;  Heath,  History,  I,  45. 


232  SEXAGESIMAL  FRACTIONS 

on.  The  Greeks  may  thus  have  been  led  to  divide  the  radius 
into  60  equal  parts  and  the  diameter  into  120  of  these  parts. 
Since  the  common  value  of  TT  was  3  in  ancient  times,  the  cir- 
cumference was  naturally  taken  as  3  x  120,  or  360. 

Such  was  the  influence  of  the  Greek  scholars  that  all  the 
medieval  astronomers,  Christian,  Jewish,  and  Mohammedan, 
used  the  sexagesimal  system;1  but  some  of  the  mathematical 
writers  referred  the  system  to  India  instead  of  Greece,  influenced 
therein  by  the  belief  that  our  numerals  came  from  the  Hindus.2 

Terms  Used.  When  the  Greeks  decided  to  take  iTil  0  of  a  circle 
as  a  unit  of  arc  measure,  they  called  this  unit  a  degree.3  They 
called  ^  of  a  degree  a  first  part,4  ^-gVu  a  second  part,5  and 
so  on. 

Multiplication  involving  Sexagesimals.  The  operations  of  ad- 
dition and  subtraction  with  sexagesimals  involved  no  difficul- 
ties, but  multiplication  and  division  were  not  so  simple.  It  is 
meaningless  to  us  to  multiply  4°  7'  38"  by  5°  6'  29",  or  even 
4  f  38"  by  5  6'  29 ",  but  to  the  medieval  scientist  it  meant 

xAs  a  noteworthy  illustration,  sec  the  Libros  del  saber  de  Astronomia  del  Rey 
Alfonso  X,  Madrid,  1863.  The  Alfonsine  astronomical  tables  date  from  c.  1254, 
but  for  argument  as  to  a  later  date,  see  A.  Wegener,  "  Die  Astronomischen  Werke 
Alfons  X,"  Bibl.  Math,,  VI  (3),  138. 

2 Thus  Johannes  Hispalensis  (c.  1140):  "placuit  tamen  Indis,  denomina- 
tionem  suarum  fractionum  facere  a  scxaginta.  Diuiserunt  enim  gradum  unum 
in  sexaginta  partes,  quas  uocauerunt  minuta"  (B.  Boncompagni,  Trattati,  II, 
49).  He  may  have  had  his  idea  from  al-Khowarizmi  (c.  825):  "Set  indi 
posuerunt  exitum  partium  suarum  ex  sexaginta:  diuiserunt  enim  unum  in  .LX. 
partes,  quas  nominauerunt  minuta"  (from  a  Cambridge  MS.  of  the  Algoritmi 
de  Numero  Indorum,  in  the  Trattati,  I,  17). 

3Mo?pa  (moi'ra);  medieval  Latin,  de  +  gradus  (step).  The  Arabs  translated 
fjioipaby  daraja  (ladder,  scale,  step),  which  led  G.  H.  F.  Nesselmann  (Die  Algebra 
der  Griechen,  p.  137  (Berlin,  1842),  hereafter  referred  to  as  Nesselmann,  Alg. 
Griechen}  to  think  that  this  word  was  the  original  form  of  the  word  "degree." 
It  may  have  influenced  the  final  form. 

*HpuTat%TiKO(rTd(pro'ta  hexekosta')',  Latin,  pars  minuta  prima  (first  small  or 
fractional  part).  From  this  came  our  "minute."  The  Greeks  also  used  \ewrd 
(lepta',  minute,  the  adjective).  In  the  i2th  century  Walcherus  (see  Volume  I, 
page  205)  spoke  of  the  minutes  as  puncta,  and  the  same  term  is  so  used  in  an 
algorism  of  c.  1200.  See  L.  C.  Karpinski,  "Two  Twelfth  Century  Algorisms," 
I  sis,  III,  396. 

5Aei/Te/3a  ^Kocrrd  (deu'tera  hexekosta') ;  Latin,  pars  minuta  secunda,  from 
which  our  "second."  See  also  Wertheim,  Elia  Misrachi,  p.  19  n. 


OPERATIONS  WITH  SEXAGESIMALS 


233 


3 

28 

54 

8 

^4 

8 

23 
16 

simply  the  finding  of  (4  +  6V  +  silo)  x  (5  +  6°o  +  a  ID-  In 
the  operation  there  is,  for  example,  7'  x  6'  ==  42",  which  means 
simply  that  ^  x  g\  =  3  tfo-1  ^n  such  WOI>k  it  became  conven- 
ient to  have  multiplication  and  division 
tables,  and  these  are  found  in  various 
medieval  manuscripts.2  Some  idea  of 
the  difficulty  of  operating  with  these 
fractions  may  be  inferred  from  a  prob- 
lem in  the  work  of  Maximus  Planudes 
(c.  i34o).3  His  multiplication  of  14°  23'  by  8°  16',  giving 
the  product  3  signs4  28°  54'  8",  is  here  shown. 

Division  involving  Sexagesimals.  In 
division  Maximus  Planudes  reduced  all 
the  terms  to  the  same  denomination. 
For  example,  the  operation 

3°  23'  54"        0      ,      „ 


or 


2  3.  _L      5  4 
(;  °         3  0  ° 


"  "86  00' 


"'60'     3  6 0 0 

is  worked  out  as  illustrated  here.5 

The  finding  of  roots  by  the  aid  of 
sexagesimals  appears  in  the  works  of 


3 

23 

54 

2 

34 

24 

12234 

i 

9264 

2970 

i  78200 

'9 

9264 

2184 

131040 

14 

9264 

in  the  translation  of  al-Khowarizmi  (c.  825)  attributed  to  Adelard  of 
Bath  (c.  1120):  "Sex  minuta  multiplicata  in  VII.  minuciis,  erunt  .XLII. 
secunda"  (Boncompagni,  Trattati,  I,  18). 

2  In  the  adaptation  of  the  Liber  Algorismi  by  Johannes  Hispalensis  (c.  1140) 
the    multiplication    table    is   given    up   to    nona   times    nona,    that    is,    up    to 

-1  x  —  =  -  -  (see  Boncompagni,  Trattati,  II,  103).  The  printed  arithmetics 
6o9  6o9  6o18 

occasionally  gave  such  tables;  e.g.,  those  of  Cardan  (1539,  cap.  38),  Fine 
(1530;  i55S  ed.,  fol.  38,  r.),  Trenchant  (1566),  and  Peletier  (iS49).  Schoner,  in 
his  De  logistica  sexagenaria  (1569;  1586  cd.,  p.  370),  calls  it  the  "abacus  logis- 
ticus,"  and  a  table  of  products  up  to  60  x  60,  for  use  with  sexagesimals,  is 
called  by  Fine  (15.30)  a  "tabvla  proportionalis."  Division  tables  are  also  given 
by  various  writers;  e.g.,  Fine  (1530)  and  Trenchant  (1566). 

3Waschke,  Planudes,  p.  34. 

4 The  "sign"  was  30°,  and  the  12  signs  of  the  zodiac  gave  360°. 

6  He  says  that  the  division  may  be  continued  farther. 


234  SEXAGESIMAL  FRACTIONS 

Theon  of  Alexandria1  (c.  390)  and  Maximus  Planudes2  (c. 
1340),  and  in  several  of  the  i6th  century  arithmetics.3  Its 
nature  may  be  inferred  from  the  work  in  division. 

Symbols.  The  symbols  (°  '  ")  are  modern.  In  medieval  and 
Renaissance  times  there  were  several  methods  used  for  desig- 
nating the  sexagesimal  orders.  Thus  in  a  manuscript  of  Leo- 
nardo of  Cremona4  (i5th  century)  we  have 

•5"     f  .  57  .  S     f°r     S°   I9'  S7"  38'"' 

and  a],      --      -»-      -~     for     46'  39"   i2flr  36iv. 

.46.     39  .   12   .   36 

Gemma  Frisius  (1540)  wrote5 

S.      g.       m.      2.        3.       4 
i.      16.     25.      17.     21.     27 

for  Is     16°     25'     17"     21'"     27iv. 

Peletier  (1571  edition  of  Gemma  Frisius)  used  m~  or  i~  for 
minutes,  2~  for  seconds,  3~for  thirds,  and  so  on.  Jean  de  Lineriis 
(c.  1340) 9  used  the  symbols  s,  g,  m,  2,  3,  4,  and  these,  with 
slight  modifications,  are  the  ones  most  commonly  seen  up  to 
the  close  of  the  i6th  century.  About  that  time  there  came 
into  use  such  forms  as 

Ilae     lae     o        I        II     III 

3    -    15-     7-     So.     34-     23. 


irrhe  process  is  given  in  Heath,  History,  I,  60,  and  in  Gow,  Greek  Math.,  p.  55- 

2 Heath,  History,  II,  547,  where  the  date  of  Planudes  is  given  a  little  earlier. 

3 E.g.,  the  Peletier  (Peletarius)  revision  (1545)  of  Gemma  Frisius. 

4In  Mr.  Plimpton's  library;  Kara  Arithmetica,  p.  474.  The  title  is  Artis 
metrice  pratice  compilatio. 

c"Circulus  12  Signis  constat:  Signum,  30  Gradibus."  The  relation  to  the 
zodiac  is  apparent. 

6  In  the  Algorismvs  de  Minutijs  appended  to  Beldamandi's  work  (1483  ed.). 


DECIMAL  FRACTIONS  235 

in  which  7  stands  for  units/  and  in  which  the  symbols  are 
evidently  the  forerunners  of  the  ones  now  in  common  use.2 

One  curious  example  of  symbolism  is  seen  in  the  multiplica- 
tion table  given  by  Fine  (1530)  for  use  in  sexagesimal 
computation,  the  product  8  x  42  being  given  as  5.36,  that  is, 
5  x  60  +  36,  the  period  being  essentially  a  sexagesimal  point. 

3.  DECIMAL  FRACTIONS 

Need  for  Decimal  Fractions.  Before  the  beginning  of  printing, 
operations  with  common  fractions  having  large  terms  are  not 
frequently  found.  In  mercantile  affairs  they  were  not  needed, 
and  in  astronomical  work  the  sexagesimal  fraction  served  the 
purpose  fairly  well.  The  elaborate  Rollandus  manuscript  of 
c.  1424  contains  the  addition  of  no  common  fractions  more  dif- 
ficult than  If  ^  and  yf-£,  and  the  work  in  multiplication  involves 
no  fraction  more  elaborate  than  ff .  There  are  exceptions  to 
the  general  rule,3  but  they  are  not  numerous.  In  the  recording 
of  results  in  division,  however,  elaborate  common  fractions 
were  frequently  used.4  By  the  advent  of  printing,  writers  were 
led  into  various  excesses.  Widman  (1489),  for  example,  used 
in  business  computations  fractions  far  beyond  any  commercial 
needs,  his  successors  were  even  more  reckless,5  and  the  theorists 
naturally  went  still  farther.0 

irThis  example  is  from  Schoner,  De  logistica  sexagenaria,  1569 ;  1586  ed.,  p.  366. 
Of  the  7  he  says,  "qui  &  partium  numerus  dicitur,  circulus,"  and  he  speaks  of 
"  7  imitates." 

2Peletier  (1549)  remarks:  "Les  Degres  dont  seront  au  milieu  de  la  numera- 
tion Astronomique :  &  seront  represented  par  °:  les  Minutes  par  i'  les  Secondes 
par  2:  .  .  .  Et  ansi  des  autres"  (L'Arithmetiqve,  p.  107  (1607  ed.)). 

3 E.g.,  in  a  Dutch  MS.  of  the  isth  century  (Boncompagni  sale  cat.,  No.  477) 
the  square  root  of  252o^YAW?\(Ws  is  required. 

4£.g.,  in  the  Svmme  Arismetiee  of  Stephano  di  Baptista  delli  Stephani  da 
Mercatello  (MS.  of  c.  1522),  a  pupil  of  Pacioli's,  there  are  results  like 
(fol.  74,O. 

•Thus  Widman  uses  88H?7»  Trenchant  (1566)  has  iQSpHoyM 
ed.,  p.  286),  and  Wentsel  (1599)  has  several  fractions  as  difficult  as 
and  all  of  these  were  commercial  writers. 

6As  when  Scheubel,  in  his  De  nvmeris,  tractatus  quintus  (i$4!>)»  gives 
3iMHiiii£?»  and  Coutereels  (Eversdyck's  edition  of  1658)  gives  a  result 
like  aHHH»m  days. 


236  DECIMAL  FRACTIONS 

Forerunners  in  the  Invention.  As  usual  in  the  case  of  an  im- 
portant invention  there  were  various  scholars  who  had  some 
intuition  of  the  need  for  such  a  device  as  the  decimal  fraction 
long  before  it  was  finally  brought  to  light.  Such  a  man  was 
Joannes  de  Muris,  or  Jean  de  Meurs,  who  wrote  early  in  the 
1 4th  century.1  The  most  interesting  of  the  early  influences 
tending  to  the  invention,  however,  was  a  certain  rule  for  the 

extraction  of  >/#,  expressed  in  modern  symbols  by  — ^— '—^ —  • 


___..„_  1Lr 

.     .         r     v  30000        V  3000000  , 

In  particular,  v  3  =  — •>  or  — ,  the  actual  process 

F  '  100  1000  ^ 

of  extracting  the  root  being  quite  like  our  present  one  with 
decimals.  It  was  known  to  the  Hindus,  to  the  Arabs,  and  to 
Johannes  Hispalensis  (c.  1140),  and  is  found  in  the  works  of 
Johann  von  Gmiinden  (c.  1430),  Peurbach  (c.  1460),  and  their 
successors  until  the  close  of  the  i6th  century.2  The  most  in- 
teresting step  from  this  rule  in  the  direction  of  the  decimal 
fraction  appears  in  certain  tables  of  square  roots,  in  connection 
with  which  the  statement  is  made  that,  the  numbers  having 
been  multiplied  by  1,000,000,  the  roots  are  1000  times  too 
large.  Such  a  table,  from  Adam  Riese's  Rechnung  auff  der 
Linlen  vnd  Federn  (Erfurt,  1522),  is  reproduced  on  page  237. 
The  same  plan  is  given  by  such  later  writers  as  Trenchant 
(1566)  and  Bartjens  (1633).  Even  after  the  decimal  fraction 
was  well  known,  the  analogous  plan  of  using  a  radius  of 
10,000,000,  in  order  to  express  the  trigonometric  functions  as 
whole  numbers,  remained  in  use  for  more  than  two  centuries. 
It  even  extended  to  the  reckoning  of  interest  "to  the  Radius 
100,000,"  as  Thomas  Willsford  says  in  his  appendix  to  the  1662 
edition  of  Recorders  Grovnd  of  Arts*  so  as  to  avoid  decimals. 

!L.  C.  Karpinski,  Science  (N.  Y.),  XLV,  663. 

2  Buckley,  for  example,  an  English  arithmetician,  who  died  0.1570,  gave  the 
rule  in  Latin  verse  as  follows: 

Quadrate  numero  senas  praefigito  ciphras 
Producti  quadri  radix  per  mille  secetur. 
Integra  dat  Quotiens,  et  pars  ita  recta  manebit 
Radici  ut  verae,  ne  pars  millesima  desit. 

Arithmetica  memorativa,  c.  1550 

3  As  spelled  in  this  edition. 


fofonimen  ioeo.  £jtann  p:epom'r  bcm  anberen 
pantren/baetffber  Siffern  s.aucfc  fcdjeo/wit 
jit  f>e  Xabirtm  quab?atSbauon/fo  Boineti  414* 
@en  batten  puna  ma<&  *a$  alfo.6^  i  .eft 


».  2JIfotj)Bfnir  alien 
Concten/fo  ttidgftubt'e  2T«fd  felber.  i£e  ift  ^ 
for  grog  muljc  onb  verb:offen  drbeyt/  Caruin 
bab  i*  btr^te  tin  £af  d  au00e$o0en  /  bte  ge  (jet 


ila  Radicum  quadratarum. 


i* 
t 


€ 

f 
9 

99 

w 

n 


1600 

4'4 


icoo 

>J4 

44f 

'45 

•» 


J«* 

44* 


if 
«9 

at 


47* 


at     »9* 


l«     477 
Jt 


4> 
41 

44 


4*   7*1 

47 
4« 


EARLY  STEPS  TOWARD  DECIMALS    (l522) 

From  Adam  Ricse's  arithmetic,  showing  a  table  of  square  roots  in  which  the 

figures  of  the  decimal  fractions  appear,  but  without  any  form  of  decimal  point. 

From  a  later  edition  of  the  work 


si 

34 

l<? 

37 


747 
•If 


too* 


«44 


4«l 

4«« 


238  DECIMAL  FRACTIONS 

Another  influence  leading  to  the  invention  of  the  decimal 
fraction  was  the  rule  for  dividing  numbers  of  the  form  a .  ion, 
attributed  by  Cardan  (1539)  to  Regiomontanus.  This  appears 
in  several  manuscripts  of  the  isth  century,1  as  in  the  case  of 
470-^-10=  47  and  503-^10=50-^0.  Borghi  (1484)  elaborates 
this  rule,  but  it  appears  in  its  most  interesting  form  in  the  rare 
arithmetic  of  Pellos  (1492),  who  unwittingly  made  use  of  the 
decimal  point  for  the  first  time  in  a  printed  work  (p.  239). 
The  use  of  the  dot  before  and  after  integers  had  been  common 
in  the  medieval  manuscripts,  as  in  the  case  of  Chuquet's  work 
already  mentioned,  but  its  use  to  separate  the  integer  from 
what  is  practically  a  decimal  fraction  is  first  seen  here.  Later 
writers  commonly  used  a  bar  for  this  purpose,  as  was  the  case 
with  Rudolff  (1530;  see  page  241),  Cardan  (1539),  Cataneo 
(1546),  and  various  other  writers.  Even  as  late  as  the  1816 
edition  of  Pike's  Arithmetick  (New  York,  1816)  46,464  is 
divided  by  7000  thus : 

7 1 ooo)  46 1 464  (6^ f| 4 

42  j 

4(464 

Pellos,  however,  did  not  recognize  the  significance  of  the  deci- 
mal point,  as  is  evident  from  the  facsimile  on  page  239,  and  no 
more  did  Cardan  appreciate  the  significance  of  the  bar  that  he 
used  for  the  same  purpose.2 

The  initial  steps  in  the  invention  of  the  decimal  fraction  were 
not  confined  to  the  West,  however ;  indeed,  the  credit  for  first 
recognizing  the  principle  of  this  type  of  fraction  may  well  be 
given  to  al-Kashi,3  the  assistant  of  the  prince  astronomer  Ulugh 
Beg  and  the  first  director  of  the  latter's  observatory  at  Sam- 
arkand. In  his  al-Risdli  al-mohittje  (Treatise  on  the  circum- 

irrhus  Chuquet  (1484) :  "Comme  qui  vouldroit  partir  .470.  par  .10.  fault  oster 
.o.  qui  est  la  pme'  figure  de  .470.  et  demeurent  .47.  et  tant  monte  la  part.  Ou  que 
vouldroit  partir  503.  par  .10.  fault  oster  .3.  et  les  raettre  dessus  .10.  et  Ion  aura 
.50.  -j3^.  pour  quotiens."  Fol.  8,  v. 

2 See  his  Practica  (1539),  cap.  38. 

3  See  Volume  I,  page  289,  n.  5.  He  died  c.  1436,  or  possibly,  as  some  writers 
assert,  c.  1424. 


tTPdrrirper  i     ol 


7     9     6     S     4.     83     y 


7 

qtwctcnt      $98*7419    - 

^    o 

J    o 

8    ?    <$    04.*? 

*    ; 
945^4  -  — 

$    o 

Ct>aiKrp<r  7    o 

9      ?      ?      7      9      I  *9 


quocicnt 


4    o    ol 

7     8     9     6     $  .7 


quodwt 


o  o  o 


$8  •  7.    9     t 

• 


quodent      i    9    *    '    9 

\_  _  ;__  ___    3000 

FROM  THE  PELLOS  ARITHMETIC  (  1492  ) 


240  DECIMAL  FRACTIONS 

ference)  he  not  only  gives  the  value  of  TT  to  a  higher  degree  of 
iccuracy  than  any  of  his  predecessors,  but  he  writes  it  (using 
Arabic  characters)  as  follows: 

sah-hah 

3  1415926535898732, 

the  word  sah-hah  meaning  complete,  correct,  integral.1  We 
have,  therefore,  a  fraction  which  we  may  express  as  follows: 

Integer 

3       14159  •"» 

the  part  at  the  right  being  the  decimal.  Manifestly  it  is,  there- 
fore, a  clear  case  of  a  decimal  fraction,  and  it  seems  to  be 
earlier  than  any  similar  one  to  be  found  in  Europe. 

The  Invention.  The  first  man  who  gave  evidence  of  having 
fully  comprehended  the  significance  of  all  this  preliminary 
work  seems  to  have  been  Christoff  Rudolff,  whose  Exempel- 
Buchlin  appeared  at  Augsburg  in  1530.  In  this  work  he  solved 
an  example  in  compound  interest,  and  used  the  bar2  precisely 
as  we  should  use  a  decimal  point  today  (see  page  241).  If  any 
particular  individual  were  to  be  named  as  having  the  best  rea- 
son to  be  called  the  inventor  of  decimal  fractions,  Rudolff 
would  seem  to  be  the  man,  because  he  apparently  knew  how  to 
operate  with  these  forms  as  well  as  merely  to  write  them,  as 
various  predecessors  had  done.  His  work,  however,  was  not 
appreciated,  and  apparently  was  not  understood,  and  it  was  not 
until  1585  that  a  book  upon  the  subject  appeared. 

The  first  to  show  by  a  special  treatise  that  he  understood  the 
significance  of  the  decimal  fraction  was  Stevin,  who  published 
a  work3  upon  the  subject  in  Flemish,  followed  in  the  same  year 

irrhe  modern  Turkish  form  is  sahih.  I  am  indebted  for  these  facts  to  Pro- 
fessor Salih  Mourad  of  Constantinople. 

2  On  the  general  question  of  notation  see  Gravelaar,  "De  Notatie  der  decimale 
Breuken,"  Nieuw  Archief  voor  Wiskunde,  IV  (2). 

^De  Thiende.  A  copy  of  this  rare  pamphlet  was  fortunately  saved  at  the  time 
of  the  destruction  of  the  Louvain  library,  having  been  borrowed  a  few  days  be- 
fore by  the  Reverend  H.  Bosnians,  S.  J.  See  the  Revue  des  Questions  Scientifiques , 
January,  1920.  There  was  an  English  translation  by  Robert  Norton,  London, 
1608. 


i>  >oi*8  9  5">  104491  r 


1  3  2  f  f  i?  1  1 


EARLY  APPROACH  TO  DECIMAL  FRACTIONS 

From  the  IS4o  edition  of  Rudolff's  Exempel-Buchlin  (1530),  showing  the  use  of 
decimal  fractions  in  compound  interest 


242  DECIMAL  FRACTIONS 

(1585)  by  a  French  translation.  This  work,  entitled  in  French 
La  Disme,  set  forth  the  method  by  which  all  business  calcula- 
tions involving  fractions  can  be  performed  as  readily  as  if  they 
involved  only  integers.1  Stevin  even  went  so  far  as  to  say  that 
the  government  should  adopt  and  enforce  the  use  of  the  deci- 
mal system,  thus  anticipating  the  modern  metric  system.2  He 
was  the  first  to  lay  down  definite  rules  for  operating  with 
decimal  fractions,  and  his  treatment  of  the  subject  left  little 
further  to  be  done  except  to  improve  the  symbolism.  Some 
idea  of  his  treatment  of  the  subject  and  of  his  symbols  may  be 
obtained  from  the  facsimile  shown  on  page  243. 

The  Symbolism.  The  decimal  fraction  had  now  reached  the 
stage  in  its  progress  when  the  symbolism  had  to  be  settled.  As 
already  stated,  Pellos  (1492)  had  used  a  period  to  separate  the 
decimal  from  the  integral  part,  but  he  had  not  comprehended 
the  nature  of  the  fraction.  This,  however,  was  hardly  more 
strange  than  that  as  good  a  computer  as  Vlacq3  (1628)  should 
use  decimals  in  his  calculations  and  tables  and  yet  give  a  result 
in  the  form  12  95-^^0^  Several  writers  had  used  the  bar 
to  mark  off  the  decimal  part,  and  Rudolff  had  probably  grasped 
the  significance  of  the  new  fraction.  Stevin  had  fully  compre- 
hended and  clearly  expounded  the  theory,  but  his  symbols  were 
not  adapted  to  use.  The  improvement  in  the  symbolism  was 
due  largely  to  Biirgi,  Kepler,  and  Beyer,  and  to  the  English 
followers  of  Napier. 

Jobst  Burgi  (1552-1632)  dropped  the  plan  used  by  Regio- 
montanus — that  of  taking  10,000,000  as  the  sinus  totus  in 
trigonometry — and  took  i  instead,  the  functions  therefore  be- 
coming decimal  fractions.  He  was  not  clear  as  to  the  best 
method  of  representing  these  fractions,  however,  and  in  his 
manuscript  of  1592  he  used  both  a  period  and  a  comma  for  the 


1<t.  .  .  facilement  expedier  par  nombres  entiers  sans  rompuz  toutes  comptes 
se  rencontrans  aux  affaires  des  Hommes." 

2Adriaen  van  Roomen  (1609)  tells  us  that  Bishop  Ernst  of  Bavaria  had 
similar  ideas  as  to  measures. 

3Arithmetica  Logarithmica,  pp.  35  et  passim  (Gouda,  1628),  evidently  thinking 
that  the  decimal  form  of  the  result  would  not  be  understood  by  most  readers. 


SECONDE     PARTIE     DE 
JLA    DISME    DE    L'OPE* 

BL  A  T  1  O  N« 

PRpPQSJTION   I,    DE 

L'A  D  D  I  T  I  O  N. 

EStant  donne*,  nombfes  dt  Difine  2  tjottjler :  Trouper  Uur 
fomme : 

Explication  du  donni.  11  y  a  ttois  ordrcs  dc  nombres  dc 
Difine,  dcfquels  le  premier  vj  @8  ©4®7(fXlc  deux- 
iefme  37  ©  8  ©  7  (2)5  ©,le  troifiefme  875  ©7  j®8@  zg), 

'Explication  du  reqw.  U  nous  fauc 
crouver  Icur  fomme  *  Cwftrnftivn. 
On  mertra  les  nombres  donncz 
ea  ordre  corame  ci  joignant ,  les 

aiouftant  felon  la  vulgaire  manierc 

d'aioaftcr nombres  entiers^cnceftc       ?  4  *    5  °  4 
forte: 

Donne  (bmme  (par  Ic  i  prdbleme  dc  rArichmeti- 
quc)  941304,  qui  font  fee  que  dctnonftrent  les  fignes 
deflus  les  nombres)  941  ®  J  ©o  @  4®-  Ic  di,  quc 
les  mefmes  fbnr  la  fomme  recjxiifc.  Demtnftration.  Les 
*7©8(T)4(D7(D donnez> font  (par  la  y  definition) 
17  -^,  i~o^*  TZZz*  enfcmble  17  ~f£/$*  &par  mcfinc 
raifon  les  57  (g)  6  ©  7  ©  5  ©  valient  37  7™^,  &  les 
875©7®8(3)4(X)  feront  875  ^^>  lefouels  trois 

nombres,comme  ^^  ~~z>  37-rp^o>  *75T^^  ^onc 
cnfemble  fpar  le  ioc  probleme  cie  rArith,J  941  -j^~~ , 
mais  autant  raut  aufli  la  fomme  941  ©  3  ©  °  ©  4  ©> 

ccff 

A  PAGE  FROM  STEVIN's   WORK,  1634   EDITION 

From  the  first  work  devoted  to  decimal  fractions.    The  first  edition  was 
published  at  Leyden,  1585 


244  DECIMAL  FRACTIONS 

decimal  point,1  and  also  wrote  1414  for  141.4.    In  his  use  of 
these  fractions  he  was  followed  by  Prsetorius,  in  a  manuscript 

of  I599-2 

In  his  tables  of  i6i23  Pitiscus  assumed  the  radius  to  be 
100,000  and  gave  sin  10"  as  4.85.  Since  this  sine  for  the  radius 
i  is  0.00004848,  the  point  after  the  4  is  possibly  intended  as  a 
decimal  point.  Occasionally  he  used  several  points,  as  when 
he  gave  sin  89°  59'  30"  as  99999-  99894-  23.  In  his  trigonometry, 
of  which  the  tables  are  a  part,  he  used  a  vertical  line  to  mark 
off  the  decimal.4  In  the  1600  edition  both  the  point  and  the 
vertical  line  are  used  for  other  purposes,  the  former  to  separate 
sexagesimals5  and  the  latter  to  separate  (as  above)  a  large 
number  into  periods,  usually  of  five  figures  each. 

It  is  unquestionably  true  that  the  invention  of  logarithms 
had  more  to  do  with  the  use  of  decimal  fractions  than  any 
other  single  influence.  When  Napier  published  his  tables  in 
1614  he  made  no  explicit  use  of  decimal  fractions,  the  sine 
and  the  logarithm  each  being  a  line  of  so  many  units.  In  the 
1616  translation  of  this  work,  however,  the  translator,  Edward 
Wright,  made  use  of  the  decimal  point.  One  line  will  serve  to 
show  the  appearance  of  the  table : 


DC 
Min. 

g.o. 

Sines. 

Logarith. 

+i- 

Differen 

Logarith. 

Sines 

30 

8/26 

4741385 

4741347 

38-1 

999961  .9 

30 

In  his  Rabdologiae  .  .  .  Libri  Dvo  of  1617  Napier  made  some 
observations  upon  the  subject  and  wrote  both  1993,273  and 
1993,  2'  7"  3'"  for  the  number  which  we  now,  in  America,  write 
1993.273.  In  the  Leyden  edition  of  this  work  (1626)  the 

iCantor,  Geschichte,  II  (2),  617.  -Ibid.,  p.  619. 

3 Canon  Triangidorum  Emendatissimus  et  ad  usum  accommodatissimus.   Per- 
tinens  ad  Trigonometriam  Bartholomaei  Pitisci  .  .  .,  Frankfort,  1612. 

4"Deinde  pro  latere  AC  nuper  invento  13(00024  assumo  13  fractione  scilicet 
Ttfoiloiy "  * '•"    Trig-  Problematum  Geod.  Liber  Unus,  p.  12. 
5  As  in  this  subtraction  (p.  67) : 

70°.  o' 
46._8__ 
23.    52' 


THE  INVENTION  OF  DECIMALS  245 

Stevin  notation  is  used,  by  which  the  above  number  would 
appear  as  1993,  ©2,  ©7,03.  l  In  any  case  it  is  evident  that 
Napier  understood  something  about  the  decimal  fraction,  that 
he  did  not  invent  our  modern  symbolism,  and  that  the  practical 
use  of  logarithms  soon  made  a  knowledge  of  decimals  essential. 

In  1616  Kepler  wrote  a  work  on  mensuration2  in  which  he 
distinctly  took  up  the  decimal  fraction,  using  both  a  decimal 
point  (comma)  and  the  parentheses  to  separate  the  fractional 
part.3  He  stated  it  as  his  opinion  that  these  fractions  were  due 
to  Biirgi,4  although  it  seems  strange  that  he  was  not  familiar 
with  the  work  of  Stevin.  In  his  edition  of  Tycho  Brahe's 
Tabulae  Rudolphinae,  published  at  Ulm  in  1627,  he  uses  (p.  25) 
the  period  for  a  decimal  point,  thus:  "29.032  valet  29^-^." 

In  the  year  1616  Johann  Hartmann  Beyer  (1563-1625) 
wrote  to  Kepler  concerning  his  work,  and  in  the  letter  he  used 
both  the  decimal  comma  and  the  sexagesimal  symbolism  for  the 
decimal,  writing  314,  i'  5"  9"'  2""  V""  5"""  +  for  314.15926  +. 
Beyer  had  before  this  (1603)  published  a  work  on  these  frac- 
tions, Logistica  decimalis,  and  on  this  account  had  laid  claim 
to  their  invention,  although  he  had  long  been  preceded  by 
Stevin.  Adriaen  Metius  (1571-1635)  took  about  the  same 
step  in  symbolism  when  he  wrote  both  47852°  iS'o'V"  and 
47852/8'o'V"  for  47852.804.  He  also  spoke5  of  ^^oVoVo  ofte 

.  /o  n  _  in  .inictt 

4  o  i    4     * 


tttut     tntn 


*C.  G.  Knott,  Napier  Memorial  Volume,  pp.  77,  182,  188,  190,  191.  Edin- 
burgh, IQI4. 

2Ausszng  auss  der  ur  alien  Messe-Kunst  Archimedis.  It  appears  in  Volume  V 
of  the  F  risen  edition  of  Kepler's  works,  1864. 

3  "Furs  ander,  weil  ich  kurtze  Zahlen  brauche,  derohalben  es  offt  Bruche  geben 
wirdt,  so  mercke,  dass  alle  Ziffer,  welche  nach  dem  Zeichcn  (,)  folgen,  die  gehoren 
zu  dem  Bruch,  als  der  zehler,  der  nenner  dazu  wirt  nicht  gcsetzt."  He  then  gives 
an  example  in  interest:  . 

6  mal 


facit     2i   (go) 

4"Dise  Art  der  Bruchrechnung  ist  von  Jost  Biirgen  zu  der  sinusrechnung 
erdacht" 

r Opera  Omnia,  1633,  PP-  T9>  31?  49>  S°-  When  De  Morgan  (Arithmetical 
Books,  p.  41)  said  of  a  1640  edition  of  Metius  that  "sexagesimal  fractions  are 
taught,  but  not  decimal  ones,"  he  may  have  confused  the  symbols. 

n 


246  DECIMAL  FRACTIONS 

There  are  numerous  examples  of  writers  of  the  same  period 
who  used  these  awkward  symbols.  Girard,  the  editor  of  Stevin, 
whose  first  edition  of  the  latter's  works  appeared  in  1625,  did 
much  to  make  known  the  works  of  his  master,  but  he  appar- 
ently added  nothing  to  the  theory  or  the  symbolism.  Even  as 
late  as  1655  we  find  the  period  used  to  separate  an  integer  and 
common  fraction,1  as  in  the  case  of  198.-^  an<3  in  the  1685  edition 
of  Casati's  work2  we  have  0.00438  represented  by 

438 

I 00000 

In  1657  Frans  van  Schooten3  used  the  symbol  17579625  ...  © 
for  17579.625.  It  had  the  advantage  that,  in  finding  the  product 
of  two  decimals,  the  indices  in  the  circles  need  only  be  added 
in  order  to  determine  the  proper  index  in  the  result. 

The  use  of  smaller  type  for  the  decimal  part  was  not  uncom- 
mon,4 and  it  is  still  seen  on  the  continent  of  Europe.  As  to  the 
development  of  these  fractions  in  England,  Professor  Cajori 
has  suggested  that  Oughtred's  (1631)  use  of  the  symbol  o  [56 
for  0.56  was  one  of  the  causes  for  delay  in  the  general  adoption 
of  the  decimal  point. 

It  should  also  be  said  that  the  symbolism  is  by  no  means 
settled  even  yet.  In  England  23^0-  is  written5  23.45,  in  the 
United  States  it  appears  as  23.45,  and  on  the  Continent  such 
forms  as  23,45  and  2345  are  common.  Indeed,  in  America 
we  commonly  write  $23.-^  or  $23^  instead  of  $23.45,  to 
avoid  forgery. 

1B.  Capra,  Vsvs  et  fabrica  circini  .  .  . ,  p.  25  (Bologna,  1655).  The  first 
edition,  however,  seems  to  have  been  1607. 

2 P.  Casati,  Fabrica  et  vso  Del  Compasso  di  Proportione,  p.  123.  Bologna, 
1685.  In  the  first  edition  (1664),  however,  he  writes  such  a  number  (p.  86)  as 
a  common  fraction,  with  the  bar  between  the  terms. 

3  Exerdtationum  Mathematicarum  Libri  quinque,  liber  primus,  p.  33.  Leyden, 
1657. 

4 E.g.,  in  some  editions  of  Vieta's  tables;  also  in  R.  Butler,  The  Scale  of 
Interest  (London,  1630),  where  i125  is  used  for  1.125. 

5 Not  always,  however.  In  a  MS.  at  the  Woolwich  Academy,  of  date  1736, 
the  decimal  point  is  always  a  comma.  Hodder  wrote  a  Decimal  Arithmetick  in 
1668,  in  which  he  used  both  the  comma  and  the  dot. 


THE  INVENTION  OF  DECIMALS  247 

Summary.  The  historical  steps  in  the  invention  of  the  decimal 
fraction  may  be  summed  up  as  follows:  Pellos  (1492)  used  a 
decimal  point  where  others  had  used  a  bar,  but  the  idea  of  the 
decimal  fraction  was  not  developed  by  him.  Rudolff  (1530) 
worked  intelligently  with  decimal  fractions,  using  a  bar  for  the 
separatrix,  but  he  did  not  write  upon  the  theory.  Stevin  (1585) 
wrote  upon  the  theory  but  had  a  poor  symbolism.  About  1600, 
several  writers  attempted  to  improve  the  symbolism,  and  Biirgi, 
in  1592,  actually  used  a  comma  for  the  decimal  point,  without 
the  common  sexagesimal  marks,  and  comprehended  the  nature 
and  advantages  of  these  fractions.  Napier  knew  something  of 
the  theory  of  decimals  and  rendered  their  use  essential,  but  did 
not  himself  contribute  to  the  symbolism.1  In  the  mere  writing 
of  the  decimal  fraction,  at  least,  all  these  efforts  had  been  an- 
ticipated by  al-Kashi  (c.  1430),  whose  symbolism  was  quite  as 
good  as  that  of  any  European  writer  for  the  next  century  and 
a  half. 

It  is  thus  difficult  to  pick  out  the  actual  inventor,  although 
Rudolff  and  Stevin  are  entitled  to  the  most  credit  for  bringing 
the  new  system  to  the  attention  of  the  world.  It  should  be 
added  that  these  fractions  were  mentioned  by  Richard  Witt 
in  his  Arithmeticall  Questions  in  1613,  and  that  Henry  Lyte 
(1619)  wrote  The  Art  of  Tens,  or  Decimall  Arithmeticke,  —  a 
work  which  did  for  England  what  the  work  of  Stevin  had  done 
for  the  Continent. 

Percentage.  Long  before  the  decimal  fraction  was  invented 
the  need  for  it  was  felt  in  computations  by  tenths,  twentieths, 
and  hundredths,  and  this  need  gave  rise  to  a  peculiar  notation 
which  took  the  place  of  the  decimal  forms  and  which  has  per- 
sisted to  the  present  time  in  the  symbol  %. 

The  computations  of  the  Romans  that  led  up  to  the  subject 
of  percentage  may  be  illustrated  by  the  vicesima  libertatis,  a 
tax  of  gV  on  every  manumitted  slave  ;  by  the  centesima  rerum 
venalium,  a  tax  of  -^^  levied  under  Augustus  on  goods  sold  at 


imperfect  his  knowledge  was  may  be  seen  by  examining  his  De  arte 
logistica,  pp.  60,  65,  75,  et  passim. 


248  DECIMAL  FRACTIONS 

auction;  and  by  the  quinta  et  vicesima  mancipiorum,  a  tax 
of  2^5  on  every  slave  sold.1  Without  recognizing  per  cents  as 
such,  the  Romans  thus  made  use  of  fractions  which  easily 
reduce  to  hundredths. 

In  the  Middle  Ages,  both  in  the  East2  and  in  the  West,  there 
was  a  gradual  recognition  of  larger  denominations  of  money 
than  the  ancients  had  commonly  known,  and  this  led  to  the 
use  of  100  as  a  base  in  computation.  In  the  Italian  manu- 
scripts of  the  isth  century  it  is  common  to  find  examples  in- 
volving such  expressions  as  20  p  100,  xp  cento,  and  vi  p  c°,  for 
our  20%,  10%,  and  6%. 

When  commercial  arithmetics  began  to  be  printed,  this  cus- 
tom was  well  established,  and  so  in  Chiarino's  work  of  1481 
there  are  numerous  expressions  like  "xx.  per  .c."  for  20%,  and 
"viii  in  x  perceto"  for  8  to  10%.  Borghi  (1484)  and  Pellos 
(1492)  made  less  use  of  per  cents  than  one  would  expect  of 
such  commercial  authorities,  although  each  recognized  their 
value.3  The  demand  was  growing,  however,  and  Pacioli  (1494), 
familiar  with  the  large  commerce  of  the  giudecca  at  Venice, 
had  much  to  say  of  it.4  Beginning  early  in  the  i6th  century 
the  commercial  arithmetics  made  considerable  use  of  per  cents 
in  connection  with  interest  and  with  profit  and  loss,  sometimes 
in  relation  to  the  Rule  of  Three,5  so  popular  with  merchants 
of  that  period,  but  more  frequently  in  relation  to  isolated 
problems.6 

1  Harper's  Diet.  Class.  Lit.,  p,  1634. 

2 E.g.,  Bhaskara  (c.  1150)  uses  per  cents  in  the  interest  problems  in  the 
Lildvati.  See  Taylor,  Lilaivati,  p.  47 ;  Colebrooke,  Lildvati,  p.  39. 

3E.g.,  Borghi:  "...  guadagno  a  rason  de  .20.  per  cento";  and  Pellos: 
U.i2.,p  .100.,"  "p  .3.  ans  a  rason  de  .16. £  cent." 

4  His  printed  forms  include  ",p  ceto"  (fol.  65),  ".10.  ,p  cento"  (fol.  66),  and 
"per  ceto"  (fol.  66). 

5Thus  Ortega  (1512;  1515  ed.)  has  a  chapter  on  "Regvla  de  tre  de  cen- 
tenare"  (fol.^i). 

6Thus  Walckl  (1536):  "Ite  einer  leihet  einem  200  fl.  3  iar  vnd  eines  ieden 
iars  nimbt  er  lofl  vo  100  ist  die  frag  wieuil  die  3  iar  thut  gwiii  vnd  gwinsgwinn  " 
(fol.  67).  So,  also,  Rudolff  (1530;  1540  ed.) :  "Wen  man  vom  hundert  zu 
jarlichem  zins  geben  sol  5  flo  .  .  ."  (Ex.  71  in  the  Exempel-Buchlin) .  He  shows 
that  the  Italian  "pro  cento"  was  not  yet  well  known  in  Germany,  for  he  says 
(Ex.  156):  ".  .  .  vnd  wieuil  pro  cento  (verstee  an  Hundert  floren) ." 


PERCENTAGE  249 

In  America  at  present  the  expression  6%  is  identical  in 
meaning  with  0.06,  per  cent  having  come  to  signify  merely 
hundredths.  This  was  not  the  original  meaning,  nor  does  it 
conform  to  the  present  usage  in  England  and  certain  other 
countries,  where  expressions  like  "£6  per  cent"  are  in  common 
use.  This  usage  is  historically  correct,  the  isth  and  i6th  cen- 
tury writers,  with  whom  percentage  begins  in  any  large  way, 
having  always  employed  it.1 

c  cUlLajnttm  mrttta  ^Uaft^coc-^ 
nelU  £fl  itiunaaJr  y  avfetftnore 
ad\  A-uvp  c 

?  in/ 


ailug4 


*4  coc 

io  nclla  $4  vnfwia  Aat- 
nc!    d  .  cu4  .  5.  mcff  -  ^>  at  ^t4  aci  mcctfnre  . 
vw  f  wetl^c/t  4-^  PC 
c 


EARLY  PER   CENT   SIGN 


From  an  Italian  MS.  of  c.  1490.   Notice  also  the  old  symbol  for  pounds,  which 
may  have  suggested  the  dollar  sign 

Chief  Use  for  Per  Cents.  The  chief  use  for  per  cents  in  the 
1  6th  century  was  in  relation  to  the  computation  of  interest, 
and  by  the  beginning  of  the  iyth  century  the  rate  was  usually 
quoted  in  hundredths.2  It  also  appears  in  computing  profit  and 
loss,  at  first  indirectly,  as  in  the  following  addition  to  Recorde 
by  John  Mellis  (1594)  :  "If  one  yard  cost  6s  --  8  pence: 

aThus  Sfortunati  (1534;  1545  ed.)  uses  "libre  .30.  per  100";  Riese  (1522) 
uses  "  10  Ib.  von  100,"  "  10  fl  zum  100,"  "  10  fl  am  100,"  and  other  similar 
forms;  and  Albert  (1534)  has  "  10  fl  mit  100  fl." 

2  .E.g.,  Trenchant  (1566)  has  "a  raison  de  12  pour  100"  with  a  12%  inter- 
est table;  Petri  (1567)  speaks  of  "8  ten  hondert"  and  "12  ten  100";  Raets 
(1580)  gives  the  rate  as  "15  ten  hondert,"  and  Wentsel  (1599)  as  "  10.  ten  100 
tsjaers,"  —  all  of  which  shows  the  high  rates  of  interest  prevailing. 


250  DECIMAL  FRACTIONS 

and  the  same  is  sold  againe  for  8  s  -  6  pence:  the  question 
is,  what  is  gayned  in  100  pounds  laying  out  on  such  commodi- 
tie."  Many  books,  however,  stated  the  problems  substantially 
as  at  present. 

The  Per  Cent  Sign.  In  its  primitive  form  the  per  cent  sign 
(%)  is  found  in  the  isth  century  manuscripts  on  commercial 
arithmetic,  where  it  appears  as  "per  c"  or  "p  c,"  a  contraction 


Ji    i^crwcxAdA)  oyuco^Ai     j4 


fi 


THE  PER   CENT   SIGN   IN  THE  17TH   CENTURY 
From  an  anonymous  Italian  MS.  of  1684 

for  "per  cento."1  As  early  as  the  middle  of  the  iyth  century 
it  had  developed  into  the  form  "per  -g-,"  after  which  the  "per" 
finally  dropped  out.  The  solidus  form  (%)  is  modern. 

Permillage.  It  is  natural  to  expect  that  percentage  will  de- 
velop into  permillage,  and  indeed  this  has  not  only  begun, 
but  it  has  historic  sanction.  Bonds  are  quoted  in  New  York 
"per  M,"  and  so  in  various  other  commercial  lines.  This  was 
already  common  in  the  i6th  century.2  At  present,  indeed,  the 
symbol  %o  is  used  in  certain  parts  of  the  world,  notably  by 
German  merchants,  to  mean  per  mill,  a  curious  analogue  to 
%  developed  without  regard  to  the  historic  meaning  of  the 
latter  symbol. 

lRara  Arithmetica,  pp.  439,  441,  458,  with  facsimiles. 
2Thus  Cardan  (1539)  says  that  "tara  coputada  est  ad  100.  vel  ad  1000." 
Arithmetica,  i$39>  capp.  57,  59. 


ANCIENT  IDEA  OF  IRRATIONALS  251 

4.  SURD  NUMBERS 

Ancient  Idea  of  Irrationals.  Proclus  (c.  460)  tells  us  that  the 
Pythagoreans  discovered  the  incommensurability  of  the  diag- 
onal and  the  side  of  a  square,1  which  is  only  a  geometric  view 
of  the  irrationality  of  V2.  Proclus  also  states  that  they  were 
led  to  study  the  subject  of  commensurability  through  their 
work  with  numbers.  Plato  says2  that  Theodorus  of  Cyrene 
(c.  425  B.C.)  discovered  that  "oblong  numbers,  3,  5,  6,  7, 
are  composed  of  unequal  sides."  He  also  states  that  "  Theo- 
dorus was  writing  out  for  us  something  about  roots,  such  as  the 
roots  of  three  or  five,  showing  that  they  are  incommensurable 
by  the  unit:  he  selected  other  numbers  up  to  seventeen — there 
he  stopped."3 

With  respect  to  other  writers  on  incommensurable  lines, 
Diogenes  Laertius  (2d  century)  tells  us  that  Democritus 
(c.  400  B.C.)  composed  a  treatise  upon  the  subject.4 

Summary  of  Greek  Ideas  on  Irrationals.  Summarizing  the 
work  of  the  Greeks,  there  seems  to  be  good  reason  for  believing 
that  the  immediate  followers  of  Pythagoras  knew  and  demon- 
strated the  incommensurability  of  the  diagonal  and  the  side  of 
a  square,  but  that  they  looked  upon  this  case  of  irrationality 
as  a  peculiarity  of  the  square.  Theodorus  seems  to  have  car- 
ried the  investigation  farther,  recognizing  that  irrationality  of 
square  roots  was  not  confined  to  Vz.  Theaetetus  (c.  3756.0.) 
appears  to  have  laid  the  foundations  for  a  general  theory  of 
quadratic  irrationals5  and  to  have  established  their  leading 

1  Heath's  Euclid,  Vol.  Ill,  p.  i  seq.,  to  the  notes  of  which  the  reader  is  re- 
ferred. See  also  Cantor,  Beitrdge,  p.  108.  The  proof  is  given  in  Euclid's  Ele- 
ments, numbered  X,  117  in  early  editions,  but  is  now  relegated  to  an  appendix. 
See  also  H.  Vogt,  "Die  Entdeckungsgeschichte  des  Irrationalen  nach  Plato  .  .  .," 
Bibl.  Math.,  X  (3),  07;  Heath,  History,  I,  65,  go,  154. 

2Thecetetus,  147  D;  Jowett  translation,  IV,  123;  Heath,  History,  I,  203. 

3  It  should  be  observed  that  the  method  of  proof  for  V3  is  quite  different 
from  that  for  V^,  and  so  for  other  surds.  See  Heath,  Euclid,  Vol.  Ill,  p.  2,  and 
History,  I,  155. 

4IIcpi  &\bywv  ypawQv  Kal  vaffr&v  /3'.  See  F.  Hultsch,  Neue  Jahrbilcher  fur 
Philologie  und  Pddagogik,  CXXIII,  578;  Heath,  History,  I,  156. 

6 Heath,  History,  I,  209. 


252  SURD  NUMBERS 

properties.  Euclid  (c.  300  B.C.)  took  the  final  important  step 
due  to  the  Greek  geometers,  classifying  square  roots  and  intro- 
ducing the  idea  of  biquadratic  irrationals.1 

This  discovery,  then,  was  the  second  noteworthy  step  in  the 
creation  of  types  of  artificial  numbers.  The  Greeks  showed  that 
all  magnitudes  are  either  rational  (fard,  rheta')  or  irrational 
(dXoya,  a'loga),  their  idea  of  an  irrational  number  being  such  a 
number  as  cannot  be  expressed  as  the  ratio  of  two  integers. 

The  geometric  treatment  of  incommensurablesx  naturally  led 
to  the  arithmetic  and  algebraic  treatment  of  ^irrationals,  the 
subject  of  the  present  discussion.2 

The  Name  "Surd."  Al-Khowarizmi  (c.  825)  spoke  of  rational 
numbers  as  "audible"  and  of  surds  as  "inaudible/'3  and  it  is 
the  latter  that  gave  rise  to  the  word  "surd"  (deaf,  mute).  So 
far  as  now  known,  the  European  use  of  this  word  begins  with 
Gherardo  of  Cremona  (c.  nso).4  The  term  was  also  used  by 
Fibonacci  (1202),  but  to  represent  a  number  that  has  no  root.5 
The  Arabs  and  Hebrews  often  called  surds  "nonexpressible  num- 
bers,"0— a  name  which  may  have  suggested  the  "inexplicable 
sides"  of  the  Renaissance  writers.7  It  is  simply  a  translation 
from  the  Euclidean  term  d\oyo$  (a 'logos,  without  ratio,  irra- 
tional, incommensurable). 

As  to  what  constitutes  a  surd,  however,  there  has  never  been 
a  general  agreement.  It  is  admitted  that  a  number  like  \/2 

aVogt,  loc.  cit.  On  Professor  Zeuthen's  discussion  of  Vogt's  conclusions,  see 
H.  Bosnians,  in  the  Revue  des  Questions  Scientifiques,  July,  1911.  See  also 
Heath,  History,  I,  402. 

-On  the  history  of  transcendental  numbers  see  the  statement  on  page  268. 

3Rosen  ed.,  p.  192.  *Bibl.  Math.,  I  (3),  516. 

5"Nam  quidem  numeri  habent  radices,  et  uocatur  [sic]  quadrati ;  et  quidarn 
non;  quorum  radices,  que  surde  dicuntur,  cum  inpossibile  sit  cas  in  numeris 
inuenire  ..."  (Liber  Abaci,  p.  353).  By  "root"  he  refers  here,  as  usual,  to 
square  root. 

6 E.g.,  al-Karkhi  (c.  1020) ;  see  Hochheim,  Kdfi  fit  Hisdb,  II,  12. 

7JS.g.,  Schoner,  in  his  De  numeris  figuratis  (1569;  1586  ed.,  p.  213),  says: 
"Explicable  latus  est,  cujus  ad  i.  ratio  explicari  potest.  Ut  latus  4  est  2,  & 
dicitur  explicabile.  .  .  .  Inexplicabile  latus  contra  est,  cujus  ad  i.  ratio  explicari 
non  potest.  Ut  3.  .  .  ." 

Stevin  (1585)  speaks  of  "nombres,  comme  A/8,  &  semblables,  qu'ils  appellent 
absurds,  irrationels,  irreguliers,  inexplicables,  sourds,  &c"  (1634  ed.,  p.  9). 


APPROXIMATE  VALUES  253 

is  a  surd,  but  there  have  been  prominent  writers  who  have 
not  included  V6,  since  V6=V2  x  V^;1  and  \/2  +>/3  is  com- 
monly excluded.2 

Approximate  Values.  An  interest  in  the  irrational  showed 
itself  strongly  among  the  ancients.  Here  was  a  mystery  to  be 
fathomed,  and  from  the  time  of  Pythagoras  to  that  of  Weier- 
strass  the  nature  of  irrationals  and  the  ability  to  work  with 
them  occupied  the  attention  of  a  considerable  part  of  the 
mathematical  world.  Among  the  noteworthy  efforts  was  the 
one  which  sought  to  find  an  approximate  value  for  an  expres- 
sion like  VT.  As  already  said  (p.  144),  the  Greeks  found  the 
square  root  of  a  number  in  much  the  same  way  as  that  which 
is  commonly  taught  in  school  today,  but  their  ignorance  of  the 
decimal  fraction  made  the  process  of  approximation  very  dif- 
ficult in  the  case  of  surds.  For  this  reason  the  ancient  and 
medieval  writers  resorted  to  various  rules  which  can  best  be 
appreciated  by  first  considering  the  principle  involved. 

Let  a  be  an  approximation  to  -\fA  by  defect.  Then  A/a 
must  be  an  approximation  by  excess,  and  the  arithmetic  mean, 

A\  N- 

is  an  approximation  of  the  second  order  by 


excess,3  and  the  harmonic  mean,  A/al9  is  an  approximation  of 
the  second  order  by  defect.  This  process  may  evidently  be 
carried  on  indefinitely.  If  A  =  a2+r,  we  have  in  particular, 


r  A 

a  +        »  --  =  a  4-  • 


2  a  a,  r 

1  2  a  -f  - 

a 


lE.g.,  Beha  Eddin  (c.  1600),  al-Karkhi  (c.  1020),  and  other  Arab  writers 
included  only  non-squares  not  divisible  by  the  digits  2,  3,  .  .  . ,  9.  See  Hoch- 
heim,  Kaji  fU  Hisab,  II,  13  n. 

2G.  Chrystal,  Algebra,  2d  ed.,  I,  203  (Edinburgh,  1889):  ".  .  .a  surd  num- 
ber is  the  incommensurable  root  of  a  commensurable  number.  .  .  .  For  example 
.  .  .  -\A  is  not  a  surd.  .  .  .  Neither  is  V(  V2  +  x )  •" 

x2  nz 

3  For  — __  =  x  -f  n  -f  — : — ,  so  that  if  we  divide  by  a  number  that  is  n  less 

x  —  n  x—  n 

than  the  square  root,  we  shall  have  a  result  that  is  more  than  «  in  excess  of 
the  square  root. 


254  SURD  NUMBERS 

and  so  on.    Recognizing  that 


a  +  — 
a  2  a 


is  an  approximation  by  excess,  the  Arabs  took 


2  a  +  I 

as  an  approximation  by  defect,  but  this  rule  is  not  found  among 
the  Greeks.1 

Of  the  various  rules  for  approximation  to  VA,  the  one  most 
commonly  used  in  the  past  may  be  expressed  as 

^JA  =  Vd2+-  r=a  +  —  > 
2  a 

as  in  v  \  o  =  V o  4-1=2+-  — - — ~  =  2  J. 

;T          ^2X3        <H> 

an  approximation  by  excess.    The  corresponding  approximation 
by  defect  is : 

^J A  ~ vV~  +  r—  a-\ > 


as  in  Vio  =  v9  +  i  =  3  +-{-  =  3y, 

which  probably  explains  why  VTo  was  so  often  used  for  TT  by 
early  writers.  Of  these  approximations,  the  one  by  excess  is 
found  in  the  works  of  Heron  (c.  50 ?).2  The  medieval  writers 
used  both  of  these  approximations,3  often  with  variations.  For 

XP.  Tannery,  "L'extraction  des  racines  carrees  d'apres  Nicolas  Chuquet," 
Bibl.  Math.,  I  (2),  17;  "Du  role  de  la  musique  grecque  dans  le  developpement 
de  la  mathematique  pure,"  ibid.,  Ill  (3),  171. 

2 A  fact  noted  by  Clavius,  Epitome,  1583;  1585  ed.,  p.  318.  It  should  oc- 
casionally be  repeated  that,  as  stated  in  Volume  I,  page  125,  this  date  is  uncer- 
tain. Heron  may  have  lived  as  late  as  the  3d  century. 

3 E.g.,  al-Hassar  (c.  1175?)   made  use  of  both  the  one  by  defect  and  the 


one  by  excess,  together  with  a  -\ — —  and  a  +  — —  -— —  for  closer  work. 

.20+2  2  0  - v 


For  other  cases  see  Hochheim,  Kdfi  ftl  Hisab,  II,  14;  Wertheim,  Elia  Misrachi, 
p.  21 ;  B.  Boncompagni,  Atti  Pontif.,  XII,  402. 


APPROXIMATE  VALUES  255 

example,  Rhabdas  (c.  1341),  following  an  Arabic  method,1  ob- 
tained a  first  approximation  to  VTo  by  using  a  rule  equivalent 

to  n         ,  A-d1 

VA  =  a  -f  --  9 
2  a 

.   .  I  —  10  —  9         1       19 

giving  V  10  -  3  +  —  ^  =  3j.  =  £  • 

Then,  since  10  -s-  -1/  =  3-^,  he  takes  the  mean  of  3^  and  3^, 
which  is  3^V    This  he  shows  to  be  a  close  approximation, 

since  (3jVff)a==IO5T9W 

A  somewhat  different  method,  also  involving  averages,  is 
given  by  Chuquet  (1484).    Let  two  approximate  values  of 


be  -~  and  -->  the  first  being  too  great  and  the  second  being  too 

?o  J\ 

small,  and  let  /'=  /<,+/!  and  #'  =  ?0  +  qr  Then  p'/qr  is  a  new 
approximation  intermediate  in  value,  and  whether  it  is  by  ex- 
cess or  defect  is  found  by  squaring.  In  the  same  way  an  ap- 
proximation is  found  between  p'/q'  and  one  of  the  others,  and 
so  on.  This  rule  was  employed  by  several  later  writers.2  De 
la  Roche  (1520),  who  plagiarized  Chuquet,  asserted  that  any 
study  of  "imperfect  roots"  was  useless,  although  custom  re- 
quired it.3  Such  approximations  are  common  in  the  works  of 
the  1  6th  century,  together  with  similar  rules  for  cube  root.4 
With  all  this  there  naturally  developed  many  evidences  of 
ignorance,  as  when  Peter  Halliman  (1688)  gave  substantially 
the  rule  „/—  r 


1P.  Tannery,  Notices  et  extraits  des  manuscrits  de  la  Bibl.  nat.,  XXXII,  185. 

2E.g.,  Ortega  (1512)  ;  substantially  by  Clavius  (Epitome,  1583;  1585  ed., 
p.  318,  where  he  gives  V2o  —  4fVVg\>  approximately);  and  substantially  by 
Metius  when  he  found  the  value  of  TT  by  interpolating  between  3  /J^  and  syVV 
the  result  being  the  Chinese  value,  •]  f  J.  See  P.  Tannery,  Bibl.  Math.,  I  (2),  17. 

3  The  study  of  "ratines  imparfaites"  was  "labeur  sans  vtile,"  but  "pour  la 
perfection  de  ce  liure"  he  gave  a  method  "par  la  regie  de  mediation  entre  le 
plus  et  le  moins,"  —  an  elementary  method  of  interpolation.   This  is  described  by 
Treutlein  in  the  Abhandlungen,  I,  66. 

4  Thus  Stevin  gives  substantially  the  rule  that 

VA  =  v'i»T7  =  a  +     —  '-—  • 

3*(*  +  i)  +  i 
CEuvres,  1634  ed.,  p.  30. 


256  SURD  NUMBERS 

celebrating  his  discovery  by  the  doggerel  verse, 

Now  Logarithms  lowre  your  sail, 

And  Algebra  give  place, 
For  here  is  found,  that  ne'er  doth  fail, 

A  nearer  way,  to  your  disgrace.1 

It  should  also  be  understood  that  such  rules  for  roots  are 
ancient.  For  example,  Heron  (c.  50?)  gives  what  may  possibly 
be  the  equivalent  of  the  formula 


where  a>^n>b,  and  a  —  6=i.     By  means  of  this  rule  we 
should  find  that  ^109  =  4-7785  instead  of  4.7769." 

^Criteria  for  Squares.  In  order  to  determine  whether  or  not 
V#  is  a  surd,  those  who  were  interested  in  number  theory  de- 
veloped from  time  to  time  criteria  for  ascertaining  whether  a 
number  is  a  square.  Such  criteria  are  found  in  various  ancient 
and  medieval  works,  both  Arab3  and  European.  A  Munich 
manuscript4  of  the  isth  century,  for  example,  states  that  if  a 
square  ends  in  an  even  number,  it  is  divisible  by  4  ;  5  that  if  it 
ends  in  zeros,  it  ends  in  an  even  number  of  zeros  ;  °  that  it  cannot 
end  in  2,  3,  7,  or  8  ;7  and  that  every  square  is  of  the  form 
3  ;/  or  3»  +  1.8  Such  rules,  often  extended,  are  found  in  various 
works  of  the  classical  and  Renaissance  periods.9 

1From  The  Square  and  Cube  Root  compleated  and  made  easie   (London, 

1688),  quoted  by  A.  De  Morgan,  Arithmetical  Books,  p.  52. 

2J.  G.  Smyly,  "Heron's  formula  for  cube  root,"   Hermathena,  XLII,   64, 

correcting  M.  Curtze,  Zeitschrift,  HI.  Abt.  (1897),  p.  119,  and  referring  to  Heron's 

Metrica,  III,  20.   See  also  the  interpretations  in  Heath,  History,  II,  341. 

y£.g.,  al-Karkhi   (c.  1020)   and  al-Qalasadi   (0.1475).    See  Hochheim,  Kaji 

jil  Hisdb,  II,  13. 

4No.  14,908,  described  by  Curtze  in  Bibl.  Math.,  IX  (2),  38. 
6"Omnis  quadratus,  cuius  prima  est  par,  est  per  4  divisibilis." 
6<tOmnis  quadratus  in  primis  locis  habet  parem  numerum  ciffrarum." 
7"Nullus  quadratus  recipit  in  primo  loco  2,  3,  7  vel  8,  sed  alios  bene."   This 

is  a  very  old  rule. 

8"Omnis  quadratus  est  simpliciter  vel  subtracte  unitate  per  3  divisibilis." 
9  Thus  Buteo  (1559)  adds  that  a  square  number  cannot  end  in  5  unless  it 

ends  in  25. 


NEGATIVE  NUMBERS  257 

Surds  in  Algebra.  The  placing  of  the  study  of  surds  in  the 
books  on  algebra  is  a  tradition  which  began  with  the  Renais- 
sance. The  books  on  logistic,  used  in  commercial  schools,  had 
no  need  for  the  subject ;  it  properly  belonged  in  the  books  on 
the  theory  of  numbers,  the  ancient  arithmetica.  Since,  how- 
ever, algebra  took  over  a  considerable  part  of  the  latter  in  the 
Renaissance  period,  surds  found  a  place  in  this  science.  Fur- 
thermore, since  these  forms  are  needed  in  connection  with  irra- 
tional equations,  they  were  usually  considered  before  that  topic 
in  the  study  of  algebra.  In  the  isth  century,  however,  they 
are  often  found  in  the  theoretical  arithmetics.1 

5.  NEGATIVE  NUMBERS 

Early  Use  of  Negative  Numbers.  No  trace  of  the  recognition 
of  negative  numbers,  as  distinct  from  simple  subtrahends,  has 
yet  been  found  in  the  writings  of  the  ancient  Egyptians,  Baby- 
lonians, Hindus,  Chinese,  or  Greeks.  Nevertheless  the  law 
of  signs  was  established,  with  the  aid  of  such  operations  as 
(10-4)  •  (8  -  2 ) ,  and  was  known  long  before  the  negative  num- 
ber was  considered  by  itself. 

The  Chinese  made  use  of  such  numbers  as  subtrahends  at  a 
very  early  date.  They  indicated  positive  coefficients  by  red 
computing  rods,  and  negative  ones  by  black,  and  this  color 
scheme  is  also  found  in  their  written  works.2  The  negative 
number  is  mentioned,  at  least  as  a  subtrahend,  in  the  K'iu- 
ch'ang  Suan-shu  (c.  200  B.C.),3  and  in  various  later  works,  but 
the  law  of  signs  is  not  known  to  have  been  definitely  stated 
in  any  Chinese  mathematical  treatise  before  1299,  when  Chu 
Shi-kie  gave  it  in  his  elementary  algebra,  the  Suan-hio-ki-mong 
(Introduction  to  Mathematical  Studies}. 

irThus  the  Rollandus  MS.  (c.  1424)  has  surds  in  the  arithmetic  just  before 
algebra  is  begun,  and  similarly  in  Pacioli's  Suma  (1494).  On  the  modern  prob- 
lem consult  the  Encyklopddie  der  mathematischen  Wissenschaften,  I,  49  (Leip- 
zig, 1898-  ) ;  hereafter  referred  to  as  Encyklopddie. 

2Mikami,  China,  pp.  18,  20,  21,  27,  89,  et  passim;  Cantor,  Geschichte, 
I  (2),  642. 

3  See  Volume  I,  page  31,  for  discussion  as  to  earlier  date  for  the  original  work. 
It  may  have  been  written  before  1000  B.C. 


258  NEGATIVE  NUMBERS 

The  first  mention  of  these  numbers  in  an  occidental  work  is 
in  the  Arithmetica  of  Diophantus  (c.  27s),1  where  the  equation 
4%  +  20  =  4  is  spoken  of  as  absurd  (aroTro?),  since  it  would 
give  #  =  —  4.  Of  the  negative  number  in  the  abstract,  Dio- 
phantus had  apparently  no  conception.  On  the  other  hand, 
the  Greeks  knew  the  geometric  equivalent  of  (a  —  b)2  and  of 
(0  +  b)  (a  —  b)',  and  hence,  without  recognizing  negative  num- 
bers, they  knew  the  results  of  the  operations  (~  b)  •(— 6) 
and  (  +  b)  •  (-  b}. 

In  India  the  negative  number  is  first  definitely  mentioned  in 
the  works  of  Brahmagupta  (c.  628).  He  speaks  of  " negative 
and  affirmative  quantities,"2  using  them  always  as  subtrahends 
but  giving  the  usual  rules  of  signs.  The  next  writer  to  treat  of 
these  rules  is  Mahavira  (c.  850),  and  after  that  time  they  are 
found  in  all  Hindu  works  on  the  subject. 

The  Arabs  contributed  nothing  new  to  the  theory,  but  al- 
Khowarizmi  (c.  825)  states  the  usual  rules,3  and  the  same  is 
true  of  his  successors. 

When  Fibonacci  wrote  his  Liber  Abaci  (1202)  he  followed 
the  Arab  custom  of  paying  no  attention  to  negative  numbers, 
but  in  his  Flos  (c.  1225)  he  interpreted  a  negative  root  in  a 
financial  problem  to  mean  a  loss  instead  of  a  gain.4  Little 
further  was  done  with  the  subject  by  medieval  writers,  but 
as  we  approach  the  Renaissance  period  we  find  the  negative 
number  as  such  receiving  more  and  more  recognition.  For 
example,  among  the  problems  set  by  Chuquet  (1484)  is  one5 
which  leads  to  an  equation  with  roots  "#1.7. f\"  and  "27.f\," 
that  is,  -  7i3T  and  27T3T. 

Modern  Usage.  The  first  of  the  i6th  century  writers  to  give 
noteworthy  treatment  to  the  negative  number  was  Cardan.  In 

1Nesselmann,  Alg.  Griechen,  p.  311;  Heath,  Diophantus,  2d.  ed.,  pp.  52,  200 
(Cambridge,  1910) ;  Cantor,  Geschichte,  I  (2),  441. 

2Colebrooke  translation,  pp.  325,  339. 

3 Rosen  translation,  p.  26. 

4"Hanc  quidem  quaestionem  insolubilem  esse  monstrabo,  nisi  concedatur, 
primum  hominem  habere  debitum."  Scritti,  II,  238. 

5  Boncompagni's  Bullettino,  XIV,  419,  Ex.  xiv.  Chuquet  adds,  "Ainsi  ce  cal- 
cule  est  vray  que  aulcuns  tiennent  Impo.hle 


NAMES  AND  SYMBOLS  259 

his  Ars  Magna  (1545)  he  recognized  negative  roots  of  equa- 
tions and  gave  a  clear  statement  of  the  simple  laws  of  nega- 
tive numbers.1 

Stifel  (1544)  distinctly  mentioned  negative  numbers  as  less 
than  zero,2  and  showed  some  knowledge  of  their  use.  By  this 
time  the  rules  of  operation  with  numbers  involving  negative 
signs  were  well  understood,  even  though  the  precise  nature  of 
the  negative  number  was  not  always  clear.  Thus  Bombelli 
(JS?2)  gave  these  rules  and  applied  them  intelligently  to  such 
cases  as  (+  15)  +  (—  20)  =  —  $.3  It  was  due  to  the  influence 
of  men  like  Vieta,  Harriot,  Fermat,  Descartes,  and  Hudde, 
however,  that  the  negative  number  came  to  be  fully  recognized 
and  understood.  The  idea  of  allowing  a  letter,  with  no  sign 
prefixed,  to  represent  either  a  positive  or  a  negative  number 
seems  due  to  Hudde  (1659). 

Names  and  Symbols  for  Negative  Numbers.  The  Hindu 
writers  who  mentioned  negative  numbers,  or  numbers  used  as 
subtrahends,  placed  a  dot  or  a  small  circle  over  or  beside  each, 
as  stated  on  page  396.  The  names  used  were  the  equivalent  of 
our  word  " negative."  The  early  European  usage  has  already 
been  mentioned,  but  it  remains  to  speak  of  the  establishing  of 
our  modern  terminology. 

As  already  stated,  the  Chinese  wrote  positive  numbers  in  red 
and  negative  numbers  in  black,  and  so  indicated  them  by  their 
stick  symbolism.  They  also  had  another  method  for  indicating 
negative  coefficients,  one  that  may  have  been  due  to  Li  Yeh 
(1259).  This  consisted  in  drawing  a  diagonal  stroke  through 
the  right-hand  digit  figure  of  a  negative  number,  as  in  the  case 
of  IOTRHI  for  -  10,724,  and  of  IO>kOO  for  -  io,2oo.4 

In  the  isth  century  the  names  "positive"  and  "affirmative" 
were  used  to  indicate  positive  numbers,  as  also  "privative" 

1Thus  on  fol.  3,  v.,  speaking  of  squares,  he  says:  "At  uero  quod  tarn  ex  3, 
quam  ex  m :  3,  fit  9,  quoniam  minus  in  minus  ductu  ^ducit  plus." 

2"  Finguntur  numeri  infra  o,  id  est,  infra  nihil."    Arithmetica  Integra,  fol.  249,  r. 

3"Piu  via  piu  fa  piu.  Meno  via  meno  fa  piu"  etc.  (p.  70).  Also:  "E  p.iS 
con  m.20  fk  m.5.  perche  se  io  mi  trouassi  scudi  15,  e  ne  fossi  debitore  20,  pagati 
li  15  restarei  debitore  5"  (p.  72). 

4Mikami,  China,  p.  82. 


260  NEGATIVE  NUMBERS 

and  " negative"  for  negative  numbers/ — a  usage  followed  by 
Scheubel  (i55i).2 

Cardan  (1545)  spoke  of  "minus  in  minus"  as  being  plus,  but 
in  general  he  called  positive  numbers  numeri  ueri3  and  negative 
numbers  numeri  ficti.4  His  symbol  for  a  negative  number  is 
simply  m : ,  as  in  the  case  of  m : 3  for  —  3.5 

Stifel  (1544)  called  these  numbers  "absurd"  and  wrote  0  —  3 
as  an  illustration.6 

Tartaglia  (1556)  spoke  of  a  negative  number  as  "the  term 
called  minus,"7  laying  down  the  usual  rules.8 

Bombelli  (1572)  used  the  word  "minus"  (meno)  as  we  do 
in  such  rules  as  "minus  times  minus  gives  plus,"  his  symbol  for 
—  5  being  111.5.  Unlike  Cardan,  he  had  a  definite  sign  for  +  5 
also,  writing  it  p. 5. 

Tycho  Brahe,  the  astronomer  (1598),  spoke  of  the  negative 
number  as  "privative"  and  indicated  it  by  the  minus  sign.9 

Napier  (c.  1600)  used  the  adjectives  abundantes  and  dejec- 
tivi  to  designate  positive  and  negative  numbers,  Sturm  (1707) 
spoke  of  Sache  and  Mangel,  and  various  other  names  and 
symbols  have  been  suggested.10 

aH.  E.  Wappler,  Zur  Geschichte  der  deutschen  Algebra  im  XV.  Jahrhundert, 
Prog.,  p.  31  (Zwickau,  1887).  2Tropfke,  Geschichte,  II  (2),  79. 

3 Or  ucri  numeri.   He  used  both  forms,  as  on  fol.  3,  v.,  of  the  Ars  Magna. 

4  So  in  speaking  of  the  roots  of  an  equation  he  says  "  una  semper  cst  rei  uera 
aestimatio,  altera  ei  aequalis,  ficta." 

6 So  he  gives  the  roots  of  x2  =  16  thus:  "res  est  4,  uel  m:  4." 

6"Finguntur  numeri  minores  nihilo  ut  sunt  0  —  3"  (Arithmetica  Integra, 
fol.  48,  r.) .  Later  (fol.  249,  v.)  he  speaks  of  zero  as  "quod  mediat  inter  numeros 
veros  et  numeros  absurdos." 

7".  .  .  il  termine  chiamato  men"  (General  Trattato,  II,  fol.  83,  r,). 

8 E.g.,  "Terzo  regola,  a  multiplicare  men  fia  men  fa  sempre  phi"  (ibid., 
fol.  85,  u.).  His  illustration  is  as  follows: 

a  multiplicar  9  in  2 

per  - — - —  — -_  -—•  -_ 8  m  3 

fa  72  men  43  piu  6 

che  sara  35  a  ponto 

9".  .  .  quod  eorum  alii  positivi  sunt  alii  privativi;  positivi  ii  quibus  vel 
nullum  signum  est  additum  vel  praengi  debet  hoc-f  ;  privativi  vero  qui  prae- 
fixum  habere  debent  signum  hoc  — "  (Tabulae  Rudolphinae,  p.  9  (Ulm,  1627)). 

10  On  the  later  treatment  of  the  negative  number  see  Cantor,  Geschichte, 
IV,  79-88. 


COMPLEX  NUMBERS  261 

6.  COMPLEX  NUMBERS 

Early  Steps  in  Complex  Numbers.  The  first  trace  of  the 
square  root  of  a  negative  number  to  be  found  in  extant  works 
is  in  the  Stereometria  of  Heron  of  Alexandria  (c.  50?),  where 
V8i  — 144  is  taken  to  be  ^144  —  81,  or  8  — TV  The  prob- 
lem involved  is  impossible  of  solution,  and  this  step  should  have 
been  left  V— 63  or  sV— 7?1  but  whether  the  error  is  due  to 
Heron  or  to  some  copyist  is  uncertain. 

The  next  known  recognition  of  the  difficulty  is  found  in  the 
Arithmetica  of  Diophantus2  (c.  275).  In  attempting  to  com- 
pute the  sides  of  a  right-angled  triangle  of  perimeter  12  and 
area  7,  Diophantus  found  it  necessary  to  solve  the  equation 
336 #2 +  24  =  1723;.  He  stated  that  the  equation  cannot  be 
solved  unless  the  square  of  half  the  coefficient  of  x  diminished 
by  24  x  336  is  a  square,  not  otherwise  seeming  to  notice  that 
this  equation  has  complex  roots. 

Mahavira  (c.  850)  was  the  first  to  state  the  difficulty  clearly, 
saying,  in  his  treatment  of  negative  numbers,  that,  "as  in  the 
nature  of  things  a  negative  [quantity]  is  not  a  square  [quan- 
tity], it  has  therefore  no  square  root."8 

Bhaskara  (c.  1150)  used  about  the  same  language  in  his 
Bija  Ganita : 

The  square  of  an  affirmative  or  of  a  negative  quantity  is  affirmative ; 
and  the  square  root  of  an  affirmative  quantity  is  two-fold,  positive 
and  negative.  There  is  no  square-root  of  a  negative  quantity:  for  it 
is  not  a  square.4 

The  Jewish  scholar  Abraham  bar  Chiia  (c.  1120)  set  forth 
the  same  difficulty  in  discussing  the  equations  xy  =  48  and 
x+y  -i4.5 

1On  this  general  topic  see  W.  W.  Beman,  "A  Chapter  in  the  History  of  Math- 
ematics," vice-presidential  address  in  Section  A,  Proc.  of  the  American  Assoc.  for 
the  Adv.  of  Set.,  1897;  E.  Study,  in  the  Encyklopddie  der  Math.  Wissensch.,  I, 
A,  4  (Leipzig,  1898)  ;  H.  Hankel,  Vorlesungen  iiber  die  complexen  Zahlen,  Leip- 
zig, 1867;  G.  Loria,  Scientia,  XXI,  101 ;  F.  Cajori,  Amer.  Math.  Month.,  XIX, 
167.  On  Heron,  see  the  Schmidt  edition  (Leipzig,  1914),  V,  35. 

2Heath,  Diophantus,  26.  ed.,  p.  244  (Cambridge,  1910). 

3  Ganita-Sdra-Sangraha,  p.  7. 

4Colebrooke's  translation,  p.  135.  5Abhandlungen,  XII,  46. 


262  COMPLEX  NUMBERS 

The  Arabs  and  Persians  seem  to  have  paid  no  special  atten- 
tion to  the  subject,  and  the  next  step  was  taken  in  Italy  and 
after  the  invention  of  printing. 

Early  European  Efforts.  Pacioli  (1494)  stated  in  his  Suma 
that  the  quadratic  equation  x2  +  c  =  bx  cannot  be  solved  un- 
less |62  =  c/  so  that  he  recognized  the  impossibility  of  finding 
the  value  of  V—  a.  About  the  same  time  Chuquet  (1484) 
seems  to  have  found  that  V—  a  represents  an  impossible  case.2 

Cardan  (1545)  spoke  of  the  equation  x*  4-  12  =  6x2  as  being 
impossible,3  referring  to  the  roots  of  such  equations  as  ficta  or 
per  in.  He  was  the  first  to  use  the  square  root  of  a  negative 
number  in  computation,  the  problem  being  to  divide  10  into 
two  parts  whose  product  is  40.*  He  found  the  number  to  be 
5  +V—  15  and  5  —V-  15,  spoke  of  the  solution  "by  the  minus 
root/'6  and  proved  by  multiplication  that  his  results  were 
correct. 

The  next  to  attack  the  problem  was  Bombelli  (1572).  In 
his  _algebra6  he  speaks  of  such  quantities  as  +V—  a  and 
—  V—  a,  but  he  made  no  advance  upon  Cardan's  theory. 

Stevin  (1585)  noted  the  difficulty  of  working  with  imagi- 
naries,  but  could  only  remark  that  the  subject  was  not  yet 
mastered.7  Girard  (1629)-  found  it  necessary  to  recognize 
complex  roots  in  order  to  establish  the  law  as  to  the  number  of 
roots  of  an  equation.8 


147,  r.  E.g.,  x2  +  7  =  *  +  5  :  "Dico  qsto  essere  impossibile."  As  in  all 
such  cases,  the  symbolism  here  shown  is  modern. 

2  Boncompagni's  Bullettino,  XIV,  444. 

3"Qu6d  si  caruerit  estimatioe  uera,  carebit  etiam  ea,  que  est  per  m:  uelut 
i  qdt!  qd'n  p:  12,  aeqtur  6  qdtu,  quia  non  potest  aequatione  ueram  habere, 
carebit  etiam  ficta,  sic  effl  uocamus  earn,  quae  debiti  est  seu  minoris"  (Ars 
Magna,  fol.  3,  v.). 

*0pera,  IV,  287.  Lyons,  1663.  6L'Algebra,  p.  294  seq.   Bologna,  1572. 

a"Per  radicem  m."  7(Euvres,  1634  ed.,  pp.  71,  72. 

8  "On  pourroit  dire  a  quoy  sert  ces  solutions  qui  sont  impossibles,  je  re- 
spond pour  trois  choises,  pour  la  certitude  de  la  reigle  generale,  et  qu'il  ny  a 
point  d'autre  solutions,  et  pour  son  utilite""  (Invention  nouvelle  en  I'algebre, 
fol.  F  i  (Amsterdam,  1629)).  The  solution  of  "i  (2)  est  esgale  a  6  (i)  —  25," 
that  is,  X1  —  6  #—25,  which  gives  *  =  3±  V—  16,  he  calls  "inexplicable" 
(p.  114).  He  places  in  the  same  category  (p.  130)  numbers  like  V—  3.  From 
this  point  on,  the  reader  may  profitably  consult  the  Encyklopadie,  I,  148. 


GRAPHIC  REPRESENTATION  263 

Approach  to  a  Graphic  Representation.  Wallis  (1673)  seems 
to  have  been  the  first  to  have  any  idea  of  the  graphic  repre- 
sentation of  these  quantities.  He  stated1  that  the  square  root 
of  a  negative  number  was  thought  to  imply  the  impossible,  but 
that  the  same  might  also  be  said  of  a  negative  number,  although 
we  can  easily  explain  the  latter  in  a  physical  application : 

These  Imaginary  Quantities  (as  they  are  commonly  called),  arising 
from  the  Supposed  Root  of  a  Negative  Square  (when  they  happen,) 
are  reputed  to  imply  that  the  Case  proposed  is  Impossible. 

And  so  indeed  it  is,  as  to  the  first  and  strict  notion  of  what  is  pro- 
posed. For  it  is  not  possible  that  any  Number  (Negative  or  Affirma- 
tive) Multiplied  into  itself  can  produce  (for  instance)  —4.  Since  that 
Like  Signs  (whether  +  or  —  )  will  produce  +  ;  and  therefore  not  —  4. 

But  it  is  also  Impossible  that  any  Quantity  (though  not  a  Supposed 
Square)  can  be  Negative.  Since  that  it  is  not  possible  that  any 
Magnitude  can  be  Less  than  Nothing  or  any  Number  Fewer  than 
None. 

Yet  is  not  that  Supposition  (of  Negative  Quantities,)  either  Unuse- 
ful  or  Absurd ;  when  rightly  understood.  And  though,  as  to  the  bare 
Algebraick  Notation,  it  import  a  Quantity  less  than  nothing:  Yet, 
when  it  comes  to  a  Physical  Application,  it  denotes  as  Real  a  Quan- 
tity as  if  the  Sign  were  -f  ;  but  to  be  interpreted  in  a  contrary  sense. 

Having  shown  that  we  may  have  negative  lines,  he  asserts 
that  we  may  also  have  negative  areas  and  that  a  negative 
square  must  have  a  side,  thus : 

Now  what  is  admitted  in  Lines  must,  on  the  same  Reason,  be 
allowed  in  Plains  also. .  • 

But  now  (supposing  this  Negative  Plain,  —  1600  Perches,  to  be  in 
the  form  of  a  Square ; )  must  not  this  Supposed  Square  be  supposed 
to  have  a  Side  ?  And  if  so,  what  shall  this  Side  be  ? 

We  cannot  say  it  is  40,  nor  that  it  is  —  40  •  •  • 

But  thus  rather  that  it  is  V—  1600,  or  •  •  •  10  V—  16,  or  20  V^~4, 
or  40  V—  i. 

*  Algebra,  cap.  LXVI;  Vol.  II,  p.  286,  of  the  Latin  edition;  but  for  his  1673 
statement,  differing  somewhat  from  that  in  his  algebra,  see  Cajori  in  Amer.  Math. 
Month.,  XIX,  167.  See  also  G.  Enestrom,  Bibl  Math.,  VII  (3),  263. 


264  COMPLEX  NUMBERS 

Where  V  implies  a  Mean  Proportional  between  a  Positive  and  a 
Negative  Quantity.  For  like  as  Vfo  signifies  a  Mean  Proportional 
between  +  b  and  -f  c ;  or  between  —  b  and  —  c ;  •  •  •  So  doth  V—  be 
signify  a  Mean  Proportional  between  +  b  and  —  c,  or  between  —  b 
and  -f  c. 

He  therefore  reached  the  position  where  he  would  be  sup- 
posed to  draw  a  line  perpendicular  to  the  real  axis  and  say  that 
this  might  be  taken  as  an  imaginary  axis,  but  although  he 
touched  lightly  upon  this  possibility,  he  did  nothing  of  conse- 
quence with  the  idea. 

Leibniz  on  Complex  Numbers.  Leibniz  was  the  next  to 
take  up  the  study  oMmaginaries.  He  showed  (1676)  that 
Vi  +V—  3  +  vi— V— 3  =  V6?  and  (1702)  that  x4  -fa4  is  equal  to 

(*•  +  a  V-V-  i  )(r  -  a  V—V—  i)(x  +  a  VV-  i)(x—a  VV—  i )/ 
He  was  much  impressed  by  the  possibilities  of  the  imaginary, 
but  he  seems  never  to  have  grasped  the  idea  of  its  graphic 
representation.2 

Modern  Analytic  Treatment.  In  1702  Jean  Bernoulli  brought 
the  imaginary  to  the  aid  of  higher  analysis  by  showing  the  rela- 
tion between  the  tan"1:*:  and  the  logarithm  of  an  imaginary 
number.3  Newton's  work  with  imaginaries  (1685)  was  con- 
fined to  the  question  of  the  number  of  roots  of  an  equation,4 
a  subject  that  was  continued  by  Maclaurin6  and  other  English 
algebraists. 

*Werke,  Gcrhardt  ed.  (Berlin,  1850),  II  (3),  12;  (Halle,  1858),  V  (3),  218, 
360.  See  Tropfke,  Geschichte,  I  (i),  171. 

2"Itaque  elegans  et  mirabile  effugium  reperit  in  illo  Analyseos  miraculo, 
idealis  mundi  monstro,  pene  inter  Ens  et  non-Ens  Amphibio,  quod  radicem 
imaginariam  appellamus"  (Werke,  V,  357). 

"Ex  irrationalibus  oriuntur  quantitates  impossibiles  seu  imaginariae,  quarum 
mira  est  natura,  et  tamen  non  contcmnenda  utilitas;  etsi  enim  ipsae  per  se 
aliquid  impossible  significant,  tamen  non  tantum  ostendunt  fontem  impossibili- 
tatis,  et  quomodo  quaestio  corrigi  potuerit,  ne  esset  impossibilis,  sed  etiam  in- 
terventu  ipsarum  exprimi  possunt  quantitates  reales"  (ibid.,  VII,  69). 

3 Opera,  I,  393  (Lausanne,  1472);  Tropfke,  Gesckichte,  II  (2),  83;  Cantor, 
Geschichte,  III,  348. 

4  Arithmetics  Universalis,  p.  242  (Cambridge,  1707),  the  imaginaries  being 
called  "radices  impossibiles." 

5 Phil.  Trans.,  XXIV  (1726),  104;  XXXVI,  59;  Algebra,  1748  ed.,  p.  275. 


GRAPHIC  REPRESENTATION  265 

The  first  important  step  in  the  new  theory  to  be  taken  by  a 
British  mathematician  was  made  by  Cotes  (c.  1710)  when  he 
stated1  that  log  (cos<£  +  i  sin  <£)  =  *<£,  corollaries  to  which  are 
the  important  formula 

€<!>*=  cos  (f>  +  i  sin  <£, 
which  bears  Euler's  name,  and  the  well-known  relation 

(cos  </>  +  i  sin  <j>)n  =  cos  n<f)  +  i  sin  n<j>, 

suggested  by  De  Moivre  in  1730  but  possibly  known  by  him 
as  early  as  1707^  Euler  (1743,  1748)  was  the  first  to  prove 
that  this  relation  holds  for  all  values  of  n,  and  also  that 


and  cos  d>  =  - 


Graphic  Representation.  Although  some  approach  to  the 
graphic  representation  of  the  complex  number  had  been  made 
by  Wallis,  and  although  the  goal  had  been  more  nearly  at- 
tained by  H.  Kiihn,  of  Danzig,  it  was  a  Norwegian  surveyor, 
Caspar  Wessel,3  who  first  gave  the  modern  geometric  theory. 
In  1797  he  read  a  paper  on  the  subject  before  the  Royal  Acad- 
emy of  Denmark.  This  was  printed  in  1798  and  appeared  in 
the  memoirs  of  the  Academy  in  1799.*  In  this  he  says: 

Let  us  designate  by  + 1  the  positive  rectilinear  unit,  by  +  c  an- 
other unit  perpendicular  to  the  first  and  having  the  same  origin ;  then 
the  angle  of  direction  of  +  i  will  be  equal  to  o°,  that  of  —  i  to  180°, 
that  of  £  to  90°,  and  that  of  —  €  to  —90°  or  to  270°. 

^Harmonia  mensurarum  (posthumous),  p.  28  (Cambridge,  1722):  "Si  quacl- 
rantis  circuit  quilibet  arcus,  radio  CE  descriptus,  sinum  habeat  CX,  sinumque 
complement!  ad  quadrantem  XE:  sumendo  radium  CE  pro  Modulo,  arcus  erit 
rationis  inter  EX  +  XC  V— i  &  CE  mensura  ducta  in  V—i."  See  also  Bibl. 
Math.,  II  (3),  442. 

2 Bibl.  Math.,  II  (3),  97-102. 

3 Born  at  Jonsrud,  June  8,  1745 ;  died  1818. 

4  See  the  French  translation,  Essai  sur  la  representation  analytique  de  la  direc- 
tion, Copenhagen,  1897. 


266  COMPLEX  NUMBERS 

The  plan  was  therefore  the  same  as  the  one  now  used  and 
until  recently  attributed  to  various  other  writers,  including  Henri 
Dominique  Truel1  (1786),  A.  Q.  Buee2  (1805),  J.  R.  Arganda 
(1806),  Gauss/  J.  F.  Frangais5  (1813),  and  John  Warren." 
W.  J.  G.  Karsten  (1732-1787),  at  one  time  professor  at  Halle, 
gave  a  method  (1768)  of  representing  imaginary  logarithms.7 

Gauss  took  as  his  four  units  the  roots  of  the  equation 
x4  —  i  =  o.  Eisenstein8  developed  a  theory  based  on  the  roots 
of  the  equation  #3  —  i  =  o,  and  Kummer9  based  his  on  the 
roots  of  the  equation  xn—  i  =  o.10 

Terms  and  Symbols.  As  already  mentioned,  Cardan  (1545) 
spoke  of  a  solution  like  5+>/— 15  as  "per  radicem  m," 
or  "  sophistic  quantities,"  and  Bombelli  called  numbers  like 
-I-V—  n  and  —V—  n  piu  di  meno  and  meno  di  meno"  abbre- 
viated to  p.  di  m.  and  m.  di  m.  Descartes  (1637)  contributed 
the  terms  "real"  and  "imaginary."12 

Most  of  the  1 7th  and  i8th  century  writers  spoke  of 
a  -f  b  V—  i  as  an  imaginary  quantity.13  Gauss  (1832)  saw  the 
desirability  of  having  different  names  for  a  V— ~T  and  a  +  b  V—  i , 

1Cauchy  mentions  this  fact,  but  nothing  is  known  of  the  man.  His  MSS.  are  lost. 

2  In  a  paper  Sur  les  Quantites  Imaginaires,  read  before  the  Royal  Society, 
London,  1805.    See  the  Philosophical  Transactions,  London,  1806. 

3  Essai  sur  une  maniere  de  representer  les  quantites  imaginaires  dans  les  con- 
structions gcometriques,  Paris,  1806;  2d  ed.,  Paris,  1874. 

4He  refers  to  the  subject  in  his  Demonstratio  nova  (1798),  but  does  nothing 
with  it.  In  his  Theoria  residuorum  biquadraticorum,  Commentatio  secunda  ( 1831 ) 
he  presents  the  theory  in  its  present  form,  evidently  ignorant  of  WessePs  work. 

5 See  Gergonne's  Annales,  IV,  61. 

6  A  Treatise  on  the  Geometrical  Representation  of  the  Square  Roots  of  Nega- 
tive Quantities,  Cambridge,  1828.  The  general  plan  is  that  of  Wessel,  but  the 
treatment  of  the  subject  is  very  abstract. 

7F.  Cajori,  Amer.  Math.  Month.,  XX,  76. 

8  On  Gauss's  estimate  of  him  see  Volume  I,  page  509. 

9 See  Volume  I,  pages  507,  508. 
10Tropfke,  Geschichte,  II  (2),  88. 

11  "Plus  of  minus"  and  "minus  of  minus."   L' Algebra,  p.  294  seq.  (Bologna, 
IS72). 

12  "Au   reste  tant  les  vrayes  racines  que  les   fausses  ne  sont  pas  toujours 
reelles,  mais  quelquefois  seulement  imaginaires."   La  Geometric,  1705  ed.,  p.  117. 

13 Thus  d'Alembert  (1746):  "Une  fonction  quelconque  de  tant  et  de  telles 
grandeurs  imaginaires,  qu'on  voudra,  peut  toujours  etre  supposee  egale  a 
p  +  ^V^"  (Hist,  de  VAcad.  d.  Berlin,  II,  19$). 


QUATERNIONS  AND  AUSDEHNUNGSLEHRE        267 

and  so  he  gave  to  the  latter  the  name  "  complex  number." x  The 
use  of  i  for  V—  i  is  due  to  Euler  ( 1 748)  ,2  Cauchy 3  ( 182 1 )  sug- 
gested the  name  "conjugates"  (conjuguees)  for  a-f  bi  and 
a  —  bi  and  the  name  "modulus"  for  vV+  &*.  Weierstrass called 
the  latter  the  "absolute  value"  of  the  complex  number  and  rep- 
resented it  by  \a+bi\.  Gauss  had  already  given  to  <r-f-62  the 
name  of  "norm."4 

Quaternions  and  Ausdehnungslehre.  The  development  of  com- 
plex numbers,  with  their  graphic  representation  in  a  plane,  nat- 
urally led  to  the  consideration  of  numbers  of  this  type  that 
might  be  graphically  represented  in  a  space  of  three  dimen- 
sions. Argand  (1806)  attempted  to  take  this  step  but  found 
himself  unable  to  do  so,  and  Servois  (1813)  also  made  the 
attempt  and  failed. 

In  1843  Sir  William  Rowan  Hamilton5  discovered  the  prin- 
ciple of  quaternions,  and  presented  his  first  paper  on  the 
subject  before  the  Royal  Irish  Academy.  His  first  complete 
treatment  was  set  forth  in  his  Lectures  on  Quaternions  (1853). 
His  discovery  necessitated  the  withdrawal  of  the  commutative 
law  of  multiplication,  the  adherence  to  which  had  proved  to 
be  a  bar  to  earlier  progress  in  this  field. 

The  most  active  of  the  British  scholars  who  first  recognized 
the  power  of  quaternions  was  Peter  Guthrie  Tait.  Becoming 
acquainted  with  Hamilton  soon  after  the  latter's  Lectures  on 
Quaternions  appeared,  Tait  began  with  him  a  correspondence 
that  was  carried  on  until  his  death.0  Tait  had  been  a  classmate 

*"  Tales  numeros  vocabimus  numeros  integros  complexes "  (Werke,  II,  102 
(Gottingen,  1876)). 

2".  .  .  formulam  V  ~~  l  Httera  i  in  posterum  designabo,  ita  vt  sit  n  =  —  i" 
(Institutionum  calculi  integrate  volumen  IV,  184  (Petrograd,  1794)).  In  his  /«- 
troductio  in  Analysin  Infinitorum  (Lausanne,  1748),  he  first  used  the  symbol: 
"  Cum  enim  numerorum  negativorum  Logarithm!  sint  imaginarii  .  .  .  erit  1.  —  n 
quantitas  imaginaria,  quae  sit  =  i."  See  also  W.  W.  Bcman  in  Bulletin  of  the 
Amer.  Math.  Soc.,  IV,  274,  551. 

3  C  ours  d' Analyse  algebrique,  p.  180  (Paris,  1821). 

4Tropfke,  Geschichte,  II  (2),  90.  For  other  attempts  at  explaining  the  imagi- 
nary number  see  Cantor,  Geschichte,  IV,  88-91,  303-318,  573,  712-715,  729-731. 

5  A.  Macfarlane,  Ten  Brit.  Math.,  p.  43  (New  York,  1916).  See  also 
P.  G.  Tait's  article  on  "Quaternions"  in  the  Encyc.  Britan.,  9th  ed.,  XX,  160. 

8  A.  Macfarlane,  "Peter  Guthrie  Tait,"  Physical  Review,  XV,  51. 


268  TRANSCENDENTAL  NUMBERS 

of  Clerk  Maxwell's  at  the  Edinburgh  Academy  and,  like  him, 
was  deeply  interested  in  physical  studies.  Partly  as  a  result 
of  this  early  training  he  soon  began  to  apply  the  theory  of 
quaternions  to  problems  in  this  field,  his  results  appearing  in  the 
Messenger  of  Mathematics  and  the  Quarterly  Journal  of  Mathe- 
matics. Recognizing  Hamilton's  wish,  if  not  at  his  request,  he 
delayed  the  publication  of  his  work  on  the  theory  until  after  the 
former's  Elements  appeared.  His  own  Elementary  Treatise  on 
Quaternions  was  therefore  not  issued  until  1866,  after  which 
he  continued  to  write  upon  the  subject  until  his  death.  In  1873 
he  published  with  Professor  Kelland  a  work  entitled  An  Intro- 
duction to  Quaternions,  which  did  much  to  make  the  subject 
known  to  physicists.  The  theory  has  not,  however,  been  as 
favorably  received  by  scientists  as  had  been  anticipated  by  its 
advocates.  It  should  be  added  that  Gauss,  about  the  year  1820, 
gave  some  attention  to  the  subject  but  without  developing  any 
theory  of  importance.1 

At  about  the  same  time  that  Hamilton  published  his  dis- 
covery of  quaternions  Hermann  Giinther  Grassmann  published 
his  great  work,  Die  lineale  Ausdehnungslehre  (1844),  although 
he  seems  to  have  developed  the  theory  as  early  as  i84o.2 

7.  TRANSCENDENTAL  NUMBERS 

Transcendental  Numbers  Considered  Elsewhere.  Among  the 
artificial  numbers  should,  of  course,  be  included  not  only  such 
types  as  surds  but  also  all  such  nonalgebraic  types  as  are  found 
in  connection  with  the  trigonometric  functions,  logarithms,  the 
study  of  the  circle,  and  the  theory  of  transcendental  numbers 
in  general.  These  may,  however,  be  more  conveniently  consid- 
ered in  connection  with  algebra,  geometry,  and  trigonometry, 
as  they  are  commonly  found  in  the  teaching  of  these  subjects. 

iTropfke,  Geschichte,  II  (2),  88. 

2V.  Schlegel,  "Die  Grassrnann'sche  Ausdehnungslehre,"  Schlomilch's  Zeit- 
schrift,  XLI.  For  A.  Macfarlane's  digest  of  the  views  of  various  writers  see 
Proceedings  of  the  American  Assoc.  for  the  Adv.  of  Sci.,  1891.  See  also 
E.  Jahnke,  in  L' Enseignement  Mathematique,  XI  (1909),  417;  F.  Engel,  in 
Grassmann's  Gesammelte  math,  und  physikal.  Werke,  III  (Leipzig,  1911). 


DISCUSSION  269 

TOPICS  FOR  DISCUSSION 

1.  The  sequence  of  development  of  artificial  numbers,  with  the 
causes  leading  to  the  successive  steps. 

2 .  General  nature  of  compound  numbers  at  various  periods  in  their 
development. 

3.  Nature  of  fractions  among  the  Egyptians,  and  the  reasons  for 
the  persistence  of  the  unit  fraction. 

4.  Greek  symbolism  for  fractions  compared  with  that  which  the 
Romans  used. 

5.  Origin  and  development  of  our  common  fractions. 

6.  The  etymology  of  terms  used  in  fractions,  with  the  change  in 
these  terms  from  time  to  time. 

7.  The  sequence  of  operations  with  fractions  in  the  early  printed 
works  on  arithmetic. 

8.  The  development  of  methods  of  performing  the  operations  with 
common  fractions. 

9.  The  origin,  development,  symbolism,  and  present  status  of 
sexagesimal  fractions. 

10.  The  origin  and  development  of  decimal  fractions,  including  the 
question  of  symbolism. 

11.  The  human  needs  that  led  to  the  development  of  the  various 
types  of  fraction. 

12.  The  origin  and  development  of  the  idea  of  per  cent,  including 
the  question  of  symbolism. 

13.  The  reason  for  the  interest  of  the  Greek  mathematicians  in  in- 
commensurable numbers. 

14.  The  development  of  surd  numbers,  particularly  among  the 
Greek  and  Arab  writers. 

15.  History  of  the  various  methods  of  approximating  the  value  of 
a  surd  number. 

1 6.  The  origin  and  development  of  negative  numbers,  including  the 
question  of  symbolism. 

17.  The  origin  and  development  of  the  idea  of  complex  numbers, 
including  the  question  of  symbolism. 

1 8.  The  origin  and  development  of  the  graphic  representation  of 
complex  numbers. 

19.  The  origin  and  development  of  the  idea  of  complex  exponents 
in  algebra. 


CHAPTER  V 

GEOMETRY 
i.  GENERAL  PROGRESS  OF  ELEMENTARY  GEOMETRY 

Intuitive  Geometry.  All  early  geometry  was  intuitive  in  its 
nature ;  that  is,  it  sought  facts  relating  to  mensuration  without 
attempting  to  demonstrate  these  facts  by  any  process  of  deduc- 
tive reasoning.  The  prehistoric  geometry  sought  merely  agree- 
able forms,  as  in  the  plaiting  of  symmetric  figures  in  a  mat.  The 
next  stage  was  that  of  the  mensuration  of 
rectangles  and  triangles,  and  geometry  was 
in  this  stage  when  the  Ahmes  Papyrus  (c. 
1550  B.C.)1  was  written^In  this  work  the  area 
of  an  isosceles  triangle  of  base  b  and  sides  s 
is  given  as  |  bs.  For  the  area  of  a  circle  of 
diameter  d  Ahmes  used  a  rule  which  may  be 
expressed  in  modern  symbols  as  (d  —  \d)2,  which  shows  that  he 
took  3.1605-  as  the  value  of  TT, — a  value  based  on  experiment.2 
In  Babylonia  the  same  conditions  existed.  The  tablets  which 
have  come  down  to  us  contain  a  few  cases  of  mensuration,3  but 
the  rules  are  based  merely  on  experiment. 

1  It  should  be  recalled  that  Professor  Peet  (Rhind  Papyrus,  p.  3)  puts  this  date 
as  probably  before  1580  B.C. 

2  On   the   general   history   of   the   development   of   geometry   see   G.   Loria, 
II  Passat o  ed  il  Presente  delle  Principali  Teorie  Geometriche,  Turin,  3d  ed.,  1907; 
hereafter  referred  to  as  Loria,  Passato -Presente  Geom.    This  work  first  appeared 
in  the  Turin  Memorie  delta  R.  Accad.,  XXXVIII  (2),  and  was  translated  into 
German   by  F.   Schutte,   Leipzig,   1888.    See   also   the   Encyklopadie,  Vol.  Ill; 
R.   Klimpert,   Geschichte   der  Geometric,  Stuttgart,   1888;   E.   F.   August,  Zur 
Kenntniss  der  geometrischen  Methode  der  Alien,  Berlin,  1843;  and  the  various 
general  histories  of  mathematics.    On  the  early  Egyptian  geometry  see  E.  Weyr, 
Veber    die    Geometric    der    alten    Aegypter,    Vienna,    1884;    Eisenlohr,    Ahmes 
Papyrus,   On  a  papyrus  which  may  be  slightly  earlier  than  that  of  Ahmes,  see 
page  293.  8Hilprecht,  Tablets. 

270 


INTUITIVE  GEOMETRY  271 

The  native  mathematics  of  China  was  also  of  this  type.  The 
Nine  Sections,  written  perhaps  c.  noo  B.C.,  contains  statements 
which  show  that  the  author  knew  the  relations  of  the  sides  of 
certain  right-angled  triangles,1  but  there  is  no  evidence  of  any 
proof  of  such  relations. 

In  the  later  Chinese  mathematics  there  are  many  ingenious 
examples  involving  mensuration,  but  nowhere  does  there  ap- 
pear any  further  idea  of  geometric  demonstration,  as  we  under- 
stand the  term,  than  is  found  in  the  earliest  works. 

In  India  the  same  conditions  existed,  the  native  geometry 
giving  us  no  evidence  of  any  approach  to  a  sequence  of  deduc- 
tive proofs.  There  was  a  large  amount  of  mensuration,2  and 
considerable  ability  was  shown  in  the  formulation  of  rules,  but 
the  basis  of  the  work  was  wholly  empirical. 

The  Romans  were  interested  in  mathematics  only  for  its  im- 
mediately practical  value.  The  measurement  of  land,  the  lay- 
ing out  of  cities,  and  the  engineering  of  warfare  appealed  to 
them;  but  for  demonstrative  geometry  they  had  no  use.  Indeed, 
it  may  be  said  that,  outside  of  those  lands  which  were  affected 
by  the  Greek  influence,  the  ancient  world  knew  geometry  only 
on  its  intuitive  side.  Demonstrative  geometry  was  Greek  in  its 
origin,  and  in  the  Greek  civilization  it  received  its  only  encour- 
agement for  more  than  a  thousand  years.  „ 

-  Demonstrative  Geometry.  The  idea  of  demonstrating  the 
truth  of  a  proposition  which  had  been  discovered  intuitively 
appears  first  in  the  teachings  of  Thales  (c.  600  B.C.).  It  is 
probable  that  this  pioneer  knew  and  proved  about  six  theo- 
rems,3 each  of  which  would  have  been  perfectly  obvious  to 
anyone  without  any  demonstration  whatever.  The  contribution 
of  Thales  did  not  lie  in  the  discovery  of  the  theorems,  but  in 
their  proofs.  These  proofs  are  lost,  but  without  them  his  work 
in  geometry  would  have  attracted  no  attention,  either  among 
his  contemporaries  or  in  the  history  of  thought. 

iSee  Volume  I,  pages  30/33. 

2 E.g.,  consider  Bhaskara's  Lilavati,  with  special  sections  on  ponds,  walls, 
timber,  heaps,  shadows,  and  excavations. 

8  See  Volume  I,  page  67 ;  Heath,  History,  I,  130. 


272          PROGRESS  OF  ELEMENTARY  GEOMETRY 

From  the  time  of  Thales  until  the  decay  of  their  ancient 
civilization  demonstrative  geometry  was  the  central  feature  of 
the  mathematics  of  the  Greeks.  The  history  of  the  general 
progress  of  the  science  has  been  sufficiently  outlined  elsewhere 
in  this  work.1 

The  Arabs  recognized  the  Greek  culture  more  completely 
than  any  other  people  until  the  period  of  the  awakening  of 
Western  Europe.  They  translated  the  Greek  classics  in  geome- 
try as  they  did  also  in  philosophy  and  natural  science,  but  they 
never  made  any  additions  of  real  significance  to  the  works  of 
Euclid  and  Apollonius. 

It  was  chiefly  through  the  paraphrase  of  Boethius  (c.  510) 
that  Euclid's  Elements  (c.  300  B.C.)  was  known  in  the  Dark 
Ages  of  Europe.  The  study  of  geometry  received  some  encour- 
agement at  the  hands  of  Gerbert  (c.  1000),  Fibonacci  (1220), 
and  a  few  other  medieval  scholars,  but  no  progress  was  made 
in  the  advance  of  the  great  discipline  which  had  been  so  nearly 
perfected  in  Alexandria  more  than  a  thousand  years  earlier. 

With  the  invention  of  European  printing  the  work  of  Euclid 
became  widely  known,  the  first  printed  edition  appearing  in 
1482. 2  Little  by  little  new  propositions  began  to  be  suggested, 
but  the  invention  of  analytic  geometry  early  in  the  iyth  cen- 
tury took  away,  for  a  considerable  period,  much  of  the  interest 
in  improving  upon  the  ancient  theory. 

The  next  advance  in  the  pure  field  was  made  in  the  iyth 
century,  when  Desargues3  (1639)  published  a  work  which 
treated  of  certain  phases  of  projective  geometry.  The  new 
analytic  treatment  of  the  subject,  however,  was  so  novel  and 
powerful  as  to  take  the  attention  of  mathematicians  from  the 
work  of  Desargues,  and  it  was  not  until  the  igth  century  that 
pure  geometry  again  began  to  make  great  progress.  <,. 

The  Greek  theory  of  conies  has  already  been  considered  suf- 
ficiently for  our  purposes  (Vol.  I,  Chap.  IV).  The  analytic  and 
modern  synthetic  geometries  are  considered  later. 

1See  Volume  I,  pages  59-146;  Heath,  History,  I,  chap.  iv. 

2  Venice,  Erhard  Ratdolt;  the  Campanus  translation.    See  Volume  I,  page  251. 

8 See  Volume  I,  page  383. 


VARIOUS  NAMES  FOR  GEOMETRY  273 

2.  NAME  FOR  GEOMETRY 

Reason  for  Uniformity.  When  we  consider  that  our  elemen- 
tary geometry  is  essentially  the  Elements  of  Euclid,  and  that 
the  subject  never  flourished  in  ancient  times  outside  the  Greek 
sphere  of  influence,  it  is  apparent  that  the  Greek  name  would 
be  the  one  generally  used  to  designate  the  science.  It  is  derived 
from  the  words  for  "earth"  and  "measure"1  and  therefore  was 
originally,  as  it  is  in  some  languages  today,  synonymous  with 
the  English  word  "surveying."  Since  the  latter  science  was 
well  developed  in  Egypt  before  the  Greeks  founded  Alexandria, 
the  name  is  probably  a  translation  of  an  Egyptian  term.  It  was 
in  use  in  the  time  of  Plato  and  Aristotle,  and  doubtless  goes 
back  at  least  to  Thales. 

Euclid  did  not  call  his  treatise  a  geometry,  probably  because 
the  term  still  related  to  land  measure,  but  spoke  of  it  merely 
as  the  Elements.2  Indeed,  he  did  not  employ  the  word  "geome- 
try" at  all,  although  it  was  in  common  use  among  Greek  writers.3 

When  Eudid  was  translated  into  Latin  in  the  i2th  century, 
the  Greek  title  was  changed  to  the  Latin  form  Elemental  but 
the  word  "geometry"  is  often  found  in  the  title-page,  first  page, 
or  last  page  of  the  early  printed  editions.5 

There  have  been,  as  would  naturally  be  expected,  various 
fanciful  names  for  textbooks  on  geometry.  In  the  i6th  cen- 
tury such  names  were  common  in  all  branches  of  learning. 
Among  the  best-known  of  these  titles  is  the  one  seen  in 
Robert  Recorders  The  pathway  to  Knowledg  (London,  1551 
and  1574). 

1F^  (ge),  earth,  and  /xerpetV  (metrein'),  to  measure. 

2  In  Greek,  crroixeta  (stoichei'a} .  So  in  the  editio  princeps  of  the  Greek  text 
(Basel,  1533)  the  title  appears  as  ETKAEIAOT  STOIXEION  BIBA>-  IE>- . 

3 Thus  Plato  (The.atetus,  173  E;  Meno,  76  A;  Republic,  546  C,  511  D), 
Xenophon  (Symposium,  6,  8,  etc.),  and  Herodotus  (II,  109)  use  the  word  in 
some  of  its  forms,  but  always  to  indicate  surveying. 

4 So  in  the  editio  princeps  (1482)  the  first  page  begins:  " Preclarissimus  liber 
elementorum  Euclidis  perspicacissimi:  in  artem  geometric  incipit  qua  foelicis- 
sime."  The  colophon  also  has  the  name  geometria. 

5 E.g.,  the  first  English  edition  (London,  1570)  has  the  title  The  Elements 
of  Geometric  of  the  most  auncient  Philosopher  Evclide  of  Megara. 


274  TECHNICAL  TERMS 

3.  TECHNICAL  TERMS  OF  EUCLIDEAN  GEOMETRY 

Point.  The  history  of  a  few  typical  terms  of  elementary  geom- 
etry will  now  be  considered.1  The  Pythagoreans  defined  a  point 
as  "a  monad  having  position/'2  and  this  definition  was  adopted 
by  Aristotle  (c.  340  B.C.).  Plato  (c.  380  B.C.)  called  a  point 
"the  beginning  of  a  line/73  and  Simplicius  (6th  century)  called 
it  "the  beginning  of  magnitudes  and  that  from  which  they 
grow,"  adding  that  it  is  "the  only  thing  which,  having  position, 
is  not  divisible."  Euclid  (c.  300  B.C.)  gave  the  definition: 
"A  point  is  that  which  has  no  part."  Heron  (c.  50?)  used  the 
same  words,  but  added  "or  a  limit  without  dimension  or  a  limit 
of  a  line."  When  Capella  (c.  460)  translated  the  definition 
into  Latin,  he  made  it  read,  "A  point  is  that  of  which  a  part  is 
nothing,"4  which  is  a  different  matter. 

Modern  writers  usually  resort  to  analogy  and  give  only  a 
quasi  definition,  or  else  they  make  use  of  the  idea  of  limit. 

{  Line.  The  Platonists  defined  a  line  as  length  without  breadth, 
and  Euclid5  did  the  same.  Aristotle  objected  to  such  a  negative 
definition,  although  Proclus  (c.  460)  observes  that  it  is  posi- 
tive to  the  extent  that  it  affirms  that  a  line  has  length.  An  un- 
known Greek  writer6  defined  it  as  "magnitude  extended  one 
way,"7  a  phrase  not  unlike  one  used  by  Aristotle.  The  latter 
defined  it  as  a  magnitude  "divisible  in  one  way  only,"8  in  con- 
trast to  a  surface,  which  is  divisible  in  two  ways,  and  to  a  solid, 
which  is  divisible  in  three  ways.  Proclus  suggested  defining 
a  line  as  the  "flux  of  a  point,"9  an  idea  also  going  back  to 

1For  further  discussion  see  J.  H.  T.  Miiller,  Beitragc  zur  Terminologie  der 
griechischen  Mathematiker,  1860;  Heath,  Euclid,  Vol.  I,  p.  155;  H.  G.  Zeuthen, 
"Sur  les  definitions  d'Euclide,"  Scientia,  XXIV,  257,  on  the  general  nature  of 
Euclid's  definitions.  See  also  Heath,  History,  on  all  such  details. 

2Moi>As  trpo<T\apov(ra  0fou>  (monad  with  position  added). 

3' Apxv  7paju/x7}j.  On  this  and  other  definitions  consult  Heath,  Euclid,  Vol.  I, 
p.  155;  H.  Schotten,  Inhalt  und  Methode  des  planimetrischen  Unterrichts,  Vol.  I 
(Leipzig,  1890) ;  Vol.  II  (Leipzig,  1893) ;  hereafter  referred  to  as  Schotten,  Inhalt. 

4"Punctum  est  cuius  pars  nihil  est." 

6  Ypa/jLfj,))  5£  /XTJKOS  drrXar^s. 

6  Alluded  to  by  al-Nairizi  (c.  910)  as  one  Heromides  or  Herundes. 


POINT  AND  LINE  275 

Aristotle,  who  remarked  that  "a  line  by  its  motion  produces 
a  surface,  and  a  point  by  its  motion  a  line."  This  occasion- 
ally appears  as  "A  line  is  the  path  of  a  moving  point." 

Straight  Line.  It  is  evident  that  certain  terms  are  so  elemen- 
tary that  no  simpler  terms  exist  by  which  to  define  them.  This 
is  true  of  "point"  and  "line,"  but  it  is  more  evidently  true  of 
terms  like  "straight  line"  and  "angle."  Plato  defined  a  straight 
line  as  "that  of  which  the  middle  covers  the  ends,"  that  is, 
relatively  to  an  eye  placed  at  either  end  and  looking  along  the 
line.  Euclid  endeavored  to  give  up  the  appeal  to  sight  and 
defined  it  as  "a  line  which  lies  evenly  with  the  points  on  itself." 
Proclus  explains  that  Euclid  "shows  by  means  of  this  that  the 
straight  line  alone  [of  all  lines]  occupies  a  distance  equal  to 
that  between  the  points  on  it,"  adding  that  the  distance  be- 
tween two  points  on  a  circumference  or  any  other  line,  and 
measured  on  this  line,  is  greater  than  the  interval  between 
them.  Archimedes  (c.  225  B.C.)  stated  this  idea  more  tersely 
by  saying  that  "of  all  lines  having  the  same  extremities  the 
straight  line  is  the  shortest,"  which  is  the  source  of  the  defini- 
tion often  found  in  textbooks,  "a  straight  line  is  the  shortest 
distance  between  two  points,"  although  "line"  and  "distance" 
are  two  radically  different  concepts.  "The  shortest  path  be- 
tween two  points"  is  an  expression  that  is  less  objectionable, 
but  it  merely  shifts  the  difficulty. 

Heron  (c.  50?)  defined  a  straight  line  as  "a  line  stretched  to 
the  utmost  toward  the  ends,"  and  Proclus  adopted  this  phrase 
with  the  exception  of  "toward  the  ends."  It  is  evidently  ob- 
jectionable, however,  because  it  appeals  to  the  eye  and  relates 
to  a  physical  object.  Heron  also  suggested  the  idea  that  "all 
its  parts  fit  on  all  [other  parts]  in  all  ways,"  a  definition  sub- 
stantially adopted  by  Proclus.  Still  another  definition  due  to 
Heron  is  "that  line  which,  when  its  ends  remain  fixed,  itself 
remains  fixed  when  it  is,  as  it  were,  turned  round  in  the  same 
plane."  This  too  was  used  with  slight  change  by  Proclus,  and 
it  appears  in  various  modern  works  as  "that  which  does  not 
change  its  position  when  it  is  turned  about  its  extremities  (or 
any  two  points  in  it)  as  poles." 


276  TECHNICAL  TERMS 

Surface.  The  Pythagoreans  used  a  word1  meaning  "skin"  or 
"color"  to  designate  a  surface.  Aristotle,  like  Plato,  used  other 
words,2  and  spoke  of  a  surface  as  extended  or  continuous  or 
divisible  in  two  ways,  and  as  the  extremity  or  the  section  of  a 
solid.  Aristotle  recognized  as  common  the  idea  that  a  line  by 
its  motion  produces  a  surface,3  Euclid  defined  a  surface  as 
"that  which  has  only  length  and  breadth." 

Plane  Surface.  The  same  difficulties  that  the  ancients  had  in 
defining  a  straight  line  were  met  when  they  attempted  to  define 
a  plane.  Euclid  stated  that  "a  plane  surface  is  a  surface  which 
lies  evenly  with  the  straight  lines  on  itself."  Heron  (c.  50?) 
added  that  it  is  "the  surface  which  is  stretched  to  its  utmost," 
this  being  analogous  to  his  definition  of  a  straight  line.  He  also 
defined  it  as  "a  surface  all  the  parts  of  which  have  the  property 
of  fitting  on"  [each  other],  and  as  "such  that  if  a  straight  line 
passes  through  two  points  on  it,  the  line  coincides  wholly  with 
it  at  every  spot,  all  ways."  Proclus  (c.  460),  adopting  an  as- 
sumption stated  by  Archimedes,  defined  it  as  "the  least  surface 
among  all  those  which  have  the  same  extremities,"  and  also 
used  a  modification  of  Euclid's  definition,  "a  surface  such  that 
a  straight  line  fits  on  all  parts  of  it,"  or  "such  that  the  straight 
line  fits  on  it  all  ways."  There  was  no  material  improvement 
on  these  statements  until  the  i8th  century,  when  Robert  Sim- 
son  (1758)  suggested  the  definition  that  "a  plane  superficies 
is  that  in  which  any  two  points  being  taken,  the  straight  line 
between  them  lies  wholly  in  that  superficies,"4  a  statement 
which  Gauss  (c.  1800)  characterized  as  redundant.  Fourier 
(c.  1810)  gave  the  definition  that  a  plane  is  formed  by  the  ag- 
gregate of  all  the  straight  lines  which,  passing  through  one 
point  on  a  straight  line  in  space,  are  perpendicular  to  that 

lXpoid  (chroia'). 

2 'E7ri0d^eta  (epipha'neia)  and  t-rrlTredov  (epi'pedon).  From  the  former,  a  word 
meaning  "appearance,"  we  have  our  word  "epiphany."  The  latter  word,  mean- 
ing a  plane  surface,  occurs  in  our  word  "parallelepiped."  Later  Greek  writers 
ilso  used  e7ri<j>dv€ia  to  indicate  any  kind  of  surface,  and  Plato  used  tirlwedov  in  the 
same  way. 

3 On  the  different  kinds  of  lines  and  surfaces,  consult  Heath,  Euclid. 

4  Compare  one  of  Heron's  definitions  above. 


SURFACE,  PLANE,  AND  ANGLE  277 

straight  line.  This  is,  of  course,  merely  putting  into  another 
form  a  well-known  theorem  of  Euclid.1  Crelle  (1834)  sug- 
gested that  a  plane  is  the  surface  containing  throughout  their 
entire  lengths  all  the  straight  lines  passing  through  a  fixed 
point  and  also  intersecting  a  straight  line  in  space.2 

Angle.  Euclid's  definitions  of  an  angle  are  as  follows: 

A  plane  angle  is  the  inclination  to  one  another  of  two  lines  in  a 
plane  which  meet  one  another  and  do  not  lie  in  a  straight  line. 

And  when  the  lines  containing  the  angle  are  straight,  the  angle  is 
called  rectilineal. 

This  excludes  the  zero  angle,  straight  angle,  and  in  general 
the  angle  WTT,  and  defines  angle  by  the  substitution  of  the  idea 
of  inclination, — in  modern  form,  the  difference  in  direction. 

Even  less  satisfactory  is  the  definition  of  Apollonius  (c.  225 
B.C.)  which  asserts  that  an  angle  is  "a  contracting  of  a  surface 
or  a  solid  at  one  point  under  a  broken  line  or  surface."  Plutarch 
(ist  century)  and  various  other  writers  defined  it  as  "the  first 
distance  under  the  point,"  which  Heath3  interprets  as  "an  at- 
tempt (though  partial  and  imperfect)  to  get  at  the  rate  of 
divergence  between  the  lines  at  their  point  of  meeting."  Per- 
haps this  idea  was  also  in  the  mind  of  Carpus  of  Antioch  (ist 
century)  when  he  said  that  the  angle  is  "a  quantity,  namely  a 
distance  between  the  lines  or  surfaces  containing  it." 

Later  writers  often  return  to  the  qualitative  idea  of  Aristotle, 
as  in  the  definition  that  an  angle  is  a  figure  formed  by  two 
lines  which  meet.  This  was  refined  by  Professor  Hilbert  of 
Gottingen4  as  follows: 

Let  a  be  any  arbitrary  plane  and  /?,  k  any  two  distinct  half-rays 
lying  in  a  and  emanating  from  the  point  O  so  as  to  form  a  part  of 
two  different  straight  lines.  We  shall  call  the  system  formed  by  these 
two  half-rays  h,  k  an  angle. 

'l  Elements,  XI,  5. 

2  For  further  consideration  of  modern  definitions  see  Heath,  Euclid,  Vol.  I, 
p.  174,  and  Schotten,  Inhalt,  II.  3  Heath,  Euclid,  Vol.  I,  p.  177. 

4 Foundations  of  Geometry,  translated  by  E.  J.  Townsend,  p.  13  (Chicago, 
1902). 

ii 


2  78  TECHNICAL  TERMS 

Circle.  The  ancient  writers  defined  a  circle  substantially  as 
Euclid  did : 

A  circle  is  a  plane  figure  contained  by  one  line  such  that  all  the 
straight  lines  falling  upon  it  from  one  point  among  those  lying  within 
the  figure  are  equal  to  one  another ; 

And  the  point  is  called  the  center  of  the  circle. 

Euclid  had  already  defined  a  figure  as  "that  which  is  con- 
tained by  any  boundary  or  boundaries/'  so  that  a  circle  is,  in 
his  view,  the  portion  of  a  plane  included  in  the  bounding 
line.  This  bounding  line  Euclid  usually  calls  the  periphery 
(7r€pi<f>ep€ia),  a  word  translated  into  the  Latin  as  circumfe- 
rentia,  whence  our  "circumference."  Euclid  is  not  consistent 
however,  for  he  speaks  of  a  circle  as  not  cutting  a  circle  in 
more  than  two  points,1  the  word  "circle"  here  referring  to  the 
bounding  line. 

This  uncertain  use  of  the  term  has  been  maintained  until  recent 
times.  The  influence  of  analytic  geometry  has  led  to  defining 
the  circle  as  a  line,  but  there  is  still  no  uniformity  in  the  matter. 

Diameter  and  Radius.  Euclid  used  the  word  "diameter"2  in 
relation  to  the  line  bisecting  a  circle  and  also  to  mean  the 
diagonal3  of  a  square,  the  latter  term  being  also  found  in  the 
works  of  Heron. 

The  term  "radius"  was  not  used  by  Euclid,  the  term  "dis- 
tance" being  thought  sufficient.  Boethius  (c.  510)  seems  to 
have  been  the  first  to  use  the  equivalent  of  our  "semidiam- 
eter."4  A  similar  use  also  appears  in  India,  in  the  writings 
of  Aryabhata5  (c.  510).  Ramus0  (1569)  used  the  term,  saying : 
"Radius  est  recta  a  centro  ad  perimetrum." 

1  Elements,  III,  10,  where  the  Greek  /O/K-AOS  (ky'klos,  circle)  is  used. 

2A«£  (dm',  through)  -f  ^erpflv(metrein',  to  measure). 

3Ata7^^o5,  from  5td  (did1,  through)  -f  yuvla  (goni'a,  angle). 

4 Thus  in  the  Ars  geometriae,  ed.  Friedlein,  p.  424  :  "  Conscribitur  age  emicyclus 
XXVIII  in  basi  et  in  semidiametro  XIIII  pedes  habens  "  (MS. of  the  nth  century). 

5L.  Rodet,  "Lemons  de  Calcul  d'Aryabhata,"  Journ.  Asiatique,  XIII  (7), 
398;  reprint  (Paris,  1879),  P-  10:  "The  chord  of  the  sixth  part  of  the  circum- 
ference is  equal  to  a  semidiameter." 

6On  the  general  question  see  Bibl.  Math.,  II  (3),  361,  and, P.  Ramus, 
Scholarvm  Mathematicarvm,  Libri  vnvs  et  triginta,  p.  155  (Basel,  1569). 


CIRCLE  AND  PARALLELS  279 

From  India  it  seems  to  have  passed  over  to  Arabia  and 
thence  to  Europe.  So  Plato  of  Tivoli  (c.  1120)  used  medlatas 
diametri  and  dimidium  diametri,  Fibonacci  (1220)  used  semi- 
dyameter?  and  Jordanus  Nemorarius  (c.  1225)  preferred  the 
form  semidiameter? 

The  early  printed  books, — such  as  those  of  Maurolico 
(1558),  Tartaglia  (1560),  and  Pedro  Nunes  (1564), — com- 
monly used  the  word  semidiameter * 

The  word  "radius"  as  used  in  this  sense  is  modern.  It  ap- 
pears, as  above  stated,  in  the  Scholarvm  Mathematicarvm,  Libri 
vnvs  et  triginta  of  Ramus  (1569),  and  a  little  later  was  used 
by  Thomas  Fincke4  (1583)  in  his  Geometria  Rotundi.  It  was 
then  adopted  by  Vieta5  (c.  1590),  and  after  that  time  it  became 
common. 

-Parallel  Lines.  The  word  "parallel"6  means  "alongside  one 
another."  Euclid  defined  parallel  straight  lines  as  "straight 
lines  which,  being  in  the  same  plane  and  being  produced  indefi- 
nitely in  both  directions,  do  not  meet  one  another  in  either 
direction."  Rather  less  satisfactory  is  the  definition  of  Posei- 
donius  (c.  100  B.C.)  as  those  lines  "which,  in  one  plane,  neither 
converge  nor  diverge,  but  have  all  the  perpendiculars  equal 
which  are  drawn  from  the  points  of  one  line  to  the  other."  This 
definition  is  substantially  that  ascribed  to  Simplicius  (6th  cen- 
tury), that  two  straight  lines  are  parallel  "if,  when  they  are 
produced  indefinitely  both  ways,  the  distance  between  them, 
or  the  perpendicular  drawn  from  either  of  them  to  the  other, 
is  always  equal  and  not  different."  The  direction  theory,  one 
of  the  least  satisfactory  of  all,  is  due  to  Leibniz.7 

lScritti,  II,  85.   See  also  dimidium  dyametri  on  page  86. 

2  See  his  "De  Triangulis"  in  the  Mitteilungen  des  Coppernicus-Vereins  .  .  . 
zu  Thorn,  VI  (1887). 

3Tropfke,  Geschichte,  IV  (2),  108. 

4 Variously  spelled.  A  Danish  mathematician  (1561-1646).  See  Volume  I, 
page  348. 

5"Posito  X  radio  seu  semidiametro  circuii."  See  Tropfke,  Geschichte,  IV 
(2),  108. 

6  Uapd\\rj\o^  (paral'lelos) . 

7  For  further  discussion,  including  the  various  bases   for  a  definition,  see 
Heath,  Euclid,  Vol.  I,  p.  192 ;  Schotten,  Inhalt,  II,  188. 


280  AXIOMS  AND  POSTULATES 

4.  AXIOMS  AND  POSTULATES 

Distinction  between  Axioms  and  Postulates.  The  Greek  writers 
recognized  the  existence  of  first  principles  "the  truth  of  which," 
as  Aristotle  affirmed,  "it  is  not  possible  to  prove. "  These,  he 
stated,  were  of  two  kinds :  ( i )  those  which  are  common  to  all 
sciences,  and  for  which  the  name  "axiom7'  was  used  by  the 
Stoic  philosophers  and  by  Aristotle  himself;  (2)  those  which 
relate  to  the  particular  science,  and  to  which  the  name  "postu- 
late" was  given  by  later  writers.1  The  distinction  was  not  com- 
pletely recognized,  even  by  Euclid,  for  his  fourth  axiom  (see 
page  281)  is  rather  a  geometric  postulate  or  a  definition. 

Aristotle  had  other  names  for  axioms,  speaking  of  them  as 
"the  common  [things]"2  or  "common  opinions."3 

Proclus  (c.  460)  states  that  Geminus  (c.  77  B.C.)  taught  that 
axioms  and  postulates  "differ  from  one  another  in  the  same 
way  as  theorems  are  also  distinguished  from  problems," — an 
opinion  which  is  quite  at  variance  with  that  of  Aristotle. 

Euclid  seems  to  have  used  the  term  "common  notion"  to 
designate  an  axiom,  although  he  may  have  used  the  term 
"axiom"  also.4 

The  word  "postulate"  is  from  the  Latin  postulare,  a  verb 
meaning  "to  demand."  The  master  demanded  of  his  pupils 
that  they  agree  to  certain  statements  upon  which  he  could 
build.  It  appears  in  the  early  Latin  translations  of  Euclid5 
and  was  commonly  used  by  the  medieval  Latin  writers. 

As  to  the  number  of  these  assumptions,  Aristotle  set  forth 
the  opinion  which  has  been  generally  followed  ever  since,  that 
"other  things  being  equal  that  proof  is  the  better  which  pro- 
ceeds from  the  fewer  postulates  or  hypotheses  or  propositions." 

1  Heath,  Euclid,  Vol.  I,  p.  117.  On  the  general  question  of  foundation  prin- 
ciples in  geometry,  see  the  Encyklopadie,  II,  i. 

2T<i  Koivd  (ta  koina'),  3Koival  86£ai  (koinai1  dox'ai) . 

4  But  not  in  his  extant  writings.  On  the  doubt  that  has  been  raised  as  to  his 
giving  a  list  of  axioms  at  all,  see  Heath,  Euclid,  Vol.  I,  p.  221. 

5"Postulata.    I.  Postuletur,  ut  a  quouis  puncto  ad  quoduis  punctum  recta 
linea  ducatur."   The  Greek  word  for  postulates  used  by  Euclid  was  al 
(aite'mata)  (Euclid,  ed.  Heiberg,  I,  8,  9) . 


TYPES  OF  ASSUMPTIONS  281 

Axioms.  Euclid  laid  down  certain  axioms,  or  "common  no- 
tions/7 probably  five  in  number,  as  follows : 

1 .  Things  which  are  equal  to  the  same  thing  are  equal  to  one  another. 

2.  If  equals  be  added  to  equals,  the  wholes  are  equal. 

3.  If  equals  be  subtracted  from  equals,  the  remainders  are  equal. 

4.  Things  which   coincide   with   one  another   are  equal   to   one 
another.1 

5.  The  whole  is  greater  than  the  part. 

The  axioms  of  inequality,  of  doubling,  and  of  halving  may 
have  been  given  by  Euclid,  but  we  are  not  certain.2 

It  will  be  observed  that  Euclid  built  up  his  geometry  on  a 
smaller  number  of  axioms  than  many  subsequent  writers  have 
thought  to  be  necessary. 

Postulates.  Euclid  does  not  use  a  noun  equivalent  to  the  Latin 
postulatunij  but  says : 

Let  the  following  be  postulated : 

1.  To  draw  a  straight  line  from  any  point  to  any  point. 

2.  To  produce  a  finite  straight  line  continuously  in  a  straight  line. 

3.  To  describe  a  circle  with  any  center  and  [any]  distance. 

4.  That  all  right  angles  are  equal  to  one  another. 

5.  That,  if  a  straight  line  falling  on  two  straight  lines  makes  the 
interior  angles  on  the  same  side  less  than  two  right  angles,  the  two 
straight  lines,  if  produced  indefinitely,  meet  on  that  side  on  which 
are  the  angles  less  than  the  two  right  angles.3 

Considerable  criticism  of  these  postulates  developed  among 
the  later  Greeks.  Zeno  of  Sidon  (ist  century  B.C.)  asserted 
that  it  was  necessary  to  postulate  that  two  [distinct]  straight 
lines  cannot  have  a  segment  in  common.  If  this  is  not  done,  he 
claimed,  one  or  more  of  the  proofs  in  Book  I  are  fallacious. 

Others  asserted  that  postulates  4  and  5  are  theorems  capable 
of  proof.  Proclus  (c.  460)  attempted  a  proof  of  postulate  4, 

1  Essentially  a  postulate  or  a  definition. 

2 For  the  evidence  pro  and  contra,  see  Heath,  Euclid,  Vol.  I,  p.  223.  Heiberg's 
edition  of  Euclid  (I,  10,  n)  numbers  these  i,  2,  3,  7,  8,  giving  in  Greek  the 
doubtful  axioms.  8Heath,  Euclid,  Vol.  I,  pp.  154,  I9S- 


282  AXIOMS  AND  POSTULATES 

but  it  was  fallacious.  He  also  claimed  that  the  converse  is  not 
necessarily  true  which  asserts  that  an  angle  which  is  equal  to  a 
c  right  angle  is  also  a  right  angle,  for  he  said 

that,  in  the  figure  given,  Z.ABC=Z.XBY 
and  yet  Z.XBY  is  not  a  right  angle.  Sac- 
cheri1  (1733)  gave  a  proof  of  the  postulate, 
but  he  assumed  other  statements  equally 
fundamental  upon  which  to  base  his  argu- 
ment. Modern  writers  often  adduce  a  simple  proof  based  upon 
the  postulate  of  the  equality  of  straight  angles,  but  this  simply 
substitutes  one  postulate  for  another. 

Postulate  5,  the  " Postulate  of  Parallels,"  has  been  frequently 
attacked.2  Ptolemy  attempted  to  prove  it,  one  of  his  arguments 
being  that  if  a  +  b  is  a  s 

straight  angle,  then  c  +  d      Q<C^ J/f>   ^^>/J 

must  be  a  straight  angle.  — -4^- ^> 

Hence  if  the  lines  meet  at  ' 

P  they  also  must  meet  at  Q,  and  in  that  case  the  two  straight 
lines  inclose  space.  Proclus  gave  a  more  seductive  argument 
relating  to  the  meeting  of  lines  in  general,  thus: 

Draw  the  lines  AK  and  CL  so  that  Z.A  +  Z.C<  two  right 

P       K  angles. 

*  Bisect  AC  at  E  and  lay  off  AF=AE 

and   CG  =  EC. 

ff  Then  AF  and  CG  cannot  meet  on  FG, 

as  at  H\  for  if  they  did  we  should  have 
AH=AE  and  Cff=  CE,  and  so  the  sum  of 
two  sides  of  a  triangle  would  be  equal  to  the  third  side. 
Bisect  EG  at  H,  make  FK=  FH=  HG  =  GL. 

^Euclides  ab  omni  naevo  vindicatus,  p.  x  (Milan,  1733). 

2Heath,  Euclid,  Vol.  I,  p.  202;  Engel  and  Stackel,  Die  Theorie  der  Parallel- 
linien,  Leipzig,  1895;  G.  B.  Halsted,  Saccheri's  Euclides  Vindicatus  (translation), 
p.  7  (Chicago,  1920)  ;  G.  Boccardini,  L'Euclide  emendato  del  P.  Gerolamo 
Saccheri,  Milan,  1904  (incomplete  translation);  R.  Bonola,  "Sulla  teoria  delle 
parallele  e  sulle  geometric  non-euclide,"  in  F.  Enriques,  Questioni  riguardanti  le 
matematiche  elementary  p.  248  (Bologna,  1912),  with  an  English  translation,  from 
the  German  edition,  by  H.  S.  Carslaw,  Chicago,  1912.  On  the  modern  theory  and 
treatment  of  postulates  see  C.  J.  Keyser,  Mathematical  Philosophy,  Lecture  II 
(New  York,  1922). 


THE  POSTULATE  OF  PARALLELS       283 

Then  FK  and  GL  cannot  meet  on  A*Z,  for  the  same  reason, 
and  so  on  however  far  we  go. 

Hence  the  lines  described  in  the  postulate  cannot  meet  at  all, 
even  though  £A  +  ZC<i8o°.1 

Further  attempts  at  a  proof  of  Postulate  5  were  made  by 
al-Tusi  (c.  1200),  Wallis  (c.  1660),  Saccheri  (1733),  Lambert 
(c.  1766),  Legendre  (1794  and  later),  and  many  others.2 

As  an  alternative  postulate  Proclus  stated  in  substance,  and 
Playfair  (1795)  made  well  known  to  the  modern  world,  the 
following : 

Through  a  given  point  only  one  parallel  can  be  drawn  to  a  given 
straight  line;  [or] 

Two  straight  lines  which  intersect  one  another  cannot  both  be 
parallel  to  one  and  the  same  straight  line. 

Playfair's  form  of  the  postulate  was  practically  given,  how- 
ever, somewhat  earlier  than  1795.  Joseph  Fenn,  in  his  edition 
of  Euclid's  Elements,  published  at  Dublin  in  I769,3  stated  it 
as  follows :  "  A  straight  line  which  cuts  one  of  two  parallel  lines 
will  necessarily  cut  the  other,  provided  this  cutting  line  is  suf- 
ficiently produced."  Substantially  the  same  assumption  was 
also  given  by  William  Ludlam4  in  1785,  and,  indeed,  was  given 
by  Proclus  (c.  460),  as  asserted  above,  in  a  note  to  Euclid,  1, 31. 

It  has  been  observed  by  various  writers  that  Euclid  tacitly 
assumed  other  postulates,  such  as  one  relating  to  the  inter- 
section of  plane  figures  and  one  which  asserts  that  space 
is  homogeneous  or  that  a  figure  may  be  transposed  without 
deformation. 


1For  the  rest  of  his  treatment  of  Postulate  5,  see  Heath,  Euclid,  Vol.  I, 
p.  207. 

2  On  the  general  question  of  the  validity  of  the  postulate,  see  page  335. 

3  First  Volume  of  the  Instructions  given  in  the  Drawing  School  established 
by   the  Dublin-Society   .  .  .  under  the  direction  of  Joseph  Fenn,  heretofore 
Professor  of  Philosophy  in  the  University  of  Nants,  Dublin,  1769.    F.  Cajori, 
"On  the  history  of  Playf air's  parallel-postulate,"  School  Science  and  Mathe- 
matics, XVIII,  778. 

4Born  c.  1718;  died  in  Leicestershire,  March  16,  1788.   He  wrote  various 
works  on  astronomy.  His  Rudiments  of  Mathematics  first  appeared  in  1785. 


284  PROPOSITIONS  OF  PLANE  GEOMETRY 

5.  TYPICAL  PROPOSITIONS  OF  PLANE  GEOMETRY 

Pons  Asinorum.  Most  of  the  basic  theorems  of  elementary 
plane  geometry  are  found  in  Euclid's  Elements.  Of  these  a 
relatively  small  number  have  any  interesting  history,  and  only 
a  few  typical  ones  need  be  considered,  the  first  being  Proposi- 
tion 5  of  Book  I.  As  given  by  Euclid,  this  reads  as  follows: 

In  isosceles  triangles  the  angles  at  the  base  are  equal  to  one  an- 
other, and,  if  the  equal  straight  lines  be  produced  further,  the  angles 
under  the  base  will  be  equal  to  one  another. 

Proclus  states  that  Thales  (c.  600  B.C.)  was  the  first  to  prove 
this  proposition.  At  any  rate  it  was  well  known  to  Aristotle 
(c.  340  B.C.),  who  discusses  one  of  the  proofs  then  possibly 
current.  Proclus  (c.  460)  says  that  Pap- 
pus (c.  300)  proved  the  theorem  without 
using  any  auxiliary  lines,  simply  taking  the 
triangle  up,  turning  it  over,  and  laying  it 
down  upon  itself.  The  question  as  to  how 
he  could  lay  the  triangle  itself  down  upon  it- 
self has  caused  a  change  in  the  phraseology 
on  the  part  of  modern  writers. 
The  proposition  represented  substantially  the  limit  of  in- 
struction in  many  courses  in  the  Middle  Ages.  It  formed  a 
bridge  across  which  fools  could  not  hope  to  pass,  and  was  there- 
fore known  as  the  pons  asinorum,  or  bridge  of  fools.1  It  has 
also  been  suggested  that  the  figure  given  by  Euclid  resembles 
the  simplest  form  of  a  truss  bridge,  one  that  even  a  fool  could 
make.  The  name  seems  to  be  medieval. 

The  proposition  was  also  called  elefuga,  a  term  which  Roger 
Bacon  (c.  1250)  explains  as  meaning  the  flight  of  the  miserable 
ones,  because  at  this  point  they  usually  abandoned  geometry.2 


term  is  sometimes  applied  to  the  Pythagorean  Theorem. 
2  "Sic  est  hie  quod  isti  qui  ignorant  alicujus  scientiae,  ut  sit  geometriae,  nisi 
sint  pueri  qui  coguntur  per  virgam,  resiliunt  et  tepescunt,  ut  vix  volunt  tres  vel 
quatuor  propositiones  scire.  Unde  ex  hoc  accidit  quod  quinta  propositio  geome- 
triae Euclidis  dicitur  Elejuga,  id  est,  f  uga  miserorum  ;  elegia  enim  Graece  dicitur,  . 
Latine  miseria',  et  elegi  sunt  miseri."  Opus  Tertium,  cap.  vi. 


CONGRUENCE  THEOREMS 


285 


Congruence  Theorems.  The  second  of  the  usual  congruence 
theorems  relates  to  the  case  of  two  angles  and  the  included  side 
of  a  triangle.  Proclus  (c.  460)  says  of  this: 

Eudemus  (c.  335  B.C.)  in  his  geometrical  history  refers  this  theo- 
rem to  Thales  (c.  600  B.C.).  For  he  says  that,  in  the  method  by 
which  they  say  that  Thales  proved  the  distance  of  ships  in  the  sea, 
it  was  necessary  to  make  use  of  this  theorem. 

How  Thales  could  have  used  this  theorem  for  the  purpose  is 
purely  a  matter  of  conjecture.  He  might  have  stood  at  T,  the 
top  of  a  cliff  TF,  and  sighted  to  the  ship 
S,  using  two  hinged  rods  to  hold  the  angle 
STF.  He  could  then  have  turned  and 
sighted  along  the  same  rod  to  a  point  P 
along  the  shore.  If  he  kept  the  angle 
constant,  he  would  then  merely  have  to 
measure  FP  to  find  the  unknown  dis- 
tance FS.  Since  in  those  days  the  ships  were  small  and  remained 
near  the  shore  in  good  weather,  this  plan  would  have  been  quite 


EARLY  METHODS  OF  MEASURING  DISTANCES 

From  Belli's  Libra  del  Misvrar  con  la  vista,  Venice,  1569,  but  representing 
essentially  the  method  probably  used  by  Thales 

feasible.    Thales  probably  had  some  simple  instrument  like  the 
astrolabe  by  which  he  could  measure  angles  when  observing  the 


286  PROPOSITIONS  OF  PLANE  GEOMETRY 

stars,  and  he  could  have  used  this.  We  shall  presently  see  that 
such  instruments,  in  primitive  form,  were  known  to  the  Baby- 
lonians before  his  time. 

Euclid  stated  the  proposition  in  a  more  complicated  form 
than  the  one  now  in  use,  but  his  proof  had  the  advantage  that 
it  did  not  employ  superposition.  The  latter  form  of  proof  is 
given  by  al-Nairizi  (c.  910),  who  ascribes  it  to  some  unknown 
predecessor. 

Renaissance  writers  often  used  the  theorem  in  practical  men- 
suration, as  Thales  is  thought  to  have  done.  The  illustration 
on  page  285,  from  BellPs  work1  of  1569,  shows  two  methods  of 
using  it,  and  there  is  a  story  that  one  of  Napoleon's  engineers 
gained  imperial  favor  by  quickly  applying  it  on  an  occasion 
when  the  army  was  held  up  by  a  river. 

Areas.  The  sources  of  such  propositions  as  those  relating  to 
the  area  of  the  triangle,  the  rectangle,  the  trapezium  (trape- 
zoid),  and  other  rectilinear  figures  are,  of  course,  unknown. 
The  theorems  relating  to  these  areas  are  found  in  Euclid  and 
apparently  were  common  property  long  before  his  time.  It  is 
interesting  to  know,  however,  that  the  Egyptian  surveyors, 
even  after  the  time  of  Euclid,  were  in  the  habit  of  finding  the 
area  of  a  field  by  taking  the  product  of  the  half-sums  of  the 
opposite  sides.  This  is  correct  in  the  case  of  a  rectangle,  but 
in  the  case  of  a  general  convex  quadrilateral  it  gives  a  result 
that  is  too  large.  The  error  was  corrected  to  a  certain  extent 
by  omitting  in  the  calculation  all  fractions  less  than  -fa  of  the 
large  unit  of  length.2 

The  rule  for  the  area  of  an  inscribed  convex  quadrilateral, 
expressed  by  the  formula 

A  -  V(s-a)(s-b)(s-c)(s-d), 

was  given  by  both  Brahmagupta  (c.  628)  and  Mahavira  (c. 
850),  but  without  the  limitation  that  it  holds  only  for  an 

aS.  Belli,  Libra  del  Misvrar  con  la  vista,  Venice,  1569. 

2 This  unit  was  the  (rxoivos  (schoi'nos},  equal  to  100  cubits.  Our  information 
comes  from  a  papyrus  in  the  British  Museum,  See  H.  Maspero,  Les  Finances  de 
I'Agypte  sous  les  Lagides,  p.  135  (Paris,  1905), 


AREAS  287 

inscribed  figure.    If  d  =  o,  the  figure  becomes  a  triangle  and  the 
formula  reduces  to 


the  rule  being  given  by  Heron  (c.  50?),  but  being  ascribed 
on  Arabic  authority  to  Archimedes.  Both  Brahmagupta  and 
Mahavira1  gave  the  equivalents  of  the  following  formulas, 
without  limitations,  for  the  lengths  of  the  diagonals  of  a 
quadrilateral :  , 

;;/  =  Ai 

n  =     foc  +  fat)(ad+bc) 
H  "  N          ab  +  c'd 

The  first  of  these  formulas  was  rediscovered  by  W.  Snell,  who 
gave  it  in  his  edition  (1619)  of  Van  Ceulen's  works. 

Among  the  other  interesting  formulas  related  to  those  con- 
cerning areas  is  one  discovered  by  Lhuilier  and  published  in 
1782.  It  gives  the  radius  of  the  circle  circumscribing  a  quad- 
rilateral and  reduces  to  the  following : 2 


_  i         b  +  cd)  (ac  +  bd)  (ad  +  ^) 
r  "  '""  A  \s  -  d)(s  -  6)~(s~^(^d) ' 

Angle  Sum.  The  fact  that  the  sum  of  the  angles  of  a  triangle 
is  equal  to  two  right  angles  has  long  been  recognized  as  one  of 
the  most  important  propositions  of  plane  geometry.  Eutocius 
(c.  560)  tells  us  that  Geminus  (c.  77  B.C.)  stated  that  "the 
ancients  investigated  the  theorem  of  the  two  right  angles  in 
each  individual  species  of  triangle, — first  in  the  equilateral, 
again  in  the  isosceles,  and  afterwards  in  the  scalene  triangle." 
Proclus  (c.  460)  says  that  Eudemus  (c.  335  B.C.)  ascribed  the 
theorem  to  the  Pythagoreans. 

There  is  also  a  possibility  that  Thales  knew  this  property  of 
the  triangle,  for  Diogenes  Laertius  (ad  century)  quotes  Pam- 
philius  (ist  century)  as  saying  that  he  was  the  first  to  inscribe 


Volume  I,  page  163. 
2 1  am  indebted  to  Professor  R.  C.  Archibald  for  this  information,  as  for  many 
other  valuable  suggestions. 


288  PROPOSITIONS  OF  PLANE  GEOMETRY 

a  right-angled  triangle  in  a  circle,  the  proof  of  which  solution 
requires  this  proposition,  at  least  in  a  special  case.  The  theorem 
was  certainly  well  known  before  Euclid,  for  Aristotle  refers  to 
it  several  times. 

Pythagorean  Theorem.  The  relation  of  the  sides  of  a  triangle 
when  these  sides  are  3,  4,  and  5  (that  is,  3"  +  4"  =  5")  was  well 
known  long  before  the  time  of  Pythagoras.  We  find  in  the 
Nine  Sections  of  the  Chinese,  perhaps  written  before  noo  B.C., 
this  statement:  "Square  the  first  side  and  the  second  side  and 
add  them  together ;  then  the  square  root  is  the  hypotenuse." 

The  Egyptians  also  knew  the  numerical  relation  for  special 
cases,  for  a  papyrus  of  the  i2th  dynasty  (c.  2000  B.C.),  dis- 
covered at  Kahun,  refers  to  four  of  these  relations,  one  being 
i2  +  (|)2  —  (i|-)2.  It  was  among  these  people  that  we  first 
hear  of  the  "rope  stretchers/71  those  surveyors  who,  it  is  usually 
thought,  were  able  by  the  aid  of  this  property  to  stretch  a  rope 
so  as  to  draw  a  line  perpendicular  to  another  line,  a  method  still 
in  use  at  the  present  time. 

Pythagorean  Numbers  in  India.  The  Hindus  knew  the  prop- 
erty long  before  the  beginning  of  the  Christian  era,  for  it  is 
mentioned  in  the  Sulvasutras,2  the  sacred  poems  of  the  Brah- 
mans.  The  Sulvasutra  of  Apastamba  gives  rules  for  construct- 
ing right  angles  by  stretching  cords  of  the  following  lengths: 
3,4,5;  12,  16,20;  15,20,25;  5,  12,  13;  15,36,39;  8,  15,  17; 
and  12,  35,  37.  Although  the  date  of  these  writings  is  uncer- 
tain,3 it  is  evident  that  the  relations  were  known  rather  early 
in  India.4 

Did  Pythagoras  prove  the  Theorem  ?  The  proof  of  the  proposi- 
tion is  attributed  to  Pythagoras  (c.  540  B.C.)  by  various  writers, 
including  Proclus  (c.  460),  Plutarch  (ist  century),  Cicero 

itApir€dopAirrai(harpedonaprtae)  (Vol.  I,  p.  81) .  See  Peet,Rhind  Papyrus,  p.  32. 

2  Curiously  the  word  is  sometimes  interpreted  to  mean  rope-stretching. 

3 Perhaps  the  4th  or  5th  century  B.C. 

4 A.  Biirk,  "Das  Apastamba-Sulba-Sutra,"  in  the  Zeitschrift  der  deutschen 
morgenldndischen  Gesellschaft,  LV,  543,  and  LVI,  327;  G.  Thibaut,  Journal  of 
the  Royal  Asiatic  Society  of  Bengal,  XLIV,  reprint  1875,  and  his  articles  in  The 
Pandit,  Benares,  1875/6  and  1880;  Heath,  Euclid,  Vol.  I,  p.  360. 


PYTHAGOREAN  THEOREM  289 

(c.  50  B.C.),  Diogenes  Laertius  (2d  century),  and  Athenaeus 
(c.  300).  No  one  of  these  lived  within,  say,  five  centuries  of 
Pythagoras,  so  that  we  have  only  a  weak  tradition  on  which 
to  rest  the  general  belief  that  Pythagoras  was  the  first  to  prove 
the  theorem.1  It  would  seem  as  if  such  an  important  piece  of 
history  would  have  some  mention  in  the  works  of  a  man  like 
Aristotle  ;  but,  on  the  other  hand,  it  is  difficult  to  see  how  such 
a  tradition  should  be  so  generally  received  unless  it  were  well 
founded.  Not  only  are  we  not  positive  that  the  proof  is  due  to 
Pythagoras  at  all,  but  we  are  still  more  in  doubt  as  to  the  line 
of  demonstration  that  he  may  have  followed. 

Hundreds  of  proofs  have  been  suggested  for  the  proposition, 
but  only  two  are  significant  enough  to  be  mentioned  at  this 
time.  Of  these  the  first  is  the  one  in  the  Elements,  a  proof 
which  Proclus  tells  us  was  due  to  Euclid  himself.  Although 
Schopenhauer,  the  philosopher,  calls  it  "a  proof  walking  on 
stilts77  and  "a  mousetrap  proof,77  it  has  stood  the  test  of  time 
better  than  any  other. 

The  second  noteworthy  proof  is  that  of  Pappus  (c.  300). 
In  this  figure  we  have  any  triangle  ABC,  with  CM  and  CN  any 
parallelograms  on  A  C  and  BC,  and  with  QR  equal  to  PC.  Then 
AT  =  CM+  CN,  a  re- 
lation that  reduces  to  the 
Pythagorean  Theorem 
when  ABC  is  a  right- 
angled  triangle  and  when 
the  parallelograms  are 
squares.2 

The  Pythagorean  The- 
orem is  not  uncommonly  called  the  pons  asinorum  by  modern 
French  writers.  The  Arabs  called  it  the  "Figure  of  the  Bride,'7 
possibly  because  it  represents  two  joined  in  one.  It  is  also  called 
the  "Bride's  Chair,77  possibly  because  the  Euclid  figure  is  not 


arguments  against  this  belief  see  H.  Vogt,  Bibl.  Math.,  VII  (3),  6,  and 
IX  (3),  15;  G.  Junge,  Wann  haben  die  Griechen  das  Irrationale  entdeckt, 
Halle,  1907. 

*Mathematicae  Collectiones,  ed.  Commandinus,  Bologna,  1660,  liber  quartus, 
p.  57;  Hultsch  ed.,  W,  177. 


290  PROPOSITIONS  OF  PLANE  GEOMETRY 

unlike  the  chair  which  a  slave  carries  on  his  back  and  in  which 
the  Eastern  bride  is  sometimes  transported  to  the  ceremony.1 
Of  the  rules  for  forming  rational  right-angled  triangles  the 
following  related  ones  are  among  the  most  important  :  2 

Pythagoras  (c.  540  B.C.) 


(2  «)2+(;/2-  l)a=(«2+  I)2  PlatO  (C.  380  B.C.) 


Proclus  (c.  460) 

Recent  Geometry  of  the  Triangle.  In  the  igth  century  the 
geometry  of  the  triangle  made  noteworthy  progress.  Crelle 
(  1816)  made  various  discoveries  in  this  field,  Feuerbach  (  1822  ) 
soon  after  found  the  properties  of  the  nine-point  circle,  and 
Steiner  set  forth  some  of  the  properties  of  the  triangle,  but  it 
was  many  years  before  the  subject  attracted  much  attention. 
Lemoine3  (1873)  was  the  first  to  take  up  the  subject  in  a  sys- 
tematic way  and  to  contribute  extensively  to  its  development. 
His  theory  of  "  transformation  continue"  and  his  "geometro- 
graphie"  should  also  be  mentioned.  Brocard's  contributions  to 
the  geometry  of  the  triangle  began  in  1877,  and  certain  critical 
points  of  the  triangle  bear  his  name. 

The  Pentagon  and  Decagon.  The  tenth  proposition  of 
Book  IV  of  Euclid  is  the  problem:  "To  construct  an  isosceles 
triangle  having  each  of  the  angles  at  the  base  double  the 
remaining  one."  This  makes  the  vertical  angle  36°  and  each 
of  the  others  72°,  and  therefore  permits  of  the  construction  of 

1  The  Greeks  are  said  to  have  called  it  the  "  theorem  of  the  married  women," 
and  Bhaskara  to  have  spoken  of  it  as  the  "chaise  of  the  little  married  women." 
E.  Lucas,  Recreations  Mathematiques,  II,  130. 

2H.  A.  Naber,  Das  Theorem  des  Pythagoras,  Haarlem,  1908;  Heath,  Euclid, 
Vol.  I,  p.  350;  the  names  are  those  of  the  authors  of  rules  approximately  repre- 
sented by  these  formulas.  The  assertion  as  to  Pythagoras  is  open  to  doubt. 

3D.  E.  Smith,  "  Emile-Michel-Hyacinthe  Lemoine,"  Amer.  Math.  Month., 
Ill,  29;  J.  S.  Mackay,  various  articles  on  modern  geometry  in  the  Proceedings 
of  the  Edinburgh  Mathematical  Society;  E.  Vigarie,  "La  bibliographic  de  1$ 
geometric  du  triangle,"  Mathesis,  XVI,  suppl.,  p.  14, 


PENTAGON  AND  DECAGON          291 

a  regular  decagon  and  of  a  regular  pentagon.  The  problem 
seems  to  have  been  known  to  the  Pythagoreans,  for  Proclus 
(c.  460)  tells  us  that  they  discovered  "the  construction  of 
the  cosmic  figures," — a  statement  anticipated  by  Philolaus 
(c.  425  B.C.)  and  lamblichus  (c.  325),  and  this  construction 
requires  the  use  of  the  problem.  Lucian  (2d  century),  the 
scholiast  to  the  Clouds  of  Aristophanes,  tells  us  that  the  penta- 
gram, the  star  pentagon,  was  the  badge  of  the  Pythagorean 
brotherhood,  and  the  construction  of  such  a  figure  depends 
upon  this  proposition.1 

The  solution  is  related  to  that  of  the  division  of  a  line  in  ex- 
treme and  mean  ratio.2  This  was  referred  to  by  Proclus  when 
he  said  that  Eudoxus  (c.  370  B.C.)  "greatly  added  to  the  num- 
ber of  the  theorems  which  Plato  originated  regarding  the  sec- 
tion" This  is  the  first  trace  that  we  have  of  this  name  for 
such  a  cutting  of  the  line. 

In  comparatively  modern  times  the  section  appears  first  as 
"divine  proportion/78  and  then,  in  the  igth  century,4  as  the 
"golden  section. " 

6.  TYPICAL  PROPOSITIONS  OF  SOLID  GEOMETRY 

Prism.  Since  the  Greeks  were  so  much  more  interested  in  the 
logic  of  geometry  than  in  its  applications  to  mensuration,  and 
since  they  found  a  sufficient  field  for  their  activities  in  the  work 
with  plane  figures,  they  did  not  develop  the  science  of  solid 
geometry  to  any  great  extent,  as  witness  the  Elements  of 
Euclid.  This  is  one  reason  why  even  the  technical  terms  were 
not  so  completely  standardized  as  those  of  plane  geometry.  Of 
the  special  types  of  prism  the  right  parallelepiped  is  naturally 

1  Heath,  Euclid,  Vol.  II,  p.  97.  2Euclid,  II,  n,  and  VI,  30. 

3  So  Pacioli  gave  to  his  work  of  1509  the  title  De  diuina  proportione.  Ramus 
(Scholarvm  Mathematicarvm,  Libri  vnvs  et  triginta,  Basel,  1569;  ibid.,  1578; 
Frankfort,  1599,  p.  191)  referred  to  it  in  these  words:  "Christianis  quibusdam 
divina  quaedam  proportio  hie  animadversa  est  .  .  ."  ;  and  Kepler  (Frisch  ed.  of 
his  Opera,  I,  377  (Frankfort,  1858))  spoke  of  it  in  the  following  terms:  "Inter 
continuas  proportiones  unum  singulare  genus  est  proportionis  divinae  .  .  .  ." 

4 The  term  seems  to  have  come  into  general  use  in  the  i9th  century.  It  is  found 
in  the  Archiv  der  Math,  und  Physik  (IV,  15-22)  as  early  as  1844. 


292  PROPOSITIONS  OF  SOLID  GEOMETRY 

the  most  important.  The  word  "parallelepiped"1  means  parallel 
surfaces.  Although  it  is  a  word  that  would  naturally  be  used 
by  Greek  writers,  it  is  not  found  before  the  time  of  Euclid. 
It  appears  in  the  Elements  (XI,  25)  without  definition,  in  the 
form  of  "parallelepipidal  solid,"  the  meaning  being  left  to  be 
inferred  from  that  of  the  word  "parallelogrammic"  as  given 
in  Book  I. 

We  have  as  yet  no  generally  accepted  name  for  a  rectangular 
solid  or  right  parallelepiped,  nor  had  the  Greeks.  The  word 
"cuboid"  is  significant  and  in  modern  times  has  had  some 
sanction. 

Euclid  used  cubus2  for  cube,  and  Heron  (c.  50?)  did  the 
same.  Heron  also  used  "hexahedron"3  for  this  purpose  and 
then  applied  cubus  to  any  right  parallelepiped. 

Although  the  Greeks  knew  that_the_diagonal  of  a  right  par- 
allelepiped of  edges  a,  6,  c  was  vV-f-  &*+  c\  strangely  enough 
the  statement  is  not  found  in  any  of  their  works.  It  first  ap- 
pears, so  far  as  now  known,  in  the  Practica  geometriae  of 
Fibonacci  (i22o).4 

The  word  "prism"  is  Greek.5  Euclid  defines  and  treats  it 
as  we  do  at  the  present  time.0 

Pyramid.  The  Greeks  probably  obtained  the  word  "pyr- 
amid"7 from  the  Egpytian.  It  appears,  for  example,  in  the 
Ahmes  Papyrus  (c.  1550  B.C.).  Because  of  the  pyramidal  form 
of  a  flame  the  word  was  thought  by  medieval  and  Renaissance 
writers  to  come  from  the  Greek  word  for  fire,8  and  so  a  pyramid 
was  occasionally  called  a  "fire-shaped  body."9 

iFrom  7rapd\\i/j\os  (paral'lelos,  parallel)  +  Mircdov  (epi'pedon,  plane  surface). 

2£tf|3os,  Latin  cu'bus.    See  Elements,  XI,  def.  25. 

8'E£deSpoj>,  from  ?£  (hex,  six)  +  $8pa(hed'ra,  seat). 

4".  .  .  ut  in  Solido  .aei.  cuius  dyameter  sit  linea  .tb."  (Scritti,  II,  163). 

tHptapa  (pris'ma),  from  irplfav  (pri'zein),  to  saw;  hence  something  sawed  off. 

6  Elements,  XI,  def.  13. 

illvpapls  (pyr amis'},  pi.  Trvpa^l^  (pyrami'des) ,  perhaps  from  the  Egyptian 
firomi,  but  also  thought  to  come  from  irvp6s (pyros',  grain),  as  if  a  granary.  On 
the  uncertainty  of  the  origin  see  Peet,  Rhind  Papyrus,  p.  98. 

8 Hvp  (pyr)y  as  in  "pyrotechnic." 

9 Thus  the  i6th  century  writer  W.  Schmid  (Das  erste  Buch  der  Geometrie, 
Nurnberg,  1539)  speaks  of  the  "feuerformige  Corper." 


THE  PYRAMID  293 

Euclid's  treatment  of  the  pyramid  has  remained  substantially 
unchanged ?  except  as  to  the  proposition  relating  to  the  equiva- 
lence of  pyramids  of  the  same  height  and  of  equivalent  bases. 
Cavalieri  (1635)  applied  to  this  proposition  his  method  of  in- 
divisibles, and  Legendre1  (1794)  gave  a  simple  proof  that  is 
now  in  common  use.  Aryabhata  (c.  510)  gave  the  volume  as 
half  the  product  of  the  base  and  height,  or  at  least  it  so  appears 
in  the  extant  manuscripts.2 

Frustum  of  a  Pyramid.  The  method  of  finding  the  volume  of 
the  frustum  of  a  pyramid  is  found  in  Heron's  Stereometry,3  his 
rule  reducing  to  the  modern  form  of  y=s%A(b  +  &r+V6jf);  but 
the  actual  rule  first  appears  in  Fibonacci's  PracMca  geometriae 
(i22o),4  unless  we  accept  a  gloss  upon  an  Arabic  manuscript 
of  the  1 2th  century  as  evidence  that  it  was  known  at  that  time.5 
The  method  itself  was  probably  known  to  the  Egyptians,  at 
least  for  a  special  case,  long  before  the  time  of  Heron ;  for  in  a 
hieratic  papyrus,  apparently  a  little  earlier  than  that  of  Ahmes, 
there  is  a  statement  which  seems  to  show  familiarity  with  the 
method  for  the  case  of  a  square  pyramid.0 

Brahmagupta  (c.  628)  also  gave  a  rule7  for  the  volume  of  a 
frustum  of  a  pyramid  with  square  base  of  sides  st  and  s0,  sub- 
stantially as  follows :  V—  \  /t(s?  +  s%  +  s^). 

