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EASY  MATHEMATICS   OF  ALL  KINDS 
VOL.  I.     CHIEFLY  ARITHMETIC 


EASY    MATHEMATICS 

CHIEFLY   ARITHMETIC 


BEING  A  COLLECTION  OF  HINTS  TO  TEACHERS,  PARENTS, 
SELF-TAUGHT  STUDENTS,  AND  ADULTS 

AND 

CONTAINING  A  SUMMARY  OR  INDICATION  OF  MOST 

THINGS  IN  ELEMENTARY  MATHEMATICS 

USEFUL  TO  BE  KNOWN 


BY 

Sm   OLIVEE  LODGE,   F.RS. 

D.SC.    LOND.   ET  OXON. 

LL.D.    ST.    ANDREWS,   GLASGOW,   AND   A'ICTORIA 

PRINCIPAL  OF   THE  UNIVERSITY   OF  BIRMINGHAM 


MACMILLAN  AND  CO.,   Limited 

NEW   YORK  :    THE   MACMILLAN   COMPANY 
1906 

All  rights  reserved 


First  Edition  1905. 
Reprinted  1906. 


GLASGOW  :    PRINTED   AT  THE   UNIVERSITY   PRES8 
BY  ROBERT  MACLKHOSB  AND  CO.  LTD. 


PA37 


"The  parent  inherits  a  primal  tendency  to  revert 
to  the  fixed  and  rooted  form,  while  the  child  is  '  free- 
swimming  ' ;  it  is  the  natural  explorer.  And  for  ages 
we  the  parents  through  the  teachers  have  been  more 
and  more  successfully  trying  to  train  and  educate  our 
'  free-swimmers '  into  fixed  and  rooted  prisoners  ;  thus 
atrophising  or  mutilating  their  discovering  and  inter- 
pretative powers  just  as  our  own  were  injured  at  the 
same  age.  Lady  "Welby. 

"  There  are  several  chapters  in  most  arithmetic  books 
that  are  wholly  unnecessary  ....  but  a  writer  of  a 
school-book  for  elementary  schools  is  not  his  own 
master  ;  he  must  comply  with  the  often  unwise  demand 
of  teachers  and  examiners."  A.  SONNKNSCHEIN. 


L5t 


PKEFACE. 

This  book  is  written  without  the  least  regard  to  any  demand 
but  those  of  children  and  of  life  and  mental  activity  generally. 
In  places  where  the  author  is  mistaken  he  cannot  plead  that 
he  has  been  hampered  by  artificial  considerations.  His  object 
in  writing  it  has  been  solely  the  earnest  hope  that  the  teaching 
of  this  subject  may  improve  and  may  become  lively  and 
interesting.  Dulness  and  bad  teaching  are  synonymous  terms. 
A  few  children  are  born  mentally  deficient,  but  a  number  are 
gradually  made  so  by  the  efibrts  made  to  train  their  growing 
faculties.  A  subject  may  easily  be  over-taught,  or  taught  too 
exclusively  and  too  laboriously.  Teaching  which  is  not  fresh 
and  lively  is  harmful,  and  in  this  book  it  is  intended  that  the 
instruction  shall  be  interesting.  Nevertheless  a  great  deal  is 
purposely  left  to  the  enterprise  of  the  student  and  the  living 
voice  of  the  teacher,  and  the  examples  given  for  practice  are 
insufficient.  The  author  has  usually  found  that  examples 
and  illustrations  are  likely  to  be  most  serviceable,  and  least 
dull,  when  invented  from  time  to  time  in  illustration  of  the 
principles  which  are  then  being  expounded;  but  a  supple- 
mentary collection  of  exercises  for  practice  is  necessary  also, 
in  order  to  consolidate  the  knowledge  and  establish  the 
principles  as  an  ingrained  habit.  Wearisome  over-practice 
and  iteration  and  needlessly  long  sums  should  be  avoided; 
because  long  sums,  other  than  mechanical  money  addition, 
seldom  occur  in  practice,  and  especially  because  many  kinds 


vm  PREFACE. 

of  future  study,  especially  the  great  group  of  sciences  called 
Natural  Philosophy,  will  be  found  to  afford  plenty  of  real 
arithmetical  practice ;  and  even  ordinary  life  affords  some, 
if  an  open  mind  is  kept.  The  cumbrous  system  of  weights 
and  measures  still  surviving  in  this  country  should  not  bo 
made  use  of  to  furnish  cheap  arithmetical  exercises  of 
preposterous  intricacy  and  uselessness.  There  is  too  much 
of  real  interest  in  the  world  for  any  such  waste  of  time 
and  energy. 

The  mathematical  ignorance  of  the  average  educated  person 
has  always  been  complete  and  shameless,  and  recently  I  have 
become  so  impressed  with  the  unedifying  character  of  much  of 
the  arithmetical  teaching  to  which  ordinary  children  are  liable 
to  be  exposed  that  I  have  ceased  to  wonder  at  the  widespread 
ignorance,  and  have  felt  impelled  to  try  and  take  some  step 
towards  supplying  a  remedy.  I  know  that  many  teachers 
are  earnestly  aiming  at  improvement,  but  they  are  hampered 
by  considerations  of  orthodoxy  and  by  the  requirements  of 
external  examinations.  If  asked  to  formulate  a  criticism  I 
should  say  that  the  sums  set  are  often  too  long  and  tedious, 
the  methods  too  remote  from  those  actually  employed  by 
mathematicians,  the  treatment  altogether  too  abstract,  didac- 
tic, and  un-experimental,  and  the  subject-matter  needlessly 
dull  and  useless  and  wearisome. 

Accordingly,  in  spite  of  much  else  that  pressed  to  be  done, 
a  book  on  arithmetic  forced  itself  to  the  front.  It  is  not 
exactly  a  book  for  children,  though  I  hope  that  elder  children 
will  take  a  lively  interest  in  it,  but  perhaps  it  may  be  con- 
sidered most  conveniently  as  one  continuous  hint  to  teachers, 
given  in  the  form  of  instruction  to  youth ;  and  it  is  hoped 
that  teachers  will  not  disdain  to  use  and  profit  by  it,  even 
though  most  of  them  feel  that  all  the  facts  were  quite  well 
known  to  them  before.     It  is  not  intended  to  instruct  them 


PREFACE.  IX 

in  subject-matter,  but  to  assist  them  in  method  of  presenta- 
tion; and  in  this  a  good  deal  of  amplification  is  left  to  be 
done  by  the  teacher.  But  it  is  of  the  first  importance  that 
the  teacher's  own  ideas  should  be  translucently  clear,  and 
that  his  or  her  feeling  for  the  subject  should  be  enthusiastic : 
there  is  no  better  recipe  for  effective  teaching  than  these  two 
ingredients. 

For  supplementary  hints  in  connexion  with  the  teaching 
of  very  small  children,  a  subject  which  occupies  the  first  four 
chapters,  a  couple  of  little  books  by  Mrs.  Boole  recently  pub- 
lished by  the  Clarendon  Press  may  be  mentioned ;  and  as  a 
convenient  collection  of  suitable  examples  for  practice  I  suggest 
a  set  by  Mr;_^;^  Tucke^j)ublished  by  Bell  and  Sons.  For 
supplementary  information  on  the  higher  parts  of  the  work 
such  a  book  of  reference  as  Chrystal's  Algebra  is  probably 
useful. 

The  author  has  to  thank  Mr.  T.  J.  Garstang,  of  Bedales 
School,  Petersfield,  Hampshire,  and  also  Mr.  Alfred  Lodge, 
of  Charterhouse,  late  Professor  of  Pure  Mathematics  at 
Coopers  Hill,  for  reading  the  proofs  and  detecting  errors 
and  making  suggestions. 


CONTENTS. 


CHAPTER  I. 

The  very  beginnings.  Counting.  Extension  or  application  of  the  idea 
of  number  to  measuring  continuous  quantity.  Introduction  of  the 
idea  of  fractions.  Practical  hints  for  teaching  the  simple  rules. 
Addition.  Subtraction.  Multiplication.  Multiplication  of  money. 
Division.     Division  of  money.     Origin  of  the  symbols  -      pp.  1-27 

CHAPTER  II. 

Further  considerations  concerning  the  Arabic  system  of  notation,  and 
extension  of  it  to  express  fractions  decimally  and  duodecimally. 
Numeration pp.  28-36 

CHAPTER  III. 

Further  consideration  of  division  and  introduction  of  vulgar  fractions. 
Extension  of  the  term  multiplication  to  fractions.  Practical 
remarks  on  the  treatment  of  fractions  -  -         "     PP-  37-44 

CHAPTER  IV. 

Further  consideration  and  extension  of  the  idea  of  subtraction.  Addi- 
tion and  subtraction  of  negative  quantities    -         -         -     pp.  45-50 

CHAPTER  V. 

Generalisation  and  extension  of  the  ideas  of  multiplication  and  division 
to  concrete  quantity.     First  idea  of  involution      -         -     pp.  51-57 

CHAPTER  VI. 
Factors  of  simple  numbers pp.  58-61 


xii  CONTENTS. 

CHAPTER  VII. 

Dealings  with  money  and  with  weights  and  measures.  Modern  treat- 
ment of  the  rule  called  ''practice."  The  practical  advantages  of 
decimalisation.  Decimalisation  of  money.  T3rpical  exercises. 
Binary  scale.  Decimal  system  of  weights  and  measures.  Decimal 
measures.     Angles  and  time.     Further  exercises  -         -     pp.  62-85 

CHAPTER  VIII. 

Simple  proportion.  Breakdown  of  simple  proportion  or  "  rule  of 
three  " pp.  86-94 

CHAPTER  IX. 
Simplification  of  fractions pp.  95-101 

CHAPTER  X. 

Greatest  common  measure  and  least  common  multiple.  Rule  for 
finding  G.C.M.  Algebraical  statement  of  the  process  for  finding 
G.C.M. pp.  102-108 

CHAPTER  XI. 

Easy  mode  of  treating  problems  which  require  a  little  thought 

pp.  109-115 
CHAPTER  XII. 
Involution  and  evolution  and  beginning  of  indices        -  pp.  116-121 

CHAPTER  XIII. 

Equations  treated  by  the  method  of  very  elementary  experiment. 
Further  consideration  of  what  can  be  done  to  equations 

pp.  122-137 
CHAPTER  XIV. 

Another  treatment  of  equations.     Introduction  to  quadratics 

pp.  138-148 
CHAPTER  XV. 
Extraction  of  simple  roots.     Surds        .         .         .         .  pp.  149-163 

CHAPTER  XVI. 

Further  consideration  of  indices.      Fractional  indices. 

Negative  indices pp.  154-^161 


I 


CONTENTS.  xiii 

CHAPTER  XVII. 

Introduction  to  logarithms pp.  162-164 

CHAPTER  XVIII. 

Logarithms.      Common    practical    base.      Examples.      Examples    for 
practice.     Fundamental  relations   ...         -        pp.  165-176 

CHAPTER  XIX. 
Further  details  about  logarithms pp.  177-183 

CHAPTER  XX. 
On  incommensurables  and  on  discontinuity   -        -        -        pp.  184-192 

CHAPTER  XXI. 

Concrete  arithmetic.     The  meaning  of  significant  figures  and  practical 
accuracy pp.  193-197 

CHAPTER  XXII. 

Practical  manipulation  of  fractions  when  decimally  expressed. 

"Order"  of  numbers pp.  198-209 

CHAPTER  XXIII. 
Dealings  with  very  large  or  very  small  numbers    -        -        pp.  210-218 

CHAPTER  XXIV. 
Dealings  with  vulgar  fractions.     Numerical  verifications      pp.  219-224 

CHAPTER  XXV. 
Simplification  of  fractional  expressions  -         -         -         .        pp.  225-229 

CHAPTER  XXVI. 
Cancelling  among  units pp.  230-234 

CHAPTER  XXVn. 
Cancelling  in  equations.     Caution  -        -        -        -        pp.  235-238 

CHAPTER  XXVIII. 

Further  cautions.     Cautious  of  a  slightly  more  advanced  character 

pp.  239-245 


XIV  CONTENTS. 

CHAPTER  XXIX. 

Illustrations  of  the  Practical  Use  of  Logarithms. 

(i).  How  to  look  out  a  logarithm,  (ii).  How  to  look  out  the  number 
which  has  a  given  logarithm.  Examples.  Logarithms  of  fractions, 
(iii).  How  to  do  multiplication  and  division  with  logs.    pp.  246-257 

CHAPTER  XXX. 

How  to  find  powers  and  roots  by  logarithms.      Exercises. 

Roots  of  negative  numbers pp.  258-264 

CHAPTER  XXXI. 

Geometrical  illustration  of  powers  and  roots.  Further  geometrical 
methods  of  finding  square  roots       ....        pp.  265-276 

CHAPTER  XXXII. 

Arithmetical  method  of  finding  square  roots  -        -        pp.  277-282 

CHAPTER  XXXIIL 

Simple  algebraic  aids  to  arithmetic,  etc.  Illustrations.  Problems. 
Proof  of  square  root  rule.     Cubes  and  cube  root.     Approximations 

pp.  283-310 

CHAPTER  XXXIV. 
To  find  any  power  of  a  binomial.     Exercise.     Examples        pp.  311-321 

CHAPTER  XXXV. 

Progressions.  Examples.  Algebraic  digression.  General  expression 
for  any  odd  number.  Arithmetical  progression.  Other  series. 
Geometrical  illustrations pp.  322-338 

CHAPTER  XXXVL 

Means.  Examples.  Mean  or  average  of  a  number  of  terms.  Weighted 
mean.    General  average.    Geometric  mean.    Example     pp.  339-351 

CHAPTER  XXXVIL 

Examples  of  the  practical  occurrence  of  progressions  in  nature  or  art. 
Interest pp.  352-358 


CONTENTS.  XV 

PART  IL 

MISCELLANEOUS  APPLICATIONS  AND  INTRODUCTIONS. 

CHAPTER  XXXVIII. 

Illustrations  of  important  principles  by  means  of  expansion  by  heat. 
Examples.     Cubical  expansion        ....         pp.  361-371 

CHAPTER  XXXIX. 

Further  illustrations  of  proportionality  or   variation. 

Inverse  variation.     Summary pp.  372-400 

CHAPTER  XL. 

Pumps  and  leaks.     Leaks.     Cooling.     Electric  leakage.     Continuously 
decreasing  G.P.     Summary pp.  401-411 

CHAPTER  XLI. 
Differentiation.     Examples pp.  412-418 

CHAPTER  XLII. 
A  peculiar  series.     Natural  base  of  logarithms       -         -        pp.  419-426 


APPENDIX. 

I.  Note  on  the  Pythagorean  numbers  (Euc.  I.  47)        -  pp.  427-430 

II.  Note  on  Implicit  dimensions pp.  430-434 

III.  Note  on  factorisation pp.  434-435 

IV.  Note  on  the  growth  of  population  ....  pp.  435-436 


CHAPTER  I. 

The  very  beginnings. 

Concerning  the  early  treatment  of  number  for  very  small 
children  the  author  is  not  competent  to  dogmatise,  but  he 
offers  a  few  suggestions,  the  more  willingly  inasmuch  as  he 
is  informed  by  teachers  that  a  great  deal  of  harm  can  be  and 
often  is  done  by  bad  teaching  at  the  earliest  stages,  so  that 
subsequently  a  good  deal  has  to  be  unlearnt.  The  principle 
of  evolution  should  be  recollected  in  dealing  with  young 
children,  and  the  mental  attitude  of  the  savage  may  often  be 
thought  of  as  elucidating  both  the  strength  and  the  weakness 
of  their  minds. 

Counting  is  clearly  the  first  thing  to  learn;  it  can  be 
learnt  in  play  and  at  meals,  and  it  should  be  learned  on 
separate  objects,  not  on  divided  scales  or  any  other  con- 
tinuous quantity.  The  objects  to  be  counted  should  be 
such  as  involve  some  childish  interest,  such  as  fruit  or  sweets 
or  counters  or  nuts  or  coins.  Beans  or  pebbles  will  also  do, 
but  they  should  not  be  dull  in  appearance,  unattractive  as 
objects  of  property,  and  so  not  worth  counting.  The  pips 
on  ordinary  playing  cards  will  also  serve,  and  they  suggest 
a  geometrical  or  regular  arrangement  as  an  easy  way  of 
grasping  a  number  at  a  glance. 

Counting  should  begin  with  quite  small  numbers  and 
should  not  proceed  beyond  a  dozen  for  some  time,  but  there 


2  EASY  MATHEMATICS.  [chap. 

is  no  object  in  stopping  or  making  any  break  at  ten.  Several 
important  facts  (the  facts  only,  not  their  symbolic  expression) 
can  now  be  realised :  such  as  that  3  +  4  =  7,  that  7  -  4  =  3, 
that  two  threes  are  6,  and  that  three  twos  are  the  same, 
without  any  formal  teaching  beyond  a  judicious  question  or 
two.  The  lessons,  if  they  can  be  called  lessons,  should  go 
on  at  home  before  school  age ;  but,  whether  this  initial  train- 
ing is  done  at  school  or  elsewhere,  formal  teaching  at  this 
stage  should  be  eschewed,  since  it  necessarily  consists  largely 
in  coercing  the  children  to  arrive  at  some  fixed  notion  which 
the  teacher  has  preconceived  in  his  mind — a  matter  usually 
of  small  importance.  The  children  should  form  their  own 
notions,  and  be  led  to  make  small  discoveries  and  inventions, 
if  they  can,  from  the  first.  Mathematics  is  one  of  the  finest 
materials  for  cheap  and  easy  experimenting  that  exists.  It 
is  partly  ignorance,  and  partly  stupidity,  and  partly  false 
tradition  which  has  beclouded  this  fact,  so  that  even  influen- 
tial persons  occasionally  speak  of  mathematics  as  "  that  study 
which  knows  nothing  of  observation,  nothing  of  induction, 
nothing  of  experiment," — a  ghastly  but  prevalent  error  which 
has  ruined  more  teaching  than  perhaps  any  other  misconcep- 
tion of  the  kind. 

As  soon  as  small  groups  can  be  quickly  counted,  and 
dimple  addition  and  subtraction  performed  with  a  few  readily 
grasped  and  interesting  objects — and  the  more  instinctively 
such  operations  can  be  done  the  better, — the  time  is  getting 
ripe  for  the  introduction  of  symbols — for  that  arbitrary  and 
conventional  but  convenient  symbolism  whereby  !•!  is  de- 
noted by  a  crooked  line,  5,  and  so  on  :  a  symbolism  which  the 
adult  is  only  relieved  from  the  necessity  of  elaborating  and  feel- 
ing difficult  because  of  the  extreme  docility  and  acquisitiveness 
of  childhood.  It  has  already  learned  26  symbols,  it  will 
patiently  absorb  nine  or  ten  more,  especially  as  they  are  soon 


I.] 


COUNTING. 


3 


found  to  be  real  conveniences;  though  if  an  adult  wishes 
to  realise  the  genuine  difficulty  of  the  process — always  a 
most  desirable  thing  to  do — he  should  set  to  work  to  learn 
the  Morse  telegraphic  alphabet,  especially  in  the  forms  used 
for  cable  telegraphy. 

I  see  no  reason  now  why  '.'.  .'.  should  not  be  written 
4  +  5  =  9,  or  soon  afterwards  why  ;  •(:)  might  not  be  written 
5-2  =  3;  but  let  no  one  suppose  that  these  steps  in  nomen- 
clature are  easy.  The  nomenclature  introduced  is  just  as 
hard  as  that  of  trigonometry  or  the  calculus,  only  adult 
persons  are  accustomed  to  the  one  and  are  often  unacquainted 
with  the  other.  A  set  of  little  blocks,  or  some  simple  cheap 
squared  paper  lends  itself  to  statements  like  the  following : 


5= 


cn 


H 

n 

B 


6  =         2+2+2    =    3+3 

Fig.  1. 


4+2 


5+1 


I  see  no  reason  for  troubling  about  the  names  "  addition " 
and  "subtraction,"  nor  yet  for  artificially  withholding  them. 
Jf  they  come  naturally  and  helpfully,  let  them  come. 
Nothing  is  gained  by  artificial  repression  at  any  stage. 
Premature  forcing  of  names  is  worse  than  artificial  with- 
holding of  them,  but  both  are  bad.  If  a  gas,  bubbling  out 
of  soda  water  and  extinguishing  a  flame,  is  familiarly  known 
as  "  carbonic  acid,"  let  it  be  called  so :  it  is  a  help  to  have 


4  EASY  MATHEMATICS.  [chap. 

a  label  with  which  to  associate  observed  properties,  just  as 
it  is  convenient  to  call  a  certain  flower  "  daisy,"  or  a  certain 
star  "Sirius."  But  to  supply  the  label  and  withhold  the 
object,  to  lecture  about  daisies  or  stars  or  numbers  before 
they  have  been  seen,  is,  let  us  politely  say,  unwise. 

It  seems  to  me  that  card  games  with  counters  may  now 
be  introduced,  to  enable  the  children  to  realise  that  their 
property  may  mount  up  beyond  the  smaller  numbers  that 
would  be  wholesome  with  sweets ;  and  they  can  learn  how  to 
group  their  counters  into  packets  of  six,  or  even  into  dozens, 
and  then  they  will  have  simply  to  count  their  packets  and  the 
odd  ones  over.  A  child  with  four  packets  of  six  and  three 
over  would  have  a  real  idea  of  his  wealth,  though  "  twenty- 
seven  "  might  still  be  a  meaningless  expression. 

Differently  coloured  counters  are  now  serviceable  to  replace 
the  packets,  and  thus  the  idea,  but  not  the  word,  of  different 
"  denominations "  will  be  imperceptibly  arrived  at :  and  it 
will  be  clinched  by  the  at  first  unexpected  discovery  that 
even  strangers  will  accept  one  white  coin  as  equivalent  to 
six  much  larger  brown  ones. 

After  this,  some  approach  toward  the  admirable  Arabic 
notation,  whereby  value  is  symbolised  by  place  or  position 
as  well  as  by  shape  of  digit,  may  be  unobtrusively  entered 
on.  The  idea  of  boxes  or  cases,  or  spaces  of  different 
value,  in  one  of  which  odd  counters  or  pennies  are  to  be 
stored,  another  one  in  which  packets,  or  silver  coins,  are  to 
be  kept ;  and  ultimately,  but  not  too  soon,  a  third  one  which 
is  to  be  occupied  by  packets  of  packets,  or  gold  coins ;  if  ever 
such  wealth  were  attained. 

While  there  is  every  advantage  in  thus  emphasising  atten- 
tion to  the  value  or  place  of  the  digit,  and  so  to  a  system  of 
numeration,  there  are  many  reasons  against  concentrating 
attention  on  the  particular  number  "  ten  "  prematurely :  it  is 


I.]  COUNTINa.  5 

not  a  specially  natural  number,  for  one  thing ;  for  another 
thing  it  is  so  large  that  ten  packets  of  ten  are  unlikely  to  occur, 
whereas  four  packets  of  four,  or  six  of  six  are  quite  possible. 
Another  reason  is  that  it  is  undesirable  to  suggest,  what  habit 
will  subsequently  only  too  erroneously  enforce,  that  there  is 
something  special  and  divine  about  the  number  ten,  so  that 
the  arrangement  of  digits  12  cannot  help  meaning  a  dozen. 
This  false  idea,  due  merely  to  habit,  will  not  occur  to  a  child, 
nor  will  he  know  intuitively  that  twelve  pence  make  a 
shilling,  or  twenty  shillings  a  sovereign;  indeed,  strange  to 
say,  he  is  usually  somewhat  callous  as  to  the  importance  of 
this  pivot  of  human  existence;  and,  though  he  soon  gets  to 
like  coins,  he  attends  chiefly  to  their  number  without  much 
regard  to  their  denomination,  unless  some  are  specially  new 
and  bright. 

Having  got  so  far,  the  conventional  symbolism,  in  which 
practice  has  been  quietly  going  on  in  the  background  during 
the  few  more  formal  school  quarter-hours,  may  be  extended, 
and  the  digit-symbols  written  in  spaces  drawn  to  represent 
the  boxes,  or  on  paper  ruled  into  quarter-inch  squares,  which 
is  cheaply  and  plentifully  accessible,  so  that  a  4  put  in  one 
box  shall  signify  4  counters,  while  a  4  put  in  another  box 
shall  signify  4  packets  of  say  ten  counters  each,  so  that  at  the 

end  of  a  game  |  0  |  3  |  shall  mean  that  the  loser  has  no  packets 
and  only  three  counters  altogether,  while  another  child  may 


have    3  I  0  ;  that  is,  three  complete  packets  and  none  over. 
A  third  may  have  two  packets  and  five  over ;  that  is  to  say 

and  another,  the  winner  at  the  game,   may  possess 
or  in  words,  1  packet  of  packets,  5  simple  packets, 


2    5 


15    2 


and  2  odd  ones. 

The   packets   may  be   represented   by  otherwise  coloured 
counters,  or  the  well  known  Tillich  bricks  or  other  Kinder- 


6  EASY  MATHEMATICS  [chap. 

garten  devices  can  be  employed  for  convenience ;  the 
important  thing  is  not  prematurely  (i.e.  not  until  the  under- 
lying reality  has  been  essentially  grasped)  to  proceed  to  the 
only  partially  expressive  symbolism  25  or  152,  which  to 
us  by  mere  habit  looks  so  living  and  significant.  Let  the 
elementary  teacher  reflect  that  to  a  mathematician  the  symbol 


i 


e~*  dx  looks  equally  living  and  significant,  and  be  not  hasty 

with  the  children. 

At  the  same  time  there  is  no  need  for  artificial  delay.  A 
child  brought  along  the  right  lines  will  jump  forward  without 
difficulty,  will  recognise  the  places  without  the  boxes,  will 
get  accustomed  to  the  savage's  mode  of  reckoning  by  tens 
without  being  encouraged  to  go  through  the  savage  process 
of  counting  on  his  fingers,  and  before  long  will  be  able 
to  interpret  such  a  complicated  symbolism  as  50327,  or 
.£175.  16s.  lid.  The  last,  indeed,  is  properly  spoken  of  as 
"  compound  "  instead  of  simple,  for  in  it  "  scales  of  notation  " 
are  badly  mixed  up.  The  reckoning  proceeds  by  tens,  by 
dozens,  and  by  scores,  sometimes  one  and  sometimes  another, 
occasionally  by  quarters  also. 

The  poor  child  who  finds  himself  able  to  master  this  and 
the  operations  which  arise  out  of  it,  need  not  be  deterred  by 
any  legitimate  obstacles  in  mathematics  until  he  comes  to  its 
really  higher  walks,  beyond  simple  differential  equations :  a 
step  which  he  will  not  be  called  upon  to  take  at  all  unless  he 
is  born  to  be  a  mathematician,  in  which  case  difficulties  of  any 
ordinary  kind  will  barely  be  felt. 

The  operations  of  addition  and  subtraction  may  now  be 
extended.  7  +  5  may  be  done  into  a  packet  of  one  dozen,  or 
into  a  packet  of  ten  and  two  over,  and  denoted  by  1/-  or  12 
according  to  which  plan  of  grouping  is  adopted. 

So  also  8  +  7  may  be  called  either  1/3  or  15,  the  former 


1.]  COUNTING.  7 

being  the  custom  if  they  are  pennies,  the  latter  if  they  are 
nuts. 

It  is  necessary  to  apologise  to  children  for  this  needless 
complication;  but  they  inherit  some  things  that  are  good, 
to  make  up  for  several  things  that  are  stupid,  and  therewith 
they  will  have  to  be  content : — 

8  +  7  +  9,  if  shillings,  will  be  grouped  differently  again,  and 
be  denoted  by  £1.  4s.;  if  pennies,  they  will  be  denoted  thus, 
2/- ;  if  ounces,  they  will  be  written  1  lb.  8  oz. ;  if  feet,  they  will 
be  called  8  yards ;  if  farthings,  they  will  be  written  6d. ;  if 
oranges,  they  will  be  called  2  dozen;  but  if  boys,  they  will 
be  written  24. 

I  do  not  recommend  anyone  to  confuse  the  minds  of  children 
by  pointing  out  these  anomalies,  or  by  quoting  a  sample  of 
them  simultaneously  as  above.  Children  will  not  detect  their 
true  character,  but  will  docilely  receive  them  as  if  all  this 
rubbish  were  part  of  the  laws  of  nature.  This  may  account 
for  their  disinclination  later  on  to  make  acquaintance  with  any 
more  of  those  laws  than  they  can  help,  but  at  this  stage  they 
are  docile  and  assimilative  enough :  they  can  at  this  stage  be 
taken  advantage  of  with  impunity.  But  I  should  very  much 
like  to  confuse  the  minds  of  some  teachers,  and  of  some  school 
inspectors — especially  some  varieties  of  school  inspector  and 
university  examiner — and  get  them  into  a  more  apologetic 
and  humble  mood  at  having  to  insist  on  filling  the  mind  of  a 
child  with  any  more  of  these  artificial  insular  conventions 
than  is  absolutely  necessary  in  the  present  stage  of  British 
political  and  commercial  wisdom. 

It  is  undesirable  to  hasten  forward  to  numbers  involving 
3  digits  too  quickly ;  they  can  be  mentioned  and  illustrated 
when  convenient,  but  real  work  should  for  some  time  be  limited 
to  2  figure  numbers,  because  in  these  the  real  principles  can 
be  recognised  and  grown  accustomed  to  in  the  simplest  way. 


6 


EASY  MATHEMATICS. 


[chap. 


The  early  operations  in  which  practice  can  be  given  are 
such  as  the  following :  Suppose  counters  are  employed  and 
that  little  cases  have  been  made  which  just  hold  six  or  ten 
or  any  convenient  number,  suppose  ten : 

Then  13  will  stand  for  one  packet  of  ten  and  three  counters 
over; 
17  added  to  it  will  amount  to  two  packets  and  ten 
counters  over;  which  the  child,  if  encouraged  by 
the  sight  of  an  unused  case  available,  may  wish 
to  make  up  into  3  whole  packets,  and  so  recognise 
the  propriety  of  denoting  the  number  by 
30 

Similarly  15  +  17  will  make  up  into  three  packets  and 
2  over,  which  may  be  shown  thus  : 


make 
which  equals 


while  25  +  37  will  equal  five  packets  and  twelve  over,  or  six 
packets  and  2  over;  29  +  37  =  66,  but  it  is  equally  per- 
missible to  keep  it  as  5  packets  and  16  counters  over,  if  it 
should  happen  to  be  convenient — as  it  sometimes  is. 

To  take    4  from  17  is  easy, 
but  to  take    9  from  1 7  will  involve  emptying  a  case ;  and  only 

8  counters  will  be  left. 

To  take  13  from  25  can  be  done  by  removing  1  case  and 
2  counters ; 

to  take  15  from  25  is  also  easy ; 
but  to  take  16  from  25  involves  the  breaking  up  of  a  packet. 


tens 
1 
1 

ones 

5 

7 

2 

12 

3 

2 

i.j  COUNTING.  d 

After  a  time  these  operations  can  be  followed  when  nothing 
concrete  is  present;  but  abstractions  are  not  natural  co 
children,  and  before  calling  upon  them  to  follow  a  difficult 
conventional  subtraction  sum  like 

82 
37 

45 

the  operation  of  breaking  up  packets  should  be  introduced 
into  the  symbolism  which  is  employed  to  faintly  shadow  the 
concrete  reality. 

It  is  perfectly  right  to  speak  of  3  packets  and  13  loose 
counters,  although  they  may  be  more  compactly  grouped  as 
4  packets  and  3  counters.  So  if  we  have  to  subtract  say  7 
from  43  we  shall  first  break  up  one  of  the  four  packets,  so 
as  to  turn  43  into  3  packets  and  13,  and  then  subtract  the 

7  without  difficulty,  leaving  what  is  abbreviated  into  36. 
Hence    before   doing   the    above    conventional  little   sum, 

8  packets  and  2  should  be  expressed  as  7  packets  and  12,  or 

From  this  3  packets  and  7  have  to  be  removed, 


7    12 


leaving  obviously  4  packets  and  5.  Wherefore  82-37  =  45 
without  any  argument. 

The  abbreviated  form  of  the  above  breaking-up  operation, 
called  borrowing,  will  now  gradually  almost  suggest  itself,  if 
many  sums  of  the  kind  are  given  to  be  done.  But  the  best 
and  easiest  method  of  subtraction  is  the  complementary 
method,  and  if  this  is  taught  from  the  first,  the  complexity 
of  borrowing  becomes  unnecessary. 

The  adult  cannot  too  clearly  realise  that  many  of  the 
operations  to  which  he  has  grown  accustomed  are  labour- 
saving  shorthand  devices  with  the  vitality  and  principle 
abbreviated  out  of  them;  quite  rightly  so  for  practical  pur- 
poses but  not  for  educational  purposes.     The  race  invented 


10  EASY  MATHEMATICS.  [chap. 

them  at  first  in  more  elaborate  shape,  and  gradually  abbre- 
viated them  into  their  present-day  form.  The  child  will 
likewise  get  accustomed  to  this  form  in  due  time,  but  he 
should  not  be  over-hurried  into  it. 

After  adding  two  numbers  for  some  time  we  may  proceed 
to  add  more  than  two, 

and  find  that  7  +  9  -f  6  =  22,  etc. ; 
also  that  7 -I- 7 +.7  =  21, 
and  it  is  natural  to  speak  of  this  as  three  sevens. 

So  also  the  fact  that  5  +  5  +  5  +  5  =  20  will  naturally  be 
quoted  as  four  fives  make  twenty  ;  and  thus  the  essential  idea 
of  multiplication  will  arrive,  as  a  shorthand  and  memorised 
summary  of  the  addition  of  a  number  of  similar  things, 
without  any  use  of  the  name  multiplication  or  any  feeling 
of  a  new  departure.  To  find  the  value  of  three  seventeens, 
that  is,  to  group  them  into  tens  and  ones,  is  a  problem  for 
an  afternoon,  and  if  it  be  done  with  counters  in  the  first 
instance,  and  ultimately  with  symbols,  the  meaning  of  the 
operations  having  been  realised  beforehand  with  the  counters, 
so  much  the  better. 

The  operation  of  adding  or  multiplying  means  grouping  the 
whole  number  into  tens  and  ones,  or  into  hundreds,  tens,  ones, 
etc.,  instead  of  in  the  given  groups. 

A  child  must  not  be  expected  to  be  able  to  formulate  his 
conception  of  the  operations,  or  to  express  them  accurately  in 
words,  at  this  stage.  It  is  a  capital  exercise  later,  but  it  is 
enough  at  first  for  him  to  realise  the  meaning  of  what  he  is 
doing  in  the  back  of  his  mind.  From  time  to  time  he  can  be 
encouraged  to  interpret  processes  into  words,  but  they  must 
have  become  familiar  first.  To  be  able  to  apply  a  rule,  from  a 
precise  statement  in  words  of  what  has  to  be  done,  is  an  adult 
accomplishment,  often  not  reached  by  adults.  To  dissect  out 
and  state  a  rule  in  words,  from  a  knowledge  of  what  the 


I.]  EDUCATION.  11 

operation  really  is,  is  perhaps  easier,  and  is  a  desirable  gift,  but 
it  is  a  training  in  the  use  of  language  rather  than  in  the  subject 
matter  of  the  craft.  It  is  most  appropriate  and  valuable  prac- 
tice for  children  at  the  proper  stage,  a  stage  reached  much 
earlier  with  some  children  than  with  others.  Children  who 
reach  the  word-expression  stage  late  are  usually  called  "stupid." 
If  this  adjective  implies  a  stigma  it  is  usually  undeserved. 
There  is  a  performance  appropriate  to  each  stage  of  develop- 
ment, and  opprobrious  epithets  are  generally  employed  by 
those  who  seek  to  force  things  several  stages  too  soon.  A 
highly  trained  and  clever  dog  would  soon  prove  himself 
"  stupid "  if  tested  by  a  formula,  or  by  words  even  of  only 
3  letters.  An  adult  who  can  hum  or  whistle  an  air  may  be 
told  that  he  ought  to  be  able  to  sit  down  and  write  it  in  the 
recognised  musical  notation.  Similarly  he  ought  to  be  able  to 
read  off  a  piece  of  music  handed  to  him.  He  might  resent 
being  called  stupid  if  he  found  it  difficult  to  do  these,  to  some, 
so  simple  things. 

"  Badness  "  of  many  kinds  may  exist  in  spoiled  children  (and 
there  are  several  ways  of  spoiling  them),  but  badness  in  un- 
spoiled children  is  rare,  and  stupidity  is  almost  non-existent 
unless  they  are  physiologically  out  of  order  and  therefore 
mentally  deficient.  Stupidity  is  however  a  product  easily 
cultivated  by  improper  feeding,  especially  improper  mental 
feeding.  The  "  badness  "  of  children  is  largely  the  effort  which 
nature  makes  at  self-preservation ;  for  inattention  and  laziness 
are  the  weapons  whereby  an  attack  of  mental  indigestion  can 
be  warded  off. 

The  only  fault  with  very  young  children  is  that  they  are 
too  good,  and  therefore  too  easily  damaged.  Later  on,  a  spirit 
of  rebellion  acts  as  a  preservative,  but  it  would  be  better  to 
dispense  both  w^ith  the  rebellious  spirit  and  with  the  causes 
which  necessitate  it. 


U  EASY  MATHEMATICS.  [chap. 

Returning  from  this  digression,  which  is  either  false  or  else 
of  very  extensive  application,  to  our  immediate  subject,  viz., 
the  introduction  of  the  fundamental  operations  to  be  performed 
on  number, — and  remember  that  what  are  called  the  first  four 
simple  rules  are  tremendously  fundamental  and  important, 
more  important  than  anything  which  follows,  until  involution, 
evolution,  and  logarithms  are  arrived  at, — we  must  exercise 
children  in  Multiplication  and  teach  them  something  of  the 
multiplication  table,  at  first  experimentally,  but  afterwards  by 
straightforward  memory  work,  for  it  is  one  of  the  things  with 
which  the  memory  may  be  rightly  loaded.  We  can  next 
recognise  that  Division  too  can  be  unceremoniously  introduced 
by  trying  to  split  up  numbers  into  equal  parts.  The  endeavour 
to  share  sweets  or  fruit  or  cards  or  counters  is  an  obvious 
beginning.  Then,  since  children  are  docile,  they  can  be  asked 
to  split  up  2  packets  and  7  into  three  equal  groups,  or  they 
can  be  asked  to  split  up  2  packets  and  4  into  eight  equal  groups, 
and  so  on  ;  for  no  reason  assigned.  But  it  must  be  recognised 
that  the  operation  of  division  in  general  is  rather  hard,  and 
involves  a  good  deal  of  tentative  procedure  or  guess  work. 
In  other  words  it  involves  the  rudiments  of  experiment  and 
verification.  Gradually,  when  the  multiplication-table  is  fairly 
known  over  some  little  range,  children  can  be  encouraged  to 
apply  theory  before  practice  and  actually  to  think  out  the 
result  before  trying  it ;  but  this  is  a  lesson  in  deductive 
reasoning,  and  represents  the  nascent  beginnings  of  a  loftier 
mode  of  procedure  than  ordinary  adults  are  accustomed  to 
apply  to  their  affairs.  When  asked  to  split  28  into  four  equal 
heaps,  it  is  an  application  of  pure  theory  to  remember  that  4 
sevens  are  28  and  then  to  count  out  seven  counters  into  each 
heap  at  once.  The  empirical  mode  would  be  a  method  of 
dealing  out  singly  into  four  groups  and  then  counting  the 
result.     It  is  easily  done  with  ordinary  playing  cards,  but 


I.] 


EDUCATION.  13 


its  value  as  training  is  much  enhanced  if  theory  is  applied 
first. 

If  for  instance  30  cards  were  given,  to  be  dealt  to  four 
players,  the  residue  that  will  not  go  round  to  be  put  in  the 
middle  or  pool,  a  decided  effort  is  required  for  a  child  to 
perceive  that  there  will  be  two  for  the  pool  and  seven  for 
each  player  :  but  if  he  could  have  time  allowed  him  so  to 
think  it  out,  and  then  to  make  the  experiment,  he  would  be 
conscious  that  his  powers  were  developing,  and  he  would  in 
reality  be  introduced  to  the  first  beginnings  of  a  mode  of 
comprehending  nature  such  as  is  in  the  higher  stages  reserved 
for  men  of  science, — using  the  term  science  in  its  most  com- 
prehensive signification. 

It  is  very  often  a  mistake  for  teachers  to  suppose  that  some 
things  are  easy  and  other  things  are  hard ;  it  all  depends  on 
the  way  they  are  presented  and  on  the  stage  at  which  they 
are  introduced.  To  ascend  to  the  first  floor  of  a  house  is 
difficult  if  no  staircase  is  provided,  but  with  a  proper  staircase 
it  only  needs  a  little  patience  to  ascend  to  the  roof.  The 
same  sort  of  steps  are  met  with  all  the  way,  only  there  are 
more  of  them.  To  people  who  live  habitually  on  the  third 
floor  it  is  indeed  sometimes  easier  to  go  on  to  the  roof  than 
to  descend  into  the  basement.  Educators  should  see  that  they 
do  not  forcibly  drive  children  in  shoals  up  an  unfinished  or  ill- 
made  stairway,  which  only  the  athletic  ones  can  climb.  It  is 
extremely  difficult  in  familiar  subjects  not  to  go  too  fast.  The 
effort  sometimes  results  in  a  process  of  going  too  slowly,  which 
is  wearisome  and  depressing  and  the  worse  fault  of  the  two. 

Extension  or  Application  of  the  idea  of  number  to 
measuring  continuous  quantity. 

So  far  we  have  been  employing  number  to  count  discrete 
objects,  and  to  perform   simple   operations  of   addition,  and 


14  EASY  MATHEMATICS.  [chap. 

the  like,  among  them.  It  is  now  appropriate  to  introduce 
the  idea  of  multiples  of  a  unit,  so  that  one  thing  can  be  twice 
as  long  or  twice  as  heavy  as  another,  without  being  in  another 
sense  "two"  at  all.  The  lines  on  ruled  paper  enable  one 
easily  to  draw  across  them  a  line  twice  or  three  times  or  six 
times  as  long  as  another.  So  also  letter-scales  can  be  used  to 
show  that  a  penny  is  twice  as  heavy  as  a  half-penny,  that  a 
half-crown  weighs  how  many  sixpences,  and  the  like. 

Given  a  foot  rule  they  can  measure  the  size  of  furniture, 
or  of  books.  Given  a  few  ounce  weights  they  can  make  very 
rough  estimates  of  the  weights  of  things  that  have  or  might 
have  to  go  by  post. 

It  is  desirable  not  to  dwell  on  these  things  at  this  stage, 
but  simply  to  accustom  a  child  to  recognise  a  rod  6  inches 
long,  and  such  like,  and  to  see  instinctively  and  without 
formula  or  expression  that  number  may  he  applied  to  con- 
tinuous magnitude  by  the  device  of  a  unit  of  measurement. 
Adults  may  realise  that  there  is  a  real  step  here,  by  remem- 
bering that  if  they  were  set  to  express  the  strength  of  an 
electric  current,  or  the  electric  pressure  on  a  main,  or  the 
strength  of  a  magnet,  numerically,  they  would  be  nonplussed, 
unless  they  knew  something  about  the  units  which  within  a 
generation  or  two  have  been  introduced  for  the  purpose, — 
the  ampere,  the  volt,  and  the  line  of  force ;  so  that  nowadays 
the  British  workman  is  able  to  speak  familiarly  of  an  electric 
current  of  so  many  amperes — (sometimes  pronounced 
"  hampers  ").  There  is  nothing  really  numerical  about  the 
length  of  a  table  or  the  height  of  a  door  or  the  weight  of 
a  sack  or  the  brightness  of  a  lamp  or  the  warmth  of  a 
room  or  the  length  of  a  day  ;  and  its  numerical  expression 
will  depend  entirely  upon  what  conventional  unit  is  em- 
ployed, and  may  vary  in  different  countries  accordingly. 
Do  not  assume  therefore  that  a  child   is   stupid  to  whom 


I.]  EARLY    OPERATIONS.  15 

an  application  of  arithmetic  to  weighing  and  measuring  is 
not  obvious. 


Introduction  of  the  idea  of  fractions 

In  the  same  way  the  idea  of  fractions  can  naturally  occur ; 
a  halfpenny  and  a  half  ounce  and  a  half  inch  being  fairly  easy 
examples :  but  not  the  easiest.  There  can  be  no  doubt  that 
just  as  numbering  ought  not  to  begin  with  continuous  quantity 
but  with  discrete  objects,  so  fractions  should  be  first  displayed 
as  actually  cut  and  broken  things. 

The  proper  fractions  to  begin  with  are  halves  and  quarters 
and  eighths;  and  apples  do  admirably  for  that.  Oranges 
suggest  further  modes  of  subdivision,  except  that  the  removal 
of  the  peel  may  constitute  an  unexpressed  but  felt  complication. 

Folding  of  a  ribbon  or  paper  easily  leads  to  thirds  and  a^ny 
other  fractions  wanted.  Any  child  can  be  sent  to  cut  off  a 
quarter  of  a  yard,  or  a  yard  and  a  half,  or  even  a  foot  and 
three  quarters,  of  tape.  But  again  do  not  be  surprised  if  this 
last  mode  of  specification  is  found  occasionally  puzzling :  it  is 
of  the  nature  of  a  problem,  and  requires  time.  The  form  of 
difficulty  which  may  properly  occur  to  some  children  is  "  a  half 
of  what  "or  "  three  quarters  of  what "  :  and  if  they  bring  the 
foot  and  the  3  quarters  all  separate,  i.e.  if  they  cut  the  tape 
into  four  pieces  altogether,  that  is  very  well  for  a  beginning. 
They  should  not  be  supervised  or  fidgeted  during  the  solution 
of  a  problem.  They  cannot  think  if  they  are.  These  expres- 
sions, 6  miles  and  a  half,  etc.,  have  a  conventional  ring,  to 
which  we  have  grown  thoroughly  accustomed,  but  they  are 
shorthand  terms  not  really  fully  expressive  :  it  might  possibly 
ambiguously  suggest  9  miles.* 

The  measure  of  time  in  half  and  quarter  hours  may  also  be 

*Cf.  George  Meredith's  "Rhoda  Fleming,"  Chap.  3. 


16  EASY  MATHEMATICS.  [chap. 

appealed  to  as  illustrative  of  fractions ;  but  in  this  form  they 
are  somewhat  abstract.  The  divisions  on  a  foot  rule  or  metre 
scale  are  easier,  and  for  further  progress  are  indeed  the  easiest 
illustration  to  be  borne  in  mind.  Afterwards,  the  halfpenny, 
the  half  crown,  the  half  sovereign,  etc.,  and  the  other  fractions 
of  money  may  be  brought  in,  whenever  they  appear  to  be 
natural. 

Practical  hints  for  teaching  the  simple  rules. 

Simultaneously  with  all  this  introduction  of  fresh  concep- 
tions, mechanical  practice  in  operations  with  symbolised  num- 
bers can  be  proceeded  with  : — 

Addition. 

About  addition  there  is  little  to  be  said  :  the  idea  of  packets 
must  have  made  everything  concerning  the  carrying-figure 
easy. 

The  principle  being  understood,  it  is  now  only  a  question  of 
practice  in  attaining  quick  and  sure  execution,  as  quick  and 
sure  as  it  is  worth  while  to  aim  at  at  this  stage. 

Addition  of  money  is  a  useful  accomplishment,  and  since  the 
packets  into  which  it  is  to  be  made  up  are  varied,  it  affords 
good  practice,  involving  a  certain  amount  of  constant  thought 
and  care.  It  is  wrong  to  try  to  force  a  child  to  acquire  the 
facility  of  a  bank  clerk  in  adding  up  long  columns  :  that  will 
come  in  due  time  and  is  quite  a  useful  faculty  :  it  is  clearly  a 
thing  to  acquire  in  commercial  schools,  but  not  while  still 
young  and  receptive. 
It  is  well  to  begin  thus  : 

£       s.         d. 

6  .  15  .     3 

5.4.9 

12  .  —  .  — . 


I.]  EARLY  OPERATIONS.  17 

where  the  packets  to  be  carried  forward  are  complete.  Then 
change  the  3  into  a  4  or  5  and  get  1  or  2  pence  over;  then 
change  the  15  into  16  or  17  and  get  some  shillings  over,  and 
so  on,  gradually.  Always  begin  with  what  illustrates  the  procedure 
in  the  simplest  form  and  gradually  complicate  it. 

There  is  one  remark  about  addition  worth  making.  In 
adding  say  43  +  8,  some  beginners  are  told  to  bethink  them- 
selves that  3  +  8  =  11,  and  so  arrive  at  the  digit  1  of  the 
result ;  while  others  are  told  to  think  of  the  sum  as  43  +  7  +  1, 
stepping  on  to  the  intermediate  stage  of  the  complete  packet 
en  route  to  51  ; 

e.g.  77  +  9  =  77  +  3  +  6  =  80  +  6  =  86. 

Perhaps  it  is  permissible  to  introduce  this  aid  as  a  temporary 
measure,  but  ultimately  addition  ought  to  proceed  by  instinct 
and  without  thought.  It  is  a  mechanical  process,  and  a  bank 
clerk  who  stopped  to  think,  while  adding,  would  be  liable  to 
make  a  mistake. 

Subtraction. 

There  appears  to  be  no  doubt  now  but  that  the  "shop 
method "  of  subtraction  is  the  handiest  and  quickest :  it  may 
as  well,  therefore,  be  acquired  almost  from  the  first. 

37  Three  and  four  make  seven. 

1^         One  and  two  make  3. 

24         Put  down  the  figures  in  black  type. 

Verify  by  adding  13  to  24.     Take  another  example : 

174 
Qo         Eight  and  six  make  fourteen. 

—^         Nine  and  one  and  seven  make  seventeen. 

I  do  not  think  that  children  need  find  this  method  hard  or 

L.E.M.  B 


18  EASY  MATHEMATICS.  [chap. 

unnatural,  but  practice  will  be  needed  before  going  on  to 
money  sums,  such  as  : 

£         s         d 

,  -        '         "  Four  and  seven  make  11. 

II  '     g  *     ^         Eight  and  eighteen  make  26. 

~T      7^        H  12  and  5  make  17. 

Verify  by  addition  of  the  two  lower  lines.  Get  the  children 
never  to  pass  and  hand  in  a  result  as  finished  unless  they  have 
taken  pains  to  assure  themselves  that  it  is  right.  This  does 
not  mean  that  they  are  not  to  hand  in  a  confessedly  unfinished 
sum  if  they  find  they  cannot  do  it  without  help. 

Multiplication. 

At  good  Kindergarten  schools,  a  step  beyond  the  first  in 
multiplication  is  often  introduced  by  some  such  questions  as 
this : 

How  many  stamps  will  three  children  have  if  each  has  14  1 
They  first  add  14  three  times,  and  they  are  allowed  to  do 
that  till  they  find  it  quicker  to  use  the  phrase  "  three  times," 
which,  if  they  know  the  multiplication  table,  they  can  hardly 
help  doing  in  the  process  of  adding ;  and  so  they  get  to  be 
able  to  give  the  answer  "3  times  14"  instantly,  without 
necessarily  having  had  time  to  realise  what  the  operation 
would  result  in  when  executed.  This  kind  of  intermediate 
answer  is  to  be  encouraged. 

In  entering  upon  multiplication,  employ  a  single  digit  as 
one  factor,  and  do  it  first  as  an  addition  sum,  e.g.  : 

142 
142 
142 
142 
142 

710 


I-] 


EARLY  OPERATIONS.  19 


then  proceed 

142 
5 

173             125 
5                8 

865           1000 

£. 
12  . 

8. 

7  . 

d. 
6 
4 

710 

49  . 

10  . 

— 

doing  this  latter  also  by  addition  first : 
£.        s.        d. 
12  .     7  .     6 
12  .     7  .     6 
12  .     7  .     6 
12  .     7  .     6 

49  .  10  .     0 
but  it  is  well  to  lead  up  to  the  last  type  of  sum  by  simple 
cases  first,  e.g.  4  x  2/6  =  10/-;   4  x  5/-  =  £1  ;    4  x  7/6  =  30/- 
=  £1    10/- ;     4  X  3d.  =  1/- ;     4  x  1/3  =  5/- ;     4  x  10/-  =  £2  ; 
4x11/3  =  £2.  5s. 

Do  not  hurry.  If  the  child  can  be  allowed  time  to  see 
a  connexion  between  the  three  last  statements,  or  the  like, 
so  much  the  better.  The  value  of  these  trifles  is  when  they 
are  discovered;  there  is  hardly  any  virtue  in  them  if  they 
are  pointed  out,  and  none  at  all  if  they  are  laboriously 
emphasised.  If  they  are  not  glimpsed  let  them  pass.  We 
all  of  us  doubtless  miss  discoveries,  most  days,  for  lack  of 
attention  and  insight. 

Next  comes  multiplication  with  two  digits  :  first  by  numbers 
like  10,  20,  70,  etc. 

Multiplying  by  ten  means  making  every  unit  into  a  packet, 
every  packet  into  a  set  of  packets,  and  so  on. 

Wherefore 


0  I  1  I  3  I  4    when  multiplied  by  ten  becomes 


13    4    0 


the  1  being  shifted  into  the  empty  compartment,  and  every 
other  digit  likewise  moved;  the  unit  box,  or  box  for  single 
counters,  being  left  empty. 


20  EASY  MATHEMATICS.  [chap. 

If  we  multiply  by  20,  the  shift  takes  place  similarly,  and 
also  every  digit  is  doubled,  yielding  2680. 

So  now  start  multiplying  a  number  like  53  by  20,  getting  1060. 
Then  a  number  with  a  carrying  figure  from  the  units  place, 

^^^®  47x20  =  940; 

then  one  involving  two  carryings,  like 

or,^  c^  57x20  =  1140, 

and  so  on.  ' 

Next  take  multiplication  by  a  number  like  23.     Let  it  be 

realised  once  more  that  23  is  short  for  20  +  3,  so  that  it  may 

be  felt  to  be  natural  to  multiply  by  20  and  by  3  successively 

and  add  the  results,  which  is  what  we  do.     At  first  let  it  be 

worked  in  this  way ;  for  instance,  to  find 

824x23 

=  20  X  824  or  16480 

and     3  x  824  or     2472 


added  together  make       18952 

but  gradually  get  it  abbreviated  into  the  usual  form 

824 
23 


1648 
2472 


18952 
without  necessarily  putting  in  the  cipher  after  the  digit  8. 

There  appears  to  be  no  doubt  now  that  it  is  best  in  mul- 
tiplication to  begin  with  the  most  important  figure,  so  that 
sums  look  thus : 

173  173  768 

56  156  107 

865  173  768 

1038  865  5376 

9688  1038  82176 


t.] 


EARLY  OPERATIONS. 


21 


a  trivial  matter  to  all  appearance,  but  helpful  in  later  stages, 
and  therefore  better  practised  from  the  first. 

[In  my  opinion  it  is  thoroughly  unwise  to  reverse  the  digits 
of  any  factor  before  multiplying  with  them,  though  some 
teachers  of  immense  experience  think  otherwise.] 

Multiplication  of  money,  at  least  of  English  money,  is 
more  difficult  of  course,  because,  in  the  specification  of  money, 
scales  of  notation  are  so  mingled;  thus,  depicting  the  com- 
partments and  labelling  them  when  necessary  : 


£ 

s. 

d. 

4 

3 

5 

1 

7 

1 

1 

at  the  double  line  the  scale  is  changed  from  ten  to  a  dozen, 
and  at  the  treble  line  it  is  changed  again  from  ten  to  a  score. 

So  if  we  have  to  double  this  sum,  even  doubling  it  is  com- 
plicated, and  results  in 

£871  .  15  .  10 
Let  no  one  suppose  that  this  is  an  easy  process,  for  a  child  or 
anyone. 

It  could  in  this  case  be  performed  more  easily  by  simple 

addition : 

£  s.  d. 

435  .  17  .  11 

435  .  17  .  11 

871  .  15  .  10 
but  that  is  hardly  applicable  to   larger  factors.     Not   only 
is  doubling  hard,  but  multiplying  even  by  10  is  hard  too. 
Take  the  amount  £5.  17s.   Ud.  and  multiply  it  by  10;  it 
becomes  the  totally  different-looking  amount 

£58.  19s.  2d. 
Multiplying   by    12    will    of   course  turn   all  the  pence  into 
shillings,    and   multiplying    by    20   will   turn   shillings   into 


22  EASY  MATHEMATICS.  [chap. 

pounds,  but  multiplying  by  any  other  factor  is  hard,  and 
is  probably  best  deferred  for  the  present. 

If  multiplication  of  money  by  a  number  like  23  is  wanted, 
not  only  must  the  23  be  divided  into  two  parts  20  +  3,  and 
the  multiplication  done  separately  as  usual,  but  it  is  generally 
needful  to  resolve  the  20  into  two  parts  also,  say  10  +  10, 
and  then  add  the  three  results  together. 

If  however  multiplication  by  24  were  desired,  it  would 
be  possible  to  split  it  into  two  factors  8x3,  and  to  multiply 
first  by  one  and  then  the  result  by  the  other,  without  any 
addition  of  results;  but  there  is  great  danger  of  confusion 
here,  and  there  are  plenty  of  what  are  considered  and  are 
really  "higher"  parts  of  arithmetic  which  are  much  easier 
than  this.  Low  class  or  unskilled  labour  is  not  necessarily 
easy :  it  may  in  some  cases  be  terribly  laborious,  like  un- 
loading a  ship.  Another  way  of  multiplying  by  20  is  to 
split  up  20  into  the  two  factors  2x10  or  4x5  and  employ 
them  successively.  In  that  case  the  result  of  multiplying 
by  23  is  ultimately  obtained  by  multiplying  the  original 
sum  by  2,  the  result  by  10,  the  original  sum  by  3,  and  then 
adding  the  last  two  results. 

The  fact  is  that  with  money  specified  in  the  customary 
English  way,  the  only  operations  that  can  comfortably  be 
performed  on  it  are  addition  and  subtraction,  and  these  are 
the  only  really  frequent  operations  in  practice. 

To  apply  multiplication  and  division  it  is  best  to  express 
the  money  differently,  in  fact  to  decimalise  it  before  commenc- 
ing operations.  This  will  be  explained  later  (Chap.  VII.), 
though  of  course  to  most  teachers  it  is  a  process  already  well 
known.  It  ingeniously  evades  the  difficulties  caused  by  our 
currency,  and  converts  its  treatment  into  almost  a  worthy 
intellectual  exercise. 


1.]  EARLY  OPERATIONS.  S3 

Division. 

First  take  simple  sums  to  introduce  the  notatioUj  such  as 

y  =  3,   or  21-r7  ^  3. 

21 
Let  it  be  realised  also  that  -5-  =  7,  and  that  3x7  =  21. 

o 

There  are  a  multitude  of  interesting  things  to  be  learnt 
before  long  about  factors,  and  criteria  for  division,  etc.,  but 
not  yet;  let  the  child  learn  how  to  perform  the  process  on 
numbers  of  which  he  knows  no  factors.  But  at  first  do  not 
trouble  him  with  remainders  :  let  him  at  first  be  given  simple 
sums  that  divide  out  completely. 

Thus  we  can  tackle  such  sums  as 

71491036        ,  .  ,    ,      ...      ,  .,,      491036       ^^,,^ 

' — ,     which  should  be  also  written  - — = —  =  70148. 

70148  7 

The  treatment  of  remainders  is  for  subsequent  consideration. 

It  is  well  to  give  the  complementary  sum  7x70148, 
especially  since  the  teacher  will  thus  have  but  little  trouble  in 
checking  results — at  least  until  the  child  finds  out  the  dodge 
—a  discovery  which  is  to  be  encouraged  like  all  other 
discoveries. 

At  good  Kindergarten  schools,  a  step  beyond  the  first  in 
division  is  often  introduced  by  some  such  plan  as  the 
following : 

To  prove  that  96  ^  4  =  24. 

Take  nine  bundles  and  six  sticks  over,  deal  out  into  four 
places,  two  bundles  in  each  place;  and  then  deal  sixteen 
sticks,  four  into  each  place,  giving  the  result  24.  And  so  on 
with  other  numbers. 

As  soon  as  short  division  is  thoroughly  understood,  long 
division  may  introduce  itself  as  an  assistance  when  more 
difficult  divisors  are  involved;    for  instance   988 -7-19.     This 


24  EASY  MATHEMATICS.  [chap. 

being  difficult  to  do  by  short  division,  where  the  multiplica- 
tion and  subtraction  have  to  be  done  in  one's  head,  it  is 
permitted  to  write  the  operations  down,  at  first  both  of  them, 
thus : 

19)988(5 
95 

3 
Afterwards,  perhaps,  only  the  result  of  them,  3,  which  in 
short  division  would  likewise  not  appear,  nothing  but  the 
quotient  being  written  in  short  division.  Long  division  is 
therefore  not  harder  than  short  division,  but  easier :  it  is  the 
identical  process,  only  written  out  more  fully,  so  as  to  be 
applicable  to  harder  sums.  It  is  the  largeness  of  the  figures 
dealt  with  that  makes  it  hard. 

For  long  division  it  appears  to  be  felt  that  by  aid  of  the 
shop  system  of  subtraction  there  is  no  undue  strain  on  the 
brain  by  the  use  of  the  abbreviated  method. 

I  would  have  it  understood  however  that  long  division 
sums  are  among  the  moderately  hard  things  of  life,  and  that 
mathematicians  seldom  trouble  themselves  to  do  them.  They 
can  be  deferred  until  many  other  things  have  been  done  and 
some  familiarity  with  figures  acquired.  It  is  a  gymnastic 
exercise  to  perform  even  so  simple  a  long  division  sum  as  the 
following,  and  if  attempted  too  early  will  involve  strain. 

72)5286456(73423 
246 
304 
165 
216 
This  is  the  process : 

Sevens  in  52  ?  guess  7  times  and  write  7  as  the  first  digit  in 
the  quotient,  then  7  x  2  =  14,  to  which  add  4  to  make  18. 
Seven  sevens  =  49,  say  50,  to  which  add  2  to  make  52 ; 
record  only  the  figures  here  printed  in  black  type;   bring 


1.]  EARLY  OPERATIONS.  25 

down  the  rest  of  the  dividend  6456  or  as  much  of  it  as  is 
wanted;  only  6  is  wanted  so  far,  and  we  guess  3  for  the 
next  digit  in  the  quotient.  Three  times  2  and  0  make  6, 
three  times  7  and  3  make  24.  Bring  down  more  of  the 
dividend,  say  456,  or  at  least  4,  and  guess  4  for  the  next 

digit. 

4x2  =  8  and  six  are  14. 
4  X  7  =  28,  say  29,  and  1  are  30. 
Bring  down   the  5,   and  guess  2  for  the  next  digit  of  the 
quotient ;  twice  2  =  4  and  1  =  5,  etc.,  and  then  finally  bring 
down  6,  and  it  goes  3  times  exactly. 

If  the  sum  is  neatly  done   the  corresponding  places  are 
vertically  under  each  other,  a  detail  of  appearance  emphasised 
by  the  presence  of  a  decimal  point. 
Let  the  result  be  written 

52^56  ^  ^3^23. 

Do  not  forget  to  set  also  the  complementary  sum 

72  X  73423. 

It   will  be  well  also  to  set  the  exercise  whose  result  is 

5286456       ^^ 
^^  .    „     =  72,  as  a  separate  sum. 

If  the  connexion  is  automatically  noticed,  it  is  well ;  it  will 
prepare  the  mind  for  the  later -on  extremely  important  and 
constantly  occurring  connected  relations, 

if  T  =  c,   then  -  =  b,  and  he  =  a, 

be 

but  refrain  from  using  this  abstract  language  at  present. 
Watch  for  the  time  when  it  can  without  strain  be  naturally 
introduced.  It  is  a  great  help  when  that  step  is  reached, 
and  it  represents  a  vital  stage  of  real  mental  progress.  The 
mind  should  be  soaked  with  particular  instances  however 
before  generalisations  can  be  usefully  and  permanently  grasped. 


26  EASY  MATHEMATICS.  [chaf. 

Division  of  money  is  of  course  difficult,  even  when  the 
divisor  is  a  small  number,  because  of  our  complex  system 
of  notation,  unless  the  money  is  first  expressed  in  decimal 
form. 

To  divide  by  23  moreover  it  is  not  correct  to  divide  by 
20  and  then  by  3  and  add  the  results,  as  it  was  with  mul- 
tiplication. A  long-division  sum  is  necessary,  and  that  is  no 
joke  with  money  as  usually  specified.  Division  by  24  can 
indeed  be  done  in  two  stages,  by  help  of  its  factors  3  and  8 
consecutively  applied,  but  that  only  masks  the  essential 
difficulty  by  a  device  applicable  only  to  special  cases. 

My  object  in  introducing  these  remarks  about  complex 
money-sums  here  (and  the  same  thing  applies  to  weights  and 
measures  sums)  is  to  urge  that  they  really  belong  to  a  later 
stage,  and  to  beg  teachers  to  defer  them  beyond  the  early 
years  at  which  they  are  too  often  introduced.  For  their 
premature  employment  has  often  resulted  in  giving  children 
an  effectual  and  lifelong  disgust  with  what  they  have  docilely 
conceived  to  be  arithmetic ;  whereas  much  of  what  they  had 
to  do  was  really  a  mechanical  and  overstraining  grind,  having 
as  much  relation  to  mathematics  as  carrying  heavy  hods  of 
bricks  all  day  up  a  ladder  has  to  architecture. 

Origin  of  the  symbols. 

It  is  amusing  to  speculate  on  the  probable  origin  of  the 
symbols  for  the  digits.  It  appears  likely  that  if  a  single 
horizontal  stroke  meant  1,  a  double  horizontal  stroke  hastily 
drawn  would  give  ~Z.  or  something  like  a  2. 

It  is  less  easy  to  make  a  sort  of  3  out  of  three  such  strokes, 
but  it  is  possible. 

The  symbol  for  four  would  seem  to  be  representative  of  a 
four-sided  figure  or  badly  drawn  square,  '-|-  ,  and  the  figure 
8  was  probably  originally  a  pair  of  such  squares  Q  • 


I.]  COUNTING.  27 

But  at  this  stage  it  appears  likely  that  some  skilled  person 
took  pains  to  design  digit  symbols  of  distinctive  form  by 
combination  of  a  stroke  and  a  semi-circle,  making  9,  set  like 
this : 

I       2.       3       y       b       1       S       10       II      etc., 

and  that  the  notion  of  the  value  of  "place"  was  a  develop- 
ment from  the  further  stages  of  this  mode  of  representation.* 
So  also  it  is  believed  that  the  Roman  symbol  X  for  ten  was 
the  result  of  counting  by  strokes  and  crossing  off  every  tenth 
stroke,  thus : 

1 1 1 1 1 1 1 1  ni  1 1 

a  practice  not  unknown  among  workmen  to  this  day. 

Two  such  crosses  would  naturally  mean  20,  etc.,  while  half 
a  cross  or  V  could  conveniently  be  used  to  denote  5. 

It  has  been  suggested  that  the  rounded  M  for  1000,  W, 
sometimes  inscribed  CIO,  if  halved,  would  give  the  D  for  500; 
and  that  a  square  C  for  100,  if  halved,  would  furnish  an  L  for 
50;  but  this  may  be  fanciful.  The  symbol  00 1 00  was  used, 
it  is  said,  for  100,000,  and  0001000  for  a  million. 


*  The  above  however  is  not  history.  The  real  history  of  the  symbols 
is  complex,  and  stages  of  it  are  given  in  Dr.  Isaac  Taylor's  learned  work 
on  the  Alphabet,  especially  Vol.  II.  pp.  263  et  seq. 

It  appears  that  our  digit  symbols  originated  in  India,  and  that  several 
of  them,  especially  7,  represent  a  corruption  of  the  initial  letters  of  the 
words  previously  employed  to  denote  the  numbers. 

"  They  were  introduced  by  the  Arabs  into  Spain,  from  whence  during 
the  12th  and  13th  centuries  they  spread  over  Europe,  not,  however, 
without  considerable  opposition.  The  bankers  of  Florence,  for  example, 
were  forbidden,  in  1299,  to  use  them  in  their  transactions,  and  the 
Statutes  of  the  University  of  Padua  ordain  that  the  stationer  should 
keep  a  list  of  the  books  for  sale  with  the  prices  marked  '  not  by  ciphers 
but  in  plain  letters '.  .  .  .  Their  use  was  at  first  confined  to  mathema- 
tical works,  they  were  then  employed  for  the  paging  of  books,  and  it 
was  not  till  the  15th  century  that  their  use  became  general." 


CHAPTEE  II. 

Further  considerations  concerning  the  Arabic  system 
of  notation,  and  extension  of  it  to  express 
fractions. 

Having  become  acquainted  with  the  fundamental  plan  of 
the  system  of  notation  in  use,  and  the  mode  of  expressing 
any  whole  number  of  things  by  a  combination  of  ten  digits 
arranged  in  places  of  different  value,  not  all  places  necessarily 
occupied — that  is,  by  means  of  nine  significant  digits  and  a 
cipher  to  express  emptiness  in  whatever  place  emptiness  may 
occur, — it  is  permissible  to  elaborate  it  further,  with  a  little 
repetition  occasionally. 

At  the  beginning  of  each  chapter  there  is  liable  to  be  a  little 
repetition  of  something  that  has  already  been  explained,  but 
in  a  slightly  different  form.  This  amount  of  repetition  is 
purposely  introduced  and  is  useful :  it  is  intended  to  link  the 
new  knowledge  on  with  the  old.  A  new  subject  should  not 
be  introduced  as  if  it  belonged  to  a  perfectly  distinct  region  of 
thought;  its  connexion  with  what  is  known  should  be  indicated, 
and  sufficient  of  the  old  should  be  reproduced  to  make  the 
connexion  secure.  Repetition  of  a  judicious  kind  is  by  no 
means  a  thing  to  be  avoided,  though  it  is  easy  to  overdo  it ; 
and  in  every  way  the  best  kind  of  repetition  is  that  which 
repeats  the  old  idea  in  a  different  form  of  words,  or  which 
looks  at  something  already  known  from  a  new  aspect. 

The  beginnings  of  each  new  chapter  should  be  easy,  and  the 


CHAP.  II.]  NUMERATION.  29 

steps  to  higher  flights  should  be  regular  and  moderate,  like  a 
staircase. 

Now  we  know  that  the  symbol  304  means  usually  that  there 
are  3  packets  of  a  hundred  things  each 

no  packets  of  tens  and 
4  single  things, 
but  the  '*  ten-system,"  though  customary,  is  not  an  essential 
part  of  this  plan  of  notation. 

40  and  4/-  are  both  constructed  essentially  on  this  plan,  both 
are  understood  to  signify  4  packets  and  no  odd  units,  though 
the  number  in  the  packets  is  not  the  same  in  the  two  cases. 

£4  .  — • .  —  signifies  again  4  of  another  variety  of  packet. 

Three  dozen  and  six  pennies  may  be  written  either  3/6  or 
42  pence.  It  would  have  been  far  more  convenient  if  the 
human  race  had  agreed  to  reckon  everything  in  dozens,  and  so 
to  express  this  number  by  the  digits  3  6  instead  of  by  the 
digits  4  2 ;  but  as  they  have  in  early  semi-savage  times  arranged 
otherwise,  we  must  now  make  the  best  of  it.  The  general 
idea  is  the  same,  only  that  whereas  in  ordinary  life  things  are 
commonly  and  conveniently  reckoned  by  dozens,  it  is  customary 
in  arithmetic  to  reckon  by  packets  of  ten,  the  symbols  being 
called  digits  because  they  used  to  be  reckoned  by  actual 
fingers :  which  by  some  simple  persons  are  so  employed  still. 
Thus  whereas  7/6  is  understood  to  mean  seven  dozen  and  six 
pence,  it  is  customary  to  mean  by  76,  seven  packets  of  ten  and 
six  units  over;  that  is  to  say,  if  the  units  were  pennies,  the 
same  as  6/4.  So  also,  instead  of  grouping  dozens  into  a  gross, 
as  in  ordinary  life,  in  arithmetic  we  group  tens  into  a  large 
packet  of  ten  tens,  which  we  denote  by  100.  The  symbol  346, 
therefore  signifies  six  single  units,  4  packets  of  ten  each,  and  3 
packets  of  a  hundred  each.  If  there  are  as  many  as  ten  sets 
of  100,  they  are  to  be  specified  by  1000,  and  so  on,  as 
ordinarily  learnt. 


80  EASY  MATHEMATICS.  [chap. 

This  system  of  notation  extends  as  far  as  we  like  to  the  left 
of  the  units  place,  and  if  six  empty  boxes  follow  the  digit  1, 
it  means  a  million.  But  we  might  suppose  boxes  added  to 
the  right  of  the  units  place ;  can  we  find  any  use  for  them  1 
Let  us  mark  the  unit  box  by  a  double  line  nearly  round  it,  so 
that  in  a  long  row 


1  1  1 

1 

there  need  be  no  hesitation  about  which  is  the  unit  box; 
then  put  the  digit  unity  into  each  box.  In  the  unit  box  it 
means  one  of  some  thing,  in  the  next  box  on  the  left  it  means 
one  packet  of  ten  of  those  same  things,  and  so  on ;  each  digit 
to  the  left  having  ten  times  the  value  of  the  one  immediately 
on  the  right.  If  this  convention  were  extended  to  the  box 
on  the  right  hand  of  the  units  place,  the  1  there  would 
signify  the  tenth  part  of  a  unit,  and  a  1  in  the  next  box  on 
the  right  would  signify  the  tenth  of  a  tenth,  and  so  on. 

For  we  know  that  not  only  can  we  group  things  together 
into  an  aggregate,  it  is  possible  also  to  cut  them  up,  or  split 
them  into  fractions. 

Thus  the  things  counted  may  be  bags  of  money,  and  each 
bag  may  be  known  to  contain  or  to  be  worth  100  sovereigns. 
In  that  case  the  figure  6  might  signify  six  bags,  and  so  stand 
for  600  sovereigns.  And  each  sovereign  might  be  called  a 
fraction  of  the  contents  of  a  bag,  viz.,  a  hundredth  part.  But 
in  some  of  the  bags  the  value  might  be  made  up  with  ten 
pound  notes,  and  each  of  them  would  likewise  be  fractions  of 
the  contents  of  a  bag,  viz.,  the  tenth  part. 

Three   such   notes   may   therefore    be   specified   either    as 

3*    ten  pound  notes 
or  30-    pounds  value 
or  three  tenths  or      -3  of  the  value  of  a  bag ; 


II.]  NUMERATION.  31 

the  stop  or  mark  or  point  being  introduced  whenever  it  is 
necessary  for  clearness.  Any  mark  will  do.  In  foreign 
countries  a  comma  is  commonly  used,  whereas  we  use  a  dot 
placed  about  the  middle  of  the  figure.  In  early  days  a  | 
mark  of  this  kind  was  used.  Thus  346  |  57  used  to  be  written 
where  we  should  now  write  346*57,  or  a  Frenchman  346,57,  the 
digits  5  7  being  partitioned  off  to  signify  that  they  represent 
fractional  parts  of  objects  or  units ;  the  digit  6  refers  to  whole 
objects  or  units,  the  digit  4  to  packets  of  ten,  the  digit  5 
to  fractions  of  one-tenth,  and  the  digit  7  to  one-tenth  part 
of  tenths,  that  is  to  say,  it  signifies  seven  hundredths  of 
a  unit. 

Suppose,  for  instance,  the  unit  was  a  bag  of  sovereigns,  as 
above  specified,  then  the  number  written  346J57  or  346 1 57  or 
346-57  would  mean  346  complete  bags  of  a  hundred  pounds 
each,  with  5  ten  pound  notes  and  7  sovereigns  loose.  The 
money  specified  would  be  equal  in  value  to 

3465-7  ten  pound  notes 
or  to  34657*  sovereigns 
or  to        34*657  thousand  pound  notes 
or  to  -034657  million  pound  notes, 

the  position  of  the  figures  being  changed  according  to  the 
unit  intended,  and  the  dot  or  other  mark  being  used  to 
signify  where  whole  numbers  end  and  fractions  begin. 

The  position  of  the  above  numbers  relative  to  each  other 
is  constant,  viz.  the  order  3,  4,  6,  5,  7 ;  but  their  absolute 
position,  or  position  relative  to  the  unit  place,  is  different  in 
the  different  cases,  and  is  specified  by  the  dot,  which  is  always 
and  invariably  placed  after  the  units  digit  whenever  it  is 
inserted  at  all.  It  is  not  always  necessary  to  insert  it.  For 
instance  the  number  3  might  be  written  more  completely  and 
equally  well  3*  or  3*0  or  3-000,  in  which  case  it  definitely 


I 


32  EASY  MATHEMATICS.  [chap. 

signifies  3  units  of  something,  and  the  0  would  indicate  the 
fact  that  there  was  no  fraction  to  be  attended  to.  If  the 
dot  is  placed  thus  30*,  it  would  mean  3  packets  of  ten 
units;  if  placed  thus  300-,  it  means  three  groups  of  ten 
packets  each ;  and  any  digit  placed  after  the  dot  thus  -3  means 
a  fraction,  viz.  three-tenths  of  a  unit.  Whereas  if  a  digit 
occurs  2  places  to  the  right  of  the  dot,  as  -03,  it  means  three 
himdredths  of  a  unit ;  as  for  instance  3  sovereigns  would  be 
3  hundredths  or  "03  of  a  bag  in  the  above  example,  or  -3  of 
a  ten-pound  note.  Similarly  a  florin  is  one-tenth  of  a  pound 
or  £0-1.  Again  it  is  the  hundredth,  or  '01,  of  a  ten-pound 
note. 

This  use  of  the  dot  is  only  a  matter  of  nomenclature,  and 
its  importance  lies  in  its  simplicity  and  convenience.  It  is 
always  possible  to  write  -03  as  ^fo-  i^  we  please,  just  as  it 
is  possible  to  denote  1864  by  MDCCCLXiv  if  we  like;  but 
it  is  not  so  simple. 

It  may  be  as  well  to  observe  that  although  there  is  no 
numerical  difference  between  6  feet  and  6*00  feet,  there  is  a 
practical  and  convenient  difference  of  signification.  In 
practice  6  feet  would  mean  something  approximately  the 
height  of  a  man,  whereas  6  00  feet  would  be  understood  to 
signify  either  that  you  had  measured  a  length  accurately  to 
the  hundredth  of  a  foot  or  something  like  the  tenth  of  an 
inch,  and  found  no  fraction;  or  else  that  you  wished  some- 
thing to  be  made  to  that  amount  of  accuracy. 

Another  way  of  reading  the  symbol  -03  is  three  per  cent., 
or  three  divided  by  one  hundred.  So  also  five  per  cent,  is 
•05;  twenty  per  cent,  is  *20;  seventy-four  per  cent,  is  -74, 
and  so  on. 

In  the  case  of  twenty  per  cent,  it  may  obviously  be  written 
•2  or  Y%  or  ^  So  also  -5  being  5-tenths  or  50  per  cent,  is  the 
same  as  | ;  and  one-half  is  often  the  neatest  way  of  speaking 


II.]  NUMERATION.  33 

of  it  and  writing  it.  Again  twenty-five  per  cent.,  or  -25,  is  the 
same  as  J,  being  25-hundredths  ;  and  '125  or  125-thousandths 
is  the  same  thing  as  ^.  Sometimes  one  specification  is 
handiest,  sometimes  the  other. 

Unfortunately  it  is  not  very  easy  to  denote  either  J  or  J  or 
§  in  any  other  very  convenient  way  on  our  decimal  system  of 
notation,  as  it  would  have  been  if  we  had  arranged  to  reckon 
in  dozens. 

One-third  of  1/6  is  easy  enough,  being  sixpence,  while 
two-thirds  is  1/0:  but  one-third  of  16  is  an  inconvenient 
number  to  write  in  the  ordinary  notation.  It  is  ^^-,  that  is 
16  divided  by  3,  that  is  5*333333...  without  end,  as  you  find 
by  simple  division. 

So  also  I  of  16  is  10-6666.... 

These  are  called  repeating  or  circulating  decimals,  and 
their  frequent  occurrence  in  ordinary  transactions  is  caused 
by  our  unfortunate  custom  of  reckoning  in  tens  instead  of 
in  dozens.  A  simple  circulating  decimal  may  always  be 
interpreted  as  so  many  ninths :  thus  whereas  '3  means  3  tenths, 
•333...  means  3  ninths,  which  is  the  equivalent  of  one-third; 
•6666...  means  6  ninths,  and  so  on. 

A  third  of  ten  is  3^333... 
A  sixth  of  ten  is  1*666  ... 
Two-thirds  of  ten  is  6*666 ... 

and  even  other  fractions  are  not  very  convenient. 

Thus  a  quarter  of  ten  is  2*5 

an  eighth  of  ten  is  1  '25 

a  sixteenth       of  ten  is    -625 
three-quarters  of  ten  is  7*5 

and  the  only  simple  things  to  specify  are  -5 

not  often  wanted,  viz.  2,  and  a  half  of  ten,  which  is  5. 

L.E.M.  C 


34  EASY  MATHEMATICS.  [chap. 

This  may  be  contrasted  with  the  convenience  of  reckoning 
in  dozens : 

a  third  of  a  dozen  is  4 

a  sixth  of  a  dozen  is  2 

two-thirds  of  a  dozen  is  8 

a  quarter  of  a  dozen  is  3 
half  a  dozen  is  6 

three-quarters  of  a  dozen  is  9 

an  eighth  of  a  dozen  is  IJ 

a  sixteenth  of  a  dozen  is  }. 

Circulating  decimals  would  not  be  avoided  by  the  duo- 
decimal notation,  but  they  would  be  rarer,  for  they  would 
then  in  the  simplest  possible  cases  signify  fifths  or  sevenths 
or  elevenths,  which  are  not  the  commonest  fractions  to  come 
across  in  practice. 

It  should  be  remarked  that  in  actual  practice  circulating 
decimals  only  occur  in  the  translation  of  numerical  fractions ; 
and  then  the  decimals  always  either  terminate  or  recur :  but  in 
real  concrete  measurement,  or  subdivision  of  continuous  magni- 
tude, circulating  decimals  never  occur,  because  such  a  specifica- 
tion would  signify  an  infinite  accuracy,  which  is  impossible. 

In  all  practical  cases  measurements  can  only  be  accurate 
to  a  certain  number  of  significant  figures,  and  though  it  may 
once  in  a  lifetime  happen  that  these  figures  are  all  the  same 
by  accident— as  for  instance  4*4444 — it  cannot  matter  in  the 
end  whether  the  last  figure  is  3  or  5  or  even  some  other 
digit.  When  the  figures  have  expressed  the  actually  attained 
accuracy,  all  subsequent  ones  are  superfluous  and  even  mis- 
leading, because  they  pretend  to  an  amount  of  accuracy  not 
really  attained. 

For  this  reason  the  doctrine  of  circulating  decimals  belongs 
rather  to  pure  than  to  applied  mathematics. 


1 
¥ 
1 
"ST 
1 

ten 

1 
eleven 

1 


II.]  NUMERATION.  35 

In  the  duodecimal  system  the  ordinary  fractions  would  be 
denoted  as  follows :  ,        ^ 

^=?     . 

^  =  '2497  or  approximately  '25 

^  =  •2 

^  =  -186^35 

16 

14 

12497  or  approximately  '125 

illlll 

twelve 

Once  we  have  realised  the  advantages  of  what  is  known  as 
the  duodecimal  system,  it  is  painful  to  have  to  return  and 
use  the  decimal  notation. 

Nevertheless  a  change  from  one  to  the  other  would  necessi- 
tate the  uprooting  of  too  deep-seated  traditions.  Among 
other  things  it  would  alter  the  multiplication  table,  that 
necessary  but  laborious  thing  to  learn.  In  teaching  children 
it  should  be  realised  by  the  teacher  that  the  multiplication 
table  is  hard  and  tedious,  and  too  much  should  not  be  ex- 
pected of  them ;  but  for  convenience  of  life  it  is  one  of  those 
things  that  it  is  best  to  know  thoroughly,  and  it  is  useful  as 
a  matter  of  discipline.  Its  rational  basis  should  be  understood, 
and  experiment  should  be  encouraged  in  the  first  instance  to 
find  out  what,  say,  four  sixes  or  seven  nines  are.  It  is  fairly 
easy  to  see  that  four  sixes  will  make  two  dozen,  it  is  not  so 
easy  to  see  that  they  will  make  two  packets  of  ten  and  four 
over,  but,  the  fact  having  been  ascertained,  it  should  be  learnt 
that  four  sixes  are  24,  or  four  times  six  are  24 — either  way, 
whichever  happens  to  be  asked,  but  not  both  ways  at  the 
same  time  so  as  to  spoil  the  rhythm. 


36  EASY  MATHEMATICS.  [chap  ii. 

Similarly  it  can  be  ascertained  that  five  sixpences  amount 
to  half-a-crown  or  2/6 ;  but  that  five  sixes  are  30,  that  is 
they  just  make  three  packets  of  ten. 

It  is  a  serious  addition  to  the  work  of  childhood  in  this 
country  that  they  have  to  learn  virtually  two  distinct  multi- 
plication tables,  viz.  the  duodecimal  pence  table  and  the 
decimal  or  ordinary  numerical  table.  There  is  plenty  of 
scope  for  discipline  in  these  things,  and  so  if  it  is  possible  to 
relieve  the  tedium  in  other  places  it  is  permissible. 

The  extent  of  multiplication  table  to  be  learnt  is  merely 
a  matter  of  convenience,  and  it  is  handy  to  learn  beyond 
12  times  12.     Especially  is  it  convenient  to  remember  that 

13x13  =  169  17x17  =  289 

14x14  =  196  18x18  =  324 

15x15  =  225  19x19  =  361 

16x16  =  256  20x20  =  400 

Also  that  9  X  16  =  12  X  12  =  144  =  1  gross. 

[The  square  numbers  may  with  advantage  be  specially 
emphasised ;  1,  4,  9,  16,  25,  36,  and  so  on  ;  and  it  is  easy  also 
as  an  exercise  to  ascertain  and  remember  the  powers  of  2, 
especially  that  32  is  the  fifth  power  of  2  :  they  are 

2,  4,  8,  16,  32,  64,   128,  256,  512,  1024,  etc., 
the  last  written  being  the  10th  power. 

A  few  of  the  powers  of  3  are  also  handy. 
3,  9,  27,  81,  243,  729. 

The  cubes  or  third  powers  of  the  simple  numbers  are  useful. 

1x1x1=       1  Cube  of  7  =     343 

2x2x2=      8  „      8  =    512 

3  X  3  X  3  =    27  „      9  =    729 

Cube  of  4  =    64  „    10  =  1000 

„       5  =  125  „    11  =  1331 

„       6  =  216  „    12  =  1728 

All  this  is  to  be  arrived  at  merely  by  simple  multiplication, 
and  the  phrase  cube  number  need  not  yet  be  used.] 


CHAPTEE  III. 

Further  consideration  of  Division,  and  introduction  of 
Vulgar  Fractions. 

Just  as  Multiplication  is  cumulative  addition,  so  Division 
may  be  regarded  as  cumulative  subtraction.  Thus,  for  instance, 
when  we  say  that  7  will  go  in  56  eight  times,  we  mean  that  it 
can  be  subtracted  from  56  eight  times.  From  59  it  can  like- 
wise be  subtracted  eight  times,  but  there  will  be  3  over. 
This  is  the  meaning  of  remainders. 

To  divide  £748.  6s.  lid.  by  £320.  2s.  4d.  we  can  proceed  if 
we  like  by  subtraction — it  happens  indeed  to  be  the  easiest 
way, — and  having  subtracted  it  twice,  we  find  that  that  is  all 
we  can  do,  and  that  there  is  .£108.  2s.  3d.  over.  So  we  say 
that  the  smaller  sum  goes  twice  in  the  bigger  one,  and 
leaves  a  certain  remainder. 

In  general  however  it  is  more  customary  to  regard  division 
as  the  inverse  of  multiplication;  and,  so  regarded,  it  leads 
straight  to  fractions  and  to  factors.  Thus  the  fact  that  3 
multiplied  by  4  equals  12,  (3x4  =  12),  may  be  equally  well 

expressed  by  saying  that  12  divided  by  3  equals  4,  (-5-  =  4), 

/12         \  ^  / 

or  that  12  divided  by  4  equals  3,  (-^-  =  3),  or  that  3  and  4 


are  corresponding  factors  of  12.     Similarly  2  and  6  are  other 
corresponding  factors,  since  12  -r  6  =  2  and  12-7-2  =  6. 

A  number  like  144,  or  one  gross,  has  a  large  number  of 
factors.     It  is  a  good  easy  problem-exercise  to  suggest  to  a 


38  EASY  MATHEMATICS.  [chap. 

child  to  find  them  all.  They  are  2,  3,  4,  6,  8,  9,  12,  16,  18, 
24,  36,  48,  72.  The  factors  of  1728  are  of  course  still  more 
numerous.  And  even  the  number  60  has  a  fair  number  of 
factors,  viz.  2,  3,  4,  5,  6,  10,  12,  15,  20,  30.  These  may  be 
contrasted  with  the  poor  show  of  factors  exhibited  by  100, 
viz.  2,  4,  5,  10,  20,  25,  50. 

Children  can  readily  be  set  to  find  the  factors  of  numbers, 
and  will  thus  incidentally  be  doing  many  simple  division  sums. 

Their  attention  must  not  however  be  too  exclusively,  i.e. 
for  too  long  together,  directed  to  integer  or  whole  number 
factors;  they  must  be  prepared  to  write  down  the  result  of 
division  when  it  is  not  a  whole  number,  but  a  fraction,  or  a 
whole  number  plus  a  fraction.  Thus  -^  for  instance  will  be 
found  to  be  28  and  four  over,  the  meaning  of  which  should 
be  carefully  explained,  being  first  thoroughly  understood  and 
led  up  to  by  the  teacher. 

To  lead  up  to  it,  it  may  be  pointed  out  that  just  as 
28  oranges  =  20  oranges  +  8  oranges 

so  28  half  oranges  =  20  half  oranges  +  8  half  oranges 
and  28  halves  =  20  halves  +  8  halves 

and  28  quarters         =  20  quarters  +  8  quarters ;  just  as  much 

as  28  farthings        =  20  farthings  +  8  farthings. 

Now  ^  =  14,  while  — -  +  ^  =  10  +  4  =  14, 

28  ^  ,.,  20  8  ^  „  ^ 
—  =  7,  while  -r  +  y  =  5  +  2=  7, 
4  4      4 

28  _  20      8   _2,.8_2-8 
28      20      8       ,8 

but  it  is  neater  to  write  it 

25       3_3„6        ^_ 
=  T+   5  =  ^-^5  =  ^  +  10  =  ''^- 


III.}  FRACTIONS.  39 

So  now  the  child  should  realise  that,  since  144  =  140  +  4,  so 
i|^  =  -|-^  +  i;  which  indicates  a  division  that  can  be  done 
and  a  division  that  cannot  be  done.  The  division  that  can  be 
done  has  the  result  28 ;  the  division  that  cannot  be  done  is 
4  -r  5,  and  it  must  be  left,  either  in  the  form  of  ^,  or  in  the 
form  —^  or  -8.     So  the  whole  result  is  expressible  as  28*8. 

A.ccordingly  a  better  way  of  saying  that  i|^  is  28  and  four 
over,  is  to  say  that  it  equals  28  +  -|>  ^^  28*8. 

To  get  it  in  the  latter  form  directly  and  easily,  the  original 
144  should  be  written  144*0,  and  then  the  sum  will  run 

51 144-0        •,        ^      11 
I  quite  naturally. 

28*8 

Take  another  example,  because  the  mind  of  a  child  is  often 
sadly  fogged  about  this  elementary  and  important  matter. 

3^  =104  =  51:«f:.=  10-333,.., 

a  result  found  by  simple  division,  a  process  which  in  this  case 
shows  not  the  slightest  sign  of  terminating  but  goes  on  for 
ever. 

Again  -^'  =  7  and  1  over,  =  7  + 1,  or  as  it  is  usually  written 
7J.  But  in  thus  writing  it  the  question  should  occur,  How 
then  would  one  write  7x11  and  why  does  not  7 J  mean  seven 
halves,  or  seven  multiplied  by  a  half,  or  3  J  ?  It  is  a  mere 
convention,  and  not  a  consistent  one,  that  7 J  shall  signify  7  + 1 
and  not  7  x  J,  and  some  confusion  is  thereby  caused.  By  no 
means  need  the  practice  be  altered :  children  must  learn  to 
accommodate  themselves  to  existing  practice,  and  must  begin 
reform  later  in  life  if  ever ;  but  the  teacher  should  realise  that 
the  simplicity  of  7  J  to  him  is  only  because  he  has  got  accus- 
tomed to  it,  that  it  is  a  confusing  thing  in  reality,  and  that  a 
child  who  is  confused  by  it  is  likely  to  be  the  bright  child  and 
not  the  dull  one. 


40  EASY  MATHEMATICS.  [chap. 

Expression  of  vulgar  fractions  as  decimals. 

There  is  nothing  new  to  be  learnt  about  expressing  a 
vulgar  fraction  in  the  decimal  notation,  it  is  only  a  question 
of  practice.  It  is  probable  that  beginners  will  find  no  diffi- 
culty, but  will  simply  divide  out.  If  any  difficulty  is  felt  it 
can  be  met  by  some  such  initial  treatment  as  the  following  : 

1    .     ,,  2         .3         4  5 

-  is  the  same  as  -  or  -  or  -  or  — , 

and  each  one  of  these  may  therefore  be  written  '5,  which  means 
5  things  in  the  tenths  place  or  compartment  devoted  to  tenths. 
A  florin  for  instance  is  the  tenth  part  of  the  value  of  a 
sovereign,  so  5  florins  =  J  a  sovereign.  £7'b  means  7  pounds 
+  5  florins  or  £7.  10s.  or  £7 J. 

So  also  1  =  ?  =  A  =  ?i,  etc., 

4      8       12       10' 

so  to  express  J  in  decimals  we  shall  have  to  put  2  J  in  the 
tenths  place ;  but  it  is  not  customary  to  place  fractions  there, 
the  J  is  best  set  down  as  5  in  the  next  place  to  the  right,  as 
•25.  In  that  place  5  will  mean  yj^ths,  and  that  is  the  same 
thing  as  I  a  tenth,  viz.  ^V*h. 

So  I  of  a  ten  pound  note  =  £2.  10s.  =  £2  J 

=  £2-5  =  -25  ten  pound  note, 
and  generally  ^  =  -25. 

So  also  f  =  -75,  I  =  -125,  etc. 

The  expression  of  any  fraction  as  a  decimal  involves  nothing 
more  than  simple  division ;  thus  ^  can  be  written  ready  for 
operating  7 1 3-00000,  and  the  quotient,  written  below,  will  be 
•42857  etc. 

In  this  particular  instance  however  there  happens  to  be  no 
simplification,  so  the  operation  is  hardly  worth  performing  in 
that  case. 


III.]  FRACTIONS.  41 

To  prove  that  ?i|^  =  57-57 

work  thus : 

6 1 345-42 
57-57' 
To  find  ^y-^  we  do  the  simple  division  sum 
8|3475-000 
434-375' 
Hence      34^75  ^  ^.3^3^^  .  3;475  ^  .^3^3^^^  ^^ 

8  0 

It  is  not  really  necessary  to  write  it  out  in  the  division  form  : 
simple  division  can  be  performed  on  the  fraction  as  it  stands. 
In  every  case  of  writing  decimal  numbers  one  under  the 
other,  the  rule  is  to  keep  the  column  of  decimal  points 
vertical ;  in  other  words,  adhere  to  your  system  as  to  which 
is  the  units  place,  which  the  tens  place,  and  which  the  tenths, 
etc.,  throughout. 

Extension  of  the  term  multiplication  to  fractions. 

i  The  ordinary  idea  of  multiplication  involves  the  repetition  of 
the  same  thing  several  times,  as  three  times  four,  or  seven  nines. 

The  adding  of  seven  nines  together  is  what  is  called  multi- 
plying nine  by  seven. 

The  payment  of  four  £5  notes  is  not  called  multiplying 
£5  by  4 :  but  if  a  conjuror  extracted  ten  apples  out  of  a  hat 
into  which  one  had  been  put,  he  might  be  said  to  have 
multiplied  it. 

So  also  seed  corn  is  multiplied  into  an  ear;  and  thus  the 
notion  of  increase  is  associated  with  the  notion  of  multiplying, 
i  But  it  is  best  to  dissociate  the  notion  of  increase  from  the 
notion  of  technical  multiplication,  and  to  be  prepared  to 
multiply  by  1  if  need  be,  leaving  it  the  same  as  before,  or  even 
by  1  leaving  it  smaller  than  before.      This  phrase  "  multiply 


42  EASY  MATHEMATICS.  [chap. 

by  a  half  "  is  not  a  simple  and  natural  one  :  it  is  a  permissible 
extension,  such  as  we  constantly  make  in  mathematics,  when 
any  operation  that  has  been  found  practically  useful  is  applied 
over  the  whole  range  within  which  it  is  possible,  and  some- 
times a  long  way  beyond  where  it  appears  possible  at  first  sight. 

Multiplication  by  J  has  some  points  in  common  with  the 
addition  of  a  negative  quantity ;  it  results  in  diminution,  and 
it  is  a  process  that  would  not  have  occurred  to  us  to  do  except 
as  an  extension  of  a  straightforward  process.  To  multiply  by 
J  and  to  divide  by  2  is  precisely  the  same  thing.  Why  not 
call  it  then  dividing  by  2  ^  Well,  we  do  very  often,  but  not 
always,  and  a  beginner  must  be  content  to  be  told  that  it  is 
useful  to  extend  the  nomenclature  of  operations  in  this  way. 
We  shall  speak  of  multiplication  by  J  if  we  choose,  when  we 
mean  division  by  3.  We  shall  occasionally  speak  of  adding 
-  4  to  a  number  when  we  really  mean  taking  4  from  it.  We 
shall  do  any  of  these  things  when  we  have  good  reason  for 
doing  so,  and  not  otherwise. 

Suppose  we  say  that  2h  sovereigns  are  equivalent  to  50s., 
we  arrive  at  the  result  by  multiplying  20  by  2J,  that  is  first 
by  2  and  then  by  J,  and  adding  the  separate  results.  It 
would  be  a  nuisance  to  be  obliged  to  say  that  we  multiply 
20  by  2  and  divide  20  by  2  and  add  the  results,  though  it 
would  be  quite  true. 

The  fact  that  the  half  of  20  is  10  may  be  written  if  we  like, 
thus  :        I  X  20  =  10  ;  or,  of  course,  %»_  or  20  -r  2  =  10  ; 
^  of  24  =  8  may  be  written  i  x  24  =  ^*-  =  8  ; 


of  J  may  be  written  J  x  ^  = 


1  . 


and  that  the  half  of  the  third  of  an  apple  or  ribbon  is  a  sixth 
of  the  apple  or  ribbon  is  easily  verified  by  experiment.  An 
experiment  need  not  always  be  performed;  after  a  time  it 
can  be  vividly  imagined,  with  advantages  on  the  side  of  clear- 
ness of  apprehension. 


III.]  FRACTIONS.  4:3 

The  natural  word  to  use  for  taking  the  fraction  of  a  thing 
is  the  word  "  of,"  like  the  half  of  an  orange  or  a  quarter  of 
a  pound  or  one-sixth  of  the  revenue ;  and  we  shall  gradually 
find  that  in  all  arithmetical  cases  the  word  "of"  has  to  be 
interpreted  as  an  instruction  to  perform  the  operation  denoted 
Jby  X  ^  that  is  to  say,  the  operation  we  have  been  accustomejj 
to  call  multiplication. 


Practical  remarks  on  the  treatment  of  fractions. 

It  so  happens  that  the  multiplication  of  vulgar  fractions 
is  easier  than  addition  and  subtraction,  and  so  it  may  take 
precedence.  One  half  of  one  quarter  is  one  eighth  :  as  can 
be  found  by  concrete  experiment,  for  instance  on  an  apple, 
or  by  looking  at  the  divisions  on  a  2-foot  rule. 

1  nf  1    —    1 

1  nf  3   —     3 
Y  01  ^  -    j-^ 

Ifvfl—      1.1    of    7^—      7.3p.f7_21 

Such  a  statement  as  the  last  must  be,  and  is,  led  up  to; 
and  gradually  the  empirical  rule  can  be  perceived,  that  in 
multiplication  of  fractions  the  numerators  must  be  multiplied 
for  the  new  numerator,  and  the  denominators  must  be  multi- 
plied for  the  new  denominator. 

[But  initial  difficulties  and  confusion  must  be  expected 
between  this  and  the  addition  of  fractions.  Thus,  for 
instance: 

4  "^8"       32       ~  32' 
This  is  set  down  here  as  a  warning. 

The  greatest  difficulty  in  dealing  with  fractions  is  felt  as 
long  as  they  are  abstract,  "f  of  what?"  is  constantly  or 
should  constantly  be  asked  by  a  child.  In  the  above  two 
sums  the  answer  to  this  question  would  be  different  : — 


44  EASY  MATHEMATICS.  [chap.  lil. 

In  one  it  is  |  of  a  fraction,  viz.  |  of  |  of  a  unit,  such  as  a 
foot,  that  has  to  be  found.  In  the  other  it  is  f  of  one  foot 
which  has  to  be  added  to  |  of  another.] 

It  is  convenient  to  ascertain  and  remember  that  ^  +  ^  =  -I 
[whereas  ^  of  ^  =  yV] ;  also  that  i  +  xV  =  h  or  J-yV  =  t- 

Exercise. 

