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UNIVERSITY  OF  CALIFORNIA. 


1  OF 

HENRY  DOUGLASS  BACON. 

1877. 

Accessions  No.  ___/?**_&/__   Shelf  No 


'••"v*: 


Wtl\> 


fVV^V;" 


TREATISE  ON  AJ 


X 


CONTENTS. 


BOOK  I. 

WHOLE    NUMBERS. 
CHAP.   L 

On  the  Ideas  of  Number,  and  their  Expression  by  Words    .  Page  1 

CHAP.  IL 
On  the  Method  of  expressing  Numbers  by  Symbols  or  Figures      .          23 

CHAP.  III. 

Addition       -  -  .  -  -  -  .50 

CHAP.  IV. 
Subtraction     ...  ...  -  .68 

CHAP.  V. 
Multiplication  -  -        '86 

CHAP.  VI. 

Division  .          -  -  -       118 

BOOK  II. 

FRACTIONS. 
CHAP.   1. 

On  the  Language  and  Notation  of  Fractions.— Various  Ways  of  expressing 
them.— Their  relative  Values.— Their  Addition  and  Subtraction         145 

CHAP.  II. 

The  Multiplication  and  Division  of  Fractions       *  171 

A 


IV  CONTENTS. 

CHAP.  III. 
Decimals      -.•-----      Page  189 

BOOK  III. 

COMPLEX    NUMBERS. 
CHAP.   I. 

Of    complex  Numbers  in  general.  —  Their  Reduction   and  Simplifies, 
tion          --.„...       209 

CHAP.  II. 

Of  the  Addition  and  Subtraction  of  Complex  Numbers     =/  -       242 

CHAP.  III. 

Of  the  Multiplication  of  Complex  Numbers    -  .       250 

CHAP.  IV. 
Of  the  Division  of  Complex  Numbers  -  .'      S72 

BOOK  IV. 

ON    PROPORTION    AND    ITS   PRACTICAL   APPLICATIONS. 

CHAP.   L 
Proportion          ......  -.286 

CHAP.  IL 

The  Rule  of  Three    -  ...       295 

CHAP.  IIL 

interest — Discount.— Profit  and  Loss.— Brokerage. — Commisaion.— insur- 
ance.—Tare  and  Tret— Partnership  .  .  3£8 


A 

TREATISE 

ON 


ARITHMETIC. 


BOOK  I. 

WHOLE    NUMBERS. 

l^  jo  •*?) 

CHAPTER   1^ 

ON    THE    IDEAS    OF    NUMBER,    AND    THEIR    EXPRESSION 
BY    WORDS. 

(1.)  THE  early  period  of  life  at  which  we  begin  to  ac- 
quire ideas  of  number,  and  the  influence  produced  on 
our  minds  by  becoming  familiar  with  the  language  of 
numeration  before  any  notions  of  the  higher  classes  of 
number  are  formed,  throw  great  difficulties  in  the  way 
of  any  attempt  to  retrace  the  steps  by  which  the  art  of 
counting  arrived  at  its  present  high  state  of  perfection. 
Under  these  circumstances  we  must  naturally  direct  our 
enquiries  to  the  state  of  arithmetic  among  people  less 
advanced  in  civilisation  than  ourselves.  But  it  is  a 
remarkable  fact,  that  while  other  sciences  are  almost 
unknown  beyond  the  limits  of  civilisation,  there  is 
no  example  of  a  people  without  a  system  of  numer- 
ation more  or  less  extensive  and  perfect,  with  the  ex- 
ception of  a  few  savage  tribes,  whose  notions  of  number 
are  singularly  limited. 

It  is,  however,  apparent,  that  before  a  people  could 
make  any  progress  in  numeration  beyond  a  few  of  the 
smaller  combinations  which  may  be  signified  by  the 
fingers,  two  things  were  indispensably  necessary  to  be 
accomplished. 


2  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

1.  To  devise  a  method  by  which  the  mind  should 
form  a  clear  and  distinct  notion  of  the  number  of  indi- 
vidual objects  in  any  assemblage,  however  great  might 
be  the  multitude  of  which  the  assemblage  consisted. 

2.  To  devise  a  system  of  names  for  numbers,  such 
that  it  might  be  possible  for  the  memory  to  retain  the 
terms  necessary  to  express  the  unlimited  variety  of  no- 
tions which  any  system  of  numeration  must  necessarily 
involve. 

(2.)  It  is,  perhaps,  one  of  the  most  curious  and  in- 
teresting facts  in  the  history  of  the  human  mind,  that 
all  the  nations  of  the  earth,  which  have  possessed  any 
system  of  numeration,  have  adopted  methods  of  solving 
these  two  problems,  which,  in  their  general  features, 
and  in  all  essential  points,  are  identical,  and  which,  in 
some  cases,  are  the  same,  even  in  their  most  minute 
details. 

Let  us  suppose  two  boxes,  containing  a  large  number 
of  individual  objects  of  the  same  kind,  such  as  counters, 
placed  before  a  person  unacquainted  with  the  received 
nomenclature  of  number,  and  who  wishes  to  form  a 
notion  of  the  number  of  counters  in  each  box,  so  as  to 
be  able  to  compare  them,  and  to  pronounce  which  col- 
lection is  the  greater.  Such  a  person  would,  probably, 
begin  by  withdrawing  the  counters  one  by  one  from 
one  of  the  boxes,  disposing  them  side  by  side,  so  that, 
upon  inspection,  he  would  have  a  distinct  notion  of  the 
number  withdrawn.  He  would,  however,  presently 
find  that  the  collection  withdrawn  would  become  so 
numerous,  that  a  mere  inspection  would  give  him  no 
clear  or  distinct  notion  of  it.  In  fact,  he  would  feel 
himself  in  that  situation  that  having  withdrawn  a 
collection  from  each  box,  and  having  each  spread  before 
him  he  would  not  immediately  perceive  which  were 
the  more  numerous. 

Instead,  therefore,  of  spreading  before  him  a  large 
number  of  the  counters,  he  will,  in  the  first  place,  col- 
lect into  one  group  only  so  many  as  he  can  form  a  clear 
idea  of  by  inspection.  Let  us  suppose  that  he  fixes 
upon  the  number  six;  he  then  withdraws  six  more,  and 


CHAP.  I.  IDEAS    OF    NUMBER.  3 

disposes  them  in  another  group,  placed  beside  the  first. 
He  proceeds  in  the  same  manner  to  form  a  third  group 
of  six,  and  continues  to  form  groups  of  six,  by  con- 
stantly withdrawing  the  counters  from  one  of  the  boxes. 
When  six  of  these  groups  are  formed.,  it  is  obviou? 
that  he  will  have  as  clear  a  conception  of  the  number  of 
counters  which  they  contain  as  he  has  of  the  number  of 
counters  contained  in  any  one  of  the  groups ;  but  for 
the  same  reason  that  he  did  not  place  more  than  six 
counters  in  each  group  in  the  first  instance  he  will  not 
now  collect  together  more  than  six  groups.  Nothing, 
however,  prevents  him  from  regarding  these  six  groups 
as  a  single  group  of  a  higher  order,  and  pursuing  his 
former  method  until  he  form  a  collection  of  six  groups 
more ;  he  will  then  have  two  collections  of  six  groups, 
six  counters  being  contained  in  each  group.  In  the 
same  manner  he  may  proceed  to  form  the  third  collec- 
tion of  six  groups,  and  so  on,  until  he  has  formed  six 
such  collections.  It  will  be  obvious  upon  the  slightest 
reflection  that  his  notion  of  the  number  of  counters 
contained  in  the  six  collections  thus  formed  is  as  clear 
and  definite  as  is  his  notion  of  only  six  counters.  If  a 
proof  of  this  be  required,  it  may  be  readily  given.  Let 
him  form  six  similar  collections  of  six  groups  from  the 
other  box :  but  let  us  suppose  that  one  of  the  groups 
in  one  collection  shall  be  deficient  by  a  single  counter ; 
he  will  then  be  immediately  conscious  that  the  number 
of  counters  withdrawn  from  one  box  exceeds  the  number 
withdrawn  from  the  other  by  a  single  counter,  and  his 
idea  of  the  number  of  counters  withdrawn  from  the  one 
box  is  as  distinct  from  his  idea  of  the  number  with- 
drawn from  the  other  box  as  his  notion  of  a  single 
counter  would  be  distinct  from  his  notion  of  the  total 
number  withdrawn. 

We  may  imagine  the  first  group  of  six  counters  with- 
drawn to  be  thus  arranged  :  — 

000 
000 

B  2 


4  A    TREATISE    ON    ARITHMETIC.  BOOK  J 

The  number  of  objects  here  is  so  limited  that  we  have 
an  immediate  and  complete  perception  of  it  the  moment 
\ve  look  at  it.  If  but  a  single  object  be  added  or  taken 
away,  we  become  immediately  conscious  that  the  num- 
ber is  increased  or  diminished.  The  first  collection  of 
six  groups  withdrawn  may  be  represented  thus  :  — 

000      000      000 
000      000      000 

000      000      000 
000      000      000 

Each  group  in  this  collection  is  similar  to  each  counter 
in  a  single  group,  and  a  single  group  withdrawn  or 
added  would  become  immediately  apparent,  as  would 
a  single  counter  deficient  or  in  excess  in  any  one  of  the 
groups.  It  will  be  equally  easy  to  imagine  six  such 
collections  of  groups  arranged  in  the  same  manner. 

The  spirit  of  this  method  consists  in  forming,  in  the 
first  instance,  a  group  of  objects  so  limited  in  number 
that  the  mind  can  picture  to  itself  so  clear  and  distinct 
a  notion  of  them  that  the  increase  or  diminution  of 
their  number  by  one  would  be  immediately  perceptible  ; 
these  groups  are  then  treated  as  the  individual  objects 
were  treated  in  the  first  group,  being  formed  into  col- 
lections, having  the  same  number  of  groups  as  there 
are  individual  objects  in  each  group.  These  collections 
of  groups  are  again  collected  in  assemblages  of  a  higher 
order,  consisting  still  of  the  same  number  of  collections 
as  there  were  individuals  in  the  first  group,  and  this 
process  has  obviously  no  limit. 

(3.)  Such  are  the  general  features  of  the  method  of 
numeration  in  which  all  nations  of  the  earth  without 
a  single  exception  have  concurred.  The  only  point  in 
the  system  which  is  arbitrary  is  the  number  of  indivi- 
duals which  may  be  adopted  to  form  the  first  group, 
and  we  accordingly  find  that  systems  of  numeration 
differ  in  this  particular ;  still  even  here  there  is  a  sur- 
prising coincidence,  even  among  people  so  far  removed 
as  to  preclude  all  possibility  of  conventional  arrange- 


OHA-P.  I.  IDEAS    OF    NUMBER.  5 

ment.  The  fingers  were  naturally  the  first  objects 
which  presented  to  the  mind  the  idea  of  number;  and 
they  furnished  also  a  set  of  natural  counters  by  which 
the  number  of  other  things  might  be  marked  and  ex- 
pressed. The  fingers  being  continually  in  view  fami- 
liarised the  mind  with  the  contemplation  of  every 
number  of  objects  not  exceeding  ten.  It  was  natural, 
therefore,  that  ten  should  be  adopted  as  the  number  of 
objects  to  form  the  first  group.  In  the  system  of  nu- 
meration, which  has  been  just  explained,  and  in  the 
example  which  we  have  already  given,  we  have  adopted 
six  ;  had  ten  been  adopted  as  the  radix  of  our  example, 
we  should  have  first  formed  a  group  of  ten  counters, 
and  then  formed  in  like  manner  nine  other  groups. 
These  ten  groups  would  then  form  a  group  of  a  higher 
order,  in  which  each  group  would  play  the  same  part 
as  the  individual  counters  did  in  the  original  groups  ; 
and  we  should  have  proceeded  in  the  same  manner  to 
form  ten  assemblages  of  the  next  order,  and  so  on. 

A  curious  example,  illustrative  of  the  universality  of 
the  method  of  forming  ideas  of  large  numbers,  which 
has  been  above  explained,  is  afforded  in  the  history  of 
the  island  of  Madagascar.  *  When  the  people  of  that 
island  wish  to  count  a  great  multitude  of  objects,  such, 
for  example,  as  the  number  of  men  in  a  large  army, 
they  cause  the  objects  to  pass  in  succession  through  a 
narrow  passage  before  those  whose  business  it  is  to  count 
them.  For  each  object  that  passes  they  lay  down  a 
stone  in  a  certain  place ;  when  all  the  objects  to  be 
counted  have  passed,  they  then  dispose  the  stones  in 
heaps  of  ten  ;  they  next  dispose  these  heaps  in  groups, 
having  ten  heaps  each,  so  as  to  form  hundreds ;  and  in 
the  same  way  would  dispose  the  groups  of  hundreds  so 
as  to  form  thousands,  until  the  number  of  stones  has 
been  exhausted. 

Although  ten  has  been  so  generally  adopted  as  the 
radix  of  systems  of  numeration  as  to  leave  no  doubt  of 

*  Histoire  de  la  Grande  Isle  de  Madagascar,  par  de  Flacourt,  ch.  xxviii. 
lool,  quoted  by  Peacock 

B    3 


6  A    TREATISE    ON    ABITH31ETIC.  BOOK  I. 

its  origin,  yet  it  is  not  the  only  one  which  has  heen 
used,  nor  is  it  the  only  radix  having  a  natural  origin. 
The  ringers  of  one  hand  rendered  the  number  five  fa- 
miliar to  the  mind  before  the  conception  of  ten  as  a 
distinct  number  presented  itself.  It  was  even  more 
natural  and  obvious  that  the  fingers  should  be  con- 
templated as  two  groups  of  five  than  as  a  single  group 
of  ten.  We  accordingly  find,  in  some  instances,  the 
number  five  taken  as  the  radix  of  numeration,  but  there 
are  inconveniences  arising  from  its  smallness  which  will 
be  more  clearly  perceived  hereafter.  The  existence  of 
the  members  and  principal  organs  of  the  human  body, 
and  of  the  bodies  of  other  animals  in  pairs,  furnished 
a  natural  foundation  for  adopting  two  as  a  radix ;  but 
the  process  of  grouping  any  considerable  number  by 
pairs,  and  by  pairs  only,  would  be  attended  with  still 
more  inconvenience  than  the  quinary  radix. 

(4.)  The  solution  of  the  second  problem  above  stated 
by  the  discovery  of  an  appropriate  system  of  names 
to  express  the  numbers,  of  which  clear  ideas  may  be 
formed  by  the  method  of  grouping  and  arrangement 
just  explained,  must  needs  have  proceeded,  pari  passu, 
with  the  solution  of  the  first.  Distinct  names  would 
be  given  to  every  collection  of  objects  not  exceeding  the 
number  selected  to  form  the  first  group.  The  same 
succession  of  names,  properly  modified,  would  express 
all  collections  of  groups  less  in  number  than  the  radix : 
thus,  in  the  first  example  above  given,  we  should  re- 
quire six  distinct  names  to  express  all  numbers  not 
exceeding  the  number  six,  which  was  selected  as  the 
radix  of  the  system.  The  same  names,  however,  which 
expressed  the  number  of  individual  objects  as  far  as 
six  would  likewise  express  the  number  of  groups  as  far 
as  six.  A  new  name  would  here  become  necessary  to 
distinguish  six  groups  from  six  individuals.  By  pro- 
perly combining  the  twelve  names,  thus  formed,  we 
should  be  enabled  to  express  all  numbers  under  six  col- 
lections of  six  groups  ;  but  here  a  new  term  must  be 
introduced.  Such  are  the  general  features  of  the 


CHAP.  I.  DECIMAL    NOMENCLATURE.  7 

method  of  naming  numbers,  which  has  been  adopted  in 
all  languages. 

(5.)  What  we  have  here  stated  will  be  more  clearly 
apprehended  when  illustrated  by  the  system  of  decimal 
numeration.  Ten  being  the  radix  in  that  system.,  the 
numbers,  as  far  as  ten  inclusive,  have  distinct  and  in- 
dependent names,  one,  two,  three,  four,  five,  six,  seven, 
eight,  nine,  ten.  The  names  eleven  and  twelve  being 
anomalous,  we  shall,  for  the  present,  substitute  for  them 
ten-one  and  ten-two.  The  names,  then,  for  all  numbers 
expressing  the  first  group,  and  part  of  the  second,  would 
be  ten-one,  ten-two,  ten-three,  ten-four,  ten-five,  ten-six, 
ten-seven,  ten-eight,  ten-nine.  Two  groups  of  ten  are 
expressed  by  two-tens,  which  abridged,  is  twenty.  All 
collections  of  individuals  consisting  of  two  groups  and 
a  part  of  a  third  are  expressed  by  twenty-one,  twenty- 
two,  &c. :  three  groups  would  be  expressed  by  three 
tens,  which,  abridged,  is  thirty,  and  so  on.  In  this 
way,  by  ten  names  and  their  combinations,  we  are  en- 
abled to  express  any  number  of  individuals  less  than 
ten  groups  of  ten.  Ten  groups  of  ten  must  be  ex- 
pressed by  ten-tens ;  but  the  reduplication  of  the  same 
term  in  such  a  compound,  if  it  can  be  called  so,  has 
rendered  it  more  convenient  to  introduce  a  new  name, 
and  accordingly  ten  groups  of  ten  are  called  a  hundred. 

(6.)  Before  we  proceed  further  in  explaining  the 
language  of  numeration,  we  may  observe  that  the  indi- 
vidual objects  which  any  number  immediately  expresses 
are  called,  with  reference  to  that  number,  units*  Thus 
the  counters  in  the  example  first  given  are  the  units 
composing  each  group;  but  in  the  collections  of  six 
groups  subsequently  formed  the  groups  themselves  are 
the  immediate  units.  In  like  manner,  in  the  decimal 
system  of  numeration,  the  immediate  units  of  such  a 
number  as  seventy,  or  seven-tens,  are  groups  of  ten, 
the  units  of  each  group  being  the  individual  objects, 
whatever  they  may  be  to  which  the  number  is  applied. 
In  the  decimal  scale,  therefore,  among  the  numbers 

*  UNUS,  one 
B    4 


8  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

from  one  to  one  hundred  there  are  two  orders  of  units> 
the  first  being  the  individual  objects,  and  the  second 
the  groups  of  ten. 

When  ten  groups  of  ten  are  collected  together,  let  us 
imagine  them  set  apart  and  expressed  by  the  word  hun- 
dred ;  another  collection  of  ten  groups  of  ten  may  be 
placed  beside  them,  and  the  number  will  be  expressed 
by  two  hundreds ;  a  third  similar  collection  will  make 
the  number  three  hundreds,  and  so  on,  until  ten  such 
collections  are  made.  It  is  evident  that  when  that 
happens  a  collection  will  be  formed  in  which  hundreds 
play  the  same  part  as  tens  did  with  respect  to  hundreds, 
and  as  individual  objects  or  original  units  did  with  re- 
spect to  tens.  A  new  name  is  imposed  on  such  a  num- 
ber, and  it  is  expressed  by  the  word  thousand. 

(7.)  These  collections  of  ten-hundreds  or  thousands 
may  be  arranged  in  the  same  manner  as  hundreds, 
until  we  obtain  ten  such  collections  :  analogy  would  then 
require  the  imposition  of  another  name,  but  in  our 
language  the  compound  term  ten-thousand  has  been 
adopted.  When  ten  similar  collections  are  made,  analogy 
would  again  require  a  new  name,  but  the  language  is 
again  in  discordance  with  the  numerical  theory,  and  the 
compound  term  hundred-thousand  is  used ;  ten  groups 
of  this  class,  which  make  up  ten  hundred-thousands,  or 
a  thousand  thousands,  is  called  a  million. 

Let  us  pause  here  and  recapitulate  the  composition  of 
this  number.  The  objects  or  units  of the  first  -  order  are 
first  conceived  to  be  arranged  in  groups  of  ten,  each  group 
being  considered  a  unit  of  the  second  order  :  these  groups 
are  then  conceived  to  be  arranged  Like  the  primary  units 
in  groups  of  ten,  each  group  being  called  a  hundred, 
these  groups  of  a  hundred  being  the  units  of  the 
third  order.  It  will  be  evident,  that  the  number  of 
units  in  the  collection  of  each  order  ascending  becomes 
rapidly  less,  the  number  of  units  of  the  second  order 
being  ten  times  less  than  the  number  of  primary  units, 
and  the  number  of  units  of  the  third  order  being  again 
ttn  times  less  than  the  number  of  units  of  the  secoi  d 


CHAP.  I.  DECIMAL    NOMENCLATURE.  9 

order.  The  groups  of  hundreds  are  now,  in  like  man- 
ner, disposed  in  collections  of  ten,  to  form  units  of  the 
fourth  order,  and  these  collections  are  called  thousands; 
these  again  are  collected  in  groups  of  ten,  to  form  units 
of  the  fifth  order,  called  ten-thousands ;  and  these  in 
groups  of  ten,  to  form  hundred-thousands.  Of  these  units 
there  are  but  ten  in  the  number  contemplated :  these  ten, 
called  a  million,  form  a  single  unit  of  the  seventh  order. 

(8.)  A  clear  and  distinct  idea  of  the  number  ex- 
pressed by  the  word  ' '  million"  being  thus  formed,  we 
may  regard  that  number  as  a  unit  in  the  formation  of 
higher  numbers,  and  we  may  apply  to  two  numbers,  the 
units  of  which  are  millions,  the  same  system  of  names 
which  have  been  applied  to  the  numbers  rising  from  the 
primary  units  to  a  million.  Accordingly,  ten  millions 
are  as  easily  conceived  and  contemplated  as  ten  units. 
The  same  may  be  said  of  a  hundred  or  a  thousand 
millions,  and  so  on,  until  we  have  to  express  a  million 
millions,  when  it  is  necessary  for  this  number  to  intro- 
duce a  new  name;  it  is  called  a  billion:  in  the  same  way, 
a  billion  being  regarded  as  a  new  unit,  the  original 
names  are  applied  until  we  reach  a  million  billions, 
which  is  called  a  trillion.  A  trillion  in  the  same  man- 
ner becoming  a  unit  is  counted  like  the  original  units 
until  we  reach  a  million  trillions,  which  is  called  a  qua- 
drillion, and  so  on. 

(9.)  In  this  system  of  naming  the  first  four  orders  of 
units  are  expressed,  by  distinct  terms,  units,  tens,  hun- 
dreds, thousands.  The  units  of  the  fifth  and  sixth  orders 
are  expressed  by  compounding  the  names  of  those  of 
the  second  and  third  orders  with  that  of  the  fourth, — 
ten-thousands,  hundred-thousands.  The  seventh  order 
of  units  would  be  expressed  by  compounding  the  name 
of  the  fourth  with  itself,  —  thousand-thousands.  To 
avoid  this  reiteration,  the  seventh  order  of  units  has 
received  a  distinct  name,  —  millions.  The  names  of 
the  several  orders  of  units  after  this  are  formed  by 
compounding  the  prev:ous  names  until  we  arrive  at  the 
thirteenth  order  of  uniis,  the  compound  name  of  which 


10  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

would  be  "  millions  of  millions."  Here  a  new  name  is 
imposed,  —  billion.  After  this  a  new  name  is  introduced 
at  every  sixth  order  of  units. 

It  is  impossible  not  to  be  struck  with  the  admirable 
simplicity  and  efficiency  of  this  system  of  names,  and 
its  beautiful  adaptation  to  that  mode  of  arrangement  by 
which  the  mind  is  enabled  to  picture  to  itself  numbers, 
however  high,  so  distinctly,  that  any  two  differing  only 
by  a  single  unit  are  as  clearly  and  unequivocally  distinct 
as  the  most  unequal  numbers ;  and  by  which  the  num- 
ber of  names  is  so  limited,  that  every  memory  can 
easily  retain  them,  and  yet  capable  by  composition  of 
expressing  a  number  of  ideas  which  considerably  exceeds 
the  number  of  words  commonly  used  in  the  most  copious 
language. 

(10.)  The  nomenclature  of  number  in  all  languages 
clearly  establishes  the  fact,  that  men  have  acquired  their 
notions  of  number  by  the  method  of  arranging  and 
grouping  individual  objects,  which  we  have  already  ex- 
plained. The  anomalies  and  irregularities  which  are 
sometimes  found  even  in  the  nomenclature  of  the  most 
civilised  nations,  and,  indeed,  often  more  in  them  than 
in  the  language  of  people  less  advanced,  are  indicative, 
not  of  any  departure  from  this  method,  but  of  an  incon- 
stancy in  the  radix  of  the  scale.  It  would  appear  that, 
in  process  of  time,  it  had  been  found  that  a  radix  in- 
conveniently great  or  inconveniently  small  had  been 
used,  and  was  by  general  consent  abandoned,  leaving, 
however,  traces  of  its  existence  in  some  parts  of  the 
numerical  nomenclature.  In  some  cases,  the  radix  ap- 
pears to  have  been  doubled  at  a  former  epoch,  and  we 
find  traces  in  the  language  of  one  radix  subordinate  to 
another. 

(11.)  In  the  English  language  the  nomenclature  of 
number  is  purely  decimal,  subject,  however,  to  a  slight 
irregularity  of  structure,  as  already  observed,  from  ten 
to  twenty.  The  words  (<  eleven'*  and  (f  twelve"  owe 
their  origin  to  the  combinations  of  the  words  left  one, 
and  two  left,  meaning  that  in  counting  a  collection 


CHAP.  I.  NOMENCLATURE.  11 

of  eleven  or  twelve  individuals  when  the  radix  was 
counted  off  one  was  left  in  the  one  case,  and  two  were 
left  in  the  other :  the  formation  of  thirteen,  fourteen, 
fifteen,  sixteen,  seventeen,  eighteen,  nineteen,  is  ob- 
vious :  twenty  is  derived  from  the  Gothic  twentig,  com- 
pounded of  twa,  two,  and  tig,  ten  ;  hence  the  formation 
of  thirty,  forty,  &c.  is  apparent.  The  names  used  for 
units  of  the  third  and  fourth  orders,  hundred  and  thou- 
sand, are  from  the  German  hundert  and  tausend :  the 
names  of  the  higher  orders  of  units  are  from  the  Italian, 
millione  signifying  a  great  thousand,  f.  e.  a  thousand 
thousand,  and  hence  billione,  trillione,  &c. 

(12.)  The  French  nomenclature  is  for  the  most  part 
purely  decimal.  From  ten  to  twenty  the  structure, 
like  the  English,  is  anomalous,  but  still  formed  from  the 
ten  preceding  names.  By  strict  analogy  the  numerals 
should  be  dix-un,  dix-deux,  dix-trois,  dix-quatre,  &c. 
instead  of  onze,  douze,  treize,  &c.  The  decimal  system 
is  observed  from  twenty  (vingt)  to  sixty  (soixante) : 
here  we  find  a  vestige  of  an  old  vicenary  scale.  Seventy, 
instead  of  being  septante,  as  the  decimal  system  would 
require,  is  soixante-dix  (sixty-ten)  ;  seventy-one,  soix- 
ante-onze  (sixty-eleven)  ;  seventy-two,  soixante-douze 
(sixty-twelve),  &c.  Eighty,  instead  of  being  octante,  is 
quatre-vingt,  or  four  twenties,  and  ninety  is  quatre-vingt- 
dix  (four  twenties  ten) ;  ninety-one,  quatre-vingt- 
onze  (four  twenties  eleven),  &c.  Thus  twenty  be- 
comes the  radix  of  the  system  from  sixty  to  a  hundred. 

(13.)  The  Greek  and  Latin  nomenclatures  are  purely 
decimal,  the  term  twoa-i,  twenty  in  Greek,  being,  how- 
ever, anomalous.  Kovra  in  Greek  and  ginta  in  Latin, 
when  compounded,  signify  ten  ;  and  we  have,  accord- 
ingly, viginti  (Wginti)  twenty,  rpaxoyra,  triginta,  thirty, 
TecrirapavtovTo,  quadraginta,  forty,  &c.  &c. 

(14.)  The  most  perfect  and  symmetrical  nomenclature 
for  decimal  numeration,  so  far  as  it  is  known  to  extend, 
is  found  in  the  language  of  THIBET.  The  first  ten 
numbers  are  expressed  in  that  language  as  usual  by  ten 
independent  terms  as  follows :  — 


12  A    TREATISE    ON    ARITHMETIC. 


Cheic  -  one. 

Gnea   -  two. 

Soom  -  three. 

Zea      -  four. 


Tru    -     -     six. 
Toon  -     -     seven. 
Ghe   -     -     eight. 
Goo    -     -     nine. 
Chutumbha  ten. 


Gna     -    five. 

In  forming  the  numbers  from  ten  to  twenty  the  last 
two  syllables  of  the  name  of  ten  are  cut  off,  and  that 
number  expressed  by  the  first  syllable  "  chu : "  to  ex- 
press eleven,  twelve,  &c.  this  syllable  is  succeeded  by  the 
names  for  one,  two,  three,  &c.,  so  that  the  numbers 
from  ten  to  nineteen  are  expressed  as  follows :  — 


Chucheic  (ten-one)  eleven. 
Chugnea  (ten-two)  twelve. 
Chusoom  (ten-three)  thirteen. 
Chuzea  (ten-four)  fourteen. 


Chutru  (ten-six)  sixteen. 
Chutoon  (ten-seven)  seventeen. 
Chughe  (ten-eight)  eighteen* 
Chugoo  (ten-nine)  nineteen. 


Chugna  .  ten-five  i  fifteen. 

Twenty  is  expressed  by  gnea  chutumbha,  two-ten  : 
the  numbers  from  twenty  to  twenty-nine  are  expressed 
by  the  name  for  two,  followed  by  the  names  for  one, 
two,  three,  &c. 

Gnea  cheic  (two-one)  twenty-one. 
Gnea  gnea  (two-two)  twenty-two. 
Gnea  soom  (two-three)  twenty-three. 
Gnea  zea  (two-four)  twenty-four. 
Gnea  gna  (two-five)  twenty-five. 
Gnea  tru  (two-six)  twenty-six. 
Gnea  toon  (two-seven)  twenty-seven. 
Gnea  ghe  (two-eight)  twenty-eight. 
Gnea  goo  (two-nine)  twenty-nine. 

It  will  be  easily  perceived  that  this  system  is,  in  fact, 
a  transcript  of  the  modern  method  of  expressing  numbers 
by  the  Arabic  digits,  and  must  certainly  be  admitted  to 
be  by  far  the  most  perfect  nomenclature  extant.  There 
is,  however,  no  evidence  of  the  continuation  of  the  no- 
menclature beyond  the  above  names :  these  are  given 
by  Turner  and  other  authorities.* 

*  Turner's  Embassy  to  Thibet,  321. ;  Klaproth,  Asia  Polyplotta,  -353. ; 
Remusat,  Recherches  sur  les  Langues  Tartares,  364.,  as  quoted  by  Peacock 
in  his  article  on  Arithmetic. 


ORIENTAL    NOMENCLATURES. 


(15.)  The  Sanskrit  nomenclature  is  even  more  purely 
decimal  than  most  of  the  modern  European  languages,, 
inasmuch  as  a  new  name  is  introduced  for  every  order 
of  units.  In  the  English  and  kindred  languages,  the 
grouping  of  number  beyond  a  thousand  is,  strictly  speak- 
ing, by  thousands,  so  that  the  immediate  radix  of  the 
system  is  a  thousand,  ten  being  subordinate  to  it.  This, 
it  is  true,  is  a  peculiarity  to  be  ascribed  rather  to  the 
names  of  high  numbers  than  to  our  method  of  con- 
ceiving them,  and  does  not  at  all  belong  to  their  ex- 
pression by  figures.  In  the  Sanskrit,  the  first  ten 
numbers  are  expressed  by  the  following  independent 
names,  to  which  the  Latin  and  its  derivatives  are  evi- 
dently related  :  — 


Eca 

-     one. 

Shata 

-     six. 

Dwau 

-     two. 

Sapta 

-     seven. 

Traya 

-     three. 

Ashta 

-     eight. 

Chatur 

-    four. 

Nova 

-     nine. 

Ponga 

-    five. 

Dasa 

-     ten. 

The  names  for  the  successive  orders  of  units  are  car- 
ried to  a  surprising  extent :   they  are  as  follows  : — 


Eca 

Dasa 

Sata 

Sahasra 

Ay  uta 

Lacsha 

Prayuta 

Coti 

Arbuda 

Abja,  or  padma 

C'harva 

Nic'harva 

Mahadpadma 

Sancu 

Jaludhi,  or  Samudra 

Antya 

Madhya 

Parard'ha 


units. 

tens. 

hundreds. 

thousands. 

tens  of  thousands. 

hundreds  of  thousands. 

millions. 

tens  of  millions. 

hundreds  of  millions. 

thousands  of  millions. 

tens  of  thousands  of  millions. 

hundreds  of  thousands  of  millions. 

billions. 

tens  of  billions. 

hundreds  of  billions. 

thousands  of  billions. 

tens  of  thousands  of  billions. 

hundreds  of  thousands  of  billions. 


14  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

(l6.)  The  Chinese,  also,  have  a  very  perfect  decimal 
system  of  numeration :  the  first  ten  numbers  are,  as 
before,  expressed  by  ten  distinct  articulate  sounds  as 
follows :  — 


Yih      - 
Irr      - 

-  one. 
-  two. 

Lyeu  - 
Ts'hih 

-  six. 
-  seven. 

San     - 

-  three. 

Pah  - 

-  eight. 

Se       - 
Ngoo     - 

-  four. 
-  five. 

Kyeu  - 
Shih  - 

-  nine. 
-  ten. 

They  have  also  distinct  names  for  every  order  of  units 
in  the  decimal  scale,  as  far  as  hundreds  of  millions 
which  are  as  follows :  — 


Yih         units. 

Shih        tens. 

Pfih        hundreds. 

Ts'hyen  thousands. 

Wan       tens  of  thousands. 


Ee  hundreds  of  thousands. 

Chad  millions. 

King  tens  of  millions. 

Kyai  hundreds  of  millions. 


(17.)  A  decimal  system  of  numeration  prevails  in  all 
the  Oriental  tongues,  in  the  languages  of  Greece  and 
Rome,  and  in  all  the  European  tongues,  including  the 
Gothic.  Mr.  Peacock,  in  his  admirable  article  on  Arith- 
metic, in  the  Encyclopaedia  Metropolitana,  quotes  various 
authorities  to  show  that  the  numeral  nomenclatures  of' 
the  native  tribes  of  America  are  far  more  complete,  both 
in  structure  and  extent,  than  could  be  expected  from 
the  low  state  of  civilisation  of  these  people.  Their 
systems  of  numeration  are  almost  invariably  the  de- 
cimal, and  seldom  extend  to  less  than  the  fourth  order 
of  units.  The  Knisteneaux,  one  of  the  principal  hunt- 
ing tribes  of  North  America,  who  inhabit  the  northern 
shores  of  Lake  Superior,  have  a  decimal  system  of  nu- 
meration which  extends  to  1000.  The  Sapibocones, 
a  South  American  tribe,  have  decimal  names  to  a  like 
extent,  but  express  a  hundred  and  a  thousand  by  ten 
tens,  and  ten  times  ten  tens,  without  introducing  any 
new  numerical  term  after  ten.  The  Algonquin s,  a 
kindred  tribe  of  the  Knisteneaux,  speaking  a  dialect  of 
the  same  language,  possess  several  numerals  in  com- 


CHAP.  I.  AMERICAN    NOMENCLATURES.  33 

mon,  but  have  distinct  names  for  hundreds  and  thou- 
sands. The  numerals  of  the  Hurons,  at  one  time  a 
numerous  tribe  of  Upper  Canada,  inhabiting  the  shores 
of  the  lake  of  that  name,  are  mentioned  in  a  rare  work 
by  a  Franciscan  monk,  G.  Sagard,  1632,  entitled  "  Lai 
Grand  Voyage  des  Hurons,"  &c.,  dedicated  to  our  Sa- 
viour. Although  their  language  was  so  rude  and  in- 
artificial as  to  be  destitute  of  adjectives,  abstract  nouns, 
or  verbs  of  action,  and  incapable  of  expressing  a  ne- 
gation without  an  absolute  change  of  the  word,  yet  it 
possessed  a  numeral  nomenclature  of  regular  structure, 
and  formed  on  the  decimal  system.  Equally  complete 
systems  of  numeral  language,  all  in  the  decimal  scale, 
are  found  among  the  Indians  of  the  Delaware,  those 
who  occupied  the  present  district  of  New  York,  the 
former  inhabitants  of  Virginia,  and  most  of  the  central 
tribes  of  North  America.* 

(18.)  The  systems  of  numeration  used  by  the  tribes 
of  South  America  are  also  generally  constructed  on  the 
decimal  radix,  but  they  are  frequently  very  limited  in 
extent.  They  also  form  the  names  of  the  higher  classes 
of  units  by  such  complex  combinations,  that  the  words 
expressing  them  appear  almost  impossible  to  be  remem- 
bered. There  is,  however,  one  instance  in  which,  in 
an  original  native  South  American  language,  viz.  the 
ancient  Peruvian,  a  decimal  system  of  numeration  exists, 
not  less  extensive  than  that  of  the  Greeks  and  Romans, 
and  one,  indeed,  which  bears  to  ours  a  very  curious 
analogy.  There  are  distinct  names  for  the  second,  third, 
and  fourth  orders  of  units, 

Chunca     -     -  ten. 

Pachac     -      -  hundred. 

Huaranca       -  thousand. 

But  there  occurs  then  no  new  term  until  we  arrive  at 
millions ;  ten  thousand,  is  chunca-huaranca ;  a  hundred 
thousand,  pachac-huaranca  ;  a  million,  hunu. 

*  Peacock  on  Arithmetic.  Richardson,  in  Franklin's  Journey.  Mac- 
kenzie's  Journey  to  the  North  Sea,  Introduction.  Humboldt,  Vues  des 
Cordilleres  et  des  Monumens  de  1'Amerique,  2ot.  Monboddo,  Origin  of 
Language,  T43. 


16  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

This  scale,  like  ours,  proceeds  after  the  first  four 
orders  of  units,,  grouping  the  superior  units  by  thou- 
sands,, and  not  by  tens  ;  or  rather  making  the  decimal 
system  subordinate  to  the  millesimal.* 

(19.)  People  have  a  natural  propensity  to  fix  upon 
the  different  orders  of  units  as  landmarks,  in  the  great 
ocean  of  number,  from  which  to  measure  points  in  their 
vicinity.  In  the  regular  schemes  which  we  have  already 
noticed,  numbers  which  terminate  between  two  such 
points  are  always  expressed  by  stating  the  number  of 
units  by  which  they  exceed  the  units  of  the  last  order. 
Thus  nineteen  expresses  nine  above  ten  ;  twenty-nine  is 
nine  above  twenty,  and  so  on.  But  it  is  obviously  more 
natural  for  a  people  not  sufficiently  refined  to  appreciate 
the  harmony  of  nomenclature,  to  express  nineteen  by 
twenty  wanting  one,  and  twenty-nine  by  thirty  wanting 
one,  and  so  on.  We  accordingly  find  this  mode  of  expres- 
sion in  languages  so  numerous  and  remote  as  almost  to 
preclude  the  possibility  of  the  forms  of  expression  being 
borrowed  by  one  from  the  other.  In  the  Sanskrit  and 
Hindostanee  nineteen  and  twenty-nine  are  expressed  by 
one  less  twenty,  and  one  less  thirty,  and  similarly  for 
higher  numbers.  In  the  Latin,  unus  de  viginti,  and 
unus  de  triginta,  are  more  elegant  than  novem  decem, 
and  viginti  novem.  The  same  idiom  prevails  in  the 
Greek.  But  what  is  more  remarkable,  we  find  a  similar 
form  of  expression  in  various  Oriental  tongues.  In  the 
Malay  language  nine  is  expressed  by  within  one  of  ten; 
and  ninety-nine  by  within  one  of  a  hundred.  In  the 
numeral  language  of  the  Sable- fur  Ostiaks  —  a  Siberian 
people  inhabiting  the  banks  of  the  Jenesei  —  nine  is 
similarly  expressed  :  —  eighteen  is  expressed  by  within 
two  of  twenty  ;  eighty,  by  within  twenty  of  a  hundred ; 
ninety-nine,  by  within  one  of  a  hundred,  and  so  on.f 

Many  other  examples  of  this  mode  of  expression,  by 
defect  from  a  complete  number  of  the  higher  units,  are 
found  in  the  numeral  terms  of  Iceland,  Denmark,  and 
several  Oriental  tongues. 

*  Peacock  on  Arithmetic.    Htimboldt,  Vues  des  Cordillfcres,  252. 
1  Klaproth,  Asia  Polyglotta,  171. 


CHAP.  I.  WELSH,    ERSE,    AND    GAELIC.  1? 

(20.)  We  are  not  aware  that  there  is  an  instance 
extant  in  any  language,  ancient  or  modern,  of  a  com- 
plete numeral  system  formed  upon  a  single  radix  other 
than  the  decimal.  We  have  abundant  examples,  as 
already  stated,  of  the  occasional  appearance  of  the  qui- 
nary and  vigesimal  radices  in  scales  which  are  chiefly 
decimal ;  and  examples  may  he  produced  from  the 
numerals  of  the  Celtic  dialects  of  a  strange  mixture  of 
the  quinary,  denary,  and  vicenary  systems.  The  Welsh, 
Erse,  and  Gaelic  numerals,  as  far  as  ten,  are  expressed 
by  independent  words.  In  the  Welsh,  eleven  appears  to 
commence  a  period  or  phase  of  the  decimal  system,  and 
is  expressed  by  words  signifying  ten  and  one ;  the 
same  formation  continues  to  fifteen  inclusive ;  but  here 
the  system  assumes  a  quinary,  or,  perhaps,  more  pro- 
perly, a  quindecimal  radix,  sixteen  being  expressed  by  a 
word  signifying  fifteen  and  one  ;  seventeen,  fifteen  and 
two,  &c. :  twenty  is  expressed  by  an  independent 
term,  and  the  scale  here  takes  the  vigesimal  radix. 
The  Erse  and  Gaelic  scale,  as  far  as  twenty,  is  purely 
decimal ;  but  at  twenty,  like  the  Welsh,  the  scale  be- 
comes vigesimal:  from  twenty  to  one  hundred  the 
scale  of  numeration  is  vigesimal  in  the  three  languages 
just  mentioned ;  taking,  however,  the  quindecimal 
radix  at  the  numbers  35,  55,  &c.  in  the  Welsh.  The 
Phenicians,  among  whom  commerce,  as  is  well  known, 
was  extensively  cultivated  at  a  very  early  period,  had  a 
scale  of  numerals  constructed  on  the  vigesimal  radix, 
/i'he  first  twenty  numbers  had  distinct  names,  and  the 
scale  was  continued  upwards  in  the  usual  manner  by 
compounding  these  names  with  twenty.*  The  inter- 
course which  the  Phenicians  are  known  to  have  had 
with  the  southern  parts  of  England,  with  Wales,  and 
with  Ireland,  affords  a  satisfactory  solution  for  the  pre- 
valence of  the  vigesimal  radix  in  the  numerals  of  these 
countries.  The  same  solution  may  be  given  for  the 
existence  of  the  vigesimal  scale  in  the  Armorican  and 
Basque  dialects. 

*  Swinton.in  Philosophical  Transactions,  PaS. 
C 


18  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

(21.)  The  vigesimal  scale  of  numeration  prevailed 
very  generally  among  the  Scandinavian  nations,  from 
which  it  is  probable  that  we  derive  our  method  of 
counting  by  scores.  This  method,  however,  not  being 
used  at  present  to  express  abstract  numbers,  but  only 
the  application  of  number  to  particular  classes  of  ob- 
jects, can  scarcely  be  considered  as  an  existing  instance 
of  the  vigesimal  radix.  The  same  may  be  said  with 
respect  to  our  method  of  counting  by  dozens,  in  which, 
however,  we  have  three  orders  of  units,  —  the  primary 
unit,  the  dozen,  and  the  gross,  which  is  a  dozen  dozen. 
Were  this  ever  applied  to  abstract  numbers,  it  would 
furnish  an  example  of  the  duodecimal  radix. 

(22.)  Next  to  the  decimal  scale,  the  quinary  is  by 
far  the  most  prevalent,  as  might  be  expected  from  its 
natural  type,  the  fingers  of  one  hand.  The  languages 
of  some  of  the  islands  of  the  New  Hebrides  afford  very 
perfect  specimens  of  numeral  systems  constructed  with 
this  radix.  The  following  are  the  numbers  as  far  as 
ten  of  three  of  these  :  — 

New  Caledonia.  Tanna.  Mallicollo. 

Parai        -  one.  Rettec           -  -  one.  Thkai     -  one. 

Pa-r'5o      -  two.  Carroo      -  -  two.  Ery         -  two. 

Par-ghen  -  three.  Kahar           -  -  three.  Erey       -  three 

Par-bai     -four.  Kafa      -  -four.  Ebats     -four. 

JPa-nim     -Jive.  Karirrom  -Jive.  Erihm    -Jive. 

Panim-gha  six.  Ma-riddee  -  six.  Tsukai   -  six. 

Panim-roo   seven.  Ma-carroo  -  seven.  Goory     -  seven. 

Panim-ghen  eight.  Ma-kahar     -  -  eight.  Goorey  -  eight. 

Panim-bai    nine.  Ma-kafa  -      -  nine.  Goodbats  nine.    ' 

Parooneek    ten.  Karirrom-harirrom  ten.  Seneam  -  ten. 

There  are,  in  each  of  these,  five  independent  terms  for 
the  first  five  numbers.  In  the  language  of  new  Cale- 
donia, the  numbers  from  five  to  ten  are  expressed  by 
five-one,  five-two,  &c.  In  the  language  of  Tanna,  they 
are  expressed  by  more  one,  more  two,  &c.,  and  in  the 
language  of  Mallicollo  they  are  expressed  by  the  words 
one,  two,  &c.  combined  with  tsu  or  goo>  the  significa- 
tion of  the  latter  not  being  known. 


CHAP.  I. 


ASIATIC    AND    AFRICAN    NAMES. 


19 

(23.)  In  general  the  quinary  system  prevails  among 
the  most  uncivilised  tribes,  and  those  who  have  little 
or  no  intercourse  with  other  nations  by  commerce  or 
otherwise.  Among  the  least  civilised  of  the  Asiatic 
tribes  abundant  examples  may  be  found  of  the  use  of 
this  radix.  Mr.  Peacock  quotes,  on  the  authority  of 
Klaproth,  the  languages  of  several  tribes  of  Kamschatka, 
which,  so  far  as  they  go,  are  purely  quinary.  The 
following  are  the  numerals  as  far  as  ten  in  three  of 
these :  — 


Onnen. 
Hyttaka. 
Ngroka. 
Ngraka. 
Myllanga. 
Onnan-myllanga 
N'jettan-myllanga. 
Ngrok-myllanga. 
Ngrak-myllanga. 
Nyngytkan. 

Ingsing. 
Gnitag. 
Gnasog. 
Gnasag. 
Monlon. 
Ingsinagasit. 
Gnitagasit. 
Gnasogasit. 
Gnasagasit. 
Damalagnos. 

Innen. 
Nirach 
N'roch. 
N'rach. 
Myllygen. 
Innan-myllygen  . 
Nirach-my  llygen  . 
Amorotkin. 
Chonatschinki. 
Myngyten. 

(24.)  When  the  quinary  scale  of  numeration  is  car- 
ried to  any  considerable  extent,  it  generally  passes  into 
the  decimal  or  vigesimal  systems.  This,  indeed,  is  a 
natural  and  necessary  consequence  of  the  inconvenience 
which  would  result  from  the  introduction  of  so  many 
new  names  as  that  system  would  require.  The  follow- 
ing example  of  the  numerals  of  the  Jaloffs,  an  African 
tribe,  will  illustrate  this  position  :  — 


Ben,  or  benna. 
Niar. 

Nyet. 

Nianet. 

Juorom. 

Juorom  ben. 

Juorom  niar. 

Juorom  nyet. 

Juorom  nianet. 

Fook. 

Fook  agh  ben. 


Fook  agh  juorom. 

Fook  agh  juorom  ben. 

Nitt,  or  niar  fook. 

Fanever,  or  nyet  fook. 

Nianet  fook. 

Juorom  fook 

Temier. 

Niar  temier. 

Djoone. 

Djoone  agh  termer. 


c  2 


20  A    TREATISE    ON    ARITHMETIC.  BOOK  f. 

The  word  here  expressing  five  signifies  the  hand, 
plainly  indicating  the  type  of  the  system.  It  will  be 
evident,  on  inspection,  that  the  system  is  decimal,  with 
the  quinary  radix  subordinate  to  it. 

(25.)  The  type  of  the  vigesimal  radix  is  indicated  in 
some  languages  by  the  circumstance  of  twenty  being 
expressed  by  the  same  word  which  signifies  a  man. 
The  fingers  and  toes  were  evidently  the  original  prac- 
tical instruments  of  numeration ;  and  after  a  number 
of  objects  had  been  counted,  corresponding  to  the  fingers 
and  toes  of  one  man,  those  of  a  second  were  referred  to. 
Thus  each  man  represented  twenty  of  the  objects  counted. 
It  will  be  easily  understood  from  this,  why  the  decimal 
and  quinary  scales  are  so  frequently  subordinate  to  each 
other  and  to  the  vigesimal ;  a  hand,  the  two  hands, 
and  the  hands  and  feet,  furnished  natural  radices  of 
the  scale. 

A  complete  examination  of  the  numeral  nomenclatures 
of  various  languages  would  afford  results  of  the  greatest 
interest,  not  only  from  the  light  which  they  would 
throw  on  the  mutual  relations  and  former  intercourse  of 
nations,  but  also  on  the  general  principles  by  which  the 
notions  of  number  are  obtained.  Such  an  investiga- 
tion, however,  would  be  unsuitable  to  the  purpose,  and 
inconsistent  with  the  necessary  limits,  of  the  present 
work.  Sufficient  has,  perhaps,  been  said  to  establish 
the  fact  that  clear  ideas  of  number  can  only  be  formed 
by  grouping  and  arrangement,  and  that  systems  of 
nomenclature  are  always  adapted  to  express  such  group- 
ing and  arrangement.  It  appears,  also,  sufficiently 
evident  that  the  radix  or  base  of  such  systems  has 
had  invariably  a  natural  origin,  and  does  not,  as  some 
have  supposed,  depend  on  any  quality  inherent  in  the 
abstract  numbers  which  have  been  taken  for  such 
radices.* 

(26.)  There  are  many  natural  objects  and  circum- 

*  Those  who  desire  to  pursue  this  curious  and  interesting  subject  fr.rtVer 
should  read  the  article  on  Arithmetic,  by  Mr.  Peacock,  already  quoieu, 
\\here  they  will  be  referred  to  numerous  original  authorities. 


CHAP.  I.     BINARY    AND    DUODECIMAL    SYSTEMS.  21 

stances,  as  has  been  already  stated,  which  would  suggest 
two  for  the  radix  of  the  numerical  scale.  There  is  no 
instance,  however,  of  any  people  adopting  a  binary  sys- 
tem of  numerals,  and  it  is  not  difficult  to  perceive  the 
reason  of  its  general  rejection.  The  number  of  inde- 
pendent terms  with  which  the  memory  must  be  bur- 
dened in  order  to  express  the  smallest  extent  of  number 
necessary  in  the  most  common  affairs  of  life,  even 
among  people  not  far  advanced  in  civilisation,  would  be 
most  inconvenient.  The  unit  of  the  second  order  would 
be  two,  that  of  the  third  order  four,  of  the  fourth  order 
eight,  and  so  on.  To  express  such  a  number  as  sixty- 
three,  which  in  the  decimal  system  is  expressed  by  two 
words,  would  in  the  binary  system  require  six,  signify- 
ing severally  thirty-two,  sixteen,  eight,  four,  two,  and 
one.  It  is  true  that  in  higher  numbers  the  names  do 
not  multiply  so  fast,  but  in  these  the  inconvenience 
would  be  less  important. 

Like  the  duodecimal  language,  the  binary  is  often 
applied  to  count  particular  objects,  although  never  used 
in  its  abstract  form.  The  words  pair,  couple,  leash, 
brace,  cannot  be  pronounced  without  calling  to  the  mind 
of  the  hearers  the  various  objects  to  which  it  is  usual  to 
apply  such  terms ;  these  are  even  less  abstract  in  their 
application  than  the  terms  dozen  and  score. 

(27.)  There  are  circumstances  which  would  have 
rendered  the  number  twelve  a  more  convenient  radix  of 
numeration  than  ten ;  and  there  can  be  no  doubt  that 
if  man  had  been  a  twelve-fingered  animal,  we  should 
now  possess  a  more  perfect  system  of  numeration  than 
we  do.  Whatever  be  the  radix  of  the  scale,  it  would 
always  be  a  convenience  to  be  able  to  subdivide  it  with 
facility  without  resorting  to  the  more  refined  expedient 
of  fractional  language ;  and  in  this  respect  twelve  pos- 
sesses much  to  recommend.  Its  half,  third,  fourth, 
and  sixth  parts  can  be  all  expressed  by  distinct  num- 
bers ;  of  course  the  same  applies  to  two  thirds,  three 
fourths,  and  five  sixths  of  the  radix.  On  the  other 
c  3 


22  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

hand,  ten  allows  only  of  its  half,  fifth,  two  fifths,  three 
fifths,  and  four  fifths  being  expressed  by  whole  num- 
bers. This  advantage,  however,  would  be  greatly 
overbalanced  by  the  inconvenience  which  would  result 
from  an  attempt  to  change  the  generally  established 
language  of  numeration. 


CHAP.  II. 

ON   THE    METHOD    OF    EXPRESSING    NUMBERS    BY    SYMBOLS    OR 
FIGURES. 

(28.)  HAVING  in  the  preceding  chapter  explained  the 
manner  in  which  clear  and  distinct  ideas  are  formed  of 
numbers,  whatever  be  their  magnitude,  and  the  principles 
by  which  names  are  affixed  to  these  ideas,  and  the 
oral  nomenclature  of  number  formed,  we  shall  now  pro- 
ceed to  consider  the  methods  by  which  numbers  are 
addressed  to  the  eye,  by  means  of  signs,  symbols,  and 
written  characters. 

(29-)  The  most  rude  and  inartificial  method  of  ex- 
pressing numbers  by  signs  would  evidently  be  by  holding 
up  as  many  fingers  as  there  are  units  in  the  number  to 
be  expressed.  The  extent  of  such  a  scale  of  signs 
would  in  the  first  instance  be  limited  to  ten.  Many 
and  obvious  contrivances  would,  however,  soon  suggest 
themselves  for  its  extension,  and  we  accordingly  find 
INDIGITATION,  or  the  art  of  expressing  number  by  the 
fingers,  practised  to  a  considerable  extent  in  dif- 
ferent ages,  and  in  various  parts  of  the  world. 

Each  finger  having  three  joints,  the  fingers  of  one 
hand  would  suffice  to  count  fifteen  ;  and  thus  both 
hands  used  even  in  the  most  simple  and  inartificial 
manner  would  serve  to  count  thirty.  But  if  a  quin- 
decimal  scale  were  adopted,  then,  after  the  joints  of  the 
finger  of  one  hand  had  been  exhausted  once,  the  superior 
unit  fifteen  might  be  expressed  by  the  first  joint  on  the 
other  hand.  Again,  when  the  fifteen  joints  of  the 
second  hand  had  been  twice  counted,  the  two  fifteens 
would  be  expressed  by  the  second  joint  of  the  other 
c  4 


24  A    TREATISE    ON    ARITHMETIC.  BOOK   I. 

hand,  and  so  on.  In  this  way  the  joints  of  the  fingers 
of  both  hands  would  enable  us  to  count  as  far  as  fifteen 
times  fifteen,  or  two  hundred  and  twenty-five. 

(30.)  A  system  of  digital  reckoning  was  used  by  the 
ancients,  by  which  they  were  enabled  to  count  on  the 
fingers  as  far  as  ten  thousand.  The  first  nine  numbers, 
and  ten,  twenty,  thirty,  &c.  to  one  hundred  inclusive, 
were  expressed  by  various  inflexions  of  the  fingers  of 
the  left  hand.  By  such  means  the  left  hand  alone  was 
sufficient  to  count  as  far  as  one  hundred :  thus,  to 
express  seventy-five,  the  two  inflexions  expressing 
seventy  and  five  should  be  exhibited.  It  is  obvious 
that  all  that  was  necessary  to  be  attended  to  in  the  form- 
ation of  such  a  system  of  signs  was,  that  each  of  the 
.  inflexions  expfessing  ten,  twenty,  thirty,  &c.  should  be 
possible  to  be  made  simultaneously  with  those  which 
expressed  the  first  nine  numbers ;  and  even  if  this 
were  not  accomplished,  the  signs  might  be  made  in 
succession.  The  fingers  of  the  right  hand  expressed 
hundreds  and  thousands  by  the  same  inflexions  as  those 
by  which  the  left  hand  expressed  units  and  tens.  It  is 
obvious  that  the  same  system  is  capable  of  almost  un- 
limited extension  by  changing  the  position  of  the  hand 
or  arm  in  making  the  signs.  Thus,  if  the  left  hand 
expressed  units  and  tens  with  the  palm  upwards,  it 
might  express  ten  thousands  and  hundred  thousands 
with  the  palm  downwards ;  and  if  the  right  hand  ex- 
pressed hundreds  and  thousands  with  the  palm  upwards, 
it  might  express  millions  and  ten  millions  with  the 
palm  downwards,  and  so  on. 

(31.)  The  Chinese  practise  a  method  of  reckoning 
on  the  fingers,  in  which  one  finger  alone  is  made  to 
express  the  first  nine  numbers,  by  placing  the  thumb 
nail  on  each  joint  of  the  little  finger  passing  upwards 
from  the  palm  of  the  hand  to  the  extremity  of  the 
finger  on  the  outside  of  the  hand,  then  down  the 
middle  of  the  finger,  returning  to  the  palm,  and,  finally, 
upwards  on  the  inside  of  the  finger.  The  tens  are 


CHAP.  II.  NUMERICAL    NOTATION.  25 

expressed  in  the  same  way  on  the  next  finger ;  the 
hundreds  on  the  succeeding  one,  and  so  on.  In  this 
way  each  finger  expresses  a  distinct  order  of  units,  so 
that  the  four  fingers  and  thumb  include  the  first  five 
orders  of  units ;  and  the  hand  is  therefore  capable  of 
expressing  all  numbers  under  one  hundred  thousand. 

The  practice  of  indigitation  prevails  generally 
through  the  East,  where  commercial  bargains  are  fre- 
quently made  in  that  way  instead  of  in  writing,  or  by 
oral  language. 

(32.)  To  the  expression  of  numbers  by  signs  sue. 
ceeded  the  more  refined  and  artificial  method  of  de- 
noting and  recording  them  by  written  characters.  The 
symbols  which  would  most  naturally  present  themselves 
for  this  purpose  were  the  letters  of  the  alphabet :  their 
form  was  familiar  to  every  eye,  and  the  order  in  which 
they  stood  being  clearly  impressed  upon  the  memory  of 
every  one  from  childhood  upwards,  furnished  an  easy 
means  of  denoting  the  amount  or  value  of  particular 
numbers.  The  use  of  such  characters  originated  in  the 
East,  where,  indeed,  the  science  of  arithmetic  may  be 
considered  as  having  had  its  birth,  and  where,  at  a  very 
early  epoch,  it  attained  an  astonishing  degree  of  per- 
fection, not  only  in  its  notation,  but  even  in  its  com- 
plex operations.  The  Hebrews,  Phenicians,  and 
kindred  nations  expressed  the  first  nine  digits,  as  the 
numbers  from  one  to  nine  are  called,  by  the  first  nine 
letters  of  their  alphabet ;  as  follows  :  — 

^  Aleph  -                -  one. 

%  Beth  -          -  two. 

J  Gimel  -               -  three. 

*T  Daleth  -  four. 

p  He  -                 -  five. 

^  Vau  -              -  six. 

*   Zain  -                     -  seven. 

H  Chet  -                -  eight. 

J2  Teth  -                        -  nine. 


26  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

The  next  nine  letters  expressed  ten,  twenty,  thirty, 
&c.  to  ninety  inclusive ;  and  the  remainder  of  their 
alphabet,  with  some  additional  symbols,  expressed  one 
hundred,  two  hundred,  three  hundred,  &c. 

(33.)  The  Greeks  adopted,  with  the  utmost  mi- 
nuteness^ this  method  of  notation :  so  closely,  indeed, 
that  where  a  Hebrew  letter  was  wanting  in  the  Greek 
alphabet,  its  place  was  supplied  by  a  symbol  con- 
trived for  the  purpose,  and  expressed  by  a  name  which 
signified  that  it  held  the  place  of  the  Hebrew  letter. 
Thus  for  the  Hebrew  letter  VAU  there  is  no  corre- 
sponding letter  in  the  Greek  alphabet,  and  the  number 
six,  which  that  letter  expresses  in  the  Hebrew,  is 
accordingly  expressed  by  the  symbol  r,  to  which  the 
name  swia-ypov  pav  is  affixed,  meaning  that  it  is  the  sign 
of  VAU.  In  like  manner,  there  are  no  Greek  letters 
corresponding  to  KOPH  and  TSADI,  and  their  places 
are  accordingly  supplied  in  the  Greek  scheme  by 
the  two  symbols  ^,  and  ^ ,  which  are  called  WT^M 
xo-TTTra,  and  ETrto-^wov  va^t,  signifying  the  sign  for 
KOPH,  and  the  sign  for  TSADI.  The  Greek  numerals 
are  as  follows :  — 

la  10  i  100  p 

2  ]3  20  x  200  (r 

3  y  30  X  300  T 

4  $  40  p  400  v 

5  e  50  v  500  Q 

6  r  60  J  600  % 
75  70  o  700  ^ 
8  v)  80  TT  800  a 

90  90  ^  900  ^ 

It  was  customary  to  distinguish  the  letters  when  em- 
ployed as  numerals  from  the  same  when  employed  in 
the  ordinary  way  by  placing  an  accent  on  them.  The 
same  letters  were  made  to  express  thousands  by  placing 
i  below  them : 


CHAP.  II.  NUMERICAL    NOTATION.  27 

1000  a  4000  a  7000  ? 

2000  j3        5000  e        8000  i\ 
3000  7        6000  r        9000  9 

(34.)  The  Greeks  also  employed  another  method 
of  expressing  number  inferior  to  the  above.  This 
method  consisted  in  denoting  the  different  classes  of 
units  by  peculiar  letters.  Thus  I  denoted  the  primary 
or  original  units,  II  denoted  fives,  A  tens,  H  hundreds, 
X  thousands,  and  M  ten  thousands.  The  last  four 
letters  being  the  initial  letters  of  the  Greek  words  signi- 
fying tens,  hundreds,  thousands,  and  ten  thousands. 
The  numeral  letter  placed  before  any  one  of  these  ex- 
pressed the  number :  thus,  %  X,  was  seven  thou- 
sand, &c. 

(35.)  The  ancient  numerals  of  Arabia,  Syria,  Per- 
sia, and  other  nations  were  formed  in  the  same  manner 
from  their  alphabets.  The  European  languages,  in- 
eluding  the  Russian,  Gothic,  Scandinavian,  and  Scla- 
vonic tongues  also  expressed  their  numerals  by  their 
respective  alphabets. 

(36.)  The  Roman  numeral  notation  is  formed  by 
combinations  of  the  following  symbols :  — 

I.  -  one. 

V.  -  five. 

X.  -  ten. 

L.  fifty. 

C.  -  one  hundred. 

I).  -  five  hundred. 

M.  -  one  thousand. 

In  some  cases  five  hundred  is  expressed  by  1 3,  and 
a  thousand  by  CIO.  The  following  table  will  explain 
the  way  in  which  these  symbols  are  combined  to  ex- 
press numbers :  — 


28                                A    TREATISE    ON    ARITHMETIC.                     BOOK  J. 

Units. 

Tens. 

Hundreds. 

Thousands. 

Tens  of 
Thousands. 

I           1 
II          2 

x             10 
xx           20 

c              100 
cc            200 

cio  or  M      1000 
MM  or  fi       2000 

ccioo  or  x     10,000 

in        3 

xxx        30 

ccc         300 

MMM  or  in  3000 

or  x"x       i  20,000 

mi  J 

xxxx  ^ 

cccc    ^ 

&c.  &c. 

or     V4 
Iv     J 

or       C40 

XL       J 

or        >400 

CD         3 

or  fv        j 

v         5 

L             50 

D  or  ID    500 

IDD  or  v       5000 

vi        6 

vn      7 

LX           60 
i,xx         70 

DC            1 

or       feoo 

IOC         J 

«cc      } 

or        r700 

IOCC      3 

IOOM  or  vi    6000 

vin      8 

LXXX         80 

nccc     ) 
or       ^800 

IOCCC    3 

'orvTir    }8000 

DCCCC  "1 

VIIIll 

or     V9 

IX      J 

LXXXX} 

or      V90 
XC      J 

or 
i3cccc  -900 
or 

IDor"MM]9000 

CM 

It  will  be  perceived  that  when  a  symbol  of  lesser 
value  is  placed  before  one  of  greater  value,  it  has  the 
effect  of  subtracting  its  own  value  from  that  which  fol- 
lows it.  Thus  IV.  signifies  five  minus  one,  and  VI. 
signifies  five  plus  one  ;  IX.  signifies  ten  minus  one  ;  and 
XI.  ten  plus  one,  &c.  When  a  line  is  drawn  above  any 
numeral,  it  makes  its  units  signify  thousands.  Thus 
while  IX.  signifies  nine  primary  units,  IX.  signifies 
nine  thousands.  It  is  also  a  curious  circumstance,  and 
worthy  of  notice,  that  while  the  numeral  language  of 
Rome  is  purely  decimal,  its  numeral  symbols  appear  to 
have  the  quinary  radix  subordinate  to  the  decimal. 
Thus  a  new  symbol  is  introduced  at  five,  which  re- 
appears at  fifteen,  twenty-five,  &c.,  another  new  symbol 
is  introduced  at  fifty,  and  another  at  five  hundred. 

(37.)  If  these  various  systems  of  numeral  symbols 
be  carefully  attended  to,  it  will  be  perceived  that  the 
chief  source  of  their  complexity  and  inconvenience  is 


CHAP.   II.  NUMERICAL    NOTATION.  29 

the  necessity  of  making  the  symbol  express  not  only 
its  actual  numeral  amount,  but  also  the  order  of  units. 
In  fact,,  a  new  set  of  symbols  becomes  necessary  for  each 
order  of  units.  The  contrivances  in  the  Greek  numerals 
for  signifying  thousands  by  the  same  set  of  numerals 
with  the  accent  placed  below  them  forms  a  slight  ap- 
proach to  what  would  have  been  a  more  symmetrical 
and  effective  system  of  notation.  It  cannot  fail  to  ex- 
cite surprise  that,  having  seized  on  the  decimal  system 
so  perfectly  in  the  mind,  the  mental  classification  which 
was  adopted  did  not  suggest  a  system  of  symbols  more 
analogous  than  those  which  were  so  long  in  general 
use.  The  notions  of  ten,  twenty,  thirty,  forty,  &c. 
would  have  naturally  suggested  a  set  of  symbols  for 
their  expression  similar  to  those  used  for  the  primary 
units  from  one  to  nine,  modified  by  an  accent  or  some 
other  similar  means.  Another  inflexion  or  mark  would 
have  enabled  the  same  nine  characters  to  express  the 
hundreds.  It  may  probably  be  said  that  our  astonish- 
ment at  the  imperfection  of  the  old  numerical  notation 
arises  from  our  extreme  familiarity  with  the  very  perfect 
system  which  has  since  been  adopted.  But  although 
something  may  be  allowed  for  this,  yet  there  is  in  the 
very  process  of  mind,  by  which  only  we  are  enabled  to 
acquire  clear  ideas  of  number,  something  which  would 
have  prompted  analogous  systems  of  names  for  the 
units  of  succeeding  orders.  The  observations  contained 
in  the  preceding  chapter  cannot,  we  think,  be  denied  to 
afford  sufficient  proof  of  this. 

(38.)  A  people  among  whom  the  arts  and  sciences 
had  been  so  highly  cultivated,  and  where  the  develope- 
ment  of  the  human  mind  was  so  advanced  as  among  the 
Greeks,  having  once  obtained  a  clear  mental  view  of 
the  decimal  system  of  numeration,  could  not,  one  would 
have  supposed,  have  failed  to  have  formed  some  system 
of  notation,  bearing  at  least  as  close  an  analogy  to  the 
idea  intended  to  be  expressed  as  the  following.  Sup- 
posing the  first  nine  digits  to  be  expressed  as  already 
^plained  by  the  nine  characters. — 


SO  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

a'    ft   y'    V   i'    r'    $'    *»'    0' 
123456789 

then  the  same  numbers,  when  their  units  are  of  the 
second  order,  or  when  they  signify  tens,  might  have 
received  a  double  accent  ;  thus,  — 

a!'     fi"    y    V    i"     r"    £"     y"     0" 
10     20     30    40     50    60     70     80     90 

Again,  when  they  signified  units  of  the  third  order,  or 
hundreds,  they  might  have  received  a  triple  accent  ; 
thus,  — 

///  Q///  "f  sv'//  ///  ///         if'ff  ///  t\f" 

a,    p    y   &    *   r   b   ^ 

100  200  300  400  500  600  700  800  900 

and  so  on.  Although  this  would  have  been  immea- 
surably inferior  to  the  system  of  notation  now  univer- 
sally adopted,  still  it  would  have  given  enormously 
increased  power  to  their  arithmetic,  by  suggesting  more 
easy  and  expeditious  methods  of  calculation. 

(39.)  In  his  investigations  respecting  the  Egyptian 
hieroglyphics,  Dr.  Young  has  explained  the  numeral 
system  used  in  them.  It  appears  to  be  exclusively 
decimal,  no  subordinate  radix  appearing  in  the  scale. 
The  primary  unit  is  expressed  by  cr  ,  and  the  first  nine 
digits  are  expressed  by  simple  repetitions  of  the  primary 
unit.  The  unit  of  the  second  order,  or  ten,  is  expressed 
by  (ft,  and  twenty,  thirty,  &c.  by  repetitions  of  this. 
In  the  same  manner,  a  hundred  and  a  thousand  are 

expressed  by  ^  and  3,  and  hundreds  and  thousands, 

as  far  as  nine,  expressed  by  simple  repetitions  of  this. 
Thus,  432  would  be  expressed  as  follows  :  — 


(ft  ffb  fa  a  a 
(40.)  But  the  triumph  of  the  art  of  calculation,  and  that 
to  which  mainly  the  modern  system  of  numeral  com. 
putation  owes  its  perfection,  consists  in  the  "  device  of 
place  ;"  by  which  all  necessity  for  distinguishing  the 
nature  of  the  units  signified  by  any  symbol  is  super- 


CHAP.  II.  NUMERICAL    NOTATION.  31 

seded.  Like  many  other  inventions  of  the  highest 
utility,  this,  when  known,  appears  to  arise  so  naturally 
and  necessarily  out  of  the  exigencies  of  the  case,  that  it 
must  excite  unqualified  astonishment  how  it  could  have 
remained  so  long  undiscovered. 

Let  us  imagine  a  person  possessing  a  clear  notion  of 
the  decimal  method  of  classifying  number,  being  de- 
sirous to  count  a  numerous  collection  of  objects  by  the 
help  of  common  counters.  He  will,  probably,  at  first 
pursue  the  method  already  described  as  practised  by 
the  savage  tribes  of  Madagascar.*  The  objects  to  be 
counted  being  passed  before  him  one  by  one,  he  places 
a  counter  in  a  box  A  for  each  object  that  passes ;  but 
presently  the  counters  in  A  become  so  numerous,  and 
form  so  confused  a  heap,  that  he  finds  it  as  difficult  to 
form  an  idea  of  their  number  as  he  would  of  the  objects 
themselves  which  he  wishes  to  count.  Being  able, 
however,  to  form  a  distinct  and  clear  notion  of  ten 
counters,  he  pauses  when  he  has  placed  the  tenth 
counter  in  the  box  A,  and  withdraws  all  the  counters 
from  it,  placing  a  single  counter  in  the  box  B,  to  denote 
that  ten  objects  have  passed.  He  then  recommences 
his  tale ;  and,  as  the  objects  continue  to  pass  before  him, 
places  counters  in  the  box  A,  and  continues  to  do  so 
until  ten  more  objects  have  passed,  and  ten  counters 
are  again  collected  in  A :  he  withdraws  this  second  col- 
lection of  ten  counters  from  A,  and  places  a  second 
counter  in  B ;  signifying  thereby  that  two  sets  of  ten 
objects  have  passed.  Recommencing  a  third  time,,  he 
proceeds  in  the  same  way,  and,  when  ten  have  passed, 
withdraws  the  counters  from  A,  and  places  a  third 
counter  in  B :  he  continues  in  this  manner,  placing  a 
counter  in  B  for  every  ten  which  he  withdraws  from  A. 
If  the  objects  to  be  counted  be  numerous,  he  finds,  after 
some  time,  that  the  counters  would  collect  in  B  so  as  to 
form  a  number  of  which  he  would  still  find  it  impos- 
sible to  obtain  a  clear  notion.  For  the  same  reason, 
therefore,  that  he  allows  no  more  than  ten  counters  to 
accumulate  in  A,  he  adopts  the  same  expedient  with 

*  See  pageS. 


A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

respect  to  the  box  B.  When  ten  sets  of  objects  have 
been  counted,  he  finds  that  ten  counters  have  collected 
in  B :  he  withdraws  them,  and  places  a  single  counter 
in  the  box  C,  that  counter  being  the  representative  of 
the  ten  withdrawn  from  B,  each  of  which  is  itself  the 
representative  of  ten  withdrawn  from  A.  The  single 
counter  in  C  will  thus  express  the  number  of  objects  in 
ten  sets  of  ten  ;  and  such  a  number  as  already  explained 
is  called  a  hundred.*  When  one  hundred  objects  have 
passed,  there  will  therefore  be  only  a  single  counter  ex- 
pressing  it  placed  in  the  box  C.  The  objects  to  be 
counted  continuing  to  pass,  the  computer  proceeds  as 
before,  placing  counters  in  the  box  A,  withdrawing  them 
by  tens,  and  signifying  the  collections  withdrawn  by 
placing  single  counters  in  B,  until  ten  counters  again 
collect  in  B  ;  these  are  withdrawn,  and  a  second  counter 
placed  in  C.  Let  us  now  conceive  the  three  boxes  in- 
scribed with  the  names  of  the  units  signified  by  the 
counters  which  they  respectively  contain.  It  will  be 
obvious  that,  by  the  aid  of  twenty-seven  counters,  all 
numbers  under  a  thousand  may  be  expressed.  Thus, 
nine  hundred  and  ninety -nine  would  be  expressed  by 
placing  nine  counters  in  each  box :  the  nine  counters  in 
the  box  C  would  stand  for  nine  hundreds  ;  those  in  the 
box  B  for  nine  tens,  and  those  in  A  for  nine  original 
units. 

It  will  be  sufficiently  evident  that  the  same  method 
may  be  continued  to  any  extent.  A  fourth  box,  D,  in- 
scribed thousands,  may  be  provided,  in  which  a  single 
counter  will  be  placed  for  every  ten  counters  withdrawn 
from  C;  and  a  fifth,  E,  inscribed  ten  thousands,  in 
which  a  single  counter  will  be  placed  for  every  ten 
withdrawn  from  D,  and  so  on.  Under  such  circum- 
stances, more  than  nine  counters  could  never  collect  in 
any  box. 

We  have  here  supposed  the  counters  to  be  all  similar 
to  each  other,  and  not  bearing  on  them  any  character  or 

*  To  demand  a  proof  that  ten  times  ten  make  a  hundred,  betrays  an  ig- 
norance of  the  true  meaning  of  numerical  terms.  The  definition  of  the 
word  hundred  is  ten  tens. 


CHAP.   II.  NUMERICAL    NOTATION.  S3 

mark  ;  but,  as  we  have  inscribed  the  several  boxes  with 
the  names  of  the  order  of  units  which  the  counters  they 
contain  express,  there  is  no  reason  why  the  counters 
themselves  may  not  be  inscribed  with  a  character  by 
which  a  single  counter  may  be  made  to  express  any 
number  of  units  from  one  to  nine.  Let  us,  then,  sup- 
pose the  computer  furnished  with  an  assortment  of 
counters,  incribed  with  the  figures  1,  2,  3,  4,  5,  6,  7, 
8,  9  •  when  he  would  express  the  number  of  units  in 
each  box,  instead  of  placing  in  it  several  individual 
counters,  the  number  of  which  might  not  be  easily  per- 
ceived, he  places  in  the  box  a  single  counter,  inscribed 
with  a  character  which  expresses  the  number  of  single 
counters  which  would  otherwise  be  placed  in  the  box. 
Thus,  instead  of  leaving  six  individual  counters  in  a 
box,  he  would  place  in  it  a  single  counter,  marked  with 
the  character  6 :  by  such  an  arrangement,  the  number 
to  be  expressed  would  be  always  evident  on  inspection, 
as  here  exhibited :  — 


Hundreds  of        Tens  of          Thousands.      Hundreds.  Tens.  Units. 

Four  hundred  and  thirty-five  thousand  seven  hundred 
and  thirty- one. 

Having  adopted  such  a  method  of  reckoning,  he 
would  naturally,  for  convenience,  always  arrange  the 
several  boxes  in  the  same  manner ;  and  very  speedily 
the  PLACE  in  which  the  box  stood  would  indicate  to  him 
the  order  of  the  units  which  it  contains :  thus  he  would 
be  at  no  loss  to  remember  that  the  second  and  third 
boxes  from  the  right  would  always  contain  tens  and 
hundreds,  and  the  like  of  the  others.  The  formal  in- 
scription, units,  tens,  &c.  would,  therefore,  become  un- 
necessary ;  and  since,  by  the  method  of  incribing  the 
counters  with  figures,  no  more  than  one  counter  need 
be  placed  in  any  box,  the  boxes  themselves  would  be 
dispensed  with,  and  it  would  be  sufficient  to  place  the 


34f  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

counters  one  beside  the  other,  the  PLACE  of  each  counter 
indicating  the  rank  of  units  which  it  signifies. 

(41.)  A  slight  difficulty  would,  however,  occasionally 
present  itself.  Suppose  that  it  should  so  happen  that, 
when  the  last  object  to  be  counted  passed,  the  tenth 
counter  was  placed  in  the  box  C,  according  to  the 
system  explained :  all  the  counters  would  be  withdrawn 
from  C,  and  a  single  counter  placed  in  D,  or  a  counter 
containing  a  figure  higher  by  one  than  that  which  was 
placed  in  it  before.  When  the  complete  number  is  ex- 
pressed, the  box  C  would,  in  this  case,  contain  no 
counter.  When  the  boxes  are  superseded,  and  the 
counters  alone  used,  the  place  of  the  third  counter 
from  the  right  would  be  unoccupied,  and  the  number 
would  be  expressed  by  the  counters  thus :  — 

000       0  © 

The  space  between  the  counters  inscribed  5  and  3  here 
shows  the  absence  of  the  counter  which  would  express 
hundreds ;  but  in  placing  the  counters,  through  negli- 
gence or  otherwise,  it  might  happen  that  the  two  counters 
which  should  thus  be  separated  by  a  space,  might  be 
brought  so  close  together,  that,  in  reading  the  number, 
the  space  might  be  overlooked;  in  which  case,  the 
counter  inscribed  5  would  erroneously  be  supposed  to 
express  5  hundreds.  To  provide  against  such  an  error, 
let  us  suppose  blank  counters  to  be  supplied,  and  one  of 
these  placed  in  the  position  which  would  be  occupied  by 
an  empty  box :  the  above  number  would  then  appear 
thus :  — 

0  0  0  O  0  © 

and  no  mistake  could  possibly  ensue. 

The  next  step  in  the  improvement  of  this  method 
would  be  to  abandon  counters  altogether,  and  imme- 
diately to  write  down  the  figures  which  would  be  in- 


CHAP.  II.     HISTORY    OF    THE    ARABIC    NOTATION.  35 

scribed  on  them  if  they  were  used  ;  these  figures  being 
written  in  the  same  order  in  which  the  counters  were 
supposed  to  be  placed.  In  this  case,  a  character  would 
become  necessary  to  signify  the  place  of  a  blank  counter, 
wherever  such  a  one  might  occur :  the  character  which 
would  be  naturally  adopted  for  this  purpose  would  be  0  ; 
and  the  above  number  would  then  be  43503]. 

Such  is  the  system  of  numerical  notation  which  has 
obtained  in  every  part  of  the  world  an  acceptance, 
the  universality  of  which  can  only  be,  attributed  to  its 
admirable  simplicity  and  efficiency. 

(42.)  If  we  examine  this  notation  in  comparison 
with  other  systems,  we  shall  find  it  distinguished  by 
two  peculiarities ;  first,  the  expression  of  the  first  nine 
numbers  by  single  characters ;  and  secondly,  by  making 
the  same  characters  express  units  of  all  orders,  by  adopt- 
ing a  certain  invariable  arrangement)  and  introducing  a 
tenth  character  (0),  to  maintain  this  arrangement  un- 
disturbed when  units  of  any  particular  order  or  orders 
happen  to  be  wanting  in  the  number  to  be  expressed. 

That  the  honour  of  the  invention  of  a  system  which 
produced  such  important  effects,  as  well  on  the  investi- 
gations of  science  as  in  the  common  concerns  of  com- 
merce, should  be  claimed  by  many  contending  nations, 
is  what  would  naturally  be  expected.  We  accordingly 
find  various  opinions  advocated,  as  well  respecting  the 
people  with  whom  this  system  originated,  as  with 
respect  to  those  who  first  had  the  honour  of  introducing 
it  into  Europe.  It  is,  however,  agreed  on  all  hands, 
that  the  method  of  expressing  number  by  nine  figures 
and  zero,  with  the  method  of  giving  value  to  these  by 
local  position,  so  as  to  enable  the  same  characters  to 
express  the  successive  orders  of  units,  was  brought  into 
Europe  immediately  from  Arabia ;  and  hence  the  com- 
mon figures  now  used  are  distinguished  from  the  Roman 
numerals  by  the  name  Arabic  figures. 

(43.)  All  Arabian  authors  on  arithmetic  appear  to 
agree  that  the  first  writer  of  that  country  upon  this 
system  of  arithmetic  was  Mohammed  ben  Muza,  the 


36  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

Khuwarezmite,  who  flourished  about  the  year  90(X 
This  writer  is  celebrated  for  having  introduced  among 
his  countrymen  many  important  parts  of  the  science 
of  the  Hindoos,  to  the  cultivation  of  which  he  was  de- 
votedly attached ;  and,  among  other  branches  of  know- 
ledge thence  derived,  there  is  satisfactory  evidence  that 
this  species  of  arithmetic  was  one.  From  the  time  of 
Mohammed  ben  Muza,  the  figures  and  modes  of  calcu- 
lation introduced  by  him  were  generally  adopted  by  scien- 
tific writers  of  Arabia,  although  a  much  longer  period 
elapsed  before  they  got  into  common  popular  use,  even  in 
that  country.  They  were  always  distinguished  by  the 
name  Hindasi;  meaning,  the  Indian  mode  of  computa- 
tion. Alkindi,  the  celebrated  Arabian  writer,  who  flou- 
rished soon  after  Ben  Muza,  wrote  a  work  on  arithmetic, 
entitled  "  Hisabu  Y  Hindi;"  meaning,  the  arithmetic  of 
the  Hindoos.  In  addition  to  this  evidence  of  its  Indian 
origin,  we  have  the  unvarying  testimony  of  all  subsequent 
Arabian  writers.  But  there  is  internal  evidence  from 
the  system  itself,  as  compared  with  the  mode  of  writing 
and  reading  Arabic,  which  furnishes  a  still  more  de- 
cisive proof  of  its  origin.  The  mode  of  writing  practised 
in  Arabia  was  like  ours,  from  left  to  right ;  whereas  in 
writing  those  figures  they  proceed  in  the  contrary  di- 
rection, in  the  manner  practised  by  the  Hindoos. 

(44.)  There  is  extant  positive  evidence  of  the  ex- 
istence of  this  arithmetic  in  India  at  least  four  centuries 
before  the  time  of  Ben  Muza.  We  possess,  in  our  own 
language,  tianslations  of  two  treatises  on  arithmetic, 
mensuration,  and  algebra,  which  are  highly  prized  in 
Hindostan  :  they  are  entitled  the  "  Lilavati"  and  "Vija- 
ganita,"  and  are  the  works  of  Bhascara.  Mr.  Cole- 
broke  has  fixed  the  age  of  Bhascara,  on  satisfactory 
evidence,  about  the  middle  of  the  twelfth  century. 
Bhascara,  in  his  work,  quotes  the  authority  of  a  former 
tvriter,  fragments  of  whose  treatise  on  arithmetic  are 
still  extant,  named  Brahmegupta.  Mr.  Colebroke  has 
also  shown  that  this  author  flourished  in  the  beginning 
of  the  seventh  century.  Brahmegupta  has  again  quoted 


CHAP.  II.  LEONARDO    PISANO.  37 

a  still  more  ancient  authority,  Arya-bhatta,  as  the 
oldest  of  the  uninspired  writers  of  that  country.  It 
appears  that  this  writer  flourished  at  a  period  not 
later  than  the  beginning  of  the  fifth  century.  Now,  it 
is  remarkable,  that  none  of  these  Hindoo  authors  claim, 
either  for  themselves  or  their  predecessors,  the  invention 
of  the  method  of  numeration  by  nine  digits  and  zero, 
with  a  method  of  giving  value  by  position,  but  always 
mention  it  as  being  received  from  the  Deity ;  from 
which  we  may  infer,  that  it  was  practised  in  that 
country  beyond  the  limits  even  of  tradition. 

(45.)  At  the  beginning  of  the  eleventh  century,  the 
use  of  the  Arabic  notation  had  become  universal  in  all 
the  scientific  works  of  Arabian  writers,  and  more  es- 
pecially in  their  astronomical  tables.  The  knowledge 
of  it  was,  of  course,  communicated  to  all  those  people 
with  whom  the  Moors  held  that  intercourse  which 
would  lead  to  a  community  of  scientific  research.  In 
the  beginning  of  the  eleventh  century,  the  Moors  were 
in  possession  of  the  southern  part  of  Spain,  where  the 
sciences  were  then  actively  cultivated :  in  this  way  the 
use  of  the  new  arithmetic  was  received  into  Europe  first 
in  scientific  treatises.  A  translation  of  Ptolemy  was 
published  in  Spain  in  1136,  in  which  this  notation  was 
used ;  and  after  this  period  it  continued  in  general  use 
for  the  purposes  of  science. 

(46.)  Notwithstanding  the  knowledge  and  practice 
of  this  superior  notation  by  scientific  men,  the  Ro- 
man numerals  continued  to  be  used  for  purposes  of 
business  and  commerce  for  nearly  three  centuries  ;  and 
it  was  only  by  slow  and  gradual  steps  that  the  improved 
notation  prevailed  over  its  clumsy  and  incommodious 
predecessor.  The  first  attempt  to  introduce  it  for  the 
purposes  of  commerce  was  made  by  a  Tuscan  merchant, 
Leonardo  Pisano ;  who,  in  1 202,  published  a  treatise  on 
arithmetic  with  a  view  to  introduce  it  among  his  coun- 
trymen. Leonardo  had  travelled  into  Egypt,  Barbary, 
and  Syria :  his  father  appears  to  have  held  some  office 
in  the  custom-house  at  Bugia  in  Barbary,  where  he 
D  3 


38  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

represented  the  interests  of  the  merchants  of  Pisa.  The 
son  there  learned  the  method  of  Hindoo  arithmetic; 
and,  struck  with  its  superiority  over  that  to  which  he 
had  been  accustomed,  he  determined  that  his  coun- 
trymen should  no  longer  be  deprived  of  the  benefit 
of  it.  He  accordingly  published  his  treatise  in  the 
Latin  language;  in  which  he  professes  to  deliver  a 
complete  doctrine  of  the  numbers  of  the  Indians :  — 
<c  Plenam  numerorum  doctrinam  edidi  Hindorum,  quern 
modum  in  ipsa  scientia  prsestantiorem  elegi." 