*  Ettmens,  ist  ed.,  VI,  17  (Paris,  1794).  The  order  differs  slightly  in  the  dif- 
ferent editions. 

2Rodet,  "Logons  de  Calcul  d'Aryabhata,"  Journal  Asiatique,  XIII  (7),  393; 
reprint,  pp.  9,  10,  20. 

3 1,  capp.33,  34.  See  Tropike,Geschichte,  H  (i),  383,  with  reference  to  the  MS. 
of  the  MerptAcd  (metrika'}  discovered  recently  and  published  by  Schone,  Leipzig, 
1903.  In  his  Stereometry  he  considers  the  pyramid  with  a  square  base;  in  the 
Metrica,  with  a  triangular  base.  *Scrittt,  II,  177. 

5  See  the  Sitzungsberichte  der  physikalisch-medizinischen  Societal  zu  Erlangen, 
5O.-5I.  Band,  p.  270,  hereafter  referred  to  as  Erlangen  Sitzungsberichte. 

8 This  was  first  published  by  B.  A.  Touraeff  (Turajev)  in  1917,  rny  attention 
being  called  to  the  fact  by  Professor  R.  C.  Archibald.  The  manuscript  is  now  in 
Moscow.  See  Ancient  Egypt,  PartJTT,  p.  TOO  (London,  1917). 

7Colebrooke  ed.,  p.  312.  On  Aryabhata  see  Rodet,  loc.  cit.,  pp.  9,  10.  On 
Bhaskara,  see  Colebrooke  ed.,  p.  97.  On  Mahavira,  see  his  work,  p.  259.  On 
the  later  treatment  of  the  tetrahedron  see  Tropfke,  Geschichte,  II  (i),  385.  For 
the  cylinder  and  cone,  which  are  closely  related  to  the  prism,  pyramid,  and  circle, 
see  ibid.,  p.  387. 


294  PROPOSITIONS  OF  SOLID  GEOMETRY 

Frustum  of  a  Cone.  The  late  Greeks  knew  how  to  find  the 
volume  of  the  frustum  of  a  cone,  deriving  it  from  the  rule 
that  Heron  used  for  the  frustum  of  a  pyramid,  and  thereafter 
the  same  method  appeared  in  various  mathematical  treatises. 
Heron,  however,  used  an  approximation  method  which  is  prob- 
ably of  Egyptian  origin,  namely,  that  of  taking  the  product  of 
the  altitude  and  the  area  of  the  circle  midway  between  the 
bases.1  An  interesting  example  of  the  use  of  this  approxima- 
tion method  has  been  found  among  the  Greek  papyri  on  arith- 
metic, being  probably  the  work  of  a  schoolboy  of  about  the 
4th  century.  The  problems  in  this  papyrus  resemble  those 
found  in  the  Akhmim  Papyrus,  which  was  written  somewhat 
later.  The  first  one  of  these  problems  relates  to  finding  the 
contents  of  a  circular  pit  of  which  the  circumference  is  20 
cubits  at  the  top  and  12  cubits  at  the  bottom,  and  of  which 
the  altitude  is  12  cubits.  The  writer  makes  an  error  in  his 
methods  as  well  as  his  calculations,  but  endeavors  to  use 
Heron's  approximation.2 

Sphere.  The  word  "sphere"  comes  to  us  from  the  Greek 
through  the  Latin.3  The  pure  geometers  of  Greece  bad  little 
interest  in  its  measurement,  although  Archimedes  tells  us  that 
"  earlier  geometers  .  .  .  have  shown  .  .  .  that  spheres  are  to 
one  another  in  the  triplicate  ratio  of  their  diameters."4 

Archimedes  also  states  that  the  volume  of  any  sphere  is  four 
times  that  of  the  cone  with  base  equal  to  a  great  circle  of  the 
sphere  and  with  height  equal  to  the  radius  of  the  sphere, — a 
statement  that  amounts  to  saying  that  V—  f 7rrs.  He  also  stated 
that  a  cylinder  with  base  equal  to  a  great  circle  of  the  sphere 
and  with  height  equal  to  the  diameter  of  the  sphere  is  equal 
to  i  J  times  the  sphere, — a  statement  that  amounts  to  the 
same  thing.5 

1  Stereometric,  ed.  Hultsch,  p.  157. 

2J.  G.  Smyly,  "Some  Examples  of  Greek  Arithmetic,"  Hermathena,  XLII 
(1920),  105. 

*24>a?/Mi  (sphai'ra) ,  Latin  sphaera. 

4Allman,  Greek  Geom.,  p.  96;  Archimedes,  ed.  Heiberg,  II,  265;  Heath, 
Archimedes,  234. 

5  Heiberg  ed.,  Vol.  I,  De  sphaera  et  cylindro. 


CONE,  SPHERE,  AND  POLYHEDRON  295 

Polyhedrons.  The  word  "  polyhedron m  is  not  found  in  the 
Elements  of  Euclid ;  he  uses  " solid,"" octahedron/'  and  "dodeca- 
hedron/7 but  does  not  mention  the  general  solid  bounded  by 
planes.  The  chief  interest  of  the  Greeks  in  figures  of  this  type 
related  to  the  five  regular  polyhedrons.  It  seems  probable  that 
Pythagoras  (c.  540  B.C.)2  brought  his  knowledge  of  the  cube, 
tetrahedron,  and  octahedron  from  Egypt,  but  the  icosahedron 
and  the  dodecahedron  seem  to  have  been  developed  in  his  own 
school.  The  Pythagoreans  assigned  the  tetrahedron  to  fire, 
the  octahedron  to  air,  the  icosahedron  to  water,  the  cube  to 
earth,  and  the  dodecahedron,  apparently  the  last  one  discov- 
ered, to  the  universe.  They  seem  to  have  known  that  all  five 
polyhedrons  can  be  inscribed  in  a  sphere.  They  passed  the 
study  of  these  solids  on  to  the  school  of  Plato  (c.  380  B.C.), 
where  they  attracted  so  much  attention  as  to  be  known  to  later 
writers  as  " Platonic  bodies "  or  "cosmic  figures."  It  is  not 
probable,  however,  that  the  early  Pythagoreans  actually  con- 
structed the  figures  in  the  sense  that  Euclid  (c.  300  B.C.)3  and 
Pappus  (c.  300) 4  constructed  them. 

We  have  specimens  extant  of  icosahedral  dice  that  date 
from  about  the  Ptolemaic  period  in  Egypt.5  There  are  also  a 
number  of  interesting  ancient  Celtic  bronze  models  of  the  regu- 
lar dodecahedron  still  extant  in  various  museums.  There  was 
probably  some  mystic  or  religious  significance  attached  to  these 
forms.  Since  a  stone  dodecahedron  found  in  northern  Italy 
dates  back  to  a  prehistoric  period,  it  is  possible  that  the  Celtic 
people  received  their  idea  from  the  region  south  of  the  Alps." 
and  it  is  also  possible  that  this  form  was  already  known  in 
Italy  when  the  Pythagoreans  began  their  teaching  in  Crotona. 

1  From  TroXtfs  (polys',  many)  +  25pa  (hed'ra,  seat) . 

2  Heath,  Euclid,  Vol.  Ill,  p.  525. 

^Elements,  XIII,  props.  13-17.  See  Heath,  Euclid,  Vol.  Ill,  pp.  467-503,  with 
notes  on  the  solutions  suggested  by  Pappus. 

4  Pappus,  ed.  Commandino,  p.  45  seq.   (Bologna,   1660);   ed.  Hultsch,  III, 
142  seq. 

5  See  the  illustration  on  page  49. 

6F.  Lindemann,  "Zur  Geschichte  der  Polyeder  und  der  Zahlzeichen,"  Sit- 
zungsberichte  der  math.-physik.  Classe  der  K.  Bayerischen  Akad,  der  Wissensch. 
zu  Miinchen,  Munich,  XXVI,  625. 


296  PROPOSITIONS  OF  SOLID  GEOMETRY 

The  five  regular  polyhedrons  attracted  attention  in  the  Mid- 
dle Ages  chiefly  on  the  part  of  astrologers.  At  the  close  of  this 
period,  however,  they  were  carefully  studied  by  various  mathe- 
maticians. Prominent  among  the  latter  was  Pietro  Franceschi, 
whose  work  De  corporibus  regularibus  (c.  1475)  was  the  first  to 
treat  the  subject  with  any  degree  of  thoroughness.  Following 
the  custom  of  the  time,  Pacioli  (1509)  made  free  use  of  the 
works  of  his  contemporaries,  and  as  part  of  his  literary  plunder 
he  took  considerable  material  from  this  work  and  embodied  it 
in  his  De  diuina  proportioned 

Albrecht  Diirer,  the  Nurnberg  artist,  showed  how  to  con- 
struct the  figures  from  a  net2  in  the  way  commonly  set  forth 
in  modern  works.  The  subject  of  stellar  polyhedrons  begins 
with  Kepler3  (1619)  and  has  attracted  considerable  attention 
since  his  time. 

Polyhedron  Theorem.  Among  the  most  interesting  of  the 
modern  formulas  relating  to  a  polyhedron  is  the  one  connecting 
the  faces,  vertices,  and  edges.  This  formula,  often  known  as 
Euler's  Theorem,  may  be  stated  as  /+  v  =  e  +  2 .  It  was  pos- 
sibly known  to  Archimedes  (c.  225  B.C.),4  but  not  until  the  i7th 
century  was  it  put  into  writing  in  a  form  still  extant.  Descartes 
(c.  1635)  was  the  first  to  state  it.5  Euler  seems  to  have  come 
upon  it  independently.  He  announced  it  in  Petrograd  in  1752, 
but  with  simply  an  inductive  proof.  General  proofs  have  been 
given  by  various  writers. 

Pappus-Guldin  Theorem.  Pappus  of  Alexandria  (c.  300) 
stated,  in  substance,  the  basis  of  the  theorem  that  the  volume 
of  a  figure  formed  by  the  revolution  of  a  plane  figure  about  an 
axis  is  equal  to  the  area  of  the  figure  multiplied  by  the  length 
of  the  line  generated  by  the  center  of  gravity.6  He  also  con- 

1  Venice,  1509. 

2  Underweyssung  der  messung  mil  dem  zirckel  un  richtscheyt,  in  Linien, 
ebnen  unnd  gantzen  corporen,  Nurnberg,  1525. 

3Frisch  edition  of  his  works,  V,  126,  where  he  speaks  of  such  a  figure  as  the 
" Stella  pentagonica"  and  carries  the  discussion  "in  solido." 

4Tropfke,  Gesckichte,  II  (i),  398. 

5  But  the  fact  was  not  made  known  until  his  CEuvres  inedites  appeared  in 
1860,  where  it  appears  on  page  218.  6Hultsch  edition  of  his  works,  p.  682. 


FAMOUS  PROBLEMS  297 

sidered  the  case  in  which  there  was  not  a  complete  revolution. 
The  proposition  was  soon  forgotten  and  so  remained  until  re- 
vived by  Kepler  (iGis),1  who  extended  the  theory  to  include 
the  study  of  the  revolution  of  various  plane  figures,  He  gave 
special  attention  to  the  torus  formed  by  the  revolution  of  a 
circle  or  an  ellipse.  When  the  axis  was  tangent  to  the  circle  or 
the  ellipse,  he  called  the  resulting  solid  an  annulus  strictus. 
Although  he  treats  of  only  special  cases,  he  doubtless  knew  the 
general  theorem. 

Various  writers  made  use  of  the  general  principle  in  the  iyth 
century,  but  it  was  brought  into  special  prominence  by  Habakuk 
Guldin2  (1577-1643),  a  Swiss  scholar.  It  appeared  in  Book  II 
of  his  Centrobaryca  (1641).  He  added  nothing  to  the  theory 
except  as  he  stated  the  general  proposition. 

In  1695  Leibniz  suggested  that  the  proposition  could  be  ex- 
tended to  include  the  case  of  a  plane  revolving  about  an  axis 
on  any  other  path  than  a  circle,  provided  it  is  always  per- 
pendicular to  this  path,3  an  idea  that  was  considered  also  by 
Euler  in  1778. 

7.  THE  THREE  FAMOUS  PROBLEMS 

Nature  of  the  Problems.  The  Greeks  very  early  found  them- 
selves confronted  by  three  problems  which  they  could  not 
solve,  at  least  by  the  use  of  the  unmarked  ruler  and  the  com- 
passes alone.4 

The  first  was  the  trisection  of  any  angle.  The  trisection  of 
the  right  angle  was  found  to  be  simple,  but  the  trisection  of  any 
arbitrary  angle  whatever  attracted  the  attention  and  baffled  the 
efforts  of  many  of  their  mathematicians.  To  this  problem  may 
be  added  the  related  ones  of  dividing  any  given  angle  into  any 
required  number  of  equal  parts  and  of  inscribing  in  a  circle 
a  regular  polygon  of  a  given  number  of  sides. 

1In  his  Stereometria  Doliorum.    See  the  Opera  Omnia,  ed.  Frisch,  W,  551-670. 

2  Known  as  Paul  Guldin  after  he  entered  the  Catholic  Church. 

3Acta  Eruditorum,  1695,  p.  493. 

4  See  any  of  the  histories  of  Greek  mathematics.  A  good  summary  is  given 
in  H.  G.  Zeuthen,  Histoire  des  Mathematiques  dans  VAntiquite  et  le  Moyen 
Age,  French  translation  by  J.  Mascart,  p.  57  (Paris,  1902). 


298  THE  THREE  FAMOUS  PROBLEMS 

The  second  problem  was  the  quadrature  of  the  circle,  that  is, 
the  finding  of  a  square  whose  area  is  the  same  as  that  of  a  given 
circle.  The  solution  would  be  simple  if  we  could  find  a  straight 
line  that  is  equal  in  length  to  the  circumference  of  the  circle ; 
that  is,  if  we  could  rectify  the  circumference.  This  is  easily 
accomplished  by  rolling  the  circle  along  a  straight  line,  but  such 
a  proceeding  makes  use  of  an  instrument  other  than  the  ruler 
and  compasses,  namely,  of  a  cylinder  with  a  marked  surface. 

The  third  problem  was  the  duplication  of  a  cube/  that  is,  the 
finding  of  an  edge  of  the  cube  whose  volume  is  twice  the  volume 
of  a  given  cube.  This  was  knnwp^3jy^J)elian.JP.roblern?  one 
story  of  its  origin  being  that  the  Athenians  appealed  to  the 
oracle  at  Delos  to  know  how  to  stay  the  plague  which  visited 
their  city  in  430  B.C.  It  is  said  that  the  oracle  replied  that 
they  must  double  in  size  the  altar  of  Apollo.  This  altar  being 
a  cube,  the  problem  was  that  of  its  duplication.  Since  problems 
about  the  size  and  shape  of  altars  appear  in  the  early  Hindu 
literature,  it  is  not  improbable  that  this  one  may  have  found 
its  way,  perhaps  through  Pythagoras,  from  the  East.  It  was 
already  familiar  to  the  Greeks  in  the  5th  century  B.C.,  for  we 
are  told  by  Eratosthenes2  that  Euripides  (c.  485-406  B.C.) 
refers  to  it  in  one  of  his  tragedies  which  is  no  longer  extant.1 

Trisection  Problem :  the  Conchoid.  There  are  various  ways  of 
trisecting  any  plane  angle,  but  it  will  suffice  at  this  time  to  give 
only  a  single  one.  Probably  the  best  known  of  the  Greek  at- 
tempts is  the  one  made  by  Nicomedes  (c.  180  B.C.).  He  used 
a  curve  known  as  the  conchoid.4  We  take  a  fixed  point  O  which 

*N.  T.  Reimer,  Historia  problematis  de  cubi  duplicatione  (Gottingen,  1798) ; 
C.  H.  Biering,  Historia  problematis  cubi  duplicandi  (Copenhagen,  1844) ;  Archi- 
medes, Opera,  ed.  Heiberg,  III,  102;  A.  Sturm,  Das  Delische  Problem  (Linz,  3 
parts,  1895,  1896,  1897),  a  critical  historical  study  with  extensive  bibliography. 

2  See  Archimedes,  Opera,  ed.  Heiberg,  III,  102. 

3 Too  small  hast  thou  designed  the  royal  tomb. 
Double  it;  but  preserve  the  cubic  form. 

4 Thus  Proclus:  "Nicomedes  trisected  every  rectilineal  angle  by  means  of 
the  conchoidal  lines,  the  inventor  of  whose  particular  nature  he  is,  and  the 
origin,  construction,  and  properties  of  which  he  has  explained.  Others  have 
solved  the  same  problem  by  means  of  the  quadratrices  of  Hippias  and  Nicomedes 


THE  TRISECTION  PROBLEM 


299 


is  d  distant  from  a  fixed  line  AB,  and  we  draw  OX  parallel  to 
AB  and  OY  perpendicular  to  OX.  We  then  take  any  line  OA 
through  O,  and  on  OA  produced  lay  off  AP~  AP  =  k,  a  con- 
stant. Then  the  locus 
of  points  P  and  P1  is  a 
conchoid.  According  as 
k~.d  we  have  O  anode, 
a  cusp,  or  a  conju- 
gate point.  The  equa- 
tion of  the  curve  is 


In  order  to  trisect  a 
given  angle  we  proceed 
as  follows : 

Let  YOA  be  the  angle 
to  be  trisected.  From  A  construct  AB  perpendicular  to  OY. 
From  O  as  pole,  with  AB  as  a  fixed  straight  line  and  2  AO  as  a 


TRAMMEL  FOR   CONSTRUCTING   THE    CONCHOID 
From  Bettini's  Apiaria  Universae  Philosophiae  Mathematicae,  Bologna,  1641 

constant  distance,  describe  a  conchoid  to  meet  OA  produced  at 
/'and  to  cut  OYat  Q.  At  A  construct  a  perpendicular  to  AB 

.  .  .;  others,  again,  starting  from  the  spirals  of  Archimedes"  (Proclus,  ed. 
Friedlein,  p.  272  (translation  by  Allman) ;  see  also  Gow,  Greek  Geom.,  p.  266; 
Heath,  History,  I,  235,  238). 


300 


THE  THREE  FAMOUS  PROBLEMS 


B 


M 


O 


Q 


A 


meeting  the  curve  at  T.    Draw  OT  and  let  it  cut  AB  at  N.    Let 
M  be  the  mid-point  of  NT. 

Then  MT=  MN=  MA. 

But  NT  =  2  OA  by  construction  of  the  conchoid. 

Hence          MA  =  OA. 

Hence    Z.AOM=  Z.AMO  =  2 /-ATM-^  2/.TOQ. 

That  is,  Z.AOM=  f  Z  ra4,  and  Z  r<9<2  =  \ •  Z  F6>.4. 

The  Quadratrix.  A  Greek  geometer  named  Hippias,  probably 
Hippias  of  Elis1  (c.  425  B.C.),  invented  a  curve  which  he  used 
in  the  trisection  of  an  angle.  In  this  figure, 
X  is  any  point  on  the  quadrant  A  C.  As 
the  radius  OX  revolves  at  a  uniform  rate 
from  the  position  O  C  to  the  position  OA, 
the  line  MN  moves  at  a  uniform  rate  from 
the  position  CB  to  the  position  OA,  always 
remaining  parallel  to  OA.  Then  the  locus 
of  P,  the  intersection  of  OX  and  MN,  is 
a  curve  CQ.  Manifestly,  when  OX  is 
one  n\h  of  the  way  from  OC  around  to  OA,MNis  one  nth  of  the 
way  from  CB  down  to  OA.  If,  therefore,  we  make  CM—  \CO, 
MN  will  cut  CQ  at  a  point  P  such  that  OP  will  trisect  the  right 
angle.  In  the  same  way,  by  trisecting  OM  we  can  find  a  point 
P1  on  CQ  such  that  OP1  will  trisect  angle  A  OX,  and  so  for  any 
other  angle.  The  method  evidently  applies  to  the  multisection 
as  well  as  to  the  trisection  of  an  angle. 

Other  Methods  of  Trisection.  The  next  prominent  investiga- 
tion of  the  problem  is  one  that  is  attributed  to  Archimedes2 
••(c.  22$  B.C.),  although  probably  not 
due  to  him  in  the  form  that  has  come 
down  to  us.    The  plan  is  as  follows: 

Produce  any  chord  AB  of  a  circle 
until  the  part  produced,  BC,  is  equal 
to  r,  the  radius.  Join  C  to  the  center 
O  and  produce  CO  to  the  circle  at  D. 

xBut  this  has  been  questioned.   For  the  arguments,  see  Allman,  Greek  Geom., 
p.  93.   See  also  Volume  I,  page  82,  n.,  and  Heath,  History,  I,  225. 

See  Heath,  Archimedes]  Allman,  Greek  Geom.r  p.  90,  with  references. 


REGULAR  POLYGONS  301 


Then  arc  AD  is  three  times  arc££]  that  is,  Z.EOB  is 
This,  however,  is  manifestly  no  solution  of  the  problem. 
Vieta  (c.  1590)  was  led  by  this  to  suggest  the  following: 
Let  Z.AOJ3  be  the  angle  to  be  trisected.     Describe  any 
circle  with  center  O.    Suppose  the  problem  solved  and  that 
Z,AOP=-  *  Z.AOB.  Through  B  sup- 
pose BR  <2  drawn   parallel  to  PO. 


2/-Q.   But 

and  hence  Z.Q  =  Z.ROQ,  and  so 
OR  =  QR.  The  problem  is  there- 
fore reduced  to  the  following: 

From  B  draw  JBRQ  so  that  the  part  RQ,  intercepted  between 
the  circle  and  the  diameter  AS  produced,  shall  be  equal  to  the 
radius,  a  construction  involving  the  use  of  a  marked  straightedge. 

Regular  Polygons.  If  we  can  trisect  an  angle  of  360°  we  can 
inscribe  a  regular  polygon  of  three  sides  in  a  circle,  and  simi- 
larly for  the  inscription  of  other  regular  polygons.  The  trisec- 
tion  problem  therefore  naturally  suggests  the  larger  problem  of 
the  inscription  of  a  polygon  of  any  given  number  of  sides.  It 
was  for  a  long  time  believed  that  the  Greeks  had  exhausted  all 
the  possibilities  in  this  line.  In  1796,  however,  Gauss  showed 
that  it  was  possible,  by  the  use  of  the  straightedge  and  com- 
passes alone,  to  inscribe  a  polygon  of  17  sides.  He  even  ex- 
tended the  solution  to  include  polygons  of  257  and  65,537 
sides.  The  general  proposition,  as  it  now  appears,  is  as  fol- 
lows :  A  regular  polygon  of  p  sides,  where  p  is  a  prime  number 
greater  than  2  ,  can  be  constructed  by  ruler  and  compasses  if  and 
only  if  p  is  of  the  form  2*  +  i.  For  t  =  o,  i,  2,  3,  and  4  the 
values  of  p  are  respectively  3,  5,  17,  257,  and  65,537;  but  if 
t  =  5,  p  =  641  x  6700417  and  hence  is  not  prime.1  Gauss  has 
left  the  following  interesting  record  of  the  discovery: 

The  day  was  March  29,  1796,  and  chance  had  nothing  to  do  with 
it.  Before  this,  indeed  during  the  winter  of  1796  (my  first  semester  in 

1For  the  theory  see  J.  W.  A.  Young,  Monographs  on  Modern  Mathematics 
(New  York,  1911),  article  by  L.  E.  Dickson,  p.  378,  with  references. 


302  THE  THREE  FAMOUS  PROBLEMS 

Gottingen) ,  I  had  already  discovered  everything  related  to  the  separa- 
tion of  the  roots  of  the  equation 

x*  —  i 

-  ss=  O 
X  —  I 

into  two  groups.  .  .  .  After  intensive  consideration  of  the  relation  of 
all  the  roots  to  one  another  on  arithmetical  grounds,  I  succeeded,  dur- 
ing a  holiday  in  Braunschweig,  on  the  morning  of  the  day  alluded  to 
(before  I  had  got  out  of  bed),  in  viewing  the  relation  in  the  clearest 
way,  so  that  I  could  immediately  make  special  application  to  the  1 7- 
side  and  to  the  numerical  verification.  ...  I  announced  this  dis- 
covery in  the  Literaturzeitung  of  Jena,  where  my  advertisement  was 
published  in  May  or  June,  I796.1 

Squaring  the  Circle.  The  second  famous  problem  of  antiquity 
was  that  of  squaring  the  circle.  The  first  attempts  were  of 
course  empirical.  They  were  made  long  before  the  scientific 
period  of  the  Greek  civilization,  and  they  naturally  resulted  in 
rude  approximations. 

The  first  definite  trace  that  we  have  of  an  approximate  value 
of  TT  is  in  the  Ahmes  Papyrus2  (c.  1550  B.C.).  There  is  given 
in  that  work  a  problem  requiring  the  finding  of  the  area  of  a 
circle,  the  method,  expressed  in  modern  symbols,  being  as 
follows:  A  =  (d-id)\ 

which  amounts  to  saying  that  TT  =3.1605  — ,  a  result  apparently 
arrived  at  empirically,  as  already  stated  on  page  270. 

It  is  probable  that  3  is  a  much  older  value  of  TT,  although  we 
have  no  extant  literature  to  prove  this  fact.  We  find  such  a 
value  in  early  Chinese  works,3  in  the  Bible,4  in  the  Talmud,5 

1R.  C.  Archibald,  "Gauss  and  the  Regular  Polygon  of  Seventeen  Sides," 
Amer.  Math.  Month.,  XXVII,  323,  with  bibliography. 

2Peet,  Rhind  Papyrus,  p.  oo;  Eisenlohr,  Ahmes  Papyrus,  p.  117;  G.  Vacca, 
"Sulla  quadratura  del  circolo  secondo  1'  Egiziano  Ahmes,"  Bollettino  di  biblio- 
grafia  e  storia  delle  scicnze  matematiche,  XI,  65. 

3 E.g.,  in  the  Chou-pei.  See  Mikami,  China,  pp.  8,  46,  135.  It  should  be  men- 
tioned again  that  there  are  doubts  as  to  the  reliability  of  ancient  Chinese  texts. 

4 "And  he  made  a  molten  sea,  ten  cubits  from  the  one  brim  to  the  other:  it 
was  round  all  about  .  .  .  and  a  line  of  thirty  cubits  did  compass  it  round  about" 
(i  Kings,  vii,  23.  See  also  2  Chronicles,  iv,  2). 

5 In  both  Mishna  and  Talmud  the  value  is  always  3,  the  reason  being  tradi- 
tional, based  upon  Solomon's  "molten  sea." 


SQUARING  THE  CIRCLE  303 

in  the  early  Hindu  works/  and  in  the  medieval  manuscripts,  so 
that  it  was  generally  accepted  in  all  countries  and  until  rela- 
tively modern  times. 

The  Greeks  were  not  content  with  results  that  were  merely 
empirical,  however,  and  so  the  rectification  of  the  circumfer- 
ence or  the  related  problem  of  the  squaring  of  the  circle 
attracted  the  attention  of  their  philosophers.  For  example,  An- 
axagoras  (c.  440  B.C.)  is  said  by  Plutarch2  to  have  been  put  in 
prison  in  Athens,  and  while  there  to  have  first  attempted  the 
solution.  The  results  of  his  work  are,  however,  unknown. 

Methods  of  Attacking  the  Quadrature.  There  are  three  meth- 
ods of  attacking  the  problem :  first,  by  the  use  of  the  ruler  and 
compasses  only ;  second,  by  the  use  of  higher  plane  curves ;  third, 
by  such  devices  as  infinite  series,  leading  to  close  approxima- 
tions. The  leading  Greek  mathematicians  seem  to  have  found 
the  futility  of  the  first  method,  although  they  did  not  prove  that 
it  is  impossible ;  with  the  second  method  they  were  successful ; 
with  the  third  method  they  were  less  skillful.3 

Method  of  Exhaustion.  Antiphon  (c.  430  B.C.)  attempted  the 
quadrature  by  inscribing  a  polygon  (some  early  writers  say  a 
square  and  others  a  triangle),  and  then  doubling  the  number 
of  sides  successively  until  he  approximately  exhausted  the  area 
between  the  polygon  and  the  circle.  By  finding  the  area  of 
each  polygon  he  was  thus  able  to  approximate  the  area  of 
the  circle.4 

Attempts  of  Hippocrates.  Hippocrates  of  Chios  (c.  460  B.C.) 
attempted  the  solution  and  was  the  first  to  actually  square  a 
curvilinear  figure.  He  constructed  semicircles  on  the  three 

1"The  diameter  and  the  square  of  the  semidiameter,  being  severally  multi- 
plied by  three,  are  the  practical  circumference  and  area.  The  square-roots  ex- 
tracted from  ten  times  the  squares  of  the  same  are  the  neat  values."  Colebrooke, 
Brahmagupta,  p.  308.  See  also  Mahavira,  p.  189;  Colebrooke,  Bhdskara,  p.  87. 

2  De  exilio,  cap.  17,  ed.  Diibner-Didot  of  the  Moralia,  I,  734  (Paris,  1885). 
See  also  the  Leipzig  edition  of  1891,  III,  573.  3 Heath,  History,  I,  220. 

4F.  Rudio,  "  Eter  Bericht  des  Simplicius  liber  die  Quadraturen  des  Antiphon  und 
des  Hippokrates,"  in  Bibl.  Math.,  Ill  (3),  7>  and  also  in  book  form,  with  Greek 
text  (Leipzig,  1907) ;  Allman,  Greek  Geom.,  pp.  64,  81.  With  respect  to  Bryson 
see  ibid.,  pp.  77,  82;  but  compare  also  Volume  I,  page  84. 


304 


THE  THREE  FAMOUS  PROBLEMS 


sides  of  an  isosceles  right-angled  triangle  and  showed  that  the 
sum  of  the  two  lunes  thus  formed  is  equal  to  the  area  of  the 

triangle  itself.  Having  a  triangle 
equal  in  area  to  a  lune,  he  had 
only  to  construct  a  square  equal 
to  the  triangle.  His  proof  in- 
volves the  proposition  that  the 
areas  of  circles  are  proportional 
to  the  squares  of  their  diameters, 
— a  proposition  which  Eudemus  (c.  335  B.C.)  tells  us  that  Hip- 
pocrates proved.1  To  the  quadrature  problem  as  such,  however, 
his  contribution  was  not  important.  His  method  of  attack  was 
substantially  as  follows : 

In  a  semicircle  ABCD,  center  O,  he  inscribed  half  of  a  regu- 
lar hexagon,  h.  On  the  three  sides  and  on  OB  he  described 
semicircles  as  here  shown.  Then  the  four  small  semicircles  are 
together  equal  to  the  large  semicircle.  Subtracting  the  common 
shaded  parts,  the  three  lunes  together  with  the  semicircle  on 
OB  are  equal  to  h,  the  half 
of  the  regular  hexagon.  Now 
take  from  h  a  surface  equal  to 
the  sum  of  the  lunes,  which 
can  be  found  by  the  method 
already  given  (and  here  is  the 
fallacy),  and  there  remains  a 
rectilinear  figure  equal  to  the 
semicircle  on  OB.  It  will  be 
observed  that  Hippocrates  as- 
sumed that  every  lune  can  be  squared,  whereas  he  has  shown, 
as  we  have  seen,  that  this  is  possible  only  in  the  special  case  of 
a  right  triangle.2 

^Eudemi  fragmenta,  ed.  Spengel,  p.  128.  The  proposition  is  equally  true  for 
any  right-angled  triangle,  but  Hippocrates  proved  it  only  for  the  isosceles  case. 
See  Allman,  Greek  Geom.,  p.  66;  W.  Lietzmann,  Der  Pythagoreische  Lehrsatz, 
p.  32  (Leipzig,  1912)  ;  E.  W.  Hobson,  Squaring  the  Circle,  p.  16  (Cambridge, 
1913) ;  Heath,  History,  I,  183,  with  a  summary  of  recent  literature  on  the  subject. 

2  For  details  relating  to  the  work  of  Hippocrates  on  lunes  in  general,  see 
Heath,  History,  I,  183-201 ;  Allman,  Greek  Geom.,  p.  69. 


SQUARING  THE  CIRCLE 


305 


The  Quadratrix.  The  next  noteworthy  attempt  was  made  by 
Deinostratus  (c.  350  B.C.).  Pappus1  makes  this  statement: 

For  the  quadrature  of  the  circle  a  certain  curve  was  employed  by 
Deinostratus,  Nicomedes,  and  some  other  more  recent  geometers, 
which  has  received  its  name  from  the  property  that  belongs  to  it; 
for  it  is  called  by  them  the  quadrat rix.2 

This  is  the  curve  used  by  Hippias  in  the  trisection  of  an  angle. 
In  the  figure  given  below  it  can  be  shown  that 

CXA  _cp  ^ 

~C<9~~<9<2' 

and  since  these  terms  are  all  straight  lines  except  the  quadrant 
CXA,  it  is  possible  to  construct  a  straight  line  equal  in  length 
to  the  quadrant,  and  hence  to  rectify  c 
the  circumference. 

To  prove  the  proposition  Pappus  states 
that  the  reductio  ad  absurdum  was  em- 
ployed. If 


B 


CO       ( 

I \ff       \ 

then  the  proportion  can  be  made  true      °         A        ®  A/  A 
by  increasing  or  by  decreasing  <9<2;  but  it  will  be  shown  that 
this  cannot  be  done  without  leading  to  an  absurdity. 

T-    .  ^  A        CXA      CO 

First,  suppose  that       —  -  — ,• 

^         .  CXA        CO      CO 

Then,  since 


it  follows  that  CPA  =  CO. 

And  since,  from  the  property  of  the  curve, 
CXA       CO 


we  have 


XA       PA" 
CXA  _  CPA 
XA  ~    PA 


CO. 


1  Pappus,  Collectiones,  IV,  cap.  xxx  (ed.  Hultsch,  I,  253) ;  Hankel,  Geschichte, 
p.  151;  Cantor,  Geschichte,  I  (2),  233;  Heath,  History,  I,  226. 
lfrvffa  (tetragoni'zousa) . 


306 


THE  THREE  FAMOUS  PROBLEMS 


whence  arc  PA'  —  PA",  which  is  impossible,  since  an  arc  cannot 
be  equal  to  its  chord,  and  since  their  halves  cannot  be  equal. 

XT     i  *u  .         CXA       CO 

Next,  suppose  that       -—r-  =  -^--^ 

Draw  the  quadrant  CnMA". 
Then 


Therefore 


But 


MA"      ~XA 

CXA          CO         CO 


C"MA"       C"0       OA" 

CXA        CO         v     ,  *  .     , 

=          .       By  hyp<       Q          A,,       QA,  A 


Hence  CO  must  be  equal  to  C"MA"  if  the  hypothesis  is  correct. 

C^MA^  _  CXA 
MA"    '~   XA  ' 


But 


and  because  CPQ  is  a  quadratrix  we  have 

CXA  =  CO  . 
XA  ~~~PA"' 
C"MA"  CO 


whence 


MA' 


But  in  the  same  way  as  it  was  shown  in  the  first  part  of  the 
proof  that  c>PA'=CO 

on  the  hypothesis  there  made,  so  it  may  be  shown  here  that 

C"MA"=CO 
on  this  hypothesis.    It  then  follows  from  this  proportion  that 


which  leads  to  the  absurdity  that  the  circumference  of  a  circle 
is  equal  to  the  perimeter  of  a  circumscribed  polygon. 
Hence  the  second  hypothesis  is  also  untenable. 

CXA      CO 
Hence  —  =  —  , 

and  CXA,  a  quadrant,  can  be  constructed. 


SQUARING  THE  CIRCLE  307 

Method  of  Archimedes.  The  next  noteworthy  contribution 
was  that  of  Archimedes  (c.  225  B.C.),  who  asserted  that: 

1.  The  area  of  a  circle  is  equal  to  the  area  of  a  right-angled 
triangle  one  of  whose  sides  forming  the  right  angle  is  equal  to 
the  circumference  of  the  circle  and  the  other  to  the  radius.1 

2.  The  ratio  of  the  area  of  a  circle  to  the  square  on  the 
diameter  is  approximately  n  114. 

3.  The  ratio  of  the  circumference  of  a  circle  to  the  diameter 
is  less  than  3^  and  greater  than  3^. 

To  prove  the  third  proposition  Archimedes  inscribed  and 
circumscribed  regular  polygons,  found  their  areas  up  to  poly- 
gons of  96  sides,  and  showed  that  the  area  of  the  circle  lies 
between  these  results.  These  limits,  expressed  in  modern  deci- 
mal form,  are  3.14285714  •  •  •  and  3.14084507  •  •  •.  If  our  pres- 
ent notation  and  our  methods  of  finding  a  square  root  had  been 
known,  the  result  would  have  been  closer,  since  the  geomet- 
ric method  permitted  of  any  desired  degree  of  approximation. 

The  Romans  were  little  concerned  with  accurate  results  in 
such  matters  as  this,  and  so  it  is  not  surprising  that  Vitruvius 
(c.  20  B.C.)  speaks  of  the  circumference  of  a  wheel  of  diameter 
4  feet  as  being  12  J  feet,  thus  taking  TT  as  3|.2 

Other  Greek  Approximations  of  TT.  After  the  time  of  Archi- 
medes the  value  3Y  became  recognized  as  a  satisfactory  ap- 
proximation and  appeared  in  the  works  of  Heron  (c.  50?), 
Dominicus  Parisiensis  (1378),  Albert  of  Saxony  (c.  1365), 
Nicholas  Cusa  (c.  1450),  and  others.  Since  one  of  the  common 
approximations  for  a  square  root  in  the  Middle  Ages  was 

V»  =  vV-f  r  =  a  H — -  , 

2a  +  i 

and  since  this  gives  Vio  =  3  H =  3^,  it  is  natural  to 

2  X  3    i    I 

expect  that  Vio,  which  is  3.1623  .  • .,  would  often  have  been 
given  as  the  value  of  TT,  and  this  was  in  fact  the  case. 

1  Heath,  Archimedes,  p.  231-233 ;  ed.  Heiberg,  I,  258;  Heath,  History,  II,  50-56. 

2  De  Architecture  X,  cap.  14.    Rose's  edition  (1889)  gives  the  diameter  as 
4  ^  feet,  which  would  make  IT  less  than  3.   See  Bibl.  Math.>  I  (3),  298. 


*  AMU  US 

Ptolemy  (c.  150)  seems  to  have  taken  the  Archimedean  limits 
and  to  have  expressed  them  in  sexagesimals,  obtaining  sub- 
stantially 3|  =38'  34-28"  and  3^  =  38'  27.04".  He  then  im- 
proved upon  the  mean  between  these  results  by  taking  3  8'  30" 
as  the  approximate  value  of  IT,  although  a  still  closer  approxi- 
mation is  3  8'  29.73355".  Since  3  8'  30"  =  3.1416,  his  result 
was  very  satisfactory.1 

Hindu  Values  of  TT.  The  Hindu  mathematicians  took  various 
values  of  TT,  and  no  writer  among  them  seems  to  have  been  uni- 
form in  his  usage. 

Aryabhata  (c.  510),  or  possibly  Aryabhata  the  Younger, 
gave  the  equivalent  of  3.1416,  his  rule  being: 

Add  4  to  100,  multiply  by  8,  add  62,000,  and  you  have  for  a 
diameter  of  tvtoayutdsthe  approximate  value  of  the  circumference.2 

Brahmagupta  (c.  628)  criticized  Aryabhata  for  taking  the 
circumference  as  3393  for  both  diameters  1080  and  1050,  which 
would  make  TT  either  3^^  or  3^Vfr>  that  is,  3.1416  or  3.2314. 

A  certain  astronomer,  Pulisa,3  to  whom  Brahmagupta  refers, 
gives  3TVsV>  which  is  3.18+,  and  Ya'qub  ibn  Tariq  (c.  775) 
mentions  certain  Hindu  astronomical  measurements  which  give 
the  same  value.  He  also  states  that  Pulisa  used  a  value  equiva- 
lent to  3.14183,  and  Brahmagupta  a  value  equivalent  to  3.162. 
For  himself  Ya'qub  ibn  Tariq  used  in  one  case  a  value  equiva- 
lent to  3. 141 1.4 

In  case  the  value  3.1416  is  due  to  either  of  the  Aryabhatas, 
it  may  have  been  obtained  from  the  Alexandrian  scholars, 

1  Heath,  History,  I,  233. 

2L.  Rodet,  "Lec.ons  de  Calcul  d'Aryabhata,"  Journal  Asiatique  as  cited,  re- 
print, p.  ii.  There  is  some  doubt  as  to  whether  this  rule  is  due  to  either  of 
the  Sryabhatas;  see  G.  R.  Kaye,  "Notes  on  Indian  Mathematics,  No.  2, 
Aryabhata,"  in  Journ.  and  Proc.  of  the  Asiatic  Soc.  of  Bengal,  IV,  reprint. 
The  word  ayutds  means  myriads,  that  is,  io,ooo's.  The  rule  is  translated  more 
simply  in  Volume  I,  page  156. 

3  The  name  appears  as  Paulisa,  Pulisa,  and  Paulisa.  For  a  discussion,  see 
Sachau's  translation  of  Alberuni's  India,  II,  304  (London,  1910) ;  hereafter  re- 
ferred to  as  Alberuni's  India.  On  Brahmagupta's  criticism  of  Xryabhata,  see 
I,  1 68.  Nothing  is  known  concerning  the  life  of  Pulisa. 

*  Alberuni's /ndfa,  II,  67. 


ORIENTAL  VALUES  OF  U  309 

by  whom  it  was  then  known  and  whose  works  may  well  have 
reached  India,  or  it  may  have  been  found  independently. 

Brahmagupta1  (c.  628)  used  3  as  the  "practical"  value  and 
Vio  as  the  "exact"  value,  and  these  values  are  also  given  by 
Mahavira2  (c.  850)  and  Sridhara3  (c.  1020). 

Bhaskara  (c.  1150)  used  f|f£  for  the  "near"  value  and  -2T2- 
in  finding  the  "gross  circumference  adapted  to  practice,"4  the 
former  being  the  same  as  the  value  3tW<5'  °f  Pulis'a.5 

Chinese  Values  of  IT.  The  Chinese  found  various  values  of  TT, 
but  the  methods  employed  by  the  early  calculators  are  un- 
known. The  value  3  was  used  probably  as  early  as  the  i2th 
century  B.C.6  and  is  given  in  the  Chou-pei  and  the  Nine  Sec- 
tions.7 Ch'ang  Hong  (c.  125)  used  VTo,  and  Wang  Fan 
(c.  265)  used  -\\2-,  which  is  equivalent  to  3.1555  •  •  •.  Liu  Hui 
(263)  gives  us  the  first  intimation  of  the  method  used  by  the 
Chinese  in  finding  the  value.  He  begins  with  a  regular  in- 
scribed hexagon,  doubles  the  number  of  sides  repeatedly,  and 
asserts  that  "if  we  proceed  until  we  can  no  more  continue  the 
process  of  doubling,  the  perimeter  ultimately  comes  to  coincide 
with  the  circle."8 

Among  other  early  Chinese  values  of  no  high  degree  of  accu- 
racy are  those  of  Men  (c.  575),  who  gave  3.14,  and  Wu 
(c.  450),  whose  value  was  3.1432  -K 

Tsu  Ch'ung-chih  (c.  470)  was  able,  by  starting  with  a  circle 
of  diameter  10  feet,  to  obtain  3.1415927  and  3.145926  for  the 
limits  of  TT,  and  from  these,  by  interpolation,  he  obtained  the 
"accurate  and  inaccurate"  values  ^||  and  -2T2-.  No  closer  ap- 
proximations were  made  in  China  until  modern  times.9 

1  Colebrooke's  translation,  p.  308.  2  Mahavira,  pp.  189,  200. 

8  Colebrooke's  translation,  p.  87. 

4  Colebrooke's  translation,  p.  87. 

5  On  the  general  subject  of  the  Hindu  quadratures  see  C.  M.  Whish,  On  the 
Hindu  Quadrature  of  the  Circle,  a  paper  read  before  the  Madras  Literary  Society, 
December  15,  1832.  6Mikami,  China,  p.  46. 

7  For  discussion  of  the  dates  of  these  works,  see  Volume  I,  page  31. 
8 For  his  computations,  see  Mikami,  China,  p.  48. 

9  On  the  work  of  the  later  writers,  after  the  introduction  of  European  mathe- 
matics, see  Mikami,  China,  p.  135.   In  none  of  these  early  approximations  was 
the  decimal  fraction  used. 

ii 


3io  THE  THREE  FAMOUS  PROBLEMS 

The  Japanese  did  no  noteworthy  work  in  this  field  until  the 
iyth  century.  They  then  developed  a  kind  of  native  calculus 
and  also  made  use  of  European  methods  which  gave  them  fair 
approximations  to  the  required  ratio.1 

Later  Approximations  of  w.  The  following  is  a  brief  summary 
of  some  of  the  later  European  approximations  of  TT,  with  the 
names  of  those  who  used  them : 

Franco  of  Liege2  (c.  1066),  7r  =  -2Y2-=  3.142857-}-. 

Fibonacci3  (1220),  TT  =  |f|  =  3.141818.  He  also  gave  the 
limits  as  3.1427  and  3.1410. 

Al-Kashi1  (c.  1430),  3-1415926535898732. 

88 
Tycho  Brahe5  (c.  1580),  ?r  = — -—  =  3.1409. 

V785 

Simon  Duchesne6  (c.  1583),  TT  ^  3^ -(11)^3.14256198. 

Vieta  (c.  1593),  3-I4i5926s35<7r<3. 1415926537. 

Adriaen  van  Roomen  (1561-1615)  gave  TT  to  17  decimal 
places. 

Ludolf  van  Ceulen  (1540-1610)  gave  TT  to  35  decimal  places, 
and  German  textbooks  still  speak  of  TT  as  the  "  Ludolphische 
Zahl." 

Adriaen  Anthoniszoon  (c.  1600)  and  his  son  Adriaen  Metius 
(1571-1635),  TT  =  f ff ,  the  Chinese  value.7 

J.  H.  Lambert8  (c.  1770), 

—  _  /  7  \2      /16\'2     /62\2      /39V-'     /218\'2     /296\2     ... 
™  —  \\)  J    \T9  /  >    V35/  »    V22/  '    U23/  »    \1~6T'  »  • 

iSmith-Mikami,  pp.  60,  63,  et  passim. 

2  Abhandlungen,  IV,  139.  The  decimal  equivalents  are  modern  in  all  these  cases. 

3"Practica  geometriae,"  in  his  Scrittij  II,  90.  See  also  H.  Weissenborn,  "Die 
Berechnung  des  Kreisumfanges  bei  Archimedes  und  Leonardo  Pisano,"  Berliner 
Studien  fur  klassische  Philologie  und  Archdologie,  XIV. 

4 See  Volume  I,  pages  289,  290;  Volume  II,  pages  238,  240. 

5  Original  name,  Tyge  Ottesen.  It  is  not  known  how  he  came  to  givelhis  curi- 
ous value.  See  F.  J.  Studnicka,  Bericht  uber  die  Astrologischen  Studien  des  .  .  . 
Tycho  Brake,  p.  49  (Prag,  1901).  <JCantor,  Geschichte,  II  (2),  592. 

7  They  took  the  approximation  3TVV<7;r<3TV^'    added    the    numerators 
(15  +  17=32)   and  the  denominators   (106+120  —  226),  took  the  means   (16 
and  113),  and  gatfe  TT~^^  =  :\'^'^  ~  3-I4IS929>  a  very  close  approximation 
for  the  time.   Priority  for  this  is  claimed  for  Valentinus  Otto  (c.  1550-1605). 

8  Vorldufige  Kenntnisse  fur  die,  so  die  Quadratur  und  Rektifikation  des  Circuls 
suchen,  II,  140  (Berlin,  1772). 


LATER  APPROXIMATIONS  OF  II  311 

The  value  of  TT  was  carried  to  140  decimal  places  (136  cor- 
rect) by  Georg  Vega  (1756-1802),  to  200  by  Zacharias  Dase 
(1824-1861),  to  500  by  Richter  (died  in  1854),  and  to  707  by 
William  Shanks  (c.  1853). 

Continued  Products  and  Series.  Vieta  (c.  1593)  gave  an- 
other interesting  approximation  for  TT,  using  continued  products 
for  the  purpose.  His  value  may  be  obtained  from  the  following 
equation : 

7T  2  " 

John  Wallis  (I65S)1  gave  the  form 


TT      2-4-4.6.6.8-8.IO.IO.I2..- 

This  is  related  to  Lord  Brouncker's  value  (c.  1658)  in  which 
use  is  made  of  continued  fractions,  as  follows : 

4      T+     T 

—  =  i  -f-  - 


7T  Q 

2  +  -  25 

2+— -*— 
4Q 


Leibniz3  (1673), 

4~    ~3      5~~7     9 
Abraham  Sharp  (c.  1717), 


.      _  _ ,  _. .. .  +  __..  _  . 

6         :s    V       3-3      32- 5      38-7      34-9 

from  which  he  found  the  value  of  TT  to  72  decimal  places. 

1Arithmetica  Infinitorum  (1655),  included  in  his  Opera,  I,  469. 

2See  L.  Euler,  Opuscula  analytica,  Vol.  I  (Petrograd,  1783);  also  (1785), 
II,  149. 

8 A  special  case  of  Gregory's  (1671)  series,  Pe  Lagny  (1682)  discovered  it 
independently. 


312  THE  THREE  FAMOUS  PROBLEMS 

John  Machin1  (c.  1706), 


__    _ 
3-53      5-56 


239      3  •  239"      5  •  2395     7  •  239' 
Matsunaga  Ryohitsu2  (1739),  a  Japanese  writer, 


4-6      4.6-8.10      4  •  6  -  8  •  10  •  12  •  14 


, 


*> 
The  Symbol  TT.  The  symbol  -  was  used  by  Oughtred  (1647; 

7T 

to  represent  the  ratio  of  the  diameter  to  the  circumference.8 
Isaac  Barrow  (from  c.  1664)  used  the  same  symbolism,  and 

77" 

David  Gregory  (1697)  used  —  for  the  ratio  of  circumference 
to  radius.  ^ 

The  first  to  use  TT  definitely  to  stand  for  the  ratio  of  c  to  d 
was  an  English  writer,  William  Jones.  In  his  Synopsis  Pal- 
mariorum  Matheseos  (1706)  he  speaks  (p.  243)  of  "Periphery 
(TT)";  but  on  p.  263  he  is  more  definite,  giving 


and 

Euler  adopted  the  symbol  in  1737,  and  since  that  time  it  has 
been  in  general  use. 

1W.  Jones,  Synopsis  Palmariorum  Matkeseos,  p.  243  (London,  1706),  gives 
IT  "True  to  above  a  100  Places;  as  Computed  by  the  Accurate  and  Ready  Pen 
of  the  Truly  Ingenious  Mr.  John  Machin." 

2 For  a  further  list  of  values  of  TT  consult  D.  E.  Smith,  "The  History  and 
Transcendence  of  ir"  on  page  396  of  J.  W.  A.  Young,  Monographs  on  .  .  . 
Modern  Mathematics  (New  York,  1911);  Tropfke,  Geschichte,  IV  (2),  195; 
F.  Rudio,  Archimedes,  Huygens,  Lambert,  Legendre.  Vier  Abhandlungen  tiber 
die  Kreismessung,  Leipzig,  1892. 

3"Si  in  circulo  sit  7.22  ::  d-  TT  ::  113.355:  erit  8-7T  ::  2R.P:  periph."  This 
symbolism  appears  first  in  the  1647  edition  of  the  Clavis  Mathematicae  (1631), 
This  quotation  is  from  the  1652  edition.  See  Cajori,  Oughtred,  p.  32. 


DUPLICATION  OF  THE  CUBE  313 

The  proof  of  the  transcendence  of  TT  was  first  given  by  F. 
Lindemann  (1882),  thus  showing  the  impossibility  of  squaring 
the  circle  by  the  use  of  ruler  and  compasses  alone.1 

Duplication  of  the  Cube.2  Hippocrates  of  Chios  (c. 460 B.C.) 
showed  that  the  problem  of  duplicating  the  cube  resolves  itself 
into  the  finding  of  two  mean  proportionals  between  two  given 
lines.  If  a:x~  x:y  =  y:b,  then  x2  =  ay,  y2  =  bx,  and  hence 
x4  =  a2y2  =  d2bx,  or  x3  =  a2b.  If  b  =  2  a,  then  x*  =  2  a3.  That 
is,  the  cube  of  edge  x  will  then  have  double  the  volume  of  a 
given  cube  with  edge  a.  Since  we  have  the  three  equations 
x2  =  ay  (parabola),  y2  =  bx  (parabola),  and  ab  =  xy  (hyper- 
bola), we  can  evidently  solve  the  problem  by  finding  the  inter- 
section of  two  parabolas  or  of  a  parabola  and  a  hyperbola. 
These  methods  are  credited  to  Menaechmus  (c.  350  B.C.).3 

Archytas  (c.  400  B.C.)  had  already  found  the  two  mean  pro- 
portionals, solving  the  problem  by  means  of  two  cylindric  sec- 
tions, for  Eratosthenes  (c.  230  B.C.)  tells  us  that  he  "is  said  to 
have  discovered  them  by  means  of  his  semicylinders."4  It  is 
possible  that  Archytas  led  Mensechmus  to  discover  a  solution 
by  means  of  conies. 

Eratosthenes  also  tells  us  that  Eudoxus  (c.  370  B.C.)  solved 
the  problem  "by  means  of  the  so-called  curved  lines,"  but 
what  these  lines  were  we  do  not  know.5  The  two  statements 
here  attributed  to  Eratosthenes  are  contained  in  a  letter 
formerly  (but  incorrectly)  credited  to  him.  In  the  main,  how- 
ever, this  letter  sets  forth  facts  with  which  he  was  familiar,  as 
is  shown  from  other  sources. 

aD.  E.  Smith,  "The  History  and  Transcendence  of  TT,"  loc.  tit.,  p.  387;  con- 
sult this  work  also  with  respect  to  transcendental  numbers  in  general. 

2J.  S.  Mackay,  Proc.  of  the  Edinburgh  Math.  Soc.,  IV,  2 ;  F.  G.  Teixeira, 
Obras  sobre  Mathematica,  VII,  283-415  (Coimbra,  1915)  ;  C.  H.  Biering,  Historia 
Problematis  Cubi  Duplicandi,  Copenhagen,  1844;  Heath,  History,  I,  244;  F.  En- 
riques,  Fragen  der  Elementar-Geometrie,  II.  Teil  (Leipzig,  1907). 

3 On  this  point  see  Heath,  History,  I,  251-255;  Allman,  Greek  Geom.,  p.  160. 

4  In  his  letter  to  Ptolemy  III.  See  Archimedes,  Opera,  ed.  Heiberg,  III,  104, 
106.  On  the  solution,  as  Eudemus  relates  it,  see  Allman,  Greek  Geom.,  p.  in. 
See  also  P.  Tannery,  "Sur  les  solutions  du  probleme  de  Delos  par  Archytas  et 
par  Eudoxe,"  in  his  Mlmoires  Scientifiques,  I,  53  (Paris,  1912) ;  Heath,  History, 
I,  246.  5  Archimedes,  Opera,  ed.  Heiberg,  III,  66. 


314 


THE  THREE  FAMOUS  PROBLEMS 


Plato  (c.  380  B.C.)  is  said  to  have  solved  the  problem  by 
means  of  a  mechanical  instrument1  but  to  have  rejected  this 
method  as  not  being  geometric. 

We  are  told  by  Joannes  Philop'onus2  that  Apollonius  (c.  225 
B.C.)  had  a  method  of  finding  the  two  mean  proportionals.  The 
construction,  however,  assumes  a  postulate  which  begs  the 
whole  question.3 

Cissoid  of  Diocles.  One  of  the  best-known  of  the  ancient 
solutions  was  that  of  Diocles  (c.  180  B.C.),  who  used  a  curve 
known  as  the  cissoid  (from  fcio-aoeiSift  (kissoeides'),  ivylike). 
In  this  figure,  if  OL  =  OT  and  A  Q  is  drawn,  then  P  is  a  point 

on  the  curve.  Similarly,  AS 
will  determine  on  TQ  pro- 
duced a  point  P  on  the  curve. 
The  cissoid  evidently  passes 
through  A,  BR  is  an  asymp- 
tote, and  the  curve  is  symmet- 
ric with  respect  to  AJ3. 

By  the  aid  of  the  curve 
two  mean  proportionals  can  be 
found  in  the  following  manner : 4 
Let  OM=  I  r,  determine  F 
by  producing  BM  to  the  curve, 
and  draw  AP  and  produce  it 
to  R,  letting  it  cut  the  circle 
at  Q. 

Through  P  and  Q  respectively  draw  SL  and  QT  perpen- 
dicular to  AB. 

Let  BL=*a,  SL^y,  AL 
a__^BO 
PL 


x. 


OM 


—  —  =  2  ;  whence  J  a  =  PL. 
I  r 


figure  and  description,  based  on  a  statement  of  Eutocius  (0.560),  see 
Tropfke,  Geschichte,  II  (i),  42;  Allman,  Greek  Geom.,p.  1 73 ;  Archimedes, Opera, 
ed.  Heiberg,  III,  66;  Heath,  History,  I,  255. 

*'lw6,wr}*  6  <£iX67roi/os  (loan'nes  ho  Philop'onos),  also  known  as  6  rpa/x^an^s 
(ho  Grammatikos'),  an  Alexandrian  scholar  of  the  7th  century;  not  a  very  reli- 
able source.  3The  proof  appears  in  Heath,  Apollonius,  p.  cxxv. 

4 For  a  somewhat  different  proof  see  Heath,  History,  I,  264. 


DUPLICATION  OF  THE  CUBE  315 

Because  A  ALP  is  similar  to  AATQ,  which  is  congruent 
to  &J3LS,  which  is  similar  to  AALS7  we  have 

a     y  __   x  _  x 
' 


We  therefore  have  two  mean  proportionals  between  a  and  |  a. 
Hence  ax=y*t  a~x*  =  y*, 


^,4 

and 


,-*.  y 


Also,  7}  ay  =  jr2  =  ~-2  ; 

whence  I  #*  =  !'* 

and  tf8=2/. 


Therefore  in  the  above  figure  we  simply  have  to  make 
PL  =  4-0,  which  we  get  by  having  made  OM—  \  r.  Then  SL  is 
the  side  of  a  cube  equal  to  -|-a3,  or  a3  =  2y\  Hence 


SL 


side  of  given  cube      side  of  required  cube 

Later  Methods.  Several  modern  writers  have  suggested 
methods  for  duplicating  the  cube.  Among  these  are  Vieta,1 
Descartes,2  Fermat,3  de  Sluze,4  and  Newton.5  Descartes  con- 
sidered not  only  the  question  of  finding  two  mean  propor- 
tionals, as  required  in  solving  the  problem,  but  also  that  of 
finding  four;  and  Fermat0  went  so  far  as  to  consider  certain 
cases  involving  n  mean  proportionals,  a  line  of  work  which  was 
later  followed  by  Clairaut. 

Viviani7  solved  the  problem  by  the  aid  of  a  hyperbola  of  the 
second  order  (xy2  —  a3).  Huygens  (1654)  gave  three  methods 

1  Opera  Mathematica,  ed.  Van  Schooten,  p.  242.   Leyden,  1646. 

2  Suggested  in  La  Geometric,  Book  III. 

3  In  his  memoir  Ad  locos  pianos  et  solidos  hagoge,  written  before  Descartes 
published  his  work,  but  not  made  known  until  after  Fermat's  death. 

4 In  his  Mesolabium,  1668. 

5Arithmetica  Universalis,  1707,  p.  309.  G(Euvres,  I,  118. 

7  Quint  o  libro  di  Euclide  o  Scienze  universale  delle  proposizioni  spiegate  colla 
dottrina  del  Galileo  (Florence,  1647). 


316  ANALYTIC  GEOMETRY 

of  solving.  Newton  (1707)  suggested  several  methods  but  pre- 
ferred one  which  made  use  of  the  limagon  of  Pascal.  One  of 
the  comparatively  recent  methods  is  that  employed  by  Mon- 
tucci,1  who  made  use  of  the  curve  defined  by  the  equation 

y  =  V#  .r  +  v ax  —  ^. 

8.  ANALYTIC  GEOMETRY 

Three  Principal  Steps.  There  are  three  principal  steps  in  the 
development  of  analytic  geometry :  ( i )  the  invention  of  a  sys- 
tem of  coordinates;  (2)  the  recognition  of  a  one-to-one  corre- 
spondence between  algebra  and  geometry;  and  (3)  the  graphic 
representation  of  the  expression  y  =  /(#)•  Of  these,  the  first 
is  ancient,  the  second  is  medieval,  and  the  third  is  modern. 

Ancient  Idea  of  Coordinates.  The  idea  of  coordinates  in  the 
laying  out  of  towns  and  lands  seems  unquestionably  to  have 
occurred  to  the  Egyptian  surveyors.  It  is  to 
them  that  Heron  was  apparently  indebted  for 
his  fundamental  principles,  and  from  them  the 
Roman  surveyors  acquired  their  first  knowledge 
of  the  science.  Indeed,  the  districts  (hesp)  into 
which  Egypt  was  divided2  were  designated  in 
hieroglyphics  by  a  symbol  derived  from  a  grid,  as  here  shown, 
quite  as  we  designate  a  survey  today.3 

Latitude  and  Longitude.  The  first  definite  literary  refer- 
ences to  the  subject  appear,  however,  in  the  works  of  the  early 
Greek  geographers  and  astronomers.  Hipparchus  (c.  140  B.C.) 
located  points  in  the  heavens  and  on  the  earth's  surface  by 
means  of  their  longitude4  and  latitude,5  the  former  being  reck- 

1  Resolution  de  I' Equation  du  cinquieme  degre,  Paris,  1869. 

2 Known  to  the  Greeks  as  vo^oL  (nomoi ',  nomes)  and  to  Pliny  as  praefecturae 
oppidorum. 

3E.  W.  Budge,  The  Mummy,  p.  8  (Cambridge,  1893).  See  also  Cantor,  Ge- 
schichte,  I  (2),  67. 

4M77/cos  (me'kos,  length;  Latin,  longitudo),  i.e.,  distance  from  east  to  west;  so 
called  because  the  length  of  the  known  world  was  along  the  Mediterranean  Sea. 

5n\dros  (pla'tos,  width;  Latin,  latitude*),  i.e.,  distance  from  north  to  south; 
so  called  because  the  width  of  the  known  world  was  north  and  south. 


GREEK  CONTRIBUTIONS  317 

oned  from  the  meridian  of  Rhodes,  where  Hipparchus  took  his 
observations.  He  also  located  the  stars  by  means  of  coordinates. 
In  the  second  century  Marinus  of  Tyre  (c>  150)  took  his 
prime  meridian  through  the  Fortunatae  Insulae?  and  perhaps 
through  the  most  western  point,  this  being  the  end  of  the  earth 
as  then  known,  and  Ptolemy  (c.  150)  used  the  same  line.2 

Ancient  Surveyors.  The  ancient  surveyors  located  points  in 
much  the  same  way  as  the  geographers.  Heron  (c.  50?),  ap- 
parently following  the  Egyptian  surveyors,  laid  out  a  field  with 
respect  to  one  axis  quite  as  we  do  at  present,  although,  strictly 
speaking,  two  coordinates  are  used.8- 

The  Romans  brought  the  science  of  surveying  to  the  highest 
point  attained  in  ancient  times.4  They  laid  out  their  towns  with 
respect  to  two  axes,  the  decimanus,  which  was  usually  from 
east  to  west,  and  the  car 'do ,  an  axis  perpendicular  to  the  deci- 
manus. They  then  arranged  the  streets  on  a  rectangular  co- 
ordinate system,  much  as  in  most  American  cities  laid  out  in 
the  i  gth  century.5 

Rectangular  Axes  in  Greek  Geometry.  In  their  treatment  of 
geometric  figures  the  Greeks  made  use  of  what  were  substan- 
tially two  rectangular  axes.  Menaechmus  (c.  350  B.C.),  for 
example,  may  have  used  that  property  of  the  parabola  expressed 
by  the  equation  y2  —  px,  and  also  that  property  of  the  rectangu- 
lar hyperbola  expressed  by  the  equation  xy  =  c2.  Archimedes 
(c.  22$  B.C.),  who  no  doubt  was  indebted  to  the  lost  work  of 
Euclid  on  conies  in  general,  used  the  same  relation  for  the  pa- 
rabola, his  results  being  expressed  as  usual  in  the  form  of  a 
proportion.0 

1  Probably,  as  stated  in  Volume  I,  including  the  Canary,  Madeira,  and  Azores 
groups. 

2Halma's  edition  of  Ptolemy,  VI,  17  (Paris,  1828). 

3  See  his  Opera  quae  supersunt  omnia,  V,  5  (Leipzig,  1899-1914),  on  his 
stereometry  and  mensuration. 

4M.  Cantor,  Die  romischen  Agrimensoren,  Leipzig,  1875. 

5  As  Frontinus  states  it  in  his  Liber  I,  "ager  .  .  .  decimanis  et  cardinibus  con- 
tinetur";  "Ager  per  strigas  [rows!  et  per  scamna  [steps]  diuisus."  He  also  used 
oblique  coordinates.  See  the  Lachmann  and  Rudorff  edition  for  diagrams. 

60n  all  this  discussion  see  Heath,  Apollonius,  p.  cxv  seq.;  History,  II,  122. 


318  ANALYTIC  GEOMETRY 

Apollonius  carried  the  method  much  farther,  as  may  be  seen 
by  the  following  statement : 

If  straight  lines  are  drawn  from  a  point  so  as  to  meet  at  given 
angles  two  straight  lines  given  in  position,  and  if  the  former  lines  are 
in  a  given  ratio,  or  if  the  sum  of  one  of  them  and  of  such  a  line  as 
bears  a  given  ratio  to  the  second  is  given,  then  the  point  will  be  on  a 
given  straight  line.1 

This  is  only  a  nonsymbolic  method  of  stating  that  the  equa- 
tion x  =  ay  —  b  represents  a  straight  line,  a  and  b  being  positive. 

Sir  Thomas  Heath  calls  attention  to  another  essential  dif- 
ference between  the  Apollonian  and  Cartesian  points  of  view: 

The  essential  difference  between  the  Greek  and  the  modern  method 
is  that  the  Greeks  did  not  direct  their  efforts  to  making  the  fixed 
lines  of  the  figure  as  few  as  possible,  but  rather  to  expressing  their 
equations  between  areas  in  as  short  and  simple  a  form  as  possible. 
Accordingly  they  did  not  hesitate  to  use  a  number  of  auxiliary  fixed 
lines,  provided  only  that  by  that  means  the  areas  corresponding  to 
the  various  terms  in  #2,  xy,  .  .  .  forming  the  Cartesian  equation 
could  be  brought  together  and  combined  into  a  smaller  number  of 
terms.  ...  In  the  case,  then,  where  two  auxiliary  lines  are  used  in 
addition  to  the  original  axes  of  coordinates,  and  it  appears  that  the 
properties  of  the  conic  (in  the  form  of  equations  between  areas)  can 
be  equally  well  expressed  relatively  to  the  two  auxiliary  lines  and  to 
the  two  original  axes  of  reference,  we  have  clearly  what  amounts  to 
a  transformation  of  coordinates.2 

Ordinate  and  Abscissa.  As  to  technical  terms,  the  Greeks  used 
an  equivalent  of  "ordinate."3  For  "abscissa"4  they  used  such 
expressions  as  "the  [portion]  cut  off  by  it  from  the  diameter 
towards  the  vertex."  Apollonius  uses  the  word  "asymptote,"5 
but  the  word  had  a  broader  meaning  than  with  us,  referring  to 
any  lines  which  do  not  meet,  in  whatever  direction  they  are 

1  On  the  work  of  Apollonius  see  Volume  I,  page  116;  Heath,  Apollonius, 
2 Heath,  Apollonius,  p.  cxviii.    See  also  his  History,  II,  126-106. 
8 That  is,  Tera.yit.4vM  (tetagmen'os,  ordinate-wise).    The  same  term  is  used  for 
the  tangent  at  the  extremity  of  a  diameter.    Ibid.,  p.  clxii. 
4Latin  ab  +  scissa,  from  ab  (off)  -j-  scindere  (to  cut). 

s  (asym'ptotos,  from  d  privative  +  <rtfi>,  together,  -f  irTwrfa,  falling) . 


COORDINATES  319 

produced.  The  names  "ellipse,"  "parabola,"  and  "hyperbola" 
are  probably  due  to  Apollonius,1  although  two  of  them  are 
found  in  late  manuscripts  of  the  works  of  Archimedes. 

Oresme's  Contribution.  In  the  Middle  Ages  Nicole  Oresme 
(c.  1360)  wrote  two  works2  in  which  he  took  a  decided  step  in 
advance.  He  considered  a  series  of  points  which  have  uni- 
formly changing  longitudines  and  latitudines,  the  first  being 
our  abscissas  and  the  second  our  ordinates.3  The  series  of 
points  determined  by  the  ends  of  the  latitudines  was  called  a 
forma,  and  the  difference  between  two  successive  latitudines 
was  called  a  gradus.  If  the  latitudines  are  constant,  the  series 
of  points  was  described  as  uniformis  eiusdem  gradus ;  but  if 
the  latitudines  varied,  the  forma  of  the  series  of  points  was 
difformis  per  oppositum.  The  difference  between  two  succes- 
sive latitudines  was  the  excessus  graduum,  and  this  might  or 
might  not  be  constant.  In  the  former  case  the  forma  was  uni- 
f  or  miter  difformis ;  in  the  latter  case,  diff  or  miter  difformis.  The 
formae  considered  were  series  of  points  arranged  in  rectilinear, 
circular,  or  parabolic  order.  Of  course  only  positive  latitudines 
were  considered.  Here,  then,  we  find  the  first  decided  step  in 
the  development  of  a  coordinate  system,  apart  from  the  locat- 
ing of  points  on  a  map  of  some  kind ;  but  we  also  find  a  lack 
of  any  idea  of  continuity  in  the  point  systems.  The  method 
was  the  subject  of  university  lectures  at  Cologne  as  early  as 
1398,*  as  witness  the  statutes  of  that  period:"  Kepler  and 
Galileo  recognized  its  value,  and  the  former  was  influenced  to 

1  Heath,  Apollonius,  p.  clxiii;  History,  II,  138. 

2  Tractatus  de  latitudinibus  formarum  and  Tractatus  de  uniformitate  et  dif- 
formitate  intensionum.    See  the  Zeitschrift   (HI.  Abt.),  XIII,  92.    The  first  of 
these  tractates  was  printed  at  Padua  in  1482  and  again  in  1486,  in  Venice  in  1505, 
and  in  Vienna  in  1515. 

•'Tropfke,  Geschichte,  II  (i),  409;  H.  Wieleitner,  Bibl.  Math.,  XIV  (3),  2:0. 

4Hankel,  Geschichte,  p.  351. 

5 "Item  statuimus  quod  Bacalarius  temptandus  debet  audivisse  libros  infra- 
scriptos  .  .  .  aliquem  tractatum  de  latitudinibus  formarum."  See  F.  J.  von 
Bianco,  Die  alte  Universitat  Koln,  I,  Anlagen,  p.  68  (Cologne,  1885) ;  S.  Gunther, 
"Die  Anfange  und  Entwickelungsstadien  des  Coordinatenprincipes,"  Abhand- 
lungen  d.  naturf.  Gesellsch.  zu  Ntirnberg,  VI  (1877);  reprint,  p.  16;  hereafter 
referred  to  as  Gunther,  Die  Anfange. 


320  ANALYTIC  GEOMETRY 

make  much  use  of  it  in  his  astronomical  work.1  Indeed,  the 
use  of  a  kind  of  coordinate  paper  for  the  graphic  representation 
of  the  course  of  the  planets  is  found  much  earlier  even  than 
Oresme,  for  Giinther2  has  called  attention  to  a  manuscript  of 
the  loth  century3  in  which  the  graphs  closely  resemble  similar 
forms  of  the  present  day. 

Relation  of  Algebra  to  Geometry.  The  second  step  in  the  de- 
velopment of  analytic  geometry  has  to  do  with  the  relation  of 
algebra  to  geometry.  If  we  consider  such  a  proposition  of 
Euclid  (c.  300  B.C.)  as  the  one  relating  to  the  square  on  the 
sum  of  two  lines,4  we  see  that  it  is  the  analogue  of  the  algebraic 
identity  (a  -f-  b)2  =  a2  +  2ab  +  b2.  Euclid,  however,  had  no 
algebraic  symbolism,  and  while  of  course  he  recognized  the 
analogy  to  the  square  of  the  sum  of  two  numbers,  it  cannot  be 
said  that  he  related  algebra  to  geometry  in  the  way  that  we  do 
with  our  modern  symbols. 

When  Archimedes  (c.  225  B.C.),  Heron  (c.  50?),  and  Theon 
of  Alexandria  (c.  390)  found  square  roots,  they  used  this  propo- 
sition of  Euclid;  but,  again,  they  can  hardly  be  said  to  have 
grasped  the  relation  that  is  so  familiar  to  us  today. 

It  is  among  the  Arab  and  Persian  writers  that  we  first  find 
geometric  figures  used  in  works  devoted  solely  to  algebra.  Thus 
al-Khowarizmi  (c.  825)  considered  numerous  cases  such  as  the 
following:  "A  square  and  ten  Roots  are  equal 
to  thirty-nine  Dirhems."r>  Here  he  uses  the 
annexed  figure,  the  square  AB  having  for  its 
side  one  of  the  roots  of  the  given  equation 

.r2+io,r=39, 

a  favorite  equation  with  subsequent  writers. 
Omar  Khayyam  (c.  noo)  made  continued  use  of  geometric 
figures  in  his  work  on  algebra,6  thus  recognizing  the  one-to-one 

*0pera  omnia,  Frisch  ed.,  IV,  610  seq.  (Frankfort  a.  M.,  1863). 
2 Die  Anf tinge,  p.  19  and  Fig.  2  in  the  plates. 

3Munich  Cod.  Lat.  14,436:  "Macrobius  Boetius  in  Isagog.  Saec.  X." 
4Elements,  II,  4.  5  Rosen's  translation,  p.  13. 

QL'Algebre  d'Omar  Alkhayydmt,  translated  by  F.  Woepcke,  Paris,  1851.  So 
Woepcke  remarks:  "II  est  une  particularite  de  cette  algebre  qui  mdrite  d'etre 


RELATION  OF  ALGEBRA  TO  GEOMETRY          321 

correspondence  between  algebra  and  geometry  even  more  than 
al-Khowarizmi  had  done  before  him.1 

The  Hindu  algebraists  also  used  geometric  figures  in  their 
work.  For  example,  Bhaskara  (c.  1150) 2  has  such  problems  as 
this:  "Tell  two  numbers,  such,  that  the  sum  of  them,  multi- 
plied by  four  and  three,  may,  added  to  two,  be  equal  to  the 
product."  In  such  cases  he  gives  two  solutions,  one  algebraic 
and  the  other  geometric. 

Europeans  relate  Algebra  to  Geometry.  Among  the  Euro- 
peans, Fibonacci  was  the  first  mathematician  of  prominence 
who  recognized  the  value  of  relating  algebra  to  geometry.  In 
his  Practica  geometriae  (1220)  he  uses  algebra  in  solving  geo- 
metric problems  relating  to  the  area  of  a  triangle.3 

In  the  early  printed  books  there  was  more  or  less  use  of 
geometric  figures  in  connection  with  algebraic  work.  Thus 
Vieta  tells  us  that  Regiomontanus  solved  algebraically  problems 
which  he  could  not  solve  by  geometry.  Pacioli  (1494)  con- 
tinually uses  geometric  figures  in  his  solution  of  quadratics,4 
and  Cardan  (1545)  does  the  same.5  After  the  publication  of 
Cardan's  work  the  recognition  of  the  relationship  became  com- 
mon. Vieta,  for  example,  generalized  the  idea  of  the  ancients 
as  to  representing  points  on  a  line,  although  adhering  to  the 
use  of  proportion  in  most  of  his  geometric  work  instead  of 
using  the  equation  form. 

The  first  textbook  on  algebraic  geometry  was  that  of  Marino 
Ghetaldi  (i63o),6  who  may  have  been  influenced  by  Vieta.7 
In  his  solution  of  geometric  problems  he  freely  brought  algebra 

remarqu6e  et  discutee  d'abord.  C'est  que  1'auteur  se  fait  une  loi,  pour  toutes 
les  equations  dont  il  s'occupe,  de  joindre  la  resolution  numerique  ou  arithm6tique 
a  la  construction  geometrique  "  (Preface,  p.  vij ) . 

1  Rosen  ed.,  p.  13  and  elsewhere.  2Colebrooke  translation,  p.  270. 

3"Quare  quadratum  lateris  .eg.  erit  J-  rei;  et  multiplicabo  .cf.  in  dimidium  .eg., 
hoc  est  radicem  rei  in  radicem  ^  rei,  ueniet  radix  11.1  census.  .  .  ."  Scritti,  II,  223. 

4 E.g.,  Suma,  1494  ed.,  fol.  146,  v.,  et  passim. 

5  For  the  use  of  geometric  figures  in  his  first  solution  of  a  cubic,  see  Ars  Magna, 
fol.  29,  v. 

6 See  E.  Gelcich,  "Erne  Studie  iiber  die  Entdeckuhg  der  analytischen  Geo- 
metric mit  Beriicksichtigung  eines  Werkes  des  Marino  GhetaldL  .  .  ."  Abhand- 
lungen,  IV,  191.  7Tropfke,  Geschichte^  II  (i),  414- 


322  ANALYTIC  GEOMETRY 

to  his  aid,  but  it  cannot  be  said  that  he  in  any  way  anticipated 
the  work  of  the  makers  of  analytic  geometry. 

Invention  of  Analytic  Geometry.1  The  invention  of  analytic 
geometry  is  commonly  attributed  to  Descartes,  he  having  pub- 
lished (1637)  the  first  treatise  on  the  subject.  There  seems 
to  be  no  doubt,  however,  that  the  idea  occurred  to  Fermat  at 
about  the  same  time  as  to  Descartes,  and  to  have  occurred  to 
Harriot  even  earlier  (c.  1600).  In  the  British  Museum  there 
are  eight  volumes  of  Harriot's  manuscripts,  and  among  these 
may  be  found  "a  well-formed  analytical  geometry,  with  rec- 
tangular coordinates  and  a  recognition  of  the  equivalence  of 
equations  and  curves."2 

Fermat  on  Analytic  Geometry.  In  a  letter  to  Roberval,  written 
September  22,  1636,  and  hence  in  the  year  before  Descartes 
published  La  Geometric,  Fermat  shows  that  he  had  the  idea 
of  analytic  geometry  some  seven  years  earlier  ;3  that  is,  in  1629. 
The  details  of  this  work  appear  in  his  Isagoge  ad  locos  pianos 
et  solidos,  which  was  published  posthumously.4  He  used  rec- 
tangular axes  and  followed  Vieta  in  representing  the  unknowns 
by  vowels  (in  this  case  only  A  and  E)  and  the  knowns  by  con- 
sonants. A  general  point  on  the  curve  was  represented  by  /, 
and  the  foot  of  the  ordinate  from  /  to  the  axis  of  abscissas  was 
represented  by  Z.  The  equation  of  a  straight  line  through  the 
origin  was  indicated  by 

D  in  A  aequetur  B  in  E? 

1G.  Loria,  Passato-Presente  Geom.\  M.  Chasles,  Aper$u  historique  sur  Vorigine 
et  le  developpement  des  methodes  en  geometrie,  3d  ed.,  Paris,  1889  (hereafter  re- 
ferred to  as  Chasles,  Aper<;u) ;  M.  Chasles,  Rapport  sur  les  Pr ogres  de  la  Geo- 
metrie, Paris,  1870;  A.  Cayley  and  E.  B.  Elliott,  "Curve,"  Encyc.  Britannica, 
nth  ed.;  Giinther,  Die  Anfdnge',  E.  Picard,  Bulletin  of  the  Amer.  Math.  Soc., 
XI,  404;  H.  Wieleitner,  Zeitschrift  fur  math,  und  naturw.  Unterr.,  XLVII,  414; 
W.  Dieck,  Mathematisches  Lesebuch,  4.  Band,  Sterkrade,  1920. 

2F.  V.  Morley,  "Thomas  Hariot,"  The  Scientific  Monthly,  XIV,  63.  These 
manuscripts  should  be  carefully  studied.  The  spelling  "Hariot"  was  used  by  some 
of  his  contemporaries.  3".  .  .  il  y  a  environ  sept  ans  etant  a  Bourdeaux." 

4In  his  Varia  Opera,  p.  2  (Toulouse,  1679)  ;  Tropfke,  Geschichte,  II  (i),  418; 
Giinther,  Die  Anfdnge,  p.  43;  CEuvres  de  Fermat,  ed.  P.  Tannery  and  Ch.  Henry, 
Vol.  I  (Paris,  1891),  Vol.  II  (Paris,  1894). 

5/.e.?  D  •  A  ~  B  '  E)  which  we  should  write  as  ax  ==  by. 


FERMAT  AND  DESCARTES  323 

and  that  of  a  general  straight  line  was  given  by  the  proportion 

ut  B  ad  D, \\&R-A  ad £.1 
The  equation  of  a  circle  appears  as 

Bq.  —  Aq.  aequetur  Eq.2 

If  the  ratio  of  Bq.  —  Aq.  to  Eq.  is  constant,  Fermat  asserted 
that  the  resulting  figure  is  an  ellipse ; 3  and  if  the  ratio  of 
Bq.  +  Aq.  to  Eq.  is  constant,  the  figure  is  a  hyperbola.4  He 
also  knew  that  xy  =  a2  is  the  asymptotic  equation  of  a  hyper- 
bola5 and  that  x2  =  ay  is  the  equation  of  a  parabola.0 

Descartes  publishes  La  Geometric.  Descartes  published  his 
Geometry  in  1637,  although  he  had  been  working  upon  it  for 
some  years, — even  as  early  as  i6ig.7  The  treatise  formed  an 
appendix  to  his  Discours  de  la  Methode  and  was  divided  into 
three  books.  The  first  book  treats  of  the  meaning  of  the  prod- 
uct of  lines.8  The  second  book  defines  two  classes  of  curves,  the 
geometric  and  the  mechanic.  We  might  now  define  the  former 
as  curves  in  which  dy/dx  is  an  algebraic  function,  and  the  lat- 
ter as  curves  in  which  it  is  a  transcendental  function.  In  this 
book  there  is  also  much  attention  given  to  tangents  and  nor- 
mals to  a  curve.  The  third  book  is  largely  algebraic,  being 
entitled,  "On  the  construction  of  solid  or  hypersolid  prob- 
lems." It  treats  particularly  of  such  topics  as  the  number  of 
roots  of  an  equation,  "false  roots,"  the  increasing  or  decreasing 
of  the  roots,  and  the  transformation  of  equations. 

*Le.y  a:  b  =  c  —  x:  y,  or  ay  —  b  (c  —  x). 
2 I.e.,  B-  —  A2  =  E2,  or  r2  —  x2  -  y2. 

3  " Bq.  —  Aq.  ad  Eq.  habeat  rationem  datam,  punctum  /  erit  ad  ellipsin."  I.e., 
f2  _  X2  —  fcy2  is  the  equation  of  an  ellipse. 

4"Si  Bq.  +  Aq.  est  ad  Eq.  in  data  ratione,  punctum  /  est  ad  hyperbolen." 
5"^4  in  E  aeq.  Z  pi.,  quo  casu  punctum  7  est  ad  hyperbolen." 

6  "Si  Aq.  aequatur  D  in  £,  punctum  7  est  ad  parabolen." 

7  J.  Millet,  Descartes.   Sa  vie,  ses  travaux,  ses  d&couvertes,  avant  1637,  p.  100 
(Paris,  1867) ;  E.  S.  Haldane,  Descartes,  p.  59  (London,  1905) ;  C.  Rabuel,  Com- 
mentaires  sur  la  Geometric  de  M.  Descartes,  Lyons,  1730;  La  Geometric,  various 
editions  from  1637.  See  the  author's  facsimile  edition  with  translation  (Chicago, 
1925). 

8"Des  Problemes  qu'on  peut  construire  n'y  employant  que  des  cercles  &  des 
lignes  droites." 


324  ANALYTIC  GEOMETRY 

'  Instead  of  using  the  name  " coordinates/'  Descartes  spoke 
of  roots  or  unknowns.  The  name  " coordinate"  is  due  to 
Leibniz,1  as  are  also  the  terms  "  abscissa "  and  "ordinate,"  al- 
though, as  we  have  seen,  the  Greeks  used  terms  that  were  simi- 
lar to  them.  Newton,  Euler,  Cramer,  and  various  other  writers 
used  "applicate"  to  represent  an  ordinate. 

Descartes  had  an  idea  of  oblique  coordinates,  but  he  used 
only  the  #-axis  and  positive  perpendicular  ordinates  in  common 
practice. 

Later  Writers.  In  1658  Jan  (Johan)  de  Witt2  wrote  a  work 
on  curve  lines3  in  which  he  set  forth  a  number  of  typical  equa- 
tions and  gave  the  geometric  character  of  each. 

Further  work  was  done  by  Lahire,4  after  which  the  elements 
of  plane  analytic  geometry  may  be  considered  as  having  be- 
come established.  The  most  noteworthy  single  contributor  to 
the  elements  of  the  subject  thereafter  was  Newton.5  In  his 
work  on  cubic  curves  he  showed  that  a  cubic  has  at  least  one 
real  point  at  infinity,  that  any  cubic  belongs  to  one  of  four 
characteristic  types,  and  that  there  are  seventy-two  possible 
forms  of  a  cubic,  a  number  since  increased  by  six.6  The  dis- 
cussion of  the  subject  was  nearly  exhaustive,  and  was  the  most 
elaborate  one  of  the  kind  that  had  been  made  up  to  that  time. 

The  idea  of  polar  coordinates  seems  due  to  Gregorio  Fontana 
(1735-1803),  and  the  name  was  used  by  various  Italian  writers 
of  the  1 8th  century.7 

lActa  Eruditorum  (1692),  p.  170. 

2 Born  at  Dordrecht,  September  12/24,  1625;  died  at  The  Hague,  August  20, 
1672.  For  biography  see  The  Insurance  Cyclopaedia,  Vol.  II  (London,  1873). 

3  Element  a  Cvrvarvm  Linearvm,  Leyden,  1659;  Amsterdam,  1683. 

4Les  Lieux  Geomttriques,  Paris,  1679;  Construction  des  Equations  Analy- 
tiques,  Paris,  1679. 

5 In  his  Principia  (London,  1687)  and  his  Arithmetica  Universalis  (Cambridge, 
1707),  but  chiefly  in  his  Enumeratio  linearum  tertii  ordinis,  which  probably  dates 
from  1668  or  1669  and  which  was  published  as  an  appendix  to  his  Optics  in  1704. 
See  also  W.  W.  R.  Ball,  "On  Newton's  classification  of  cubic  curves,"  Transac- 
tions of  the  London  Mathematical  Society  (1891),  p.  104. 

6G.  Loria,  Ebene  Kurven,  Theorie  und  Geschichte,  p.  20  (Leipzig,  1902;  2d 
ed.,  1910-1911),  hereafter  referred  to  as  Loria,  Kurven. 

1  For  a  discussion  of  the  later  types  of  coordinates  see  the  Encyklopadie,  III, 
596,  656;  Cantor,  Geschichte,  IV,  513. 


LATER  DEVELOPMENTS  325 

Solid  Analytic  Geometry.  Descartes  clearly  mentioned  solid 
analytic  geometry,  but  he  did  not  elaborate  it.  Frans  van 
Schooten  the  younger  suggested  the  use  of  coordinates  in  three- 
dimensional  space  (1657),  and  Lahire  (1679)  also  had  it  in 
mind.  Jean  Bernoulli  (1698)  thought  of  equations  of  surfaces 
in  terms  of  three  coordinates,  but  published  nothing  upon  the 
theory  at  that  time. 

The  first  work  on  analytic  geometry  of  three  dimensions 
was  written  by  Antoine  Parent  and  was  presented  to  the  French 
Academic  in  i7oo.1  A.  C.  Clairaut  (1729)  was  the  first  to  write 
on  curves  of  double  curvature.2  The  third  great  contributor  to 
the  theory  was  Euler  (1748),  with  whose  work  the  subject  ad- 
vanced beyond  the  elementary  stage. 

Euler  also  laid  the  foundations  for  the  analytic  theory  of 
curvature  of  surfaces,  attempting  to  do  for  the  classification  of 
surfaces  of  the  second  degree  what  the  ancients  had  done  for 
curves  of  the  second  order.  Monge  introduced  the  notion  of 
families  of  surfaces  and  discovered  the  relation  between  the 
theory  of  surfaces  and  the  integration  of  partial  differential 
equations,  enabling  each  to  be  advantageously  viewed  from  the 
standpoint  of  the  other. 

Modern  Theory.  Mobius  began  his  contributions  to  geometry 
in  1823,  and  four  years  later  published  his  Barycentrische 
Calcul.  In  this  great  work  he  introduced  homogeneous  coordi- 
nates. Of  modern  contributors  to  analytic  geometry,  however, 
Plucker  stands  easily  foremost.  In  1828  he  published  the  first 
volume  of  his  Analytisch-geometrische  Entwickelungen,  in 
which  there  appeared  the  modern  abridged  notation.  In  the 
second  volume  (1831)  he  set  forth  the  present  analytic  form  of 
the  principle  of  duality.  To  him  is  due  (1833)  the  general 
treatment  of  foci  for  curves  of  higher  degree,  and  the  complete 
classification  of  plane  cubic  curves  (1835)  which  had  been  so 
frequently  attempted  before  him.  He  also  gave  (1839)  an 

iaDes  effections  des  superficies."  This  appears  in  his  Essais  et  Recherches, 
Paris,  1705  and  1713. 

2  Recherches  sur  les  courbes  a  double  courbure,  printed  in  1731.  It  was  pre- 
sented to  the  Academic  when  Clairaut  was  only  sixteen  years  old. 

ii 


326  ANALYTIC  GEOMETRY 

enumeration  of  plane  curves  of  the  fourth  order.  In  1842  he 
gave  his  celebrated  "six  equations/'  by  which  he  showed  that 
the  characteristics  of  a  curve  (order,  class,  number  of  double 
points,  number  of  cusps,  number  of  double  tangents,  and  num- 
ber of  inflections)  are  known  when  any  three  are  given.  To 
him  is  also  due  the  first  scientific  dual  definition  of  a  curve,  a 
system  of  tangential  coordinates,  and  an  investigation  of  the 
question  of  double  tangents.  The  theory  of  ruled  surfaces, 
begun  by  Monge,  was  also  extended  by  him.  Possibly  the 
greatest  service  rendered  by  Pliicker  was  the  introduction  of 
the  straight  line  as  a  space  element,  his  first  contribution 
(1865)  being  followed  by  his  well-known  treatise  on  the  subject 
(I868-I86Q).1 

Certain  Well-Known  Curves.  There  are  certain  curves  that  are 
so  frequently  met  in  textbooks  on  analytic  geometry  as  to  de- 
serve mention  in  an  elementary  history.  Several  of  these  have 
been  considered  elsewhere  in  this  work,  and  a  few  others,  with 
additional  notes  on  those  already  given,  will  now  be  men- 
tioned,2 and  for  convenience  will  be  given  alphabetically. 

Brachistochrone*  the  curve  of  quickest  descent,  was  studied 
by  Galileo,  Leibniz,  Newton,  and  the  Bernoullis,  and  was 
shown  to  be  the  cycloid.  The  name  is  due  to  the  Bernoullis.4 

Cardioid,  the  epicycloid  (x2  +  y2  —  2  ax)2  =  4a2(x2  +  y2). 
The  name  is  due  to  Giovanni  Francesco  M.  M.  Salvemini,  called 
from  his  birthplace  de  Castillon  (1708-1791),  De  curva  cardi- 
oide  (1741).  It  had  already  been  studied  by  Ozanam.5 


the  various  coordinate  systems,  see  the  Encyklopadie,  III,  221,  596. 

2H.  Brocard,  Notes  de  Bibliographic  des  Courbes,  lith.  autog.,  Bar-le-Duc, 
1897;  Partie  complementaire,  1899.  See  also  Chasles,  Aper^u,  and  the  Encyklo- 
padie,  III,  185,  457;  E.  Pascal,  Repertorium  der  hoheren  Mathematik,  German 
translation  by  A.  Schepp,  Leipzig,  1902,  especially  Vol.  II,  chap,  xvii;  Loria, 
Kurven  ;  Joaquin  de  Vargas  y  Aguirre,  "  Catalogo  General  de  curvas,"  Memorias 
de  la  Real  Acad.  de  Ciencias  exactas,  XXVI,  Madrid,  1908;  F.  G.  Teixeira,  Traite 
des  courbes  spttiales  remarquables  planes  et  gauches,  3  vols.,  Coimbra,  1908, 
1909,1915. 

8From  /Spdxwros  (brach'istos,  shortest)  and  xpfoo*  (chron'os,  time).  Formerly 
spelled  br  achy  stochr  one  by  a  confusion  of  the  superlative  /fydxto-ros  with  its  posi- 
tive ppaxh- 

4  Cantor,  Geschichte,  III,  chap.  92.  6  It  is  a  special  case  of  the  limac,on. 


CERTAIN  SPECIAL  CURVES  327 

(  *      --^ 
Catenary ',  the  French  chamette,y=\ a\e«+  e  <*).  The  name 

of  the  curve  (catenaria)  and  the  discovery  of  the  equation  and 
its  properties  are  due  to  Leibniz.1 

Cissoid  of  Diodes,  the  "ivy-shaped"  curve,  y2  =  x*/(2a  —  x\ 
due  to  Diodes  (c.  180  B.C.).2 

Cochlioidy*  r  =  (a  sin  0)/0,  a  spiral  curve  discussed  by 
J.  Perk,  Phil.  Trans.,  1700,  this  form  being  a  late  one,  due  to 
J.  Neuberg,  a  Belgian  geometer.  The  name  originated  (1884) 
with  two  recent  writers,  Bentham  and  Falkenburg. 

(2              2  \ 
2  a  r— tf2  =  o, 
a  muuern  name,  uue  10  oyivesier.-                ? 

Conchoid  of  Nicomedes  (c.  180  B.C.),  the  "shell -shaped" 
curve.6  The  Cartesian  equation  is  (x—  a)2(x2  +  y2)  —  b2x2  =  o 
and  the  polar  equation  is  r  =  a/cosd  +  b. 

Conchoid  oj  deSluze,  the  cubic  curve  a  (r  cos#  —  a)=k2  cos20 
or  a  (x  —  a)  (**  +  y2)—  k'2x2  =  o,  first  constructed  by  Rene  de 
Sluze  (1662). 

Curve  of  Pursuit,  French  courbe  du  chien  as  a  special  case. 
The  name  ligne  de  pour  suite  seems  due  to  Pierre  Bouguer6 
(1732),  although  the  curve  had  been  noticed  by  Leonardo  da 
Vinci. 
Cycloid f  the  transcendental  curve 


x  =  a  arc  cos  • 


This  curve,  sometimes  incorrectly  attributed  to  Nicholas  Cusa 
(c.  1450),  was  first  studied  by  Charles  de  Bouelles  (1501).  It 
then  attracted  the  attention  of  Galileo  (1599),  Mersenne 

10n  the  history  of  this  curve  see  C.  A.  Laisant,  Association  Fran$aise  pour 
I'avancement  desSdences^CongresdeToulouse^.^  (1887) ;  Loria, Jfttrven,!!,  204. 

2  See  Volume  I,  page  118;  Volume  II,  page  314. 

3  From  K-oxXtes  (cochli'as,  snail)  and  eWos  (ei'dos,  form). 
4 See  also  Educational  Times,  quest.  12,978  (Matz). 

5  See  Volume  I,  page  118. 

6Born  at  Croisic,  Brittany,  February  16,  1698;  died  in  Paris,  August  15,  1758. 
He  was  one  of  the  French  geodesists  sent  to  Peru  in  1735  to  measure  an  arc  of 
a  meridian.  See  articles  by  F.  V.  Morley,  R.  C.  Archibald,  H.  P.  Manning,  and 
W.  W.  Rouse  Ball,  Amer.  Math.  Month.,  XXVIII. 


328  ANALYTIC  GEOMETRY 

(1628),  and  Roberval  (1634).  Pascal  (1659)  called  it  the 
"roulette,"  completely  solved  the  problem  of  its  quadrature, 
and  found  the  center  of  gravity  of  a  segment  cut  off  by  a  line 
parallel  to  the  base.1  Jean  and  Jacques  Bernoulli  showed  that  it 
is  the  brachistochrone  curve,  and  Huygens  (1673)  showed  how 
its  property  of  tautochronism  might  be  applied  to  the  pendulum. 
Devil's  Curve,  French  courbe  du  diable,  in  general  repre- 
sented by  the  equation  /  —  x*  4-  ay2  +  bx2  =  o,  and  in  particular 
by  y  —  #4  —  96  a*yz  + 100  a2x2  —  o.  The  polar  equation  is 

r—  2  #V(25  —  24  tan2#)/(i  —  tan2#). 


It  was  studied  by  G.  Cramer  (lyso)2  and  Lacroix  (i8io)3  and 
is  given  in  the  Nouvelles  Annales  (1858),  p.  317. 

Elastic  Curve,  French  courbe  Mastique,  the  differential  equa- 
tion of  which  is 


It  was  first  studied  by  Jacques  Bernoulli  (1703). 

Epicycloid,  literally  "epicycle-shaped,"  a  curve  traced  by  a 
point  on  a  circle  which  rolls  on  the  convex  side  of  a  given  circle. 
The  equation  is  (x2  +  y2  -  02)2  =  40  [(*-  a)2  +y2].  The 
curve  was  recognized  by  Hipparchus  (c.  140  B.C.)  in  his  as- 
tronomical theory  of  epicycles.  Albrecht  Diirer  (1525)  was 
the  first  to  describe  it  in  a  printed  work.  It  was  next  studied 
by  Desargues  (1639),  but  it  first  received  noteworthy  con- 
sideration by  Lahire  (1694)  and  Euler  (1781). 

Folium  of  Descartes,  a  curve  represented  by  the  equation 
x*  +  y*  =  $axy.  The  problem  was  proposed  to  Roberval  to 
determine  the  tangent  to  this  curve,  and  through  an  error  he 
was  led  to  believe  that  the  curve  had  the  form  of  a  jasmine 
flower,  and  hence  he  gave  it  the  name  fleur  de  jasmin,  which 
was  afterwards  changed.  It  is  also  known  as  the  noeud  de  ruban. 


1H.  Bosnians,  "Pascal  et  les  premieres  pages  de  P'Histoire  de  la  Roulette,"* 
Archives  de  Philosophic,  1  (1923),  cah.  3. 

2 Introduction  a  V analyse  des  lignes  courbes  alg&briques,  p.  19  (Geneva,  1750). 
3  Traiti  du  calcul  differential  et  ,  ,  ,  integral,  I,  391  (Paris,  1797;  1810  ed.), 


CERTAIN  SPECIAL  CURVES  329 

Helix,  the  name  given  by  Archimedes  (c.  225  B.C.)  to  a 
spiral  already  studied  by  his  friend  Conon.1  It  is  now  known  as 
the  spiral  of  Archimedes.  The  equation  is  r  =  ad,  or  tan  <f>  =  0. 
It  is  one  of  the  class  of  which  the  general  equation  is  r  =  a6n. 
The  name  is  now  usually  applied  to  a  curve  traced  upon  a 
cylinder  and  cutting  the  generatrices  under  a  constant  angle. 
There  are  also  the  conical  helix,  the  spherical  helix  (or  loxo- 
drome),  and  other  types. 

Lemniscate2  a  curve  first  mentioned  by  Jacques  Bernoulli 
(i694).3  Its  principal  properties  were  discovered  by  Fagnano 
(1750).  The  analytic  theory  of  the  curve  is  due  to  Euler 
(1751,  1752).  The  general  lemniscate  has  for  its  equation 
(x2  +  y2)2  =  2  a2  (x2  -  y2)  +  b4  -  a4,  while  that  of  Bernoulli  is 
represented  algebraically  by  (x2  +  y2)2  =  2a2(x2  —  y2),  and  is 
called  the  hyperbolic  lemniscate.  The  general  lemniscate  is  also 
known  as  Cassini's  oval,  after  Giovanni  Domenico  (Jean  Domi- 
nique) Cassini,  who  described  it  in  1680. 

Lima^on,  French  limagon  (a  snail),  Italian  lumaca,  from 
Latin  Umax,  called  also  by  the  French  the  concho'ide  du  cercle. 
The  curve  is 


-  axf  =  t*  (x*  +/2),     or     p  = 

Roberval  called  it  the  limagon  of  Pascal,  Etienne  Pascal  (father 
of  Blaise)  having  discovered  it.  German  writers  speak  of  it 
as  the  Pascal'sche  Schnecke. 

Lituus  (the  Latin  word  for  an  augur's  staff),  the  curve 
r20  =  a2.  The  name  is  due  to  Cotes  (c.  1710). 

Logarithmic  or   Equiangular  Spiral,  the  curve  r  =  aekQ  ,  or 
^- 

k9  =  log->  studied  by  Jacques  Bernoulli  (1692),  who  spoke  of 
ct 

it  as  a  spira  mirabilis.  It  is  still  to  be  seen,  in  rude  form,  upon 
his  tomb  in  Basel.  The  logarithmic  spiral  was  the  first  non- 
algebraic  plane  curve  to  be  rectified. 

1But  see  Volume  I,  page  107. 

2  From  \i)/jivl<TKos  (lemnis'kos,  Latin  lemniscus),  a  ribbon  on  which  a  pendant 
is  hung. 

3".  ,  .  formam  refert  jacentis  notae  octonarii  oo,  seu  complicitae  in  nodum 
fasciae,  sive  lemnisci."  See  F.  Cajori,  Hist,  of  Math.,  2d  ed.,  p.  221. 


330  ANALYTIC  GEOMETRY 

Pearls,  a  name  given  by  Pascal  and  de  Sluze  to  the  curves 
whose  equation  is     a*+9-ryr  =  x*(a  _  ^ 
or,  in  particular,  ^  =  ^(a  _  *). 

De  Sluze  proposed  their  consideration  to  Huygens  (1658),  and 
the  latter  made  a  careful  study  of  them. 

Roseate  Curve,  Rosace,  or  Rhodonea,  the  curve  whose  gen- 
eral polar  equation  is  r  —  a  cosmO.  The  name  Rhodonea  is  due 
to  Guido  Grandi  (1713).  The  Rosace  a  quatre  jeuilles,  or 
Quadrifolium,  has  for  its  polar  equation  r=a  sin  26,  and  for 
the  Cartesian  form  (x2  -f  >r)3  =  4a2x2y2. 

Semicubk  Parabola,  or  Neile's  Parabola,  y*  =  ax2.  It  was 
the  second  curve  to  be  rectified.  William  Neile  discovered  the 
curve  in  1657.  The  method  of  rectification  was  published  by 
Wallis  in  1659,  credit  being  given  by  him  to  his  pupil,  Neile, 
although  there  is  still  some  dispute  as  to  whether  it  was  due  to 
him,  to  Fermat,  or  to  the  Dutch  writer  Van  Heuraet.1 

Serpentine  Curve,  a  name  proposed  by  Newton  for  the  curve 


Spiral  of  Archimedes,  the  curve  r  =  a6,  mentioned  under 
Helix. 

Spiral  of  Fermat,  the  curve  r  =  0,  proposed  by  Fermat  in  a 
letter  to  Mersenne,  June  3,  1636. 

Strophoid,  French  stropho'ide?  a  name  proposed  by  a  modern 
writer,  Montucci  (1846),  for  the  curve  y~x^J(a--x)/(a  +  x). 
Lehmus  had  already  proposed  (1842)  the  name  kukumae'ide 
(cucumber  seed),  and  various  other  names  have  been  used.  The 
curve  has  been  studied  by  Barrow,  Jean  Bernoulli,  Agnesi 
(1748),  James  Booth  (1858),  and  various  others. 

Tractrix,  the  tractoria  of  Huygens  (1693).  The  differential 
equation  is  , 

~ 


1Hendrik  van  Heuraet,  born  at  Haarlem  in  1633.   His  brief  Epistolae  de 
curvarum  linearum  in  rectos  transmutatione  was  published  by  Van  Schooten  in 

1659- 

2  From  <rr/>60os  (stroph'os,  a  twisted  band,  a  cord)  +  efSos  (ei'dos,  form). 


PERIODS  OF  GEOMETRY  331 

Witch  of  Agnesij  Versiera,  Cubique  d'Agnesi,  or  Agnhienne, 
the  curve  y*x  +  r\x  —  r)  =  o,  discussed  by  Maria  Gaetana 
Agnesi  in  1748  in  her  Istituzioni  Analitiche.1 

9.  MODERN  GEOMETRY 

Four  Periods  of  Geometry.  In  order  to  appreciate  the  histori- 
cal setting  of  modern  geometry  it  is  well  to  remember  that  the 
history  of  geometry  in  general  may  be  roughly  divided  into 
four  periods:  (i)  the  synthetic  geometry  of  the  Greeks,  in- 
cluding not  merely  the  geometry  of  Euclid  but  the  work  on 
conies  by  Apollonius  and  the  less  formal  contributions  of  nu- 
merous other  writers;  (2)  the  birth  of  analytic  geometry,  in 
which  the  synthetic  geometry  of  Desargues,  Kepler,  Roberval, 
and  other  writers  of  the  iyth  century  merged  into  the  coordi- 
nate geometry  already  set  forth  by  Descartes  and  Fermat; 

(3)  the  application  of  the  calculus  to  geometry, — a  period  ex- 
tending from  about  1650  to  1800,  and  including  the  names  of 
Cavalieri,  Newton,  Leibniz,  the  Bernoullis,  1'Hospital,  Clairaut, 
Euler,  Lagrange,  and  d'Alembert,  each  one,  especially  after 
Cavalieri,  being  primarily  an  analyst  rather  than  a  geometer ; 

(4)  the  renaissance  of  pure  geometry,  beginning  with  the 
igth  century  and  characterized  by  the  descriptive  geometry  of 
Monge,  the  projective  geometry  of  Poncelet,  the  modern  syn- 
thetic geometry  of  Steiner  and  Von  Staudt,  the  modern  analytic 
geometry  of  Pliicker,  the  non-Euclidean  hypotheses  of  Lobachev- 
sky,  Bolyai,  and  Riemann,  and  the  foundations  of  geometry.  v 

Descriptive  Geometry.  Descriptive  geometry  as  a  separate 
science  begins  with  Monge.  He  had  been  in  possession  of  the 
theory  for  over  thirty  years  before  the  publication  of  the 
Geometric  Descriptive  (1794),* — a  delay  due  to  the  jealous 

!G.  Loria,  Bibl.  Math.,  XI  (2),  7.  See  the  English  translation  by  J.  Hellins, 
I,  222  (London,  1801).  See  also  Volume  I,  page  519. 

2G.  Loria,  Storia  delta  Geometria  Descrittiva  (Milan,  1921),  the  leading 
authority  on  the  subject;  Chr.  Wiener,  Lehrbuch  der  darstellenden  Geometrie, 
Leipzig,  1884-1887;  Geschichte  der  darstellenden  Geometrie,  ibid.,  1884.  See  En- 
cyklopadie,  III,  517;  F.  J.  Obenrauch,  Geschichte  der  darstellenden  und  pro- 
jectiven  Geometric,  Briinn,  1897. 


332  MODERN  GEOMETRY 

desire  of  the  military  authorities  to  keep  the  valuable  secret. 
Certain  of  its  features  can  be  traced  back  to  Frezier,  Desargues, 
Lambert,  and  other  writers  of  the  preceding  century,  but  it  was 
Monge  who  worked  it  out  in  detail  as  a  science,  although  La- 
croix  (1795),  inspired  by  Mongers  lectures  in  the  Ecole  Poly- 
technique,  published  the  first  work  on  the  subject.  After 
Mongers  work1  appeared,  Hachette  (1812,  1818,  1821)  added 
materially  to  the  theory.  ^ 

Period  of  Projective  Geometry.  It  is  also  in  this  period  that 
projective  geometry  has  had  its  development,  even  if  its  origin 
is  more  remote.  The  origin  of  any  branch  of  science  can  al- 
ways be  traced  far  back  in  human  history,  and  this  fact  is 
patent  in  the  case  of  this  phase  of  geometry.  The  idea  of  the 
projection  of  a  line  upon  a  plane  is  very  old.  It  is  involved  in 
the  treatment  of  the  intersection  of  certain  surfaces,  due  to 
Archytas  (c.  400  B.C.),  and  appears  in  various  later  works  by 
Greek  writers.  Similarly,  the  invariant  property  of  the  an- 
harmonic  ratio  was  essentially  recognized  both  by  Menelaus 
(c.  100)  and  by  Pappus  (c.  300).  The  notion  of  infinity  was 
also  familiar  to  several  Greek  geometers  and  to  the  Latin  writer 
Lucretius  (c.  100),  so  that  various  concepts  that  enter  into  the 
study  of  projective  geometry  were  common  property  long  be- 
fore the  science  was  really  founded.  v~ 

Desargues,  Pascal,  Newton,  and  Carnot.  One  of  the  first  im- 
portant steps  to  be  taken  in  modern  times,  in  the  development 
of  this  form  of  geometry,  was  due  to  Desargues.  In  a  work 
published  in  1639  Desargues  set  forth  the  foundation  of  the 
theory  of  four  harmonic  points,  not  as  done  today  but  based 
on  the  fact  that  the  product  of  the  distances  of  two  conjugate 
points  from  the  center  is  constant.  He  also  treated  of  the  theory 
of  poles  and  polars,  'although  not  using  these  terms.  In  the 
following  year  (1640)  Pascal,  then  only  a  youth  of  sixteen  or 
seventeen,  published  a  brief  essay  on  conies  in  which  he  set 
forth  the  well-known  theorem  that  bears  his  name. 

1  Essais  sur  les  plans  et  les  surfaces,  Paris,  1795;  Complement  des  Siemens  de 
Geometric  ou  ILlemens  de  Geometrie  descriptive,  Paris,  1796;  Essais  de  Geo- 
metrie sur  les  plans  et  les  surfaces  courbes,  Paris,  1812. 


PROJEGTIVE  GEOMETRY  333 

In  the  latter  part  of  the  i7th  century  Newton  investigated 
the  subject  of  curves  of  the  third  order  and  showed  that  all 
such  curves  can  be  derived  by  central  projection  from  five 
fundamental  types.  In  the  i8th  century  relatively  little  atten- 
tion was  given  to  the  subject,  but  at  the  close  of  this  period, 
as  already  stated,  the  descriptive  geometry  of  Monge  was 
brought  into  prominence, — itself  a  kind  of  projective  geometry, 
although  not  what  is  technically  known  by  this  name. 

Inspired  by  the  general  activity  manifest  in  the  i8th  century, 
and  following  in  the  footsteps  of  Desargues  and  Pascal,  Carnot 
treated  chiefly  of  the  metric  relations  of  figures.  In  particular 
he  investigated  these  relations  as  connected  with  the  theory  of 
transversals, — a  theory  whose  fundamental  property  of  a  four- 
rayed  pencil  goes  back  to  Menelaus  and  Pappus,  and  which, 
though  revived  by  Desargues,  was  set  forth  for  the  first  time 
in  its  general  form  by  Carnot  in  his  Geometric  de  Position 
(1803),  and  supplemented  in  his  ThSorie  des  Transver  sales 
(1806).  In  these  works  Carnot  introduced  negative  magni- 
tudes, the  general  quadrilateral,  the  general  quadrangle,  and  nu- 
merous other  similar  features  of  value  to  elementary  geometry. 

Poncelet  on  Projective  Geometry.  The  origin  of  projective 
geometry  as  we  know  it  today  is  generally  ascribed  to  Ponce- 
let.1  A  prisoner  (1813-1814)  in  the  Russian  campaign,  con- 
fined at  Saratoff  on  the  Volga,  with  no  books  at  hand,2  he  was 
able  in  spite  of  all  such  discouragement  to  plan  the  great  work3 
which  he  published  in  1822.  In  this  work  he  made  prominent 
for  the  first  time  the  power  of  central  projection  in  demonstra- 
tion and  the  power  of  the  principle  of  continuity  in  research. 
His  leading  idea  was  the  study  of  projective  properties,  and  as 
a  foundation  principle  he  introduced  the  anharmonic  ratio,  a 

10n  the  whole  question  consult  the  Encyklopadie,  III,  389. 

2"Priv6  de  toute  espece  de  livres  et  de  secours,  surtout  distrait  par  les  mal- 
heurs  de  ma  patrie  et  les  miens  propres." 

3  J.  V.  Poncelet,  Traite  des  proprietts  projectives  des  figures,  Paris,  1822 ;  ibid., 
1865-1866;  Applications  d> 'analyse  et  de  geometric,  ed.  Mannheim  and  Moutard, 
2  vols.,  Paris,  1862,  1864.  On  the  general  subject  of  the  development  of  modern 
geometric  methods  see  J.  G.  Darboux,  Bulletin  of  the  Amer.  Math.  Soc.,  XI,  £17. 
See  also  Volume  I,  page  496. 


334  MODERN  GEOMETRY 

concept  which  possibly  dates  back  to  the  lost  porisms  o 
and  which  Desargues  (1639)  had  used.  The  anharmonic  point- 
and-line  properties  of  conies  have  since  then  been  further  elab- 
orated by  Brianchon,  Chasles,  Steiner,  Pliicker,  Von  Staudt, 
and  other  investigators.  To  Poncelet  is  also  due  the  theory  of 
"figures  homologiques,"  the  perspective  axis  and  perspective 
center  (called  by  Chasles  the  axis  and  center  of  homology),  an 
extension  of  Carnot's  theory  of  transversals,  and  the  "cordes 
ideales"  of  conies  which  Pliicker  applied  to  curves  of  all  orders. 
Poncelet  also  considered  the  circular  points  at  infinity  and  com- 
pleted the  first  great  principle  of  modern  geometry,  the  principle 
of  continuity.  Following  upon  the  work  of  Poncelet,  Mobius 
made  much  use  of  the  anharmonic  ratio  in  his  Barycentri- 
sche  Calcul  (1827),  but  he  gave  it  the  name  "Doppelschnitt- 
Verhaltniss  (ratio  bisectionalis),"  a  term  now  in  common  use 
under  Steiner's  abbreviated  form  "Doppelverhaltniss."  The 
name  "anharmonic  ratio"  or  "anharmonic  function"  ("rap- 
port anharmonique "  or  "fonction  anharmonique")  is  due  to 
Chasles,  and  "cross-ratio"  was  suggested  by  Clifford. 

Gergonne,  Steiner,  and  Von  Staudt.  Joseph-Diez  Gergonne1 
(1813)  introduced  the  term  "polar"  in  its  modern  geometric 
sense,  although  Servois  (1811)  had  used  the  expression  "pole." 
Gergonne  was  the  first  (1825-1826)  to  grasp  completely  the 
principle  to  which  he  gave  the  name  of  "Principle  of  Duality," 
the  most  important  principle,  after  that  of  continuity,  in  modern 
geometry.  He  used  the  word  "class"  in  describing  a  curve, 
explicitly  defining  class  and  degree  (order)  and  showing  the 
duality  between  them.  He  and  Chasles  were  among  the  first  to 
study  surfaces  of  higher  order  by  modern  methods. 

Jacob  Steiner,  the  most  noted  of  the  Swiss  geometers  of  the 
ipth  century,  gave  the  first  complete  discussion2  of  the  projec- 
tive  relations  between  rows,  pencils,  etc.  and  laid  the  founda- 
tion for  the  subsequent  development  of  pure  geometry.  For 
the  present,  at  least,  he  may  be  said  to  have  closed  the  theory  of 
conic  sections,  of  the  corresponding  figures  in  three-dimensional 

fSee  Volume  I,  page  495. 

2 Systematische  Entwkkelungen  .  .  .,  Berlin,  1832.   See  Volume  I,  page  524. 


NON-EUCLIDEAN  GEOMETRY  335 

space,  and  of  surfaces  of  the  second  order,  and  hence  there 
opens  with  him  the  period  of  the  special  study  of  curves  and 
surfaces  of  higher  order. 

Between  1847  and  1860  Karl  Georg  Christian  von  Staudt 
set  forth  a  complete  system  of  a  pure  geometry1  that  is  inde- 
pendent of  metrical  considerations.  All  projective  properties 
are  here  established  independently  of  number  relations,  number 
being  drawn  from  geometry  instead  of  conversely,  and  im- 
aginary elements  being  systematically  introduced  from  the 
geometric  side.  A  projective  geometry,  based  on  the  group 
containing  all  the  real  projective  and  dualistic  transformations, 
is  developed,  and  imaginary  transformations  are  introduced. 

Non-Euclidean  Geometry.  The  question  of  Euclid's  fifth  pos- 
tulate, relating  to  parallel  lines,  has  occupied  the  attention  of 
geometers  ever  since  the  Elements  was  written.2  The  first 
scientific  investigation  of  this  part  of  the  foundation  of  geome- 
try was  made  by  Girolamo  Saccheri3  (1733),  a  work  which  was 

1  Geometric  der  Lage,  Nurnberg,  1847;  Beitrdge  zur  Geometric  der  Lage, 
3  parts,  Nurnberg,  1856,  1857.  See  M.  Noether,  Zur  Erinnerung  an  K.  G.  C. 
von  Staudt ,  Erlangen,  1901,  and  Volume  I,  page  505. 

2F.  Engel  and  P.  Stackel,  Die  Theorie  der  Parallellinien  von  Euklid  bis  auf 
Gauss,  Leipzig,  1895;  G.  B.  Halsted,  various  contributions,  including  "Bibliog- 
raphy of  Hyperspace  and  Non-Euclidean  Geometry,"  American  Journal  of 
Mathematics,  Vols.  I,  II;  Amer.  Math.  Month.,  Vol.  I;  translations  of 
Lobachevsky's  Geometry,  Vassilief's  address  on  Lobachevsky,  Saccheri's  Geom- 
etry, Bolyai's  work  and  his  life;  "Non-Euclidean  and  Hyperspaces,"  Mathe- 
matical Papers  of  Chicago  Congress,  p.  92 ;  G.  Loria,  Die  hauptsdchlichsten 
Theorien  der  Geometric,  Leipzig,  p.  106;  A.  Karagiannides,  Die  Nichteuklidische 
Geometric  vom  Alterthum  bis  zur  Gegenwart,  Berlin,  1893;  E.  McClintock,  "On 
the  Early  History  of  Non-Euclidean  Geometry,"  Bulletin  of  New  York  Mathe- 
matical Society,  II,  144;  W.  B.  Frankland,  Theories  of  Parallelism,  Cambridge, 
1910  (particularly  valuable) ;  H.  Poincar6,  "Non- Euclidean  Geometry,"  Nature, 
XLV,  404;  P.  Stackel,  Wolfgang  und  Johann  Bolyai,  Geometrische  Unter- 
suchungen,  2  vols.,  Leipzig,  1913.  See  also  Volume  I,  Chapter  X,  under  the  sev- 
eral names  mentioned.  On  the  general  question  of  the  modern  synthetic  treatment 
of  elementary  geometry,  see  the  Encyklopddie,  III,  859;  for  the  analytic  treat- 
ment, ibid.,  771.  See  also  C.  J.  Keyser,  Mathematical  Philosophy,  p.  342  (New 
York,  1922).  For  an  excellent  bibliography  up  to  the  time  it  was  printed  see 
D.  M.  Y.  Sommerville,  Bibliography  of  Non-Euclidean  Geometry,  London,  1911. 

3  Born  at  San  Remo,  September  4  or  5,  1667 ;  died  at  Milan,  October  25,  1733. 
The  work  was  Euclides  ab  omni  naevo  vindicatus,  Milan,  1733;  English  transla- 
tion by  G.  B.  Halsted,  Chicago,  1920.  Saccheri  was  a  Jesuit  and  taught  mathe- 
matics in  Turin,  Pavia,  and  Milan. 


336  MODERN  GEOMETRY 

not  looked  upon  as  a  precursor  of  Lobachevsky,  however,  until 
Beltrami  (1889)  called  attention  to  the  fact.  Johann  Heinrich 
Lambert  (1728-1 777)*  was  the  next  to  question  the  validity  of 
Euclid's  postulate,  in  his  Theorie  der  Parallellinien  (posthu- 
mous, Leipzig,  1786),  the  most  important  treatise  on  the  sub- 
ject between  the  publication  of  Saccheri's  work  and  the  works 
of  Lobachevsky  and  Bolyai.  Legendre  (1794)  also  contrib- 
uted to  the  theory,  but  failed  to  make  any  noteworthy  advance. 
During  the  closing  years  of  the  i8th  century  Kant's2  doc- 
trine of  absolute  space,  and  his  assertion  of  the  necessary  pos- 
tulates of  geometry,  were  the  object  of  much  scrutiny ''and 
attack.  At  the  same  time  Gauss  was  giving  attention  to  the 
fifth  postulate,  although  at  first  on  the  side  of  proving  it.  It 
was  at  one  time  surmised  that  Gauss  was  the  real  founder  of 
the  non-Euclidean  geometry,  his  influence  being  exerted  on 
Lobachevsky  through  his  friend  Bartels,3  and  on  Janos  Bolyai 
through  the  father  Farkas,  who  was  a  fellow  student  of  Gauss, 
and  it  will  presently  be  seen  that  he  had  some  clear  ideas  of 
the  subject  before  either  Lobachevsky  or  Bolyai  committed 
their  theories  to  print. 

Lobachevsky.  Bartels  went  to  Kasan  in  1807,  and  Lobachev- 
sky was  his  pupil.  The  latter's  lecture  notes  fail  to  show  that 
Bartels  ever  mentioned  the  subject  of  the  fifth  postulate  to 
him,  so  that  his  investigations,  begun  even  before  1823,  seem 
to  have  been  made  on  his  own  motion,  and  his  results  to  have 
been  wholly  original.  Early  in  1826  he  set  forth  the  principles 
of  his  famous  doctrine  of  parallels,  based  on  the  assumption 
that  through  a  given  point  more  than  one  straight  line  can  be 
drawn  which  shall  never  meet  a  given  straight  line  coplanar 
with  it.  The  theory  was  published  in  full  in  1829-1830,  and 
he  contributed  to  the  subject,  as  well  as  to  other  branches  of 
mathematics,  until  his  death.  ^ 

aD.  Huber,  Lambert  nach  seinem  Leben  und  Wirken,  Basel,  1829.  See 
Volume  I,  page  480.  2  E.  Fink,  Kant  als  Mathematiker,  Leipzig,  1889. 

8  Johann  Martin  Christian  Bartels,  born  at  Braunschweig,  August  12,  1769; 
died  at  Dorpat,  December  19,  1836.  He  was  professor  of  mathematics  at  Kasan 
and  later  at  Dorpat. 


NON-EUCLIDEAN  GEOMETRY  337 

The  Bolyais  and  Gauss.  Janos  Bolyai  received,  through  his 
father,  Farkas,  some  of  the  inspiration  to  original  research 
which  the  latter  had  received  from  Gauss.  When  only  twenty- 
one  he  discovered,  at  about  the  same  time  as  Lobachevsky,  the 
principles  of  non-Euclidean  geometry,  and  he  refers  to  them 
in  a  letter  of  November,  1823.  They  were  committed  to  writ- 
ing in  1825  and  were  published  in  1832.  Gauss  asserts  in  his 
correspondence  with  Schumacher1  (1831-1832)  that  he  had 
thought  out  a  theory  along  the  same  lines  as  Lobachevsky  and 
Bolyai,  but  the  publication  of  their  works  seems  to  have  put 
an  2nd  to  his  investigations.  His  statement  on  the  subject  is 
as  follows : 

I  will  add  that  I  have  recently  received  from  Hungary  a  little 
paper  on  non-Euclidean  geometry  in  which  I  rediscover  all  my  own 
ideas  and  results  worked  out  with  great  elegance.  .  .  .  The  writer 
is  a  very  young  Austrian  officer,  the  son  of  one  of  my  early  friends, 
with  whom  I  often  discussed  the  subject  in  1798,  although  my  ideas 
were  at  that  time  far  removed  from  the  development  and  maturity 
which  they  have  received  from  the  original  reflections  of  this  young 
man.  I  consider  the  young  geometer  Von  Bolyai  a  genius  of  the 
first  rank.2 

This  was  not,  however,  the  first  statement  of  Gauss  upon 
the  subject,  for  in  a  letter  written  on  November  8,  1824,  he 
remarked : 

The  assumption  that  the  sum  of  the  3  angles  is  smaller  than  180° 
leads  to  a  new  geometry  entirely  different  from  ours  [the  Euclidean] 
— a  geometry  which  is  throughout  consistent  with  itself,  and  which  I 
have  elaborated  in  a  manner  entirely  satisfactory  to  myself,  so  that 
I  can  solve  every  problem  in  it  with  the  exception  of  the  determining 
of  a  constant  which  is  not  a  priori  obtainable.8 

1Heinrich  Christian  Schumacher  (1780-1850),  the  astronomer. 

2Sedgwick  and  Tyler,  A  Short  History  of  Science,  p.  338  (New  York,  1917). 

3  P.  Stackel,  Wolfgang  und  Johann  Bolyai,  I,  95  (Leipzig,  1913).  The  letter 
was  written  to  one  Taurinus,  who,  two  years  later,  published  a  Geometriae  prima 
elementa  (1826),  in  which  he  gives  evidence  of  having  thought  upon  a  non- 
Euclidean  trigonometry.  See  Volume  I,  page  527. 


338  PERSPECTIVE  AND  OPTICS 

Riemann's  Theory.  Of  all  the  contributions  which  appeared 
after  Bolyai's  publication  the  most  noteworthy,  from  the  scien- 
tific standpoint,  is  that  of  Georg  Friedrich  Bernhard  Riemann. 
In  his  Habilitationsschrift  (1854)  he  applied  the  methods  of 
analytic  geometry  to  the  theory  and  suggested  a  surface  of 
negative  curvature,  which  Beltrami  called  "pseudo-spherical," 
thus  leaving  Euclid's  geometry  on  a  surface  of  zero  curvature 
midway  between  his  own  and  Lobachevsky's.  He  thus  set 
forth  three  kinds  of  geometry,  Bolyai  having  noted  only  two. 
These  Klein  (1871)  called  the  elliptic  (Riemann's),  parabolic 
(Euclid's),  and  hyperbolic  (Lobachevsky's)  geometry.,^- 

10.  PERSPECTIVE  AND  OPTICS 

Relation  of  Perspective  to  Mathematics.  While  all  painters 
seek  to  secure  proper  perspective  in  their  pictures,  the  most  suc- 
cessful of  the  painters  of  the  Renaissance  made  an  effort  to 
base  their  treatment  of  the  subject  on  mathematical  prin- 
ciples. Of  late  these  principles  have  interested  architects  more 
than  painters,  but  in  any  case  the  subject  is  largely  a  mathe- 
matical one.1 

The  Greeks  included  perspective  in  their  science  of  optics, 
and  the  Arabs  in  their  science  of  appearances,  their  title  being 
translated  into  Medieval  Latin  as  De  aspectibus.2  Therefore, 
while  there  is  a  manifest  difference  between  perspective  and 
optics  as  we  consider  these  terms  today,  it  is  necessary  to  treat 
of  them  as  closely  related. 

Ancient  Works.  While  several  Greek  writers  wrote  on  the 
subject  of  perspective,  the  earliest  mathematical  work  that  has 
come  down  to  us  is  the  Optics  of  Euclid.3  In  this  work  Euclid 

1On  the  history  of  the  subject  a  beginning  can  be  made  with  N.  G.  Poudra, 
Histoire  de  la  Perspective,  Paris,  1864,  a  rather  poorly  arranged  work  with  no 
index. 

2  The  first  translation  (1505)  of  Euclid's  Optics,  however,  used  the  term 
perspectiva. 

8 The  latest  Latin  edition  of  the  Optics  is  that  of  J.  L.  Heiberg,  in  Euclidis  opera 
omnia,  Vol.  VII,  Leipzig,  1895.  There  are  various  translations  of  the  text  from 
Greek  into  Latin.  The  first  is  that  of  Zamberto  (Venice,  1505),  in  the  collected 
werks  of  Euclid;  the  second,  that  of  J.  Pena  (Pena,  de  la  Pene),  Paris,  1557,  or 


ANCIENT  WORKS  339 

lays  down  a  series  of  axioms,1  quite  as  he  does  in  his  Elements, 
the  first  being:  "Therefore  it  is  assumed  that  [visual]  rays 
emitted  from  the  eye  are  carried  in  a  straight  line,  whatever 
may  be  the  distance."2 

On  the  axioms  Euclid  bases  his  propositions,  sixty-one  in 
number,  proving  them  geometrically  after  the  plan  used  by 
him  in  the  Elements. 

There  is  also  a  work  on  catoptrics  containing  thirty-one 
propositions  and  attributed  to  Euclid,  but  it  is  doubtful  if  the 
text  published  by  Gregory3  and  Heiberg  is  his. 

Some  idea  of  the  nature  of  Euclid's  work  may  be  obtained 
from  a  single  proposition  in  his  Optics:  "If  from  the  center  of 
a  circle  a  line  be  drawn  at  right  angles  to  the  plane  of  the  circle, 
and  the  eye  be  placed  at  any  point  on  this  line,  the  diameters 
of  the  circle  will  all  appear  equal."4 

Later  Classical  Writers.  The  only  Roman  writer  who  paid  any 
attention  to  the  subject  is  Vitruvius  (c.  20  B.C.),  who,  in  his 
work  on  architecture,  has  something  to  say  on  the  plans  and 
elevations  of  buildings.  He  seems  to  have  had  the  idea  of  two 
projections,  these  being  on  two  planes  perpendicular  to  each 
other  and  arranged  as  in  descriptive  geometry. 

Heron  of  Alexandria  (c.  50?)  is  known  to  have  written  on 
dioptrics,  but  only  a  fragment  of  the  work  exists.5  His  theory 
of  light  involved  the  usual  error  of  most  of  the  Greek  scientists, 
that  the  rays  of  light  proceed  from  the  eye  to  the  object  instead 
of  from  the  object  to  the  eye. 

that  of  Dasypodius  which  appeared  at  Strasburg  in  the  same  year.  See  also 
G.  Ovio,  L'ottica  di  Euclide,  Milan,  1918;  D.  Gregory,  Eudidis  quae  supersunt 
omnia,  Oxford,  1703,  p.  [599],  with  parallel  Greek  and  Latin  texts;  La  pro- 
spettiva  di  Evclide,  .  .  .  tradotta  dal  R.  P.  M.  Egnatio  Danti,  Florence,  1573; 
La  perspective  d'Euclide,  traduite  en  fran$ais  .  .  .  par  R.  Freart  de  Chante- 
loup,  Mans,  1663. 

xln  the  Gregory  edition  (1703,  p.  604)  06rew  (the'seis)  and  positiones. 

2  That  the  eye  emitted  the  visual  rays  was  Plato's  idea.  Aristotle  held  a  view 
more  in  accordance  with  our  own,  asking  why,  if  the  older  idea  were  correct,  we 
cannot  see  in  the  dark.  3Loc.  cit.,  p.  643.  4Prop.  XXXV. 

5  Opera  quae  supersunt  omnia,  Leipzig,  1899-1914;  Traiti  de  la  dioptre,  ed. 
A.  J.  H.  Vincent,  Paris,  1858,  in  the  BiblioMque  Nationale,  Notices  et  extraitsf 
XIX,  Pt.  2,  pp.  157-347- 


340  PERSPECTIVE  AND  OPTICS 

Ptolemy  (c.  150)  is  said  to  have  written  upon  the  subject, 
but  it  is  not  certain  that  he  did  so.1  The  work 'attributed  to 
him  contains  five  books,  the  first  dealing  with  the  properties  of 
light,  the  second  with  the  nature  of  vision,  the  third  with  re- 
flection, the  fourth  with  concave  mirrors  and  with  two  or  more 
mirrors,  and  the  fifth  with  refraction. 

The  next  Greek  writer  on  the  subject  was  Heliodorus  of 
Larissa,2  whose  date  is  uncertain  but  who  lived  after  Ptolemy. 
His  work  is  little  more  than  a  commentary  on  Euclid.3 

Medieval  Writers.  One  of  the  greatest  of  the  medieval  writers 
on  perspective  was  the  Arab  scholar  Alhazen  (c.  iooo).4  His 
work  was  the  basis  of  Peckham's  Perspectiva  mentioned  below. 
The  following  well-known  problem  relating  to  optics  bears  his 
name:  "From  two  given  points  within  a  circle  to  draw  to  a 
point  on  the  circle  two  lines  which  shall  make  equal  angles  with 
the  tangent  at  that  point."5 

Of  the  European  writers  the  first  one  of  importance  was 
Roger  Bacon  (c.  1250).  In  his  Opus  Ma  jus  he  devotes  Part  V 
(De  scientla  perspectlva)  to  perspective,6  dividing  it  into  three 
parts.  Part  I  explains  the  general  principles  of  vision,  Part  II 
deals  with  direct  vision,  and  Part  III  discusses  reflection  and 
refraction.  In  the  Opus  Tertium  there  is  also  a  brief  tractatus 

1  There  is  a  MS.  in  Paris  beginning:    Incipit  Liber  Ptholemaei  de  Opticis  she 
Aspectibus  translatus  ab  Ammiraco   [or  Ammirato]   Eugenio  Siculo,  consisting 
originally  of  five  books.    For  a  discussion,  see  W.  Smith,  Diet,  of  Greek  and 
Roman  Biog.,  Ill,  573  (London,  1864).    See  also  N.  G.  Poudra,  Histoire  de  la 
Perspective,  p.  28  (Paris,  1864). 

2  Possibly  his  name  was  Damianus.   At  any  rate  some  of  the  MSS.  bear  the 
title  AajuiapoO  0iX<xr6</>ou  rou  'HXioSwpou  Aapur <ra,lov  trepl  OITTIK&V  UTro^cretov  /3i/3X{a  /3'. 

3  La    Prospettiva    di   Eliodoro    Larisseo,    Tradotto    Dal    Reverendo    Padre 
M.  Egnatio  Danti,  Florence,  1573,  bound  with  La  Prospettiva  di  Evclide.   There 
are  other  translations. 

4Al-Hasan  .  .  .  ibn  al-!iai£am.  See  Volume  I,  page  175.  A  Latin  trans- 
lation, under  the  title  Opticae  Thesauri  Libri  VII,  was  published  at  Basel  in  1572. 

5  For  a  discussion  of  the  problem  see  American  Journal  of  Mathematics, 
IV,  327- 

QRogerii  Baconis  angli,  viri  eminentissimi,  Perspectiva,  Frankfort,  1614.  This 
is  best  found,  however,  in  the  editions  of  the  Opus  Ma  jus  by  S.  Jebb  (London, 
1.733;  Venice,  1750)  and  J.  H.  Bridges  (2  vols.,  Oxford,  1897;  suppl.  vol.,  Lon- 
don, 1900).  See  also  E.  Wiedemann,  "  Roger  Bacon  und  seine  Verdienste  um  die 
Optik,"  in  A.  G.  Little,  Roger  Bacon  Essays,  p.  185  (Oxford,  1914). 


MEDIEVAL  WRITERS  341 

de  perspectives.1  Besides  this,  Bacon  wrote  two  other  brief  trea- 
tises2 on  the  subject,  and  still  others  are  attributed  to  him 
without  historic  sanction.3 

The  work  that  had  the  greatest  influence  upon  the  subject  of 
perspective  in  the  Middle  Ages  was  the  Perspectives  communis 
of  John  Peckham4  (c.  1280).  This  work  was  the  recognized 
standard  for  three  hundred  years.  It  was  edited  and  published 
by  Cardan's  father  and  went  through  various  editions.  As 
already  stated,  Peckhani  drew,  largely upon  Alhazen's  work. 
The  work  is  divided  into  three  parts,  the  second  containing* 
fifty-six  propositions  on  reflection,  and  the  third  containing 
twenty-two  on  refraction. 

About  the  same  time  as  Peckham,  the  German  (or  possibly 
Polish)  scholar  Witelo  (c.  1270) 5  was  called  to  Rome  and  there 
became  conversant  with  the  works  of  the  ancients  as  well  as 
those  of  the  Arabs  in  the  science  of  perspective.  Georg  Tan- 
stetter  von  Thannau0  (1480-1530)  and  Apianus7  prepared  edi- 
tions of  his  work  which  were  published  at  Niirnberg  in  1533 
and  1551.  The  treatise  is  divided  into  ten  books,  the  first  four 
being  a  summary  of  the  works  of  earlier  writers ;  the  fifth,  a 
treatment  of  reflection ;  the  sixth,  reflection  by  convex  spheric 
mirrors;  the  seventh,  cylindric  and  compound  mirrors;  the 
eighth,  concave  spheric  mirrors;  the  ninth,  concave  conic 
mirrors  and  irregular  mirrors;  and  the  tenth,  refraction. 

Among  the  other  medieval  writers  on  perspective  were  Wil- 
liam of  Moerbecke8  (£.'1250)  and  Campanus9  (c.  1260). 

Renaissance  Writers.  The  first  writers  of  the  Renaissance  to 
take  up  the  subject  were  the  painters  and  engravers.  Pietro 

1This  in  a  Paris  MS.,  formerly  attributed  to  Alpetragius,  discovered  by  Duhem 
and  not  yet  printed.  See  Little,  loc.  cit.,  p.  390. 

2  De  speculis  combiirentibus  and  Notulae  de  speculis,  both  published  at  Frank- 
fort (1614)  in  Combach's  Specula  mat hematic 'a ,  pp.  168-207. 

3  See  Little,  loc.  cit.,  p.  409  seq. 

4  See  Volume  I,  page  224.   It  was  often  printed.   For  editions,  see  Kastner, 
Geschichte,  II,  264;  for  Kastner's  history  of  optics  in  general,  ibid.,  p.  237. 

5 See  Volume  I,  page  228.  On  his  work  at  Padua  see  A.  Birkenmajer,  Witelo  e 
lo  Sti^dio  di  Padova,  reprint,  Padua,  1922. 

6  Professor  of  astronomy  at  Vienna.  7  See  Volume  I,  page  333. 

8  William  Fleming.   See  Volume  I,  page  229.  9See  Volume  I,  page  218. 

ii 


342  PERSPECTIVE  AND  OPTICS 

Franceschi  (or  Delia  Francesca),  for  example,  who  died  in 
1492,  wrote  the  work  De  corporibus  regularibus  and  a  work 
De  perspectiva  pingendi*  which  is  still  extant  in  manuscript, 
and  in  which  he  takes  up  the  theory  of  perspective.2  There 
were  also  such  artists  as  Leonardo  da  Vinci3  (c.  1500),  many 
of  whose  ideas  on  perspective,  and  particularly  on  the  nature 
of  vision  and  the  camera  obscura,  were  a  distinct  advance  in 
knowledge ;  Benvenuto  Cellini/  whose  work  on  perspective  was 
largely  taken  from  Leonardo ; 5  and  Albrecht  Diirer,  whose 
work  on  drawing6  includes  some  treatment  of  perspective. 

One  of  the  first  men  in  this  period  to  write  a  work  of  any 
note,  devoted  solely  to  optics,  was  Ramus7  (c.  1550).  This 
work  was  published  by  his  pupil,  Friedrich  Risner8  (died  1580), 
who  also  published  the  works  of  Alhazen  (c.  1000)  and  Witelo 
(c.  i27o).9  The  work  of  Ramus  is  in  four  books,  but  it  con- 
tains little  that  Witelo  did  not  give. 

Optics  in  the  i7th  Century.  In  the  1 7  th  century  the  science  of 
optics  took  a  great  step  forward,  notably  through  the  efforts  of 
Kepler.  These  efforts  first  appear  in  his  unpretentious  work 
of  1604,  the  Paralipomena  ad  Vitellionem,  this  Vitello  (Witelo) 
being  the  German  or  Polish  scholar  already  mentioned.  In 
this  little  work  Kepler  explained  the  mechanism  of  the  eye, 
comparing  the  retina  to  the  canvas  on  which  images  were  de- 
picted. He  showed  that  imperfect  vision  is  caused  by  the  failure 
of  the  rays  of  light  to  converge  properly  on  the  retina.  In  1611 
he  published  a  work  on  dioptrics  in  which  he  set  forth  his  ideas, 

aG.  Pittarelli,  "Intorno  al  libro  'de  perspectiva  pingendi'  di  Pier  dei  Frances- 
chi," Atti  del  Congresso  internazionale  di  scienze  storiche,  XII  (Rome,  1904),  262. 

2H.Wieleitner,  "  Zur  Erfindung  der  verschiedenen  Distanzkonstruktionen  in  der 
malerischen  Perspektive,"  Repertorium  jur  Kunstwissenschaft,  XLII  (1920),  249. 

sTrattato  delta  pit  turn,  Paris,  1651.   See  Volume  I,  page  294. 

4  Born  1500;  died  c.  1571.  Various  dates  of  his  death  are  given,  ranging  from 
December  13,  1569,  to  February  25,  1571. 

5 P.  Duhem,  fitudes  sur  Leonard  de  Vinci,  s£r.  I,  p.  225  (Paris,  1906) ;  G.  P. 
Carpani,  Memoirs  of  B.  Cellini,  English  translation  by  Roscoe,  London,  1878; 
(Euvres  completes  de  Benvenuto  Cellini,  2d  ed.,  2  vols.,  Paris,  1847. 

*Underweysung  der  messung,  Nurnberg,  1525;  see  Volume  I,  page  326. 

7  See  Volume  I,  page  309. 

sOpticae  libri  quatuor,  ex  voto  Petri  Kami  novissimo,  per  Fr.  Risnerum  .  .  ., 
Cassel,  1606  (posthumous).  9 Basel,  1572.  See  page  341. 


RENAISSANCE  WRITERS 


343 


imperfect  though  they  were,  upon  the  law  of  refraction.  He 
also  gave  a  scientific  explanation  of  the  telescope,  then  recently 
invented.  In  the  same  year  (1611)  Antonio  de  Dominis,  arch- 
bishop of  Spalato,  published  his  De  Radiis  Lucis  in  Vitris  Per- 
spectiva  et  hide,  in  which  he  explained  more  fully  than  his 


^li^f-'^^ 


DESCARTES'S  EXPLANATION  OF  THE  RAINBOW 
From  his  Meteor  a,  1656  ed.,  p.  214 

predecessors  the  phenomena  of  the  rainbow,  basing  them  upon 
principles  of  refraction.  It  was  Descartes,  however,  who  in  his 
Dioptrica  (1637)  gave  the  law  that  the  sine  of  the  angle  of 
incidence  has  a  constant  ratio  to  the  sine  of  the  angle  of  refrac- 
tion, the  ratio  being  a  function  of  the  medium.  The  law  was, 
in  fact,  known  to  Snell  twenty  years  earlier,  but  he  had  failed 
to  set  it  forth  in  print,  although  he  had  taught  it.  Nevertheless, 


344  INSTRUMENTS  IN  GEOMETRY 

Descartes  was  living  in  Holland  at  that  time,  and  there  is 
some  suspicion  that  he  had  there  heard  of  SnelPs  discovery. 
In  his  Dioptrica,  Descartes  completed  the  theory  of  the  rain- 
bow by  giving  an  explanation  of  the  outer  bow. 

Just  before  Descartes  produced  his  work,  Francois  Aguillon 
(1566-1617),  a  Belgian  Jesuit,  published  a  treatise1  of  some 
importance.  In  this  he  used  the  term  "stereographic  projec- 
tion," although  the  idea  was  known  to  the  Greeks. 

Frans  van  Schooten  the  Younger  published  in  1656-1657  a 
book  of  mathematical  exercises2  in  which  he  treated  of  perspec- 
tive,8 but  it  contained  little  that  was  original. 

Newton's  Work.  Newton  began  to  work  seriously  on  optics 
about  1666.  In  his  treatise4  of  1704  he  states  that  part  of  the 
treatise  was  written  in  1675;  an(l  in  his  posthumous  work  the 
editor  states  that  Newton  a  first  found  out  his  Theory  of  Light 
and  Colours"  as  early  as  1666,  lecturing  upon  it  in  1669.  By 
this  time  the  elementary  theory  of  optics  was  well  established. 

ii.    INSTRUMENTS  IN  GEOMETRY 

Early  Instruments.  Before  the  invention  of  the  telescope, 
microscope,  and  vernier  there  can  hardly  be  said  to  have  been 
any  instruments  of  precision.  For  practical  land  measure, 
however,  for  leveling,  and  for  the  measuring  of  heights,  the 
world  developed  several  interesting  instruments  worthy  of 
mention. 

In  general,  the  ancient  surveyors  measured  distances  by  the 
use  of  a  rope  or  a  wooden  rod,  the  units  of  measure  varying  in 
different  localities.  They  laid  off  right  angles  by  the  use  of  an 

^Francesci  Aguilonii  e  societate  Jesu  Opticorum  libri  VI,  Antwerp,  1613. 

2 Exercitationum  Mathematicarum  libri  F,  Amsterdam,  1656-1657;  Dutch  edi- 
tion, ibid.,  1659. 

3<tEen  korte  verhandeling  van  de  Fondementen  der  Perspective."  It  was  also 
separately  printed,  Amsterdam,  1660. 

4Opticks :  or,  a  Treatise  of  the  Reflexions,  Refractions,  Inflexions,  and  Colours 
of  Light  .  .  .  ,  London,  1704,  with  various  later  editions  and  translations.  His 
second  work  in  point  of  publication,  but  not  of  composition,  was  his  Optical 
Lectures  Read  in  .  .  .  i66g,  published  posthumously,  London,  1728. 


EARLY  INSTRUMENTS 


345 


instrument  resembling  the  carpenter's  square  of  the  present 
time,  by  a  kind  of  cross  placed  horizontally  on  a  staff,  or  by 
the  3-4-5  relation  applied  to  a  stretched  cord.  For  finding  a 
level  they  ordinarily  used  a  right-angled  isosceles  triangle  with 
a  plumb  line.  Illustrations  of  such  instruments  are  found  on 
monuments  to  certain  ancient  surveyors.1 

Early  Printed  Books.  The  early  printed  books  give  us  much 
information  as  to  the  nature  of  the  instruments  inherited  from 
the  Middle  Ages.  Of  these  there  may  be  mentioned  the  mirror 
for  the  measuring  of  heights  by  the  forming  of  similar  triangles, 
the  geometric  square  (quadratum  geometricum) ,  the  quadrant, 


THE  QUADRATUM  GEOMETRICUM 

From  Oronce  Fine's  De  re  &  praxi  geomelrica,  Paris,  1556.   The  two  triangles 
being  similar,  AB  is  easily  found  from  the  distances  AC  and  AF 

the  astrolabe,  and  the  cross-staff  (baculum,  also  called  the 
baculus).  The  method  of  using  most  of  these  instruments  is 
evident,  but  a  brief  description  of  some  of  them  will  be  helpful.2 

The  Square.  The  simplest  of  all  the  instruments  of  this  class 
was  the  ordinary  carpenter's  square,  known  in  some  of  the 
works  on  mensuration  as  the  geometric  square.  Its  use  in  find- 
ing short  distances  by  means  of  the  principle  of  similar  tri- 
angles will  be  easily  understood  from  the  above  illustration. 

1  See  page  357. 

2\V.  E.  Stark,  "Measuring  Instruments  of  Long  Ago,"  School  Science  and 
Math.,  X,  48,  126;  M.  Curtze,  "Ueber  die  im  Mittelalter  zur  Feldmessung 
benutzten  Instrumente,"  Bibl.  Math.,  X  (2),  65;  M.  Cantor,  Die  Romischen 
Agrimensoren,  Leipzig,  1875 ;  E.  N.  Legnazzi,  Del  Catasto  Romano,  Verona,  1887 ; 
G.  Rossi,  Groma  e  squadro  ovvero  Storia  dell'  Agrimensura  Italiana  dai  tempi 
antichi  al  secolo  XVII0,  Turin,  1877. 


346  INSTRUMENTS  IN  GEOMETRY 

The  Baculum.  In  its  simplest  form  the  baculum,  arbalete 
(crossbow),  geometric  cross,  cross-staff,  or  Jacob's  staff1  was 
a  rod  about  4  feet  long,  of  rectangular  cross  section,  and  having 
a  crosspiece  that  could  slide  upon  it  and  always  remain  perpen- 
dicular to  it.  The  staff  ,was  marked  off  in  sections  each  equal 
in  length  to  the  crosspiece.  In  actual  use  the  crosspiece  was 


PRACTICAL  MATHEMATICS  IN  THE    17TH  CENTURY 
From  Simon  Jacob's  work  of  1560  (1565  ed.) 

first  placed  at  one  of  the  division  marks  of  the  staff,  the  ob- 
server then  facing  approximately  the  mid-point  of  a  line  that 
he  wished  to  measure  and  standing  at  a  distance  such  that, 
when  he  sighted  along  the  staff,  the  crosspiece  should  be  parallel 
to  the  line  and  just  cover  it.  The  crosspiece  was  then  moved  to 
the  next  division  on  the  staff,  the  observer  taking  a  position 
where  the  first  process  of  covering  the  line  with  the  crosspiece 
could  be  repeated,  as  shown  in  the  illustration  on  page  347. 

1  This  name  had  various  other  uses,  however. 


THE  BACULUM 


347 


The  length  of  the  line  to  be  measured  was  then  the  same  as  the 
distance  between  the  two  positions  of  the  observer.  There  were 
also  various  other  methods  of  using  the  instrument.1 

Sector  Compasses.  About  the  year  1597  Galileo  invented  the 
proportional  compasses,2  or  sector  compasses,  an  ingenious  de- 
vice for  solving  a  variety  of  problems  often  met  by  architects, 
engineers,  and  others  who  have  much  to  do  with  applied  mathe- 
matics. The  instrument  consists  ordinarily  of  two  brass  rules 


THE  BACULUM,  OR  CROSS-STAFF 

From  Oronce  Fine's  De  re  &  praxi  geometrica,  Paris,  1556,  showing  the  methods 
of  measuring  distances 

hinged  at  one  end.  There  are  usually  six  pairs  of  lines,  three 
on  each  face,  radiating  from  the  pivot.  One  pair  might,  for 
example,  represent  equal  parts;  another,  squares;  and  the 
third,  lines  of  polygons ;  but  this  varied  according  to  the  pur- 
pose of  the  particular  instrument. 

To  give  a  single  illustration  of  its  use,  suppose  that  each  line 
of  equal  parts  is  divided  into  200  equal  segments,  numbered 
by  tens,  beginning  at  the  pivot.  Then,  to  divide  any  given  line 

aFor  a  brief  resume  see  G.  Bigourdan,  U  Astronomic,  Evolution  des  idees  et  des 
methodes,  p.  116  (Paris,  1911;  1920  ed.),  hereafter  referred  to  as  Bigourdan, 
Astronomic.  2Le  operazioni  del  compasso  geometrico  e  militate,  Padua,  1606. 


INSTRUMENTS  IN  GEOMETRY 


segment  into  any  number  of  equal  parts,  say  nine,  open  a  pair 
of  ordinary  dividers  to  the  length  of  the  segment,  then  open 
the  sector  compasses  so  that  one  point  of  the  dividers  rests  on 


ASTROLABE   OF    CHAUCER  S   TIME 

Fine  piece  of  medieval  workmanship  now  in  the  British  Museum.   It  may  well  be 
that  Chaucer  himself  made  use  of  this  in  preparing  his  treatise  on  the  astrolabe 

90  on  one  face  and  the  other  point  rests  on  90  on  the  other 
face;  then  the  distance  from  the  10  on  one  face  to  the  10  on 
the  other  is  one  ninth  of  the  length  of  the  given  line  segment. 

Astrolabe.    Of  all  the  early  astronomico-mathematical  instru- 
ments none  was  better  known  than  the  astrolabe.    The  name 


ITALIAN  ASTROLABE  OF  1558 

It  bears  the  inscription  "  Patavii  Bernardinvs  Sabevs  faciebat  MDLVIII." 
From  the  author's  collection 


THE  ASTROLABE  IN  SIMPLE  MENSURATION 

From  Bartoli's  Del  Modo  di  Misvrare,  Venice,  1589,  showing  simple  work  in  a 
crude  kind  of  trigonometry 


350 


INSTRUMENTS  IN  GEOMETRY 


is  Greek  and  means  the  taking  of  the  stars.1  Hence  any  instru- 
ment for  measuring  the  angles  by  which  a  star  was  " taken" 
(as  a  sailor  today  speaks  of  "  taking "  the  sun)  was,  strictly 

speaking,  an  astrolabe. 
One  of  the  early  forms 
was  the  armillary  sphere, 
so  called  from  the  armil- 
lae?  or  rings,  which  were 
so  arranged  as  to  form 
two,  or  sometimes  three, 
circles,  ordinarily  placed 
at  right  angles  to  one 
another.  One  ring  usu- 
ally corresponded  to  the 
plane  of  the  equator  and 
the  other  to  the  plane  of 
the  meridian.  By  these 
two  circles  the  ancients 
determined  the  two  co- 
ordinates of  a  star.  The 
astrolabe  described  by 
Ptolemy  the  astronomer 
is  a  kind  of  armillary 
sphere,3  and  furthermore 
these  spheres  are  first 
heard  of  in  connection 
with  the  school  with  which  he  was  associated.  It  is  asserted 
by  early  writers  that  Eratosthenes,  through  his  interest  in  geod- 
esy and  astronomy,  induced  King  Ptolemy  III  to  have  such 
instruments  made  and  placed  in  the  museum  at  Alexandria. 

aFrom  Affrpov  (as'tron,  a  heavenly  body)  +  Xa/u/3(£m*>,  \afitiv  (lamba'nein,  la- 
beinf,  take) .  Ptolemy  spoke  of  the  two  circles  that  he  used  in  locating  a  star  as 
d<rrpoX(i/3ot  KJL>K\OI  (astrola'boi  ky'kloi)  and  spoke  of  the  whole  instrument  as 
&<rTpo\d(Bov  ftpyavov  (astrola'bon  or'ganon}  or,  commonly,  as  6  dcrrpoXd/Sos  (ho 
astrolabes}.  See  J.  Frank,  "Zur  Geschichte  des  Astrolabs,"  Erlangen  Sitzungs- 
berichte,  50-51.  Band,  p.  275;  R.  T.  Gunther,  Early  Science  in  Oxford,  II,  181. 

2Armilla  means  an  armlet,  bracelet,  hoop,  or  ring.  It  is  probably  a  diminutive 
of  armus,  the  shoulder  or  upper  arm. 

3  Almagest,  VII,  2,  4. 


CHAMPLAIN'S  ASTROLABE 

Found  near  the  Ottawa  River  about  1870.  It 
was  made  in  Paris  in  1603.  This  is  the  type  of 
astrolabe  known  as  the  planisphere.  From  the 
collection  of  Samuel  V.  Hoffman,  New  York 


THE  PLANISPHERE 


351 


Planisphere.  Another  ancient  and  common  form  of  the  as- 
trolabe consisted  simply  of  a  disk  upon  the  rim  of  which  were 
marked  the  units  of  angle  measure.  Such  instruments  were 
probably  well  known  in  ancient  times  among  all  who  made  any 
scientific  study  of  the  stars.  That  they  were  familiar  in  ancient 
Babylon  we  have  definite  proof.1  Fragments  of  several  such 
instruments  have  been  found  and  the  inscriptions  interpreted. 
They  go  back  to  the  2d  millennium  B.C.,  which  goes  to  show  that 
the  early  Greeks  undoubtedly  knew  of  their  value  and  made 


THE  QUADRANT 
From  the  Protomathesis  of  Oronce  Fine,  Paris,  1530-1532 

use  of  them  in  angle  measure.  These  astrolabes  are  in  the  form 
of  planispheres  and  are  made  of  clay,  baked  like  the  tablets.2 
A  planisphere  may  be  defined  as  a  stereographic  projection  of 
the  celestial  sphere  either  upon  the  plane  of  the  equator  or 
upon  the  plane  of  the  meridian. 

Such  instruments  were  used  in  various  practical  ways  in 
which  angle  measure  was  the  chief  purpose,  and  this  use  con- 
tinued until  recent  times.  Even  now  they  are  seen  in  the 
Orient  in  the  hands  of  the  astrologers. 

*£.  F.  Weidner,  Handbuch  der  Babylonischen  Astronomie,  Lieferung  I,  62; 
with  bibliography,  Leipzig,  1915. 

2 For  a  photographic  reproduction,  see  Weidner,  loc.  cit.,  p.  107,  from  A.  Jere- 
mias,  Handbuch  der  altorientalischen  Geisteskultur,  Leipzig,  1913.  There  is  a 
good  specimen  in  the  British  Museum. 


3S2 


INSTRUMENTS  IN  GEOMETRY 


The  planisphere  in  common  use  in  later  times  represents  the 
stereographic  projection  of  the  celestial  sphere  upon  the  plane 
of  the  equator,  the  eye  being  at  the  pole.  Planispheres  of 


BRASS  QUADRANT 

Austrian  work  of  the  i8th  century.   The  original  is  29.5  cm.  square.   From  the 

author's  collection 


various  types  were  used  by  early  navigators  for  the  purpose 
of  finding  the  elevation  of  the  north  star,  or  for  other  angle 
measurements,  and  were  often  furnished  with  several  plates 
which  could  be  so  adjusted  as  to  allow  the  instrument  to  be 
used  in  different  latitudes. 


THE  ASTROLABE  IN  THE  EAST 


353 


The  Astrolabe  in  the  East.  From  Babylon1  the  astrolabe 
may  have  passed  to  China  and  India,  or  vice  versa.  At  any 
rate,  Mesopotamia  seems  to  have  been  the  source  from  which 
the  Greeks  derived  their  knowledge  of  the  instrument.  It  is 
probable  that  Thales  used 
it  in  measuring  the  dis- 
tances of  ships,  since 
the  Babylonian  astron- 
omy was  already  becom- 
ing known  in  the  Greek 
civilization.  It  may  be 
inferred  from  Plato's 
Timceus  that  some  such 
instrument  was  in  use 
in  his  day,  but  in  any 
case  an  astrolabe  of  some 
type  was  known  to  Era- 
tosthenes, Hipparchus, 
and  other  Greek  as- 
tronomers even  before 


Ptolemy    described 
armillary  sphere.2 


the 


Arab  Treatises  on  the 


SMALL  IVORY  QUADRANT 


A    ,     ,   ,  T      ,  ,       ,,     .      Italian  work  of  the  i8th  century.   If  we  sight 

Astrolabe.     Led  by  their  through  holes  in  the  two  projections  on  the 

Study  of  Greek  astron-  upper  right-hand  edge,  the  angle  of  elevation  is 
Omy  the  Arabs  begin  indicated  by  the  plumb  line  and  the  arc.  The 

ning  in  the  9th  century,  original  is  5  cm'  bycosu^on  Frora  the  author's 
wrote  numerous  works 

upon  the  astrolabe,  and  these,  in  turn,  influenced  the  medieval 
scholars  of  Europe.  Thus  we  find  Messahala  (c.  800)  compos- 
ing a  work  upon  the  subject,  which  formed  the  basis  of  two 
manuscripts  by  Rabbi  ben  Ezra  (c.  1140).  From  one  of  these 
manuscripts  Chaucer  (c.  1380)  seems  to  have  drawn  his  in- 
formation for  his  treatise  upon  the  astrolabe. 

*A.  H.  Sayce  and  R.  H.  M.  Bosanquet,  "Babylonian  Astronomy,"  Monthly 
Notices  of  the  Royal  Astron.  Society,  XL,  No.  3,  with  illustrations. 
2R.  Wolf,  GeschichU  der  Astronomic,  p.  160  (Munich,  1877). 


*4 


INSTRVCTIO 

CAPVT    XII. 
OVA  'RATIONS  EL  1C  1  END  A  SIT  At* 

^s£,~vnn.  4UCV1VS  RMI,   gyAE   i^fCCESSrtf  KPN 
tdmittitiVtfMt  ijMtruiitm,  *rtiHm,&c. 

mil  ii  l«i  n.i. III! 


S'untje  Uet^  rgjt^nnm  Vmlrr#,v$Jiil>(!rr4ftu>»ie  nut 
nor  is  a  ittcuori,  ad  Jz,itaje  ht&efjfeatiii  inter 

ad  tottf  altitiidmtm-.auoA  tiia  p 
£ewijbition'i  Its Jiic  profatur. 


EXPLANATION   OF   THE   QUADRANT 

From  De  Quadrante  Geometrico,  usually  referred  to  Cornelius  de  Judeis,  Niirn- 

berg,  1594,  but  in  fact  written  by  Levinus  Hulsius.    Cornelius  made  the  drawings 

with  the  help  of  Martin  Geet 


THE  QUADRANT 


3SS 


Quadrant.   Closely  related  to  the  astrolabe  is  the  quadrant, 
an  instrument  in  which  only  a  quarter  of  a  circle  is  used.   It 


USE  OF  THE  QUADRANT 
From  Ottavio  Fabri's  V  Uso  della  Squadra  Mobile,  Trent,  1752 

appears  in  various  forms,  sometimes  without  an  arc,  the  angles 
being  read  on  the  sides  of  a  square.  The  earliest  description 
that  we  have  is  given  in  the  Almagest,  and  on  this  account  the 


DRUMHEAD  TRIGONOMETRY 

A  common  method  of  triangulating  in  the  i6th  century.   From  Belli's  Libro  del 
Misvrar,  Venice,  1569 


G      t/riji 


EARLY  APPROACH  TO  THE  PLANE  TABLE 

The  plane  table  in  various  forrns  was  probably  developed  from  such  an  instrument 

as  the  one  here  shown.  In  this  case  the  table  was  used  merely  for  taking  horizontal 

angles.   From  Cosimo  Bartoli,  Del  Modo  di  Misvrare,  Venice,  1589 


USE  OF  THE  SHADOW  AND  THE  MIRROR  IN  MEASURING  HEIGHTS 
From  Giovanni  Pomodoro's  La  Geometria  Prattica,  Rome,  1624 


THE  QUADRANT 


357 


MAEBVMM-L    \ 
MACEDOPAM 

M<AEBvnvs.A/M: 

CALLlSTRATVSf 


honor  of  its  invention  is  usually  awarded  to  Ptolemy.  He  used 
a  stone  cube,  on  one  of  the  faces  of  which  the  quadrant  was  cut. 
On  this  was  mounted  a  small  cylindric  pipe,  as  we  should  mount 
a  telescope,  and  by  this  device  he  was  able  to  take  the  height  of 
the  sun,  evidently  by  means  of  the  ray  of  light  which  shone 
through  the  cylinder.  There  is  no 
indication  of  the  size  of  Ptolemy's 
quadrant,  but  if  we  judge  by  the 
later  specimens  in  use  in  the  East, 
and  by  the  incomplete  records,  it 
was  probably  a  large  one.  He  says 
that  he  used  his  quadrant  in  taking 
many  astronomical  observations; 
but  he  gives  no  results,  and  it  is 
rather  doubtful  whether  he  did 
more  than  suggest  the  instrument, 
depending  upon  the  results  se- 
cured by  Eratosthenes  and  others 
of  his  predecessors. 

The  quadrant  is  described  in 
many  works  of  the  i6th,  lyth,  and 
1 8th  centuries,  but  with  the  inven- 
tion of  the  telescope  all  devices  of 
this  kind  gradually  gave  way  to  the 
transit  in  astronomical  work.  The 
sextant  was  invented  by  Thomas 
Godfrey,  of  Philadelphia,  in  1730. 

Drumhead  Trigonometry.  The 
continual  warfare  of  the  Renais- 
sance period  shows  itself  in  many  ways  in  the  history  of  mathe- 
matics. Some  of  these  manifestations  are  mentioned  from  time 
to  time  in  this  work,  and  one  of  them  is  related  to  the  subject 
now  under  consideration.  Several  writers  of  the  i6th  century 
give  illustrations  of  the  use  of  the  drumhead  as  a  simple  means 
of  measuring  angles  of  elevation  in  computing  distances  to  a 
castle  or  in  finding  the  height  of  a  tower.  Such  an  illustration 
is  shown  on  page  355  and  is  self-explanatory. 

ii 


IVLlALLHESVO 
POMPONIALtSEL 


DIADLAKTIo 

2.—L  ____  -1 


ANCIENT  LEVELS  AND  SQUARE 

From  the  tomb  of  Marcus  /£bu- 

tius  Macedo.     Like  the  tomb  of 

Lucius    ^Ebutius    Faustus    (page 

361),  it  is  of  uncertain  date 


358 


INSTRUMENTS  IN  GEOMETRY 


Somewhat  related  to 
this  crude  instrument  is 
one  for  taking  horizontal 
angles,  as  illustrated  on 
page  356.  From  this  de- 
vice the  plane  table  was 
probably  developed. 

The  Mirror.  In  the 
early  printed  works  on 
applied  geometry  there 
are  frequent  references 
to  the  speculum,  a  hori- 
zontal mirror  used  in 
measuring  heights  by  the 
aid  of  similar  triangles. 
The  method  is  still  in  use 
for  certain  purposes,  but 
in  the  i6th  and  iyth  cen- 
turies it  seems  to  have 
been  extensively  employed.  On  account  of  the  difficulty  of 
obtaining  a  satisfactory  level,  and  the  fact  that  one  triangle  was 


EARLY  METHODS  OF  LEVELING 

From  Pomodoro's  La  Geometria  Prattica,Rome, 
1624.   This  was  an  early  Egyptian  method  and 
was  transmitted  through  the  Greek  and  Ro- 
man surveyors 


JAPANESE  LEVELING  INSTRUMENT 
From  Murai  Masahiro's  Ryochi  Shinan,  a  work  on  surveying,  about  1732 


THE  SPECULUM  AND  LEVEL 


359 


small  and  not  easily  measured  with  accuracy,  the  method  was 
not  of  much  value.  The  plan  of  using  the  speculum  was  based 
upon  the  principle  of  similar  triangles  and  is  illustrated  on 
Page  356. 

Leveling  Instruments.  The  common  leveling  instrument  of 
ancient  times  was  the  isosceles  triangle  with  a  plumb  line  from 
the  vertex.  This  is  found  in  Egyptian  remains,  is  represented 
on  the  monuments  of  Roman  surveyors,1 
is  referred  to  by  medieval  writers,  and 
is  still  in  general  use  in  various  parts  of 
the  world.  Until  the  invention  of  the 
telescope,  and  the  consequent  increase 
in  accuracy  of  observation,  it  satisfied 
all  ordinary  needs.  There  are  many 
reasons  for  believing  that  the  early 
Egyptian  surveyors  who  laid  out  the 
pyramids  made  use  of  this  instrument 
for  establishing  their  levels.  An  inter- 
esting variant  of  this  instrument  is  seen 
in  the  quadrilateral  which  the  Japanese 
scholars  developed  before  the  free  in- 
flux of  Western  mathematics.  Such  a 
device  is  shown  on  page  358. 