Find  the  third  plus  half  the  third  of  eight.  The  answer  is 
4,  but  the  decimal  notation  confuses  the  matter : 

J  of  8  is  2-6666...  and  half  this  is  1-333... 
so  the  sum  is  3-9999...,  that  is  3|  or  4. 
So  also  a  third  +  half  a  third  of  ten  would  seem  troublesome, 
though  it  results  simply  in  five.     But  a  third  +  half  a  third  of 
a  dozen  is  simple  enough,  being   4  +  2  =  6.      And  always 
1  .  1  _  1 

Division  of  fractions  may  be  exhibited  thus : 
Suppose  we  have  to  find  what  |-  -f  f  amounts  to, 

write  it  thus,  seven  eighths 

three  tilths 

seven  x  five  fortieths       7x5      35       7     5 

=  7:X^;       . 


three  x  eight  fortieths       3x8       24       83' 
wherefore  instead  of  dividing  by  |,  we  find  we  may  multiply 

The  idea  underlying  the  above  process  is  that  things  called 
eighths  have  to  be  divided  by  things  called  fifths,  and  that 
to  make  it  possible  they  must  be  expressed  in  the  same 
denomination,  which  in  this  case  is  fortieths.  Thus  we  get 
the  rule,  invert  the  divisor  and  multiply.  Or  otherwise 
expressed :  to  divide  by  a  number  multiply  by  its  reciprocal. 
Division  by  ^  is  the  same  thing  as  multiplication  by  2.  The 
symbol  -f  J  is  equivalent  to  the  symbol  x  4. 


CHAPTEE  IV. 

Further  consideration  and  extension  of  the  idea  of 
subtraction. 

If  a  man  gains  £21.  6s.  5d.  and  loses  £15.  4s.  4d.,  his  nett 
gain  is  found  by  subtraction,  and  is  called  the  "difference,"  viz. 
£6.  2s.  Id. ;  the  total  money  which  has  changed  hands  being 
the  "  sum,"  viz.  £36.  10s.  9d.  A  loss  may  be  called  a  negative 
gain ;  thus  a  gain  of  £10  minus  £6,  would  mean  a  gain  of 
£10  accompanied  by  a  loss  of  £6,  or  a  nett  positive  gain  of 
£4.  This  leads  us  to  discriminate  between  positive  and 
negative  quantities,  and  to  regard  subtraction  as  negative 
addition.  Subtracting  a  positive  quantity  is  the  same  as 
adding  an  equal  negative  one. 

Geometrically  it  is  sometimes  convenient  to  discriminate 
between  the  journey  A  to  B,  or  AB,  and  the  journey  B  to  Aj 
or  BAj  just  as  a  French-English  dictionary  is  not  the  same  as 
an  English-French  dictionary.  When  expressed  numerically 
a  length  AB  may  be  denoted  by  its  value,  say  3  inches,  or 
3  miles ;  and  the  reverse  journey  may  be  denoted  by  -  3 
inches  or  -  3  miles,  because  this  when  added  to,  or  performed 
subsequently  to,  the  direct  journey,  will  neutralise  it  and 
leave  the  traveller  where  he  started.  The  two  opposite  signs 
cancel  each  other  in  this  sense,  and  the  two  quantities  added 
together  are  said  to  amount  to  zero  algebraically — that  is 
when  their  signs   are   attended  to,  and  as  regards  the  end 


46  EASY  MATHEMATICS.  [chap. 

result  only ;  but  the  traveller  will  himself  be  conscious  that 
although  he  is  where  he  started  from,  he  has  really  walked 
6  miles ;  so  that  for  some  purposes  such  quantities  may  be 
added,  and  they  are  then  said  to  be  "arithmetically"  or 
better  "  numerically  "  added ;  for  other  purposes  they  are  to 
be  "numerically  subtracted,"  or,  as  it  is  called,  "algebraically 
added,"  that  is  with  their  signs  attended  to,  and  with  "minus" 
neutralising  an  equal  "  plus." 

If  a  height  above  sea  level  is  reckoned  positive,  a  depth 
below  may  be  reckoned  negative ;  so  that  a  well  may  be 
spoken  of  either  as  60  feet  below  or  as  -  60  feet  above  the 
sea  level. 

The  latter  mode  of  specification  sounds  absurd,  but  one 
should  gradually  accustom  one's  self  to  it,  for  practical  pur^ 
poses  later  on. 

If  children  feel  a  difficulty  with  these  negative  quantities, 
as  they  have  every  right  to,  they  can  be  accustomed  to  them 
gently,  as  a  horse  to  a  motor  car.  Mathematicians  found 
some  difficulty  with  them  once  upon  a  time,  so  the  difficulty 
is  real,  though  like  so  many  others  it  rapidly  disappears  by 
custom.  Debts,  return  journeys,  fall  of  thrown-up  stones, 
losings,  apparent  weights  of  balloons  or  of  corks  under  water, 
dates  of  reckoning  B,C.,  and  many  other  things  will  serve  as 
illustrations;  not,  however,  to  be  taken  all  at  once. 

Time  is  the  one  thing  that  never  goes  backwards;  but 
nevertheless  intervals  of  time  may  be  considered  negative  if 
they  date  back  to  a  period  antecedent  to  the  era  of  reckoning. 
In  a  race,  for  instance,  it  would  be  an  ordinary  handicap 
if  one  of  the  competitors  was  set  12  yards  behind  scratch,  or 
if  he  was  made  to  start  from  scratch  3  seconds  late.  In  either 
case  he  could  be  said  to  have  a  negative  start. 

In  golf  handicaps  it  is  customary  to  denote  these  positions 
behind  scratch  as  positive,  because  they  are  added  to  the 


IV.]  NEGATIVE  QUANTITIES.  47 

score.     This  is  because  the  object  in  golf  is  to  get  as  low  a 
score  as  possible,  not  a  high  one  as  at  cricket. 

Addition  and  subtraction  of  negative  quantities. 

Suppose  a  man  inherited  a  lot  of  debts,  his  property 
would  be  diminished  by  their  acquisition.  The  addition  to 
it  would  be  negative,  and  would  be  indistinguishable  from 
subtraction. 

A  debt  of  £300  added  to  a  possession  of  £500  would  result 
in  nett  property  of  £200 ;  which  we  might  express  by  saying 
that  -3  +  5  =  +  2. 

Or  of  course  the  debt  might  exceed  the  possession  and 
leave  a  balance  of  debt.  For  instance  -8  +  5  =  -  3 ;  where 
the  unit  intended  by  these  digits  might  be  a  hundred  or  a 
thousand  pounds.  This  may  be  taken  as  an  illustration  of  the 
gain  of  a  negative  quantity.     Take  another. 

An  axe-head  at  the  bottom  of  a  river  weighs  3  lbs.  Some 
corks,  which,  when  submerged,  pull  upwards  with  a  force 
equal  to  the  weight  of  49  ounces,  are  attached  to  the  mass 
of  iron.  Its  weight  is  thus  more  than  counteracted,  and  it  is 
floated  upwards  with  a  force  equal  to  the  weight  of  1  ounce, 
,  because  48  -  49  =  - 1. 

I  .  A  raisin  at  the  bottom  of  a  champagne  glass,  or  a  speck  of 
grit  in  a  soda-water  bottle,  can  often  be  seen  to  accumulate 
bubbles  on  itself  till  it  floats  to  the  surface  and  gets  rid  of 
some,  when  it  sinks  again,  and  so  on  alternately. 

The  negative  or  upward  weight  of  the  corks,  or  of  the 
bubbles,  counteracts  and  overbalances  the  positive  weight  of 
the  iron  or  of  the  fruit.  It  may  be  said  that  we  have  sub- 
tracted more  weight  from  it  than  it  itself  possessed,  and  so 
left  it  with  a  negative  weight — like  a  balloon.  The  weight 
of  a  balloon  is  not  really  negative,  but  it  superficially  appears 
to  be ;  because  the  surrounding  air  buoys  it  up  with  a  force 


48  EASY  MATHEMATICS.  [chap. 

equal  to  the  weight  of  the  air  it  displaces,  which  represents  a 
greater  weight  than  its  own. 

When  we  have  to  subtract  a  bigger  number  from  a  smaller, 
we  must  not  always  merely  say  we  cannot  do  it.  It  is  con- 
venient in  subtraction  sums  to  say  so,  and  to  "  borrow  "  from 
the  digit  in  the  next  higher  place  (i.e.  to  undo  one  of  the 
available  packets  and  bring  the  contents  one  step  down),  so 
long  as  there  is  something  there  to  be  "  borrowed,"  but  if  we 
perceive  that  at  the  end  of  the  sum  there  will  be  a  manifest 
deficiency  we  must  proceed  differently. 

Suppose  we  were  told  to  collect  £S  from  a  man  who  had 
only  £Sj  we  could  not  really  do  it;  but  we  might  report  to 
our  chief,   "if   we   do   we   shall   leave   him  £5   in   debt   to 
somebody,"  which  could  be  expressed  arithmetically  thus  : 
3  -  8  =  -  5. 

Suppose  we  were  told  to  pull  5  feet  of  a  gate-post  out  of 
the  ground,  and  when  we  came  to  try  we  found  that  it  had 
only  2  feet  buried;  we  might  at  first  say  that  it  could  not 
be  done;  but  on  second  thoughts  we  could  say  that  it  was 
hard  to  do,  and  that  the  only  plan  we  could  see  would  be  to 
pull  it  minus  3  feet  out  first,  that  is  to  get  a  mallet  and  drive 
it  3  extra  feet  in,  before  pulling  at  it  at  all. 

Suppose  a  stone  were  30  feet  above  the  ground,  and  we 
were  told  to  drop  it  36  feet,  that  is  to  subtract  36  feet  from 
its  height  of  30  feet.  It  would  not  be  easy  to  do,  but  it 
could  be  done,  for  we  might  dig  a  hole  6  feet  deep;  or  it 
might  even  be  sufficient  if  we  dropped  it  over  a  pond  of  that 
depth.  In  either  case  it  would  afterwards  be  6  feet  below 
the  surface  of  the  ground,  for  30  -  36  =  -  6 ;  it  would  then 
be  at  an  elevation  of  -  6  feet,  which  means  the  same  as  a 
depression  of  6  feet. 

To  speak  of  a  depth  of  6  feet  as  a  negative  height,  in 
ordinary  conversation,  would  be  absurd ;  but  to  interpret  an 


IV.]  NEGATIVE  QUANTITIES.  49 

arithmetical  answer,  which  gives  a  height  as  -  6  feet,  to  mean 
that  a  thing  is  not  elevated  at  all  but  is  depressed  6  feet, 
would  be  quite  right  and  in  accordance  with  commonsense. 
Hence  the  following  examples  are  correct : 
4-      9  =  -      5 
17-    39  =  -    22 
546-827  =  -281 

But  now  here  is  a  necessary  caution.  Take  the  last  case. 
We  see  that  it  is  right,  for  if  we  add  281  to  546  we  get  827 ; 
but  suppose  we  had  put  it  down  like  an  ordinary  subtraction 
sum  and  noticed  nothing  wrong  with  it,  it  would  have 
looked  like  this 

546 

827  (example  of  the  way  not  to  do  it). 

-319 

We  should  have  said  in  the  old-fashioned  way  7  from  6  we 
cannot,  so  borrow  10  from  the  next  place;  7  from  16  is  9,  put 
it  down.  Now  we  have  either  2  from  3,  or  what  is  more 
commonly  said,  and  comes  to  the  same  thing,  3  from  4, 
leaving  1,  which  we  put  down;  and  then  we  have  to  take 
8  from  5.  There  is  nothing  more  to  borrow,  so  we  must  set 
it  down  as  -  3.  Well  that  is  not  wrong,  but  it  requires 
interpreting,  and  it  is  not  convenient.  The  minus  sign  only 
applies  to  the  3,  which,  being  in  the  third  place,  means  300 ; 
the  other  figures,  the  19,  were  positive.  Hence  the  meaning 
is  -300  +  19,  or  in  other  words  -281.  It  might  be  written 
319,  with  the  minus  sign  above  and  understood  to  apply  only 
to  the  digit  3,  but  it  could  not  properly  be  written  -  319. 

The  above  is  therefore  a  very  troublesome  way  of  arriving 

at  the  result.     The  convenient  way  is  not  to  begin  performing 

the  impossible  subtraction,   but  to  perceive  the  threatening 

dilemma,  and  invert  it  at  once ;    then  subtract  the  smaller 

I4.E.M,  D 


50  EASY  MATHEMATICS.  [chap.  iv. 

number  from  the  bigger  in  the  ordinary  way,  labelling  the 
result  however  as  negative.  This  is  of  course  what  we  really 
do  when  we  say  5-8  =  -  3.  We  do  not  begin  saying  "  8  from 
5  we  cannot,  so  borrow"  from  nowhere,  for  there  is  nowhere 
to  borrow  from.  We  stop,  invert  the  operation,  and  record 
the  result  as  negative ;  because  a-h  =  -  (b-a). 

One   more   case   we   must   take   however,   viz.  where  the 

quantity  to  be  subtracted  is  itself  negative :  and  its  subtrac- 

\.  tion  therefore  represents  a  gain.     The  loss  of  an  undesirable 

•  burden  was  esteemed  by  Bunyan's  Pilgrim  to  be  a  clear  gain. 

A  negative  subtraction  is  a  positive  addition. 

6-(-3)  =  9;     7-(-9)  =  16. 

This  is  sometimes  expressed  by  saying  that  two  minuses 
make  a  plus.  The  effect  of  a  minus  is  always  to  reverse  the 
sign  of  any  quantity  to  which  it  is  prefixed,  so  if  applied  to  a 
negative  quantity  it  turns  it  into  a  positive  quantity.  It  is 
equivalent  to  more  than  the  removal,  or  subtraction,  of  a  debt, 
which  would  be  effected  by  an  equal  sum  added.  A  loss  is 
more  than  neutralised  by  a  negative  sign,  it  is  reversed. 

Add  -  31  to  114,  the  result  is  83  ;  but  subtract  -  31  from 
114,  and  the  result  is  145. 

No  more  words  are  necessary.  Familiarity  and  practice 
will  come  in  due  course  as  we  proceed.  A  surviving  puzzle 
may  occasionally  be  felt,  and  can  from  time  to  time  be 
removed.  It  is  a  mistake  to  hammer  at  a  simple  thing  like 
that  till  it  becomes  wearisome ;  for  trifling  puzzles  or  foggi- 
nesses  evaporate  during  sleep,  and  in  a  few  years  have 
automatically  disappeared,  from  children  properly  taught. 
They  continue  to  trouble  too  many  adults  at  present. 


CHAPTEE  V. 

Generalisation  and  extension  of  the  ideas  of  multi- 
plication and  division  to  concrete  quantity. 

The  idea  of  multiplication  arose  as  a  convenient  summary 
of  a  special  kind  of  addition,  viz.  the  addition  of  several 
things  of  the  same  magnitude  to  each  other.  Thus  four  sixes 
added  together,  if  counted,  make  24,  and  so  it  is  summarised 
and  remembered  as  4  sixes  are  24,  or  4  times  6  =  24 ;  and  4 
and  6  are  called  '  factors '  of  24. 

Originally  therefore  the  two  factors  in  multiplication 
signified,  one  of  them  the  size  of  the  quantity  of  which  several 
are  to  be  added  together,  and  the  other  the  number  of  times 
it  was  to  be  so  added. 

Thus  3x6,  read  3  times  6,  meant  a  summarised  addition 
sum,  6  +  6  +  6.  But  if  read  6  times  3  it  meant  the  addition 
sum  3  +  3  +  3  +  3  +  3  +  3.  That  the  result  is  the  same  may  be 
treated  as  a  matter  of  experience,  and  may  be  demonstrated 

by  grouping,  but  it  is  not  to  be  regarded  as 

self-evident.     Nevertheless  the  diagram  (fig.  2)     ••.... 

demonstrates   that  3  rows  of   6   each  is  the    •••••• 

same   as   6    columns    of    3    each.      And    the  ^^'  ^* 

counting  of  window  panes  and  postage 
stamps  are  illustrations  of  practically  the 
same  thing. 
^ — ' — ' — '  Thus   we   get   led   to   the  area   of  a 

Fio.  3.  rectangle  of  length  a  and  breadth  h  as 

a  X  J,  or  briefly  written  ah  (fig.  3). 


52 


EASY  MATHEMATICS. 


[chap. 


But  the  idea  of  multiplication  soon  generalises  itself,  and 
the  expression  ah  gets  applied  to  a  number  of  things  to  which 
a  simple  numerical  idea  like  3  times  6,  or  a  times  h,  would 
hardly  apply. 

It  may  be  worth  showing  however  that  the  numerical 
notion  will  apply  further  than  might  have  been  anticipated, 
for  instance  the  rectangle  (fig.  4)  is  built  up  of  5  equal  staves 
each  of  them  say  3  inches  long  and  an  inch  wide.  The  area 
of  each  stave  is  thus  3  inches  x  1  inch,  or  3  square  inches. 
And  by  adding  5  of  the  staves  together  (or  multiplying  one 
of  them  by  5)  we  get  the  total  area. 


Fig.  4. 


Fio.  5. 


And  the  same  area  could  be  equally  well  obtained  by 
putting  together  3  staves  each  of  5  square  inches  area  (fig.  5). 

The  number  12  can  be  resolved  into  two  factors  3  and  4,  as 
is  shown  by  the  annexed  group  which  consists  of  3  rows  of 
four  dots  each,  or  of  4  columns  of  3  dots  each,  proving  that 
3  times  4  -=  4  times  3. 


Fig.  6. 


Fig.  7. 


A  dozen  can  equally  well  be  grouped  as  in  fig.  7  :  its  large 
number  of  factors  confers  distinction  on  the  number  12. 

The  number  10  has  only  two  factors,  viz.  2  and  5,  since 
the  name  "factor"  is  usually  limited  to  whole  numbers.  It  is 
possible  to  say  that  3|  is  a  factor  of  10,  because  if  it  be 


v.]  CONCRETE  QUANTITY.  53 

repeated  3  times  the  number  ten  results ;  as  is  shown  by  the 
following  set  of  3^  disks  repeated  3  times,  where  the  central 
sectors  have  each  of  them  an  angle  1 20°  or  q 

1^  of  a  revolution,  and  so  make  up  a  disk  q 

when  put  together.     But  the  name  "  factor  "  q 

is  not  usually  applied  to  fractions.  O  O  O  ® 

Again,  a  slab  of  any  given  area  and  unit  Q 

thickness  will  have  a  bulk  which,  measured  O 

in  cubic  inches,  is  numerically  equal  to  its  Q 

area  in  square  inches.    If  such  a  slab  is  mul-  ^^°"  ^' 

tiplied  or  repeated,  each  slab  being  piled  up  on  similar  ones, 
say  7  times,  then  7  times  its  bulk  will  give  the  volume  of  a 
rectangular  block ;  or  the  volume  of  a  block  may  be  said  to 
be  obtained  by  multiplying  its  length,  breadth,  and  height. 
There  is  no  reason  to  take  one  of  these  factors  as  numerical 
more  than  another,  and  the  truth  is  that  none  of  them  need 
be  numerical. 

When  we  say  volume  =  Ibh,  or  length  x  breadth  x  height, 
we  may  and  should  mean  by  I  the  actual  length, 
hy  b    „       „      breadth, 
and  by  h    „       „      height, 
— not  the  number  of  inches  or  centimetres  in  each — and  the 
resulting  product  is  then  the  actual  volume,  and  not  any 
numerical   estimate  of   it.      [If   anyone   disagrees   with   this 
they  are  asked  to  withhold  their  disagreement  for  the  present. 
This  is  one  of  the  few  things  on  which  presently  I  wish  to 
dogmatise.     See  Chap.  XXVI.  and  Appendix  II.] 

From  this  point  of  view  the  symbols  of  algebra  are  concrete 
or  real  physical  quantities,  not  symbols  for  numbers  alone,  and 
algebra  becomes  more  than  generalised  arithmetic. 

In  such  cases  however  the  old  original  definition  of  multipli- 
cation requires  generalisation,  and  a  good  deal  can  be  written 
on  it ;  but  no  difficulty  arises,  and  the  question,  being  inter- 


54  EASY  MATHEMATICS.  [chap. 

esting  chiefly  from  the  philosophic  point  of  view,  does  not  in 

this  book  concern  us. 

"We  may  proceed  without  compunction  to  multiply  together 

all  sorts  of  incongruous  things  if  we  find  any  convenience  in  so 

doing.     Thus,  a  linear  foot  multiplied  by  a  linear  foot  gives  a 

square  foot, 

6  feet  X  3  feet  gives  18  square  feet, 

4  feet  X  3  feet  x  2  feet  gives  24  cubic  feet. 

In  all  these  cases  something  real  and  intelligible  results; 
but  if  we  multiply  square  feet  by  square  feet,  nothing  intelli- 
gible results ;  consequently  such  a  process  will  never  appear 
in  a  correct  end-result,  though  we  shall  find  that  it  often 
appears  as  a  step  in  a  process  without  any  detriment. 

Again  we  may  multiply  a  weight  by  a  length,  say  3  lbs.  by 
7  feet,  and  get  what  is  called  21  foot-lbs.,  where  the  unit  has 
a  meaning  which  can  be  interpreted,  viz.  the  work  done  in 
raising  a  3  lb.  weight  7  feet  high  against  gravity,  or  else  the 
moment  of  a  force  round  an  axis.  But  if  we  try  to  multiply 
3  lbs.  by  7  lbs.,  we  should  get  21  square  lbs.,  which  has  no 
intelligible  meaning  and  is  nonsense.  There  is  nothing  in  the 
symbols  to  tell  us  whether  it  is  sense  or  not :  operations  can 
be  consistently  performed  even  on  meaningless  symbols. 
To  discriminate  sense  from  nonsense,  appeal  must  be  made 
to  reality  and  to  actual  life  and  instructed  experience. 

Division  is  merely  the  inverse  of  multiplication,  and  similar 
considerations  apply  to  it. 

If  we  divide  1  by  any  quantity  we  get  what  is  called  the 
reciprocal  of  that  quantity. 

Thus  J  is  the  reciprocal  of  2.     xir  is  the  reciprocal  of  10. 

— P —  is  the  reciprocal  of  a  length,  and  could  be  read 
o  leet 

1  per  yard. 


v.]  CONCRETE  QUANTITY.  55 

-— — -^  might  represent  the  number  of  telegraph  posts 

per  mile. 

— -.  is  the  reciprocal  of  a  time,  and  might  be  read 

^^  'once  every  tenth  of  a  second';   or  it  could  be 

simplified  into  a  repetition  of  something  ten 
times  a  second,  or  10  per  second.  It  is  what 
is  called  a  '  frequency,'  and  is  in  constant  use  for 
vibrations. 

-^ is  a  slow  frequency,  the  frequency   with  which 

a   cycle   of    astronomical    eclipses    approximately 
recurs. 

6000  revolutions 
5  minutes 

wheel  of  a  small  engine,  and  may  be  read  as 
1 200  revolutions  per  minute,  or  20  revolutions 
per  second. 

If  we  divide  a  length  by  a  time,  as  for  instance 

. ^^  ^^,  we  get  a  velocity;  e.g.  the  speed  of  an  express 

l^""""     train. 

is  exactly  the  same  velocity. 


is  a  frequency  of  rotation,  as  of  the  fly- 


1  second 

^^  ^^,  or  approximately      ^^^  ^  ,  is  a  walking  pace. 

No  hesitation  must  be  felt  at  thus  introducing  the  units 
into  the  numerator  or  denominator  of  fractions.  If  they  are 
left  out,  the  residue  becomes  a  mere  numerical  fraction,  the 
ratio  of  two  pure  numbers  ;  whereas  with  the  units  inserted 
they  are  real  physical  quantities  with  a  concrete  meaning,  and 
are  capable  of  vaTied  numerical  specification. 


56  EASY  MATHEMATICS.  [cha1». 

Thus  the  velocity  of  sound  in  air  at  the  freezing  point  is 
1090  feet         33000  centimetres  1  mile 


or 5 =, or 


1  second  1  second  5  seconds 

•33  kilometres         1  kilometre  •      .  i 

or  — . —  or  -^; -^ —  approximately 

1  second  3  seconds 

10  minutes' walk         240000  miles 
3  seconds  a  fortnight 


First  idea  of  involution. 

When  a  number  of  the  same  things  were  added  together 
many  times,  the  process  was  specially  treated  and  called 
multiplication.  When  a  number  of  the  things  are  multiplied 
together  several  times,  the  process  is  likewise  worthy  of  special 
treatment,  and  is  called  "  involution  "  or  the  raising  of  a  thing 
to  a  certain  "power." 

The  raising  to  a  power  is  compressed  or  summarised  multi- 
plication. The  expression  4x3  meant  four  added  to  itself 
3  times  (or  12),  whereas  4^  is  understood  to  mean  4  multiplied 
by  itself  3  times  (or  64). 

So  25  =      32,     63  =  216, 
103  =  1000,  106  =  a  million, 

122  =  144^  and  can  be  read  12  square,  for  short ;  though 
really  a  square  number  is  an  absurdity.  It  is  called  "  twelve 
square"  because  if  the  12  represented  inches,  12^  would  mean 
a  square  foot. 

If  ft  is  a  length,  a^  is  truly  a  square  whose  side  is  of  length 
a,  and  a^  is  truly  a  cube  whose  side  is  of  length  a.  So  4^  is 
read  "4  square,"  and  6^  is  often  read  "six  cube,"  by  analogy. 
It  is  also  true  that  2*  =  16,  but  here  there  is  no  geometrical 
analogy,  and  it  is  read  "  2  to  the  fourth  power  "  simply,  the 
word  "power"  being  often  omitted  in  practice.  Similarly  a 
million  is  "  ten  to  the  sixth  "  or  10^. 


v.]  CONCRETE  QUANTITY.  57 

A  length  divided  by  a  time  is  a  velocity  (v)y  and  a  velocity 
divided  by  a  time  is  an  acceleration  (a). 

V 

So  in  mechanics  we  find  such  an  expression  as 

where  f^  is  often  read  as  the  square  of  the  time,  although 
strictly  speaking  such  an  expression  is  nonsense.  We  can 
have  a  square  mile,  but  not  a  square  fortnight;  there  is  no 
meaning  to  be  attached  to  the  term ;  time  cannot  be  multi- 
plied by  time  with  any  intelligible  result.  Whenever  such  an 
expression  occurs,  it  is  to  be  understood  as  an  abbreviation 
for  something :  in  the  above  case  for  this 

s  =  ■^{at)t, 

where  the  at  is  v,  and  is  a  real  and  simple  physical  quantity. 

s  is  a  velocity  multiplied  by  a  time,  and  the  double 
reference  to  time  is  caused  by  the  introduction  of  the  specially 
defined  quantity  "acceleration,"  which  is  often  expressed 
correctly  as  so  many  feet  per  second  per  second;  the  two 
units  of  time  in  the  denominator  being  conveniently  spoken  of 
as  the  square  of  the  time — by  analogy  with  geometry  again — 
without  thought  and  without  practical  detriment,  though 
confusing  to  anyone  who  seeks  a  real  philosophic  meaning  in 
the  expression. 


CHAPTEE  VI. 
Factors  of  simple  numbers. 

A  CHILD  should  be  encouraged  who  notices  that  no  factor  is 
ever  greater  than  half  the  number;  for  though  there  is  nothing 
in  that  but  what  is  obvious,  yet  that  is  the  type  of  noticing 
which  frequently  leads  to  observations  of  interest.  An  even 
number  always  has  this  largest  factor,  but  an  odd  number 
can  never  have  a  factor  greater  than  a  third  its  value ;  and 
frequently  its  largest  factor  is  less  than  this.  Some  numbers 
have  no  factors  at  all;  like  7  and  11  and  13  and  29  and  131. 
These  are  called  prime  numbers,  and  a  child  should  make  a 
small  list  of  them  as  an  exercise.  But  do  not  attempt  to 
make  it  learn  them  or  anything  of  this  kind  by  heart.  Ease 
and  quickness  of  obtaining  when  wanted  is  all  that  is 
practically  needed. 

A  child  should  be  encouraged  to  discover  criteria  for  the 
existence  of  simple  factors;  but  is  hardly  likely  to  be  able 
to  notice  the  facts  without  aid. 

Any  number  (written  in  the  decimal  notation)  which  is 
divisible  by  3  (i.e.  which  has  3  as  a  factor)  has  the  sum  of  its 
digits  also  divisible  by  3.  But  this,  though  convenient  as  a 
rule,  is  in  no  sense  fundamental :  it  depends  merely  on  our 
habit  of  grouping  in  tens.  In  the  duodecimal  system  every 
number  ending  in  0  would  necessarily  be  divisible  by  3  as  well 
as  by  4  and  by  6 ;  and  extremely  convenient  the  fact  would  be. 


CHAP.  VI.]  FACTORS.  59 

For  instance,  1/-  and  2/-  and  4/-  and  5/-,  or  any  number  of 
shillings,  can  be  divided  by  3,  4,  or  6 ;  that  is,  can  be  parcelled 
out  exactly  into  a  whole  number  of  pennies. 

By  reason  of  the  system  of  reckoning  12  pence  to  a 
shilling,  any  sum  of  money  can  be  subdivided  into  three  or 
six  equal  parts  without  halfpence  or  farthings ;  thus  J  of  a 
pound  is  6s.  8d.,  two-thirds  is  13s.  4d.,  one-sixth  is  3/4,  one 
8th  is  2/6,  and  one-twelfth  is  1/8. 

In  the  decimal  notation  a  number  has  to  end  in  00  in  order 
to  be  certainly  divisible  by  4;  and  in  000  in  order  to  be 
certainly  divisible  by  8.  And  the  division  is  seldom  worth 
doing  even  then,  because  it  hardly  results  in  simplification. 

The  number  5  in  the  decimal  system  has  an  artificial  sim- 
plicity conferred  upon  it,  but  it  is  not  often  that  we  should 
naturally  group  things  in  5,  except  for  the  accident  of  our 
5  fingers  :  and  one  of  them  is  a  thumb. 

The  advantage  of  working  in  at  least  two  diff'erent  scales 
of  notation  is  that  it  becomes  thereby  easy  to  discriminate 
what  is  essential  and  fundamental  from  what  is  accidental 
and  dependent  on  the  scale  of  notation  employed.  Thus  the 
curious  properties  of  the  number  nine  or  eleven  are  artificial, 
and  in  the  duodecimal  scale  are  transferred  to  eleven  and 
thirteen  respectively. 

The  well-known  criterion  for  divisibility  by  3  or  9,  viz. 
whether  the  sum  of  the  digits  is  so  divisible,  is  accidental 
again,  and  disappears  in  another  scale  of  notation — for  instance 
when  units  are  grouped  in  dozens  instead  of  tens, — to  give 
place  however  to  a  much  simpler  rule. 

The  rule  about  divisibility  of  the  sum  of  the  digits  applies 
to  eleven  in  the  duodecimal  scale,  and  indeed  would  always 
apply  to  the  number  which  is  one  less  than  the  group  number 
artificially  selected. 

But  the  existence  and  identity  of  prime  numbers  is  not 


60  EASY  MATHEMATICS.  [chap. 

accidental  at  all,  but  fundamental,  and  so  also  is  the  existence 
of  any  given  numbers  of  factors  to  a  number — however  it  be 
specified. 

Thus  one  gross  can  be  parcelled  out  into  factors  or  equal 
groups  in  a  given  number  of  ways,  whether  it  be  denoted  by 
1/0/0  or  by  144  or  by  any  other  system  of  notation. 

So  also  the  number  one-hundred  has  only  six  factors 
whether  it  be  denoted  by  8/4  or  by  100  (one  nought  nought), 
and  its  factors  are  (in  the  duodecimal  scale) : 

4/2         2/1         1/8         t        5         2, 

that  is  these  actual  numbers,  however  they  are  denoted.  In 
the  duodecimal  scale  it  is  needful  to  have  single  symbols  for 
ten  and  eleven ;  and  the  initial  letters  serve  the  purpose. 

An  actual  number  is  easily  exhibited  by  means  of  counters 
or  coins  or  marbles :  its  expression  in  digits  is  an  artificial 
arrangement  and  is  adopted  simply  for  convenience :  it  is 
analogous  to  sorting  the  marbles  into  bags  of  which  each 
must  contain  an  equal  number — whatever  number  may  be 
chosen  as  suitable  and  fixed  upon  for  the  purpose. 

It  may  be  interesting  to  write  down  the  numbers  in  the 
duodecimal  scale  which  would  be  divisible  by  5. 

5,  t,  1/3,  1/8,  2/1,  2/6,  2je,  3/4,  3/9,  4/2,^4/7,  5/0,  ..., 
and  the  even  numbers  in  the  above  are  divisible  also  by  ten. 
The  above  numbers  should  be  read  five,  ten,  one  and  three, 
one  and  eight,  two  and  one,  two  and  six,  two  and  eleven,  etc., 
meaning  one  dozen  and  three,  one  dozen  and  eight,  two  dozen 
and  six,  two  dozen  and  eleven,  etc. 

Numbers  which  have  the  factor  7  are 
7,  1/2,  1/9,  2/4,  2/^,  3/6,  4/1,  4/8,  5/3,  5/^,  6/5,  7/0,  ..., 
and  the  even  ones  are  divisible  also  by  fourteen. 

Numbers  which  have  the  factor  eleven  {e)  are 
e,  Ijt,  2/9,  3/8,  4/7,  5/6,  6/5,  7/4,  8/3,  9/2,  tjl,  e/0,  ..., 


VI.]  FACTORS.  61 

namely  eleven,  one  and  ten,  two  and  nine,  and  so  on :    the 
last  one  written  being  read  eleven  dozen. 

Numbers  divisible  by  thirteen  (1/1)  are 

1/1,  2/2,  3/3,  4/4,  5/5,  6/6,  7/7,  8/8,  9/9,  %  e/e,  1/1/0,  .... 

In  the  last  two  cases  a  law  or  order  among  the  digits  is 
manifest,  but  in  all  four  cases  it  may  be  noticed  that  every 
digit  makes  its  appearance  in  the  units  place,  though  only  in 
the  last  two  cases  do  they  appear  in  a  simple  order. 

Numbers  divisible  by  3  are 

3,  6,  9,  1/0,  1/3,  1/6,  1/9,  2/0,  2/3,  2/6,  2/9,  3/0,  .... 
and  the  even  ones  are  divisible  by  6.     Every  third  one  of  the 
above  series,  viz.  those  in  thick  type,  are  divisible  by  9. 

Numbers  divisible  by  4  are 
4,  8,  1/0,  1/4,  1/8,  2/0,  2/4,  2/8,  3/0,  3/4,  3/8,  4/0,  .... 
Alternate  ones  are  divisible  by   8,  and  those  in  thick  type 
are  divisible  by  sixteen. 

Numbers  divisible  by  twelve,  that  is  arrangeable  in  dozens, 
are  of  course, 

1/0,  2/0,  3/0,  4/0,  etc.,  1/0/0,  ..., 
the  last  written  being  the  symbol  for  a  dozen  dozen  or  one 


CHAPTEE  VII. 
Dealings  with  money  and  with  weights  and  measures. 

In  the  British  Isles  it  is  customary  to  count  pennies  by  the 
dozen,  the  value  of  which  when  coined  in  silver  is  called  a 
"  shilling  " ;  and  shillings  are  counted  by  the  score,  the  value 
of  which  is  called  a  "pound  sterling,"  or  when  coined  in  gold 
a  "  sovereign."  Five  dozen  pence,  or  a  quarter  of  a  pound 
sterling,  when  in  a  single  silver  piece  used  also  to  be  called  a 
"crown."  And  these,  together  with  the  half-sovereign,  half- 
crown,  half-penny,  etc.,  are  the  chief  names  in  vogue ;  except 
the  "guinea"  and  the  "farthing,"  neither  of  which  need  much 
concern  us.  The  "  florin  "  is  an  attempt  at  a  decimal  coinage, 
being  the  tenth  of  a  pound;  and  the  double-florin  is  an 
attempt  at  an  international  currency  or  equivalence  with  the 
dollar  and  the  five-franc  piece. 

The  addition  of  money  is  a  practical  operation  in  constant 
use,  and  plenty  of  practice  in  addition  is  obtainable  by  its 
means.  No  other  addition  sums  are  worth  attention  for  their 
own  sake :  but  in  addition  of  money  it  is  worth  while  taking 
pains  to  acquire  a  fairly  quick  and  accurate  style.  At  the 
same  time  it  is  to  be  remembered  that  it  is  a  purely  mechanical 
process — one  that  in  large  offices  is  better,  more  rapidly  and 
more  accurately,  performed  by  a  machine,  into  which  the 
figures  are  introduced  by  pressing  studs,  and  then  the  addition 
performed  instantaneously  by  turning  a  handle. 


CHAP.  VII.]  MONEY.  63 

Nothing  that  can  be  performed  by  turning  a  handle  can  be 
considered  an  element  in  a  liberal  education  :  it  can  only  be  a 
practical  and  useful  art.  That  however  it  is ;  partly  because 
a  machine  is  seldom  available,  partly  because  it  is  ignominious 
to  be  helpless  without  a  tool  of  this  kind,  chiefly  because 
addition  of  money  is  an  operation  which  is  called  for  by 
commonplace  daily  life  more  often  than  any  other. 

Nothing  much  need  here  be  said  about  it.  The  columns 
of  an  actual  account  hook  are  the  best  addition  sums  to  set  for 
practice.  Also,  in  writing  figures  down,  it  is  well  to  take 
care  to  place  the  unit  digits  under  each  other,  leaving  a  place 
for  a  left-hand  digit  whenever  such  occurs  in  the  pence  and 
shillings  columns,  and  to  be  equally  careful  to  write  the 
pounds  with  the  corresponding  places  vertical.  Also  to  write 
all  figures  very  plainly.  This  last  always,  and  for  all  purposes  : 
A  good  clear  style  of  figure- writing  should  be  cultivated. 

Subtraction  of  money  is  greatly  facilitated  by  the  use  of 
the  "  shop  "  method  :  the  old-fashioned  process  of  "  borrowing" 
was  troublesome,  and  moreover  only  enabled  one  row  of 
figures  to  be  subtracted  from  one  row,  whereas  with  the  shop 
or  complementary  method  any  number  of  rows  may  be  sub- 
tracted from  another  row,  and  the  process  is  practically  only 
addition.  For  instance  suppose  it  is  wished  to  subtract  all 
the  smaller  amounts  from  the  larger  in  the  annexed  statement: 


£ 

s. 

d. 

341 

8 

7 

less  19 

5 

9 

and  14 

0 

3 

and  36 

17 

5 

271 

5 

2 

The  process  is,  to  say,  5  and  3  and  9  make  1/5  and  2  make 
1/7,  put  down  the  2d.  and  carry  1/-;  then  18  and  0  and  5 
make  23  and  5  make  28,  put  down  5/-  and  carry  £1 ;  then 


64  EASY  MATHEMATICS.  [chap. 

7  and  4  and  9  make  20  and  1  make  21,  put  down  1  and  carry 
2;  then  5  +  1  +  1  =  7  and  7  make  14;  and  finally  1  and  2 
make  3.     In  reading,  emphasise  all  the  black  figures. 

Verify  by  adding  the  four  lower  lines. 

As  to  multiplication  and  division  of  money  or  of  weights 
and  measures  we  will  deal  briefly  with  them :  the  old- 
fashioned  practice  in  such  matters  was  tedious  and  was 
pushed  in  childhood  into  needless  intricacies.  Dulness  is 
apt  to  line  all  this  region,  unless  skill  is  expended  on  it  and 
due  care  taken,  and  no  more  practice  should  be  enforced  in 
it  than  is  required  for  ordinary  life.  Discipline  and  punish- 
ment lessons  might  possibly  with  advantage  be  confined  to  this 
region.  Even  for  punishment  it  is  however  hardly  necessary 
to  inflict  sums  dealing  with  acres,  furlongs,  poles  or  perches ; 
or  with  bushels,  pecks,  scruples,  quarters,  pennyweights,  and 
drams.  Hogsheads,  kilderkins,  and  firkins  may  perhaps  at 
length  be  considered  extinct,  except  for  purposes  connected 
with  the  study  of  folk-lore.  There  are  plenty  of  real  and 
living  units  to  be  learnt  in  Physics :  we  need  not  ransack 
old  libraries  and  antique  country  customs  for  them.  And, 
though  the  humanity  involved  in  and  represented  by  old 
names  has  been  a  relief  to  some  children,  during  their  dismal 
lessons,  far  too  much  has  been  made  of  the  trivial  and  dull 
operations  suggested  by  tables  of  British  weights  and  measures. 
The  sooner  most  of  them  are  consigned  to  oblivion  the 
better. 

Real  living  arithmetic  is  the  same  in  any  country ;  and 
the  most  important  of  all  is  that  which  must  necessarily 
be  the  same  on  any  planet. 

The  units  that  are  at  present  worthy  of  terrestrial  attention 
are  the  following : 

Units   of  length — inch,   foot,   yard,  mile,  millimetre, 
centimetre,  metre,  kilometre. 


VII.]  UNITS.  66 

units    of    time — second,   minute,    hour,  day,    week 

year, 
units    of    area — square   mile,    square    foot,    square 

centimetre,  etc. 
units  of  volume — cubic  foot,  cubic  inch,  cubic  yard, 
cubic  centimetre,  cubic  metre ; 
occasionally     also     litres,    gallons 
and  pints, 
units    of    mass — pound,  ton  (ounce,  grain,  hundred- 
weight   occasionally),    gramme, 
kilogramme,  milligramme, 
units  of  money — pence,    shillings,    pounds,    francs, 
marks,  dollars. 
But  conversion  from  one  to  the  other  of  the  last-mentioned 
denominations   should   in   every   case   be   only  approximate. 
Accurate  work  when  wanted  is  done  by  tables,  and  the  rate 
of  exchange  is  constantly  varying. 

For  division  of  money,  and  of  weights  and  measures,  the 
orthodox  school  rule  is  called  "practice";  and  it  sometimes 
happens  that  by  excessive  practice  children  are  able  to  do 
this  kind  of  sum  much  better  than  adults — better  even 
than  mathematicians;  but  since  school  time  is  limited,  such 
extravagant  facility  in  one  direction  is  necessarily  balanced 
by  extreme  deficiency  in  many  others,  and  is  therefore  to  be 
deprecated.  The  world  is  too  full  of  interest  to  make  it 
legitimate  to  exhaust  the  faculties  of  children  over  quite 
needless  arithmetical  gymnastics,  which  confer  no  mathe- 
matical facility,  but  engender  dislike  of  the  whole  subject. 

Modern  treatment  of  the  rule  called  "  Practice."    The 
practical  advantages  of  decimalisation. 

In  old  days  some  very  long  sums  used  to  be  invented  for 
British  children  whereby  our  insular  system  of  coinage  and  of 


66  EASY  MATHEMATICS.  [chap. 

weights  and  measures  was  pressed  into  the  service  to  make 
difficult  exercises.     The  form  was  usually  something  like  this : 

Find  the  cost  of  131  tons  5  cwt.  3  qrs.  24  lbs.  5  oz.  at 
£4.  13s.  9id.  a  ton. 

Mr.  Sonnenschein  led  the  way,  I  think,  towards  taking  all 
the  sting  out  of  these  outrageous  problems,  and  reducing  them 
to  useful  though  unimpressive  and  essentially  insular  exercises, 
by  introducing  the  chief  advantage  of  the  decimal  system  into 
the  working,  before  it  had  been  embodied  by  Parliament  in 
a  legal  system  of  weights  and  measures  and  coinage  itself. 

If  such  sums  have  to  be  done,  and  a  moderate  amount  of 
"  practice  "  in  that  direction  is  quite  legitimate,  decimalisation 
of  at  least  one  of  the  quantities  specified,  that  is,  expressing  it 
in  terms  of  one  denomination,  is  undoubtedly  the  proper 
initial  step  to  take ;  and  then  if  we  are  asked  the  cost  of  so 
much  goods  at  a  given  price,  the  matter  becomes  a  mere 
straightforward  multiplication ;  while  if  we  are  asked  to  find 
the  price  of  a  given  amount  of  goods  which  have  cost  so  much 
money,  or  the  amount  of  goods  which  can  be  obtained  for  a 
given  sum  of  money  at  a  given  price,  we  have  only  a  straight- 
forward division  sum  to  do ; — once  the  complication  of  many 
denominations,  that  is  to  say  the  "compound"  nature  of  the 
specification,  with  scales  of  notation  mixed  up,  is  by  an  initial 
process  got  rid  of.  It  is  always  possible,  and  sometimes 
advocated,  to  reduce  everything  to  the  lowest  denomination, 
e.g.,  in  the  sum  above  to  halfpennies  and  ounces ;  but  that  is 
terribly  long  and  tedious.  Expression  in  terms  of  the  highest 
denomination  is  much  neater.  The  initial  process  is  as 
follows : 

Decimalisation  of  money. 

To  express  any  sum  of  money  in  terms  of  a  single  unit,  say 
£1,  which  is  the  best  unit  for  the  purpose,  it  is  sufficient  to 


viT.]  PRACTICE.  67 

notice  and  remember  a  few  simple  convenient  facts.  They 
are  all  painfully  insular,  and  are  not  an  essential  part  of  real 
arithmetic  at  all,  but  if  properly  and  lightly  treated  they 
afford  to  British  children  an  amount  of  easy  practice  which 
foreign  children  are  destitute  of.  It  is  only  when  trivial  facts 
and  insignificant  sums  are  laboured  at,  till  they  kill  all  interest 
of  the  British  child  in  real  arithmetic,  that  they  become  deadly 
and  deserving  of  the  harshest  epithets. 

The  decimalisation  of  money  in  terms  of  a  pound  is  easy, 
since  a  florin  is  the  tenth  of  a  sovereign ;  so  any  number  of 
shillings  is  easily  expressed  in  decimals  of  a  pound. 

2/-  =  <£•!,  1/-  =  £-05, 

3/-  =  £-15,  6d.  =  £-025, 

4/-  =  £%  1/6  =  £-075, 

5/-  =  £-25,  2/6  =  £-125, 

6/-  =  £-3,  3/6  =  £-175, 

7/-  =  £-35,  4/6  =  £-225, 

and  so  on.  etc. 

A  penny  is  2To*^  ^^  ^  pound,  but  that  is  not  specially 
convenient  when  expressed  as  a  decimal ;  a  farthing  is  -g^^^th 
of  a  pound,  and  that  is  approximately  iwuo  ^^  £'001. 

Since  money  is  never  needed  closer  than  to  the  nearest 
farthing,  except  in  the  price  of  cotton  per  lb.  and  a  few  rare 
cases,  the  approximation  of  £-001,  sometimes  called  a  mil,  for 
1  farthing,  or  the  writing  of  a  farthing  instead  of  £'001,  often 
suffices  ;  especially  in  interpreting  results. 

The  following  expressions  are  all  equivalent  in  value : 
^  a  sovereign  =  10/-  =  5  florins  =  £-5. 

So  also  are  the  following,  each  row  among  themselves : 
£7.  10s.  =  £7i  =  £7-5  =  £7  +  5  florins. 
J  of  a  ten-pound  note  =  '25  ten-pound  note  =  £2*5  =  £2. 10s. 
15/-  =  £|  =  £-75. 


68  EASY  MATHEMATICS.  [chap. 

150/-  =  75  florins  =  £7'5  =  £7.  10s. 
18/-  =     9  florins  =  <£-9. 
12/-  =    6  florins  =  £'6. 
£1.  12s.  =  <£l-6. 
£4:.  18s.  =  £4-9. 
£7.  19s.  =  £7-95. 

All  these  expressions  should  be  read  backwards  as  well  as 
forwards. 
So  also 
£b.    2s.  6d.     =  £5-125. 
£3.    Is.  6d.     =  £3-075. 
£3.  lis.  6d.     =  £3-575. 
£3.  lis.  ^d.  =  £3-577,  almost  exactly. 

Take  a  few  examples  of  the  interpretation  of  decimals  of  a 
pound  into  ordinary  coinage  : 
£1-2       =  £1.  4s. 
£4-25    =  £4.  5s. 

£7-904  =  £7.  18s.  Id.,    the   four  mils  being  practically   a 

penny. 
£13-127  =  £13.  2s.  6Jd.,  the  -125  being  2/6,  and  the  2  extra 

mils  ^d. 
£1-178  --=  £1.  3s.  6|d.,  the  -15  being  3/-,   -025  =  6d.,  and 

there  being  3  mils  more. 
£-025  =  6d. 

£-026  =  6|d.  almost  exactly. 
£-027  =  6Ld. 

£-028  =  6|d.        „  „      . 

£-029  =  7d. 

£•030  =  7 |d.        „  „         or  exactly  7ld. 

We  are  now  ready  to  do  any  number  of  sums  like  the 
following : 

Find  the  cost  of  324  horses  at  £17.  9s.  6d.  a  horse. 
Now  9/6  =  8/-+ 1/6  =  £-4 +  £-075  =  £-475, 


VII.]  PROBLEMS.  69 

so  the  answer  is  merely 

324x<£l7-475  =  £5561-9  =  £5561.  18s. 

Find  the  cost  of  900  things  at  £9.  7s.  4Jd.    (Sonnenschein.) 
Answer  is £9-36875x900  =  £8431-875  by  simple  multiplication 
=  £8431.  17s.  6d. 

How  much  a  year  is  £31.  9s.  9d.  per  day? 

Answer  365  x  £31-4875  =  £11492-9375  =  £11492.  18s.  9d. 

How  much  interest  must  be  paid  for  43  days'  loan  of  a  sum 
of  £543.  17s.  6d.  at  the  rate  of  3 J  per  cent,  per  annum"? 

(Sonnenschein.) 

Here  £3|  must  be  paid  for  each  hundred  pounds  lent  for  a 
year,  so  for  43  days  only  ^^g^^ths  of  that  sum  has  to  be  paid. 

Now  17/6  =  8  florins  + 1/6  (or,  otherwise,  seven-eighths  of  a 
pound)  =  £-875  ;  so  the  amount  to  be  paid  is : 

that  is  to  say,  x  £5-43875. 

This  yields  £2-243  =  £2.  4s.  lOJd.,  the  answer. 


Typical  exercises. 

There  are  certain  time-honoured  exercises  of  a  type  such  as 
the  following,  in  which  a  fair  amount  of  practice  is  desirable. 
[Type  only  here  given.] 

If  3  peaches  cost  a  shilling,  what  will  20  cost  *? 

If  I  have  to  pay  15  workmen  at  lOd.  an  hour  for  8  hours, 
how  much  money  do  I  need  1 

If  the  butcher  supplies  7 J  lbs.  of  meat  for  5s.,  what  has  he 
charged  per  pound  1 


70  EASY  MATHEMATICS.  [cHAf. 

and  so  on.  The  last  being  a  troublesome  kind  of  sum  fre- 
quently occurring  to  housekeepers,  but  usually  and  most  easily 
done  by  tables.* 

Examples  like  these  are  quite  harmless  and  give  needful 
practice,  but  when  they  become  complicated  a  little  of  them  is 
sufficient,  except  for  discipline,  and  the  more  concrete  and 
amusing  they  can  be  kept  for  ordinary  purposes  the  better. 

A  slight  further  development,  not  quite  so  harmless,  is  of 
the  following  type : 

Find  the  cost  of  6  lbs.  11  oz.  9  dwt.  at  1 7s.  8 Jd.  per  ounce. 

In  British  schools  there  is  far  too  great  a  tendency  to  limit 
all  exercises  to  pseudo-commercial  matters.  In  real  business 
this  kind  of  sum  hardly  occurs ;  and  besides,  greater  interest 
can  be  obtained  by  opening  up  fresh  ground  for  the  sub- 
ject matter  of  examples. 

A  few  specimens  may  be  here  suggested.  A  great  deal  of 
what  has  to  be  laboriously  taught  later  as  physics  is  nothing 
but  simple  arithmetic,  and  could  easily  be  assimilated  uncon- 
sciously while  doing  sums. 

1.  If  the  sound  of  thunder  takes  10  seconds  to  reach  our 
ears,  how  far  has  it  come  %  (See  p.  56  for  velocity  of  sound  : 
it  travels  approximately  a  mile  in  five  seconds.  For  more 
accurate  specification  the  temperature  would  have  to  be  known.) 


*  Answers  to  these  sums  are  as  follows  :  Each  peach  costs  the  third 
of  Is.,  so  twenty  peaches  will  cost  the  third  of  £1,  or  6s.  8d. 

The  15  workmen's  wages  will  amount  to  £5,  since  80d.  is  the  third 
of  a  pound. 

The  price  is  8d.  a  lb.,  since  7id.  doubled  three  times  makes  60d. 
or  5s. 

But  it  will  be  observed  that  in  each  case  some  accidentally  convenient 
relation  is  seized  and  utilised.  That  is  the  essence  of  mental  arithmetic: 
it  is  a  training  in  quickness  and  ingenuity,  not  in  mathematics  ;  and  its 
merits  can  be  appraised  accordingly. 


VII.]  PROBLEMS.  71 

2.  If  a  pistol  shot  is  heard  across  an  estuary  15  seconds 
after  the  pistol  was  fired  (which  can  be  told  by  observing  the 
flash),  how  wide  is  the  estuary  ? 

3.  If  light  reaches  the  earth  from  the  sun  in  8  minutes, 
what  is  its  velocity?  (The  distance  of  the  sun  being  93 
million  miles.) 

4.  How  long  does  it  take  to  come  from  the  moon  1 

5.  How  long  would  it  take  to  travel  a  distance  equal  to 
seven  times  the  circumference  of  the  earth  1 

6.  If  it  takes  5  years  to  arrive  from  a  star,  how  far  off  is 
that  star  1 

7.  If  a  locomotive  could  be  run  60  miles  an  hour  day  and 
night,  how  long  would  it  take  to  go  round  the  earth  ? 

8.  How  long  to  reach  the  sun?  etc.* 

Answers  should  be  given  in  weeks  or  years  or  whatever 
unit  is  appropriate  and  most  suggestive.  This  is  a  good  rule 
always,  and  is  the  real  use  of  units  to  which  people  are  accus- 
tomed. Conversion  of  miles  into  inches  is  tedious  and  use- 
less :  but  stating  a  big  result  in  miles,  a  small  result  in 
inches,  and  a  moderate  result  in  feet  or  yards,  is  right  and 
illuminating. 


■'^Answers  : 

1.  2  miles. 

2.  About  3  miles. 

3.  93  million  miles  -f  8  minutes 

93000 


— ^—  thousand  miles  per  second 
8  X  bO 

-j^  =  nearly  194  thousand  miles  per  second. 


4.  About  a  second  and  a  half. 

5.  About  1  second. 

6.  5  X  365^  X  86400  x  194,000  miles. 

7.  Nearly  17  days. 

8.  About  180  years. 


72  EASY  MATHEMATICS.  [chap. 

9.  If  a  pistol  shot  fired  in  a  valley,  at  a  spot  which  is 
distant  from  the  summit  of  a  mountain  by  an  amount  which  is 
represented  by  a  length  of  4  inches  on  an  ordnance  map  of 
scale  1  inch  to  the  mile,  is  heard  on  that  mountain  top  25 
seconds  after  the  flash,  how  high  is  the  mountain  above  the 
valley  1  (Ans. :  3  miles.) 

This  is  perhaps  hard :  it  can  be  done  by  drawing  and 
measuring,  after  it  has  been  perceived  that  the  sound  has 
travelled  5  miles  in  a  straight  line. 

10.  If  a  motor  car  is  travelling  21  miles  an  hour,  how  long 
will  it  take  to  go  100  yards  1 

Ans. :  9*74  or  9 J  seconds. 

11.  If  the  estimate  of  time  were  f  second  out,  what  error 
would  be  made  in  reckoning  the  speed  from  the  measured 
distance  1 

Ans. :  I  sec.  is  j\th  of  -^^  sec,  so  the  error  in  estimate  of 
speed  would  be  about 

f  1^  miles  an  hour,  or  about  7^  per  cent. 

12.  If  a  volunteer  corps  of  84  members  shoots  160  rounds  a 
day  each  for  5  weeks,  and  if  each  bullet  weighs  |  of  an  ounce, 
what  weight  of  lead  will  they  have  expended  1 

13.  If  each  bullet  needed  one  halfpennyworth  of  powder  to 
propel  it,  and  if  lead  cost  17/-  a  cwt.,  what  would  be  the  cost, 
in  powder  and  shot,  for  a  regiment  of  12  such  corps,  in  the 
course  of  5  weeks  ? 

14.  If  an  iron  rod  expands  J  per  cent,  of  its  length  when 
warmed  200  degrees,  what  allowance  must  be  made  for  the 
expansion  of  a  bridge  girder,  ^  mile  long,  between  a  winter 
temperature  of  -  40°  and  a  summer  temperature  of  110°? 

15.  With  the  above  data  how  much  will  an  iron  rod  a  foot 
long  expand  if  warmed  one  degree  'i 

16.  If  a  snail  crawl  half  an  inch  each  minute,  how  far  will 
it  go  in  3  hours  'i 


vir.]  PROBLEMS.  73 

17.  If  sound  goes  a  mile  in  5  seconds,  how  long  would  it 
take  to  go  a  foot  ^ 

18.  If  sound  reverberated  between  two  walls  10  feet  apart, 
how  many  excursions  to  and  fro  will  it  make  per  second  1 

19.  If  light  takes  8  minutes  to  travel  93  million  miles,  how 
long  would  it  take  to  go  one  yard"?  How  many  kilometres 
would  it  travel  per  second?  How  many  centimetres  per 
second  1  * 


*  Answers  to  the  above  : 

12.  84  X  160  X  35  X  f  ounces  =  about  10  tons. 

13.  In  shot,  about  £170  ;  since  a  shilling  per  cwt.  is  a  pound  per  ton. 
In  powder,  84  x  160  x  35  x  ^  pence  =  7  x  80  x  35  shillings  =  28  x  35 
pounds  per  corps. 

14.  The  range  of  temperature  is  150°;  for  this  range  iron  expands  | 
of  J  per  cent,  of  its  length  ;  that  is, 

»"ile  =  t^-^  feet  =  ^^  =  1  '2375  feet,  or  nearly  15  inches. 
Id  X  oUU  ovv 

—-^th  of  i  per  cent,  of  its  length  ;  which  is  so.ooo^^  ®^  ^  ^°°*' 
or  "0000125  expansion  per  unit  length  per  degree  ;  which  is  about  the 
riglit  value  for  iron. 

17.  1  second -7- 1056,  or  about  the  thousandth  of  a  second. 

18.  In  each  excursion  to  and  fro  it  will  have  to  travel  20  feet ;  but  it 
can  travel  1056  feet  in  a  second,  therefore  it  has  time  to  make  52 "8 
excursions  per  second.  If  the  walls  were  only  2  feet  apart  instead  of 
ten,  the  rate  of  reverberation  would  be  5  times  as  rapid,  and  would 


correspond  to  the  note    /k      i —     This  therefore  is  the  musical  note 


heard  if  a  short  sharp  noise,  like  a  blow  or  clap,  be  made  between  two 
walls  two  feet  apart. 

19.  To  travel  1  mile,  light  would  take  8  minutes -r  93  million  ;  there- 
fore to  travel  1  yard  it  would  take  l/1760th  part  of  this. 

^"'^  '•         ^^  =  sTxTTo  =  Ml  "'""°"*'^^ "' "  ^^'=°"'*- 

Light  travels  300,000  kilometres  per  second  or  3  x  10^"  centimetres 
per  second  ;  as  nearly  as  experiment  at  present  enables  us  to  eay. 


74  EASY  MATHEMATICS.  [chaf. 

20.  If  you  buy  a  large  number  of  oranges  at  three  a  penny, 
and  an  equal  number  at  two  a  penny,  and  then  sell  them  all  at 
five  for  twopence,  how  much  have  j^ou  lost  on  the  transaction  1 

(Ans.  :  a  penny  for  every  5  dozen  sold. 

The  buying  price  per  couple  is  Jd.  +  ^d. ; 

2 
the  selling  price  per  couple  is  -yd. 

So  the  loss  per  couple  is^  +  l^-^  =  |-|  =  wu^-) 
There  are  many  ways  of  doing  this  problem,  and  it  should 
not  be  left  till  it  is  fully  realised.     Other  problems  depend 
on   the   same   principle,   which   is  an   important   one.      For 
instance : 

21.  An  oarsman  rows  a  boat  a  certain  distance  up  a  river  and 
back,  and  then  across  the  river,  or  on  a  lake,  the  same  distance 
and  back.     Which  will  be  the  quickest  to  and  fro  journey  1 

22.  If  a  steamer  travels  down  a  river  at  a  rate  of  19  miles 
per  hour,  and  up  the  same  river  with  the  same  engine-exertion 
at  7  miles  an  hour,  what  is  the  speed  of  the  river  1  How  long 
would  the  steamer  take  to  go  a  journey  of  65  miles  and  back  1 

(Ans. :  The  speed  of  the  boat  in  stagnant  water  is  the 
half-sum,  viz.  13,  the  speed  of  the  river  is  the  the  half- 
difference,  viz.  6  miles  per  hour.  The  journey  of  130  miles 
would  take  ten  hours  in  stagnant  water,  but  up  and  down 
the  river  it  will  take  nearly  thirteen  hours.) 

The  general  principle  is  that  whereas  {l+x)  +  (\  -x)  =  2, 

+  - does  not  equal  2 

1  +a:     1  -« 

9 

but  does  equal  — ^,  which  is  greater  than  2 ;  though  not 

much  greater  when  x  is  small.    This  applies  to  (20)  (21)  and  (22). 

23.  If  a  couple  of  travellers  sharing  expenses  are  found  to 
be  out  of  pocket  in  the  course  of  the  day.  A,  £2,  4s.  6d.,  and 


VII.]  PROBLEMS.  75 

B,  £1.  3s.  4d.,  what  sum  must  be  transferred  from  one  to  the 
other  to  equalise  matters  % 

(Ans. :  Half  the  difference,  viz.  10s.  7d. ;  and  the  cost  to 
each  has  been  half  the  sum,  viz.  .£1.  13s.  lid.) 

24.  If  three  travellers  on  a  tour  have  expended  when 
they  return 

^  £17  .  4  .  6 

B    £4.3.2 

C    £7  .  5  .  4 
how  can  they  best  arrange  to  share  expenses  equally  1 

{Ans.  Find  the  mean  expenditure  by  adding  the  items 
together  and  dividing  by  3 ;  and  then  take  the  difference 
between  this  mean  and  the  expenditure  of  each.  B  and  C 
will  then  have  to  pay  their  respective  differences  to  A.  Their 
two  deficiencies  from  the  mean,  added  together,  should  equal 
^'s  excess  expenditure  over  the  mean ;  if  this  is  not  the  case 
a  mistake  has  been  made.) 

The  same  rule  would  apply  to  any  number  of  travellers. 
Observe  how  it  works  for  the  couple  of  last  question. 

These  exercises  do  not  contain  examples  of  so  many  quarts, 
pecks,  pennyweights,  and  drams.  Such  sums  have  no  business 
to  occur.  If  artificial  complexities  of  that  sort  are  set,  any 
way  of  dealing  with  them  will  do :  the  simplest  way  is  the 
best  way. 

If  a  pupil  is  constrained  to  bethink  himself  of  how  the 
teacher  intends  him  to  do  a  sum,  it  destroys  originality.  His 
effort  should  always  be  devoted  to  find  the  best  and  simplest 
way.  This  a  teacher  can  help  him  to  find,  but  a  self-found 
way  is  more  wholesome  in  many  respects  than  a  coerced  way, 
even  though  the  latter  is  neater.  Originality  should  always 
be  respected  :  it  is  rather  rare.  Perhaps  docility  is  made  too 
much  of,  and  budding  shoots  of  originality  are  frozen. 


76  EASY  MATHEMATICS.  [chap. 

Binary  scale. 

Although  the  natural  method  of  dealing  with  multiples  of 
a  unit  is  to  employ  the  same  system  of  notation  as  is  in  vogue 
in  arithmetic,  and  although  therefore  it  is  natural  to  specify 
large  numbers  of  things  by  powers  of  ten,  there  is  a  natural 
tendency  also  to  deal  with  fractions  on  a  different  basis,  viz. 
to  proceed  by  powers  of  J.  We  see  this  on  a  foot  rule,  where 
the  inches  are  first  halved,  then  quartered,  then  divided  into 
eighths,  then  into  sixteenths,  and  sometimes  even  into  thirty- 
second  parts  of  an  inch. 

The  same  method  of  dealing  with  fractions  is  found  in 
prices,  as  for  instance  of  cotton,  or  any  commodity  which 
requires  a  penny  to  be  subdivided.  Below  the  halfpenny  and 
the  farthing  we  find  the  eighth,  sixteenth,  thirty-second,  and 
sixty-fourth  of  a  penny  in  use  for  quotations;  and  these 
ungainly  figures  are,  or  used  to  be,  even  telegraphed  and 
automatically  printed  on  tape.  So  also  a  carpenter  will 
understand  a  specification  in  sixteenths  of  an  inch,  while  a 
decimal  subdivision  would  puzzle  him. 

A  thousandth  of  an  inch  is  sometimes  used  however  in 
fine  metal  fitting  work,  and  the  thickness  of  a  rod  wanted 
may  be  specified  to  a  fitter  as  the  thousandth  of  an  inch 
greater  than  2j^^  inch. 

These  peculiarities  are  insular  and  not  to  be  encouraged, 
having  originated  in  laziness  and  ignorance ;  but  they  are 
not  nearly  so  bad  as  the  weights  and  measures  which  people 
who  ought  to  know  better  still  require  that  children  shall 
be  taught. 

It  is  quite  possible  to  word  arithmetic  itself  on  the  binary 
scale,  counting  in  pairs  only;  thus  10  (read  one  nought)  may 
be  understood  to  mean  1  pair;  100  may  mean  1  pair  of  pairs 
(or  4),  1000  on  the  same  plan  will  mean  2x2x2  or  eight, 
and  so  on.     And  on  this  scale  *1  would  mean  J,  -01  a  quarter, 


VII.]  COINAGE.  77 

•001  an  eighth;  so  that  one  and  a  quarter  plus  an  eighth 
would  be  written  1*011. 

The  natural  tendency  to  this  kind  of  subdivision  is  apparent 
in  coins,  even  in  countries  with  a  decimal  currency.  For 
instance  in  America  you  find  the  half  and  the  quarter  dollar, 
beside  the  dime  and  the  cent.  In  France  you  find  the  double 
franc,  franc,  half-franc,  and  quarter-franc.  So  in  Germany  we 
have  as  a  drink-measure  the  halh-Uter  and  viertel-liter.  And 
in  England  we  have  half-sovereigns,  half-crowns,  also  three- 
penny bits,  sixpences,  shillings,  florins,  and  double  florins, 
each  double  the  preceding;  the  double  florin  being  roughly 
equivalent  to  a  dollar  or  to  a  five-franc  piece. 

So  also  the  commonest  gold  piece  in  France  is  the  Napoleon 
or  20-franc  piece;  not  the  ten-franc,  or  the  hundred-franc 
piece,  though  they  both  exist. 

This  natural  tendency  is  the  chief  difficulty  in  introducing 
a  purely  decimal  coinage ;  another  is  the  convenience  of  the 
penny  and  the  shilling.  If  a  decimal  system  is  to  be  intro- 
duced, one  or  other  of  these  coins  must  give  way.  If  the 
Bhilling  gives  way,  we  can  have  an  approximation  to  the  franc, 
and  much  inconvenience  or  grumbling  in  connexion  with  cab 
fares,  etc.  If  the  penny  gives  way,  and  is  made  the  tenth 
of  a  shilling,  we  approach  closely  to  the  German  system ; 
and  many  commodities  used  by  poor  people  will  automatically 
rise  in  price. 

In  Austria  an  attempt  is  being  made  to  replace  the  gulden 
and  kreutzer  by  their  respective  halves,  called  krone  and  heller, 
which  correspond  approximately  with  the  franc  and  centime ; 
but  the  older  denominations  persist,  and  it  is  quite  likely  that 
the  two  will  co-exist  and  be  convenient. 

It  may  be  asked  "why  mention  these  things  in  a  book  of 
this  kind  "  1  And  the  answer  is  because  children  can  take  an 
intelligent  interest  in  them,  and  because  it  is  instructive  for 


78  EASY  MATHEMATICS.  [chap. 

them  to  realise  that  our  present  coinage  is  not  a  heaven-sent 
institution,  but  is  susceptible  of  change, — change  too  in  which, 
when  adult,  some  of  them  can  take  their  part,  either  in  pro- 
moting or  opposing.  There  is  therefore  a  reality  about  these 
things,  and  arithmetical  ideas  can  inculcate  themselves  in 
connexion  with  them  without  labour. 

Decimal  system  of  weights  and  measures. 

Although  the  present  division  of  money  is  so  deep-rooted 
that  decimal  coinage  is  difficult  of  introduction,  and  although 
the  decimal  system  in  arithmetic  is  not  the  best  that  could 
have  been  devised ;  yet  its  advantages  over  most  other  systems 
are  so  enormous  that  in  connexion  with  weights  and  measures 
it  undoubtedly  ought  speedily  to  be  introduced. 

The  first  and  easiest  place  to  introduce  it  is  in  connexion 
with  weights.  No  one  really  wants  to  reckon  in  ounces  and 
pennyweights  and  grains  and  scruples  and  drachms.  Ounces 
used  to  be  perpetuated  and  popularised  by  the  Post  Office 
regulations;  but  now  that  a  quarter  of  a  pound  will  go 
for  a  penny,  and,  under  certain  restrictions,  an  eighth  of  a 
pound  for  a  halfpenny,  the  necessity  for  ounces  has  really 
disappeared.  It  would  be  quite  easy  to  make  the  halfpenny 
postal  regulations  refer  to  a  tenth  of  a  pound  instead  of 
an  eighth,  and  to  construct  ten-pound  weights,  hundred- 
pound  weights,  and  their  convenient  doubles  and  halves  and 
quarters. 

There  is  however  this  fundamental  question  to  be  considered : 
shall  the  British  pound  be  adhered  to,  or  shall  we  adopt  the 
unit  of  our  neighbours  and  employ  the  kilo  (short  for  kilo- 
gramme) or  the  demi-kilo  1 

The  kilo  is  too  big  for  many  ordinary  purposes.  In  France 
small  marketing  is  still  done  by  the  demi-kilo,  because  it 
represents  a  reasonable  and  commonly-needed  amount  of  stufE 


VII,] 


MEASURES.  79 


It  is  altogether  handier  than  the  kilo.  A  demi-kilo  might  be 
introduced,  and  with  us  might  still  be  called  a  pound,  or,  for 
a  time,  an  "  imperial  pound,"  though  its  value  would  have  to 
be  increased  by  ten  per  cent,  above  our  present  pound.  The 
kilo  is  approximate!}^  2-2  lbs.,  so  the  new  pound  or  demi-kilo 
would  be  one  and  a  tenth  old  pounds.  The  gramme  would  be 
•002  new  pound. 

The  disadvantages  of  any  change  are  obvious.  The  advan- 
tage would  be  that  we  should  then  be  using  practically  the 
same  unit  as  our  neighbours. 

All  other  denominations  could  be  swept  away ;  except,  for 
occasional  rough  use,  the  ounce  and  the  ton,  which  continue 
useful ;  for  the  ton  would  be  2000  of  the  new  pounds,  and 
would  correspond  exactly  with  the  French  tonne ;  and  the 
ounce,  slightly  changed,  would  be  yY  of  the  new  pound,  or  it 
might  be  changed  so  as  to  be  one-tenth  of  it.  The  grain  or 
TWO"  P^^  ^^  ^^®  ^^^  pound  might  easily  give  place  to  a  new 
grain  yxjj^jtr  P^^*  ^^  *^®  ^^^  <^^®- 

These  handy  names  are  useful  for  common  purposes  and  for 
speech.  All  accurate  specifications  should  be  made  in  terms 
of  the  pound,  and  of  that  alone.  Thus  1*4903  lbs.  would  be 
a  specification  accurate  to  the  nearest  grain  of  a  weighing  of 
something  like  a  pound  and  a  half. 

3 '01 4  tons  would  be  a  statement,  intended  to  be  accurate  to 
the  nearest  pound,  of  the  weighing  of  a  3-ton  mass. 

Let  me  emphasise  what  may  be  regarded  as  one  of  the 
special  advantages  of  this  simple  and  easily  introduced  change. 
Children  could  then  be  practised  in  weighing  at  once  :  to  the 
vast  advantage  of  their  education.  At  present  an  apothecary's 
scales  are  an  abomination,  and  no  child  can  weigh  satisfactorily 
with  the  weights  of  a  letter  balance,  which  are  all  in  the  binary 
scale ;  though,  as  aforesaid,  these  serve  as  an  introduction  to 
ideas  of  weighing,  etc,  in  quite  early  stages.    Letter  weights  go 


80  EASY  MATHEMATICS.  [chap. 

down  too  rapidly ;  there  are  not  enough  subdivisions ;  and  the 
result  cannot  easily  and  quickly  be  specified,  except  as  an 
awkward  series  of  vulgar  fractions,  or  else  in  the  binary  scale 
of  arithmetical  notation. 

The  only  way  in  which  school  weighings  can  be  satisfactorily 
done  now  is  by  the  use  of  grammes  and  kilogrammes :  and 
there  is  a  foreign  feel  about  these  things;  which  those  who 
learn  chemistry  indeed  get  over,  but  which  gives  it  a  flavour 
distinct  from  ordinary  life. 

What  we  want  is  that  children  shall  weigh  and  measure  all 
sorts  of  things,  and  do  a  large  part  of  their  arithmetic  in 
terms  of  their  own  weighings  and  measurings  :  thus  making 
it  real  and  concrete  and  if  possible  interesting. 

Weighings  of  plants  and  of  growing  seeds,  of  rusting  iron 
and  of  burning  candles,  of  dissolving  salts  and  of  evaporating 
liquids,  can  all  be  made  interesting  and  instructive. 

Weighings  in  air  and  water,  and  finding  thereby  the  specific 
gravity  or  the  volume  of  irregular  solids,  can  easily  be  over- 
done and  made  tedious,  but,  short  of  this,  such  operations  are 
quite  instructive. 

G-auging  and  measuring  of  regular  solids  is  an  equally  in- 
structive way  of  arriving  at  their  specific  gravity,  or,  as  it  may 
be  more  scientifically  called,  "  density."  The  approximate 
relative  densities  of  such  things  as  stone,  lead,  iron,  gold, 
copper,  platinum,  cork,  air,  referred  to  water,  are  worth 
remembering  :  stone  say  2*5,  lead  11,  iron  7,  gold  19,  copper  8, 
platinum  21,  cork  ^,  air  -g^. 

Decimal  measures. — Continued. 

The  introduction  into  commerce  of  "the  decimal  system" 
is  a  more  difficult  matter  however.  The  admirable  duo- 
decimal division  of  the  foot  into  inches  (like  that  of  the 
shilling  into   pence)  stands  in  the  way.     The  foot  and   the 


VII.]  MEASURES.  81 

inch  and  the  yard  seem  ingrained  in  the  British  character, 
and  will  give  place  to  the  metre  and  the  centimetre  only 
with  difficulty. 

The  fact  is  that  the  introducers  of  the  "  metre"  made  a  great 
mistake  by  not  adopting  the  yard  or  the  foot  or  some  other 
*existing  unit  as  its  value  :  they  would  also  have  been  wise 
if  they  had  adopted  the  pound  as  their  kilogramme,  and  left 
the  dimensions  of  the  earth  alone.  It  is  the  magnitude  of 
the  human  body  which  really  and  scientifically  specifies  and 
confers  any  meaning  on  absolute  size  :  our  bodily  dimensions 
and  time  relations  must  be  the  basis  of  all  our  measures  and 
ideas  of  absolute  magnitude.  To  abandon  the  human  body 
and  to  attend  to  the  dimensions  of  the  earth  was  essentially 
unscientific  or  un  philosophical :  it  has  all  the  marks  of  faddism 
and  self-opinionatedness.  However  these  unwisdoms  of  sections 
of  the  human  race  we  have  to  put  up  with,  and  at  any  rate 
the  French  evolved  a  better  system  on  the  whole  than  that 
which  had  come  down  to  us  by  inheritance  and  tradition  from 
uncivilised  times. 

If  we  were  at  liberty  to  adopt  the  foot  as  our  standard,  and 
to  call  its  decimal  subdivisions  inches,  or  if  a  new  foot  were 
made  ten  inches  long,  the  change  would  not  be  so  very 
difficult.  If  it  had  been  extensively  customary  to  divide  the 
inch  too  into  twelfths  (called  lines)  the  change  would  be 
harder  ;  but  divisions  of  the  inch  in  the  binary  scale  have  been 
customary,  and  these  are  not  really  convenient :  a  decimal 
system  is  better  than  that ;  and  foot  rules  decimally  divided 
and  subdivided  could  easily  be  supplied  and  used. 

But  then,  as  in  the  case  of  our  present  pound,  we  should  be 
using  an  insular  measure  different  from  all  the  rest  of  Europe, 
and  amid  the  stress  of  industrial  and  engineering  competition 
this  is  a  serious  handicap. 

A  metre  scale  is  a  rather  unwieldy  thing :    a  half-metre 


82  EASY  MATHEMATICS.  [chap. 

scale  is  handier  for  many  purposes,  and  might  be  made  like 
a  folding  two-foot  rule. 

There  is  no  help  for  it :  we  must  get  used  to  metres  and 
centimetres,  and  the  sooner  we  begin  the  better. 

Angles  and  Time.  • 

There  are  two  things  which  have  not  yet  been  subdivided 
decimally  with  any  considerable  consensus  of  agreement :  they 
are  Angles  and  Time. 

The  division  of  the  right  angle  into  90  equal  parts  is 
convenient.  The  subdivision  of  the  degree  into  sixtieths 
and  again  into  sixtieths  (called  respectively  partes  minutas  and 
paries  minutae  secundae,  now  abbreviated  into  "  minutes  "  and 
"seconds")  is  peculiar  and  sometimes  troublesome  but  not 
exactly  inconvenient,  though  a  decimal  subdivision  of  the 
degree  would  be  simpler. 

As  to  time,  the  fundamental  unit  is  the  day  or  period  of 
the  earth's  rotation  (this  being  the  most  uniformly  moving 
thing  we  know).  Its  subdivisions  (into  24  parts,  and  then 
into  sixtieths,  etc.)  are  curious,  but  too  deep-rooted  for 
anyone  to  attempt  to  alter;  and  fortunately  they  are  the 
same  in  all  countries.*  Legitimate  practice  in  dealing  with 
different  denominations  can  therefore  be  afforded  to  children 
by  our  large  admixture  of  universally  understood  measures  of 
time;  including  weeks,  months  of  different  kinds,  years  of 
different  kinds,  and  centuries.  All  other  weight  and  measure 
complications,  especially  those  of  a  merely  insular  and  boorish 
character.,  should  forthwith  cease  to  be  instilled  into  children. 

Further  exercises. 

It  is  worth  noticing  and  remembering  that  a  kilometre  =  10^ 
centimetres. 


■^A  third  subdivision,  the  sixtieth  part  of  a  secoud,  is  sometimes 
known  as  a  "  trice." 


yii.]  MEASURES.  83 

It  is  also  ten  minutes'  walk,  or  very  roughly  two-thirds  of 
a  mile. 

A  cubic  metre  is  a  million  cubic  centimetres. 

A  cubic  kilometre  is  a  trillion  cubic  millimetres ;  meaning 
by  "trillion"  a  million  million  million,  after  the  English 
custom.  (But  the  French  use  the  term  "  billion  "  to  signify  a 
thousand  million ;  and  a  million  million  they  accordingly  call 
a  trillion ;  while  the  above  number  would  by  them  be  desig- 
nated a  quintillion :  in  any  case  it  is  1  followed  by  eighteen 
ciphers). 

A  cubic  centimetre  is  1000  cubic  millimetres,  and  is  xoVo  ^^ 
a  litre. 

A  gallon  of  cold  water  weighs  10  lbs.,  by  definition  of  a 
gallon  ;  a  pint  therefore  weighs  a  pound  and  a  quarter. 

A  cubic  metre  of  water  is  a  tonne,  and  very  approximately, 
though  accidentally,  equals  an  English  ton  also. 

A  cubic  centimetre  of  water,  at  its  temperature  of  maximum 
density,  weighs  a  gramme  exactly,  from  the  definition  of  a 
gramme. 

The  speed  of  an  express  train,  60  miles  an  hour,  is  only  15 
times  a  walking  pace. 

The  speed  of  a  bullet,  say  1800  feet  a  second,  is  twenty 
times  that  of  a  train. 

The  speed  of  sound  is  comparable  with  that  of  bullets. 

The  speed  of  light  is  a  million  times  the  speed  of  sound  in 
air. 

Four  miles  an  hour  is  2  yards  a  second,  approximately, 
or  accurately  60  miles  an  hour  is  88  feet  a  second. 

It  is  an  instructive  exercise  to  let  a  boy  find  out  the 
sizes  and  distances  of  the  planets  of  the  solar  system,  and  cal- 
culate a  numerical  model  illustrating  them  on  any  convenient 
scale. 


84  EASY  MATHEMATICS.  [chap. 

I  have  myself  found  a  local  topographical  scale  the  most 
convenient :  one  on  which  the  earth  was  about  the  size  of  a 
football,  and  the  sun  the  size  of  some  public  building  a  mile 
or  two  distant.  The  other  planets  distribute  themselves 
about  the  town  and  county;  some  of  them  extending  into 
more  distant  counties. 

It  is  instructive  to  try  to  place  the  nearest  fixed  star  in 
such  a  scale,  and  to  find  that  it  will  not  come  on  to  the  earth 
at  all. 

The  price  of  a  railway  ticket  to  the  nearest  fixed  star,  at 
Id.  per  hundred  miles,  can  also  be  calculated;  and  found  to 
approach  or  exceed  the  National  debt. 

The  earth  takes  a  year  to  go  round  the  sun  in  a  circle  of 
93  million  miles  radius  :  how  fast  does  it  go  ? 

(Ans. :  About  19  miles  a  second.) 
Light  goes  10000  times  as  fast  as  this. 
How  fast  would  a  train  have  to  run  on  the  equator  if  it 
were  to  keep  up  with  the  apparent  motion  of  the  sun,  so  that 
it  should  continue  the  same  time  of  day  1 

(Ans. :  About  1000  miles  an  hour.) 
How  far  from  the  North  Pole  could  the  same  thing  be 
accomplished  by  a  man  walking  4  miles  an  hour  ? 

(Ans. :  About  30  miles  away.) 
If  a  man  walked  30  miles  South  from  the  North  Pole,  and 
then  walked  40  miles  due  West,  how  far,  and  in  what  direc- 
tion, would  he  have  to  go  to  get  back  to  the  Pole  1 

(Ans. :  30  miles  due  North.) 
What  is  the  density  of  a  rectangular  block  whose  height  is 
5  inches,   length   11   inches,   breadth    8   inches,    and   weight 
82  J  lbs.  ^ 

(Ans.  :  3  ounces  per  cubic  inch.) 


VII.]  EXERCISES.  85 

Directly  the  elements  of  mechanics  and  of  heat  and  of 
chemistry  have  been  begun,  any  number  of  useful  and  fairly 
interesting  examples  can  be  constructed.  They  afford 
practice  in  arithmetic  of  the  best  and  most  useful  kind; 
quick  and  ingenious  computation  being  what  is  wanted, 
not  laborious  dwelling  upon  long  artificial  sums.  Long  sums 
are  never  done  in  adult  practice :  there  are  always  grown-up 
methods  of  avoiding  them. 

It  is  cruel  to  subject  children  to  any  such  disciplinary 
process,  as  part  of  what  might  be  their  happy  and  stimulating 
education.  Before  they  have  been  subjected  to  it  they  are 
often  eager  to  have  lessons;  but  experience  of  the  average 
lesson,  as  often  administered,  soon  kills  off  any  enthusiasm, 
and  instils  the  fatal  habits  of  listlessness  and  inattention 
which  check  the  sap  of  intellectual  growth  for  a  long  time. 

If  the  teacher  of  arithmetic  knows  arithmetic  and  nothing 
else,  he  is  not  fit  to  teach  it.  His  mind  should  be  alive  with 
concrete  and  living  examples ;  and  it  is  well  to  utilise  actual 
measurings,  weighings,  surveyings,  laboratory-experiments, 
and  the  like,  to  furnish  other  opportunities  for  arithmetical 
exercises. 

Arithmetical  exercise  can  be  obtained  unconsciously,  as 
bodily  exercise  is  obtained  by  playing  an  outdoor  game. 
The  mechanical  drill  or  constitutional-walk  form  of  exercise 
has  its  place  doubtless,  but  its  place  among  children  is  limited. 
There  used  to  be  too  much  of  it,  and  too  little  spontaneity 
of  bodily  exercise,  in  girls'  schools.  Now  the  spontaneity  and 
freshness  is  permitted  to  the  body,  but  too  often  denied  to 
the  mind. 

The  same  kind  of  reform  is  called  for  in  both  cases.  The 
object  of  this  book  is  to  assist  in  hastening  this  vital  reform. 


CHAPTEE  VIII. 

Simple  proportion. 

Any  number  of  sums  are  of  the  following  character  : 

If  3  sheep  cost  £20,  what  will  100  cost  ? 

Now  the  so-called  "rule  of  three"  method  of  dealing  with 
sums  of  this  kind,  though  permissible,  is  not  really  a  good 
method,  because  it  leads  to  nothing  beyond,  and  employs  an 
antiquated  system  of  notation. 

The  answer  is  one  hundred  thirds  of  twenty  pounds 
=  ioo  x£20  =  ^%^-^  =  X666-6  =  £666f  =  .£666.  13s.  4d. 

If  the  answer  is  not  obvious,  it  can  be  arrived  at  by  the 
intermediate  step  of  considering  one  sheep,  which  will  cost 
the  third  of  .£20,  namely,  £6.  13s.  4d.* 

And  so  a  hundred  sheep  will  cost  600  pounds,  1300 
shillings,  and  400  pence. 

The  1300  shillings  reduce  to  65  pounds,  since  100  shillings 
is  five  pounds;  and  the  400  pence  make  £1.  13s.  4d.,  since 
240  pence  is  a  pound,  and  so  400  pence  is  thirty  shillings 
and  40  pence  (or  3s.  4d.)  over. 

This  is  not  an  orthodox  way  of  doing  the  sum,  but  it  is 
just  as  good  as  any  other,  and  it  is  one  that  a  boy  might 

*  [It  would  not  come  out  even  so  well  as  this  but  for  the  fortunate 
duodecimal  division  of  the  shilling  into  pence,  so  that  one-third  of  a 
pound,  viz.  6s.  8d.,  and  two-thirds,  viz.  13s.  4d.,  can  be  exactly 
specified  without  fractions.  These  amounts  are  worth  remembering  as 
one-third  and  two-thirds  of  a  sovereign.] 


CHAP,  vin.]  SIMPLE  PROPORTION.  87 

scheme  for  himself.  There  would  be  no  need  to  snub  him 
for  it.  Everything  which  is  troublesome  about  such  a  sum 
results  from  the  miserable  property  of  the  number  ten,  that 
it  is  not  divisible  by  3. 

If  we  had  set  the  following  very  similar  question : 

If  3  sheep  cost  £24,  what  would  100  cost  ? 
An  infant  could  answer  £800,  doing  it  in  its  head.      But  it 
would  clearly  do  it  by  the  same  process,  viz.  the  process  of 
considering  the  price  per  single  sheep,  and  that  is  therefore 
the  natural  and  simplest  method. 

To  summarise :    The  childish   method   is   the  i^iethod   by 
units,  and  may  be  written  out  at  length ;  the  adult  method  is 
the  method  by  ratio ;   what  place   is  there  for  the  rule  of 
three  1     The  rule  of  three  with  its  symbols         :         :  :         : 
is  reserved  for  antiquated  school  instruction. 

Observe,  there  is  no  harm  in  writing  a  ratio  as  2  :  3  or  a  :  6, 
and  occasionally  it  may  be  convenient  to  do  so,  though  2  -r  3, 

9 

or  a  -r  6  is  precisely  the  same  thing,  and  usually  the  form  ^  or 
-,  or  a/b,  is  in  every  way  better.     So  the  symbol  : :  is  needless, 

because  replaced  by  =.  The  fact  is  that  :  connotes  the 
theoretical  idea  of  ratio,  while  -f  indicates  the  practical 
operation  of  division,  which  is  the  actual  means  of  working 
a  ratio  out.  The  vulgar-fraction  form  may  be  used  instead 
of  either  of  these  signs  and  is  usually  best.  The  division 
may  or  may  not  be  actually  performed,  as  we  please. 

I  feel  inclined  to  illustrate  good  and  bad  methods  at  this 
stage  a  little  further,  by  taking  a  few  more  very  simple 
examples.     For  instance  : 

If  twenty  dogs  pulling  equally  at  a  sledge  exert  a  hori- 
zontal force  of  1  cwt.,  what  force  do  any  three  of  them 
exert? 


88  EASY  MATHEMATICS.  [chap. 


Adult  method  : 

J.ths  of  1  cwt.  =  ^ 

!x  112  lbs. 

20 

20 

Good  cMldish  method  : 

20  dogs  pull 

112  lbs. 

10  dogs  pull 

56      „ 

1  dog  pulls 

5-6     „ 

3  dogs  pull 

3x5-6  =  1 

=  16- 


16-8  lbs. 

If  it  be  asked  why  not  stop  at  g^^ths  of  a  cwt.  and  give  the 
answer  as  '15  cwt.,  I  reply,  no  reason  against  it  at  all;  but 
children  should  be  accustomed  to  realise  forces  and  other 
things,  in  p^jtual  homely  units  that  they  can  feel  and  appre- 
ciate ;  and  a  cwt.  is  too  big  for  them. 

Mechanical  method : 

20  :  3  ::  112  :  the  answer. 
Rule.     Multiply  the  means  and  divide  by  one  extreme  and  you  get 
the  other  extreme. 

.'.  the  answer  is,  etc. 

British  Method: 

There  is  indeed  a  barbarous  way  of  complicating  the  sum,  which 
is  typical  of  much  that  goes  on  in  these  islands  at  inferior  schools  : 

lbs.      oz.  drachms 
20 1  336.    0.    0 

16  .  12  .  12| 

which  is  done  thus : 

Twenty  into  336  goes  16  and  16  over,  that  is  16  lbs.  over,  which 
equals  256  ounces.  Twenty  into  this  goes  12  times  and  16  over,  that 
is  16  ounces  or  256  drachms  ;  into  which  twenty  again  goes  12  times 
and  Jf  ths  over,  which  last  equals  f  ths,  that  is  fths  of  a  drachm. 

So  the  answer  is  16  lbs.  12  oz.  12f  drachms. 

On  this  one  has  to  remark  that  since  the  unfortunate  ^  has 
to  appear  (as  it  happens)  sooner  or  later,  why  should  it  not 
appear  at  firsts  Why  is  |ths  of  a  drachm  easier  to  understand 
than  Iths  of  a  pound?  The  fact  is  that  it  is  not  easier  to 
understand,  and  by  children  is  not  understood  :  the  "  4  over  " 
which  remains   at  the  end  is  a  continual  puzzle  to  them. 


VIII.]  SIMPLE  PROPORTION.  89 

They  have  been  so  accustomed  to  getting  rid  of  fractions  by 
reducing  to  a  lower  denomination,  that  at  the  end,  when 
lower  denominations  unaccountably  fail  them,  they  are  non- 
plussed. Quite  rightly  so  ;  the  fault  is  not  with  the  children. 
Whenever  an  attentive  child  finds  a  persistent  difficulty, 
teachers  should  be  sure  that  there  is  something  wrong  with 
their  mode  of  presenting  it,  probably  with  their  own  compre- 
hension of  it.  Nothing  is  difficult  when  properly  put.  The 
whole  art  of  teaching  should  be  so  to  lead  on  that  everything 
arrives  naturally  and  easily  and  happily,  like  fruit  and  flowers 
out  of  seeds. 

Another  British  method.  Usually  however  the  sum  is  not  recorded 
80  briefly  as  this,  but  is  written  out  in  what  is  known  as  the  long- 
division  plan ;  and  it  is  perhaps  the  safest  mode  of  getting  the  right 
answer  if  the  answer  is  required  to  be  thus  barbarously  specified,  for 
it  certainly  shirks  nothing.     This  is  the  way  of  it  : 

To  divide  336  lbs.  av.  into  20  equal  parts 

lbs.      oz.       dr. 

2,0)33,6(16  .  12  .  12| 
20 

136 
120 

16 
16* 

96 

256  oz. 
240 

16 
_16* 

96 
16 

256  dr. 
240 

16  remainder,  and  ^^  =  -|  dr. 


*  If  any  mathematician  glances  through  this  book,  as  I  hope  he  may, 
he  will  require  at  these  stages  to  be  reminded  if  British,  to  be  informed 


90  EASY  MATHEMATICS.  [chap. 

This  may  look  like  a  parody,  but  it  is  soberly  the  way  in 
which  innumerable  children  have  been  taught  in  the  past  to 
do  such  a  sum.  And  the  fact  that  they  have  been  so  taught 
can  easily  be  tested  by  setting  it  to  people  who  were  children 
a  few  years  ago. 

Another  method.  If  the  factor  plan  of  division  is  adopted 
there  is  great  danger  of  confusion  and  error  about  the  carrying 
figure.  For  instance,  in  dividing  336  lbs.  into  20  equal  parts, 
a  child  as  sometimes  now"  taught  will  proceed  thus  : 

2  [336  lbs. 
1  0 1 16,8 

16  and  8  over. 

8  what  over  ?  They  are  apt  to  take  it  as  8  lbs.  over,  and  so 
interpret  it  as  128  ounces,  and  proceed  to  divide  these  again 
by  20  by  the  same  process 

2 1 128 

10  I    64 

6  and  4  over 

apt  to  be  called  4  ounces  over,  which  are  interpreted  as  64 
drachms,  and  so  on. 


if  Foreign,  that  in  these  islands  a  drachm  is  defined  to  be  the  sixteenth 
of  an  ounce,  and  that  an  ounce  avoirdupois  is  one  sixteenth  of  an 
avoirdupois  pound  ;  moreover  that  a  drachm  is  the  lowest  recognised 
denomination  of  avoirdupois  weight :  after  that  fractions  are  permitted. 
Pennyweights  and  grains  belong  to  a  system  of  measures  to  which  the 
name  of  "  Troy "  is  (for  some  to  me  unknown  reason,  perhaps  from 
Troyes  in  France)  prefixed.  There  is  a  "Troy  pound"  and  a  "Troy 
ounce,"  for  "metallurgical"  use,  but  they  differ  from  their  "grocery" 
cousins  which  are  explicitly  asserted  "to  have  some  weight."  Then 
between  grains  and  Troy  ounces  there  are  other  denominations  used  by 
"apothecaries,"  called  scruples  and  drams.  This  dram  is  not  the  same 
as  the  grocery  drachm.  There  appears  however  to  be  only  one  kind  of 
"grain,"  and  7000  of  these  make  1  lb.  avoirdupois,  while  5760  of  them 
make  1  lb.  Troy. 


VIII.]  SIMPLE  PROPORTION.  91 

This  is  quite  wrong.  The  8  over  in  the  first  little  sum  was 
really  8  double-pounds,  and  so  the  second  little  sum  is  all 
wrong.  If  it  had  been  right,  the  4  over  could  not  have  been 
4  ounces,  but  4  double-ounces ;  but  what  needless  trouble  and 
risk  of  error  is  introduced  by  having  to  perceive  this ! 

Again  let  many  children  be  asked  to  divide  £336  by  25, 
they  will  few  of  them  have  been  taught  to  proceed  thus : 

%Y-  =  3-36x4  =  £13-44 

=  £13.  8-8s. 

=  £13.  8s.  9-6d.  or  about  9Jd. 

but  they  will  proceed,  either  by  long  division  on  much  the 
same  lines  as  in  the  last  example,  which  is  long  to  write,  or 
else  by  short  division,  dividing  by  5  twice  over,  which  is  not 
too  long  to  write, 

£        s.      d. 
5  I  336  .0.0 
5 1  67  .  4  .  0 
13  .  8  .  9| 

short  to  write,  but  rather  hard  to  do.  Such  trivial  sums 
should  not  call  for  so  much  brain  power  as  is  involved  in 
various  and  complicated  carryings. 

Money  sums  however  are  the  best  examples  of  the  kind. 
If  it  was  336  tons  that  had  to  be  divided  into  25  equal  parts, 
grown  people  would  be  satisfied  to  say  that  each  part  must  be 
13-44  tons;  but  at  some  schools  it  would  have  to  be  done 
thus, — if  not  by  a  still  longer  process  equally  liable  to  acci- 
dental error : 

tons.  cwt.  qrs.  lbs.   ozs.   dr. 

5  I  336  .  0  .  0  .  0  .  0  .  0 

^  167.4.0.0.0.0 

13  .  8  .  3  .  5  .   9  .  9j  Ans. 


92  EASY  MATHEMATICS.  [chap. 

Breakdown  of  simple  proportion  or  "rule  of  three." 

Simple  proportion,  or  the  rule  of  three,  is  by  some  teachers 
regarded  as  a  kind  of  fetish ;  moreover  its  extreme  simplicity 
makes  it  9-  rather  favourite  rule  with  children  and  they  will 
naturally  do  many  exercises  in  it.  Not  always,  it  is  to  be 
hoped,  by  the  same  mechanical  method. 

But  there  is  all  the  more  necessity  for  bringing  home  to 
them  the  fact  (strange  if  it  is  unknown  to  any  teacher),  that 
it  does  not  always  work.     For  instance  : 

A  stone  dropped  down  an  empty  well  16  feet  deep  reaches 
the  bottom  in  one  second.  How  deep  will  a  well  be  if  a 
stone  takes  two  seconds  to  reach  the  bottom  1 

The  answer  expected  is  of  course  32  feet;  but  it  is  not 
right.     The  correct  answer  is  64  feet. 

If  a  stone  drops  16  feet  in  one  second,  how  far  will  it  drop 
in  J  second  ]  (Ans. :  1 2  inches.) 

Again,  if  a  stone  dropped  over  a  cliff  descends  64  feet  in 
2  seconds,  how  far  will  it  drop  in  the  next  second  1 

(Ans. :  80  feet.) 

A  steamer  is  propelled  at  the  rate  of  8  knots  by  its  engines 
exerting  themselves  at  the  rate  of  1000  horse  power.  What 
power  would  drive  it  at  12  knots? 

Probably  no  one  would  expect  the  answer  1500  to  this  ;  for 
on  that  principle  10000  horse  power  would  propel  it  at 
80  knots. 

An  initial  velocity  of  1600  feet  a  second  will  carry  a  rifle 
bullet  3  miles.     What  velocity  would  carry  it  6  miles  1 

An  ounce  weight  drops  4  feet  in  half  a  second.  How  far 
will  a  pound  weight  drop  in  the  same  time  ? 

(Ans. :  By  experiment,  4  feet  likewise.  A  most  important 
fact,  discovered  by  Galileo,  and  illustrated  from  the  tower  of 
Pisa.) 


VIII.]  NON-SIMPLE  PROPORTION.  93 

Let  it  not  be  dogmatised  on,  but  illustrated  by  dropping 
things  together ;  and  if  it  appears  puzzling,  so  much  the  better. 
Ignoring  or  eliminating  the  resistance  of  the  air  everything 
falls  at  the  same  pace.  The  air  has  very  slight  influence 
on  the  drop  of  smooth  spheres  through  a  moderate  height. 
Cotton  wool  and  feathers  and  bits  of  paper  will  drop  more 
slowly,  but  the  reason  is  obvious :  a  bullet  will  drop  more 
slowly  in  treacle  than  in  air.  That  is  because  the  air  resistance 
is  small :  it  is  not  zero,  and  if  a  bullet  and  a  pea  were  dropped 
from  too  great  a  height,  air  friction  would  begin  perceptibly 
to  retard  the  lighter  body.  So  it  is  that  big  rain-drops  fall 
quicker  than  little  ones ;  and  these  small  drops  quicker  than 
mist  and  cloud  globules.  So  also  does  heavy  fine  powder, 
even  gold  powder,  fall  slowly  in  water,  not  because  it  is 
buoyed  up,  but  because  it  is  resisted.  Eemove  the  air,  and  in 
a  vacuum  a  coin  and  a  feather  will  fall  at  the  same  rate.  The 
statement  does  not  explain  the  fact.  The  full  explanation  of 
the  fact  is  not  even  yet  known.  But  a  very  great  deal  more 
is  known  about  the  whole  subject  than  is  or  can  be  here 
expressed.  That  is  characteristic  of  elementary  books  through- 
out, and  the  object  of  the  learners  should  be  to  get  through 
all  this  easy  stuff,  and  get  on  into  more  exciting  matters 
beyond :  matters  which  the  majority  of  the  human  race  never 
have  the  least  knowledge  of,  because  their  early  education 
has  been  neglected. 

A  balloon  18  feet  in  diameter  can  carry  a  load  equal  to  one 
man.  What  load  can  a  similar  balloon  carry  which  is  36  feet 
in  diameter.  (Simplest  rough  answer,  8  men.) 

A  rope  stretches  half  an  inch  when  loaded  with  an  extra 
hundredweight. 

How  much  would  it  stretch  if  loaded  with  an  extra  ton  1 


94  EASY  MATHEMATICS.  [chap.  viii. 

A  half  crown  is  ten  times  the  value  of  a  threepenny  bit. 
How  many  threepenny  bits  can  lie  flat  on  a  half-crown 
without  overlapping  the  edge*?       (Ans. :  By  experiment,  one.) 

A  boy  slides  20  yards  with  an  initial  run  of  10  feet.  What 
initial  run  would  enable  him  to  slide  half  a  mile  ? 

If  2  peacocks  can  waken  one  man,  how  many  can  waken 
six? 

If  a  diamond  is  worth  ten  thousand  pounds,  what  would 
950  similar  diamonds  be  worth  ? 

If  a  camel  can  stand  a  load  of  5  cwt,  for  6  hours,  for  how 
long  could  he  stand  a  load  of  ten  tons  ? 

These  things  cannot  be  done  by  simple  proportion.  They 
require  something  more  to  be  known  before  they  can  be  done 
at  all ;  and  accordingly  it  would  appear  as  if  generations  of 
teachers  had  discreetly  shied  at  them  all,  indiscriminately, 
and  had  excluded  them  from  arithmetical  consideration 
altogether.  It  is  just  as  if  in  geometry,  finding  straight  lines 
simpler  than  curves,  they  had  agreed  to  found  all  their 
examples  upon  straight  lines. 


CHAPTER   IX. 

Simplification  of  fractions. 