(47.)  The  date  of  this  work  has  been  disputed,  and 
it  has  been  contended  that  it  is  the  production  of  a 
later  age.  This  supposition  is,  however,  attended  with 
some  difficulties.  It  is  evident,  from  the  work  itself, 
that  at  the  time  it  was  written  Algorithm  (the  name 
given  to  the  Indian  arithmetic)  was  not  known  or 
practised  in  Italy;  yet,  it  is  certain  that  treatises 
on  arithmetic  with  Arabic  numerals  were  common 
in  Italy,  and  well  known  during  the  whole  of  the 
fourteenth  century.  Those  who  dispute  the  date  of 
Leonardo's  work,  refer  it,  nevertheless,  to  a  period 
so  late  as  the  fifteenth  century.  But  there  is  an- 
other argument  still  more  decisive  against  such  an 
hypothesis.  In  the  fifteenth  century,  to  which  the 
treatise  of  Leonardo  is  referred,  the  Italian  language 
had  long  superseded  the  Latin.  In  all  ordinary  works, 
indeed,  the  general  use  of  the  Italian  language  instead 
of  the  Latin,  commenced  about  the  middle  of  the  thir- 
teenth century.  It  is  therefore  impossible  to  suppose 
that  Leonardo  would  write,  for  the  avowed  purpose  of 
benefiting  his  countrymen  engaged  in  commerce,  a 
treatise  on  arithmetic  in  a  language  of  which  they 
would  probably  understand  as  little  as  an  Englishman 
of  the  present  day  would  have  understood  the  language 
used  before  the  time  of  Chaucer. 

(48.)  The  work  of  Leonardo  is  referred  to  by  Lucas 
de  Burgo,  in  1484,  and  by  all  subsequent  writers,  as 
being  the  first  means  of  introducing  the  Arabic  notation 
into  Italy.  A  considerable  period,  however,  was  neces- 


CHAP.  II.  ROMAN    ABACUS.  39 

sary  to  introduce  this  system  into  the  common  business 
of  life.  The  extensive  commerce  maintained  by  the 
Italian  states  directed  their  attention  to  the  subject  at 
an  earlier  period  than  other  nations  ;  and  although,  for 
scientific  purposes,  the  date  of  the  introduction  of  the 
Arabic  numeration  into  Spain  is  earlier  than  that  of  its 
appearance  in  Italy,  yet  its  use  for  the  common  business 
of  life  prevailed  at  a  much  earlier  period  among  the 
Italian  states  than  in  any  other  nation  of  Europe.  To 
the  exigencies  of  Italian  trade,  we  owe  the  formal  sub- 
division of  arithmetic  under  the  various  heads  of  the 
Rule  of  Three,  Profit  and  Loss,  Exchange,  Interest, 
Discount,  &c.  &c. 

(49.)  Although  the  artifice  of  expressing  the  suc- 
cessive orders  of  units  by  the  same  signs,  arranged  in 
different  positions,  was  undoubtedly  the  invention  of  the 
Hindoos,  and  to  the  combination  of  this  principle  with 
the  adoption  of  the  nine  digits  and  zero  to  express  the 
digital  numbers,  is  due  the  great  perfection  and  efficiency 
of  the  present  system,  yet  the  old  method  of  computation, 
practised  among  the  Romans,  and  subsequently  used  in 
Europe  until  the  introduction  of  the  Arabic  notation, 
contained  traces  of  this  principle.  The  application  of 
it  was,  unquestionably,  very  inferior,  because  it  was 
used  merely  for  computation,  and  not  for  recording  or 
expressing  numbers. 

(50.)  Among  the  Greeks  the  elements  of  arithmetical 
knowledge  were  taught  upon  a  board  called  an  ABAX. 
On  this  board  progressive  rows  of  counters  were  placed, 
which  consisted  of  pebbles,  pieces  of  ivory  or  coins. 
The  Greek  word  for  pebble  is  psephos,  and  hence  the 
word  PSEPHIZEIN,  to  compute  or  reckon :  the  Latin 
word  for  pebble  is  CALCULUS,  and  hence  calculare  to 
reckon,  and  our  term  to  calculate. 

(51.)  From  the  abax  of  the  Greeks  the  Romans  de- 
rived their  ABACUS:  this  was  in  iifce  manner  a  board 
on  which  pebbles  (calculi)  were  placed,  and  which  by 
various  arrangements  were  made  the  instruments  of 
calculation.  "  The  use  of  the  abacus,"  says  Professor 
D  4 


40  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

Leslie,"  formed  an  essential  part  of  the  education  of 
every  noble  youth.  A  small  box  or  coffer  called  a  /ocw- 
lus,  having  compartments  for  holding  the  calculi  or 
counters,  was  considered  a  necessary  appendage.  In- 
stead of  carrying  a  slate  and  satchel,  as  in  modern  times, 
the  Roman  boy  was  accustomed  to  trudge  to  school 
loaded  with  those  ruder  implements,  his  arithmetical 
board,  and  his  box  of  counters." 

(52.)  The  form  of  the  abacus  was  subsequently  im- 
proved :  instead  of  the  perpendicular  lines  or  bars,  of 
which  it  first  consisted,  the  board  had  its  surface  divided 
by  sets  of  parallel  grooves,  by  extended  wires,  or  by  suc- 
cessive rows  of  holes.  It  was  easy  to  move  small 
counters  in  the  grooves,  to  slide  perforated  beads  along 
the  wires,  or  to  stick  large  knobs  or  round-headed  nails 
in  the  different  holes.*  A  representation  of  such  an 
instrument  is  here  given,  fig.  1.  There  are  seven  long 
parallel  grooves  in  the  lower  row,  over  which  are  seve- 
rally written  the  names  of  the  units,  which  the  counters 
they  contain  express.  To  prevent  the  necessity  of  using 
a  great  number  of  counters  another  set  of  shorter  grooves 
are  placed  above  the  former,  in  which  a  single  counter 
is  equivalent  to  five  in  the  groove  below.  Thus,  four 
counters  in  the  groove  below,  and  one  in  the  groove 
above,  count  nine ;  three  below  and  one  above  count 
eight,  and  so  on.  The  first  groove  proceeding  from  the 
right  to  the  left,  expresses  the  primary  units,  the  second 
tens,  the  third  hundreds,  and  so  on ;  so  that  the  last  of 
seven  grooves  expresses  millions.  It  will  be  evident, 
therefore,  that  any  number  expressed  by  the  common 
Arabic  figures  will  be  expressed  by  such  an  instrument 
in  exactly  the  same  manner,  only  inserting  in  the  suc- 
cessive grooves  the  number  of  counters  corresponding 
to  the  digits  which  occupy  the  places  severally.  Thus, 
if  we  would  express  the  number  7,580648,  we  should 
do  it  in  the  following  manner:  —  In  the  lower  groove 
of  units  we  should  place  three  counters,  and  in  the 
upper  groove  one ;  in  the  lower  groove  of  tens,  four, 

J*  Leslie's  Philosophy  of  Arithmetic,  p.  95. 


CHAr.  II. 


ROMAN    ABACUS. 


41 


and  none  in  the  upper ;  in  the  lower  groove  of  hun- 
dreds, one,  and  one  in  the  upper;  in  the  grooves  of 
thousands,  none;  in  the  lower  groove  of  ten  thou- 
sands, three,  and  one  in  the  upper ;  in  the  lower  groove 
of  hundred  thousands,  none,  and  one  in  the  upper ;  in 
the  lower  groove  of  millions,  two,  and  one  in  the  upper. 

Fig.  1. 


(53.)  It  will  be  evident,  that  this  method  of  express- 
ing numbers  is  subject  to  no  other  limit  than  the  num- 
ber of  grooves  which  may  be  provided  in  the  abacus. 
That  number  should  always  be  the  same  as  the  number 
of  digits  by  which  the  number  could  be  expressed  in  the 
Arabic  notation.  Thus,  to  express  ten  millions  would 
require  eight  figures  in  the  Arabic  notation,  and  eight 
grooves  in  the  abacus.  In  fact,  the  inscription  of  I,  X, 
C,  &c.,  over  the  grooves  successively,  is  no  more  neces- 
sary in  the  abacus  than  the  inscription  of  units,  tens, 
hundreds,  &c.,  over  the  successive  figures  of  a  common 
number.  By  general  consent,  the  first  groove  on  the 
right  being  used  for  units,  the  second  for  tens,  &c.  the 
inscription  of  the  value  of  the  grooves  would  be  unne- 
cessary, and  their  position  would  become  the  indication 
of  that  value. 


A    TREATISE    ON    ARITHMETIC. 


BOOK  I. 


0=0 


==00000 
000=00 

ooooo 


O=O 
00= 
O0= 


We  have  here,  then,  the  principle  of  value  by  position 
distinctly  practised ;  and  if  the  Romans  had  thought  of 
expressing  hy  nine  characters  in  writing  what  in  cal- 
culation they  here  expressed  by  counters  placed  in  the 
grooves,  they  would  undoubtedly  have  hit  upon  the 
Arabic  method  of  notation  and  computation. 

(54.)  A  curious  coincidence  is  observable  between 
the  Roman  abacus  and  a  calculating  instrument  called 
the  SWAN-PAN,  used  by  the  Chinese.  A  representation 
of  this  instrument  of  calculation  is  given  in  fig.  2.  It 
Fig.  2.  consists  of  ten  parallel  wires 

unequally  divided  :  there  are  five 
beads  on  each  of  the  longer,  and 
two  on  each  of  the  shorter  di- 
visions. In  representing  num- 
bers on  this  instrument,  it  is 
held  so  that  the  wires  are  hori- 
zontal ;  and  the  values  of  the 
beads  increase  in  decuple  pro- 
gression downwards.  The  beads 
on  the  top  wire  express  the 
primary  units;  on  the  second 
the  tens,  and  so  on.  This  in- 
strument is  admirably  adapted 
for  Chinese  calculation,  since 

the  subdivision  of  their  mea- 
sures, weights,  and  money  is  made  on  the  decimal 
system  :  hence  the  calculator  may  select  at  pleasure  any 
bar  for  the  primary  units;  in  which  case,  the  bars 
above  it  will  express  the  subdivisions  by  tenth  parts. 
"  Those  arithmetical  machines,"  says  professor  Leslie, 
"  have  been  adopted  in  China  by  all  ranks,  from  the 
man  of  letters  to  the  humblest  shopkeeper,  and  are  con- 
stantly used  in  all  the  bazaars  and  booths  of  Canton 
and  other  cities,  being  handled,  it  is  said,  by  the  native 
traders  with  a  rapidity  and  address  which  quite  astonish 
the  European  factors."* 

(55.)  From  the  abacus  of  the  Romans  was  derived 

*  Leslie's  Phil,  of  Arith.  p.  98. 


O=0 

00= 


OO  OOO 


CHAP.  II. 


COMPUTATION    BY    COUNTERS. 


the  methods  of  reckoning  by  counters  practised  through- 
out the  whole  of  Europe  during  the  middle  ages,  and 
the  use  of  which  was  continued  until  it  was  superseded 
by  the  improved  arithmetic  now  in  use.  The  calcu- 
lating board  or  frame  was  abandoned,  and  in  its  place 
a  number  of  parallel  lines,  drawn  on  a  board  or  other 
surface,  were  substituted.  The  lowest  of  these  lines 
was  the  line  of  primary  units ;  the  second,  proceeding 
upwards,  expressed  tens ;  the  third  hundreds,  and  so 
on.  A  plan  of  this  kind  being  placed  before  the  cal- 
culator, he  expressed  the  number  by  placing  counters 
on  the  parallel  lines.  Thus,  to  express  7^48,  he  would 
arrange  his  counters  as  follows :  — 

Fig.  3. 


•     A      A      A 

The  difficulty  of  catching  at  a  glance  the  number  of 
counters  placed  on  the  several  lines,  when  they  are  nu-< 
merous,  led  to  the  adoption  of  an  expedient,  probably 
suggested  by  the  use  of  the  Roman  numeral  V :  five 
counters  on  any  line  was  expressed  by  a  single  counter 

Fig.  4. 


*     •     9 


44  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

placed  immediately  above  it.  The  number,  therefore, 
otherwise  expressed  in  Jig.  3.,  would,  according  to  this 
arrangement,  be  expressed  as  in^r.  4. 

On  the  lower  line,  the  counter  immediately  above 
counts  five ;  which,  added  to  the  three  on  the  line,  makes 
eight  for  the  units ;  the  tens  being  less  thanjzw,  remain 
as  before;  the  six  hundreds  are  expressed  by  one 
counter  on  the  third  line  and  one  above  it;  and  the 
seven  thousands  are  expressed  by  two  counters  on  the 
fourth  line  and  one  above  it. 

(56.)  The  methods  of  performing  the  various  arith- 
metical operations  by  these  means  were  extremely 
simple  and  obvious,  and  addressed  themselves  to  the 
understanding  even  more  plainly  and  forcibly  than  do 
the  rules  of  the  arithmetic  practised  at  the  present  day. 
We  shall  hereafter  refer  to  them  more  fully. 

During  the  middle  ages,  it  was  usual  for  mer- 
chants, accountants,  and  judges,  who  arranged  matters 
of  revenue,  to  appear  on  a  covered  bane,  so  called 
from  an  old  Saxon  word  signifying  a  seat*  Before  them 
was  placed  a  flat  surface,  covered  by  a  black  cloth, 
divided  by  parallel  white  lines  into  perpendicular  co- 
lumns, and  these  again  divided  transversely  by  lines 
crossing  the  former,  so  as  to  separate  each  column  into 
squares.  This  table  was  called  an  EXCHEQUER,  from  its 
resemblance  to  a  chess  board ;  and  the  calculations  were 
made  by  counters  placed  on  its  several  divisions  in  the 
manner  and  according  to  the  principles  which  we  have 
just  explained. 

(57.)  A  money-changer's  office  or  shop  was  com- 
monly indicated  by  a  sign  of  this  checquered  board  sus- 
pended. This  sign  afterwards  came  to  indicate  an  inn 
or  house  of  public  entertainment,  probably  from  the 
circumstance  of  the  innkeeper  also  following  the  trade 
of  a  money-changer ;  a  coincidence  which  is  still  very 
common  in  sea-port  towns. 

(58.)  The  transition  from  the  method  of  expressing 
numbers  and  making  computations  upon  them  by 
counters  to  the  improved  method  now  in  use,  seems  to 


OHAP.  II. 


ARRANGEMENT    IN    PERIODS. 


45 


us  easy  and  natural,  although  the  change  took  more 
than  three  centuries  in  being  effected.  Let  us  suppose 
the  rows  of  counters  expressing  the  different  orders  of 
units,  instead  of  being  placed  horizontally,  as  we  have 
just  described,  to  be  placed  in  an  upright  or  vertical 
position,  the  units'  column  standing  in  the  first  place 
on  the  right,  the  tens'  succeeding  it  towards  the  left, 
and  so  on.  Such  a  scheme  would  differ  in  no  respect 
from  the  arrangement  of  the  abacus  of  the  Romans,  and 


Fig.  5. 


i 
< 
< 
< 
i 

> 
1 
j 
I 

1 

( 

1       ( 

)       ( 

4 

1       1 

)       ( 

>      < 

>       4 

)       ( 

4.      8      3      3 


would  differ  only  in  position  from  the 
computing  board  of  more  modern  times. 
The  number  1833  would  be  expressed 
as  in  fig.  5. 

The  difference  between  the  two  methods 
would  thus  be  reduced  to  the  device 
of  expressing  all  collections  of  units 
less  than  nine  by  single  characters,  and 
of  marking  a  blank  line  by  0. 

(59.)  From  what  has  been  explained 
in  the  preceding  chapter  respecting 
the  nomenclature  of  number,  no  dif- 
ficulty will  be  found  in  appropriating  the  names  of  the 
successive  orders  of  units  to  the  successive  places  of 
figures,  beginning  from  the  right.  When  the  figures 
are  numerous,  however,  some  inconvenience  and  diffi- 
culty may  be  found  in  perceiving  at  a  glance  the  order 
of  units  expressed  by  the  first  figure  on  the  left.  If 
it  were  customary  to  announce  in  spoken  language 
the  amount  of  numbers,  by  beginning  with  the  primary 
units,  and  then  ascending  through  tens,  hundreds,  &c., 
a  number  expressed  in  figures  could  be  read  without 
any  difficulty,  however  numerous  its  figures  might  be. 
But  as  the  first  figure  to  be  announced  is  the  highest 
order  of  units  in  the  number,  before  we  can  express  it, 
we  must  perceive  the  total  number  of  figures,  and  ascer- 
tain the  order  of  units  which  the  first  figure  on  the  left 
expresses.  For  the  purpose  of  facilitating  this,  it  has 
been  usual,  when  high  numbers  are  expressed,  to  point 
or  distribute  them  into  periods  of  three  or  six,  some- 


4.5  A    TREATISE    ON    ARITHMETIC.  BOOK  I« 

times  marked  by  leaving  a  wider  space  between  every 
third  or  sixth  figure,  and  sometimes  by  introducing  a 
comma. 

(60.)  The  distribution  into  periods  of  three  places 
corresponds  to  the  nomenclature  used  for  high  numbers 
in  France,  where  a  thousand  million  is  called  a  billion,  a 
thousand  billion  a  trillion,  and  so  on ;  but  according  to 
our  nomenclature,  a  billion,  as  already  explained,  sig- 
nifies a  million  millions,  a  trillion  a  million  billions,  and 
so  on.  It  is,  therefore,  more  consistent  with  our  no- 
menclature to  distribute  the  figures  expressing  numbers 
into  periods  of  six.  The  advantage  of  this  will  be  seen 
by  attempting  to  read  the  following  number  without 
such  distribution :  — 

3576200137961024689. 

In  its  present  state  this  number  cannot  be  read  with- 
out counting  off  the  figures  from  the  units'  place,  calling 
them  units,  tens,  &c.,  until  we  arrive  at  the  first  figure 
on  the  left ;  but  if  they  be  distributed  into  periods  of 
six  as  below,  we  perceive  at  once  that  the  first  figure 
standing  in 

3  576200  137961  024689 

the  fourth  period,  here  consisting  of  one  digit,  signifies 
trillions,  and  the  number  would  be  read  thus ;  —  3  tril- 
lions, 576200  billions,  137961  millions,  024689- 

(6l.)  This  system  of  numeral  notation  being  well 
understood,  it  will  be  perceived  that  every  digit,  of 
which  a  number  is  composed,  has  two  distinct  values, 
which  it  will  be  convenient  to  denominate  the  absolute 
and  the  local  value.  The  absolute  value  of  a  digit  is 
that  value  which  it  has  when  it  occupies  the  place  of 
the  primary  units,  and  is,  therefore,  the  number  of  in- 
dividuals which  it  expresses ;  the  local  value  is,  as  the 
term  intimates,  that  value  which  it  derives  from  its 
position  or  place  in  the  number  to  which  it  belongs, 
Thus,  in  the  number  365,  the  absolute  value  of  3  is 
three  units,  the  local  value  is  3  hundreds. 


CHAP.  II.  USES    OF    CIPHER.  47 

(62.)  The  object  of  0  being  merely  to  fill  a  place, 
so  as  to  mark  the  position  of  other  digits,  and  thereby 
to  give  them  their  local  value,  this  character"  is  distin- 
guished from  the  other  digits  which  possess  absolute 
value.  The  digits  which  possess  absolute  value,  viz. 
1,  2,  3,  4,  5,  6,  7y  8,  9*  are  called  significant  digits,  to 
contradistinguish  them  from  0. 

(63.)  The  character  0  is  called  nought  (nothing*), 
and  is  also  called  cipher,  a  term  which  is  derived  from 
the  Arabic  word  tsaphara,  which  signifies  a  blank  or 
void.  The  uses  of  this  character  in  numeration  are  so 
important  that  its  name  cipher  has  been  extended  to  the 
whole  art  of  arithmetic,  which  has  been  called  to  cipher, 
meaning,  to  work  with  figures. 

(64.)  If  a  0  be  added  on  the  right  of  any  number, 
its  effect  will  be  to  remove  each  digit  of  that  number 
one  place  to  the  left.  Now,  since  the  local  value  of 
the  digits  increases  from  right  to  left  in  a  decuple  pro- 
portion, this  effect  will  be  equivalent  to  increasing  the 
value  of  every  digit  tenfold,  and  therefore  multiplying 
the  number  by  10.  Suppose  the  number  in  question 
is  999;  if  we  add  0  to  this,  and  make  it  9990,  the  9 
which  before  occupied  the  units'  place  is  transferred  to 
the  tens,  and  signifies  9  tens  instead  of  9  units ;  the  9 
which  filled  the  place  of  tens  is  transferred  to  the  place 
of  hundreds,  and  signifies  9  hundreds  instead  of  9  tens  ; 
and  the  9  which  filled  the  place  of  hundreds  is  passed 
to  the  place  of  thousands.  Thus  the  local  value  of  each 
of  the  three  digits  is  increased  10  times,  and  since  the 
0  itself  has  no  value,  the  latter  number  is  exactly  10 
times  the  former. 

(65.)  For  like  reasons  the  addition  of  two  ciphers  to 
the  right  will  multiply  the  number  by  100,  since  it 
transferred  the  digit  which  occupied  the  units'  place  to 
the  hundreds'  place,  that  which  occupied  the  tens'  place 
to  the  thousands',  and  so  on. 

(66.)  In  general,  the  addition  of  three  ciphers  will 
multiply  a  number  by  1000,  the  addition  of  four  by 
10,000,  and  so  on.  On  the  other  hand,  ciphers  pre- 


48  A    TREATISE    ON    ARITHMETIC.  BOOK  L 

fixed  to  a  number,  or  placed  to  the  left  of  it,  produce 
no  effect  on  its  value,  because  they  do  not  change  the 
place  of  any  of  its  digits ;  the  digit  which  before  oc- 
cupied the  place  of  units  still  retains  the  same  position 
and  the  same  may  be  said  of  the  others.  Thus,  999, 
0999,  00999  have  all  the  same  value.  We  must  infer 
therefore,  that  the  first  figure  on  the  left  of  a  number 
can  never  be  0,  since  in  that  position  the  nought  has  no 
signification. 

(67.)  A  significant  digit  placed  to  the  right  of  a 
number  has  the  same  effect  in  multiplying  it  by  10  that 
a  cipher  would,  and  for  the  same  reason ;  but,  besides 
this,  the  value  of  the  significant  digit  is  added  to  the 
number ;  thus,  if  we  place  9  to  the  right  of  any  number, 
that  number  will  be  multiplied  by  10,  and  9  will  be 
added  to  the  product.  Let  3  be  the  number ;  9  an- 
nexed to  the  right  of  this  makes  the  3  signify  30, 
and  adds  9-  In  the  same  manner  9  annexed  to  the 
right  of  30  makes  the  3  signify  3  hundreds  instead  of 
3  tens,  and  also  adds  9,  making  the  number  3  hundreds 
and  9  (309).  For  the  same  reason  two  or  more  sig- 
nificant digits  added  to  the  right  of  a  number  have  the 
same  effect*  in  multiplying  it  by  100,  1000,  &c.  as  the 
same  number  of  ciphers  would  have  ;  but  they  also  add 
their  own  value  to  the  number  so  multiplied.  Thus,  if 
we  annex  99  to  the  right  of  1,  we  shall  change  the  1 
into  1 00,  just  as  we  should  do  by  annexing  two  ciphers ; 
but  to  this  number  will  be  also  added  99- 

(68.)  A  significant  digit  added  to  the  left  of  a  number 
makes  no  change  in  the  value  of  any  other  digit  in  the 
number,  but  it  adds  to  the  number  the  local  value  which 
that  digit  acquires  from  the  position  in  which  it  is  placed. 
Thus,  if  to  99  we  prefix  1,  the  99  will  stiU  retain  the 
same  value  as  before,  but  the  1  prefixed  will,  in  the 
position  given  to  it,  signify  a  hundred,  and  thus  100 
will  be  added  to  the  former  number. 

Having  thus  developed,  with  some  detail,  the  prin- 
ciples on  which  the  nomenclature  and  notation  of  arith- 


CHAP.  II.  NOTATION.  49 

metic  depend,  we  shall  proceed  in  the  following  chapters 
to  explain  the  various  processes  by  which  numbers  are 
combined  or  separated.  The  most  simple  of  these  oper- 
ations are  ADDITION  and  SUBTRACTION,  out  of  which 
will  naturally  arise  two  other  operations,  MULTIPLICA- 
TION and  DIVISION. 


50  A    TREATISE    ON    ARITHMETIC.  BOOK  L 


CHAP.  III. 


(69.)  WHEN  two  or  more  numbers  are  added  together, 
the  number  which  is  obtained  by  such  addition  is  called 
their  SUM.  Thus,  if  3,  5,  and  2,  added  together,  give 
10,  then  10  is  said  to  be  the  sum  of  3,  5,  and  2. 

If  the  method  of  performing  arithmetical  operations 
practised  before  the  adoption  of  the  present  arithmetic 
were  incommodious  in  practice,  limited  in  its  powers, 
and  inadequate  to  the  wants  of  a  highly  advanced  state 
of  society,  it  was  not  nevertheless  destitute  of  other  ad- 
vantages, which,  even  at  the  present  day,  should  recom- 
mend it  as  an  instrument  at  least  of  illustration.  The 
operations  of  arithmetic  performed  by  it  did  not  require 
the  results  of  previous  calculations  in  the  form  of  tables 
to  be  committed  to  memory,  as  is  necessary  in  all  the 
elementary  operations  of  arithmetic  now  practised.  But 
what  is  of  greater  importance  to  our  present  purpose  is, 
that  the  rationale  of  the  process  was  so  palpable  that  it 
could  not  fail  to  be  perceived  by  any  one  capable  of 
managing  the  counters.  In  fact,  the  rules  required  no 
proofs,  the  reasons  of  the  process  being  self-evident. 
So  far  is  this  from  being  the  case  with  our  present 
arithmetic,  that  many  hold  (in  which,  however,  we  do 
not  concur)  that  the  reasons  of  the  rules  of  arithmetic 
are  incapable  of  being  made  intelligible  to  children  at 
the  early  period  of  life,  at  which  circumstances  render 
it  necessary  that  they  should  learn  the  practice  of  them. 
Although,  however,  we  feel  assured  that  a  skilful  teacher, 
or  even  an  ordinary  teacher,  when  assisted  by  a  well 


CHAP.  III.  ADDITION.  51 

written  work  of  instruction,  would  be  capable  of  making 
the  reasons  of  the  arithmetical  operations  understood  by 
a  child  of  common  capacity,  at  the  age  at  which  these 
operations  are  usually  taught,  yet  it  is  certain  that 
such  a  task  would  be  incomparably  more  difficult  than 
to  teach  the  same  child  the  reason  of  the  operations 
when  performed  with  counters.  Indeed,  the  best  method 
of  making  the  reasons  of  the  rules  of  computation  ap- 
parent seems  to  be  this  :  —  to  teach  the  rules  first  by 
the  old  method  of  counters  placed  upon  lines,  and  then 
to  represent  the  same  process  in  figures.  This  would 
at  least  be  necessary  in  the  first  rules,  viz.  addition  and 
subtraction.  When  these  are  well  understood,  the  minds 
of  most  children  would,  perhaps,  be  enabled  to  com- 
prehend the  inference  of  the  process  of  multiplication 
and  division  from  them  without  the  further  use  of  illus- 
tration by  counters.  Nevertheless,  we  cannot  too  strongly 
recommend  such  a  method  to  teachers ;  it  renders  the 
ideas  of  numbers  and  their  mutual  relations  clear  and 
definite,  and  will  disabuse  the  learner  of  the  pernicious 
habit,  so  commonly  contracted,  of  acquiring  a  flippancy 
in  terms  where  the  mind  has  no  corresponding  ideas 
whatever,  or  no  distinct  ones. 

(70.)  Let  us  suppose  that  it  is  required  to  add  to- 
gether the  following  numbers,  176'8,  2804,  9999,  and 
5407.  As  there  are  four  orders  of  units  in  these  num- 
bers, we  shall  mark  upon  paper  four  vertical  columns, 
included  by  parallel  lines  marked  A,  B,  C,  and  D,  in- 
tended to  receive  counters  expressive  of  the  number  of 
units  of  each  order,  in  the  number  to  be  added.  In  the 
column  of  primary  units,  marked  A,  we  first  put  eight 
counters  for  the  eight  units  in  the  first  number ;  in  the 
column  marked  B,  we  put  six  counters  for  the  tens  in 
that  number ;  in  the  column  marked  C,  we  put  seven 
counters  for  the  hundreds ;  and  in  the  column  marked 
D,  we  put  a  single  counter  for  the  thousands.  We  then 
cut  off  each  number  by  a  line,  and  in  the  same  way  ex- 
press the  second  and  succeeding  numbers. 
E  2 


52  A    TREATISE    ON    ARITHMETIC.  BOOlx 

From  the  column  A  we  now  withdraw  ten  counters, 
and  put  a  single  counter  into  the  column  B ;  this  evi- 


0 

000 
000 

000 

ooo 

oooo 
oooo 

1768 

0 

0   0 

oooo 

oooo 

0   0 
0   0 

2804 

000 

000 

000 

000 

000 

o  o  o 

000 

000 

9999 

000 

ooo 

000 

ooo 

0   O 

0   0 

000 

o 

000 

5407 

0   0 

0   0 

o 

E 

0   0 

0 

tf  0 

* 

000 
000 

ooo 
ooo 

ooo 
ooo 

oooo 
oooo 

19978 

000 

ooo 

0 

dently  makes  no  change  in  the  total  number  expressed 
by  the  counters,  since  one  counter  in  the  column  B  is 
equivalent  to  ten  in  the  column  A.  We  next  withdraw 
ten  more  counters  from  A,  and  put  another  counter  into 
B ;  there  will  then  remain  eight  counters  in  A,  two 
counters  being  added  to  the  column  B  :  the  eight  counters 
remaining  in  A  are  moved  down  and  placed  in  the  same 
column  under  the  line  X  Y ;  the  two  counters  added  to 
the  column  B  are  marked  thus  tf. 

We  now  withdraw  ten  counters  from  B,  and  place  a 
single  counter  in  C,  which  is  distinguished  from  the 
others  by  the  same  mark  d :  seven  counters  remain  in 
B,  which  are  in  like  manner  moved  down  and  placed  in 


CHAP.  III.  ADDITION.  53 

the  column  B,  under  the  line  X  Y.  Two  sets  of  ten 
counters  are  now  withdrawn  from  the  column  C,  and 
two  additional  counters  placed  in  the  column  D,  marked 
as  before:  the  remaining  counters  in  C  will  be  nine,, 
which  are  moved  down  and  placed  as  before  under  the 
line  X  Y,  in  the  same  column :  1 0  counters  are  now 
withdrawn  from  D,  and  a  single  counter  placed  under 
the  line  X  Y,  in  an  additional  column  E  :  the  9  counters 
which  remain  in  D  are  brought  down  as  before,  and 
placed  under  the  line  X  Y. 

It  will  be  perceived  that  for  every  10  counters  which 
are  withdrawn  from  any  column  in  this  process,  a  single 
counter  is  placed  in  the  next  column  to  the  left,  a  cir- 
cumstance which  makes  no  change  in  the  amount  of  the 
total  number  expressed  by  the  counters,  since  10  coun- 
ters in  any  column  are  equivalent  to  a  single  counter  in 
the  column  next  on  its  left.  The  counter  thus  added 
to  the  column  on  the  left  is  said  to  be  carried.  Thus 
for  20  counters  withdrawn  from  A,  we  carried  2  to  the 
column  B;  and  for  10  withdrawn  from  B,  we  carried  1 
to  the  column  C ;  and  for  20  counters  withdrawn  from 
C,  we  carried  2  to  the  column  D ;  and,  finally,  for  1 0 
withdrawn  from  the  column  D,  we  carried  1  to  the  ad- 
ditional column  E.  The  counters  which  remain  in  each 
column,  after  as  many  tens  as  possible  are  withdrawn, 
being  always  less  than  10  in  number,  are  brought  down, 
and  occupy  the  same  column  in  the  sum  as  they  did  in 
the  numbers  themselves.  In  the  present  case,  therefore, 
the  sum  is  19978. 

(71.)  It  will  be  evident,  upon  the  slightest  consider- 
ation, that  if  a  calculator  be  furnished  with  a  sufficient 
number  of  counters,  he  can  by  this  method  add  together 
any  collection  of  numbers,  however  great,  without  the 
possibility  of  error,  and  without  previously  committing 
to  memory  any  table  of  addition.  Let  us  now  see  how 
the  same  object  would  be  attained,  if  the  several  num- 
bers to  be  added  were  expressed  by  figures  instead  of 
counters.  For  the  sake  of  clearness,  we  shall  preserve 
E  3 


54 


A    TREATISE    ON    ARITHMETIC. 


the  same  scheme  of  parallel  lines,  placing  the  figures  in 
the  squares  previously  occupied  by  the  counters. 
D        c       B        A 


1 

7 

6 

8 

2 

8 

0 

4 

9 

9 

9 

9 

Jii 

5 

4 

0 

7 

1 

9 

9 

7 

8 

Sum 

It  will  be  necessary,  in  the  first  place,  to  withdraw  the 
tens  from  the  column  A,  but  to  do  this  is  not  so  easy 
a  matter  as  it  was  when  counters  were  used.  We  must 
now,  in  the  first  place,  ascertain  the  total  amount  of  the 
four  figures  in  the  column  A  ;  and  this  can  only  be  done 
by  having  previously  calculated,  by  the  aid  of  counters, 
or  some  other  such  means,  the  sum  which  would  be 
obtained  by  the  addition  of  every  two  single  figures. 
In  fact,  a  TABLE  OF  ADDITION  has  become  necessary, 
which  must  either  be  referred  to  for  the  purpose  of  as- 
certaining the  sum  of  every  two  digits,  or  must  be  com- 
mitted to  memory,  so  that  such  sum  may  be  recollected 
when  the  knowledge  of  it  is  needed.  Not  to  interrupt 
our  process  of  investigation,  we  shall  assume,  for  the 
present,  that  such  a  table  has  been  committed  to  memory : 
the  computist  then  knows  that  7  added  to  9  gives  16: 
he  withdraws  the  10,  and  bears  in  recollection  that  1  is 
to  be  carried  to  the  column  B ;  the  other  6  he  adds  to 
4,  which  gives  another  10,  so  that  2  must  be  carried  to 
B  ;  the  remaining  8  he  writes  in  the  units'  place  of  the 
sum  immediately  under  7-  He  now  adds  to  9  *n  the 
column  B  the  2  which  he  has  carried ;  this  gives  1 1  ; 
he  sets  apart  10,  bearing  in  mind  that  1  is  to  be  carried 
to  C,  and  he  adds  the  remaining  1  to  6,  placing  7  in 
the  tens'  place  of  the  sum.  The  1  which  he  carries  to 
C  he  adds  to  4,  which  gives  him  5 ;  this  added  to  9 


CHAP.  III.  ADDITION.  55 

gives  14 ;  he  sets  apart  the  10,  and  adds  4  to  8,  which 
gives  12  ;  he  sets  apart  the  second  10,  and  adds  2  to  7, 
which  gives  9  j  this  9  he  places  in  the  hundreds'  place 
of  the  sum.  Having  reserved  2  tens  in  the  column  C, 
he  carries  2  to  the  column  D ;  these  2  added  to  5  give 
7,  which  added  to  9  gives  16;  reserving  10,  he  adds  6 
to  2,  which  gives  8,  and  this  he  adds  to  1,  which  gives 
9;  he  writes  9  *n  the  thousands'  place  of  the  sum. 
Having  reserved  one  ten  from  the  column  D,  he  writes 
1  in  the  ten  thousands'  place  of  the  sum,  and  the  cal- 
culation is  complete. 

(72.)  It  is  clear  that  in  this  case  there  are  two 
sources  of  possible  error,  which  do  not  exist  in  the 
method  of  counters.  One  is  the  possibility  of  not  per- 
fectly remembering  the  true  sum  of  every  two  digits. 
Considering  how  small  the  number  of  pairs  of  digits,  the 
sums  of  which  it  is  necessary  to  remember,  is,  this,  per- 
haps, can  scarcely  be  regarded  after  a  little  practice  as 
a  source  of  probable  error :  the  other  is  the  probability 
of  forgetting  the  number  of  tens  reserved  in  each  column, 
and  therefore  of  carrying  a  wrong  number  to  the  suc- 
ceeding one.  When  the  numbers  to  be  added  are  con- 
siderable, this  is  a  source  of  very  frequent  error,  even 
with  practised  arithmeticians,  and  different  computers 
adopt  different  means  of  registering  the  number  to  be 
carried.  If  the  column  of  figures  to  be  added  be  not 
great,  the  fingers  of  the  left  hand  will  serve  as  a  register 
for  each  1 0  which  is  reserved,  provided  the  addition  of 
the  column  does  not  amount  to  60 ;  and  by  using  the 
fingers  a  second  time,  this  method  may  enable  him  to 
register  them  as  far  as  100.  Other  computers  find  it 
more  convenient,  instead  of  reserving  and  registering  the 
tens,  to  add  the  column  directly  up,  performing  each 
addition  mentally:  thus  in  the  units'  column  of  the 
number  just  given,  the  computer  would  say  7  and  9  are 
16,  and  4  are  20,  and  8  are  28.  By  practice  this  be- 
comes no  very  difficult  matter ;  but  still  it  is  liable  to 
error  upon  a  momentary  relaxation  of  attention  in  the 
computer. 

E  4 


56  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

(73.)  Whatever  method  the  computer  may  adopt  for 
the  purpose  of  registering  the  carriages  from  column  to 
column,  he  must  at  all  events  commit  to  memory  the 
sums  of  every  possible  pair  of  single  digits  which  can  be 
required  to  be  combined  by  addition. 

(74.)  The  sign  +  is  used  to  express  the  operation 
of  addition,  and  when  it  occurs  between  two  numbers  it 
is  intended  to  express  their  sum :  thus  5  -f  2  means 
that  number  which  is  found  by  adding  2  to  5. 

(75.)  The  sign  =  means  equal,  or  is  equal  to,  or 
makes  ;  thus  5  -f  2  =  7  means  that  2  added  to  5  makes 
7 ',  or  that  the  sum  of  2  and  5  is  7-  The  reader  will 
find  it  convenient  to  render  himself  familiar  with  the 
use  of  these  abbreviations. 

(76.)  In  the  opposite  table  of  addition  the  single 
digits  are  combined  by  pairs  in  every  possible  manner ; 
so  that,  if  it  be  committed  to  memory,  the  sums  of  every 
pair  of  digits  will  be  known,  and  the  computer  will  be 
in  a  condition  to  solve  any  questions  whatever  in  mere 
addition. 

(77.)  In  the  first  column  of  this  table  are  expressed 
the  sums  of  the  different  pairs  which  follow  the  num- 
bers in  the  cross  rows : — thus  6  is  obtained  by  the  ad- 
dition of  1  and  5,  2  and  4,  or  3  and  3 :  9  by  the 
addition  of  1  and  8,  2  and  7,  3  and  6,  or  4  and  5 ;  and  the 
same  with  the  others.  Although  it  is  probable  that  the 
results  here  tabulated  will  be  already  familiar  to  the 
mind  of  every  reader  of  the  present  treatise,  yet  we 
consider  it  right,  in  this  and  other  cases,  distinctly  to 
explain  the  principle  or  rationale  by  which  such  results, 
however  familiar,  are  verified  and  proved,  and  to  dis- 
tinguish clearly  between  what  depends  on  the  con- 
ventional nomenclature  of  number,  and  that  which  is 
matter  of  inference ;  or,  in  other  words,  to  mark  out 
clearly  the  boundary  where  the  province  of  definition 
terminates,  and  the  operation  of  demonstration  begins. 
That  1  and  1  make  2  is  a  matter  of  definition  ;  it  is,  in 
fact,  the  meaning  of  the  word  or  figure  2.  In  the  same 
manner  that  1  and  2  make  3,  is  the  same  as  saying 


CHAP.   III. 


ADDITION. 


TABLE    OF   ADDITION. 


2 

1  +  1 

3 

1+2 

4 

H-3 

2  +  2 

5 

1+4 

2+3 

6 

1+5 

2  +  4 

3  +  3 

7 

1+6 

2  +  5 

3  +  4 

8 

1+7 

2  +  6 

3  +  5 

4  +  4 

9 

1  +  8 

2  +  7 

3  +  6 

4  +  5 

10 

1+9 

2  +  8 

3  +  7 

4+6      5+5 

11 

2+9 

3  +  8 

4  +  7 

5  +  6 

12 

3  +  9 

4  +  8 

5  +  7 

6  +  6 

13 

4  +  9 

5  +  8 

6  +  7 

14 

5  +  9 

6  +  8 

7+7 

15 

6+9 

7  +  8 

16 

7  +  9 

8  +  8 

17 

8  +  9 

18 

9  +  9 

•58  A    TREATISE    ON    ARITHMETIC.  BOOK  I* 

that  3  is  the  number  next  above  2,  and,  therefore,  this  is 
a  definition.  In  the  same  way  it  will  appear  that  the 
second  column  of  the  above  table  as  far  as  1  +  9  inclu- 
sive tells  nothing  that  was  not  already  made  known  in 
the  explication  of  the  nomenclature  and  notation  of 
number.  The  next  term  of  the  column,  however,  is  a 
matter  of  inference :  the  sum  of  2  and  9,  whatever  it  be, 
exceeds  the  sum  of  1  and  9  hy  a  single  unit ;  but  the 
sum  of  1  and  9  is  10,  by  definition :  therefore,  the  sum 
of  2  and  9>  being  1  greater  than  10,  must  be  11. 
In  the  same  manner,  the  sum  of  3  and  9,  being  1 
greater  than  the  sum  of  2  and  9>  must  be  12.  For  a 
like  reason  the  sum  of  4  and  9  must  be  13,  and  so  on. 
Thus  the  remainder  of  the  second  column  is  derived  by 
an  easy  and  obvious  inference  from  the  preceding  part 
of  it. 

(78.)  The  third  column  may  be  derived  from  the 
second,  by  very  simple  reasoning.  It  will  be  observed, 
that  the  two  numbers  added  in  the  third  column  are  one 
of  them  greater,  and  the  other  less,  than  the  two  which 
are  added  in  the  second :  thus,  opposite  6  we  have  1  -f  5  ; 
succeeding  1  -f  5  in  the  same  row  we  have  2  +  4.  Now, 
if  we  suppose  the  numbers  to  express  counters,  2+4 
may  be  derived  from  1  +  5  by  transferring  a  single 
counter  from  the  5  to  the  1 ;  this  transfer  can  make  no 
change  in  the  total  amount  of  the  counters,  and,  there- 
fore, whatever  be  the  sum  of  1  and  5,  the  same  must 
be  the  sum  of  2  and  4.  But  we  have  already  seen  that 
the  sum  of  1  and  5  is  6,  and,  therefore,  6  must  also 
be  the  sum  of  2  and  4.  The  same  reasoning  will  show 
that  the  sum  of  1  and  6  is  the  same  as  the  sum  of  2 
and  5;  that  1+7  =  2  +  6,  and  so  on.  Thus  the 
third  column  is  directly  inferred  from  the  second. 

(79-)  By  a  like  process  of  reasoning  the  fourth  co- 
lumn may  be  inferred  from  the  third,  for  there,  in  like 
manner,  of  the  two  numbers  added,  the  first  exceeds  and 
the  other  falls  short  of  the  preceding  numbers  by  one ; 
opposite  12,  for  example,  we  find  4  +  8  and  5  +  7:  if 
we  consider  them  as  representing  heaps  of  counters,  the 


GHAP.  III.  ADDITION.  59 

latter  is  evidently  derived  from  the  former  by  trans- 
ferring a  single  counter  from  the  heap  of  8  to  the  heap 
of  4. 

The  fifth  column  is  derived  from  the  fourth,  and  the 
sixth  from  the  fifth,  by  exactly  the  same  reasoning ;  and 
thus  the  results  of  this  table,  however  familiar,,  are  re- 
duced to  the  rigorous  test  of  demonstration,  and  receive 
the  same  validity  and  certainty  as  the  conclusions  of 
geometry. 

(80.)  In  teaching  the  first  principles  of  arithmetic 
to  children,  who  must  needs  at  a  very  early  age  com- 
mit to  memory  the  computations  of  the  above  table, 
how  much  more  effectual  would  the  instruction  be  ren- 
dered if  a  teacher  would  occasionally  take  the  trouble 
(pleasure  we  would  rather  say)  of  addressing  himself  to 
the  understanding  of  his  little  pupil,  as  well  as  to  the 
memory  !  What  could  be  more  easy  than  to  provide  a 
parcel  of  counters,  and  dispose  them  in  heaps  or  in 
rows,  and  make  the  child  verify  with  them  all  the  above 
results  ?  the  very  playthings  of  the  child  might  thus  be 
made,  as  they  ought  always  to  be,  instruments  of  in- 
struction. 

(81.)  A  computer,  as  has  been  already  observed, 
finds  it  frequently  convenient  to  be  able  to  add  with 
facility  and  despatch  a  single  digit  to  a  number  con- 
sisting of  two  or  more  digits ;  when  the  above  table  of 
addition  has  been  committed  to  memory,  such  an  ad- 
dition is  almost  as  easy  as  the  addition  of  only  two 
digits,  and,  indeed,  in  some  cases  is  quite  as  easy. 
Thus,  suppose  we  wish  to  add  4  to  35,  we  have  only 
to  add  4  to  the  units  in  35,  which  gives  39 ;  a  process 
which  is  as  easy  and  expeditious  as  adding  4  to  5. 
If,  however,  the  addition  of  the  single  digit  to  the  units 
produce  a  sum  greater  than  9>  it  will  then  be  necessary 
to  carry  1  to  the  tens ;  but  as  such  an  addition  never 
can  produce  a  sum  greater  than  18,  it  never  can  be 
necessary  in  this  case  to  carry  more  than  1  to  the  tens. 
Thus,  for  example,  if  we  wish  to  add  9  to  35,  we  should 
get  30  and  14;  the  4  would  take  the  units'  place,  and 


60  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

1  would  be  carried  to  the  3,  so  that  the  sum  would  be 
44.  In  such  additions,  therefore,  whenever  the  sum  of 
the  units  exceeds  9,  the  last  figure  only  is  to  be  retained 
for  the  units,  and  1  is  to  be  added  to  the  tens.  The 
young  student  must  always  be  made  to  practise  additions 
of  this  kind,  so  as  to  perform  them  with  facility  and 
despatch,  before  he  is  introduced  to  more  complex 
questions. 

(82.)  As  in  all  arithmetical  operations  as  at  present 
performed,  there  is  a  certain  liability  to  error, — a  lia- 
bility which  is  great  in  proportion  to  the  complexity  of 
the  question  and  the  number  of  numerical  quantities 
which  it  involves, — it  is  in  the  last  degree  desirable,  if 
not  absolutely  necessary,  that  the  computer  should  pos- 
sess some  means  of  verifying  his  work,  and  that  those 
who  employ  the  results  of  the  computation  should  be 
able  to  discover  whether  they  be  erroneous.  When 
computers  are  used  for  any  practical  purpose,  as  in  the 
calculations  made  for  almanacks,  tables  of  insurance, 
logarithms,  trigonometry,  &c.,  probably  one  of  the  most 
effectual  and  best  methods  of  verification  is  to  make 
different  computers  work  independently  of  one  another, 
and  to  compare  their  results :  if  they  obtain  the  same 
result  by  the  same  set  of  operations,  it  may  be  assumed 
to  be  all  but  mathematically  certain  that  the  conclusion 
is  correct ;  for  to  suppose  it  erroneous,  it  would  be  ne- 
cessary to  assume  that  the  different  computers  com- 
mitted the  same  number  of  errors,  that  these  errors 
were  exactly  the  same  in  magnitude,  and  fell  upon  the 
same  figures:  such  a  supposition  would  outrage  all  the 
rules  of  probability. 

(83.)  The  same  practical  principle  may  be  applied 
in  the  instruction  of  youth,  whether  in  numbers  or  in- 
dividually. If  a  number  of  pupils  be  instructed  to- 
gether, let  the  same  question  be  proposed  to  them  all, 
or  to  any  convenient  number  of  them,  and  take  care 
that  those  who  are  working  the  same  question  shall  not 
have  communication  with  each  other.  If  their  answers 
agree,  it  may  be  assumed  that  all  are  correct ;  if  not 


CHAP.  III.  ADDITION. 


61 


the  discretion  of  the  master  and  his  knowledge  of  the 
pupil  will  prohably  direct  him  in  the  selection  of  those 
on  whom  it  may  be  advisable  to  impose  the  labour  of 
revision. 

(84.)  Even  if  the  instruction  is  conveyed  to  an  indi^ 
vidual  pupil,  it  will  not  be  difficult  to  propound  the 
same  question  at  different  times,  under  forms  so  differ- 
ent that  it  will  not  be  known  to  be  the  same :  the  re- 
sults may  then  be  compared,  and  if  they  are  identical, 
the  solution  may  be  assumed  to  be  correct.  Thus  if  it 
be  proposed  to  add  together  several  numbers,  the  figures 
which  occupy  the  places  of  any  order  of  units  may  be 
transposed  at  pleasure,  so  as  in  appearance,  and  indeed 
in  reality,  to  vary  the  numbers  added  together,  but  at 
the  same  time  to  make  no  change  in  their  sum.  We 
have  here  stated  several  varieties  which  may  be  given 
to  the  question  in  addition  which  has  been  already 
propounded. 

1904  2808  146? 

9899  1407  5988 

5467  9969  2829 

2708  5794  9694 


19978     19978     19978 


In  one  the  column  of  units  consists,  reading  downwards, 
of  4978  :  in  the  original  number  it  consists  of  8497,  the 
same  figures  differently  arranged,  but  of  course  making 
up  by  addition  the  same  number  of  primary  units.  In 
the  tens'  column  the  only  significant  digits  are  9  and  6, 
and  the  same  may  be  said  of  the  tens'  column  of  the 
original  number.  If  the  column  of  hundreds  and 
thousands  be  likewise  examined,  they  will  be  found  in 
like  manner  to  consist  of  the  same  figures  as  in  the 
original  number,  varied  only  in  their  order. 

(85.)  This  mode  of  testing  questions  in  addition 
proposed  simultaneously  to  several  pupils,  or  at  different 
times  to  the  same  pupil,  may  be  further  varied  by  in- 


02  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

creasing  any  figure  in  any  column,,  and  at  the  same  time 
decreasing  another  figure  in  the  same  column  by  the 
same  amount:  thus  in  the  units'  column  of  the  first 
example,,  instead  of  8  and  4  we  may  write  7  and  5,  and 
instead  of  9  and  7  we  may  write  8  and  8,  which  will 
convert  the  units'  column  into  7588,  reading  downwards. 
instead  of  8497  >  and  the  same  principle  will  apply  to 
the  columns  of  units  of  superior  orders.  In  this  way 
the  same  question  may  be  disguised  under  such  a  variety 
of  forms  that  it  is  impossible  the  pupils  can  recognise 
its  identity. 

(86.)  But  this  method  of  verification  by  the  master 
may  be  carried  still  further :  besides  varying  the  ar- 
rangement of  the  figures  in  each  column,  he  may  also 
add  1  or  2  to  some  one  figure  in  every  column,  so  that 
the  answers  furnished  by  one  set  of  pupils  could  be 
found  by  adding  1  or  2  to  every  digit  in  the  answer  fur- 
nished by  another  set. 

(87-)  These  methods  afford  the  teacher  easy  and 
expeditious  means  of  ascertaining  whether  the  work  of 
his  pupils  be  correct,  without  the  trouble  and  loss  of 
time  attendant  on  performing  the  same  calculation  him- 
self. As  occasional  errors  are,  however,  incidental  on 
all  arithmetical  calculations  made  with  figures,  it  is 
desirable  that  the  computer  himself  shall  possess  some 
means  of  verifying  his  work.  Several  methods  of 
effecting  this  present  themselves,  adapted  to  the  different 
classes  of  questions  to  which  the  work  is  applied :  these 
may  generally  be  resolved  into  performing  the  work 
twice  by  different  methods,  and  trying  the  coincidence 
of  the  results ;  if  they  be  both  correct  they  must  be 
both  identical ;  if  not  identical,  one  or  other  must  be 
erroneous,  and  the  work  requires  revision. 

(88.)  Let  us  take  the  first  example  already  given 
for  the  addition  of  numbers  :  it  appears  that  by  adding 
the  several  columns  in  the  ordinary  way  the  sum  was 
found  to  be  19978  ;  now  let  the  addition  be  performed, 
beginning  in  each  column  at  the  top  instead  of  at  the 
bottom,  and  try  whether  the  same  sum  will  be  obtained ; 


CHAP.  III.  ADDITION.  6S 

or  cut  off  the  top  line,  and  add  the  three  lines  below  it  ; 
then  add  the  top  line  to  the  sum  thus  obtained :  the 
result  should  be  still  19978.  The  numbers  to  be 
added  may  also  be  divided  into  two,  the  first  two  lines 
being  first  added,  and  then  the  last  two  lines  ;  the  two 
sums  thus  obtained  added  together  should  produce  the 
same  final  sum.  We  have  here  subjoined  several  num- 
bers to  be  added  together,  and  their  addition  in  the 
usual  way  gives  us  as  a  sum  4519;  we  then  add  the 
first  three,  and  find  their  sum  to  be  1750 ;  the  next 
three  added  gives  837  ;  the  next  three  gives  542,  and 
the  last  two  give  1 390  :  these  four  numbers  added  to- 
gether ought  to  give  the  same  sum  as  the  number  first 
produced,  and  they  accordingly  do  so.  Had  any  error 
been  committed  in  making  the  original  addition  it  is  in 
the  last  degree  improbable  that  the  same  error  exactly 
should  be  made  in  performing  the  several  partial  ad- 
ditions, or  in  adding  the  partial  sums  together. 


357 
431 
962 

479 
301 
57 
9 
200 
333 

1750 

~837 

489 
901 

542 

1390 

4519 

4519 

(89-)  It  is  the  invariable  practice  in  modern  arith- 
metic to  add  upwards,  and  from  right  to  left.  This  is 
not,  however,  necessarily  connected  with  the  principle 
of  the  operation,  nor  was  it  always  the  mode  of  prac- 
tising it.  In  the  Hindoo  arithmetic,  as  given  in  the 
Lilavati,  the  method  pursued  is  different  in  detail, 


b4  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

though  identical  in  principle.  It  may  not  be  unin- 
structive  to  recur  occasionally  to  these  old  and  obsolete 
methods,  since  it  will  assist  us  in  perceiving  what  it  is 
in  the  modern  processes  which  is  arbitrary,  and  what 
essential.  We  shall  thereby  the  more  clearly  perceive 
and  remember  the  principle  which  is  involved  in  every 
operation.  Let  us  suppose  that  the  numbers  to  be 
added  are  4,  12,  37,  8,  64,  and  201  ;  the  process 
would  be  as  follows  :  — 

Sum  of  the  hundreds,  2  -     2 

Sum  of  the  tens,  1,  3,  6     -  -     10 

Sum  of  the  units,  4,  2,  7,  8,  4,  1     -       26 

326 

Or,  if  we  proceed  from  the  inferior  to  the  superior 
orders  of  units,  it  would  be  as  follows  :  — 

Sum  of  the  units,  4, 2,  7,  8,  4,  1     -       26 
Sum  of  the  tens,  1,  3,  6     -  -     10 

Sum  of  the  hundreds,  2      -  -     2 

326 


(90. )  It  is  evidently  unimportant  in  what  order  the 
process  of  addition  is  conducted,  if  only  the  numbers 
carried  be  duly  attended  to.  Let  us  suppose  that  the 
following  numbers  are  required  to  be  added  together : — 

24605 

68979 
30895 
47638 
32756 
87104 


68747 
22323 

291977 


CHAP.  III.  ADDITION.  65 

We  shall  commence  our  proceedings  at  any  proposed 
column,  suppose  the  column  of  hundreds :  adding  up 
this  column  we  find  that  it  makes  37 ;  we  place  7 
under  the  column  of  hundreds,  and  three  under  the 
column  of  thousands,  leaving  above  it  a  space  to  receive 
the  number  obtained  by  the  addition  of  that  column : 
let  us  next  add  the  column  of  ten  thousands  ;  we  find 
that  its  sum  is  26,  and  we  accordingly  place  6  under 
it,  and  2  in  the  place  of  hundred  thousands,  leaving  a 
space  above  :  in  the  same  manner  add  the  units'  column ; 
we  find  that  the  sum  is  37,  and  we  accordingly  place 

7  under  the  units'  column  and  3  under  the  tens',  with  a 
space  above ;  the  tens  being  now  added  give  24,  and 
we  place  a  4  under  the  column  of  tens,  and  a  2  under 
the  column  of  hundreds :  the  column  of  thousands  are 
now  added,  which  give  us  28,  and  we  place  accordingly 

8  under  the  column  of  thousands,  and  two  under  the 
ten  thousands.     The  lower  line  now  exhibits  the  seve- 
ral numbers  carried  from  the  units  to  the  tens,  from  the 
tens  to  the  hundreds,  &c. ;  these  numbers  being  added 
to  the  numbers  above  give  the  total  sum. 

(91.)  It  is  obvious  that  in  the  ordinary  way  of  per- 
forming the  process  of  addition,  the  lower  line  is  added 
mentally  to  the  upper :  thus  when  we  add  the  units' 
column  we  carry  3,  which  being  added  to  the  sum  of 
the  tens'  column^  gives  the  7  for  the  tens'  place,  and 
leaves  2  to  be  carried,  and  so  on ;  the  actual  process 
being  an  abridgment  by  mental  calculation  of  that 
which  is  here  written  down. 

(92.)  If  it  should  so  happen  that  the  sum  of  all 
the  digits  forming  each  column  of  units,  those  of  the 
highest  order  excepted,  should  be  less  than  9>  it  is  a 
matter  of  absolute  indifference  whether  the  process  of 
addition  begins  from  the  column  of  the  highest  order 
of  units,  and  proceeds  to  the  lowest,  or  begins  at  the 
lowest  and  proceeds  to  the  highest ;  or,  in  fact,  in  what 
order  it  may  be  taken.  In  the  following  example  we 
may  obtain  the  same  sum  with  the  same  facility  and 
F 


66  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

expedition,  whether  we  proceed  from  left  to  right  or 
from  right  to  left ;  or  begin  at  the  middle  column  and 
proceed  first  to  the  left  and  then  to  the  right,  or  per. 
form  the  operation  in  any  other  order. 

93041 
72130 
62323 

227494 


Although  it  is  probable  that  what  has  been  already 
said  will  make  the  principle  of  addition  sufficiently  in- 
telligible, yet  it  may  not  be  useless  in  this  and  similar 
cases  to  lay  down,  in  formal  terms,  the  several  steps  of 
the  process,  which  require  to  be  attended  to. 


General  Rule  for  Addition. 

(93.)  In  order  to  add  several  numbers  together, 
write  each  of  these  numbers  one  under  the  other  in 
such  a  manner,  that  the  units  of  the  same  order  shall 
stand  in  the  same  vertical  column ;  that  is,  that  the 
units  of  one  number  shall  be  immediately  under  the 
units  of  another,  the  tens  under  the  tens,  the  hundreds 
under  the  hundreds,  and  so  on :  then  add  together  the 
digits  found  in  the  units'  column ;  if  their  sum  be  ex- 
pressed by  a  single  digit,  write  this  figure  under  the 
units'  column,  and  commence  the  same  process  with  the 
tens'  column.  But  if  the  sum  of  the  digits  in  the  units' 
column  be  greater  than  9>  it  must  in  that  case  be  ex- 
pressed by  more  than  one  figure ;  write  the  last  figure 
only  under  the  units'  column,  and  carry  to  the  column 
of  tens  as  many  units  as  are  expressed  by  the  remaining 
figures ;  this  number  must  be  added  to  the  column  of 
tens.  Proceed  in  the  same  manner  with  the  column  of 
tens,  and  so  with  every  column,  proceeding  from  right 


CHAP.  III.  ADDITION.  67 

to  left.  When  the  column  of  the  highest  order,  which 
is  always  the  first  on  the  left,  has  heen  added,  including 
the  number  carried  from  the  preceding  column,  if  th« 
sum  be  expressed  by  a  single  figure,  place  that  figure 
under  the  column,  but,  if  it  be  expressed  by  more  figures 
than  one,  write  these  figures  in  their  proper  order,  the 
last  under  the  column,  and  the  other  preceding  it. 


i  2 


68  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 


CHAP.  IV. 

SUBTRACTION. 

(94.)  WHEN  two  unequal  numbers  are  proposed,  there 
is  a  certain  number,  which,  being  added  to  the  lesser 
would  make.it  equal  to  the  greater ;  and  it  is  evident 
that  if  the  amount  of  the  lesser  be  taken  from  the 
greater,  the  same  number  would  remain.  The  arithme- 
tical process  by  which  such  a  number  is  discovered,  is 
called  SUBTRACTION.  This  operation,  therefore,  may 
be  considered  under  two  points  of  view,  either  as  the 
means  of  diminishing  the  greater,  by  the  amount  of 
the  less,  or  of  increasing  the  less,  until  it  becomes 
equal  to  the  greater.  It  was  evidently  under  the  former 
point  of  view  that  the  operation  received  the  name  of 
SUBTRACTION  ;  and  the  same  way  of  considering  the 
process  gave  the  name  MINUEND,  or  "  number  to  be 
diminished,"  to  the  greater  of  the  two  proposed  numbers; 
and  SUBTRAHEND,  or  "  number  to  be  taken  away," 
to  the  less.  In  order,  however,  to  obtain  a  full  and 
clear  perception  of  the  principle  of  the  operation,  it  will 
be  necessary  that  we  should  consider  it  under  both 
points  of  view. 

(95.)  In  commercial  arithmetic,  the  result  of  the 
operation  is  commonly  called  the  REMAINDER,  a  term 
which  also  implies  that  the  operation  is  one  by  which 
the  greater  number  is  diminished  by  the  amount  of  the 
less:  in  the  mathematical  sciences,  the  result  of  the 
operation  is  called  the  DIFFERENCE  of  the  two  numbers, 
thereby  implying  merely  that  it  is  that  by  which  the 
greater  exceeds  the  less. 

(96.)  Let  us  suppose  the  minuend  A  and  subtrahend 
B  to  be  expressed  by  counters,  there  being  as  many 
counters  in  the  row  A,  as  there  are  units  in  the  minuend, 
and  as  many  in  the  row  B  as  there  are  units  in  the  sub- 
trahend. 


CHAP.  IV.  SUBTRACTION.  69 

A  :..  000000000000 
B 00000000 

If  we  remove  the  counters  by  pairs,  beginning  from  the 
right,  until  all  the  counters  of  the  subtrahend  are  taken 
away,  and  also  the  counters  immediately  above  them  in 
the  minuend,  the  counters  which  remain  will  be  evi 
dently  the  difference  between  the  original  numbers,  and 
will  be  the  remainder  which  would  be  obtained  in  the 
process  of  subtraction.  In  the  present  case,  this  re- 
mainder is  4,  Now,  suppose  that,  instead  of  removing 
the  counters  by  pairs,  we  add  counters  to  the  subtra- 
hend B,  until  we  fill  all  the  vacant  places  below  the 
counters  in  A ;  the  two  numbers  A  and  B  will  then  be 
equalised  by  increasing  B,  and  the  number  of  counters 
necessary  to  effect  this  will  be  4. 

(97«)  Before  we  pass  to  the  consideration  of  more 
complex  cases,  we  shall  take  occasion  to  observe  that 
by  adding  the  same  number  of  counters  to  A  and  B, 
their  difference  will  not  be  changed,  and  of  course  their 
remainder,  obtained  by  the  process  of  subtraction,  will 
continue  the  same.  This  will  be  evident  if  we  observe 
the  effect  of  adding  a  counter  to  A  and  B  on  the  right 
At  present  eight  of  the  counters  in  the  row  A,  hav* 
eight  immediately  below  them  in  the  row  B  :  if  we  add 
another  counter  to  the  right  in  the  row  A,  and  another 
also  to  the  right  in  the  row  B,  the  number  in  each  row 
will  be  increased  by  one ;  but  still  the  counters  to  the 
left  in  A,  which  have  none  below  them  in  B,  continue 
the  same,  and  these  are  the  counters  which  form  the 
remainder  in  the  process  of  subtraction.  We,  there- 
fore, infer,  that,  without  affecting  the  result  of  the  oper- 
ation, we  may  always  add  the  same  number  to  the 
subtrahend  and  minuend,  and  for  like  reasons  we  may 
also  deduct  the  same  number  from  both. 

(98.)  Let  us  now  suppose  that  it  is  required  to  sub- 
tract 2345  from  4689 :  we  shall  consider  the  units 
of  each  order  expressed  by  counters  in  the  subjoined 
scheme :  — 

p  3 


70 


A    TREATISE    ON    ARITHMETIC. 

D  C  B  A 


0  0 
0  0 

000 
000 

0000 
0000 

000 
000 
000 

4689 
2345 

0  0 

000 

0  0 
0  0 

0  0 
0 
0  0 

0  0 

000 

0  0 
0  0 

0  0 
0  0 

2344 

If  we  consider  that  the  object  of  the  operation  is  to 
take  from  the  superior  number  the  amount  of  the  infe- 
rior, and  to  place  the  remainder  in  the  third  line,  we 
shall  proceed  as  follows :  —  Remove  5  counters  from 
the  9  in  the  column  A  of  the  minuend,  and  transfer  the 
remaining  counters  of  the  column  to  the  remainder :  we 
have  thus  taken  from  the  units  of  the  minuend  as  many 
as  are  contained  in  the  units'  place  of  the  subtrahend. 
In  the  same  manner  we  shall  take  4  from  the  tens  of 
the  minuend,  and  transfer  the  remaining  4  to  the  re- 
mainder. Of  the  8  tens  in  the  minuend,  we  have, 
therefore,  removed  4 :  in  the  same  manner,  from  the  6 
hundreds  of  the  minuend  we  take  3  hundreds,  which  is 
the  number  in  the  subtrahend,  and  place  the  remaining 
3  counters  in  the  hundreds  of  the  remainder.  In  the 
minuend  there  are  4  thousands,  and  in  the  subtrahend 
2 ;  taking  2  from  the  former,  we  place  the  remaining 
2  in  the  remainder :  the  remainder,  therefore,  is  2344. 
(99-)  If  we  consider  the  operation  under  another 
point  of  view,  we  should  obtain  the  same  remainder  in 
the  following  manner  :  —  Place  counters  in  the  units' 
place  of  the  remainder,  until  the  number  placed  there, 
added  to  the  number  in  the  units  of  the  subtrahend, 
shall  make  up  the  units  of  the  minuend :  thus,  there 
are  5  counters  in  the  column  A  of  the  subtrahend ;  if 
we  put  4  in  the  same  column  of  the  remainder,  these 


CHAP.  IV. 


SUBTRACTION. 


71 


4,  added  to  the  5,  will  make  up  9:,  which  is  the  num- 
ber of  counters  in  the  column  A  of  the  minuend.  In 
the  same  manner,  if  we  place  4  in  the  column  B  of  the 
remainder,  these  4,  added  to  the  4  in  the  same  column 
of  the  subtrahend,  will  make  8,  which  is  the  number 
contained  in  the  same  column  of  the  minuend.  There 
are  3  counters  in  the  column  C  of  the  subtrahend ; 
and  if  3  more  be  placed  in  the  same  column  of  the 
remainder,  the  two  added  together  will  make  6,  which 
is  the  number  of  counters  in  the  same  column  of  the 
minuend.  In  the  same  manner,  2  counters  placed  in 
the  column  D  of  the  remainder,  added  to  2  in  the 
same  column  of  the  subtrahend,  give  4,  the  num- 
ber of  counters  in  the  same  column  of  the  minuend. 
Thus,  the  units  of  each  order  in  the  remainder,  added 
to  the  units  of  the  same  order  in  the  subtrahend,  pro- 
duce a  sum  or  total  equal  to  the  number  of  the  units  of 
the  same  order  in  the  minuend. 

(100.)  The  example  just  given  is  attended  with 
circumstances  which  afford  a  facility  in  the  operation 
of  subtraction,  which  are  not  found  generally  to  exist : 
A  difficulty  frequently  presents  itself,  which  will  be 
perceived  in  the  following  example. 

(101.)  Let  it  be  required  to  subtract  2987  from 
4345. 

D  C  B  A 


0  0 
0  0 

000 

0  0 
0  0 

0  0 
0 
0  0 

4345 
2987 

0  0 

000 
000 
000 

0000 
0000 

000 
000 
0 

0 

000 

0  0 
0 
0  0 

0000 
0000 

1358 

F    4 


72  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

If  we  consider  the  operation  as  one  by  which  the  lesser 
number  must  be  taken  from  the  greater,  we  must  pro- 
ceed as  follows:  —  Since  the  number  of  counters  in  the 
column  A  of  the  minuend  is  less  than  the  number  in 
the  same  column  of  the  subtrahend,  it  is  evident  that 
the  latter  cannot  be  taken  from  the  former  :  to  remove 
this  inconvenience,  let  us  take  away  one  of  the  4  coun- 
ters in  B,  and  place  10  more  counters  in  A.  It  is 
evident  that  this  will  make  no  real  change  in  the  value 
of  the  number  expressed  by  the  counters  in  the  first  row, 
since  1  counter  in  B  is  equivalent  to  10  in  A :  we 
may  thus  have  15  counters  in  A,  and  7  under  it  in  the 
subtrahend.  Taking  7  from  the  15,  there  will  remain 
8,  and  8  counters  are  accordingly  put  in  the  column  A 
of  the  remainder :  3  counters  now  remain  in  B,  under 
which  are  8,  in  the  subtrahend ;  since  it  is  impossible 
to  take  8  from  3,  we  must  have  recourse  to  a  contrivance 
similar  to  that  just  resorted  to.  A  single  counter  is 
taken  from  C,  and  1 0  counters  placed  in  B :  there  are 
now  13  counters  in  B,  and  8  under  them  in  the  sub- 
trahend :  taking  away  8  of  the  13  in  B,  there  will 
remain  5  ;  and  5  counters  are  accordingly  placed  in  the 
column  B  of  the  remainder.  There  are  now  2  counters 
in  C,  under  which  are  9  m  the  subtrahend :  as  no 
subtraction  can  be  performed  in  this  case,  the  same 
expedient  is  adopted  as  before ;  a  single  counter  is 
withdrawn  from  D,  and  10  counters  more  placed  in  C  : 
there  are  now  12  counters  in  C,  from  which  9  must  be 
taken  ;  3  will  remain,  and  3  counters  are  accordingly 
placed  in  the  column  C  of  the  remainder.  There  are 
now  3  counters  in  the  column  D,  having  2  counters 
under  them  in  the  subtrahend :  2  being  taken  from  the 
3,  one  will  remain ;  and  1  is  accordingly  placed  in  the 
column  D  of  the  remainder :  the  remainder  is,  there- 
fore, 1358. 

(102.)  Now,  it  will  be  observed,  that  the  difficulty 
which  here  presents  itself,  arises  from  this  circumstance  : 
although  the  subtrahend,  as  a  whole,  is  a  less  number 
than  the  minuend,  and  therefore  may  be  subtracted  from 


CHAP.  IV.  SUBTRACTION.  73 

it,  yet  the  units  of  inferior  orders  in  the  former  happen 
to  be  greater  than  those  of  the  same  orders  in  the  lat- 
ter ;  consequently  it  is  impossible  to  subtract  the  units 
of  each  order  in  the  subtrahend  from  those  of  the  same 
order  in  the  minuend.  In  the  first  example  above  given 
this  difficulty  did  not  present  itself,  because  the  number 
of  units  of  each  order  in  the  minuend  was  greater  than 
the  number  of  units  of  the  same  order  in  the  subtra- 
hend. The  expedient  by  which  this  difficulty  is  over- 
come, consists  in  borrowing,  as  it  were,  a  unit  of  a 
superior  order,  and  adding  its  value  to  the  units  of 
the  inferior  order,  which  are  deficient  in  number. 

(103.)  If  we  consider  the  operation  in  another  point 
of  view,  the  question  will  be,  to  discover  the  number, 
which,  being  added  to  the  subtrahend,  would  produce  a 
sum  equal  to  the  minuend.  Since  no  number  of  coun- 
ters added  to  7  would  produce  5,  we  suppose  such  a 
number  added  to  it  as  would  produce  15  ;  that  number 
being  8,  we  place  8  counters  in  the  column  A  of  the  re- 
mainder ;  but  since  the  addition  of  these  two  produce  15, 
it  will  be  necessary  to  carry  1  to  the  tens'  place :  we  must, 
therefore,  add  1  counter  in  the  column  B  of  the  sub- 
trahend; there  will  thus  be  9  counters,  to  which  a 
number  must  be  added,  which  would  give  4  for  the 
sum :  as  this  is  impossible,  we  add  a  number  which 
would  give  14;  that  number  being  5,  we  place  5 
counters  in  the  column  B  of  the  remainder,  and  carry  1 
to  the  column  C.  There  will  now  be  10  counters  in 
the  column  C  of  the  subtrahend,  to  which  such  a  num- 
ber must  be  added  as  would  cause  3  counters  to  be 
placed  in  the  sum ;  that  number  being  3  itself,  3  coun- 
ters are  placed  in  the  remainder,  and  1  is  carried  to  the 
column  D  :  there  will  then  be  3  counters  in  the  column 
D  of  the  subtrahend,  which  would  require  the  addition 
of  1  to  make  up  the  number  in  the  minuend ;  1  is, 
therefore,  placed  in  the  remainder,  and  the  result  is 
1358. 

(104.)  The  student  will  now  have  no  difficulty  in 
applying  the  above  reasoning  to  any  numbers  expressed 


74 


A    TREATISE    ON    ARITHMETIC. 


in  figures.  Let  it  be  required  to  subtract  2431  from 
4679-  The  units  of  each  order  are  here  regarded  sepa- 
rately, and  subtracted  one  from  the  other :  thus  taking 
1  unit  from  9,  8  remain ;  taking  3  tens  from  7  tens, 
4  tens  remain.  In  the  same  manner,  taking  4  hun- 
dreds and  2  thousands  from  6'  hundreds  and  4  thou- 
sands, 2  hundreds  and  2  thousands  remain :  the  total 
remainder  is  then  2248,  and  the  work  stands  as 
below :  — 


4 

6 

7 

9 

2 

4 

3 

1 

2 

2 

4 

8 

It  may  be  observed,  that  in  this  case  the  minuend 
and  subtrahend  are  each  regarded  as  four  separate  num- 
bers, each  column  of  units  being  separately  subtracted 
independently  of  the  others ;  and  it  matters  not  whether 
we  begin  from  the  thousands,  and  proceed  from  left  to 
right,  or  begin  from  the  units  as  usual,  and  proceed  from 
right  to  left :  we  may  even  begin  at  the  hundreds  or 
tens,  since  the  subtraction  of  any  one  order  of  units  is 
quite  independent  of  the  others. 

(105.)  The  same  remainder  would  be  obtained,  with 
equal  facility,  if  we  sought  the  digits,  which,  added  to 
those  of  the  subtrahend,  would  produce  the  digits  of  the 
minuend :  thus,  it  is  obvious  that  8  is  the  number 
which,  added  to  1,  would  produce  9j  4  added  to  3 
would  produce  7  j  2  added  to  4  would  produce  6,  and 
2  to  2  would  produce  4.  Thus  the  same  remainder 
would  be  obtained  by  either  method  with  the  same  ease 
and  expedition,  and  might  be  obtained  with  equal  faci- 
lity in  whatever  order  we  should  proceed,  whether  from 


CHAP.  IV.  SUBTRACTION.  75 

right  to  left,  or  from  left  to  right.     In  the  following 
example,  however,  we  do  not  find  the  same  facilities  : — 


§        -3         .        A 
S          c         w        .*• 

X              3            §J          *3 

9 

2 

4 

3 

3 

8 

7 

6 

We  cannot  subtract  6  from  3,  and  therefore  we  borrow 
1  from  the  4  occupying  the  tens'  place  ;  and,  instead  of 
subtracting  6  from  3,  we  subtract  6  from  13  :  in  the 
same  manner,  since  we  cannot  subtract  7,  the  next 
figure  from  3,  which  remains  after  1  has  been  borrowed 
from  4>,  we  borrow  I  from  the  2  which  fills  the  place  of 
hundreds,  and  we  subtract  7  tens  from  13  tens.  It 
becomes  necessary,  for  the  same  reason,  to  transfer  1 
from  the  thousands'  place  to  the  hundreds'  place  in  the 
minuend.  Instead  of  writing  the  minuend  thus,  in  the 
ordinary  way,  9243,  we  write  it  as  below :  — 


o5             ^ 

""§ 

1             §           «          1 

fi         W        H        £ 

8 

11 

13 

13 

3 

8 

7 

6 

5 

3 

6 

7 

It  will  easily  be  perceived,  that  the  two  methods  of  ex- 


76 


A    TREATISE    ON    ARITHMETIC. 


BOOK  I. 


pressing  the  number  are  equivalent  to  each  other :  sub- 
tracting 6  from  13,  we  now  obtain  a  remainder  7: 
subtracting  7  tens  from  13  tens,  we  obtain  a  remainder 
6  tens.  In  the  same  manner,  subtracting  8  hundreds 
from  1 1  hundreds,  we  obtain  a  remainder  3  hundreds  ; 
and  subtracting  3  thousands  from  8  thousands,  we  get 
the  remainder  5  thousands. 

(106.)  Hence  it  may  be  perceived,  that  when  any 
digit  of  the  subtrahend  is  greater  than  the  digit  of  the 
minuend  immediately  above  it,  we  should  add  10  to  the 
latter,  and  subtract  the  former  from  the  sum,  to  com- 
pensate for  which  it  will  be  necessary  to  diminish  by  1 
the  next  figure  of  the  minuend.  It  will  be  always  pos- 
sible to  practise  this  except  in  the  particular  case  when 
the  next  figure  of  the  minuend  is  0,  as  in  the  following 
example :  — 


I   i   -  - 

LJiJiJ 

9 

8 

0 

2 

5 

7 

4 

8 

The  units  of  the  minuend  in  this  case  being  less  than 
those  of  the  subtrahend,  it  is  necessary  to  borrow  a  unit 
from  the  tens  of  the  minuend  ;  but  the  tens'  place  being 
occupied  by  0,  this  is  impossible :  we  may,  therefore, 
borrow  1  from  the  hundreds  ;  taking  1,  then,  from  the 

8  hundreds,  this  1  will  be  equivalent  to  10  units  and 

9  tens:  we    add  the  10  units  to  the  2  units  in  the 
units'  place,  and  instead  of  0  in  the  tens'  place  we  sub- 
stitute 9.    The  statement,  therefore,  takes  the  following 
form :  — 


CHAP.  IV. 


SUBTRACTION. 


77 


1     §      g     -a 

&        ffl        H       &  . 

9 

7 

9 

12 

5 

7 

4 

8 

4 

0 

5 

4 

The  hundreds  being  diminished  by  1,  the  tens  re- 
placed  by  9,  and  10  being  added  to  the  units,  the  sub- 
traction will  now  be  performed  without  difficulty,  and 
we  obtain  the  remainder  4054. 

(107.)  It  appears,  therefore,  that  when  a  figure  of 
the  minuend  is  less  than  that  immediately  below  it  in 
the  subtrahend,  and  at  the  same  time  the  preceding 
figure  of  the  minuend  is  0,  then  we  must  add  1 0  to  the 
figure  of  the  minuend  which  is  less  than  that  below  it : 
conceive  9  to  be  substituted  for  the  0,  and  diminish  by 
1  the  following  figure  of  the  minuend. 

(108.)  There  is,  however,  an  easier  way  of  encoun- 
tering this  difficulty,  derived  from  an  observation  al- 
ready made,  —  that  if  the  minuend  and  subtrahend  be 
increased  by  the  addition  of  the  same  number,  no 
change  will  be  made  in  the  remainder.  Giving  due  at- 
tention to  this  observation,  let  us  suppose,  in  the  above 
example,  that  we  add  10  units  to  the  units  of  the  minu- 
end. The  effect  of  such  an  addition  would  be  neutral- 
ised by  adding,  at  the  same  time,  10  units  to  the  units 
of  the  subtrahend.  But  as  this  would  still  leave  the 
units  of  the  subtrahend  greater  than  those  of  the  minu- 
end, we  make  the  same  addition  to  the  subtrahend  by 
another  method,  viz.  by  adding  1  to  its  tens ;  so  that 
the  tens,  instead  of  being  4,  are  increased  by  1,  and 
become  5.  The  statement  of  the  question  will  then 
be  as  follov/s :  — 


7S 


A    TREATISE    ON    ARITHMETIC. 


9 

8 

10 

12 

5 

8 

5 

8 

4 

0 

5 

4 

We  have  here  added  10  to  the  units  of  the  minuend, 
and  1  to  the  tens  of  the  subtrahend.  These  two  ad- 
ditions neutralise  each  other,  on  the  principle  just  re- 
ferred  to.  We  have  also  added  10  to  the  tens  of  the 
minuend;  and  at  the  same  time  added  1  to  the  hundreds 
of  the  subtrahend.  These  two  additions  in  the  same 
way  neutralise  each  other.  In  fact,  by  the  two  addi- 
tions which  have  been  made,  the  minuend  has  been  in- 
creased by  the  addition  of  11  tens,  or  110,  and  the 
subtrahend  has  been  increased  also  by  the  addition  of 
110:  the  two  additions,  therefore,  being  equal,  leave 
the  remainder  the  same. 

(lOp.)  The  following  general  rule  will  then  serve  for 
all  cases  in  which  it  is  required  to  subtract  the  lesser 
number  from  the  greater  :  — 

General  Rule  for  Subtraction. 

To  subtract  a  lesser  number  from  a  greater,  place 
the  lesser  number  under  the  greater,  so  that  the  units  of 
the  same  order,  in  the  two  numbers,  shall  stand  in  the 
same  vertical  column,  the  units  under  the  units,  the  tens 
under  the  tens,  and  so  on.  Then  subtract  the  units  of 
the  subtrahend  from  the  units  of  the  minuend,  and 
write  the  remainder  under  them  in  the  same  column, 
placing,  in  the  same  way,  the  tens  under  the  tens,  and 
the  hundreds  under  the  hundreds,  and  so  on,  for  every 
column,  from  right  to  left. 


CHAP.  IV.  SUBTRACTION.  79 

If  it  should  happen  that  any  figure  of  the  minuend 
should  be  less  than  the  figure  of  the  subtrahend  imme- 
diately below  it,  then  increase  the  former  by  1C),  and 
proceed  as  before ;  but  in  that  case  it  will  be  necessary 
to  increase  the  next  figure  of  the  subtrahend  by  1,  or 
to  carry  1  to  it. 

(110.)  To  be  enabled  to  perform  the  operation  of 
subtraction  with  ease  and  expedition  by  the  above  rule, 
it  will  be  necessary  that  the  student  should  retain  in  his 
memory  the  remainders  which  are  found  by  subtracting 
the  single  digits  from  each  other,  and  from  all  numbers 
between  10  and  19  inclusive  ;  but  this  knowledge  he 
must  needs  possess  if  he  has  learned  the  table  of  ad- 
dition given  in  the  last  chapter.  Thus,  if  it  be  required 
to  know  what  is  left  when  8  is  subtracted  from  17,  the 
question  is,  what  number  added  to  8  will  produce  17- 
This,  it  will  be  perceived,  forms  a  part  of  the  table  of 
addition. 

(111.)  The  operation  of  subtraction  is  indicated  by 
the  sign  — ,  minus,  placed  between  the  minuend  and 
subtrahend.  Thus,  7  —  2  means  that  2  is  to  be  sub- 
tracted from  7 ;  and  it  expresses  the  remainder  which 
is  obtained  by  that  operation.  Thus,  7—2  =  5,  means 
that  2  subtracted  from  7  leaves  the  remainder  5. 

(112.)  Since  no  more  than  two  numbers  can  be  con- 
cerned in  any  question  of  subtraction,  the  operation  is 
in  general  more  simple  than  in  questions  of  addition, 
where  several  numbers  may  be  concerned.  Subtrac- 
tion, therefore,  stands  less  in  need  of  verification,  be- 
ing less  liable  to  error;  but  still  it  is  desirable  that 
the  computer  should  possess  some  means  of  checking 
his  work.  An  easy  and  obvious  method  of  doing  so  is 
suggested  by  the  fact,  that  the  remainder,  added  to  the 
subtrahend,  must  make  a  total  equal  to  the  minuend. 
When,  therefore,  any  subtraction  has  been  performed, 
add  the  subtrahend  and  remainder,  and,  if  the  sum  be 
the  same  as  the  minuend,  the  work  may  be  considered 
as  correct. 

(113.)  The  teacher  may  check  the  work  of  his  pupils 


t, 

80  A    TREATISE    OX    ARITHMETIC.  BOOK  I. 

without  the  labour  of  re-calculation,  by  giving  to  them, 
as  already  explained,  questions  in  which  the  numbers 
are  different,  but  which  he  knows  must  give  the  same 
remainder.  He  may  always  accomplish  this  by  in- 
creasing any  digit  of  the  minuend  by  1,  and  increas- 
ing the  corresponding  digit  of  the  subtrahend  also  by 
1,  or  increasing  both  by  2,  or  by  any  other  number. 
This  contrivance  rests  upon  the  principle  already  ex- 
plained, that,  when  two  numbers  are  equally  increased 
their  difference  will  remain  the  same.  He  may  also 
increase  any  figure  of  the  minuend  alone,  observing 
that  the  corresponding  figure  in  the  one  remainder  must 
be  as  much  greater  than  ihe  same  figure  in  the  other 
remainder.  He  may  also  increase  any  figure  in  the 
subtrahend,  observing  that  the  corresponding  figure  in 
the  remainder  must  be  equally  diminished. 

(114.)  The  following  example  will  illustrate  these 
observations  :  —  Let  it  be  required  to  subtract  37b'9 
from  4354,  and  to  verify  the  process. 

M.  4354 

S.   -         -     -     3769 

R         -         -       585 


M.-         -     -     4354 

We  say  9  cannot  be  subtracted  from  4,  but,  adding  10, 
and  taking  it  from  14,  we  get  the  remainder  5.  To 
compensate  for  the  1 0  added  to  the  4,  we  carry  1  to  the 
6  ;  but  7  tens  cannot  be  taken  from  5  tens  :  we  there- 
fore add  10  to  the  5,  and,  taking  7  tens  from  15  tens, 
we  set  down  the  remainder,  which  is  8  tens.  For  the 
10  added  to  the  5  above,  we  carry  1  to  the  7  below, 
which  makes  it  8  :  8  hundreds  cannot  be  deducted  from 
3  hundreds  ;  we  therefore  add  1 0  to  the  3,  and,  deduct- 
ing 8  from  the  13,  obtain  the  remainder  5  hundreds. 
For  the  10  added  to  the  3  above,  we  carry  1  to  the  3 
below,  which  makes  it  4 :  4  thousands  being  deducted 
from  the  4  thousands  above,  leaves  no  remainder.  The 


CHAP.  IV.  SUBTRACTION.  81 

total  remainder  is,  therefore,  585.  To  verify  the  pro- 
cess, this  remainder  is  now  added  to  the  subtrahend, 
and  we  find  that  by  such  addition  the  minuend  is  re- 
produced :  consequently  the  process  is  correct.  If  we 
wish  to  propose  questions  to  different  computers  which 
will  produce  the  same  remainder,  585,  we  have  only  to 
increase  or  diminish  equally  the  digits  which  occupy 
any  vertical  column  in  the  minuend  and  subtrahend. 
Thus,  if  we  add  1  to  the  hundreds  and  tens,  the  ques- 
tion will  be  stated  and  proved  as  follows :  — 

M        -         -     4464 

S    -         -     -     3879 


R         -         -       585 


M  -         -     -     4464 

In  like  manner,  if  the  thousands  and  units  be  both 
diminished  by  1,  while  the  tens  or  hundreds  are  each 
increased  by  2,  the  question  will  take  the  following 
form,  with  the  same  remainder :  — 

M  3573 

S    -         -     -     2988 


R         -         -       585 


M  -         -     -     3573 

If  we  increase  each  of  the  digits  of  the  minuend  by  1, 
the  remainder  will  likewise  have  its  digits  increased 
by  1:  — 

M  5465 

S    -         -     -     3769 

R  -     1696 

M  -         -     -     5465 


82  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

In  this  case  a  0  is  understood  to  precede  the  5  in  the 
hundreds'  place  of  the  first  remainder ;  and  this  0  be- 
ing increased  by  1,  brings  1  into  the  thousands'  place  in 
the  above  remainder.  If  each  of  the  digits  of  the 
subtrahend  be  diminished  by  1,  the  digits  of  the  re- 
mainder will  likewise  be  increased  by  1,  and  the  same 
change  will  be  made  as  in  the  last  case :  — 

M        -         -     4354 

S    -         -     -     2658 


R         -         -     1696 
M  -         -     -     4354 

It  is  unnecessary  to  pursue  these  examples  further. 
Teachers  will  easily  see  how  they  may  modify  examples 
with  unlimited  variety,  so  as  either  to  cause  them  all  to 
have  the  same  remainder,  or  to  make  any  proposed 
change  in  one  or  more  figures  of  the  remainder,  so  that 
by  the  correspondence  of  the  results  with  these  princi- 
ples, they  may  be  assured  that  the  work  of  their  pupils 
is  correct  without  the  labour  of  recalculation. 