The  principles  underlying  the  later 
forms  of  leveling  instruments  were  not 
numerous ;  in  fact,  the  fundamental  ones 
were  only  two  in  number,2  the  older 
one  depending  upon  the  plumb  line, 
and  the  later  one  upon  the  state  of 


EARLY  JAPANESE  SUR- 
VEYING INSTRUMENT 

From  a  drawing  in  a  manu- 
script    of     a     work     (see 
page  358)  by  Murai  Masa- 
hiro,  about  1732 


10n  the  leveling  instruments  of  the  Romans  see  C.  G.  de  Montauzan,  Essai  sur 
la  science  et  Vart  de  I'ingenieur  aux  premiers  siecles  de  I' Empire  Romain,  pp.  46, 
62,  74  (Paris,  1908).  On  Greek  and  Roman  engineering  instruments  see  Pauly- 
Wissowa ;  R.  C.  Skyring  Walters,  Transactions  of  the  Newcomen  Society,  II,  45, 
and  T.  East  Jones,  ibid.,  p.  61 ;  E.  N.  Legnazzi,  Del  Catasto  Romano,  Verona,  1887. 

2  For  a  discussion  of  the  subject  see  N.  Bion,  Traite  de  la  Construction  et  des 
principaux  Usages  des  Instrumens  de  Mathematique,  p.  285  (Paris,  1713;  ed.  of 
The  Hague,  1723).  This  is  the  best  of  the  early  classical  treatises  upon  the  sub- 
ject of  mathematical  instruments,  and  is  profusely  illustrated. 


360  INSTRUMENTS  IN  GEOMETRY 

equilibrium  of  some  kind  of  liquid.  The  former  was  used  in 
the  ancient  triangle  illustrated  on  page  358,  and  in  related  types, 
while  the  latter  is  still  seen  in  the  ordinary  level  used  by  car- 
penters and  in  the  leveling  instrument  used  by  engineers, — the 
niveau  a  Veau  described  by  Bion  (1713).  The  triangle  level 
had  various  special  forms,  such  as  an  inverted  T  (that  is,  _L) 
with  a  plumb  line  along  the  vertical  arm.  In  this  form  it  was 
called  by  French  writers  the  niveau  d'air.  The  horizontal  part 
was  usually  a  tube  through  which  the  observer  could  sight 
when  running  a  level  line. 

After  the  telescope  was  invented  the  tube  was  fitted  with 
lenses,1  and  the  instrument  became,  either  with  the  plumb  line 
or  with  the  water  level  attached,  not  unlike  the  instrument  in 
common  use  today.  Sometimes  the  plumb  line  and  the  level 
were  attached  to  the  same  instrument.  Huygens  invented  a 
level  in  the  form  of  a  cross  on  which  an  inverted  T  was  hung 
by  a  ring  at  the  top,  the  telescope  being  kept  horizontal  by 
means  of  a  weight.2 

Until  the  advent  of  a  new  type  of  engineering,  made  possible 
by  the  commercial  use  of  structural  steel,  the  level  was  used 
chiefly  for  two  purposes.  The  first  of  these  was  the  construc- 
tion of  canals  for  purposes  of  irrigation,  particularly  in  Meso- 
potamia and  Egypt,  and  of  aqueducts  as  a  result  of  the  Roman 
demand  for  pure  water.  The  second  use  was  seen  in  the  build- 
ing of  fortifications,  particularly  during  and  as  a  result  of  the 
wars  of  the  iyth  century.  The  textbooks  of  that  century  on 
applied  geometry  (mensuration)  gave  much  attention  to  the 
subject.  The  general  practice  in  leveling  was  not  unlike  that 
of  the  present  time,  the  chief  difference  being  in  the  degree 
of  precision  of  the  instruments  used.  It  is  evident  that  in  con- 
struction work  of  any  extent  the  level  was  always  necessary,  but 
its  elaborate  use  in  modern  engineering — as  in  railway  gradi- 
ents, tunnels,  and  bridges,  and  as  in  the  erecting  of  modern 
office  buildings  of  great  height — surpasses  anything  conceived 
of  in  ancient  times. 

1The  niveau  d'air  a  lunette  of  Bion's  treatise  of  1713. 
2 Bion,  loc.  cit.,  p.  no. 


ROMAN  SURVEYING  INSTRUMENTS 


Other  Surveying  Instruments. 
It  is  not  possible,  in  the  limited 
space  that  should  be  allowed  the 
subject  in  a  work  of  this  kind,  to 
mention  all  the  simple  surveying 
instruments  and  devices  that  have 
come  down  to  us  from  the  Egyp- 
tian, Greek,  and  Roman  civiliza- 
tions. The  simple  staff,  with  a 
crude  diopter  through  which  to 
sight  in  running  a  line,  is  found 
in  all  parts  of  the  world  and  is 
probably  very  ancient. 

From  such  a  humble  origin 
sprang  the  groma  used  by  the 
Roman  surveyors  and  here  illus- 
trated from  the  tomb  of  Lucius 
^butius  Faustus.  He  is  men- 
tioned in  the  third  line  as  a  men- 
sor, but  the  term  more  commonly 
used  in  the  case  of  a  land  sur- 
veyor was  agrimensor  (field  meas- 
urer) or  gromaticus  (one  who 
used  the  groma}.  The  groma 
(cruma,  gruma)  consisted  of  the 
stella  (the  star-shaped  part)  and 
the  pondera  (the  plumb  lines). 
Surveyors  in  the  time  of  the  em- 
pire often  spoke  of  the  machina 
or  machinula  which  they  used 
and  which  consisted  of  two  parts, 
the  groma  or  stella  and  the  fer- 
r amentum  (the  iron  standard). 
Hence  Hyginus  (c.  120)  says 
"ferramento  groma  superpona- 
tur"  (let  the  groma  be  placed 
upon  the  iron  standard). 


KEBVTIV.V 

OTWMENS 

<rp  vm  tf  .mi'  ET 


,     V/ORF-  FT'JVS'  E 
2EPVRE  •      JUBERT 


ANCIENT   SURVEYING 
INSTRUMENT 

From  the  tomb  of  Lucius  ^Ebutius 
Faustus,  a  mensor  (agrimensor,  sur- 
veyor, or  perhaps  a  measurer  for 
architects  and  builders).  The  entire 
inscription  was  as  follows :  Tribv 
Clavdia  Lvcivs  Aebvtivs  Lvcii 
libertvs  Favstvs  mensor  sevir  sibi 
et  Arriae  Qvinti  libertae  avctae 
vxori  et  svis  et  Zepyre  libertae 
vivvs  fecit.  The  instrument  shown 
is  the  groma 


362  INSTRUMENTS  IN  GEOMETRY 

In  the  i  yth  and  i8th  centuries,  stimulated  by  the  metal  work 
of  the  Italian  artists,  the  instrument-makers  of  France  and 
Italy  produced  many  beautiful  pieces  of  workmanship  designed 
with  much  ingenuity.  These  pieces  are  occasionally  seen  in 
museums,  and  one  is  shown  in  the  following  illustration.1 


ELABORATE   MATHEMATICAL   INSTRUMENT 

Showing  artistic  metal  work  of  the  i8th  century.   Now  in  the  Metropolitan 
Museum  of  Art,  New  York 

Such  instruments  were  often  elaborately  engraved  and  some 
of  those  apparently  made  for  the  noble  patrons  of  the  sciences 
were  even  gold-plated.  One  of  the  elaborate  forerunners  of  the 
range  finder  is  shown  on  page  363. 

!R.  T.  Gunther,  Early  Science  in  Oxford  (Vols.  I  and  II,  Oxford,  1922,  1923), 
with  a  catalogue  of  the  early  mathematical  instruments  belonging  to  the  Univer- 
sity and  colleges  of  Oxford.  See  particularly  II,  192-233. 


EARLY  FORM  OF  RANGE  FINDER 

From  Danfrie's  Declaration  de  I'Vsage  du  Graphometre,  Paris,  1597 ;  appendix  on 

trigonometry,  p.  n 


364 


INSTRUMENTS  IN  GEOMETRY 


Other  Astronomical  Instruments.  While  this  work  is  only 
indirectly  concerned  with  astronomy,  many  astronomical  instru- 
ments are  distinctly  mathematical,  and  some  of  them  are  partic- 
ularly interesting  as  works 
of  art.  Such  are  the  elab- 
orate bronze  pieces  on  the 
walls  of  the  city  of  Peking, 
mostly  due  to  the  Jesuit  in- 
fluence which  began  about 
1600,  but  partly  native  in 
their  design  and  general 
plan.  It  was  through  the 
devising  and  use  of  instru- 
ments like  these  that  such 
missionaries  as  F.  Verbiest 
and  J.  A.  Schall  von  Bell  (c. 
i66a)  were  able  to  make 
observations  that  demon- 
strated, even  to  the  hostile 
critics,  the  superiority  of 
European  astronomy  over 
that  of  the  Chinese.  One 
of  these  pieces  is  shown  in 
the  illustration.1 

In  Persia  and  India  there 
are  still  to  be  found  celes- 
tial spheres  of  great  beauty, 
generally  dating  from  the 
iyth  century.  These  are 
usually  of  bronze,  some- 
times with  silver  stars. 
The  Hindus,  Persians,  and  Arabs  have  also  left  many  astro- 
labes of  beautiful  workmanship,  some  of  them  with  constella- 
tions or  particular  stars  represented  in  silver.  Until  the 
invention  of  the  telescope  their  smaller  types  of  astronomical 
instruments  were  unsurpassed  both  in  beauty  and  in  accuracy. 


BRONZE   QUADRANT   ON   THE  WALLS 
OF  PEKING 

One  of  several  elaborate  bronze  instru- 
ments, most  of  them  made  under  the 
influence  of  Jesuit  missionaries 


aSee  Volume  I,  page  272. 


HINDU  OBSERVATORIES 


365 


The  most  interesting  of  the  Hindu  instruments  are  found  in 
the  five  observatories  built  by  the  Maharajah  Jai  Singh  be-f 
tween  1728  and  I734-1  These  observatories  were  located  at' 
Delhi,  Jaipur,  Benares,  Ujjain,  and  Mathura,  and  represent 
the  Arab  astronomico-astrological  science  instead  of  the  native 
Hindu  or  the  European.  Jai  Singh  was  a  Sikh  by  birth  and: 
was  so  interested  in  astronomy  that  he  translated  Ulugh  Beg's 


HINDU  CELESTIAL  SPHERE 

This  piece  is  of  bronze,  the  stars  being  inlaid  in  silver.   It  was  made  c.  1600. 
From  the  author's  collection 

catalogue  of  the  stars  (c.  1435).  He  was  of  the  opinion  that 
the  small  brass  instruments  used  in  Samarkand  were  not  accu- 
rate enough,  and  hence  he  determined  to  construct  pieces  so 
large  and  substantial  as  to  leave  no  doubt  about  the  validity  of 
the  observations.  The  results  were  monumental  and  are  still 
the  object  of  admiration  to  those  interested  in  the  science 
of  India.  An  illustration  showing  one  of  the  most  elaborate 
and  carefully  preserved  of  these  observatories  (the  one  at 
Jaipur)  will  be  found  on  page  366. 

1G.  R.  Kaye,  The  Astronomical  Observatories  of  Jai  Singh,  Calcutta,  1918; 
see  also  the  review  in  the  Journal  of  the  Royal  Asiatic  Soc.,  July,  1919,  p.  427. 


366 


INSTRUMENTS  IN  GEOMETRY 


The  Jaipur  observatory  was  constructed  by  the  Maharajah 
Jai  Singh  about  1734.  The  Jesuit  missionary  Joseph  Tieffen- 
thaler,  in  a  work  published  in  1785,  speaks  of  it  as  follows: 


OBSERVATORY  AT  JAIPUR,  INDIA 

Showing  the  kinds  of  instruments  generally  used  before  the  days  of  the  telescope. 
This  observatory,  although  relatively  modern,  is  based  upon  ancient  models 

It  is  such  a  work  as  is  never  seen  in  this  part  of  the  world  and,  by 
the  novelty  and  grandeur  of  the  instruments,  strikes  one  with  astonish- 
ment. .  .  .  What  attracts  most  attention  is  a  gnomon  (axis  mundi), 
remarkable  for  its  height  of  70  Paris  feet.  .  .  .  There  are  three  very 
large  astrolabes,  cast  in  copper,  suspended  by  iron  rings. 

This  is  all  quite  as  impressive  to  the  visitor  now  as  it  was 
then.  The  instruments,  which  had  become  damaged  through 
age  and  neglect,  were  restored  in  1902. 


mmf 


JAPANESE  CELESTIAL  SPHERE 
From  a  wood  engraving  in  Baba  Nobutake's  Shogaku  Tenmon  (1706) 


368 


THE  PROBLEM  OF  EARTH  MEASURE 


The  Chinese  influence  shows  itself  in  the  Japanese  works  of 
the  iyth  and  i8th  centuries,  as  is  seen  in  the  illustration  from 
Baba  Nobutake's  work  of  1706.  We  also  find  in  Japan  in  this 
same  period  the  use  of  the  pierced  sphere  in  astronomical  ob- 
servations and  in  the  work  of  the  astrologers.  This  device  was 

common  in  Europe  in  the  latter 
part  of  the  Middle  Ages  and  is 
found  in  various  printed  works 
of  the  1 6th  century. 

12.   THE  PROBLEM  OF  EARTH 
MEASURE 

Need  for  Instruments  of  Pre- 
cision. The  need  for  instru- 
ments of  a  high  degree  of 
precision  was  first  felt  in  con- 
nection with  astronomy  and  the 
measure  of  the  earth.  The 
subject  is  too  extensive  to  be 
considered  at  length  in  a  work 
of  this  kind,  but  its  general 
nature  will  be  understood  by 
a  brief  reference  to  the  history 
of  the  measure  of  the  earth's 
circumference  and  density. 

Circumference  determined 
from  Arc.  It  should  first  be 
understood  that  the  solution  of 
this  problem  did  not  involve  the  ratio  of  the  circumference  to 
the  diameter ;  it  required  the  finding  of  the  circumference  when 
the  diameter  was  unknown.  When  first  undertaken  it  had 
nothing  to  do  with  navigation,  economics,  or  military  conquest; 
it  developed  as  a  purely  abstract  contribution  to  human  knowl- 
edge. The  plan  adopted  by  the  Greeks  was  the  same,  in  basic 
principle,  as  the  one  used  today,  namely,  that  of  measuring  the 
amplitude  and  the  length  of  an  arc  of  a  great  circle  (generally 


JAPANESE  FIGURE  OF  AN 
ASTRONOMER 

Caricature  in  ivory.  From  the  author's 
collection 


EARLY  ATTEMPTS  369 

a  meridian)  and  from  these  data  computing  the  circumference. 
This  led  to  one  of  the  many  branches  of  geodesy,  a  subject  into 
the  history  of  which  we  cannot  enter  at  length  in  this  work. 

Application  of  Circle  Measure  to  Geodesy.  Pythagoras  (c.  540 
B.C.)  was  the  first,  so  tradition  asserts,  to  teach  that  the  earth 
is  a  sphere  and  that  it  is  situated  in  the  center  of  the  universe.1 
This  idea  was  accepted  by  various  Greek  philosophers,  and 
Aristotle  (c.  340  B.C.)  states  that  "the  mathematicians  who 
have  attempted  to  calculate  the  circumference  of  the  earth  say 
that  it  may  be  forty  myriads  of  stadia/'2  that  is,  400,000  stadia. 
The  stadium  varied  so  much  with  ancient  writers  that  this  does 
not  give  us  any  very  satisfactory  information.3  Taking  a  rough 
approximation,  however,  say  ten  stadia  to  an  Anglo-American 
mile,  this  makes  the  circumference  40,000  miles.  Aristotle 
gives  us  no  information  as  to  the  names  of  the  mathematicians 
who  made  the  calculations,  and  none  as  to  the  method  em- 
ployed, but  it  has  been  thought  that  the  approximation  is  due 
to  Eudoxus  (c.  370  B.C.).  It  is  evident,  however,  that  the  cir- 
cumference was  found  by  multiplying  the  length  of  a  known 
arc,  and  not  by  using  the  ratio  of  the  circumference  to  the 
diameter. 

Four  Greek  Computations  of  the  Earth's  Circumference.  From 
the  time  of  Aristotle  to  that  of  Ptolemy  (c.  150)  there  were 
four  noteworthy  attempts  at  measuring  the  earth's  circum- 
ference. Of  these  the  first  is  referred  to  by  Archimedes 
(c.  22$  B.C.),  who  speaks  of  certain  writers  as  having  stated 
that  the  circumference  is  30  myriads  of  stadia,  say  about 
30,000  miles.  He  does  not  mention  the  writers,  and  it  is  pos- 
sible that  he  may  have  referred  to  some  of  the  earlier  attempts 
made  by  his  friend  Eratosthenes  (c  230  B.C.).  In  his  com- 
putation of  the  number  of  grains  of  sand  in  the  universe,  how- 
ever, he  takes  the  circumference  as  ten  times  this  distance,  so 
as  to  be  on  the  safe  side.4 

aOn  this  entire  subject  see  Bigourdan,  Astronomic,  p.  144  seq. 
2De  Ccelo,  II,  16. 

•^One  of  the  stadia  was  125  paces  (double  steps),  or,  say,  625  Roman  feet, 
equal  to  6o6j  Anglo-American  feet.  4  Archimedes,  ed.  Heiberg,  I,  221, 


370  THE  PROBLEM  OF  EARTH  MEASURE 

The  third  important  attempt  at  the  measure  of  the  earth's 
circumference  is  definitely  known  to  have  been  made  by  Eratos- 
thenes, and  the  fourth  by  Poseidonius  (c.  100  B.C.). 

Eratosthenes  on  the  Measure  of  the  Earth.  The  first  attempt 
of  which  we  have  any  details  is  this  third  one,  —  the  one  briefly 
described  as  due  to  Eratosthenes.  Supplementing  the  descrip- 
tion given  in  Volume  I,  page  no,  it  may  be  said  that  Eratos- 
thenes used  the  arc  of  a  great  circle  extending  from  Syene 
(the  modern  Assouan)  to  Alexandria.  He  took  the  length  of 
this  arc  as  5000  stadia,  but  how  this  length  was  ascertained 
is  not  stated  in  any  ancient  writings.  It  is  probable  that  the 
official  pacers,1  employed  by  Alexander  and  other  military 
leaders  in  planning  their  campaigns,  had  made  reports  of  all 
such  standard  distances,  and  Eratosthenes,  as  librarian  at  Alex- 
andria, doubtless  had  access  to  their  records. 

It  was  well  known  that  on  the  day  of  the  summer  solstice  the 
sun's  rays  lighted  up  completely  the  wells  of  Syene  at  noon- 
time, and  that  a  body  like  an  obelisk  cast  no  shadow.  On  the 
other  hand,  Eratosthenes  found  that  the  zenith  distance  of  the 
sun  on  this  day,  as  measured  at  Alexandria,  was  -^  of  the  cir- 
cumference. It  is  not  known  how  this  angle  was  found,  but  it 
has  been  thought  that  Eratosthenes  used  certain  armillary 
spheres  which  tradition  says  were  furnished  by  the  king, 
Ptolemy  Euergetes.  We  are  told  by  Cleomedes  (c.  40  B.C.), 
however,  who  wrote  on  the  Circular  Theory  of  the  Heavenly 
Bodies,  that  he  used  the  sca'phe,2  a  concave  sundial,  and  this 
may  easily  have  been  the  case.  Since  the  zenith  distance  of 
the  sun  changed  ^  of  the  circumference  in  5000  stadia,  Eratos- 
thenes concluded  that  the  circumference  was  50  x  5000  stadia, 
or  250,000  stadia,  roughly  equivalent  to  25,000  miles,  or  40,000 
kilometers.  This  would  make  694!  stadia  to  a  degree;  and 
since  Eratosthenes  was  naturally  aware  that  his  measurements 
were  merely  approximate,  he  felt  it  allowable  to  take  700 


r  iff  rat  (bematistai')  ,  singular  pinjLarumjs(bematistes'), 
a  step. 

2  SK(10i7,  originally  anything  dug  out;  hence  a  bowl,  and  then  the  bowl  of  a 
hemispherical  sundial, 


ERATOSTHENES  AND  POSEIDONIUS  371 

stadia  as  a  more  convenient  measure  for  i  ° .  He  had  crude  in- 
struments with  which  to  work,  he  did  not  take  into  considera- 
tion the  difference  of  longitude  of  his  two  stations,  and  the 
stadium  was  a  varying  unit  at  best,  so  that  his  assumption  of 
700  stadia  was  not  an  unreasonable  one. 

Poseidonius  on  the  Measure  of  the  Earth.  Poseidonius  (c.  100 
B.C.)  was,  as  we  have  seen,1  a  Stoic  philosopher,  well  known  as 
an  astronomer,  a  geographer,  a  historian,  and  a  statesman. 
After  having  traveled  extensively  in  all  the  Mediterranean 
countries,  he  opened  a  school  at  Rhodes  and  had  among  his 
pupils  both  Cicero  and  Pompey.  Although  his  works  are  lost, 
Cleomedes  (c.  40  B.C.)  has  given  us  a  certain  amount  of  in- 
formation as  to  his  method  of  measuring  the  circumference  of 
the  earth.  Like  Eratosthenes  he  took  a  known  arc,  selecting 
the  one  from  Rhodes  to  Alexandria  and  estimating  its  length 
as  5000  stadia.  He  then  observed  that  when  the  star  Canopus 
was  on  the  horizon  at  Rhodes,  it  was  \  of  a  sign  (that  is, 
|  of  30°,  or  ^g-  of  360°)  above  the  horizon  at  Alexandria.  He 
concluded  that  the  circumference  is  48  x  5000  stadia,  or  240,000 
stadia.  This  made  the  length  of  the  degree  666  f  stadia. 

It  is  hardly  probable  that  Poseidonius  considered  these  re- 
sults as  close  approximations,  since  neither  the  length  of  his 
arc  nor  the  elevation  of  the  star  could  be  measured  with  any 
approach  to  accuracy  by  instruments  then  available. 

Ptolemy  on  the  Measure  of  the  Earth.  The  last  of  the  note- 
worthy attempts  of  the  ancient  Greeks  to  find  the  circumfer- 
ence of  the  earth  was  made  by  Ptolemy  (c.  150).  He  took  a 
degree  as  500  stadia,  thus  finding  the  circumference  to  be 
180,000  stadia.  He  asserted  that  it  was  unnecessary  to  take 
the  arc  of  a  meridian,  an  arc  of  any  other  great  circle  being 
sufficient.  We  are  without  information,  however,  as  to  his 
method  of  measuring  the  arc  selected.  It  will  be  noticed  that 
his  result  is  to  that  of  Poseidonius  as  3  14 ;  and  since  this  is  the 
ratio  between  two  of  the  stadia  employed  by  the  ancients,  it 
is  possible  that  he  simply  used  the  latter's  computations. 

*Vol.  I,  p.  118.    See  O.  Viedebantt,  "Poseidonius,"  Klio,  XVI,  94- 


372  THE  PROBLEM  OF  EARTH  MEASURE 

The  theory  has  been  advanced  that  all  these  results  set  forth 
by  the  Greeks  were  due  to  Egyptian  or  other  measurements 
which  are  no  longer  extant,  but  there  is  no  scientific  basis  for 
the  conjecture. 

Arab  Measure  of  the  Earth.  It  was  some  seven  centuries  after 
the  last  of  the  Greek  geodesists  that  the  Arabs  engaged  in  the 
work  of  measuring  the  circumference  of  the  earth.  By  order 
of  al-Mamun  certain  mathematicians  of  Bagdad  undertook  the 
necessary  surveys  on  the  plain  of  ^ujar.in^Mesopotamia.  They 
formed  two  groups,  one  party  going  to  the  north  and  one  to 
the  south,  each  proceeding  to  a  point  at  which  the  elevation  of 
the  pole  changed  i°  from  that  of  the  base  station.  They  then 
measured  the  respective  distances,  one  being  found  to  be  57 
miles  and  the  other  56^  miles;  this  mile  was  given  as  4000 
"black  cubits,"  but  the  length  of  this  cubit  is  now  unknown. 
The  difference  in  the  two  measurements  illustrates  the  lack  of 
the  necessary  instruments  of  precision,  even  among  a  people 
who  had  brought  the  construction  of  such  instruments  to  the 
highest  degree  of  perfection  known  at  that  time. 

Invention  of  the  Telescope.  With*  respect  to  instruments  used 
in  astronomy  and  geodesy  the  greatest  improvement  is  due  to 
the  invention  of  the  telescope,  although  much  is  also  due  to 
modern  technique  in  manufacture  and  to  the  use  of  such  de- 
vices as  the  vernier  and  the  micrometer. 

Roger  Bacon  (c.  1250)  stated  that  it  was  possible  to  con- 
struct tubes  by  means  of  which  distant  objects  could  be  seen 
as  if  they  were  near  at  hand,1  but  we  have  no  evidence  that  this 
was  other  than  a  prophetic  statement  by  a  man  who  seemed 
peculiarly  gifted  in  this  respect.  Possibly  he  was  led  to  this 
prophecy  by  a  knowledge  of  spectacles,  which  appeared  some- 
time in  the  i3th  century,  for  they  were  certainly  known  in 
1299,  and  a  certain  Salvino  degli  Armati,  a  Florentine  (died 
in  1317),  is  mentioned  as  their  inventor. 

1"Ita  ut  in  incredibili  distancia  videremus  arenas  et  litteras  minias  minutas,  et 
ut  altissima  videantur  infima  ct  e  contrario."  Sloane  Ms.,  fol.  84,  a,  2.  Cf.  Opus 
Majus  (ed.  Bridges),  ii,  164;  Opus  Tertium  (ed.  Little),  41. 


THE  TELESCOPE  373 

The  possibility  of  the  telescope  is  also  mentioned  by  Fra- 
castorius,  who,  in  his  Homocentricorum  seu  de  stellis  Liber 
Unus  (1538),  speaks  of  using  two  superposed  lenses  in  looking 
at  a  distant  object.  Somewhat  similar  statements  were  made  by 
Giambattista  della  Porta  in  his  Magia  Naturalis  (Naples,  1558) 
and  by  Kepler  in  his  Paralipomena  (Frankfort,  1604). 

The  invention  seems  due,  however,  not  to  the  work  of  scien- 
tists like  those  mentioned,  but  largely  to  chance.  It  is  uncer- 
tain who  was  the  actual  inventor,  the  claims  of  three  artisans 
being  about  equal.  These  men  are  Zacharias  Janszoon  (Jan- 
sen),  Johann  Lippersheim  (Lippershey,  Lipperseim,  Laprey, 
c.  1608),  and  Jacob  Metius  Adriaenszoon1  (c.  1608). 

Janszoon  lived  at  Middelburg,  was  a  grinder  of  lenses,  and, 
apparently  with  the  aid  of  his  father,  improved  the  microscope 
in  1590  and  is  known  to  have  had  a  telescope  in  1610. 

Lippersheim  was  also  a  lens  grinder  of  Middelburg.  He  is 
known  to  have  asked  for  a  patent  in  1608  for  an  instrument 
intended  to  see  distant  objects,  the  lenses  being  of  rock  crystal. 

Descartes  and  others  attributed  the  invention  to  Jacob 
Metius  Adriaenszoon  (c.  1608),  who  happened  to  make  the 
necessary  combination  of  certain  lenses  and  burning  mirrors. 
He  also  asked  for  a  patent  in  1608,  a  few  days  after  Lip- 
persheim had  made  his  request.2 

The  invention  is  known  to  have  been  made  public  in  October, 
1608,  and  the  knowledge  of  the  instrument  spread  all  through 
Europe  with  astonishing  rapidity.  Even  in  1608,  and  still  more 
in  1609,  instruments  were  made  in  France,  England,  Italy, 
and  Germany.  Hearing  of  the  new  device,  Galileo,  in  1609, 
invented  an  instrument  of  his  own  and  by  its  aid  at  once  made 
remarkable  discoveries  in  astronomy ;  and  from  this  time  on  a 
precision  of  observation  unknown  to  earlier  scientists,  although 
the  instrument  was  not  accepted  by  all  astronomers,  became 
possible.  It  was  only  after  the  invention  of  achromatic  lenses, 
however,  that  satisfactory  results  were  obtained. 

1He  was  a  brother  of  the  Adriaen  Metius  (1571-1635)  mentioned  in  Volume  I, 
page  340.    He  was  born  at  Alkmaar  and  was  interested  in  the  grinding  of  lenses. 
2Bigourdan,  Astronomic,  p.  124. 
11 


374  THE  PROBLEM  OF  EARTH  MEASURE 

Modern  Measures  of  the  Earth.  The  first  noteworthy  modern 
attempt  at  measuring  the  earth  was  made  by  Jean  Fernel 
(c.  1528),  physician  to  Henri  II  of  France.  Fernel  took  the 
arc  determined  by  Paris  and  Amiens,  two  stations  being  taken 
on  the  same  meridian.  Knowing  the  altitude  of  the  sun  at  Paris, 
he  proceeded  northward  to  a  point  where  the  altitude  was  30' 
less  than  that  at  Paris.  He  then  measured  the  arc  by  taking 
the  number  of  revolutions  of  a  wheel  of  known  circumference. 
No  record  is  available  as  to  his  method  of  allowing  for  errors, 
but  certain  compensations  were  made  and  the  conclusion  was 
reached  that  i°  =  57,099  toises,  1000  toises  being  a  little  more 
than  a  geographic  mile.  This  result  is  remarkable,  the  mean 
afterwards  obtained  by  Lacaille1  and  Delambre  in  the  latter 
part  of  the  i8th  century  being  57,068  toises. 

In  1617  Snell  undertook  the  measurement  of  an  arc  by  an 
elaborate  system  of  triangulation,  and  although  his  results 
were  satisfactory  as  to  length  of  arc,  they  were  not  so  as  to 
amplitude. 

Further  French  Attempts.  In  1669  and  1670  Jean  Picard2  car- 
ried on  an  elaborate  system  of  triangulation,  measured  an  arc 
from  a  point  near  Corbeil  to  one  near  Amiens,  and  found  that 
i°2i'54"  corresponded  to  68,347  toises  3  pieds,  which  gave 
57,060  toises  to  i°.  He  estimated  that  the  amplitude  was  cor- 
rect to  within  2"  or  3". 

In  1686  Newton  proved  that  the  earth  is  an  oblate  spheroid, 
a  result  not  generally  accepted  by  French  scientists,  chiefly 
owing  to  the  conclusions  reached  by  Jacques  Cassini  (Cassini  II) 
as  mentioned  below.  It  was  therefore  decided  that  France 
should  undertake  a  more  elaborate  and  careful  survey,  not  con- 
fined to  that  country  alone,  but  including  arcs  nearer  to  and 
more  remote  from  the  equator. 

1  Nicolas  Louis  de  Lacaille  (La  Cattle) ;  born  at  Rumigny,  May  15,  1713;  died 
in  Paris,  March  21,  1762.  He  wrote  upon  mathematics  and  physics,  but  chiefly 
upon  astronomy. 

2 Born  at  La  Fleche,  Anjou,  July  21,  1620;  died  in  Paris,  July  12,  1682.  He 
wrote  upon  physics  and  astronomy  and  was  particularly  well  known  for  his  work 
on  the  measure  of  the  earth. 


MODERN  MEASUREMENTS  375 

The  degree  of  accuracy  reached  by  Picard  was  increased 
through  the  efforts  of  Giovanni  Domenico  (Jean  Dominique) 
Cassini  (Cassini  I),  who  extended  Picard's  meridian  in  1701 
southward  to  the  Pyrenees.  It  was  afterwards  extended  north- 
ward to  Dunkirk  (Dunkerque),  although  the  results,  pub- 
lished by  Jacques  Cassini  (Cassini  II)  in  1720,  provoked 
great  opposition  because  of  their  lack  of  precision  and  the 
incorrect  conclusions  reached  with  respect  to  the  elongated 
form  of  the  earth.  In  1735  France  sent  a  mission  to  Peru,  and 
an  elaborate  survey  was  made  for  the  purpose  of  measuring  an 
arc.  This  work  was  carried  on  under  the  direction  of  Bouguer,1 
Condamine,2  and  Godin.3  By  1745  they  had  completed  the 
measurement  of  an  arc  of  3°.  D'Alembert  spoke  of  the  work 
as  the  greatest  scientific  enterprise  that  had  thus  far  been  under- 
taken. In  the  following  year  another  mission,  including  such 
French  scientists  as  Maupertuis  and  Clairaut,  and  the  Swedish 
scientist  Celsius,4  began  a  similar  work  in  Lapland.  The  result 
of  this  survey  was  the  measurement  of  an  arc  of  i°.  The  con- 
clusions reached  in  Peru  and  Lapland  confirmed  Newton's 
assertion  of  the  flattening  of  the  earth  at  the  poles  and  led  to 
Voltaire's  reference  to  Maupertuis,  against  whom  he  had  a 
personal  grudge,  as  the  "great  flattener"  (grand  aplatisseur}. 
The  form  of  the  earth  appears  from  the  fact  that  degrees  of  lati- 
tude increase  in  length  as  we  approach  the  poles. 

In  1739  and  1740,  owing  chiefly  to  the  work  of  Lacaille,  an 
arc  of  the  meridian  was  again  measured  in  France,  the  result 
being  a  correction  of  the  errors  published  in  1720  and  a  new 
confirmation  of  Newton's  theory  of  the  shape  of  the  earth. 

Toward  the  close  of  the  i8th  century  France  undertook  a 
third  great  survey,  this  time  for  the  purpose  of  determining  the 


1See  page  327. 

2  Charles  Marie  de  la  Condamine,  born  in  Paris,  January  28,  1701;  died  in 
Paris,  February  4,  1774.  He  wrote  extensively  on  geodesy. 

3 Louis  Godin,  born  in  Paris,  February  28,  1704;  died  at  Cadiz,  September  n, 
1760.  He  wrote  chiefly  on  astronomy. 

4 Anders  Celsius;  born  at  Upsala,  November  27, 1701 ;  died  at  Upsala,  April  25, 
1744.  He  was  professor  of  astronomy  at  Upsala,  but  spent  some  years  in  France, 
Germany,  and  Italy. 


376  THE  PROBLEM  OF  EARTH  MEASURE 

length  of  the  standard  meter.  In  this  undertaking  a  number  of 
the  greatest  French  scientists  were  engaged,  but  for  the  geodetic 
work  Delambre  and  Mechain  were  chiefly  responsible. 

In  the  i  Qth  and  20th  centuries  extensive  triangulations  have 
been  made,  and  with  the  methods  employed  there  have  been 
connected  such  prominent  names  as  those  of  Biot,  Arago, 
Schumacher,  Legendre,  Laplace,  Gauss,  and  Bessel.  The  Ord- 
nance Survey  of  Great  Britain,  begun  in  1783  and  completed 
in  1858,  resulted  in  the  measurement  of  an  arc  of  10°  13',  ex- 
tending from  the  Isle  of  Wight  to  one  of  the  Shetland  group ; 
the  triangulation  of  India  (1790-1884)  gave  an  arc  of  about 
24° ;  and  the  Russo-Scandinavian  measurements,  begun  in  1817, 
resulted  in  an  arc  of  25°  20'.  The  arc  recently  measured  in 
Africa,  extending  over  a  distance  of  about  65°,  will,  joined  to  the 
Russo-Scandinavian  arc,  give  an  arc  of  about  106°. 

Mass  of  the  Earth.  The  determination  of  the  earth's  density 
depends  on  the  law  of  gravitation,  and  so  it  began  in  the  work  of 
Newton,  who  estimated  it  as  five  or  six  times  that  of  water. 

The  first  of  the  later  methods  depends  upon  the  deflection  of 
a  plumb  line  due  to  the  attraction  of  a  mountain.  This  was  first 
used  by  Pierre  Bouguer,  in  Peru  (c.  1740).  By  this  plan  Maske- 
leyne1  (1774-1776)  placed  the  density  between  4.5  and  5. 

The  second  method  is  based  upon  a  comparison  of  the  vibra- 
tions of  a  pendulum  at  sea  level  with  those  at  the  top  of  a  high 
mountain.  Francesco  Carlini,  the  Italian  astronomer,  used  the 
method  in  1821  and  obtained  a  density  of  4.84. 

The  third  method  is  due  to  Henry  Cavendish2  (1798)  and  is 
based  upon  the  mutual  attraction  of  known  masses.  Francis 
Baily3  (1843)  obtained  the  result  of  5.67  by  this  method. 

The  fourth  method  uses  a  finely  graduated  balance  to  de- 
termine the  attraction  of  known  masses.  By  its  use  results  of 
5.69  were  obtained  by  Von  Jolly  in  1881,  and  5.49  by  Poynting 
in  1891.  The  latest  experiments  give  the  result  as  about  5.53. 

1Nevil  Maskeleyne,  born  in  London,  October  5  (O.  S.)>  1732;  died  at  Green- 
wich, February  9,  1811.  He  became  astronomer  royal  in  1765. 

2  Born  at  Nice,  October  10,  1731;  died  in  London,  February  24,  1810. 

3  Born  at  Newbury,  Berkshire,  April  28,  1774;  died  in  London,  August  30, 1844. 


DISCUSSION  377 

TOPICS  FOR  DISCUSSION 

1.  Intuitive  geometry  as  it  shows  itself  in  the  primitive  decoration 
used  by  various  peoples. 

2.  Intuitive  geometry  as  it  shows  itself  in  the  early  stages  of 
mathematics  in  various  countries. 

3.  The  rise  of  demonstrative  geometry  and  the  six  most  impor- 
tant contributors  to  the  science  in  ancient  Greece. 

4.  The  various  names  used  for  geometry  and  the  special  signifi- 
cance of  each. 

5.  The  development  of  the  terminology  of  elementary  geometry, 
especially  ^mong  the  Greeks. 

6.  The  development  of  the  postulates  and  axioms  of  elementary 
geometry  before  the  ipth  century. 

7.  Propositions  of  elementary  geometry  of  which  the  origin  is 
known  or  which  have  any  history  of  special  interest. 

8.  The  various  methods  of  solving  each  of  the  Three  Famous 
Problems  of  antiquity. 

9.  The  historical  development  of  methods  for  rinding  the  ap- 
proximate value  of  IT. 

10.  The  principal  steps  taken  by  the  Greeks  in  the  development 
of  geometric  conies. 

1 1 .  The  principal  steps  in  the  development  of  plane  analytic  geome- 
try, with  special  reference  to  the  iyth  century. 

12.  A  discussion  of  the  history  of  solid  analytic  geometry. 

13.  The  history  of  the  most  important  higher  plane  curves  com- 
monly found  in  the  study  of  elementary  analytic  geometry,  together 
with  the  applications  of  these  curves. 

14.  The  nature,  purpose,  and  history  of  descriptive  geometry. 

15.  The  relation  of  the  fine  arts  to  geometry  in  the  isth  century. 

1 6.  The  development  of  projective  geometry. 

17.  The  development  of  the  non-Euclidean  geometries,  with  special 
reference  to  the  work  of  Bolyai,  Lobachevsky,  and  Riemann. 

1 8.  The  development  of  perspective   and   optics  considered   as 
mathematical  subjects. 

19.  A  study  of  the  most  interesting  of  the  primitive  instruments. 

20.  The  general  development  of  geodesy,  particularly  among  the 
Greeks  and  in  modern  times,  and  with  reference  to  the  measure  of  the 
circumference  of  the  earth. 


CHAPTER  VI 

ALGEBRA 
i.    GENERAL  PROGRESS  OF  ALGEBRA 

Nature  of  Algebra.  When  we  speak  of  the  early  history  of 
Algebra  it  is  necessary  to  consider  first  of  all  the  meaning  of 
the  term.  If  by  algebra  we  mean  the  science  which  allows  us 
to  solve  the  equation  ax2  -f-  bx  -h  c  =  o,  expressed  in  these  sym- 
bols, then  the  history  begins  in  the  iyth  century;  if  we  remove 
the  restriction  as  to  these  particular  signs,  and  allow  for  other 
and  less  convenient  symbols,  we  might  properly  begin  the  his- 
tory in  the  30!  century;  if  we  allow  for  the  solution  of  the  above 
equation  by  geometric  methods,  without  algebraic  symbols  of 
any  kind,  we  might  say  that  algebra  begins  with  the  Alexandrian 
School  or  a  little  earlier ;  and  if  we  say  that  we  should  class  as 
algebra  any  problem  that  we  should  now  solve  by  algebra  (even 
though  it  was  at  first  solved  by  mere  guessing  or  by  some  cum- 
bersome arithmetic  process),  then  the  science  was  known  about 
1800  B.C.,  and  probably  still  earlier.1- 

\  A  Brief  Survey  proposed.  It  is  first  proposed  to  give  a  brief 
survey  of  the  development  of  algebra,  recalling  the  names  of 
those  who  helped  to  set  the  problems  that  were  later  solved  by 
the  aid  of  equations,  as  well  as  those  who  assisted  in  establishing 
the  science  itself.  These  names  have  been  mentioned  in  Vol- 
ume I  and  some  of  them  will  be  referred  to  when  we  consider 
the  development  of  the  special  topics  of  algebra  and  their  appli- 
cation to  the  solution  of  elementary  problems.  J\ 

xFor  a  brief  study  of  the  early  history  see  H.  G.  Zeuthen,  "Sur  1'origine  de 
TAIgebre,"  in  the  KgL  Danske  Videnskab ernes  Selskab,  Math.-fysiske  Meddelelser, 
II,  4,  Copenhagen,  1919;  M.  Chasles,  "Histoire  de  1'Algebre,"  Comptes  rendus, 
September  6,  1841 ;  but  the  subject  is  treated  of  in  any  general  history  of  mathe- 
matics and  in  the  leading  encyclopedias.  * 

378 


EARLY  TRACES  OF  ALGEBRA        379 

It  should  also  be  stated  as  a  preliminary  to  this  discussion 
that  Nesselmann1  (1842)  has  divided  the  history  of  algebra 
into  three  periods:  the  rhetorical,  in  which  the  words  were 
written  out  in  full ;  the  syncopated,  in  which  abbreviations  were 
used ;  and  the  symbolic,  in  which  the  abbreviations  gave  place 
to  such  symbols  as  occur  in  statements  like  \Jx  —  x*~cfr. 
There  are  no  exact  lines  of  demarcation  by  which  to  establish 
these  divisions,  Diophantus,  for  example,  having  made  use  of 
certain  features  of  all  three;  but  the  classification  has  some 
advantages  and  the  student  will  occasionally  find  the  terms 
convenient. 

It  should  be  borne  in  mind  that  most  ancient  writers  outside 
of  Greece  included  in  their  mathematical  works  a  wide  range 
of  subjects.  Ahmes  (c.  1550  B.C.),2  for  example,  combines  his 
algebra  with  arithmetic  and  mensuration,  and  even  shows 
some  evidence  that  trigonometry  was  making  a  feeble  start. 
There  was  no  distinct  treatise  on  algebra  before  the  time  of 
"Hiophantus  (c.  275). 

Algebra  in  Egypt.  The  first  writer  on  algebra  whose  works 
nave  come  down  to  us  is  Ahmes.  He  has  certain  problems  in 
linear  equations  and  in  series,  and  these  form  the  essentially 
new  feature  in  his  work.  His  treatment  of  the  subject  is  largely 
rhetorical,  although,  as  we  shall  see  later,  he  made  use  of  a 
small  number  of  symbols. 

There  are  several  other  references  to  what  may  be  called 
algebra  in  the  Egyptian  papyri,  these  references  consisting 
merely  of  problems  involving  linear  or  quadratic  equations. 
There  is  no  good  symbolism  in  any  of  this  work  and  no  evidence 
that  algebra  existed  as  a  science.  - 

i  Algebra  in  India.  There  are  only  four  Hindu  writers  on  alge- 
"bra  whose  names  are  particularly  noteworthy.  These  are 
Aryabhata,3  whose  Aryabhatiyam  (c.  510)  included  problems  in 

1G.  H.  F.  Nesselmann,  Alg.  Griechen,  p.  302. 

2  As  already  stated,  the  period  may  have  been  c.  1600  B.C.  or  earlier. 

3  See  Volume  I,  page  153,  and  remember  that  there  were  two  Aryabhatas  and 
that  we  are  not  certain  which  one  of  them  is  entitled  to  the  credit  for  various 
contributions. 


380      GENERAL  PROGRESS  OF  ALGEBRA 

series,  permutations,  and  linear  and  quadratic  equations; 
Brahmagupta,  whose  Brahmasiddhdnta  (c.  628)  contains  a 
satisfactory  rule  for  solving  the  quadratic,  and  whose  problems 
include  the  subjects  treated  by  Aryabhata;  Mahavira,  whose 
Ganita-Sdra  Sangraha  (c.  850)  contains  a  large  number  of 
problems  involving  series,  radicals,  and  equations ;  and  Bhas- 
kara,  whose  Bija  Ganita  (c.  1150)  contains  nine  chapters  and 
extends  the  work  through  quadratic  equations.1 

Algebra  in  China.  It  is  difficult  to  say  when  algebra  as  a 
science  began  in  China.  Problems  which  we  should  solve  by 
equations  appear  in  works  as  early  as  the  Nine  Sections2  and 
so  may  have  been  known  by  the  year  1000  B.C.  In  Liu  Hui's 
commentary  on  this  work  (c.  250)  there  are  problems  of  pur- 
suit, the  Rule  of  False  Position,  explained  later  in  this  chapter, 
and  an  arrangement  of  terms  in  a  kind  of  determinant  notation.3 
The  rules  given  by  Liu  Hui  form  a  kind  of  rhetorical  algebra. 

The  work  of  Sun-tzi'4  (perhaps  of  the  ist  century,  but  the 
date  is  very  uncertain  and  may  be  several  centuries  earlier) 
contains  various  problems  which  would  today  be  considered 
algebraic.  These  include  questions  involving  indeterminate 
equations  of  which  the  following  is  a  type : 

There  are  certain  things  whose  number  is  unknown.  If  they  are 
divided  by  3  the  remainder  is  2  ;  by  5,  the  remainder  is  3  ;  and  by  7, 
the  remainder  is  2.  Find  the  number. 

Sun-tzi  solved  such  problems  by  analysis  and  was  content 
with  a  single  result,  even  where  several  results  are  admissible. 

The  Chinese  certainly  knew  how  to  solve  quadratics  as  early 
as  the  ist  century  B.C.,  and  rules  given  even  as  early  as  the 
K'iu-ch'ang  Suan-shu  above  mentioned  involve  the  solution  of 
such  equations. 

^H.  T.  Colebrooke,  Algebra  with  Arithmetic  and  Mensuration,  from  the  San- 
scrit, pp.  129-276  (London,  1817).  For  the  various  spellings  of  Bija  Ganita  see 
Volume  I,  page  278. 

2K'iu-ch'ang  Suan-shu. 

8Mikami,  China,  pp.  19,  23. 

4  Sun-tzi  Suan-king. 


CHINA  AND  GREECE  381 

Liu  Hui  (c.  250)  gave  various  rules  which  would  now  be 
stated  as  algebraic  formulas  and  seems  to  have  deduced  these 
from  other  rules  in  much  the  same  way  as  we  should  deduce 
formulas  at  the  present  time.1 

By  the  yth  century  the  cubic  equation  had  begun  to  attract 
attention,  as  is  evident  from  the  Ch'i-ku  Suan-king  of  Wang 
Hs'iao-t'ung  (c.  625). 

The  culmination  of  Chinese  algebra  is  found  in  the  i3th  cen- 
tury. At  this  time  numerical  higher  equations  attracted  the 
special  attention  of  scholars  like  Ch'in  Kiu-shao  (c.  1250), 
Li  Yeh  (c.  1250),  and  Chu  Shi-kie  (c.  1300), 2  the  result  be- 
ing the  perfecting  of  an  ancient  method  which  resembles  the 
one  later  developed  by  W.  G.  Horner  (1819). 

With  the  coming  of  the  Jesuits  in  the  i6th  century,  and  the 
consequent  introduction  of  Western  science,  China  lost  interest 
in  her  native  algebra  and  never  fully  regained  it.  u 

"'-•'Algebra  in  Greece.  Algebra  in  the  modern  sense  can  hardly 
be  said  to  have  existed  in  the  golden  age  of  Greek  mathematics.3 
The  Greeks  of  the  classical  period  could  solve  many  algebraic 
problems  of  considerable  difficulty,  but  the  solutions  were  all 
geometric.  Hippocrates  (c.  460  B.C.),  for  example,  assumed 
a  construction  which  is  equivalent  to  solving  the  equation 
x2  +  \/| •- •  ax  =  a2,  and  Euclid  (c.  300  B.C.),  in  his  Data, 
solved  problems  equivalent  to  the  following : 

1.  xy^k*,  x-y  =a    (Prob.  84). 

2.  xy  =  6*9  x+y  =a    (Prob.  85). 

3.  xy^k\  x*-f  =  a*  (Prob.  86). 

In  his  Elements  (II,  n)  Euclid  solved  the  equivalent  of 
x2  -f  ax  =  a,  and  even  of  x2  +  ax  =  b2,  substantially  by  com- 
pleting the  geometric  square  and  neglecting  negative  roots. 

After  Euclid  there  came  a  transition  period  from  the  geo- 
metric to  the  analytic  method.  Heron  (c.  50?),  who  certainly 

1Mikami,  China,  pp.  35,  36.  2Mikami,  China,  pp.  63,  79,  89. 

3Nesselmann,  Alg.  Griechen;  Heath,  Diophantus.  On  the  "application  of 
areas"  see  Heath,  History,  and  R.  W.  Livingstone,  The  Legacy  of  Greece,  p.  in 
(Oxford,  1922). 


382       GENERAL  PROGRESS  OF  ALGEBRA 

solved  the  equation  144  x  (14  —  x)  =  6720,  may  possibly  have 
used  the  analytic  method  for  the  purpose  of  finding  the  roots  of 


With  Diophantus  (c.  275)  there  first  enters  an  algebraic 
symbolism  worthy  of  the  name,  and  also  a  series  of  purely 
algebraic  problems  treated  by  analytic  methods.  Many  of  his 
equations  being  indeterminate,  equations  of  this  type  are  often 
called  Diophantine  Equations.  His  was  the  first  work  devoted 
chiefly  to  algebra,  and  on  this  account  he  is  often,  and  with 
much  justice,  called  the  father  of  the  science,  v 

Algebra  among  the  Arabs  and  Persians.  The  algebraists  of 
special  prominence  among  the  Arabs  and  Persians  were  Mo- 
hammed ibn  Musa,  al-Khowarizmi,  whose  al-jabr  w'al  muqd- 
balah  (c.  825)  gave  the  name  to  the  science  and  contained  the 
first  systematic  treatment  of  the  general  subject  as  distinct 
from  the  theory  of  numbers;  Almahani  (c.  860),  whose  name 
will  be  mentioned  in  connection  with  the  cubic  ;  Abu  Kamil 
(c.  900),  who  drew  extensively  from  al-Khowarizmi,  and  from 
whom  Fibonacci  (1202)  drew  in  turn;  al-Karkhi  (c.  1020), 
whose  Fakhri  contains  various  problems  which  still  form  part 
of  the  general  stock  material  of  algebra  ;  and  Omar  Khayyam 
(c.  noo),  whose  algebra  was  the  best  that  the  Persian  writers 
produced.  v 

Medieval  Writers.  Most  of  the  medieval  Western  scholars 
who  helped  in  the  progress  of  algebra  were  translators  from  the 
Arabic.  Among  these  were  Johannes  Hispalensis  (c.  1140), 
who  may  have  translated  al-Khowarizmi's  algebra;  Gherardo 
of  Cremona  (c.  1150),  to  whom  is  also  attributed  a  translation 
of  the  same  work;  Adelard  of  Bath  (c.  1120),  who  probably 
translated  an  astronomical  work  of  al-Khowarizmi,  and  who 
certainly  helped  to  make  this  writer  known;  and  Robert  of 
Chester,  whose  translation  of  al-Khowarizmi  's  algebra  is  now 
available  in  English.1  ^ 

*L.  C.  Karpinski,  Robert  of  Chester's  Latin  Translation  .  .  .  of  al- 
Khowarizmi.  New  York,  1915. 


FIRST  PAGE  OF  AL-KHOWARIZMI7S  ALGEBRA 

From  a  MS.  of  1456.  It  begins,  "Liber  mahucmeti  de  Algebra  et  almuchabala." 
In  Mr.  Plimpton's  library 


384       GENERAL  PROGRESS  OF  ALGEBRA 

The  greatest  writer  on  algebra  in  the  Middle  Ages  was 
Fibonacci,  whose  Liber  Quadratorum  (c.  1225)  and  Flos  both 
relate  to  the  subject.  The  former  work  includes  the  treatment 
of  such  problems  as  x2  +  y2  =  z~  and  other  well-known  types, 
and  shows  great  ingenuity  in  the  solution  of  equations. 

Of  the  German  algebraists  in  the  Middle  Ages  the  leading 
writer  was  Jordanus  Nemorarius  (c.  1225).  His  De  Numeris 
Datis,  already  described  (Vol.  I,  p.  227),  contains  a  number  of 
problems  in  linear  and  quadratic  equations  of  the  type  still 
familiar  in  our  textbooks.  In  general,  however,  the  medieval 
writers  were  more  interested  in  mathematics  as  related  to  as- 
tronomy than  in  mathematics  for  its  own  sake.1 

The  Renaissance.  Algebra  in  the  Renaissance  period  received 
its  first  serious  consideration  in  Pacioli's  Siima  (1494),  a  work 
which  summarized  in  a  careless  way  the  knowledge  of  the  sub- 
ject thus  far  accumulated.  By  the  aid  of  the  crude  symbolism 
then  in  use  it  gave  a  considerable  amount  of  work  in  equations. 

The  next  noteworthy  work  on  algebra,  and  the  first  to  be 
devoted  entirely  to  the  subject,  was  RudolfPs  Coss  (1525). 
This  work  made  no  decided  advance  in  the  theory,  but  it  im- 
proved the  symbolism  for  radicals  and  made  the  science  better 
known  in  Germany.  StifePs  edition  of  this  work  (1553-1554) 
gave  the  subject  still  more  prominence. 

The  first  epoch-making  algebra  to  appear  in  print  was  the 
Ars  Magna  of  Cardan  (1545).  This  was  devoted  primarily  to 
the  solution  of  algebraic  equations.  It  contained  the  solution 
of  the  cubic  and  biquadratic  equations,2  made  use  of  complex 
numbers,  and  in  general  may  be  said  to  have  been  the  first 
step  toward  modern  algebra. 

The  next  great  work  on  algebra  to  appear  in  print  was  the 
General  Trattato  of  Tartaglia  (1556-1560),  although  his  side 
of  the  controversy  with  Cardan  over  the  solution  of  the  cubic 
equation  had  already  been  given  in  his  Qvesiti  ed  invenzioni 
diverse  (1546). 

1  On  the  general  topic  see  P.  Cossali,  Origine,  trasporto  in  Italia,  primi  pro- 
gressi  in  essa  delV  Algebra,  2  vols.,  Parma,  1797-1799. 

2  In  chapters  xi  and  xxxix  et  seq. 


ALGEBRA  IN  THE  RENAISSANCE  385 

Algebra  in  the  New  World.  As  already  stated,1  the  first  mathe- 
matical work  published  in  the  New  World  was  the  Sumario 
Compendioso  of  Juan  Diez.  This  appeared  in  the  City  of 
Mexico  in  1556  and  contains  six  pages  on  algebra.  Some  idea 
of  its  general  nature  may  be  obtained  from  two  of  the  prob- 
lems relating  to  the  subject.  Of  these  the  first,  literally  trans- 
lated and  requiring  the  solution  of  the  quadratic  equation 
x*  —  1 5 1  =  x,  is  as  follows : 

Find  a  square  from  which  if  15^  is  subtracted  the  result  is  its  own 
root. 

Let  the  number  be  cosa  [x].  The  square  of  half  a  cosa  is  equal  to 
^  of  a  zenso  [x2\.  Adding  15  and  |  to  i  makes  16,  of  which  the 
root  is  4,  and  this  plus  ^  is  the  root  of  the  required  number. 

Proof:  Square  the  square  root  of  16  plus  half  &  cosa,  which  is  four 
and  a  half,  giving  20  and  -J,  which  is  the  square  number  required. 
From  2o|  subtract  15  and  £  and  you  have  4  and  £,  which  is  the  root 
of  the  number  itself. 

The  second  problem,  also  literally  translated,  requires  the 
solution  of  the  quadratic  equation  ,r2-f  #=1260: 

A  man  takes  passage  in  a  ship  and  asks  the  master  what  he  has  to 
pay.  The  master  says  that  it  will  not  be  any  more  than  for  the  others. 
The  passenger  on  again  asking  how  much  it  would  be,  the  master 
replies:  "It  will  be  the  number  of  pesos  which,  multiplied  by  itself 
and  added  to  the  number,  gives  1260."  Required  to  know  how  much 
the  master  asked. 

Let  the  cost  be  a  cosa  of  pesos.  Then  half  of  a  cosa  squared  makes 
£  of  a  zenso,  and  this  added  to  1260  makes  1260  and  a  quarter,  the 
root  of  which  less  £  of  a  cosa  is  the  number  required.  Reduce  1260 
and  |  to  fourths ;  this  is  equal  to  A°44Jl  >  the  root  of  which  is  71  halves ; 
subtract  from  it  half  a  cosa  and  there  remains  70  halves,  which  is 
equal  to  35  pesos,  and  this  is  what  was  asked  for  the  passage. 

Proof:  Multiply  35  by  itself  and  you  have  1225 ;  adding  to  it  35, 
you  have  1260,  the  required  number. 

*See  Volume  I,  page  353.  D.  E.  Smith,  The  Sumario  Compendioso  of  Brother 
Juan  Diez,  Boston,  1921, 


386  NAME  FOR  ALGEBRA 

First  Teachable  Textbooks  in  Algebra.  The  first  noteworthy 
attempt  to  write  an  algebra  in  England  was  made  by  Robert 
Recorde,  whose  Whetstone  of  witte  (1557)  was  an  excellent 
textbook  for  its  time.  The  next  important  contribution  was 
Masterson's  incomplete  treatise  of  1592-1595,  but  the  work  was 
not  up  to  the  standard  set  by  Recorde. 

The  first  Italian  textbook  to  bear  the  title  of  algebra  was 
Bombelli's  work  of  1572.  In  this  book  the  material  is  arranged 
with  some  attention  to  the  teaching  of  the  subject.1 

By  this  time  elementary  algebra  was  fairly  well  perfected, 
and  it  only  remained  to  develop  a  good  symbolism.  As  will  be 
shown  later,  this  symbolism  was  worked  out  largely  by  Vieta 
(c.  1590),  Harriot  (c.  1610),  Oughtred  (c.  1628),  Descartes 
(1637),  and  the  British  school  of  Newton's  time  (c.  1675). 

So  far  as  the  great  body  of  elementary  algebra  is  concerned, 
therefore,  it  was  completed  in  the  i7th  century. 

2.  NAME  FOR  ALGEBRA 

Early  Names.  The  history  of  a  few  of  the  most  familiar  terms 
of  algebra  not  elsewhere  discussed  will  now  be  considered,  and 
of  these  the  first  is  naturally  the  name  of  the  science  itself. 

Ahmes  (c.  1550  B.C.)  called  his  treatise  " Rules  for  inquiring 
into  nature,  and  for  knowing  all  that  exists,  [every]  mys- 
tery, .  .  .  every  secret/72  and  this  idea  is  not  infrequently  ex- 
pressed by  later  writers.  Thus  Seki  (c.  1680)  called  a  certain 
part  of  algebra  the  kigen  seiho,  meaning  a  method  for  revealing 
the  true  and  buried  origin  of  things,  and  we  find  the  same  idea 
in  the  titles  of  algebras  by  Follinus  (i622)3  and  Gosselin 
(i577),4  and  in  a  note  on  Ramus  written  by  Schoner  in  is86.5 

^L' Algebra  parte  maggiore  dell'  arimetica  .  .  .  ,  Bologna,  1572.  There  is  a 
second  edition,  differing  only  in  the  title-page,  Bologna,  1579. 

2 He  adds:  "Behold,  this  roll  was  written  .  .  .  [under  .  .  .  the  King  of  Upper] 
and  Lower  Egypt,  Aauserre.  ...  It  was  the  scribe  Ahmose  who  wrote  this  copy." 
Peet,  Rhind  Papyrus ,  p.  33.  Professor  Peet  gives  the  probable  date  as  between 
1788  and  1580  B.C. 

3  Algebra  sive  liber  de  rebus  occultis. 

4.  .  .  de  occulta  parte  numerorum. 

5"  ,  ,  ,  Almucabalam,  hoc  est,  librum  de  rebus  occultis"  (p.  322). 


EARLY  NAMES  387 

Since  the  Greeks  gave  the  name  "arithmetic"  to  all  the 
theory  of  numbers,  they  naturally  included  their  algebra  under 
that  title,1  and  this  explains  why  the  algebra  of  Diophantus 
went  by  the  name  of  arithmetic. 

The  Hindu  writers  had  no  uniform  name  for  the  science. 
Aryabhata  (c.  510)  included  algebra  in  his  general  treatise,  the 
Aryabhatiyam]  Brahmagupta  (c.  628)  placed  it  in  his  large 
treatise,  giving  a  special  name  (kutaka,  the  pulverizer)2  to  his 
chapter  on  indeterminate  equations.  Mahavira  (c.  850)  included 
it  in  his  Gayita-Sara-Sangraha,  a  title  meaning  a  brief  exposi- 
tion of  the  compendium  of  calculation.  Bhaskara  (c.  1 1 50)  had  a 
name  for  general  arithmetic,  Bija  Ganita*  meaning  the  calcula- 
tion of  seeds,  that  is,  of  original  or  primary  elements,4  and  a 
special  name  for  algebra,  Avyakta  ganita?  or  Avyakta-kriya, 
the  former  referring  to  the  calculation  with  knowns  and  the 
latter  to  that  with  unknowns. 

The  Chinese  used  various  fanciful  titles  for  their  books  con- 
taining algebra  and  spoke  of  the  method  of  the  t'ien-yuen 
(celestial  element)/  meaning  the  algebra  that  made  use  of  cal- 
culating rods  (the  Japanese  sangi},  to  indicate  coefficients.7 
Similar  fanciful  names  were  used  in  Japan,  as  also  the  name 
yendan  jutsu  (method  of  analysis),  and  the  name  kigen  seiho 
already  mentioned. 


1  So  Euclid's  Elements,  II,  devoted  to  arithmetic,  includes  a  considerable  part 
of  algebra,  such  as  the  geometric  proofs  about  (a  ±  b}2  and  (a  +  b)  (a  —  b). 
The  fact  was  recognized  by  Ramus  (1569)  when  he  stated  that  "Algebra  est  pars 
arithmeticae "  (1586  ed.,  p.  322). 

2Colebrooke  (pp.  112,  325)  transliterates  this  as  cuiidcdra,  culia,  cutiaca, 
cuttaca-vyavahdra,  and  cuiiacdd' hydra,  meaning  the  determination  of  a  pulveriz- 
ing multiplier  p  such  that,  if  n^,  w.,,  and  n^  are  given  numbers,  then  pnl  -f  n2 
shall  be  divisible  by  w8. 

3  Or  VI ja  Ganita,  Bee)  Gunnit.  The  spelling  as  given  in  the  first  printed  edi- 
tion (Calcutta,  1846)  is  Beej  Guntta.  The  name  Vija-kriyd,  meaning  seed  analy- 
sis, is  also  used. 

4Sir  M.  Monier-WiUiams,  Indian  Wisdom,  4th  ed.,  p.  174  (London,  1803). 

5  Nesselmann,  loc.  cit.,  p.  44;  Colebrooke  translation,  p.  129.  Bhaskara  also 
used  sama-sodanam  (transposition)  to  include  the  two  terms  which  had  been  used 
by  al-Khowarizmi. 

*T'ien-yuen-shu,  celestial-element  method.  The  Japanese  called  it  tengen  jutsu. 

7Mikami,  China,  p.  157. 


388  NAME  FOR  ALGEBRA 

Algebra  at  one  time  stood  a  fair  chance  of  being  called 
Fakhri,  since  this  was  the  name  given  to  the  work  of  al-Karkhi 
(c.  1020) ,  one  of  the  greatest  of  the  Arab  mathematicians.  Had 
his  work  been  translated  into  Latin,  as  al-Khowarizmi's  was, 
the  title  "might  easily  have  caught  the  fancy  of  the  European 
world.  Al-Karkhi  relates  that  he  was  long  and  sorely  hindered 
in  his  attempts  to  complete  his  work,  because  of  the  tyranny 
and  violence  endured  by  the  people,  until  "God,  may  his  name 
be  hallowed  and  exalted,  sent  to  their  aid  our  protector,  the 
vizir,  the  illustrious  lord,  the  perfect  one  in  government,  the 
vizir  of  vizirs,  clothed  with  double  authority,  Abu  Galib," 
whose  familiar  name  was  Fakhr  al-Mulk.  In  honor  of  this 
patron  the  name  Fakhr  gave  rise  to  the  title  of  the  book, 
al-Fakhri. 

The  Name  w  Algebra."  Our  real  interest  in  the  name  centers 
around  the  word  algebra,  a  word  appearing,  as  we  have  seen, 
in  the  title  of  one  of  the  works  by  al-Khowarizmi  (c.  825), — 
al-jabr  w'al-muqabalah*  It  also  appears  in  the  early  Latin  trans- 
lations under  such  titles  as  Ludus  algebrae  almucgrabalaeque 
and  Gleba  mutabilia.  In  the  i6th  century  it  is  found  in  English 
as  algiebar  and  almachabel,  and  in  various  other  forms,  but 
was  finally  shortened  to  algebra.1  The  words  mean  restoration 
and  opposition,2  and  one  of  the  clearest  explanations  of  their 
use  is  given  by  Beha  Eddin  (c.  1600)  in  his  Kholasat  al-Hisab 
(Essence  of  Arithmetic) :  ^/ 

The  member  which  is  affected  by  a  minus  sign  will  be  increased  and 
the  same  added  to  the  other  member,  this  being  algebra ;  the  homoge- 
neous and  equal  terms  will  then  be  canceled,  this  being  al-muqdbala. 

That  is,  given  bx  +  2  q  =  ,-r2  +  bx  —  q, 

al-jabr  gives  bx+2q  +  q~ x*  +  bx, 

and  al-muqdbalah  gives  3  9  =  •**• 

xAn  English  translation  of  al-Khowarizmi's  work  by  F.  Rosen  appeared  in 
London  in  1831.  A  Latin  version  was  published  by  Libri  in  his  Histoire,  Vol.  I 
(Paris,  1835),  and  by  Karpinski  (1915)  from  a  Scheubel  (Scheybl)  MS.  at 
Columbia  University,  as  already  stated. 

2  Or  redintegration  and  equation.  Jabr  is  from  jabara  (to  reunite  or  con- 
solidate), possibly  allied  to  the  Hebrew  gdbar  (make  strong). 


AL-JABR  W'AL-MUQABALAH  389 

This  statement  was  put  into  verse,  as  was  usual  in  the  East, 
and  thus  became  generally  known  in  the  Arab  schools.  It  may 
be  crudely  translated  thus: 

Cancel  minus  terms  and  then 
Restore  to  make  your  algebra ; 
Combine  your  homogeneous  terms 
And  this  is  called  muqabalah? 

In  a  general  way  we  may  say  that  al-]abr  or  al-jebr  has  as 
the  fundamental  idea  the  transposition  of  a  negative  quantity, 
and  muqabalah  the  transposition  of  a  positive  quantity  and  the 
simplification  of  each  member.2  Al-Khowarizmi's  title  was 
adopted  by  European  scholars,3  appearing  both  in  the  Arabic, 
with  many  curious  variants,  and  in  Latin.  The  Moors  took  the 
word  al-jabr  into  Spain,  an  algebrista  being  a  restorer,  one  who 
resets  broken  bones.4  At  one  time  it  was  not  unusual  to  see 
over  the  entrance  to  a  barber  shop  the  words  "Algebrista  y 
Sangrador"  (bonesetter  and  bloodletter),  and  both  the  striped 
pole  which  is  used  in  America- as  a  barber's  sign  and  the  metal 
basin  used  for  the  same  purpose  in  Europe  today  are  relics 
of  the  latter  phase  of  the  haircutter's  work.  From  Spain  the 
word  passed  over  to  Italy,  where,  in  the  i6th  century,  algebra 
was  used  to  mean  the  art  of  bonesetting.5  Thence  it  found  its 
way  into  France  as  algebre,  and  so  on  to  England,  where  one 
writer  (1541)  speaks  of  "the  helpes  of  Algebra  &  dislocations," 


1From  a  Persian  algebra  written  probably  after  the  i2th  century.  Nesselmann 
(p.  50)  put  it  into  German  verse,  and  the  above  English  quatrain,  taken  from  his 
translation,  gives  only  a  general  idea  of  the  wording. 

2Rollandus  (c.  1424)  has  De  arte  dolandi,  the  art  of  chipping  off  or  cutting 
with  an  ax,  probably  meaning  the  chipping  off  or  subtracting  of  equals  from 
both  members. 

3  Thus  Robert  of  Chester's  translation  (c.  1140)  begins  his  Liber  Algebrae  et 
Almucabola  thus :  "  In  nomine  Dei  xpij  et  misericordis  incipit  liber  Restaura- 
tionis  et  Oppositioriis  numeri  .  .  .  filius  Mosi  Algourizim  dixit  Mahometh." 

4So  in  Don  Quixote  (II,  chap.  15),  where  mention  is  made  of  "tw  algebrista 
who  attended  to  the  luckless  Samson." 

5Libri,  Histoire,  1838  ed.,  II,  80.  The  question  of  the  connection  of  al-jabr 
with  the  Hebrew  root  sh-b-r  (from  which  comes  tiskboreth,  fracture),  and  with 
the  Hindu  word  for  pulverizer,  is  worthy  of  study. 

ii 


390  NAME  FOR  ALGEBRA 

and  another  (1561)  says:  "This  Araby  worde  Algebra  sygni- 
fyeth  as  well  fractures  of  the  bones,  etc.  as  sometyme  the 
restauration  of  the  same."1 

As  already  said,  the  name  was  much  distorted  by  the  Latin 
translators.  Thus  Guglielmo  de  Lunis  (c.  1250?)  gives  it  as 
gleba  mutabilia,  and  Roger  Bacon  (c.  1250)  speaks  of  the 
science  as  algebra  .  .  .  et  almochabala?  A  i  sth  century  manu- 
script testifies  both  to  the  mystery  of  the  subject  and  to  the 
uncertainty  of  name  when  it  speaks  of  the  subtleties  of  largibra.3 
In  the  early  printed  books  it  appeared  in  equally  curious  forms, 
such  as  Gebra  vnd  Almuthabola.4 

Some  of  the  late  Latin  writers  attributed  the  name  to  one 
Geber,5  an  Arab  philosopher,  whom  they  supposed  to  be  the 
inventor  of  the  science;6  and  certain  Arab  writers  speak  of  a 
Hindu  named  Argebahr  or  Arjabahr,  a  name  which  may  have 
influenced  the  Latin  translators.7  Even  as  good  a  scholar  as 
Schoner  went  far  astray  in  his  interpretation  of  the  title.8 


1See  Oxford  Dictionary  under  algebra. 

2 "Algebra  quae  est  negotiatio,  et  almochabala  quae  est  census."  Opus  Majus, 
ed.  Bridges,  I,  p.  Ivii. 

3  Anon.  MS.  in  Boncompagni's  library:  "  .  .  .  di  subtili  .  R; .  di  largibra." 
Narducci  Catalogo  (2d  ed.,  1892),  No.  397  (2). 

4 A.  Helmreich,  Rechenbuch,  1561  (1588  ed.). 

5  There  was  an  Arab  scholar,  Jabir  ibn  Aflah,  Abu  Mohammed,  of  Seville 
(c.  1145),  whose  astronomy  was  translated  by  Gherardo  of  Cremona,  his  con- 
temporary, and  was  printed  in  1534. 

6The  name  appears  as  Greber  in  Heilbronner's  Hist.  Math.,  p.  340  (1742). 
Ghaligai  (1521)  spoke  of  it  as  "composta  da  uno  home  Arabo  di  grade  intelli- 
gentia,"  adding  that  "  alcuni  dicono  essere  stato  uno  il  qua!  nome  era  Geber." 

7Libri,  Histoire  (I,  122),  thinks  that  this  writer  was  Aryabhata. 

8Thus  in  a  note  on  Ramus  (1586  ed.,  p.  322)  he  says:  "Nomen  Algebrae 
Syriacum  putatur  significans  artem  &  doctrinam  hominis  excellentis.  Nam  geber 
Syris  significat  virum  ...  ut  apud  nos  Magister  aut  Doctor  .  .  .  &  ab  Indis 
harum  artium  perstudiosis  dicitur  Aliabra  item  Alboret,  tametsi  proprium  autoris 
nomen  ignoretur." 

As  an  example  of  still  more  uncertain  history,  A.  Helmreich  (Rechenbuch, 
1561 ;  1588  ed.,  fol.  b  2,  r.,  of  the  Vorrede)  asserts  that  algebra  was  due  to  "  Ylem/ 
der  grosse  Geometer  in  Egyptcn/zur  zeit  desz  Alexandri  Magni,  der  da  war  ein 
Praeceptor  oder  vorfahrer  Euclidis,  desz  Fursten  zu  Megarien."  We  also  find 
such  forms  as  Agabar,  Algebra  muchabila,  Reghola  della  raibre  mochabiln?',  regola 
del  acabrewp\  ghabile,  dellacibra  e  muchabile,  lacibra  umachabille,  all  in  MSS.  of 
the  i5th  century;  and  in  the  i6th  century,  such  forms  as  arcibra. 


FIRST   PAGE   OF    AN    ALGEBRA   MANUSCRIPT   OF    C.   1460 

Possibly  by  Raffaele  Canacci,  a  Florentine  mathematician.  On  the  second  line 
may  be  seen  the  name  for  algebra—  Algebra  amucabak.  In  Mr.  Plimpton's  library 


392  NAME  FOR  ALGEBRA 

Other  Names.  Because  the  unknown  quantity  was  called  res 
by  the  late  Latin  writers,  which  was  translated  into  Italian  as 
cosa*  the  early  Italian  writers  called  algebra  the  Regola  de  la 
Cosa,  whence  the  German  Die  Coss  and  the  English  cossike 
arte.2 

The  Italians  of  the  isth  and  i6th  centuries  often  called 
algebra  the  greater  art,  to  distinguish  it  from  commercial  arith- 
metic, which  was  the  lesser  art,  just  as  we  speak  of  higher 
arithmetic  and  elementary  arithmetic.  This  distinction  may 
have  been  suggested  by  the  seven  arti  maggiori  and  the  fourteen 
arti  minori  recognized  by  the  merchants  of  medieval  Florence.  a 
Thus  we  have  such  names  as  Ars  Magna,  used  by  Cardan 
(i545),4  V  Arte  Maggiore,  used  by  various  other  Italian  writers, 
and  Varte  mayor,  used  by  Juan  Diez,  whose  book  has  been 
mentioned  as  having  appeared  in  Mexico  in  1556.  The  title 
as  given  in  the  Mexican  book  is  as  follows  : 


Vieta  (c.  1590)  rejected  the  name  "algebra"  as  having 
no  significance  in  the  European  languages,  and  proposed  to 
use  the  term  "analysis,"  and  it  is  probably  to  his  influence 
that  the  popularity  of  this  term  in  connection  with  higher 
algebra  is  due. 

Of  the  other  names  for  algebra  the  only  one  that  we  need 
consider  is  "logistic."  Since  the  term  had  dropped  out  of  use 
as  a  name  for  computation  about  1500,  it  was  employed  to 

iFrom  the  Latin  causa]  compare  the  French  chose. 

2Thus  Pacioli  (1494  ed.,  fol.  67,  r.}:  "Per  loperare  de  larte  magiore  :  ditta 
dal  vulgo  la  regola  de  la  cosa  ouer  algebra  e  amucabala."  It  will  be  recalled  that 
Rudolff's  title  for  his  book  (1525)  on  algebra  was  Die  Coss.  Helmreich  says: 
"...  vnd  wird  bey  den  Welschen  [the  Italians]  genent  das  buch  Delacosa/welchs 
wir  Deutsche  die  Reguld  Cos  oder  Algebra  nennen."  Rechenbuch,  1561  (1588  ed.), 
fol.  b2. 

3E.  G.  Gardner,  The  Story  of  Florence,  p.  42.    London,  1900. 

4  Also  by  Gosselin,  De  arte  magna,  sen  de  occulta  parte  numerorum  quae  & 
Algebra  &  Almucabala  vulgo  dicitur,  libri  IV,  Paris,  1577.  The  name  was  used, 
however,  for  various  purposes,  as  in  Kircher's  Ars  Magna  Lucis  et  Umbrae, 
Amsterdam,  1671,  and  numerous  other  works. 


TECHNICAL  TERMS  IN  ALGEBRA  393 

designate  a  higher  branch,  just  as  "calculus"  was  appropriated 
a  century  or  so  later.  Thus  we  find  it  used  by  Buteo  and 
others1  to  cover  advanced  arithmetic  and  algebra,  although  it 
never  became  popular.2 

3.  TECHNICAL  TERMS 

Coefficient.  Of  the  terms  commonly  used  in  algebra,  it  is  pos- 
sible at  this  time  to  mention  only  a  few  typical  ones. 

The  coefficient  was  called  by  Diophantus  (c.  275)  the 
ple'thos*  (multitude),  and  by  Brahmagupta  (c.  628)  the  anca, 
or  prakriti,4  but  most  early  writers  used  no  special  name.  The 
term  "coefficient"  and  the  use  of  literal  coefficients  are  late 
developments,  the  former  being  due  to  Vieta. 

Unknown  Quantity.  The  unknown  quantity  was  called  by 
Ahmes  (c.  1550  B.C.)  ahe*  or  hau  ("mass,"  "quantity,"  or 
"heap  ") .  Diophantus  called  it  "an  undefined  number  of  units."6 
Brahmagupta  called  it  the  yavat-tdvat,1  and  possibly  this  sug- 
gested to  the  Arabs  the  use  of  shd  (sei,  chat,  meaning  "thing"  or 
"anything"),  whence  the  medieval  use  of  res  (thing)  for  this 
purpose.8  The  Chinese  used  yuen  (element),9  as  already  stated, 
but  they  also  used  a  word  meaning  "thing."10 

Powers.  Before  the  invention  of  a  satisfactory  symbol  like 
x2  it  became  necessary  to  have  a  special  name  for  the  square  of 
the  unknown,  and  in  the  Greek  geometric  algebra  it  was  called 

1H.  Vitalis,  Lexicon M athematicvm,  p.  25  (Rome,  1690) ;  Nesselmann, /oc.  cit., 

p.  57- 

2 There  are  many  other  names  used  for  this  purpose.  -E.g.,  'Ali  ibn  Veli  ibn 
Hamza,  a  western  Arab,  then  (1590/91)  living  in  Mecca,  wrote  a  work  entitled 
Tuhfet  al-ardad  li-davi  al-roshd  ve'l  sadad  (The  Gift  of  Numbers  for  the  Posses- 
sors of  Reason  and  Correct  Insight},  relating  to  elementary  algebra.  See 
E.  L.  W.  M.  Curtze,  Abhandlungen,  XIV,  184.  The  name  reminds  one  of 
Recorde's  Whetstone  of  witte,  London,  1557. 

3nXii0os.  4  Colebrooke's  translation,  pp.  246,  348. 

KBy  Egyptologists,  *h*w.   See  Peet,  Rhind  Papyrus,  p.  61. 

6  nx?)0os  (jLoixiSwv  &\oyov  (ple'thos  mona'don  a'logon) . 

7  Also  given  as  jabut  tabut,  literally  the  "so  much  as,"  "as  far,  so  far,"  "as 
much, so  much,"  or  "  however  much."  Compare  Bombelli's  use  oitanto  (so  much) . 

8  The  Arabs  also  used  jidr  (dyizr,  root) ,  whence  the  Latin  radix. 
9Mikami,  China,  pp.  81,  91.  10  Compare  res  and  cosa,  page  392. 


394  TECHNICAL  TERMS 

a  "  tetragon1  number"  or  a  "power."2  Diophantus  called  the 
third  power  a  cube,3  the  fourth  power  a  "power-power,"4  the 
fifth  power  a  "power-cube,"5  and  the  sixth  power  a  "cube- 
cube,"6  using  the  additive  instead  of  the  multiplicative  principle. 
The  Arab  writers  called  the  square  of  the  unknown  a  mal,  a 
word  meaning  "wealth,"  whence  the  medieval  Latin  census 
(evaluation  of  wealth,  tax)  was  used  for  the  same  purpose, 
appearing  in  the  early  Italian  algebras  as  censo,  sometimes  in- 
correctly written  as  zenso.  Therefore  algebra  was  not  uncom- 
monly called  Ars  rei  et  census  as  well  as  Ars  rei1  and  Regola  o 
I'arte  delta  cosa. 

Equation.  The  word  "equation/'  while  generally  used  as  at 
present  ever  since  the  medieval  writers  set  the  standard,  has 
not  always  had  this  meaning.  It  is  used  by  Ramus  in  his 
arithmetic  and  by  his  commentator  Schoner  to  denote  a  con- 
tinued proportion,  although  in  speaking  of  algebra  Ramus 
(i567)8  and  Gosselin  (1577)  follow  the  ordinary  usage. 

Absolute  Term.    In  the  equation 


we  speak  of  an  as  the  absolute  term.    There  have  been  various 
names  for  it,  Diophantus   (c.  275)  calling  it  monads.9    The 


dpiOfj,6s  (tetra'gonos  arithmos',  four-angled  number).  So  in  Euclid 
VII,  def.  18. 

2  Aura/us  (dy'namis),  from  the  same  root  as  "dynamo,"  "dynamic,"  and 
"dynamite."  So  in  Plato  (Timccus,  31)  ;  but  he  also  uses  the  term  (Theo&tetus, 
147  D)  to  mean  the  square  root  of  a  non-square  number.  When  Diophantus 
speaks  of  any  particular  square  number,  he  uses  rerpdyuvos  d/w0/x6s,  but  otherwise 
dvva/j.is.  Heath,  Diophantus,  2d  ed.,  p.  38. 

3Ki5/?os  (ku'bos).  5  AwafjAKvpos  (dynamo'  kubos}  . 

4  Avvafj.odvvafj.LS  (dynamo  dy'namis}  .  fiKu/36»cw/3os  (kubo'kubos}  . 

7Nesselmann,  Alg.  Griechen,  pp.  55,  56. 

8Thus  Ramus,  in  his  arithmetic  (1567),  says:  "AEquatio  est  quando  con- 
tinuatae  rationes  continuantur  iterum,"  Schoner  giving  as  an  example 

10.        15.        12. 

20.  30.  24. 

meaning  10:15:12  =  20:30:24  (1586  ed.,  p.  188). 

*MovA8cs(mona'des)  ,  with  the  abbreviation  ju5.  Heath,  Diophantus,  2d  ed.,  p.  39. 


SYMBOLS  395 

Hindus  called  it  rupa  or  rw,1  and  the  Chinese  gave  it  the  name 
tai,2  an  abbreviation  of  tai-kieh  (extreme  limit). 

Commutative  and  Distributive  Laws.  The  use  of  the  terms 
" commutative"  and  " distributive"  in  the  usual  algebraic  sense 
is  due  to  the  French  mathematician  Servois  (1814).  The  use  of 
the  term  "associative"  in  this  sense  is  due  to  Sir  William  Rowan 
Hamilton.3 

4.    SYMBOLS  OF  ALGEBRA 

Symbols  of  Operation.  The  symbols  of  elementary  arithmetic 
are  almost  wholly  algebraic,  most  of  them  being  transferred  to 
the  numerical  field  only  in  the  igth  century,4  partly  to  aid  the 
printer  in  setting  up  a  page  and  partly  because  of  the  educa- 
tional fashion  then  dominant  of  demanding  a  written  analysis  for 
every  problem.  When  we  study  the  genesis  and  development 
of  the  algebraic  symbols  of  operation,  therefore,  we  include  the 
study  of  the  symbols  used  in  arithmetic.  Some  idea  of  the 
status  of  the  latter  subject  in  this  respect  may  be  obtained 
by  looking  at  almost  any  of  the  textbooks  of  the  iyth  and 
1 8th  centuries.  Hodder,5  for  example,  gives  no  symbols  before 
page  201,°  then  remarking:  "Note  that  a  -f-  thus,  doth  signifie 
Addition,  and  two  lines  thus  £  Equality,  or  Equation,  but  a 
x  thus,  Multiplication,"  no  other  symbols  being  used.  Even 
Recorde,  who  invented  the  modern  sign  of  equality,  did  not  use 
it  in  his  arithmetic,  the  Ground  oj  Aries  (c.  1542),  but  only  in 
his  algebra,  the  Whetstone  of  witte  (1557). 

Earliest  Symbols.  The  earliest  symbols  of  operation  that 
have  come  down  to  us7  are  Egyptian.  In  the  Ahmes  Papyrus 
(c.  1550  B.C.)  addition  and  subtraction  are  indicated  by 

1  Colebrookc's  VI ja  Ganita,  p.  i86n. ;  E.  Strachey,  Bija  Ganita,  p.  117  (Lon- 
don, n.d.),  where  it  is  given  as  roop. 

2Mikami,  loc.  cit.,  p.  81.   The  word  is  also  transliterated  tae. 

3  Cajori,  Hist,  of  Math.,  20!  ed.,  p.  273.   New  York,  1919. 

4  There  are,  of  course,  exceptions.   The  Greenwood  arithmetic  (1729),  for  ex- 
ample, used  the  algebraic  symbols. 

5  His  was  the  first  English  arithmetic  to  be  reprinted  in  the  American  colonies 
(Boston,  1719).  6i672ed. 

7  Excepting  those  connected  with  notation,  as  in  the  subtractive  principle  of 
the  Babylonians,  already  mentioned. 


396  SYMBOLS  OF  ALGEBRA 

special  symbols,  but  these  are  simply  hieratic  forms  from  the 
hieroglyphics  and  are  not  symbols  in  the  sense  in  which  we  use 
the  term.  The  symbol  ^4  was  used  to  designate  addition.  It 
appears  in  the  Ahmes  Papyrus  as  h..1  The  symbol  A.  was 
used  to  designate  subtraction.  It  appears  in  the  Ahmes  Papyrus 
as  _/J.2 

Diophantus  (c.  275)  represented  addition  by  simple  juxta- 
position, as  in  KYaAYi7  for  .a^+iS^2.  For  subtraction  he 
seems  to  have  used  the  symbol  ^ ,  although  we  are  not  certain 
as  to  its  precise  form.  Since  we  have  no  manuscript  of  his 
Arithmetica  earlier  than  the  Madrid  copy  of  the  i3th  century, 
we  are  also  uncertain  as  to  the  authenticity  of  the  following 
passage : 

"  Minus  multiplied  by  minus  makes  plus,  and  minus  by  plus 
makes  minus.  The  sign  of  negation  is  -^  turned  upside  down,  W 3 
It  is  fully  as  probable  that  the  symbol  is  a  deformed  A 
(lambda},  the  Greek  letter  L  and  the  initial  for  a  word  indi- 
cating subtraction.4 

The  Hindus  at  one  time  used  a  cross  placed  beside  a  number 
to  indicate  a  negative  quantity,  as  in  the  Bakhshali  manuscript 
of  possibly  the  loth  century.  With  this  exception  it  was  not 
until  the  i2th  century  that  they  made  much  use  of  symbols 
of  operation.5  In  the  manuscripts  of  Bhaskara  (c.  1150)  a 
small  circle  or  a  dot  is  placed  above  a  subtrahend,  as  in  6  or  6 
for  —  6,6  or  the  subtrahend  is  inclosed  in  a  circle,  just  as 
children,  in  scoring  a  game,  indicate  6  less  than  zero7  by  the 
symbol  ® 

1  Ahmes  wrote,  as  usual,  from  right  to  left,  but  hieroglyphic  sentences  are 
generally  printed  from  left  to  right.  See  the  Brit.  Mus.  facsimile,  PI.  IX,  row  5. 

2Peet,  Rhind  Papyrus,  p.  64;  Eisenlo'hr,  Ahmes  Papyrus,  p.  47. 

8 Tannery,  Diophantus,  I,  13,  and  Bibl.  Math.,  V  (3),  5;  Heath,  Diophantus, 
2d  ed.,  p.  130.  On  the  relation  of  this  symbol  to  the  symbol  c^,  which  is  used  in 
the  Ayer  Papyrus  (c.  200-400),  see  Amer.  Journ.  of  Philology,  XIX,  25. 

4Ai7r6vres  (lipon'tes,  diminished  by),  or  kdirew  (lei'pein,  to  be  missing). 

Of  course  a  further  exception  is  also  to  be  made  of  the  representing  of  sums 
by  juxtaposition  and  of  division  by  means  of  fractions.  For  the  Bakhshali  MS, 
see  Volume  I,  page  164. 

6 See  Colebrooke's  edition,  p.  131;  Taylor's  edition,  Introduction,  p.  u. 

7  C.  I.  Gerhardt,  Etudes  historiques,  p.  8.    Berlin,  1856. 


PLUS  AND  MINUS  397 

European  Symbols  for  Plus  and  Minus.  The  early  European 
symbol  for  plus,  used  in  connection  with  the  Rule  of  False 
Position,  was  p,1  P,2  or  p,  the  last  being  the  most  common  of 
the  three.8  The  word  plus,  used  in  connection  both  with  addi- 
tion and  with  the  Rule  of  False  Position,  was  also  employed ; 
but,  strange  to  say,  it  is  much  later  than  the  word  minus  as 
indicating  an  operation.  The  latter  is  found  in  the  works  of 
Fibonacci  (1202),  while  the  use  of  plus  to  indicate  addition  is 
not  known  before  the  latter  part  of  the  isth  century.4 

Since  p  or  p  was  used  for  plus,  in  or  m  was  naturally  used 
for  minus,  and  this  usage  is  found  in  many  works  of  the  isth 
and  1 6th  centuries.  As  usual,  the  bar  simply  indicated  an 
omission,  as  in  Suma  for  Summa,  in  the  title  of  Pacioli's  work.5 
In  the  isth  century  the  symbol  79  was  often  used  for  minus" 
but  most  writers  preferred  the  m. 

Racial  Preferences.  We  now  come  to  one  of  the  many  cases  of 
racial  habit  in  determining  mathematical  custom.  In  the  i6th 
century  the  Latin  races  generally  followed  the  Italian  School, 
using  p  and  in  or  their  equivalents,7  while  the  German  School 

1  So  in  the  Rollandus  MS.  (c,  1424),  where  the  terms  are  so  arranged  as  to 
require  no  minus  sign.   The  Rule  of  False  Position  is  explained  on  pages  437-441. 

2  Clavius  (1583)  uses  P  and  M  for  plus  and  minus  in  his  Rule  of  False  Posi- 
tion, which  he  gives  in  his  arithmetic;  but  in  his  algebra  (1608)  he  uses  the  cross, 
saying:  "Plerique  auctores  pro  signo  +  ponunt  literam  P,  .  .  .  sed  placet  nobis 
uti  nostris  signis."   It  must  be  remembered  that  the  symbols  in  the  Rule  of  False 
Position  are  hardly  symbols  of  operation  in  the  ordinary  sense  of  the  term. 

3  E.g.,  Chuquet  (1484)  and  Pacioli  (1494),  the  latter  first  using  it  in  his  Rule 
of  False  Position  (fol.  106). 

4Bibl.  Math.,  XIII  (2),  p.  105.  Fibonacci  used  it,  however,  in  the  Rule  of 
False  Position. 

5  Compare  the  French  hotel  for  hostel  or  hospital,  and  the  German  iiber  for 
ueber.   See  also  Cajori,  "Varieties  of  Minus  Signs,"  Math.  Teacher,  XVI,  295. 

6  This  is  the  case  in  Mr.  Plimpton's  MS.  of  al-Khowarizmi,  written  in  1456. 
See  also  the  Regiomontanus-Bianchini  correspondence  (c.  1464)  in  the  Abhand- 
lungen,  XII,  233,  279;  and  Curtze  in  the  Bibl.  Math.,  I  (3),  506. 

7  E.g.,  Pacioli  (1494)  writes  m  for  minus  in  his  algebra,  but  he  follows  the 
general  custom  of  using  de  in  cases  like  "7.  de  9*';  Cardan   (1539)    writes 
R  .  V  .  7  .  p  R  .  4  for    v  7  -f  "v/4  ;  Feliciano  Lazesio  (1526)  writes  6.  piu.  R  .  16 
for  6  +  1 6  x,  and  10.  mR  .  4  for  10  —  4  x',  Tartaglia  (1556)  and  Cataneo  (1546) 
write  piu  and  men  for  plus  and  minus  in  the  Rule  of  False  Position;  Santa- 
Cruz  (1594)  uses  the  equivalent  Spanish  words  mas  and  menos;  and  Peletier 


3Q8  SYMBOLS  OF  ALGEBRA 

preferred  the  symbols   +    and    — ,  neither  of  which  is  found 
for  this  purpose,  however,  before  the  isth  century.1 

Origin  of  our  Plus  and  Minus  Signs.  In  a  manuscript  of  1456, 
written  in  Germany,2  the  word  et  is  used  for  addition  and  is 
generally  written  so  that  it  closely  resembles  the  symbol  +  • 
The  et  is  also  found  in  many  other  manuscripts,  as  in  "5  et  7" 
for  5  +  7,  written  in  the  same  contracted  form,3  as  when  we 
write  the  ligature  &  rapidly.  There  seems,  therefore,  little 
doubt  that  this  sign  is  merely  a  ligature  for  et. 

The  origin  of  the  minus  sign  has  been  more  of  a  subject  of 
dispute.  Some  have  thought  that  it  is  a  survival  of  the  bar  in 
1~9  or  in  m,  but  it  is  more  probable  that  it  comes  from  the  habit 
of  early  scribes  of  using  it  as  the  equivalent  of  ra,  as  in  Suma 
for  Summa.  Indeed,  it  is  quite  probable  that  the  use  of  X  for 
10  thousand  (X  mille}  is  an  illustration  of  the  same  tendency, 
the  bar  ( — )  simply  standing  for  m  (mille).  In  the  uncial 
writing  we  commonly  find  —  for  m,  and  in  the  Visigothic  we 
find  —  for  the  same  purpose.  It  is  quite  reasonable,  therefore, 
to  think  of  the  dash  ( — )  as  a  symbol  for  m  (minus),  just  as 
the  cross  (  +  )  is  a  symbol  for  et.  It  is  also  possible  that  its 
use  in  this  sense  may  have  come  from  the  habit  of  merchants  in 
indicating  a  missing  number  in  a  case  like  2  yd.  —3  in.,  where 
the  number  of  feet  is  missing.  We  have  the  same  habit  in  writ- 
ing certain  words  today,  using  either  a  dash  or  a  series  of  dots. 


(1549)  uses  the  French  plus  and  moins,  while  Gosselin  (1577)  uses  P  and  M. 
There  are,  of  course,  exceptions,  as  when  Trenchant  (1566)  uses  +  and  —  in  his 
work  in  the  Rule  of  False  Position,  and  when  Ramus  (1569)  writes:  "At  si  falsa 
conjectura  sit,  notatur  excessus  cum  signo  plus  sic  -f,  vel  defectus  cum  signo 
minus  —  "  (Schol.  Math.,  1569,  p.  138).  When  he  comes  to  algebra,  the  plus 
takes  the  form  — h—  (ibid.,  p.  269).  Vieta  (c.  1590)  wrote  #4*.  px2  for  #4  +  px2, 
frequently  using  the  dot  as  a  sign  of  addition.  He  also  used  —  for  subtraction  in 
certain  cases.  The  asterisk  denoted  an  absence  of  some  power  of  x. 

1  Libri's  surmise  that  they  are  due  to  Leonardo  da  Vinci  is  not  warranted.  See 
Bibl.  Math.,  XIII  (2),  52,  and  his  MSS.  as  published  by  Boncompagni.  Similarly, 
Treutlein's  idea  that  they  are  due  to  Peurbach  (Abhandlungen,  II,  29)  is  not 
substantiated.  2The  al-Khowarizmi  MS.  in  Mr.  Plimpton's  library. 

3 E.g.,  the  Regiomontanus-Bianchini  correspondence,  Abhandlungen,  XII, 
PP.  233,  279;  Bibl.  Math.,  I  (3),  506.  See  also  I.  Taylor,  The  Alphabet,  I,  8 
(London,  1883) ;  J.  W.  L.  Glaisher,  Messenger  of  Math.,  LI,  1-148. 


PLUS  AND  MINUS  SIGNS 


399 


4  -J- 

4  -  1  *     fcit  O&er  fcejjgcy* 

3  +  30    cbw/Qofutme* 

4  —  —  1  9    fcie  setter  ner  xwt> 
$  4-  44  8>*mn&BMe<w£ 
3  4-  az  —  t'ft/Oaet'ff  mi* 

Seitttter  ?  --  1 

3  4~  S$ 

4  -  1  6 
3-4-44  ^u  We 
3 

3  — 

3  -f-  p  -f  &«0  tjl  meet 
b<trj(12(t>t>icreil>nO?Tmmu0.  Hun 
folc  Ou  ftf  r  ^o!  g  abfd)U^«it  «Uwee«  foe 
m»ile3eU 

»mt>mad)t  3  i  z  tfc  ^ 
^rt6ijl>^  tt>t)rtt>tt?e 
tranter  won  4T3  p-XJrtD  6le?bert  4 
' 


The  signs  -f  and  —  first  appeared  in  print  in  an  arithmetic, 
but  they  were  not  employed  as  symbols  of  operation.  In  the 
latter  sense  they  appear  in  algebra 
long  before  they  do  in  arithmetic. 
Their  first  appearance  in  print  is 
in  Widman's  arithmetic  (1489),  the 
author  saying:  "Was  —  ist  /  das  ist 
minus  .  .  .  vnd  das  +  das  ist  mer."1 
He  then  speaks  of  "4  centner  +5 
pfund"  and  also  of  "4  centner  —  17 
pfund,"  thus  showing  the  excess  or 
deficiency  in  the  weight  of  boxes  or 
bales.  He  does  not  use  the  symbols 
to  indicate  operations,  but  writes,  for 
example,  "f  |  f  adir  fa  |  ist  i|," 
as  we  write  if  instead  of  i  4-  |, 
juxtaposition  signifying  addition. 

Manifestly  the  minus  sign  was 
more  important  as  a  warehouse  mark 
than  the  plus  sign,  since  mere  juxta- 
position serves  to  express  excess.2 

The  first  one  to  make  use  of  the 
signs  +  and  —  in  writing  an  algebraic 
expression  was  the  Dutch  mathema- 
tician Vander  Hoecke  (isi4_),3  who 

gave  R  f  -  R  f  for  Vf  -  Vf,  and  B  3  +  5  for  V3  +  5-  The 
next  writer  to  employ  them  to  any  extent  was  Grammateus 
(1518).  He  first  used  them  in  the  Rule  of  False  Position, 
where,  as  already  stated,  they  expressed  excess  and  deficiency 


p:o  4  ff  j  tvtc  f  nmcit  4  1  S"  *  tt>  »nt>  f  «m2 


FIRST  USE  OF  THE  SYMBOLS 

+    AND    —  ,    1489 

First  printed  use  of  these  sym- 
bols, from  Widman's  Behennde 
vnd  hupsche  Rechnung,  Leipzig, 
1489.  This  facsimile  is  from  the 
Augsburg  edition  of  1526 


1iSo8  ed.,  fol.  59;  in  the  1526  edition,  "was  auss  —  ist  /  das  ist  minus  .  .  . 
vnnd  das  /  +  das  ist  meer." 

2E.g.,  Albert  (1534)  writes:  "Item/Wie  komen  n  cent.  3  stein  18  pfund 

Zien,"  and  "Item  /  12  centner  4  stein 6  pfund  Talg."  He  frequently  uses  the 

long  bar  to  indicate  deficiency,  but  never  uses  the  plus  sign. 

3  Such  a  statement  is  likely  to  be  invalidated  at  any  time,  and  it  simply  means 
that  no  case  is  known  to  the  author  that  can  be  placed  earlier  than  that  in  Vander 
Hoecke's  work.  See  the  facsimile  on  page  401.  There  is  a  copy  of  the  1514 
edition  in  the  British  Museum.  For  the  1537  edition  see  Kara  Arithmetica,  p.  183. 


400  SYMBOLS  OF  ALGEBRA 

instead  of  operations  to  be  performed.1    When  he  wrote  upon 
algebra,  however,  he  used  them  in  the  modern  sense.2 

These  symbols  seem  to  have  been  employed  for  the  first  time 
in  arithmetic,  to  indicate  operations,  by  Georg  Walckl  ( 1-536), 8 
who  used  -f-  J  230  to  indicate  the  addition  of  $  of  230,  and 
—  1 460  to  indicate  the  subtraction  of  I  of  460.  The  algebraist 
who  did  the  most  to  bring  them  into  general  use  was  Stifel, 
to  whom  the  credit  for  their  invention  was  formerly  given.4 
For  3  x  -f  2  he  wrote  "3  sum:  +2.,"5  and  similarly  for  polyno- 
mials involving  the  minus  sign.0  From  this  time  on  the  two  sym- 
bols were  commonly  used  by  both  German  and  Dutch  writers, 
the  particular  forms  of  the  signs  themselves  not  being  settled 
until  well  into  the  i8th  century.  Thus,  for  example,  the  1752 
edition  of  Bar tj ens  has 

xx -^—  -. 2375  x^  1785000 

for  ,r'2  =  -  2375  ,r  +  1,785,000. 

1"Ist  zu  vil  /  setze -f  1st  aber  zu  wenig  /  setze —"  (1535  ed.,  fol.  E  3). 
Riese  (Rechnung  aufj  der  Linien  vnd  Federn,  1522)  used  the  symbols — I —  and 
— ; — ,  the  latter  being  also  used  by  various  other  writers  to  indicate  subtraction. 
Thierfelder  (1587),  for  example,  has  "25  fl.  •*•  232  gl."  (pp.  no,  229).  There  are 
numerous  variants,  such  as  HH  (Coutereels,  1599, 1690  edition  of  the  Cyffer-Boeck) 
and  — ~-  (Wilkens,  1669).  On  the  present  use  of  -*-  for  — ,  see  R.  Just,  Kauf- 
mannisches  Rechnen,  Leipzig,  1901. 

2 "Vnd  man  brauchet  solche  zeichen  als  +  ist  mehr/vnd — /minder."  He 
illustrates  by  adding  6*  -f  6  and  12^  —  4,  thus: 

6  pri.  +  6  N 
12  pri.  —  4N 
18  pri.  -f  2  N 

In  one  sense  6  x  +  6  means  an  excess  of  6  over  6  x,  and  we  evidently  find  here  the 
transition  from  the  excess  stage  to  the  addition  stage. 

3  Die  Walsch  practica,  Strasburg  (Nurnberg?),  1536. 

4  Probably  because  of  this  expression :    "  Darumb  so  gedenck  nur  nicht,  das 
disc  ding  schwer  seyen  zu  lernen,  oder  zubehalten,  und  ist  doch  die  gantz  sach 
diser  meiner  zeichen  hiemit  gantz  auszgericht  unnd  an  tag  gebracht."   Deutsche 
Arithmetica,  1545.   For  a  facsimile  from  his  work,  see  page  403. 

5"  ...  das  zeichen  +  /welches  ich  setzg  muss  zwischen  sie/als  2  zu  3  sum: 
machen  3  sum:  +  2.  das  machstu  denn  also  lesen/3  summen  vnd  2."  Ibid., 
fol.  21. 

6 "Denn  wo  du  discs  zeichen  —  findest/magstu  darfur  lesen/Weniger  oder 
Minder." 


foab 

Hem  fjcpfcc  tic  quatoafnt  toft  i  T~  toieritamuW 
litccrtD'CficuuaO.:acc  incttctt  Au&crmcoemti  &(t 


C-Oif  hratfonatcjcat  uct  —  oft  mct+itacr  Orn 


oft  —  uan  4-  aDDccrt  iubcfubfracri'c  fo  ucr* 
re  ate  fi  ai)0mlif  fun  eft  fuUtratotrlif  . 
__  C  fftultipliranc  mOru  B;  oanrattoua;cn« 
firr  E»i!Dt  multtpiif  rmt  utttrn  ^c  foe  niece  tat  ohp 
^-^tnnetfTcllcnalUOcnommcrouait  cnjfiu  naru 
re  aloft  tnnultiplumn  met  fimpclcit  notnnurfcc 
tnoct  al'i  Den  nommnr  multipluc  rt  n  iiac  fcc  rjualur  vt 
tco  lx  .M  to  wiltu  multipltrtrtu  (V  Q  met  4  fo  frt  4  in 
Cucn  lx  nuiltipltcccrt  4  tit  tiacr  fclutn  (ociitt  i<c  1  4  I:Q 
inulttplicrcvtomctio  cofe  144  bier  mt  trcrtftroct 
12,  foe  Dccl  io  (x  9  flhcmulrtpiircf  rf  met  4/  tuant  (V  9 
f  03  Dtt  mulf  iplicecrt  met  4  roemf  a  afo  oo;ai. 
Wtltitmu!tjpl(tcrHV*8 


VANDER  HOECKE'S  USE  OF  THE  PLUS  AND  MINUS  SIGNS 

Early  use  of  these  signs  in  Belgium  and  Holland  in  1514.   This  facsimile  is  from 

the  1537  edition 


402  SYMBOLS  OF  ALGEBRA 

England  adopts  the  Symbols.  England  early  adopted  the 
Teutonic  forms,  and  Recorde  (c.  1542)  says  "thys  fygure-f, 
whiche  betokeneth  to  muche,  as  this  lyne,  —  plaine  without  a 
crosse  lyne,  betokeneth  to  lyttle."1  Baker  (1568)  made  a  vain 
attempt  to  change  the  plus  sign,  saying:  "This  Figure  x, 
betokeneth  more :  and  this  plaine  line  - — ,  signifieth  lesse."2  All 
this  was  in  connection  with  the  Rule  of  False  Position,  and  not 
in  connection  with  arithmetic  operations.  As  symbols  of  opera- 
tion most  of  the  English  writers  of  this  period  reserved  the 
+  and  —  for  algebra.3 

Variants  of  the  Symbols.  The  variants  of  the  plus  sign  (  4- ) 
were  naturally  many,  partly  because  the  early  printers  had  to 
make  up  the  sign  by  combining  lines  that  they  had  in  their 
fonts.  Occasionally,  however,  the  religious  question  enters,  as 
in  certain  Hebrew  works  of  the  igth  century,  in  which  the 
Christian  symbol  of  the  cross  is  changed4  to  J~. 

The  expression  "plus  or  minus"  is  very  old,  having  been 
in  common  use  by  the  Romans  to  indicate  simply  "more 
or  less."  It  is  often  found  on  Roman  tombstones,  where 
the  age_pf  the  deceased  was  given  in  some  such  form  as 
AN  •  DODCXllTl  -  P  •  M;  that  is,  "94  years  more  or  less." 

Symbols  of  Multiplication.  Symbols  of  multiplication  were 
more  slow  in  their  development  than  symbols  of  addition  and 
subtraction,  the  reason  being  the  need  for  the  latter  as  ware- 
house marks  and  in  the  popular  Rule  of  False  Position.  The 
absence  of  a  sign  as  in  £5  and  3  ft.  led  naturally  in  the  i6th 
century  to  a  similar  usage  in  such  algebraic  forms  as  6  Pri.  for 
6x  and  73  (7  zenzo  or  7  censo)  for  *jx2.  The  late  medieval 

1  Ground  of  Aries,  ed.  1558,  fol.  Z  6. 

2  Ed.    1580,    fol.    184    (numbered    194).     Thierfelder    (1587)    uses    x    twice 
through  a  mistake  of  the  printer  (pp.  194,  246),  and  Wilkens  (1669)  uses  it  pur- 
posely in  connection  with  -f  (pp.  190,  191). 

•i  Thus  Digges  (1572),  in  his  treatment  of  algebra:  "Then  shall  you  ioyne 
them  with  this  signe  4-  Plus";  and  Hylles  (1600)  says:  "The  badg  or  signe  of 
addition  is  + ,"  stating  the  sum  of  3  and  4  as  "3  more  4  are  7,"  and  writing 
10 3  for  "  TO  lesse  3." 

4  This  is  found  in  several  such  works.  Among  the  latest  writers  to  use  the 
symbol  was  G.  J.  Lichtenfeld,  Yedeeotk  ha-Sheurim,  Warsaw,  1865. 


ritbmum, 


»ttt>~.          VII. 

£  id?  von  Jepcfcw  rrten  wr&e/fottu  nucfc  wffc&n 
von  Mfm  iftdjw  -f-  »n&  —  y&cff  foHic&e  v<r*<i$ 
nte/  dutmobcr  @«m:  X  otxr  (?.  if.  SSfcr&e  icfe 


nuns  txr  Jdlcn.lBa  Kfj  nu  rcfc  ven  glcicljcn  ictcfie/ 


em 


if^  hie  (UUen  trtU  auff  4  Ovco;c(n.  £)cnn  er 

>oa  fe* 
fum* 


D^  addict  aUee  |t(  Ixr  /  «(^  vntcr  ctn  cmigm 


VIII. 


big  $ctd)cii/  tm  2l6bi«n  vtt  £3ubtt:af)it:cn/ 
ol)«  rtllem  fo  bu  tm  fubt  rol)ircn  bie  $al  / 


t:  -i-  7. 
i*@um:  -+-   n. 


S   (Sum: 
3    @«m: 


*o@um;  ^-  is.  n  @uni; 


STIFEL'S  USE  OF  THE  SIGNS  +  AND  —  IN  ALGEBRA 

From  Stifel's  Deutsche  Arithmetica.  Inhaltend.  .  .  .  Die  Deutsche  Coss, 
NUrnberg,  1545 


404 


SYMBOLS  OF  ALGEBRA 


17 
17 

18 
19 

306 
323 

writers  usually  arranged  their  multiplication  tables  for  com- 
mercial use  in  columns,  as  in  the  two  cases  which  follow: 

2  •  43  .  86 
2-44-88 

In  this  arrangement  no  symbols  of  operation  or  equality  were 
used,1  the  dot  serving  for  both  purposes,  being  really  nothing 
but  a  symbol  of  separation,  like  the  ruled  lines. 

In  the  first  printed  books  no  such  symbols  appear,  the  Treviso 
arithmetic,  for  example,  giving  the  multiplication  table  in  the 

form  2  via  5  fa  10. 

Development  of  the  Symbol  x  .  The  common  symbol  x  was 
developed  in  England  about  1600.  In  the  second  edition  of 
Edward  Wright's  translation  of  Napier's Mirificilogaritkmorum 
canonis  descriptio  (London,  i6i8)2  is  "An  Appendix  to  the 


5  7 

6 

X 

2  3 

X' 

13  i  i 

6 

Logarithmes,"  and  this  contains  the  statement  (p.  4) :  "The 
note  of  Addition  is  (  +  )  of  subtracting  (— )  of  multiplying 
(x)/'  a  statement  that  is  very  likely  due  to  Samuel  Wright. 
The  larger  symbol  (  x  )  is  probably  due  to  Oughtred.3  It  was 
not  a  new  mathematical  sign,  having  long  been  used  in  cross 

xThe  first  of  these  examples  is  from  a  MS.  of  Benedetto  da  Firenze  writ- 
ten c.  1460,  and  the  second  from  one  of  Luca  da  Firenze  (c.  1475),  both  in 
Mr.  Plimpton's  library. 

2  But  not  in  the  1616  edition. 

3 See  F.  Cajori,  in  Nature,  (December  3)  1914,  p.  364;  William  Oughtred,  p.  27 
(Chicago,  1916) ;  "A  List  of  Oughtred's  Mathematical  Symbols,"  University  of 
California  Publications  in  Mathematics,  I,  171.  This  monograph  should  be  con- 
sulted on  the  entire  question  of  symbols.  It  contains  a  careful  study  of  various 
algebraic  signs.  Samuel  Wright  was  the  son  of  Edward  Wright.  He  entered 
Caius  College,  Cambridge,  in  1612  and  died  c.  1616. 


SYMBOL  OF  MULTIPLICATION  405 

multiplication,  in  the  check  of  nines,1  in  connection  with  the 
multiplication  of  terms  in  the  division2  or  addition3  of  frac- 
tions, for  the  purpose  of  indicating  the  corresponding  products 
in  proportion,4  and  in  the  "multiplica  in  croce"  of  algebra  as 
well  as  in  arithmetic.5  It  was  probably  because  of  this  last  use 
that  the  symbol  was  suggested  for  multiplication,  but  we  have 
no  positive  evidence  on  the  subject.  It  was  not  readily  adopted 
by  arithmeticians,  However,  being  of  no  practical  value  to  them. 
In  the  1  8th  century  some  use  was  made  of  it  in  numerical  work, 
but  it  was  not  until  the  second  half  of  the  igth  century  that  it 
became  popular  in  elementary  arithmetic.  On  account  of  its 
resemblance  to  x  it  was  not  well  adapted  to  use  in  algebra,  and 
so  the  dot  came  to  be  employed,  as  in  2-3  =  6  (America)  and 
2.3=6  (Europe).  This  device  seems  to  have  been  suggested 
by  the  old  Florentine  multiplication  tables  ;  at  any  rate  Vlacq, 
the  Dutch  computer  (1628),  used  it  in  some  of  his  work,  thus: 

Factores  Faci 

7.17  119 

although  not  as  a  real  symbol  of  operation.6  Clavius  (1583) 
had  an  idea  of  the  dot  as  a  symbol  of  multiplication,  for  he 
writes  f  •  f  for  f  x  |;7  and  Harriot  (posthumous  work  of  1631) 
actually  used  the  symbol  in  a  case  like  2.aaa  for  2  a3.  The  first 

xln  this  connection  Hylles  (1600)  speaks  of  it  as  the  "byas  crosse." 

2  As  in  2  V  3  JL,  for  -?-*-?  =  -?_.    See  page  226. 
3          5  10  5      3       10 


56  56  30 

4  As  in  the  case  of  2:3=4:6,  as  shown  at  the  right.    See  Buteo,      2y4 
De  Qvadratvra  circuit,  p.  67,  et  passim  (i5S9).  3^6 

6  Thus  Ghaligai  (1521;  1552  ed.,  fol.  76)  gives 

7  Pi"  1%    48 

X 
7  piu  r&   48 

to  indicate  (7  +  V^S)  (7  +  V^S). 

6  In  his  text  he  uses  a  rhetorical  form,  thus:   "3041  per  10002  factus  erit 
30416082." 

7  "...  minutia  minutiae  ita  scribeda  est  £  •  7  pronuciaturque  sic.   Tres  quin- 
tae  quatuor  septimaru  vnius  integri"  (Epitome,  1583). 

ii 


4o6  SYMBOLS  OF  ALGEBRA 

writer  of  prominence  to  employ  the  dot  in  a  general  way  for 
algebraic  multiplication  seems  to  have  been  Leibniz  (who  also 
used  the  symbol/^)1  or  possibly  his  contemporary,  Christian 
Wolf,  and  subsequent  algebraists  have  commonly  used  it  where 
the  absence  of  a  sign  does  not  suffice.2 

The  Symbol  -r-.  The  Anglo-American  symbol  for  divis'ion 
(  -s- ),  as  already  stated,  has  long  been  used  on  the  continent 
of  Europe  to  indicate  subtraction.  Like  most  elementary  com- 
binations of  lines  and  points,  the  symbol  is  old,3  and  toward  the 
close  of  the  isth  century  the  Lombard  merchants  used  it  to  in- 
dicate a  half,  as  in  4  -*-,  4  ~ ,  and  similar  expressions.4  There 
is  even  a  possibility  that  it  was  used  by  some  Italian  algebraists 
to  indicate  division,5  but  it  first  appeared  in  print  in  the 
Teutsche  Algebra,  by  Johann  Heinrich  Rahn6  (1622-1676), 
which  appeared  in  Zurich  in  1659.  John  Pell  had  been  Crom- 
well's political  agent  in  Switzerland  (1654-1658),  and  Aubrey7 
tells  us  that  "Rhonius  was  Dr.  PelPs  pupil  at  Zurich."  He  fur- 
ther asserts  that  "Rhonius's  Algebra,  in  High  Dutch,  was  in- 
deed Dr.  Pell's."  At  any  rate,  Rahn  used  the  symbol  and  Pell 
made  it  known  in  England  through  his  translation  (London, 
1688)  of  the  work. 

Symbol  ( :)  for  Ratio.  The  symbol  ( : )  to  indicate  ratio  seems 
to  have  originated  in  England  early  in  the  iyth  century.  It 
appears  in  a  text  entitled  Johnsons  Arithmetick ;  In  two 

1  Gerhardt's  edition  of  his  works,  II,  239 ;  VII,  54. 

2  Wolf  (1713)   makes  frequent  use  of  the  dot  in  cases  like  1.2.3.4    and 
m ;  —  2  .  m  —  3,  for  4!  and  (m  —  2)  (m  —  3)  respectively.    See  the  second  edition 
of  his  Elementa  Matheseos,  I,  322  (Halle,  1730) ;  also  the  facsimile  of  Leibniz's 
letter,  Volume  I,  page  420. 

3  It  was  used  for  est  as  early  as  the  loth  century,  as  in  i  -=- for  id  est  and 
it—  for  interest.   If  used  in  a  case  like  divisa-s-,  for  divisa  esty  it  might  possibly 
have  suggested  its  independent  use  as  a  symbol  of  division. 

4  A.  Cappelli,  Dizionario  di  abbreviature  latine  ed  italiane,  2d  ed.,  pp.  415, 
425.    Milan,  1912. 

r'In  a  MS.  in  Mr.  Plimpton's  library,  the  Aritmetica  et  Prattica,  by  Giacomo 
Filippo  Biodi  (Biondi)  dal  Anciso,  copied  in  1684,  the  symbol  _4~  stands  for 
division,  so  that  various  forms  of  this  kind  were  probably  used. 

6  Latin  Rhonius;  see  Volume  I,  page  412. 

7  Brief  Lives,  Oxford  edition  of  1898,  II,  121. 


RADICAL  SIGN  407 

Bookes,1  but  to  indicate  a  fraction,  f  being  written  3:4.  To  indi- 
cate a  ratio  it  appears  in  an  astronomical  work,  the  Harmonicon 
Coeleste  (London,  1651),  by  Vincent  Wing  and  an  unknown 
writer,  "R.  B."  In  this  work  the  forms  A\  B::  C:Z>  anA 
A.  B'.'.C.D  appear  frequently  as  equivalent  in  meaning.2  It  is 
possible  that  Leibniz,  who  used  it  as  a  general  symbol  of  divi- 
sion in  i684,3  took  it  from  these  writers.  The  hypothesis  that 
it  came  from  -5-  by  dropping  the  bar  has  no  historical  basis. 
Since  it  is  more  international  than  -^  ,  it  is  probable  that  the 
latter  symbol  will  gradually  disappear. 

Various  other  symbols  have  been  used  to  indicate  division, 
but  they  have  no  particular  interest  at  the  present  time. 

The  Radical  Sign.  The  ancient  writers  commonly  wrote  the 
word  for  root  or  side,4  as  they  wrote  other  words  of  similar 
kind  when  mathematics  was  still  in  the  rhetorical  stage.  The 
symbol  most  commonly  used  by  late  medieval  Latin  writers  to 
indicate  a  root  was  R,5  a  contraction  of  radix,  and  this,  with 
numerous  variations,  was  continued  in  the  printed  books  for 
more  than  a  century.6  The  symbol  was  also  used  for  other  pur- 

1  Title  as  in  F.  Cajori,  "Oughtred's  Mathematical  Symbols,"  Univ.  of  Calif. 
Pub.  in  Math.,  I,  181.  De  Morgan  (Arith.  Books,  p.  104)  gives  it  as  Johnson's 
Arithmatick  In  2  Bookes,  2d  ed.,  London,  1633. 

2F.  Cajori,  "Oughtred's  Math.  Symbols,"  loc.  cit.,  p.  181.  See  also  W.  W.  Be- 
man,  in  L' Intermediate  des  math.,  IX,  229;  F.  Cajori,  William  Oughtred,  p.  75- 
In  his  Clavis  Mathematicae  (1631)  Oughtred  used  a  dot  to  indicate  either  division 
or  ratio,  but  in  his  Canones  Sinuum  (1657)  the  colon  (:)  is  used  for  ratio,  pos- 
sibly by  some  editor  or  assistant.  It  appears  in  the  proportion  62496 : 34295  : : 
i  :  0/54.9  — .  Oughtred  ordinarily  used  the  dot  for  ratio,  as  in  A.  B  :  :  C.  D. 

3Gerhardt,  edition  of  his  works,  3.  Folge,  V,  223  :  "jc:  y  quod  idem  est  ac  x 

,.  .  x  „ 

divis.  per  v  seu  -. 

y 

4  As  in  Euclid,  X,  96.  Schoner  used  /  for  the  square  root :  "Quadrati  latus  in- 
explicabile  retextum  significatur  praenota  litera  /"  (De  numeris  figuratis  liber, 
1569;  1586  ed.,  p.  263).  On  the  Egyptian  symbol  see  Peet,  Rhind  Papyrus,  p.  20. 

ftThus  Chuquet  (1484)  used  both  \\  and  II2  for  square  root,  R3  for  cube 
root,  R4  for  fourth  root,  and  so  on.  See  Boncompagni's .Bullettino,  XIII,  655. 
Regiomontanus  (e.  1464)  has"2/'  »9  U  de  G-| !  "for  -./'  —  v  (1|J ,  as  in  the  Abhand- 
lungen,  XII,  234. 

«Thus  Pacioli  (1494)  has  "E  cosi  la  .R.  de  .20].  e  .4^"  (fol.  45,  v.).  He  also 
uses  R.  2*  for  square  root  and  R .  3a  for  cube  root,  as  on  fol.  46,  r.  E.  de  la  Roche 
(1520)  used  R  and  R2  for  square  root,  RQ  and  R3  for  cube  root,  and  I-R  and  R  l 


408  SYMBOLS  OF  ALGEBRA 

poses,  including  response?  res?  ratio*  rex*  and  the  familiar 
recipe  in  a  physician's  prescription.5 

Meanwhile  the  Arab  writers  had  used  various  symbols  for 
expressing  a  root,  among  them  .?»,  as  in  the  case  of 


for  x/4o  +  \/U  r,6  but  none  of  these  signs  seem  to  have  in- 
fluenced European  writers! 

European  Symbols  for  Roots.  The  symbol  V  first  appeared  in 
print  in  RudolfPs  Coss  (i525),7  but  without  our  modern  in- 
dices. When  Stifel  edited  this  work,8  in  1553,  he  varied  this 
symbolism,  using  ^/  for  V,  £/  for  -\/,  I/  for  -\/,  and  so  on. 
It  is  frequently  said  that  Rudolff  used  V  because  it  resembled 
a  small  r,  for  radix,  but  there  is  no  direct  evidence  that  this  is 
true.  The  symbol  may  quite  as  well  have  been  an  arbitrary 
invention.  It  is  a  fact,  however,  that  in  and  after  the  i4th 
century  we  find  in  manuscript  such  forms  as  **•  f  r^/t^and  r* 
used  for  the  letter  r? 

It  was  a  long  time  after  these  writers  that  a  simple  method 
was  developed  for  indicating  any  root,  and  then  only  as  a  result 
of  many  experiments.  For  example,  Vlacq10  used  V  for  square 
root,  VCD  for  cube  root,  Wfor  fourth  root,  and  so  on;  Rahn11 

for  fourth  root.  See  the  Abhandlungen,  I,  63.  Cardan  (1539)  and  Tartaglia  (1556) 
used  lifor  square  root  and  li  cu.  for  cube  root,  while  Ghaligai  (1521)  usedl^D 
and  RQ3,  and  Bombelli  (1572)  usedR  .q  and  RJ  .c.  respectively  for  the  same  pur- 
poses. There  were_also  the  usual  run  of  eccentricities,  as  illustrated  by  the  use  of 
Ract.  300  for  V^oo  by  an  Italian  arithmetician,  Bonini,  in  1517. 

1  Trenchant  (1566). 

2  For  the  unknown  quantity,  as  in  the  Rollandus  MS.  (c.  1424).   As  represent- 
ing res  in  general,  it  is  found  as  early  as  the  8th  century. 

3  As  early  as  the  8th  century.  4  As  early  as  the  i4th  century. 

5  Also  as  early  as  the  i4th  century. 

6  F.  Woepcke,  Recherches,  p.  15.   The  Arabic  forms  are  read  from  right  to  left. 
7"  .  .  .  vermerkt  von  kiirtz  wegen  radix  quadrata  mit  solchem  character  V 

.  .  .  radix  cubica  wiirt  bedeut  durch  solchen  character  C  V    V  ." 

8  Die  Coss  Christoph  Rudolff  s,  fols.  61,  62  (Konigsberg  i.  Pr.,  1553).  The  title- 
page  bears  the  date  1553;  the  colophon,  1554. 

9  For  these  and  other  forms  consult  A.  Cappelli,  Dizionario,  26.  ed.,  p.  318. 

10  Arithmetica  Logarithmica,  p.  4.   Gouda,  1628.         n  Teutsche  Algebra,  1659. 


RADICAL  SIGN  409 


(1622-1676)  used  V,  Vc;  \/V,  ^J  CC,  and  VW  for  the  square, 
cube,  fourth,  sixth,  and  eighth  roots  respectively,  and  various 
writers  used  V.  3.,  V.  cc.,  VS. 5.,  V^S,  and  V.5.  cc.  for  the 
square,  cube,  fourth,  fifth,  and  sixth  roots  respectively.1 

French,  English,  and  Italian  writers  of  the  i6th  century  were 
slow  in  accepting  the  German  symbol,  and  indeed  the  German 
writers  themselves  were  not  wholly  favorable  to  it.  The  letter 
/  (for  latiiSj  side;  that  is,  the  side  of  a  square)2  was  often  used. 
Thus  we  find  the  Ramus-Schoner  work  of  1592  using  £4  for  \/4, 
Ics  for  -^5,  Isq6  and  116  for  A/6,  1  /  3  for  -^3,  and  other 
similar  forms,  and  using  the  related  forms  i  1.,  i  q.,  i  c.,  i  bq., 
and  i  qc.  for  a,  a2,  a3,  a4,  and  a5  respectively.  For  the  binomial 
I2+V32  the  work  has  b  12  +132,  and  for  the  residual 
12  —  \/32  it  has  r  ?2  — 132.  In  a  somewhat  similar  way  Gosse- 
lin,  in  his  De  Arte  Magna  (1577),  uses  L  9  for  VQ,  LC  8  for 

•v'S,  LL  16  for  -v/76,  and  LV  24  PL  9  for  ^24  +•  Vg  (the  V 
standing  for  universale  and  the  P  for  plus). 

General  Adoption  of  the  Radical  Sign.  In  the  1 7th  century  our 
common  square-root  sign  was  generally  adopted,  of  course  with 
many  variants.  Thus  Stevin3  has  substantially  the  same  sym- 
bols as  those  used  by  Rudolff,  but  with  V(3)  for  cube  root, 
\V©  for  the  fourth  root  of  the  cube  root,  and  so  on,  with 
•\/3)(2  for  V3  •  x2  and  Vs(2)  for  Vslx^.  Antonio  Biondini, 
whose  algebra  appeared JnJ/enice  injc689,  has  such  symbols 
as  V8  x  for  V8  x  and  ¥24  xx  for  ^24  x2.  The  different  vari- 
ants of  the  root  sign  are  too  numerous  to  mention  in  detail  in 
this  work,  particularly  as  they  have  little  significance.  Such 

forms  as  ,—  /- 

v#  x  100     r       V#  x  100 

for    — -^~==r-. 

I     100  vioo 

84  _ 

are  not  uncommon.    Newton  _used  V8?   Vi6?  .  .  .  for    ^8, 

•VTfi,  .  .  .  ,4  but  he  also  used  -\/a. 

1  JB.g.,  Cardinael,  Arithmetka,  Bk.  I.   Amsterdam,  1659. 

2  See  page  407,  n.  4.   3  Arithmetiqve  (1585),  Girard  edition  of  1634,  pp.  10,  19. 
*Arithmetica  Universcdis,  p.  37  (Cambridge,  1707).   Among  other  statements 

he  has  "quod  V6  valeat  V2  x  3."  Later,  as  on  page  273,  he  has  V  —  4  ±  Vs. 


410  SYMBOLS  OF  ALGEBRA 

By  the  close  of  the  1  7th  century  the  symbolism  was,  there- 
fore, becoming  fairly  well  standardized.  We  have,  however, 
in  Ozanam's  Dictionnaire  Mathematiquc  (Paris,  I691)  J>uch 
forms  as  ^C.aab  for  ^a2b  and  VC.a3  +  3  abb  for  -\/a3  +  3^2> 
so  that  there  still  remained  some  work  to  be  done.  The  i8th 
century  saw  this  accomplished,  and  it  also  saw  the  negative  and 
fractional  exponent  come  more  generally  into  use.  The  early 
history  of  these  forms  is  considered  later. 

Symbols  of  Relation.  One  of  the  earliest  known  symbols  of 
algebra  is  a  sign  of  equality.  This  may  be  said  to  have  appeared 
in  the  Ahmes  Papyrus  (c.  1550  B.C.),  although  Ahmes  simply 
used  a  hieratic  form  for  a  hieroglyphic.  He  commonly  wrote 
-*-  for  the  hieroglyphic  difa,  temt,  meaning  "together,"  the  re- 
sult of  addition.  In  hieroglyphics,  for  example,  we  should  have 
0  IciSbfiJ  for  10  +  i  =  ii.1  The  Egyptians  also  used  ,»,  er, 
meaning  "it  makes,"  as  in 


<> 

T-P  iiuii      n    nno 

meaning  f     £     TV     ^V     er    *> 

or  f  +  i+^  +  ^j.' 

There  is  no  evidence  of  the  use  of  a  generally  recognized 
symbol  for  equality  until  the  Greeks  employed  the  initials 
i°  or  la  for  ?<ro9  (i'sos}?  equal.  This  symbol  is  found  in  the 
Arithmetica  of  Diophantus  (c.  27S).4  The  Arabs,  contrary  to 
the  Greek  custom,  used  for  this  purpose  the  final  letter  of  their 
word  for  equality. 

In  general  the  classical  and  medieval  writers  used  the  full 
word.5  In  the  Middle  Ages  a  general  shorthand  was  adopted 

1Eisenlohr  ed.,  p.  39.   But  on  all  this  consult  Peet,  Rhind  Papyrus. 
2Eisenlohr,  loc.  cit.,  p.  41.    They  also  had  other  forms,  for  which  consult 
both  Peet  and  Eisenlohr.    The  symbols  used  in  equations  are  given  on  page  422. 

3  As  in  isosceles,  isoperimetry,  isogonal,  etc. 

4  He  also  used  fcros  fan.   For  a  discussion  of  the  symbol  see  Heath,  Diophan- 
tus, 2d  ed.,  p.  47.  The  small  Greek  letters  here  shown  are  modern. 

5  E.g.,  Fibonacci,,  in  his  Flos  (c.  1225),  used  such  forms  as  equantur  and 
equabitur.   Scritti,  II,  235. 


SYMBOLS  OF  RELATION  411 

by  university  students  in  copying  their  texts.  Partly  as  a 
result  of  this  movement  there  slowly  developed  a  set  of  mathe- 
matical symbols,  other  contributing  causes  being  a  commercial 
shorthand  and  the  advantage  of  expressing  an  equation  in  a 
form  easily  held  by  the  eye.  Thus  we  have  such  symbols  as 
-*-  for  est,  p  for  per,  &  for  cento,  and  oc  or  x  for  equality.1 
This  symbol  for  equality,  oc  or  CQ,  was  used  by  Descartes 
(1637)  and  is  found  in  various  manuscripts  of  his  period.  It 
has  generally  been  thought  to  come  from  ae,  for  aequalis, 
acquales,  aequalia,  or  aequantur.  This  may  be  the  case,  al- 
though it  is  by  no  means  certain.2 

Various  other  symbols  were  used  for  the  same  purpose.  Thus 
Buteo  (1559)  used  [;  Xylander  (1575),  II;  and  Herigone 
(1634),  2/2.  Leibniz  (c.  1680)  used  =,  1—1,  and  other  sym- 
bols with  nearly  the  same  meaning. 

Modern  Symbol  of  Equality.  As  a  printed  symbol  our  sign 
(  =  )  is  due  to  Recorde,3  who  says:  "I  will  sette  as  I  doe  often 
in  woorke  vse,  a  paire  of  paralleles,  or  Gemowe4  lines  of  one 
lengthe,  thus:  =  ,  bicause  noe  .2.  thynges,  can  be  moare 
equalle."  If  he  had  used  shorter  lines  (  =  ),  there  might  be 
some  reason  for  thinking  that  the  symbol  was  suggested  by  the 
medieval  use  of  —  for  esse*  but  Recorders  clear  statement  of 
its  arbitrary  invention  in  the  form  =rr:  is  conclusive. 

The  symbol  was  not  immediately  popular.  When  Rahn 
(1622-1676)  wrote  his  algebra,  a  century  later,  he  felt  obliged 


bar,  -  ,  indicating  equality,  as  used  in  the  correspondence  of  Regio- 
montanus,  can  hardly  be  considered  a  symbol  in  the  ordinary  sense.  See  Bibl. 
Math.,  I  (3),  So6. 

2  E.g.,  in  the  algebra  of  Clavius  (Rome,  1608,  p.  39  seq.)  there  are  expres- 
sions like  "aequatio  inter  f  ^(-f  7,  &  i%"  and  "sit  aequatio  inter  4^,  &  72  —8^," 
so  that  the  20  may  possibly  have  come  from  &.   It  is  quite  as  reasonable  to  think 
that  it  was  a  purely  arbitrary  invention. 

3  Whetstone  of  witte,  London,  1557.   See  the  facsimile  on  page  412. 

4  From  O.  F.  gemeus,  twins,  from  Lat.  gemellus,  twin.   Recorde  uses  gemowe 
in  his  Pathewaie  to  knowledg  (1551)  to  mean  parallel,  speaking  of  "Paralleles, 
or  Gemowe  lynes."    The  various  zodiacal  signs  for  the  gemini  may  have  sug- 
gested all  these  forms. 

5  But  not  for  est,  where  -f-  was  commonly  used.   We  find  it  also  in  compounds 
like  =nt  for  essent.  See  A.  Cappelli,  Dizionario,  2d  ed.,  p.  407. 


412  SYMBOLS  OF  ALGEBRA 

to  explain  its  meaning  as  not  familiar  to  mathematicians/  and 
the  use  of  so  continued  until  well  along  in  the  i8th  century. 

Tbejftte 

as  tfjcir  foo;ftea  Doe  ejrtett&e  )  to  m'8i  IMC  it  cmcty  Atto 
ttooo  parted  Whereof  tbcfirtlete,  vbnonenomteris 
cqtulle  »nto  one  other.  0nD  tbe  fecon&e  id  >*&f  »  one  »  m? 
ttris  comfareJ  as  ejudle  Tnttt.wtbcr  nomben, 

aitoatc*  lulling  pott  to  remiber,  tftat  pou  retiuce 
journomber*,  totijeirleaOe  Denominations  ^  antt 
fmallctte  fo;me0,bcfo?e  rou  p:occoc  anp  farther. 

0nD  agatn,f  f  pour  ^«4^»  be  Cocbc,  tljat  tljc  grca^ 
tette  Denomination  G/%>  be  toineD  to  anp  parte  of  9 
compounDc  nombet  9  pou  fl^all  tourne  it  To  ,  tl>at  tl)e 
nombcroftticgrcatcttc  Cgne  alone, 


ano  tfr*  t5  all  tbat  neaDctb  to  be  taugljte  , 
npngtl)i*tooo;tte. 

^otubctt.fo;  eafie  altcratf  o  of  *  j/wf/w  .3  Mil  pw* 
pounDe  a  fetoe  craple0,bf  caufe  tbe  extraction  of  tljetc 
roote0,maie  tbe  mo;e  aptlp  bee  tujougbte,  ^nD  to  v 
uotoe  tbe  teDtoufe  repetition  of  tbefe  luoo^Dcs  :  tee* 
qualle  to  :  3  Urill  fette  a^  3  Doe  often  tn  iuoo;be  Dfe^a 
pairc  of  parallele0,oz  dCfemotoe  lines  of  one  lengrtbc, 
tbu0:=—  —  ,bicaufe  noe.2*  tbpnges,can  be  moare 
equalle. 


1  —  •  i  y-f  =«—  7  hf  . 

•!  8.f  =*««=«*  I  o  2* 
26.5*  —  I  —  1 
1  9.2£—  H  —  1  92.5=—  1  05 


6.      545—  -  i2^-*=4o^—  f—  48of  —  9.5- 

RECORDERS  SIGN  OF  EQUALITY 
From  Recorded  Whetstone  of  witte  (1557) 

1  tf  Bey  disem  anlaasz  hab  ich  das  namhaf  te  gleichzeichen  =  zum  ersten 
gebraucht,  bedeutet  ist  gleich"  (Teutsche  Algebra,  1659).  It  was  probably  sug- 
gested to  him  by  Pell,  who  was  familiar  with  Recorded  works. 


PROPORTION  AND  INEQUALITY  413 

Symbol  of  Proportion.  The  symbol  for  the  equality  of  ratios 
( : : ) ,  now  giving  way  to  the  common  sign  of  equality,  was  in- 
troduced by  Oughtred  (c.  I628),1  and  Dr.  Pell  gave  it  still 
more  standing  when  he  issued  Rahn's  algebra  in  English 
(1668).  It  seems  to  have  been  arbitrarily  chosen. 

The  symbol  ^  for  continued  proportion  was  used  by  Eng- 
lish writers  of  the  i7th  and  i8th  centuries2  and  is  still  com- 
monly seen  in  French  textbooks. 

Symbols  of  Inequality.  The  symbols  >,  <,  for  greater  and 
less,  are  due  to  Harriot3  (1631).  They  were  not  immediately 
accepted,  for  many  writers  preferred  [T"~  and  D,  symbols  which 
Oughtred  (1631)  had  suggested.4 

The  symbols  =£,  «£,  and  *$>  are  modern  and  are  not  inter- 
national, but  in  the  1647  edition  of  Oughtred's  Clavis  the  some- 
what analogous  symbols  C7~  and  _.*"!  appear  for  non  majus  and 
non  minus  respectively.  On  the  Continent  the  symbols  =  and 
~,  or  some  of  their  variants,  apparently  invented  by  Pierre 
Bouguer5  (1734),  are  commonly  used. 

Symbol  for  Infinity.  The  symbol  for  infinity  (  oo  )  is  first 
found  in  print  in  the  Arithmetica  Infinitorum  published  by 
Wallis  in  1655,°  and  may  have  been  suggested  by  the  fact  that 
the  Romans  commonly  used  this  symbol  for  a  thousand,  just  as 
we  use  "myriad"  for  any  large  number,  although  in  the  Greek 
it  meant  ten  thousand. 

1  In  his  Elementi  decimi  Euclidis  declaratio,  added  to  the  1648  edition  of  his 
Clavis,  he  gives  the  symbol  for  "proportio,  sive  ratio  aequalis  :  :."  F.  Cajori, 
William  Oughtred,  p.  26;  "Oughtred's  Math.  Symbols,"  loc.  cit.,  p.  181,  n.  8.  It 
appears  also  in  the  1631  edition  of  the  Clavis  itself. 

2 E.g.,  Barrow  (Lectiones  Mathematicae,  Lect. XXVII,  London,  1683) .  J.Ward 
(c.  1706)  says:  "The  character  made  Use  of  to  signify  continued  Proportionals 
isH"  (The  Young  Mathematician's  Guide,  London,  i2th  ed.,  1771,  p.  77).  It 
also  appears  in  the  American  Greenwood  arithmetic  (1729). 

BArtis  Analyticae  Praxis.   London,  1631  (posthumous) . 

4  E.g.,  Barrow:  "AQ~"~B.   A  major  est  quam  B.   A     H  B  A  minor  est  quam 
B"  (Lectiones  Opticae  &  Geometricae   (London,  1674),  preface;  and  English 
edition  (1735),  p.  310). 

5  See  biographical  note  on  page  327. 

6 This  is  seen,  for  example,  in  such  expressions  as  "jam  numerus  incre- 
mentorum  est  <*>"  (Opera,  I,  453  (1695)). 


414  SYMBOLS  OF  ALGEBRA 

Integral  Exponents.  Our  present  integral  exponents  may  be 
said  to  have  begun  with  Descartes  (1637),  although  Herigone ' 
(1634)  had  nearly  anticipated  him.  Since  the  early  methods 
of  indicating  powers  relate  naturally  to  the  writing  of  equa- 
tions, these  are  more  appropriately  considered  in  connection 
with  that  topic  (page  421).  It  may  simply  be  said  at  this  time 
that  Harriot  (who  died  in  1621),  in  the  transition  period  from 
the  use  of  forms  like  Aq  to  forms  like  x2,  used  aa  for  a2  and 
aaa  for  a3.  This  symbolism  was  commonly  employed  until  well 
into  the  i8th  century,  even  in  writing  a  polynomial  involving 
a4]  that  is,  before  c.  1750  it  was  common  to  find  expressions 
like  a5  4-  a4  +  aaa  -f  aa  -f  i,  or  even  a5  +  aaaa  +  aaa  -h  aa  +  i . 

In  his  Cursus  Mathematicus  (1634-1637)  Herigone  used 
02,  03,  and  a 4  for  a2,  a3,  a*,  no  doubt  influenced  by  the  fact  that 
Girard  (1629)  used  forms  like  5  (2)  for  s*2;  and  some  of  his 
contemporaries,  like  Dechales  (c.  1660)  and  Jacques  de  Billy 
(1602-1679),  did  the  same.  Descartes  (1637),  however,  wrote 
the  exponents  in  the  present  manner;1  but  even  without  this 
symbolism  Stevin  (1585)  had  already  given  a  systematic  dis- 
cussion of  integral  exponents.2 

General  Exponents.  The  general  exponent  was  known  in 
theory  long  before  it  came  into  practical  use.  Oresme  (c.  1360) 
wrote 


and 

and  used  other  similar  forms,  as  already  stated.    He  also  gave 
rules  for  fractional  exponents. 

Chuquet  (1484)  used3  12°  for  12,  12  *  for  12  times  a  "nom- 
bre  linear,"  12 2  for  12  £2,  and  so  on.  For  gx~3  he  wrote4  .g.3  m, 

!"Et  aa,  ou  a2,  pour  multiplier  a  par  soy-meme;  Et  a3,  pour  le  multiplier  en- 
core une  fois  par  a,  &  ansi  a  1'infini"  (1705  ed.,  p.  4). 

2L'Artikmetiqve,  Girard  edition  of  1634,  p.  53.  See  also  the  general  discussion 
by  H6rigone,  loc.  tit. 

3  As  he  says,  "coe  nobre  simplemt  pris  sans  aulcune  denomlacion  ou  dont  sa 
denolacio  est  .o."  (Roncompagni's  Bullettino,  XIII,  737). 

llbid.y  p.  742.  See  also  Ch.  Lambo,  "Une  Algebre  Franchise  de  1484.  Nicolas 
Chuquet,"  Revue  des  Questions  Scientifiques,  October,  1902. 


EXPONENTS  415 

thus  showing  that  he  had  an  idea  of  negative  exponents,  but 
it  was  more  than  two  centuries  before  the  theory  was  under- 
stood. As  to  fractional  exponents,  certain  evidences  show  that 
the  idea  was  developing  during  the  i6th  century.  This  is  seen 
especially  in  StifePs  Arithmetica  Integra*  (1544),  where  there 
is  given  what  amounts  to  the  relation 

<¥)"*  =  (¥)*  =  f  I- 

Albert  Girard2  ( 1629)  employed  the  fractional  exponent,  rep- 
presenting  it  by  such  forms  as  (|)  2000  for  \/2QOO,  and  (f )  49 
for  49  * ;  and  the  study  of  logarithms  from  the  standpoint  of 
exponents,  undertaken  at  about  the  same  time,  tended  to  bring 
these  general  forms  into  wider  use. 

Wallis  on  General  Exponents.  The  first  of  the  writers  of  this 
period  to  explain  with  any  completeness  the  significance  of 
negative  and  fractional  exponents,  however,  was  Wallis  (1655). 
He  showed  that  x°  should  signify  i,  and  established  relations 
of  the  following  nature : 3 


x" 

*  =  ~f  =  V 


Newton  supplemented  the  work  of  Wallis  and  in  1669  made 
use4  of  such  forms  as  x*  and  or3,  and  after  this  time  the  sym- 
bolism became  universally  recognized. 

!H.  Wieleitner,  "Gebrochene  Exponenten  bei  Michael  Stifel,"  Unterrichtsblat- 
ter  fur  Mathematik  und  Naturwissenschaften,  1922,  No.  5. 

2  Invention  nouvelle  en  I'algebre,  pp.  97-101.    Amsterdam,  1629. 

s  Thus  he  speaks  of  -  "cujus  index  —  3,"  that  is,  2~3  =  -  ;  of  —p  "cujus  in- 
o  o          V2 

dex  —    '"  that  is,  2"^^  =  —=  ("Arithmetica  Infinitorum,^  in  the  Opera  (1695), 

2  V2 

I,  410,  459,  and  in  the  earlier  edition) . 

4  In  the  "De  Analysi  per  aequationes  numero  terminorum  infinitas"  sent  by 
Collins  to  Barrow,  July  31,  1669.  See  the  Commercium  Epistolicum,  p.  67 
(London,  1725).  For  interesting  comments  on  Newton's  use  of  exponents  see 
G.  A.  Lecchi,  Arithmetica  Univer salts  Isaaci  Newtoni,  Liber  II,  Pars  III,  p.  118 
(Milan,  1752), 


416  FUNDAMENTAL  OPERATIONS 

Symbols  of  Aggregation.  Symbols  of  aggregation  first  devel- 
oped to  any  considerable  extent  in  the  i6th  century,  and  in 
connection  with  the  study  of  radicals.  Tartaglia  (1556)  writes 
"22  men  (22  men  ft  6"  for  22  —(22  — \/6).  Bombelli  (1572) 
used  L,  for  legato,  as  a  kind  of  symbol  of  aggregation,  as  in  the 
squaring  of  2  +  x  +  \/2O  —  6x  +  x2,  which  appears  as 

1  1  2 

2.  p.  Y.  p.  R.  q.  L  20.  m.  6.  p.  Y.  J, 
the  result  being  given  as 

2  ^  t  ^  <L^ 

2.  p.  24.  m.  2.  p.  R.  q.  L  4.  m.  8.  p.  224.  p.  320  J, 

in  which  the  L  and  the  reversed  L  are  clearly  symbols  of  ag- 
gregation and  may  naturally  have  suggested  our  square  paren- 
theses, first  used  by  Girard  (1629)  for  this  purpose. 

Other  Italian  writers  frequently  employed  the  letter  V,  the 
initial  of  universalis,  to  indicate  that  a  root  sign  applied  to  all 
the  expression  which  followed.  Thus  Cardan,  in  his  first  printed 
solution  of  the  cubic  equation,  has 

R  V  :  cu.  Ei  108  p  :  10 
m  :  R  V  :  cu.  B  108  m  :  10 

for  V-VioS  -f  10  -  V -\/io8  -  10. 

By  the  time  Clavius  published  his  algebra  (1608)  the  paren- 
theses had  apparently  become  common,  for  he  uses  them  freely 
without  any  explanation. 

5.  FUNDAMENTAL  OPERATIONS 

Number  of  Operations.  While  there  were  certain  operations  in 
arithmetic  that  were  looked  upon  as  fundamental,  the  number 
varying  from  time  to  time,  this  was  not  the  case  in  the  early 
printed  algebras.  It  was  only  when  textbooks,  based  upon  the 
early  arithmetics,  came  into  use,  that  such  operations  as  ad- 
dition and  subtraction  were  given  as  distinct  topics.  For 
example,  Pacioli  (1494)  begins  his  work  on  algebra1  by  consid- 

a,  fol.  in,  v. 


NUMBER  OF  OPERATIONS  417 

ering  a  few  definitions,  then  the  laws  of  signs,  and  then  the 
operations  with  monomials,  taking  up  the  operations  with  poly- 
nomials somewhat  incidentally  as  they  arise;1  and  the  same 
may  be  said  of  the  other  Italian  algebraists2  of  the  i6th  cen- 
tury. Clavius  was  one  of  the  first  to  consider  the  subject 
somewhat  as  we  do  at  present.3  He  early  introduces  a  chapter 
De  additione  et  svbtractione  numerorum  Cossicorum,  this 
being  followed  by  De  mvltiplicatione  &  diuisione  numerorum 
Cossicorum. 

The  reason  for  this  early  neglect  was  that  algebra  was  looked 
upon  as  a  study  for  mathematicians,  not  for  boys  and  girls  in 
their  school  years.  For  any  mature  mind  that  is  interested  in 
mathematics  these  operations  are  too  simple  to  require  any 
special  attention. 

Amount  of  Work.  For  this  reason  the  amount  of  work  as- 
signed to  topics  of  this  kind  by  those  algebraists  who  gave  them 
any  attention  was  very  slight.  For  example,  Pacioli  gives  no 
examples  involving  numerical  cases  like  (—2)  (  —  3)  ==  +  6, 
except  a  few  that  are  completely  worked  out,4  and  similarly 
when  he  comes  to  surds.5  An  illustration  of  his  problems  is 
seen  in  the  following : 

via.  4.   p.  Tfc.  6. 
4.  m.  B.  6. 
16.  m.  6. 
Productum  10 

meaning  that  (4  +  *>/6)  (4  —  >/6)  =  16  —  6  =  io.6  Similarly, 
Tartaglia7  solves  a  few  typical  problems  involving  signs,  but 
gives  no  exercises  for  original  work. 

Difficulties  Due  to  Poor  Symbolism.  A  good  idea  of  the  general 
difficulties  which  characterized  this  period  of  poor  symbolism 
is  seen  in  the  algebra  of  Pedro  Nunes,  of  which  the  second 

1£.g.,  fol.  127,  v.,  seq. 

2Tartaglia,  General  Trattato,  II,  fol.  81  (1556) ;  Bombelli,  Algebra,  libro  primo 
(1572) ;  Cardan,  Ars  Magna,  cap.  i  (1545).  *  Algebra,  p.  16  (Rome,  1608). 

4 Fol.  in,  v.,  seq.  5Fol.  115,  i>.,  seq.  6  Fol.  123,  r. 

1  General  Trattato,  La  seconda  parte,  fol.  81  (Venice,  1556). 


418  CONTINUED  FRACTIONS 

edition  appeared  at  Antwerp  in  1567.    His  multiplication  of 
3  x~  +  2\  x  -f  5^  by  4  x  -f  3  appears  as  follows : 

3  •  ce  •  p  •    2  •  co  •  \  •  p  •  5^ 

4  •  co  •  p  •  3-     


1 2  •  cu  •  p  •    9  •  ce  •  p  •  20  •  co  •  ^ 

9  •  ce  •  p  •    6  •  co  •  2  •  p  •  1 5-^g 
12  •  cu  •  p  •  18  •  ce  •  p  •  27  •  co  •       p  -  I5y*6 

Bombelli  (1572)  sets  forth  the  work  more  after  the  modern 
plan,  but  gives  no  cases  to  be  solved  independently.  The  fol- 
lowing is  a  type : 

i     I  p  2 

i     I  p  2 

2  I  p    4   i  p     4 

4  f  p     $   3  p  24  2  p  32  i  p   16 

IIP       *    ~ 

5  I  p  10  4  p  40  3  p  80  2  p  80  j:   p  32, 

meaning  that    (x  +  2)'2  =  ;r2  -f  4  *  +  4, 

(.r2  4-4^  +  4)2  =  ^  4-  8  .r3  -f  24  ,r2  -f  32  .r  -f-  16, 
and  this  multiplied  by  x  -f  2  gives  the  fifth  power  of  x  4-  2.1 

6.  CONTINUED  FRACTIONS 

Early  Ideas.  It  is  not  necessary  to  speak  of  the  history  of 
simple  algebraic  fractions,  since  these  forms  were  transferred 
from  arithmetic.  When  Euclid  found  the  greatest  common 
measure  of  two  lines,2  or  when  the  same  principle  was  applied 
to  the  finding  of  the  greatest  common  divisor  of  two  numbers,3 

1 1572  ed.,  p.  69.   The  I  seems  to  have  been  the  coefficient  of  the  highest  power, 

2  Elements,  X,  3  and  4,  for  commensurable  magnitudes  in  general. 

3  Elements,  VII,  i  and  3. 


BEGINNING  OF  THE  THEORY  419 

a  process  was  used  that  is  similar  to  that  of  converting  a  frac- 
tion into  a  continued  fraction,  as  is  evident  from  the  following : 

12)38(3 

36  £2  __   6_  __       I 

2)12(6  38"  i9~~3+i' 

12, 

This  is  the  earliest  important  step  in  the  theory  of  continued 
fractions.1  Further  traces  of  the  general  idea  are  found  occa- 
sionally in  the  Greek  and  Arab  writings. 

Beginning  of  the  Modern  Theory.  Although  the  Greek  use  of 
continued  fractions  in  the  case  of  greatest  common  measure 
was  well  known  in  the  Middle  Ages,  the  modern  theory  of  the 
subject  may  be  said  to  have  begun  with  Bombelli  (1572).  In 
his  chapter  relating  to  square  root2  he  considered  the  case  of 
A/i3.  Substituting  our  modern  symbolism,  he  showed  that  this 
number  is  equal  to 

4 


3  + 


644 


6+  •••. 
In  other  words,  he  knew  essentially  that 


=  a  + 

2  a 


2a-\ 


The  next  writer  to  consider  these  fractions,  and  the  first  to 
write  them  in  substantially  the  modern  form,  was  Cataldi3 
(1613),  and  to  him  is  commonly  assigned  the  invention  of  the 
theory.  His  method  was  substantially  the  same  as  Bombelli's, 

1On  the  history  in  general,  see  S.  Gunther,  Beitrage  zur  Erfindungsgeschichte 
der  Kettenbruche,  Prog.,  Weissenburg,  1872;  Italian  translation,  Boncompagni's 
Bullettino,  VII. 

2  "  Modo  di  formare  il  rotto  nella  estrattione  delle  Radici  quadrate,"  Algebra, 
P-  35- 

3  Trattato  del  modo  brevissimo  di  trovare  la  radice  quadra  delli  numeri 
Bologna,  1613. 


420  CONTINUED  FRACTIONS 

but  he  wrote  the  result  of  the  square  root  of  1  8  in  the  fol- 
lowing form  : 


This  he  then  modified,  for  convenience  in  printing,1  into  the 
form2 

4&!.&t.&!.- 

The  third  writer  to  take  up  the  theory  was  Daniel  Schwenter 
(1618).  In  attempting  to  find  approximate  values  for  |||  he 
found  the  greatest  common  divisor  of  177  and  233,  and  from 
this  he  determined  the  convergents3  as-jW  ^  |,  |,  j,  and  -£. 

The  next  writer  of  prominence  to  use  these  forms  was  Lord 
Brouncker,4  who  transformed  the  product 

4      3-  3-  5-  5  •  7-  7-" 


which  had  been  discovered  by  Wallis,  into  the  fraction 
i  —  i 

7T~  O 

2  +  ^,25      _ 


as  already  stated  on  page  311.  He  made  no  further  use  of 
these  forms,  Wallis  then  taking  up  the  work  and  using  the 
name  "continued  fraction."5 

ltlNotisi,  che  no  si  potendo  comodamete  nella  stampa  formare  i  rotti  .  .  .  ." 
See  Tropfke,  GescMchte,  II  (i),  362. 

2".  .  .  facendo  vn  punti  all'  8.  denominatore  de  ciascun  rotto,  a  significare, 
che  il  sequente  rotto  e  rotto  d'  esso  denominatore"  (p.  70). 

8For  details  see  Tropfke,  Geschichte,  II  (i),  363. 

4J.  Wallis,  Opera  Mathematics,  I,  469  (Oxford,  1695).  See  also  Commercium 
Epistolicum,  London,  1725  ed.,  p.  215. 

5".  .  .  quae  denominatorem  habeat  continue  fractum"  (Opera,  I,  469).  His 
symbolism  is 


See  also  Euler,  Introditctio  in  Analysin  Infinitorum,  ed.  nova,  1,  305  (Lyons,  1797)  • 


THE  MODERN  THEORY  421 

The  next  advance  was  made  by  Huygens  in  his  work  on  the 
description  of  a  planetarium,1  the  ratio  2,640,858  :  77,708,431, 
for  example,  being  written  as  a  continued  fraction. 

In  some  manner,  perhaps  through  the  missionaries  in  China, 
the  idea  of  the  continued  fraction  found  its  way  to  Japan  at 
about  this  time.2  Takebe  Hikojiro  Kenko  (1722)  used  such 
forms  for  the  value  of  TT,  stating  that  the  plan  was  due  to  his 
brother,  Takebe  Kemmei.  The  first  few  convergents  given  by 

him  ar&    3      22      333      355      onrl    103993 

mm  are  T,  -y-,  y^,  ITS>  ana   $3^2-. 

Euler  founds  the  Modern  Theory.  The  first  great  memoir  on 
the  subject  was  Euler's  De  jractionlbus  continuis  (1737),  and 
in  this  work  the  foundation  for  the  modern  theory  was  laid. 
Among  other  interesting  cases  Euler  developed  e  as  a  continued 
fraction,3  thus: 

,=  2+-'- 


Of  the  later  contributors  to  the  theory,  special  mention 
should  be  made  of  Lagrange4  (1767)  and  Galois.5 

7.  THE  WRITING  OF  EQUATIONS 

Equations  in  One  Unknown.  In  speaking  of  the  symbols  for 
unknown  quantities  we  are  brought  directly  in  touch  with  the 
symbols  for  integral  exponents  and  with  the  writing  of  equa- 
tions, and  so  it  is  convenient  to  treat  of  these  topics  in  their 
relation  to  one  another. 

1  Descriptio  auiomati  planetarii,  The  Hague,  1698  (posthumous). 

2  Smith-Mikami,  p.  145. 

8  Comm.  Acad.  Petrop.  for  1737,  IX,  120  (Petrograd,  1744)  .  See  Tropfke,  Ge- 
schichte,  II  (i),  337.  See  also  Euler's  Introductio,  I,  293,  "De  fractionibus  con- 
tinuis," especially  the  forms  on  pages  117,  307  ;  and  his  "De  formatione  fractionum 
continuarum,"in  iheActa  I'etrop.  for  1779,  1,3  (Petrograd,  1782  \  and  other  essays. 

4  See  Serret's  edition  of  his  works,  II,  539,  and  VII,  3  (Paris,  1868). 

5Gergonne's  Annales  de  Math.  Pures  et  Appliques,  XIX,  294;  posthumous 
(1828-1829).  CEuvres  mathtmatiques  d'Evariste  Galois,  pp.  1-8  (Paris,  1897). 


422  THE  WRITING  OF  EQUATIONS 

As  already  stated,  the  Egyptians  called  the  unknown  quan- 
tity ahe  or  hau,  meaning  "mass."1    This  word  was  represented 

in  the  hieroglyphic2  as  $    \"  "ppf  . 

For  example,   the   equation   x  (  f  +  |  •  +  1  4-  1  )  =  37    would 
appear  in  hieroglyphics  as 

?-*  —  ^  i  ^  i  n          /  —  <=>  _        £>i    w  «       £k     nnn 
A     V"T~i  <p>H^  MM^-^_V      w  *V^.  )g\    .  1  1.1.1  1, 

ZA     MII  *y^     in  *^^     O  JF^  1  1  III  II 

and  in  the  hieratic  of  the  Ahmes  Papyrus3  (c.  1550  B.C.)  as 


It  will  be  observed  that,  although  Ahmes  knew  of  symbols  for 
plus,  minus,  and  equality,  they  are  not  commonly  used  in  his 
equations.  They  are  found,  however,  in  No.  28  of  both  the  Feet 
and  the  Eisenlohr  translation. 

Symbolism  of  Diophantus.  The  first  writer  to  make  much 
effort  toward  developing  a  symbolism  for  the  powers  of  alge- 
braic expressions  was  Diophantus  (c.  275).  He  used  the  fol- 
lowing abbreviations  for  the  various  powers  of  the  unknown: 

MODERN  DioPHANxus4  LATE  EDITIONS 

o 

x*  M  fjiovdSes,  units  /i° 

,rl  S  apidpos,  number  g'  or  ^ 

x2  AT  Svva/MS,  power  S1' 

,rs  KT  /cv/3o9,  cube  /c1 

x*  ATA  Svva/jioSvvaiJiis,  power-power     SSU 

xb  AKT  Swafjidfcvftos,  power-cube  8/cu 

x*  KTK  /cu/3o'/cu/3o9,  cube-cube  KKV 


1  On  this  term  see  page  393,  n.  5. 

2  There  are  several  variants.    See  Eisenlohr,  Ahmes  Papytus,  p.  42. 

3  Eisenlohr,  ibid.,  p.  54,  No.  33;  in  the  British  Museum  facsimile,  Plate  X, 
row  5. 

4  See  Heath,  Diophantus,  2d  ed.,  p.  32.   Much  lias  been  written  about  the 
symbol  €  for  x,  and  Heath  gives  a  careful  discussion  of  the  various  theories  and 
a  statement  of  the  various  forms  for  the  symbol  as  they  occur  in  different  MSS. 
He  concludes  that  the  original  symbol  was  a  contraction  of  the  initial  letters 
ap  of  aptOv-fa  (arithmos1,  number),  instead  of  being  the  final  sigma.    Originally 
the  capitals  AY  were  used  for  5U,  and  similarly  KY  instead  of  *u,  and  so  on. 


SYMBOLISM  OF  DIOPHANTUS  423 

Diophantus  wrote  his  equations  quite  as  we  do,  except  for 
the  symbols ;  thus  the  equation 

/  ,->    ^      I       T  \2  A     -V2   __!_    A     <v  __L    T 

(2  x  +  i )  =  4  ,r  -}-  4  ^  ~i   * 
appears,  in  modern  Greek  letters,  as 

$  $  M  a  Dos  eVr/  AT  8  ^  8  M  3, 
and  the  equation  8  x*  —  16  x*  =  ^ 

appears1  as  KT  rj  /\  AT  IF  la-  KT  a. 


DIOPHANTUS  ON  EQUATIONS 

From  a  manuscript  of  the  i4th  century  showing  the  symbolism  then  in  use.  It 
begins:  "We  call  the  square  Stfi/a/xts,  and  it  has  for  its  symbol  a  delta  (A)  sur- 
mounted by  a  upsilon  (T) ."  (This  character  is  seen  in  the  middle  of  line  2.1  The 
symbol  AAU  (for  ,v4)  appears  in  line  5,  AKV  (for  xr>)  in  line  8,  and  KKV  (for  #B) 
in  the  last  line.  From  Rodet,  Sur  les  Notations  Numfriques,  Paris,  1881 

In  speaking  of  Diophantus,  however,  it  should  again  be 
stated  that  the  most  ancient  manuscript  of  his  Arithmetica  now 
extant  was  written  in  the  i3th  century, — about  a  thousand 
years  after  the  original  one  appeared.  We  are  therefore  quite 

1In  the  first  equation  earrl  (esti')  stands  for  "is  equal  to,"  and  in  the  second 
case  ta  stands  for  t<ro$(i'sos,  equal) .  See  Tannery's Diophant us,  I,  230-231,258-259. 


424  THE  WRITING  OF  EQUATIONS 

uncertain  as  to  the  symbols  used  by  Diophantus  himself  and  as 
to  the  various  interpolations  that  may  have  been  made  by  the 
medieval  copyists. 


x  s  *+ 


•  M* 

•  ^T  i 


ALGEBRAIC  SOLUTION  ACCORDING  TO  DIOPHANTUS 

From  a  manuscript  of  the  i4th  century.   The  problem1  is  to  find  two  numbers 
such  that  their  sum  is  equal  to  20  and  the  difference  of  their  squares  to  80 

Oriental  Symbols.  It  was  from  this  Greek  method  of  express- 
ing the  equality  of  the  two  members  that  the  Arabs  seem  to 
have  derived  theirs,  as  in  the  case  of 


r    A    j*    /<? 
for  38  =19 

the  equation  being  written  from  right  to  left  after  the  Semitic 
custom,  which  obtains  in  writing  Arabic.2  From  this  form, 
reversed  in  order  of  writing,  came  the  one  that  we  use. 

The  Chinese  and  Hindus,  however,  had  methods  of  writing 
their  equations  that  were  very  different  from  those  which  the 

1  The  solution  reads  substantially  as  follows  : 

Let  there  be  x  +  TO  10  —  x. 

Squares,  x2  +  20  x  +  TOO      jr2  +  100  —  20  x. 

Difference  of  squares,  40^  —80. 

Division,  x  —  2, 

whence  x  +  10  =  12         10  —  x  =  8. 

From  Rodet 

2  The  above  is  from  al-Qalasadi.    See  L.  Matthiessen,  Grundzuge  der  antiken 
und  modernen  Algebra,  2d  ed.,  p.  269  (Leipzig,  1896)  ;  hereafter  referred  to  as 
Matthiessen,  Grundzuge. 


ORIENTAL  SYMBOLS  425 

Arabs  and  Persians  adapted  from  the  Greek  works.  The  Far 
East  depended  more  upon  position.- The  Chinese  commonly 
represented  the  coefficients  by  sticks,  their  so-called  "  bamboo 
rods"  which  they  used  in  calculating,  and  these  they  placed 
in  squares  on  a  ruled  board.  Ch'in  Kiu-shao  (c.  1250),  for 
example,  represented  the  equation  x*  +  1 5  X*  +  66  x  —  360  as 
here  shown ;  and  if  he  needed  to  write  the  equation,  he  did  so 
in  the  same  manner.1  The  positive  terms  were  represented  by 
red  sticks  or  marks,  and  the  nega- 
tive terms  either  by  black  ones  or 
(as  in  the  illustration)  by  a  stick 


placed  diagonally  across  some  part  — j  1 1  [  | 

of  the  numeral.    The  system  is  pF                   ,  .        , 

,                £  ,   ,      ,      ,         ~    .      ,  wen     (element) 

simply  one  of  detached  coefficients,  -*-' 


the  place  values  of  the  coefficients     1  1  |^v  O 


tai       (extreme) 

being  indicated  sometimes  by  the 
squares  running  horizontally,  but  ordinarily  as  shown  in  the 
illustration.  The  native  Japanese  mathematicians  used  the  same 
method,  having  imported  it  from  China.2 

The  Hindu  method  was  better  than  the  Chinese,  and  in  one 
respect  was  the  best  that  has  ever  been  suggested.  Bhaskara 
(c.  1150)  represented  the  equation  18^=  16**  +  9^-4-  18  as 
follows: 


This  may  be  transliterated  as 

ya  v  1  8              ya  o  m    o 

ya  v  1  6               ya  9  ru  1  8 

which  means               i8^          ^  Q 

\6x*           Q,r  18 

or  18^  +  0^  +  0=  i6,r2  +  9*+  18, 

or  2x* 


1The  Chinese  word  tai  (from  tai-kieh  or  tai-chi,  extreme  limit,  or  great 
extreme)  means  the  absolute  term,  and  yuen  (element)  means  the  first  power  of 
the  unknown.  See  Mikami,  China,  pp.  81,  82,  91  ;  L.  Vanh6e,  "La  notation  alge*- 
brique  en  Chine  au  XIIIe  siecle,"  Revue  des  Questions  Scientifiques,  October,  1913. 