Vulgar  fractions  are  much  harder  to  deal  with  than 
decimals ;  but  as  sometimes  several  have  to  be  added  together 
it  is  desirable  to  know  how  to  do  it.  Besides,  the  exercise  so 
afforded  is  of  a  right  and  wholesome  kind. 

Consider  the  following  addition  :  J  +  J.  Small  children  can 
see  (by  experiment  on  an  apple)  that  the  result  is  |,  and  they 
can  also  be  taught  to  regard  it  as  f  +  ^  =  f^,  which  should  be 
read  in  words — two  quarters  added  to  one  quarter  make 
three  quarters. 

Thus,  it  can  be  realised  that  when  the  denominators  are  all 
the  same,  addition  of  fractions  becomes  simple  addition  of  the 
numerators. 

For  just  as  5  oranges  +  6  oranges  =11  oranges,  so 

5      I      6      _     11 
TT'^TT   ~   TT' 

reading  "  seventeenths  "  instead  of  "  oranges." 

When  denominations  differ,  therefore,  the  first  thing  to  do 
is  to  make  them  the  same. 

Thus,  for  instance,  3  apples  +  4  oranges,  is  an  addition  which 
can  only  be  performed  by  finding  some  denomination  which 
includes  both,  say  "pieces  of  fruit." 

So  also  7  horses +  3  pigs  =10  quadrupeds.  5  copies 
of  Bobinson  Crusoe  +  3  copies  of  Ivanhoe  =  8  prize-books, 
perhaps. 


96  EASY  MATHEMATICS.  [chap. 

Eeduction  to  the  same  denomination  cannot  always  be 
done,  when  denominations  are  anything  whatever,  except  by 
using  the  vague  term  "objects"  or  "things";  but  with 
numerical  denominators  it  can  always  be  done,  and  the  method 
of  doing  it  has  to  be  learnt,  y  +  tt  =  xi'  ^^^  ^^^h  like,  are 
easy  examples.     |-  +  y 2  =  ^  is  a  slightly  harder  one. 

It  is  done  by  saying  -^^  +  iV  =  A  =  J- 

So  also  ^  +  i  =  I,  being  equal  to  ^. 

A  harder  example  is  |  4- 1,  which  can  be  written 

21,10    _    3  1    _    Kl 

In  the  decimal  notation  this  would  appear  thus : 

3-5  +  1-666...  =  5-1666.... 
A  still  harder  example  can  be  worked  out  thus : 

9i5    _     6. -J,    40    _    103    _    147 
8"+'T    —    S'F"'"  "56 "5  6"   -^    -^TF' 

though  the  final  step  is  one  that  need  not  always  be  made. 

Now  it  is  evident,  or  at  least  it  will  gradually  be  found  true, 
that  in  a  mechanical  process  of  this  kind  there  is  always  some 
simple  rule  by  which  the  result  can  be  obtained  without  thought. 
What  is  that  rule  ?  If  the  child  can  find  it  out  for  himself,  by 
experimenting  on  lots  of  pairs  of  fractions,  so  much  the  better. 
A  week  is  none  too  much  to  give  him  to  try,  for  if  he  finds  it 
out  himself  he  will  never  forget  it. 

The  rule  is  :  cross-multiply  for  the  numerators,  and  multiply 
the  denominators. 

1     1  ^6  +  2  _  8       2 
2'^6  ~    12    "  12~3* 
1     !_&  +  « 
a    b  ~    ah  ' 
3    4  _  27  +  28  _  55 
T'^g"      63      ~"63' 
a     c  _ad+hc 
h^d~'~hdr' 

but  it  would  be  a  pity  to  spoil  this  by  premature  telling. 


IX.]  FRACTIONS.  97 

The  fact  that  the  sum  of  two  reciprocals  is  the  sum  of 
the  numbers  divided  by  their  product,  is  worth  illustrating 
fully  and  remembering :  remembering,  that  is,  by  .growing 
thoroughly  accustomed  to  it,  not  exactly  learning  by  heart. 
There  is  hardly  any  need  to  learn  easy  things  like  that  by 
heart:  nevertheless  it  is  a  very  permissible  operation,  whenever 
the  fact  to  be  learnt  is  really  worth  knowing. 

11      7    .,    ,  .    .,       sum 

0  +  T  =  Ts.  that  IS  the 3 — -• 

3    4      12  product 

1  1  _a  +  h 
a     h~    ah  ' 

}_l_  28 
23    5      115* 

2  +  49  "98  — 100-   ^''• 
the  symbol  aCh  meaning  "  approximately  equals." 

[The  approximation  is  seen  to  be  true  because  adding  1  to 
50  makes  the  same  proportional  difference  as  adding  2  to  100. 
If  this  is  too  hard,  it  can  be  postponed.  It  is  unimportant, 
but  represents  a  kind  of  thing  which  it  is  often  handy  to 
do  in  practice.] 

But  this  rule  of  cross-multiplication  hardly  serves  for  the 
addition  of  three  or  more  fractions,  at  least  not  without 
modification.     Take  an  example, 

1^2    7_14    21_26i 
6'''3"*'2~6"''6"^  6  ~  6  -  ^* 

Take  another,  1  ^  1  +  1  =  i±|±l  =  7, 

where  the  three  fractions  |,  f,  and  \^  all  having  the  same 

denominator,  are  written  all  together,  with  the  addition  of  the 

numerators  indicated,  and  subsequently  performed, 

r.  111      3  +  4^  +  1      8j      17 

One  more,  +    +     = — —  =  -^  =  r^. 

3     2     9  9  9        18  ^ 

This  might  hardly  be  considered  a  legitimate  procedure,  but 

L.E.M.  G 


98  EASY  MATHEMATICS.  [chap. 

there  is  nothing  the  matter  with  it.  You  might,  instead, 
proceed  thus  : 

1    1    1  _  18^27    A^^l^lZ 

;V^2"^9  ~  54 "^54 "^54  ~  54  ~  18' 

and  that  is  equally  a  correct  method. 

But  neither  of  these  plans  is  quite  the  grown  up  plan.  Let 
a  better  plan  be  found;  but  first  let  the  above  plans  be 
formulated  and  expressed.  Is  it  not  plain  that  the  numerator 
of  each  particular  fraction  is  found  by  multiplying  two  of 
the  denominators  together,  while  the  common  denominator  of 
all  the  fractions  is  found  by  multiplying  all  the  denominators 
together  1     Apply  this  rule  : 

1     1     1  _  20  +  24  +  30  _  7£_37 
6"^5"^4~         120         "120  "60* 

For  instance,  a  sixth  of  an  hour  +  a  fifth  of  an  hour  +  a 
quarter  of  an  hour  =  37  minutes,  a  minute  being  the  sixtieth 
of  an  hour.  Now  a  sixth  of  an  hour  is  ten  minutes,  a  fifth  is 
12  minutes,  and  a  quarter  of  an  hour  is  15  minutes:  conse- 
quently the  neatest  way  of  doing  the  sum  would  be 

1     1     1      10  +  12  +  15      37 
6"^5^4~         60  60* 

.      ^,  ,  111      180  +  36  +  720 

Another  example,  ^  +  ^^  +  3  =   -^f^^.-J-^ 

but  here  every  term  in  numerator  and  denominator  can  be 
divided  by  3  and  by  1 2,  so  that  the  above  may  be  written 

12^60    3  60  60      30 

And  it  would  have  been  neater  to  write  it  so  at  first — neater 
but  not  essential,  and  sometimes  not  even  the  most  rapid 
plan. 

To  illustrate  the  above  example  : 

i\th  of  a  day  is  2  hours. 
^^0  th  of  a  day  is  24  miuutes. 
^rd  of  a  day  is  8  hours. 


IX.]  FRACTIONS.  99 

Consequently  the  sum  of  these  fractions  of  a  day  is  10  hours  and 
24  minutes, 

which  is  lOf^  of  an  hour  [  =  lOyV  =  10'4  hours]  or  ^j  +  ^V  of  a  day, 
which  again  may  be  written  iriJ  +  FT7  =  FIT  =^  ^iyths  =  •043,  as  before. 
The  form  of  the  general  rule,  then,  is  given  by 
I      1     1  _  bc  +  ca  +  ab  ^ 
a     b     G  ahc        ' 

but  in  practice  it  is  possible  to  abbreviate  this  in  some  cases, 
when  one  of  the  denominators  contains  the  others  as  factors, 
or  when  some  simple  relation  of  the  kind  exists  between  them. 
This  is  what  was  made  use  of  in  the  early  simple  cases, 

such  as  y*2  +  2^ ;   we  did  not  proceed  to  write  ""  ^  and 

then  simplify  it,  but  we  wrote  at  once  ^t  +  2T  ^  tt  y  ^^^^  i^ 
to  say  we  perceived  that  24  would  do  for  the  new  denomin- 
ator, and  we  adjusted  the  numerators  accordingly. 

Perhaps  we  had  better  display  this  algebraically.  Let  each 
denominator  contain  a  common   factor,   say  r?,   so   that   the 

reciprocals  to  be  added  are h  ^  H — ,  then  if  we  applied  the 

^  na     nb     nc  ^^ 

,       ,             111        -^     '^^^<^  + 1^'^<^<^  +  i^^o^b  , 
mere  general  rule  we  should  write  g-y ,  but  the 

repetition  of  the  powers  of  n  is  manifestly  needless,  since  they 

cancel   out ;    and  it   is   much   neater    to   write  for  the  new 

denominator  an  expression  which  contains  the  common  factor 

,              , ,         bc  +  ca-\-ab 
n  only  once,  thus  :  ^ 

The  denominator  so  obtained  is  called  the  least  common 
multiple  of  the  three  denominators ;  and  it  is  frequently,  in 
examination  papers,  denoted  by  the  letters  L.C.M.  It  is  not 
an  important  idea  at  all.  Sums  can  be  done  quite  well 
without  it,  but  its  introduction  affords  some  scope  for  neat- 
ness and  ingenuity.     Easy  processes  can  be  given  for  finding 


100  EASY  MATHEMATICS.  [chap. 

it,  but  they  are  hardly  worth  giving,  as  in  real  practice  they 
are  seldom  used :    they  are   of   most   educational  service  if 
employed  as  an  exercise  for  the  student's  invention.      They 
will  be  dealt  with  sufficiently  in  the  next  chapter. 
Now  take  a  numerical  example  : 

Add  together  |  +  J  +  f  +  t\  +  ^2- 

Here  32  is  evidently  the  l.c.m.  of  the  denominators,  since  it 
contains  all  the  others  as  factors.  So  that  will  serve  as  the  simplest 
common  or  combined  denominator.  The  first  numerator  accordingly 
will  be  16,  the  second  8,  the  third  4  but  taken  5  times  and  therefore 
20,  the  next  2  taken  3  times,  and  the  last  1  taken  7  times. 

Consequently  the  sum  is  written  as  follows  : 

1     1     5    ^    J  _  16  +  8  +  20  +  6  +  7  _57 

2'^4"^8"^16'*"32"  32  32* 

Take  another  example  of  addition  : 

1      1      1     J^  _  72  +  9  +  56  +  8  _  145 
7"^56'^9'''63~  504  "504 

Here  7  is  plainly  a  factor  of  both  the  larger  denominators,  and  8  and 
9  are  the  other  factors,  so  the  least  common  denominator  will  only 
contain  7  ard  9  once,  and  will  equal  7x8x9  =  504,  and  this  being  the 
smallest  common  multiple  possible,  no  further  simplification  can  be 
eflfected ;  beyond  of  course  expressing  the  result  as  a  decimal  if  we  so 
choose.  To  express  it  as  a  decimal  we  must  eflfect  the  division 
indicated  ;  the  result  happens  to  equal  '2877  almost  exactly. 

It  is  worth  noticing  that  the  series  of  powers  of  -J,  viz.  : 

add  up  very  nearly  to   1  ;    and  the  more  nearly  the  more 
terms  of  the  series  are  taken. 

It  can  be  shown,  not  by  trial  indeed,  but  by  simple  reason- 
ing, that  if  an  infinite  sequence  of  this  series  are  added 
together  the  result  is  exactly  1.  Thus  the  first  term  con- 
stitutes half  of  the  whole  quantity,  say  a  loaf,  the  second 
term  added  to  it  gives  us  three  quarters,  the  third  term  gives 
us  |-th  more,  and  we  only  need  another  eighth  to  get  the 


IX.]  FRACTIONS.  101 

whole.  The  next  term  gives  us  half  of  the  deficiency,  and 
now  we  need  the  other  sixteenth  to  make  the  whole.  We 
do  not  get  it  however:  we  get  half  of  it  in  the  next  term, 
and  thus  still  fall  short,  but  this  time  only  by  ^^ ;  and  so  at 
the  end  of  the  above  series,  as  far  as  written,  our  deficiency 
is  6T*^-  Each  term  therefore  itself  indicates  the  outstanding 
deficiency,  and  as  the  terms  get  rapidly  smaller  and  smaller, 
so  does  the  deficiency  below  1  get  rapidly  diminished  till 
it  is  imperceptible.     (Compare  p.  325.) 

It  is  convenient  to  plot  these  fractions  as  lengths  (setting 
them  up  at  equal  distances  along  a  horizontal  line),  say  half  a 
foot,  then  a  quarter,  then  an  eighth,  and  so  on.  Then  joining 
their  tops  we  get  a  curve  which  has  the  remarkable  property 
of  always  approaching  a  straight  line,  but  never  actually 
meeting  or  coinciding  with  it,  or  at  least  not  meeting  it  till 
infinity ;  when  at  length  it  has  become  quite  straight. 


There  are  many  curves  with  such  a  property,  but  fig.  9  may 
be  the  first  a  child  has  met.  He  can  of  course  continue  the 
curve  in  the  other  direction — the  direction  of  whole  numbers, 
or  powers  of  two,  and  observe  how  rapidly  it  tilts  upwards ; 
but  there  is  no  straight  line  in  this  direction  to  which  it  tends 
to  approach ;  this  end  proceeds  to  infinity  both  upwards  and 
sideways,  not  only  upwards,  though  it  proceeds  far  more 
rapidly  in  the  vertical  direction  than  in  the  horizontal ;  and 
this  end  of  it  never  becomes  straight. 


CHAPTER   X. 

Greatest    Common    Measure    and    Least    Common 
Multiple. 

Another  name  of  slight  importance,  which  is  usually  paired 
off  with  Least  Common  Multiple  (page  99),  is  Greatest 
Common  Measure  or  Highest  Common  Factor :  often  denoted 
by  G.C.M.  or  by  H.C.F. 

The  two  numbers  24  and  16  have  several  factors  common 
to  both  of  them,  for  instance  8 ;  and  this  as  it  happens  is  the 
greatest  common  factor,  the  others  which  they  possess  in 
common  being  4  and  2. 

The  numbers  20  and  35  have  5  as  the  largest  factor  common 
to  both  of  them.  The  numbers  72  and  84  have  12 ;  while  72 
and  96  have  24  as  their  g.C.m. 

The  numbers  23  and  38  have  no  factor,  above  unity, 
common  to  both.     In  fact  23  has  no  factor  at  all. 

The  word  "common"  so  used  does  not  mean  "ordinary," 
as  children  sometimes  think,  nor  does  it  mean  vulgar,  but  it 
has  the  signification  which  it  possesses  in  "common  friend," 
or  in  vulgar  phrase  "  mutual  friend,"  or  when  people  are  said 
to  own  property  "in  common." 

To  find  common  factors  of  two  numbers,  one  way  is  to 
arrange  all  the  factors  of  each  in  two  rows  one  under  the 
other  and  see  how  many  correspond.  Inspection  will  then 
readily  show  which  pair  is  the  biggest. 


CHAP.  X.]  GREATEST  COMMON  MEASURE.  103 

Suppose  the  two  numbers  given  were  40  and  60 ;  the 
following  are  the  factors  of  60, 

2,  3,  4,  5,  6,   10,   12,   15,  20,  30, 
and  the  following  are  the  factors  of  40, 

2,  4,  5,  8,   10,  20. 
Of  these,  the  2,  4,  5,  10,  20  are  common  to  both,  and  20  is 
the  largest  of  them. 

In  old-fashioned  language,  factors  were  called  "measures," 
and  the  largest  common  factor  was  called  the  "greatest 
common  measure,"  and  abbreviated  into  g.c.m. 

What  is  the  use  of  it  1  Very  little ;  but  the  meaning  is 
perfectly  simple  and  should  be  understood.  It  can  be  utilised 
for  finding  the  Least  Common  Multiple  of  a  set  of  numbers, 
that  is  to  say  the  smallest  number  which  contains  them  all 
as  factors ;  for  the  G.C.M.  represents  what  may  be  struck  out, 
once  at  least  and  sometimes  more  than  once,  from  the  product 
of  a  set  of  numbers,  in  order  to  leave  behind  the  smallest 
number  which  they  are  able  to  divide  without  a  remainder. 

Thus  take  the  numbers  40  and  60,  their  product  is  2400, 
and  of  course  they  will  both  divide  that ;  but  their  G.C.M.,  20, 
may  be  cancelled  out  of  it,  leaving  1 20 ;  and  both  40  and  60 
will  divide  that  too.  It  is  the  least  number  which  they  can 
both  divide  exactly,  i.e.  it  is  the  least  number  of  which  they 
are  both  factors,  it  is  in  fact  their  least  common  multiple. 

Example.— Of  the  numbers  12,  20,  24  what  is  the  g.c.m. 
and  L.C.M.  ?  Of  these,  12  need  not  be  attended  to  in  finding 
the  largest  common  factor,  because  it  is  itself  a  factor  of  24. 

Of  the  numbers  20  and  24,  4  is  a  common  factor ;  so  divide 
all  by  that,  and  we  get  left  with  3,  5,  6. 

No  factor  will  divide  all  these,  so  4  was  the  g.c.m.  of  the 
original  numbers. 

Their  least  common  multiple  is  not  3  x  5  x  6  =  90,  because 
that  would  have  omitted  the  factor  4  which  they  possess  in 


104  EASY  MATHEMATICS.  [chap. 

common.  The  common  factor  need  not  be  repeated  more  than 
once,  (for  if  it  is,  though  you  get  a  common  multiple,  you  do 
not  get  the  least  common  multiple),  but  it  must  not  be  omitted 
altogether,  or  you  will  not  get  a  common  multiple  at  all. 

The  L.C.M.  accordingly  is  3  x  5  x  6  x  4  =  360,  and  of  that  it 
will  be  found  that  the  given  numbers  12,  20,  24,  are  factors. 
The  product  of  those  numbers  is  5760,  and  out  of  that  the 
G.C.M.  4  can  be  struck  twice  before  arriving  at  the  L.C.M. 

Anyone  therefore  can  invent  a  rule  for  finding  the  L.C.M. 
of  a  set  of  numbers;  it  is,  find  their  g.c.m.  and  divide  or 
cancel  it  out  of  all  the  numbers  but  one,  then  multiply  the 
quotients  together. 

But  a  rule  for  finding  the  g.c.m.  is  by  no  means  so  easy  to 
invent :  it  is  an  ingenious  process,  and  the  whole  subject  is 
essentially  a  little  bit  of  rudimentary  pure  mathematics ;  it 
has  no  practical  importance  or  application  except  when  dealing 
with  the  properties  of  numbers. 

The  proof  of  the  rule  is  an  interesting  and  easy  exercise  in 
the  application  of  reason  and  commonsense  to  arithmetic,  but 
perhaps  it  is  better  deferred. 

Rule  for  finding  G.C.M. 

The  rule  depends  on  the  demonstration  that  any  factor  of 
two  numbers  is  likewise  necessarily  a  factor  of  the  remainder 
left  when  one  is  divided  by  the  other. 

Thus  consider  the  two  numbers  40  and  24.  Divide  one 
by  the  other,  we  get  1  and  16  over.  The  above  sentence  in 
black  type  assumes  or  asserts  that  every  factor  of  40  and  24 
must  also  be  a  factor  of  16.     In  this  case,  as  a  matter  of  fact, 

40  =  24  +  16 
and  it  is  manifest  that  a  number  which  divides  24  and  does 
not  divide  16  cannot  divide  40. 

Well  that  is  the  whole  idea. 


X.]  GREATEST  COMMON  MEASURE.  105 

If  we  were  told  to  find  the  G.c.M.  of  40  and  24,  we  could 
by  this  means  reduce  the  problem  to  finding  the  G.c.M.  of  24 
and  16.  And  then,  repeating  the  division  process,  we  should 
observe  that  24=16  +  8, 

so  that  the  problem  becomes  reduced  still  further  into  finding 
the  G.C.M.  of  16  and  8.  There  is  no  question  but  that  this  is 
8 ; — as  indeed  we  might  have  guessed  at  first  if  our  object 
had  been  attainment  of  a  result,  instead  of  explication  of  a 
process — and  the  way  to  clinch  that  is  to  perform  the  division 
again  and  to  find  that  there  is  now  no  remainder  at  all. 

The  matter  can  be  stated  algebraically,  but  beginners  can 
skip  the  algebra  and  come  to  the  "  illustration  "  which  follows. 

Algebraic  proof  of  the  process  for  finding  G.C.M. 

To  find  a  common   factor  of   two  numbers  P  and  Q,  of 

which  P  is  the  bigger, 

let  X  be  one  common  factor, 

P  Q 

then  —  and  -  will  be  the  complementary  factors. 

An  extreme  case  is  when  P  is  divisible  by  Q  without  a 
remainder,  in  that  case  x  =  Q.  Suppose  however  that  when 
P  is  divided  by  Q  the  remainder  is  i?, 

Q)P{n 

nQ 

R 

so  that  P  =  iiQ  +  R;  then  if  i?  is  a  factor  of  Q  it  must  be  one 

of  P  also  (because  P  equals  a  multiple  of  Q  plus  E),  so  try 

if  i?  is  a  factor  of  Q. 

If  it  is,  it  is  the  common  factor  required ;  but  if  not,  work 
out  a  division  again,  and  let  the  remainder  be  >S^, 

R)Q{m 
mR 

S 
so  that  Q  =  mR  +  S. 


106  EASY   MATHEMATICS.  [chap. 

Then  if  aS'  is  a  factor  of  R  it  must  be  one  of  Q  too,  and  so  also 
of  P,  and  in  that  case  S  will  be  the  common  factor  required. 

But  if  not,  we  must  repeat  the  process  and  see  what  the 
remainder  is  when  R  is  divided  by  S,     Call  it  2\ 

S)R{1 
IS 

so  that  i?  =  IS-hT.  ^' 

Now  once  more  if  2^  is  a  factor  of  aS^  it  is  necessarily  a  factor 
of  R,  and  therefore  of  Q,  and  therefore  also  of  F,  and  so  2' 
is  the  common  factor  required. 

If  not,  the  process  must  go  on  until  there  is  no  further 
remainder;  and  then  the  last  remainder  (or  divisor)  is  a 
common  factor  of  the  two  original  numbers  P  and  Q.  Let  us 
assume  that  T  divides  S  without  a  remainder,  then  2'  is  the 
common  factor  of  all  the  numbers  P,  Q,  R,  S,  T. 

It  is  likewise  the  largest  common  factor  which  exists.  Why? 
because  it  has  to  be  a  factor  not  only  of  P  and  Q  but  also  of 
R,  of  aS',  and  of  T ;  and  certainly  T  is  the  largest  factor  of  Ty 
therefore  it  is  likewise  the  largest  common  factor  of  the  others. 

Statement  in  another  form. 

The  whole  process  can  be  written  thus : 

To  find  the  g.c.m.  of  P  and  Q^  work  successive  division 

sums  thus :  ■        P  R 

Q  =  "■'? 

Q         .  S 

R^'^'-R^ 

S  ~    ^^aS' 

p  1  1  1 

or  ^  =  rt  4-  7Tn\  —  n-\ ;—  =  7i  + 


g  ■  Q/i?  '       ^    1  '        ^       1      ' 

^^aS/T 
the  process  terminating  only  when  S\T  is  an  integer. 


X.]  GREATEST  COMMON  MEASURE.  107 

The  r  is  a  factor  of  all  the  numbers  P,  Q,  R,  S,  T;  and 
since  it  must  satisfy  this  condition  if  it  is  to  be  a  factor  of 
P  and  Q  at  all,  it  is  necessarily  the  greatest  common  factor 
of  F  and  Q,  and  indeed  of  the  others  too. 

Or  the  whole  process  may  be  written  (as  usually  performed) 
in  one  sura  thus : 

Q)P(n 
nQ 

B)Q(m 
mil 

'~S)R{1 
IS_ 

T)S{k 
TcT 

Then  the  last  remainder  (or  divisor)  T  is  the  G.C.M.  of  P 
and  Q. 

Illustration  (modified  from  Kirkman  and  Field). 

Let  the  two  numbers  be  492  and  228.  Go  through  a 
process  of  successive  divisions. 

228)492(2 
456 
36)228(6 
216 

^)36(3 
36 


Hence  12  is  the  G.C.M.  of  the  two  original  numbers,  and  it 
likewise  is  a  factor  of  the  intermediate  divisor,  viz.  36. 

The  argument  runs  as  follows  : 

The  common  factor  of  492  and  228  must  also  be  a  factor 
of  the  remainder  when  492  is  divided  by  228,  for  in  fact 
492  =  (2x228)4-36, 


108  EASY  MATHEMATICS.  [chap.  X. 

so  that  anything  which  divides  228  and  fails  to  divide  36 
cannot  possibly  divide  492. 

Hence  the  problem  reduces  itself  to  finding  the  common 
factor  of  228  and  36. 

But  now  228  =  (6  x  36)  +  12, 

hence  the  factor  required  must  likewise  divide  12,  as  well  as 
36.  The  numbers  2,  3,  4,  6,  12  all  satisfy  that  condition,  and 
hence  all  these  are  factors  of  both  the  original  numbers,  but 
of  them  12  is  the  biggest. 

Therefore  12  is  the  G.C.M.  of  the  two  given  numbers  492 
and  228.  (Verify  this  by  actual  division.  The  quotients 
are  41  and  19,  and  neither  of  these  have  any  factors  at  all.) 


CHAPTER   XL 

Easy  mode  of  treating  problems  which   require  a 
little  thought. 

Many  of  the  problems  set  for  purposes  of  arithmetic  are 
best  done  in  the  first  instance  by  rudimentary  algebra,  that  is 
by  the  introduction  of  a  symbol  for  the  unknown  quantity, 
so  that  it  can  be  tangibly  dealt  with.  This  introduction  and 
manipulation  of  a  symbol  for  an  unknown  quantity  need  not 
be  discouraged,  even  from  the  first.  It  confers  both  power 
and  clearness.  Many  arithmetical  sums  are  needlessly  hard 
because  x  is  forbidden.  There  is  a  certain  amount  of  sense  in 
the  artificial  restriction,  but  in  complicated  sums  and  in  physics 
the  symbolic  treatment  of  unknown  quantities  is  essential, 
and  the  sooner  children  are  accustomed  to  it  the  better. 

The  introduction  of  a  symbol  for  an  unknown  quantity  is  a 
device  to  enable  a  sum  to  be  clearly  and  formally  stated. 
After  the  sum  has  been  solved  by  this  aid,  it  is  well  to  try 
and  express  it  so  that  it  can  be  grasped  and  understood 
without  such  assistance.  The  fear  of  those  who  object  to  x 
in  arithmetic  is  that  this  final  step  may  be  omitted.  The  grasp 
is  clearer  when  an  auxiliary  symbol  can  be  dispensed  with;  but 
that  is  not  possible  always  at  first.  The  x  is  to  be  thought  of 
as  a  kind  of  crutch :  but  sometimes  it  is  like  a  leaping-pole 
and  enables  heights  to  be  surmounted  which  without  it  would 
be  impossible. 


110  EASY  MATHEMATICS.  [chap. 

Example. — How  soon  after  twelve  o'clock  will  the  hour  and 
minute  hand  of  a  clock  again  be  superposed  ? 

It  is  plain  that  it  is  soon  after  1  o'clock,  and  that  it  is  an 
amount  which  has  been  traversed  by  the  hour  hand  while  the 
minute  hand,  travelling  twelve  times  as  quickly,  has  gone 
that  same  distance  and  5  minutes  more ;  but  it  is  not  easy  to 
think  out  the  required  fraction  in  one's  head,  though  ex- 
ceptional children  can  do  it. 

But  let  it  be  postulated  as  n.  minutes  after  1 ;  the  hour  hand 
travels,  starting  from  mark  I,  a  distance  n,  while  the  minute 
hand,  starting  from  mark  XII  five  minutes  further  back,  has 
to  travel  5  +  w  in  order  to  catch  it  up ;  so  the  relative  speeds 
of  the  two  hands  are  as  (w.  +  5)  :  w,  and  are  also  as  12  :  1 ; 
wherefore  7j,  +  5       12 


or  12w  =  71  +  5,  • 

or  Wn  =  5, 

5 

"  =  n' 

and  so  the  time  required  (or  the  answer)  is  five  minutes  and 
five  elevenths  of  a  minute  {i.e.  yy  hour)  past  one  o'clock. 

Take  another  question. — Start  with  a  clock  face  indicating 
9  o'clock,  and  ask  when  the  hands  will  for  the  first  time  be 
superposed. 

The  slow-moving  hand  has  forty-five  minutes'  start;  so,  how- 
ever many  minutes  it  goes,  the  quick  one  has  to  go  45  minutes 
more,  at  twelve  times  the  pace.  Wherefore  .t  -f  45  =  1 2aj,  or 
the  meeting  point  is  yy  =  4y\  minutes  after  the  mark  IX; 
or  ^fths  of  45  minutes,  i.e.  j^ths  of  an  hour,  since  9  o'clock. 
The  start  in  this  case  is  nine  times  as  great  as  was  allowed 
after  one  o'clock,  in  the  previous  question,  and  accordingly  the 
distance  before  overtaking  occurs  is  likewise  nine  times  as 
great :  in  accordance  with  common  sense. 


XI.]  PROBLEMS.  Ill 

The  constant  occurrence  of  11  in  such  sums  shows  that  11 
must  have  a  decipherable  meaning :  it  means  the  excess  pace, 
or  relative  velocity,  of  the  quick  hand  over  the  slow.  And 
when  this  has  been  perceived,  the  easiest  way  to  do  such 
sums  in  the  head  is  self -suggested,  viz.  to  treat  it  as  a  case  of 
relative  velocities,  with  the  hour  hand  stationary,  and  simply 
ask  how  soon  the  minute  hand  will  move  to  where  the  hour 
hand  was^  if  it  (the  minute  hand)  went  at  -54ths  of  its  real 
speed. 

The  interval  between  successive  overlaps  is  therefore  always 
yfths  of  an  hour,  or  65^^  minutes. 

Exercise. — The  hands  make  a  straight  line  at  6  o'clock, 
when  will  they  be  at  right  angles  1  Ans. :  One  has  to  gain 
relatively  15  minutes  on  the  other,  and  since  its  relative 
speed  is  jjths  of  an  hour  per  hour,  the  time  required  is 
15x|f  minutes,  that  is  to  say  lj\-  minutes  more  than  a 
quarter  of  an  hour. 

Pains  should  always  be  taken  to  express  an  answer  com- 
pletely and  intelligibly.  If  any  joy  is  taken  in  work,  it 
should  be  decorated  and  embroidered,  so  to  speak,  not  left 
with  a  minimum  of  bare  necessity. 

Moreover,  never  let  it  be  taught  (as  Todhunter  taught) 
that  the  x  or  other  symbol  so  employed  is  always  necessarily 
only  a  pure  number.  When  we  say  "  let  a;  be  the  velocity  of 
the  train,"  or  "the  weight  of  the  balloon,"  etc.,  we  should 
mean  that  x  is  to  stand  for  the  actual  velocity,  the  actual 
weight:  however  they  be  numerically  specified.  (Appendix  11.) 

Some  teachers  of  importance  will  demur  to  this.  I  assert 
with  absolute  conviction  that  it  is  the  right  plan,  and  will 
justify  it  hereafter.  But  it  is  a  matter  for  adults  to  consider, 
and  is  only  incidentally  mentioned  here. 

The  dislike  felt  by  teachers  of  arithmetic  to  the  intro- 
duction  of   z   prematurely,   is   because  there  is   a   tendency 


112  EASY    MATHEMATICS.  [chap. 

thereafter  to  do  arithmetical  problems  so  easily  that  their 
features  are  not  grasped,  and  so  some  useful  perceptions  are 
missed.  If  this  were  a  necessary  consequence  it  would  be 
a  valid  argument  against  the  introduction  of  an  algebraic 
symbol,  but  it  is  not  a  necessary  consequence. 

For  instance,  in  examples  about  the  supply  of  a  cistern  by 
pipes,  or  the  work  of  men  per  day,  it  is  admittedly  desirable 
to  realise  that  we  are  here  often  dealing  with  the  reciprocals 
of  the  specified  quantities  ;  and  this  may  be  masked  by  the 
use  of  algebra,  possibly,  but  it  need  not.  I  suggest  that 
algebra  is  the  right  way  of  discovering  the  fact,  but  that  after 
its  discovery  the  fact  itself  may  be  properly  dwelt  on,  and 
thereafter  directly  applied.  There  is  indeed  too  much  ten- 
dency to  hurry  away  from  an  example  when  its  mere 
"answer"  has  been  obtained,  without  staying  to  extract  its 
nutriment  and  learn  all  that  it  can  teach  :  sometimes  without 
even  trying  whether  the  answer  found  will  really  fit  or 
satisfy  the  data  in  question.  That  is  altogether  bad.  The 
full  discussion  of  a  sum,  in  all  its  bearings,  after  the  answer 
is  known,  is  often  the  most  interesting  and  instructive  part 
of  the  process. 

Children  should  always  be  encouraged  to  do  this,  and  to 
invent  fresh  ways  of  putting  things,  or  detect  or  devise  a 
generalisation  of  their  own  for  any  suitable  special  case.  Here 
is  afforded  a  first  scope  for  easy  kinds  of  originality  of  a 
valuable  kind. 

Girls  especially  would  find  the  benefit  of  being  encouraged 
to  seek  the  general  under  the  mask  of  the  special.  It  seems 
to  fail  to  come  to  them  naturally. 

Illustrative  Examples,  showing  the  advantage  of  intro- 
ducing symbols  for  unknown  quantities. 

Three  pipes  supply  a  cistern  which  can  hold  144  gallons. 


XI.]  PROBLEMS.  113 

One  supplies  a  gallon  a  minute,  another  2  gallons,  and  the 
third  3  gallons  per  minute.     How  soon  will  the  cistern  be  full  1 

Let  t  be  the  number  of  minutes  before  the  cistern  is  full 
after  the  pipes  are  all  turned  on  simultaneously ;  then  in  t 
minutes  the  first  pipe  will  have  supplied  t  gallons,  the  second 
2^  gallons,  and  so  on, 
hence  t  +  'it  +  U  =  144. 

So  t  =  U. 

This  is  easy  enough,  but  I  think  even  this  is  made  easier 
by  the  introduction  of  a  symbol  for  the  unknown  quantity. 

Take  however  the  following  variation  of  the  same  problem : 

A  cistern  is  to  be  filled  by  three  pipes  labelled  A^  B,  and  G; 
Pipe  A  alone  would  fill  the  cistern  in  2  hours  24  minutes. 
Pipe  B  alone  in  1  hour  12  minutes. 
Pipe  C  alone  in  48  minutes. 
How  soon  would  they  all  three  fill  it  1 

This  form  of  statement  evidently  makes  the  problem  harder, 
and  it  is  clearly  desirable  to  simplify  it  by  ascertaining  the 
rate  of  supply  of  each  pipe.  This  can  be  done  at  once  if  we 
say,  let  n  be  the  number  of  gallons  corresponding  to  the 
contents  of  the  cistern,  for  then  the  data  give  us  that 

Pipe  A  supplies  at  the  rate  of  n  gallons  in  144  minutes 

or  — —  gallons  per  minute, 

B  supplies  at  the  rate  --  gallons  per  minute, 
72 

and  C  supplies  at  the  rate  --  gallons  per  minute. 

48 

So  the  set  of  pipes  together  supply,  at  the  combined  rate, 
n        n       n   _  n 
l44"^72'^48  ^  T 
that  is  to  say,  n  gallons  in  the  unknown  time  t,  which  time  is 
the  thing  to  be  found. 

L.E.M.  H 


114  EASY  MATHEMATICS.  [chap. 

We  now  see  that  the  contents  of  the  cistern  is  immaterial, 
when  the  data  are  thus  specified,  for  n  cancels  out  of  the 
equation,  and  leaves  us  with  the  relation 
11       1        1 
t  ~  48  "^72  "^144" 

We  have  thus  discovered  the  mode  of  dealing  with  problems 
of  this  kind,  viz.  to  take  the  reciprocals  of  the  times  given. 
In  other  words,  to  say  that  the  rate  of  supply  is  inversely  as 
the  time  taken,  or  that  it  is  proportional  to  the  reciprocal  of 
that  time ;  and  hence,  writing  the  combined  rate  as  the  sum 
of  the  rates,  we  get  the  equation  directly  as  last  written. 

Now  it  is  true  that  a  mathematician  would  have  seen  this 
at  once,  and  written  the  equation  as  above  without  appearing 
to  think  about  it;  but  a  child  cannot  be  expected  to  think 
out  such  a  relation,  at  least  not  for  a  long  time,  unless  he  is 
encouraged  to  consider,  either  tacitly  or  explicitly,  the 
contents  of  the  cistern ;  when  it  at  once  becomes,  not  exactly 
easy  but,  possible. 

The  above  equation  may  be  called  "the  solution"  of  the 
problem,  so  far  as  it  involves  reasoning  or  thought;  the 
subsequent  arithmetical  working  necessary  to  obtain  a 
numerical  result  is  comparatively  mechanical,  but  it  should 
not  be  omitted. 

J_    J^       1     ^  3  +  2  +  1  ^6^1 
48  ^72  "^144"       144      ~  144  ~  24' 
This  is  the  reciprocal  of  the  time ;  and  thus  the  time  required 
for  the  conjoint  filling  is  24  minutes,  as  we  found  in  the  first 
or  easy  mode  of  statement,  where  the  rates  were  explicitly 
specified  among  the  data. 

Another  question  of  the  same  kind :  If  A  can  build  a  wall 
in  30  days,  B  in  40  days,  and  C  in  50  days,  how  soon  can 
they  all  build  it,  if  they  can  all  work  together  without 
interfering  with  each  other  | 


XL]  PROBLEMS.  115 

Answer  in  x  days,  where 

-1     i-     i_  =  1- 

30'^40"^50       x' 

because,  during  each  day,  A  does  ^^^th  of  the  wall,  B  does 
^th,  and  G  does  ^^^th ;  so  the  three  together  do,  each  day, 
what  is  represented  by  these  fractions  added  together.  Hence 
the  number  of  days  will  be  the  reciprocal  of  the  sum  of  these 
fractions. 

It  is  probably  undesirable  to  assist  a  beginner  to  so  easy  a 
solution  of  this  class  of  problem  prematurely,  or  until  he  has 
been  afforded  an  opportunity  of  expending  some  thought 
upon  it ;  for  it  is  difficult  to  get  a  good  grip  of  a  thing  which 
is  too  smooth  and  slippery. 


CHAPTER   XII. 

Involution  and  evolution  and  beginning  of  indices. 

Because  6x6=  36,  which  may  be  called  6^  (six  square), 
and         6x6x6=     216,  and  may  be  called  6^  (six  cube), 

6x6x6x6  =  1296  =  64,  (six  to  the  fourth  power), 
and  so  on, 

it  is  customary  to  call  6  the  square  root  of      36 ; 
it  is  also  the  cube  root  of    216, 
the  fourth  root  of  1296, 
and  so  on ; 

and  the  process  of  finding,  or  extracting,  the  root  of  any 
number  is  called  evolution, — though  the  name  is  of  small 
importance. 

The  idea  of  roots  and  powers  however  is  of  great  importance 
and  it  is  necessary  to  know  how  to  find  them. 

The  square  root  of  49  is  7 ;  as  we  know  from  the  multipli- 
cation table.  So  also  we  know  in  the  same  way,  that  is  by 
direct  experiment,  that  the  square  root  of  64  is  8 ;  because 
this  is  only  another  way  of  saying  that  8  square,  8^,  or  8  x  8, 
equals  64. 

The  statement  that  9^  =  81  is  identical,  in  everything  except 
in  form,  with  the  statement  that  the  square  root  of  81  is  9. 
The  square  root  of  100  is  10, 
that  of  144  is  12, 
and  of  400  is  20. 


CHAP.  XII.]  POWERS  AND  ROOTS.  117 

A  notation  or  mode  of  writing  is  necessary  for  roots,  to 
avoid  having  constantly  to  write  words,  and  for  compactness ; 
just  as  3  is  handier  to  deal  with  than  "three,"  though  it 
means  the  same  thing. 

The  notation  employed  in  involution  or  raising  to  powers 
we  have  already  stated  (p.  56),  viz.  little  figures  or  indices 
placed  after  the  main  figure,  as  for  instance  4^  =  1 6,  the  index 
denoting  how  many  fours  are  to  be  multiplied  together. 

So  6^  means  that  three  sixes  are  to  he  multiplied  together ; 
and  that  is  all  that  the  index  shows. 

9^  means  that  five  nines  are  to  be  multiplied  together ;  and 
the  result  is  a  big  number,  which  a  child  may  at  once  be  set  to 
calculate.  He  might  also  calculate  such  numbers  as  2^,  3^,  4^, 
55,  ...,  99,  1010. 

Moreover,  he  should  at  once  write  down  the  values  of  the 
following:  102,  10M0M0M0«,  .... 

and  perceive  that  in  each  case  the  number  of  ciphers  following 
the  one  is  indicated  by  the  index.  So  he  can  write  down  in 
full  10^^  without  consideration,  and  can  be  told  that  the  short 
form  is  a  compact  and  handy  and  universally  adopted  method 
of  expressing  large  numbers. 

From  all  this,  if  a  sharp  child  were  asked  to  invent  a 
notation  for  roots,  he  might  perhaps,  though  it  is  much  to 
expect  if  really  ignorant  of  the  convention,  but  he  might 
suggest  that  since  4^  =  16,  perhaps  16^  =  4;  or  perhaps 
he  might  suggest  16 "^  as  a  suitable  notation.  In  either 
case  he  should  be  much  encouraged. 

Of  the  two  notations,  thus  suggested,  the  first  is  correct 
and  is  employed.  The  second  is  employed  for  something 
also,  but  for  something  totally  different  from  a  root,  viz.  a 
reciprocal. 

Let  us  get  used  to  the   notation  for  roots  by  fractional 


118  EASY   MATHEMATICS.  [chap. 

indices,  and  at  the  same  time  justify  it  as  a  consistent  and 
convenient  method. 

First  of  all  it  must  be  admitted  as  not  easy  to  put  into 
words.  The  index  2  signifies  that  the  number  is  to  be  raised 
to  the  second  power,  or  multiplied  by  itself,  so  that 

42  =  4x4  =  16; 
hence  we  might  say  that  16^  means  that  the  number  is  to  be 
raised  to  the  half  power,  or  multiplied, — how  *?     It  is  hardly 
an  interpretable  phrase ;  so  we  must  proceed  more  gradually. 

First  of  all,  it  is  simple  to  suppose  that  if  the  index  is  unity 
it  should  be  understood  to  leave  the  figure  unaltered,  so  that 

41  ==  4   and    I61  =  16 ; 
therefore  we  may  write  indices  on  both  sides,  thus,  16^  =  4^; 
let  us  next  suppose  that  we  may  halve  the  index  on  each  side 
getting  16^  =  4^,  and  read  this,  root  16  equals  4.     We  might 
halve  the  indices  again,  and  get  16*  =  4*;  which  equals  the 
square  root  of  four,  or  2 ;  so  that  we  may  surmise  that  the 
fourth  root  of  16  is  2  ;  and  verify  it  thus, 
2x2x2x2  =  16. 
Similarly,  27  =  33, 

27^  =  31, 
which  agrees  with  the  fact  that  the  cube  root  of  27  is  3, 
(27  =  3  X  3  X  3). 
Again,  81^  =  9, 

8ii  =  9*  =  3. 

All  that  we  have  here  assumed  (and  it  is  a  large  assumption) 
is  that  in  an  equation  involving  terms  with  indices,  if  we 
perform  an  operation  on  the  indices — provided  we  perform 
the  same  operation  on  both  sides, — the  equality  remains 
undisturbed. 

This  is  an  assumption,  a  guess,  an  expeetation,  to  be  justified 
or  contradicted  experimentally  by  results.     We  shall  find  that 


Xii.l  POWERS  AND  ROOTS.  119 

its  truth  depends  entirely  on  the  kind  of  operation  so  per- 
formed. We  happen  to  have  hit  first  upon  trying  multipli- 
cation and  division  as  applied  to  indices,  and  that  seems  to 
work  correctly.  But  we  shall  try  other  operations  shortly 
and  will  find  them  fail. 

Those  who  imagine  or  assert  that  experiment  has  no  place 
in  mathematics  do  not  know  anything  about  mathematics. 
Sometimes  results  are  arrived  at  by  theory,  sometimes  by 
experiment,  sometimes  by  a  mixture  of  the  two ; — either 
theory  first  and  confirmation  by  experiment,  or  experiment 
first  and  justification  by  theory :  just  as  in  Physics  or  any 
other  developed  science. 

Let  us  now  press  our  assumption  to  extremes  and  experi- 
ment on  it  in  various  ways  so  as  to  see  whither  it  will  lead  us. 
Start  with  any  equation,  such  as 

42  =  161. 

Double  each  index,  and  we  get 

44  ^  162  ^  256. 

So  the  fourth  root  of  256  is  given  as  4,  and  the  eighth  root 
will  accordingly  be  2,  or  256  is  asserted  to  be  the  eighth 
power  of  two ;  which  is  the  fact :  eight  twos  multiplied 
together  do  yield  256. 

Treble  the  index,  and  it  becomes 

46  =  163  =  256  X  16  =  4096, 
or  conversely,  4096^  =  4. 

Hence  the  sixth  root  of  4096  is  given  as  4,  and 
4096tV  =  4*  =  2, 
that  is,  its  twelfth  root  is  2.     Again  fact  agrees  with  theory. 
2  multiplied  by  itself  12  times  does  equal  4096. 

Hence  it  appears  that  the  operation  of  multiplying  indices 
by  any  the  same  factor  on  each  side  of  an  equation  may  be 
trusted  to  give  true  results. 


120  EASY  MATHEMATICS.  [chap. 

So  also  division  of  indices  by  any  the  same  number  may  be 
trusted  too ;  thus  starting  as  before  with 
161  =  42^ 

quarter  each  index  16^  =  4^  =  2, 

halve  each  again  16^  =  4^  =  2^. 

How  are  we  to  calculate  2^  1  That  is  not  an  easy  matter : 
we  will  leave  it  unvalued  for  a  time  and  merely  call  it  the 
square  root  of  2.  It  is  often  denoted  by  a  sort  of  badly- 
written  long-tailed  -^  in  front  of  the  digit,  thus,  4/'!  or  ^2. 

Try  again,  27  =  33, 

27*  =  3, 

27*  =  3*, 

there  is  the  same  difficulty  about  interpreting  3^ ;  no  whole 

number  will  serve.     We  can  call  it  the  square  root  of  3,  or 

briefly  "  root  3,"  and  can  denote  it  by  writing  ^3  as  before. 

^16  means  the  same  as  16^,  namely  4 ;  and  ^4  =  4^  =  2 ; 
but  whereas  the  fractional  index  contains  an  important  and 
valuable  idea,  which  remains  to  be  developed,  the  symbol  J 
is  nothing  but  shorthand  for  the  word  "root,"  and  is  itself 
trivial  and  inexpressive,  though  quite  harmless  and  of  constant 
service. 

What  we  have  learnt  from  the  above  examples  resulting  in 
^2  and  ^3  is  that  when  employing  fractional  indices  we  can 
arrive  at  something,  easy  of  interpretation  indeed,  but  not 
easy  of  numerical  evaluation ;  there  is  no  need  to  mistrust  the 
result  but  only  to  wait  till  more  light  can  fall  upon  it. 

Now  try  some  other  operations  applied  to  indices,  we  shall 
find  that  wariness  is  necessary,  and  that  mere  guesses  and 
surmises  as  to  what  it  is  permissible  to  do  to  equations  are 
not  worth  much.  Everything  must  be  tested.  Suppose  we 
try  squaring  them  on  both  sides  as  thus  :  Starting  with 

42  =  161, 


xii.]  POWERS  AND  ROOTS.  121 

squaring  indices  would  give  us 

44  ^  161, 
since  the  square,  or  any  other  power,  of  1  is  1, 

1x1x1x1  =  1. 
The  result,  that  16  is  both  the  square  and  the  fourth  power 
of  4,  is  false  and  absurd :    and  hence  the  sham   equation  is 
erased. 

So  we  learn  that  whereas  multiplication  of  indices  by  any 
factor  is  an  operation  that  can  be  trusted  to  give  true  results, 
and  division  of  indices  by  a  factor  can  probably  be  trusted 
too,  since  one  operation  is  the  inverse  of  the  other,  yet  that 
involution  is  not  an  operation  that  can  legitimately  be  per- 
formed upon  indices,  but  only  upon  the  numbers  themselves. 

Suppose  we  try  addition,  equal  additions  to  the  indices  on 
each  side;  add  1  for  instance,  we  get  4^  ^  16^,  which  is  a 
falsehood  if  the  equality  sign  is  left  unerased. 

It  is  time  we  began  to  consider  what  operations  are  really 
legitimate  and  what  are  not ;  and  gradually  in  both  cases  we 
must  proceed  to  ask.  Why  ? 


CHAPTER  XIII. 

Equations  (treated  by  the  method  of  very  elementary  experiment). 

It  is  therefore  convenient  at  this  stage  to  introduce  the  idea 
of  an  expressed  equality,  which  is  called  an  equation,  and  to 
consider  what  are  the  operations  to  which  an  equation  can  be 
subjected  without  destroying  the  equality. 

It  is  customary  to  postpone  this  subject  to  Algebra,  but  we  do 
not  wish  to  perpetuate  any  sharp  distinction  between  algebra 
and  arithmetic,  and  it  is  useful  to  begin  experimenting  with 
equations  while  still  they  are  expressed  in  terms  familiar  to 
beginners. 

Typical  equations  are  of  many  kinds,  of  which  we  may 
now  consider  the  following  : 

The  addition  kind,  3  +  2  =  5. 

The  subtraction  kind,         3-2  =  1. 

The  multiplication  kind,    3x2  =  6. 

The  division  kind,  3  -r  2  =  1-5. 

The  involution  kind,  3^  =  9. 

The  evolution  kind,  9*  =  3. 

There  are  plenty  of  others,  but  these  will  do  to  begin  with. 
Every  equation  has  two  sides,  called  respectively  the  left-hand 
side  and  the  right-hand  side;  the  symbol  =  is  the  barrier. 
It  is  not  an  impassable  barrier,  but  terms  get  reversed  when 
they  are  taken  across  it ;  positive  becomes  negative,  and  vice 
versa.  In  order  to  find  out  what  may  be  done  to  equations 
we  can  experiment. 


CHAP.  XIII.]        EXPERIMENTS  ON  EQUATIONS.  123 

Take  any  of  these  equations  and  try  experiments  on  it. 
For  instance,  add  something  to  or  subtract  something  from 
each  side.     So  long  as  we  add  the  same  thing  to  each  side  no 
harm  is  done  :  the  equality  persists.     For  instance,  start  with 
the  first  two  of  the  above  equations  and  modify  them  by 
addition  or  subtraction  in  various  ways  : — 
3  +  2  +  7  =  5  +  7  =  12; 
3-2-1  =  1-1=0; 
3  +  2-6  =  5-6=   -I, 
3-2  +  ft  =  l+a, 
x-h3-  2-«  =  x-a  +  lj 
3-2  +  2  =  1+2  =  3, 
3  +  2  +  1  =  51 
So  far  everything  is  very  simple  and  safe. 
Not  only  may  we  add  the  same  thing  to  each  side,  but  we 
may  add  equal  things  to  each  side  (which  may  be  regarded  as 
an  illustration  of  the  axiom,  that  if  equals  be  added  to  equals 
the  wholes  are  equal). 

Thus  3  +  2  =     5,| 

and  also  7  +  6  =  13.  J 

So  3  +  2  +  7  +  6  =  5  +  13  =  18. 

Or  again,  3-2 

and  32 

So  3  -2  +  32  =1+9  =  10. 

Also  take  the  following  : 

3-2  =  1,      1 
and  5-6  =   -l;j 

.-.    3-2  +  5-6  =  1-1=0. 
But  take  an  equation  of  the  multiplication  kind, 
3x2  =  6, 
a  little  caution  is  necessary  in  adding  anything  to  the  left- 
hand  side. 


:l-} 


124  EASY  MATHEMATICS.  [chap. 

We  might  have     (3  x  2)  +  1  =  6  +  1  =  7, 
or  we  might  have      3  x  (2  +  1)  =  3  x  3  =  9. 

If  we  only  write  3x2  +  1,  without  brackets,  it  is  ambiguous  ; 
for  the  value  depends  on  whether  the  addition  or  the  multi- 
plication is  performed  first:  that  is,  on  whether  the  2  is 
grouped  along  with  the  3  or  with  the  1,  but  the  brackets 
enable  us  to  indicate  the  grouping  clearly. 

Take  another  example, 

7  X  8  =  56, 
(7  X  8)  -  4  =  52, 
although  7  X  (8  -  4)  =  28  ; 

but  the  last  is  quite  a  different  equation,  and  is  not  deduced 
by  simple  subtraction  of  4  from  both  sides. 

About  the  other  forms  of  equations  there  is  no  difficulty  ; 
we  will  just  write  them,  with  something  either  added  to  or 
subtracted  from  each  side  : 

1-1  =  1-5-1  =  -5  =  I, 
32  -  1  =  9  -  1  =  8, 
9^+2  =  3  +  2  =  5. 

Incidentally  we  here  observe  the  advantage  of  the  fractional 
notation  over  the  +  notation.  If  we  had  written  3  +  2-1  we 
should  have  had  to  avoid  ambiguity  by  the  use  of  brackets,  as 
was  necessary  in  multiplication ;  but  f  -  1  is  unambiguous. 
Unity  is  subtracted  from  the  whole  fraction,  not  from  either 
numerator  or  denominator.  If  unity  were  subtracted  from 
the  numerator  it  would  not  be  right, 

^^1-5-1  =  -5; 

nor  will  it  do  to  subtract  from  the  denominator,  nor  from  both. 

So  much  at  present  for  addition  and  subtraction ;  now  try 
multiplication  and  division  :  start  with 
3  +  2  =  5; 


XIII.]  EXPERIMENTS  ON  EQUATIONS.  125 

double  each  terra,  6  +  4  =  10  ; 

treble  each  term,  9  +  6  =  15; 

halve  each  of  these  terms, 

41  +  3  =  7i. 
So  here  we  are  safe. 

Proceed  now  to  the  factor  or  multiplication  form  of  equation : 
3x2  =  6; 
double  each  digit,  6x4  ^  12, 

and  we  get  wrong. 

We  learn  that  we  must  not  double  each  factor  in  a  product, 
though  we  must  double  each  term  of  a  sum  ;  hence  the  expres- 
sion 3  +  2  is  commonly  spoken  of  as  containing  two  terms,  but 
3  X  2  is  spoken  of  as  a  single  term. 

To  double  the  single  term  it  is  sufficient  to  double  one  of  its 
factors  ;  so  if  we  write 

6x2  =  12 
we  get  right  again. 

Similarly  we  must  halve  one  of  the  factors  only, 
11  X  2  =  3, 
or  else  3x1  =  3. 

Now  attend  to  the  quotient  form, 

double  every  digit,  f  ^  3, 

and  we  get  wrong. 

Double  the  denominator  only, 

it  is  still  wrong. 

Double  the  numerator  only,  f  =  3, 

and  we  get  right. 
^    So  in  multiplication  and  division  of  a  quotient  by  a  whole 
*  number,    the    factor    has    to   be   applied   to   the   numerator 

only. 


126  EASY   MATHEMATICS.  [chap. 

Take  another  example,       ^  =  4, 

6  _  4    _    9 

3  _  4    _    1 
^-T   -    ^» 

-V-  =  4x2  =  8. 
Finally,  take  the  involution  form, 

32  =  9. 
What  are  we  now  to  double  if  we  want  to  double  both 
sides  1  62  =  18  is  wrong, 

3^  ==  18  is  also  wrong. 
We  cannot  do  it  quite  so  simply ;  so  we  must  write  merely 

2  X  32  =  18, 

which  leaves  the  step  really  undone  and  only  indicated. 
But  take  another  example, 

3  X  32  =  27. 
This  could  be  written  3^ 

Again,  33  x  32  =  27  x  9  =  243  =  3^, 

and  a  rule  of  extraordinary  interest  and  usefulness  is  suggested. 
Think  it  over,  we  shall  return  to  it  in  Chapter  XVI. 

Further  consideration  of  what  can  be  done  to  equa- 
tions. 

A  sentence  like  the  following : 

"  If  both  sides  of  an  equation  be  treated  alike,  the  equality 
will  persist,"  might  easily  be  considered  axiomatic  ;  but  so 
much  caution  is  required  before  we  can  be  sure  that  both 
sides  have  been  really  treated  alike,  that  it  is  highly  dangerous 
to  employ  such  an  axiom.  We  have  already  come  across 
some  cases  of  the  danger,  but  the  subject  is  very  important 
and  will  bear  fuller  treatment. 

The  general  doctrine  may  be  laid  down  that  before  we 
understand  properly  what  can  be  done,  or  what  it  is  per- 
missible to  do,  in  any  subject  whatever,  we  should  take  pains 


XIII.]  EXPERIMENTS  ON  EQUATIONS.  127 

to  ascertain  also  what  cannot  be  done  under  the  same  circum- 
stances, i.e.  what  it  is  not  possible  to  do  without  error. 
This  latter  part  should  not  be  too  long  dwelt  upon,  because 
error  is  most  simply  excluded  by  attention  to  and  familiarity 
with  the  correct  processes,  so  that  presently  all  others  instinc- 
tively feel  wrong ;  but  once  at  least  we  should  examine  the 
whole  matter,  and  learn,  if  we  can,  why  one  set  of  things  are 
wrong  and  another  set  right.  This  remark  applies  also  to 
other  things  than  arithmetic. 

An  equation  consists  of  two  sides,  and  each  side  consists  of 
terms.  Frequently  the  right  hand  side  is  zero,  especially  in 
algebra  and  in  higher  mathematics.  Sometimes,  instead  of 
being  zero,  it  is  some  constant  or  other  independent  quantity, 
and  is  called  "  the  absolute  term,"  because  it  is  undetermined 
by  anything  on  the  left  hand  side :  to  which  however  it  is 
equated. 

An  equation  is  the  most  serious  and  important  thing  in 
mathematics.  The  assertion  that  two  quantities  or  two  sets 
of  quantities  are  equal  to  each  other,  whether  it  is  meant 
that  they  are  always  equal,  or  only  that  they  are  equal 
under  certain  circumstances  which  have  to  be  specified,  is  a 
very  definite  assertion  and  may  carry  with  it  extraordinary 
and  at  first  unsuspected  consequences. 

The  equations  we  are  now  using  as  illustrations  are  by  no 
means  of  this  high  character ;  they  are  usually  mere  identities, 
and  depend  on  the  truism  that  a  combination  of  things  grouped 
or  expressed  in  one  way  are  unchanged  in  number  when 
grouped  or  expressed  in  another  way."*^     But  although  it  may 


*  We  call  this  a  truism ;  but  it  is  a  dangerous  term  to  employ,  and 
when  we  come  to  Chemistry  we  must  be  on  our  guard  against  assuming 
that  the  volume  of  Hg  +  O  is  equal  to  that  of  HgO,  under  the  same 
external  circumstances.  It  is  true  of  weights  (as  nearly  as  we  can  tell) 
but  it  is  not  even  approximately  true  of  volumes, 


128  EASY   MATHEMATICS.  [chap. 

be  some  time  before  they  realise  the  vast  importance  attaching 
to  equations,  children  will  take  it  on  trust  that  they  are  now 
entering  the  central  arcana  of  the  subject,  and  will  be  willing 
to  give  the  needful  attention  to  the  processes  which  have 
constantly  to  be  employed.  An  initial  account  of  them  is 
given  in  the  following  chapter,  parts  of  which  may  be  read 
before  the  whole  of  the  following  introductory  matter. 

When  a  number  of  quantities  are  multiplied  together,  they 
are  held  to  constitute  one  term.  Whenever  the  sign  +  or  - 
intervenes,  it  interrupts  the  term,  and  each  such  sign  has  a 
term  on  either  side  of  it. 

Thus  a  +  bj  70-6,  are  each  two  terms ;  but  ab,  and  70  x  6, 

and  abc,  and  10^,  and  5aJ'2,  and  are  all  single  terms. 

X 

What  about  such  an  expression  as 
5ab-x 
abx 
where   there   is   one  (or  more  than   one)   addition   or   sub- 
traction sign  in  the  numerator  ? 

Answer :  So  long  as  it  is  kept  all  together  it  can  be  called 
one  term,  but  it  can  easily  be  split  into  two,  viz. 

X     ab^ 
and   for  some  purposes  its  terms  can  be  considered  plural 
without  re-writing. 

The  long  line  of  division  in  the  original  expression  however 
may  be  held  to  weld  the  whole  into  one  term ;  and  brackets 
have  the  same  effect.     Thus, 

(a  +  b),  (70-6),  5{a  +  b)x,  J  (ax -by) 
are    all   single   terms    once   more ;     until    the    brackets   are 
removed.      And  removal  of  brackets  is  an  operation  to  be 
performed  cautiously.     Rubbing  them  out  is  not  a  legitimate 
way  of  removing  them. 


XIII.]  EXPERIMENTS  ON  EQUATIONS.  129 

For  instance,  3(7-4)=  9; 

but  37-4,  and  3x7-4,  and  3-7-4 

are  all  different. 

Again,  ^(16  +  9)  =  5, 

but  V16+V9  =  7, 

and  ^^6+    9  =  13, 

while  16+^9  =  19; 

the  three  are  entirely  distinct  statements  from  the  first,  and 
are  not  deducible  from  it. 

So  we  learn  that  the  right  removal  of  brackets  is  a  matter 
to  be  studied. 

When  we  assert  that  the  same  operation  can  properly  be 
applied  to  each  side  of  an  equation  then,  we  must  be  careful  to 
interpret  it  always  as  an  operation  applied  to  the  whole  side, 
and  not  to  any  part  of  it.  We  may  not  tamper  with  one 
term  and  leave  the  others  alone,  nor  must  we  tamper  with  a 
part  of  a  term  only.  Nor  must  we  repeat  the  operation  for 
each  of  the  factor  components  of  a  single  term. 

This  must  be  illustrated  : 

Given  that  a  +  b  =  c, 

it  is  correct  to  say  that 

2a  +  2b  =  2c, 
or  that  2(«  +  &)=  2c; 

but  given  that  axb  =  c, 

it  is  not  correct  to  say  that 

2a  X  2b  =  2c. 
For  here  the  term  ab  is  one,  and  it  only  needs  doubling  once. 

Given  that  a^  =  J^, 

it  is  true  that  5^2  =  553^  (1) 

but  it  is  not  true  that         (5a)2  =  (55)3, (2) 

for  that  would  mean  26a^  =  125b^, 

which,  subject  to  the  given  data,  is  absurd,  unless  a  and  h  are 

both  zero. 

L.E.M.  I 


130  EASY  MATHEMATICS.  [chap. 

In  reading  the  two  lines  labelled  (1)  and  (2)  it  is  customary 
to  read  them  carefully  in  order  to  discriminate  what  otherwise 
would  sound  quite  similar.  The  former  of  the  two  lines  is 
read  t(  five  a-square  =  five  b-cnhe  " ; 

the  latter  of  the  two  lines  is  read 

"  five  a,  squared  =  five  b,  cubed  " ; 
and  these  are  quite  different.     They  cannot  under  any  circum- 
stances be  both  true  (unless  indeed  a  and  b  are  both  zero). 
They  are  therefore  called  "  inconsistent "  equations 

(like  X  =  y,  and  x  =  2y,  which  cannot  both  be  true). 

To  illustrate  the  inconsistency,  take  an  example  : 
82  =  43,  both  being  64, 
and  so  also         5  x  8^  =  5  x  4^,  both  being  320, 
but  (5  X  8)2  ^  (5  X  4)8,  the  one  being  1600 

and  the  other  8000. 
Read  the  sign  ^  as  "  does  not  equal. " 

Given  again  that  a^  =  6^, 

it  does  not  follow  that  a^  =  b\ 

although  we  have  done  the  same  thing,  that  is  added  1,  to  the 
index  on  each  side. 

Nor  would  it  be  true  to  say  that 

although  we  have  now  squared  the  index  on  each  side. 

But  it  does  turn  out  true  that  if  we  double  or  treble  the 
index,  the  equality  persists  :  given  that  a^  =  b^ 
it  is  true  that  a^  =  ¥ 

and  that  a^  =  b^, 

so  that  it  appears  as  if  it  were  permissible  to  multiply  the 
index  on  each  side  by  any  the  same  factor.  We  must  examine 
this  later,  but  at  present  we  will  merely  verify  the  truth  of 
these  last  assertions  by  an  arithmetical  example  : 


XIII.]  EXPERIMENTS  ON  EQUATIONS.  131 

For  instance  8^  =  4^, 

whereas  8^  ^  4^ 

one  being  512,  and  the  other  256 ; 
but  84  =  46, 

both  being  (64)2  ^r  4096. 

Likewise  8^  =  4^, 

both  being  (64)3  or  262,144. 

Now  take  a  slightly  more  general  type  of  equation. 
ax  =  hy^ 
it  follows  that  (aa;)^  =  (5y)^, 

but  it  is  by  no  means  necessary  that  aa:^  si^^ll  equal  li^. 

For  instance,  7x4  =  14x2, 

and  (7x4)3  =  (14x2)3; 

but  7x43^14x23, 

for  one  equals  448,  and  the  other  equals  112. 

Take,  as  given,  the  equation  db  —  xy,  and  let  us  multiply, 
add,  and  divide  on  both  sides,  so  as  to  illustrate  legitimate 
and  illegitimate  operations;  the  pupil  being  left  to  devise 
numerical  illustrations  and  tests  for  himself. 

First  multiply  or  divide  by  any  quantity  whatever,  say  c. 
abc  =  cxy ; 
or,  assuming  another  quantity  z  =  c,  we  may  write  it 
abc  =  xyz. 

So  also  ^  =  % 

c        z 

axb-^c  =  xxy~z  =  yxx-i-z, 

-(ao)  =  -xo  =  ax-  =  xx-  =  yx-  =  ~(xy). 
c^    ^      c  c  z      ^     z      z^  ^' 

Next  add  or  subtract  something  to  or  from  each  side. 

ab-\-c  =  xy  +  c, 

ab-b  =  xy-b  ^  xy-y, 

ab-x  =  xy-x, 

ab-xy  =  xy-xy  =  0. 


132  EASY  MATHEMATICS.  [chap. 

This  last  is  worth  attention.  The  result  has  been  to 
transfer  a  term  from  one  side  of  the  equation  to  the  other, 
its  sign  being  changed  in  the  process.  This  is  important  and 
demands  further  illustration. 

Let  JB  =  6. 

We  can  equally  well  write  it,  by  subtracting  6  from  both 
sides,  a;  -  6  =  0, 

and  the  6  has  been  transferred,  with  change  of  sign. 

Or  let  X  =  y. 

Subtract  y  from  both  sides, 
then  x-y  —  0. 

Again  let  a;  =  -  y, 

then  add  y  to  both  sides  and  we  get 
x  +  y  =  0. 

Or  let  ax  +  by  =  -Cj 

add  c  to  both  sides,  ax  +  by  +  c  =  0. 

This  kind  of  simple  operation  has  constantly  to  be  per- 
formed. 

One  more  illustration  therefore: 
Let  ax-{-by  —  cx  +  dy. 

We  can  subtract  the  right  hand  side  from  both  sides ;  in 
other  words,  transfer  it  to  the  left,  with  change  of  sign; 
getting  ax  +  by-cx-  dy  =  0, 

which  is  more  neatly  written 

{a-c)x  +  {b-d)y  —  0; 
or  again,  (a-c)x  =  {d-  b)y. 

Here  the  last  mode  of  expression  is  deserving  of  attention. 
We  will  arrive  at  it  more  directly. 

To  this  end  start  again  with 

ax  +  by  =  cx  +  dy; 
transfer  by  to  the  right,  and  ex  to  the  left ;  thus  we  get 
ax -ex  =  dy-by, 


XIII.  ]  EXPERIMENTS  ON  EQUATIONS.  1^3 

or  what  is  the  same  thing, 

(a-c)x  =  (d-b)y. 
Divide   each   aide/^y   the   product  {a-c)(d-b)  and   the 
equation  becomes  /  \ 

/(a-c)x      _      (d-  l)y 
(a-c){d-h)  ~  (a-c)(d'-by 
In   each   of   these    terms   there   is    a  common   factor   in 
numerator  and  denominator,  so  we  can  cancel  them,  and  are 

left  with  -^  =  -y.. 

d-o      a-c 

Or  we  might  have  divided  otherwise,  and  arrived  at  any  of 

the  following : 

X  _  d-b 

y  ~  a-c 

x:y  =  (d-b):{a-c), 

y:x  =  (a-c):{d-b\ 

d-b 
X  = y, 

a-c 

(i  (a-c)x  _  - 

(a-c)x        _ 
{d-b)y     '-"' 

^1^-2^  =  0 
d-b    X 

All  these  are  entirely  equivalent  forms ;  and  as  an  exercise 
they  should  all  be  deduced,  any  one  from  any  other. 

And  all  can  be  numerically  illustrated,  by  attributing  to 
the  symbols  some  particular  values ;  for  instance  by  taking 
a;  =  24,  y  =  2,  a  =  5,   6  =  - 19,   c  =  1,  d  =  29. 

Let  this  be  done,  as  an  exercise. 


134  EASY  MATHEMATICS.  [chap. 

Among  things  that  can  legitimately  be  done  to  equations 
are  certain  operations  which  are  by  no  means  obvious,  and 
demand  attention. 

Suppose  we  are  told  that 

?  =  ^ (1) 

We  are  not  allowed  to  say  — ^  =  —I— ;  but  we  are  entitled 
to  say  that  ^ 

x-y  _a-h  .. 

~V~~V' ^^^ 

because  this  is   equivalent  to  subtracting   unity  from  both 
sides,  Le.  is  equivalent  to 

?-l=^l. 

y  0 

So  also  we  might  have  truly  written 

^±i'  =  ^ (3) 

But  from  the  truth  of  these  two  operations  it  follows  that 
we  might  also  have  written 

x-y_^a-h  .^. 

x  +  y      a  +  b  ^  ^ 

For  this  would  be  obtained  if  we  had  divided  each  side  of 
equation  (2)  by  the  corresponding  side  of  equation  (3) ;  for  if 
equals  be  divided  by  equals  the  quotients  are  equal. 
Let  us  illustrate  this  important  result  arithmetically. 

Start  with  -^  =  —-,  which  can  be  easily  proved  true,  and 

may  be  taken  as  corresponding  to  (1). 

Then  it  follows  that 

14  _  6      49-21     .       ,,  ,8      28       ,  .  , 

— - —  =  — — — ;   in  other  words  -  =  r^;  which 
6  21      '  6       2r 

corresponds  to  the  form  numbered  (2) ; 


XIII.]  EXPERIMENTS  ON  EQUATIONS.  135 

1      1-1     /Qx  ..i>  ^  14  +  6      49  +  21    .     ..    .  20      70. 
also,  like  (3),  that  =  — ^r — ,  *•«•  that  -g-  =  gj ' 

A  VI     fA\  ^v,  ^  14  +  6       49  +  21   ^^  ...  20       70 
and,  like  (4),  that  ^^^^-^  =  ^^-^,  or  that  _  =  -. 

Or  each  member  of  any  of   them  may  be  inverted :   for 
instance  the  last :  A  -  ?? 

20  "  70' 

Starting  once  more  with 

X  _  a 

we  might  equally  well  write  it 

1  =  1 (5) 

a      0 

for  this  is  the  result  of  multiplying  both  sides  by  ^ ;  so  there- 
fit 

fore  it  is  true  to  say  that 

a^-^  ^  IzA  (6) 

and  — —  =  ^-T-,  (7) 

a  0 

also  that  ^:i^  =  ^,  (8) 

x  +  a      y  +  b 

and  that  is  really,  though  by  no  means  obviously,  the  same 
thing  as  equation  (4). 

Illustrate  this  too,  numerically,  with  the  same  numbers  as 

^^^^^«  '-  M  =  1.      corresponds  to  (5) 

35  15 

-  -—  =  -  ^—  corresponds  to  (6) 
49  21 

where  the  minus  signs  may  equally  well  be  omitted  or  can- 
celled by  multiplying  each  side  by  -  1 ; 

35       15 
and  _  =  -^      corresponds  to  (8) 


136  EASY  MATHEMATICS.  [chap. 

or  again  any  of  them  may  be  inverted,  e.g.: 
49       21     ^ 

Now  let  us  apply  the  so-called  involutional  operations  to 
both  sides  of  an  equation,  and  ascertain  what  we  may  do  and 
what  we  may  not  do. 

Begin  with  ah  =  xy. 

Square  both  sides,         {ahY  =  {xyy. 

Square  each  factor,         a^h^  =  z^y^. 

Square  one  of  them  only,  and  we  get  wrong, 
a&2  ^  xy'^. 

Take  the  square  root  of  both  sides, 
J{ab)  =  J{xy\ 
also  of  each  factor,  Jajh  =  Jxjy, 

or  what  is  the  same  thing, 

ar¥      x^y^. 
So  far  we  are  all  right  except  in  the  one  marked  instance  : 
as  can  be  tested  by  giving  suitable  numerical  values  to  the 
four  symbols. 

But  now  take  an  equation  with  more  than  one  term  on  a 
side,  say  x  +  y  ==  c. 

Square  both  sides     (x  +  yY  =  c^. 
Square  each  term,      x^  +  y^  ^  c\ 
and  we  get  wrong. 

This  is  a  mistake  constantly  being  made  by  beginners,  and 
it  must  be  further  emphasised.     As  an  example, 
4  +  5  =  9, 
(4  +  5)2  =  92  =,  81, 
but  42  +  52  =  16  +  25  =  41  ^  92. 

The  following  fallacy  may  serve  as  an  illustration : 
^25  =  J(U  +  9),     .'.   5  =  JU+J9  =  7. 


XIII.]  EXPERIMENTS  ON  EQUATIONS.  137 

Observe  that  these  numerical  instances,  if  they  lead  to 
error,  show  quite  decidedly  that  the  operation  tested  is  wrong. 
They  do  not  prove  with  equal  validity  that  it  is  right, 
if  they  turn  out  correctly :  certainly  a  single  instance  of  correct- 
ness is  insufficient.  They  render  its  rightness  probable,  but 
the  rationale  of  it  has  to  be  further  investigated.  A  single 
instance  of  real  error  however  is  sufficient  to  invalidate  any 
operation  under  test. 

Exercises. — Test  the  correctness  of  the  following  horizontally 
juxtaposed  statements  : 


2x3  = 

6. 

2  +  3  =  5. 

22  X  32  = 

62. 

22  +  32  ^  52. 

5x6  = 

30. 

5  +  6  =  11. 

52  X  62  = 

302. 

52  +  62  i?  112. 

23  X  63  = 

123. 

23  +  63  ^  123. 
23  +  63  ^  83. 

4x9  = 

36. 

4  +  9  =  13. 

4^  X  9^  = 

36*  =  6. 

4^  +  9*  =  5. 

9x144  = 

:  1296. 

9  +  144  =  153. 

^9x^144  =  V(1296).  ^9  +  ^144  =  15. 

27  X  216  =  5832.  27  +  216  =  243. 

27^  X  216^  =  5832i  27^  +  216^  =  9. 
or  3  X  6  =  18. 
But  now  it  must  be  admitted  that  this  experimental  mode 
of  treatment  may  not  be  considered  the  best  mode  of  beginning 
the  experience  of  equations :  and  it  is  certainly  not  the  most 
conducive  to  rapid  progress;  it  may  be  better  therefore  to 
apply  treatment  like  that  of  the  present  chapter  at  a  rather 
later  stage  and  to  use  it  as  a  cautionary  and  salutary  exercise. 
The  importance  of  the  subject  is  so  great  that  it  can  hardly 
be  over- emphasised,  nor  is  one  mode  of  approach  sufficient. 
In  the  next  chapter  a  somewhat  more  orthodox  and  quite 
effective  mode  of  procedure  is  adopted. 


CHAPTER  XIV. 

Another  treatment  of  Equations. 

Equations  may  be  classified  in  various  ways:  there  are  such 
things  as  differential  equations,  there  are  quadratic  equations 
and  equations  of  the  fifth  degree,  etc.,  but  for  the  present  we 
will  classify  them  under  three  simple  heads  : — 

1st.  Statements  of  specific  or  particular  fact,  such  as : 
3  +  4  =  7, 
or  9,7(144)  =  108; 

these  involve  only  definition  and  re-grouping. 

2nd.  Statements  of  general  or  universal  truth,  such  as  : 
n^-l  =  (n  +  l){n-l) 
log  a'  =  xloga; 
these  are  called  identities,  and  are  frequently  denoted  by  a 
triple  sign  of  equality  = ,  for  instance  a  +  6  =  6  +  a,  whenever  it 
is  desired  to  emphasise  their  distinction  from  the  third  class. 

3rd.  Equations  proper,  or  statements  of  condition  or  in- 
formation such  as : 

17a;  =  34, 

or  6x^  =  40  ; 

statements  which  are  not  by  any  means  generally  true,  but 
are  only  satisfied  by  some  implicit  datum,  such  as,  in  the 
above  instances,  x  =  2. 

4th.  There  is  also,  from  this  point  of  view,  a  fourth  class  of 


CHAP.  XIV.]  EQUATIONS.  13d 

equations,  expressive  of  a  relation  between  two  quantities, 

such  as  3x  +  iy  =  12, 

or  x^  +  y^  =  25; 

which  are  satisfied  not  by  all  possible  values  of  x  and  ?/,  as 

an  identity   is  satisfied,  but  by  an   exclusive  and  definite 

though  infinite  series  of  values. 

The  first  is  satisfied,  on  a  certain  geometrical  convention,  by 
all  the  points  which  lie  on  a  specific  straight  line,  the  second 
by  all  the  points  which  lie  on  a  definite  circle. 

If  the  equations  are  given  simultaneously,  they  are  satisfied 
together  by  two  and  only  two  points,  viz.  the  points  where 
the  straight  line  cuts  the  circle. 

With  this  fourth  class  we  have  nothing  to  do  just  yet :  it 
opens  up  a  large  and  exhilarating  subject. 

With  the  first  kind  of  equation  we  have  constantly  had  to 
do  already :  all  purely  arithmetical  equations  are  necessarily  of 
this  kind. 

The  second  kind  is  constantly  encountered  throughout 
algebra  and  trigonometry ;  identities  represent  the  skeleton 
or  framework  of  mathematical  science,  all  its  universal  and 
undeniable  truths  can  be  thus  expressed. 

The  third  kind  of  equation,  or  equation  proper, — equations 
which  have  a  definite  solution,  equations  which  convey  specific 
information  about  an  unknown  quantity,  and  express  it  in 
terms  of  numbers  or  known  quantities  of  some  kind — those 
are  the  equations  with  which  we  deal  in  this  chapter,  and 
that  is  the  kind  which  gives  immediate  practical  assistance 
towards  the  solving  of  problems. 

The  process  of  "  solving  "  an  equation  is  simply  the  act  of 
reducing  it  to  its  simplest  possible  form.  Written  in  any 
form  the  equation  conveys  the  same  information,  but  in  some 
forms  it  is  not  easy  to  read ;  the  solving  of  it  is  analogous  to 


140  EASY  MATHEMATICS.  [chap. 

the  interpretation  of  a  hieroglyph  or  the  translation  of  an 
unknown  phrase. 

For  instance  the  following  equations 

13a;  =  65 

x=  b. 
all  express  the  same  fact  and  convey  the  same  information 
concerning  x,  but  the  last  obviously  conveys   it  in  simplest 
form,  and  it  is  called  the  "  solution  "  of  the  first,  the  second 
being  an  intermediate  step. 

The  two  sides  of  an  equation  may  be  likened  to  the  two 
pans  of  a  balance,  containing  equal  weights  of  different 
materials  or  of  the  same  material  differently  grouped.  It 
is  permissible  to  take  from  or  to  add  equal  quantities  to 
both  pans,  the  balance  or  equality  being  still  preserved ;  but 
a  weight  must  not  be  taken  out  of  one  pan  and  added  to  the 
other,  unless  its  force  be  reversed  in  direction  and  made  to 
act  upwards  instead  of  downwards ;  which  can  be  actually 
managed  by  hanging  it  to  a  string  over  a  fixed  pulley,  the 
other  end  of  the  string  being  attached  to  the  pan. 

This  fact  is  most  simply  expressed  by  saying  that  if  any 
term  or  quantity  is  transferred  from  one  side  of  an  equation 
to  the  other,  it  must  be  reversed  in  sign,  if  the  equality  is  still 
to  persist,  i.e.  if  the  equation  is  to  remain  true. 

This  is  a  simple  but  important  matter  of  constant  practical 
use,  and  it  requires  illustration  : 

Let  the  equation  be  given 

a; -2  =  3. 
We  can  transfer   -  2  to  the  other  side  of  the  equation,  where 
it  will  become    +  2,  giving  us  a;  =  3  +  2  or  in  other  words 
X  =  b.     (We  may  consider  that  we  have  added  2  to  each  side.) 

The  value  5  obviously  satisfies  the  equation  in  its  original 
form,  because  it  is  true  that  5-2  =  3 ;  and  the  substitution 


XIV.]  EQUATIONS.  141 

of  the  found  value  in  the  original  equation  and  then  seeing  if 
it  fits  or  holds  good,  is  called  '  verifying '  the  solution. 
Take  another  case : 

3x2+17  =  4a;2- 8. 
Getting  the  unknown  quantities  on  one  side,  and  the  known 
on  the  other,  it  becomes 

3x^-ix^  =  -8-17; 
if  we  like  we  may  now  reverse  the  sign  of  every  term,  which 

will  give  us 

4a;2-3a;2  =  17  +  8 

or  fl;2  =  25 

or  ic  =  5. 

Thus  all  the  equations  we  have  written  recently  happen  to 
be  expressive  of  the  same  fact :  namely  that  the  particular  x 
denoted  by  them  is  merely  the  number  5.     Substituting  this 
number  in  the  above  equation,  it  becomes 
3x25  +  17  =  4x25-8 
or  75  +  17  =  100-8 

which  we  perceive  to  be  an  arithmetical  identity,  since  both 
sides  =  92, 

thus  the  solution  is  verified,  or  the  value  a;  =  5  is  proved  to 
satisfy  the  given  equation. 

We  do  not  know  for  certain  that  it  is  the  only  value  that 
will  satisfy  it,  but  at  any  rate  it  is  one  solution.  It  so 
happens  that  the  equation  last  written  will  also  be  satisfied 
by  the  solution  x  =  -6 ;  and  this  is  characteristic  of  square 
or  quadratic  equations  in  general,  that  there  are  two  answers 
instead  of  only  one. 

An  equation  of  the  third  degree,  that  is  an  equation 
involving  x^,  will  in  general  have  three  answers ;  and  so  on. 

Take  one  more  quite  simple  example,  for  practice : 
given  7a;- 12  =  5a; +  6,  to  find  a;. 


142  EASY  MATHEMATICS.  [chap. 

Subtract  5x  from  both  sides,  or,  what  is  equivalent,  transfer 

5x  over  to  the  other  side  with  change  of  sign, 

we  get  2a;- 12  =  6. 

Now  add  12  to  both  sides,  or,  what  is  the  same  thing,  transfer 

-  12  over  to  the  other  side,  and  it  becomes 

2a;  =  6  +  12  =  18; 
wherefore  a;  =  9  is  the  solution. 

Try  it  in  the  original  equation  in  order  to  verify  it  and  we 
get  63-12  =  45  +  6, 

which  is  an  arithmetical  identity. 

As  to  algebraic  identities,  it  is  probably  needful  to  remind 
young  beginners  occasionally  even  of  such  simple  facts  as 
these :  at  the  same  time  making  mysterious  hints  that  there 
are  possible  interpretations,  to  be  met  with  hereafter,  wherein 
even  these  simple  statements  lack  generality  and  are  open  to 
reconsideration,  a  +  b  =  b  +  a^ 

and  ab  =  ba; 

and    they    should    be   frequently   reminded  of  such   useful 
identities  as  (a  +  bf  =  a^  +  2ab  +  b\ 

\a-hf  =  a^-2ab  +  b^, 
{a  +  b){a-b)  =  a^-b\ 
Oral  questions  should  be  asked  at  odd  times  concerning 
equivalent  expressions  for  such  things  as 

{V  +  q)\  {^-W,  {x  +  y){x-y\ 

{X+\){X-1\  (71+1)2,  (^_l)2, 

(l  +  a)2,  {\-a)\  (a; +  3)2, 

{x  -  5)2,  (3  +  72)2,  (7  4.  ia,)2^  etc.,  etc., 

since  a  pupil's  knowledge  of  such  fundamental  things  should 

be  ready  for  immediate  application — like  a  well  constructed 

machine. 

We  have  not  yet  taken  an  example  of  an  equation  involving 

a;2  as  well  as  a;,  because  they  are  not  quite  so  easy  to  solve ; 


XTV.]  EQUATIONS.  143 

but  a  parenthetical  remark  may  be  introduced  even  at  this 
stage.  We  know  that  quantities  of  different  kinds  do  not 
occur  in  one  expression;  in  other  words,  that  all  the  terms 
of  an  expression  must  refer  to  the  same  sort  of  thing,  if  they 
are  to  be  dealt  with  together  or  equated  to  any  one  value. 
Nevertheless  an  expression  like  a^  +  6x^  +  2x -h  6  is  common, 
and  X  may  be  a  length;  which  looks  as  if  we  could  add  to- 
gether a  volume,  an  area,  a  length,  and  a  pure  number. 
Not  so,  really,  however :  see  Appendix. 

The  equation  (x  -  3)  (re  -  4)  =  0, 

written  out,  becomes     ic^  _  7^;  + 12  =  0, 
or  x^  =  7a;- 12. 

We  may  guess  at  numbers  which  will  satisfy  this  equation, 
and  we  have  been  told  there  must  be  two,  because  it  is  a 
quadratic  :  it  contains  x^.  By  trial  and  error  it  will  be  found 
that  the  number  3  and  the  number  4  will  both  satisfy  it ;  for 
insertion  of  the  first  gives  the  identity  9  =  21-12,  and  inser- 
tion of  the  second  gives  the  identity  16  =  28-12;  but  no 
other  number  whatever,  when  substituted,  will  result  in  an 
identity,  that  is  to  say  no  other  number  will  satisfy  the 
equation ;  the  equation  has  two,  and  only  two,  solutions,  or, 
as  they  are  often  called,  "roots." 

Looking  at  the  factor  form  of  the  equation  with  which  we 
started,  (x  -  3)  (a;  -  4)  =  0, 

it  is  obvious  that  either  3  or  4  will  satisfy  it;  because  the  value 
3  makes  the  first  factor  zero,  and  the  value  4  makes  the  second 
factor  zero.  It  is  not  necessary  that  both  factors  shall  be  zero 
— either  will  do — hence  the  useful  answer  is  not  necessarily 
both  3  and  4,  but  either  3  or  4,  or  possibly  both. 

The  factor  form  of  writing  the  equation,  therefore,  contains 
the  solution  in  so  obvious  a  manner,   that  it  is  sometimes 
spoken  of  as  "  the  solution  " ;  and  if  an  equation  like 
3a;2  +  7a;-31  =  11 -8a; 


144  EASY  MATHEMATICS.  [chap. 

were,  by  any  process  of  manipulation,  reduced  to  the  form 

(a;-2)(a;  +  7)  =  0 
it  would  be  considered  solved ;  because  it  is  then  obvious  that 
the  values  +  2  and  -  7,  that  is  to  say  either  a;  =  2  or  a;  =  -  7, 
or  both,  satisfy  the  equation.     Inserting  them  successively, 
for  the  purpose  of  verification,  we  get  for  the  value  x  =  2 

12  +  14-31  =  11-16 
which  is  an  identity ; 
and  for  the  value  a;  =  -  7 

147-49-31  =  11+56 
which  is  another  identity. 

In  collecting  the  terms  of  the  given  equation  the  two  x 
terms  can  be  put  together,  making  15a;,  and  the  two  absolute 
terms  can  be  put  together,  making  42,  but  neither  of  these 
pairs  can  be  merged  in  the  other,  nor  in  the  term  3^^ .  there 
are  essentially  three  distinct  kinds  of  terms  in, the  equation, 
and  they  must  be  kept  distinct. 

Introduction  to  Quadratics. 

When  beginning  quadratic  equations,  it  is  a  good  plan 
to  give  them  first  of  a  kind  that  can  easily  be  resolved  into 
simple  factors,  so  as  to  remove  the  appearance  of  difficulty, 
and  yet  to  suggest  a  real  method  of  solution. 

For  instance,  a;^  -  7a;  +  1 2  =  0 

has  roots  3  and  4,  for  these  numbers  add  together  to  7  and 
multiply  together  to  12.  So  the  expression  on  the  left  hand 
side  can  be  resolved  into  factors  as 

(x-S){x-4) 
and  the  equation  can  be  re-written 

(a;-3)(a;-4)  =  0. 

Again  a;2-5a;+6  =  0 

is  plainly  satisfied  by  the  values  a;  =  3  and  a;  =  2. 


XIV.]  QUADRATIC  EQUATIONS.  145 

Once  more,  a;^- 11a; +  30  =  0 

has  the  roots  5  and  6,  and  is  equivalent  to 
(x-5){x-6)  -  0. 

If  we  had  chosen  the  equation 

a;2  +  lla;  +  30  =  0 
the  roots  would  have  been  -  5  and  -  6,  and  the  equation 
written  in  the  factor  form  would  have  been 

{x  +  5){x  +  6)  =  0. 
And  so  on,  according  to  innumerable  examples  given  in  every 
text  book  of  algebra. 

When  a  quadratic  expression  equated  to  0  is  solved,  it  is 
always  really  resolved  into  two  factors,  for  it  is  always 
virtually  expressed  in  the  form 

(x  -a){x-h)  =  0, 
where  a  and  h  are  the  two  numbers  which  satisfy  the  equa- 
tion, its  two  "  roots  "  as  they  are  called ;  a  term  which  is  thus 
used  in  a  new  sense,  having  no  reference  to  square  or  cube 
root. 

Multiplying  out  the  above  expression,  it  takes  the  form 
x^-{a-{-h)x  +  db  =  0 
so  that  the  coefficient  of  the  middle  term  is  the  sum  of  the 
roots,  and  the  absolute  term  is  their  product;  provided  that 
the  coefficient  of  the  quadratic  term  is  unity,  and  the  sign  of 
the  middle  term  is  negative. 

The  process  of  solving  the  equation  is  the  same  as  that  of 
resolving  the  above  expression  into  factors,  and  one  way  of 
achieving  it  is  to  think  of  two  numbers  which  add  together 
into  the  middle  term  and  multiply  together  into  the  absolute 
term,  provided  the  coefficient  of  the  quadratic  term  is  unity. 

Suppose,  for  instance,  the  equation  given  were  a  general 
quadratic  in  a;, 

Ax^  +  Bx  +  C  =(i. 


146  EASY  MATHEMATICS.  [chap. 

Divide  everything  by  A  in  order  to  reduce  the  coefficient  of 
the  quadratic  term  to  unity,  getting 

A      A 

Here  we  know  that  the  sum  of  the  roots  of  the  equation  must 
be  equal  to  the  ratio  -  BjA  and  that  the  product  of  the  roots 
must  be  equal  to  the  ratio  GjA.     (See  also  Appendix  III.) 

In  the  above  cases  it  was  easy  to  guess  the  roots,  but  it  is 
by  no  means  always  easy.  A  process  must  be  used  for  find- 
ing them,  and  as  far  as  possible  the  pupil  should  be  left  to 
find  it  out:  with  guidance,  but  no  more  actual  telling  than 
may  be  found  necessary.  But  time  and  perseverance  will  be 
required.  If  the  child  has  no  head  for  it  the  attempt  may  be 
useless,  and  should  not  be  persisted  in  unduly;  nor  should  any 
disgrace  attach  to  failure;  success  is  a  triumph  rather  than 
otherwise. 

If  the  equation  x^-\-\^x  =  24  is  given,  it  happens  to  be 
rather  obvious  that  12  and  2  must  be  the  numbers  concerned, 
if  the  signs  are  properly  attended  to ;  but  the  rule  for  finding 
them  in  general  will  have  to  be  evolved  from  a  consideration 
of  the  chief  quadratic  identity, 

{x  +  of  =  ic2  +  lax  +  a2. 

Suggest  a  trial  of  this  to  the  pupil,  and  if  necessary  suggest 
trying  the  value  5  for  the  auxiliary  and  gratuitously  intro- 
duced symbol  a,  because  that  will  give 

3;2+l0a:  +  25, 

which  imitates  the  only  hard  part,  the  left-hand  side,  of  the 

given  equation;    for  a  little  complication  of  the   easy,  the 

numerical,  part  on  the  right-hand  side,  does  not  matter.  So 
we  might  write  the  given  equation  now  if  we  like, 

.iB2  +  10a;-t-25  =  24  +  25  =  49; 


XIV.]  QUADRATIC  EQUATIONS.  147 

but  directly  we  have  done  that,  the  equation  is   practically 
solved,  for  it  is  plainly  equivalent  to 

(a; +  5)2=  72, 
and  therefore  to  a:  +  5  =  ±  7  ; 

that  is  to  say  x  =  either  2  or  - 12,  as  the  case  may  be;  for 
either  will  satisfy  or  solve  the  equation. 

Wherefore  the  given  equation,  with  the  roots  wrapped  up 
fl;2  +  l0a;-24  =  0, 
may  likewise  be  written  (a;  -  2)  (a;  + 12)  =  0, 
with  the  roots  visible. 

Another  example  or  two  to  clinch  the  matter : 
let  it  be  given  that         a;2  +  14a;  =  15, 

here  if  we  try  to  throw  the  left-hand  side   into   the  form 
{x  +  a)2,  the  auxiliary  number  a  is  given  by 

2a  =  14,     so     a^  =  id; 
and  the  equation  becomes 

a;2+14a;  +  49  =  15  +  49  =  64, 
or  (a;  +  7)2  =  82; 

whence  x  =  -  7  ±  8, 

wherefore  x  =  either  1  or  -  15. 

One  more  plainly  numerical  example : 
a;2-6a;  =  20, 
here  a  is  manifestly  3,  and  the  equation  becomes 

a;2-6a;+9  =  29, 
or  (a;-3)2  =  29; 

wherefore  a;  =  3  ±  J(29) 

and  it  can  only  be  carried  further  by  extracting  the  numerical 
and  incommensurable  root. 

Now  a  slightly  more  general  touch : 
given  a;2-12a;  =  n. 

To  reduce  this  to  the  form  (x  -  6)2,  we  must  add  36  to  each 
side,  getting  a;2  -  1 2a;  +  36  =  ?i  +  36 

whence  the  solution  is     {x-%)  =  ±  J{n  +  36) 


148  EASY  MATHEMATICS.  [chap.  xiv. 

Finally  let  the  given  equation  be 
ic^  _  2ax  =  71^. 

Complete  the  square  on  the  left  hand  side, 
x^-2ax  +  a^  =  n^  +  a^ 
it  becomes  {x  -  of  ==  n'^  +  a? 

or  a;  =  a  ±  J{n^  +  a?) ; 

which  is  essentially  a  general  result. 

The  form  of  this  result  is  easy  to  remember,  and  it  is 
really  general ;  for  if  the  quadratic  equation  had  been  given 
in  the  manifestly  general  form 

Ax^  +  Bx  +  C  =  0, 
where  the  coefficients  A,  B,  C,  stand  for  any  known  quantities 
of  any  kind  whatever,  it  can  be  reduced  to  the  above  form  by 
first  dividing  by  A^  and  then  instituting  comparisons  between 
it  and  the  above ;  for  we  then  see  that  correspondence  requires 
the  following  identities : 

71^  =   _--    and    2a  =   --J, 

B^ 


so  that 

a^  = 

'  4.A^' 

wherefore  the  solution  of  the 

general  quadratic  is 

B 
^=  -2A 

-\iA^    a) 

"  2A^ 

-B±J{B^-4AC)}. 

But  this  should  not  be  given  to  pupils  for  a  long  time  yet, 
and  perhaps  we  have  already  been  attracted  a  little  further 
than  in  the  present  book  is  legitimate.  The  pupil  should  by 
no  means  be  thus  hurried.  A  month's  practice  at  the  numerical 
and  factor  forms  of  expression  may  be  desirable  before  passing 
to  even  slightly  more  general  forms. 


CHAPTER   XV. 

Extraction  of  Simple  Roots. 

The  last  arithmetical  lines  of  Chapter  XIII.  practically 
asserted  that  18  =  5832^;  and  it  can  easily  be  verified  by 
multiplication  that 

18x18x18  =  5832 
or  that  18  is  the  cube  root  of  5832. 

Here  then  is  a  method,  automatically  suggested,  for  finding 
cube  and  other  roots : — Analyse  the  number  into  factors 
whose  roots  are  known,  as  5832  was  analysed  into  27  and 
216,  at  the  end  of  the  chapter  referred  to.  It  cannot  always 
or  often  be  done,  but  whenever  it  can  it  is  quite  the  best  way. 

But  to  be  able  to  apply  this  method  we  must  cultivate  an 
eye  for  factors,  and  we  must  also  recognise  or  know  by  heart 
a  certain  collection  of  cube  and  square  numbers. 

Thus  1728  =  12x12x12 

or  the  cube  root  of  1728  is  12. 

This  is  easy  to  remember  because  it  represents  the  number 
of  cubic  inches  in  a  cubic  foot.  In  a  country  with  a  purely 
decimal  system  of  measures,  this  fact  would  not  be  known 
with  the  same  ease.     They  would  know  well  however  that 

1000  =  10x10x10 
or  that  the  cube  root  of  a  thousand  is  ten ;  and  so  do  we. 

We  may  also  know  that  729  has  9  as  its  cube  root,  since  it 
evidently  equals  81  x  9,  that  is  9^  x  9  or  9^.  The  cube  root  of 
343  is  7. 


150  EASY  MATHEMATICS.  [chap. 

The  square  root  of  10,000  is  100,  and  the  square  root  of 
this  is  10 ;  but  what  its  cube  root  is  is  not  so  easy  to  say. 

The  cube  root  of  1000  is  10 ;  but  what  its  square  root  is 
is  not  so  easy  to  say. 

The  fifth  root  of  32  is  2,  but  neither  its  square  root  nor 
cube  root  is  simple. 

It  is  valuable  to  remember  thoroughly  that  2  is  the  cube 
root  of  8. 

The  square  root  of  a  million  is  1000, 
the  cube  root  of  a  million  is  100 ; 

the  sixth  root  of  a  million  is  10. 

But  there  is  no  need  to  trouble  about  remembering  any 
more  than  a  few  ordinarily  occurring  square  roots  and  cube 
roots ;  for  the  sixth  and  higher  roots  are  seldom  wanted,  and 
they  can  usually  be  derived  from  square  and  cube  roots. 

A  number  like  64  can  resolve  itself  into  8x8 

or  into  4  x  16. 

Its  cube  root  is  therefore  easily  stated  as  2x2,  viz.  4,  and 
its  square  root  as  2  x  4,  viz.  8. 

144  again  =  12x12  and  also  =  9x16, 
and  either  pair  of  factors  gives  its  square  root  but  not  its  cube 
root. 

Surds. 

Now  let  us  proceed  to  ask  what  is  the  root  of  a  number 

like  12  [where  the  word  "root"  is  used  alone,  square  root  is 

understood].     We  can  resolve  it  into  factors  and   find  the 

root  of  each 

12  =  4x3, 

so  V12  =  74x^3  =  2x^3. 

So  the  result  may  be  stated  that 
VI 2  =  2^3. 
Similarly  JS  =  2J2. 


XV.]  ROOTS.  151 

Again,  let  us  find  the  cube  root  of  say  24. 
24  =  8x3   and   4/8  =  2; 
so  24^,  which  is  often  written  4/24,    =  24/3. 
Note  the  following : 

732  =  V16  J2  =  4^2, 


•^'-V©=f=l- 


8 
Thus  it  would  appear  that  4^2  must  equal  -j-.      If  we 

\/2 
multiply  or  divide  each  of  these  numbers  by  J2  we  can  easily 
verify  this  asserted  quality.     For  multiplication  by  J 2  makes 
them  both  8  ;  division  by  J2  makes  them  both  4. 
Verify  the  following  statements  : 

4/56  =  2^7. 

v/20  =  2^5. 

718  =  372. 

V27  =  3^3. 

772  =  3^8  =  6J2. 

v/50  =  5^2. 
V(200)  =  1072. 

^216  =  6J6. 
^(360)  =  6710. 
V(810)  =  VIO. 
V(490)  =  7^10. 
V(125)  =  5^5. 

V(iooo)  =  10710. 

V(1728)  =  12^12. 
7343  =  777. 
V512  =  8^8  =  16^2. 
The  last  five  are  all  cube  numbers,  and  their  value  suggests  a 
rule  for  expressing  the  square  roots  of  any  cube  number ;  e.g. 
V27  =  3^3,    or   >3  =  ^Jn. 


152  EASY  MATHEMATICS.  [chai*. 

N.B. — The  best  way  of  interpreting  the  word  "verify"  at 
the  head  of  the  above  set  of  examples,  or  in  any  similar  place, 
is  for  the  pupil  to  take  the  left-hand  expression,  and  try  as  an 
exercise,  independently,  to  simplify  or  otherwise  express  it, 
and  see  if  he  can  reduce  it  to  the  form  given  on  the  right- 
hand  side.  He  will  thus  perceive  that  numbers  which  have 
two  factors  can  have  the  expression  for  their  roots  put  into 
another  form,  which  is  often  a  more  simple  form ;  and  that 
a  large  number  of  roots  could  be  found  numerically  if  the 
roots  of  a  few  prime  numbers  were  known. 