(115.)  As  addition  becomes  the  means  of  verifying 
subtraction,  so  likewise  we  may  use  subtraction  as  a 
means  of  verifying  addition. 

Let  several  numbers,  marked  A,  B,  C,  D  and  E,  be 
added  together  by  the  rule  for  addition,  so  that  the  sum 
S  shall  be  found  :  — 

A  3579 


B  - 
C 
D  - 
E 

S  - 
S' 

-  2684 
-  3761 
-  2007 
-  9889 

-  21920 
-  18341 

A  -        -    -     3579 


CHAP.  IV.  SUBTRACTION.  83 

Let  the  top  line  A  be  now  cut  off,  and  let  the  four  lines 
E}  C,  D  and  E  be  added  ;  write  their  sum,  S',  under  the 
total  sum  S.  It  is  evident  that  since  S'  is  the  sum  of 
all  the  proposed  numbers  except  A,  we  ought  to  get  the 
number  A  if  we  subtract  S'  from  S ;  and  we  accord- 
ingly find  that  the  remainder  of  such  subtraction'  is  the 
number  A.  Had  it  been  otherwise,  the  inference  would 
be,  that  the  addition  was  performed  incorrectly,  and 
required  revision. 

(116.)  The  principle  of  this  method  of  verification 
may  be  extended  and  varied,  so  as  to  afford  exercise  to 
the  judgment  and  understanding  of  the  pupils.  After 
the  addition  has  been  performed,  let  the  master  strike 
one  figure  out  of  each  vertical  column,  and  direct  the 
pupil  to  add  the  numbers,  omitting  the  figures  thus 
crossed  out.  The  sum  thus  obtained  being  subtracted 
from  the  total  sum,  the  remainder  should  be  a  number 
consisting  of  the  same  figures  as  were  struck  out  of  the 
numbers  added.  The  following  example  will  illustrate 
this :  — 

A    -    -  1^57 

B  -    -  -  $642 

C         76^4 

D  _   -  -  9870 

E    ~    -  7530 

S  -    -  -  35059 
S'         26703 

R  -    -  -  8356 


Here  the  numbers  A,  B,  C,  D  and  E,  being  added  toge- 
ther, produce  the  total  S.  A  figure  is  then  struck 
out  of  each  vertical  column,  and  the  numbers  are  again 
added,  omitting  those  figures.  The  sum  S'  is  thus  ob- 
tained. S'  being  then  subtracted  from  S,  we  obtain  a 
remainder  composed  of  the  same  figures  as  were  struck 
out  of  the  original  numbers,  and  occurring  in  the  same 
places.  The  reason  of  this  is  sufficiently  obvious  :  by 
G  2 


84-  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

striking  out  8  in  the  number  13,  3  in  A,  5  in  C,  and  6 
in  D,  we  diminished  the  amount  of  those  numbers 
respectively,  —  viz.  that  of  B  by  8000,  that  of  A  by 
300,,  that  of  C  by  50,  and  that  of  D  by  6.  The  total 
amount  of  the  whole  was,  therefore,  diminished  by 
8356 :  consequently  S',  which  is  the  sum  of  what  re- 
mained, ought  to  be  less  than  S,  the  total  sum,  by  8356, 
which  is  accordingly  the  remainder. 

(117.)  Another  mode  of  verifying  the  results  of 
subtraction,  and  indeed  also  of  addition,  is  by  proceed- 
ing from  left  to  right,  instead  of  from  right  to  left. 
This,  although  not  the  most  convenient  in  practice,  yet 
should  be  occasionally  resorted  to  by  teachers,  with  a 
view  to  familiarise  the  pupils  with  the  reasons  of  the 
arithmetical  processes.  In  the  following  number  the 
subtrahend  has  been  deducted  from  the  minuend  first, 
in  the  usual  way,  by  proceeding  from  right  to  left,  and 
thus  the  remainder  has  been  found.  It  may  be  sub- 
sequently verified  by  the  following  process  :  — 

M        -         -     5641 
S    -         -     -     3760 


R  1881 


Commencing  from  the  left,  we  shall  subtract  from  the 
minuend  both  the  remainder  and  the  subtrahend.  As 
these  two  numbers  together  must  be  equal  to  the  minu- 
end, the  remainder  in  this  subtraction  ought  to  be 
nothing.  Subtracting,  then,  3  thousands  from  5  thou- 
sands, we  get  the  remainder,  2  thousands  ;  and  sub- 
tracting 1  thousand  from  this,  1  thousand  remains ; 
this  being  added  to  the  next  figure,  6,  of  the  minuend, 
makes  16  hundreds.  Subtracting  from  this  the  7  hun- 
dreds of  the  subtrahend,  there  remain  9  hundreds  ; 
and  from  this  subtracting  the  8  hundreds  of  the  re- 
mainder, there  remains  1  hundred ;  this  being  added  to 
the  succeeding  4  of  the  minuend,  gives  14  tens.  The 
6  tens  of  the  subtrahend  being  taken  from  this,  leaves 


CHAP.  IV.  SUBTRACTION. 

8  tens ;  and  the  8  tens  of  the  remainder  being  taken 
from  this,  leaves  nothing.  The  1  in  the  units'  place 
of  the  remainder  being  then  taken  from  the  1  in  the 
units'  place  of  the  minuend,  leaves  0.  Consequently 
the  subtrahend  and  remainder,  being  successively  sub- 
tracted from  the  minuend,  leave  no  remainder. 

(118.)  Upon  the  same  principle  questions  in  addi- 
tion may  be  verified.  In  the  following  example  the 
numbers  are  first  added  by  proceeding  from  right  to  left 
in  the  usual  way  :  — 

A  -    376 

B     -        -     -     489 

C  -     768 


S     -        -     -  1633 


To  verify  the  work  by  proceeding  from  left  to  right  we 
add  the  first  column,  and  find  14>  hundreds  ;  subtract. 
ing  this  from  the  16  hundreds  in  the  sum,  2  hundreds 
are  left  :  we  write  therefore  2  under  the  6.  Adding  the 
tens'  column  we  find  the  sum  2  1  ;  this  being  taken  from 
the  3  tens  of  the  sum  and  the  2  hundreds  which  re- 
mained, or,  what  is  the  same,  from  23  tens,  2  tens 
remain  :  we  therefore  strike  out  the  2  under  the  6,  and 
write  2  in  the  tens'  place  under  the  3.  The  units* 
column  being  now  added  gives  23  ;  but  there  remain  of 
the  sum  2  tens  and  3  units,  from  which  the  23  ob- 
tained by  the  addition  of  the  units'  column  being  taken, 
there  is  no  remainder  :  thus  the  successive  columns 
being  added,  from  left  to  right,  and  subtracted  from 
the  total,  leave  no  remainder,  and  the  work  is  therefore 
correct. 


o  3 


86  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 


CHAP.  V. 

MULTIPLICATION, 

BEFORE  we  enter  into  the  details  of  the  arithmetical 
operations  of  multiplication  and  division,  it  will  be  use- 
ful to  explain  some  properties  by  which  certain  classes 
of  number  are  distinguished,  relatively  to  their  compo- 
sition by  the  addition  of  other  numbers. 

(lip.)  We  have  seen  that  all  numbers  whatever  may 
be  composed  by  the  addition  of  units.  Thus  the  com- 
position of  the  successive  numbers,  1,  2,  3,  &c.  may  be 
exhibited  in  the  following  manner  :  — 


1+1+1+1=4 
&c.  &c. 

This  property  of  being  formed  by  the  continued  addition 
of  the  number  1,  is  common  to  all  numbers  whatever  ; 
but  certain  numbers  are  distinguished  by  a  like  property 
with  respect  to  others  ;  —  that  is  to  say,  of  being  formed 
by  the  continued  addition  of  other  numbers,  such  as 
2,  3,  4,  &c  The  numbers  which  are  formed  by  the 
continued  addition  of  2  are  as  follows  :  — 

2=2 

2+2=4 
2+2+2=6 
2+2+2+2=8 
&c.  &c. 

In  like  manner,  the  numbers  formed  by  the  continued 
addition  of  3,  4,  &c.  are  expressed  as  follows  :  — 


MULTIPLICATION. 
3  =  3 

3  +  3=6 
3+3+3=9 


&c.  &c.  £c. 


=12 

4+4+4+4=16 
&c.  &c.  &c. 

It  will  be  perceived  that  in  the  series  of  numbers 
proceeding  upwards  from  1,  some  will  be  found  which 
cannot  be  formed  by  the  continued  addition  of  any  other 
number  except  1  ;  the  others  may  be  formed  by  the  con- 
tinued addition  of  2,  3,  or  some  other  higher  number. 
Thus  the  numbers  3,  5  and  7  cannot  be  produced  by 
the  continued  addition  of  any  other  number,  1  excepted, 
while  the  intermediate  numbers  4  and  6  admit  of  being 
formed,  the  first  by  the  addition  of  2,  and  the  second 
by  the  addition  of  either  2  or  3. 

(120.)  Those  numbers  which  cannot  be  formed  by 
the  continued  addition  of  any  number  except  1,  are  dis- 
tinguished by  the  name  of  prime  numbers  :  thus  the 
numbers  11,  13,  17,  &c.  are  prime  numbers. 

(121.)  All  other  numbers  are  called  multiple  num- 
bers, and  they  are  said  to  be  multiples  of  those  lesser 
numbers  by  the  continued  addition  of  which  they  may 
be  formed  :  thus  6  is  a  multiple  of  2,  because  it  may  be 
formed  by  the  addition  of  3  twos.  It  is  likewise  a 
multiple  of  3,  because  it  may  be  formed  by  the  addition 
of  2  threes.  In  the  same  manner  12  is  a  multiple  of 
2,  of  3,  of  4,  and  of  6,  because  it  may  be  formed  by  the 
addition  of  6  twos,  4  threes,  3  fours,  or  2  sixes. 

(122.)  In  the  ascending  series  of  numbers  every  al- 
ternate number  is  a  multiple  of  two.  This  will  easily 
be  perceived  when  it  is  considered  that  each  successive 
number  in  the  series  is  formed  by  adding  1  to  the  pre- 
ceding number  :  beginning  then  at  the  second  number 
G  4 


88  A    TREATISE    ON    ABITH31ETIC.  BOOK  I. 

of  the  series,  namely  2  itself,  we  proceed  to  the  next  but 
one  by  adding  2,  and  pass  to  the  next  alternate  number 
by  adding  2  more,  and  so  on.  From  the  very  nature 
of  the  series,  therefore,  every  alternate  number  beginning 
from  the  second  is  formed  by  the  constant  addition  of  2, 
and  is  therefore  a  multiple  of  2.  Such  numbers  are 
called  even  numbers,  and  the  intermediate  numbers  of 
the  series  are  called  odd  numbers :  thus  the  series  of 
even  numbers  are  2,  4,  6,  8,  10,  &c.,  and  the  series  of 
odd  numbers  1 ,  3,  5,  1,  9,  &c. 

(123.)  Since  every  even  number  is  a  multiple  of  2, 
it  is  evident  that  no  even  number,  except  2  itself,  can 
be  prime,  and  therefore  every  prime  number,  except  2 
itself,  must  be  an  odd  number.  We  must  not,  however, 
infer  on  the  other  hand  that  every  odd  number  is  prime  : 
9  is  a  multiple  of  3,  15  of  5,  &c. 

(124.)  Since  in  the  series  of  numbers  an  odd  number 
always  stands  between  two  even  ones,  it  follows  that  an 
even  number  will  be  obtained  either  by  subtracting  one 
from  an  odd  number,  or  adding  1  to  it. 

(125.)  In  like  manner,  since  an  even  number  always 
stands  between  two  odd  ones,  we  shall  get  an  odd  num- 
ber either  by  subtracting  1  from  an  even  number,  or 
adding  1  to  it. 

(126.)  Two  numbers  are  said  to  be  the  same  mul- 
tiples of  two  others,  when  they  are  formed  by  the  ad- 
dition of  those  two  others  the  same  number  of  times  : 
thus  12  being  formed  by  the  addition  of  4  threes,  and 
20  by  the  addition  of  4  fives,  12  is  the  same  multiple  of 
3  as  20  is  of  5. 

(127.)  The  number,  by  the  continued  addition  of 
which  another  is  formed,  is  called  a  sub-multiple  of  that 
other  :  thus  1 2  being  formed  by  the  continued  addition 
of  2, 3, 4  or  6,  these  numbers  are  severally  sub-multiples  of 
12.  In  fact,  if  one  number  be  a  multiple  of  another,  then 
the  latter  must  always  be  a  sub-multiple  of  the  former. 

(128.)  When  a  number  is  not  a  multiple  of  anothei 
number,  it  is  frequently  necessary  to  consider  the  two 
successive  multiples  of  that  other  between  which  it  is 


CHAP.  V.  MULTIPLICATION  8.9 

placed  in  the  numerical  series,  and  to  mark  its  distance 
from  the  one  or  the  other  of  these.  Let  us  suppose  that  it 
is  required  to  determine  whether  40  be  a  multiple  of  9, 
and  if  not,  to  determine  how  it  stands  with  respect  to 
the  two  successive  multiples  of  9j»  between  which  it  is 
placed  in  the  numerical  series.  By  adding  4  nines  and 
5  nines  respectively  we  obtain  the  following  results, 


=45 

It  appears,  then,  that  40,  the  number  in  question,  is 
greater  than  that  which  is  produced  by  the  addition  of 
4  nines,  and  less  than  that  which  results  from  the  ad- 
dition of  5  nines.  The  composition  of  the  number  40, 
so  far  as  it  can  be  made  by  the  addition  of  nines,  would 
then  be  expressed  in  this  manner, 


The  greatest  multiple  of  9  contained  in  40  is  the  sum  of 
4  nines,  and  the  number  40  exceeds  this  by  4. 

In  the  same  manner,  if  we  enquire  how  far  the  com- 
position of  the  number  53  can  be  effected  by  the  conti- 
nued addition  of  8,  we  shall  find  the  following  results, 

84-8  +  8  +  8  +  8  +  8  =  48 
8+8  +  8  +  8  +  8  +  8-^8  =  56. 

Now  it  appears  that  the  number  53  is  placed  in  the  nu- 
merical series  between  the  two  multiples  of  8,  formed 
by  the  addition  of  6  eights  and  7  eights.  The  compo- 
sition of  the  number  53  is  then  expressed  as  follows, 

8+8+8+8+8+8+5=53 

(129.)  When  it  is  necessary  then  to  investigate  the 
composition  of  any  number  by  the  continued  addition 
of  any  other  number,  the  result  of  the  investigation  will 
always  be,  either  that  the  former  is  a  multiple  of  the 
latter,  —  in  which  case  it  will  be  necessary  to  state  what 
multiple  it  is,  —  or  that  the  former  exceeds  a  certain 
multiple  of  the  latter,  by  a  number  which  is  always  less 
than  the  latter  number.  In  fact,  we  must  always  de- 


99  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

clare  what  multiple  the  number  is,  if  it  be  a  multiple,  or 
if  not  ive  must  determine  the  greatest  multiple  of  the 
proposed  number  which  it  contains,  and  its  excess  above 
that  multiple. 

It  is  necessary  that  these  points  should  be  distinctly 
understood  by  all  who  expect  to  form  clear  and  distinct 
conceptions  of  the  processes  of  MULTIPLICATION  and 
DIVISION. 

(130.)  The  object  of  multiplication  is  to  furnish  a 
method  of  discovering  the  number  which  would  be  pro- 
duced by  adding  the  same  number,  any  proposed  num- 
ber of  times,  more  concise  and  expeditious  than  the 
ordinary  methods  of  addition  already  explained.  M ul- 
tiplication,  is,  therefore,  that  arithmetical  operation  by 
which  we  may  with  brevity  and  facility  discover  any  pro- 
posed multiple  of  any  proposed  number. 

(131.)  The  data  necessary  in  such  a  question  are  the 
number  whose  multiple  is  to  be  found,  and  the  number 
of  times  that  number  is  to  be  repeated  in  order  to  form 
the  proposed  multiple. 

The  number  by  the  continued  addition  of  which  the 
sought  multiple  is  formed,  is  called  the  multiplicand, 
or  the  number  to  be  multiplied;  the  number  of  times 
which  that  number  must  be  repeated  in  order  to  form 
the  sought  multiple  is  called  the  multiplier,  and  in  this 
case  the  result  of  the  operation,  which  is  the  multiple 
sought,  is  called  the  product. 

Thus  if  it  be  required  to  discover  what  number  would 
be  produced  by  the  repetition  of  9  six  times,  then  9  is 
the  multiplicand,  and  6  the  multiplier.  If  we  proceeded 
by  the  method  of  addition,  we  should  write  down  9  m  a 
column  6  times,  and  we  should  add  the  column  by 
the  ordinary  rules  of  addition,  and  should  find  that  the 
addition  would  produce ^4 ;  this  number  54  would  then 
be  the  product. 

The  brevity  and  expedition  which  are  obtained  by  the 
methods  of  multiplication  commonly  practised,  depend 
partly  on  the  student  committing  to  memory  the  results 
of  certain  simple  multiplications,  so  as  to  apply  them 


CHAP.  V.  MULTIPLICATION.  91 

-when  necessary,  and  partly  on  certain  properties  of 
number,  which  we  shall  now  proceed  to  explain. 

(132.)  If  we  add  together  5  sevens,  we  shall  find 
that  their  sum  is  35  :  now,  if  we  add  together  7  fives, 
we  shall  find  that  their  sum  is  the  same.  In  the  one 
case  7  is  the  multiplicand,  and  5  the  multiplier ;  in  the 
other  5  is  the  multiplicand  and  7  the  multiplier.  In 
this  case,  therefore,  it  appears  that  the  product  will  be 
the  same  if  the  multiplicand  be  changed  into  the  multi- 
plier, and  the  multiplier  into  the  multiplicand.  This  is 
a  general  property  of  all  numbers,  and  it  will  he  found 
universally  that  when  any  number  multiplied  by 
another  gives  a  certain  product,  the  same  product  will 
be  obtained  if  the  latter  be  multiplied  by  the  former. 
The  reason  of  this  will  become  apparent  if  we  suppose 
the  numbers  in  question  represented  by  counters*  Tak- 
ing the  example  already  given,  let  us  express  by  counters 
5  times  7 ;  we  shall  place  7  counters  in  a  row,  and  re- 
peat that  row  5  times :  the  arrangement  will  then  be  as 
follows  ;— 

o  o  o  o  o  o  o 

ooooooo 

o  o  o  o  o  o  o 

ooooooo 

ooooooo 

We  have  here  five  cross  rows  of  7>  and,  therefore,  the 
total  number  of  counters  is  5  times  7  j  hut  if  we  con- 
sider-the  same  collection  of  counters  in  another  point  of 
view,  we  shall  see  that  they  are  also  7  times  5.  It  will 
be  perceived  that  the  collection  consists  of  7  upright  co- 
lumns, each  column  containing  5  counters:  the  total 
number  is  therefore  7  times  5. 

The  same  illustration  will  be  applicable  to  any  mul- 
tiplier and  multiplicand,  and  will  therefore  establish  the 
general  conclusion,  that  the  multiplier  and  multiplicand 
may  be  interchanged  without  affecting  the  product. 

(133.)  Since  there  is  no  real  distinction,  then,  between 
the  multiplier  and  multiplicand  in  their  relation  to  the 
product,  there  is  no  reason  why  they  should  be  called  by 


92  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

different  names,  and  accordingly  they  are  sometimes 
denominated  by  the  common  name  factors.  Thus  48 
is  a  product  whose  factors  are  6  and  8,  or  4  and  12,  &c. 
(134.)  A  number  may  be  multiplied  by  another  by 
resolving  it  into  several  parts,  multiplying  each  of  those 
parts  separately,  and  then  adding  the  products.  This 
will  become  evident  when  we  apply  it  to  an  example. 
Let  us  suppose  that  we  wish  to  multiply  6*  by  4  ;  6  may 
be  formed  by  the  addition  of  1,  2,  and  3,  so  that 
1+2+3=6 

To  multiply  6  by  4  we  should  add  together  4  sixes : 
following  the  simple  method  of  addition,  the  process 
would  be  as  follows, 

142+3  =  6 

1+2+3  =  6 

1+2+3  =  6 

1+2+3  =  6 


4  +  8  +  12=24 

Here  we  have  obtained  the  product  in  two  distinct  ways, 
either  by  adding  the  4  sixes,  or  by  adding  separately  the 
three  columns  in  which  1,2  and  3  are  four  times  repeated. 
In  the  latter  case  we  obtain  the  3  products,  4,  8  and  12, 
and  by  the  addition  of  these  three  we  must,  of  course, 
get  the  same  result  as  we  have  obtained  by  adding  the 
column  of  four  sixes.  The  reader  will  find  no  -diffi- 
culty in  generalising  this  result,  and  he  will  perceive 
that,  universally,  the  multiplicand  may  be  resolved  into 
parts,  each  part  being  multiplied  by  the  multiplier,  and 
the  total  product  will  be  obtained  by  adding  together  these 
partial  products. 

(135.)  It  is  not  difficult  to  show,  that  the  principle 
just  explained  involves  another  proof  that  the  multiplier 
and  multiplicand  are  interchangeable.  Let  us  suppose  that 
we  are  required  to  multiply  8  by  6 :  we  may  resolve  the 
multiplicand  8  into  eight  component  units,  so  that  we 
should  have  its  composition  expressed  thus:  — 


CHAP.  V.  MULTIPLICATION.  9^ 

1+1+1+1+1+1+1+1=8 

Now,  by  the  principle  just  established,  we  shall  multiply 
8  by  6,  providing  that  we  multiply  each  of  the  com. 
ponent  parts  of  8  by  6,  and  add  together  the  pro- 
ducts ;  but  if  we  add  these  we  shall  obtain  8  sixes, 
since  each  unit,  multiplied  by  6,  will  give  the  product  6 : 
thus  we  perceive  that  6  eights  are  equivalent  to  8 
sixes. 

(136.)  As  we  have  already  proved  that  the  multiplier 
and  multiplicand  are  interchangeable  without  affecting 
the  product,  it  follows  from  what  has  been  just  estab- 
lished, that  multiplication  may  be  effected  by  resolving 
the  multiplier  into  smaller  numbers,  and  multiplying 
the  multiplicand  by  each  of  those  numbers  separately. 
Thus,  if  the  multiplier  be  8,  we  may  perform  the  mul- 
tiplication by  multiplying  the  multiplicand  by  5  and  3 
separately,  and  adding  together  the  products.  Multi- 
plication may,  therefore,  be  always  performed  either  by 
resolving  the  multiplicand  into  smaller  numbers,  and 
multiplying  each  of  these  by  the  multiplier,  or  resolving 
the  multiplier  into  smaller  numbers,  and  multiplying 
the  multiplicand  by  each  of  these.  In  either  case  the 
several  partial  products  being  added  together,  the  total 
product  will  be  the  result. 

(137-)  The  principle  just  explained  facilitates  the 
process  of  multiplication  by  making  the  multiplication 
of  large  numbers  depend  on  that  of  small  ones ;  but 
still  more  by  a  circumstance  which  will  be  explained 
more  fully  hereafter,  namely,  that  there  are  some  num- 
bers whose  multiplication  involves  no  difficulty  beyond 
a  knowledge  of  the  notation  of  number.  We  have  al- 
ready seen  that  a  number  may  be  multiplied  by  10,  100, 
1000,  &c.  by  merely  annexing  noughts  to  it. 

(138.)  If  the  multiplier  be  not  a  prime  number  (120.), 
it  will  always  be  itself  the  product  of  two  other  num- 
bers. In  such  a  case  the  multiplication  may  be  per- 
formed by  using  these  two  numbers  successively  as 
multipliers,  instead  of  the  given  multiplier.  Thus_j  if 
the  given  multiplier  be  6,  we  may  first  multiply  by  £, 


9^  A    TREATISE    ON    ARITHMETIC.  BOOK  !• 

and  then  multiply  the  product  thus  found  by  3 :  the 
final  product  will  he  the  same  as  if  we  had  multiplied 
in  the  first  instance  by  6.  To  understand  this  process, 
it  is  only  necessary  to  refer  to  an  example.  Let  us 
suppose  that  we  wish  to  multiply  8  by  6 :  by  what  has 
been  just  established,  if  we  resolve  6  into  3  parts,  each 
of  which  shall  be  2,  the  multiplication  will  be  effected 
by  multiplying  8  by  its  three  parts  severally,  and  then 
adding  the  products,  so  that  in  the  final  result  we  shall 
have  the  product  of  8  and  2  three  times  repeated ;  that 
is,  we  shall  have  that  product  multiplied  by  3.  It  is 
evident,  then,  that  the  final  product  is  3  times  the  pro- 
duct of  8  and  2.  There  can  be  no  difficulty  in  per- 
ceiving that  this  reasoning  is  universally  applicable. 

(139.)  In  order  to  multiply  with  ease  and  despatch 
any  numbers  exceeding  9?  it  is  necessary  that  the  pro- 
duct of  every  pair  of  single  digits  should  be  first 
investigated  and  committed  to  memory.  The  products 
of  the  smaller  digits  can  only  be  found  by  addition. 
Those  of  the  greater  ones  may  be  determined  by  com- 
bining addition  with  the  multiplication  of  inferior  digits 
by  some  of  the  principles  already  explained.  We  shall 
presently  explain  these  more  fully ;  meanwhile  we 
shall  exhibit  in  the  following  table  the  product  of  every 
pair  of  single  digits. 

The  operation  of  multiplication  is  expressed  by  the 
sign  x  placed  between  the  two  numbers  to  be  multi- 
plied :  thus  2x3  =  6  means  that  the  product  of  2  and 
3  is  6  :  — 

MULTIPLICATION  TABLE.* 


2x2=  4 
2x«3=  6 
'2X4=  8 
2x5  =  10 
2x6=12 
2x7=14 
2x8=16 
2X9=18 

3x3=  914X4=16 
3x4  =  124x5  =  20 
3x5=1514x6=24 
3x6=18!4x7=2S 
3x7=21(4x8=38 
3X8=244X9=36 
3X9=271 

5x5=25|6x6=36i7x7=49|8x8=ft 
5x6=306x7=427x8=568x9=73 
5x7=35  6X8=487X9=631 
5X8=406X9=541 
5X9=45| 

19X9=81 

*   As  the  multiplication  table  is  most  commonly  given  it  is  twice  the 
length  of  the  above :  this  arises  from  the  circumstance  of  the  product  of 


BHAP.  V.  MULTIPLICATION.  95 

The  same  results  may  be  exhibited  in  a  still  more 
convenient  form  as  follows :  — 


2 

3 

4 

£ 

6 

7 

8   9 

9 

18 

27 

36 

45 

54 

63 

72   81 

8 

16 

24 

32 

40 

48 

56 

64 

7 

14 

21 

28 

35 

42 

49 

6 

12 

18 

24 

30 

36 

5 

10 

15 

20 

25 

4 

8 

12 

16 



3 

6 

9 

2 

4 

To  find  the  product  of  two  numbers  by  this  table,  we 
must  look  for  the  greater  number  in  the  first  upright 
column  on  the  left,  and  for  the  lesser  in  the  highest 
cross  row.  The  product  of  the  two  numbers  will  be 
found  in  the  same  cross  row  with  the  greater  number, 
and  in  the  same  upright  column  with  the  less.  Thus, 
if  we  wish  to  find  the  product  of  8  and  5,  we  look 
along  the  cross  row  from  8  until  we  find  the  number 
which  is  directly  under  5 :  this  number,  40,  is  the  pro- 
duct of  8  and  5. 

(140.)  All  the  products  contained  in  this  table  may 
be  found  directly  by  addition,  and,  as  the  numbers  to  be 
added  cannot,  in  any  case,  exceed  9,  such  a  process  of 

the  same  pair  of  numbers  being  given  in  two  different  forms  ;  thus  it  is 
usual  to  insert  "  twice  3  make  6,"  and  also  "  3  times  2  make  6."  The  pupil, 
however,  having  once  understood  that  the  multiplicand  and  multiplier  are 
interchangeable,  this  repetition  becomes  quite  unnecessary,  and  the  greater 
length  of  the  table  has  a  tendency  only  to  confuse  the  pupil,  and  needlessly 
to  burthen  his  memory. 


96  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

addition  will  not  in  any  case  be  attended  with  labour  or 
difficulty.  Nevertheless,,  by  bearing  in  mind  some  of 
the  principles  respecting  multiplication  which  have 
been  already  established,  the  larger  products  contained  in 
the  multiplication  table  may  be  easily  inferred  from  the 
smaller  ones.  If  we  wish  to  prove,  for  example,  that 
the  product  of  9  and  8  is  72,  we  may  resolve  9  into  4 
and  5  (134.),  and  multiply  each  of  these  by  8.  We 
find,  by  the  other  parts  of  the  table,  that  the  product  of 
8  and  4  is  32,  and  that  the  product  of  8  and  5  is  40 : 
these  numbers  added  together  obviously  make  72,  which 
is  therefore  the  product  of  8  and  9- 

Again,  if  we  wish  to  prove  that  the  product  of  6  and 
7  is  42,  we  may  resolve  7  into  the  parts  2,  3,  and  2  : 
these  three  severally  multiplied  by  6  give  12,  18,  and 
12,  which  added  together  make  42.  In  the  same  man- 
ner all  the  higher  products  in  the  table  may  be  inferred 
from  the  lower  ones. 

Having  committed  to  memory  the  products  contained 
in  the  above  table,  the  student  will  be  prepared  to  prac- 
tise and  comprehend  the  process  of  multiplication  in 
those  cases  in  which  one  or  both  of  the  numbers  multi- 
plied consist  of  two  or  more  digits.  In  these  cases,  how- 
ever, the  more  complex  questions  require  a  previous 
knowledge  of  the  methods  of  solution  for  the  more 
simple  ones.  In  unfolding  the  principles  on  which  the 
general  rule  for  the  multiplication  of  any  numbers  de- 
pends, it  will  be  necessary  that  we  should  proceed  step 
by  step,  from  the  most  simple  questions  which  can  be 
proposed,  through  intermediate  degrees  of  complexity, 
in  order  to  arrive  at  a  general  method. 

(141.)  We  shall  first  consider  the  case  in  which  one 
of  the  factors  is  a  single  digit,  and  the  other  a  number 
consisting  of  several  places.  As  it  is  a  matter  of  in- 
difference which  of  the  two  factors  is  considered  as  the 
multiplier  (132.),  we  shall,  for  convenience,  consider  the 
latter  number  as  the  multiplicand,  and  the  former  as  the 
multiplier.  Let  the  multiplicand,  then,  be  23789,  and 
the  multiplier  6.  The  question,  if  solved  by  the 


CHAP.  V.  MULTIPLICATION.  97 

direct  method  of  addition,  would  be  worked  as  fol- 
lows :  — 

23789 

23789 

23789 

23789 

23789 

23789 

142734 

Having,  however,  previously  committed  to  memory  the 
multiplication  table,  this  process  of  addition  admits  of 
being  considerably  expedited.  Instead  of  adding  together 
the  six  nines  in  the  units  column,  we  recall  that  6  times 
9  make  54,  which  being  the  sum  of  the  units  column,  we 
write  down  4  in  the  units  place  of  the  sum,  and  carry 

5  to  the  tens  column  by  the  rule  for  addition  (93.). 
In  like  manner,  instead  of  adding  the  numbers  in  the 
tens  column,  we  learn  from  the  multiplication  table  that 

6  eights  are  48,  to  which  adding  the  5  carried,  we  get 
53 :  by  the  rule  for  addition,  we  put  3  in  the  tens 
place,  and  carry  5  to  the  hundreds.     We  proceed  in  the 
same  way,  rinding  the  sum  of  each  column,  not  by  ad- 
dition, but  by  the  multiplication  table,  adding,  however, 
to  the  result,  thus  found,  the  number  carried  from  the 
preceding  column.     In  this  process,  it  will  be  perceived 
that  we  depart  in  no  respect  from  the  practice  esta- 
blished in  addition,  except  merely  in  our  way  of  obtain- 
ing the  knowledge  of  the  number  which  would  be  found 
by  adding  the  figures  of  each  column.     It  will,  there- 
fore, be  evident  that  such  a  process  is  nothing  but 
addition  somewhat  abridged,  or  rather  expedited.     The 
figures  of  every  column  being  the  same,  it  is  not  ne- 
cessary that  the  numbers  should  be  written  one  under 
another,  provided  that  we  have  any  way  of  expressing 
the  number  of  times  which  each  figure  occurs  in  the 
upright  columns.     In  the  present  case,  the  units  column 
consists  of  6  nines,  the  tens  column  of  6  eights,  the 
hundreds  column  of  6  sevens,  and  so  on.     The  process, 


98  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

may,  therefore,  be  expressed  without  repeating  the  mul- 
tiplicand, as  follows  :  — 

23789 


142734 

In  this  case  the  multiplier,  6,  written  under  the  units 
place  of  the  multiplicand,  signifies  the  number  of  times 
the  multiplicand  is  understood  to  be  repeated  ;  and  it 
will  be  evident  that  the  process  of  multiplication  will 
consist  in  multiplying  each  digit  of  the  multiplicand, 
beginning  from  the  units  place  and  proceeding  from 
right  to  left,  the  product  of  the  multiplier  by  each  digit 
of  the  multiplicand  to  be  treated  in  the  same  manner  as 
the  number  found  by  adding  the  successive  columns  in 
addition. 

The  rationale  of  the  process  should  be  strongly  im- 
pressed upon  the  mind  of  the  young  pupil  by  every 
teacher,  and  this  may  be  always  effected  by  causing  the 
pupil,  as  he  works  each  question,  to  repeat  the  results 
of  his  work  in  the  following  manner  :  — 

"  Six  nines  are  54,  or  5  tens  and  4  units  :  I  write 
4  in  the  units  place,  and  I  reserve  5  to  be  added  to  the 
tens.  Six  eights  are  48,  to  which  the  5  tens  carried 
being  added,  make  53  tens,  or,  what  is  the  same,  5 
hundreds  and  3  tens  :  in  the  tens  place  I  write  3,  and 
reserve  the  5  hundreds  to  be  added  to  the  hundreds. 
Six  sevens  are  42,  which  being  hundreds  are  42  hun- 
dreds ;  adding  the  5  hundreds  carried,  I  obtain  47 
hundreds,  or,  what  is  the  same,  4  thousands  and  7 
hundreds  :  in  the  hundreds  place  I  write  1,  and  reserve 
the  4  thousands  to  be  added  to  the  thousands.  Six 
threes  are  18,  which  being  in  the  thousands  place, 
are  18  thousands;  adding  the  4  carried,  we  have  22 
thousands,  or  2  ten  thousands  and  2  thousands:  I 
write  2  in  the  thousands  place,  and  reserve  2  for  the 
ten  thousands.  Six  twos  are  12,  which  being  in  the 
ten  thousands  place,  are  12  ten  thousands  ;  adding  to 
this  the  2  carried,  I  have  14  ten  thousands,  or  1  hun- 


CHAP.  V.  MULTIPLICATION.  99 

dred  thousand,  and  4  ten  thousands :  I  write  4  in  the 
ten  thousands  place,  and  1  in  the  place  of  hundred 
thousands." 

When  the  pupil  has  heen  made  to  work  a  sufficient 
number  of  examples  in  this  manner,  he  will  distinctly 
perceive  and  rememher  the  reasons  for  the  several  steps 
of  the  process.  It  may  then  be  useful  to  cause  him  to 
commit  to  memory  the  following  rule,  which  is  nothing 
more  than  a  statement  in  general  terms  of  what  he 
will  have  already  applied  in  the  various  particular  ex- 
amples :  — 

Rule. 

(142.)  When  the  multiplier  is  a  single  digit,  the 
product  is  found  by  multiplying  the  several  digits  of 
the  multiplicand  by  the  multiplier,  proceeding  from 
right  to  left.  The  figures  of  the  product  are  to  be 
written  as  in  addition,  and  the  numbers  carried  from 
place  to  place,  determined  in  the  same  way. 

(143.)  It  is  evident  from  what  has  been  explained 
above,  that  if  the  multiplicand  end  in  one  or  more 
ciphers,  the  product  will  also  terminate  in  ciphers,  be- 
cause, if  the  multiplicand  be  obtained  by  the  direct 
method  of  addition,  the  columns  of  ciphers  when  added 
will  give  a  cipher  for  the  same  place  in  the  product.  If 
we  would,  therefore,  multiply  such  a  number  as  23000 
by  5,  the  operation  would  be  as  follows :  — 
23000 

5 

115000 


This  process  may  often  be  abridged  by  omitting  the 
ciphers  of  the  multiplicand  in  the  first  instance,  multi- 
plying the  other  figures  only,  the  ciphers  omitted  being 
subsequently  annexed  to  the  product.  Thus,  in  the 
example  just  givenj  we  should  multiply  23  by  5,  and 
should  obtain  the  product  115.  Having  done  this,  to 
get  the  final  product  we  have  only  to  annex  three  noughts. 
When  the  multiplicand  is  a  high  number,  terminating 
H  2 


100  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

in  a  great  number  of  ciphers,  the  process  is  materially 
abridged  by  this  means. 

(144.)  Let  us  next  consider  the  case  in  which  the 
multiplier  is  a  number  consisting  of  two  places,  the  lat- 
ter place  being  filled  by  a  cipher  :  thus,  let  it  be  required 
to  multiply  23789  by  60.  It  has  been  already  shown, 
that  we  can  multiply  a  number  by  6'0  by  first  multi- 
plying it  by  6,  and  then  multiplying  the  product  by  1 0. 
(138.)  But  a  number  is  multiplied  by  10  by  merely 
annexing  to  it  a  cipher:  hence,  to  multiply  by  60  we 
multiply  by  6,  and  annex  a  cipher  to  the  product :  the 
process  is  as  follows :  — 

23789 
60 


1427340 

Hence  it  appears  that  when  the  multiplier  is  a  number 
consisting  of  a  significant  digit  followed  by  a  cipher,  the 
operation  is  performed  with  as  much  ease  and  expe- 
dition as  if  the  multiplier  were  a  single  digit.  The 
same  will  be  found  to  be  true  when  the  multiplier  is  a 
single  digit  followed  by  two  or  more  ciphers.  Let  us 
take  the  same  example,  the  multiplier  being  600 :  the 
multiplication  may  be  performed,  in  this  case,  by  first 
multiplying  by  6,  and  then  multiplying  the  product  by 
100.  (138.)  But  since  a  number  is  multiplied  by  100 
by  annexing  two  ciphers  to  it  (65.),  the  multiplication 
by  600  is  performed  by  multiplying  by  6,  and  annexing 
two  ciphers  to  the  product.  The  operation,  in  the  pro- 
posed example,  is  as  follows  :  — 

23789 
600 


14273400 

If  the  multiplier  be  6000,  it  may  be  shown,  by  the 
same  reasoning,  that  the  product  will  be  found  by  mul- 
tiplying by  6,  and  annexing  three  noughts;  and  the  same 
for  multipliers  ending  in  a  greater  number  of  noughts. 


CHAP.  V.  MULTIPLICATION.  101 

(145.)  These  principles  being  clearly  fixed  in  the 
mind  of  the  pupil,  he  will  be  prepared  to  understand 
the  details  of  the  more  complex  questions  in  multipli- 
cation. Let  us  take  the  case  where  the  multiplier  con- 
sists of  two  significant  digits:  retaining  the  same 
multiplicand,  let  the  multiplier  be  67.  We  have  seen 
(136.)  that  multiplication  may  be  performed  by  resolv- 
ing the  multiplier  into  two  parts,  multiplying  the  multi- 
plicand by  each  of  these  parts  separately,  and  adding 
together  the  products  thus  found.  In  the  present  case, 
let  us  conceive  the  number  67  resolved  into  two  parts, 
60  and  7 :  we  shall  multiply  the  multiplicand  by  each 
of  these  parts  separately. 

Multiply     -     -     -     23789 

by       -     -     -     - 7 

Product     -     -     -    166523 

Multiply     -     -     -~~23789 
by       ...     - 60 

Product     -     -        1427340 

Add     -     -     -     -    166523 

to      -     -     -      1427340 

Total  product     -    1593863 

Such  is  the  product  of  the  two  numbers  required ;  but 
the  process  may  be  written  in  a  much  more  abridged 
form.  Instead  of  actually  resolving  the  multiplier  into 
two  parts,  we  shall  imagine  it  to  be  so  resolved,  and  shall 
perform  the  multiplication  as  if  it  were  so  :  the  process 
which  has  just  been  written  at  length  may  be  expressed 
more  shortly  in  the  following  manner:  — 

23789 
67 


166523 
1427340 

1593863 
H  3 


102 


A    TREATISE    ON    ARITHMETIC. 


BOOK  I. 


(146.)  Let  us  next  suppose  the  multiplier  to  consist 
of  three  digits,  such  as  673.  In  this  case  we  shall  con- 
ceive it  resolved  into  three  parts,  600,  70,  and  3  :  by 
each  of  these  it  i§  necessary  to  multiply  the  multiplicand, 
and  when  the  products  are  obtained,  they  must  be  added 
together  to  get  the  total  product.  The  process,  stated 
at  full  length,  would  be  as  follows :  — 

Multiply 

by        -    - 
Product     -     - 


Multiply     - 

by         - 
Product 

Multiply     - 

by      - 
Product 


-     23789 

__- 70 

1665230 

23789 
-       600 

14273400 


Add     -       - 

Total  product    -     16009997 


71367 
1665230 
14273400 


As  in  the  former  case,  this  process  may  be  abridged 
as  follows  :  — 


A 

B  - 
C 
P 


23789 
673 

-  71367 

-  1665230 
14273400 

-  16009997 


In  this  case,  the  first  line,  A,  is  the  partial  product  found 
\)y  multiplying  the  multiplicand  by  3 ;  the  line  B  is 


CHAP.  V.  MULTIPLICATION.  103 

found  by  multiplying  the  multiplicand  by  70  ;  and  th£ 
line  C  by  multiplying  it  by  600.  The  total  product,  P, 
is  found  by  adding  together  these  partial  products.  To 
find  the  line  B,  it  is  only  necessary  to  place  a  nought 
under  the  units  place  of  the  first  product  A,  and  then 
multiply  the  multiplicand  by  7,  placing  the  successive 
figures  found  under  the  succeeding  places  of  the  pro- 
duct A,  proceeding  from  right  to  left.  It  is  evident 
that  this  is  equivalent  to  multiplying  the  multiplicand 
by  7  and  annexing  a  nought  to  the  product,,  or,  in  other 
words,  to  multiplying  the  multiplicand  by  70.  In  the 
same  manner,  by  writing  noughts  in  the  units  and  tens 
place  of  the  third  product,  and  then  multiplying  the 
multiplicand  by  6,  we  in  fact  multiply  by  6,  and  annex 
two  noughts  to  the  product,  which  is  equivalent  to  mul- 
tiplying by  600. 

But,  with  the  slightest  attention,  it  must  be  apparent 
that  the  introduction  of  the  noughts  into  the  products  B 
and  C,  is  of  no  other  use  than  to  throw  the  figures  of 
these  partial  products  so  many  places  to  the  left.  The 
nought  which  terminates  the  product  B  has  no  other 
effect  than  to  remove  the  figure  3  from  the  units  co- 
lumn to  the  tens  column,  and  every  other  figure  in  the 
same  product  one  place  farther  to  the  left.  In  the  same 
manner,  the  addition  of  the  two  noughts  to  the  product 
C  has  the  effect  only  of  removing  each  of  the  figures 
two  places  to  the  left,  by  which  the  units  figure  is  trans- 
ferred to  the  hundreds  column,  the  tens  to  the  thousands, 
and  so  on.  That  the  ciphers  which  occur  in  the  oper- 
ation, as  written  above,  have  no  other  effect  than  those 
just  mentioned,  will  be  quite  apparent  if  it  be  considered 
that  no  effect  will  be  produced  upon  the  total  product 
P,  by  expunging  the  ciphers  annexed  to  the  partial  pro- 
ducts B  and  C,  provided  that  the  other  figures  of  B  and 
C  are  allowed  to  retain  the  places  assigned  to  them  in  the 
above  method  of  writing  the  process.  If  the  ciphers  were 
omitted,  subject  to  this  condition,  the  operation  would 
be  written  in  the  following  manner :  — 

B  4 


104  A    TREATISE    ON    ARITHMETIC.  JtOOK.  I. 

23789 
673 


71367 
166523 
142734 

16009997 

(J47-)  Let  us  take  another  example.  Suppose  it  re- 
quired to  multiply  65379  by  47853.  Placing  the 
multiplier  under  the  multiplicand,  as  before,  we  shall 
consider  the  former  as  consisting  of  five  distinct  parts, 
expressed  by  the  local  values  of  its  several  digits  ;  these 
parts  will  be  the  following :  — 

3 
50 

800 

7000 

40000 

and  by  these,  successively,  we  must  multiply  the  mul- 
tiplicand. When  the  five  partial  products  are  obtained, 
the  total  product  will  be  found  by  adding  them  together  : 
the  process,  expressed  at  length,  is  as  follows: — 

65379 
47853 

A    -  -      196137 

B  -  -   3268950 

C  52303200 

D   -  457653000 

E  -    -  -  2615160000 

P          312^8581287 

In  the  line  A  we  have  the  product  of  the  multiplicand 
multiplied  by  3  :  the  line  B  is  found  by  multiplying  the 
multiplicand  by  5,  and  annexing  a  nought  to  the  pro- 
duct, which  is  in  effect  multiplying  it  by  50  ;  and,  con- 
sequently, the  number  B  is  the  second  partial  product. 
In  the  same  manner  the  number  C  is  found  by  multi- 


CHAP.  V.  MULTIPLICATION.  105 

plying  the  multiplicand  by  8,  and  annexing  two  ciphers 
to  the  product,  which  is  equivalent  to  multiplying  it  by 
800  :  the  number  C  is,  therefore,  the  third  partial  pro- 
duct. In  the  same  way  it  may  be  shown  that  the 
numbers  D  and  E  are  the  two  remaining  partial  pro- 
ducts :  these  five  numbers,  being  added  together,  give 
the  total  product,  P. 

As  in  the  former  example,  it  must  be  apparent 
that,  provided  the  places  of  the  significant  digits  in  the 
products  B,  C,  D,  and  E  are  preserved,  the  presence 
of  the  ciphers  produces  no  effect  in  the  addition  by 
which  the  total  product  P  is  obtained.  It  is,  there- 
fore, unnecessary  to  write  these  ciphers  in  the  process ; 
but,  in  omitting  them,  care  must  be  taken  to  maintain 
the  other  figures  in  those  places  which  they  would  have  if 
the  ciphers  were  inserted. 

As  the  ciphers,  then,  produce  no  effect  on  the  total 
product,  it  is  usual  to  omit  them,  and  to  write  down 
the  process  thus  :  — 

65379 
47853 

A  196137 

B  326895 

C  523032 

D  .  .  .  457653 
E  .  ..261516 

P  .  .  .  3128581287 

(148.)  The  principle  already  explained  in  the  case 
where  the  multiplier  is  a  single  digit,  followed  by  one 
or  more  ciphers,  will  be  equally  applicable  where  the 
multiplier  is  a  number  consisting  of  several  places  ter- 
minating in  one  or  more  ciphers.  In  that  case,  the  final 
ciphers  of  the  multiplier  may  be  omitted  in  the  first 
instance :  and  after  the  multiplication  has  been  thus 
performed,  the  same  number  of  ciphers  should  be  an- 
nexed to  the  product.  The  proof  of  this  is  precisely 
the  same  as  the  proof  given  in  the  case  where  the  mul- 
tiplier was  a  single  digit  followed  by  ciphers.  Let  us 


106  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

suppose  that  the  multiplier  is  47000 ;  in  that  case,  the 
multiplier  being  the  product  of  47  and  1000,  we  shall 
obtain  the  true  product  by  first  multiplying  by  47,  and 
then  multiplying  the  result  by  1000.  (138.)  But  the 
latter  operation  is  performed  by  merely  annexing  three 
noughts  to  the  product  first  obtained.  (65.)  To  mul- 
tiply 65379  by  47000,  the  process  would  then  be  as 
follows :  — 

65379 
47 

457653 
261516 

3072813 


3072813000 

(149.)  From  what  has  been  just  proved,  combined 
with  what  was  formerly  proved  in  (65.)  it  will  follow, 
that,  when  the  multiplicand  and  multiplier  both  termi- 
nate in  one  or  more  ciphers,  the  multiplication  may  be 
performed  by  omitting  the  ciphers  altogether  in  the  first 
instance,  and  annexing  to  the  number  which  results 
from  the  operation  as  many  ciphers  as  were  omitted  in 
the  multiplicand  and  multiplier  taken  together.  Thus, 
if  the  multiplicand  terminated  in  three  ciphers,  and  the 
multiplier  in  two,  we  should  annex  five  to  the  product. 
For  example,  let  it  be  required  to  multiply  65000  by 
3300  :  the  process  would  be  as  follows  :  — 

65 
33 

195 


2145 


214500000 


( 1 50.)  If  one  or  more  figures  of  the  multiplier  happen 
to  be  ciphers,  these  figures  not  being  final,  the  process 


CHAP.  V.  MULTIPLICATION.  107 

is  somewhat  modified ;  but  the  principles  on  which  it 
rests  are  the  same.  In  the  example  already  given,  let 
us  suppose  that  the  8  is  removed  from  the  hundreds 
place,  and  replaced  by  a  nought :  it  is  plain  that  in  that 
case  the  third  partial  product,  C,  which  was  obtained  by 
multiplying  the  multiplicand  by  800,  can  have  no  place 
in  the  process  while  the  other  4  partial  products  re- 
main unchanged.  The  operation  will,  therefore,  stand 
as  before,  with  the  exception  that  the  product  C  will 
be  omitted,  and  the  process  will  be  expressed  as  fol- 
lows :  — 

65379 
47053 

A 196137 

B  326895 

D .  .  .  457653 
E..  261516 

3076278087 

It  will  be  observed  here,  that  the  units  figure  of  the 
third  partial  product,  D,  is  not  placed  under  the  tens  of 
the  product  B,  as  it  would  be  in  ordinary  cases,  but  is 
placed  under  the  hundreds  figure  of  the  product  B, 
and  under  the  thousands  figure  of  the  product  A. 
The  reason  of  this  will  be  easily  understood :  the  par- 
tial product  D  is  obtained  by  multiplying  the  multi- 
plicand by  7,  and  annexing  three  noughts  to  the  result ; 
these  three  noughts,  if  inserted,  would  stand  under  the 
last  three  figures  of  the  product  A,  and  consequently  the 
figure  3  would  stand  in  the  thousands  column  of  the 
numbers  to  be  added.  When  a  nought  occurs,  therefore,  in 
the  multiplier,  it  should  be  omitted  in  the  multiplication; 
but  the  units'  figure  of  the  following  partial  product 
must  be  placed  under  the  hundreds  figure  of  the  preceding 
one  instead  of  the  tens. 

(151.)  Let  us  take  the  case  where  two  or  more  noughts 
occur  in  succession  in  the  multiplier,  not  occupying  the 
final  places.  Let  the  multiplier,  for  example,  be  40003  ; 
the  process  will  be  as  follows  :  — 


108  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

65379 
40003 


A  196137 

E...  261516 

2615356137 

In  this  case,  the  partial  products,  B,  C,  and  D,  disappear, 
and  the  figures  of  the  product  E  maintain  the  same 
places,  with  respect  to  those  of  A,  as  they  had  when  the 
intermediate  products  existed.  The  reason  of  this  will 
be  quite  apparent,  when  it  is  considered  that  the  product 
E  is  understood  to  he  followed  by  four  ciphers,  which 
are  merely  omitted  for  the  sake  of  brevity.  These  four 
ciphers  would  stand  under  the  last  four  places  of  the 
partial  product  A. 

(152.)  From  all  that  has  been  now  explained  we, 
may  derive  the  following  general  rule  for  multiplication. 

GENERAL  RULE. 

I.  Place  the  multiplier  under  the  multiplicand,  as  in 
Addition. 

(It  will  be  convenient  always  to  consider  the  smaller 
of  the  two  numbers  to  be  multiplied  together  as  the 
multiplier,  and  the  greater  as  the  multiplicand.) 

II.  Multiply  the  multiplicand  separately  by  every  sig- 
nificant digit  which  is  found  in  the  multiplier,  by  which 
you  will  obtain  as  many  partial  products  as  there  are 
significant  digits  in  the  multiplier. 

III.  Write  these  products  one  under  the  other,  so  that 
the  last  figure  of  each  shall  be  under  that  figure  of  the 
multiplier  by  which  it  was  produced. 

IV.  Add  the  partial  products  thus  placed,  and  their 
sum  will  be  the  total  product. 

(153.)  Although  the  above  rule  will  serve  for  the 
solution  of  every  possible  question  which  can  occur  in 
multiplication,  yet,  in  particular  cases,  other  methods 
may  be  applied,  by  which  the  process  may  occasionally 
be  abridged,  and  which,  as  they  are  illustrations  of  pro- 


CHAP.  V.  MULTIPLICATION.  109 

perties  of  number  which  are  otherwise  useful,  it  may 
not  be  improper  to  notice  here. 

(154.)  It  sometimes  happens  that  the  multiplier  is 
obviously  the  product  of  two  or  more  smaller  numbers, 
in  which  case  the  operation  may  be  performed  by  suc- 
cessive mutiplications  without  addition.  Thus,  if  the 
multiplier  be  72,  we  may  obtain  the  product  by  first 
multiplying  by  9  and  then  by  8  (138.);  or,  since  9 
is  the  product  of  3  and  3,  we  may  obtain  the  product 
by  multiplying  successively  by  3,  3,  and  8.  Again, 
since  the  product  of  8  is  2  and  4,  we  may  obtain  the 
product  by  multiplying  successively  by  3,  3,  2,  and  4. 
Let  the  multiplicand,  for  example,  be  86 :  if  we  mul- 
tiply by  72,  by  the  general  rule,  the  process  is  as 
follows :  — 

Multiply     .     .     86 
by      ...     72 


Product 
If  we  multiply  by  9  and  8 

Multiply 
by      . 

Multiply 
by     . 
Product 

172 
602 

.     .    6192 

successively,  we  shall  have 
.     .     86 
.     •      9 

.     .  774 
.     .       8 

.     .    6192 

If  we  multiply  by  3,  3,  and  8,  we  have — 

Multiply  .  .     86 

by      .  .  . £ 

Multiply  .  .  258 

by      .  .  . 3 

Multiply  .  .  774 

by      .  .  .       8 

Product  .     .  6192 


110  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

Finally,  if  we  multiply  by  3,  3,  2,  and  4,  we  have — 

Multiply     .     .     86 
by      .     .     . 3 

Multiply     .     .  258 

by  .  .  . 3 

Multiply  .  .  774 

by  .  .  . 2_ 

Multiply  .  .  1548 

by    ...  _     4 

Product     .     .  6192 

(155.)  We  bave  seen  that  the  multiplier  may  always 
be  resolved  into  parts,  and  the  total  product  obtained 
by  adding  the  partial  products.  Analogous  to  this  is 
another  method,  which  sometimes  furnishes  the  means 
of  considerable  brevity  in  the  process.  We  may  first 
take  a  greater  multiplier  than  that  proposed ;  and, 
having  obtained  the  product,  we  may  subtract  from  it 
the  partial  product  obtained  by  multiplying  the  multi- 
plicand by  that  number  by  which  the  assumed  multi- 
plier exceeds  the  proposed  one.  This  will  be  easily 
understood  when  applied  to  an  example.  Let  us  sup- 
pose that  a  number  is  required  to  be  multiplied  by  8 : 
if  we  first  multiply  it  by  10  and  then  by  2,  and  sub- 
tract the  latter  product  from  the  former,  it  will  be 
evident  that  the  remainder  will  be  the  product  which 
would  be  obtained  by  multiplying  it  by  8.  This  amounts 
to  no  more  than  stating,  that  if  from  ten  times  any 
thing  we  subtract  twice  that  thing,  eight  times  the  same 
thing  will  remain,  which  is  self-evident. 

The  application  of  this  principle  frequently  presents 
great  facility  and  brevity  in  the  process  of  multipli- 
cation. For  example,  suppose  it  is  required  to  multi- 
ply 387  by  299 :  we  shall  first  multiply  387  by  300, 
and  then  subtract  from  the  product  387.  By  the  first 
process  we  have  taken  the  multiplicand  300  times,  which 
is  once  too  much ;  if  we  subtract  from  it  the  multi- 
plicand, we  shall  therefore  get  the  true  product :  the 
process  would  be  as  follows  :  — 


CHAP.  V.  MULTIPLICATION.  Ill 

Multiply     .     .     387 
by     ...     300 

116100 

Subtract    .     .       387 
Product     .     115713 

Again  :  let  it  be  required  to  multiply  49687  by  99999 : 
by  the  ordinary  method  this  would  require  five  multi- 
plications,, and  the  addition  of  five  lines  of  figures.  If, 
however,  we  multiply  the  multiplicand  first  by  100000, 
which  is  done  by  annexing  five  ciphers  to  it,  and  subtract 
from  the  number  thus  found  the  multiplicand,  the  re- 
mainder will  be  the  product  sought :  the  process  would 
be  as  follows :  — 

4968700000 

Subtract  j*9687 

Product      .     4968650313 

(156.)  We  have  hitherto  confined  our  attention 
chiefly  to  the  products  formed  by  the  multiplication  of 
two  factors  only.  Products  may,  however,  be  formed  by 
the  continued  multiplication  of  three  or  more  factors.  The 
operation  which  is  expressed  in  the  following  manner : — 
2  X  3  x  4  is  the  continued  multiplication  of  the  fac- 
tors 2,  3,  and  4;  and  means  that  2  is  to  be  first  multi- 
plied by  3,  and  the  product  thus  obtained  to  be  then 
multiplied  by  4.  The  result  of  such  a  process  would 
be  24,  which  is,  therefore,  the  continued  product  of  2, 
3,  and  4 ;  which  fact  is  expressed  thus  : — 

2  x  3  x  4  =  24. 
In  like  manner, 

2  X  3  x  4  x  5  =  120, 

means  that  the  continued  product  of  2,  3,  4,  and  5, 
that  is,  the  product  of  2  and  3  multiplied  by  4,  and  the 
result  multiplied  by  5,  produces  120. 

(157.)  The  name  factor  is  extended  to  the  numbers 
by  the  continued  multiplication  of  which  any  other 
number  is  formed.  Thus,  2,  3,  and  4,  are  factors  of 
24.  The  prime  factors  of  any  number  are  those 


112  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

prime  numbers,  by  the  continued  multiplication  of 
which  the  number  in  question  is  formed :  since  24  is 
formed  by  the  continued  multiplication  of  2,  3,  and  4, 
and  since  4  itself  is  formed  by  the  multiplication  of 
2  and  2, 

24=2x2x2x3. 

Thus  the  prime  factors  of  24  are  2  and  3 ;  but  2  is  3 
times  repeated  in  the  continued  multiplication. 

(158.)  When  a  product  is  formed  by  the  continued 
multiplication  of  the  same  factors,  it  is  called  a  power :  thus, 

2x2  =  4 
2x2x2=8 
2x2x2x2  =  16,  &c, 

The  numbers  4,  8,  16,  £c.  are  powers  of  2  :  4  is  called 
the  square  of  2,  or  second  power ;  8  is  called  the  cube 
of  2,  or  third  power  ;  16  is  called  the  fourth  power  of  2 ; 
and  all  products  in  which  2  is  repeated  by  continued 
multiplication,  are  in  like  manner  called  powers  of  2, 
the  numerical  order  of  the  power  being  determined  by 
the  number  of  times  which  2  occurs  as  a  factor  in  the 
continued  product. 

The  powers  of  other  numbers  are  determined  in  the 
same  manner :  thus, 

3x3  =  9 
3x3x3  =  27 
3x3x3x3  =  81,&c. 

9  is,  therefore,  the  square  of  3,  27  its  cube,  81  its 
fourth  power,  &c.  £c. 

The  first  power  of  any  number  is,  therefore,  the 
number  itself. 

(159.)  There  are  various  ways  by  which  the  teacher 
may  verify  or  prove  the  work  of  his  pupil  in  multipli- 
cation ;  but  the  best  are  those  by  which  the  pupil  is 
made  unconsciously  to  verify  his  own  work,  while  he 
at  the  same  time  is  further  exercised  in  the  practice  of 
the  rule.  This  end  may  be  attained  in  various  ways, 
which  will  suggest  them  selves  to  the  mind  of  every  teacher. 
The  same  or  different  pupils  may  be  made  to  solve  the 
same  question  by  different  methods,  and  the  coincidenre 


CHAP.  V.  MULTIPLICATION.  113 

of  the  results  will,  in  general,  prove  their  correctness. 
Thus,  the  number  which  is  given  to  one  pupil  as  the 
multiplicand,  may  be  given  to  another  as  the  multiplier, 
and  vice  versa.  A  method  of  verification  may  also  be 
derived  from  the  fact,  that  if  one  factor  be  doubled  and 
the  other  halved,  the  product  will  remain  unchanged, 
being  as  much  increased  by  the  one  operation  as  it  is 
diminished  by  the  other.  This  method  will  always  be 
applicable  when  one  of  the  factors  is  an  even  number, — 
a  circumstance  for  which  it  is  always  in  the  power  of 
the  teacher  to  provide.  Thus,  let  the  following  question 
be  proposed  to  one  pupil :  — 

Multiply          -     34765 
By  -        7564 

At  the  same  time  propose  to  another  pupil  the  follow- 
ing question,  in  which  the  multiplicand  is  double  the 
former  multiplicand,  and  the  multiplier  half  the  former 
multiplier :  — 

Multiply          -     69530 

By  -       3782 

The  products  must  needs  be  the  same.  If  they  are  found 
to  differ,  therefore,  one  or  the  other  must  be  wrong,  and 
the  pupils  should  be  made  to  revise  their  work. 

(160.)  Two  questions  may  be  proposed  successively 
to  the  same  pupil,  or  at  the  same  time  to  different  pu- 
pils, in  which,  with  the  same  multiplicand,  one  multiplier 
may  be  double  the  other,  or,  with  the  same  multiplier, 
one  multiplicand  will  be  double  the  other.  In  such  case, 
one  product  must  be  double  the  other;  and  if  not,  the 
work  must  be  wrong,  and  should  be  revised. 

In  the  next  chapter  we  shall  have  occasion  to  point 
out  various  methods  by  which  the  processes  of  multi- 
plication and  division  may  be  used  to  verify  each  other. 

(l6l.)  The  most  easy  and  expeditious  method  of 
verifying  complex  questions  in   multiplication,  is  that 
which  is  commonly  called  the  method  of  casting  out  the 
nines.     It  is  performed  as  follows :  — 
i 


114  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

Add  the  figures  which  occur  in  the  multiplicand,  and 
which  are  less  than  9-  In  the  progress  of  the  addition, 
when  the  sum  surpasses  9>  omit  the  9,  and  only  carry 
on  the  remainder.  You  will  then  have,,  finally,  a  re- 
mainder less  than  9-  D°  the  same  with  the  multiplier, 
and  then  multiply  the  two  remainders  together ;  divide 
this  product  by  9,  and  find  the  remainder.  In  like 
manner,  add  the  figures  of  the  product,  casting  out  the 
nines  —  a  remainder  will  be  obtained  less  than  9-  If 
this  be  the  same  as  the  remainder  found  by  dividing 
the  product  of  the  remainders  in  the  multiplicand  and 
multiplier,,  then  it  may  be  considered,  generally,  that 
the  work  is  correct.  To  make  the  above  explanation 
intelligible,  let  us  suppose  that  the  multiplicand  and 
multiplier  are  23707  and  4567,  the  product  being 
108269869.  Adding  the  figures  of  the  multiplicand, 
we  have  7  and  7  make  14 ;  rejecting  9>  the  remainder 
is  5.  This  added  to  3  and  2  gives  10;  rejecting  the 
9,  the  remainder  is  1.  Proceeding  in  the  same  way 
with  the  multiplier,  we  have  7  and  6  make  13;  re- 
jecting the  9>  we  have  a  remainder  4,  which,  added  to 
5,  makes  9 ;  rejecting  this,  we  have  the  last  figure,  4, 
remaining.  The  remainders,  therefore,  in  the  multipli- 
cand and  multiplier,  are  1  and  4,  which  multiplied 
together  give  4 :  this  being  less  than  9,  has  no  nines 
to  be  rejected.  Proceed  in  the  same  manner  with  the 
product.  The  first  figure,  9^  is  neglected:  6  and  8 
are  14,  which  is  5  above  9.  The  next  figure,  9, 
is  neglected,  and  5  is  added  to  6,  which  gives  1 1 . 
Carrying  the  2  above  9  to  the  next  figure,  we  obtain  4, 
which  added  to  8  gives  12.  Carrying  the  3  above  9 
to  the  1,  we  obtain  4,  which  is  the  same  remainder  as 
that  obtained  from  the  multiplier  and  multiplicand.* 

*  The  truth  of  this  method  may  be  established  in  the  following  man- 
ner :  — 

1.  It'  a  number  be  divided  by  9,  the  same  remainder  wilt  be  obtained  as  if 
the  sum  of  its  digits  were  divided  by  *j. 

Suppose  the  number  is  2376.  It  may  be  resolved  into  the  following 
parts :  — 

2  X  (999  +  1) 

3  X  (9<-»  +  1) 
7  X  (9  +  1) 


CHAP.  T.  MULTIPLICATION,  1  1  3 

If  the  remainder  which  is  obtained  after  casting  out 
the  nines  from  the  product,  he  not  the  same  as  the  re- 
mainder obtained  as  above  described  from  the  factors, 
then  it  is  certain  that  the  work  must  be  incorrect ;  but, 
on  the  other  hand,  if  the  remainder  be  the  same  in  both 
cases,  still  it  is  not  absolutely  certain,  though  in  the 
highest  degree  probable,  that  the  work  is  correct.  It 
may  happen  by  possibility  that  two  errors  may  occur  in 
the  product  which  will  compensate  each  other,  so  far  as 
they  produce  any  effect  on  the  remainder  after  casting 
out  the  nines.  Thus,  if  one  figure  of  the  product  be 
less  than  it  ought  to  be  by  1,  while  another  figure  is 
greater  than  it  ought  to  be  by  1,  then  the  sum  of  the 
digits  will  remain  the  same  as  if  the  product  were 
correct ;  and  therefore  the  remainder,  after  casting  out 
the  nines,  will  not  be  affected.  It  is  true  that  such  a 
coincidence  of  errors  as  would  produce  this  compens- 
ation is  highly  improbable,  and  therefore  the  method 
may  be  used  in  teaching  as  a  sufficiently  certain  means 
of  verification ;  but  when  calculations  are  made  for 

or,  what  is  the  same, 

2x999+3x99+7x9+2+3+7+6. 

Now  it  is  evident,  t hit  2X999,  3x  99,  and  7x9,  are  severally  multiples  of  9'; 
and,  consequently,  when  divided  by  9  would  leave  no  remainder.  When 
the  entire  number  2376  is  divided  by  9,  the  remainder  must  therefore  be  the 
same  as  it  would  be  if  2+3+7+6  were  divided  by  9. 

2.  If  the  two  factors  be  considered  as  consisting  of  multiples  of  9  and  re- 
mainders,  the  product  will  consist  of  a  multiple  of  9,  and  the  product  of  the 
same  remainders. 

Let  the  factors  be  357  and  254.  The  greatest  multiple  of  9  contained  in 
the  former  is  351,  and  in  the  latter  252.  The  numbers  to  be  multiplied  are 
then  351+6  and  i'52+2.  We  must  multiply  the  former,  first  by  252,  which 
gives  the  products  351x252+6x252,  and  next  by  2,  which  gives  the  pro- 
ducts 351 X  2+6x2.  The  total  product  is  then  as  follows  :  — 

351X252+6X252+351X2+6X2. 

Now,  since  351  and  252  are  each  of  them  multiples  of  9,  it  follows  that  351  x 
£52, 6  X  252,  and  35 1  x  2,  are  severally  multiples  of  9.  The  sum  of  these,  there- 
fore, is  a  multiple  of  9;  and  therefore  the  whole  product  consists  of  a 
multiple  of  9,  and  the  product  of  the  remainders  (6  and  2)  found  by  dividing 
the  factors  by  9. 

3.  If  the  factors  be  divided  by  9  respectively,  and  the  remainders  be  mul- 
tiplied together  and  divided  by  9,  the  same  remainder  will  be  obtained  as 
if  the  product  of  the  factors  were  divided  by  9. 

It  has  been  just  proved  that  the  product  of  the  factors  consists  of  a 
multiple  of  9  and  the  product  of  the  remainders.  The  former  divided  by 
9  has  no  remainder  ;  therefore,  if  the  whole  product  be  divided  by  9,  the 
same  remainder  will  be  found  as  if  the  product  of  the  two  remainders  were 
divided  by  9. 

4.  But  the  same  remainders  will  be  found  if  the  sums  of  the  digits  of  the 
factors  and  the  product  be  divided  by  9,  as  if  these  numbers  themselve* 
were  divided  by  9. 

12 


116 


A    TREATISE    ON    ARITHMETIC. 


BOOK  I. 


practical  purposes,  no  method  of  verification  should  be 
relied  upon  except  the  coincidence  of  the  results  of 
different  computers,  between  whom  no  communication 
is  permitted  during  the  calculation. 

(162.)  Although  the  table  of  products  of  the  single 
digits,  combined  by  multiplication  in  pairs,  is  sufficient 
for  the  solution  of  all  questions  in  complex  multipli- 
cation, yet  the  student  will  find  it  a  great  convenience 
to  commit  to  memory  the  products  of  the  numbers  above 
10  to  a  certain  extent.  It  is  usual  for  students  in 
arithmetic  to  commit  to  memory  the  products  of  num- 
bers combined  in  pairs  as  far  as  1 2  inclusive ;  but  those 
who  practise  arithmetic  will  find  it  convenient  to  re- 
gister in  their  memory  the  products  of  numbers  thus 
combined  to  a  greater  extent.  We  here  subjoin  a  mul- 
tiplication table  which  goes  as  far  as  20  x  20  :  we  also 
subjoin  another  table  of  the  squares  and  cubes  of  num- 
bers as  far  as  100. 


)  —  MULTIPLICATION  TABLE. 


2 
2040 

3 

4J  5 

6 

7  8 

9 

10  j  11  j  12 

13  14  15 

16  i  17 

18 

19i  20 

GO 

80 

ioo 

130 

140 

160 

180 

200  j  220  240 

S60 

280300 

320 

340 

360380 

400 

1938 
1836 

57 

76 

95 

114 

133  152 

171 

190209 

228 

247 

266(285 

304 

323 

342 

361 

54 

?2 

90 
~85 

106 

126 

144 

162 

180  198 

216 

234 

252  270 

288 

306 

324 

1734 

51 

48 

GS 
64 

102 

119 

13G 

153 

170!187 

204 

221 

238;  255 

272 

289 

1632 

SO 

96 

112 

128 

144 

160  176  192 

208  224  240 

256 

1530 
1428 

45 

42 

60  75 

56!  70 

90 
"84 

105  j  120 
~98!Tl2 

135 
126 

150  165  180 
140  154  168 

195 

fb:2 

210 
196 

225 

1326 

39 

52  65 

78!  91 

104 

117 

130  ;  143  156 

169 

12;24 
1122 

S6 

33 

48 

44 

_GO 
55 

721  84 
~e"6i~77 

96 
~88 

108 
99 

120  132,144 
110121 

1020 

SO 

40 

50 

60  70 

80 

90 

100 

918 

27 

36 

w 

54  63 

72 

81 

8  16 

24 

'52 

40 

48 

56 

64 

7  14 

21 

28 

S5 

42 

49 

_612 

5  10 

1824  SO 
15  20  |  25 

36 

4  8 

12 

16  ! 

3  "6 

9 

2:» 

CHAP.  V. 


MULTIPLICATION. 


(164.)  —  TABLE  OF  SQUARES  AND  CUBES. 


No. 

Square. 

Cubes. 

No. 

Square. 

Cubes. 

No. 

Square. 

Cubes. 

2 

4 

8 

35 

1225 

42875 

68 

4624 

314432 

3 

9 

27 

36 

1*96 

46656 

69 

4761 

328509 

4 

16 

64 

37 

1869 

50653 

70 

4900 

343000 

5 

25 

125 

38 

1444 

54872 

71 

5041 

35791  1 

6 

36 

216 

39 

1521 

59319 

72 

5184 

373248 

7 

49 

343 

40 

1600 

64000 

73 

5329 

38i»01  7 

8 

64 

512 

41 

1681 

68921 

74 

5476 

405224 

9 

81 

729 

42 

1764 

74088 

75 

5625 

421875 

10 

100 

1000 

43 

1849 

79507 

76 

5776 

438976 

11 

121 

1331 

44 

1936 

85184 

77 

5929 

456533 

12 

144 

1728 

45 

2025 

91125 

78 

6084 

474552 

13 

169 

2197 

46 

2116 

97336 

79 

6241 

493039 

14 

196 

2744 

47 

2209 

103823 

80 

6400 

512000 

15 

225 

3375 

48 

2304 

110592 

81 

6561 

531441 

16 

256 

4096 

49 

2401 

117649 

82 

6724 

551368 

17 

289 

4913 

50 

2500 

125COO 

83 

6889 

571787 

18 

324 

5832 

51 

2601 

132651 

84 

7056 

592704 

19 

361 

6859 

52 

2704 

140608 

85 

7225 

614125 

20 

400 

8000 

53 

2809 

148877 

86 

7396 

636056 

21 

441 

9261 

54 

2916 

157464 

87 

7569 

658503 

22 

484 

10648 

55 

3025 

166375 

88 

7744 

681472 

23 

529 

12167 

56 

3136 

175616 

89 

7921 

704969 

24 

576 

13824 

57 

3249 

185193 

90 

81«0 

729000 

25 

625 

15625 

58 

3364 

195112 

91 

8281 

753571 

26 

676 

17576 

59 

3481 

205379 

92 

8464 

778688 

27 

729 

19686 

60 

3600 

216000 

93 

8649 

804357 

28 

784 

21H52 

61 

3721 

226981 

94 

8836 

850584 

29 

841 

243891 

62 

3844 

238328 

95 

9025 

857375 

30 

900 

27000 

63 

3969 

250047 

96 

9216 

884736 

31 

961 

29791 

64 

4096 

262144 

97 

9409 

9  i  2673 

32 

1024 

327R8 

65 

4225 

274625 

98 

9604 

941192 

33 

1089 

35937 

ei6 

4356 

287496 

99 

9801 

970299 

34 

1156 

59304 

67 

4489 

300763 

100 

10000 

1000000 

i  3 


118  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 


CHAP.    VI. 


(165.)  IN  division  there  are  two  numbers  given  ;  one  of 
which  is  called  the  dividend)  or  number  to  be  divided, 
and  the  other  the  divisor,  or  number  by  which  the  divi- 
dend is  to  be  divided.  The  result  of  the  operation,  or 
number  sought,  is  called  the  quotient,  or  quote.  This 
arithmetical  operation  may  be  considered  under  several 
different  points  of  view. 

(166.)  The  dividend  being  regarded  as  the  product 
of  two  factors,  one  of  which  is  the  divisor,  the  object 
of  division  is  to  discover  the  other  factor ;  when  found, 
the  other  is  the  quotient.  Thus,  if  48  be  the  dividend, 
and  8  the  divisor,  then  that  number  which,  being  mill, 
tiplied  by  8,  gives  48  as  the  product,  is  the  quotient ;  — 
that  number  is  6. 

(16'7.)  Hence  it  follows,  that  by  whatever  means  the 
quotient  is  found,  the  process  may  be  verified  by  mul- 
tiplying the  divisor  by  the  quotient.  The  product  should 
be  the  dividend ;  and  if  it  be  not,  the  operation  must 
have  been  incorrectly  performed. 

(l68.)  Since  the  product  of  the  divisor  and  quotient 
is  the  dividend,  it  follows  that  the  quotient,  repeated  as 
often  as  there  are  units  in  the  divisor,  wiil  make  up  the 
dividend.  Hence  the  process  of  division  is  presented 
under  another  point  of  view;  being  that  process  by 
which  the  dividend  is  divided  into  as  many  equal  parts 
as  there  are  units  in  the  divisor,  one  of  those  equal 
parts  being  the  quotient.  It  is  from  this  mode  of  con- 
sidering the  operation  that  it  has  received  the  name 


(169.)  It  likewise  follows,  that  the  divisor  being 
repeated  as  often  as  there  are  units  in  the  quotient,  a 
number  will  be  obtained  equal  to  the  dividend :  hence 


the  process  is  exhibited  under  another  aspect,  being  the 
operation  by  which  we  may  discover  how  often  the 
divisor  must  be  repeated  in  order  to  make  up  the  divi- 
dend, or,  as  it  is  commonly  expressed,  how  often  the 
divisor  is  contained  in  the  dividend. 

Under  this  point  of  view,  division  may  be  considered 
as  continued  subtraction,  in  the  same  manner  as  multi- 
plication is  continued  addition.  If  we  wish  to  find  how 
often  8  is  contained  in  48,  we  shall  subtract  8  from  48, 
and  then  subtract  8  from  the  remainder,  and  so  continu- 
ally subtract  8  from  every  remainder  until  the  subtra- 
hend is  exhausted.  The  number  of  times  which  the 
divisor  8  has  then  been  subtracted  from  the  dividend  is 
the  same  as  the  number  of  times  which  the  divisor  is 
contained  in  the  dividend. 

(170.)  But  from  these  ways  of  considering  the  pro- 
cess of  division,  it  would  appear  that  the  operation  could 
never  be  performed,  except  in  the  particular  case  in 
which  the  dividend  happens  to  be  the  product  of  two 
numbers,  one  of  which  is  the  divisor.  Now^  in  by  far 
the  greater  number  of  cases  which  can  be  proposed, 
this  circumstance  will  not  happen.  If  the  dividend  bt- 
50,  and  the  divisor  8,  there  is  no  number  (so  far  as  the 
definitions  of  number  already  given  extend)  which, 
being  multiplied  by  the  divisor,  would  produce  the  divi- 
dend ;  and  therefore  the  division,  in  the  sense  in  which 
that  operation  has  been  just  defined,  would  be  impossi- 
ble. We  shall  see  hereafter,  however,  that  by  enlarging 
our  ideas  of  number,  we  shall  be  enabled  to  perform 
division  even  in  this  case :  meanwhile  it  is  usual  to  ef- 
fect a  partial  division  in  those  cases  in  which  the 
dividend  is  not  an  exact  multiple  (121.)  of  the  divisor. 
The  quotient  which  is  obtained  in  such  cases  is  a  partial 
quotient,  expressing  merely  the  number  of  times  which 
the  divisor  is  contained  in  the  dividend  (129.).  In 
such  cases  there  will  always  be  a  remainder,  being  that 
part  of  the  dividend  which  would  remain  if  the  divisor 
were  subtracted  as  often  as  possible  from  it. 

Let  us  suppose,  for  example,  that  the  dividend  is  50: 
i  4 


120  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

by  successively  subtracting  8  from  the  dividend,  we 
should  find  that  after  subtracting  it  6'  times,,  there  would 
be  a  remainder  2  :  from  this,  8 ,  being  a  greater  number, 
could  not  be  subtracted.  When  it  is  proposed,  then,  to 
divide  50  by  8.,  we  say  that  the  quotient  is  6,  and  the 
remainder  2  ;  meaning  by  the  word  quotient,  not  the 
complete  quotient  of  50  divided  by  8,  but  the  exact 
quotient  of  48  divided  by  8,  48  being  the  greatest  mul- 
tiple of  8  contained  in  50.  The  incompleteness  of  the 
division  is  always  signified  by  declaring  what  is  the  re- 
mainder at  the  same  time  that  the  quotient  is  expressed. 
Although,  in  such  cases,  the  quotient  is,  strictly  speak- 
ing, a  partial  quotient,  yet  it  is  customary  to  call  it 
simply  the  quotient,  since  no  mistake  can  arise,  provided 
that  the  remainder  at  the  same  time  be  declared. 

(171.)  From  what  has  been  here  explained,  it  will 
follow,  that  when  the  dividend  is  not  an  exact  multiple 
of  the  divisor,  the  product  of  the  quotient  and  divisor 
is  the  greatest  multiple  of  the  divisor  contained  in  the 
dividend  :  it  is  also  the  greatest  multiple  of  the  quotient 
contained  in  the  dividend.  It  also  follows,  that  when 
the  product  of  the  quotient  and  divisor  is  subtracted 
from  the  dividend,  the  remainder  will  be  the  same  as 
the  remainder  obtained  in  the  ordinary  process  of  di- 
vision; and  that,  if  to  the  product  of  the  quotient  and 
divisor  the  remainder  be  added,  the  sum  will  be  the 
product. 

Let  the  dividend  be  68,  and  the  divisor  9  ;  the  quo- 
tient will  be  in  this  case  7,  and  the  remainder  5.  Now 
Q  X  7  is  the  greatest  multiple  of  9  or  °f  7  contained  in 
6*8.  If  this  product  be  subtracted  from  68,  the  re- 
mainder will  be  5  ;  for  if  5  be  added  to  this  product,  the 
sum  will  be  the  dividend  68. 

(172.)  Hence,  to  verify  the  process  of  division,  mul- 
tiply the  divisor  by  the  quotient,  and  to  this  product 
add  the  remainder:  the  sum  should  be  the  dividend; 
and  if  it  be  not,  there  must  be  some  error  in  the  work. 

(173.)  The  work  may  also  be  verified  in  the  follow- 
ing manner: — Subtract  the  remainder  from  the  divi- 


CHAP.  VI.  DIVISION.  121 

dend ;  the  number  remaining  should  be  the  exact  pro- 
duct of  the  divisor  and  quotient.  If  this  number  be 
divided  by  the  quotient,  the  result  should  be  the  divisor; 
or  if  it  be  divided  by  the  divisor,  the  result  should  be 
the  quotient ;  or,  finally,  if  the  divisor  and  the  quotient 
be  multiplied  together,  this  same  number  should  be  the 
result.  Any  of  these  methods  may  be  used  for  veri- 
fication. 

Having  stated  and  proved  the  general  principles  on 
which  the  process  of  division  rests,  we  shall  now  pro. 
ceed  to  explain  the  practical  methods  by  which,  when  the 
divisor  and  dividend  are  given,  the  quotient  may  be  dis- 
covered. 

(174.)  If  the  divisor  be  a  single  digit,  and  the  divi- 
dend do  not  exceed  100,  then  the  quotient  and  remain- 
der may  generally  be  found  by  the  knowledge  of  the 
multiplication  table  carried  as  far  as  10  times  10. 

If  the  dividend  in  this  case  happen  to  be  an  exact 
multiple  of  the  divisor,  then  it  will  be  remembered  as 
one  of  the  products  in  the  multiplication  table,  and  it 
will  be  recollected  what  that  number  is,  by  which  the  di- 
visor being  multiplied,  the  product  is  the  dividend.  Thus, 
suppose  the  dividend  is  72,  and  the  divisor  8  ;  the  ques- 
tion is,  what  number  multiplied  by  8  produces  72  ?  and 
the  knowledge  of  the  multiplication  table  immediately 
suggests  that  this  number  is  9  :  the  quotient,  therefore, 
is  in  this  case  Q,  there  being  no  remainder. 

The  discovery  of  the  quotient  is  not  more  difficult 
when  the  dividend  is  not  an  exact  multiple  of  the  di- 
visor. '  Let  us  suppose  that  the  dividend  is  60,  and  the 
divisor  8  ;  the  question  then  is,  what  is  the  greatest 
number  which,  being  multiplied  by  8,  will  give  a  product 
less  than  60  ?  Now,  we  know  from  the  multiplication 
table,  that  7  multiplied  by  8  gives  the  product  56,  and 
that  8  multiplied  by  8  gives  the  product  64 :  conse- 
quently 56  is  the  greatest  multiple  of  8  contained  in  60. 
If  60,  therefore,  be  divided  by  8,  the  quotient  will  be  7, 
and  the  remainder  4. 

(175.)  We  have  seen,  that  if  the  dividend  be  an  ex- 


122  A    TREATISE    ON    ARITHMETIC.  BOOK  I- 

act  multiple  of  the  divisor,  the  division  will  be  com- 
pleted without  a  remainder,  and  the  quotient  will  express 
the  number  of  times  which  the  divisor  is  contained  in 
the  dividend.  If  in  this  case  the  dividend  be  multiplied 
by  10,  the  quotient  must  also  be  multiplied  by  10,  since 
the  divisor  will  be  contained  10  times  as  often  in  a 
dividend  10  times  as  great.  In  the  same  manner,  if 
the  dividend  be  multiplied  by  100,  the  quotient  must 
also  be  multiplied  by  100,  for  a  like  reason,  and  so  on. 
Thus,  it  appears  that  in  cases  where  there  is  no  remain- 
der, the  division  will  remain  correct,  if  we  annex  the 
same  number  of  ciphers  to  the  dividend  and  quotient 
(65.).  Thus,  if  the  divisor  be  2  and  the  dividend  8, 
the  quotient  will  be  4,  since  2  is  contained  4  times  in 
8  :  if  we  annex  a  0  to  the  dividend,  we  multiply  it  by 
10,  and  2  is  contained  in  80  ten  times  as  often  as  it  is 
contained  in  8  ;  therefore  it  is  contained  40  times  in  80, 
and  the  quotient  will  be  40,  being  the  former  quotient 
with  a  cipher  annexed.  In  like  manner,  if  we  annex  2 
ciphers  to  the  dividend,  we  multiply  it  by  100,  and  the 
divisor  2  will  be  contained  in  800  a  hundred  times 
as  often  as  it  is  contained  in  8,  and  the  quotient  will  ac- 
cordingly be  400,  which  is  the  first  quotient  with  2 
ciphers  annexed.  We  shall  presently  perceive  the  use 
of  these  inferences. 

(176.)  When  the  dividend  is  a  large  number,  it  is 
necessary  in  the  practical  process  of  division  to  resolve 
it,  or  to  imagine  it  resolved,  into  several  smaller  ones, 
and  the  division  is  in  that  case  effected  by  dividing  each 
of  the  smaller  numbers  separately  by  the  divisor,  and 
then  adding  together  the  quotients.  Thus  we  might 
divide  1 8  by  3,  by  resolving  it  into  3,  6,  and  9,  which 
added  together  would  make  up  18.  3  is  contained  in 
3  once,  in  6  twice,  and  in  9  three  times :  the  three  quo- 
tients are,  therefore,  1,  2,  and  3  ;  and  these  added  to- 
gether make  6,  which  is  the  total  quotient  of  18  divided 
by  3. 