2  Smith-Mikami,  p.  50. 


426 


THE  WRITING  OF  EQUATIONS 


rrzr 


The  word  for  the  first  power  of  the  unknown,  the  yavat- 
tdvat,  already  explained,  is  abridged  to  ya ;  and  the  word  for 

the  second  power,  ydvat- 
varga,  to  ya  v.1 

Such  a  plan  shows  at  a 
glance  the  similar  terms  one 
above  another,  and  permits 
of  easy  transposition. 

When  the  Arabic  alge- 
bras were  translated  into 
Latin,  the  rhetorical  form 
was  used.  Thus  Robert  of 
Chester  (c.  1140),  in  his 
translation  of  al-Khowa- 
rizmi,  wrote  "Substantia  et 
10  radices  39  coaequantur 
drachmis"  for  "A  square 
and  10  roots  are  equal  to  39 
units";  that  is, 

x2  +  IQX  =  39. 2 

Al-Khowarizmi  himself 
wrote  his  equations  in  rhe- 
torical form,  thus:  "A 
square,  multiply  its  root  by 
four  of  its  roots,  and  the 
product  will  be  three  times 
the  square,  with  a  surplus 
of  fifty  dirhems."3 

Medieval  Manuscripts.  In 
the  manuscript  period  of 
the  Middle  Ages  we  find 

1Colebrooke,  loc.  cit.,  p.  140;  E.  Strachey,  Bija  Ganita  (often  bound  with 
Colebrooke),  p.  117. 

2  In  the  Scheubel  MS.,  translated  by  Karpinski,  pp.  70-73.  Somewhat  the  same 
form  is  used  in  the  transcription  in  Libri's  Histoire,  I,  255,  although  the  exact 
wording  is  "  Census  et  decem  radices  equantur  triginta  novem  dragmis." 

3  Rosen  translation,  p.  56 ;  Arabic  text  in  the  Rosen  edition,  p.  40. 


inrt 


PAGE  FROM  BHASKARA'S  BIJA  GANITA 

From  the  first  printed  edition,  showing 
the  method  of  writing  equations.  Lines 
8  and  9  give  the  equation  shown  in  the 
text.  The  next  equation  is  interesting  for 
the  use  of  the  dot  above  the  Sanskrit  9  to 
indicate  subtraction,  thus  : 

ya  v  2     ya  9     ru    o 

ya  v  o    ya  o     nt  18 


MEDIEVAL  FORMS  427 

letters  coming  into  use  to  represent  algebraic  as  well  as  geometric 
quantities.  This  is  seen  in  the  work  of  Jordanus  Nemorarius 
(c.  1225),  a  contemporary  of  Fibonacci.1  This  was  not  com- 
mon, however,  most  writers  preferring  to  use  some  such  symbols 
as  R  for  res  (thing,  the  unknown),  ce.  for  census  (the  second 
power  of  the  unknown),  and  cu.  for  cubus  (the  third  power  of 
the  unknown),  with  other  shorthand  abbreviations.  Such  sym- 
bolism is  seen  in  the  manuscripts  of  Regiomontanus  (c.  1463), 
one  problem  from  which  is  reproduced  in  facsimile  on  page  429. 
In  an  Italian  manuscript  of  about  the  same  period2  the  quadratic 
equation  x2  +  iox—39  appears  in  the  rhetorical  form  as  follows: 
"lo  censo  e-io-sue  cose  cioe-io-sue  ra.  sono  igualj  a*39* 
draine." 

Equations  in  Printed  Form.  The  following  examples  will  suf- 
fice to  show  the  general  development  of  the  symbolism  of  the 
equation  from  the  first  printed  work  containing  algebra  to  the 
time  when  our  present  symbolism  was  fairly  well  settled  : 

Pacioli  (1494)  3  :  "Trouame  .1.  n°.  che  gioto  al  suo  qdrat0  facia 
.12."  Modern  form:  x  +  x2  =  12. 

Vander  Hoecke  (i5i4)4:  4  Se.  —  51  Pri.  —30  N.  dit  is 
ghelijc  45f  .  Modern  form:  $x2  —  51^  —  30  =  45!- 

Ghaligai  (1521)  r>  :  iDe32C°  —  320  numeri.  Modern  form: 
x2  +  32X  =  320. 


example,  in  the  second  problem  in  his1  De  Numeris  Datis  he  says: 
"Datus  numerus  sit  .a.  qui  diuidatur  in  .b.c.d.e.  .  .  ."  Abhandlungen,  II,  135. 
See  also  A.  Favaro,  Boncompagni's  Bullettino,  XII,  129;  M.  Curtze,  Abhand- 
lungen,  XII;  A.  Witting  and  M.  Gebhardt,  Beispiele  zur  Gesch.  der  Math.,  II,  26 
(Berlin,  1913)- 

2  In  Mr.  Plimpton's  library.  See  Kara  Arithmetical  p.  459.  The  equation  is 
from  the  MS.,  fol.  279,  v. 

3Fol.  145,  r.  In  his  solutions,  but  not  in  his  problems,  he  used  .co.  (cosa, 
thing,  as  already  explained)  for  x;  .ce.  (census  or  zensusj  in  Italian,  censo,  evalua- 
tion of  wealth,  tax)  for  x2;  .cu.  (cubus)  for  x3;  .ce.cc.  (census  census)  for  re4; 
and  .p°.r°.  (primo  relato)  for  x5.  The  Latin  census  (a  registering  of  citizens  and 
property)  was  conducted  by  the  censors,  who  gave  censura,  censure,  to  those  who 
incurred  their  disfavor,  census  coming  probably  from  cent  ere,  to  number  by  the 
centum,  hundred. 

4  He  used  Pri.,  Se.,  3",  4",  and  5"  for  x,  x2,  x*,  x*,  and  xr>  respectively.  The 
system  failed  because  of  the  difficulty  in  writing  coefficients.  See  his  1537  edition, 
fol.  64,  v,  5  Fol.  96,  v, 


428  THE  WRITING  OF  EQUATIONS 

Rudolff  (1525)"  :  Sit  i  5  aequatus  12  X.  —  3  6-  Modern  form: 

X2  =  12  £-36. 

Cardan  (iS4S)2:  cubp  p:  6  reb*  aeqlis  20.    Modern  form: 
x*+  6x  =  20. 

Scheubel  (issi)3:  4  sex.  aequantur  108  ter.    Modern  form: 


Tartaglia  (iss6)4:  "Trouame  uno  numero  che  azontoli  la 
sua  radice  cuba  uenghi  ste,  cioe  .6."  Modern  form  :  x  +  -\fx  =  6. 

Buteo  (1559)':  i{)  P6/>P9Ci(}P3/3P24.  Modern  form: 
x2+  6x  +  g  =  x2  -r-3#  +  24. 

6  3 

Bombelli  (1572)  e:  Y.  p.  §•  Eguale  a  20.  Modern  form: 
x"+  8x*=  20.  In  the  text  proper  he  would  write  this  equation 
i£p.  S^eguale  &  20.  In  the  same  way  he  would  write  i^  eguale 
a  R.  q.  1  08.  p.  10,  for  x3  =  Vio8  +  10,  which  may  be  compared 
with  Cardan's  form  on  page  463. 


1From  StifeFs  edition  of  1553  (1554),  fol.  243,  v.  The  symbols  for  the  first 
five  powers  of  the  unknown,  beginning  with  the  first,  are  \  (contraction  of 
radix?),  J  (zensus},  *-£  (contraction  of  cs,  for  cubus),  §§  (zensus  zensus),  fy 
(sursolidus) .  The  German  writers  in  general  used  this  system  until  well  into  the 
1 7th  century.  Although  the  symbol  for  the  unknown  is  usually  taken  as  a  con- 
traction for  radix,  it  is  quite  as  probable  that  it  is  the  common  ligature  of  the 
Greek  7  and  p.  This  stood  forgram'ma  (ypd^a) ,  a  letter,  or  for  gramme'  (ypawfj] , 
a  line.  In  the  medieval  works  it  was  a  common  thing  to  represent  the  unknown 
by  a  line.  This  is  seen  in  an  algebra  as  early  as  al-Khowarizmi's  (c.  825)  and  in 
one  as  late  as  Cardan's  (1545).  For  evidence  of  the  frequent  use  of  the  symbol 
for  gramma,  see  Michael  Neander,  STNO^IS  mensvrarvm  et  Pondervm,  Basel, 
1555.  It  may,  indeed,  have  suggested  to  Descartes  the  use  of  x,  this  being  the 
letter  most  nearly  resembling  it. 

2Ars  Magna,  1545  ed.,  fol.  30,  r.  His  names  or  abbreviations  for  #,  x2,  #3,  #4, 
and  x5  are  res,  qd  (quadratum) ,  cu'  or  cub9  (cubus),  qd'  qd',  qd  qd,  or  Tjd'*  qdm 
(quadrati  quadratum),  and  relatum  primum,  with  necessary  variants  of  these 
forms.  For  a  facsimile,  see  pages  462,  463. 

3 Also  i  pri.  +  12  N  aequales  8  ra.,  for  x2  +  12  =  8*.  See  Abhandlungen, 
IX,  455- 

4 La  Nona  Scientia,  1554  ed.,  fol.  114,  r.  His  problems  are  in  rhetorical  form, 
and  his  symbolism  is  substantially  that  of  Cardan  and  other  Italian  contempora- 
ries. For  Recorde's  equations  (1557)  see  the  facsimile  on  page  412. 

5The  next  step  is  3pfisJ,  sometimes  with  both  brackets,  sometimes  with  only 
the  first.  Buteo,  Logistica,  quae  &  Arithmetica  vulgd  dicitur,  Lyons,  1559. 

6  Algebra,  p.  273.  He  indicated  x,  x2,  #8,  ...  by  ^,3, !,....  It  is  in  this 
work  that  there  appears  for  the  first  time  in  Italian  any  important  approach  to 
the  modern  symbolism  of  the  equation. 


r, 

j4%n»»*to  * 


•**          I*. 


1*0***- 


z- 
SYMBOLISM  OF  THE  EQUATION  AS  USED  BY  REGIOMONTANUS, 

C.  1463 

From  a  letter  written  by  Regiomontanus.   The  problem  is  to  find  a  number  x 

such  that 

100        100  0 

—  +  __^  -  4o    or    x2  +  3  x  =  20, 
^        ^r  +  8 


from  which  x  =  V9^  —  |,  as   stated  in  the  third  line  from  the  bottom.   The 
conclusion  is  that  the  first  divisor  is  "V/2zJ  minus  ij.   See  page  427 


430  THE  WRITING  OF  EQUATIONS 

Gosselin    (1577)  *:    i2LMiQP48    aequalia    I44M24LP2Q. 
Modern  form:  i2X  —  xz+  48  ==  144  —  243;  +  2x2. 
Stevin  (  1585)  2  :   3  (D  +  4  egales  a  2  ®  +  4.    Modern  form  : 


Ramus  and  Schoner  (i586)3:  iq  —  I  —  81  aequatus  sit  65. 
Modern  form:  x2  +  &x  =  65. 

Vieta  (c.  i59o)4:  i  Q  C  -  15  Q  Q  +85  C  -  225  Q  4-  274  N, 
aequatur  120.  Modern  form: 

x*  —  i$x*  4-  &sx3  —  22$x2  +  274X  =  120. 


Clavius(i6o8)5:  "  Sit  aequatio  inter  1.3**  &  800-6-156751." 
Modern  form:  x6  =  8oox3  —  156,751. 

Girard  (1629)°:  i  (4)  +  35  (»  +  24  =  10  (3)  +50  (i), 
or  with  the  several  exponents  inclosed  in  circles.  Modern  form  : 

#4+35:r  +  24  =  I0#3  +  5°*- 

Oughtred  (1631)  7:  i  Z  ±  Vq  :  |  Zq  -  AE  -  A.  Modern 
form  :  \Z  ±  V|  z*  -  AE  =  ^. 

Harriot  (  1  63  1  )  g  :  aaa  -  3  -  bbai^^~  4-  2  •  ccc.  Modern 
form  :  x3  —  3  b2x  =  2  c3. 


iGuillaume  Gosselin  was  a  native  of  Caen,  but  we  know  almost  nothing  of 
his  life.  He  published  an  algebra,  De  arte  magna,  sen  de  occulta  parte  numerorum, 
quae  &  Algebra,  &  Almucabala  vulgo  dicitur ;  Libri  QVATVOR,  Paris,  1577;  and 
a  French  translation  of  Tartaglia's  arithmetic,  Paris,  1578.  See  H.  Bosnians, 
Bibl.  Math.,  VII  (3),  44.  In  the  above  equation  he  uses  L  for  latus  (the  side  of 
the  square),  Q  for  quadratics  (square);  and  P  and  M  for  plus  and  minus. 

2UArithmetiqve,  p.  272.  See  also^jjfo  (Euvres,  Girard  ed.,  p.  69  (Leyden,  1634) . 
He  used  ©,  ©,®,  •  •  •  for  *,  *2,  x9,  **'-. 

3  Algebrae  Liber  Primus,  1586  ed.,  p.  349. 

4  For  a  discussion  of  the  dates  of  his  monographs,  see  Cantor,  Geschichte,  II 
(2),  582.  He  also  used  capital  vowels  for  the  unknown  quantities  and  capital  con- 
sonants for  the  known,  thus  being  able  to  express  several  unknowns  and  several 
knowns.   The  successive  powers  of  A  were  then  indicated  by  A,  Aq,  Acu,  Aqq, 
Aqcu,  and  so  on,  the  additive  principle  of  exponents  being  followed.   The  above 
example  is  from  his  Opera  Mathematica,  ed.  Van  Schooten,  p.  158  (Leyden,  1646) . 

5  Algebra,  p.  62.   For  his  symbols,  see  ibid.,  p.  n. 

6  Invention  nouvelle  en  I'algebre,  p.  131  (Amsterdam,  1629),  with  rules  for 
the  symmetric  functions  of  the  roots. 

1  Clavis,  p.  50;  Cajori,  William  Oughtred,  p.  29. 

8  See  his  Artis  Analyticae  Praxis,  London,  1631.  He  represented  the  successive 
powers  of  the  unknown  by  a,  aa,  aaa,  .  .  .  ;  Tropfke,  Geschichte,  III  (2),  143, 
a  work  which  should  be  consulted  (pp.  119-148)  on  this  entire  topic. 


TYPICAL  FORMS  431 

Herigone(i634)1:  i54a^7ia2  — f—  I4a3-~a4  2/2  120.  Mod- 
ern form:  1540  —  7ia2-f-  i4«a  —  a4  —  120. 

ex 
Descartes  ( 1637)":  yy  *>  cy~-  —  y-f-ay  — ac.  Modern  form: 

2  CX  t 

y2  =  Cy  —  —  y  -f-  ay  —  ac. 
b 

Wallis  (1693) :  x4  +  bx*  +  cxx  +  dx  -f-  e  —  o,  which  is  the 
modern  form,  with  the  exception  of  xx ;  this,  as  already  stated, 
was  commonly  written  for  x2  until  the  close  of  the  i8th  century.3 

Equating  to  Zero.  It  is  difficult  to  say  who  it  was  who  first 
recognized  the  advantage  of  always  equating  to  zero  in  the 
study  of  the  general  equation.  It  may  very  likely  have  been 
Napier,  for  he  wrote  his  De  Arte  Logistica  before  1594  (al- 
though it  was  first  printed  in  Edinburgh  in  1839),  and  in  this 
there  is  evidence  that  he  understood  the  advantage  of  this  pro- 
cedure.4 Biirgi  (c.  1619)  also  recognized  the  value  of  making 
the  second  member  zero,  Harriot  (c.  1621)  may  have  done  the 
same,  and  the  influence  of  Descartes  (1637)  was  such  that  the 
usage  became  fairly  general.5 

Several  Unknowns.  The  ancients  made  little  use  of  equations 
with  several  unknown  quantities.  The  first  trace  that  we  find 
of  problems  involving  such  equations  is  in  Egypt.  There  are 

^Cursus  Mathematicus,  5  vols.,  Paris,  1634-1637;  2d  ed.,  6  vols.  in  4  (Paris, 
1644),  Vol.  II,  chap.  xiv.  He  represented  the  successive  powers  of  the  unknown 
by  a,  0.2,  0,3,  #4,  .... 

2 La  Geometric,  1637;  I7°5  ed.,  p.  36.  It  will  be  seen  that  this  form  does  not 
differ  much  from  our  own.  Descartes  used  the  last  letters  of  the  alphabet  for  the 
unknown  quantities  and  the  first  letters  for  the  known,  and  this  usage  has  per- 
sisted except  in  the  case  of  those  formulas  in  which  the  initial  letter  serves  a  bet- 
ter purpose.  That  it  was  not  immediately  accepted,  however,  is  seen  by  the  fact 
that  Rahn  (Rhonius)  used  final  letters  for  unknowns  and  large  letters  for  knowns, 
as  in  his  algebra  of  1659  (English  translation,  1688). 

8 E.g.,  in  Euler's  Algebra,  French  ed.,  Petrograd,  1798,  where  such  forms  as 
xx  +  yy  =  n  are  found. 

4  For  example,  on  page  156  he  takes  the  equation  4"$  —  6  ~  5  R  --  20  and  re- 
duces it  to  —  IJ  +  14  =  o,  "quae  aequatio  ad  nihil  est."  In  general  all  his  higher 
equations  have  zero  for  the  second  member.  See  also  Enestrb'm,  Bibl.  Math., 
Ill  (3),  145. 

5For  a  discussion  see  Tropfke,  Geschichte,  III  (2),  26;  Kepler's  Opera,  ed. 
Frisch,  V,  104.  The  credit  is  often  claimed  for  Stifel  (c.  1525),  but  he,  like  Harriot, 
made  no  general  practice  of  equating  to  zero. 


432  THE  WRITING  OF  EQUATIONS 

three  papyri1  of  the  Middle  Kingdom  (c.  2160-1700  B.C.)  which 
contain  problems  of  this  nature.  One  of  these  problems  is  to 
divide  100  square  measures  into  two  squares  such  that  the  side 
of  one  of  the  squares  shall  be  three  fourths  the  side  of  the  other  ; 


/if    —  -    O      -J- 

y  —  4*- 

The  other  problems  also  involve  quadratics,  one  found  in 
1903  being  substantially2 


Simultaneous  Linear  Equations.  The  earliest  of  the  Greek 
contributions  to  the  subject  of  simultaneous  linear  equations 
are,  according  to  the  testimony  of  lamblichus  (c.  325)  in  his 
work  on  Nicomachus,  due  to  Thymaridas  of  Pares  (  c.  3  80  B.  c.  ?  )  . 
He  is  said  to  have  given  a  rule  called  eiravOj^ia  (eparithema, 
flower),  which  he  seems  to  have  used  in  solving  n  special  types 
of  equations,  namely,  ,ro  +  ^  +  ^  +  •  •  •  +  xn_^  =  s,  XQ  +  x^  a^ 
XQ+  x^  #2,  •  •  •  x^+  xn_l  =  an_^  the  method  being  the  ordinary 
one  of  adding.  lamblichus  applied  the  rule  to  other  cases.3 

Some  use  of  simultaneous  linear  equations  is  also  found  in  the 
work  of  Diophantus  (c.  275),  who  spoke  of  the  unknowns  as 
the  first  number,  the  second  number,  and  so  on,4  a  method  that 
was  too  cumbersome  to  admit  of  any  good  results. 

Chinese  and  Japanese  Methods.  The  subject  was  greatly  ex- 
tended by  the  Chinese.  Using  the  "bamboo  rods"  as  calculat- 
ing sticks,  they  placed  these  in  different  squares  on  the  table  so 
as  to  represent  coefficients  of  different  unknowns,  and  hence 

1The  Petrie  Papyrus,  published  by  F.  LI.  Griffith  in  1897;  the  Berlin  Papyrus 
No.  6619,  published  by  H.  Schack-Schackenburg,  Zeitschrift  fur  dgyptische 
Sprache,  XXXVIII  (1900),  135;  and  the  Kahun  Papyrus,  also  studied  by  Schack- 
Schackenburg  in  1903. 

2M.  Simon,  Geschichte  der  Mathematik  im  AUertumy  pp.  41,  42  (Berlin,  1909)  ; 
hereafter  referred  to  as  Simon,  Geschichte. 

3Heath,  in  R.  W.  Livingstone's  The  Legacy  of  Greece,  p.  no  (Oxford,  1922). 

4  That  is,  6  Trpwros  dpt0/^s,  6  Setrepos  d/>i0/*6j,  and  so  on.  E.g.,  Book  II,  Prop. 
17;  Book  IV,  Prop.  37. 


SIMULTANEOUS  LINEAR  EQUATIONS  433 

they  needed  no  special  symbols.1  Indeed,  we  are  quite  justified 
in  saying  that  the  first  definite  trace  that  we  have  of  simultane- 
ous linear  equations  is  found  in  China.  In  the  Arithmetic  in 
Nine  Sections2  there  are  various  problems  that  require  the  solu- 
tion of  equations  of  the  type  y=  ax  —  b,  y  =  a'x  +  b'.  A  rule 
is  given  for  the  solution  which  amounts  substantially  to  the 
following :  , 

A  CC.     '         4.  a          a 

Arrange  coefficients,  .      7. 

b      b 


aV      a'b 
b         b' 
ab'  +  a'b 


Multiply  crosswise,  7         ;, 


Add, 
Result, 
and 


The  method  of  reasoning  is  not  stated,  but  the  work  was  prob- 
ably done  by  the  aid  of  bamboo  rods  to  represent  the  coefficients.3 

The  next  step  of  which  we  have  evidence  was  taken  much 
later  by  Sun-tzi",  the  date  being  uncertain  but  probably  in  the  ist 
century.  He  solved  what  is  equivalent  to  the  system  2x+y=g6, 
2X  +  $y  =  144,  and  his  method  of  elimination  was  substantially 
first  to  multiply  the  members  of  each  equation,  when  necessary, 
by  the  coefficient  of  x  in  the  other  equation.4 

From  this  time  on,  the  solution  of  simultaneous  linear  equa- 
tions was  well  known  in  China.  The  only  improvement  made 
upon  the  early  methods  consisted  in  the  arrangement  of  the 
bamboo  rods  in  such  a  way  as  to  allow  for  a  treatment  of 
the  coefficients  similar  to  that  found  in  the  simplification  of  de- 
terminants. This  was  finally  carried  over  to  Japan  and  was 
amplified  by  Seki  Kowa  (1683)  into  what  may  justly  be  called 
the  first  noteworthy  advance  in  the  theory  of  these  forms.5 


i,  China,  p.  73. 

2  K'iu-ch'ang  Suan-shu,  of  uncertain  date,  possibly  as  early  as  noo  B.C.,  and 
certainly  pre-Christian.   See  Volume  I,  page  31. 

8  Mikami,  China,  p.  16.  *Ibid.,  p.  32.  5  Ibid.,  p.  191. 


434  THE  WRITING  OF  EQUATIONS 

Hindu  Symbolism.  The  Hindus  represented  the  various  un- 
knowns by  the  names  of  colors,  calling  them  "black,"1  "blue,"2 
"yellow,"3  "red/'  and  so  on.  They  wrote  the  coefficients  at  the 
right  of  the  abridged  words  and  represented  a  negative  term  by 
a  dot  placed  above  the  coefficient.  For  example,4 

ya  5     ka  8     m  7     ru  90 
ya  7     ka  9     m  6     ru  62 

means        $x  +  8y  +  ?z  +  90  =  jx  +  gy  4-  6z  4-  62 

_    .         .  ka  i  m  i  ru  28 

and  gives  rise  to  --------  > 

ya2 

as  it  appears  in  the  Colebrooke  version,  which  means 


Problems  involving  several  unknowns  did  not  possess  much 
interest  for  the  Arab  and  Persian  writers,  as  may  be  seen  from 
the  algebras  of  al-Khowarizmi  r>  and  Omar  Khayyam. 

Early  European  Symbolism.  The  algebraists  of  the  i6th  cen- 
tury gave  relatively  little  attention  to  simultaneous  linear  equa- 
tions. The  use  of  x,  y,  and  z  for  unknown  quantities  was  not 
suggested  until  the  lyth  century,  and  so  it  was  the  custom  of 
some  writers  to  use  ordinary  capital  letters.  For  example,  we 
find  Buteo  (1559)  using 


for  what  we  should  now  write  as  x  +  i  y  4-  \  z  =17,  etc.    By 
multiplication  he  reduces  these  to  three  equivalent  equations  in 


lKdlaca>  abridged  to  ka  (^T)-  ^ee  Colebrooke's  translation  of  Bhaskara, 
pp.  184  n.,  227.  The  known  number  was  rtipa,  abridged  to  ru. 

2Ntlaca,  abridged  to  ni  (T  Y)- 

tPitaca,  abridged  to  fit  C^T^).  ^Colebrooke's  translation,  p.  231. 

5  But  see  E.  Wiedemann,  Sitzungsberichte  der  Physikalisch-medizinischen 
Sozietat  zu  Erlangen,  50-51.  Bd.  (1920),  p.  264. 


LINEAR  EQUATIONS  435 

which,  however,  the  original  symbolism  changes  slightly,  the 
period  replacing  the  comma  to  indicate  addition,  thus: 

2  A  •  i  B  •  i  C  [  34, 


He  then  eliminates  in  the  usual  manner.1    Gosselin,  in  his  De 
arte  magna  (Paris,  1577),  uses  a  similar  arrangement.2 

Literal  Equations.  The  equations  considered  by  the  ancient 
and  medieval  writers  were  numerical.  Even  the  early  Renais- 
sance algebraists  followed  the  same  plan,  their  crude  symbolism 
allowing  no  other.  It  was  not  until  the  close  of  the  i6th  century 
that  the  literal  equation  made  its  appearance,  owing  largely  to 
the  influence  of  the  new  symbolism  invented  by  Vieta  and  his 
contemporaries.  For  example,  Adriaen  van  Roomen  published 
in  1598  a  commentary  on  the  algebra  of  al-Khowarizmi  3  in 
which  he  distinguished  between  two  types  of  equation,  the 
numerosa  and  the  figurata.  The  former  was  applied  to  prob- 
lems with  numerical  data,  while  the  latter  resulted  in  general 
formulas.4  Van  Roomen  asserts  that  writers  on  algebra  up  to 
his  time  used  the  numerosa  method  only,  whereas  he  was  the 
first  to  use  the  figurata  one,  although  as  a  matter  of  fact  Vieta 
seems  to  have  preceded  him.  The  actual  dates  of  invention, 
but  not  of  publication,  are,  however,  obscure. 

8.  THE  SOLUTION  OF  EQUATIONS 

Linear  Equations.  The  earliest  solutions  of  problems  involv- 
ing equations  were  doubtless  by  trial.  In  the  time  of  Ahmes 
(c.  1550  B.C.),  however,  the  methods  of  making  the  trials 

i  G.  Wertheim,  "Die  Logistik  des  Johannes  Buteo,"  Bibl.  Math.,  II  (3),  213. 

2H.  Bosnians,  "Le  'De  arte  magna'  de  Guillaume  Gosselin,"  Bibl.  Math., 
VII  (3),  44- 

8H.  Bosnians,  "Le  fragment  du  Commentaire  d'Adrien  Romain  sur  Falgebre 
de  Mahumed  ben  Musa  El-chowarezmi,"  Annales  de  la  Societe  Scientifique  de 
Bruxelles,  XXX  (1906),  second  part,  p.  266. 

4  "Differentia  igitur  inter  has  duas  talis  statui  potest,  quod  figurata  inveniat 
regulam  solvendi  problema  propositum;  numerosa  vero  duntaxat  regulae  illius 
exemplum." 


436  THE  SOLUTION  OF  EQUATIONS 

were  fairly  well  simplified.    Thus,  his  equation1 

is  solved  substantially  as  follows:    Assume  7  as  the  number. 

Then,  to  use  the  form  of  the  text, 

"Once        gives        7 

$  gives         i 

i  \  gives        8 

"As  many  times  as  8  must  be  multiplied  to  make  19,  so  many 
times  must  7  be  multiplied  to  give  the  required  result. 

"Once  gives  8 

Twice  gives  16 

I  gives  4 

-5-  gives  2 

i  gives  i 


"Together,     2,  |,   J-  gives        19  [in  which  he  selects  the 

addends  making  19]. 

"Multiply  2,  |,  \  by  7  and  obtain  the  required  result. 


'Once 
Twice 
4  times 

gives 
gives 
gives 

2, 

4, 
9, 

i.     f 

"Together,  7  gives      16,     |,     f,  the  result."2 

The  Greek  methods  are  discussed  later  in  connection  with 
the  quadratic. 

The  chief  contribution  to  the  solution  of  linear  equations 
made  by  the  Arab  writers  was  the  definite  recognition  of  the 
application  of  the  axioms  to  the  transposition  of  terms  and  the 
reduction  of  an  implicit  function  of  x  to  an  explicit  one,  all  of 
which  is  suggested  by  the  name  given  to  the  science  by  al- 
Khowarizmi  (  c.  825). 

aln  this  discussion  all  equations  will  be  given  in  the  modern  form.  On  the 
general  history  of  solutions  two  of  the  best  works  for  the  student  to  consult  are 
A.  Favaro,  "Notizie  storico-critiche  sulla  Costruzione  delle  Equazioni,"  Atti  delta 
R.  Accad.  dt  Scienze,  Lettere  ed  Arti  in  Modena,  Vol.  XVIII,  206  pages  with  ex- 
tensive  bibliography;  Matthiessen,  Grundziige. 

2 For  the  translation  I  am  indebted  to  Dr.  A.  B.  Chace,  of  Providence,  Rhode 
Island.  For  a  slightly  different  version  see  Peet,  Rhind  Papyrus,  p.  61. 


FALSE  POSITION  437 

False  Position.  To  the  student  of  today,  having  a  good 
symbolism  at  his  disposal,  it  seems  impossible  that  the  world 
should  ever  have  been  troubled  by  an  equation  like  ax  +  b  =  o. 
Such,  however,  was  the  case,  and  in  the  solution  of  the  problem 
the  early  writers,  beginning  with  the  Egyptians,  resorted  to  a 
method  known  until  recently  as  the  Rule  of  False  Position. 
The  ordinary  rule  as  used  in  the  Middle  Ages  seems  to  have 
come  from  India,1  but  it  was  the  Arabs  who  made  it  known 
to  European  scholars.  It  is  found  in  the  works  of  al-Khowariz- 
mi  (c.  825),  the  Christian  Arab  Qosta  ibn  Luqa  al-Ba'albeki 
(died  c.  912/13),  Abu  Kamil  (c.  900),  Sinan  ibn  al-Fath  (loth 
century),  Albanna  (c.  i30o);2  al-Iias§ar  (c.  i2th  century),8 
and  various  others.  The  Arabs  called  the  rule  the  hisab  al- 
Khataayn*  and  so  the  medieval  writers  used  such  names  as 
elchataym?  When  Pacioli  wrote  his  Suma  (1494)  he  used  the 
term  el  cataymf  probably  taking  it  from  Fibonacci.  Following 
Pacioli,  the  European  writers  of  the  i6th  century  used  the  same 
term,  often  with  a  translation  into  the  Latin  or  the  vernacular.7 

1  There  is  a  medieval  MS.,  published  by  Libri  in  his  Histoire,  I,  304,  and 
possibly  due  to  Rabbi  ben  Ezra.   It  refers  to  this  rule,  "  quern  Abraham  com- 
pilavit  et  secundum  librum  qui  Indorum  dictus  est  composuit."   See  M.  Stein- 
schneider,  Abhandlungen,  III,  120;  F.  Woepcke,  "M£moire  sur  la  propagation  des 
chiffres  indiens,"  Journal  Asiatique  (Paris,  1863),  I  (6),  34,  180;  Matthiessen, 
Grundzuge,  p.  275;  C.  Kost'al,  Regula  falsae  positionis,  Prog.,  Braunau,  1886. 

2  See  Volume  I,  page  211.   For  the  original  and  a  translation  of  his  process  see 
F.  Woepcke,  Journal  Asiatique,  I  (6),  511. 

8  See  Volume  I,  page  210.  For  a  translation  of  his  arithmetic  see  H.  Suter, 
Bibl.  Math.,  II  (3),  12 ;  on  this  rule  see  page  30. 

4  Rule  of  Two  Falses.   There  are  various  transliterations  of  the  Arabic  name. 

5 Leonardo  Fibonacci,  in  the  Liber  Abaci,  cap.  XIII,  under  the  title  De  regulis 
elchatayn  says:  "  Elchataieym  quidem  arabice,  latine  duarum  falsarum  posi- 
cionum  regula  interpretatur.  .  .  .  Est  enim  alius  modus  elchataym;  qui  regula 
augmenti  et  diminucionis  appelatur."  See  the  Boncompagni  edition,  p.  318; 
M.  Steinschneider,  Abhandlungen,  III,  122  ;  G.  Enestrom,  Bibl.  Math.,  IV  (3),  205. 

6  He  speaks  of  it  as  a  "certa  regola  ditta  El  cataym.  Quale  (secondo  alcuni) 
e  vocabulo  arabo."  Fol.  98,  v. 

7 Thus  we  have  "...  per  il  Cataino  detto  alcuni  modo  Arabo"  (Cataneo, 
Le  Pratiche,  Venice,  1567  ed.,  fol.  58);  "Delle  Regole  del  Cattaino  ouero  false 
positioni"  (Pagani,  1591,  p.  164) ;  "Regola  Helcataym  (vocabulo  Arabo)  che  in 
nostra  lingua  vuol  dire  delle  false  Positioni"  (Tartaglia,  General  Trattato,  I, 
fol.  238,  v.  (Venice,  1556));  "La  Reigle  de  Faux,  que  les  Arabes  appellent  la 
Reigle  Catain"  (Peletier,  1549;  1607  ed.,  p.  253);  "La  regola  del  Cataino" 
(G.  Ciacchi,  Regole  generali  d'  abbaco,  p.  278  (Florence,  1675)). 


438  THE  SOLUTION  OF  EQUATIONS 

This  name  was  not,  however,  the  common  one  in  the  Euro- 
pean books,  and  in  the  course  of  the  i6th  century  it  nearly  dis- 
appeared. In  general  the  method  went  by  such  names  as  Rule 
of  False,1  Rule  of  Position,2  and  Rule  of  False  Position.3 

Rule  of  Double  False  explained.  The  explanation  of  this  rule, 
as  related  to  the  equation  ax  +  b  =  o,  is  as  follows : 

Let  gl  and  g2  be  two  guesses  as  to  the  value  of  x,  and  let  fl  and 
/2  be  the  failures,  that  is,  the  values  of  agl  4-  b  and  ag,2  -f  b, 
which  would  be  equal  to  o  if  the  guesses  were  right.  Then 

^•1+*=/1  (i) 

and  ^a+*=/a;  (2) 

whence  a  (^  -  £»)  =/!  -/a-  (3) 

From  ( i ) ,  ag^  +  bgtl  = 

and  from  ( 2 ) ,  ag^  +  bgl  = 

whence  *(^2~^1)=71^8-72^r  (4) 


Dividing  (4)  by  (3),    --  =  ^f- 

a  fi~Jz 

But,  since  —  -  =  x, 

we  have  here  a  rule  for  finding  the  value  of  #.4 

lf'La  Reigle  de  Faux"  (Trenchant,  1566;  1578  ed.,  p.  213)  ;  "Falsy"  (Van  der 
Schuere,  1600,  p.  185) ;  "Regula  Falsi"  (Coutereels,  1690  edition  of  the  Cyffer- 
Boeck,p.  541). 

2"Auch  Regula  Positionum  genant"  (Suevus,  1593,  p.  377) ;  "Reigle  de  Faux, 
mesmes  d'une  Position"  (Peletier,  1549;  1607  ed.,  p.  269). 

8  "Rule  of  falshoode,  or  false  positions"  (Baker,  1568;  1580  ed.,  fol.  181); 
"False  Positie-"  and  "Fausse  Position"  (Coutereels,  Dutch-French  ed.,  1631, 
p.  329) ;  "Valsche  Positie"  (Eversdyck's  Coutereels,  1658  ed.,  p.  360) ;  "Reghel 
der  Valsches  Positien"  (Wilkens,  1669  ed.,  p.  353). 

4  The  formula  is  more  elegantly  derived  by  taking  the  eliminant  of 

ax  -f  b  +  o  =  o 
og\  +  b  -/i  =  o 

%  +  *-/*  =  °> 

=  o, 


which  is 


~f\ 


by  the  expansion  of  which  the  result  at  once  appears. 


RULE  OF  DOUBLE  FALSE  439 

Suppose,  for  example,  that 

5^—10  =  0. 

Make  two  guesses  as  to  the  value  of  x,  say  g1  =  3  and  g,2  =  i . 
Then  5  .3-10=5=^, 

and  5  .  i -io  =  - 5  =/2. 

Then      *  =  £&^=*l±^p^  =  ™  =  * 
/!-/,  5  -(-5)          10 

Awkward  as  this  seems,  the  rule  was  used  for  many  centuries, 
a  witness  to  the  need  for  and  value  of  a  good  symbolism.  We 
have  here  placed  two  false  quantities  in  the  problem,  and  from 
these  we  have  been  able  to  find  the  true  result. 

Recorde's  Rule  in  Verse.  From  the  above  formula  for  x  it 
will  be  possible  to  interpret  the  doggerel  rule  given  by  Robert 
Recorde  in  his  Ground  of  Artes  (c.  1542) : 

Gesse  at  this  woorke  as  happe  doth  leade. 
By  chaunce  to  truthe  you  may  precede. 
And  firste  woorke  by  the  question, 
Although  no  truthe  therein  be  don. 
Suche  falsehode  is  so  good  a  grounde, 
That  truth  by  it  will  soone  be  founde. 

From  many  bate  to  many  mo, 
From  to  fewe  take  to  fewe  also. 
With  to  much  ioyne  to  fewe  againe, 
To  to  fewe  adde  to  manye  plaine. 
In  crossewaies  multiplye  contrary  kinde, 
All  truthe  by  falsehode  for  to  fynde.1 

Recorde  thought  highly  of  the  rule,  and  it  was  appreciated  by 
writers  generally  until  the  igth  century.2 

1  Ground  of  Artes,  1558  ed.,  fol.  Z  4. 

2Thus  Thierfelder  (1587,  p.  226)  says:  "Fur  alien  Regeln  der  gantzen  Arith- 
metic (ohn  allein  die  Regel  Cosz  auzgenommen)  ist  sie  die  Kunstreichste  / 
weytgegreifflichste  vn  schonste";  and  Peletier  (1549;  1607  ed.,  p.  269)  remarks: 
"  Gemme  Phrissien  a  inuente  Partifice  de  soudre  par  la  Reigle  de  Faux,  mesmes 
d'une  Position,  grand'  partie  des  exemples  subjects  a  PAlgebre."  Even  as  late  as 
1884,  in  the  Instruction  fur  den  Unterricht  an  den  Gymnasien  in  Osterrekh 
(Vienna,  1884,  pf  315),  the  rule  is  recommended. 


440  THE  SOLUTION  OF  EQUATIONS 

Method  of  the  Scales.  The  Arabs  modified  the  rule  by  what 
they  called  the  Method  of  the  Scales,1  a  name  derived  from  the 
following  figure,  used  in  the  solution : 


X 


Suppose,  for  example,  that  we  wish  to  solve  the  equation 
x  4- 1  x  +  i  =  10,  a  problem  set  by  Beha  Eddin  (c.  1600).  We 
may  make  as  our  guesses  gl  =  9,  whence  fl  =  6 ;  g2  =  6,  whence 
/8  =  i .  Then  place  the  figures  thus : 


The  lines  now  aid  the  eye  to  write  the  result  according  to  the 
rule  already  set  forth,  as  follows  :2 

6-6-1  -9__27__ 
6-1       ~  5  ~    *' 

Rule  of  Single  False.  Thus  far  we  have  considered  the  Rule 
of  Double  False,  where  a  double  guess  was  made,  but  there  was 
also  a  modification  of  the  method  known  as  the  Rule  of  Single 
False.3  Albanna  (c.  1300)  gives  the  latter  in  the  form  of  a  rule 
which,  worked  out  in  modern  symbols,  is  as  follows  :  Given  that 

ax  +  b  =  o. 
Make  a  guess,  g,  for  the  value  of  x,  the  failure  being  /  ;  that  is, 


lfAlm  bi'l  kaffatain.  This  name  was  translated  into  Latin  as  the  Regula 
lancium  or  Regula  bilancis. 

2  For  variations  of  the  method  see  Matthiessen,  Grundzuge,  p.  278;  for  the 
Arab  proof  by  geometry  see  ibid.,  p.  281. 

sTartaglia  called  it  "Position  Sempia,"  as  distinct  from  "Position  Doppia" 
(General  Trattato,  I,  fol.  239,  v.,  and  266,  r.)  ;  the  Spanish  had  rules  "De  vna 
falsa  posicion"  and  "De  dos  falsas  posiciones"  (Santa-Cruz,  1594;  1643  ed.,  f°ls- 
210,  212);  Clavius  (1586  Italian  ed.,  pp.  195,  203)  has  "Regola  del  falso  di 
semplice  positione"  and  "di  doppia  positione";  and  Chuquet  (1484,  MS.,  fols.  32, 
42)  has  "De  la  Rigle  de  vne  posicion"  and  "de  deux  positions." 


RULE  OF  SINGLE  FALSE  441 

Then  to  obtain  the  required  rule  we  may  proceed  as  follows  : 


fx 

whence  —  -----  h  b  =  o 

g  -  x 


—  —  —  • 

b  -f  ag  f~  b 

this  last  indicating  the  rule  used. 

For  example,  in  the  equation  \x  +  I  x  =  20,  if  we  take  g  =  30, 


*•  30  +  J-  30  =  11, 
which  is  9  too  small,  whence  /  =  —  9.    Then 


20 


"      ~54TT> 


Apologies  for  the  Name  of  Rule  of  False.  The  name  "  Rule  of 
False"  was  thought  to  demand  an  apology  in  a  science  whose 
function  it  is  to  find  the  truth,  and  various  writers  made  an  effort 
to  give  it.  Thus  Humphrey  Baker  (1568)  says: 

The  Rule  of  falsehoode  is  so  named  not  for  that  it  teacheth  anye 
deceyte  or  falsehoode,  but  that  by  fayned  numbers  taken  at  all  aduen- 
tures,  it  teacheth  to  finde  out  the  true  number  that  is  demaunded,  and 
this  of  all  the  vulgar  Rules  which  are  in  practise)  is  y  most  excellence.1 

Besides  the  "Rule  of  False"  the  method  was  also  called  the 
"Rule  of  Increase  and  Diminution/72  from  the  fact  that  the 
error  is  sometimes  positive  and  sometimes  negative.  Indeed,  as 
already  stated,  in  the  i6th  century  the  symbols  4-  and  —  were 
much  more  frequently  used  in  this  connection  than  as  symbols 
of  operation. 

1iS8o  ed.,  fol.  181.  Similar  excuse  is  offered  by  Thierfelder  (1587,  p.  225): 
"Darum  nicht  dasz  sie  falsch  oder  vnrecht  sey  ";  by  Apianus  (1527)  :  "Vnd  heisst 
nit  darum  falsi  dass  sie  falsch  vnd  unrecht  wehr,  sunder,  dass  sie  auss  zweyen 
falschen  vnd  vnwahrhaftigen  zalen,  vnd  zweyen  lugen  die  wahrhaftige  vnd 
begehrte  zal  finden  lernt."  Like  explanations  are  given  by  many  other  writers. 

2<tRegula  augment!  et  decrement!"  or  "  diminutionis." 


442  THE  SOLUTION  OF  EQUATIONS 

Regula  Infusa.  Rabbi  ben  Ezra  ( c.  1 140)  tells  us  of  a  substi- 
tution method  due  to  another  Hebrew  writer,  Job  ben  Salomon, 
of  unknown  date,  which  was  called  in  Latin  translation  the 
Regula  infusa.1  This  may  be  illustrated  as  follows: 

Given  m  (ax  +  6)  +  c  =  o, 

let  ax  +  b=yy 

and  then  my  +  c  =  o, 

whence  y  —  ~  c/m, 

and  so  ax  +  6  =  —  c/m, 

which  can  now  be  solved.  Rabbi  ben  Ezra  illustrates  this  by 
taking 

x  -  \  x-4~  I  (*  -  i  *  -  4)  =  20 
and  letting  x—\x  —  ^=y ; 

whence  y  —  \y  =  20 

and  y  =  26|, 

and  so  x  -  \  x  -  4  =  26f , 

which  can  now  be  solved.2  Although  the  method  is  very  artifi- 
cial, it  is  occasionally  found  in  the  algebras  of  today,  especially 
in  connection  with  radical  equations. 

Classification  of  Equations.  Our  present  method  of  classify- 
ing equations  according  to  their  degree  is  a  modern  one.  The 
first  noteworthy  attempt  at  a  systematic  classification  is  found 
in  the  algebra  of  Omar  Khayyam  (c.  noo),  but  the  classifica- 
tion there  given  is  not  our  present  one.  Omar  considers  equa- 
tions of  the  first  three  degrees  as  either  simple  or  compound.  The 
simple  equations  are  of  the  type  r  =  #,  r  =  x2,  r  =  #8,  ax  —  x2, 
ax  =  XB,  ax2  =  x3.  Compound  equations  are  first  classified  as 
trinomials,  and  these  include  the  following  twelve  forms:  (i) 
x2  +  bx  —  c,  x2  +  c  =  bx,  bx  +  c  =  x2 ;  (2 )  x3  4-  bx2  =  ex, 

1".  .  .  secundum  regulam  que  vocatur  infusa.  Et  ipsa  est  regula  Job,  filii 
Salomonis."  Libri,  Histoire,  1838  ed.,  I,  312.  There  is  some  doubt,  however,  as 
to  whether  Libri  was  right  in  referring  this  work  to  Rabbi  ben  Ezra. 

2  Matthiessen,  Grundzuge,  p.  272. 


CLASSIFICATION  OF  EQUATIONS  443 


x3,x*  +  cx  =  d,x*  4-  d  =  cx,  ex-}-  d^x*; 
x3  +  bx*==d,x3  +  d  =  bx2,  bx*  +  d  =  x\  They  are  then  classi- 
fied as  quadrinomials,  as  follows:  (3)  x3  +  bx2  4-  ex  =  d, 
x3  +  bx2  +  d  =  cx\  (4)  r3  +  bx2  =-cx  +  d,  x*  +  cx  =  bx2  4-  d, 
x*  4-  d  =  &#2  4-  ex.  Of  this  early  plan  of  classifying  equations 
by  the  number  of  terms  we  still  have  a  trace  in  our  chapter  on 
binomial  equations. 

Classification  according  to  Degree.  Such  was  the  general 
method  of  classifying  equations,  naturally  with  variations  in  de- 
tails, until  after  books  began  to  be  printed.  Pacioli  (1494),  for 
example,  has  a  similar  system.1  It  was  not  until  about  the  be- 
ginning of  the  iyth  century  that  the  classification  according  to 
degree,  with  a  recognition  that  a  literal  coefficient  might  be  either 
positive  or  negative,  was  generally  employed,  and  this  was  due 
in  a  large  measure  to  the  influence  of  such  writers  as  Stevin 
(r58s),  Vieta  (c.  1590),  Girard  (1629),  Harriot  (1631,  posthu- 
mous) ,  Oughtred  (  1  63  1  )  ,  and  Descartes  (  1  63  7  )  .  In  particular, 
Descartes  set  forth  in  his  Geometric  the  idea  of  the  degree  of  an 
equation,  or,  as  he  says,  of  the  dimensions  of  an  equation,2  re- 
serving the  word  "degree"  for  use  with  respect  to  lines.3 

Quadratic  Equations.  The  first  known  solution  of  a  quadratic 
equation  is  the  one  given  in  the  Berlin  Papyrus  mentioned  on 
page  432.  The  problem  reduces  to  solving  the  equations 


and  the  solution  is  substantially  as  follows  : 

Make  a  square  whose  side  is  i  and  another  whose  side  is  \  . 
Square  f,  giving  TV  Add  the  squares,  giving  f  f  ,  the  square 
root  of  which  is  f  .  The  square  root  of  100  is  10.  Divide  10  by  f  , 
giving  8,  and  f  of  8  is  6.  Then 

8*4-  62=  100     and     6  =  f  of  8, 


a,  1494  ed.,  fol.  145  seq. 

2  "  Sgachez  done  qu'en  chaque  Equation,  autant  que  la  quantit£  inconnue  a  de 
dimensions,  autant  peut-il  y  avoir  de  diverses  racines,  c'est  a  dire  de  valeurs  de 
cette  quantite"  (1705  ed.,  p.  106).  He  then  speaks  of  tc*3  —  gxx  4  26*  —  24  *>  o, 
...  en  laquelle  x  ayant  trois  dimensions  a  aussi  trois  valeurs  qui  sont  2,  3,  &  4." 

3".  .  .  distingue  divers  degrez  entre  ces  lignes"  (ibid.,  p.  27). 


444  THE  SOLUTION  OF  EQUATIONS 

so  that  the  roots  of  the  two  implied  equations  are  6  and  8.    The 
solution  is  therefore  a  simple  case  of  false  position.1 

The  Greeks  were  able  to  solve  the  quadratic  equation  by 
geometric  methods.  As  already  stated,  Euclid  (c.  300  B.C.)  has 
in  his  Data  three  problems  involving  quadratics.  Of  these  the 
first  (Prob.  84)  is  as  follows: 

If  two  straight  lines  include  a  given  area  in  a  given  angle  and  the 
excess  of  the  greater  over  the  less  is  given,  then  each  of  them  is  given. 

Expressed  in  algebraic  form  with  reference  to  the  rectangle, 
if  xy  =  k2  and  x  —  y  ~  a,  then  x  and  y  can  be  found.  Euclid 
solves  the  problem  geometrically.2  He  also  gives  in  the  Elements 
such  geometric  problems  as  the  following : 

To  cut  a  given  straight  line  so  that  the  rectangle  contained  by  the 
whole  and  one  of  the  segments  shall  be  equal  to  the  square  on  the 
remaining  segment.3 

This  may  be  represented  algebraically  by   the   equation 

a(a  —  x)  —  x2  or  by  x2  +  ax  =  a2. 

Quadratics  among  the  Hindus.  It  is  possible  that  the  altar 
constructon  of  the  Hindus  involved  the  solution  of  the  equation 
ax2  +  bx  =  c,  and  this  may  date  from  the  Sulvasutra  period 
(roughly  speaking,  say  500  B.C.)  ;  but  whether  or  not  this  is  the 
case,  we  have  no  record  of  the  method  of  solution.4 

When  we  come  to  the  time  of  Aryabhata  (c.  510),  we  find  a 
rule,  relating  to  the  sum  of  a  geometric  series,  which  shows  that 
"the  solution  of  the  equation  ax2  +  bx  +  c  =  o  was  known,  but  we 
have  no  rule  for  the  solution  of  the  equation  itself.5 

It  should  be  repeated,  however,  that  up  to  the  i  yth  century 

an  equation  of  the  type  x2  +  px  =  g,  for  example,  was  looked 

.* 

1  Schack-Schackenburg,  Zeitschrift  fur  dgyptische  Sprache,  XXXVIII,  135; 
XL,  65.    See  also  Cantor,  Geschichte,  I  (3),  95,  and  Simon,  Geschichte,  p.  41. 

2  The  other  two  have  already  been  given  on  page  381. 
8 Elements,  II,  n.    See  also  VI,  28,  29. 

4  G.  MUhaud,  "  La  Geometric  d'Apastamba,"  in  the  Revue  gtntmle  des 
sciences,  XXI,  512-520. 

GThe  rule  for  the  summation  is  No.  XX  in  Rodet's  Lemons  de  Calcul  d'Ar- 
yabhata,  pp.  13,  33  (Paris,  1879) .  In  all  such  cases  the  possibility  of  the  younger 
Aryabhata  must  be  considered. 


HINDU  RULES  FOR  QUADRATICS  445 

upon  as  distinct  from  one  of  the  type  x2  —  px  =  q  ;  the  idea  that 
p  might  be  either  positive  or  negative  did  not  occur  to  alge- 
braists until  some  time  after  the  invention  of  a  fairly  good  sym- 
bolism. This  accounts  for  the  special  rules  for  different  types 
that  are  found  in  the  Middle  Ages  and  the  early  Renaissance. 

Brahmagupta's  Rule.     Brahmagupta  (c.  628)  gave  a  definite 
rule  for  the  quadratic.    For  example,  he  gave  the  equation 

ya  v  I  ya  16 


that  is,  r2  —  10  x  =  —  g,1  with  the  solution  substantially  as  follows  : 

Here  absolute  number  (9)  multiplied  by  (i)  the  [coefficient  of 
the]  square  (9),  and  added  to  the  square  of  half  the  [coefficient  of 
the]  middle  term,  namely,  25,  makes  16  ;  of  which  the  square  root  4, 
less  half  the  [coefficient  of  the]  unknown  (5),  is  9;  and  divided  by 
the  [coefficient  of  the]  square  (i)  yields  the  value  of  the  unknown  9. 

Expressed  in  modern  symbols, 


Mahavira's  Rule.    Mahavlra  (c.  850)  gave  no  rule  for  the 
quadratic,  but  he  proposed  a  problem  involving  the  equation 


adding  the  following  statement: 

In  relation  to  the  combined  sum  [of  the  three  quantities]  as  multi- 
plied by  12,  the  quantity  thrown  in  so  as  to  be  added  is  64.  Of  this 
[second]  sum  the  square  root  diminished  by  the  square  root  of  the 
quantity  thrown  in  gives  rise  to  the  measure  .  .  . 

Expressed  in  modern  symbols,  this  means  that 

-f-64  —  V64, 


which  shows  that  Mahavlra  had  substantially  the  modern  rule 
for  finding  the  positive  root  of  a  quadratic.2 

1  Colebrooke,  p.  347,  his  transliteration  being  followed. 

2  See  his  work,  p.  192. 


446  THE  SOLUTION  OF  EQUATIONS 

The  Hindu  Rule.  Sridhara  (c.  1025)  was  the  first,  so  far  as 
known,  to  give  the  so-called  Hindu  Rule  for  quadratics.  He  is 
quoted  by  Bhaskara  (c.  1150)  as  saying: 

Multiply  both  sides  of  the  equation  by  a  number  equal  to  four 
times  the  [coefficient  of  the]  square,  and  add  to  them  a  number  equal 
to  the  square  of  the  original  [coefficient  of  the]  unknown  quantity. 
[Then  extract  the  root.] x 

This  rule,  although  stated  by  Bhaskara,  is  not  the  first  one 
given  by  him.  He  begins  by  saying : 

[Its  re-solution  consists  in]  the  elimination  of  the  middle  term,  as 
the  teachers  of  the  science  denominate  it.  ...  On  this  subject  the 
following  rule  is  delivered.  .  .  .  When  a  square  and  other  [term]  of 
the  unknown  is  involved  in  the  remainder;  then  after  multiplying 
both  sides  of  the  equation  by  an  assumed  quantity,  something  is  to  be 
added  to  them,  so  as  the  side  may  give  a  square-root.  Let  the  root 
of  the  absolute  number  again  be  made  equal  to  the  root  of  the  un- 
known ;  the  value  of  the  unknown  is  found  from  that  equation. 

It  will  be  observed  that  this  is  simply  a  more  general  form  of 
Sridhara's  rule.  The  method  has  been  the  subject  of  much  dis- 
cussion by  the  various  commentators  on  Bhaskara.2 

Al-Khowarizmi's  Rules.  Al-Khowarizmi  (c.  825)  used  two 
general  methods  in  solving  the  quadratic  of  the  form 
x2+px  =  q,  both  based  upon  Greek  models. 
Given  x2  +  iox  =  39,  he  constructed  a  square 
as  here  shown.  Then  the  unshaded  part  is 
x2  +  px,  and  is  therefore  equal  to  q.  In  order 
to  make  it  a  square  we  must  add  the  four  shaded 
squares,  each  of  which  is  (\py  and  the  sum  of 


ita,  p.  209.    That  is,  given  ax*  +  bx  =  c,  we  have  first  4  a2*2  + 
4abx  =  4ac.   Then  4 a2*2  +  4abx  +  b2  =  b2  +  4ac,  whence 

2  ax  +  b  =  V^2  +  4  act 

the  negative  root  being  neglected.   The  purpose  of  the  multiplication  by  4 a  was 
to  avoid  fractions.  , 

2  See  the  Vija-Gariita,  pp.  207-209. 


ARABIC  RULE  FOR  QUADRATICS  447 

which  is  lp2,  which  in  this  case  is  25.    Since  25  +  39  =64, 
wehave  *+J/  =  8; 

whence  x  -I-  5  =  8 

and  ^=3- 

His  statement  is  as  follows  : 

You  halve  the  number  of  the  roots,  which  in  the  present  instance 
yields  five.  This  you  multiply  by  itself  ;  the  product  is  twenty-five. 
Add  this  to  thirty-nine;  the  sum  is  sixty-four.  Now  take  the  root 
of  this,  which  is  eight,  and  subtract  from  it  half  the  number  of  the 
roots,  which  is  five  ;  the  remainder  is  three.  This  is  the  root  of  the 
square  which  you  sought  for  ;  the  square  itself  is  nine.1 

The  negative  root  was  neglected,  as  was  regularly  the  case 
until  modern  times. 

His  second  method  was'  similar  to  our  common  one.  In  the 
figure  the  unshaded  part  is  x2  +  px,  and  he  adds  the  square  of  i  p. 

He  then  has  x?  +px  +  \  /  =  \  /  +  g, 

whence  x  =  \/|  p'+q  —  \p> 

of  which  he  takes  only  the  positive  root.2 

Al-Khowarizmi  also  considers  other  forms,  his  solution3  of 
the  type  x2+q~-px  being  based  upon  the  identity 


from  which  it  follows  that 


Omar  Khayyam's  Rule.  Omar  Khayyam's  rule  (c.  nob)  for 
solving  the  quadratic  x'2  +  px  —  q  is  as  follows  : 

Multiply  half  of  the  root  by  itself  ;  add  the  product  to  the  number 
and  from  the  square  root  of  this  sum  subtract  half  the  root.  The  re- 
mainder is  the  root  of  the  square.4 

1  Rosen  ed.,  p.  8. 

2  For  a  discussion  of  his  methods  see  Matthiessen,  Grundzuge^  p.  299. 

3  For  discussion   and   for  the   geometric   proof   see   Rosen's  edition,   p.   16; 
Matthiessen,  Grundzuge,  p.  304;  Libri,  Histoire,  I,  236. 

4  That  is,  x=  V^/2  +  q  —  \p.    By  "  half  the  root"  is  meant  \p,  and  by  "  the 
number"  is  meant  q.   He  used  the  equation  x2  +  iooc  =  39,  which  was  the  one 


448  THE  SOLUTION  OF  EQUATIONS 

He  also  gave  rules  for  other  types,  that  for  x*  +  q  =  px  being 
based  upon  the  identity 


and  that  for  px  +  q  —  x2  upon  the  identity1 


Chinese  Work  in  Quadratics.  The  Chinese  gave  some  atten- 
tion to  quadratic  equations  in  the  Middle  Ages,  including  those 
of  the  form 


but  how  far  they  were  original  in  their  work  has  not  yet  been 
scientifically  determined.2 

With  respect  to  the  quadratic  equation  the  medieval  alge- 
braists added  nothing  of  importance  to  the  work  of  the  Arabic 
writers  from  whom  they  derived  their  inspiration,  and  the 
Renaissance  algebraists  did  little  except  in  their  improvement 
of  the  symbolism.  It  was  not  until  the  close  of  the  i6th  century 
that  the  next  noteworthy  contribution  was  made. 

Harriot  treats  of  Equations  by  Factoring.  The  first  important 
treatment  of  the  solution  of  quadratic  and  other  equations  by 
factoring  is  found  in  Harriot's  Artls  Analyticae  Praxis  (1631). 
He  takes  as  his  first  case  the  equation 

aa  —  ba  +  ca~  +  be 
and  writes  it  in  the  form 


also  used  by  al-KhowHrizmi  and  was  apparently  a  favorite  problem  of  the 
schools.  He  also  considered  the  arithmetically  impossible  solutions.  For  a  dis- 
cussion of  his  methods  and  proofs  see  Woepcke's  translation,  p.  17;  Matthiessen, 
Grundzuge,  p.  301. 

aSee  Woepcke's  translation,  pp.  20,  23;  Matthiessen,  Grundzuge,  pp.  305,  309. 

2L.  Vanh£e,  in  Toung-pao,  XIII,  291;  XII,  559;  XV,  in. 


FACTORING  PROCESS  449 

where  the  first  member  stands  for  (a  —  b)  (a  -f  c)  and  the  equa- 
tion becomes  (a  —  b)  (a  +  c)  =  o.  From  this  fact  he  finds 
that  b  =  a.1 

In  a  similar  way  the  equation 

aaa  -f-  ###  —  cda  — 


is  factored  into  (a  +  b)  (aa  —  cd)  =  o,  and  the  solution  is  given 
that  aa  =  £d.2 

Vieta  advances  the  Theory.  In  the  work  of  Vieta  the  analytic 
methods  replaced  the  geometric,  and  his  solutions  of  the  quad- 
ratic equation  were  therefore  a  distinct  advance  upon  those 
of  his  predecessors.  For  example,  to  solve  the  equation3 
x2  -f  ax  -f-  b  =  o  he  placed  u  +  z  f  or  x.  He  then  had 


b)  =  o. 
He  now  let  2  z  +  a  -  o,  whence  z  =  —  |-  a,  and  this  gave 


and  ;F 

ltf  Nam  si  ponatur  a  ----  6  erit  a—  b  r-  —  :  o,"  p.  16.  The  relation  a  4-  c  =  o 
is  neglected. 

2P.  19.   The  relation  a  =  —  b  is  neglected. 

3  The  symbolism  used  here  is,  of  course,  modern.  Vieta's  own  solution  is 
as  follows  : 

"Si  A  quad.  +  Ba  in  A,  aequatur  Z  piano.  A  +  B  esto  E.  Igitur  E  quad. 
aequabitur  Z  piano  -f  B  quad.  ____  ____  _._  ___  __ 

tc  Consectarium.  Itaque  VZ  plani  -f  B  quad.  —  B  fit  A,  de  qua  primum 
quaerebatur.  — 

"Sit  B  i.    Z  planum  20.  A  i  N.  i  Q  +  2  N,  aequatur  20.  et  fit  i  N  V2i  -  i." 

That  is,  if  A2  -f  2  B  A  =  Z,  we  may  represent  this  in  modern  form  as 
x2  +  tax  =  6,  where  A  =  #,  B  =  a,  Z  =6. 

Let  A  +  B  =  E,  that  is,  let  x  +  a  =  u. 

It  follows  that  w2  =  x2  +  2  ax  4-  a2  =  6  4-  a2,  and  so  x  =  V<*  +  a2  —  a. 

In  particular,  he  says,  Jet  B  =  i  and  Z  =  20.  The  equation  is  then 
x2  +  2*  =  20,  whence  re  =V2i  —  i. 

He  has  similar  solutions  for  the  following  : 

"Si  A  quad.  —  B  in  A  2,  aequatur  Z  piano," 
and  "Si  D  2  in  A  —  A  quad.,  aequatur  Z  piano," 

showing  that  at  this  time  in  his  work  he  had  not  grasped  the  idea  of  such  a 
general  quadratic  equation  as  x2  +  a^x  +  az  =  o.  In  his  PC  nmnerosa  potestatum 


450  THE  SOLUTION  OF  EQUATIONS 

Modern  Methods.  Of  the  modern  methods 1  for  obtaining  the 
formula  for  the  solution  of  the  quadratic,  interesting  chiefly 
from  the  standpoint  of  theory,  a  single  one  may  be  mentioned. 
This  method  uses  determinants  and  is  due  to  Euler  and  Bezout, 
but  was  improved  by  Sylvester  (1840)  and  Hesse  (1844). 

Given  x*.+px  +  q  =  oy 

]pt  r  —  u  4-  <*  ' 

1CL  "V   —  ll  -p  **>   y   •• 

whence  x*  =  (u  +  z}x 

Then      -  x*    +px*    +  qx  =  o, 

and          x*  —  (//  4-  z)  x*        =  o  ; 

i     / 
whence        o     i     —  (;/ 

I    -  (U  +  S)  O 

Expanding,         —p(u  +  s)  —  (u  4-  ^)2  —  q  =  o, 
and  hence       if  -f  (2  z+p}  u  +  (£+pz  +  q)  =  o. 
Letting  2 


we  find  that  //  =  ±  i  V/1J-  4  ^ 

and  x=—  —  ±  \^p*—  4<7- 

Simultaneous  Quadratic  Equations.  Problems  involving  the 
combination  of  a  linear  and  a  quadratic  equation  were,  as  we 
have  seen,  familiar  to  the  Egyptians,  and  the  Greeks  were 
fully  able  to  apply  their  geometry  to  such  cases.  The  algebraic 
treatment  of  two  quadratics  was  not  seriously  considered,  how- 
ever, until  it  was  taken  up  by  Diophantus  (c.  275)  for  indeter- 
minate forms.  He  speaks  of  equations  like 


and  f^aW+b'x  +  c' 

.  .  .  resolutione  tractatus  (Paris,  1600),  however,  he  uses  the  terms  "affected" 
and  "pure"  with  respect  to  quadratic  equations.  See  Volume  I,  page  311.  See 
also  his  De  aequationum  recognitione  et  emendatione  libri  duo.  Tract.  II,  cap.  i 
(Paris,  1615) ;  Matthiessen,  Grundzuge,  p.  311. 

!For  a  list  of  modern  methods  consult  Matthiessen,  Grundzuge,  p.  315  seq. 


SIMULTANEOUS  QUADRATICS  451 

as  "double  equations."1    Among  his  more  difficult  equations  of 
this  type  is  the  pair 

x2  +  x  -  I  =  u 


which  Diophantus  expresses  as  follows  : 

To  find  three  numbers  such  that  their  solid  content  minus  any  one 
gives  a  square.2 

The  subject  never  interested  the  medieval  writers  particu- 
larly, and  not  until  the  1  7th  century  do  we  find  much  attention 
paid  to  it.  By  that  time  the  symbolism  was  such  that  the  only 
question  involved  was  that  of  stating  the  cases  in  which  a  solu- 
tion is  possible. 

Indeterminate  Quadratic  Equations.  The  study  of  indeter- 
minate quadratic  equations  begins  with  such  cases  as  xz  +  y2  =  z2. 
The  finding  of  formulas  for  these  sides  of  a  Pythagorean  Tri- 
angle occupied  the  attention  of  various  Greek  writers.  Proclus 
(c.  460)  tells  us  that  Pythagoras  (c.  540  B.C.)  himself  gave  a 
rule,  and  tradition  says  that  it  was,  as  expressed  in  modern 
symbo,s, 

ft    ~T  I 


where  n  is  an  odd  number.    Plato  (c.  380  B.C.)  gave  the  rule 

(2  w)«+(^-  !)»=(,,"+!)«, 

which,  like  the  one  attributed  to  Pythagoras,  is  connected  with 
Euclid's  proposition3  to  the  effect  that 


a  relation  that  forms  the  basis  of  the  theory  of  quarter  squares. 


?,  8nr\y  icrbTijs,  dur\ri  fou<ru.  See  Heath,  Diophantus,  2d  ed.,  p.  73. 
2  Book  IV,  23.   That  is,  the  first  number  is  x,  the  second  is  i,  and  the  third 
is  x  4-  i,  the  "solid  content"  being  x  •  i»  (x  +  i).   The  results  are  ^g7-,  i,  and  ^. 
For  further  explanation  see  Heath,  loc.  cit.,  p.  184. 
-     8  Heath,  Diophantus,  2d  ed.,  pp.  116,  242  n. 


452  THE  SOLUTION  OF  EQUATIONS 

Diophantus  on  Indeterminate  Equations.  It  was  Diophantus, 
however,  who  may  properly  be  called  the  father  of  the  study  of 
indeterminate  equations,  which  were  generally  limited  in  his 
Arithmetica  to  quadratic  types.  With  these  equations  the  ob- 
ject was  to  obtain  rational  results,  while  with  indeterminate 
equations  of  the  first  degree  the  object  was  usually  to  obtain 
integral  results.  The  problem  proposed  by  Diophantus  is  that 
of  solving  either  one  or  two  equations  of  the  form 


His  simpler  types  may  be  represented  by  the  following  : 

To  add  the  same  [required]  number  to  two  given  numbers  so  as 
to  make  each  of  them  a  square.1 

One  of  the  more  difficult  problems  is  as  follows  : 

To  find  three  numbers  such  that  their  sum  is  a  square  and  the  sum 
of  any  pair  is  a  square.2 

Pell  Equation.   One  of  the  most  famous  indeterminate  quad- 
ratic equations  is  of  the  form 

4  i  =  *2. 

This  form  is  commonly  attributed  to  John  Pell  (1668)  but  is 
really  due  to  Fermat  (c.  1640)  and  Lord  Brouncker3  (1657). 
The  problem  itself  is  apparently  much  older  than  this,  however, 
for  it  seems  involved  in  various  ancient  approximations  to  the 
square  roots  of  numbers.  Thus  the  Greek  approximation  £  for 
the  ratio  of  the  diagonal  to  the  side  of  a  square  goes  back  to 
Plato's  time  at  least,  and  7  and  5  are  the  roots  of  the  equation 


1III,  ii.  That  is,  if  the  given  numbers  are  2  and  3,  then  #  4-  2  and  x  +  3 
must  both  be  squares.  He  finds  that  x  =  JJ. 

2  III,  6.  His  results  are  80,  320,  41.  For  solution  see  Heath,  Diophantus, 
2d  ed.,  pp.  68,  158. 

8E.  E.  Whitford,  The  Pell  Equation,  New  York,  1912;  H.  Konen,  Geschichte 
der  Gleichung  t2  —  Du2  -  i,  Leipzig,  1901  ;  G.  Wertheim,  "Ueber  den  Ursprung 
des  Ausdruckes  'Pellsche  Gleichung/"  Bibl.  Math.,  II  (3),  360;  Heath,  Dio- 
phantusy  2d  ed.,  p.  286. 


INDETERMINATE  EQUATIONS  453 

Theon  of  Smyrna  (c.  125)  considered  a  relation  that  would 
now  be  written  as  the  equation 

X*-  2/=±I, 

carrying  his  computations  as  far  as  the  case  of 


and  stating  a  rule  for  finding  the  solutions. 

The  special  case  of  the  Cattle  Problem,  doubtfully  attributed 
to  Archimedes;  requires  the  number  of  bulls  of  each  of  four 
colors,  white  (W),  blue  (B),  yellow  (Y),  and  piebald  (P),  and 
the  number  of  cows  of  the  same  colors  (w,  b,  y,  p)  such  that 


Reduced  to  a  single  equation,  the  problem  involves  the  solution 
of  the  indeterminate  quadratic  equation 


and  the  number  of  yellow  bulls,  for  example,  has  68,848  periods 
of  three  figures  each.1 

The  general  problem  may  have  been  discussed  in  the  lost 
books  of  Diophantus,2  perhaps  in  the  form 


and  its  equivalent  is  clearly  stated  in  the  works  of  Brahma- 
gupta3  (c.  628). 

Fermat  (c.  1640)  was  the  first  to  state  that  the  equation 
x2  —  Ay2  =  i7  where  A  is  a  non-square  integer,  has  an  un- 
limited number  of  integral  solutions,4  and  from  that  time  on  the 
problem  attracted  the  attention  of  various  scholars,  among  the 
most  prominent  being  Euler  (1730),  who  stated  that  the  solu- 

1  Heath,  History,  II,  97;  Whitford,  loc.  cit.,  p.  20,  with  bibliography. 

2  P.  Tannery,  "  L'arithme'tique  des  Grecs  dans  Pappus,"  in  the  Memoires  de  la 
Soc.  des  sci.  de  Bordeaux,  III  (2),  370. 

3Colebrooke  ed.,  p.  363. 

4(Euvres,  ed.  Tannery  and  Henry,  II,  334  (Paris,  1894)  >  Whitford,  loc.  cit., 
p.  46. 
ii 


454  THE  SOLUTION  OF  EQUATIONS 

tion  of  the  equation  ax2  4-  bx  4-  c  =  y2  requires  the  solution  of 
the  equation  x2  —  Ay*  =  i.1  It  was  he  who,  through  an  error, 
gave  to  the  general  type  the  name  of  the  Pell  Equation. 

Cubic  Equation.  The  oldest  known  cubic  equation  of  the 
form  x3  =  k  is  possibly  due  to  Menaechmus  (c.  350  B.C.),  al- 
though tables  of  cubes  had  been  worked  out  by  the  Baby- 
lonians two  thousand  years  earlier.  It  had  been  recognized 
since  the  time  of  Hippocrates  (c.  460  B.C.)  that  the  solution  of 
the  problem  of  the  duplication  of  the  cube  depended  on  the 
finding  of  two  mean  proportionals  between  two  given  lines. 
Algebraically  this  means  the  finding  of  x  and  y  in  the  equations 

a  _  x  __  y 

x     y      b 

From  these  relations  it  is  evident  that 

y2  =  bx  (a  parabola) 

and  xy  =  ad  (an  equilateral  hyperbola) ; 

whence  y'3  =  ab2  (a  cubic  equation). 

Menaechmus  is  said  to  have  solved  the  cubic  by  finding  the 
intersection  of  the  two  conies.  If  b  =  2  a,  then  y3  —  2  a3,  and 
the  problem  becomes  the  well-known  one  of  the  duplication  of 
the  cube,  which  interested  so  many  Greek  writers.2 

The  next  reference  to  the  cubic  among  the  Greeks  is  in  a  cer- 
tain problem  of  Archimedes,  to  cut  a  sphere  by  a  plane  so  that 
the  two  segments  shall  have  a  given  ratio.3  This  reduces  to  the 
proportion 

c  -  x  __  c* 

b       ~  X* 

and  to  the  equation        x*  +  <*b  =  ex*. 

1P.  H.  von  Fuss,  Correspondance  mathematique  et  physique  de  quelques  ctle- 
bres  gtometres  du  XV Illume  siecle,  I,  37  (Petrograd,  1843). 

2  For  a  partial  list  of  these  writers  see  Woepcke,  translation  of  Omar  Khayyam, 
p.  xiij.   The  reference  to  Menaechmus  is  not  certain. 

3  De  sphaera  et  cylindro,  Lib.  II.   See  also  Heath's  Archimedes,  chap.  vi. 


THE  CUBIC  EQUATION  455 

Eutocius  (c.  560)  tells  us  that  Archimedes  solved  the  prob- 
lem by  finding  the  intersection  of  two  conies,  namely, 

*2  =  —y  (a  parabola) 

c 

and  y(c  —  x)  =  be  (a  hyperbola). 

Diophantus  solved  a  single  cubic  equation,  x34-  #  =  4JC2H-  4. 
This  equation  arises  in  connection  with  the  following  problem : 

To  find  a  right-angled  triangle  such  that  the  area  added  to  the 
hypotenuse  gives  a  square,  while  the  perimeter  is  a  cube.1 

His  method  is  not  given,  the  statement,  expressed  in  modern 
language,  being  that  £  "is  found  to  be"  4.  Possibly  Diophan- 
tus saw  that  x  (x2  +  i )  =  4  (x*+  i ) ;  whence  x  =  4. 

The  Cubic  among  the  Arabs  and  Persians.  Nothing  more  is 
known  of  the  cubic  equation  among  the  Greeks,  but  the  prob- 
lem of  Archimedes  was  taken  up  by  the  Arabs  and  Persians  in 
the  Qth  century.  In  a  commentary  on  Archimedes,  Almahani 
(c.  860)  considered  the  question,  but  so  far  as  known  he  con- 
tributed nothing  new.  He  brought  the  problem  into  such  promi- 
nence, however,  that  the  equation  xs  +  a2b  =  ex9  was  known 
among  the  Arab  and  Persian  writers  as  Almahani's  equation.2 

One  of  his  contemporaries,  Tabit  ibn  Qorra  (c.  870),  con- 
sidered special  cases  of  cubic  equations,  as  in  the  duplication  of 
the  cube.  These  equations  he  solved  by  geometric  methods,  but 
he  was  unable  to  contribute  to  the  general  algebraic  theory. 

A  little  later  Abu  Ja'far  al-Khazin  (c.  960),  a  native  of 
Khorasan,  considered  the  problem  and,  as  Omar  Khayyam  tells 
us,  "solved  the  equation  by  the  aid  of  conic  sections."3 

The  last  of  the  Arabs  to  give  any  particular  attention  to  the 
solution  was  Alhazen4  (c.  1000).  Omar  Khayyam5  refers  to 

1Bk.  VI,  prob.  17.    See  also  Heath's  Diophantus,  2d  ed.,  p.  66. 
2  Matthiessen,  Grundzuge,  p.  367 ;  Cantor,  Geschichte,  I,  chap.  xxxv. 
sWoepcke  translation,  p.  3. 
V  4  Al-^Jasan  ibn  al-Iiasan  ibn  al-Haitam. 

5  Woepcke's  translation,  p.  73,  with  discussion;  Matthiessen,  Grundzuge,  p.  367- 


456  THE  SOLUTION  OF  EQUATIONS 

his  method.    Alhazen  solved  the  equation  by  finding  the  inter- 
section of  x^ay  (a  parabo]a) 

and  y(c  —  *)—  ab  (a  hyperbola), 

a  method  not  unlike  the  one  attributed  to  Archimedes. 

The  last  of  the  Persian  writers  to  consider  the  cubic  equation 
with  any  noteworthy  success  was  Omar  Khayyam1  (c.  noo). 
In  his  list  of  equations  he  specified  thirteen  forms  of  the  cubic 
that  had  positive  roots,  this  being  a  decided  advance  in  the  gen- 
eral theory.  He  solved  equations  of  the  type  #3  +  b2x  =  b2c  by 
finding  the  intersection  of  the  conies  x2  —  by  and  /  =  *(<;  — *)  ; 
of  the  type  x3  +  ax2  =  c3  by  finding  the  intersection  of  xy  =  c2 
and  y2  =  c(x  4-  a} ;  and  of  the  type  x3  ±  ax2  -f-  b2x  =  b2c  by 
finding  the  intersection  of  y2~(x±  a)  (c—  x)  and  x(b  ±  y)  =  be. 

It  is  said,  but  without  proof  from  the  sources,  that  Omar 
Khayyam  stated  that  it  was  impossible  to  solve  in  positive  inte- 
gers the  equation  x3  -f-  y3  =  z3,  the  simplest  of  the  family  of 
equations  of  the  type  xn  4-  yn  —  zn  with  which  Fermat's  name 
is  connected. 

In  general  it  may  be  said  that  the  Arab  writers  believed  that 
the  cubic  equation  was  impossible  of  solution.2 

Chinese  anil  Hindu  Interest  in  the  Cubic.  The  Chinese  alge- 
braists did  nothing  worthy  of  note  with  the  general  cubic  equa- 
tion. Their  interests  lay  in  applied  problems,  and  these  all  led 
to  numerical  equations.  The  numerical  cubic  first  appears  in  a 
work  by  Wang  Hs'iao-t'ung,  about  62  5.3 

He  gave  the  following  problem  : 

There  is  a  right-angled  triangle  the  product  of  the  sides  of  which  is 
706^  and  the  hypotenuse  of  which  is  greater  than  one  side  by  36^. 
Find  the  lengths  of  the  three  sides. 

Wang  used  a  numerical  equation  of  the  form  x3  +  ax2  —  b  =  o 
and  stated  the  answer  incorrectly  as  14^,  49^,  and  51], 
although  there  is  doubt  as  to  the  validity  of  the  copy. 

1Ball,  Hist,  of  Math.,  6th  ed.,  p.  159;  Woepcke's  translation,  p.  25  seq. 

2Cantor,(  Geschichte,  I  (2),  736. 

8  In  the  Ch'i-ku  Suan-king.   See  Mikami,  China,  p.  54. 


MEDIEVAL  INTEREST  IN  THE  CUBIC  457 

Various  later  Chinese  algebraists  treated  of  numerical  equa- 
tions, but  it  was  not  until  the  i8th  century,  when  European  in- 
fluences were  powerful,  that  any  attempt  was  made  by  them  to 
classify.  equations  of  the  third  degree.  In  a  work  prepared  under 
the  direction  of  Emperor  Kanghy,  who  ruled  China  from  1662 
to  1722,  nine  types  are  given  : 

x*  ±  bx  =c  x*  ±  ax*  =  c 


but  in  every  case  the  solution  is  numerical  and  only  a  single 
positive  root  is  given.1 

The  Hindus  paid  little  attention  to  cubic  equations  except  as 
they  entered  into  relatively  simple  numerical  problems  relating 
to  mensuration.  Bhaskara  (c.  1150)  gave  one  example, 

x3  +  i2x  =  6x2  +  35, 

the  root  being  5,2  but  such  a  result  is  easily  found  by  trial,  the 
equation  being  made  for  this  purpose. 

Medieval  Interest  in  the  Cubic.  In  the  Middle  Ages  various 
sporadic  attempts  were  made  by  European  scholars  to  solve  the 
cubic  equation.  Fibonacci,  for  example,  attacked  the  problem 
in  his  Flos  of  c.  1225.  He  states  that  one  Magister  Johannes,  a 
scholar  from  Palermo,  proposed  to  him  the  problem  of  finding  a 
cube  which,  with  two  squares  and  ten  roots,  should  be  equal  to 
2o.3  That  is,  the  problem  is  to  solve  the  equation 

X3  +  2X2  +  IOX  =  20, 

a  numerical  equation  discussed  later  (p.  472).  Another  attempt 
was  made  by  an  anonymous  writer  of  the  i3th  century  whose 
work  has  been  described  by  Libri.4  He  took  two  cubics,  one 
of  the  type  ax3  =  ex  +  k  and  the  other  of  the  type  ax3  =  bx2  -f  k. 

1The  work  was  the  Lii-li  Yuan-yuan.   See  Mikami,  China,  pp.  117-119. 

2Colebrooke,  loc.  cit.,  p.  214. 

8"Altera  uero  questio  a  predicto  magistro  lohanne  proposita  fuit,  vt  in- 
ueniretur  quidam  cubus  numerus,  qui  cum  suis  duobus  quadratis  et  decem 
radicibus  in  unum  collectis  essent  uiginti"  (Flos,  Boncompagni  ed.,  p.  228). 

*Histoire,  1838  ed.,  II,  213,  214.   The  MS.  is  probably  Florentine. 


458  THE  SOLUTION  OF  EQUATIONS 

In  each  case  he  displayed  great  ignorance,  possibly  because  he 
was  unable,  on  account  of  his  unfamiliarity  with  radicals  to 
check  his  results.  It  is  also  possible  that  he  sought  only  approxi- 
mate results,  although  this  is  not  stated.  His  method  in  the  first 
case  was  as  follows : 

Given  ax*~cx  +  k, 

8 


we  have  ^ 

a         a 

whence  he  assumed  that      x=* h  A  ( — )  +  - , 

2#      \\2aJ      a 

which  is  the  root  of  ax2  ==  ex  +  k  but  not  of  the  given  equation. 

His  method  in  the  second  case  was  equally  fallacious. 

Slight  attempts  at  numerical  cubics  were  also  made  by  Regio- 
montanus,1  who  gave,  for  example,  the  equation 


but  he  contributed  nothing  of  value  to  the  theory. 

The  Cubic  in  Printed  Books.  Pacioli  (1494)  asserted  substan- 
tially that  the  general  solution  is  impossible.2 

Of  the  early  German  writers  only  one  made  any  noteworthy 
attempt  at  the  solution,  and  this  was  a  failure.  Rudolff  (1525) 
suggested  three  numerical  equations,  each  with  one  integral  root 
and  each  being  easily  solved  by  factoring.3  His  method  in  con- 
nection with  one  of  these  equations  is  interesting.  In  modern 
symbols  it  is  substantially  as  follows : 

Given  #*=  10^  +  20^  +  48, 

wehave  #s  +  8==  10^  +  20^+  56; 


;r+2 


whence  ^r  - 

1  Cantor,  Geschichte,  II,  chap.  Iv. 
2Fol.  149,  r.,  has  the  following: 

Impossibile.   Censo  de  censo:  e  ceso  equale.  a  cosa. 
Impossibile.   Censo  de  censo  e  cosa.  equale.  a  censo. 

That  is,  the  solution  of  equations  like  ax4  +  ex2  =  dx  and  ax*  -f  dx  —  ex2  is 
impossible.  3  Die  Cossy  1553  ed.,  fol.  477,  r. 


ITALIAN  TREATMENT  OF  THE  CUBIC  459 

all  of  which  is  correct.   He  now  assumes  that  he  can,  in  general, 
split  the  two  members  and  say  that 

X1  —  2  X~  IOX 

56 


and  4  — 

Both  of  these  equations  are  satisfied  if  x  =  12,  but  the  method 
is  not  otherwise  general. 

Similar  solutions  of  special  cases  are  found  in  various  works 
of  the  1 6th  century,  notably  in  a  work  by  Nicolas  Petri  of 
Deventer,1  published  at  Amsterdam  in  1567.  This  writer  was 
highly  esteemed  by  his  contemporaries.2 

A  few  special  cases,  such  as 

*3- 381^-90, 

he  solves  by  factoring,  and  he  then  proceeds  to  a  more  elaborate 
discussion  of  certain  cases  that  are  mentioned  later. 

The  Italian  Algebraists  and  the  Cubic.  The  real  interest  in 
the  cubic  lies,  however,  in  the  work  of  the  Italian  algebraists  of 
the  1 6th  century,  and  notably  in  the  testimony  of  Cardan  and 
Tartaglia.  Cardan  (1545)  says  that  Stipio  del  Ferro  discovered 
the  solution  of  the  type  x3  +  bx  =  c  thirty  years  earlier  (c.  1 5 1 5 ) , 
revealing  the  secret  to  his  pupil  Antonio  Maria  Fior  (Florido).3 
The  source  of  the  solution  is  unknown.  Ferro  may  have  re- 

1  Arithmetica.  Practicque  omne  cortelycken  te  lere  chijphere  .  .  .  Door  my 
Nicolaum  petri  F.  Daitentriensem,  Amsterdam,  1567.  The  name  also  appears  as 
Nicolas  Peetersen  or  Pietersz  (Pieterszoon),  Petri  F.  meaning  Petri  Filius  (son 
of  Peter). 

2H.  Bosnians,  "La  'Practiqve  om  te  leeren  cypheren'  de  Nicolas  Petri  de 
Deventer,"  Annales  de  la  Societe  scientifique  de  Bruxelks,  XXXII,  2e  Partie, 
Reprint,  1908. 

3"Verum  temporibus  nostris,  Scipio  Ferreus  Bononiensis,  capitulum  cubi  & 
rerum  numero  aequalium  inuenit,  rem  sane  pulchram  &  admirabilem.  .  .  .  Huius 
emulatioe  Nicolaus  Tartalea  Brixellensis,  amicus  noster,  cu  in  certame  cu  illius 
discipulo  Antonio  Maria  Florido  uenisset,  capitulum  idem,  ne  uinceretur,  inuenit, 
qui  mihi  ipsum  multis  precibus  exoratus  tradidit"  (Ars  Magna,  fol.  3,  r,} .  On  the 
general  work  of  the  Italians  with  respect  to  the  development  of  algebra  see 
E.  Bortolotti,  "Italiani  scopritori  e  promotori  di  teorie  algebriche,"  in  the  An- 
nuario  delta  R.  Universita  di  Modern,  Anno  1918-1919. 


460  THE  SOLUTION  OF  EQUATIONS 

ceived  it  from  some  Arab  writer,  or  he  may  have  discovered  it 
himself  in  spite  of  his  apparent  lack  of  mathematical  ability. 
Tartaglia  agrees  with  Cardan's  statement  except  as  to  time,  plac- 
ing it  somewhat  earlier  (in  I5O6),1  a  matter  of  little  conse- 
quence. Cardan  further  says  that  Florido  had  a  contest  with 
Tartaglia  which  resulted  in  the  latter's  discovery  of  the  method 
for  solving  this  particular  type,  and  that  Tartaglia,  at  Cardan's 
request,  revealed  it  to  him. 

Tartaglia  states  his  side  of  the  case  rather  differently  and 
more  explicitly.  He  says  that  Zuanne  de  Tonini  da  Coi2  (see 
Volume  I,  page  295)  sent  him,  in  1530,  two  problems,  namely, 


and  x3  +  6x2  +  8x  =  1000, 

neither  of  which  he  could  solve  ;  but  that  in  IS3S3  he  found  the 
method  of  solving  any  equation  of  the  type  x*  +  ax2  =  c.  Tar- 
taglia further  states  that  he  had  a  contest  with  Florido  in  1535 
and  knew  that  he  had  only  to  set  problems  of  this  type  to  defeat 
his  opponent,  provided  he  could  first  find  the  latter's  method  of 
solving  problems  of  the  type  r3  +  bx  =  c.  He  therefore  exerted 
himself  and  succeeded  in  discovering  it  just  before  the  contest/ 
thus  being  able  to  solve  anything  that  Florido  could  set,  and 
being  able  to  propose  problems  that  the  latter  could  not  master. 

Tartaglia  and  Cardan.  Da  Coi  now  importuned  Tartaglia 
to  publish  his  method,  but  the  latter  declined  to  do  so.  In  1539 
Cardan  wrote  to  Tartaglia,  and  a  meeting  was  arranged  at  which, 

1<l.  .  .  se  auantaua  che  gia  trenta  anni  tal  secreto  gli  era  stato  mostrato 
da  un  gran  mathematico."  From  Qvesito  XXV,  dated  December  10,  1536.  See 
the  1554  edition  of  the  Qvesiti,  fol.  106,  v. 

2  Also  known  as  Giovanni  Colle  and  Joannes  Colla. 

3  In  his  statement  of  December  10,  1536  (Qvesito  XXV),  he  says:  "...  & 
questo  fu  Panno  passato,  cioe  del  .1535.  adi  .12.  di.  Febraro  (uero  e  in  Venetia 
ueneua  a  esser  del  .1534.)  ..."   See  also  A.  Oliva,  Sulla  soluzione  dell'  equazione 
cubica  di  Tartaglia,  Milan,  1909. 

4  "Per  mia  bona  sorte,  solamente  .8.  giorni  auanti  al  termine  .  .  .  lo  haueua 
ritrouata  la  regola  generate."    Qvesiti,  libro  nono,  Qvesito  XXV;    1554  ed., 
fol.  106,  v. 


TARTAGLIA  AND  CARDAN  461 

Tartaglia  says,  having  pledged  Cardan  to  secrecy,  he  revealed 
the  method  in  cryptic  verse1  and  later  with  a  full  explanation.2 
Cardan  admits  that  he  received  the  solution  from  Tartaglia, 
but  says  that  it  was  given  to  him  without  any  explanation.3  At 
any  rate,  the  two  cubics  #3  +  ax2  =  c  and  x8  +  bx  =  c  could  now 
be  solved.  The  reduction  of  the  general  cubic  x*  +  ax2  +  bx  =  c 
to  the  second  of  these  forms  does  not  seem  to  have  been  con- 
sidered by  Tartaglia  at  the  time  of  the  controversy.  When 
Cardan  published  his  Ars  Magna  (1545),  however,  he  trans- 
formed the  types  x3  =  ax2  +  c  and  x3  +  ax2  =  c  by  the  substitu- 
tions x  =  y  +  5  a  and  x  =  y  —  ^  a  respectively,  and  transformed 
the  type  Xs  +  c  —  ax2  by  the  substitution  x  =  $~c*/y,  thus  free- 
ing the  equations  of  the  term  in  x2.  This  completed  the  general 
solution,  and  he  applied  the  method  to  the  complete  cubic  in  his 
later  problems. 

Cardan's  Originality.  Cardan's  originality  in  the  matter 
seems  to  have  been  shown  chiefly  in  four  respects.  First,  he 
reduced  the  general  equation  to  the  type  x3  +  bx  =  c  ;  second,  in 
a  letter  written  August  4,  1539,  he  discussed  the  question  of  the 
irreducible  case  ;  third,  he  had  the  idea  of  the  number  of  roots 
to  be  expected  in  the  cubic  ;  and,  fourth,  he  made  a  beginning 
in  the  theory  of  symmetric  functions.4 

1  Quando  chel  cubo  con  le  cose  appresso 

Se  aggualia  a  qualche  numero  discreto  ^3  +  bx  —  c 

Trouan  dui  altri  different!  in  esso.  u  ~  v  =  c 

Dapoi  terrai  questo  per  consueto  /M  8 

Che  '1  lor  produtto  sempre  sia  eguale  uv  =  (    \ 

Al  terzo  cubo  delle  cose  neto,  3 

El  residue  poi  suo  generale 
Belli  lor  lati  cubi  ben  sostratti 

Varra  la  tua  cosa  principale.  x  ~  V«  —  v  v 

Qvesiti,  1554  ed.,  fol.  120,  v.   There  are  sixteen  lines  more.   See  also  Gherardi 

in  Grunert's  Archiv,  LII,  143  seq.  and  188. 


2  Substantially   this:    If  *8  +  bx  =  c,   let    u  —  v-c    and    uv  —  \-.    Then 


x=zWi--\/v>  for  (^-V^*+b(yu-Vv)  =  u-v.  See  the  second  part 
of  his  Qvesito  XXXV,  Qvesiti,  1554  ed.,  fol.  121,  v. 

3".  .  .  ut  Nicolaus  inuenerit  &  ipse,  qui  cum  nobis  rogantibus  tradidisset, 
suppressa  demonstratione  ..."  Ars  Magna,  1545  ed.,  fol.  29,  v.,  shown  in  fac- 
simile on  pages  462,  463. 

4See  also  Enestrom's  summary  in  Bibl  Math.,  VII  (3),  293. 


relinqiifturprimatf  m:m  ?of,h*  autem  quamitates  proportional 
fjncAquadraiuiTi  fecundac  eft  arqualc  duplo  product  fecundx  m 
pnniam,cum  quadruple  primor,ut  proponebatur, 

DC  cubo  &  rebus  a?qua!ibus  numcro.       Cap*    X  I, 

Cipio  Fcrrcus  Bononicnfis  jam  annis  ab  hinc  triginta  fer- 
ine capitulum  hoc  inucnit ,  tradidit  ucro  Anthonio  Ma* 
ria:  Flondo  Vcneto,qui  cii  in  cerramcn  cu  Nicolno  Tar* 
_^_^_____  ralea  Hnxcllcnfc  aliquando  uemfler,  occafioncm  dcdir,  ut 
Nicolaus  inuenen't  &  ipfc,qui  cum  nobis  rogantibus  tradidiffer,  fup 
prcffa  dcmonftratione,  freti  hoc auxilio,  dcmonftrarioncm  quxlitii* 
imis,eamcjjmmodos,quoddirficillmujmfuir>  rcda<flam  fie  fubicci* 
mus.  DEMONSTRATIO. 

Sit  igittir  exempli  caufa  cubus  G  H  cV  fcxcuplum  latcn's  c  H  xqua 
le  20,6^  ponam  duos  cubes  A  E  Sf  c  L,quorum  diftcrcntia  lie  zo ,  ita 
c^iodproductum  A  c  latcns,  in  c  K  latus, 
fit  i,  tertia  icilicet  mimcri  rcrum  pars ,  & 
abfcindam  c  B,xqualcm  c  K,dico,  quod  Q 
ita  fucrir,Iincam  A  B  rcfiduum ,  eife  arqua* 
1cm  G  H,&:  ideo  rci  a^ftimacionein,  nam  dc 
G  H  lam  fupponebatur,quod  ita  c/Tet,  per* 
haam  igicur  per  modum  primi  fuppofm 
.^'capirulihuiuslibri,  corpora  D  A, DC,DE  ^  "  w 

,D  F,u,r  per  D  c  intelligamus  cubum  B  c,pcr 

p  F  cubum  A  E,per  D  A  rrjplum  c  B  in  quadrarum  A  B,per  D  K  triplum 
A  B  in  quadratu  B  c.quia  jgirur  ex  A  c  in  c  K  fit  i,ex  A  c  in  c  ic  ter  hce 
^  niimcrus  rcrum,  igimr  c^  A  B  yi  triplum  A  c  in  c  ic  fiunt^  res  A  B, 
feu  fcxcuplum  A  B,quarc  rriplum  produc^i  ex  A  B,  B  C,A  c,  eft  fexcu* 
plum  A  B,at  uero  differentia  cubi  A  c ,  a^ubo  c  K  ,  ck  exiftenri  a  aibo 
B  c  01  nrqlecx  fuppofito,ert  io,Kex  fuppofito  pnmo  6*  capituh  ,  eft 
agoj-cgatum  corporum  D  A,D  E,D  F,triaigitur  hxc  corpora  func  20, 
pofua  uero  B  c  m:cubus  A  B,xquahs  eft  cubo  A  c,&  mplo  A  c  in  qua 
jdiatum  c  B,5^  cubo  B  c  m:6V  tripfo  t  c  in  quadratum  A  c  m:  per  de# 
monftrata  ilhc^ifTercntia  aurem  tripli  B  c  in  quadratum  A  c,  a  triplo 
A  c  in  quadratum  B  ceft  producTum  A  B,B  c,A  CjquarccumhoCjUt  dc 
nionftratum  cft,orqualc  fit  fcxcuplo  A  B,  igitur  addito  fexcuplo  A  B, 
ad  id  quod  fit  ex  A  c  in  quadratum  B  c  tcr,fict  triplum  B  c  in  quadra* 
i  um  A  c,cum  igitur  B  c  fit  m:iam  oftcnfum  cft,quod  produclum  c  B 


C, 

D 

C 

1               £ 
L 

JC 

111 


CARDAN'S  SOLUTION  OF  THE  CUBIC 


First  page  of  the  solution  as  given  in  the  first  edition  of  Cardan's  Ars  Magna, 
Nurnberg,    1545.    The  solution   was   slightly   expanded   in   the  second   edition, 

Basel,  1570 


DE    ARITHMETIC*       Life.     5c.  20 

in  quadratum  A  c  ter,eft  m:cV  reliquum  quod  ei  xquatur  eft  piigirui 
triplum  c  B  in  qdratum  A  B,cV  rriplum  A  c  in  qdratu  c  H,  &  fexcuphl 
A  B  nihil  faciunt.  Tanta  igitur  eft  diflferentia^x  comuni  animi  fcntcn* 
tia^pfius  cubi  A  c,i  cubo  B  c, quantum  eft  quod  cofiatur  ex  cubo  A  c, 
&  triplo  A  c  in  quadratum  c  B,&  triplo  c  B  in  quadratum  A  c  m:Ss.'  cu 
bo  B  c  m:&  fexcuplo  A  B,hoc  igitur  eft  2o,quia  differentia  cubi  A  c»a 
cubo  C  B,fuit  2o,quare  per  fecundum  fuppofitum  6l  captuli ,  pofira 
B  cmrcubus  A  B  xquabitur  cubo  A  c  >  &  tripfo  A  c  in  quadi  aitim  B  c, 
&cuboB  c  m:&  triplo  B  c  in  quadratum  A  £  m: cubus  igitur  A  0,01111 
fexcuplo  A  B,pcrcommuncm  animi  fcntcntiam,  cum  arqucrur  cubo 
A  c  &  triplo  A  c  in  quadratum  c  B,  cV  triplo  c  B  in  quadratum  A  B  in: 
ck  cubo  c  sm:cV  fexcuplo  A  B  ,  quariam  icquarur  20  ,  ur  probatum 
eft,acquabuntur  etiam  io,aim  igitur  cubus  ABcV  fexciiplum  A  B  a-* 
quentur  2o,ck cubus  G  H,cum  fexcuplo  G  H  arqticntur  2o,erit  ex  com 
muni  animi /ententia,cV  ex  dicT;is,m  $  5-*  p'cx:  51*  undecimi  clemenro* 
rum,G  Harqualis  A  B,igitur  G  H  eft  differentia  A  c  &  c  B  ,  funt  auttiTi 
A  C  ck  C  B,ucl  A  C  ck  c  K,numeri  feu  liniac continences  fuperficicni ,  ic- 
qualem  tertix parti  numeri  rcrum,quarumcubi  diiFea-unt  in  nuinero 
scquationis>^uarc  habebimus  regulam* 
RE  G  v  L  A. 

Deducito  tertiam  partem  numeri  rcrum  ad  cubum ,  cui  addes . 
quadratum  dimidrj  numeri  acquationis.ck  totius  accipc  radicem,  (cili 
cet  quadratam,quam  feminabis,unicj?  dimidium  numeri  quod  iam 
in  fcduxcras,adrjcies,ab  altera  dimidium  idem  mimics, liabebisc^Bi 
nomitim  cum  fua  Apotome,  indc  detracla  i^  cubica  Apotomtr  ex  RJ 
cubica  fui  Binomij>refiduu  quod  ex  hoc  rclinquiturjcft  rei  cftimatio. 
Exemplum. cubus  cV  ^pofltiones,  xquan* 
tur  20,ducito  2  ,  tertiam  partem  <? ,  ad  cu* 
bum,fit£>ducio  dimidium  numeri  infr, 
fit  i  oo,iunge  i  oo  cV  8, fit  1 08 ^ccipe  radi# 
cem  qu*c  ell  R2  I  oS,  cV  earn  gcminabis,alte 
riaddcs  io,dimidium  numeri,ab  altero  mi 
nues  tantundem,habebis  Binomiu  RZ  1 08 
p:io,ck  ApotomenRz  108  m.'io  ,  horiim 
accipe  RZ'*  cub**  cV  minuc  illam  quc  eft  Apo 
tomd^ab  ca  qux  eft  Binomi),  habebis  rci  xftimanoncm,  RI  v:  cub:  i# 
toS  p: f om:R: v: cubica Rt  loS m.io. 

Aliud^cubus  p:$  rebus xquetur  io,duc  I, tertiam  partcm  :,  ad 
cubum,fit  i  ,duc  c»dimidium  i  o,ad  quadratum ,fit  2c,iungc  2^  ex'  i , 

H    2  fiiinr 

CARDAN'S  SOLUTION  OF  THE  CUBIC 

Continuation  of  the  solution  as  given  on  page  462 .    For  the  meaning  of  the 
symbols  see  page  428 


464  THE  SOLUTION  OF  EQUATIONS 

With  respect  to  the  irreducible  case,  his  solution  of  the  type 

x3  +  bx  =  c  is 


and  if  b  is  negative  and  is  such  that  \c~  4-  ^y  63  is  also  negative, 
then  we  have  the  cube  root  of  a  complex  number,  thus  reaching 
an  expression  that  is  irreducible  even  though  all  three  values 
of  x  turn  out  to  be  real.1 

With  respect  to  the  number  of  roots  to  be  expected  in  the 
cubic,  he  gave2  the  equations  x3  4-  lox  =  6x2  +  4  with  roots 
2,  2  ±  Va  ;  x3  +  2ix  =  gx2-\- 5  with  roots  5,  2  ±  V3 ;  and  x*  +  2 6x 
=  i2X2  +  i2  with  roots  2,  5  ±V^9;  but  before  this  time  only 
two  roots  were  ever  found,3  negative  roots  being  generally 
rejected. 

As  to  the  question  of  symmetric  functions,  he  stated  that  the 
sum  of  the  roots  is  minus  the  coefficient  of  x2* 

Cardan's  solution,  with  part  of  his  explanation,  is  shown  in 
facsimile  on  pages  462  and  463.  In  the  solution  he  states  that 
the  root  of  the  equation  x3  +  6x  =  20  is 

x  =v  V 108  4-  io—VVio8  —  10. 

He  also  gave  thirteen  forms  of  the  cubic  which  have  positive 
roots,  these  having  already  been  given  by  Omar  Khayyam. 

^he  reality  of  the  roots  for  this  case  was  shown  by  Kastner  (1745)  and 
A.  C.  Clairaut  (1746).  As  an  example  of  the  irreducible  case,  in  the  equation 
Xs  —  63*  —  162  =  o  the  rule  gives 


x  =  "^81  +  30  V—  3  -f  "^81  —  30  V  —  3, 
which  we  cannot  reduce,  although  as  a  matter  of  fact  the  solution  is 

x  =  (-  3  +  2  V-  3)  +  (-  3  -  2  V-  3)  =  -  6. 

2  Cap.  XVIII,  Exemplum  quintum:  "Cubus  &  10  res,  aequatur  6  quadratis 
p:4  (1545  ed.,  fol.  39,  r.).   The  roots  are  "2  p :  IJs  2,  uel  2  m:  I$2,  potest  etiam 
esse  2."   The  folio  is  incorrectly  numbered  36. 

3  It  was  Euler  (1732)  who  gave  the  first  noteworthy  modern  discussion  of 
the  cubic,  insisting  on  the  recognition  of  all  three  roots  and  stating  how  these 
roots  were  found.    "De  formis   radicum   aequationum  cuiusque   ordinis   con- 
jectatio,"  in  Comment.  Petropol.  ad  annos  1732-1733,  printed  in  1738;  VI,  217. 

4".  .  .  uelut  in  quinto  exemplo,  2p:Ij2,  &2,  &2m:  I$2,  componunt  6, 
numerum  quadratorum,"  and  so  for  other  cases.  Tha_t  is,  in  the  case  of 
*3  +  10*  =  6*2  4-  4  the  sum  of  2  +  V'z,  2,  and  2  —^/2  is  6  (fol.  39,  v.). 


THE  CUBIC  EQUATION  465 

Nicolas  Petri  and  the  Cubic.  In  his  work  of  1567  Nicolas 
Petri  of  Deventer,  as  already  mentioned,  gave  some  attention  to 
the  cubic  equation.  This  is  found  in  a  subdivision  on  Cubicq 
Coss,1  in  which  he  gives  eight  cubic  equations  such  as 

x*  =  9^  +  28, 
23  x3  +  32  x  =  905!,     and     x3  =  $x2  +  $x  +  16, 

all  of  which  he  solves  by  Cardan's  method. 

In  the  same  year  that  Petri's  work  appeared,  Pedro  Nunez 
(to  take  the  form  of  his  name  used  in  the  treatise  here  mentioned) 
published  his  Libro  de  algebra  en  arithmetica  y  geometria  at 
Antwerp.2  In  this  work  he  considers  such  equations  as 

x3  +  3*  =  36     and     r3  +  gx  =  54, 

and  seeks  to  show  that  Tartaglia's  rule  is  not  practical  where 
one  root  is  easily  found  by  factoring.  He  shows  a  familiarity 
with  the  works  of  both  Tartaglia  and  Cardan. 

Vieta  generalizes  the  Work.  Although  Cardan  reduced  his 
particular  equations  to  forms  lacking  a  term  in  x2,  it  was  Vieta3 
who  began  with  the  general  form 

x*  +  px2  +  qx  +  r  =  o 

and  made  the  substitution  x  =  y  —  %py  thus  reducing  the  equa- 

tion to  the  form  .,  ,      , 

y  +3&y  =  2c. 

He  then  made  the  substitution 


which  led  to  the  form       6 

Z* 

a  sextic  which  he  solved  as  a  quadratic. 

1MVolgen  sommighe  exempelen  ghesolueert  deur  die  Cubicq  Coss." 
2H.  Bosnians,  "Sur  le  ^ibro  de  algebra'  de  Pedro  Nunez,"  Bibl.  Math., 
VIII  (3),  154.   The  original  name  is  Nunes.   See  Volume  I,  page  348. 

3  Opera  mathematica.  IV.  De  aequationum  recognitione  et  emendatione  libri 
duo,  Tract.  II,  cap.  vii  (Paris,  1615).  His  equation  is  stated  thus:  "Proponatur 
A  cubus  -f  B  piano  3  in  A,  aequari  Z  solido  2";  that  is,  A3  -f  3  BA  =  2  Z, 
or,  in  our  symbols,  y3  +  3  by  —  zc.  The  problem  as  worked  out  by  Vieta  is 
given  in  Matthiessen,  GnmdzUge,  p.  371. 


466  THE  SOLUTION  OF  EQUATIONS 

He  also  gave  two  or  three  other  solutions,  but  the  one  here 
shown  is  particularly  clear  and  simple.  In  his  work  in  equa- 
tions he  was  greatly  aided  by  his  new  symbolism  (p.  430). 

Hudde's  Contribution.  Although  Descartes  contributed  to  the 
solution  of  the  cubic  equation  by  his  convenient  symbolism  and 
by  his  work  on  equations  in  general,  he  made  no  specific  con- 
tribution of  importance.  The  next  writer  to  materially  simplify 
the  work  of  Vieta  was  Hudde  (c.  1658).  Taking  advantage  of 
Descartes's  symbolism,  he  brought  the  theory  of  the  cubic  equa- 
tion to  substantially  its  present  status.  He  is  also  the  first 
algebraist  who  unquestionably  recognized  that  a  letter  might 
stand  for  either  a  positive  or  a  negative  number.1 

His  method  of  solving  the  cubic  equation  is  to  begin  with 

,rs  =  qx  +  r    ' 

and  let  x  —  y  -f  z, 

so  that     /  -f  3  /-  +  3  y?  +?*=  qx  +  r. 

He  then  lets  /+^3=r 

and  3  zf  -f  3  ^y  =  qx, 

which  gives  y  =  \  q/z. 

Hence  /-r-^-^A3, 


and  so  z*=\r±  V}  r1-  ^  q*  =A 

and  /  =  \r  T  V|  ?^~V7-  B. 

Hence  x  -  ^~A  +  WB> 

which  satisfies  both  his  assumptions.2 

Equation  of  the  Fourth  Degree.  After  the  cubic  equation  had 
occupied  the  attention  of  Arab  scholars,  with  not  very  signifi- 
cant results,  the  biquadratic  equation  was  taken  up.  Abu'l- 
Faradsh3  completed  the  Fihrist  c.  987,  and  in  this  he  refers  to 

^nestrom,  in  Bibl.  Math.,  IV  (3),  pp.  208,  216. 

2 The  problem,  as  worked  out  by  Hudde,  is  given  in  Matthiessen,  Gmndzuge, 
P-  374- 

3  Abu'l-Faradsh  (Faraj)  Mohammed  ibn  Ishaq,  known  as  Ibn  Abi  Ya'qub  al- 
Nadim.  The  title  is  Kitdb  al-Fihrist  (Book  of  Lists) .  See  the  Abhandlungen,  VI,  i. 


THE  BIQUADRATIC  EQUATION  467 

the  following  problem  by  Abu'1-Wefa  (c.  980) :  "On  the  method 
of  finding  the  root  of  a  cube  and  of  a  fourth  power  and  of  ex- 
pressions composed  of  these  two  powers."1  The  last  means 
that  we  are  to  solve  the  equation  x4  +  px?  =  q.  The  equation 
could  have  been  solved  by  the  intersection  of  the  hyperbola 
y2  +  axy  +  £  ==  o  and  the  parabola  x2  -  y  =  o,  but  the  work  in 
which  Abu'l-Wefa's  problem  appeared  is  lost  and  we  do  not 
know  what  he  did  in  the  way  of  a  solution. 

Woepcke,  a  French  orientalist  (c.  1855),  has  called  attention 
to  an  anonymous  MS.  of  an  Arab  or  Persian  algebraist  in  which 
there  is  given  the  biquadratic  equation 

(100  —  x2}  (10  —  xY  =  8100. 

This  is  solved  by  taking  the  intersection  of  (10  —  x)y  =  90  and 
x2  +  y*  =  ioo,  but  there  is  no  evidence  that  the  author  was  con- 
cerned with  the  algebraic  theory.2 

It  may  therefore  be  said  that  the  Arabs  were  interested  in  the 
biquadratic  equation  only  as  they  were  in  the  cubic,  that  is,  from 
the  standpoint  of  the  intersection  of  two  conies. 

The  Italian  Algebraists  and  the  Biquadratic.  The  problem  of 
the  biquadratic  equation  was  laid  prominently  before  Italian 
mathematicians  by  Zuanne  de  Tonini  da  Coi?  who  in  1540  pro- 
posed the  problem,  "Divide  10  into  three  parts  such  that  they 
shall  be  in  continued  proportion  and  that  the  product  of  the  first 
two  shall  be  6."  He  gave  this  to  Cardan3  with  the  statement 
that  it  could  not  be  solved,  but  Cardan  denied  the  assertion,  al- 
though himself  unable  to  solve  it.  He  gave  it  to  Ferrari,  his 

1  Abhandlungen,  VI,  73,  note  253.  See  also  Matthiessen,  Grundzuge,  p.  543; 
F.  Woepcke,  Recherches  .  .  .  Constructions  giom.  par  About  Wafa,  p.  36,  8,  n.  2 
(Paris,  1855). 

2Woepcke's  translation  of  Omar  Khayyam,  Addition  D,  p.  115.  The  prob- 
lem was  to  construct  an  isosceles  trapezium  (trapezoid)  ABCD  such  thai 
AB  =  AD  =  BC  =  10,  and  the  area  is  90. 

8 Cardan  states  it  thus:  "Exemplum.  Fac  ex  10  tres  partes  proportionales 
ex  quarum  ductu  primae  in  secundam,  producantur  6.  Hanc  proponebat  Ioanne< 
Colla,  &  dicebat  solui  non  posse,  ego  uero  dicebam,  earn  posse  solui,  modurr 
tanie  ignorabam,  donee  Ferrarius  eum  inuenit."  Ars  Magna>  cap.  xxxix,  qvaestic 
v;  1545  ed.,  fol.  73,  v. 


468  TllE  SOLUTION  OF  EQUATIONS 

pupil  (Vol.  I,  p.  300),  and  the  latter,  although  then  a  mere  youth, 
succeeded1  where  the  master  had  failed. 

Ferrari's  method2  may  be  summarized  in  its  modern  form  as 
follows  :   Reduce  the  complete  equation 


4  =  o 

to  the  form  x*  +px*  +  qx  +  r=o 

and  thence  to  x*  +  2px*  +  /  =/^2  -qx-r  +/2, 

or  (J  +/)2  =/**  -  qx  +  /  -  r. 

Write  this  as 


Now  determine  y  so  that  the  second  member  shall  be  a  square. 
This  is  the  case  when 


which  requires  the  solution  of  a  cubic  in  y,  which  is  possible. 
The  solution  then  reduces  to  the  mere  finding  of  square  roots. 
This  method  soon  became  known  to  algebraists  through 
Cardan's  Ars  Magna  ,  and  in  1567  we  find  it  used  by  Nicolas 
Petri  in  the  work  already  mentioned.  Petri  solves  four  equa- 
tions, the  first  being 

x*  +  6  x*  =  6  x2  +  30  x  +  1  i. 


Of  this  he  gives  only  the  root  i+vX  neglecting  the  roots  i—  v'2, 
—  4  ±  V~5  because  they  are  negative. 

1The  proportion  is  -  :  x  =  x  :  -#8,  and  the  other  condition  is  that 


the  two  conditions  reducing  to  x4  +  6#2  +  36  =  6o#.  Ferrari's  method  makes 
this  depend  upon  the  solution  of  the  equation  y3  +  i$y2  +-$6y  =  450,  or,  as 
Cardan  (Ars  Magna,  fol.  74,  r.)  states  the  problem,  "i  cubum  p:  15  quadratis 
p:  36  positionibus  aequantur  450." 

2  Cardan,  Ars  Magna,  1545,  cap.  xxxix,  qvaestio  v,  fol.  73,  v.;  Bombelli,  Alge- 
bra, 1572,  p.  353  ;  Matthiessen,  Grundzuge,  p.  540.  Bombelli's  first  special  case  is 
"i  i  p.  20-1  eguale  a  21";  that  is,  x4  +  2ox  =  21. 


EQUATION  OF  THE  FIFTH  DEGREE  469 

Vieta  and  Descartes.  Vieta  (c.  1590)  was  the  first  algebraist 
after  Ferrari  to  make  any  noteworthy  advance  in  the  solution 
of  the  biquadratic.1  He  began  with  the  type  x*  +  2  gx2  4-  bx  =  c, 
wrote  it  as  x4  4-  2  gx2  =  c  -  bx,  added  g2  4-  \  y'2  4-  yx2  -h  gy  to 
both  sides,  and  then  made  the  right  side  a  square  after  the  man- 
ner of  Ferrari.  This  method  also  requires  the  solution  of  a 
cubic  resolvent. 

Descartes2  (1637)  next  took  up  the  question  and  succeeded 
in  effecting  a  simple  solution  of  problems  of  the  type 


a  method  considerably  improved  (1649)  by  his  commentator 
Van  Schooten.4  The  method  was  brought  into  its  modern  form 
by  Simpson  (i745).5 

Equation  of  the  Fifth  Degree.  Having  found  a  method  differ- 
ing from  that  of  Ferrari  for  reducing  the  solution  of  the  gen- 
eral biquadratic  equation  to  that  of  a  cubic  equation,  Euler 
had  the  idea  that  he  could  reduce  the  problem  of  the  quintic 
equation  to  that  of  solving  a  biquadratic,  and  Lagrange  made 
the  same  attempt.  The  failures  of  such  able  mathematicians 
led  to  the  belief  that  such  a  reduction  might  be  impossible. 
The  first  noteworthy  attempt  to  prove  that  an  equation  of 
the  fifth  degree  could  not  be  solved  by  algebraic  methods  is 
due  to  Ruffini  (1803,  i8o5),6  although  it  had  already  been 
considered  by  Gauss. 

The  modern  theory  of  equations  in  general  is  commonly  said 
to  date  from  Abel  and  Galois.  The  latter's  posthumous  (1846) 
memoir  on  the  subject  established  the  theory  in  a  satisfactory 
manner.  To  him  is  due  the  discovery  that  to  each  equation  there 


aequationum  recognitions  et  emendatione  libri  duo,  Tract.  II,  cap.  vi, 
prob.  iii  (Paris,  1615).   For  solution  see  Matthiessen,  Grundzuge,  p.  547. 
2  La  Geometric,  Lib.  Ill;  1649  ed.,  p.  79;  1683  ed-»  p.  71  ;  1705  ed.,  p.  109. 
8  For  examples  see  Matthiessen,  Grundmge,  p.  549. 

4  1649  ed.,  p.  244. 

5  For  the  various  improvements  see  Matthiessen,  Grundzuge,  p.  $45  seq. 

6  "Delia  insolubilita  delle  equazioni  algebraiche  generali  di  grado  superiore  al 
quarto,"  Mem.  Soc.  Hal.,  X  (1803),  XII  (1805). 

II 


470  THE  SOLUTION  OF  EQUATIONS 

corresponds  a  group  of  substitutions  (the  " group  of  the  equa- 
tion " )  in  which  are  reflected  its  essential  characteristics.  Galois's 
early  death  left  without  sufficient  demonstration  several  im- 
portant propositions,  a  gap  which  has  since  been  filled. 

Abel1  showed  that  the  roots  of  a  general  quintic  equa- 
tion cannot  be  expressed  in  terms  of  its  coefficients  by  means 
of  radicals. 

Lagrange  had  already  shown  that  the  solution  of  such  an 
equation  depends  upon  the  solution  of  a  sextic,  "Lagrange's 
resolvent  sextic/'  and  Malfatti  and  Vandermonde  had  investi- 
gated the  construction  of  resolvents. 

The  transformation  of  the  general  quintic  into  the  trinomial 
form  x5  +  ax  +  b  =  o  by  the  extraction  of  square  and  cube  roots 
only  was  first  shown  to  be  possible  by  Bring  (1786)  and  in- 
dependently by  Jerrard2  (1834).  Hermite  (1858)  actually 
effected  this  reduction  by  means  of  a  theorem  due  to  Tschirn- 
hausen,  the  work  being  done  in  connection  with  the  solution 
by  elliptic  functions.3 

Symmetric  Functions.  The  first  formulas  for  the  computation 
of  the  symmetric  functions  of  the  roots  of  an  equation  seem  to 
have  been  worked  out  by  Newton,  although  Girard  (1629)  had 
given,  without  proof,  a  formula  for  a  power  of  the  sum,  and 
Cardan  (1545)  had  made  a  slight  beginning  in  the  theory.  In 
the  i8th  century  Lagrange  (1768)  and  Waring  (1770,  1782) 
made  several  valuable  contributions  to  the  subject,  but  the 
first  tables,  reaching  to  the  tenth  degree,  appeared  in  1809  in 
the  Meyer-Hirsch  Aufgabensammlung.  In  Cauchy's  celebrated 
memoir  on  determinants  (1812)  the  subject  began  to  assume 
new  prominence,  and  both  he  and  Gauss  ( 1816)  made  numerous 
and  important  additions  to  the  theory.  It  is,  however,  since  the 
discoveries  by  Galois  that  the  subject  has  become  one  of  great 


sur  les  Equations  algebriques,  Christiania,  1824,  and  Crellc's 
Journal ',  1826. 

2R.  Harley,  "A  contribution  to  the  history  ...  of  the  general  equation  of 
the  fifth  degree  .  .  . ,"  Quarterly  Journal  of  Mathematics,  VI,  38. 

3  For  a  bibliography  of  much  value  in  the  study  of  the  history  of  equations 
see  G.  Loria,  in  Bibl.  Math.,  V  (2),  107. 


NUMERICAL  HIGHER  EQUATIONS  471 

significance.  Cayley  (1857)  gave  a  number  of  simple  rules  for 
the  degree  and  weight  of  symmetric  functions,  and  he  and 
Brioschi  simplified  the  computation  of  tables. 

Harriot's  Law  of  Signs.  The  law  which  asserts  that  the  equa- 
tion ^Y=o,  complete  or  incomplete,  can  have  no  more  real 
positive  roots  than  it  has  changes  of  sign,  and  no  more  real 
negative  roots  than  it  has  permanences  of  sign,  was  apparently 
known  to  Cardan1 ;  but  the  first  satisfactory  statements  relating 
to  the  matter  are  due  to  Harriot  (died  162 1)2  and  Descartes.3 

Numerical  Higher  Equations.  The  solution  of  the  numerical 
higher  equation  for  approximate  values  of  the  roots  begins,  so 
far  as  we  know,  in  China.  Indeed,  this  is  China's  particular 
contribution  to  mathematics,  and  in  this  respect  her  scholars 
were  preeminent  in  the  i3th  and  i4th  centuries.4  In  the  Nine 
Sections,  written  apparently  long  before  the  Christian  era,  there 
is  found  the  "celestial  element  method."5  This  was  a  method 
of  solving  numerical  higher  equations;  it  is  found  in  various 
early  Chinese  works,  reaching  its  highest  degree  of  perfection 
in  the  works  of  Ch'in  Kiu-shao  (1247).  Here  it  appears,  as 
already  stated,  in  a  form  substantially  equivalent  to  Horner's 
Method  (i8ig).6 

Fibonacci  on  Numerical  Equations.  The  first  noteworthy  work 
upon  numerical  higher  equations  done  in  Europe  is  due  to 
Fibonacci  (1225),  and  relates  to  the  case  of  the  cubic  equation 

1  Cantor,  Geschichte,  II  (2),  539;  Enestrom,  BiblMath.,  VII  (3),  293. 

2Artis  analyticae  praxis.  Ad  aequationes  Algebraicas  .  .  .  resolvendas,  Lon- 
don, 1631  (posthumous) ;  Matthiessen,  Grundziige,  26.  ed.,  pp.  18,  268. 

*La  Geometrie,  1637;  *649  ed.,  p.  78;  1705  ed.,  p.  108,  with  the  statement: 
"On  connoit  aussi  de  ceci  combien  il  peut  y  avoir  de  vrayes  racines,  &  combien 
de  fausses  en  chaque  Equation;  a  s^avoir,  il  y  en  peut  avoir  autant  de  vrayes 
que  les  signes  +  &  — •  s'y  trouvent  de  fois  etre  changez,  &  autant  de  fausses  qu'il 
s'y  trouve  de  fois  deux  signes  -f  ,  ou  deux  signes  —  qui  s'entresuivent."  The  law 
usually  bears  the  name  of  Descartes. 

4Y.  Mikami,  China,  25,  53,  76,  et  passim]  L.  Matthiessen,  "Zur  Algebra  der 
Chinesen,"  in  Zeitschrift  fur  Math,  und  Phys.,  XIX,  HI.  Abt.,  270.  For  doubts 
as  to  the  originality  of  this  work  and  as  to  the  authenticity  of  the  text  of  the 
Nine  Sections  see  G.  Loria,  "  Che  cosa  debbono  le  matematiche  ai  Cinesi,"  Bollet- 
tino  della  Mathesis,  XII  (1920),  63. 

5  T'ien-yuen-shu,  the  Japanese  tengen  jutsu. 

6 See  Volume  I,  page  270.   For  a  detailed  solution  see  Mikami,  China,  p.  76  seq. 


472  THE  SOLUTION  OF  EQUATIONS 

#3  -f  2x2  4-  lox  =  20,  already  mentioned.  His  method  of  at- 
tack was  substantially  as  follows : 

Since  x9  +  2  x1  +  i  o  x  =  20, 

we  have  i  o  (x  -h  TV  .r3  -f  j  -r2 )  =  20, 
or  .r  4-  -/o  Jl"8  +  5  -r~  =  2> 

so  that  -t'<  2. 

But  I  +2  +  10-13  <  20, 

and  so  .r  >  i . 

But  x  is  riot  fractional ;  for  if  x  =  a/6,  then 

£      io/73      5  If 

cannot  be  integral,  and  so  x  must  be  irrational. 

Further,  x  cannot  be  the  square  root  of  an  integer ;  for,  from 
the  given  equation,  20-2  _r2 

X  ==  ~  y 

and  if  x  were  equal  to  ^ a  we  should  have 

/-  _  20  —  2  a 
10  +  a 

which  is  impossible. 

Fibonacci  here  closes  his  analysis  and  simply  makes  a  state- 
ment which  we  may  express  in  modern  symbols  as 

x  =  i°  22'  f  42"'  33iv  4V  40vi, 

a  result  correct  to  ijvi;  that  is,  the  value  is  only  3-fT7RlUoFro 
too  large.  How  this  result  was  obtained  no  one  knows,  but  the 
fact  that  numerical  equations  of  this  kind  were  being  solved  in 
China  at  this  time,  and  that  intercourse  with  the  East  was 
possible,  leads  to  the  belief  that  Fibonacci  had  learned  of  the 
solution  in  his  travels,  had  contributed  what  he  could  to  the 
theory,  and  had  then  given  the  result  as  it  had  come  to  him. 

Vieta  and  Newton  contribute  to  the  Theory.  About  the  year 
1600  Vieta  suggested  that  a  particular  root  of  a  numerical  equa- 
tion could  be  found  by  a  process  similar  to  that  of  obtaining 


NUMERICAL  HIGHER  EQUATIONS  473 

a  root  of  a  number.  By  substituting  in  f(x)  a  known  approx- 
imate root  of  /(#)  =  n  he  was  able  to  find  the  next  figure  by 
division.1 

Newton  (1669)  simplified  this  method  of  Vieta's,  and  the 
plan  of  procedure  may  be  seen  in  his  solution  of  the  equation 
yj  —  2y  —  5  =  o.  He  first  found  by  inspection  that  2  <  y  <  3. 
He  then  let  2  +  p  =  y ; 

whence         f-2y-  5  =-  i  +  io/  +  6/+/8=  o, 

and  p  =  o.i,  approximately. 
Letting  o.i  4-  #  =  />,  we  have 

0.06 1  +  1 1.23  q  +  6.3  (f+  £3=  o  ; 

whence  #  =  —  0.0054,  approximately. 
Letting  —  0.0054  +  r  =  g,  we  have 

0.000541708  + 1 1. 16196?- +6. 3  ^=0; 

whence  r  =  —  0.00004854,  approximately. 

Similarly,  we  could  let  —  0.00004854  +  s  =  r,  and  proceed 
as  before.  We  could  then  reverse  the  process  and  find  p. 

In  this  way  he  finds2  the  approximate  value 

y=  2.0945  5 147. 

As  already  stated  (page  471),  in  1819  William  George  Horner 
carried  this  simplification  still  farther,  the  root  being  developed 
figure  by  figure.  The  process  terminates  if  the  root  is  commen- 
surable, and  it  may  be  carried  to  any  required  number  of  deci- 
mal places  if  it  is  incommensurable.3 

Fundamental  Theorem.  The  Italian  algebraists  of  the  i6th 
century  tacitly  assumed  that  every  rational  integral  equation 
has  a  root.  The  later  ones  of  that  century  were  also  aware  that 
a  quadratic  equation  has  two  roots,  a  cubic  equation  three  roots, 

iBurnside  and  Panton,  Theory  of  Equations,  4th  ed.,  I,  275.   Dublin,  1899. 

2"De  analysi  per  aequationes  numero  terminorum  infmitas,"  extract  of 
1669  in  the  Commercium  Epistolkum,  p.  76  (London,  1725).  Wallis  also  gave 
an  approximation  method  in  1685. 

3For  a  simple  presentation  see  Burnside  and  Panton,  loc.  cit.,  I,  227,  and 
consult  that  work  (I,  275)  for  further  information  on  the  subject. 


474  THE  SOLUTION  OF  EQUATIONS 

and  a  biquadratic  equation  four  roots.  The  first  writer  to 
assert  positively  that  every  such  equation  of  the  wth  degree  has 
n  roots  and  no  more  seems  to  have  been  Peter  Roth,  a  Niirn- 
berg  Rechenmeister,  in  his  Arithmetica  philosopkica  (Niirn- 
berg,  1 60S).1  The  law  was  next  set  forth  by  a  more  prominent 
algebraist,  Albert  Girard,  in  i62Q.2  It  was,  however,  more 
clearly  expressed  by  Descartes  (1637),  w^o  not  only  stated  the 
law  but  distinguished  between  real  and  imaginary  roots  and 
between  positive  and  negative  real  roots  in  making  the  total 
number.3  Rahn  (Rhonius),  also,  gave  a  clear  statement  of  the 
law  in  his  Teutschen  Algebra  (i659).4 

After  these  early  steps  the  statement  was  repeated  in  one  form 
or  another  by  various  later  writers,  including  Newton  (c.  1685) 
and  Maclaurin  (posthumous  publication,  1748).  D'Alembert 
attempted  a  proof  of  the  theorem  in  1746,  and  on  this  account 
the  proposition  is  often  called  d'Alembert's  Theorem.  Other 
attempts  were  made  to  prove  the  statement,  notably  by  Euler 
(1749)  and  Lagrange,  but  the  first  rigorous  demonstration  is 
due  to  Gauss  (1799,  with  a  simple  treatment  in  1849). 

Trigonometric  Solutions.  In  the  i6th  century  Vieta5  suggested 
(1591)  the  treatment  of  the  numerical  cubic  equation  by  trigo- 
nometry, and  Van  Schooten  later  elaborated  the  plan.  GirardG 
(1629)  was  one  of  the  first,  however,  to  attack  the  problem 
scientifically.  He  solved  the  equation  i  ©so  13  ©+12,  that 
is,  #3  =  130;  +  12,  by  the  help  of  the  identity 

cos  3  <£  =  4  cos8  </>  —  3  cos  <£. 

xln  modern  works  the  name  also  appears  as  Rothe.  See  Tropfke,  Geschichte, 
III  (2),  95,  with  a  quotation  from  the  original  work.  Roth  died  at  Niirnberg  in 
1617.  See  Volume  I,  page  421. 

2"Toutes  les  equations  d'algebre  resolvent  autant  de  solutions,  que  la  denomi- 
nation de  la  plus  haute  quantit6  le  demonstre."  Invention  nouvelle  en  I'algebre, 
Amsterdam,  1629;  quoted  in  Tropfke,  Geschichte,  III  (2),  95,  to  which  refer  for 
further  details. 

3"Au  reste  tant  les  vrayes  racines  que  les  fausses  ne  sont  pas  toujours  r&lles, 
mais  quelquefois  seulement  imaginaires."  La  Geometrie  (1705  ed.),  p.  117. 

4  English  translation,  London,  1668.    See  Volume  I,  page  412. 

5  See  Van  Schooten's  edition  of  his  Opera,  p.  362  (Leyden,  1646). 

6  Invention  nouvelle  en  I'algebre,  Amsterdam,  1629.   On  the  primitive  Arab 
method  see  Matthiessen,  Grundziige,  p.  894 ;  on  Girard,  see  ibid.,  p.  896. 


TRIGONOMETRIC  SOLUTIONS  475 

A  single  solution  of  a  quadratic  equation  by  trigonometric 
methods  will  show  the  later  development  of  the  subject.1 

Fischer's  Solution.   Let 

x1 — px  +  q  =  o.  p*^<\q 

Then  let  .r^/cos2^ 

and  ;r2=/sin'2<£. 

Then  ^  +  *a =/  (cos2  0  +  sin2  <£)  =/ 

and  xjc^  — /2  (cos  </>  sin  <£)2 

=  I /sin2  20. 
The  angle  $  can  now  be  found  from  the  relation 

sin  2  c/>  =  2  V«^. 

For  example,  given  the  equation 
x*  -  937062  jr  +  198474  =  O, 
we  find  20=71°  5/44.6", 

whence  </>  =  35°  5$' 52.3"  ; 

and  hence  ^=61.3607 

and  *a  =  32.3454. 

Such  methods  have  been  extensively  used  with  the  cubic  and 
biquadratic  equations.2 

9.  DETERMINANTS 

Among  the  Chinese.  The  Chinese  method  of  representing  the 
coefficients  of  the  unknowns  of  several  linear  equations  by 
means  of  rods  on  a  calculating  board  naturally  led  to  the  dis- 
covery of  simple  methods  of  elimination.  The  arrangement  of 
the  rods  was  precisely  that  of  the  numbers  in  a  determinant. 
The  Chinese,  therefore,  early  developed  the  idea  of  subtracting 
columns  and  rows  as  in  the  simplification  of  a  determinant.3 

irThis  is  due  to  Fischer,  Die  Auflosung  der  quadratischen  und  kubischen 
Gleichungen  durch  Anwendung  der  goniometrischen  Functionen,  Elberfeld,  1856 
See  Matthiessen,  Grundzuge,  p.  885,  and  consult  this  work  for  a  detailed  history 
of  the  subject.  2For  a  list  of  writers  see  Matthiessen,  Grundzuge,  p.  888  seq. 

i,  China,  pp.  30,  93. 


476  DETERMINANTS 

Among  the  Japanese.  It  was  not  until  Chinese  science  had 
secured  a  firm  footing  in  Japan,  and  Japanese  scholars  had  be- 
gun to  show  their  powers,  that  the  idea  of  determinants  began 
to  assume  definite  form.  Seki  Kowa,  the  greatest  of  the  Japanese 
mathematicians  of  the  iyth  century ,  is  known  to  have  written  a 
work  called  the  Kai  Fukudai  no  Ho  in  1683.  In  this  he  showed 
that  he  had  the  idea  of  determinants  and  of  their  expansion.  It 
is  strange,  however,  that  he  used  the  device  only  in  eliminating 
a  quantity  from  two  equations  and  not  directly  in  the  solution 
of  a  set  of  simultaneous  linear  equations.1 

Determinants  in  Europe.  So  far  as  Western  civilization  is  con- 
cerned, the  theory  of  determinants  may  be  said  to  have  begun 
with  Leibniz2  (1693),  who  considered  these  forms  solely  with 
reference  to  simultaneous  equations,  as  the  Chinese  had  al- 
ready done. 

It  was  Vandermonde  (1771)  who  first  recognized  determi- 
nants as  independent  functions.  To  him  is  due  the  first  con- 
nected exposition  of  the  theory,  and  he  may  be  called  its  formal 
founder.  Laplace  (1772)  gave  the  general  method  of  expanding 
a  determinant  in  terms  of  its  complementary  minors,  although 
Vandermonde  had  already  considered  a  special  case.  Immedi- 
ately following  the  publication  by  Laplace,  Lagrange  (1773) 
treated  of  determinants  of  the  second  and  third  orders  and  used 
them  for  other  purposes  than  the  solution  of  equations. 

The  next  considerable  step  in  advance  was  made  by  Gauss 
(1801).  He  used  determinants  in  his  theory  of  numbers,  in- 
troduced the  word  "determinant"3  (though  not  in  the  present 
signification,4  but  rather  as  applied  to  the  discriminant  of  a 
quantic),  suggested  the  notion  of  reciprocal  determinants,  and 
came  very  near  the  multiplication  theorem. 

*T.  Hayashi,  "The  Fukudai  and  Determinants  in  Japanese  Mathematics,"  in 
the  Proc.  of  the  Tokyo  Math,  Soc.,  V  (2),  257;  Mikami,  Isis,  II,  9. 

2  Sir  Thomas  Muir,  Theory  of  Determinants  in  the  Historical  Order  of  De- 
velopment (4  vols.,  London,  1890,  1911,  1919;  2d  ed.,  1906,  1911,  1920,  1923), 
which  consult  on  the  whole  question;  M.  Lecat,  Histoire  de  la  theorie  des  Deter- 
minants a  plusieurs  dimensions,  Ghent,  1911.  3Laplace  had  used  "resultant." 

4"Numerum  bb  —  ac,  cuius  indole  proprietates  formae  (a,  b,  c)  imprimis 
pendere  in  sequentibus  docebimus,  determinantem  huius  uocabimus." 


CONCEPT  OF  RATIO  477 

The  next  great  contributor  was  Jacques-Philippe-Marie 
Binet,1  who  formally  stated  (1812)  the  theorem  relating  to  the 
product  of  two  matrices  of  m  columns  and  n  rows,  which  for  the 
special  case  of  m  —  n  reduces  to  the  multiplication  theorem. 

On  the  same  day  (November  30,  1812)  that  he  presented  his 
paper  to  the  Academic,  Cauchy  presented  one  on  the  same  sub- 
ject. In  this  paper  he  used  the  word  "  determinant"  in  its 
present  sense,  summarized  and  simplified  what  was  then  known 
on  the  subject,  improved  the  notation,  and  gave  the  multiplica- 
tion theorem  with  a  proof  more  satisfactory  than  Binet's.  He 
may  be  said  to  have  begun  the  theory  of  determinants  as  a 
distinct  branch  of  mathematics. 

Aside  from  Cauchy,  the  greatest  contributor  to  the  theory 
was  Carl  Gustav  Jacob  Jacobi.2  With  him  the  word  "determi- 
nant" received  its  final  acceptance.  He  early  used  the  functional 
determinant  which  Sylvester  has  called  the  Jacobian,  and  in  his 
famous  memoirs  in  Crelle's  Journal  for  1841  he  considered 
these  forms  as  well  as  that  class  of  alternating  functions  which 
Sylvester  has  called  alternants. 

About  the  time  of  Jacobi's  closing  memoirs  Sylvester  (1839) 
and  Cay  ley  began  their  great  work  in  this  field.  It  is  impossible 
to  summarize  this  work  briefly,  but  it  introduced  the  most 
important  phase  of  the  recent  development  of  the  theory. 

10.   RATIO,  PROPORTION,  AND  THE  RULE  OF  THREE 

Nature  of  the  Topics.  It  is  rather  profitless  to  speculate  as  to 
the  domain  in  which  the  concept  of  ratio  first  appeared.  The 
idea  that  one  tribe  is  twice  as  large  as  another  and  the  idea 
that  one  leather  strap  is  only  half  as  long  as  another  both  in- 
volve the  notion  of  ratio  ;  both  are  such  as  would  develop  early 
in  the  history  of  the  race,  and  yet  one  has  to  do  with  ratio  of 
numbers  and  the  other  with  ratio  of  geometric  magnitudes.  In- 
deed, when  we  come  to  the  Greek  writers  we  find  Nicomachus 
including  ratio  in  his  arithmetic,  Eudoxus  in  his  geometry,  and 


at  Rennes,  February  2,  1786;  died  in  Paris,  May  12,  1856. 
2  See  Volume  I,  page  506. 


478      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

Theon  of  Smyrna  in  his  chapter  on  music.1  Still  later,  Oriental 
merchants  found  that  they  could  easily  secure  results  to  certain 
numerical  problems  by  a  device  which,  in  the  course  of  time, 
became  known  as  the  Rule  of  Three,  and  so  this  topic  found 
place  in  commercial  arithmetics,  although  fundamentally  it  is 
an  application  of  proportion.  Since  ratio,  proportion,  and  vari- 
ation are  now  considered  as  topics  of  algebra,  however,  it  is 
appropriate  to  treat  of  these  subjects,  as  well  as  the  Rule  of 
Three,  in  the  present  chapter.2 

Technical  Terms.  The  word  "ratio"  as  commonly  used  in 
school,  while  sanctioned  by  ancient  usage,3  has  never  been  a 
favorite  outside  the  mathematical  classroom.  It  is  a  Latin  word4 
and  was  commonly  used  in  the  arithmetic  of  the  Middle  Ages 
to  mean  computation.  To  represent  the  idea  which  we  express 
by  the  symbols  a :  b  the  medieval  Latin  writers  generally  used 
the  word  proportioj  not  the  word  ratio ;  while  for  the  idea  of 
an  equality  of  ratio,  which  we  express  by  the  symbols  a\b~c:dy 
they  used  the  word  proportionalitas.5  That  these  terms  were 
thoroughly  grounded  in  the  vernacular  is  seen  today  in  the 
common  use  of  such  expressions  as  "divide  this  in  the  propor- 
tion of  2  to  3,"  and  "your  proportion  of  the  expense,"  and  in 

1P.  Tannery,  "Du  role  de  la  musique  grecque  dans  le  developpement  de  la 
mathematique  pure,"  BibL  Math.,  Ill  (3),  161. 

2  It  may  be  said  that  medieval  writers  looked  upon  ratio  and  proportion  as  a 
branch  of  mathematics  quite  distinct  from  geometry  and  arithmetic.    See  also 
S.  Gunther,  Geschichte  der  Mathematik,  p.  180  (Leipzig,  1908)  ;  hereafter  referred 
to  as  Gunther,  Geschichte. 

3  For  a  discussion  of  the  terms  \6yos  (lo'gos},  ratio,  and  proportio,  see  Heath, 
Euclid,  Vol.  II,  pp.  116-129.   See  also  Boethius,  ed.  Friedlein,  p.  3  et  passim. 

4  From  the  verb  reri,  to  think  or  estimate ;  past  participle,  ratus.   Hence  ratio 
meant  reckoning,  calculation,  relation,  reason. 

5  Thus  Boethiu&  (£.  510) :   "  Proportionalitas  est  duarum  vel  plurium  propor- 
tionum  similis  ha&tudo,"  ed.  Friedlein,  p.  137;  Jordanus  Nemorarius  (c.  1225): 
"Proportionalitas  est  si'litudo  ^pportionu"  (1496  ed.,  Lib.  2). 

In  the  Biblioteca  Laurenziana  at  Florence  is  a  MS.  (Codex  S.  Marco  Florent. 
184)  of  Campanus  (0.1260)  De  proportione  et  proportionalitate  with  the  inscrip- 
tion Tractatus  Campani  de  proportione  et  proporcionabilitate.  Included  with  it 
is  a  MS.  of  al-Misri  (£.900),  Epistola  Ameti  filii  Joseph  de  proportione  et  pro- 
portionalitate. See  BibL  Math.,  IV  (3),  241;  II  (2),  7-  The  title  of  Pacioli's 
work  affords  another  example:  Siima  de  Arithmetics  Geometria  Proportioni  & 
Proportionality  Venice,  1494. 


TECHNICAL  TERMS  479 

the  occasional  use  of  an  expression  like  "the  proportionality  of 
the  cost  is  the  same  as  that  of  the  amount." 

That  the  word  " proportion"  was  commonly  used  in  medieval 
and  Renaissance  times  to  mean  ratio  is  seen  in  most  mathe- 
matical works  of  those  periods.1  It  was  so  used  by  the  Ameri- 
can Greenwood  (1729)*  and  has  by  no  means  died  out  in 
our  language.3 

The  use  of  proportio  for  ratio  was  not  universal  in  the  early 
days  of  printing,  however,  for  various  writers  used  both  terms 
as  we  use  them  today.4 

General  Types  of  Ratio.  From  the  time  of  the  Greeks  to  the 
iyth  century  the  writers  on  theoretical  arithmetic  employed  a 
set  of  terms  and  ideas  in  connection  with  ratio  that  seem  to 
mathematicians  of  the  present  time  unnecessarily  complicated. 
A  few  of  these  have  survived  in  our  algebra,  most  of  them  have 
disappeared,  and  all  of  them  had,  under  ancient  conditions, 
good  reasons  for  being.  Of  those  which  are  still  found  in  some 
of  our  textbooks  there  may  be  mentioned  three  general  types  of 
ratio  of  integers:  namely,  a  ratio  of  equality,5  like  a :  a ;  a  ratio 

1Thus  Campanus  (c.i26o):  "Proportio  est  duarum  quantitatum  eiusdem 
generis  ad  inuicem  habitudo"  (Codex  S.  Marco  Florent.  184);  Jordanus 
Nemorarius  (c.  1225):  "Proportio  est  dual/  quatitatum  eiusdem  generis  vnius 
ad  alteram  certa  in  quatitate  relatio"  (1496  ed.,  Lib.  2) ;  Leonardo  of  Cremona: 
"la  proporcion  del  diametro  a  la  circonferentia"  (original  MS.  in  Mr.  Plimpton's 
library;  see  Kara  Arithmetic^  p.  474) ;  Chuquet  (1484):  "Proporcion  cest  labi- 
tude  qui  est  entre  deux  nobres  quant  est  compare  (lung)  a  laultre"  (the  Marre 
MS.  in  the  author's  possession,  used  by  Boncompagni,  Bullettino,  XIII,  621) ; 
Rudolff  (1526):  "Die  proporcion  oder  schickligkeit  der  ersten  gegg  der  andern" 
(1534  ed.,  fol.  Eviij).  Barrow  (1670)  used  the  expression  in  his  lectures  on 
geometry,  and  most  other  writers  of  the  period  did  the  same. 

2"  ...  the  Proportion  that  each  Figure  bears  to  its  neighbouring  Figure" 
(p.  So). 

3 E.g.,  Alison  and  Clark,  Arithmetic,  chap,  xxi  (Edinburgh,  1903). 

4 Thus  Fine  (Fhueus) :  "Ratio  igitur  ...  est  duarum  quantitatum  eiusdem 
speciei  adinuicem  comparataru  habitudo.  .  .  .  Proportio  est,  contingens  inter  com- 
paratas  adinuicem  quantitates  rationum  similitude"  (Protomathesis,  IS3O-I532; 
1555  ed.,  fols.  38  and  57).  See  also  L.  L.  Jackson,  The  Educational  Significance 
of  Sixteenth  Century  Arithmetic,  p.  119  (New  York,  1906). 

5 The  "aequalitatis  proportio"  of  the  Latin  writers;  e.g.,  Scheubel  (iS45)-  In 
a  numerical  ratio  like  a:  6,  both  a  and  b  were  generally  considered  integral  unless 
the  contrary  was  stated,  but  the  incommensurable  ratio  of  lines  was  recognized 
by  the  Pythagoreans  and  by  all  subsequent  geometers. 


480      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

of  greater  inequality,  like  a :  b  when  a>b\  and  a  ratio  of  lesser 
inequality,1  like  a:  b  when  a<b.  Of  the  last  two  there  were 
recognized  various  subspecies,  such  as  multiple  ratio,2  like  ma :  a, 
where  m  is  integral;  superparticular  ratio,  like  (m  +  i)  :  m 
(which  had  several  types,  such  as  sesquialteran,  as  in  the  case 
of  3  :  2,  sesquitertian,  as  in  the  case  of  4 : 3,  and  so  on) ;  super- 
partient,  like  (m  +  n) :  n,  where  m  >  n  >  i,  as  in  the  cases  of 
5  : 3  and  7:3;  multiple  superparticular,  like  (mn  +  i) :  m,  as  in 
the  cases  of  7 : 3  and  15:7;  and  multiple  superpartient,  like 
(mn  -\-  k)  :  m,  where  m>  k  >i;  as  in  the  cases  of  14:3  and 
19:5.  These  terms  were  capable  of  a  large  number  of  combi- 
nations and  were  essentially,  from  our  present  point  of  view,  the 
result  of  an  effort  to  develop  a  science  of  general  fractions  at  a 
time  when  the  world  had  no  good  symbolism  for  the  purpose. 
With  the  introduction  of  our  common  notation  and  the  invention 
of  a  good  algebraic  symbolism  such  terms  disappeared.3  This 
disappearance  was  hastened  by  such  writers  as  Stifel  (1546), 
who  spoke  out  plainly  against  their  further  use,  although  his 
own  acts  were  not  always  consistent  with  this  statement.4  When 
T9g  had  to  be  called  "suboctupla  subsuperquadripartiens  nonas" 
by  a  writer  as  late  as  i6oo,5  it  was  evident  that  the  ancient  usage 
must  give  way,  and  that  ratios  must  be  considered  with  respect 
to  the  modern  fractional  notation  instead  of  depending  upon 
the  ancient  Roman  method. 

1  Boethius,  ed.  Friedlein,  p.  238. 

2  Boethius  (ed.  Friedlein,  p.  46)  speaks  of  such  relations:  "Maioris  vero  in- 
aequalitatis  .V.  sunt  partes.   Est  enim  una,  quae  vocatur  multiplex,  alia  super- 
particularis,  ..."  It  should  be  observed  that  the  Greeks  did  not  consider  ratios 
as  numbers  in  the  way  that  we  do ;  that  is,  they  did  not  consider  6 : 3  as  identical 
with  2  but  as  a  relation  of  6  to  3,  this  relation  being  a  multiple  ratio. 

3 For  a  full  treatment  of  the  subject  see  Pacioli,  Siima,  1494  ed.,  fol.  72. 

4 In  his  Rechenbuch  (p.  35)  he  says:  "Von  den  Proportzen.  Zvm  ersten|des 
Boetius  |  Stapulensis  |  Apianus  |  Christoff  Rudolff  |  vnd  andere  gelerte  Leuth  |  die 
proportiones  leren  mit  solche  worten  Multiplex  Duplex  |  Tripla  |  Superparticu- 
laris  |  Sesquialtera  .  .  .  vnnd  der  gleichen  wort  ohn  zal  |  ist  wol  recht  vnd  nutzlich 
gelert  Aber  das  man  ein  Teutschen  Leser  |  dem  die  Lateinisch  sprach  ist  vnbekant  | 
will  man  solchen  worten  beladen  |  das  ist  ohn  not  vnnd  ohn  nutz."  In  his  Coss 
Christoff s  Rudolff s  (chap.  12),  however,  he  gives  "die  fiinferley  proportionirte 
zalen,"  the  multiple,  superparticular,  superpartient,  multiplexsuperparticular,  and 
the  multiplexsuperpartient. 

5  Van  der  Schuere,  1624  ed.,  fol.  193. 


GREEK  IDEAS  481 

Other  Greek  Ideas  of  Ratio.  Certain  other  Greek  ideas  have 
come  down  to  us  and  still  find  a  place  in  our  algebras.  For  ex- 
ample, we  speak  of  or :  b'2  as  the  duplicate  ratio1  of  a  to  6,  al- 
though to  double  a :  b  would  give  2  a :  b.  To  the  Greek,  however, 
the  ratio  a± :  an  was  considered  as  compounded  or  composed2  of 
the  ratios  0i :  cz2, 02 : 03  •  •  • ,  <z,t  - 1 : a» !  and  since  a2 :  b2  is  similarly 
compounded  of  a2 :  ab  and  ab :  b2  ot  of  a :  b  and  a :  6,  it  was 
called3  the  duplicate  of  a:  b. 

In  like  manner  we  have  from  the  Greeks  the  idea  of  ratios 
compounded  by  addition  when  as  a  matter  of  fact  they  have 
been,  according  to  our  conception,  multiplied.4 

In  the  Middle  Ages  the  distinction  between  ratios  and  frac- 
tions, or  ratio  and  division,  became  less  marked,  and  in  the 
Renaissance  period  it  almost  disappeared  except  in  cases  of 
incommensurability.5  An  illustration  of  this  fact  is  seen  in  the 
way  in  which  Leibniz  speaks  of  " ratios  or  fractions."6 

Proportion  as  Series.  The  early  writers  often  used  proportio 
to  designate  a  series,7  and  this  usage  is  found  as  late  as  the  i8th 
century.8  The  most  common  use  of  the  word,  however,  limited 
it  to  four  terms.  Thus  the  early  writers  spoke  of  an  arithmetic 
proportion,  meaning  b  —  a  =  d  —  c,  as  in  2,  3,4,  5 ;  and  of  a 
geometric  proportion,  meaning  a :  b  =  c :  d,  as  in  2,  4,  5,  10.  To 

1  Euclid's  SnrAcurtai/  (diplasi'on),  but  commonly  given  by  other  Greek  writers 
as  5nr\<£0-ios  \6yos (dipla'sios  lo'gos).  See  Heath,  Euclid,  Vol.  II,  p.  133. 

2Heath,  Euclid,  Vol.  II,  p.  133. 

8  Euclid,  Elements,  VI,  def.  5,  apparently  an  interpolation.  See  Heath,  Euclid, 
Vol.  II,  p.  189. 

4See  Heath,  Euclid,  Vol.  II,  p.  168.  Similarly,  Scheubel  (1545,  Tract.  II) 
speaks  "  de  proportionum  Additione  .  .  .  siue  ut  alij  Compositione,"  saying  that 
the  ratios  9  : 4  and  5  : 3  "componunt"  45  : 12. 

5 A  modern  Arab  arithmetic,  published  at  Beirut  in  1859,  remarks:  "This 
division  is  called  by  the  Magrebiner  [West  Arabs]  'the  denomination/  but  the 
Persians  call  it  al-nisbe  [the  ratio]."  H.  Suter,  Bibl  Math.,  II  (3),  17. 

°"  .  .  .  aut  in  rationibus  vel  Fractionibus."    Letter  to  Oldenburg,  1673. 

7Thus  Pacioli  (1494):  "che  tu  prendi  )i  numeri  ...  i.  2.  3.  4.  5.  6.  7.  8.  9.  10 
.  .  .  hauerai  la  prima  specie  de  la  proportion"  (Suma,  fol.  72,  r.). 

•'When  Vitalis  (Geronimo  Vitale)  published  his  Lexicon  Mathematicvm 
(Rome,  1690),  the  usage  was  apparently  unsettled.  He  says:  "Igitur  Proportio 
Arithmetica  est  cum  tres,  vel  plures  numeri  per  eandem  differentiam  progrediun- 
tur;  vt  4.  7.  10.  13.  16.  19.  22.  &  sic  procedendo  in  infmitum"  (p.  681) ;  but  he 
also  uses  proportio  in  the  modern  sense  (p.  732). 


482      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

these  proportions  the  Greeks1  added  the  harmonic  proportion 

i  _£__£_  i 

bade 

as  where  a=^j  b  =  $,  c  —  \,  and  rf=|.  These  three  names  are 
now  applied  to  series.  To  them  the  Greeks  added  seven  others, 
all  of  which  go  back  at  least  to  Eudoxus  (c.  370  B.C.).2 
The  Renaissance  writers  began  to  exclude  several  of  these,3 
and  at  the  present  time  we  have  only  the  geometric  proportion 
left,  and  so  the  adjective  has  been  dropped  and  we  speak  of 
proportion  alone. 

Types  of  Proportion.  The  fact  that  geometric  proportion  has 
survived,  in  algebra  at  least,  is  largely  due  to  Euclid's  influence, 
since  algebraically  a  proportion  is  nothing  more  than  a  frac- 
tional equation  and  might  be  treated  as  such.  Especially  is  this 
true^of  such  expressions  as  "by  alternation,"  "by  inversion," 
"by  composition,"  "by  division,"  and  "by  composition  and 
division,"  three  of  which  are  now  misnomers  in  the  modern  use 
of  the  words.  They  come  to  us  directly  from  the  Arabs,4  who 
received  them  from  Greek  sources.5  There  were  also  various 
other  types  of  geometric  proportion  besides  the  one  commonly 
seen  in  textbooks,6  but  most  of  these  types  are  now  forgotten, 

1On  the  theory  in  Euclid,  see  Heath,  Euclid,  Vol.  I,  p.  137;  Vol.  II,  pp.  113, 
119,  292. 

2Boethius  and  certain  of  his  predecessors  gave  all  ten  forms,  those  besides  the 
three  above  mentioned  being  as  follows : 

a:c  =  b—  c:a  —  c,  a:c  ~  a  —  c:b  —  c, 

b:c  =;  b  —  c:a  —  b,  a:c  =  a  —  c:a  —  b, 

a:b  =b  —  c:a  —  b,  b:c  =  a—  c:b  —  c, 

b:c  =  a—  c:a—  b. 

See  his  Arithmetica,  ed.  Friedlein,  p.  137  seq.;  Gunther,  Math.  Unterrichts, 
p.  85 ;  Cantor,  Geschichte,  I,  chap,  xi,  for  the  earlier  knowledge  of  these  forms. 

3Thus  Ramus:  "Genera  aute  proportionis  duo  tantum  instituimus,  quia  haec 
sola  simplicia  &  mathematica  sunt.  Nicomachus  fecit  decem.  Jordanus  addidit 
undecimam."  Scholarvm  Matkematicarvm,  Libri  vnvs  et  triginta,  p.  134 
(Basel,  1569). 

4 E.g.,  see  al-Karkhi  (c.  1020),  the  K&ft  ftl  Hisdb,  ed.  Hochheim,  II,  15. 

5Heath,  Euclid,  Vol.  I,  p.  137;  Vol.  II,  pp.  113,  "9,  133,  168,  189,  292. 

°£.g.,Scheubel  (1545) :  "  Sex  sunt  species  proportionalitatis,permuta,conuersa, 
coniuncta,  disiuncta,  euersa,  &  aequa"  (Tractatus  II). 


TYPES  OF  PROPORTION  483 

although  "  continued  proportion,"  but  with  a  change  in  the  older 
meaning,1  has  survived  both  in  algebra  and  in  geometry. 

Terms  Used  in  Proportion.  The  terms  "  means,"  "antecedent," 
and  "consequent"  are  due  to  the  Latin  translators  pf  Euclid.2 
There  have  been  attempts  at  changing  them,  as  when  the  ante- 
cedent was  called  a  leader  and  the  consequent  a  comrade,3 
but  without  success.  It  would  be  quite  as  simple  to  speak  of 
them  as  the  first,  second,  third,  and  fourth  terms.4 

Rule  of  Three.  The  mercantile  Rule  of  Three  seems  to  have 
originated  among  the  Hindus.  It  was  called  by  this  name  by 
Brahmagupta  (c.  628)  and  Bhaskara5  (c.  1150),  and  the  name 
is  also  found  among  the  Arab  and  medieval  Latin  writers. 

Brahmagupta  and  Mahavira  state  the  Rule.  Brahmagupta 
stated  the  rule  as  follows:  "In  the  Rule  of  Three,  Argument, 
Fruit,  and  Requisition  are  the  names  of  the  terms.  The  first 
and  last  terms  must  be  similar.  Requisition  multiplied  by  Fruit, 
and  divided  by  Argument,  is  the  Produce."6  Mahavira  (c.  850) 
gave  it  in  substantially  the  same  form;  thus :  "Phala  multiplied 
by  Icchd  and  divided  by  Pramdna  becomes  the  answer,  when  the 
Icchd  and  Pramdna  are  similar."7 

For  example:  "A  lame  man  walks  over  \  of  a  krosa  [32,000 
feet]  together  with  ^  [thereof]  in  y|  days.  Say  what  [dis- 
tance] he  [goes  over]  in  3^  years  [at  this  rate]," — a  very  good 
illustration  of  the  absurdity  of  the  Oriental  problem. 

*E.g.,  Fine  (Finaeus,  1530)  defines  a  proportio  continua  as  one  like  "8/4/2/1 : 
ut  enim  8  ad  4,  sic  4  ad  2,  atq$  2  ad  i"  (De  Arithmetica  Practica,  1555  ed., 
fol.  59),  and  a  proportio  disjuncta  as  (to  use  his  symbolism)  one  like  8/4,  6/3. 

2 Euclid  used  /ie<r6rr;Tes  (mesot'etes,  means),  yyotineva  (hegou'mena,  leading 
[terms],  antecedents),  and  cirbneva  (hepom'ena,  following  [terms],  consequents), 
but  he  had  no  need  for  "extremes."  See  Elements,  VII,  19. 

3  Thus  Scheubel  (1545):  "...  alter  antecedens  uel  dux,  alter  consequens  uel 
comes  appellatur."  The  use  of  dux  (duke,  leader)  comes  from  Euclid's  term. 

4 E.g.,  see  Clavius  (1583),  Epitome  Arithmeticae  Practicae,  chap.  xvii. 

5 One  of  the  scholiasts  of  Bhaskara  called  it  the  Trairdsica,  the  "three  rule." 
See  Colebrooke's  translation,  pp.  33,  283.  On  the  general  subject  see  Taylor's 
translation,  p.  41.  6  Colebrooke's  translation,  p.  283. 

7  Ganita-Sdra-Sangraha,  p.  86.  The  phala  is  the  given  quantity  corresponding 
to  what  is  to  be  found ;  the  pramdna  was  a  measure  of  length,  but  in  proportion 
it  is  the  term  corresponding  to  the  icchd ;  the  icchd  is  the  third  term  in  the  rule. 


484      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

Bhaskara  (c.  1150)  gave  the  rule  in  much  the  same  form  as 
that  used  by  Brahmagupta,  thus:  "The  first  and  last  terms, 
which  are  the  argument  and  requisition,  must  be  of  like  denomi- 
nation ;  the  fruit,  which  is  of  a  different  species,  stands  between 
them;  and  that,  being  multiplied  by  the  demand  [that  is,  the 
requisition]  and  divided  by  the  first  term,  gives  the  fruit  of 
the  demand  [that  is,  the  Produce]."1 

As  an  example,  Bhaskara  gave  the  following :  "Two  palas  and 
a  half  of  saffron  are  purchased  for  three  sevenths  of  a  niska : 
How  many  will  be  purchased  for  nine  niskas?" 

His  work  appears2  as  2  c  a 

721 

In  our  symbolism  it  might  be  represented  as  |  N.  2 1  P.  9  N. 
It  is  thus  seen  that  the  idea  of  equal  ratios  is  not  present,  as 
would  be  the  case  if  we  should  write  x  :  2\  —  9  :  |- .  Proportion 
was  thus  concealed  in  the  form  of  an  arbitrary  rule,  and  the 
fundamental  connection  between  the  two  did  not  attract  much 
notice  until,  in  the  Renaissance  period,  mathematicians  began 
to  give  some  attention  to  commercial  arithmetic.  One  of  the 
first  to  appreciate  this  connection  was  Widman3  (1489),  and 
in  this  he  was  followed  by  such  writers  as  Tonstall4  (1522), 
Gemma  Frisius5  (1540),  and  Trenchant6  (1566). 

Names  for  the  Rule  of  Three.  Recorde  (c.  1542)  calls  the 
Rule  of  Three  "the  rule  of  Proportions,  whiche  for  his  excellency 
is  called  the  Golden  rule/'7  although  his  later  editors  called  it 
by  the  more  common  name.8  Its  relation  to  algebra  was  first 
strongly  emphasized  by  Stifel9  (1553-1554)- 

1  Colebrooke's  translation,  p.  33.  2See  Taylor's  translation,  p.  41. 

3"Sy  ist  auch  recht  genat  regula  proportionQ/wa  in  d>  regel  werde  erkat  vn 
erfunde  alle  pportiones"  (1508  ed.,  fol.  50).  4De  Arte  Supputandi,  Lib.  III. 

5  He  calls  his  chapter  "De  Regvla  Proportion vm,  siue  Trium  Numerorum" 
(1575  ed.,  fol.  C6). 

6 "La  regie  de  troys,  qui  est  la  regie  des  proportions  ou  proportionaux "  (1578 
ed.,  p.  120).  7  1558  ed.,  fol.  M4.  8 •£•£•>  John  Mellis,  1594  ed.,  p.  449. 

9 "  Gar  wunderbarlich  wickeln  vnd  verkniipffen  sich  zusammen  die  Detri  vnd 
die  Coss  also  dass  die  Coss  im  grund  auch  wol  mochte  genennt  werden  die  Detri. 
.  .  .  Vn  steckt  also  die  gantz  Coss  in  der  Regel  Detri/widervmb  steckt  die  Gantz 
Detri  in  der  Coss."  See  the  Abhandlungen,  I,  86. 


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RULE   OF   THREE   IN   THE   SIXTEENTH   CENTURY 

From  an  Italian  MS.  of  1545.   Notice  the  arrangement  of  terms;  also  the  early 

per  cent  sign  as  given  at  the  end  of  the  fifth  line.    From  a  manuscript  in 

Mr.  Plimpton's  library 


486      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

When  the  rule  appeared  in  the  West,  it  bore  the  common 
Oriental  name,1  although  the  Hindu  names  for  the  special  terms 
were  discarded.  So  highly  prized  was  it  among  merchants,  how- 
ever, that  it  was  often  called  the  Golden  Rule,2  a  name  ap- 
parently in  special  favor  with  the  better  mathematical  writers.3 
Hodder,  the  popular  English  arithmetician  of  the  i  yth  century, 
justifies  this  by  saying:  "The  Rule  of  Three  is  commonly 
called,  The  Golden  Rule ;  and  indeed  it  might  be  so  termed  ;  for 
as  Gold  transcends  all  other  Mettals,  so  doth  this  Rule  all  others 
in  Arithmetick."4  The  term  continued  in  use  in  England  until 
the  end  of  the  i8th  century  at  least,5  perhaps  being  abandoned 
because  of  its  use  in  the  Church. 

1Thus  Pacioli  (1494)  calls  it  the  "regula  trium  rerum  la  regola  ditta  dl  .3. 
ouer  de  le  .3.  cose,"  and  "la  regola  del  .3.";  the  Treviso  arithmetic  (1478),  "La 
regula  de  le  tre  cose";  Pellos  (1492),  the  "Regula  de  tres  causas."  Chuquet 
(1484)  remarks,  "La  rigle  de  troys  est  de  grant  recomandacion.  ...  La  rigle  de 
troys  est  ansi  appellee  pource  quelle  Requiert  tousiours  troys  nombres";  Gram- 
mateus  (1518)  speaks  of  the  "Regula  de  tre  in  gantze"  and  "in  prikhen,"  and 
Rudolff  (1526)  of  "Die  Regel  de  Tri,"  a  term  often  abridged  by  German  writers 
into  "Regeldetri,"  as  in  the  work  of  Licht  (1500).  Klos,  the  Polish  writer  of 
1538,  also  calls  it  the  "ReguTa  detri." 