The  number  ten,  as  usual,  has  an  unfortunate  disability,  in 
that  neither  of  its  factors  is  a  perfect  square,  as  one  of  the 
factors  of  12  is.  All  we  can  do  with  ^10  therefore  is  to 
say  that  it  equals  J2  Jb,  which  is  of  extremely  little  use. 
It  is  better  kept  as  ^^10  and  considered  to  be  one  of  the 
things  to  be  found.  Since  the  sq.  root  of  9  is  3,  while  4  is 
the  sq.  root  of  16,  the  pupil  may  make  a  guess  and  try 
whether  3"1  approximates  to  the  sq.  root  of  10  or  not.  He 
can  easily  do  this  by  multiplying  3-1  by  itself.  By  this 
means  he  can  gradually  correct  its  value.  He  can  in  a 
similar  way  make  guesses  also  at  ^20  and  ^30  and  ^^50.  Let 
him  try. 

No  simplification,  by  resolution  into  factors,  can  be  made 
with  any  such  numbers  as 

^/2,  JS,  A  v/7,  jn,  J13,  J17,  719,  V23, 
and  so  on;    that  is,  no   simplification   of  this  kind  can  be 
applied  to  any  root  of  any  prime  number,  naturally. 

The  roots  of  even  numbers  may  always  have  a  ^2  exhibited: 
^6  may  be  written  ^3  ^2, 

JU         „         „        J7J2, 
but  it  is  seldom  useful  to  express  them  in  this  way. 


XV.]  ROOTS.  153 

Let  us  see  if  we  must  draw  a  distinction  between  (^9)^  and' 
^(9^),  that  is  between  the  cube  of  root  nine  and  the  root  of 
nine  cubed.     Now 

(^9)3  =  3^  =  27, 

while  J{9^)  =  V(729)  =  9J9  =  27  likewise.. 

So  they  turn  out  to  be  the  same. 

The  cube  of  a  root  appears  to  be  equal  to  the  root  of  a  cube-. 
That  is  curious,  and  may  well  be  unexpected.  It  is  not  the 
sort  of  thing  at  all  safe  to  assume.  Plausible  assumptions 
are  always  to  be  mistrusted  and  critically  examined ;  occasion- 
ally, as  in  this  instance,  they  turn  out  true. 

Let  us  consider  the  fact  more  generally,  and  see  whether  it 
is  always  true  that 

J{n')  =  Unf. 

The  other  and  more  expressive  notation  for  roots  will  here 
come  to  our  aid. 

J(n^)  may  be  written  (n^)^, 
and  (Jn)^  may  be  written  (71^)^, 
so  it  looks  as  if  both  could  be  written  as  n^  or  n^^,  or,  more 
properly,  ?i^  +  ^. 

This  last  is  a  thing  we  have  not  yet  learnt  how  to  interpret. 
We  may  assume  however,  as  an  experimental  fact,  that 

J(n^)  =  Jn^sjn  =  njn, 
hence  the  interpretation  nJn,  that  is  ?i  x  ?i^,  suggests  itself 
for  n}^  or  ?i^  +  ^;  and  it  is  the  right  interpretation. 

Here  again  (as  on  page  126)  we  have  arrived  at  a  striking 
circumstance  about  indices,  which  is  now  well  worthy  of 
examination. 


CHAPTER  XVI. 

Further  consideration  of  indices. 

There  are  two  things  to  which  we  might  now  appropriately 
turn  our  attention :  one  is  the  numerical  calculation  of  all 
manner  of  roots,  for  instance,  ^2,  ^3,  ^10,  4/2,  ^3,  etc.,  IJ2^ 
yXOOy  and  so  on;  evidently  a  large  subject,  since  we  may 
require  to  find  any  root  of  any  number;  the  other  is  the 
discussion  of  that  curious  property  of  indices,  which  has  been 
dimly  suggested  by  certain  of  the  examples  chosen,  viz.  the 
suggestion  that 

and  that  (a;**)'"  =  («*")"  =  a;^». 

Of  these  two  directions  along  which  we  could  now  continue 
the  discussion,  the  latter  is  undoubtedly  the  easier,  and  so  we 
will  proceed  this  way  first;  and  incidentally  we  shall  find 
ourselves  led  to  a  very  practical  and  grown-up  way  of  dealing 
with  the  former  more  difficult  line  of  advance. 

What  we  found  experimentally  (on  page  1 26)  was  that 
3x32  =  27  =  33; 
also  that  32  x  3^  =  9  x  27  =  243  =  3^. 

And  so  we  might  have  taken  other  instances : 
22x24  =  4x16  =  64  =  2^, 
2  X  25  =  2  X  32  =  26, 
23  X  23  =  8  X  8  =  26, 
2  X  22  X  23  =  2  X  4  X  8  =  26. 


CHAP,  XVI.]         HORSE-SHOE-NAILS  PROBLEM.  155 

What  does  all  this  look  like  1 

Manifestly  it  looks  as  if  to  effect  a  product  among  the 
powers  of  a  given  number,  we  must  add  the  indices  of  the 
several  powers.     It  looks  like 
23  X  24  =  27, 
23  X  28  =  211, 
68  X  62  =  65, 

2i  X  22  =  2^  =  (^2)5  =  ^(26)  =  ^32  =  4^2, 
2*  X  2^  =  2*^  =  2^  =  ^(29)  =  (72)9, 

=  V(51 2)  =  7(2x256)  =  16^2. 
Now  when  the  indices  are  whole  numbers  it  is  very  easy  to 
see  the  reason  of  this  simple  rule.  What  does  2^  mean  1  It 
means  that  four  factors  each  of  them  2  are  to  be  multiplied 
together.  The  index  is  only  an  indication  of  how  many  times 
the  similar  multiplication  is  to  be  performed. 

2^  means  simply  2x2x2x2;  the  number  of  multiplication 
signs  being  one  less  than  the  index,  i.e.  one  less  than  the 
number  of  factors  of  course.  Similarly  2^  is  merely  an 
abbreviation  for  2x2x2.     Hence 

2^x23  =  2x2x2x2x2x2x2, 
that  is  seven  2's  are  to  be  multiplied  together;  and  so  it  is 
naturally  indicated  by  2''. 

The  index  counts  the  number  of  similar  factors;  hence 
when  the  factors  are  increased  in  number  the  index  shows 
the  simple  increase;  but  the  effect  of  the  continued  multi- 
plication on  the  resulting  number  may  be  prodigious. 

The  anecdote  about  the  nails  in  the  horse's  shoes  here 
appropriately  comes  in : 

A  man,  who  objected  to  the  price  asked  for  a  horse,  was 
offered  the  horse  as  a  free  gift,  thrown  into  the  bargain,  if  he 
would  buy  merely  the  nails  in  its  shoes,  of  which  there  were 
6  in  each  foot;  at  the  price  of  a  farthing  for  the  first  nail, 


156  EASY  MATHEMATICS.  [ghap. 

2  farthings  for  the  second,  4  farthings  for  the  third,  8  for  the 
fourth,  and  so  on.  The  offer  being  accepted,  he  had  to  pay 
.£17,476  5s.  3|d.  for  the  nails;  and  he  did  not  consider  the 
horse  cheap. 

The  number  of  farthings  in  this  sum  is  very  great,  but  it  is 
simply  one  less  than  22^. 

If  a  beginner  wishes  to  verify  the  above  by  multiplying  2 
by  itself  23  times,  he  can  easily  do  it,  though  it  will  take  a 
little  time;  and  he  can  then  reduce  the  result  to  pounds 
shillings  and  pence,  as  he  has  been  no  doubt  so  well  taught 
how  to  do.  It  is  not  a  grown-up  way  of  ascertaining  22-*,  but 
it  serves.     (Reference  to  pp.  166  and  259  may  be  convenient.) 

If  he  is  properly  sceptical  about  the  magnitude  and  correct- 
ness of  the  above  sum,  he  should  do  it.  It  is  good  practice  in 
easy  multiplication ;  and  sums  which  are  set  by  the  pupil  to 
himself  are  likely  to  secure  greater  attention  from  him  than 
those  enforced  from  outside.  It  is  probably  desirable  that 
children  should  often  set  sums  as  well  as  work  at  them.  I 
would  even  sometimes  encourage  them  to  set  examination 
papers.     It  is  a  good  way  of  getting  behind  the  scenes. 

As  regards  the  verification  of  a"*  x  a"  =  a"*"^"  therefore,  the 
idea  is  very  simple,  so  long  as  m  and  n  are  whole  numbers ; 
because  it  is  a  mere  matter  of  counting  the  number  of  similar 
factors. 

"When  we  say  that  five  sixes  multiplied  together  equal  7776, 
we  are  employing  the  number  five  in  this  very  way.  The 
expression  6^  does  not  mean  five  sixes  added  together,  or  30 ; 
but  it  means  five  sixes  multiplied  together,  yielding  a  much 
larger  result. 

So  also  six  tens  multiplied  together  make  a  million,  whereas 
added  they  only  make  60.  In  fact,  as  we  said  before,  page  56 
and  Chap.  XII.,  while  multiplication  is  abbreviated  addition, 
involution  is  abbreviated  multiplication. 


XVI.]       ^^,>><^  INDICES.  157 


Fractional  indices. 

When  the  indices  m  and  n  are  fractions,  the  idea  they  express 
is  not  so  simple,  and  the  above  relation  a"*  x  a"  =  ft*"^"  is  not 
so  easily  justified;  but  we  may  be  willing  to  accept  it  by 
analogy  and  see  how  it  works. 

If  asked  wherein  the  proof  consists  for  fractional  indices, 
we  must  answer  in  "  consistency,"  constant  coherence  and 
agreement  with  results  so  obtained,  and  in  corresponding 
convenience  of  manipulation. 

At  one  time  2^  and  such  like  were  called  irrational  quan- 
tities because  it  was  difficult  to  attach  a  commonsense  signifi- 
cance to  "2  multiplied  by  itself  half  a  time";  and  it  is 
certainly  not  to  be  interpreted  as  half  2  multiplied  by  itself, 
for  that  would  be  unity. 

There  is  nothing  irrational  about  this  quantity  however : 
it  has  a  value  approximately  1*4142  ...  though  it  will  hereafter 
be  found  that  it  will  not  express  itself  exactly  by  a  finite 
series  of  digits  in  any  system  of  notation  whatever. 

It  may  rightly  be  styled  "incommensurable"  therefore, 
but  it  is  in  no  sense  irrational. 

"Irrational"  however  was  a  term  at  one  time  applied  to 
any  power  of  a  number  whose  index  was  not  a  positive  integer. 

The  thing  has  to  be  mentioned,  for  historical  reasons,  but 
the  term  "irrational"  should  now  cease  to  be  used.  The  term 
"  surd,"  being  meaningless,  may  be  employed  if  we  like,  but  it 
is  never  really  wanted :  it  only  serves  as  a  heading  to  a 
chapter  to  indicate  its  contents. 

Negative  indices. 

But  let  us  go  on  and  ask  how  shall  we  interpret  the  ex- 
pression if  one  of  the  indices  be  not  fractional  but  negative  ? 
For  instance,  how  shall  we  interpret 
3-2  or  2-6. 


158  EASY  MATHEMATICS.  [chap. 

Suppose  for  instance  we  had 

or  X  a-", 
we  should  naturally  say  that  the  result  must  be  a*""**. 

Very  well,  let  m  =  2  and  n  =  3, 

then  0?  X  a~^  =  a~^. 

What  does  fl~^  mean  ? 

How  can  we  multiply  a  number  by  itself  a  negative  number 
of  times  ?  At  first  the  term  "  irrational "  was  applied  to  such 
quantities  as  these :  but  a  consistent  interpretation  was  soon 
found  for  them.  If  addition  of  indices  means  multiplication, 
it  is  natural  that  subtraction  of  indices  shall  mean  division. 

Make  the  hypothesis  therefore  that  a*""**  can  be  interpreted 
as  a""  -f  a",  and  let  us  see  how  that  works. 

Suppose  we  had  2^  ~  2^,  it  could  be  written  out  in  full , 

2x2x2x2x2 
2x2x2   ' 

and  the  result  after  cancelling  would  be  2  x  2, 
that  is  22  =  25-3. 

The  whole  thing  is  therefore  quite  simple. 

Take  other  examples : 


3(6-5) 

= 

3« 
3^^  ~ 

3, 

a'-* 

= 

a' 
a'  ~ 

a\ 

^«-. 

= 

x^  x"  ' 

7  X  T'' 
74-2+1  =  1^  =  73  =  343^ 


XVI.]  INDICES.  159 


7*        1 

na-2   _    _   _    _!_7« 

'       -  72  -  49  '  ' 


7* 

This  last  is  a  most  interesting  and  useful  result. 

If  the  index  is  zero,  the  quantity,  whatever  it  may  be,  is 
reduced  to  unity ;  for 

a—  =  ^  =  1 ; 

it  equals  1  whatever  a  may  be. 

a<^  =  1  is  the  brief  summary  of  this  important  consequence 
of  our  notation.  The  index  0  would  have  been  hard  to 
interpret,  just  as  fractional  and  negative  indices  were  hard  to 
interpret,  but  fortunately  it  thus  interprets  itself. 

A  negative  sign  applied  to  an  index  turns  out  therefore  to 
have  the  effect  of  giving  the  reciprocal  of  the  quantity; 
for  since  „» 

we  have  only  to  take  the  case  where  m  is  zero,  in  order  to  get 
-n  _t  -  L 


Hence  2~'^  = 

2-2  = 
3-2  = 

3-3  = 

2-6  = 


1 

2' 

1 
? 

1 
9' 

27' 

J_ 
32' 


160  EASY  MATHEMATICS.  [chap. 

Hence  while  2^  means  ^2,  we  thus  find  that  2"^  means  ^ ; 

or,  in  general,  i         .       ,       1 

X     =  —  and  x~^  =  — 

«"  X 

Take  the  last  simple  and  useful  mode  of  expression.     To 
verify  it,  simply  multiply  both  sides  by  x,  thus 

a;-i  X  a;+i  =  a;-i+i  =  a;0  =  -  =  1 

X 


Similarly       x-^  =  i  ^  g)'  =  {x-% 


And  this  suggests  powers  of  powers;  like  ( 10^)2,  that  is  the 
square  of  a  thousand,  which  is  a  million,  1  followed  by  6  ciphers, 
or  106. 

So  also  (10^)2  =  10^2  =  a  billion;  the  indices  being  in  this 
case  multiplied  to  give  the  result. 

So  now  we  leave  addition  and  subtraction  among  indices, 
which  merely  meant  multiplication  and  division  among  the 
quantities  themselves,  and  begin  to  study  multiplication  among 
indices. 

Consider  for  instance  what  the  meaning  should  be  of  4^^^ ; 
it  equals  4^  =  (43)2  =  (42)3  =  4096, 

So  multiplication  among  indices  means  involution  among 
the  quantities  themselves. 

So  also  division  among  indices  will  signify  evolution  among 
quantities,  thus 

7t  =  (73)^  =  (7^)3, 

the  order  of  the  factors  (which  in  this  case  are  m  and  - j 
being  indifferent.  ^ 


XVI.]  INDICES.  161 

If  it  were  worth  while  we  might  proceed  further,  and  con- 
sider what  would  be  the  meaning  of  the  process  "  involution  " 
applied  to  indices;  how  would  that  affect  the  quantities 
themselves  ?  What  for  instance  is  the  meaning  of  2^  1  but  it 
is  a  mere  curiosity  and  is  hardly  worth  while.  Suffice  it  to 
say  that  the  numbers  so  reached  become  rapidly  prodigious. 
10^0  is  a  number  with  ten  ciphers  after  the  1,  or  ten  thousand 
million ;  but  10^^  possesses  a  hundred  cyphers,  and  represents 
a  number  far  greater  than  that  of  all  tTie  atoms  of  matter  in 
the  whole  solar  system — earth,  sun,  and  all  the  planets, — not- 
withstanding the  fact  that  a  speck  containing  a  million-million 
atoms  is  only  visible  in  a  high  power  microscope. 


CHAPTER  XVII. 

Introduction  to  Logarithms. 

The  equation  y  =  a;",  that  is,  the  ti*^  power  of  «,  may  be 
equally  expressed  as  a;  =  yn^  that  is,  the  v>'^  root  of  y ;  this  is 
not  an  inverse  expression,  but  the  same  in  inverse  form. 

So  also  the  equation  xy  =  \,  which  represents  a  multiplication 
sum,  can  also  be  written  y=l/Xj  which  represents  a  division 
sum;  and  x^y^  ■■=  c^  can  appear  as  c  =  ±xy,  the  double  sign 
representing  an  ambiguity  or  double  solution,  because  either 
+  c  or  -c  would  when  squared  give  the  right  result. 

If  y  is  the  n}^  power  of  x,  it  is  easy  to  say  that  x  is  the  rt*^ 
root  oi  y ;  we  can  also  say  that  n  is  the  index  or  exponent  of 
X  which  yields  the  value  y ;  but  how  are  we  to  express  the 
relation  that  n  bears  to  y1 

It  is  a  thing  we  have  not  yet  come  across. 

It  is  called  a  logarithm;  it  involves  a  reference  to  both 
X  and  y ;  it  is  called  the  logarithm  of  y  to  the  base  x. 

Let  us  understand  this  matter. 

Write  down  100  =  IO2, 

2  is  called  the  logarithm  of  a  hundred  to  the  base  ten. 

Conversely  10  =  100*  so  J  might  be  called  the  logarithm 
of  ten  to  the  base  a  hundred. 

Write  down  25  =  52 

2  is  the  logarithm  of  25  to  the  base  5. 


CHAP.  XVII.]  LOGARITHMS.  163 

The  logaritlim  of  a  number  is  defined  as  the  index  of  the 
power  to  which  the  base  must  be  raised  in  order  to  equal 
the  given  number. 

Thus  if  we  are  told  that  3  is  the  logarithm  of  a  thousand  to 
the  base  ten,  it  is  another  mode  of  stating  that  10^  =  1000. 

So  3  is  the  logarithm  of      8  to  the  base  2, 

2  is  the  logarithm  of    49  to  the  base  7, 
5  is  the  logarithm  of    32  to  the  base  2, 

3  is  the  logarithm  of  216  to  the  base  6, 

4  is  the  logarithm  of    81  to  the  base  3, 
and  so  on. 

It  looks  a  cumbrous  and  roundabout  mode  of  expressing 
what  is  more  neatly  expressed  by  the  index  notation,  but  it  is 
an  exceedingly  practical  and  convenient  mode  of  statement  all 
the  same,  and  is  a  great  help  in  practical  computation. 

What  is  the  logarithm  of  343  to  the  base  7  ?     Answer,  3. 

What  is  the  logarithm  of  a  million  to  the  base  10  ?  Answer,  6. 

What  is  the  logarithm  of  64  *?  It  is  6  to  the  base  2,  and  2  to 
the  base  8. 

What  is  it  to  the  base  10?  Answer,  something  less  than  2 
and  more  than  1. 

What  is  the  logarithm  of  10  to  the  base  10,  or  of  any 
number  to  its  own  base  ?     Answer,  unity,  for  a  =  a^. 

What  is  the  logarithm  of  unity  itself  ? 

The  answer  is  0,to  any  base,  because  1  ^a". 

What  is  the  logarithm  of  a  fraction,  say  J,  to  the  base  2  ? 
Answer,  a  negative  quantity,  in  this  instance  -  2,  because 
J  =  2-2. 

So  also  -  2  is  the  logarithm  of  y^^j-  to  the  base  10,  because 
yJ^  =  10-2.  And  the  fact  that  ^^  =  10-^  can  be  expressed 
by  saying  that  -  6  is  the  log  of  a  millionth  to  the  base  10. 

It  appears  therefore  that  the  logarithms  of  reciprocals  or  of 
numbers  less  than  1  are  negative,  the  log  of  1  itself  being  0. 


164  EASY  MATHEMATICS.  [chap.  xvii. 

This  is  satisfactory.  Everything  greater  than  1  has  a  positive 
logarithm,  everything  less  than  1  has  a  negative  logarithm; 
provided  always  that  the  base  itself  is  greater  than  one. 
The  further  a  number  is  removed  from  1  both  ways,  whether 
in  the  direction  of  greatness  or  of  smallness,  the  larger 
numerically  is  the  logarithm ;  but  it  is  positive  bigness  in  the 
one  case,  negative  bigness  in  the  other.  It  is  natural  there- 
fore that  the  logarithm  of  1,  to  any  base,  should  be  zero. 

Mathematicians  know  how  to  calculate  the  log  of  any 
number,  no  matter  how  complicated,  and  they  have  recorded 
the  results  in  a  book  called  a  table  of  logarithms;  just  as 
grammarians  and  scholars  know  how  to  translate  any  foreign 
word,  and  have  recorded  the  results  in  books  called  dic- 
tionaries. A  Table  of  Logarithms  is  to  be  used  like  a 
dictionary.  It  can  be  readily  used,  and  is  used  every  day, 
by  those  who  would  find  it  difficult  to  construct  it.  It  should 
puzzle  children  sometimes  how  the  meaning  of  words  in  dead 
foreign  languages  were  ascertained ;  they  mostly  take  it  for 
granted  and  do  not  think  about  it.  So  also,  for  a  time,  and 
until  they  make  some  approach  to  becoming  budding  mathe- 
maticians, they  need  not  learn  how  to  compute  a  table  of 
logarithms;  but  they  must  imbibe  a  clear  idea  as  to  their 
meaning.  They  must  also,  and  that  is  an  easier  matter  still, 
learn  their  practical  use,  and  be  able  to  use  a  table  as  they 
have  learnt  how  to  use  a  dictionary. 


CHAPTER  XVIII. 

Logarithms. 

When  we  express  a  number  thus : 
64  =  82, 
1000  =  108 
32  =  25, 
or,  in  general,  n  =  a*, 

we  are  said  to  express  it  "  exponentially,"  that  is,  by  means 
of  the  index  or  "  exponent "  of  the  power  to  which  a  certain 
other  number  called  a  base  is  to  be  raised  in  order  to  be  equal 
to  the  given  number. 

In  the  above  equation  n  stands  for  the  number,  a  for  the 
base,  and  x  for  the  index  or  exponent  of  that  base. 

The  question  naturally  arises,  what  relation  does  x  bear  to 
n,  for  it  manifestly  depends  upon  both  n  and  a  1  If  the  base 
has  been  specified  and  kept  constant,  then  x  will  vary  only 
as  n  varies.  It  is  plain  that  x  will  increase  as  n  increases,  but 
not  nearly  so  fast. 

Take  a  few  examples,  and  first  take  the  number  2  as  base : 

2  =  21, 

4  =  22, 

8  =  23, 

16  =  24, 

32  =  25 

64  =  26, 


166  EASY  MATHEMATICS.  [chap. 

1024  =  210, 

16,777,216  =  224. 

Here  the  index  runs  up  slowly,  1,  2,  3,  4,  etc.,  according  to 
what  are  called  the  "  natural  numbers  "  ;  whereas  the  number 
on  the  left-hand  side  runs  up  very  quickly.  The  index  is  said 
to  progress  "arithmetically,"  that  is,  by  equal  additions;  the 
number  on  the  other  hand  is  said  to  progress  "  geometrically  " 
(a  curious  use  of  the  word),  that  is,  by  equal  multiplications. 
There  is  evidently  some  law  connecting  the  index  and  the 
number,  when  a  base  is  given;  and  the  following  nomenclature 
is  adopted : 

5  is  called  the  logarithm  of  32  to  the  base  2 ; 

3  is  the  logarithm  of  8  to  the  base  2 ; 

4  is  the  log  of  16  to  base  2  ; 

6  =  log  64  (base  2), 

which  is  usually  abbreviated  still  further : 
6  =  log2  64  ; 
10  =  logg  1024, 
the  base  being  indicated  as  a  small  suffix  to  the  word  log. 

Make  now  a  more  complete  table ;  first  of  powers  : 
1  =  2°, 


2  =  21,       -         -         -       1 


4  =  22. 

8  =  23 

16  =  24, 

32  =  25 


i  =  2-2, 

i  =  2-3, 

1  _    0-4 

T¥  -    ^      ' 

1  '  _    0-5 

^2"  -    '^      ' 


1024  =  210,     .         .  _^_  =  2-10, 

and  then  of  the  corresponding  logarithms  : 


XVIII.]  LOGARITHMS.  167 

From  the  above  table  it  follows  that  (with  the  base  2) 

logl  =  0, 

log  2  =  1,  logi=  -1, 

log4  =  2,  logi=  -2, 

logS  =  3,  logl  =  -3, 

log  16  =  4,  logTV=  -4, 

log  32  =  5,  log^V=  -5. 

It  would  be  a  good  thing  to  plot  both  these  tables  on 
squared  paper,  representing  for  the  first  the  indices  1,  2,  3,  4 
as  horizontal  distances,  and  the  numbers  2,  4,  8,  16  as  vertical 
distances;  and  for  the  second  measuring  distances  to  repre- 
sent the  2,  4,  8,  16  numbers  horizontally,  and  the  logarithm 
numbers  1,  2,  3,  4  vertically. 

The  first  is  called  an  exponential  curve,  or  curve  of  exponents 
or  indices ;  the  second  is  called  a  logarithmic  curve,  or  curve  of 
logarithms.  The  two  curves  turn  out  to  be  identically  the  same, 
only  differently  regarded, — to  make  their  identity  apparent, 
the  paper  can  be  turned  round  and  looked  through  at  the  light. 

If  drawn  on  the  same  sort  of  squared  paper  the  curves 
will  fit.  They  may  either  of  them  be  said  to  represent  the 
relation  between  Geometrical  and  Arithmetical  progression : 
in  one  direction  distances  proceed  arithmetically,  or  by  equal 
differences ;  in  the  other  geometrically,  or  by  equal  factors. 

These  curves  will  do  for  any  base,  if  their  scale  is  suitably 
interpreted.  The  divisions  we  have  labelled  2,  4,  8,  etc.,  may 
equally  well  be  considered  to  represent  3,  9,  27,  etc.,  or 
a,  a%  a^,  etc.,  or  10,  100,  1000,  etc. 

That  is  the  advantage  of  a  curve.  Once  drawn,  it  represents 
to  the  eye  a  general  kind  of  relationship ;  and  nothing  but  an 
interpretation  of  its  scale  is  necessary  to  make  it  fit  any 
required  instance  of  that  relationship.  The  shape  of  the  above 
suggested  curve  is  drawn  on  pages  101  and  179. 


168  EASY  MATHEMATICS.  [CHAf. 

Verify  the  following  statements  : 

3  =  log  27  to  the  base  3, 

4  =  log3  81, 
6  =  logs  729, 

.2  =  log,  16, 

4  =  log,  256, 

5  =  log,  1024, 

2  =  logs  36, 

3  =  loge216, 
3  =  logy  343, 
3  =  log,  729, 
2  =  log,  81, 

but  9  ^  log2  81, 

2  =  log,2 144, 

3  =  logi2  1728. 

This  last  seems  a  curious  and  roundabout  way  of  expressing 
the  fact  that  12x12x12  =  1728,  and  if  it  did  not  turn  out 
practically  very  convenient  there  would  be  no  justification  for 
introducing  such  a  complication  as  the  logarithmic  notation 
instead  of  the  index  notation;  but  it  is  constantly  to  be 
noticed,  when  a  new  notation  has  been  introduced  into  mathe- 
matics, that  it  confers  on  us  an  extraordinary  power  of 
progress,  and  enables  difficulties  further  on  to  be  dealt  with 
which  were  before  intractable. 

Any  complication  which  is  of  no  use — or  let  us  say  of  no 
obvious  and  well-known  use — anything  which  should  not  be 
familiar  to  every  educated  person — is  not  treated  of  in  this 
book ;  the  justification  of  any  notation  is  that  though  for  the 
expression  of  simple  and  already  well-known  facts  it  may  look 
cumbrous  and  inexpressive,  yet  when  we  want  to  express 
harder  and  at  present  unknown  ideas  it  becomes  helpful  and 
luminous. 


I 


xviii.]  LOGARITHMS.  l6& 

Common  practical  base. 

The  case  when   the  logarithmic  base  is  the  same  as  the 
base  adopted  for  our  system  of  numerical  notation,  is  worth 
special  attention,  because  it  is  the  one  most  frequently  used 
in  practice.    What  the  base  for  numerical  notation  may  be,  is, 
as  we  know,  a  pure  convention ;  and,  as  we  have  explained,  it 
is  perhaps  an  unfortunate  but  now  irremediable  convention 
that  the  base  of  notation  is  ten.     It  does  not  follow  that  the 
logarithmic  base  must  also  be  ten :  it  is  perhaps  possible  to 
find  a  natural  base,  involving  no  convention.     If  so,  such  a 
base   would   of   course   be   important  and  interesting;    but 
meanwhile  we  will  take  ten  as  the  base  also  of  a  practical 
system  of  logarithms. 
Let  us  first  make  a  table  of  powers  of  ten. 
1  =  100, 
10  =  101,  _i^  =  10-1, 

100  =  102,  ^  ^  10-2, 

1000  =  103,  ^V^  =  10-3, 

1,000,000  =  10^         ..ooi.ooo  =  10-^ 

Whence  it  follows  that  (with  base  10) 
to  ^^l  logl  =  0, 


log  10  =  1 

log  100  =  2 

log  1000  =  3 

log  1,000,000  =  6 


log-rV  =  log-1  =  -1, 

logTk  =  log-01  =  -2, 

logToVo  =log-001         =  -3, 

log,.ooi.ooQ  =  log -000001  =  -6, 


Hence  (with  base  10)  the  logarithms  of  numbers  between 
10  and  100  lie  between  1  and  2,  that  is  to  say  consist  of  1 
and  a  fraction :  the  log  of  1 1  will  be  1  and  a  small  fraction, 


170  EASY  MATHEMATICS.  [chap. 

the  log  of  99  will  be  1  and  a  large  fraction — very  near  to  2  in 
fact.  Consequently  with  all  double-digit  numbers  the  char- 
acteristic property  of  the  logarithm  is  that  it  begins  with  1. 

All  numbers  which  consist  of  three  figures  lie  between  100 
and  1000,  and  these  have  the  characteristic  2 ;  that  is  to  say 
they  all  consist  of  2  +  a  fraction.  This  is  true  even  of  such  a 
number  as  999-99,  provided  the  9's  are  not  repeated  for  ever; 
because  although  the  log  of  such  a  number  is  very  nearly  3, 
it  is  not  quite  3  until  1000  is  reached. 

A  number  consisting  of  five  digits  will  have  a  log  whose 
characteristic  is  4,  and  so  on;  the  characteristic  is  always 
equal  to  the  number  of  digits  on  the  left  of  the  unit  digit, 
which  is  taken  as  a  zero  of  reckoning.  Thus  the  characteristic 
of  the  log  of  any  of  the  following  numbers  (1200,  1728,  5760, 
9898,  1431-8,  1696-25)  is  3. 

The  logarithm  of  every  fraction  between  0  and  '1  will  be  a 
negative  fraction  :  it  will  not  be  quite  equal  to  -  1,  but  it  may 
be  put  equal  to  -  1  plus  a  positive  fraction. 

The  logarithm  of  every  number  between  *!  and  '01  will  lie 
between  -  1  and  -  2,  and  therefore  may  be  expressed  either 
as  —  1  minus  a  fraction,  or  as  -  2  plus  a  fraction ;  and  the 
latter  is  the  usual  plan. 

The  rule  for  the  characteristic  therefore  is  to  count  always 
to  the  first,  i.e.  the  most  important  significant  figure,  starting 
from  the  units  place  as  zero.  On  this  method  of  expression 
it  is  easy  to  write  down  the  characteristic  of  the  logarithm  of 
any  number  at  sight. 

The  best  plan  is  to  employ  the  term  "  order,"  connoting  by 
the  (yrder  of  the  number  the  index  of  the  power  to  which  the 
base  say  ten  must  be  raised  in  order  to  give  a  number  with 
that  number  of  digits.  E.g.  the  order  of  100  is  2,  because  it 
equals  10^,  and  all  the  numbers  121,  256,  780,  900  may  be 
technically    designated  as   of   the   same    "order";    because. 


XVIII.]  LOGARITHMS.  171 

though   greater   than   10^,  they  are  less  than  10-^ ;   and  the 
amount  by  which  they  exceed  10-  is  shown  by  the  fractional 
part  of  the  logarithm,  not  by  its  integer  part  or  characteristic. 
But  1000  is  of  the  order  3,  and  so  likewise  is  1728,  etc. 
17  is  of  the  order  1,  and  so  is  14-58, 
4  is  of  the  order  0,  and  so  is    4*6, 
•3  is  of  the  order  -  1,  and  so  are  -35  and  -78, 
•02  and  -035  and  -016  are  of  the  order  -  2. 
Accepting  this  nomenclature,  which  is  useful  in  quite  rudi- 
mentary arithmetic,  e.g.  in  long  division  and  the  like,  we  are 
able  to  say  simply  that  the  characteristic  of  a  logarithm  is  the 
"  order  "  of  its  number. 

Let  there  be  no  confusion  between  the  table  on  page  169 
and  the  one  on  page  167  to  base  2.  They  involve  different 
bases ;  and  though  the  base  is  not  expressed  every  time,  but 
only  in  the  heading,  that  is  merely  because  of  the  needless 
trouble  of  frequently  printing  or  writing  suffixes,  like  this : 
3  =  log2  8  =  logs  216  =logio  1000. 

Examples. 

The  characteristic  of  the  logarithm  of  the  following  numbers 
(also  called  the  "  order  "  of  the  number  itself)  is  as  here  given: 
The  logarithm,  to  base  10,  of  each  of  the  numbers 
5,  8-7,  1-23,  9-99,  Mil  has  the  characteristic  0 ; 
and  this  is  the  '*  order  "  of  each  number. 
Of  each  of  the  numbers 

11  1  the  order  is      1 
of  300,        981-4,       101-01,  it  is      2 

it  is  -  1 
it  is  -2 
it  is  -  3 


17, 

94, 

17-65, 

300, 

981-4, 

101-01, 

•17, 

-94, 

-11101, 

-08, 

•01, 

•0999, 

•002, 

-0056, 

-009846, 

108-001, 

17-9909, 

5-0652, 

8  X  106, 

1-0056, 

10-001, 

256000, 

•0000256. 

ly^  Easy  mathematics.  [chap. 

Examples  for  Practice. 

Write  down  the  characteristic  of  the  logarithm  of  each  of 
the  following  numbers,  (in  other  words  express  the  "  order  "  of 
each  number) : 

56,  108,  56-75, 

8-3,         8300,        5065-2, 

-56,         -0056,       -008309, 

99-9,       9-9,  -099, 

Fundamental  relations. 

There  are  a  few  fundamental  properties  appropriate  to 
logarithms  belonging  to  any  base  whatever. 

One  of  them  is  that  log  1=0, 

and  another  is  that        log  (base)  =  1, 

but  there  are  others  which  we  have  already  several  times 
hinted  at. 

Let  us  recollect  once  more  what  an  index   or   exponent 
signifies.      It  signifies  the   number  of  similar  factors  which 
have  to  be  multiplied  together.     For  instance, 
25  =  2x2x2x2x2, 
64  =  6x6x6x6, 
or,  in  general,  a^  =  axaxa, 

a**  =  axaxax...  to  n  factors. 
So  now  if  we  write  a  number  N  equal  to  any  of  these,  as  for 
instance  N  =  a^, 

the  index,  or  exponent,  which  we  shall  now  call  the  logarithm 
of  i\r  to  the  base  a,  simply  counts  the  number  of  times  the 
base  occurs  as  a  factor  in  the  number  N. 
N  =  axax.a. 

Suppose  now  we  took  some  number  not  quite  so  easy  to 
deal  with  as  those  in  the  examples  we  have  hitherto  considered. 


XVIII.]  LOGARITHMS.  173 

a  number  which  cannot  be  represented  as  any  simple  power 
of  any  integer,  say  for  instance  the  number  30 ;  and  ask  what 
will  the  logarithm  of  30  be  to  the  base  10. 

First  of  all  we  see  that  it  must  be  between  1  and  2,  because 
1  is  the  logarithm  of  10,  while  2  is  the  logarithm  of  100;  and 
the  logarithm  increases  with  the  number,  but  arithmetically 
instead  of  geometrically.  So  as  30  lies  roughly  half  way 
geometrically  between  10  and  100,  it  may  be  expected  that 
its  logarithm  will  be  somewhere  about  halfway  arithmetically 
between  1  and  2.  It  will  be  1  +  a  fraction ;  and  what  that 
fraction  is  can  be  approximated  to  more  or  less  closely  by 
examining  and  measuring  the  logarithmic  curve  which  we 
ought  to  have  carefully  drawn,  as  indicated  on  p.  179,  and 
specially  labelled  so  as  to  suit  the  base  10.  Measuring  that 
curve  for  the  logarithm  of  30,  it  suggests  a  value  something 
like  IJ  or  1'5.  This  would  be  the  result  to  "two  significant 
figures,"  but  if  the  curve  has  been  carefully  drawn,  it  might 
give  us  3-figure  accuracy,  that  is,  would  enable  us  to  express 
the  result  correctly  to  3  significant  figures ;  in  that  case  we 
might  estimate  log  30  to  be  about  1 48. 

The  number  which  really  lies  geometrically  half-way 
between  10  and  100  would  be  V(IOOO),  since  10:n/1000  = 
a/IOOO  :  100;  and  ^(1000)  is  accordingly  called  the  geometrical 
mean  of  ten  and  100.  Hence  the  logarithm  of  VlOOO  is 
exactly  1*5  or  1^.  Similarly  the  logarithm  of  mJIO  is  '5  or  J. 
It  is  a  curious  thing  that  though  we  do  not  yet  know  how  to 
calculate  the  root  of  10,  we  know  its  logarithm;  and  this 
suggests — what  frequently  happens — that  the  logarithm  of 
the  result  of  an  arithmetical  operation  is  easier  to  perceive 
than  the  result  itself. 

We  can  examine  the  curve  again,  to  see  if  it  will  show 
us  what  number  has  a  logarithm  exactly  0*5;  we  shall  see 
that  it  indicates  something  like  3-1   or  3*2,  and  if  it  were 


iT4  EASY  MATHEMATICS,  [chap. 

drawn  carefully  it  might  indicate  3*16.  This  is  one  of  the 
values  that  we  ought  already  to  have  arrived  at  by  trial  and 
error,  as  recommended  on  p.  152  ;  taking  different  numbers 
between  3'1  and  3 "2  and  squaring  them,  to  see  how  nearly 
the  square  would  approach  10. 

No  number  that  we  can  select  will,  when  squared,  exactly 
equal  ten.  It  has  no  square  root  that  can  be  expressed 
numerically  with  exactness.  Nor  has  any  number  except  the 
square  numbers,  1,  4,  9,  16,  25,  etc.  These  numbers,  i.e.  the 
group  ordinarily  denoted  by  these  symbols,  are  square 
numbers  in  any  system  of  notation,  their  square  roots  can  be 
numerically  expressed  precisely  ;*  and  for  no  other  numbers 
can  the  same  be  done,  however  many  fractions  or  combina- 
tions of  fractions,  or  however  many  decimal  places,  are 
employed.  Nor  can  it  be  done  in  any  other  system  of 
notation.     In  other  words, 

x/2,  V3,  VS,  J7,  n/1000,  etc.,  etc., 

are  all  incommensurable. 

But  their  logarithms  are  easily  expressed,  to  any  base 
whatever,  in  terms  of  the  logarithm  of  the  number  itself  to 
the  same  base.     Thus 

log   V2  =  J   log  2, 

log   V3  =  J   logs, 

log   x/5  =  J   log  5, 

logVlO  =  JloglO. 

*  Caution. — It  is  not  intended,  and  it  is  not  true,  that  the  above  digits 
express  square  numbers  when  interpreted  in  accordance  with  any  scale 
of  notation ;  for  instance,  the  amount  of  money  represented  by  2/5  is 
not  a  square  number  of  pennies,  but  the  number  we  are  accustomed 
to  designate  by  25  is  a  square  number,  and  25  coins  can  be  easily 
arranged  to  form  a  square. 


XVIII.]  LOGARITHMS.  175 

Hence  to  the  base  10, 

logViO  =  J  log  10  =  0-5, 

logVlOO  =         J  log  100  =  1-0, 

logv/1000  =       J  log  1000  =  1-5, 

logv/10000  =    i"iog  10000  =  2-0, 

logVlOOOOO  =  J  log  100000  =  2-5. 

Similarly  we  may  guess  that 

log^lO  =  -J  log  10  =  -3333..., 
and  so  we  may  refer  to  the  curve  and  see  what  number  has 
the  logarithm  J,  for  that  will  be  the  cube  root  of  ten.  We 
find  that  it  is  2-1544347,  or  approximately  2*14;  and  if  we 
multiply  this  by  itself  three  times,  2'14x2-14x2-14,  we  shall 
get  a  number  not  far  off  ten — a  trifle  greater  than  ten. 

Similarly  21*4  will  be  approximately  the  cube  root  of  ten 
thousand,  and  214  of  ten  million. 

No  exact  numerical  specification  of  the  cube  roots  of  any 
number  can  be  given,  except  of  the  cube  numbers,  that  is, 
those  numbers  which,  given  in  the  form  say  of  marbles,  can 
be  built  up  to  represent  cubes ;  namely  such  numbers  as 

1,  8,   27,   64,   125,  and  so  on. 
For   the   cube   root   of  any   other    number,    if   it   could   be 
expressed,  would  be  a  fraction ;  and  a  fraction  multiplied  by 
itself  necessarily  remains  a  fraction;   it  can  never  yield  an 
integer.     You  cannot  fractionate  a  fraction  into  a  whole. 
This  remark  is  further  developed  in  Chapter  XX. 
We  now  know  how  to  find  the  logarithm,  to  the  base  ten, 
of  any  power  of  ten,  whether  integral,  negative,  or  fractional. 
Examples : 

log  103  =  3^     log  10-3  =   -  3,     log  10*  =  J ; 
log  104  =  4^     log  10-4  =   -  4,     log  10*  =  J ; 
so  generally  log^^lO*  =  a;, 

and  this  may  be  easily  generalised  so  as  to  apply  to  any  base. 


176  EASY  MATHEMATICS.  [chap,  xviii. 

For  log  ft*  =  X  log  ft, 

but  we  know  that  log  (base)  =  1, 

so  we  see  that  the  logarithm  of  any  power  of  the  base  is  equal 
to  the  index  or  exponent  of  the  power,  or 
log  (base)*  =  x. 

We  have  thus  arrived  at  the  original  definition  of  a 
logarithm  from  which  we  started, — having  reasoned  *'in  a 
circle." 

The  advantage  of  reasoning  in  a  circle  is  that  we  thereby 
check  and  verify  to  some  extent  the  intermediate  steps,  for  if 
any  of  them  had  been  inconsistent  we  could  not  have  worked 
round  to  our  starting  point ;  unless  indeed  we  had  happened 
to  make  a  pair  of  errors  which  cancelled  each  other :  a  thing 
which  is  sometimes  done — especially  when  the  conclusion  is 
consciously  in  our  minds.  Working  round  a  circle  of  reasoning 
is  in  that  case  no  adequate  check.  It  is  not  possible  to  get 
round  by  any  odd  number  of  errors,  but  with  an  even  number 
of  errors  it  is  possible  though  not  very  probable;  unless 
indeed  we  know  our  destination  too  well  beforehand. 

The  real  test  of  truth  is  that  it  shall  turn  out  to  be 
consistent  with  everything  else  which  we  know  to  be  true^ 
No  one  chain  of  reasoning,  however  apparently  cogent,  is  to 
be  absolutely  trusted, — for  there  is  always  the  danger  of 
oversight  due  to  defective  knowledge.  Complete  consistency 
i-s  the  ultimate  test  of  truth ;  and  convergence  of  a  number  of 
definite  lines  of  reasoning  is  an  admirable  practical  test. 


CHAPTER   XIX. 

Further  details  about  logarithms. 

Involution  and  evolution  become  easy  directly  we  employ 
logarithms : 

To  obtain  any  root,  say  the  r*^  root,  of  any  number : 

Find   the   logarithm   of   the   given   number   to   any  base, 

calculate  -th  of  this  logarithm,  then  find  the  number  which 
r 

has  this  value  for  its  logarithm  to  the  same  base;  that 
number  is  the  ?'*^  root  of  the  given  number. 

Or  put  it  thus :  utilising  the  logarithmic  curve,  page  179. 

Take  a  length  on  the  horizontal  line  as  representing  the 
given  number;  find  its  logarithm,  as  the  vertical  distance  to 

the  curve  at  this  point ;  calculate  -th  of  this  length,  and  find 

on  the  curve  a  point  whose  vertical  height  is  equal  to  it; 
then  the  foot  of  the  perpendicular  from  this  point  marks  out 
on  the  horizontal  line  a  length  which  represents  the  Hh  root 
of  the  given  number,  on  the  same  scale  as  the  number  itself 
was  represented.  Thus,  for  instance,  \y6.  of  the  height  of  the 
curve  at  division  8,  projected  back  horizontally,  should  meet 
the  curve  above  the  division  2  ;  because  2  is  the  cube  root  of  8. 
You  see  it  is  worth  while  to  draw  the  curve  neatly  and 
carefully,  so  that  fairly  correct  measurements  may  be  made 
upon  it.  Besides,  accurate  drawing  is  a  useful  art,  and  it 
takes  a  little  time  to  employ  drawing  instruments  accurately 
so  as  to  make  no  blots  or  smudges,  and  to  get  all  lines 
uniformly  thick  and  accurately  passing  through  the  points 

L.E.M.  M 


178  EASY  MATHEMATICS.  [chap. 

intended.  It  is  an  art  worthy  of  cultivation  for  future  use. 
Much  information  can  be  gained  from  such  curves,  not  only 
in  science  but  even  in  business  and  in  politics. 

It  may  be  said  that  by  this  process  of  drawing  and 
measuring,  a  logarithm  or  a  root  can  after  all  only  be  attained 
approximately.  Yes,  but  the  same  is  true  of  any  process, 
so  far  as  accurate  expression  is  concerned.  A  logarithm  or 
a  root  in  general  requires  an  infinite  series  of  digits  to  express 
it ;  all  finite  expression  is  approximate. 

I  do  not  however  say  that  a  mathematician  would  calculate 
logarithms  or  roots  by  such  a  curve:  he  would  know  plenty  of 
other  contrivances  for  such  things,  and  perhaps  we  may  know 
some  of  them  later  on  ;  but  he  would  not  despise  the  curve 
method,  at  least  in  more  really  difficult  investigations.  He 
would  use  it  frequently.  But  for  mere  logarithms  he  would 
use  a  table,  somewhat  as  indicated  in  Chapters  XXIX.  and  XXX. 

Now  let  us  see  if  we  can  calculate  a  few  other  logarithms. 
We  can  obtain  any  we  want  from  the  curve,  but  if  we  could 
obtain  a  few  once  for  all,  and  label  them,  and  then  be  able 
to  express  the  logarithms  of  other  numbers  in  terms  of  these, 
it  might  save  us  time  and  trouble ;  and  besides  it  is  a  de- 
sirable and  useful  thing  to  be  able  to  do. 

We  have  managed  to  find  the  logarithm  of  any  power  of 
ten  (p.  175),  let  us  see  if  we  can  manage  the  logarithm  of  any 
product  containing  ten  as  one  of  its  factors. 

We  have  indeed  already  tried  one  of  them,  viz.  30  (see 
p.  173);  let  us  try  another,  say  50,  likewise  to  the  base  10. 

We  want  to  find  to  what  power  10  must  be  raised  in  order 
to  equal  50. 

Let  X  be  the  power,  then 

50  =  10* 
is  an  equation  from  which  we  have  to  find  x. 

This  is  only  another  mode  of  stating  that  x  =  log^Q  50. 


X 

■   -    -                                           J_ 

it                         J" 

2                 J: 

Ui 

^ 

h-                                        ^  ' 

t                                    x^ 

T                                       °'                                          9      a 

t                                                                       ><      T3^ 

4                                            M'  -i-T 

1                                                op^-C 

"               "    "I^X 

gi_|:X 

1                                        L     X  n' 

t                           3  5::t: 

V                             r    ir-s 

I               f-            4a  XX 

4                           :s  T3^ 

1                       ^  i:r 

ii                         _g  ^^ 

+             CO           _o  j:_|_ 

\                                                0      T    '52 

4                                   B     -fcX 

J                                       S     ^  5 

JL                «                _x    It  -^ 

L                              C    ^3^ 

t                              C   XX 

4                      2  ill: 

4                               S     S— ii 

JL           *            _j5  js-4: 

C                       ^  ^X 

V                              IS    4fl^i 

A                                    19      g^T 

4                          -r  r 

"                      ^  S- 

^                       is"ir 

4                          i|  x 

V-                                   1  -X- 

4t                      l-i: 

1         <N                         2r  cj- 

r 

5 

V 

^ 

;s^ 

5 

s 

^ 

^^=^^ 

y            "6                 in|                    g~         in                 o                 in                 o 

oj-               ^            6                ^                -                pj 

XIX.]         .  LOGARITHMS.  181 

But  now  resolve  50  into  two  factors,  and  write 

5  X  10  =  10% 

10* 
then  5=i^=W-^; 

hence  x-1  =  log  5, 

or  a;  =  1  +log5  =  1-7  about  (by  the  curve). 

Thus  log  50  =  log  5  + log  10, 

which  is  a  special  case  of  a  general  assertion  that 

log  nm  =  log  n  +  log  m. 
Examine  this  : 

Let  n  =  a%  so  that  x  =  log  w, 
and  let  m  =  a^,  so  that  y  =  log  m ; 
then  nm  =  a*o?  =  (f^^-, 
.'.  log  nm  =  %-\-y  =  log  n  +  log  m. 
Hence  by  using  logarithms,  multiplication  is  turned  back 
into  addition,  just  as  involution  was  turned  back  into  multi- 
plication.    So  also  division  is  turned  into  subtraction,  just  as 
evolution  was  turned  into  division. 

The  fundamental  relations  are  as  follows ;  and  although  we 
have  stated  them  several  times  before,  they  are  supremely 
important  and  will  bear  repetition. 

Let  a*  =  n  and  a"  =  m, 

so  that  X  =  logn  and  y  =  logm, 

then  nm  =  a*ft*  =  a"'^" ; 

.*.   lognm  =  x  +  y  =  logn  +  logm. 

Furthermore,  —  =  —  =  a*-*; 

.*.   log£  =  o^-y  =  logn-logm. 

Moreover,  n"^  =  {ay  =  a^  -, 

.'.   logTi"*  =  xy  =  mlogw. 


182  EASY  MATHEMATICS.  ,      [chap. 

Likewise  log  %"'"  =  -  m  log  w, 

„  log  71^  =  log'^n  =  -log  71. 

Apply  these  ideas.    We  can  write  at  once  that  to  the  base  10 
log  5000  =  log  5  +  log  1000  =     3  +  log  5, 
log   500  =  log  5  + log   100=     2  + log  5, 
log     50  =  log  5  + log     10=     l+log5, 
log       5  =  log  5  + log       1  =     0  +  log5, 
log  ^jj  =  log      -5  =  log  5  -  log     10  =  -  1  +  log  5, 
log     -05  =  log  5  -  log    100  =  -  2  +  log  5, 
log  -005  =  log 5 -log  1000  =  -3  +  log5. 
If  we  know  log  5  therefore  we  should  know  the  logarithm 
of  five  times  any  power  of  ten,  or  even  of  five  times  any  root 
of  ten ;  for 

log  5^10  =  log  5  + 1  log  10  =  -5  +  log  5, 
log  571000  =  logs  4-i  log  1000  =--  1-5  + log  5, 
log  vfo  =  Iog5-ilogl0  =--5  +  log5, 
log  5^10  =  Iog5  +  Jlogl0  =  •3  +  log5; 
but  this  is  perhaps  hardly  worth  stating. 

How  are  we  to  find  log  5?    We  can,  if  we  choose,  express 
it  by  means  of  log  2,  thus  : 

log5  =  log-V-  =  Iogl0-log2  =  1  -log2, 
or  log   2  + log   5  =  1. 

Similarly  log  20  +  log  50  =  3, 

log  20  + log   5  =  2. 
So  also  log    2  + log   6  =  log     12, 

log  3  + log  4  =  log  12, 
log  7  + log  9  =  log  63, 
log  8  + log  8  =  log  64, 
log  9  + log  9  =  log  81, 
log  17  + log  13  =  log   221, 


XIX.]  LOGARITHMS.  183 

log  6  -  log  2  =  log  3, 
log  9  -  log  3  =  log  3, 
log  4 -log  3  =  log  l'^, 
log  5 -log  2  =  log  1-5, 
log  5  -  log  3  =  log  1  '6, 
log  7  -  log  5  =  log  1  -4, 
log   9+ log  16  =  log   144, 

2  log  12  =  log   144, 

3  log  12  =  log  1728, 
ilogl2  =  logV12  =  log  2^3  =  log2  +  logV3 

=  log2  +  Jlog3, 
ilogl6  =  log4, 
Jlog    8  =  log2, 
-1- log  49  =  log  7, 
ilog25  =  log5, 

Jlog72  =  -Uog(36x2)  =  log6H-Jlog2 
=  Jlog(  9x8)  =  log3  +  log2V2 

=  log  3  +  log  2  +  J  log  2. 
This  might  have  been  set  as  an  exercise.     Prove  that 

I  log  72  =  log  3 +  1-5  log  2. 
One  way  to  prove  it  would  be  to  double  both  sides, 
log  72  =  2log3  4-31og2 
=  log  32  + log  23 
=  log  9  +  log  8 
=  log(9x8) 
=  log  72.  Q.E.D. 

Exercises. — Verify,  by  means  of  the  curve  in  this  chapter, 
the  following  approximate  statements, 

log  2  =  -3,   log  4  =  -6,   log  8  =  -9, 
710  =  3-16...,  710  =  2-15.,., 

711-6  =  3-4...,  7115  =  4-86..., 

77  =  2-6...,  7841  =  29-0. 


CHAPTER  XX. 
On  incommensurables  and  on  discontinuity. 

By  this  time  it  should  have  struck  pupils  with  any  budding 
aptitude  for  science,  and  for  such  alone  is  this  particular 
chapter  written,  that  it  is  strange  and  rather  uncanny,  un- 
expected and  perhaps  rather  disappointing,  that  magnitudes 
should  exist  which  cannot  be  expressed  exactly  by  any  finite 
configuration  of  numbers :  not  only  that  they  should  exist, 
but  that  they  should  be  common.  Draw  two  lines  at  right 
angles  from  a  common  point,  each  an  inch  long;  then  join 
their  free  ends,  and  measure  the  length  of  the  joining  line 
(which  is  often  called  the  hypotenuse  of  the  right-angled 
isosceles  triangle  that  has  been  constructed) :  that  is  one  of 
the  quantities  that  cannot  be  expressed  numerically  in  frac- 
tions of  an  inch,  i.e.  in  terms  of  the  sides.  Its  value  can  be 
approximated  to  and  expressed,  say  in  decimal  fractions  of  an 
inch,  to  any  degree  of  accuracy  we  please;  but  the  more 
carefully  it  is  measured  the  more  figures  after  the  decimal 
point  will  make  their  appearance :  the  decimal  is  one  that 
never  stops  and  never  recurs.  An  infinite  number  of  digits 
are  necessary  for  theoretical  precision,  though  practically  six 
of  them  would  represent  more  accuracy  than  is  attainable  by 
the  most  careful  and  grown-up  measurement.  It  is  therefore 
incommensurable,  and  can  only  be  expressed  exactly  by  another 
incommensurable  quantity,  viz.  in  this  case  the  square  root 


CHAP.  XX.]  INCOMMENSURABLES.  185 

of  2.  The  length  is  ^2  times  an  inch,  or  about  1-4142  ... 
inches.  Draw  a  square  upon  it  and  it  will  be  found  to  be 
two  square  inches  in  area.  That  is  just  the  fact  which  (when 
proved)  enables  us  to  assert  that  each  of  its  sides  is  of  length 
J^ ;  since  that  is  the  meaning  of  the  phrase  "  square  root." 
It  may  be  proved  by  the  annexed  figure : 


Fig.  11. 


Where  the  shaded  area  ABC  is  an  isosceles  right-angled 
triangle,  the  area  of  which  is  repeated  several  times  in  the 
figure;  four  times  inside  a  square  drawn  on  the  hypotenuse 
AB,  and  twice  inside  a  square  drawn  on  one  of  the  sides  AC. 
Wherefore  the  square  on  ^^  is  twice  the  square  on  AC. 

Observe  however  that  there  is  nothing  necessarily  incom- 
mensurable about  a  hypotenuse  itself :  it  is  only  incommen- 
surable when  the  sides  are  given.  It  is  easy  to  draw  a 
hypotenuse  of  any  specified  length,  say  IJ  inches  long,  and  to 
complete  an  isosceles  right-angled  triangle ;  but  now  it  is  the 
sides  that  will  be  incommensurable.  The  real  incommensura- 
bility is  not  a  length,  but  a  ratio,  that  is  a  number  from 
which  dimensions  have  cancelled  out.  No  length  is  incom- 
mensurable, but  it  may  be  inexpressible  in  terms  of  an 
arbitrarily  chosen  unit,  i.e.  it  may  be  incommensurable  with 
the  unit  selected,  and  the  chances  are  infinity  to  one  that  any 
length  pitched  upon  at  random  will  be  in  this  predicament. 
It  will  not  be  precisely  expressible  in  feet  or  metres,  nor  even 
in  fractions  of  them,  though  it  can  be  expressed  with  any 
degree  of  accuracy  required. 


186  EASY  MATHEMATICS.  [chap. 

The  hypotenuse  of  most  right-angled  triangles  will  be 
incommensurable  with  both  the  sides,  but  there  are  a  few 
remarkable  exceptions ;  one  in  especial,  known  to  the  ancients, 
viz.  the  one  where  the  sides  are  in  the  ratio  of  three  to  four. 
If  such  a  triangle  be  drawn,  with  the  sides  respectively  three 
inches  and  four  inches  long,  the  hypotenuse  will  be  found  to 
be  five  inches  long ;  the  more  accurately  it  is  measured  the 
nearer  it  approaches  to  5.  It  can  indeed  be  shown  theoreti- 
cally, it  is  shown  in  Euclid  I.  47,  that  it  equals  5  exactly :  a 
surprising  and  interesting  fact. 

With  an  isosceles  right-angled  triangle  however,  no  such 
simple  relation  holds :  the  hypotenuse  is  ^2  one  of  the  sides, 
and  ^2  is  incommensurable;  for,  as  we  have  previously 
suspected  and  may  now  see,  every  root,  whether  square  or 
cube  or  fourth  or  any  other  root,  of  every  whole  number,  is 
incommensurable,  unless  the  number  be  one  of  the  few  and 
special  series  of  squares  or  cubes  or  higher  powers.  Cf.  p.  1 75. 

To  prove  this  we  have  only  to  observe  that : 

The  square  or  any  higher  power  of  a  fraction  can  never  be 
other  than  a  fraction ;  for  you  cannot  fractionate  a  fraction 
into  a  whole. 

The  square  of  a  fraction  cannot  be  an  integer.  Hence 
no  integer  can  have  a  fraction  as  its  square  root.*  Yet  every 
integer  must  have  a  square  root  of  some  kind,  that  is  a 
quantity  which,  squared  or  multiplied  by  itself,  will  equal  the 
given  number;  but  this  quantity,  though  it  may  be  readily 
exhibited  geometrically  and  otherwise,  can  never  be  exhibited 
as  a  fraction,  i.e.  it  cannot  be  expressed  numerically  by  any 
means,  either  in  vulgar  fractions  or  in  decimals  or  in  duo- 
decimals or  in  any  system  of  numerical  notation;  in  other 
words,  every  root  of  every  integer  except  unity  is  incom- 
mensurable (incommensurable,  that  is,  with  unity  or  any 
*  Attend  here.     It  is  easy  to  miss  the  meaning. 


XX.]  INCOMMENSURABLES.  187 

other  integer),  except  of  those  few  integers  which  are  built 
up  by  repeating  some  one  and  the  same  integer  as  a  factor ; 
for  instance  the  following  set : 

4  =  2x2 
8=2x2x2 
9  =  3x3 
16  =  4x4 
25  =  5  X  5 
27  =  3  X  3  X  3 
32  =  2x2x2x2x2 
36  =  6  X  6 
49  =  7  X  7 
and  so  on ; 
which  class  of  numbers  are  therefore  conspicuous  among  the 
others  and  are  called  square  and  cube  numbers,  etc.     Every 
root  of  every  other  number  is  incommensurable,  and  most 
roots  of  these  are  too. 

Not  roots  alone  but  many  other  kinds  of  natural  number  are 
incommensurable  :  circumference  of  circle  to  diameter,  natural 
base  of  logarithms,  etc.,  etc. ;  everything  in  fact  not  already 
based  upon  or  compounded  of  number,  like  multiples,  etc. 

Incommensurable  quantities  are  therefore  by  far  the  com- 
monest, infinitely  more  common  in  fact,  as  we  shall  find,  than 
the  others :  "  the  others "  being  the  whole  numbers  and 
terminable  fractions  to  which  attention  in  arithmetic  is 
specially  directed,  which  stand  out  therefore  like  islands  in 
the  midst  of  an  incommensurable  sea;  or,  more  accurately, 
like  lines  in  the  midst  of  a  continuous  spectrum. 

What  is  the  meaning  of  this  ?     The  meaning  of  it  involves 
the  difference  between  continuity  and  discontinuity.     There 
is  something  essentially  jerky  and  discontinuous  about  number. 
Numerical  expression  is  more  like  a  staircase  than  a  slope :  it  V 
necessarily  proceeds  by  steps  :  it  is  discontinuous. 


188  EASY   MATHEMATICS.  [chap. 

A  row  of  palings  is  discontinuous :  they  can  be  counted, 
and  might  be  labelled  each  with  its  appropriate  number. 
Milestones  are  also  discontinuous,  but  the  road  is  continuous. 
The  divisions  on  a  clock  face  are  discontinuous  and  are 
numbered,  and,  oddly  enough,  the  motion  of  the  hands  is 
discontinuous  too  (though  it  need  not  theoretically  have  been 
so,  and  is  not  so  in  clocks  arranged  to  drive  telescopes).  The 
hands  of  an  ordinary  clock  proceed  by  jerks  caused  by  the 
alternate  release  of  a  pair  of  pallets  by  a  tooth  wheel — an 
ingenious  device  called  the  escapement,  because  the  teeth  are 
only  allowed  to  escape  one  at  a  time ;  and  so  the  wheels 
revolve  and  the  hands  move  discontinuously,  a  little  bit  for 
every  beat  of  the  pendulum,  which  is  the  real  timekeeper. 
The  properties  of  a  pendulum  as  a  timekeeper  were  discovered 
by  Galileo ;  an  escapement  of  a  primitive  kind,  and  a  driving 
weight,  were  added  to  it  by  Huyghens,  so  that  it  became  a 
clock. 

Telegraph  posts  are  discontinuous,  but  telegraph  wires  are 
continuous.  They  are  discontinuous  laterally  so  as  to  keep 
the  electricity  from  escaping,  but  they  are  continuous  longi- 
tudinally so  that  it  may  flow  along  to  a  destination. 

But,  now,  are  we  so  sure  about  even  their  longitudinal 
continuity  1  The  pebbles  of  a  beach  are  discontinuous,  plainly 
enough ;  the  sand  looks  a  continuous  stretch ;  but  examine  it 
more  closely,  it  consists  of  grains ;  examine  it  under  the 
microscope,  and  there  are  all  sorts  of  interesting  fragments  to 
be  found  in  it :  it  is  not  continuous  at  all.  The  sea  looks 
continuous,  and  if  you  examine  that  under  the  microscope  it 
will  look  continuous  still.  Is  it  really  continuous  1  or  would 
it,  too,  appear  granular  if  high  enough  magnifying  power  were 
available  1  The  magnifying  power  necessary  would,  indeed, 
be  impossibly  high,  but  Natural  Philosophers  have  shown 
good  reason  for  believing  that  it,  too,  is  really  discontinuous. 


XX.]  DISCONTINUITY.  189 

that  it  consists  of  detached  atoms,  though  they  are  terribly 
small,  and  the  interspaces  between  them  perhaps  equally 
small,  or  even  smaller.  But  even  so,  are  they  really  dis- 
continuous? Is  there  nothing  in  the  spaces  between  them, 
or  is  there  some  really  continuous  medium  connecting 
theml 

The  questions  are  now  becoming  hard.  Quite  rightly  so  ;  a 
subject  is  not  exhausted  till  the  questions  have  become  too 
hard  for  present  answer. 

There  are  several  curious  kinds  of  subterranean  or  masked 
continuity  possible,  which  may  be  noted  for  future  reference. 
Look  at  a  map  of  the  world ;  the  land,  or  at  least  its  islands, 
are  after  a  fashion  discontinuous,  the  ocean  is  continuous ;  but 
the  land  is  continuous  too,  underneath,  in  a  dimension  not 
represented  on  the  map,  but  recognisable  if  we  attend  to 
thickness  and  not  only  to  length  and  breadth. 

Human  beings  are  discontinuous :  each  appears  complete 
and  isolated  in  our  three-dimensional  world.  If  we  could 
perceive  a  fourth  dimension,  should  we  detect  any  kind  of 
continuity  among  them  % 

The  questions  have  now  become  too  hard  altogether;  we 
have  left  science  and  involved  ourselves  in  speculation.  It  is 
time  to  return.  A  momentary  jump  into  the  air  is  invigorat- 
ing, but  it  is  unsupporting,  and  we  speedily  fall  back  to  earth. 

But  how,  it  may  be  asked,  does  this  discontinuity  apply  to 
number?  The  natural  numbers,  1,  2,  3,  etc.,  are  discontinuous 
enough,  but  there  are  fractions  to  fill  up  the  interstices  ;  how 
do  we  know  that  they  are  not  really  connected  by  these  frac- 
tions, and  so  made  continuous  again  ?  Well,  that  is  just  the 
point  that  deserves  explanation. 

Look  at  the  divisions  on  a  foot  rule ;  they  represent  lengths 
expressed  numerically  in  terms  of  an  arbitrary  length  taken  as 


190  EASY  MATHEMATICS.  [chap. 

a  unit:  they  represent,  that  is  to  say,  fractions  of  an  inch; 
they  are  the  terminals  of  lengths  which  are  numerically 
expressed ;  and  between  them  lie  the  unmarked  terminals 
of  lengths  which  cannot  be  so  expressed.  But  surely  the 
subdivision  can  be  carried  further;  why  stop  at  sixteenths 
or  thirty  seconds  1  Why  proceed  by  constant  halving  at  all  ? 
Why  not  divide  originally  into  tenths  and  then  into  hun- 
dredths, and  those  into  thousandths,  and  so  onl  Why  not 
indeed  ?  Let  it  be  done.  It  may  be  thought  that  if  we  go  on 
dividing  like  this  we  shall  use  up  all  the  interspaces  and  have 
nothing  left  but  numerically  expressible  magnitudes.  Not  so, 
that  is  just  a  mistake  ;  the  interspaces  will  always  be  infinitely 
greater  than  the  divisions.  For  the  interspaces  have  all  the 
time  had  evident  breadth,  indeed  they  together  make  up  the 
whole  rule  ;  the  divisions  do  not  make  it  up,  do  not  make  any 
of  it,  however  numerous  they  are.  For  how  wide  are  the 
divisions'?  Those  we  make,  look,  when  examined  under  the 
microscope,  like  broad  black  grooves.  But  we  do  not  wish  to 
make  them  look  thus.  We  should  be  better  pleased  with  our 
handiwork  if  they  looked  like  very  fine  lines  of  unmagnifiable 
breadth.  They  ought  to  be  really  lines — length  without 
breadth ;  the  breadth  is  an  accident,  a  clumsiness,  an  unavoid- 
able mechanical  defect.  They  are  intended  to  be  mere 
divisions,  subdividing  the  length  but  not  consuming  any  of 
it.  All  the  length  lies  between  them ;  no  matter  how  close 
they  are  they  have  consumed  none  of  it ;  the  interspaces  are 
infinitely  more  extensive  than  the  barriers  which  partition 
them  off  from  one  another ;  they  are  like  a  row  of  compart- 
ments with  infinitely  thin  walls. 

Now  all  the  incommensurables  lie  in  the  interspaces;  the 
compartments  are  full  of  them,  and  they  are  thus  infinitely 
more  numerous  than  the  numerically  expressible  magnitudes. 
Take  any  point  of  the  scale  at  random :  that  point  will  cer- 


XX.]  INCOMMENSURABLES.  191 

tainly  lie  in  an  interspace  :  it  will  not  lie  on  a  division,  for  the 
chances  are  infinity  to  1  against  it. 

Let  a  stone — a  meteor — drop  from  the  sky  on  to  the  earth. 
What  are  the  chances  that  it  will  hit  a  ship  or  a  man  ]  Very 
small  indeed,  for  all  the  ships  are  but  a  small  fraction  of  the 
area  of  the  whole  earth ;  still  they  are  a  finite  portion  of  it. 
They  have  some  size,  and  so  the  chances  are  not  infinitesimal ; 
one  of  them  might  get  struck,  though  it  is  unlikely.  But  the 
divisions  of  the  scale,  considered  as  mathematically  narrow, 
simply  could  not  get  hit  accidentally  by  a  mathematical  point 
descending  on  to  the  scale.  Of  course  if  a  needle  point  is 
used  it  may  hit  one,  just  as  if  a  finger-tip  is  used  it  will  hit 
several ;.  but  that  is  mere  mechanical  clumsiness  again. 

If  the  position  is  not  yet  quite  clear  and  credible,  consider 
a  region  of  the  scale  quite  close  to  one  of  the  divisions  already 
there,  and  ask  how  soon,  if  we  go  on  subdividing,  another 
division  will  come  close  up  against  the  first,  and  so  encroach 
upon  and  obliterate  the  space  between  them.  The  answer  is 
never.  Let  the  division  be  decimal,  for  instance,  and  consider 
any  one  division,  say  5.  As  the  dividing  operation  proceeds, 
what  is  the  division  nearest  to  it  1 
At  first         4         of  course, 

then  4-9, 

then  4-99, 

then  4-999, 
and  so  on. 
But  not  till  the  subdivision  has  been  carried  to  infinity,  and 
an  infinite  number  of  9's  supplied  after  the  decimal  point,  will 
the  space  between  be  obliterated  and  the  division  5  be  touched. 
Up  to  that  infinite  limit  it  will  have  remained  isolated,  stand- 
ing like  an  island  of  number  in  the  midst  of  a  .  blank  of 
incommensurableness.  And  the  same  will  be  true  of  every 
other  division. 


192  EASY  MATHEMATICS.  [chap.  xx. 

Whenever,  then,  a  commensurable  number  is  really  associ- 
ated with  any  natural  phenomenon,  there  is  necessarily  a 
noteworthy  circumstance  involved  in  the  fact,  and  it  means 
something  quite  definite  and  ultimately  ascertainable. 

For  instance : 

The  ratio  between  the  velocity  of  light  and  the  inverted 
square  root  of  the  electric  and  magnetic  constants  was  found 
by  Clerk  Maxwell  to  be  1 ;  and  a  new  volume  of  physics  was 
by  that  discovery  opened. 

Dal  ton  found  that  chemical  combination  occurred  between 
quantities  of  different  substances  specified  by  certain  whole  or 
fractional  numbers ;  and  the  atomic  theory  of  matter  sprang 
into  substantial  though  at  first  infantile  existence. 

The  atomic  weights  are  turning  out  to  be  all  expressible 
numerically  in  terms  of  some  one  fundamental  unit;  and 
strong  light  is  thrown  upon  the  constitution  of  matter  thereby. 

Numerical  relations  have  been  sought  and  found  among  the 
lines  in  the  spectrum  of  a  substance ;  and  a  theory  of  atomic 
vibration  is  shadowed  forth. 

Electricity  was  found  by  Faraday  to  be  numerically  con- 
nected with  quantity  of  matter ;  and  the  atom  of  electricity 
began  its  hesitating  but  now  brilliant  career. 

On  the  surface  of  nature  at  first  we  see  discontinuity,  objects 
detached  and  countable.  Then  we  realise  the  air  and  other 
media,  and  so  emphasise  continuity  and  flowing  quantities. 
Then  we  detect  atoms  and  numerical  properties,  and  discon- 
tinuity once  more  makes  its  appearance.  Then  we  invent  the 
ether  and  are  impressed  with  continuity  again.  But  this  is 
not  likely  to  be  the  end ;  and  what  the  ultimate  end  will  be, 
or  whether  there  is  an  ultimate  end,  are  questions,  once  more, 
which  are  getting  too  hard. 


CHAPTER  XXL 

Concrete  Arithmetic. 

It  is  highly  desirable  that  arithmetical  practice  should  be 
gained  in  connexion  with  laboratory  work,  for  then  the 
sums  acquire  a  reality,  and  interest  is  preserved.  It  is 
absolutely  essential  that  all  concrete  subject-matter  be  based 
upon  first-hand  experience,  for  unless  that  can  be  appealed  to, 
abstractions  have  no  basis,  but  are  floating  unsupported  in 
air.  It  far  too  frequently  happens  that  a  child,  constrained 
to  do  sums  expressed  in  terms  of  weights,  has  never  weighed 
a  thing  in  its  life.  It  is  the  same  mistake  as  is  made  when 
a  child  is  drilled  in  the  formal  grammar  of  a  language  about 
which  it  knows  absolutely  nothing.  In  every  case  concrete 
experience  should  be  the  first  thing  provided,  and  abstractions 
may  follow.  The  teacher  is  apt  not  to  realise  this,  because 
grown  persons  have  necessarily  acquired  some  first-hand 
experience  in  the  ordinary  course  of  life ;  but  a  teacher  who 
is  really  educated  all  round  and  has  a  living  acquaintance 
with  a  great  number  of  subjects  should  be  able  to  enliven  a 
lesson  into  something  quite  exciting,  if  only  he  or  she  can 
cultivate  the  patience  necessary  to  allow  time  for  the  indi- 
viduals of  a  class  to  attain  some  first-hand  experience  for 
themselves. 

L,.E.M.  N 


194  EASY  MATHEMATICS.  [chap. 

This  is  the  real  object  of  school  laboratory  work,  and  the 
mathematical  teacher  should  seek  to  keep  in  touch  with,  and 
to  be  aware  of,  what  the  pupils  are  doing  under  other 
teachers,  so  as  to  illuminate  his  abstractions  with  concrete 
instances  and  examples.  By  far  the  best  kind  of  examples 
are  not  those  contained  in  books,  but  those  which  arise 
naturally  or  are  invented  by  a  stimulating  teacher  in  the 
course  of  his  exposition,  or  as  a  result  of  actual  manipulation 
on  the  part  of  the  taught. 

The  result  of  a  laboratory  measurement  is  always  an  incom- 
mensurable number;  for  the  mere  counting  of  a  number  of 
distinct  objects  is  not  to  be  called  a  laboratory  measurement. 
No  measurement  of  length,  for  instance,  could  ever  be 
expressed  as  a  whole  number  of  inches,  nor  yet  as  a  whole 
number  plus  a  definite  fraction  of  an  inch.  No  measurement 
that  ever  was  made  could  be  expressed  by  either  a  termi- 
nating or  a  recurring  decimal,  nor  by  a  vulgar  fraction; 
for  any  of  these  modes  of  specification  would  imply  infinite 
accuracy. 

Suppose  that  an  astronomical  measurement  is  expressible 
by  the  number  17 "4673,  it  is  absolutely  certain  that  3  cannot 
be  the  last  digit  of  the  series  if  it  is  to  be  expressive  of 
absolute  fact.  It  may  be  that  the  next  is  0,  and  perhaps  the 
next  also,  but  unless  you  can  guarantee  that  all  the  digits 
to  infinity  are  0,  the  only  reason  for  stopping  at  3  (and  it 
is  a  good  reason)  is  that  we  can  measure  no  more. 

So  a  decimal  expressing  the  result  of  measurement  cannot 
terminate,  neither  can  it  recur.  For,  suppose  the  result, 
as  nearly  as  we  could  get  it,  were  4*6666,  how  do  wc  know 
that  the  next  digit  is  going  to  be  6,  and  the  next  and  the 
next  also  1    We  cannot  know  it. 

If  it  did  recur  it  would  be  the  vulgar  fraction  4|;  hence, 
this  also  is  strictly  an  impossibly  accurate  result  of  measure- 


XXI.]  SIGNIFICANT  FIGURES.  195 

mcnt.     The  same  with  every  vulgar  fraction:  it  may  be  an 
approximate  result,  but  no  more. 

The  phrase  a  quarter,  or  a  half,  or  seven-eighths,  is 
appropriate  therefore  to  rough  specifications  of  approximate 
magnitude,  but  is  inappropriate  to  precise  specification  of 
anything  beyond  counting  of  objects  and  fractions  of  an  object. 
Measurements  should  be  expressed  in  decimal  notation,  and 
the  number  of  significant  figures  given  should  be  characteristic 
of  the  order  of  accuracy  of  the  work. 

The  meaning  of  significant  figures   and  practical 
accuracy. 

Rough  workshop  measurements  are  accurate,  let  us  say,  to 
3  significant  figures.  Students'  measurements  in  Physics, 
which  are  naturally  more  difficult  than  those  of  the  workshop, 
if  of  the  schoolboy  kind,  do  well  if  they  are  accurate  to  two 
significant  figures.  For  instance,  if  the  latent  heat  of  melting 
ice  came  out  79  or  80,  it  is  quite  as  good  as  can  be  expected. 
A  great  deal  of  trouble  is  necessary  to  get  a  third  figure  right, 
for  of  course  it  means  just  ten  times  the  accuracy.  A  good 
student  would  however  try  to  get  the  third  figure  right,  and 
might  succeed,  if  it  were  not  too  complicated  a  measurement. 
The  Demonstrator,  and  senior  students  who  give  some  months 
to  the  work,  would  aim  at  4  figure  accuracy,  and,  if  they 
attained  it,  would  do  well.  A  few  exceptionally  skilled 
experimenters  with  a  genius  for  the  work,  devoting  a  year  to 
a  research,  might  attain  5  figure  accuracy,  but  such  accuracy 
as  this  is  generally  limited  to  the  astronomical  observatory, 
where  the  measurements  are  fairly  simple  and  the  theory  of 
the  errors  to  which  instruments  are  necessarily  liable  has  been 
studied  for  centuries.  In  taking  the  mean  of  a  number  of 
astronomical  observations,  even  6  figure  accuracy  is  attainable, 
but  beyond  this  it  is  extremely  difficult  to  go. 


196  EASY  MATHEMATICS.  [chap. 

The  fundamental  measurements  that  have  to  be  made  are 
the  following : 

length 

time 

angle 

mass 
and  of  these,  oddly  enough,  length  is  by  far  the  hardest  to  do 
accurately,  though  the  easiest  to  do  approximately. 

Time  is  measured  with  considerable  accuracy,  even  by  a  pocket 
watch.  Suppose  the  watch  were  uncertain  by  3  seconds  a  day, 
it  would  not  be  bad.  If  it  lost  or  gained  regularly  it  would 
be  a  perfect  time  keeper,  for  a  regular  loss  can  be  estimated 
and  allowed  for ;  but  that  is  not  feasible  except  in  elaborate 
chronometers  carefully  preserved.  What  is  meant  by  the 
above  is  that  having  allowed  for  any  known  regular  loss,  it 
may  lose  or  gain  3  seconds  a  day  irregularly,  so  that  to  be 
quite  safe  we  might  consider  it  uncertain  to  the  amount  of 
plus  or  minus  3  seconds,  or  6  seconds  altogether.  There  are 
86,400  seconds  in  a  day,  so  the  outstanding  possible  error 
would  be  6  parts  in  86,400,  or  1  part  in  14,400,  or  7  parts  in 
a  hundred  thousand,  or  '007  per  cent.,  and  it  would  therefore 
be  liable  to  cause  a  bad  error  in  the  fifth  significant  figure — an 
error  which  even  slightly  affects  the  fourth.  Still  for  a  cheap 
watch  that  is  good  performance,  and  means  long  hereditary 
skill  on  the  part  of  makers  of  watches. 

You  could  not  hope  to  measure  a  mile  with  the  same 
accuracy  as  you  can  measure  the  length  of  a  day. 

Angles  are  not  very  difficult  to  measure,  because  a  number 
of  disturbing  causes  have  no  effect  on  the  divisions  of  a  circle. 
If  the  weather  gets  warmer  or  colder,  your  yard  and  other 
measures  change,  and  clock-pendulums  and  watch  hair-springs 
change  too;  but  though  a  circle  expands  and  its  divisions 
grow  wider  with  heating,  their  number  is  not  affected ;   the. 


XXI.]  SIGNIFICANT  FIGURES.  197 

expanded  circle  is  still  divided  into  360  equal  parts  or 
degrees.  There  is  something  essentially  numerical  about  the 
divisions  of  a  circle ;  and  measurement  of  angle  is  subject  to 
fewer  disturbing  causes  than  measurement  of  length. 

But  the  really  easy  thing  to  determine  accurately  is  mass  or 
weight.  For  it  never  changes  whatever  you  do  to  it.  The 
weight  of  a  piece  of  matter  is  constant,  whether  it  be  hot  or 
cold,  or  whether  it  be  evaporated  to  a  gas,  or  dissolved  in  a 
liquid,  or  whether  it  be  molten,  or  boiled,  or  vaporised,  or 
chemically  decomposed,  or  burnt  up,  or  subjected  to  any  other 
operation.  So  far  as  is  known,  its  weight  continues  absolutely 
unchanged;  although  in  combustion  it  appears  to  increase  in 
weight,  because  it  combines  with  other  things.  Moreover  the 
balance  is  an  easy  and  an  accurate  instrument.  Even  a 
beginner  can  weigh  on  a  reasonably  delicate  balance  to  4 
significant  figures,  that  is,  he  could  weigh  ten  grammes  to  the 
nearest  milligramme.     He  could  hardly  do  better  than  that. 

It  is  possible,  however,  with  elaborate  care  to  weigh  to  6 
significant  figures,  i.e.  to  weigh  10  grammes  to  the  hundredth 
of  a  milligramme ;  but  it  needs  a  good  balance  and  precaution 
against  currents  of  air,  dust,  warmth  of  observer's  body,  acci- 
dental electrification,  in  some  cases,  and  other  disturbing 
causes.  These  things  do  not  really  disturb  either  the  weights 
or  the  thing  weighed,  but  they  disturb  the  balance. 

However,  this  is  a  digression,  so  as  to  make  clear  what 
is  meant  by  a  reasonable  number  of  significant  figures.  We 
see  that  the  number  that  is  properly  to  be  recorded  will  de- 
pend upon  circumstances,  that  every  additional  figure  expressed 
is  a  claim  to  greater  accuracy,  and  that  it  is  always  better  to 
aim  at  too  many  than  too  few ;  but  we  should  cultivate  an 
instinct  for  knowing  when  we  have  recorded  as  many  as  the 
experiment,  or  the  observation,  or  the  circumstances  will 
justify. 


CHAPTER  XXII. 

Practical  manipulation  of  fractions  when  decimally 
expressed. 

Since  the  results  of  all  actual  measurements  yield  incom- 
mensurable numbers,  it  is  desirable  to  be  able  to  deal  with 
them  freely.  The  present  chapter  will  be  considered  very 
elementary,  but  it  is  inserted  thus  apparently  out  of  place  in 
order  to  emphasise  the  desirability  of  reintroducing  familiar 
matter  with  variations,  and  also,  more  particularly,  to  uphold 
the  doctrine  that  the  other  things  treated  are  equally  easy : 
ease  is  only  a  matter  of  use  and  custom. 

In  the  easy  manipulation  of  fractions  there  is  much  to  be 
learnt,  and  considerable  practice  is  necessary  to  attain  facility. 
It  is  not  worth  while  to  exaggerate  this  practice,  because  the 
resulting  art  is  not  an  accomplishment  capable  of  giving  plea- 
sure to  other  people,  like  some  other  arts  which  can  be 
attained  by  practice  ;  nevertheless,  some  practice  in  arith- 
metic is  essential,  and  on  this  part  of  the  subject  some  of  the 
time  which  has  been  saved  from  hogsheads  and  drachms 
can  be  usefully  and  interestingly  expended. 

First  of  all,  we  may  notice  that  the  manipulation  of  frac- 
tions is  much  simplified  when  they  are  stated  in  the  ordinary 
arithmetical  notation,  utilising  the  same  system  as  is  em- 
ployed for  whole  numbers.  The  ease  conferred  is  similar  to 
that  gained  by  abolishing  strange  denominations  of  every  kind. 

Thus  it  is  simpler  to  deal  with  17 '34  cwts.  than  it  is  to  deal 


CHAP.  XXII.]  DECIMALS.  199 

with  it  when  expressed  as  17  cwts.  1  quarter,  10  pounds,  1 
ounce,  4^f  drachms,  which  is  the  way  that  helpless  children 
are  constrained  to  deal  with  it. 

So  also  it  is  simpler  to  deal  with  a  fraction  when  expressed 
as  4-4  inches,  than  when  expressed  as  25  "mils"  more  than 
4  inches  and  3  eighths,  or  4  +  f  +  y|^o  inches,  which  is,  how- 
ever, the  way  the  British  workman  seems  to  prefer  to  have 
it  expressed — to  the  detriment  of  international  engineering 
operations. 

In  other  words,  it  is  always  simpler  to  express  a  thing 
numerically  in  a  single  denomination  than  to  employ  a  multi- 
tude of  denominations  or  denominators. 

Even  such  a  thing  as  1  + 1^  +  J  +  ^  +  yV  is  more  simply 
expressed  as  1*9375,  though  still  better  as  2  -  y\,  or  f  J.  Sim- 
plicity is  attained  by  use  of  a  single  denominator,  whether 
sixteenths  or  tenths,  or  whatever  it  may  be.  It  is  the  admix- 
ture of  denominations  or  denominators  that  is  troublesome. 

So  also  the  manipulation  of  fractions  when  expressed 
decimally  is  as  easy  as  the  manipulation  of  whole  numbers. 
Care  has  to  be  taken  about  the  position  of  the  digits  in  either 
case,  and  the  explicit  writing  of  the  decimal  point  almost  makes 
the  matter  easier.  The  essential  rule  is,  keep  the  decimal 
points  under  one  another,  and  they  will  then  keep  the 
places  of  the  digits  right. 

Thus,  add  together 

4-375  + -025 +  53-1. 

The  sum  is  written  4*375 

•025 
53-1 

57-500 

and  the  result  is  verbally  expressed  as  57  J. 

For  just  as  various  denominations,  inches,  weeks, 
months,  ounces,  tons,  gallons,  are  handy  in  speech  and  for 


200  EASY   MATHEMATICS.  [chap. 

realising  and  speaking  of  magnitudes  after  they  have  been 
calculated,  so  vulgar  fractions  are  often  handy  enough  to 
express  a  result  at  the  end.  When  they  are  complicated, 
however,  they  should  only  be  used  to  quickly  express  approxi- 
mate results.  For  instance,  5*12  inches  might  be  spoken  of 
as  about  5Jth  inches.  So  also  35*9  inches  might  be  spoken  of 
as  about  a  yard.  And  the  number  14 '34  might  be  spoken 
of  as  about  14 J.  For  instance,  if  it  expressed  a  length  in 
feet,  the  length  should  be  called  14  feet  4  inches  if  we  were 
speaking  to  a  carpenter.  And  similarly  5*67  feet  would 
be  approximately  5f  feet,  or  5  feet  8  inches. 

In  subtraction  just  the  same  rule  holds :  keep  the  decimal 
points  vertical.     E.g.  to  subtract  15*43  from  304, 

write  it  304-00 

15-43 

288-57 

and  there  is  nothing  more  to  be  said. 

In  countries  with  decimal  coinage,  this  is  all  the  arith- 
metic that  book-keeping  clerks  have  to  employ.  Although 
they  may  use  the  terms  dollars,  and  quarters,  and  dimes,  and 
cents  in  ordinary  speech,  they  do  not  express  a  sum  of  money 
after  our  fashion,  as 

Dollars.  Quarters.  Cents. 

17  3  18 

but  they  express  it  simply  as  17*93  dollars. 

So,  also,  if  another  amount  of  3  dollars,  2  quarters,  and 
17  cents  has  to  be  added,  it  is  never  expressed  in  that  way, 
but  as  3-67  dollars;  17'93 

3-67 


the  addition  is  then  quite  easy,  viz.  21*60  dollars.  All  addi- 
tion becomes  simple  addition ;  and  compound  addition  no 
more  exists. 

To  express  the  resulting  amount  in  the  form  given  by  com- 


XXII.]  DECIMALS.  201 

pound  addition  (which  try),  viz.  as  21  dollars,  2  quarters,  and 
10  cents  would  be  unnatural ;  but  it  might,  of  course,  be 
spoken  of  as  21  dollars  and  60  cents,  for  that  etymologically 
means  precisely  the  same  as  21  and  60  hundredths,  i.e.  21*60. 

The  use  of  variegated  and  picturesque  units,  like  weeks,  and 
fortnights,  and  centuries,  and  acres,  and  hundredweights,  and 
quarts,  is  to  relieve  the  monotony  of  conversation  ;  they  should 
not  be  introduced  into  the  workings  of  arithmetic.  The  end 
result  can  be  interpreted  into  them,  for  vivid  realisation,  as  occa- 
sion arises,  and  the  instructed  person  should  always  be  able  to 
speak  to  the  uninstructed  person  in  his  own  language.  For 
an  instructed  youth  to  expect  workmen  and  others,  who  have 
not  had  his  advantages,  to  appreciate  his  scholastic  mode  of 
expression,  is  barbarous,  and  shows  a  pitiful  lack  of  sense  on 
his  part. 

So  long  as  popular  units  exist  they  should  be  employed  in  the 
proper  place  :  they  are  part  of  folk-lore,  and  are  often  inter- 
esting enough  ;  it  is  only  when  they  are  allowed  to  get  out  of 
their  proper  place  and  spoil  the  lives  of  children  that  they  are 
to  be  condemned.     In  arithmetic  proper  they  are  out  of  place. 

Now  take  multiplication.  It  is  a  little  more  troublesome, 
of  course,  but  not  much. 

Keep  the  points  vertical,  as  before ;  in  other  words,  keep 
the  digits  expressive  of  the  same  denomination  under  each 
other,  i.e.  the  units  under  the  units,  the  tenths  under  the 
tenths,  etc. ;  then  the  denomination  of  the  answer  looks  after 
itself  without  any  trouble. 

For  instance,  multiply  30*57  by  4*3.     Write  it  thus: 

30*57 
4*3 


122*28 
9*171 

131*451 


202  EASY  MATHEMATICS.  [chap. 

I  need  not  have  written  the  last  figure  of  the  result,  for 
most  purposes  ;  for  since  the  data  are  only  given  to  two  places 
of  decimals,  an  appearance  of  three  decimal  places  in  the  result 
may  give  a  notion  of  spurious  and  deceptive  accuracy,  and  so  is 
often  better  eschewed. 

But  this  idea  of  approximate  accuracy  does  not  apply  to 
results  in  pure  mathematics,  such  as  the  properties  of  numbers, 
and  things  like  that :  it  is  the  results  of  practical  measurement 
that  are  not  wanted  to  impossible  accuracy,  just  as  the  price  of 
a  ship,  or  a  railway,  or  a  war,  is  not  wanted  closer  than  the 
nearest  penny,  if  indeed  so  close. 

I  may  say,  however,  that  when  we  are  dealing  with  the 
results  of  practical  measurement,  it  is  the  number  of  significant 
figures  in  the  whole  specification,  rather  than  the  number  of 
decimal  places,  which  is  the  thing  to  be  attended  to.  In  the 
above  sum  the  data  involved  four  significant  figures,  and  so  a 
sixth  significant  figure  in  the  result  would  be  without  meaning, 
and  ought  not  to  be  written. 

Now  take  a  further  example  in  multiplication :  suppose 
we  had  5-4306  grammes  to  multiply  by  70*2  :  the  Avhole  sum 
would  stand  thus : 

5  "4306    grammes 
70-2 


380-142 
1-08612 


381-22812  grammes 

The  weighing  was  only  given  to  5  figure  accuracy,  so  any- 
thing more  is  delusive  in  the  result.  Six  figures  may  perhaps 
be  permitted,  that  is  as  far  as  381*228,  but  the  last  two  figures 
after  this,  the  12,  which  are  really  -00012,  have  no  useful 
meaning,  and  need  never  have  been  written.  And  even  the 
8  is  quite  uncertain,  so  that  the  way  to  state  the  result  with 
the  same  accuracy  as  the  data  is  381*23  grammes.     Three  being 


XXII.]  dp:cimals.  •  203 

put  as  the  last  digit  instead  of  two,  because  the  next  digit, 
viz.  8,  carries  it  more  than  half  way  to  the  higher  figure. 

We  observe  then  that  when  we  multiply  by  a  figure  in  the 
units  place,  we  place  the  digits  of  the  product  under  the  cor- 
responding digits  of  the  multiplicand.  When  we  multiply  by 
a  figure  in  the  ten's  place,  we  shift  each  digit  one  place  to  the 
left.  If  we  multiplied  by  a  figure  in  the  hundred's  place  we 
should  shift  them  two  places  to  the  left.  Whereas  when  we 
multiply  by  a  digit  in  the  tenth's  place,  that  is  one  place  to  the 
right  of  the  decimal  point,  we  shift  the  resulting  figures  of  the 
product  one  place  to  the  right,  instead  of  writing  them  im- 
mediately under  the  corresponding  digits  of  the  multiplicand. 

The  rule  about  division  is  similar.  Let  us  divide  470-82  by 
5*7.     Write  it  in  its  first  stage : 

5-7)  470-82  (8 
456- 

Now,  here  8x5-7  =  45*6,  whereas  in  order  to  perform  the 
subtraction  we  really  require  456,  else  the  decimal  points 
would  not  be  in  the  right  position :  hence  the  8  is  not 
really  8,  but  80 ;  that  is  it  is  not  in  the  units'  place,  but  in  the 
ten's  place,  and  so  the  decimal  point  is  to  be  placed  after  the 
next  digit. 

Performing  the  subtraction  indicated  above,  we  see  that  the 
next  digit  of  the  quotient  is  a  2,  and  so  the  sum  goes  on 
without  any  further  trouble  or  attention  : 

5-7)  470-82  (82-6 
456- 


14-8 
11-4 

3-42 
3-42 


And  there  happens  to  be  no  remainder.     But,  if  there  were,  it 


204  EASY  MATHEMATICS.  [chap. 

would  give  no  trouble  ;  we  should  not  take  it  up  and  express 
it  as  a  vulgar  fraction,  but  should  continue  the  sum  in  the 
same  way  as  before,  bringing  down  ciphers  as  long  as  we  chose, 
that  is  until  we  had  got  the  quotient  to  the  required  degree  of 
accuracy. 

Dealing  with  fractions  then  in  the  decimal  notation  is  just 
as  easy  as  dealing  with  whole  numbers  in  the  same  notation. 
The  process  is  just  the  same,  only  we  must  be  careful  to  put 
the  decimal  point  in  the  right  place.  So,  however,  we  must 
with  whole  numbers,  only  we  do  not  have  to  actually  write  a 
decimal  point  in  their  case  (except  in  the  quotient  perhaps) ; 
but  we  always  have  to  be  careful  to  interpret  the  quotient  as 
meaning  hundreds  or  thousands,  or  whatever  it  is,  correctly, 
and  that  is  essentialy  the  same  thing  as  attending  to  the 
position  of  the  decimal  point. 

For  instance,  divide  729  by  14. 


14 

)  729- (52-07143 
70 

29- 

28- 

1-00 
•98 

•020 
014 

60 
.56 

40 

The  last  figure  in  the  quotient  is  not  exactly  3,  but  that  is  the 
nearest,  and  it  is  quite  time  to  stop,  as  we  have  already  reached 
the  extravagant  accuracy  of  seven  significant  figures.  If  we 
wanted  to  go  on,  however,  there  is  not  the  slightest  difficulty. 
We  simply  go  on  till  the  remainder  is  negligible,  not  because 
it  is  itself  numerically  small,  but  because  it  occurs  so  many 


xxii.]  DECIMALS.  205 

decimal  places  away  from  the  left  hand  significant  figure  that 
only  an  utterly  insignificant  fraction  is  left.  For  instance,  in 
the  above  sum,  the  last  remainder  which  is  indicated,  as  giving 
the  quotient  3  in  the  fifth  decimal  place  of  the  quotient,  is 
really  '00040,  and  the  multiplication  of  the  divisor  by  3  would 
give  "00042,  which  leaves  a  remainder  of  -  2  in  the  fifth 
decimal  place,  to  be  divided  by  14;  with  a  result  wholly 
trifling. 

In  the  above  sum  the  decimal  points  and  a  few  preceding 
ciphers  are  indicated  to  show  where  they  really  occur,  and  to 
show  how  they  might  be  indicated  all  the  way  along,  if  we 
chose;  but  there  is  no  real  need  to  indicate  them  anywhere 
except  in  the  quotient.  At  the  same  time  it  sometimes  helps 
to  keep  us  right  and  clear  to  put  all  the  points  into  the 
process,  where  they  ought  by  rights  to  be,  and  always  to  see 
that  they  keep  strictly  vertical. 

"  Order  "  of  Numbers. 

As  has  been  said  before,  in  another  connexion,  p.  171,  an 
extremely  useful  idea  is  the  "order"  of  a  number,  that 
is  to  say  the  index  of  its  order  of  magnitude  :  in  other  words, 
the  power  of  10  which  it  represents.  This  can  be  definitely 
specified  by  the  distance  of  its  highest  significant  figure  to  the 
left  or  the  right  of  the  unit's  place  :  distances  to  the  left  being 
called  positive,  to  the  right  negative ;  the  unit's  place  itself 
being  characterised  by  the  order  0,  and  everything  being 
reckoned  from  that  as  the  zero  position. 

For  instance,  any  single  digit,  like  6,  would  be  of  the  order 
0;  26  would  be  of  the  order  1 ;  526  of  the  order  2;  8526  of  the 
order  3,  and  so  on ;  the  order  being  given  by  the  position  of 
the  highest  significant  figure,  and  by  nothing  else.  Thus 
8526-79  would  still  be  of  the  order  3;  so  would  8000,  or  7000, 
or  1000. 


206  EASY  MATHEMATICS.  [chap. 

26*79  is  of  the  order  1. 
6*79  is  of  the  order  0. 
•79  is  of  the  order  -  1. 
•09  is  of  the  order  -  2. 

What,  then,  is  the  order  of  -00058  ?  Here  the  highest  sig- 
nificant figure  is  5,  and  its  position  is  4  places  to  the  right 
of  the  unit's  place  ;  hence  the  order  of  this  number  is  -  4.  So 
also  the  numbers  '0001  and  "0009578  are  of  the  minus-fourth 
order;  but  1-0009  is  of  the  order  0  again,  and  27-0009  is  of 
the  order  1. 

In  the  example  -00058  it  is  right  to  say  that  the  digit  5 
is  of  the  order  -  4,  the  digit  8  of  the  order  -  5 ;  and  it  is 
right  to  say  that  the  number  58,  which  it  contains,  is  also  of 
the  order  -  5.  Again,  in  the  number  525,  we  may  say  that 
the  52  which  it  contains  is  of  the  order  1,  that  is  to  say, 
that  it  occurs  one  place  to  the  left  of  the  unit's  place. 

It  is  often  in  practice  convenient  thus  to  attend  to  the  order 
of  particular  digits,  or  pair  of  digits. 

The  rule  for  multiplication  and  division  can  now  be  given 
thus  : 

For  multiplication  of  two  numbers,  take  the  highest  signifi- 
cant figure  of  each,  multiply  them  together,  and  give  the 
resulting  product  a  position  representing  the  sum  of  the 
orders  of  the  two  digits  taken.  For  instance,  multiply  36  by 
745.  You  take  the  two  highest  digits,  3  and  7,  the  sum  of 
whose  orders  is  1  +  2  =  3.  The  product,  which  is  21,  has  to  be 
placed  so  that  it  shall  have  the  order  3,  that  is  to  say,  the 
unit's  figure  of  the  21  is  to  be  3  places  to  the  left  of  the  unit's 
place. 

Or  take  this  example  : — Multiply  081  by  -742.  We  say  8 
times  7  is  56,  and  this  is  to  have  the  order  compounded 
of  -  2  and  -  1,  that  is  to  say  -  3.  Hence  the  56  is  to  be  placed 
so  that  its  unit's  digit  is  3  places  to  the  right  of  the  unit's 


XXII.]  DECIMALS.  207 

place  ;  or  in  other  words,  there  will  be  one  0  between  the  5 
and  the  decimal  point. 

The  rule  for  division  is  to  be  stated  similarly  : — 

Take  the  first  significant  figure  of  the  divisor,  and  the  first 
one  or  two  of  the  dividend :  enough,  that  is,  to  be  able  to 
effect  a  division.  Then  the  resulting  quotient  will  have  the 
order  of  this  part  of  the  dividend  minus  the  order  of  the 
figure  taken  in  the  divisor. 

For  instance,  if  we  had  to  divide  81  by  742,  there  would  be 
no  difficulty.  We  should  take  the  7  from  the  divisor,  which 
is  of  the  order  2,  and  8  from  the  dividend  of  the  order  1 ;  and 
the  quotient,  has  an  order  equal  to  the  difference  of  the  two 
orders,  viz.  -  1. 

But  if,  on  the  other  hand,  we  had  to  divide  742  by  81,  we 
should  take  8  from  the  divisor,  where  it  is  of  order  1 ;  but  it 
would  be  useless  to  take  7  from  the  dividend :  we  must  take 
74,  its  place  being  also  of  the  order  1 ;  so  that  the  resulting 
quotient  will  be  of  the  order  0. 