(177-)  Let  us  now  consider  the  case  in  which  the 
divisor  is  a  single  digit,  and  the  dividend  a  number 


CHAP.  VI.  DIVISION.  123 

consisting  of  several  digits.  Let  the  divisor,  for  ex- 
ample, be  3,  and  the  dividend  963  :  we  shall  conceive 
this  dividend  resolved  into  3  parts,  —  900,  60,  and  3. 
In  writing  down  the  process  of  division,  it  is  usual  to 
place  the  divisor  on  the  left  of  the  dividend,  separated 
by  a  line,  and  to  write  the  quotient  either  under  the  di- 
vidend, or  separated  by  a  line  to  the  right  of  the  divi- 
dend. Taking  the  three  partial  dividends,  into  which 
we  have  resolved  the  total  dividend  963,  the  three  par- 
tial divisions  would  be  written  as  follows ;  — 

3  )  900  (  300  3  )  60  (  20  3)3(1 

We  know  that  3  is  contained  3  times  in  9>  without  a 
remainder ;  and,  therefore,  by  what  has  been  proved 
(175.),  it  is  contained  300  times  in  900.  For  the  same 
reason,  since  3  is  contained  twice  in  6,  without  a  re- 
mainder, it  is  contained  20  times  in  60.  It  is  evident, 
then,  that  the  three  partial  quotients  are  300,  20,  and  1, 
and  the  total  quotient  321. 

In  order  to  make  the  principle  on  which  the  process 
rests  intelligible,  we  have  here  written  it  at  greater 
length  than  is  necessary  in  practice:  all  the  three 
partial  divisions  may  be  written  and  performed  as 
follows :  — 

3)963 

_s2i|in 

The  figure  9  of  the  dividend  here  expresses  900,  and 
3  being  contained  in  it  300  times  without  a  remainder, 
we  place  3  in  the  hundreds'  place  of  the  quotient ;  we 
do  not  annex  two  ciphers  to  it,  because  the  places  occu- 
pied by  those  ciphers  are  to  be  filled  up  by  the  other 
two  partial  quotients.  In  the  same  manner  6  in  the 
dividend  signifies  60,  which  divided  by  3  gives  the 
quotient  20 :  we  therefore  place  2  in  the  tens'  place  of 
the  quotient,  leaving  the  units'  place  still  unoccupied, 
to  be  filled  by  the  third  partial  quotient,  which  in  this 
case  being  1,  the  division  is  completed  as  above. 

(178.)  It  might  so  happen  that  the  third  partial  dividend 
might  be  wanting,  and  that  the  dividend  would  only  be 


124<  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

resolved  into  the  first  two  partial  dividends :  this  would 
happen  if  the  last  figure  of  the  dividend  were  0.  In 
that  case,  the  two  partial  dividends  would  be  900  and 
60,  and  the  two  partial  quotients  would  be  300  and  20; 
these  would  be  expressed  by  placing  3  and  2  in  the 
hundreds'  and  tens'  place  of  the  quotient ;  but  as  no 
third  partial  quotient  would  in  that  case  be  furnished 
to  fill  the  units'  place  of  the  total  quotient,  it  would  be 
necessary  to  fill  that  place  by  0,  otherwise  the  3  and  2 
would  express  30  and  2,  and  not  300  and  20. 

(179-)  Again:  it  might  so  happen,  that  the  second 
partial  dividend  would  be  wanting,  the  first  and  third 
remaining:  this  would  happen  if  the  dividend  had 
been  903  ;  in  that  case  it  would  be  resolved  into 
the  partial  dividends  900  and  3,  and  the  two  partial 
quotients  would  be  300  and  1.  In  that  case  it  would 
be  necessary  to  write  the  3  in  the  hundreds'  place,  and 
the  1  in  the  units' ;  but  it  would  be  obviously  necessary 
to  fill  the  tens'  place  by  a  0. 

(180.)  In  the  examples  which  we  have  here  given,  it 
will  be  observed  that  each  of  the  digits  of  the  dividend 
is  an  exact  multiple  of  the  divisor.  This  circumstance 
gives  a  simplicity  to  the  example,  which  renders  it 
proper  as  a  first  step  in  the  explanation  of  the  practical 
process.  Let  us  now,  however,  take  the  more  general 
case,  in  which  the  digits  of  the  dividend  are  not  exact 
multiples  of  the  divisor. 

Let  the  dividend  be  7^2,  and  let  the  divisor,  as .  be- 
fore, be  3 :  following  the  steps  taken  in  the  former 
example,  we  should  first  naturally  resolve  this  dividend 
into  three  parts :  — 

700  60  2 

But  700  not  being  an  exact  multiple  of  the  divisor,  we 
adopt  another  expedient :  the  number  next  less  than  7 
which  is  divisible  by  3  is  6 ;  3  is  contained  in  6  twice, 
and  200  times  in  600  :  instead,  therefore,  of  taking  700 
as  the  first  partial  dividend,  we  shall  take  600,  and  add 
the  remaining  100  to  the  second  partial  dividend;  the 


CHAP.  VI.  DIVISION.  125 

three  parts  into  which  the  original  dividend  is  resolved 
will  then  be 

600  160  2 

The  second  of  these,  160,  is,  however,  not  an  exact 
multiple  of  3 ;  the  greatest  multiple  of  3  contained  in 
16  is  15;  and  since  3  is  contained  5  times  in  15,  it 
will  be  contained  50  times  in  150  :  we  shall  therefore 
take  150  as  the  second  partial  dividend,  and  transfer 
the  remaining  10  to  the  third  part  2,  by  which  it  becomes 
12,  and  the  original  dividend  will  then  be  resolved  into 
the  following  parts  :  — 

600  150  12 

these  being  severally  divided  by  3,  give  the  following 
quotients :  — 

200  50  4 

The  total  quotient  will  be  obtained  by  adding  together 
.these  three  partial  quotients  :  it  is,  therefore,  254. 

But  in  practice  this  process  may  be  written  in  a  very 
abridged  form.  To  obtain  the  first  quotient,  200,  it  is 
only  necessary  to  divide  7  by  3,  and  put  the  quotient  2 
in  the  hundreds'  place,  reserving  the  remainder  1.  Con- 
ceive this  remainder  prefixed  to  the  digit  6,  which  fills 
the  tens'  place,  and  divide  the  number  16  thus  found 
by  3 ;  place  the  quotient  5  in  the  tens'  place  of  the 
quotient,  and  prefix  the  remainder  1  to  the  figure  2, 
which  fills  the  units'  place  of  the  dividend :  divide  12, 
the  number  thus  obtained,  by  3,  and  put  the  quotient  4 
in  the  units'  place  of  the  quotient.  It  must  be  evident 
that  in  this  way  the  same  process  is  performed  as  that 
which  was  more  fully  developed  above. 

In  general,  then,  when  the  dividend  consists  of  seve- 
ral digits,  the  divisor  being  a  single  digit,  proceed  by 
the  following 

RULE. 

(181.)  Divide  the  first  figure  of  the  dividend  by  the 
divisor,  and  place  the  quotient  under  the  same  figure  of 
the  dividend  ;  prefix  the  remainder  to  the  next  figure  of 
the  dividend,  and  divide  the  number  thus  obtained  by  thv 
divisor ;  place  the  quotient  under  the  second  figure  of 


126  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

the  dividend,  and  prefix  the  remainder  to  the  third  figure 
of  the  dividend ;  divide  the  number  thus  obtained  by  the 
divisor,  and  proceed  as  before,  continuing  this  process 
until  you  arrive  at  the  units'  place  of  the  dividend,  when 
the  division  will  be  complete. 

(182.)  When  it  happens  that  the  first  figure  of  the 
dividend  is  less  than  the  divisor,  the  first  figure  of  the 
quotient  would  he  0,  since  the  divisor  is  not  contained 
in  a  number  less  than  itself ;  but  as  0  standing  on  the 
left  of  a  number  does  not  affect  its  value,  it  is  unneces- 
sary to  insert  it.  The  following  example  will  illustrate 
this :  —  Let  the  dividend  be  162,  the  divisor  being  3  ; 
by  following  the  above  rule  strictly,  the  process  of  di- 
vision would  be  as  follows  :  — 
3  )  162 
"054 

In  performing  the  division,  we  should  say,  the  quotient 
of  1  by  3  is  0,  with  a  remainder  1,  which  being  pre- 
fixed to  6,  we  divide  16  by  3,  &c.  But  since  the  0 
prefixed  to  54?  has  no  signification,  it  is  omitted,  and 
we  begin  the  process  of  division,  not  by  expressing  the 
division  of  1  by  3,  but  of  16  by  3.  Whenever,  then, 
the  first  figure  of  the  dividend  is  less  than  the  divisor, 
the  first  step  of  the  process  should  be  to  divide  the  first 
two  figures  of  the  dividend  by  the  divisor,  and  in  that 
case  the  quotient  should  be  placed  under  the  second 
figure  of  the  dividend.  The  above  example  would, 
therefore,  be  written  thus :  — 
3  )  162 
54 

(183.)   The  rule  above  given  will  still  be  applicable 
if  any  of  the  places  of  the  dividend  happen  to  be  filled 
by  ciphers.    Take  the  following  example :  — 
5  )  980700 
196HO 


The  first  two  figures  of  the  quotient  are  found  by  the 
rule  already  explained.  When  9  is  divided  by  5  there  is 
a  remainder  4^  which,  prefixed  to  8,  gives  48 ;  this  di- 


CHAP.  VI.  DIVISION,  127 

vided  by  5  gives  9^  with  a  remainder  3  ;  this  remainder 
must  be  prefixed  to  the  following  0 :  the  30  thus  ob- 
tained, divided  by  5,  gives  6  for  the  next  figure  of  the 
quotient :  the  2  which  remains  when  7  is  divided  by  5 
must  be  prefixed  to  the  following  0,  which  gives  20  ; 
this  divided  by  5  gives  the  quotient  4,  without  a  re- 
mainder. But  as  there  is  another  final  0  in  the  divi- 
dend, a  0  must  be  also  annexed  to  the  quotient.  (178.) 

(184.)  Let  us  now  consider  the  most  complex  case 
of  division,  in  which  the  divisor  and  dividend  are  both 
numbers  which  consist  of  several  places.  As  in  the 
former  cases,  the  division  is  here  effected  by  resolving 
the  dividend  into  a  number  of  parts,  each  of  which  is 
an  exact  multiple  of  the  divisor.  We  shall  first  explain 
the  method  of  thus  resolving  the  dividend  into  parts, 
which  will  be  easily  understood  by  its  application  to 
an  example.  Let  the  dividend  be  4967398,  and  the 
divisor  37  :  it  is  required  to  resolve  the  former  into 
several  numbers  each  of  which  will  be  an  exact  multiple 
of  37.  To  effect  this,  let  it  be  considered  that  if  any 
number  be  a  multiple  of  another,  the  same  number  will 
still  be  a  multiple  of  the  other  when  any  number  of  ci- 
phers are  annexed  to  it :  thus,  if  8  be  a  multiple  of  4, 
80,  800,  8000,  &c.  will  also  be  multiples  of  4  (175.). 
This  is  sufficiently  evident. 

The  practical  process  by  which  the  dividend  is  re- 
solved into  multiples  of  the  divisor  may  be  written  as 
follows :  — 

From     -  -  A  -  4967398 

Subtract  -  B  -  3700000 

From      -  -  C  -  1267398 

Subtract  -  D  -  111000O 


From     -  -  E  -  157398 

Subtract  -  F  -  148000 

From     -  -  G  -  9398 

Subtract  -  H  -  7400 

From      -  -  I  -  1998 

Subtract  -  K  -  1850 

L  -  148 


128  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

The  number  A  is  the  dividend  itself :  we  can  find  one 
multiple  of  the  divisor  which  is  contained  in  it  by  find- 
ing the  greatest  multiple  of  the  divisor  contained  in  49, 
and  annexing  to  it  as  many  noughts  as  there  are  remain- 
ing places  in  the  dividend.  Now,  37  is  contained  only 
once  in  49 ;  if,  therefore,  to  37  we  annex  five  ciphers, 
we  shall  obtain  a  multiple  of  the  divisor  (175.)  less  than 
the  dividend  A,  and  which  is  therefore  contained  in  it  : 
this,  then,  is  one  of  the  parts  into  which  we  wish  to  resolve 
the  dividend.  We  accordingly  subtract  this  first  mul- 
tiple, B,  of  the  divisor  from  A,  and  we  get  the  remainder 
C  ;  this  number  C  is  now  what  remains  of  the  dividend 
to  be  resolved  into  multiples  of  the  divisor.  We  shall 
obtain  another  multiple  of  the  divisor  contained  in  C  if 
we  can  ascertain  how  often  the  divisor  is  contained  in 
the  initial  figures  of  C  :  the  first  two  figures  of  C  form- 
ing a  number  less  than  the  divisor,  we  must  take  the 
first  three  figures ;  and  the  question  is,  how  often  37 
is  contained  in  126:  this  can  only  be  ascertained  by 
trial ;  but  we  may  form  a  near  guess  at  it  by  enquiring 
how  often  3  is  contained  in  12.  Since  12  divided  by 
3  gives  the  quotient  4,  we  must  try  whether  4  times  37 
are  less  than  1 26 :  we  find,  however,  that  they  are  greater, 
and  therefore  infer  that  37  is  contained  less  than  4  times 
in  126.  We  next  try  if  it  be  contained  3  times:  we 
find  that  3  times  37  are  111,  which,  being  less  than  126, 
is  the  greatest  multiple  of  37  contained  in  that  number. 
Now,  since  111  is  a  multiple  of  37,  1110000  is  also  a 
multiple  of  37,  and  this  number  is  less  than  the  first 
remainder  C  :  it  is,  therefore,  a  multiple  of  37  contained 
in  C.  We  have  thus  found  two  multiples  of  37,  B  and 
D,  which  form  parts  of  the  dividend  A :  by  subtracting 
D  from  C  we  shall  obtain  what  remains  of  the  dividend 
to  be  resolved  into  multiples  of  37  :  this  remainder  is  E, 
and  we  proceed  with  it  as  in  the  former  cases,  ascertain- 
ing how  often  37  is  contained  in  157-  Observing  that 
3  is  contained  5  times  in  15,  we  try  whether  37  is  con- 
tained 5  times  in  157  ;  but  we  find  that  5  times  37  are 
greater  than  1^7 :  we  next  try  whether  37  is  contained 


CHAP.  VI.  DIVISION.  129 

4  times  in  157;  and  finding  that  4  times  37  are  1 48,  we 
infer  that  148  is  the  greatest  multiple  of  37  contained 
in  15?.  Annexing  to  148  as  many  ciphers  as  there  are 
remaining  places  in  E,  we  obtain  a  multiple  of  37  which 
is  contained  in  E  :  we  have  thus  obtained  3  multiples  of 
37,  B,  D,  and  F,  which  form  parts  of  the  original  di- 
vidend, and  we  shall  obtain  what  remains  of  that  di- 
vidend to  be  resolved  into  parts  by  subtracting  F  from 
E ;  the  remainder  is  the  number  G ;  we  proceed  with  it 
in  the  same  manner,  and  enquire  how  often  37  is  con- 
tained in  93  :  since  3  is  contained  3  times  in  Q,  we  en- 
quire if  37  be  contained  3  times  in  93 ;  but  finding  that 
3  times  37  are  greater  than  Q3,  we  take  twice  37  or  74, 
which,  being  less  than  Q3,  is  contained  in  it.  Annexing 
as  many  ciphers  to  this  as  there  are  remaining  places  in 
the  number  G,  we  obtain  the  number  H,  which  is  a 
multiple  of  37-  We  have  thus  obtained  four  multiples 
of  37,  B,  D,  F,  and  H,  which  are  parts  of  the  original 
dividend :  the  remaining  part  of  the  dividend  not  yet 
resolved  into  multiples  of  the  divisor  is  I,  found  by  sub- 
tracting H  from  G.  We  proceed  with  this  as  in  the 
former  case,  enquiring  how  often  37  is  contained  in  199* 
we  find  that  it  is  contained  5  times;  and  185  being  a 
multiple  of  37,  the  same  number  with  a  cipher  annexed 
is  also  a  multiple  of  37.  We  thus  obtain  K,  another 
multiple  of  37,  which  is  a  part  of  the  original  dividend : 
the  remainder  L  is  itself  4  times  37,  and  is  therefore  a 
multiple  of  the  divisor.  We  have  thus  resolved  the 
original  dividend  into  the  parts  B,  D,  F,  H,  K,  L,  which 
are  respectively  multiples  of  the  divisor,  and  which 
when  added  together,  must  reproduce  the  original  di- 
vidend :  — 

B  -  -                       3700000 

D  -     1110000 

F  -  148000 

H  -  7400 

K  -           1850 

L  -  148 

A  4967898 


ISO  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

In  order  to  find  how  often  the  divisor  is  contained  in 
the  original  dividend  A,  it  is  now  only  necessary  to  find  how 
often  it  is  contained  in  the  several  parts  B,  IX,  F,  H,  K, 
and  L  of  that  dividend ;  in  other  words,  taking  these 
numbers  severally,  as  partial  dividends,  and  dividing 
them  by  37,  we  shall  obtain  as  many  partial  quotients; 
which  being  added  together,  will  give  the  total  quotient 
of  the  dividend  A,  when  divided  by  the  divisor  37.  The 
process  at  length  would  be  as  follows :  — 

Divisor.    Dividend.     Quotient. 


37 

3700000 

100000 

37 

1110000 

30000 

37 

148000 

4000 

37 

7400 

200 

37 

1850 

50 

37 

148 

4 

37  )  4967398  (  134254 

With  a  view  to  explain  in  the  clearest  manner  the 
principles  on  which  this  process  rests,  we  have  here  ex- 
pressed it  at  much  greater  length  than  is  necessary  in 
its  practice  after  those  principles  have  been  understood. 
It  will  be  observed,  that  each  of  the  partial  divisions 
furnishes  the  successive  figures  of  the  total  quotient  fol- 
lowed by  ciphers.  Thus  the  last  partial  quotient  would 
be  expressed  by  placing  4  in  the  units'  place  of  the  first, 
the  preceding  one  by  placing  5  in  the  tens'  place,  and  so 
on.  In  practice,  therefore,  instead  of  writing  the  se- 
veral partial  quotients  separately,  we  omit  the  ciphers  in 
the  first,  and  in  their  places  put  the  first  figures  of  all 
the  others.  It  is  likewise  obviously  unnecessary  to  write 
the  divisor  before  each  of  the  partial  dividends,  as  its 
presence  may  be  understood  after  being  once  written 
before  the  first..  The  process  of  resolving  the  dividend 
into  parts  which  are  severally  multiples  of  the  divisor, 
and  the  actual  process  of  the  several  partial  divisions, 
may  also  be  combined  in  the  same  written  arrangement. 
Observing  these  abridgments,  the  practical  process  of 
division  would  take  the  following  form :  — 