2  Thus  Petzensteiner  (1483) :  "  Vns  habn  die  meyster  der  freyn  kunst  vo  d'  zal 
ein  regel  gefunde  die  heist  gulden  regel  Dauo  das  sie  so  kospar  vnd  nucz  ist.  .  .  . 
Sie  wirdet  auch  genenet  regula  d'  tre  nach  welsischer  [i.e.,  Italian]  zungen.  .  .  . 
Sie  hat  auch  vile  ader  name";  and  Kobel  (Augsburg  edition  of  1518)  speaks  of 
"Die  Gulden  Regel  (die  von  den  Walen  de  Try  genant  wirt)."  In  Latin  it  often 
appears  as  regula  aurea.  The  Swedish  savant,  Peder  Mansson,  writing  in  Rome 
f.  1515,  speaks  of  the  rule  "quam  nonnulli  regulam  auream  dixere:  Itali  vero 
regulam  de  tri"  (see  Bibl.  Math.,  II  (2),  17). 

The  French  writers  used  the  same  expression.  Thus  Peletier  (1549):  "La 
Reigle  de  Trois  .  .  .  vulgairement  ansi  dite.  .  .  .  Les  ancients  Pont  appellee  la 
Reigle  d'or :  parce  que  1'invention  en  est  tres  ingenieuse,  &  1'usage  d'icelle  infini " 
(1607  ed.,  p.  68). 

3Thus  Ciacchi  (1675):  "La  Regola  del  Tre  cosi  chiamata  da'  Practici  vulga- 
ri,  e  da'  Mattematici  regola  d'  oro,  o  pure  delle  quattro  proporzioni  e  principalis- 
sima,  ed  apporta  vn'  inestimabile  benefizio,  ed  vna  gran  comodita  a'  Mercanti." 
Regole  generate  dr  abbaco,  p.  121  (Florence,  1675). 

4  See  the  tenth  edition,  1672,  p.  87.  This  simile  was  a  common  one  with 
writers;  thus  Petzensteiner  (1483):  "als  golt  vbertrifft  alle  ander  metall." 

Vitalis  (Geronimo  Vitale),  in  his  Lexicon  Mathematicvm,  p.  748  (Rome,  1690), 
says:  "Quare  merito  Aurea  appellata  est;  namque  plus  auro  valet:  &  non 
Arithmeticis  modo,  Geometris  .  .  .  necessaria  est ;  sed  &  vniuerso  hominum  generi, 
in  commercijs  ineundis,  .  .  ." 

B  E.g.,  the  1771  edition  of  Ward's  Young  Mathematician's  Guide  (p.  85)  speaks 
"of  Proportion  Disjunct;  commonly  called  the  Golden  Rule." 


19 


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RULE    OF    THREE,    OR    THE    GOLDEN    RULE,    IN    VERSE 
From  Lautenschlager's  arithmetic  (1598) 


488      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

The  Merchants'  Rule.  Its  commercial  uses  also  gave  to  the 
Rule  of  Three  the  name  of  the  Merchants'  Key  or  Merchants' 
Rule,1  and  no  rule  in  arithmetic  received  such  elaborate  praise 
as  this  one  which  is  now  practically  discarded  as  a  business  aid.2 

A  Rule  without  Reason.  The  rule  was  usually  stated  with  no 
explanation;  thus  Digges  (1572)  merely  remarked,  "Worke  by 
the  Rule  ensueing.  .  .  .  Multiplie  the  last  number  by  the 
seconde,  and  diuide  the  Product  by  the  first  number/7  and  sim- 
ilar statements  were  made  by  most  other  early  arithmeticians, 
occasionally  in  verse.3 

The  arrangement  of  the  terms  was  the  same  as  in  the  early 
Hindu  works,  the  first  and  third  being  alike.  As  Digges  ex- 
pressed it,  "In  the  placing  of  the  three  numbers  this  must  be 
observed,  that  the  first  and  third  be  of  one  Denomination.7'4 
This  custom  shows  how  completely  these  writers  failed  to  recog- 
nize the  relation  between  lie  Rule  of  Three  and  proportion. 

iThus  Licht  (1500,  fol.  9)  says:  "Regula  Mercatorum.  [Q]uam  detri.  quia 
de  trib\  per  apocopam  appellamus.  Regula  Aurea  docte  ac  perite  ab  omnV 
appellari  videt  deberi";  Clavius  (1583)  calls  it  the  "clavis  mercatorum  " ;  Peletier 
(1549)  says,  "Mesmes,  aucuns  Font  nominee  la  Clef  des  Marchands"  (1607  ed., 
p.  68)  ;  Wentsel  (i.?99)  speaks  of  the  "Regvla  avrea  mercatoria/  regvla  de  tri, 
Regvla  van  dryen/  regie  a  trois,  &c.";  and  Lautenschlager,  in  his  arithmetic  in 
rime  (1598),  has,  as  shown  in  the  facsimile  on  page  487, 

REGULA  DE  TRI  |  ODER  GULDEN  REGUL. 
REgnla  de  Tri  MERCATORum  | 
Genannt  aurea  proportionum. 

Of  the  various  other  names,  Schlussrechnung  has  continued  among  the  Ger- 
mans. See  R.  Just,  Kaufmannisches  Rechnen,  I.  Teil,  p.  75  (Leipzig,  1901). 

2  A  few  of  the  hundreds  of  eulogies  given  to  the  rule  are  as  follows :  Gemma 
Frisius  (1540),  "Res  breuis  est  &  facilis,  vsus  immensus,  cum  in  vsu  communi, 
turn  in  Geometria  ac  reliquis  artibus  Mathematicis "  (1563  ed.,  fol.  18) ;  Adam 
Riese  (1522),  "1st  die  furnamcste  vnder  alle  Regeln"  (1550  ed.  of  Rechenung, 
fols.  13,  59) ;  Clavius  (1583;  Opera,  1611,  II,  35),  "Primo  autem  loco  sese  offert 
regula  ilia  nunquam  satis  laudata,  quae  ob  immensam  vtilitatem,  Aurea  dici  solet, 
vel  regula  Proportionum,  propterea  quod  in  quatuor  numeris  proportionalibus, 
quorum  priores  tres  noti  stint  .  .  .  vnde  &  regula  trium  apud  vulgus  appellata 
est,"  showing  that  he,  like  Stifel,  recognized  the  relation  to  proportion;  Van  der 
Schuere  (1600),  "Den  Regel  van  Drien,  die  van  vele  ten  rechte  den  Gulden  Regel 
genaemd  word/  overmits  zyne  weerdige  behulpzaemheyd  in  alle  andere  Regelen" 
(1634  ed->  fol.  12). 

3 E.g.,  Lautenschlager  (1598)  and  Sfortunati  (i534J  XS45  ed.,  fol.  33), 

4 1579  ed.,  p.  29. 


ARRANGEMENT  OF  TERMS 


489 


Once  set  down,  it  was  the  custom  to  connect  the  terms  by 
curved  lines,  as  in  the  following  cases : 1 

2  4 


2 

3 

4 
6 

The  Arabs,  however,  used  such  forms  as 


and 


to  indicate  a  proportion,  paying aio  attention  to  the  labels  on 
the  numbers.2 

Arrangement  of  Terms.  The  rule  being  purely  arbitrary,  it 
became  necessary  to  have  this  arrangement  in  the  proper  order, 
and  the  early  printed  books  gave  much  attention  to  it.  Borghi 
(1484)  gave  a  whole  chapter  to  this  point,3  and  Glareanus 
(1538)  arranged4  an  elaborate  scheme  to  help  the  student.5 
Later  writers,  however,  recognized  that  if  the  rule  were  to  be 
considered  as  a  case  of  proportion,  it  would  be  necessary  to  re- 
arrange the  terms  so  that  the  first  two  should  be  alike.  Thus  in 
place  of  a  form  like 

12  yards — 205. 6  yards, 

xThe  first  of  these  is  from  a  i7th  century  MS.  in  the  author's  library;  the 
second  is  from  Werner's  Rechenbuch,  1561,  fol.  62. 

2E.  Wiedemann,  "Uber  die  Wage  des  Wechselns  von  Chazini  und  liber  die 
Lehre  von  den  Proportionen  nach  al-Biruni,"  Sitzungsberichte  der  Physik.-med. 
Societal  zu  Erlangen,  48.  u.  49.  Bd.,  p.  4. 

3"flComo  le  tre  cose  contenute  in  delta  regola  sono  ordinate,  e  quale  debbi 
esser  prima,  e  qual  seconda,  e  qual  terza"  (1540  ed.,  fol.  36). 
4 1543  ed.,  fol.  20. 
6 He  arranged  his  rule  thus: 

Sinistra  Medius  locus  Dextera 

Res  empta  Numer9  pretij  Numer*  qstionis 

Diuisor  Multiplicadus  Multiplicator 

2  36  7 


490      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

as  given  in  Hodder's  i7th  century  work,1  we  find  Blassiere 
(1769)-  and  others  of  the  i8th  century  using  such  forms  as 

Ellen  Ellen  Guld.  Guld. 

3  :  36        =         4  :  x 

In  the  old  Rule  of  Three  the  result  was  naturally  written  at 
the  right,  and  for  this  reason  the  unknown  quantity  came  to  be 
placed  at  the  right  in  the  commercial  problems  in  proportion.3 

Inverse  Proportion.  Of  the  various  special  forms  of  the  Rule 
of  Three  the  one  known  as  inverse  proportion  is  the  simplest.  It 
results  when  the  ratio  of  two  quantities  is  equal  to  the  reciprocal 
of  the  ratio  of  two  quantities  which  seem  to  correspond  to  them. 
Bhaskara  (c.  1150)  gives  an  illustration :  "Bullocks  which  have 
plowed  four  seasons  cost  four  niskas :  what  will  bullocks  which 
have  plowed  twelve  seasons  cost  ?  "4 

This  rule  went  by  such  names  as  the  inverse,  converse,  or 
everse  Rule  of  Three.5  Recorde  (c.  1542)  used  a  name  that 
became  quite  common  in  England,  remarking:  "But  there  is 
a  contrarye  ordre  as  thys :  That  the  greater  the  thyrde  summe  is 
aboue  the  fyrste,  the  lesser  the  fourthe  summe  is  beneth  the 
second,  and  this  rule  you  maye  call  the  Backer  rule."6 

1  Arithmetick,  loth  ed.  (1672),  p.  89.  2i7QO  ed.,  p.  149. 

3 Thus  Rabbi  ben  Ezra  (c.  1140)  wrote 

47  63 
7    o 

for  47:7  —  63:*,  the  o  standing  for  the  unknown.  In  the  translation  of  certain 
Arab  works  the  unknown  is  placed  first,  as  hi  AT: 84  =  12:7,  because  the  Arabs 
wrote  from  right  to  left. 

4  Taylor's  Lilawati,  p.  42,  with  spelling  as  given  by  Taylor.  The  result  is 
written  "niska  i,  and  fraction^." 

6Thus  Kobel  (1514),  "Die  Regel  de  Tri  verkert/  im  Latein  Regula  conuersa 
genant"  (1549  ed.,  fol.  68);  Gemma  Frisius  (1540),  "Regvla  Trivm  Euersa"; 
Albert  (1534),  "Regula  Detri  Conuersa ";  Thierf elder  ( 1587),"  Regula  Conversa/ 
oder  vmbekehrte  Regel  de  Try";  Van  der  Schuere  (1600),  "Verkeerden  Reghel 
van  Drien";  Digges  (1572),  "The  Rule  of  Proportion  Inuersed." 

°i558  ed.,  fol.  M  6.  In  some  later  editions  it  is  called  "the  Backer  or  Reverse 
Rule"  (1646  ed.,  p.  180).  See  also  Baker  (1568;  1580  ed.,  fol.  46).  In  French  it 
appeared  as  "La  regie  de  troys  rebourse"  (Trenchant,  1566;  1578  ed.,  p.  155), 
"La  Reigle  de  Trois  Reuerse  ou  Rebourse"  (Peletier,  1549;  1607  ed.,  p.  74),  and 
"La  Regie  Arebourse"  (Coutereels,  Dutch  and  French  work,  1631  ed.,  p.  204). 
It  was  occasionally  called  the  bastard  Rule  of  Three.  Thus  Santa-Cruz  (1594): 
"  Exemplo  de  la  regla  de  tres  bastarda." 


INVERSE  AND  COMPOUND  PROPORTION         491 

In  this  rule  it  was  the  custom  to  leave  the  terms  as  in  simple 
proportion  but  to  change  the  directions  for  solving.  Hylles 
( I S92 )  gives  the  rule  as  follows  : 

The  Golden  rule  backward  or  conuerst, 

Placeth  the  termes  as  dooth  the  rule  direct : 

But  then  it  foldes1  the  first  two  termes  rehearst, 
Diuiding  the  product  got  by  that  effect. 

Not  by  the  first,  but  onely  by  the  third, 

So  is  the  product  the  fourth  at  a  word.2 

Compound  Proportion.  What  has  been  called,  for  a  century 
or  two,  by  the  name  of  compound  proportion  originally  went  by 
such  names  as  the  Rule  of  Five3  when  five  quantities  were  in- 
volved, the  Rule  of  Seven  if  seven  quantities  were  used,  and  so 
on.  Bhaskara,  for  example,  gives  rules  of  five,  seven,  nine,  and 
eleven.4  Peletier  (1549)  speaks  of  Ptolemy  as  the  inventor  of 
the  Rule  of  Six,  referring,  however,  to  the  proposition  in 
geometry  relating  to  a  transversal  of  three  sides  of  a  triangle.5 
The  names  beyond  that  for  five  were  rarely  used;6  indeed, 
all  beyond  that  for  three  were  more  commonly  called  by  the 
general  name  of  Double  Rule  of  Three,7  Compound  Rule  of 

TAn  interesting  translation  of  plicare  as  found  in  multiplicare  (to  manifold). 

2Arithmeticke,  1600  ed.,  fol.  135. 

3It  appears  in  Bhaskara  (c.  1150)  as  pancha-rdsica  (five-rule).  See  Taylor's 
translation,  p.  43;  but  the  spellings  in  Colebrooke's  translation,  p.  37,  are  here 
followed.  In  Europe  we  find  such  names  as  "Regvle  de  cinqve  parte"  (Ortega, 
1512;  1515  ed.) ;  "Regula  Quinque/oder  zwyfache  Regel  de  Try"  (Thierfelder, 
1587);  "Regel  von  fiinffen"  (Rudolff,  1526);  "Die  Regel  von  fiinff  zalen" 
(Kobel,  1514;  1531  ed.) ;  "  Regvla  dvplex  Auch  Regula  Quinque  genant  ...  die 
zwyfache  Regel  .  .  .  von  funff  Zalen"  (Suevus,  1593);  "den  Zaamengestelden 
Reegel  van  Drien.  Anders  Genaamd  Den  Reegel  van  Vyven"  (Blassiere,  1769) ; 
and  "  Den  Regel  van  Vyven,  of  anders  genaamt  den  Dobbelen  Regel  van  Drien " 
(Bartjens,  1792  ed.). 

4  Pancha-rdsica,  sapta-rdsica,  nava-rasica,  and  tcddasa-rdsica,  as  spelled  by 
Colebrooke. 

r'"La  Reigle  de  six  Quantites  a  est6  inuentee  par  Ptolemee"  (1607  ed.,  p.  220). 
On  this  see  also  Cardan's  Practica  (1539)  with  its  "Caput  46.  de  regula  6.  quati- 
tatum." 

"Thus  Ciacchi  (1675)  says:  "Non  e  molto  vsitata  da'  practici  Arrimetici  la 
regola  del  sette." 

7  As  by  Recorde  (c.  1542).  The  name  also  appears  under  such  forms  as 
"  Regula  duplex  "  ( Gemma Frisius,  1540)  and  "  la  regie  double  "( Trenchant,  1566) . 


492      RATIO,  PROPORTION,  THE  RULE  OF  THREE 

Three,1  Conjoint  Rule,2  Plural  Proportion,3  and,  finally,  Com- 
pound Proportion,  a  term  which  became  quite  general  in  the 
1 8th  century. 

Artificial  Nature  of  the  Problems.  The  artificial  nature  of  the 
problems  in  compound  proportion  has  been  evident  from  the 
beginning.  Thus  Mahavira  (c.  850)  gives  this  case:  "He  who 
obtains  20  gems  in  return  for  100  gold  pieces  of  16  varnas — 
what  [will  he  obtain]  in  return  for  288  gold  pieces  of  10 
varnas?"4  And  Bhaskara  (c.  1150)  has  this  type:  "If  eight 
best,  variegated  silk  scarfs  measuring  three  cubits  in  breadth 
and  eight  in  length  cost  a  hundred  [nishcas] ;  say  quickly,  mer- 
chant, if  thou  understand  trade,  what  a  like  scarf  three  and  a 
half  cubits  long  and  half  a  cubit  wide  will  cost.775 

Practice.  There  appears  in  certain  English  arithmetics  of  the 
present  Hay  a  chapter  on  Practice,  a  kind  of  modification  of  the 
Rule  of  Three.  In  the  manuscripts  of  the  later  Middle  Ages  and 
in  the  early  printed  books  of  Italy  the  word  is  used  to  mean 
simply  commercial  arithmetic  in  general,  whence  possibly  the 
origin  of  our  phrase  "commercial  practice"  today.6  When 
northern  writers  of  the  i6th  century  spoke  of  Italian  practice 
they  usually  referred  merely  to  Italian  commercial  arithmetic 
in  general.7  In  the  i?th  century  the  Dutch  writers  generally 
used  the  term  Practica  (Practijcke)  to  mean  that  part  of  arith- 

1"Regula  Trivm  Composita"  (Clavius,  1583) ;  "The  Golden  Rule  compound" 
(Recorde,  c.  1542;  1646  ed.,  p.  195)  ;  "Den  menichvuldigen  Regel"  (Coutereels, 
1631  ed.,  p.  213);  " Gecomposeerden  dubbelden  Regel"  (Houck,  1676,  with  a 
distinction  between  this  and  the  mere  "Dubbelden  Regel"  and  "Regel  van  Con- 
juncte"). 

2"Den  Versamelden  Reghel"  or  "La  Regie  Conjoincte"  (Coutereels,  1631  ed., 
p.  219). 

s"  The  Double  Rule  of  Three  .  .  ,  Under  this  Rule  is  comprehended  divers 
Rules  of  Plural  Proportion"  (Hodder,  Arithmetick,  loth  ed.,  1672,  p.  131). 

4  English  translation,  p.  91. 

5 Colebrooke's  translation,  p.  37. 

<>So  Tartaglia  (1556)  has  a  chapter  "Delia  Prattica  Fiorentina"  for  Florentine 
commercial  arithmetic,  and  Riese  (1522)  has  "Rechenung  ...  die  practica 
genandt." 

7So  Stifel  (1544) :  "Praxis  Italica  Praxis  ilia  quam  ab  Italis  ad  nos  devolutam 
esse  arbitramur."  Even  as  late  as  1714  there  was  a  chapter  in  Starcken's  arithme- 
tic on  "Die  Italianische  Practica/  oder  Kurtze  Handels-Rechnung." 


PRACTICE  493 

metic  relating  to  financial  problems/  and  they  also  used  the  ex- 
pression "Italian  Practice,"  as  in  the  work  by  Wentsel  (1599) 
of  which  part  of  the  title  is  here  shown  in  facsimile.  The  term 
"  Welsh  practice"  had  a  similar  meaning*,  the  word  "  Welsh" 
("Welsch")  signifying  foreign.2  This  expression  is  often  found 
in  the  German  arithmetics  of  the  i6th  century. 

I7FONDAMENT 


hamfcge  ^aettjefe  /  mtntf&atiers 

noottomiricijtte  ttucfccn  tern  fcenltegliel  ban 


2UICJ& 

MARTINVM    VVE'N  C  E  SL  A  VM, 
AQVISGRANENSEML 

WELSH    OR   ITALIAN    PRACTICE 
Part  of  the  title-page  of  the  arithmetic  of  Wentsel  (Wenceslaus) 


In  England  practice  came  to  mean  that  part  of  commercial 
arithmetic  in  which  short  processes  were  used.  Baker  (1568) 
mentions  it  in  these  words  : 

Some  there  be,  whiche  doe  call  these  rules  of  practise,  breefe  rules  : 
for  that  by  them  many  questions  may  bee  done  with  quicker  expedi- 
tion, then  by  the  Rule  of  three.  There  be  others  which  call  them  the 
small  multiplication,  for  because  that  the  product  is  alwayes  lesse 
in  quantity,  than  the  number  whiche  is  to  be  multiplyed. 

'Thus  Eversdyck's  edition  of  Coutereels  (1658  ed.,  p.  QI),  Stockmans  (1676 
ed.,  p.  173),  Van  der  Schuere  (1600,  fol.  49),  and  Mots  (1640,  fol.  G  5). 

2  Modern  German  walsch,  foreign;  particularly  Gallic,  Roman,  Italian.  So  we 
find  Rudolff  (1526)  speaking  of  "Practica  oder  Wellisch  Rechnung"  and  Helm- 
reich  (1561)  having  a  chapter  on  "De  Welsche  Practica  oder  Rechnung."  Dutch 
writers  commonly  used  "foreign"  instead  of  "Welsch,"  and  so  we  often  find 
chapters  on  "  Buy  ten-lantsche  Rekeninghe." 


494  SERIES 

This  rather  indefinite  statement  gave  place  to  clearer  defi- 
nitions as  time  went  on,  and  Greenwood  (1729)  speaks  of 
practice  as  follows : 

THIS  Rule  is  a  contraction  or  rather  an  Improvement  of  the  Rule 
of  Three]  and  performs  all  those  Cases,  where  Unity  is  the  First 
Term ;  with  such  Expedition,  and  Ease,  that  it  is,  in  an  extraordinary 
manner,  fitted  to  the  Practice  of  Trade,  and  Merchandise ;  and  from 
thence  receives  its  Name. 

A  single  example  from  Tartaglia  (1556)  will  show  the  resem- 
blance of  the  Italian  solution  to  that  which  was  called  by 
American  arithmeticians  the  "unitary  method,"  especially  in  the 
1 9th  century: 

If  i  pound  of  silk  costs  9  lire  18  soldi ,  how  much  will  8  ounces  cost  ? T 
The  solution  is  substantially  as  follows : 

i  Ib.  costs  9  lire  18  soldi, 

4  oz.  cost  \  of  this,  or  3  lire  6  soldi, 

8  oz.  cost  twice  this,  or  6  lire  12  soldi. 

ii.  SERIES 

Kinds  of  Series.  Since  the  number  of  ways  in  which  we  may 
have  a  sequence  of  terms  developing  according  to  some  kind  of 
law  is  limitless,  like  the  number  of  laws  which  may  be  chosen, 
there  may  be  as  many  kinds  of  series  or  progressions 2  as  we  wish. 
The  number  to  which  any  serious  attention  has  been  paid  in 
the  development  of  mathematics,  however,  is  small.  The  arith- 
metic and  geometric  series  first  attracted  attention,  after  which 
the  Greeks  brought  into  prominence  the  harmonic  series.  These 
three  were  the  ones  chiefly  studied  by  the  ancients.  Boethius 
(c.  510)  tells  us  that  the  early  Greek  writers  knew  these  three, 
but  that  later  arithmeticians  had  suggested  three  others  which 
had  no  specific  names.3 

1 1  lb.  =  12  oz.,  i  lira  =  20  soldi. 

2  For  our  purposes  we  shall  not  distinguish  at  present  between  these  terms. 
3"Vocantur  aute  quarta:    quinta:    vel  sexta"    (Arithmetica,   ist  ed.,   1488, 
II,  cap.  41 ;  ed.  Friedlein,  p.  139) . 


MEDIEVAL  TREATMENT  495 

Occasionally  some  special  kind  of  series  is  mentioned,  as  when 
Stifel  speaks  of  the  "astronomical  progression"  i,  ^,  ^Vo, 
•  •  •  ,  r  one  of  the  few  instances  of  a  decreasing  series  in  the 
early  European  books. 

Most  of  the  Hindu  writers  used  only  two  elementary  series, 
but  Brahmagupta  (c.  628),  Mahavira  (c.  8so),2  and  Bhaskara 
(c.  1150)  all  considered  the  cases  of  the  sums  of  squares  and 
cubes.3  The  Arab4  and  Jewish5  writers  also  gave  some  atten- 
tion to  these  several  types. 

Medieval  Treatment  of  Series.  In  the  medieval  works  a  series 
was  generally  considered  as  ascending,  although  descending 
series  had  been  used  by  Ahmes,  Archimedes,  and  certain  Chi- 
nese writers  long  before  the  time  in  which  these  works  were 
written.0  The  same  custom  was  followed  by  the  early  Renais- 
sance writers.7 

Somewhat  better  known  than  the  classification  of  series  as 
arithmetic,  geometric,  and  harmonic,  at  least  before  the  lyth 
century,  was  the  classification  into  natural,  nonnatural,  con- 
tinuous, and  discontinuous,  these  terms  being  used  rather  loosely 

1  Arithmetica  Integra,  1544,  fol.  64,  the  name  being  astronomica  progress^.  It 
is  simply  the  natural  series  of  "astronomical  fractions." 

2  English  translation,  p.  170.    His  rule  for  the  sum  of  the  squares  is  sub- 
stantially 


3  Colebrooke's  translation,  p.  52.  Bhaskara  remarks:  "Former  authors  have 
stated  that  the  sum  of  the  cubes  of  the  terms  one,  &c.  is  equal  to  the  square  of 
the  summation";  that  is,  Sw3  =  (S«)2.  (Taylor's  translation,  p.  60.) 

4E.g.,  al-Hassar  (c.  ii?S?);  see  Bibl.  Math.,  II  (3),  32.  They  are  alsc 
given  by  al-Qalasadi  (c.  1475)  ;  see  Boncompagni's  Bullettino,  XIII,  277. 

6  E.g.,  Rabbi  ben  Ezra  (c.  1140),  in  the  Sefer  ha-Mispar,  Silberberg  transla- 
tion, p.  120. 

6  So  Fibonacci  (1202)  says:  "...  colligere  numeros  quotcumque  ascendente< 
ab  ipso  dato  numero  equaliter,  ut  per  ascensionem  unitatis,  uel  binarii,  uel  ter- 
narii  .  .  ."  (Liber  Abaci,  p.  166  (fol.  70,  r.)). 

7  Thus  Stifel,  in  his  edition  of  Rudolffs  Coss  (1553)  :    "Es  1st  aber  Progressic 
(eygentlich  zu  reden  nach  der  Arithmetica)  ein  ordnung  vieler  zalen  so  nacl: 
einander  auffsteygen  oder  absteygen  nach  eyner  rechten  richtigen  Regei"   (fol 
7,  v.).   So  Trenchant  (1566)  states  definitely  that  the  terms  must  increase,  anc 
Chuquet   (1484)    says:    "Progression  est  certaine  ordonnance  de  nombre  pai 
laquelle  le  premier  est  surmonte  du  second  dautant  que  le  second  est  surmont< 
du  tiers  et  p  sequement  les  ault's  se  plus  en  ya"  (fol.  20,  r.). 


496  SERIES 

by  early  writers.  For  example,  the  series  i,  2,  3,  .  .  .  was 
called  a  natural  series,1  from  which  we  have  the  expression 
"natural  series  of  numbers."2  A  discontinuous,  or  intercised,3 
progression  was  one  in  which  the  difference  was  not  unity.4 

Name  for  Series.  The  Greek  name  for  a  series,  as  used  first 
by  the  early  Pythagoreans,  was  aefco-t?  (ek'  thesis}?  literally 
a  selling  out,  and  the  name  for  a  term  of  the  series  was  opo? 
(hor'os)f  literally  a  boundary.  Boethius  (c.  510),  like  the  other 
Latin  writers^  used  the  word  progressio,7  and  this  was  generally 
the  custom  until  modern  times. 

The  Teutonic  writers  followed  their  usual  plan  of  avoiding  in- 
ternational names  based  upon  the  Latin,  and  so  we  find  various 
terms  used  by  the  Dutch8  and  German9  mathematicians. 

1Thus  Chuquet  (1484):  "Et  doit  on  sauoir  que  progression  se  fait  en 
plusieurs  et  diuerses  manieres.  Car  aulcunesfoiz  elle  comance  a  .1.  et  progredyst 
par  .1.  come  .1.  2.  3.  4.  £c.  tellc  est  appellee  par  les  anciens  progression 
naturelle  ou  continue  pgression  "  (fol.  20,  r.).  Similarly,  Pellos  (1492)  speaks  of 
"  Egression  natural"  (i,  2,  3,  .  .  .  ),  "  pgression  no  natural"  (i,  3,  5,  .  .  .  ), 
and  "^gression  ni  part  natural  £  ni  part  no  natural"  (8,  9,  10,  .  .  .  ). 

Van  der  Schuere  (1600),  however,  calls  any  series  like  i,  2,  3,  ...  or  i,  3,  5, 
...  a  "natuerlikke  overtredinghe  /oft  Aritmetische  Progressio,"  speaking  of  a 
geoinetric  series  as  "  onnatuerlikke  overtredinghe." 

«his  is  found  in  Stifel  (1544):  "naturalis  numerorum  Progressio,  est  Pro- 
gressio Arithmetica  progrediens  ab  unitate  per  binarium  ad  reliquos  numeros 
secundum  differentiam  unitatis.  ut 


.  1. 


4. 


He  also  speaks  "de  Progressione  naturali  numerorum  imparium,"  viz.,  i,  3, 
5,  7,  ...  (Arithmetica  Integra,  fols.  20,  21). 

BIntercissa  (Huswirt,  i$oi),int*cise  (Chuquet,  1484),  vnderschnitten  (Kobel, 


4"Alcune  comace  a  .1.  mais  el  progredist  par  aultre  nombre  que  .1.  coe  i.  3.  5. 
"£c.  ou  .1.  4.  7.  £c.  et  est  ceste  appellee  Int^cise  progression  ou  prog^ssion  dis- 
continuee"  (Chuquet  (1484),  fol.  20,  r.\  Boncompagni's  Bullettino,  XIII,  617). 

Santa-Cruz  (1594)  says  that  any  progression  w  comengando  de  la  vnidad 
dicha  continua."  5  A  word  also  meaning  exhibition  or  exposition. 

6  A  word  also  meaning  a  limit,  marking  stone,  rule,  standard,  or  boundary 
between  two  objects.  7  Ed.  Friedlein,  I,  pp.  9,  10,  et  passim. 

8  Dutch  writers  of  the  i6th  century  used  progressio  together  with  such  terms  as 
overtredinghe  (stepping  over)  and  opklimminge  (ascending;  literally,  upclimbing). 

9  Although  modern  writers  use  Reihe.   Kobel  (1514),  for  example,  says  :  "Die 
acht  species  ist  Progressio  zu  Latein/vnnd  ist  Furzelen  geteutscht,"  in  later  Ger- 
man Filrzdhlung.  The  terms  Aufsteigung,  Fortgehung,  Reihe,  and  Progression 
are  also  used. 


RELATION  TO  PROPORTION  497 

The  change  to  the  name  "series"  seems  to  have  been  due 
to  writers  of  the  lyth  century.  James  Gregory,  for  example, 
writing  in  1671,  speaks  of  "  infinite  serieses,"  and  it  was  in  con- 
nection with  infinite  sequences  that  it  was  at  first  used  by  the 
British  algebraists.  Even  as  late  as  the  1693  edition  of  his 
algebra,  however,  Wallis  used  the  expression  "infinite  progres- 
sions" for  infinite  series. 

Extent  of  Treatment.  Although  series  was  commonly  looked 
upon  as  one  of  the  fundamental  operations,1  it  was  rarely  ac- 
corded much  attention  in  the  early  printed  books.  Tzwivel 
(1505),  for  example,  gives  only  32  lines  to  both  arithmetic 
and  geometric  progressions,  including  all  definitions  and  rules ; 
while  Huswirt  (1501)  allows  only  one  page  and  Digges  (1.572) 
only  two  pages  to  the  subject. 

Nearly  all  the  early  writers  limited  the  work  to  finding  the 
sum  of  the  series,2  although  a  few  gave  a  rule  for  finding  the 
last  term  of  an  arithmetic  or  a  geometric  series.  With  these 
writers  there  was  no  attempt  to  justify  the  rule,  the  mere  state- 
ment sufficing.  It  was  only  through  the  influence  of  a  better 
algebraic  symbolism  in  the  1 7th  century  that  the  various  cases 
could  easily  be  discussed  and  the  development  of  rules  for  all 
these  cases  made  simple. 

Relation  to  Proportion.  The  ancient  writers  commonly  con- 
nected progression  with  proportion,  or  rather  with  proportion- 
ality, to  use  a  name  which,  as  already  stated,  was  at  one  time 
popular ;  and  they  applied  the  names  "arithmetic,"  "geometric," 
and  "harmonic"  to  each.  Some  of  the  early  printed  books  call 
attention  to  this  relation,  saying  that  a  proportion  is  merely  a 
progression  of  four  terms.3 

irThus  Pacioli  (1494) :  "la  sexta  e  penultla  specie  dilla  pratica  p?  arithca.  laqle 
e  chiamata  pgrcssioe"  (Suma,  fol.  37). 

2  Johann  Albert  (1534)  distinctly  states  that  this  is  the  sole  purpose  of  the 
work:   "Progredirn  leret/wie  man  viel  zaln  (welche  nach  naturlicher  ordnung 
oder  durch  gleiche  mittel/nach  einander  folgen)  in  eine  Summa/auffs  kurtzest  vnd 
behendest  bringen  sol"  (1561  ed.,  fol.  E  4).   See  also  Treutlein,  Abhandlungen, 
I,  60. 

3  Thus  Trenchant   (1566):   ".  .  .  car  Progression  n'est  qu'vne  continuation 
des  termes  d'vne  proportion"  (1578  ed.,  p.  274). 


4Q8  SERIES 

Arithmetic  Series.  The  first  definite  trace  that  we  have  of  an 
arithmetic  series  as  such  is  in  the  Ahmes  Papyrus  (c.  1550  B.C.), 
where  two  problems  are  given  involving  such  a  sequence.  The 
first1  of  the  problems  is  as  follows  :  "  Divide  100  loaves  among 
five  persons  in  such  a  way  that  the  number  of  loaves  which  the 
first  two  receive  shall  be  equal  to  one  seventh  of  the  number 
that  the  last  three  receive." 

The  solution  shows  that  an  arithmetic  progression  is  under- 
stood, in  which  n  =  5,  $6=  100,  and 

+  2d)^        (         |  ^ 


Then,  by  modern  methods,  2  d=  na. 

Therefore          100  ==  s  =  —  —  —  •  5  =  60  a, 

2 

whence  #  =  i|        and       d—<)^. 

Therefore  the  series  is  if,  io|,  20,  29^,  38^,  although  the 
method  here  given  is  not  the  one  followed  by  Ahmes. 

The  second  problem,2  with  its  solution  as  given  by  Ahmes, 
reads  as  follows  : 

Rule  of  distributing  the  difference.  If  it  is  said  to  thee,  corn 
measure  10,  among  10  persons,  the  difference  of  each  person  in  corn 
measure  is  i.  Take  the  mean  of  the  measures,  namely  i.  Take  i 
from  10,  remains  9.  Make  one  half  of  the  difference,  namely,  rV 
Take  this  9  times.  This  gives  to  thee  ^  TV  Add  to  it  the  portion  of 
the  mean.  Then  subtract  the  difference  ^  from  each  portion,  [this  is 
in  order]  to  reach  the  conclusion.  Make  as  shown  : 

1  i  iV>       i   4  I-  iV       i  i  T?T.       *  i  iV>       i  rV» 
i  i  i  iV,       i  i  iV>       i  1  TV,       I  yV.       i  i  iV 

This  may  be  stated  in  modern  form  as  follows  :  Required  to 
divide  10  measures  among  10  persons  so  that  each  person  shall 
have  |  less  than  the  preceding  one. 

1  Problem  40  in  the  Eisenlohr  translation,  p.  72  ;  Peet,  Rhind  Papyrus,  p.  78. 

2  No.  64  of  the  Eisenlohr  translation.    The  version  here  given  is  furnished 
by  Dr.  A.  B.  Chace.   For  another  translation  see  Peet,  loc.  cit.,  p.  107. 


ARITHMETIC  SERIES  499 

That  is,  n  =  10,  sn=  10,  d  —  ~  £,  so  that 

^  =  10=  -  «  =  (2*-f).  5, 

whence  a  =  i-^,  and  the  series  is  the  descending  progression 
TG>        T6»        T6"     *  *  ">      T(P      1(6* 

Connection  with  Polygonal  Numbers.  The  Greeks  knew  the 
theory  of  arithmetic  series,  but  they  usually  treated  it  in  con- 
nection with  polygonal  numbers.  For  example,  the  following 
are  the  first  four  triangular  numbers : 


&    •  •          •••          •  •  •  •          ••••• 

/       3  6  10  15 

It  is  evident  that  each  triangular  number  is  the  sum  of  the 

« 

series  ]^«,  and  the  Greeks  were  well  aware  of  the  rule  for  this 

i 
summation.1 

Chinese  Work  in  Series.  Nowhere  in  the  very  early  Chinese 
works  do  we  find  any  attempt  to  sum  either  an  arithmetic  or  a 
geometric  series.2  In  the  Wu-ts'ao  Suan-king,  written  about 
the  beginning  of  the  Christian  era,  or  possibly  earlier,  we  find 
the  following  problem : 

There  is  a  woman  who  weaves  5  feet  the  first  day,  her  weaving 
diminishing  day  after  day  until,  on  the  last  day,  she  weaves  i  foot. 
If  she  has  worked  30  days,  how  much  has  she  woven  in  all  ? 

The  unknown  author  then  gives  this  rule : 

Add  the  amounts  woven  on  the  first  and  last  days,  take  half  the 
sum,  then  multiply  by  the  number  of  days. 

It  is  interesting  to  see  that  this  earliest  Chinese  problem  that 
we  have  yet  found  on  the  subject  is,  like  the  second  case  in  the 
Ahmes  Papyrus,  one  involving  a  descending  series.3 

1  Heath,  Diophantus,  26.  ed.,  247;  Gow,  loc.  cit.,  p.  103;  Nesselmann,  Alg. 
Griechen,  chap.  xi.  2Mikami,  China,  p.  18.  3Mikami,  China,  p.  41. 


500  SERIES 

In  Europe  the  rule  for  the  sum  was  naturally  the  same  as  in 
the  East,  allowing  for  the  difference  in  language,1  and  was  oc- 
casionally put  in  verse  for  easy  memorizing.2 

The  rule  for  finding  any  specified  term  is  given  by  Cardan 
in  his  Practica  (1539)  and  by  Clavius  in  his  Epitome  (1583). 

Geometric  Series.  The  first  examples  of  a  geometric  series 
yet  found  are  due  to  the  Babylonians,  c.  2000  B.C.,  and  tablets 
containing  such  examples  are  still  extant.3  In  Egyptian  mathe- 
matics the  first  problem  on  this  subject  thus  far  found  is  in 
the  Ahmes  Papyrus  (c.  1550  B.C.)*  and  reads  as  follows: 

The  one  scale.                              Household  7 

Once  gives                  2801      Cats  49 

Twice  gives                5602      Mice  343 

Four  times  gives        11204      Barley  [spelt]  2301  [sic] 

Together                  ^9607      Hekt  measures  1^6807 

Together  19607 

The  left-hand  column  seems  to  be  intended  as  a  deduction  of 
a  rule  for  summing  a  geometric  progression.  Probably  Ahmes 
saw  that  if  the  ratio  is  equal  to  the  first  term,  sn  =  (sn~i  +  i)r. 
Thus  he  found  the  sum  of  four  terms  to  be  2800,  and  to  this 
he  added  i  and  multiplied  the  result  by  7  in  order  to  obtain  the 
sum  of  five  terms.  Possibly  this  is  the  significance  of  the  ex- 
pression "The  one  scale."  Similarly,  in  the  right-hand  column 

1  Thus  in  an  old  MS.  at  Munich :  «  Addir  albeg  zesam  daz  erst  vnd  das  leczt, 
vnd  daz  selb  multiplicir  mit  dem  halben  der  zal  des  posicionum"  (Curtze,  Bibl. 
Math.,  IX  (2),  113). 

2  Thus  Huswirt  (1501) : 

Si  primus  numerus  cum  postremo  faciat  par 
Eius  per  mediu  loca  singula  multiplicabis 
Ast  impar  medium  vult  multiplicari  locorum. 

That  is,  s  =  \n  (a  +  /)  if  (a  -f-  /)  is  odd,  but  s  =  n-  \  (a  +  /)  if  (a  +  I)  is 
even.  The  rule  for  the  two  cases  goes  back  at  least  to  Fibonacci  (1202) .  See  the 
Boncompagni  edition,  I,  166.  By  the  time  of  Stifel  (1544)  a  single  rule  answered 
for  both  cases.  3Hilprecht,  Tablets,  p.  17. 

4  Eisenlohr  translation,  p.  184,  No.  79.  The  author  has  used  a  MS.  translation 
from  the  hieratic,  by  Dr.  A.  B.  Chace.  On  this  section  consult  Tropfke,  Geschichte, 
II  (i),  315 ;  Peet,  Rhind  Papyrus,  p.  121.  As  in  all  such  cases,  reference  to  Ahmes 
means  to  the  original  from  which  he  copied. 


GEOMETRIC  SERIES  501 

it  is  quite  possible  that  Ahmes  added  four  terms,  then  added  i, 
making  the  2801  of  the  left-hand  column,  and  finally  multiplied 
by  7  ;  but  all  this  is  merely  conjectural. 

The  problem  suggests  the  familiar  one  of  the  seven  cats,  al- 
though here  stated  quite  differently.  There  is  some  doubt  as  to 
the  word  "household/'  the  original  word  pir  (pr)  possibly  having 
a  different  meaning.  The  hekt  (hckat)  was  a  measure  of  capacity  i 
Essentially,  therefore,  Ahmes  uses  a  rule  based  upon  the  for- 
mula s  =  a  (rn  —  i)/ (r  —  i).  It  is  interesting  to  observe  that 
a  similar  problem  is  given  by  Fibonacci  (1202)  and  is  solved 
in  much  the  same  way.1 

The  Greeks  had  rules  for  summing  such  a  series,2  and  Euclid 
gave  one  that  may  be  expressed  as  follows: 


an— a, 


which  amounts  to  saying  that 

arn—  a  __ar—  a 
sn  a 

whence  would  come  our  common  formula 

__  arn  -  a 

sn  — 

r  —  i 

The  Hindus  showed  their  interest  in  geometric  series  chiefly 
in  the  summation  problems.  The  following  typical  problem  is 
taken  from  Bhaskara  (c.  1150): 

A  person  gave  a  mendicant  a  couple  of  cowry  shells  first;  and 
promised  a  twofold  increase  of  the  alms  daily.  How  many  nishcas 
does  he  give  in  a  month?3 

*Scritti,  I,  311;  Tropfke,  Geschichte,  VI  (2),  15. 

2Nesselmann,  Alg.  Griechen,  p,  160;  Euclid,  Elements,  IX,  35,  36;  Heath, 
Euclid,  Vol.  II,  p.  420. 

3  The  wording  and  spelling  is  that  of  Colebrooke,  Bhaskara,  §128,  p.  55.  A 
niska  (to  take  the  better  spelling)  is  16  X  16  x  4  x  20  cowry  shells.  The  cowry 
shell  was  then  used  as  a  small  unit  of  value.  The  answer  given  in  the  transla- 
tion i*  2,147,483,646  cowry  shells  =  104,857  nishcas,  9  drammas,  9  pahas,  2  cacinis, 
and  6  shells.  See  also  ibid.,  p.  291. 

w 


502  SERIES 

The  Arabs  apparently  obtained  the  rule  for  summation  from 
the  Greeks,  and  it  appears  in  an  interesting  form  in  the  chess- 
board problem  in  the  works  of  Alberuni  (c.  1000). 

Medieval  European  Rule.  The  medieval  writers  apparently 
obtained  the  rule  from  the  Arabs,  for  it  appears  in  the  Liber 
Abaci  of  Fibonacci  (  1202  )/  The  first  modern  treatment  of  the 
case  is  found  in  the  Algorithmus  de  Integris  (1410)  of  Pros- 
docimo  de'  Beldamandi.2  Prosdocimo's  treatment  is  as  follows  : 


a  -f  ar  -f  err"-}-  ...-)-  arn~l  =  ar"'1  4- 


r-~  I 


which  is  but  little  more  complicated  than  our  ordinary  formula.3 
The  same  rule  is  given  by  Peurbach4  (c.  1460).  It  is  given 
by  Chuquet  (1484)  in  the  form 


mrn~l  —  a 
s  — > 


and  this  is  the  plan  used  by  Simon  Jacob  (1560),  Clavius 
(1583),  and  others.  Stifel  (1544)  gave  the  rule  in  the  awk- 
ward form  of 

(rarn~l—  a)a 

s  —  — — > 

ar  —  a 

a  method  used  by  Tartaglia5  (1556),  although  he  ordinarily 
preferred  the  one  given  by  Prosdocimo  de'  Beldamandi. c 

The  ordinary  type  of  puzzle  problem  in  series,  running 
through  all  the  literature  of  the  subject  from  the  time  of  the 
Hindus  to  the  igth  century,  may  be  illustrated  by  the  following 
from  Baker  (1568) :  "A  Marchante  hath  solde  15  yeardes  of 
Satten,  the  firste  yarde  for  is,  the  second  2§,  the  thyrd  45,  the 

1  Boncompagni  ed.,  I,  309,  under  "  de  duplicatione  scacherii." 

2  First  printed  at  Padua  in  1483. 

3  As  in  all  such  cases  it  is  to  be  understood  that  the  rule  is  stated  rhetorically 
in  the  original  work,  the  modern  algebraic  notation  being  then  unknown. 

4  It  appears  in  his  Element  a  Arithmetices  Algorithmvs  de  numeris  integrity 
Vienna,  1492.  5 General  Trattato,  II,  fol.  6,  r. 

*lbid.\  see  the  last  problem  on  the  same  page. 


MEDIEVAL  RULES  503 

fourth  8s,  and  so  increasing  by  double  progression  Geometri- 
call  .  .  .  ,"  the  total  cost  being  then  required.1 

Other  problems  relate  to  the  buying  of  orchards  in  which  the 
value  of  the  trees  increases  in  geometric  series,  or  to  buying  a 
number  of  castles  on  the  same  plan.  Problems  of  this  kind  are 
mentioned  later. 

The  rule  for  the  sum  of  n  terms  is  given  by  Clavius2  (1583) 
and  was  undoubtedly  known  to  various  earlier  writers.  If  we 
designate  the  elements  by  a,  r,  n,  I,  and  s,  and  if  any  three  of 
these  elements  are  known,  then  the  others  can  be  found.  This 
general  problem  was  first  stated  by  Wallis3  (1657)  and  was 
solved  for  all  cases  not  requiring  logarithms.  His  formula*  for 
S,  one  of  the  earliest  stated  in  a  form  analogous  to  the  one 
used  at  present,  is  v  __ 


R  -  i 


r» 


The  first  infinite  geometric  series  known  to  have  been 
summed  is  the  one  given  by  Archimedes  (c.  225  B.C.)  in  his 
quadrature  of  the  parabola.5  The  series  summed  is 


The  general  formula  for  summing  the  infinite  series  a,  ar, 
ar,  •  •  •,  arn,  •  •  •,  where  r  <  i,  was  given  by  Vieta  (c.  1590). 

Harmonic  Series.  Pythagoras  and  his  school  gave  much  at- 
tention to  the  cultivation  of  music,  not  only  as  a  means  of 
exciting  or  subduing  the  passions  but  as  an  abstract  science. 
This  led  to,  or  at  any  rate  was  connected  with,  the  important 

1  1580  ed.,  fol.  40.  Substantially  the  same  problem  is  given  in  Trenchant  (1566  ; 
1578  ed.,  p.  292). 

2"Detrahatur  primus  terminus  ab  vltimo,  &  reliquus  mimerus  per  numerum, 
qui  vna  unitate  minor  sit,  quam  denominator,  diuidatur.  Si  enim  Quotient! 
vltimus  terminus,  siue  maius  extremum  adiiciatur,  componetur  summa  omnium 
terminorum"  (Opera,  1611,  II,  68,  of  the  Epitome  Arithmeticae  Practicae)  ;  that 
is,  j  =  (/-*)/(/•-!)+/. 

B  Opera,  I,  cap.  xxxi,  p.  158  seq. 

4"  ...  si  terminus  primus  seu  minimus  diaitur  A,  maximus  V,  communis 
rationis  Exponens  R,  &  progressionis  summa  S"  (p.  158). 

6  Heath,  Archimedes,  chap,  vii;  Kliem  translation,  p.  137. 


504  SERIES 

discovery  of  the  relation  of  the  tone  to  the  length  of  the  vibrat- 
ing string,  and  hence  to  the  introduction  of  harmonic  propor- 
tion,1 which  later  writers  developed  into  harmonic  series. 

Higher  Series.  The  first  instances  of  the  use  of  arithmetic 
series  of  higher  order  were  confined  to  special  cases.  The  series 
of  squares  was  the  earliest  to  attract  attention.  Archimedes2 
used  geometry  to  show  that 


=  (;/  +  i)  (mtf  +  a(a  +  2a  +  3al  ----  -f  na}. 
For  a  =  i  this  reduces  to 


which  appears  substantially  in  the  Codex  Arcerianus  (6th  cen- 
tury). It  is  also  found  in  the  Hindu  literature  as  shown  by  the 
works  of  Mahavira  (c.  850).  3 

The  sum  of  the  cubes  appears  in  the  Codex  Arcerianus  in 
the  form 

i8-j-28  +  38+  •  •  •  +  io8  =  (£  •  io.  ii)2. 

The  Hindus  had  rules  for  finding  this  sum,  and  they  appear 
in  the  works  of  Brahmagupta  (c.  628)  ,4  Mahavira  (c.  850);' 
and  Bhaskara  (c.  1150).° 

Among  the  Arabs  similar  rules  are  found,  as  in  the  works  of 
al-Karkhi  (c.  1020)  /  where 


and 


1T.  Gomperz,  Les  penseurs  de  la  Grece,  p.  112  (Lausanne,  1904) ;  H.  Hankel, 
Geschichte,  p.  105;  Gow,  Greek  Math.,  p.  68. 

2 See  Tropfke,  Geschichte,  II  (i),  318,  on  the  entire  topic.  On  this  point  see 
the  Heiberg  edition  of  Archimedes,  II  (i),  34. 

3  English  translation,  p.  170.  4  Colebrooke's  translation,  p.  293. 

6  P.  171.  °  Colebrooke's  translation,  p.  53, 

7  See  Woepcke's  translation  of  the  Fakhri,  pp.  60,  61. 


HIGHER  SERIES  505 

Fibonacci1  (1220)  and  various  other  medieval  scholars  gave 
the  same  treatment  of  the  subject.  In  the  Liber  Quadratorum2 
(1225)  Fibonacci  also  gave  the  related  forms 

12  (i2  f  32  +  52  4-  •  •  •  +  *?}  =  n(n  +  2)  (2  n  +  2)  when  n  is  odd 
and 
1  2  (22  +  42  +  62  +  ----  \-  n2)  =  «  («  -f  2)  (2  «  +  2)  when  w  is  even. 

That  the  sum  of  the  cubes  may  be  found  by  adding  the  odd 
numbers  is  apparent  from  the  following  relations  : 


23=  3  +  5, 
38=7  +  9  +  u, 

n 

and  so  on.    This  method  of  finding  ]>V3  was  known  to  Nicom- 

i 
achus  (c.  100).   The  general  formula 


appears  in  substance   in   Pacioli's  Suma3    (1494),   but  was 
already  known. 

A  rule  for  summing  the  fourth  powers,  which  may  be  ex- 
pressed by 


appears  in  the  Key  of  Computation  of  al-Kashi  (c.  1430). 4 

Bernoulli  Numbers.  The  case  of  ]>V"  attracted  attention  in 
the  i yth  century,  but  the  rule  is  first  found  in  ih&Ars  Conjec- 
tandi5  (1713)  of  Jacques  Bernoulli  and  involves  Vhat  Euler6 

lScritti,  1, 167  (fol.  70,  v.) .  2Scritti,  II,  263,  264.  8 Fol.  44,  r.,  1.  29. 

4  The  Miftdh  al-hisab  of  Jemshid  ibn  Mes'ud  ibn  Mahmud  Giyat  ed-din  al- 
Kashi  (died  c.  1436). 

5 II,  cap.  3,  p.  97;  Tropfke,  Geschichte,  VI  (2),  24. 

6 Institutiones  calculi  differentialis,  II,  §  122  (Petrograd,  1755)-  Euler's  words 
are  "ab  inventore  Jacobo  Bernoulli  vocari  so  lent  Bernoulliani." 


So6  SERIES 

designated  as  the  "  Bernoulli  Numbers.7'  These  numbers  (A, 
B,  C,  D)  appear  in  the  following  summation  of  powers  as  given 
by  Bernoulli : L 


/• 


I 


C  -f  I  2  2  2.3.4 


2.3.4.5.6 


2.3.4.5.6.7.8 
where  A  DQ    ,     />'  DO  —  .    , 


and  where  oo  expresses  equality;  and  the  method  of  deriving 
these  values  is  also  given. 

Revival  of  Infinite  Series.  The  interest  in  the  infinitesimal  as 
an  element  in  analysis,  which  manifested  itself  about  the  be- 
ginning of  the  i  yth  century,  carried  with  it  the  notion  of  an 
infinite  number  of  elements.  Partly,  no  doubt,  on  this  account 
the  study  of  series  with  an  infinite  number  of  terms,  already 
known  to  the  Greeks,  was  revived,  and  the  idea  of  products  with 
an  infinite  number  of  factors  was  suggested. 

The  first  of  these  products  of  any  special  interest  has  al- 
ready2 been  mentioned  as  due  to  Vieta  (1593).  It  may  be 
expressed  in  modern  form3  as 


and  this,  with  others  of  the  same  nature,  has  already  been  con- 
sidered in  this  work/ 

There  .are  three  general  periods  in  the  later  development  of 
infinite  series:5  (i)  the  period  of  Newton  and  Leibniz,  —  that 


Conjectandi,  p.  97.  2Vol.  I,  p.  312. 

3See  the  Van  Schooten  edition  of  Vieta  's  works,  p.  300. 

4  For  logarithmic  series,  see  page  513  and  Volume  I,  page  434. 

6  R.  Reiff,  GescMchte  der  unendlichen  Reihen,  Tubingen,  1889.  See  also,  for 
comparison,  H.  Wieleitner,  "Zur  Geschichte  der  unendlichen  Reihen  im  christ- 
lichen  Mittelalter,"  Bibl.  Math.,  XIV  (3),  150;  Tropfke,  Geschichte,  VI  (2),  54. 


INFINITE  SERIES  507 

of  its  introduction  ;  (2  )  the  period  of  Euler,  —  the  formal  stage  ; 
(3)  the  modern  period,  —  that  of  the  scientific  investigation  of 
the  validity  of  infinite  series.  This  third  period,  which  may  be 
designated  as  the  critical  one,  began  in  1812  with  the  publica- 
tion of  Gauss's  celebrated  memoir  on  the  series 

(ft  +i)    a 

X       l      "    *    *  • 


1.7  I  .  2  .7  .  (7+  I) 

Euler  had  already  considered  this  series,  but  Gauss  was  the 
first  to  master  it,  and  under  the  name  of  "hypergeometric 
series/7  due  to  Pfaff  (1765-1825),  it  has  since  occupied  the  at- 
tention of  a  large  number  of  mathematicians.  The  particular 
series  is  not  so  important  as  the  standard  of  criticism  which 
Gauss  set  up,  embodying  the  simpler  criteria  of  convergence 
and  the  questions  of  remainders  and  the  range  of  convergence. 

Cauchy  (1821)  took  up  the  study  of  infinite  series  and  elabo- 
rated the  theory  of  convergence  which  James  Gregory  (1668) 
had  already  begun  and  to  which  Maclaurin,  Euler,  and  Gauss 
had  made  noteworthy  contributions.1  The  term  "convergent 
series"  is  due  to  Gregory  (1668)  and  the  term  "divergent 
series"  to  Nicolas  (I)  Bernoulli  (i7i3).2 

Abel  (1826)  gave  careful  study  to  the  series 

;;/         m  (m  —  I  )    „  , 
i+  —  x  +  -'  —  :  —  -  x"  H  ----  , 
I  2  ! 

correcting  certain  of  Cauchy's  conclusions  and  giving  a  scien- 
tific summation  of  the  series  for  complex  values  of  m  and  x. 

Binomial  Theorem.  The  development  of  (a  +  b)u  for  any 
integral  value  of  n,  or  at  least  a  device  for  finding  the  coeffi- 
cients, was  known  in  the  East  long  before  it  appeared  in  Europe. 
The  case  of  n  =  2  was  also  known  to  Euclid  (c.  300  B.C.),3  but 
any  evidence  of  the  generalization  of  the  law  for  other  values 

1  On  the  history  of  criteria  of  convergence  see  F.  Cajori,  in  the  Bulletin  of  the 
New  York  Math.  Soc.,  II,  i  ;  see  also  III,  186. 

2  F.  Cajori,  Bulletin  of  the  Amer.  Math.  Soc.,  XXIX,  55. 

3  Elements,  II.   For  a  summary  of  his  work  on  algebraic  identities  see  Nessel- 
mann,  Alg.  Griechen,  p.  154. 


508  SERIES 

of  n  first  appears,  so  far  as  we  know,  in  the  algebra  of  Omar 
Khayyam  (c.  noo).  This  writer  did  not  give  the  law,  but  he 
asserted  that  he  could  find  the  fourth,  fifth,  sixth,  and  higher 
roots  of  numbers  by  a  law  that  he  had  discovered  and  which 
did  not  depend  upon  geometric  figures.1  He  states  that  this 
law  was  set  forth  by  him  in  another  work,  but  of  this  work  there 
seems  to  be  no  copy  extant. 

Pascal  Triangle.  In  one  of  the  works  of  Chu  Shi'-kie  (1303), 
the  greatest  of  the  Chinese  algebraists  of  his  time,  the  triangular 
arrangement  of  the  coefficients  is  given  in  the  following  form, 

i 
i      i 

I        2        I 

i33i 

14641 

i     5    10  10    5     i 


a  form  now  commonly  known  as  the  Pascal  Triangle.2 

This  triangular  array  first  appeared  in  print  on  the  title-page 
of  the  arithmetic  of  Apianus  ( 1527),  as  shown  in  the  illustration 
on  page  sog.3  In  the  form 

i     2     i 

i33i 
14641 
i     5    10   10     5      i 

1"J'ai  compost  un  ouvrage  sur  la  demonstration  de  1'exactitude  de  ces 
me'thodes.  .  .  .  J'en  ai,  en  outre,  augmente"  les  especes,  c'est-a-dire  que  j'ai 
enseign6  a  trouver  les  cotes  du  carre"-carre",  du  quadrato-cube,  du  cubo- 
cube,  etc.,  a  une  etendue  quelconque,  ce  qu'on  n'avait  pas  fait  precedemment. 
Les  demonstrations  que  j'ai  donnees  a  cette  occasion  ne  sont  que  des  demonstra- 
tions arithmetiques."  Translated  by  F.  Woepcke,  L'Algebre  d'Omar  Alkhdyyami, 
p.  13  (Paris,  1851). 

2  Mikami,  China,  p.  106. 

3On  the  general  subject  see  H.  Bosmans,  "Note  historique  sur  le  Triangle 
arithme'tique,  dit  de  Pascal,"  Annales  de  la  Societe  scientifique  de  Bruxelles, 
XXXI,  October,  1906;  Tropfke,  GeschicMe,  VI  (2),  37. 


••^tfM 

Tfile  we 


t>n&ern?eyflmg  alter  Rastftmafif* 
j  niwg  in  ozeven  frichcrit/irut  fdionctt 
i  adit  vn  fragflucFcit  6ectriffcit  *  © 
Udb  was  foirl  t?iint)  6e5«nt>igfai 


irt  bcr 


Oceglevdoert  fu'wmlj?  tvt&er  in 
rtcdb  tn  VOcIfcber  (biarf)  ntc 
*  turcb  pctrtim  2(pmnu 
2f  |?ronomet 


ttu/wrfcrrigcr. 


PASCAL    TRIANGLE    AS    FIRST    PRINTED,  1527 

Title-page  of  the  arithmetic  of  Petrus  Apianus,  Ingolstadt,  1527,  more  than  a 
century  before  Pascal  investigated  the  properties  of  the  triangle 


SERIES 


it  is  first  found  in  StifePs  Arithmetica  Integra  (1544),  appear- 
ing a  year  later  in  the  De  Nvmeris  et  Diver  sis  Rationibvs  of 
Scheubel  (1545).  It  also  appears  in  the  various  editions  of 
Peletier's  arithmetic  (Poitiers,  1549  and  later).  Tartaglia 
(1556)  gave  it  as  his  own  invention,1  and  soon  after  his  time 
it  became  common  property.  Bombelli  (1572),  for  example, 
gave  the  coefficients  for  all  powers  of  a  +  b  up  to  the  seventh, 
using  them  in  finding  corresponding  roots,2  and  Oughtred 
( 1631 )  gave  them  up  to  the  tenth  power.3  The  triangular  array 
was  investigated  by  Pascal  (1654)  under  a  new  form,  sub- 
stantially as  follows:4 

123456789     10 


8 

9 
10 


I 

I 

i 

I 

i 

i 

I 

i      i      i 

I 

o 

3 

4 

5 

6 

7 

8      9 

I 

3 

6 

10 

15 

21 

28 

^5     Rangs  Paralleles 

I 

4 

10 

20 

35 

56 

84 

Rangs  Perpendiculaires 

I 

5 

IS 

35 

70 

126 

I 

6 

21 

56 

126 

I 

7 

28 

84 

I 

8 

36 

I 

9 

I 

He  made  numerous  discoveries  relating  to  this  array  and  set 
them  forth  in  his  Traite  du  triangle  arithmitique?  published 

^General  Trattato,  II, fols.  69,  v.\  71,  v.  (Venice,  1556).  2 Algebra,  p.  64. 

3F.  Cajori,  William  Oughtred,  p.  29  (Chicago,  1916). 

4 This  is  from  the  plate  in  Pascal's  (Euvr.es,  Vol.  V  (Paris,  1819) .  The  descrip- 
tion is  given  on  pages  1-56.  In  the  original  there  are  diagonals  in  the  above 
figure.  See  also  Tropfke,  Geschichte,  VI  (2),  37. 

5"Le  nombre  de  chaque  cellule  est  egal  a  celui  de  la  cellule  qui  la  precede  dans 
son  rang  perpendiculaire,  plus  a  celui  de  la  cellule  qui  la  precede  dans  son  rang 
parallele"  ((Euvres,  V,  3)  (Paris,  1819). 


PASCAL  TRIANGLE 


posthumously  in  1665,  and  among  these  was  essentially  our 
present  Binomial  Theorem  for  positive  integral  exponents. 
After  this  time  the  triangular  _ 

array  was  common  in  the  East 
as  well  as  in  the  West. 


(D®1 
AQS 


Generalization  of  the  Bino- 
mial Theorem.  The  generaliza- 
tion of  the  binomial  theorem 
for  negative  and  fractional 
values  of  n  is  due  to  Newton, 
who  set  it  forth  in  letters 
which  he  wrote  to  Oldenburg 
on  June  13, 1676,  and  October 
24,  1676. ' 

The  proof  of  the  Binomial 
Theorem  was  slowly  devel- 
oped by  later  writers.  Among 
those  who  contributed  to  a 
satisfactory  demonstration 
were  Maclaurin2  for  rational 
values  of  n,  Giovanni  Fran- 
cesco M.  M.  Salvemini  (de 
Castillon ) 3  and  Kastner 
(i74S)4  for  integral  values, 
Euler5  (1774)  for  fractional  exponents,  and  Abel0  (c.  1825) 
for  general  values  of  72,  taking  n  as  a  complex  number. 

xSee  Commercium  Epistolicum,  London,  1712;  1725  ed.,  pp.  131,  142.  In  his 
letter  of  October  24  he  proceeds  from  (i  —  .*'2)2,  (i  —  x2)*,  (i  —  x'2)*,  •  •  •  to 
(i  —  ,r2)2  anci  (r_  #2^  «vei  generaiiter  i  —  xx\''*  "  and  finds,  for  example, 

that  "F—  xx  Y  valeret  i  —  A-**2—  i^4  —  jV^  &c-"   ^n  the  doubtful  assertion 
that  Pascal  may  have  anticipated  this  discovery,  see  G.  Enestrom,  Bibl.  Math., 

V(3),72. 

2  Treatise  of  Fluxions,  p.  607  (1742). 

3 Born  at  Castiglione,  1708;  died  1791.  See  Phil.  Trans.,  XLII  (1742),  91.  He 
used  the  theory  of  combinations.  See  page  326.  4  Cantor,  Geschichte,  III,  660. 

5Novi  comment.  Petrop.,  XIX,  103 ;  see  Tropfke,  Geschichte,  II  (i),  331-  See 
also  the  English  translation  of  Euler's  Algebra,  I,  172, 177  (London,  2d  ed.,  1810). 

6The  article  appeared  posthumously  in  Crelle's  Journal,  I  (1826),  311.  See 
also  Abel's  (Euvres,  I,  219  (Christiania,  1881). 


PASCAL   TRIANGLE    IN    JAPAN 

From  Murai  Chuzen's  Sampo  Doshi-mon 

(1781),  showing  also  the  sangi  forms  of 

the  numerals 


512  SERIES 

The  generalization  of  the  Binomial  Theorem  into  the  Poly- 
nomial Theorem  was  due  chiefly  to  Leibniz  (1695),  Jacques 
Bernoulli,  and  De  Moivre.1 

Finite  Differences.  The  treatment  of  series  by  the  method  of 
finite  differences  appeared  in  the  i7th  century.  In  1673  Leibniz 
wrote  to  Oldenburg  concerning  the  following  scheme  of  treating 
the  series  of  cubes  : 

o  o  o 

6666 
6  12          18          24          30 

.1  7  19         37         61          9i 

o  i  8          27         64         125        216 

He  said  that  John  Pell  attributed  the  discovery  to  Gabriel 
Mouton,  of  Lyons.2 

Taylor's  Formula  and  Maclaurin's  Formula.  In  1715  Brook 
Taylor  published  the  formula  which  bears  his  name,  and  which 
we  now  express  as  follows  : 

f(*  +  A)  =/(*) 


It  was  not  until   1742   that  Colin  Maclaurin  published  the 
corresponding  formula 

f(.r)  =/(o)  +  xf(o)  +  ^/"(o)  +  .  .  ., 

a  relation  that  is  easily  derived  from  the  preceding  one.4 

Trigonometric  Series.  The  development  of  trigonometric  func- 
tions in  series  first  attracted  the  attention  of  mathematicians 


*De  Moivre's  articles  appeared  in  the  Phil.  Trans.,  XIX  (1697),  619;  XX,  190. 

2Commercmm  Epistolicum,  p.  109  (London,  1712  ;  1725  cd.).  Gabriel  Mouton, 
born  at  Lyons,  1618;  died  at  Lyons,  September  28,  1694.  He  suggested  (1670)  a 
system  of  measures  not  unlike  the  metric  system. 

*Methodus  Increment  orum  directa  et  inversa,  prop.  7  (London,  1715).   The 
series  had  already  been  announced  by  him  in  1712. 
A  Complete  System  of  Fluxions,  Edinburgh,  1742. 


INFINITE  SERIES  513 

in  the  iyth  century.   To  James  Gregory  (1671)  are  due  the 
following  :  * 

x  =  tan  x  —  ?,  tan8  x  +  ^  tan5  x  —  \  tan7  x  +  •  •  •, 
tan  x  =  x  +  1  x*  +  -^  •  a5  +  sYir  #7  H  ----  , 


He  also  gave  the  important  series 

arc  tan  x~x  —  ^x3  +  ^x5—  •  •  •, 

but  this  is  easily  deduced  from  the  one  given  above  for  tan  x. 
Newton2  gave  (c.  1669)  the  anti  trigonometric  series  for  arc 
sin  x,  essentially  as  follows: 

arc  sin  x  =  sin"1  x  =  #  +  -J-  x*  -f-  430  #f)  f  T|2  x1  +  •  •  •• 

Logarithmic  Series.  The  idea  of  expressing  a  logarithm  by 
means  of  a  series  seems  to  have  originated  with  Gregory  and  to 
have  been  elaborated  by  Nicolaus  Mercator3  (1667),  who  dis- 
covered, for  a  special  case  at  least,  the  relation 

log  (i+a)  =  a-  \  a*  -f  £  a*  -  I  a4  +  •  •  •, 
where  i  ^  a  >  —  i  . 

The  value  of  Mercator's  and  Gregory's  contributions  was 
recognized  by  Wallis  in  reviews  which  he  wrote  of  their  works.'1 

12.  LOGARITHMS 

Technical  Terms.  The  word  "logarithm"5  means  "ratio  num- 
ber77 and  was  an  afterthought  with  Napier.  He  first  used  the 
expression  "artificial  number/7  but  before  he  announced  his 
discovery  he  adopted  the  name  by  which  it  is  now  known.6 

1  These  were  communicated  to  Collins  in  a  letter  from  Gregory.  See  the 
Commercium  Epistolicum,  London,  1712,;  1725  ed.,  pp.  98,  210  n. 

>2Commercium  Epistolicum,  pp.  97,  126;  Tropfke,  Geschichte,  VI  (2),  46. 

3  Logarithmotechnia  sive  methodus  construendi  logarithmos  nova,  London, 
1668.   The  theory  was  worked  out  the  year  before. 

4  Phil.  Trans.,  1668,  pp.  640,  753. 

5  From  the  Greek  \6yos  (log'os),  ratio,  -f  dpi0[j.6s  (aritfimQs'}  ,  number. 

6  This  fact  is  evident  from  his  Descriptio,  1619  ed. 


514  LOGARITHMS 

Briggs  introduced  (1624)  the  word  "mantissa."  It  is  a  late 
Latin  term  of  Etruscan  origin,  originally  meaning  an  addition, 
a  makeweight,  or  something  of  minor  value,  and  was  written 
mantisa.  In  t^fc  i6th  century  it  came  to  be  written  mantissa 
and  to  mean  "appendix,"1  and  in  this  sense  it  was  probably 
considered  by  Briggs.  The  name  also  appears  in  connection 
with  decimals  in  Wallis's  Algebra  (1685),  but  it  was  not  com- 
monly used  until  Euler  adopted  it  in  his  Introductio  in  analysin 
infinitorum  (1748).  Gauss  suggested  using  it  for  the  fractional 
part  of  all  decimals.2 

The  term  " characteristic"  was  suggested  by  Briggs  (1624) 
and  is  used  in  the  1628  edition  of  Vlacq.3 

The  characteristic  was  printed  in  the  early  tables,  and  it  was 
not  until  well  into  the  i8th  century  that  the  custom  of  printing 
only  the  mantissas  became  generally  established. 

Napier's  Invention.  So  far  as  Napier's  invention  is  concerned, 
Lord  Moulton  expressed  the  fact  very  clearly  when  he  said : 4 

The  invention  of  logarithms  came  on  the  world  as  a  bolt  from  the 
blue.  No  previous  work  had  led  up  to  it,  foreshadowed  it  or  heralded 
its  arrival.  It  stands  isolated,  breaking  in  upon  human  thought 
abruptly  without  borrowing  from  the  work  of  other  intellects  or  fol- 
lowing known  lines  of  mathematical  thought. 

Napier  worked  at  least  twenty  years  upon  the  theory.  His 
idea  was  to  simplify  multiplications  involving  sines,  and  it  was 
a  later  thought  that  included  other  operations,  applying  loga- 
rithms to  numbers  in  general.  He  may  have  been  led  to  his 
discovery  by  the  relation 


sin  A  sin  B  =  l  (cos  A—B—  cos 


iWith  this  meaning  it  appeared  as  late  as  1701  in  J.  C.  Sturm,  Mathesis 
juvenalis. 

2  "  Si  fractio  communis  in  decimalem  convertitur,  seriem  figurarum  decimalium 
.  .  .  fractionis  mantissam  vocamus  .  .  ."  See  E.  Hoppe,  "Notiz  zur  Geschichte 
der  Logarithmentafeln,"  Mittheilungen  der  math.  Gesellsch.  in  Hamburg,  IV,  52. 

3".  .  .  prima  nota  versus  sinistram,  quam  Characteristicam  appellare  poteri- 
mus  .  .  ."  It  again  appeared  in  Mercator's  Logarithmotecknia  (1668). 

4  "Inaugural  Address:  The  Invention  of  Logarithms,"  Napier  Tercentenary 
Memorial  Volume,  p.  i  (London,  1915). 


NAPIER'S  INVENTION  515 

for,  as  Lord  Moulton  says,  in  no  other  way  can  we  "  conceive 
that  the  man  to  whom  so  bold  an  idea  occurred  should  have 
so  needlessly  and  so  aimlessly  restricted  himself  to  sines  in 
his  work,  instead  of  regarding  it  as  applicable  to  numbers 
generally." 

Napier  published  his  Descriptio^  of  the  table  of  logarithms 
in  1614.  This  was  at  once  translated  into  English  by  Edward 
Wright,"  but  with  the  logarithms  contracted  by  one  figure. 

In  Napier's  time  sin  </>  wras  a  line,  not  a  ratio.  The  radius 
was  called  the  sinus  totus,  and  when  this  was  equal  to  unity  the 
length  of  the  sine  was  simply  stated  as  sin  $.  If  r  was  not 
unity,  the  length  was  r  sin</>.  With  this  statement  we  may 
consider  Napier's  definition  of  a  logarithm : 

The  Logarithme  therefore  of  any  sine  is  a  number  very  neerely 
expressing  the  line,  which  increased  equally  in  the  meane  time,  whiles 
the  line  of  the  whole  sine  decreased  proportionally  into  that  sine, 
both  motions  being  equal-timed,  and  the  beginning  equally  swift.3 

From  this  it  follows  that  the  logarithm  of  the  sinus  totus  is 
zero.  Napier  saw  later  that  it  was  better  to  take  log  i  =  ex4 

Napier  then  lays  down  certain  laws  relating  to  proportions, 
which  may  be  stated  symbolically  as  follows : 

1 .  If  a :  b  —  c :  d,  then  log  b  —  log  a  =  log  d  —  log  c. 

2 .  If  a :  b  =  b  :  c,  then  log  c  —  2  log  b  —  log  a. 

3.  If  a :  b  =  b :  c ,  then  2  log  b  =  log  a  +  log  c. 

4.  If  a :  b  =  c :  d,  then  log  d  =  log  b  4-  log  c  —  log  a. 

5.  If  a :  b  —  c :  d,  then  log  b  +  log  c  =  log  a  +  log  d. 

6.  If  a :  b  =  b  :  c  =  c :  d,  then  3  log  b  =  2  log  a  +  log  d  and 

3  log  c  =  log  a  +  2  log  d. 

^Mirifid  Logarithmorum  Canonis  Descriptio,  Edinburgh,  1614. 
2 A  Description  of  the  Admirable  Table  of  Logarithmes,  London,  1616,  pub- 
lished after  Wright's  death. 

3  Wright's  translation  of  the  Description  pp.  4,  5- 

4  As  to  the  priority  of  this  idea,  see  G.  A.  Gibson,  "Napier's  logarithms  and 
the  change  to  Briggs's  logarithms,"  in  the  Napier  Tercentenary  Memorial  Volume, 
p.  114  (London,  1915)  ;  this  volume  should  be  consulted  on  all  details  of  this 
kind.   See  also  Dr.  Glaisher's  article  on  logarithms  in  the  eleventh  edition  of  the 
Encyclopaedia  Britannica. 


Si6  LOGARITHMS 

The  system  was,  therefore,  designed  primarily  for  trigonom- 
etry, but  would  also  have  been  valuable  for  purposes  of  ordi- 
nary computation  had  not  a  better  plan  been  suggested. 

Napier  also  wrote  a  work  on  the  construction  of  a  table,1 
which  was  published  posthumously  as  part  of  the  1619  edition 
of  the  Descriptio. 

Napier's  logarithms  are  not  those  of  the  so-called  Napierian, 
or  hyperbolic,  system,  but  are  connected  with  this  system  by 
the  relation  logn  a  =  io7  •  log^  io7  —  io7  •  log^  a.  The  relation 
between  the  sine  and  its  logarithm  in  Napier's  system  is 


sin<£  =  io7 

so  that  the  sine  increases  as  its  logarithm  decreases. 

Briggs's  System.  Henry  Briggs,  professor  of  geometry  at 
Gresham  College,  London,  and  afterwards  Savilian  professor 
of  geometry  at  Oxford,  was  one  of  the  first  to  appreciate  the 
work  of  Napier.  Upon  reading  the  Descriptio  (1614)  he  wrote  : 

Naper,  lord  of  Markinston,  hath  set  my  head  and  hands  at  work 
with  his  new  and  admirable  logarithms.  I  hope  to  see  him  this  sum- 
mer, if  it  please  God  ;  for  I  never  saw  a  book  which  pleased  me 
better,  and  made  me  more  wonder. 

He  visited  Merchiston  in  1615  and  suggested  another  base, 
of  which,  however,  Napier  had  already  been  thinking.  In 
Briggs's  Arithmetica  Logarithmic  a  the  preface,  written  by 
Vlacq,  contains  the  following  statement2  by  the  author  of  the 
work  itself: 

That  these  logarithms  differ  from  those  which  that  illustrious  man, 
the  Baron  of  Merchiston  published  in  his  Canon  Mirificus  must  not 
surprise  you.  For  I  myself,  when  expounding  their  doctrine  publicly 
in  London  to  my  auditors  in  Gresham  College,  remarked  that  it  would 
be  much  more  convenient  that  o  should  be  kept  for  the  logarithm  of 

^Mirifici  ipsius  canonis  construct™. 

2  Arithmetica  Logarithmica  sive  Logarithmorum  Chiliades  Triginta  (London, 
1624),  preface.  The  original  is  in  Latin;  the  translation  of  the  statement  is  from 
the  Napier  Tercentenary  Memorial  Volume. 


LATER  SYSTEMS  517 

the  whole  sine  (as  in  the  Canon  Mirificus).  .  .  .  And  concerning 
that  matter  I  wrote  immediately  to  the  author  himself ;  and  as  soon 
as  the  season  of  the  year  and  the  vacation  of  my  public  duties  of  in- 
struction permitted  I  journeyed  to  Edinburgh,  where,  being  most  hos- 
pitably received  by  him,  I  lingered  for  a  whole  month.  But  as  we 
talked  over  the  change  in  logarithms  he  said  that  he  had  for  some 
time  been  of  the  same  opinion  and  had  wished  to  accomplish  it.  ... 
He  was  of  the  opinion  that  .  .  .  o  should  be  the  logarithm  of  unity. 

The  real  value  of  the  proposition  made  by  Briggs  at  this  time 
was  that  he  considered  the  values  of  log  10"  a  for  all  values  of  n. 
The  relation  between  the  two  systems  as  they  first  stood  may 
be  indicated  as  follows : 

Napier,  log 3^=  r(loger  —  logey),  where  r  =  io7; 

Briggs,  logy  =  io10(io-  Iog10;y)  ; 

Napier  (later  suggestion),  logy  =  io9log10y. 

The  first  table  of  logarithms  of  trigonometric  functions  to 
the  base  io  was  made  by  Gunter,  a  colleague  of  Briggs  at 
Gresham  College,  and  was  published  in  London  in  I62O.1 

The  Base  e.  In  the  1618  edition  of  Edward  Wright's  trans- 
lation of  the  Descriptio  there  is  printed  an  appendix,  probably 
written  by  Oughtred,  in  which  there  is  the  equivalent  of  the 
statement  that  log^io  =  2.302584,  thus  recognizing  the  base  e. 

Two  years  later  (1620)  John  Speidell'2  published  his  New 
Logarithmes,  also  using  this  base.  He  stated  substantially  that 

log  n  =  icr1  (nap  log  i  —  nap  log  ft), 
or  logn=  i  o5  (io -f- log,  io~5x). 

Continental  Recognition.  The  same  year  (1624)  that  Briggs 
published  his  Arithmetlca  Logarithmica  Kepler's  first  table  ap- 
peared. A  year  later  Wingate's  Arithmetiqve  Logarithmiqve 
(Paris,  1625)  gave  the  logarithms  of  numbers  from  i  to  1000, 
together  with  Gunter's  logarithmic  sines  and  tangents. 

1  Canon  Triangulorum,  sive  Tabulae  Sinuum  et  Tangentium. 

2  See  Napier  Tercentenary  Memorial  Volume,  pp.  132,  221;  F.  Cajori,  History 
of  Elem.  Math.,  p.  164,  rev.  ed.  (N.Y.,  1917),  and  History  of  Math.,  p.  153, 
rev.  ed.  (N.Y.,  1919). 

n 


Si8  LOGARITHMS 

Holland  was  the  third  Continental  country  to  recognize  the 
work  of  Napier  and  Briggs.  In  1626  there  was  published  a 
work1  by  Adriaen  Vlacq,2  assisted  by  Ezechiel  de  Decker.  In 
1628  Vlacq  republished  Briggs's  tables,3  filling  the  gap  from 
20,000  to  90,000.  The  tables  in  this  work  were  reprinted  in 
London  by  George  Miller  in  i63i.4  It  is  interesting  to  note  that 
the  next  complete  edition  of  Vlacq's  tables  appeared  iix  China.5 

In  Germany  the  theory  was  first  made  known  by  Johann 
Faulhaber6  (1630). 

Logarithms  in  Arithmetic.  By  the  middle  of  the  seventeenth 
century,  logarithms  found  their  way  into  elementary  arith- 
metics, as  is  seen  in  HartwelPs  (1646)  edition  of  Recorders 
Ground  of  Artes?  where  it  is  said  that  "for  the  extraction  of  all 
sorts  of  roots,  the  table  of  Logarithmes  set  forth  by  M.  Briggs 
are  most  excellent,  and  ready."  Thereafter  they  were  occa- 
sionally found  in  textbooks  of  this  kind,  both  in  Great  Britain 
and  on  the  Continent. 

Forerunners  of  Biirgi.  Napier  approached  logarithms  from 
the  standpoint  of  geometry,  whereas  at  the  present  time  we 
approach  the  subject  from  the  relation  aman  —  am*n.  This 
relation  was  known  to  Archimedes8  and  to  various  later  writers. 

More  generally,  if  we  take  the  two  series 

o         i         2.3         4         $         6         7 
and  i         2         4         8       16       32       64     128, 


lEerste  Deel  van  de  Nieuwe  Telkonst,  Gouda,  1626.  See  D.  Bierens  de  Haan 
in  Boncompagni's  Bullettino,  VI,  203,  222 ;  J.  W.  L.  Glaisher,  "Notice  respecting 
some  new  facts  in  the  early  history  of  logarithmic  tables,"  Philosoph.  Mag., 
October,  1872. 

2  Born  at  Gouda,  c.  1600;  died  after  1655.  The  common  Dutch  spelling  is  now 
Vlack. 

8Arithmetica  Logarithmica,  Gouda,  1628.  It  was  also  published  with  a  French 
title-page.  *  Logarithmicall  Arithmetike,  London,  1631. 

5  Magnus  Canon  Logarithmorum   .  .  .   Typis  Sinensibus  in  Aula  Pekinensi 
.  .  .,  1721. 

6Inginieurs-Schul,  erster  Theil,  darinen  durch  den  Canonem  Logarithmicum 
.  .  .  ,  Frankfort,  1630.  Faulhaber  was  born  at  Ulm,  May  5,  1580;  died  at 
Ulm,  1635.  7  Also  the  editions  of  1662  and  1668. 

6  Opera  omnia,  ed.  Heiberg,  2d  ed.,  II,  243;  Heath,  Archimedes,  p.  230. 


FORERUNNERS  OF  BURGI  519 

the  first  one  being  arithmetic  and  the  second  one  being  geo- 
metric, we  see  that  the  latter  may  be  written  as  follows : 

2°  21  22  23  24  25  26  27. 

From  this  it  is  evident  that 

23'24:=27,  (22)3=26, 

27:28=24,  (24)'=22, 

which  are  the  fundamental  laws  of  logarithms. 

Most  writers1  refer  to  Stifel  as  the  first  to  set  forth  these 
basal  laws,  and  we  shall  see  that  he  did  set  them  forth  very 
clearly ;  but  he  was  by  no  means  the  first  to  do  so,  nor  did  they 
first  appear  even  in  his  century.  Probably  the  best  of  the  state- 
ments concerning  them  which  appeared  in  the  isth  century 
were  those  of  Chuquet  in  Le  Triparty  en  la  Science  des  Nom- 
bres,  written  in  1484,  from  which  Estienne  de  la  Roche  copied 
so  freely  in  his  Larismethique  of  1520.  Chuquet  expressed 
very  clearly  the  relations 

a man =  am  +  n 
and  (am)H=aMn 

in  connection  with  the  double  series  to  which  reference2  has 
been  made,  calling  special  attention  to  the  latter  law  as  "a 
secret "  of  proportional  numbers.3 

1  Among  them  is  Kastner,  Geschichte  der  Mathematik,  I,  119,  who  has  been 
generally  followed  in  this  matter.   See  also  Th.  Miiller,  Der  Esslinger  Mathematiker 
Michael  Stijel,  Prog.,  p.  16  (Esslingen,  1897),  where  the  author  states:  "'Dies  ist 
das  alteste  Buch,'  sagt  Strobel,  'in  welchem  die  Vergleichung  des  arithmetischen 
Reihe  mit  der  geometrischen  als  der  Grund  der  Logarithmen  vorkommt.'" 

Much  of  the  work  on  this  topic  appeared  in  the  author's  paper  published  in 
the  Napier  Tercentenary  Memorial  Volume,  p.  81  (London,  1915)- 

2  "II  convient  poser  pluses  nobres  ^porcional}  comancans  a  i.  constituez  en 
ordonnance  continuee  come  1.24.8.16.32.  &c.  ou  .1.3.9.27.  &c.   (fl.  Maintenant  con- 
uient  scauoir  que  .1.  represente  et  est  ou  lieu  des  nombres  dot  ler  denolaonest  .o./2. 
represente  et  est  ou  lieu  des  premiers  dont  leur  denomiacion  est  .i./4-  tient  le  lieu 
des  second}  dont  leur  denomiacion  est  .2.  Et  .8.  est  ou  lieu  des  tiers  .16.  tient  la 
place  des  quartz7'  (fol.  86,  v.,  of  the  Triparty) .   This  is  taken  from  the  copy  made 
by  A.  Marre  from  the  original  manuscript.   Boncompagni  published  it  in  the 
Bullettino,  XIII,  593  seq.,  fol.  86,  v.t  being  on  page  740. 

3"  (H.Seml51ement  qui  multiplie  .4.  qui  est  nombre  second  par  .8.  qui  est  nombre 
tiers  montent  .32.  qui  est  nombre  quint  ...  <H.  En  ceste  consideration  est  malfeste 


520  LOGARITHMS 

It  is  difficult  to  say  when  a  plan  of  this  kind  first  appears  in 
print,  because  it  is  usually  hinted  at  before  it  is  stated  defi- 
nitely. Perhaps  it  is  safe,  however,  to  assign  it  to  RudolfPs 
Kunstliche  rechnung  of  1526,  where  the  double  series  is  given 
and  the  multiplication  principle  is  clearly  set  forth ;  *  and  inas- 
much as  this  work  had  great  influence  on  Stifel,  who  in  turn 
influenced  Jacob,  Clavius,  and  Biirgi,  it  was  somewhat  epoch- 
making. 

The  next  writer  to  refer  to  the  matter  was  probably  Apianus 
(1527),  who  followed  Rudolff  so  closely  as  to  be  entitled  to 
little  credit  for  what  he  did. 

Following  Apianus,  the  first  arithmetician  of  any  standing 
who  seems  to  have  had  a  vision  of  the  importance  of  this  rela- 
tion was  Gemma  Frisius  (1540),  who  gave  the  law  with  rela- 
tion to  the  double  array 

3         9       27       8 i      243     729 
012345, 

saying  that  the  product  of  two  numbers  occupies  a  place  indi- 
cated by  the  sum  of  their  places  (3X9  occupying  the  place 
indicated  by  i  4-  2,  or  3),  and  that  the  square  of  a  number  in 
the  fifth  place  occupies  the  2  x  5th  place.2 

The  first  arithmetician  to  take  a  long  step  in  advance  of 
Rudolff  was  Stifel  (1544),  the  commentator  (1553)  on  Die 

vng  secret  qui  est  es  nombres  ^porcionalz.  Cest  que  qui  multiplie  vng  nombre 
^porcional  en  soy  II  en  viet  le  nombre  du  double  de  sa  denomiacion  come  qui 
mltiplie  .8.  qui  est  tiers  en  soy  II  en  vient  .64.  qui  est  six6.  Et  .16.  qui  est  quart 
multiplie  en  soy.  II  en  doit  venir  256.  qui  est  huyte.  Et  qui  multiplie  .128.  qui 
est  le  .7e .  jpporcional  par  .512.  qui  est  le  Qe.  II  en  doit  venir  65536.  qui  est  le  i6e  " 
(ibid.,  p.  741). 

J"Nun  merck  wenn  du  zwo  zalen  mit  einander  multiplicirst/  wiltu  wissen  die 
stat  des  quocients/  addir  die  zalen  der  natiirlichen  ordnung  so  ob  den  zweyen  mit 
einander  gemultiplicirten  zalen  gefunden/  d}  collect  bericht  dich.  Als  wen  ich 
8  multiplicir  mit  16.  muss  komen  128.  darumb  das  3  vnd  4  so  vber  dem  8  vnnd  16 
geschriben  zusamen  geaddirt  7  machen."  He  gives  several  examples,  but  goes  no 
farther  with  the  law. 

2  "Si  enim  duos  quoscunque  ex  his  numeris  inuicem  multiplicaueris,  produc- 
tumque  per  primum  diviseris,  producetur  numerus  eo  loco  ponendus,  que  duo  facta 
indicabunt .  .  ."  (ed.  1553,  fol.  17,  r.,  and  note  by  Peletier  (Peletarius),  fol.  78,  v.). 
The  relation  is  not  so  clear  as  in  some  of  the  other  texts,  on  account  of  the 
arrangement  of  the  series. 


FORERUNNERS  OF  BURGI  521 

Coss.  It  is  not,  however,  in  this  work  that  the  theory  is  set 
forth,  but  in  the  Arithmetica  Integra  of  1544.  Stifel  here  re- 
fers several  times  to  the  laws  of  exponents.  At  first  he  uses 
the  series 

012345678 
i         2         4         8       16       32       64     128     256, 

distinctly  calling  the  upper  numbers  exponents,  and  saying 
that  the  exponents  of  the  factors  are  added  to  produce  the  ex- 
ponent of  the  product  and  subtracted  to  produce  the  exponent 
of  the  quotient.1  Moreover,  he  expressly  lays  down  four  laws, 
namely,  that  addition  in  arithmetic  progression  corresponds 
to  multiplication  in  geometric  progression,  that  subtraction  cor- 
responds to  division,  multiplication  to  the  finding  of  powers, 
and  division  to  the  extracting  of  roots.  Furthermore,  Stifel  not 
only  set  forth  the  laws  for  positive  exponents  but  also  saw  the 
great  importance  of  considering  the  negative  exponents  of  the 
base  which  he  selected,  using  the  series 

-3-2-1         o         i         2         3         4         5         6 
I        ||i248       16       32       64 

and  making  the  significant  remark:  "I  might  write  a  whole 
book  concerning  the  marvellous  things  relating  to  numbers, 
but  I  must  refrain  and  leave  these  things  with  eyes  closed."2 
What  these  mysteries  were  we  can  only  conjecture. 

1  "Qualicunq3  facit  Arithmetica  progressio  additione,  &  subtractione,  talis  facit 
progressio  Geometrica  multiplicatione,  &  diuisione.  ut  plene  ostendi  lib.  i.  capita 
de  geomet.  progres.   Vide  ergo, 

0.  i.          2.          3.  4.  5.          6.          7.          8. 

1.  2.          4.          8.  16.          32.        64.       128.       256. 

Sicut  ex  additione  (in  superiore  ordine)  3  ad  5  fiunt  8,  sic  (in  inferiore  ordine) 
ex  multiplicatione  8  in  32  fiunt  256.  Est  autem  3  exponens  ipsius  octonarij,  &  5 
est  exponens  32  &  8  est  exponens  numeri  256.  Item  sicut  in  ordine  superior!,  ex 
subtractione  3  de  7,  remanent  4,  ita  in  inferior!  ordine  ex  diuisione  128  per  8, 
fiunt  16"  (fols.  236,  237). 

It  will  be  noticed  that  he  speaks  of  8  as  "exponens  numeri  256,"  and  not  as  the 
exponent  of  2,  but  this  has  no  significance  with  respect  to  the  theory. 

2  "Posset  hie  fere  nouus  liber  integer  scribi  de  mirabilibus  numerorum,  sed 
oportet  ut  me  hie  subduca,  &  clausis  oculis  abea." 


522  LOGARITHMS 

A  number  of  French  writers  of  this  period  were  also  aware 
of  the  law,  and  Peletier1  (1549)  stated  it  clearly  for  the  case  of 
multiplication.  Five  years  later  Claude  de  Boissiere  elaborated 
this  treatment  and  spoke  of  the  " marvellous  operations"  which 
can  be  performed  by  means  of  the  related  series.  Two  years 
after  Boissiere's  work  was  published  the  theory  was  again  given 
by  Forcadel  (1565),  with  a  statement  that  the  idea  was  due  to 
Archimedes,  that  it  was  to  be  found  in  Euclid,  and  that  Gemma 
Frisius  had  written  upon  it.  Ramus  recognized  its  value  but 
added  nothing  to  it  or  to  its  possible  applications.  When, 
however,  Schoner  came  to  write  his  commentary  on  the  work 
of  Ramus,  in  1586,  a  decided  advance  was  made,  for  not  only 
did  he  give  the  usual  series  for  positive  exponents,  but,  like 
Stifel,  he  used  the  geometric  progressions  with  fractions  as  well, 
although,  as  stated  above,  not  with  negative  exponents.  Further, 
he  used  the  word  "index"  where  Stifel  had  used  "exponent," 
and,  like  this  noteworthy  writer,  gave  evidence  of  an  apprecia- 
tion of  the  importance  of  the  law.  In  general  the  French 
writers  already  named  (and  in  the  list  should  also  be  included 
the  name  of  Chauvet)  paid  no  attention  to  any  of  the  laws 
except  that  of  multiplication,  while  the  German  writers,  fol- 
lowing the  lead  of  Stifel,  took  the  broader  view  of  the  theory. 
This  was  not  always  the  case,  for  Sigismund  Suevus,  a  German 
arithmetician  who  wrote  as  late  as  1593,  did  not  go  beyond  the 
limits  set  by  most  of  the  French  arithmeticians;  but  in  general 
the  German  writers  were  in  the  lead.  This  is  particularly  true 
of  Simon  Jacob  (1565),  who  followed  Stifel  closely,  recognizing 
all  four  laws,  and,  as  is  well  known,  influencing  Jobst  Biirgi. 
These  writers  did  not  use  the  general  exponents  essential  to 
logarithms,  but  the  recognition  of  the  four  laws  is  significant. 

1  The  extract  here  given  is  from  the  1607  edition  of  L'Arithmetiqve,  p.  67.  In 
speaking  of  the  series 

3          6        12        24        48        96 
012345 

he  says :  "le  sc.auoir  qui  est  le  nobre  qui  eschet  au  neufieme  lieu  en  ceste  Progres- 
sion Double  le  diuise  48,  qui  est  sur  4,  par  le  premier  nombre  de  la  Progression, 
3 :  prouiennent  16 :  lesquels  je  multiplie  par  96,  qui  est  sur  5 :  (car  4  &  5  font  9) 
prouiendront  1536,  qui  sera  le  nombre  a  mettre  au  neufieme  lieu." 


BURGFS  TABLES 


S23 


Biirgi  and  the  Progress  Tabulen.  In  1 620  Jobst  Burgi  published 
his  Progress  Tabulen,  a  work  conceived  some  years  earlier.  As 
stated  above,  it  is  well  known  that  he  was  influenced  by  Simon 
Jacob's  work.  The  tables  were  printed  at  Prag  and  are  simply 
lists  of  antilogarithms  with  base  i.oooi.  The  logarithm  is 
printed  in  red  in  the  top  line  and  the  left-hand  column,  and  the 
antilogarithms  are  in  black,  and  hence  Biirgi  calls  the  logarithm 
Die  Rot  he  Zahl.  The  first  part  of  his  table  is  as  follows : 


0 

500 

1000 

1500 

2000 

0 

IOOOOOOOO 

100501227 

101004966 

101511230 

102020032 

10 

....  IOOOO 

....11277 

....15067 

21381 

....30234 

20 

....  2OOOI 

....21328 

25168 

..-.31534 

....40437 

30 

—  30003 

....31380 

...,35271 

41687 

50641 

The  manuscripts  of  Biirgi  are  at  the  Observatory  at  Pulkowa, 
but  none  seem  to  be  of  a  date  later  than  1610,  so  that  he  prob- 
ably developed  his  theory  independently  of  Napier.  It  is  evi- 
dent that  he  approached  the  subject  algebraically,  as  Napier 
approached  it  geometrically.1 

The  only  extensive  table  of  antilogarithms  is  due  to  James 
Dodson  (London,  1742). 

Logarithms  in  the  Orient.  Logarithms  found  their  way  into 
China  through  the  influence  of  the  Jesuits.  The  first  treatise 
upon  the  subject  published  in  that  country  was  a  work  by  one 
Sie  Fong-tsu,  a  pupil  of  the  Polish  Jesuit  John  Nicolas  Smogo- 
lenski  (1611-1656).  This  treatise  was  published  about  1650, 
although  Smogolenski  had  already  mentioned  the  theory  in  one 
of  his  works.2  Vlacq's  tables  (1628)  were  reprinted  in  Peking, 
as  already  stated,  in  1713. 

1Thus  Kepler  says:  "...  qui  etiam  apices  logistic!  Justo  Byrgio  multis  annis 
ante  editionem  Neperianam,  viam  praeiverunt,  ad  hos  ipsissimos  Logarithmos. 
Etsi  homo  cunctator  et  secretorum  suorum  custos,  foetum  in  partu  destituit,  non 
ad  usus  publicos  educavit"  (Opera  Omnia,  VII,  298)  (Frankfort  a.  M.,  1868). 

2 The  Tien-pu  Chen-yuan,  as  stated  in  Volume  I,  page  436. 


524    PERMUTATIONS,  COMBINATIONS,  PROBABILITY 

13.  PERMUTATIONS,  COMBINATIONS,  PROBABILITY 

Permutations  and  Combinations.  The  subject  of  permuta- 
tions may  be  said  to  have  had  a  feeble  beginning  in  China  in 
the  I-king  (Book  of  Changes),  the  arrangements  of  the  mystic 
trigrams,  as  in  ="=.  furnishing  the  earliest  known  example.1 
It  is  not  improbable  that  it  was  the  I-king  that  suggested  to  a 
certain  Japanese  daimyo  of  the  i2th  century  that  he  write  a 
book,  now  lost,  upon  permutations." 

Greek  Interest  in  the  Subject.  The  subject  received  some  slight 
attention  at  the  hands  of  certain  Greek  writers.  Plutarch8  (ist 
century)  tells  us4  that  Xenocrates  (c.  350  B.C.),  the  philosopher, 
computed  the  number  of  possible  syllables  as  1,002,000,000,000, 
but  it  does  not  seem  probable  that  this  represents  an  actual  case 
in  combinations.*  Plutarch  also  states  that  Chrysippus  (c.  280- 
c.  207  B.C.),  a  Stoic  philosopher,  found  the  number  of  combi- 
nations of  ten  axioms  to  be  more  than  1,000,000,  and  that 
Hipparchus  (c.  140  B.C..)  gave  the  number  as  ioi;o496  if  ad- 
mitted and  310,925  if  denied;  but  we  have  no  evidence  of  any 
theory  of  combinations  among  the  Greeks.7 

Interest  of  Latin  Writers  in  the  Subject.  The  Latin  writers, 
having  little  interest  in  any  phase  of  mathematics  except  the 
practical,  paid  almost  no  attention  to  the  theory  of  combina- 
tions. The  leading  exception  was  Boethius  (c.  510).  He  gives 
a  rule  for  finding  the  combinations  of  n  things  taken  two  at  a 
time  which  we  should  express  as  \n(ti—  i ) .8 

aSee  Volume  I,  page  25. 

2 The  theory  is  referred  to  as  Keishizan  in  Volume  I,  page  274. 

*Quaestiones  Conviv.,  Lib.  VIII,  9,  iii,  12 ;  ed.  Dubner,  II,  893  (Paris,  1877). 

4Tropfke,  Geschichte,  II  (i),  351. 

5Gow,  Greek  Math.,  pp.  71  n.,  86;  Tropfke,  Geschichte,  II  (i),  351. 

6"Centena  millia  atque  insuper  mille  et  quadraginta  novem."  The  number  in 
Tropfke  is  incorrect. 

7  With  respect  to  the  single  possible  case  in  Pappus,  see  ed.  Hultsch,  II,  646- 
649 :  "  Nam  ex  tribus  dissimilibus  generis  triades  diversae  inordinatae  existunt 
numero  decem." 

There  is  a  slight  trace  of  interest  in  the  subject  in  the  works  of  Plato  and 
Aristotle,  but  not  enough  to  be  worthy  of  discussion  in  this  chapter.  See  J.  L. 
Heiberg,  Philologus,  XLIII,  475,  with  references.  8  J.  L.  Heiberg,  ibid. 


EARLY  WORKS  525 

Hindu  Interest  in  the  Subject.  The  Hindus  seem  to  have  given 
the  matter  no  attention  until  Bhaskara  (c.  1150)  took  it  up  in 
his  Lilavati.  In  this  work  he  considered  the  subject  twice.  He 
asserted  that  an  idea  of  permutations  "serves  in  prosody  .  .  . 
to  find  the  variations  of  metre ;  in  the  arts  [as  in  architecture] 
to  compute  the  changes  upon  apertures  [of  a  building]  ;  and 
[in  music]  the  scheme  of  musical  permutations;  in  medicine, 
the  combinations  of  different  savours."1  He  gave  the  rules  for 
the  permutations  of  n  things  taken  r  at  a  time,  with  and  without 
repetition,  and  the  number  of  combinations  of  n  things  taken  r  at 
a  time  without  repetition.2 

Early  European  Interest  in  the  Subject.  Early  in  the  Christian 
Era  there  developed  a  close  relation  between  mathematics  and 
the  mystic  science  of  the  Hebrews  known  as  the  cabala.  This 
led  to  the  belief  in  the  mysticism  of  arrangements  and  hence  to 
a  study  of  permutations  and  combinations.  The  movement 
seems  to  have  begun  in  the  anonymous  Sejer  Jezira  (Book  of 
Creation),  and  shows  itself  now  and  then  in  later  works. 

It  seems  to  have  attracted  the  attention  of  the  Arabic 
and  Hebrew  writers  of  the  Middle  Ages  in  connection  with 
astronomy.  Rabbi  ben  Ezra  (c.  1140),  for  example,  considered 
it  with  respect  to  the  conjunctions  of  planets,  seeking  to  find 
the  number  of  ways  in  which  Saturn  could  be  combined  with 
each  of  the  other  planets  in  particular,  and,  in  general,  the 
number  of  combinations  of  the  known  planets  taken  two  at  a 
time,  three  at  a  time,  and  so  on.  He  knew  that  the  number  of 
combinations  of  seven  things  taken  two  at  a  time  was  equal  to 
the  number  taken  five  at  a  time,  and  similarly  for  three  and 
four  and  for  six  and  one.  He  states  no  general  law,  but  he 
seems  to  have  been  aware  of  the  rule  for  finding  the  combina- 
tions of  n  things  taken  r  at  a  time.3 

1Colebrooke  translation,  p.  49.  2Ibid.,  p.  123. 

3D.  Herzog,  Zophnath  Paneach  (in  Hebrew),  Cracow,  1911.  It  is  an  edition 
of  Josef  ben  Eliezer's  supercommentary  (that  is,  a  commentary  on  a  commen- 
tary by  Rabbi  ben  Ezra)  on  the  Bible.  The  passage  occurs  in  an  extract  from 
Rabbi  ben  Ezra's  astrological  manuscript  ha-Olam,  now  in  Berlin.  The  title  of 
the  book  means  "the  revealer  of  secrets."  See  also  J.  Ginsburg,  "Rabbi  ben  Ezra 
on  Permutations  and  Combinations,"  Mathematics  Teacher,  XV,  347. 


526     PERMUTATIONS,  COMBINATIONS,  PROBABILITY 

Levi  ben  Gerson,  in  his  Maassei  Choscheb  (Work  of  the 
Computer},  written  in  1321,  carried  the  subject  considerably 
farther.  He  gave  rules  for  the  permutation  of  n  things  taken 
all  together  and  also  taken  r  at  a  time,  and  for  the  combination 
of  n  things  taken  r  at  a  time.1 

A  few  years  later  Nicole  Oresme  (c.  1360)  wrote  a  work2  in 
which  he  gave  the  sum  of  the  numbers  representing  the  com- 
binations of  six  things  taken  i,  2,  3,  4,  and  5  at  a  time.  He 
also  gave  these  combinations  in  detail,  as  that  2^  —  15, 
SC6=  20,  and  so  on,  of  course  in  the  rhetorical  form,  and  seems 
to  have  known  the  general  law  involved,  although  he  did 
not  state  it. 

First  Evidence  of  Permutations  in  Print.  The  first  evidence  of 
an  interest  in  the  subject  to  be  found  in  the  printed  books  is 
given  in  PaciolPs  Suma  (1494),  where  he  showed  how  to  find 
the  number  of  permutations  of  any  number  of  persons  sitting 
at  a  table.3  In  England  the  subject  was  touched  upon  by 
W.  Buckley  (c.  1540),  who  gave  special  cases  of  the  combina- 
tions of  n  things  taken  r  at  a  time.  Tartaglia  (1523)  seems 
first  to  have  applied  the  theory  to  the  throwing  of  dice.4 

In  the  1 6th  century  the  learned  Rabbi  Moses  Cordovero5 
wrote  the  Pardes  Rimmonim  (Orchard  of  Pomegranates) ,6  in 
which  he  gave  an  interesting  treatment  of  permutations  and 
combinations  and  showed  some  knowledge  of  the  general  laws.7 

1  Enestrom,  Bibl.  Math.,  XIV  (3),  261;  G.  Lange,  German  translation  of  the 
treatise,  published  at  Frankfort  a.  M.,  1909;  Tropfke,  Geschichte,  VI  (2),  64. 

2  Tractatus  de  figuratione  potentiarum  et  mensurarum  difformitatum.    See 
H.  Wieleitner,  "Ueber  den  Funktionsbegriff  und  die  graphische  Darstellung  bei 
Oresme,"  BibL  Math.,  XIV  (3),  193. 

3Fol.  43,  v.  He  gives  the  results  for  n  =  i,  2,  .  .  .  ,  n,  and  adds  "Et  sic  in 
infinitum." 

4 In  the  General  Trattato,  II,  fol.  17,  r.,  he  states  that  he  discovered  the  rule : 
"Regola  generale  del  presente  auttore  ritrouata  il  primo  giorno  di  quarasima 
1'anno  1523.  in  Verona,  di  sapere  trouare  in  quanti  modi  puo  variar  il  getto  di 
che  quantita  di  dati  si  voglia  nel  tirar  quelli."  See  also  L'Enseignement  Mathe- 
matique,  XVI  (1914),  92. 

KBorn  at  Safed,  Palestine,  in  1522;  died  at  Safed,  June  25,  1570. 

6 Salonika,  1552,  with  later  editions. 

7M.  Turetsky,  "Permutations  in  the  i6th  century  Cabala,"  Mathematics 
Teacher,  XVI,  29. 


EARLY  PRINTED  WORKS 


S27 


At  about  the  same  time  Buteo  not  only  discussed  the  ques- 
tion of  the  number  of  possible  throws  with  four  dice1  but  took 
up  the  problem  of  a  combination  lock  with  several  movable 
cylinders  like  those  shown  in  the  illustration  of  the  lock  below. 


EARLY   COMBINATION    LOCK 
From  Buteo  's  Logistka,  1560  ed.,  p.  313 

As  would  naturally  be  expected,  special  cases  of  combina- 
tions of  various  kinds  occur  in  the  works  of  the  iyth  century. 
An  illustration  is  found  in  the  Arils  Analyticae  Praxis  (p.  13) 
of  Harriot,  where  the  following  symbolism  is  used  for  the 
product  of  binomials: 

—  aaaa  —  baaa  +  bcaa 

—  caaa  +  bdaa 

—  daaa  +  cdaa  —  bcda 

—  faaa  +  bfaa  —  bcfa 

+  cfaa  —  bdfa 
+  dfaa  -  cdfa  +  bcdf 
The  first  writer  to  give  the  general  rule  that 


a-b 
a  —  c 
a-d 
a-f 


r 


was  Herigone2  (1634). 


1  "  Ludens  aleator  lessens  quatuor,  quaero  quibus  &  quot  modis  inter  se  diuersis 
iacere  possit?"   Logistica,  Lyons,  1559;  1560  ed.,  p.  305. 
2Cursus  mathematicus,  II,  102.  Paris,  1634. 


528     PERMUTATIONS,  COMBINATIONS,  PROBABILITY 

In  his  work  on  the  arithmetic  triangle1  Pascal  showed  the 
relation  between  the  formation  of  the  binomial  coefficients  and 
the  theory  of  combinations,  a  subject  also  treated  of  by  Per  mat 
and  others.  Among  the  early  writers  upon  the  theory  were 
Huygens,  Leibniz,2  Frenicle,3  and  Wallis,4  and  there  is  a  brief 
tract  on  the  subject  which  is  thought  to  be  due  to  Spinoza 


The  first  work  of  any  extent  that  is  devoted  to  the  subject 
was  Jacques  Bernoulli's  Ars  Conjectandi?  This  work  contains 
the  essential  part  of  the  theory  of  combinations  as  known  to- 
day. In  it  appears  in  print  for  the  first  time,  with  the  present 
meaning,  the  word  "permutation."7  For  this  concept  Leibniz 
had  used  variationes  and  Wallis  had  adopted  alternationes. 
The  word  "combination"  was  used  in  the  present  sense  by  both 
Pascal  and  Wallis.8  Leibniz  used  complexiones  for  the  general 
term,  reserving  combinationes  for  groups  of  two  elements  and 
conternationes  for  groups  of  three,  —  words  which  he  general- 
ized by  writing  con2natio,  consnatio,  and  so  on. 

Probability.9  The  theory  of  probability  was  mentioned  in 
connection  with  the  throwing  of  dice  by  Benvenuto  d'  Imola,  a 
commentator  on  Dante's  Divina  Commcdia,  printed  in  the 


1  Written  c.  1654  but  printed  posthumously  in  1665.    Beginning  at  this  point, 
the  reader  may  profitably  consult  the  Encyklopddie  der  math.  Wissensch.,  I,  29. 

2  Ars  combinatoria^  1666. 

3"Abrege  des  combinaisons "  (1676),  published  in  the  Mem.  de  I'acad.  royale 
des  sciences,  Paris,  V  (1729),  167. 

4  De  combinationibus,  alternationibus,  et  partibus  aliqotis,  tractatus  (1685),  in 
his  Opera,  II,  483  (Oxford,  1693)  - 

5D.  Bierens  de  Haan,  "Twe  zeldzame  Werken  van  Benedictus  Spinoza,"  Nieuw 
Archief  voor  Wiskunde,  Amsterdam,  XI  (1884),  49-  The  title  of  the  tract  is 
Reeckening  van  Kanssen,  and  the  work  appeared  in  1687,  ten  years  after 
Spinoza's  death. 

6  Posthumously  printed  at  Basel  in  1713.  There  is  an  English  edition  of  1795 
under  the  title:  Permutations  and  Combinations:  Being  an  Essential  and  Funda- 
mental Part  of  the  Doctrine  of  Chances. 

7"De  Permutationibus.   Permutationes  rerum  voco  variationes.  .  .  ." 

8In  the  latter's  De  Combinationibus,  English  ed.,  1685;  Opera  (1693),  II, 
483.  His  definition  of  combinations  is  on  page  489  of  that  work. 

9 1.  Todhunter,  History  of  the  Mathematical  Theory  of  Probability,  Cam- 
bridge, 1865;  C.  Gouraud,  Histoire  du  Calcul  des  Probability,  Paris,  1848. 


PROBABILITY  529 

Venice  edition  of  I47?.1  The  gambling  question  first  appears 
in  a  mathematical  work,  however ;  in  Pacioli  Js  Suma2  (1494). 
Here  two  gamblers  are  playing  for  a  stake  which  is  to  go  to  the 
one  who  first  wins  n  points,  but  the  play  is  interrupted  when 
the  first  has  made  p  points  and  the  second  q  points.  It  is  re- 
quired to  know  how  to  divide  the  stakes.  The  general  problem 
also  appears  in  the  works  of  Cardan3  (1539)  and,  as  already 
stated,  of  Tartaglia4  (1556).  It  first  attracted  wide  attention 
in  connection  with  the  question  proposed  to  Pascal  (c.  1654) 
and  by  him  sent  to  Fermat.  The  statement  was  substantially 
the  one  given  in  Pacioli  to  the  effect  that  two  players  of  equal 
skill  left  the  table  before  completing  the  game.  The  stakes, 
the  necessary  score,  and  the  score  of  each  person  being  known, 
required  to  divide  the  stakes.  Pascal  and  Fermat  agreed  upon 
the  result,  but  used  different  methods  in  solving.  As  a  result 
of  the  discussion  so  much  interest  was  aroused  in  the  theory 
that  the  doctrine  of  probability  is  generally  stated  to  have  been 
founded  by  Pascal  and  Fermat. 

The  first  printed  work  on  the  subject  was  probably  a  tract 
of  Huygens  that  appeared  in  1657.°  There  also  appeared  an 
essay  upon  the  subject  by  Pierre  Remond  de  Montmort  in 
1708.°  The  first  book  devoted  entirely  to  the  theory  of  prob- 
ability was  the  Ars  Conjectandi  (1713)  of  Jacques  Bernoulli, 
already  mentioned.  The  second  book  upon  the  subject  was 
De  Moivre's  Doctrine  of  Chances :  or,  A  Method  of  Calculat- 
ing the  Probability  of  Events  in  Play  (1718) ;  and  the  third, 

1  Cantor,  Geschichte,  II  (2),  327;  Tropfke,  Geschichte,  II  (i),  356.  This  is  the 
fifth  or  sixth  printed  edition,  Hain  5942 ;  Copinger,  I,  185,  No.  5942. 

2 "Una  brigata  gioca  apalla  a  .60.  el  gioco  e  .10  p  caccia.  e  fano  posta  due  .10. 
acade  p  certi  accideti  che  no  possano  fornire  e  lima  jpte  a  .50.  e  laltra  .20.  se 
dimanda  che  tocca  p  pte  de  la  posta."  Fol.  197,  r. 

zpractica,  Milan,  1539,  "Caput  61.  De  extraordinariis  &  ludis,"  No.  17  of  the 
chapter. 

^General  Trattato  (1556),  I,  fol.  265,  r.,  where  he  quotes  Pacioli  under  the  title 
"Error  di  fra  Luca  dal  Borgo."  On  a  trace  of  the  theory  in  a  writing  by  Giovanni 
Francesco  Peverone  (c.  1550),  see  L.  Carlini,  //  Pitagora,  VII,  65. 

5"De  ratiociniis  in  ludo  aleae,"  in  Van  Schooten's  Exercitationum  mathe- 
maticarum  libri  quinque,  Leyden,  1657.  See  also  the  pars  prima  of  Bernoulli's 
Ars  Conjectandi. 

*Essai  d'analyse  sur  les  jeux  d'hasard,  Paris,  1708;  2d  ed.,  ibid.,  1714- 


530     PERMUTATIONS,  COMBINATIONS,  PROBABILITY 

Thomas  Simpson's  Laws  of  Chance  (1740).  One  of  the  best- 
known  works  on  the  theory  is  Laplace's  Theorie  analytique  des 
probability •,  which  appeared  in  1812.  In  this  is  given  his  proof 
of  the  method  of  least  squares. 

The  application  of  the  theory  to  mortality  tables  in  any  large 
way  may  be  said  to  have  started  with  John  Graunt,  whose 
Natural  and  Political  Observations  (London,  1662)  gave  a  set 
of  results  based  upon  records  of  deaths  in  London  from  1592. 
The  first  tables  of  great  importance,  however,  were  those  of 
Edmund  Halley,  contained  in  his  memoir  on  Degrees  of  Mor- 
tality of  Mankind,1  in  which  he  made  a  careful  study  of  an- 
nuities. It  should  be  said,  however,  that  Cardan  seems  to  have 
been  the  first  to  consider  the  problem  in  a  printed  work,  al- 
though his  treatment  is  very  fanciful.  He  gives  a  brief  table 
in  his  proposition  "Spatium  vitae  naturalis  per  spatium  vitae 
fortuitum  declarare,"  this  appearing  in  the  De  Proportionibvs 
Libri  F,2  p.  204. 

Although  a  life-insurance  policy  is  known  to  have  been  under- 
written by  a  small  group  of  men  in  London  in  1583,  it  was  not 
until  1699  that  a  well-organized  company  was  established  for 
this  purpose. 

Besides  the  early  work  of  Graunt  and  Halley  there  should  be 
mentioned  the  Essai  sur  les  probabilites  de  la  vie  humaine  (Paris, 
1746 ;  supplementary  part,  1760)  by  Antoine  Deparcieux  the  el- 
der (1703-1768).  The  early  tables  were  superseded  in  the 
1 8th  century  by  the  Northampton  Table.  Somewhat  later  the 
Carlisle  Table  was  constructed  by  Joshua  Milne  (1776-1853). 
In  1825  the  Equitable  Life  Assurance  Society  of  London  began 
the  construction  of  a  more  improved  table,  since  which  time 
other  contributions  in  the  same  field  have  been  made  by  the 
Institute  of  Actuaries  of  Great  Britain  in  cooperation  with 
similar  organizations,  by  Sheppard  Romans  (c.  1860)  of  New 
York, — the  so-called  American  Experience  Table, — and  by 
Emory  McClintock  (1840-1916),  also  of  New  York. 

1PhU.  Trans.,  London,  1693. 

2  Basel,  1570.  For  a  sketch  of  the  later  tables  see  the  articles  on  Life  Insur- 
ance in  the  encyclopedias. 


DISCUSSION  531 

TOPICS  FOR  DISCUSSION 

1.  Leading  steps  in  the  development  of  algebra. 

2.  General  racial  characteristics  shown  in  the  early  development 
of  algebra. 

3.  The  early  printed  classics  on  algebra. 

4.  Various  names  for  algebra,  with  their  origin  and  significance. 

5.  Development  of  algebraic  symbolism  relating  to  the  four  fun- 
damental operations  and  to  aggregations. 

6.  Development  of  symbolism  relating  to  powers  and  roots. 

7.  Development  of  symbolism  relating  to  the  equality  and  to  the 
inequality  of  algebraic  expressions. 

8.  Methods  of  expressing  equations,  with  a  discussion  of  their 
relative  merits. 

9.  Methods  of  solving  linear  equations. 

10.  Methods  of  solving  quadratic  equations, 
n.  History  of  the  discovery  of  the  method  of  solving  cubic  and 
biquadratic  equations. 

12.  History  of  continued  fractions  and  of  their  uses. 

13.  General  steps  in  the  development  of  the  numerical  higher 
equation. 

14.  History  of  the  Rule  of  False  Position,  with  the  reasons  for  the 
great  popularity  of  the  rule. 

15.  Development  of  the  idea  of  classifying  equations  according  to 
degree  instead,  for  example,  according  to  the  number  of  terms. 

1 6.  Development  of  the  indeterminate  equation. 

17.  General  steps  in  the  application  of  trigonometry  to  the  solution 
of  the  quadratic  and  cubic  equations. 

1 8.  General  steps  in  the  early  development  of  determinants. 

19.  History  of  the  Rule  of  Three  and  of  its  relation  to  proportion. 

20.  General  nature  of  series  in  the  early  works  on  mathematics. 

21.  History  of  infinite  products  in  the  I7th  century. 

2  2 .  The  historical  development  of  the  Binomial  Theorem. 

23.  History  and  applications  of  Taylor's  and  Maclaurin's  formulas. 

24.  History  of  the  Pascal  Triangle  and  of  its  applications. 

25.  The  invention  of  logarithms  and  the  history  of  their  various 
applications. 

26.  History  of  permutations,   combinations,   and   the  theory  of 
probability. 


CHAPTER  VII 

ELEMENTARY  PROBLEMS 
I.  MATHEMATICAL  RECREATIONS 

Purpose  of  the  Study.  In  this  chapter  we  shall  consider  a 
few  of  the  most  familiar  types  of  problems  that  have  come 
down  to  us.  Some  of  these  types  relate  to  arithmetic,  while 
others  have  of  late  taken  advantage  of  algebraic  symbolism, 
although  at  one  time  they  were  solved  without  the  modern  aids 
that  algebra  supplies. 

Mathematical  Recreations.  Ever  since  problems  began  to  be 
set,  the  mathematical  puzzle  has  been  in  evidence.  Without 
defining  the  limits  that  mark  the  recreation  problem  it  may  be 
said  that  the  Egyptians  and  Orientals  proposed  various  ques- 
tions that  had  no  applications  to  daily  life,  the  chief  purpose 
being  to  provide  intellectual  pleasure.  The  Greeks  were  even 
more  given  to  this  type  of  problem,  and  their  geometry  was  de- 
veloped partly  for  this  very  reason.  In  the  later  period  of  their 
intellectual  activity  they  made  much  of  indeterminate  problems, 
and  thereafter  this  type  ranked  among  the  favorite  ones. 

In  the  Middle  Ages  there  developed  a  new  form  of  puzzle 
problem,  one  suggested  by  the  later  Greek  writers  and  modified 
by  Oriental  influences.  This  form  has  lasted  until  the  present 
time  and  will  probably  continue  to  have  a  place  in  the  schools. 

Problems  of  Metrodorus.  So  far  as  the  Greeks  were  concerned, 
the  source  book  for  this  material  is  the  Greek  Anthology  .^ 
This  contains  the  arithmetical  puzzles  supposed  to  be  due  to 

^^The  first  noteworthy  edition  was  that  of  Friedrich  Jacobs,  Leipzig,  1813- 
1817.  There  is  an  English  translation  by  W.  R.  Paton,  London,  1918,  being 
Volume  V  of  the  Loeb  Classical  Library.  In  this  translation  the  arithmetic  prob- 
lems begin  on  page  25,  and  from  these  the  selections  given  here  have  been  made. 

532 


PROBLEMS  OF  METRODORUS  533 

Metrodorus  about  the  year  500 ( ?).   A  few  of  these  problems 
will  serve  to  show  the  general  nature  of  the  collection. 

Polycrates  Speaks :  "  Blessed  Pythagoras,  Heliconian  scion  of  the 
Muses,  answer  my  question :  How  many  in  thy  house  are  engaged  in 
the  contest  for  wisdom  performing  excellently?" 

Pythagoras  Answers:  "I  will  tell  thee,  then,  Polycrates.  Half  of 
them  are  occupied  with  belles  lettres ;  a  quarter  apply  themselves  to 
studying  immortal  nature ;  a  seventh  are  all  intent  on  silence  and  the 
eternal  discourse  of  their  hearts.  There  are  also  three  women,  and 
above  the  rest  is  Theano.  That  is  the  number  of  interpreters  of  the 
Muses  I  gather  round  me." 

The  following  problem  relates  to  a  statue  of  Pallas: 

"I,  Pallas,  am  of  beaten  gold,  but  the  gold  is  the  gift  of  lusty 
poets.  Christians  gave  half  the  gold,1  Thespis  one  eighth,  Solon  one 
tenth,  and  Themison  one  twentieth,  but  the  remaining  nine  talents 
and  the  workmanship  are  the  gift  of  Aristodicus." 

The  following  relates  to  the  finding  of  the  hour  indicated  on  a 
sundial  and  still  appears  in  many  algebras,  modified  to  refer  to 
modern  clocks : 

"Best  of  clocks,2  how  much  of  the  day  is  past?" 
"  There  remain  twice  two  thirds  of  what  is  gone." 

The  next  problem  involves  arithmetic  series,  as  follows: 

Croesus  the  king  dedicated  six  bowls  weighing  six  minae,3  each 
[being]  one  drachma  heavier  than  the  other.4 

A  type  that  has  long  been  familiar  in  its  general  nature  is  seen 
in  the  following: 

A.  " Where  are  thy  apples  gone,  my  child?" 

B.  "Ino  has  two  sixths,  and  Semele  one  eighth,  and  Autonoe  went 
off  with  one  fourth,  while  Agave  snatched  from  my  bosom  and  carried 

1  It  should  be  recalled  that  this  was  written  probably  about  the  time  of  such 
Christian  scholars  as  Capella  and  Cassiodorus. 

2  Literally,  hour  indicator. 

3  A  mina  contained  100  drachmas. 

4That  is,  than  the  one  next  smaller.   Find  the  weight  of  each, 
ii 


534  MATHEMATICAL  RECREATIONS 

away  a  fifth.    For  thee  ten  apples  are  left,  but  I,  yes  I  swear  it 
by  dear  Cypris,  have  only  this  one."1 

The  following  problem  has  more  of  an  Oriental  atmosphere : 

"After  staining  the  holy  chaplet  of  fair-eyed  Justice  that  I  might 
see  thee,  all-subduing  gold,  grow  so  much,  I  have  nothing ;  for  I  gave 
forty  talents  under  evil  auspices  to  my  friends  in  vain,  while,  O 
ye  varied  mischances  of  men,  I  see  my  enemy  in  possession  of  the 
half,  the  third,  and  the  eighth  of  my  fortune."2 

One  of  the  remote  ancestors  of  a  type  frequently  found  in  our 
algebras  appears  in  the  following  form: 

"Brick-maker,  I  am  in  a  great  hurry  to  erect  this  house.  Today 
is  cloudless,  and  I  do  not  require  many  more  bricks,  but  I  have  all  I 
want  but  three  hundred.  Thou  alone  in  one  day  couldst  make  as 
many,  but  thy  son  left  off  working  when  he  had  finished  two  hundred, 
and  thy  son-in-law  when  he  had  made  two  hundred  and  fifty.  Work- 
ing all  together,  in  how  many  days  can  you  make  these  ?  " 

This  collection  of  puzzles,  now  attributed  entirely  to  Metro- 
dorus,  contains  numerous  enigmas,  one  of  which  is  numerical 
enough  to  deserve  mention : 

If  you  put  one  hundred  in  the  middle  of  a  burning  fire,  you  will 
find  the  son  and  slayer  of  a  virgin.3 

Comparison  with  Oriental  Problems.  Such  problems  seem 
more  Oriental  than  Greek  in  their  general  form,  but  if  we  could 
ascertain  the  facts  we  should  probably  find  that  every  people 
cultivated  the  somewhat  poetic  style  in  the  recreations  of 
mathematics.  It  happens,  however,  that  we  have  more  evidence 
of  it  in  India  and  China  than  we  have  in  the  Mediterranean 
countries,  and  hence  we  are  led  to  believe  it  was  more  frequently 
found  among  the  higher  class  of  mathematicians  in  the  East 
than  among  those  of  the  West. 

1  There  were  120,  for  120  =  40  4-  15  -f  30  +  24  -f-  10  -f  i. 

2480  +  320  -f  120  +  40  =  960. 

8  The  answer  is  Pyrrhus,  son  of  Deidameia  and  slayer  of  Polyxena;  for  if  p, 
the  Greek  symbol  for  100,  is  inserted  in  the  middle  of  the  genitive  form  irup6s 
(fire),  it  becomes  irvpp6s  (Pyrros,  Pyrrhus).  This  is  the  mythological  Pyrrhus 
(Neoptolemus) ,  son  of  Achilles  and  Deidameia. 


EARLY  PROBLEMS  535 

Medieval  Collections.  The  first  noteworthy  collection  of 
recreations,  after  the  one  in  the  Greek  Anthology,  is  the  Propo- 
sitiones  ad  acuendos  iuvenes,  of  which  there  is  extant  no  manu- 
script written  before  the  year  1000.  This  collection  is 
attributed  to  Alcuin  of  York  (c.  775),  who  is  known  to  have 
sent  a  list  of  such  recreations  to  Charlemagne.1  It  contains 
many  stock  problems  such  as  those  of  the  hare  and  hound,  and 
the  cistern  pipes.  Rabbi  ben  Ezra  (c.  1140),  Fibonacci  (1202), 
Jordanus  Nemorarius  (c.  1225),  and  many  other  medieval 
writers  made  use  of  these  standard  types. 

Printed  Books.  The  first  noteworthy  collection  of  recreative 
problems  to  appear  in  print  was  that  of  Claude-Caspar  Bachet 
(i6i2).2  While  not  so  popular  as  various  later  works,  and 
containing  much  that  is  trivial,  it  was  a  pioneer  and  is  much 
better  than  some  of  those  that  went  through  many  more 
editions. 

From  the  bibliographical  standpoint  the  most  interesting  of 
the  printed  collections  is  that  of  a  Jesuit  scholar,  Jean  Leure- 
chon  (i624).3  He  published  his  work  under  the  name  of 
H[endrik]  van  Etten  at  Pont-a-Mousson  in  1624.  It  was  a  poor 
collection  of  trivialities,4  but  it  struck  the  popular  fancy  and 
went  through  at  least  thirty-  four  editions  before  1  700,  some  of 
these  being  published  under  other  names. 

The  next  writer  of  note  was  Jacques  Ozanam  (1640-1717), 
a  man  who  was  self-taught  and  who  had  a  gift  for  teaching 
others.  He  had  faith  in  the  educational  value  of  recreations, 
and  this  fact,  together  with  his  familiarity  with  the  subject 
and  his  success  as  a  teacher,  enabled  him  to  write  one  of  the 
most  popular  works  on  the  subject  that  has  ever  appeared 


aliquas  figuras  Arithmeticae  subtilitatis  laetitiae  causa*'  (Cantor 
Geschichte,  I  (2),  784). 

2Problemes  plaisans  et  delectables,  qui  se  font  par  Us  nombres.  Partie  re- 
cueillis  de  diuers  autheurs,  &  inuentez  de  nouueau  auec  leur  demonstration,  Lyons 
1612.  There  is  a  copy  of  this  edition  in  the  Harvard  Library.  Later  editions: 
Lyons,  1624;  Paris,  1874,  J879,  1884. 

8Born  at  Bar-le-Duc,  c.  1591  ;  died  at  Pont-a-Mousson,  January  17,  1670.  He 
wrote  on  astronomy. 

4Montucla,  in  his  revision  of  Ozanam,  speaks  of  it  as  "  une  pitoyable  rapsodie.' 


536  TYPICAL  PROBLEMS 

The  work  was  first  published  in  1692  or  1694*  and  since  then 
there  have  been  at  least  twenty  different  editions. 

There  have  been  many  other  works  on  the  subject,2  but  none 
of  them  has  had  the  popular  success  of  those  of  Leurechon 
and  Ozanam. 

Japanese  Geometric  Problems.  The  Japanese  inherited  from 
the  Chinese  a  large  number  of  curious  geometric  problems,  and 

by  their  own  ingenuity 
and  perseverance  elabo- 
rated these  tests  of  skill 
until  they  far  surpassed 
their  original  teachers. 
Some  of  these  problems 
were  mentioned  in  Volume 
I,  and  the  circle  problem 
will  be  referred  to  in 
A  FAN  PROBLEM  FROM  JAPAN  Chapter  X  of  this  volume ; 

From  TakedaShingen's5a^^^,  1824       but   1H   tWs   Connection   it 

is  proper  to  refer  to  one 

type  of  interesting  problems  frequently  found  in  the  early 
Japanese  works.  These  problems  refer  to  the  inscribing  and 
measuring  of  circles  inscribed  in  various  figures  such  as  semi- 
circles, fans,  and  ellipses. 

2.  TYPICAL  PROBLEMS 

Pipes  filling  the  Cistern.  Few  problems  have  had  so  extended 
a  history  as  the  familiar  one  relating  to  the  pipes  filling  a  cis- 
tern,3 and  the  traveler  who  is  familiar  with  the  Mediterranean 

aThe  date  1692  is  on  the  testimony  of  Montucla,  in  his  1790  edition  of  Ozanam. 
It  is  probable  that  he  was  in  error  on  this  point.  See  L'Intermediare  des  Matht- 
maticiens,  VI,  112,  and  various  histories  of  mathematics. 

2  Bibliographies  that  are  fairly  complete  may  be  found  in  E.  Lucas,  Recreations 
Mathtmatiques,  4  vols.,  I,  237  (Paris,  1882-1894) ;  W.  Ahrens,  Mathematische 
Unterhaltungen  und  Spiele,  p.  403  (Leipzig,  1901 ;  2d  ed.,  1918) .  These  are  the 
leading  modern  contributors  to  the  subject,  the  works  of  Lucas  being  probably 
the  best  that  have  as  yet  appeared. 

3 See  the  author's  article  in  the  Amer.  Math.  Month.,  XXIV,  64,  from  which 
extracts  are  here  made. 


PROBLEM  OF  TANGENT  CIRCLES 
From  a  manuscript  by  Iwasaki  Toshihisa  (c.  1775) 


538  TYPICAL  PROBLEMS 

lands  cannot  fail  to  recognize  that  here  is  its  probable  origin. 
Not  a  town  of  any  size  that  bears  the  stamp  of  the  Roman 
power  is  without  its  public  fountain  into  which  or  from  which 
several  conduits  lead.  In  the  domain  of  physics,  therefore, 
this  would  naturally  be  the  most  real  of  all  the  problems  that 
came  within  the  purview  of  every  man,  woman,  or  child  of  that 
civilization.  Furthermore,  the  elementary  clepsydra1  may  also 
have  suggested  this  line  of  problems,  the  principle  involved 
being  the  same. 

The  problem  in  definite  form  first  appears  in  the  Mer/o^cm? 
(metre' 'seis)  of  Heron  (c.  50?),  and  although  there  is  some  ques- 
tion as  to  the  authorship  and  date  of  the  work,  there  is  none  as 
to  the  fact  that  this  style  of  problem  would  appeal  to  such  a 
writer  as  he.  It  next  appears  in  the  writings  of  Diophantus 
(c.  275) 2  and  among  the  Greek  epigrams  of  Metrodorus 
(c.  500?),  and  soon  after  this  it  became  common  property  in 
the  East  as  well  as  the  West.  It  is  found  in  the  list  attributed 
to  Alcuin  (c.  775) ;  in  the  Lildvati  of  Bhaskara3  (c.  1150) ;  in 
the  best-known  of  all  the  Arab  works  on  arithmetic,  the  Kho- 
lasat  al-Hisab  of  Beha  Eddin  (c.  1600);  and  in  numerous 
medieval  manuscripts.  When  books  began  to  be  printed  it  was 
looked  upon  as  one  of  the  standard  problems  of  the  schools,  and 
many  of  the  early  writers  gave  it  a  prominent  position,  among 
them  being  men  like  Petzensteiner  (1483),  Tonstall  (1522), 
Gemma  Frisius  (1540),  and  Robert  Recorde  (c.  1542 ).4 

1  Attributed  to  Plato  (c.  380  B.C.)  but  improved  by  Ctesibius  of  Alexandria 
in  the  second  century  B.C.  On  the  subject  of  clepsydrae  see  Chapter  IX  of  this 
volume. 

2 In  Bachet's  edition  (the  Fermat  edition  of  1670,  p.  271)  appears  this  metrical 
translation : 

Totum  implere  lacum  tubulis  e  quatuor,  uno 
Est  potis  iste  die,  binis  hie  &  tribus  ille, 
Quatuor  at  quartus. 
Die  quo  spatio  simul  omnes. 

8See  Taylor's  translation,  p.  50;  Colebrooke  translation,  p.  42. 

4  In  Recorde  it  appears  for  the  first  time  in  English :  "  Ther  is  a  cestern  with 
iiij.  cocks,  conteinyng  72  barrels  of  water,  And  if  the  greatest  cocke  be  opened, 
the  water  will  auoyde  cleane  in  vj  howers,"  etc.  (Ground  of  Artes,  1558  ed., 
fol.A7,v.). 


THE  CISTERN  PROBLEM  539 

Variants  of  the  Problem.  Such,  then,  was  the  origin  of  what 
was  once  a  cleverly  stated  problem  of  daily  life.  This  problem, 
like  dozens  of  others,  went  through  many  metamorphoses,  of 
which  only  a  few  will  here  be  mentioned. 

In  the  isth  century,  and  probably  much  earlier,  there  ap- 
peared the  variant  of  a  lion,  a  dog,  and  a  wolf,  or  other  animals, 
eating  a  sheep,1  and  this  form  was  even  more  common  in  the 
1 6th  century.2 

In  the  1 6th  century  we  also  find  in  several  books  the  variant 
of  the  case  of  men  building  a  wall  or  a  house,  and  this  form 
has  survived  to  the  present  time.  It  appeared  in  TonstalPs  De 
Arte  Supputandi  (iS22)3  and  in  Cataneo's  work  (i546),4  and 
in  due  time  became  modified  to  the  form  beginning,  "If  A  can 
do  a  piece  of  work  in  4  days,  B  in  3  days,"  and  so  on. 

The  influence  of  the  wine-drinking  countries  shows  itself  in 
the  variant  given  by  Gemma  Frisius  (1540),°  who  states  that 
a  man  can  drink  a  cask  of  wine  in  20  days,  but  if  his  wife 
drinks  with  him  it  will  take  only  14  days,  from  which  it  is  re- 
quired to  find  the  time  it  would  take  his  wife  alone. 

The  influence  of  a  rapidly  growing  commerce  led  one  of  the 
German  writers  of  1540  to  consider  the  case  of  a  ship  with 


1Johann  Widman  (1489)  under  the  chapter  title  "Eyn  fasz  mit  3  zapffen." 
His  form  is:  "Lew  Wolff  Hunt  Itm  des  gleichen  i  lew  vnd  i  hunt  vh  i  wolff 
diese  essen  mit  einander  i  schaff.  Vnd  der  lew  esz  das  schaff  allein  in  einer  stund. 
Vnd  d'  wolf  in  4  stunden.  Vnd  der  hunt  in  6  stunden.  Nun  ist  die  frag  wan  sy 
dass  schaff  all  3  mit  einader  essen/  in  wie  lager  zeit  sy  das  essen"  (1509  ed., 
fol.  92  ;  1519  ed.,  fol.  112) . 

2Thus  Cataneo,  Le  Pratiche,  1546;  Venice  edition  of  1567,  fol.  59,  v.:  "Se  un 
Leone  mangia  in  2.  hore  una  pecora,  &  1'  Orso  la  mangia  in  3.  hore,  &  il  Leopardo 
la  mangia  in  4.  hore,  dimandasi  cominciando  a  mangiare  una  pecora  tutti  e  3.  a 
un  tratto  in  quanto  tempo  la  fmirebbono." 

This  form  is  also  found  in  J.  Albert's  work  of  iS34  (T56i  ed.,  fol.  N  viii),  in 
Coutereels  (1631  ed.,  p.  352),  and  in  the  works  of  numerous  other  writers. 

In  this  chapter  a  few  authors  of  textbooks  will  be  mentioned  whose  names  are 
not  of  sufficient  importance  to  entitle  them  to  further  attention.  The  dates  will 
serve  to  show  their  relative  chronological  position.  For  names  of  major  im- 
portance consult  Volume  I. 

sWith  the  statement  that  it  is  similar  to  the  one  about  the  cistern  pipes: 
"Questio  haec  similis  est  illi  de  cisterna  tres  habete  fistulas:  et  simili  modo 
soluenda"  (fol.  f  i).  *See  fol.  60,  v.,  of  the  Venice  edition  of  1567. 

1*1563  edition  of  his  arithmetic,  fol.  38. 


540 


TYPICAL  PROBLEMS 


3  sails,  by  the  aid  of  the  largest  of  which  a  voyage  could  be  made 
in  2  weeks,  with  the  next  in  size  in  3  weeks,  and  with  the 
smallest  in  4  weeks,  it  being  required  to  find  the  time  if  all 


Llucondocro  cmpieu 
nafontc  w^dirqun 
do  e  picna  non  mcrrcn 
do  ilcondocto  t  fttir.i 
doiluoracoiofmorcrc 
beladccra  foiue  in  \  \ 
di :  do  fapere  efiendo 
uora  la  fonre  r  mctren 
do  i!cond.oceo  t  fturn 
doiluoratoio  tfiquiui 
di  fara  plena  la  decra 
fonte 


Qnoferpenrcein 
po>o 


. 

uolendoufhrc  fuoraoj 
gnidi  falc^  dtbraccio" 
crdipot  bnocre/cead: 
^  dibrarcfo  :  no 
inqaari  dt  fara 


FROM    CALANDRI'S   WORK   OF   1491 

The  problems  of  the  pipe  filling  the  cistern  and  of  the  serpent  crawling  out  of  the 
well.    Calandri's  was  the  first  arithmetic  printed  with  illustrations 

three  were  used.  Unfortunately  several  factors  were  ignored, 
such  as  that  of  one  sail  blanketing  the  others  and  the  fact  that 
the  speed  is  not  proportional  to  the  power.1 

1"Item/  i  ein  Schiff  mit  3  Siegeln  gehet  vom  Sund  gen  Riga/  Mit  dem 
grosten  allein/  in  2  wochen/  Mit  dejn  andern/  in  3  wochen/  Vnnd  mit  dem 
kleinsten/in  4  wochen,"  etc.  (J.  Albert  (1540;  1561  ed.,  fol.  N  vii)). 


THE  TURKS  AND  CHRISTIANS  541 

The  agricultural  interests  changed  the  problem  to  that  of 
a  mill  with  four  "  Gewercken,"  *  and  other  interests  continued 
to  modify  it  further  until,  as  is  usually  the  case,  the  style  of 
problem  has  tended  to  fall  % 

from  its  own  absurdity.    Its  \    *  *  •  * 

varied  history  may  be  closed  Q  *  O 

by  referring  to  a  writer  of  the          O  O 

early  iQth  century,2  moved  by        *  * 

a  bigotry  which  would  hardly  ^ 
be  countenanced  today,  who  O 
proposed  to  substitute  a  prob-  ° 
lem  relating  to  priests  praying  o 

for  souls  in  purgatory.  *  0 

/  O  e 

Turks  and  Christians.  There  *     *  °   •  ° 

is  a  well-known  problem  which         THE  TURKS  AND  CHRISTIANS 
relates  that  fifteen  Turks  and     From  Buteo,s  Logistic(ly  LyonSj  I559 
fifteen  Christians  were  on  a     (1560  ed.,  p.  304).  The  problem  be- 


ship  which  was  in  danger,  and     &*}  /'In  f  au^  vecton*  . 

,,,<.,      ,         .  .i       i        Chnstiam     totideq;     ludei,     suborta 

that  half  had  to  be  Sacrificed.  tepestate  magna" 

It  being  necessary  to  choose 

the  victims  by  lot,  the  question  arose  as  to  how  they  could 
be  arranged  in  a  circle  so  that,  in  counting  round,  every  fifteenth 
one  should  be  a  Turk. 

It  is  probable  that  the  problem  goes  back  to  the  custom  of 
decimatio  in  the  old  Roman  armies,3  the  selection  by  lot*  of 
every  tenth  man  when  a  company  had  been  guilty  of  cowardice, 
mutiny,  or  loss  of  standards  in  action.  Both  Livy  (II,  59)  and 
Dionysius  (IX,  50)  speak  of  it  in  the  case  of  the  mutinous  army 
of  the  consul  Appius  Claudius  (471  B.C.),  and  Dionysius  fur- 
ther speaks  of  it  as  a  general  custom.  Polybius  (VI,  38)  says 
that  it  was  a  usual  punishment  when  troops  had  given  way  to 


^'Ein  Mlilmeister  hat  ein  Mule  mit  vier  Gewercken/  Mit  dem  ersten  mehlt 
er  in  23  studen  35  Scheffel/Mit  dem  andern  39  Scheffel/  Mit  dem  dritten  46 
Scheffel/Vnnd  mit  dem  vierten  52  Scheffel,"  etc.  The  question  then  is,  How  long 
it  will  take  them  together  to  grind  19  Wispel  (i  Wispel  =  24  Scheffel)  (ibid.). 

2R.  Hay,  The  Beauties  of  Arithmetic,  p.  218  (1816). 

3E.  Lucas,  Arithmetique  Amwante,  p.  17  (Paris,  1895). 


542 


TYPICAL  PROBLEMS 


panic.  The  custom  seems  to  have  died  out  for  a  time,  for  when 
Crassus  resorted  to  decimation  in  the  war  of  Spartacus  he  is 
described  by  Plutarch  (Crassus,  10)  as  having  revived  an  an- 

_      cient  punishment.  It  was  ex- 
^i     tensively   used   in   the   civil 

'">      wars  and  was  retained  under 

/-.» 

^     the    Empire,    sometimes    as 
\^i 


\ 


vicesimatio  (every  twentieth 
man  being  taken),  and  some- 
times as  centesimatio  (every 
hundredth  man). 

Now  it  is  very  improbable 
that  those  in  charge  of  the 
selection  would  fail  to  have 
certain  favorites,  and  hence 
it  is  natural  that  there  may 
have  grown  up  a  scheme  of 
selection  that  would  save  the 
latter  from  death.  Such  cus- 
toms may  depart,  but  their 
influence  remains. 

In  its  semimathematical 
form  the  problem  is  first  re- 
ferred to  in  the  work  of  an 
unknown  author,  possibly 
Ambrose  of  Milan  (£.370), 
who  wrote,  under  the  nom  de 
plume  of  Hegesippus,  a  work 
De  hello  iudaico.1  In  this 
work  he  refers  to  the  fact  that 

Josephus  was  saved  on  the  occasion  of  a  choice  of  this  kind.2 
Indeed,  Josephus  himself  refers  to  the  matter  of  his  being  saved 
by  lucky  chance  or  by  the  act  of  God.3 

i  Edited  by  C.  F.  Weber  and  J.  Caesar,  Marburg,  1864.  See  W.  Ahrens,  Math. 
Unterhaltungen  und  Spiele,  p.  286  (Leipzig,  1901 ;  2d  ed.,  1918). 

2"Itaque  accidit  ut  interemtis  reliquis  losephus  cum  altero  superesset  neci" 
(quoted  from  Ahrens,  loc.  tit.). 

8KaTaAehreTcu  8£  OUTOS,  efrc  virb  rtixw  XP^l  Mycivctre  virb  OeoO  irpovotas,  <rbv  trtpy. 


THE  JOSEPHUS  PROBLEM  IN 
JAPAN 

From   Muramatsu    Kudayu   Mosei's 
Mantoku   Jinko-ri    (1665) 


THE  JOSEPHUS  PROBLEM 


543 


The  oldest  European  trace  of  the  problem,  aside  from  that 
of  Hegesippus,  is  found  in  a  manuscript  of  the  beginning  of 
the  icth  century.  It  is  also  referred  to  in  a  manuscript  of  the 
nth  century  and  in  one  of  the  i2th  century.  It  is  given  in 


THE  JOSEPHUS  PROBLEM  IN  JAPAN 

From  Miyake  Kenryu's  Shojutsu  Sangaku  Zuye  (1795  ed.),  showing  the  problem 
of  the  stepmother,  referred  to  on  page  544 


the  To^j^  (c.  1140),  and  indeed  it  is  to 

this  writer  that  Elias  Levita,  who  seems  first  to  have  given  it 
in  printed  form  (1518),  attributes  its  authorship. 

The  problem,  as  it  came  to  be  stated,  related  that  Josephus, 
at  the  time  of  the  sack  of  the  city  of  Jotapata  by  Vespasian, 
hid  himself  with  forty  other  Jews  in  a  cellar.  It  becoming 
necessary  to  sacrifice  most  of  the  number,  a  method  anal- 
ogous to  the  old  Roman  method  of  decimatio  was  adopted,  but 
in  such  a  way  as  to  preserve  himself  and  a  special  friend.  It  is 


544  TYPICAL  PROBLEMS 

on  this  account  that  German  writers  still  call  the  ancient 
puzzle  by  the  name  of  Josephsspiel. 

Chuquet  (1484)  mentions  the  problem,  as  does  at  least  one 
other  writer  of  the  i5th  century.1  When,  however,  printed 
works  on  algebra  and  higher  arithmetic  began  to  appear,  it 
became  well  known.  The  fact  that  such  writers  as  Cardan2 
and  Ramus8  gave  it  prominence  was  enough  to  assure  its  com- 
ing to  the  attention  of  scholars.4 

Like  so  many  curious  problems,  this  one  found  its  way  to  the 
Far  East,  appearing  in  the  Japanese  books  as  relating  to  a 
stepmother's  selection  of  the  children  to  be  disinherited.  With 
characteristic  Japanese  humor,  however,  the  woman  was  de- 
scribed as  making  an  error  in  her  calculations,  so  that  her 
own  children  were  disinherited  and  her  stepchildren  received 
the  estate. 

Testament  Problem.  There  is  a  well-known  problem  which 
relates  that  a  man  about  to  die  made  a  will  bequeathing  ^  of 
his  estate  to  his  widow  in  case  an  expected  child  was  a  son,  the 
son  to  have  f ;  and  f  to  the  widow  if  the  child  was  a  daugh- 
ter, the  daughter  to  have  |.  The  issue  was  twins,  one  a  boy 
and  the  other  a  girl,  and  the  question  arose  as  to  the  division 
of  the  estate. 

The  problem  in  itself  is  of  no  particular  interest,  being  legal 
rather  than  mathematical,  but  it  is  worthy  of  mention  because 
it  is  a  type  and  has  an  extended  history.  Under  both  the 
Roman  and  the  Oriental  influence  these  inheritance  problems 
played  a  very  important  role  in  such  parts  of  analysis  as  the 
ancients  had  developed.  In  the  year  40  B.C.  the  lex  Falcidia 
required  at  least  \  of  an  estate  to  go  to  the  legal  heir.  If  more 
than  I  was  otherwise  disposed  of,  this  had  to  be  reduced  by  the 
rules  of  partnership.  Problems  involving  this  "Falcidian 

1  Anonymous  MS.  in  Munich.   See  Bibl.  Math.,  VII  (2),  32;  Curtze,  ibid., 
VIII  (2),  116;  IX  (2),  34;  Abhandlungen,  III,  123. 

2  In  his  Practica  of  1539. 

3 See  his  edition  of  1569,  p.  125. 

4It  is  also  in  Thierfelder's  arithmetic  (1587,  p.  354),  in  Wynant  van  Westen's 
Mathemat.  Vermaecklyckh  (1644  cd.,  I,  16),  in  Wilkens's  arithmetic  of  1669 
(P-  39S)>  and  in  many  other  early  works. 


THE  TESTAMENT  PROBLEM  545 

fourth"  were  therefore  common  under  the  Roman  law,  just  as 
problems  involving  the  widow's  dower  right  were  and  are  com- 
mon under  the  English  law. 

The  problem  as  stated  above  appears  in  the  writings  of 
Juventius  Celsus  (c.  75),  a  celebrated  jurist  who  wrote  on 
testamentary  law;  in  those  of  Salvianus  Julianus,  a  jurist  in  the 
reigns  of  Hadrian  (117-138)  and  Antoninus  Pius  (138-161 ) ; 
and  in  those  of  Csecilius  Africanus  (c.  100),  a  writer  who  was 
celebrated  for  his  knotty  legal  puzzles.1 

In  the  Middle  Ages  it  was  a  favorite  conundrum,  and  in  the 
early  printed  arithmetics2  it  is  often  found  in  a  chapter  on  in- 
heritances which  reminds  one  of  the  Hindu  mathematical  col- 
lections. It  went  through  the  same  later  development  that 
characterizes  most  problems,  and  finally  fell  on  account  of  its 
very  absurdity.  That  is,  Widman  (1489)  takes  the  case  of 
triplets,  one  boy  and  two  girls,3  and  in  this  he  is  followed  by 
Albert  (1534)  and  Rudolff  (1526).  Cardan  (1539)  compli- 
cates it  by  supposing  4  parts  to  go  to  the  son  and  i  part  to 
the  mother,  or  i  part  to  the  daughter  and  2  parts  to  the  mother, 
and  in  some  way  decides  on  an  8,  7,  i  division.4  Texeda  (1546) 
supposes  7  parts  to  go  to  the  son  and  5  to  the  mother,  or  5  to 
the  daughter  and  6  to  the  mother,  while  other  writers  of  the 
1 6th  century  complicate  the  problem  even  more.5  The  final 
complications  of  the  "swanghere  Huysvrouwe"  or  "donna 
grauida"  are  found  in  some  of  the  Dutch  books,  and  these  and 

*  Coutereels  (Eversdyck  edition  of  1658,  p.  382)  traces  the  problem  back  to  lib. 
28,  title  2,  law  13,  of  the  Digest  of  Julianus.  He  gives  the  usual  4,  2,  i  division 
as  followed  by  Tartaglia,  Rudolff,  Ramus,  Trenchant, Van  derSchuere,and  others. 
Coutereels,  however,  argues  for  the  4,  3,  2  division,  and  in  this  he  has  the  support 
of  various  writers.  Peletier  gives  2,  2,  i,  and  others  give  9,  6,  4.  Brief  historical 
notes  appear  in  other  books,  as  in  the  Schoner  edition  of  Ramus  (1586  ed.,  p.  186) . 

2 Thus  we  have  "  Ein  Testament"  (Widman) , " Erbteilung  vnd  vormundschaft" 
(Riese),  " Erf-Deelinghe "  (Van  der  Schuere),  and  "Erbtheilugs-Rechnung" 
(Starcken). 

3Edition  of  1558,  fol.  07.  He  then  divides  the  property  in  the  proportion  4, 
2,  i,  i.  4Practica,  cap.  66,  ex.  87. 

5Ghaligai  (1521),  Kobel  (1514),  Riese  (Rechnung  nach  lenge,  1550  ed.),  Tren- 
chant (1566),  Van  der  Schuere  (1600),  Peletier  (1607  ed.,  p.  244),  Coutereels 
(1631  ed.,  p.  358),  Starcken  (1714  ed.,  p.  444),  Tartaglia  (Tvtte  I'opere  d'arit- 
metica,  1592  ed.,  II,  136). 


546  TYPICAL  PROBLEMS 

the  change  in  ideas  of  propriety  account  for  the  banishment  of 
the  problem  from  books  of  our  day.1  The  most  sensible  remark 
about  the  problem  to  be  found  in  any  of  the  early  books  is 
given  in  the  words  of  the  "Scholer"  in  Robert  Recorders 
Ground  of  Artes  (c.  1542) :  "If  some  cunning  lawyers  had  this 
matter  in  scanning,  they  would  determine  this  testament  to  be 
quite  voyde,  and  so  the  man  to  die  vntestate,  because  the  testa- 
ment was  made  vnsufficient."2 

Problems  of  Pursuit.  Problems  of  pursuit  are  among  the  most 
interesting  elementary  ones  that  have  had  any  extended  his- 
tory. It  would  be  difficult  to  conceive  of  problems  that  seem 
more  real,  since  we  commonly  overtake  a  friend  in  walking,  or 
are  in  turn  overtaken.  It  would  therefore  seem  certain  that 
this  problem  is  among  the  ancient  ones  in  what  was  once  looked 
upon  as  higher  analysis.  We  have  a  striking  proof  that  this  must 
be  the  case  in  the  famous  paradox  of  Achilles  and  the  Tortoise.3 

It  is  a  curious  fact,  however,  that  the  simplest  case,  that  of 
one  person  overtaking  another,  is  not  found  in  the  Greek  col- 
lections, although  it  appears  in  China4  long  before  it  does  in 
the  West.  It  is  given,  perhaps  for  the  first  time  in  Europe, 
among  the  Propositiones  ad  acuendos  juvenes  attributed  to 
Alcuin,  in  the  form  of  the  hound  pursuing  the  hare.5  There- 
after it  was  looked  upon  as  one  of  the  necessary  questions  of 
European  mathematics,  appearing  in  various  later  medieval 
manuscripts.  It  is  given  in  Petzensteiner's  work  of  1483, 
Calandri6  used  it  in  1491,  Pacioli  has  it  in  his  Suma7  (1494), 

^'Soo  ontfangt  sy  ter  tijdt  haerder  baringhe  eenen  Sone  met  een  Dochter/ 
en  een  Hermaphroditus,  dat  is/half  Man /half  Vrouwe."  Van  der  Schuere,  1600, 
fol.  98.  In  this  case  he  divides  3175  guldens  thus:  d.  254,  m.  508,  s.  1524,  h.  889. 
The  same  problem  appears  in  Clausberg,  Demonstrative  Rechen-Kunst,  1772. 

2 1558  ed.,  fol.  X8. 

8  For  a  study  of  this  problem  see  F.  Cajori,  Amer.  Math,  Month.,  XXII,  i  seq. 

4For  example,  in  the  Nine  Sections  (c.  1105  B.C.?)  and  in  Liu  Hui's  com- 
mentary (c.  263)  on  this  classic.  See  also  Volume  I,  page  32. 

e"De  cursu  canis  ac  fuga  leporis." 

6 "Una  lepre  e  inanzi  aun  chane  3000  passi  et  ogni  5  passi  delcane  sono  p  8 
diquegli  della  lepre  uosapere  inquanti  passi  elcane  ara  giuto  lalepre." 

7"  Vna  lepre  e  dinanqe  a  vn  cane  passa  .60.  e  per  ogni  passa  .5.  che  fa  el  cane 
la  lepre  ne  fa  .7.  e  finalmente  el  cane  lagiongni  [la  giongi  in  the  edition  of  1523, 


PROBLEMS  OF  PURSUIT 


547 


and  most  of  the  prominent  writers  on  algebra  or  higher  arith- 
metic inserted  it  in  their  books  from  that  time  on.1 

In  those  centuries  in  which  commercial  communication  was 
chiefly  by  means  of  couriers  who  traveled  regularly  from  city 
to  city  (a  custom  still  determining  the  name  of  correo  for  a 
postman  in  certain  parts  of  the  world)  the  problem  of  the  hare 


PROBLEM  OF  THE  HARE  AND  HOUND 

From  a  MS.  of  Benedetto  da  Firenze,  c.  1460.    It  begins,  "Vna  lepre  e  inanzi 

a  .1°.  cane" 

and  hound  naturally  took  on  the  form  of,  or  perhaps  paralleled, 
the  one  of  the  couriers.  This  problem  was  not,  however,  always 
one  of  pursuit,  since  the  couriers  might  be  traveling  either  in 

from  la  giugnere,  to  overtake  her]  dimando  in  quanti  passa  el  cane  giogera  la 
lepre."  Fol.  42,  v.  He  says  that  the  problem  is  not  clear,  because  we  do  not  know 
whether  the  "passa  .60."  are  leaps  of  the  dog  or  of  the  hare,  showing  that 
he  felt  bound  to  take  the  problem  as  it  stood,  without  improving  upon  the 
phraseology. 

irThus  Rudolff  (Kunstliche  rechnung,  1526;  1534  ed.,  fol.  N  vj) ;  Kobel  (Re- 
chenbuch,  1514 ;  1549  ed.,  fol.  88,  under  the  title  "Von  Wandern  uber  Landt,"  with 
a  picture  in  which  the  hare  is  quite  as  large  as  the  hound) ;  Cardan  (Practica, 
1539,  cap.  66)  ;  Wentsel  (1599,  p.  51) ;  Ciacchi  (Regole  generali  d'Abbaco,  p.  130, 
Florence,  1675) ;  Coutereels  (Cyffer-Boeck,  1690  ed.,  p.  584),  and  many  others. 


548  TYPICAL  PROBLEMS 

the  same  direction  or  in  opposite  directions.1  This  variant  of 
the  problem  is  Italian,  for  even  the  early  German  writers  gave 
it  with  reference  to  Italian  towns.2  As  a  matter  of  course,  it 
was  also  varied  by  substituting  ships  for  couriers.3 

It  was  natural  to  expect  that  the  problem  should  have  a  fur- 
ther variant,  namely,  the  one  in  which  the  couriers  should  not 
start  simultaneously.  In  this  form  it  first  appeared  in  print  in 
Germany  in  I483,4  in  Italy  in  1484,°  and  in  England  in  1522,° 
although  doubtless  known  much  earlier. 

The  invention  of  clocks  with  minute  hands  as  well  as  hour 
hands  gave  the  next  variant,  as  to  when  both  hands  would  be 
together, — a  relatively  modern  form  of  the  question,  as  is  also 
the  astronomical  problem  of  the  occurrence  of  the  new  moon. 
One  of  the  latest  forms  has  to  do  with  the  practical  question 
of  a  railway  time-table,  but  here  graphic  methods  naturally 
take  the  place  of  analysis,  so  that  of  all  the  variants  those  of 
the  couriers  and  the  clock  hands  seem  to  be  the  only  ones  that 
will  survive.  Neither  is  valuable  per  se,  but  each  is  interesting, 
each  is  real  within  the  range  of  easy  imagination,  and  each 
involves  a  valuable  mathematical  principle, — a  fairly  refined 
idea  of  function. 


1See  Pacioli's  Suma,  1494,  fol.  39,  for  various  types. 

2Thus  Petzensteiner  (1483,  fol.  53),  in  his  chapter  "Von  wandern,"  makes  the 
couriers  go  to  "rum"  (Rome),  thus:  "Es  sein  zween  gesellen  die  gand  gen  rum. 
Eyner  get  alle  tag  6  meyl  der  ander  geth  an  dem  ersten  tage  i  meyl  an  dem 
andern  zwue  etc.  unde  alle  tag  eyner  meyl  mer  dan  vor.  Nu  wildu  wissen  in 
wievil  tagen  eyner  als  vil  hat  gangen  als  der  ander."  Gunther,  Math.  UnterrichtSj 
p.  304- 

3 Thus  Calandri  (1491)  says:  "Una  naue  ua  da  Pisa  a  Genoua  in  5  di:  unaltra 
naue  uiene  da  genoua  a  pisa  in  3  di.  uo  sapere  partendosi  in  nun  medesimo  tempo 
quella  da  Pisa  per  andare  a  Genoua  et  quella  da  Genoua  ,p  andare  a  pisa  in  quahti 
di  siniscon  terrano  insieme." 

4Petzensteiner's  arithmetic,  printed  at  Bamberg. 

5Borghi's  arithmetic. 

6TonstalFs  De  Arte  Supputandi,  fol.  4,  "Cvrsor  ab  Eboraco  Londinvm  pro- 
ficiscens,"  etc.  See  also  Cardan  (Practica,  1539,  cap.  66,  with  various  types) ; 
Ghaligai  (1521;  1552  ed.,  fol.  64) ;  Albert  (1540;  i$6i  ed.,  fol.  Pi) ;  Baker  (1568; 
1580  ed.,  fol.  36) ;  Coutereels  (1631  ed.,  p.  371,  and  Eversdyck  edition  of  1658, 
p.  403);  Trenchant  (1566;  1578  ed.,  p.  280);  Wentsel  (1599,  p.  51);  Peletier 
(1549;  1607  ed.,  p.  290)  ;  Van  der  Schuere  (1600,  fol.  179) ;  Schoner  (notes  on 
Ramus,  1586  ed.,  p.  174),  and  many  others. 


THE  CHESSBOARD  PROBLEM 


549 


dettiwftcit  medctige* 
I;  mMbtt  <Cott$  vott  tCrebcr  was  jwitg  \wt> 
ffortf/tot  modjt  dncn  tag  tj.wctlcrt  gc^ctt/ 
g^oit  i&cytm'd?  ncwn  tag 


The  Chessboard  Problem.  One  of  the  best-known  problems  of 
the  Middle  Ages  is  that  relating  to  the  number  of  grains  of 
wheat  that  can,  theoreti- 
cally speaking,  be  placed 
upon  a  chessboard,  one 
grain  being  put  on  the 
first  square,  two  on  the 
second,  four  on  the  third, 
and  so  on  in  geomet- 
ric progression,  the  total 
number  being  264  —  i,  or 
18,446,744,073,709,551,- 
615.  The  problem  is 
Oriental.  A  chessboard 
problem  of  a  different 
character  appeared  in  the 
writings  of  one  I  Hang, 
a  Chinese  Buddhist  of 
the  T'ang  Dynasty  (620- 
907),*  so  that  games  on 
a  checkered  board  had 

already  begun  to  attract         H*n«flWefra<j/tot»fev<f 
the  attention  of   mathe-     W»*«*«0«n«<ytitt*«ivbcrgd«0cii/ 
maticians  in  the  East.  WtowrtwrntomMfM 

Ibn  Khallikan;2  one  of 
the  best  known  of  the 
Arab  biographers  (1256), 
relates"  that  when  Sissah  ibn  Dahir  invented  the  game  of 
Chess,  the  king,  Shihram,  was  filled  with  joy  and  commanded 
that  chessboards  should  be  placed  in  the  temples.  Further- 
more, he  commanded  Sissah  to  ask  for  any  reward  he  pleased, 
Thereupon  Sissah  asked  for  one  grain  of  wheat  for  the  first 
square,  two  for  the  next,  and  so  on  in  geometric  progression. 

1G.  Vacca,  Note  Cinesi,  p.  135  (Rome,  1913).  This  problem  is  rather  one  of 
permutations. 

2Or  Challikan.    Born  September  22,  1211;  died  October  29,  1282. 

3 In  his  Biographical  Dictionary  (translation  from  the  Arabic  by  Mac  Guckin 
de  Slane,  4  vols.,  Paris  and  London,  1843-1871),  III,  69. 


PROBLEM   OF  THE   COURIERS 

From  Kobel's  Rechenbuch  (1514),  the 
edition  of  1564 


550 


TYPICAL  PROBLEMS 


The  result  of  the  request  is  not  recorded,  but  as  an  old  German 
manuscript  remarks,  "Daz  mecht  kain  kayser  bezalen."1 

The  problem  goes  back  at  least  as  far  as  Mas(udi's  Meadows 
of  Gold2  of  the  loth  century.  It  also  appeared  in  the  works 
of  various  other  Arab  writers,3  and  thence  found  its  way  into 

Europe  through  the 
Liber  Abaci4  (1202)  of 
Fibonacci.  It  is  found  in 
numerous  manuscripts 
of  the  i3th,  i4th,  and 
1 5th  centuries  and  in 
various  early  printed 
books.5  The  problem 
was  much  extended  by 
later  writers.6  It  found 
a  variant  in  the  problem 
of  the  horseshoe  nails 
which  appears  in  sev- 
eral manuscripts  of  the 
1 5th  and  1 6th centuries. 
A  Dutch  arithmeti- 
cian, Wilkens,7  takes  the 
ratio  in  the  chessboard 
problem  as  three  instead 
of  two,  and  considers  not  only  the  number  of  grains  but  also  the 
number  of  ships  necessary  to  carry  the  total  amount,  the  value 
of  the  cargoes,  and  the  impossibility  that  all  the  countries  of 
the  world8  should  produce  such  an  amount  of  wheat.9 

1<(No  emperor  could  pay  all  that."    Curtze,  Bibl.  Math.,  IX  (2),  113. 