These  matters  are  not  particularly  easy,  they  can  be  much 
simplified  by  employing  powers  of  10,  as  we  will  soon  show ; 
but  meanwhile  we  will  do  sums  of  this  kind  on  commonsense 
principles,  as  follows :  Divide  742  by  -081.  A  simple  and 
favourite  way  of  doing  such  sums,  is  to  get  rid  of  the 
decimals  as  much  as  we  please  by  shifting  the  decimal  point 
in  both  equally,  that  is,  multiplying  them  both  by  the  same 
power  of  ten,  so  that  it  would  be  transformed  into  742  -f  81 
simply.     The  answer  comes  out  about  9'1605. 

One  more  example.     Take  the  inverse  of  this  sum. 

■742  )  -081  ( -1 

i  .  — : 

Here  the  first  product  742  is  required  shifted  one  place  to  the 
right  in  order  to  come  under  the  proper  digits  of  the  dividend, 


i 


208  EASY  MATHEMATICS.  [chap. 

so  the  quotient  must  be  not  unity,  but  one  tenth,  or  -1.     That 
once  determined,  the  rest  is  quite  ordinary. 

•742) -0810  (-109164 
•0742 

6800 
6678 


1220 

742 

4780 
4452 

3280 

Now,  here  it  must  be  admitted  that  people  clever  at  arith- 
metic do  not  write  long  division  sums  in  so  full  and  lengthy  a 
manner.  They  do  both  the  multiplication  and  the  subtraction 
in  their  head,  and  write  down  the  remainder  only ;  so  that 
the  sum  just  done  would  look  like  this  when  people  have  done 
it  by  aid  of  the  "  shop  "  method  of  subtraction  : 

•742) -0810  (-109164 
6800 
1220 
4780 
328 

I  can  do  it  this  way  if  I  am  put  to  it,  but  it  seems  to  me  a 
needless  tax  upon  the  brain,  at  least  when  grown  up ;  and  I 
am  more  likely  to  make  mistakes  and  am  less  able  to  check 
them  when  made.  Consequently  for  myself,  I  prefer  the 
longer  method,  for  it  is  the  same  sum  in  reality,  the  only 
difference  is  in  the  amount  of  it  recorded  on  paper.  I  suppose 
that  very  clever  people  indeed  would  record  nothing  of  it 
except  the  quotient :  all  the  rest  they  would  do  in  their  head, 
as  if  it  were  a  short  division  sum,  or  would  even  perceive,  in- 
tuitively as  it  were,  that  --;=p  = -109164,      Boys  have  been 


XXII.]  DECIMALS.  209 

known  to  be  able  to  do  things  like  this,  and  they  are  called 
calculating  boys.  They  are,  however,  rather  rare.  Never- 
theless, people  when  young  are  much  cleverer  at  learning 
things  than  old  folk,  so  perhaps  they  will  get  used  to  the 
abbreviated  method  of  recording,  if  they  begin  young  enough, 
and  may  like  it  better  than  the  other.  It  is,  I  believe, 
found  so. 

One   other  point,  however,  I  must  not  forget  to  mention 
here,  and  that  is  that  if  I  had  a  sum  like  '742 -^ '081  to  do, 

742 
I  should  first  write  it  thus :  -^y,  and  then  proceed  to  look  for 

factors.  If  they  do  not  occur  easily,  it  is  not  worth  while  to 
spend  time  in  hunting  for  them ;  still  less  is  it  worth  while 
to  go  through  the  farce  of  finding  G.C.M.  or  H.C.F.,  or  what- 
ever it  is  called :  one  might  as  well  be  doing  the  long 
division  sum  as  that.  And  then  I  should  proceed  to  look 
out  logarithms,  and  so  turn  it  into  simple  subtraction.  In  the 
particular  instance  I  have  chosen,  however,  it  is  hardly  worth 
while  taking  even  this  trouble,  for  directly  you  write  81,  you 
see  that  you  can  divide  by  9  in  two  stages ;  and  although  this 
might  be  found  a  little  unsafe  in  old-fashioned  times,  when 
one  had  remainders  to  express  as  vulgar  fractions,  now  that 
we  know  how  never  to  be  troubled  with  remainders,  we 
proceed  to  divide  numerator  and  denominator  by  9  twice 
over,  as  follows : 

742     82-444 

-^  =  ^^^4^*"' =9-16049382716, 

that  is,  for  all  practical  purposes,  9*1605,  as  we  found  before 
(p.  207)  by  long  division. 


CHAPTER  XXIII. 

Dealings  with  very  large  or  very  small  numbers. 

But  there  is  a  mode  of  dealing  with  all  these  sums  which  is 
of  great  simplicity  and  service,  and  is  more  particularly  useful 
when  the  figures  to  be  dealt  with  are  nowhere  near  the  region 
of  unity.  In  ordinary  life  we  usually  have  to  deal  with  a 
moderate  number  of  things,  or  a  few  simple  fractions  of  things ; 
we  seldom  have  to  deal  with  billions  or  trillions,  or  with 
billionths  or  trillionths ;  but  in  science  there  is  no  restriction 
of  this  kind :  we  may  have  quantities  of  every  order  of  magni- 
tude to  deal  with.  The  human  body  is  our  natural  standard 
of  size,  and  on  it  our  measuring  units  are  or  ought  to  be 
based.  Everything  much  bigger  than  our  body  requires  a 
large  number  to  express  it;  so  also  anything  incomparably 
smaller  requires  a  very  minute  fraction  to  express  it.  We 
must  be  prepared  to  deal  easily  and  familiarly  with  very  large 
and  very  small  numbers,  and  we  need  never  suppose  that  a 
large  number  requires  a  great  number  of  significant  figures  to 
express  it ;  for  by  that  means  it  would  not  be  of  any  different 
size,  it  would  only  be  expressed  with  preposterous  accuracy. 
A  number  like  17,199,658  is  for  most  purposes  quite  suflS- 
ciently  expressed  as  17*2  millions  or  17,200,000. 

So  also  our  lifetime  constitutes  a  natural  human  standard  of 
time,  and  our  walking  and  other  movements  are  standards  of 
velocity ;  but,  to  express  the  facts  of  nature  in  general,  these 


CHAP.  xxiiT.]  ORDER  OF  MAGNITUDE.  211 

magnitudes  may  have  to  be  multiplied  or  subdivided  to  almost 
any  extent.  The  distance  of  the  fixed  stars,  and  the  velocity 
of  light,  and  the  age  of  the  earth,  are  examples  of  one  kind  of 
magnitude.  The  size  of  atoms  and  the  duration  of  their 
collisions  lie  towards  the  other  end  of  the  scale. 

In  many  cases  the  precise  numerical  specification  is  of  less 
importance  than  is  the  order  of  magnitude ;  sometimes  because 
it  is  not  accurately  known,  sometimes  because  it  may  be 
variable  within  certain  limits.  The  "order  of  magnitude" 
may  roughly  be  said  to  be  given  by  the  number  of  digits 
involved  in  its  specification ;  in  other  words,  by  the  power  of 
ten  concerned,  without  much  regard  to  the  particular  figures 
that  precede  that  power.  Thus,  for  instance,  in  3  x  10^^  it  is 
the  index  ten  which  gives  the  order  of  magnitude;  the  numbers 
4  X  10^0  and  5  x  lO^o  and  even  8  x  10^^  or  i  x  iQio  ^re  of  the 
same  "  order,"  viz.  '  ten.' 

So  also  the  numbers  30  and  70  are  of  the  same  order  of 
magnitude,  viz.  '  one,'  though  one  of  the  two  numbers  is  more 
than  double  the  other. 

The  closeness  of  specification  required  depends  upon  the 
subject  matter  and  the  object  for  which  it  is  wanted.  Occasion- 
ally, though  not  often,  it  would  be  possible  to  consider  ten  and 
a  thousand  as  practically,  though  not  technically,  of  the  same 
order  of  magnitude  :  they  would  be  roughly  alike  as  compared 
with  either  a  billion  or  a  billionth. 

Now  let  us  take  some  examples  of  the  index  method  of 
dealing  with  figures.  Take  first  mere  numbers  of  different 
orders  of  magnitude.  For  instance,  divide,  multiply,  add,  and 
subtract  the  following  pair  of  numbers  in  every  way  : 

a  =17,400,000,  J  =-0015; 

which  may  be  called  17-4  millions,  and  1*5  thousandths,  or 
17 '4  X  10^,  and  1-5  x  10"^  respectively. 


212  EASY  MATHEMATICS.  [chap. 

First  we  notice  that  when  numbers  differ  greatly  in  magni- 
tude, addition  and  subtraction  are  operations  that  are  useless; 
a  +  b  and  a~b  are  to  all  intents  and  purposes  the  same  as  a  in 
the  above  case ;  the  larger  magnitude  dominates  the  smaller,  so 
far  as  addition  or  subtraction  is  concerned.     A  million  plus  or 
minus  three  is  practically  the  same  as  a  million.     So  no  finite 
quantity  added  to  infinity  makes  the  smallest  difference  to  it. 
This  is  a  frequently  useful  fact :  small  quantities  can  be  neg- 
lected when  added  to  or  subtracted  from  large  ones. 
1+0=1,  when  z  is  small  enough. 
a^  -x^  =  a^,  when  x  is  small  compared  with  a, 
which  may  happen  either  when  a  is  very  big  or  when  x  is 
very   small,   or  even  when  both  are  big  or  both   small   so 
long  as  a  is  much  bigger  than  x ;  in  other  words,  so  long  as 

X 

the  ratio  -  is  small.     The  term  "small,"  so  used,  signifies  small 
a 

compared  with  the  other  quantities  concerned  in  the  expression; 
or  sometimes,  as  in  this  case  of  the  ratio,  small  compared  with 
unity. 

But  when  we  proceed  to  multiplication  or  to  division,  we  find 
a  very  different  state  of  things  ;  there  is  then  no  domination 
of  a  big  quantity  over  a  small  one ;  the  bigness  may  be 
exaggerated,  or  it  may  be  partially  destroyed,  by  the  influence 
of  the  small  one. 

Take  the  example  suggested  above  : 
a&  =  17-4  X  106x1-5x10-3  =  26-1x103  =  26100 (i) 

1  =  17-4  X  106  ^  (1-5  X  10-8)  =  1 X  17-4  x  109=  11-6  ^  10^ (2) 

&_  1-5  xl0-3_  10-^  _  10-^0  _ 

a~  17-4x106"  11-6"  i-ie  -  ^^-""^^     ^'^) 

These  results  are  numbered  (1)  (2)  (3)  for  reference.  The 
three  results  are  of  different  orders  of  magnitude.  The 
middle  result  is  about  half  a  million  times  the  first ;  it  is  so 
much  greater  because  a  number  has  been  divided  by  the  small 


XXIII.]  ORDER  OF  MAGNITUDE.  .  213 

quantity  b  instead  of  being  multiplied  by  it.  The  ratio  between 
results  (1)  and  (2)  is  therefore  exactly  equal  to  //^,  that  is 
(1-5)2  X  10-6  or  2-25  x  IQ-^  or  -00000225. 

The  first  result  is  more  than  a  hundred  billion  times  the 
third ;  the  ratio  between  them  being  a^,  that  is 

(17-4)2  X  1012  =  302-76  x  10^2  =  3  x  lO^^  approximately, 
an  enormous  number,  but  not  bigger  than  what  we  have 
frequently  to  deal  with  in  physics.  The  particles  in  a  candle 
flame  are  quivering  with  about  this  number  of  vibrations  per 
second,  otherwise  we  should  not  be  able  to  see  the  light. 
Everything  self-luminous  must  be  quivering  at  this  or  at  a 
somewhat  greater  rate,  consequently  such  rates  of  vibration 
are  quite  common. 

The  result  (2)  compared  with  result  (3)  shows  a  still  greater 
difference   in   order   of    magnitude.      To   express   the   ratio, 

11-2 
which  is  tq^t^  x  lO^'-*,  a  number  of  21  digits  is  required,  viz. 

the  number  l-3xl02o,  more  accurately  1-2993x1020,  a 
number  which  is  of  the  same  order  of  magnitude  as  the 
number  of  atoms  in  a  drop  of  water. 

Now  take  another  example.  If  light  travels  a  distance 
equal  to  seven  and  a  half  times  round  the  world  in  a  second, 
how  long  does  it  take  to  come  from  the  sun,  a  distance  of  93 
million  miles  1  How  long  does  it  take  to  travel  1  foot,  or 
say  30  centimetres.  And  how  long  to  travel  from  molecule 
to  molecule  in  glass,  supposing  that  they  are  the  ten  millionth 
of  a  millimetre  apart  1 

•  The  circumference  of  the  earth  is  just  40  million  metres, 
by  the  definition  of  a  metre.  It  therefore  equals  4  x  10^ 
centimetres.  7 J  times  this  equals  3  x  lO^^^  centimetres ;  and 
this  distance  traversed  per  second  gives  the  velocity  of  light. 

A  mile  is  about  1  -6  kilometres,  so  the  distance  of  the  sun  is 
i-io  X  1-6  =  149  million  kilometres,  as  nearly  as  it  is  at  present 


214  .  EASY  MATHEMATICS.  [chap. 

known;  in  other  words  it  is  149x10^x10^  centimetres  == 
1-49  X  10^^  centimetres.  Hence  the  time  required  by  light  for 
its  journey  from  the  sun  is 

1  -49  X  1013  centimetres     1 490 


„     ^^,„     cenUmetres 
3  X  10^^ 


=  497  seconds. 


second, 
or  about  8  minutes  and  a  quarter. 

This  begins  to  illustrate  the  right  method  of  dealing  with 
units.  We  shall  have  occasion  to  illustrate  and  emphasise  it 
later  at  much  greater  length ;  but  it  will  be  seen  already 
that  the  "centimetres"  in  numerator  and  denominator  cancel 
out,  and  that  the  "  seconds  "  in  a  denominator  of  the  denomi- 
nator come  up  to  the  top,  and  gives  Us  the  units  of  the  answer. 

If  this  is  not  clear,  never  mind,  we  shall  return  to  it  and  to 
much  more  like  it.     We  might  have  written  the  whole  working 

thus : 

93  million  miles  _  93 miles 

7*5  X  40  million  metres  per  second  ""300  metres  per  second 
=  •31  X  1600  seconds  =  496  seconds 
=  SJ  minutes  approximately, 

and  that  is  really  the  best  and  safest  way  to  do  it.  We  have 
here  put  the  actual  data  into  the  fraction,  and  then  cancelled 
out  the  "  millions  "  ;  next  expressed  numerically  the  ratio  of 
miles  to  metres,  which  is  1600,  since  1*6  kilometre  is  a  mile ; 
and  then  we  bring  the  "  per  second  "  out  of  the  denominator, 
and  call  it  "  seconds  "  in  the  numerator. 

In  so  far  as  the  two  answers  are  not  identical  to  the  nearest 
second,  that  is  simply  because  of  the  approximate  working, 
which  is  justified  by  reason  of  the  uncertainty  of  the  data. 
If  the  result  were  expressed  as  496-666  seconds  it  would  be 
merely  dishonest.  The  velocity  of  light  and  the  distance  of 
the  sun  are  both  quantities  which  have  had  to  be  experi- 
mentally determined,  and  neither  is  known  with  more  tha?:-' 


xxm.]  LARGE  AND  SMALL  NUMBERS.  215 

three  figure  accuracy.  In  fact,  the  latter  is  not  known  quite 
so  closely  as  this.  Moreover,  it  is  at  best  only  an  average  value : 
the  sun  is  not  always  at  the  same  distance  from  the  earth, 
since  the  earth's  orbit  is  not  circular ;  the  distance  we  have 
chosen  is  an  approximation  to  the  mean  or  average  distance. 

Now  take  the  latter  parts  of  the  question,  viz.  the  time 
required  by  light  to  travel  a  foot,  or  say  30  centimetres  ;  and 
the  time  required  to  travel  a  molecular  distance  of  the  ten 
millionth  of  a  millimetre,  or  10"^  centimetre. 

These  are  quite  easily  ascertained,  since  the  velocity  is 
given  as  3  X  10^^  centimetres  per  second.     To  travel  30  centi- 

30 
metres  liffht  takes  ^     ■,^,^  second,  that  is  10"^  second,  which 
°  ox  10^" 

means  the  thousand  millionth  part  of  a  second. 

10-8 
To  travel  from  molecule  to  molecule,  it  takes  ^ — i7uo~  J  10"^^, 

or  say  the  third  part  of  the  trillionth  of  a  second.  Here 
the  digit  3  is  quite  unimportant.  The  order  of  magni- 
tude is  all  that  is  of  any  use,  and  that  is  the  trillionth  or 
10  "1^  of  a  second.  Molecular  magnitudes  are  not  known  more 
accurately  than  that.  It  may  be  considered  remarkable  that 
they  have  been  measured  at  all.  The  way  they  are  obtained, 
a  way  necessarily  indirect,  can  only  be  understood  later. 
With  attention  to  these  early  stages,  this  and  much  else  can 
presently  be  understood  by  everybody.  At  present,  grown 
people  are  ignorant  of  all  these  things,  because  they  have  not 
prepared  their  minds. 

Now  take  a  more  childish  example,  akin  to  the  horseshoe 
nails,  page  155,  and  perhaps  equally  surprising. 

A  country  the  size  of  England  was  being  besieged  by  a 
hostile  fleet,  and  its  inhabitants  were  in  danger  of  starvation 
because  they  did  not  grow  their  own  corn.  Under  these 
circumstances    the    captain    of  a   merchant   steamer   craved 


216  EASY  MATHEMATICS.  [chap. 

permission  from  the  enemy  to  run  the  blockade  with  a  chess 
board  full  of  wheat  for  his  starving  wife  and  family,  the  board 
to  contain  a  single  grain  of  wheat  on  the  first  square,  two 
grains  on  the  next,  four  on  the  next,  and  so  on. 

But  when  the  enemy's  admiral  had  had  the  necessary 
calculation  made,  by  a  Japanese  sailor  who  happened  to  be  on 
board,  and  was  informed  that  the  corn  thus  to  be  passed 
through  his  lines  was  sufficient  not  only  to  feed  but  to 
smother  every  living  soul  in  the  country,  in  fact  to  cover 
the  whole  land  with  a  layer  of  grain  more  than  a  dozen  yards 
thick,  he  declined  to  grant  the  request  unless  the  whole 
supply  were  delivered  at  one  operation. 

To  do  the  sum,  proceed  as  follows  : — The  number  of  grains 
is  2^\  or,  strictly  speaking,  one  grain  less  than  this  number. 
A  mode  of  arriving  at  this,  if  it  is  not  obvious,  will  be  given 
below,  but  it  could  be  reasoned  out  by  an  intelligent  beginner. 

Call  the  number  71. 
then  log  71  =  64  log  2, 

and  log  2,  either  from  the  curve  (p.  179)  or  from  a  table  of 
logs,  is  approximately  -3,  more  accurately  -30103 ; 
hence  logTi  =  19*266. 

n  is  therefore  a  number  with  twenty  digits  of  "  order  "  19  ; 
in  other  words,  it  is  approximately  eighteen  trillion;  more 
accurately  it  is  1*845  x  10^^. 

This  number  is  not  so  great  as  the  number  of  atoms  in  a 
drop  of  water,  but  it  is  a  large  number.  To  see  what  it 
means :  buy  half  a  pound  of  wheat  as  imported,  without  its 
husk,  etc. — it  costs  only  a  penny — and  devise  a  plan  of  practi- 
cally counting  the  grains  in  say  a  cubic  inch,  without  actually 
counting  so  many  individually.  This  should  not  be  beyond  a 
youth's  ingenuity. 

I  find  that  on  the  average  a  grain  is  J  inch  long  and  }  of  an 
inch  broad.     So  if  they  were  regularly  arranged,  in  what  is 


XXIII.]  GRAINS  OF  WHEAT  PROBLEM.  217 

called  square  order,  there  would  be  28  of  them  lying  in  a 
square  inch ;  and  if  piled  up  an  inch  high,  also  in  regular  order, 
there  would  be  7x28  =  196  in  the  cubic  inch  so  constructed. 

An  allowance  for  irregularity  should  doubtless  be  made, 
but  it  is  uncertain ;  it  is  not  even  quite  clear  whether  more  or 
fewer  could  be  got  into  a  given  space  by  a  higgledy  piggledy 
arrangement  than  regular  packing  in  artificially  square  order. 
It  will  be  near  enough  if  we  take  it  as  about  the  same,  and  so 
estimate  200  grains  of  wheat  to  the  cubic  inch. 

We  are  now  prepared  to  go  on  with  the  sum  set.  The  area 
of  a  country  the  size  of  England  is  given  in  the  geography 
books,  or  the  Penny  CyclopaBdia,  as  50,000  square  miles.  A 
mile  is  1760  x  36  inches  =  6-336  x  lO'*  inches,  so  a  square  mile 
is  the  square  of  this,  viz.  40-145  x  10^  square  inches,  of  which 
the  first  two  digits  are  sufficient  for  our  purpose. 

Hence  the  area  of  a  country  as  big  as  England  is  5  x  10*  x 
40  X  10^  =  2  X  10^*  square  inches.  Now  the  number  of  grains 
which  are  to  be  distributed  over  this  area  is  given  by  our 
previous  working  as  1*845x10^^,  and  we  have  ascertained 
that,  roughly  speaking,  200  of  the  grains  will  occupy  a  cubic 
inch.  Hence  the  number  of  cubic  inches  which  have  to  be 
provided  to  hold  all  the  corn,  is  the  200th  part  of  1-845  x  lO^^, 
that  is  to  say,  -922  x  lO^'' ;  or  just  less  than  the  tenth  of  a 
trillion  cubic  inches.  To  provide  this  capacity  on  the  surface 
of  the  country,  the  grain  would  have  to  be  spread  all  over  it 
in  a  uniform  layer,  of  thickness 

•922  X  lOi'' cubic  inches       ._,      ^^o  t  .     i 

—^ — Tmi = — c — =  '4ol  X  10"^  linear  inches. 

2x  10^^  square  inches 

In   other  words,  the   corn   would   flood   the   whole  country 

to  a  depth  of   461   inches,  or  38-4   feet,   which  is  as  high 

as    an    ordinary    house.      All   cottages   would   therefore   be 

completely   submerged   by    the   chess    board    full    of  grain 

distributed  uniformly  over  the  face  of  the  country. 


218  EASY   MATHEMATICS.  [chap. 

Perhaps,  however,  our  initial  step,  that  the  number  of 
grains  is  precisely  2^^  -  1,  was  not  obvious.  It  can  easily  be 
seen  thus.  The  number  on  any  square  will  be  one  grain  more 
than  all  those  on  the  preceding  squares  added  together. 
Thus,  for  instance,  the  number  on  the  third  square  is  4,  and 
the  two  previous  squares  contain  one  grain  and  two  grains 
respectively,  or  3  grains  together ;  which  is  one  less  than  the 
number  on  the  next.  The  number  on  the  next  following 
square  is  8,  and  the  three  previous  squares  together  hold  7,  or 
again  one  grain  less,  and  so  on.  Hence  the  number  on  the 
tenth  square  would  be  the  number  on  the  nine  previous 
squares  added  together,  plus  one.  The  number  on  the  tenth 
square  is  2^,  so  the  number  on  the  9  previous  squares  added 
together  will  be  2^  -  1.  The  number  on  the  sixth  square  is 
2^  =  32,  hence  the  number  on  the  five  previous  squares  added 
together  will  be  31  ;  and  so  it  is,  viz.  1  +  2  +  4  +  8  +  16,  which 
equals  31.     Compare  page  323. 

Now  the  total  number  of  squares  in  a  chess  board  is  8^,  or 
64 ;  the  number  of  grains  on  a  65th  square,  if  there  were  one, 
would  be  2^^,  hence  the  number  on  the  64  previous  squares 
added  together  (which  is  just  what  we  want)  is  2^*  -  1. 

This  peculiar  result  of  continued  doubling,  that  the  product 
each  time  just  exceeds  the  sum  of  all  the  preceding  products, 
has  suggested  a  plan  of  what  is  called  "breaking  the  bank,"  at 
a  place  where  you  stake  on  one  of  two  events,  either  of  which 
is  equally  probable,  say  red  or  black,  and .  win  back  double 
your  stake  if  you  win,  that  is  receive  your  stake  and  another 
added  to  it  by  the  "  bank."  The  simplest  rule  for  "  breaking 
the  bank  "  is  simply  this  :  Begin  small,  and  double  your  stake 
every  time  you  lose ;  whenever  you  win,  begin  again. 

If  it  were  feasible  to  continue  this  process  you  could  never 
really  lose,  because  your  stake  would  always  just  exceed  the 
sum  of  your  previous  losses,  so  that  whenever  you  won  you 


XXIII.]  LARGE  NUMBERS.  219 

would  get  them  all  back  plus  1  counter  more.  Winning  would 
be  slow  but  sure.  To  work  the  process  you  must  be  prepared, 
however,  with  a  considerable  number  of  counters  to  stake 
with,  if  you  happen  to  lose  many  times  in  succession.  And  as 
a  matter  of  fact  every  '  bank '  protects  itself  against  so  simple 
an  arithmetical  device  by  declining  to  receive  more  than  a 
certain  maximum  stake.  If  therefore  you  have  staked  the 
maximum  and  lost,  you  have  no  way  of  getting  your  losses 
recouped ;  and  so  it  is  universally  conceded,  even  by  gamblers, 
that  there  are  more  profitable,  as  well  as  more  useful,  ways 
of  earning  a  living. 

The  operation  of  constant  doubling  is  a  particular  case  of 
what  is  generally  called  geometrical  progression,  and  it 
is  remarkable  how  rapidly  we  can  thus  reach  enormous 
magnitudes. 

Of  course,  if  instead  of  doubling  we  treble  or  quadruple 
each  time,  the  large  result  will  be  reached  still  sooner,  but,  as 
a  matter  of  fact,  any  constant  factor  greater  than  1,  repeated 
often  enough,  will  grow  to  any  magnitude ;  whereas  any 
constant  factor  less  than  1  repeated  often  enough  will  obliterate 
or  reduce  to  insignificance  any  initial  magnitude. 

Take  an  example.  Let  the  factor  be  1*1,  that  is  one  and  a 
tenth.  Multiply  it  by  itself  20  times,  so  that  the  result  is 
(1-1)20,  whose  value  can  be  found  by  logarithms  easily  enough, 
thus : 

Call  it  X,  then  log  x  =  20  log  1  -1  =  20  x  -0414  =  -828, 
so:c  =  6*6;  showing  that  the  initial  rate  of  increase  is  slow. 
Look,  for  instance,  at  the  geometrical  progression  or  compound- 
interest  curve  on  page  357,  which  is  the  same  as  the 
exponential  curve  on  page  101  taken  backwards,  and  note 
that  it  begins  slowly.  But  continue  the  process  until  we  have 
reached  (l-l)^^^,  whose  logarithm  will  be  100  x  -0414,  or  4-14, 
and  already  the  result  is  13,800.     To  get  really  large  numbers 


220  EASY   MATHEMA1 ICS.  [chap,  xxiir. 

with  such  a  factor  as  1-1,  we  should  therefore  have  to  repeat 
the  operation  very  often  indeed. 

So  also  to  reduce  to  insignificance  by  means  of  such  a  factor 
as  '9,  we  should  have  to  repeat  the  multiplication  very  often  : 
for  let  (-9)100  =  x 
then       \ogx  =  100  log -9  =  100  log ^^ 

=  100  (log  9  -  log  10)  =  100  log  9  -  100 
=  95-42-100  =  -4-58  =  -5  + -42  =5-42, 
wherefore  a;  =  -0000263, 
a  number  which  is  of  the  "  order  "  -  5. 

Illustrations  of  excessively  rapid  multiplication  by  geometri- 
cal progression  occur  in  Natural  History,  where  certain 
organisms  are  known  to  increase  at  a  prodigious  rate,  this  rate 
of  increase  being  the  cause  of  plagues,  like  a  plague  of 
locusts,  or  blight,  or  like  certain  kinds  of  disease.  For  sup- 
pose a  parent  insect  laid  and  hatched  a  thousand  eggs 
(which  is  indeed  a  very  moderate  number),  and  suppose  each 
of  these  also  hatched  a  thousand,  and  suppose  each  genera- 
tion only  required  a  month  to  come  to  maturity,  and  lived  for 
a  year ;  the  number  of  descendants  in  the  course  of  twelve 
months  would  be  a  thousand  raised  to  the  twelfth  power, 
that  is  to  say  a  number  of  the  "  order "  36,  or  1  followed 
by  36  ciphers,  or  a  trillion-trillion. 

Some  diseases  are  caused  by  the  fission  or  splitting  up 
of  cells  into  two  or  more,  which  rapidly  grow  and  split  up 
again.  In  such  case  the  rapidity  of  increase  can  be  still  more 
prodigious,  because  the  time  which  need  elapse  between 
the  splitting  and  the  re-splitting  of  cells  may  be  short. 

It  does  not  follow  that  these  geometrical-progression  rates 
of  increase  apply  without  qualification  to  every  kind  of 
population,  nor  to  one  whose  needs  are  in  excess  of  available 
supply.     For  some  actual  tacts,  however,  see  Appendix  IV. 


CHAPTER  XXIV. 

Dealingfs  with  Vulgar  Fractions. 

Having  now  exhibited  the  easy  mode  of  dealing  with  frac- 
tions, we  must  proceed  to  the  more  difficult  method  where  the 
division  operation  has  not  been  performed,  but  is  only  indi- 
cated :  the  same  sort  of  indication  as  has  been  used  in  algebra. 
For  instance,  to  divide  a  hy  b  you  cannot  really  do  it  in 

algebra,  you  can  only  indicate  it,  as  ct  -r  &  or  t.      So    also    in 

arithmetic  if  one  has  to  divide  3  by  4,  we  can,  if  we  choose, 
do  it,  and  write  '75  simply,  but  for  many  homely  purposes  it 
is  sufficient  to  indicate  it  only,  and  leave  it  in  the  form  of 
3  -r  4  or  f .  Fractions  left  like  this  are  not  so  easy  to  deal 
with,  but  they  usually  apply  to  such  simple  magnitudes  that 
they  are  simple  enough.  For  anything  complicated,  however, 
they  are  unsuitable,  and  they  must  be  simplified ;  moreover, 
as  we  have  seen  in  Chapter  XXL  on  concrete  arithmetic,  they 
seldom  occur  in  practical  measurements;  nevertheless  we 
must  learn  to  perform  the  fundamental  operations  upon 
vulgar  fractions  without  having  necessarily  to  reduce  them 
first  to  a  simpler  form.  One  way  of  dealing  with  mixed 
units,  such  as  cwts.,  quarters,  and  lbs.,  or  pounds,  shillings, 
and  pence,  is  to  reduce  them  all  to  some  one  denomina- 
tion; but  it  would  be  rather  stupid  if  we  did  not  know 
how  to   treat   them   in   any  other  way.     So   also   one  way 


222  EASY   MATHEMATICS.  [chap.  xxiv. 

of  treating  collections  of  vulgar  fractions  is  to  reduce  them 
all  to  some  one  denomination,  or  to  decimals ;  but  we  ought 
to  learn  how  to  manage  them  without  this  preliminary- 
operation. 

So  we  will  proceed  to  illustrate  by  example  some  simplifying 
processes,  first  reminding  ourselves  of  the  fundamental  opera- 
tions of  addition,  subtraction,  multiplication,  division,  involu- 
tion, and  evolution,  applied  to  vulgar  fractions. 

No  further  explanation  is  needed  beyond  what  has  gone 
before,  Chaps.  III.,  IX.,  etc.  For  addition,  make  a  common 
denominator  and  cross  multiply. 

X    y     hx  +  ay 


or  another  example 


a     0 


X 

;7  +  y  = 


For  subtraction,  the  same. 


ah     ■ 

> 

x  +  ay 

a 

hx-ay 

ah    ' 

uals  — 

ay 

a     b 

or  if  given  --y,  it  equals 

For   multiplication,    multiply   numerators    together    and 

denominators  together. 

x^y^xy 

a     b     ab^ 

or,  in  the  common  case  when  one  denominator  only  appears, 

X  nx 

-xn=—. 
a  a 

For  division,  invert  the  divisor,  and  multiply 

X    y    X    h  _hx 

a  '  b    a    y    ay' 

XX 

or  -^n=--. 

a  na 


XXIV.]  VULGAR  FRACTIONS.  223 

For  involution,  operate  similarly  on  both  numerator  and 
denominator.  /a^\«     r,^ 


41J      a" 

For  evolution,  just  the  same  ;  only  we  may  write  ?i  as  —  if 
it  is  a  fraction,  if  we  like, 


Numerical  Verifications. 

The  simplest  fractions  of  all,  to  deal  with,  are  those  which 
are  not  really  fractions,  but  integers  in  disguise,  like  ^^-  or  -%*-, 
and  these  serve  for  testing  any  operation  easily  and  quickly. 

If  these  two  are  added,  for  instance,  the  result  according  to 


the  above  rule  is      12     24 

96  +  72 
24 

168 
~  24  "  ^• 

Subtracted,  the  result  is 

96-72 
24 

=  ?-^  =  l 
24     ^• 

Multiplied,  the  result  is 

288 
24 
Divided,  they  become 

=  12. 

12      8 
3  ""24 

=  1=1-333.... 

Squared,  the  results  are 

g*  =  16,   and   - 

576  _ 
64      ^' 

respectively. 

Square-rooted,  we  get 

n/(12)     2^3 
v/3        V3 

2  for  one. 

^"^  ^  =  l72  =  72  =  N/3f-'''«°*''- 


224  EASY  MATHEMATICS.  [chap. 

Of  course,  in  practice  we  should  get  rid  of  such  pretence  or 
imitation  fractions,  by  expressing  them  as  whole  numbers  at 
once,  before  beginning  operations  on  them.  But  we  may  often 
have  fractions  not  unlike  them ;  composed,  that  is  to  say,  of  a 
whole  number  and  a  proper  fraction  in  addition.  Those 
fractions,  larger  than  unity,  are  sometimes  called  "improper 
fractions,"  and  when  expressed  as  an  integer  +  a  fraction,  they 
are  called  "mixed  numbers";  but  these  terms  are  hardly  ever 
used  out  of  the  schoolroom. 

A  pretended  fraction  like  y-  cannot  be  concretely  exhibited 
to  small  children  unless  there  are  several  things  to  be  cut  up. 
With  several  apples  we  can  do  it ;  for  if  we  cut  them  all  up 
into  thirds,  and  then  pick  out  12  of  the  thirds,  we  shall  find 
that  we  can  build  up  with  them  4  apples. 

So  also  -^/-  would  give  us  4  apples  and  J  of  another ;  and  we 
could  not  exhibit  it  properly  unless  we  had  5  apples  to  start 
with. 

These  fractions  are  called  improper  fractions  because  they 
are  not  fractions  of  one  thing,  but  fractions  of  a  lot  of  things. 

To  exhibit  a  proper  fraction  like  |  is  easy,  for  we  have  only 
to  cut  an  apple  into  half  quarters  and  then  remove  five  of 
these,  leaving  f  ths  of  the  apple  behind. 

.It  is  from  this  point  of  view  that  vulgar  fractions  are  simpler 
than  decimals.  There  is  always  some  good  reason  why  a 
popularly  employed  nomenclature  has  been  hit  upon.  For 
these  simple  things  it  is  excellent :  it  is  only  when  we  come  to 
complicated  things  that  we  find  it  rather  difficult.  In  practice, 
however,  difficult  sums  never  occur  in  this  form,  and  there  is 
no  reason  for  wasting  the  time  and  brains  of  children  in 
simplifying  unwieldy  artificial  complications  ;  for  these  things 
may  give  them  much  trouble  when  young,  whereas  later,  if 
they  ever  learn  mathematics,  they  will  experience  none  at  all, 
even  if  they  come  across  the  most  complicated  of  them. 


CHAPTER   XXV. 

Simplification  of  fractional  expressions. 

We  Mall  attend  now  to  certain  simple  operations  which 
constantly  occur  in  practice ;  it  is  easy  to  get  accustomed  to 
them  and  to  take  an  interest  in  them,  as  in  any  natural 
exercise  of  intelligence.  First  of  all  we  will  take  "cancelling," 
that  is,  striking  out  of  common  factors,  a  process  in  which 
useful  ingenuity  can  be  trained.  Examples  are  better  than 
precept,  so  try  the  following :  Simplify 
36  X  108  X  -91 
17-28  X  65  x8-r 

Here  we  see  at  a  glance  that  a  lot  of  factors  can  be  struck 
out,  because 

36  =  3x12,         108  =  9x12,         91  =  7x13, 
though   that   last   is  not  so  likely  to   be  known,  unless  an 
extended  multiplication-table  has  been  learnt — a  very  useful 
accomplishment ;  moreover 

1728  =  123,  65  =  5x13,  and  81  =  92. 
As  to  the  position  of  the  decimal  point,  that  is  a  matter 
that  gives  no  trouble  at  all.  The  decimal  point  must  always 
occur  somewhere  ;  it  is  understood  and  not  written  at  the  end 
of  integers,  but  it  is  there  all  the  time ;  and  its  influence  can 
be  attended  to  after  the  cancelling  has  been  done.  Of  course 
we  might  shift  it  equally  to  the  right  in  both  numerator  and 

JL.E.M,  P 


226  EASY   MATHEMATICS.  [chap. 

denominator,  and  so  get  rid  of  its  explicit  appearance,  and 

we  shall  do  this  ultimately,  but  we  will  not  do  that  at  present 

since  we  want  to  use  it  as  an  example ;  we  will  cancel  factors 

as  they  stand,  and  leave  the  decimal  points  unchanged  in 

position  till  the  end ;  and  the  result,  written  down  in  practice 

in  a  few  seconds  without  all  this  talk,  is 

3x9x-Q7  _3x-07_-07      7 

•12x5x8-l~-6x-9  "OS"  18' 

This  will  not  simplify  further,  because  7  is  a  prime  number 

and  does  not  go  into  18. 

It  would   be   very   seldom   useful   to   write  the  result  as 

1  1 

or 


2|         2-5714...' 
but  it  would  often  be  useful  to  write  it  as 

3*5 

-Q-  =  -3888  ...  or  approximately  '4. 

Take  another  example  :  one  less  likely  to  occur  however, 
one  of  a  double  fraction. 

8       w     2  1       • 
TT¥  ^  T(J(T 

Here  we  may  cancel  out  factors  among  the  numerator  frac- 
tions, and  likewise  among  the  denominator  fractions;  but 
we  must  not  cancel  a  factor  in  the  upper  numerator  against  a 
factor  in  the  lower  denominator.  119  contains  the  factor 
17,  and  also  the  factor  7,  being  equal  to  7  x  17  ;  and  obviously 
543  contains  the  factor  3,  since  its  digits  divide  by  3.  100, 
as  usual,  is  useless  for  factor  purposes. 

So  we  re-write  the  fraction  (with  needless  elaboration) 
1     4     181  4x181 

1 ""  9  ""  ~r  ~9~ 

or 


1 ""  100  100 


XXV.]  FRACTIONS.  227 

the  middle  line  being  drawn  rather  longer  and  stronger  than 
the  other  two,  so  as  to  show  that  the  upper  of  the  two 
fractions  is  to  be  divided  by  the  lower. 

We  have  now  to  multiply  the  extremes  and  divide  by  the 
means,  a  convenient  rule  to  remember,  giving  us 
400x181 
63       • 
But  a  rule  of  this  kind  should  never  be  given,  it  should  be 
ascertained  and,  if  possible,  invented  by  the  pupil.    To  invent 
a  handy  rule  involves  a  little  bit  of  original  thought,  and  the 
opportunity  for  exercising  that  vital  power  should  never  be 
lost. 

Hence  we  should  not  at  first  make  the  above  convenient 
short  cut,  but  proceed  thus  : 

rpi.  4x181        7 

This  means  —9—^100. 

and  is  an  example  of  division  of  fractions,  so  invert  the 
second  number  and  multiply 

724     100     72400 
9    ""    7   ~    63    • 

To  express  this  in  decimals  we  might  proceed  thus,  for 
although  9  is  not  a  factor  of  the  numerator  it  will  be  approxi- 
mately one,  and  can  at  any  rate  be  divided  out,  leaving 

^^^  =  1 1 49-206349206349 .... 

Digression  on  recurring  decimals. — No  importance  attaches 
to  the  notation  of  the  superposed  dot  or  dots  for  circulating  or 
recurring  decimals.  Children  may  write  as  many  of  the  recur- 
ring places  as  it  amuses  them  to  write.  In  practice,  the  result 
would  not  usually  be  wanted  beyond  the  first  '2,  which  is  nearly 
equivalent  to  six  figure  accuracy,  since  the  next  figure  is  a  0. 

The  interpretation  of  recurring  decimals  as  vulgar  fractions, 


228  EASY   MATHEMATICS.  [chap. 

with  a  certain  number  of  nines  and  noughts  in  the  denomi- 
nator, is  of  no  practical  moment.  It  should  be  reserved  for 
the  more  intelligent  and  irrepressible  children,  and  by  them  it 
might  be  found  out,  with  great  advantage.  Children  who 
delight  in  finding  out  such  things  are  on  the  way  to  acquire 
some  of  the  powers  and  tastes  of  the  pure  mathematician. 

The  simplest  case  may,  however,  be  known,  and  perhaps 
this  amount  of  hint  would  be  necessary  even  to  sharp  boys. 
A  recurring  decimal  is  a  geometrical  progression,  with  fractional 
common  ratio,  and  extending  to  infinity. 

Thus  the  commonest  of  all 

•3  or  -3333 

roVim^ • 

the  common  ratio  being  Yuth. 

Hence  its  sum,  by  the  rule  for  G.P.  [see  Chap.  XXXV.]  is 
a    _     '3        3     1 
l-r~l-TV~9"3- 
The  precise  value  of  the  answer  on  p.  227  expressed  as  a 
vulgar  fraction,  though  never  really  wanted,  is  well  known  by 
everybody  (needlessly  well  known  for  so  trivial  and  useless  a 
thing)  to  be  11 49f  f  ||||.  (End  of  digressim.) 


The  fact  that  ^  equals  % 

y 

as  is  proved  by  writing  it 

a     h    a    y 
—  —  —  =  —  X  — 

x'  y    X     V 

is  worth  remembering  :  most  easily  remembered  as  worded  in 
the  rule,  multiply  the  extremes  for  the  new  numerator  and 
the  means  for  the  new  denominator. 


XXV.]  CANCELLING.  229 

The  y  may  be  called  a  double  denominator,  and  we  observe 
that  it  comes  up  into  the  ultimate  numerator. 

The  rule  for  cancelling  may  also  be  similarly  illustrated : 


na 

a 

In 

X 

y 

or  in 

~mx 
my 

e  Ti's 

and  the 

m's  cancel  out. 

But 

in  such  a 

\.  fraction  as 

this 

ra 

X 

T 

ry 
the  r's,  so  far  from  cancelling,  appear  in  the  result  twice  over, 
that  is,  squared ;  for  it  equals 

r^ay 

bx  ' 

The  rule  for  cancelling  in  the  case  of  double  fractions  there- 
fore is :  cancel  common  factors  from  alternate  members  in  the 
double  fraction,  then  deal  with  the  extremes  and  means  to 
attain  the  simplified  result. 

It  may  be  preferred,  and  it  may  be  safer,  to  perform  the 
latter  operation  first,  and  so  keep  all  the  cancelling  for  an 
expression  in  simple  fractional  form ;  but  either  is  a  correct 
procedure. 


CHAPTER  XXVI. 

Cancelling  among  units. 

It  is  not  only  numbers  that  can  so  be  cancelled :  we  may 
and  often  do,  have  fractions  composed  of  concrete  or  physical 
quantities — quantities  with  length,  and  breadth,  and  thickness, 
and  weight,  and  velocity,  and  other  things.  It  will  be  found 
that  cancelling  can  conveniently  go  on  among  these  also. 

Suppose  we  had  the  following  ratio  to  interpret: 

330  yards  x  16  square  yards  x  77  lbs. 
4  inches  x  J  mile  x  '14  ton  x  5  minutes' 

an  experienced  eye  would  see  at  once  that  the  result  was  a 
velocity,  i.e.  that  it  could  be  expressed  as  so  many  miles  an 
hour,  or  feet  per  second.  And  the  working  is  on  the  following 
lines,  though  again  the  actual  operation  is  much  speedier  than 
is  the  explanation  of  it : — 

First  we  have  the  ratio  of  yards  to  inches,  which  is  36,  and 
this  is  most  conveniently  and  safely  recorded  by  erasing  the 
word  **yard  "  and  replacing  it  by  "  36  inches." 

Next  in  the  numerator  we  have  square  yards,  and  in  the 
denominator  we  have  a  linear  mile,  which  is  1760  linear 
yards,  and  that  value  is  therefore  conveniently  substituted  for 
"mile." 

Then  we  have  the  ratio  of  tons  to  pounds,  which  is  2240 ; 
and  we  get  left  with  one  of  the  "  yard  "  factors  of  the  square 
yards   uncancelled  in  the   numerator,  and   with    "minutes" 


CHAP.  XXVI.]  CONCRETE  QUANTITIES.  231 

uncancelled  in  the  denominator.    The  result,  before  any  cancel- 
ling is  done,  will  be  the  following  : 

330  X  36  inches  x  16  yards  x  yards  x  77  lbs. 
4  inches  x  ^  x  1760  yards  x  -14  x  2240  lbs.  x  5  minutes' 

Now  we  can  strike  out  a  number  of  units  common  to  both 
numerator  and  denominator,  and  can  at  the  same  time  do 
some  numerical  cancelling,  of  which  we  will  indicate  the 
steps  sufficiently,  noting  that  11  is  a  factor  of  1760,  because 
the  sums  of  its  alternate  digits  are  equal. 

30x9x32x11  yards 

160  X  -02  X  2240  x  5  minutes* 

Now,  we  see   that    16   will   cancel   out   with   32,  and   of 
course  the  ciphers  can  go  from  the  30  and  the  160. 
So  we  get  it  thus 

27x2x11       yards 
•02  X  2240  X  5  minutes* 

So   many   yards  by   so  many  minutes  ;  in   other   words,   a 
velocity  of  so  many  yards  per  minute.     How  many  1 

27x11      297 


•1x1120     112 


yards  per  minute. 


Here,  perhaps,  it  is  simplest  to  resort  to  long  division,  since 
no  more  factors  are  obvious  :  so  we  might  leave  the  answer  as 
2-6518  yards  per  minute,  which  is  a  sort  of  racing  snail's 
pace ;  or  we  might  reduce  it  to  other  units.  This  last  is  a 
thing  which  often  has  to  be  done,  and  so  no  opportunity  for 
showing  the  right  way  to  do  it  must  be  lost  at  this  early 
stage. 

This  is  the  easiest  and  only  safe  way : 

^97  y^^ds  ^     297  X  3  feet     ^  297  ^^^^^^ 

112  minutes     112  x  60  seconds     2240  ^ 


232  EASY  MATHEMATICS.  [chap. 

or  again 

miles 

297  yards  ^  297   neo         297  x  60  miles 
112  minutes"  112  ^«~  112x1760  hours 

27x3       81       ., 
=  56716  =  896  ^^^"^P"^^^^"- 

It  may  be  said  that  simple  reductions  like  that  can  easily  be 
done  without  writing  them  down  fully.  So  they  can,  but 
they  can  easily  be  done  wrong.  Change  of  units  is  a  subject 
extraordinarily  easy  to  make  a  slip  in,  especially  by  multiplying 
where  one  ought  to  divide.  It  is  at  best  a  mechanical  process, 
and  it  should  be  done  mechanically ;  that  is  by  a  straight- 
forward method  which  involves  no  delicate  thought,  and  affords 
no  loopholes  for  mistakes  to  creep  in. 

To  check  the  above  result,  we  can  recollect  that  4  miles  an 
hour  is  about  2  paces  or  6  feet  per  second ;  so  that  the  ratio  of 
the  above  two  specifications  for  the  same  thing  should  be 
roughly  as  3  to  2.  And  so  it  is ;  for  the  first  is  very  roughly 
4-  foot  per  second,  while  the  second  is  roughly  yV  mile  per 
hour;  and  the  ratio  of  7  to  11  is  not  very  different  from  that 
of  6-7  to  10,  which  is  frds. 

This  rough-and-ready  checking,  in  terms  of  anything  that 
comes  handy,  and  with  quite  rough  approximation  to  the 
figures,  is  very  useful,  and,  in  real  practice,  wise  ;  else  we  exhibit 
the  ridiculous  result  of  academic  correctness  in  minutiae,  and 
commercially  hopeless  error  in  the  order  of  magnitude ;  so 
that,  for  instance,  a  quantity  pretending  to  be  accurate  to  four 
or  five  significant  figures  may  be  all  the  time  a  thousand 
times  too  great  or  too  small. 

This  is  the  kind  of  thing  that  always  moves  the  practical 
man  to  legitimate  and  sarcastic  mirth,  because  he  could  get 
nearer  than  that  by  his  own  untutored  instinct  and  common- 
sense.     People  who  have  been  elaborately  tutored,  but  have 


XXVI.]  DEALma  WITH  UNITS.  233 

not  taken  care  for  themselves  of  their  own  birthright  of 
coramonsense,  are  denominated  "prigs,"  and  their  existence 
tends  to  bring  education  into  contempt. 

We  must  take  another  example  of  this  cancelling  of  units, 

and  we  will  take  an  instance  of  the  occurrence  of  a  double 

denominator  in  them.     Suppose  the  following  given : 

15  cwt.  X  (32  ieety 

1080  grammes  x  400  centimetres  per  second  x  |  yard' 

Here  the  experienced  eye  will  see  that  the  result  must  be  a 

time,  for  every  kind  of  unit  will  cancel  out  except  the  "per 

second "  in  the  denominator.     This  is  what  I  call  a  double 

denominator,  for  the  per  alone  would  put  it  in  a  denominator ; 

so  the  result  is  that  it  comes  up  into  the  ultimate  numerator. 

To   work   the    sum,   proceed    thus   (with   full   elaboration 

shown,  because  it  is  an  illustration)  : 

15  X  112  lbs.  X  32x32  x  (feet)^ 

-  ^oA  ^  metres     ,  ^  r    / 

1080  fframmes  x t-  x  1*5  feet 

°  second 

1120  lbs.  x8x  32  feet 

1AOA  1  metre' 

1080  grammes  x r 

°  second 

18     454  erammes  x  256  feet  , 

■=-=  X ^  -„  - — -  seconds 

17         grammes     x  3*28  feet 

18x454x256  , 

-      17x3-28       ^^^^^^- 

To  work  this  out,  either  a  slide-rule  or  logarithm-table  would 

be  advantageous.     Suppose  we  take  this  as  an  opportunity  for 

utilising  a  table  of  four-figure  logarithms,  and  see  what  we  get. 

log    18  =  1-2553  log     17  =  1-2304 

log  454  =  2-6571  log  3-28=    -5159 

log  256  =  2-4082  

6-3206  1-7463 


234  EASY  MATHEMATICS.  [chap.  xxvi. 

Subtracting,  we  get  4*5743, 
which  is  the  logarithm  of  37530  to  four  significant  figures ; 
and  37,530  seconds  is  therefore  the  answer. 

This  is  equal  to  10  hours  25 J  minutes;  and  so  the  com- 
plicated expression,  involving  many  kinds  of  units,  with 
which  we  started,  represents  really  nothing  more  elaborate 
than  the  length  of  a  working  day. 

These  examples  are  rather  dull  and  artificial ;  but  to  take 
a  real  example,  which  would  lead  to  this  kind  of  concrete 
result,  would  assume  some  knowledge  of  mechanics  or  physics. 
Suffice  it  to  say  that  plenty  of  quite  similar  examples  will 
occur  when  we  get  to  real  subjects  like  those,  and,  meanwhile, 
all  that  we  can  show  is  that  they  involve  no  difficulty 
of  dealing  with  and  interpreting.  No  admixture  of  units 
involves  anything  the  least  difficult :  it  only  wants  disen- 
tangling; and,  in  order  to  disentangle  it  securely  and 
easily,  the  best  plan  is  not  to  be  afraid  of  writing  out  the 
thing  at  length,  with  all  the  factors  present — both  the 
numerical  and  the  concrete  units,  or  standards — and  so 
gradually  boil  it  down  by  a  mechanical  process  involving 
no  troublesome  thought. 

Whenever  thought  is  necessary,  it  is  to  be  exercised  vigor- 
ously, but  it  should  not  be  wasted  over  simple  mechanical 
operations.  Take  thought  once  for  all,  learn  good  methods, 
and  so  economise  thought  in  future.  This  is,  indeed,  the 
principle  of  any  mathematical  machine.  Machines  can  be 
constructed,  and  are  used,  for  performing  really  intricate 
mathematical  operations;  for  analysing  out  the  harmonic  | 
constituents  of  a  tidal  or  other  irregularly  periodic  curve,  for 
instance.  To  devise  such  a  machine  required  thought,  and 
indeed  genius,  of  the  highest  order:  to  work  it,  requires 
nothing  beyond  what  an  intelligent  office  boy  can  learn. 


CHAPTER  XXVII. 

Cancelling  in  Equations. 

One  more  detail  concerning  cancelling  may  here  be 
mentioned.  It  relates  to  cancelling  on  either  side  of  an 
equation.  One  side  of  an  equation  may  be  considered  as 
divided  by  the  other,  and  the  result  equated  to  unity,  so  that 
the  rules  for  cancelling  are  easily  deduced. 

r  or  instance  m  —  =  -^ 

a       0 

the  n's  may  be  cancelled,  for  it  is  equivalent  to 

nx 

ny 
t 

nxb     ^       bx     ^        , 
or  • —  =  l,or  —  =  1,  or  ox  =  ay : 

nya  ay 

as  might  have  been  seen  at  once  by  cross  multiplying. 
Suppose,  for  instance,  that  it  had  been  written  thus  : 
nx     f^y  _r. 

it  would  have  been  the  same  thing ;  and  the  left  hand  of  this 
equation  might  be  reduced  to  a  common  denominator,  with 
the  subtraction  performed  as  far  as  possible,  by  writing 
hnx  —  nay_^ 
ab 


236  EASY  MATHEMATICS.  [chap. 

How  comes  it  that  this  is  the  same  thing  as  bx  =  ay1 
Because  it  may  be  written 

^(bx-ay)  =  0, 

and,  in  order  that  this  may  be  true,  one  of  the  two  factors 
must  vanish,  that  is,  must  itself  equal  0.  For  you  cannot  get 
zero  by  multiplying  two  finite   quantities   together.     Hence 

either  -r  mubt   equal   0,    which   is   in   some   cases  possible, 

but  is  clearly  not  here  intended ;  or  else  bx-ay  =  0.  And  the 
latter  cannot  happen  unless  ay  and  bx  are  equal. 

So  we  get  this  simple  rule,  that  when  an  expression  is 
equated  to  0  any  factor  can  be  struck  out,  without  having 
to  be  accounted  for,  provided  always  that  that  factor  is  not  itself 
zero. 

This  last  is  a  most  tremendously  important  proviso,  and  its 
neglect  may  land  you  in  the  utmost  absurdity.  If  we  strike 
out  a  factor  zero,  from  an  expression  equated  to  zero,  we  may 
be  striking  out  the  very  and  only  factor  which  made  it  zero  ; 
the  factor  left  behind  may  have  any  value  whatever:  the 
equation  declines  to  tell  us  for  certain  what  that  value  is, 
and  we  must  not  proceed  to  work  on  the  assumption  that  it 
does.  Similarly,  if  a  zero  factor  is  cancelled  on  either  side 
of  an  equation,  we  can  make  no  deduction  concerning  the 
equality  or  otherwise  of  the  residual  factors. 

Caution. 

This  inequality  of  zeros  is  a  matter  of  great  importance,  and 
I  must  proceed  to  illustrate  it  even  at  this  stage,  though  we 
shall  find  plenty  of  instances  later  on. 

Suppose  an  expression  like  this  were  given,  from  which  to 
find  X.  iej(n^-4)(x^  +  ¥) 


XXVII.]  CANCELLING  IN  EQUATIONS.  237 

we  should  be  quite  safe  in  striking  out  16  and  likewise  ^  (viz. 
the  3  in  the  denominator),  for  these  numerical  factors  are 
certainly  not  zero,  so  we  should  get  left  with 

VK- 4)^  =  0. 

Now,  if  we  strike  out  the  factor  J(n^  -  4)  and  the  factor 
-,  we  shall  be  left  with  the  impossible  result  x'^  +  b'^  =  0. 

Why  impossible  ?    Because  it  means  that 

x^=  -h\ 

and  the  square  of  a  real  number,  whether  that  number  be 

positive  or  negative,  cannot  possibly  be  negative ;   for  two 

similar  signs  multiplied  together  give  a  positive  sign  always ; 

-3x  -3=+9  just  as  much  as  3  x  3  does. 
What  the  equation  suggests  is  that,  under  the  circumstances, 
n  must  equal  2.  It  is  the  {n  -  2)  component  of  the  (w^  -  4) 
factor,  and  not  the  factor  containing  x,  which  is  responsible 
for  the  zero  value  of  the  whole;  and  the  equation  tells  us, 
therefore,  nothing  at  all  about  the  value  of  x. 

I  do  not  say  that  that  is  all  that  can  be  deduced  from  the 
equation,  but  that  is  all  that  lies  on  the  surface. 

To  clinch  the  danger  of  striking  out  a  factor,  without  at  the 
same  time  recollecting  the  possibility  that  it  may  be  itself  the 
essentially  zero  factor,  the  following  absurdity  may  be  given. 
To  "  prove  "  algebraically  that  2  =  1. 

Let  ic  =  1,  so  that  a;  -  1  =  0, 

then  a;2  =  1^  and  ic^  —  1  =  0. 

So  «2  -  1  =a; -  1,  since  both  equal  zero,       * 

wherefore  (a;  +  1)  (a;  -  1)  =  (a;  -  1). 

Cancel  out  the  factor  (a;  -  1)  from  both  sides,  and  we  get  left 

a;+l  =  l; 
but  we  knew  all  the  time  that  a;=l,  therefore  the  left  hand 
side  is  2,  and  so  2  =  1. 


258  EASY   MATHEMATICS.  [chap,  xxvii. 

Instead  of  going  through  the  above  farce,  it  would  be 
briefer  to  say  2x0  =  0; 

divide  both  sides  by  0,  hence 

2=1; 
or  instead  of  2  you  may  put  any  quantity  you  please. 

It  is  a  point  that  may  possibly  require  emphasis,  so  we  will 
put  it  still  more  evidently  : 

It  is  undeniable  that  7  x  0  is  0, 

and  also  that  6  x  0  is  0, 

if  then  it  be  argued  that  .*.  7x0  =  6x0, 
and  that  the  0  factor  may  be  cancelled  out,  it  seems  to  follow 
that  7  =  6. 

It  is  unsafe  then  to  press  the  axiom  that  things  which  are 
equal  to  the  same  thing  are  equal  to  one  another,  to  cover  the 
case  when  "  the  same  thing  "  is  zero. 

It  is  a  question  whether  we  have  a  right  to  say  that  7x0  = 
6  X  0  at  all,  although  they  are  both  zero.  It  rather  depends 
on  what  we  mean  by  0.  It  is  certainly  untrue  to  say  that  ^  =  J 
always,  because  clearly  any  numbers  might  be  substituted 
for  the  7  and  the  6.  Do  not,  however,  assume  that  J  is 
gibberish.  A  meaning  can  be  found  for  everything  if  you 
are  patient  and  persevering.  At  any  rate,  we  have  no  right 
to  cancel  out  the  zero  factor  which  alone  is  responsible  for  the 
pretended  equality  6x0  =  7x0.  Of  course  the  expres- 
sion does  not  in  practice  occur  in  this  crude  form,  but  it 
occurs  in  some  masked  form,  such  as 

18  («2-4)  =  39  (a;-2), 
whence,  cancelling  out  the  common  factor  3  {x-2),  we  get 

6  (a;  +  2)  =  13,  ori)j  =  J; 
which  may,  however,  be  quite  false,  and  is  not  at  all  a  necessary 
consequence  of  the  equation  from  which  it  is  supposed  to 
be  deduced;    it  is  a  possible  consequence,  or  "solution,"  but 
« =  2  is  another,  and  may  be  the  only  real  one. 


CHAPTER  XXVIII. 

Further  Cautions. 

Before  leaving  the  subject  of  "cancelling,"  it  may  be  well 
to  append  a  caution  concerning  a  small  point  which  does 
sometimes  give  trouble  to  a  beginner.  The  fractions  so  far 
chosen  for  simplification  had  both  numerator  and  denominator 
composed  of  factors ;  in  other  words,  numerator  and  denomi- 
nator was  each  really  a  single  "  term " :  they  were  not 
composed  of  a  number  of  terms  united  by  the  sign  +  or  -  . 
Compound  fractions  of  this  latter  kind  are  more  troublesome. 
In  arithmetic  they  do  not  often  present  themselves  in  this 
form  for  simplification,  because  when  they  occur,  the  addition 
or  subtraction  can  be  so  easily  performed  that  naturally  it  is 
done  before  any  process  of  simplification  is  thought  of.  But, 
in  algebra,  addition  and  subtraction  are  operations  that  cannot 
be  done,  they  are  only  indicated.  Indeed  that  is  one  of  the 
chief  advantages  of  algebra,  that  the  operations  to  be  per- 
formed are  preserved  intact  and  evident,  and  are  not  masked 
by  the  poor  achievement  of  performance. 

Suppose  then  we  had       , 

^^  2inx 

the  whole  thing  is  full  of  factors,  but  we  may  not  cancel  any. 

If  only  the  -\-  were  replaced  by  x  we  could  cancel  everything, 

and  leave  nothing  but  unity;    but  as  it  is,  the  fraction  is 

already  in  its  simplest  form,  unless  indeed  we  choose  to  split 

it  up  into  two  fractions. 


240  EASY  MATHEMATICS.  [chap. 

Why  may  we  not  cancel  anything  1     Because  a  factor,  in 
order  to  be  cancelled,  must  apply  to  the  whole  denominator 
and  to  the  whole  numerator.     In  the  above,  there  is  no  factor 
which  applies  to  the  whole  of  the  numerator.     So  there  we  are 
stopped.     Let  us  however  resolve  it  into  two  fractions 
371        8a; 
2inx     24:nx 
and  from  each  of  these  cancelling  is  easy,  yielding 

Sx^Sn 
This  form  may  be  preferable  to  the  first  given  form,  or  it  may 
not.     It  depends  on  what  we  want  to  do  with  it. 

Suppose  however  we  take  another  example,  very  like  the 
first,  but  upside  down, 

36my 

4m  +  9y 
Still  no  factors  can  be  cancelled,  for  there  is  no  factor  common 
to  the  two  terms  of  the  denominator ;  but  now  we  cannot  even 
separate  it  into  two  fractions.     The  attempt  is  often  made  by 
beginners ;  they  try  to  write  it 

a  splendid  simplification  certainly,  but  bearing  no  resemblance 
whatever  to  the  originally  given  fraction  of  which  it  is 
supposed    by   the   mistaken   beginner   to  be   a   counterpart. 

The  mistake  is  so  often  made  that  it  is  worth  numerical 
illustration. 

Example  (i).     ^^.  Example  (ii).       "^^^ 


144  ^     '  ■'      24  +  7 

The  first  can  be  split  into  two  fractions 
24      _7^_1     _^ 
144  "^144     6  "^144' 


XXVIII.]  CANCELLING  IN  FRACTIONS.  241 

The  second  can  not  be  split  up  at  all.     It  could  be  written, 
if  it  were  worth  while, 

6  6 

1  + 2V  ~  1-2916 
Of  course  both,  being  arithmetical,  can  be  written 

l44    ^^^    "sT    ^^^P®^^^^®^^'' 
and  that  is  just  why  these  forms  do  not  occur  in  arithmetic  as 
they  do  constantly  in  algebra. 

Is  no  cancelling   ever   to  be  done  when  a  numerator  or 
denominator  contains  more  than  one  term^     Certainly  there 
is,  if  each  term  has  a  common  factor.     For  instance, 
Tia  +  nb  21  +  51 

-MT  °'  '^^   -TOTT- 

If  the  +  were  replaced  by  x  the  n^  would  cancel  out  alto- 
gether ;  but  as  it  is,  only  one  n  cancels  out,  and  the  result  is 

a  +  b  7  +  17 

1 or    ■ . 

nab  357 

I  have  found  beginners  who  thought  that  if  they  used  the 

factor  in  the  denominator  to  cancel  one  of  the  terms  in  the 

numerator,  they  could  not  use  it  likewise  to  cancel  the  other 

term;   who  would  wish  therefore   to  divide   the   1071   by  9 

instead  of  by  3,  and  to  write  it  119  in  the  result,  because  a  3 

has  been  cancelled  out  of  each  term  in  the  numerator,  and 

therefore  it  looks  as  if  a  9  should  be  cancelled   from   the 

denominator.     But  there  is  every  difference  between  striking 

out  a  factor  from  each  of  two  terms,  and  striking  it  out  from 

each   of   two   factors.      The   mistake   arises   in   fact   from   a 

momentary  confusion  between  +  and  x  . 

When  the  expression  is      — — - — ^, 

the  result  is  ^  : 

mz 

L.E.M.  Q 


242  EASY  MATHEMATICS.  [chap. 

but  if  the  expression  had  been 

rnxx^my 

the  result  would  be  ^  : 

z 

but  just  as  the  former  expression  with  the  +  sign  would 
hardly  occur  as  such  in  arithmetic,  so  the  latter  with  the 
X  sign  would  hardly  occur  as  such  in  algebra ;   it  would  be 

written  — -^ 

m^z 

and  no  shadow  of  doubt  could  arise.  The  doubt  seems  to 
occur  only  when  there  are  several  terms. 

Take  the  case  of  more  than  one  term  in  both  numerator 
and  denominator,  like 

a  +  b 

x  +  y' 

Can  we  split  this  up  into  two  fractions  1     Certainly,  but  not 

ah 
into  -  +  -  j 

X    y 

the  two  fractions  into  which  it  splits  up  are 

a  b 

x  +  y    x  +  y^ 

the  whole  denominator  occurring  in  both. 

Cautions  of  a  slightly  more  advanced  character. 

There  is  another  mistake  often  made  by  beginners  later  on ; 
and  we  may  as  well  mention  it  here,  along  with  the  other 
cautions.  When  we  have  a  simple  factor  applied  to  two  terms, 
like  n(a  +  b), 

we  may  take  away  the  brackets  and  apply  it  to  each  term, 
getting  the  equivalent  form 

vu  +  nb, 


XXVIII.]  OPERATIONS.  243 

But  although  this  is  legitimate  with  a  factor,  it  is  not  legiti- 
mate with  everything  that  can  occur  outside  a  bracket, — not 
legitimate  with  a  symbol  of  operation  for  instance ;  neither 
is  it  legitimate  with  a  square  root  or  a  logarithm. 

Thus:  n{a  +  b)  =  na-\-nby 

but  J{a  +  b)\Ja  +  Jb, 

and  log  (a  +  b)^ log  a  +  log  b. 

The  sign  ^  is  to  be  read  "  is  not  equal  to  "  or  "  does  not 
equal." 

The  two  root  expressions  are  quite  different,  and  each  is 
already  in  its  simplest  form.     To  illustrate  numerically  : 

7(4  +  9)^2  +  3, 
for  J\3,  so  far  from  being  5,  is  something  between  3  and  4 ; 
because  3^  =  9  and  4^=  16,  so  ^13  lies  between  them,  and  as  a 
matter  of  fact  is  3-6055513... . 

(Never  imagine  from  the  accidental  repetition  of  some 
figure,  like  the  5  in  this  number,  that  it  is  going  to  "  recur." 
A  root  cannot  possibly  be  a  recurring  decimal,  for,  if  it  could, 
it  would  be  a  fraction,  and  therefore  commensurable ;  and  a 
root  is  always  incommensurable,  except  when  it  is  an  integer. 
See  Chapter  XX.). 

So  again  of  course     (a  +  b)^ ^a^  +  b% 
and  (x  -  yf  ^^  -  y^. 

As  to  log  a  +  log  &,  so  far  from  equalling  log  {a  +  b\  we  know 
that  it  equals  log  (axb),  that  is  log  ab. 

So  also  ft^+^Vja^  +  a", 

but,  instead,  a'^^  =  a*  x  a". 

Ja  +  Jb  is  by  no  means  equal  to  J{ab\  although  log  a  +  log  h 
does  equal  log  ab  ;  nor  does  cos  x  +  cos  y  equal  either  cos  xy  or 
cos  (x  +  y) ;  they  are  all  different.  So  we  learn  to  be  cautious 
with  symbols  of  operation  and  not  to  treat  them  as  factors 
nor  to  treat  them  all  alike.  We  have  to  be  very  cautious 
about  the  removal  of  brackets  in  their  case,  and  must  always 


244  EASY  MATHEMATICS.  [chap. 

be  sure  that  we  understand  the  meaning  and  value  of  the 
symbol  outside  them.     Some  operations  can  be  treated  in  this 
way,  and  some  cannot ;  and  we  must  learn  to  discriminate. 
Before  long  we  shall  find  that  a  highly  important  operator, 

denoted  by  -y-,  can  be  treated  in  this  way ;  so  that 
d  ,        .      d        d 

(uJ^'V)=-j-U  +  -r-V. 

dx^        '     dx       dx 
And  another  operation  denoted  by   Idx  can  likewise  be  so 
treated,  so  that 

l{u  +  v)dx=  ludx+  jvdx; 

but  these  things  have  to   be   proved,   they  must   never   be 
assumed ;  and  the  time  for  discussing  them  is  not  yet. 

We  may  notice  however  that  the  familiar  symbol  of  opera- 
tion X  is  one  that  can  be  treated  in  this  way 
7x(4  +  6)  =  (7x4)  +  (7x6), 
whereas  the  symbol  -r  c&nnot  be  so  treated 

7-f(4  +  6)H(7-^4)  +  (7-6). 
Nor  can  the  symbol  +  be  so  treated.     Anything  which  can  be 
so  treated  is  said  to  be  subject  to  "  the  distributive  law,"  that 
is  it  may,  and  indeed  must,  be  distributed  among  all  the  terms. 

There  is  another  law,  spoken  of  as  the  "  commutative  law," 
which  is  sometimes  applicable  and  sometimes  not ;  that  is  to 
say  it  applies  to  some  things  and  not  to  others.     It  applies 
when  the  order  can  be  inverted  ;  for  instance, 
axb  has  the  same  value  as  6  x  a. 

3   times  4   gives   the   same   number,   though   it  does  not 
suggest  the  same  grouping,  as  4  times  3. 

Similarly         a-{-b    is  the  same  as    b  +  a, 
but  a  -  Z>  is  not  the  same  as  b-a; 

it  is  numerically  equal  but  is  opposite  in  sign :  an  important 
distinction. 


XXVIII.]  OPERATIONS.  245 

Nor  is  a^h  the  same  as  6  ^  a,  not  even  numerically  equal. 
It  is  not  opposite,  but  "reciprocal." 
Again  there  is  a  permutative  law : 

cxab  is  the  same  as  a  x  c&  or  bx  ac, 
so  also  x  +  y-hz  =  y  +  x  +  z  etc.  ; 

and  under  certain  circumstances,  though  not  invariably, 
d         d  .     . 

but  ajh  is  not  the  same  as  J{ab),  nor  the  same  as  bja ;  the 
three  things  are  in  fact  equal  to  J((i^b),  J{ab),  and  J{ab^) 
respectively. 

The  expression  nlogx  is  not  the  same  as  \ognx,  it  equals 
log  (a;"). 


CHAPTER  XXIX. 


ILLUSTRATION  OF  THE  PRACTICAL  USE  OF 
LOGARITHMS. 

(i).  How  to  look  out  a  logarithm. 

Below  is  given  the  simplest  table  of  logarithms  that  can  be 
used.  You  can  buy  four-figure  logarithms  conveniently  printed 
on  a  card,  and  perhaps  you  may  prefer  to  use  them  at  once, 
because  four-figure  logarithms  are  accurate  enough  for  many 
practical  purposes,  and  are  handy  in  actual  work.  But  to  ex- 
plain the  method  of  using  a  table  and  the  principle  of  it,  without 
niceties  and  details,  the  annexed  table  will  serve  quite  well. 

You  will  find  this  table  repeated  at  the  end  of  the  book, 
folded  in  such  a  way  that  it  is  easy  to  refer  to. 

Table  of  3-figiire  Logarithms. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

000 

004 

009 

013 

017 

021 

025 

029 

033 

037 

11 

041 

045 

049 

053 

057 

061 

065 

068 

072 

075 

1-2 

079 

083 

086 

090 

093 

097 

100 

104 

107 

111 

1-3 

114 

117 

121 

124 

127 

130 

134 

137 

140 

143 

1-4 

146 

149 

152 

155 

158 

161 

164 

167 

170 

173 

1-5 

176 

179 

182 

185 

188 

190 

193 

196 

199 

201 

1-6 

204 

207 

210 

212 

215 

218 

220 

223 

225 

228 

1-7 

230 

233 

236 

238 

241 

243 

246 

248 

250 

253 

1-8 

255 

258 

260 

263 

265 

267 

270 

272 

274 

277 

1-9 

279 

281 

283 

286 

288 

290 

292 

295 

297 

299 

2 

301 

322 

342 

362 

380 

398 

415 

431 

447 

462 

3 

477 

491 

505 

519 

532 

544 

556 

568 

580 

591 

4 

602 

613 

623 

634 

644 

653 

663 

672 

681 

690 

5 

699 

708 

716 

724 

732 

740 

748 

756 

763 

771 

6 

778 

785 

792 

799 

806 

813 

820 

826 

833 

839 

7 

845 

851 

857 

863 

869 

875 

881 

887 

892 

898 

8 

903 

909 

914 

919 

924 

929 

935 

940 

945 

949 

9 

954 

959 

964 

969 

973 

978 

982 

987 

991 

996 

6HAP.  XXIX.]        USE  OF  LOGARITHM  TABLES.  247 

The  triple  digits  which  occur  throughout  this  table  are  the 
decimal  parts  of  the  logarithms  of  the  numbers  on  the  left  and 
above.  The  decimal  point  is  not  printed,  but  it  is  always  to 
be  understood,  and  on  taking  out  the  triple  figures  a  decimal 
point  must  always  be  written  in  front  of  them. 

Now  let  us  use  the  table  to  find  a  few  logarithms.  The 
most  obvious  record  in  the  table  is  that 

logl     =0,  log  M  = -041,     log  1-2  =  -079, 

logl-3  =  -114,  etc.  logl-9  =  -279, 

log  2     =-301,     log  3    =-477,     log  4    ='602, 
and  so  on. 

Next  we  have,  by  the  use  of  the  top  row  of  figures  combined 
with  the  left-hand  column, 

log  2-1  =  -322,     log  2-2  =  -342,     log  2-3  =  '362,  etc. 
log  3-1  =  -491,     log  3-2  =  -505,  etc. 
log  4-1  =  -613,  etc. 
and  so  on. 

For  all  these  figures  there  is  nothing  more  to  do  than  just 
extract  the  logarithms  from  the  table  as  they  stand. 
But  now  suppose  we  wanted  the  logarithm  of  20  or  30. 
We  know  that  log  30  =  log  10  +  log  3  =  1+  log  3,  hence  look 
out  log  3,  and  write  log  30  =  1*477. 
Similarly  log  20  =  1-301, 

log  70  =  1  -845,     and  so  on. 
So  all  we  have  to  do  in  that  case  is  to  prefix  a  1  to  the 
decimal  point. 

If  we  wanted  the  logarithm  of  11  or  12  or  13  it  would  be 
just  the  same,  we  must  prefix  a  1  to  the  decimal  point,  so  that 
log  11  =  1-041,     log  12  =  1-079,     log  13  =  1114. 
Similarly 

log  21  =  1-322,     log  22  =  1-342,     log  31  =  1-491,  etc. 
1  is  called  the  "  characteristic  "  of  any  number  of  "  order  "  1. 


248  EASY  MATHEMATICS.  [chap. 

9 
Further  if  we  want  the  logs  of  100  or  200  or  300,  we  must 
prefix  a  2  to  the  decimal  point  because 

log  300  =  log  100  +  log  3  =  2  +  log  3  =  2-477. 
Similarly      log  110  =  2*041,     log  120  =  2'079, 
log  210  =  2-322,     log  220  =  2-342, 
log  310  =  2-491,     log  320  =  2-505. 

So  the  logarithm  of  any  number  consisting  of  two  signifi- 
cant figures  can  be  readily  obtained  from  the  table,  and  the 
"  characteristic  "  or  integer  part  of  the  logarithm  is  given  by 
the  "order"  of  the  number.  "Characteristic"  and  "order" 
are  in  fact  two  names  for  the  same  thing,  except  that  the  first 
is  appropriate  to  a  logarithm,  and  the  second  is  applied  to 
the  original  number. 

The  logarithm  of  every  number  with  only  two  significant 
figures  is  therefore  directly  contained  in  the  little  table  printed 
above,  no  matter  how  big  the  number  may  be.     For  instance, 

log  98,000,000  =  7-991. 

But  suppose  the  number  had  3  significant  figures.  What  is 
the  logarithm  of  215  for  instance"?  Well,  it  will  lie  approxi- 
mately half-way  between  log  210  and  log  220.  Not  exactly 
half-way  because  the  number  grows  in  G.P.  while  the  logarithm 
grows  in  A.P.,  but  half-way  is  near  enough  for  most  practical 
purposes.  So  we  can  see  that  approximately  log  215  =  2-332, 
because  that  is  half-way  between  2-322  and  2*342. 

But  suppose  the  number  whose  log  was  wanted  did  not  lie 
half-way  between  others,  but  only  one-tenth  of  the  way ; 
suppose  for  instance  log  211  was  wanted,  we  should  have  to 
take  one  tenth  of  the  difference  between  322  and  342,  which 
difference,  being  20,  the  tenth  of  it  is  2 ;  and  this  would  have 
to  be  added  on,  as  representing  one-tenth  of  the  interval. 
So  log  211  would  equal  2-324. 


xxix.]  USE  OF  LOGARITHM   TABLES.  249 

We  can  in  fact  make  an  extension  of  the  table  for  any  third 
significant  figure  in  the  number  whose  log  is  required,  thus, 
log  210  =  2-322,     log  211  =  2-324, 
log  212  =  2-326,     log  213  =  2-328, 
log  214  =  2-330,  and  so  on,  up  to 
log  220  =  2-342. 
Take  a  few  more  illustrations  of  this. 
Wanted  log  2-35. 
From  the  table,  log  2-30  =  -362  and  log  2-40  =  -380, 

so     log  2-35  =  -371. 
Wanted  log  3-41. 

log  3-40  =  -532,         log  3-50  =  544, 
so     log  3-41  =  -533  approximately. 
Similarly  log  3*42  =  -534  approximately. 

Wanted  log  5-63. 

log  5-6  =  -748,         log  5-7  =  -756, 

.*.   log  5-63  =  -750, 

being  three  tenths  of  the  interval  added  on  to  the  smaller  one. 

Wanted  log  5-67.     We  might  add  on  seven  tenths  of  the 

interval  to  the  smaller  one,  or,  rather  better,  subtract  three 

tenths  of  the  interval  from  the  bigger  one,  getting 

log  5-67  =  -754. 

But  the  table  contains  more  than  I  have  at  present  described 
and  used.  The  first  half  of  the  table  gives  the  logarithms  of 
numbers  near  to  unity,  so  we  can  get  out  logarithms  of  1  -01 
or  r02  etc.  up  to  1*99,  the  numbers  being  expressed  to  3 
significant  figures  and  all  the  logarithms  recorded.  It  is  a 
help  to  have  this  given,  as  a  sort  of  extra,  because  for  these 
small  numbers  the  logarithms  change  so  rapidly  that  the  jump 
is  too  great  for  easy  and  safe  treatment  by  attending  to  the 
diff'erences,  and  when  we  come  to  look  out  anti-logs  (see  next 
p^e)  they  will  fall  in  gaps  of  too  large  size. 


250  EASY  MATHEMATICS.  [cha^. 

Using  this  part  of  the  table  we  see  that 

log  1  -01  =  -004,  log  1  -02  =  -009, 

log  Ml  =  -045,  log  1-25  =  -097, 

log  1-53=  -185,  log  1-99  =  -299. 

And  consequently 

log  10-2  =1-009,         log  102-   =2009, 

login-   =2-045,         log  125-   =2-097, 

log  12-5  =  1  -097,  log  19-9  =  1  -299, 

log  199-   =2-299,       log  1990-   =3-299. 

The  characteristic  of  the  logarithm  is  always  the  "  order " 

of  the  number. 

(ii).  How  to  look  out  the  number  which  has  a  given 
logarithm. 

To  look  out  the  number  which  possesses  a  given  log  we 
have  only  to  use  the  table  backwards.  It  is  quite  simple  and 
obvious  in  idea,  the  only  trouble  is  that  we  shall  not  usually 
find  the  given  logarithm  actually  in  the  table.  If  it  is  an 
extensive  table  we  are  more  likely  to  find  it,  and  that  saves 
thought,  but  involves  the  turning  over  of  many  pages  ;  with  a 
little  compressed  table  like  the  one  given,  we  are  not  likely  to 
find  a  number  exactly  entered,  and  a  trifle  of  thought  is 
necessary.  That  is  no  defect  however  for  our  present  purpose, 
which  is  not  immediately  to  facilitate  practice,  but  to  furnish 
instruction  which  shall  facilitate  practice  by  and  bye. 

The  phrase  "  number  which  possesses  the  logarithm  "  so  and 
so,  is  rather  long  and  unwieldy,  and  it  is  commonly  shortened 
into  anti-log. 

Thus  log  2  =  -301,  or  2  is  the  anti-log  of  '301. 

Given  then  the  following  logarithm,  -380,  what  is  its  anti-log  ? 
Referring  to  the  table,  we  see  that  it  is  2-4. 

Given  -663,  the  anti-log  is  4-6,  and  so  on. 

But  suppose  the  given  logarithm  had  been  1  -380,  what  then  ? 


XXIX.] 


USE  OF  LOGARITHM   TABLES. 


251 


We  should  look  in  the  table  for  the  decimal  part  only,  for  it 
is  only  the  decimal  part  which  is  ever  there  recorded.  The 
prefix  1  before  the  decimal  point  tells  us  the  power  of  ten  in 
the  result,  shows  in  fact  that  the  result  lies  between  ten  and  a 
hundred.     The  antilog  is  therefore  not  now  2 '4  but  24* 

So  also  the  antilog  of  1-663  is  46-  not  4*6. 

The  antilog  of    2-663   is     460', 
of    3-663   is   4600-, 
and  so  on.     It  is  safer  to  actually  write  the  decimal  points  at 
the  end  of  whole  numbers  in  this  sort  of  case. 

The  integer  part  of  the  logarithm,  often  called  its  "  charac- 
teristic," has  simply  the  efiect  of  determining  the  order  of 
magnitude  of  the  result  (p.  171).  Surely  however  a  most 
important  effect,  and  one  not  to  be  slurred  over. 


Examples. 

What  is  the  antilog  of  1*672  'i    Answer  47*. 
What  is  the  antilog  of  I'SOl  1     Answer  20*. 
What  is  the  antilog  of  2-041  ?     Answer  110*. 
What  is  the  antilog  of  3-699  1     Answer  5000'. 
Employing  the  upper  part  of  the  table  we  see  that 


log  1-34=    -127, 
log  1-01=    -004 
log  101       =2-004, 
log  1-09=    -037, 
log  109       =2-037. 
Likewise  the  antilog  of    -111 
1-111 
2-111 


log  1-18  = -072, 
log  10-1    =1-004, 
log  1010      =3-004, 
log  10-9   =1037, 


IS  1-29 

is  12-9 

is  129- 

•097  is  1-25. 

1-097  is  12-5. 

•196  is  1-57. 

2-196  is  157-. 


252  EASY   MATHEMATICS.  [chap. 

So  far  the  logarithms  have  been  found  in  the  table,  because 
we  chose  numbers  of  only  two  significant  figures.  The  case 
when  a  logarithm  does  not  occur  exactly  in  the  table  causes 
no  difficulty,  it  only  gives  a  little  more  trouble. 

What  is  the  antilog  of  '389?  Answer  2*45;  because  it  lies 
about  half-way  between  '380  and  '398,  so  the  answer  lies 
half-way  between  2-4  and  2-5. 

What  is  the  antilog  of  2-389  ?     Answer  245. 

What  is  the  antilog  of  1-675 1     Answer  47*3. 

Why?  because  the  given  log  lies  one-third  of  the  way  be- 
tween 672  and  681 ;  so  its  antilog  will  lie  about  one-third  of 
the  way  between  47  and  48.  As  to  the  position  of  its  decimal 
point,  that  is  determined  by  the  "characteristic"  or  integer 
part  of  the  given  logarithm,  which  was  unity. 

Logarithms  of  fractions. 

So  far  all  the  antilogs  have  turned  out  greater  than  1, 
because  all  the  logarithms  chosen  have  been  positive.  The 
characteristics  have  all  been  either  1  or  2  or  3  or  0 ;  for  the 
logarithm  -672  has  the  characteristic  0.  It  might  be,  and 
often  is,  written  0-672. 

But  now  suppose  it  had  a  negative  characteristic ;  for 
instance  1*672,  where  the  minus  sign  is  placed  above  the  1 
instead  of  in  front  of  it,  in  order  that  it  may  not  be  applied  to 
the  whole  number,  but  only  to  the  1 ;  which  is  a  conventional 
but  convenient  mode  of  representing  an  important  distinction. 

The  meaning,  written  out  fully,  is  -  1  -i-  -672. 

Naturally  this  might  be  written,  if  we  liked,  -  -328,  but  if 
we  did  that  we  should  want  another  table  full  of  negative 
numbers  wherein  to  look  out  the  logarithms  of  fractions.  By 
the  above  device  we  can  use  the  same  table  all  the  time,  and 
only  adjust  the  position  of  the  decimal  point  in  accordance 
with  the  characteristic,  so  that  it  fixes  the  "  order  "  as  usual. 


XXIX.]  USE  OF  LOGARITHMS.  253 

For  instance, 

antilog  of    '672  from  the  table  is  4*7, 
so  antilog  of  1-672  will  be  '47, 

and  antilog  of  2-672  will  be  '047, 

antilog  of  3-672  will  be  '0047, 

the   negative   characteristic   indicating    the    position   of    the 
highest  significant  figure  counting  from  the  units'  place. 
The  antilog  of  1*672  is  of  course      47- 
of  2-672  470- 

and  of  3-672  4700- 

and  so  on ;  the  positive  characteristic  counting  the  number  of 
places  to  the  left  of  the  units'  place. 

(iii).  How  to  do  Multiplication  and  Division  with 
Logs. 

We  know  that  log  ah  =  log  a  +  log  b, 

and  that  log  -  =  log  b  -  log  c. 

c 

So  we  know  that  log  —  =  log  a  +  log  b  -  log  c, 

c 

and  likewise  log  —r  =  log  a  +  log  b  -  log  c  -  log  d  -  log  e 

=  (log  a  +  log  b)  -  (log  c  +  logd  +  log  e). 
In  other  words,  whenever  we  have  a  fraction  consisting  of  a 
number  of  factors  in  numerator  and  denominator,  we  must 
look  out  the  logarithm  of  each  factor.  All  those  in  the 
numerator,  arrange  in  one  column,  and  add ;  all  those  in  the 
denominator,  arrange  in  another  column,  and  add ;  then 
subtract  one  addition  from  the  other  so  as  to  get  the  logarithm 
of  the  quotient.  We  have  then  only  to  refer  to  the  table  to 
find  the  number  which  possesses  this  logarithm,  and  the 
quotient  is  found. 


254  EASY  MATHEMATICS.  [chap. 

For  instance,  take  this  fraction 

6-7  X  43  X  170 
74x3-2xl-3" 

Look  out  the  logs  of  the  factors  as  stated : 

log  6-7=    -826  log  74-   =l-869 

log  43-   =1-634  log  3-2=    -505 

log  170-   =  2-230  log  1-3=    -114 

add     4-690  add     2-488 

subtract     2-488 

2-202 

2-202  is  therefore  the  log  of  the  resulting  quotient.  Eeferring 
to  the  table,  we  find  that  the  number  possessing  this  log  is 
159-2.     Hence  that  is  the  answer. 

It  may  be  asked,  why  do  it  this  way  when  we  could  easily 
do  it  by  simple  multiplication  and  division  1 

Reply  :  Very  little,  if  any,  advantage  in  such  a  simple  case 
as  that.  No  advantage  at  all  if  you  can  easily  see  factors 
which  may  be  struck  out. 

But  people  who  often  have  to  do  such  sums  get  rather  tired 
of  frequent  multiplication  and  division,  and  they  usually  prefer 
logarithms  as  a  quicker  and  surer  way.  It  becomes  quicker 
and  surer  with  practice.  Engineers  usually  employ  what  is 
really  the  same  process,  but  they  have  their  table  of  logarithms 
constructed  in  wood;  and  instead  of  looking  out  the  logarithms, 
they  slide  a  slip  along  this  rule,  till  a  mark  on  it  points 
to  the  number  printed  where  its  logarithm  ought  to  be, 
and  so  attain  the  result  in  an  ingenious  manner,  without 
actually  recording  or  thinking  of  any  logarithm  at  all.  They 
shift  the  pointers,  of  which  there  are  a  pair,  alternately  to  one 
factor  after  another,  taking  numerator  and  denominator  factors 
alternately,  and  then  at  the  end  they  read  off  the  result  as 
indicated  by  one  of  the  pointers. 


XXIX.]  USE  OF  LOGARITHMS.  255 

The  instrument  is  called  a  "slide-rule";  it  is  in  fact  a 
mechanical  table  of  logarithms  arranged  ingeniously  for  quick 
and  practical  use,  and  it  gives  you  about  3-figure  accuracy  if 
it  is  of  a  simple  and  well  made  pocket  kind.  More  elaborate 
and  larger  instruments  can  give  6-figure  accuracy.  The  in- 
genuity belongs  to  the  devising  and  making  of  the  rule  :  the 
use  of  it  is  quite  simple,  but  it  has  to  be  learnt.  It  should  not 
be  learnt  as  a  substitute  for  other  methods,  but  as  a  supplement. 
Pupils  are  not  recommended  to  learn  the  slide-rule  till  they 
can  use  a  numerical  table  of  logarithms.  Nor  are  they  recom- 
mended to  use  logarithms  till  they  can  multiply  and  divide 
with  facility.  In  other  words,  these  aids  to  rapidity  should 
be  kept  in  their  proper  place, — not  to  make  people  helpless 
without  them,  but  to  assist  people  who  can  work  quite  well 
without  them  to  obtain  results  more  quickly  and  with  less 
labour. 

Another  example.     Find  the  value  of 

27-1  X  -16  X  -089 
•00055x3430  * 

Look  out  logs  and  record  them  as  below : 

log  27-1      =2 '^^^  log  '^0^^^  =  5'*^^^ 

log -16    =1-204  log  3430-  =3-535 

log -089  =  2-949  0»275 

T-586 
subtract     0-275 

1-311 

Antilog  of  this  is  -205,  which  is  therefore  the  result,  and  may 
be  recorded  as  equal  to  the  above  fraction  to  something  like 
3-figure  accuracy.  This  should  be  checked  by  actual  multi- 
plication. Indeed  for  some  time,  and  especially  when  there 
are  negative  characteristics,  it  is  safest  to  check  over  the 
result  by  other  means  than  the  inere  logarithms.     It  is  the 


256  EASY   MATHEMATICS.  [chap. 

order  of  magnitude  that  runs  great  risk  of  going  wrong :  the 
actual  digits  can  easily  be  got  right. 

A  few  more  examples  for  practice  : 
What  is  the  log  of  -05 1 
And  what  is  the  antilog  of  I'89  ? 
The  log  of  5  in  the  table  is  '699, 

so  the  log  of  -05  is  2-699. 
The  antilog  of  '89  estimated  from  the  table  is  about  7*77, 
so  the  antilog  of  T'89  is  -777. 
What  is  the  antilog  of  -049  ? 
We  find  this  number  in  the  upper  part  of  the  table,  and 
the  antilog  required  is  1-12. 

What  is  the  antilog  of  2-049 1     Answer  112. 

T-0491     Answer        -112. 
2-0491     Answer        -0112. 
What  is  the  antilog  of  -023 1     It  does  not  occur  even  in  the 
upper  part  of  the  table,  but  it  lies  half-way  between  two 
numbers  which  we  find  there;    so  we  estimate  the  antilog 
as  1-055. 

The  antilog  of  3-023  =  1055' 

ofT023=  -1055. 

This  is  the  use  of  the  upper  part  of  the  table,  as  previously 
half  explained,  that  it  gives  us  the  logarithms  of  numbers  only 
slightly  greater  than  1  in  greater  profusion ;  and  it  is  just 
here  that  profusion  is  necessary,  for  in  other  parts  of  the 
table  logarithms  lie  much  closer  together  in  value  than  they 
do  here.  Consequently  what  would  naturally  be  the  first  row 
of  the  table  is  spread  out  into  nine  rows,  the  first  row  itself 
becoming  thereby  a  column,  reaching  from  Ito  2,  and  giving 
all  the  tenths  of  this  interval. 

If  a  beginner  likes  to  think  out  the  reason  and  meaning  of 
the  different  closeness  of  distribution  in  various  parts  of  a 


XXIX.]  USE  OF  LOGARITHMS.  257 

logarithm  table,  he  should  by  all  means  do  so,  but  he  need 
not  be  made  to  do  it.  The  reasonableness  of  it  can  be  put 
thus: 

All  the  900  integers  between  100  and  1000  have  logarithms 
lying  between  2  and  3 :  this  unit  difference  of  logarithms  is 
therefore  spread  over  all  that  range ;  while  the  same  loga- 
rithmic interval,  viz.  that  between  0  and  1,  has  to  be  squeezed 
between  the  numbers  1  and  10,  covering  only  nine  consecutive 
integers. 

The  logarithmic  interval  between  1  and  2  has  to  serve  for 
the  90  whole  numbers  between  10  and  100,  while  the  same 
logarithmic  interval,  viz.  between  3  and  4,  is  all  that  can  be 
allowed  to  cover  the  9000  numbers  between  1000  and  10000. 

Hence  manifestly  the  logarithms  of  integers  between  1  and 
10  must  be  few,  and  the  intervals  between  must  be  great, 
though  they  may  be  conveniently  filled  up  with  the  logarithms 
of  intervening  fractions;  but  the  logarithms  of  integers 
between  1000  and  10000  are  close  together,  their  value 
increasing  only  slightly  for  each  addition  of  unity  to  the 
number.  In  other  words,  the  logs  of  integers  take  9000  steps 
to  go  from  3  to  4 ;  they  only  take  9  steps  to  go  from  0  to  1. 
The  one  is  a  trip,  the  other  is  a  straddle. 


L.E.M.  R 


CHAPTER   XXX. 

How  to  find  powers  and  roots  by  logarithms. 

The  finding  of  any  power,  or  any  root,  is  now  an  extremely 
simple  operation. 

We  know  that  log  x"  =  ?i  log  x,  and  this  holds  whether  n  be 
an  integer  or  any  fraction. 

In  other  words,  as  said  before, 

log  x^  =  2  log  X, 

log  x^  =  3  log  X, 

logJx  =  logx^=^^\ogx, 

log  ^a;  =  log  a;^  =  J  log  a;, 

and  so  on.     Hence  the  method  suggests  itself,  and  we  need 

only  proceed  to  examples. 

To  find  the  value  of  ^2,  the  logarithm  of  it  will  be  half  the 
logarithm  of  2,  and  that  we  look  for  in  the  table,  and  find  to 
be  '301,  so  half  of  it  is  -1505.  This  we  do  not  find  exactly  in 
the  table,  but  we  see  that  it  is  the  logarithm  of  a  number 
lying  between  1-41  and  1-42,  and  Ave  estimate  the  number  as 
being  1'414.  This  is  of  course  only  an  approximation, 
because  no  arithmetical  specification  of  it  can  be  anything 
but  approximate.  If  calculated  more  elaborately,  it  comes 
out  1*4 142136...,  but  it  can  neither  stop  nor  circulate. 

Similarly  ^3  =  1*732  approximately, 

or  more  nearly  1  -7320508 . . . , 

again  without  either  recurrence  or  termination. 


CHAP.  XXX.]  USE  OF  LOGARITHMS.  259 

Now  take  a  case  of  a  power.  Suppose  we  want  to  calculate 
the  2^4  involved  in  an  example  on  page  156.  Its  logarithm 
will  be  24  log  2  =  24  x  -301  =  7*224  and  we  have  only  to  look 
out  the  number  which  has  this  logarithm,  that  is  look  for  the 
antilog  of  7-224.  We  shall  find  225  in  the  table  (p.  246),  and 
that  is  really  better  than  224,  because  when  we  multiply  a 
number  by  so  big  a  factor  as  24  there  must  probably  be  some 
carried  forward  figure  to  be  attended  to.  Anyway  we  find 
that  '225  is  the  logarithm  of  1  •68.  This  is  not  the  result,  of 
course,  since  we  have  not  yet  attended  to  the  characteristic, 
which  is  7.  The  characteristic  is  indeed,  in  these  big  numbers, 
usually  the  most  important  thing  to  notice.  The  charac- 
teristic 7  shows  that  the  number  is  of  the  order  seven,  i.e.  that 
it  lies  between  10'^  and  10^;  in  other  words,  that  it  requires 
eight  digits  to  express  it,  and  so  it  is  approximately 

16,800,000, 
that  is  sixteen  million  eight  hundred  thousand,  so  far  as  we 
can  express  it  with  3-figure  accuracy.     There  are  eight  digits 
in  this  result,  but  only  three  of  them  are  "significant,"  the 
others  are  mere  ciphers  to  indicate  the  order  of  magnitude. 

The  neatest  way  of  recording  such  a  result  is  therefore 
1-68x107, 
and  the  characteristic  of  the  logarithm  will  always  give  us  the 
index  of  the  power  of  ten  when  the  number  is  so  written. 

For  instance,  antilog  19  '330  =  2  -1 4  x  1 O^^, 
antilog  6 -552  =  3  -56  X  1 0-6, 
antilog  2-950  =  8-92  x  10-2=  0892. 

The  operation  of  finding  a  root  will  look  thus  : 
To  find  the  fifth  root  of  1930. 
log  1930  =  3-286, 
ilogl930  =  0-657  =  logof  4-54... 
wherefore  4-54...  =  ^(1930). 


260  EASY  MATHEMATICS.  [chap. 

Observe  that  when  dividing  a  logarithm  the  characteristic 
is  to  be  included  in  it  and  divided  with  the  rest  of  it.  It  is 
only  in  dealings  with  the  "  table  "  that  the  characteristic  does 
not  appear.  It  should  however  always  be  supplied  and  should 
not  be  forgotten  or  ignored. 

Thus  if  we  had  wanted  the  fifth  root  of  193  or  of  19-3  or 
of  1  93  we  should  have  obtained  a  totally  different  number : 
not  the  same  number  with  the  decimal  point  shifted,  but  a 
different  number  altogether.     For  instance, 
log  193  =  2-286, 
1  log  193=   -457  =  log  of  2-865. 
log  19-3  =  1-286, 
I  log  19-3=   •257  =  logof  1-81. 

log  1-93  =  0-286, 
^  log  1-93=    -057  =  log  of  1-U. 

But  now  suppose  we  required  the  root  of  a  fraction,  i.e.  of 
something  whose  logarithm  was  negative.  We  must  think 
how  to  proceed  in  that  case.  Suppose  for  instance  we  want 
the  fifth  root  of  -193, 

log -193  =  1-286, 
that  is  to  say  a  negative  part  and  a  positive  part ;  it  means 

-  1  +  -286. 
In  order  to  divide  this  by  5  conveniently,  it  is  best  to  increase 
both  the  negative  and  the  positive  parts  by  any  convenient 
equal  amounts :  in  this  case  the  convenient  amount  is  4. 
Add  -  4  to  the  negative  part,  and  add  +  4  to  the  positive  part: 
the  value  will  thereby  be  unaltered,  but  it  will  be  written  as 

-5  +  4-286 
and  now  it  is  quite  easy  to  divide  by  5,  yielding 

-1  +  -857   or  T-857, 
which  is  the  log  of  -72.     Wherefore 
•72=^-193. 


i 


XXX.]  USE  OF  LOGARITHMS.  261 

The  root  is  bigger  than  the  number.  That  is  universal  with 
roots  of  proper  fractions.  When  we  square  a  fraction  we 
diminish  it ;  when  we  square-root  a  fraction,  consequently,  we 
increase  it.  Think  it  out;  it  is  all  in  accordance  with  common- 
sense. 

But  we  must  take  another  example. 
Let  us  find  the  square  root  of  •0054. 
log -0054  =  3-732 

=  -4  +  1-732, 
Jlog-0054= -2+    -866 

=  2-866  =  logof -0735, 
wherefore  -0735  =  ^-0054. 

We  might  have  made  an  approximate  guess  at  this,  because 
^•0049  could  have  been  written  down  as  '07  by  inspection, 
and  so  ^'0054  will  be  a  little  bigger ;  how  much  bigger  it  is 
not  so  easy  to  guess. 

But  suppose  we  had  wanted  J'Obiy  we  should  have  found 
nothing  like  a  7  in  the  root.     Let  us  do  it : 
log -054  =  2-732, 
1  log -054  =1-366  =  log  of  ;232. 

So  -232    =^054,       whereas     -0735  =  ^-0054, 

-0232  =  7-00054,  -735   =^-54, 

2-32     =75-4,  7-35     =^54, 

23-2       =V540,  73-5        =^5400. 

A  little  easy  repetition  on  this  point  may  be  useful  so  as  to 
emphasise  it. 

^49  =7,    and  ^100=  10; 

so  V4900       =  70, 

and  ^490000  =  700  ; 

V-49         =-7, 
V0049     =  -07, 
V-000049  =  -007 ; 


262  EASY  MATHEMATICS.  [chap. 

but  if  instead  of  two  ciphers  we  had  suffixed  or  prefixed  only 
one  cipher,  we  should  have  had  quite  different  results,  and  not 
so  easy  to  ascertain,  viz.  the  following : 

74-9         =     2-214, 
V'490  =    22-14, 

V'49000  =221-4, 

^•049     =        -2214, 
^•00049=        -02214.      . 

Exercises. 