CHAP.  VI.  DIVISION.  131 

37  )  4967398  (  134254 
3700000 
1267398 
1110000 


157398 
148000 

~~~9398 
7400 


1998 
1850 

~~I4S 
148 

It  will  be  further  observed,  that  in  each  successive 
remainder  the  latter  figures  remain  unchanged  through- 
out the  process,  the  initial  figures  only  being  affected  by 
the  subtractions.  It  is,  therefore,  unnecessary  to  subtract 
in  any  case  more  than  three  figures  of  each  remainder 
each  time,  provided  that  the  next  figure  of  the  dividend 
is  always  annexed  to  the  remainder.  The  process  ac- 
cording to  this  abridgment  would  be  thus  expressed:  — 

37  )  4967398  (  134254 

87 

~126 
111 

157 

148 

93 

74 

199 


148 
148 


By  placing  37  under  49,  the  former  is  here  made  to 
express  3700000,  which  is  the  same  number  as  it  ex- 
pressed when  the  process  was  written  down  in  full.  We 
subtract  37  from  49,  and  to  the  remainder,  1 2,  we  annex 
the  next  figure,  63  of  the  dividend :  by  placing  the  second 


132  A    TREATISE    ON    ARITHMETIC.  BOOK   I. 

dividend,  111,  under  126,  or  under  496  of  the  original 
dividend,  it  is  made  to  express  1110000.  In  the  same 
manner  the  third  dividend,  148,  being  placed  under  96'7 
of  the  original  dividend,  expresses  148000;  and  so  on. 
To  fix  the  ideas  of  this  arithmetical  process  clearly 
and  firmly  in  the  mind  of  the  pupil,  we  shall  now  take 
another  example,  in  which  the  divisor  shall  consist  of 
3  figures :  — 

256  )  890368  (  3478 
768 

1223 
1024 


1996 
1792 

2048 
2048 


We  begin  by  taking  the  first  3  figures  of  the  di- 
vidend, and,  considering  them  as  one  number,  we  en- 
quire how  often  the  divisor  is  contained  in  them :  this 
can  only  be  determined,  as  before,  by  trial;  but  we  shall 
in  some  measure  be  led  to  the  knowledge  of  the  number 
sought  by  trying  how  often  the  first  figure  of  the  divisor 
is  contained  in  that  of  the  dividend.  Since  2  is  con- 
tained 4  times  in  8,  we  shall  first  try  whether  256  is 
contained  4  times  in  890 ;  but  by  multiplying  256  by  4 
we  obtain  a  product  greater  than  890  :  therefore  256  is 
not  contained  4  times  in  890.  We  next  try  whether 
it  is  contained  3  times  in  890  ;  and  by  multiplying  256 
by  3  we  obtain  the  product  768,  which,  being  less  than 
890,  must  be  the  greatest  multiple  of  256  contained  in 
that  number :  we  place  3  as  the  first  figure  of  the  quo- 
tient, and  subtract  768  from  890  ;  but,  in  doing  this,  it 
is  evident  that  we  have,  in  fact,  subtracted  768000  from 
890000,  such  being  the  local  values  of  the  digits  en- 
gaged in  the  operation.  The  3  which  is  placed  in  the 
quotient  should,  therefore,  signify  3000,  and  we  should 
accordingly  write  three  ciphers  after  it ;  but  it  is  unne- 
cessary to  write  these,  because  their  places  will  be  filled 


CHAP.  VI.  DIVISION.  133 

by  the  other  figures  of  the  quotient,  which  will  be  dis- 
covered in  the  course  of  the  operation.  To  the  remain- 
der 122  we  annex  the  succeeding  figure  of  the  dividend, 
so  that  the  second  partial  dividend  will  be  1223.  We 
must  now  try  how  often  256  is  contained  in  this  num- 
ber; and  since  2  is  contained  in  12  six  times,  we  should 
try  whether  256  be  contained  in  1223  six  times:  we  find, 
however,  that  by  multiplying  256  by  6,  or  even  by  5, 
we  obtain  a  product  greater  than  1223.  If,  however, 
we  multiply  256  by  4,  we  obtain  a  product  less  than 
1223 :  4  is,  therefore,  the  next  figure  of  the  quotient; 
and  we  obtain  the  corresponding  partial  dividend,  1 024, 
by  multiplying  the  divisor  by  4.  By  continuing  the 
process  in  the  same  manner,  we  obtain  the  total  quotient 
as  above. 

In  the  examples  which  we  have  given  above,  it  hap. 
pens  that  every  figure  of  the  quotient  is  a  significant 
digit.  Although  the  same  principles  will  be  applicable 
in  the  cases  in  which  ciphers  occur,  yet  the  student 
might  be  embarrassed  if  he  were  not  practically  pre- 
pared for  that  circumstance.  The  following  example 
will  illustrate  the  mode  of  proceeding  in  such  a  case :  — 

2465  )  123262325  (  50005 
12325 

12325 
12325 


The  first  4  figures  of  the  dividend  forming  in  this  case 
a  less  number  than  the  divisor,  we  must  try  how  often 
the  divisor  is  contained  in  the  number  formed  by  the 
first  5  figures.  Since  2  is  contained  in  12  six  times,  we 
shall  try  if  the  divisor  is  contained  in  that  number  6 
times :  we  find,  however,  that  it  is  not,  but  that  it  is 
contained  in  it  5  times.  Multiplying  the  divisor, 
then,  by  5,  the  first  figure  of  the  quotient,  we  write  the 
product  under  the  first  5  figures  of  the  dividend  ;  and 
performing  the  subtraction,  we  obtain  the  remainder  1. 
To  this  remainder,  according  to  the  process  already  ex- 
it 3 


134?  A    TREATISE    ON    ARITHMETIC.  BOOK  i. 

plained,  we  should  annex  the  succeeding  figure  of  the 
dividend;  but  this  gives  12,  a  number  much  less  than 
the  divisor.  Under  these  circumstances,  we  must  con. 
tinue  to  annex  to  the  remainder  the  successive  figures  of 
the  dividend,  until  we  obtain  a  number  which  is  not  less 
than  the  divisor.  To  accomplish  this  in  the  present 
instance.,  it  is  necessary  to  annex  all  the  remaining 
figures  of  the  dividend;  and  a  number  is  thus  obtained 
which  is  exactly  5  times  the  divisor.  This  number 
is,  then,  the  second  partial  dividend,  and  the  corre- 
sponding quotient  is  5,  which,  as  it  should  express  units, 
ought  to  stand  in  the  units'  place  of  the  quotient.  The 
first  quotient,  however,  should  express  50,000,  and 
should  therefore  stand  in  the  ten  thousands'  place. 
Now,  in  order  to  express  the  proper  values  of  these  two 
digits,  it  is  necessary  to  place  between  them  3  ciphers ; 
and  the  quotient  is  accordingly  the  number  above  writ- 
ten. In  general,  in  such  cases,  the  method  of  proceed- 
ing is  as  follows :  —  When,  on  annexing  a  figure  from 
the  dividend  to  any  remainder,  you  find  that  that  re- 
mainder is  less  than  the  divisor,  you  must  immediately 
annex  a  cipher  to  the  quotient,  and  annex  the  next 
figure  of  the  dividend  to  the  remainder.  If  the  re- 
mainder still  continue  to  be  less  than  the  divisor,  you 
must  annex  another  0  to  the  quotient ;  and  you  must 
continue  to  do  this  until,  by  successively  annexing 
figures  of  the  dividend  to  the  remainder,  you  obtain  a 
number  greater  than  the  divisor. 

The  following  example  will  illustrate  these  observ- 
ations :  — 

6437  )  19343313785059  (  3005020007 
19311 

32313 
32185 


12878 
12874 


45059 
45059 


Proceeding  in  the  usual  way,  the  first  remainder  is  32. 
Annexing  to  this  the  next  figure,  3,  of  the  dividend,  the 
remainder  is  still  less  than  the  divisor ;  we  must  there- 
fore annex  a  0  to  the  quotient,  and  bring  down,  as  it  is 
called,  the  next  figure,  1 ,  of  the  dividend.  It  will  be  found 
convenient  in  working  questions  of  this  kind,  and,  in 
general,  in  long  division,  to  mark  with  a  dot  each  figure 
of  the  dividend  which  is  successively  annexed  to  the 
remainders  :  by  this  means  no  mistake  can  be  made  in 
selecting  the  figures  to  be  annexed. 

In  order  to  render  the  first  remainder  in  the  present 
example  greater  than  the  divisor,  it  is  necessary  to  bring 
down  3  figures  from  the  dividend.  The  first  2  of  these 
make  it  necessary  to  add  noughts  to  the  quotient,  and, 
when  the  third  is  annexed,  the  divisor  is  contained  in 
the  number  so  formed.  5  is  then  the  next  figure  of 
the  quotient :  the  next  remainder  requires  2  figures  to 
be  brought  down  to  render  it  greater  than  the  divisor  ; 
one  0  is  therefore  added  to  the  quotient,  and  the  divisor 
being  contained  in  the  number  thus  formed  twice,  the 
succeeding  figure  of  the  quotient  is  2.  The  next  re- 
mainder is  4,  and  the  4  remaining  figures  of  the  divi- 
dend must  be  brought  down  in  order  to  get  a  number 
greater  than  the  divisor :  3  noughts  are  therefore  an- 
nexed to  the  quotient,  and  the  divisor  is  found  to  be 
contained  exactly  7  times  in  the  number  thus  formed  : 
7  is  therefore  the  last  figure  of  the  quotient. 

(185.)  In  certain  particular  cases  the  process  of  divi- 
sion may  be  greatly  facilitated  by  the  peculiar  nature  of 
the  divisor  or  dividend.  We  have  already  seen  that  a 
number  may  be  multiplied  by  10,  100,  1000,  &c.  by 
annexing  to  it  a  corresponding  number  of  ciphers.  It 
follows,  therefore,  that  a  number  terminating  in  ciphers 
may  be  divided  by  10,  100,  1000,  &c.  by  expunging 
a  corresponding  number  of  ciphers :  that  is,  it  may  be 
divided  by  10  by  cutting  off  1  cipher,  by  100  by  cut* 
ting  off  2  ciphers ;  and  so  on. 

Whether  a  number  terminates  in  ciphers  or  not,  how- 
ever, the  division  by  10,  100,  1000,  &c.  is  not  more 
K  4 


136*  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

difficult.  Let  us  suppose  that  we  wish  to  divide  3567 
by  10.  We  first  subtract  7  from  it,  and  it  becomes 
356*0  :  the  latter  is  divided  by  10  by  omitting  the  0  : 
therefore,  if  3567  be  divided  by  10,  the  quotient  will 
be  356,  with  a  remainder  7-  In  the  same  manner,  if 
we  would  divide  it  by  100,  we  shall  suppose  6*7  in  the 
first  instance  subtracted  from  it.  The  remainder,  3500, 
will  be  divided  by  100,  by  omitting  the  2  ciphers  :  thus, 
if  3567  be  divided  by  100,  the  quotient  will  be  35,  with 
a  remainder  67.  In  general,  then,  it  will  be  easily  per- 
ceived, that  to  divide  any  number  by  10,  100,  1000,  &c. 
we  must  cut  off  as  many  figures  on  the  right  as  there 
are  ciphers  in  the  divisor.  The  figures  thus  cut  off 
will  be  the  remainder  in  the  division,  and  the  other 
figures  will  be  the  quotient. 

(186.)  As  multiplication  maybe  performed  by  mul- 
tiplying successively  by  the  factors  of  the  multiplier 
(138.),  so  division  may  be  also  performed  by  dividing 
successively  by  the  factors  of  the  divisor.  By  attend- 
ing to  this  circumstance,  the  process  of  division  may 
very  often  be  considerably  abridged.  Thus,  if  we  wish 
to  divide  by  such  a  number  as  72,  we  may  divide  first 
by  9,  and  then  divide  the  quotient  thus  obtained  by  8. 
If  this  process  be  compared  with  the  general  method 
already  explained,  it  will  be  perceived  that  it  is  much 
more  expeditious  and  concise.  Let  the  dividend  be 
62000478,  and  the  divisor  81.  Instead  of  dividing 
directly  by  81,  we  shall  divide  the  dividend  by  9>  and 
the  quotient  thus  obtained  also  by  9-  The  process  will 
be  as  follows  :  — 

9  )  62000478 

9  )    6888942 
765438 


In  the  example  just  given  there  is  no  remainder.  In  ap- 
plying this  method  to  cases  in  which  there  is  a  remain- 
der, the  process  is  still  very  simple  and  brief.  Let  it 
be  required  to  divide  3763 by  72  :  we  shall  first  divide 
it  by  9j  and  then  divide  the  quotient  by  8 ;  the  process 
will  be  as  follows  :  — 


CHAP.  VI.  DIVISION.  137 

9  )  3763 

8  )    418       -       1  remainder 
52-2  ditto 

The  quotient  is  52  ;  but  neither  of  the  above  remain- 
ders would  be  the  actual  remainder  in  the  division  by 
72.  The  actual  remainder,  however,  may  be  easily  found 
by  the  following  rule  :  — Multiply  the  second  remainder 
above  by  the  first  divisor,  and  to  the  product  add  the  first 
remainder  ;  the  sum  will  be  the  remainder  in  the  actual 
division;  which,  in  the  present  case,  would  be  19»* 

(187-)  The  operations  of  multiplication  and  division 
may  frequently  be  brought  in  aid  of  each  other,  so  that, 
by  combining  both,  the  result  may  be  obtained  more 
concisely  than  it  could  be  by  either  separately.  For  ex- 
ample, if  we  wish  to  multiply  a  number  by  25,  we  may 
proceed  thus  :  —  First  annex  2  ciphers  to  it,  and  then 
divide  by  4.  Let  the  number  to  be  multiplied  be 
56738.  According  to  the  ordinary  method  of  multipli- 
cation, the  process  would  be  as  follows  :  — 

56738 
25 


2813690 
113476 

1418450 

According,  to  the  method  above  explained,  the  process 
would  take  the  following  more  abridged  form  :  — 
4  )  5673800 
1418450 


*  The  proof  of  this  will  be  easily  understood  by  those  who  have  some 


d"  q  +  r"  ~  q'  ; 

therefore      -     d1  d"  q  +  d'r"  +  r'=V: 
but  since       -     d'  d"  =  dt 
we  have        -    dq  +  d'r"  +  r1  =  D  ; 
also       -        -    dq+r  =  T): 
therefore      -    d'  r"  +  r'  —  r  j 
which  is  the  rule  expressed  in  the  text 


138  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

The  principle  on  which  this  method  rests  is  easily  un- 
derstood. We  are  required  to  multiply  by  25  ;  but  in 
multiplying  by  100  in  the  first  instance,  which  we  do 
by  annexing  2  ciphers,  we  have  obtained  a  product  four 
times  too  great,  since  100  is  4  times  25 :  wherefore  it 
is  necessary  to  divide  the  product  thus  obtained  by  4,  in 
order  to  get  the  true  product. 

Again,  suppose  it  is  required  to  multiply  67389  by 
125,  the  ordinary  process  would  be  as  follows  :  — 

67389 
125 


336945 
134778 
67389 
8423625 

This  question  may,  however,  be  more  briefly  solved  in 
the  following  manner :  —  Annex  3  ciphers  to  the  mul- 
tiplicand, and  divide  by  8.  This  process  will  then 
be  — 

8  )  67389000 
8423625 

In  this  case,  by  annexing  3  ciphers,  we  multiply  the 
multiplicand  by  1000,  which  is  8  times  the  proposed 
multiplier;  consequently  the  product  thus  obtained,  being 
divided  by  8,  gives  us  the  true  product. 

(188.)  In  similar  cases  oi  division  like  methods  may 
be  adopted  for  abridging  the  process.  If  we  wish  to 
divide  a  number  by  25,  we  may  first  divide  it  by  100, 
and  then  multiply  the  quotient  by  4  :  for,  by  dividing 
by  100,  we  divide  by  a  number  4  times  greater  than 
that  proposed,  and  therefore  must  multiply  the  quotient 
by  4,  in  order  to  obtain  the  true  quotient.  In  the  same 
manner,  to  divide  by  125  we  may  divide  by  1000,  and 
multiply  the  result  by  8  for  like  reasons. 

If  the  dividend  ends  in  2  or  more  ciphers,  this  pro- 
cess is  extremely  short  and  simple.  Let  the  dividend 
be  76*37500,  and  the  divisor  25 :  we  divide  by  100,  by 
cutting  off  the  2  ciphers,  and,  multiplying  the  remain- 


CHAP.  VI.  DIVISION.  139 

ing  number  by  4,  we  obtain  305500,  which  is  the 
quotient  sought. 

In  the  same  manner,  to  divide  678375000  by  125, 
it  is  only  necessary  to  cut  off  the  3  ciphers,  and  to 
multiply  the  remaining  number  by  8 ;  the  result  is 
5427000,  which  is  the  quotient  sought. 

But  if  the  dividend  do  not  terminate  in  ciphers,  the 
process  is  scarcely  less  simple.  Let  it  be  proposed  to 
divide  634782  by  25:  we  shall  first  subtract  from 
it  the  last  2  digits,  resolving  it  into  the  following 
parts :  — 

634700  82 

Each  of  these  must  be  divided  by  25.  The  first  may 
be  so  divided  by  cutting  off  the  ciphers  and  multiplying 
the  remaining  number  by  4  :  the  product,  25388,  is  a 
part  of  the  quotient  sought.  The  second  number  above 
must  now  be  divided  by  25 ;  but  that  number  being  less 
than  100,  its  quotient  by  25  is  always  readily  perceived. 
In  the  present  case  25  is  contained  3  times  in  82,  with 
a  remainder  7»  This  quotient  3  must  be  added  to  the 
former,  25388,  and  the  total  quotient  is  25391,  the 
remainder  being  7- 

In  like  manner,  if  it  were  required  to  divide  634782 
by  125,  the  process  would  be  as  follows  :  — 

634000  125  )  782  (  6 

Multiply    634  75° 

By  4  Remainder      32 

2536 
Add  (> 


Quotient    2542 


It  is  evident  that  this  process  consists  in  resolving  the 
dividend  into  two  parts,  and  dividing  these  parts  se- 
parately by  125,  by  different  methods,  the  partial  quo- 
tients being  added  together  to  obtain  the  total  quotient. 

A  like  method  may  be  extended  to  any  divisor  which 
is  a  sub-multiple  of  100,  1000,  10000,  &c. 

(189.)  We  have  already  explained  several  methods 
of  proving  Division  by  Multiplication  (l67.et  seq.)  ;  but 


140  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

when  the  divisor  and  dividend  consist  of  several  places, 
these  methods  are  often  tedious,  and  it  is  desirable  that 
we  should  possess  some  short  means  of  checking  arithme- 
tical computations  of  this  kind.  The  method  of  casting 
out  the  nines,  already  explained  (l6l.),  may  be  used  for 
this  purpose.  It  has  been  shown,  that  if  the  remainder, 
after  the  process  of  division  has  been  performed,  be 
subtracted  from  the  dividend,  we  shall  obtain  a  number 
which,  if  the  work  be  correct,  should  be  the  product  of 
the  divisor  and  quotient.  Whether  it  is  so  or  not  may 
be  determined  by  applying  to  it  the  method  of  casting 
out  the  nines,  considering  it  as  the  result  of  the  multi- 
plication of  the  divisor  and  quotient.  This  is,  perhaps, 
the  shortest  method  of  verification  which  can  be  used 
for  questions  in  Division.  It  is,  however,  as  in  Multi- 
plication, liable  to  failure,  though  in  rare  cases. 

(190.)  The  process  of  division  may  be  used  as  a 
means  of  verifying  that  of  multiplication,  if  the  product 
obtained  in  multiplication  be  taken  as  the  dividend,  and 
either  the  multiplier  or  multiplicand  as  divisor.  The 
quotient  obtained  should  be  the  multiplicand  when  the 
multiplier  is  taken  as  divisor,  and  the  multiplier  when  the 
multiplicand  is  taken  as  divisor.  The  teacher  may  check 
the  work  of  his  pupils  by  proposing  to  one  pupil,  as  a 
question  in  Division,  the  result  obtained  by  another  in 
Multiplication. 

(191.)  After  what  has  been  explained  in  the  present 
chapter,  the  following  general  directions  for  the  solution 
of  questions  in  Division  will  be  easily  understood. 

GENERAL  RULES. 

1 .  Write  the  divisor  on  the  left  of  the  dividend,  se- 
parated from  it  by  a  line :  place  another  line  on  the  right 
of  the  dividend  after  the  units'  place,  to  separate  the  quo- 
tient from  the  dividend;  the  quotient  being  afterwards 
written  on  the  right  of  that  line. 

2.  Count  off  from  the  left  of  the  dividend,  or  from  its 
highest  place,  as  many  digits  as  there  are  places  in  the  divi- 
sor :  if  the  number  formed  by  these  be  less  than  the  divisor 


CHAP.  VI.  DIVISION.  141 

then  count  off  one  more;  consider  these  digits  as  forming 
one  number,  and  find  how  often  the  divisor  is  contained  in 
that  number :  it  will  be  always  contained  in  it  less  than 
10  times,  as  we  shall  presently  perceive,  and  therefore  the 
quotient  of  the  division  will  always  be  a  single  digit :  place 
this  single  digit  as  the  first  figure  of  the  quotient. 

3.  Multiply  the  divisor  by  the  same  digit,  and  place 
the  product  under  those  figures  of  the  dividend  which  were 
taken  off  on  the  left,  and  then  subtract  such  product  from 
the  number  above  it,  by  which  you  will  obtain  the  first 
remainder  :   this  remainder  must  be  less  than  the  divisor, 
for  the  digit,  placed  as  the  first  figure  of  the  quotient, 
expressed  the  greatest  number  of  times  which  the  divisor 
was  contained  in  the  number  cut  off  from  the  dividend. 
If  the  remainder  were  greater  than  the  divisor,  this  would 
not  be  the  case. 

4.  On  the  right  of  the  first  remainder  place  that  figure 
of  the  dividend  which  next  succeeds  those  which  were  cut 
off  to  the  left :  find  how  often  the  divisor  is  contained  in 
the  number  thus  formed.     It  will  be  contained  in  it  less 
than  10  times,  and  therefore  the  quotient  will  be  a  single 
digit :  place  this  digit  as  the  next  figure  of  the  quotient. 

5.  Multiply  the  divisor  by  the  same  digit,  and  place 
the  product  under  the  first  remainder,  with  the  digit  of 
the  dividend  annexed :  subtract  it  from  that  number,  and 
you  will  obtain  the  second  remainder. 

6.  On  the  right  of  the  second  remainder  place  the  sue- 
ceeding  figure  of  the  dividend,  and  proceed  with  the  num- 
ber thus  formed  as  in  the  former  case,  and  the  third 
digit  of  the  quotient  will  be  obtained. 

7.  Continue  to  annex  the  succeeding  figures  of  the  di- 
vidend to  the  succeeding  remainders  until  every  figure  of 
the  dividend  has  been  thus  brought  down :  the  division 
will  then  be  complete;  and  the  last  remainder  being  sub" 
traded  from  the  dividend  would  leave  a  remainder,  which 
should  be  equal  to  the  product  of  the  quotient  and  divisor. 

It  has  been  stated  above,  that  if  a  digit  be  annexed 
to  the  right  of  a  number  less  than  the  divisor,  the  divi- 


142  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

sor  will  be  contained  in  that  number  less  than  10  times. 
To  perceive  the  truth  of  this,  we  have  only  to  consider 
that  when  the  digit  is  so  annexed  to  the  preceding  figures 
of  the  number,  forming  a  number  less  than  the  divisor, 
we  should  obtain  a  greater  number  by  annexing  a  0  to 
the  divisor  than  we  could  by  annexing  any  digit  to  the 
number  in  question.  This  will  be  easily  understood 
when  applied  to  an  example.  Let  the  divisor  be  1534, 
and  let  the  number  which  is  less  than  the  divisor,  and 
to  which  a  digit  is  to  be  annexed,  be  1527  :  the  greatest 
digit  which  can  be  annexed  to  this  is  9.>  by  which  it 
will  become  15279  :  this  will,  evidently,  be  less  than  10 
times  the  divisor,  because,  if  we  annex  a  0  to  the  latter, 
we  shall  obtain  a  number  consisting  of  the  same  number 
of  places,  but  of  which  the  first  4  places  form  a  greater 
number.  Since,  then,  10  times  the  divisor  is  greater 
than  the  number  found  by  annexing  the  digit,  the  di- 
visor must  be  contained  in  that  number  less  than  10 
times. 

If  the  first  or  any  succeeding  remainder,  with  the 
digit  brought  down  from  the  dividend  annexed,  form  a 
number  less  than  the  divisor,  then  a  cipher  must  be 
written  in  the  quotient,  and  another  figure  brought 
down.  If  the  number  thus  obtained  be  still  less  than 
the  divisor,  another  0  must  be  written  in  the  quotient, 
and  the  next  figure  of  the  dividend  brought  down  ;  and 
this  must  be  continued  until  a  number  is  obtained  greater 
than  the  divisor.  If,  after  all  the  figures  of  the  divi- 
dend are  brought  down,  the  number  should  still  be  less 
than  the  divisor,  then  that  number  must  be  taken  as  the 
remainder  in  the  division,  and  the  quotient  will  be  a 
number  terminating  in  ciphers. 

(192.)  When  there  is  no  remainder  in  the  process  of 
division,  if  the  divisor  and  dividend  be  both  multiplied 
by  the  same  number.,  the  quotient  will  remain  unaltered. 
This  follows  immediately  from  what  has  been  already 
proved  in  Multiplication,  that  when  the  multiplicand  is 
increased  any  number  of  times,  the  multiplier  remaining 
the  same,  the  product  will  be  increased  the  same 


CHAP.  VI.  DIVISION.  143 

number  of  times.  Since  the  dividend  must  be  the 
product  of  the  divisor  and  quotient,  it  follows,  that  if 
we  increase  the  divisor  any  number  of  times,  the  quo- 
tient remaining  the  same,  we  must  increase  the  dividend 
the  same  number  of  times.  It  is  evident,  that  if  the 
divisor  be  contained  in  the  dividend  6  times,  10  times 
the  divisor  will  be  contained  in  10  times  the  dividend 
also  6  times. 

(193.)  We  may  hence  infer  also,  that  if  the  divisor 
and  dividend  be  both  divided  by  the  same  number,  the 
quotient  will  remain  the  same;  since  this  may  be  consi- 
dered as  merely  doing  away  with  a  previous  multipli- 
cation of  both  by  the  same  number.  If  the  divisor  be 
contained  6  times  in  the  dividend,  the  tenth  part  of  the 
divisor  will  be  contained  6  times  in  the  tenth  part  of 
the  dividend. 

(194.)  From  this  principle  we  may  derive  a  method 
of  abridging  the  process  of  division  in  the  case  where 
the  divisor  and  dividend  both  terminate  in  ciphers.  We 
may  in  such  case  cut  off  the  same  numbers  of  ciphers 
from  both,  because  by  so  doing  we  divide  them  both  by 
the  same  number,  such  as  10,  100,  1000,  &c.  according 
to  the  number  of  ciphers  cut  off. 

(195.)  In  general,  if  it  be  apparent  on  inspection 
that  both  divisor  and  dividend  are  divisible  by  any 
number,  the  process  of  division  may  be  abridged  by 
previously  dividing  both  of  them  by  that  number.  Thus, 
if  we  would  divide  72  by  24,  we  may  previously 
divide  each  of  them  by  8,  and  the  question  will  be 
reduced  to  the  division  of  9  by  3,  the  quotient  of  which 
is  3. 

(196.)  When  the  dividend  is  not  an  exact  multiple 
of  the  divisor,  and  there  is  therefore  a  remainder,  we 
may  still  multiply  the  divisor  and  dividend  by  the  same 
number  without  altering  the  quotient.  But  it  will  be 
necessary  in  this  case  to  multiply  the  remainder  by  that 
number.  This  will  be  apparent  from  considering  that 
the  remainder  constitutes  part  of  the  dividend ;  and  thei  e- 
fore  the  multiplication  of  the  dividend  by  the  proposed 


144  A    TREATISE    ON    ARITHMETIC.  BOOK  I. 

number  necessarily  infers  the  multiplication  of  the 
remainder  by  that  number. 

For  example:  if  we  divide  79  by  8,  we  get  the 
quotient  9»  with  a  remainder  7-  Let  us  now  multiply 
the  divisor,  dividend,  and  remainder  by  10,  and  we  shall 
find,  that  by  dividing  790  by  80,  we  shall  still  get  the 
quotient  9^  but  will  have  the  remainder  70;  and  a  similar 
result  would  be  obtained  by  whatever  number  we  might 
multiply  the  divisor  and  dividend. 

(197.)  By  reversing  this  process,  it  follows,  that  we 
may  divide  the  divisor  and  dividend  by  the  same  num- 
ber without  altering  the  quotient ;  but  that  having  done 
so,  the  remainder  which  will  be  obtained  in  the  division 
will  be  as  many  times  less  than  the  remainder  which 
would  be  obtained  had'  the  divisor  and  dividend  re- 
mained unaltered,  as  the  original  divisor  and  dividend  are 
greater  than  those  obtained  by  division. 

In  general,  then,  the  process  of  division  may  le 
abridged,  when  the  divisor  and  dividend  are  both  obvi- 
ously divisible  by  the  same  number  without  a  remainder. 
"We  may  in  that  case  divide  them  by  the  same  number, 
and  operate  on  the  quotients  thus  obtained  by  the  usual 
process  of  division.  It  will  only  be  necessary  to  mul- 
tiply the  remainder  by  that  number  by  which  the 
original  divisor  and  dividend  were  divided. 


145 


BOOK  II. 

FRACTIONS. 

CHAPTER  I. 

ON   THE    LANGUAGE    AND    NOTATION     OF    FRACTIONS,  —  VARIOUS 

WAYS    OF    EXPRESSING   THEM.  THEIR    RELATIVE   VALUES.  — — 

THEIR    ADDITION    AND    SUBTRACTION. 

(198.)  WE  have  seen  that,  except  in  the  particular 
case  in  which  the  dividend  happens  to  be  an  exact  mul- 
tiple of  the  divisor,  it  is  impracticable,  by  means  of  the 
language  and  notation  of  number  hitherto  explained,  to 
complete  the  process  of  division.  When  the  dividend, 
as  generally  happens,  is  not  an  exact  multiple  of  the 
divisor,  the  quotient  which  we  obtain  is  not  the  actual 
quotient  of  the  dividend  divided  by  the  divisor  ;  because 
the  remainder,  being  a  part  of  that  dividend,  has  not  been 
so  divided.  The  quotient  found  in  this  case  is,  in  fact, 
the  true  quotient  which  would  be  obtained  by  using  as 
a  dividend  not  the  actual  dividend,  but  the  number 
which  would  be  obtained  by  subtracting  the  remainder 
from  the  proposed  dividend.  To  complete  the  process, 
and  obtain  the  true  quotient,  it  would  be  necessary  to 
divide  the  remainder  by  the  divisor :  but  the  remainder 
being  less  than  the  divisor,  the  ordinary  process  of 
division,  so  far  as  we  have  yet  explained  that  opera- 
tion, becomes  inapplicable. 

It  will  be  recollected  that  the  process  of  division  was 
presented  under  two  distinct  points  of  view :  first,  as 
that  operation  by  which  it  is  discovered  how  many  times 
the  divisor  is  contained  in  the  dividend  ;  and  secondly, 
as  the  process  by  which  the  dividend  is  resolved  into 
as  many  equal  parts  as  there  are  units  in  the  divisor 

L 


146  A    TREATISE    ON    ARITHMETIC.  BOUH  II. 

Now,  according  to  the  first  view  of  this  operation,  it  is 
clearly  impracticable  when  the  divisor  is  greater  than 
the  dividend ;  for  in  that  case  the  divisor  is  not  con- 
tained in  the  dividend  at  all.  Under  the  second  aspect, 
however,  we  arrive  at  a  somewhat  different  conclusion. 
When  the  dividend  is  less  than  the  divisor,  if  we  were 
to  attempt  to  divide  the  former  into  as  many  equal  parts 
as  there  are  units  in  the  divisor,  it  is  evident  that  the 
number  of  those  parts  would  be  greater  than  the  num- 
ber of  units  which  the  dividend  itself  contains,  and 
consequently  each  of  the  parts  must  be  less  than  a  unit. 
So  far  as  the  language  and  notation  of  number  hitherto 
explained  go,  they  are  incapable  of  expressing  any 
thing  less  than  the  unit;  but  there  is  evidently  no 
absurdity  or  difficulty  in  conceiving  quantities  less  in 
any  degree  than  the  unit,  and  therefore  the  absence  of 
proper  means  of  expressing  such  quantities  is  a  de- 
fect in  the  nomenclature  of  number  which  must  be 
removed. 

If,  instead  of  using  them  in  the  abstract,  we  apply 
numbers  to  express  any  particular  species  of  quantity, 
we  shall  have  no  difficulty  in  perceiving  the  necessity  of 
providing  means  for  expressing  numbers  less  than  the 
unit.  Let  us  suppose  that  the  dividend  in  any  question 
of  Division  expresses  a  certain  number  of  inches,  which 
are  to  be  divided  into  as  many  equal  parts  as  there  are 
units  in  the  divisor.  Now,  suppose  it  so  to  happen 
that,  the  divisor  being  17, there  is  a  remainder  10  inches: 
to  complete  the  division  it  would  be  necessary  that  these 
10  inches  should  be  divided  by  17;  that  is,  that  they 
should  be  divided  into  17  equal  parts.  Now,  whatever 
difficulty  we  may  have  in  conceiving  the  division  of  10 
abstract  units  into  17  equal  parts,  we  certainly  can  have 
none  whatever  in  conceiving  the  division  of  a  line  10 
inches  in  length  into  17  equal  parts.  It  will  be  evi- 
dent that  each  of  these  parts  is  less  than  an  inch,  the 
ivhole  17  of  them  making  up  10  inches.  We  shall  now 
proceed  to  explain  the  system  of  language  and  notation 
by  which  quantities  which  are  either  less  than  the  unit. 


CHAP,  I.  FRACTIONS.  14? 

or  which  are  not  an  exact  multiple  of  unity,  may  be  ex- 
pressed hy  numbers. 

(199.)  Such  numbers  are  called  FRACTIONS;  while 
the  name  INTEGERS,  or  WHOLE  NUMBERS,  is  applied  to 
unity  and  its  multiples. 

(200.)  Recollecting  that  the  unit  of  number  in  its 
practical  application  always  expresses  some  quantity 
which  admits  of  subdivision,  let  us  conceive  the  unit 
divided  into  any  proposed  number  of  equal  parts,  as  10 : 
each  of  these  parts  is  called  a  tenth,  meaning  a  tenth 
part  of  the  unit.  In  like  manner,  if  the  unit  be  sup 
posed  to  be  divided  into  8  equal  parts,  each  part  is 
called  an  eighth ;  if  into  9  equal  parts,  a  ninth;  and 
so  on. 

The  unit  being  always  a  multiple  of  such  parts,  these 
parts  are  called  sub-multiples  of  the  unit. 

(201.)  We  have  frequently  occasion  to  express  two 
or  more  of  those  parts  into  which  the  unit  is  supposed 
to  be  divided.  In  such  cases  we  first  name  the  number 
of  parts  to  be  expressed,  and  next  the  number  of  parts 
into  which  the  unit  is  divided.  Thus,  if  we  suppose 
the  unit  divided  into  10  equal  parts,  and  that  we  wish 
to  express  7  of  these  parts,  we  call  them  seven  tenths : 
if  the  unit  be  supposed  to  be  divided  into  9  equal  part? 
and  that  we  wish  to  express  5  of  them,  we  call  them 
five  ninths;  and  so  on. 

(202.)  In  the  case  of  the  remainders  in  incom- 
plete divisions,  a  number  greater  than  unity  is  generally 
required  to  be  divided  into  a  greater  number  of  parts 
than  the  units  it  contains.  Thus,  if  5  be  the  remainder 
and  9  the  divisor,  to  complete  the  division  it  would  be 
necessary  to  divide  5  into  9  equal  parts.  To  effect  this, 
let  us  imagine  each  of  the  units  which  compose  the 
number  5  to  be  divided  into  9  equal  parts.  It  is  evi- 
dent that  by  taking  the  ninth  part  of  each  unit,  and  ad- 
ding such  parts  together,  we  shall  obtain  the  ninth  part  of 
the  whole  :  we  shall  thus  have  five  ninth  parts  of  a  single 
unit ;  from  whence  it  appears  that  five  ninths  of  a  single 
unit  is  the  same  as  the  ninth  part  of  five  units.  In  the 
L  2 


148  A    TREATISE    ON    ARITHMETIC.  BOOK  IT. 

same  manner,  if  it  be  required  to  divide  10  into  13 
equal  parts,  we  should  imagine  each  of  the  units  which 
compose  1 0  to  be  divided  into  1  3  equal  parts,  and  one 
of  such  parts  taken  from  each  unit  to  make  up  the  thir- 
teenth part  of  the  whole:  the  thirteenth  part,  there- 
fore, of  10  is  the  same  as  ten  thirteenth  parts  of  the  unit. 

(203.)  From  what  has  been  here  explained,  it  will 
be  perceived  that  in  order  to  express  a  fraction  two 
numbers  are  necessary, — one,  which  expresses  the  num- 
ber of  equal  parts  into  which  the  unit  is  supposed  to  be 
divided ;  and  the  other,  the  number  of  those  parts  in. 
tended  to  be  taken  in  the  fraction.  That  which  ex- 
presses the  number  of  parts  into  which  the  unit  is 
supposed  to  be  divided  is  called  the  DENOMINATOR  of  the 
fraction  ;  and  that  which  expresses  the  number  of  those 
parts  which  compose  the  fraction  is  called  its  NUME- 
RATOR. Thus,  if  the  fraction  be  five  ninths,  the  unit  is 
supposed  to  be  divided  into  nine  equal  parts,  five  of 
which  compose  the  fraction :  therefore  the  denominator 
is  in  this  case  nine,  and  the  numerator  five.  In  the 
same  manner  if  the  fraction  be  ten  thirteenths,  the  unit 
is  supposed  to  be  divided  into  thirteen  equal  parts,  and 
ten  of  these  compose  the  fraction  :  the  denominator  is, 
therefore,  thirteen,  and  the  numerator  ten. 

Such  is  the  nomenclature  by  which  numbers,  which 
are  either  less  than  the  unit,  or  which  are  not  an  exact 
multiple  of  that  unit,  are  expressed.  We  shall  now  ex- 
plain the  notation  by  which  the  same  are  expressed  by 
the  aid  of  figures. 

(204.)  The  figures  expressing  the  numerator  are 
usually  placed  above  a  line,  and  those  expressing  the 
denominator  below  it.  Thus  the  fractions  five  ninths 
and  ten  thirteenths  are  expressed  in  figures  in  the  fol- 
lowing manner, — 

4  «• 

(205.)  It  is  not  necessary  that  the  language  and  no. 
tation  of  fractions,  here  explained,  should  be  confined 
to  numbers  less  than  the  unit.  On  the  contrary,  in 
practical  calculations  it  is  often  convenient,  if  not  ne- 


CHAP.  I.  FRACTIONS.  149 

cessary,  to  express  fractions  not  only  greater  than  the 
unit,  but  greater  than  numbers  of  more  considerable 
magnitude.  There  is  no  difficulty  in  conceiving  the  ap- 
plication of  the  nomenclature  just  explained  to  such 
numbers.  Having  supposed  the  unit  divided  into  any 
number  of  parts,  we  can  easily  conceive  those  parts  re- 
peated much  oftener  than  the  number  of  times  that  they 
are  contained  in  the  unit.  Thus,  if  an  inch  be  divided 
into  10  equal  parts,  it  is  as  easy  for  us  to  conceive  11 
of  these  parts  as  9  of  them  ;  and  we  can  go  on  increas- 
ing their  number  to  any  extent  whatever :  thus  35  or 
99  tenths  of  an  inch  is  just  as  intelligible  as  35  or  99 
inches.  It  is  true  that  such  numbers  may  be  expressed 
otherwise  and  more  briefly  ;  for  since  every  10  tenths 
make  an  inch,  35  tenths  will  be  3  inches  and  5  tenths, 
and  99  tenths  will  be  9  inches  and  9  tenths.  Never- 
theless it  is  often  more  convenient  to  express  them  in 
the  purely  fractional  form. 

(206.)  Fractions  greater  than  the  unit  are  frequently 
distinguished  from  those  less  than  the  unit  by  the 
name  improper  fractions,  —  a  proper  fraction  being  one 
which  is  less  than  the  unit,  and  all  others  being  called 
improper. 

(207.)  From  what  has  been  explained  above,  it  will 
appear  that  a  fraction  may  be  considered  under  two  points 
of  view  ;  first,  either  as  a  certain  number  of  sub-multi- 
ples of  the  unit ;  or,  secondly,  as  a  quotient  whose  divi- 
dend is  the  numerator,  and  divisor  the  denominator. 

(208.)  It  follows  also  that,  by  the  aid  of  the  notation 
of  fractions,  we  can  at  once  render  complete  all  those 
processes  of  division  explained  in  the  last  chapter  in 
which  there  is  a  remainder.  It  is  only  necessary  to  add 
to  the  integral  part  of  the  quotient  a  fraction  whose  nu- 
merator is  the  remainder,  and  whose  denominator  is  the 
divisor,  such  a  fraction  expressing  the  quotient  arising 
from  the  division  of  the  remainder  by  the  divisor. 
Thus,  if  we  would  divide  11  by  3,  we  shall  find  the  in- 
tegral part  of  the  quotient  to  be  3,  with  a  remainder  2  ; 
this  remainder  being  divided  by  3,  gives  the  quotient 
L  3 


150  A    TREATISE    ON    ARITHMETIC.  BOOK  II, 

-J,  and  therefore  the  total  quotient  is  3-|,  which  means  3 
and  2  thirds. 

(209.)  One  integer,  or  whole  number,  may  he  con- 
sidered as  a  fraction  of  another,  which,  in  that  case,  re- 
latively to  it  is  taken  as  the  unit.  Thus  we  say,  that  3 
is  the  fourth  part  of  12,  that  9  is  3  fourths  of  12,  that 
4  is  the  fifth  of  20,  that  3  is  3  fifths  of  5,  and  so  on. 

(210.)  The  numerator  of  a  fraction  is  always  the 
same  fraction  of  its  denominator  as  the  fraction  itself  is 
of  the  unit.  This  is  evident  from  the  explanations  al- 
ready given.  In  the  fraction  J  the  numerator  is  3 
fourths  of  the  denominator ;  and  the  fraction  itself  is 
3  fourths  of  the  unit.  In  the  same  manner  in  the  frac- 
tion J,  the  numerator  is  5  sixths  of  the  denominator, 
and  the  fraction  itself  5  sixths  of  the  unit.  Since  there 
is  an  unlimited  variety  of  whole  numbers,  which  are 
the  same  fractions  of  other  whole  numbers,  it  follows 
that  the  same  fractional  numbers  may  be  expressed  in 
an  unlimited  variety  of  ways.  Thus,  for  example,  since 
3,  6,  and  9  are  respectively  3  fourths  of  4,  8,  and  12, 
the  fractions  -J-  -|  -^  are  equal ;  because  the  numer- 
ator of  these  being  3  fourths  of  its  denominator,  the  frac- 
tions must  each  be  3  fourths  of  the  unit.  Let  us  con- 
sider how  this  may  be  reconciled  with  the  nature  of 
fractions  above  explained.  In  the  fraction  -|  the  nu- 
merator and  denominator  are  respectively  double  the 
numerator  and  denominator  of  J :  when  we  double  the 
denominator  of  f ,  we  double  the  number  of  parts  into 
which  the  unit  is  divided,  and  each  part  will  therefore 
have  only  half  its  former  magnitude.  To  make  up  the 
same  amount  it  would  be  necessary,  therefore,  to  take 
twice  the  former  number  of  parts ;  but  that  is  done  by 
doubling  the  numerator.  If  we  increased  the  denomi- 
nator alone  in  a  twofold  proportion,  we  should  diminish 
the  magnitude  of  the  parts  composing  the  fraction  in 
that  proportion.  If  we  increased  the  numerator  alone  in 
a  twofold  proportion,  we  should  double  the  number  of 
parts  composing  the  fraction  without  changing  the 
magnitude  of  those  parts.  By  the  one  process  we  should, 


CHAP.  I.  FRACTIONS.  151 

reduce  it  to  half  its  former  magnitude;  and  by  the 
other,  we  should  increase  it  to  double  its  former  mag- 
nitude. When  both  these  operations,  however,  are  per- 
formed at  once,  the  two  effects  neutralise  each  other,  and 
the  magnitude  of  the  fraction  remains  unaltered. 

Let  us  apply  this  reasoning  to  an  example.  —  Take  a 
line  a  foot  long:  if  we  divide  it  into  4  equal  parts, 
and  take  3  of  these,  we  shall  have  3  fourths  of  a  foot  ; 
but  if  we  divide  the  same  line  into  8  equal  parts,  each 
of  these  parts  will  be  only  half  the  magnitude  of  the 
former  ones :  3  of  them,  therefore,  would  be  half  the 
length  of  3  fourths  of  a  foot ;  and  6  of  them  would, 
consequently,  be  equal  to  3  fourths  of  a  foot.  Thus, 
-J  of  a  foot  and  -|  of  a  foot  are  the  same  length.  In  the 
same  manner  it  may  be  shown,  that  -f^  of  a  foot  is  also 
the  same  length. 

(211.)  Since  the  numerator  and  denominator  of  a 
fraction  may  be  multiplied  by  the  same  number  with- 
out changing  the  value  of  the  fraction,  they  may  also  be 
divided  by  the  same  number  without  changing  its 
value ;  for  this  is  only  undoing  the  previous  multiple 
cation.  Thus,  -|  is  reduced  to  -§-  by  dividing  its  ira- 
merator  and  denominator  by  3,  and  these  fractions  have 
evidently  the  same  value. 

From  all  that  has  been  just  stated,  it  appears  that  the 
value  of  a  fraction  does  not  depend  on  the  absolute  mag- 
nitudes of  the  numbers  forming  its  numerator  and  de- 
nominator, since  these  magnitudes  may  be  subject  to 
unlimited  variation;  the  value  of  the  fraction  remaining 
unchanged.  The  value  of  the  fraction,  however,  de- 
pends on  the  relative  or  proportional  magnitudes  of  its 
numerator  and  denominator.  So  long  as  the  numerator 
and  denominator  retain  the  same  relative  or  proportional 
values,  so  long  will  the  fraction  remain  unaltered  in  its 
magnitude*  The  following  fractions,  differing  very 
much  in  the  magnitude  of  their  numerators  and  deno- 
minators, are,  nevertheless,  themselves  of  the  same  mag- 
nitude :  — 

t   t»   if   it    n   u   « 

L   4 


152  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

In  the  same  manner,  the  following  are  of  equal  value :  — 

4   -14   M   i*   if   is-   n 

In  this  series  each  succeeding  form  of  the  fraction  is 
found  by  multiplying  the  numerator  and  denominator  of 
the  first  form  by  the  same  number ;  and,  on  the  other 
hand,  the  first  form  in  each  series  may  be  found  from 
any  of  the  others,  by  dividing  both  numerator  and  de- 
nominator by  the  same  number. 

(212.)  The  numerator  and  denominator  of  a  fraction 
are  called  its  terms. 

(213.)  From  what  has  been  stated,  it  appears  that 
the  terms  of  a  fraction  may  be  increased  without  limit ; 
because  there  is  no  limit  to  the  magnitude  of  the  num- 
ber by  which  we  may  multiply  them.  But  since  they 
may  not  be  both  capable  of  being  divided  by  the  same 
number,  or,  if  so,  that  number  may  be  limited  in  mag- 
nitude, there  is  a  limit  to  the  extent  to  which  its  terms 
may  be  diminished. — Take  the  example  -£-J.  We  may 
multiply  both  terms  of  this  fraction  by  any  number, 
however  great,  and  we  shall  get  an  equivalent  fraction 
whose  terms  are  proportionally  great;  but  the  same 
terms  cannot  be  divided  by  any  number  greater  than  5, 
and,  consequently,  the  terms  of  the  fraction  cannot  be 
reduced  lower  than  J.  When  the  terms  of  a  fraction 
are  thus  divided  by  the  greatest  number  which  exactly 
divides  them  both,  the  fraction  is  therefore  reduced  to 
its  least  or  lowest  terms. 

Owing  to  the  convenience  of  using  small  numbers, 
fractions  are  generally  expressed  in  their  lowest  terms, 
unless  in  particular  cases,  which  will  be  explained, 
hereafter.  Thus,  we  do  not  commonly  use  the  fraction 
Kfj-f},  but  the  equivalent  fraction  -f,  which  is  in  its 
lowest  terms. 

(214.)  From  what  has  been  already  explained,  it  ap- 
pears that,  provided  the  denominator  of  a  fraction  remain 
unaltered,  the  fraction  will  be  increased  or  diminished 
in  whatever  proportion  its  numerator  is  increased  or 
diminished  (2  J  0).  Hence  a  fraction  may  always  be  mul- 


CHAP  I.  FRACTIONS.  153 

tiplied  or  divided  by  A,  whole  number,  by  multiplying  or 
dividing  its  numerator  by  that  number:  thus,  if  we  would 
divide  ^  by  2,  we  must  divide  its  numerator  by  2,  pre- 
serving the  same  denominator,  and  the  quotient  is  -J. 
If  we  would,  on  the  other  hand,  multiply  the  fraction 
by  2,  we  must  multiply  its  numerator  by  2,  and  the 
product  is  ^.  It  is  evident  that  half  of  4  ninths  is  2 
ninths,  and  twice  4  ninths  is  8  ninths. 

(215.)  Since  it  has  been  proved  that  in  whatever 
proportion  the  denominator  is  increased  the  fraction  is 
diminished,  and  in  whatever  proportion  the  denomi- 
nator is  diminished  the  fraction  is  increased,  it  follows 
that  by  multiplying  the  denominator  by  any  number 
we  divide  the  fraction  by  the  same  number,  and  by 
dividing  the  denominator  by  any  number  we  multiply 
the  fraction  by  that  number.  —  Let  the  fraction,  for 
example,  be  -J.  If  we  multiply  the  denominator  by  2, 
it  becomes  f^.  Now,  since  a  sixteenth  part  of  the 
unit  is  half  of  an  eighth  part,  3  eighths  must  be  double 
3  sixteenths ;  that  is,  T%  is  half  of  -|.  Thus,  by  dou- 
bling the  denominator  of  -|  we  actually  divide  the 
fraction  by  2.  Let  us  next  suppose  that  we  divide  the 
denominator  of  -|  by  2,  and  obtain  -J-.  We  have  here 
really  multiplied  the  fraction  by  2 ;  for  since  a  fourth 
of  the  unit  is  twice  the  magnitude  of  an  eighth,  3 
fourths  will  be  twice  the  magnitude  of  3  eighths ;  and, 
therefore,  |  is  double  -|. 

When  fractions  occur  in  arithmetical  calculations  in 
terms  which  are  not  their  lowest,  it  is  often  required  to 
reduce  them  to  their  lowest  terms.  To  accomplish  this, 
it  is  necessary  to  find  the  greatest  number  which  divides 
exactly  both  numerator  and  denominator. 

(216.)  A  number  which  divides  exactly  2  or  more 
other  numbers  without  a  remainder  is  called  a  common 
measure  of  these  numbers ;  and  the  greatest  number 
which  divides  2  or  more  others  without  a  remainder  is 
called  the  greatest  common  measure  of  these  others. 
Thus,  2  is  a  common  measure  of  8  and  12,  because  it 
divides  both  without  any  remainder  ;  but  it  is  not  the 


154  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

greatest  common  measure  of  8  and  12,  because  4  also 
divides  8  and  12  without  any  remainders.  No  number 
greater  than  4  divides  8  and  12  ;  and,  therefore,  4  is 
their  greatest  common  measure.  If  it  be  required  to 
reduce  the  fraction  -fz2  to  its  lowest  terms,  we  must 
divide  its  numerator  and  denominator  by  their  greatest 
common  measure  4  ;  after  which  the  fraction  will  be- 
come |-,  which  is  evidently  in  its  lowest  terms,  since  no 
other  whole  number  divides  2  and  3. 

(2 1 7.)  It  has  been  already  explained,  that  a  prime 
number  is  one  which  has  no  divisor  save  the  unit.  In 
like  manner,  2  numbers  are  said  to  be  relatively  prime, 
or  prime  to  each  other,  when  they  have  no  common 
measure  greater  than  the  unit.  Thus,  7  and  9  are 
prime  to  each  other,  although  the  latter  is  not  a  prime 
number,  being  measured  by  3. 

Two  numbers  may  be  prime  to  each  other  although 
neither  by  itself  is  prime.  Thus,  8  and  9  are  relatively 
prime,  although  neither  of  them  is  a  prime  number. 

If  either  of  2  numbers  be  absolutely  a  prime  number, 
they  must  be  prime  to  each  other ;  for  since  that  which 
is  absolutely  prime  is  not  measured  by  any  other  num- 
ber greater  than  the  unit,  the  2  numbers  can  have  no 
common  measure. 

(218.)  From  these  circumstances  we  may  infer,  that 
whenever  either  of  the  terms  of  the  fraction  is  a  prime 
number,  the  fraction  is  in  its  least  terms  ;  and  a  frac- 
tion is  always  in  its  least  terms  when  its  terms  are  rela- 
tively prime. 

(219.)  Two  even  numbers  cannot  be  relatively  prime, 
since  2  is  always  a  common  measure  of  them :  hence 
a  fraction,  having  both  its  terms  even,  can  be  reduced 
to  lower  terms  by  dividing  the  numerator  and  deno- 
minator by  2. 

(220.)  If  a  number  ends  in  either  5  or  0,  it  is 
always  divisible  by  5  without  a  remainder :  hence,  if 
both  terms  of  a  fraction  end  in  5  or  0,  or  one  in  5  and 
the  other  in  0,  the  fraction  may  be  reduced  to  lower 
terms  by  dividing  both  numerator  and  denominator  by  5. 


CHAP.  I.  FRACTIONS.  155 

(221.)  If  both  terms  of  a  fraction  terminate  in 
ciphers,  the  same  number  of  those  ciphers  may  be 
struck  off  from  both  ;  for,  by  so  doing,  both  terms  of  the 
fraction  are  divided  by  10,  100,  1000,  &c.  according 
to  the  number  of  ciphers  struck  off. 

(222.)  If  the  digits  of  a  number  added  together 
give  a  sum  which  is  an  exact  multiple  of  9,  the  number 
itself  will  then  be  an  exact  multiple  of  9  (l6'l  note).  If 
this  happen  with  both  terms  of  a  fraction,  we  may  infer 
that  they  are  both  divisible  by  9>  and  in  this  way  the  frac- 
tion may  be  reduced  to  lower  terms.  Thus,  in  the 
fraction  |  -||4i,  we  find  that  the  digits  of  the  numerator 
added  together  make  27,  which  is  a  multiple  of  9; 
and  the  digits  of  the  denominator  added  together  make 
1  8,  which  is  also  a  multiple  of  9-  We  therefore  infer 
that  the  numerator  and  denominator  may  both  be 
divided  by  9  without  a  remainder  :  performing  this 
division,  we  find  the  quotient  to  be  532  and  759«  The 
proposed  fraction  is,  therefore,  equivalent  to  -£~f  -|  . 

(223.)  This,  and  such  methods  of  reduction,  will 
serve,  however,  only  in  particular  cases.  To  obtain  a 
general  method  of  reducing  a  fraction  to  its  lowest  terms, 
it  is  necessary  that  we  should  be  able  to  find  the  greatest 
common  measure  of  its  numerator  and  denominator. 

Let  the  numerator  and  denominator  be  376  and  788. 
To  find  the  greatest  common  measure  of  these,  we  shall 
proceed  in  the  following  manner  :  — 

B        A 

376  )  788  (  2 
C    -752 

D  -  36)  376  (  10 
E       -     360 

F     -       16  )  36  (  2 
G     -     -        32 

H     -     -     "1)  16  (4 


In  explaining  the  process  here  written,  we  shall,  for 
brevity,  express  the  different  numbers  by  the  letters  pre- 


156  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

fixed  to  them.  Of  the  2  numbers  A  and  B,  whose  com- 
mon measure  is  to  be  found,  we  divide  the  greater  A  by 
the  lesser  B ;  and  we  find  the  quotient  2,  with  the  remain- 
der D.  We  then  take  B  for  the  dividend,  and  D  for 
divisor,  and  make  another  division,  where  we  find  the 
quotient  10,  and  the  remainder  F.  Again,  taking  D  for 
the  dividend,  and  F  for  the  divisor,  we  make  another 
division,  in  which  we  get  the  quotient  2,  and  the  re- 
mainder H.  Taking  H  in  like  manner  as  divisor,  and 
F  as  dividend,  we  get  the  quotient  4,  without  a  re- 
mainder: the  last  remainder,  H,  is  the  greatest  com- 
mon measure  of  the  proposed  numbers  A  and  B. 

We  may  prove  that  the  number  H  is  a  common 
measure  of  A  and  B  in  the  following  manner :  — 
By  the  last  division,  it  appears  that  H  measures  F 
four  times,  and  by  the  preceding  division  it  appears  that 
G  is  twice  F.  Since  H  measures  F  four  times,  it  will 
therefore  measure  twice  F  (that  is  G)  eight  times.  Now, 
D  is  obtained  by  adding  G  and  H  together  ;  and  since  H 
measures  G  eight  times,  it  measures  G  -f-  H  nine 
times;  that  is,  it  measures  D  nine  times.  By  the 
second  division  it  appears  that  E  is  ten  times  D ;  and 
since  H  measures  D  nine  times,  it  measures  E  ninety 
times  ;  but  it  measures  F  four  times,  and,  consequently, 
it  measures  E  -f  F  ninety-four  times.  Now,  B  is 
obtained  by  adding  E  to  F ;  and,  consequently,  H 
measures  B  ninety-four  times.  Now,  since  H  measures 
B  ninety-four  times,  it  must  measure  twice  B  (that  is  C) 
twice  ninety-four  times,  that  is,  188  times;  but  it 
measures  D  nine  times :  therefore  it  measures  C  -f  D, 
that  is,  A,  197  times.  It  follows,  then,  that  4  measures 
B  94  times,  and  A  197  times ;  and,  consequently, 
is  a  common  measure  of  A  and  B. 

"We  shall  now  show  that  H  is  the  greatest  common 
measure  of  A  and  B,  by  proving  that  if  A  and  B  have 
any  other  common  measure,  that  common  measure  must 
measure  H,  and  must  be  therefore  less  than  it. 

If  such  a  common  measure  exist,  since  it  measures 
B  it  measures  C9  which  is  twice  B ;  and  since  it  mea- 


CHAP.  I.  FRACTIONS.  1 57 

sures  A  and  C,  it  must  measure  their  difference  D. 
This  will  be  easily  understood  by  an  example. — Thus, 
if  it  measured  A  ten  times  and  C  eight  times,  it  would 
measure  D  twice.  Since,  then,  it  measures  D  it  must 
measure  E,  which  is  ten  times  D ;  and  since  it  mea- 
sures B  and  E,  it  must,  as  before,  measure  their  difference 

F.  In  like  manner,   since  it  measures  F  it  must  mea- 
sure twice  F,  which  is  G ;  and  since  it  measures  D  and 

G,  it  measures  their  difference  H,  and  is  therefore  less 
than  H.    Hence,  if  A  and  B  have  any  other  common 
measure  besides  H,  that  common  measure  must  mea- 
sure H,  and  therefore  be  less  than  it.     By  generalising 
the  above  process,  we  shall  find  a  rule  for  obtaining  the 
greatest  common  measure  of  two  numbers. 


RULE. 

Divide  the  greater  by  the  less,  and  divide  the  less  by  the 
remainder.  If  there  be  any  remainder,  divide  this  first 
remainder  by  the  next  remainder;  and  so  continue 
dividing  each  remainder  by  the  following  remainder, 
until  a  remainder  is  found  which  exactly  divides  the  pre- 
ceding remainder.  Such  a  remainder  is  the  greatest 
common  measure  of  the  two  numbers. 

If  the  last  remainder  obtained  by  this  process  happen 
to  be  1,  then  1  is  the  greatest  common  measure  of  the 
two  proposed  numbers,  and  they  are  therefore  relatively 
prime,  and  the  fraction  whose  numerator  and  denomi- 
nator they  are  is  in  its  least  terms. 

(224.)  Since  a  fraction  may  always  be  regarded  as  a 
mode  of  expressing  the  quotient  of  the  numerator  di- 
vided by  the  denominator,  it  follows  that  an  improper 
fraction  may  always  be  reduced  to  a  number  partly  in- 
tegral and  partly  fractional,  called  a  mixed  number ,  by 
performing  the  process  of  division  by  the  rule  given  in 
the  last  chapter.  In  that  case  the  integral  part  of  the 
number  will  be  the  quotient  of  the  incomplete  division, 
and  the  fractional  part  will  be  a  proper  fraction,  whose 


158  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

numerator  is  the  remainder,  and  whose  denominator  is 
the  divisor.  This  will  be  easily  understood  from  an 
example. — Let  the  improper  fraction  be  -J^-:  the  quotient 
in  the  process  of  division  would  be  3,  and  the  remainder 
4 ;  but,  to  make  the  division  complete,  it  would  be 
necessary  to  divide  this  remainder  4  by  31,  and  add 
the  quotient  to  the  3  already  found ;  but  the  quotient 
of  4  by  3 1  is  expressed  -fa,  and  therefore  the  complete 
quotient  of  the  division  is  3-fa. 

(225.)  An  improper  fraction  may  happen  to  be  equi- 
valent to  an  integer.  This  will  occur  whenever  the 
numerator  is  a  multiple  of  the  denominator.  Thus,  the 
fraction  *-£  is  equivalent  to  the  integer  9- 

(226.)  Hence  it  is  apparent,  that  an  integer  may 
always  be  converted  into  an  improper  fraction  having 
any  proposed  denominator,  by  multiplying  the  integer 
by  the  proposed  denominator,  and  subscribing  the  same 
denominator  below  it.  Thus,  9  may  be  converted  into 
an  improper  fraction  whose  denominator  is  4,  by  mul- 
tiplying it  by  4,  and  writing  4  under  it,  3^6. 

(227.)  The  number  which  stands  between  proper 
and  improper  fractions  is  a  fraction  whose  numerator  and 
denominator  are  the  same  number,  and  such  a  fraction 
is  obviously  equal  to  the  unit.  If  we  regard  it  as  a 
fraction,  we  consider  the  unit  to  be  divided  into  a 
certain  number  of  parts,  and  the  fraction  to  consist  of 
the  same  number  of  those  parts.  If  we  regard  it  as  a 
quotient,  the  divisor  is  contained  in  the  dividend  once 
without  a  remainder. 

It  is  sometimes  necessary  to  consider  the  unit  under 
such  a  point  of  view,  as  we  shall  see  hereafter. 

(228.)  Since  the  relative  magnitudes  of  fractions 
depend  upon  two  numbers,  while  those  of  integers  de- 
pend on  only  one,  the  latter  is  much  more  evident  on 
inspection  than  the  former.  It  is  not  always  easy  at 
first  view  to  pronounce  which  of  two  fractions  is  the 
greater,  especially  if  their  numerators  and  denominators 
are  high  numbers.  It  is,  therefore,  of  considerable  im- 
portance to  possess  the  means  of  at  once  determining 


CHAP.  I.  FRACTIONS.  159 

the  relative  amount  or  value  of  any  two  fractions  which 
can  be  proposed. 

The  difficulty  which  attends  the  perception  of  the 
comparative  value  of  fractions  is  one  which  attends  all 
quantities  which  are  expressed  by  numbers  having  dif- 
ferent units.  Let  us  suppose  that  two  sums  of  money 
are  expressed,  one  in  pounds  and  the  other  in  shillings; 
and  that  we  are  either  required  to  declare  which  is  the 
greater,  or  to  express  their  proper tional  value,  or  to  add  or 
subtract  them,  so  as  to  express  their  sum  or  difference  by 
a  single  number.  We  should  find  it  impossible  to  do 
so,  unless  the  numbers  in  question  were  first  submitted 
to  some  change,  so  as  to  render  them  directly  com- 
parable, or  capable  of  addition  or  subtraction.  Let  the 
two  numbers  proposed  be  479  shillings  and  23  pounds. 
We  are  required  to  declare  the  proportional  value  of 
these  sums,  or  to  add  them  together  so  as  to  express 
their  amount  by  a  single  number.  It  is  evident  that 
their  addition  would  not  be  effected  by  adding  23  to 
479  j  for  in  that  case  the  sum,  which  would  be  502, 
would  neither  express  shillings  nor  pounds,  some  of  its 
units  being  of  one  kind  and  some  of  another.  In 
order  to  enable  us  to  compare  the  values  of  the  two 
sums,  or  to  add  or  subtract  them  so  as  to  express 
their  sum  or  difference  by  a  single  number,  we  must,  in 
the  first  instance,  effect  such  a  change  in  one  or  both  of 
them  that  their  component  units  shall  be  the  same.  This 
may  be  done  by  discovering  the  number  of  shillings 
which  are  contained  in  23  pounds.  Since  there  are 
20  shillings  in  a  pound,  there  are  23  times  that  number 
in  23  pounds :  hence  23  pounds  are  equivalent  to  460 
shillings.  We  can  now  express  both  sums  in  shillings, 
the  one  being  460  and  the  other  479-  It  is  manifest 
that  their  comparative  values  are  expressed  by  these 
numbers ;  also,  that  by  adding  these  numbers  we  may 
get  a  single  number,  9^9^  which  shall  express  the 
number  of  shillings  in  their  sum,  and  by  subtracting 
the  lesser  from  the  greater  we  shall  get  a  number,  19, 


1(30  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

which  will  express  the  number    of   shillings  in  their 
difference. 

From  this  example  it  will  he  apparent  that,  in  cases 
where  two  numbers  composed  of  different  units  are 
required  to  be  compared,  added  or  subtracted,  they 
must  be  previously,  by  some  means,  converted  into  other 
equivalent  numbers  which  are  composed  of  the  same 
units. 

(229.)  The  operation  by  which  a  number  composed 
of  units  of  one  kind  is  converted  into  another  equivalent 
number  composed  of  units  of  another  kind  is  called,  in 
arithmetic,  Reduction.  Such,  for  example,  is  the  oper- 
ation by  which  a  number  whose  units  are  pounds  is 
converted  into  an  equivalent  number  whose  units  are 
shillings. 

We  have  already  stated,  that  a  fraction  may  be  re- 
garded as  a  whole  number  if  the  parts  of  which  it 
consists  be  considered  as  units.  Thus,  f  may  be  ex- 
pressed by  the  whole  number  3,  if  it  be  at  the  same 
time  declared  that  its  component  units  are  fourths. 
Considered  in  this  point  of  view,  fractions  having  the 
same  denominator  are  numbers  composed  of  the  same 
units,  and  may  be  expressed  by  their  numerators  taken 
as  whole  numbers,  if  the  value  of  their  component 
units  be  at  the  same  time  declared.  Thus,  for  ex- 
ample, the  fractions  T^,  -fa,  -£%,  -f-%  may  be  expressed 
as  follows :  — 

twelfths  twelfths  twelfths         twelfths 

9753 
The  relative  magnitudes  of  these  fractions  will  obvi- 
ously be  expressed  by  their  numerators :  their  total 
amount  would  be  obtained  by  adding  those  numerators, 
still  bearing  in  mind  that  the  units  they  express  are 
twelfths ;  and  any  one  of  them  may  be  subtracted  from 
another  in  the  same  manner. 

(230.)  But  if,  instead  of  writing  over  the  numerator 
the  nature  of  the  unit  of  which  the  number  is  composed, 
we  subscribe  in  the  usual  way  the  denominator,  then  the 
same  operations  may  be  still  performed  in  the  same 


CHAP.  I.  FRACTIONS.  l6l 

manner,  the  denominator  being  written  under  the 
result,  to  signify  the  magnitude  of  the  units  of 
which  the  number  is  composed.  From  these  consi- 
derations we  shall  deduce  the  following  general  con- 
clusions :  — 

1.  Fractions  having  the  same  denominator  have  the 
same  relative  magnitudes  as  their  numerators. 

2.  Fractions  having  the  same  denominator  may  be 
added  by  adding  their  numerators,   still  preserving  the 
same  denominator  ;  thus,  -f-g-  +  iV  —  15* 

3.  Fractions  having  the  same  denominator  may  be 
subtracted  by  subtracting  their  numerators,  still  preserv- 
ing the  same  denominators;  thus,  -f%— iS  =  -T?' 

(231.)  Since,  then,  the  addition  and  subtraction  of 
fractions  can  only  be  performed,  and  their  relative  values 
expressed,  when  they  have  the  same  denominator,  it  is 
necessary,  in  order  to  be  enabled  to  perform  these  oper- 
ations, and  to  express  the  relative  magnitude  of  fractions, 
that  we  should  possess  some  means  of  converting  any 
proposed  fractions  having  different  denominators  into 
other  equivalent  fractions  having  the  same  denominator. 

We  have  already  shown  that  the  same  fraction  may 
be  expressed  in  a  great  variety  of  forms,  either  by  mul- 
tiplying or  dividing  its  numerator  and  denominator  by 
the  same  number.  Now,  if  two  fractions  have  different 
denominators,  let  us  suppose  all  the  possible  fractions 
equivalent  to  each  of  them  with  other  numerators  and 
denominators  found :  among  these  may  be  discovered 
two  that  have  the  same  denominator ;  the  numerators 
of  these  two  will  then  express  the  relative  values  of  the 
two  fractions,  and  either  addition  or  subtraction  may  be 
performed  by  the  rule  just  given.  Let  the  two  fractions 
|-  and  f  be  proposed,  and  let  it  be  required  to  discover 
2  fractions  equivalent  to  these  having  the  same  deno- 
minator. Since  the  terms  of  neither  of  these  fractions 
admit  of  being  both  divided  by  any  number,  we  shall 
find  all  the  equivalent  fractions  by  multiplying  their 
numerators  and  denominators  successively  by  2,  3,  4, 
£c.  They  will  be  as  follows :  — 


162  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

f  4*2    —     8_          4X3    __    j  o          4X4    _  J^6. 
4  — J    5X2    ~  10          5x3    "'   17          5X4    ~~    20 


f= 

5X3  __  15    5X4  _  OQ. 

6X3  —  18    6X4  -"  24 


{5X_2  —  !Q. 
6X2     12 


On  examining  this  series  of  equivalent  fractions,  we 
find  that  -J^  and  -J^  have  the  same  denominator.  These 
equivalent  fractions  have  been  obtained  by  multiplying 
the  numerator  and  denominator  of  the  first  of  the  pro- 
posed fractions  by  6,  and  those  of  the  second  by  5.  Now, 
it  will  be  observed,  that  6  is  the  denominator  of  the 
second,  and  5  of  the  first;  and  we  therefore  infer  that 
this  reduction  has  been  made  by  multiplying  both  terms 
of  the  first  fraction  by  the  denominator  of  the  second, 
and  both  terms  of  the  second  by  the  denominator  of  the 
first. 

Let  us  now  enquire  whether  this  method  is  general 
in  its  nature,  or  accidental  on  the  particular  fractions 
taken  in  the  above  example.  Let  the  proposed  fractions 
be  §  and  -&-•  If  we  multiply  both  terms  of  the  first  by 
7,  the  denominator  of  the  new  fraction  will  be  the  pro- 
duct of  9  and  7  j  and  if  we  multiply  both  terms  of  the 
second  by  9,  the  denominator  of  the  new  fraction  will  be, 
in  like  manner,  the  product  of  9  and  7-  In  this  case, 
therefore,  also  the  equivalent  fraction  obtained  in  this 
way  will  have  the  same  denominator.  But,  in  general, 
if  we  multiply  both  terms  of  the  first  fraction  by  the 
denominator  of  the  second,  the  new  denominator  will 
evidently  be  the  product  of  the  denominators  of  the  pro- 
posed fractions ;  and,  in  like  manner,  if  we  multiply 
both  terms  of  the  second  by  the  denominator  of  the  first, 
the  denominator  of  the  fraction  so  obtained  will  be  like- 
wise the  product  of  the  denominators  of  the  proposed 
fractions.  Thus  we  perceive  that  the  application  of  this 
principle  will,  in  general,  enable  us,  whenever  two  frac- 
tions are  proposed  with  different  denominators^  to  find 


CHAP.  I.  FRACTIONS.  163 

two  other  fractions  equivalent  to  those  with  the  same 
denominator. 

(232.)  When  it  is  required  to  convert  three  or  more 
fractions  having  different  denominations  into  equivalent 
fractions  having  the  same  denominator,  it  is  only  neces- 
sary first  to  apply  the  principle  just  explained  to  any  two 
of  them,  and  then  to  repeat  its  application  to  these  two 
and  a  third,  and  so  on.  The  practical  process  may, 
however,  be  abridged  by  observing  the  following 

RULE. 

Multiply  loth  terms  of  each  fraction  by  the  continued 
product  of  the  denominators  of  the  others,  and  equivalent 
fractions  will  be  obtained  having  the  same  denominator , 
which  denominator  will  be  the  continued  product  of  the 
denominators  of  the  proposed  fractions. 

For  example,  let  the  proposed  fractions  be  -|,  -f,  -|,  |-; 
multiply  both  terms  of  the  first  by  4x5x6;  multiply 
both  terms  of  the  second  by  3  X  5  X  6  ;  both  terms  of 
the  third  by  3x4x6;  and  both  terms  of  the  fourth  by 
3x4x5.  We  shall  obtain  the  following  equivalent 
fractions : — 

2X4X5X6    040  3X3X5X6    £7^0. 

3x4x5x6  —  irecr       3x4x5x6  —  3  (To 

4X3X4X6  __  28&    5X3X4X5:  __  ^QQ 
3X4X5X6  —  3~60     3X4X5X6  —  360* 

It  will  be  perceived  that,  in  these  equivalent  fractions, 
the  common  denominator  is  the  continued  product  of  the 
denominators  of  the  4  fractions  proposed  ;  and  also,  that 
the  numerators  of  each  are  found  by  multiplying  the 
numerator  of  each  fraction  by  the  continued  product  of 
the  denominators  of  the  other  fractions. 

Though  not  indispensably  necessary,  yet  it  will  be 
found  expedient,  before  we  proceed  to  reduce  fractions 
to  a  common  denominator,  to  reduce  them  to  their  lowest 
terms,  since  otherwise  we  shall  have  to  deal  with  higher 
numbers  in  the  calculation  than  would  be  necessary. 

(233.)  To  find  numbers  which  would  express  the 
M  2 


164  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

relative  values  of  any  proposed  fractions,  it  is  only  ne- 
cessary to  find  the  numerators  of  equivalent  fractions 
having  the  same  denominator.  These  numerators  are, 
as  we  have  seen  above,  the  products  of  the  numerator  of 
each  fraction,  and  the  denominator  of  the  other.  Thus, 
if  the  fractions  proposed  be  •§-  and  J,  the  numerators  of 
the  equivalent  fractions  would  be  2  x  4  and  3x3;  and, 
consequently,  the  two  fractions  will  be  in  the  proportion 
of  8  to  9-  In  the  same  manner,  if  the  fractions  pro- 
posed were  |  and  -J-,  the  numerators  of  the  equivalents 
would  be  5  X  8  and  7x6;  and  the  proportion  of  the 
proposed  fractions  would  be  that  of  40  and  42. 

(234.)  From  all  that  has  been  explained,  the  truth 
of  the  following  rule  for  the  addition  and  subtraction  of 
fractions  will  be  apparent. 

RULE. 

To  add  or  subtract  fractions. 

1 .  If  they  have  the  same  denominator,  add  or  subtract 
their  numerators,  and  subscribe  their  denominator. 

2.  If  they  have  not  the  same  denominator,  find  equi- 
valent fractions  which  have,   and  proceed  in  the  same 
manner. 

We  have  seen  that  a  whole  number  may  be  converted 
into  an  equivalent  fraction  having  any  proposed  deno- 
minator, by  multiplying  it  by  that  denominator,  and 
writing  under  it  the  same  denominator.  By  this  means, 
and  by  the  rule  just  given,  we  are  enabled  to  add  a 
fraction  to  a  whole  number,  or  to  subtract  the  one  from 
the  other,  so  as  to  express  their  sum  or  difference  by  a 
fraction.  Let  it  be  proposed  to  add  f  to  4 :  we  shall 
reduce  4  to  a  fraction  whose  denominator  is  3,  by  mul- 
tiplying it  by  3,  and  subscribing  the  denominator  $  ; 
this  fraction  is  M.  This  being  added  to  |-  gives  y, 
which  is  the  sum  of  4  and  J .  If  it  be  required  to  sub- 
tract •§•  from  4,  we  get  the  remainder  \£y  which  is  the 
difference  between  4  and  §.  In  this  way,  a  mixed 
number  may  always  be  converted  into  an  improper  frac- 


CHAP.  I.  FRACTIONS.  l65 

tion.  Thus/  7f  may  be  converted  into  an  improper 
fraction  by  adding  7  to  -j?-  by  the  above  method.  To  do 
so  we  must  multiply  7  by  6,  and  subscribe  6,  and  then 
add  the  fractions  having  the  common  denominator  6'  : 
the  process  would  be  as  follows  :  — 


(235.)  The  following  is  the 

GENERAL  RULE. 

To  convert  a  mixed  number  into  an  improper  fraction 
—  Multiply  the  integral  part  by  the  denominator  of  the 
fractional  party  and  to  the  product  add  the  numerator  of 
the  fractional  part.  Talcing  this  as  numerator,  subscribe 
the  denominator  of  the  fractional  part. 

(236.)  Although  the  rule  already  given  for  reducing 
fractions  to  the  same  denominator  is  the  only  method 
which  is  universally  applicable,,  yet  it  will  happen  in  par- 
ticular cases  that  more  abridged  processes  may  be  found, 
arising  from  the  peculiar  relations  of  the  numbers  under 
consideration.  Suppose,  for  example,  that  the  fractions 
to  be  reduced  to  a  common  denominator  were  |  and  -J. 
If  we  followed  the  general  rule  in  this  case,  we  should 
obtain  the  equivalents  -|  and  -|  ;  but  a  slight  attention  to 
the  particular  case  proposed  will  show  us  that  the  deno- 
minator  of  the  first  fraction  is  twice  that  of  the  second, 
and  that,  consequently,  if  we  multiply  both  terms  of  the 
second  by  2,  we  shall  obtain  an  equivalent  fraction  -|, 
having  the  same  denominator  as  the  first.  The  equiva- 
lent fractions  thus  obtained  are  evidently  in  lower  terms, 
and  therefore  more  convenient  than  those  which  would 
result  from  the  application  of  the  general  rule. 

Again,  if  it  were  required  to  reduce  the  following 
fractions  to  a  common  denominator,  — 

i  i  i  -h  e» 

we  should  observe  that  the  denominator  of  the  last  is  a 
multiple  of  each  of  the  other  denominators;  and  that, 
consequently,  we  can  obtain  a  fraction  equivalent  to 
each  of  the  others,  having  the  same  denominator  as  the 
M  3 


166  A    TREATISE    OX    ARITHMETIC.  BOOK  II. 

last,  by  multiplying  both  numerator  and  denominator  by 
that  number  which  would  be  found  by  dividing  the 
last  denominator  by  the  denominator  of  the  proposed 
fraction.  Thus,  since  36  is  3  x  12,  we  shall  obtain  an 
equivalent  for  the  first  by  multiplying  both  terms  by 
12.  Again,  since  36  is  4  x  9,  we  shall  obtain  an  equi- 
valent for  the  second  by  multiplying  both  terms  by  9. 
In  the  same  manner,  it  will  be  perceived  that  we  shall 
obtain  equivalents  for  the  third  and  fourth  by  multiply- 
ing their  terms  respectively  by  6  and  3.  The  5 
equivalents  with  a  common  denominator  thus  obtained 

are  -H  U.  U  U  H-  If  we  aPPlied  the  general 
rule  in  this  case,  the  common  denominator,  instead 
of  being  36,  would  be  31104,  and  the  numerators 
would  be  proportionally  great.  The  method  here  ex- 
plained may  be  practised  in  every  case  where  the 
greatest  of  the  proposed  denominators  is  a  multiple  of 
each  of  the  others ;  but,  although  that  may  not  happen, 
still  it  may  happen  that  the  greatest  denominator  mul- 
tiplied by  some  smaller  number  may  be  a  multiple  of 
all  the  others  ;  and,  in  such  a  case,  the  reduction  to  a 
common  denominator  may  not  only  be  considerably 
abridged,  but  the  equivalent  fractions,  as  in  the  above 
case,  may  be  obtained  in  comparatively  low  terms. 

Let  the  proposed  fractions  be  J,  $,  -f|,  j|,  \\,  f£. 
The  greatest  denominator,  36,  is  not  in  this  case  a  mul- 
tiple of  each  of  the  others.  Let  us  try,  however, 
whether  36  X  2  is  so.  We  find  that  72=24  x  3  =  18 
X4=12  x6=8  X9=4x  18.  Hence  it  is  apparent 
that  we  can  obtain  equivalent  fractions  with  the  deno- 
minator 72,  by  multiplying  the  terms  of  the  first  fraction 
by  18,  those  of  the  second  by  9,  of  the  third  by  6,  of 
the  fourth  by  4,  of  the  fifth  by  3,  and  of  the  sixth  by  2. 
These  fractions  axe  the  following :  — 

4f   ff   43    H   4i   43- 

It  appears  then,  that  the  same  fractions  may  be  re- 
duced to  a  great  variety  of  different  common  denomin- 
ators ;  and,  indeed,  this  is  sufficiently  evident  when 


CHAP.  I.  FRACTIONS.  16? 

we  consider  that,  after  having  found  any  set  of  equi- 
valent fractions  with  the  same  denominator,  these  may 
be  infinitely  varied  by  multiplying  all  their  terms  suc- 
cessively by  different  numbers.  We  may  by  this  means 
obtain  an  unlimited  number  of  sets  of  equivalent  frac- 
tions, the  common  denominator  of  each  set  being  a 
multiple  of  the  original  common  denominator. 

(237.)  Upon  the  general  principle  of  convenience  on 
which  we  prefer,  in  general,  to  use  fractions  in  their 
least  terms,  so,  HI  the  present  instance,  whenever  it  is 
necessary  to  convert  any  set  of  fractions  with  different 
denominators  into  another  equivalent  set  having  a  com- 
mon denominator,  we  should  select  among  all  the  pos- 
sible common  denominators  that  which  is  the  least. 
This  may  be  easily  done. 

Let  us  suppose  equivalent  fractions,  with  a  common 
denominator,  to  be  found  by  the  rule  already  given. 
We  are  required  to  ascertain  whether  the  common 
denominator  thus  obtained  is  the  least  possible;  and, 
If  not,  to  find  what  is  the  least  possible.  To  ac- 
complish this,  find  the  greatest  common  measure  of 
the  several  numerators  of  the  equivalent  fractions,  and 
their  common  denominator.  This  common  measure 
will  divide  both  numerator  and  denominator  of  each  of 
the  fractions,  and  will  convert  them  into  another  equi- 
valent set,  which  will  still  have  a  common  denominator. 
But  since  the  divisor  thus  used  is  the  greatest  common 
divisor,  the  set  of  equivalent  fractions  obtained  will  be 
in  the  least  terms  which  are  consistent  with  having  the 
same  denominator. 

To  apply  this  rule,  let  it  be  required  to  reduce  the 
fractions  3,  f,  and  -|  to  equivalent  fractions  having  the 
least  possible  common  denominator.  By  the  general 
rule  (232.)  we  should  obtain  the  folio  wing  results*:  — 

.3X6X8         5X4  8         7X4X6 
4X0X8        6X4X8        8X4X6* 

-m      w      m- 

By  the  rule  for  finding  the  greatest  common  measure, 

M  4 


168  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

we  find  that  the  greatest  common  measure  of  144, 160, 
168,  and  192  is  8.  Dividing  the  numerator  and  de- 
nominator of  each  fraction  by  8,  we  get  the  following 
equivalent  fractions :  — 

is.     SLQ.     2 1 

24        24        24* 

If  the  several  numerators  and  common  denominator 
of  the  first  set  of  equivalent  fractions  obtained  by  the 
rule  (232.)  have  no  common  measure  greater  than 
unity,  then  their  common  denominator  is  the  least  pos- 
sible. This  is  evident,  since  the  only  way  they  could 
be  reduced  to  lower  terms,  preserving  the  same  deno- 
minator, would  be  by  dividing  their  several  numerators 
and  their  common  denominator  by  the  same  number ; 
and  that  number  must,  therefore,  be  a  common  measure. 

(238.)  Although  the  least  possible  common  denominator 
may  always  be  found  by  the  above  rule,  yet,  in  practice, 
we  may  sometimes  abridge  the  process  in  the  following 
manner  :  —  If  there  be  but  two  fractions  given,  find, 
in  the  first  instance,  the  greatest  common  measure  of 
their  denominators.  If  the  denominators  be  not  great, 
this  number  will  frequently  be  found  by  mere  inspec- 
tion. Multiply  the  two  denominators  together,  and 
divide  their  product  by  their  greatest  common  measure. 
The  quotient  will  always  be  a  multiple  of  each  of  the 
denominators,  or  a  common  multiple  of  them.  Each 
of  the  fractions  may  be  reduced  to  an  equivalent  one 
having  this  multiple  as  the  denominator.  The  two 
fractions  thus  obtained  will  have  the  least  possible  com- 
mon denominator.  This  process  will  be  easily  under- 
stood by  means  of  an  example. 

Let  the  proposed  fractions  be  f  and  -|  :  the  greatest 
common  measure  of  their  denominators  is  2.  Now,  if 
we  multiply  the  denominators  together,  and  divide  the 
product  by  2,  we  must  necessarily  obtain  the  same 
result  as  if  we  first  divided  either  of  the  denominators 
by  2,  and  then  multiplied  the  quotient  by  the  other. 
It  is  evident  that  the  same  arithmetical  operations  are 
performed  in  each  case,  only  in  a  different  order.  If 


CHAP.  I.  FRACTIONS.  169 

we  divide  4  by  2,  and  multiply  the  quotient  by  6,  we 
shall  obtain  12,  which  is  a  multiple  of  6  :  or  if  we 
divide  6  by  2,  and  multiply  the  quotient  by  4t,  we 
shall  obtain  a  multiple  of  4,  being  the  same  number, 
12.  Thus,  in  each  case  the  result  will  be  the  same, 
and  will  be  at  once  a  multiple  of  each  denominator. 
But  we  should  obtain  precisely  the  same  result  if  we 
performed  the  same  operations  in  another  way,  viz.  by 
multiplying  the  denominators  together,  and  dividing  the 
product  by  2. 

To  find  two  equivalent  fractions  having  the  common 
denominator  12,,  it  is  only  necessary  to  find  how  often 
each  denominator  is  contained  in  12.  The  first  being 
contained  in  12  three  times,  we  multiply  both  terms  of 
the  first  fraction  by  3,  by  which  means  it  is  converted 
into  the  equivalent  fraction  -?•».  The  denominator  of 
the  second  fraction  being  contained  in  12  twice,  we 
multiply  both  terms  of  it  by  2,  by  which  means  it  is 
converted  into  the  equivalent  fraction  J-£. 

(239.)  If  it  be  required  to  reduce  3  or  more  frac- 
tions to  the  least  possible  common  denominator,  we 
may  extend  the  application  of  this  method  without 
difficulty.  Find  the  least  common  multiple  of  their  com- 
mon denominator  and  the  denominator  of  the  third 
fraction ;  next  find  the  least  common  multiple  of  this 
and  the  denominator  of  the  fourth,  and  so  on.  The 
common  multiple  last  found  will  be  the  least  common 
denominator  to  which  the  fractions  can  be  reduced. 
They  may  be  reduced  to  this  denominator  by  finding  how 
often  the  denominator  of  each  fraction  is  contained  in  the 
proposed  common  denominator,  and  then  multiplying 
both  terms  of  the  fraction  by  that  number. 

Let  it  be  proposed  to  find  the  least  common  deno- 
minator to  which  the  fractions  -f,  |-,  -|,  and  -J  may  be 
reduced.  First,  find  the  greatest  common  measure  of 
the  first  2  denominators ;  in  this  case  that  common 
measure  is  1  ;  the  least  common  multiple  of  these 
two  is,  therefore,  12  (238.).  Next  find  the  greatest 
common  measure  of  12  and  8,  which  is  4.  Divide 


170  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

12  by  4,  and  multiply  the  quotient  by  8  :  the  result  is 
24,  which  is  the  least  common  denominator  of  the  first 
3  fractions.  Next  find  the  greatest  common  measure  of 
24  and  9^  which  is  3.  Divide  24  by  3,  and  multiply 
the  quotient  by  9 :  the  result  is  72,  which  is  the  least  pos- 
sible common  denominator  of  all  the  proposed  fractions. 

To  find  the  equivalents,  find  how  often  3  is  con. 
tained  in  72  :  it  is  contained  24  times.  Multiply  both 
terms  of  the  first  fraction  by  24,  and  the  result  is  4-&. 
In  like  manner,  find  how  often  4  is  contained  in  72  :  it 
is  contained  in  72  eighteen  times.  Multiply  both  terms 
of  the  second  fraction  by  18  :  the  result  is  •!•§-.  In 
like  manner,  8  being  contained  9  times  in  72,  multiply 
the  terms  of  the  third  fraction  by  9 :  and  9  being  con- 
tained 8  times  in  72,  multiply  the  terms  of  the  fourth 
fraction  by  8.  The  series  of  equivalent  fractions  will 
then  be  -|f ,  -1%  -f ! ,  ^%  ',  and  these  are  the  equivalent 
fractions  which  have  the  least  possible  common  deno- 
minator. 

(240.)  It  is  sometimes  required  to  convert  a  given 
fraction  into  an  equivalent,  having  some  proposed 
number  as  its  denominator,  the  given  fraction  being 
reduced  to  its  least  terms.  It  is  only  possible  to  effect 
this  when  the  denominator  proposed  is  an  exact  multiple 
of  the  denominator  of  the  fraction.  If  it  be  so,  mul- 
tiply both  terms  of  the  proposed  fraction  by  the  quotient 
found  by  dividing  the  proposed  denominator  by  the 
denominator  of  the  fraction.  For  example  :  let  it  be 
proposed  to  convert  J  into  an  equivalent  fraction  having 
16  as  a  denominator  :  4  divides  16  four  times;  there- 
fore, multiply  both  terms  of  J  by  4,  and  we  shall  ob- 
tain J-|,  which  is  the  equivalent  sought.  Had  it  been 
required  to  convert  f  into  an  equivalent  fraction,  having 
18  for  its  denominator,  the  solution  of  the  problem 
would  be  impossible ;  since  there  is  no  whole  number  by 
which  the  denominator  of  |  could  be  multiplied  which 
would  produce  18 


171 


CHAP.  II. 

THE    MULTIPLICATION   AND    DIVISION    OP   FRACTIONS. 

(241.)  IN  the  first  book  we  have  contemplated  multi- 
plication as  an  abridged  method  of  addition,  by  which 
we  are  enabled  to  determine,  by  a  short  process,  what 
number  would  be  obtained  by  the  addition  of  the  same 
number  repeated  any  proposed  number  of  times.  The 
extension  which  has  been  given  to  the  language  and 
notation  of  number  in  the  last  chapter,  renders  it  ne- 
cessary that  we  should  enquire  what  effect  the  same 
extension  will  produce  upon  the  operations  of  multipli- 
cation as  already  described. 

If  we  are  required  to  determine  the  product  of  12 
multiplied  by  8,  the  thing  to  be  ascertained  is  the  sum 
which  would  be  obtained  by  the  addition  of  12  eight 
times  repeated.  Again :  if  we  are  required  to  deter- 
mine the  product  of  12  multiplied  by  9,  it  is  necessary 
to  find  what  sum  would  be  obtained  by  12  nine  times 
repeated.  In  fact,  the  multiplicand  must  in  this  case 
be  repeated  in  the  addition  as  many  times  as  there  are 
units  in  the  multiplier.  Now,  suppose  that  the  multi- 
plier, instead  of  being  either  8  or  9,  were  8f ,  shall  we 
repeat  the  multiplicand  eight  times  or  oftener  ?  How 
many  units  are  contained  in  8  J  ?  More  than  8  units 
are  contained  in  it,  but  less  than  9 :  the  multiplicand 
ought  therefore  to  be  repeated  more  than  8  times, 
but  less  than  9  times.  The  multiplier  contains  only 
a  part  of  the  ninth  unit,  therefore  the  multiplicand  must 
be  repeated  only  a  part  of  the  ninth  time;  that  is 
to  say,  the  whole  multiplicand  must  not  be  repeated  the 
ninth  time,  but  only  as  much  of  it  as  there  is  of  the 
ninth  unit  contained  in  the  multiplier.  Now,  if  the 


172  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

ninth  unit  be  imagined  to  be  divided  into  four  equal 
parts,  three  of  these  are  taken  in  the  multiplier.  We 
must,,  therefore,,  to  find  the  product  required,  first  repeat 
the  multiplicand  8  times,  and  then,  instead  of  re- 
peating the  whole  multiplicand  the  ninth  time,  we  must 
divide  it  into  4  equal  parts,  and  repeat  only  3  of 
these.  The  total  process  by  addition  would  then  be  as 
follows :  — - 

12 

12 

12 

12 

12 

12 

12 

12 
3 
3 
3 

105 

The  multiplicand  being  divided  into  4  equal  parts, 
each  of  these  parts  is  3 :  after  repeating  the  multipli- 
cand 8  times,  we  therefore  repeat  its  fourth  part,  3, 
three  times.  Adding  the  whole,  we  obtain  105,  which 
is  the  product  sought. 

So  much  of  this  process  as  consists  in  multiplying 
12  by  8  is  nothing  more  than  the  ordinary  process  of 
multiplication,  as  explained  in  our  first  book.  But  the 
analogous  process  of  multiplying  12  by  J  is  an  exten- 
sion of  multiplication  to  the  new  species  of  multiplier 
introduced  to  our  notice  in  the  last  chapter.  If  we  wish 
to  multiply  12  by  3,  we  have  only  to  repeat  it  three 
times,  and  add :  if  we  wish  to  multiply  12  by  f ,  we 
have  only  to  repeat  it  three  fourths  of  a  time,  or,  what 
is  the  same,  we  have  only  to  repeat  its  fourth  part  3 
times,  and  add ;  the  product  would  be  9* 

The  constant  habit  which  persons  are  given,  of  con- 
sidering multiplication  only  as  applied  to  integers, 
causes  the  fact  of  finding  a  product  less  than  the  multi. 


CHAP.  II.  MULTIPLICATION    OF    FRACTIONS.  1  IS 

plicand  to  have  a  startling  effect.  That  the  operation 
of  multiplication  should  diminish  and  not  increase  that 
to  which  it  is  applied,  is  a  matter  which  at  first  view 
seems  paradoxical.  It  is  hoped,  however,  that  the  above 
illustration  will  convince  any  one  who  gives  the  neces- 
sary attention  to  it  that  there  is  nothing  contradictory 
or  paradoxical  in  the  matter.  The  word  "  multipli- 
cation" has  received  an  extended  meaning,  which  is 
perfectly  consistent  and  analogous  with  its  more  popular 
acceptation,  as  applied  exclusively  to  whole  numbers. 
Without  such  an  extension,  the  theory  and  practice  of 
fractions  would  fall  into  inextricable  complexity  and 
confusion ;  and,  indeed,  it  would  be  difficult  to  frame  a 
system  of  language  by  which  those  arithmetical  operations 
could  be  expressed  in  which  mixed  numbers  are  engaged. 
We  should  have  one  nomenclature  for  the  operations 
performed  on  the  integral  parts  of  them,  and  another 
for  the  like  operations  performed  on  the  fractional  parts. 
Such  a  complicated  phraseology  would  be  equally  per- 
plexing and  absurd.  As,  however,  minds  habituated 
to  the  consideration  of  the  arithmetic  of  whole  numbers, 
and  of  whole  numbers  only,  commonly  find  some  diffi- 
culty in  this  extension  of  the  meaning  of  the  term  multi- 
plication, we  shall  further  illustrate  the  perfect  consistency 
of  the  application  of  the  term  to  fractions  with  its  sense 
as  applied  to  whole  numbers. 

Let  us,  as  before,  suppose  that  we  are  required  to 
multiply  a  number  by  f;  and  let  us  take  the  terms 
multiplication  and  division  in  the  sense  in  which  they 
are  usually  understood  when  applied  to  whole  num- 
bers, viz.  multiplication,  as  an  operation  by  which 
something  is  increased  in  a  certain  proportion ;  and  divi- 
sion, an  operation  by  which  it  is  diminished  in  a  certain 
proportion. 

We  have  already  explained  that  J  may  be  considered 
as  expressing  the  fourth  part  of  3.  Being  required  to 
multiply  by  -|,  we  are  then  required  to  multiply  by 
the  fourth  part  of  3.  Let  us  suppose  that  we  proceed 
by  first  multiplying  the  number  proposed  by  3.  What- 


174?  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

ever  the  product  may  be  which  is  obtained  by  this  pro- 
cess, it  is  evidently  greater  than  that  which  we  seek ; 
because  we  have  multiplied  by  3,  instead  of  multiplying 
by  the  fourth  part  of  3.  But  it  is  further  evident  that 
it  is  not  only  greater  than  the  product  required,  but 
greater,  in  exactly  the  same  proportion,  as  3  is  greater 
than  its  fourth  part :  the  product  found  is,  therefore, 
4  times  too  great.  To  diminish  it  to  its  just  mag- 
nitude, it  will,  therefore,  be  necessary  to  divide  it  by  4  : 
the  quotient  of  such  division  will  be  the  true  product 
sought.  Apply  this  to  the  case  in  which  the  multipli- 
cand is  1 2 :  we  are  required  to  multiply  J  2  by  the 
fourth  part  of  3  :  we  multiply  it  by  3,  and  obtain  36* ; 
but  having  multiplied  12  by  a  number  four  times  too 
great,  the  product  36  is  four  times  the  product  which 
we  seek.  We  therefore  divide  36  by  4,  and  obtain  9> 
which  must  be  the  product  sought :  when  12  is  multi- 
plied by  %,  the  product  then  is  9- 

Under  this  point  of  view,  the  multiplication  of  a 
number  by  a  fraction  is  a  two  fold  operation ;  a  multi- 
plication by  its  numerator,  and  a  division  by  its  deno- 
minator. We  must  not,  however,  infer  that  the  mul- 
tiplication by  a  fraction,  in  its  own  essential  nature,  is 
a  more  complex  process  than  the  multiplication  by  a 
whole  number ;  or,  to  speak  more  strictly,  we  must  not 
conclude  that  the  product  bears  to  its  factors  a  different 
relation  when  one  of  those  factors  is  a  fraction,  from  the 
relation  it  has  to  them  when  both  are  integers.  The 
greater  complexity  is  merely  in  the  method  adopted  for 
practising  the  operation:  the  arithmetical  relation  of 
the  numbers  engaged  in  the  question  is,  in  both  cases, 
precisely  the  same. 

There  is  one  circumstance  more,  which  offers  a 
forcible  proof  of  the  consistency  of  the  extension  of  the 
the  term  multiplication  here  contemplated.  Taking  the 
example  of  the  multiplication  of  12  by  f,  it  will  not  be 
doubted  that  whether  we  multiply  12  by  f,  or  f  by  12, 
we  must  needs  obtain  the  same  product.  Now  let  us 
take  the  latter  view  of  the  question,  and  let  us  suppose 


CHAP.  II.         MULTIPLICATION    OP    FRACTIONS.  175 

that  we  choose  to  perform  the  operation  by  multiplying 
J  by  12  :  we  are  then  to  repeat  f  twelve  times,  and  add. 
Since  all  the  fractions  to  be  added  in  this  case  have  the 
same  denominator,  4,  we  shall  add  them  by  merely  adding 
their  numerators  (234?.),,  retaining  the  same  denominators : 
3,  twelve  times  repeated,  must  therefore  be  added  ;  and 
the  result  is  36,  the  product  sought  being  316.  But, 
by  what  has  been  already  proved  (225.),  3-f  =  9  *  the 
product,  then,  of  f  multiplied  by  12  is  9*  which  is  the 
same  as  the  product  already  found  by  other  reasoning 
when  12  was  multiplied  by  ^.  If,  then,  it  be  admitted 
that  f  ought  to  be  multiplied  by  12,  by  repeating  | 
twelve  times,  and  adding,  we  must  needs  also  admit  that 
12  should  be  multiplied  by  f,  by  dividing  it  into  four 
equal  parts,  and  repeating  one  of  these  parts  three  times, 
the  result  being  the  same  in  both  cases.  If  this  was 
not  conceded,  we  should  be  forced  into  the  absurdity  of 
maintaining  that  12  multiplied  by  f  is  a  different  num- 
ber, and  the  operation  a  different  operation  from  f  mul- 
tiplied by  12. 

(242.)  From  all  that  has  been  above  stated,  we  may 
infer  the  following 

RULE. 

To  multiply  any  number  by  a  fraction,  we  should 
multiply  that  number  by  the  numerator  of  the  multiplier, 
and  divide  the  result  by  the  denominator  of  the  multiplier. 
The  quotient  will  be  the  product  sought. 

(243.)  A  fraction  may  be  multiplied  by  a  whole 
number,  by  either  of  two  methods ;  one  of  which  is  al- 
ways practicable,  and  is  therefore  preferable  as  a  general 
rule,  and  the  other  more  simple  when  it  is  practicable, 
and  therefore  advantageous  as  an  occasional  rule.  By 
what  has  been  proved  above,  it  appears  that  a  fraction 
may  always  be  multiplied  by  a  whole  number,  by  mul- 
tiplying its  numerator  by  that  number,  preserving  the 
denominator.  This  rule  is  perfectly  general.  When 
it  happens,  however,  that  the  denominator  is  divisible 


1?  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

exactly  by  the  whole  number,  then  the  fraction  may  be 
multiplied  by  the  whole  number,  by  dividing  its  deno- 
minator by  that  number  (210.). 

(244.)  The  multiplication  of  any  number  by  a  pro- 
per fraction  will  diminish  it  in  the  proportion  of  the 
numerator  of  the  multiplier  to  its  denominator;  and 
the  multiplication  by  an  improper  fraction  will  increase 
it,  in  the  proportion  of  the  numerator  of  the  multiplier 
to  its  denominator.  This  will  be  easily  perceived  by  con- 
sidering attentively  what  has  been  already  proved.  If 
a  number  be  multiplied  by  -i,  the  product  will  be  four 
fifths  of  the  multiplicand  (241.)  :  consequently,  the  pro- 
duct will  be  less  than  the  multiplicand  in  the  proportion 
of  4  to  5.  If  a  number  be  multiplied  by  f ,  the  product 
will  be  five  fourths  of  the  multiplicand,  and  conse- 
quently will  be  greater  than  the  multiplicand  in  the 