2Mas'udi  died  at  Cairo  in  956.  A  French  translation  in  nine  volumes,  with 
Arabic  text,  appeared  in  Paris,  1861-1877.  See  also  Boncompagni's  Bullettino , 
XIII,  274. 

3H.  Suter,  Bibl.  Math.,  II  (3),  34. 

4Boncompagni  ed.,  I,  309. 

5 E.g.,  Pacioli's  Suma  (1494),  fol.  43;  Cardan's  Practica  (1539),  cap.  66. 

6 As  by  Clavius,  Epitome  (1585),  p.  297. 

7  1669  ed.,  p.  112. 

*"A1  de  Provintien  van  de  gheheele  werelt." 

9  For  further  historical  notes  see  J.  C.  Heilbronner,  Historia  Matheseos  Uni- 
versa,  p.  440  (Leipzig,  1742). 


CHESSBOARD  PROBLEM,  C.  1400 

From  an  Italian  manuscript  of  c.  1400,  now  in 
the  Columbia  University  Library 


THE  PROBLEM  OF  THE  HORSESHOE  NAILS 

From  an  anonymous  MS.  written  in  Italy  c.  1535.    As  the  problem  is  usuall> 
stated,  the  blacksmith  receives  one  penny  for  the  first  nail,  two  pence  for  the 
second,  four  for  the  third,  and  so  on,  and  there  are  twenty-four  nails.  This  manu- 
script is  in  Mr.  Plimpton's  library. 


552  COMMERCIAL  PROBLEMS 

The  Mule  and  the  Ass.  Among  the  recreational  problems  that 
have  come  down  to  us  there  is  one  which  appears  in  the  form 
of  an  epigram  with  the  name  of  Euclid  attached.  Rendered 
in  English  verse  it  is  as  follows : 

A  mule  and  an  ass  once  went  on  their  way  with  burdens  of  wine-skins ; 
Oppressed  by  the  weight  of  her  load,  the  ass  was  bitterly  groaning. 
The  mule,  observing  her  grievous  complaints,  addressed  her  this 

question : 

"  Mother,  why  do  you  murmur,  with  tears,  for  a  maiden  more  fitting  ? 
For  give  me  one  measure  of  wine,  and  twice  your  burden  I  carry ; 
But  take  one  measure  from  me,  and  still  you  will  keep  our  loads  equal." 
Tell  me  the  measure  they  bore,  good  sir,  geometry's  master.1 

3.  COMMERCIAL  PROBLEMS 

Economic  Problems.  For  the  student  of  economics  there  is  an 
interesting  field  in  the  problems  of  the  isth  and  i6th  cen- 
turies, as  may  be  seen  from- a  few  illustrations.  The  manu- 
scripts and  early  printed  books  on  arithmetic  tell  us  that 
Venice  was  then  the  center  of  the  silk  trade,  although  Bologna, 
Genoa,  and  Florence  were  prominent.  Florence  was  the  chief 
Italian  city  engaged  in  the  dyeing  of  cloth.  "Nostra  magnifica 
Citta  di  Venetia,"  as  Tartaglia  so  affectionately  and  appro- 
priately called  her,  carried  on  her  chief  trade  with  Lyons,  Lon- 
don, Antwerp,  Paris,  Bruges,  Barcelona,  Montpellier,  and  the 
Hansa  towns,  besides  the  cities  of  Italy.  Chiarino  (Florence, 
1481)  indicates  the  following  as  the  most  important  cities  with 
which  Florence  had  extensive  trade,  his  spelling  being  here 
preserved:  Alessandria  degypto,  Marsilia,  Mompolieri,  Lis- 
bona,  Parigi,  Bruggia,  Barzalona,  Londra,  Gostatinopoli,  and 
Dommasco,  with  the  countries  of  Tunizi,  Cypri,  and  Candia. 
Tartaglia  gives  Barcelona,  Paris,  and  Bruges  as  the  leading 
cities  connected  with  Genoa  in  trade  a  half  century  later. 

lEuclidis  Opera,  ed.  Heiberg  and  Menge,  VIII,  286  (Leipzig,  1916).  The 
translation  is  by  Professor  Robbins,  University  of  Michigan.  See  The  Classical 
Journal ,  XV,  184. 

2 See  the  author's  article  in  the  Amer.  Math.  Month.,  XXIV,  221,  from  which 
extracts  are  here  made. 


ECONOMIC  PROBLEMS 


553 


We  also  know  from  Chiarino  the  most  important  commodi- 
ties of  Florentine  trade  in  the  decade  before  America  was 
discovered.  These  were  rame  (brass),  stoppa  (tow),  zolphi 
(sulphur),  smeriglio  (emery),  lana  (wool),  ghalla  (gall),  tre- 
mentina  (turpentine),  sapone  (soap),  risi  (rice),  zucchari 
(sugar),  cannella  ( cinna- 
mon ),piombo  (lead),lini 
(flax),  pece  (pitch),  ac- 
ciai  ( thread ) ,  canapa 
(hemp),  incenso  (in- 
cense), indachi  (indigo), 
mace  ( mace ) ,  cubeba 
(cubebs), borage  (borax), 
and  the  ever-present  saf- 
fron, the  "king  of  plants," 
then  everywhere  used  as  fKn&awM&'&wQmKtotKgebetauff 
a  sine  qua  non  in  daily  ^cnmardrV  ^uffrvbctbawptaiV^trblftt 
life  and  now  almost  for-  witt»cbnci:byrrt/t)a«tmb0ibtfiertcbQcbetl 
otten  pfrmtfng/  fofcbrfm  eoropt/fttoctfiefm 

The  problems  also  tell  fM$/wtevilbyttnfievtnbttnvftnnin$bfi' 
us  the  cost  of  the  luxuries  bcl TClw/alsobsdM/fbVomytbttwbtn/ 
and  the  necessities  of  life.  2Hfo  vil byi-ctt b<u  fi'cVrob cfttenpfcit* 

nfiig/  Xmotft  wolfcyt 
turumb. 


THE  PROBLP:M  OF  THE  MARKET  WOMAN 

From  Kobel's  Rechenbuchlein  of  1514  (1564 
edition) 


Spanish  linen  was  worth, 
for  example,  from  94  to 
1 20  ducats  per  hundred- 
weight, while  Italian  linen 
ran  as  high  as  355  ducats 
and  Saloniki  linen  as  high  as  380  ducats.  French  linen  was 
much  cheaper  than  the  latter,  selling  for  140  ducats.  The  arith- 
metics tell  us  that  the  linen  was  baled  and  sent  from  Venice  to 
towns  like  Brescia  on  muleback. 

The  problems  "delle  pigione"  tell  us  that  the  houses  of  the 
bourgeoisie  rented  in  Siena,  in  1540,  at  about  25  to  30  lire  per 
year,  while  a  century  later  they  rented  in  Florence  for  from 
120  to  300  lire.  We  also  have  the  prices  of  sugar,  ginger,  pep- 
per, and  other  commodities,  showing  that  these  three,  for  ex- 
ample, were  only  within  the  reach  of  the  wealthy. 


554  COMMERCIAL  PROBLEMS 

Hotel  life  in  a  grand  establishment  is  also  revealed  in  various 
problems,  of  which  this  one,  printed  in  1561,  is  a  fair  type: 

Item/Wenn  in  einem  Gasthause  weren  8  Kamern/in  jglicher 
Kamer  stiinden  12  Bette/in  jglichem  Bette  legen  3  Geste/vnd  ein 
jglicher  Cast  gebe  dem  Hausgesinde  6  fr  trinckgelt/Wie  viel  thuts  in 
einer  Summa? 

That  these  conditions  of  12  beds  in  a  room  and  3  guests  in  a 
bed  are  not  exaggerated,  many  travelers  in  remote  parts  of  the 
world  today  can  testify. 

Partnership.  There  are  three  historic  stages  in  the  conduct 
of  mercantile  business:  (i)  that  of  individual  enterprise, 
(2)  that  of  partnership,1  and  (3)  that  of  corporations.2  The 
first  of  these  has  always  existed,  but  in  extensive  business  af- 
fairs it  early  gave  way  to  partnerships  in  which  the  profits  were 
divided  according  to  the  money  invested,  the  time  that  it  was 
employed  in  the  business,  or  both.  As  business  operations  be- 
came still  more  extensive  the  partnership  generally  gave  place 
to  the  corporation.  Although  the  corporation  has  only  recently 
come  into  great  prominence,  there  were  societates  publicano- 
rum3  in  Rome,  each  directed  by  the  magister  societatis  and 
made  up  of  members  who  received  shares  of  the  profits  in  pro- 
portion to  their  investments.  These  societies  were  not  formed 
for  the  conduct  of  general  business,  however,  but  only  for  col- 
lecting taxes  for  the  censors.4  The  division  of  profits  according 
to  amounts  invested  goes  back  to  the  Babylonian  merchants 
and  is  frequently  mentioned  in  ancient  records.5 

Partnerships  and  Usury.  Aside  from  the  necessity  of  joining 
capital  in  large  business  enterprises  there  was  another  reason 

*" Partner"  is  from  the  Latin  partionarius,  from  partitio,  a  share  or  part.  It 
comes  through  the  Old  French  parsonnier  and  Middle  English  parcener. 

2 Latin  corporatio,  from  corpus,  a  body.  Compare  "corporeal,"  "corps," 
"corse,"  and  "corpse." 

3  That  is,  societies  of  the  farmers-general  of  the  revenues. 

4 From  censere,  to  value  or  tax,  whence  our  "census." 

CA.  H.  Sayce,  Social  Life  among  the  Assyrians  and  Babylonians,  p.  63 
(London,  n.  d.). 


PARTNERSHIP  555 

why  partnerships  flourished  so  extensively  in  the  Middle  Ages. 
The  laws  and  the  popular  prejudice  in  Christendom  against 
taking  interest  on  money  placed  the  "pope's  merchants"1  at  a 
disadvantage  with  respect  to  the  Jews.  Merchants  in  need  of 
money  were  generally  helped  by  their  guilds,  ordinary  borrow- 
ing being  resorted  to  only  in  cases  of  emergency,  as  in  the 
Merchant  of  Venice.2  Hence,  if  a  man  had  money  lying  idle 
for  a  time,  it  was  natural  that  he  should  join  with  others  in 
some  temporary  venture  and  take  his  share  of  the  profit.  He 
thus  secured  interest  on  his  capital  without  incurring  popular 
odium.  A  man  might  even  be  taken  into  partnership  for  a 
limited  time  only,  or  he  might  be  compelled  by  his  partners 
to  withdraw ; 3  in  these  cases  it  became  necessary  to  divide  the 
profits  according  to  the  amount  invested  and  the  time. 

Various  Names  for  Partnership.  There  is  hardly  a  medieval 
writer  on  business  arithmetic  who  does  not  give  this  subject 
an  important  place,4  and  nearly  every  printed  commercial 
book  for  a  period  of  four  hundred  years  devoted  a  chapter  to 
the  topic.  The  Latin  arithmeticians  called  it  the  Regula  de 
societate5  or  Regula  consortuf  while  the  Italian  writers  com- 
monly used  the  plural  term  compagnie.7  When  the  services  of  a 

1W.  Cunningham,  The  Growth  of  English  Industry  and  Commerce  during  the 
Early  and  Middle  Ages,  pp.  329,  364  (London,  1896) . 

2  The  Christian  laws  had  forced  the  business  of  money-lending  into  the  hands 
of  the  Jews,  as  in  the  case  of  Shylock. 

3  An  interesting  case  is  told  in  the  records  of  the  famous  business  house  of 
Kress,  in  Niirnberg:  "Und  do  die  rechnung  geschah,  do  zalt  man  Paulus  Forchtel 
sein  gelt  und  wolten  sein  nit  langer  in  unser  gesellschaft  haben."   The  records 
also  relate :  "  Item  wir  haben  gantze  rechnung  gemacht  an  sant  barbara  obent  do 
man  zelt  von  gotes  gepurt  1395  yar  und  es  westund  [belonged  to]  yeden  Ic 
XXXI  gld.  zu  gewinn."  G.  von  Kress,  Beitrdge  zur  Nurnberger  Handelsgeschichte 
aus  den  Jahren  1370  bis  1430.  See  Gunther,  Math.  Unterrichts,  p.  291  n. 

4  E.g.,  Fibonacci  (1202),  Liber  Abaci,  I,  139;  Johannes  Hispalensis  (£.1140), 
Liber  algorismi  (No.  II  of  Boncompagni's  Trattati),  p.  in;  and  many  others. 

5 Thus  Huswirt  (1501)  :  " Regula  de  societate  mercatorum  et  lucro"  and  "De 

societate  et  intercessione  t^is"   (temporis).    Cardan   (i539>  cap.  52)   has  "De 

societatibus." 

°Thus  Gemma  Frisius  (1540)  :  "Regula  consortij,  siue,  vt  dicunt,  Societatis." 
7  Thus  Feliciano  (1526)  has  a  chapter  Dele  compagnie  (1545  ed.,  fol.  30).   In 

Spanish  the  word  appears  as  compania  and  in  French  as  compagnie,  but  the  word 

societe  was  also  used,  as  by  Peletier  (1549). 


556  COMMERCIAL  PROBLEMS 

partner  were  considered  instead  of  any  money  contribution 
that  he  might  have  made,  they  used  the  term  soccite.1 

English  writers,  following  the  Italian  practice,  often  used 
the  word  "company,"2  although  in  general  the  word  "  fellow- 
ship"3 was  preferred. 

5  j8  Two  marchants  made  a  companic,  A  put  in  300  pound  for  % 
inonethes,  and  then  puttcth  yet  in  100  pound  ,  and  6monethes 
after  that  takcth  out  200  poud,and  with  the  reft  rcmaincth  vn- 
till  the  yearcs  cnd.B  put  in  i  oo  pound  for  one  moneth,and  then 
putteth  yet  in  700  pound  ,  and  6  moncthes  after  that  takcth  out 
d  certaine  (hmme  of  money  ,  and  with  the  reft  remaineth  vntill 
the  yea'res  end.and  then  finde  to  haue  gained  together  400  poud, 
whereofB  muft  haue  80  pound  more  then  A  ,  the  queftion  is 
Iipw  much  money  B  tookc  out  of  the  companie,  without  recko- 
ning intcrcft  vpon  intercft. 

3  oo  .  •  2  .  .    c»oo  400 

1  o  o  _  80 
400  .  .  6  .  .  2400                 3  20 

2  oo  i  60 


200  .   .  4  .  .„ 

160  «   .   3800.  .240 


OO  «  «.I    .   .       XOO 


_ 

800*  •  ^jj  4800 

4900 

800 

1(50  If 

PROBLEM  IN  PARTNERSHIP 
From  Masterson's  Arithmetike  (1592) 

iThus  Cataneo  (1546)  follows  his  chapter  Delle  Compagnie  by  one  Delle 
Soccite,  saying:  "lequali  son  simili  alle  compagnie,"  but  that  the  latter  con- 
sider "il  capitale  e  non  la  persona  &  1'  altro  mette  solo  la  persona  senza  altro 
capitale."  Practically,  he  says,  these  problems  have  to  do  with  the  case  in  which 
some  "gentiP  huomo"  puts  in  his  cattle  and  some  "uillano  o  soccio  minore"  puts 
in  his  time.  The  i6th  century  books  also  use  the  form  soccide.  The  modern  form 
is  soccio  ,  soccita. 

2  "Two  men  Company,  and  make  a  Stock  of  700!,"  in  Hodder's  Arithmetick, 
loth  ed.,  p.  152  (1672)  ;  but  he  calls  the  subject  "The  Rule  of  Fellowship." 

3Thus  Recorde  (c.  1542)  speaks  of  "the  rule  of  Fellowshyppe  ...  or  Com- 
pany" (1558  ed.,  fol.  Ni),  and  Baker  (1568)  gives  "the  rule  of  Felowship." 
The  term  is  used  by  the  American  Greenwood  (1729)  and  in  Pike's  well-known 
arithmetic. 


PARTNERSHIP  557 

The  Germans  ordinarily  preferred  the  term  Gesellschaft1 
and  the  Dutch  writers  followed  their  lead.2 

Pasturage  Problems,  Akin  to  partnership  problems,  and  often 
classified  with  them,  are  pasturage  problems.3  These  may 
have  begun  with  the  custom  of  the  Roman  publicani  of  renting 
to  stock  owners  sections  of  the  estates  which  the  government 
had  farmed  out  to  them,  payment  being  made  in  proportion 
to  the  number  of  cattle.4  It  is  probable  that  the  early  use  of 
commons  by  the  shepherds  was  regulated  according  to  the 
principles  inherited  from  the  early  Roman  conquerors. 

The  importance  attached  to  the  subject  in  the  i6th  and  iyth 
centuries  may  be  inferred  from  the  fact  that  Clavius  (1583) 
devotes  thirty-two  pages  to  it  and  Coutereels  (1599)  allows 
forty  pages.5 

Profit  and  Loss.  The  expression  "  profit  and  loss,"  still  found 
in  our  arithmetics,  although  not  always  used  in  commercial 
parlance  in  quite  the  same  sense,  is  an  old  Italian  one.  The 
books  written  in  the  vernacular  used  the  term  guadagni  e 
perdite?  while  those  written  in  Latin  called  their  chapter  on 
the  subject  De  lucris  &  damnis.1  The  term  passed  over  into 
German  as  Gewin  und  Verlust*  into  Dutch  as  Winst  ende 

1Thus  Kobcl  (1514)  has  Gesellschaft  der  Kaufleit,  and  Albert  (1534)  has 
Gesellschaft  /  oder  der  Kauffleut  Regel  /  von  eigelegtem  Gelde.  Suevus  (1593) 
gives  the  Latin  form  also,  Regvla  societatis.  Regel  der  Gesellschaft. 

2  Thus  Van  der  Schuere  (1600)  has  Reghel  van  Gheselschap. 

3  Cardan  (1539)  speaks  of  them  under  the  head  De  societatibus  bestiarum,  and 
Ortega  (1512,  the  Rome  edition  of  1515)  speaks  of  compagnia  pec  or  aria.    In  the 
Dutch  books  of  the  iyth  century  the  subject  commonly  went  under  the  name 
Vee-Weydinghe. 

4  Ramsay  and   Lanciani,  Manual  of  Roman   Antiquities,   i7th   ed.,  p.   548 
(London,  1901). 

5  Similarly,  Pagani   (1591),  twenty-four  pages;  Werner    (1561),  twenty-six 
pages;  Van  der  Schuere  (1600),  twenty-six  pages;  and  Cardinael   (1674  ed.), 
twenty-five  pages. 

«"  Gains  and  losses,"  as  in  Sfortunati  (1534),  Cataneo   (1546),  and  Pagani 


7  As  in  Cardan,  1539,  cap.  59. 

8  Thus  Rudolff  (1526)  gives  an  "Exepel  von  gewin  vii  verlust,"  Riese  (1522) 
has  "Vom  gewin  vnd  vorlust,"  and  Kobel   (1514)  has  "Regel  vnd  frag  /von 
gewin  der  Kauffleut  angekaufftet  wahr  /  Regula  Lucri." 


558  COMMERCIAL  PROBLEMS 

verlies,1  and  into  French  as  gain  &  perte.2  The  English  writers 
used  "loss  and  gain,"  Recorde  (c.  1542)  saying  that  "the 
fourth  Chapter  treateth  of  Losse  and  Gaine,  in  the  trade  of 
Merchandise." 

The  early  American  texts  followed  the  English  phraseology, 
speaking  of  "loss  and  gain."  Thus  Greenwood  (1729)  remarks 
that  "the  Intention  of  this  Rule  is,  to  discover  what  is  Lost,  or 
Gained  per  Cent,  in  the  Sale  or  Purchase  of  any  Quantity  of 
Goods :  in  Order  to  raise,  or  fall  the  Price  thereof  accordingly." 

The  popularity  of  the  subject  in  the  i6th  century  may  be 
inferred  from  the  fact  that  Werner's  Rechenbuch  (1561)  de- 
votes forty-seven  pages  to  it,  and  that  other  commercial  arith- 
metics were  similarly  generous. 

Commission  and  Brokerage.  Although  the  subject  of  commis- 
sion and  brokerage  is  not  new,  these  terms  are  relatively 
modern.  The  early  printed  books  use  such  terms  as  "factor- 
age"3 and  "factorie,"4  from  "factor,"5  a  middleman  in  the  pur- 
chase and  sale  of  products.  The  term  "factor"  was  used  in  this 
sense  in  the  Middle  Ages,  when  the  father  of  Fibonacci  was 
(£.1175)  a  factor  in  Bougie,  and  in  the  Renaissance  period.6 
It  is  still  used,  although  less  commonly,  in  America  and  Great 
Britain,  and  warehouses  for  goods  to  be  exported  are  still  called 
"factories"  in  various  parts  of  the  world. 

*As  in  Van  der  Schuere  (1600).  It  also  appears  as  Winningh  en  verlies,  as  in 
Bart jens  (1633). 

2  So  the  Dutch-French  work  of  Coutereels  (1631)  has  "Comptes  de  gain  & 
perte." 

8  Even  as  late  as  the  i9th  century  Pike's  arithmetic  (8th  ed.,  p.  204  (New 
York,  1816))  has  the  definition:  "Factorage,  Is  an  allowance  of  so  much  per 
cent,  to  a  Factor  or  Correspondent,  for  buying  and  selling  goods."  It  defines  a 
broker  as  a  merchant's  assistant  in  buying  or  selling. 

4  Thus  Rudolff   (1526)    and  Werner   (1561)    have  Factorey.    Of  the  Dutch 
writers,  Bartjens  (1633)  has  Factorie;  Raets  (1580),  Rekeninghen  van  Facteuri- 
jen\  and  Van  der  Schuere  (1600),  Facteur-Rekeninghe. 

5  That  is,  operator,  from  the  Latin  facere,  to  act  or  do.    Compare  the  factor 
of  a  number. 

°Thus  Werner  (1561) :  "Item  ein  Kauffman  macht  seinem  Factor  ein  geding"; 
Trenchant  (1566):  "Aux  compagnies  d'entre  marchans  &  facteurs";  Recorde 
(c.  1542):  "A  Merchant  doth  put  in  800  pound  into  the  hands  of  his  Factor" 
(1646  ed.,  p.  519). 


EQUATION  OF  PAYMENTS  559 

The  word  "broker"  is  not  so  common  as  "f actor "  in  prob- 
lems before  the  igth  century,  although  it  appears  in  Middle 
English1  to  designate  one  who  does  business  for  another  or 
acts  as  his  agent.2 

The  term  "  commission,"  as  now  used  to  indicate  a  percent- 
age, is  relatively  modern. 

Equation  of  Payments.  The  absence  of  banking  facilities  to 
the  extent  now  known  in  America,  the  difficulties  in  transmitting 
money,  and  the  scarcity  of  currency  before  the  great  improve- 
ments in  gold-mining  in  the  igth  century  rendered  necessary 
until  very  recently  an  extensive  credit  system.  Importing  houses 
bought  on  credit  and  exported  goods  on  credit  to  those  from 
whom  they  bought,  balancing  their  accounts  from  time  to 
time.  The  process  of  finding  the  balance  due,  so  that  neither 
party  should  lose  any  interest,  was  the  problem  of  the  equation 
of  payments.  The  subject  is  found  in  many  manuscripts  of 
the  i4th  and  i5th  centuries,  and  when  textbooks  began  to  ap- 
pear in  print  it  was  looked  upon  as  of  great  importance.  Thus 
Recorde  (c.  1542)  says: 

Rules  of  Payment,  which  is  a  right  necessarie  Rule,  and  one  of 
the  chiefest  handmaydes  that  attendeth  vpon  buying  and  selling.3 

The  subject  went  by  various  names,4  but  the  later  English 
and  American  writers  generally  used  the  expression  "Equation 
of  Payments." 

Interest.  The  taking  of  interest  is  a  very  old  custom,  going 
back  long  before  the  invention  of  coins,  to  the  period  in  which 
values  were  expressed  by  the  weight  of  metal  or  by  the  quan- 
tity of  produce.  The  custom  of  paying  interest  was  well  known 

1  Brocour  or  broker.  It  probably  came  from  the  Anglo-Saxon  brucan,  to  use 
or  employ.  The  root  is  found  in  the  Scandinavian  and  Teutonic  languages,  re- 
ferring to  business  in  general. 

2 A  word  coined  in  the  i6th  century  from  the  Latin  agere,  to  act  or  to  do. 

sMellis  ed.,  p.  478  d594). 

4  Thus  Hodder  (1672  ed.,  p.  163)  calls  the  chapter  "Of  Equation,"  and  the 
Dutch  works  have  such  names  as  "Den  Regel  van  Paeyement  of  Betalinghe" 
(Eversdyck's  Coutereels,  1658  ed.,  p.  181)  or  simply  "Reghel  van  Payementen" 
(Stockmans,  1609  ed.,  fol.  Q4;  Houck,  1676  ed.,  p.  108), 


560  COMMERCIAL  PROBLEMS 

in  ancient  Babylon.  In  Sumerian  tablets  of  the  period  before 
2000  B.C.  the  rate  is  often  given  as  varying  from  the  equivalent 
of  20  per  cent  to  that  of  30  per  cent,  according  to  whether  it 
was  paid  on  money,  that  is,  on  precious  metals,  or  on  produce. 
In  general,  in  the  later  Babylonian  records,  the  rate  ran  from 
S|  per  cent  to  20  per  cent  on  money  and  from  20  per  cent  to 
33!  per  cent  on  produce,  although  not  expressed  in  per  cents.1 
Even  princes  engaged  in  trade  and  insisted  upon  their  interest, 
for  one  of  the  tablets  relates  the  following : 

Twenty  manehs  of  silver,  the  price  of  wool,  the  property  of  Bel- 
shazzar,  the  son  of  the  king.  .  .  .  All  the  property  of  Nadin-Mero- 
dach  in  town  and  country  shall  be  the  security  of  Belshazzar,  the  son 
of  the  king,  until  Belshazzar  shall  receive  in  full  the  money  as  well  as 
the  interest  upon  it.2 

Tablets  of  Nineveh  as  old  as  the  7th  century  B.C.  have  the 
following  records : 

The  interest  [may  be  computed]  by  the  year. 
The  interest  may  be  computed  by  the  month. 

The  interest  on  ten  drachmas  is  two  drachmas. 

Four  manehs  of  silver  .  .  .  produce  five  drachmas  of  silver  per 
month.3 

Interest  in  Ancient  India.  The  custom  was  also  known  in  an- 
cient India,  appearing  in  the  early  legal  writings  of  the  Sutra 
period,  some  centuries  before  the  beginning  of  our  era.4  In  the 

1M.  Jastrow,  Jr.,  The  Civilization  of  Babylonia  and  Assyria,  pp.  323,  326, 
338  (Philadelphia,  1915);  A.  H.  Sayce,  Zeitschrift  jur  Assyriologie,  V  (1890), 
276;  T.  G.  Pinches,  ibid.,  I,  198,  202;  A.  H.  Sayce,  Social  Life  among  the  As- 
syrians and  Babylonians,  chap,  v,  p.  67  (London,  n.  d.)  (hereafter  referred  to  as 
Sayce,  Social  Life}  ;  G.  Billeter,  Geschichte  des  Zinsfuss.es  im  griechisch-rom. 
Altertum  (Leipzig,  1898), —  the  leading  authority.  2 Sayce,  Social  Life,  p.  65. 

3 Since  four  manehs  was  about  $180,  and  five  drachmas  was  about  $2,  the  in- 
terest on  $180  was  $24  a  year,  the  rate  being  13  per  cent.  See  J.  Menant,  La 
Bibliotheque  du  palais  de  Ninive,  p.  71  (Paris,  1880). 

4  Thus  the  Dharma-sastras  state  that  "5  Mashas  for  every  20  [Kdrshdpanas] 
may  be  taken  every  month."  Since  20  mdshds  were  probably  equal  to  a  kdr- 
shdpana,  the  rate  was  1}  per  cent  per  month,  or  15  per  cent  annually.  See  R.  C. 
Dutt,  A  History  of  Civilization  in  Ancient  India,  I,  174,  237  (London,  1893). 


INTEREST  PROBLEMS  561 

medieval  period  there  are  many  evidences  of  the  taking  of 
interest.  For  example,  Mahavlra  (c.  850)  has  various  prob- 
lems of  the  following  type : 

O  friend,  mention,  after  calculating  the  time,  by  what  time  28  will 
obtain  as  interest  on  80,  lent  out  at  the  rate  of  3^  per  cent  [per 
month]. 

Bhaskara  (c.  1150)  also  paid  much  attention  to  the  subject, 
giving  such  problems  as  the  following : 

If  the  interest  of  a  hundred1  for  a  month  be  five,  say  what  is  the 
interest  of  sixteen  for  a  year. 

If  the  interest  of  a  hundred  for  a  month  and  one  third  be  five  and 
one  fifth,  say  what  is  the  interest  of  sixty-two  and  a  half  for  three 
months  and  one  fifth. 

If  the  principal  sum,  with  interest  at  the  rate  of  five  on  the  hun- 
dred by  the  month,  amount  in  a  year  to  one  thousand,  tell  the  prin- 
cipal and  interest  respectively.2 

Interest  Customs  in  Greece.  The  rate  in  Greece  seems  not  to 
have  been  restricted  by  law  and  to  have  varied  from  12  per 
cent  to  1 8  per  cent.  In  the  time  of  Demosthenes  12  per  cent 
was  thought  to  be  low.  There  were  two  general  plans  for  com- 
puting interest:  (i)  at  so  much  per  month  per  mina,  and 
(2)  at  such  a  part  of  the  principal  per  year.  Interest  was 
usually  paid  at  the  end  of  each  month.3 

Interest  in  Rome.  In  Rome  the  rate  of  interest  was  at  first 
unrestricted.4  The  Twelve  Tables5  limited  the  interest  charged 

'That  is,  the  rate  per  cent. 

2  These  extracts  show  that  the  rate  of  interest  in  India  in  the  i2th  century 
was  about  60  per  cent,  and  that  interest  was  computed  on  a  percentage  basis. 
See  Colebrooke's  translation  of  the  Lttdvati,  pp.  36,  39. 

3F.  B.  Jevons,  A  Manual  of  Greek  Antiquities,  p. 397  (London,  1895)  ;  Harper's 
Diet,  of  Class.  Lit.  and  Antiq.,  p.  665. 

Interest  was  called  faenus,  or  fenus,  a  later  term  being  usura  (from  uti,  to 
use),  commonly  expressed  in  the  plural,  usurae.  So  Cicero  has  "pecuniam  pro 
usuris  auferre."  From  this  came  the  French  usure  and  our  "usury."  Capital  was 
caput  (head,  originally  a  head  of  cattle)  or  occasionally  sors  (lot  or  chance) . 

*Duodecim  Tabulae,  the  first  code  of  Roman  law,  451-449  B.C.,  and  the 
foundation  of  that  law  up  to  the  time  of  the  Corpus  luris  of  Justinian,  c.  530. 


562  COMMERCIAL  PROBLEMS 

to  Romans  to  one  twelfth  (8|-  per  cent)  of  the  capital,  and 
later  (c.  100  B.C.)  this  limitation  was  extended  to  aliens  as  well. 
The  Lex  Genucia  (342  B.C.)  prohibited  the  taking  of  interest 
altogether,1  but  like  the  medieval  canon  law  this  seems  not  to 
have  been  enforced. 

In  later  Roman  times  the  Eastern  custom  of  monthly  in- 
terest came  into  use,  the  ordinary  rate  being  i  per  cent  per 
month,  payable  in  advance,  or  12  per  cent  per  year.  In  Cicero's 
time  48  per  cent  per  year  was  allowed,  and  under  the  first 
emperors  25  per  cent  was  common.  A  little  later  12  per  cent 
per  year  was  made  the  maximum ;  Justinian  reduced  this  rate 
to  ^  per  cent  per  month,  which  gave  rise  to  the  common  rate 
of  6  per  cent.  In  classical  Latin  works  the  rates  of  interest  are 
usually  mentioned  either  as  jenus  unciarum2  or  as  usurae 
centesimae? 

Interest  in  the  Middle  Ages.  In  medieval  Europe  the  canon 
law  forbade  the  taking  of  usury,  that  is,  the  payment  in  ad- 
vance for  the  use  of  money.  The  time  had  not  come  for  bor- 
rowing money  for  such  remunerative  purposes  as  extensive 
manufacturing  or  as  building  railways  and  steamships,  and  so 
the  principal  was  often  consumed  by  usury  instead  of  being 
increased.  Usury  would  therefore  have  speedily  resulted  in 
the  enslavement  of  the  peasants,  who  were  without  money  or 
financial  ability.  Hence  the  Church  came  to  recognize  a  dis- 
tinction between  loans  for  production,  which  might  reasonably 
have  carried  some  remuneration,  and  those  for  consumption, 
which  were  contrary  to  public  policy.4 

1M.  Cantor,  Politische  Arithmetik,  p.  2  (Leipzig,  1898). 

2  Uncial  interest,  that  is,  interest  by  twelfths,  ^  being  the  common  rate.  This 
was  Ti^  per  month  when  the  ancient  year  consisted  of  ten  months.  When  the 
year  was  later  divided  into  twelve  months  the  rate  was  still  \^  per  month  or 
TYff  per  year.  Since  interest  was  paid  by  the  month,  this  made  the  former  rate 
83  per  cent  and  the  latter  10  per  cent  per  year.  See  Ramsay  and  Lanciani, 
Manual  of  Roman  Antiquities,  i7th  ed.,  p.  472  (London,  1901). 

3 Hundredth  interest,  or  i  per  cent  a  month.  This  was  the  ancient  "per  cent." 
If  the  security  was  poor,  this  was  raised  to  binae  centesimae  (2  per  cent  per 
month)  or  even  to  quaternae  centesimae  (4  per  cent  per  month). 

4See  the  decree  of  the  fifth  Lateran  Council  (1512-1517)  in  Janet,  Le  capital, 
la  speculation,  et  la  finance,  au  XIX*  siecle,  p.  81  (Paris,  1892). 


INTEREST  563 

There  was,  however,  another  reason  which  was  not  so  openly 
stated,  namely,  the  desire  of  the  Church  and  of  the  ruling 
classes  to  prevent  the  dangerous  rivalry  to  authority  which 
would  have  resulted  from  the  accumulation  of  too  large  for- 
tunes ;  in  other  words,  to  avoid  the  dangers  of  capitalism. 

Origin  of  the  Term  "Interest."  To  overcome  this  restriction 
there  accordingly  developed  a  new  economic  custom.  The  bor- 
rower paid  nothing  for  the  use  of  the  money  if  it  was  repaid 
at  the  time  specified.  If,  however,  he  failed  so  to  pay  the  prin- 
cipal, he  was  held  to  compensate  his  creditor  by  a  sum  which 
represented  the  difference,  or  "that  which  is  between"  ("id 
quod  interest")  the  latter 's  position  because  of  the  delay  and 
what  his  position  would  have  been  had  he  been  paid  promptly. 
Id  quod  interest  was  recognized  by  the  Roman  law,  but  as  a 
certain  per  cent  agreed  to  in  advance  it  first  appears  in  the 
1 3th  century,  possibly  suggested  from  the  East.1  Speaking  of 
this  method,  Matthew  Paris  (1253)  tells  us  that  in  his  time 
10  per  cent  was  exacted  every  two  months,  and  adds  that  in 
this  way  unscrupulous  men  "circumvented  the  needy  in  their 
necessities,  cloking  their  usury  under  the  show  of  trade."2 

Among  the  economic  movements  of  the  Renaissance  period 
was  a  serious  questioning  of  the  validity  of  the  canon  law 
against  usury  and  a  determination  to  recognize  a  new  type  of 
interest,  namely,  usury  paid  at  the  end  of  the  term  of  borrow- 
ing.3 As  a  result  of  this  feeling  the  subject  of  interest  found 
place  in  many  of  the  early  printed  books,  particularly  in  Italy,4 

1  Compare  Bhaskara's  Lildvati  (c.  1150),  Colebrooke's  translation,  p.  39.  Fibo- 
nacci (1202)  gives  problems  involving  20  per  cent  interest,  but  the  Hindu  works 
give  rates  as  high  as  60  per  cent. 

2Chronica  Majora,  III,  329,  published  in  the  Rolls  Series.  See  also  W.  Cun- 
ningham, The  Growth  of  English  Industry  and  Commerce  .  .  .  ,  p.  329  (London, 
1896).  One  of  the  best  historical  sketches  to  be  found  in  the  early  arithmetics 
is  given  by  Sfortunati,  Nvovo  Lvme,  1534  (1544-1545  ed.,  fol.  60). 

8 See  such  works  as  F.  de  Platea,  Opus  restitutionum  usurarum  et  excom- 
municationum,  Venice,  c.  1472  (de  Platea  lived  c.  1300) ;  J.  Nider,  Tractate 
de  contractibus  mercatorum  (s.  1.  a.,  but  Cologne,  with  at  least  seven  editions 
before  1501) ;  and  many  other  similar  works  of  the  period. 

4  So  Calandri  (1491),  who  uses  thirty  days  to  the  month,  sometimes  using 
per  cent  ("per  3  anni  a  10  per  cento  lanno")  and  sometimes  stating  the  rate  as 
the  equivalent  of  so  many  pence  in  the  pound. 


564  COMMERCIAL  PROBLEMS 

although  sometimes  against  the  protest  of  the  author.1    This 
spirit  of  protest  showed  itself  in  the  people's  literature  of  Eng- 
land, as  in  Francis  Thynne's  (i6th  century)  epigram: 
Stukelie  the  vsurer  is  dead,  and  bid  vs  all  farwell, 
who  hath  a  lourney  for  to  ride  vnto  the  court  of  hell. 

A  similar  testimony  is  found  in  Lauder  (1568) : 

Credit  and  frist  [delay]  is  quyte  away, 

No  thing  is  let  but  for  Usure ; 
For  euerie  penny  thay  wyll  haue  tway: 

How  long,  Lord,  will  this  warld  indure?2 

In  spite  of  these  protests  the  English  parliament  in  1545 
sanctioned  the  taking  of  interest,3  fixing  the  maximum  rate  at 
10  per  cent.  The  protest  was  such  that  the  law  was  repealed 
in  1552,  but  it  was  reenacted  in  1571,*  and  since  that  time  all 
works  on  commercial  mathematics  have  included  the  topic. 

In  Germany  the  opposition  to  interest  was  also  very  strong, 
and  Martin  Luther  published  a  sermon  on  the  subject  in  15 19.* 

Compound  Interest.  The  compounding  of  interest  was  known 
to  the  Romans  and  was  not  forbidden  until  rather  late.8  The 
late  medieval  and  the  Renaissance  Italians,  from  whom  we  de- 
rive so  much  of  our  modern  business  arithmetic,  used  the  word 
merito1  for  interest  in  general,  and  where  it  was  computed 
"simply  by  the  year"8  it  was  called  simple  interest.9 

1So  Cataneo  (Le  Pratiche,  1546),  under  the  title  De  semplici  meriti  vsvreschi, 
speaks  of  the  practice  as  often  "diabolical,"  and  Pagani  (1591)  calls  it  "Cosa  in 
vero  molto  biasmeuole,  &  diabolica." 

2W.  Lauder,  The  Lamentationn  of  the  Pure,  twiching  the  Miserabill  Estait 
of  this  present  World,  published  by  the  Early  Eng.  Text  Soc.,  ,p.  28  (London, 
1870).  337  Hen.  VIII,  c.9. 

4E.  P.  Cheyney,  Industrial  and  Social  History  of  England,  p.  172  (New  York, 
190*)-  ^Eyn  Sermon  von  den  Wucher,  Wittenberg,  1519. 

6 Harper's  Diet,  of  Class.  Lit.  and  Antiq.,  p.  665. 

7  It  passed  into  the  French  as  merite,  although  the  word  interest  was  also 
used.  Thus  Trenchant  (1566):  "A  calculer  les  merites  ou  interestz"  (1578  ed., 
p.  299).  The  Italians  also  used  the  term  usura.  The  1515  Italian  edition  of  Ortega 
(1512)  has  Regula  de  lucro. 

8"Simplicemente  all'  anno,"  as  Tartaglia  (1556)  says  (1592  ed.,  II,  fol.  95). 
See  also  Cardan's  Practica,  1539,  capp.  57  and  58. 

9Ciacchi  (1675,  pp.  80,  228),  a  later  writer,  speaks  De'  meriti  semplici. 


COMPOUND  INTEREST  565 

Compound  interest  among  the  early  Italians  was  computed 
from  the  beginning  of  each  year1  or  period2  and  was  called 
by  the  English  writers  of  the  iyth  century  "  interest  upon 
interest."8  The  taking  of  such  interest  was  frequently  charged 
against  the  Jews,4  although  unjustly  so,5  and  is  even  character- 
ized by  their  name.6 

In  the  1 5th  and  i6th  centuries  interest  was  usually  computed 
either  on  a  percentage  basis  or  at  so  many  pence  to  the  pound.7 
The  rate  varied  from  the  60  per  cent  mentioned  by  Bhaskara 
(c.  1150)  and  his  European  contemporaries  to  smaller  limits.8 

The  difficulty  in  computing  interest  gave  rise  in  the  i6th  cen- 
tury to  the  use  of  tables.  These  were  extended  in  the  i?th 
century,  a  table  of  compound  interest  appearing  in  Richard 
Witt's  Arithmeticall  questions  (London,  1613). 

Discount.  The  computing  of  discount9  for  the  payment  of 
money  due  at  a  future  time  is  relatively  modern.  It  is  found 

1MA  capo  d'  anno,"  as  Tartaglia  (1556)  describes  it  (1592  ed.,  II,  fol.  95). 

2"A  capo  d'  alcun  tempo,"  as  Cataneo  (1546)  describes  it  (1567  ed.,  fol.  53). 
Similarly  Tartaglia  (150^  ed.,  II,  fol.  119) :  "Del  meritar  a  capo  d'  anno,  6  altro 
termine  che  d'  alcuni  e  detto  vsura."  The  expression  passed  over  into  French  as 
"merite  a  chef  de  terme"  (Trenchant,  1566;  1578  ed.,  p.  299). 

3So  in  Hodder,  loth  ed.,  1672,  p.  139.  The  Dutch  commonly  called  it  "in- 
terest op  interest"  or  " Wins-ghewin  (VVinsts-Gewin)." 

4Thus  Pagani  (1591,  p.  147):  "e  questo  modo  di  meritare  e  communemente 
vsitato  da  gP  hebrei  ne  suoi  Banchi." 

5Gunther,  Math.  Unterrichts,  p.  290.  Pagani  also  says  that  the  Christians 
were  equally  to  blame. 

6 "Ma  d'  altra  sorte  e  la  ragion  dell'  usura,  che  chiamano  Guidaica"  (sic,  for 
"Giudaica"),  as  in  the  Italian  edition  (1567,  fol.  32)  of  Gemma  Frisius,  but  not 
in  the  Latin  edition.  Similarly  Van  der  Schuere  (1600,  fol.  127)  speaks  of  "een 
loodtsch  profijt." 

7  "  Meriter  est  baillcr  ses  deniers  pour  profiter  a  raison  d'vn  tant  pour  -h  ou 
pour  100 :  par  an"  (Trenchant,  1566;  1578  ed.,  p.  298). 

8 For  example,  Sfortunati  (1534)  gives  the  rate  "a  denari  .2.  la  libra  il  mese," 
which  is  10  per  cent  a  year,  and  goes  even  as  high  as  4  pence  per  pound  per 
month,  or  20  per  cent  a  year;  Trenchant  (1566)  gives  one  problem  at  12  per 
cent;  Tartaglia  (1556)  gives  10  per  cent,  16  per  cent,  20  per  cent,  and  other  rates; 
one  of  the  Dutch  writers,  Raets  (1580)  gives  from  8  per  cent  to  14  per  cent,  10 
per  cent  being  stated  thus:  "Soo  100  winnen  in  een  iaer  10";  and  Cardinael 
(1674  ed.)  gives  rates  ranging  from  10  per  cent  to  20  per  cent. 

9  Formerly  "discompt,"  from  the  Old  French  descompter,  to  reckon  off,  from 
des-j  away,  compter,  to  count. 

ii 


566  COMMERCIAL  PROBLEMS 

in  some  of  the  i6th  century  arithmetics1  but  is  more  common 
in  the  century  following,  appearing  under  various  names.2 

Assize  of  Bread.  One  of  the  standard  problems  of  the  i6th 
century  books  related  to  the  variation  in  size  of  a  loaf  of  bread 
as  the  wheat  varied  in  value.  For  example,  if  a  ic-cent  loaf 
weighs  14  oz.  when  wheat  is  worth  $1.80  a  bushel,  how  much 
should  it  weigh  when  wheat  is  worth  $2.20  a  bushel?8 

The  problem  had  its  genesis  in  real  conditions.  Loaves  were 
formerly  of  two  kinds:  (i)  "assized  bread/'  always  sold  at 
the  same  price  but  varying  in  weight  according  to  the  price  of 
wheat,  and  (2)  "prized  bread,"  always  of  the  same  weight  but 
varying  in  price.4  The  legal  regulation  for  the  assized  bread 
goes  back  at  least  to  794,  being  found  at  that  time  in  a  Frank- 
fort capitulary,  and  is  probably  of  Roman  origin.  London 
regulations  are  found  as  early  as  the  i2th  century,  and  in  the 
" assize  of  bread"  of  Henry  II  (1154-1189)  these  are  worked 
out  by  inverse  proportion.5  As  a  result  of  these  regulations, 
tables  of  the  assize  of  bread  were  prepared  and  their  use  was 

1Thus  Cataneo  (1546)  has  a  chapter  "Del  semplice  sconto"  and  one  (corre- 
sponding to  compound  interest)  "Dello  sconto  a  capo  d'  alcvn  tempo."  Tren- 
chant (1566)  discounts  an  amount  due  in  four  years  "a  raison  de  12  pour  100 
par  an." 

2Thus  in  Coutereels's  Cyffer-Boeck  (1690  ed.,  p.  289)  it  appears  as  "Rabat- 
teeren,  Disconteeren,  of  af-korten,"  and  in  Hodder's  Arithmetick  (1672  ed.,  p.  175) 
as  "The  Rule  of  Rebate,  or  Discount." 

8Thus  Ortega  (1512;  1515  ed.,  fol.  59):  "  (fl.  Si  de  vno  misura  de  grano  die 
costa  10  carlini-mi  dano  4  vnze  de  pane  per  vno  dinaro  si  voi  sapere  se  de  vna 
altra  misura  che  costera  20  carlini  quante  vnze  ne  darano  per  uno  medesimo 
dinaro."  See  also  Gemma  Frisius  (1540;  1555  ed.,  fol.  66),  Rudolff  (Kunstliche 
rechnung,  1526;  1534  ed.,  fol.  K4),  Albert  (1534;  1561  ed.,  fol.  N  i,  under  Regula 
Detri  Conuersa),  Suevus  (1593,  p.  320,  with  two  pages  "Vom  Brodgewichte  in 
thewren  vnd  wolfeihlen  Jaren"),  and  many  other  writers  of  the  i6th  century 
and  later. 

4J.  Nasmith,  An  examination  of  statutes  .  .  .  the  assize  of  bread,  Wisbech, 
1800;  S.  Baker,  Artachthos  Or  a  New  Booke  declaring  the  Assise  or  Weight  of 
Bread,  London,  1621. 

6"Quando  quartierium  frumewti  se  vendit  pro  sex  sol.;  tune  debet  panis  esse 
bonus  et  albus  et  ponderare  sexdedm  sol.  de  xxli  lores  [i.e.,  20  d.  to  i  oz.].  .  .  . 
Qwando  pro  qwatuor  solidis  tune  debet  ponderare  tnginta  sex  sol.  et  alius  quad- 
raginta  sex  sol.  .  .  .  ,"  and  so  on  for  different  weights.  W.  Cunningham,  The 
Growth  of  English  Industry  and  Commerce  during  the  Early  and  Middle  Ages, 
p.  568  (London,  1896) . 


TARE  AND  TRET  567 

made  obligatory.1  This  problem  of  the  size  of  loaves  was  a 
common  one  in  the  early  printed  books  and  is  often  found  as 
late  as  the  second  half  of  the  ipth  century.2  The  following, 
from  the  1837  edition  of  DabolPs  well-known  American  arith- 
metic, illustrates  the  type:  "If  when  wheat  is  73.  6d.  the  bushel, 
the  penny  loaf  will  weigh  9  oz.  what  ought  it  to  weigh  when 
wheat  is  6s.  per  bushel?" 

Tare  and  Tret.  Until  the  middle  of  the  igth  century  the  sub- 
ject of  "tare  and  tret"  was  found  in  most  of  the  English  and 
American  commercial  arithmetics.  "Tare"  meant  an  allow- 
ance of  a  certain  weight  or  quantity  from  the  weight  or  quan- 
tity of  a  commodity  sold  in  a  box,  cask,  bag,  or  the  like.  The 
word  came  from  the  Arabic  tarha*  (what  is  thrown  away) 
through  the  Spanish  tar  a  and  the  French  tare,  and  shows  the 
commercial  influence  exerted  by  the  Arabs  in  Spain. 

"Tret"  meant  about  the  same  thing,  but  the  word  shows  the 
Italian  influence,  meaning  originally  an  allowance  on  things 
transported.4  In  England  it  was  an  allowance  of  4  Ib.  in 
every  104  Ib. 

There  was  also  a  third  term  that  was  related  to  "tare"  and 
"tret"  and  is  commonly  found  in  the  English  books  of  the  i6th 
century.  This  term  is  "cloff,"  meaning  an  allowance  of  2  Ib. 
made  on  every  3  cwt.  of  certain  goods  in  order  that  the  weight 
might  hold  out  in  retailing.  Thus  Recorde  (c.  1542)  has  prob- 
lems of  this  type: 

"Item  at  35  4d  the  pound  weight,  what  shal  254^  be  worth, 
in  giuing  4  1  weight  vpon  euery  100  for  treate." 

"Item  if  100  1  be  worth  363  8d,  what  shall  800  1  be  worth  in 
rebating  4  pound  upon  euery  100  for  tare  and  cloff  e."r> 

1Such  a  table,  from  the  Record  Book  of  the  city  of  Hull,  is  reproduced  in 
facsimile  in  E.  P.  Cheyney,  Industrial  and  Social  History  of  England,  p.  67  (New 
York,  i 901). 

2  A  rare  and  interesting  tract  on  the  subject  is  that  of  J.  Powel,  Assize  of 
Bread,  London,  1615,  a  guide  for  those  who  had  to  interpret  the  old  law. 

3  From  tar  aha,  he  threw  down. 

4The  word  is  from  the  Latin  trahere,  to  draw  or  pull,  whence  tractus,  Italian 
tratto,  and  French  trait.   From  the  same  root  we  have  "tract"  and  "traction.** 
5 Ground  of  Artes,  1594  ed.,  p.  487.  The  origin  of  the  term  is  uncertain. 


5 68  COMMERCIAL  PROBLEMS 

In  Baker's  arithmetic  (1568)  "the  eyght  chapter  treateth  of 
Tares  and  allowances  of  Marchandise  solde  by  weight,"  and 
other  arithmetics  of  the  period  also  presented  the  subject  at 
considerable  length.1 

Cutting  of  Cloth.  Problems  relating  to  the  cutting  of  cloth 
correlated  so  closely  with  the  needs  of  merchants  that  the  com- 
mercial schools  seem  generally  to  have  included  them  in  the 
1 6th  century.  Thus  Grammateus  (1518)  has  problems  on  the 
cutting  of  cloth  by  tailors,2  and  Tartaglia  (1556)  also  devoted 
considerable  attention  to  them.3  No  attention  was  paid  to  the 
pattern,  and  the  problems  show  that  drapers  had  flexible  con- 
sciences with  respect  to  advising  as  to  the  amount  needed  for  a 
garment.4  Baker  (1568)  says  that  "the  5  Chapter  treateth  of 
lengthes  and  bredthes  of  Tapistrie,  &  other  clothes/75  and 
John  Mellis  has  a  similar  chapter  in  his  addition  (1582)  to 
Recorders  Ground  of  Aries. 

The  custom  of  carpeting  rooms,  which  reached  its  highest 
point  in  the  igth  century,  led  to  the  inclusion  of  problems 
relating  to  this  subject.  The  return  to  rugs  in  the  2oth  century 
is  leading  to  a  gradual  elimination  of  the  topic  in  America. 

Barter.  Of  the  applications  of  arithmetic  none  has  had  a  more 
interesting  history  than  barter,6  a  subject  now  very  nearly 
obsolete  in  textbooks,  although  temporarily  revived  among 
nations  as  a  result  of  the  World  War  of  1914-1918,  owing  to 
conditions  of  exchange.  There  are  three  fairly  well  defined 
periods  in  the  exchange  of  products.  The  first  is  that  of  pure 
barter,  seen  today  in  the  exchange  of  guns  and  ammunition 
for  a  tusk  of  ivory  in  remote  parts  of  Africa, — a  period  lasting 
throughout  the  era  of  savage  life.  This  is  also  seen  in  the 
ancient  method  of  paying  taxes  "in  kind,"  so  many  fowls  out 

iThus  Ortega  (1512;  1515  ed.,  fol.  53),  "Regvla  de  tre  de  tara";  Stockmans 
(1589),  "Reghel  van  Tara";  Ciacchi  (1675),  "Delle  tare  a  vn  tanto  per  cento"; 
Coutereels  (1690  ed.),  "Tara-Rekeningh." 

s'^Schneider  regel"  (1535  ed.,  fol.  C6). 

3 1592  ed.,  II,  fol.  79.  4Tartaglia,  loc.  tit.,  fol.  81.  5i58o  ed.,  fol.  126. 

6  Possibly  from  the  Old  French  barat,  barate,  barete,  whence  bareter,  to  cheat 
or  beguile.  It  appears  in  Italian  as  baratto  (Ortega,  1512  ;  1515  Italian  ed.,  fol.  78) 
or  baratti  (Feliciano  da  Lazesio,  1536). 


BARTER  569 

of  a  dozen,  or  one  cow  out  of  a  given  number.1  The  second  is 
that  wherein  a  fixed  value  was  assigned  to  certain  products, 
such  as  grain  or  dates,2  these  products  acting  as  media  of 
exchange  or  as  bases  for  determining  values  in  bartering  other 
products, — a  period  lasting  until  money  was  invented  and  in- 
deed until  currency  became  common.  The  third  period  is  that 
of  the  adoption  of  money  as  a  medium  of  exchange,  this 
medium  taking  such  forms  as  wampum,  shells,  coins,  ingots, 
and  government  certificates. 

Two  influences  perpetuated  barter  long  after  the  first  of 
these  periods  and  indeed  down  to  the  present  time,  namely, 
the  scarcity  of  currency*  and  the  international  fairs.  In  these 
fairs4  the  merchants  found  that  barter  was  a  necessity  on 
account  of  the  scarcity  and  diversity  of  money.5 

Various  Names  for  Barter.  Barter  also  went  by  the  name  of 
" exchange,"  quite  as  we  use  the  word  " trade"  at  present.  Thus 
an  English  writer  of  1440  has  the  expression  "Bartyrn  or 
changyn  or  chafare6  oone  thynge  for  a  othere,  cambio"1  and 

1  For  example,  in  Egypt.    See  H.  Maspero,  Les  Finances  de  VEgypte  sous  les 
Lagides,  p.  29  (Paris,  1905). 

2  A.  H.  Sayce,  Social  Life  among  the  Assyrians   and  Babylonians,  chap,  v 
(London,  n.  d.) ;  W.  Cunningham,  The  Growth  of  English  Industry  and  Com- 
merce, p.  114  (London,  1896). 

3Ciacchi  mentions  this  effect  in  his  Regale  generali  d'abbaco,  p.  114  (Florence 
1675).  It  should  be  observed  that  the  output  of  gold  from  1850  to  1900  was 
greater  than  that  of  the  preceding  three  hundred  and  fifty  years,  which  accounts 
in  part  for  the  greater  amount  of  currency  now  available. 

4 Compare  the  fair  of  Nijni  Novgorod  and  the  smaller  fairs  of  Leipzig,  Munich, 
and  Lyons,  all  of  which  still  continue,  and  the  various  international  expositions 
which  are  modern  relics  of  the  ancient  gatherings  of  merchants.  In  a  MS.  on 
arithmetic,  written  in  Italy  in  1684,  nine  pages  are  given  to  a  list  of  great  fairs, 
mostly  European,  which  Italian  merchants  of  that  time  were  in  the  habit  of 
attending. 

5 Thus  Cataneo  (1546),  speaking  "De  baratti,"  says:  "E  Necessario  al  buon 
mercante  non  uolendo  receuer  danno  esser  molto  experto  nel  barattere"  (1567  ed., 
fol.  49). 

6  Middle  English  chaff  are,  chepefare,  from  the  Anglo-Saxon  ceapian  (to  buy) 
-\-jare  (to  go).  There  is  the  same  root  in  the  word  "cheap,"  originally  a  bargain, 
and  in  "Cheapside,"  the  well-known  London  street. 

7 Italian  for  "exchange."  So  Ghaligai  (1521)  speaks  of  "Barattare,  ouer 
cambiare  una  Mercantia  a  un'  altra";  and  Pellos  (1492)  has  "lo  .xiij.  capitol  qui 
ensenha  cabiar  aut  baratar  vna  causa  per  lautra." 


570  COMMERCIAL  PROBLEMS 

Baker  (1568)  gives  fifteen  pages  to  his  chapter  which  "treateth 
of  the  Rules  of  Barter :  that  is  to  say  to  change  ware  for  ware." 
The  early  German  writers  had  a  similar  usage.1 

The  French  writers  of  the  i6th  century  often  used  the  in- 
teresting word  troquer*  a  word  meaning  to  barter,  the  chapter 
being  called  Des  Troques.  From  this  word  we  have  the  Eng- 
lish "truck,"  the  material  bartered,  a  word  which  came  to  mean 
the  most  common  objects  of  exchange,  such  as  garden  truck, 
and  the  cart  (truck)  in  which  the  dealer  (truckman)  carried 
it,  and  finally  to  mean  worthless  material  in  general. 

Since  merchandise  was  often  bartered  for  future  delivery,  as 
in  the  case  of  goods  from  Damascus  or  China,3  the  question  of 
interest,  or  its  equivalent,  often  had  to  be  considered.  This 
gave  rise  to  the  distinction  between  barter  without  time  and 
barter  with  time.4 

Barter  in  America.  In  American  colonial  life  the  subject  of 
barter  played  an  important  part.  A  diary  of  1704,  kept  by 
one  Madam  Knight  of  Boston,  gives  an  idea  of  the  arithmetic 
involved,  as  the  following  extract  will  show : 

They  give  the  title  of  merchant  to  every  trader;  who  Rate  their 
Goods  according  to  the  time  and  spetia  they  pay  in:  viz.  Pay,  mony, 
Pay  as  mony,  and  trusting.  Pay  is  Grain,  Pork,  Beef,  &c.  at  the 
prices  sett  by  the  General  Court  that  Year ;  mony  is  pieces  of  Eight, 
Ryalls,  or  Boston  or  Bay  shillings  (as  they  call  them,)  or  Good  hard 
money,  as  sometimes  silver  coin  is  termed  by  them ;  also  Wampom, 
vizt.  Indian  beads  wch  serves  for  change.  Pay  as  mony  is  provisions, 
as  aforesd  one  Third  cheaper  than  as  the  Assembly  or  Genel  Court 
sets  it ;  and  Trust  as  they  and  the  mercht  agree  for  time. 

1  Using  their  word  Stick,  meaning  exchange.  Thus  Petzensteiner  (1483):  "Nu 
merck  hubsch  rechnung  von  stich."  Rudolff  (1526)  tells  how  "zwen  stechen 
mit  einander,"  a  phrase  now  used  with  respect  to  dice,  and  Albert  (1534)  relates 
that  "Zween  wollen  miteinander  stechen." 

2Thus  Trenchant  (1566):  "Deux  marchans  veulent  troquer  leurs  marchan- 
dises."  Compare  the  Dutch  Mangelinge  (Manghelinge,  Mangelingh)  as  a  syn- 
onym for  troques  and  change  in  the  i6th  and  iyth  century  arithmetics. 

3 These  two  cases  are  mentioned  in  a  MS.  of  1684,  written  at  Ancona,  now  in 
the  library  of  Mr.  Plimpton. 

4Thus  Ortega  (1512;  1515  ed.,  fol.  78):  "...  baratto  .  .  .  p  tempo  como 
senza  tepo." 


BARTER  AND  TAXES  571 

Now,  when  the  buyer  comes  to  ask  for  a  commodity,  sometimes 
before  the  merchant  answers  that  he  has  it,  he  sais,  is  Your  pay  redy  ? 
Perhaps  the  Chap  Reply's  Yes:  what  do  You  pay  in?  say's  the  mer- 
chant. The  buyer  having  answered,  then  the  price  is  set ;  as  suppose 
he  wants  a  sixpenny  knife,  in  pay  it  is  i2d — in  pay  as  money  eight 
pence,  and  hard  money  its  own  price ;  viz.  6d.  It  seems  a  very  Intri- 
cate way  of  trade  and  what  Lex  Mercatoria  had  not  thought  of. 

Another  diary,  kept  by  one  Jeremiah  Atwater,  a  New  Haven 
(Connecticut)  merchant,  about  1800,  had  various  entries  of  a 
similar  nature,  among  which  is  the  following : 

To  5  yds  Calico  at  2S  6d  per  yard. 

To  be  paid  in  turnips  at  is  6d  and  remainder  in  shoes.  As  far  as 
the  turnips  pay,  the  calico  is  to  be  2  s  6d  and  the  remainder  toward 
shoes  at  2s8d. 

Taxes.  Of  all  the  applications  of  arithmetic,  taxation  is  one 
of  the  oldest.  The  tax  collector  is  mentioned  in  the  ancient 
papyri  of  Egypt,1  in  the  records  of  Babylon,2  in  the  Bible,3 
and,  indeed,  in  the  histories  of  all  peoples.  His  methods  reached 
the  extreme  of  cruelty  among  the  Saracens,  and  a  decided  trace 
of  this  cruelty  is  still  seen  among  some  of  their  descendants. 

In  Greece  the  tax4  was  levied  directly  on  property  or  in- 
directly by  tolls  or  customs.  Resident  aliens  paid  a  poll5 
tax,  and  an  indirect  tax0  of  2  per  cent  was  levied  at  the  custom- 
houses. 

Rome  had  an  elaborate  system  of  taxation,7  and  this  was  the 
source  of  our  present  systems.  It  included  the  tariff,8  the 

1  See  Volume  I,  page  45.  2Sayce,  Social  Life,  p.  68. 

3  E.g.,  2  Kings,  xxiii,  35;  Luke,  ii,  i.  4TVXos  (tel'os). 

5 Head,  from  the  Danish  bol,  a  ball,  bowl,  bulb,  or  head.  Hence  the  "polls," 
where  the  heads  of  the  electors  are  counted.  The  Greek  term  was  perolKiov 
(metoi'kion). 

6The  Trevn/jKoarrj  (pentecoste'} ,  fiftieth.  This  is  the  same  word  as  our  "pen- 
tecost,"  which  refers  to  the  fiftieth  day  after  the  Passover  in  the  Jewish  calendar. 
In  Greek  taxation  the  word  referred  to  the  tax  of  ^,  or  2  per  cent,  on  exports 
and  imports. 

7  Latin  taxatio,  from  taxare,  to  estimate  or  evaluate. 

8 Spanish  tarifa,  a  price  list  or  book  of  rates;  from  the  Arabic  ta'rlf,  giving 
information,  from  the  root  rarf,  knowing.  The  word  shows  the  Arab  influence, 
through  the  Spanish,  upon  modern  business.  The  Spanish  town  Tariffa  was 


572  COMMERCIAL  PROBLEMS 

ground  tax/  the  poll  tax,2  the  tithes  (still  familiar  in  certain 
parts  of  the  world),3  and,  in  later  times,  the  tax  on  traders.4 

It  is  a  curious  and  interesting  fact  that  the  subject  of  tax- 
ation commanded  but  little  attention  in  the  early  textbooks. 
It  is  possible  that  authors  hesitated  to  touch  upon  such  a  sensi- 
tive spot  because  of  the  necessity  for  receiving  an  imprimatur 
from  the  taxing  powers.  Although  it  is  occasionally  found  in 
the  1 6th  century,5  it  was  not  common  until  textbook  writers 
were  more  free  in  their  offering. 

Banking  and  Exchange.  In  the  days  when  Europe  was  made 
up  of  a  large  number  of  small  principalities,  each  with  its  own 
system  of  coinage,  the  subject  of  exchange  was  much  more 
familiar  to  the  average  business  man  than  it  is  today.  How 
recently  this  was  the  case  may  be  seen  in  a  remark  of  Met- 
ternich's  in  1845,  that  Italy  "  represents  simply  a  group  of  in- 
dependent states  united  under  the  same  geographic  term." 
The  European  traveler  gets  some  idea  of  the  early  situation 
today,  for  at  the  railway  stations  on  the  borders  he  finds  the 
exchange  office,6  where  he  may  exchange  the  money  of  the 
country  he  is  leaving  for  that  of  the  country  he  is  entering. 

So  important  was  the  subject  considered  that  the  1594  edi- 
tion of  Recorde's  Ground  of  Artes  devotes  twenty-one  pages  to 
it,  saying  that  it  is  of  great  value  to  the  merchants  dealing  with 
Lyons,  inasmuch  as  "  there  are  4  faires  in  a  yeere,  at  which 
they  do  commonly  exchange,777 

named  from  the  fact  that  it  was  the  leading  customhouse  at  one  time.  The 
Latin  term  was  portorium,  from  portare,  to  carry,  as  in  "import,"  "export,"  and 
"  transport." 

1  Tributum  soli,  tribute  of  the  land,  tributum  coming  from  tribuere,  to  bestow 
or  pay.  2Tributum  capitis.  ^Decimae. 

*Collatio  lustralis.  This  was  especially  prominent  in  the  5th  century,  when 
the  great  social  upheaval  led  to  the  aggrandizement  of  the  aristocracy.  Sec  S.  Dill, 
Roman  Society  in  the  Last  Century  under  the  Western  Empire,  p.  204  (London, 
1898). 

5As  in  Savonne  (1563;  1571  ed.,  fol.  41),  with  the  name  "Reigles  des  im- 
positions." 

6  Bureau  des  changes.  Weeks  elbureau,  Wissel  Bureau,  Cambto. 

7  The  editor,  Mellis,  gives  a  long  list  of  the  leading  fairs  which  an  English 
merchant  might  attend. 


EXCHANGE 


573 


Chain  Rule.  In  the  days  when  the  value  of  coins  varied 
greatly  from  city  to  city  as  well  as  from  country  to  country, 
money  changers  employed  a  rule,  probably  of  Eastern  origin, 
which  was  known  by  various  names1  but  was  most  commonly 


RENAISSANCE    DEALER    IN    EXCHANGE 
From  the  1500  edition  of  Widman's  arithmetic  (1489) 

called  the  chain  rule2  or  continued  proportion.  The  following 
problem  is  adapted  from  Widman  (1489)  and  illustrates  the 
type  and  the  solution: 

A  man  went  to  a  money  changer  in  Vienna  with  30  Nurnberg 
pence  and  asked  that  they  be  exchanged  for  Viennese  money. 
Since  the  money  changer  was  ignorant  of  their  value,  he  pro- 
ceeded thus  :  7  pence  of  Vienna  are  worth  9  pence  of  Linz,  and 
8  of  Linz  are  worth  n  of  Passau,  and  12  of  Passau  are  worth 


example,  Regula  conjuncta,  Re  gel  conjoinct,  Te  Zamengevoegden  Regel, 
Regel  van  Vergelykinge,  and  De  Gemenghde  Regel,  these  terms  being  taken  from 
various  Dutch  and  Dutch-French  books  of  the  i;th  and  i8th  centuries. 

2  Den  Kettingh-Regel,  Den  Ketting  Reegel,  in  the  early  Dutch  books.  "Gleich- 
sam  wie  die  Glieder  einer  'Kette,'"  as  R.  Just,  a  modern  German  writer,  has 
it  in  his  Kaujmannisches  Rechnen,  I,  81  (Leipzig,  1901). 


574  COMMERCIAL  PROBLEMS 

13  of  Wilsshof,  and  15  of  Wilsshof  are  worth  10  of  Regensburg, 
and  8  of  Regensburg  are  worth  18  of  Neumarkt,  and  5  of 
Neumarkt  are  worth  4  of  Niirnberg.  Then 


7  •  8  •  1 2  •  i  s  •  8  •  5  -  30  0  Q 

so  that  - — -— — — ^--—  =  1 3  M)  > 

9  •  ii  •  13  -  10  •  18  -  4  4"9 

the  value  of  the  Niirnberg  money  in  pence  of  Vienna.1 

Like  many  Eastern  problems  it  is  found  in  the  works  of 
Fibonacci  (i2O2),2  and  thereafter  it  was  common  until  the 
latter  part  of  the  iQth  century. 

Early  Banks.  The  early  banks  were  established  in  places  of 
greatest  relative  safety,  and  these  were  usually  the  temples. 
All  kinds  of  valuables  were  thus  protected  from  the  depreda- 
tions of  thieves,  both  private  and  governmental,  civil  and  mili- 
tary. This  is  seen,  for  example,  in  the  great  business  interests 
carried  on  within  the  precincts  of  the  temples  in  the  Ur  Dynasty 
(c.  2450-2330  B.C.)  of  Babylon.3  On  the  tablets  of  this  period 
may  be  found  the  records  of  loans,  receipts,  promissory  notes, 
leases,  mortgages,  taxes,  and  other  commercial  activities.  A 
little  later,  in  the  first  millennium  B.C.,  drafts  appear4  in  quite 
the  form  used  even  today. 

For  the  reason  above  stated,  the  priests  in  the  Greek  temples 
were  frequently  money  lenders.5  It  was  also  on  this  account 


1The  rule  closes:  "Vn  multiplicir  in  krcucz  durchauss  auff  2  teyl  vn  dividir" 
(fol.  152). 

2"De  baractis  monetarum  cum  plures  monete  inter  similes"  (Liber  Abaci, 
p.  126). 

3M.  Jastrow,  Jr.,  The  Civilization  of  Babylonia  and  Assyria,  p.  318  (Phila- 
delphia, 1915). 

4  A.  T.  Clay,  Babylonian  Records  in  the  Library  oj  J.  Pierpont  Morgan,  Part 
I  (New  York,  1912). 

5  J.  P.  Mahaffy,  Old  Greek  Life,  p.  38  (London,  1885) ;  hereafter  referred  to 
as  Mahaffy,  Greek  Life. 


EARLY  BANKING  575 

that  coins  were  struck l  at  the  temple  of  Juno  Moneta'2  in  Rome. 

The  later  Greek  bankers  and  money  changers  were  called 
rpaTrellra^  a  word  derived  from  T/oa7re£a,4  a  table,  just  as 
"bank"  comes  from  "bench."5  Their  tables  were  placed  in 
the  agorae  (public  places)  and  the  finding  of  "the  tables  of  the 
money  changers"6  in  the  Temple  at  Jerusalem  was  not  at  all 
unusual.  Indeed,  one  may  see  similar  sights  in  the  temples  of 
Southern  India  today,  or  in  the  entrance  to  the  great  pagoda  at 
Rangoon,  Burma. 

The  business  of  the  trapezitai  included  buying  foreign  money 
at  a  discount  and  selling  it  at  a  premium,  paying  interest  on 
deposits,  acting  as  pawnbrokers,  and  performing  the  duties 
of  modern  notaries.7 

Banks  in  Rome.  Bankers  are  mentioned  by  Livy  (IX,  40, 
1 6)  as  carrying  on  business  as  early  as  the  4th  century  B.C. 
At  first  a  private  banker  was  called  an  argentarius  (silver 
dealer),  an  officer  connected  with  the  mint  being  a  nummu- 
larius.8  Somewhat  later  these  terms  were  used  along  with 
mensarius9  and  collectarius™  to  represent  any  kind  of  banker. 


iln  early  days  they  were  stamped  by  the  stroke  of  a  hammer,  and  the  word 
has  remained  in  use. 

2 Juno  the  Admonisher  (Adviser,  Instructor).  On  this  account  the  Romans 
used  the  word  moneta  to  mean  money,  whence  also  our  word  "mint." 

'•^Trapezi'tai,  literally,  "tablers."   The  Hebrew  usage  is  the  same. 

*Tra'peza,  whence  our  "trapezium,"  a  figure  representing  a  table  (originally 
with  two  parallel  sides),  and  "trapezoid,"  a  figure  shaped  like  a  trapezium  (origi- 
nally with  no  parallel  sides) .  The  bankers  were  also  called  dpyvpafjLoifiol,  argyra- 
moiboi',  money  changers.  See  Harper's  Diet.  Class.  Lit.,  p.  1597;  Mahaffy,  Greek 
Life,  p.  38;  F.  B.  Jevons,  Manual  of  Greek  Antiq.,  p.  395  (London,  1895). 

5 Late  Latin  bancus,  a  bench;  French  bane,  a  long  seat  or  table.  So  we  have 
a  bench  of  judges  and  the  bank  of  a  stream.  "Banquet,"  simply  a  diminutive 
form,  came  to  mean  a  feast  instead  of  the  table.  In  Italy  banca  came  to  mean 
a  tradesman's  stall,  a  counter,  and  a  money  changer's  table,  as  well  as  a  bank. 

6 As  late  as  1567  we  find  English  writers  telling  of  how  "Christ  overthrew  the 
Exchaungers  bankes." 

7M.  S.  Koutorga,  Essai  historique  sur  les  trapezites  ou  banquiers  d'Athenes, 
Paris,  1859. 

8  Coin-man.   Originally  an  officer  of  the  mint  who  tested  the  silver  before  it 
was  coined. 

9  Or  mensularius,  from  mensa,  table,  influenced  by  the  Greek  name  for  banker. 
10 Late  usage,  found  in  Justinian's  Institutes. 


576  COMMERCIAL  PROBLEMS 

The  exchange  bureau  was  called  the  permutatio?  In  the  bank- 
ing department  the  funds  of  the  creditor  were  called  the  deposi- 
tum? whence  our  "deposit."  This  was  subject  to  a  perscriptio,3 
a  check,  quite  as  it  is  at  present.  The  depositum  drew  no 
irtterest,4  being  like  our  common  open  accounts;  but  there 
developed  also  a  kind  of  savings-bank  department  in  which  a 
deposit  known  as  a  creditum*  drew  interest.0 

Letters  of  Credit.  The  ancient  bankers  issued  letters  of  credit 
quite  like  those  issued  at  present/  and  also  made  drafts  on  one 
another.8  The  idea  is  therefore  without  foundation  that  it  was 
the  Jews  who,  driven  from  France  to  Lombardy  in  the  yth 
century,  first  made  use  of  foreign  drafts.9 

Stockholders.  Among  the  " seven  greater  arts"  recognized 
in  medieval  Florence  was  that  of  the  money  changers,10  As 
early  as  1344  the  city  government,  finding  itself  unable  to  pay 
some  $300,000  that  it  owed,  formed  a  bank11  and  issued  shares 
of  stock12  which  were  transferable  as  in  modern  corporations.13 

1  Compare  our  "permutation."  J.  Marquardt,  La  vie  privee  des  Romains, 
French  translation,  p.  15  (Paris,  1893). 

2De  -f  ponere,  to  place.    A  depositum  is  that  which  is  placed  down. 

3 Per  +  scribere,  to  write;  a  written  order. 

4 It  was  vacua  pecunia,  unproductive  money. 

5 From  credere,  to  trust,  to  have  confidence  in,  to  believe,  whence  our  "creed"; 
in  banking  business  a  sum  held  in  trust. 

fi  Because  of  this  interest  feature,  the  claims  of  the  depositarii  were  legally  pre- 
ferred to  those  of  the  credit  ores  in  case  of  the  failure  of  a  bank.  See  the  Digest, 
XVI,  3,  7,  2. 

7Mahaffy,  Greek  Life,  p.  38.  They  were  called  by  the  Greeks  o-uo-rariKal 
eVto-roXa/  (systatikai'  epistolai'),  letters  of  introduction. 

8  Cicero  uses  "permutare  Athenas"  to  mean  "to  draw  on  Athens";  and  "ab 
Egnatio  solvat"  to  mean  "to  pay  by  draft  on  Egnatius." 

9 linger,  Die  Methodik,  p.  90. 

i°Arte  del  cambio  or  del  cambiatori.  The  other  six  arti  maggiori  were  those 
of  (i)  dressers  of  foreign  cloth,  (2)  dealers  in  wool,  (3)  judges  and  notaries,  (4) 
physicians  and  apothecaries,  (5)  dealers  in  silk,  (6)  furriefs.  E.  G.  Gardner, 
The  Story  of  Florence,  p.  28  (London,  1900) . 

11  Monte,  mount,  bank,  money;  compare  the  French  mont-de-piete. 
12 Originally  the  word  meant  a  thing  that  was  stuck  or  fixed,  and  hence  a  post, 
the  Anglo-Saxon  stocc,  as  in  "stockade."    The  same  root  is  found  in  "etiquette" 
(Old  French  estiquet),  a  little  note  "stuck  up"  on  the  gate  of  a  court.   From  the 
same  source  we  have  "stack"  and  "ticket."    See  also  page  194. 

18C.  A.  Conant,  History  of  Modern  Banks  of  Issue,  p.  21  (New  York,  1896). 


KINDS  OF  EXCHANGE  577 

The  idea  of  issuing  notes  payable  in  coin  but  only  partly 
covered  by  a  reserve  was  a  development  of  the  i7th  century, 
beginning  in  Amsterdam  (1609)  an(^  developing  into  more 
modern  form  in  Stockholm  (I66I).1 

Bills  of  Exchange.  Probably  the  first  bill  of  exchange  to  ap- 
pear in  a  printed  work  on  mathematics  is  the  one  given  by 
Pacioli  (1494),  the  form  being  substantially  the  same  as  the 
one  now  in  use.2 

Four  Kinds  of  Exchange.  In  the  early  printed  arithmetics 
there  were  four  kinds  of  exchange,  of  which  we  have  preserved 
two.  The  four  types  were  as  follows  : 

1.  Common  exchange,  the  mere  interchange  of  coins,  the 
work  of  the  "  money  changers."3 

2.  Real  exchange,  by  means  of  drafts.4 

3.  Dry  exchange,5  a  method  of  evading  usury  laws  by  means 
of  fictitious  bills0  of  exchange,  —  drafts  that  bore  no  fruit.7 


y  p.  24. 

2  1494  adi  9  agosto  I  va. 

Pagate  per  questa  prima  nostra  a  Lodouico  de  francesco  da  fabriano  e  com- 
pagni  once  cento  doro  napolitane  insu  la  proxima  fiera  de  fuligni  per  la  valuta 
daltretanti  receuuti  qui  rial  Magninco  homo  miser  Donate  da  legge  quonda  miser 
Priamo.  E  ponete  ;p  noi.  Jdio  da  mal  ve  guardi. 

vostro  Paganino  de  paganini  da  Brescia  ss.  .  .  . 
Domino  Alphano  de  Alphanis  e  copagni  in  peroscia.  .  .  . 
Suma,  1494  ed.,  fol.  167,  v. 

For  later  examples,  see  Cardan's  Practice  1539*  cap.  56;  Trenchant's  arith- 
metic of  1566  (1578  ed.,  p.  350),  and  other  commercial  works. 

8  The  Italians  called  it  "cambio  menuto,  ouer  commune,"  as  in  Tartaglia's 
work  (1592  ed.,  II,  fol.  174).  In  Spanish  it  appeared  as  "cambio  por  menudo" 
(Saravia,  1544),  and  in  French  as  "change  menu  ou  commun"  (Trenchant, 
1566).  It  is  this  form  that  is  referred  to  in  1335,  in  the  English  act  of  9  Edw. 
Ill,  stat.  2,  c.  7:  "Et  que  table  dcschangc  soit  a  Dovorri  &  aillours,  ou  &  qant 
il  semblera  a  nos  &  notre  consail  per  faire  eschange,"  —  our  "bureau  des  changes." 

4  Saravia  (1544)  rnakes  two  divisions  of  this  type.    See  his  Italian  translation 
of  1561,  fols.  108,  no. 

5  Italian,  cambio  secco',  French,  change  sec;  German,  trockener  Wechsel,  as 
the  terms  appear  in  the  i6th  century. 

6From  the  Latin  bulla,  a  bubble,  a  leaden  seal  that  looked  like  a  bubble,  and 
hence  the  sealed  document,  like  a  papal  bull.  From  the  same  root  we  have 
"bullet,"  "bulletin,"  "bowl,"  and  "bullion"  (a  mass  of  sealed  or  stamped  metal). 

7  So  called,  as  Saravia  (Italian  translation,  1561)  says,  from  their  resemblance 
to  an  "albero  secco,  il  quale  non  ha  humore,  ne  foglie  ne  frutto." 


S78 


COMMERCIAL  PROBLEMS 


Such  exchange  was  placed  under  the  ban  by  an  English  statute 
of  1485/6:  "enybargayne  .  .  .  by  the  name  of  drye  exchaunge 
...  be  utterly  voide."  English  writers  sometimes  spoke  of  it 
as  "sick"  exchange/  confusing  the  French  sec  (dry)  with  an 

English  word  of  different' 
meaning. 

4.  Fictitious  exchange,2 
the  plan  of  collecting  a  debt 
by  drawing  on  the  debtor. 

As  coinage  came  to  be 
better  settled  a  definite  par 
of  exchange  was  recognized, 
and  so,  beginning  in  the 
arithmetics  of  the  latter 
part  of  the  i6th  century, 
we  find  various  rules  relat- 
ing to  this  subject.  Thus 
Recorde  (c.  1542)  says  that 
"as  touching  the  exchange, 
it  is  necessary  to  vnderstand 
or  know  the  Pair,  which  the 
Italians  call  Pan."3 


FROM  THE  MARGARITA  PHYLOSOPHICA 

(1503) 

Showing  geometry  as  largely  concerned 
with  gaging  and  similar  practical  work 


Days  of  Grace.  The  Ital- 
ian cities  had  fixed  rules  as 
to  the  number  of  days  after 
sight  or  after  date  at  which 
drafts  should  be  paid.  Drafts  between  Venice  and  Rome  were 
payable  ten  days  after  sight ;  between  Venice  and  London,  three 
months  after  date ;  of  Venice  on  Lyons,  at  the  next  succeeding 
quarterly  fair,  and  so  on,  thus  giving  the  payer  time  to  obtain 
money.  In  these  customs  is  to  be  found  the  origin  of  the 


!So  T.  Wilson,  writing  on  usury  in  1584,  speaks  of  "sicke  and  drie  exchange." 
2 The  cambio  fittitio  of  the  Italian  writers  of  the  i6th  century,  and  the  change 

fict  of  the  French. 

8  1594  ed.,  p.  557.   We  also  have  such  Dutch  terms  as  Rekeninghe  vander  Pary 

(Raets,  1580)  and  Den  Reghel  Parij  (Stockmans,  1676  ed.),  and  such  French 

terms  as  le  per  (Savonne,  1563). 


BANK  CHECKS 


579 


"days  of  grace,"  formerly  allowed  in  England  and  America, 
but  generally,  owing  to  improved  banking  facilities,  abandoned 
in  the  latter  country  about  the  opening  of  the  2oth  century. 


\\\^^ 


THE   GAGER   (GAUGER) 
From  KobeFs  Vysirbuch,  1515,  showing  the  tools  of  the  art 

Meaning  of  a  Check.  At  present  the  check  (in  England, 
cheque)  is  extensively  used  instead  of  a  bank  draft.  The  word 
has  an  interesting  history,  coming  from  the  Persian  shah,  a 
king.  In  the  game  of  chess  the  player  called  out  "shah" 


S8o  COMMERCIAL  PROBLEMS 

when  the  king  was  in  danger,  and  "shah-mat"  ("the  king 
is  dead")  when  the  king  could  no  longer  move.  From  this 
we  have  our  "check"  and  "checkmate"  in  chess,  "check" 


THE   GAGER   AT    WORK 
From  Johann  Prey's  Bin  new  Visier  buchlein,  Niirnberg  [1543] 

being  thought  to  mean  simply  "stop."  Hence  we  have  the  Mid- 
dle English  ckek,  French  echec  (a  check  or  defeat) ,  Italian 
scacco  (a  chess  board),  "checkers,"  and  "check"  (a  stop  in 
one's  account  at  a  bank). 

Gaging.  Before  the  size  of  casks  was  standardized  as  the 
result  of  manufacturing  in  quantities  or  of  general  laws  affect- 
ing large  territories,  the  subject  of  gaging1  (gauging)  played 

1The  Middle  English  was  gagen  or  gawgen,  to  gage.  The  u  came  in  through 
the  Old  French  ganger.  In  medieval  Latin  ga-ugia  meant  a  standard  wine  cask, 
but  the  origin  of  the  word  is  uncertain. 


GAGING  581 

an  important  role  in  applied  mathematics.  The  word  relates 
to  the  finding  of  the  capacity  of  casks  and  barrels.  In  Ger- 
many a  gager  was  called  a  Visierer,1  and  in  the  isth  century 
there  appeared  numerous  manuals  with  such  titles  as  Vysirbuch, 
Vysierbuch,  and  Visyrbuechleynn?  The  custom  was  carried 
over  to  England,  and  even  in  our  early  American  arithmetics 
there  were  chapters  on  gaging.  The  first  notable  German  book 
on  the  subject  was  KobePs  work  (see  page  579)  of  1515,  al- 
though the  Margarita  phylosophica  of  1503  pictured  geometry 
(see  page  578)  as  chiefly  concerned  with  such  measurements. 

Other  Applications.  It  would  be  difficult  to  give  a  satisfac- 
tory list  of  all  the  applications  of  elementary  mathematics  to 
the  manifold  interests  of  man  that  have  developed  in  the  cen- 
turies past.  These  applications  include,  besides  those  already 
mentioned,  such  topics  as  the  adulteration  of  goods/  account- 
ancy and  exchange,4  small  commerce  from  town  to  town,6  the 
leather  trade,6  grazing,  and  baking, — a  list  that  might  be 
extended  to  include  many  other  topics  and  that  illustrates 
the  way  in  which  arithmetic  has  met  human  needs. 

iprom  visieren,  to  vise,  to  indorse  a  standard,  to  show  that  it  has  been  seen 
(subjected  to  vision)  and  approved. 

2E.g.,  Grammateus,  1523. 

3 Thus  Kb'bcl :  "Die  Regel  Fusci  mit  jhrer  Erklarung/folget  hernach.  .  .  . 
Das  wort  Fusci/bcdcut  nicht  anders/dann  ein  zerbrochen  gut  gemiilb/oder  andere 
vnreynigkeit/so  in  der  Specerei  funden  wirt/als  under  den  Negelin/Imber/Saf- 
fran/&c.  Auch  Silber  vnderm  Golt/Kupffer  vnderm  Silber."  Rechenbuchlein, 
1514;  1549  ed.,  fol.  77.  See  also  other  German  works  of  the  period,  such  as 
Albert  (1534;  1561  ed.,  fol.  D  viii)  and  Thierfelder  (1587,  p.  116).  The  word 
also  appears  as  Fusti. 

4  Frequently  found  in  the  early  Dutch  books  as  "Rekeninghe  voor  cassiers" 
(Van  der  Schuere,  1600,  fol.  65),  "Reductio,  ofte  Cassiers  Rekeninge"  (Cock, 
1696  ed.,  p.  96),  "Cassiers  Rekeninghe"  (Mots,  1640,  fol.  i),  "Den  Regel  vander 
Munte  Oft  den  Reghel  gheheeten  Regula  Cassiers"  (Stockmans,  1676  ed.,  p.  205). 

5  A  common  topic  in  the  French,  Dutch,  and  German  books  of  the  i6th  and 
1 7th  centuries  under  such  titles  as  "Rechnung  vberland"  (Rudolff,  Kunstliche 
rechnung  (1526;  1534  ed.,  fol.  ki)) ;  " Overlantsche  Rekeninghe"  and  "Comptes 
de  Voyage"  (Cou tercels,  Dutch  and  French  arithmetic,  1631,  p.  283) ;  "Uytlandt- 
sche  Rekeningen"   and  "Voyages"    (Eversdyck's  edition   of   Coutereels,   1658, 
p.  299) ;  and  "Voyagien"  (Houck,  1676  ed.,  p.  148). 

6  Particularly  in  such  centers  as  Leipzig,  where  we  find  arithmetics  of  about 
1700  with  such  chapters  as  "Leder  und  Rauchwahrcn-Rechnung." 


582  APPLICATIONS  OF  ALGEBRA 

4.  APPLICATIONS  OF  ALGEBRA 

General  Nature.  The  first  applications  of  algebra  were  in  the 
nature  of  number  puzzles.  Such  was  the  first  algebraic  prob- 
lem1 of  Ahmes  (c.  1550  B.C.),  already  mentioned, — "Mass,  its 
whole,  its  seventh,  it  makes  19."  Such  are,  in  general,  the 
problems  of  Diophantus  (c.  275),  and  the  problems  which  still 
form  the  large  majority  of  those  given  in  current  textbooks. 
When  a  pupil  is  called  upon  today  to  solve  the  equation 
x  +  $-x  =  19,  he  is  really  solving  the  first  problem  of  Ahmes, 
and  all  our  abstract  work  in  equations  is  a  development  of 
this  type. 

The  second  general  application  is  geometric,  and  this  char- 
acterizes the  works  of  the  Greek  writers,  with  the  exception 
of  Diophantus. 

The  third  general  application  is  to  fanciful  problems  relating 
to  human  affairs,  and  this  is  essentially  Oriental  in  spirit. 

The  fourth  type  is  characterized  by  the  attempt  to  relate 
algebra  actually  to  the  affairs  of  life.  The  first  steps  in  this 
direction  were  taken  when  algebra  was  more  or  less  a  part  of 
arithmetic,  problems  often  being  given  that  were  essentially 
algebraic  but  were  solved  without  any  further  symbolism  than 
that  afforded  by  the  medieval  algorism.  It  is  with  this  type  that 
we  are  working  at  present  and  are  making  some  advance. 

We  shall  now  consider  the  first  three  of  these  stages  in  detail. 

The  Number  Puzzle.  It  was  with  the  number  puzzle  that  al- 
gebra seems  to  have  taken  its  start.  The  desire  of  the  early 
philosophers  to  unravel  some  simple  numerical  enigma  was 
similar  to  the  child's  desire  to  find  the  answer  to  some  question 
in  the  puzzle  column  of  a  newspaper.  A  few  types  will  be 
selected  involving  linear  equations,  the  quadratic  being  con- 
sidered later. 

Ahmes  (c.  1550  B.C.)  gave  numerous  problems  like  "Mass, 
its  f ,  its  |,  its  |,  its  whole,  it  makes  33. "2 

1No.  24  in  Peet,  Rhind  Papyrus,  p.  61,  with  slightly  different  translation. 
2lbid.,  No.  31.    In  modern  symbols, 

X  +    ~  X  +    I  X  -f    7  X  =  33. 


NUMBER  PUZZLES  583 

The  problems  of  Diophantus  (c.  275)  were  often  of  this 
type,  as  in  the  first  one  given  in  Book  I:1  "To  divide  a  given 
number  into  two  having  a  given  difference,"  with  the  particular 
case  of  2x  +  40  =  100. 

In  the  Middle  Ages  in  Europe  the  standard  algebraic  prob- 
lem was  of  the  same  general  nature.  This  is  seen,  for  example, 
in  the  De  Numeris  Datis  of  Jordanus  Nemorarius  (c.  1225), 
where  all  the  problems  are  abstract,  as  in  the  following  case : 

"If  there  should  be  four  numbers  in  proportion  and  three  of 
ttiem  should  be  given,  the  fourth  would  be  given.'72 

From  early  times  to  the  present  the  number  puzzle  has 
played  a  leading  part  in  algebra,  and  under  current  conditions, 
when  algebra  is  required  as  a  school  subject,  this  is  not  alto- 
gether fortunate. 

Simultaneous  Linear  Equations.  Problems  involving  simul- 
taneous linear  equations  were  more  numerous  in  the  Orient  in 
early  times  than  they  were  in  Europe.  Thus  we  find  in  India 
a  considerable  number  of  such  problems,  together  with  rules 
that  amount  to  directions  for  solving  various  types  of  simulta- 
neous equations.  For  example,  Mahavira  (c.  850)  has  the 
following  problem : 

The  mixed  price  of  9  citrons  and  7  fragrant  wood-apples  is  107 ; 
again,  the  mixed  price  of  7  citrons  and  9  fragrant  wood-apples  is  101. 
O  you  arithmetician,  tell  me  quickly  the  price  of  a  citron  and  of  a 
wood-apple  here,  having  distinctly  separated  those  prices  well. 

His  rule  for  the  solution  is  similar  to  the  one  used  today  for 
eliminating  one  unknown.3 

Another  of  Mahavira's  problems,  evidently  suggested  by  one 
that  appeared  in  the  Greek  epigrams  and  was  there  attributed 
to  Euclid,  is  as  follows : 

Three  merchants  saw  [dropped]  on  the  way  a  purse  [containing 
money].  One  [of  them]  said  [to  the  others],  "If  I  secure  this  purse, 

1  Heath,  Diophantus,  2d  ed.,  p.  131. 

2 Problem  30:   "Si  fuerit  [sic]  IIII  numeri  proporcionales  et  tres  eorum  dati 
fuerint  et  quartus  datus  erit"  (Abhandlungen,  II,  143). 
3  See  his  work,  p.  130. 


584  APPLICATIONS  OF  ALGEBRA 

I  shall  become  twice  as  rich  as  both  of  you  with  your  moneys  on 
hand."  Then  the  second  [of  them]  said,  "I  shall  become  three  times 
as  rich."  Then  the  other  [,  the  third,]  said,  "I  shall  become  five 
times  as  rich."  What  is  the  value  of  the  money  in  the  purse,  as  also 
the  money  on  hand  [with  each  of  the  three  merchants]?1 

Indeterminate  Problems.  The  indeterminate  problem,  leading 
to  what  is  now  called  the  indeterminate  equation,  is  very  old. 
It  is  probable  that  it  formed  a  type  of  recreation  long  be- 
fore the  time  of  Archimedes,  since  in  the  problems  assigned  to 
him  there  appears  one  of  very  great  difficulty.  This  problem 
has  already  been  discussed  on  page  453."' 

Although  Diophantus  (c.  275)  proposed  many  indeterminate 
equations,  they  were  not  in  the  form  of  applied  problems. 
Cases  of  the  latter  kind  seem  to  have  come  chiefly  from  the 
Orient,  at  least  in  early  times. 

In  the  Greek  Anthology  (c.  500?)  there  are  two  problems  in- 
volving indeterminate  linear  equations.  The  first  (XIV,  48)  is 
as  follows: 

The  three  Graces  were  carrying  baskets  of  apples  and  in  each  was 
the  same  number.  The  nine  Muses  met  them  and  asked  them  for 
apples,  and  they  gave  the  same  number  to  each  Muse,  and  the  nine 
and  three  had  each  of  them  the  same  number.  Tell  me  how  many 
they  gave  and  how  they  all  had  the  same  number.3 

The  second  (XIV,  144)  is  a  dialogue  between  two  statues: 

A.  How  heavy  is  the  base  on  which  I  stand,  together  with  myself ! 

B.  My  base  together  with  myself  weighs  the  same  number  of 
talents. 

A.  I  alone  weigh  twice  as  much  as  your  base. 

B.  I  alone  weigh  three  times  the  weight  of  yours.4 

1See  his  work,  p.  155. 

2 The  problem  was  discovered  by  the  German  dramatist  G.  E.  Lessing,  in  1773, 
while  he  was  serving  as  court  librarian  at  Wolf enbuttel .  On  the  history  of  the 
problem  see  B.  Lefebvre,  Notes  d'Histoire  des  M athtmatiques ,  p.  33  n.  (Louvain, 
1920).  On  a  few  indeterminate  problems  due  to  Heron,  see  Heath,  History,  II, 
444. 

3 The  equations  reduce  to  x  =  4y.   There  were  12 n  apples  in  all. 

4  The  equations  are  #  +  y  =  w  +  v,  #  =;  2  v,  «  =  3  y . 


INDETERMINATE  PROBLEMS  585 

With  their  usual  desire  to  give  a  fanciful  but  realistic  touch 
to  algebra,  the  Chinese  applied  the  indeterminate  equation  to  a 
problem  commonly  known  as  that  of  the  Hundred  Fowls.  This 
problem  goes  back  at  least  to  the  6th  century1  and  differs  so 
greatly  in  its  nature  from  those  of  the  Greeks  that  it  seems  to 
have  originated  in  the  East.  The  problem  is  as  follows : 

If  a  cock  is  worth  5  sapeks ;  a  hen,  3  sapeks ;  and  3  chickens  to- 
gether, i  sapek,  how  many  cocks,  hens,  and  chickens,  100  in  all,  will 
together  be  worth  100  sapeks? 

From  China  the  problem  apparently  found  its  way  to  India, 
for  it  appears  in  Mahavira's  work  (c.  850)  in  the  following 
form: 

Pigeons  are  sold  at  the  rate  of  5  for  3  [panas],  sdrasa  birds  at  the 
rate  of  7  for  5  [panas],  and  peacocks  at  the  rate  of  3  for  9  [panas]. 
A  certain  man  was  told  to  bring  at  these  rates  100  birds  for  100 
panas  for  the  amusement  of  the  king's  son,  and  was  sent  to  do  so. 
What  [amount]  does  he  give  for  each  [of  the  various  kinds  of  birds 
that  he  buys]  ? 

Mahavlra  gave  a  method  for  solving  such  problems  that  was 
sufficient  to  satisfy  those  who  were  interested  in  puzzles,  but 
which  had  little  merit  otherwise.2 

This  fanciful  type  of  problem  was  probably  made  known  in 
Europe  at  the  time  of  the  general  penetration  of  Oriental  ideas, 
and  here  it  developed  into  a  form  somewhat  like  this : 

20  persons,  men,  women,  and  girls,  have  drunk  20  pence  worth  of 
wine ;  each  man  pays  3  pence,  each  woman  2  pence,  and  each  girl 
^  penny ;  required  the  number  of  each. 

Algebraically,  m  +  w+g=2O 

and  3m  +  2w  +  %g—20> 

aKaye,  Indian  Math,,  p.  40;  L.  Vanhee,  "Les  cents  volailles  ou  1'analyse  in- 
d6terminee  en  Chine,"  Toung-pao,  XIV,  and  reprint;  L.  Matthiessen,  "Verglei- 
chung  der  indischen  Cuttaca-  und  der  chinesischen  Tayen-Regel,  unbestimmte 
Gleichungen  und  Congruenzen  ersten  Grades  aufzulosen,"  Sitzungsberichte  der 
math.-naturwiss.  Section  in  der  Verhandl.  der  Philol.  Vers.  zu  Rostock,  1875. 

2  See  his  work,  pp.  133-135. 


586  APPLICATIONS  OF  ALGEBRA 

which  would  be  indeterminate  if  it  were  not  for  the  fact  that 
there  were  some  of  each  and  that  the  result  must  be  in  positive 
integers,  thus  admitting  of  only  one  set  of  answers,  namely, 
i,  5,  and  14.  In  general,  all  problems  of  this  type  are  of  the 


ax  +  by  +  cz  =  ;/, 
reducing  to  px  +  qy  —  r, 

thus  being  indeterminate  unless  the  physical  conditions  are 
such  as  to  exclude  all  but  one  set  of  values. 

The  Regula  Coecis.  Such  problems  are  known  in  European 
works  as  early  as  the  gth  century,  and  thereafter  they  become 
common.1  In  the  isth  century  they  begin  to  mention  the  per- 
sons as  being  at  a  cecha?  and  hence  the  rule  for  solving  such 
problems  became  known  to  i6th  century  writers  as  the  Regula 
Coecis.  There  has  been  much  dispute  as  to  the  origin  of  the 
term  coecis?  but  from  such  historic  evidence  as  we  now  have  it 
seems  to  relate  to  the  fact  above  stated,  namely,  that  the  per- 
sons were  at  a  cecha.4  The  problem  seems  to  relate  to  drinking 
where  each  paid  his  own  share,5  very  likely  from  the  fact  that 
zeche  meant  originally  the  money0  paid  for  the  drinks. 


a  i4th  century  MS.  in  Munich  (cod.  lat.  Monac.  14684):  "Sint  hie 
milites,  pedites  et  puellc,  et  sint  in  universe  12,  ct  habeant  12  panes,  parciendos, 
et  quilibet  miles  accipiat  duos,  quilibet  pedes  quartam  partcm  panis,  quilibet 
puella  medietatem  panis :  queritur,  quot  erunt  milites,  pedites  et  puelle."  Bibl. 
Math.,  IX  (2),  79- 

2Curtze,  for  example,  found,  in  a  MS.  of  1460,  a  problem  beginning:  "ponam 
casus,  quod  sint  20  persone  in  una  cecha"  (Abhandlungen,  VII,  35). 

3  Which  also  appears  in  such  forms  as  zekis,  zeches,  cekis,  ceci,  coed,  caeci,  and 
caecis.    Bibl.  Math.,  XIII  (2),  54,  and  VI  (3),  112. 

4  We  find  similar  expressions  in  the  i6th  century.   Thus  Rudolff  (Kunstliche 
rechnung,  1526)  has  a  topic  "Von  mancherley  person  an  einer  zech,"  and  says 
further:  "Es  sitzen  20  person  an  einer  zech/man/frawen/vnd  jungfrawen  .  .  . ," 
adding  that  this  is  the  "  regel/welche  sie  nennen  Cecis  oder  Virginu/"  (1534  ed., 
fol.  Nvij). 

6L.  Diefenbach,  in  his  Novum  Glossarium  (Frankfort  a.  M.,  1867,  p.  339), 
says:  "symbolum  .  .  .  vulgo  zecha  quo  quisque  suam  portionem  confert."  In 
American  slang,  a  zecha  was  a  "Dutch  treat." 

6Italian  zecca;  compare  zecchiere,  a  mint  master,  as  in  Ciacchi,  Regole  gene- 
rali  dy  Abbaco  (Florence,  1675,  p.  247). 


REGULA  COECIS  587 

There  have  been  various  other  speculations  as  to  the  word. 
One  of  these  is  that  it  comes  from  an  Arabic  expression  signi- 
fying not  content  with  one  but  demanding  many,  referring  to 
the  many  possible  solutions.1  It  has  also  been  thought  to  come 
from  the  Latin  caccus  (blind),  with  such  fanciful  explanations 
as  that  a  problem  of  this  kind  was  solved  by  the  blind  Homer, 
or  that  the  solver  went  blindly  to  work  to  find  the  solution. 
Indeed,  it  was  often  called  the  Blind  Rule  in  the  i6th  and 
1 7th  centuries.2 

Another  name  for  the  rule  was  Regula  potatorum  (rule  of 
the  drinkers),  and  this  gives  added  reason  for  the  interpreta- 
tion of  zecha  as  a  drinking  bout.3  A  still  more  common  name 
was  Regula  virginum,  rule  of  the  girls,  usually  explained  by 
the  fact  that  the  solutions  show  more  girls  than  men  or  women.4 

In  spite  of  the  fact  that  this  style  of  problem  is  interesting, 
the  arithmeticians  often  discouraged  its  use  because  of  its  in- 
determinateness,5  although  there  were  found  others  who  in- 
creased the  difficulty  by  using  more  than  three  unknowns. 

Alligation.  Beginning  apparently  in  the  Renaissance  period 
as  an  application  of  indeterminate  equations,  the  Regula  Alii- 

1 "  Cintu  Sekis,  hoc  est  adulteram  indigetarunt :  propterea,  ut  opinor,  quod 
uno  ac  legitimo  quaestionis  enodatu  non  contenta,  plures  plerumque  admittat 
solutioncs."  J.  W.  Lauremberg,  Arithmetica  (Soro,  1643),  quoted  by  Zeuthen  in 
L' Intermediate  des  Mathematitiens,  p.  152  (Paris,  1896)  (hereafter  referred 
to  as  U  Intermediate}  ;  Bibl.  Math.,  X  (2),  96.  Carra  de  Vaux  (BibL  Math.,  XI 
(2),  32)  says  that  Lauremberg's  expression  should  have  the  word  sikkir,  which 
means  toper,  and  this  is  more  reasonable. 

2  Thus  Thierfelder  asks :  "  Warumb  wirdt  disc  Regel   Cecis  genannt  ? "  and 
explains  that  the  problem  is  indeterminate  and  that  "ein  Ungeiibter  nicht  bald 
finden  kan/darumb  ist  es  jm  ein  blinde  Regel"  (1587,  p.  211).    Cardinael  speaks 
of  it  as  "Den  Blinden-Reghel"  (1674  ed.,  p.  88). 

3  The  Dutch  writer,  Bartjens,  also  speaks  of  it  as  "  Bachus-rekeninge "  (1752 
ed.,  p.  213).   See  also  Unger,  Die  Methodik,  p.  101. 

4This  origin  is  stated  by  Jacobus  Micyllus  (1555):  "regula  quam  ab  eo,  ut 
videtur,  appellarunt,  quod  virginum  personae  ac  nomen  inter  exempla  illius  sub- 
inde  repetuntur."  So  we  find  "Coeds  oft  Virginum"  (Van  der  Schuere,  1600, 
fol.  174),  "Rekeninge  Coecis,  ofte  Virginum"  (Bartjens,  1633),  and  similar  forms, 
expecially  among  the  early  Dutch  writers. 

5  Thus  Stockmans :  "  Desen  regel  niet  soo  seker  en  is  in  zijn  werckinge  als  de 
ander  voorgaede"  (1676  ed.,  p.  380);  and  Coutereels  remarks  that  it  is  "meer 
vermakelijkheyd  als  sekerheyd"  (1690  edition  of  the  Cyfier-Boeck,  p.  559). 


588  APPLICATIONS  OF  ALGEBRA 

gationis,1  or  Rule  of  Alligation,  attracted  considerable  attention 
for  nearly  three  hundred  years.  Problems  in  alligation  were 
sometimes  indeterminate  and  sometimes  not.  The  following, 
for  example,  is  indeterminate : 

How  many  Raisins  of  the  Sun,  at  7d.  per  Ib.  and  Malaga  Raisins 
at  4d.  per  Ib.  may  be  mixed  together  for  6d.  per  Ib.?2 

Such  a  problem  becomes  determinate  by  the  addition  of  some 
further  appropriate  condition,  as  in  the  following  case: 

A  tobacconist  mixed  36  Ib.  of  tobacco,  at  is.  6d.  per  Ib.  12  Ib. 
at  2S.  a  pound,  with  12  Ib.  at  is.  tod.  per  Ib. ;  what  is  the  price  of  a 
pound  of  this  mixture  ? y 

In  general,  such  problems  were  simply  ingenious  efforts  to 
make  algebra  seem  real,  but  they  were  usually  solved  without 
the  aid  of  algebraic  symbols,  and  hence  they  found  place  in 
higher  arithmetics  until  the  close  of  the  iQth  century,  when 
they  generally  disappeared  except  as  a  few  remained  in  the 
form  of  mixture  problems4  in  the  elementary  algebras.  Indeed, 
Recorde  (c.  1542)  remarked  that  the  rule  of  alligation  "might 
be  well  called  the  rule  of  Myxture."5  He  was  the  first  English 
writer  to  suggest  other  applications  than  those  referring  to 
alloys,  saying:  "it  hath  great  vse  in  composition  of  medicines, 

1From  ad,  to,  +  ligare,  to  bind.  From  the  same  roots  come  the  French  al- 
liage,  and  our  words  alloy  and  ally.  So  we  have  in  French  "La  regie  des  aliages" 
(Trenchant,  1566;  1578  ed.,  p.  191)  and  "La  Reigle  d'Alligation "  (Peletier,  1549; 
1607  ed.,  p.  247). 

2T.  Dilworth,  The  Schoolmasters  Assistant,  new  ed.,  p.  97  (London,  1793). 
This  was  one  of  the  most  celebrated  English  arithmetics  of  the  period;  it  had 
great  influence  on  American  textbooks. 

3N.  Daboll,  Schoolmaster's  Assistant,  1837  ed.,  p.  177.  This  was  one  of  the 
most  celebrated  of  the  early  American  arithmetics. 

4 The  Dutch  writers  called  them  problems  solved  by  "Den  Reghel  van 
Menginghe"  (Cardinael,  1674  ed.,  p.  66,  with  23  pages  to  the  subject).  So  we 
find  "Allegatio,  Menginghe"  (Van  der  Schuere,  1600),  "Rekeninghen  van  Men- 
gelingen"  (Raets,  1580  ed.,  fol.  KS),  "Alligationis,  ofte  Menginghe"  (Bartjens, 
1676  ed.,  p.  165),  and  "Alligatio,  Alliage,  of  Mengingh"  (Coutereels,  Cyffer- 
Boeck,  1690  ed.,  p.  484) .  The  Italian  writers  sometimes  had  a  chapter  De'  mescoli 
(on  mixtures),  and  this  may  have  suggested  the  German  Mischungsrechnung. 
For  a  recent  use  of  the  topic  in  Germany,  see  R.  Just,  Kaufmannisches  Rech- 
nen,  I,  p.  86  (Leipzig,  1901).  5  Ground  of  Artes,  1558  ed.,  fol.  ¥3. 


MINT  PROBLEMS  589 

and  also  in  myxtures  of  metalles,  and  some  vse  it  hath  in  myx- 
tures  of  wines,  but  I  wshe  it  were  lesse  vsed  therin  than  it  is  now 
a  daies."  These  practices  rendered  the  subject  so  popular  that 
Baker  (1568)  gave  forty-eight  pages  to  it  in  his  1580  edition. 

Mint  Problems.  One  of  the  leading  applications  of  alligation 
was  found  in  the  general  need  for  the  mixing  of  chemicals  and 
metals  by  the  alchemists  of  the  Renaissance/  by  bell  founders, 
and  by  mint  masters.  As  to  the  coining  of  money,  it  should  be 
remembered  that  this  was  not  in  general  a  government  monop- 
oly in  the  Middle  Ages,  so  that  it  was  looked  upon  as  some- 
thing unheard  of  that  Ferdinand  and  Isabella  should  assert  this 
right  in  I4Q6.2  The  privilege  belonged  rather  to  cities  or  dis- 
tricts and  even  in  a  single  small  country  was  often  claimed  by 
several  people,3  often  as  an  inherited  right.4  Add  to  this  fact 
the  great  awakening  in  the  mining  industry  of  Germany  in  the 
latter  part  of  the  isth  century  and  the  extensive  importations 
of  gold  and  silver  from  the  Americas  in  the  first  half  of  the 
1 6th  century,  and  it  will  be  seen  that  the  subject  of  alligation 
naturally  had  at  that  time  a  new  and  popular  field  in  the  mixing 
of  alloys  for  purposes  of  coinage.5  This  explains  the  interest 
in  the  subject  of  coinage  in  the  i6th  century.6 

ltfH  consolare  oro,  ed  argento  non  e  altro,  che  vn'  allegazione  di  que'  due 
metalli,  per  li  quali  la  maggior  parte  degli  Alchimisti  son  diuenuti  miseri,  e  men- 
dichi,  per  volere  inuestigare  la  congelazione  del  Mercuric  in  vera,  ed  ottima  Luna, 
o  Sole,  la  quale  senza  il  diuino  aiuto  in  vano  da  gli  Alchimisti  vien  tentata." 
Ciacchi,  Regole  generali  d'  Abbaco,  p.  244  (Florence,  1675). 

2H.  B.  Clarke,  in  the  Cambridge  Mod.  Hist.,  I,  chap.  xi. 

8W.  Cunningham,  in  the  Cambridge  Mod.  Hist.,  I,  chap.  xv. 

4E.  P.  Cheyney,  Documents  Illustrative  of  Feudalism,  p.  34  (Philadelphia,  1898) . 

5 Of  course  alloys  for  this  purpose  had  been  known  and  used  to  some  extent 
ever  since  the  early  coinage  from  natural  electron  in  Asia  Minor  and  from  bronze  in 
Rome.  On  the  latter,  see  F.  Gnecchi,  Monete  Romane,  26.  ed.,  p.  86  (Milan,  1900). 

6  So  Rudolff  (1526;  1534  ed.,  fol.  M  6)  and  Grammateus  (1518;  1535  ed., 
fols.  C  8,  D  4,  etc.)  have  chapters  on  Muntzschlag,  and  various  writers  speak  of 
the  problems  of  the  muntzmeister  (Rudolff),  Afunt-meester  (Van  der  Schuere, 
1600),  and  mint-master  (Hodder,  1672  ed.).  The  Germans  also  had  chapters  on 
Silber  Rechnung,  Goldt  Rechnung,  and  Kupfer  Rechnung  (Riese,  1522;  1550  ed., 
fols.  37-40)  ;  the  Dutch,  on  Rekeninghe  van  Goudt  end  Silber,  with  Comptes 
d'or  &  d} argent  (Coutereels's  Dutch  and  French  editions  of  1631,  p.  298,  and  other 
dates)  ;  and  the  Italians,  on  Del  consolare  dell'  oro  et  dell'  argento  (Cataneo, 
1546;  Tartaglia,  1556). 


590  APPLICATIONS  OF  ALGEBRA 

Problem  of  Hiero's  Crown.  Closely  related  to  this  subject  is 
the  problem  of  Hiero's  crown,  which  Robert  Recorde  (c.  1542) 
states  in  quaint  language  as  follows: 

Hiero  kynge  of  the  Syracusans  in  Sicilia  hadde  caused  to  bee  made 
a  croune  of  golde  of  a  wonderfull  weight,  to  be  offered  for  his  good 
successe  in  warres:  in  makynge  wherof,  the  goldsmyth  fraudulently 
toke  out  a  certayne  portion  of  gold,  and  put  in  syluer  for  it.  [Recorde 
then  relates  the  usual  story  of  Archimedes  and  the  bath,  telling  how 
the  idea  of  specific  gravity  came]  as  he  chaunced  to  entre  into  a 
bayne  full  of  water  to  washe  hym,  [and]  reioycing  excedingly  more 
then  if  he  had  gotten  the  crown  it  self,  forgat  that  he  was  naked,  and 
so  ranne  home,  crying  as  he  ranne  evprjKa,  tvprjKa,1  I  haue  foud,  I 
have  found. 

Cardan2  asserts  that  the  story  is  due  to  Vitruvius,  who  sim- 
ply transmitted  the  legend.  It  appears  in  various  books  of  the 
1 6th  century  and  is  still  found  in  collections  of  algebraic 
problems. 

First  Problems  in  the  New  World.  To  those  especially  who 
live  in  the  New  World  some  local  interest  attaches  to  the  prob- 
lems thaiv appear  in  the  first  mathematical  work  (1556)  to  be 
printed  there.  Among  these  problems  are  the  following:3 

I  bought  10  varas  of  velvet  at  20  pesos  less  than  cost,  for  34  pesos 
plus  a  vara  of  velvet.  How  much  did  it  cost  a  vara?  Add  20  pesos 
to  34  pesos,  making  54  pesos,  which  will  be  your  dividend.  Subtract 
one  from  10  varas,  leaving  9.  Divide  this  into  54,  giving  6,  the  price 
per  vara. 

I  bought  12  varas  of  velvet  at  30  pesos  less  than  cost,  for  98  pesos 
minus  4  varas.  How  much  was  the  cost  per  vara  ?  The  following  is 
a  short  method:  add  the  30  pesos  and  the  98  pesos,  making  128; 
add  the  number  of  varas,  12  and  4,  making  16;  divide  16  into  128, 
giving  8,  the  price  per  vara. 

lEu'reka,  eu'reka,  more  precisely,  heu'reka. 

tPractica,  1539,  cap.  66,  ex.  45.  He  relates  that  Archimedes  "nudus  e  balneo 
exultas  domu  reuertebatur,"  adding,  "nescio  an  ob  amore  veritatis  potius  lau- 
dandus  qua  ob  importuna  &  impudica  nuditate  vituperadus." 

3  See  also  pages  385,  392,  and  Volume  I,  pages  353-356. 


SPANISH-AMERICAN  PROBLEMS  591 

I  bought  9  varas  of  velvet  for  as  much  more  than  40  pesos  as  13 
varas  at  the  same  price  is  less  than  70  pesos.  How  much  did  a  vara 
cost?  Add  the  number  of  pesos,  40  and  70,  making  no.  Add  the 
number  of  varas,  9  and  13,  making  22.  Dividing  no  by  22,  the 
quotient  is  5,  the  price  of  each  vara. 

A  man  traveling  on  a  road  asks  another  how  many  leagues  it  is  to 
a  certain  place.  The  other  replies:  " There  are  so  many  leagues  that, 
squaring  the  number  and  dividing  the  product  by  5,  the  quotient  will 
be  80."  Required  to  know  the  number  of  leagues. 

A  man  is  selling  goats.  The  number  is  unknown  except  that  it  is 
given  that  a  merchant  asked  how  many  there  were  and  the  seller 
replied:  " There  are  so  many  that,  the  number  being  squared  and 
the  product  quadrupled,  the  result  will  be  90,000."  Required  to  know 
how  many  goats  he  had. 

The  work  contains  other  and  more  difficult  problems  in  al- 
gebra and  the  theory  of  numbers,  but  the  above  are  types  of  the 
ordinary  puzzles  which  the  author  places  before  his  readers. 

5.  MAGIC  SQUARES 

Oriental  Origin.  The  magic  square  seems  unquestionably  to 
be  of  Chinese  origin.  The  first  definite  trace  that  we  have  of 
it  is  in  the  I-king,  where  it  appears  as  one  of 
the  two  mystic  arrangements  of  numbers  of 
remote  times.  This  particular  one,  the  lo-shu, 
is  commonly  said  to  have  come  down  to  us 
from  the  time  of  the  great  emperor  Yu,  c.  2200 
B.C.  The  tradition  is  that  when  this  ruler  was 
standing  by  the  Yellow  River  a  divine  tortoise 
appeared,  and  on  its  back  were  two  mystic 
symbols,  one  being  the  lo-shu  already  described  in  Volume  I, 
Chapter  II.  As  may  be  seen  from  the  illustration  there  given, 
it  is  merely  the  magic  square  here  shown. 

This  particular  square  is  found  in  many  recent  Chinese 
works,  and  every  fortune  teller  of  the  East  makes  use  of  it  in 
his  trade.  Little  by  little  the  general  knowledge  of  magic 
squares  seems  to  have  been  extended,  and  when  Ch'eng  Tai-wei 


dJirtjlLi1  SiJM  &  ^iB-^t^F  'J 
Sill  I  ^liir^l'ilCljImillii 

=?--  I  ~T"7T"    TT  !  "^  1 7T  I  "FT  i  -^    "^T^PA-^  ^'-rrSn  -=•  i  •£•  I  -^K 


MAGIC  SQUARES  IN  JAPAN 
Half  of  a  magic  square  as  given  in  Hoshino  Sanenobu's  Ko-ko-gen  Sho,  1673 

•      ±      M      IB       IL 


MAGIC   CIRCLE   FROM   SEKI  S  WORKS 

From  the  reprint  of  the  works  of  the  great  Japanese  mathematician,  Seki  Kowa 

(c.i66i) 


JAPANESE  SQUARES  AND  CIRCLES 


593 


wrote  his  Systematized  Treatise  on  Arithmetic1  in  1593,  he 
included  not  only  a  discussion  of  magic  squares  but  also  one  of 
magic  circles. 

Japan.    The  Japanese  became  particularly  interested  in  the 
subject  in  the  1 7th  century,  as  the  illustrations  from  the  works 


MAGIC  CIRCLE  OF  129  NUMBERS 

From  Muramatsu  Kudayu  Mosei's  Mantoku  Jinko-ki,  1665.  The  numbers  in  each 
radius  add  to  524,  or  525  with  the  center  i 

of  some  of  their  leading  writers  show.2  Among  the  prominent 
scholars  who  gave  attention  to  these  forms  were  Muramatsu 
Kudayu  Mosei,  who  wrote  several  works  on  arithmetic  and 

^Suan-fa  Tong-tsung.    See  Volume  I,  page  352. 

2 For  their  treatment  of  the  subject  see  Smith-Mikami,  pp.  57,  69,  71,  73,  79, 
116,  120,  177. 


594 


MAGIC  SQUARES 


7 

12 

i 

14 

2 

J3 

8 

1  1 

16 

3 

10 

5 

9 

6 

1S 

4 

geometry,  beginning  in  1663 ;  Hoshino  Sanenobu,  whose  Trian- 
gular Extract^  appeared  in  1673;  Isomura  Kittoku,  whose 
Ketsugi-sho  appeared  in  1660;  and  the  great  Seki  Kowa,  who 

devoted  one  of  his  Seven  Books2  to 
the  theory  of  magic  squares  and  magic 
circles. 

India.  From  China  the  magic  square 
seems  to  have  found  its  way  into  India 
and  the  adjacent  southern  countries, 
but  whether  this  was  direct  or  through 
the  Arab  influence  we  can  only  conjec- 
ture. It  appears  in  a  Jaina  inscription 
in  the  ancient  town  of  Khajuraho,  India,  where  various  ruins 
bear  records  of  the  Chandel  dynasty  (870-1200),  and  is  prob- 
ably not  older  than  the  nth  or  1 2th  century.  This  Indian  square, 
shown  above,  displays  a  somewhat  ad- 
vanced knowledge  of  the  subject,  for 
not  only  has  it  an  even  number  of 
squares  on  a  side,  but  each  of  the  four 
minor  squares  has  a  relation  to  the 
others,  as  may  be  seen  by  the  illustra- 
tion at  the  right. 

This  is  perhaps  the  earliest  trace 
of  such  fantastic  elaborations  of  the 
magic  square,3  although  no  careful  study  of  the  history  has  yet 
been  made.4  It  is  probable  that  the  astrologers  carried  such 
ideas  to  the  West,  where  their  influence  upon  the  medieval 
mathematics  of  Europe  is  apparent. 

Today  the  magic  square  is  used  as  a  charm  all  through  India, 
being  found  in  fortune  bowls,  in  medicine  cups,  and  in  amulets. 
In  Thibet  it  is  particularly  in  evidence,  being  found  in  the 

1  Ko-ko-gen  Sho. 

2  Or  Shichibusho.   The  particular  book  is  the  Hojin  Vensan,  revised  in  MS.  in 
1683. 

8F.  Schilling,  Jahresbericht  der  deutschen  Math.  Verein.,  XIII,  383. 

4 But  see  such  works  as  W.  S.  Andrews,  Magic  Squares  (Chicago,  1907),  and 
subsequent  articles  in  The  Open  Court.  For  Roman  and  Egyptian  claims  see 
E.  Falkener,  Games,  Ancient  and  Oriental,  p.  277  (London,  1892), 


i9 

'5 

9    25 

9    25 

i5 

19 

I9 

15 

25     9 

25     9 

15 

19 

THE  ASTROLOGERS  AND  ALCHEMISTS 


595 


"wheel  of  life"1  and  worn  as  an  amulet  to  ward  off  evil.  It  is 
also  seen  in  Sumatra,  in  the  Malay  Peninsula,  and  in  the  other 
countries  which  have  had  close  relations  with  India  and  China. 

The  squares  are 
not  always  of  the 
pure  type,  however; 
that  is,  the  sums  of 
the  rows,  columns, 
and  diagonals  are  not 
always  constant.  In 
some  of  them,  for  ex- 
ample, the  columns 
add  successively  to 
300,  200,  100,  and 
the  like,2  and  in  many 
of  them  the  numbers 
are  repeated  when- 
ever it  was  necessary 
to  make  the  sums 
come  as  desired. 

Connection  with  Al- 
chemy. It  seems  that 
the  numbers  must 
often  have  been  con- 
nected with  the  old 
alchemistic  idea  of 
the  planets  and  the 
metals, — an  idea  that 
permeated  the  doctrines  of  many  of  the  medieval  mystics.  Of 
the  three  triads  made  up  of  the  nine  digits,  the  first  had  the 
following  relations : 

1  =  gold  =  the  sun,  O 

2  =  silver  =  the  moon,  D 

3  =  tin  =  Jupiter,  1L  (the  hand  grasping  the  thunderbolt) 

1See  the  illustration  in  Volume  I,  page  27. 

2S.  S.  Stitt,  "Notes  on  some  Maldivian  Talismans,"  Journ.  Royal  Asiatic  Soc., 
pp.  121,  130  (London,  1906). 


THIBETAN  TALISMAN 

With  signs  of  the  zodiac,  the  ancient  pa-kua  or 
trigrams,  and  the  lo-shu  in  Thibetan  numerals 


5Q6  MAGIC  SQUARES 

The  second  triad  was  as  follows: 

4  =  gold  again  —  the  sun,  O 

5  =  mercury  =  Mercury,  £ 

6  =  copper  =  Venus,  9 

The  third  triad  was  as  follows : 

7  ==  silver  again  =  the  moon,  D 

8  =  lead  —  Saturn,  "fy 

9  =  iron  =  Mars,  $ 

With  such  a  relationship  it  is  possible  to  understand  such 
talismans  as  one  found  in  the  Maldivian  Islands,  in  the  Indian 
Ocean,  where  the  charm  to  protect  a  virgin  sums  to  18,  whose 
digits  sum  to  9,  the  number  of  Mars,  the  protector.  In  these 
numbers  there  also  enters  the  idea  of  congruence,  particularly 
to  the  modulus  9.  Thus  a  talisman  to  keep  out  Satan  has  its 
rows  80  =  8,  1600  =  7,  180  =  9,  •  •  •,  which  have  for  their 
sum  69  =  15  =  6  =  Venus, — a  vagary  that  can  be  explained 
only  by  conjecture.1 

Hebrews.  The  magic  square  played  an  important  part  in  the 
cabalistic  writings  of  the  Hebrews.  Rabbi  ben  Ezra  (c.  1140) 
mentions  it,  although  it  had  been  used  by  Hebrew  writers  long 
before  his  time.2  The  Eastern  Jews  early 
found  in  the  ancient  Chinese  lo-shu,  with 
its  constant  sum  15,  a  religious  symbol,  in- 
asmuch as  15  in  Hebrew  is  naturally  m 
(10  -I-  5),  which  is  made  up  of  the  first  two 
letters  of  Jahveh  (Jehovah),  that  is,  mrr>, 
although,  lest  they  should  be  guilty  of  pro- 
fanity, they  always  wrote  9  +  6  for  15.  If 
the  corner  numbers  of  the  common  form  of  square  are  sup- 
pressed, the  even  (feminine)  elements  are  eliminated  and  there 
remain  only  the  odd  numbers,  this  cruciform  arrangement 
serving  as  a  charm  among  various  Oriental  peoples.3 

1Stitt,  loc.  cit.y  p.  144.  2M.  Steinschneider,  Abhandlungen,  III,  98. 

8D.  Martines,  Origine  e  progressi  dell'  Aritmetica,  p.  39  (Messina,  1865),  with 
considerable  information  upon  this  subject. 


AMULETS  597 

Arabs.  Whether  the  magic  square  reached  the  Arabs  from 
India,  from  China,  or  from  Persia,  we  do  not  know.  It  might 
readily  have  come  from  any  one  of  these  countries  through 
either  of  the  others,  but  at  any  rate  it  was  well  known  to 
various  late  Arab  writers,  appearing,  for  example,  in  the  works 
of  the  philosopher  Gazzali1  about  noo. 

Christians.  In  general  the  magic  square  found  no  recognition 
as  a  Christian  symbol,  although  the  occult  writers  naturally 
made  use  of  it.  When  it  appears  in  medieval  mathematical 
works  it  is  usually  in  the  form  of  a  problem  whose  solution 
requires  the  arrangement  of  the  ordinary  g-celled  square 
shown  below.2 

The  textbook  writers  of  the  i6th  century  paid  considerable 
attention  to  the  subject.  Cardan,  for  example,  gives  seven 
different  squares  bearing  respectively  the  names  of  the  sun,  the 
moon,  and  the  five  planets  then  known,  and  gives  some  direc- 
tions for  the  formation  of  such  squares.3 

In  art  the  first  instance  of  the  use  of  these  forms  is  probably 
the  one  in  the  well-known  Melancholia  by  Diirer. 

One  of  the  most  elaborate  examples  found  in  architectural 
decoration  is  cut  in  the  wall  in  the  Villa  Albani  at  Rome.4  It 
contains  eighty-one  cells  and  is  dated  1766. 

1  Mohammed  ibn  Mohammed  ibn  Mohammed,  Abu  Hamid,  al-Gazzali,  born 
at  TUS,  1058/59;  died  at  Tus,  mi.  The  Arabic  name  of  the  magic  square  is 
shakal  turdbi.  See  E.  Rehatsek,  "Explanations  and  Facsimiles  of  eight  Arabic 

Talismanic  Medicine-cups,"  Journ.  of  the  Bombay  Branch      

of  the  Royal  Asiatic  Soc.,  X,  150. 

2For  example,  Gunther  reports  a  i$th  century  MS. 
with  this  problem:  "Tres  erant  fratres  in  Colonia,  ha- 
bentes  9  vasa  vini.  Primum  vas  continet  i  amam,  secun- 
dum  2,  tertium  3,  •  •  •  nonum  9.  Divide  vinum  illud 


5 


aequaliter,  inter  illos  tres,  vassis  inconfractis." 

The  division  is  i  +  5  +  9,  3  +  4  +  8,  2  +  6  +  7,  but 
that  this  is  indeterminate  is  seen  by  the  magic  square 
here  represented.  3Practica,  1539,  capp.  42,  66. 

4  The  inscription  is  "Caetanus  Gilardonus  Romanus  philotechnos  inventor. 
A.D.  MDCCLXVL"  On  the  recent  mathematical  investigations  the  reader  will 
do  well  to  consult  such  works  on  mathematical  recreations  as  those  of  Lucas, 
Ball,  Schubert,  and  Ahrens;  the  work  by  W.  S.  Andrews,  mentioned  on  page  $94; 
and  the  references  in  G.  A.  Miller,  Historical  Introduction  to  Mathematical  Litera- 
ture, p.  20  (New  York,  1916). 

II 


DURER'S  "  MELANCHOLIA/'  WITH  MAGIC  SQUARE 

One  of  the  first  magic  squares  to  appear  in  print.    Chiefly  interesting  because  it  is 
an  even-celled  square.    Diirer  (1471-1528)  also  wrote  on  higher  plane  curves  in 

connection  with  art 


DISCUSSION  599 

TOPICS  FOR  DISCUSSION 

1.  The  general  progress  of  mathematical  recreations,  with  a  con- 
sideration of  the  leading  works  upon  the  subject. 

2.  The  influence  of  mathematical  recreations  upon  the  develop- 
ment of  mathematics. 

3 .  Traces  of  the  ancient  puzzle  problems  in  the  elementary  mathe- 
matical literature  of  the  present  time,  with  a  consideration  of  the 
value  of  these  problems. 

4.  Types  of  puzzle  problems  dependent  upon  indeterminate  equa- 
tions, with  a  study  of  their  history. 

5.  Mathematics  as  an  aid  to  the  study  of  the  history  of  economics 
and  commerce. 

6.  The   historical   development   of   the   corporation,   as   traced 
through  the  problems  of  arithmetic. 

7.  Racial  influences  upon  business  customs,  as  seen  in  commercial 
problems. 

8.  The  standard  applications  of  arithmetic  from  remote  times 
to  the  present,  with  a  consideration  of  their  important  changes. 

9.  Commercial  problems  at  one  time  important  and  now  nearly 
obsolete,  with  a  consideration  of  the  causes  of  their  rise  and  fall. 

10.  Certain  commercial  problems  and  customs,  at  one  time  im- 
portant but  now  nearly  or  quite  obsolete,  left  their  impress  upon  our 
language.    Consider  a  few  such  cases. 

11.  Ancient  problems  of  the  bank  compared  with  those  of  the 
present  time. 

12.  The  need  for  barter  in  ancient  times  and  the  reasons  for  its 
retention,  even  in  highly  civilized  countries,  until  recently. 

13.  The  history  of  various  types  of  taxation  in  ancient,  medieval, 
and  modern  times. 

14.  The  reason  why  gaging  was  considered  as  an  important  branch 
of  mathematics  in  the  early  printed  arithmetics  and  geometries,  and 
why  it  lost  its  standing  in  the  igth  century. 

15.  Reasons  for  the  study  of  magic  squares  in  ancient  times,  in 
the  Middle  Ages,  and  at  the  present  time,  with  typical  illustrations. 

1 6.  Instances  of  the  use  of  the  magic  square  in  the  East  as  a  talis- 
man or  amulet. 

17.  The  relation  of  magic  squares  to  alchemy,  and  the  relation  of 
alchemy  to  modern  science. 


CHAPTER  VIII 

TRIGONOMETRY 
i..  GENERAL  DEVELOPMENT  OF  TRIGONOMETRY 

Meaning  of  the  Term.  If  we  take  trigonometry  to  mean  the 
analytic  science  now  studied  under  this  name,  we  might  prop- 
erly place  its  origin  in  the  i  yth  century,  after  the  development 
of  a  satisfactory  algebraic  symbolism.  If  we  take  it  to  mean 
the  geometric  adjunct  to  astronomy  in  which  certain  functions 
of  an  angle  are  used,  we  might  look  for  its  real  origin  in  the 
works  of  Hipparchus  (c.  140  B.C.),  although  there  are  earlier 
traces  of  its  use.  If  we  take  it  to  mean  literally  "triangle 
measurement/71  the  origin  would  naturally  be  placed  much 
earlier,  say  in  the  second  or  third  millennium  B.C.  Since  this 
third  phase  is  considered  under  geometry,  we  may  property 
confine  our  work  to  the  development  of  the  idea  of  the  func- 
tions of  an  angle,  giving  first  a  brief  sketch  of  the  rise  of  the 

science  and  then  the  history  of  certain 

of  its  details. 

Egypt.  In  the  Ahmes  Papyrus  (c. 
1550  B.C.)  there  are  five  problems2  re- 
lating to  the  mensuration  of  pyramids, 
and  four  of  these  make  mention  of  the 
seqt*  of  an  angle.  Ahmes  is  not  at  all 
clear  in  expressing  the  meaning  of  this 
word,  but  from  the  context  it  is  thought  that  the  seqt  of 


ov  (tri'gonon,  triangle)  -f-  ^Tpov  (met'ron,  measure). 
~Nos.  56-60  in  the  list.    See  Pcet,  Rkind  Papyrus,  p.  97;  Eisenlohr,  Ahmes 
Papyrus,  pp.   134-148.    Throughout   this  chapter  much  use  has  been  made  of 
A.  Braunmiihl,  Geschichte  der  Trigonometric,  2  vols.,  Leipzig,  1900,  1903  ;  here- 
after referred  to  as  Braunmiihl,  Geschichte. 

3  Eisenlohr  takes  the  word  to  mean  ratio  number.     It  is  also  transliterated 
skd  and  seqet.    It  may  be  significant  that  the  Hebrew  sgd  means  "bowing." 

600 


EARLY  STEPS  IN  TRIGONOMETRY  60 1 

the  regular  pyramid  shown  on  page  600  is  probably  equivalent 
to  cot  Z  OMV^  The  Egyptian  pyramids  were  generally  con- 
structed so  that  Z-OMV  was  approximately  constant  (about 
52°)  and  Z.OAV  was  about  42°.  At  present  we  are  without 
means  of  knowing  what  use  was  made  of  this  function. 

Babylon.  The  relation  between  the  mathematical  knowledge 
of  the  Egyptians  and  that  of  the  Babylonians  in  the  third 
millennium  B.C.,  as  seen  in  the  unit  fraction  (p.  210),  leads  us 
to  suppose  that  the  latter  people  may  have  known  of  the  primi- 
tive Egyptian  trigonometry.  We  have,  however,  no  direct 
knowledge  that  this  was  the  case.  There  are  evidences  of 
angle  measure  at  a  very  early  date,  as  witness  fragments  of 
circles  which  seem  to  have  been  used  for  this  purpose  and 
which  have  come  down  to  us.  These  fragments  seem  to  have 
been  parts  of  primitive  astrolabes,  as  stated  on  page  348. 

There  is  also  extant  an  astrological  calendar  of  King  Sargon, 
of  the  28th  century  B.C.,  and  a  table  of  lunar  eclipses  beginning 
747  B.C.,  so  that  evidence  of  an  interest  in  astronomy  is  not 
lacking  throughout  a  long  period  of  Babylonian  history.  All 
this  involved  a  certain  amount  of  angle  measure,  but  there  is 
no  direct  evidence  of  any  progress  in  what  we  commonly  un- 
derstand as  trigonometry.  ^ 

The  Gnomon.  Herodotus  (c.  450  B.C.)  tells  us  that  the 
Greeks  obtained  their  sundial  from  Babylon.  This  is  very 
likely  true,  for  we  know  that  the  Egyptians  used  a  sun  clock 
as  early  as  1500  B.C.,  and  the  Baby- 
lonians  could  hardly  have  been  behind 
them  in  the  knowledge  of  such  a  device. 
The  relation  of  the  sundial  to  trigonom- 
etry is  seen  in  the  fact  that  it  is  an 
instrument  for  a  form  of  astronomical 
observation.  A  staff  GN,  called  by  the  Greeks  a  gnomon  (p.  16), 
is  erected  and  the  shadow  AN  observed.  It  is  longest  at  noon 
when  Sy  the  sun,  is  farthest  south,  this  being  at  the  winter 

1  For  discussion  see  Braunmiihl,  Geschichte,  I,  2 ;  Eisenlohr,  Ahmes  Papyrus, 
p.  137.  Peet,  Rhind  Papyrus,  p.  98,  gives 


602     GENERAL  DEVELOPMENT  OF  TRIGONOMETRY 

solstice,  and  shortest  when  it  is  farthest  north,  at  the  summer 
solstice;  and  hence  an  examination  of  its  limits  enables  the 
observer  to  measure  the  length  of  the  year.  The  daily  lateral 
motion  of  the  point  A  allows  for  the  measure  of  diurnal  time, 
quite  as  the  motion  of  noon  along  AN  allows  for  the  measure  of 
annual  time.  The  gnomon  being  constant,  the  length  of  AN  at 
noon  varies  with  Z.A,  and  to  us  this  means  a  recognition  that 
AN,  orAN:GN,  is  a  function  ofZ^4,  namely,  the  cotangent. 
We  have  no  trace,  however,  of  any  name  (except  the  seqt)  for 
such  a  relation  in  the  period  of  which  we  are  speaking. 

,  China.  In  the  Chou-pei  Suan-king  (c.  1105  B-C.)1  the  right- 
angled  triangle  is  frequently  used  in  the  measure  of  distances, 
heights,  and  depths,  and  it  is  quite  probable  that  the  ratios  of 
the  sides  were  recognized.  One  passage  reads,  "The  knowledge 
comes  from  the  shadow,  and  the  shadow  comes  from  the 
gnomon,"  so  that  possibly  a  primitive  plane  trigonometry  was 
known  in  China  in  the  second  millennium  B.C.  Aside  from  this 
there  is  no  evidence  that  the  early  Chinese  had  names  for  any 
functions  of  an  angle.  The  early  astronomical  interests  of  the 
Chinese,  however,  like  those  of  other  ancient  peoples,  necessi- 
tated some  kind  of  angle  measure. 

Greece.  When  Thales  measured  the  height  of  a  pyramid  by 
means  of  its  shadow,  he  used  what  was  already  known,  prob- 
ably in  various  parts  of  the  world,  as  "shadow  reckoning."2 
In  his  "Banquet  of  the  Seven  Wise  Men"  Plutarch  speaks  of 
Nilax,  one  of  the  guests,  as  saying  to  Thales : 

Whereas  he3  honors  you,  he  particularly  admires  you  for  divers 
great  accomplishments  and  particularly  for  the  invention  whereby,  with 
little  effort  and  by  the  aid  of  no  mathematical  instruments,  you  found 
so  accurately  the  height  of  the  pyramids.  For,  having  fixed  your  staff 
erect  at  the  point  of  the  shadow  cast  by  the  pyramid,  two  triangles 

1But  see  Volume  I,  page  30;  also  Mikami,  China,  p.  4. 

2 We  have  this  term,  substantially  at  least,  in  the  Chdu-pe'i  Suan-king;  in 
the  works  of  Brahmagupta  (c.  628),  under  "Measure  by  Shadow,"  p.  317; 
in  Mahavlra  (c.Sso),  under  "Calculations  relating  to  Shadows,"  p.  275;  and  in 
Bhaskara,  under  "Ch'haya-vyavahara"  (determination  of  shadow),  p.  106. 

sThe  king  of  Egypt,  called  by  the  Greeks  Amasis,  c.  570  B.C. 


SHADOW  RECKONING  603 

were  formed  by  the  tangent  rays  of  the  sun,  and  from  this  you  showed 
that  the  ratio  of  one  shadow  to  the  other  was  equal  to  the  ratio  of  the 
[height  of  the]  pyramid  to  the  staff. 

Essentially  the  measure  of  heights  by  means  of  shadows  in- 
volves the  knowledge  that,  in  this  figure,  EC  \  AB—B'C  :  AB1. 
To  us  it  seems  as  if  tan  A  would  be  suggested  by  such  a 
relation,  but  we  have  no  evidence  that  this  was  the  case  in 
the  time  of  Thales.  We  only  know 
that,  centuries  later,  AB  was  called 
the  umbra  recta  (right  shadow),  show- 
ing that  the  relation  of  AB  to  BC 
entered  trigonometry  through  shadow 
reckoning. 

It  is  said  that  Anaximander  (c.  575  B.C.)  erected  near  Sparta 
the  first  gnomon  in  Greece.  It  was  probably  in  the  form  of 
an  obelisk,  a  mere  post  placed  perpendicular  to  the  apparent 
plane  of  the  earth's  surface,  and  not  the  triangular  form  later 
in  use.  It  could  have  been  used  for  determining  the  meridian 
line,  and  tradition  says  that  this  was  done ;  but  besides  this  it 
served,  as  it  probably  did  in  Egypt  and  Babylon,  to  measure  the 
year,  the  seasons,  and  the  time  of  day.  v 

Relation  to  Astronomy.  In  this  early  work  of  Anaximander, 
as  in  similar  cases  among  the  Babylonians  and  Egyptians,  it  is 
evident  that  the  real  purpose  in  view  was  the  study  of  astron- 
omy, the  unraveling  of  the  mysteries  of  the  universe.  This  led 
to  the  study  of  the  celestial  sphere,  the  triangles  being,  there- 
fore, spherical  figures.  This  accounts  for  the  fact  that  the 
study  of  spherical  triangles  kept  pace  with  that  of  plane  tri- 
angles in  the  Greek  trigonometry.  We  find,  however,  no  tan- 
gible evidence  of  the  definition  or  even  of  the  idea  of  a  spherical 
triangle  before  the  appearance  of  the  work  of  Menelaus  on 
spherics  (c.  loo).1 

Early  Works  on  Spherics.  The  oldest  extant  works  on  spher- 
ics, and  indeed  the  oldest  Greek  mathematical  texts  that  have 
come  down  to  us,  are  two  astronomical  treatises  by  Autol'ycus2 

1  Heath,  History,  II,  262.  2 Atfr6XuKos.    See  Heath,  History,  I,  348. 


604    GENERAL  DEVELOPMENT  OF  TRIGONOMETRY 

of  Pitane  (c.  330  B.C.).  The  first  is  on  a  moving  sphere1  and 
consists  of  twelve  elementary  propositions  relating  to  the  prin- 
cipal circles.  The  second  work  was  on  the  risings  and  settings 
of  the  fixed  stars,2  in  two  books.  Neither  of  these  works  shows, 
however,  any  knowledge  of  spherical  trigonometry. 

Aristarchus.  The  next  important  step  in  the  development 
uf  trigonometry  was  taken  by  the  astronomer  Aristarchus 
of  Samos  (c.  260  B.C.).3  He  attempted  to  find  the  distances 
from  the  earth  to  the  sun  and  the  moon,  and  also  the  diame- 
ters of  these  bodies.  His  geometric  reasoning  was  accurate, 
but  his  instruments  were  so  crude  that  he  could  come  no 
nearer  the  ratio  of  the  distance  of  the  moon  to  that  of  the  sun 
than  to  say  that  it  was  between  ^  and  ^V  In  his  proof  he 
makes  use  of  ratios  which  are  suggestive  of  the  tangent  of 
an  angle.4 

Hipparchus.  In  his  commentary  on  the  Almagest*  Theon  of 
Alexandria  (c.  390)  asserts  that  Hipparchus  (c.  140  B.C.),  the 
greatest  of  the  Greek  astronomers,  wrote  twelve  books  on  the 
computation  of  chords0  of  angles,  but  of  these  books  we  have 
no  further  trace.  Hipparchus  himself,  in  the  fragment  of  his 
work  that  has  come  down  to  us,  leads  us  to  believe  that  he  was 
engaged  in  such  computations  and  in  the  graphic  solution  of 
spherical  triangles.7  It  therefore  seems  reasonable  to  assert, 

1  Hepl  K(.vovfjL€V7)s  cr<pa.tpas. 

2IIepi  iiriToKCjv  /cat  dvcrewv.  The  two  works  were  edited  by  Hultsch  in  1885. 

3  P.  Tannery  thinks  that  this  step,  which  is  usually  attributed  to  Aristarchus, 
was  taken  by  Eudoxus  (c. 370  B.C.).    See  his  "Aristarque  de  Samos,"  Mem.  de 
la  Soc.  des  sciences  de  Bordeaux,  V  (2),  241;  Memoires  Scientifiqiies,  I,  371; 
Heath,  History,  II,  i. 

4  This  proof  is  given  in  Braunmiihl,  Geschichte,  I,  8,  and  by  Tannery,  Me- 
moires Scientifiques,  I,  376.    See  also  the  Commandino  edition  of  Aristarchus, 
1572;  R.  Wolf,  Geschichte  der  Astronomie,  p.  172  (Munich,  1877).    The  work  of 
Aristarchus,  Hcpl  /jieyeO&v  Kal  dTrocrrTjfjLdTwv  ijXtov  Kal  (reX^^s,  was  translated  by 
A.  Nokk  and  published,  with  a  commentary,  in  a  Programm,  Freiburg,  1854.   See 
also  Heath,  History,  IT,  4. 

5 Raima's  French  translation,  p.  no  (Paris,  1821). 

6The  Greeks  called  the  chord  etf0e?a  (euthei'a),  the  Latin  chorda  being  from 
the  Greek  x°P^  (chorde1,  intestine),  whence  it  meant  a  string  made  of  dried  in- 
testine used  in  a  lyre,  and  hence  a  straight  chord  of  a  bow  (arc). 

7Braunmtihl,  Geschichte,  I,  10;  Heath,  History,  II,  257. 


GREEK  CONTRIBUTIONS  605 

from  the  evidence  that  we  have,  that  the  science  of  trigonome- 
try begins  with  Hipparchusl"  It  has  been  asserted,  but  the 
proof  is  unsatisfactory,  that  the  formulas  for  sin  (A  ±  £)  and 
cos  (A±£),  and  for  the  radius  of  the  circumscribed  circle 

( Jt  =  —r ) ,  were  essentially  known  to  him.2  In  order  to  solve  a 
\  4-A/ 

triangle  Hipparchus  and  other  early  writers  always  supposed 
it  inscribed  in  a  circle.  The  sides  were  then  considered  as 
chords,  and  these  were  computed  as  functions  of  the  radius. 
In  this  way  the  table  of  chords  was  of  special  value.  Triangles 
on  a  sphere  were  always  decomposed  into  right-angled  triangles, 
and  these  were  solved  separately.  Although  not  mentioning  the 
subject  of  spherical  triangles  in  any  of  his  works  now  extant, 
Hipparchus  solves  a  certain  problem  in  which  he  must  have 
used  the  equivalent  of  the  formula  tan  b  =  cos  A  tan  c,  where 
C  —  go0,  and  both  he  and  Ptolemy  (c.  150)  knew  the  relation 
which  we  express  by  the  equation  sin2  A  +  cos2  A  =  i.3 

The  treatise  of  Theodosius  of  Tripoli  (c.  100)  on  the  sphere4 
may  be  passed  with  mere  mention,  since  it  contains  no  work 
on  trigonometry. 

Heron  of  Alexandria.  Although  Heron  (c.  50  ? ) 5  showed  much 
ingenuity  in  his  mensuration  of  the  triangle,  and  was  thoroughly 
conversant  with  the  art  of  surveying  as  practiced  in  Egypt,  it 
cannot  be  said  that  he  gave  any  evidence  of  appreciating  the 
significance  of  trigonometry.  He  made  use  of  certain  rules 
which  we  should  express  in  formulas  for  finding  the  area  of 
regular  polygons,  giving  in  each  case  the  product  of  the  square 
of  a  side  by  a  certain  number,  and  these  rules  afford  some  evi- 
dence of  a  kind  of  prognosis  of  trigonometric  functions.  That 


1On  his  astronomical  work,  see  J.  B.  J.  Delambre,  Histoire  de  Vastronomie 
ancienne,  I,  106  (Paris,  1817) ;  P.  Tannery,  Recherches  sur  Vhistoire  de  Vastro- 
nomie ancienne,  Paris,  1893. 

2  That  they  are  essentially  involved  in  Euclid's  Elements,  VI,  16,  is  shown  in 
the  Simson  additions,  Props.  C  and  D.    See  Heath's  Euclid,  Vol.  II,  pp.  224,  225. 

3  Heath,  History,  II,  259. 

4Latin  ed.,  Paris,  1529;  Greek  ed.,  Beauvais,  1558.   See  Volume  I,  page  125. 
5  Or  possibly  as  late  as  c.  200.   See  Volume  I,  page  125. 


6o6     GENERAL  DEVELOPMENT  OF  TRIGONOMETRY 

is,  taking  A  as  the  area  of  a  regular  w-gon  and  sn  as  the  side,  he 
stated  that  the  following  relations  exist: 


or     Y-T?»  A9=^l-s$,     or 

^      __  J   5_j.   <2 


I  80° 

Since  Af~\ns]t  cot — ->  it  might  be  inferred  that  Heron  had 
?/ 

T  80° 

some  knowledge  of  cot >  but  there  is  nothing  in  the  coeffi- 
cients to  indicate  this  knowledge.1 

Menelaus.  About  100  A.D.  the  astronomer  Menelaus  of  Alex- 
andria, then  living  in  Rome,  took  up  the  study  of  spherical 
triangles,  a  subject  which,  as  we  have  seen,  may  have  occu- 
pied the  attention  of  Hipparchus.  He  wrote  a  work  in  six 


E 


books  on  chords,  and  although  this  is  lost  we  have  his  treatise 
on  spherics,2  which  not  only  forms  the  oldest  known  work  on 
spherical  trigonometry  but  reveals  a  remarkable  knowledge  of 
geometry  and  trigonometry  in  general. 

*P.  Tannery,  "  Arithmetique  des  Grecs  dans  l'H6ron  d'Alexandrie,"  Mi- 
moires  de  la  SOG.  des  sciences  de  Bordeaux,  IV;  Memoires  Scientifiques,  I,  189; 
Heath,  History,  II,  326. 

2Menelai  Sphaericorum  Libri  III,  translated  by  Maurolycus  from  Arabic  and 
Hebrew  sources  and  published  at  Messina  in  1558.  Mersenne  (c.  1630)  published 
an  edition  in  1644,  and  Halley's  edition  appeared  posthumously  at  Oxford  in 
1758.  For  other  editions  see  A.  A.  Bjornbo,  "Studien  iiber  Menelaos'  Spharik," 
Abhandlungen,  XIV,  i,  especially  p.  17.  See  Heath,  History,  II,  261. 


MENELAUS  AND  PTOLEMY  607 

In  the  plane  and  spherical  triangles  shown  on  page  606  he 
proved  the  following  relations: 


Plane  Triangle 
CE      CF    DB 

Spherical  Triangle 
cd2CE       cd2CP     cd2DB 

AE 
CA 

DP 
CD 

AB 
PB 

cd2AE 
cd2CA 

cd2 
cd2 

DP 
CD 

cd2AB 
cd2PB 

AE 

DP 

BE 

cd2  AE 

Cd2 

DP 

cd2BE 

where  cd  2  CE  stands  for  the  chord  of  twice  the  arc  CJS,  that 
is,  for  what  we  call  2  sin  CE.  Since  six  quantities  are  involved 
in  each  equation,  this  was  known  in  the  Middle  Ages  as  the 
regula  sex  quantitatum  and  was  looked 
upon  as  the  fundamental  theorem  of 
the  Greek  trigonometry.  Whether  it  is 
due  to  Menelaus,  to  Hipparchus,  or 
possibly  to  Euclid  is  a  matter  of  dis- 
pute, but  it  is  found  first  in  definite 
form  in  the  Spherics  of  Menelaus,1  the 

proposition  on  the  plane  triangle  being  a  lemma  for  the  other  one. 
Menelaus  also  gave  a  regula  quatuor  quantitatum,  as  fol- 
lows : 2    If  the  two  triangles  ABC  and  DEP  have  Z  A  =  Z  D 

and  /-C=-£P  then  ,    ^ 

cd2AB      cdzDE 

cd2BC~  cd2EP' 

Ptolemy.  The  original  contributions  of  Ptolemy  (c.  150)  to 
trigonometry  are  few,  if  any;  but  we  are  greatly  indebted  to 
him  for  his  summary,  in  the  Almagest,  of  the  theorems  known 
to  Hipparchus.3  Like  other  Greek  writers,  he  used  chords  of 
angles  instead  of  sines,  but  the  idea  of  the  sine  seems  to  have 
been  in  his  mind.4  He  extended  the  table  of  chords  begun  by 
Hipparchus,  and  it  is  quite  probable  that  this  is  the  source  of 
the  table  of  sines  used  by  the  early  Hindu  writers. 

1On  this  point  consult  M.  Chasles,  Aper$u  historique,  2d  ed.,  291;  Delambre, 
Histoire  de  V astronomic  ancienne,  I,  245  (Paris,  1817) ;  A.  A.  Bjornbo,  Abhand- 
lungen,  XIV,  96,  99;  Heath,  History,  II,  266. 

2  For  other  features  see  Braunmiihl,  Geschichte,  I,  17;  A.  A.  Bjornbo,  loc.  cit.t 
p.  124.  3Heath,  History,  II,  276.  4Ibid.,  II,  283. 


6o8    GENERAL  DEVELOPMENT  OF  TRIGONOMETRY 

Hindu  Trigonometry.  Although  the  Hindus  had  already  pro- 
duced the  Surya  Siddhdnta  (c.  400),  and  although  this  work 
treated  of  the  ancient  astronomy,  gave  a  table  of  half  chords 
apparently  based,  as  stated  above,  upon  Ptolemy's  work,  and 
showed  some  knowledge  of  trigonometric  relations,1  it  was  not 
until  Aryabhata  wrote  his  Aryabhafiyam  (c.  510)  that  we  had 
in  Oriental  literature  a  purely  mathematical  treatise  containing 
definite  traces  of  the  functions  of  an  angle.  In  this  work  he 
speaks  of  the  half  chord,  as  the  Surya  Siddhdnta  had  done  be- 
fore him.2 

The  subsequent  work  of  the  Hindus  was  concerned  chiefly 
with  the  construction  of  tables,  and  this  will  be  mentioned  later.  ' 

Arab  and  Persian  Trigonometry.  The  chief  interest  that  the 
Arab  and  Persian  writers  had  in  trigonometry  lay,  as  with  their 
predecessors,  in  its  application  to  astronomy.  On  this  account 
we  find  a  growing  appreciation  of  the  science,  beginning  with 
the  founding  of  the  Bagdad  School  and  extending  to  the  close 
of  the  Mohammedan  supremacy  in  scientific  matters. 

The  chief  Arab  writer  on  astronomy  was  Albategnius4 
(c.  920),  who  ranked  as  the  Ptolemy  of  Bagdad.  Like  the 
Hindus,  he  used  half  chords  instead  of  chords.  He  also  gave 
the  rule  for  finding  the  altitude  of  the  sun,  which  we  express 

by  the  formula  .  .    ,     0      Jx 

J  I  sin  (90°  -  <fr) 

-  __ 


_ 

sm<p 

which  is  simply  equivalent  to  saying  that 

x  =  I  cot  $, 

but  there  is  no  evidence  to  show  that  he  had  any  real  knowledge 
of  spherical  trigonometry. 

!See  Volume  I,  pages  34,  145.  There  is  a  translation  by  Burgess  in  the  Journal 
of  the  American  Oriental  Society,  VI. 

2*L.  Rodet,  Lemons  de  Calcul  d'Aryabhata,  pp.  n,  24  (Paris,  1879). 

3Colebrooke,  Aryabhata,  pp.  90  n.,  309  n. 

4Al-Battani.  His  work  on  the  movements  of  the  stars  was  translated  by 
Plato  of  Tivoli  (c.  1120)  under  the  title  De  motu  slellarum.  It  is  known  to  us 
through  the  writings  of  Regiomontanus,  and  was  published  at  Niirnberg  in  1537. 
There  was  also  a  Bologna  edition  of  1645. 


ARAB  INFLUENCE  609 

Abu'1-Wefa  (c.  980)  did  much  to  make  the  Almagest  known, 
computed  tables  with  greater  care  than  his  predecessors,  and 
began  a  systematic  arrangement  of  the  theorems  and  proofs  of 
trigonometry.  With  him  the  subject  took  on  the  character  of 
an  independent  science. 

It  was,  however,  Na§ir  ed-din  al-Tusi  (c.  1250),  a  Persian 
astronomer,  who  wrote  the  first  work1  in  which  plane  trigo- 
nometry appears  as  a  science  by  itself. 

Ulugh  Beg  (c.  1435)  of  Samarkand  was  better  known  as  an 
astronomer  than  as  a  writer  on  trigonometry,  but  the  tables  of 
sines  and  tangents  computed  under  his  direction  helped  to 
advance  the  science. 

Arab  Influence  in  Europe.  With  the  decline  of  Bagdad  the 
study  of  trigonometry  assumed  greater  importance  in  Spain, 
particularly  as  related  to  those  spherical  triangles  needed  in 
the  work  in  astronomy.  The  most  important  writers  were  the 
astronomers  Ibn  al-Zarqala2  (c.  1050),  who  constructed  a  set 
of  tables,  and  Jabir  ibn  Aflah3  (c.  1145).  In  the  i3th  century 
Alfonso  X  (c.  1250)  directed  certain  scholars  at  Toledo  to  com- 
pute a  new  set  of  tables,  chiefly  for  astronomical  purposes ;  these 
Alfonsine  Tables  were  completed  c.  1254*  and  were  long  held  in 
high  esteem  by  later  astronomers. 

Fibonacci  (1220)  was  acquainted  with  the  trigonometry  oi 
the  Arabs  and,  in  his  Practica  Geometriae,  applied  the  subject 
to  surveying.  Vv 

Peurbach  and  Regiomontanus.  By  the  i4th  century  England 
knew  the  Arab  trigonometry,  and  in  the  isth  century,  thanks 
largely  to  Peurbach  (c.  1460),  who  computed  a  new  table 
of  sines,  and  to  his  pupil  Regiomontanus  (c.  1464),  European 
scholars  in  general  became  well  acquainted  with  it.  The  work 

^Shakl  al-qatta*  (Theory  of  Transversals}.  There  was  a  French  translatior 
published  at  Constantinople  in  1891. 

2  Ibrahim  ibn  Yahya  al-Naqqash,  Abu  Ishaq,  known  as  Ibn  al-Zarqala,  or,  ir 
the  translations,  as  Arzachel.  He  lived  in  Cordova. 

3 Or  Jeber  (Geber)  ibn  Aphla,  of  Seville.  Tne  German  transliteration  ii 
Dschabir  ibn  Aflah.  His  astronomical  work  was  published  at  Nurnberg  in  1543 

4 See  page  232  and  also  Volume  I,  page  228, 


6 10    GENERAL  DEVELOPMENT  OF  TRIGONOMETRY 

of  Regiomontanus1  had  great  influence  in  establishing  the 
science  as  independent  of  astronomy.  He  computed  new  tables 
and  may  be  said  to  have  laid  the  foundation  for  the  later 
works  on  plane  and  spherical  trigonometry.  In  this  general 
period  there  were  also  various  minor  writers,  like  Leonardo 
of  Cremona  (c.  1425),  but  they  contributed  little  of  value.2 
Copernicus  (c.  1520)  completed  some  of  the  work  left  un- 
finished by  Regiomontanus  and  embodied  it  in  a  chapter  De 
Lateribus  et  Angulis  Triangulorum,  later  (1542)  published  sepa- 
rately by  his  pupil  Rhaeticus. 

Influence  of  Printed  Books.  The  first  printed  work  on  the 
subject  may  be  said  to  be  the  Tabula  directionum  of  Regiomon- 
tanus, published  at  Niirnberg  before  1485. 3 

The  first  book  in  which  the  six  trigonometric  functions  were 
defined  as  functions  of  an  angle  instead  of  an  arc,  and  sub- 
stantially as  ratios,  was  the  Canon  doctrinae  triangulorum  of 
Rhaeticus  (Leipzig,  1551),  although  it  gives  no  names  for 
sin  0,  cos</>,  and  csc</>  except  perpendiculum,  basis,  and  hypo- 
tenusa.4  Rhaeticus  was  the  first  to  adopt  the  semiquadrantal 
arrangement  of  the  tables,  giving  the  functions  to  45  °  and  then 
using  the  cofunctions.  He  found  sin  n$  in  terms  of  sin  $, 
sin  (n  —  i)$,  and  cos  (n  —  2)<f>,  a  subject  elaborated  by  Jacques 
Bernoulli5  (1702). 

Vieta  and  his  Contemporaries.  Vieta  (c.  1580)  added  materi- 
ally to  the  analytic  treatment  of  trigonometry.  He  also  com- 
puted sin  i'  to  thirteen  figures  and  made  this  the  basis  for  the 
rest  of  the  table.  With  him  begins  the  first  systematic  develop- 
ment of  the  calculation  of  plane  and  spherical  triangles  by  the 

lDe  triangulis  omnimodis  Libri  V,  written  c.  1464,  first  printed  at  Niirnberg  in 
1533.  He  also  edited  Ptolemy's  Almagest,  the  first  edition  appearing  at  Venice  in 
1496.  On  his  indebtedness  to  Nasir  ed-din  and  others  see  A.  von  Braunmiihl, 
"Nassir  Eddin  Tusi  und  Regiomontan,"  Abh.  der  Kaiserl.  Leop. -Carol.  Deut- 
schen  Akad.  der  Naturjorscher,  LXXI,  p.  33  (Halle,  1897). 

2 See  J.  D.  Bond,  his,  IV,  295. 

3 Second  ed,  Venice,  1485;  3d  ed.,  Augsburg,  1490.   See  Hain,  13,799. 

4  On  his  double  use  of  this  term  for  secant  and  cosecant,  see  Braunmiihl, 
Geschichte,  I,  147. 

BSee  page  629;  Tropfke,  Geschichte,  II  (i),  229. 


EARLY  PRINTED  BOOKS  6n 

aid  of  all  six  functions.  In  one  of  his  tracts  there  appears  the 
important  formula 

*_±1*  -  tan-*-(^+.g) 

a  —  b~~  tan  |  (A  —  B} ' 

which  had  already  been  discovered  by  Fincke,  as  mentioned 
below. 

Albert  Girard  published  at  The  Hague  in  1626  a  small  but 
noteworthy  work  on  trigonometry,  and  in  this  he  made  use  of 
the  spherical  excess  in  finding  the  area  of  a  spherical  triangle. 
This  was  also  given  in  his  algebra  of  1629.  It  also  appeared 
at  about  the  same  time  in  Cavalieri's  Directorium  generate 
(Bologna,  1632)  and,  a  little  later,  in  his  Trigonometria  plana 
et  spherica^  (Bologna,  1643). 

Thomas  Fincke,2  a  Danish  mathematician,  published  an  im- 
portant work,  the  Geometria  Rotundi,  in  Basel  in  1583  (2d  ed., 
1591).  He  gave  the  law  relating  to  a  +  b  :  a—  b,  expressing  it  as 

tan  -i(i 80°  -  C) 


tan[£(i8o°-  C)-Jff] 

The  equivalent  of  our  present  form  is  due  to  Vieta,  as  already 
stated. 

Pitiscus  (1595)  published  an  important  trigonometry  in 
which  he  corrected  the  tables  of  Rhaeticus  and  modernized  the 
treatment  of  the  subject.  In  this  work  the  word  "  trigonometry  " 
appears  for  the  first  time  as  the  title  of  a  book  on  the  subject. 

British  Writers.  Besides  his  invention  of  logarithms,  which 
has  already  been  considered,  Napier  replaced  the  rules  for 
spherical  triangles  by  one  clearly  stated  rule,  the  Napier  Anal- 
ogies,3 published  posthumously  in  his  Construct™  (Edinburgh, 
1619). 

Oughtred's  trigonometry  appeared  in  1657.  In  this  work  he 
attempted  to  found  a  symbolic  trigonometry ;  and  although 

1"In  omnibus  vero  triangulis  sphaericis  tres  eorum  anguli  simul  sumpti  supe- 
rant  duos  rectos.  Et  excessus  eorum  est  .  .  ."  (p. 29).  He  adds:  "quod  ego  pro- 
baui  in  meo  Directorio  P.  3,  Cap.  8." 

2 See  N.  Nielsen,  Matematiken  i  Danmark,  1528-1800  (Copenhagen,  1912). 


612     GENERAL  DEVELOPMENT  OF  TRIGONOMETRY 

algebraic  symbolism  was  now  so  advanced  as  to  make  this  pos- 
sible, the  idea  was  not  generally  accepted  until  Euler's  influence 
was  exerted  in  this  direction  in  the  i8th  century. 

John  Newton  (1622-1678)  published  in  1658  a  treatise  on 
trigonometry1  which,  while  based  largely  on  the  works  of  Gelli- 
brand  and  other  writers,  was  the  most  complete  book  of  the 
kind  that  had  appeared  up  to  that  time.  Newton  and  Gellibrand 
even  went  so  far  as  to  anticipate  our  present  tendency  by  giving 
tables  with  centesimal  divisions  of  the  angle. 

The  greatest  contribution  to  trigonometry  made  by  John 
Wallis  (1616-1703)  was  probably  his  encouragement  of  the 
statement  of  formulas  by  equations  instead  of  by  proportions, 
and  his  work  on  infinite  series.  The  former  advanced  the  ana- 
lytic feature  and  the  latter  made  possible  the  calculation  of 
functions  by  better  methods. 

Sir  Isaac  Newton2  (1642-1727)  made  many  improvements 
in  trigonometry,  as  in  all  other  branches  of  mathematics.  He 
expanded  sin"1^,  or  arc  sin  x,  in  series,  and  by  reversion  he 
then  deduced  a  series  for  sin  x.  He  also  communicated  to  Leib- 
niz the  general  formulas  for  sin  nx  and  cos  nx. 