^407  =  7-41 
because  log  40  7  =  2  -6 1 0, 

and  one-third  of  it  is  -870  =  log  of  7*41. 

Similarly  work  out  the  following  : 

^•407       =      -741, 
^407000-  =74-1, 

^-000407=     -0741, 
so  that  whereas  for  square  roots  the  noughts  can  be  added  in 
pairs  to  leave  the  digits  unaltered,  for  cube  roots  the  ciphers 
must  be  added  in  triplets  if  they  are  to  make  no  change  in  the 
digits.     This  is  an  immediate  consequence  of  the  fact  that 
^1000  =  10. 
The  last  case,  for  instance,  works  out  thus : 
log  -000407  =  4-61  =  -  6  +  2-61, 
of  which  one-third  is         2-870  =  log  of  -0741. 

The  simplest  way  of  dealing  with  these  things  however  is 
to  express  them  in  powers  of  ten. 

Thus  -000407  =  407  xlO-«, 

so  its  cube  root  is 

4/407  X  10-2  =  7-41  ^  100  =  -0741. 
But  now  suppose  the  digits  had  not  been  added  in  triplets. 
Find  cube  root  of  40-7. 

log  40-7  =  1-61, 


XXX.]  ROOTS  OF  NEGATIVE  NUMBERS.  263 

a  third  of  that  is  '537,  which  is  the  logarithm  of  3-44...  which 
is  therefore  the  root  required. 

Again,  to  find  ^4-07. 

log  4-07  =  -61, 
one-third  is  2033  and  the  number  corresponding  to  this  log  is 
nearly  TG. 

So       (4070)^=16  nearly;  more  accurately  15*966... 
Also  (-00407)^  =  -16. 

Find  the  cube  root  of  '0078. 

We  may  ^vrite  it  as        7*8x10"^, 
and  so  express  its  cube  root  as 

1-98x10-1  = -198. 

Find  the  cube  root  of  -000000081. 

Express  it  as  81  x  10"^. 

Its  cube  root  is  4-33  x  lO'^  =  00433. 

Roots  of  negative  numbers. 

Perhaps  it  is  not  likely  to  occur  often  in  elementary  practice, 
but  it  is  worth  noticing  that  the  cube  root  of  a  negative 
number  is  by  no  means  impossible.  What,  for  instance,  is  the 
cube  root  of  -  8 ;  that  is,  what  number  multiplied  twice  by 
itself  will  make  -  8  ]    The  answer  is  -  2,  for 

-2x-2=+4   and    +4x-2=-8. 
So  y-27=-9,    y- 1728= -12,   and  so  on. 

Also  y- 407  =-7-41,   see  above. 

The  square  root  of  a  negative  number  has  no  simple  meaning. 
If  we  tried  to  find  the  square  root  of  -  9  or  -  25  we  could  not 
do  it,  for  -  3  X  -  3  =  +  9  and  -  5  x  -  5  =  +  25.  Hence  nega- 
tive numbers  have  no  square  roots,  but  they  have  cube  roots. 
Having  no  square  roots  of  course  they  cannot  have  fourth 
roots,  for  a  fourth  root  is  simply  the  square  root  of  a  square 
root.  But  they  have  fifth  roots  and  seventh  roots  and  any 
odd  numbered  roots,  because  an  odd  number  of  minus  signs 


264  EASY  MATHEMATICS.  [chap.  xxx. 

multiplied  together  make  minus.  Negative  numbers  have  no 
even  roots. 

This  is  not  all  that  can  be  said  concerning  the  roots  of 
negative  numbers,  by  any  means  :  Pure  mathematicians  know 
a  great  deal  more  than  that  about  them ;  and  later,  children 
who  like  the  subject  may  learn  some  of  it,  but  not  yet.  In 
order  however  to  prepare  them  for  a  convenient  way  of 
dealing  with  the  matter,  I  will  point  out  that  any  negative 
number  can  be  said  to  have  -  1  as  a  factor ;  for  instance, 

-  8=   8x  -1, 
-16  =  16x  -1, 

-  27  =  27  X  -  1,  and  so  on. 

Hence  any  root  of  any  negative  number  is  equal  to  the  same 
root  of  the  same  positive  number  multiplied  by  the  appropriate 
root  of  -  1.     For  instance, 

y-8  =  ^8x4/-l  =  2^-l, 
^-27=y27x4/-l  =  34/-l, 
^-32  =  2^-1,  and  so  on. 
[Rememher  thai,tj(xi/)  =  jxjy  or  that  (a5)"  =  a"6^] 

But  the  same  method  may  be  extended  to  even  roots,  thus 

V-16  =  V16xV-l  =  V-l, 
V-9  =  3V-1, 

4/- 81  =  3^-1, 

4/-64  =  2«/-l,  and  so  on. 
It  is  true  that  we  do  not  yet  know  what  to  make  of  J-  1 
or  4/-  1  or  y-  1 ;  it  is  an  impossible  or  imaginary  quantity ; 
but  though  we  think  that  we  do  know  what  to  make  of  I/-  1 
or  ^ -I,  viz.  although  we  know  that  they  =  -  1,  do  not  let 
us  be  too  sure  that  we  know  all  about  even  these.  It  is  at 
any  rate  true  that  -Ix  -Ix  -1=  -1,  and  that  is  all  that 
need  now  concern  us ;  but  it  is  not,  strange  to  say,  the  whole 
truth  concerning  even  the  odd  roots  of  minus  one. 


CHAPTER  XXXI. 


Geometrical  Illustration  of  Powers  and  Roots. 

Geometrical  illustration,  or  illustration  of  number  by- 
simple  diagrams,  cannot  be  pressed  very  far  with  advantage  for 
elementary  purposes.  But  for  simple  things  the  illustrations 
are  so  vivid  and  useful  and  interesting  that  they  should 
often  be  employed,  and  especially  be  set  as  exercises  so  as 
to  infuse  life  and  interest  into  what  might  otherwise  be  dull. 

The  simplest  illustration  of  all  relates  to  the  squares  or 
cubes  of  integers.  That  the  square  of  3  is  9  is  illustrated  in 
the  most  conspicuous  manner  by  the  diagram. 


Fia.   12. 

So  also  that  the  square  of  4  is  16,  and  the  square  of  5  is  25. 

That  the  cube  of  2  is  8  is  illustrated  thus, 
but  the  best  plan  of  dealing  with  solids  is 
to   use    cubical   wooden    blocks   and   build 
them  up. 
8  blocks  will  build  a  cube  whose  side  is  2 
27  „  „  „  3 

64  »  »  »  4 

and  so  on. 


/   / 

/ 

/  /  / 

/ 

/ 

/ 

y 

Fio.  13. 


266  EASY  MATHEMATICS.  [chap. 

The  same  blocks  laid  flat  on  the  table  will  serve  con- 
veniently for  squares  and  rectangles  and  commensurable  areas 
generally.  They  will  also  serve  to  outline  commensurable 
triangles  :  with  conspicuous  advantage  in  some  cases. 

By  this  kind  of  practice  a  reality  about  square  and  cube 
numbers  is  attained  which  can  be  got  in  no  other  way. 

Naturally  also  the  area  of  any  rectangle  can  be  thus 
illustrated  as  the  product  of  length  and  breadth ;  and  the 
volume  of  rectangular  solids  as  the  product  of  length,  breadth, 
and  height. 

If  we  try  to  illustrate  fourth  or  higher  powers  in  this  way 
we  shall  find  ourselves  helpless.  Space  is  only  of  3  dimensions. 
There  are  length  and  breadth  and  thickness,  and  no  more. 
Some  have  tried  to  imagine  what  a  fourth  dimension  would 
be  like,  but  for  the  present  we  will  be  content  with  an  actually 
experienced  and  familiar  three  dimensions. 

So  much  for  powers ;  now  what  about  roots  ? 

The  few  commensurable  roots  that  exist  must  all  be  whole 
numbers,  and  they  will  be  represented,  so  far  as  square  and 
cube  roots  are  concerned,  by  the  length  of  the  sides  or  edges 
of  the  squares  or  cubes  which  have  so  far  been  drawn  or  built 
up.  Thus,  for  instance,  the  square  root  of  16  is  4,  and  the 
cube  root  of  27  is  3.  But  this  fact,  which  is  experimentally 
obvious  in  the  commensurable  case,  where  the  square  or  the 
cube  can  be  built  of  blocks,  is  true  also  in  the  general  case. 
The  length  of  a  side  of  a  square  is  the  square  root  of  its  area 
always,  and  the  length  of  the  edge  of  a  cube  is  the  cube  root  of 
its  volume  always.  This  represents  the  geometrical 
notion  of  a  root  so  far  as  geometry  can  illustrate  it. 
We  will  now  proceed  a  little  further. 
Suppose  we  take  a  square  and  draw  a  diagonal 

Fio.  14.  across  it,  what  is  the  length  of  that  diagonal  1  It 
is  evidently  greater  than  a  side,  and  not  so  great  as  two  sides. 


XXXI.] 


GEOMETRICAL  ILLUSTRATIONS. 


267 


Fig.  15. 


I 


If  we  measure  it  carefully  we  shall  find  it  rather  less  than 
a  side  and  a  half.  It  will  be  about  one  and  two-fifths  or  1*4 
times  a  side. 

Now  construct  a  square  on  the  diagonal,  i.e.  a  fresh  square 
with  the  old  diagonal  for  one  of  its  sides.  We  may  not  know 
how  to  do  it  accurately  on  blank  paper,  but  it  is  quite  easy 
to  do  if  we  use  paper  ruled  faintly  in  squares,  such  as  can 
easily  be  obtained  in  copy-books.  Or  the 
figure  may  be  constructed  by  folding  over 
the  tongue  of  a  sort  of  square  envelope. 
In  any  case  it  is  quite  easy  to  see  that 
the  square  on  the  hypothenuse  is  twice 
the  area  of  the  square  on  either  side  of 
the  isosceles  rt.-angled  triangle.  For 
produce  the  sides  along  the  dotted  lines. 
The  larger  square  is  thereby  cut  up  into  four  parts  each  of 
which  is  half  of  the  smaller  square :  see  fig.  15.  Therefore 
the  areas  of  the  squares  are  as  2  to  1. 

But  the  side  of  a  square  is  the  square  root  of  its  area,  hence 
a  side  of  the  new  square  is  ^2  times  a  side  of 
the  old  one.  In  other  words,  the  diagonal  of  a 
square  is  ^2  times  the  length 
of  one  of  the  sides. 

Or,  expressing  it  otherwise, 
the  hypothenuse  of  an  isosceles  right-angled 
triangle  is  J'2  times  either  of  the  sides. 

If  we  were  to  draw  a  square  on  one  of 
the  sides  and  a  square  on  the  hypothenuse, 
the  two  squares  would  be  as  2  to  1. 

(The  area  of  the  triangle  itself  is  evi- 
dently i  on  the  same  scale.) 

Drawn  thus,  we  might  not  see  how  to 
prove  it,  but  drawn  as  in  the  previous  figure  the  proof  is 


268  EASY  MATHEMATICS.  [qbA^. 

obvious.  To  be  sure  that  there  is  no  mistiness  about  it,  a 
beginner  should  write  the  proof  out  for  himself,  expressing  it 
as  well  as  he  possibly  can.  The  inventing  and  writing  out  of 
proofs  is  good  exercise,  and  to  do  it  really  well  demands  some 
thought  and  a  little  skill.  The  skill  so  cultivated  is  of  a 
useful  kind  in  life. 

An  example  is  necessary ;  but  the  danger  of  an  example 
is  that  it  is  apt  to  become  stereotyped.  It  may  "be  varied 
in  innumerable  ways,  and  a  way  invented  by  the  pupil  is 
better  than  one  which  he  has  to  learn.  If  there  are  actual 
errors  in  his  proof  they  can  be  pointed  out,  but  defect  of 
taste  and  style,  though  much  to  be  deprecated  in  adult 
persons,  must  be  eliminated  gradually  from  a  beginner.  He 
cannot  be  expected  to  concoct  a  proof  in  finished  style  from 
the  first. 

Something  like  the  following  would  be  good  enough : — 
To  prove  that  the  square  on  the  hypothenuse  of  an  isosceles 
right  angled  triangle  is  double  of  the  square  on  either  of  the 
sides. 

Construction. — Draw  the  triangle  ABC  with  right-angle  at  C\ 
so  that  AC^  BC  are  the  equal  sides,  and  AB  the  hypothenuse. 
P  g  Now  draw  a  square  on  AC,  and  draw 

it  so  that  the  equal  sides  of  the  triangle 
shall  serve  as  two  of  the  equal  sides  of 
the  square.  That  is  draw  the  square 
ACBR 

Next  draw  a  square  on  AB,  and  draw 
it  so  that  C  lies  in  the  middle  of  it, 
which  is  best  done  by  producing  AC  bh 
equal  length  to  E,  and  producing  BC 
an  equal  length  to  F,  and  then  joining  up  so  as  to  make  the 
square  ABEF,  which,  being  a  quadrilateral  figure  with  equal 
diagonals  at  right  angles  to  each  other,  must  be  a  square. 


f ""'7 

^ 

\ 

\ 

1/ 

^_ 

\\^ 

C           /E 

^x    j  ^' 

F 

Fig 

18. 

XXXI.]  GEOMETRICAL  ILLUSTRATIONS.  269 

Proof. — The  square  so  constructed  contains  the  area  of  the 
original  triangle  four  times,  while  the  former  square  contains 
it  only  twice.  Therefore  the  square  on  the  hypothenuse  is 
double  the  square  on  one  of  the  equal  sides  of  an  isosceles 
right-angled  triangle,     q.e.d. 

Beginners  can  and  should  realise  the  fact,  immediately  and 
without  words,  by  having  given  to  them  small  triangles  in 
wood,  and  by  then  piecing  them  together  so  as  to  make  the 
above  figure.  In  a  short  time,  left  to  themselves,  the  realisa- 
tion becomes  vivid. 

Now  proceed  to  a  right-angled  triangle  with  unequal  sides. 
Suppose  as  a  special  case  the  hypothenuse  is  double  one  of 
the  sides.  It  is  not  difficult  to  devise  a  way 
of  drawing  this  figure  if  we  use  a  pair  of 
compasses. 

For  let  AB  be  one  of  the  sides.  Double 
it  and  you  get  AC.  With  centre  A  and 
radius  AC  mark  off  a  circle.  This  gives  the 
length  of  the  hypothenuse. 

At  B  draw  a  line  perpendicular  to  AB 
till  it  meets  the  circle  at  D;  then  join  A  and  D.    The  triangle 
ABD  is  the  triangle  required,  viz.  a  right-angled  triangle  with 
its  hypothenuse  double  one  of  the  sides. 

If  we  were  to  draw  a  square  on  AD  and  another  on  AB^ 
the  area  of  the  one  square  would  be  quadruple  that  of  the 
other;  because  the  sides  are  as  2  to  1,  therefore  the  squares  of 
the  sides  will  be  as  4  to  1. 

What  about  a  square  drawn  on  BD%  If  drawn  and 
measured  it  will  be  found  to  be  about  f  of  the  big  square. 
It  can  be  shown  by  geometry  that  its  area  is  exactly  three 
quarters  of  the  big  one.  In  other  words,  that  the  middle 
sized  square  and  the  small  one  added  together  exactly  equal 
the  big  square  in  area.     This  is  a  most  curious  and  important 


870  EASY   MATHEMATICS.  [chap. 

fact :  about  as  important  as  anything  we  have  come  across,  if 
it  applies,  as  it  does,  to  all  plane  right-angled  triangles  without 
exception.  But  they  must  be  plane  triangles ;  i.e.  the  sides 
must  be  straight. 

We  are  not  yet  supposed  to  know  how  to  prove  it.  We 
can  verify  it  approximately  by  drawing  the  squares  carefully 
and  cutting  them  out  in  wood  or  cardboard  or  sheet  lead, 
and  then  weighing  them.  The  two  smaller  squares  will  be 
found  just  to  balance  the  big  one,  if  they  are  cut  out  neatly 
and  if  the  sheet  was  uniform  in  thickness  and  material.  This 
is  not  a  proof  that  they  are  mathematically  equal,  but  it  is  a 
verification  that  they  are  approximately  equal,  equal  "within 
the  limits  of  error  of  experiment." 

That  is  a  kind  of  equality  by  no  means  to  be  despised.  In 
some  difficult  cases  it  is  all  the  equality  that  can  be  ascertained. 
In  the  present  case  it  is  by  no  means  all ;  but  no  proof  of 
exact  equality  can  be  obtained  by  empirical  or  experimental 
processes,  no  matter  how  carefully  they  are  carrried  out. 
Exactness  is  a  prerogative  of  mathematical  reasoning,  that  is 
reasoning  on  pure  abstractions  from  which  all  flaws  and  imper- 
fections and  approximations  are  by  hypothesis  eliminated. 

The  fact  that  the  squares  on  the  sides  of  any  right-angled 
triangle  are  together  equal  to  the  square  on  the  hypothenuse, 
was  known  to  the  ancients.  It  was  called  the  theorem  of 
Pythagoras ;  and  a  classical  proof,  a  fine  example  of  ingenious 
reasoning,  is  given  as  the  47th  proposition  of  the  collection  of 
geometrical  propositions  made  by  the  Greek  Geometer  Euclid 
in  his  first  book.  Translations  of  that  ancient  treatise  are 
sometimes  still  learnt  by  schoolboys  in  this  country,  and  may 
be  considered  a  part  of  classical  education.  It  is  an  antiquated 
and  slow  way  of  learning  geometry  however,  and  in  fact  can 
hardly  be  intended  seriously  for  that  purpose.  Nevertheless 
it  is  a  delightful  literary  work  and  pleasant  for  reference. 


XXXI.] 


GEOMETRICAL  ILLUSTRATIONS. 


271 


People  who  are  not  acquainted  with  it  are  hardly  educated  in 
the  usual  English  sense. 

Many  proofs  can  be  given  of  any  proposition,  and  the  fact 
itself  is  of  more  importance  than  any  one  proof  of  it. 

That  does  not  for  a  moment  mean  that  proofs  can  be  dis- 
pensed with,  for  without  a  proof  we  should  not  really  know 
the  fact.  We  could  know  it  approximately  but  not  rigorously 
and  exactly;  and  it  should  be  always  a  joy  to  feel  that  a 
theorem  or  a  statement  can  be  made  without  limitation  or 
approximation.  Such  statements  are  the  only  ones  that  can 
be  pressed  into  extreme  cases,  with  perfect  confidence  that 
whenever  applicable,  that  is  whenever  the  postulated  data  are 
satisfied,  they  will  be  always  precisely  true. 

What  we  are  doing  at  present  however  does  not  necessarily 
demand  extreme  accuracy.  We  have  been  finding  roots,  which 
we  can  only  do  approximately,  and  we  now  want  to  illustrate 
them.  It  will  suffice  for  our  present  purpose  if  we  assume 
Pythagoras's  theorem  as  experimentally  verifiable  with  suffi 
cient  accuracy  for  our  present  purpose,  and  proceed  to  use  it. 

The  most  remarkable  of  all  right-angled  triangles  is  the 

one  whose  sides  are  all  commensurable,  namely  3,  4  and  5. 

The  square  of  5  is  equal  to  the  sum  of  the  squares  of  4  and  3. 

25  =  16  +  9.     See  fig.  20. 

Of  course  the  sides  might  equally 

well  be  6,  8,  and  10  ; 

also   9,  12,  and  15; 

12,  16,  and  20; 

and  so  on. 

Also  they  could  be    1-5,  2,  2*5 ; 
•75,  1,  1-25; 
and  so  on. 

So  long  as  the  proportion  holds,  the  absolute  length  of  the 
sides  is  only  a  matter  of  "scale." 


Kyv 

)C^4 

/     3 

Fio.  20. 


272 


EASY  MATHEMATICS. 


[OHAP. 


There  is  no  other  commensurably-sided  right-angled  triangle 
until  we  come  to  the  one  with  sides  5,  12,  13 ;  and  the  next 
one  has  sides  8,  15,  17.     [See  Appendix.] 

Triangles  with  commensurable  sides  can  be  outlined  by 
children  by  surrounding  them  with  square  blocks  or  slabs; 
and  it  is  especially  instructive  to  outline  right-angled  triangles 
in  this  way,  because  then  the  squares  on  the  three  sides  can, 
after  suggestion,  be  completed,  and  the  number  of  blocks  in 
each  counted :  When  it  will  be  perceived  that 

9  +  16  =  25,    144  +  25  =  169,     225  +  64  =  289; 
a    fact    which    ought    to    arouse    some    curiosity,    since    it 
represents    the    first    inkling   of    one    of    the    most    simple 
fundamental  and  universal  truths  in  existence. 

What  we  have  learnt  by  assuming  Pythagoras's  proposition, 
so  far,  enables  us  to  say  that  in  a  right-angled  triangle  with 
the  hypothenuse  double  the  base  the  vertical  side  is  ^3  times 
the  base.  For  the  squares  on  each  are  as  1:3:4;  therefore 
the  sides  are  as  ^1  :  ^^3  :  ^4,  that  is  as  1  :  ^3  :  2. 

If  the  hypothenuse  is  treble  the  base,  the  squares  will  be 
as  9  to  1,  and  so  the  square  on  the  vertical 
side  will  be  represented  by  8  on  the  same 
scale,  and  the  vertical  side  itself  will  be  ^8, 
which  equals  2J2. 

This  should  be  examined  and  verified. 
It  will  be  easy  for  a  beginner  to  devise  a 
verification  of  it.     For  instance,  thus  : 

Draw  any  vertical   AB.     Draw   half   a 

square  on  it,  as  shown,  ADB, 

Take  half  one  of  these  sides,  and  lay  it  off  horizontally,  BC. 

Then,  this  being  called  1,  BD  ov  AD  will  be  2,  and  AB 

will  be  2^2  or  ^8 ;  and  so  therefore  AC  should  equal  3  on 

the  same  scale,  because  8  +  1  =  3^.    See  if  ^C  does  measure  3. 


Fig.  21. 


XXXT.3  GEOMETRICAL  ILLUSTRATIONS.  273 

Further  geometrical  methods  of  finding  square  roots. 

Let  us  now  attend  a  little  more  carefully  to  the  important 
statement  that  the  square  root  of  the  area  of  a  square  is  the 
length  of  one  of  the  sides.  We  have  seen  that  it  is  true 
numerically,  now  see  if  it  is  true  and  sensible  physically.  The 
point  to  attend  to  is  that  the  square  of  a  length  is  an  area, 
and  the  square  root  of  an  area  is  a  length,  not  proportional  to 
a  length  or  numerically  represented  by  a  length,  but  actually 
and  physically  a  length. 

Ja'^  =  a. 
a  being  a  length,  a^  is  an  area,  and  Ja^  is  therefore  a  length 
again.     But  there  is  no  reason  why  the  area  need  be  square. 
Suppose  it  were  oblong,   and  given  as  axh;    if  J{ah)  =  a-, 
X  would  still  be  a  length.     What  length  1 

Answer.     The  length  whose  square  is  equal  to  the  product 
a6,  the  geometric  mean  of  the  two  lengths  a  and  h ;  for  if 
Jab  =  c, 
ah  =  c^, 

and  so  -  =  -   ot   a:c  =  c:b, 

c      b 

or  the  three  quantities  a,  c,  h  are  in  geometrical  progression, 

for  they  dififer  by  a  constant  factor,  viz. 

r  =  -  =  - 
a       c 

They  might  be  written  -,  c,  cr. 
r 

The  term  c  is  the  mean  of  the  other  two  terms  in  the  G.P., 
so  it  is  called  their  geometric  mean. 

Can  it  be  found  geometrically  1  It  can,  and  this  is  another 
most  interesting  proposition  known  to  the  ancients  and 
recorded  by  Euclid,  It  is  called  the  14th  proposition  of  his 
second  book.  Though  perhaps  not  easy  to  prove,  it  is 
pxtremely  easy  to  state.     We  will  state  it  now  and  prove  it 


274 


EASY  MATHEMATICS. 


[chap. 


Fig.  22. 


later.  The  statement  without  a  proof  is  a  poor  thing,  but  the 
statement  as  a  prelude  to  the  proof — a 
statement  which  shall  provide  a  niche 
for  the  proof  in  the  mind  of  a  beginner 
and  cause  him  to  welcome  it  when  it 
comes — is  an  excellent  thing. 

Construction   for   finding  the  geo- 
metric mean  of  two  lengths. 

Lay  off  the  lengths  end  to  end  as 
JB  +  BC. 

Draw  a  circle  on  the  combined  lengths  as  diameter,  and 
erect  a  perpendicular  at  the  junction-point  B  till  it  meets  the 
circle  in  D ;  then  BD  is  the  geometric  mean  of  JB  and  BC. 

The  figure  shall  be  repeated  below,  with  the  lengths  labelled, 
and  the  rectangle  ab  shown.    (Fig.  23). 
"Geometric mean"  is  an  arithmetical 
or  algebraical  sort  of  term.    What  will 
it  mean  geometrically^     It  will  mean 
that  the  square  on  BD  has  to  equal 
in  area  the  so-called   rectangle  AB. 
BOf  which  means  the  real  rectangle 
AB  .  BE.     That  is  to  say  the  square 
on  the  length  Jab  has  to  equal  the  area  ab. 
That  is  precisely  what  remains  to  be  proved. 
D        If  c2  =  ab 

then  c  is  the  geometric  mean  of  a  and  J, 

a      c 

or     -  =  V. 

c       b 

The  only  practical  difficulty  is  how  to  find 

the  length  c,  and  that  is  overcome  in  a  very 

simple  manner  by  the  circle  in  the  above 

construction. 

If  any  one  has  got  as  far  as  the  35th  proposition  of  Euclid's 


FiQ.  23. 


XXXI.] 


GEOMETKICAL  ILLUSTRATIONS. 


275 


Fio.  25. 


third  book,  they  can  devise  a  proof  of  this  curious  and  very 
important  property  of  the  circle  for  themselves ;  in  fact  the 
figure  annexed  suggests  it  at  once,  as  soon 
as  we  know  that  the  rectangles  contained 
by  the  segments  of  two  chords  are  equal. 

Given  this  simple  and  beautiful  con- 
struction, we  can  at  once  find  a  length 
numerically  representing  the  square  root 
of  any  given  number  n ;  for  we  can  take 
the  two  initially  given  lengths  as  n  and 
1  respectively,  so  that  their  product  is  n,  and  the  geometric 
mean  will  then  represent  the  square  root  of  w,  because  it 
will  be  equal  to  J(n  x  1).     (Fig.  26). 

For  instance  to  construct  ^4. 
Take  a  line  5  inches  long  as  the  dia- 
meter of  the  circle,  mark  off  4  inches 
and  draw  a  perpendicular  to  meet  the 
circle;  this  will  be  ^4,  and  if  measured 
will  be  found  to  equal  2  inches. 

To  find  *y5  geometrically. 

Draw  a  circle  of  radius  3  inches,  so  that  its  diameter  is 
6  inches.  At  the  first  inch  draw  a  perpendicular  and  measure 
its  length.  That  will  be  the  root  of  5. 
It  should  equal  2*236  inches  if  carefully 
drawn  and  measured. 

For  the  root  of  7  the  same  construc- 
tion exactly  is  to  be  carried  out,  only 
the  circle  will  be  4  inches  in  radius. 

For  the  root  of  2  the  circle  will  be 
1*5  inches  radius,  or  3  inches  diameter. 

And  for  the  root  of  any  number  whatever,  ??,  the  radius  of 


Fig.  26. 


Fio.  27. 


the  circle  will  be 


n+l 


276 


EASY  MATHEMATICS. 


[OHAP.  XXXI.] 


For  roots  of  large  numbers,  this  method  will  not  be  con- 
venient, but  for  roots  of  fractions  not  too  far  removed  from 
unity  it  serves  well. 

For  instance,  to  find  the  root  of  3-6.  Take  a  circle  of 
2*3  inches  radius,  and  make  the  construction,  erecting  a 
perpendicular  to  the  diameter  at  the  end  of  the  first  inch. 
Its  length  gives  the  root,  and  should  equal  1  -9  inches. 


Fia.  29. 


To  find  the  root  of  -75. 

Take  a  circle  J  x  1-75  =  -875  inch  radius,  and  at  the  first 
inch  of  it  erect  the  perpendicular. 

Its  length  will  be  greater  than  "75,  as  necessary  for  the 
root  of  a  proper  fraction,  and  it  should  equal  -866. 

This  particular  result  could  however  have  been  still  more 
easily  calculated,  or  at  least  expressed  in  terms  of  ^^3;  for 
•75  =  }. 


so 


7-75  =  ^=  J73  =  1x1-732...  =  -866.... 


It  must  be  understood  then  that  a  geometrical  construction 
in  these  cases,  though  it  may  be  regarded  as  a  simple  method 
of  arriving  at  the  result,  is  more  particularly  an  illustration  of 
a  result  otherwise  arrived  at.  This  is  however  not  always 
the  case,  and  sometimes  by  construction  results  can  be  found 
which  it  would  be  extremely  difficult  to  get  in  any  other  way. 
Engineers  and  building  constructors  know  this  well :  and 
graphical  methods  are  in  constant  practical  use. 


CHAPTER  XXXII. 

Arithmetical  method  of  finding  Square  Roots. 

We  now  know  three  methods  of  finding  a  square  root. 

1.  The  factor  method,  when  it  is  applicable,  which  it  seldom 
is ;  whenever  it  is  easily  applicable  it  should  be  used.  Often 
it  becomes  a  matter  of  guessing  and  trial  and  error,  with 
the  error  gradually  corrected  or  diminished. 

2.  The  logarithm  method,  which  is  the  real  practical  plan, 
and  is  frequently  done  with  a  slide  rule. 

3.  The  graphical  method. 

4.  There  is  however  another,  an  arithmetical  method,  which 
is  usually  learnt,  though  seldom  really  employed.  It  is  an 
ingenious  plan  and  is  not  at  all  bad  for  finding  square 
roots.  For  cube  roots  it  gets  complicated,  and  for  higher 
roots  like  fifth  and  seventh  it  would  be  altogether  too 
difficult  for  anyone  but  a  mathematician,  and  he  would 
never  think  of  employing  it. 

To  find  a  really  high  root,  for  instance  a  9th  root,  the 
logarithm  method  is  the  only  reasonable  one;  though  we 
might  take  the  cube  root  twice  over.  A  sixth  root  is  the 
square  root  of  a  cube  root.  An  eighth  root  is  the  result  of  a 
square  root  operation  three  times  repeated.  An  eleventh  root 
I  could  only  do  by  logarithms,  and  with  them  it  is  so  easy 


278  EASY  MATHEMATICS.  [chap. 

that  nothing  better  is  needed.     Let  us  see,  for  instance,  what 
is  the  eleventh  root  of  2, 

^^••'  =  -0273664  ...  =  log  of  1-06503  ... , 

which  is  therefore  the  root  required. 

[If  any  part  of  such  an  answer  as  the  above  pretended  to 
"  circulate,"  we  should  know  that  the  recurrence  was  spurious, 
and  only  due  to  the  fact  that  not  enough  digits  in  log  2  had 
been  taken  into  account.  Roughly  speaking  we  may  say 
that  all  numbers  are  incommensurable,  except  those  specially 
selected  to  be  otherwise.] 

Why  then  learn  any  arithmetical  method  for  finding  square 
roots,  other  than  the  logarithm  method  1 

Answer.  Because  we  might  not  have  a  table  of  logarithms 
handy,  and  because  it  is  ignominious  to  be  dependent  on 
material  tools  except  in  operations  which  are  complicated. 

To  find  a  cube  root  by  direct  process  is  rather  complicated, 
and  I  do  not  recommend  its  being  learnt  except  by  enthusiasts : 
and  they  will  forget  it  again.  But  the  rule  for  square  root 
is  fairly  easy  and  often  useful.  It  will  however  be  the 
hardest  thing  we  have  attempted  yet,  and  the  proof  will  be 
deferred  to  the  next  chapter.  It  is  not  usually  considered 
hard,  but  all  the  things  before  this  have  been  easier  in 
reality,  though  people  often  shy  at  them.  I  hope  they  will 
do  so  no  longer. 

To  find  the  square  root  of  256  by  direct  arithmetic.  Set 
it  down  like  a  long  division  sum,  but  with  the  digits  marked 
out  in  pairs,  by  dots  or  commas  or  other  marks,  as  shown, 
beginning  with  the  units  place,  then  work  as  follows : 

1)256(16 

1 

26  )  156 
166 


xxxu.]  RULE  FOR  SQUARE  ROOT.  2*79 

First  guess  the  square  root  of  2,  or  the  integer  smaller. 

It  is  1,  so  put  it  in  two  places,  and  multiply  and  subtract  as 
in  long  division.  Then  double  the  1,  and  place  it  on  the  left 
as  2,  and  see  how  many  times  it  will  go  into  something  less 
than  15 ;  guess  6. 

Set  down  6  in  two  places  as  shown,  multiply  and  subtract, 
and  there  is  no  remainder.  The  sum  comes  to  an  end :  the 
root  is  16. 

If  we  had  guessed  7  instead  of  6,  as  might  seem  natural, 
the  product  treated  as  above  would  have  been  189,  and  been 
too  big. 

If  we  had  been  given  the  number  2560,  it  would  have  been 
dotted  off  in  pairs  as  follows : 

2566, 
and  the  result  would  have  been  quite  different.     We  should 
now  have  to  guess  the  root,  not  of  2,  but  of  25,  which  is  very 
easily    done.      The    process    would   then   have   looked    like 
this : 

5  )  2S66  (  50-6 
25 
1006  )      6000 
6036 
-36 
so  that  50-6  is  approximately  the  square  root  of  2560. 

The  small  remainder  shows  that  the  result  is  not  quite 
accurate,  and  its  negative  value  shows  that  the  result  is 
slightly  in  excess. 

(Observe  that  ciphers,  like  the  other  figures,  are  always 
brought  down  in  pairs.  If  it  were  a  cube  root  we  were  finding 
they  would  be  brought  down  in  triplets.) 

It  is  natural  to  put  6  in  the  second  stage,  after  the  0,  as  we 
have  done  above,  because  it  is  very  nearly  right.  It  is  a  little 
too  big  however,  and  if  we  wanted  to  work  the  root   out 


%80  EASY   MATHEMATICS.  [chap. 

further,  we  should  put  5  and  be  sure  that  the  next  figure 
would  be  9. 

A  more  exact  result  is  50-5964426.... 

To  find  the  square  root  of  6241. 

Set  it  down,  and  again  partition  off  the  figures  in  pairs,  be- 
ginning with  the  units  place,  by  dots  or  other  marks,  as  shown 

7  )  6241  (  79 
49 
149  )  1341 
1341 

Guess  the  root  of  62  or  the  next  lower  integer;  guess  7. 
Set  it  down  in  2  places,  multiply,  and  subtract.  Double  7, 
and  see  how  many  times  it  will  go  in  134 ;  guess  9  times. 

Set  down  9  in  2  places,  multiply  as  shown,  and  subtract. 

There  is  no  remainder :  the  root  required  is  79  exactly. 

We  might  have  guessed  this.  Looking  at  the  number  we 
see  that  the  root  will  be  less  than  80,  for  80^  =  6400.  But 
it  will  not  be  much  less  than  80,  because  a  moderate 
difference  in  a  square  is  but  a  small  difference  in  the  root. 
So  we  might  try  78.  Multiplying  out,  we  should  find 
78  X  78  =  6084,  which  is  about  as  much  too  small  as  the  other 
was  too  big.  Hence  we  know  that  it  is  either  79  or  something 
very  near  to  79. 

Take  another  instance  of  guessing  :  choosing  a  number  quite 
at  random,  say  ^(596).     We  know  that 

242  =  4x122  =  4x144  =  576, 
while  252  =  625.     So  here  again  the  number  lies  about  half 
way  between  24  and  25,  but  a  little  nearer  the  smaller  of  the 
two ;  and  we  might  see  how  24*4  would  answer. 

Multiplying  24-4  x  24-4  we  should  get  595*36,  which  is  very 
close.     As  a  matter  of  fact, 

V(596)  =  24-4131112... 


XXXII.] 


RULE  FOR   SQUARE  ROOT. 


281 


which  you  can  proceed  to  ascertain  by  the  arithmetical  process 
worked  out  at  length,  as  thus  : 


44 


596  I  24-41311123 
4 

196 
176 


484 

2000 
1936 

0 

1 

900 
469 

48^ 
4885 

!3 

640 

488 

151 
146 

48826 
48826 

)1  5 
4 

43100 

88261 

121 

5483900 
4882621 

48826221 
488262222 

60127900 
48826221 

1130167900 
976524444 

153643456 
We  see  that  the  next  digit  will  be  3,  and  have  placed  it 
in  position,  but  we  consider  that  as  we  have  now  obtained 
ten  significant  figures,  we  have  gone  far  enough,  especially 
as  we  know  that  there  can  be  no  end. 

If  we  have  to  find  the  square  root  of  a  decimal,  we  can  mark 
it  oif  into  pairs,  as  before,  always  beginning  with  the  units 
place.  Thus  17*8534  is  marked  off  properly  for  the  purpose 
of  extracting  its  square  root,  which  is  plainly  4  decimal 
something. 

So  also  6-060576  is  properly  marked  off,  and  its  root  is  '024. 

The  marking  off  in  pairs  is  manifestly  connected  with  the 
fact  that  7100  =  10.     It  is  to  get  the  power  of  ten  in  the 


282  EASY  MATHEMATICS.  [chap,  xxxit. 

answer  right.  The  number  of  dots  gives  the  number  of  figures 
in  the  answer,  if  the  units  place  is  included  in  it.  To  find  a 
cube  root,  the  dots  would  be  placed  on  every  third  digit, 
but  always  beginning  with  the  units  place,  because  any  root 
of  1  is  1. 

There  is  not  much  more  than  this  to  be  learnt  about  this 
ingenious  and  practical  process,  until  we  are  able  to  prove  it 
and  see  the  reason  of  the  successive  steps :  this  will  be  fully 
attended  to  in  the  next  chapter,  pages  296  to  299.  There  are 
however  a  great  number  of  far  more  important  things,  and  I 
only  place  this  brief  record  of  the  process  here,  because 
I  by  no  means  wish  to  extrude  it;  moreover  it  is  an  in- 
teresting thing  to  prove.  It  is  essentially  a  limited  process, 
however,  since,  for  any  useful  purpose,  it  only  applies  to 
square  roots ;  though  a  complication  of  it,  on  the  same  principle, 
will  apply  to  cube,  and  even  to  higher,  roots.  At  the  same 
time  it  is  undeniable  that  square  roots  and  cube  roots  occur 
much  more  frequently  than  do  others,  just  as  second  and 
third  powers  do ;  partly  because  they  cover  the  actual 
dimensions  of  our  space. 


CHAPTER   XXXIIT. 

Simple  Algebraic  Aids  to  Arithmetic,  etc. 

A  VERY  little  knowledge  of  algebra  enables  us  to  make 
better  estimates,  and  to  approximate  as  closely  as  we  please, 
l)oth  to  powers  and  to  roots ;  and  it  is  worth  while  to  show 
this  now :  this  chapter  being  chiefly  one  for  exercise  and 
practice.  It  may  be  regarded  as  a  chapter  of  miscellaneous 
worked  out  examples,  rather  than  as  a  progressive  chapter; 
though  it  contains  the  proof  or  explanation  of  the  ordinary 
square  root  rule. 

First  of  all  consider  the  multiplication  of  two  binomials,  that 
is  two  factors  each  consisting  of  two  terms,  say  {a  +  b){c-\-d). 
Every  term  will  have  to  be  multiplied  by  every  other,  for  it 
means  a(c-{-d)-\-b(c  +  d),  that  is  ac  +  ad-hbc  +  bd. 

So  for  instance  (3  +  ^2)  (4  +  ^3) 

will  equal  (3  x  4)  +  3  ^3  +  4  ^2  +  ^2  ^3 

=  12  +  3^3  +  4^2+^6. 
Or  take  this  example, 

(J2  +  2){J3-S) 
multiplied  out  it  becomes  ^6  +  2^3-3^2-6. 
But  take  a  more  easily  verifiable  example,  say 
(17-5)(13-10) 
=  221-170-65  +  50 
=  271-235  =  36; 
rather  an  absurd  way  to  do  such  simple  arithmetic  as  12x3. 


284  fiASY  MATHEMATICS.  [chap. 

Very  well,  now  take  the  case  of  {a  +  hy. 
It  means  (a  +  h){a-\- h). 

And  this  multiplied  out  equals  a^  +  ab  +  ba  +  b^ ; 
but  ab  +  ba  =  2ab, 

so  (a+b)2  =  a2+2ab+b2. 

Similarly  (a-b)2  =  a2-2ab+b'». 

Now  let  us  use  these  results  to  obtain  powers  and  to  ap- 
proximate to  roots.     Suppose  we  want  lOS'^,  work  it  out  thus: 
(103)2  =  (100  +  3)2 

=  1002  +  6x100  +  32 
=  10,000  +  600  +  9 
=  10609. 
Again,  to  find  (998)2 ;  write  it  as  (1000  -  2)2 

=  (1000)2-4000  +  4 
=  one  million  less  3996 
=  996004. 
Similarly:  (125)2  =  (120  +  5)2  =  14400  +  1200  +  25 

=  15625 
or  (125)2  =  (130-5)2  =  16900-1300  +  25 
=  15625. 
(79-2)2  =  (80 --8)2   =  6400 -1-6x80 +  -64 
=  6400-64-128 
=  6272-64. 
(5-11)2  =  (5 +  -11)2   =  25  +  1-1  +  -0121 
=  26-1121. 
(39)2  =  (40-1)2    =  1600-80  +  1 
=  1521. 

A  further  algebraical  aid  is  often  of  great  use,  especially  in 
preparing  for  logarithmic  calculations. 

The  value  of  {a  +  b)(a  -  b)  when  multiplied  out  is 
a^-ab  +  ba-b'^: 


XXXIII.] 


ALGEBRAIC  AIDS. 


285 


the  two  middle  terms  destroy  each  other,  and  so  only  «2  _  52 
is  left. 

This  is  a  most  useful  fact  to  remember, 

a2-ba  =  (a+b)(a-b). 
For  instance     92  -  42  =  (9  +  4)  (9  -  4)  =  13  x  5  =  65, 
(17-31)2 -(2-69)2  =  20x14-62  =  292-4, 
(•019)2  _  (.^08)2  =  -027  X  -Oil  =  2-97  x  10"*, 
(1-05)2 -(-95)2  =,  2x-l  =  -2. 
The  fact  is  so  important  that  it  is  worth  learning  in  words. 
The  difference  of  two  squares  is  equal  to  the  product  of 
sum  and  difference. 

Expressed  thus  it  suggests  a  geometrical  way  of  putting  it : 


K 


C! 


M 


G^. 


Fia.  30. 


Let  AB  and  ^C  be  any  two  given 
lengths. 

Erect  a  square  on  each,  viz.  the  square 
AD  and  the  square  AE,  drawing  them 
so  that  they  are  superposed. 

The  difference  of  the  two  squares  is 
shown  by  the  irregular  six-sided  rect- 
angular figure  with  what  is  called  a 
"  re-entrant "  angle  at  E. 

We  have  to  show  that  this  area  is 
equal  to  that  of  a  rectangle  bounded  by 
lengths  representing  the  sum  and  the  difference  respectively 
of  the  two  given  lengths. 

To  construct  such  a  rectangle  in  a  convenient  position, 
produce  CE  both  ways  to  F  and  G,  making  CG  =  CA.  Then 
FG  is  equal  to  the  sum  of  the  two  given  lengths,  viz.  AB  +  AC; 
and  GH,  which  is  the  same  as  CB,  is  equal  to  their  difference. 

Therefore  the  area  of  the  rectangle  GHDF  exhibits  the 
product  of  the  sum  and  difference.  Hence  we  have  to  show 
that  this  rectangle  is  equal  to  the  area  of  the  irregular  figure 
CELKDBC,  the  difference  of  the  two  squares. 


286  EASY  MATHEMATICS.  [chaf. 

Now  the  two  areas  have  a  great  part  common,  viz.  the 
rectangle  BF;  so  we  have  only  to  show  that  the  residues  LF 
and  GB  are  equal. 

By  producing  LE  to  M,  another  rectangle  EB  is  constructed 
equal  to  GB ;  and  this  rectangle  is  plainly  equal  to  LF,  because 
the  height  and  base  of  the  one  correspond  to  the  base  and 
height  of  the  other. 

The  proof  is  therefore  completely  indicated.  It  has  been 
rather  long  and  not  particularly  neat,  but  it  is  such  a  proof  as 
could  be  invented  by  an  industrious  beginner  for  himself. 
The  proposition  is  really  an  ancient  one,  and  is  established 
with  due  ceremony  in  Euclid  Book  II.,  Propositions  5  and  6. 

We  observe  from  this  example  that  a  geometrical  proof  is 
or  may  be  hard,  while  an  algebraic  proof  of  the  same  thing  is 
absurdly  easy :  so  it  often  is,  though  not  always.  As  usual 
there  are  plenty  of  ways  of  proving  a  proposition ;  the  pro- 
position itself  is  more  important  than  any  one  proof  of  it. 

The  geometrical  illustration  has  been  introduced  here  to  em- 
phasise the  extreme  importance  and  usefulness  of  the  fact  that 
{x  +  y)(x-y)  =  x^-y^. 

Now  let  us  proceed  to  show  how  it  is  employed  for  adapting 
things  to  logarithmic  calculation. 

Suppose  we  had  to  find  the  value  of  the  following : 
(8-131)2 -(4-026)2. 
We  might  look  out  the  logarithm  of  each,  double  it,  find  the 
antilog  of  each,  and  then  subtract  them. 

But  on  the  other  hand  we  might  first  throw  it  into  the  form 
12-157  X  4-105, 
look  out  the  logarithms  of  these  two  numbers,  add  them, 
and  find  the  antilog  of  the  sum.     And  this  is  a  shorter  process 
than  the  preceding. 

In  general,  sums  and  differences  are  awkward  for  logarithmic 
calculation,  while  products  and  quotients  are  convenient. 


xxiiii.]  ALGEBRAIC  AIDS.  287 

Take  another  example  of  finding  the  value  of  a  difference  of 
two  squares : 

(15)  ~  V35  j    ^  \T5'^  35  j  Vr5  ~  35; 

~    105  ^  105 

_40_ 
(105)2* 

And  it  is  easy  to  look  out  the  necessary  logarithms : 
log  40  =  1-6021  ;    log  105  =  2-0212. 
2  log  105  =  4-0424 
difference  3-5597  =  log  of  -003625. 
•003625  is  therefore  the  result. 

We  might  indeed  have  done  the  above  differently,  because 
we  happen  to  see  a  common  factor  in  the  given  expressions, 
and  can  take  it  outside  brackets,  thus, 

V15;      \35/    "  W  \V3/      \7/  J      25  •    21    •    21 
40  160 


~  25  X  (21)2     44100' 

log  16  =  1-2041 
log  4410  =  3-6444 

difference  3-5597  =  log  of  -003625  as  before. 

This  therefore  serves  as  a  check,  and  is  itself  instructive. 

Sums  of  this  kind,  given  as  exercises,  will  call  out  nascent 
ingenuity  and  will  furnish  much  better  and  more  real 
arithmetical  practice  than  a  quantity  of  routine  examples 
without  much  variety. 

In  so  far  as  the  actual  arithmetical  operations  to  be 
performed  are  usually  simple  and  short,  that  is  a  peculiarity 
pharacteristic  pf  nearly  all  the  real  sums  that  have   to   be 


288  EASY  MATHEMATICS.  [chap. 

done  in  practice;  always  excepting  the  long  and  intricate 
operations  occasionally  undertaken  for  special  purposes  by 
pure  mathematicians — a  matter  with  which  children  have 
nothing  whatever  to  do. 

Sometimes  the  converse  use  of  the  proposition 

^2-52  =  (a  +  b)(a-h) 

is  convenient.    For  instance,  suppose  we  had  to  find  the  value  of 

(N/3  +  y2)(V3-V2). 

It  would  be  very  clumsy  to  interpret  it  arithmetically  thus  : 

(1-732 +  l-4U2)(l-732- 1-4142) 

=  3-1462x0-3178, 

whose  logarithm  is     -4977 

plus  1-5022 

equals  1-9999 

which  is  the  log  of  something  extremely  near  to  unity,  and 
perhaps  unity  itself  if  we  had  taken  more  places  in  the 
logarithms. 

I  say  this  would  be  an  extremely  clumsy  way. 

The  neat  and  direct  way  is  to  write  the  product  as  the 
diflference  of  the  two  squares,  thus  : 

(73  +  J2)  U3  -  V2)  =  USy  -  (72)2  =  3  -  2  =  1, 
which  shows  that  it  is  unity  exactly. 

Take  other  examples  of  {a  -\-b){a-b)  =  o?  -l)^: 
(Vl4-V8)(V14  +  ^8)  =  14-8  =  6. 
(x/7-x/3)(77  +  V3)=    7-3  =  4. 
(n/5  +  1)(x/5-1)  =    5-1  =  4. 
(v/57-l)(V57  +  l)  =  57-1  =  56. 
(1+V17)(1-^17)  =  1-17  =-16. 
(V3T4T59  +  l)(x/3T4T59-l)  =  2-14159. 


cxiii.]  ALGEBRAIC  AIDS.  289 

{1  +  (-0012)^}  {1- (-0012)^"}  =  1--0012  =  -9988. 

This  last  might  have  been  done  thus  : 
3  +  1    3-1  ^  8 
JS    '  J'S    ~  3' 

(6 +^20)  (6 -^20)  =  36-20  =  16. 
(v/5-2)(v/5  +  2)  =  5-4  =  1. 

(m  +  Jn)(m-Jn)  =  rn^-n. 

2      a%^-n^ 


62     dP-' 


2      x^  -  a^ 


(M)(M) 

/ m      n\  /m      n\  _m'^    n^ 

(  X       a  \  f  X       a\  _x'^     a^  _ 
\sjci    Jx)  \Ja ~  Jx)  ~  a~~x  ~ 

(30  +  iV^)(30-i»  =  900-^. 

\x^    6;W'*'6;~ir4    3e 

^  576_-^o 
36a;4    * 

{ab-Jab)(ab  +  Jah)  =  a^^  -  ab. 

(^a-hi/b){^a-l'b)^J-b^. 

{a'^b){a'-b)  =  cv^-b\ 

(a"  +  ft-")((t"-a-«)  =  a^-a-^ 


L.E.M. 


290 


EASY  MATHEMATICS. 


[chap. 


(a*" +  «")(«"' -a")  =  a" 


{ajx  +  hjy){ajx-hjy)  =  a^x-ll^y. 

(l+a")(l-«")  =  l-d"^. 

\  _x-\ 

X  X     ' 

3\/.    ,        3\        _       9      16a;2 


(.v.+|)(v-j.)  - 


16.T 


a;  X 

_  (4a; +  3)  (42; -3) 


(1  +  sjlog  n)  (1  -  x/log  n)  =  1  -  log  71. 
{jT^  +  sfl3y)(JPf3i-sF73y)  =  l-73j/--73y  =  y. 
(J(l  +m) .  u -  Jm  .  u){J(l  +m) .  u  +  Jm .  u)  =  ul 
{JJ+l  +  76)  {Ja  +  h  -  Jh)  =  «. 

If  we  have  now  driven  home  the  important  fact  that 

{a  +  h){a-h)  =  a'--b^  (1) 

sufficiently,  we  will  proceed  to  illustrate  geometrically  those 
other  equally  important  truths,  viz.  that 

(fl  +  6)2  =  a^  +  2ab  +  b^, (2) 

{a-bf  =  a^-2ab  +  b\... (3) 

or,  expressed  in  words,  the  square  of  a  binomial  is  the  sum  of  the 
squares  of  its  terms  plus  twice  their  product. 
Or  expressed  geometrically.  (2)  The 
square  on  a  line  made  up  of  two  parts  is  the 
sum  of  the  squares  on  the  parts  plus  twice 
the  rectangle  contained  by  the  parts. 

The  annexed  figure  makes  this  obvious. 
For  the  base  of  the  big  square  is  made  up 
of  two  parts  labelled  a  and  b. 
And  we  see  that  it  is  built  up  of  the  square  on  a,  plus  a 


ab 

b' 

a= 

ba 

ci  b 

Fio.  31. 


D             a 

ab 

(a-bf 

b"" 

A      a-b     C 

XXXIII.]  ALGEBRAIC  AIDS.  291 

square  virtually  on  h,  plus  two  rectangles  each  equal  to  the 
product  ah. 

It  is   a   quite   ancient  proposition   known   as   the   fourth 
proposition  of  Euclid's  second  book. 

(3)  The  statement  for  the  squared  difference  {a  -  by  expres- 
sion may  be  worded  geometrically  thus  : 

If  a  straight  line  is  divided  into  two  parts,  the  sum  of 
the  squares  on  the  whole  line  and  one  of 
the  parts  is  equal  to  twice  the  rectangle 
contained  by  the  whole  line  and  that  part 
together  with  the  square  on  the  other 
part. 

The  same  figure  serves,  differently 
labelled ;  but  a  separate  figure  may  make 
it  clearer.  The  square  of  AB,  which  is  a'^, 
together  with  the  square  on  BC,  which  is  F   <^ 

b'^j  exceeds  the   square   on  AC,  which   is 
(a  -  by,  by  twice  the  rectangle  ab,  that  is  by  the  two  rectangles 
DE  and  EF. 

This  proposition,  which  asserts  that 

a^  +  b'^  =  (a-by  +  2ab, 
is  known  as  the  7th  proposition  of  the  second  book  of  Euclid. 
It  may  be  illustrated,  like  the  preceding,  by  the  folding  of  paper. 

The  process  of  putting  these  propositions  into  proved 
geometric  form  is,  we  see,  liable  to  be  rather  troublesome  and 
long.  Algebraically  they  are  quite  easy.  Geometry  illustrates 
the  algebra,  but  it  does  not  in  this  latter  instance  illustrate  it 
strikingly;  and  it  is  quite  possible  to  spend  too  much  time 
over  such  geometrical  illustrations,  unless  they  are  made  out 
by  pupils  for  themselves,  which  is  an  admirable  exercise.  A 
great  deal,  though  not  all,  of  Euclid's  second  book  is  of  this 
character,  and  represents  an  antique  method  of  expressing 
algebraic  results  without  employing  algebra.     For  good  reason 


292  EASY  MATHEMATICS.  [chap. 

in  those  days, — because  algebra  was  not  then  invented. 
Children  need  not  be  dosed  with  too  much  of  this  rather  con- 
fusing and  nearly  useless  kind  of  geometry  at  the  present  time. 

Illustrations. 

Let  us  write  down  some  illustrations  of  the  use  of  these 
results  in    simplifying   algebraic  expressions,  and  in  finding 
roots.     Write  the  results  compactly  thus, 
(a±by  =  a^±2ab  +  h\ 
and  then  illustrate  them  : 

{Ja  +  Jhf  =  a  +  2J{ab)  +  h 
{x^-y^Y  =  x  +  y-2jxi/. 
{6+xy  =  36  +  l2x  +  x^ 
(a;-l)2  =  a;2-2x+l. 


("-;) 


2  1 

=  a;24-2  + 

X 


2 

+  a;2_2. 


(5-^2)2  =  25-10^2  +  2  =  27-14-142... 

=  12-858.... 
(1-73)2  =  4-2^3  =  2(2 -V3)  =  '536.... 
Notice  that  although  ^3   is  greater  than   1   the  squared 
difference  cannot  help  being  positive. 

/  1  \2 


W3 

-TV^y 

—   -j-r^  -rt 

J     —      W    tJ, 

(.. 

=   5-2  +  i 

I  =  3-2. 

(121)2 

=  (120)2  + 

240  +  1  = 

14641. 

(119)2 

=  (120)2- 

240  +  1  = 

14161. 

(1-5)2 

=  (1+^)^ 

= 1+1+1 

-  =  2-25. 

(1-3)2 

=  i-6  +  -i  = 

=  1-7. 

XXXIII.]  M.GEBRAIC  AIDS.  293 

Problems. 

1.  If  any  diagram  has  all  its  linear  dimensions  increased 
by  one-sixth,  by  how  much  is  the  whole  area  of  the  figure 
increased  ? 

The  answer  liable  to  be  given  is  one  thirty-sixth,  but  it  is 
not  right.  The  right  answer  is  Jfths,  or  a  little  more  than 
one-third  of  the  original  area.  The  first  answer  attends  only 
to  the  little  corner  squares  and  neglects  the  two  strips,  for 

{a  +  hf-a^  =  &2  +  2a&; 
the  2ah  being  much  bigger  than  J^. 

The  simplest  solution  is  to  say  that  in  the  linear  dimensions 
throughout,  6  has  become  7,  hence,  in  the  area,  36  has  become 
49  ;  wherefore  the  superficial  increase  is  13  of  the  same  parts, 
that  is  13/36ths  of  the  original. 

2.  If  a  block  is  reduced  in  the  ratio  of  3  :  2  linear,  that  is  if 
its  length,  breadth,  and  thickness  are  all  made  two-thirds  of 
what  they  were,  the  shape  being  preserved,  what  change  has 
been  made  in  the  surface  or  superficial  area  and  in  the  volume 
or  cubical  contents  ? 

Answer.  The  linear  dimensions  being  reduced  by  one-third, 
or  from  3  to  2,  the  superficial  are  reduced  by  five-ninths,  or 
from  9  to  4 ;  and  the  cubical  are  reduced  by  nineteen  twenty- 
sevenths,  or  from  27  to  8.  In  other  words  the  surface  is  less 
than  half  what  it  was,  and  the  volume  is  less  than  a  third 
what  it  was. 

3.  If  every  linear  foot  becomes  13  inches,  every  square  foot 
becomes  169  square  inches,  and  every  cubic  foot  becomes  2197 
cubic  inches.  So,  while  the  linear  increase  is  y^^th  of  the 
original,  the  superficial  increase  is  y^^^ths,  or  a  little  more 
than  Jth  of  the  original  area ;  the  volume  increase  is  yYA*^^' 
or  distinctly  more  than  ^th  of  the  original  volume. 

4.  If  one  per  cent,  is  docked  ofi"  linear  dimensions,  about 


294  EASY  MATHEMATICS.  [chap. 

two  per  cent,  are  thereby  taken  from  area,  and  about  three 
per  cent,  from  bulk. 

Now  use  the  same  equation  to  find  soLuare  roots. 

Suppose  we  want  the  square  root  of  50.     We  see  instantly 
that  it  is  a  little  more  than  7,  let  us  call  it  7  +  x,  then  write 

50  =  {7  +  xf  =  4:9  +  Ux-hx\ 
or,  subtracting  49  from  both  sides  (i.e.  transferring  49  over  to 
the  left  with  change  of  sign), 

1  =  Ux  +  x\ 
wherefore  x  =  j^  is  a  first  approximation,  for  the  x^  is  a  very 
small  number,  almost  negligible,  x  is  really  a  trifle  less 
than  y'j,  though  not  so  much  less  as  ^^  would  be,  for  its  defect 
is  yj  of  x^,  which  is  approximately  ^wois  ^^^7-  ^^  i^  impossible 
to  express  the  root  accurately,  and  the  result  obtained  by 
neglecting  x^  is  usually  a  quite  sufficiently  close  approximation. 

So  the  root  is  l  +  ^t  =  7-0714.     The  error  is  in  the  last 
place ;  the  4  is  too  big,  it  ought  to  be  a  1. 

So  also  to  guess  the  root  of  143 
Write  it  12-«. 

143  =  (12-a;)2  =  144-24a;  +  a;2, 

neglect  x%  and  ^  =  aV  =  *^^  ^  7» 

so  approximately         JUS  =  12 --0417  =  11-9583. 
Its  real  value  is  11-9582607  .... 

What  is  the  square  root  of  99  ? 

99  =  (10-x)2  =  100-20x  +  a;2^ 

so  X  =    217  =   '^^ 

and  ^99  =  9-95.... 

What  is  the  root  of  395 1 

Let  it  be  20  -  x. 

395  =  400  -  40a;  +  a;2 
or  a:  =  /^  =  i  =  -125, 

so  7395  =  19  875.... 


xxxiii.]  ALGEBRAIC  AIDS.  295 

What  is  the  square  root  of  1,000,015'? 
that  is  of  10<5  +  15. 

Call  it  103 +  a;; 

then  10«  +  15  =  (W  +  x)^  =  10^  + 2000a;  +  x^ ; 

15 
whence  x  =  ^^^  =  -0075, 

and  so  the  required  root  is  1000*0075. 

In  such  a  case  the  extra  quantity  is  extremely  small,  and 
we  see  that  in  the  root  it  is  just  half  the  value  of  the  corre- 
sponding quantity  in  the  given  square. 

This  is  a  handy  approximation  which  may  be  generalised 
and  recollected.  It  is  an  immediate  consequence  of  neglecting 
a;2  and  writing  (1  +xY  =  l  +  2x  approximately  when  x  is  small. 

So  \/l  +  2x  =  1+x    approximately. 

For  instance  ^^(1*008)  will  equal  1-004  ; 
and  ^(100-084)  =  10^1*00084  =0. 10  x  1-00042 

=  10-0042  ; 
or  the  equation  may  be  written 

J{l+x)  =  1  +  |a;  approximately. 

The  following  relation, 

^(10"  +  a;)^10^"  +  10~^\|a:, 
when  X  is  moderately  small,  is  a  general  result;  but  for 
memory  it  is  best  to  make  the  first  term  unity ;  and  so  in  the 
numerical  example  just  above,  the  factor  100  was  first  taken 
outside  the  root,  where  of  course  its  value  is  10.  If  the  factor 
had  been  1000  instead  of  100,  that  is,  if  there  had  been  an 
odd  number  of  ciphers  in  it,  this  could  not  have  been  done  so 
easily  :  we  should  then  have  had  a  ^10  to  deal  with,  and  that 
would  destroy  the  advantage  of  the  process. 

The  process  applies  most  obviously  to  numbers  which  can  be 
separated  into  two  very  unequal  portions,  one  of  which  has  a 
known  square  root.     If  they  are  not  very  unequal,  the  neglect 


296  EASY  MATHEMATICS.  [chap. 

of  x^  becomes  of  more  consequence,  and  the  same  sort  of 
process  must  be  continued  further,  before  the  square  of  an 
outstanding  error  is  neglected. 

Suppose  for  instance  we  wanted  the  square  root  of  72 ;  we 
could  write  it  as  (8  +  «)2  or  we  could  write  it  as  (9  -  yf^ 
so  that  72  =  64  +  16a:4-a;2, 

or  72  =  81 -18«/  +  ?/2, 

whence    approximately     x  =  ^^  =  i,    and    we     should     be 
neglecting  \\ 
or  2/  =  tJ^  ^^^  w®  should  be  neglecting  a  trifle  less. 

So  the  answer  would  be  roughly  8*5,*  but  this  would  be  a 
little  too  big,  and  the  process  must  be  continued,  by  successive 
approximations,  beyond  8*4,  in  such  a  case;  the  process 
develops,  in  fact,  into  the  ordinary  arithmetical  method  of 
finding  a  square  root,  as  described  but  not  explained  in  the 
last  chapter.  We  can  now  explain  it,  for  it  all  depends  on 
what  we  have  just  been  doing ;  it  involves  an  ultimate  ignoring 
of  an  Qi?,  but  it  carries  the  process  of  surmising  the  root  to  any 
desired  degree  of  approximation,  before  the  inevitable  out- 
standing error  is  considered  so  minute  that  its  square  may 
safely  be  neglected. 

To  illustrate  the  process  arithmetically,  and  at  the  same 
time  display  its  rationale  algebraically,  take  any  simple 
number  at  random,  say  for  instance  33,  call  it  iV,  and  proceed 
to  approximate  to  its  square  root. 

(1)  First  guess  the  nearest  lower  integer  root,  namely  5, 
call  it  a  in  general,  and  write  x  for  the  unknown  necessary 
complement  to  be  found,  so  that 

*In  the  particular  example  chosen  it  happens  to  be  very  easy  to 
calculate  the  square  root,  because  the  'factor  method'  would  apply. 
Beginners  may  be  reminded  always  to  keep  an  eye  open  for  the  simple 
and  satisfactory  factor-method,  such  as  this : 

72  =  9  X  8, 
80  ^72  =  3x^8  =  3x2^2  =  6^/2=8-48528.... 


XXXIII.]  PROOF  OF  SQUARE  ROOT  RULE.  297 

i\^  =  {a  +  xy  =  a^  +  2ax  +  x'^, 
or  33  =  (5  +  x)2  =  25  +  lOx  +  xK 

From  this  we  deduce  that  the  deficiency  N-a^  =  x{1a-\rx), 

or  that  8  =  x{\0^-x). 

This  gives  us  our  first  approximation  to  the  required  com- 
plement, or  error  in  our  rough  estimate  of  the  root,  namely 

X  =  -TK ,  or  say  "7  as  the  first  di^it  of  it. 

(2)  Thus  we  can  now  make  a  closer  guess  at  the  root, 
namely  5*7,  and  start  afresh  for  a  second  approximation,  x\ 
writing  33  =  (5*7  +  xj  =  32-49  +  11  -^x'  +  x'% 

so  the  second  deficiency  is  '51  =  x'{\\'4:  +  x'), 
which  gives,  as  the  second  outstanding  error, 

*51 

x'  =  ■— — ,  =  -04  as  the  first  diijjit  of  that. 

ll-4  +  «  ^ 

(3)  Our  approximation  to  the  root  has  now  become  5*74, 
and  we  start  ofi"  a  third  time  to  write 

33  =  (5-74  +  a;72  =  32-9476  + ll-48a;"  +  a;"^ 
whence  the  third  deficiency  -0524  =  ic"(ll-48  +  ic"),  which  gives 
us  x"  =  -004  as  the  next  digit   of   the   rapidly   diminishing 
outstanding  error. 

(4)  The  approximation  is  now  getting  closer,  being  5-744, 
and  so  we  start  again,  saying 

33  =  (5-744 +  a;"')2, 
whence  the  fourth  deficiency  comes  out 

•006464  =  a;'"(ll-488  +  a;'"), 
yielding  x'"  =  -0005  ...  as  the  error  still  remaining. 

(5)  We  have  now  arrived  at  ^33  =  5*7445  . . . ,  and  we  can 
continue  the  process  as  long  as  we  like  ;  but,  at  this  (or  at  any 
other)  stage,  we  can  take  refuge  in  simple  division,  to  get  at 
once  a  still  closer  approximation.  For  hitherto  we  have  not 
neglected  the  square  of  any  small  quantity  :  everything  so  far 


298  EASY  MATHEMATICS.  [chap. 

has  been  exact ;  but  sooner  or  later  exactness  will  have  to  be 
abandoned,  because  we  know  that  a  number  really  has  no 
exact  numerical  root.  It  was  considered  too  inaccurate  to 
neglect  the  gquare  of  x,  but  we  might  perhaps  have  neglected 
the  square  of  x\  or  at  least  of  x'\  We  did  not  neglect  even 
this  however,  but  we  are  now  going  to  neglect  the  square  of 
x'" ;  so  after  reckoning  the  present  deficiency,  -00071975, 
instead  of  saying  ^^^^  _     -00071975 

^      ~  11-4890  +  x'"" 
which  would  be  continuing  the  process,  we  will  say  simply 

""  _  '00071975 
""     "     11-4890"' 
very  nearly,  and  divide  straight  out,  getting  -00006265  as  the 
result.       Wherefore   finally  the  approximation  at  which  we 
have  arrived  is         ^/33  =  5-74456265  .... 

If  the  process  thus  elaborated  be  compared  with  the 
operation  as  ordinarily  performed,  a  little  thought  will  make 
everything  clear  without  more  words. 

The  only  thing  that  can  require  explanation  is  the  actual 
mode  of  reckoning  the  successive  outstanding  deficiencies,  viz. : 
N-a^',  N~{a  +  xf;  N -  {a ■\- x ^^ x'f ;  a,ndiN-{a  +  x  +  x' -{■x"f. 
The  original  number  N  is  not  in  practice  thus  manifestly 
reverted  to  for  the  purpose  of  getting  these  values — which  in 
the  above  numerical  example  are  successively 

8;  -51;  -0524;  and  -006464,— 
but  exactly  the  same  result  is  obtained  by  the  successive 
subtractions  as  ordinarily  performed :  the  value  of  an  ex- 
pression like  (N -  a^)  -  x{2a -\- x)  being  practically  employed, 
each  time,  instead  of  the  equivalent  N  -{a  +  xY,  because 
(having  already  found  N-  a^)  it  is  quicker  to  reckon. 

The  well  known  ordinary  process  is  here  exhibited  for  the 
same  number,  in  order  that  it  may  be  compared  with   the 


XXXIII.] 


PROOF  OF  SQUARE  ROOT  RULE. 


299 


above  fully  explained  treatment.     To  find  the  square  root  of 

33,  write 

r 

33- 1  5-7445 

■ 

25- 

10-7 

8-00 

i- 

7-49 

11-44 

•5100 

-4576 

11-484 

•052400 

•045936 

t 

11-4885 

•00646400 

f 

•00574425 

11-4890 

•00071975 

and  the  outstanding  error  in  the  root  is  very  closely  indeed 
equal  to  the  residual  deficiency  divided  by  twice  the  root 
so  far  found,  that  is  to  say,  -00071975  -^  ir4890,  or  00006265. 

The  advantage  of  the  approximation  we  noted  on  p.  295, 

is  so  great  that  even  when  the  first  number  is  conspicuously 
not  unity,  it  is  often  convenient  to  make  it  so  by  division. 
For  instance  to  find  ^85,  it  equals  J{^l-\-  4) 

=  9V(1+A)=^*9a+A)  =  9  +  1  =  9-2. 
And  so  with  some  of  the  other  examples,  they  too  may  be 
done  this  way..    We  will  therefore  repeat  them. 

750  =  ^(49  +  1)  =  1JU^^^1(1+^) 
=  7x1-0102  =  7-0714. 
In  this  case  the  approximate  value  -0102  is  obtained  thus. 
98  is  two  per  cent,  less  than  100,  so  —  is  two  per  cent,  greater 
than  -01. 
7143  =  7(144-1)  =  127(1 -tJ^)- 12(1 -^1^)  =  12-^V 

*At  this  stage  the  second  term  is  halved  and  the  root  sign  dropped. 


300  EASY  MATHEMATICS.  [chap. 

^99  =  10V(1-tJo)=^10(1-^1^)  =  10 --05  =  9-95. 
V375  =  V(400  -  5)  =  20J{1  -  ^\)  -  20(1  -  ^i^) 
=  20-i  =  20 --125  =  19-875. 
Perhaps  decimals  might  be  preferred  throughout.     Some- 
times they  would  be  handier,  sometimes  not. 

^396  =  7(400  -  4)  =  207(1  -  -01)  ii=  20(1  -  -005) 
=  20--1  =  19-900. 
The  result  of  this  convenient  approximation  is  always  to  give 
slightly  too  big  a  value  for  the  root,  and  this  whether  terms 
under  the  root  are  separated  by  a  negative  or  a  positive  sign. 

Thus  for  instance  the  approximation  to  JlOl  namely  10-050, 
and  to  799  namely  9-950  are  both  of  them  a  trifle  too  big. 

The  error  itself  can  be  estimated  by  a  further  stage  of 
approximation,  and  so  gradually  we  can  get  as  nearly  accurate 
as  we  please,  but  we  leave  it  there  for  the  present. 

The  error  in  either  case  is  about  -000125,  so  the  digits  as 
they  stand  above  are  fairly  near  the  truth. 

Cubes  and  Cube  Root. 

Now  let  us  see  what  we  can  get  of  the  same  kind  to  help  us 
in  other  cases.     Suppose  we  cube  a  binomial,  what  shall  we  get  ? 
First  notice  that 

{a  +  b){c  +  d){e+f)  =  (a  +  h){ce  +  cf+de-]-df) 
=  ace  +  acf  +  ade  +  adf 
+  bee  +  bcf+  bde  +  bdf, 
eight  terms  altogether. 

So  take  the  three  factors  all  alike. 

{a  +  by  =  (a  +  b){a  +  b)(a  +  b) 
=  acut  +  aab  +  aba  +  abb 

+  baa  +  bab  +  bba  +  bbb 
=  a^+a%  +  a%  +  ab^ 

+  a^b  +  ab^  +  ab^  +  b^ 
=  a^  +  3a%  +  Sab'^  +  b^ 


XXXIII.]  ALGEBRAIC  AIDS.  301 

Not  a  very  simple  expression  at  first  sight,  but  quite  simple 
when  you  get  accustomed  to  it,  and  very  easy  to  remember 
and  write  down. 

Notice  first  that  every  term  is  of  the  same  "dimension," 
that  is  to  say  it  involves  three  letters  multiplied  together, 
no  more  and  no  less.  There  is  no  term  involving  only  a^j 
nor  only  b^,  nor  a  alone,  nor  is  there  anything  like  a\  The 
expression  is  a  cube,  and  every  term  is  of  the  nature  of  a  cube. 
If  a  and  b  were  lengths,  the  cube  is  a  volume,  and  every  term 
is  necessarily  a  volume.  You  cannot  with  any  sense  add  an 
area  like  a^  to  a  volume  like  a^,  but  you  can  add  a  volume  like 
a^b  or  like  ab^  to  another  volume  like  a^,  and  you  can  add  each 
more  than  once,  in  fact  3  times  if  you  choose. 

Notice  next  that  the  power  of  a  decreases  by  one  each  term, 
and  the  power  of  b  increases.  We  might,  if  we  liked,  introduce 
the  index  0,  because  we  know  that 

ao  =  1  =  60. 

So  the  more  fully  written  expression 

would  represent,  with  needless  explicitness,  the  truth  that  the 
sum  of  the  indices  of  each  term  is  3. 

As  to  the  big  3's  prefixed  to  the  two  middle  terms,  they  are 
styled  coefficients,  or  numerical  factors ;  we  have  seen  exactly 
how  they  arise,  simply  because  we  had  to  add  three  equal 
terms.  They  take  the  place  of  the  2  in  the  middle  term  when 
we  were  squaring  a  binomial. 

We  illustrated  the  square  of  a  binomial  by  fig.  31, — where 
the  a^  and  the  b^  and  the  two  rectangles  each  equal  to  ab 
are  obvious,  and  plainly  make  up  the  (a  +  by. 

So  also  we  can  geometrically  illustrate  the  cube  of  a 
binomial ;  taking  a  cube  whose  every  edge  is  divided  into  any 
two  parts,  respectively  a  and  b,  we  get  a  figure  like  33,  which 
is  more  easily  realised  when  built  up  or  sawn  out  of  wood. 


302 


EASY  MATHEMATICS. 


[chap. 


Such  of  the  portions  as  are  visible  are  labelled  with  their 
respective  volumes.  There  is  first  a  big  cube  a^^  then  there 
are  three  slabs  each  of  area  a^  and  thickness  h,  but  one  of  them 
in  the  figure  is  invisible  at  the  back;   there  are  3  rods  or 


'^b^-... 


prisms  each  of  the  length  a  and  sectional  area  h"^ ;  and  lastly 
there  is  a  little  cube  b^  diagonally  opposite  the  big  one ;  and 
these  make  up  the  8  pieces,  out  of  which  the  whole  cube  has 
been  built  up,  (a  +  hf. 

This  then  is  a  solid  figure  illustrating  the  cube  of  a  bi- 
nomial in  the  same  sort  of  way  that  Euclid  11.  4  illustrates 
the  square  of  the  same  quantity. 

Suppose  we  wished  to  illustrate  the  fourth  power  of  a 
binomial  by  geometry.  We  could  not  possibly  do  it  in  any 
natural  fashion,  for  we  have  already  exhausted  all  the  dimen- 
sions of  space.  Hence  geometrical  propositions  on  involution 
are  not  only  complicated  and  wordy,  but  are  feeble  and  limited. 

Algebra  is  not  limited  at  all;  we  can  raise  a  binomial 
to  the  fourth,  fifth,  fifteenth,  or  any  other  power  that  we 
please,  and  presently  we  will  do  it.  But  first  we  will  take 
a  few  examples  and  applications  of  what  we  have  learnt 
about  the  cube  or  third  power. 


XXXIII.]  ALGEBRAIC  AIDS.  303 

First  a  mere  numerical  illustration  or  verification  : 
(5  +  2)3  _    53  +  (3x25x2) +  (3x5x4) +  23 
=  125+        150        +        60        +8 
=  343. 
Then  take  a  case  where  the  first  terra  is  unity  : 
{l+xf  =  l  +  3x  +  3x^  +  x^, 
and  then  one  with  the  second  term  negative : 
(l-xf  =  l-3x  +  3x^-xP. 
Notice  in  this  case  that  the  signs  in  the  expansion  are 
alternate,  because  the  powers  of  {-x)  are  alternately  odd 
and  even :  the  odd  will  all  be  negative,  and  the  even  will  be 
positive.     The  general  case,  with  the  negative  sign  to  the 
second  member  of  the  binomial,  ought  also  to  be  recorded  : 
(a  -  bf  =  a3  _  Sa%  +  Sab'^  -  b% 
(x-lf  =  .'c3-3a;2  +  3a:-l, 
but  this  is  just  the  same  as  (1  -xy  with  the  sign  of  every 
term  reversed. 

It  is  worth  obtaining  the  general  result  for  the  third  power 
of  a±b  in  another  way,  by  help  of  what  we  know  about  its 
second  power. 

{a±by  =  (a±b)(a^±2ab  +  b^) 
=  a^±3a%  +  3ab^±b^ 
Observe   that   the   alternative   sign   affects  only  alternate 
terms,  viz.  those  which  involve  the  odd  powers  of  the  possibly 
negative  quantity  b.     Among  its  even  powers  there  is  never 
any  variety. 

Another  special  case  is 

x  +  ~)    =  x^  +  3x  +  -  +  -^. 
x)  XX? 

This  is  rather  a  curious  case,  considered  from  the  point 
of  view  of  the  '  dimensions '  of  each  term,  x^  looks  like  a 
volume,  and  would  be   a  volume   if  x  were   a  length,  and 


304  EASY  MATHEMATICS.  [chap. 

3a;  would  be  merely  treble  that  length ;  then  come  reciprocals. 
How  can  this  be  possible  1  Answer  : — It  is  never  possible  to 
have  different  dimensions  in  different  terms  of  an  expression. 
It  is  quite  easy  and  common  to  have  factors  of  different 
dimensions,  as  components  of  a  single  term,  united  by  the 
sign  X ,  but  different  terms  united  by  the  sign  +  or  -  are 
always  of  the  same  dimensions. 

Apply  that   to   the  case  of  lx  +  -j    and  we  see  that  x^ 

cannot  possibly  be  a  volume,  nor  can  a;  be  a  length,  if  1  is 
a  pure  number.  It  can  in  that  case  only  be  of  the  same 
dimension  as  its  reciprocal. 

Length  and  volume  are  all  very  well  as  illustrations,  but 
it  would  be  a  great  mistake  to  suppose  that  algebraic  sym- 
bols can  express  nothing  else.  The  terms  "square"  and 
"cube"  suggest  geometrical  signification,  and  that  doubtless 
was  their  original  meaning,  but  now  they  have  been  so 
generalised  that  the  original  geometrical  signification  is 
almost  forgotten.  Cube  is  still  used  merely  as  short  for 
"third  power,"  and  square  is  short  for  "second  power,"  but 
the  things  that  we  raise  to  powers  may  be  anything  whatever 
that  we  find  convenient.  Often  they  are  mere  numbers, 
like  a  number  of  oranges.  If  we  speak  of  3^  oranges  meaning 
27  oranges,  it  may  be  a  pedantic  mode  of  statement,  but  it 
is  not  incorrect.  Even  if  we  spoke  of  a  cube  of  3  oranges, 
or  3  oranges  cubed,  we  might  possibly  be  understood,  as 
meaning  a  cubical  box  full  of  oranges  with  3  in  each  edge, 
9  in  each  face,  and  27  in  the  box. 

But  this  expression  would  not  bear  close  examination, 
unless  we  put  it  in  brackets,  thus,  (cube  of  3)  oranges,  and 
then  it  does  express  more  than  merely  27.  For  27  might 
be  lying  about  anyhow,  but  (cube  of  3)  signifies  that  they 
are  packed  in  a  certain  compact  arrangement.. 


XXXIII.]  ALGEBRAIC  AIDS.  305 

Why  is  cube  of  (3  oranges)  wrong?     Because  that  would 
mean   3^  x  oranges^ ;   and  the  latter  factor  has  no  meaning. 
Cube  of  (3  feet)  is  perfectly  right,  for  that  means 
33  X  feet3  =  27  cubic  feet. 

You  can  have  a  cubic  foot,  but  you  cannot  have  a  cubic 
orange  ;  or  rather  perhaps  you  cannot  have  anything  linear  or 
superficial  in  oranges,  as  you  can  with  feet  or  metres  or  inches. 

Eeturning  to  the  expression  x  +  -  then,  what  can  x  mean  1 

Only  a  thing  whose  dimensions  are  the  same  as  its  reciprocal, 
that  is  to  say,  a  thing  which  has  no  "  dimension,"  not  a  con- 
crete thing  at  all,  but  an  abstract  number,  a  number  of  things 
abstracted  from  "  things  "  altogether  and  contemplated  alone. 
That  is  what  we  mean  by  an  "abstract  number"  or  "pure 
number."  It  is  the  simplest  kind  of  "abstraction"  there  is, 
and  the  first  we  arrive  at ;  later  we  shall  employ  plenty  more. 
If  It  is  a  pure  number,  like  2, 

-  is  likewise  a  pure  number  like  n- 

n^  is  also  a  pure  number,  and  n^,  and  any  power. 
Jn  or  any  root  is  also  a  pure  number. 

So  is  log  n. 

We  cannot  assert  that  a*"  is  a  pure  number  for  certain, 
because  it  depends  entirely  on  what  a  is. 

a  might  be  a  length,  in  which  case  a^  would  be  an  area,  and 
a^  be  a  volume,  a^  would  in  that  case  have  no  assignable 
physical  meaning,  but  it  would  certainly  not  be  a  pure 
number. 

There  is  therefore  no  difficulty  about  an  expression  like 

3?~        ic3      ' 


(■ 

-S' 

=  x' 

-Zx  +  ~- 

X 

every 

term  must 

be  a 

pure 

number. 

L 

E.M. 

u 

306  EASY  MATHEMATICS.  [chap. 

This  is  not  necessary  with  the  next  example,  because  there 
all  the  terms  in  the  expansion  have  the  same  dimensions; 
provided  always  that  a  and  b  are  quantities  of  similar  kind. 

(a2  -  &2)3   =  ^6  _  3ft4J2  +  3^254  _  J6, 

In  the  next  case,  however,   a  must  be   a  pure   number, 
because  of   the  term  unity.      If  1  means   1  something,  the 
something  cubed  can  go  outside  the  brackets  :   it  must  apply 
equally  to  both  a^  and  1. 
(^2-1)3  =  a^-3a^  +  3a^-l. 
(1-^2)3=  l-3J2  +  (Sx2)-{j2f 
=1+6-372-272 
=  7-572  =  7-7-071  =  --071. 
(7-1)3  =  73 +  -3x72 +  21  X -01 +001 
=  343  +  3x4-9  + -211 
=  357-911. 
(57)3  =  (50  +  7)3  =  125000 +  (21x2500) +  (150x49) 
+  343  =  185193; 
but   in   this   case   it  would   be   easier   to   do   it   by    simple 
multiplication,  57  x  57  x  57,  or  perhaps  by  logarithms.     The 
worst   of   logarithms  for  finding  a  positive  integer  power  is 
that  they  only   give   it  approximately,    unless   you   take   a 
considerable  number  of  places ;  and  an  integer  power  never  is 
approximate,  it  can  always  be  numerically  expressed,  because 
we  start  with  a  number  and  only  multiply  it  by  itself. 

By  "integer  power"  or  "integral  power"  I  do  not  mean  a 
power  of  an  integer,  I  mean  any  number  raised  to  a  power 
whose  index  is  a  whole  number  and  not  a  fraction.  If  the 
index  is  fractional  it  represents  a  root.  The  case  is  entirely 
different  with  a  root,  for  then  we  are  endeavouring  to  find 
something  which  multiplied  by  itself  will  produce  a  given 
number :  and  the  result  is  usually  incommensurable. 


XXXIII.]  FINDING  CUBE  ROOTS.  307 

But  for  integer  indices,  whether  positive  or  negative,  we 
can  always  get  an  exact  result  by  straightforward  multiplication; 
for  instance  2^^,  or  2~%  or  (1'2)^. 

(1-2)3  =  1  +  (3  X  -2)  X  (3  X  -04)  +  '008 
=  1  + -6 +  -12  + -008 
=  1-728, 
which  is  a  familiar  number — expressing  the  thousandth  part 
of  a  cubic  foot,  if  1"2  means  the  tenth  of  a  foot  in  inches. 

Now  find  a  cube  root  or  two  by  the  approximation  method, 
choosing  numbers  which  are  not  very  different  from  a  perfect 
cube. 

Say  we  want  the  cube  root  of  65,  call  it  4  +  a;. 

65  =  (i  +  xf  =  64  +  48x+12a;2  +  ic3; 

So  the  first  approximation  to  x  is  ■^^. 

This  however  is  a  trifle  too  big,  because  12a;2  has  been 
neglected.  So  we  might  call  it  ^g  or  even  -^q,  at  a  shot, 
and  say  that  the  answer  is  4-02  ... .  As  to  neglecting  x^  it 
is  of  slight  consequence.  This  process,  elaborated,  is  the  basis 
of  the  arithmetical  cube-root  rule. 

Take  only  one  more  example  of  finding  cube  roots,  because 
they  are  usually  done  most  easily  by  logarithms. 

To  find  ^341. 

341  =  (7  -  xf  =  343  -  3  X  i9x  +  21a:2  -  a;^; 

2 
.*.    approximately   x  =  ^ — j^,  or,  as  this  is  a  trifle  too  small, 

2  1 

say  - — —  =  —  =  -0139.     So  approximately 

v^341  =  7 --0139  =  6  9861, 

which  is  still  a  trifle  too  small  in  the  last  place.     The  digit 
1  ought  to  be  a  3  or  a  4. 


308  EASY  MATHEMATICS.  [chap. 

As  an  exercise  it  would  be  desirable  to  establish  a  method 
akin  to  the  square  root  approximation,  like  this, 
(341)^  =  (343 -2f  =  7(1-^1^)^ 
=  7(1  -  Y^\y-)  approximately 
=  7  X  y^l^ ;  which  equals  -2  per  cent,  less  than  7, 
=  6-986  roughly ; 
or  generally,  when  a;  is  a  small  quantity, 

^{l±x)  =  l±lx  approximately ; 
which  is  equivalent  to  neglecting  squares  and  cubes  and  all 
higher  powers  of  x. 

Approximations. 

The  fact  that  the  square  of  a  small  quantity  is  very  small, 
and  the  cube  of  it  extremely  small,  is  easy  enough  to  under- 
stand ;  and  since  it  is  extremely  useful  in  application,  it  should 
be  thoroughly  understood  and  remembered.  Let  the  small 
quantity  be  1  per  cent.,  for  instance,  or  "01  or  Yho-  ^^^  square 
is  X(tJo  oj  0"6  ten-thousandth ;  and  its  cube  is  a  millionth ; 

If  then  we  have  to  find  (I'Ol)^,  it  will 

=  1  + -03 +  -0009  + -000001 
=  1  030901, 
of  which  the  first  significant  digit  of  the  decimal  is  decidedly 
the  most  important,  the  second  is  sometimes  worth  attention, 
denoting  a  value  about  ^^rd  of  the  previous  one,  and  the  last 
is  utterly  trivial,  except  for  exact  mathematical  purposes. 

A  cube  of  a  foot  and  one  inch  (or  13  inches  cubed), 
(13  inches)^,  is  decidedly  bigger  than  a  cubic  foot;  but  never- 
theless a  cubic  inch  is  almost  negligible  in  comparison  with  a 
cubic  foot :  it  is  only  the  ytVf^^  P^^^  ^^  i*- 

Let  us  examine  this,  because  beginners  often  make  mistakes 
here. 


XXXIII.]  INCREASES  IN  BULK  AND  AREA.  309 

( 1  foot  + 1  inch)^  they  incline  to  write  down  as  a  cubic  foot 
plus  a  cubic  inch :  which  is  just  the  mistake  of  thinking  that 
(a +  6)3  equals  a^  +  b^;  in  other  words  it  is  the  mistake  of 
altogether  ignoring  Sci^b  +  Sab'^,  three  slabs  and  three  rods,  and 
attending  only  to  the  little  insignificant  corner  cube  of  the 
small  quantity  b  (supposing  6  to  be  a  small  quantity)  in  fig.  33. 

The  true  value  is 
(1  foot+1  inch)3  =  (1  foot)3  +  3(feet)2xl  inch 

+  3feetx(inch)2  +  (linch)3 
=  1  cubic  foot 

+  3  slabs  a  foot  square  and  an  inch  thick 
+  3  rods  a  foot  long  and  a  square  inch  section 
+  a  cubic  inch. 
The  last  term  is  the  most  trivial  of  the  eight  terms,  and  the 
3  slabs  are  the  most  important  after  the  cubic  foot  itself. 
Translating  to  inches,  we  see  that 

(13inches)3  =  1728  +  (3  x  144)  +  (3  x  12)  + 1 
=  2197  cubic  inches, 
which  is  otherwise  very  easily  arrived  at. 

If  instead  of  a  foot  and  an  inch  we  had  taken  a  yard  and  an 
inch,  the  smallness  of  everything  except  the  slabs  would  have 
been  accentuated ;  and  if  we  take  a  metre  and  a  millimetre  we 
shall  see  it  still  more  forcibly  : 

(1  metre  +  1  millimetre)^  =  1  cubic  metre  +  3  slabs  a  metre 

square  and  a  millimetre  thick 
+  3  lines  a  metre   long  and  a 
square  millimetre  cross  section 
+  a  millimetre  cube ; 
or  expressing  it  all  in  cubic  centimetres 

=  1  million  c.c.  +  three  hundred  thousand  c.c.  +  three  c.c. 

+  a  thousandth  of  a  c.c. 
=  1,300,003-001  c.c. 


310  EASY  MATHEMATICS.  [chap,  xxxiii. 

When  things  expand  by  heat,  the  expansion  is  usually  very 
small ;  the  increase  of  bulk  is  not  so  small  as  the  increase  of 
length  however.  If  the  edge  of  a  cube  expands  1  per  cent, 
the  volume  of  it  expands  just  about  3  per  cent.,  and  the  area 
of  one  of  its  faces  about  2  per  cent.  This  follows  from  what 
we  have  been  saying.     Compare  page  293,  No.  4. 

It  is  sometimes  expressed  by  saying  that  the  proportional 
superficial  expansion  is  twice  the  linear,  while  the  cubical 
expansion  is  three  times  the  linear.  We  will  employ  the 
subject  of  expansion  to  furnish  us  with  a  few  interesting 
arithmetical  examples  of  an  easy  and  uncommercial  kind  in  a 
future  chapter,  but  first  we  will  do  some  algebraic  expansions. 


CHAPTER  XXXIV. 

To  find  any  power  of  a  Binomial. 

Suppose  we  have  to  find  {a  +  &)^  we  have  only  to  multiply 
a  +  ?>  by  itself  four  times,  and  write  down  the  result.  We 
might  write  it  thus 

(a  +  hf{a  +  hf 
=  (a2  +  2ab  +  62)  (^2^  ^ah  +  h'^) 
=  a^  +  2a%+   a%-^ 

+  2a%  +  4a%^  +  2ab^ 

+   a%^  +  2ab^  +  b^ 

=  a^  +  ^a%  +  6fl2/>2  +  4a  j3  _^  j4 

Now  here  we  see  the  same  sort  of  law  as  was  observed  in 
the  expansion  of  (a  +  b)^ ;  the  indices  of  a  decrease  regularly, 
and  those  of  b  increase  regularly,  so  that  every  term  is  of  the 
fourth  degree.  The  numerical  coefficients  follow  a  less 
obvious  law.  Let  us  write  them  down  for  the  cases  that  we 
know. 

for  (a  +  b)  1  1 

„    (a  +  6)2  1  2  1 

„    (a  +  bf  1  3  3  1 

„    {a  +  by  1  4  6  4  1 

The  law  is  fairly  plain,  and  we  might  guess  the  coefficients 
for  the  next  sets  : 

(a  +  bf  1  5  10  10  5  1 

(a  +  bf         1  6  15  20  15  6  1 


312  EASY  MATHEMATICS.  [chap. 

and  then  we  can  verify  them,  by  direct  multiplication,  thus 
(a +  5)5  =  {a  +  bf(a  +  by 

=  a^  +  ba^b  + 10^^362  +  lOa^^  +  5a¥  +  b\ 
(a  +  bf  =  (a  +  bfia  +  bf 

=  ft6  +  6^56  +  15a452  +  20a3^>3  + 15^2^4  +  Qab^  +  jo, 

A  guessing  process  like  the  above,  which  is  subsequently 
verified  and  obviously  extensible  to  the  case  of  any  positive 
integer  as  index,  is  a  method  of  frequent  and  considerable  use 
in  order  to  first  ascertain  a  rule  or  law  or  method  of  pro- 
cedure ;  but  one  should  not  rest  satisfied  without  perceiving 
the  rationale  of  it,  and  so  to  say  "  proving  "  it  or  reasoning  it 
out;  otherwise  it  remains  what  is  called  an  "empirical"  law, 
meaning  a  law  ascertained  by  experiment  and  observation 
without  a  full  knowledge  of  the  reason.  Some  laws  have  to 
remain  of  this  character,  when  the  subject  matter  is  difficult 
or  obscure;  but  that  is  not  the  case  with  little  calculations 
like  the  present :  the  reasonableness  of  the  result  can  always 
be  made  out,  and  it  is  a  most  wholesome  exercise.  In  the 
present  instance  the  method  of  expanding  any  binomial  as  an 
empirical  process  seems  to  have  occurred  to  Isaac  Newton 
while  still  quite  young ;  and  the  reasoned  proof  of  this  process 
is  what  we  now  know  as  "  the  binomial  theorem." 

We  will  not  go  into  this  fully  just  at  present,  nor  at  all 
more  fully  than  is  needed  for  practical  purposes,  but  for  a 
positive  integer  the  empirical  process  itself  is  easy  and  worth 
while  for  anybody  to  know. 

First  write  down  what  we  have  observed,  for  any  positive 
integer  index  n,  concerning  (a  +  by  : — 

We  know  that  the  powers  of  a  will  begin  with  a",  and 
decrease  by  one  each  time  down  to  a^  or  unity. 

The  powers  of  b  will  begin  with  b%  or  unity,  and  climb 
by  one  each  time  up  to  b" ;  so  that  as  regards  the  algebraic 
part  of  the  expansion,  the  terms  will  be 


XXXIV.]  BINOMIAL  EXPANSION.  313 

or,  or-%  ar-w,  ar-n)\ a%''-\  a}f-\  b'\ 

the  sura  of  the  indices  of  a  and  b  always  adding  up  to  n, 
which  may  be  called  the  "  order  "  or  "  degree  "  of  the  whole. 

Now  what  about  the  numerical  coefficients  1  We  can  obtain 
them  as  follows.  Take  the  coefficient  of  any  term,  multiply  it 
by  the  index  of  a  in  that  term,  and  divide  by  the  number 
of  terms  preceding  the  next  term,  the  result  will  give  the 
coefficient  for  that  next  term.  This  is  what  we  have  ascer- 
tained empirically,  though  we  did  not  word  or  express  it 
before,  but  it  is  what  we  did  or  might  have  done ;  because, 
take  the  case  of  {a  +  6)^, 

the  first  term  is  a', 
so  the  coefficient  of  the  next  term  is  5,  giving 
a^  +  ba^b. 

Now  take  the  5  and  the  4,  multiply  them  together,  and 
divide  by  2 ;  we  get  10,  which  is  the  coefficient  of  the  next 
term,  carrying  us  as  far  as  three  terms, 
a^  +  ba^b  +  lOa^^. 
Then  take  the  10  and  the  3,  multiply  them,  and  divide  by 
the  number  of  terms ;  thus  we  get  the  next  coefficient,  viz.  10, 

a^  +  5a'^b+10a%^  +  l0a^b^ 
Now  take  the  10  and  the  2,  multiply  them,  and  divide  by  4, 
and  we  get  the  coefficient  of  the  next  term,  viz.  5. 

Then  take  5  and  1,  multiply,  and  divide  by  5,  and  we  get 
the  coefficient  of  the  last  term,  viz.  1,  giving  the  whole  ex- 
pansion, with  six  terms  in  all, 

a^  +  ba^b  +  lOa^b^  +  I0a%^  +  bab^  +  b^ 

In  the  last  term  the  index  of  a  is  zero,  hence  a  does  not 
appear,  because  a^  =  1 ;  and  if  we  apply  the  rule  further  it 
will  give  us  zero  as  a  factor  of  the  next  and  of  every  succeeding 
term ;  which  therefore  all  vanish,  so  the  series  terminates. 


314  EASY  MATHEMATICS.  [chap. 

Try  this  rule  also  for  {a  +  b)^  and  (a  +  hf,  getting  the  result 
in  the  latter  case, 

a^  4-  7a%  +  2la^b^  +  35a^b^  +  3ba%^  +  2U%^  +  lab^  +  b\ 
and  then  apply  it  to  (a  +  6)", 

a**  +  na"  ^b  +  — ^r — '  a"  ^^  +  -^ ^r-^-^ ^  a**  ^b^ 

+"(»-^)("-^)<"-3V-'y  +  etc. 

Now  this  is  a  most  interesting  example  of  a  very  important 
algebraic  thing  called  a  '  series.'  It  appears  to  go  on  for  ever, 
but,  as  we  have  seen,  it  does  not  go  on  very  long  when  n  is  a 
positive  integer,  for  sooner  or  later  there  will  come  the  index 
n-n,  whose  value  is  0 ;  and  as  this  quantity  n-n  will  enter 
as  a  factor  into  every  subsequent  coefficient,  they  all  vanish, 
and  the  term  with  index  0  applied  to  a  is  the  last  term. 

Thus  for  (ft  +  by*  there  were  six  terms  in  the  series,  and  no 
more.  For  {a  +  b)^  there  were  seven  terms,  and  no  more ;  and 
for  (a  +  by-  there  will  be  n  +  1  terms,  and  no  more,  provided  n 
is  a  positive  integer.  All  subsequent  terms  are  zero,  because 
they  all  contain  the  factor  n-n.  But  if  the  index  n  \%  z, 
fraction,  or  if  it  has  a  negative  value,  even  a  negative  integer 
value,  the  cause  of  stoppage  will  no  longer  occur ;  for, 
naturally,  a  numb&r-of-terms  can  never  be  a  fraction  or  negative. 
There  will  therefore  never  be  an  index  n-n-,  there  will  be 
71-7,  71-8.  71-9,  etc.,  but  none  of  these  can  possibly  be  zero 
unless  n  itself  is  a  positive  integer. 

Consequently  in  these  cases  the  series  does  not  stop,  but 
goes  on  for  ever,  extending  to  infinity.  It  may  happen 
however  that  its  later  terms  become  insignificantly  small,  and 
that  all  after  a  certain  number  can  be  neglected  for  practical 
purposes.  This  a  point  to  which  attention  must  be  specially 
directed,  because  it  is  exceedingly  useful  in  practice. 


XXXIV.]  FACTORIALS.  315 

Notice  that  we  have  not  yet  established  or  proved  the  above 
series  for  the  expansion  or  power  of  a  binomial,  even  for  the 
case  of  n  a  positive  integer.  We  will  defer  the  proof  for  the 
present :  so  far  we  have  only  arrived  at  it  by  experiment. 
The  proof  is  not  difficult  for  n  a  positive  integer,  but  it  will 
come  better  later.  Mathematicians  know  how  to  prove  it  for 
a  fractional  and  a  negative  index,  that  is  for  the  case  of  an 
infinite  series,  which  however  is  exactly  of  the  same  algebraic 
form  as  the  one  we  have  written. 

The  method  of  experiment  and  observation  is  quite  a  good 
practical  method,  only  it  might  in  some  cases  lead  us  wrong 
unless  it  can  be  checked  over  and  reasoned  out  by  some  more 
intellectual  process. 

For  the  present  we  will  accept  the  series  and  study  it. 
Notice  first  the  denominators  of  the  several  terms.  They  consist 
of  a  series  of  consecutive  natural  numbers  1  .2.3.4.5,  etc., 
multiplied  together.  This  sort  of  product  often  occurs,  and  it 
is  convenient  to  have  a  symbol  for  it.  [_5  is  the  way  it  is 
written,  5  !  is  the  Way  it  is  printed,  and  it  is  called  "  factorial 
5."  They  all  mean  the  same  thing,  viz.  1x2x3x4x5,  that 
is  to  say  120. 

So  4  !  has  the  value  24,  since  it  means  1x2x3x4 


M 

>>                     5> 

6, 

\1 

))                     J  J 

2, 

\1 

"                    " 

1, 

11 

»                     J) 

720, 

7!  or  [7 

equals 

5040, 

8! 

= 

40,320, 

and  so  on. 

So  that  factorial  20  is  an  enormous  number. 

Now  look  at  the  numerators.  They  too  are  factorials  of  a 
kind,  but  they  do  not  begin  at  1,  they  begin  at  the  other  end ; 
they  represent  a  part  of  factorial  w,  with  the  early  part  cut  off. 


316  EASY  MATHEMATICS.  [chap. 

" ) 

\n       ' 
They  might  be  denoted  by     —  ^ ;  for 

-V,    » 

71  .  71  -  1  .  71  -  2  .  71  -  3  . . .  w  -  r  +  1   = 


(71  -  r) ! 
So  the  successive  numerators  are  as  follows : 

[n       \ii^         \n  I 


etc., 


\n    1 71-1    1 71-2    1 71 -3' 

being  1,  ti,  n{n-l\  n{n-l){n-2),  etc.,  respectively. 
Hence  any  of  the  coefficients  may  be  written  in  this  form 


\L' 


while  as  to  the  ab  part  corresponding  to  this  general  coefficient 
it  will  be  a'-'"^''. 