proportion  of  5  to  4. 

When  a  number  is  multiplied  by  a  fraction,  it  is 
submitted  successively,  as  we  have  observed,  to  the 
two  operations  of  multiplication  by  the  numerator,  and 
division  by  the  denominator.  It  is  immaterial,  in  so 
far  as  regards  the  result,  in  which  order  these  operations 
are  performed;  but  it  is  frequently  more  convenient, 
first,  to  divide  the  multiplicand  by  the  denominator,  and 
then  to  multiply  the  quotient  by  the  numerator.  This 
method  is  always  preferable  when  the  multiplicand, 
being  a  whole  number,  is  a  multiple  of  the  denominator. 
In  the  example  above  given,  we  are  required  to  multi- 
ply 12  by  ~.  The  most  simple  method  of  proceeding 
is  first  to  divide  12  by  4,  and  then  multiply  the  quo- 
tient by  3.  In  this  case  we  shall  have  small  numbers 
to  deal  with,  whereas  in  multiplying  by  3,  and  dividing 
by  4,  we  shall  have  higher  numbers.  In  the  present  in- 
stance, the  numbers  concerned  in  the  question  being 
small,  this  advantage  is  not  so  apparent ;  but  in  ques- 
tions involving  high  numbers  it  is  of  some  importance. 
Let  us  suppose  that  we  are  required  to  multiply  376625 
by  fj-r.  If  we  first  multiplied  by  37*  and  then  divided 
by  125,  the  process  would  be  of  considerable  length  : 


CHAP.  II.         MULTIPLICATION    OF    FRACTIONS.  177 

but  by  dividing  the  multiplicand  first  by  125,  we  get 
the  quotient  3013,  which  multiplied  by  37  gives  the 
true  quotient,  111481. 

It  is,  perhaps,  scarcely  necessary  to  observe,  that  in 
multiplication,  as  in  most  other  operations  in  which 
fractions  are  engaged,  it  is  convenient,  in  the  first  in- 
stance, to  reduce  the  fractions  to  their  least  terms,  in 
order  to  avoid  the  introduction  of  numbers  unnecessarily 
large. 

(245.)  When  the  multiplicand  is  a  whole  number, 
and  the  multiplier  a  fraction,  the  product  will  always  be 
an  integer,  when  the  multiplicand  is  a  multiple  of  the 
denominator  of  the  multiplier.  This  will  be  evident 
from  what  has  been  just  proved ;  for  if  we  divide  the 
multiplicand  by  the  denominator  of  the  multiplier,  we 
shall  obtain  a  whole  number  for  the  quotient,  and 
this  whole  number,  being  multiplied  by  the  nume- 
rator of  the  multiplier,  will  give  a  whole  number  for  the 
product.  But  if  the  multiplicand  be  not  an  exact  mul- 
tiple of  the  denominator  of  the  multiplier,  the  latter  be- 
ing reduced  to  its  least  terms,  then  the  division  of  the 
multiplicand  by  the  denominator  of  the  multiplier  will 
give  a  quotient  which  is  either  a  fraction  or  a  mixed 
number ;  and  this  being  multiplied  by  the  numeiator, 
it  will  still  be  either  a  fraction  or  a  mixed  number. 

(246.)  From  what  has  been  explained,  it  is  easy  to 
discover  methods  by  which  one  fraction  may  be  multi- 
plied by  another.  It  has  been  proved  (242.),  that  when 
any  number  is  required  to  be  multiplied  by  a  fraction, 
we  have  only  to  multiply  that  number  by  the  numera- 
tor of  the  fraction,  and  to  divide  the  result  by  its  deno- 
minator. Let  the  multiplicand,  then,  be  a  fraction 
we  must  first  multiply  it  by  the  numerator  of  the  mul- 
tiplier ;  that  numerator  being  a  whole  number,  and  the 
multiplicand  being  a  fraction,  we  must  multiply  the 
numerator  of  the  multiplicand  by  that  whole  number, 
preserving  the  denominator  (214.).  It  is  next  neces- 
sary to  divide  this  result  by  the  denominator  of  the 
multiplier  (242.).  Now,  it  has  been  proved  that  a 


178  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

fraction  is  divided  by  a  whole  number,  by  multiplying 
its  denominator  by  that  whole  number  (215.).  We 
must,  therefore,  in  the  present  case,  multiply  the  deno- 
minator of  the  result  we  have  just  obtained,  by  the 
denominator  of  the  multiplier :  the  fraction  which  we 
shall  thus  find  will  be  the  product  sought. 

Let  us  apply  this  reasoning  to  an  example. — Let  the 
multiplicand  be  4,  and  the  multiplier  %  :  we  must  first 
multiply  i  by  3,  and  then  divide  the  result  by  7 
(242.).  To  multiply  ^  by  3,  we  multiply  4  by  3,  pre- 
serving the  denominator  (214.),  and  obtain  ^  :  this 
must  be  divided  by  7  ;  but  this  division  may  be  effected 
by  multiplying  its  denominator  by  7  (215.),  and  we 
shall  thus  obtain  -^-J-,  which  is  therefore  the  product 

r-x> 

(247.)  Hence  we  may  infer  the  following 

RULE. 

To  find  the  product  of  two  fractions  multiply  the  nu 
meratorsfor  a  numerator,  and  the  denominators  for  a 
denominator. 

Whatever  may  be  the  terms  in  which  the  fractions  arc 
expressed,  and  whatever  may  be  their  mutual  relation, 
this  rule  is  always  applicable,  and  therefore  perfectly 
general.  In  particular  instances,  however,  the  multi- 
plication may  be  performed  more  simply,  and  the  product 
obtained  in  lower  terms,  by  other  methods,  which  are 
founded  on  the  principles  explained  (243.). 

(248.)  Since  the  multiplicand  may  be  multiplied  by 
the  numerator  of  the  multiplier,  either  by  multiplying 
its  numerator,  or  dividing  its  denominator,  we  may 
adopt  the  latter  method  with  advantage,  whenever  the 
denominator  of  the  multiplicand  happens  to  be  an  exact 
multiple  of  the  numerator  of  the  multiplier.  Suppose, 
for  example,  the  multiplicand  is  -|,  and  the  multiplier 
£.  In  this  case,  instead  of  following  the  general  rule, 
we  shall  multiply  J-  by  2,  by  dividing  its  denominator 
by  2 ;  the  result  is  |  :  this  being  divided  by  5,  by 


CHAP.  II.         MULTIPLICATION     OP    FRACTIONS.  179 

multiplying  its  denominator  by  5,  we  obtain  -*$,  the 
product.  Had  we  proceeded  by  the  general  rule  we 
should  have  obtained  the  product  in  the  terms  •£$, 
which  should  be  afterwards  reduced  to  its  least 
terms,  ;,3^. 

Since  there  is  no  real  distinction  between  the  multi- 
plicand and  multiplier,  we  may  therefore  infer,  gene- 
rally, that  when  the  denominator  of  one  fraction  is 
an  exact  multiple  of  the  numerator  of  the  other,  instead 
of  multiplying  the  numerators,  we  may  divide  the  deno- 
minator of  the  one  fraction  by  the  numerator  of  the 
other,  and  multiply  the  quotient  by  the  other  denominator. 

(249.)  If  the  denominator  of  one  fraction  happen  to 
be  the  same  number  as  the  numerator  of  the  other,  then 
the  product  maybe  obtained  without  either  multiplication 
or  division,  by  merely  retaining  the  numerator  and  deno- 
minator, which  are  dissimilar.  Let  the  fractions  to 
be  multiplied  be  -|  and  -|.  If  the  latter  be  multiplied 
by  4,  we  have  only  to  omit  its  denominator :  the  result 
being  3,  must  be  divided  by  5,  which  gives  the  frac- 
tion %  as  the  true  product.  If  we  apply  the  general 
rule  in  this-  case,  we  should  first  obtain  Jg,  which 
reduced  to  its  lowest  terms  would  be  -J. 

(250.)  If  it  happen  that  the  numerator  of  one  of 
the  fractions  is  an  exact  multiple  of  the  denominator  of 
the  other,  the  process  may  also  be  abridged.  Let  the 
fractions  to  be  multiplied  be  J  and  -fj.  It  is  necessary  to 
multiply  the  latter  by  3,  and  to  divide  it  by  4.  Now, 
it  may  be  divided  by  4,  by  dividing  its  numerator  by 
4 ;  the  result  would  be  T2T :  this  multiplied  by  3  will 
give  T6T,  the  complete  product.  Hence,  when  the  nu- 
merator of  one  fraction  is  a  multiple  of  the  denomina- 
tor of  the  other,  divide  the  numerator  of  the  former  by 
the  denominator  of  the  latter,  and  multiply  the  quotient 
by  the  remaining  numerator,  preserving  the  remaining 
denominator, 

(251.)  It  may  so  happen  that  the  numerator  of  one 
factor  may  be  a  multiple  of  the  denominator  of  the 
other,  and  also  its  denominator  a  multiple  of  the  nu- 
N  2 


1  80  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

merator  of  the  other.  In  such  case  the  process  of 
multiplication  may  be  abridged. 

Let  the  fractions  to  be  multiplied  be  |  and  -f-J.  It 
is  necessary  to  multiply  the  latter  by  3,  and  to  divide 
the  result  by  5.  We  may  multiply  it  by  3,  by  divid- 
ing its  denominator  by  3  (243.)  :  the  result  will  be  f|. 
This  must  be  divided  by  5  (242. ),  which  may  be  done 
by  dividing  its  numerator  by  5  (214.)  :  the  result  wiL1 
be  -^7,  which  is  the  product  sought.  Had  we  proceeded 
in  this  case  by  the  general  rule,  we  should  obtain  the 
product  under  the  form  -{'-£$,  which,  reduced  to  its  least 
terms,  would  be  T5-0. 

(252.)  A  similar  means  of  abridgment  may  be 
adopted  if  the  numerator  of  each  factor  be  a  multiple 
of  the  denominator  of  the  other.  Let  it  be  required  to 
multiply  *g  by  -^  :  we  shall  first  divide  $f  by  3  (242.), 
the  result  will  be  1-/.  This  must  be  multiplied  by  25; 
but  we  may  multiply  25  by  l-/,  by  first  dividing  it  by 
5,  and  then  multiplying  by  12  (242.) :  the  result  will 
be  60.  When  the  numerators,  therefore,  are  multiples 
of  their  alternate  denominators,  divide  each  by  the  alter- 
nate denominator,  and  multiply  the  quotients  together. 
The  product  will  in  this  case  be  always  a  whole  number. 

It  would  be  equally  tedious  and  unnecessary  to  ex- 
plain the  details  of  the  various  expedients  which  may 
be  adopted  for  the  abridgment  of  the  multiplication  of 
fractions  in  cases  where  the  factors  have  peculiar  nu- 
merical relations.  It  will  be  sufficient  to  observe  that 
all  such  methods  depend  immediately  on  the  principles 
proved  in  (214.)  (215.). 

(253.)  If  it  be  required  to  multiply  a  mixed  number 
by  a  whole  number,  the  most  expeditious  method  is, 
first,  to  multiply  the  fractional  part  by  the  whole  num- 
ber, converting  the  product,  if  it  be  an  improper 
fraction,  into  a  mixed  number ;  and  next  to  multiply 
the  integral  part  by  the  whole  number  ;  and,  finally,  add 
the  results.  Thus,  if  we  are  required  to  multiply  7-§ 
by  5,  we  first  multiply  -§  by  5,  which  gives  y>  ;  this 
reduced  to  a  mixed  number  (235.)  gives  3k  The 


CHAP.  II.  DIVISION    OF    FRACTIONS.  181 

integral  part,  7>  being  multiplied  by  5,  gives  35,  which 
being  added  to  3^  gives  38  J-,  the  total  product. 

(254.)  When  the  two  factors  are  the  same  fraction,, 
the  product,  as  in  the  case  of  integers,  is  called  the 
square  or  second  power  of  the  fraction ;  and  the  terms 
cube  or  third  power,  fourth  power,  &c.  are  applied  to 
fractions  in  the  same  sense  as  they  are  applied  to  whole 
numbers.  It  appears,  therefore,  that  any  power  of  a 
fraction  is  found  by  taking  the  same  powers  of  its  nu- 
merator and  denominator  for  the  numerator  and  deno- 
minator of  the  required  power  of  the  fraction.  This 
wiU  be  evident  by  applying  the  rule  for  the  multipli- 
cation of  fractions  to  the  determination  of  the  powers 
of  any  proposed  fraction.  Let  the  fraction  be  -J,  its 
powers  will  be  found  in  the  following  manner  :  — 

2d  power     -  *Xf= 
Sdpower     - 
4th  power    - 

5th  power   -  fxf  Xf  X  f  x|  =  | 
&c.  &c.  £c. 

It  is  evident  that  the  numerators  are  the  same  powers  of 
2,  and  the  denominators  the  same  powers  of  3,  as  the 
fractions  themselves  are  of  J- . 

(255.)  All  that  has  been  explained  respecting  the 
multiplication  of  fractions  will  render  the  process  of 
division  of  fractions  more  easily  comprehended.  The 
same  anomalous  and  paradoxical  circumstance  noticed 
in  multiplication  reappears  in  division.  According  to 
the  common  popular  notion  of  division,  derived  exclu- 
sively from  the  case  in  which  the  divisor  is  a  whole 
number,  the  effect  which  is  produced  upon  the  divi- 
dend is  to  diminish  its  amount;  and  it  is  accord- 
ingly expected  that  the  quotient  in  this  operation 
will  be  less  than  the  dividend.  It  requires,  however, 
very  little  attention  to  perceive,  that  when  once  the 
notions  and  language  of  fractions  are  introduced,  there 
is  no  reason  why  the  quotient  should  be  less  than  the 
N  3 


1  82  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

dividend.  Let  it  be  recollected  that  the  quotient  ex. 
presses  the  number  of  times  the  divisor  is  contained  in 
the  dividend.  Now,  if  the  divisor  be  a  proper  fraction, 
it  will  be  contained  in  the  dividend  a  greater  number  of 
times  than  1  is  contained  in  the  dividend,  because,  in 
that  case,  the  divisor  will  be  smaller  than  1.  Since, 
then,  1  is  contained  in  the  dividend  as  often  as  there 
are  units  in  it,  a  proper  fraction  will  be  contained  in  it 
a  greater  number  of  times  than  there  are  units  in  it ; 
and,  consequently,  in  such  a  case,  the  quotient  must 
needs  be  greater  than  the  dividend.  Let  us  suppose, 
for  example,  that  the  divisor  is  %.  This  divisor  will 
be  contained  4  times  in  every  unit  of  the  dividend.  If 
the  dividend,  then,  be  a  whole  number,  the  quotient 
will  be  another  whole  number  4>  times  as  great.  Let 
the  dividend,  for  example,  be  5  :  in  each  of  the  5  units 
which  compose  the  dividend,  the  divisor  is  contained 
4  times ;  consequently  it  is  contained  in  the  whole 
dividend  20  times.  The  quotient  is,  therefore,  20,  and 
is  4  times  as  great  as  the  dividend.  It  will  be  easily 
understood  from  this  example,  that  there  is  nothing  in- 
consistent or  absurd  in  the  supposition  that  the  process 
of  division  should  give  a  quotient  greater  than  the 
dividend. 

(256.)  But  let  us  take  the  question  under  another 
point  of  view.  Suppose  that  a  certain  number  is 
required  to  be  divided  by  f ,  or,  what  is  the  same,  by 
the  fourth  part  of  3.  If  we  divide  it,  in  the  first 
instance,  by  3,  we  shall  evidently  get  too  small  a  quo- 
tient, because  we  have  used  too  great  a  divisor :  we 
have  divided  by  3  instead  of  its  fourth  part.  The 
divisor,  then,  being  4  times  too  great,  the  quotient 
will  be  four  times  too  small.  In  order  to  compensate 
for  this  error,  we  should  multiply  the  quotient  by  4, 
and  the  product  will  be  the  true  quotient  sought.  By 
generalising  this  reasoning,  it  will  be  apparent,  that 
when  it  is  required  to  divide  any  number  by  a  fraction, 
it  is  only  necessary,  first,  to  divide  that  number  by  the 
numerator  of  the  fraction,  and  then  to  multiply  the 


CHAP.  II,  DIVISION    OF    FRACTIONS.  183 

result  by  its  denominator.  It  may  be  observed,  also, 
that,  as  the  result  must  be  the  same  in  whatever  order 
these  operations  are  performed,  we  may  invert  their 
order,  and  first  multiply  by  the  denominator  of  the 
traction,  and  then  divide  the  result  by  its  numerator. 

(257.)  This  rule  for  division  by  a  fraction  might 
have  also  been  discovered  from  the  consideration  that 
division  is  a  process  exactly  the  reverse  of  multipli- 
cation ;  that,  in  fact,  the  dividend  being  imagined  to  be 
the  product  of  the  divisor  and  another  number,  the 
effect  of  the  division  is  to  undo  the  previous  multipli- 
cation by  the  divisor.  Since,  therefore,  the  multipli- 
cation by  a  fraction  was  performed  by  multiplying  by 
the  numerator  and  dividing  by  the  denominator,  the 
reverse  process  of  division  must  naturally  be  inferred 
to  be  executed  by  dividing  by  the  numerator  and  mul- 
tiplying by  the  denominator. 

The  whole  practice  of  the  division  of  fractions 
follows  as  an  easy  and  obvious  consequence  of  the 
principle  just  proved. 

(258.)  If  the  dividend  be  a  whole  number,  we 
should,  in  general,  first  multiply  it  by  the  denominator 
of  the  divisor,  and  then  divide  the  product  by  the  nu- 
merator of  the  divisor. 

(259.)  But  if  it  happen  that  the  dividend  is  a  mul- 
tiple of  the  numerator  of  the  divisor,  then  we  should 
first  divide  by  the  numerator  of  the  divisor,  and  mul- 
tiply the  quotient  by  its  denominator.  Let  the  divisor 
be  |  and  the  dividend  22  :  first  multiply  22  by  5,  and 
divide  the  product  110  by  4.  The  quotient,  2  7^  is 
the  quotient  of  22  divided  by  -f . 

If  the  dividend  had  in  this  case  been  24,  we  should 
first  have  divided  it  by  4,  and  then  multiplied  the 
quotient,  6,  by  5.  The  product,  30,  would  then  be  the 
quotient  of  24  divided  by  4. 

(260.)  If  the  dividend  be  a  fraction,  the  method  ol 

effecting  the  division  will  easily  follow  from  what  has 

been  already  proved.     We  must  first  divide  it  by  the 

numerator  of  the  divisor  (257.).     This  may  be  always 

N  4 


184  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

done  by  multiplying  its  denominator  by  that  numerator. 
This  done,  we  must  next  multiply  the  result  by  the 
denominator  of  the  divisor.  This  may  be  always 
effected  by  multiplying  its  numerator  by  that  deno- 
minator. For  example,  let  the  divisor  be  f  and  the 
dividend  %.  To  divide  ^  by  3,  we  must  multiply  its 
denominator  by  3  :  the  result  is  7,5-r  This  must  be 
multiplied  by  4,  which  is  done  by  multiplying  its  nu- 
merator by  4  :  the  result  is  |^_,  which  is  the  quotient 
sought.  Hence  we  obtain  the  following  general 

RULE. 

To  divide  one  fraction  by  another,  multiply  the  deno- 
minator efthe  dividend  by  the  numerator  of  the  divisor, 
and  the  numerator  of  the  dividend  by  the  denominator 
of  the  divisor. 

(26r.)  When  two  fractions  consist  of  the  same 
terms,  but  in  an  inverted  position,  one  is  called  the 
reciprocal  of  the  other  :  thus  J  is  the  reciprocal  of  -J, 
\  the  reciprocal  of  4-,  &c. 

The  term  reciprocal,  thus  explained,  being  understood, 
the  rule  for  the  division  of  fractions  rnay  shortly  be 
expressed  thus :  — 

(262.)  Multiply  the  dividend  by  the  reciprocal  of  the 
divisor. 

(263.)  There  are  various  expedients  by  which,  in 
particular  cases,  the  division  of  fractions  may  be  faci- 
litated, all  depending  on  the  same  principles  as  those 
from  which  similar  methods  of  abridgment  were  in- 
ferred in  multiplication.  Since  the  division  of  one 
fraction  by  another  is  performed  by  multiplying  the 
dividend  by  one  integer,  and  dividing  it  by  another, 
and  since  there  are  two  methods  by  which  each  of  these 
operations  may  in  certain  cases  be  performed,  we  may 
frequently  choose  among  these  methods  that  which  is 
most  convenient.  In  particular  cases,  then,  instead  of 
multiplying  the  denominator  of  the  dividend  by  the  nu- 
merator of  the  divisor,  we  may  produce  the  same  effect  by 


CHAP.  II.  DIVISION    OP    FRACTIONS.  185 

dividing  the  numerator  of  the  dividend  by  the  numer- 
ator of  the  divisor.  This,  however,,  can  only  he  done 
when  the  numerator  of  the  dividend  happens  to  he  an  ex- 
act multiple  of  the  numerator  of  the  divisor.  Suppose  it 
is  required  to  divide  -|  by  |  ;  instead  of  multiplying 
5  by  2,  we  should  divide  4  by  2  :  the  result  would  be 
f .  We  multiply  the  numerator  of  this  by  3,  and  ob- 
tain -|,  which  is  the  quotient  sought. 

(26*4.)  Again,  instead  of  multiplying  the  numerator 
of  the  dividend  by  the  denominator  of  the  divisor,  we 
may  produce  the  same  effect  by  dividing  the  denomin- 
ator of  the  dividend  by  the  denominator  of  the  divisor. 
Let  it  be  required  to  divide  f  by  •§-..  We  should 
divide  6  by  3,  and  multiply  the  quotient  by  2  :  the  re- 
sult would  be  J,  the  quotient  sought. 

(265.)  If  it  should  so  happen  that  the  numerator 
and  denominator  of  the  dividend  should  both  be  exact 
multiples  of  the  numerator  and  denominator  of  the 
divisor,  we  may,  by  combining  both  the  above  me- 
thods, proceed  by  dividing  the  numerators  for  a  numer- 
ator, and  the  denominators  for  a  denominator.  Let  it 
be  required  to  divide  f^  by  -| :  we  should,  in  this  case, 
divide  6  by  2  and  15  by  3,  and  the  result  would  be  |, 
the  true  quotient. 

If  the  divisor  and  dividend  have  the  same  denomin- 
ator, the  division  may  be  performed  by  dividing  their 
numerators.  In  this  case  the  quotient,  which  would  be 
obtained  by  the  general  rule,  would  be  a  fraction,  both 
whose  terms  would  be  divisible  by  the  common  deno- 
minator ;  and,  being  divided  by  it,  would  be  reduced  to 
a  fraction  whose  terms  would  be  the  numerators  of  the 
dividend  and  divisor.  Let  the  divisor  be  ^,  and  the 
dividend  J-.  By  the  general  rule,  the  quotient  would 
be  -|£|.  Dividing  both  numerator  and  denominator 
by  5,  it  will  become  '-J. 

(266.)  If  a  fraction  be  multiplied  by  its  reciprocal, 
the  product  will  always  be  1  ;  for  in  that  case  the  nu- 
merator and  denominator  of  the  product  will  be  the 
same  :  thus,  |  X  f =|-f =1. 


186  A    TREATISE    ON    ABITH3IETIO.  BOOK.  II. 

(267.)  On  the  other  hand,  a  fraction  divided  by  its 
reciprocal  will  give  a  quotient  which  is  the  square  of 
the  dividend  (254.)  ;  for  in  that  case  the  numerator  will 
be  the  square  of  the  numerator  of  the  dividend,  and 
the  denominator  the  square  of  the  denominator  of 
the  dividend :  thus  J  divided  by  %  gives  as  a  quotient 
^,  which  is  the  square  of  -J. 

(268.)  When  it  is  necessary  to  multiply  one  mixed 
number  by  another,  or  a  mixed  number  by  a  fraction, 
the  best  general  method  of  proceeding  is,  first  to  convert 
the  mixed  number  into  an  equivalent  fraction  (235.), 
and  then  proceed  by  the  rule  for  the  multiplication  of 
fractions.  For  example,  let  it  be  required  to  multiply 
7f  by  5J.  These  being  converted  into  equivalent 
fractions,  they  become  ^3  and  4J,  which,  being  mul- 
tiplied (247.),  give  the  product  ^-Jf-1.  This,  being 
an  improper  fraction,  may  be  reduced  to  an  equivalent 
mixed  number  (224.),  by  dividing  the  numerator  by 
the  denominator  :  it  is  thus  reduced  to  45  ^. 

(269.)  When  mixed  numbers  are  required  to  be 
divided,  either  by  each  other  or  by  fractions,  they  must, 
as  in  multiplication,  be  converted  into  equivalent  frac- 
tions :  the  division  may  then  be  performed  by  the 
general  rule  for  the  division  of  fractions.  For  example, 
let  it  be  proposed  to  divide  12|  by  6f:  converting 
these  into  equivalent  fractions,  they  become  5^  and  ^p  ; 
these  being  divided  by  the  rule  (260.),  we  obtain  the 
quotient  '-//,  which,  reduced  to  a  mixed  number 
(224.),  becomes  1$J. 

(270.)  Most  of  the  methods  of  verification  or  proof 
of  the  arithmetical  operations  in  whole  numbers  will 
equally  extend  to  fractions.  Thus,  multiplication  may 
be  verified  by  division,  and  division  by  multiplication 
The  product,  when  found,  divided  by  the  multiplier, 
ought  to  give  the  multiplicand;  or  divided  by  the 
multiplicand,  ought  to  give  the  multiplier.  In  like 
manner,  in  division,  the  quotient,  multiplied  by  the 
divisor,  ought  to  give  the  dividend.  Methods  of  proof 
may  also  be  resorted  to,  which  consist  in  effecting  two 


CHAP.  II.  COMPOUND    FRACTIONS.  J#7 

operations,  which  ought  to  have  the  same  result  if 
correct.  Thus,  if  it  be  required  to  multiply  two  frac- 
tions, we  know  that  the  same  product  should  be  ob- 
tained if  one  of  them  be  doubled  and  the  other  halved. 
After  performing  the  multiplication  by  the  rule,  we  may 
then  double  or  halve  the  numerator  of  one  fraction,  and 
the  denominator  of  the  other,  and  then  perform  the  mul- 
tiplication again  on  the  fractions  thus  altered.  If  the 
product  be  the  same,  the  result  will  be  verified. 

To  verify  division,  we  may  double  the  divisor  and 
dividend,  for  in  that  case  the  quotient  should  remain 
the  same  (211.).  If  the  result,  therefore,  of  the  divi- 
sion, when  the  divisor  and  dividend  are  so  altered,  be 
the  same  as  before,  the  verification  will  be  obtained. 

(271.)  In  the  practical  operations  of  arithmetic,  we 
have  sometimes  occasion  to  consider  one  fraction  as  the 
whole  to  which  another  is  referred.  The  fraction  con- 
templated is  then  called  a  fraction  of  a  fraction,  or  a 
compound  fraction.  In  such  cases,  it  is  generally  neces- 
sary to  express  the  fractional  number  by  a  simple  frac- 
tion related  to  the  original  unit.  Let  us  suppose  that 
we  are  required  to  express  |-  of  4  by  a  simple  fraction 
referred  to  the  original  unit;  or,  in  other  words,  to 
express  the  actual  value  of  two  thirds  of  five  sevenths 
in  a  simple  fraction. 

Considering  5  sevenths  as  the  whole  of  which  we 
wish  to  take  2  thirds,  we  must  first  take  its  third  part, 
and  then  multiply  that  by  2  :  in  other  words,  we  must 
divide  %  by  3,  and  multiply  the  quotient  by  2.  Thus, 
of  the  two  fractions  concerned  in  the  question,  we  must 
multiply  the  numerators  for  a  numerator,  and  the  deno- 
minators for  a  denominator:  the  third  part  of  ^  is  g5T, 
and  twice  this  is  -J^.  It  is  evident,  then,  that  |  of  4  is 
the  same  as  the  product  of  §  and  ^ ;  and,  in  general,  a 
fraction  of  a  fraction  is  the  same  as  the  product  of  the 
two  fractions  (247.). 

(272.)  Compound  fractions  may  be  removed  from  the 
original  unit  by  more  than  two  steps  ;  in  other  words, 
we  may  be  required  to  find  the  fraction  of  a  fraction  oi  a 


188  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

fraction,  &c.  Let  it  be  required  to  find  |-  of  4  of  -fl : 
by  what  has  been  just  proved,  |-  of  -5  is  the  product  -J  <* 
of  these  two  fractions :  we  are,,  therefore,  to  find  -1°-  of 
T^r,  and  this  is,  in  like  manner,  the  product  of  these  two, 
and  is  T^°T  ;  but  90  is  the  continued  product  of  2,  5, 
and  9j>  and  231  is  the  continued  product  of  3,  7,  and 
11.  Hence  a  compound  fraction  of  the  third  order 
is  the  continued  product  of  the  fractions  of  which  it  is 
composed. 

By  the  same  reasoning  it  will  easily  appear  that  a 
compound  fraction  of  the  fourth  or  any  higher  order  is 
the  continued  product  of  its  component  fractions.  To 
find  f  of  -f  of  -ft  of  -1^,  we  must  first  find  f  of  4, 
which  is  the  product  of  these  fractions  :  we  must  next 
find  the  last  fraction  of  T9T,  which  is  the  continued  pro- 
duct of  the  first  three  component  fractions ;  we  must 
finally  determine  this  last  fraction  of  -if.?  the  result  of 
which  is  the  compound  fraction  required,  and  is  evi- 
dently the  continued  product  of  the  component  fractions. 

(273.)  A  whole  number  or  a  mixed  number  may  be 
one  of  the  component  parts  of  a  compound  fraction.  In 
such  a  case  it  is  combined  with  the  other  parts  by  mul- 
tiplication, in  the  same  manner  as  if  it  were  a  fraction. 
If  it  be  a  whole  number,  it  may  be  considered  as  a 
fraction  having  1  as  its  denominator :  if  it  be  a  mixed 
number,  it  may  be  reduced  to  an  improper  fraction, 


189 


CHAP.  III. 

DECIMALS. 

(274.)  THE  decimal  nomenclature  of  number,  as  ex- 
plained in  the  first  Chapter  of  our  first  Book,  has  a 
minor  but  no  major  limit :  in  descending,  it  stops  at 
units;  but  in  ascending,  a  regular  system  of  names  is 
contrived,  which  may  be  continued  indefinitely ;  and 
there  is,  accordingly,  no  number  so  great  that  it  may 
not  be  expressed  by  appropriate  terms.  But,  on  the 
other  hand,  no  number  less  than  1  is  capable  of  ex- 
pression by  the  system  of  language  there  explained. 
In  that  stage  of  our  progress,  however,  we  limited  our 
notions  to  that  class  of  numbers  which  are  multiples  of 
the  unit.  Having  extended  our  views,  and  increased 
the  range  of  our  notions  of  number,  in  the  present  Book, 
so  as  to  imagine  and  reason  about  numbers  less  in 
magnitude  than  unity,  it  is  natural  to  enquire  whether 
the  admirable  nomenclature,  which  has  been  so  univer- 
sally and  successfully  adopted  to  express  numbers  in  the 
ascending  scale  above  unity,  may  not  admit  of  such  a 
modification  as  may  render  it  equally  unlimited  in  the 
descending  scale  below  that  term.  We  should  thus 
consider  unity  as  the  centre  of  the  numerical  system, 
with  an  infinite  ascending  and  descending  series  above 
and  below  it.  It  would  evidently,  also,  contribute  to 
the  beauty  and  perfection  of  such  a  nomenclature,  if  it 
could  be  so  contrived  that  the  descending  series  below 
unity  should  harmonise  in  its  nomenclature  with  the 
ascending  series  above  it. 

It  requires  but  little  attention  to  the  nature  of  the 
decimal  nomenclature  contrived  to  express  whole  num- 
bers, to  discover  a  corresponding  nomenclature  in 
perfect  keeping  with  it,  by  which  numbers  less  than 
the  unit  may  be  expressed.  As  the  successive  or- 


J90  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

ders  of  units  in  the  ascending  series  are  decimal  mul- 
tiples of  the  original  unit,  we  should,  by  a  parity  of 
reasoning,  adopt,  for  the  units  of  the  successive  orders 
in  the  descending  series,  decimal  submultiples  of  the 
original  unit.  In  descending  through  the  nomenclature 
of  whole  numbers  towards  the  original  unit,  each  order 
of  units  is  a  tenth  part  of  the  order  immediately  abov^ 
it.  Thus,  hundreds  are  a  tenth  of  thousands,  tens  of 
hundreds,  and  units  of  tens.  If  we  proceed  by  a  con- 
tinuance of  the  same  method  below  the  units,  the  next 
order  of  units  would  be  tenths  ;  and  the  following  one, 
being  a  tenth  part  of  this,  would  be  hundredths,  the 
next  thousandths,  and  so  on.  The  numerical  no- 
menclature, departing  in  both  directions  from  the  units, 
would  then  be  as  follows  :  — 

2 

*  i  I  * 

*    *  «  * 


"^  ASCENDING  DESCENDING  83f 

The  transition  will  now  be  natural  and  easy  to  a  cor- 
responding extension  of  the  decimal  notation  of  num- 
ber explained  in  the  second  chapter  of  our  first  Book. 
As  in  the  decimal  nomenclature  there  was  a  minor  limit, 
so  also  in  the  decimal  notation  the  line  of  number  ter- 
minated at  the  units'  place  in  the  descending  direction, 
—  that  is,  from  left  to  right,  —  while  it  was  subject  to 
no  limit  whatever  in  the  ascending  direction,  from  right 
to  left.  Following  out,  however,  the  above  extension, 
we  should  consider  the  series  of  figures  as  equally  capa- 
ble of  extension  from  left  to  right  below  the  units' 
place,  as  in  the  other  direction.  In  the  descending 
direction  the  value  of  the  units  of  each  place  is  a  tenth 
part  of  those  immediately  on  its  left.  If  this  principle 
be  continued  below  the  units'  place^  a  digit  immediately 


CHAP.  HI.  DECIMALS.  101 

to  the  right  of  that  place  would  express  tenths,  the 
next  digit  to  the  right  would  express  hundredths,  and 
the  following  one  thousandths,  and  so  on.  The  system 
of  numeration  thus  extended  is  exhibited  in  the  follow, 
ing  example :  — 


It  is  necessary,  however,  here  to  observe,  that  so  long 
as  the  line  of  number  always  terminated  in  descending 
at  the  units'  place,  the  first  digit  on  the  right  was  al- 
ways known  to  express  the  original  units  •  and  the  orders 
of  units  expressed  by  the  other  digits  were  always 
known  by  the  relative  positions  of  these  digits  with 
respect  to  the  first.  By  the  extension  now  proposed, 
the  series  of  places  would  be  equally  unlimited  on  the 
right  and  on  the  left,  and  there  would,  therefore,  be  no 
means  of  designating  the  units'  place,  with  reference  to 
which  the  value  of  all  other  places  on  both  sides  must  be 
determined.  To  remedy  this  inconvenience,  some  means 
should  be  adopted  by  which  the  units'  place  would  be 
marked.  To  write  over  it  the  word  units,  is  the  first 
method  which  would  suggest  itself ;  but  it  is  evident 
that  any  other  shorter  mark  would  equally  answer  the 
purpose,  provided  its  adoption  were  universal.  It  has 
been  accordingly  agreed  by  all  modern  nations,  who 
have  cultivated  arithmetic  to  any  considerable  extent,  to 
mark  the  units'  place  by  a  dot  between  it  and  the  place 
of  tenths.  Thus,  the  above  number,  in  which  the 
values  of  the  several  digits  are  indicated  by  the  terms 
denoting  those  values  written  above  them,  would  be 
expressed  with  equal  clearness,  disencumbered  of  the 
words  inscribed,  in  the  following  manner  :  — 


»92  A    TREATISE    ON    ARITHMETIC.  BOOK  H. 

97865-3247. 

The  dot*  placed  on  the  right  of  5  is  the  means  of  ex- 
pressing that  5  is  the  units'  place.  That  being  once 
understood^  all  the  other  places,  both  to  the  left  and  to 
the  right;  become  known  by  their  relative  position  with 
respect  to  the  units'  place ;  and  after  a  little  practice,  the 
number  may  be  read  with  as  much  facility  as  if  the 
names  of  the  units  were  written  above  the  several 
digits. 

(275.)  The  dot  is  called  the  decimal  point ;  the 
numbers  expressed  by  digits  to  the  right  of  that  point 
are  called  decimal  fractions,  or,  shortly,  decimals. 

(276.)  A  number  is  said  to  have  so  many  decima 
places,  as  there  are  digits  in  it  to  the  right  of  the  deci- 
mal point.  Thus,  the  number  86'40032  has  five  deci- 
mal places. 

(277.)  When  the  point  does  not  appear  in  a  number, 
the  number  is  always  understood  to  be  a  whole  number, 
and  the  proper  place  for  the  point  would  be  immediately 
after  the  last  figure  on  the  right. 

A  number  which  consists  solely  of  decimal  places, 
must  always  have  the  decimal  point  before  the  first 
figure  on  the  left.  Thus,  -2376  is  such  a  number,  and 
signifies  2  tenths,  3  hundredths,  7  thousandths,  and  6 
ten-thousandths.  In  this  case  the  presence  of  the  dot 
is  necessary,  because  otherwise  the  number  would  be 
understood  to  be  a  whole  number.  To  prevent  the 
possibility  of  the  dot  escaping  the  eye  in  such  cases,  it 
is  sometimes  usual  to  place  before  it  a  nought,  thus 
0*2376  ;  the  nought  indicating  the  absence  of  all  signi- 
ficant digits  from  the  integral  places, 

(278.)  From  what  has  been  stated,  it  will  be  evident 
that  the  local  value  of  every  digit  in  a  number  will 
depend  upon  its  place  with  respect  to  the  decimal  point. 
If,  therefore,  the  decimal  point  be  removed  from  one 
position  to  another,  the  local  value  of  every  digit  will 
undergo  a  change,  and  this  change  will  be  the  same  for 
all  the  digits,  since  their  distances  from  the  decimal 

•  A  comma  is  sometimes  used  instead  of  the  dot,  thus  : — 97865,3247. 


CHAP.  III.  DECIMALS.  1Q 

point  will  be  equally  increased  or  decreased.  Let  us 
suppose  that,  in  the  following  number, 

376-531, 

the  point  is  transferred  between  the  7  and  6,  thus; 
37-6531.  The  3,  which  before  expressed  hundreds,  now 
expresses  tens,  being  one  place  nearer  to  the  point ;  the 
7,  which  before  expressed  tens,  now  expresses  units,  be- 
ing next  the  point ;  the  6,  which  before  expressed  units, 
now  expresses  the  tenths,  being  removed  from  the  left  to 
the  right  of  the  point.  In  the  same  manner,  each  of 
the  decimal  places,  being  one  place  farther  from  the 
point,  is  diminished  to  a  tenth  of  its  former  value. 
Thus  the  local  values  of  all  the  digits  are  reduced  to  a 
tenth  of  their  former  values,  and  the  number  is,  in  fact, 
divided  by  10.  For  like  reasons,  if  we  had  removed  the 
point  one  place  to  the  right,  the  local  values  of  the  digits 
would  be  increased  in  a  tenfold  proportion,  and  the 
number  would  be  multiplied  by  10.  It  appears,  then, 
that  a  number  may  be  multiplied  or  divided  by  10, 
merely  by  changing  the  position  of  the  point  one  place 
to  the  right  or  one  place  to  the  left. 

(2790  ^ty  pursuing  this  reasoning,  it  will  appear 
that,  to  multiply  a  number  by  100,  we  have  only  to  move 
the  point  two  places  to  the  right ;  and  that  we  may 
divide  it  by  100  by  removing  it  two  places  to  the  left. 
In  the  same  manner  the  number  may  be  multiplied  or 
divided  by  1000,  by  removing  the  point  three  places  to 
the  right  or  to  the  left,  and  so  on. 

(280.)  Since  the  point  is  always  understood  after  the 
last  figure  of  an  integer,  the  annexing  of  ciphers  to  the 
right  is  equivalent  to  the  removal  of  the  point  so  many 
places  to  the  right ;  and,  therefore,  the  consistency  of 
the  above  results  with  what  has  been  already  proved 
(64.  et  seq.)  will  be  apparent. 

(281.)  When  there  are  no  places  on  the  left  of  the 
point,  the  point  may  still  be  removed  any  number  of 
places  to  the  left,  by  interposing  so  many  ciphers  be- 
tween it  and  the  first  digit  of  the  decimal.     Thus,   if 
0 


194  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

we  would  remove  the  point  in  the  number  -2376,  three 
places  to  the  left,  we  have  only  to  interpose  three 
ciphers  between  the  point  and  2.  Thus  the  number 
would  be  expressed  '0002376.,  which  is  the  former 
number  divided  by  1 000. 

(282.)  It  has  been  formerly  observed,  that  ciphers 
placed  on  the  left  of  a  whole  number  produce  no 
effect  upon  its  value,  because  they  have  themselves  no 
absolute  values,  and  do  not  change  the  position  of  any 
significant  digit  with  respect  to  the  units'  place.  The 
same  observation  will  apply  with  equal  force  to  ciphers 
annexed  to  the  right  of  a  decimal.  Such  ciphers  hav- 
ing themselves  no  absolute  values,  and  the  significant 
digits  of  the  decimal  holding  the  same  place  with  re- 
spect to  the  point  as  before,  they  must  retain  their 
former  values.  Thus,  if  to  the  number  '2376  we  annex 
3  ciphers,  it  will  become  '2376000.  Each  of  the  digits 
2,  3,  7,  and  6  will  here  have  the  same  local  value  as 
before :  the  2  will  express  tenths,  the  3  hundredths,  and 
so  on ;  the  ciphers  annexed  will  express  nothing.  If, 
in  the  course  of  any  arithmetical  operation,  therefore, 
we  should  obtain  a  decimal  terminating  in  ciphers,  such 
ciphers  may  be  omitted.  Also,  if  any  arithmetical 
process  should  be  facilitated  by  annexing  ciphers  to  a 
decimal,  such  ciphers  may  be  annexed,  since  the  value 
of  the  decimal  is  not  changed  by  their  presence. 

(283.)  Since  the  decimal  point  is  always  understood 
to  be  placed  on  the  right  of  the  units'  place  in  an  in- 
teger, it  will  follow  from  what  has  been  already  proved 
(278.),  that  an  integer  may  always  be  divided  by  10  by 
introducing  the  decimal  point  between  the  units'  and 
tens'  place.  On  the  same  principle  it  may  be  divided 
by  100  by  interposing  the  decimal  point  between  the 
place  of  tens  and  hundreds,  and  by  1000  by  inter- 
posing the  decimal  point  between  the  place  of  hundreds 
.  and  thousands ;  and,  in  general,  it  may  be  divided  by  a 
number  consisting  of  1  followed  by  any  number  of  ci- 
phers, by  cutting  off  to  the  right  as  many  places  of 
decimals  as  there  are  ciphers  in  such  divisor.  For  ex- 


CHAP.  III.  DECIMALS.  19^ 

ample,  the  number  76453  divided  by  10  would  be 
7645-3  ;  divided  by  100  it  would  be  764-53;  divided 
by  1000  it  would  be  76-453.  It  appears,  therefore, 
that  the  following  numbers  are  equivalent :  — 

7645 '3  =  76r4o53  764  «53  =  7f$p 

76 -453  =  ^3  7-6453=^53 

•76453  =  J&$£         -076453  = 


In  general,  therefore,  a  number  consisting  either  of 
decimal  places  only,  or  of  places  partly  integral  and 
partly  decimal,  may  be  converted  into  an  equivalent 
fraction  by  writing  the  number  itself  without  any  de- 
cimal places  as  numerator,  and  1  followed  by  as  many 
ciphers  as  there  are  decimal  places  as  denominator. 

(284.)  It  appears,  from  these  observations,  that  any 
number  containing  decimal  places  may  be  considered 
under  different  points  of  view,  and  expressed  in  words, 
or  in  vulgar  fractions,  in  different  ways.  According  to 
the  principles  on  which  the  nomenclature  and  notation  of 
decimals  have  been  explained,  we  should  read  such  a 
decimal  as  76-453  in  the  following  manner: — 76  units, 
4  tenths,  5  hundredths,  and  3  thousandths,  or  should 
express  it  in  figures  as  follows :  — 

™  +  TU  +  TM  +  Tow- 
In  this  point  of  view,  the  number  is  regarded  as  an 
integer,  followed  by  a  series  of  fractions  having  the 
denominators  10,  100,  1000,  &c.  But  from  what  has 
been  stated  above,  we  may  also  consider  the  number  to 
express  a  single  fraction  whose  denominator  is  1000, 
and  under  this  point  of  view  it  would  be  expressed  in 
words  thus:  —  76453  thousandths,  or  in  figures,  Vo'W' 
We  may  also  consider  the  integral  part  separated  from 
the  decimal,  and  the  latter  as  a  single  fraction  whose 
denominator  is  1000,  in  which  case  it  would  be  ex- 
pressed in  words  thus :  —  76  and  453  thousandths;  or 
in  figures  thus,  76  ^<jV 

(285.)  Any  common  fraction  having  a  denominator 
consisting  of  1  followed  by  ciphers,  may  be  immediately 
converted  into  a  decimal  by  writing  down  the  nume- 
o  2 


196  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

rator  alone,  and  cutting  off  as  many  places  to  the  right 
as  there  are  ciphers  in  the  denominator.  This  follows 
evidently  from  what  has  been  just  stated.  For  ex~ 
ample,  let  the  fraction  he  7T6(j%V  5  we  ma7  om^  ^e 
denominator,  and  write  it  as  a  decimal,  thus,  76*453. 

If,  in  this  case,  there  should  be  more  ciphers  in  the 
denominator  than  there  are  places  in  the  numerator,  the 
decimal  places  must  be  supplied  by  ciphers  placed  on 
the  left :  thus,  if  the  fraction  be  To7o6o4o¥o  o*  the  e(lui- 
valent  decimal  will  then  be  0-0076453. 

It  appears,  therefore,  that  every  number,  including 
decimal  places,  may  immediately  be  converted  into  a 
vulgar  fraction,  and  every  vulgar  fraction  whose  deno- 
minator is  1  followed  by  ciphers  may  be  immediately 
converted  into  a  decimal. 

(286.)  Having  shown  that  decimals  are  always  re- 
ducible to  equivalent  vulgar  fractions,  we  shall  be 
enabled  to  deduce  from  the  established  properties  of  the 
latter,  methods  by  which  the  various  elementary  arith- 
metical operations  may  be  performed  on  decimals. 

(287-)  When  decimals  have  the  same  number  of 
decimal  places,  their  equivalent  fractions  have  the  same 
denominators  ;  and  since,  by  what  has  been  already 
proved  (282.),  we  may  annex  ciphers  to  the  right  of  a 
decimal  without  changing  its  value,  so  as  to  increase  the 
number  of  decimal  places  at  pleasure,  we  may  always  by 
this  means  equalise  the  number  of  decimal  places  in 
several  decimals,  and  thereby  reduce  their  equivalent 
fractions  to  a  common  denominator. 

For  example,  let  the  proposed  decimals  be  12-506, 
0-34,  6-0356,  23-4.  The  greatest  number  of  decimal 
places  here  is  four :  we  shall  reduce,  therefore,  all  the 
decimals  to  four  decimal  places,  by  supplying  the  neces- 
sary number  of  ciphers  in  all  that  are  deficient ;  the 
decimals  will  thus  become  12-5060,  0*3400,  6-0356, 
23-4000.  All  these  are  equivalent  to  fractions  having 
the  denominator  10000. 

(288.)  It  will  be  remembered  that  vulgar  fractions 
are  added  and  subtracted  by  first  reducing  them  to  the 


CHAP.  III.  DECIMALS.  197 

same  denominator,  adding  or  subtracting  their  numer- 
ators, and  then  subscribing  their  common  denominator, 
(234.)  This  rule  may  be  at  once  transferred  to  de- 
cimals:— "  To  add  or  subtract  decimals,  equalise  their 
decimal  places,  add  or  subtract  them  as  if  they  were 
whole  numbers,  and  take  in  the  result  the  same  number 
of  decimal  places."  The  reason  of  this  rule  is  obvious : 
by  equalising  the  decimal  places  we  reduce  the  equivalent 
fractions  to  the  same  denominator  (28?.) ;  by  adding 
or  subtracting  as  whole  numbers  the  decimals  thus  mo- 
dified, we  add  or  subtract  the  numerators  of  the  equi- 
valent fractions,  and  by  taking  in  the  result  the  same 
number  of  decimal  places,  we  subscribe  their  common 
denominator.  (284.)  The  rule  is  thus  brought  strictly 
under  that  established  for  the  addition  or  subtraction  of 
vulgar  fractions. 

Let  it  be  required  to  add  the  following  decimals, 
32-4036,  245-379,  12-0476',  9'38,  459*2375.  We 
shall  equalise  the  decimal  places  in  these  by  annexing 
one  cipher  to  the  second  and  two  to  the  fourth  :  this 
being  done,  let  the  decimals  be  added  as  whole  numbers  ; 
the  result  will  be  as  follows :  — 


758-4497 


We  have  here  pointed  off  four  places  in  the  total  ob- 
tained, the  equivalent  fraction  thus  having  10000  for 
its  denominator,  which  is  the  same  as  the  denominator 
of  the  several  decimals  which  are  added  together. 

In  practice  it  is  not  necessary  or  usual  to  annex  the 
ciphers  to  fill  the  deficient  places :  it  will  be  sufficient 
so  to  range  the  numbers  one  under  the  other,  that  the  deci- 
mal point  of  one  number  shall  be  immediately  under  that 
of  the  other.  It  is  evident  that,  if  this  is  attended  to,  the 
result  of  the  operation  will  be  the  same  as  if  the  ciphers 
o  3 


198  A    TREATISE    ON    ARITHMETIC.  BOOK  H. 

were  supplied:    the  above  number   would  then  stand 
thus : — 

32-4056 
245-379 
12-0476 

9-38 
459-2375 


758-4497 

Let  it  be  required  to  subtract  23-0784  from  62-09  : 
placing  the  numbers  as  above,  the  operation  will  stand 
thus :  — 

62  09 

23-0784 

39-0116 


The  subtraction  is  here  performed  as  if  the  deficient 
places  above  the  8  and  4  of  the  subtrahend  were  sup- 
plied by  ciphers,  and  the  number  of  decimal  places  in 
the  remainder  is  the  same  as  it  would  be  in  the  minuend 
and  subtrahend  had  the  places  been  supplied.  Thus 
the  minuend  and  subtrahend  are  virtually  reduced  to  the 
same  denominator,  and  that  denominator  is  preserved  in 
the  remainder. 

(289-)  We  shall  with  equal  facility  derive  the  rule 
for  the  multiplication  of  decimals  from  that  established 
for  the  multiplication  of  vulgar  fractions.  If  the  deci- 
mals be  multiplied  as  whole  numbers,  we  shall,  in  fact, 
multiply  the  numerators  of  the  equivalent  fractions  ; 
but  to  obtain  the  denominator  of  the  product,  it  is 
necessary  to  multiply  the  denominators  of  the  equivalent 
fractions :  these  denominators  are  two  numbers  ex- 
pressed by  1,  followed  by  as  many  ciphers  as  there  are 
decimal  places  ;  and  the  product  of  two  such  numbers 
will  be  1  followed  by  as  many  ciphers  as  there  are  in 
both  denominators  taken  together:  thus,  if  there  be 
three  decimal  places  in  one  of  the  numbers,  and  four  in 
the  other,  the  denominators  of  the  equivalent  fractions 
will  be  1000  and  10000 ;  the  product  of  these  will  be 
10000000 :  this  being  the  denominator  of  the  product, 


CHAP.  III.  DECIMALS.  199 

it  will  follow  that  we  must  take  in  the  product  7 
decimal  places ;  that  is,  as  many  decimal  places  as  are 
in  the  two  numbers  taken  together.  To  multiply  deci- 
mals, therefore,  we  have  the  following  rule :  —  "  Mul- 
tiply them  as  whole  numbers,  and  take  in  the  product 
as  many  decimal  places  as  are  in  the  multiplicand  and 
multiplier  taken  together." 

Let  it  he  required  to  multiply  35'407  by  12-54 ;   the 
process  will  be  as  follows :  — 

35-407 
12-54 


1416-28 
177035 
70814 
35407 

444-00378 

In  this  example  there  are  3  decimal  places  in  the  mul- 
tiplicand, and  2  in  the  multiplier :  we,  therefore,  take 
5  in  the  product. 

The  above  rule  for  the  multiplication  of  decimals 
may  also  be  explained  as  follows :  —  If  we  remove  the 
dot  from  the  multiplicand  in  the  above  number,  we  mul- 
tiply it  by  1000  (279.),  and  by  removing  the  dot  from 
the  multiplier,  we  multiply  it  by  100 ;  the  figures  are 
thus  converted  into  whole  numbers,  and  their  product 
found.  But  since  one  was  previously  multiplied  by 
1000  and  the  other  by  100,  the  product  will  be  100000 
times  too  great ;  consequently,  to  reduce  it  to  its  true 
value,  we  must  divide  it  by  100000;  but  this  is  done 
by  taking  in  it  five  decimal  places. 

If  one  only  of  the  numbers  to  be  multiplied  includes 
decimal  places,  we  must  then  take  in  the  product  as 
many  decimal  places  as  it  contains  :  thus,  if  we  are  re- 
quired to  multiply  23  by  4* 5 7,  we  multiply  it  by  457, 
considered  as  a  whole  number;  the  product  will  be 
100  times  too  great,  and  consequently  will  be  reduced 
to  its  true  value  by  taking  in  it  two  decimal  places. 

(2^0.)  It  may  happen  that,  after  the  product  has 
o  4 


200  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

been  found,  the  total  number  of  places  it  contains  may 
be  less  than  the  decimal  places  in  the  two  numbers  mul- 
tiplied. Jn  this  case,  it  will  be  necessary,  in  supplying 
the  decimal  point,  to  fill  the  deficient  places  on  the  left 
by  ciphers. 

Let  it  be  required  to  multiply  0-03054  by  0-023  ; 
the  process  will  be  as  follows  :  — 

3054 
23 


9162 
6108 


70242 


0-00070242 

We  have  here,  in  the  first  instance,  multiplied  the 
decimals,  considered  as  whole  numbers ;  but  their  pro- 
duct, 70242,  had  only  five  places,  while  the  number  of 
decimal  places  in  the  numbers  to  be  multiplied  amounted 
to  eight :  it  was  necessary  to  take  eight  decimal  places 
in  the  product,  and  therefore  three  ciphers  were  inter- 
posed between  the  decimal  point  and  the  first  figure  of 
the  product  found. 

(291.)  It  has  been  proved  that  two  fractions  may  be 
divided  one  by  the  other  when  they  have  the  same 
denominator,  by  merely  dividing  their  numerators, 
omitting  altogether  their  denominators.  (265.)  This 
principle  renders  the  process  for  the  division  of  deci- 
mals extremely  simple.  Their  equivalent  fractions  may 
be  always  reduced  to  the  same  denominator  by  equalising 
their  decimal  places.  The  general  rule,  therefore,  for 
the  division  of  decimals  is,  to  "  Equalise  the  decimal 
places,  expunge  the  decimal  points,  and  divide  them  as 
whole  numbers ;"  this  being  obviously  equivalent  to 
reducing  them  to  the  same  denominator,  and  dividing 
their  numerators. 

Let  it  be  required  to  divide  43-047  by  2*53698. 
The  number  of  decimal  places  will  be  equalised  by  an- 
nexing two  ciphers  to  the  dividend:  this  being  done,  and 


CHAP.  III.  DECIMALS.  201 

the  decimal  point  removed,  the  process  of  division  will 
be  as  follows  :  — 

253698)4304700(16 
253698 


245532 

The  integral  part  of  the  quotient  is  then  l6}  and  the 
remainder  is  245532  :  this  remainder  being  less  than 
the  divisor,  the  division  can  be  carried  no  farther  in 
whole  numbers  ;  but  the  total  quotient  may  be  made  up 
by  adding  the  fraction  |f£f $f,  to  16  (208.);  the  total 
quotient  is  then  16  f  f|-fM' 

It  will  be  observed  that  the  quotient  found  in  this 
way  is  not  itself  a  decimal,  but  is  a  mixed  number,  one 
part  being  a  whole  number,  and  the  other  a  vulgar  frac- 
tion. If  it  be  required  to  exhibit  the  quotient  as  a  de- 
cimal, it  will  be  necessary  to  convert  the  vulgar  fraction 
into  an  equivalent  decimal  fraction :  the  method  of 
doing  this  we  shall  presently  explain. 

If,  after  equalising  the  decimal  places  and  expunging 
the  point,  the  divisor  be  greater  than  the  dividend,  the 
division,  as  whole  numbers,  cannot  be  effected;  and  the 
quotient  can  only  be  expressed  by  a  fraction  whose  nu- 
merator and  denominator  are  the  dividend  and  divisor 
thus  changed.  For  example,  suppose  it  is  required  to 
divide  0'13  by  4*7 ;  equalising  the  decimal  places  and 
expunging  the  point,  the  numbers  will  become  13  and 
470;  the  quotient  will  be  expressed  by  the  fraction 

*vv 

The  quotient  can  only  be  expressed  in  decimals,  after 
we  have  obtained  a  method  of  converting  vulgar  frac- 
tions into  decimals. 

(292.)  It  has  been  proved  (175.)  that  any  increase 
which  takes  place  in  the  dividend  produces  a  corre- 
sponding increase  in  the  quotient,  the  divisor  being  sup- 
posed to  remain  the  same :  thus,  if,  preserving  the 
divisor,  we  multiply  the  dividend  by  100  or  1000,  we 


202  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

multiply  the  quotient  also  by  100  or  1000.  If  we 
would,  in  such  a  case,  restore  the  quotient  to  that  value 
which  it  would  have  had  if  no  increase  had  taken  place 
in  the  dividend,  we  must  obviously,  in  such  a  case,  di- 
vide it  by  100  or  1000.  This  observation  will  lead  us 
to  an  easy  method  of  converting  a  vulgar  fraction  into 
an  equivalent  decimal. 

Let  it  be  proposed  to  reduce  the  fraction  J£  to  an 
equivalent  decimal.  It  has  been  shown  that  -J--J-  means 
the  quotient  which  would  be  obtained  by  dividing  13  by 
25.  Now,  if  we  multiply  13  by  100,  and  then  perform 
the  division  by  25,  we  shall  obtain  a  quotient  100  times 
greater  than  that  which  we  should  have  had  if  we  di- 
vided 13  by  25  without  any  previous  change.  Per- 
forming the  division  as  above  stated,  we  obtain  the 
following  result :  — 

25)  1300(52 
125 

50 
50 


The  quotient  of  1300  divided  by  25  is  then  52 ;  but 
this  quotient  is  necessarily  100  times  greater  than  the 
quotient  of  13  by  25,  since  the  dividend  has  been  mul- 
tiplied by  100  :  if  we  divide  52  by  100  we  shall  there- 
fore obtain  the  true  quotient  of  13  by  25 ;  but  52  may 
be  divided  by  100,  by  placing  the  decimal  point  before 
the  5.  (283.)  The  decimal  0*52  is,  therefore,  the  true 
quotient  of  13  by  25,  and  is  equal  to  the  fraction  ^-|. 

If  this  process  be  examined,  it  will  be  perceived  that, 
in  order  to  reduce  the  fraction  -J-J  to  an  equivalent  de- 
cimal, we  have  annexed  two  ciphers  to  the  numerator, 
have  then  divided,  as  we  should  with  integers,  by  the 
denominator,  and  have  taken  two  decimal  places  in  the 
quotient.  By  this  means  the  numerator  of  the  proposed 
fraction  is  converted  into  a  number  which  is  exactly 
divisible  by  the  denominator,  without  a  remainder.  If 
we  had  annexed  only  one  cipher,  so  as  to  convert  the 


CHAP.  III.  DECIMALS.  203 

numerator  into  130,  we  should  have  had  a  remainder, 
so  that  the  quotient  would  not  he  complete.  If,  on  the 
other  hand,  we  had  annexed  more  than  two  ciphers,  there 
would  be  no  remainder  after  the  division  had  proceeded 
as  far  as  the  second  cipher ;  and  the  third  and  all  the 
succeeding  figures  of  the  quotient  would  he  ciphers : 
the  equivalent  decimal  would  still  he  obtained,  but  it 
would  terminate  in  one  or  more  useless  ciphers. 

Let  us  suppose  that  we  had  annexed  4<  ciphers  to  the 
numerator ;  the  quotient  would  then  have  been  5200 : 
having  multiplied  the  numerator  by  10000,  this  quo- 
tient will  be  1 0000  times  greater  than  the  true  quotient 
of  13  divided  by  25  :  to  reduce  it  to  its  true  value,  we 
must,  therefore,  divide  it  by  10000,  which  will  be  done 
by  taking  in  it  4  decimal  places ;  the  true  quotient  will 
then  be  0*5200;  but  the  two  final  ciphers  here  are  insig- 
nificant and  useless  (282.),  and  the  equivalent  decimal, 
as  before,  is  0'52. 

(293.)  From  this  we  may  collect  the  following  ge- 
neral rule  for  converting  a  vulgar  fraction  into  an  equi- 
valent decimal. 

Place  the  numerator  as  dividend,  and  the  denominator 
as  divisor,  and  annex  a  cipher  to  the  former  ;  divide  the 
numerator,  with  the  cipher  so  annexed,  by  the  denomi- 
nator, and  write  down,  as  in  division,  the  first  figure  of 
the  quotient ;  to  the  remainder,  if  there  be  any,  annex 
another  cipher,  and  divide  as  before  to  obtain  the  second 
figure  of  the  quotient,  and  continue  to  annex  a  cipher  to 
every  remainder,  writing  down  the  successive  figures  of 
the  quotient,  as  in  division.  When  a  remainder  is  ob~ 
tained,  which,  with  a  cipher  annexed,  is  exactly  divisible 
by  the  denominator,  the  operation  will  be  complete*  Point 
off  as  many  decimal  places  in  the  quotient  as  there  were 
ciphers  annexed,  and  the  decimal  thus  obtained  will  be 
equivalent  to  the  vulgar  fraction. 

It  will  be  perceived  in  this  process,  that  annexing  the 
ciphers  to  the  several  remainders  is  equivalent  to  annex- 
ing them  to  the  original  numerator ;  so  that,  in  fact,  the 
numerator  has  been  by  this  means  multiplied  by  1  fol- 


204  A    TREATISE    ON   ARITHMETIC.  BOOK  II. 

lowed  by  as  many  ciphers  as  have  been  annexed  ;  and 
by  taking  the  same  number  of  decimal  places  in  the 
quotient,  the  quotient  is  divided  by  that  number.  These 
two  processes,  as  already  explained,  neutralise  each 
other  :  the  ciphers  annexed  to  the  dividend  would  ren- 
der the  quotient  too  great,  in  the  proportion  of  a  num- 
ber expressed  by  1  followed  by  the  ciphers  to  unity. 
By  taking  the  same  number  of  decimal  places  in  the 
quotient,  it  is  diminished  in  exactly  the  same  proportion, 
and  therefore  restored  to  its  true  value. 

(294.)  If  the  numerator  of  the  fraction  be  greater 
than  the  denominator,  the  first  step  in  the  division  will 
be  effected  without  annexing  ciphers.  In  that  case, 
when  the  division  is  complete,  the  quotient  will  contain 
a  greater  number  of  places  than  the  number  of  ciphers 
which  have  been  annexed,  and  the  equivalent  decimal 
will,  therefore,  have  one  or  more  places  on  the  left  of 
the  point.  This  is  a  circumstance  which  might  be 
easily  anticipated  ;  for  if  the  numerator  of  the  fraction 
be  greater  than  its  denominator,  the  fraction  must  be 
greater  than  1,  and  therefore  a  part  of  its  decimal  ex- 
pression must  consist  of  whole  numbers. 

Let  the  fraction  be  lfg  ;  the  process  for  converting 
this  into  a  decimal  will  be  as  follows  :  — 

16)  197(12-3125 
16 

~37 
32 

50 
48 


16 


32 

80 
80 


CHAP.  III.  DECIMALS.  205 

We  have  here  taken  four  decimal  places  in  the  quo- 
tient, having  annexed  a  cipher  to  each  of  the  last  four 
remainders. 

(295.)  It  must  be  evident  that  every  case  in  which 
the  division  of  one  whole  number  by  another  is  incom- 
plete, in  consequence  of  the  last  remainder  being  less 
than  the  divisor,  may  by  these  means  be  continued  so 
as  to  obtain  the  remainder  of  the  quotient  in  decimals. 
It  is  only  necessary  to  annex  a  cipher  to  the  last  re- 
mainder, and  continue  the  division  in  the  same  manner, 
annexing  ciphers  until  a  remainder  is  found  which,  with 
a  cipher  annexed,  is  exactly  divisible  by  the  divisor. 
Let  as  many  decimal  places  be  then  taken  in  the  quo- 
tient as  there  were  ciphers  annexed. 

(296.)  It  sometimes  happens  that,  in  converting  a 
fraction  into  an  equivalent  decimal,  it  is  necessary  to 
annex  two  or  more  ciphers  to  the  numerator,  before  any 
figure  of  the  quotient  is  obtained :  the  rule  already 
given  will  still  apply  to  this  case  ;  but  as  the  number  of 
places  in  the  quotient  will,  under  these  circumstances, 
be  less  than  the  number  of  ciphers  annexed,  it  will  be 
necessary  to  place  ciphers  to  the  left  of  the  quotient,  in 
order  to  make  up  the  necessary  number  of  decimal 
places. 

Let  it  be  proposed  to  convert  the  fraction  T|-^  into 
an  equivalent  decimal ;  the  process  will  be  as  follows : — 

125)  300  (.024 
250 

500 
500 


The  quotient,  in  this  case,  being  24,  and  having  only  two 
places,  while  three  ciphers  have  been  annexed,  it  is  ne- 
cessary to  place  a  cipher  to  the  left  of  the  2,  in  order  to 
make  up  three  decimal  places,  which  it  is  necessary  to 
take  in  the  quotient. 

(297.)  In  any  number,  whether  integral  or  decimal. 


206  A    TREATISE    ON    ARITHMETIC.  BOOK  H. 

the  local  value  of  a  unit,  occupying  any  place,  is  greater 
than  the  total  value  of  the  digits  to  the  right  of  that 
unit,  to  whatever  number  of  places  such  digits  may  ex- 
tend. This,  which  is  evidently  true  with  whole  num- 
bers, is  not  less  apparent  in  decimals.  It  is  evident  that 
a  unit  in  the  place  of  thousands  must  have  a  greater 
value  than  the  total  amount  of  any  digits  which  can 
fill  the  inferior  places  of  hundreds,  tens,  and  units: 
the  greatest  digits  which  can  occupy  these  places  are 
nines,  and  if  they  were  all  filled  by  nines,  their  total 
amount,  999,  would  still  be  1  less  than  the  value  of  a 
single  unit  in  the  thousands'  place.  The  same  reason- 
ing will  hold  good  for  any  other  place  in  the  line  of  in- 
tegers. 

It  is  the  same  with  decimals.  In  the  decimal 
O1 54376  a  unit  in  the  second  place  from  the  point  is 
greater  than  the  aggregate  value  of  all  the  digits  which 
succeed  it ;  for  if  all  those  digits  were  nines,  they  would 
have  the  greatest  value  which  by  possibility  could  be 
conferred  on  them,  and  yet  they  would  still  be  less  than 
a  single  unit  in  the  second  place  from  the  point,  as  may 
be  easily  proved.  The  total  value  of  the  nines  filling 
the  places  just  mentioned  would  be  0-00999 :  now  if 
to  this  number  we  add  O'OOOOl,  we  shall  obtain  the 
number  O'Ol 000;  but  this  number  is,  in  fact,  a  unit 
in  the  second  place  from  the  point,  and  since  it  is  ob- 
tained by  adding  a  certain  number  to  O1 00999*  it  must 
be  a  greater  number  than  the  latter.  The  same  reason- 
ing will  apply  in  every  case,  and  we  therefore  infer  that 
a  single  unit  occupying  any  decimal  place,  is  of  greater 
value  than  the  total  amount  of  all  the  places  to  the  right 
of  it,  however  numerous  those  places  may  be.  Thus, 
in  the  number  0-54376,  a  single  unit  of  the  5,  that  is, 
a  tenth,  is  greater  than  the  value  of  all  the  succeeding 
figures.  In  the  same  manner  the  total  value  of  0'0037§ 
is  less  than  a  hundredth,  and  the  value  of  0-000376  is 
less  than  a  thousandth,  and  so  on.  We  shall  presently 
see  the  importance  of  this  conclusion. 


CHAP.  III.  DECIMALS.  207 

(298.)  In  the  process  for  converting  a  fraction  into  a 
decimal,  or  for  continuing  the  operation  of  division 
where  a  remainder  is  found  less  than  the  divisor,  by 
annexing  decimal  places  to  the  quotient,  we  have  sup- 
posed that  hy  continuing  the  operation  a  remainder  will 
at  length  he  found,  which,  with  a  cipher  annexed,  will 
be  exactly  divisible  by  the  divisor.  This,  however,  is 
frequently  not  the  case,  and  it  will  happen  that  the 
division  may  be  continued  without  end,  remainders  con- 
tinually arising,  none  of  which  are  divisible  exactly  by 
the  divisor.  In  sucji  a  case  it  is  impossible  to  express 
the  exact  quotient  by  decimals ;  nevertheless  we  may 
obtain  a  number  expressed  in  decimals  differing  from 
the  exact  quotient  by  as  small  a  quantity  as  may  be 
desired.  This  will  be  easily  understood  when  it  is 
remembered  that  a  single  unit  in  any  decimal  place  is  of 
greater  value  than  the  total  amount  of  all  the  decimal 
places  which  can  follow  it,  however  numerous  these  may 
be.  Suppose  that  the  process  of  division  be  continued  by 
annexing  ciphers  until  ten  ciphers  have  been  annexed ; 
it  would  be  necessary  then  to  take  ten  decimal  places 
in  the  quotient :  had  the  operation  been  continued,  the 
remainder  of  the  quotient  would  be  expressed  by  digits 
occupying  decimal  places  to  the  right  of  the  tenth 
place.  The  total  value  of  such  digits,  however  nu- 
merous they  might  be,  could  never  amount  to  the  value 
of  a  single  unit  in  the  tenth  place  of  decimals ;  the  de- 
ficient part  of  the  quotient  would  therefore  be  less  than 
the  1000000000th  part  of  the  original  unit.  To 
whatever  extent,  therefore,  the  division  be  carried  by 
annexing  ciphers,  the  remainder  of  the  quotient  will  al- 
ways be  a  proportionally  small  part  of  the  unit ;  and  as 
there  is  no  limit  to  the  extent  to  which  we  may  carry 
the  operation,  so  there  is  no  limit  to  our  approximation 
to  the  true  quotient. 

Let  it  be  required  to  divide  294  by  7*356,  and  to 
obtain  a  decimal  differing  from  the  true  quotient  by  less 
than  the  10000th  part  of  the  unit;  the  process  is  as 
follows :  — 


208  A    TREATISE    ON    ARITHMETIC.  BOOK  II. 

7356)  294000(39-9673 
22068 


73320 
66204 

71160 
66204 

49560 
44136 

54240 
51492 


27480 
22068 

5412 

We  have  first  equalised  the  decimal  places  by  annex- 
ing three  ciphers  to  the  dividend ;  the  decimal  points 
are  then  removed,  and  the  numbers  are  treated  as  whole 
numbers  (291.)-  The  first  figures  of  the  quotient  are 
obtained  without  annexing  any  more  ciphers  to  the 
dividend,  and  there  are  subsequently  ciphers  annexed 
to  four  remainders :  four  decimal  places  are  therefore 
taken  in  the  quotient  (293.).  But  there  is  still  a  final 
remainder,  and  therefore,  if  the  process  were  continued, 
the  quotient  would  contain  decimal  places  to  the  right 
of  its  last  figure.  The  total  value  of  these  decimal 
places,  however,  would  be  less  than  the  value  of  a  single 
unit  in  the  last  place  of  the  quotient :  that  value  being 
the  10,000th  part  of  the  unit,  it  follows  that  the  quo- 
tient above  found  is  less  than  the  true  quotient  by  a 
number  less  than  the  10,000th  part  of  the  unit. 


209 


BOOK  III. 

COMPLEX    NUMBERS. 

CHAPTER  I. 

OF    COMPLEX    NUMBERS    IN    GENERAL.  THEIR    REDUCTION    AND 

SIMPLIFICATION. 

(299.)  SIMPLE  numbers  are  those  which  are  formed  by 
the  aggregation  of  the  same  primary  or  original  units, 
and  to  such  the  investigations  in  the  preceding  Book 
have  been  confined ;  for  although  the  units  expressed 
by  digits  occupying  different  places  possess  different 
values,  yet  the  relations  which  these  values  bear  to  the 
primary  or  original  unit  are  always  explicitly  denoted 
by  the  position  of  the  digit.  We  shall  now  direct  our 
attention  to  another  class  of  numbers,  into  which  units  of 
different  magnitudes  enter,  and  which  are  thence  called 
complex  or  compound  numbers. 

Complex  numbers  owe  their  origin  to  the  inconve- 
nience and  difficulty  found  in  the  ordinary  affairs  of 
life  in  the  use  of  very  high  numbers.  When  quantities 
are  to  be  expressed  which  would  require  high  numbers, 
the  difficulty  is  therefore  avoided  by  adopting  a  large 
unit ;  but  if  such  a  unit  alone  were  adopted,  a  similar 
inconvenience  would  arise  when  very  small  quantities 
of  the  same  kind  are  expressed,  for  in  that  case  very 
complex  and  inconvenient  fractions  would  be  unavoid- 
able. Several  units  of  different  magnitudes  are  there- 
fore employed;  and  the  quantity,  instead  of  being 
expressed  by  one  number  consisting  of  many  places  of 
figures,  is  expressed  by  several  numbers,  each  having 
different  units. 

Let  us  suppose  that  a  penny  were  adopted  as  the  unit 
of  numbers  expressing  money ;  a  pound  sterling  would 
then  be  expressed  by  the  number  240,  ten  pounds  by 
p 


210  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

2400,  and  so  on.  These,  however,  being  sums  which 
it  is  necessary  frequently  to  express,  the  use  of  such 
high  numbers  would  be  attended  with  manifest  incon- 
venience. If,  on  the  other  hand,  a  pound  were  taken  as 
the  pecuniary  unit,  such  a  sum  as  a  penny,  which  it  is 
likewise  necessary  frequently  to  express,  could  only  be 
denoted  by  the  fraction  ^-1^ ;  and  all  intermediate  sums 
between  a  penny  and  a  pound  could  only  be  expressed  by 
fractions  of  proportionate  value^  the  numerators  and  de- 
nominators of  which  would  frequently  be  high  numbers. 

Such  inconvenience  would  be  unavoidable  under  any 
circumstances  in  which  one  unit,  and  one  only,  could  be 
adopted.  They  are  avoided  by  the  use  of  several 
units  of  different  magnitudes ;  so  that  when  small  sums 
are  to  be  expressed,  small  units  are  used,  the  higher  units 
being  resorted  to  for  higher  sums.  Also  in  the  higher 
sums,  where  it  is  necessary  to  express  fractions  of 
the  higher  units,  instead  of  such  fractions,  numbers 
of  equivalent  value  composed  of  the  inferior  units  are 
used.  These  observations,  which  are  applicable  gene- 
rally to  all  complex  numbers,  will  be  more  clearly 
apprehended  when  we  proceed  further  with  this  subject. 

The  complex  numbers  which  are  used  in  the  ordinary 
affairs  of  life  are  those  which  are  necessary  to  express 
TIME,  MONEY,  SPACE,  and  WEIGHT. 

Space  is  expressed  by  different  kinds  of  complex  num- 
bers, according  to  the  way  in  which  it  is  considered. 
There  are  measures  of  lengthy  measures  of  surface,  and 
measures  of  capacity. 

We  shall  now  proceed  to  explain  the  different  classes 
of  units,  and  their  mutual  relation  as  to  magnitude  in 
these  several  species  of  complex  numbers. 

MEASURES  OF  TIME. 

(300.)  A  DAY  is  that  interval  of  time  which  elapses 
between  two  successive  massages  of  the  centre  of  the  sun 
over  the  same  point  of  the  heavens.  This  interval  be- 
ing divided  into  24?  equal  parts,  each  of  these  parts  is 
called  an  HOUR. 


CHAP.  I.  COMPLEX    NUMBERS.  211 

An  HOUR  being  divided  in  60  equal  parts,  each  of 
these  parts  is  called  a  MINUTE. 

A  MINUTE  being  divided  into  60  equal  parts,  each  of 
these  parts  is  called  a  SECOND. 

A  SECOND  is  the  smallest  subdivision  of  time  used 
for  the  ordinary  purposes  of  life,  but  for  the  more  exact 
purposes  of  science,  a  second  is  again  supposed  to  be 
subdivided  into  tenths. 

Hours  are  expressed  by  placing  the  letter  h  above  the 
number ;  minutes,  by  placing  m  or  '  above  it ;  and  se- 
conds, by  s  or  " :  thus,  22  hours  35  minutes  and  56 
seconds  would  be  expressed  —  22h  35m  56s,  or22h  35' 
56''. 

(301.)  Hence  it  appears,  that  when  a  number  ex- 
presses days,  it  can  be  converted  into  an  equivalent 
number  of  hours  by  multiplying  it  by  24  ;  and  a  num- 
ber expressing  hours  may  be  converted  into  days,  or 
fractions  of  a  day,  by  dividing  it  by  24. 

(302.)  When  a  number  expresses  hours,  it  may  be 
converted  into  an  equivalent  number  expressing  minutes, 
by  multiplying  it  by  60 ;  and  a  number  expressing 
minutes  may  be  converted  into  an  equivalent  number 
expressing  hours,  or  fractions  of  an  hour,  by  dividing 
it  by  60. 

In  the  same  manner,  numbers  expressing  minutes 
may  be  converted  into  numbers  expressing  seconds,  and 
vice  versa. 

(303.)  Since  there  are  24  hours  in  a  day,  and  50  mi- 
nutes in  an  hour,  we  shall  find  the  number  of  minutes  in 
a  day  by  multiplying  24  by  60  ;  the  product  is  1440. 

A  number  expressing  days,  therefore,  may  be  con- 
verted into  an  equivalent  one  expressing  minutes,  by 
multiplying  it  by  1 440  ;  and  a  number  expressing  mi- 
nutes may  be  converted  into  an  equivalent  one  express- 
ing days,  or  fractions  of  a  day,  by  dividing  it  by  1440. 

(304.)  Since  there  are  60  minutes  in  an  hour,  and 
60  seconds  in  a  minute,  we  shall  find  the  number  of 
seconds  in  an  hour  by  multiplying  60  by  60 ;  the  pro- 
duct is  3600. 

p  2 


212  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

A  number  expressing  hours  may  therefore  be  con- 
verted into  an  equivalent  one  expressing  seconds,  by  mul- 
tiplying it  by  3600;  and  a  number  expressing  seconds 
may  be  converted  into  an  equivalent  one  expressing  hours, 
or  fractions  of  an  hour,  by  dividing  it  by  3600. 

(305.)  Since  there  are  1440  minutes  in  a  day,  and 
60  seconds  in  a  minute,  we  shall  find  the  number  of 
seconds  in  a  day  by  multiplying  1440  by  60;  the  pro- 
duct is  86400. 

A  number  expressing  days  may  therefore  be  converted 
into  an  equivalent  one  expressing  seconds,  by  multiply- 
ing it  by  86400;  and  a  number  expressing  seconds  may 
be  converted  into  an  equivalent  one  expressing  days,  or 
fractions  of  a  day,  by  dividing  it  by  86400. 

(306.)  A  common  year  consists  of  365  days,  and  a 
leap  or  bissextile  year  of  366  days.  Every  fourth  year, 
commencing  from  the  birth  of  Christ,  is  a  leap  or  bis- 
sextile year,  and  every  other  a  common  year.  Thus,  if 
the  number  expressing  any  year  from  the  birth  of 
Christ  be  divided  by  4,  that  year  will  be  a  leap  year  if 
there  be  no  remainder,  but  otherwise  it  will  be  a  com- 
mon year.  Thus  1832,  being  divided  by  4,  gives  the 
quotient  458  without  a  remainder;  the  year  1832 
was  therefore  a  leap  year.  If  1834  be  divided  by  4, 
there  will  be  a  remainder,  2  ;  1834  is  therefore  a  com- 
mon year.  There  are,  however,  exceptions  to  this  rule  : 
the  years  which  complete  centuries  from  the  birth  of 
Christ  are  leap  years  only  when  their  first  two  figures 
are  divisible  by  4  without  a  remainder.  Thus,  of  the 
following,  those  only  which  are  marked  *  are  leap  years  : 
1600*,  1700,  1800,  1900,  2000*,  2100,  2200,  2300, 
2400*,  &c.  &c. 

(30?.)  The  results  of  the  above  calculations  are  ex- 
pressed in  the  following 

TABLE  OP  TIME. 
1  minute  =  60  seconds. 
1  hour  =  60  minutes  =  3600  seconds. 
1  day  =  24  hours  =  1440  minutes  =  86400  seconds. 
1   common  year  =  365  days. 
1  leap  year  =  366  days. 


CHAr.  I.  TIME. MONEY.  213 

Besides  the  above  divisions  of  time,  there  are  some 
others  in  ordinary  use,  such  as  weeks,  months,  centu- 
ries. A  week  is  composed  of  7  days,  and  a  number 
expressing  weeks  is  therefore  converted  into  one  ex- 
pressing  days  by  multiplying  it  by  7;  a  number  express- 
ing days  is  converted  into  one  expressing  weeks,  and 
fractions  of  a  week,  by  dividing  it  by  7. 

The  word  month  is  used  in  different  senses :  it  is 
sometimes  used  to  express  4  weeks ;  in  this  sense  it 
is  distinguished  from  the  calendar  month,  which  is  an 
interval  of  time  varying  in  length.  The  year  consists 
of  12  calendar  months,  some  of  30,  and  others  of  31 
days  ;  one  month  alone  having  28  days,  except  in  leap 
year,  when  it  has  29-  The  months  which  have  30 
days  are  April,  June,  September,  and  November.  Feb- 
ruary has  28  days  in  common  years,  and  29  in  leap 
years.  The  remaining  months  have  31  days.* 

MONEY. 

(308.)  The  classes  of  units  by  which  sums  of  money 
are  expressed  are  denominated  pounds,  shillings,  and 
pence :  the  fractions  of  a  penny  in  use  are  the  half  and 
the  fourth,  called  the  halfpenny  and  farthing.  The  rela- 
tive value  of  these  units  is  expressed  in  the  following 

TABLE  OF  MONEY. 

1  penny  =  2  halfpence  =  4  farthings. 

1  shilling  =  12  pence  =  24  halfpence  =  48  farthings. 

1  pound  =  20  shillings  =  240  pence  =  480  halfpence  -  960  farthings. 

The  gold  coin  called  a  sovereign  has  the  value  of  a 
pound,  or  20  shillings;  it  contains  123.274  grains  of 
standard  gold :  the  purity  of  standard  gold  is  in  the 
proportion  of  1 1  parts  of  pure  gold  to  1  of  alloy :  thus, 
a  sovereign  contains  113*001  grains  of  pure  gold  and 
10-273  grains  of  alloy. 

*  The  following  well-known  lines  serve  as  a  help  to  the  memory  to  retain 
the  length  of  the  several  months  :  — 

Thirty  days  hath  September, 
April,  June,  and  November ; 
February  hath  twenty-eight  alone, 
And  all  the  rest  have  thirty-one. 

p  3 


214  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

The  alloy  contained  in  coin  has  no  sensible  value 
compared  with  the  value  of  the  coin  of  which  it  forms 
a  part.  It  is  used  in  the  coin,  or  rather  allowed  to  re- 
main mixed  with  the  precious  metal,  merely  to  save  the 
expense  which  would  he  incurred  in  rendering  the 
metal  perfectly  pure  hy  refining.  It  is  also  of  some  use 
in  rendering  the  metal  harder,  and  more  slow  to  wear. 

A  pound  Troy  weight  of  standard  gold  would,  ac- 
cording to  the  weight  just  assigned  to  the  sovereign,  be 
coined  into  46  ^  sovereigns,  which  would  be  worth 
46/.  14s.  6d.  The  value  of  an  ounce  Troy  weight  of 
standard  gold  will  therefore  be  found  by  taking  the 
twelfth  part  of  this  sum,  which  is  31.  IJs.  W±d. 

The  SHILLING  is  a  silver  coin  containing  80'727  grains 
of  fine  silver  and  6'543  grains  of  alloy. 

PENCE.  HALFPENCE,  and  FARTHINGS  are  copper 
tokens  not  possessing  the  intrinsic  value  of  the  sums 
for  which  they  are  legally  exchangeable.  Thus,  a  silver 
shilling  has  a  value  72  per  cent,  greater  than  12  copper 
pennies.  The  evil  effects  which  would  result  from  this 
circumstance  are  counteracted  by  the  manufacture  of 
these  tokens  being  confined  to  the  government,  and  the 
restrictive  condition  that  they  are  not  a  legal  tender  to 
an  extent  beyond  one  shilling  in  any  single  payment. 

Formerly  was  current  a  gold  coin  denominated  a 
guinea,  and  others  of  one  half  and  one  third  of  its 
value.  The  guinea  was  worth  21s.  and  the  other  two 
coins  worth  10s.  6d.  and  7*-  These  coins  are  now, 
however,  out  of  use. 

MEASURES  OF  SPACE. 

(309.)  To  establish  a  uniform  system  of  measures 
and  weights,  and  to  give  such  a  system  permanency,  is 
an  object  of  such  general  convenience  and  utility,  that 
it  could  not  fail  to  attract  the  attention  of  every  nation 
at  all  advanced  in  civilisation ;  and  we  accordingly  find 
that  there  is  hardly  a  country  in  which  some  attempts 
to  accomplish  this  have  not  been  made.  The  attain- 
ment of  such  an  end  is,  however,  attended  with  many 


CHAP.  I.  WEIGHTS    AND    MEASURES.  215 

practical  difficulties,  arising  out  of  the  very  circumstances 
which  render  it  desirahle.  The  denominations  of  mea- 
sure and  weight  have  been  necessarily  in  such  constant 
and  early  use  in  domestic  economy  and  in  commerce, 
that  each  local  system  becomes  deeply  rooted  from  the 
effects  of  ancient  custom,  and  is  so  intimately  associated 
with  the  daily  habits  of  life,,  that  any  attempt  to  change 
it  is  attended  with  almost  as  much  difficulty  as  to 
change  the  general  habits,  manners,  or  language  of  a 
country.  Until  very  recently,  in  different  parts  of 
Great  Britain,  the  greatest  confusion  and  inconvenience 
arose,  from  the  total  want  of  uniformity  in  the  systems 
of  weights  and  measures  in  common  use.  Different  de- 
nominations were  used  in  different  parts  of  the  kingdom; 
and  still  more  frequently,  and  with  greater  inconve- 
nience, the  same  denomination  was  used  to  express  dif- 
ferent quantities  in  different  places  ;  nay,,  even  the  same 
expressions  were  not  unfrequentlyused,  in  the  same  place, 
to  denote  different  quantities  of  different  commodities. 
Thus,  a  stone  of  one  commodity  had  a  different  weight 
from  a  stone  of  another,  a  gallon  of  one  liquid  had  a 
different  measure  from  a  gallon  of  another,  and  so  on. 

It  was  attempted  in  various  acts  of  parliament  to 
remedy  this  inconvenience,  but  without  effect,  until  the 
statute  passed  in  5  George  IV.,  by  which  a  uniform 
system  of  weights  and  measures  was  established,  under 
the  denomination  of  IMPERIAL  WEIGHTS  AND  MEA- 
SURES, and  their  use  enforced  under  severe  penalties. 
This  act  has  been  generally  enforced;  and  the  system 
used  throughout  the  British  islands  is  now  approaching 
to  the  desired  uniformity. 

To  render  this  system  permanent,  standards  of  length 
and  weight  have  been  selected,  which  can  at  any  future 
period  be  verified;  so  that  if  the  weights  or  measures  in 
use  underwent  in  process  of  time  any  gradual  alteration, 
however  small,  the  amount  of  such  variation  might  be 
ascertained,  and  a  correction  made  with  certainty  and 
accuracy. 

The  standard  or  original  unit  of  measure  selected  for 


216  A    TREATISE    OX    ARITHMETIC.  BOOK  III. 

this  purpose  was  the  YARD  ;  and  its  length  was  defined 
by  expressing  its  numerical  proportion  to  the  length  of 
a  pendulum,  which  in  the  latitude  of  London  and  at 
the  level  of  the  sea  vibrates  seconds.  Since  the  length 
of  such  a  pendulum  can  always  be  ascertained,  the 
length  of  the  yard,  which  bears  to  it  a  fixed  proportion, 
may  always  be  verified. 

Suppose  the*  length  of  the  seconds'  pendulum  to  be 
divided  into  391392  equal  parts;  360000  of  these 
parts  are  taken  as  the  length  of  the  yard.  This  length 
was  selected,  from  the  circumstance  of  its  being  as  nearly 
as  possible  equivalent  to  that  of  the  standard  yard  used 
previously  to  the  statute  now  alluded  to,  by  which  means 
the  original  measure  to  which  the  public  was  accus- 
tomed underwent  no  apparent  change ;  the  effect  being 
merely  to  provide  the  means  of  correcting  its  length 
at  all  future  times,  in  the  event  of  the  original  or 
standard  yard  having  been  destroyed,  lost,  or  injured. 

The  foot  is  defined  by  the  statute  to  be  one  third  part 
of  this  standard  yard  ;  the  inch,  one  twelfth  of  the  foot. 
The  pole,  or  perch,  is  defined  to  be  5^  such  yards ;  the 
furlong,  220  yards;  and  the  mile,  1760  yards. 

(310.)  The  following  table  exhibits  the  relative 
values  of  the  different  units  in  the  system  of  Imperial 
measures  of  length  :  — 

IMPERIAL  MEASURES  OP  LENGTH. 

1  foot  =  12  inches. 

1  yard  =  3  feet  =  36  inches. 

1  perch  =  5|  yards  =  16|  feet  =  198  inches. 

1  furlong  =.  40  perches  =  220  yards  =  .»60  feet  =  7920  inches. 

I  mile  =  8  furlongs  =  320  perches  =  1760  yards  =  5280  feet  =  63360  inches. 

To  which  we  may  add  the  following :  — 

1  league  =  3  miles. 
1  degree  =  69^  common  miles. 
1  geographical  mile  =  Jjth  of  a  degree. 
12  lines  =  3  barleycorns  =  I  inch. 
1  palm  =  3  inches. 
)  hand  =  4  inches. 
1  span  =  9  inches. 
1  fathom  =  6  feet. 

The  inch  is  sometimes  divided  Into  tenths,  but  more 


CHAP.  1.  WEIGHTS    AND    MEASURES.  2 1 7 

commonly  into  twelfths,  or  lines.  A  further  subdivision 
is  sometimes  made ;  the  line  being  divided  into  1 2  equal 
parts  called  seconds,  and  the  seconds  again  into  12 
equal  parts  called  thirds.  This  duodecimal  division  is 
generally  used  in  the  measurement  of  artificers'  work. 

(311.)  The  extent  of  surfaces  is  measured  by  square 
feet,  square  yards,  square  miles,  &c.  A  SQUARE  is  a 
figure  formed  by  four  straight  lines  equal  in  length,  and 
placed  so  as  to  form  right  angles ;  the  four  straight  lines 
forming  such  a  figure  are  called  the  sides  of  the  square, 
arid  the  magnitude  of  the  square  is  expressed  by  the 
length  of  its  side.  Thus,  a  square,  whose  side  measures 
1  inch,  is  called  a  square  inch  ;  one  whose  side  mea- 
sures 1  foot,  is  called  a  square  foot,  and  so  on. 

As  any  length  is  expressed  by  the  number  of  inches, 
feet,  or  miles  it  contains,  so  any  surface  is  expressed  by 
the  number  of  square  inches,  square  feet,  or  square  miles 
it  contains. 


12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 



— 

— 

— 

| 

If  12  square  inches  be  placed  one  beside  the  other  in 
a  horizontal  row,  AB,  as  here,  twelve  such  rows,  1,  2, 
3,  4,  &c.,  placed  one  over  the  other,  will  form  a  square 
foot.  This  will  be  evident,  since  the  height  of  the 
figure  is  equal  to  its  breadth ;  each  being  12  inches :  the 
number  of  square  inches  in  a  square  foot  will  therefore 
be  found,  by  taking  12  square  inches  12  times,  or  by 


218  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

multiplying   12  by  12.      One   square   foot,   therefore, 
contains  12  times  12,  or  144,  square  inches. 

Since  a  yard  consists  of  three  feet,  the  same  reason- 
ing will  show  that  there  are  9  square  feet  in  a  square 
yard  ;  and  by  pursuing  a  similar  reasoning,  we  may  ob- 
tain the  results  contained  in  the  following  table :  — 

SQUARE  MEASURE. 

Square.  Square. 

1  toot  =  144  inches. 

1  yard  =  9  feet  =  1296  inches. 

1  perch  =  30±  yards  =  272*  feet  =  39204  inches. 

1  rood  =  40  perches  =  1SUO  yards  -  10890  feet  =  1568160  inches. 

1  acre  =  4  roods  =  160  perches  =  4840  yards  =  43560  feet  =  6272640  inches. 

1  mile  =  640  acres. 

(312.)  If  a  space  be  bounded  by  four  straight  lines 
at  right  angles  to  each  other,  we  shall  find  the  number 
of  square  inches  it  contains,  by  multiplying  the  number 
of  inches  in  its  length  by  the  » amber  of  inches  in  its 
breadth.  This  will  be  easily  understood  by  reasoning 
similar  to  that  already  applied  to  a  square  foot.  Sup- 
pose that  the  length  is  10  inches^  and  the  breadth  8  ; 
let  a  row  of  8  square  inches  be  formed,  and  let 
10  such  rows  be  placed  one  above  the  other,  as  in  the 
figure :  — 


We  shall  thus  have  a  space  bounded  by  two  straight 
lines  at  right  angles  to  each  other,  one  measuring  10 
and  the  other  8  inches,  and  the  whole  consisting  of  10 
rows  of  8  square  inches.  The  total  number  of  square 
inches  is  evidently  10  times  8,  or  80. 

The  methods  of  finding  the  superficial  magnitudes  of 


CHAP.  I.  WEIGHTS    AND    MEASURES.  21$ 

spaces  bounded  by  straight  lines  not  at  right  angles,  or  by 
curved  lines,,  are  explained  in  Geometry  ;  but  are,,  in  ge- 
neral, too  complicated  and  difficult  to  be  introduced  here. 

(313.)  When  we  wish  to  express  the  magnitude  of 
any  solid  body,  if  its  sides  be  flat  and  at  right  angles  to 
each  other,  we  do  so  by  declaring  its  length,  breadth, 
and  thickness  ;  but  if  it  be  of  any  other  shape,  its  mag- 
nitude cannot  be  expressed  in  this  way.  It  is  not 
always  sufficient  to  declare  its  length,  breadth,  and 
thickness,  because  by  so  doing,  without  some  further 
calculation,  the  relative  magnitudes  of  different  bodies 
would  not  be  known 

The  magnitudes  of  solids,  in  general,,  are  expressed 
in  a  manner  analogous  to  that  already  explained  for  ex- 
pressing the  magnitude  of  surfaces. 

A  CUBE  is  a  figure  of  the  shape  of  a  common  die 
with  which  we  play  in  games  of  chance  :  it  is  bounded 
by  6  flat  sides,  forming  12  rectangular  edges,  all  of  the 
same  length  :  its  magnitude  is  denominated  according  to 
the  length  of  its  edge :  if  the  length  measure  an  inch, 
the  solid  is  called  a  cubic  inch  ;  if  it  measure  a  foot,  it 
is  called  a  cubic  foot,  &c.  It  is  evident  that  each  of  the 
6  sides  of  a  cube  is  a  square ;  if  it  be  a  cubic  inch,  its 
sides  are  square  inches ;  if  it  be  a  cubic  foot,  its  sides 
are  square  feet,  &c. 

From  what  has  been  already  shown,  it  will  follow 
that  12  rows  of  12  cubic  inches,  placed  on  the  same  flat 
surface,  will  cover  a  square  foot :  now,  if  12  layers  of 
these  be  piled  one  upon  the  other,  we  shall  have  a  solid 
12  inches  in  height,  12  inches  in  breadth,  and  12  inches 
in  thickness  ;  we  shall,  in  fact,  have  a  cube  whose  edge 
measures  a  foot.  A  cubic  foot,  therefore,  may  be  con- 
sidered as  consisting  of  12  layers  of  cubic  inches,  each 
layer  consisting  of  12  rows,  having  12  cubic  inches  in 
each  row. 

It  is  evident,  therefore,  that  to  find  the  number  of 
cubic  inches  which  form  a  cubic  foot,  we  must  first 
multiply  12  by  12,  and  then  multiply  the  product  by 
12  ;  the  result  will  be  the  cube  of  12,  which  is  1728. 
In  the  same  manner,  if  th.3  edge  of  a  cube  consist  of  3 


220  A    TREATISE    ON    ARITHMETIC.  BOOK   III. 

feet,  it  will  contain  a  number  of  cubic  feet  expressed  by 
the  cube  of  3  :  a  cubic  yard,  therefore,  consists  of  27 
cubic  feet. 

If  a  solid  body  be  bounded  by  flat  surfaces  at  right 
angles  to  each  other,  we  shall  always  find  the  number  of 
cubic  inches  it  contains,  by  ascertaining  the  number  of 
inches  in  its  length,  breadth,  and  height :  multiply  the 
number  of  inches  in  its  length  by  the  number  of  inches 
in  its  breadth,  and  multiply  the  product  by  the  number 
of  inches  in  its  height ;  the  result  will  be  the  number  of 
cubic  inches  in  the  solid.  This  may  be  easily  under- 
stood, by  considering  how  the  solid  may  be  built  with 
cubic  inches.  Let  us  suppose  that  its  length  is  10 
inches,  its  breadth  8,  and  its  height  5  ;  place  8  cubic 
inches  in  a  horizontal  row,  and  then  place  10  of  these 
rows  one  beside  the  other :  we  shall  thus  form  a  solid, 
whose  length  is  10  inches,  breadth  8  inches,  and  height 
1  inch ;  let  5  such  layers  be  piled  one  upon  the  other, 
and  we  shall  form  a  solid  of  the  same  length  and 
breadth,  but  with  the  height  of  5  inches.  Now,  since 
each  layer  consists  of  10  rows  of  8  cubes,  the  number 
of  cubes  in  each  layer  will  be  found  by  multiplying  10 
by  8  ;  and  since  there  are  5  such  layers  in  the  solid  so 
formed,  we  shall  get  the  total  number  of  cubes  by  mul- 
tiplying the  number  in  each  layer  by  5.  We  thus  mul- 
tiply the  length  by  the  breadth,  and  then  multiply  the 
product  by  the  height. 

(314.)  The  following  table  exhibits  the  relative  values 
of  the  different  units  of  cubic  measure. 

CUBIC  MEASURE. 

1728  cubic  inches      -  -     =  1  cubic  foot. 

27  cubic  feet     -  =  I  cubic  yard. 

40  feet  of  rough  timber,  or  j  =  x  ]oad  Qr  ^ 
sO  reet  hewn  ditto        -         J 
42  cubic  feet          -  -      =  1  ton  of  shipping. 

By  the  denominations  in  this  table,  all  artificers'  solid 
work  is  measured,  such  as  carpentry,  masonry,  and,  in 
general,  all  works  estimated  by  length,  breadth,  and 
thickness. 


CHAP.  I.  WEIGHTS    AND    MEASURES.  221 

(315.)  Liquids  in  general,  and  all  substances  which 
are  in  a  state  like  that  of  powder,  so  as  to  he  capable  of 
filling  a  hollow  vessel,  are  measured  by  a  system  of  mea- 
sures denominated  "  the  Imperial  liquid  and  dry  mea- 
sure." This  system  has  for  its  basis,  or  primary  unit,  the 
Imperial  gallon,  which  is  a  vessel,  the  capacity  of  which 
is  277'274<,  or  277|->  cubic  inches  very  nearly.  The 
manner  in  which  the  magnitude  of  this  vessel  is  denned 
by  the  act,  is  that  it  shall  be  such  as  to  contain  10 
pounds  avoirdupois  weight  of  distilled  water,  weighed 
in  air  at  the  temperature  of  62°  of  Fahrenheit,  the  ba- 
rometer being  at  30  inches.  It  is  declared  that  such 
Imperial  standard  gallon  shall  be  the  only  standard 
measure  of  capacity  to  be  used  for  wine,  beer,  and  all 
sorts  of  liquids,  and  for  such  dry  goods  as  are  measured 
like  liquids  and  not  heaped;  that  from  this  all  other 
measures  shall  be  derived ;  the  quart  to  be  a  fourth  part 
of  the  gallon,  the  pint  an  eighth  part ;  and  2  such  gal- 
lons to  be  a  peck,  8  to  be  a  bushel,  and  8  bushels  to 
form  a  quarter  of  corn,  or  other  dry  goods  not  sold  by 
heaped  measure.  The  following  table  exhibits  the  re- 
lative and  absolute  values  of  the  different  classes  of 
units  forming  the  Imperial  liquid  and  dry  measure  not 
heaped :  — 

IMPERIAL  LIQUID  AND  DRY  MEASURE. 

Gills.  Pint. 

4.  =  1  Quart. 

8  =  2    =  1        Pottle. 

16  =  4    =  2  =        1        Gallon. 

S2  =  8=  4  =        2=1       Peck. 

64  =  16    =  8  =        4=2=1      Bushel 

256  =  6*    =  32  =  16    =      8    =      4    =     1       Coom. 

1024  =  256    =  128  =  64    =    32    =     16    =    4     =     1  Quarter. 

2048  =  512    =  256  =  128    =     64    =    32    =    8    =    2    =     1 

When  heaped  measure  is  used,  the  bushel  is  directed  by 
the  act  to  be  constructed  in  the  following  manner  :  — 
It  shall  contain  80  pounds  avoirdupois  of  distilled 
water,  being  made  cylindrical,  and  having  a  diameter 
not  less  than  double  its  depth;  that  the  goods  measured 
shall  be  heaped  in  the  form  of  a  cone^  the  height  of 
which  shall  be  at  least  3  fourths  of  the  depth  of  the 
measure,  and  the  outside  of  the  bushel  to  be  the  ex- 


£22  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

tremity  of  the  base  of  the  cone  :  3  such  bushels  shall 
be  a  sack  ;  and  12  sacks,  a  chaldron.  The  contents  of 
the  Imperial  heaped  bushel  amount  to  2815*4887  cubic 
inches. 

(3  16.)  As  there  is  frequently  occasion  to  refer  to  the 
old  systems  of  measures,  which  have  been  superseded 
by  the  Imperial  system,  the  following  tables  exhibiting 
the  relative  and  absolute  values  of  those  measures  may 
be  useful. 

OLD  WINE  MEASURE. 

4  gills  =  1  pint. 

2  pints  =  1  quart. 

4  quarts  =  1  gallon. 

42  gallons  =  1  tierce. 

2  tierces  =  1  puncheon. 

63  gallons  =  1  hogshead. 

2  hogsheads  =  1  pipe  or  butt. 

2  pipes  =  1  tun. 

From  this,  and  the  table  of  Imperial  measure,  we 
derive  the  following  results  :  — 

(317-) 


)                  WINE  MEASURE. 

1  tierce          = 

Imp.  galls. 
34-99066    = 

Imp.  galls. 
35    nearly. 

1  puncheon  = 
1  hogshead   = 
1  pipe 
1  tun             ™ 

69-98132    = 
52-48599    = 
104-97198    = 
209-94396   = 

70    nearly. 
52^  nearly. 
105    nearly. 
210    nearly. 

(318.)      OLD  ALE  AND  BEER  MEASURE. 


4 
2 
4 
8 
9 
2 
2 

H 
2 

gills 
pints 
quarts 
gallons 
gallons 
firkins 
kilderkins 
:  barrel 
barrels 

= 

1  pint. 
1  quart. 
1  gallon. 
1  firkin  of  ale. 
1  ditto  of  beer. 
1  kilderkin. 
1  barrel. 
1  hogshead. 
1  puncheon. 

2    hogsheads    =    1  butt. 
2    butts  =   1  tun. 


CHAP.  I.  WEIGHTS  AND  MEASURES.  223 