The  first  to  derive  general  formulas  for  tan  nx  and  sec  nx 
directly  from  the  right-angled  triangle  was  a  French  writer, 
Thomas-Fantet  de  Lagny  (c.  1710).  He  was  also  the  first  to 
set  forth  in  any  clear  form  the  periodicity  of  the  functions. 
The  word  "goniometry"  was  first  used  by  him  (1724),  although 
more  in  the  etymological  sense  of  mere  angle  measure  than  is 
now  the  case. 

The  Imaginary  recognized  in  Trigonometry.  The  use  of  the 
imaginary  in  trigonometry  is  due  to  several  writers  of  the  first 
half  of  the  i8th  century.  Jean  Bernoulli  discovered  (1702) 
the  relation  between  the  arc  functions3  and  the  logarithm  of  an 
imaginary  number.  In  his  posthumous  work  of  1722  Cotes 

showed  that 

<f>i  =  log  (cos  </>  +  i  sin  <£), 

1  Trigonometria  Britannica,  or  the  doctrine  of  triangles  in  two  books,  London, 
1658. 

2Braunmiihl,  Geschichte,  II,  chap.  Hi.  3Such  as  arc  sin  x,  or  sin-1  x. 


IMAGINARY  UNITS  613 

although  no  writers  at  that  time  used  this  particular  symbolism. 
As  early  as  1707  De  Moivre  knew  the  relation 

!_  1 

cos  (f>  =  ^  (cos  ;/(/>  4-  i  sin  n$)n  4-  J  (cos  ;/$  —  z  sin  n<j>)u9 
which  is  obviously  related  to  the  theorem 

(cos  <f>  +  /sin  <f>)*  =  cos  n<f>  -f-  /sin  ;/</>, 

published  in  1722  and  usually  called  by  his  name.1 
Euler  gave  (1748)  the  equivalent  of  the  formula 


but  this  was  no  longer  new.  His  use  of  i  for  V—  i  (1777)  was, 
however,  a  welcome  contribution.  Lambert  (1728-1777)  ex- 
tended this  phase  of  trigonometry  and  developed  the  theory  of 
hyperbolic  functions  which  Vincenzo  Riccati  had  already 
(c.  1757)  suggested  and  which  Wallace2  elaborated  later. 

Functions  as  Pure  Number.  The  first  writer  to  define  the 
functions  expressly  as  pure  number  was  Kastner3  (1759),  al- 
though they  had  already  been  used  as  such  by  various  writers.4 

Trigonometry  becomes  Analytic.  As  already  stated,  through 
the  improvements  in  algebraic  symbolism  European  trigonome- 
try became,  in  the  i?th  century,  largely  an  analytic  science, 
and  as  such  it  entered  the  field  of  higher  mathematics. 

In  the  Orient,  however,  the  science  continued  in  its  primitive 
form,  largely  that  of  shadow  reckoning,  until  the  Jesuits  carried 
European  methods  to  China,  beginning  about  the  year  1600. 
From  that  time  on  the  Western  influence  generally  prevailed, 
not  merely  in  such  centers  as  Nanking  and  Peking  but  also, 
somewhat  later,  in  Japan. 

1Braunmiihl,  Geschichte,  II,  76.  2See  Volume  I,  page  458. 

3  A.  G.  Kastner,  Anfangsgrunde  der  Arithmetik  Geometric  ebenen  und  sph&- 
rischen  Trigonometrie  und  Perspectiv,  Gottingen,  ist  ed.,  1759;  2d  ed.,  1764;  3d 
ed.,  1774.   He  remarks  :  "Bedeutet  also  nun  x  den  Winkel  in  Graden  ausgedruckt, 
so  sind  die  Ausdruckungen  sin  x\  cos  #;  tang  x  u.  s.  w.  Zahlen,  die  fur  jeden 
Winkel  gehoren"  (3d  ed.,  p.  380). 

4  Thus  Regiomontanus  (c.  1463)  speaks  of  the  tangents  as  numeri.  This  occurs 
in  his  tabula  foecunda,  prepared  for  astronomical  purposes,  and  so  called  "  quod 
multifariam  ac  mirandam  utilitatem  instar  foecundae  arboris  parare  soleat." 

ii 


614 


TRIGONOMETRIC  FUNCTIONS 


With  this  brief  summary  of  the  development  of  the  science 
we  may  proceed  to  a  consideration  of  a  few  of  the  special  fea- 
tures which  the  teacher  will  meet  in  elementary  trigonometry. 


IftlffaiHfM 

13  %   IC^'*  t    ni5  ^  U    '?, 


JAPANESE  TRIGONOMETRY  OF  C.  1700 

From  Murai  Masahiro's  Riochi  Shinan,  early  in  the  i8th  century,  showing 
European  influence 

2.  TRIGONOMETRIC  FUNCTIONS 

Sine.  The  most  natural  function  for  the  early  astronomer  to 
consider  was  the  chord  of  an  arc  of  a  circle  having  some  arbi- 
trary radius.  Without  any  good  notation  for  fractions  it  was 
not  convenient  to  take  a  radius  which  would  give  difficult  frac- 
tional values  for  the  approximate  lengths  of  the  chords.  A  con- 
venient radius,  such  as  60,  being  taken,  the  chord  of  the  arc, 
being  considered  purely  as  a  line,  was  the  function  first  studied 
by  the  astronomers. 

The  first  table  of  chords  of  which  we  have  any  record  was 
computed  by  Hipparchus  (c.  140  B.C.),  but  this  table  is  lost 
and  we  have  no  knowledge  as  to  its  extent  or  its  degree  of  accu- 
racy. The  next  table  of  chords  of  which  we  have  good  evidence 


THE  SINE  615 

was  that  of  Menelaus  (c.  100),  but  this  is  also  lost,  although 
his  work  on  spherics  shows  his  use  of  the  function.  The  third 
important  table  of  chords  is  that  of  Ptolemy  (c.  150).  He 
divided  the  circle  into  360°  and  the  diameter  into  120  equal 
parts,1  a  relation  doubtless  suggested  both  by  the  numerous 
factors  of  120  and,  since  3  x  120  =  360,  by  the  ancient  use  of 
3  for  TT.  Influenced  like  Hipparchus  by  Babylonian  precedents, 
he  used  sexagesimal  fractions,  the  radius  consisting  of  60  moirai, 
each  moira  of  60  minutes,  and  so  on.2 

Origin  of  the  Sine.  A  special  name  for  the  function  which  we 
call  the  sine  is  first  found  in  the  works  of  Aryabhata  (c.  510). 
Although  he  speaks  of  the  half  chord,3  he  also  calls  it  the  chord 
half4  and  then  abbreviates  the  term  by  simply  using  the  name 
jyd  or  jtva  (chord).  He  follows  Ptolemy  in  dividing  the  circle 
into  360°,  and  gives  a  table  of  sines,  of  which  a  portion  is 
shown  on  page  626. 

It  is  further  probable,  from  the  efforts  made  to  develop 
simple  tables,  that  the  Hindus  were  acquainted  with  the  princi- 
ples which  we  represent  by  the  formulas 


sin2  (f>  -f-  cos2  <f>  —  i , 

sin2  <f>  -f-  versin2  <£  —  4  sin2  — 


,  .    <f>          i  —  cos  cf> 

and  sin  —  =  \  —  ----  -  > 

2^2 

the  last  two  of  these  appearing  in  the  Panca  Siddhdntikd  of 
Varahamihira  (c.  505). 

The  table  of  sines  given  by  Aryabhata  was  reproduced  by 
Brahmagupta  (c.  628),  but  he  did  nothing  further  with  trigo- 
nometry. Bhaskara  (c.  1150),  however,  in  his  Siddhdnta 
Siromdni,  gave  a  method  of  constructing  a  table  of  sines  for 
every  degree.5 


(tre'mata,  literally  "holes,"  and  hence  the  holes,  or  pips,  of  dice). 
These  parts  were  also  called  /-totpcu  (moi'rai,  parts),  usually  translated  as  degrees. 

2  The  minutes  were  ^KoerrA  irpuTa  (hexekosta'  pro'ta),  first  sixtieths;   the 
seconds  were  ^Koo-rd  defocpa  (hexekosta'  deu'tem),  second  sixtieths. 

3  Ardha-jya,  ardhajyd,  or  ardhd-djyd. 

*Jyd-ardhd.  5Tropfke,  Geschkhte,  II  (i),  192. 


6 1 6  TRIGONOMETRIC .  FUNCTIONS 

Name  for  Sine.  The  jyd1  of  Aryabhata  found  its  way  into  the 
works  of  Brahmagupta  as  kramajya,  that  is,  straight  sine,  or 
sinus  rectus,  as  distinguished  from  the  sinus  versus,  the  versed 
sine.  This  was  changed  to  karaja  when  it  went  over  into 
Arabic,  and  as  such  appears  in  the  Bagdad  School  of  the  gth 
century.  In  particular,  al-Khowarizmi  used  it  in  the  extracts 
which  he  made  from  the  Brahmasiddhdntd  of  Brahmagupta, 
probably  the  work  known  as  the  Sindhind.  It  is  also  found, 
with  natural  variants  in  form,  in  the  writings  of  the  Spanish 
Arab  Ibn  al-Zarqala2  (c.  1050). 

The  sine  also  appears  in  the  Panca  Siddhdntikd  of  Varaha- 
mihira  (c.  505),  where  a  table  is  computed  with  the  Greek 
diameter  of  120.  Indeed,  the  probability  of  Greek  influence 
upon  the  methods  used  by  the  Hindus  is  very  strong. 

The  Arabs  used  the  meaningless  word  jiba,  phonetically  de- 
rived from  the  Hindu  jyd.  The  consonants  of  the  word  per- 
mitted the  reading  jaib,  which  means  bosom,  and  so  this  was 
adopted  by  later  Arabic  writers. 

Sine  in  Latin  Works.  When  Gherardo  of  Cremona  (c.  1150) 
made  his  translations  from  the  Arabic3  he  used  sinus  for  jaib) 
each  word  meaning  a  fold,4  and  this  usage,  possibly  begun  even 
earlier,  was  followed  by  other  European  scholars.  The  word 
"chord"  was  also  used  for  the  same  purpose.5 

alt  has  such  forms  as  djya,  dschyd  (German  transliteration),  fiva  and  fiba. 

2  Or  al-Zarkala,  the  Latin  Arzachel.  In  the  Latin  translation  there  is  a  chapter 
"  De  inventione  sinus  et  declinationis  per  Kardagas."  On  his  use  of  kardagas  see 
Braunmiihl,  Geschichte,  I,  78;  on  such  variants  as  gardaga  and  cada,  see  ibid., 
page  102;  and  on  such  special  uses  as  cardaga  for  arc  15°,  see  ibid.,  pages  no, 
120. 

3 E.g.,  the  Canones  sive  regulae  super  tabulas  Toletanas  of  al-Zarqala  :  "Sinus 
cuius  libet  portionis  circuli  est  dimidium  corde  duplicis  portionis  illius."  See  also 
the  Astronomia  Gebri  filii  Affla  Hispalensis,  which  Apianus  edited  and  published 
at  Niirnberg  in  1533. 

4 Jaib  means  bosom,  breast,  bay;  and  sinus  means  bosom,  bay,  a  curve,  the 
fold  of  the  toga  about  the  breast,  the  land  about  a  gulf,  a  fold  in  land. 

5  Thus  Plato  of  Tivoli:  "...  sive  mentione  cordaru  de  medietatis  cordis 
opportere  intelligi,  nisi  aliquo  proprio  nomine  signauerimus,  quod  &  corda  integram 
appellabimus."  On  this  subject,  which  has  caused  much  controversy,  see  Braun- 
miihl, Geschichte,  I,  49,  with  bibliography.  For  the  absurd  suggestion  that  sinus 
=  s.  ins.  =  semissis  inscriptae  [chordae],  with  bibliography,  see  Tropfke,  Ge-< 
schichte,  II  (i),  212. 


THE  SINE  617 

Abu'1-Wefa  (c.  980)  defined  clearly  the  chord,  sine,  and 
versed  sine  (sinus  versus).    He  showed  that 

sin  <f>  =  J  cd  2  $, 

_,  9 

2r-cd(i8o0-<f>)          2  .  2<f>  . 
I—. LL  ~ ,     our     2  sin  —  =  i  —  cos  9, 

j  9  ^  2 

cd~ 


.    d>       <f> 

our      sm  9  =  2  sm  T-  cos  — , 
22 


and  sin  (<£>  ±  (/>')  =  Vsin2  </>  —  sin2  $  sin2  <£'  ±  Vsin2  <£'  —  sin2  <f>  sin2  <£'. 

He  also  constructed  a  table  of  sines  for  every  15'. 

Ibn  al-Zarqala,  mentioned  on  page  616,  computed  a  table1  of 
sines  and  versed  sines,  using  as  his  arbitrary  radius  150'  and 
also,  following  Ptolemy,  6cA  where  n  stands  for  polpai  (moir- 
rai) 2  and  is  here  used  so  as  not  to  confuse  units  of  line  measure 
with  degrees  of  angle  measure.  It  is  probable  that  such  tables 
were  known  to  Rabbi  ben  Ezra  (c.  1140). 

In  his  Practica  Geometriae  (1220)  Fibonacci  defines  the 
sinus  rectus  arcus  and  sinus  versus  arcus?  and  from  that  time 
on  the  terms  were  generally  recognized  in  the  Middle  Ages. 
Tables  of  sines  were  given  in  various  works  thereafter,4  so  that 
their  use  became  common. 

Other  Names  for  Sine.  The  term  "sine"  was  not,  however, 
universally  recognized,  for  Rhaeticus  (c.  1560)  preferred  per- 
pendiculum.  Of  the  special  terms  which  appeared  from  time 
to  time  there  may  be  mentioned  the  sinus  lotus  and  sinus  per- 
fectus,  both  of  which  were  used  for  sin  go0.5 

1See  page  616,  n.  2.  2See  page  232,  11.3. 

3".  .  .  .be.  uocatur  sinus  rectus  utriusque  arcus  .ab.  et  .be.;  et  recta  .ae. 
uocatur  sinus  uersus  arcus  .ab."  (Scritti,  II,  94) . 

4£.g.,  Johannes  de  Lineriis  (c.  1340).  Ulugh  Beg's  tables  (c.  1435)  were  com- 
puted for  every  minute  of  arc. 

5 E.g.,  by  Johann  von  Gmiinden  (c.  1430).  Regiomontanus  (1463)  used  sinut 
totus  rectus.  Rhaeticus  (c.  1550)  used  sinus  totus,  as  did  most  other  writers  of 
the  time. 


618  TRIGONOMETRIC  FUNCTIONS 

Abbreviations  for  Sine.  The  first  writer  to  make  any  general 
use  of  a  satisfactory  abbreviation  for  sine  was  Girard  (1626). 
He  designated  the  sine  of  A  by  A,  and  the  cosine  of  A  by  a.1 
As  early  as  1624  the  contraction  sin  appears  on  a  drawing 
representing  Gunter's  scale,  but  it  does  not  appear  in  Gunter's 
work  published  in  that  year.2  In  a  trigonometry  published  by 
Richard  Norwood  (London,  1631)  the  author  states  that  "in 
these  examples  s  stands  for  sine :  t  for  tangent :  sc  for  sine  com- 
plement :  tc  for  tangent  complement :  sec  for  secant"  The  first 
writer  to  use  the  symbol  sin  for  sine  in  a  book  seems  to  have 
been  the  French  mathematician  Herigone  (1634).  Cavalieri 
(1643)  suggested  Si,  and  in  the  1647  edition  of  Oughtred  the 
symbol  6*  is  used.  In  1654,  Seth  Ward,  Savilian  professor  of 
astronomy  at  Oxford,  himself  a  pupil  of  Oughtred's,  used  s, 
taking  S3  for  the  sinus  complement.  Oughtred's  symbol  was 
adopted  by  various  English  writers  of  the  iyth  century.3  The 
symbols  sin"1:*;,  cos"1*,  •  •  •,  for  arc  sin  x,  arc  cos  x,  •  •  •,  were 
suggested  by  the  astronomer  Sir  John  F.  W.  Herschel  (1813). 

Versed  Sine.  The  next  function  to  interest  the  astronomer 
was  neither  the  cosine  nor  the  tangent,  but,  strange  as  it  may 
seem  to  us,  the  versed  sine.  This  function,  already  occasionally 
mentioned  in  speaking  of  the  sine,  is  first  found  in  the  Surya 
Siddhanta  (c.  400)  and,  immediately  following  that  work,  in 
the  writings  of  Aryabhata,  who  computed  a  table  of  these  func- 
tions. A  sine  was  called  the  jya ;  when  it  was  turned  through 
90°  and  was  still  limited  by  the  arc,  it  became  the  turned 
(versed)  sine,  utkramajya?  so  that  the  versin  <f>  =  i  —  cos  </>. 

From  India  it  passed  over  to  the  Arab  writers,  and  Albateg- 
nius  (al-Battani,  c.  920),  for  example,  expressly  states  that  he 
uses  the  expression  "turned  chord"5  for  the  versed  sine. 

Since  the  early  writers  were  given  to  fanciful  resemblances 
and  spoke  of  the  bow  (ACE)  and  string  (AS),  or  the  arcus 

iCantor,  Geschichte,  II  (2),  709;  Tropfke,  Gesckichte,  II  (i),  217. 
2F.  Cajori,  "Oughtred's  Mathematical  Symbols,"  Univ.  of  Calif.  Pub.  in  Math., 
I,  185.    Consult  this  article  also  for  the  rest  of  this  topic. 

3£.g.,  Sir  Charles  Scarburgh  (i6i6-c.  1696),  a  name  also  given  as  Scarborough. 
4  Or  utramadjyd.          5In  some  of  the  Latin  translations,  chorda  versa. 


VERSED  SINE  AND  COSINE  619 

and  chorda,  it  was  natural  for  them  to  speak  of  the  versed  sine 
as  the  arrow.  So  the  Arabs  spoke  of  the  sahem,  or  arrow,  and 
the  word  passed  over  into  Latin  as  sagitta,  a  B 

term  used  by  Fibonacci  (1220)^0  mean  versed 
sine  and  commonly  found  in  the  works  of  other 

medieval    writers.2     Among    the    Renaissance     

writers  there  was  little  uniformity.    Maurolico     ° 
(1558)  used  sinus  versus  major  of  <£  to  desig- 
nate versin  (180°  —  <£),  but  others  preferred 
the  briefer  term  sagitta.  •* 

Cosine.  Since  the  Greeks  used  the  chord  of  an  arc  as  their 
function,  they  had  no  special  use  for  the  chord  of  the  comple- 
ment. When,  however,  the  right-angled  triangle  was  taken  as 
the  basis  of  the  science,  it  became  convenient  to  speak  of  the 
sine  of  a  complement  angle.  Thus  there  came  into  use  the 
kotijya  of  Aryabhata  (c.  510), 3  although  the  sine  of  90°  —  </> 
commonly  served  the  purpose  then  as  it  did  later  with  the 
Arabs.4  Even  when  a  special  name  became  necessary  it  was 
developed  slowly.  Plato  of  Tivoli  (c.  1120)  used  chorda  residui 
or  spoke  of  the  complement  angle.5  Regiomontanus  (c.  1463) 
used  sinus  rectus  complementi.  Rhseticus  (1551)  preferred 
basis,  Vieta  (1579)  used  sinus  residuae,  Magini  (1609)  used 
sinus  secundus,  while  Edmund  Gunter  (1620)  suggested 
co. sinus,  a  term  soon  modified  by  John  Newton  (1658)  into 
cosinus,  a  word  which  was  thereafter  received  with  general 
favor.  Cavalieri  (1643)  used  the  abbreviation  Si.2]  Oughtred, 
s  co  arc]  Scarburgh,  c.s. ;  Wallis,6  2;  William  Jones  (1706), 
^;  and  Jonas  Moore7  (1674),  Cos.,  the  symbol  generally 
adopted  by  later  writers. 

iScritti,  II,  94,  11.  10,  16,  18. 

2 -E.g.,  Levi  ben  Gerson  (c.  1330),  in  his  De  sinibus,  ehordis,  et  arcubus,  where 
his  translator  also  uses  sinus  versus;  and  Johann  von  Gmlinden  (c.  1430) . 

3The  possible  seqt,  skd,  or  seqet  of  the  Egyptians  (p.  600).  Also  transliter- 
ated kptidjyd. 

4  As  with  Albategnius  (al-Battani)  and  others. 

5 "Quod  ad  perficiendum  90  deficit." 

6"S,  co-sinus,  seu  sinus  complementi"  (Opera>  1693,  H»  S91)- 
t.Math.,1  (3),  69. 


620  TRIGONOMETRIC  FUNCTIONS 

Tangent  and  Cotangent.  While  the  astronomers  found  the 
chord  and  sine  the  functions  most  useful  in  their  early  work, 
and  so  developed  them  first,  the  more  practical  measurements 
of  heights  and  distances  first  required  the  tangent  and  cotan- 
gent,— the  gnomon  and  shadow  respectively.  It  is  possible  that 
Ahmes  (c.  1550  B.C.)  knew  the  tangent,  but  in  any  case  we 
know  that  shadow  reckoning  was  an  early  device  for  finding 
heights,  and  that  it  was  related  to  the  sundial  which  Anaxi- 
mander  (c.  575  B.C.)  introduced  into  Greece.  Unlike  the  sine 
and  cosine,  the  tangent  and  cotangent  developed  side  by  side, 
the  reason  being  that  the  gnomon  and  shadow  were  equally 
important,  the  complementary  feature  playing  no  part  at  first. 
The  Greeks,  however,  made  no  use  of  these  functions  of  an 
angle,  so  far  as  we  know,  except  as  Thales  measured  the  heights 
of  pyramids  by  means  of  shadows  and  similar  triangles. 

The  Umbra  Recta  and  Umbra  Versa.  The  Surya  Siddhdnta 
(c.  400)  and  other  Hindu  works  speak  of  the  shadow,  particu- 
larly in  connection  with  astronomical  rules,  but  it  was  the  Arabs 
who  first  made  any  real  use  of  it  as  a  function. 

It  was  Ahmed  ibn  'Abdallah,1  commonly  known  as  Habash 
al-Hasib,  "the  computer"  (c.  860),  who  constructed  the  first 
g   table  of  tangents  and   cotangents,2  but  it 
exists  only  in  manuscript.    The  Arab  writers 
distinguished  the  straight  shadow,  translated 
by  the  later  medieval  Latin  writers  as  umbra, 


Umbra  recta  umbra  recta,  or  umbra  extensa,  and  the 
turned  shadow,  the  umbra  versa  or  umbra  stans,  the  terms  vary- 
ing according  as  the  gnomon  was  perpendicular  to  a  horizontal 
plane,  as  in  ordinary  dials,  or  to  a  vertical  wall,  as  in  sundials  on 
a  building.  They  were  occasionally  called  the  horizontal  and 
vertical  shadows.3  The  shadow  names  were  also  used  by  most  of 
the  later  Latin  authors  and  by  writers  in  general  until  relatively 
modern  times,  being  frequently  found  as  late  as  the  i8th  century. 

1Or  al-Mervazi.   See  Volume  I,  page  174. 

2  These  are  given  in  a  MS.  of  his  astronomical  tables  preserved  at  Berlin.   See 
Suter,  Abhandlungen,  X,  209. 

3 E.g.,  by  Abft'l-yasan  AH  ibn  'Omar  al-Marrakoshf,  Of  Morocco  (c.  1260). 


TANGENT  AND  COTANGENT         621 

These  functions  do  not  seem  to  have  interested  the  western 
Arab  writers,  no  trace  of  either  umbra  being  found  in  the 
works  of  Jabir  ibn  Aflah  (c.  1145). 

The  terms  umbra  recta  and  umbra  versa  were  not  used  by 
Gerbert,  but  Robertus  Anglicus  (c.  1231)  speaks  of  the  umbra, 
so  that  by  his  time  it  had  come  to  be  somewhat  recognized. 
Thereafter  the  names  umbra  recta  and  umbra  versa  were  in 
fairly  common  use. 

Table  of  Shadows.  The  first  writer  whose  table  of  shadows 
is  generally  known  is  Albategnius  (al-Battani,  c.  920),  the 
table  giving  the  cotangents  for  each  degree  of  the  quadrant.1 

Abu'1-Wefa  (c.  980)  constructed  a  table  of  tangents  for 
every  15',  the  first  table  of  tangents  that  is  known  to  us;  and 
about  this  time  there  was  computed  a  table  of  cotangents  for 
every  10'.  Under  the  direction  of  Ulugh  Beg  (c.  1435)  there 
was  prepared  a  table  of  tangents  for  every  i'  from  o°  to  45° 
and  of  every  5'  from  45°  to  90°,  but  his  table  of  cotangents  was 
constructed  only  for  every  i°.2 

Names  and  Symbols.  Although  Rhaeticus  (1551)  did  not  use 
these  common  names  for  tangent  and  cotangent,  he  defined  each 
as  a  ratio  and  gave  the  most  complete  table  that  had  appeared 
up  to  that  time. 

Vieta  (c.  1593)  called  the  tangent  the  sinus  foecundarum 
(abridged  to  foecundus3)  and  also  the  amsinus  and  pro  sinus. 
It  was  not  until  Thomas  Fincke  wrote  his  Geometria  Rotundi 
(1583)  that  the  term  "tangent"  appeared  as  the  equivalent  of 
umbra  versa*  The  name  was  adopted  by  Pitiscus  (1595),  and 
the  reputation  of  this  great  writer  gave  it  permanent  stand- 
ing. Magini  (1609)  used  tangens  secunda  for  cotangent.  The 
term  cotangens  was  first  used  for  this  function  by  Edmund 
Gunter  (1620). 

!That  is,  he  gave  the  value  of  u  =  /  -  — — •  for  <f>  =  i°,  2°,  •  •  •,  /  being 
the  length  of  the  gnomon.  sin  ^ 

2Braunmuhl,  Geschichte,  I,  75. 

3 On  the  origin  of  the  term  see  page  613  and  Braunmiihl,  Geschichte,  I,  161  n.; 
Tropfke,  Geschichte,  II  (i),  210. 

4"  Recta  sinibus  connexa  est  tangens  peripheriae,  aut  earn  secans"  (Geometria 
Rotundi,  p.  73  (Basel,  1583)). 


622  TRIGONOMETRIC  FUNCTIONS 

As  abbreviations  for  tangent  and  cotangent,  Cavalieri  (1643) 
used  Ta  and  Ta.2  ;  Oughtred  (1657),  t  arc  and  t  co  arc]  Sir 
Charles  Scarburgh,  t.  and  ct.\  and  Wallis  (1693),  T  and  r. 
The  abbreviation  tan,  as  in  A,  was  first  used  by  Girard  (1626), 
and  Cot.  was  suggested  by  Jonas  Moore  (1674),  but  even  yet 
we  have  no  generally  accepted  universal  symbols  for  tangent 
and  cotangent. 

Secant  and  Cosecant.  Since  neither  the  astronomer  nor  the 
surveyor  of  early  times  had  any  need  for  the  secant  and  cose- 
cant, except  as  the  hypotenuse  of  a  right-angled  triangle,  these 
functions  were  developed  much  later  than  the  others.  The 
secant  seems  first  to  have  been  considered  by  al-Mervazi 
(Habash,  c.  860) ,  although  the  two  functions  first  appear  in  def- 
inite form  in  the  works  of  Abu'l  Wefa  (c.  980),  but  without 
special  names.  Little  was  done  with  them  by  the  Arabs,  how- 
ever, and  it  was  not  until  tables  for  navigators  were  prepared 
in  the  isth  century  that  secants  and  cosecants  appeared  in  this 
form.1  Although  Copernicus  (1542)  knew  the  secant,  speaking 
of  it  as  the  hypotenusa  and  computing  a  set  of  values  of  these 
functions,  it  was  his  pupil  Rhaeticus  who  first  included  secants 
in  a  printed  table.  The  secant  and  cosecant  appear  with  the 
other  four  functions  in  his  Canon  doctrinae  triangulorum 
(Leipzig,  1551),  although  Rhaeticus  speaks  of  each  in  that  work 
as  a  hypotenuse.  The  name  "cosecant77  seems  to  have  appeared 
first  in  his  posthumous  Opus  Palatmum  (1596).  Maurolico 
(1558)  included  in  his  tables2  the  secants  from  o°  to  45°. 

Names  for  Secant  and  Cosecant.  The  name  "secant"  was 
first  used  by  Fincke  (1583)  and,  although  Vieta  (1593)  called 
this  function  the  transsinuosa,  the  more  convenient  and  sug- 
gestive name  soon  came  into  general  use.  The  cosecant  was 
called  the  secans  secunda  by  Magini  (1592)  and  Cavalieri 
(1643).  Pitiscus  (1613)  gave  the  secants  and  cosecants  in 
his  tables,  and  since  then  they  have  been  commonly  found  in 
similar  publications. 

iBraunmuhl,  Geschichte,  I,  114;  115  n. 
2  Tabula  benefica,  in  his  work  on  spherics. 


RELATION  BETWEEN  FUNCTIONS  623 

By  way  of  abbreviations  for  secant  and  cosecant,  Cavalieri 
(1643)  used  Se  and  Se.2]  Oughtred  (1657),  se  arc  and  sec  co 
arc\  Wallis  (1693),  s  and  cr;  but  the  more  convenient  symbol 
sec,  suggested  by  Girard  (1626)  in  the  form  A  soon  came  into 
general  use.  There  is  as  yet  no  international  symbol  for  cose- 
cant, cosec  and  esc  both  being  used. 

Relation  between  Functions.  Although  the  functions  them- 
selves were  not  specifically  named,  various  early  writers  make 
statements  which  involve  in  substance  many  of  the  relations 
that  we  now  recognize.  Thus  the  formula 

sin  <£  =  Vi  —  cos'2  </> 
or  sin2  $  -f  cos2  $  =  i 

is  essentially  the  Pythagorean  Theorem  and  as  such  was  known 
to  the  Greeks. 

Abu'1-Wefa  (c.  980)  knew  substantially  the  formulas 

tan  <f> :  i  =  sin  </> :  cos  </>, 

cot  <£  :  i  =  cos  (/>  :  sin  <jb, 

sec  </>  =  V  i  -+-  tan'2  <f>, 

and  esc  <£  =  V  i  +  cot2  </>. 

Rhseticus  (1551)  knew  the  relations 

sec  </> :  i  =  i  :  cos  $ 

and  esc  <£  :  i  =  i  :  sin  $. 

Vieta  (1579)  gave  the  following  proportions:1 

i  :  sec  $  =  cos  $  :  i  =  sin  ^> :  tan  (/>, 
esc  (/> :  sec  <£  =  cot  <£  :  I  =  i  :  tan  </), 
and  i  :  esc  <£  =  cos  <f> :  cot  0  =  sin  c/> :  i. 

3.  TRIGONOMETRIC  TABLES 

Early  Methods  of  Computing.  The  more  important  of  the 
earliest  trigonometric  tables  have  been  mentioned  in  connec- 
tion with  the  several  functions.  A  brief  statement  will  now  be 

iTropfke,  Geschichte,  II  (i),  225. 


624  TRIGONOMETRIC  TABLES 

made  as  to  the  general  methods  of  computing  these  tables 
and  as  to  the  early  printed  tables  themselves. 

The  first  methods  of  which  we  have  definite  knowledge  are 
those  of  Ptolemy  (c.  iso).1  His  computation  of  chords  de- 
pends on  four  principles : 

I.  From  the  sides  of  the  regular  inscribed  polygons  of  3,  4, 
5,  6,  and  10  sides  he  obtained  the  following: 

cd  36°  =  37*  4' 5 5", 
cd  72°  =  70*32'  3", 
cd  60°  =  60*, 

°  =  84*  51'  10", 


and          cd  120°  =  103^  55'  23". 

In  a  semicircle  as  here  shown,  £c*  -h  6^2  =  AB* ,  and  so 
cd(i8o°  —  36°)  =  cd  144°  =  Vi202  — cd2  36°  =  114^  7'  37". 

II.  In  an  inscribed  quadrilateral  the  sum  of  the  rectangles 
of  the  two  pairs  of  opposite  sides  is  equal  to  the  rectangle  of 
the  two  diagonals.    This  is  known  as  Ptolemy's  Theorem  and 
is  found  in  the  Almagest2 

III.  The  chord  of  a  half  arc  can  be  found  from  the  chord 
of  the  arc;  that  is,  from  ccl  12°  it  is  possible  to  find  cd  6°,  and 
then  cd  3°,  and  so  on.' 

IV.  By  a  scheme  of  interpolation  it  is  possible  to  approxi- 
mate the  chord  of  \ <£,  when  cd  <£  is  known.4 

With  the  help  of  these  principles  Ptolemy  was  able  to  find 
the  chords  of  all  angles  to  a  fair  degree  of  approximation.  Thus 
he  found  that  cd  i°o'=  i"  2' 50", 

which  would  make 

sin  30'  =  £cd  i°  o'  =  o^  31'  25"  =  0.0087268, 

1  The  tables  are  given  at  the  end  of  Lib.  I,  cap.  ix,  of  the  Almagest. 
2Halma  ed.,  I,  29;  Heiberg  ed.,  p.  36;  Braunmuhl,  Geschkhte,  I,  19. 

3 Essentially  he   has  sin-  =\/ — —     For  the  mathematical  discussion, 

2          \  2 

see  Braunmuhl,  Geschichte,  I,  20. 

4 For  mathematical  discussion,  see  Braunmuhl,  Geschichte,  I,  21;  Tropfke, 
Geschichte,  II  (i),  296. 


METHODS  OF  COMPUTING 


625 


whereas  our  seven-place  tables  give  it  as  0.0087265. 

Wefa  (c.  980)  computed  this  result  as  o^  31'  24"  55'"  54 

—  a  value  which  is  correct  as  far  as  the  tenth  decimal  place. 


Abu'l- 
*      v 


%  3  f\irr  i 


Vt 


it     „ 


ift 


5«  «?W 


FROM  THE  FIRST  PRINTED  EDITION  OF  THE   SURYA  SIDDHANTA 

Printed  at  Meerut,  India,  c.  1867.   This  is  the  oldest  Hindu  work  on  astronomy. 
It  shows  the  table  of  which  a  portion  is  given  on  page  626 

A  table  of  sines  is  given  in  the  Surya  Siddhdnta  (c.  400), 
and  Aryabhata  (c.  510)  gives  a  table  of  sines  and  versed  sines. 
The  following  portion  of  the  table  of  sines,  substantially  as  in 


TRIGONOMETRIC  TABLES 

some  of  the  manuscripts  of  the  Surya  Siddhdnta  and  as  given 
also  in  Aryabhata's  work,  will  serve  to  show  the  degree  of 


accuracy : 


A  KC 

SINK 

MODKRN  VALUE 

3°  45' 

225' 

224.84' 

7°  30' 

449' 

448.72' 

11°  15' 

67i' 

670.67' 

•5° 

890' 

889.76' 

Aryabhata's  method  of  working  out  his  table  was  to  take 
sin  3°  45'1  as  equal  to  arc  3°  45',  and  from  this  to  find  the 
sines  of  multiples  of  this  angle  by  the  rule  already  given  in 
the  Surya  Siddhdnta2 

sin  (;/  -f  i)  </>  —  sin  n$  +  sin  ;/$  —  sin  (;/  —  i)  $ — ~> 

which  is  correct  except  for  the  last  term. 

Arab  Methods.  The  early  Arabs  used  the  Hindu  results,  but 
later  scholars  developed  original  methods  of  attack.  Of  these, 
one  of  the  best  known  is  given  by  Miram  Chelebi  (c.  i52o)3 
in  his  commentary  on  Ulugh  Beg  (c.  1435). 4  He  gives  two 
methods,  the  first  being  somewhat  similar  to  the  one  used  by 
Ptolemy.  The  second  method  is  interesting  because  it  involves 
the  approximate  solution  of  a  cubic  equation  of  the  form 

ax  —  b  =  x?. 

European  Computers.  Of  the  later  computers  of  the  Middle 
Ages  and  early  Renaissance,  Regiomontanus  (1546)  stands  at 
the  head,  but  his  methods  were  not  new.  Indeed,  there  was  no 
particular  originality  shown  in  the  computations  from  the  time 
of  Ptolemy  to  the  invention  of  the  modern  methods  based  on 
series. 

1  Known  by  the  special  name  kramajyd.        2Braunmuhl,  Geschichte,  I,  35. 

3Musa  ibn  Mohammed  ibn  Mahmud  ibn  Qadizadeh  al-Rumi,  a  teacher  in 
Gallipoli,  Adrianople,  and  Brusa;  died  1524/25. 

4M.  Woepcke,  "Discussion  de  deux  methodes  arabes  pour  determiner  une 
value  approchee  de  sin  i°,"  Journal* de  math,  pures  et  appliquees,  XIX  (1854), 
153;  A.  Sedillot,  Prolegomenes  des  Tables  astronomiques  d'Ouloug-Beg,  Paris, 
1853;  Braunmiihl,  Geschichte,  I,  72,  with  incorrect  date. 


EARLY  TABLES  627 

In  general,  all  ancient  tables  were  constructed  with  Ptolemy's 
radius  of  60;  that  is,  the  sinus  totus,  or  sin  90°,  was  60.  This 
was  due  to  the  necessity  of  avoiding  fractions  in  the  period 
before  the  invention  of  decimals.  The  first  to  adopt  the  simpler 
form,  sin  90°  =  i,  was  Jobst  Biirgi  (c.  1600),  but  his  tables 
computed  on  this  basis  are  not  extant.  Although  the  invention 
of  decimal  fractions  had  now  made  the  use  of  unity  possible  for 
the  sinus  totus,  this  idea  was  not  fully  appreciated  until  a 
memoir  by  de  Lagny  was  written  in  i Tig.1  It  was  nearly  thirty 
years  later  that  the  plan  received  its  first  great  support  at  the 
hands  of  Euler.2 

Early  Printed  Tables.  Of  the  early  printed  tables  there  may 
be  mentioned  as  among  the  more  important  the  table  of  sines 
with  the  radius  divided  decimally,  published  by  Apianus  in 
*533 !  the  table  of  all  six  functions  based  on  a  semiquadrantal 
arrangement,  published  by  Rhaeticus  in  1551,  calculated  to 
every  10'  and  to  seven  places;  Vieta's  extension  of  the  tables  of 
Rhaeticus  to  every  minute  (1579,  but  the  printing  began  in 
1571);  the  table  of  tangents  by  Reinhold  (1511-1553)  to  every 
minute,  printed  in  1554;  the  table  of  all  six  functions,  published 
in  England  by  Blundeville  in  1594;  the  Opus  Palatinum,  with 
the  functions  for  every  10"  to  ten  decimal  places,  with  tables 
of  differences,  compiled  by  Rhaeticus  and  published  by  Valentin 
Otto  (or  Otho)  in  1596.  Dr.  Glaisher,  referring  to  the  work  of 
Rhaeticus,  speaks  of  him  as  "  by  far  the  greatest  computer  of  pure 
trigonometrical  tables"  and  as  one  " whose  work  has  never  been 
superseded."  The  Opus  Palatinum  was  so  named  in  honor  of 
the  elector  palatine,  Friedrich  IV,  who  paid  for  its  publication.3 

The  serious  use  of  tables  based  upoi>  the  centesimal  division 
of  the  angle  was  a  result  of  the  movement  that  led  to  the  metric 
system.  An  elaborate  set  of  such  tables  was  prepared  in  Paris 
at  about  the  close  of  the  i8th  century,  and  little  by  little  the 
plan  found  favor.  Such  a  set  of  tables  appeared  even  in  Japan 

iHistoire  et  Mtmoires  de  I'Acad.  d.  sci.,  Paris,  1721,  p.  144;  1726,  p.  292 ;  1727, 
p.  284;  1729,  p.  121. 

2Introductio  in  analysin  infinitorum,  I,  §  127.   Lausanne,  1748. 

3 For  a  summary  of  such  tables  see  the  Encyc.  Brit.,  nth  ed.,  XXVI,  325. 


628 


TYPICAL  THEOREMS 


as  early  as  1815,  but  it  was  not  until  the  close  of  the  igth 
century  that  the  idea  took  any  firm  hold  upon  the  mathematics 
of  Europe,  and  then  with  the  French  schools  still  in  the  lead. 


5.  a.  /L 


-MJ 


ooooooo— 


A-.  —  A 


CENTESIMAL  TABLES  OF  JAPAN 

From  a  manuscript  of  a  work  on  trigonometry,  by  Miju  Rakusai,  written  in  1815, 

showing  a  table  of  natural  functions  on  the  decimal  division  of  the  angle.    This 

page  shows  the  cotangents  and  cosines,  beginning  at  the  top  with  o° 


4.  TYPICAL  THEOREMS 

Addition  Theorem  of  Sines.  It  is  impossible,  in  the  space  al- 
lowed, to  mention  more  than  a  few  of  the  important  theorems 
of  trigonometry,  and  these  will  now  be  considered. 

The  Greeks  knew  essentially  that 

sin  (<£  ±  cf>f)  =  sin  <f>  cos  <f>f  ±  cos  <f>  sin  <f>'. 


FUNCTIONS  OF  ANGLES  629 

Stated  as  a  proposition  involving  chords,  it  is  probable  that 
Hipparchus  (c.  140  B.C.)  knew  it.  It  was  certainly  known  to 
Ptolemy  (c.  150),  and  it  often  bears  his  name.  Bhaskara 
(c.  1150)  also  gives  the  theorem.  As  already  stated  on  page  617, 
Abu'1-Wefa  (c.  980)  gave  it  essentially  under  the  form 

sin(<£  ±  <f>f)  =  Vsin2^  —  sin2</>  sm2(£'  ±  Vsin2<//  —  sin2</>  sin2<£'. 

Functions  of  Multiple  Angles.    The  formula 

sin  2  (f>  =  2  sin  $  cos  <f> 

is  a  corollary  of  the  general  case  of  sin  (<f> 4-  </>').    It  is  first 
expressly  given  as  a  rule  by  Abu'1-Wefa,  the  form  being,  as  al- 
ready stated,  chord  <f> :  chord  |  ^  =  chord  ( 180°—  |  <f>) :  r. 
Vieta1  (1591)  first  gave  the  formulas 

sin  3  $  —  3  cos'2<£  sin  $  —  sin3</>, 
cos  30  —  cos3</>  —  3  sin2c/>  cos  </>, 

and  connected  sin;/$  with  sin0  and  cos  <f>. 
Rhseticus  (1569)  found  the  relation 

cos;/</>  =  cos(;/  —  2) </>  —  2  sin  ^  sin(;/  —  i)</>. 
Newton  (1676)  gave  the  well-known  relation 

i           -i      ( 1  —  #2) n  -  *  i 
sin  7/9  =  ;/  sin  9  -h    -  sin  9  -f  .  .  ., 

and  Jacques  Bernoulli  (1702)  showed  that 

sin  n$  =  cosw<£  — -L— ~J  cosw~2<#>  sin2^  -h  •  •  •, 

i       w       ,    i  ,    •     ,       n  (n  —  i)  (;/  —  2)  fl  .    .  ft  , 

cos  n(j>  =  ~  cos""1^  sin  ^ ^~ ^-v— '-  cosn~*<f>  sm8</>  -f  •  •  •. 

Functions  of  Half  an  Angle.  Ptolemy2  (c.  150)  knew  substan- 
tially the  sine  of  half  an  angle,  expressed  as  half  a  chord,  and 
it  is  probable  that  Hipparchus  (c.  140  B.C.)  and  certain  that 
Varahamihira  (c.  505)  knew  the  relation  which  we  express  as 


_       ;  —  cos  <f> 
2 

1Tropfke,  Geschichte,  II  (2),  57-61,  with  bibliography. 

2  See  the  Heiberg  edition  of  Ptolemy,  p.  39. 

ii 


630  TYPICAL  THEOREMS 

After  the  development  of  analytic  trigonometry  in  the  lyth 
century,  these  relations  were  greatly  extended.  Four  others 
may  be  mentioned  as  typical,  the  first  two,  due  to  Euler  (1748), 

being 

5  .         2  tan  6 

tan  2$  =  — — -TT' 
i  -  tan2c/> 

cot  6  —  tan  6 

cot  2  (/>  =  ~—T—^- — r ; 

and  the  others,  due  to  Lambert  (1765),  being 

2  tan  <£> 

sin  2  <£  =  — —  ~j> 
i-r-tairc/> 

.       i  — tan'2<f> 
cos  2  d>  =  - —  — 0  7  • 
^      i-f-tair(/> 

Theorem  of  Sines.    The  important  relation  now  expressed  as 

a     __     b     __     c 

sin  A      sin  B     sin  C 

was  known  to  Ptolemy    (c.  150)   in  substance,  although  he 
expressed  it  by  means  of  chords.1 

While  recognized  by  Alberuni  and  other  Oriental  writers,  it 
was  Nasir  ed-din  (c.  1250)  who  first  set  it  forth  with  any 
clearness.  A  little  later  Levi  ben  Gerson  (c.  1330)  stated  the 
law  in  his  work  De  sinibus,  chordis,  et  arcubus ; 2  but  the  first 
of  the  Renaissance  writers  to  express  it  with  precision  was 
Regiomontanus 3  (writing  c,  1464). 

i-Thus,  if  C  is  a  right  angle  in  triangle  ABC,  then 

__  c  chord  2  A 

I2OM 

where  120  /xo?pcu  is  the  diameter  of  the  circumcircle. 

2"  .  .  .  omnium  triangulorum  rectilineorum  talem  proportionem  una  linea  habet 
ad  aliam,  qualem  proportionem  unus  sinus  angulorum,  quibus  dictae  lineae  sunt 
subtensae,  habet  ad  alium."  See  Braunmiihl,  Geschichte,  I,  106. 

3 "In  omni  triangulo  rectilineo  proportio  lateris  ad  latus  est,  tamquam  sinus 
recti  anguli  alterum  eorum  respicientis,  ad  sinum  recti  anguli  reliquum  latus 
respicientis"  (Lib.  II,  prop.  i).  See  Tropfke,  Geschichte,  V  (2),  74. 


SINES  AND  COSINES  631 

Theorem  of  Cosines.   The  fact  that 


is  essentially  a  geometric  theorem  of  Euclid.1  In  that  form  it 
was  known  to  all  medieval  mathematicians.  In  the  early  printed 
books  it  appears  in  various  forms,  Vieta  (1593)  giving  it  sub- 

stantially as  .  ,,  „     ,„      0,         /  •    ,     o 

J  2  a&/(a*  +  b2-  c")  =  I  /sin  (90°  -  C), 

and  W.  Snell  (1627)  as 

2  abl\f-  (a  -  l>)2]  =•-  I  /(I  -  cos  C). 

Theorem  of  Tangents.  The  essential  principle  of  the  law  of 
tangents,  which  was  given  by  Vieta2  (p.  611)  and  improved  by 
Fincke  (1583),  was  known  to  Ptolemy  (c.  150).  Regiomon- 
tanus  (c.  1464)  expressed  it  by  a  rule  which  we  should  state  as 

sin  A  ~h  sin  13      tan  -J-  (A  +  B) 
sin  A  -  sin  /?  "  tan  \  (A  —  1$) 

Areas.  The  first  evidence  of  the  rule  which  resulted  in  the 
formula  for  the  area  of  a  triangle,  which  we  know  as 

A  =  I  ab  sin  Cy 

is  found  in  the  trigonometry  of  Regiomontanus  (c.  I464),3  but 
the  theorem  is  not  explicitly  stated  by  him.  Snell  (1627)  gave 
it  in  the  form  I:sin^  =  ^:2A. 

Right-angled  Spherical  Triangle.  The  Greek  mathematicians 
made  use  of  the  right-angled  spherical  triangle  in  their  com- 
putations, but  nowhere  do  we  find  a  systematic  treatment  of 
the  subject.  Taking  the  hypotenuse  as  c,  we  have  the  follow- 
ing six  cases  : 

i  .  cos  c  =  cos  a  cos  b.  4.  cos  A  =  tan  b  cot  c. 

2.  cos  c  =  cot  A  cot  B.  5.  sin  b  —  sin  c  sin  B. 

3.  cos  A  —  cos  ft  sin  B.  6.  sin#  =  tan  a  cot  A. 

1  Elements,  II,  12,  13. 

2"Ut  aggregatum  crurum  ad  differentiam  eorundem,  ita  prosinus  dimidiae 
angulorum  ad  basin  ad  prosinum  dimidiae  differentiae"   (Opera,  Schooten  ed., 
p.  402).    See  Tropfke,  Geschichte,  V  (2),  80. 
triangulis  omnimodis,  Niirnberg,  1533. 


632  TYPICAL  THEOREMS 

In  his  astronomical  problems  Ptolemy  (c.  150)  makes  use 
essentially  of  the  first,  fourth,  fifth,  and  sixth  of  these  cases,1 
although  without  the  functions  he  could  not  give  the  rules. 

The  third  case  is  essentially  given  by  Jabir  ibn  Aflah 
(c.  1145),  and  so  it  was  commonly  known  as  Jabir's  Theorem.2 

The  first  writer  to  set  forth  essentially  all  six  cases  was  Nasir 
ed-din  (c.  1250).  Napier's  Rules  for  the  right-angled  spherical 
triangle  appeared  in  his  tables  of  i6i4.3 

Oblique-angled  Spherical  Triangle.  The  oblique-angled  spher- 
ical triangle  was  not  seriously  studied  by  itself  until  the  Arabs 
began  to  consider  it  in  the  loth  century. 

The  Theorem  of  Sines, 

sin  a  __  sin  b  _  sine 
sin  A      sin  B     sin  C ' 

and  the  Theorem  of  Cosines  of  Sides, 

cos  a  —  cos  b  cos  c  -\-  sin  b  sin  c  cos  A, 

may  have  been  known  to  them,  but  they  are  first  found  in  print 
in  the  De  triangulis  written  by  Regiomontanus  c.  I464.4 
The  Theorem  of  Cosines  of  Angles, 

cos  A  =  —  cos  ,#  cos  C+  sin  .#sin  <7cos  a, 

was  given  in  substance  by  Vieta  in  1593,  although  he  had  used 
it  before  this  date.5   It  was  first  proved  by  Pitiscus  in  1595. 
The  Theorem  of  Cotangents, 

I         __  cot  c  +  cos  A  cot  b 
sin  b  esc  A  cot  C 

was  also  given  in  substance  by  Vieta,  but  was  modified  by 
Adriaen  van  Roomen  (1609)  and  proved  by  Snell  (1627).° 

1Tropfke,  Geschichte,  V  (2),  131,  with  references.  Consult  this  work  and 
Braunmiihl's  Geschichte  for  further  details. 

2  It  was  possibly  known  to  Tabit  ibn  Qorra  (c.  870). 

3Mirifici  logarithmorum  canonis  descriptio,  1614,  Lib.  II,  cap.  iv. 

4 On  the  Theorem  of  Sines  see  Tropfke,  V  (2),  133 ;  on  the  Theorem  of  Cosines 
see  ibid.,  p.  139. 

5For  the  priority  question  see  Tropfke,  Geschichte,  V  (2),  139. 

6 For  the  general  literature  on  this  subject  see  Braunmiihl,  Geschichte,  I  (i), 
25;  Tropfke,  Geschichte^  V  (2),  137  seq.,  especially  p.  143. 


DISCUSSION  633 

TOPICS  FOR  DISCUSSION 

1.  The  etymology  of  the  words   " trigonometry,"   "geometry," 
"mensuration,"  "agrimensor,"  "survey,"  "geodesy,"  and  other  terms 
having  a  related  meaning. 

2.  Primitive  needs  that  would  naturally  tend  to  the  development 
of  trigonometry. 

3.  The  relation  of  shadow  reckoning  to  plane  trigonometry  in 
various  countries  and  at  various  times. 

4.  The  influence  of  astronomy  upon   the  development  of  the 
science  of  trigonometry. 

5.  The  Greek  astronomers  who  contributed  most  to  the  study  of 
trigonometry,  the  function  which  they  developed,  and  the  reason  why 
this  function  was  selected. 

6.  The  contributions  of  Menelaus  to  the  study  of  trigonometry. 

7.  The  Hindu  contributions  to  the  science. 

8.  The  assertion  that  the  chief  contribution  to  mathematics  made 
by  the  Arab  scholars  was  to  the  science  of  trigonometry,  and  that 
this  contribution  was  important. 

9.  The  Arab  and  Persian  writers  on  trigonometry,  and  the  im- 
portant features  of  their  work. 

10.  Influence  of  Peurbach  and  Regiomontanus. 

1 1 .  The  change  of  trigonometry  from  being  essentially  geometric 
to  being  largely  analytic,  and  the  influence  of  this  change  upon  the 
later  development  of  the  science. 

12.  Development  of  the  concept  of  the  sine  of  an  angle,  and  the 
origin  of  the  name. 

13.  Development  of  the  concept  and  name  of  the  cosine. 

14.  Development   of   the  concepts  of   the  versed   sine  and  the 
coversed  sine,  with  reasons  for  their  gradual  disappearance. 

15.  The  favorite  functions  in  astronomy  and  those  used  in  prac- 
tical mensuration. 

1 6.  Development  of  the  tangent  and  cotangent. 

17.  Development  of  the  secant  and  cosecant,  and  the  causes  lead- 
ing to  their  gradual  disuse  in  modern  times. 

1 8.  The  relation  between  trigonometric  functions. 

19.  Development  of  the  leading  methods  of  computing  trigono- 
metric tables. 

20.  The  history  of  typical  and  important  theorems  of  trigonometry. 


CHAPTER  IX 


MEASURES 
i.  WEIGHT 

Measures  in  General.    The  subject  of  metrology  is  so  exten- 
sive that  it  is  impossible,  in  a  work  like  this,  to  do  more  than 

give  a  few  notes  relating  to 
the  measures  in  common  use. 
The  purpose  of  this  chapter, 
therefore,  is  simply  to  lay  be- 
fore the  student  some  of  the 
points  of  interest  in  the  his- 
tory of  the  most  familiar  of 
our  several  units  of  measure, 
to  suggest  the  significance  of 
the  names  of  these  units,  and 
to  indicate  some  of  the  works 
on  the  subject  to  which  he  may 
go  for  further  information.1 

Egypt-  The  use  of  the  bal- 
ance for  purposes  of  weigh- 
ing is  doubtless  prehistoric, 
for  weights  are  found  in  re- 
mains of  the  first  dynasty  of 

A  porphyry  weight  found  near  the  North       Egypt   (c.  34OO  B.C.)  .2     The 
Pyramid  at  Lisht  and  now  in  the  Metro- 
politan Museum.    The  inscription  reads, 
"Senusert,  giving  life  eternally,  70  gold 
debens."    It  was  used  for  weighing  gold 


EGYPTIAN  WEIGHT 


first  inscribed  weight  that 
has  been  found  is  of  the  4th 
dynasty,  the  time  of  the  Great 


1An  excellent  summary  of  the  history  is  given  under  "Weights  and  Measures" 
in  the  Encyc.  Brit.,  nth  ed.  The  measure  of  angles  and  arcs  has  been  already 
considered  in  Chapter  VIII. 

2  See  W.  M.  Flinders  Petrie,  Proceedings  of  the  Soc.  of  Biblical  Archceol.,  Lon- 
don, XXIII,  385.  See  also  Bulletin  of  the  Metrop.  Mus.  of  Art,  New  York,  XII,  85. 

6-u 


MEASURES  IN  GENERAL 


635 


Pyramid.1  The  earliest  Egyptian  scales  were  simple  balances, 
either  held  in  the  hand  or  supported  on  a  standard.  These  are 
frequently  illustrated  in  the  temple  wall  pictures.  The  steelyard 
with  its  sliding  weight 
and  fixed  fulcrum  was 
used  as  early  as  1350  B.C. 
The  Egyptian  weights 
of  which  the  names  and 
values  are  known  with 
certainty  were  the  deben 
(dbn,  formerly  read  uten, 
about  13.6  grams,  but 
commonly  taken  as  15 
grams )  and  the  kidet  ( kdt, 
kite,  o.i  of  a  deben).2 

Babylonia.  The  Baby- 
lonians used  a  cubic  foot 
of  rain  water  to  establish 
their  unit  of  weight,  the 
standard  talent.3  The 
chief  subdivision  of  the 
talent  was  called  a  maneh 
and  was  -£$  of  a  talent. 
They  also  had  a  unit  of 
weight  known  as  the  she, 
about  45  mg.  Our  knowl- 
edge of  the  Babylonian  measures  in  general  is  derived  from  a 
number  of  inscribed  tablets  such  as  the  one  here  shown. 

Hebrews.  The  Hebrew  standards  were  kept  in  the  temple, 
as  was  also  the  case  in  other  nations.  Thus  we  read4  of  "the 
shekel  of  the  Sanctuary,"  that  is,  the  standard  shekel,  about  a 
quarter  of  an  ounce  in  early  times,  or  a  half  ounce  after  the 

1  Proceedings  of  the  Soc.  of  Biblical  Archceol.,  XIV,  442. 

2Peet,  Rhind  Papyrus,  p.  26;  Encyc.  Brit.,  nth  ed.,  XXVIII,  480. 

3Mahaffy,  Greek  Life,  p.  67.  See  also  J.  Brandis,  Das  Miinz-  Mass-  und 
Gewichtswesen  in  Vorderasien,  pp.  i,  41  (Berlin,  1866)  ;  G.  A.  Barton,  Haver ford 
Library  Collection  of  Cuneiform  Tablets,  Philadelphia,  1905,  1909. 

4  Exodus,  xxx,  13. 


TABLE  OF  BABYLONIAN  MEASURES  OF 
CAPACITY  AND  WEIGHT 

A  fragment  of  a  clay  tablet  found  at  Nippur 
and  dating  from  c.  2200  B.C.  The  reverse  side 
contains  tables  of  weight,  length,  and  area. 
Courtesy  of  the  University  of  Pennsylvania 


636  WEIGHT 

time  of  the  Maccabees  (ist  century  B.C.).  This  shekel  of  the 
Hebrews  was  the  sicilicus  of  the  Romans.1  The  Hebrew  maneh 
was  100  shekels,  or  ^  of  a  talent.  The  shekel  was  also  used 
as  a  unit  of  capacity,  and  with  the  Babylonians  it  was  equiva- 
lent to  0.07  liter.2 

Greece.  The  Greek  unit  of  weight  in  Homer's  time  was  the 
talent,3  a  standard  that  varied  from  country  to  country.  The 
ancient  Greek  talent  weighed  about  57  pounds,  but  the  Hebrews 
used  the  term  for  a  unit  of  about  93!  pounds.  It  was  also  used 
as  a  unit  of  value,  generally  the  value  of  a  talent  of  silver,  this 
being  about  $1180  in  Greece  and  from  about  $1650  to  about 
Si 900  among  the  Hebrews,  according  to  present  standards. 

For  a  smaller  weight  the  Greeks  used  the  drachma,4  origi- 
nally "a  handful"  but  used  by  the  ancients  to  designate  both  a 
weight  and,  as  in  the  case  of  the  talent,  a  unit  of  value.  In 
modern  Greece  it  is  a  coin  identical  in  value  with  the  franc  at 
the  normal  rate  of  exchange. 

The  later  Greek  weights  may  be  thus  summarized:  i  talan- 
ton  =  6o  mnai  =  6000  drachmai  =  36,000  060/0^  =  288,000 
chalkoiJ1 

Rome.  The  Roman  unit  of  weight  was  the  pound.  This 
was  divided  into  twelfths  (unciae}?  The  usual  Roman  weights 
may  be  summarized  as  follows : 

i  libra  — 12  unciae~4&  sicilici  =  2&8  scripula  =  576  oboli 
=  1728  siliquae. 

The  ounce  was  about  1.09  oz.  avoirdupois,  or  412  grains. 

1Or  siclus',  Greek  <rty\os.  2  Barton,  loc.  cit.,  II,  18. 

3TdAai/rop  (tal'anton).  It  was  originally  smaller  than  the  later  talent.  See 
F.  Hultsch,  Griechische  und  romische  Metrologie,  p.  104  (Berlin,  1882)  ;  Harper's 
Diet.  Class.  Lit.;  Pauly-Wissowa ;  A.  Bockh,  Metrologische  Untersuchungen, 
Berlin,  1838. 

4Apax^  (drachme'}.  The  Lydian  drachma  of  the  7th  century  B.C.  was  \  of 
a  shekel. 

5  In  Greek,  T&Kavrov ;  yii»>a,  /ivcu;  S/oax/UT;,  5/oa^/aat ;  <5/3oX6s,  6(3o\ol ;  ^aX/cous,  ^aX/cot. 
The  chalkous  was  about  0.091  g.   As  a  measure  of  value  it  was  a  copper  coin 

worth  i  of  an  obol,  somewhat  less  than  \  of  an  American  cent.  The  talanton 
was  about  26, 196 g. 

6  Hultsch,  loc.  cit.,  p.  144;  Ramsay  and  Lanciani,  Manual  of  Roman  Antiqs., 
i7th  ed.,  p.  461  (London,  1901).   There  are  many  works  on  the  subject.   Among 


GREEK  AND  ROMAN  WEIGHTS  637 

The  Romans  had  a  table  known  as  the  mensa  ponderaria,  in 
the  stone  top  of  which  were  cavities  like  washbasins,  with  a 
plug  in  the  bottom  of  each  cavity.  These  were  standards  of 
capacity,  or  of  capacity  with  respect  to  weight.1 

Far  East.  In  India  and  other  parts  of  the  Far  East  the 
weights  and  currencies  were  commonly  based  upon  the  weights 
of  certain  seeds.  The  favorites  were  the  abrus  precatorius,  a 
creeper  having  a  small,  bright-red  seed  with  a  black  spot  on  it,2 
and  the  adenanthera  pavonina,  a  large  pod-bearing  tree  with  a 
bright-red  seed  which  is  conventionally  taken  as  weighing  twice 
an  abrus  seed.3 

England.  In  England  the  grain  was  originally  the  weight  of 
a  barleycorn,  a  barley  grain.  The  Latin  granum  has  the  same 
root  (gar)  as  our  word  "corn."4 

the  earlier  ones  are  A.  Alciatus,  Libellvs.  De  Ponderibvs  et  mensuris,  Copenhagen, 
1530;  L.  Portius,  De  sestertio  pecvniis  ponderibvs  et  mensvris  antiqvis  libri  duo, 
s.  1.  a.  (Venice?,  c.  1500),  with  editions  at  Florence  (1514?)  and  Basel  (1520 
and  1530) ;  G.  Budaeus,  De  asse,  et  partibvs  eivs,  libri  V,  Paris,  1514  (title  as  in 
Lyons  edition,  1551)  ;  G.  Agricola,  Libri  quinque  de  Mensuris  &  Ponderibus,  Paris, 
1533;  Venice,  1533  and  1535;  Basel,  1549  and  1550;  an  epitome,  Lyons,  1552; 
H.  Uranius,  De  re  nvmaria,  mensvris  et  ponderibus  Epitome  ex  Budaeo,  Portia, 
.  .  .,  Solingen,  1540;  M.  Neander,  STNO^IS  mensvrarvm  et  pondervm,  .  •  • 
Basel,  1555.  These  show  the  interest  taken  in  the  subject  in  the  i6th  century. 
Of  the  1 8th  century  works  one  of  the  best  is  J.  Arbuthnot,  Tables  of  Antient 
Coins,  Weights,  and  Measures,  2d.  ed.,  London,  1754. 

*A  Naples  specimen  is  illustrated  in  Mau's  Pompeii,  Kelsey's  second  edition, 
New  York,  1902. 

2  Of  ten  seen  for  sale  in  European  and  American  shops.  The  name  precatorius 
(from  precator,  one  who  prays)  comes  from  the  fact  that  certain  Buddhists  use 
these  as  beads  for  their  rosaries. 

3R.  C.  Temple,  "Notes  on  the  Development  of  Currency  in  the  Far  East," 
Indian  Antiquary,  1899,  P-  IO2-  Other  seeds  were  used,  as  is  shown  by  H.  T. 
Colebrooke,  "On  Indian  Weights  and  Measures,"  Asiatic  Researches,  V  (1799), 
91,  with  tables.  See  also  his  Lildvati,  p.  i,  §2. 

4 Whence  also  "garner,"  to  gather  grain;  "pomegranate,"  from  the  French 
pomme  (apple)  and  grenate  (seeded);  "granite,"  a  grained  or  spotted  stone; 
"garnet";  "grange";  and  the  Spanish  granada.  The  Scotch  statute  required  that 
the  inch  be  "iii  bear  cornys  gud  and  chosyn  but  tayllis"  (tailless).  The  Lathi 
statute  of  England  read :  "  Tria  grana  ordei  sicca  et  rotunda  f aciunt  pollicem." 
On  the  history  of  British  measures  in  general  see  F.  W.  Maitland,  Domesday  Book 
and  Beyond,  Cambridge,  1897,  P-  368;  J.  H.  Ramsay,  The  Foundations  of  Eng- 
land, London,  1898, 1,  533 ;  F.  Seebohm,  The  English  Village  Community,  4th  ed., 
p.  383  (London,  1896) ;  R.  Potts,  Elementary  Arithmetic,  London,  1886. 


638  WEIGHT 

The  word  "pound"  comes  from  the  Latin  pondo  (by 
weight),1  and  the  ounce,  as  already  stated,  from  the  Latin 
uncia,  a  twelfth  of  the  Roman  pound.2 

England  had  developed  a  system  of  weights  before  the  Troy 
weight  was  introduced  from  the  French  town  of  Troyes,  one  of 
the  many  places  in  which  fairs  were  held  in  the  Middle  Ages. 
This  introduction  seems  to  have  taken  place  as  early  as  the 
second  half  of  the  i3th  century,  for  Graf  ton's  Chronicles*  has 
this  to  say  of  the  matter : 

About  this  tyme4  was  made  the  statute  of  weightes  and  measures, 
that  is  to  say,  that  a  sterlyng  penny  should  waye  .xxxij.  graynes  of 
wheate  drie  and  round,  and  taken  in  the  middes  of  the  eare,5  and  .xx. 
of  those  pence  shoulde  make  an  ounce,  and  .xij.  ounces  make  a  pound 
Troy:  And  .viij.  pound  Troy  weight  make  a  gallon  of  Wine,  and 
.viij.  wyne  galons  to  make  a  London  bushell,  which  is  the  .viij.  part 
of  a  quarter.  Also  three  barly  Cornes  dry  and  round  should  make  an 
ynch,  &  .xij.  ynches  a  foote,  and  thre  foote  a  yard,  a  fiue  yards,  halfe 
a  perch,  or  poll,  &  ,xl.  pol  in  length  &  thre  in  bredth  an  acre  of  land. 

And  these  standardes  of  weight  and  measures  were  confirmed  in 
the  .xv.  yere  of  king  Edward  the  thirde,  and  also  in  the  tyme  of 
Henry  the  sixt  and  of  Edward  the  fourth,  and  lastly  confirmed  in  the 
last  yere  of  Henry  the  seuenth.  But  in  the  time  of  king  Henry  the 
sixt  it  was  ordeyned  that  the  same  ounce  should  be  deuided  into 
.xxx.  pence,  and  in  the  tyme  of  king  Edward  the  fourth,  into  .xl. 
pence,  and  in  the  tyme  of  king  Henry  the  eight  into  .xliiij.  pence: 
But  the  weight  of  the  ounce  Troy,  and  the  measure  of  the  foote  con- 
tinued alwayes  one. 

In  due  time  the  Troy  weight  was  replaced  by  the  avoirdupois 
for  general  purposes  and  was  thenceforth  limited  chiefly  to 

1From  pendere,  to  weigh.  From  the  same  root  we  have  such  words  as  "de- 
pend," "spend,"  and  "pendulum,"  and  the  French  poids  and  our  "poise." 

2  Lack  of  space  precludes  any  discussion  of  the  relation  of  the  apothecaries' 
weight  to  the  ancient  Greek  and  Roman  systems  and  symbols.   There  is  an  ex- 
tensive literature  on  the  subject. 

3  1569  ed.;  1809  reprint,  p.  277.       4"The  LIJ  Yere  of  Henry  III,"  i.e.,  1268. 
5So  Recorde  (0.1542)  says:  "Graine,  meaninge  a  erayn  of  corne  or  wheat 

drye,  and  gathered  out  of  the  myddle  of  the  eare."    Ground  of  Aries,  1558  ed., 
fol.L4. 


ENGLISH  WEIGHTS  639 

the  use  of  goldsmiths.1  These  goldsmiths  also  used  in  this  con- 
nection the  carat,  a  weight  consisting  of  12  grains.2  The  word 
had  a  variety  of  meanings,  being  commonly  used  to  express  the 
purity  of  gold,  "22  carats  fine"  meaning  an  alloy  that  is  f-f 
"fine  gold."  It  appears  in  various  forms,3  and  its  meaning  in 
this  sense  comes  from  the  fact  that  a  gold  mark  was  24  carats, 
so  that  a  mark  that  had  only  18  carats  of  gold  was  only  •£-£ 
pure.  So  Recorde  (c.  1542)  says :  "The  proofe  of  gold  is  made 
by  Caracts,  whereof  24  maketh  a  Marke  of  fine  gold:  the 
Caract  is  24  graines."4 

Avoirdupois  Weight.  The  word  "avoirdupois"  is  more  prop- 
erly spelled  "averdepois,"  and  it  so  appears  in  some  of  the 
early  books.  It  comes  from  the  Middle  English  aver  de  polzf 
meaning  "goods  of  weight."  In  the  i6th  century  it  was  com- 
monly called  "Haberdepoise,"  as  in  most  of  the  editions  of 
Recorders  (c.  1542)  Ground  of  Aries.  Thus  in  the  Mellis  edi- 
tion of  1594  we  have: 

At  London  &  so  all  England  through  are  vsed  two  kinds  of  waights 
and  measures,  as  the  Troy  waight  &  the  Haberdepoise. 

1So  the  Dutch  arithmetics  of  the  iyth  century  speak  of  it  as  Assay -gewicht. 
E.g.,  Coutereels's  Cyffer-Boeck,  1690  ed.,  p.  16.  The  Dutch  writers  also  called  it 
Trois  gewicht,  as  in  Bartjens's  arithmetic,  1676  ed.,  p.  155. 

2 So  the  Dutch  arithmetics  of  Petrus(  1567), Van  der  Schuere(i6oo),  and  others 
give  12  grains  —  i  karat,  24  karats  =  i  marck  (for  gold),  and  20  angels—  i 
ounce,  8  ounces  —  i  marck  (for  silver).  Trenchant  (1566)  says:  "Per  ansi  le 
marc  d'or  sans  tare  est  a  24  kar.  de  fin  aloy."  In  this  sense  it  comes  from  the 
Arabic  qtrdt,  a  weight  of  4  barleycorns;  but  the  Arabs  derived  it  from  the  Greek 
Kepdrtov  (kera'tion),  the  fruit  of  the  locust  tree,  L.  Latin  cerates.  Perhaps  the 
Arabic  use  is  responsible  for  the  carat  weight's  being  4  diamond  grains,  now  taken 
as  200  milligrams. 

3Italian  carato,  French  carat,  and  Spanish  quilate.  So  Texada  (1546):  "24. 
quilates  son  de  puro  oro";  Sfortunati  (1534) :  "lo  mi  trouo  oro  di  .24.  charatti"; 
Trenchant  (1566):  "18  karats  de  fin";  and  Rudolff  (1526):  "fein  18  karat." 

4  Compare  "4  marcx  d'Or  a  14  Carats  de  fin,"  in  Coutereels's  Dutch-French 
arithmetic,  1631  ed.,  p.  309. 

5  Aver  de  pois,  pels,  etc.  The  English  aver,  from  the  Old  French  aveir  or  avoir, 
meant  goods,  and  poiz  was  the  French  pets  or  pois,  Latin  pensum,  from  pendere, 
to  weigh.    About  the  year  1500  the  old  Norman  pels  was  superseded  by  the 
modern  pois.   The  incorrect  du,  for  de,  came  in  about  1650.   Even  as  late  as  1729 
the  American  Greenwood  used  "averdupois." 


640  LENGTH 

The  system  was  introduced  into  England  from  Bayonne 
c.  1300,  but  is  essentially  Spanish.  The  name  is  limited  to  the 
English-speaking  countries,  the  pound  of  16  ounces  being  called 
on  the  Continent  by  various  names,  such  as  the  pound  mer- 
chant.1 Troy  weight  was  the  more  popular  until  the  i6th 
century,  when,  as  Digges  (1572)  tells  us,  "Haberdepoyse" 
became  the  "more  vsuall  weight."  Even  a  century  later,  how- 
ever, the  Troy  weight  was  given  first  and  was  used  for  weigh- 
ing such  commodities  as  figs  and  tobacco  and  even  lead 
and  iron.2  There  was  also  the  Tower  pound  of  11.25  Troy 
ounces,  but  this  was  abandoned  about  the  year  1500.  In  the 
latter  part  of  the  i8th  century  a  popular  writer3  thus  refers 
to  the  matter : 

When  Averdupois  Weight  became  first  in  Use,  or  by  what  Law  it 
was  at  first  settled,  I  cannot  find  out  in  Statute  Books ;  but  on  the 
contrary,  I  find  that  there  should  be  but  one  Weight  (and  one 
Measure)  used  throughout  this  Realm,  viz.  that  of  Troy,  (Vide  14 
Ed.  Ill,  and  17  Ed.  III).  So  that  it  seems  (to  me)  to  be  first  intro- 
duced by  Chance,  and  settled  by  Custom,  viz.  from  giving  good  or 
large  weight  to  those  Commodities  usually  weighed  by  it,  which  are 
such  as  are  either  very  Coarse  and  Drossy,  or  very  subject  to  waste ; 
as  all  kinds  of  Grocery  Wares. 

2.  LENGTH 

Babylonia  and  Egypt.  The  Babylonian  measures,  like  those 
of  most  early  peoples,  were  derived  to  a  considerable  extent 
from  the  human  body.  For  example,  one  of  the  world's  primi- 
tive measures  was  the  cubit,4  the  length  of  the  ulna,  or  forearm, 
whence  the  English  ell  and  French  aune,  but  applied  to  various 
lengths.  This  standard  is  found  among  the  Babylonians,  the 

1  Thus  Trenchant  (1566)  says,  "  La  liure  marchande  vaut  16  onces."  The  Dutch 
writers  sometimes  called  it  "Holland  weight,"  as  in  Coutereels's  Cyffer-Boeck, 
1690  ed.,  p.  17,  where  i  pound  =  "2  Marck"  or  "16  once"  or  "32  loot." 

2Hodder's  arithmetic,  1672  ed.,  pp.  15,  66,  68. 

3J.  Ward,  The  Young  Mathematician's  Guide,  i2th  ed.,  p.  32.   London,  1771. 

4 Latin  cubitum,  elbow.  Sir  Charles  Warren,  The  Ancient  Cubit,  London, 
1003 ;  a  scholarly  and  extended  treatment  of  the  subject. 


ANCIENT  UNITS  641 

length  varying  from  525  mm.  to  530  mm.1  It  was  known  some- 
what earlier  in  Egypt  and  numerous  specimens  are  still  extant.2 

Greece  and  Rome.  The  method  of  fixing  standards  by  meas- 
urement of  the  human  body  naturally  led  to  many  variations. 
Thus  the  Attic  foot3  averaged  295.7  mm.;  the  Olympic,  320.5 
mm. ;  and  the  ^Eginetan,  330  mm.  A  similar  variation  is  found 
in  Western  Europe,  the  Italian  foot  being  275mm.;  the  Ro- 
man, 296  mm.  (substantially  the  same  as  the  Attic) ;  and  the 
pes  Drusianus,  333  mm.4  The  foot  was  not  a  common  measure 
until  c.  280  B.C.,  when  it  was  adopted  as  a  standard  in 
Pergamum. 

The  fingerbreadth5  was  used  by  both  Greeks  and  Romans, 
as  was  also  the  palm6  of  four  digits.  The  cubit  was  six  palms, 
or  twenty-four  digits,  the  Roman  foot  was  13^  digits,  and  the 
fathom7  was  the  length  of  the  extended  arms.  The  mile8  was, 
as  the  name  indicates,  a  thousand  units,  the  unit  being  a 
double  step. 

In  general,  the  most  common  Roman  measures  of  length  may 
be  summarized  as  follows:  the  pes  (foot)  was  0.296  m.  long, 
and  5  pedes  (feet)  made  i  passus\  125  passus  made  i  stadium, 
about  185  m. ;  and  8  stadia  made  i  mile,  about  1480  m.9 

XJ.  Brandis,  Das  Mtinz-  Mass-  und  Gewichtswesen  in  Vorderasien  (Berlin, 
1866),  p.  21 ;  Hilprecht,  Tablets,  p.  35.  The  Babylonian  name  was  ammatu,  and 
this  unit  was  divided  into  30  ubdnu  (ubdne) .  To  use  our  common  measures, 
we  may  say  that  the  average  Roman  cubit  was  174  inches;  the  Egyptian, 
20.64  inches;  and  the  Babylonian,  20.6  inches.  See  Peet,  Rhind  Papyrus,  p.  24. 

2  As  the  mahij  three  of  which  made  the  xylon,  the  usual  length  of  a  walking 
staff,  about  61.5  inches,  and  40  of  which  made  the  khet.  Other  measures  are  also 
known,  such  as  the  foot,  which  was  equivalent  to  about  12.4  of  our  inches. 

3noOs  (POMS).  The  general  average  as  given  by  Hultsch  (loc.  cit.,  p.  697)  is 
308  mm.  4K.  R.  Lepsius,  Langenmasse  der  Alien,  Berlin,  1884. 

5  MKTv\os(dak'tylos) ;  Latin,  digitus. 

6  AOXM^  (dochme'} ;  Latin,  palmus.   This  is  our  "hand,"  used  in  measuring  the 
height  of  a  horse's  shoulders.   Homer  speaks  of  handbreadths  (d&pov,  do'ron)  and 
cubits  (irvydv ,  pygon'} . 

7 Anglo-Saxon  fcethm,  embrace.  The  Greek  word  was  tipyvia  (or'guia},  the 
length  of  the  outstretched  arms;  Latin,  tensum,  stretched. 

8  Mills  passuum  (colloquially  passum).  The  pace  was  a  double  step,  and  hence 
a  little  over  5  Anglo-American  feet. 

9 The  Greek  stadium  (<rrd5iov,  sta'dion)  varied  considerably  in  different  cities. 
The  Athenian  stadium  was  about  603-610  Anglo-American  feet. 


642  LENGTH 

Far  East.  The  finger  appears  in  India  as  "  eight  breadths  of 
a  yava"  (barleycorn),  four  times  six  fingers  making  a  cubit,1 
as  in  Greece.  The  other  Oriental  units  have  less  immediate 
interest. 

England.  In  England  there  was  little  uniformity  in  standards 
before  the  Norman  Conquest.  The  smaller  units  were  deter- 
mined roughly  by  the  thumb,,2  span,3  cubit,  ell,4  foot,  and  pace. 

A  relic  of  this  primitive  method  is  seen  in  the  way  in  which 
a  woman  measures  cloth,  taking  eight  fingers  to  the  yard,  or 
the  distance  from  the  mouth  to  the  end  of  the  outstretched  arm. 

For  longer  distances  and  for  farm  areas  it  was  the  custom  to 
use  time-labor  units,  as  in  a  day's  journey  or  a  morning's  plow- 
ing, such  terms  being  still  in  use  in  various  parts  of  the  world. 
The  furlong  (40  rods,  or  an  eighth  of  a  mile)  probably  came 
from  the  Anglo-Saxon  furlang,  meaning  "  furrow  long." 

The  word  "yard"  is  from  the  Middle  English  yerd  and  the 
Anglo-Saxon  gyrd,  meaning  a  stick  or  a  rod,  whence  also  a 
yardarm  on  a  ship's  mast.  That  the  standard  was  fixed  in  Eng- 
land by  taking  the  length  of  the  arm  of  Henry  I  (1068-1135)  is 
not  improbable.  Thus  an  old  chronicle  relates:  "That  there 
might  be  no  Abuse  in  Measures,  he  ordained  a  Measure  made 
by  the  Length  of  his  own  Arm,  which  is  called  a  Yard." 

The  words  "rod"  and  "rood"  may  have  had  a  common  ori- 
gin. The  rod  was  used  for  linear  measure  and  the  rood  came  to 
be  used  for  a  fourth  of  an  acre.5 


et,  cara,  the  forearm.    Colebrooke's  Lildvati,  p.  2,  §35.    The  word  cubit 
appears  in  India  and  Siam  as  covid,  in  Arabia  as  covido,  and  in  Portugal  as  covado. 

2  Latin  pollex,  whence  the  French  ponce,  an  inch.    The   word  "inch,"  like 
"ounce,"  is  (as  already  stated)  from  the  Latin  uncia,  the  twelfth  of  a  foot  or  the 
twelfth  of  a  pound.    Originally  the  word  meant  a  small  weight  and  is  allied  to 
the  Greek  oyKos(on'kos),  bulk,  weight.    The  old  Scotch  inch  was  averaged  from 
the  thumbs  of  three  men,  "hat  is  to  say,  a  mekill  man  and  a  man  of  messurabill 
statur  and  of  a  lytill  man."    See  Maitland,  loc.  cit.,  p.  369. 

3  The  distance  spanned  by  the  open  hand,  from  thumb  to  little  finger;  finally 
taken  as  9  inches. 

4  The  ell  has  varied  greatly.    In  England  it  is  45  inches,  that  is,  ij  yards.   The 
old  Scotch  ell  was  37.2  inches,  and  the  Flemish  ell  was  27  inches. 

5  For  a  bibliography  of  the  subject  of  measures  of  length  consult  the  encyclo- 
pedias.  Among  the  most  ingenious  studies  of  the  subject  is  W.  M.  F.  Petrie, 
Inductive  Metrology,  London,  1877. 


MEASURES    OF    LENGTH 

European  of  the  i8th  century,  showing  the  general  appearance  of  the  common 
measuring  sticks  of  the  period.   The  three  shortest  pieces  are  ells 


644  CAPACITY 

3.  AREAS 

Acre.  The  common  unit  of  land  measure  known  as  the  acre1 
has  varied  greatly  in  different  countries  and  at  different 
periods.2  It  was  commonly  taken  to  mean  a  morning's  plow- 
ing,3 a  strip  of  land  4  rods  wide  and  one  furrow  long,  that  is, 
4  rods  by  i  furlong,  4  rods  by  40  rods,  or  160  square  rods. 

The  rood  was  a  fourth  of  an  acre  and  was  also  called  a 
perch.4  It  is  thus  described  by  Recorde  (c.  1542) : 

5  Yardes  and  a  halfe  make  a  Perche  .  .  .  [and]    i   Perche  in 
bredth  &  40  in  length,  do  make  a  Rodde  of  land,  which  some  cal  a 
Rood,  some  a  Yarde  lande,  and  some  a  Forthendale.5 

4.  CAPACITY 

Modern  Times.  The  subject  of  measures  of  capacity  is  so  ex- 
tensive as  to  make  it  impossible  to  mention  more  than  a  few 
facts  concerning  our  modern  British  and  American  units. 

The  gill  was  the  Old  French  gdle,  a  sort  of  wine  measure, 
from  the  Middle  Latin  gillo  or  gellus,  a  wine  vessel.6 

The  quart7  is,  of  course,  simply  the  quarter  of  a  gallon.8 

1  Anglo-Saxon  cecer.   The  ancient  units  of  area  have  no  particular  significance 
at  the  present  time.    It  is,  therefore,  sufficient,  merely  to  mention  the  Greek 
plethron,  about  0.235  of  an  acre,  and  the  Roman  jugerum^  about  0.623  of  an  acre. 

2  For  some  of  these  variations  see  F.  W.  Maitland,  Domesday  Book  and  Be- 
yond, p.  374  (Cambridge,  1897). 

3  The  cattle  used  in  plowing  in  the  morning  were  put  out  to  pasture  in  the 
afternoon.    Compare  the  German  Morgen. 

4  Latin  pertica,  a  pole,  staff,  or  rod.   The  word  has  various  other  uses,  as  in 
the  case  of  a  perch  of  stone  or  masonry,  the  contents  of  a  wall  18  inches  thick, 
i  foot  high,  and  i  rod  long,  or  24!  cubic  feet.    The  perch  as  a  unit  of  length  was 
the  same  as  the  rod.  5 1558  ed.,  fol.  L  6. 

6  It  may  come  from  the  same  root  as  "gallon."   The  United  States  gill  contains 
7.217  cu.  in.,  or  118.35  cu.  cm.   The  British  gill  contains  142  cu.  cm. 

7  French  quarte,  from  the  Latin  quartus,  fourth,  which  is  related  to  quattuor, 
four,  and  to  such  words  as  "quadrilateral"  (four-sided),  "quarry"   (a  place 
where  stones  are  squared),  "quarantine"  (originally  a  detention  of  forty  days), 
"quarto,"  "quire"  (Low  Latin  quaternum,  a  collection  of  four  leaves),  "square" 
(probably  Low  Latin  ex  (intensive)    -f-   quadrare,  to  square),  "squad,"  and 
M  squadron." 

8The  United  States  gallon  contains  231  cu.  in.,  like  the  old  English  wine  gallon. 
The  imperial  (British)  gallon  contains  277.274  cu.  in. 


VALUE 


645 


The  pint  may  possibly  receive  its  name  from  the  Spanish 
pinta?  a  mark,  referring  to  a  marked  part  of  a  larger  vessel.2 

The  word  "bushel"3  means  a  small  box,  but  the  origin  of  the 
word  "peck,"  as  applied  to  a  measure,  is  obscure. 

5.  VALUE 

Early  Units.  In  the  measure  of  value  it  became  necessary  at 
an  early  period  to  develop  media  of  exchange  of  one  kind  or 
another.  The  primitive  pas- 
toral people  naturally  used 
cattle  of  some  kind,  whence 

the     Latin     noun pecunia4 

(money)  and  the  English  ad- 
jective "pecuniary." 

For  media  of  exchange  the 
Greeks  often  used  copper  uten- 
sils, and  ingots  of  silver  and 
gold.  The  Babylonians  and 
Egyptians  also  made  use  of 
ingots  and  rings  of  the  precious 
metals,  selling  these  by  weight,  whence  came  the  aes  infectum* 
of  the  Latins.  From  this  relation  of  money  to  value  came  the 
double  use  of  such  measures  as  the  talent  and  the  pound.  Even 

1  Latin  picta,  marked  or  painted,  whence  "  picture."   The  Middle  English  form 
is  pynte.   The  origin  is,  however,  uncertain. 

2  Of  the  British  measures  whose  names  are  still  heard  in  the  colonies  and  in 
America,  kilderkin  was  the  Dutch  kindeken  (German  Kinderchen),  a  babekin, 
that  is,  a  mere  baby  in  bulk  as  compared  with  a  tun  or  vat.   "  Tun  "  and  "  ton  " 
are  the  same  word  (Middle  English  tonne,  Low  Latin  tunna),  meaning  a  large 
barrel  and  hence  also  a  great  weight. 

8  Middle  English  buschel  or  boischel;  Low  Latin  bussettus,  or  bustellus,  di- 
minutive related  to  Old  French  boiste,  a  box.  Compare  "pyx,"  Greek  irv%ls 
(pyxis') ,  a  box,  particularly  one  made  of  TT^OJ  (pyx'os,  Lat.  buxus],  boxwood. 
The  imperial  (British)  bushel  contains  2218.192  cu.  in.,  and  the  Winchester  bushel 
(which  became  the  legal  standard  in  the  United  States)  contains  2150.42  cu.  in. 

4  From  Latin  pecus,  sheep,  cattle.  For  discussion,  see  T.  Gomperz,  Les  pen- 
seurs  de  la  Grece,  French  translation,  p.  8  (Lausanne,  1904) ;  F.  Hultsch,  Grie- 
chische  und  romische  Metrologie,  p.  162  (Berlin,  1882). 

5 Aes,  bronze,  copper,  money;  infectum,  in  -f  facere,  to  make;  that  is,  uncoined 
money.  See  F.  Gnecchi,  Monete  Romane,  2d  ed.,  p.  86  (Milan,  1900). 


EARLY  ROMAN  MONEY   (PECUNIA) 

Showing  how  a  coin  was  stamped  to 
represent  the  value  of  an  ox,  4th  cen- 
tury B.C.    From  Breasted's  Survey  of 
the  Ancient  World 


646  VALUE 

at  present,  in  certain  mining  districts,  the  ounce  of  gold  is  com- 
monly spoken  of  as  a  unit  of  value.  The  pound1  became  the 
libra2  in  most  Latin  countries. 

From  the  aes  injectum  as  pieces  of  metal  came  the  large  and 
heavy  metal  disks  of  the  early  Romans.  When  these  were 
stamped  they  became  the  aes  signatum.3 

Coins.  The  earliest  stamped  coins  found  in  the  Mediterra- 
nean countries  were  probably  struck  in  Lydia  in  the  7th  century 
B.C.,4  or  possibly  in  ^gina  in  the  8th  century.  They  seem  to 
have  appeared  in  China  at  about  the  same  time. 

The  first  silver  money  coined  in  Rome  (268  B.C.)  was  based 
upon  the  relation  of  10  asses  to  the  denarius,  but  the  number 
was  changed  at  a  later  date.  Pliny  tells  us  that  the  first  gold 
money  coined  in  Rome  appeared  in  217  B.C."  The  aureus,  or 
gold  denarius,  was  first  coined  under  Augustus  (31  B.C.- 
14  A.D.)  as  ^V  °f  a  pound,  but  it  underwent  gradual  changes 
until,  under  Constantine  (306-337),  it  became  y1^  of  a  pound, 
then  taking  the  name  solidus? 

Great  Britain.  When  Caesar  went  to  Britain  (c.  55  B.C.)  he 
found  the  natives  using  certain  weights  of  metal  as  media 
of  exchange.  Coinage  was  introduced  soon  thereafter,  based 
on  Roman  values.  The  figure  of  Britannia,  which  is  still  seen 

1  Latin,  pondo  libra,  "a  pound  by  weight."    From  the  same  root  as  pondus, 
a  weight,  we  also  have  such  words  as  "ponder,"  and  such  units  as  the  Spanish 
peso-,  see  also  page  638,  note  i. 

2  Latin  libra,  a  balance,  a  pound  weight,  from  the  Greek  \irpa  (li'tra),  a 
pound,  whence  litre,  liter.    The  constellation  Libra  has  for  its  symbol    ^,  the 
scale  beam.    From  the  same  root  we  have  such  words  as  "deliberate,"  to  weigh 
our  thoughts.   The  libra  appears  in  French  as  livre  and  in  Italian  as  lira,  the  old 
pound  in  weight  as  well  as  in  value.   The  French  livre  was  also  called  a  franc,  as 
in  Trenchant  (1566) :  "la  liure  autremet  appellee  frdc" 

3Gnecchi,  loc.  cit.,  p.  89. 

4 Herodotus,  I,  94.  Judging  from  the  museum  pieces  the  early  coins  seem  to 
have  been  both  of  gold  and  of  silver,  and  both  circular  and  oblong.  See  an 
illustration  in  Volume  I,  page  56. 

5Gnecchi,  loc.  cit.,  chap,  xiv  and  p.  145. 

6  /.  e.,  a  solid  piece  of  money.  From  this  we  have  the  word  "  soldier,"  a  man  who 
fought  for  money,  and  such  words  as  the  Italian  soldo  and  the  French  sol  and 
sou.  The  English  symbol  for  shilling  (/)  comes  from  the  old  form  of  s  (f)  and 
was  the  initial  for  solidus,  just  as  £  is  the  initial  of  libra  (pound)  and  as  d  (for 
penny)  is  the  initial  of  denarius. 


BRITISH  COINS  647 

on  British  coins  as  mistress  of  the  seas;  is  not  at  all  modern. 
It  appears  on  one  of  the  pieces  of  Hadrian's  time  (c.  130). x 

The  most  ancient  coin  of  the  Saxon  period  (c.  600)  is  the 
sceat,  a  silver  coin  weighing  about  i  gram.  The  word  stilling, 
for  shilling,  appears  at  this  time,  the  word  possibly  meaning  a 
little  scar.2  The  origin  is  doubtful,  and  the  word  may  mean 
the  "clinking  coin."3 

The  word  "penny"  may  have  come  from  the  Latin  pannus,  a 
cloth,  and  hence  is  the  value  of  a  certain  piece ;  but,  as  in  many 
similar  cases,  the  origin  is  obscure.  Since  a  piece  of  cloth  was 
a  convenient  pledge  for  money  borrowed,  the  word  "pawn"  may 
have  come  from  the  same  source.4 

The  word  "farthing,"  the  Anglo-Saxon  feorthling,  is  simply 
the  "little  fourth"  of  a  penny.5 

The  Saxon  coins  were  regulated  by  the  pound  weight.  This 
pound  was  commonly  known  as  the  Cologne  pound,  having 
been  brought  from  that  city,  and  was  probably  the  same  as  the 
weight  known  after  the  Conquest  as  the  Tower  pound,  so  called 
because  the  mint  of  London  was  in  the  Tower.  A  pound  ster- 
ling was  this  weight  of  silver  coins. 

United  States.  The  word  "dollar"  comes  from  the  Low  Ger- 
man Daler,  German  Thaler,  from  Joachims  thaler,  since  these 
large  silver  pieces  first  appeared  in  the  Thalfi  of  St.  Joachim, 

1  For  a  brief  resume  of  the  history  of  British  coins,  see  R.  Potts,  Elementary 
Arithmetic  with  brief  notices  of  its  history,  §  II  (London,  1886). 

2  Skil  means  to  divide  and  comes  from  skal  or  skar,  to  cut ;  so  that  scar-ling 
means  a  little  cut  on  a  tally  stick,  to  distinguish  the  mark  from  the  larger  scar 
(score)   which  indicated  20  shillings,  or  a  pound.    Skar  is  the  root  of  Greek 
Kelpeiv  (kei'rein),  to  cut  close,  and  is  allied  to  the  Anglo-Saxon   sccer  and  the 
German  scheren,  whence  our  "shear."  The  English  "jeer"  may  be  from  the  same 
source  through  the  Dutch  phrase  den  gheck  sheer  en,  "to  shear  the  fool."   A 
"  share  "  of  stock  is  from  the  same  root. 

3  Compare  the  German  schellen,  to  sound  or  tinkle.    See  Greenough  and  Kit- 
tredge,  Words  and  their  Ways,  p.  140  (New  York,  1901). 

^Similarly  "panel,"  a  piece  of  anything  marked  off.  The  Middle  English  of 
"penny"  is  peni,  plural  penies  and  pens.  Compare  the  Anglo-Saxon  pening  and 
the  German  Pfennig. 

5 The  word  is  substantially  the  same  as  "firkin,"  from  the  Dutch  vier  (four) 
+  kin  (diminutive,  as  in  "lambkin"),  once  a  fourth  of  a  barrel. 

6 English  "dale"  and  "dell."  Coined  there  because  of  the  silver  mines  in  the 
valley. 


648  METRIC  SYSTEM 

Bohemia,  in  the  i6th  century.  "Dime"  is  from  the  French 
disme  and  Latin  decem  (ten),  "cent"  is  from  centum  (hun- 
dred), and  "mill"  is  from  mille  (thousand).  It  took  over  fifty 
years  to  replace  the  English  system  by  the  "Federal"  in  the 
United  States.  The  origin  of  the  symbol  $  is  uncertain.  It 
seems  to  have  first  appeared  in  print  in  Chauncey  Lee's  The 
American  Accomptant  (Lansingburgh,  1797)  (although  in  a 
form  very  different  from  the  one  familiar  at  the  present  time), 
but  it  was  used  in  manuscripts  before  that  date.  The  Italian 
and  British  merchants  had  long  used  Ib  for  pounds,  writing  it 
quite  like  our  dollar  sign  (see  Volume  I,  page  233),  and  it  is  pos- 
sible that  our  merchants  in  the  closing  years  of  the  i8th  century 
simply  adopted  this  symbol,  just  as  we  have  adopted  the  English 
word  "penny"  to  mean  a  cent,  which  is  only  a  halfpenny.1 

6.  METRIC  SYSTEM 

Need  for  the  System.  The  ancient  systems  of  measures  were 
open  to  two  serious  objections:  (i)  they  were  planned  on  a 
varying  scale  instead  of  the  scale  of  ten  by  which  the  civilized 
world  always  counts,  and  (2)  they  were  not  uniform  even  in 
any  single  country.  Before  the  metric  system  was  adopted 
there  were,  in  northern  France  alone,  eighteen  different  aunes,2 
and  in  the  entire  country  there  were  nearly  four  hundred  ways 
of  expressing  the  area  of  land. 

This  condition  was  not  unique  in  France ;  it  was  found  in  all 
European  countries.  Before  the  days  of  good  roads  and  easy 
communication  from  place  to  place  the  difference  in  standards 
was  not  very  troublesome,  but  by  the  end  of  the  i8th  century 
it  became  evident  that  some  uniformity  was  essential. 

Early  Attempts  at  Reform.  As  early  as  650  there  was  an  effort 
made  at  uniformity  in  France,  a  standard  of  measure  being 
kept  in  the  king's  palace.  Under  Charlemagne  (c.  800)  there 

1See  Kara  Arithmetica,  p.  470.  There  are  various  hypotheses  as  to  the  origin 
of  the  symbol  $,  most  of  them  obviously  fanciful. 

2  The  cloth  measure,  the  old  English  ell,  as  already  given.  On  the  metric  system 
as  a  whole,  see  the  excellent  historical  work  of  G.  Bigourdan,  Le  systeme  metrique, 
Paris,  1901. 


EARLY  ATTEMPTS  AT  REFORM        649 

was  nominal  uniformity  throughout  the  kingdom,  the  standards 
of  the  royal  court  being  reproduced  for  use  in  all  leading  cities. 
After  Charlemagne's  death,  however,  the  numerous  feudal  lords 
adopted  scales  to  suit  their  own  interests.  Attempts  were  made 
at  various  other  times,  as  in  864,  1307,  and  1558,  to  unify  the 
systems  in  France,  but  none  of  these  were  successful. 

Rise  of  the  Metric  System.  In  1670,  Gabriel  Mouton,  vicar  of 
the  church  of  St.  Paul,  at  Lyons,  proposed  a  system  which 
should  use  the  scale  of  10,  and  which  took  for  its  basal  length 
an  arc  i'  long  on  a  great  circle  of  the  earth.  This  unit  he 
called  a  milliare  or  mille,  o.ooi  of  a  mille  being  called  a  virga 
and  o.i  of  a  virga  being  called  a  virgula.  It  will  be  seen  that 
this  was,  in  general  plan,  not  unlike  the  metric  system. 

In  England,  Sir  Christopher  Wren  (c.  1670)  proposed  as  the 
linear  unit  the  length  of  a  pendulum  beating  half  seconds.  In 
France,  Picard  suggested  (1671)  the  length  of  a  pendulum  beat- 
ing seconds,  and  Huygens  (1673)  approved  of  this  unit. 

In  order  to  avoid  the  difficulty  involved  in  the  varying  length 
of  the  second  pendulum  in  different  latitudes*,  La  Condamine 
suggested  (1747)  the  use  of  a  pendulum  beating  seconds  at  the 
equator,  a  proposal  which  would,  if  adopted,  have  given  a 
standard  approximating  the  present  meter.  In  1775  Messier 
determined  with  great  care  the  length  of  the  second  pendulum 
for  45°  of  latitude,  and  an  effort  was  made  to  adopt  this  as  the 
linear  unit,  but  it  met  with  no  success. 

France  works  out  the  Metric  System.  In  1789  the  French 
Academic  des  Sciences  appointed  a  committee  to  work  out  a 
plan  for  a  new  system  of  measures,  and  the  following  year  Sir 
John  Miller  proposed  in  the  House  of  Commons  a  uniform 
system  for  Great  Britain.  About  the  same  time  Thomas 
Jefferson  proposed  to  adopt  a  new  system  in  the  United  States, 
taking  for  a  basal  unit  the  length  of  the  second  pendulum  at 
38°  of  latitude,  this  being  the  mean  for  this  country.  In  1790 
the  French  National  Assembly  took  part  in  the  movement,  and 
as  a  result  of  the  widespread  agitation  it  was  decided  to  pro- 
ceed at  once  with  the  project  of  unification.  The  second  pen- 


650  METRIC  SYSTEM 

dulum  was  given  up  and  an  arc  of  one  ten-millionth  of  a 
quarter  of  a  meridian  was  selected  as  the  basal  unit.  A  careful 
survey  was  made  of  the  length  of  the  meridian  from  Barcelona 
to  Dunkirk,  but  troubles  with  the  revolutionists  (1793)  delayed 
the  work.  The  committees  which  began  and  carried  on  the 
enterprise  were  changed  from  time  to  time,  but  they  included 
some  of  the  greatest  scientists  of  France,  such  as  Borda,  La- 
grange,  Lavoisier,  Tillet,  Condorcet,  Laplace,  Monge,  Cassini, 
Meusnier,  Coulomb,  Haiiy,  Brisson,  Vandermonde,  Legendre, 
Delambre,  Berthollet,  and  Mechain.  Owing  to  a  slight  error  in 
finding  the  latitude  of  Barcelona,  the  original  idea  of  the  unit 
was  not  carried  out,  but  a  standard  meter  was  fixed,  and  from 
this  copies  were  made  for  use  in  all  civilized  countries. 

The  system  was  merely  permissive  in  France  until  1840, 
when  it  was  made  the  only  legal  one.  The  expositions  held  in 
London  (1851)  and  Paris  (1855,  1867)  aided  greatly  in  mak- 
ing the  system  known  outside  of  France.  In  1919  the  system 
was  the  only  legal  one  in  thirty-four  countries,  with  a  popula- 
tion of  about  450,000,000;  was  optional  in  eleven  countries, 
with  a  population  of  about  730,000,000 ;  and  was  recognized 
by  twenty-six  countries,  with  a  population  of  about  690,000,000, 
this  recognition  taking  the  form  of  assisting  in  the  support  of 
the  International  Bureau  of  Weights  and  Measures  at  Paris. 

The  modern  plan  of  determining  such  standards  adopts  as 
the  unit  the  length  of  a  light  wave  of  a  defined  type.1 

1  Owing  to  the  importance  of  the  subject  and  the  necessity  for  condensing  the 
treatment  in  this  chapter,  the  following  bibliographical  references  are  added : 
W.  S.  B.  Woolhouse,  Measures,  Weights,  and  Moneys  of  all  Nations,  6th  ed., 
London,  1881 ;  F.  Hultsch,  Griechische  und  romische  Metrologie,  Berlin,  1882; 
A.  Bockh,  Metrologische  Untersuchungen,  Berlin,  1838;  F.  W.  Clarke,  Weights, 
Measures,  and  Money  of  all  Nations,  New  York,  1875;  William  Harkness,  "The 
Progress  of  Science  as  exemplified  in  the  Art  of  Weighing  and  Measuring,"  Bul- 
letin of  the  Philosophical  Society  of  Washington,  X,  p.  xxxix;  E.  Noel,  Science  of 
Metrology)  London,  1889;  C.  E.  Guillaume,  Les  recents  progres  du  systeme 
mitrique,  Paris,  1913.  See  also  the  various  Proces-verbaux  des  seances  of  the 
Comite  international  des  poids  et  mesures,  Paris,  1876  to  date;  W.  Cunningham, 
The  Growth  of  English  Industry  and  Commerce,  p.  118  (London,  1896);  A. 
De  Morgan,  Arithmetical  Books,  p.  5  (London,  1847) ;  Alberuni's  India,  trans- 
lated by  E.  C.  Sachau,  2  vols.,  London,  1910,  for  the  metrology  of  India  c.  1000 
(see  particularly  Volume  I,  chapter  xv). 


THE  COMPUTUS  651 

7.  TIME 

Problem  Stated.  Before  the  time  of  printed  calendars,  when 
astronomical  instruments  were  crude  affairs  and  the  astrono- 
mer was  merely  a  court  astrologer,  and  when  the  celebration 
on  a  movable  feast  day  of  the  anniversary  of  a  fixed  religious 
event  did  not  seem  to  arouse  adverse  criticism,  even  among  the 
enemies  of  the  various  faiths,  the  regulation  of  the  calendar1 
naturally  ranked  as  one  of  the  chief  problems  of  mathematics. 

The  Computus.  Accordingly  there  arose  in  all  ecclesiastical 
schools  of  any  standing  in  the  Christian  church  the  necessity 
for  instructing  some  member  or  group  of  the  priestly  order  in 
the  process  of  computing  the  dates  of  Easter  and  the  other  holy 
days.  For  this  purpose  there  were  prepared  short  treatises  on 
the  subject.  A  book  of  this  kind  was  generally  known  as  a  Com- 
putus Paschalis,  Computus  Ecclesiasticus,  or,  more  commonly, 
a  Computus  or  Compotus.2 

General  Nature  of  the  Computi.  Briefly  stated,  the  modern 
form  of  the  ancient  computus  begins  with  the  assertion  that 
Easter  day,  on  which  the  other  movable  feasts  of  the  Church 
depend,  is  the  first  Sunday  after  the  full  moon  which  happens 
upon  or  next  after  March  21.  If  the  full  moon  happens  upon  a 
Sunday,  Easter  day  is  the  Sunday  following.  The  full  moon 

1From  the  Latin  kalendarium,  a  list  of  interest  payments  due  upon  the  first 
days  of  the  months,  the  kalendae.  The  word  was  not  used  in  its  present  sense  by 
the  Romans.  They  used  fasti  to  indicate  a  list  of  days  in  which  the  holidays 
were  designated. 

Of  the  many  works  on  the  calendar,  one  of  the  latest  and  most  extensive  is 
F.  K.  Ginzel,  Handbuch  der  math,  und  techn.  Chronologic.  Das  Zeitrechnungs- 
wesen  der  V biker,  3  vols.,  Leipzig,  1906-1914.  See  also  J.  T.  Shotwell,  "The  Dis- 
covery of  Time,"  Journal  of  Philosophy,  Psychology,  and  Scientific  Methods, 
XII,  Nos.  8,  10,  12,  and  Records  of  Civilization,  Sources  and  Studies,  chapter  iv 
(New  York,  1922).  For  a  popular  essay  see  M.  B.  Cotsworth,  The  Evolution  of 
Calendars,  Washington,  1922. 

2Hieronymus  Vitalis  (Girolamo  Vitali),  Lexicon  Mathematicvm,  Paris,  1688 
(Rome  edition  of  1690,  p.  173),  thus  defines  computus'.  "Significatio  pressius 
accepta  est,  atque  antonomastice  haesit  annorum  ratiocinio,  &  temporum  distri- 
bution!, quod  proprie  Chronologos,  &  Astronomos  spectat."  The  spelling  com- 
potus  was  at  one  time  the  more  common, — possibly  a  kind  of  pun  upon  the 
convivial  habits  of  the  computers,  a  compotatio,  the  Greek  symposium 
sympos'ion},  meaning  a  "drinking  together." 


652  TIME 

is  taken  as  the  i4th  day  of  a  lunar  month,  "  reckoned  according 
to  an  ancient  ecclesiastical  computation  and  not  the  real  or 
astronomical  full  moon." 

In  order  to  use  this  ancient  computation  it  becomes  neces- 
sary to  be  able  to  find  the  Golden  Number  of  the  year.  This 
is  done  by  adding  i  to  the  number  of  the  year  and  dividing  by 
19,  the  remainder  being  the  number  sought.  If  the  remainder 
is  o,  the  Golden  Number  is  19.  Thus  the  Golden  Number  of 
1930  is  found  by  taking  the  remainder  of  (1930  +  i)  -*-  19, 
which  is  12. 

Taking  the  seven  letters  A,  B,  C,  D,  E,  F,  G,  the  letter  A 
belongs  to  January  i,  B  to  January  2,  and  so  on  to  G,  which 
belongs  to  January  7,  after  which  A  belongs  to  January  8,  and 
so  on.  If  January  2  is  Sunday,  the  Dominical  letter  of  the  year 
is  B.  By  means  of  the  Dominical  letter  it  is  possible  to  find  the 
day  of  the  week  of  any  given  date.  The  finding  of  this  letter 
depends  upon  a  few  simple  calculations  connected  with  tables 
given  in  the  computi.1 

Universality  of  the  Problem.  The  problem  was  not  confined 
to  the  Christian  church.  Since  most  early  religions  were  con- 
nected with  sun  worship  or  with  astrology,  work  somewhat 
similar  to  that  of  preparing  the  computus  was  needed  in  all 
religious  organizations.  Hence  we  find  a  problem  analogous 
to  the  Christian  one  in  the  routine  work  of  the  Hebrew, 
Mohammedan,  Brahman,  and  ancient  Roman  priests,  all  of 
whom  acted  as  guardians  of  the  calendar.2  We  shall  now  con- 
sider some  of  the  astronomical  difficulties  in  the  way  of  making 
a  scientific  calendar. 

1For  a  full  discussion  see  A.  De  Morgan,  "On  the  Ecclesiastical  Calendar," 
Companion  to  the  Almanac  for  1845,  p.  i  (London,  n.d.),  and  "On  the  Earliest 
Printed  Almanacs,"  Companion  to  the  Almanac  lor  1846,  p.  i. 

One  of  the  best  studies  of  the  computus  is  C.  Wordsworth,  The  Ancient 
Kalendar  of  the  University  of  Oxford,  Oxford,  1904.  This  represents  the  calendar 
as  it  stood  c.  1340. 

2  There  is  extant  an  Egyptian  papyrus  of  about  the  beginning  of  the  Christian 
Era  that  evidently  was  intended  to  serve  the  same  purpose  as  the  later  computi. 
See  W.  M.  Flinders  Petrie,  "The  geographical  papyrus  (an  almanack),"  in  Two 
Hieroglyphic  Papyri  from  Tanis,  published  by  the  Egypt  Exploration  Fund, 
London,  1889. 


UNITS  OF  MEASURE  653 

The  Day.  Of  the  chief  divisions  of  time  the  most  obvious  one 
is  the  day.  This  was,  therefore,  the  primitive  unit  in  the  meas- 
ure of  time  and  the  one  which  for  many  generations  must  have 
been  looked  upon  as  unvarying.  As  the  race  developed,  how- 
ever, various  kinds  of  day  were  distinguished.  First  from  the 
standpoint  of  invariability  is  the  sidereal  day,  the  length  of 
time  of  a  revolution  of  the  earth  as  shown  by  observations  on 
the  fixed  stars,  namely,  23  hours  56  minutes  4.09  seconds1  of 
our  common  time.  First  from  the  standpoint  of  the  casual  ob- 
server, however,  is  the  true  solar  day,  the  length  of  time  be- 
tween one  passage  of  the  sun's  center  across  the  meridian  and 
the  next  passage.  This  varies  with  the  season,  the  difference 
between  the  longest  and  shortest  days  being  51  seconds;  but 
for  common  purposes  the  solar  day  sufficed  for  thousands  of 
years,  the  sundial  being  the  means  by  which  it  was  most  fre- 
quently measured.  As  clocks  became  perfected  a  third  kind 
of  day  came  into  use,  the  artificial  mean  solar  day,  the  average 
of  the  variable  solar  days  of  the  year,  equal  to  24  hours 
3  minutes  56.56  seconds  of  sidereal  time.  In  addition  to  these 
general  and  obvious  kinds  of  day,  writers  on  chronology  dis- 
tinguish others  which  do  not  concern  the  present  discussion.2 

The  day  began  with  the  Babylonians  at  sunrise;  with  the 
Athenians,  Jews,  and  various  other  ancient  peoples,  and  with 
certain  Christian  sects,  at  sunset ;  with  the  Umbrians,  at  noon ; 
and  with  the  Roman  and  Egyptian  priests,  at  midnight.3 

The  Month.  The  next  obvious  division  of  time  was  the  month, 
originally  the  length  of  time  from  one  new  moon  to  the  next, 
and  one  that  served  as  the  greater  unit  for  many  thousands  of 
years.  As  science  developed,  however,  it  became  apparent,  as 
in  the  case  of  the  day,  that  there  are  several  kinds  of  months. 
There  is  the  sidereal  month,  the  time  required  for  a  passage  of 
the  moon  about  the  earth  as  observed  with  reference  to  the 

1  All  such  figures  are  approximations,  varying  slightly  with  different  authorities. 

2 See,  for  example,  A.  Drechsler,  Kalenderbuchlein,  p.  19  (Leipzig,  1881). 

8Thus  Pliny:  "Ipsum  diem  alii  aliter  observavere.  Babylonii  inter  duos  solis 
exortus;  Athenienses  inter  duos  occasus;  Umbri  a  meridie  ad  meridiem;  .  .  . 
Sacerdotes  Romani,  et  qui  diem  difftniere  civilem,  item  Aegyptii  et  Hipparchus, 
a  media  nocte  in  mediam"  (Hist.  Nat.,  II,  cap.  79). 


654  TIME 

fixed  stars,  namely,  2  7  days  7  hours  43  minutes  11.5  seconds. 
There  is  also  the  synodical  month,  from  one  conjunction  of  the 
sun  and  moon  to  the  next  one,  averaging  2  9  days  12  hours 
44  minutes  3  seconds,  or  2  days  5  hours  o  minutes  51.5  seconds 
more  than  the  sidereal  month.  This  is  the  month  of  those  who 
use  a  lunar  calendar,  and  is  the  basis  of  the  artificial  month, 
twelve  of  which  make  our  common  year.1 

The  Year.  Less  obvious  than  the  day  or  the  month  was  the 
year,  a  period  observable  only  about  one  three  hundred  sixty- 
fifth  as  often  as  the  day  and  about  one  twelfth  as  often  as  the 
month.  It  took  the  world  a  long  time  to  fix  the  length  of  the 
year  with  any  degree  of  accuracy,  and  the  attempt  to  har- 
monize time-reckoning  by  days,  months,  and  years  has  given 
rise  to  as  many  different  calendars  as  there  have  been  leading 
races.  First  of  all  there  has  to  be  considered  what  constitutes 
a  year.  The  sidereal  year  is  the  period  of  revolution  of  the 
earth  about  the  sun,  namely,  365  days  6  hours  9  minutes  9.5 
seconds  (365.256358  days).  The  tropical  year  is  the  period  of 
apparent  revolution  of  the  sun  about  the  earth  from  the  instant 
of  one  vernal  equinox  to  the  next,  and  would  be  the  same  as  the 
sidereal  year  if  it  were  not  for  the  slight  precession  of  the 
equinox,  amounting  to  about  50"  a  year.  This  precession 
makes  the  length  of  the  tropical  year  365  days  5  hours  48  min- 
utes 46.43  seconds  (365.242204  days).2  There  is  also  the 
anomalistic  year  of  365  days  6  hours  14  minutes  23  seconds, 
measured  from  the  time  when  the  earth  is  nearest  the  sun  to 
the  next  time3  that  they  are  in  the  same  relative  position, — a 
year  that  is  slightly  longer  than  the  sidereal.  There  is  also  the 
lunar  year  of  twelve  synodical  months,  probably  the  first  one 
recognized  by  the  primitive  observers  of  nature,  and  in  addi- 
tion to  this  there  are  various  other  periods  which  have  gone  by 
the  same  general  name.4 

1  For  other  types  of  month  see  Drechsler,  lac.  cit.,  p.  24. 

2  This  was  the  length  in  the  year  1800.    It  varies  about  0.59  seconds  a  century. 
B.  Peter,  Kalenderkunde,  2d  ed.,  p.  20  (Leipzig,  1901). 

3 From  perihelion  to  perihelion. 

4  For  the  list,  see  Drechsler,  loc.  cit.,  p.  26.  On  the  cycle,  see  ibid.,  p.  30;  on 
the  era,  p.  44. 


EARLY  CALENDARS  655 

The  Week.  The  week  was  less  obvious  than  the  day,  the 
month,  or  even  the  year,  having  no  astronomical  events  by 
which  to  mark  its  limits.  It  seems  very  likely  that  it  arose 
from  the  need  for  a  longer  period  than  the  day  and  a  shorter 
period  than  the  month.  Hence  we  have  the  half  month,  known 
as  the  fortnight  (fourteen  nights),  and  the  quarter  month, 
known  as  the  week. 

Early  Attempts  in  making  a  Calendar.  Of  the  various  at- 
tempts to  perfect  a  calendar  only  a  few  will  be  mentioned,  and 
in  general  those  that  had  some  bearing  upon  the  Christian 
system.1 

The  Babylonians,  whose  relation  to  the  invading  Chaldeans 
was  such  as  to  make  their  later  calendars  substantially  identi- 
cal, seem  to  have  been  the  first  of  the  world's  noteworthy 
astronomers.  Aristotle  relates  that  before  2200  B.C.  they  at- 
tempted scientific  observations  of  the  heavens,  and  Porphyrius 
(c.  275)  tells  us  that  Callisthenes  (c.  330  B.C.)  took  to  the 
Greek  sage  the  results  of  a  series  of  Chaldean  observations  ex- 
tending over  1903  years.  The  Chaldeans  knew  the  length  of 
the  year  as  365  days  6  hours  n  minutes,  but  used  both  the 
lunar  month  and  lunar  year  for  civil  purposes.  They  divided 
both  the  natural  day  and  the  natural  night  into  twelve  hours 
each,  and  in  quite  early  times  the  sundial  and  water  clock  were 
known,  the  latter  for  use  at  night.  For  astronomical  purposes 
the  clay  was  divided  into  twenty- four  equal  hours.  They  prob- 
ably very  early  used  a  fourth  of  a  month  as  a  convenient 
division  of  time,  or  rather  the  half  of  the  half,  as  was  the 
customary  way  of  thinking  in  the  ancient  world. 

The  Chinese  Calendar.  We  are  ignorant  of  the  nature  of  the 
primitive  Chinese  calendars.2  Certainly  earlier  than  2000  B.C. 

*A  good  treatment  of  the  subject  is  given  by  L.  Ideler,  Handbuch  der  math, 
und  techn.  Chronologic,  2  vols.,  1825;  2d  ed.,  Breslau,  1883,  a  facsimile  of  the 
first  edition.  Good  resumes  are  given  by  Drechsler,  loc.  tit.,  p.  56;  Peter,  loc,  tit., 
p.  5  (chiefly  on  the  Christian  calendar).  From  all  these  sources  information  has 
freely  been  taken.  On  the  general  question  of  chronology  see  also  J.  B.  Biot, 
"Resume  de  Chronologie  Astronomique,"  Mimoires  del' Academic  des  Sciences, 
XXII,  209-476  (Paris).  See  especially  Ginzers  work  mentioned  on  page  651  n, 

2Mikami,  China,  pp.  5,  45. 


656  TIME 

the  subject  occupied  the  attention  of  the  astrologers.  It  is, 
however,  very  difficult  to  unravel  a  system  which  changed  with 
each  emperor,  and  only  a  few  general  principles  can  be  set 
forth.  Under  the  emperor  Yau  (c.  2357-^.  2258  B.C.)  an 
effort  was  made  to  establish  a  scientific  calendar  for  the  whole 
country,  and  possibly  this  was  done  even  earlier,  under  the 
emperor  Huang-ti  (c.  2700  B.C.).1  In  accordance  with  a  decree 
of  Wu-wang  (1122  B.C.)2  the  day  seems  to  have  begun  with 
midnight,  although  before  this  time,  under  the  Shang  dynasty 
(1766-1122  B.C.),  it  began  at  noon.  The  civil  day  has  twelve 
hours,  and  the  middle  of  the  first  hour  is  midnight.3  Each  hour 
is  divided  into  eight  parts  (khe),  each  being  our  quarter,  and 
each  of  these  into  fifteen  fen,  each  therefore  being  our  minute. 
In  modern  times  the  jen  has  been  divided  into  sixty  miao 
(seconds)  under  European  influence.  At  present  also  the 
American  clock  is  becoming  common  in  China,  so  that  the 
ancient  system  seems  destined  soon  to  pass  away.  Neverthe- 
less the  connection  between  the  old  Chinese  calendar  and  that 
of  Europe  seems  apparent.  The  Chinese  days  were  named  in 
such  a  way  as  to  give  seven-day  periods  corresponding  to  our 
weeks,  and  the  month  began,  as  was  so  often  the  case  in  early 
times,  with  the  new  moon.  The  twelve  lunar  months  were 
supplemented  in  such  a  way  as  to  harmonize  the  lunar  and 
solar  years,  the  Chang  Cycle4  being  used  by  the  Chinese  before 
the  Greeks  knew  of  the  same  system  under  the  name  of  the 
Metonic  Cycle. 

The  Egyptian  Calendar.  In  the  ancient  Egyptian  calendar, 
which  influenced  all  the  Mediterranean  countries  beginning 
with  Crete  and  the  mainland  of  Greece,  the  business  day  in- 
cluded the  night,  the  natural  day  and  night  being  each  divided 
into  twelve  hours,  these  hours  varying  in  length  with  the 

xOr  Hoang-ti.  His  reign  seems  to  have  begun  in  the  year  2704  B.C.  See 
Volume  I,  page  24;  Mikami,  China,  p.  2. 

2 Founder  of  the  Chau  (Cheu,  Choi)   dynasty,  1122-225  B.C. 

3These  hours  (shi)  are  each  120  European  minutes  in  length. 

4  For  details  of  the  complex  system,  and  for  the  influence  of  China  on  Japan, 
see  Drechsler,  loc.  cit.,  pp.  71,  88. 


CHINESE  AND  EGYPTIAN  CALENDARS  657 

season.  The  civil  day  seems  to  have  commonly  begun  at  sun- 
set, although  the  priests  are  said  by  Pliny  to  have  begun  theirs 
at  midnight.  In  later  times,  for  astronomical  purposes,  the  day 
began  at  noon1  and  was  divided  into  twenty-four  equal  hours. 
The  Romans  considered  the  planets  as  ruling  one  hour  of  each 
day,  in  the  following  order,  beginning  with  the  first  hour  of 
Saturday :  Saturn,  Jupiter,  Mars,  Sun,  Venus,  Mercury,  Moon, 
the  sun  and  moon  being  placed  among  the  "wanderers.7'2  From 
these  planets  they  named  the  days  by  the  following  plan: 
Taking  Saturn  for  the  first  hour  of  Saturday  and  counting  the 
hours  forward,  it  will  be  seen  that  the  second  hour  is  ruled  by 
Jupiter,  and  so  on  to  the  twenty-fourth,  which  is  ruled  by 
Mars.  Then  the  next  hour,  the  first  of  Sunday,  is  ruled  by  the 
Sun,  the  first  hour  of  the  next  day  by  the  Moon,  and  so  on. 
Thus  the  days  of  the  week  were  named  by  the  ruling  planets  of 
their  first  hours,  and  we  have  Saturn's  day,  Sun's  day,  Moon's 
day,  Mars's  day,3  Mercury's  day,4  Jupiter's  day,5  Venus's  day,6 
a  system  that  has  come  down  to  our  time  and  seems  destined  to 
continue  indefinitely,  in  spite  of  the  fact  that  we  are  using  names 
of  heathen  gods  in  the  calendars  of  various  religions. 

Each  month  except  the  last  (Mesori)  in  the  native  Egyptian 
calendar  contained  thirty  days,  five  days  being  added  to  Mesori 
so  as  to  make  the  year  one  of  three  hundred  sixty-five  days. 
Since  this  gave  an  error  of  about  one  fourth  of  a  clay,  the  year 
was  a  changing  one,  coming  back  to  its  original  position  with 
respect  to  the  heavenly  bodies  once  in  4  x  365  common  years, 
or  1460  years  (1461  Egyptian  years).  The  year  began  with 
the  first  day  of  the  month  of  Thoth,  the  god  who,  according  to 
Plato's  Phcedrus,  introduced  the  calendar  and  numbers  into 
Egypt.  As  early  as  the  i4th  century  B.C.  the  Egyptians  recog- 
nized the  value  of  a  fixed  year,  but  the  changing  one  was  too 
strongly  implanted  in  the  religious  canons  of  the  people  to  be 
given  up.  The  fixed  year  was  used  to  the  extent  of  a  division 

xSo  with  Ptolemy  the  astronomer,  c.  150. 

2 "Planet"  is  from  the  Greek  TrXa^TTjs  ( plane' 't e s) ,  originally  a  wanderer. 

3  French,  Mardi.  4  French,  Mercredi. 

5  In  the  Northern  lands,  Thor's  day. 

6 In  the  Northern  lands,  Frigg's  day,  Frigg  being  the  goddess  of  marriage. 


658  TIME 

into  three  seasons,  regulated  by  the  river, — the  Water  Season, 
the  Garden  Season,  and  the  Fruit  Season,1 — these  being  easily 
determined  by  the  temple  observers.  From  the  temple,  too, 
came  the  announcements  of  the  turn  in  the  rise  or  fall  of  the 
river,  the  nilometers  being  under  the  observation  of  the  priests. 
These  early  nilometers  may  be  seen  in  the  temples  today ;  they 
were  concealed  from  the  observation  of  the  common  people,  the 
water  being  admitted  by  subterranean  channels. 

The  Alexandrian  Calendar.  After  Egypt  became  a  Roman 
province  (c.  30  B.C.)  the  Alexandrian  calendar,  including  the 
fixed  year,  was  introduced,  although  the  varying  year  of  the 
ancients  remained  in  popular  use  until  the  4th  century.  The 
Alexandrian  system  was  used  until  the  first  half  of  the  yth 
century,  when  the  country  yielded  (638)  to  the  Mohammedan 
conqueror,  with  an  attendant  change  of  the  calendar  except  in 
Upper  Egypt,  where  the  Coptic,  Abyssinian,  or  Ethiopian 
churches  maintained  their  supremacy.  Since  1798,  when  the 
French  obtained  brief  control  of  the  country,  the  European 
system  has  been  used  side  by  side  with  the  Mohammedan. 

The  Athenian  Calendar.  The  Athenian  calendar  followed  the 
Egyptian  in  beginning  the  new  day  at  sunset  and  in  dividing 
both  day  and  night  into  twelve  hours.  The  seven-day  week 
was  not  used,  however,  the  lunar  month  being  divided  into 
three  parts.  Of  these  the  first  consisted  of  ten  days,  numbered 
in  order,  the  "5th  day  of  the  beginning  of  the  month"  being 
the  fifth.  Then  followed  nine  days,  numbered  as  before,  but 
with  the  designation  "over  ten."  From  there  to  the  end  of  the 
month  the  numbers  were  20,  i  over  20,  and  so  on,  these  days 
also  being  numbered  backwards  from  the  end  of  the  month. 
In  the  popular  calendar  the  month  began  with  the  new  moon, 
and  twelve  of  these  months  made  three  hundred  fifty-four 
days,  requiring  the  insertion  of  a  new  month2  every  three 
years.3  Meton  (432  B.C.)  constructed  a  nineteen-year  cycle 

1  Namely,  June  21  to  October  20;  October  21  to  February  20;  February  21  to 
June  20.  2  A  second  month  of  Poseideon,  known  as  Poseideon  II. 

8  The  detailed  variations  of  this  plan  need  not  be  considered  here.  See  any 
work  on  the  calendar. 


GREEK  AND  ROMAN  CALENDARS  659 

in  which  the  third,  fifth,  eighth,  eleventh,  thirteenth,  sixteenth, 
and  nineteenth  years  should  contain  the  extra  month,1  a  plan 
which  Callippus,  a  century  later  (325  B.C.),  modified  to  include 
four  nineteen-year  cycles.2  Still  later  Hipparchus  (150  B.C.) 
suggested  the  use  of  four  of  the  cycles  of  Callippus,  less  a  day, 
or  110,036  days  in  all,  but  neither  of  the  last  two  calendars 
came  into  popular  use. 

Roman  Calendar.  The  oldest  of  the  Roman  calendars  seems 
to  have  been  the  one  attributed  to  Romulus.  The  year  prob- 
ably consisted  of  ten  months  of  varying  length,  or  of  304  days, 
beginning  with  March.  Numa  Pompilius  (715-672  B.C.)  is 
said  to  have  added  two  other  months,  January  and  February, 
and  his  year  was  probably  lunar.  The  Decemvirs  (sth  century 
B.C.)  decreed  a  solar  year,  the  regulation  of  which  was  left  in 
the  hands  of  the  priests.  The  calendar  was  so  mismanaged, 
however,  that  by  the  time  of  Julius  Caesar  each  day  was  eighty 
days  out  of  its  astronomical  place,  and  radical  measures  were 
necessary  for  its  reform.  Caesar  therefore  decreed  that  the  year 
46  B.C.  should  have  four  hundred  forty-five  days3  and  that 
thereafter  the  year  should  consist  of  three  hundred  sixty-five 
days,  with  a  leap  year  every  fourth  year.4 

Names  of  the  Months.  Following  the  older  custom  as  to  the 
beginning  of  the  year  in  March,  Caesar  at  first  used  the  follow- 
ing plan  for  the  calendar : 

1.  Martius,  31  days.          7.  Septembris,  30  days. 

2.  Aprilis,  30  days.          8.  Octobris,  31  days. 

3.  Maius,  31  days.          9.  Novembris,  30  days. 

4.  Junius,  30  days.  10.  Decembris,  31  days. 

5.  Quintilis,  31  days.  n.  Januarius,  31  days. 

6.  Sextilis,  31  days.  12.  Februarius,  2  8  days. 

1 19  years  =  235  months  —  6939  J  days.  The  months  as  arranged,  however,  con- 
tained 6940  §  days. 

2  4  x  19  years  =  76  years  =  940  months.  The  months  were  29  or  30  days  and 
totaled  37>759  days.  3 Hence  called  annus  confusionis. 

4  A  calendar  of  c.  100  B.C.  was  recently  found  at  Anzio,  in  the  Campagna,  based 
upon  a  lunar  year  of  353  days  with  an  intercalary  month  of  27  days  on  alter- 
nate years. 


660  TIME 

This  accounts  for  our  names  "September"  (;th  month), 
" October,"  " November,"  and  " December."  On  his  original 
plan  every  alternate  month,  beginning  with  March,  had  thirty- 
one  days,  the  others  having  thirty  days,  except  that  February  re- 
ceived its  thirtieth  day  only  once  in  four  years.  Caesar  later 
decreed  that  the  year  should  begin  with  January,  and  finally, 
but  during  his  life,  the  name  of  Quintilis,  the  month  in  which 
he  was  born,  was  changed  to  Julius.  He  also  changed  the  num- 
ber of  days  in  certain  months,  and  the  result  appears  in  our 
present  calendar.  After  his  death,  in  the  second  year  of  his 
calendar,  a  further  confusion  arose,  apparently  through  a  mis- 
understanding on  the  part  of  the  priests  as  to  the  proper  date  for 
leap  year.  This  was  corrected  by  Augustus,  and  in  his  honor 
the  name  of  Sextilis  was  changed  to  bear  his  name.  From  that 
time  on  the  Julian  calendar  remained  in  use  until  its  reformation 
under  Gregory  XIII  in  1582,  and  it  was  used  by  the  Greek 
Catholics,  including  the  Russians,  until  the  World  War  of 
1914-1918,  the  dates  until  that  time  differing  by  thirteen  days 
from  those  of  the  calendar  of  Western  Europe. 

Christian  Calendar.  The  indebtedness  of  the  present  Euro- 
pean calendar  to  those  already  described  is  apparent,  and  it  is 
also  evident  that  our  calendar  has  had  an  extensive  history. 
The  beginning  of  the  year,  for  example,  has  not  been  uniform 
from  time  to  time  and  in  different  countries.  In  the  early  cen- 
turies the  year  usually  began  with  April  in  the  East1  and  with 
March  in  the  West,  although  sometimes  with  the  Feast  of  the 
Conception,  Christmas  day,2  Easter,  or  Ascension  day,  or  at 
other  times  according  to  the  fancy  of  the  popes.  Finally 
Innocent  XII  again  decreed  that  the  year  should  begin  on 

1  Although  the  Byzantine  calendar  began  with  September  i. 

2  As  in  Spain  until  the  i6th  century  and  in  Germany  from  the  nth  century. 
March  i  and  March  25  (the  Annunciation)  were  favorite  dates,  although  Advent 
Sunday  (the  fourth  Sunday  before  Christmas)  has  generally  been  recognized  as 
the  beginning  of  the  ecclesiastical  year.    March  i  was  used  generally  in  medieval 
France,  in  Oriental  Christendom,  and  (until  1707)  in  Venice.   March  25  Was  used 
by  the  medieval  Pisans  and  Florentines.    Most  of  the  Italian  states  adopted 
January  i  in  1750.   For  further  details  consult  Ginzel  (see  page  651  n.)  or  such 
works  as  A.  Drechsler,  Kalenderbuchlein,  p.  77  (Leipzig,  1881). 


THE  CHRISTIAN  CALENDAR  66 1 

January  i,  beginning  with  1691,  as  Philip  II  had  done  for  the 
Netherlands  in  I57S,1  and  as  Julius  Caesar  had  done  before 
the  Christian  Era. 

Numbering  the  Years.  Following  the  Roman  custom,  the 
years  in  the  early  centuries  of  Christianity  were  dated  from 
the  accession  of  the  emperor  or  consul.  We  have  a  relic  of  this 
in  the  dating  of  acts  of  parliament  in  England  and  of  presi- 
dential proclamations  in  America.2  It  was  not  until  the  abbot 
Dionysius  Exiguus3  (533)  arranged  the  Christian  calendar  that 
the  supposed4  date  of  the  birth  of  Christ  was  generally  taken 
for  the  beginning  of  our  era,  Christmas  day  being  therefore 
appropriately  selected  as  the  first  day  of  the  year  i .  This  cal- 
endar was  adopted  in  Rome  in  the  6th  century,5  in  various 
other  Christian  countries  in  the  yth  century,  and  generally 
throughout  Europe  in  the  8th  century.0 

Changes  in  Easter.  Not  only  the  beginning  of  the  year  but 
the  determination  of  Easter  has  been  the  subject  of  much 
change.  We  now  consider  Easter  as  the  first  Sunday  after  the 
first  full  moon  following  the  vernal  equinox,7  as  decreed  in  325 
by  the  Council  of  Nice  (Nicaea).  Formerly  it  fell  on  the  date 

1  England  adopted  January  i  in  1752. 

2 As  "in  the  i5oth  year  of  our  independence." 

3 Dionysius  the  Little.    He  went  to  Rome  c.  500  and  died  there  in  540. 

4  He  considered  the  birth  of  Christ  as  taking  place  in  the  year  754  of  the 
founding  of  Rome,  although  early  Christians  placed  it  in  the  year  750. 

5  There  are,  however,  no  extant  inscriptions  of  the  6th  century  which  bear 
dates  in  the  Christian  Era.  See  M.  Armellini,  Archeologia  Cristiana,  p.  479  (Rome, 
1898).    Sporadic  efforts  had  been  made  before  the  6th  century  to  use  a  Christian 
calendar.    The  oldest  known  specimen  of  such  a  calendar  dates  from  354.   See 
B.  Peter,  Kalenderkunde,  26.  ed.,  p.  4  (Leipzig,  1901). 

(J  The  exceptions  were  the  Spanish  peninsula  and  Southern  France.  Charle- 
magne was  the  first  great  ruler  to  use  (783)  the  Dionysian  calendar. 

7  In  Rome.  It  is  possible  to  have  a  difference  of  a  week  between  this  Sunday 
in  Rome  and  in  (say)  Honolulu,  the  full  moon  occurring  on  Sunday  in  Rome 
when  it  is  still  Saturday  in  some  places  to  the  west.  This  has  occasionally  occurred 
as  an  astronomical  fact  although  not  as  an  ecclesiastical  one.  It  should  be  under- 
stood that,  for  Church  purposes,  March  21  is  taken  as  the  date  of  the  vernal 
equinox,  and  that  the  full  moon  is  not  determined  by  modern  astronomy  but  by 
certain  rules  as  laid  down,  say,  in  the  Book  of  Common  Prayer.  Easter,  there- 
fore, now  varies  from  March  22  to  April  25.  For  a  good  r£sum6  of  the  Easter 
problem  see  Peter,  loc.  cit.,  p.  58. 


662  TIME 

of  the  Jewish  Passover,  but  in  order  to  avoid  this  coincidence 
the  Church  readjusted  its  calendar.  Justinian,  with  this  in 
view,  decreed  (547)  that  Easter  should  be  21  days  (instead  of 
14  days)  after  the  first  new  moon  after  March  7.  In  this  way, 
in  general,  the  Passover  and  Easter  do  not  come  together, 
although  occasionally  they  synchronize.1 

The  Gregorian  Calendar.  The  present  calendar  of  Western 
Europe  and  the  Americas,  the  so-called  Gregorian  calendar, 
was  necessitated  by  the  fact  that  the  year  is  not  365^  days 
long,  as  recognized  by  Caesar,  but  is  about  n  minutes  14  sec- 
onds shorter  than  this.  Therefore  once  in  128  years  the  Julian 
calendar  receded  one  day  from  the  astronomical  norm,  and  by 
the  close  of  the  i4th  century  the  departure  of  Easter  day  from 
its  traditional  position  became  so  noticeable  that  it  was  the 
subject  of  much  comment.  It  was  not  until  Gregory  XIII, 
however,  consulting  with  such  scientists  as  Aloysius  Lilius2  and 
Christopher  Clavius  (c.  1575),  determined  on  a  reform,  that 
anything  was  really  accomplished.  He  decreed  that  October  4, 
1582,  should  be  called  October  15,  and  that  from  the  total 
number  of  leap  years  there  should  be  dropped  three  in  every 
four  centuries.  In  particular  he  decreed  that  only  such  cen- 
tennial years  as  are  divisible  by  400  (1600,  2000,  2400,  etc.) 
should  be  leap  years.  This  requires  no  further  adjusting  of  the 
calendar  for  over  3000  years.  Italy,  Spain,  Portugal,  Poland, 
France,  and  a  part  of  the  Netherlands  adopted  this  calendar  in 
1582.  In  1583  it  was  recognized  in  part  of  Germany,  the  old 
style  being  also  used  until  1 700.  Part  of  Switzerland  adopted 
it  in  1584,  and  the  other  part,  together  with  Denmark  and 
the  rest  of  the  Netherlands,  in  1700.  It  was  also  adopted  in 
Hungary  in  1587,  in  Prussia  in  1610,  in  England  in  1752,  and 
in  Sweden  in  1753.  So  fixed  had  the  Julian  calendar  become  in 
the  minds  of  the  people,  however,  that  even  as  late  as  the  open- 
ing of  the  igth  century  O.  S.  (old  style)  and  N.  S.  (new  style) 

!As  in  1805,  1825,  1828,  1832,  and  early  in  the  present  century,  on  April  12, 
1903.  Among  the  many  suggestions  for  Easter  is  that  of  Jean  Bernoulli  that  it 
should  be  the  first  Sunday  after  March  21,  without  reference  to  the  moon. 

2  Ludovico  Lilio,  Luigi  Lilio  Ghiraldi  (1510-1576). 


GREGORIAN  AND  FRENCH  CALENDARS  663 

were  used  in  dating  letters  in  America,  while  writers  on  arith- 
metic felt  it  necessary  to  include  a  description  of  the  Julian  and 
Gregorian  calendars  as  late  as  the  close  of  that  century.  The 
changes  brought  about  by  the  World  War  of  1914-1918  led  to 
a  more  general  adoption  of  the  Gregorian  calendar  in  the  few 
countries  which  had  continued  to  use  the  Julian  or  other  types. 

The  Calendar  in  the  French  Revolution.  In  the  early  days  of 
the  French  Revolution  an  attempt  was  made  to  impose  a  new 
calendar  upon  the  country,  partly  as  a  protest  against  the 
Christian  church.  It  was  hoped  that  this  reform,  like  that 
which  resulted  in  the  metric  system,  would  receive  international 
recognition.  The  new  era  was  to  begin  with  the  autumnal 
equinox  which  occurred  on  September  22,  1792.  There  were 
twelve  months  of  thirty  days  each,  and  these  months  were 
divided  into  decades  in  which  the  days  were  named  numeri- 
cally,— Primidi,  Duodi,  and  so  on.  The  extra  five  or  six  days 
of  the  year  were  grouped  at  the  end  as  holidays.  The  months 
were  named  according  to  natural  conditions,  thus :  In  autumn, 
Vendemiaire  (vintage),  September  22-October  21;  Brumaire 
(fog),  October  22-November  20;  Frimaire  (sleet),  November 
2i-December  20.  In  winter,  Nivose  (snow),  December  21- 
January  19;  Pluviose  (rain),  January  2O-February  18;  Ven- 
tose  (wind),  February  ig-March  20.  In  spring,  Germinal 
(seed),  March  2i-April  19;  Floreal  (blossom),  April  20- 
May  19;  Prairial  (pasture),  May  2o-June  18.  In  summer, 
Messidor  (harvest),  June  ig-July  18;  Fervidor  or  Thermi- 
dor  (heat),  July  19- August  17;  Fructidor  (fruit),  August  18- 
September  16.  As  might  have  been  known,  the  scheme  failed, 
and  on  August  30,  1805,  a  decree  was  signed  reestablishing  the 
Gregorian  calendar,  beginning  January  i,  I8O6.1 

Other  Calendars.  The  other  calendars  are  of  no  special  in- 
terest in  the  history  of  mathematics.  The  Hindus  began  their 
year  with  the  day  of  the  first  new  moon  after  the  vernal 
equinox.  The  Jews  begin  their  day  at  sunset,  their  week  on 
Saturday  night  (i.e.,  when  their  holy  day  ends  and  Sunday  be- 

1  This  decree  may  still  be  seen  in  the  Musee  des  Archives  Nationales,  in  Paris. 


664  TIME 

gins),  and  their  year  with  Tishri  i.1  Their  calendar,  more 
lunar  than  ours,  is  quite  complicated. 

The  Maya  civilization2  had  a  curious  system,  the  year  begin- 
ning with  the  winter  solstice  and  being  divided  into  eighteen 
months,  entirely  independent  of  astronomical  considerations. 
Scholars  have  recently  asserted  that  their  calendar  goes  back  as 
far  as  the  34th  century  B.C. 

The  Mohammedans  begin  their  day  with  sunset,  and,  like 
many  other  Eastern  peoples,  divide  both  daytime  and  night- 
time into  twelve  hours,  the  length  of  the  hour  varying  with 
the  season.  The  week  begins  on  Sunday,  and  Friday  is  the  day 
of  rest.  Their  month  begins  with  the  new  moon,  and  the  year 
is  purely  lunar,  of  354  or  355  days.  The  year  1343  A.H.  began 
on  August  2 ,  1924,  of  the  Christian  calendar.  The  era  began  with 
the  Hejira,  the  flight  of  Mohammed  from  Mecca  on  July  15  or 
1 6,  622.  On  account  of  frequent  references  to  the  Mohammedan 
calendar  in  literature,  it  may  be  added  that  a  simple  rule,  ac- 
curate enough  for  practical  purposes,  for  translating  a  year  of 
the  Hejira  into  a  Christian  year  is  as  follows :  To  97  per  cent  of 
the  number  of  the  year  add  622  ;  the  result  is  the  Christian  year. 
Thus  1326  A.H.  =  97  per  cent  x  1326  -I-  622  =  1908  A.D. 

Early  Christian  Computi.  The  first  noteworthy  Christian 
work3  on  the  calendar  was  that  of  Victorius  of  Aquitania 
(457).  About  a  century  later  a  second  Computus  Paschalis 
appeared,  probably  written  by  Cassiodorus  (562).  In  the  next 
century  the  question  of  Easter  had  become  so  complicated  as 
to  cause  (664)  a  dispute  between  the  church  in  England  and 
the  authorities  in  Rome.  The  best  of  the  early  works  on  the 

*!.€..,  the  first  new  moon  after  the  autumnal  equinox.  Their  year  formerly 
began  with  Nisan,  their  seventh  month,  thus  using,  like  that  of  the  Hindus,  the 
vernal  equinox.  In  1908  Tishri  i  was  September  26  of  the  Christian  calendar.  See 
S.  B.  Burnaby,  Elements  of  the  Jewish  and  Muhammedan  Calendars,  London, 
1901. 

2S.  G.  Morley,  An  Introduction  to  the  Study  of  the  Maya  Hieroglyphs, 
Washington,  1915;  C.  P.  Bowditch,  The  Numeration,  Calendar  Systems,  and 
Astronomical  Knowledge  of  the  Mayas,  Cambridge,  Massachusetts,  1910. 

3  On  this  work  and  the  works  of  later  scholars  on  the  same  subject  see  B. 
Lefebvre,  Notes  d'Histoire  des  Mathematiques,  p.  39  (Louvain,  1920). 


CHRISTIAN  COMPUTI  665 

computus  is  the  one  written  by  Bede1  in  the  8th  century.  This 
contains  a  precise  statement  as  to  the  method  of  finding  the 
date  of  Easter  in  any  year. 

In  the  pth  century  both  Hrabanus  Maurus  (c.  820)  and 
Alcuin  (c.  775)  wrote  upon  the  problem,  and  Charlemagne 
thought  the  subject  so  important  that  he  urged  that  it  be  con- 
sidered in  every  monastery.2 

Medieval  Works.  Lectures  were  held  upon  the  subject  in  the 
i3th  and  i4th  centuries  in  the  various  European  universities. 
Sacrobosco  (c.  1250)  wrote  a  work  on  it,3  such  a  practical  cal- 
culator as  Paolo  Dagomari  (c.  1340)  did  not  hesitate  to  do  the 
same,  and  even  the  Jewish  scholars  contributed  treatises  on  the 
Christian  calendar  as  well  as  their  own.4  It  is  to  a  commentary 
by  Andalo  di  Negro  (c.  1300)  on  a  work  by  Jacob  ben  Machir 
(d.  1307)  that  we  owe  the  first  prominent  use  in  Europe  of  the 
Arabic  word  almanac*  later  brought  into  general  use  by  such 
writers  as  Peurbach  (c.  1460)  and  Regiomontanus  (c.  1470). 

Printed  Works.  The  first  printed  computus  was  that  of  Ani- 
anus.0  In  this  work  there  appears  the  original  of  the  familiar 
rime  beginning  "Thirty  days  hath  September."7 

^De  temporum  ratione. 

2D.  C.  Munro,  Selections  from  the  Laws  of  Charles  the  Great,  p.  15  (Phila- 
delphia, 1900),  "Admonitio  generalis,"  789.  See  also  T.  Ziegler,  Geschichte  der 
Padagogik,  p.  28  (Munich,  1895)  J  Giinther,  Math.  Unterrichts,  p.  66. 

3Libellus  de  anni  ratione,  sen  ut  vocatur  vulgo  computus  ecclesiasticus. 

4 There  is  a  MS.  now  in  Petrograd,  written  by  Jechiel  ben  Josef  (1302),  under 
the  title  Injan  Sod  ha-Ibbur,  with  a  chapter  on  the  Christian  computus.  See 
M.  Steinschneider,  "Die  Mathematik  bei  den  Juden,"  Bibl.  Math.,  XI  (2),  16, 
38,  74;  XII  (2),  5,33. 

r>Heb.  *ptt?tf,  from  the  Arab,  al-mandkh.  The  word  is  not  pure  Arabic,  how- 
ever, and  the  real  origin  is  unknown.  See  Boncompagni's  Bullettino,  IX,  595 ; 
Giinther,  Math.  Unterrichts,  p.  190  n. 

^Cdputus  manualis  magri  aniani.  metricus  cu?meto  (Strasburg,  1488).  There 
is  said  to  have  been  an  edition  printed  at  Rome  in  1486.  For  bibliography  of 
Anianus  see  C.  Wordsworth,  The  Ancient  Kalendar  of  the  University  of  Oxford, 
p.  113  (Oxford,  1904).  See  also  the  facsimile  on  page  668. 

7  The  Latin  form  as  given  by  Anianus  is  as  follows : 

Junius  aprils  September  et  ipse  nouember 

Dant  triginta  dies  reliquis  su^padditur  vnus, 

De  quorum  numero  februarius  excipiatur. — Fol.  B  8 

See  Kara  Arithmetica,  p.  33. 


666 


TIME 


This  was  not  original  with  Anianus,  however,  for  it  is  found 
in  various  medieval  manuscripts.1  It  first  appeared  in  English 
verse  in  1590. 


w -n*fV8 


urt«^poC« 

A  COMPUTUS  OF  1393 
In  this  MS.  there  appears,  in  Italian,  the  verse  "Thirty  days  hath  September'' 

lE.g.,  in  the  above  anonymous  Italian  MS.  of  1393,  beginning: 

"Trenta  di  a  nouembre  a^ile  giugno  &  settembre." 
The  MS.  is  in  Mr.  Plimpton's  library.   See  Kara  Arithmetica,  p.  443. 


FROM  A  COMPUTUS  OF  1476 

This  page  shows  the  usual  verses  beginning  "  Sunt  aries,  thaurus,  gemini,  cancer." 
The  work  also  contains  the  verses,  in  Latin,  beginning  "Thirty  days  hath  Septem- 
ber," which  are  found  as  early  as  the  isth  century.  From  Mr.  Plimpton's  library 


liber  cjol  £ompota0  i  ntafoitwtyw  cum  ffeorfa 
nibu0  necdferiitmrofo  fate  locte 


^je  mta  eft  iufto  WataHfta$fta  ver 

tapofluntD4pUdter  confiderari  .  prtmopfit 
Did  oc  oeo  cj  eft  lujt  vera.  ioeo  Dicebat  Dauid* 
latottaeftiuflo.tetDeillaiuce  Diaf  Joba.f* 
j£mr  liijc  vera  3  iliumiat  omne  bomine  venic/ 
^(nbunc  mmtdu^0ecudopom{  Deraenna. 
Ct  Mcff  Itit  (IttalT  famtc  reddcte  luddiLqma  fade  bomtite  fcie/ 
tern  eflc  loddu^u  qUibus  vcrbfe  ad  comendatf  one  frtc  DUO  me 
uftcr  tangiitur*pr(mo  cti  tangif  fdcticalritudo  pzcdofa  j>  boc 
quod  Dint  lu)c.  ©ecundo  largf  tudo  gloziofa  per  boc  <p  Dtat  oz/ 

fie  Diccnas/^iffia^l  fbne  indefidena  J>on<tatfe  via.fai  fol/ 
uatoae  cognirto,  Ba:(6e  fic.illud  ell  vabdl  ec  pzedofum  quod 
De  inoaltdo  i  imperfeoo  fade  validum  i  pfectum*fadia  e  botufi 
modi.ergo  zc*  matoz  til  matuftlla.  mi  ttoz  Declaranir  p  p5m  rcr/ 
do  DC  aia  lie  tnccnte:  Sia  in  pndpio  fue  crcationio  e  taiu^  tabu 
la  rafa  m  qua  mba  Depicta  eft.Depmgibrtfe  tn  fdeafe  i  virairt/ 
bus.-p  zimamjg?pbaX^uctontace  boeni  z  ratoe  in  pzologo  arff 
memce/0aentiire  co?t  que  vera  funt  ^  impmutabtto  cflentfe  no 
riraqscompbenfionewitarts.Batioe  ficnlludeltranqp  fiimfi 
bonu  quod  babet  larsirtone  ^oriofam  fcia  e  brnot.  ergo  tc*mi. 
loz  dl  vcra*mmo2  ^)bawr  p  Diffimcione  fcietie  q  tali'8  c353^ 
<jda  babitus  aieronalie  no  innatus  fed  accifitueoanbuana?  re 
riimlfndagatnjc  *  toouebumanevitegubernam'jc/^  fcfofitba/ 
bitos  pt5.q:  fcia  eft  afiquod  e]riftcn0  in  aia.ftd  omne  tllud  quod 
et  m  aia  aut  c  bCtiie  aut  poterta  aut  paflio^tbcc  teltaE  Srifto 
tn  feciido  erbicou.^  fcia  no  fie  pa?1io  pt5  qz  palTionea  funt  in  va 
luntate  fcia  no  I  bm5i.ergo  tc.cp  n5  fit  potetia  ptj.qz  6l!b5  po/ 
ten  1a  fit  a  natura.  ficotirafcibite^  cocupifdbilfe.z  fi'c  retfnquf 
wrno  f(totm«afitbi0ae 


*    M 

THE   COMPUTUS   OF  ANIANUS  (l488> 

First  page  of  the  edition  of  c.  1495.    The  work  of  Anianus  was  the  first  one 
printed  on  the  computus.   See  page  665 


PRIMITIVE  TIMEPIECES  669 

Among  the  prominent  computi  printed  in  the  i6th  century  is 
that  of  Arnaldo  de  Villa  Nova  (c.  I275).1  Many  of  the  early 
arithmetics  also  gave  a  brief  treatment  of  the  computus.2 

The  computus  finally  found  a  place  in  various  liturgical 
works,  and  at  present  can  be  conveniently  studied  in  pages  pre- 
fixed to  most  editions  of  the  Book  of  Common  Prayer. 

Early  Timepieces.  One  of  the  general  problems  in  connection 
with  the  calendar  has  to  do  with  the  finding  of  the  hours  of  the 
day.  For  this  purpose  the  shadow  cast  by  some  obstruction  to 
the  sun's  rays  was  probably  used  by  all  primitive  peoples.  At 
first  it  is  probable  that  a  prominent  tree,  a  rock,  or  a  hill  was 
selected,  but  in  due  time  an  artificial  gnomon3  was  erected  and 
lines  were  drawn  on  the  earth  to  mark  off  the  shadows.  Since 
the  hour  shadow  is  longer  when  the  sun  is  near  the  horizon, 
either  concave  surfaces  or  curve  lines  on  a  plane  were  placed 
at  the  foot  of  the  gnomon. 

Hours.  The  ancients  usually  had  twelve  hours  in  the  day  and 
the  same  number  in  the  night.  There  have  been  various  specu- 
lations as  to  why  twelve  was  selected  for  this  purpose,  among 
them  being  one  which  referred  the  custom  to  the  Babylonian 
knowledge  of  the  inscribed  hexagon.4  It  is  probable,  however, 
that  twelve  was  used  in  measuring  time  for  the  same  reason 
that  it  was  used  for  measuring  length  and  weight, — because 
the  common  fractional  parts  (halves,  thirds,  and  fourths)  were 
easily  obtained.  The  day  hours  were  longer  than  the  night 
hours  in  the  summer  and  shorter  in  the  winter,  a  fact  referred 
to  by  several  ancient  writers.5 

1  Computus  Ecclesiasticus  &  Astronomicus,  Venice,  1501.  There  was  another 
work  by  this  name  printed  at  Venice  in  1519. 

2£.g.,  the  Treviso  arithmetic  (1478),  fol.  57.  KobePs  Rechenbuchlin  (1531 
ed.)  devotes  ten  pages  to  the  subject. 

3Herodotus  uses  this  term  yvAnuv  (gno'mon},  and  it  is  common  in  Greek 
literature.  Later  writers  sometimes  called  it  a  horologe  (wpo\6yi.ov,  horolog'ion} 
when  used  for  the  sundial  specifically.  In  still  later  times  it  was  called  the  pole 
(7r6Xoy,  pol'os} . 

4  G.  H.  Martini,  Abhandlung  von  den  Sonnenuhren  der  Alien,  p.  18  (Leipzig, 

1777). 

5 So  Vitruvius:  "Brumalis  horae  brevitates";  St.  Augustine:  "Hora  brumalis 
aestivae  horae  comparata,  minor  est." 


670  TIME 

Although  Herodotus  (II,  109)  speaks  of  the  "12  parts7'"  of 
the  day  among  the  Babylonians  and  the  Greeks,  the  word 
"hour"1  was  not  used  either  by  him  or  by  Plato  or  Aristotle. 
It  was  apparently  a  later  idea  to  give  these  divisions  a  special 
name. 

Early  Dials.  The  sundial  seems  to  have  been  used  first  in 
Egypt,  but  it  is  found  also  at  an  early  date  in  Babylonia. 
Herodotus  (II,  109)  says  that  it  was  introduced  from  Babylon 
into  Greece,  and  tradition  says  that  this  was  done  by  Anaxi- 
mander  (c.  575  B.C.),  the  gnomon  being  placed  at  the  center  of 
three  concentric  circles.  The  early  Egyptian  dial  has  already 
been  mentioned  in  Volume  I,  page  50.  The  first  concave  dial 
to  be  used  in  Greece  is  said  to  have  been  erected  on  the  island 
of  Cos  by  Berosus.2  Several  such  dials  have  been  found  in  the 
Roman  remains,3  and  the  early  ones  have  no  numerals  on  the 
hour  lines,  these  lines  being  easily  distinguished  without  such 
aids.4 

Besides  the  plane  dials  and  the  concave  spherical  dials  there 
were  both  concave  and  convex  cylindric  forms.  Vitruvius  (c.  20 
B.C.)  tells  us  that  one  Dionysodorus 5  invented  the  cylindric 


t"ftpa(ho'ra) .  There  may  be  some  relation  between  the  word  and  the  name 
of  the  Egyptian  Horus,  god  of  the  rising  sun,  and  the  Hebrew  or  (light) . 

2 A  priest  of  Belus  at  Babylon.  The  name  was  probably  Bar  (Ber)  Oseas  or 
Barosus,  that  is,  son  of  Oseas.  Fl.  0.250  B.C.  Vitruvius  (IX,  4;  X,  7,  9)  says 
that  Berosus  went  to  Cos  in  his  later  years,  founded  a  school  of  astrology,  and 
invented  what  seems  to  have  been  a  hemispherical  sundial.  For  a  general  descrip- 
tion of  early  dials  see  G.  H.  Martini,  loc.  cit.,  pp.  24,  70.  On  Anaximander's  con- 
nection with  the  gnomon,  as  recorded  by  Diogenes  Laertius,  Favorinus,  and 
Herodotus,  see  W.  A.  Heidel,  "Anaximander's  Book,  the  earliest  known  geograph- 
ical treatise, "Proceedings  of  the  American  Academy  of  Arts  and  Sciences,  LVI,  239. 

3  The  first  one  was  discovered  (1741)  in  a  Tusculan  villa.    For  early  descrip- 
tions see  G.  L.  Zuzzeri,  D'una  antica  villa  scoperta  sul  dosso  del  Tuscttlo,  e  d'un 
antico  orologio  a  sole,  Venice,  1746.    Boscovich  also  described  it  in  the  Giornale 
de'  Letterati  per  .  .  .  1746,  art.  14.   The  second  one  was  found  (1751)  at  Castel- 
nuovo,  near  Rome;  the  third,  also  near  Rome;  and  the  fourth  (1764),  at  Pompeii, 
although  apparently  made  in  Egypt. 

4  Thus  Persius  (Satires,  III,  4)  :  "...  quinta  dum  linea  tangitur  umbra," 
the  shadow  resting  on  the  fifth  line  of  the  day,  an  hour  before  noon. 

5  Also  spelled  Dionysiodorus.   He  is  said  by  Pliny  to  have  found  the  radius  of 
the  earth  to  be  c.  5000  miles,  but  nothing  is  known  of  his  life.   He  lived  c.  50  B.C. 


THE  SUNDIAL  671 

form  of  dial,  but  we  do  not  know  whether  it  was  convex  or 
concave.  We  also  know  from  Vitruvius  that  there  were  various 
other  forms  in  use  by  the  Romans.1 

Difficulties  with  the  Gnomon.  One  difficulty  that  was  expe- 
rienced with  the  large  sundials  of  the  ancients  was  that  the 
gnomon  did  not  cast  a  distinct  shadow.  The  size  of  the  sun  is 
such  as  to  have  the  shadow  terminate  in  a  penumbra  which 
rendered  the  determination  of  the  solstice,  for  example,  a  diffi- 
cult matter.  This  is  one  reason  why  it  is  not  probable  that 
the  Egyptian  obelisks  were  used  by  scientific  observers  as 
gnomons.  To  overcome  the  difficulty  the  Greeks  often  used  a 
column  with  a  sphere  on  top,  the  center  of  the  sphere  corre- 
sponding to  the  center  of  the  shadow  and  the  center  of  the  sun. 
Such  gnomons  are  found  on  medals  of  the  time  of  Philip  of 
Macedon,  and  it  is  possible  that  this  is  the  explanation  of 
the  column  on  the  coin  of  Pythagoras  shown  on  page  70 
of  Volume  I.  Dials  of  this  type  were  introduced  in  Rome 
by  Menelaus  (c.  100),  or  at  least  were  improved  by  him.2 

As  would  naturally  be  expected,  there  were  many  special 
forms  of  dials.  The  "dial  of  Ahaz"  (Isaiah,  xxxviii),  for 
example,  was  probably  a  flight  of  stairs,  very  likely  curved, 
upon  which  a  ray  of  sunlight  fell.  This  dates  from  about  the 
8th  century  B.C. 

It  is  impossible  to  do  more  at  this  time  than  to  refer  thus 
briefly  to  the  use  of  dials  among  the  ancients.  The  literature 
of  the  subject  is  very  extensive. 

Hourglasses  and  Clepsydrae.  The  need  was  early  felt  for 
some  kind  of  device  to  tell  the  hours  at  night  as  well  as  during 
the  day,  and  in  cloudy  weather  as  well  as  when  the  sun's  direct 
rays  gave  their  aid.  Various  methods  were  employed,  such  as 
burning  tapers,  hourglasses,  and  water  clocks.  The  hourglass 
was  known  probably  as  early  as  250  B.C.  Plato  (c.  380  B.C.) 
gave  much  thought  to  the  matter,  and  his  conclusions  may  have 

^'Aliaque  genera  et  qui  supra  scripti  sunt,  et  alii  plures  inventa  reliquerunt." 
2  On  this  type  see  Bigourdan,  L 'Astronomic,  p.  91. 


672  TIME 

suggested  to  Ctesib'ius  (c.  150  B.C.)  1  the  idea  of  a  water  clock, 
the  clepsydra,2  which  the  second  P.  Cornelius  Scipio  Nasica 
("Scipio  with  the  pointed  nose")  is  said  to  have  introduced 
into  Rome  (c.  159  B.C.).  In  the  early  forms  of  these  clocks 
the  water  trickled  from  one  receptacle  to  another  in  a  given 
time,  much  as  the  sand  flows  in  an  hourglass,  but  the  later 
forms  were  more  complicated.  It  was  such  an  instrument  that 
Harun  al-Rashid  sent  to  Charlemagne  in  807.  A  clepsydra  of 
the  primitive  type  was  until  recently  in  operation  in  one  of  the 
ancient  towers  of  Canton,  China. 

Influence  on  Later  Timepieces.  Since  the  priesthood,  which 
composed  the  learned  class,  kept  account  of  the  official  time  in 
the  early  days  of  civilization,  the  Church  continued  to  under- 
take this  task  until  the  modern  period.  The  priest  tolled  the 
hour  as  determined  by  the  dial  or  hourglass.  The  dial  was 
put  in  a  conspicuous  place  in  the  town,  on  the  church  tower, 
and  hence  in  modern  times  the  clock  is  often  seen  in  the  church 
tower  and  the  hours  are  struck  on  the  church  bell.  Because  of 
this  fact  we  have  our  name  "clock,"  a  word  probably  derived 
from  the  Celtic  and  meaning  bell,  whence  the  French  cloche, 
a  bell. 

When  the  hour  lines  were  marked  on  the  dial,  Roman  nu- 
merals were  used,  always  with  IIII  instead  of  IV  for  four,  and 
hence  we  see  the  same  markings  upon  modern  clocks.  The 
ancient  gnomon  was  under  the  care  of  the  priests,  and  brass 
plates  are  still  to  be  seen  in  the  floors  of  some  of  the  churches  in 
the  Mediterranean  countries,  the  sun  shining  through  a  certain 
window  and  telling  the  seasons  as  marked  upon  them.3 


s,  a  native  of  Alexandria.  None  of  his  works  are  extant,  but  he  is 
said  to  have  invented  not  only  a  water  clock  but  also  a  hydraulic  organ  and 
other  machines. 

2K\e$ijdpa  (klepsy'dra)  .  See  Volume  I,  page  69  n.j  Vitruvius,  De  Architec- 
tura,  IX,  cap.  9  ;  Pliny,  Hist.  Nat.,  VII. 

3  Those  who  wish  to  obtain  further  information  upon  the  subject  should  con- 
sult such  encyclopedias  as  the  Britannica  and  such  works  as  the  following  :  A. 
Fraenkel,  "Die  Berechnung  des  Osterfestes,"  Crelle's  Journal,  CXXXVIII,  133; 
E.  M.  Plunket,  Ancient  Calendars  and  Constellations  ,  London,  1903  ;  C.  P.  Bow- 
ditch,  "Memoranda  on  the  Maya  Calendars,"  American  Anthropologist,  III 
(N.S.),  129. 


INVENTION  OF  CLOCKS  673 

Clocks.  It  should  not  be  thought  that  clocks  of  the  general 
form  known  at  the  present  time  date  only  from  Galileo's  dis- 
covery of  the  isochronal  property  of  the  pendulum.  As  a 
matter  of  fact,  wheel  clocks  go  back  to  Roman  times,  and 
Boethius  is  said  to  have  invented  one  (c.  510).  Such  clocks 
are  known  to  have  been  used  in  churches  as  early  as  612.  The 
invention  of  those  driven  by  weights  is  ascribed  to  Pacificus, 
archdeacon  of  Verona,  in  the  gth  century,  although  a  similar 
claim  is  made  on  behalf  of  various  others.  Clocks  involving 
an  assemblage  of  wheels  are  medieval  in  origin,  and  one  was 
set  up  in  St.  Paul's,  London,  as  early  as  1286.  Small  portable 
clocks  were  in  use  in  the  i5th  century,  as  witness  a  letter  of 
1469,  written  by  Sir  John  Paston  and  containing  the  following 
admonition : 

I  praye  you  speke  wt  Harcourt  off  the  Abbeye  ffor  a  lytell  clokke 
whyche  I  sent  him  by  James  Gressham  to  amend  and  yt  ye  woll  get  it 
off  him  an  it  be  redy. 

The  oldest  mechanical  clock  of  which  we  have  any  complete 
description  was  made  by  a  German  named  Heinrich  De  Vick 
and  was  set  up  in  the  tower  of  the  palace  of  Charles  V  of  France 
in  I37Q.1  The  principle  employed  was  that  of  a  weight  sus- 
pended by  a  cord  which  was  wound  about  a  cylinder.  This 
cylinder  communicated  power  to  a  train  of  geared  wheels  which, 
in  turn,  transformed  by  means  of  a  " scape  wheel"  the  rotary 
motion  to  a  backward-and-forward  motion  controlling  the 
hands.  The  tendency  of  the  weight  to  descend  too  rapidly  was 
checked  by  a  device  for  regulating  the  action  of  the  wheels. 

The  pendulum  clock  was  introduced  about  1657  and  seems 
chiefly  due  to  Huygens.  The  principle  of  the  pendulum,  prop- 
erly attributed  to  Galileo,  had  been  observed  as  early  as  the 
1 2th  century  by  Ibn  Yunis  (c.  1200),  and  had  been  employed 

iOn  the  general  topic  see  W.  I.  Milham,  Time  and  Timekeepers,  New  York, 
1923,  with  full  bibliography;  E.  von  Bassermann-Jordan,  Die  Geschichte  der 
Zeitmessung  und  der  Uhren  (Berlin,  1920-  ) ;  R.  T.  Gunther,  Early  Science  in 
Oxford,  Vol.  II  (Oxford,  1923);  H.  T.  Wade,  "Clocks,"  New  International 
Cyclopaedia,  V,  470  (2d  ed.,  New  York,  1914). 