Hence  the  whole  series  may  be  neatly  written  as  the  sum 
of  a  number  of  terms  all  of  this  kind,  for  every  value  of  r  from 
0  to  7i;  and  such  a  summation  is  usually  expressed  by  the 
capital  letter  sigma  ;  hence 

^        ^        ^Q  1(71-7')!  r!  J 

which  means  that  you  write  down  all  the  terms  of  this  form 
in  regular  order  from  r  =  0  up  to  r  =  tj,  and  then  add  them 
together.  Try  to  do  this,  for  different  values  of  n,  for  instance 
3  or  4  or  5  or  6,  and  see  that  you  get  the  series  already  obtained. 
The  only  thing  that  requires  explanation,  until  we  come  to 
fractional  and  negative  indices,  is  how  to  interpret  "  factorial 
nought."  To  common  sense  such  an  expression  sounds  meaning- 
less ;  and  to  understand  it  fully,  together  with  the  factorials  of 
negative  and  fractional  numbers,  a  good  deal  of  mathematics 
must  be  conquered.  It  is  easy  however  to  show  that  1 0  must 
be  interpreted  as  unity,  that  is  to  say  that  [J_and  [O^are  alike 
equal  to  1. 


XXXIV.]  BINOMIAL  EXPANSION.  317 

Proof.  \n  =  n\'n  -\  ;  but  in  the  special  case  when  n  =  1 
n  and  n  are  the  same  thing ;  hence  in  that  case  \n-\  is  unity, 
but  it  is  also  factorial  nought. 

Exercise. — Make  a  table  of  binomial  coefficients  up  to  say 
the  index  12  as  the  finish.     For  answer,  see  p.  334. 

A  special  case  of  frequent  occurrence  is  when  one  of  the 
terms  of  the  binomial  is  unity,  as  for  instance  (1  +xY. 

Consider  this  case.  Any  binomial  can  be  thrown  into  this 
form  by  an  obvious  process,  as  follows  : 

{a  +  hy  =  «"(l+^y  =  a'\\+x)\ 

where  a"  is  a  factor  taken  outside  brackets,  and  the  ratio  hja 
is  treated  as  a  single  quantity,  a  pure  number,  and  called  x. 

Observe  that  a  and  h  might  be  anything,  so  long  as  they 
are  the  same  thing,  but  that  x  must  be  a  number,  in  order 
that  it  may  be  added  to  1 ;  and  being  a  ratio  of  similar 
quantities,  it  is  a  number.  The  most  important  case,  with 
fractional  and  negative  indices,  is  when  a;  is  a  small  number, 
for  then  the  series,  or  expansion  in  powers  of  x,  will  rapidly 
diminish,  and  all  beyond  a  few  terms  can  be  neglected.  The 
meaning  of  this  will  become  clearer  soon. 

First  apply  the  ordinary  rule  for  the  expansion,  observing 

that  1""^  1""'^,  etc.,  need  not  be  written,  because  they  are  all 

mere  unity  factors.    We  have  nothing  therefore  to  write  but  the 

successive  binomial  coefficients  and  the  ascending  powers  of  x. 

/,       v_       ,             71.71-1    o    n  .n-\  .n-2   ^ 
{l+xf  =  1  +nx  +  — 2l — ^  + 13 ^  +  •••  » 

a  very  useful  expansion  ;  and  if  x  is  really  small,  so  that  x^  may 

be  neglected,  it  gives  us  this  extremely  handy  approximation, 

(1  +  j)"  sQb  1  +  wa;   when  x  is  very  small. 

As  a  matter  of  fact  we  have  used  this  already  for  extracting 

approximate  roots  (p.  308),  arriving  at  it  by  a  different  process. 


318  EASY  MATHEMATICS.  [chap. 

Thus  to  find  J{1  i-x)  when  x  is  small  we  have  only  to  put 


»  =  i 

^(1  +a:)  =  (1  -\-xy  ^  1  +^a;  approximately. 
E.g.        v/(l'01)  =  1-005  approximately. 
^(1-008)  =  1-004 
VlOO-6  =  10^(1  +  006)  ^  10  X  1-003  =  10-03. 
So  also  for  cube  or  other  roots. 

^(l+x)  =  (l+x)^^l+^x, 
i/1003  =  10yi003  =  10(1  003)^  =£3^10x1-001  =  10-01. 

^33  =  2V(UA)  =  2(1+^V)'^2(1+^) 
=  2  +  Jlg.  =  2-0125. 
Or  take  an  example  of  a  negative  index. 

-^  =  (i+.r^-^i-i.. 

But  this  case  of  a  negative  index  will  bear  examining  more 
fully. 

Let  us  write  n  =  -m,  and  then  interpret  the  general 
expression  for  the  special  case  of  a  negative  index.  Observe 
that  it  is  no  new  expansion,  only  the  old  one  re-written  with 
the  sign  of  the  index  changed,  but  it  looks  different : 

(a  +  b)-"^  =  a-^  +  {-m)a-^-'b+  ~^^[~'^^~\-'^-'b^+ ... 

i  .  A 

ffl(l+m)(2  +  m)^_(^^3)^3 

=  J_      ^^      m.m  +  1.62     m(m  +  l)(m  +  2)73  , 

a^  ~  a"*+^  "^     |_2  a"*-^"  |_3  qT'^^ 

the  terms  having  alternate  signs. 

(a  — 6)""*  would  be  similar  but  have  all  the  terms  positive. 


XXXIV.  ]  BINOMIAL  EXPANSION.  319 

Hence  also  • 

/I  ^  \-m       1-  m.m  +  1    „_m. 771  +  1. m  +  2  „  ,      " 

(l±x)-"'  =  1+mx-^ — r-^ ^  + ■  I  Q    ■ ^  +  +  etc., 

where  it  will  be  observed  that  with  the  -H  sign  on  the  left, 
the  terms  on  the  right  are  alternately  +  and  -  ;  but  with 
the  —  sign  on  the  left  they  are  all  +  on  the  right. 

The  series  is  infinite,  but  if  x  is  small  a  few  terms  practically 
sufiice. 

Examples. 

Take  some  examples  or  special  cases  : 

(l+ic)-i  =  I-x  +  yttX^ r^—x^+... 


=     1     -X-{-X'^-X^+X*-X^+     ...     y 

of  which  only  a  few  terms  are  important  if  x  is  small,  e 
p^  =  (l-Ol)-i  =  1-01  + -0001- -000001+    . 
=  -990099...  ib-9901, 
7YJ^  =  (l+a;)-2  =  l-2x  +  Sx'^-4:X^  +  5x^-..., 

J(l-hx)  =  (l+a:)^=  i  +  ix  +  i^^x'' 

+ [^  ^+- 

v/r^  =  1  -  la:  - 1^2  -  ^^x^  -  ^^x^  -  ... , 

-i(-i-2)(-i-2)^ 

il 


320  EASY  MATHEMATICS.  [chap. 

A  curious  case  is  afforded  when  both  terms  of  the  binomial 
are  unity,  like  (1  +  1)".  When  the  index  n  is  not  a  positive 
integer  the  series  is  divergent  and  useless;  but  when  ti  is  a 
positive  integer  it  is  simple  enough,  for  the  sum  is  finite.  It 
is  a  mere  curiosity,  but  we  may  as  well  find  a  power  in 
this  way. 

For  instance  to  find  2^, 

,,      ,,,       ,      .     5.4     5.4.3     5.4.3.2     |5 

(1  +  1)^  =  1+54-^-  +  -^^+—^^—  +  !^ 

=  1+5  +  10  +  10  +  5  +  1  =  32. 

Similarly  2^  =  1+6  +  15  +  20  +  15  +  6  +  1  =  64. 

This  set  of  numbers,  as  tabulated  in  their  early  stages  on 
pages  311  and  334,  are  called  the  binomial  coefficients;  and 
you  observe  that  each  set  of  them  adds  up  to  a  power  of  2. 
We  had  not  noticed  this  before. 

Now  what  is  the  good  of  an  expansion  generally  1    Is  it  of 
any  practical  use?     Well  it  is,  but  it  is  the  first  few  terms 
which  are  the  most  useful.     The  expansion  of  some  power  of 
(1  +a;)  is  specially  useful  when  x  is  very  small,  for  then 
^(l+a:)^l  +  |a;. 


This  approximation  is  said  to  be  correct  to  the  first  order  of 
small  quantities,  or  to  be  an  approximation  of  the  first  order. 

To  be  correct  to  the  second  order  of  small  quantities  we 
must  introduce  the  terms  involving  «2,  and  so  on. 

When  X  is  only  moderately  small,  third  and  even  fourth 
terms  may  have  to  be  employed,  and  the  more  terms  intro- 
duced the  more  accurate  will  the  result  be. 


XXXIV.]  INFINITE  SERIES.  321 

If  X  is  greater  than  1,  the  series  becomes  hopeless,  but  if  x 
is  only  slightly  less  than  1,  it  can  always  be  approximated  to 
sufficiently,  by  taking  enough  terms,  though  it  is  not  then 
really  useful. 

The  series  is  said  to  be  convergent  or  converging  when  x  is 
less  than  1.  A  converging  series  is  one  whose  terms  con- 
tinually decrease  in  such  a  way  that  the  sum  of  an  infinite 
number  of  them  is  finite. 

For  instance,  l+J  +  |-  +  J^  +  yV+'-is  ^  converging  series, 
and  its  value,  to  an  infinite  number  of  terms,  is  2  ; 

but  1  +  1  +  T.  +  J+1+    ... 

happens  not  to  be  convergent,  for  though  the  terms  keep  on 
diminishing,  they  do  not  diminish  with  sufficient  rapidity  to 
be  able  to  stop  at  any  point  and  say  'we  will  neglect  the  rest.' 
Those  which  we  neglected  would  in  fact  amount  to  more  than 
those  we  took  into  account,  for  the  sum  of  an  infinite  number 
of  terms  of  such  a  series  is  infinite.  It  is  not  a  convergent 
series  at  all,  although  each  term  is  smaller  than  the  preceding 
one.     A  curious  case. 

The  first  is  called  a  geometrical  progression,  the  second  is 
called  a  harmonic  progression,  because  it  gives  the  series  of 
the  harmonics  or  simplest  overtones  in  music.  The  time  of 
vibration  of  the  fundamental  note  being  called  1,  a  trained  ear 
can  hear,  when  a  string  is  struck  or  plucked  or  bowed,  or 
when  an  open  organ  pipe  is  blown,  other  superposed  notes, 
with  their  times  of  vibration  \^  ^,  J,  etc.,  of  the  first ;  and 
these  superposed  or  secondary  tones  are  called  harmonics.  So 
the  series  is  called  a  harmonic  series. 

An  arithmetical  series  is  one  whose  terms  proceed  by  simple 
addition.  In  a  harmonic  series  it  is  the  denominators  or 
reciprocals  of  the  terms  which  proceed  in  this  way.  For- 
tunately we  seldom  or  never  want  to  sum  a  harmonic  series. 


L.E.M. 


CHAPTER  XXXV. 

Progressions. 

We  have  now,  in  the  last  chapter,  arrived  at  an  example  of 
a  series  or  progression.  The  subject  of  'series'  is  immense 
and  endless,  but  there  are  a  few  simple  ones  which  are  excep- 
tionally easy  to  deal  with. 

Of  these,  three  are  commonly  treated  quite  early,  viz.  the 
three  called  Arithmetical,  Geometric,  Harmonic,  respectively. 

In  an  arithmetical  series  the  terms  proceed  by  a  common 
difference. 

In  a  geometric  series  the  terms  proceed  by  a  common  factor. 

In  a  harmonic  series  the  reciprocals  proceed  by  a  common 
difference. 

Thus  1,  2,  3,  4,    5,  ...  is  the  simplest  example  of  an  a.p. 
1,  2,  4,  8,  16,  ...  „  „  „  G.P. 

11111  TT    p 

But  the  common  difference  may  be  negative,  or  the  common 
factor  less  than  1,  so  that 

7,  6,  5,  4,  3,  2,  1,  0,  -  1,  -  2,  ...  is  an  example  of  A.P. 

1111  n  T> 

1111—1-1  TTT> 

T>  T>  ^>   ^J        ^J        T»    •••  "  >»  "•^• 

Also  1,  1-25,  1-5,  1-75,  2  ...  is  an  a.p. 

1000,  100,  10,  1,  -1,  -01,  ...  is  a  G.P. 

^  +  A  +  F+ 1  +  6  +  36  +  216  ...  is  also  a  G.P. 

-1,  -1-6,   -5,  5,  1-6,  1,  -714285,  -5,  -45  is  an  H.P. 


CHAP.  XXXV.]  PROGRESSIONS.  323 

The  latter  is  perhaps  too  much  disguised  for  a  beginner,  but 
if  the  terms  be  written  as  vulgar  fractions  it  is  plain  enough  : 
the  denominators  are  in  A.p.  with  common  difference  2,  for  it 
is  the  same  as 


5 

5 

5       5     5     5     5     5 

5 

-5' 

-3' 

-1'    1'    3'    5'    r    9' 

11' 

I 


The  thing  that  generally  has  to  be  done  with  a  series  is  to 
evaluate  the  sum  of  its  terms ;  and  the  most  important  are 
those  whose  terms  decrease,  so  that  an  infinite  number  of 
them  have  a  finite  sum,  which  can  be  ascertained.  Otherwise 
we  must  know  how  many  terms  we  are  intended  to  add  up. 

Another  thing  that  may  be  necessary  to  do  with  a  series, 
especially  those  which  do  not  converge,  or  which  actually 
increase  as  they  go  on,  is  to  find  the  value  of  the  nth  term. 

Thus  in  the  horse-shoe  nail  question,  page  155,  we  had 
really  to  add  24  terms  of  a  G.P.,  beginning  with  1  and  proceed- 
ing by  a  common  factor  2 ;  but  the  finding  of  the  25th  term 
of  that  series  was  sufficient,  because  we  could  see  experi- 
mentally that  each  term  was  almost  precisely  equal  to  the 
sum  of  all  that  had  preceded  it,  being,  as  a  matter  of  fact, 
just  one  in  excess.     (Compare  page  218.) 

1  -h  2    with  1  added  made  the  next  term  4. 
1+2  +  4             „  „  „  „  8. 

1  +  24-4-J-8  „  „  „  „  16. 

1  +  2  +  4  +  8  +  16  „  „  „  „  32. 

etc. 

So  all  the  first  24  terms,  with  1  added,  were  equal  to  the 
25th  term  of  the  series. 

But  the  25th  term  was  2-%  therefore  the  whole  24  terms  of 
the  series,  added  up,  was  one  less  than  2^* ;  that  is  to  say  the 
series  equalled  2^4-1. 


324  EASY   MATHEMATICS.  [chap. 

Of  the  three  progressions  we  have  mentioned,  G.P.  is 
certainly  the  most  commonly  occurring  and  the  most  useful. 
Let  us  take  it  first. 

Let  the  common  factor  be  called  r,  and  the  first  term  a,  so 
that  the  terms  run  thus 

a-\-ar  +  ar^  +  ar^  +  . . . , 

r  being  any  number  whole  or  fractional. 

If  ?•,  when  interpreted  arithmetically,  is  a  negative  quantity 
the  terms  will  have  alternately  opposite  signs,  and  the  result 
will  be  a  combination  of  alternate  addition  and  subtraction ; 
which  however  can  conveniently  be  called  the  algebraic  sum, 
meaning  the  sum  when  written  algebraically  with  sign  implied 
but  unexpressed,  but  of  course  subtracting  from  the  series 
those  terms  with  negative  signs  when  arithmetical  interpre- 
tation is  entered  upon. 

One  sees  at  once  that  since  the  second  term  is 

ar 
and  the  third  a?-2, 

the  fourth  ar^^ 

the  nth.  term  must  be  a?'""^ 

The  sum  of  the  first  n  terms  will  therefore  be 
-:    -    a(l +r  +  r^-l- ... +r""^). 

Now  this  is  a  thing  we  have  already  come  across ;  it  was 

when  r  was  small,  that  is  to  say 

=  1  +  r  +  r2  +  r^ . . . , 
but  r  must  be  less  than  1,  or  the  series  will  not  converge; 
every  term  will  get  bigger  than   the  preceding  one  if  r  is 
greater  than  1,  and  there  would  be  no  meaning  except  infinity 
in  an  infinite  number  of  such  terms.     But  the  expansion  is 


XXXV.]  PROGHESSIONS.  325 

only  true  for  an  infinite  number  of  terms;  consequently  it 
is  only  serviceable  when  r  is  less  than  1. 

However,  that  is  a  very  important  case :  the  most  important 
case.    Let  us  apply  that  to  a  few  examples  before  we  go  further. 

Find  the  sum  1  +  i  +  i  + 1  •  ■  •  • 

Here  a  =  1    and   r  =  i, 

the  sum  then  will  be  1  x  ( j  =  - — -  =  2,  which  we  already 

knew.    (pp.  321  and  100.)^^  "^'^       ^~2 

This  series  can  be  well  illustrated  by  cutting  up  an  apple  or 
a  loaf  of  bread ;  for  if  such  an  object  be  taken  and  first  a  half 
cut  off",  then  a  quarter,  then  an  eighth,  then  a  sixteenth,  and 
so  on,  all  the  cutting  can  be  performed  on  a  single  object,  and 
however  long  the  cutting  be  continued  the  single  unit  will 
not  be  exhausted  :  and  yet  if  the  cutting  be  continued  ad 
infinitum  the  apple  will  be  all  exactly  used  up.  In  other 
words,  although  the  sum  of  any  finite  number  of  terms  of 
the  series  J,  J,  \^  etc.,  is  less  than  unity,  the  sum  of  the 
infinite  series  of  these  fractions  is  exactly  one  whole,  no  more 
and  no  less,  that  is  to  say 

i  +  T  +  i  +  TV+---«^*V-  =  1- 
As  another  example  take 

1  +  -1  +  -01  +  -001  +  ...; 
-.1  1  1       10       ,  . 


It  equals 

1-1       -9 

T=^^' 

as  is  otherwise  obvious  by  simp] 

le  addition  of  the  terms. 

Again 

12  +  444  + 

4       4 

27  +  8T+-- 

The  sum 

A-^        ^ 

=  12x|=18. 

Another 

way  of  putting  it  is 

to  say  that 

1     1      1 

3'^9'^27"^"" 

=  r  =  1. 

326  EASY  Mathematics.  [chap. 

The  series  10  +  9  +  8*1 +7*29  + ...,  to  infinity,  is  a  G.P.  that 
does  not  decrease  very  fast,  but  it  converges  nevertheless,  and 
the  value  towards  which  it  converges,  constantly  approaching 

though  never  actually  reaching,  is  — —  as  usual,  that  is  100 ; 

the  sum  to  infinity  =  :j ^  =  j^ — ^  =  100  exactly. 

In  general,  so  long  as  r  is  less  than  1,  it  matters  not  how 
little  less,  the  series  will  converge,  and  we  can  find  the  sum  of 
an  infinite  number  of  terms.  Suppose  the  common  ratio  were 
•999  for  instance,  and  the  first  term  were   1,  the   sum   to 

infinity  would  be ^-^^,  that  is  to  say  1000.     If  the  first 

1  ~  To  IT (7 

term  of  this  series  had  been  anything  else  than  1,  say  56  for 
instance,  the  sum  would  have  been  merely  56000.  Or  if  the 
first  term  had  been  4*35782,  or  any  number  you  please, 
the  sum  would  have  been  4357*82,  if  the  common  factor  were 
*999  as  supposed. 

The  first  term  therefore  causes  no  difficulty,  it  is  the  com- 
mon ratio  or  factor  that  requires  attention  when  a  finite 
number  of  terms  is  wanted;  and  a  finite  number  of  terms 
always  is  required  whenever  the  common  factor  is  greater 
than  1,  and  often  is  when  it  is  less. 

How  are  we  to  find  the  sum  of  n  terms  then  ? 

It  can  be  done  by  a  contrivance  :         . 

Write  down  the  series,  and  then  write'  it  down  again  with 
every  term  multiplied  by  r,  and  then  subtract  the  two  series, 
thus  : 

Call  *S^  the  sum  of  n  consecutive  terms  of  the  series. 


,S^  = 

1  +  r  +  7*2  +  r^  +  . 

..7-1; 

.*.  • 

r^  = 

r  +  r2  +  ?-3  +  . , 

..r"-l  +  r^ 

Now  subtract 

^- 

-rS=  1 

_,.n 

because  all  the  other  terms 

go 

out; 

XXXV.]  PROGRESSIONS.  327 


I  _  7-"         if 
therefore  S  =  -^ ,  or 


1-7'        r-V 
which  is  the  same  thing. 

If  the  first  term  is  (X,  then  the  above  expression  has  to  be 
multiplied  by  a ;  so  that  in  general,  whatever  r  may  be,  the 
sum  of  n  terms  of  a  geometrical  progression  is 

5  =  a r-. 

r-  1 

If  n  should  be  00  there  is  no  finite  meaning  in  the  series 
unless  r  is  less  than  1 ;  in  that  case  r"  =  0,  because  higher 
powers  of  a  proper  fraction  keep  on  diminishing,  so  an  infinite 
power  must  disappear  altogether ;  we  then  get  the  case  which 

we  already  know,  viz. 

Examples. 

Apply  this  to  the  sum  of  24  horse-shoe  nails  with  one 
farthing  for  the  first,  and  with  common  factor  2.     (p.  156.) 
224  _  \ 
Ans. :  The  price  is  a  -^ — 3-  =  2^4-1  farthings. 

Find  the  sum  of  six  terms  of  the  series 

100 +  200 +  400  + etc. 

06  _  1 

Jns.:  It  equals  lOOx-^r — -  =  6300. 

Z  —   L 

Find  the  sum  of  1  +  3  +  9  +  27  +  etc.  to  six  terms. 

.         ^,  36-1       728       ^^, 

Ans.:  The  sum  =  -^ — ^  =  -^  =  364. 

Find  the  value  of  64  + 16  +  4  + 1 +i  + y^  +  ^V 
Ans. :  This  is  a  G.P.  of  seven  terms  with  common  ratio  J 
and  first  term  64. 

s.  .  =  ..-:iffi.J.e..(-^.) 

256 
~   3 


328  EASY  MATHEMATICS.  [chap. 

1  —  r** 
The  numerator  of  the  fraction ,  in  a  case  of  many 

terms  with  a  fractional  ratio,  is  of  small  significance :  it  is 
nearly  unity. 

Algebraic  Digression. 

The  result  we  have  arrived  at,  as  the  sum  of  a  G.P.,  may  be 

regarded  as  an  expansion  for  an  algebraical  division. 

1  -?•" 

T =  l+r-\-r^  +  r^+  ... . 

1  -r 

This  might  be  generalised  hypothetically  thus, 

x~y 
which  could  be  verified  by  direct  division,  or  more  easily 
by  multiplication,  and  could  be  led  up  to  experimentally  thus  : 

x^  -y^ 


x-y 

—  =  x'^  +  xij  +  y^j 

X  —  y 

—  ^  x^-\-  x^y  +  xy"^  +  «/3, 

and  so  on. 

If  we  try  the  positive  sign  between  the  terms  on  the  left, 
the  matter  is  a  little  more  troublesome. 

Try  it  first  in  the  denominator  only  : 

—  =  X-y, 

x  +  y  ^ 

— ^1^  will  not  go  without  a  remainder, 

x  +  y  ^ 

t^  =  x^-x'^y  +  xf--y\ 
0^  -y^ 


X  +  y 
and  so  on.   • 


will  not  go  again. 


XXXV.]  ALGEBRAIC  DIVISIONS.  329 

Now  try  the  positive  sign  in  numerator  only: 

^      -^   will  not  fifo,  i.e.  will  have  a  remainder ; 
x-y 

- — ^  will  not  ffo  either. 

x-y  ^ 

Now  try  positive  signs  in  both  numerator  and  denominator: 

x^  +  y- 


x  +  y 
x^  +  y^ 

x^  +  y^ 

x  +  y 

x^  +  y^ 


will  not  go, 


xy  +  y'^; 


will  not  go, 


c*  -  x^y  +  a^y  -  xy^  +  y*. 

So  it  makes  all  the  difference  whether  the  indices  are  even 
or  odd.  All  the  above  can  easily  be  verified  by  direct 
operation ;  and  the  reason  of  the  failure  to  divide  out,  when 
they  do  fail,  will  also  be  manifest  on  trial.  The  reason  is  that 
the  last  term,  the  y^  or  yS  etc.,  would  have  the  wrong  sign. 
To  sum  up  what  we  have  observed : 

a;" - ?/"  is  divisible  hy  x-y  whatever  n  is, 
and  likewise  hy  x  +  y  when  n  is  even. 

«"  +  y'"  is  divisible  hy  x  +  y  when  n  is  odd, 
but  is  not  divisible  by  x-y,  whatever  n  is;  understanding 
by  "  divisible,"  divisible  without  a  remainder,  that  is,  that  the 
denominator  is  a  fadm-  of  the  numerator ;  and  understanding 
by  n  always  a  positive  integer. 

Another  way  of  putting  it  is  as  follows : 

x-y  and  x  +  y  are  both  factors  of  «" -  y"  if  n  is  even, 
x-y  only  is  a  factor  of  x" -y"  ii  n  is  odd, 
x  +  y  only  is  a  factor  of  xT"  +  y'^  ii  n  is  odd, 
neither  is  a  factor  of  x"  +  y^  if  n  is  even. 


330  EASY  MATHEMATICS.  [chap. 

Or  thus,  which  forms  an  easy  way  of  remembering  the  facts  : 
^.3  _|.  yZ  ig  divisible  by  aj  +  y, 
a;3  _  yZ  ig  divisible  by  «  -  y, 
x^  -  y2  is  divisible  by  both, 
01?  4-  ip'  is  divisible  by  neither. 

General  expression  for  any  odd  number. 

It  is  handy  to  be  able  to  discriminate  between  an  odd  and 
an  even  number  algebraically. 

It  is  done  thus  : 

2/1  is  always  even,  if  n  is  an  integer. 

In  ±  1  is  always  odd,  again  if  n  is  an  integer. 

The  wth  odd  number  is  ^n-\  (hence  this  is  commonly  the 
expression  used  for  an  odd  number) ; 

e.g.  5  is  the  third  odd  number  and  is  equal  to  (2  x  3)  -  1 ; 
11  is  the  sixth  odd  number  and  is  equal  to  (2  x  6)  -  1 ; 
and  so  on. 

The  hundredth  odd  number  is  therefore  199  and  the  365th 
is  729. 

Arithmetical  Progression. 

Now  take  some  examples  of  A. P. 

An   interesting   case   is   to   find   the   sum   of   the    first    n 
consecutive  odd  numbers  added  together,  that  is  to  find  the 

value  of 

1+34-5  +  . ..  +  (271-1). 

This  sum  might  be  found  by  experiment,  thus  : 

1  +  3  =    4  =  22, 

1+3  +  5  =    9  =  32, 

1+3  +  5  +  7  =  16  =  42. 

So  the  sum  of  the  first  four  odd  numbers  is  42  and  of  the 

first  five  will  be  found  to  be  52  =  25. 


XXXV.] 


PROGRESSIONS. 


331 


9 

7 

5 

3 

1 

Fig.  84. 


Trying  a  few  more  we  come  to  the  experimental  conclusion 
that  the  sum  of  the  first  n  odd  numbers  will  be  n^. 
1  +  3  +  5  +  7  +  .. .  +  27Z-1  =  n\ 
The  annexed  diagram  illustrates  to  the  eye  the  facts  that 
1  +  3  =  the  square  of  2, 
1  +  3  +  5  =  the  square  of  3, 
1+3  +  5  +  7  =  the  square  of  4, 
and  so  on. 

The  sum  of  the  first  n  natural  num- 
bers is  not  so  simple,  but  is  a  good 
problem  to  solve  experimentally,  thus  : 

1  +  2  =  3, 
1+2  +  3  =  6, 

1  +  2  +  3  +  4  =  10. 
So  we  have  a  series,  rather  a  notable  series,  with  differences 
increasing  by  1  each  time. 
For  the  sum  is 

one  term    two  terms    three  terms    four  terms    five  terms    six  terms 
13  6  10  15  21,    etc. 

2x5  21x6     3x7 

What  would  be  the  number  thus  to  be  placed  under  n  terms  % 

The  answer  is  \n{n-{-\)',  and  it  is  possible  but  not  quite 
easy  to  guess  that. 

It  is  worth  remembering  however :  the  sum  of  the  first  n 
natural  numbers  is  \n{n-\- 1). 

As  for  the  sum  of  the  even  numbers,  that  is  a  very  simple 
matter,  for  it  is  merely  double  the  preceding ;  or  it  may  be 
regarded  as  the  sum  of  the  odd  numbers  plus  n ; 

2  +  4  +  6  +  8  +  10...  +  2n  =  7i2  +  ^ 

=  w(7i+l). 
This  method  of  guessing  and  verifying  will  not  carry  us 
far :    a  reasoned  process  of  arriving  at  a  result  is  far  more 


332  EASY  MATHEMATICS.  [chat. 

powerful  and  effective.     Algebra  enables  us  to  reason  things 
out;   and  the  customary  method  for  the  sum  of  an  A. P.  is 
as  follows  : 

Let  the  general  arithmetical  progression  be  the  following, 
to  n  terms, 

a,  a  +  b,  a-\-2b,         a  +  Sb ...  a  +  {n-l)b; 

write  it  again,  but  backwards, 

a  +  {n-l)bj  a  +  (n-2)b,   a  +  {n-3)b,   .......  a. 

Now  add  the  two  series  together,  term  by  term,  as  they 
stand  one  under  the  other ;  and  the  result  will  be  2a+{n-l  )b 
every  time. 

Hence,  since  there  are  n  terms,  the  result  of  the  double 
series  added  together,  if  *S'  is  the  sum  of  a  single  series,  will  be 
2,8' =  n{2a  +  {n-l)b}; 
.'.   S  =  na  +  ln(n-l)b. 
This  is  the  general  result  for  an  A.  P. 
For  example,  to  test  it  by  speci-al  cases  : 
In  the  case  of  the  first  n  natural  numbers  a  =  b  =  1,  and  so 
*S'  =  ni-^n{n-l) 
=  in(n  +  l), 
as  we  have  already  found  by  experiment. 

In  the  case  of  the  first  n  odd  numbers,  a  is  1  and  b  =  2  ; 
S  =  n  +  n(n-  1)  =  n^, 
as  we  also  found  experimentally. 

It  is  very  instructive  and  pleasing  to  see  how  a  general 
formula  thus  gives  special  cases,  and  it  is  one  of  the  verifications 
by  which  a  general  formula  should  always  be  tested. 
The  following  is  interesting  for  practice  : 
1  =  P 
3  +  5  =23 
7  +  9  +  11        =33 
13  +  15+17  +  19  =  43 
etc. . . 


XXXV.]  SERIES.  333 

Other  Series. 

The  number  of  series  or  progressions  that  can  be  dealt  with 

is  enormous,  is  indeed  infinite;  and  is  too  large  a  subject  for  us 

to  enter  upon  in  this  book.     Suffice  it  to  say  that  many  others 

occur  in  practice  besides  the  simple  ones  which  are  best  known. 

This  series,  for  instance, 

124-22  +  32  +  42 

is   neither   a  geometric   nor  an   arithmetic   nor   a  harmonic 
progression.     Something  like  it  occurs  in   che  overtone  fre- 
quencies of  vibration  of  plates  and  bars. 
Manifestly  we  might  have 

12,     32,     52,     ..    , 

or  P,    23,    43,    83, 

and  so  on ;  any  number  of  such  series  could  be  invented. 

There  is  one  simple  series  that  we  came  across  recently 
on  p^ge  331,  the  difference  of  whose  terms  was  constantly 
and  steadily  increasing:  the  series  1,  3,  6,  10,  15,  etc. 

If  we  started  with  this  series  and  took  the  differences  we 
should  get  an  A. p.   series,  and  this  is  a  process  we  might 
continue ;  thus : 
Start  with  this, 

0       1       3       6      10     15     21     28.... 
Take  differences, 

12       3       4       5       6       7       ...anA.P. 
Take  second  differences, 

111111  a  series  of  constants. 

Take  third  differences, 

0       0       0       0       0  a  series  of  zeros. 

Suppose  we  start  with  a  different  series,  say  the  natural 
series  of  square  numbers, 

0,   1,  4,  9,   16,  25, 
the  differences  of  these  will  give  the  series  of  odd  numbers ; 
while  the  second  differences  would  be  constant. 


334  EASY  MATHEMATICS.  [chap. 

If  we  took  any  geometrical  series  the  differences  would  be 
the  same  series  again,  multiplied  by  a  factor,  the  factor  being 
one  less  than  the  common  ratio. 

Hence  the  differences  of  the  powers  of  2,  viz.  1,  2,  4,  8,  16,  32, 
would  be  the  same  series  over  again. 

The  binomial  coefficients  can  be  obtained  by  interjecting  a 
single  1  into  the  middle  of  a  row  of  noughts  and  then  adding 
adjacent  terms  to  make  a  term  of  the  next  series,  as  thus, 

00000  10  0000 
0000          1            10000 

0        0  0  1  2  1  0  0        0 

0  0  13  3  10         0 

0  14  6  4  10 
1          5          10         10         5  1 

1  6  15        20         15         6  1 

1  7         21        .35         35        21         7         1 
18         28        56        70        56        28         8        1, 

19         36        84        126      126       84        36         9       1 

1  10       45       120      210      252      210       120       45       10       1 

1       11      55       165      330      462      462      330       165       55      11       1 
1      12      66      220      495      792      924      792      495      220      66      12      1 

The  simplest  illustration  of  an  arithmetical  progression  is 
the  natural  series  of  numbers — the  ordinary  counting  of  a 
child.  The  most  important  instance  of  an  arithmetical 
progression  that  occurs  in  nature  is  afforded  by  time.  It  is 
true  that  it  progresses  continuously  and  not  by  jerks,  but  the 
motion  of  a  clock  hand  is  a  jerky  motion,  and  the  succession 
of  days,  weeks,  and  years  divide  the  continuum  into  units  for 
measuring  purposes,  and  represent  a  perfectly  uniform  and 
inexorable  constant  rate  of  progress. 

A  set  of  numbers  are  said  to  be  in  geometrical  progression 
when  their  logarithms  are  in  arithmetical  progression.  The 
notes  of  a  piano  are  in  this  predicament,  when  estimated  by 
their  vibration  frequencies.    The  chromatic  scale,  on  a  tempered 


XXXV.] 


PROGRESSIONS. 


335 


instrument  like  a  piano,  proceeds  by  equal  musical  intervals, 
but  these  intervals  are  characterised  by  equal  ratios  of 
vibration  frequency,  every  octave  having  double  the  vibration 
rate  of  its  predecessor ;  in  other  words  the  factor  2  carries  us 
over  the  interval  of  an  octave,  the  factor  f  gives  successive 
fifths,  and  so  on ;  so  that  the  same  musical  interval,  in 
different  parts  of  the  register,  is  characterised  by  a  constant 
difference  of  logarithm. 

A  set  of  numbers  are  said  to  be  in  harmonic  progression 
when  their  reciprocals  are  in  arithmetical  progression. 

The  series  of  square  numbers  have  their  roots  proceeding 
in  A. P.;  another  series  we  have  encountered  has  consecutive 
differences  in  A.  P. ;  another  series  might  have  successive  ratios 
in  A. P. ;  and  so  on. 

Geometrical  Illustrations. 

The  heights  of  a  row  of  palings  may  be  used  to  illustrate  the 
three  best  known  modes  of  progression,  if  their  tops  all  reach 
a  sloping  straight  line.  If  they  are  spaced  simply  at  equal 
intervals,  they  of  course  form  an  A. P. ;  if  they  are  spaced  so 
that  lines  drawn  from  the  foot  of  each  to  the  top  of  the  next 
are  all  parallel,  they  will  form  a  g.p.  ;  but  if  spaced  so  that  a  line 
from  the  foot  of  each  to  the  top  of  the  next-but-one  bisects  the 
intermediate  one,  they  form  an  H.p.  The  three  figures  annexed 
illustrate  this. 


Arithmetical 

Fig.  35. 


Concerning  fig.  35  there  is  nothing  to  be  said  but  what  is 
obvious. 


336 


EASY   MATHEMATICS. 


[chap. 


Concerning  fig.  36  it  can  be  pointed  out  that  the  triangles 
formed  are  all  similar,  that  the  lengths  of  the  slant  lines  are 
in  G.P.  as  well  as  the  vertical  lines ;  and  so  are  the  areas  of  the 


Geometrical 

Fig.  36. 

triangles.    They  may  be  said  to  illustrate  the  successive  heights 
attained  by  a  bouncing  ball :  which  heights  are  also  in  G.P. 

Fig.  37  is  the  most  notable;   it  may  be  regarded  as  the 
perspective  view  of  a  series  of  equal  rectangles  or  parallelo- 


grams— the  perspective  view  in  fact  of  a  uniform  fence.    Hence 
it  is  useful  in  drawing  metrical  perspective  figures. 

Proofs. — The  proof  that  fig.  36  represents  a  geometrical  pro- 
gression is  almost  obvious,  since  by  construction  the  triangles 
are  similar,  their  sides  being  parallel ;  hence 

h^b=:yi  =  h==h    etc 

^1    h    y^    h    yi 

The  fact  that  fig.  37  gives  a  harmonic  progression  can  be 
established  thus : 

Let  a,  Z>,  c  be  three  verticals  erected  so  that  a  line  from  the 
foot  of  a  to  the  top  of  c,  or  from  the  foot  of  c  to  the  top  of  a, 
bisects  the  intermediate  height  &,  which  therefore  divides  the 


XXXV,] 


PROGRESSIONS. 


337 


base  in  some  ratio  m :  n,  then  it  can  be  shown  that 
harmonic  mean  of  a  and  c ;  for  by  similar  triangles 


is  the 


ib_     n 
a       m  +  n 

and  ^*  _     ™     ; 
c       m  +  n 

therefore 

a       c 

or 

1     1       2 

FiQ.  38. 

wherefore  the  reciprocals  are  in  arithmetical  progression. 

In  fig.  37  it  is  convenient  to  call  the  slant  lines  transversals, 
and  to  say  that  the  transversal  from  the  foot  of  each  passes 
through  the  mid-point  of  the  next  to  the  top  of  the  next-but-one. 

Another  geometrical  illustration  of  a  g.p.  is  the  following — 


a  sort  of  straight  line  spiral;  the  inclinations  being  any  constant 
angle  other  than  45°,  the  vertex  angles  being  90°. 

L.E.M.  Y 


338 


EASY  MATHEMATICS. 


[chap.  XXXV. 


Successive  sides  of  the  spiral  are  in  g.p.,  and  so  are  the 
distances  of  successive  vertices  from  the  centre. 

To  convert  fig.  35  or 
fig.  41  into  the  repre- 
sentation of  a  G.P.  as 
it  stands,  the  roof  must 
be  made  of  a  logarith- 
mic or  exponential 
curve  instead  of  a 
straight  line. 

Thus  fig.  9  and 
fig.  47  already  repre- 
sent a  G.P. ;  each  verti- 
cal height  is  the  Geo- 
metric Mean  of  any 
pair  of  heights  equi- 
distant on  either  side 
of  it. 

The  'amplitudes'  of 
the  swings  of  a  dying- 
out  pendulum  consti 
tuteaG.R:  the  'periods' 
of  successive  swings 
constitute  an  A.  P. 
See  (fig.  40.) 

The  temperatures  of 
a  cooling  body,  read 
every  minute,  consti- 
tute an  approximate 
G.P.,  and  if  plotted  would  give  a  logarithmic  curve :  looking 
like  fig.  9  or  fig.  78,  or  the  dotted  line  in  fig.  40,  or  part  of 
the  figure  on  page  179- 


CHAPTER   XXXVI. 
Means. 

A  THING  of  some  interest  and  use  is  the  mean  or  average  of 
a  set  or  a  pair  of  terms  in  a  progression.  In  an  A.  P.  the  mean 
can  be  found  by  adding  and  halving  the  two  extreme  terms. 
Thus  for  instance  in  the  progression 

7     9     11     13     15 
1 1  is  the  mean  term,  and  it  can  be  found  as  the  half  sum  of  9 
and  13,  or  the  half  sum  of  7  and  15. 

The  arithmetic  mean  of  a  and  c 
is  \{a  +  c);  for  calling  this  h,  it 
makes  h  -a  =  c-h,  that  is,  it  gives 
a  common  difference  in  the  pro- 
gression a,  b,  c;  and  it  is  illustrated 
by  the  figure,  where  b  is  the  mean 
height  of  the  trapezium  shown,  whose  area  is  therefore  h 
times  the  base. 


Fig.  41. 


The  arith 

metic  mean  of      1  and      7  is      4 

of      0  and  100  is    50 

of      0  and      9  is      4J 

of      6  and    16  is    11 

of  -  1  and  +  1  is      0 

of  -  6  and  +  8  is      1 

of  -  3  and  +  9  is  +3 

of  -  9  and  +  3  is  -  3 

of    12  and    90  is    51 

because  51  - 

- 12  =  '39,  and  90  -  51  =  39  ;  or  because 

51  =  6  +  45. 

340  EASY  MATHEMATICS.  [chap. 

In  general  then  the  arithmetic  mean  is  the  half  sum,  the 
sum  being  understood  as  the  algebraic  sum,  paying  attention 
to  sign. 

The  geometric  mean  of  two  terms  is  the  square  root  of  their 
product,  because  this  would  give  a  common  ratio;  thus  if 
three  terms  a^  b,  c  are  in  a. p.,  b  must  equal  J{ac)f  because 

^(ac)  :a  =  G  :  sj(ac)  =  \l[~),  the  common  ratio ; 

*  be  If 

in  other  words  the  common  ratio  -  or  r  must  be  equal  to  ^/-. 

In  the  progression  a,  ar^  ar^  the  middle  term  is  plainly  the 
square  root  of  the  product  of  the  end  terms. 

The  Geometric  Mean  is  also  called  a  "  mean  proportional." 
To  illustrate  a  geometric  mean  it  is  customary  to  use  either 
a  right-angled  triangle  or  a  circle.  Thus  if  the  two  lengths 
whose  geometric  mean  is  required  are  CA  and  CB,  any  circle 
drawn  through  A  and  B  has  the  property  that  its  tangent 
drawn  from  C  is  equal  to  the  geometric  mean  required; 
for  by  Euclid  III.  36,  CF^  =  CA.CB;  hence  incidentally 
we  arrive  at  the  proposition  that 
all  the  circles  that  can  be  drawn 
through  the  two  points  A  and  B 
can  be  cut  at  right  angles  by  a 
certain  circle  drawn  round  C  as 
centre;  because  the  length  CP  is 
constant.  If  only  the  points  A  and 
B  had  been  initially  given,  then  a 
number  of  such  points  C  could  be  found,  each  with  its  ap- 
propriate length  of  radius,  by  drawing  a  tangent,  or  a  series 
of  tangents,  to  any  one  of  the  circles. 

If  a  right-angled  triangle  ABC  be  drawn,  and  a  per- 
pendicular be  let  fall  from  the  right  angle  C  on  to  the  opposite 


XXXVI.] 


MEANS. 


341 


side,  the  length  of  this  perpendicular  is  a  mean  proportional 
between  the  segments  of  the  base  : 

CD  =  J  AD  .  DB,   since  -tttt  =  ^Diy 

Similarly  ^C  is  a  mean  proportional  between  AB  and  AD, 
and  BC  „  ,, 

The  same  thing  is  true  for 
a  semicircle,  since  the  angle 
in  a  semicircle  is  a  right  angle. 
Euclid  III.  31. 

Hence  an  easy  construction 
for  finding  the  Geometric 
Mean  of  two  lengths  is  to 
place  them  end  to  end,  as 
AD,  DB,  construct  a  semicircle  on  the  whole  length  AB  thus 
compounded,  and  erect  a  perpendicular  DC  at  the  junction 
point  of  the  two  lengths.    This  is  the  g.m.  required.    (Cf.  p.  274.) 

Or  if  the  two  given  lengths  had  been  A B  sand  AD,  then  the 
distance  AC  would  be  their  G.M. 


The  harmonic  mean  of  two  terms 
is  su(5h  that  it  would  be  the  arithmetic 
mean  of  the  two  terms  inverted. 

For  instance,  J,  J,  J  are  in  H.P., 


Fig.  43. 


Fig.  44. 


and 


is   the   harmonic   mean   be- 


tween ^  and  J. 


arithmetical  progression,  and  -  =  -  (  - 

0         A  \0j 


Let  a,  b,  c  be  in  harmonic  progression,  then  -^  j-^ 


are  in 


wherefore 


2ac 

a  +  c 


The  harmonic  mean  can  therefore  be  described  as  twice  the 
product  divided  by  the  sum. 


342 


EASY  MATHEMATICS. 


[chap. 


Geometrically  it  could  be  represented  by  setting  up  the  two 
given  numbers  as  parallel  measured  lengths,  like  a  and  c,  and 

joining   their   ends   both   direct 
and  crosswise. 

Then  the  parallel  drawn 
through  the  crossing  point  is 
the  harmonic  mean  of  a  and  c. 
It  is  represented  by  a  dotted 
line  in  the  figure. 
The  proof  of  this  construction  is  given  above  in  connexion 
with  figs.  37  and  38. 

If  the  two  outer  lines  of  the  figure  are  continued  till  they 
meet,  the  fourth  position  thus  determined  forms  a  H.P.  with 
the  three  positions  determined  by  the  crossings  depicted  in  the 
figure ;  and  they  are  familiar  in  elementary  geometrical 
optics. 


Examples. 

The  student  should  cover  up 

the 

right-hand 

column  and 

reckon  the  entries  in  it.     They  are  all  intended  to  be  done  in 

the  head. 

The  geometric  mean    of 

1 

and 

9 

is 

3. 

of 

9 

and 

9 

is 

9. 

of 

1 

and 

100 

is 

10. 

of 

0 

and 

100 

is 

0. 

of 

0 

and 

n 

is 

0. 

of 

9 

and 

36 

is 

18. 

of 

4 

and 

25 

is 

10, 

or  generally,  of 

W2 

and 

^2 

is 

mn. 

of 

40-5  and 

24-5 

is 

31-5, 

that  is  to  say  of 

|-m2 

and 

|7l2 

is 

\mn. 

of 

1 

and 

81 

is 

4-5. 

of 

7 

and 

I 

is 

1. 

xxxvi.]  MEANS.  34S 

is  1-3. 

is  6. 

is  2-3, 

is  2-3. 


the  G.M.  of 

1 

and 

16 

of 

1 
T 

and 

252 

of 

147 

and 

1 

2T 

of 

49 

and 

1 
9 

or  generally,  of 

am'^ 

and 

1 

— ,       IS 


m 


of      am     and         —     is        a, 
m 

e.g.  of      72  and  8  is      24. 

of  - 1 6  and  - 1  is     -  4. 

of  -16  and  -100  is  -40. 

of     -i  and  -72  is     -6. 

of  -14  and  -7  is  -7^2. 

of       ab  and  a  is  ajb. 

To  find  the  harmonic  mean  b  of  two  terms  a  and  c,  we  can 

write  down  that  —  t  =  i j 

a    0      b     c 

^,    ^  b-a      c-b 

or  that  — J—  =  -^ — , 

ab  be 

,    ,  b-a      a 

or  that 


c-b       c 

The  two  most  usual  expressions  for  a  three-term  harmonic 

1     1      2 
progression  are  _  +  -  =  -, 

J  7       2ac 

and  b  =  — ■ — ; 

a  +  c 

these  are  simple  and  useful,  and  we  come  across  them  con- 
stantly in  elementary  optics.  In  reflexions  from  a  small 
curved  mirror  the  object  and  image  and  the  centre  of  curvature 
of  the  mirror  are  situated  at  points  whose  distances  from  the 
mirror  itself  run  in  a  harmonic  progression. 

So  also  in  lenses,  if  a  is  the  distance  of  object  from  the  lens, 
and  c  the  distance  of  the  image,  ^b  is  its  focal  length. 

This  is  set  down  here  merely  as  a  reminder. 


344 


EASY   MATHEMATICS. 


[chap. 


Illustrations. — The  harmonic  mean  of  J  and  |^  must  be  i ; 
and  twice  the  product  by  the  sum  of  the  two  numbers  is 

if      120      5* 


2x1x1. 


1  4-  1 

T  +  F 


The  harmonic  mean  of  4  and  6  is 

2x4x6      48       ,  ^ 

-TT6-  =  io==^'^- 

2  X  99 
The  harmonic  mean  of  1  and  99  =     ..^^     =  1-98. 


100 


The  harmonic  mean  of  0  and  1  is  0. 

The  harmonic  mean  of  17  and  13  is 

34x13      442 

30   ~  ~   30 


The  harmonic  mean  of 
of 


14-7i 

-2and+6  =  ^ 

2  and  +2  =   -oo. 
70 


24 


+  4 


=  -6. 


of  -  5  and  -  7  = 


of  -  1  and  -  9  = 


-12 

18 


10 


—        f^s 


=   -1-8. 


of      1  and    00  =  2. 


The  geometric  mean  of  6  and  20  is  Jb  x  20 
ratio  being  2. 

The  geometric  mean  of 

of 

of 

of 

of 

of 


10,  the  common 


4. 


1  and    16  is 

2  and  32  is  8. 
4  and  9  is  6. 
8  and  2  is  4. 
8  and  -  2  is  -  4. 
8  and  +  2  is  imaginary. 


of  -  1  and  -  9  is  -  3. 
of     a  and      b  is  J(ab). 
Comparing  the  three  means  of  the  same  two  quantities  a  and  by 


XXXVI.]  •         MEANS.  345 

the  arithmetic  mean  is  J  {a  +  h)  and  is  the  biggest  of  the  three, 
the  geometric  mean  is  sfabj 

the  harmonic  mean  is f  and  is  the  smallest  of  the  three. 

a  +  h 

The  H.M.  may  be  considered  as  ^^-^ — —.  that  is  the  ratio  of 

^  A.M.    ' 

the  square  of  the  G.M.  to  the  A.M. ;  or  it  is  equal  to  the  G.M. 
multiplied  by  the  proper  fraction  g.m./a.m. 

Examples. 

Take  any  two  numbers,  say  4  and  9  : 

the  arithmetic  mean  is  6*5, 

the  geometric  mean  is  6*0, 

the  harmonic  mean  is  ^f  =  5'53846... . 
Let  the  two  numbers  be  1  and  25  : 

the  A.M.  is  13, 

the  G.M.  is  5, 

the  H.M.  is  fl  =  1-923...  . 
Let  the  two  numbers  be  49  and  36  : 

the  A.M.  is  42-5, 

the  G.M.  is  42-0, 

/42)2         49 

theH.M.is^  =  ~x42, 

which  is  necessarily  less  than  the  G.M. 
Let  the  two  numbers  be  0  and  4  : 

the  A.M.  is  2, 

the  G.M.  is  0, 

the  H.M.  is  0, 
but  not  the  same  0,  it  is  half  the  G.M  squared. 
Let  the  two  numbers  be  6  and  -  6  ; 

the  A.M.  is  0  : 

the  G.M.  is  imaginary, 

the  H.M.  is  -  00  . 


346  EASY  MATHEMATrcS.  [chap. 

Let  the  two  numbers  be  -  3  and  ~  6  : 
the  A.M.  is  -  4-5, 
the  G.M.  is  -  372  =  -  4-2426, 
the  H.M.  is  -  4-0. 

In  some  cases  the  order  of  numerical  magnitude  is  inverted ; 
but,  when  compared  with  positive,  the  smallest  negative 
quantity  is  represented  by  the  largest  number.  If  heights  of 
mountains  were  reckoned  from  sky  instead  of  from  earth,  as 
by  dropping  a  plummet  from  a  balloon,  the  lowest  mountain 
would  need  the  longest  plumb-line  to  reach  it.  The  lowest 
parts  of  the  solid  earth  are  beneath  the  sea  and  require  a  long 
sounding  line  to  reach  them. 

The  A.M.  is  of  course  always  half  way  between  the  two 
numbers. 

The  G.M.  is  nearer  to  the  smaller  one,  and  ultimately 
coincident  with  the  smaller  one  when  the  other  is  infinitely 
bigger.- 

The  H.M.   is  less  than  the  G.M.  in  the  ratio  of  — — '  \  or 

A.M. 

H.M.  X  A.M.  =  G.M.2,  or  the  G.M.  is  a  mean  proportional  between 
the  other  two. 

Mean  or  Average  of  a  Number  of  Terms. 

In  taking  a  mean  of  terms  there  is  no  need  whatever  that 
those  terms  should  form  any  sort  of  progression  or  ordered 
series.  Hitherto  we  have  only  taken  the  mean  of  two  terms, 
and  two  terms  cannot  possibly  determine  any  kind  of  pro- 
gression, any  more  than  two  points  can  determine  a  curve. 
But  we  can  reckon  the  arithmetical  or  the  geometrical  mean  of 
any  number  of  terms  as  follows : 

Suppose  we  want  the  mean  of  a  set  of  observations  of 
temperature,  taken  at  every  hour  of  the  day,  so  as  to  determine 
the  mean  temperature  during  the  day  of  12  hours,  say  from 


XXXVI.] 


MEANS. 


347 


8  a.m.  to  8  p.m.  Let  the  thermometer  readings  be  the 
following — there  will  be  13  readings,  because  of  the  beginning 
and  end  points  of  time  between  which  the  twelve  hours  lie : 


Add  them  up  and  divide  by  the  number  of 
them,  that  is  by  13.  This  is  the  mean  or 
average  of  the  readings,  and  is  found  to  be 
67 "58.  It  is  apparently  a  summer  day  with  a 
warm  and  probably  cloudy  morning  giving  place 
to  a  clearer  sky  and  cooler  evening. 

If  the  temperature  readings  were  plotted  and 
joined,  the  result  would  be  a  curve  (fig.  46) ; 
and  the  average  height  of  this  curve  would  be 
the  mean  temperature. 

The  average  height  must  be  approximately 
67*58;  but  when  the  curve  is  drawn  by  a 
recording  thermometer,  so  as  to  give  the 
temperature  not  only  at  every  hour,  but  at  every  instant,  a 
more  exact  determination  of  the  average  can  be  made. 


60-5 
65-0 
67-2 
67-3 
71-25 
75-0 
79-0 
77-0 
74-6 
70-3 
62-4 
553 
53-7 
13  I  878-5 
67-58 


■--- 

zzz 

.^^ 

-^^ 



^ 



-  — 

"^ 

X 

^ 

Mean 
Temp. 

8 

a.m.  £ 

)     1 

0        1 

1      1 

2 

p.nui 

Fig 

2        C 

46. 

}         < 

I         I 

3      e 

i 

T         i 

i 

The  average  or  mean  height  of  such  a  curve  is  the  height  of 
a  rectangle  with  the  same  base  which  shall  equal  the  curve  in 
area,  as  shown  in  the  figure  by  the  dotted  line. 

One  way  of  finding  out  the  area  of  such  a  curve,  and  in 
that  way  of  obtaining  its  mean  height,  is  to  cut  the  above  figure 
out  in  cardboard  or  tinfoil  or  sheet  lead,  and  then  weigh  it ; 


348  EASY  MATHEMATICS.  [chap. 

weighing  at  the  same  time  a  rectangle  or  square  of  known 
dimensions  cut  out  of  the  same  sheet. 

Thus  suppose  the  curve  carefully  drawn  on  such  a  uniform 
sheet,  on  a  scale  which  gave  1  horizontal  centimetre  to  each 
hour  and  1  vertical  millimetre  to  each  degree,  and  that  the 
figure  bounded  as  above  was  carefully  cut  out  and  found  to 
weigh  4'98  grammes;  while  a  rectangle  of  the  same  base  12 
centimetres,  and  height  7  centimetres,  was  found  to  weigh 
5*16  grammes. 

We  should  know  that  the  area  of  the  curve-bounded  figure 

was  ^7j-^x  7  X  12  sq.  centimetres,  and  that  its  average  height 

was  — -—  X  70  millimetres. 
516 

This  process  would  give  the  average  height,  and  therefore 
the  number  of  degrees  in  the  average  temperature,  as  67 '56  ; 
and,  if  carefully  carried  out,  it  should  be  more  correct  than 
merely  averaging  numerical  readings  taken  each  hour,  for  it 
averages  the  temperature  recorded  from  instant  to  instant. 

To  follow  a  process  of  this  kind  profitably,  the  best  plan  is 
actually  to  do  it,  and  then  the  method  of  working  will 
naturally  occur  to  you  with  a  little  thought;  and  a  good 
result  can  be  obtained  with  some  handicraft  skill.  It  is  a 
practical  method  of  experimentally  performing  the  operation 
known  as  integrating;  it  is  integration  between  definite 
limits,  or  the  finding  of  a  definite  integral  of  a  function 
represented  by  a  curve. 

Weighted  Mean. 

Generally  the  word  "  mean  "  implies  the  simple  arithmetic 
mean,  and  the  niean  of  several  numbers,  say  Wj,  n^  and  n^,  is 
yii  +  yi2  +  %  ^^.^j^  .g  ^i^g^  written  n  and  read  "ri  bar." 


XXXVI.]  MEANS.  349 


The  average  of  a^,  a^.-.a^  is  a  =-  -(ttj  +  a2+  •••  +<^n)  ^    but 


sometimes  we  have  to  do  with  a  weighted  mean.  One  case  is 
when  a  series  of  observations  of  the  same  thing  are  taken 
under  different  circumstances,  and  some  of  the  circumstances 
are  more  favourable  than  others;  for  instance  if  the  height 
of  a  flagstaff  were  being  measured  by  the  length  of  its  shadow, 
at  noon  on  successive  days  at  the  time  of  the  summer  solstice, 
and  suppose  the  record  of  the  shadow  measurements  was 
entered  thus : 

15-46  feet  day  fair  observer  W.  Smith. 

15*30    „  cloudy  „  „ 

15'47    „  day  bright        observer  E.  Jones. 

15 '50    „  weather  hazy  ,,  „ 

25-6      „  day  bright        observer  J.  Williams, 

and  the  most  probable  result  were  required. 

First  of  all  the  last  observation  would  have  to  be  thrown 
aside  altogether,  because  Williams  has  evidently  made  a 
mistake  in  the  very  first  significant  figure,  and  although  his 
observation  may  be  correct  in  the  last  figure  and  almost 
certainly  means  15*6,  it  is  hardly  safe  to  begin  doctoring 
results ;  it  is  safer  to  reject  any  that  thus  show  obvious  signs 
of  carelessness. 

The  other  observations  may  have  different  weights  attached 
to  them,  and  to  know  how  to  attach  weights  satisfactorily 
needs  considerable  experience,  and  experience  too  with  these 
same  observers,  because  it  may  happen  that  Jones  is  known  to 
be  a  more  trustworthy  and  exact  observer  than  Smith ;  each 
person  has  what  is  called  a  "  personal  equation,"  some  always 
tend  to  read  slightly  too  large,  others  slightly  too  small,  while 
others  cannot  be  trusted  to  more  than  say  three  significant 
figures. 

Let  us  suppose  that  an  experienced  person  decides  to  attach 


350  EASY  MATHEMATICS.  [chap. 

the  weight  3  to  the  first  observation,  the  weight  1  to  the 
second  because  of  the  clouds,  the  weight  6  to  the  next  because 
Jones  is  a  good  observer,  and  the  weight  2  to  the  next 
because  of  the  haze,  the  weighted  mean  of  the  set  of  obser- 
vations would  be  obtained  as  follows,  attending  only  to  the 
decimal  places  because  the  15  is  common  to  all : 

(3  X  -46)  +  (1  +  -30)  +  (6  X  -47)  +  (2  x  -50) 
3+1+6+2 

1-38 +  -30 +  2-82 +  1-00       5-50        ,^„^ 
= ^ .=  -^  =  -4583; 

wherefore  the  result  as  thus  determined  would  be  given  as 
15'458,  or  say  15*46  feet  to  four  significant  figures:  some 
probable  error  affecting  the  last  place. 

General  Average. 

An  average  in  general  may  be  better  expressed  thus  : 
Let  Tij  observations  give  a  result  x^ 

let    Tin  ,,  ,,  .,  Xa 


"^G  "  "  "  "^C' 

then  the  total  number  of  observations  is  Wj  +  ?i2+  •••^6'  ^^^ 
the  appropriate  weighted  mean  or  average  of  the  whole 
number  of  observations, 

W^  +  ^2  +  •  •  •  ^6 

1         vx  -       ^(nx) 

commonly  written  as  ic  =   ^\   /, 

2(71) 

the  2  being  read  "  sum." 

This  is  a  most  important  and  commonly  occurring  form  of 

average,  or  arithmetic  mean,  of  any  number  of  like  and  unliko 

quantities. 


XXXVI.]  MEANS.  351 

Geometric  Mean  of  several  Numbers. 

To  find  the  g.m.  of  two  quantities  we  multiply  them 
together  and  extract  the  square  root.  Similarly  to  find  the 
G.M.  of  three  quantities  we  should  multiply  them  together  and 
extract  the  cube  root;  and  to  find  the  g.m.  of  say  six 
quantities,  multiply  them  all  and  extract  the  sixth  root.  But 
this  would  have  to  be  done  by  logarithms ;  so  the  process  is 
better  put  into  logarithmic  form  from  the  first. 

To  find  the  geometric  mean  of  n  quantities 

Find  the  arithmetic  mean  of  their  logarithms : 

the  resulting  a  as  thus  calculated  from  its  logarithm  will  be 
the  G.M.  required. 

Example. — To  find  the  g.m.  of 

92,  100,  121,  and  89; 
look  out  their  logarithms,  add,  and  divide  by  four, 

1-9638 
2-0000 
2-0828 
1-9494 

4  I  7-9960 

1-9990  =  log  of  99-8 
Therefore  the  g.m.  required  is  99-8. 
The  A.m.  would  have  been  100*5. 

As  to  the  H.M.  of  more  than  two  quantities  I  do  not  re- 
member that  it  is  often  required  :  it  can  be  got  by  taking  the 
arithmetic  mean  of  the  reciprocals  of  the  given  set  of  numbers 
and  then  the  reciprocal  of  that. 


CHAPTER  XXXVIL 

Examples  of  the  practical  occurrence  of  Progressions 
in  Nature  or  Art. 

In  illustrating  a  subject  by  examples  there  is  a  great 
advantage  in  selecting  natural  examples  in  place  of  artificial 
ones.  Natural  examples  may  not  be  quite  so  easy  as 
artificial  ones,  but  they  are  vastly  better  worth  studying. 
Artificial  examples  are  often  easy  and  handy  for  practice, 
and  many  of  them  can  be  done  in  a  short  time,  but  a  real 
or  naturally  occurring  example  will  take  you  into  the  essence 
of  the  subject  and  is  worth  dwelling  on  long  and  steadily. 
Every  such  example  is  more  than  an  example :  it  opens  a 
chapter,  and  sometimes  needs  a  treatise. 

The  chief  instance  of  the  natural  occurrence  of  a  geometrical 
progression  is  in  the  theory  of  "  leaks  " — a  leak  of  steam  out 
of  a  boiler,  or  of  compressed  air  or  water  out  of  a  reservoir, 
of  heat  out  of  a  cooling  body,  of  electricity  out  of  a  charged 
conductor;  these  and  many  other  instances  are  all  subject 
to  the  same  mathematical  law — the  law  of  a  decreasing 
geometric  progression.     See  Chapters  XL.  to  XLIL 

A  commercial  example  of  the  occurrence  of  G.P.  is  the 
institution  known  as  compound  interest,  when  money  in- 
vested in  some  business  undertaking  is  allowed  to  supply  the 
necessities    and    the    supplementary    accessories   of   effective 


CHAP.  XXXVII.]  INTEREST.  353 

human  labour ;  on  the  strength  of  which,  under  good  manage- 
ment, the  sum  invested  increases  in  value  at  an  ascertainable 
or  arbitrarily  specified  rate,  and  may  accumulate  until  it 
becomes  a  very  large  fortune. 

Let  us  take  this  case  first,  for  it  may  perhaps  seem  simpler 
than  an  example  from  Physics. 

Interest. 

Suppose  ,£1000,  invested  in  machinery  and  wages,  enables 
a  workman  to  produce  fifty  pounds  worth  of  goods  every 
year  more  than  need  be  expended  on  advertising  the  goods, 
carrying  them  to  their  destination,  feeding  and  clothing  the 
workman,  patching  up  his  shed  and  repairing  the  machinery ; 
it  is  called  capital,  and  is  said  to  increase  at  the  rate  of 
5  per  cent,  per  annum,  the  increase  being  called  interest. 
If  the  fifty  pounds  is  taken  away  and  otherwise  utilised, 
so  that  the  original  capital  of  £1000  remains  what  it  was, 
without  increase  or  diminution  as  the  years  go  by,  it  is  called 
simple  interest,  and  is  an  example  of  arithmetical  progression. 
But  if  the  fifty  pounds  is  invested  in  improved  machinery 
and  in  extra  assistance,  in  such  a  way  that  it  too  brings  in  a 
profit  at  the  same  rate ;  and  if  this  is  steadily  done  each  year 
as  interest  accumulates,  so  that  it  is  always  added  to  the 
capital,  which  thus  goes  on  increasing ;  it  is  called  compound 
interest,  and  is  governed  by  the  law  of  geometrical  pro- 
gression. 

Every  year  the  capital  is  increased  in  the  ratio  of  yg^j  ^^ 
every  pound  at  the  beginning  of  the  year  becomes  a  guinea 
at  the  end  of  it.  If  this  is  supposed  to  go  steadily  on,  what 
will  be  the  result  after  say  1 5  years  1 

The  result  will  be  that  the  original  capital  has  been 
increased  f^ths  in  one  year,  and  this  new  capital  has  been 
increased  f  Jths  in  the  second  year,  or  the  original  capital 


354  EASY  MATHEMATICS  [chap. 

f  J  X  f  J-  in  two  years.  In  three  years  the  original  capital 
will  have  increased  in  the  ratio  ( ^ )  >  and  in  1 5  years  ( —  j    . 

Now  M  +  — J     could  be  found  by  the  binomial  theorem  if 

we  liked :  and  we  know  that  as  a  first  approximation  it  will 
bel+i|. 

Introducing  the  second  order  approximation 

-  n  .n-1    „ 

l+nx  +  — j-^ —  x% 

we  shall  find  its  value  to  this  order  of  approximation  as 

1.5     15xU      1  7x15 

^20  2         400  "^   400 

-  1-3.  105  _ 

which  equals  approximately  2 ;  that  is  to  say  the  capital  in 
15  years  will  be  by  this  operation  a  little  more  than  doubled, 
and  will  have  become  rather  more  than  .£2000. 

Now  a  quantity  which  is  frequently  doubled  becomes,  as 
we  know,  extraordinarily  large  after  the  operation  has  been 
repeated  several  times.  If  it  doubles  every  15  years,  in  60 
years  it  will  have  become  £16000,  and  in  a  century  and  a  half 
it  will  have  to  be  multiplied  by  2^^,  viz.  1024;  that  is  to  say 
the  capital  will  have  swelled  to  more  than  a  million. 

A  "  penny  "  put  out  to  5  per  cent,  compound  interest  in  the 
time  of  Caesar  would  now  theoretically  far  exceed  all  the 
material  wealth  of  the  world  in  value. 

But  though  an  approximate  calculation  of  compound  interest 
is  instructive,  there  is  no  need  to  make  it  approximate,  we  can 
calculate  it  exactly  if  we  choose.  We  have  only  to  raise  the 
ratio  of  the  G.P.,  viz.  1*05  or  1^^^,  to  the  15th  power  in 
order  to  find  the  value  after  1 5  years. 


xxxvii.]  INTEREST.  355 

This  we  should  naturally  do  by  logarithms, 
log  1-05  =  -0211893, 
15  X  log  1-05  =  -3178365  =  log  of  2-078914. 
So  the   original   capital   of  .£1000   becomes  increased   to 
£2078.  18s.  3d.  in  fifteen  years,  at  five  per  cent,  per  annum 
compound  interest. 

Suppose  we  wished  to  find  at  what  rate  of  compound 
interest  a  sum  of  money  would  exactly  double  itself  in  any 
given  time,  say  for  instance  in  12  years,  we  should  have  to 
proceed  thus: 

Let  X  be  the  rate  of  interest, 
then  (l+icp  has  to  equal  2, 

wherefore  1  +  «  =  ^^2. 

We  must  use  logarithms  to  find  the  twelfth  root  of  2. 
log  2  =  -301030, 
TVlog2  =  -025086  =  log  of  1-05945, 
which  equals       l+x,   wherefore   x  =  -05945, 
or  the  rate  of  interest  must  be  5-945,  or  nearly  six,  per  cent., 
in  order  that  doubling  may  occur  every  dozen  years. 

Let  us  see  at  what  rate  doubling  will  occur  in  thirty  years. 
^i_log2  =  -0100343 

=  log  of  1-0241, 
or  a  little  more  than  2-4  per  cent.,  about  £2.  8s.  2d.  added  to 
every  hundred  pounds  per  year. 

Any  rate  of  interest  will  double  property  if  sufficient  time 
be  allowed  to  it;  but  if  we  wanted  to  double  capital  every 
six  years,  we  should  need  a  high  rate  of  interest : 
(l+xf  =  2, 
ilog2  =  -050172  =  log  of  1-12247, 
indicating  about  12  J  per  cent. 

Approximately  therefore  the  doubling  time  and  the  rate 
of  interest  vary  inversely.     If  one  is  increased,  the  other  can 


356  EASY  MATHEMATICS.  [chap. 

be  decreased  in  roughly  something  like  the  same  proportion, 
especially  when  the  rate  is  small. 

.     That  is  apparent  at  once  if  we  expand  by  the  binomial  and 
put  it  to  double  itself  in  n  years : 

{1+xy  =*  l+nx  approximately  =  2,  say, 
wherefore  nx  =  1  approximately, 

or  n  and  a;  vary  inversely  as  one  another,  to  a  first  approxi- 
mation.    (See  Chap.  XXXIX.) 

But  then  this  first  approximation  is  exactly  simple  interest, 
it  ignores  the  x^  and  x^  and  higher  terms ;  and  it  is  just  in 
the  presence  of  those  terms  that  the  virtue  of  compound 
interest  consists. 

If  X  is  added  to  each  pound  every  year,  but  the  interest  is 
not  allowed  to  become  part  of  the  capital,  so  as  to  increase,  the 
amount  becomes  at  the  end  of  n  years  simply  (l-{-nx)  times  its 
original  value.  Thus  at  5  per  cent,  simple  interest,  so  that 
X  =  -05,  .£1000  in  1  year  will  become  1-05  times  £1000, 
or  £1050.  In  three  years  it  will  be  £1150,  and  in  20  years  it 
will  be  multiplied  by  1  +  (20  x  -05)  =  2  ;  that  is  to  say  it  will  be 
just  exactly  doubled  in  twenty  years. 

So  whereas  compound  interest  at  5  per  cent,  doubles  an 
amount  in  about  15  years,  simple  interest  does  the  same 
thing  in  20  years ;  and  the  interest  could  have  been  drawn  and 
otherwise  utilised  all  the  time. 

At  10  per  cent,  simple  interest  a  sum  would  be  doubled  in 
ten  years,  and  at  1  per  cent,  simple  interest  a  sum  would 
be  doubled  in  a  century;  whereas  at  1  per  cent,  compound 
interest,  in  a  century,  it  would  have  increased  in  the  pro- 
portion (l-01)i<>o  =  2-705,  that  is  would  have  considerably 
more  than  doubled,  though  it  would  not  have  trebled. 

The  advantage  of  compound  interest  over  simple  interest  tells 
more  at  high  rates,  for  then  the  higher  powers  in  the  expansion 
become  important;  and  therein  lies  the  difference      It  is  in- 


XXXVII. 


INTEREST. 


357 


structive  to  plot  the  two  things.  Simple  interest  increase  would 
be  represented  by  a  straight  line  law,  compound  interest  by  an 
exponential  curve,  the  two  starting  off  together,  but  ultimately 
separating  with  greater  and  greater  rapidity  as  time  passes. 


Corr 


our 
Sim 


Irtereit 


7        8        9 

Time 

Fig.  47. 


10 


12 


13        14       15 


One  is  an  A. P.,  the  other  a  G.P. ;  one  is  a  straight  line  law, 
the  other  a  compound  interest  law.  One  proceeds  by  constant 
increment,  the  other  by  constant  factor.     (See  pages  404,  etc.) 

As  an  example  consider  interest  at  the  rate  of  ten  per  cent, 
per  annum,  and  tabulate  the  value  of  <£1000  after  successive 
years  on  the  two  systems. 


Value  at  10  %  Interest. 

Time. 

Factor. 

Simple. 

Compound. 

at  start 

£1000 

£1000 

1 

1  year 

1100 

£1100 

11 

2  years 

1200 

£1210 

(1-1)2=1  + -2+ •01  =  1-21 

3  years 

1300 

£1331 

(1-1)»=1-331 

4  years 

1400 

£1464.  2s. 

(1-1)4=1-4641 

5  years 

1500 

£1610.  10s.  2id. 

(l-l)5=l  +  -5+-10+-01+0005 

6  years 

1600 

(1-1)6           [+-00001  =  1-61051 

10  years 

2000 

£2593.  16s. 

(1-1)10 

15  years 

2500 

20  years 

3000 

50  years 

6000 

100  years 

11,000 

£13,781,000 

(1.1)100 

200  years 

21,000 

£189,910,000,000 

(1-1)200 

358  EASY  MATHEMATICS.  [chap,  xxxvii. 

The  rate  of  interest  plotted  in  fig.  47  is  arbitrary,  and 
depends  upon  the  original  sum  or  'principal';  this  would 
appear  below  the  base  line,  and  the  diagram  represents  only 
its  growth.  The  curve  on  page  179  is  really  the  same  curve, 
but  to  get  the  right  aspect  it  must  be  looked  at  through  the 
paper ;  and  the  scale  of  plotting  is  unsuitable. 

The  right  hand  column  of  the  above  tabulated  numbers 
contains  the  factors  by  which  the  original  sum  must  be 
multiplied  to  give  the  amount  at  compound  interest.  It  is 
hoped  that  the  binomial  coefficients  will  be  recognised. 

It  may  be  noticed  that  during  the  length  of  an  active  life- 
time the  difference  between  simple  and  compound  interest  is 
not  extravagant,  even  at  so  high  a  rate  as  ten  per  cent ;  but 
that  if  a  sum  is  locked  up  during  a  long  minority,  or  if 
otherwise  the  interest  be  left  to  accumulate  for  a  long  time 
without  being  contemporaneously  expended,  the  growth  by 
compound  interest  becomes  enormous. 

The  operation  of  a  "  sinking  fund  "  for  the  annihilation  of 
great  debts  can  thus  be  illustrated.  But  it  is  not  to  be 
supposed  that  interest  accumulates,  without  effort,  auto- 
matically. It  is  the  result  of  human  skill,  brains,  labour, 
and  management. 


PART   11. 

MISCELLANEOUS  APPLICATIONS  AND 
INTEODUCTIONS. 


CHAPTER   XXXVIII. 

Illustrations   of  important   principles   by  means   of 
expansion  by  heat. 

Everyone  has  seen  a  telegraph  wire  by  the  side  of  a 
railway  and  observed  the  peculiar  effect  of  its  sag,  as  the 
train  passes  along,  when  such  a  wire  is  near  the  window;  it 
seems  to  be  moving  up  and  down.  A  wire  or  rope  or  chain 
stretched  between  two  posts  cannot  be  perfectly  straight,  but 
sags,  something  like  the  top  of  a  lawn  tennis  net. 

In  hot  weather  the  sag  of  a  given  span  of  wire  is  greater, 
in  cold  weather  it  is  less;  because  the  material  expands  by 
heat  and  contracts  by  cold.  Suppose  the  length  of  a  wire  on 
a  certain  span  is  I  during  a  night  of  light  frost,  then  by  noon, 
when  the  sun  has  been  up  some  time,  it  will  have  increased  to 
V  :  the  increase  of  length  being  l'  - 1. 

This  notation  for  the  same  kind  of  quantity  under  different 
circumstances,  by  means  of  the  same  letter  with  a  dash  affixed 
to  it,  is  in  constant  use,  and  must  be  grown  accustomed  to ; 
the  new  length  should  be  read  "  Z-dash  " ;  the  increase  should 
be  treated  as  a  single  quantity  and  should  be  read  "/-dash 
minus  I."  And  here  it  is  desirable  to  remark  that  all  mathe- 
matics is  intended  to  be  read,  and  that  it  is  good  and  necessary 
practice  to  read  it. 

Algebra  is  a  language — a  very  expressive  language ;  and 
although  it  appeals  primarily  to  the  eye,  it  should  be  made  to 


362  EASY   MATHEMATICS.  [chap. 

appeal  to  both  eye  and  ear,  that  is  it  should  always  be  "  read," 
if  only  to  oneself.  It  is  a  great  mistake  to  treat  it  as  a  silent 
language  and  only  to  look  at  it,  beginners  must  learn  to  read 
it ;  and  it  would  be  well  now  to  turn  back  and  read  aloud  all 
the  equations  and  other  algebraical  expressions  we  have 
employed  so  far.  It  cannot  be  done  properly  without  a  good 
deal  of  practice.  Everything  written  on  a  blackboard  by  a 
teacher  should  be  spoken  also. 

Now  consider  what  the  increase  in  length  depends  on.  In 
the  first  place  it  depends  on  the  original  length  of  the  wire. 
No  one  would  expect  a  wire  a  few  inches  long  to  elongate  as 
■much  as  one  a  few  hundred  yards  long.  It  is  only  common 
sense  to  realise  that  every  yard  of  the  wire  will  elongate  an 
equal  amount  under  the  same  conditions,  and  that  therefore 
20  yards  will  lengthen  20  times  as  much  as  one  yard ;  so  I' -I 
is  proportional  to  /.  Next  it  will  depend  on  the  change  of 
temperature,  which  we  may  treat  as  a  single  quantity  and  call 
T'  -  Tf  where  T  was  the  original  temperature  and  T'  the  new 
temperature. 

We  have  no  guarantee  that  the  lengthening  is  proportional 
to  the  rise  of  temperature,  but  it  is  a  natural  assumption  to 
begin  with,  and  will  have  to  be  corrected  if  necessary  later. 
We  can  assume  that  it  expands  twice  as  much  for  two  degrees' 
rise  as  it  does  for  one  degree,  and  ten  times  as  much  for  ten 
degrees'  rise.  There  are  only  a  few  substances  for  which  this 
is  really  and  precisely  true,  but,  for  all,  it  is  a  rough  approxi- 
mation to  the  truth,  and  will  do  for  the  present. 

Lastly,  the  lengthening  will  depend  on  the  material  of  which 
the  wire  is  composed.  If  it  were  a  copper  wire  it  would  be 
found  to  expand  more  than  if  it  were  of  iron  Every  material 
has  its  own  "expansibility,"  by  which  is  meant  the  rate  of 
expansion,  or  increase  per  unit  length  per  degree  rise  of 
temperature. 


xxxviii.]  EXPANSION  BY  HEAT.  363 

If  we  denoted  the  "  expansibility  "  of  the  material  by  k,  we 
should  be  able  to  express  in  one  line  all  we  have  so  far  said, 

thus:  I' -I  =  U{r-T) (1) 

for  this  asserts  that  the  lengthening  is  dependent  on  and 
proportional  to  three  factors,  viz.  (i)  a  constant  representing 
the  properties  of  the  material,  (ii)  the  length  of  the  piece 
of  that  material  which  is  under  consideration,  and  (iii)  the 
rise  of  temperature  or  the  warming  to  which  it  has  been 
subjected. 

Of  the  three  factors,  k  may  be  styled  a  "  constant "  to  be 
determined  by  experiment  in  the  laboratory,  a  thing  depending 
on  the  properties  of  a  material,  and  beyond  our  control,  except 
in  so  far  as  we  can  select  the  material ;  Z  is  an  arbitrary  con- 
stant entirely  in  our  control,  depending  only  on  what  we 
choose  to  attend  to.  We  might  observe  the  lengthening  of 
the  whole  of  a  span  of  wire,  or  of  any  portion  of  it,  or  we  might 
select  spans  of  different  length,  or  we  might  cut  off  a  bit  and 
attend  only  to  that.  T'  -T  represents  the  change  or  variation 
of  a  variable  quantity,  in  this  case  temperature;  it  is  some- 
times called  the  independent  variable,  for  its  changes  go  on 
independently  of  any  of  the  other  things  we  have  considered, 
and  the  change  of  length  is  dependent  on  it.  One  could  hardly 
make  the  converse  statement  and  say  that  change  of  tempera- 
ture was  caused  by  change  of  length,  even  though  the  length 
was  that  of  a  thermometer  column  (though  it  might  be  rash 
to  stigmatise  even  this  statement  as  absurd  under  all  circum- 
stances :  there  are  things  which  get  warm  when  and  by  reason 
of  being  stretched),  but  it  is  extremely  natural  to  say  that 
the  change  of  length  is  caused  by  change  of  temperature; 
so  the  cause  is  called  the  independent,  and  the  effect  the 
dependent,  variable.  By  some  people  the  names  "  principal  " 
and  "  subordinate  "  variables  are  preferred  to  "  independent " 
and  "dependent." 


364  EASY  MATHEMATICS.  [chap. 

A  change  of  temperature  might  be  caused  out  of  doors  by 
the  appearance  and  disappearance  of  the  sun,  or  by  a  change 
of  wind ;  in  a  laboratory  it  could  be  caused  by  the  application 
and  removal  of  flame,  or  of  an  electric  current,  or  of  some 
other  means  of  heating.  Anyway  it  is  to  be  observed  by 
a  thermometer  of  some  kind,  and  T  -T  may  be  considered 
as  being  measured  by  the  rise  of  the  thermometer. 

/'  —  Zis  the  change  of  the  dependent  or  subordinate  variable ; 
and  its  dependence  might  be  conveniently  indicated  by  putting 
the  two  variables  on  one  side  of  the  above  equation,  and  the 
constants  on  the  other,  as  for  instance  : 

ji'  _  ji  —  "'^ (2) 

To  emphasise  the  fact  that  the  two  terms  T'  -T  represent  a 
thing  which  is  really  one  quantity  viz.  a  warming,  a  difference 
of  temperature,  it  is  convenient  to  have  a  single  symbol  for  it; 
and  the  symbol  usually  chosen  is  an  abbreviation  for  difference 
of  temperature,  namely  87'  or  dT ;  meaning 

diff.  of  temp.  =  T'-T, 
or  diff.  T  =  T- T,   or  simply  dT  =  T -  T. 

This  mode  of  expression  is  very  handy  and  extraordinarily 
convenient.  It  can  be  applied  of  course  to  all  kinds  of 
quantities,  so  V  -I  may  be  written  " diff.  length,"  or  dl ; 
wherefore  our  two  above  numbered  forms  of  a  proportionality 
statement  become  abbreviated  into 

dl  =  kldT (1) 

and  ^  =  kl (2) 

respectively. 

These  equations  are  labelled  like  the  previous  ones  because 
they  say  precisely  the  same  thing  as  the  others  did,  and  in 
the  same  way.     It  is  generally  understood  that  the  symbol  d 


XXXVIII.]  EXPANSION  BY  HEAT.  365 

is  used  for  a  difference  only  when  it  is  an  infinitesimal 
difference.  For  finite  differences  delta  I  is  used,  or  simply  V  -  I. 
Form  (1)  gives  explicitly  the  change  of  length  in  terms  of  the 
original  length  and  the  change  of  temperature ;  form  (2)  gives 
the  ratio  of  the  changes  of  the  two  variables  in  terms  of  the 
constants — viz.  the  expansion-property  of  the  material,  and 
the  length  selected  for  observation. 

Another  way  of  writing  the  equation  is  often  useful,  in 
which  the  expansion  per  unit  length  is  explicitly  attended  to  : 
that  is  the  lengthening  by  heat  of  any  one  yard  or  foot  or 
metre  of  the  wire,  without  regarding  the  whole  wire.  To  get 
this  we  have  only  to  divide  the  length  out,  and  so  get 

j  =  kdT, (3) 

a  quantity  which  is  often  technically  referred  to  as  "  the  ex- 
pansion " ;  it  is  defined  as  the  ratio  dl/l,  and  it  is  equal  to  the 
expansibility  multiplied  by  the  rise  of  temperature. 

Let  no  beginner  suppose  that  these  various  forms  of  the 
equation  are  different.  They  are  all  essentially  the  same,  but 
they  emphasise  features  differently;  just  as  in  any  language 
a  sentence  may  be  recast  so  as  to  say  the  same  thing  with 
various  emphases.  Never  forget  to  regard  algebra  as  a 
language,  in  which  statements  of  singular  definiteness  and 
precision  can  be  compactly  made. 

Now  suppose  the  temperature  fell  instead  of  rose :  the  ex- 
pression T'  -T  would  be  negative,  and  we  might  sometimes 
choose  to  denote  the  fall  of  the  temperature  hjT-  T.  At 
the  same  time  most  substances  contract  with  cold,  and  so 
V  -I  would  also  be  negative,  and  the  contraction  could  be 
written  l-V  or  -dl;  but  usually  dl  and  dT  would  not  have 
the  negative  sign  actually  prefixed  to  them,  it  would  be 
sufficient  to  say  that  dT  and  usually  dl  are  both  negative,  for 
the  case  of  a  fall  of  temperature. 


366  EASY  MATHEMATICS.  [chap. 

Now  let  us  begin  again,  and  look  at  the  matter  afresh  and 
in  a  still  simpler  manner.  Take  a  rod  of  length  1,  that  is  to 
say  1  foot  or  one  metre  or  one  inch  in  length,  at  a  temperature 
7*,  and  warm  it  one  degree.  Its  length  will  now  be  increased 
by  an  amount  we  will  call  ^,  meaning  Tc  feet  or  metres  or 
inches,  according  to  our  choice  of  unit  length. 

Warm  it  2  degrees  and  its  increase  of  length  will  be  2^, 
and  so  on,  as  shown  thus : 

its  length  being  1  at  temperature  T, 

its  length  is         1+^  „         „         T+1 

i>         >j  1  +  2a;        „         „         T+2f 

l  +  Sk      „       „       r+3, 

l+nk       „         „         T  +  n. 

This  temperature  T  +  n  we  will  call  T'  so  that  n  =  T'-T. 

If  the  original  rod  had  been  originally  of  length  I  instead  of 
length  1  and  had  been  all  of  it  treated  alike,  every  unit  would 
have  expanded  by  the  same  amount,  so  the  final  length  would 
in  that  case  be  /(I  +nk),  which  we  may.  call  l. 

Hence  I'  =  /(1+n^) 

or  V  -I  =  Ink 

=  kl{r-T\ 
thus  arriving  at  the  same  result  as  before,  which  we  will  now 
write  in  any  one  of  four  equivalent  ways,  e.g. 

dl  =  kUT, (1) 

S=*' (^) 

f  =  WT,  (3) 

i    ^-ifc  (4) 


XXXVIII.]  EXPANSION  BY  HEAT.  367 

The  last  may  be  taken  as  a  definition  of  the  expansibility  h, 
and  shows  the  principle  of  what  we  must  do  in  the  laboratory 
in  order  to  measure  it. 

We  must  take  a  rod  or  wire  or  something  convenient  of 
the  given  material,  and  measure  either  its  whole  length  /, 
or  a  length  I  between  two  marks  or  scratches  on  it ;  then  we 
must  subject  it  to  a  measured  rise  of  temperature,  and  observe 
the  increase  in  length  of  the  chosen  portion  carefully,  with 
a  microscope  or  micrometer  by  preference. 

Then  it  is  best  to  cool  it  down  again  and  see  that  the  rod 
recovers  its  original  length ;  and  then  the  warming  can  be 
repeated,  and  the  increase  in  length  observed  again,  and  so 
on  several  times,  to  avoid  accidental  errors  and  to  get  the 
true  reading  as  nearly  as  we  can. 

Thus  the  three  required  quantities  dl,  dTy  and  /,  are  all 
measured;  and,  dividing  dl  by  /  and  by  dT^  we  get  the 
expansibility  of  the  material  as  a  result. 

These  are  not  laboratory  instructions,  and  accordingly  little 
or  nothing  shall  here  be  said  about  the  practical  mode  of  over- 
coming difficulties.  Suffice  it  to  say  that  the  readiest  mode  of 
securing  measurable  differences  of  temperature,  is  by  making 
use  of  properties  of  substances  designated  by  such  phrases  as 
boiling  oil,  molten  lead,  melting  ice,  boiling  water,  condensing 
steam,  and  the  like ;  and  that  the  chief  precautions  needed,  in 
order  to  measure  with  precision  the  expanded  length,  are 
those  which  shall  guard  the  measuring  scale,  or  standard  of 
length,  from  being  likewise  affected  by  the  high  temperature 
of  the  rod  to  which  it  has  in  some  sort  to  be  applied. 

With  this  hint  the  somewhat  elaborate  arrangements  de- 
picted in  text  books  of  Physics  can  be  appreciated. 

This  matter  has  been  gone  into  at  some  length,  because 
it  is  typical :  it  is  always  worth  while  to  master  a  type,  and 
nothing  is  gained  by  haste. 


368  EASY  MATHEMATICS.  [chap. 

Examples. 

A  bar  of  iron  10  yards  long  expands  '444  inch  when  taken 
out  of  ice  and  put  into  steam  or  boiling  water ;  what  is  its 
expansibility^  i.e.  what  is  the  increase  per  unit  length  per 
degree  for  iron;  meaning  by  a  degree  Centigrade  the  hun- 
dredth part  of  the  interval  between  freezing  and  boiling 
water,  and  by  a  degree  Fahrenheit  the  180th  part  of  that 
same  interval. 

•444 
Answer.    -— — — -  =  -0000123  per  degree  Centigrade, 

or  -00000683  per  degree  Fahrenheit. 

The  numerical  result  is  worth  remembering  as  specified  in 
the  Centigrade  scale  of  thermometry,  which  is  the  most  used 
for  scientific  purposes.  Observe  that  there  are  4  ciphers 
before  the  significant  figures,  which  happen  to  be  the  first 
three  natural  numbers,  and  so  are  quite  easy  to  remember  ;  the 
amount  is  about  IJ  in  a  hundred  thousand,  or  about  12  parts 
in  a  million :  meaning  that  iron  expands  this  fraction  of  its 
length  for  each  Centigrade  degree  rise. 

Brass  would  give  a  number  about  18  instead  of  12;  and 
zinc,  which  is  one  of  the  most  expansible  metals,  would  expand 
nearly  25  parts,  or  just  double  as  much  as  iron.  Platinum  how- 
ever, and  glass,  would  have  been  found  to  expand  only  about 
8  or  9  parts  in  a  million,  per  unit  length  per  degree  Centigrade. 

If  a  material  expanded  1  per  cent,  of  its  length  for  a  rise 
of  100°,  its  expansibility  would  be  -0001.  If  it  expanded  J 
per  cent,  for  a  rise  of  250°,  its  expansibility  would  be  -00002, 
which  lies  between  that  of  brass  and  that  of  zinc. 

Cubical  Expansion. 

It  would  be  a  mistake  to  suppose  that  a  rod  increases  in 
only  length  when  heated :  it  swells  in  every  direction,  just 


XXXVIII.]  EXPANSION  BY  HEAT.  369 

as  if  it  were  slightly  magnified.  Its  increase  in  length  is 
most  noticeable,  because  that  was  originally  its  greatest 
dimension,  but  its  increase  in  thickness  is  proportionately  as 
much.  Thus  if  a  bar  were  a  yard  long  and  an  inch  thick  it 
would  expand  in  length  36  times  its  increase  in  thickness, 

but  its  proportional  expansion,  its  y,  would  be  the  same  in 
every  direction. 

Consider  an  iron  plate  10  metres  long,  1  metre  broad,  and 
1  millimetre  thick,  and  let  it  be  warmed  406  degrees.  Its 
linear  expansion  is 

406  X  -0000123  =  -00500, 

or  five  parts  in  a  thousand  (or  the  half  of  1  per  cent.). 
So  its  increase  in  length  is  5  centimetres ; 
its  increase  in  breadth  is  5  millimetres ; 
its  increase  in  thickness  is  -005  millimetre, 

or  5  millionths  of  a  metre, 
or  5  mikrons, 

a  mikron  (sometimes  spelt  micron)  being  a  convenient  unit 
for  microscopic  work,  and  being  sometimes  mconveniently 
denoted  in  biological  books  by  the  symbol  /x.  Units  or 
standards  should  be  expressed  in  words;  symbols  are  never 
used  for  them  by  mathematicians. 

What  is  the  increase  in  area  and  in  bulk  of  such  a  plate 
when  so  heated  1 

The  first  thing  to  learn  is  that  we  must  not  take  the 
increases  and  multiply  them  together.     (Cf.  p.  293.) 

The  increase  in  area  is  not  5  centimetres  x  5  millimetres. 

The  increase  in  volume  is  not  5  centimetres  x  5  millimetres 
X  5  mikrons. 

But  the  new  area  is  1005  x  100-5  sq.  centimetres,  whence  the 
increase  in  area  is  1002-5  sq.  centimetres. 

^.B.M.  2  A 


370  EASY  MATHEMATICS.  [chap. 

The  new  volume  is  1005  x  100*5  x  -1005  cubic  centimetres 

=  103  X  102  X  10-1  X  (1-005)3 

=  104x1-015075 

=  10150-75  c.c. 
The   old   volume    was    lO^xlO^xlO"!  =  10*  c.c,   so   the 
increase  in  volume  is  150'75  c.c. 

But,  as  usual,  there  is  a  quicker  and  better  mode  of  making 
the  numerical  calculation,  by  first  treating  it  algebraically. 
Let  V  =  l{\  +kt)  be  the  new  length, 

b'  =  b(l+kt)       the  new  breadth, 
and  z'  =  z{l+  U)       the  new  thickness, 
where  t  stands  for  the  rise  of  temperature  T'  -  T. 
Then  the  new  volume  is 

I'h'z'  =  lhz{l+ktf; 
that  is,  calling  the  old  volume  V  and  the  new  volume  V\ 
V  =  V{\-\-Uf 

=  r(i  +  3^^+3F^2+^/3). 

but,  since  kt  is  a  small  quantity,  this  is,  to  a  first  approxi- 
mation, V'  =  r(l  +  Zkt\ 

or  F'-F=  3kFt  =  3kF(T'  -  T), 

or  -j^  =  3kFy  or  -pr  =  3kt; 

a  result  which  can  be  expressed  by  saying  that  the  cubical 
expansibility,  viz.  3^,  is  three  times  the  linear  expansibility. 

Similarly  the  coefficient  of  superficial  expansion  is  twice 
the  linear. 

This  is  equivalent  to  neglecting  squares  and  cubes  of  small 
quantities ;  and  for  most  purposes  that  can  safely  be  done 
in  cases  of  solid  expansion.  Hence  to  do  the  above  sum, 
very  approximately,  all  that  is  necessary,  after  observing  that 


xxxviiT.]  EXPANSION  BY  HEAT.  371 

the  linear  proportional  increase  is  '005,  is  to  say : — The 
original  area  =  10  square  metres,  so  the  increase  of  area  is 
2  X  "005  X  10  square  metres 

=  0-1  sq.  metre  ^  1000  sq.  centimetres  approximately. 
The  original  volume  is 

10  metres  x  1  metre  x  '001  metre  =  -01  cubic  metre 

=  10,000  c.c. 
so  the  increase  of  volume  is  3  x  -005  x  10,000  c.c. 
=  150  c.c.  approximately 


CHAPTER  XXXIX. 

Further  Illustrations  of  Proportionality  or  Variation. 

One  of  the  most  important  things  to  understand,  in  order 
to  be  able  to  apply  elementary  mathematics  to  simple 
engineering  facts,  is  the  law  of  simple  proportion.  Two 
quantities  are  said  to  be  proportional,  or  to  vary  as  each  other, 
if  they  are  both  doubled  when  one  is  doubled  and  if  they 
vanish  together. 

Thus  for  instance  the  stretch  of  an  elastic  and  the  force  of  its 
pull  are  proportional.  For  first  of  all  they  vanish  together  :  if 
the  elastic  is  not  pulled  it  is  not  stretched.  Secondly,  if  it  has 
been  stretched  with  a  certain  pull,  and  you  double  the  pull, 
the  stretch  also  will  be  found  to  be  doubled.  You  can  try  this 
by  hanging  up  an  elastic  or  a  spiral  spring  and  loading  it  with 
different  weights.  As  the  weight  increases,  the  stretch 
increases,  and  in  a  simply-proportional  manner.  This  is 
the  principle  of  a  spring  balance. 

Take  all  the  load  off,  and  the  pointer  returns  to  zero, 
indicating  no  stretch.  Observe,  it  is  not  the  length  of  the 
elastic  that  is  proportional,  for  that  does  not  become  zero,  nor 
is  it  doubled  when  the  load  is  doubled,  but  it  is  the  stretch  or 
increase  of  length  that  is  proportional  to  the  load.  Suppose 
however  there  had  been  some  irremovable  load  on  all  the  time, 
as  indeed  there  is,  for  the  spring  or  elastic  itself  has  a  trifle  of 
weight  of  its  own,  how  do  we  allow  for  that  %     Answer :  by 


CHAP.  XXXIX.]  VARIATION.  373 

always  attending  to  the  variable  part  of  the  load  only,  just  as 
we  attend  to  the  variable  part  of  the  length  only.  The  "  load  " 
must  really  signify  the  load  added  or  subtracted ;  and  the 
stretch  corresponding  thereto  signifies  the  increase,  or  it  may 
be  the  decrease,  of  length  which  accompanies  that  variation  of 
load  ;  so  that  instead  of  saying  the  length  I  varies  as  w  the 
load,  which  is  not  true,  we  ought  to  say,  change  of  length 
varies  as  change  of  load,  or  dl  varies  as  dw^  which  is  quite 
true ;  unless  indeed  the  spring  is  overloaded  and  permanently 
strained  or  injured  so  that  it  cannot  recover;  or,  in  other 
words,  unless  it  is  not  perfectly  elastic.  So  long  as  it  is 
perfectly  elastic,  the  law  of  simple  proportion  holds ;  and  the 
test  of  whether  it  is  perfectly  elastic  or  not  is  to  see  if  it  can 
completely  recover  when  the  load  is  removed. 

Some  substances  stand  a  great  deal  of  loading,  such  as  steel ; 
some  stand  only  a  little  without  giving  way,  like  glass  or 
copper ;  and  some  stand  hardly  any,  or  none,  like  lead  or 
straw  or  dough. 

There  are  two  methods  of  giving  way,  one  by  breaking,  like 
glass,  the  other  by  permanently  bending,  like  lead.  There  is 
plenty  to  be  learnt  about  all  these  things,  but  the  time  for 
learning  them  will  come  later.  All  that  we  have  to  note  at 
present  is  that  the  law  of  proportion  is  not  to  be  expected  to  be 
verified  when  the  substance  experiences  a  permanent  set,  or 
deformation  of  any  kind,  from  which  it  cannot  recover ;  and  of 
course  not  when  it  is  broken. 

The  law  holds  "within  the  limits  of  elasticity,"  and  it  is 
known  as  "Hooke's  law,"  because  that  great  and  ingenious 
man  Robert  Hooke  experimented  on  it  and  emphasised  its 
simplicity  and  convenience  more  than  two  centuries  ago. 

You  may  think  that  it  is  so  simple  as  not  to  be  much  of  a  law 
of  nature;  but  you  will  find  that  all  the  most  fundamental  laws 
are  simple.      Simplicity  and  importance  may  quite  well  go 


374 


EASY  MATHEMATICS. 


[chap. 


together,  though  there  is  perhaps  no  necessary  connexion 
between  them. 

Now  take  another  example  of  simple  proportion.  Let  a 
balloon  ascend  at  a  perfectly  steady  pace  from  the  ground. 
Its  height  is  proportional  to  the  time  which  has  elapsed  since 
it  started.  In  one  second  it  went  up  let  us  say  a  yard.  In  a 
minute  it  will  go  up  60  yards,  and  in  an  hour  3600  yards, 
provided  the  same  upward  speed  was  all  the  time  maintained. 
If  it  went  sometimes  faster  and  sometimes  slower,  the  simple 
proportionality  would  not  hold.  The  height  and  the  time 
vanish  together,  for  we  began  to  reckon  time  at  the  instant  it 
was  let  go,  and  we  were  careful  to  measure  height  always  to 
that  same  point  of  the  balloon  which  touched  the  earth  at  the 
moment  of  letting  go. 

But  now  take  an  example  where  simple  proportion  does  not 
hold  between  two  connected  variable  quantities. 


20    Years 


Fia.  48. 


The  height  of  a  child  depends  upon  his  age,  and  increases 
with  his  age,  but  it  is  not  proportional  to  it :  in  other  words, 
his  height  does  not  vary  with  his  age,  in  the  technical  sense. 


XXXIX.] 


STRAIGHT  LINE   LAW. 


375 


For  first  of  all  they  do  not  vanish  together :  the  child  had  some 
length  when  he  was  born ;  and  next,  they  are  not  doubled 
together.  A  child  has  not  twice  the  height  at  6  that  he 
had  at  3. 

If  we  were  to  plot  age  and  height  together,  it  would  be 
instructive,  and  the  result  might  be  something  like  fig.  48, 
where  age  is  plotted  horizontally  and  height  vertically. 

The  point  0  is  called  the  origin,  and  represents  the  epoch 
of  birth.  If  the  curve  passed  through  this  point  0,  zero 
initial  length  would  be  signified ;  so  the  curve  does  not  pass 
through  it,  but  starts  above  it  at  the  infant's  length  at  birth  : 
say  fourteen  inches 

Such  a  curve  is  not  simple  proportion  at  all.  It  is  easy 
enough  to  understand,  but  the  law  represented  by  the  curve  is 
not  a  simple  one. 

Simple  proportion  would  be  represented,  on  the  same  plan, 
by  a  straight  line  through  the  origin ;  as  for  instance  if  the 
stretch  and  the  load  of  an  elastic  thread  or  spiral  spring  were 
plotted  :  they  vanish  together.     (Fig.  49.) 


But  suppose  the  two  quantities  did  not  vanish  together,  we 
might  still  have  them  plotted  as  a  straight  line.     For  instance. 