(319.)  OLD  DRY  OR  WINCHESTER  MEASURE. 


4  gills 

r= 

I  pint. 

2  pints 

= 

quart. 

2  quarts 

= 

pottle. 

2  pottles 

= 

gallon. 

2  gallons 

= 

peck. 

2  pecks 

= 

bushel. 

4  bushels 

= 

coom. 

2  cooms 

=: 

quarter. 

5  quarters 

= 

wey,  or  load. 

2  weys 

= 

I  last. 

(320.)  Since  the  old  wine  gallon  contains  231  cubic 
inches,,  and  the  Imperial  gallon  277*274,  any  number 
expressing  wine  gallons  will  be  converted  into  an  equi- 
valent number  of  Imperial  gallon s,  by  diminishing  it  in 
the  proportion  of  277*274  to  231,,  or,  what  is  the  same, 
in  the  proportion  of  277274  to  231000:  this  reduc- 
tion may  be  made  by  multiplying  the  number  of  wine 
gallons  by  the  fraction  \)^\®%®>  or  by  its  equivalent 
decimal,  0-83311. 

To  convert  a  number  expressing  Imperial  gallons 
into  an  equivalent  one  expressing  wine  gallons,  we 
should,  on  the  contrary,  increase  it  in  the  proportion  of 
231000  to  277274,  which  may  be  done  by  multiplying 
it  by  the  fraction  v^ooo.>  or  ^7  *ts  equivalent  decimal, 
1-20032. 

When  extreme  accuracy  is  not  required,  a  number 
expressing  wine  gallons  may  be  reduced  to  an  equivalent 
number  of  Imperial  gallons,  by  multiplying  it  by  5, 
and  dividing  the  product  by  6  ;  and  a  number  express- 
ing Imperial  gallons  may  be  reduced  to  an  equivalent 
number  of  wine  gallons  by  the  contrary  process,  viz. 
multiplying  by  6,  and  dividing  by  5. 

Since  the  old  ale  gallon  contains  282  cubic  inches, 
and  the  Imperial  standard  gallon  277*274  cubic  inches, 
any  number  expressing  ale  gallons  will  be  converted 
into  an  equivalent  number  expressing  Imperial  gal- 
lons, by  diminishing  it  in  the  proportion  of  282  to 
277'277>  or,  what  is  the  same,  in  the  proportion  of 


224  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

282000  to  277274 :  this  would  be  done  by  multiplying 
it  by  the  fraction  ~fi°-y°,  or  by  the  equivalent  deci- 
mal, 1 '01 70445.  To  convert  a  number  expressing 
Imperial  gallons  into  an  equivalent  number  expressing 
ale  gallons,  it  would  be  necessary  to  diminish  it  in  the 
proportion  of  282000  to  277274,  which  would  be  done 
by  multiplying  it  by  the  fraction  |^|f^,  or  by  the 
equivalent  decimal,  0*9882411. 

When  extreme  accuracy  is  not  necessary,  the  number 
expressing  ale  gallons  may  be  reduced  to  an  equivalent 
number  expressing  Imperial  gallons,  by  multiplying  it 
by  60,  and  dividing  the  product  by  59  ;  and  a  number 
expressing  Imperial  gallons  may  be  converted  into  an 
equivalent  one  expressing  ale  gallons,  by  multiplying  it 
by  59  and  dividing  the  product  by  60. 

Since  the  Winchester  bushel  contains  2150*420  cubic 
inches,  and  the  Imperial  bushel  2218-198  cubic  inches, 
any  number  expressing  Winchester  bushels  may  be 
converted  into  an  equivalent  one  expressing  Imperial 
bushels,  by  diminishing  it  in  the  proportion  of  2218192 
to  2150420,  which  will  be  done  by  multiplying  it  by 
the  fraction  ||4fiM^  or  by  its  equivalent  decimal, 
0-969447.  A  number  expressing  Imperial  bushels  may 
be  converted  into  an  equivalent  one  expressing  Win- 
chester bushels,  by  multiplying  it  by  the  fraction 
I  Ti"oT$tb  or  kv  its  equivalent  decimal,  1-0315157. 

When  great  accuracy  is  not  required,  a  number  ex- 
pressing Winchester  bushels  may  be  reduced  to  one 
expressing  Imperial  bushels,  by  multiplying  it  by  31, 
and  dividing  the  product  by  32  ;  and  a  number  ex' 
pressing  Imperial  bushels  may  be  converted  into  one 
expressing  Winchester  bushels,  by  multiplying  it  by 
32,  and  dividing  the  product  by  31. 

It  would  be  much  more  convenient  and  equitable,  if 
grain,  seeds,  and  all  substances  which  are  usually  sold 
by  heaped  measure,  were  sold  by  weight :  the  weight  of 
a  substance  is  always  proportional  to  its  quantity ;  but 
the  same  measure  of  different  specimens  of  the  same 
commodity  will  differ  in  quantity  The  average  bushel 


CHAP.  I.  WEIGHTS    AND    MEASURES.  225 

of  wheat  is  generally  considered  as  equivalent  to  60 
pounds  weight;  but  wheat  of  different  qualities  will 
vary  from  this  medium,  a  bushel  being  sometimes  less 
and  sometimes  greater  than  60  pounds:  the  average 
weight  of  a  bushel  of  barley  is  reckoned  at  47  pounds  ; 
of  oats,  38  pounds ;  of  peas,  64  pounds ;  of  beans,  63 
pounds ;  of  clover,  68  pounds  ;  of  rye  and  canary,  53  ; 
and  of  rape,  48. 

Coals,  which  were  formerly  sold  by  measure,  are  now 
sold  by  weight. 

MEASURES  OF  WEIGHT. 

(321.)  We  have  seen  that  the  system  of  measures 
of  length,  surface,  and  capacity  are  all  derived,  by  de- 
finite numerical  proportions,  from  the  length  of  a 
pendulum  which  vibrates  seconds,  by  reference  to  which 
the  system  of  measures  may  be  verified  and  corrected 
at  all  future  times.  In  order  to  obtain  a  similar  test 
for  the  verification  of  weights,  the  imperial  standard 
pound  troy  weight  was  compared  with  the  weight  of  a 
cubic  inch  of  distilled  water,  at  a  fixed  temperature 
and  under  a  fixed  atmospheric  pressure.  The  weight 
of  such  quantity  of  water  being  supposed  to  be  divided 
into  505  equal  parts,  each  of  these  parts  would  be  half 
a  grain  troy  weight.  The  pound  troy  *  was  declared  to  be 
composed  of  5?60  grains  thus  determined,  and  the 
pound  avoirdupois  t  to  be  composed  of  7000  such 
grains. 

*  *'  There  are  reasons  to  believe  that  the  word  troy  has  been  derived 
from  the  monkish  name  given  to  London  of  Troy  novant,  founded  on  the 
legend  of  Brute.  Troy  weight,  therefore,  according  to  this  etymology,  is  in 
fact  London  weight.  We  were  induced,  moreover,  to  preserve  the  troy 
weight,  because  all  the  coinage  has  been  uniformly  regulated  by  it,  and  all 
medical  prescriptions  or  formulae  now  are,  and  always  have  been,  estimated 
by  troy  weight  under  a  peculiar  subdivision,  Which  the  College  of  Phy- 
sicians have  expressed  themselves  most  anxious  to  preserve."—  Report 
of  Commissioners  of  Weights  and  Measures. 

f  "  We  find  the  avoirdupois  weight  by  which  all  heavy  goods  hare 
been  for  a  long  time  weighed  (probably  derived  from  avoirs  (averia)  the 
ancient  name  for  GOODS  or  CHATTELS,  and poids,  WEIGHT,)  to  be  universally 
used  throughout  the  kingdom.  This  weight,  however,  seems  not  to  have 
been  preserved,  with  such  scrupulous  accuracy  as  troy  weight,  by  which  the 
most  precious  articles  have  been  weighed;  but  we  have  reason  to  believe 
that  the  pound  cannot  differ  by  more  than  1, 2,  or  3  grains  from  7000  grains 
troy.  It  therefore  occurred  to  us  that  we  should  be  offering  no  violence  to 

Q 


226  A    TREATISE   ON   ARITHMETIC.  BOOK  III. 

The  relative  values  of  the  units  of  imperial  troy 
weight  are  expressed  in  the  following  table :  — 

(322.)  —  TABLE  OF  TROY  WEIGHT. 

1  pennyweight  (dwt.)    =   24  grains. 

1  ounce  (oz. )  =   20  pennyweights. 

1  pound  (ft.)  =    12  ounces. 

This  denomination  of  weight  is  used  in  estimating 
the  quantity  of  the  precious  metals,  jewels,  &c.  Jt  is 
a'so  used  in  determining  specific  gravities,  and  in  ge- 
neral in  philosophical  investigations.  For  the  purposes 
of  excise  it  is  used  in  determining  the  strength  of 
spirituous  liquors. 

The  troy  weight,  differently  divided,  is  used  by  apo- 
thecaries for  compounding  prescriptions. 

(323.)  —  APOTHECARIES'  WEIGHT. 

1  scruple  9   =   20  grains. 
1  dram      3=3  scruples. 
1  ounce     *   =     8  drams. 
1  pound     ft  =    12  ounces. 

(324.)  A  peculiar  division  of  weight  is  used  in  esti- 
mating the  value  of  diamonds :  3^  grains  troy  weight 
make  1  carat ;  1  carat  is  divided  into  4  equal  parts 
called  grains,  and  the  grain  is  resolved  into  1 6  equal 
divisions  called  parts  :  thus  one  part  diamond  weight  is 
the  twentieth  part  of  a  grain  troy  weight.  The  rela- 
tive value  of  the  units  avoirdupois  weight  are  expressed 
in  the  following  table :  — 

(325.)  —  AVOIRDUPOIS  WEIGHT. 


scruple 

=   10  grains. 

dram 

=     3  scruples. 

ounce 

=    16  drams. 

if 

pound 

=   16  ounces. 

quarter 

=   28  pounds. 

cwt. 

=     4  quarters. 

.] 

L  ton 

=   20  cwt. 

this  system  of  weights,  if  we  declared  that  7000  grains  troy  should  hereafter 
be  considered  as  the  pound  avoirdupois."  —  Same  Report. 


CHAP.  I.  WEIGHTS    AND    MEASURES.  227 

From  a  comparison  of  this  table  with  the  table  of 
troy  weight,  it  will  appear  that  7680  grains  avoirdu- 
pois =  7000  grains  troy,  each  of  these  being  equal  to 
a  pound.  One  grain  troy  weight  is,  therefore,  equal  to 
1-097  grains  avoirdupois. 

It  follows,  also,   that  while  the  ounce  avoirdupois 
is  less  than  the  ounce  troy,  in  the  proportion  of  175  to 
192,  the  pound  avoirdupois  is  greater  than  the  pound 
troy,  in  the  proportion  of  175  to  144.     Thus 
1 44  frs.  avoirdupois     •=    175  tbs.  troy. 
192  oz.      =   175  oz.     - 

In  general  a  stone  weight  is  14  pounds  avoirdupois, 
but  for  butcher's  meat  or  fish  it  is  8  pounds  :  the  hun- 
dred weight,  therefore,  is  8  stone  of  14  pounds,  or  14 
stone  of  8  pounds.  In  weighing  glass  the  stone  is  5 
pounds ;  and  a  denomination  of  weight  is  used  called  a 
seam,  which  is  24  stone,  or  120  pounds.  In  weighing 
hay,  the  truss  weighs  half  a  hundred  weight,  or  66 
pounds.  If  the  hay  be  new,  which  it  is  considered  to 
be  until  the  1st  of  September  in  the  year  in  which  it  is 
grown,  the  truss  is  taken  to  be  60  pounds.  The  truss 
of  straw  is  36  pounds. 

(326.)  The  divisions  of  avoirdupois  weight  by  which 
wool  is  estimated  are  expressed  in  the  following  table:  — 

WOOL  WEIGHT. 

7  pounds  =  1  clove. 
2    cloves  =  1  stone. 
2    stone  =  1  tod. 
6J  tods  =  1  wey. 
2    weys  =  1  sack. 

12     sacks       =       1  last. 
1    pack       =  240  pounds. 

(327.)  In  weighing  cheese  and  butter,  the  following 
denominations  are  used :  — 

CHEESE  AND  BUTTER  WEIGHT. 

8  pounds     =   I  clove. 

32  cloves       =   1  wey  (Essex). 
42  cloves       =    1  wey  (Suffolk). 
56  pounds     =    1  firkin. 
Q  2 


228  A   TREATISE  ON   ARITHMETIC.       BOOK  III* 

FRENCH  SYSTEM. 

(328.)  No  nation  has  succeeded  in  establishing  a 
system  of  weights,  measures,  and  money,  at  once  so 
simple  and  uniform  as  that  which  has  been  established 
ince  the  revolution  in  France.  The  basis  on  which  this 
system  is  founded  is  the  magnitude  of  a  quadrant,  or 
fourth  part  of  the  meridian ;  that  is,  the  distance  from 
the  equator  to  the  pole,  as  it  would  be  measured  upon 
the  surface  of  the  sea  uninterrupted  by  the  irregularities 
of  land.  This  measurement  was  made  with  the  utmost 
accuracy  by  observations  on  arches  of  the  meridian  in 
different  latitudes,  by  which  the  exact  figure  of  the  me- 
ridian was  ascertained,  and  the  actual  length  of  an  arch 
of  considerable  magnitude,  extending  from  north  to 
south  between  Dunkirk  and  Barcelona.  The  whole 
length  of  the  meridional  quadrant  being  divided  into 
1 0000000  equal  parts,  one  of  these  parts  was  taken  as  the 
primary  and  original  unit  from  which  all  weights  and 
measures  should  be  derived :  it  was  called  the  METRE, 
and  its  length  corresponded  very  nearly  with  the  ancient 
French  yard,  being  equal  to  3 '07844  French  feet,  or 
3-281  English  feet,  or  39'3?08  English  inches.* 

The  object  kept  in  view  in  the  formation  of  the  French 
weights  and  measures  was  to  enable  all  quantities  to  be 
expressed  by  whole  numbers  and  decimals,  without  the 
separation  into  classes  of  units  distinct  from  the  local 
values  which  the  digits  have  in  the  ordinary  arithmetical 
notation :  this  was  accomplished  by  forming  the  classes 
of  units  of  weight,  measure,  and  money  according  to  the 
decimal  scale. 

The  metre  being  the  original  unit,  the  next  superior 
unit  was  the  decametre,  which  was  equivalent  to  10 
metres ;  next  above  this  was  the  hectometre,  or  100 
metres,  and  then  followed  the  kilometre,  or  1000  metres, 
and  the  myriametre,  or  10000  metres.  In  general, 

*  By  an  accidental  correspondence,  the  metre  is  very  nearly  equal  to  th* 
length  of  the  seconds'  pendulum. 


CHAP.   I.  FRENCH    METRICAL    SYSTEM.  229 

the  successive  units  rising  in  a  decuple  progression  above 
the  metre  were  expressed  by  Greek  prefixes ;  Latin  pre- 
fixes were  adopted  to  express  the  classes  of  units  inferior 
to  the  metre:  thus  the  tenth  part  of  the  metre  was 
called  a  decimetre,  the  hundredth  part  the  centimetre,  the 
thousandth  the  millimetre. 

The  convenience  of  this  system  will  be  evident  if  we 
attempt  to  express  by  it  any  length  in  which  units  of 
several  classes  occur :  thus,  suppose  we  have  to  express 
6  myriametres,  5  kilometres,  3  decametres,  4  metres, 
6  decimetres,  8  centimetres,  and  9  millimetres.  Had 
the  relative  values  of  these  units  not  been  formed  on  the 
decimal  scale,  we  could  only  express  them  as  so  many 
distinct  numbers,  writing  the  name  of  each  unit  above 
the  numbers  respectively,  as  we  do  in  English  measures, 
with  miles,  furlongs,  perches,  &c.  But  the  relation  be- 
ing formed  on  the  decimal  system,  the  above  length  may 
be  expressed  thus  :  —  65034'689  metres. 

it  is  evident,  that  all  lengthsj  however  numerous  the 
classes  of  units  they  contain,  can  be  expressed  by  sim- 
ple numbers,  the  units  inferior  to  the  metre  occupying 
the  decimal  places,  and  the  superior  units  those  of  tens, 
hundreds,  &c.  All  arithmetical  operations  may,  there- 
fore, be  performed  on  such  numbers  according  to  the 
rules  already  established  for  whole  numbers  and  decimals. 

(329.)  The  lengths  of  the  several  denominations  of 
French  measures  of  length  are  expressed  in  English 
measures  in  the  following  table :  — 

Millimetre    =  0-03937  inches. 

Centimetre    =  0-393708  

Decimetre     =  3-937079  

("39-37079  

Metre     -      = -j  3-2808992  feet. 

(_  1-093633  yards. 

Myriametre  =  6-2138  miles. 

In  the  following  table,  the  English  measures  are  ex- 
pressed in  terms  of  the  French  measure :  — 
Q  3 


230  A    TREATISE    ON    ARITHMETIC.          BOOK  III. 

1  inch     »  =  2-539954  centimetres. 

1  foot     -            -  =  3-0479449  decimetres, 

imperial  yard  =  0-91438348  metres. 

fathom  (2  yards)  =  1-82876696  

perch               -  =  5-02911         

furlong             -  =  201-16437         

mile       -         -  =  1609-3149 

(330.)  The  French  system  of  superficial  measures  is 
derived  from  that  of  linear  measure ;  the  are  is  the  unit 
of  superficial  measure,  and  is  equal  to  100  square  metres; 
the  centiare  is  1  square  metre,  and  the  hectare  is  100 
ares,  or  10000  square  metres.  The  following  tables 
will  serve  for  the  reduction  of  French  to  English  mea- 
sures, and  vice  versa:  — 

Square  metre  =     1*196033  square  yards. 
Are  -      =     0-098845  roods. 

Hectare     -     =     2*473614  acres. 

Square  yard    =     0*836097  square  metres. 

perch  =  25-291939  

Rood       -       =  10-116775  ares. 
Acre        -       =    0'404671  hectares. 


(331.)  The  units  of  solid  measure  are  the  stere,  or 
cubic  metre,  and  the  decistere,  which  is  a  tenth  of  the 
former.  There  are  three  denominations  of  liquid  mea- 
sure, the  litre,  which  is  a  cubic  decimetre ;  the  decalitre, 
which  is  10  decimetre  cubes,  and  the  decilitre,  which  is 
the  tenth  part  of  a  decimetre  cube. 

The  measures  for  dry  goods  are  the  litre,  or  decimetre 
cube,  the  decalitre,  hectolitre,  and  kilolitre,  which  are 
respectively  for  1 0,  1 00,  and  1 000  decimetre  cubes.  The 
relation  between  these  and  English  measures  is  exhibited 
in  the  following  tables :  — 

T  .  __  f  1  -760773    pints. 

~  |  0-^200967  gallons. 

Decalitre       -      =  2*2009667  gallons. 

Hectolitre          =  22-009667  gallons. 


CHAP.  I.  FRENCH    METRICAL    SYSTEM.  231 

Pint     -         -     =  0-567932       litres. 

Quart  -     =  1-135864         — 

Imperial  gallon  =  4-54345794    — 

Peck    -         -     =  9-0869159       — 

Bushel          -     =  36-347664         — 

(332.)  The  basis  of  the  French  system  of  weights  is 
the  kilogramme,  which  is  the  weight  of  a  decimetre  cube 
of  distilled  water  at  the  temperature  of  40°  of  Fahren- 
heit's thermometer  :  the  thousandth  part  of  this,  or  the 
gramme,  is  the  unit  of  weight  :  the  decagramme)  hecto- 
gramme, and  kilogramme  are  respectively  10,  100,  and 
1000  grammes.  The  quintal  is  100  kilogrammes,  and 
the  milier  1000  kilogrammes;  the  decigramme  is  the 
tenth  part  of  the  gramme  ;  the  centigramme  the  hun- 
dredth part  of  the  gramme,  and  so  on.  The  following 
tables  express,  very  nearly,  the  relation  between  the 
French  and  English  weights  :  — 

15'438          grains  troy. 
0*643  pennyweights. 

0-03216      ounces  troy. 


"Troy  wt. 

1  grain  -  =  0*06477      grammes. 

1  pennyweight  =  1*55456          - 

1  ounce    -         -  =  31*0913  - 

1  imperial  pound  =  0-3730956  kilogrammes. 

Avoirdupois. 

1  dram  -  =         1-7712        grammes. 

1  ounce    -  =       28-3384  - 

1  imperial  pound  =         0-4534148  kilogrammes. 

1  hd.  weight     -  =       50-78246          - 

1  ton         -         -  =  1015-649  - 

(333.)  The  unit  of  French  money  is  the  silver  coin 
called  a  FRANC:  it.  consists  of  9  parts  of  pure  silver 
and  1  of  alloy.  The  weight  of  a  5  franc  piece  is  25 
grammes  ;  so  that  5  grammes  of  standard  silver  repre- 
sent the  value  of  1  franc  :  the  franc  is  supposed  to  be 
divided  into  10  parts  called  decimes;  the  decime,  again, 
Q  4 


232  A    TREATISE    ON    ARITHMETIC.          BOOK  III. 

is  divided  into  10  parts,  called  centimes.  Any  sum  of 
money,  however  great,  is  expressed  in  francs  and  deci- 
mal parts  of  a  franc,  so  that  all  calculations  of  money 
are  made  by  the  rules  established  for  whole  numbers 
and  decimals. 

An  English  pound  sterling,  when  the  exchange  is  at 
par,  is  equivalent  to  25*2  francs ;  thus  the  value  of  a 
franc  is  9*523  pence,  or  very  nearly  9|^ 

(334.)  The  operation  which,  in  treatises  on  arithmetic, 
is  generally  distinguished  by  the  name  REDUCTION  is 
that  by  which  complex  numbers  are  converted  into  sim- 
ple numbers,  or  vice  versa ;  or  more  generally,  by  which 
the  numbers  expressing  one  class  of  units  are  converted 
into  equivalent  numbers  expressing  other  classes.  The 
relative  values  of  the  different  classes  of  the  units  of 
complex  numbers  exhibited  in  the  preceding  part  of  this 
chapter  furnish  all  the  data  necessary  for  such  reduc- 
tions ;  but  as  these  operations  afford  a  useful  exercise 
in  the  arithmetical  principles  already  developed,  and  in 
many  cases  give  rise  to  rules  and  methods  of  calculation 
which  are  useful  in  the  ordinary  affairs  of  life,  we  shall 
here  enter  into  the  details  of  some  of  the  most  necessary 
and  useful  of  such  calculations.  The  spirit  of  the  method 
by  which  they  are  conducted  will  be  readily  seized  by  the 
student,  and  applied  to  other  cases  of  complex  numbers, 
for  which  we  cannot  conveniently  afford  space. 

Let  it  be  required  to  convert  the  sum  of  17 1-  l6s.  9%d. 
into  an  equivalent  number  of  farthings.  We  may  pro- 
ceed to  accomplish  this  by  multiplying  the  pounds  by 
960,  the  shillings  by  48,  and  the  pence  by  4 ;  these 
being  respectively  the  number  of  farthings  contained  in 
a  pound,  a  shilling,  and  a  penny :  but  it  is  generally 
more  convenient  to  convert  the  pounds  into  shillings  in 
the  first  instance,  then  add  the  shillings  to  the  result, 
and  convert  the  total  number  of  shillings  into  pence:  to 
the  pence  thus  obtained,  add  the  pence  in  the  given  sum, 
and  convert  the  whole  into  farthings,  adding  the  farthings 
in  the  given  sum.  The  process  would  be  as  follows : — 


CHAP.  I.  REDUCTION.  233 

£    s.    d. 

Multiply  17  16  9j 

by  20 

340  shillings. 
Add  16     

Multiply        356     

by  12 

4272  pence. 
Add  9   

Multiply      4281   ' 

by  _4 

17124  farthings. 
Add  3      

17127  farthings  =  £17   16s.  9|rf. 

By  multiplying  17  pounds  by  2  Owe  reduce  them  to  shil- 
lings, to  which  we  add  the  1 6  shillings  of  the  given  sum : 
we  thus  find  that  the  given  sum  consists  of  356  shillings 
and  9  pence  3  farthings ;  we  then  convert  the  shillings 
into  pence  by  multiplying  by  12 ;  and  adding  the  9 
pence  of  the  given  sum,  we  find  that  the  given  sum  con- 
sists of  4281  pence  3  farthings:  converting  this  num- 
ber of  pence  into  farthings  by  multiplying  it  by  4,  and 
adding  the  3  farthings  of  the  given  sum,  we  find  that 
the  total  number  of  farthings  is  17127- 

The  practical  process  may  be  somewhat  abridged  by 
adding  at  once  the  shillings  in  the  process  of  multiply- 
ing by  20,  and  in  the  same  manner  adding  the  pence 
and  farthings  in  the  process  of  multiplying  by  12  and  4. 
It  is  also  unnecessary  to  write  the  multipliers  20, 12, 
and  4,  since  they  are  always  well  known  :  the  written 
process  would  then  be  as  follows :  — 

£      s.    d. 
17     16     9J 

356~ 

4287 

17127 


234  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

To  convert  a  simple  number  expressing  farthings  into 
a  complex  one  expressing  pounds,  shillings,  &c.  we  di- 
vide successively  by  4, 12,  and  20.  If  the  given  num- 
ber be  17127,  the  process  would  be  as  follows  :  — 

4)17127 

12)4281     j 


20  )  356     9| 
£~T7      16     9j 

the  number  is  divided  by  4,  the  quotient  will  be 
pence,  and  the  remainder  farthings.  This  number  of 
pence,  divided  by  12,  will  give  a  quotient  expressing 
shillings,  and  a  remainder  expressing  pence.  This 
number  of  shillings,  divided  by  20,  will  give  a  quotient 
expressing  pounds,  and  a  remainder  expressing  shillings. 
We  thus  obtain  the  pounds,  shillings,  pence,  and  far- 
things in  the  proposed  sum. 

(335.)  It  is  sometimes  required  to  express  a  sum  of 
money  in  pounds  and  decimals  of  a  pound. 

Since  a  shilling  is  the  twentieth  part  of  a  pound,  it 
will  be  expressed  by  5  in  the  second  decimal  place,  since 
TL  is  T^  :  thus,  1  shilling  =  ^0-05.  We  shall, 
therefore,  find  the  decimal  of  a  pound  which  is  equiva- 
lent to  any  number  of  shillings,  by  multiplying  0*05 
by  the  number  of  shillings  ;  or,  what  will  produce  the 
same  result,  let  the  number  of  shillings  be  multiplied 
by  5,  and  take  2  decimal  places  in  the  product.  Thus, 
if  the  number  of  shillings  be  12,  the  equivalent  decimal 
of  a  pound  will  be  0*60  :  if  the  number  of  shillings  be 
9,  the  equivalent  decimal  of  a  pound  will  be  0'45- 
The  following  is  an  easy  practical  rule  for  making  this 
calculation :  — 

If  the  number  of  shillings  be  even,  place  half  that 
number  in  the  first  decimal  place;  and  if  it  be  odd, 
place  in  the  first  decimal  place  half  the  number  which 
is  one  less  than  the  shillings,  and  5  in  the  second. 
Thus,  if  the  number  of  shillings  be  6,  the  equivalent 
decimal  is  0'3;  if  the  number  of  shillings  be  16,  the 
equivalent  decimal  is  0'8 ;  if  the  number  of  shillings 


CHAP.  I.  REDUCTION.  235 

be  17,  the  equivalent  decimal  is  0*85  ;  and  so  on.  If 
the  number  of  shillings  proposed  exceed  20,  it  should 
first  be  reduced  to  pounds  and  shillings,  and  then  the 
shillings  reduced  to  decimals. 

Let  it  be  required  to  express  375  shillings  in  de- 
cimals of  a  pound.  To  reduce  shillings  to  pounds,  it 
is  necessary  to  divide  by  20.  An  abridged  method  of 
performing  this  operation  may  be  obtained  in  the  fol- 
lowing manner :  —  We  may  divide  by  20  by  dividing  by 
10  and  2  successively.  We  shall  divide  by  10  by  cut- 
ting off  the  units'  figure  of  the  shillings,  and  taking  it 
as  remainder,  and  the  other  figures  as  the  quotient 
(185.).  In  the  present  case  this  quotient  would  be  37* 
and  the  remainder  5;  but  since  we  have  divided  the 
shillings  by  10,  each  unit  of  this  quotient,  37>  signifies 
10  shillings.  If  we  divide  it  by  2,  we  shall  obtain  the 
quotient  1 8  and  the  remainder  1  :  the  quotient  1 8  will 
signify  pounds,  and  the  remainder  1  will  express  10 
shillings.  Thus,  375  shillings  will  be  equivalent  to 
18  pounds,  10  shillings  and  5  shillings,  or  181.  15s.  Od. 

In  general,  then,  to  reduce  shillings  to  pounds,  cut 
off  the  last  figure,  and  divide  the  remaining  figures  by 
2.  The  quotient  will  be  the  number  of  pounds,  and  if 
there  be  no  remainder  the  figure  cut  off  will  be  the 
number  of  shillings ;  but  if  there  be  a  remainder  of  1, 
that  1  must  be  prefixed  to  the  figure  cut  off  to  express  the 
shillings.  In  the  present  example,  when  37  was  di- 
vided by  2,  there  was  a  remainder  1,  which,  prefixed  to 
the  5  cut  off,  gave  15  shillings.  Having  obtained  the 
number  of  pounds  and  shillings,  we  may  now  convert 
the  shillings  into  a  decimal  of  a  pound  by  the  rule  al- 
ready given,  and  we  find  that  15s.  is  equal  to  j£0*75; 
and,  therefore,  375s.  =  <€l8'75. 

(336.)  Since  there  are  960  farthings  in  a  pound,  we 
shall  find  the  decimal  of  a  pound,  which  is  equivalent  to 
a  farthing,  by  cowjerting  the  fraction  -g^  into  an  equi- 
valent decimal.  This  decimal  is  0-001041 6.  In  the 
same  manner,  we  shall  find  the  decimal  of  a  pound 
equivalent  to  2,  3,  4,  &c.  farthings,  by  converting  the 
fractions  ^|^,  -gf  ^,  -g^,  &c.  into  equivalent  decimals. 


236  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

It  will  be  found  that  in  all  such  decimals  the  figures 
which  fill  the  first  three  decimal  places  are  those  which 
express  the  number  of  farthings,  provided  that  number 
be  less  than  24.  For  24  farthings  the  equivalent  frac- 
tion of  a  pound  is  -f^  or  -J^  ;  and  the  equivalent  de- 
cimal is,  therefore,  0*025.  Whenever  the  number  of 
farthings  exceeds  23,  then,  it  will  be  found  that  the 
figures  which  fill  the  first  three  places  express  a  number 
which  exceeds  the  proposed  number  of  farthings  by  1 . 
This  observation  extends  to  every  number  of  farthings  less 
than  48;  and,  since  48  farthings  are  equivalent  to  a  shil- 
ling, it  is  never  necessary  to  seek  the  equivalent  decimal 
for  a  greater  number  of  farthings  than  47.  It  appears, 
therefore,  that  we  can  always  find  the  first  3  places  of 
the  decimal  of  a  pound,  which  is  equivalent  to  any  num- 
ber of  farthings,  by  the  following  rule  :  — 

If  the  number  of  farthings  be  less  than  24,  place 
the  figure  or  figures  which  express  them  in  the  third, 
or  in  the  second  and  third,  decimal  places.  If  the  num- 
ber be  not  less  than  24,  then  add  1  to  it,  and  proceed  in 
the  same  manner.  Since  the  decimal  places  beyond  the 
third  express  fractions  less  than  the  thousandth  part  of  a 
pound,  the  quantity  they  express  is  less  than  a  farthing. 
Such  places  may,  therefore,  be  omitted  in  all  calculations 
where  quantities  less  than  a  farthing  are  disregarded. 

Example.  —  Let  it  be  required  to  reduce  lOf  d.  to  an 
equivalent  decimal  of  a  pound.  The  number  of  farthings 
will  be  43,  and  the  equivalent  decimal  will  be  0-044. 
If  the  sum  be  8^d.,  the  number  of  farthings  will  be  34, 
and  the  equivalent  decimal  will  be  0*035,  &c. 

By  combining  this  rule  with  that  established  for 
shillings,  we  shall  be  able,  without  difficulty,  to  find  the 
decimal  of  a  pound  equivalent  to  any  number  of  shil- 
lings and  pence.  Let  the  sum  be  18s.  7fd.,  we  find 
the  decimal  as  follows  :  — 

£    « 

18*.  Od.      =  0-90 
7|d.    =  0-032 

0-932 


CHAP.  I.  REDUCTION.  237 

To  convert  251.  14s.  3Jd.  into  an  equivalent  decimal 
of  pounds : — 

£     3.    d.  £ 

25     14     0     =   25-70 
3§   =     0-014 

25-714 


(337.)  By  retracing  this  process  in  a  reverse  order, 
we  shall  obtain  a  rule  for  converting  any  decimal  of  a 
pound  into  shillings  and  pence. 

To  find  the  number  of  shillings,  double  the  number 
which  occupies  the  first  decimal  place,  and  if  the  num- 
ber occupying  the  second  place  of  decimals  be  not  less 
than  5,  add  1  to  the  number  so  found.  This  will  be 
the  number  of  shillings.  If  the  number  occupying  the 
second  decimal  place  be  greater  than  4,  subtract  5  from 
it.  If  the  number  which  remains  prefixed  to  the 
figure  in  the  third  decimal  place  be  less  than  24,  that 
number  will  be  the  number  of  farthings  in  the  required 
number.  If  it  be  not  less  than  24,  then  it  will  be  1 
more  than  the  number  of  farthings.  The  number  of 
farthings  being  thus  found  and  divided  by  4  will  give 
the  number  of  pence. 

Let  it  be  required  to  convert  the  decimal  a^?  75*876 
into  pounds,  shillings,  &c.  We  double  the  first  deci- 
mal figure,  8,  and  obtain  16.  The  second  place  being 
greater  than  5,  we  add  1  to  l6,  which  gives  17  for  the 
number  of  shillings.  Subtracting  5  from  the  second 
place,  we  prefix  the  remainder  to  the  third  place,  which 
gives  26.  This  being  greater  than  24,  we  subtract  1 
from  it,  and  find  25  for  the  number  of  farthings.  This, 
divided  by  4,  gives  6%d.  The  number  sought  is 
therefore,  75/.  17*.  6$d. 

To  convert  a  complex  number,  expressing  a  distance 
in  miles,  furlongs,  perches,  yards,  feet,  and  inches,  into 
a  simple  number  expressing  inches,  we  might  multiply 
the  units  of  each  denominator  by  the  number  of  inches 


238  A    TREATISE    ON    ARITHMETIC.          BOOK  III. 

which  they  respectively  contain  (310.),  and  then  add 
together  the  several  products ;  but  the  process  gene- 
rally used  is,  to  reduce  the  miles  to  furlongs  by  multiply- 
ing them  by  8,  and  add  to  the  product  the  furlongs  in  the 
proposed  length.  The  furlongs  thus  found  are  reduced 
to  perches  by  multiplying  them  by  40,  to  which  the 
number  of  perches  are  added ;  and  the  reduction  is 
continued  in  the  same  way,  upon  the  principle  already 
applied  to  sums  of  money. 

Let  it  be  required  to  convert  into  inches  the  following 
distance :  — 

miles,    furl,   perch,  yds.      ft.      inch. 
17         6         22         4         2         7 

the  process  will  be  as  follows :  — 

m.     f.     p.  yds.  ft.  in. 
17     6     22     427 
8 

142     furlongs. 
40 

5702     perches. 
5-5 

28510 
28510 
4 


31365-O 


By  multiplying  by  8,  and  adding  6  to  the  product,  we 
obtain  the  number  of  furlongs.  Multiplying  this  by 
40,  and  adding  22,  we  obtain  the  number  of  perches. 
To  reduce  this  to  yards,  we  should  multiply  by  5^,  or 


CHAP.  I.  REDUCTION.  239 

by  the  equivalent  decimal  5'5.  Performing  this  multi- 
plication, and  adding  4  in  the  units'  column  for  the 
yards  in  the  given  sum,  we  obtain  the  number  of  yards, 
omitting  the  0  which  fills  the  decimal  place.  We 
obtain  the  feet  by  multiplying  the  yards  by  3,  and  add- 
ing the  2  feet  in  the  given  sum;  and  we  obtain  the  inches 
by  multiplying  the  feet  by  12,  and  adding  the  7  inches 
in  the  given  sum. 

By  reversing  this  process,  we  may  convert  a  simple 
number,  expressing  inches,  into  an  equivalent  one,  ex- 
pressing miles,  furlongs,  &c. 

Let  the  proposed  number  be  1129171  inches.  To 
reduce  this  to  feet  divide  it  by  12  :  we  obtain  the  quo- 
tient 94097,  with  the  remainder  7  '•  the  quotient  is 
here  feet,  and  the  remainder  inches.  To  reduce  this 
number  of  feet  to  yards,  we  divide  by  3,  and  obtain  the 
quotient  31365,  with  the  remainder  2,  which  remainder 
is  yards.  In  the  same  manner  we  proceed  dividing 
successively  by  5*5,  40,  and  8,  and  we  obtain  quotients 
and  remainders  corresponding  to  the  numbers  in  the 
above  process,  until,  finally,  we  reproduce  the  complex 
number  with  which  we  commenced. 

To  convert  a  number  of  tons,  hundred  weights, 
quarters,  &c.  into  an  equivalent  number  of  ounces,  we 
proceed  in  a  manner  altogether  analogous  to  the  me- 
thods already  applied  to  other  complex  numbers ;  the 
relative  values  of  the  different  classes  of  units  being  de- 
rived from  the  table  (325.). 

Let  the  proposed  complex  number  be  15  tons,  7  cwt. 
3  qrs.  231bs.  15oz.  We  reduce  the  tons  to  hundred 
weights  by  multiplying  by  20,  adding  7  to  the  product. 
We  reduce  the  hundred  weights  to  quarters  by  multi- 
plying by  4.  In  the  same  manner  the  quarters  are 
reduced  to  pounds  by  multiplying  by  28,  and  the  pounds 
to  ounces  by  multiplying  by  16,  adding  to  the  several 
products  the  sums  taken  from  the  given  complex  num- 
ber. The  process  at  length  will  be  as  follows :  — 


A    TREATISE    ON    ARITHMETIC.          BOOK  JIT. 


Multiply 

tons.   cwt.  qrs.    Ibs.      oz. 
15       7       3       23       15 
20 

Add 

300     cwt. 

7      — 

Multiply 

307      — 
4 

Add 

1228     qrs. 
3      — 

Multiply 

1231      — 
28 

Add 

9848 
2462 

34468     Ibs. 
23      — 

Multiply 

34491      — 
16 

206946 
34491 

Add 

551856     oz. 
15     — 

551871     — 


The  reverse  process  of  converting  a  number  express- 
ing ounces  into  an  equivalent  number  expressing  tons 
hundred  weights,  &c.  would  be  as  follows :  — 

16)551871  oz. 


28)  34491  ft.    15  oz. 
4)  1231  qrs.   23ft.    15oz. 
20  )  307  cwt.   3  qrs.   23ft.    15  oz. 

15  tons,  7  cwt.   3  qrs.   23  ft.    15  oz. 

In  dividing  by  16,  there  is  a  remainder  15,  which 
signifies  15  ounces,  the  quotient  giving  the  number 
of  pounds.  Dividing  this  by  28,  we  get  the  quarters, 
with  a  remainder  23,  which  signifies  23  pounds  remain- 


CHAP.  I.  REDUCTION.  241 

ing  over  and  above  quarters.  The  first  remainder,  15 
ounces,  must  still  be  brought  down.  Dividing  again  by 
4t,  we  get  the  number  of  hundred. weights,  with  3  quar- 
ters remaining  over  ;  and  dividing  the  hundred- weights 
by  20,  we  get  the  tons,  with  a  remainder  over  of  7 
hundred- weights.  This  process  is  exactly  the  reverse 
of  the  first  one. 

It  is  unnecessary  to  pursue  such  reductions  farther, 
since  the  methods  applied  to  different  classes  of  complex 
numbers  have  no  other  difference  than  that  which  arises 
from  the  difference  between  the  relative  values  of  their 
several  units. 


24-2 
CHAP.  II. 

OF    THE    ADDITION    AND   SUBTRACTION   OP    COMPLEX    NUMBERS. 

(338.)  THE  method  by  which  the  several  arithmetical 
operations  are  performed  on  complex  numbers  rests  upon 
principles  essentially  identical  with  those  which  govern 
the  same  operations  performed  on  simple  numbers.  In 
these  numbers,  as  expressed  by  the  common  arithmetical 
notation,  there  is,  in  fact,  a  regular  succession  of  dis- 
tinct classes  of  units.  The  difference  between  them  and 
complex  numbers  only  consists  in  this,  that  in  simpje 
numbers  each  superior  class  of  units  has  the  same  nu- 
merical relation  to  the  class  below  it,  the  proportional 
values  being  always  in  a  decuple  progression ;  whereas, 
in  complex  numbers,  that  regular  relation  is  not  found 
to  exist  between  the  successive  classes  of  units. 

The  operations  of  addition  and  subtraction,  in  simple 
numbers,  are  effected  by  performing  them  successively 
on  each  order  of  units,  commencing  from  the  units'  co- 
lumn, and  proceeding  from  right  to  left.  The  same 
method  precisely  is  observed  in  complex  numbers ;  and 
the  difference  between  the  two  operations  only  arises 
from  the  way  in  which  numbers  must  be  carried  from 
one  order  of  units  to  another.  These  general  observ- 
ations will  be  easily  understood,  when  we  attempt  to 
perform  the  operations  on  a  few  examples  in  complex 
numbers. 

Let  it  be  proposed  to  add  together  the  several  sums  of 
money  here  expressed :  — 

£       s.       d. 

25       17       6 
6       13       5§ 
4         0       3£ 

10       11       7J 

12       14       6f 

769 

17 


CHAP.  II.  COMPOUND    ADDITION.  243 

To  perform  this  addition,  we  shall  first  add  together 
all  the  farthings  which  occur  after  the  pence,  counting 
each  halfpenny  as  2  farthings;  the  total  number  of  far- 
things which  we  shall  obtain  is  13;  but  since  every  4 
farthings  is  equivalent  to  1  penny,  13  farthings  will  be 
equivalent  to  3  pence  and  1  farthing :  we  therefore  write 
down  1  farthing,  and  carry  3  to  the  pence.  In  general, 
therefore,  when  the  farthings  in  the  sums  to  be  added 
are  added  together,  we  must  divide  their  number  by  4, 
put  down  the  remainder  as  farthings,  and  carry  the 
quotient. 

We  now  add  the  number  3  carried  from  the  farthings 
to  the  column  of  pence,  and  adding  that  column  we  find 
the  total  number  to  be  50  pence ;  but  since  every  12 
pence  makes  1  shilling,  we  shall  find  the  number  of 
shillings  in  50  pence  by  dividing  it  by  12 :  50  pence, 
therefore,  are  4s.  2d.;  we  write  2  in  the  pence  place,  and 
carry  4  to  the  shillings  column.  In  general,  therefore, 
when  the  pence  column  is  added  together  with  the  pence 
carried  from  the  farthings,  we  must  divide  the  sum  by 
12,  write  the  remainder  as  pence,  and  carry  the  quotient 
to  the  shillings. 

We  next  add  the  shillings*  column,  including  the  4» 
carried  from  the  pence,  and  we  find  the  sum  to  be  82 : 
dividing  this  by  20,  we  find  it  equal  to  4/.  2s.;  we  write 
2  in  the  shillings'  column,  and  carry  4  to  the  pounds* 
In  general,  therefore,  when  the  shillings'  column  is 
added,  we  divide  by  20,  write  the  remainder  in  the 
shillings'  place,  and  carry  the  quotient  to  the  pounds. 

The  process  of  adding  the  shillings  may  be  facilitated 
in  the  following  manner :  —  Add  the  digits  only  which 
occupy  their  units'  places  in  the  first  instance ;  write  the 
figure  as  you  would  in  simple  numbers  in  the  units' 
place  of  the  shillings  of  the  sum  ;  carry  to  the  tens  in 
the  same  manner  as  for  simple  numbers,  and  add  the 
tens'  column.  If  the  sum  be  an  even  number,  carry  half 
of  it  to  the  pounds,  and  write  no  figure  in  the  tens'  place 
of  the  shillings;  but  if  the  sum  be  an  odd  number, 
write  1  in  the  tens'  place  of  the  shillings,  and  carry  half 
R  2 


244  A    TREATISE    ON    ARITHMETIC.  BOOK  III* 

the  remainder  to  the  pounds :  thus,  if,  by  adding  the 
units  of  the  shillings  column  we  obtain  75,  we  write  5 
in  the  units'  place  of  shillings,  and  carry  7  to  the  tens  ; 
if,  after  adding  the  tens  with  the  7  carried,  the  sum  be 
12,  we  carry  6  to  the  pounds;  but  if  the  sum  be  13, 
we  write  1  in  the  tens'  place  of  the  shillings,  and  carry 
6  to  the  pounds. 

By  generalising  the  method  adopted  in  this  example, 
we  shall  perceive  that  all  complex  numbers  may  be  added 
together  by  the  following 

RULE. 

(339.)  Place  the  complex  numbers  one  under  another, 
so  that  the  same  classes  of  units  shall  stand  in  the  same 
vertical  column,  the  smallest  units  occupying  the  first  co- 
lumn on  the  right,  and  the  units  of  succeeding  orders 
being  placed  in  successive  columns  from  right  to  left.  Add 
the  first  column  on  the  right,  and  having  found  its  sum, 
divide  it  by  that  number  which  expresses  the  number  of 
units  of  that  order  contained  in  a  single  unit  of  the  next 
order  above  it ;  the  quotient  will  give  the  number  to  be 
carried,  and  the  remainder  will  give  the  number  to  be 
placed  under  the  first  column  to  the  right :  add  the  next 
column  on  the  left  together  with  the  number  carried,  and 
having  obtained  the  sum,  divide  it  by  that  number  which 
expresses  how  often  the  unit  of  the  column  added  is  con- 
tained in  the  next  superior  unit;  write  the  remainder 
under  the  column,  and  carry  the  quotient  to  the  next  co- 
lumn on  the  left,  and  proceed  in  the  same  way  until  all 
the  columns  of  units  have  been  added. 

This  rule  will  be  better  understood  when  applied  to 
the  following  examples.  Let  it  be  required  to  add  to- 
gether the  following  times  :  — 

years,  weeks  days,  hours,  min.   seconds. 

3         27         5         22         51         37 

21          15         6         14         32         25 

41         49         4  0         ,0         59 

38          6          23         47         42 

67          28          2          13          T2         43 


CHAP.  II.  COMPOUND    ADDITION.  245 

By  adding  the  seconds  column  we  find  the  total  num- 
ber 163 ;  this  divided  by  6'0  gives  the  quotient  2,  with 
the  remainder  43 ;  we  write  43  under  the  column,  and 
carry  2  to  the  minutes.  The  addition  of  the  minutes' 
column  with  the  2  carried  gives  132 ;  this  divided  by 
60  gives  the  quotient  2  and  a  remainder  12:  we  write 
12  under  the  minutes'  column  and  carry  2  to  the  hours. 
Adding  the  hours  with  the  2  carried,,  we  find  the  sum 
6l :  this  divided  by  24  gives  the  quotient  2  and  the 
remainder  13:  we  write  13  in  the  hours'  place  and  carry 
2  to  the  days.  The  addition  of  the  column  of  days  gives 
23,  which,  divided  by  7,  gives  the  quotient  3  with  a 
remainder  2 :  writing  2  under  the  days  and  carrying  3 
to  the  weeks,  we  find  the  total  weeks  1 32 ;  this  divided 
by  52  gives  a  quotient  2  and  a  remainder  28  :  we  write 
28  under  the  weeks  and  carry  2  to  the  years,  by  the  ad- 
dition of  which  we  get  67. 

The  process  of  adding  complex  numbers  may  some- 
times be  facilitated  by  omitting  the  numbers  carried  in 
the  first  instance,  and  adding  each  successive  column  as 
simple  numbers.  We  shall  proceed  thus  in  the  following 
example.  Let  it  be  required  to  add  the  following  dis 
tances :  — 

miles,     furl,  perch,  yds.  ft.  inches. 

11         6         29         4  2  11 

7         38         3  0  8 

22         5         27         0  1  10 

8         4         35         5  2  9 


41        22      129        12         5          38 

We  have  here  obtained  the  true  sum  of  the  distances, 
but  the  several  classes  of  units  are  not  expressed  in  their 
least  terms :  38  inches  may  be  reduced  to  feet  and  inches 
by  dividing  it  by  12,  and  we  find  that  it  is  equivalent  to 
3  feet  2  inches;  we,  therefore,  instead  of  38  inches, 
write  2  inches  and  add  2  to  the  number  of  feet.  We 
should  thus  have  7  feet  in  the  second  column ;  but  7 
feet  are  equivalent  to  2  yards  1  foot :  we  write,  there- 
fore, 1  under  the  column  of  feet,  and  carry  2  to  the 


246  A    TREATISE    ON    ARITHMETIC.  BOOK  III- 

yards  ;  the  number  of  yards  thus  becomes  14:  but  since 
1 1  yards  are  2  perches,  14  yards  are  2  perches  3  yards  ; 
we  therefore  write  3  under  the  yards,  and  carry  2  to  the 
perches :  the  number  of  perches  thus  becomes  131.  Di- 
viding this  by  40,  it  will  be  reduced  to  furlongs  and 
perches,  and  is  equivalent  to  3  furlongs  1 1  perches :  we 
therefore  write  1 1  under  the  perches  and  carry  3  to  the 
furlongs ;  the  number  of  furlongs  will  thus  be  25  :  to 
reduce  this  to  miles  and  furlongs,  we  divide  it  by  8.  It 
is,  consequently,  equivalent  to  3  miles  1  furlong :  we 
therefore  write  1  under  the  furlongs  and  carry  3  to  the 
miles.  The  total  sum  is  therefore  expressed  as  follows: — 

miles.         furlong,     perches,     yds.      ft.  inches. 

44  1  11  3  1  2 

(340.)  One  complex  number  is  subtracted  from  an- 
other by  placing  the  subtrahend  under  the  minuend, 
the  units  of  the  same  classes  being  placed  in  the  same 
vertical  columns.  The  units  of  each  class  of  the  subtra- 
hend are  then  subtracted  from  those  of  the  same  class 
in  the  minuend,  and  the  results  are  written  under  them 
in  the  remainder.  If  the  number  of  units  of  each  class 
in  the  subtrahend  be  less  than  those  of  the  same  class  in 
the  minuend,  the  process  will  be  evident,  as  in  the 
following  examples.  Subtract  31.  7s.  6±d.  from 
11.  10*.  8ld.  :  — 

£         s.         d. 

7        10        8J 

3  7         6§ 


Subtract  7  cwt.   2  qrs.  8  Ibs.  6oz.   from  12cwt.  3  qrs. 
lOlbs.  11  oz.:-— 

cwt.     qrs.       fts.        oz. 

12         3          10         11 
7286 


1 


If  the  units  of  any  class  in  the  subtrahend  be  greater 
in  number  than  the  units  of  the  same  class  in  the  minu- 
end, the  subtraction  cannot  be  immediately  performed: 


CHAP.  II.  COMPOUND    SUBTRACTION.  247 

the  difficulty,  however,  is  removed  by  the  same  artifice 
as  was  explained  in  the  like  case  in  simple  numbers.  A 
unit  of  a  higher  order  is  carried  to  the  next  column,  and 
an  equivalent  number  of  units  of  the  same  order  is 
added  to  the  number  in  the  minuend  :  thus  equal  quan- 
tities are  added  to  the  minuend  and  subtrahend,  and 
therefore  their  difference  remains  the  same.  (97.)  (108.) 
Let  it  be  required  to  subtract  81.  17*.  10cZ.  from 
111.5s.  4d. 

£       5.        d. 

1154 

8       17       10 


The  number  of  pence  in  the  minuend  being  less  than 
the  number  of  pence  in  the  subtrahend,  the  latter  can- 
not be  subtracted  from  the  former:  the  class  of  units 
next  superior  to  pence  being  shillings,  and  1  shilling 
being  equal  to  12  pence,  we  add  12  to  the  pence  in  the 
minuend,  by  which  we  obtain  16  pence:  subtracting  10 
from  this,  we  write  the  remainder  6  under  the  pence. 
To  compensate  for  the  12  pence  added  to  the  minuend, 
we  now  add  1  shilling  to  the  subtrahend,  or  carry  1  to 
the  shillings  :  we  have  then  to  subtract  18  from  5  ;  but 
that  being  impossible,  and  the  class  of  units  next  supe- 
rior being  pounds,  and  1  pound  being  equal  to  20  shil- 
lings, we  add  20  shillings  to  the  minuend,  by  which  we 
obtain  25  shillings  :  subtracting  18  from  this,  we  write 
7  in  the  remainder.  To  compensate  for  the  20  shillings 
added  to  the  minuend,  we  add  1  pound  to  the  subtra- 
hend, or  carry  1  to  the  pounds :  we  accordingly  sub- 
tract 9  from  11,  and  obtain  the  remainder  2 :  the  total 
remainder  is  then  2/.  7*.  6d. 

Let  it  be  required  to  subtract  3  cwt.  3  qrs.  24  Ibs, 
12  oz,  from  7  cwt.  1  qr.  5  Ibs. 

cwt.  qrs.  Ibs.  oz. 

7150 

3       3     24     12 
~3       I       8       4~ 

R  4 


248  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

Since  there  are  no  ounces  in  the  minuend,  and  the  unit 
next  superior,  1  pound,  being  equal  to  16  ounces,  we 
subtract  12  ounces  from  16,  and  write  4  in  the  remain- 
der. To  compensate  for  the  l6  ounces  added  to  the 
minuend,  we  add  1  pound  to  the  subtrahend ;  but  since 
25  is  greater  than  5,  and  the  unit  next  superior,  1  quar- 
ter, being  equal  to  28  pounds,  we  add  28  to  the  minu- 
end, and  subtract  25  from  33,  and  write  8  in  the 
remainder.  To  compensate  for  the  28  pounds  added  to 
the  minuend,  we  add  1  quarter  to  the  subtrahend,  or 
carry  1  to  the  quarters:  we  have  then  to  subtract  4 
from  1 ;  but  1  being  less,  and  the  next  superior  unit,  1 
hundred-weight,  being  equal  to  4  quarters,  we  add  4  to 
the  quarters,  and,  subtracting  4  from  5,  write  1  in  the 
remainder.  To  compensate  for  the  4  quarters  added  to 
the  subtrahend,  we  add  1  hundred- weight  to  the  minu- 
end, or  carry  1  to  the  hundred- weights ;  we  write  3  in 
the  remainder,  and  the  total  remainder  is  3  cwt.  1  qr. 
8  Ibs.  4  oz. 

(341.)  When  these  examples  are  duly  considered, 
the  student  will  perceive,  that  when  the  number  of  units 
of  any  class  of  the  subtrahend  is  greater  than  the  num- 
ber of  units  of  the  same  class  in  the  minuend,  instead  of 
using  the  given  minuend  and  subtrahend,  we  use  a  dif- 
ferent minuend  and  subtrahend,  but  of  such  a  nature  as 
will  give  by  subtraction  the  same  remainder :  in  fact, 
we  add  equal  quantities  to  the  minuend  and  subtrahend, 
by  which  their  difference  will  remain  unchanged.  (97.) 
Thus,  in  the  above  numbers,  the  given  minuend  and  sub- 
trahend in  each  have  been,  in  fact,  changed  in  the  fol- 
lowing manner,  previous  to  subtraction : — 

£      s.      d.  cwt.  qrs.  Ibs.   oz. 

11     25     16  75     33     16 

9     18     10  4     4     25     12 


3184 


In  the  first  example,  we  have  added  to  the  minuend  and 
subtrahend  the  sum  of  I/.  Is.;  but  we  have  added  it  in 
a  different  manner  to  each.  In  the  minuend  we  have 


CHAP.  II.  COMPOUND    SUBTRACTION.  249 

added  20  to  the  shillings  and  12  to  the  pence;  and 
in  the  subtrahend  we  have  added  1  to  the  pounds  and 
1  to  the  shillings. 

In  the  second  example,  we  have  added  1  cwt.  1  qr. 
1  Ib.  to  the  minuend  and  subtrahend ;  but,  as  before, 
we  have  made  this  addition  in  a  different  manner  in 
each  case.  In  the  minuend  we  have  added  4  to  the 
quarters,  28  to  the  pounds,  and  16  to  the  ounces ; 
while  in  the  subtrahend  we  have  added  1  to  the  hun- 
dred-weights, 1  to  the  quarters,  and  1  to  the  pounds. 

The  student  will  feel  no  difficulty  in  generalising 
these  ideas,  so  as  to  apply  them  to  the  subtraction  of  any 
complex  numbers. 

(342.)  The  same  methods  of  verification  already  ex- 
plained in  the  addition  and  subtraction  of  simple  num- 
bers are  applicable  likewise  to  complex  numbers.  These 
operations  may  be  used,  therefore,  to  verify  each  other. 
In  addition,  if  from  the  sum  be  subtracted  the  sum  of 
all  the  numbers  added,  except  one,  that  one  must  be  the 
remainder,  if  the  work  be  correct ;  and,  in  subtraction, 
if  the  remainder  be  added  to  the  subtrahend,  the  sum 
should  be  the  minuend.  Also,  if  the  remainder  be  sub- 
tracted from  the  minuend,  the  new  remainder  should  be 
the  subtrahend.  These  consequences  are  so  evident, 
that  it  is  unnecessary  to  illustrate  them  by  examples. 


250 
CHAP.  III. 

OP   THE    MULTIPLICATION   OF    COMPLEX   NUMBERS. 

(343.)  THE  multiplication  of  complex  numbers  is  an 
operation  which,  in  many  cases,  is  attended  with  con- 
siderable complexity  and  difficulty.  The  methods  of 
performing  it  will  be  most  clearly  understood  by  practical 
examples  of  its  application  :  we  shall,  therefore,  proceed 
to  give  a  series  of  examples,  ascending  from  the  most 
simple  to  the  most  complex  cases ;  and  shall  subjoin  to 
each  example  such  observations  as  will  enable  the  stu- 
dent to  generalise  the  methods  which  it  suggests. 

(344.)  FIRST  CASE.  Let  the  multiplier  be  a  simple 
number,  not  exceeding  12. 

Example  1.  A  piece  of  a  certain  cloth  costs  2/.  ?*•  9d.; 
what  will  be  the  price  of  7  pieces  of  the  same  cloth  ? 
To  solve  this  question,  it  is  necessary  to  multiply 
2/.  7#»  9^'  hy  7  '•  the  process  is  as  follows :  — 

£      *.     d. 
•279 

7 


14     49     63 
16     14       3 

We  have  here  multiplied  the  pounds,  shillings,  and 
pence  severally  by  7}  arid  obtained  the  three  products 
1 4  pounds,  49  shillings,  and  63  pence :  but  since  63 
pence  admits  of  being  reduced  to  shillings,  and  49  shil- 
lings to  pounds,  we  divide  63  by  12,  and,  writing  the 
remainder,  3,  carry  the  quotient,  5,  to  the  shillings, 
which  makes  the  number  of  shillings  54.  Dividing  this 
by  20,  we  write  the  remainder,  14,  in  the  shillings,  and 
carry  the  quotient,  2,  to  the  pounds :  the  product  is, 


CHAP.  III.  COMPOUND    MULTIPLICATION.  251 

therefore,  reduced  to  the  form  l6l.  14s.  Sd.}  which  is 
the  answer  to  the  proposed  question. 

It  is  not  necessary,,  however,  to  write  the  product  in 
the  first  form,  since  the  reduction  may  be  made  at  the 
same  time  with  the  process  of  multiplication  :  thus,  we 
say,  7  times  9  are  63  pence,  which  are  5s.  3d. ;  we  write 
3  in  the  pence,  and  carry  5  to  the  shillings :  7  times  7 
are  49  shillings,  which,  with  5  carried,  make  54 ;  this 
is  equivalent  to  %l.  14s.;  we  write  14  in  the  shillings, 
and  carry  2  to  the  pounds:  7  times  2  are  14,  and  2 
carried  are  16;  we  write  16  in  the  pounds. 

Example  2.  An  engineer  contracts  to  make  12  miles 
of  road,  at  367^  15s.  4>^d.  a  mile;  what  will  be  the 
cost  of  the  whole  length  ?  We  must  multiply  the  cost 
of  1  mile  by  the  number  of  miles  :  — 

£       s.       d. 

367     15      4i 
12 


4413       4       6 

We  say  12  halfpence  are  6  pence,  carry  6  to  the  pence  : 
12  times  4  are  48,  and  6  are  54,  which  is  equivalent  to 
4s.  6d. ;  write  6  in  the  pence,  and  carry  4  to  the  shil- 
lings :  12  times  5  are  60  and  4  are  64,  write  4  in  the 
units'  place  of  the  shillings,  and  carry  6  to  the  tens  : 
12  times  1  are  12,  and  6  carried  are  18 ;  this  being  18 
ten  shillings,  is  equivalent  to  9  twenty  shillings,  or  Ql. ; 
carry  9  to  the  pounds.  12  times  7  are  84,  and  9  are 
93 ;  write  3  in  the  units  of  the  pounds  and  carry  9  to 
the  tens ;  12  times  6  are  72  and  9  are  81,  write  1  in 
the  tens  and  carry  8  to  the  hundreds  ;  12  times  3  are 
36  and  8  are  44,  write  4  in  the  hundreds  and  4  in  the 
thousands. 

In  these  examples,  it  will  be  perceived  that  the  me- 
thod adopted  is  to  multiply  each  class  of  units  in  the 
multiplicand  separately  by  the  multiplier,  beginning 
with  the  inferior  classes  of  units  and  proceeding  to  the 
superior.  When  the  product  of  each  class  of  units  is 
found,  it  is  reduced  to  the  superior  class  by  dividing  it 


252  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

by  the  number  which  expresses  how  many  inferior  units 
are  contained  in  the  superior.  Thus  we  find  first  the 
product  of  the  farthings  or  halfpence,  if  there  be  any,  and 
to  find  the  number  to  be  carried,  reduce  that  to  pence ; 
we  then  find  the  product  of  the  pence,  adding  the 
number  carried,  and  reduce  that  to  shillings  ;  we  then 
find  the  product  of  the  shillings,  adding  the  number 
carried,  and  reduce  that  to  pounds  ;  finally,  we  find  the 
product  of  the  pounds,  adding  the  number  carried. 

This  method  will  always  be  sufficient  for  every 
multiplier  which  does  not  exceed  the  extent  of  our  know- 
ledge of  the  multiplication  table.  Since  the  multipli- 
cation table  is  usually  committed  to  memory  for  num- 
bers as  far  as  12,  the  process  of  multiplication  may 
commonly  be  performed  in  this  way,  when  the  multi- 
plier does  not  exceed  12 ;  but  the  same  method  will 
serve  for  higher  numbers  with  those  who  have  com- 
mitted to  memory  the  multiplication  table  to  a  greater 
extent. 

(345.)  SECOND  CASE.  When  the  multiplier  is  a 
number  which  is  the  product  of  two  figures  neither  of 
which  exceeds  12. 

Example  1.  A  certain  cloth  costs  2J.  5s.  7d.  per  yard: 
what  is  the  price  of  72  yards  ?  72  being  the  product  of 
9  and  8,  we  may  multiply  the  multiplicand  first  by  9* 
and  then  multiply  the  product  by  8  :  the  product  finally 
obtained  will  be  that  which  is  sought.  (140.) 

£     s.    d. 

257 

9 


20     10     3 
8 

164       2     0 


Example  2.  The  weight  Sqrs.  171bs.  lloz.  of  a  cer- 
tain grain  is  purchased  for  11. ;  how  much  of  the  same 
grain  can  be  purchased  for  96/.? 


CHAP.  III.  COMPOUND    MULTIPLICATION.  253 

96  being  the  product  of  12  and  8,  we  shall  obtain 
the  product  sought  by  first  multiplying  the  multiplicand 
by  12,  and  then  multiplying  the  product  by  8. 


cwt.  qrs.    Ibs.   oz. 
0       3       17     11 

12 

10       3       15       9 
8 

87       0       12       8 


The  numbers  to  be  carried  from  each  class  of  units  to 
the  superior  class,  are  determined  here  in  the  same  man- 
ner as  already  explained  in  Addition. 

(346.)  THIRD  CASE.  When  the  multiplier  is  a  whole 
number  which  cannot  be  conveniently  resolved  into 
small  factors. 

Example  1 .  Let  it  be  required  to  multiply  784/.  1 5s.  §d. 
by  857:  the  process  would  be  performed  as  follows:  — 


£ 

s. 

d. 

784 

15 

9 

857 

5488 

3920 

s. 

d. 

6272 

10 

6 

428 

10 

0 

5 

0 

214 

5 

0 

0 

6 

21 

8 

6 

0 

3 

10 

14 

3 

£672562     17     9 


We  first  multiply  the  pounds  of  the  multiplicand  by 

the  multiplier,  according  to  the  rule  for  the  multipli- 

'  cation  of  whole  numbers ;  but  we  postpone  the  addition 

of  the  three  partial  products  until  the  shillings  and  pence 

of  the  multiplicand  have  been  likewise  multiplied  by  the 


254  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

multiplier.  We  might  proceed  to  multiply  the  shillings 
by  the  multiplier  by  the  rules  for  whole  numbers,  and 
obtain  the  products,  which  in  that  case  would  be  ex- 
pressed in  shillings,  and  might  be  subsequently  re- 
duced to  pounds ;  but  we  obtain  the  result  by  a  more 
abridged  though  less  direct  process:  we  consider  the  15 
shillings  of  the  multiplicand  to  be  resolved  into  two  parts, 
viz.  1 0 shillings,  and  5  shillings :  10  shillings  being  the  half 
of  a  pound,  we  should  obtain  by  multiplying  it  by  the 
multiplier  half  as  many  pounds  as  are  expressed  by  that 
multiplier  j  we  have,  therefore,  only  to  consider  the  mul- 
tiplier as  expressing  pounds,  and  to  divide  it  by  2,  in 
order  to  obtain  the  product  of  10  shillings  multiplied  by 
it.  Dividing  857  by  2,  we  obtain  the  quotient  428, 
with  a  remainder  1,  which  expresses  half  a  pound,  or 
10  shillings  :  the  product,  therefore,  of  10  shillings  by 
857  is  428/.  10s.  It  now  remains  to  multiply  5  shil- 
lings by  the  multiplier,  but  this  product  will  evidently 
be  half  the  preceding  product  in  which  the  multiplicand 
was  10  shillings  :  dividing  the  preceding  product  by  2, 
we  obtain  214/.  5s. ,  which  is,  therefore,  the  product  of 
5  shillings  by  the  multiplier.  It  now  remains  to  mul- 
tiply 9  pence  by  the  multiplier,  and  we  accomplish  this 
by  considering  9  pence  to  be  resolved  into  two  parts,  6 
pence,  and  3  pence :  6  pence  being  the  tenth  part  of  5 
shillings,  we  shall  obtain  the  product  of  6  pence  by  the 
multiplier  by  taking  the  tenth  part  of  the  preceding  pro- 
duct; dividing  the  pounds  by  10,  we  get  21  with  a  re- 
mainder 4 ;  this,  being  4  pounds,  is  equivalent  to  80 
shillings,  to  which  the  5  shillings  being  added,  we  ob- 
tain 85  shillings  ;  this,  divided  by  10,  gives  the  product 
8,  with  a  remainder  5 ;  this  5  shillings  is  equivalent  to 
60  pence,  which  being  divided  by  10  gives  the  quotient 
6:  the  partial  product  is  therefore  2 11.  8s.  6d.  The 
product  of  3  pence  by  the  multiplier  is,  evidently,  half 
the  last  found  product ;  to  obtain  it,  therefore,  we  have 
only  to  divide  the  last  product  by  2 :  dividing  the  pounds 
by  2,  we  get  the  quotient  10,  with  a  remainder  1:  this 


CHAP.  III.  COMPOUND    MULTIPLICATION.  255 

1  pound  being  equivalent  to  20  shillings,  and  added  to 
the  8  shillings,  we  divide  28  by  2,  and  get  the  quotient 
14.  We  next  divide  6  by  2,  and  get  the  quotient  3  : 
the  last  partial  product  is  therefore  101.  I4>s.  3d.  All 
these  partial  products  being  now  added  together,  we  ob- 
tain the  total  product. 

By  the  method  here  pursued,  we  multiply,  in  the  first 
instance,  the  highest  class  of  units  in  the  multiplicand 
by  the  multiplier,  following  the  rules  established  for 
simple  numbers:  the  shillings  are  then  resolved  into 
parts,  one  or  more  of  which  are  aliquot  parts,  or  sub- 
multiples  of  a  pound.  In  the  present  case,  the  number 
being  15,  we  resolve  it  into  2  parts,  one  of  which,  10,. 
is  the  half  of  a  pound,  and  the  other  is  the  half  of  that, 
or  the  fourth  of  a  pound :  had  the  number  of  shillings 
been  17,  we  should  have  resolved  it  into  10,  5,  and  2, 
the  first  being  half  of  a  pound,  the  second  half  the  first, 
and  the  third  a  fifth  of  the  first.  Having  found  the  first 
partial  product,  its  half  would  be  the  second,  and  its 
fifth  the  third. 

The  pence  are  resolved  into  such  parts  that  one  of 
them  shall  be  an  aliquot  part  of  one  of  the  parts  into 
which  the  shillings  have  been  resolved.  In  the  present 
case,  6  pence  is  the  tenth  of  5  shillings,  and  we,  accord- 
ingly, find  the  product  corresponding  to  6  pence  by  di- 
viding the  product  corresponding  to  5  shillings  by  10. 
Had  the  number  of  shillings  been  17,  we  should  have 
had  a  product  corresponding  to  2  shillings,  in  which 
case  we  should  have  found  the  product  corresponding  to 
6  pence  by  dividing  the  latter  by  4. 

The  spirit  of  this  method  consists  in  resolving  the 
shillings  and  pence  into  a  series  of  parts,  each  of  which 
shall  be  an  aliquot  part  or  submultiple  of  some  preced- 
ing part,  so  that  the  successive  partial  products  may  be 
derived  one  from  another  by  dividing  by  single  digits. 

Example  2.  Let  it  be  required  to  multiply  67  fathoms, 
5  feet,  6  inches,  5  lines,  by  59  (1  fathom  =  6  feet)  :  the 
process  will  be  as  follows  :  — 


256  A    TREATISE    ON    ARITHMETIC. 


fath. 

ft. 

ins. 

lines. 

67 

5 

6 

5 

59 

603 

3  feet  , 

335 
29 

3 

2  feet  

19 

4 

4 

g 

0 

jj 

yy 

4  lines  ...... 

0 

7T 
1 

•*/* 
7 

8 

1  line........ 

0 

0 

4 

11 

4007       267 

As  before,  we  begin  by  multiplying  the  highest  class 
of  units  by  the  multiplier,  as  in  whole  numbers,  and 
obtain  the  first  two  partial  products,  which  are  arranged 
according  to  the  rule  for  simple  multiplication.  The 
partial  products  corresponding  to  the  inferior  classes  of 
units  are  next  found :  3  feet  being  half  a  fathom,  we 
resolve  the  5  feet  of  the  multiplicand  into  3  and  2 ;  we 
may  obtain  the  partial  product  corresponding  to  3  feet 
by  dividing  the  multiplier  by  2  :  the  quotient  is  29,  with 
a  remainder  1 :  this  being  a  fathom,  when  divided  by 
2  we  obtain  the  quotient  3  feet.  The  partial  product  is 
therefore  29  fathoms  3  feet :  2  feet  being  the  third  part 
of  a  fathom,  we  obtain  the  fourth  partial  product  by 
dividing  59  fathoms  by  3 ;  the  quotient  is  19,  with  a 
remainder  2,  which,  divided  by  3,  gives  two  thirds  of  a 
fathom,  or  4  feet :  6  inches  being  the  fourth  part  of  2 
feet,  we  obtain  the  fifth  partial  product  by  dividing  the 
fourth  by  4:  19  fathoms  divided  by  4  gives  the  quo- 
tient 4,  with  a  remainder  3  fathoms,  or  18  feet:  this 
added  to  4  feet  gives  22  feet,  which  divided  by  4  gives 
the  quotient  5,  with  a  remainder  2  feet,  or  24  inches : 
this  divided  by  4  gives  the  quotient  6  inches.  The  fifth 
partial  product  is,  therefore,  4  fathoms  5  feet  6  inches. 
To  facilitate  the  remaining  partial  products,  we  first  ob- 
tain that  which  would  correspond  to  1  inch,  which  is 
done  by  dividing  the  fifth  partial  product  by  6 :  4  fa- 
thoms 5  feet  being  equivalent  to  29  feet,  when  divided 
by  6  gives  the  quotient  4  feet,  with  a  remainder  5  feet, 
or  60  inches ;  this  added  to  6  inches  gives  66  inches, 


CHAP.  III.  COMPOUND  MULTIPLICATION.  257 

which,  divided  by  6,  gives  the  quotient  11.  This  num- 
ber, 4  feet  1 1  inches,  would  be  the  partial  product,  there- 
fore, corresponding  to  1  inch,  or  12  lines.  Let  the  5 
lines  in  the  multiplicand  be  supposed  to  be  resolved  into 
4  and  1 :  the  partial  product  corresponding  to  4  lines 
will  be  found  by  dividing  4  feet  1 1  inches  by  3,  since 
4  is  the  third  part  of  12:  dividing  4  by  3,  we  get  the 
quotient  1,  with  a  remainder  1  foot,  or  12  inches,  to 
which  1 1  being  added,  we  get  23  inches ;  which,  divided 
by  3,  gives  the  quotient  7,  with  the  remainder  2  inches, 
or  24  lines.  Dividing  24  lines  by  3,  we  get  the  quo- 
tient 8  lines :  the  sixth  partial  product  is,  therefore,  1 
foot  7  inches  8  lines ;  the  seventh  partial  product,  cor- 
responding to  1  line,  will  be  the  fourth  part  of  this : 
dividing,  therefore,  1  foot  7  inches,  or  19  inches,  by  4, 
we  get  the  quotient  4,  with  a  remainder  3  inches,  or  36 
lines :  this,  added  to  8  lines,  gives  44  lines,  which,  di- 
vided by  4,  gives  the  quotient  1 1  lines.  The  last  par- 
tial quotient  is,  therefore,  4  inches  1 1  lines. 

The  product  corresponding  to  1  inch  being  intro- 
duced merely  for  the  purpose  of  facilitating  the  process 
by  which  the  remaining  partial  quotients  are  found,  and 
not  forming  a  part  of  the  actual  multiplication,  is  omitted 
in  the  addition,  and,  to  indicate  this,  lines  have  been 
drawn  across  the  figures. 

(347.)  When  the  multiplicand  is  a  sum  of  money, 
the  process  of  multiplication  may  frequently  be  sim- 
plified by  converting  it  into  decimal  parts  of  a  pound, 
by  the  method  already  explained  (336.  et  seq.). 

Example  I.  Let  it  be  required  to  find  the  annual 
amount  of  31.  13s.  6d.  per  day:  31.  13s.6d.=  3'675. 
We  must  therefore  multiply  the  latter  number  by  365, 
the  number  of  days  in  the  year:  the  process  will  be 
as  follows  :  — 

3-675 
365 


£1341-375  =  £1341     7     6 


258  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

The  decimal  found  by  this  process  of  multiplication 
is  here  reconverted  into  pounds,  shillings,  and  pence,  by 
the  method  explained  in  (337). 

Example  2.  The  wages  of  an  artisan  are  2/.  10s.  6d. 
per  week ;  what  are  his  annual  wages  ?  The  weekly 
wages  must  here  be  multiplied  by  52,  the  number  of 
weeks  in  a  year.  The  process  is  as  follows :  — 

£    s.    d. 
2     10     6  =  2'525 
52 

5050 
12625 


£131-300  =  £131     5     0 

(848.)  When  the  multiplicand  is  a  sum  of  money, 
the  process  of  multiplication  is  frequently  facilitated 
by  resolving  the  shillings  and  pence  into  sub-multi- 
ples, or  aliquot  parts,  of  a  pound.  To  do  this  with 
facility,  it  would  be  necessary  that  the  computer  should 
commit  to  memory  the  principal  subdivisions  of  the 
pound  sterling,  which  are  expressed  in  the  following 
table  :  — 

£        s.    d.  £        s.     d. 

i   =    10     0  i   =    1     0 

£=68  ,£   =   010 

£=50  sb   =   0     8 

£=40  &   =   0     74 

I  =     3     4  J,   =   0     6 

£=26  ^=05 

^=20  ^  =   04 

A  *•    i   a         A  =  o   sj 


It  will  be  also  advantageous  to  commit  to  memory  the 
following  table  of  aliquot  parts  of  a  shilling  :  — 

9.        d.         s.       d. 


14 


CHAP.  III.  COMPOUND  MULTIPLICATION.  259 

The  use  of  these  tables  will  be  perceived  in  the  follow- 
ing examples. 

Example  1.  The  wages  of  a  labourer  are  II.  7s.  8a. 
per  week  ;  what  will  be  his  yearly  receipt  ?  It  is  neces- 
sary to  multiply  the  weekly  wages  by  52 ;  the  process 
is  as  follows :  — 

£    s.     d. 
1     7     8 
52 

s.    d.  ~52~" 

68=  j 17     6     8 

1     0   =      2  12     0 


18     8 


We  first  multiply  the  pounds  by  52,  and  obtain  the 
first  partial  product ;  we  then  resolve  7*.  Sd.  into  6*.  8d., 
and  Is.,  which  are  respectively  -J-  and  ^  of  a  pound. 
Since  II.  a  week  would  be  52/.  a  year,  ^  and  ^  °f  a 
pound  will  be  respectively  ^  and  ^  of  52/.  a  year :  we 
therefore  divide  52J.  by  3  and  by  20,  and  we  obtain  the 
second  and  third  partial  products,  which  are,  in  fact,  the 
products  of  6s.  Sd.  and  Is.  multiplied  by  52. 

Example  2.  What  is  the  annual  amount  of  17*.  1  Id. 
per  day  ?  It  is  necessary  to  multiply  this  sum  by  365  :  to 
do  so,  we  shall  resolve  it  in  such  a  manner  that  the  first 
part  shall  be  an  aliquot  part  of  a  pound,  and  each  suc- 
ceeding part  an  aliquot  part  of  the  preceding  one.  If 
we  take  10  shillings  for  the  first  part,  5  shillings  for 
the  second,  2  shillings  and  6  pence  for  the  third,  and 
5  pence  for  the  fourth,  this  will  be  accomplished.  The 
first  will  then  be  half  of  a  pound,  the  second  half  the 
first,  the  third  half  the  second,  and  the  fourth  the  sixth 
part  of  the  third,  and  the  four  partial  products  will  have 
corresponding  relations.  We  shall  find  the  amount  of 
10*.  a  day  by  dividing  3651.  by  2;  the  quotient  is 
1 82/.  10*. :  the  remaining  partial  products  are  found  as 
follows :  — 

s  2 


260  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

s.  d.  £      s.  d. 

10  0  per  day  =  182  10  0  per  annum. 

5     0    =  91     5  0       

2     6    =  45   12  6       

0     5    =  7  12  1       


£•326  19     7 

In  this  case,  the  second  partial  product  is  found  by 
dividing  the  first  by  2 ;  the  third  by  dividing  the  se- 
cond by  2 ;  and  the  fourth,  by  dividing  the  third  by  6. 

(349.)  This  method  of  multiplication,  by  resolving 
the  multiplicand  into  a  number  of  parts,  which  are 
either  sub-multiples  of  each  other,  or  sub-multiples  of 
one  of  the  principal  units  of  the  multiplicand,  is  usually 
delivered,  in  treatises  on  Arithmetic,  as  a  distinct  rule, 
under  the  name  of  PRACTICE.  It  is,  however,  as  will 
be  easily  perceived,  nothing  but  a  peculiar  method  of 
multiplication,  and  the  facility  with  which  it  may  be 
performed  will  depend  on  the  expertness  and  ingenuity 
with  which  complex  numbers  may  be  resolved  into 
parts  having  the  peculiar  relation  to  each  other  just 
mentioned.  No  general  rule  can  be  given  for  such  a 
resolution,  and  there  are  many  ways  in  which  the  same 
number  may  be  thus  resolved.  Thus^  the  above  sum 
might  have  been  resolved  into  1 0$.,  6s.  Sd.,  and  1  s.  3d., 
in  which  case  the  three  partial  products  would  have  been 
found  by  dividing  365  by  2,  3,  and  16  successively: 
the  process  would  be  as  follows  :  — 

s.  d.  £  s.  d. 

10  0  per  day  =  182  10  0  per  annum. 

6  8    =  121  13  4       

1  3    =     22  16  3       


£326     19     7 

In  this  case,  the  parts  are  all  sub-multiples  of  the  prin- 
cipal unit  of  the  multiplicand ;  but  the  process  is  not 
as  concise,  and  is  more  liable  to  error  in  calculation  than 
when  the  parts  are  taken  BO  as  to  be  sub-multiples  of 
each  other. 


CHAP.  III.  COMPOUND  MULTIPLICATION.  26l 

Example.  What  is  the  price  of  165  pieces  of  silk  at 
71.  1 3s.  7%d. :  we  shall  resolve  this  sum  into  the  fol- 
lowing parts,  71;  10s.,  3s.  4<d.,  3d.,  and  ^d.:  the  process 
will  be  as  follows  :  — 


£    s.     d. 
700  

£      s.     d. 
1155     0     0 

0  10     0  

82  10     0 

034  

27   10     0 

0     1      0  ...  . 

003 

213 

0     0     O1  .... 

O     6  10^ 

£1261     8     li 

The  first  product  is  the  price  of  l6'5  yards  at  11.  per 
yard;  the  second  at  10s.  per  yard  is  found  by  divid- 
ing 165  by  2,  10*.  being  half  of  a  pound;  the 
third  is  found  by  dividing  the  second  by  3,  for  3s.  4>d, 
is  the  third  part  of  10*.  In  order  to  facilitate  the 
discovery  of  the  other  partial  products,  we  now  find 
the  product  which  would  correspond  to  1  s.  Od.,  which 
is  done  by  dividing  the  product  corresponding  to 
10*.  by  10:  the  product  corresponding  to  1*.  being 
thus  found,  we  find  the  product  corresponding  to 
3d.  by  dividing  it  by  4? ;  and,  as  the  former  product 
does  not  constitute  a  part  of  the  question,  but  is  merely 
introduced  to  facilitate  the  calculation,  we  strike  it  out. 
The  last  partial  product  corresponding  to  \d.  is  found 
by  dividing  the  preceding  product  by  6,  since  \d.  is  the 
sixth  part  of  3d. 

The  same  question  solved  by  the  decimal  method  ex- 
plained in  (336.),  would  be  as  follows  :  —  71.  13s.  7^d. 
=  7-6811.  Multiplying  this  by  165,— 

7-681 
165 


38405 
46086 
681 


.£1267-365  =  £1267     7 


262  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

It  will  be  observed  that  this  result  is  less  by  §\d.  than 
the  sum  found  by  the  other  method  :  this  has  not  re- 
suited  from  any  error  in  the  work,  but  has  arisen  from 
another  circumstance,  which  admits  of  easy  explanation. 
It  will  be  recollected  that  the  method  of  finding  the 
equivalent  decimal  for  any  sum  of  money,  explained 
in  (336.),  is  accurate  only  as  far  as  the  third  place  of 
decimals,  which  is  all  that  is  necessary  when  it  is  not 
required  to  express  fractions  less  than  a  farthing ;  but 
when  the  sum  is  to  be  multiplied  by  any  high  number, 
as  in  the  present  case,  where  the  multiplier  is  16'5,  the 
figures  which  would  fill  the  places  beyond  the  third 
place  of  decimals  would,  after  the  multiplication,  affect 
the  second  and  third  places,  because  numbers  would  then 
be  carried  to  these  places.  In  the  present  instance,  the 
decimal  places,  instead  of  being  -365,  would  have  been 
•406 :  the  error  is,  however,  trifling  in  amount ;  and, 
therefore,  where  extreme  accuracy  is  not  sought,  the 
rule  may  still  be  applied. 

(350.)  In  ordinary  cases,  it  seldom  happens  that  the 
sums  which  fall  under  the  hands  of  the  computer  in- 
clude parts  of  a  shilling  less  than  6d.y  especially  where 
they  are  prices  of  goods  sold  per  piece,  or  of  pay  per  week 
or  day.  If  the  sum  to  be  computed  have  6d.  in  the 
pence,  then  the  method  just  referred  to  will  still  give 
an  accurate  result,  however  high  the  multiplier  may  be. 
We  shall  apply  both  methods  to  the  following  exam  pie: — 

Example. — A  certain  stuff  costs  31.  1 3s.  6d.  a  piece; 
what  is  the  price  of  376"  pieces  ? 

We  shall  resolve  the  given  sum  into  the  following 
parts  S/..,  12*.,  and  1*.  6d. :  the  calculation  will  be 
as  follows :  — 

£    s.    d.  £      s.    d. 

300  1128     O     0 

0  12     0  225  12     0 

016  28     4     0 


£1381   16     0 


CHAP.  III.  COMPOUND  MULTIPLICATION.  263 

The  first  product  is  found  by  multiplying  376  by  3, 
since  12  shillings  is  the  fifth  part  of  31.;  the  second  pro- 
duct is  found  by  dividing  the  first  by  5;  and  since  Is.  6d. 
is  the  eighth  part  of  12*.,  the  third  product  is  found  by 
dividing  the  second  by  8. 

The  same  question   will  be  solved  by  the  decimal 
method  as  follows :  — 

£    s.     d. 

3     13     6   =  £3-675 

Multiply  this  by  376 :  — 


£1381-800  =  £1381     16     0 


(351.)  FOURTH  CASE.  When  the  multiplier  is  a 
fraction,  we  have  already  explained  that,  to  multi- 
ply any  quantity  by  a  fraction,  it  is  only  necessary 
to  multiply  that  quantity  by  its  numerator,  and  to 
divide  the  product  by  its  denominator.  Thus,  to  mul- 
tiply by  a  fraction  involves,  in  fact,  the  multiplication 
and  division  by  whole  numbers.  When  the  denomi- 
nator of  the  fraction  does  not  exceed  12,  the  process  is 
sufficiently  simple,  and  has  been  applied  in  the  preceding 
examples ;  but  if  the  denominator  be  a  large  number, 
the  case  must  be  reserved  for  investigation  in  the  fol- 
lowing chapter. 

If  the  numerator  of  the  fraction  be  1,  the  multipli, 
cation  is  performed  by  merely  dividing  by  the  deno- 
minator: in  fact,  multiplying  by  such  a  fraction  is 
equivalent  to  division.  Thus,  to  multiply  by  -^i  and 
to  divide  by  10,  are  one  and  the  same  operation.  In 
practice,  it  is  frequently  convenient  when  a  fraction 
occurs  having  a  numerator  greater  than  1,  to  resolve  it 
into  several  fractions,  whose  numerator  shall  be  1,  and 
s  4 


264  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

to  obtain  several  partial  products,  or  rather  quotients, 
by  dividing  the  multiplicand  by  their  denominators 
severally :  thus,  if  the  multiplier  were  -^0-,  we  should 
resolve  it  into  three  fractions,  -£•§,  -fyy  and  -f-^y  or,  what 
is  the  same,  \,  ^,  and  -£.  We  should  in  this  case  ob- 
tain the  three  partial  products  by  dividing  the  multipli- 
cand, first  by  2,  then  by  5,  and  again  by  5.  Again,  if 
the  multiplier  was  -J-J-,  we  should  resolve  it  into  -fy,  -f§9 
and  -f§ ;  or  £,  J,  and  £.  We  should  obtain  the  first 
partial  product,  by  dividing  the  multiplicand  by  2  ;  and 
since  -fa  is  the  half  of  T%-,  we  should  obtain  the  second 
partial  product  by  dividing  the  first  by  2  :  and  since 
/2  is  the  third  of  -fyf  we  should  obtain  the  third  par- 
tial product  by  dividing  the  first  by  3. 

(352.)  FIFTH  CASE.  When  the  multiplier  is  a 
mixed  number. 

In  this  case,  we  must  multiply  first  by  the  integral 
part  of  the  multiplier,  and  then  multiply  by  the  frac- 
tional part,  according  to  the  method  explained  in  the 
preceding  case :  the  two  products  being  added  together, 
the  total  product  will  be  found. 

Example.  Let  it  be  required  to  find  the  total  amount 
of  657.  17*.  1 1^.  per  annum  for  39  g  years  :  the  process 
is  as  follows :  — 

£  s.  d. 
65  17  11 
S9J 

585~ 

x.  d.  195 

10  0 19  10     0 

5  O 9  15     0 

2  0 3  18     0 

O  6 0196 

O  3 099 

O  2 066 

O  f 32  18   1H 

0  | 16     9     5| 

O  J 8     4     8| 

£2627   11    HI 


CHAP.  III.  COMPOUND  MULTIPLICATION.  265 

As  in  the  former  examples,  we  first  multiply  the 
pounds  by  39;  we  then  resolve  17*.  into  10*., 
5s. 3  and  2s.  :  to  multiply  10s.  by  39,  we  consider 
39  as  pounds,,  and  divide  it  by  2,  and  obtain  the 
third  partial  product ;  to  find  the  next,  we  divide  this 
latter  by  2,  since  5  is  the  half  of  10;  and,  to  find  the 
following  one,  we  divide  it  by  5.  Since  2  is  the  fifth  of 
10,  we  resolve  lid.  into  6d.,  3d.,  and  2d. :  we  find  the 
product  corresponding  to  6d.  by  dividing  the  product 
corresponding  to  5  shillings  by  10;  we  find  the  pro- 
duct corresponding  to  3d.  by  dividing  that  correspond- 
ing to  6d.  by  2,  and  we  find  the  product  corresponding 
to  2d.  by  dividing  it  by  3.  We  have  thus  obtained  all  the 
partial  products  necessary  to  compose  the  product  of  the 
multiplicand  by  39  :  but  it  still  remains  to  multiply  the 
multiplicand  by  the  fraction  | :  to  accomplish  this,  we 
have  resolved  the  fraction  into  three  parts,  viz.  |j,  -|,  and  -^ : 
since  -|  is  ^  to  find  the  product  corresponding  to  it,  we 
have  only  to  divide  the  multiplicand  by  2,  by  which  we 
obtain  the  corresponding  partial  product :  to  find  the  pro- 
duct corresponding  to  -f,  which  is  the  half  of  -J,  we 
have  only  to  divide  the  last  product  by  2.  By  again 
dividing  the  product  thus  found  by  2,  we  find  the  pro- 
duct corresponding  to  -l.  The  addition  of  all  these 
sums  gives  the  total  product  sought. 

The  same  question  might  be  solved  by  the  decimal 
method  in  the  following  manner: — Let  us  first  find  the 
value  of  651.  18s.  Od.  per  annum  for  39|-  years,  and 
then  subtract  from  the  result  the  amount  of  a  penny  a 
year  for  39^  years;  that  is,  subtract  the  39|  pence : 
the  process  will  be  as  follows :  — 

£65  18     0  =  £65-9, 

\Ve  shall  first  find  the  product  corresponding  to  39, 
and  shall  afterwards  obtain  the  product  corresponding 


266  A    TREATISE  ON    ARITHMETIC.  BOOK  HI. 

659  8)65-9000 

OQ 

8-2375 

5931  7 

1977 


57-6625 


25701 

39  years  .........  2570-1 

|    —     ,  ........      57-6625 


2627-7625  =  £2627  15     3 
Subtract  39?  ........................  3     S| 

£2627  11   llj 


(353.)  The  artifice  used  here  for  the  simplification 
of  the  process  is  one  of  which  we  may  frequently  avail 
ourselves  with  advantage.  It  consists  in  taking  a  multi- 
plicand somewhat  greater  than  the  one  proposed ;  but 
which,  being  expressed  more  in  whole  numbers,  renders 
the  process  of  multiplication  more  expeditious.  The 
excess  is  afterwards  compensated  for  by  subtraction. 
The  same  artifice  may  be  always  used  in  all  cases  where 
the  multiplicand  is  a  little  less  than  a  round  number. 

Example.  Let  it  be  required  to  find  the  total 
amount  of  9^  19^.  U^-  payable  yearly  for  100  years. 

Had  the  sum  been  101.  the  amount  would  evidently 
be  WOOL  ;  but  the  sum  proposed  is  Id.  less  than  1 01., 
and  therefore  the  sum  sought  will  be  100  pence  less  than 
10001.  We  shall  find  the  sum  sought  by  subtracting 
8s.  4>d.  from  WOOL :  the  result  is  999/.  11*.  8d. 

(354.)  SIXTH  CASE.  When  the  multiplier  is  a  com- 
pound number. 

Example  1.  What  is  the  expense  of  repairing  a 
road,  the  length  of  which  is  69  miles,  6  furlongs,  25 
perches,  at  25/.  19*.  5d.  per  mile  ? 

We  shall  first  ascertain  the  price  for  69  miles,  which 
will  be  done  by  multiplying  25/.  19*.  5d.  by  69,  by  the 
methods  explained  in  the  preceding  cases :  we  must  next 
resolve  the  furlongs  and  perches  into  sub-multiples,  or 
aliquot  parts,  of  a  mile  or  of  each  other :  we  shall  re- 
solve the  furlongs  into  4  furlongs,  which  is  half  a  mile 


CHAP.  III.  COMPOUND    MULTIPLICATION.  267 

and  2  furlongs,  which  is  half  the  latter.  We  shall  re- 
solve  the  perches  into  20  perches,  which  is  half  a  fur. 
long,  or  the  fourth  part  of  2  furlongs,  and  5  perches, 
which  is  the  fourth  part  of  20  :  the  process  will  be  as 
follows :  — 


£25  19     5 
mil.  fur.  per. 
69     6  25 

225 

s.     d. 

150 

10     0 

34  10 

0 

50 

17     5 

0 

20 

6  18 

0 

20 

6  18 

0 

04 

1     3 

0 

0     1 

0     5 

9 

4  furlongs 

-     12  19 

8 

2     — 

-       6     9 

10 

20  perches 

-       1   12 

5J 

5     — 

-       0     8 

If 

.£1813     9     9| 

The  first  8  products  above  are  the  partial  products 
found  by  multiplying  the  multiplicand  by  6*9;  by  their 
addition  we  should  obtain  the  cost  for  69  miles :  the  re- 
maining 4  products  are  the  cost  for  the  furlongs  and 
perches  in  the  multiplier ;  the  first  is  the  cost  for  4 
furlongs,  or  half  a  mile,  and  is  found  by  taking  half  the 
multiplicand ;  the  next  product  is  half  this,  being  the 
cost  of  2  furlongs ;  the  following  one,  the  fourth  of  the 
latter,  being  the  cost  of  half  a  furlong ;  or  20  perches  ; 
and  the  last  product  is  the  fourth  part  of  this  sum,  being 
the  cost  of  5  perches.  These  all  added  together  give 
the  total  cost  sought. 

(355.)  In  the  multiplication  of  simple  numbers,  and 
even  in  complex  numbers,  when  one  of  the  factors  is  a 
simple  number,  the  multiplier  and  multiplicand  may  in- 
terchange places  without  producing  any  change  in  the 
product :  this  has  been  fully  explained  with  respect  to 
simple  ii umbers  in  the  former  book,  and  due  attention 
to  the  examples  given  in  the  present  chapter  will  show 


268  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

its  truth  when  one  of  the  factors  is  a  complex  and 
the  other  a  simple  number.  In  this  last  case  the  effect  of 
the  multiplication  will  he  to  add  together  the  complex 
number,  repeated  as  often  as  there  are  units  in  the  simple 
number,  if  the  latter  be  a  whole  number :  if  it  be  a 
mixed  number,  or  a  fraction,  then  the  operation  will  be 
equivalent  to  taking  as  much  of  it  as  is  proportionate 
to  the  value  of  the  fraction  (241.).  In  every  such  case, 
it  is  evident  that  the  product  will  be  a  complex  number 
of  the  same  kind  as  the  complex  factor  :  thus,  if  the 
complex  factor  express  money,  the  product  will  also  ex- 
press money  ;  if  the  complex  number  express  weight, 
the  product  will  also  express  weight,  and  so  on. 

When  both  factors,  however,  are  complex,  they  lose 
their  quality  of  being  interchangeable,  and  the  multi- 
plier becomes  essentially  distinct  from  the  multiplicand, 
and  must  be  regarded  in  a  totally  different  sense.  In 
the  example  just  given,  the  object  of  the  operation  is, 
first  to  repeat  the  given  sum  of  money  as  often  as  there 
are  miles  in  the  proposed  distance,  then  to  add  the  same 
fractional  parts  of  that  sum  of  money  as  the  furlongs 
and  perches  in  the  given  distance  are  of  1  mile  ;  we 
might  then  have  proceeded  by  first  multiplying,  as 
above,  the  sum  of  money  by  69  :  since  6  furlongs  are  3 
of  a  mile,  we  might  then  have  added  to  the  result  -J  of 
the  proposed  sum ;  and,  again,  since  25  perches  are  | 
of  a  furlong,  and  therefore  654  of  a  mile,  we  might  have 
annexed  65r  of  the  cost  of  1  mile.  The  result  of  the 
calculation  would  have  been  the  same,  but  the  process 
more  complex. 

It  will  then  be  perceived  that  in  this  process  the  prin- 
cipal units  of  the  multiplier  are  used  as  a  whole  num- 
ber would  be  if  the  multiplier  was  simple,  and  the 
units  of  inferior  orders  are  used  merely  as  fractional 
parts  of  the  principal  units.  By  this  means  the  multi- 
plier, though  complex,  is  implicitly  reduced  to  a  simple 
number,  and  the  process  is  the  same  as  if  we  expunged 
all  the  classes  of  units  inferior  to  the  principal  units, 
and  substituted  in  their  places  the  equivalent  frac- 
tions of  the  principal  unit.  In  the  above  number  the 


CHAP.  III.  COMPOUND  MULTIPLICATION;  29 

multiplier  is  69  miles,  6'  furlongs,,  25  perches.  This 
might  be  converted  into  a  simple  number  by  expressing 
it  6'9|  •  •  -iftV  or,  what  is  the  same,  69f  .  .  .  B\. 

Let  us  now  consider  what  the  meaning  of  the  process 
would  be,  if  the  places  of  the  multiplicand  and  multiplier 
were  interchanged.  In  that  case,  the  effect  of  the  operation 
would  be,  in  the  first  place,  to  repeat  the  given  distance 
25  times,  and  we  should  accordingly  find  the  distance 
which  would  cost  25/.  if  the  given  distance  cost  11. 
We  should  next  take  the  same  fraction  of  the  given  dis- 
tance as  19*.  is  of  a  pound,  which  would  be  the  distance, 
the  repair  of  which  would  cost  IQs.,  the  whole  distance 
being  supposed  to  cost  a  pound.  In  the  same  manner, 
we  should  next  take  the  same  fraction  of  the  given  dis- 
tance as  5  pence  is  of  a  pound,  and  we  should,  as  before, 
find  the  distance  which  would  cost  5  pence,  the  whole 
given  distance  being  supposed  to  cost  1  pound.  The 
question,  therefore,  to  which  such  an  operation  would 
give  the  answer  would  be  the  following.  If  the  repair 
of  69  miles,  6  furlongs,  25  perches  cost  I/.,  what  dis- 
tance can  be  repaired  for  25/.  19*.  5d.  The  result  of 
the  operation  will  in  this  case  be  expressed  in  units 
similar  to  those  of  the  multiplicand,  viz.  miles,  fur. 
longs,  &c.  The  actual  process  would  be  as  follows : — 

mil.  fur.  per. 
69     6     25 
£    s.     d. 
23  19     5 


345 

138 

4  furlongs 

12 

4     0 

2      — 

6 

2     0 

20  perches 

1 

4  20 

5      — 

0 

3     5 

10«.  Oct. 

34 

7  12$ 

50 

17 

3  26f 

20 

6 

7  34i 

20 

6 

7  34J 

0     4 

1 

1    12* 

0     1 

0 

2  IS* 

1813     3 


2?0  A    TREATISE    ON    ARITHMETIC.          BOOK  III. 

(356.)  It  is  sometimes  more  convenient  and  expe- 
dient to  convert  the  multiplier  into  a  simple  number. 
Jn  the  example  just  given,  we  might  proceed  by  first 
taking  26/.  as  a  multiplier,  and  then  subtracting  from 
the  result  the  product  which  would  correspond  to  7 
pence,  the  excess  of  the  assumed  multiplier  above  the 
given  one.  Proceeding  by  this  method,  the  process 
would  be  as  follows  :  — 


mil.  fur.  perc.  mil.  fur.  per. 

69       6  25 


69 
26 

6 
0 

25 
0 

4  furlongs 
2        — 
20  perches 
5        — 

Subtract  7d. 

414 
138 
13 
6 
1 
0 

0 
0 
0 
4 
5 
3 

O 
0 
0 
0 
0 
10 

1815 
2 

4 
0 

10 

6d.        1        5  38f 
Id.        0       2   13,| 


1813     3 


We  have  here  found  the  product  corresponding  to  the 
multiplier  26,  by  the  method  explained  in  (346).  To 
facilitate  the  determination  of  the  product  corresponding 
to  7  pence,  we  first  find  the  product  corresponding 
to  ls.}  which  is  done  by  dividing  the  multiplicand  by 
20.  Half  this  product  gives  the  product  correspond- 
ing to  6(?.,  and  the  sixth  part  of  the  latter  is  the 
product  corresponding  to  Id. ;  adding  the  last  two,  we 
obtain  the  product  corresponding  to  Jd.;  this  subtracted 
from  the  product  corresponding  to  26/.,  gives  the  pro- 
duct corresponding  to  25£  19*-  5d.,  which  is  sought. 

(357.)  The  results  of  the  multiplication  of  complex 
numbers  may  be  verified,  as  in  simple  numbers,  by  di- 
viding the  product  by  the  multiplier.  The  quotient 
should  in  that  case  be  the  multiplicand  ;  but  it  is  gene- 
rally more  expeditious  to  verify  the  work  by  working 


CHAP.  III.  COMPOUND  MULTIPLICATION.  271 

it  twice  by  different  methods,  which  may  always  be 
done  by  resolving  one  or  both  factors  into  a  different 
series  of  aliquot  parts. 

We  may  also  verify  by  doubling  one  factor  and  halv- 
ing the  other.  The  product,  after  this  change,  should 
remain  the  same ;  or,  if  one  factor  only  be  doubled,  we 
should  get  a  product  equal  to  double  the  original 
product. 


A    TREATISE    ON    ARITHMETIC. 


CHAP.  IV. 


OF    THE    DIVISION    OF    COMPLEX    NUMBERS. 

IN  the  same  manner  as  in  the  last  chapter,  we  shall 
consider  successively  the  different  cases  of  division, 
taking  the  most  simple  first  in  order. 

(358.)  FIRST  CASE.  When  the  divisor  is  a  whole 
number. 

Example  1.  The  sum  of  S5469/.  19$.  lid.  is  paid 
for  568  pipes  of  wine ;  what  is  the  cost  per  pipe  ? 

This  question  will  be  solved  by  dividing  the  sum  pro- 
posed into  568  equal  parts  :  one  of  these  parts  will  be 
the  price  of  a  single  pipe.  The  dividend,  therefore,  is 
the  sum  proposed,  and  the  divisor  the  number  of  pipes  ; 
the  quotient  will  be  the  price  of  one  pipe.  The  pro- 
cess is  as  follows  :  — 

£         s.     d.      £     s.     d. 
568)25469     19     11     (44     16     9j 
2272 

2749 
2272 

477  pounds 
Multiply  20 

9540  shillings 
Add         -  19 


568)9559(16 
568 

3879^ 
3408 

471  shillings 
Multiply        12 

5652  pence 


CHAP.  IV.  COMPOUND    DIVISION.  273 

Brought  forward  5652 

Add         -          11 


568)5663(9 
5112 

551  pence 
Multiply         4 

568)2204" farthings  (3 
1704 

500 


We  first  divide  the  pounds  by  568,  in  the  same 
manner  as  in  simple  numbers,  and  find  the  quotient  44, 
with  a  remainder  447.  This,  being  less  than  the  divi- 
sor, does  not  admit  of  a  further  division.  In  order  to 
continue  the  process,  it  is  converted  into  shillings,  the 
equivalent  number  of  which  is  9540.  The  number  of 
shillings  in  the  dividend  being  added  to  this,  the  sum 
which  remains  to  be  divided  by  the  divisor  is  9559  shil- 
lings. This  number  of  shillings  being  now  taken  as 
dividend,  we  obtain  the  quotient  16,  with  a  remainder 
471  shillings.  This  latter  number,  being  less  than  the 
divisor,  is  converted  into  pence,  the  equivalent  number 
of  which  is  5652.  The  pence  of  the  dividend  being 
added  to  this,  we  obtain  5663  pence,  which  is  the  sum 
which  remains  to  be  divided  by  the  divisor.  This 
being  taken  as  dividend,  we  obtain  the  quotient  9  pence, 
and  the  remainder  551  pence.  The  last  is  reduced  to 
farthings  by  multiplying  by  4,  by  which  we  obtain 
2204  farthings.  This,  divided  by  the  divisor,  gives 
the  quotient  3,  with  a  remainder  500.  The  whole 
quotient,  therefore,  is  44/.  l6s.  9|^-^  with  a  remainder 
500,  which  divided  by  the  divisor  would  give  the  frac- 
tion of  a  farthing  |^~|.  This  fraction,  being  very  little 
less  than  1,  is  nearly  equal  to  1  farthing.  Thus  the 
quotient  44/.  l6s.  Wd.  is  in  excess  of  the  true  quotient 
by  a  very  minute  fraction  of  a  farthing. 


274*  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

If  it  be  required  to  express  the  quotient  with  the 
exact  fraction  of  a  penny,  we  might  stop  the  process  of 
division  after  the  quotient  9  had  been  obtained  in  the 
pence.  The  remainder  obtained  at  that  point  was  551 
pence;  this  divided  by  568  gives  the  fraction  of  a 
penny  -|-|-|-  >  anc^  ^e  exact  quotient  is,  therefore,  44/. 

16*.  9m*- 

(359.)  The  method  adopted  in. this  example  is,  first 
to  divide  the  principal  units  of  the  dividend  by  the 
divisor,  then  to  reduce  the  remainder  to  units  of  the 
order  next  inferior,  and  add  to  it  the  units  of  the  same 
order  in  the  dividend.  Taking  the  number  thus  ob- 
tained as  dividend,  the  quotient  will  be  the  units  of  the 
next  inferior  order  in  the  quotient  sought;  and  the  re- 
mainder is  reduced,  as  before,  to  the  units  of  the  next 
order.  The  units  of  that  order  in  the  dividend  are 
added  to  it,  and  the  sum  taken  is  the  next  partial  divi- 
dend. The  same  process  would  be  continued  until 
every  class  of  units  of  which  the  complex  number  is 
susceptible  has  been  successively  obtained  by  reduction. 
If  there  still  is  found  a  remainder,  then,  by  taking  that 
remainder  as  numerator,  and  the  divisor  as  denomin- 
ator, we  obtain  the  fraction  of  the  last  unit,  which  is 
necessary  to  complete  the  quotient.  These  observations 
will  be  further  illustrated  in  the  following  example  :  — 

Example.  The  sum  of  ?65/.  is  paid  for  47  tons, 
12  cwt.  3  qrs.  57  Ibs.  12  oz.  of  a  certain  commodity  ; 
what  is  the  quantity  which  could  be  obtained  for  I/.? 

To  solve  this  question,  it  is  necessary  to  divide  the 
proposed  weight  into  as  many  equal  parts  as  there  are 
pounds  in  the  given  sum  of  money,  and  one  of  these 