674  TIME 

by  astronomers  to  estimate  intervals  of  time  elapsing  during  an 
observation,  but  it  had  not  been  applied  to  a  clock.  It  was 
made  known  in  England  through  Ahasuerus  Fromanteel,  a 
Dutch  clockmaker,  about  1662. 

John  Harrison's1  great  construction  of  a  ship's  chronometer 
with  a  high  degree  of  precision  was  made  in  the  second  half 
of  the  1 8th  century  and  finally  secured  for  him  the  prize  of 
£20,000  offered  by  the  British  government  in  1714  for  a  method 
of  ascertaining  within  specified  limits  the  longitude  of  a  ship 
at  sea."  At  the  present  time  the  noon  of  Greenwich  mean  time 
(G.  M.  T.)  is  communicated  to  ships  by  wireless,  and  so,  in  the 
case  of  the  larger  vessels,  the  finding  of  longitude  no  longer  de- 
pends upon  the  chronometer  alone. 

Harrison's  contribution  to  practical  navigation  was  so  impor- 
tant as  to  warrant  a  brief  statement  about  the  nature  of  his 
work.  Although  he  was  by  trade  a  carpenter,  his  mechanical 
tastes  led  him  to  experiment  with  clocks.  Having  observed  the 
need  for  a  pendulum  of  constant  length,  he  devised  (1726)  the 
"gridiron  pendulum,"  in  which  the  downward  expansion  of  the 
steel  rods  compensated  for  the  upward  expansion  of  the  brass 
ones.  After  the  British  government  (1714)*  had  offered  the  prize 
already  mentioned,  Harrison  gave  his  attention  to  the  perfection 
of  a  watch  that  should  serve  to  assure  Greenwich  time  to  a  ship 
at  sea.  By  1761  he  had  constructed  one  that,  after  a  voyage  of 
several  months,  had  lost  only  i  min.  54^  sec.  and  assured  the 
longitude  within  18  miles.  The  government  paid  him  £10,000 
in  1765,  and  a  like  sum  in  1767, — a  modest  reward  for  an  in- 
vention of  such  great  value  to  the  world,  even  though  the  de- 
gree of  accuracy  would  now  be  considered  very  unsatisfactory. 

1  Born  at  Foulby,  parish  of  Wragby,  Yorkshire,  early  in  1693 ;  died  in  London, 
March  24,  1776. 

2 For  a  list  of  such  prizes  see  Bigourdan,  L*  Astronomic,  p.  166. 

3  The  original  act  reads,  "At  the  Parliament  to  be -Held  at  Westminster,  the 
Twelfth  Day  of  November,  Anno  Dom.  1713,"  but  was  printed  (and  doubtless 
enacted)  in  1714.  It  is  entitled  "An  Act  for  Providing  a  Publick  Reward  for 
such  Person  or  Persons  as  shall  Discover  the  Longitude  at  Sea,"  there  being 
nothing  "so  much  wanted  and  desired  at  Sea,  as  the  Discovery  of  the  Longitude." 
On  the  entire  topic  see  R.  T.  Gould,  The  Marine  Chronometer,  its  History  and 
Development,  London,  1923. 


DISCUSSION  675 

TOPICS  FOR  DISCUSSION 

1.  Additional  information  concerning  weight,  length,  area,  and 
volume  as  found  in  the  various  encyclopedias. 

2.  The  cubit,  shekel,  talent,  and  various  other  measures  referred 
to  in  Biblical  literature. 

3.  The  development  of  measures,  including  weights,  in  accord 
with  human  needs. 

4.  An  etymological  study  of  such  words  as  metric,  groschen,  doub- 
loon, measure,  watch,  day,  month,  and  year. 

5.  Influence  of  the  Roman  system  of  measures,  including  weights 
and  values,  upon  other  European  systems. 

6.  The  universality  of  certain  primitive  units  of  measure  such  as 
the  cubit  and  inch. 

7.  Primitive  customs  as  related  to  units  of  measure,  such  as  "a 
day's  journey,"  "a  watch  in  the  night/'  and  a  "Morgen." 

8.  Primitive  measures  still  found  in  various  parts  of  the  country, 
having  been  transmitted  from  generation  to  generation  like  the  folk- 
lore of  the  people. 

9.  Supplementary  information  on  the  calendar,  as  found  in  the 
various  encyclopedias. 

10.  Meaning  and  use  of  the  "golden  number"  and  "dominical 
letter"  as  set  forth  in  various  encyclopedias  or  in  works  on  the  Church 
calendar. 

11.  Methods  of  finding  the  date  of  Easter  as  given  in  the  books 
referred  to  in  the  preceding  topic. 

12.  The  reform  of  the  calendar  under  Gregory  XIII,  including  a 
study  of  earlier  attempts  at  reform. 

13.  Significance  of  such  technical  terms  as  computus  Paschalis, 
calendar,  a  red-letter  day,  and  almanac. 

14.  History  of  the  discovery  of  the  approximate  length  of  the  year, 
and  the  method  of  ascertaining  it. 

15.  Relation  among  calendars  used  by  various  peoples  of  ancient 
and  medieval  times. 

1 6.  Influence  of  Rome  upon  the  calendars  of  European  countries, 
and  the  effort  of  France,  during  the  Revolution,  to  break  away  from 
tradition  with  respect  to  the  divisions  of  the  year  and  the  names  of 
the  days  and  months. 

17.  The  mathematics  of  the  sundial. 


CHAPTER  X 

THE  CALCULUS 
i.  GREEK  IDEAS  OF  A  CALCULUS 

General  Steps  Described.  There  have  been  four  general  steps 
in  the  development  of  what  we  commonly  call  the  calculus,  and 
these  will  be  mentioned  briefly  in  this  chapter.  The  first  is 
found  among  the  Greeks.1  In  passing  from  commensurable 
to  incommensurable  magnitudes  their  mathematicians  had  re- 
course to  the  method  of  exhaustion,  whereby,  for  example,  they 
"exhausted"  the  area  between  a  circle  and  an  inscribed  regular 
polygon,  as  in  the  work  of  Antiphon  (c.  430  B.C.). 

The  second  general  step  in  the  development,  taken  two  thou- 
sand years  later,  may  be  briefly  called  the  method  of  infinitesi- 
mals. This  method  began  to  attract  attention  in  the  first  half 
of  the  1 7th  century,  particularly  in  the  works  of  Kepler  (1616) 
and  Cavalieri  (1635),  and  was  used  to  some  extent  by  Newton 
and  Leibniz. 

The  third  method  is  that  of  fluxions  and  is  the  one  due 
chiefly  to  Newton  (c.  1665).  It  is  this  form  of  the  calculus 
that  is  usually  understood  when  the  invention  of  the  science  is 
referred  to  him. 

The  fourth  method,  that  of  limits,  is  also  due  to  Newton,  and 
is  the  one  now  generally  followed. 

Contributions  of  the  Greeks.  As  stated  above,  the  Greeks 
developed  the  method  of  exhaustion  about  the  sth  century  B.C. 
The  chief  names  connected  with  this  method  have  already  been 

1Sir  T.  L.  Heath,  "Greek  Geometry  with  special  reference  to  infinitesimals," 
Mathematical  Gazette,  March,  1923 ;  D.  E.  Smith,  Mathematics,  in  the  series 
"Our  Debt  to  Greece  and  Rome,"  Boston,  1923;  G.  H.  Graves,  "Development 
of  the  Fundamental  Ideas  of  the  Differential  Calculus,"  The  Mathematics  Teacher, 
III,  82. 

676 


METHOD  OF  EXHAUSTION  677 

mentioned,  but  a  few  details  of  their  work  and  that  of  their 
contemporaries  will  now  be  given. 

Zeno  of  Elea  (c.  450  B.C.)  was  one  of  the  first  to  introduce 
problems  that  led  to  a  consideration  of  infinitesimal  magni- 
tudes. He  argued  that  motion  was  impossible,  for  this  reason : 

Before  a  moving  body  can  arrive  at  its  destination  it  must  have 
arrived  at  the  middle  of  its  path ;  before  getting  there  it  must  have 
accomplished  the  half  of  that  distance,  and  so  on  ad  infinitum:  in 
short,  every  body,  in  order  to  move  from  one  place  to  another,  must 
pass  through  an  infinite  number  of  spaces,  which  is  impossible.1 

Leucippus  (c.  440  B.C.)  may  possibly  have  been  a  pupil  of 
Zeno's.  Very  little  is  known  of  his  life  and  we  are  not  at  all 
certain  of  the  time  in  which  he  lived,  but  Diogenes  Laertius 
(ad  century)  speaks  of  him  as  the  teacher  of  Democritus 
(c.  400  B.C.).  He  and  Democritus  are  generally  considered  as 
the  founders  of  the  atomistic  school,  which  taught  that  magni- 
tudes are  composed  of  indivisible  elements2  in  finite  numbers. 
It  was  this  philosophy  that  led  Aristotle  (c.  340  B.C.)  to  write 
a  book  on  indivisible  lines.3 

Democritus  is  said  to  have  written  on  incommensurable  lines 
and  solids,  but  his  works  are  lost,  except  for  fragments,  and  we 
are  ignorant  of  his  method  of  using  the  atomic  theory^; 

Method  of  Exhaustion.  Antiphon  (c.  430)  is  one  of  the 
earliest  writers  whose  use  of  the  method  of  exhaustion  is  fairly 
well  known  to  us.  In  a  fragment  of  Eudemus  (c.  335  B.C.), 
conjecturally  restored  by  Dr.  Allman,4  we  have  the  following 
description : 

Antiphon,  having  drawn  a  circle,  inscribed  in  it  one  of  those  poly- 
gons5 that  can  be  inscribed:  let  it  be  a  square.  Then  he  bisected 
each  side  of  this  square,  and  through  the  points  of  section  drew 
straight  lines  at  right  angles  to  them,  producing  them  to  meet  the  cir- 

1  Allman,  Greek  Geom.,  p.  55. 

2  "Aro^tot   (a'tomoi) .   Allman,  Greek  Geom.y  p.  56. 

3  Hcpl  Ar6/uwi/  ypaw&v  (first  edition,  Paris,  1557) .   The  work  is  also  attributed 
to  Theophrastus.  *  Allman,  Greek  Geom.,  p.  65. 

5  That  is,  according  to  the  usage  of  the  time,  regular  polygons, 
n 


678  GREEK  IDEAS  OF  A  CALCULUS 

cumf erence ;  these  lines  evidently  bisect  the  corresponding  segments 
of  the  circle.  He  then  joined  the  new  points  of  section  to  the  ends 
of  the  sides  of  the  square,  so  that  four  triangles  were  formed,  and  the 
whole  inscribed  figure  became  an  octagon.  And  again,  in  the  same 
way,  he  bisected  each  of  the  sides  of  the  octagon,  and  drew  from  the 
points  of  bisection  perpendiculars ;  he  then  joined  the  points  where 
these  perpendiculars  met  the  circumference  with  the  extremities  of 
the  octagon,  and  thus  formed  an  inscribed  figure  of  sixteen  sides. 
Again,  in  the  same  manner,  bisecting  the  sides  of  the  inscribed  poly- 
gon of  sixteen  sides,  and  drawing  straight  lines,  he  formed  a  polygon 
of  twice  as  many  sides;  and  doing  the  same  again  and  again,  until 
he  had  exhausted  the  surface,  he  concluded  that  in  this  manner  a 
polygon  would  be  inscribed  in  the  circle,  the  sides  of  which,  on  ac- 
count of  their  minuteness,  would  coincide  with  the  circumference  of 
the  circle. 

We  have  in  this  method  a  crude  approach  to  the  integration 
of  the  i  yth  century. 

Bryson  (c.  450  B.C.),  who  seems  to  have  lived  just  before 
Antiphon's  period  of  greatest  activity,  was  at  one  time  thought 
to  have  used  a  method  that  had  the  merit  of  circumscribing  as 
well  as  inscribing  regular  polygons  and  exhausting  the  area  be- 
tween them.  This  was  probably  not  the  case  (Vol.  I,  p.  84),  al- 
though the  method  was  used  by  some  of  his  successors.  There 
is  also  no  reliable  evidence  to  prove  the  assertion  that  Bryson 
assumed  that  the  area  of  the  circle  is  the  arithmetic  mean  be- 
tween the  areas  of  two  similar  polygons,  one  circumscribed  and 
the  other  inscribed. 

The  Contribution  of  Eudoxus.  Eudoxus  of  Cnidus  (c.  370  B.C.) 
is  probably  the  one  who  placed  the  theory  of  exhaustion  on 
a  scientific  basis.  It  is  uncertain  just  how  much  reliance  is  to 
be  placed  upon  the  tradition  which  asserts  that  Book  V  of 
Euclid's  Elements  (the  book  on  proportion)  is  due  to  him,  but 
it  is  thought  that  the  fundamental  principles  there  laid  down 
are  his.  The  fourth  definition  in  Book  V  is:  " Magnitudes  are 
said  to  have  a  ratio  to  one  another  which  are  capable,  when 
multiplied,  of  exceeding  one  another,"  and  this  excludes  the 
relation  of  a  finite  magnitude  to  a  magnitude  of  the  same  kind 


EARLY  INTEGRATION  679 

which  is  either  infinitely  great  or  infinitely  small.1  It  is  in  this 
definition  and  the  related  axiom  that  Dr.  Allman  finds  a  basis 
for  the  scientific  method  of  exhaustion  and  discerns  the  prob- 
able influence  of  Eudoxus.  According  to  Archimedes,  this 
method  had  already  been  applied  by  Democritus  (c.  400  B.C.) 
to  the  mensuration  of  both  the  cone  and  the  cylinder. 

It  is  known  that  Hippocrates  of  Chios  (c.  460  B.C.)  proved 
that  circles  are  to  one  another  as  the  squares  on  their  diameters, 
and  it  seems  probable  that  he  also  used  the  method  of  ex- 
haustion,— a  subject  which  was  evidently  much  discussed 
about  that  time.  Archimedes  tells  us  that  the  " earlier  geom- 
eters" had  proved  that  spheres  have  to  one  another  the  tripli- 
cate ratio  of  their  diameters,  so  that  the  method  was  probably 
used  by  others  as  well. 

Archimedes  and  Integration.  It  is  to  Archimedes  himself 
(c.  22$  B.C.)  that  we  owe  the  nearest  approach  to  actual  in- 
tegration to  be  found  among  the  Greeks."  His  first  noteworthy 
advance  in  this  direction  was  concerned  with  his  proof  that  the 
area  of  a  parabolic  segment  is  four  thirds  of  the  triangle  with 
the  same  base  and  vertex,  or  two  thirds  of  the  circumscribed 
parallelogram.  This  was  shown  by  continually  inscribing  in 
each  segment  between  the  parabola  and  the  inscribed  figure  a 
triangle  with  the  same  base  and  the  same  height  as  the  segment. 
If  A  is  the  area  of  the  original  inscribed  triangle,  the  process 
adopted  by  him  leads  to  the  summation  of  the  series 
A  +  \A  +  (\)*A+..., 

or  to  finding  the  value  of 

^  [i +i +(!)*+ (£)•+•••], 

so  that  he  really  finds  the  area  by  integration  afcd  recognizes, 
but  does  not  assert,  that 

(-4-)*  """*"  °  as  ;/  """*"  °°> 

this  being  the  earliest  example  that  has  come  down  to  us  of  the 
summation  of  aji  infinite  series. 

1  Heath,  Euclid,  Vol.  II,  p.  120;  see  also  his  Archimedes,  p.  xlvii. 

2  Heath,  Archimedes,  p.  cxlii. 


680  GREEK  IDEAS  OF  A  CALCULUS 

Area  of  the  Parabola.  In  his  proof  relating  to  the  quad- 
rature of  the  parabola  Archimedes  first  proves  two  propositions 
numbered  14  and  15  in  his  treatise  on  this  curve.  These  assert 
that,  with  respect  to  the  figure  here  shown,1 


and 

He  then  states  (Prop.  16)  that  the  area  of  the  segment  of 
the  parabola  is  equal  to  ^&EgQ-  The  proof  is  by  a  reductio 
ad  absurdum  and  is  given  by  Heath  substantially  as  follows: 


I.   Suppose  that  the  area  of  the  segment  is  greater  than 
Then  the  excess  can,  if  continually  added  to  itself,  be  made  to  exceed 
&EqQ.    And  it  is  possible  to  find  a  submultiple 
of  the  triangle  EqQ  less  than  the  said  excess  of 
the  segment  over  \  &EqO. 

Let  the  triangle  FqQ  be  such  a  multiple  of  the 
triangle  EqQ.  Divide  Eq  into  equal  parts  each 
equal  to  qF^  and  let  all  the  points  of  division 
including  F  be  joined  to  Q  meeting  the  parabola 
in  jRv  A'2,  •  •  •,  Rn  respectively.  Through  A^, 
A'a,  •  •  •  ,  RH  draw  diameters  of  the  parabola  meet- 
ing qQ  in  Ov  6>2,  •  •  •  ,  On  respectively. 

Let  OlRl  meet  QR^  in  Fv 

Let  6>2AJ2  meet  QRl  in  Dl  and  QA\  in  *\- 

Let  <98A'8  meetr  QR^  in  Z>2  and  QR^  in  F#  and  so  on. 

We  have,  by  hypothesis, 

&FqQ<  area  of  segment  —  \&EqQ, 
or  area  of  segment  —  A/-^g>  \  l\EqO.  (i) 

Now,  since  all  the  parts  of  qE,  such  as  qF  and  the  rest,  are  equal,  we 
have  O^  =  Rf^  Of>^=D^R^R^\,  and  so  on;  therefore 


HQ.  (2) 

But  area  of  segment  <FO^  +  F{)^+  •  •  •  +  FH^OH  +  ^EHOnQ.  (3) 


the  proof,  see  Heath,  Archimedes,  p.  241,  preferably  F.  Kliem's  German 
translation,  pp.  361-365  (Berlin,  1914).  The  proof  of  the  next  proposition  is 
taken  from  the  same  work,  p.  244. 


AREA  OF  THE  PARABOLA  68  1 

Subtracting  the  equation  (2)  from  the  inequality  (3),  we  have 
area  of  segment  -AFqQ<  A\O2  +  /?2O8  H 

whence,  a  fortiori,  by  (i), 


But  this  is  impossible,  since  [Props.  14,  15] 


Therefore  area  of  segment  J>  \  A  EqQ. 

II.  If  possible,  suppose  the  area  of  the  segment  less  than 

Take  a  submultiple  of  the  triangle  EqQ  (as  the  triangle  FqQ),  less 

than  the  excess  of  j-  A  EqQ  over  tne  area  °f  tne  segment,  and  make  the 

same  construction  as  before. 

Since  AFqQ  <  J  A  EqQ  —  area  of  segment,  it  follows  that 
+  area  of  segment  <  % 


[Props.  14,  15] 

Subtracting  from  each  side  the  area  of  the  segment,  we  have 
A  FqQ  <  sum  of  spaces  qFR^  /?1/r1^2,  •  •  •  ,  EnRnQ, 

<  FOi+FlDl+  -  •  -  -f-^_iA,_i+  &EnRnQ,  a  fortiori-, 
which  is  impossible,  because,  by  (2)  above, 


Hence  area  of  segment  <  J  A  EqQ. 

Since,  then,  the  area  of  the  segment  is  neither  greater  than  nor  less 
than  \  A  EqQ,  it  is  equal  to  it. 

The  Method  of  Archimedes.  As  to  the  working  of  the  mind 
of  Archimedes  in  arriving  at  the  conclusion  in  regard  to  the 
area  of  the  parabola  (a  conclusion  which  led  to  the  above 
proof)  we  have  some  interesting  evidence.  In  a  manuscript 
discovered  in  Constantinople  in  1906  by  Professor  Heiberg,  the 
editor  of  the  works  of  Archimedes,  the  latter's  method  of  ap- 
proach to  certain  propositions  is  set  forth.  In  particular  the 
first  proposition  relates  to  the  steps  taken  in  arriving  at  the 
conclusion  with  respect  to  the  quadrature  of  the  parabola.1 

1The  Heiberg  edition  was  translated  by  Lydia  G.  Robinson,  Chicago,  1909, 
and  by  Sir  Thomas  L.  Heath,  Cambridge,  1912. 


682 


GREEK  IDEAS  OF  A  CALCULUS 


The  following  is  the  translation  as  given  in  Heath's  edition : 

LetABC  be  a  segment  of  a  parabola  bounded  by  the  straight  line  A  C 
and  the  parabola  ABC,  and  let  D  be  the  middle  point  of  AC.  Draw  the 
straight  line  DBE  parallel  to  the  axis  of  the  parabola  and  join  AB,  BC. 

Then  shall  the  segment  ABC  be  f  of  the  triangle  ABC. 

From  A  draw  AKF  parallel  to  DE,  and  let  the  tangent  to  the  parabola 
at  C  meet  DBE  in  E  and  AKF  in  F.  Produce  CB  to  meet  AF  in  K, 
and  again  produce  CK  to  //,  making  JfCff  equal  to  CK. 

Consider  CH  as  the  bar  of  a  balance,  K  being  its  middle  point. 


Let  MO  be  any  straight  line  parallel  to  ED,  and  let  it  meet  CF,  CK, 
AC  in  M,  N,  O,  and  the  curve  in  P. 

Now,  since  CE  is  a  tangent  to  the  parabola  and  CD  the  semiordinate, 

EB  =  Bl)  ; 

for  this  is  proved  in  the  Elements  \pf  Conies].1 
Since  FA,  MO  are  parallel  to  ED,  it  follows  that 


and 


MN=NO. 


1/.e.,  the  works  on  conies  by  Aristaeus  and  Euclid.   See  the  similar  expression 
in  On  Conoids  and  Spheroids,  Prop.  3,  and  Quadrature  of  Parabola,  Prop.  3. 


METHOD  OF  ARCHIMEDES  683 

Now,  by  the  property  of  the  parabola,  which  is  proved  in  a  lemma, 
MO  :OP=CA\AO        [Quadrature  of  Parabola,  Prop.  5] 
=  CK :  KN  =  HK :  KN.  [Eucl.  V I.  2] 

Take  a  straight  line  TG  equal  to  OP,  and  place  it  with  its  center  of 
gravity  at  //,  so  that  TH  =  HG ;  then,  since  N  is  the  center  of  gravity 
of  the  straight  line  MO,  and  MO :  TG  =  HK\  KN,  it  follows  that  TG  at 
H  and  MO  at  N  will  be  in  equilibrium  about  K. 

[On  the  Equilibrium  of  Planes,  I,  6,  7] 

Similarly,  for  all  other  straight  lines  parallel  to  DE  and  meeting  the 
arc  of  the  parabola,  (i)  the  portion  intercepted  between  FC,  AC  with  its 
middle  point  on  KC  and  (2)  a  length  equal  to  the  intercept  between  the 
curve  and  AC  placed  with  the  center  of  gravity  at  H  will  be  in  equilib- 
rium about  K. 

Therefore  A' is  the  center  of  gravity  of  the  whole  system  consisting  (i)  of 
all  the  straight  lines  as  MO  intercepted  between  FC,  A  C  and  placed  as 
they  actually  are  in  the  figure  and  (2)  of  all  the  straight  lines  placed  at  // 
equal  to  the  straight  lines  as  PO  intercepted  between  the  curve  and  AC. 

And,  since  the  triangle  CFA  is  made  up  of  all  the  parallel  lines  like 
MO,  and  the  segment  CBA  is  made  up  of  all  the  straight  lines  like  PO 
within  the  curve,  it  follows  that  the  triangle,  placed  where  it  is  in  the 
figure,  is  in  equilibrium  about  K  with  the  segment  CBA  placed  with  its 
center  of  gravity  at  //. 

Divide  KC  at  W  so  that  CK  -  3  KW\  then  W  is  the  center  of  gravity 
of  the  triangle  ACF\  for  this  is  proved  in  the  books  on  equilibrium. 

[On  the  Equilibrium  of  Planes,  I,  15] 

Therefore  A  A  CF :  segment  A  BC  =  HK :  KVV 

=  3:1. 

Therefore  segment  A  BC  =  J  A  A  CF. 

But  &ACF=*&ABC. 

Therefore  segment  ABC=$&A BC. 

Now  the  fact  here  stated  is  not  actually  demonstrated  by  the  argu- 
ment used,  but  that  argument  has  given  a  sort  of  indication  that  the 
conclusion  is  true.  Seeing,  then,  that  the  theorem  is  not  demonstrated, 
but  at  the  same  time  suspecting  that  the  conclusion  is  true,  we  shall  have 
recourse  to  the  geometrical  demonstration  which  I  myself  discovered  and 
have  already  published. 


684  MEDIEVAL  IDEAS  OF  THE  CALCULUS 

Archimedes  anticipates  Modern  Formulas.  In  his  treatment 
of  solids  bounded  by  curved  surfaces  he  arrives  at  conclusions 
which  we  should  now  describe  by  the  following  formulas  :  i 


Surface  of  a  sphere,  4  ira*  •  \  {    si 

"Jo 

Surface  of  a  spherical  segment, 

ra 

Tra'2  I     2  sin  0  d6  =  2  Tra2  (  I  —  cos  a). 

Jo 

Volume  of  a  segment  of  a  hyperboloid  of  revolution, 
f  (ax  +  x*)dx=b*§a  +  %b). 

Jo 

Volume  of  a  segment  of  a  spheroid, 

Xb 
x*dx^\l?. 
' 

7T      Ca 

Area  of  a  spiral,        —  I    x*dx  =  ^  Tra*. 
a  Jo 

Area  of  a  parabolic  segment, 


v  .,      2.  MEDIEVAL  IDEAS  OF  THE  CALCULUS 

Relation  to  Mensuration.  The  only  traces  that  we  have  of  an 
approach  to  the  calculus  in  the  Middle  Ages  are  those  relating 
to  mensuration  and  to  graphs.  The  idea  of  breaking  up  a  plane 
surface  into  infinitesimal  rectangles  was  probably  present  in  the 
minds  of  many  mathematicians  at  that  time  in  the  West  as  well 
as  in  the  East,  but  it  was  never  elaborated  into  a  theory  that 
seemed  worth  considering.  For  example,  a  Jewish  writer, 
Jehudah  Barzilai,  living  in  Barcelona  in  the  i3th  century,2  as- 

1Heath,  Archimedes,  p.  cxlvi  scq.;   G.  Loria,  Le  scienze  esatte  nelV  antica 
Grecia,  2d  ed.,  p.  108  (Milan,  1914)  ;  Heath,  Method  of  Archimedes,  p.  8  (1912). 
2Sefer  Jezira,  commentary  by  Judah  ben  Barzilai,  p.  255  (Berlin,  1884). 


FORERUNNERS  OF  MODERN  THEORY  685 

serts  that  "it  has  been  said  that  there  is  no  form  in  the  world 
except  the  rectangle,  for  every  triangle  or  rectangle  is  composed 
of  rectangles  too  small  to  be  perceived  by  the  senses." 

The  next  important  step  in  the  preparation  for  the  calculus 
taken  in  the  Middle  Ages  is  the  one  already  described  in  con- 
nection with  the  geometric  work  of  Oresme  (c.  1360).  His 
method  of  latitudes  and  longitudes  gave  rise  to  what  we  should 
now  call  a  distribution  curve  or  graph, — a  step  that  is  funda- 
mental to  the  modern  method  of  finding  the  area  included  be- 
tween a  curve  and  certain  straight  lines.  v 

3.  MODERN  FORERUNNERS  OF  THE  CALCULUS 

Early  Writers.  As  is  usual  in  such  cases,  it  is  impossible  to 
determine  with  certainty  to  whom  credit  belongs,  in  modern 
times,  for  first  making  any  noteworthy  move  in  the  calculus,  but 
it  is  safe  to  say  that  Stevin  is  entitled  to  serious  consideration. 
His  contribution  is  seen  particularly  in  his  treatment  of  the  sub- 
ject of  the  center  of  gravity  of  various  geometric  figures,  antici- 
pating as  it  did  the  work  of  several  later  writers.1  Other  writers, 
even  in  medieval  times,  had  solved  various  problems  in  mensu- 
ration by  methods  which  showed  the  influence  of  the  Greek 
theory  of  exhaustion  and  which  anticipated  in  some  slight  degree 
the  process  of  integration.  Among  them  may  be  mentioned 
the  name  of  Tabit  ibn  Qorra  (c.  870),  who  found  the  volume 
of  a  paraboloid.  Soon  after  Stevin  wrote,  Luca  Valerio2  pub- 
lished his  De  quadrature,  parabolae  (Rome,  1606),  using  a 
method  of  attack  that  was  essentially  Greek  in  its  spirit. 

Kepler.  Among  the  more  noteworthy  attempts  at  integration 
in  modern  times  were  those  of  Kepler  (1609).  In  his  notable 
work  on  planetary  motion  he  asserted  that  a  planet  describes 
equal  focal  sectors  of  ellipses  in  equal  times.  This  naturally 
demands  some  method  for  finding  the  areas  of  such  sectors, 

*De  Beghinselen  der  Weeghconst,  Leyden,  1686.  For  a  summary  see  H.  Bos- 
mans,  "Le  calcul  infinitesimal  chez  Simon  Stevin,"  Mathesis,  XXXVII  (1923). 

2  Born  c.  1552 ;  died  in  1618.  He  was  professor  of  mathematics  and  physics  at 
Rome. 


686     MODERN  FORERUNNERS  OF  THE  CALCULUS 

and  the  one  invented  by  Kepler  was  called  by  him  the  method 
of  the  "sum  of  the  radii,"  a  rude  kind  of  integration.  He  also 
became  interested  in  the  problem  of  gaging,  and  published  a 
work  on  this  subject  and  on  general  mensuration  as  set  forth 
by  Archimedes.1  Far  from  being  an  elementary  treatment  of 
gaging,  this  was  a  scientific  study  of  the  measurement  of  solids 
in  general.  Kepler  considers  solids  as  composed  "as  it  were" 
(veluti)  of  infinitely  many  infinitely  small  cones  or  infinitely 
thin  disks,  the  summation  of  which  becomes  the  problem  of 
the  later  integration. 

Cavalieri.  It  was  Kepler's  attempts  at  integration  that  led 
Cavalieri  to  develop  his  method  of  indivisibles,2  a  subject  which 
may  also  have  been  suggested  to  him  by  Aristotle's  tract  De 
lineis  insecabilibus,  to  take  the  common  Latin  title.3  It  may 
also  have  been  suggested  by  one  of  the  fragments  of  Xenocrates 
(c.  350  B.C.),  an  Athenian,  who  wrote  upon  indivisible  lines.4 

Cavalieri's  Lack  of  Clearness.  Cavalieri  was  not  always  clear 
in  his  statements  respecting  the  nature  of  an  indivisible  magni- 
tude. In  general,  however,  he  seems  to  have  looked  upon  a 
solid  as  made  up  practically  of  superposed  surfaces,  a  surface 
as  made  up  of  lines,  and  a  line  as  made  up  of  points,  these  com- 
ponent parts  being  the  ultimate  possible  elements  in  the  decom- 

lAusszug  auss  der  vralten  Messe-Kunst  Archimedes,  .  .  .  Erkldrung  vnd 
Bestdttigung  der  Oesterreichischen  Weinvisier-Ruthen,  Linz,  1616;  ed.  Frisch, 
V,  497,  614  (Frankfort  a.  M.,  1864).  Kepler's  letters  (ibid.,  p.  626)  show  that 
he  was  working  on  the  subject  as  early  as  1605.  On  this  entire  period  see  C.  R. 
Wallner,  Bibl.  Math.,  V  (3),  113. 

2  Geometric,  indivisibilibus  continiwrum  nova  qiiadam  ratione  promota, 
Bologna,  1635;  2d  ed.,  ibid.,  1653;  Exercitationes  geometricae  sex,  Bologna,  1647. 

3H.  Vogt  referred  to  this  tract  in  the  Bibl.  Math.,  X  (3),  146,  and  F.  Cajori 
called  attention  to  it  more  prominently  in  Science  (U.S.),  XLVIII  (N.S.),  577. 
See  also  Heath,  History,  I,  346. 

*u  .  .  .  in  infinitum  vero  dividi  non  posse,  sed  in  atomos  quasdam  desinere: 
has  porro  atomos  non  esse  partium  expertes  et  minimas,  sed  pro  quantitate  et 
materia  dividi  posse  et  partes  habere :  caeteroqui  specie  atomos  et  prima  naturae 
statuens  esse  primas  quasdam  lineas  insecabiles  et  ex  his  facta  plana  et  solida 
prima." 

This  Latin  translation  by  Simplicius  is  given,  with  the  original  Greek,  in 
the  "Xenocratis  Fragmenta,"  F.  W.  A.  Mullach,  Fragmenta  Philosophorum 
Graecorum,  III,  118,  §21  (Paris,  1881). 


CAVALIERI'S  METHOD  687 

position  of  the  magnitude.  He  then  proceeded  to  find  lengths, 
areas,  and  volumes  by  the  summation  of  these  "  indivisibles," 
that  is,  by  the  summation  of  an  infinite  number  of  infinitesimals. 
Such  a  conception  of  magnitude  cannot  be  satisfactory  to 
any  scientific  mind,  but  it  formed  a  kind  of  intuitive  step  in 
the  development  of  the  method  of  integration  and  undoubtedly 
stimulated  men  like  Leibniz  to  exert  their  powers  to  place  the 
theory  upon  a  scientific  foundation.1 

Illustration  of  Cavalieri's  Method.  Some  idea  of  Cavalieri's 
method  may  be  obtained  by  considering  his  comparison  of  a 
triangle  with  a  parallelogram  having  the  same  base  and  the 
same  altitude.  Calling  the  smallest  element  of  the  triangle  i, 
the  next  will  be  2,  the  next  3,  and  so  on  to  n,  the  base.  The 
area  is  therefore  i+2+3  +  -*-  +  w,  or  \n(n  +  i).  But  each 
element  of  the  parallelogram  is  n,  and  there  are  n  of  them,  as 
in  the  triangle,  and  so  the  area  is  n2.  Then  the  ratio  of  the  area 
of  the  triangle  to  the  area  of  the  parallelogram  is 

-n(n+ 1)  :;/2     or     - 

2     V  7  2 

But  -fi +-)->-     as     ;/  — >  oo, 

2\         ;//        2 

and  so  the  triangle  is  half  the  parallelogram. 

By  means  of  his  method  Cavalieri  was  able  to  solve  various 
elementary  problems  in  the  mensuration  of  lengths,  areas,  and 
volumes,  and  also  to  give  a  fairly  satisfactory  proof  of  the 
theorem  of  Pappus  with  respect  to  the  volume  generated  by  the 
revolution  of  a  plane  figure  about  an  axis.2 

1For  a  discussion  of  Cavalieri's  work  and  its  relation  to  the  calculus,  see 
H.  Bosnians,  "Sur  une  contradiction  reprochee  a  la  theorie  des  indivisibles'  chez 
Cavalieri,"  Annales  de  la  Societe  scientifique  de  Bruxelles,  XLII  (1922),  82.  For 
his  work  on  the  center  of  gravity  see  E.  Bortolotti,  "Le  prime  applicazioni  del 
calcolo  integrate  alia  determinazione  del  centre  di  gravita  di  figure  geometriche," 
Rendiconto  .  .  .  della  R.  Accad.  delle  Scienze,  Bologna,  1922,  reprint. 

2  For  a  translation  of  Cavalieri's  Theorem  relating  to  the  volumes  of  solids, 
see  G.  W.  Evans,  in  Amer.  Math.  Month.,  XXIV,  447.  On  the  method  in  general 
see  also  H.  Bosnians, "  Un  chapitre  de  1'oeuvre  de  Cavalieri,"  Mathesis,  XXXVI,  365. 


688      MODERN  FORERUNNERS  OF  THE  CALCULUS 

Roberval.  At  the  same  time  that  Cavalieri  was  working  on 
the  problem  of  indivisibles  Roberval1  was  proceeding  upon  a 
similar  hypothesis.  He  considered  the  area  between  a  curve 
and  a  straight  line  as  made  up  of  an  infinite  number  of  infinitely 
narrow  rectangular  strips,  the  sum  of  which  gave  him  the  re- 
quired area.  In  the  same  way  he  attacked  the  problems  of 
rectification  and  of  cubature.  He  also  found  the  approximate 

value  of  /   xmdx,  m  being  a  positive  integer,  by  finding  the 

Jo 

value  of  »*  __  i >n  __    m  __  . . .  __      _  i\fft 


asserting  that  this  approaches  i/(m  +  i)  as  «  — ^  oo. 

Fermat.  Fermat  (1636)  reached  the  same  conclusion,  basing 
his  treatment  upon  a  method  set  forth  by  Archimedes,  and  also 
extended  the  proof  to  include  substantially  the  cases  in  which 
ra  is  fractional  (1644)  or  negative  (1659),  although  not  using 
either  the  fractional  or  the  negative  exponent  in  his  work.  He 
also  attacked  (1636)  the  problem  of  maxima  and  minima,  that 
is,  of  finding  the  points  on  a  curve  at  which  the  tangent  is 
parallel  to  the  #-axis.  It  was  probably  because  of  this  step  that 
Lagrange  expressed  himself  as  follows : 2 

One  may  regard  Fermat  as  the  first  inventor  of  the  new  calculus. 
In  his  method  De  maximis  et  minimis  he  equates  the  quantity  tif 
which  one  seeks  the  maximum  or  the  minimum  to  the  expression  of 
the  same  quantity  in  which  the  unknown  is  increased  by  the  indeter- 
minate quantity.  In  this  equation  ...  he  divides  ...  by  the  in- 
determinate quantity  which  occurs  in  them  as  a  factor ;  then  he  takes 
this  quantity  as  zero  and  he  has  an  equation  which  serves  to  deter- 
mine the  unknown  sought.  .  .  .  His  method  of  tangents  depends 
upon  the  same  principle.3 

lTraite  des  indivisibles,  m£moire,  Paris,  1634.  See  A.  E.  H.  Love,  "Infinitesi- 
mal Calculus,"  Encyc.  Brit.,  nth  ed. 

2CEuvres  de  Lagrange,  ed.  Serret,  X,  294.  See  Cajori  in  Amer.  Math.  Month., 
XXVI,  16. 

3 For  confirmation  of  this  opinion  by  Laplace  and  Tannery,  see  Cajori,  loc. 
cit.,  p.  17.  On  the  work  De  maximis  et  minimis  consult  the  Supplement  to 
Volumes  I-IV  of  the  CEuvres  de  Fermat,  edited  by  C.  de  Waard,  Paris,  1922,  and 
the  review  by  H.  Bosnians,  Revue  des  Questions  Scientifiques,  Brussels,  April,  1923. 


ROBERVAL,  FERMAT,  AND  WALLIS  689 

With  his  name  should  be  joined  that  of  a  later  writer, 
Antonio  di  Monforte1  (1644-1717),  a  Neapolitan  mathema- 
tician who  worked  along  similar  lines. 

Problem  of  Tangents.  The  problem  of  tangents,  the  basic 
principle  of  the  theory  of  maxima  and  minima,  may  be  said  to 
go  back  to  Pappus  (c.  300 ).2  It  appears  indirectly  in  the 
Middle  Ages,  for  Oresme  (c.  1360)  knew  that  the  point  of 
maximum  or  minimum  ordinate  of  a  curve  is  the  point  at  which 
the  ordinate  is  changing  most  slowly.  It  was  Fermat,  however, 
who  first  stated  substantially  the  law  as  we  recognize  it  today, 
communicating  (i638)3  to  Descartes  a  method  which  is  es- 
sentially the  same  as  the  one  used  at  present,  that  of  equating 
/'(/)  to  zero.  Similar  methods  were  suggested  by  Rene  de 
Sluze4  (1652)  for  tangents,  and  by  Hudde5  (1658)  for  max- 
ima and  minima. 

Other  Writers.  From  then  until  Newton  finally  brought  the 
work  to  a  climax  various  efforts  were  made  in  the  same  direc- 
tion by  such  writers  as  Huygens,  Torricelli,  Pascal,6  and 
Mersenne.  The  fact  that  the  area  of  the  hyperbola  xy  —  i , 
found  by  Gregoire  de  Saint-Vincent7  (1647),  is  related  to 
logarithms  was  recognized  by  Fermat,  and  Nicolaus  Mercator8 
made  use  of  the  principle  in  his  calculation  of  these  functions. 

Wallis.  The  first  British  publication  of  great  significance 
bearing  upon  the  calculus  is  that  of  John  Wallis,  issued  in  1655. 

XF.  Amodeo,  "La  Regola  di  Fermat-Monforte  per  la  ricerca  del  massimi  e 
minimi,"  Periodico  di  Matematica,  XXIV,  fasc.  VI. 

2  There  is  a  good  summary  of  the  history  of  tangents  as  related  to  the  calculus 
in  a  work  by  Anibal  Scipiao  Gomes  de  Carvalho,  A  Teoria  das  Tangentes  antes 
da  Inven$ao  do  Cdlculo  Diferencial,  Coimbra,  1919. 

3  Opera  varia,  Toulouse,  1679. 

4 See  also  "A  short  and  easy  method  of  drawing  tangents  to  all  geometrical 
curves,"  Phil.  Trans.,  1672. 

5De  reductione  aequationum  et  de  maximis  el  minimis,  in  a  letter  published 
in  1713. 

6H.  Bosmans,  Archivio  di  Storia  delta  Scienza,  IV,  369. 

7  Opus  geometricum  quadraturae  circuli  et  sectionum  coni,  2  vols.,  Antwerp, 

1647- 

8 Logarithmotechnia,  London,  1668  and  1674. 


690     MODERN  FORERUNNERS  OF  THE  CALCULUS 

It  is  entitled  Arithmetica  Infinitorum,  sive  Nova  Methodus  In- 
quirendi  in  Curvilineorum  Quadraturam,  aliaque  difficiliora 
Matheseos  Problemata,  and  is  dedicated  to  Oughtred.  By  a 
method  similar  to  that  of  Cavalieri  the  author  effects  the  quad- 
rature of  certain  surfaces,  the  cubature  of  certain  solids,  and 
the  rectification  of  certain  curves.  He  speaks  of  a  triangle,  for 
example,  "as  if"  (quasi)  made  up  of  an  infinite  number  of  paral- 
lel lines  in  arithmetic  proportion,  of  a  paraboloid  "as  if"  made 
up  of  an  infinite  number  of  parallel  planes,  and  of  a  spiral  as  an 
aggregate  of  an  infinite  number  of  arcs  of  similar  sectors,  apply- 
ing to  each  the  theory  of  the  summation  of  an  infinite  series. 
In  all  this  he  expresses  his  indebtedness  to  such  writers  as  Torri- 
celli  and  Cavalieri.  He  speaks  of  the  work  of  such  British  con- 
temporaries as  Seth  Ward  and  Christopher  Wren,  who  were 
interested  in  this  relatively  new  method,  ancl,  indeed,  his  dedi- 
cation to  Oughtred  is  the  best  contemporary  specimen  that 
we  have  of  the  history  of  the  movement  just  before  Newton's 
period  of  activity.1  All  this,  however,  was  still  in  the  field  of 
integration,  the  first  steps  dating,  as  we  have  seen,  from  the 
time  of  the  Greeks. 

Barrow.  What  is  considered  by  us  as  the  process  of  differ- 
entiating was  known  to  quite  an  extent  to  Barrow  (1663).  In 
his  Lectiones  opticae  et  geometricae2  he 
gave  a  method  of  tangents  in  which,  in  the 
annexed  figure,  Q  approaches  P  as  in  our 
present  theory,  the  result  being  an  indefi- 
nitely small  (indefinite  parvum)  arc.  The 
triangle  PRQ  was  long  known  as  "Bar- 
row's differential  triangle," 3  a  name  which, 
however,  was  not  due  to  him.  It  is  evident  that  this  method, 
and  the  figure  as  well,  must  have  had  a  notable  influence  upon 
the  mathematics  of  his  time. 


1The  work  also  appears  in  his  Opera  Mathematica,  I,  255-470  (Oxford,  1695). 

2London,  1669.  *  The  work  seems  to  have  been  written  in  1663  and  1664. 
Love,  loc.  cit.\  J.  M.  Child,  Geometrical  Lectures  of  Isaac  Barrow,  Chicago,  1916. 

3  On  this  close  approach  to  the  later  calculus  see  Whewell's  edition  of  Barrow's 
Mathematical  Works,  p.  xii  (Cambridge,  1860),  and  Child,  loc.  cit. 


BARROW'S  INFLUENCE  691 

It  is  quite  probable  that  Barrow  had  advised  Newton  of  his 
work  on  this  figure  as  early  as  I664.1  Pascal  had  already  pub- 
lished a  figure  of  somewhat  the  same  shape,2  so  that  the  study 
of  triangles  of  the  general  nature  illustrated  above  was  being 
undertaken  and  discussed  at  this  time  in  both  England  and 
France.  The  triangles  given  by  both  Barrow  and  Pascal  were 
apparently  known  to  Leibniz,  and  they  assisted  him  in  develop- 
ing his  own  theory.3 

Barrow  also  recognized  the  fact  that  integration  is  the  in- 
verse of  differentiation,  but  he  did  not  use  this  relation  to  aid 
him  in  solving  the  quadrature  problem. 

Period  of  the  Invention  of  the  Newtonian  Method.  We  now 
approach  the  period  which  is  popularly  thought  to  be  the  one 
in  which  the  calculus  was  invented.  It  is  evident,  however,  that 
a  crude  integral  calculus  was  already  in  use  and  that  some 
approach  had  been  made  to  the  process  of  differentiation.  It 
is  also  evident  that  the  lines  of  approach  to  the  calculus  in 
general  have  been  two  in  number,  one  representing  the  static 
phase  as  seen  in  the  measurement  of  fixed  lengths,  areas,  or 
volumes,  and  in  the  making  use  of  such  ideas  as  those  of  infini- 
tesimals and  indivisibles;  the  other  representing  the  dynamic 
phase  as  seen  in  the  motion  of  a  point.  To  the  former  belong 
such  names  as  Kepler  and  Cavalieri  and,  in  general,  Archime- 
des ;  to  the  latter  belong  the  great  leaders  in  the  mathematics  of 
the  time  of  Newton  and  Leibniz.4 

1 J.  M.  Child,  The  Early  Mathematical  Manuscripts  of  Leibniz,  p.  n  (Chicago, 
1920),  a  work  which  students  of  the  history  of  the  calculus  should  consult,  not 
merely  for  its  translations  but  for  its  notes.  On  the  figures  used  by  Barrow, 
Pascal,  and  Leibniz,  see  ibid.,  p.  15.  This  work  is  hereafter  referred  to  as  Child, 
Leibniz  Manuscripts. 

2 In  his  Lettres  de  A.  Dettonville  (Paris,  1659),  the  part  relating  to  the  triangle 
having  been  written  in  1658. 

3  Child,  Leibniz  Manuscripts,  p.  16.  See  Leibniz's  admission  as  to  Barrow  in 
his  letter  to  Jacques  Bernoulli  (1703),  ibid.,  p.  20. 

*On  the  general  history  of  the  development  of  the  calculus  in  the  i7th  century 
the  following  works  may  be  consulted  with  profit : 

W.  W.  R.  Ball,  History  of  Mathematics,  loth  ed.,  London,  1922,  the  treatment 
of  the  calculus  being  particularly  complete ;  A.  De  Morgan,  On  a  point  connected 
with  the  dispute  between  Keill  and  Leibniz  about  the  Invention  of  Fluxions, 
London,  1846;  also  the  Companion  to  the  British  Almanack,  1852,  and  Philo- 


692  NEWTON  AND  LEIBNIZ 

4.  NEWTON  AND  LEIBNIZ 

Newton.  Newton's  great  contribution  to  the  theory  consists 
in  part  in  his  extension  of  the  method  to  include  the  other 
functions  then  in  common  use,  in  his  recognition  of  the  fact 
that  the  inverse  problem  of  differentiation  could  be  used  in 
solving  the  problem  of  quadrature,  in  his  introduction  of  a 
suitable  notation,  and  in  his  wide  range  of  applications  of 
the  subject.  Starting  with  the  knowledge  already  acquired  by 
Barrow,  he  developed,  beginning  in  1665,  his  method  of 
"  fluxions."  This  he  afterward  set  forth  in  three  tracts,1  which, 
in  accordance  with  his  unfortunate  plan  of  avoiding  publicity 
in  his  discoveries,  were  not  printed  until  many  years  later. 

Newton's  Three  Types.  Newton  recognized  three  types  of 
the  calculus.  In  his  Principia  (1687)  he  made  some  use  of 

sophical  Magazine,  June  and  November,  1852;  G.  J.  Gerhardt,  Die  Entdeckung 
der  Differenzialrechnung  durch  Leibniz,  Halle,  1848;  J.  Raphson,  The  History  of 
Fluxions,  London,  171$;  Latin  edition  the  same  year;  H.  Sloman,  Leibnizens 
Anspriiche  auf  die  Erfindung  der  Differenzialrechnung,  Leipzig,  1857;  English 
translation  with  additions,  Cambridge,  1860;  M.  Cantor,  Geschichte  der  Mathe- 
matik,  III,  chap.  97 ;  W.  T.  Sedgwick  and  H.  W.  Tyler,  A  Short  History  of  Science, 
chap,  xiii  (New  York,  1917) ;  H.  Weissenborn,  Die  Principien  der  ho  her  en 
Analysis,  als  hutorisch-kritischer  Beitrag  zur  Geschichte  der  Mathematik,  Halle, 
1856;  D.  Brewster,  Memoirs  of  the  Life,  Writings,  and  Discoveries  of  Sir  Isaac 
Newton,  2  vols.,  Edinburgh,  1885;  2d  ed.,  1860;  J.  Collins,  Commercium  Epis- 
tolicum  de  Varia  Re  Mathematica,  London,  1712;  2d  ed.,  London,  1722  (two 
editions) ;  3d  ed.,  London,  1725;  French  ed.,  Paris,  1856;  G.  Vivanti,  //  concetto 
d'  infinitesimo  e  la  sua  applicazione  alia  matematica,  Mantua,  1894;  2d  ed., 
Naples,  1901 ;  J.  M.  Child,  The  Early  Mathematical  Manuscripts  of  Leibniz, 
Chicago,  1920. 

On  the  history  of  fluxions  in  Great  Britain,  beginning  with  Newton,  the  best 
work  is  F.  Cajori,  A  History  of  the  Conceptions  of  Limits  and  Fluxions  in  Great 
Britain  from  Newton  to  Woodhouse,  Chicago,  1919.  On  the  general  history  of 
the  later  development  of  the  calculus,  see  J.  A.  Serret  and  G.  Scheffers,  Lehrbuch 
der  Differential-  und  Integralrechnung,  II,  581-626  (5th  ed.,  Leipzig,  1911),  and 
III,  694-720  (Leipzig,  1914);  Sedgwick  and  Tyler,  loc.  cit.,  chap,  xv;  E.  W. 
Brown,  "Mathematics,"  in  The  Development  of  the  Sciences,  New  Haven,  1923. 

1  (i)  De  Analyst  per  Equationes  numero  terminorum  infinitas,  written  in  1666 
and  sent  to  Barrow,  who  made  it  known  to  John  Collins,  who  allowed  Lord 
Brouncker  to  copy  it;  it  was  not  published,  however,  until  (London)  1711. 
(2)  Method  of  Fluxions  and  Infinite  Series,  written  in  1671  but  not  printed  until 
(London)  1736,  and  then  in  John  Colson's  translation  with  this  title.  (3)  Trac- 
tatus  de  Quadratura  Curvarum,  apparently  written  in  1676,  but  not  published 
until  (London)  1704  (appendix  to  his  Opticks). 


METHOD  OF  FLUXIONS  693 

infinitely  small  quantities,1  but  he  apparently  recognized  that 
this  was  not  scientific,  for  it  is  not  the  basis  of  his  work  in 
this  field. 

Method  of  Fluxions.  His  second  method  was  that  of  fluxions. 
For  example,  he  considered  a  curve  as  described  by  a  flowing 
point,  calling  the  infinitely  short  path  traced  in  an  infinitely 
short  time  the  moment  of  the  flowing  quantity,  and  designated 
the  ratio  of  the  moment  to  the  corresponding  time  as  the 
"fluxion"  of  the  variable,  that  is,  as  the  velocity.  This  fluxion 
of  x  he  denoted  by  the  symbol  x.  In  his  Method  of  Fluxions2 
he  states  that  "the  moments  of  flowing  quantities  are  as  the 
velocities  of  their  flowing  or  increasing," — a  statement  which 
may  be  expressed  in  the  Leibnizian  symbolism  as 

dy  _  dy    dx 
dx  ~  dt  '  dt  ' 

His  treatment  of  fluxions  may  be  illustrated  by  the  following 
extract  from  his  work : 3 

If  the  moment  of  x  be  represented  by  the  product  of  its  celerity  x 
into  an  indefinitely  small  quantity  o  (that  is  ico),  the  moment  of  y 
will  be  yo,  since  xo  and  yo  are  to  each  other  as  x  and  y.  Now  since 
the  moments  as  xo  and  yo  are  the  indefinitely  little  accessions  of  the 
flowing  quantities,  x  and  y,  by  which  these  quantities  are  increased 
through  the  several  indefinitely  little  intervals  of  time,  it  follows  that 
these  quantities,  x  and  y,  after  any  indefinitely  small  interval  of  time, 
become  x  +  xo  and  y  -f-  yo.  And  therefore  the  equation  which  at  all 
times  indifferently  expresses  the  relation  of  the  flowing  quantities  will 
as  well  express  the  relation  between  x  -f  xo  and  y  +  yo  as  between 
x  and  y ;  so  that  x  4-  xo  and  y  4-  yo  may  be  substituted  in  the  same 
equation  for  those  quantities  instead  of  x  and  y. 

Therefore  let  any  equation 

xs  —  aoP  -f-  axy  —  y*  =  o 

1F.  Cajori,  in  Amer.  Math.  Month.,  XXIV,  145;  ibid.,  XXVI,  15. 

2  Colson  translation,  p.  24  (London,  1736). 

3 Pp.  24,  25.    See  also  G.  H.  Graves,  loc.  cit.,  Ill,  82. 


694  NEWTON  AND  LEIBNIZ 

be  given,  and  substitute  x  +  xo  for  x  and  y  +  yo  for  y,  and  there 
will  arise 

•**  -H  3  x*xo  H~  3  xxxoo  +  #808 
—  ax*  —  2  axxo  —  ax*oo 


-f-  axy  +  axyo  +  axoy  + 
—  ys  —  $yoy*  —  $y*ooy  —  }W  =  o. 
Now,  by  supposition, 

x9  —  ax2  +  axy  —  jr3  =  o, 

which  therefore  being  expunged  and  the  remaining  terms  being 
divided  by  o,  there  will  remain 

3  xx*  H~  3  x*ox  +  3?  oo  —  2  axx  —  ax*o  +  ^T  +  &xy 

+  «jc>  -  3  yf  ~  3  >  V  ~  /^  =  °- 

But  whereas  o  is  supposed  to  be  infinitely  little  that  it  may  represent 
the  moments  of  quantities,  the  terms  which  are  multiplied  by  it  will 
be  nothing  in  respect  to  the  rest.  Therefore  I  reject  them  and  there 
remains  : 

3  XX*  —  2  axx  +  ayx  -f-  axy  —  -$yy*  =  o. 

Method  of  Limits.  Newton's  third  method,  that  of  limits, 
appears  in  his  Tractatus  de  Quadratura  Curvarum  (1704).  In 
the  introduction  he  says  : 

Let  a  quantity  x  flow  uniformly  and  let  it  be  required  to  find  the 
fluxion  of  xn.  In  the  time  in  which  x  by  flowing  becomes  x+  o,  the 
quantity  xn  becomes  sT-fcT]*;  i.e.,  by  the  method  of  infinite  series, 


nn  —  n 
xn  -f  noxM"l+  ------------  ooxn~~*  -f  ,  etc., 


and  the  increment  o  and 


nn  —  n 
noxn~l  H  ---  -  -------  ooxn~*+,  etc., 


are  to  each  other  as  i  and 


nn  —  n 
ft~l  -f  —  -  oxn~*+,  etc. 

• 


Now  let  the  increment  vanish  and  their  last  ratio  will  be  i  to  nx"~l. 


METHOD  OF  LIMITS  695 

He  also  gives  the  interpretation  of  these  ratios  as  the  slopes 
of  a  secant  through  two  points  on  a  curve  and  of  the  tangent 
which  is  the  limiting  position  of  this  secant.1  He  adds : 

If  the  points  are  distant  from  each  other  by  an  interval  however 
small,  the  secant  will  be  distant  from  the  tangent  by  a  small  interval. 
That  it  may  coincide  with  the  tangent  and  the  last  ratio  be  found, 
the  two  points  must  unite  and  coincide  altogether.  In  mathematics, 
errors,  however  small,  must  not  be  neglected. 

In  the  Principia  (Section  I)  Newton  set  forth  his  idea  of 
these  ultimate  ratios  as  follows : 

Ultimate  ratios  in  which  quantities  vanish,  are  not,  strictly  speak- 
ing, ratios  of  ultimate  quantities,  but  limits  to  which  the  ratio  of 
these  quantities,  decreasing  without  limit,  approach,  and  which, 
though  they  can  come  nearer  than  any  given  difference  whatever, 
they  can  neither  pass  over  nor  attain  before  the  quantities  have 
diminished  indefinitely. 

In  the  fluxional  notation  Newton  represented  the  fluent  of 
x  by  xo,  or  simply  by  x.  The  fluent  of  x  he  represented  by  x, 
and  so  on,"  a  notation  first  published  in  the  Algebra  of  John 
Wallis  (1693). 

Summary  of  Newton's  Method.  Ball  has  clearly  summarized 
Newton's  general  method  of  treatment  as  follows : 

There  are  two  kinds  of  problems.  The  object  of  the  first  is  to 
find  the  fluxion  of  a  given  quantity,  or,  more  generally,  "the  relation 
of  the  fluents  being  given,  to  find  the  relations  of  their  fluxions."  This 
is  equivalent  to  differentiation.  The  object  of  the  second,  or  inverse, 
method  of  fluxions  is,  from  the  fluxion  or  some  relations  involving  it, 
to  determine  the  fluent;  or,  more  generally,  "an  equation  being  pro- 
posed exhibiting  the  relation  of  the  fluxions  of  quantities,  to  find  the 
relations  of  those  quantities,  or  fluents,  to  one  another."  This  is 

1  Graves,  loc.  cit. 

2"Sint  v,  x,  y,  2  fluentes  quantitates,  &  earum  fluxiones  his  notis  i>,  #,  y,  2,  de- 
signabuntur  respective.  .  .  .  Qua  ratione  v  est  fluxio  quantitatis  v,  &  v  fluxio 
ipsius  ?),  &  v  fluxio  ipsius  v"  (Opera,  II,  392). 


696  NEWTON  AND  LEIBNIZ 

equivalent  either  to  integration,  which  Newton  termed  the  method  of 
quadrature,  or  to  the  solution  of  a  differential  equation  which  was 
called  by  Newton  the  inverse  method  of  tangents.1 

Leibniz.  Leibniz  (1684)  was  well  aware  of  the  work  of  men 
like  Barrow,  Huygens,  Gregoire  de  Saint- Vincent,  Pascal,  and 
Cavalieri.  He  was  in  London  in  1673,  and  there  he  prob- 
ably met  with  scholars  who  were  perfectly  familiar  with  the 
discoveries  of  Barrow  and  Newton,  and  with  Barrow  himself 
he  had  extended  correspondence.  After  leaving  England  he 
set  to  work  upon  the  problems  of  tangents  and  quadratures  and 
invented  a  notation  which  was  original  and  at  the  same  time 
was  generally  more  usable  than  that  of  Newton, — the  "dif- 
ferential notation."  He  proposed  to  represent  the  sum  of 
Cavalieri's  indivisibles  by  the  symbol  /,  the  old  form  of  5,  the 
initial  of  summa,  using  this  together  with  Cavalieri's  omn.  (for 
omnia),  and  to  represent  the  inverse  operation  by  d.  By  1675 
he  had  settled  this  notation,2  writing  fydy  =  \y~  as  it  is  written 
at  present. 

Leibniz  published  his  method  in  i6843  and  i686,4  speaking 
of  the  integral  calculus  as  the  calculus  summatorius,  a  name 
connected  with  the  summa  (/)  sign.  In  1696  he  adopted  the 
term  calculus  integralis,  already  suggested  by  Jacques  Ber- 
noulli in  1690. 

His  Conception  of  the  Differential.  Some  idea  of  his  concep- 
tion of  the  differential  may  be  obtained  from  a  statement  in  a 
letter  written  by  him  to  Wallis  on  March  30,  1699 : 

It  is  useful  to  consider  quantities  infinitely  small  such  that  when 
their  ratio  is  sought,  they  may  not  be  considered  zero,  but  which  are 
rejected  as  often  as  they  occur  with  quantities  incomparably  greater. 

*W.  W.  R.  Ball,  Hist,  of  Math.,  6th  ed.,  p.  344  (London,  1915),  to  which  the 
reader  is  referred  for  further  details,  Mr.  Ball  having  given  special  attention  to 
the  work  of  Newton.  See  also  A.  von  Braunmtihl,  Bibl.  Math.,  V  (3),  355. 

2  But  published  in  1686.   Love,  loc.  cit. 

3 "Nova  methodus  pro  maximis  et  minimi's,  itemque  tangentibus  .  .  .  ,"  in  the 
Acta  Eruditorum. 

4"De  geometria  recondita  et  analyst  indivisibilium  atque  infinitorum,"  also  in 
the  Acta  Eruditorum.  On  an  early  case  of  integration  (1599)  before  the  symbol- 
ism appeared,  see  F.  Cajori,  in  Bibl.  Math.,  XIV  (3),  312. 


THE  WORK  OF  LEIBNIZ  697 

Thus  if  we  have  x  +  dx,  dx  is  rejected.  But  it  is  different  if  we  seek 
the  difference  between  x  -f  dx  and  x,  for  then  the  finite  quantities 
disappear.  Similarly  we  cannot  have  xdx  and  dxdx  standing  to- 
gether. Hence  if  we  are  to  differentiate  xy  we  write : 

(x  4-  dx)  (y  -f-  dy)  —  xy  —  xdy  +  ydx  -f-  dxdy. 

But  here  dxdy  is  to  be  rejected  as  incomparably  less  than  xdy  -{-ydx. 
Thus  in  any  particular  case  the  error  is  less  than  any  finite  quantity.1 

As  to  the  approximate  period  at  which  he  began  to  arrive  at 
his  laws  for  the  differentiation  of  algebraic  functions,  we  have 
a  manuscript  of  his  which  was  written  in  November,  1676,  and 
in  which  he  gives  the  following  statements : 2 

dx  =  i ,  dx*  —  2  x,  dx*  —  3  x*,  etc. ; 


d -  =  —    0 >  d -^  =  —  -"o »  d  —5  =  —5 >  etc. ; 
x          x2       x*          x*       x*      x1 


^x=        ',  etc.; 


/—      x**1 
x?  =  -------------- 


T  ___    ____  rj  _  ____  . 

Hence  d~^=-  dx~*  will  be  —  2  x~*  or  —  —^  and  d^fx  or  dx*  will  be 
x 

I    _1  I 

—  -  x    2  or 

2 


- 

i 

/I 
-V/— 

^JC 


Some  of  these  results  are  incorrect,  probably  because  of  care- 
less writing,  and  some  appear  in  his  earlier  manuscripts,  but 
they  all  serve  to  show  how  the  mind  of  Leibniz  was  working  in 
this  period.  By  the  end  of  the  year  1676  he  had  developed  the 
rule  for  differentiating  a  product,  and  by  July,  1677,  he  had 
the  differentiation  of  algebraic  functions  well  in  hand.8 

*  Leibnitzens  Mathematische  Schriften,  Gerhardt  ed.,  TV,  63  (Series  III,  in 
Leibnitzens  Gesammelte  Werke,  Pertz  ed.,  Halle,  1859)  (this  portion  translated 
by  Mr.  Graves) . 

2  J.  M.  Child,  The  Early  Mathematical  Manuscripts  of  Leibniz,  p.  124  (Chicago, 
1920),  the  results  being  as  there  stated,  including  errors. 

3  Child,  loc,  tit.,  p.  116. 


698  NEWTON  AND  LEIBNIZ 

His  notation  for  differentiation  was  used  in  England  by  John 
Craig  as  early  as  I6Q3,1  and  the  same  writer  used  his  sign  for 
integration  ten  years  later.2  Both  symbols  were  somewhat 
familiar  to  English  mathematicians  throughout  the  i8th  cen- 
tury, although  it  was  not  until  the  igth  century  that  their  use 
in  Great  Britain  became  general. 

Priority  Dispute.  The  dispute  between  the  friends  of  Newton 
and  those  of  Leibniz  as  to  the  priority  of  discovery  was  bitter 
and  rather  profitless.  It  was  the  subject  of  many  articles3  and 
of  a  report  by  a  special  committee  of  the  Royal  Society.4 

English  readers  of  the  i8th  century  were  so  filled  with  the 
arguments  respecting  the  controversy  as  set  forth  in  the  Com- 
mercium  Epistolicum  (1712)  and  Raphson's  History  of  Flux- 
ions (1715),  that  they  gave  Leibniz  little  credit  for  his  work. 
It  was  not  until  De  Morgan  (1846)  reviewed  the  case  that  they 
began  generally  to  recognize  that  they  had  not  shown  their 
usual  spirit  of  fairness.  On  the  other  hand,  Leibniz  was  so 
stung  by  the  accusations  of  his  English  critics  that  he  too 
showed  a  spirit  that  cannot  always  be  commended. 

Leibniz  states  his  Case.  It  is  interesting  to  read  the  words  of 
Leibniz  in  his  own  defense,  as  presented  in  his  Historia  et  Origo 
Calculi  Differentiates : 5 

Since  therefore  his6  opponents,  neither  from  the  Commercium 
Epistolicum  that  they  have  published,  nor  from  any  other  source, 
brought  forward  the  slightest  bit  of  evidence  whereby  it  might  be 
established  that  his  rival  used  the  differential  calculus  before  it  was 
published  by  our  friend;7  therefore  all  the  accusations  that  were 
brought  against  him  by  these  persons  may  be  treated  with  contempt 

^Methodus  Figurarum  (London,  1693).          2Tractatus  Mathematicus. 

3  Beginning  with  a  publication  by  a  Swiss  scholar,  Nicolas  Fatio  de  Duillier 
(1664-1753),  whose  Lineae  brevissimi  descensus  investigatio  geometrica  duplex 
appeared  in  London  in  1699.   See  Child,  Leibniz  Manuscripts,  pp.  22,  23. 

4  The  report  appeared  in  1712.    See  Collins,  Commercium  Epistolicum.   It  was 
also  edited  by  Biot  and  Lefort  and  published  at  Paris  in  1856. 

5Found  in  MS.  by  Dr.  C.  I.  Gerhardt  in  the  Royal  Library  at  Hannover  and 
published  in  Latin  in  1846;  English  translation  by  Child,  Leibniz  Manuscripts, 
PP.  22,  57. 

6/.e.,  Leibniz's,  the  work  being  written  in  the  third  person. 

7 I.e.,  himself. 


PRIORITY  DISPUTE  699 

as  beside  the  question.  They  have  used  the  dodge  of  the  pettifogging 
advocate  to  divert  the  attention  of  the  judges  from  the  matter  on 
trial  to  other  things,  namely  to  infinite  series.  But  even  in  these  they 
could  bring  forward  nothing  that  could  impugn  the  honesty  of 
our  friend,  for  he  plainly  acknowledged  the  manner  in  which  he  made 
progress  in  them;  and  in  truth  in  these  also,  he  finally  attained  to 
something  higher  and  more  general. 

Brief  Summary  of  the  Dispute.  The  facts  are  that  Leibniz 
knew  of  Barrow's  work  on  the  "differential  triangle"  before  he 
began  his  own  investigations,  or  could  have  known  of  it,  and 
that  he  was  also  in  a  position  to  know  something  of  Newton's 
work.  The  evidence  is  also  clear  that  Newton's  discovery  was 
made  before  Leibniz  entered  the  field ;  that  Leibniz  saw  some 
of  Newton's  papers  on  the  subject  as  early  as  1677;  that  he 
proceeded  on  different  lines  from  Newton  and  invented  an 
original  symbolism;  and  that  he  published  his  results  before 
Newton's  appeared  in  print.  With  these  facts  before  us,  it 
should  be  possible  to  award  to  each  his  approximate  share  in 
the  development  of  the  theory.1 

Successors  of  Newton  and  Leibniz.  Most  of  the  British  writ- 
ers of  the  period  1693-1734,  failing  to  comprehend  Newton's 
position,  considered  a  fluxion  as  an  infinitely  small  quantity." 
The  first  noteworthy  improvement  in  England  is  due  to  Bishop 
Berkeley,  who,  in  his  Analyst  (1734),  showed  the  fallacy  of 
this1  method  of  approach  and  attempted  to  prove  that  even 
Newton  was  at  fault  in  his  logic.  Berkeley  provoked  great 
discussion  in  England,  and  the  result  was  salutary,  not  that  it 
affected  Newton's  standing,  but  that  it  put  an  end  to  much  of 
the  lax  reasoning  of  his  followers.3 

30n  the  general  controversy  see  the  summary  given  in  Ball,  Hist,  of  Math., 
6th  ed.,  pp.  356-362;  H.  Sloman,  The  Claim  of  Leibniz  to  the  Invention  of  the 
Differential  Calculus,  English  translation,  London,  1860. 

2F.  Cajori,  Amer.  Math.  Month.,  XXIV,  145;  XXVI,  15;  to  these  articles 
the  reader  is  referred  for  valuable  details  relating  to  this  period. 

3  On  the  gradual  improvement  of  the  Leibniz  theory  through  the  laying  of  a 
scientific  foundation  for  the  doctrine  of  limits,  see  F.  Cajori,  "Grafting  of  the 
theory  of  limits  on  the  calculus  of  Leibniz,"  Amer.  Math.  Month.,  XXX,  223, 
with  excellent  bibliography. 


700  NEWTON  AND  LEIBNIZ 

Cauchy's  Contribution.  Perhaps  the  one  to  whom  the  greatest 
credit  is  due  for  placing  the  fundamental  principle  of  the  cal- 
culus on  a  satisfactory  foundation  is  Cauchy.1  He  makes  the 
transition  from  ,  r/  ,  -\  ^/  \ 


dx  i 

to  dy=f(x)dx 

as  follows  : 

Let  y  =  f(x)  be  a  function  of  the  independent  variable  jc;  i,  an 
infinitesimal,  and  h,  a  finite  quantity.  If  we  put  t  =  ah,  a  will  be  an 
infinitesimal  and  we  shall  have  the  identity 

/(*  +  Q  -  /(*)  __  f(x  +  ah)  -/(*)  ^ 

i  ah 

whence  we  derive 

/N  f(x  +  aK)-f(x)      f(x  +  i)  -/(*) 

_      _     _    ___ 


The  limit  toward  which  the  first  member  of  this  equation  converges 
when  the  variable  a  approaches  zero,  h  remaining  constant,  is  what 
we  call  the  ((  differential"  of  the  function  y  =  /(#).  We  indicate  this 
differential  by  the  characteristic,  d,  as  follows  : 

dy  or  df(x). 

It  is  easy  to  obtain  its  value  when  we  know  that  of  the  derived  func- 
tion, y'  or  }'(x).  In  fact,  taking  the  limits  of  both  members  of  equa- 
tion (  i  )  ,  we  have  in  general  : 

(2)  <^(*)  =  */(*). 

In  the  particular  case  where  /(#)  =  r,  equation  (2)  reduces  to 

dx  =  h. 

Thus  the  differential  of  the  independent  variable,  x,  is  simply  the 
finite  constant,  h.  Substituting,  equation  (2)  will  become 

df(x)=f(x)dx, 
or,  what  amounts  to  the  same  thing, 

dy  =  y*dx. 

1R6sum6  des  Lemons  sur  le  Calcul  Infinitesimal,  Quatrieme  Le$on,  Paris,  1823; 
CEuvres  Completes,  Str.  77,  Tome  IV,  Paris,  1899. 


THE  YENRI  PROCESS 
5.  JAPAN 


701 


The  Yenri.  There  developed  in  Japan  in  the  i  yth  century  a 
native  calculus  which  may  have  been  the  invention  of  the  great 
Seki  Kowa  (1642-1708),  as  tradition  asserts,  although  we  have 
no  positive  knowledge  that  he  ever  wrote  upon  the  subject. 
This  form  of  the  calculus  is  known  as  the  yenri,  a  word  mean- 
ing " circle  principle"  or  "theory  of  the  circle"  and  possibly 


EARLY  STEPS  IN  THE  CALCULUS  IN  JAPAN 

Crude  integration,  from  Sawaguchi  Kazuyuki's  Kokon  Sampo-ki,  1670.   Sawa- 
guchi  was  a  pupil  of  Seki  Kowa,  the  Newton  of  Japan 

suggested  by  an  earlier  Chinese  title  or  by  the  fact  that  the 
method  was  primarily  used  in  the  measurement  of  this  figure. 
The  mensuration  of  the  circle  by  crude  forms  of  integration 
is  found  in  various  works  of  the  i8th  century,  such  as  the  one 
illustrated  above  and  the  one  shown  on  page  702,  published  by 
Mochinaga  and  Ohashi  in  1687.  A  similar  use  of  the  theory  is 
found  in  connection  with  the  mensuration  of  the  sphere  in 


702 


JAPAN 


Isomura's  work  of  1684,  and  thereafter  it  appeared  in  numerous 
works  in  the  closing  years  of  the  lyth  century  and  the  early 
part  of  the  century  following. 

In  a  general  way  it  may  be  said  that  the  yenri  was  an  ap- 
plication of  series  to  the  ancient  method  of  exhaustion.  For 
example,  Takebe  Kenko  (1722)  found  the  approximate  value 
of  TT  by  inscribing  regular  polygons  up  to  1024  sides,  and  prob- 
ably more,  giving  the  value  to  upwards  of  forty  decimal  places. 
In  this  work  Takebe  states  that  his  method  of  approximation 


EARLY  STEPS  IN  THE  CALCULUS  IN  JAPAN 

From  the  Kaisan-ki  Komoku,  by  Mochinaga  and  Ohashi,  representatives  of  the 
Seki  School.  The  work  was  published  in  1687.  The  method  is  essentially  that 

of  Sawaguchi 

was  not  the  one  used  by  Seki  Kowa.  In  fact  we  know  that  the 
latter  found  an  approximate  value  of  IT  by  computing  successive 
perimeters,  whereas  Takebe  based  his  work  upon  the  squares 
of  the  perimeters,  ?r2  being  taken  as  the  square  of  the  perimeter 
of  a  regular  polygon  of  512  sides.  The  value  of  TT  is  expressed 
as  a  continued  fraction,  a  plan  which  he  states  was  due  to  his 
brother,  Takebe  Kemmei.  Some  of  the  formulas  and  series  used 
by  Takebe  were  very  ingenious.1 

iSmith-Mikami,  p.  143. 


DISCUSSION  703 

TOPICS  FOR  DISCUSSION 

1.  General  steps  in  the  development  of  the  calculus  from  the  time 
of  the  Greeks  to  the  present. 

2.  Zeno's  paradoxes,  their  purpose,  their  fallacies,  and  their  rela- 
tion to  the  calculus. 

3.  The  study  of  indivisible  elements  among  the  Greek  philoso- 
phers, and  its  influence  upon  mathematics. 

4.  The  atomistic  philosophy  of  the  Greeks,  its  founder  and  advo- 
cates ;  its  bearing  upon  ancient  mathematics  and  its  relation  to  the 
modern  calculus. 

5.  The  relation  of  the  method  of  exhaustion,  especially  as  devel- 
oped by  Archimedes,  to  the  integral  calculus. 

6.  The  various  Greek  writers  on  the, method  of  exhaustion,  to- 
gether with  a  consideration  of  its  results. 

7.  The  contributions  of  Archimedes  to  the  making  of  the  calculus. 
His  methods  of  proof. 

8.  The  method  employed  by  Archimedes  in  discovering  his  geo- 
metric propositions. 

9.  Formulas  of  the  modern  calculus  anticipated  by  Archimedes 
and  any  other  Greek  writers. 

10.  The  contributions  of  the  Greeks  to  the  subject  of  mechanics, 
and  especially  those  of  Aristotle  and  Archimedes. 

11.  Influence  of  Oresme  with  respect  to  the  calculus. 

12.  Causes  leading  to  Kepler's  study  of  the  problem  of  the  calculus, 
together  with  a  statement  of  the  results  of  his  work. 

13.  General   nature   of   Cavalieri's   contribution;    the   problems 
studied ;  the  weakness  of  his  method ;  the  special  results  that  he  ac- 
complished ;  and  his  influence  upon  Leibniz. 

'14.  Fermat's  contributions  to  the  calculus  compared  with  those  of 
Cavalieri,  Barrow,  and  Roberval. 

15.  The  contributions  of  Roberval,  Barrow,  and  other  immediate 
predecessors  of  Newton. 

1 6.  Newton's  discoveries  in  the  calculus,  with  particular  reference 
to  the  fundamental  principles  employed  by  him. 

17.  Leibniz's  discoveries  and  the  question  of  priority. 

1 8.  General  nature  of  the  developments  in  the  calculus  after  New- 
ton and  Leibniz. 

19.  General  nature  of  the  early  Japanese  calculus. 


INDEX 


Since  certain  proper  names  are  mentioned  many  times  in  this  volume,  only  such  page 
references  have  been  given  as  are  likely  to  be  of  considerable  value  to  the  reader,  the 
first  reference  being  to  the  biographical  note  in  case  one  is  given.  In  general,  the  biog- 
raphies and  bibliographies  are  to  be  found  in  Volume  I.  As  a  rule,  the  bibliographical 
references  give  only  the  page  on  which  some  important  book  or  reference  is  first  men- 
tioned.  Except  for  special  reasons  (such  as  a  quotation,  a  discovery,  or  a  contribution 
to  which  a  reader  may  be  likely  to  refer),  no  references  are  given  to  elementary  text- 
books or  to  the  names  of  authors  which  are  already  given  in  Volume  I  and  are  men- 
tioned only  incidentally  in  Volume  II.  Obsolete  terms  are  usually  indexed  only  under 
modern  forms.  For  further  information  consult  the  index  to  Volume  I. 


Aahmesu.    See  Ahmes 

Abacus,  7,  86,  156,  177;  arc,  181; 
Armenian,  174;  Babylonian,  160; 
Chinese,  168;  dust,  157;  Egyptian, 
160;  in  France,  191;  Gerbert's,  180; 
in  Germany,  183,  190;  Greek,  161 ; 
grooved,  166;  Japanese,  170;  Korean, 
171,  174;  line,  181,  186;  Mohamme- 
dan, 174;  Polish,  176;  Pythagorean 
table,  177;  Roman,  165;  Russian, 
175,  176;  Turkish,  174;  Western 
European,  177 

Abbreviations  of  fractions,  221 

Abel,  N.  H.  (c.  1825),  469 

Abhandlungen,  15 

Abraham  ben  Ezra  (c.  1140),  353,  437, 
442,  543 

Abscissa,  318,  324 

Absolute  number,  12 

Absolute  term,  394 

Absolute  value,  267 

Abstract,  n 

Abu  Bekr  Mohammed.    See  al-Karkhi 

Abu  Ja'far  al-Khazin  (c.  960),  455 

Abu'l-Faradsh  (c.  987),  466 

Abu'l-Hasan  (c.  1260),  620 

Abu'1-Wefa  (c.  980),  467,  609,  617, 
622,  623 

Abundant  number,  20 

Achilles  problem,  546 

Acre,  644 

Acts  (operations),  36 

Adams,  G.  (c.  1748),  206 

Addend,  88 

Addition,  88,  184;  of  fractions,  223; 
symbols  of,  395 


Adelard  of  Bath  (c.  1120),  12,  382 

Adriaan.    See  Adriaen 

Adriaen  Anthoniszoon  (c.  1600),  310 

Adriaen  Metius  (c,  1600),  310 

Adriaenszoon,  J.  M.  (c.  1608),  373 

/Ebutius  Faustus,  L.,  361 

/Kbutius  Macedo,  M.,  357 

Aethelhard.    See  Adelard 

Affected  quadratics,  450 

Afghanistan,  72 

Aggregation  symbols,  416 

Agnesi,  M.  G.  (c.  1748),  331 

Agricola,  G.,  637 

Agrimensor,  361 

Aguillon,  F.  (1613),  344 

Ahmed      ibn      'Abdallah      al-Mervazi 

\c.  860),  620 
Ahmes   (c.  1650-1550  B.C.),  210,  386, 

498,  500 

Ahmose.    See  Ahmes 
Ahrens,  W.,  536,  542 
Akhmim,  212 
Albategnius  (c.  920),  608 
al-Battani.    See  Albategnius 
Alberuni  (c.  1000),  73,  308 
Alchemy,  595 
Alciatus,  A.  (1530),  637 
Alcobatiensis,  Codex ,    198 
Alcuin  (c.  775) >  535 
Alessandro  (c.  1714),  116 
Alexandre  de  Villedieu   (c.   1240),  14, 

80 

Alexandrian  calendar,  658 
al-Fazari  (c.  773),  72 
Alfonsine  Tables  (c.  1250),  609 
Alfonso  X,  el  Sabio,  609 


705 


706 


INDEX 


Algebra,  378;  applications,  582 ;  Arabic, 
382;  Chinese,  380;  Egyptian,  379; 
Greek,  381;  Hindu,  379;  medieval, 
382 ;  name,  386 ;  Persian,  382 ; 
powers,  393;  related  to  geometry, 
320;  symbols,  382,  395;  unknown 
quantity,  393 

Algorism  (algorithm),  9,  78,  88 

Algus,  9,  78 

al-IIaitam  of  Basra  (c.  1000),  455 

al-ljlasan   (Alhazen).   See  al-Haitam 

al-ljassar  (c.  i2th  century),  118 

'AH  ibn  Veli  (c.  1590),  393 

al-Karkhi  (c.  1020),  388,  504 

al-Kashi  (c.  1430),  310,  505 

al-Khayyami.   See  Omar  Khayyam 

al-Khowarizmi  (c.  825),  9,  72,  382, 
388,  446 

Alliage,  588 

Alligation,  587 

Allman,  G.  J.  (c.  1880),  677 

Allotte  de  la  Fuye,  38 

Alloy,  588 

Almagest.   See  Ptolemy 

Almahani  (al-Mahani)    (c.  860),  382, 

455 

al-Mamun  (c.  820),  372 
Almanac,  665 
al-Mervazi     (Habash    al-Hasib)     (c. 

860),  622 

al-Rashid  (c.  800),  672 
al-Rumi  (c.  1520),  626 
al-Zarqala  (Zarkala)  (c.  1050),  609, 

616 

Ambrose  of  Milan  (c.  370),  542 
Amicable  (amiable)  numbers,  23 
Amodeo,  F.,  689 
Analytic    geometry,    316,    322,    324; 

solid,  325 

Anatolius  (c.  280),  5 
Anaximander  (c.  575  B.C.),  603 
Andalo  di  Negro  (c.  1300),  665 
Anderson,  G.,  80 
Andres,  M.  J.  (c.  1515),  200 
Andrews,  W.  S.,  594 
Angle,  277;  sum  of  angles  in  a  triangle, 

287;  trisection,  297,  298 
Anharmonic  ratio,  333,  334 
Anianus  (1488),  665,  668 
Annotio,  Perito.   See  Cataldi 
Antecedent,  483 
Anthology,  Greek,  532 
AntOogarithms,  523 
Antiphon  (c.430  B.C.),  677 
Antonio  de  Dominis  (1611),  343 
Apianus,  P.  (c.  1527),  341,  441,  508,  509 


Apices,  75 

Apollonius  (c.  225  B.C.),  318 

Approximate  roots,  253 

Arabic  numerals,  69,  70.  See  Hindu- 
Arabic  numerals 

Arabs,  achievements,  272,  455,  467; 
algebra,  382 ;  computation  of  tables, 
626;  in  Europe,- 609;  geometry,  272; 
magic  squares,  597;  measure  of  the 
earth,  372;  trigonometry,  608.  See 
Arabic  numerals,  Hindu-Arabic  nu- 
merals 

Arbalete,  346 

Arbuthnot,  J.,  637 

Arcerianus,  Codex,  504 

Archibald,  R.  C.,  21,  30,  287,  293,  302 

Archimedes  (c.  225  B.C.),  5,  454,  679, 
681,  684;  cattle  problem,  453,  584; 
on  the  circle,  307;  cubic  of,  80 

Arcus  PythagoreuSy  177 

Area,  of  a  circle,  298,  302 ;  of  a  poly- 
gon, 606;  of  a  triangle,  631 

Areas,  286,  644 

Arenarius,  5 

Argand,  J.  R.  (c.  1810),  266 

Argus,  9,  10 

Aristarchus  (c.  260  B.C.),  604 

Aristotle  (c.  340  B.C.),  2 

Arithmetic,  7,  8.  See  Calculate,  Cal- 
culating machines,  Logistic,  Nu- 
merals, Problems,  Series,  and  the 
various  operations  and  rules 

Arithmetica,  i,  7 

Arithmetics,  American,  86 

Armillary  sphere,  350,  370 

Arnaldo  de  Villa  Nova  (c.  1275),  669 

Arnauld,  A.  (c.  1650),  28 

Arnauld  de  Villeneuve  (c.  1275),  669 

Arnold,  Sir  E.,  80 

Ars  Magna,  461-464 

Ars  supputandi)  u 

Articles,  12,  14 

Artificial  numbers,  208 

Aryabhata  the  Elder  (c.  510),  379,  387, 

__  444,  608,  615,  6216 

Aryabhata  the  Younger,  379 

Arzachel.   See  al-Zarqala 

As,  208 

Asoka  (3d  century  B.C.),  65-68 

Assize  of  bread,  566 

Astrolabe,  348,  601 

Astrology,  73 

Astronomical  fractions,  229 

Astronomical  instruments,  348,  364 

Astronomical  progression,  495 

Astronomy,  601-607 


INDEX 


707 


Asymptote,  318 

Athelhard.   See  Adelard 

Athenaeus  (£.300?),  289 

Athenian  calendar,  650 

Atomic  theory,  677 

Augrim.   See  Algorism 

August,  E.  F.  (c.  1850),  270 

Augustine  of  Hippo  (c.  400),  200 

Aurelius  Clemens  Prudentius  (c.  400), 

166 

Ausdehnungslehre,  268 
Autolycus  (c.  330  B.C.),  603 
Aventinus,  J.  (c.  1552),  200 
Avoirdupois  weight,  639 
Axioms,  280,  281 
Ayer  Papyrus,  396 
Ayutas,  308 

Baba  Nobutake  (c.  1700),  367 
Babbage,  C.  (c.  1840),  204 
Babylonians,  calendar,  655;  geometry, 

270;  measures,  635,  640;  numerals, 

36;  trigonometry,  601 
Bachet,  C.  G.  (c.  1612),  535 
Backer  Rule,  490 
Backgammon,  166 

Bacon,  Roger  (c.  1250),  282,  340,  372 
Bagdad,  72 

Baily,  F.  (1843),  376    , 
Baker,  H.  (1568),  493,  502 
Baker,  S.,  566 
Bakhshall  manuscript,  71 
Ball,  W.  W.  Rouse,  324,  691,  695,  696 
Bamboo  rods,  169-171,  432 
Banerjee,  G.  N.,  64 
Bank,  187 
Banking,  572,  574 
Baraniecki,  M.  A.,  176 
Barbaro,  E.  (c.  1490),  186 
Barbieri,  M.,  197 
Barcelona,  75 
Barclay,  A.  (c.  1500),  187 
Barnard,  F.  P.,  157 
Bar  Oseas  (c.  2508.0.),  670 
Barozzi,  F.  (c.  1580),  82 
Barrow,  I.  (c.  1670),  413,  690 
Bartels,  J.  M.  C.  (c.  1800),  336 
Barter,  568 

Bartoli,  C.  (c.  1550) ,  349,  356 
Barton,  G.  A.,  38,  635 
Base  line,  376 

Bassermann- Jordan,  E.  von,  673 
Bastard  Rule,  490 
BattHni.  See  Albategnius 
Baumeister,  A.,  162 
Bayley,  Sir  E.  C.,  157,  158,  197 


Bechtel,  E.  A.,  200 

Bede  the  Venerable  (c.  710),  200 

Beer,  R.,  75 

Beha  Eddin  (c.  1600),  388 

Beldamandi  (c.  1410),  502 

Bell,  J.  D.,  168 

Belli,  S.  (c.  1570),  285,  286,  355 

Beman,  W.  W.,  261,  267,  407 

Ben.   See  Ibn 

Benedetto  da  Firenze  (c.  1460),  547 

Benedict.   See  Benedetto 

Benedict,  S.  R.,  32 

Ben  Ezra.   See  Abraham  ben  Ezra 

Ben  Musa.   See  al-Khowarizmi 

Bentham,  327 

Benvenuto  d'  Imola,  528 

Berkeley,  G.  (c.  1740),  699 

Berlin  Papyrus,  432,  443 

Bernoulli,  Jacques  (I)   (c.  1690),  505, 

528,  629 
Bernoulli,  Jean  (I)  (c.  1700),  612,  662; 

on  complex  numbers,  264 
Bernoulli  numbers,  505 
Berosus  (c.  2508.0.),  670 
Beyer,  J.H.  (1616),  245 
Bezout,  E.  (c.  1775),  450 
Bhandarkar,  70 
Bhaskara  (c.  1150),  380,  425,  426,  446, 

484,  501,  525,  615 
Bianco,  F.  J.  von,  319 
Bierens  de  Haan,  D.   (c.  1870),  518, 

528 

Biering,  C.  H.,  298 
Bigourdan,  G.,  347,  648 
Bija  Ganita,  380,  426 
Bill  of  exchange,  577 
Billeter,  G.,  560 
Billion,  84 

Binet,  J.  P.  M.  (c.  1812),  477 
Binomial  Theorem,  507,  511 
Bion,  N.  (c.  1713),  359 
Biot,  J.  B.  (c.  1840),  655 
Biquadratic  equation,  466 
Birkenmajer,  A.,  341 
Bissaker,  R.  (1654),  205 
Bjornbo,  A.  A.,  606 
Bloomfield,  M.,  71 
Blundeville,  T.  (c.  1594),  627 
Bobynin,  V.,  34,  213,  219 
Boccardini,  G.,  282 
Bockh,  A.,  636 
Boklen,  £.,17 

Boethius  (Boetius)  (c.  510),  6,  73,  524 
Bolyai,  F.  (c.  1825),  337 
Bolyai,  J.  (c.  1825),  335,  337 
Bombelli,  R.  (1572),  19*  386,  428 


7o8 


INDEX 


Boncompagni,  B.  (c.  1870),  15,  27,  34, 

71,  104,  108,  114,  153,  254 
Bond,  J.  D.,  610 
Bonola,  R.,  282 
Borghi  (Borgi),  P.  (1484),  81 
Borrowing  process,  99 
Bortolotti,  E.,  26,  459,  687 
Bosanquet,  R.  H.  M.,  230 
Bosnians,  H.,  240,  252,  328,  430,  43$, 

459,  465,  508,  685,  687,  688,  689 
Bouelles,  Charles  de  (c.  1500),  22,  327 
Bouguer,  P.  (1734).  327,  376 
Bouvelles.   See  Bouelles 
Bowditch,  C.  P.,  664,  672 
Bowring,  J.,  166 
Brachistochrone,  326 
Brahe,  Tycho,  310 
Brahmagupta  (c.  628),  380,  387;  on 

quadratics,  445 
Brahml  forms,  67,  69 
Brandis,  J.,  635 

Braunmiihl,  A.  von  (c.  1900),  600,  610 
Breasted,  J.  H.,  645 
Brewster,  D.,  692 
Bridges,  J.  H.,  340 
Briggs,  H.  (c.  1615),  516 
Bring  (c.  1786),  470 
Brocard,  H.  (c.i9oo),  326 
Broken  numbers,  217 
Broker,  558 

Brouncker,  W.  (c,  1660),  420,  452,  692 
Brown,  E.  W.,  692 
Brown,  R.,  58 
Brugsch,  H.  K.,  34 
Bryson  (c.  450 B.C.),  678 
Buckley,  W.  (c.  1550) ,  236 
Bude  (Budaeus),  G.   (c.   1516),  209, 

637 

Budge,  E.  A.  W.,  316 
Bu6e,  Abbe  (1805),  266 
BUhler,  G.,  71 

Burgi,  J.  (c.  1600),  431,  523,  627 
Biirk,  A.,  288 
Burgess,  E.,  608 
Burnaby,  S.  B.,  664 
Burnam,  J.  M.,  75 
Burnell,  A.  C.,  70 
Burnside,  W.  S.,  473 
Bushel,  645 
Buteo  (Buteon),  J.  (c.  1525),  428,  434, 

S4i 

Butler,  R.,  246 
Butler,  W.,  n 

Cabala,  525 
Cabul,  72 


Caecilius  Africanus  (c.  100),  $45 
Caesar,  Julius  (c.  468.0.),  659,  660 
Caesar,  J.  (1864),  S42 
Cajori,  F.,  5,  64,  88,  205,  231,  246,  283, 
397,   404,   507,   546,   618,   686,   688, 
692,  693,  699 
Calandri,  P.  (1491),  142 
Calculate,  166 
Calculating  machines,  202 
Calculators,  166 
Calculi,  166.   See  also  Counters 
Calculones,  166 
Calculus,  676 

Calendar,  651,  655-664;  Athenian,  658; 
Babylonian,  655;  Chinese,  655; 
Christian,  660;  Egyptian,  656; 
French  Revolution,  663;  Gregorian, 
662 ;  Roman,  659 

Callippus  (Calippus)  (c.  3253.0.),  659 
Cambien,  183 
Cambio,  569 

Canacci,  R.  (c.  1380),  391 
Canon  Paschalis.  See  Calendar 
Cantor,  M.  (c.  1900),  10,  160,  177,  345, 

562 

Capacity,  644 
Capella  (c.  460),  3,  200 
Cappelli,  A.,  62 
Capra,  B.  (1655),  246 
Caracteres,  75 
Carat,  639 

Cardan,  H.  (or  J.)  (c.  1545),  384,  428, 
459-464,  467,  530 

Cardinal  numbers,  26 

Cardioid,  326 

Cardo,  317 

Carlini,  F.,  376 

Carlini,  L.,  529 

Carmen  de  Algorismo,  78 

Carnot,  L.  N.  M.  (c.  1800),  333 

Carpeting  problems,  568 

Carra  de  Vaux,  64,  229,  587 

Carrying  process,  93,  183 

Carslaw,  H.  S.,  282 

Cartesian  geometry,  318.  See  also 
Analytic  geometry 

Carvalho,  A.  S.  G.  de,  689 

Casati,  P.  (1685),  246 

Cassini's  oval,  329 

Cassiodorus  (c.  502),  3 

Castellum  nucum,  26 

Castillon  (Castiglione),  G.  F.  M.  M. 
Salvemini,  de  (c.  1750),  326,  511 

Casting  accounts,  98 

Casting  out  nines,  151 

Cataldi,  P.  A.  (c.  1590),  419 


INDEX 


709 


Catenary,  327 

Catoptrics,  339 

Cauchy,  A.  L.  (c.  1830),  477,  700 

Cavalieri,  B.  (c.  1635),  686 

Cayley,  A.  (c.  1870),  322,  477 

Celestial  sphere,  364,  365,  367 

Cellini,  B.,  342 

Celsius,  A.  (c.  1740),  375 

Celsus,  J.  (c.  75),  545 

Censo,  394,  427 

Census,  554.   See  also  Censo 

Cent,  648 

Centesima  rerum  venalium,  247 

Centesimal  angle  division,  627 

Ceulen,  L.  van  (c.  1580),  310 

Chace,  A.  B.,  436,  498,  500 

Chain  rule,  573 

Chaldea.   See  Babylonians 

Chalfant,  F.  H.,  40 

Challikan,  Ibn  (1256),  549 

Chamberlain,  B.  H.,  171 

Champlain's  astrolabe,  350 

Chances,  doctrine  of,  529 

Characteristic,  514 

Charlemagne  (c.  780),  648,  672 

Chasles,  M.  (c.  1850),  9,  13,  80,  109, 

157,  i7S»  I77,  322,  378,  607 
Chassant,  L.  A.,  57 
Chaucer,  9,  188 
Check,  579;  of  elevens,  154;  of  nines, 

151.   See  also  Checks 
Checkered  board,  187 
Checks  on  operations,  151 
Chelebi  (c.  1520),  626 
Chessboard  problem,  549 
Cheyney,  E.  P.,  564 
Child,  J.  M.,  690,  691 
Chinese,  algebra,  425,  432,  457,  475; 

calendar,    655 ;    determinants,    475 ; 

geometry,  271;  numerals,  39,  67,  68; 

series,     499 ;      trigonometry,      602 ; 

values  of  TT,  309 
Ch'in  Kiu-shao  (c.  1250),  42,  381 
Chords,  table  of,  604,  607,  614,  624 
Choreb,  174 

Chdu-pe'i  Suan-king,  215,  602 
Christian  of  Prag  (c.  1400),  77,  95 
Christian  calendar,  660 
Chrysippus  (c.  240  B.C.),  524 
Chrystal,  G.,  253 
Chuquet,    N.    (1484),    84,    414,    502, 

5i9 

Chu  Shi-kie1  (c.  1299),  257,  381 
ChQzen.   See  Murai 
Ciacchi,  29 
Ciermans,  J.  (c.  1640),  203 


Ciphering,  n 

Circle,  278;  quadrature  of,  298,  302. 
See  also  TT 

Circumference,  278;  of  the  earth,  369 

Cissoid,  314,  327 

Cistern  problem,  536 

Clairaut,  A.  C.  (c.  1760),  325,  464 

Clairaut,  J.  B.  (c.  1740),  206 

Clarke,  F.  W.,  650 

Clarke,  H.  B.,  589 

Classification,  of  numbers,  n  ;  of  equa- 
tions, 442 

Clavius,  C.  (c.  1583),  430,  662 

Clay,  A.  T.,  574 

Clepsydra,  538,  671 

Cloche,  672 

Clock,  671,  672,  674 

Clock  problem,  548 

Clodd,  E.,  195 

Cloff,  567 

Cloth,  cutting  of,  568 

Cochlioid,  327 

Cocked  hat   (curve),  327 

Codex  Alcobatiensis,  198 

Codex  Arcerianus,  504 

Coefficient,  393 

Coins,  646 

Colebrooke,  H.  T.,  91,  380,  637 

Collins,  J.  (c.  1700),  415,  692 

Colmar,  T.  de  (c.  1820),  204 

Colson,  J.,  693 

Columna  rostrata,  60 

Combination  lock,  527 

Combinations,  524 

Commercial  problems,  552 

Commercium  Epistolicum,  511 

Commission  and  brokerage,  558 

Commutative  law,  395 

Company.   See  Partnership 

Compasses,  sector,  347 

Complement,  98 

Complex  numbers,  261,  267 

Composites,  12,  14 

Compotus  Reinherij  62 

Compound  interest,  564 

Compound  numbers,  14 

Compound  proportion,  491 

Comptroller,  186 

Computing  table.   See  Abacus 

Computus,  651,  664.  See  also  Calendar, 
Compotus 

Conant,  C.  A.,  576 

Conant,  L.  L.,  $9 

Conchoid,  298,  327 

Concrete,  n 

Condamine,  La,  649 


7io 


INDEX 


Cone,  frustum  of,  294 

Congruence  theorems,  285 

Congruent  figures,  285 

Congruent  numbers,  30 

Conic  sections,  317,  454,  679 

Conjoint  rule,  492 

Conjugate  numbers,  267 

Consequent,  483 

Contenau,  G.,  37 

Continued  fractions,  311,  418 

Continued  products,  311 

Continuous  magnitude,  26 

Convergence,  507 

Coordinate  paper,  320 

Coordinates,  316,  324 

Copernicus,  610,  622 

Cordovero,  M.  (c.  1560),  526 

Cornelius  de  Judeis  (i594)»  354 

Corporation,  576 

Corpus  Inscriptionum  Etruscarum,  58 

Corpus  Inscriptionum  Latinarum,  57 

Corssen,  W.,  58 

Cortese,  G.  (c.  1716),  115 

Cosa.   See  Coss 

Cosecant,  622 

Cosine,  619,  631,  632 

Cosmic  figures,  295 

Coss,  392 

Cossali,  P.,  108,  384 

Cotangent,  620-622,  632 

Cotes,  R.,  265,  613 

Cotsworth,  M.  B.,  651 

Coulba,  174 

Counter,  181,  188 

Counters,  158,  165,  166,  186,  190.   See 

also  Abacus 

Counting.   See  Numerals 
Counting  rods,  169 
Counting  table,  174 
Court  of  the  Exchequer,  188 
Cowry  shells,  501 
Crajte  of  Nombryng  (c.  1300),  32,  78, 

92,  98,  102,  104 
Craig,  J.  (1693),  698 
Cramer,  G.  (c.  1740),  328 
Credit  (creditum),  576 
Credit,  letter  of,  576 
Cretan  numerals,  48 
Cross  ratio,  334 
Cross  staff,  346 
Cruma,  361 
Ctesibius   of  Alexandria    (2d  century 

B.C.),  538,  672 

Cube,  292;  duplication  of,  298,  313 
Cube  numbers,  19 
Cube  root,  144,  148 


Cubes,  sum  of,  504 

Cubic  curves,  324 

Cubic  equation,  454-467 

Cubit,  640 

Cuboid,  292 

Cubus,  427 

Cuento,  82,  88 

Cuneiform  numerals,  36,  68 

Cunningham,  W.,  555 

Currency,  569 

Curtze,  E.  L.  W.  M.,  123,  256,  345,  393, 
544,  550,  586 

Curve  surfaces,  325 

Curves,  algebraic,  324;  characteristics 
of,  326;  cubic,  324,  325;  of  descent, 
326;  of  double  curvature,  325  ;  plane, 
324;  of  pursuit,  327;  tautochronous, 
328;  transcendental,  324;  well- 
known,  326.  See  also  Cissoid  and 
other  names 

Cusa,  Nicholas  (c.  1450),  327 

Cushing,  F.  H.,  59 

Cycloid,  327 

Cypriote  numerals,  48,  49 

Daboll,  N.,  588 

Dagomari,  Paolo  (c.  1340),  81,  665 

D'Alembert's  Theorem,  474 

Damianus,  340 

Danfrie,  363 

Danti,  E.  (c.  1573),  339)  34° 

Danzel,  T.  W.,  59 

Darboux,  J.  G.,  333 

Daremberg,  C.,  209 

Darius  vase,  161 

Dase,  Z.  (c.  1860),  311 

Day,  653 

Days,  names  of,  657 

Days  of  grace,  578 

Decagon,  290 

Decimal  point,  238.   See  also  Fractions 

Decimanus,  317 

Decimatio,  541 

Decker,  E.  de  (1626),  518 

De  Colmar.   See  Colmar 

Decourdemanche,  J.  A.,  64 

Decussare  principle,  56 

Deficient  number,  20 

Degree,  232,  374,  443 

De  Haan.   See  Bierens  de  Haan 

Delamain,  R.  (c.  1630),  205 

Delambre,  J.  B.  J.  (c.  1800),  605 

De  latitudinibus  formarum,  319 

Democritus  (c.  400  B.C.),  677 

De  Moivre,  A.  (c.  1720),  265,  529,  613 

Demonstrative  geometry,  271 


INDEX 


711 


De  Morgan,  A.  (c.  1850),  148,  652,  691, 

698 

Demotic  writing,  47,  68 
Denarius,  208 
Denominate  number,  12 
Denominator,  220 
Density  of  the  earth,  376 
Deparcieux,  A.  de  (1746)  >  53° 
Deposit  (depositum),  576 
Desargues,  G.  (c.  1640),  332 
Descartes,  R.  (c.  1637),  322,  328,  343, 

431,  443,  469,  4?i,  689 
Descriptive  geometry,  331 
Determinant,  433,  475 
Develey,  E.  (c.  1800),  85 
De  Vick,  H.  (c.  1379),  673 
Devil's  curve,  328 
De  Witt,  J.  (1658),  324 
Dhruva,  H.  H.,  70 
Dial  of  Ahaz,  671.   See  also  Sundial 
Diameter,  in  geometry,  278;  as  num- 
ber, 6 

Diaz.   See  Diez 
Dickson,  L.  E.,  2,  29,  301 
Dickstein,  S.,  n,  176 
Dieck,  W.,  322 
Diefenbach,  L.,  586 
Diego  de  Landa,  43 
Diez,  J.  (c.  1550),  385,  392;  problems 

by,  590 
Difference,  97 
Differences,  finite,  512 
Differential,  696 
Differential  notation,  696-698 
Differential  triangle,  690 
Differentiation,  691 
Digges,  L.  and  T.  (c.  1572),  488 
Digits,  12,  13,  15 
Dill,  S,  572 
Dilworth,  T.,  588 
Dime,  647 

Dimension  of  an  equation,  443 
Dionysodorus  (c.  50  B.C.),  670 
Diophantus  (c. 275),  422-424,  450, 

452,  455 
Dioptrics,  339 
Discount,  565 
Discrete  magnitude,  26 
Distances,  285 
Distributive  law,  395 
Dividend,  131 
Divine  proportion,  291 
Divisibility,  221 
Division,  128;  a  danda,  141;  batello, 

136;    of    common    fractions,    226; 

complementary,  134;  definition,  128; 


ferrea,  aurea,  and  permixta,  135 ;  gal- 
ley method,  136;  Gerbert's  method, 
134;  Greek,  133;  long,  140,  142; 
repiego,  135;  scapezzo,  136;  sexagesi- 
mal, 233 ;  short,  133 ;  symbol,  406 

Divisor,  131,  222;  advancing,  139 

D'Ocagne,  M.,  203 

Dodson,  J.  (1742),  523 

Dollar,  647 

Dominical  letter,  652 

Dominis,  A.  de  (1611),  343 

Double  False,  438 

Drachma,  636 

Dragoni,  A.,  198 

Draughts,  166 

Drechsler,  A.,  653 

Drieberg,  F.  von,  16 

Drumhead  trigonometry,  357 

Duality,  325 

Duchesne,  S.  (c.  1583),  310 

Durer,  A.  (c.  1510),  296,  328,  342,  597, 
598 

Duhem,  P.  (c.  1900),  342 

Duillier.   See  Fatio 

Duplation,  33 

Duplication  of  the  cube,  298,  313 

Duranis,  72 

Dust  table,  177 

Dutt,  R.  C.,  213 

Dynamis,  394 

e,  logarithmic  base,  517 

Earth,  density,  376;  form,  369,  374; 

measure,  368-376 

Easter,  651,  659.   See  also  Calendar 
Eastlake,  F.  W.,  197 
Economic  problems,  552 
Egypt,  45,  68,  270,  379,  600 
Egyptians,    calendar,    656;    equations, 

43 J>  435J  fractions,  210;  geometry, 

270;  measures,  634;  numerals,  45; 

symbols,  410 
Elastic  curve,  328 
Elchatayn.   See  False  Position 
Elefuga  proposition,  284 
Elevens,  check  of,  154 
Ell,  640,  643 
Ellipse,  317,  454 
Elworthy,  F.  T.,  200 
Encyklopadie,  257 
Enestrom,   G.,  13,  96,   120,   139,  263, 

431,  437,  461,  466,  511,  526 
Engel,  F.,  267,  335 
English  measures,  640,  642 
Enriques,  F.,  282 
Epanthema  of  Thymaridas,  432 


712 


INDEX 


Epicycloid,  326,  328 

Equality,  symbol  of,  395 

Equating  to  zero,  431 

Equation,  394;  dimension  of,  443;  of 
payments,  559 

Equations,  Arab  forms,  424,  434,  436; 
biquadratic,  384,  466 ;  Chinese  forms, 
425,  432,  457;  classification  of ,  442 ; 
cubic,  384,  454-467 ;  Egyptian  forms, 
43i,  435;  factoring,  448;  fifth  de- 
gree, 469;  fundamental  theorem,  473; 
Hindu  forms,  425,  434,  458;  indeter- 
minate, 451-453,  584;  Japanese 
forms,  433;  linear,  432,  435,  583; 
literal,  435;  number  of  roots,  473; 
numerical  higher,  471;  in  printed 
form,  426-432;  quadratic,  443; 
simultaneous,  431,  432,  583;  solu- 
tion of,  435 

Equiangular  spiral,  329 

Eratosthenes,  5,  370 

Erlangen,  Sitzungsberichte,  293 

Erman,  J.  P.  A.,  130 

Etruscans,  58,  64 

Etten,  H.  van  (1624),  535 

Euclid  (c.  300  B.C.),  4,  338;  first  edi- 
tions, 272,  273;  geometric  terms, 
274;  on  quadratics,  444 

Eudoxus  (0.370  B.C.),  678 

Euler,  L.  (c.  1750),  265,  311,  431,  45°, 
453,  464,  469,  613,  627,  629 

Euler's  Theorem,  296 

Evans,  A.  J.,  50 

Evans,  G.  W.,  687 

Even  numbers,  16,  18 

Exchange,  569,  572,  577 

Exchequer,  188 

Exhaustion,  method  of,  303,  677-679 

Exponents,  414 

Eyssenhardt,  22 

Fabri,  O.  (1752),  355 

Factor  (broker),  558 

Factors,  30;  of  equations,  448 

Fairs,  569 

Fakhri,  382,  388 

Falkener,  E.,  594 

False  Position,  437-4 

Famous  problems,  three,  297 

Farthing,  647 

Fathom,  641 

Fatio  de  Diallier,  N.  (c.  1700),  698 

Faulhaber,  J.  (c.  1620),  518 

Faustus,  L.  ^butius,  361 

Favaro,  A.,  126,  206,  427,  436 

Fegencz,  H.  G.,  195 


Fellowship,  See  Partnership 

Fenn,  J.  (1769),  283 

Fermat,  P.  de  (c.  1635),  452,  453,  688, 

689 

Fermat,  S.  (1679),  322 
Fermat's  Numbers,  30 
Fermat's  Theorem,  30 
Fernel  (Fernelius),  J.  (c.  1535),  374 
Ferramentum,  361 
Ferrari,  L.  (c.  1545),  467 
Ferro,  Scipio  del  (c.  1500),  459 
Fibonacci,  Leonardo  (c.  1202),  6,  310, 

382,  384,  437,  457,  47i,  505,  609 
Figurate  numbers,  24,  170 
Fihrist,  Kitdb  al-  (Book  of  Lists),  466 
Finaeus.     See  Fine 
Fincke    (Fink,    Finke,    Finchius),    T. 

(c.  1583),  611,  621 
Fine,  Oronce  (c.  1525),  345,  347 
Finger  notation,  196 
Finger  reckoning,  12,  120,  196 
Finite  differences,  512 
Fink,  E.,  336 

Fior,  Antonio  Maria  (c.  1506),  459 
Fischer,  475 

Fitz-Neal  (c.  1178),  188 
Fleet,  J.  F.,  70 
Fleur  de  jasmin  (curve),  328 
Floridus.   See  Fior 
Fluxions,  693.   See  also  Calculus 
Foecundus,  613,  621 
Folium  of  Descartes,  328 
Fontana,  G.  (0.1775),  324 
Foot,  641 

Fortunatae  Insulae,  317 
Fracastorius,     H.      (Fracastoro,     G.) 

(c.  1540),  373. 
Fraction,  definition,  219;  name,  217; 

terms,  220 

Fractional  exponent,  414 
Fractions,  addition  of,  223;  astronomi- 
cal,    229;     bar     in     writing,     215; 

Chinese,    215;    common,   215,    219; 

complex,  219;  continued,  311,  418; 

decimal,     235;     division     of,     226; 

Egyptian,  210;  general,  213;  Greek, 

214,  231;  multiplication  of,  224,  232; 

operations  with,  222;  periodic,  30; 

physical,    229;    Roman,    208,    214; 

sexagesimal,     228;     subtraction    of, 

223;  unit,  210,  212;  vulgar,  219 
Francesca.  See  Franceschi 
Franceschi,  Pietro  (c.  1475),  296,  342 
Franco  of  Liege  (c.  1066),  310 
Frank,  J.,  350 
Frankland,  W.  B.,  335 


INDEX 


713 


Freigius,  J.  T.,  61,  179 

Frey,  J.,  580 

Friedlein,  G.,  16,  19,  50,  126,  127 

Frisius.   See  Gemma 

Frizzo,  G.,  123 

Fromanteel,  A.  (c.  1662),  673 

Frustum,  of  a  cone,  294 ;  of  a  pyramid, 

293 

Fujita  Sadasuke  (c.  1780),  41 
Fundamental  operations,  32,  35,  416 
Furlong,  642 
Fuss,  P.  H.  von,  454 

Gaging,  580 

Galileo  (c.  1600),  347,  373,  673 

Gallon,  644 

Galois,  E.  (c.  1830),  469 

Gambling  and  probability,  529 

Games,  number,  16 

Garbe,  R.,  71 

Gardner,  E.  G.,  392 

Gardthausen,  V.,  52 

Gauging.   See  Gaging 

Gauss,  C.  F.  (c.  1800),  337,  469,  474, 
476,  S07 

Geber,  390.   See  also  Jabir 

Gebhardt,  M.,  427 

Geet,  M.,  354 

Geiler  of  Kaiserberg  (c.  1500),  190 

Gelcich,  E.,  321 

Gellibrand,  H.  (c.  1630),  612 

Gematria,  54,  152 

Gemma  Frisius  (c.  1540),  520 

Gemowe  lines,  411 

Geometric  cross,  346 

Geometric  series.   See  Series 

Geometric  square,  345 

Geometry,  270;  analytic,  316;  Baby- 
lonian, 270;  Chinese,  271;  demon- 
strative, 271 ;  descriptive,  331 ;  Egyp- 
tian, 270;  elliptic,  338;  Greek,  271; 
hyperbolic,  338 ;  instruments  of,  344 ; 
intuitive,  270;  modern,  331;  name 
for,  273;  non-Euclidean,  331,  336; 
parabolic,  338;  projective,  272,  332; 
Roman,  271 

Gerbert  (c.  1000),  74 

Gergonne,  J.  D.  (c.  1810),  334 

Gerhardt,  C.  I.,  10,  92,  396,  692,  698 

Gernardus    (i3th  century  ?),  34,   100 

Gersten,  C.  L.  (1735),  204 

Ghaligai,  F.  (c.  1520),  427 

Gherardo  Cremonense  (c.  1150),  382, 
616 

Ghetaldi,  M.  (c.  1600),  321 

Giambattista  della  Porta  (1558),  373 


Gibson,  G.  A.,  515 

Gill,  644 

Ginsburg,  J.,  525 

Ginzel,  F.  K.,  651 

Girard,  A.  (c.  1630),  415,  430,  474,  618, 

622,  623 

Glaisher,  J.  W.  L.,  515,  518 
Gnecchi,  F.,  589 
Gnomon,  16,  601,  603,  669,  671 
Gobar  numerals,  73,  175 
Golden  Number,  652 
Golden  Rule,  484,  486,  491 
Golden  Section,  291 
Gomperz,  T.,  504 
Goniometry,  612 
Goschkewitsch,  J.,  168 
Gosselin,  G.  (c.  1577),  43O,  392,  435 
Gould,  R.  T.,  674 
Gouraud,  C.  (1848),  528 
Gow,  J.,  50,  145 
Grafton's  Chronicles,  638 
Grain  (weight),  637 
Gramma  symbol,  428 
Grassmann,  H.  G.  (c.  1850),  268 
Graunt,  J.  (1662),  530 
Gravelaar,  240 
Graves,  G.  H.,  676 
Great  Britain,  640,  642,  646 
Greatest  common  divisor,  222 

Greek  Anthology,  532,  584 

Greeks,  algebra,  381;  astronomy,  603; 
geometry,  271;  measures,  636,  641; 
numerals,  47 ;  trigonometry,  602 

Green,  J.  R.,  189 

Greenough,  J.  B.,  193,  647 

Greenwood,  I.  (1729),  86,  494 

Gregorian  calendar,  662 

Gregory,  D.  (c.  1700),  339 

Gregory  XIII,  662 

Griffith,  F.  L.,  432 

Groma,  361 

Gromaticus,  361 

Group  of  an  equation,  470 

Gruma,  361 

Gunther,  S.,  126,  319,  419,  478 

Guillaume,  C.  E.,  650 

Gunter,  E.  (c.  1620),  619,  621 

Gunther,  R.  T.,  362,  673 

Gupta  forms,  67 

Gwalior  inscription,  69 

Gyula  v.  Sebestyen,  194 

Haan.   See  Bierens  de  Haan 
tjabash  al-JJasib  (c.  860),  620 
Hager,  J.,  501 
Haldane.  E.  S.,  323 


714 


INDEX 


Half-angle  functions,  629 

Hall,  H.,  188 

Halley,  E.  (c.  1690),  530 

Halliman,  P.  (1688),  255 

Halliwell,  J.  0.,  100 

Halsted,  G.  B.,  282,  335 

Hamilton,  W.  R.  (c.  1850),  267 

Hanai  Kenkichi  (c.  1850),  203 

Hankel,  H.  (c.  1870),  118,  261 

Hare-and-hound  problem,  546 

Harkness,  W.,  650 

Harley,  R.,  470 

Harmonic  points,  332 

Harmonic  series,  503 

Harpedonaptae,  288 

Harper's  Dictionary  of  Classical  Lit" 
erature  and  Antiquities,  162 

Harriot  (Hariot),  T.  (c.  1600),  322, 
430,  431,  471,  527 

Harrison,  J.  (c.  1750),  674 

Harun  al-Rashid  (c.  800),  672 

Haskins,  C.  H.,  27,  189 

IJassar,  al-  (c.  i2th  century),  118 

Havet,  J.,  15 

Hay,  R.,  541 

Hayashi,  T.,  476 

Heath,  Sir  T.  L.,  676,  680,  681 ;  edi- 
tion of  Euclid,  14;  History,  16 

Hebrews,  measures,  635;  mysticism, 
596;  numerals,  53,  59 

Hegesippus  (c.  370),  542 

Heiberg,  J.  L.,  5,  80,  338,  524,  681 

Heidel,  W.  A.,  670 

Heilbronner  J.  C.  (c.  1740),  17 

Heliodorus  of  Larissa,  340 

Helix,  329 

Hellins,  J.,  331 

Helmreich,  A.,  390 

Henderson,  E.  F.,  188 

Henry,  C.,  10,  14,  322 

H6rigone,  P.  (c.  1634),  431,  618 

Hermann,  J.  M.  (c.  1814),  206 

Hermite,  C.  (c.  1870),  470 

Herodianic  numerals,  49 

Heromides,  274 

Heron  (Hero)  (c.  50,  or  possibly  c. 
200),  605 

Herschel,  J.  F.  W.  (c.  1840),  618 

Herundes,  274 

Herzog,  D.,  525 

Heteromecic  numbers,  18 

Heuraet,  H.  van  (c.  1659),  330 

Hexagonal  number,  24 

Hiao-tze  (0.350 B.C.),  169 

Hieratic  writing,  47,  68 

Hieroglyphics,  45,  68 


Higher  series,  504 

Hilbert,  D.,  277 

Hill,  G.  F.,  64,  76 

Killer,  E.,  17 

Hilprecht,  H.  V.,  37 

Hindasi,  64,  118 

Hindu-Arabic  numerals,  42 

Hindus,  algebra,  379;  astronomy,  625; 
equations,  434;  instruments,  365; 
measures,  637,  642;  observatories, 
365;  quadratics,  444;  trigonometry, 
608,  615,  625,  629;  values  of  ir,  308 

Hipparchus  (c.  1403.0.),  524>  604,  614, 

659 

Hippias  of  Elis  (c.  425  B.C.),  300,  305 
Hippocrates     of     Chios     (6.460  B.C.), 

679;  lunes  of,  304 
Hippolytus   (3d  century?),  152 
Hisab  al-Khataayn.  See  False  Position 
Hobson,  E.  W.,  304 
Hoccleve  (1420),  188 
Hochheim,  A.,  123 
Hock,  C.  F.,  7 
Hoffman,  S.  V.,  350 
Homans,  S.  (c.  1860),  530 
Homology,  334 
Hoppe,  E.,  230,  514 
Horace,  16 

Horner,  W.  G.  (1819),  381 
Horsburgh,  E.  M.,  203 
Horseshoe  problem,  551 
Hoshino  Sanenobu  (1673),  592,  594 
Hostus,  M.,  55 
Ho-t'u,  196 
Houel,  S.,  165 

Hound-and-hare  problem,  546 
Hour,  669 
Hourglass,  671 
Howard,  H.  H.,  68 
Huber,  D.,  336 
Hudalrich  Regius  (1536),  181 
Hudde,  J.  (1659),  466,  689 
Hiibner,  M.,  157 
Hulsius,  L.,  354 
Hultsch,  F.,  209,  251,  636 
Hunger,  K.  G.,  18 
Hunrath,  K.,  10,  145 
Huswirt,  J.  (1501),  83 
Huygens,  C.  (c.  1670),  673 
Hylles,  T.  (1592),  491 
Hyperbola,  317,  454,  689 
Hyperbolic  functions,  613 


i  ,  613 

lamblichus  (c.  325),  432 
Ibn  al-Zarqala  (c.  1050),  609,  616 


INDEX 


Ibn  Khallikan  (1256),  549 

Ibn  Yunis  the  Younger  (c.  1200),  673 

Ibrahim  ibn  Yahya.   See  Zarqala 

Ideler,  L.,  655 

I  Hang  (c.  800  ?),  549 

I-king,  524,  591 

Imaginary  numbers,  261 ;  graphic  rep- 
resentation, 263;  in  trigonometry, 
612 

Inch,  642 

Incommensurable  lines  and  numbers, 

251 

Incomposite  numbers,  20 
Indeterminate  equations,  451-453,  584 
India,  364.   See  also  Algebra,  Hindus, 

Numerals 

Indivisibles,  677,  686 
Infinite  products,  420,  506 
Infinite  series,  506,  679 
Infinitesimal.   See  Calculus 
Inscribed  quadrilateral,  286 
Instruments  in  geometry,  344,  368 
Integral  sign,  696 
Integration,  679,  684,  691 
Interest,    555,    559;    compound,    564; 

origin  of  term,  563 
Inverse  proportion,  490 
Inverted  fractional  divisor,  227 
Irrationals,  251 

Irreducible  case  in  cubics,  461,  464 
Italian  practice,  492 
Iwasaki  Toshihisa  (c.  1775),  537 

Jabir  ibn  Aflah  (c.  1130),  390,  609,  632 

Jackson,  L.  L.,  479 

Jacob,  Simon  (c.  1550),  346 

Jacob  ben  Machir  (c.  1250),  665 

Jacobi,  C.  G.  J.  (c.  1830),  477 

Jacob's  staff,  346 

Jacobs,  F.,  532 

Jacobus.   See  Jacopo 

Jacopo  da  Firenze  (1307),  71 

Jahnke,  E.,  268 

Jaipur  observatory,  366 

Jai  Singh  (c.  1730),  365 

Janet,  P.,  562 

Janszoon  (Jansen),  Z.  (c.  1610),  373 

Japan,  421,  614,  701 

Jastrow,  M.,  560 

Jebb,  S.,  340 

Jeber.   See  Jabir 

Jechiel  ben  Josef  (1302),  665 

Jefferson,  T.  (c.  1790),  649 

Jellen,  M.  (c.  1779),  118 

Jemshid.   See  al-Kashi 

Jenkinson,  C.  H.,  194 


Jeremias,  A.,  351 

Jerrard  (c.  1834),  470 

Jesuit  missionaries,  364 

Jetons  (jettons),  192.  See  also  Counters 

Jevons,  F.  B.,  561 

Jews,  555,  565 

Joannes  Philoponus  (c.  640?),  314 

Job  ben  Salomon,  442 

Johannes  Hispalensis  (c.  1140),  382 

Johnson,  G.,  6 

Jolly,  Von  (1881),  376 

Jones,  T.  E.,  359 

Jones,  W.  (c.  1706),  312 

Jordanus  Nemorarius   (of  Namur,  de 

Saxonia)  (c.  1225),  384 
Josephus  problem,  541-544 
Judah  ben  Barzilai,  684 
Junge,  G.,  289 
Just,  R.,  400 
Jya  (jiva),  615,  616 

Kabul,  72 

Kastner,  A.  G.  (c.  1770),  464,  613 

Kalinga  numerals,  67 

Kant,  L,  336 

Karagiannides,  A.,  335 

Karat,  639 

Karkhi,  al-  (c.  1020),  382,  388,  504 

Karosthi  numerals,  65 

Karpinski,  L.  C.,  64,  93,  188,  212,  232, 

236,  382 

Kaye,  G.  R.,  152,  158,  308,  365 
Kelland,  P.,  268 
Kenyon,  F.  G.,  50 
Kepler,  J.  (c.  1610),  342,  431,  685 
Kerbenrechnung,  194 
Keyser,  C.  J.,  282,  335 
Khallikan,  Ibn  (1256),  549 
Khayyam.   See  Omar  Khayyam 
Khowarizmi  (c.  825),  9,  72,  382,  388, 

446 

Kilderkin,  645 
Kircher,  A.  (c.  1650),  392 
Kittredge,  G.  L.,  193,  647 
K'iu-ch'ang   Suan-shu    (Arithmetic  in 

Nine  Sections),  257,  380,  433 
Kliem,  F.,  679 
Klimpert,  R.,  270 
Klos,  T.  (1538),  176 
Knight,  Madam,  570 
Knossos,  48 
Knott,  C.  G.,  157,  245 
Knotted  cords,  59 
Kobel,  J.  (c.  1520),  549,  553 
Koelle,  S.  W.,  197 
Konen,  H.,  452 


7i6 


INDEX 


Kost'al,  C.,  437 
Koutorga,  M.  S.,  575 
Kowa.  See  Seki 
Kress,  G.  von,  555 
Ksatrapa  numerals,  67 
Kubitschek,  W.,  162 
Kuckuck,  A.,  1 01,  165 
Kiihn,  H.  (1756),  265 
Kusana  numerals,  67 

Lacaille,  N.  L.  de  (c.  1750),  374 
La  Condamine,  649 
Lacouperie,  A.  T.  de,  157 
Lacroix,  S.  F.  (c.  1800),  328 
Lagny,  T.  F.  de  (c.  1710),  612,  627 
Lagrange,  J.  L.    (c.   1780),  469,  470, 

476,  688 

Lahire,  P.  de  (c.  1690),  324 
Laisant,  C,  A.,  327 
Lambert,  J.  H.   (c.  1770),  310,  336, 

613,  629 

Lambo,  Ch.,  414 
Lanciani,  R.  A.,  165 
Landa,  D.  de,  43 
Lange,  G.,  526 
Langland,  W.,  80 
Lao-tze,  195 

Laplace,  P.  S.  (c.  1800),  476 
Lapland,  survey  in,  375 
La  Roche.     See  Roche 
Latitude,  316 
Latitudines,  319 
Latus,  407,  409,  430 
Lauder,  W.  (1568),  564 
Lauremberg,  J.  W.,  587 
Lautenschlager,  J.  F.  (c.  1598),  487 
Law  of  signs  (equation),  471 
Leap  year.   See  Calendar 
Least  squares,  530 
Lecat,  M.,  476 
Lecchi,  G.  A.,  415 
Lefebvre,  B.,  58 
Legendre,  F.  (c.  1725),  192 
Legge,  J.,  40 
Legnazzi,  E.  N.,  345 
Leibniz,  G.  W.  Freiherr  von  (c.  1682), 

476,  691,  696;  on  complex  numbers, 

264;  priority  dispute,  698 
Lemniscate,  329 

Lemoine,  &  M.  H.  (c.  1873),  290 
Leonardo  of  Cremona  (c.  1425),  610 
Leonardo    Fibonacci    (of    Pisa).  See 

Fibonacci 

Leonardo  da  Vinci  (c.  1500),  327,  342 
Lepsius,  K.  R.,  641 
Lessing,  G.  E.,  584 


Letter  of  credit,  576 

Leucippus  (c.  4408.0.),  677 

Leupold,  J.  (c.  1720),  204 

Leurechon,  J.  (1624),  535 

Levels,  357-360 

Levi  ben  Gerson  (c.  1330),  526,  630 

Lex  Falcidia,  544 

Leybourn,  W.  (c.  1670),  202 

Lichtenfeld,  G.  J.,  402 

Liebermann,  F.,  188 

Lietzmann,  W.,  304 

Lilius  (Lilio),  A.  (c.  1560),  662 

Limagon,  326,  329 

Limits,  13,  694 

Lindemann,  F.,  26,  295 

Line,  274 

Lippersheim   (Lippershey),  J.  (1608), 

373 

Liter  (litre),  646 
Little,  A.  G.,  340 
Lituus,  329 
Liu  Hui  (c.  263),  380 
Livingstone,  R.  W.,  381 
Livre,  646 

Li  Yeh  (c.  1250),  381 
Lobachevsky,  N.  I.  (c.  1825),  335,  336 
Locke,  L.  L.,  195 
Loftier,  E.,  230 
Logarithmic  spiral,  329 
Logarithms,  513 
Logistic,  7,  10,  392 
Longitude,  316,  673 
Longitudines,  319 
Loria,  G.,  211,  212,  261,  270,  324,  331, 

335,  470,  471,  684 
Lo-shu,  196,  591 
Love,  A.  E.  H.,  688 
Lowell,  P.,  174 
Lucas,  E.,  290,  536,  541 
Luchu  Islands,  171 
Lucky  numbers,  17 
Lucretius  (c.  100),  332 
Ludlam,  W.  (1785),  283 
Ludolf  (Ludolph)  van  Ceulen  (c.  1580), 

310 

Ludus  duodedm  scriptorum,  166 
Ludus  latrunculorum,  166 
Lunes  of  Hippocrates,  304 
Lutz,  H.  F.,  210 
Lyte,  H.  (1619),  247 

McClintock,  E.  (c.  1890),  335,  530 
Macedo,  M.  ^Ebutius,  357 
Macfarlane,  A.  (c.  1900),  268 
Mac  Guckin  de  Slane,  549 
Machin,  J.  (c.  1706),  312 


INDEX 


717 


Machina,  361 

Machinula,  361 

Mackay,  J.  S.,  290,  313 

Maclaurin's  Theorem,  512 

Macrobius,  22 

Magic  circles,  592,  594 

Magic  squares,  591 

Magister  Johannes,  457 

Mahaffy,  J.  P.,  50,  160 

Mahavlra.   See  Mahavlracarya 

Mahavlracarya  (c.  850),  108,  380,  387; 

on  quadratics,  445 
Mahmoud  Bey,  34 
Mahmud    ibn    Mohammed    al-Rumi 

(c.  1520),  626 
Man,  E.  H.,  18 
Mannheim,  A.  (c.  1850),  206 
Mannheim  and  Moutard,  333 
Mansson,  P.  (c.  1515),  8,  486 
Mantissa,  514 

Margarita  phylosophica,  578 
Marquardt,  J.,  165 
Marre,  A.,  84,  128,  200 
Martin,  Th.,  12 
Martines,  D.,  157 
Martini,  G.  H.,  669 
Masahiro.    See  Murai 
Mascart,  J.,  297 
Maskeleyne,  N.  (c.  1770),  376 
Maspero,  G.,  193,  569 
Maspero,  H.,  286 
Mass  of  earth,  376 
Massoretes,  53 

Masterson,  T.  (c.  1590),  386,  556 
Mas'udi  (c.  950),  550 
Mathieu,  E.,  144 
Matthiessen,  L.,  424,  471,  585 
Maupertuis,  P.  L.  M.  de   (1746),  375 
Maurolico    (Maurolycus),   F.    (1558), 

622 

Maximus  Planudes  (c.  1340),  81 
Maya  numerals,  43 
Meadows  of  Gold,  550 
Mean  proportionals,  454,  483 
Measures,  634 
Mechanical  calculation,  156 
Mediation,  33 

Mei  Wen-ting  (c.  1675),  170 
Mellis,  J.  (1594),  249 
Menaechmus   (c.  350  B.C.),  454 
Menant,  J.,  560 

Menelaus  (c.  100),  603,  606,  615 
Mensa  geometricalis,  177 
Mensa  Pythagorica,  177 
Mensor,  361 
Mercatello  (c.  1522),  235 


Merchants'  Rule  (or  Key),  488 

Messahala  (c.  800),  353 

Messier,  C.  (i775)>  649 

Metius.   See  Adriaen 

Meton  (c.  432  B.C.),  658 

Metric  system,  242,  376,  648 

Metrodorus  (c.  500?),  532 

Metropolitan  Museum,  48,  49,  634 

Miju  Rakusai  (1815),  628 

Mikami,  Y.,  40,  124,  215 

Mile,  641 

Milham,  W.  I.,  673 

Milhaud,  G.  (£.1900),  444 

Miller,  G.  (1631),  518 

Miller,  G.  A.,  597 

Miller,  J.  (1790),  649 

Millet,  J.,  323 

Milliard,  85 

Million,  80 

Milne,  Joshua  (c.  1830),  530 

Mint  problems,  589 

Minuend,  96 

Minus  sign,  396,  397;  in  the  Rule  of 

False,  397,  441 
Minute,  218,  232 
Miram  Chelebi  (c.  1520),  626 
Mirror,  356,  358 
Misrachi,  Elia  (c.  1500),  33 
Mixed  numbers,  14 
Mixtures,  588 

Miyake  Kenryu  (c.  1715),  173,  543 
Mochinaga  (1687),  702 
Mocnik,  101 
Modern  geometry,  331 
Modulus,  267 

Moirai  (/Aotpcu),  232,  615,  617 
Moivre.    See  De  Moivre 
Mommsen,  T.  (c.  1850),  55 
Money  changers,  575.   See  also  Bank, 

Check,  Currency,  Exchange 
Monforte,  A.  di  (c.  1700),  689 
Monge,  G.  (c.  1800),  332 
Monier- Williams,  M.,  387 
Montauzan,  C.  G.  de,  359 
Month,  653 

Months,  names  of,  659 
Montucci  (1846),  330 
Moore,  Jonas  (1674),  622 
Morland,  S.  (c.  1670),  204 
Morley,  F.  V.,  322 
Morley,  S.  G.,  43 
Morse,  H.  B.,  40 
Mortality  table,  530 
Moulton,  Lord,  514 
Mouton,  G.  (c.  1670),  512,  649 
Moya,  J.  P.  de  (c.  1562),  198 


7i8 


INDEX 


Miiller,  J.  H.  T.,  274 

Muller,  T.,  519 

Muir,  T.,  476 

Mule-and-ass  problem,  552 

Multiple  angles,  functions,  629 

Multiplication,  101;  by  aliquot  parts, 
123;  per  bericocoli,  107;  Bhaskara's 
plan,  107;  cancellation,  118;  per 
casteluccio,  in;  per  colonna,  124; 
column,  112;  of  common  fractions, 
224;  complementary,  119-122;  con- 
tracted, 123;  per  coppa,  119;  cross, 
112;  per  gelosia,  114;  Greek  and 
Roman,  106;  left-to-right,  118;  per 
organetto,  108;  Pacioli's  plans,  107; 
Polish,  120;  process  of,  106;  quad- 
rilateral, 114;  quarter  squares,  123; 
per  repiego,  117;  Russian,  106,  120; 
per  scaccherOj  108;  per  scapezzo, 
117;  of  sexagesimals,  232;  short 
methods,  119,  122;  sign  of,  114; 
Spanish,  107;  symbols,  402;  table, 
123 

Munro,  D.  C.,  665 

Murai  ChQzen  (c.  1765),  511 

Murai  Masahiro  (c.  1732),  358,  359, 
614 

Muramatsu  Kudayu  Mosei  (c.  1663), 
542,  593 

Muratori,  L.  A.,  200 

Myriad,  308 

Naber,  H.  A.,  290 

Nagari  numerals,  67 

NagI,  A.,  109,  158,  162 

Name,  R.  van,  168 

Nana  Ghat  inscriptions,  65 

Napier,  J.  (c.  1614),  431,  514,  611,  632 

Napier's  rods,  202 

Napier's  Rules,  632 

Narducci,  E.,  3,  8 

Nasik  numerals,  06,  67 

Nasir  ed-din  (c.  1250),  609,  630,  632 

Nasmith,  J.,  566 

Neander,  M.  (c.  1570),  428 

Negative  exponents,  414 

Negative  numbers,  257,  396 

Neile,  W.  (c.  1665),  330 

Nepal  numerals,  67 

Nesselmann,  G.  H.  F.,  232 

Neuberg,  J ,  327 

Newton,  Sir  Isaac  (c.  1680),  324,  344, 

472,  511,  612,  692;  priority  dispute, 

698 

Newton,  John  (c.  1658),  612,  619 
Nicholas  Cusa  (c.  1450),  327 


Nicholson,  W.  (1787),  206 

Nick  sticks,  194 

Nicolas  Petri  (1567),  459,  465,  468 

Nicomachus  (c.  100),  5 

Nicomedes,  conchoid  of,  298,  327 

Nider,  J.,  563 

Nielsen,  N.,  6n 

Nine  Sections,  215,  380,  432 

Nines,  casting  out,  151 

Ninni,  A.  P.,  64 

Nobutake.   See  Baba 

Noel,  E.,  650 

Noether,  M.,  335 

Nokk,  A.,  604 

Non-Euclidean  geometry,  335 

Norm,  267 

Norton,  R.,  240 

Norwood,  R.  (1631),  618 

Notation,  33.    See  also  Numerals 

Number  puzzles,  582 

Number  theory,  4.  See  also  Arithmetica 

Numbers,  artificial,  208;  complex,  261; 
composite,  12,  14;  compound,  14; 
conjugate,  267;  cube,  19;  even,  16, 
18;  heteromecic,  18;  imaginary,  261, 
263 ;  irrational,  251 ;  large,  86 ;  mixed, 
14;  negative,  257;  oblong,  251;  odd, 
16,  18;  perfect,  20;  plane,  18;  prime, 
5,  20,  30;  reading,  36,  86;  square, 
18,  24;  writing,  36,  86 

Numerals,  Arabic,  68,  69,  70 ;  Attic,  49  ; 
Babylonian,  36,  68;  Chinese,  39, 
67,  68;  Cretan,  48;  Cuneiform,  36, 
68;  Cypriote,  48;  Egyptian,  45,  68; 
Etruscan,  58,  64 ;  Gobar,  73 ;  Greek, 
47,  49,  164;  Hebrew,  53,  59;  Hero- 
dianic,  49;  Hindu-Arabic,  42-88; 
Hindu  variants,  67,  70,  71;  Roman, 
54;  Sanskrit,  42,  70;  Spanish,  86; 
Sumerian,  67 

Numeration,  33 

Numerator,  220 

Numerical  higher  equations,  471 

Nunes,  P.  (c.  1530),  465 

Nunez.   See  Nunes 

Obenrauch,  F.  J.,  331 

Oblique  coordinates,  324 

Oblong  numbers,  251 

Odd  numbers,  16,  18 

Ohashi  (1687),  702 

Oliva,  A.,  460 

Omar  Khayyam   (c.  noo),  382,  426, 

442,  447,  456,  508 

Operations,  fundamental,  32,  35,  416 
Oppert,  G.,  64,  165 


INDEX 


719 


Optics,  338 

Ordinal  numbers,  26 

Ordinate,  318,  324 

Oresme  (c.  1360),  319,  414,  526,  689 

Otto  (Otho),  V.  (c.  1573),  310,  627 

Oughtred,  W.  (c.  1630),  205,  413,  430, 

611 

Ounce,  636 
Oval  of  Cassini,  329 
Ovio,  G.,  339 
Ozanam,  J.  (1691),  326,  535 

•w,  270,  307-313,  702;  the  symbol,  312 

Pacioli,  L.  (c.  1494) ,  384,  427,  443 

Pan  chu  tsih,  168 

Panton,  A.  W.,  473 

Paolo  Dagomari,  dell'  Abaco  (c.  1340) , 

123,  136,  216 
Papias  (c.  1050),  178 
Pappus  (0.300),  689 
Pappus-Guldin  theorem,  296 
Parabola,  317,  454,  679,  680 
Paraboloid,  685 
Parallelepiped,  291 
Parallels,  279,  335,  336;  postulate  of, 

282 

Parent,  A.  (c.  1710),  325 
Parentheses,  416 
Partnership,  554 
Pascal,  B.  (c.  1650),  203,  332,  508,  528, 

529,  691 
Pascal,  E.,  326 
Pascal's  Triangle,  508 
Passions  (operations),  36 
Pasturage  problems,  557 
Paton,  W.  R.,  532 
Paul  (Paolo)  of  Pisa,  81 
Pauly-Wissowa,  209 
Pearls  (curve),  330 
Peck,  645 

Peckham,  John  (c.  1280),  341 
Pecunia,  645 
Peet,  T.  Eric,  34 
Peetersen,  N.  (1567),  459 
Peking,  instruments  at,  364 
Peletier  (Peletarius) ,  J.  (c.  1560),  439 
Pell,  J.  (c.  1650),  406,  413 
Pell  Equation,  452 
Pellos  (Pellizzati)   (c.  1492),  238 
Pena  (Pena,  de  la  Pene),  J,  (c.  1557), 

338 

Penny,  647 
Pentagon,  290 
Pentagonal  number,  24 
Per  cent  sign,  250 
Percentage,  247 


Perch,  644 

Perez  (modern  Perez).  See  Moya 

Perfect  numbers,  20 

Periodic  fractions,  30 

Periods  in  notation,  86 

Periphery,  278 

Permillage,  250 

Permutations,  524,  528 

Persia,  364,  455,  608 

Perspective,  338 

Peru,  mission  to,  375 

Peter,  B.,  654 

Petri,  Nicolas  (1567),  459,  465,  468 

Petrie,  W.  M.  F.,  293,  634,  642,  652 

Petrie  Papyrus,  432 

Peurbach,  G.  von  (c.  1460),  609 

Picard,  J.  (c.  1670),  322,  374 

Pinches,  T.  G.,  560 

Pint,  645 

Pitiscus,  B.  (c.  1595),  611,  622 

Pittarelli,  G.,  342 

Place  value,  43,  44 

Plane  numbers,  18 

Plane  surface,  276 

Plane  table,  356 

Planets,  657 

Planisphere,  351 

Planudes,  Maximus  (c.  1340),  81 

Platea,  F.  de  (c.  1300) ,  563 

Plato  (c.  380  B.C.),  2,  5 

Platonic  bodies,  295 

Playfair,  J.  (c.  1795),  283 

Plimpton,  G.  A.,  383,  391,  397,  404, 

406,  427,  485,  551,  571,  666,  667 
Plucker,  J.  (c.  1850),  325 
Plunket,  E.  M.,  672 
Plural  proportion,  492 
Plus  and  minus  signs,  397,  398,  402 
Plus  sign,  variants,  402;  in  the  Rule 

of  False,  397,  441 
Plutarch,  602 

Poincare,  H.  (c.  1900),  335 
Point,  2.74 

Points,  harmonic,  332 
Polar  coordinates,  326 
Poleni,  G.  (c.  1740),  204 
Poll  tax,  572 

Polygonal  numbers,  24,  27,  499 
Polygons,  area  of,  606 ;  regular,  301 
Polyhedron  theorem,  296 
Polyhedrons,  295 ;  regular,  296 ;  stellar, 

296 

Pomodoro,  G.  (1624),  356,  358 
Poncelet,  J.  V.  (c.  1830),  333 
Pondera,  361 
Pons  asinorum,  284,  289 


72O 


INDEX 


Poole,  R,  L.,  189 

Porta,  G.  della  (1558),  373 

Portius,  L.,  637 

Poseidonius  (Posidonius)   (c.  IOOB.C.), 

5,  37i 

Position.    See  False  Position 

Postulate  of  parallels,  282 

Postulates,  280,  281 

Pott,  A.  F.,  200 

Potts,  R.,  637 

Poudra,  N.  G.,  338 

Pound,  636,  638,  646,  647 

Powel,  J.,  567 

Powers,  393 

Poynting,  J.  H.  (1891),  376 

Practica  (pratica,  pratiche),  n 

Practice,  492 

Prayer  sticks,  196 

Prime  meridian,  317 

Prime  number,  5,  20,  30 

Printing,  effect  on  numerals,  77 

Priscian  (6th  century),  54 

Prism,  291 

Probability,  528 

Problem,  chessboard,  549;  cistern,  536; 
hare-and-hound,  546;  of  Hiero's 
crown,  590;  horseshoe,  551;  Jose- 
phus,  542;  testament,  544;  Turks- 
and-Christians,  541.  See  also 
Problems 

Problems,  algebraic,  582 ;  commercial, 
552;  economic,  552;  elementary, 
532;  famous  and  fanciful,  297,  501, 
S32>  536;  of  Metrodorus,  532;  of 
pursuit,  546;  typical,  536 

Product,  90 

Profit  and  loss,  557 

Progressions,  494,  496 

Projectiles,  192 

Projective  geometry,  331,  332 

Proportion,  413,  477,  479;  arrange- 
ment of  terms  in,  483,  488,  489; 
compound,  491 ;  divine,  291 ;  inverse, 
490;  relation  to  series,  497;  terms  of 
a,  483;  types  of  a,  482.  See  also 
Rule  of  Three 

Proportional  compasses,  347 

Proportionality,  478 

Propositions  of  geometry,  typical,  284 

Prosdocimo  de'  Beldamandi  (c.  1410), 
502 

Psammites,  5 

Ptolemy,  Claude  (c.  150),  371,  607, 
615,  624,  629,  631,  632 

Ptolemy's  Theorem,  624 

Pulisa,  308 


Pulverizer,  387 

Pure  quadratic,  450 

Pursuit,  curve  of,  327;  problems  of, 

546 

Puzzles,  582.   See  also  Problems 
Pyramid,  292;  frustum  of,  293 
Pyramidal  number,  25 
Pythagoras   (c.  540  B.C.),  4 
Pythagorean  numbers,  288,  451 
Pythagorean  table,  124 
Pythagorean  Theorem,  288 

Quadrant,  352-357 

Quadratic   equation,   443-451 ;    Hindu 

rules  for,  444-446 
Quadratrix,  300,  305 
Quadratum  geometricum,  345 
Quadrature,  298,  302.    See  also  Circle 
Quadrivium   (Quadruvium) ,  3 
Quart,  644 
Quarter  squares,  123 
Quaternions,  267 
Quentos,  88 
Quipu,  195 
Quotient,  131 

Rabuel,  C.,  323 
Radical  sign,  408,  409 
Radius,  with   abacus,   138,   178;   geo- 
metric, 278 

Radulph  of  Liege  (c.  1010),  177 
Rahn.,  J.  H.  (c.  1660),  406,  411,  431, 

474 

Rainbow,  343 
Ramsay,  J.  H.,  189 
Ramsay  and  Lanciani,  165 
Ramus,  P.  (c.  1550),  342,  43O 
Range  finder,  363 
Raphson    (Ralphson),    J.    (c.    1715), 

692,  698 

Rara  Arithmetica,  34 
Ratio,  477,  478,  678;  anharmonic,  333, 

334 

Ratios  compounded,  481 
Rechenmeisters,   190 
Rechenpfennig,  191 
Reckoning  on  the  lines,  183 
Recorde,  R.  (c.  1542),  386,  411,  412, 

439 
Recreations,    mathematical,    532.    See 

Problem  and  Problems 
Rectifications,  330 
Refraction,  343 
Regiomontanus    (c.    1470),  427,   429, 

609,  626,  630 
Regius,  H.  (1536),  181 


INDEX 


721 


Regula,  augmenti  et  decrement!,  441 ; 

bilancis,  440;  coecis,  586;  falsi,  437- 

442  ;  inf  usa,  442  ;  lancium,  440 ;  posi- 

tionis,    437-442;    potatorum,    587; 

quatuor     quantitatum,     607;      sex 

quantitatum,  607;  virginum,  587 
Regular  polygons,  301 
Rehatsek,  E.,  597 
Reiff,  R.,  506 
Reimer,  N.  T.,  298 
Reinach,  S.,  50 
Reinaud,  J.,  10 
Relation,  symbols  of,  410 
Remainder,  132 
Requeno,  V.,  200 
Res,  408,  427 
Rhaeticus,  G.  J.   (c.  1550),  610,  621, 

622,  623,  627,  629 
Rhind,  A.  H.  (papyrus),  34 
Rhonius.   See  Rahn 
Riccati,  V.  (c.  1750),  613 
Richardson,  L.  J.,   198 
Richter  (c.  1850),  311 
Riemann,  G.  F.  B.  (c.  1850),  338 
Right-angled  triangle,  288 
Risner,  F.  (c.  1570),  342 
Robbins,  F.  E.,  5,  552 
Robert    of     Chester     (c.  1140),    382, 

426 

Roberts,  E.  S.,  50 
Robertus  Anglicus  (c.  1231),  621 
Roberval,  G.  P.  de  (c.  1640),  688 
Robinson,  L.  G.,  681 
Roche,  E.  de  la  (c.  1520),  407 
Rod,  642 

Rod  numerals,  40,  45 
Rodet,  L.,  168,  278,  423 
Rogimbold  (c.  1010),  177 
Rollandus  (c.  1424),  77 
Romans,  calendar,  659;  fractions,  208; 

measures,  636,  641 ;  numerals,  54 
Rood,  644 

Roomen,  A.  van  (c.  1593),  310 
Roots,  144;  abbreviated  methods,  150; 

approximate,   253;   cube,   144,   148; 

higher,   149;   meaning  of  the  term, 

150;  square,  144,  253 
Rosaries,  196 
Roscoe,  translation,  342 
Rosen,  F.,  388 
Rosenhagen,  G.,  193 
Rossi,  G.,  345 
Roth,  P.  (c.  1610),  474 
Roulette,  328 
Round,  J.  H.,  189 
Rudio,  F.,  303,  312 


Rudolff,  C.   (c.  1525),  384,  408,  428, 

458,  520 

Ruffini,  P.  (c.  1800),  469 
Rule  of  False  Position,  437-442.  See 

also  Regula 

Rule  of  Five,  Seven,  etc.,  491 
Rule  of  Mixtures,  588 
Rule  of  Three,  477,  483;  compound, 

491 

Ruled  surfaces,  326 
Rumi,  al-  (c.  1520),  626 

Sa'adia  ben  Joseph  (c.  930),  212 

Saalfeld,  G.  A.,  167 

Saccheri,  G.  (1733),  335 

Sachau,  E.  C.,  650 

Sagitta,  619 

Saglio,  E.,  209 

Saint- Vincent,  G.  de  (c.  1650),  689 

£aka  forms,  67 

Salami's  abacus,  162 

Salomon,  J.,  101 

Salvemini.   See  Castillon 

Salvianus  Julianus  (c.  125),  545 

Salvino  degli  Armati  (1317),  372 

Sanchu,  171 

Sand  Reckoner,  5,  80 

Sand  table,  156 

Sang  Hung  (c.  nSB.c.),  170 

Sangi,  41,  171-173 

Santa  Maria  de  Ripoll,  75 

Sarada  numerals,  71 

Sargon  (c.  2750  B.C.),  601 

Sato  Shigeharu  (1698),  171 

Sawaguchi  Kazuyuki  (c.  1665),  701 

Sayce,  A.  H.,  230,  554,  560 

Scales,  method  of  the,  440;  of  count- 
ing, 9,  4i 

Scaphe,  370 

Scarburgh  (Scarborough),  C.  (c.  1660), 
618 

Schack-Schackenburg,  H.,  432,  444 

Schepp,  A.,  326 

Scheubel  (Scheybel),  J.  (c.  1550),  428 

Schilling,  F.,  594 

Schisare,  221 

Schlegel,  V.,  268 

Schmid,  W.  (1539),  292 

Schmidt,  J.  J.,  23 

Schoner,  A.  (c.  1560),  430 

Schooten,  F.  van,  the  Younger  (c. 
1656),  344,  469,  474 

Schotten,  H.,  274 

S'choty,  176 

Schubert,  H.,  191 

Schiilke,  A.,  231 


722 


INDEX 


Schiitte,  F.,  270 

Schulenberg,  W.  von,  194,  195 

Schumacher,  H.  C.  (c.  1830),  337 

Scipio  Nasica  (c.  1593.0.),  672 

Score,  192 

Sebestydn,  194 

Sebokht  (c.  650),  64,  72 

Secant,  622 

Second,  232 

Sector  compasses,  347 

Sedgwick,  W.  T.,  337 

Se~dillot,  L.  P.  E.  Am.,  626 

Seebohm,  F.,  637 

Sefer  Jezira,  684 

Seki  Kowa  (c.  1680),  433,  476,  592,  701 

Semidiameter,  279 

Seqt  (seqet,  skd),  600,  619 

Series,  494;  antitrigonometric,  513; 
arithmetic,  498;  convergency  of, 
507 ;  extent  of  treatment  of,  497 ; 
geometric,  500;  Gregory's,  513;  har- 
monic, 503;  higher,  504;  infinite, 
506,  679 ;  kinds  of,  495 ;  logarithmic, 
513;  names  for,  496;  relation  to 
proportion,  497;  sum  of,  17,  497- 
505;  trigonometric,  512 

Serret,  J.  A.  (c.  1865),  692 

Seven,  2 ;  check  of,  154 

Seven  liberal  arts,  3 

Sexagesimal  fractions,  228;  symbols  of, 

234 

Shadows,  17,  602,  620;  tables  of,  621 
'Shanks,  W.  (c.  1853),  311 
Sheffers,  G.,  692 
Shekel,  636 
Shilling,  647 
Shotwell,  J.  T.,  651 
Sie"  Fong-tsu  (c.  1650),  523 
Sieve  of  Eratosthenes,  5 
Sign,  233 

Significant  figures,  15 
Signs,  law  of,  396 
Silberberg,  M.,  100,  152 
Simon,  M.  (c.  1890),  432 
Simson,  R.  (c.  1750),  469 
Simultaneous  equations,  431 
Simultaneous  quadratic  equations,  450 
Sine,  614;  abbreviations  for,  618;  name 

for,  616 
Sines,  addition  theorem  of,  628;  tables 

of,  626;  theorem  of,  630,  632 
Single  False,  440 
Sinus  totus,  627 
Sissah  ibn  Dahir  (c.  1250),  549 
Slane,  Mac  Guckin  de,  549 
Slate,  179 


Sloman,  H.,  692 

Sluze,  R.  F.  W.,  Baron  de  (c.  1660), 

689;  conchoid  of,  327 
Smethurst,  168 
Smith,  A.  H.,  168 
Smith,  D.  E.,  64,  124,  182,  210,  290, 

312,  385,  536,  552,  676 
Smith,  W.,  340 
Smogolenski  (c.  1650),  523 
Smyly,  J.  G.,  53,  164,  256,  294 
Snell,  W.  (1627),  631 
Snelling,  T.,  182 
Solid  analytic  geometry,  325 
Solid  geometry,  291 
Solid  numbers,  19,  25 
Solidus,  646 

Sommerville,  D.  M.  Y.,  335 
Soreau,  R.,  156 
Spaces  (spacia),  183 
Species,  35 
Speculum,  356,  358 
Speidell,  J.  (1620),  517 
Sphere,  294;  astronomical,  603-608 
Spherical  numbers,  25 
Spherical  triangle,  631 
Spinoza,  B.  (c.  1670),  528 
Spirals,  329 

Square,  geometric,  345,  355 
Square  numbers,  18,  24 
Square  root,  144,  253 
Squares,   criteria   for,   256;   sums   of, 

17,  504 

Squaring  the  circle,  298,  302 
Sridhara  (c.  1020),  446 
Stadium,  340,  641 
Stackel,  P.,  335,  337 
Stark,  W.  E.,  345 
Steele,  R.,  32,  78 
Steiner,  J.  (c.  1840),  334 
Steinschneider,  M.   (c.  1850),  34,  36, 

127,  200,  437,  665 
Stephano  Mercatello  (c.  1522),  235 
Stereographic  projection,  344,  351 
Stevin,  S.  (c.  1590),  430 
Stifel,  M.  (c.  1525),  384,  403,  502,  519, 

520 

Stitt,  S.  S.,  595 
Stock,  194,  576 
Stone,  E.  (1740),  326 
Stoy,  H.,  197 
Straight  line,  275 
Studnicka,  F.  J.,  77,  310 
Study,  E.,  261 
Sturm,  A.,  298 
Sturm,  L.  C.  (c.  1710),  85 
Suan-hio-ki-mong,  257 


INDEX 


723 


Suan-pan,  168,  203 

Substractio,  95 

Subtraction,  94,  184;  of  fractions,  223; 
methods  of,  98 ;  symbols  of,  395 

Subtrahend,  96 

Suevus,  12 

Sum,  89,  90;  of  a  series,  17,  497-505 

Sumario  Compendioso,  385,  392,  590 

Sundial,  370,  601,  620,  669-671 

Sun-tzi  (ist  century  ?),  380,  433 

Superstitions,  17 

Supputandi  ars,  n 

Surds,  251,  252,  257 

Surface,  276 

Surfaces,  325;  curvature  of,  325;  ruled, 
326 

Surveying,  317,  344,  363 

Su-shuh  ki-i,  168 

Suter,  H.  (c.  1890),  34,  74,  92,  118, 
164,  437)  481,  550  » 

Swan  pan.   See  Suan-pan 

Sylvester,  J.  J.  (c.  1850),  477 

Sylvester,  Pope.   See  Gerbert 

Symbolism,  poor,  417.  See  also  Sym- 
bols 

Symbols,  of  addition,  395-398,  402 ;  of 
aggregation,  416;  of  Diophantus, 
422-424;  of  division,  406;  of  equal- 
ity, 410;  of  equations,  421,  434;  of 
imaginary  numbers,  266;  of  inequal- 
ity, 413;  of  multiplication,  402;  of 
negative  number,  259,  395~3975 
Oriental,  424;  of  per  cent  and  per 
mill,  250;  of  proportion,  413;  of 
ratio,  406;  of  relation,  410;  of  roots, 
407 ;  in  the  Rule  of  False,  397,  441 ; 
of  subtraction,  395-397;  of  the  un- 
known, 422,  428;  of  Vieta,  430,  449 

Tabit  ibn  Qorra  (c.  870),  455,  685 
Tables,    609    seq.    See    also    Chords, 
Mortality,    Multiplication,    Powers, 
Roots,    Shadows,    Sines,    Tangents, 
Trigonometric  functions 
Tabula  geometricalis  (abaci),  177 
Tait,  P.  G.  (c.  1880),  267,  268 
Takebe  Hikojiro  Kenko  (c.  1722),  702 
Takebe  Kemmei  (c.  1722),  702 
Takeda  Shingen  (1824),  536 
Talent,  635,  636 
Tally  sticks,  171,  192 
Tangent,  620 

Tangential  coordinates,  326 
Tangents,  abbreviations  for,  622 ;  prob- 
lem of,  689 ;  tables  of,  624 ;  theorem 
of,  611,  631 


Tangutans,  169 

Tannery,  P.  (c.  1900),  3,  101,  124,  126, 
254,  255,  313,  322,  453,  478,  604,  605 

Tanstetter,  G.  (c.  1520),  341 

Tanto,  393 

Tare  and  tret,  567 

Tariff,  571 

Tartaglia,  Nicolo  (c.  1545),  26,  384, 
428,  460,  494 

Taurinus,  337 

Tautochronous  curve,  328 

Taxes,  571 

Taylor,  I.,  398 

Taylor,  J.,  92 

Taylor's  Theorem,  512 

Teixeira,  F.  G.,  313,  326 

Telescope,  372 

Temple,  R.  C.,  637 

Tessera,  195 

Testament  problem,  544 

Thales  (c.  600  B.C.),  4,  602;  measuring 
distances,  285 

Theologumena,  27 

Theon  of  Smyrna  (c.  125),  5,  6,  453 

Theophrastus  (c.  350 B.C.),  677 

Theory  of  numbers,  2,  29 

Thibaut,  G.,  288 

Thibetan  "wheel  of  life,"  595 

Thierfelder,  C.  (1587),  439 

Thirteen,  fear  of,  17 

Three,  2 

Thymaridas    (0.380  B.C.),  432 

Thynne,  F.,  564 

Tieffenthaler,  J.  (c.  1750),  366 

Time,  651 

Tithes,  572 

Titulus,  61 

Tod,  M.  N.,  50,  162 

Todhunter,  I.   (c.  1850),  528 

Ton,  645 

Topics  for  Discussion,  31,  155,  207, 
269,  377,  53i,  599,  633,  675,  703 

Touraeff  (Turajev),  B.  A.,  293 

Townsend,  E.  J.,  277 

Tractatus  de  uniformitate,  etc.,  319 

Transcendental  numbers,  268.  See  also 
Logarithms,  TT,  Trigonometric  func- 
tions 

Transversals,  theory  of,  333 

Treichel,  A.,  195 

Tret,  567 

Treutlein,  P.,  197,  255,  497 

Triangle,  288,  290;  arithmetic  (Pas- 
cal's), 508;  right-angled,  288;  spheri- 
cal, 604-608,  631 

Triangular  numbers,  24 


724 


INDEX 


Trigonometric  functions,  623.  See  also 

Sine,  Cosine,  etc. 
Trigonometric  solutions  of  equations, 

474 

Trigonometry,  357,  600;  analytic,  613 
Trisection  of  an  angle,  297,  298 
Trivium,  3 
Tropfke,  J.,  32 
Troy  weight,  638 
Truck  (troquer),  570 
Truel,  H.  D.  (1786),  266 
Tschirnhausen,   E.  W.,   Graf  von   (c. 

1690),  470 
Tseu  pan  tsih,  168 
Tun,  645 

Turajev.   See  Touraeff 
Turchillus  (c.  1200),  177 
Turetsky,  M.,  526 
Turks-and-Christians  problem,  541 
Tycho  Brahe,  310 
Tyler,  H.  W,  337 
Tylor,  E.  B.,  16,  195 

Ulugh  Beg  (c.  I43S),  609 

Umbra  recta,  17,  602,  620 

Umbra  versa,  620 

Uncial  interest,  562 

Unger,  F.,  84 

Unit  fraction,  210,  212 

Unitary  method,  494 

United  States,  arithmetic  in,  86 

United  States  money,  647 

Unity,  13,  26 

University  of  Pennsylvania,  635 

Unknown  quantity,  393 

Uranius,  H.,  637 

Vacca,  G.,  302,  549 

Vakataka  numerals,  67 

Valerianus,  J.  P.  (Bellunensis),  200 

Valerio,  Luca  (1606),  685 

Valhabi  numerals,  67 

Value,  645 

Vander  Hoecke   (c.   1515),  399,  401, 

427 
Vandermonde,  A.  T.    (c.  i775)>  47°, 

476 

Vanh6e,  L.,  40,  42,  425,  448,  585 
Varro  (c.  60  B.C.),  3,  215 
Vassilief,  335 

Vaux,  Carra  de,  64,  229,  587 
Vega,  G.  (c.  1775),  311 
Veratti,  B.,  3 

Verse,  rules  in,  439,  487,  488,  491,  500 
Versed  sine,  618 
Verses  in  notation,  53 


Vicesima  libertatis,  247 

Vicesimatio,  542 

Viedebantt,  O.,  371 

Vieta,  F.  (c.  1580),  310,  392,  430,  449, 

465,  469,  472,  474,   503,  610,  623, 

627,  629,  631 
Vigarie,  E,,  290 
Villicus,  F.,  194 

Vincent,  A.  J.  H.,  162,  165,  339 
Vinculum,  60 
Visierer,  581 
Vissiere,  42 

Vitale,  G.  (Vitalis,  H.)   (c.  1690),  393 
Vitello.   See  Witelo 
Vivanti,  G.,  692 

Vlacq  (Vlack),  A.  (c.  1650),  518 
Vogt,  H.,  251,  289,  686 
Volusius  Maecianus  (2d  century),  215 
Von  Schulenberg,  W.,  194,  195 
Vysierer.  See  Visierer 

Waard,  C.  de,  688 

Wade,  H.  T.,  673 

Waschke,  H.,  81 

Wallace,  W.  (c.  1810),  613 

Wallis,  J.  (c.  1650),  7,  263,  311,  413, 

415,  420,  431,  503,  612,  689,  695 
Wallner,  C.  R.,  686 
Walters,  R.  C.  S.,  359 
Wampum,  196 

Wang  Jung  (3d  century),  170 
Wappler,  H.  E.,  260 
Ward,  J.,  29 
Ward,  S.  (1654),  618 
Warren,  C.,  640 
Wattenbach,  W.,  194 
Wax  tablet,  178 
Weber,  C.  F.,  542 
Week,  655 
Wegener,  A.,  232 
Weidner,  E.  F.,  351 
Weight,  634 

Weissenborn,  H.,  27,  123,  164,  692 
Welsh  (Welsch)  practice,  493 
Wenceslaus  (Wentsel),  M.  (1599),  493 
Wertheim,  G.,  33,  435,  452 
Wessel,  C.  d797>,  265 
Weyr,  E.,  270 
Whipple,  F.  J.  W.,  203 
Whish,  C.  M.,  309 
Whitford,  E.  E.,  452 
Widman  (Widmann),  J.  (1489),  573 
Wiedemann,  E.,  340,  434,  489 
Wieleitner,  H.,  322,  342,  415,  506,  526 
Wiener,  C.,  331 
Wilkens,  M.,  12 


INDEX 


725 


William  of  Malmesbury,  175 

Williams,  S.  W.,  40 

Willichius   (1540),  19 

Willsford,  T.  (c.  1662),  236 

Wilson,  T.,  578 

Wilson's  Theorem,  29 

Wing,  V.  (c.  1648),  407 

Wingate,  E.,  205 

Witelo  (c.  1270),  341 

Witt,  J.  de  (1658),  324 

Witt,  R.,  247,  565 

Witting,  A.,  427 

Woepcke,  F.,  10,  34,  74,  115,  118,  157, 

160,  175,  320,  437,  467,  508,  626 
Wolf,  R.,  604 
Woolhouse,  W.  S.  B.,  650 
Wordsworth,  C.,  652 
Wren,  Sir  C.  (c.  1670),  649 
Wright,  E.  (c.  1600),  404 
Wright,  S.  (c.  1614),  404 
Writing  material,  36,  45 
Wu-ts'aoSuan-king  (c.ist  century), 499 
Wylie,  A.,  42 


Xenocrates  (c.  350  B.C.),  $24 

Yard,  642 

Year,  654,  661 

Yenri  (circle  principle),  701 

Young,  J.  W.  A.,  301 

Zamberto  (c.  1505),  338 

Zangemeister,  K.,  56 

Zarqala,  Ibn  al-  (c.  1050),  609.  616 

Zeitschrift,  109 

Zeno  of  Elea   (c. 450 B.C.),  677 

Zeno    of    Sidon    (ist    century    B.C.), 

281 

Zenso.    See  Censo 
Zero,  44,  69,  71,  74,  78;  equating  to, 

43i 

Zeuthen,  H.  G.  (c.  1900),  274,  296,  378 
Ziegler,  T.,  665 
Zuanne  de  Tonini  da  Coi   (c,   1530), 

460,  467 
Zuni  Indians,  59 
Zuzzeri,  G.  L.,  670