376 


EASY  MATHEMATICS. 


[chap. 


the  length  of  a  rod  at  different  temperatures  (or  the  total 
length  of  a  piece  of  elastic  under  different  loads) : 


Temperature 
Fig.  60. 

and  this,  though  it  is  not  exactly  simple  proportion,  is  the 
next  thing  to  it,  and  is  sometimes  called  "a  straight  line 
law."     (Fig.  50.) 

It  can  be  made  simple  proportion  by  deducting  a  constant, 
by  deducting  the  original  length  for  instance,  I'  -I  =  kit, 
whereas  if  the  length  I  had  not  been  deducted  it  would  have 
been  expressed  by  I'  =  1  +  Ikt, 

and    this    is    characteristic    of    a 
straight  line  law. 

In  general  a  straight  line  law  is 
represented  by 

y  =  a  +  hx, 
whereas  simple  proportion  would  be 

y  =  bx 
without  the  constant  a. 

So  by  subtracting  the  constant  a 
a  straight  line  law  can  always  be 
expressed  as  simple  proportion.  So  it  can  if  we  attend  to 
changes  only.  For  let  y  become  y',  and  x  become  x,  while  a 
and  h  remain  constant  all  the  time,  we  should  have 


Fio.  61. 


XXXIX.]  PARABOLIC   LAW.  377 

y  =  a  +  hx, 

y'=  a  +  hx\ 
y'-y  =  b{x'-x), 
or  dy  =  hdXf 

and  the  a  has  disappeared.  The  two  differerices,  dy  and  dxy  are 
simply  proportional;  for  they  increase  together  in  a  constant 
ratio,  and  they  vanish  together :  one  cannot  become  zero 
without  the  other. 

The  weight  of  a  boy  is  not  represented  by  a  straight  line, 
even  if  his  weight  at  birth  is  deducted.  His  law  of  growth  is 
not  a  straight  line  law  but  a  more  complicated  law :  it  could 
be  plotted  as  a  curve,  from  successive  observations. 

Any  law  can  be  expressed  by  a  curve  ;  thus  we  might  have 
a  parabolic  law,  meaning  that  the  curve  of  plotting  is  a  para- 
bola as  nearly  as  can  be  told. 

A  parabolic  law  is  expressible  in  algebra  thus, 
y  =  a-hbx  +  cx'^j 
and  it  would  be  instructive  for  children  to   plot   the   curve 
represented  by  this  equation,  and  see  what  it  looks  like.     To 
carry  out  the  plotting  we  must  be  told  the  values  which  are 
to  be  attributed  to  a,  b,  and  c,  and  can  arrange  the  scale  to  suit. 

Thus  let  a  =  4,  b  =  1,  and  c  =  J. 

Make  a  table  of  corresponding  successive  values  of  x  and  y. 


When  X  =  0, 

y  =  a,             or  in 

this  case  4 ; 

a:=  1, 

y  =  a  +  b  +  c,  or  in 

this  case  5 J ; 

x  =  2, 

y  =  a  +  2b  +  4c, 

or         7; 

x  =  S, 

y  =  a  +  3b  +  % 

or         91; 

a:  =  4, 

y  =  a  +  46+16r. 

or         12; 

x  =  5, 

y  =  a  +  5b  +  25r, 

or         15J; 

x  =  6, 

y  =  a  +  6b  +  3Qc, 

or         19; 

etc. 


378  EASY  MATHEMATICS. 

Plot  this,  and  it  looks  somewhat  thus  : 


[chap. 


15- 


10- 


3        4 

Fig.  52. 


But  we  need  not  necessarily  limit  ourselves  to  the  positive 
side  of  the  vertical  axis ;  we  might  ascertain  and  plot  values  of 
y  when  x  is  negative,  otherwise  the  curve  is  incomplete.  You 
could  hardly  tell  it  was  a  parabola  from  its  appearance  so  far. 


When  X  = 

-1, 

y  =  a-b  +  c        = 

3^; 

X  = 

_  9 

-'5 

y  =  a-2b  +  4:C    = 

3; 

X  = 

-3, 

y  ==  a-Sb  +  9c    = 

3i; 

X  = 

-4, 

y  =  a  -  4b  +  16c  = 

4; 

X  = 

-5, 

y  =  a-bb  +  25c  = 

5i; 

X  = 

-6, 

2/  =  a- 66  + 36c  = 

7; 

X  = 

-7, 

y  = 

9i; 

X  = 

-8, 

y  = 

12; 

X  = 

-9, 

y  = 

15i; 

X  = 

-10, 

y  = 

19;  etc., 

XXXIX.] 


PLOTTING  A  CURVE. 


379 


every  value  on  this  side  being  equal  to  a  certain  value  for 
another  x  on  the  other  side.  The  whole  curve  is  symmetrical 
about  a  vertical  axis  through  x  =  -2. 


The  dotted  line  is  called  the  axis  of  the  parabola.     (Fig.  53.) 

Another  example  of  what  may  be  a  straight-line-law  is  the 
slope  or  gradient  of  a  railway.  At  a  certain  place  it  may  be 
said  to  rise  1  in  30,  for  instance ;  meaning  that  if  you  go 
30  feet  along  the  railway  you  have  ascended  1  vertical  foot  j  if 
you  go  30  yards  you  have  ascended  1  vertical  yard,  and  so  on. 

A  uniform  gradient  is  naturally  plotted  by  a  straight  line, 
and  if  the  vertical  height  is  called  y  while  the  horizontal 
distance  is  called  a:,  the  gradient  is  approximately  denoted  by 
— .  Not  exactly,  because  the  denominator  is  usually  measured 
along  the  sloping  railway,  and  not  horizontally. 


380 


EASY  MATHEMATICS. 


[chap. 


slope  are  -^  and 
dx 


Either  way  of  measuring  the  gradient  is  a  good  method; 
and  sometimes  one  is  used,  sometimes  the  other.  If  the 
slant  distance  is  called  ds^  the  two  chief  ways  of  measuring 

-,  respectively. 

Except  when  the  slope  is  steep,  the  difference  between  these 

two  methods  of  measuring  it 
is  not  marked  : — the  connexion 
between  the  two  methods  is 
easily  shown  by  a  diagram. 

A  considerable   but   feasible 
slope  for  an  ordinary  railway 

would  be  a  gradient  of  1  in  30,  that  is  to  say  -^  =  — , 

Often  it  is  not  more  than  1  in  100.  The  actual  gradient 
is  frequently  written  up  on  low  posts  by  the  side  of  the 
line. 

But  take  the  case,  impossible  for  a  practical 
railway,  where  the  slope  is  45°,  which  would 
be  a  steep  mountain  side,  dy  and  dx  are  in 
that  case  equal,  and  ds  =  ^2  either  of  them, 

dy       _      ,  dy        1 

so  j^  =  1,  whereas  -5-  =  —p.. 
dx  ds      J'2 

Take  the  case  of  a  precipice  or  a  steeple,  almost 
vertical,  so  that  dx  is  extremely  small. 

dy  and  ds  are  now  nearly  equal  and  dx  is  nearly 
0  ;  in  the  limit  quite  zero. 

So  approximately,  for  an  angle  nearly  90°, 


dy  _  , 


and  zr-  =  7^ 
dx      0 


or   at   any  rate   a   very  great  number;   in  the   limit   quite 
infinite. 


XXXIX.] 


GRADIENT. 


381 


Take  the  case  of  that  famous  right-angled  triangle  with 
commensurable  sides,  and  express  the  slope  of  its  hypothenuse  : 
^  _  3         dy  _Z 
dx  ~  4'        ^s  ~  5' 


5X 

3                                    . 

4                                                 dx 
Fio.  57.                                                Fia.  58. 

irse  always  d&^  =  d^  +  dy'^^ 

dy_ 
dy              dy                      dx 

ds      J(dx^  +  dy^)         ji+f^l)' 

so  that 


It  is  often  convenient  to  measure  slope  in  yet  another  way, 
viz.  by  the  angle  of  slope,  which  we  denote  by  a  or  ^ ;  then  the 

ratio  of  the  height  to  the  slant  length,  -j-^  is  called  the  sine  of 

the  angle,  and  is  written  sine  0  or  sin  ^,  while  the  ratio  of  the 

height  to  the  base,  -^,  is  called  the  tangent  of  0,  and  is  written 

tan  Q,     So  we  see  from  the  above  that  an  angle  whose  sine 
is  I,  or  -6,  has  a  tangent  whose  value  is  f  or  -75 ; 

also  that  the  sine  of  45'  is  -p, 

while  the  tangent  of  45"  is  1, 

and  that  sin  90°  =  1,  while  tan  90°  =  oo.     (Cf.  fig.  56.) 

On  the  other  hand,  when  the  slope  is  very  slight,  the 
sine  and  tangent  are  about  equal,  and  it  does  not  much 
matter  which  measure  of  the  slope  we  employ.     (Cf.  fig.  54.) 


382 


EASY   MATHEMATICS. 


[chap. 


Ultimately  when  the   slope  vanishes   they  also  vanish,  and 
vanish  on  terms  of  equality,  so  that  sin  0  =  tan  0  =  0. 

Here  is  a  case  where  two  things  vanish  together  but  are  by 
no  means  proportional ;  they  are  approximately  proportional, 
or  indeed  equal,  for  small  angles,  but  the  tangent  increases 
faster  than  the  sine ;  and  as  the  angle  grows,  it  increases  very 
much  faster ;  so  that,  by  the  time  the  sine  has  reached  unity, 
the  tangent  has  gone,  with  a  rush,  to  infinity. 

Plotting  them  they  would  look  thus  : 
where  one  curve  represents  the  sines, 
and  the  other  the  tangents,  of  angles 
from  0°  to  90° 


Fia.  59. 


It  may  be  worth  showing,  even  at  this  stage,  what  suggested 
these  curious  names.     The  name  "  tangent "  especially  sounds 


XXXIX.] 


MEASURES  OF  ANGLE. 


383 


curious  when  applied  to  a  ratio.  The  idea  arose  from  drawing 
a  circle  round  an  angle  and  seeing  all  the  different  ways  in 
which  it  might  be  measured. 

In  this  figure  the  angle  is  at  C,  and 
AD  is  a,  bit  of  a  circle  with  centre  C. 
This  figure  long  ago  suggested  a  bow 
and  arrow,  hence  EB  was  called  the 
sagitta,  and  ABD  was  called  the  arc  :  the 
string  A  ED  is  called  the  chord,  and  half 
of  it  under  certain  restrictions  is  called 
the  sine,  presumably  because  the  point  E 
of  the  string  is  pulled  to  the  breast  before 
releasing  the  bow.  The  tangent  can  then 
be  measured  as  part  of  a  line  drawn  through  B,  touching  the 
circle,  when  the  circle  is  drawn  of  unit  radius. 

Let  CA  or  CB  equal  1  on  any  arbitrary  scale;  then  C  or 
ACB  is  the  angle,  a, 

AE  is  its  sine, 
AB  is  its  arc, 
EB  is  its  sagitta, 
and  BF  is  its  tangent, 

always  provided  CA  or  OB  are  equal  to 
1,  that  is  taking  the  radius  of  the  circle 
as  unity. 

The  size  of  the  circle  is  quite  arbitrary : 
any  length  whatever  may  be  chosen  as  a 
unit  of  measurement. 

But  it  is  desirable  to  bear  in  mind 
that  angles  are  not  measured  by  length 
but  by  ratio;  and  accordingly  the  state- 
ment that 

AB 


Fio.  61. 


angh 


arc 


radius       AC' 


384  EASY   MATHEMATICS.  [chap. 

or  intercepted  part  of  arc  -f  radius,  is  a  better  statement  than 

to  say  that  i  ,  i-  i 

''  angle  =  arc   when    radius  =  1. 

And  to  say  that 

,         intercepted  perpendicular      AE 

sinanffle  =  ~ ^^—- =  —r-y 

°  radius  A  C 

is  better  than  saying  that  it  equals  AE  when  AC  equals  1. 
So  also 

f    i  p  -  intercepted  portion  of  tangent  _  FB 
°  ~  radius  ~  BC' 

It  may  be  worth  while  also  to  state  here  that  the  length  of 
that  boundary  line  of  the  angle  which  cuts  the  circle  and  is 
produced  to  meet  the  tangent,  viz.  CF,  is  called  the  "  secant " 
when  the  circle  is  of  unit  radius ;  or  in  general,  dividing  by 
the  radius,  ^r» 

secant  of  angle  (7,  or  sec  C,  =  y^. 

These  different  fractions  or  ratios  are  all  measures  of  the 
angle :  they  are  quite  independent  of  the  size  of  the  circle 
or  of  any  linear  dimension  whatever.  They  indicate  angular 
magnitude  alone. 

In  any  right-angled  triangle,  if  the  length  of  the  hypothenuse 
is  called  r,  and  the  angle  at  the  base  be  called  d,  then  the 
length  of  the  base  is,  by  definition  of  cosine,  equal  to  r  cos  6 ; 
that  is  to  say,  it  is  the  length  r  multiplied  by  a  proper  fraction, 
which  fraction  is  called  the  cosine  of  the  angle  0. 


The  base  may  be  thought  of  as  the  projection  of  r  on  to  a 
direction  inclined  at  the  angle  ^  to  it ;  the  shadow  as  it  were 


XXXIX.] 


MEASURES  OF  ANGLE. 


385 


of  a  slant  rod  of  length  r  thrown  by  vertical  rays  of  light  on  a 
horizontal  ground  (Hg.  64). 

If  the  projection  were  made  in  a  direction  at  right  angles  to 
the  first,  then  the  projection  is  r  sin  6^  provided  6  still  means 
the  same  angle  as  before.    So  that  the  sides  in  any  right  angled 


^ 


Fio.  65. 

triangle  are  related  as  in  the  diagram  (fig.  62);  and  it  is 
obvious,  by  definition  of  the  tangent  ratio,  that 

sin  <9  ^ 

7,  =  tan  u. 

cos  t) 

It  also  follows,  by  Euc.  I.  47,  that 

(r  sin  (9)2 +  (r  cos  (9)2  =  r\ 

or  that  (sin  Bf  +  (cos  df=  1, 

a  fundamental  identity. 

One  peculiarity  of  angular  magnitude  is  that  it  is  un- 
magnifiable.  Look  at  an  angle  through  a  magnifying  glass, 
and  though  its  sides  lengthen,  the  angle  continues  constant. 
A  right-angle,  for  instance  the  corner  of  a  book,  continues  a 
right-angle,  and  45°  continues  45°.  A  degree  is  always  the 
360th  part  of  a  circle,  however  big  the  circle ;  a  quarter  of 
a  revolution,  or  a  right  angle,  is  always  90° ;  and  so  on.  The 
size  of  the  divisions  of  a  circle  change  when  it  is  magnified, 
but  their  number  remains  constant. 

Number  is  another  thing  which  is  unmagnifiable.  Magnify 
3  oranges,  the  oranges  look  bigger,  but  they  still  look  3. 

So  when  a  plate  expands  with  heat,  if  it  is  uniform,  any 
L.E.M.  2  b 


386  EASY   MATHEMATICS.  [chap. 

angles  it  may  have  remain  constant.  If  there  were  a  hole 
in  the  plate,  the  hole  would  expand  just  as  if  it  had  been 
filled  with  solid ;  its  boundary  line  might  have  been  drawn 
with  ink  on  a  similar  solid  plate.  Everything  expands 
with  heat  as  if  it  were  looked  at  through  a  weak  mag- 
nifying glass.  So  a  hollow  space  is  not  encroached  upon 
by  expanding  walls,  but  is  enlarged  as  if  it  were  full  of 
substance.  A  hollow  bulb,  for  instance,  has  a  greater  capacity 
when  heated  than  it  had  before.  It  may  not  necessarily  hold 
more  fluid,  not  more  weight  or  mass  of  fluid,  for  the  fluid 
might  expand  still  faster  than  the  solid,  but  it  holds  a  greater 
volume.  A  thermometer  bulb  containing  mercury  is  in  this 
predicament ;  and  what  we  observe,  when  the  thermometer 
rises,  is  the  apparent  expansion  of  the  whole  bulb-full  of 
mercury  swelling  in  the  only  direction  open  to  it,  viz.  in  the 
narrow  stem.  It  is  called  "apparent"  expansion  because  it 
represents  what  is  visible,  viz.  the  excess  of  the  expansion 
of  the  mercury  over  that  of  the  vessel  which  contains  it. 
If  the  vessel  expanded  more  than  the  liquid,  the  rise  in  the 
stem,  indicating  the  apparent  expansion,  would  be  negative ; 
but  this  is  hardly  a  possible  case  in  practice,  for  all  liquids 
expand  more  than  any  solid  Nevertheless  the  true  or 
absolute  expansion  of  a  liquid  is  always  greater  than  it 
appears  to  be,  unless  we  could  observe  it  in  a  vessel  which 
did  not  expand  with  heat. 

Let  V  be  the  volume  of  the  vessel,  and  a  the  expansibility 
of  its  material,  that  is  to  say  its  increase  in  bulk  per  unit 
original  bulk  per  degree  rise  of  temperature,  then  the  total 
swelling  for  any  rise  of  temperature  I"  -  T  is 
v'  -V  ==  av{T'  -T\ 
or  dv  =  avdT, 

or  —  =  adT; 


xxxTX.]  VARIATION.  387 

or,  in  words,  the  proportional  expansion  equals  the  change 
of  temperature  multiplied  by  the  expansibility.  If  we  assume 
simple  proportionality  between  change  of  volume  and  change 
of  temperature,  it  is  the  same  thing  as  assuming  that  a  is  a 
constant ;  and  in  that  case  the  expansion  is  said  to  vary  with 
the  temperature. 

The  term  "  varies  with  "  or  "  varies  as  "  is  a  technical  term, 
and  is  understood  to  mean  "is  proportional  to."  The  latter 
is  really  the  better  expression,  for  in  common  language  two 
things  can  vary  or  change  together  without  being  pro- 
portional, like  the  age  of  a  child  and  its  weight,  or  the 
amount  of  sunshine  and  the  cheapness  of  corn,  or  the  height 
of  a  look-out  man  on  board  ship  and  the  distance  of  his 
horizon,  or  the  amount  of  oil  consumed  in  a  lamp  and  the 
brightness  of  the  consequent  light. 

But  the  term  "varies  as"  or  "varies  with"  is  understood 
to  indicate  more  than  merely  changing  together:  it  means 
that  they  vary  in  a  simply  proportional  manner,  so  that 
if  one  is  doubled  or  trebled  or  increased  1  per  cent., 
the  other  is  doubled  or  trebled  or  increased  1  per  cent, 
too. 

When  y  varies  as  x,  in  the  technical  sense,  it  can  be  written 
y  ^  x;  where  oc  is  a  mere  symbol,  not  much  used,  to  denote 
"  varies  as." 

Or  it  may  be  written  y  :'.x^  and  read  y  is  proportional  to  x ; 
or  it  may  be  written,  the  ratio  «/ :  a;  is  constant ;  or  y  -=- «  or 

^  is  constant,  equal  to  h  say  ;  or  y  =  hx. 

This  last  is  the  simplest  and  most  satisfactory  mode  of 
stating  simple  proportionality,  h  being  understood  to  be  a 
constant,  that  is  something  not  at  all  dependent  on  the  value 
of  X  and  y,  which  are  the  variables. 

A  straight   line   law  is  slightly  more  general  than  this : 


388  EASY   MATHEMATICS.  [chap. 

it  includes  the  main  idea  of  proportionality,  but  it  exempts 
from  the  necessity  of  vanishing  together ; 

y  =  a  +  hx 
is  the  typical  straight  line  law. 
Either  can  be  expressed  as 

dy  =  bdx, 

for  when  we  attend  to  variations,  the  constant  term  a  has 
no  influence,  and  so  disappears.  The  constant  factor  b,  which 
is  only  part  of  a  term,  by  no  means  disappears ;  b  represents 
the  rate  of  increase  of  y  with  respect  to  x;  for  instance  it 
represents  the  slope  of  the  line,  being  the  change  of  elevation 
per  unit  step  along  the  base;  it  is  in  fact  the  'tangent'  of 
the  slope. 

The  next  more  general  law  is  the  parabolic  law. 

y  =  a  +  bx  +  cx^j (68  i.) 

or  it  might  be  y  =  a-bx  +  cx^, (66.) 

or  y  ^  a  +  bx- cx\ (68  ii.) 

or  y  =  a-bx-  cx^, (68  iii.) 

all  of  them  parabolic,  with  different  appearances. 
The  slope  of  such  a  curve  is  of  course  not  constant. 
To  find  an  expression  for  the  slope,  we  must  take  two 
points  near  together  and  compare  the  vertical  with  the 
horizontal  step,  that  is  find  the  dy  corresponding  to  a  given 
small  dx.  To  do  this  we  let  x  change  to  x'  and  y  to  y',  and 
then  write  the  relation  once  again  for  the  changed  values 

y'  =  a  +  bx'  +  cx'^, 
and  now  subtract  the  old  value  from  the  new,  so  as  to  get  the 
difference,        y  -y  =  {a-a)  +  b{x'  -x)  +  c{x'^ - x^) 
=  {b  +  c(x'  +  x)}(x'  -x); 
.    diff.  y       1       /  ,      . 

•   .        -..rn  =    b  +  C(X    +X). 

diff.  X  ^         ' 


XXXIX.] 


SLOPE. 


389 


Now  if   the  step   is  made  quite  small,   the  x  and  ic'  are 
extremely  nearly  equal,  so  that  it  matters  little  whether  we 


Fio.  66. 


write  y^  ■\-x  or  2ic  or  2a;' ;   and  in  the  limit  when  the  step  is 
infinitesimal  they  become  actually  equal,  and  then 
dy 


dx 


=  J  +  '2cx, 


It  is  not 


and  this  is  the  gradient  of  the  curve  at  any  point, 
constant,  but  it  follows  a 
straight  line  law. 

The  rate  of  change  of 
the  angle  of  slope  may 
be  called  the  curvature. 
When  the  slope  is  constant, 
as  in  a  straight  line,  the 
curvature  is  nothing,  but 
when  the  slope  changes,  as  in  the  last  case,  the  curvature  can 
be  measured  as  the  rate  of  change  of  the  angle  of  slope  per 
unit  step  along  the  curve.     Suppose  the  gradient  or  angle  of 


Fig.  67. 


390  EASY  MATHEMATICS.  [chap. 

slope  at  any  point  is  denoted  in  angular  measure  by  <^,  then  the 

curvature  could  be  defined  as  -^,  and  that  is  its  usual  measure. 

as 

Another  method  of  measuring  it,  which  seems  simpler  but 
is  not  so  satisfactory,  is  to  indicate  the  slope  by  the  tangent  of 

the  angle  <^,  that  is  by  the  gradient  -j-,  which  we  may  denote 

by  a  single  letter  g,  and  then  to  denote  the  curvature  by  the 

rate  of  change  of  gradient  per  horizontal  step,  that  is  by  -t-; 

On  this   plan   the   value   of   the   gradient   for   the   above 
parabola  is  g  =  b  +  2cx, 

and  the  rate  of  change  of  gradient  is 

dx  -  ^'^ 
because  if  x  changes  to  x'j  g  changes  to  ^'  =  J  +  lex',  and  when 
you  come  to  reckon  the  difference  ratio,  the  6's  go  out : 

g'  -g  ^  2ca;^  -  2ct  ^  ^^ 

X  —  X  X       X 

It  is  not  customary  or  necessary  thus  to  introduce  a  new 

d^-y 

symbol  g -,  it  is  neater  to  express  -^  as  — — ,  that  is  to  say  as 

70  O/X  OjX 

y|;  and  this  it  is  which  is  equal  to  2  c. 

CLX 

So,  in  the  original  parabolic  expression 
y  =  a  +  hx  +  cx'^, 
the  meaning  of  the  constant  c  is  half  the  rate  of  change  of 
gradient,  which  is  a  principal  term  in  the  curvature;  the 
meaning  of  the  constant  h  is  the  slope,  or  gradient  itself, 
especially  the  slope  at  the  place  where  x  is  0,  that  is  to  say 
where  it  cuts  the  axis  of  y  ;  and  the  meaning  of  a  is  the  height 
at  which  the  curve  cuts  the  axis  of  y,  that  is  to  say  the  inter- 
cept on  that  axis.  Compare  the  equations  on  page  388  with 
the  curves  drawn. 


XXXIX.] 


ALGEBRA  AND  GEOMETRY. 


391 


392 


EASY  MATHEMATICS. 


[chap. 


Observe,  with  regard  to  slope,  that  when  the  curve  slopes 
upward  where  it  cuts  the  vertical  axis,  as  in  fig.  68  i.  or  ii.,  the 
coefficient  b,  which  measures  the  slope  there,  is  positive;  but 
when  it  slopes  downward  as  in  fig.  68  iii.  the  term  involving  b  is 
negative.  If  the  curve  cut  the  vertical  axis  without  slope,  or 
horizontally,  the  term  involving  b  would  disappear,  and  in 
such  a  case  the  parabola  would  be  represented  by  y  =  ai- cx^ 
(fig.  69). 


Pio.  69, 


If  the  curve  cut  off  no  length  on  the  axis  of  y,  that  would 
be  indicated  by  the  non-existence  of  the  constant  a,  so  that 
that  parabola  would  be  written  y  =  cx^  (fig.  70), 


Fio.  70. 


and  this  is  the  simplest  expression  for  a  parabola  possible. 


XXXIX.]  ALGEBRA  AND  GEOMETRY.  393 

Observe  that  it  is  the  same  curve  all  the  time ;  it  is  only 
shifted  with  respect  to  the  axes  by  the  different  values  which 
can  be  attributed  to  the  constants. 

c  is  the  curvature  term,  and  when  c  is  positive  it  curves 
upwards,  like  68  i. ;  when  c  is  negative  it  curves  downwards, 
like  68  ii.  or  iii. 

y  =  -cx^  would  be  like  this  (fig.  71) : 


Fio.  71. 

If  wo  want  the  parabola  to  look  like  this  (fig.  72) : 


FiQ.  72. 


we  have  only  to  turn  it  through  a  right-angle,  that  is  to  say, 
interchange  the  axes  of  x  and  y,  and  write 


X  =  cy\ 


394  EASY  MATHEMATICS. 

or  if  we  wrote  x  =  -cy'^  it  would  look  like  this 


[chap. 


Fig.  73. 


If  we  wrote  x  =  a -{-cy^  it  would  become  like  this 


Fio.  74. 


where  the  x  intercept  is  a. 


ALGEBRA  AND  GEOMETRY. 


395 


If  we  wrote  ic  =  a  +  &y  +  c/  it  would  slope  up  where  it  cuts 
the  axis. 


Fig.  76. 


If  we  had  x  =  a-hy  +  cff^  it  would  slope  down  at  the  foot, 
like  this : 


Fig.  76. 


396  EASY  MATHEMATICS.  [chap. 

li  X  =  -a  +  cy^  the  curve  would  cut  the  x  axis  on  the  left : 


Pig.  77. 

If  we  introduce  a  term  containing  x^  as  well  as  a  term 
containing  y^,  and  if  necessary  a  term  xy,  we  can  tilt  the 
parabola  in  any  desired  direction  and  place  it  anywhere  in  the 
plane  :  though  in  these  cases  there  is  a  risk  that  it  may  cease 
to  be  a  parabola;  and  if  we  introduced  a  term  x^  or  y^  its 
parabolic  character  is  bound  to  be  spoilt ;  just  as  the 
introduction  of  either  x^  or  y^  destroyed  the  straightness  of  the 
line  y  =  a  +  bx. 

The  next  step  towards  algebraic  complexity  is  what  is  called 
the  cubic  parabola     y  =  a  +  bx  +  cx'^  +  dx^, 
where  again  the  coefficients  or  constants,  a,  J,  c,  d,  may  be  any 
of  them  positive  or  any  of  them  negative. 

The  beginner  can  plot  this  and  see  what  it  looks  like, — 
proceeding  after  the  same  fashion  as  before  ;  that  is  attributing 
any  arbitrary  value,  positive  or  negative,  to  the  four  constants 
ahcdj  and  then  reckoning  the  value  of  y  for  different  values  of  x ; 
afterwards  plotting  them  to  any  convenient   scale — remem- 


XXXIX.]  INVERSE   VARIATION.  397 

bering  that  the  horizontal,  scale  and  the  vertical  scale  need  not 
necessarily  be  the  same,  but  may  be  chosen  independently,  to 
suit  the  convenience  of  the  draughtsman. 

Inverse  Variation. 

It  frequently  happens  that  two  quantities  are  connected  in 
such  a  way  that  one  increases  when  the  other  decreases.  For 
instance,  the  plentifulness  of  a  commodity,  say  corn,  and  its 
price.  In  a  year  of  good  harvest  the  price  of  wheat  drops. 
During  a  famine  the  price  rises.  It  might  happen  that  the 
total  money  to  be  obtained,  for  the  produce  of  a  certain  farm 
acreage,  was  constant,  whether  the  crop  was  plentiful  or  sparse. 
Such  a  simple  relation  as  that  is  not  likely  to  hold  exactly ;  but 
if  it  did,  the  two  quantities — the  price  and  the  supply — would 
be  said  to  vary  inversely  as  one  another,  that  is  to  vary  in 
inverse  simple  proportion,  so  that  their  product  remained 
constant ;  whereas  if  they  varied  in  direct  simple  proportion 
it  would  be  their  ratio  which  remained  constant. 

It  does  not  follow  that  this  law  of  simple  inverse  variation 
holds  because  one  quantity  decreases  and  the  other  increases ; 
all  manner  of  complicated  relations  may  hold  between  such 
quantities ;  the  law  of  inverse  proportion  is  the  simplest 
possible,  and  there  are  a  great  many  cases  where  it  holds, 
or  holds  very  approximately. 

Take  a  piece  of  india-rubber  cord  or  tubing,  and  pull  it  out 
longitudinally ;  as  the  length  increases,  the  sectional  area 
diminishes,  and  it  is  a  matter  of  measurement  to  ascertain 
what  relation  holds  between  these  two  things. 

If  the  tube  were  filled  with  water  and  were  then  pulled  out, 
the  behaviour  of  the  water  would  furnish  a  test  of  how  the 
sectional  area  varied  with  the  length.  If  the  water  continued 
to  fill,  or  to  stand  at  the  same  level  in,  the  tube  (which 
might    terminate    at    one    end  in  a   piece    of    glass    tubing, 


398  EASY  MATHEMATICS.  [chap. 

for  convenience  of  observation,  and  be  closed  with  a  solid 
plug  at  the  other),  that  would  mean  that  the  sectional  area 
and  the  length  varied  inversely  as  one  another;  in  other 
words  that  their  product  was  constant;  for  the  product  of 
length  and  sectional  area  is  the  volume,  and  it  is  the  volume 
which  the  water  measures.  Try  the  experiment.  As  a  matter 
of  fact  the  water  will  be  found  to  sink  a  little  as  the  tube  is 
stretched,  showing  that  the  volume  increases  slightly :  the 
law  of  simple  inverse  proportion  does  not  hold  in  this  case. 

Consider  another  case  then.  Take  a  vessel  of  variable 
capacity,  for  instance  a  cylinder  and  piston ;  or  a  tube  open  at 
one  end,  which  can  be  plunged  mouth  downward  under  a 
liquid,  like  a  long  diving  bell,  and  can  be  lowered  or  raised  so 
that  the  air  in  it  shall  be  compressed  or  expanded  at  pleasure. 

If  a  pressure  gauge  is  attached,  it  will  be  possible  to  read 
how  the  pressure  increases  as  the  volume  diminishes ;  and  it 
will  be  found  that  the  two  vary  inversely  as  one  another, 
provided  one  is  careful  to  take  the  whole  dry-air  pressure  and 
the  whole  volume  into  account.  The  product  of  pressure  and 
volume  will  be  found  experimentally  constant ;  because  if  one 
is  halved,  the  other  will  be  doubled ;  if  one  is  trebled,  the 
other  becomes  one-third  of  its  original  value;  and  so  on,  a  law 
which  is  written  : 

1 

pec-, 

V 

or  vac  -,        or  pv  =  constant. 
P 

These  are  all  statements  of  the  same  fact ;  p  standing  for 

the  pressure,  and  v  for  the  volume  of  a  given  quantity  of  dry  air 

at   constant   temperature.     If   the   vessel  leaks,   so  that  the 

actual  amount  of  air  under  observation  changes,  the  law  will 

not  apply.     Nor  will  it  hold  if  the  temperature  is  allowed  to 

change.     For  if  air  is  warmed  it  expands,,  that  is,  it  increases 


XXXIX.]  VARIATION  DIRECT  AND  INVERSE.  399 

in  volume  or  in  pressure  or  in  both  together ;  and  there  is  no 
necessity  at  all  for  the  pressure  to  decrease  as  the  volume 
increases  unless  the  temperature  is  maintained  constant. 
Hence  a  complete  statement  is  that  pv  =  constant  if  T  is 
constant;  or,  if  we  choose  so  to  express  it,  jpv  =  const,  pro- 
vided dT  =  0,  i.e.  provided  the  difference  of  T  is  zero,  which 
is  only  another  way  of  saying  "  if  T  is  constant." 

Suppose,  subject  to  dT  =  0,  the  pressure  was  increased  by 
a  small  amount  dp,  and  the  volume  thereby  decreased  a  small 
amount  -  dv,  we  should  have  the  new  product  of  pressure  and 
volume  expressed  thus : 

p'v  =  (p  +  dp)(v  +  dv\ 
and  this  product  must  equal  the  old  product,  pv,  because  of 
the  law  that  the  product  of  pressure  and  volume  is  constant ; 
so,  multiplying  out,  we  get 

pv  +p  dv  +  vdp  +  dpdv  =  pv. 

Wherefore,  ignoring  the  second-order  small  quantity  dpdv, 

we  have  vdp+pdv  =  0 

dp          dv 
or  -^  = 

p  V 

as  another  statement  of  the  law  of  inverse  variation. 

Summary. 

Direct  Variation 

y  =  kx, 

or  di/  =  kdx, 

dy  _  1  _1 
dx~      ~  X 


or  xdy-ydx  =  0, 
dy    da 


dy    dx      ^ 
or  -^ =  0, 


400  EASY  MATHEMATICS.  [chap,  xxxrx. 

Inverse  Variation. 

xy  =^k, 
or  xdy  +  ydx  =  0, 

dy  y  k 

dx  X  a;2' 

di/    dx 
or  —  +  —  =  0. 
y      X 

^o  that  if  J  =   ±|,  that  is  if  the  ratio  of  differences  is 

numerically  the  same  as  the  ratio  of  the  quantities  themselves, 
it  is  a  case  of  simple  proportion ;  but  two  distinct  cases  are 
given  by  the  alternative  sign  : 

if  the  sign  is  +  it  is  direct  proportion ; 
if  the  sign  is  -  it  is  inverse  proportion. 


CHAPTER   XL. 

Pumps  and  Leaks. 

When  you  pump  water  out  of  a  reservoir,  taking  a  barrel 
full  of  water  out  at  each  stroke,  the  quantity  of  water 
remaining  decreases  in  an  arithmetical  progression,  of  which 
the  first  term  was  the  contents  of  the  well,  and  the  com- 
mon difference  is  the  contents  of  the  pump  barrel.  If  one 
were  called  F,  the  other  v  (read  big  F  and  little  v),  the 
level  in  the  well  would  fall  after  successive  strokes  in  the 
following  series : 

Aq,  /ij,  W-g,  ...  /«-„, 

where  h^  is  the  height  of  the  water  before  pumping  began,  and 
h„  is  the  height  after  n  strokes  of  the  pump, 

such  that       ^  =  A-  =  -A_  =  _^A__  =  ..., 
F      V-v       F-2v       F-3v 

a  mode  of  writing  which  is  called  a  continued  proportion. 

The  quantity  of  water  remaining  in  the  well  descends  in  a 
decreasing  Arithmetical  Progression, 

F,  F-  V,  F-  2v,  etc.  ...  V-  nv, 

and  the  well  is  empty  when  nv  =  F;  or  the  number  of  strokes 
required  to  empty  it  is  the  ratio  of  the  capacities  concerned,  F/v. 
The  height  or  level  of  the  water  in  the  well  goes  in  the 
same  sort  of  progression,  and  h„  is  zero  after  F/v  strokes. 
L.E.M.  2c 


402  EASY  MATHEMATICS  [cuap. 

But  now  consider  an  air  pump  instead  of  a  water  pump. 
The  peculiarity  of  air  or  any  other  gas  is  that  it  always  fills 
the  vessel  which  contains  it,  and  does  not  accumulate  in  one 
part  as  a  liquid  does.  A  bottle  may  be  said  to  be  "  full "  of  air, 
whether  it  contains  much  or  little,  in  the  sense  that  all  parts 
are  equally  full.  It  is  always  full  in  this  sense,  and  it  can 
never  be  full  in  any  other  sense ;  because  however  much  air 
is  in,  some  more  can  always  be  pumped  in :  the  only  limit  is 
the  bursting  and  destruction  of  the  bottle.  Or,  if  it  were 
made  of  porous  material,  it  could  be  said  to  be  as  full  as  it 
would  hold  when  the  rate  of  leak  was  equal  to  the  rate  at 
which  air  was  being  pumped  in;  but  even  that  could  be 
exceeded  by  beginning  to  pump  a  little  faster. 

With  a  liquid,  on  the  other  hand,  a  bottle  may  be  properly 
said  to  be  "  half-full " ;  it  can  also  be  completely  full,  for  you 
cannot  pump  more  than  a  certain  quantity  into  a  closed  vessel. 
If  it  is  an  open  vessel  the  rate  of  leak  at  a  certain  definite 
point  becomes  suddenly  equal  to  the  rate  of  supply,  and  the 
vessel  overflows ;  which  is  a  good  practical  method  for  main- 
taining a  constant  level. 

There  is  no  such  easy  method  for  providing  a  constant  air 
or  gas  or  steam  pressure,  though  something  of  the  kind  is 
attempted  by  means  of  a  leak  so  adjusted  as  to  suddenly 
change  from  near  zero  to  something  considerable,  at  a  certain 
critical  pressure, — such  an  arrangement  being  called  a  "  safety- 
valve."  Locomotive  boilers  are  usually  filled  with  steam  to 
this  pressure  before  a  train  starts  on  a  long  journey,  and  any 
excess  steam  which  the  furnace  generates  blows  off  noisily  in 
a  visible  cloud. 

If  you  were  to  pump  air  into  a  closed  chamber,  a  barrel  full 
of  atmospheric  air  would  be  injected  at  every  stroke,  and  the 
pressure  would  rise  in  an  increasing  arithmetical  progression. 


XL.]  PUMPS.  403 

Pq  being  the  initial  pressure  before  pumping,  and  j?„  the 
pressure  after  n  strokes, 

such  that        %  =     P-l-  =  -=.&-  =  ...  =  -^^, 

the  pressure  being  proportional  to,  and  a  measure  of,  the  extra 
quantity  of  air  injected.  But  if  a  pump  is  used  to  eject  the 
air,  that  is  to  say,  to  draw  out  from  a  closed  chamber  a  barrel 
full  of  air  at  every  stroke,  the  law  of  decreasing  pressure  is 
different :  it  then  forms  a  geometrical  progression. 

For  the  same  quantity  of  air  is  not  removed  each  time. 
The  same  volume  is  removed,  but  it  is  removed  from  air  of 
gradually  diminishing  density.  The  air  keeps  on  getting 
rarefied,  and  this  rarefied  air  it  is  which  has  to  supply  the 
pump  barrel ;  so  that  during  every  direct  stroke  the  air  which 
at  first  occupied  F"  expands  to  occupy  V+v,  and  then  the 
excess  is  ejected  into  the  atmosphere  at  the  return  stroke  of 
the  pump,  ready  for  the  expanding  operation  to  begin  again. 

Thus,  assuming  the  temperature  to  remain  constant,  we 
have  the  pressure  diminished  at  every  stroke  in  the  constant 

ratio  -j= —  ;   and  the  series  p^,  p^,  p^,  ...  p^  is  a  decreasing 

geometric  progression  with  the  common  ratio  F/(F+v). 
F 


So  that  p^  =  F+^^«' 


_     F        _  f    V  \ 
\FVv) 


2 

Pi)' 


Ps  -  yj^^P^  ~\f+v)  ^«' 
etc., 

the  ratio  of  the  pressure  at  beginning  and  end  of  any  stroke 

being  constant,  viz. 

Pn  V  ^^  . 

Hence  the  operation  of  an  ordinary  exhausting  air  pump  is 


404  EASY   MATHEMATICS.  [chap. 

governed  by  the  law  of  a  decreasing  geometrical  progression ; 
and  an  infinite  number  of  strokes  would  be  necessary  completely 
to  empty  the  vessel,  that  is  to  reduce  the  pressure  to  zero. 

Leaks  and  Compound  Interest. 

Now  suppose  instead  of  being  pumped  out  the  vessel  were 
full  of  compressed  air  and  were  to  leak ;  or  suppose  there 
were  a  cistern  full  of  water  with  a  crack  in  the  bottom  of 
it;  the  pressure  in  the  one  case  and  the  level  in  the  other 
would  fall  according  to  a  certain  law.  If  the  leakage  rate 
were  constant,  that  is  to  say  if  the  same  amount  of  material 
escaped  every  second,  the  law  would  be  a  decreasing  A.P. ; 
but  that  is  never  the  case  in  fact.  The  size  and  circumstance 
of  the  leak-orifice  being  constant,  the  amount  of  matter  which 
escapes  through  it  depends  on  the  force  with  which  it  is 
urged,  that  is  to  say  on  the  pressure  behind  it.  A  high 
pressure  reservoir,  or  a  tall  full  cistern,  would  leak  fast,  the 
air  or  water  rushing  out  of  the  leak  with  violence ;  whereas 
towards  the  end,  when  the  vessel  was  nearly  empty,  the  rush 
would  have  degenerated  into  a  mere  dribble  or  ooze,  unless  of 
course  it  had  worn  the  hole  larger — which  we  will  not  suppose 
to  be  the  case.  With  a  constant  sized  orifice  the  rate  of  leak 
is  therefore  proportional  to  that  which  causes  the  leak,  viz. 
the  pressure  ;  and  so  the  pressure  keeps  on  falling,  at  a  rate 
depending  on  itself :  a  curious  and  important,  because,  in  one 
form  or  another,  a  frequent  case. 

When  you  come  to  think,  it  is  just  the  compound-interest 

case,  but  inverted.     Capital  increases  at  a  rate  proportional  to 

itself :  when  small  it  grows  slowly,  that  is  by  small  additions, 

when  large  it  grows  quickly.     If  we  call  the  capital  at  any 

dx 
moment  aj,  its  rate  of  growth  will  be  — ,  since  dx  means  an 

CLl 

increase  of  capital,  and  dt  the  time  during  which  this  increase 


XL.]  LEAKS.  405 

occurs.  In  the  case  of  capital  the  increase  is  somewhat 
discontinuous :  the  interest  is  added  every  year,  or  it  may  be 
every  month,  or  perhaps  every  day,  but  not  every  instant. 
liCt   us   assume   that   it   is   continuous  however,    so   that  it 

increases  from  moment  to  moment  at  the  rate  -=- ;  this  rate  of 

Cit 

increase  will  be  proportional  to  z  itself,  and  of  course  to  the 
percentage  which  is  granted. 

Suppose  for  instance  it  was  5  per  cent.,  or  -05,  the  law  of 
increase  would  be  dx       ^- 

the  interest,  dx,  is  proportional  to  the  rate,  *05,  to  the  capital 
on  which  it  is  paid,  x,  and  to  the  time  during  which  it  has 
accumulated,  dt;  or  dx  =  'Obxdt 

If  it  were  4  per  cent.,  or  3  per  cent ,  or  2 J  per  cent.,  of 
course  we  should  substitute  '04,  or  -03,  or  -025,  for  the  '05 
numerical  coefficient. 

So  with  the  leak,  we  have  similarly  to  express  that  -  dx^  the 
loss  of  pressure,  is  proportional  to  the  pressure,  and  to  the 
time,  and  to  a  leak-aperture  constant  which  we  will  call  k  ;  so 

dx  =  -kxdt; 
for  to  express  that  it  is  a  loss  and  not  a  gain,  a  decrease  not  an 
increase,  we  must  apply  to  it  a  negative  sign.  The  x  might  be 
pressure,  or  it  might  be  level  or  "head," — the  two  are 
proportional  in  the  case  of  water ;  but  level  has  no  meaning  in 
the  case  of  gas,  so  we  will  take  "pressure"  as  the  more 
general  term,  and,  denoting  it  by  p  instead  of  x,  write  the 
law  of  leak,  in  the  simplest  possible  case  of  a  constant  orifice,  as 

dt~     "P' 
the  rate  of  fall  of  pressure  is  proportional  to  the  pressure  from 
instant  to  instant,  diminishes  as  it  diminishes,  and  does  not 
reach  zero  till  the  pressure  reaches  zero.     The  pressure  in  fact 


406 


EASY   MATHEMATICS. 


[chap. 


decreases  as  a  geometric  progression.  But  it  is  a  geometric 
progression  with  one  curious  feature  about  it,  it  is  continuous, 
not  discontinuous  like  numbers ;  it  does  not  go  in  steps  or  jumps, 
like  compound  interest  added  every  year  or  every  day,  but  it 
is  like  compound  interest  added  or  rather  subtracted  every 
instant,  with  complete  continuity,  according  to  a  smooth  curve, 
the  logarithmic  or  g.p.  curve,  see  page  357  or  101  or  179. 

Cooling. 

The  cooling  of  a  hot  body  under  simplest  conditions  follows 
just  the  same  law ;  the  rate  of  fall  of  temperature  is  propor- 
tional to  the  actual  excess  of  temperature  above  surrounding 
objects.     If  we  denote  this  temperature  by  6  and  time  by  t^ 
dd 


dt 


=   -kd 


expresses  the  simplest  possible   law  of  cooling  as  the  heat 
escapes  or  leaks  from  the  body  into  surrounding  air  or  space. 

It  is  instructive  to  put  a  thermometer  into  a  flask  or  pan  of 
very  hot  water,  and  read  the  thermometer  from  time  to  time ; 
at  first  every  half-minute,  or  oftcner,  then  every  minute,  and 
then  as  it  cools  more  and  more  slowly,  it  will  suffice  to  read  it 
every  five  minutes  ;  finally  plotting  the  result  thus  : 


Atmospheric 
Temperature 


Fio.  78. 


XL.]  LEAKS.  407 

By  choosing  different  vessels,  say  one  black  and  one  bright, 
or  by  choosing  similar  vessels  and  filling  them  with  different 
liquids,  one  water  and  another  turpentine  say,  many  instructive 
observations  can  be  made;  but  a  discussion  of  these  would 
carry  us  too  far  at  present. 

The  curve  of  cooling  is  identical  with  the  curve  of  leaking ; 
and  the  curve  of  leaking  might  be  plotted  by  reading  a 
pressure  gauge,  or  by  reading  the  level  of  a  leaking  water- 
reservoir,  from  time  to  time.  And  both  are  curves  of  de- 
creasing G.P.  or  are  logarithmic  curves. 

Electric  Leakage. 

Experiments  in  electricity  are  more  difficult,  but  if  it  were 
possible  to  read  satisfactorily  by  means  of  an  electrometer  the 
potential  of  an  electrified  body  or  Ley  den  jar  or  condenser 
which  was  steadily  leaking,  it  would  be  found  to  obey  the 
same  law. 

Continuously  decreasing  G.P. 

Now  see  how  to  express  a  quantity  which  decreases  geo- 
metrically with  perfect  continuity,  and  not  by  steps,  as  time 
goes  on.  Notice  that  time  is  a  continuous  progression  ;  there 
is  no  means  of  hurrying  it ;  one  day  is  like  another,  and  they 
follow  on  with  absolute  regularity.  Time  is  an  inexorable 
arithmetical  progression,  an  increasing  one  if  you  think  of 
your  birth,  a  decreasing  one  if  you  attend  to  the  other  end  of 
your  life.  Whatever  can  be  varied,  time  cannot :  at  least  not 
by  us. 

Now  when  a  vessel  is  leaking,  the  pressure  is  to  be  multi- 
plied by  a  constant  factor,  some  proper  fraction,  at  each 
successive  equal  interval  of  time. 

SoPq  at  the  start,  when  time  is  0,  or  say  at  12  o'clock  noon, 
becomes  let  us  suppose  ^p^  =  Pj  after  the  lapse  of  1  hour,  or 


408  EASY  MATHEMATICS.  [chap. 

at  1  p.m.     If  so,  then  in  another  hour  it  will  have  fallen  to  J^j, 
or  what  is  the  same  thing  J^q  ;  in  yet  another  hour,  that  is  at 

3  o'clock,  it  will  be  ^p^,  and  in  n  hours  it  will  be  —Pq  ;  hence 

it  will  have   fallen   continuously  down  the   decreasing  geo- 
metrical-progression-curve depicted  on  page  101. 

But  why  should  we  suppose  it  halved  in  each  unit  of  time  1 
We  can  be  more  general  than  that,  and  say  that  it  is  reduced 

to  -Pq  (read,  '  one  r-th  of  jo-nought ')  after  the  lapse  of  one  hour, 

where  r  is  some  number  greater  than  unity ;  then  in  another 

hour  the  pressure  will  have  become  -p-^,  or  what  is  the  same 

thing  ijPo- 

So  the  pressure  at  2  p.m.  is    p^^  =  r'-p^, 

at  3  p.m.         jOg  =  r->o, 

and  at  n  hours  after  noon  p^  =  '^~"Po' 

Or  we  might  say  that,  at  any  time  t  after  the  start,  the 
pressure  is  p  =  t~*Pq. 


This  then  is  the  law 


p  =  p^r 


P         -t 
or  —  =  r  \ 

Po 

or  log^  -   -t\ogr, 

or     log/?o-log^  =  nogr, 

log.  =  l^il^Y^i?, 

all  expressive  of  the  very  same  fact. 

Now  r  is  a  constant  depending  on  the  size  of  the  opening  the 
viscosity  of  the  escaping  fluid  (or  on  the  covering  and  contents 


XL.]  LEAKS.  409 

of  the  cooling  body),  and  any  other  circumstance  which  can 
affect  the  rate  of  leak,  other  than  pressure  (or  temperature)  and 
time.  And  the  log  of  r  is  the  diminution  of  the  log  of  the 
pressure,  during  any  lapse  of  time,  divided  by  the  time  which 
has  elapsed.  It  is  the  ratio  of  the  logarithmic  diminution,  or 
decrement,  to  the  time ;  it  is  the  decrement  of  the  logarithm 
of  the  pressure  per  unit  time,  and  is  technically  known  as 
the  "  logarithmic  decrement "  of  the  pressure  (or  of  the 
temperature  in  the  case  of  a  cooling  body,  or  of  the  potential 
in  the  case  of  an  electric  leak,  or  of  the  level  in  the  case  of  a 
leaking  cistern). 

To  measure  log?-,  all  we  have  to  do  is  to  read  the 
pressure  (or  temperature,  etc.)  at  any  one  instant,  and  then 
read  it  again  some  time  later,  observing  the  interval  of 
time. 

Let  the  two  readings  be  denoted  by  p^  and  ^„,  and  let  the 
intervening  time  be  n  seconds,  then 

log;?o  -  log  j^n 
n 

is  the  logarithmic  decrement  per  second,  and  is  a  measure  of 
the  constant  we  have  called  log  r. 

Thus  the  law  which  was  at  first  expressed  in  diiferential 

form  as  ^  ==    _  K,  or   -^  =    -  kdt, 

at  ^  p 

can  also  be  expressed  in  integral  form  as 

P  "=  Po^~'   OT  \ogpQ  -\ogp  =  tlogr; 
and  it  now  becomes  necessary  to  ascertain  and  express  the 
relation  between  the  two  constants  k  and  r,  which  evidently 
refer  to  the  same  sort  of  thing,  viz.  the  fixed  circumstances 
of  the  leak. 

Now  remembering  what  we  know  of  exponentials,  let  us  see 
if  we  can  puzzle  out  the  connexion  between  these  constants. 


410  EASY  MATHEMATICS.  [chap. 

The  law  that  we  have  written  expresses  the  fact  that 
pressure  decreases  geometrically  as  time  increases  arithmeti- 
cally :  a  constant  factor  is  characteristic  of  one  progression, 
while  a  constant  difference  characterises  the  other. 

We  know  that  if  p^  is  the  pressure  at  the  era  of  reckoning, 
that  is  at  the  instant  from  which  time  is  to  be  reckoned,  then 
at  any  time  t  the  pressure  is  ^  =  i'o^ "^  ^^^  ^^  ^^7  ^^^^^ 
time  f  the  pressure  is^  =  PqV'^',  therefore 


p  =  dp  and 


P 

Now 

let 

the   change 

be   small. 

so   that  jt/ 

f 

-t  = 

dt; 

then  the  last 

equation  is 

p  +  dp  _ 
P 

-f=' 

-dt 

or 

dp^ 

=  r-'^'-l  = 

-dt.  logr. 

p 

The  last  step  we  are  not  supposed  to  know  enough  yet  to 
justify ;  but,  assuming  it  and  deferring  its  justification  to 
page  425,  we  see  that 

J-  =,  -  log r .  dt, 
p  ^         ' 


|=-^log 


and  this  we  can  compare  with  the  equation  at  which  we 
started  (p.  405),  ^jp 

Thus  the  relation  between  the  constants  k  and  r  is  simply 

k  =  logr, 

and  accordingly  k  is  itself  the  logarithmic  decrement  of  the 
pressure  per  second. 


XL.] 


LEAKS. 


411 


The  physical  meaning  of  ^"  is   —  -4^, 


the  physical  meaning  of  log  r  is 


log;?- log  jp' 


Summary. 

These  two 
things  turn 
out  to  be 
mathemati- 
cally the 
same  thing. 

But  when  logr  is  thus  written,  what  base  is  intended  for 
the  logarithm?  There  is  nothing  to  say  that  the  base  is  10, 
and  indeed  no  explicit  assertion  has  been  made  about  any 
base  whatever :  all  that  has  been  asserted  is  that  p  is  to  be  a 
quantity  whose  rate  of  change  is  proportional  to  itself,  or 
equal  to  itself  when  multiplied  by  the  constant  h  or  log  r. 

There  is  evidently  something  here  worth  investigation  from 
the  purely  mathematical  point  of  view.  It  is  a  definite 
mathematical  question  to  put  "What  is  that  quantity  whose 
rate  of  change  shall  be  proportional  to  itself?  how  is  such  a 
quantity  to  be  expressed  in  generaU"  To  investigate  this 
question,  we  can  study  the  rates  of  variation  of  various 
algebraic  expressions. 


^^ 


-^  ^,„  CHAPTER   XLL 

Differentiation. 

Take  the  area  of  a  square,  and  ask  how  it  varies  with  the 
side  which  contains  it,  when  the  square  slightly  expands.  We 
already  know,  but  we  will  go  through  the  process,  especially 
for  a  very  small  or  infinitesimal  increment  of  the  side.  Let 
the  side  be  x,  and  the  area  of  the  square  be  called  «/,  so  that 
y  =  a;2,  then  when  the  square  is  warmed  a  little,  x  increases  by 
the  amount  dx^  and  y  increases  by  the  amount  dy,  such  that 
y  +  dy  =  (x  +  dxy 

=  x^  +  2xdx-h{dxy. 

Now  let  dx  be  so  small  that  the  square  of  dx  may  be  utterly 
neglected  in  comparison  with  dx  itself.  In  the  limit  suppose 
dx  actually  infinitesimal,  so  that  {dxY,  being  dx  x  dx,  is  again 
or  still  further  infinitesimal,  even  in  comparison  with  dx ;  then 

y  +  dy  =  x^  +  2xdx; 
but  y  =  x^,  therefore,  subtracting,  there  remains  dy  =  2xdx, 

1  =  ^- 

whence  the  rate  of  change  of  area  of  an  expanding  square,  per 
unit  expansion  of  edge,  is  twice  the  length  of  one  of  the  sides  : 
a  very  elementary  statement,  but  not  obvious.  It  is  of  course 
a  general  analytic  or  algebraic  result,  and  in  no  way  depends 
upon  any  geometrical  meaning  attached  to  y'^.    The  geometrical 


CHAP.  XLi.]  RATES  OF  CHANGE.  413 

square   is   only   a   special   case,   and  it  is  convenient  as   an 

illustration ;    but   it   would   be    equally  true  for   any   other 

variation   of   one   quantity   as   the   square    of    another;    for 

instance,  the  relation  between  the  velocity  of  a  falling  body 

and  the  height  it  has  fallen,  so  well  known  in  mechanics,  is 

written  v^  =  2^/i,  and  this  we  can  re- write  in  differential  form 

d{v^)  =  Ivdv  =  "Igdh, 

dv       a 
or  —  =  - 

dh       v' 

which  gives  us  the  extra  speed  gained  for  each  additional  foot 
or  centimetre  or  other  small  unit  of  height. 

Suppose  for  instance  the  height  already  fallen  were  100  feet: 
a  dropped  stone  would  have  acquired  a  speed  of  80  feet  a 
second.  By  the  time  it  has  dropped  a  foot  more,  the  above 
equation  asserts  that  its  speed  will  have  increased  by  the 
amount  f f  =  f  =  '4  feet  per  sec. 

We  might  also  get  the  above  relation  thus : 
v^  =  2ghj 
v'^  =  2g{h+l); 
:.  v'^-v^  =  2g, 

2(7  2q  a 
V  +v  2v  V 
but  in  this  case  there  is  an  approximation,  because  1  foot 
added  to  100  is  by  no  means  infinitesimal  though  it  is 
moderately  small.  Consequently  a  sort  of  average  or  mean  has 
to  be  taken  between  v  and  v\  which  in  the  limit  would  be 
ultimately  equal. 

The  expression  dy  =  2xdx  we  long  ago  illustrated  by  the 
two  strips,  each  equal  to  xdx  in  area,  which  went  to  form  the 
increase  of  surface  in  a  square  plate  x'^  expanded  by  heat 
(page  369) ;  the  little  corner  bit  {dxf  being  ignored,  because 
when  the  strips  themselves  are  infinitesimal,  the  infinitesimal 


414  EASY  MATHEMATICS.  [chap. 

bit  of  each  at  the  ends  is  nought  in  comparison,  or  is  said  to  be 
an  infinitesimal  of  the  second  order. 

Similarly  we  may  deal  with  the  expansion  or  variation 
of  a  cubical  block  of  side  x. 

Denote  its  volume  by       y  =  a;^, 
then  when  it  expands  infinitesimally 
y  +  dy  =^  {x  +  dxf 

=  x^  +  Sx^dx  +  infinitely  smaller  quantities ; 
.*.   dy  =  3x^dx, 

dx 
or  the  rate  of  expansion  of  a  cubical  volume,  per  unit  increase 
of  a  side,  is  three  times  the  area  of  one  of  its  faces. 

Observe  that  the  rate  of  increase  of  an  area  is  a  length, 
while  that  of  a  volume  is  an  area;  but  that  is  because  the 
rate  of  increase  is  taken  per  unit  of  length.  If  it  were  taken 
per  unit  of  time  or  of  temperature,  and  if,  as  before,  we  write 
y  =  x^,  we  could  say  that 

dt  ~  '^'^  dt' 
or  the  rate  of  variation  of  the  volume  with  respect  to  any 
outside   or  independent  variable,  such  as  temperature,  (-^\ 
(i.e.   the  cubical  expansion  per  degree),   is  greater  than  the 
rate  at  which  each  edge  expands  for  the  same  variable,  (j-j 

(the  linear  expansion  per  degree),  in  the  ratio  of  three  times 
the  area  of  one  of  its  faces. 

Stated  thus  it  is  perhaps  hardly  geometrically  evident, 
nor  need  it  be  made  so.  What  is  geometrically  capable  of 
illustration  is  the  fact  that 

dy  or  d(x^)  =  Sx^dx. 


xLi.]  DIFFERENTIATION.  415 

Other  expressions  of  the  same  kind  of  fact  are  best 
treated  as  mere  analytic  or  algebraic  statements,  without  any 
necessary  geometrical  signification. 

So  we  learn  that  to  get  the  small  change  of  any  quantity 
we  have  only  to  attend  to  the  early  terms  of  a  binomial 
expansion  :  two  only,  if  the  change  is  infinitesimal. 

For  instance,  to  find  d{x%  that  is  to  express  it  in  terms 
of  dx,  we  let  x  increase  by  dx,  and  then  expand  and  neglect 
all  beyond  the  first  power  of  dx  \  thus 

{x  +  dxf  =  x'^  +  ioi^dx  +  higher  powers ; 
but  x  +  dx  =  x\   and   d(x^)  means  x'^-x\ 

therefore  d{x!^)  =  A:X^dx. 

Similarly  d{:i^)  =  bx^dxy 

d{x^)  =  Qx^dXy  and  so  on,  until 
f/(«i2)  =  I2x^idx, 
and  dx""  =  nx'^'^dx. 

So  also  with  fractional  indices  : 

For  instance,  to  find  d  Jx.  Expand,  and  ignore  all  higher 
powers  of  the   infinitesimal  quantity  dx. 

(x -^  dxy  =  x^  +  Ix  ^^ic  +  higher  powers, 

but  (x  +  dxy  =  Jx', 

so  djx  =  Jx'  -Jx  =  ^x'^dx  =  j^. 

3  3  1 

Again     {x-\-dxy=  x^  +  ^x^dx, 

3  1 

therefore        d  Jx^  =  dx^  =  ^x^dx  =  ^Jx.dx. 
Or  take  negative  indices  : 

To  find  d(-\  or  dx~'^;  expand  again, 

{x  +  dx)~'^  =  x~'^  -  x'^dx, 


416  EASY  MATHEMATICS.  [chap 


but  {x  +  dxy^  =  '-,, 

X 


\X/         X       X 


di^]  =±,-t-^-x-'^dx  =  -~. 


And  in  general,  whatever  n  may  be, 
dx«»  =  nx^-^dx, 

a  perfectly  general   result,   worth   thoroughly  learning   and 
applying  to  special  cases. 

Even  in  the  case  when  n  =  -0  it  holds  good ;  for  then  it  says 
that  dx^  =  0, 

which  we  know  is  true,  because  x^  =  1  =  constant,  and  so  its 
differences  or  variations  must  be  zero. 

If  71  =  1   it  gives   dx  =  1  .  dx,  which  is  a  mere  identity. 

If  w  =  2,  it  gives  dx^  =  2xdx; 
if  w  =  3,  „  da^  =  3x^dx, 
which  we  separately  verified ;  and  so  on. 

Examples. 

Check  the  following  statements  : — 

dxy  =  7afidx;  ^  -  7x^;  ~  =  Sx^ ; 

dax  =  adx;  d{ax^)  =2axdx;  daa^  =  Sax^dx. 

d7x^ 

—1 —  =  35.^* ;        d(x  +  y)  =  dx  +  dy;     d{ax  +  by)  =  adx  +  hdy  ; 

d{ax  +  hx'^)  =  adx  +  2bxdx  =  {a  +  2bx)dx; 

d 

^{cty  +  by^)  =  a  +  2by;  d(a  +  bx)  =  bdx; 


XLi.]  DIFFERENTIATION.  417 

-^(a  +  bx)  =  b;  d(a  +  bx  +  cx^)  =  {b  +  2cx)dx; 

^{a  -{■bx  +  cx^  +  x^)  =  b->r  2cx  +  3x^ ;       ~(A  +  Bx"")  =  nBx''-^ ; 
dx  dx^  ' 

^(A+Bx+Cx^-hD:^  +  Eo^+   ..+Zx^') 
dx 


B  +  2CX  +  Wx^  +  iEx^  +  . . .  +  nZx''-^ ; 


dx,  d  (l\  _       I 

y?  '  dx\x)  ~~      x^ 


dx\xy  ~  x^'  dx\x^)  ~      x^ 

A.(—\  -  — '  —(4-     E\  -  -A 

dx\x )  ~  x^'  dx\x        )  ~      x' 

d  /  a\  a  d  fA      r,     rt  \          ^     n 

dx\Fx)  =-  -w^'  dxK^^^^^V  =  -^2+^; 

4 
dxy. 


^^  +  i  +  C+Dx  +  Ex^  =  -'^-^,  +  D  +  2Ex; 
X  )  xy      x^ 


dx'-^x^dx;  dx^""  '-^  2jx'^ 

dx^  =  ^x^dx;  ^(»'  =  W^; 

-j^a^x  +  b  +  -j-^j  -  j-j-^-^^-j^  =  2jxV~x)'- 
~(a  +  bxf  =  ^(a+2abx  +  bV}  =  2a6  +  2i%  =  2b{a  +  bx); 

L.E.M.  2d 


418  EASY  MATHEMATICS.  [chap.  xli. 

dii^)  =  2vdv;  d(2gh)  =  2gdh',  j^{v^)  =  2vj^  ; 

j-(au^  +  bu  +  c)  =  2au  +  b',  j-av^  =  2av~r-  ; 

du^  '  '  du  du 

d  f     o     t        ^       c^     du     ,du       ,»         ,.du 
j^iau^  +  bn  +  c)  =  2a«^  +  6^  =  <2««  +  *)^<  ' 


CHAPTER  XLII. 


A  Peculiar  Series. 

We  are  now  able  to  write  down  a  set  of  algebraic  terms, 
each  of  which  shall  be  the  differential-coefficient  of  the  one 
following  it : 

0  +  l+x  +  ix^. 

Of   this   we   might   make  a   regular  series,   for   just  as  ^x^ 
differentiated  gives  x,  so  ^x^  differentiated  would  give  x^,  and 

-1—  would  give  ^x^.    So  also  - — - — -  differentiated  would  give 

X? 

- — -,  and  so  on ;  hence  the  series 

a;2     a;3     x^     x^ 
l+^  +  2l  +  3!  +  4!  +  5!  +  - 
is  a  series  which,  when  differentiated,  gives  as  result 

/g2       /g3       ^4 

0  +  l+»'  +  2!  +  3!  +  ¥!  +  -' 
the   very  same   series, — provided   both   extend   to    infinity : 
a  very  curious  case,  the  rate  of  variation  of  the  series  is 
equal  to  itself.     (Cf.  p.  411.) 

Such  a  series  must  therefore  be  appropriate  for  use  in  the 
theory  of  leaks,  that  is  for  dealing  with  a  quantity  whose 
rate  of  change  is  proportional  to  or  equal  to  itself.  We  can 
guess  therefore  that  such  a  series  must,  when  plotted,  give 
a  curve  of  the  nature  of  the  exponential  or  logarithmic  or 

I..E,M,  2  D  2 


420  EASY   MATHEMATICS.  [chap. 

geometrical-progression  or  compound-interest  curve.  If  we 
call  its  value  ?/,  it  satisfies  the  equation  -f-,  =  y  (cf.  page  405). 

It  is  a  notable  series.  It  is  plainly  convergent  if  x  is  less 
than  1 ;  but  it  is  convergent  even  when  x  is  equal  to  1  or 
greater  than  1,  because  the  denominators  increase  so  fast; 
they  increase  so  fast  indeed  that  a  moderate  number  of  terms 
are  generally  sufficient  to  evaluate  it  fairly.  The  powers  of  x 
grow  fast  in  size  when  x  is  greater  than  1,  but  the  factorials 
of  the  corresponding  index-number  grow  still  faster,  and  so 
must  ultimately  get  bigger ;  for  x  stays  as  a  constant  factor 
while  being  raised  to  any  power,  while  in  'factorials'  the 
factor  keeps  on  increasing.     See  page  315. 

Let  us  try  what  the  value  of  this  series  is  when  x  —  1: 

I  +  I+-I-  +  J+2V+  xhs  +  tI^  +  Wtct  +  — 
Greater  than  2  and  apparently  less  than  3,  because 

1  +  1  +  J  +  J  + 1  +  xV  +  •  •  •  would  equal  3. 
With  patience  its  value  can  be  reckoned  to  any  desired 
degree  of  accuracy,  and  it  comes  out 

2-71828..., 
a  remarkable  number,  usually  called  e. 

So  now  we  can  reckon  what  the  series  is  when  x  has  any 
other  value  than  unity.  If  we  try  it  arithmetically  f or  a;  =  2 
we  shall  get 

where  we  observe  that  though  at  first  the  numerators  are 
bigger  than  the  denominators,  afterwards,  in  spite  of  the  well- 
known  rapid  increase  of  the  powers  of  2,  the  factorials  in  the 
denominators  soon  overpower  them;  for  2^2  =  4096,  whereas 
1 12  =  479,001,600,  and  is  thus  a  hundred  thousand  times  as 
great. 

To  get  a  good  value  for  this  last  series  we  must  take  a  fair 


XLii.]  EXPONENTIAL  SERIES.  421 

number  of  terms,  ten  or  a  dozen,  into  account ;  and  if  we  do 
we  find  the  result 

7-389..., 
which  is  e^. 

Similarly  if  we  put  a;  =  3  we  shall  get  20-09... , 
which  is  e^ ; 

whereas  if  we  put  a;  =  i  we  get  1-6467... , 
which  is  ^e. 

Thus  we  suspect  that  the  series 

is  in  fact  e* ;  which  is  true,  and  it  is  called  the  exponential 
series  accordingly. 

It  has  the  singular  and  very  useful  property  that  its  rate  of 
change  is  equal  to  itself,  that  is  to  say  that 

as  we  have  already  proved  by  differentiating  each  term  of  its 
series  separately  and  observing  that  the  series  is  unchanged  by 
the  process,  being  simply  repeated  over  again. 

Natural  base  of  logarithms. 

We  can  now  apply  this  to  logarithms ; 
Let  y  =  (^, 

so  that  log  1/  =  a;  log  ^,  or  log^  y  =  a; ; 

we  have  just  learnt  that  in  this  case 

dx      ^' 

wherefore  d  log^  y  ==  dx  =  —. 

That  is  to  say,  the  rate  of  change  of  the  logarithm  of  a 
yariable  number  js  ec^ual  to  the  rate  of  change  of  the  number 


422  EASY  MATHEMATICS.  [chap. 

itself  divided  by  that  number ;  provided  the  base  of  logarithms 
is  e. 

If  we  take  any  other  base  than  e  we  shall  not  get  quite  so 
neat  a  result. 

For  let  u  =  ?-*,  where  r  is  any  number  whatever, 

then  log  M  =  a;  log  r  (or  log^  u  =  x\ 

and  so  —  =  c?a; .  log  r ; 

wherefore      d  loff,  u  ^  dx  =  —. , 

o*-  li  log  r 

which  only  reduces  to  the  above  simple  form  when  r  =  e; 
otherwise  it  requires  the  natural  logarithm  of  r  to  appear. 

For  instance,  suppose  we  put  u  =  lO^'^isosos^  ^g  g,  re- 
presentation of  r*,  and  change  the  index  by  a  small  amount, 
say  to  2*9180408;  then,  by  referring  to  an  ordinary  seven- 
figure  table  of  logarithms,  we  shall  see  that  the  corresponding 
change  in  the  number  u  is  -02 

since  u  =  828-00  and  v!  =  828-02. 

Now   our  assertion  is   that   the  change   in   the   logarithm 

(x'-x  or  dx,  viz.   -0000105)  would  have  been  equal  to  the 

change  in  the  number  (u'  -u  or  du)  divided  by  the  number 

'02 
{u  or  u'),  that  is  to  say  would  be  practically  equal  to  — — ,  if  the 

828 

base  had  been  e;  but  since  the  base  is  10  this  result  has  to  be 
divided  by  the  further  fixed  quantity — the  natural  logarithm 
of  our  artificial   base  10  (which  is  a  number   approximately 
equal  to  2-3),  in  order  to  give  the  right  result. 
And  it  will  be  found  accordingly  that 

-02 
ooQ — o^  =  -0000105,  almost  exactly ; 

which  illustrates  the  last  algebraic  line  above. 


XLiL]  BASE  OF  LOGARITHMS.  423 

Let  us  illustrate  the  occurrence  of  this  natural  logarithm  of 
10  by  another  numerical  example,  and  at  the  same  time  make 
an  estimate  of  its  value. 

Suppose  we  put  10"2  to  represent  r*,  and  then  allow  the  index 
X  to  increase  somewhat,  say  to  2-01 ;  what  will  be  the  corre- 
sponding change  in  r*  % 

We  might  write   u  =  lO^oo,  v!  =  lO^oi ; 
so  that        <fo^H:^«^10^°;-10-^IO.o._l (1) 

But  in  general  when  w  =  r*,    _  =  ?•*  log  ?',    wherefore 

nil 

—  =  dxAogr  =  -OlloglO;   (2) 

and  from  these  two  expressions  for  the  same  thing  we  can 
approximately  evaluate  the  natural  log  10.  For  equating  (1) 
and  (2),  we  get 

lO'Oi  -  1 
log  10  =       ,Q^       =  100  X  (1001  -  1) 

=  100(^710-1)  =  100(1-0232-1)  =  2-32, 
the  last  digit  being  affected  by  an  error  caused  by  the  increase 
in  X  not  being  infinitesimal. 

This  2*3...  is  approximately  the  logarithm  of  10  to  a  certain 
base  which  has  not  been  artificially  specified,  and  which  there- 
fore must  have  entered  automatically  and  "naturally"  without 
convention  or  artifice.     What  is  that  base  % 

It  is  a  number  such  that  if  raised  to  the  2-3...  power  it  will 
equal  10.     Call  it  w,  then 

2-3...1ogio7i  =  logiolO  =  1, 

or  logioTi  =  2^37^7=  -434..., 

wherefore  n  =  2-717...  ; 

which  plainly  points  to  e,  with  a  deficiency  of  one  part  in  two 

thousand,  or  a  twentieth  of  one  per  cent.,  due  to  approximations. 


424  EASY  MATHEMATICS.  [chap. 

Clearly  therefore  there  is  something  peculiar  about  e  as  the 
base  of  an  exponential  system :  it  is  simpler  than  any  other, 
and  it  occurs  automatically  or  naturally,  unless  we  force  some 
other  base  in ;  for  when  one  finds  that 

whereas  -y-  =  e*, 

it  becomes  apparent  that  the  base  here  automatically  indicated 
is  such  as  to  make  loge  =  1. 

The  fact  is  that  logr,  wherever  it  naturally  occurs,  means 
log^r,  and  not  a  logarithm  to  any  base  at  random.  There 
appears  therefore  to  be  a  natural  base  for  logarithms  ;  in  this 
respect  differing  entirely  from  the  base  or  radix  of  the  scales  of 
notation  in  ordinary  counting.  Ten,  or  twelve,  or  any  number, 
might  be  used  for  that — it  was  a  pure  convention ;  but  though, 
as  soon  as  we  have  adopted  ten  as  the  numeration  base,  ten 
becomes  specially  convenient  for  practical  calculations  by  aid 
of  logarithms  also,  yet  ten  is  not  the  natural  base  of  logarithms ; 
nor  is  it  the  simplest  base  for  an  exponent.  That  property 
specifically  belongs  to  the  incommensurable  number  called  e. 

The  expansion  of  any  exponential,  such  as  r*,  is  now  easily 
managed  in  terras  of  e;  for  r  may  be  expressed  as  e*,  whence 
r*  =  e** ;  and  we  already  know  that 

But  since  r  =  e^  it  follows  that  k  =  log/,  hence  the  above 
expansion  may  also  be  written  as 

7^=  l+a;logr  +  ^    ,|   ^  +..., 

where  the  logarithms  are  all  to  the  base  e. 


XLii.]  LEAKS.  425 

For  the  special  case  when  x  is  infinitesimal,  say  dt, 

that  gives  us  r*"  =  1  +  ^^ .  log  r, 

wherefore  j*^'  -  1  =  dt .  log  r, 

which   justifies   a   step   assumed   above   (page   410) ;   where, 
however,  it  happened  that  the  dt  had  a  negative  sign. 

The  whole  theory  of  leaks  or  cooling  is  now  quite  easy,  after 
this  incursion  into  the  elements  of  pure  mathematics :  for 
given  that  any  quantity  'p  (say  pressure  or  temperature  or 
potential)  changes  at  a  rate  proportional  to  itself,  we  can 
write  down  instantly  the  following  equivalent  expressions 
{t  meaning  time) : 

dj) 

It 


kp, 


P 
d\ogp  =   -Tcdt, 
log/ -log;?  =  k{t-t'\ 

\ogp  +  U  =  constant  =  logj?Q, 

p  =p^e'^. 

All  these  are  different  modes  of  expressing  the  same 
physical  fact :  the  law  of  a  cooling  body,  or  a  leaking  reser- 
voir, or  any  other  of  the  many  cases  where  rate  of  change 
of  a  quantity  is  proportional  to  the  quantity  itself;  and  the 
last  gives  explicitly  the  value  of  that  quantity  at  any 
instant,  in  terms  of  the  initial  value,  the  logarithmic  decre- 
ment, and  the  time. 

And  this  must  be  regarded  as  typical  of  the  way  in  which 
general  facts  in  Physics  are  simplified,  summarised,  and  com- 
pactly treated,  by  aid  of  more  or  less  easy  mathematics. 


426  EASY  MATHEMATICS.  [chap.  xlii. 

So  ends  the  present  book,  but  in  a  subject  like  this  there 
can  be  no  termination;  every  avenue  leads  out  into  infinity 
and  must  be  left  with  its  end  open.  In  no  science  are  there 
any  real  boundaries.  In  an  advanced  book  a  subjective 
boundary  may  be  reached,  viz.  the  boundary  of  our  present 
knowledge ;  but  in  an  elementary  book  like  the  present  that 
is  immensely  far  away,  and  the  only  terminus  that  can  here 
be  reached  is  a  terminus  of  print  and  paper. 


APPENDIX. 


I.    Note  on  the  Pythagorean  Numbers  (Euc.  I.  47). 
See  Chapter  XXXI. 

By  the  Pythagorean  numbers  I  mean  simply  those  triplets  of 
integers  which  serve  to  express  the  relative  lengths  of  the  sides  of 
commensurate  right-angled  triangles  :  numbers  therefore  which 
satisfy  the  conditions  of  Euclid  I.  47,  that  any  two  of  them  are 
greater  than  the  third,  and  that  the  sum  of  the  squares  of  two  of 
them  equals  the  square  of  the  third. 

The  only  numbers  mentioned  in  the  text,  page  272  are  : 
3,  4,  5  ;     5,  12,  13  ;     and  8,  15,  17  ; 
but  there  are  innumerable  others. 

The  subject  is  of  no  practical  importance,  and  is  only  mentioned 
here  as  an  example  of  an  easy  kind  of  investigation  in  pure 
mathematics  which  an  enthusiastic  and  advanced  pupil  might  be 
encouraged  to  undertake,  and  which  might  lead  him  to  take  some 
interest  in  less  simple  parts  of  the  theory  of  numbers.  The  result 
of  the  investigation,  in  this  case,  might  be  worded  thus  : 

In  general  the  sides  and  hypothenuse  of  a  right-angled  triangle 
are  incommensurable,  but  an  infinite  number  of  such  triangles 
exist  in  which  the  three  sides  may  be  represented  by  integers. 
These  are  of  some  interest,  and  the  simplest  of  them,  when  the 
sides  are  in  the  ratio  of  the  numbers  3,  4,  5,  is  frequently  used  by 
surveyors. 

A  formula  from  which  all  such  sides  may  be  calculated  is  the 

identity  .«,.  +  (^J.(2!|*^)V 


428  EASY   MATHEMATICS. 

meaning  that  ah  and  \{a'^-h'^)  represent  the  sides  containing  the 
right  angle,  and  that  \{a^  +  h^)  represents  the  hypothenuse. 

To  get  a  number  of  these  triangles  rapidly,  without  repetition 
of  shape,  i.e.  without  obtaining  mere  multiples  of  other  sets, 
it  is  sufficient  to  choose  as  the  auxiliary  integers  a  and  6  any  odd 
numbers  which  are  prime  to  each  other.  The  reason  for  choosing 
the  auxiliary  integers,  a,  6,  as  odd  numbers  prime  to  each  other,  is 
simply  that  if  they  contained  a  common  factor  the  triplets  obtained 
from  them  would  be  merely  a  multiple  set  representing  the  same 
shape  as  a  simpler  set ;  whereas  if  one  was  even  while  the  other 
was  odd,  then  a^-W  would  be  odd,  and  \{a^-¥)  would  not  be  an 
integer ;  or  if  everything  were  doubled  it  would  be  merely 
repeating  the  sides  of  some  previous  shape  in  another  order. 

Excluding  multiple  sets,  one  of  the  sides  and  the  hypothenuse 
are  always  represented  by  an  odd  number,  and  the  other  side  by 
an  even  number. 

It  is  easy  to  prove  that  one  of  the  sides  containing  the  right  angle 
must  always  be  a  multiple  of  4,  that  one  of  them  (it  may  be  the 
same)  must  be  a  multiple  of  3,  and  that  one  of  the  three  sides  (again 
it  may  be  the  same)  must  be  a  multiple  of  5  : 

One  of  the  two  sides  must  be  a  multiple  of  3. 

Of  course  a,  or  ?>,  may  itself  be  a  multiple  of  3,  thus  satisfying 
the  condition  for  the  side  ah.  If  neither  of  them  is,  then  a  6  is  not 
a  multiple  of  3,  but  in  that  case  their  squares  must  be  of  the  form 
3m  +  l,  3w+l  [or  rather  of  the  form  6m +  1,  6w  +  l,  since  they  are 
odd  numbers],  and  so  the  other  side,  viz.  \{a^-b^)  is  then  of  the 
form  3(m-w). 

One  of  the  two  sides  must  be  a  multiple  of  4. 

Taking  a  =  2m +  1, 

h  =  2^1  +  1, 
^(a2_62)  ^  2{m-n){m  +  n  +  \\ 
and  either  m-  ,^or  m  +  n-\-\  must  be  an  even  number,  since  their 
difference  is  an  odd  number. 

One  of  the  three  sides  must  be  a  multiple  of  5. 

If  a  or  &  is  a  multiple  of  5,  one  of  the  sides,  viz,  the  odd  side, 
is  the  required  multiple.  If  not,  its  square  must  be  of  the  form 
40m  +  9  or  40m  +  1.    If  the  squares  of  both  have  the  same  remainder, 


APPENDIX. 


429 


the  even  side  is  a  multiple  of  5.     If  one  has  remainder  9  and  the 
other  1,  the  hypothenuse  is  a  multiple  of  5,  of  the  form  20m +  6. 

So  if  neither  a  or  6  is  a  multiple  of  either  3  or  5  it  follows  that 
the  number  representing  the  even  side  has  all  three  of  the  factors, 
3,  4,  5  ;  i.e.  that  it  is  a  multiple  of  60. 

Moreover  it  can  be  shown  that  the  hypothenuse  is  always  itself 
the  sum  of  two  square  numbers,  one  odd  and  one  even,  and  that  the 
odd  side  is  the  difference  of  those  same  squares.  Thus,  writing  the 
odd  side  as 

(2m  +  l)(2ri+l)  =  {m  +  n-\-\f-{m-n)\ 
the  even  side  is  '2,{m-n){m-\-n  +  \ ), 

and  the  hypothenuse  is  {m  —  7if  +  {m-\-n-\-  Vf. 

The  following  is  a  table  of  the  Pythagorean  triplets,  with  the 
mode  of  obtaining  them  displayed. 


Auxiliary  pair 

Odd 

Even 

of  numbers. 

side. 

side. 

Hypothenuse. 

3,1 

3  =  4-1 

4 

5  = 

4  +  1 

5,1 

5  =  9-4 

12 

13  = 

9  +  4 

7,1 

7 

24 

25  = 

16  +  9 

9,1 

9 

40 

41  = 

25  +  16 

11,1 

11 

60 

61  = 

36  +  25 

13,1 

13 

84 

85  = 

49  +  36 

15,1 

15 

112 

113  = 

64  +  49 

17,1 

17 

144 

145  = 

81+64 

19,  1 

19  =  100- 

81  180 

181  = 

100  +  81 

5,3 

15 

8 

17  = 

16  +  1 

7,3 

21 

20 

29  = 

25  +  4 

9,3 

27 

36 

45 

etc. 

11,3 

33 

56 

65 

13,3 

39 

80 

89 

15,3 

45 

108 

117 

17,3 

51 

140 

149 

19,3 

57 

176 

185 

7,5 

35 

12 

37 

9,5 

45 

28 

53 

11,5 

55 

48 

73 

13,5 

65 

72 

97 

15,5 

75 

100 

125 

17,5 

85 

132 

157 

19,5 

95 

168 

193 

430  EASY  MATHEMATICS. 


Auxiliary  pair 

Odd 

Even 

of  numbers. 

Bide. 

side. 

Hypothenuse. 

9,7 

63 

16 

65 

11,7 

77 

36 

85 

13,7 

91 

60 

109 

15,7 

105 

88 

137 

17,7 

119 

120 

169 

19,7 

133 

156 

205 

11,9 

99 

28 

101 

13,9 

117 

44 

125 

15,9 

135 

72 

153 

17,9 

153=  169- 

-16  104 

185  =  169-hl6 

19,9 

171 

140 

221 

13,  11 

143 

24 

145 

15,  11 

165 

52 

173 

17,  11 

187 

84 

205 

19,11 

209 

120 

241 

The  left-hand  column  is  simply  a  series  of  pairs  of  odd  numbers, 
mainly  prime  to  each  other  (but  a  few  are  included,  for  the  sake  of 
systematic  completeness,  which  are  not  prime,  and  therefore  involve 
repetition)  ;  the  second  column  is  their  product ;  the  third  column 
half  the  difference  of  their  squares  ;  and  the  fourth  column  half  the 
sum  of  their  squares ;  the  incipient  columns  merely  illustrate  the  fact 
that  the  hypothenuse  is  the  sum  of  two  square  numbers,  one  odd, 
one  even,  whose  difference  is  equal  to  the  odd  side. 

The  identity        (m^-ny  +  {%n7if  =  {m^+'nFf 

represents  the  facts  most  simply,  where  m  and  n  are  any  integers. 
One  of  these  integers  must  be  even  and  the  other  odd,  with  no 
common  factor,  if  mere  multiples  or  repetitions  of  shape  are  to 
be  avoided. 

II.    Note  on  Implicit  Dimensions  {see  pp.  53, 1 1 1, 143, 230). 

The  treatment  of  algebraical  symbols  as  representing  concrete 
quantities,  with  all  the  simplification  and  increased  interest  which 
this  treatment  involves,  was  first  effectively  called  attention  to  by  my 
brother,  Alfred  Lodge,  at  that  time  Professor  of  Pure  Mathematics 
at  Coopers  Hill,  and  now  a  Mathematical  Master  at  Charterhouse. 
See  Nature  for  July,  1888,  vol.  38,  p.  281,  which  was  the  sequel 


APPENDIX.  431 

to  a  pioneer  paper  read  by  him  before  the  Association  for  the 
Improvement  of  Geometrical  Teaching,  in  January,  1888. 

The  subject  was  subsequently  and  independently  developed  by 
Mr.  W.  Williams  of  South  Kensington,  now  of  Swansea ;  and, 
whether  it  has  received  full  recognition  or  not,  it  has  undoubtedly 
justified  itself  in  the  eyes  of  all  who  have  put  it  to  the  test  of 
practical  experience.  The  whole  subject  is  too  large  for  this  place, 
but  a  few  elementary  remarks  are  appropriate  : 

Quantities  of  different  kinds  do  not  occur  in  one  expression  ; 
in  other  words,  the  terms  of  an  expression  must  all  refer  to  the 
same  sort  of  things,  if  they  are  to  be  dealt  with  together  or 
equated  to  any  one  thing.     Nevertheless  an  expression  like 

is  common,  and  ^  may  be  a  length  ;  which  looks  as  if  we  could  add 
together  a  volume,  an  area,  a  length,  and  a  pure  number.  Not  so, 
really,  however  ;  suppressed  or  implicit  or  unexpressed  or  masked 
dimensions  must  in  that  case  exist  in  the  numerical  coefficients; 
the  coefficient  5  must  implicitly  or  tacitly  refer  to  a  length,  2  to  an 
area,  and  6  to  a  volume,  if  ^  is  a  length  ;  and  thus  all  the  terms  are 
really  of  the  same  kind.  So  they  always  will  be  in  every  real 
problem. 

When  an  equation  contains  terms  of  essentially  different  kind,  it 
must  really  consist  of  two  or  more  equations  packed  together  into 
the  apparent  form  of  one.  Thus  ^/(  -  1)  is  a  quantity  of  essentially 
different  kind  from  1  or  (v'  +  l) ;  the  former  being  imaginary,  the 
latter  real.  Hence  if  ever  they  occur  together  in  an  equation,  as 
for  instance  in  such  an  equation  between  complex  quantities  as 

«v/(  +  l)  +  6V(-l)  =  c^{  +  l)  +  d^(-l\ 
or  what  is  the  same  thing  (writing  /^(  -  1)  as  i,  for  short,  and  s/{  +  l) 
as  an  unexpressed  unity  factor) 

a  +  bi  =  c  +  di, 
it  must  be  regarded  as  a  double  equation,  unless  some  of  the 
quantities  a,  6,  c,  d  are  themselves  complex  ;  for  it  can  only  be 
interpreted  and  satisfied  by  the  two  separate  equations 

a  =  c  and   h  =  d. 
In  other  words  it  is  really  two   equations  packed  together  for 
brevity  into  a  single  statement, 


432  EASY  MATHEMATICS. 

For  if  either  of  these  conditions  is  not  satisfied,  if  for  instance  b 
is  less  than  d,  it  is  impossible  to  fill  up  the  deficit  by  any  increase 
in  the  value  of  a,  since  that  refers  to  a  quantity  of  totally  different 
kind.  A  deficiency  of  oxygen  in  the  atmosphere  cannot  be  com- 
pensated by  a  surplus  of  gold  in  a  bank  ;  nor  can  deficiency  of 
beauty  be  effectively  counterbalanced  by  excess  of  size. 

The  group  met  with  in  a  German  philosophical  treatise  (according 
to  a  writer  in  the  Hibhert  Journal)^  as  representing  the  class  which 
does  not  "  count "  for  moral  and  intellectual  purposes, 
"  cows,  women,  sheep,  Christians,  dogs, 
Englishmen,  and  other  democrats," 
cannot  be  regarded  as  classified  according  to  a  satisfactory  system, 
any  more  than  can  the  somewhat  similar  group  of  tax-payers  which 
is  at  present  disfranchised  by  Act  of  Parliament. 

So  that  any  conclusions,  inferences,  or  results  due  to  the 
aggregation  of  such  individuals  in  a  community  must  be  separable 
into  a  series  of  independent  conclusions,  inferences,  or  results,  except 
in  so  far  as  some  of  these  things  are  themselves  complex,  partaking 
more  or  less  of  each  other's  characteiistics. 

Sometimes  we  have  equations  among  integers  or  other  commen- 
surable numbers,  with  incommensurables  likewise  involved,  such  as 

If  now  m,  ?i,  ^,  y  are  all  to  be  considered  integers  or  any  vulgar 
fractions  or  terminating  or  recurring  decimals,  i.e.  unless  some  of 
the  quantities  m,  ?i,  .r,  y  are  in  whole  or  in  part  themselves  surds, 
it  must  follow  from  the  above  statement  that 

X  =  m  and  y  =  n, 
otherwise  the  equation  cannot  be  satisfied. 

Again  suppose  x  means  a  distance  measured  horizontally,  and  y 
a  distance  measured  vertically,  and  the  equation  given  is 

ax-hby  =  cx  +  dy  ; 
it  consists  of  two  distinct  and  independent  equations,  unless  a,  b, 
c,  d  are  themselves  directional  quantities  and  not  mere  numbers  ; 
in  that  case,  however,  i.e.  in  case  a,  c  are  vertical  lengths  and  6,  d 
are  horizontal  lengths,  the  equation  is  quite  homogeneous  and 
satisfactory,  and  denotes  certain  relations  among  rectangular  areas. 
Or  a,  c  may  be  reciprocals  of  horizontal  lengths,  and  b,  d  re- 
ciprocals of  vertical  lengths  ;  and  so  on.     But  if  a,  6,  c,  d  are  mere 


APPENDIX.  433 

numbers,  we  are  bound  as  before  to  equate  the  coefficients,  that  is 
to  say  to  admit  that  a  =  c  and  h  =  d ;  for  no  amount  of  horizontal 
travel  is  equivalent  to  a  rise,  nor  can  horizontal  dimension  make 
up  for  a  deficiency  in  height. 

In  any  single  equation  therefore,  like  v^  =  2gh  for  instance,  where 
one  side  is  plainly  the  square  of  a  velocity,  the  other  side  must 
also  be  really,  though  not  obviously,  the  square  of  a  velocity. 
And  since  g  is  an  acceleration  and  A  is  a  height,  those  who 
know  any  mechanics  will  realise  that  the  necessary  condition  is 
thoroughly  satisfied. 

But  when  g  is  interpreted  as  32  or  981,  the  fact  is  masked,  as 
facts  often  are  masked  by  the  incomplete  method  of  arithmetical 
or  numerical  specification.  If  32  or  981  is  regarded  as  a  pure 
number,  which  is  all  of  g  that  it  is  customary  actually  to  express  in 
writing,  then  the  equation  becomes  an  absurdity,  since  it  appears 
to  assert  that  a  velocity  multiplied  by  itself  results  in  a  certain 
multiple  of  an  elevation. 

But  when  it  is  remembered  that  the  32  means  really  32  feet  per 
second  per  second,  everything  is  perfectly  right ;  for,  the  height 
being  expressed  in  feet,  the  right-hand  side  of  the  above  equation 
is  so  many  square  feet  per  second  per  second,  or  square  feet 
divided  by  square  seconds,  which  is  the  square  of  a  velocity,  in 
perfect  agreement  with  the  left-hand  side. 

So  also  in  the  equation  to  a  parabola,  y  =  a-\-hx-\-cx^^  the 
convention  is  that  y  is  a  vertical  height,  and  x^  the  square  of  a 
horizontal  length  ;  but,  since  all  the  terms  must  really  be  alike  in 
kind,  it  follows  that  a  must  be  a  vertical  height  (and  it  is  :  viz.  the 
intercept  on  the  vertical  axis),  that  h  must  be  a  ratio  of  vertical  to 
horizontal  (and  it  is  the  tangent  of  an  angle  accordingly,  namely 

the  value  of  -^  at  the  place  where  the  curve  cuts  the  axis  of  y)  : 

and  further  that  c  must  be  a  sort  of  curvature,  a  quantity  in- 
volving a  vertical  direction  once  in  the  numerator  and  a  horizontal 

dimension  twice  in  the  denominator.     It  is  in  fact  half   .  .  •;„  ;   it 

{dxy 

represents  the  rate  at  which  the  tangent  to  the  curve  swings  round 
as  the  ordinate  travels  uniformly  along  the  axis  of  x  ;  and  this 
rate,  when  measured  by  changes  in  the  tangent  of  the  angle  of 
slope,  is  constant.     Compare  Chapter  XXXIX, 


434  EASY  MATHEMATICS. 

But  2^  =  ;r  is  also  a  possible  equation,  and  looks  as  if  a  vertical 
height  could  be  equivalent  to  a  horizontal  length.  But  it  is  only 
an  appearance,  due  to  suppressed  quantities.  The  coefficient  1, 
not  written,  is  really  the  tangent  of  an  angle  of  45°,  and  involves 
the  ratio  of  vertical  to  horizontal  required  to  restore  the  balance 
and  common  sense. 

So  also  when  y  —  x^  there  is  an  unwritten  unity  coefficient  which 
is  not  a  pure  number,  but  an  actual  quantity,  the  ratio  y  :  x^^  which 
the  equation  asserts  has  in  this  case  the  magnitude  1. 

Or  when  ^  =  6,  if  x  is  a  length,  it  follows  that  the  6  is  a  numeri- 
cal abbreviation  for  6  feet  or  6  centimetres  or  6  miles,  measured  in 
the  same  direction  as  x.  See  Article  in  "  School  World  "  for  July, 
1904. 

It  is  frequently  best  to  express  these  units  fully,  and  not  to  get 
too  exclusively  into  the  habit  of  writing  a  length  as  50  without 
saying  whether  inches  or  centiinetres  is  intended,  or  an  age  as  15 
without  saying  years  or  months,  or  a  price  as  42  without  saying 
shillings  or  pounds,  or  whether  it  is  per  hundredweight  or  per  ton 
(compare  pages  232,  4).  For  though  these  and  other  less  customary 
abbreviations  are  permissible  among  experts,  beginners  who  get  too 
used  to  them  are  apt  to  degenerate  into  slovenly  incompleteness 
and  inaccuracy,  and  to  suffer  by  finding  difficulties  hereafter  where 
none  exist. 

III.    Note  on  Factorisation  (see  Chapter  XIV.). 

A  quadratic  expression  ax'^ -^-hx-^-c  can  be  resolved  into  factors  if 
the  middle  term  hx  can  be  separated  into  two  parts  such  that  when 
multiplied  together  the  product  is  acx'^. 

Thus  take  3:pH10^+7, 

and  write  it  3jp2  +  ^x + 3.r  +  7  ; 

it  becomes  at  once  (3.r  +  7)  (.r  + 1 ). 

Again  take  bx"^  +  27.r + 28, 

and  write  it  hx"^  +  7a; + 20a;  +  28  ; 

it  becomes  (5a? +7) (a? +  4). 

When  a  quadratic  expression  is  thus  written  in  four  terms,  such 
that  the  product  of  the  means  is  equal  to  the  product  of  the 
extremes,  the  four  terms  are  necessarily  proportional ;  and  if  such 
proportionality  does  not  hold,  you  cannot  factorise. 


APPENDIX. 


435 


When  they  are  proportional,  as  in  the  above  case,  and  their  sum 
equated  to  zero,  {5x^-h7a:)  and  (20.r  +  28)  must  have  a  common 
factor  ;  so  also  must  {5x'^  +  20.r)  and  (7^  +  28)  have  another  common 
factor. 

If  we  write  such  four  proportional  terms  with  the  common  factors 
displayed,  they  must  have  the  form 

ac  +  ad+bc  +  bd ; 
which  terms  geometrically  represent  themselves  thus  : 

a  h 


ac 

Ic 

ad 

bd 

bj:^- 

7. 

20^ 

28 

In  the  above  example  either  aisx  and  b  is  4,  which  are  equivalent  ; 
or  a  is  5.r  and  b  is  7,  which  are  also  equivalent.  For  instance  the 
diagram  explicitly  applied  to  the  above  case  would  look  thus  :  no 
scale  being  implied  in  the  drawing,  but  simply  a  framework. 

(5x)       (7) 

(X) 

(4) 

IV.    Note  on  the  Growth  of  Population  (page  220). 

In  spite  of  what  has  been  said  in  the  text  as  to  the  danger  of 
applying  the  geometrical  law  of  increase,  or  indeed  any  fixed  law 
of  increase,  to  a  given  country  or  to  any  assemblage  without  taking 
into  account  all  the  circumstances,  nevertheless  the  growth  of  the 
population  of  England  and  Wales  during  the  last  century  illustrates 
with  remarkable  closeness  the  geometrical-progression  law. 

The  following  is  a  table  of  the  common  logarithms  (to  base  10)  of 
the  population  of  England  and  Wales  for  each  decade  from  1801  to 
1901  ;  together  with  their  differences.  If  the  geometrical  law  held 
precisely,  these  differences  would  all  be  constant ;  as  it  is,  they 
hover  about  a  mean  value,  except  in  some  of  the  early  years  of 
the  century,  when  they  are  abnormally  big, — apparently  as  a 
reaction  from  the  Napoleonic  wars,  but  doubtless  also  on  account 
of  some  applications  of  science,  and  other  economic  conditions. 


436  EASY   MATHEMATICS. 

Population  table  of  England  and  Wales  for  last  century. 


Logs. 

Diffs. 

1801 

6-949026 

-058050 

1811 

7-007076 

-072114 

1821 

7-079190 

-063724 

1831 

7-142914 

-058869 

1841 

7-201783 

-051739 

1851 

7-253522 

-048944 

1861 

7-302466 

-053794 

1871 

7-356260 

•058286 

1881 

7-414546 

-047890 

1891 

7-462436 

•049796 

1901 

7-512232 

From  these  data  the  curve  of  population  might  be  plotted,  and 
it  will  be  seen  that  from  1841  onwards  it  would  be  fairly  steady, 
the  mean  or  average  diflFerenee  for  10  years  during  this  period 
being  -051741.    So  the  difference  of  logarithms  for  one  year  is  -00517, 

or  to  base  e,  -0119,  or  say  -012.     But  we  know  that  dlogep  =  -^ 

(see  Chap.  XLII.)  ;  hence  -012,  or  12  per  thousand  per  annum,  is 
the  average  rate  of  increase  of  the  population  since  1841.  The 
curve  is  mainly  a  geometrical-progression  or  exponential  curve, 
with  this  value  as  the  common  ratio. 

All  the  fluctuations  noticed  in  such  a  curve  could  doubtless  be 
explained  instructively,  though  to  some  extent  hypothetieally,  by 
a  Historian.  For  instance  there  is  an  excessive  rate  of  growth  in 
the  decade  1871  to  1881,  which  probably  includes  a  period  of  good 
trade ;  but  even  that  is  not  equal  to  the  rates  of  increase  nearer  the 
beginning  of  the  century,  when  presumably  the  population  was 
emerging  out  of  extreme  poverty. 


GLASGOW  :   PRINTED  AT  THE  UNIVERSITY  PRESS  BY  ROBERT  MACl.BHOSE  AND  CO.   LTD. 


OF  3-FIGURE  LOGARITHMS 

2 

3 

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5 

6 

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8 

9 

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021 

025 

029 

033 

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090 

093 

097 

100 

104 

107 

111 

7 

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137 

140 

143 

9 

152 

155 

158 

161 

164 

167 

170 

173 

9 

182 

185 

188 

190 

193 

196 

199 

201 

)7 

210 

212 

215 

218 

220 

223 

225 

228 

13 

236 

238 

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243 

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248 

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253 

»8 

260 

263 

265 

267 

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274 

277 

n 

283 

286 

288 

290 

292 

295 

297 

299 

!2 

342 

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380 

398 

415 

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447 

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505 

519 

532 

544 

556 

568 

580 

591 

3 

623 

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681 

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)8 

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964 

969 

973 

978 

982 

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996 

14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 


LOAN  DEPT. 


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