parts  will  be  the  quantity  which  may  be  obtained  for  11. 
We  must,  therefore,  take  the  proposed  weight  as  divi- 
dend, and  765  as  divisor,  and  the  quotient  will  be  the 
weight  sought. 


CHAP.  IV.  COMPOUND    DIVISION.  2?5 

tons.  cwt.  qrs.  Ibs.  oz.     cwt  Ibs.  oz. 
765)47     12     3     57     12     (1      27     9^ 
20 

940 
12 


765 

~187  cwt. 
4 

748 
3 


751 
28 

6008 
1502 

21028  Ibs. 
57 


5785 
5355 


6880  oz. 
12 

6892 
6885 


The  number  of  tons,  the  principal  units  of  the  divi- 
dend being  less  than  the  divisor,"1  does  not  admit  of 
division ;  we  therefore  convert  it  to  hundred. weights, 
by  multiplying  by  20,  and  add  to  the  product  the  hun- 
dred-weights of  the  dividend.  Taking  the  number  of 
T  2 


276*  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

hundred-  weights  thus  obtained  as  the  first  dividend,  we 
obtain  the  quotient  1  and  the  remainder  187.  This 
remainder  is  reduced  to  quarters,  and  the  quarters  of 
the  dividend  added,  by  which  we  obtain  751  quarters  ; 
but,  this  being  less  than  the  divisor,  we  convert  it  into 
pounds,  and  add  the  pounds  of  the  dividend.  We 
thus  obtain  21085  pounds,  which,  being  taken  as  a  di- 
vidend, we  obtain  27  for  the  pounds  of  the  quotient, 
with  a  remainder  430  pounds.  This  remainder  is  con»- 
verted  into  ounces,  and  the  ounces  of  the  dividend 
added,  which  gives  68Q2  ounces  for  the  next  partial 
dividend.  We  then  obtain  9  for  the  ounces  of  the 
quotient,  with  a  remainder  7-  This  remainder  being 
taken  as  numerator,  and  the  divisor  as  denominator,  we 
get  the  fraction  of  an  ounce,  -r^,  which  completes  the 
division. 

(360.)  SECOND  CASE.  When  the  divisor  is  a  frac- 
tion. (This  case  embraces  that  in  which  the  divisor  is 
a  mixed  number,  for  such  a  number  may  always  be  re- 
duced to  an  equivalent  fraction.) 

It  has  been  explained  that,  in  order  to  divide  any 
quantity  by  a  fraction,  it  is  necessary  to  divide  it  by 
the  numerator,  and  multiply  the  quotient  by  the  deno- 
minator of  the  fraction.  Thus,  the  present  case  may 
always  be  performed  by  combining  the  method  explained 
in  the  first  case  with  that  explained  in  the  first  case  of 
the  preceding  chapter. 

Example.  If  25-f-^-  pieces  of  a  certain  cloth  cost 
417/.  13*.  7d.y  what  will  be  the  price  of  one  piece? 
In  this  case  it  is  necessary  to  find  such  a  sum  as,  if  re- 
peated 25  times,  and  to  the  result  the  fractional  part 
expressed  by  -f-^  of  the  same  sum  added,  the  total  would 
be  41  11.  13*.  Id.  This  will  be  evidently  obtained  by 
dividing  the  sum  of  money  by  25|y. 


We  must,  therefore,  multiply  41  11  13*.  Id.  by  31,  and 
divide  the  quotient  by  79& 


CHAP.  IV.  COMPOUND    DIVISION.  277 

£      s.     d. 
417     18     7 
31 

"417 

5.  d.  1251 
10  0  -  15  10  0 
20-  3  20 
1  0  -  1  11  0 
0  6  -  0  15  6 
01-027 

-  -  £  s.  d. 
796  )  12948   1  1  (  16  5  3g| 
796 
4988 
4776 

212 
20 

4240 
_  1 

4241 
3980 

261 
12 

3132 

1 

3133 
2388 

745 

(361.)  THIRD  CASE.  When  the  divisor  is  a  complex 
number  of  the  same  kind  as  the  dividend. 

Example.  To  cut  a  certain  canal  costs  at  the  rate  of 
47/.  19*.  5d.  per  perch  ;  how  many  perches  can  be  cut 


It  is  evident  that  the  number  of  perches  and  parts  of 
a  perch  sought  is  the  same  as  the  number  of  times  and 
parts  of  a  time  that  4?7.  1  9s.  5d.  are  contained  in 
2728/.  17s.  Wd.  :  we  must,  therefore,  divide  the  latter 
nnmber  by  the  former,  and  the  quotient  will  express 
the  number  of  perches  and  parts  of  a  perch  sought.  To 
effect  this  division,  it  is  only  necessary  to  reduce  the 
T  3 


278  A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

divisor  and  dividend  to  units  of  the  same  kind,  and 
then  divide  the  one  by  the  other  as  simple  numbers : 
thus,  if  both  sums  of  money  be  converted  into  pence,  it 
is  only  necessary  to  find  how  often  the  lesser  number  of 
pence  is  contained  in  the  greater,  which  may  be  done 
by  the  rule  for  the  division  of  whole  numbers.  The 
process  is  as  follows :  — 

£     *.     d.  £       s.    d. 

47     19     5  2728    17    10 

20  20 

959  54577 

I?  per.  yds.  ft.    inch. 

U511  11513)654934(56     4     2 

57565 

79284 
69078 


10206 

_£? 
51030 
5103 

56133 
46052 


10081 
3 

30243 
23026 

7217 
12 

86604 

80591 

6013 

We  find  the  quotient  56,  which  is  the  number  of 
perches,  with  the  remainder  10206:  this  remainder 
divided  by  the  divisor  would  be  the  fraction  of  a  perch 
necessary  to  complete  the  quotient ;  but,  instead  of  ex- 
pressing the  fraction  of  a  perch,  we  may  express  the 
remainder  of  the  quotient  in  yards,  feet,  inches,  and 
fractions  of  an  inch.  Since  there  are  5J  yards  in  a 


CHAP.  IV.  COMPOUND    DIVISION.  279 

perch,  we  multiply  the  remainder  by  5j,  and  then 
divide  the  product  by  the  divisor ;  we  obtain  the  quo- 
tient, four  yards,  with  the  remainder  10081.  We  then 
multiply  this  by  3,  and  divide  the  product  by  the  divi- 
sor, by  which  we  obtain  the  quotient  2  for  the  number 
of  feet.  Multiplying  the  remainder  by  12,  and  dividing 
by  the  quotient,  we  obtain  the  number  of  inches,  which 
is  7  •'  the  final  remainder,  divided  by  the  divisor,  gives 
the  fraction  of  an  inch  necessary  to  complete  the 
quotient. 

We  have  here  reduced  both  the  divisor  and  dividend 
to  the  units  of  the  lowest  class  which  they  contain. 
This  is  not  necessary :  we  might  express  them  in  units  of 
any  class,  provided  that  they  are  both  expressed  in 
units  of  the  same  class.  Thus,  we  might  reduce  them 
both  to  pounds  and  decimals  of  a  pound,  and  then 
divide  according  to  the  rules  for  the  division  of  deci- 
mals by  the  method  of  reduction  explained  in  (335.). 

£      s.     d.  £         s.      d. 

47     19     5  =  £  47-970         2728     17     10  »  £  2728-891. 

The  object,  then,  is,  to  divide  the  latter  decimal  by  the 
former ;  but,  as  they  both  contain  the  same  number  of 
decimal  places,  the  decimal  point  may  be  omitted,  and 
the  numbers  may  be  treated  as  whole  numbers  (291»). 
The  process  of  division  would  then  be  as  follows :  — 

47970)  2728891  (  56-887 
239850 


357400 
835790 
21610 


280  A    TREATISE    ON    ARITHMETIC.          BOOK  III. 

We  have  here  continued  the  division  after  the  re- 
mainder has  been  found  as  far  as  the  third  place  of 
decimals.  Since  a  unit  in  each  place  expresses  the 
thousandth  of  a  perch,  which  is  less  than  the  fifth  of  an 
inch,  it  is  unnecessary  to  continue  the  division  farther, 
since  the  purposes  of  such  computation  do  not  require 
smaller  fractions  of  an  inch.  The  integral  places 
here  express  perches,  and  the  decimal  places  fractions 
of  a  perch.  If  it  be  required  to  convert  these  fractions 
of  a  perch  into  yards,  feet,  &c.,  it  may  be  done  as 
follows:  —  Since  there  are  5£  yards  in  a  perch,  multiply 
•887  by  5*5,  and  we  shall  find  the  yards  and  fractions 
of  a  yard :  — 

•885 
5-5 

4425 
4425 


4-8675 

The  integer  4  here  expresses  the  yards,  and  the 
decimal  places  fractions  of  a  yard.  Since  it  is  not 
required,  however,  to  attain  accuracy  beyond  the  hun- 
dredth part  of  a  yard,  we  shall  neglect  the  decimal 
places  after  the  second,  and  shall  reduce  the  decimals  of 
a  yard  to  feet  by  multiplying  by  3  :  we  have  then 

•86  x  3  =  2-58. 

The  integer  2  here  expresses  the  feet,  and  the  decimal 
•58  the  fraction  of  a  foot:  this  may  be  reduced  to 
inches  by  multiplying  it  by  12 ;  we  have,  then, 

•58  +12  =6-96. 

The  total  quotient  is  then  56  perches,  4  yards,  2  feet, 
6'96  inches.  The  number  of  inches  is  not  equivalent 
to  the  number  found  by  the  preceding  method,  because 
of  the  omission  of  the  decimal  places  beyond  the  third, 
in  the  fractions  of  a  pound ;  but  in  practice  this  small 
quantity  is  altogether  unimportant. 


CH4P.  IV.  COMPOUND    DIVISION.  281 

Example.  We  pay  I/,  for  30  yards,  4  feet,  7  inches 
of  a  certain  stuff;  what  sum  must  be  paid  for  658  yards, 
5  feet,  1 1  inches,  8  lines  ? 

It  is  evident  that,  if  we  could  find  how  often  30 
yards,  4  feet,  7  inches  are  contained  in  658  yards, 
5  feet,  11  inches,  8  lines,  we  should  then  know  the 
number  of  pounds  and  fractions  of  a  pound  necessary 
to  be  paid.  We  have  then  to  divide  the  latter  length 
by  the  former,  and  this  may  be  done  by  reducing  both 
to  units  of  the  same  class,  and  dividing  them  as  whole 
numbers.  In  this  case,  we  shall  reduce  both  lengths  to 
lines :  — 

yds.  ft.  ins.  lines,     yds.  ft.  inches. 

658    5    11    8  30    4    7  13620)  2851 16  (20'933=20Z.  18*.  8rf. 

'  3  __3  27240 

1979  feet       ~94  feet  127160 

12         ~  12  122580 


23759  inches     1135  inches          45800 
12  12  40860 

285116  lines     13620  lines  49400 
40860 

8540 


Having  reduced  the  two  lengths  to  lines,  we  divide 
the  greater  by  the  less,  and  continue  the  division  after 
the  remainder  has  been  found,  by  annexing  ciphers 
until  we  obtain  three  decimal  places :  the  quotient  found 
is  20'933,  which,  by  the  method  explained  in  (337-)* 
we  find  to  be  equivalent  to  20/.  1 8*.  Sd. 

(362.)  FOURTH  CASE.  When  the  divisor  and  di- 
vidend are  complex  numbers  of  different  kinds. 

Example.  258  Ibs.  15oz.  10  drams,  of  a  certain 
alloy  are  bought  for  3259/.  17*.  I0d.;  what  is  the 
price  of  1  pound  ?  To  solve  this  question,  it  is  neces- 
sary to  divide  the  given  sum  of  money  into  the  same 
number  of  equal  parts  and  fractions  of  a  part  as  there 
are  pounds  and  fractions  of  a  pound  in  the  proposed 
weight :  we  must,  therefore,  first  reduce  the  proposed 
weight  to  a  fraction  of  a  pound,  and  then  divide  the 
given  sum  by  that  fraction. 


282  'A    TREATISE    ON    ARITHMETIC.  BOOK  III. 

To  reduce  a  complex  number  to  an  equivalent  frac- 
tion of  its  principal  unit,  it  is  only  necessary  first  to 
reduce  it  to  an  equivalent  number  of  units  of  the  lowest 
denomination  which  it  contains,  and  then  to  divide  it 
by  the  number  of  those  units  contained  in  the  principal 
unit.  In  the  present  case,  we  shall  first  reduce  the  pro- 
posed weight  to  drams,  and  then  divide  the  number  of 
drams  so  found  by  the  number  of  drams  in  a  pound : 
the  quotient  will  express,  in  the  form  of  a  mixed  num- 
ber, the  number  by  which  it  is  necessary  to  divide  the 
given  sum  of  money,  and  the  division  may  be  performed 
by  the  methods  explained  in  the  second  case  of  this 
chapter  :  the  process  will  be  as  follows :  — 

Ibs.    oz.  drams. 
258     15     10 
16 


1548 
258 
15 

4143 
16 

24858 
4143 
10 

66298 

Since  there  are  16  ounces  in  a  pound,  and  16  drams 
in  an  ounce,  we  shall  find  the  number  of  drams  in  a 
pound  by  multiplying  16  by  16;  the  product  is  256. 
To  reduce  the  number  of  drams  in  the  proposed  weight 
to  pounds  and  fractions  of  a  pound,  we  have  then  only 
to  divide  it  by  256:  hence  we  find  258  Ibs.  15  oz. 
10  Ibs.  =  66?9S  Ibs.  This  fraction  may  be  reduced 

256 

to  lower  terms,  by  dividing  both  numerator  and  deno- 
minator by  2,  after  which  it  becomes  3  3  *  4  9  :  we  must 
then  divide  the  proposed  sum  of  money  by  this  fraction, 
which  is  done  by  multiplying  it  by  128,  and  dividing 
the  product  by  33149. 


CHAP.  IV.  COMPOUND    DIVISION.  283 

£        S-     d. 
3259  17  1O 
128 

26072 
6518 

5.    d.  3259 

10    0     -     -  64 

5     0     -     -  32 

2     0     -     -  12  16 

6     -    -  34 

3     -     -  1    12 

1     -     -   10     8 

417266     2     8 


£         S.    d. 

66298)417266     28(65  10|||§| 
397788 

19478 
20 


389562 
331490 

58072 
12 

696872 
66298 
33892 

The  fraction  being  less  than  a  farthing,  may  be 
neglected,  and  the  answer  to  the  question  is  6l.  5s.  Wd. 

(363.)  When  the  several  examples  which  we  have 
given  in  this  chapter  are  considered,  it  will  be  perceived 
that  in  some  cases  the  species  of  complex  number,  which 
the  quotient  must  be,  will  be  apparent*  from  the  very 
nature  of  the  division ;  but  in  other  cases  it  can  only  be 
known  from  the  conditions  of  the  question  proposed. 

If  the  divisor  be  a  simple  or  abstract  number,  then 
the  quotient  must  be  a  complex  number  similar  to  the 
dividend ;  for  the  effect  of  the  operation,  in  that  case, 
would  be,  to  find  such  a  number  as,  when  repeated  as 
often  as  there  are  units  and  parts  of  a  unit  in  the  divisor, 
would  make  up  the  dividend.  Since,  then,  the  dividend 


284  A    TREATISE    ON   ARITHMETIC.  BOOK  III. 

would  be  made  up  by  the  repetition  of  the  quotient,  it 
is  evident  that  the  quotient  must  be  a  complex  number 
of  the  same  kind  as  the  dividend. 

If  the  divisor  be  a  complex  number  of  the  same  kind 
as  the  dividend,  then  the  effect  of  the  division  is  to 
ascertain  how  often  the  divisor  should  be  repeated  in 
order  to  make  up  the  dividend.  So  far  as  the  mere 
operation  of  division  is  concerned,  the  quotient  would, 
in  this  case,  be  merely  an  abstract  or  simple  number, 
and  the  nature  of  its  unit  can,  therefore,  be  only  dis- 
covered by  the  peculiar  nature  of  the  question  from 
which  it  arises.  In  the  first  example  given  in  the  third 
case,  we  divided  2728J.  17*.  10d.  by  47/.  19*.  5d. : 
we  should  obtain  by  such  division  the  quotient  56^-^f  £-|; 
but  it  would,  so  far  as  the  mere  process  of  division  is 
concerned,  be  impossible  to  say  what  is  the  nature  of 
the  units  of  this  quotient :  in  fact,  the  quotient  would 
only  express  the  number  of  times  that  the  divisor  must 
be  repeated  to  make  up  the  dividend.  But  on  ex- 
amining the  question  which  gave  rise  to  this  division, 
we  find  that  the  divisor  is  the  price  of  cutting  one  perch 
of  a  canal,  and  that  the  number  sought  is  the  price  of 
cutting  as  many  perches  of  the  same  canal  as  there  are 
units  in  the  number  which  expresses  how  often  the 
divisor  is  contained  in  the  dividend:  that  number  is 
the  quotient,  and,  therefore,  the  principal  units  of  the 
quotient  express  perches.  Had  the  question  from 
which  this  division  arose  stated  that  the  divisor  was 
the  price  of  one  hundred-weight  of  a  certain  commodity, 
then  the  principal  units  of  the  quotient  would  be  hun- 
dred-weights ;  but  in  other  respects  the  question  would 
remain  unaltered.  It  is  the  principal  units  only,  how- 
ever, which  are  independent  of  the  nature  of  the  ques- 
tion; for  when  we  convert  the  fractional  part  of  the 
principal  unit  into  inferior  units,  the  method  of  pro- 
ceeding will  entirely  depend  upon  the  nature  of  the 
principal  units. 

If  the  divisor  and  dividend  be  complex  numbers  of 
different  kinds,  then  the  object  of  the  division  is  to 


CHAP.  IV.  COMPOUND    DIVISION.  285 

find  such  a  number  as  shall  be  contained  in  the  divi- 
dend, the  same  number  of  times  and  parts  of  a  time 
that  the  principal  unit  of  the  divisor  is  contained  in  it : 
hence  it  follows  that  in  this  case  the'quotient  must  be 
a  complex  number  of  the  same  kind  as  the  dividend. 
We  may  therefore  infer,  generally,  that  the  dividend  and 
quotient  are  always  complex  numbers  of  the  same  kind, 
except  in  the  case  in  which  the  divisor  and  dividend 
are  complex  numbers  of  the  same  kind. 


286 
BOOK  IV. 

PROPORTION^    AND   ITS   PRACTICAL    APPLICATIONS. 

CHAPTER  I. 

PROPORTION. 

(364.)  THERE  is  no  mathematical  term  in  more  fami- 
liar use,  and  about  the  meaning  of  which  less  douht 
exists  in  its  common  acceptation,  than  the  word  PRO- 
PORTION, and  yet  there  is  no  term  in  its  scientific  use 
respecting  the  exact  definition  of  which  more  numer- 
ous and  perplexing  disputes  have  been  maintained. 
These  disputes,  however,  rest  more  upon  the  geometri- 
cal than  the  arithmetical  use  of  the  term.  In  its  latter 
application,  we  shall  not  encounter  much  difficulty  in 
fixing  its  precise  meaning. 

When  we  say  that  all  the  parts  of  a  building,  or  of 
a  piece  of  furniture,  or  any  other  structure,  are  in  pro- 
portion, it  is  evidently  meant  that  when  its  different 
parts,  or  its  measurements  in  different  ways,  are  com- 
pared together,  no  one  will  be  found  too  great  or  too 
small  for  another.  We  say  the  human  figure  in  a 
dwarf  is  out  of  proportion,  because  his  breadth  or  thick- 
ness is  too  great  for  his  height ;  or,  what  amounts  to  the 
same,  his  height  is  too  small  for  his  breadth  or  thick., 
ness.  All  this  implies  that  there  exists  a  certain  known 
relation,  with  respect  to  height,  breadth,  and  thickness 
in  the  human  figure,  of  such  a  nature  that,  when  any 
one  of  these  dimensions  is  increased,  the  other  is  simi- 
larly increased. 

The  common  acceptation  of  the  word  SCALE  will 
illustrate  the  notion  of  Proportion.  A  map  exhibits  a 
copy  of  the  outline  of  a  country,  but  on  a  reduced 


CHAP.  I.  PROPORTION.  28? 

scale.  What  is  the  meaning  of  this  ?  If  every  mile 
in  the  length  of  the  country  correspond  to  a  tenth  of 
an  inch  in  the  length  of  the  picture  of  that  country  ex- 
hibited on  the  map,  it  is  clearly  intended  that  every 
mile  in  the  breadth  shall  be  also  expressed  by  the  tenth 
of  an  inch  on  the  map ;  or,  in  other  words,  if  in  the 
length  of  the  country  on  the  map  there  be  as  many 
tenths  of  an  inch  as  there  are  miles  in  the  actual  length 
of  the  country,  then  there  must  be  also  in  the  breadth 
of  the  country  on  the  map  as  many  tenths  of  an  inch 
as  there  are  miles  in  its  actual  breadth ;  and,  in  gene- 
ral, in  whatever  direction  the  country  be  measured,  there 
must  be  as  many  tenths  of  an  inch  measured  in  the 
same  direction,  on  the  map,  as  there  are  miles  in  the 
country  itself.  In  such  a  case,  the  map  is  said  to  ex- 
hibit a  representation  of  the  country,  but  on  "  a  re- 
duced scale  : "  every  part  of  the  map  is  said  to  be  "  in 
the  same  proportion  "  as  the  corresponding  parts  of  the 
country. 

If  two  similar  buildings  be  erected,  having  their 
parts  in  the  same  proportion,  but  one  having  double 
the  height  of  the  other,  then  it  must  follow  that  the 
breadth  and  depth  of  the  one  will  also  be  double  the 
breadth  and  depth  of  the  other ;  that  every  room  in  the 
one  shall  have  double  the  height,  double  the  breadth, 
and  double  the  depth  of  the  corresponding  room  in  the 
other,  and  so  on. 

(365.)  From  these  considerations  it  appears  that 
objects  are  said  to  be  in  the  same  proportion  when  the 
dimensions  of  the  smaller  are  all  the  same  fractions  of 
the  dimensions  of  the  greater.  Let  us  apply  this  to 
abstract  numbers. 

When  two  numbers,  such  as  7  and  8,  are  proposed, 
there  is  a  certain  relation  between  them,  which  is  called 
their  RATIO  :  this  relation  is  expressed  by  the  fractional 
part  of  the  second  number,  considered  as  unity,  which 
is  equal  to  the  first.  In  the  present  case,  if  we  regard 
8  as  the  unit,  then  7  eighths  of  it  will  be  equal  to  the 
first  number,  and  accordingly  the  RATIO  of  7  to  8  is 


288  A    TREATISE    ON    ARITHMETIC.  BOOK  IV- 

expressed  by  the  fraction  -|.  In  the  same  manner  the 
ratio  of  8  to  7  would  be  expressed  by  the  fraction  -|. 

(366.)  The  word  "  ratio/'  therefore,  always  implies 
the  comparison  of  two  numbers  in  a  certain  determi- 
nate order  :  when  the  order  is  reversed  the  ratio  is  said 
to  be  inverted,  and  is  sometimes  called  "  the  inverse 
ratio. ''  Thus  the  ratio  of  8  to  7  is  the  "  inverse 
ratio"  of  7  to  8. 

(367.)  Since  the  ratio  of  two  numbers  is  always  esti- 
mated by  the  fraction  formed  by  taking  the  first  of  the 
two  numbers  as  the  numerator,  and  the  second  as  the 
denominator,  it  follows  that  two  ratios  are  equal  when 
the  two  fractions  thus  formed  are  equal.  Thus,  the 
ratio  of  7  to  8,  and  that  of  14  to  16,  are  equal,  because 
the  fractions  -J-  and  -^  are  equal. 

(368.)  When  two  ratios  are  equal,  the  two  pairs  of 
numbers  are  said  to  be  in  proportion  :  proportion  is 
then  the  equality  of  ratios.  Thus  we  say  the  four 
numbers  7,  8,  14,  and  16  are  in  proportion,  and  this 
fact  is  usually  expressed  by  saying  that  "  7  is  to  8  as 
14  is  to  16:"  the  arithmetical  notation  by  which  this 
is  expressed  is  as  follows,  7  '•  8  :  :  14  :  16. 

The  sign  :  between  two  numbers,  therefore,  expresses 
the  ratio ;  and  the  sign  : :  expresses  the  equality  of  ratios. 

The  ordinary  sign  for  equality  is  sometimes  used  for 
ratios  in  the  same  manner  as  for  numbers :  thus  the 
above  proportion  would  be  expressed, 

7  :  8  =  14:  16; 

and  this  method  is  perhaps  to  be  preferred. 

(369.)  A  ratio  is  also  frequently  expressed  by  the 
notation  for  fractions,  since  the  fraction  formed  by  its 
terms  is  the  number  by  which  the  ratio  is  estimated : 
thus  7  :  8  would  be  expressed  -J,  and  the  above  pro- 
portion would  then  be  expressed  thus,  -J  =  -i-|-.  It  ap- 
pears, therefore,  that  "  ratio"  is  only  another  name  for 
a  fraction,  and  "  proportion"  expresses  the  equality  of 
two  fractions. 

Since  the  value  of  a  fraction  is  not  changed  by  muL. 


CHAP.  I.  PROPORTION.  289 

tiplying  or  dividing  both  its  terms  by  the  same  number, 
(210.)  (211.).,  we   may  also  multiply  or   divide  both 
terms  of  a  ratio  by  the  same  number  without  changing 
the  ratio  :  thus,  the  following  ratios  are  equal :  — 
5:    6 

10  :  12 

15:  18 

20  :  24 

25  :  30 

&c.  &c. 

(370.)  Tb*»  first  term  of  a  ratio  is  called  its  ante- 
cedent, and  the  second  its  consequent.  The  antecedent, 
therefore,  corresponds  to  the  numerator  of  the  fraction, 
and  the  consequent  to  its  denominator. 

(371.)  Since  a  fraction  is  increased  in  proportion  as 
its  numerator  is  increased,  or  its  denominator  diminished 
(210.),  a  ratio  is  increased  according  as  its  antecedent 
is  increased,  or  its  consequent  diminished. 

Since  two  equal  fractions  are  equally  increased  by 
multiplying  their  numerators  by  the  same  numbers, 
or  by  dividing  their  denominators  by  the  same  number 
(214.)  (215.),  two  equal  ratios  are  equally  increased 
by  multiplying  their  antecedents,  or  dividing  their  con- 
sequents by  the  same  number. 

(372.)  Hence  two  equal  ratios  will  continue  to  be 
equal  when  their  antecedents  are  multiplied  by  the  same 
number,  or  their  consequents  divided  by  the  same 
number,  or  if  the  antecedent  of  one  be  multiplied  by  a 
certain  number,  and  the  consequent  of  the  other  divided 
by  the  same  number.  For  like  reasons  we  may  infer 
that  two  ratios  will  continue  equal  when  the  antecedents 
are  divided  by  the  same  number,  or  the  consequents 
multiplied  by  the  same  number,  or  the  antecedent  of 
one  divided  by  a  certain  number,  and  the  consequent  of 
the  other  multiplied  by  the  same  number. 

(373.)  A  proportion  which  expresses  the  equality  of 
two  ratios  will  therefore  not  be  disturbed,  but  will  con. 
tinue  to  be  a  true  proportion,  though  it  may  have  been 
submitted  to  any  of  the  above  changes  :  the  following 
examples  will  illustrate  this:  — 


20 

20 
20 
40 
10 


20 


40 


290  A    TREATISE    ON    ARITHMETIC.  BOOJ 

8       10=16 

Multiply  antecedents  by  2  -  -  16  10  =  32 

Divide  antecedents  by  2  -410=8 

Multiply  consequents  by  2  8  20  =  16 

Divide  consequents  by  2  -85=16 

Multiply  one  antecedent  by  2,  and  divide 

the  other  consequent  by  2  -  -  8  5  =  32 
Divide  one  antecedent  by  2,  and  multiply 

the  other  consequent  by  2  -             -       410=16 

(374.)  It  is  frequently  necessary  to  determine  whe- 
ther four  proposed  numbers  are  in  proportion  or  not;  or, 
in  other  words,  whether  the  ratio  of  the  first  to  the 
second  be  equal  to  the  ratio  of  the  third  to  the  fourth. 
To  determine  this,  it  is  only  necessary  to  enquire  whe- 
ther the  fractions  which  are  equivalent  to  these  ratios  be 
equal.  This  will  be  known  by  reducing  the  two  fractions 
to  the  same  denominator,  when,  if  they  be  equal,  their 
numerators  will  be  equal,  but  otherwise  not.  Let  the 
four  numbers  proposed  be  23,  27.,  54,  and  65.  If  these 
numbers  be  in  proportion,  then  the  fractions  |-^,  |^, 
must  be  equal.  These  fractions  will  be  reduced  to  the 
same  denominator  by  multiplying  both  terms  of  the  first 
by  the  denominator  of  the  second,  and  both  terms  of 
the  second  by  the  denominator  of  the  first  (232.)  :  the 
fractions  will  then  become  - — 

23x65  and  54x27 
27x65          27x65. 

Now,  since  these  have  the  same  denominator,  if  they 
be  equal,  their  numerators  must  be  equal,  and  we  shall 
ascertain  that  by  actually  multiplying  the  numbers 
which  are  here  only  connected  by  the  sign  of  multipli- 
cation. But  before  we  proceed  to  this,  we  shall  observe 
that  the  numerators  of  these  fractions  are  the  products 
of  the  first  and  fourth,  and  of  the  second  and  third 
terms,  in  the  proposed  series  of  four  numbers,  and  hence 
we  may  derive  the  following  important 

RULE. 
(375.)  Four  numbers  will  be  in  proportion  if  the 


CHAP.  I.  PROPORTION.  291 

product  of  the  first  and  fourth  be  equal  to  the  product  of 
the  second  and  third  ;  but,  if  these  products  be  unequal, 
the  numbers  will  not  be  in  proportion. 

(In  the  series  of  four  numbers,  the  first  and  fourth 
are  called  extremes,  and  the  second  and  third  means.) 

Performing  the  actual  multiplications  above  indi- 
cated, we  find  that  the  product  of  the  extremes  is  1495, 
and  the  product  of  the  means  is  1458:  the  four  numbers 
proposed  are  therefore  not  in  proportion,  the  first  ratio 
being  greater  than  the  second.  Let  it  be  required  to 
determine  whether  the  series  23,  27?  184,  2 16  are  in 
proportion:  multiplying  the  extremes  and  means  we 
find 

23  x  216  =  4968  27  x  184  =  4968. 

The  products  of  the  extremes  and  means  are  therefore 
equal,  and  the  four  numbers  are  in  proportion,  so  that 

23  :  27  =  184  :  216. 

(376.)  Since  the  product  of  the  means  in  a  propor- 
tion is  equal  to  the  product  of  the  extremes,  it  follows 
that  the  product  of  the  means,  divided  by  one  extreme, 
will  give  a  quotient,  which  is  the  other  extreme.  Thus 
the  fourth  term  of  a  proportion  may  always  be  found 
by  dividing  the  product  of  the  second  and  third  by  the 
first.  For  the  same  reason,  if  the  product  of  the  second 
and  third  be  divided  by  the  fourth,  the  quotient  must 
be  the  first. 

Tn  the  proportion  23  :  27  =  184  :  21 6,  the  product 
of  the  second  and  third  is  4968  ;  this,  divided  by  23, 
will  give  the  quotient  21 6,  which  is  the  fourth;  also, 
if  divided  by  21 6,  it  will  give  the  quotient  23,  which  is 
the  first. 

(377.)  For  similar  reasons,  if  the  product  of  the  first 
and  fourth  terms  of  a  proportion  be  divided  by  the 
second,  the  quotient  will  be  the  third ;  and  if  it  be  di- 
vided by  the  third,  the  quotient  will  be  the  second. 

Tn  the  proportion  23  :  27  =  184  :  216  the  product 
of  the  first  and  fourth  is  4968  :  if  this  be  divided  by 
u  2 


292  A    TREATISE    ON    ARITHMETIC.         BOOK  IV. 

27,  the  quotient  will  be  184;  and  if  it  be  divided  by 
184,  the  quotient  will  be  27. 

(378.)  From  these  observations,  it  follows  that  if  any 
three  terms  of  a  proportion  be  known,  the  fourth  term 
may  always  be  found,  for  of  the  three  known  terms  two 
must  either  be  means  or  extremes.  In  the  proportion, 
let  these  two  be  multiplied  together,  and  their  product 
found :  this  product  divided  by  the  other  known  term 
will  always  give  a  quotient,  which  is  the  remaining  term. 

If  the  first,  second,  and  third  terms  of  a  proportion 
be  23,  27,  and  184,  the  fourth  term  will  be  found,  by 
multiplying  184  by  27,  and  dividing  the  product  by 
23 ;  the  quotient  will  be  21 6,  as  we  have  already  seen. 

In  the  same  manner,  if  the  first,  second,  and  fourth 
terms  be  23,  27,  and  21 6,  the  third  will  be  found  by 
multiplying  23  by  21 6,  and  dividing  the  product  by 
27  :  the  quotient  will  be  184,  as  already  found. 

(379-)  Since  four  numbers,  which  are  in  proportion, 
will  continue  to  be  in  proportion  so  long  as  the  pro- 
duct of  the  means  is  equal  to  the  product  of  the  ex- 
tremes, it  follows  that  we  may  make  any  of  the  following 
changes  in  the  order  of  the  four  terms,  without  destroying 
their  proportion. 

1.  The  means  and  extremes  may  interchange  places  : 
thus,  if  23  :  27  =  184  :  216,  then  we  shall  also  have 
27  :  23  =  216  :  184,  or  27 :  216  =  23  :  184,  or  184  : 
23  =  216  :  27,  or  184  :  216  =  23  :  27- 

2.  The  places  of  the  extremes  may  be  transposed : 
thus,  if  we  have  23  :  27  =  184  :  216,  we  may  infer 
216:  27=  184  :  23. 

3.  The  means  may  be  transposed :  thus,  if  we  have 
23 :  27  =  184  :  216,  we  may  infer  23  :  184  =  27  :  216. 

4.  Both  means  and  extremes  may  be  transposed  : 
thus,  if  we  have  the  same  proportion  as  above,  we  may 
infer  216  :  184  =  27  :  23. 

(380.)  Any  change  may  be  made  in  the  means  or 
extremes,  which  does  not  alter  the  value  of  their  pro- 
duct, and  such  change  will  not  destroy  the  proportion  : 
thus,  one  mean  or  extreme  may  be  doubled,  and  the 


GHAP.  1.  PROPORTION.  293 

other  halved,  or,  in  general,  one  may  he  multiplied  by 
any  number  if  the  other  be  divided  by  the  same  num- 
ber. Also,  if  one  mean  be  doubled,  and  one  extreme 
also  doubled,  the  proportion  will  be  preserved,  because 
in  that  case  the  product  of  the  means  and  extremes  will 
be  both  doubled  ;  and  in  general,  if  one  mean  and  one 
extreme  be  multiplied  or  divided  by  the  same  number, 
the  proportion  will  be  preserved,  because  in  that  case 
the  products  of  the  means  and  extremes  must  continue 
to  be  equal. 

These  observations  will  be  found  useful  in  abridging 
the  computation  of  many  arithmetical  questions,  involv- 
ing the  consideration  of  proportions. 

(381.)  If  two  or  more  proportions  be  given,  another 
proportion  may  be  obtained  by  multiplying  the  corre- 
sponding terms  of  the  given  proportions,  —  that  is,  by 
multiplying  all  the  first  terms  together  for  a  first  term, 
all  the  second  terms  for  a  second  term,  and  so  on. 
Thus  if  the  following  proportions  be  given  :  — 

3  :  5  =  18  :  30 
2:1  =  4  :  2 
4:3=  8  :  6 

we  shall  have  the  following  proportion  :  — 

3x2x4:  5x1x3  =  18x4x8:  30x2x6. 

That  this  last  is  a  true  proportion,  may  be  ascertained 
by  actually  performing  the  multiplications  indicated, 
and  then  obtaining  the  products  of  the  extremes  and 
means,  which  will  be  found  to  be  equal  ;  but  we  may 
infer  the  truth  of  the  general  principle,  of  which  this  is 
merely  an  example,  by  expressing  the  proportions  as 
fractions  in  the  following  manner  :  — 


Since  the  three  fractions  in  the  first  column  are  respec- 

tively equal  to  the  three  fractions  in  the  second  column 

the  continued  product  of  the  former  must  be  equal  to  the 

u  3 


A    TREATISE    ON    ARITHMETIC.         BOOK  IV. 

continued  product  of  the  latter;  but  the  continued  pro- 
duct of  the  former  is, 

3x2x4 

5x1x3 
and  that  of  the  latter, 

18x4x8 


30  x  2  x  6 

These  fractions  being  equal,  the  ratio  of  the  numerator 
to  the  denominator  of  the  first,  must  be  the  same  as  the 
ratio  of  the  numerator  to  the  denominator  of  the  second : 
hence  we  infer  : — 

3x2x1 :  5x1x3  =  18x4x8  :  30x2x6. 


CHAP.  II.  RULE  OP  THREE.  295 

CHAP.  II. 

THE    RULE   OF   THREE. 

(382.)  THE  method  explained  in  the  last  chapter,  by 
which,  when  three  terms  of  a  proportion  are  given,  the 
fourth  term  may  be  found,  is  commonly  called,  in  trea- 
tises on  arithmetic,  THE  RULE  OF  THREE  ;  and,  from 
its  extensive  usefulness  in  the  solution  of  arithmetical 
questions,  was  formerly  called  the  GOLDEN  RULE.  This 
rule,  as  well  as  the  whole  doctrine  and  application  of 
proportion,  is  indeed  nothing  more  than  a  peculiar 
manner  of  expressing  fractional  relations,  a  fact  which 
will  be  still  more  clearly  illustrated  in  the  questions,  the 
solutions  of  which  we  shall  now  proceed  to  explain. 

The  rules  and  methods  of  computation  which  have 
been  explained  hitherto  in  this  treatise,  are  sufficient  for 
the  solution  of  all  questions  whatever,  which  can  be 
proposed  in  arithmetic ;  but  these  rules  always  suppose 
that  the  various  arithmetical  operations  to  be  performed 
on  the  given  numbers,  are  distinctly  stated,  and  that  the 
computist  is  only  required  to  execute  them  so  as  to  ob- 
tain correct  results ;  but  in  the  practical  application  of 
arithmetic,  there  is  another  difficulty  which  constantly 
presents  itself,  and  which  cannot  easily  be  met  by  any 
very  general  rules.  When  a  question  is  proposed,  it  is 
frequently  a  matter  of  some  difficulty,  especially  to  an 
unpractised  computist,  to  collect  from  the  conditions 
given,  what  the  arithmetical  operations  are,  which  must 
be  performed  on  the  numbers  proposed  in  the  question, 
in  order  to  obtain  its  solution.  To  discover  from  these 
conditions  the  series  of  arithmetical  operations  necessary 
to  the  solution,  is  called  the  analysis  or  resolution  of  the 
problem,  and  a  facility  in  effecting  such  resolution  can. 
only  be  obtained  by  extensive  practice  in  arithmetical 
questions,  aided  by  such  general  comments  on  particular 
u  4 


296  A    TREATISE    ON    ARITHMETIC*        BOOK  IV. 

questions  as  their  circumstances  and  conditions  may  en- 
able a  judicious  teacher  to  make. 

The  different  circumstances  and  conditions  which  at- 
tend the  resolution  of  arithmetical  questions  involving 
the  consideration  of  proportion,  have  led  to  a  classifi- 
cation of  such  questions  under  the  titles  of  "  THE  DI- 
RECT RULE  OF  THREE,  THE  INVERSE  RULE  OF  THREE, 
THE  COMPOUND  RULE  OF  THREE/'  &c.  &c. 

THE  DIRECT  RULE  OF  THREE. 

(383.)  Example  1.  —  If  25  bales  of  goods  cost 
6501.,  what  will  be  the  price  of  384  bales  of  the  same 
goods  ? 

ANALYSIS.  —  In  this  question  the  price  is  supposed 
to  increase  or  decrease  in  the  same  proportion  as  the 
number  of  bales  increases  or  decreases.  Now  there  are 
two  numbers  of  bales  proposed,  the  price  of  one  num- 
ber being  given,  while  the  price  of  the  other  number  is 
sought.  It  is  clear,  that  whatever  the  sought  price  be, 
the  given  price  must  have  the  same  proportion  to  it  as 
the  number  of  bales  to  be  obtained  for  the  given  price 
has  to  the  number  of  bales  obtained  for  the  sought 
price ;  that  is,  25  bales  will  be  to  384  bales  in  the  same 
proportion  as  650/.  is  to  the  price  of  384  bales.  Let 
us  suppose  that  this  unknown  price  of  384  bales  is  ex- 
pressed by  the  letter  x  :  a  proportion  must  exist  between 
25,  384,  650,  and  x. 

COMPUTATION.  —  We  have  then  — 
25  :  384  =  650  :  x. 

We  find  x  the  fourth  term  of  the  proportion,  by  mul- 
tiplying together  the  second  and  third,  — 

384x650  =  249600 

and  dividing  the  product  by  the  first  (378.).     The 
price,  therefore,  of  384  bales  is  — 
25  )  249600 
.£9984. 

If  the  student  should  find  any  difficulty  in  compre- 


CHAP.  II.  DIRECT  RULE  OF  THREE.  29? 

bending  the  reasoning  on  which  this  process  rests,  it 
will  be  made  still  more  evident  by  proceeding  in  the 
following  manner :  —  First,  find  the  price  of  a  single 
bale.  This  is  easily  done :  the  price  of  25  bales  is 
6501.,  and  the  price  of  a  single  bale  must  therefore  be 
the  twenty-fifth  part  of  this.  If  we  divide  6'50  by  25, 
we  find  the  quotient  26:  the  price  of  a  single  bale 
is,  therefore,  26/.  To  find  the  price  of  384  bales,  we 
must  multiply  the  price  of  a  single  bale  by  384.  The 
product  of  26  and  384  is  9984  :  the  price  of  384  bales 
is,  therefore,  998  4^. 

It  will  easily  appear  that  these  two  processes  are,  in 
fact,  identical.  In  the  first  we  multiply  the  third  term 
of  the  proportion  by  the  second,  and  divide  the  product 
by  the  first.  According  to  the  second  method,  we  di- 
vide the  third  by  the  first,  and  multiply  the  quotient  by 
the  second.  The  same  operations  are,  therefore,  per- 
formed in  each  case,  but  are  performed  in  a  different 
order. 

The  computation  may  also  be  simplified  by  recollect- 
ing that  we  may  divide  the  first  and  third  terms  in  a 
proportion  by  the  same  number  without  destroying  the 
proportion  (373.).  In  the  present  case  it  is  obvious, 
upon  inspection,  that  the  first  and  third  are  both  divi- 
siule  by  25.  If  this  division  be  effected,  the  propor- 
tion becomes  — 

1  ;  384  =  26  :  z, 

The  fourth  term,  #,  is  then  found,  by  merely  mul- 
tiplying the  second  and  third, 

(384.)  Example  2.  —  87  yards,  2  feet,  4  inches  of  a 
certain  canal  cost  743J.  15s.  8d.,  what  will  be  the  cost 
of  155  yards,  0  feet,  8  inches  of  the  same  canal f 

ANALYSIS.  —  This  question  is  evidently  one  of  the 
same  nature  as  the  last,  the  numbers  proposed  being, 
however,  complex  numbers.  There  are  three  numbers 
given  in  order  to  find  a  fourth :  two  of  the  given  num- 
bers express  certain  quantities  of  work,  the  third  ex- 
presses the  price  of  one  of  these  quantities,  and  the 


298  A    TREATISE    ON    ARITHMETIC.         BOOK  IV, 

number  which  is  sought  is  the  price  of  the  other.  It 
is  evident  from  the  conditions  of  the  question,  that  the 
prices  of  the  two  quantities  of  work  will  be  in  the  same 
proportion  as  the  quantities  themselves.  The  quantity 
of  work  whose  price  is  given  is  to  the  quantity  of  work 
whose  price  is  sought,  in  the  same  proportion  as  the 
given  price  is  to  the  sought  price. 

COMPUTATION.  —  We  have  then  the  following  pro- 
portion, the  fourth  or  sought  number  being  expressed 
by  x :  — 

yds.  ft.  in.   yds.   ft.  in.       £    s.     d. 
87     2     4  :  155     0     8  =  743  15     8  :  x. 

We  must  here  proceed  to  find  x  by  multiplying  the 
second  by  the  third,  and  dividing  by  the  first,  accord- 
ing to  the  rules  established  for  these  operations  on 
complex  numbers,  in  Book  III.  chap.  iii.  iv.  But  we 
may  abridge  the  process  if,  in  the  first  instance,  we  re- 
duce the  first  and  second  terms  to  simple  numbers,  by 
converting  the  yards  and  feet  into  inches.  Proceeding 
according  to  the  methods  explained  in  Book  III.  chap.  i. 
we  find  that 

yds.  ft.  in.      in. 

87     2     4  =  3160 

155     0     8  =  5588. 

Putting  these  numbers  of  inches  as  the  first  and  se- 
cond terms  of  the  above  proportions,  it  becomes  — 

£     s.     d. 
3160  :  5588  =  743  15     8  :  x. 

The  computation  of  the  fourth  term  may  still  further 
be  simplified  by  observing,  that  the  first  and  second 
terms  may  be  divided  by  4  (380.),  by  which  the  pro- 
portion becomes  — 

£     s.  d. 
790:  1397  =  743  15  8  :  a?. 

Multiplying  743/.  1 5s.  Sd.  by  1397  (346.),  we  find 
the  product  1039065/.  6*.  4A 

Dividing  this  product  by  790,  we  obtain  the  quo- 


CHAP.  II.  DIRECT  RULE  OF  THREE.  299 


tient  1315/.  5s.  53$$d.,  which  is,  therefore,  the  price 
of  155  yards,  0  feet,  8  inches. 

(385.)  Example  3.  —  If  SO  quarters  of  corn  are 
bought  for  751.  10s.,  what  will  be  the  price  of  180 
quarters? 

ANALYSIS.  —  In  this  case  two  quantities  of  corn  are 
given,  and  the  price  of  one  of  those  quantities  is  also 
given,  while  the  price  of  the  other  is  required.  It  is 
evident  that  the  prices  are  in  the  same  proportion  as  the 
quantities  ;  and,  consequently,  we  can  state  that  the  first 
quantity  is  to  the  second  quantity  as  the  price  of  the 
first  is  to  the  price  of  the  second,  which  is  the  number 
sought. 

COMPUTATION.  —  Let  x  express  the  price  of  180 
quarters  :  we  have  — 

£    s. 
SO:  180  =  75   10:  x. 

In  this  proportion  the  second  term  is  exactly  divi- 
sible by  the  first.  The  process  of  calculation  will  be 
abridged  by  first  dividing  the  second  by  the  first,  and 
then  multiplying  the  quotient  by  the  third.  The  quo- 
tient of  the  second  by  the  first  is  6.  We  shall,  there- 
fore, find  the  price  sought  by  multiplying  J51.  Ws.  by 
6.  The  product  is  45S/. 

(386.)  We  may  observe,  generally,  that  when  the 
first,  second,  and  third  terms  of  a  proportion  are  given 
to  find  the  fourth,  although  the  operations  by  which  it 
is  found  must  always  be  the  same,  yet  they  may  be 
performed  in  different  orders,  and  the  process  may 
sometimes  be  rendered  more  expeditious  by  a  proper 
selection  of  the  order  in  which  these  operations  are  per- 
formed. 

First,  we  may  multiply  the  second  and  third  toge- 
ther, and  divide  by  the  first. 

Second,  we  may  divide  the  second  by  the  first,  and 
multiply  the  quotient  by  the  third. 

Third,  we  may  divide  the  third  by  the  first,  and  mul- 
tiply the  quotient  by  the  second. 


SOO  A    TREATISE    ON    ARITHMETIC.  BOOK  IV* 

It  is  evident  that  the  same  operations  are  performed 
in  each  case,  but  are  taken  in  a  different  order.  If  it 
happen  that  either  the  second  or  the  third  is  exactly  di- 
visible by  the  first,  then  the  second  or  third  of  the  above 
methods  is  preferable  to  the  first. 

In  every  case  we  should  observe  whether  there  is  any 
number  which  will  divide  both  the  first  and  second,  or 
the  first  and  third,  in  which  case  such  division  may  be 
made  without  disturbing  the  proportion,  and  its  effect 
will  be  to  make  the  calculation  depend  on  smaller  num- 
bers. 

(387.)  Example  4.  —  If  the  pendulum  of  a  clock 
vibrate  180  times  in  3  minutes,  how  often  will  it  vibrate 
in  an  hour  and  a  quarter  ? 

ANALYSIS.  —  In  this  case  there  are  two  portions  of 
time  given,  3  minutes  and  1  hour  and  15  minutes;  and 
we  are  also  given  the  number  of  swings  of  the  pendu- 
lum in  the  first  time,  to  find  the  number  of  swings  in 
the  second.  It  is  evident  that  the  number  of  swings 
or  vibrations  will  be  greater  or  less  in  proportion  as  the 
time  is  greater  or  less  ;  and,  consequently,  3  minutes  is 
to  1  hour  and  15  minutes  as  the  number  of  swings  in 
3  minutes,  i.  e.  1 80,  is  to  the  number  of  swings  in  1 
hour  and  15  minutes. 

COMPUTATION.  —  In  1  hour  and  15  minutes  there  are 
75  minutes.  We  have,  therefore,  the  following  pro- 
portion :  — 

3  :  75  =  180  :  x. 

In  this  case  the  third  term  is  divisible  by  the  first. 
We  accordingly  find  the  quotient,  which  is  60,  and 
multiply  it  by  75  (386,),  by  which  we  find  the  fourth 
term  to  be  4500, 

(388.)  Example  5.  —  If  56  men  are  able  to  make 
7 20 -feet  of  a  road  in  a  week3  how  many  feet  of  the  same 
road  would  24  men  make  in  the  same  time  ? 

ANALYSIS. — We  are  here  given  two  troops  of  workmen, 
one  consisting  of  56  labourers,  and  the  other  of  24. 
We  are  also  given  the  length  of  road  which  the  first 
troop  can  make  in  a  week,  and  we  are  required  to  find 


CHAP.  II.  DIRECT  RULE  OF  THREE.  SOI 

the  length  of  road  which  the  other  troop  can  make  in  a 
week.  It  is  evident  that  the  two  lengths  of  road  would 
be  greater  or  less  in  proportion  as  the  numher  of  la- 
bourers in  each  troop  is  greater  or  less.  A  troop  of 
double  or  half  the  number,  would  perform  double  or 
half  the  work,  and  so  on.  Hence,  if  we  express  by  x 
the  number  of  feet  of  road  which  24  men  would 
construct  in  a  week,  we  shall  have  the  following  pro- 
portion :  — 

COMPUTATION.  — 

56  :  24  =  720  :  x. 

In  this  proportion  the  first  and  second  terms  may  be 
both  divided  by  8,  by  which  the  proportion  will  be  re- 
duced to  — 

7  :  3  =  720  :  x. 

To  find  the  fourth  term,  we  multiply  the  third  by  the 
second,  and  divide  the  product  by  the  first,  by  which 
we  obtain  308-1}-  feet,  which  is  the  length  of  road  re- 
quired. 

(389-)  If  tne  preceding  examples  be  considered,  and 
attentively  compared  one  with  another,  they  will  be 
found  to  agree  in  certain  general  features.  In  each  of 
them  there  are  four  quantities,  or  numbers,  contemplated, 
two  of  which  are  of  a  certain  kind,  and  two  others  of 
another  kind,  each  of  the  latter  being  related  to  each  of 
the  former  in  the  same  manner.  Thus,  in  some  of  the 
examples,  there  are  two  quantities  of  work,  which  are  the 
two  quantities  of  the  first  kind,  and  the  two  prices  of 
these  quantities  of  work,  which  are  the  two  quantities  of 
the  second  kind.  It  is  evident  that  the  first  price  has 
the  same  relation  to  the  first  quantity  of  work  as  the 
second  price  has  to  the  second  quantity  of  work.  In 
the  same  manner,  in  another  example,  two  troops  of 
workmen  are  the  quantities  of  the  first  kind,  and  two 
lengths  of  a  road  which  they  can  construct  in  a  week 
are  the  two  quantities  of  the  second  kind.  In  this  case, 
also,  the  first  length  of  road  has  the  same  relation  to  the 
first  troop  of  men  as  the  second  length  of  road  has  to 


302  A    TREATISE    ON    ARITHMETIC.          BOOK  IT. 

the  second  troop  of  men.  Again,  in  another  example, 
the  two  quantities  of  the  first  kind  are  two  parcels  of 
bales  of  goods,  and  the  two  quantities  of  the  second 
kind  are  the  two  prices  of  these  parcels.  The  first  of 
these  prices  has  the  same  relation  to  the  first  parcel  as 
the  second  price  has  to  the  second  parcel. 

The  same  observation  will  be  applicable  to  every  ex- 
ample which  can  be  proposed  in  this  class.  The  two 
terms  of  the  first  kind  are  given,  and  the  first  term  of 
the  second  kind  is  also  given,  while  the  second  term  is 
sought.  In  every  case,  also,  the  ratio  of  the  first  term 
of  the  first  kind  to  the  second,  is  the  same  as  the  ratio 
of  the  first  term  of  the  second  kind  to  the  second  or 
sought  number.  The  two  terms  of  the  second  kind 
may  be  called  the  correspondents  of  those  of  the  first 
kind,  to  which  they  are  related  by  the  conditions  of  the 
question.  Thus,  when  two  quantities  of  goods  are 
given,  and  the  price  of  one  of  them,  to  find  the  price 
of  the  othei,  the  two  quantities  of  goods  are  the  two 
quantities  of  the  first  kind,  and  the  prices  are  their  cor- 
respondents ;  and  the  two  quantities  of  the  first  kind, 
are  evidently  proportional  to  their  correspondents. 
Again,  if  two  quantities  of  work  are  given,  and  the 
number  of  men  necessary  to  execute  the  first  in  a  day, 
to  find  the  number  necessary  to  execute  the  second  in  a 
day,  the  two  terms  of  the  first  kind  are  the  two  num- 
bers of  men,  and  their  correspondents  are  the  quantities 
of  work  they  can  respectively  finish  in  a  day.  In  this 
case,  also,  the  terms  of  the  first  kind  are  in  the  same 
proportion  as  their  correspondents. 

(390.)  In  general,  then,  in  the  class  of  questions  to 
which  the  above  examples  belong  we  may  observe, — 

First,  That  the  two  terms  of  the  first  kind  are  in  the 
same  ratio  as  their  correspondents  ;  the  first  term  being 
to  the  second  term  as  the  correspondent  of  the  first  term 
is  to  the  correspondent  of  the  second  term. 

Second,  The  correspondent  of  the  second  term,  which 
is  the  number  sought,  is  found  by  one  of  the  three 
methods  explained  in  (386.). 


CHAP.  II.        DIRECT  RULE  OF  THREE.  303 

(391.)  The  analysis  of  questions  of  this  class  seldom 
presents  much  difficulty.  In  such  questions  three  quan- 
tities are  always  given,  and  one  sought ;  of  the  three 
given  quantities  two  are  always  of  the  same  kind,  while 
the  third  is  of  a  different  kind,  and  connected  with  one 
of  the  two  former  in  some  manner  which  is  distinctly 
expressed  in  the  conditions  of  the  question.  Thus  of 
the  three  given  quantities,  two  may  he  parcels  of  goods, 
while  the  third  is  the  price  of  one  of  those  parcels ;  or 
two  may  be  quantities  of  work,  while  the  third  is  the 
price  of  performing  one  of  them ;  or,  again,  two  may 
he  quantities  of  work,  while  the  third  is  the  time  which 
a  man  would  take  to  perform  one  of  them,  and  so  on. 
The  two  quantities  of  the  same  kind  which  are  given, 
are  what  we  have  called  the  two  terms  of  the  first 
Kind ;  but  that  quantity  is  to  be  taken  first  in  order, 
to  which  the  third  given  quantity  is  related.  Thus  if 
two  parcels  of  goods  be  given,  together  with  the  price 
of  one  of  them,  we  must  take,  as  the  first  term  of  the 
first  kind,  that  parcel  whose  price  is  given,  while  its 
correspondent,  which  is  its  price,  will  be  the  first  term 
of  the  second  kind.  The  student  may  facilitate  the 
analysis  by  first  writing  down  the  two  given  quantities  of 
the  same  kind,  placing  first  in  order  that  which  is  re- 
lated to  the  third  given  quantity.  Expressing  the  sought 
quantity  by  x9  he  may  then  write  under  these  two  terms 
of  the  first  kind  their  correspondents.  Thus,  in  the 
analysis  of  Example  I.,  he  would  first  write  the  quan- 
tities as  follows :  — 

Bales      -     25     -     384 
Prices     £650     -         a% 

If  it  is  apparent,  from  the  conditions  of  the  question, 
that  the  first  term  of  the  first  kind  is  to  the  second  as 
the  correspondent  of  the  first  is  to  the  correspondent  of 
the  second,  the  analysis  will  be  completed  by  stating  the 
proportion  in  that  form.  Thus  the  above  example  would 
be  stated  thus  :  — 

25  :  384  =  650  :  x. 


304  A  TREATISE  ON  ARITHMETIC.        BOOK  IV. 

THE  INVERSE  RULE  OF  THREE. 

(392.)  The  analysis  of  the  following  example  will 
lead  us  to  consider  another  class  of  problems. 

Example  6. — If  21  men  take  18  days  to  perform  a 
certain  work,  in  how  many  days  will  7  men  perform  the 
same  work,  working  at  the  same  rate  ? 

ANALYSIS. —  In  this  case,  the  same  quantity  of  work 
which  21  men  are  able  to  execute  in  18  days,  is  re- 
quired to  be  done  by  only  7  men.  It  is  evident  that,  in 
proportion  as  the  number  of  labourers  is  diminished,  the 
time  required  for  the  performance  of  the  work  will  be 
increased;  one  man  will  take  two  days  to  perform  what 
two  men  will  do  in  one  day.  In  the  same  manner,  one 
man  would  take  3,  4,  or  5  days  to  perform  what  3,  4, 
or  5  men  would  do  in  one  day.  The  two  terms  of  the 
first  kind  are  in  this  case  the  two  numbers  of  men,  and 
their  correspondents  are  the  number  of  days  in  which 
they  would  respectively  perform  the  work.  The  cor- 
respondent of  the  first  troop  is  18  ;  let  us  call  the  cor- 
respondent of  the  second  troop,  which  is  sought,  a. 
Labourers  21  -  7 
Days'  work  18  -  x. 

From  what  has  been  just  stated,  it  appears  that  the 
first  term  of  the  first  kind  is  to  the  second  as  the  cor- 
respondent of  the  second  is  to  the  correspondent  of  the 
first;  or,  what  amounts  to  the  same,  the  second  term  of 
the  first  kind  is  to  the  first  as  the  correspondent  of  the 
first  is  to  the  correspondent  of  the  second.  We  have, 
therefore,  the  foUowing  proportion  :  — 

COMPUTATION. —  . 

7  :  21  =  18  :  x. 

In  this  proportion  the  second  term  is  divisible  by  the 
first.  Performing  this  division,  and  multiplying  the 
quotient  by  the  third  (386.),  we  find  the  fourth  term 
to  be  54 ;  7  men  would,  therefore,  take  54  days  to 
perform  the  work. 

Example  7. — A  steam  engine  moving  on  a  level  raiL 
way  is  capable  of  transporting  carriages  at  the  rate  of  32 
miles  an  hour,  while  carriages  drawn  by  horses  on  turn 


GHAP.  II.       INVERSE  RULE  OF  THREE.  305 

pike  roads  cannot  exceed  the  average  rate  of  9  miles  an 
hour.  If  the  distance  from  London  to  Liverpool,  at  this 
rate,  on  the  turnpike  roads,  is  performed  at  present  in 
24  hours,  in  what  time  would  the  same  distance  be  per- 
formed  on  a  level  rail-road  by  steam  power  ? 

ANALYSIS. —  The  two  quantities  of  the  first  kind  are 
the  rates  of  travelling,  and  their  two  correspondents  are 
the  times  of  performing  the  journey. 

Miles  per  hour     -  9     -  32 

Hours  from  London  to  Liverpool     24     -     jr. 

It  is  evident  that,  in  proportion  as  the  rate  of  tra- 
velling is  increased,  the  time  necessary  to  perform  the 
journey  will  be  diminished.  The  second  term  of  the 
first  kind  will,  therefore,  be  to  the  first  as  the  corre- 
spondent of  the  first  is  to  the  correspondent  of  the 
second. 

COMPUTATION  :  — 

32  :  9  =  24  :  x. 

In  this  proportion  the  first  and  third  may  be  divided 
by  8  (380.),  by  which  the  proportion  becomes  — 

4  :  9  =  3  :  x; 

from  whence  we  find  the  fourth  term  to  be  6|.  The 
time,  therefore,  on  a  level  rail-road,  would  be  6  hours 
and  3  quarters. 

(393.)  In  considering  these  examples,  we  find  that, 
as  in  the  first  class,  we  still  have  under  contemplation 
two  terms  of  the  first  kind  and  their  correspondents ; 
but  in  the  proportion  which  subsists  between  these 
terms  and  their  correspondents,  there  is  a  striking  dif- 
ference. In  the  examples  of  the  first  class,  the  first 
term  of  the  first  kind  always  had  the  same  ratio  to  the 
second  as  its  correspondent  had  to  the  correspondent  of 
the  second.  The  terms  of  the  first  kind  in  the  former 
examples,  therefore,  always  occupied  the  same  order  in 
the  proportion  as  their  correspondents.  However,  these 
circumstances  are  now  reversed :  the  first  term  of  the 
first  kind  is  to  the  second,  not  as  the  correspondent  of 
x 


306  A  TREATISE    ON    ARITHMETIC.  BOOK  IV. 

the  first  to  the  correspondent  of  the  second,  but  as  the 
correspondent  of  the  second  to  the  correspondent  of  the 
first.  The  terms  of  the  first  kind  are  still  proportional 
to  their  correspondents,  but  they  are  proportional  to 
them  only  when  taken  in  a  contrary  order.  When  the 
proportion  subsists  under  these  circumstances,,  the  terms 
are  said  to  be  inversely  or  reciprocally  proportional  to 
their  correspondents  ;  and  the  class  of  questions  which 
come  under  this  condition  is  placed  under  the  Inverse 
Rule  of  Three. 

(394.)  In  general,  whenever  a  quantity  of  one  kind 
increases  in  the  same  proportion  as  another  quantity 
connected  with  it  by  the  conditions  of  the  question 
diminishes,  the  question  comes  under  this  rule.  Thus, 
the  quantity  of  goods  which  can  be  bought  for  a  given 
sum  of  money  increases  in  the  same  proportion  as  the 
price  diminishes.  If  cotton  were  1  shilling  a  yard,  we 
should  only  obtain  half  the  quantity  for  a  given  sum 
which  we  would  obtain  at  6  pence  a  yard.  The  num- 
ber of  days  necessary  to  perform  a  given  quantity  of 
work  will  diminish  in  proportion  as  the  number  of  men 
employed  at  it  is  increased.  In  the  one  case  we  say, 
that  the  quantity  of  cotton  to  be  obtained  for  a  given 
sum  is  inversely  as  the  price  per  yard ;  and,  in  the  other 
case  we  say,  that  the  number  of  days  necessary  for  the 
completion  of  a  given  work,  is  inversely  as  the  number 
of  labourers  employed  at  it. 

If,  on  the  other  hand,  one  quantity  increases  in  the 
same  proportion  as  another  quantity  connected  with  it 
by  the  condition  of  the  question  increases,  the  one  quan- 
tity is  said  to  be  directly  proportional  to  the  other ;  and 
all  questions  involving  such  a  condition  belong  to  the 
direct  rule  of  three.  Thus,  the  quantity  of  work  which 
a  given  number  of  men  performs  is  directly  proportional 
to  the  time  or  number  of  days  during  which  they  are 
employed :  the  quantity  of  goods  which  can  be  bought 
at  a  given  price  is  directly  proportional  to  the  sum  of 
money  to  be  expended  on  them,  and  so  on. 

(395.)  Fractions  which  have  the  same  denominator 


CHAP.  II.  RULE  OF  THREE.  SO? 

are  directly  proportional  to  their  numerators ;  for,  since 
their  denominators  are  the  same,  they  consist  of  the 
same  aliquot  parts  of  the  unit,  and  their  magnitude* 
must  be  greater  or  less,  in  proportion  as  they  contain  a 
greater  or  less  number  of  these  parts.  But  their  nu- 
merators express  the  numbers  of  those  parts  which  they 
respectively  contain  (203.)  ;  the  fractions  are,  therefore, 
directly  proportional  to  those  numerators.  The  frac- 
tions •£%  and  T72-  express,  respectively,  5  and  7  twelfth 
parts  of  the  unit,  and  are  evidently  in  the  proportion  of 
5  to  7)  or  directly  as  their  numerators. 

If  two  fractions  have  the  same  numerator,  they  will 
consist  of  the  same  number  of  aliquot  parts  of  the  unit, 
and  their  magnitudes  will  be  directly  proportional  to 
the  magnitude  of  the  parts  of  which  they  consist.  Thus, 
if  the  two  fractions  be  J-  and  1IIT,  each  will  consist  of  7 
parts  of  the  unit,  but  one  of  these  is  a  ninth  part,  and 
the  other  is  a  tenth :  the  magnitudes  of  the  fractions 
will,  .therefore,  be  in  the  direct  proportion  of  a  ninth  to 
a  tenth,  or  of  -^  to  -J^.  But  it  is  evident  that  the  mag- 
nitude of  any  aliquot  part  of  the  unit  will  be  greater  in 
proportion  as  the  number  of  parts  into  which  the  unit 
is  divided  is  less.  Thus,  if  the  unit  be  divided  into  3 
parts,  every  part  will  be  twice  as  great  as  if  it  were 
divided  into  6  parts  :  thus,  -^  will  be  double  J,  or, 

}t  1-8:1, 
or,  what  is  the  same, 

t  '  i  -  6  '  »• 
111  the  same  manner, 

£:^=10:  9; 

that  is,  the  ninth  part  of  the  unit  is  greater  than  its 
tenth  part,  in  the  same  proportion  as  10  is  greater  than 
9.  In  general,  then,  the  aliquot  parts  of  the  unit  are 
inversely  as  the  number  of  parts  into  which  the  unit  is 
divided,  or  inversely  as  the  denominators  of  the  fractions 
by  which  these  aliquot  parts  are  expressed.  But  it  has 
been  proved  above,  that  fractions  which  have  the  same 
numerator  are  in  the  direct  proportion  of  the  aliquot 
x  2 


3 OS  A     TREATISE    ON     ARITHMETIC.  BOOK  IV, 

parts  of  unity  of  which  they  are  composed  ;  but  these 
aliquot  parts  being  inversely  as  the  denominators  of  the 
fractions,  it  follows  that  the  fractions  themselves  ave 
inversely  as  their  denominators.  In  general,  therefore, 
fractions  which  have  the  same  numerators  are  inversely 
as  their  denominators. 

(39&)  In  stating  the  proportion  which  results  from 
the  analysis  of  a  question  in  the  rule  of  three,  it  is 
customary  so  to  arrange  the  terms,  that  the  fourth  or 
last  shall  always  be  the  number  which  is  sought.  Al- 
though this  arrangement  is  not  necessary,  yet  it  is  so 
usual,  that  it  will  be  advantageous  to  point  out  to  the 
student  the  manner  in  which  it  may  in  every  case  be 
conducted.  If  the  question  be  one  in  the  direct  rule  of 
three,  the  first  term  of  the  proportion  should  be  that 
term  of  the  first  kind  whose  correspondent  is  given,  and 
the  second  term  should  be  the  other  term  of  the  first 
kind  whose  correspondent  is  the  sought  quantity  :  the 
third  term  will  then  be  the  given  correspondent  of  the 
first  term.  Thus,  in  the  first  example,  the  first  and 
third  terms  are  25  bales  of  goods,  and  650/.,  the  price 
of  these  bales  ;  the  second  term  is  384  bales  ;  and  the 
fourth  term  is  its  correspondent,  that  is,  its  price. 

If  the  question  belong  to  the  inverse  rule  of  three, 
the  first  term  in  the  proportion  must  be  that  term  of 
the  first  kind  whose  correspondent  is  sought,  and  the 
second  term  that  whose  correspondent  is  given  ;  the  third 
term  will  then  be  the  correspondent  of  the  second  term, 
and  the  fourth  the  correspondent  of  the  first  term,  which 
is  the  sought  quantity.  Thus,  in  the  Example  6.,  the 
term  of  the  first  kind  whose  correspondent  is  sought  is 
7  men,  it  being  required  to  determine  the  number  of 
days  in  which  they  can  perform  the  work ;  the  second 
term  of  the  first  kind  whose  correspondent,  18  days,  is 
given,  is  2 1  men :  the  order  of  the  proportion,  there- 
fore, is  — 

7  :  21  =  18  :  x. 

(397.)  In  some  treatises  on  arithmetic,  the  student 
is  directed  to  take,  as  the  first  and  second  terms  of  the 


CHAP.  II.  RULE  OF  THREE.  309 

proportion,  the  first  term  of  the  first  kind  and  its  cor- 
respondent. This,  however,  is  not  a  correct  method  of 
proceeding,  although,  in  many  cases,  the  right  result 
will  he  obtained.  A  ratio  can  only  subsist  between  two 
quantities  of  the  same  kind :  thus,  we  can  conceive  a  ratio 
subsisting  between  two  weights,  since  one  may  be  double 
or  triple  another,  or  may  have  any  other  assigned  pro- 
portion to  it ;  but  we  cannot  conceive  any  ratio  subsist- 
ing between  a  weight  and  a  sum  of  money,  which  will 
be  perceived  when  we  consider  the  absurdity  implied  in 
the  statement,  that  a  certain  number  of  pounds,  ounces, 
and  grains,  are  double  or  triple  a  number  of  shillings, 
pence,  and  farthings.  It  is  true  that  a  ratio  may  sub- 
sist between  the  abstract  numbers  expressing  money  and 
weight,  but  that  ratio  can  only  be  conceived  when  the 
numbers  themselves  are  considered  apart  from  the  quan- 
tities which  they  express.  Thus,  the  number  1 0  has  a 
certain  ratio  to  the  number  5,  but  10  shillings  has  no 
ratio  to  5  ounces.  In  the  first  example  given  in  this 
chapter,  the  proportion  which  results  from  its  analysis 
is,  that  the  same  ratio  which  subsists  between  25  bales 
and  384  bales,  also  subsists  between  650/.  and  the  price 
of  384  bales.  Here,  the  first  ratio  is  between  two  weights, 
and  the  second  ratio  between  two  sums  of  money ;  but  we 
should  divest  the  statement  of  all  propriety  and  distinct 
meaning,  if  we  stated  that  the  ratio  of  25  bales  to  65  O/. 
was  the  same  as  the  ratio  of  384  bales  to  the  price  of 
384  bales.  The  absurdity  of  such  a  statement  will  be 
apparent,  if,  instead  of  using  the  general  term  ratio,  we 
state  some  specific  ratio  :  thus,  to  say  that,  25  bales  being 
three  fourths  of  65 01.  sterling,  384  bales  must  also  be 
three  fourths  of  its  price,  is  manifestly  absurd. 

It  will  be  asked,  then,  if  such  an  absurdity  exists  in 
such  statements  of  questions  in  the  rule  of  three,  how 
it  happens  that  the  operations  performed  under  these 
statements  produce  right  results  ?  The  answer  is  ob- 
vious. When  the  numerical  statement  is  made,  the 
numbers  are  frequently  divested  of  their  compound  cha- 
racter, and,  in  fact,  become  abstract  numbers :  this 
x  3 


310  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

always  happens  when  only  one  species  of  units  enter 
each  term  of  the  proportion.  Thus,  in  the  first  ex- 
ample, the  units  of  the  first  and  second  terms  express 
bales  of  goods,  and  those  of  the  third,  pounds  sterling. 
AVhen  the  numerical  statement  is  made,  it  is  not  ne- 
cessary to  consider  the  nature  of  these  several  units,  but 
only  to  recollect  that  the  units  of  the  fourth  term  must 
express  pounds  sterling.  In  this  case,  therefore,  so  far 
as  the  mere  process  of  computation  is  concerned,  the 
numbers  are  treated  as  abstract  numbers,  and  the  result 
will  be  the  same  whatever  their  units  are  supposed  to 
express. 

Example  8.-^  A  vessel  at  sea  has  provisions  sufficient 
to  supply  the  passengers  and  crew  with  their  usual  daily 
rations  for  20  days;  but  it  cannot  complete  its  voyage  in 
less  than  35  days.  It  is  required  to  determine  what 
diminution  must  be  made  in  the  daily  ration  of  each  in- 
dividual, so  as  to  make  the  provisions  last  till  the  voyage 
has  been  completed. 

ANALYSIS. —  In  this  case,  the  same  quantity  of  pro- 
visions which,  at  the  ordinary  rate,  would  last  only  20 
days,  is  required  to  last  for  35  days.  It  is  evident  that 
the  daily  rations  must,  therefore,  be  diminished,  and 
also  that  they  must  be  diminished  in  exactly  the  same 
proportion  as  the  number  of  days  they  are  required  to 
last  is  increased.  Thus,  if  they  were  required  to  last 
double  the  time,  each  individual  should  only  be  supplied 
with  half  the  quantity  per  day ;  in  other  words,  the 
magnitude  of  the  daily  rations  will  be  inversely  as  the 
times  they  are  required  to  last :  hence  we  have  the  fol- 
lowing statement :  — 

Days          -  20-35 

Rations  per  day         1         -          x. 

Here  we  have  used  1  to  express  the  ordinary  ration, 
and  x  to  express  the  diminished  ration  which  we  seek ; 
and,  from  what  has  been  just  explained,  it  is  clear  that 
x  is  less  than  1,  in  the  same  proportion  as  20  is  less 
than  35  :  hence  we  have  the  following  proportion :  — 

S5  :    20  =   1    :  x. 


CHAP.  II.  COMPOUND    RULE    OF    THREE.  311 

COMPUTATION. —  The  first  two  terms  of  this  propor- 
tion are  divisible  by  5,  by  which  it  is  reduced  to  — 

7  :   4  =  1   :  x. 

The  sought  number  is,  therefore,  i-,  and  the  daily  ration 
will  be  four  sevenths  of  the  ordinary  ration.  This  ques- 
tion might  also  be  solved  without  the  consideration  of 
proportion.  The  actual  quantity  of  provisions  being 
20,  its  supply  may  be  expressed  by  the  number  20; 
and,  since  this  must  last  35  days,  the  daily  consumption 
must  be  its  35th  part :  the  fraction,  therefore,  of  the 
ordinary  daily  consumption  which  must  be  used  in  order 
to  last  35  days,  is  |-g-  —  \. 

COJIPOUND  RULE  OF  THREE. 

(398.)  In  the  examples  which  we  have  hitherto  ex- 
plained, the  number  of  given  terms  which  occurs  does 
not  exceed  3,  from  which  a  fourth  is  required  to  be  found 
by  computation.  We  shall  now  investigate  some  ques- 
tions in  which  a  greater  number  of  given  numbers  occurs. 

Example  9«  —  If  20  men  require  1 8  days  to  build 
500  feet  of  wall,  in  how  many  days  can  *j6  men  build 
126'5  feet  of  a  similar  wall? 

ANALYSIS.  —  In  this  question  there  are  3  pairs  of 
quantities  of  different  kinds  contemplated  :  there  are, 
first,  two  troops  of  workmen;  second,  two  pieces  of 
work  ;  third,  two  numbers  of  days.  The  first  and  se- 
cond pairs  of  numbers  are  given,  and  one  of  the  numbers 
of  the  third  pair  is  given,  while  the  other  is  sought. 
Expressing  the  sought  number  of  days  by  x,  the  num- 
bers contemplated  are  as  follows  :  — 

Men         -  20  76 

Feet  of  wall      500         -       1265 
Days         -  18         -  x. 

It  will  be  perceived,  that  the  three  numbers  in  the  first 
column  are  related  to  each  other  in  the  same  manner  as 
the  three  numbers  in  the  second  column  :  in  each  of 
them  there  are  expressed  a  number  of  men,  the  work 
x  4 


312  A    TREATISE    OX    ARITHMETIC.  BOOK  IV. 

they  can  perform,  and  the  time  of  performing  it ;  but 
in  the  first  column  all  the  numbers  are  given,  whereas 
in  the  second  column  one  of  the  numbers  is  sought. 
From  the  first  column  we  shall  be  able  to  derive  the 
means  of  finding  the  sought  number  in  the  second 
column. 

Since  500  feet  of  wall  are  built  in  18  days  by  the 
first  troop  of  builders,  one  day's  work  of  this  troop  will 
be  found  by  dividing  500  by  18.  Having  thus  found 
one  day's  work  of  the  first  troop,  we  shall  find  the  daily 
work  of  a  single  builder  by  dividing  the  work  of  the 
first  troop  by  20.  The  daily  work  of  a  single  builder 
being  thus  found,  we  shall  find  the  daily  work  of  the 
second  troop  by  multiplying  the  daily  work  of  one 
builder  by  the  number  of  builders  in  the  second  troop, 
which  is  76.  The  daily  work  of  the  second  troop  being 
thus  found,  we  shall  find  how  often  it  is  contained  in 
1265  feet  by  dividing  the  latter  by  it;  the  quotient  will 
evidently  be  the  number  of  days'  work  for  the  second 
troop  in  1265  feet  of  wall,  and  will,  therefore,  be  the 
number  sought. 

COMPUTATION.  — I.  Divide  500  by  18;  the  quotient 
will  be  27J-J,  or  27-J  feet:  this  is  the  daily  work  of  the 
first  troop. 

II.  Divide  27^  the  daily  work  of  the  first  troop,  by 
20 ;  the  quotient  will  be  -j-|g,  or  -J-j  feet,  which  is  the 
daily  work  of  one  man. 

III.  Multiply  4^|  feet,  the  daily  work  of  one  man, 
by  76,   the  number  of  men  in  the  second  troop  ;   the 
product  is  105§  feet,  which  is  the  daily  work  of  the 
second  troop. 

IV.  Divide  1265  feet  by  105^  feet;  the  quotient  is 
11-^-gJ,  which  is,  therefore,  the  number  of  days'  work 
for  the  second  troop  in  1265  feet  of  wall.     This  num- 
ber is  within  a  minute  fraction  of  12  days. 

The  preceding  analysis  of  the  question  has  been  made 
without  the  immediate  consideration  of  proportion.  We 
shall  now  consider  it  under  another  point  of  view. 

ANALYSIS. — In  the  question  as  announced,  three  ratios 


CHAP.  II.      COMPOUND  RULE  OF  THREE.          313 

are  contemplated ;  first,  the  ratio  between  the  number 
of  labourers  in  the  first  and  second  troop ;  secondly,  the 
ratio  between  the  magnitudes  of  the  two  pieces  of  work ; 
and,,  thirdly,  the  ratio  between  18  days  and  the  number 
of  days  sought.  To  simplify  the  investigation,  we  shall 
first  enquire  what  number  of  days  the  second  troop  would 
require  to  complete  the  first  piece  of  work.  Since  the 
number  of  days  necessary  to  perform  any  work  increases 
in  the  same  proportion  as  the  number  of  labourers  di- 
minishes^ this  will  be  a  question  in  the  inverse  rule  of 
three,  and  will  resolve  itself  into  the  following  propor- 
tion :  —  As  the  number  of  workmen  in  the  second  troop 
is  to  the  number  in  the  first  troop,  so  is  18  days  to  the 
number  of  days  which  the  second  troop  would  require 
to  complete  500  feet  of  the  work. 

Having  found  this  number,  we  shall  easily  discover 
the  number  of  days  which  the  second  troop  would  take 
to  complete  1265  feet  of  work.  In  this  case  the  num- 
ber of  days  necessary  to  perform  the  work  is  propor- 
tional to  the  work  to  be  done  :  hence  we  infer,  that 
500  feet  is  to  1265  feet  as  the  number  of  days  necessary 
to  complete  500  feet  to  the  number  of  days  necessary  to 
complete  1265  feet.  This  is  obviously  a  question  in 
the  direct  rule  of  three. 

From  this  analysis  it  will  be  perceived  that,  in  order 
to  solve  the  question  proposed,  it  is  necessary  to  solve 
another  question  ;  viz.,  to  discover  the  time  which  the 
second  troop  would  take  to  perform  the  first  piece  of 
work. 

COMPUTATION.  —  I.  Let  x  express  the  number  of  days 
in  which  76  men  would  complete  500  feet  of  work :  we 
have  the  following  proportion :  — — 

76  :   20  =  18  :  x. 

The  first  and  second  terms  of  this  proportion  being 
divisible  by  4,  it  is  reduced  to  — 

19  :   5  =  18  :  x. 

Multiplying  the  second  and  third  together,  and  dividing 
by  the  first,  we  find  the  quotient  4>±±. 


314  A    TREATISE    ON    ARITHMETIC.  Bl/OK  IV. 

II.  Expressing  now  by  x  the  number  of  days  which 
the  second  troop  would  take  to  perform  1265  feet,  we 
have  the  following  proportion  :  — 

500  :    1265  =  4$  :  x. 

The  first  and  second  terms  of  this  proportion  being 
divisible  by  5,  it  is  reduced  to  — 

100  :    253  =  4JJ  :  x. 

Reducing  the  third  term  to  an  improper  fraction  (235.), 
and  multiplying  by  the  second,  we  find  g  ' 7  7  °  ;  this 

must  be  divided  by  100,  which  gives  ^-gVo0  =  ll^fj, 
the  same  result  as  found  in  the  preceding  analysis,  and 
which  expresses  the  number  of  days  and  fractions  of  a 
day  necessary  for  the  second  troop  of  workmen  to  exe- 
cute 1265  feet  of  work. 

(399-)  Example  10. — In  6  days  126  acres  of  meadow 
are  mowed  by  14  mowers.  It  is  required  to  determine 
how  many  acres  can  be  mown  in  3  days  by  16  mowers. 

ANALYSIS. — We  have  here  under  consideration  two 
troops  of  mowers  working  for  two  distinct  times,  in 
which  one  of  the  troops  mow  126  acres,  and  the  other 
a  number  of  acres  which  we  are  required  to  discover. 
The  numbers  involved  in  the  question  are  as  follows, 
expressing  the  sought  number  of  acres  by  x :  — 

Mowers     -       14     -     16 

Acres        -     126     -      x 

Days         -         6-3. 

As  in  the  former  example,  we  shall  find  from  the  first 
column  the  daily  work  of  one  labourer ;  by  dividing  the 
number  of  acres  by  the  number  of  mowers,  we  shall 
find  the  work  of  one  mower  in  6  days ;  and,  by  di- 
viding this  work  by  6,  we  shall  find  the  daily  work  of 
one  mower. 

Having  found  the  daily  work  of  one  mower,  we  shall 
find  the  daily  work  of  the  second  troop,  consisting  of  ifi 
mowers,  by  multiplying  the  daily  work  of  one  mower  by 
16;  and  we  shall  find  their  work  in  3  days  by  multiply- 
ing their  daily  work  by  3. 


CHAP.   II.  COMPOUND    RULE    OF    THREE.  315 

COMPUTATION.  —  I.  Divide  126  by  14,  and  we  find 
the  quotient  9,  which  is,  therefore,  the  number  of  acres 
mown  in  6  days  by  one  mower. 

II.  Divide  9  by  6,  and  the  quotient  is  32  =  1 J.     The 
daily  work  of  each  mower  is,  therefore,  an  acre  and  a  half. 

III.  Multiply  l£  by  16;  the  product  is  24,  which 
is  therefore  the  daily  work  of  16  mowers. 

IV.  Multiply  24  by  3 :  the  product  is  72,  which  is 
the  work  of  16  mowers  in  3  days,  and  is  the  answer 
to  the  question. 

Example  11.  —  A  regiment,  consisting  of  9#9  men, 
consume  351  quarters  of  wheat  in  336  days  ;  how  many 
men  will  consume  1404  quarters  in  112  days? 

ANALYSIS.  —  Expressing  the  number  of  men  required 
by  x,  we  shall  have  the  following  statement :  — 
Soldiers      -  -     939     -        x 

Quarters  of  corn  351     -     1404 

Days  -  -     336     -        112. 

From  the  first  column  we  shall  find  the  quarters  of  corn 
consumed  by  one  man  in  336  days,  by  dividing  the  total 
number  of  quarters  consumed  by  the  number  of  soldiers  ; 
we  shall  next  find  the  daily  consumption  of  one  soldier 
by  dividing  the  quotient  last  found  by  the  number  of 
days,  336. 

Since  the  required  number  of  men  consume  1404 
quarters  in  112  days,  we  shall  find  their  daily  consump- 
tion by  dividing  1404  by  112. 

Having  found  the  daily  consumption  of  the  sought 
number  of  men,  we  shall  find  that  number  by  dividing 
it  by  the  daily  consumption  of  one  man. 

COMPUTATION. — I.  Divide  351  by  93$;  the  quotient 
is  ^f -i,  which,  reduced  to  its  least  terms,  becomes  ^rJ, 
which  is  the  fraction  of  a  quarter  consumed  by  one  man 
in  336  days. 

II.  To  find  the  fraction  consumed  by  one  man  in 
one  day,  we  must  divide  this  by  336,  which  is  done  by 
multiplying  its  denominator  by  that  number  (215.); 
the  result  is  -j^^VW.     This^  then,  is  the  fraction  of  & 
quarter  daily  consumed  by  a  single  soldier. 

III.  Divide  1404  by   112;  the  quotient  is 


316 


A    TREATISE    ON    ARITHMETIC. 


which  is,  therefore,  the  number  of  quarters  daily  con- 
sumed by  the  number  of  men  required. 

IV.  Divide  12£f  by  ToVrVs  ^  the  quotient  is  1 1268  *, 
which  is,  therefore,  the  number  of  men  required. 

(400.)  If  each  of  the  preceding  examples  be  atten- 
tively considered,  we  shall  perceive  that,  although  they 
differ  in  words,  and  in  the  particular  numbers  involved 
in  them,  they  are  still  identical  in  their  essential  parts,  and 
all  belong  to  the  same  class :  in  all  of  them  certain  agents 
are  contemplated,  which  are  employed  in  producing 
certain  effects,  in  a  certain  time.  In  each  there  are  two 
numbers  of  agents,  two  effects,  and  two  times,  con- 
sidered, so  that  six  quantities  or  numbers  enter  each 
question.  In  every  case  five  of  these  numbers  are  given, 
and  it  is  the  object  of  the  question  to  determine  the  sixth. 

In  the  Example  9-  the  agents  are  two  troops  of 
builders;  the  effects  proposed  are  certain  numbers  of 
feet  of  wall  built  by  these  troops ;  and  the  times  are  the 
numbers  of  days  which  the  troops  respectively  require  to 
produce  the  effects.  The  numbers  of  men  in  the  troops 
are  given ;  the  work  which  they  perform  is  also  given, 
as  well  as  the  number  of  days  in  which  one  of  them 
performs  the  work ;  while  the  number  of  days  in  which 
the  other  performs  it  is  sought.  The  data  here  are, 
therefore,  the  two  numbers  of  agents,  the  two  effects, 
and  the  time  of  performing  one  of  them ;  while  the 
sought  quantity  is  the  time  of  performing  the  other. 

In  Example  10.  two  bands  of  mowers,  the  number 
of  acres  they  mow  respectively,  and  the  number  of 
days  in  which  they  perform  the  work  are  considered. 
The  number  of  mowers  in  each  band,  the  number  of 
days  which  each  band  works,  and  the  number  of  acres 
which  one  band  mows,  are  severally  given ;  while  the 
number  of  acres  mown  by  the  other  band  is  sought.  In 
this  case,  therefore,  the  two  sets  of  agents,  the  times 
which  they  take  to  produce  the  effects,  and  the  effect 
produced  by  one  of  them,  are  given;  while  the  effect  pro- 
duced by  the  other  is  sought. 

»  The  exact  quotient  is  11267  a??6    being  less  than  11268  by  a  small 
fraction. 


CHAP.  II.  COMPOUND    RULE    OF    THREE.  Si 7 

In  Example  1 1 .  the  agents  are  soldiers  ;  the  effects, 
the  consumption  of  certain  numbers  of  quarters  of  corn  ; 
and  the  times,  the  days  in  which  these  quarters  are 
consumed.  In  this  case  the  sought  quantity  is  one  of 
the  numbers  of  soldiers  ;  the  effects,  times,  and  the 
other  number  of  soldiers  are  given. 

As  this  class  of  questions  is  very  extensive,  and  em- 
braces a  great  number  of  the  examples  which  are  usually 
given  for  exercise  in  the  Compound  Rule  of  Three,  it 
may  be  worth  while  here  to  investigate  a  general  rule 
for  the  solution  of  all  such  questions. 

Let  us  take  Example  10.  Supplying  the  place  of  the 
sought  quantity  by  72,  which  it  was  found  to  be  by  the 
computation,  the  statement  would  be  as  follows :  — 

Agents  -  14-16 
Effects  -  126  -  72 
Times  -  6  -  3; 

by  which  we  mean  to  express  that  14  agents  produce 
the  effect  126  in  the  time  6 ;  and,  also,  that  16  agents 
produce  the  effect  72  in  the  time  3.  If  we  divide  the 
effect  produced  by  the  first  number  of  agents  by  14,  we 
obtain  the  fraction  1T-46,  which  is  the  effect  produced  by 
a  single  agent  in  the  time  6*.  If  we  divide  this  again 
by  6,  which  is  done  by  multiplying  its  denominator  by 
6,  we  shall  find  the  effect  produced  by  1  agent  in  the 
time  1 :  this  effect  is,  therefore,  -^6-« 

In  like  manner,  from  the  second  column  we  find,  that 
the  work  performed  by  1  agent  in  3  days  is  ^-|,  and 
therefore,  the  work  performed  by  1  agent  in  1  day  is 

3X1  6* 

Since,  then,  the  fractions  -126   and     7^     both  ex- 

14X6  3X1 6 

press  the  effect  produced  by  1  agent  in  the  time  1,  they 
must  be  equal ;  and,  therefore,  the  numerator  is  to  the 
denominator  of  the  first  as  the  numerator  to  the  deno- 
minator of  the  second;  and  we  have  the  following 
proportion :  — 

126  :  14  x  6  =  72  :  3  x  16. 
In   this  proportion  the  product  of  the  means  is  equal 


318  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

to  the  product  of  the  extremes,  and,  therefore,  we 
have  — 

126x3  x  16  =  72  x  14x6. 

It  appears,  then,  that  the  continued  product  of  the 
first  and  last  numbers  of  the  first  column,  and  the  middle 
number  of  the  second,  is  equal  to  the  product  of  the 
first  and  last  numbers  of  the  second  column,  and  the 
middle  number  of  the  first.  The  same  reasoning  will 
show  that  this  will  always  be  the  case  in  questions  where 
agents ,  effects,  and  times  are  considered. 

The  manner  in  which  the  numbers  should  be  con- 
nected by  multiplication  may  always  be  shown  by  the 
following 

RULE. 

(401.)  First,  arrange  the  numbers  expressing  the 
agents,  effects,  and  times  in  two  columns,  as  already  ex. 
plained ;  then  transpose  the  numbers  expressing  the 
effects,  placing  the  first  in  the  second  column,  and  the 
second  in  the  first.  After  this  change,  the  product  of  the 
three  numbers  in  the  first  column  will  be  equal  to  the 
product  of  the  three  numbers  in  the  second  column. 

Thus,  in  the  preceding  example,  the  two  columns  will 
first  stand  thus:  — 

Agents  -  14-16 
Effects  -  126  -  72 
Days  -  6-3. 

Transposing  the  numbers  expressing  the  effects,  the 
arrangement  becomes  — 

Agents  -  14  -  16 
Effects  -  72  -  126 
Days  6  -  3. 

The  product  of  the  three  numbers  in  the  first  column 
is  now  equal  to  the  product  of  the  three  numbers  in  the 
second. 

(402.)  We  have  here  assumed  the  six  quantities  as 
all  known.  Now,  any  one  of  the  six  may  be  the  number 
sought  in  the  question,  while  the  other  five  are  given ; 


CHAP.  II.  COMPOUND    RULE    OF    THREE.  319 

and  the  method  by  which  such  sought  quantity  may  be 
found  is  perceivable  at  once,  by  considering  the  equality 
of  the  two  products  above.  In  the  second  question  from 
which  we  have  taken  the  numbers  used  in  this  illus- 
tration, the  quantity  sought  was  the  effect  produced  by 
the  second  set  of  agents.  We  know,  then,  by  what  has 
been  just  proved,  that  this  effect  must  be  expressed  by 
a  number  which,  if  multiplied  by  the  product  of  the 
two  given  numbers  in  the  first  column,  viz.  14  and  6, 
would  give  a  product  equal  to  the  continued  product  of 
the  three  numbers  in  the  second  column,  viz.  16,  126, 
and  3.  The  statement  would  be  as  follows : — 

Agents  -t  14  -  16 
Effects  -  x  -  126 
Days  -6-3. 

The  products  which  are  equal  are  as  follows :  — 
16  x  126x3  =  14x6  xx. 

Since  x  is  such  a  number  that,   if  multiplied  by 
1 4  X  6,  we  should  obtain  a  product  equal  to  16  X  126  x  3, 
it  follows  that  the  latter  number,  divided  by  14x6, 
should  give  a  quotient  which  will  be  the  number  go. 
To  find  x  we  have,  therefore,  only  to  find  the  continued 
product  of  16,  126,  and  3,  and  to  divide  it  by  14  x  6. 
This  process  is  expressed  thus :  — 
16  x  126  x  3 
14x6 

By  generalising  the  above  results,  we  shall  find  that 
all  questions  involving  the  consideration  of  agents, 
effects,  and  times, — provided  they  involve  nothing 
more,  —  may  be  solved  by  the  following 

RULE. 

(403.)  Write  in  two  columns,  as  usual,  the  agents, 
effects,  and  times,  transposing,  however,  the  place  of  the 
effects,  so  that  the  effects  of  each  set  of  agents  shall  be 
found  in  the  column  with  the  other  set.  Supply  the  place 
of  the  number  required  by  the  letter  x :  there  will  then 


320  A     TREATISE    ON    ARITHMETIC.  BOOK  IV. 

be  three  given  numbers  in  one  column,  and  two  in  the 
other.  Find  the  continued  product  of  the  former,  and 
divide  it  by  the  product  of  the  latter;  the  quotient  will  be 
the  number  sought. 

Example  12.  —  If  the  capital,  1001.,  invested  in  trade, 
gain  l6/.  in  12  months,  what  capital  would  gain  20/.  in 
1  9  months  ? 

ANALYSIS.  —  In  this  case  the  capital  is  the  agent,  and 
the  profit  the  effect.  According  to  the  above  rule,  the 
statement  would  be  as  follows  :  — 

Agents  -  £100  -  x 
Effects  -  £  20  -  16 
Times  -  12-19. 

COMPUTATION  :  — 

100  x  20  x  12       24OOO 

x  =  -  .  --  =  -  =  £78   18   1  li. 
16x19  304 

Example  13.  —  If  60  bushels  of  oats  are  consumed 
by  24  horses  in  40  days,  in  how  many  days  will  48  horses 
consume  30  bushels? 

ANALYSIS.  —  In  this  case  the  horses  are  the  agents, 
the  oats  consumed  the  effects,  and  the  number  of  days 
the  times.  According  to  the  rule,  the  statement  is  as 
follows  :  — 

Agents  -  24  -  48 
Effects  -  30  -  60 
Times  -  40  -  x. 

COMPUTATION  :  — 

24  x  30  x  40 


=       48x60 

(404.)  When  a  question  of  this  class  has  been  solved, 
and  the  six  numbers  engaged  in  it  all  become  known,  the 
teacher  can  form  out  of  them  six  distinct  questions,  in- 
cluding that  originally  proposed  ;  for  it  is  evident  that 
anyone  of  the  six  numbers  may  be  sought,  the  other  five 
being  given.  In  Example  10.,  taking  successively  as  the 
sought  number,  the  six  numbers  engaged  in  it,  in  the 
following  order:  —  14,  126,  6,  16,  72,  3,  we  should  have 
the  following  six  questions  :  — 


CHAP.  H.  COMPOUND    RULE    OF    THREE.  321 

QUESTION  I.  —  If  16  mowers  mow  72  acres  in  3 
days,  how  many  mowers  will  mow  126  acres  in  6  days? 
Answer  14. 

QUESTION  II.  —  If  16  mowers  mow  72  acres  in  3 
days,  how  many  acres  will  14  mowers  mow  in  6  days  ? 
Answer  126. 

QUESTION  III.  — If  16  mowers  mow  72  acres  in  3 
days,  how  many  days  will  be  required  for  14  mowers  to 
mow  126  acres  ?  Answer  6. 

QUESTION  IV.  —  If  14  mowers  mow  126  acres  in  6 
days,  how  many  mowers  would  mow  72  acres  in  3  days? 
Answer  16. 

QUESTION  V.  — If  14  mowers  mow  126  acres  in  6 
days,  how  many  acres  will  1 6  mowers  mow  in  3  days  ? 
Answer  72. 

QUESTION  VI.  —  If  14  mowers  mow  126  acres  in  6 
days,  how  many  days  will  be  required  for  1 6  mowers  to 
mow  72  acres  ?  Answer  3. 

By  observing  this,  the  teacher  will  have  the  advan- 
tage not  only  of  multiplying  numbers  for  the  exercise 
of  his  pupils,  but  also  of  making  the  results  of  differ- 
ent pupils  verify  each  other. 

(405.)  In  questions  of  this  kind,  when  the  numbers 
proposed  are  complex  numbers,  they  should  be  reduced 
to  simple  numbers  previously  to  the  calculation ;  and 
those  of  the  same  kind  should  always  be  reduced  to  the 
same  class  of  units.  Thus,  if  two  numbers  expressing 
time  enter  the  question,  they  should  be  both  expressed 
either  in  days  or  in  hours,  &c. ;  but  one  must  not  be 
expressed  in  days  while  the  other  is  expressed  in  hours  ; 
and  the  same  observation  will  extend  to  other  complex 
numbers. 

(406.)  Many  questions  may  be  brought  under  the 
preceding  rule,  although  the  consideration  of  time  may 
not  be  immediately  involved  in  them.  Sometimes  dis- 
tance travelled  over  is  considered;  in  which  cases  such 
distances  may  be  conceived  as  taking  the  places  of  the 
times  in  the  rule. 

Example  14.  —  A  carrier  transports  60  cwt.  to  the 

Y 


322  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

distance  of  20  miles  for  14/.  10*.  ;  what  weight  should 
lie  carry  30  miles  for  51.  8s.  $d.? 

ANALYSIS.  —  In  this  case  the  money  may  be  consi- 
dered as  the  effect  produced  by  the  transport,  the  goods 
as  the  agent,  and  the  distance  which  it  is  carried  as  the 
time,  since  it  is  evident  that,  if  the  rate  of  transport  be 
uniform,  we  might  substitute  the  time  for  the  distance. 
Thus,  if  the  transport  was  effected  at  the  rate  of  a  mile 
every  half  hour,  we  might  substitute  20  and  30  half 
hours,  instead  of  20  and  30  miles,  in  the  question.  The 
statement,  therefore,  according  to  the  rule,  will  be  as 
follows :  — 

Agents  60  x 

Effects     -    £14  10  0     ,£589 
Miles       -        20     -  30. 

It  will  be  necessary  to  reduce  the  two  sums  of  money 
to  the  same  denomination.  The  first  sum  is  equal  to 
290  shillings,  and  the  second  to  108  shillings  and  9 
pence ;  but  9  pence  being  f  of  a  shilling,  it  will  be  equal 
to  the  decimal  0'75:  therefore,  51.  8s.  9^.  =  108*75. 
Hence,  transposing  the  effects,  the  statement  becomes-— 

Agents     -       60 

Effects     -     108-75     290 

Miles       -       20  30 

COMPUTATION  :  — 

_  20  x  60  x  108-75  _ 

30~x"290 
The  number  required  is,  therefore,  1 5  cwt. 

(407.)  It  sometimes  happens  that  questions  of  this 
class  are  rendered  apparently  complex,  by  involving  in 
them  the  condition  that  the  effects  produced  by  the 
agents  shall  be  produced  at  so  many  hours  per  day, 
making  the  two  sets  of  agents  work  different  numbers 
of  hours  per  day.  This  complexity,  however,  may  im- 
mediately be  removed  by  previously  finding  the  total 
number  of  hours  which  each  set  of  agents  works,  and 
then  omitting  altogether  the  consideration  of  the  num- 
ber of  days.  Thus,  if  one  set  of  agents  works  10  days 


CHAP.  II.  COMPOUND    RULE    OP    THREE.  323 

at  6  hours  a  day,  while  the  other  set  works  8  days  at  10 
hours  a  day,  it  will  be  sufficient  to  state,  that  the  first 
set  of  agents  works  for  60  hours,  and  the  second  set  for 
80  hours.  Such  questions  may,  therefore,  be  reduced 
to  the  above  class,  provided  the  number  of  days  and 
hours  per  day  are  given  for  both  sets  of  agents. 

Thus,  Example  10.  might  have  been  announced  in 
the  following  manner  :  — 

If  14  mowers  mow  126  acres  in  6  days,  working  at 
the  rate  of  12  hours  per  day,  how  many  acres  would  be 
mown  by  ]  6  mowers,  working  for  4  days  at  the  rate  of 
9  hours  a  day  ? 

The  first  set  would,  in  this  case,  work  for  72  hours, 
and  the  second  set  for  36  hours,  and  the  statement 
would  be  made  as  follows  :  — 

Mowers      -     14     -      16 

16  x  126x36 

Acres          -     x     -    126    x  =  — =  72. 

14x72 

Hours         -     72     -      36. 

If  it  happen,  however,  that  in  such  a  question  the 
hours  per  day,  or  the  number  of  days,  be  the  number 
required,  we  cannot  adopt  this  method,  and  the  question 
will  belong  to  a  class  in  which  there  will  be  a  greater 
number  of  given  quantities  than  in  the  examples  already 
explained. 

Example  15.  — If  14  mowers  mow  126  acres  in  6 
days,  at  12  hours  per  day,  how  many  hours  a  day  must 
16  mowers  work  in  order  to  mow  72  acres  in  9  days? 

ANALYSIS.  —  The  statement  will  be  as  follows :  — 

Agents         -  14         -         16 

Effects          -  126         -         72 

Days             -  6-3 

Hours  per  day  12         -           x. 

We  shall  find,  from  the  first  column,  the  effect  which 
one  agent  can  produce  in  an  hour.  If  we  divide  126 
by  14,  we  shall  find  the  number  of  acres  mown  by  one 
mower  in  6  days  at  12  hours  a  day.  If  we  divide  this 
again  by  72,  or  by  6  X  12,  we  shall  find  the  quantity 
Y  2 


324"  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

mown  in  one  hour  :  this  will,  therefore,  be       1 


_ 

1  4  X  6  X  1  i 

Proceeding  in  the  same  way  with  the  second  column,  we 
shall  divide  72  by  16,  to  find  the  work  of  one  man  in 
the  whole  time,  and  again  divide  this  by  the  total  num- 
ber of  hours  ;  to  find  the  work  of  one  man  in  one 
hour.  Supposing,  then,  the  number  of  hours  per 
day  during  which  the  second  band  are  employed,  to  be 
expressed  by  x)  the  total  number  of  hours  would  be  9 
times  this  number,  and  would,  therefore,  be  expressed 
by  9  X  x.  The  work  of  one  man  in  one  hour  will  be 
found  by  dividing  ^|  by  9  x  x  ',  but  this  division  is 
made  by  multiplying  the  denominator  of  the  fraction  by 
the  multiplier.  Hence  the  work  of  one  man  in  one 
hour,  as  deduced  from  the  second  column,  would  be 
expressed  by  —  1-2  —  x.  Since  the  fractions  found  to  ex- 

3      1  6  X  0  X 

press  the  work  of  one  man  in  one  hour  from  each  column 
must  be  equal,  their  numerators  must  have  the  same 
ratio  to  their  denominators,  and  hence  we  have  the  fol- 
lowing proportion  :  — 

126  :  14  x  6  x  12  =  72  :  16  x  9  x  jr. 

The  product  of  the  means  must  be  equal  to  the  pro- 
duct of  the  extremes  :  therefore  we  have  — 

14  x  6  x  12  x  72  =  126  x  16  x  9  x  x. 

The  required  number  is,  therefore,  such  a  number 
that  if  it  were  multiplied  by  126  x  16  x  9>  it  would 
give  a  product  equal  to  14  x  6  X  12  x  72  ;  and, 
therefore,  the  latter  product,  divided  by  the  former, 
must  give  a  quotient  equal  to  at.  Hence  we  have  — 
14X  6  x  12  x  7*  _ 

126  x  72  x  9 

The  required  number  of  hours  per  day  is,  there- 
fore, 4. 

It  appears,  therefore,  that  to  questions  of  this  kind, 
the  rule  (403.)  already  explained  may  be  extended. 

RULE. 

(408.)    All  questions  in  which  the  effects  produced 


CHAP.  II.      COMPOUND  RULE  OP  THREE.          325 

by  agents  working  a  certain  number  of  days  and  hours 
per  day  are  considered,  may  be  solved  in  the  following 
manner  :  —  Place  the  numbers  expressing  the  agents, 
effects,  days,  and  hours  per  day  in  two  columns,  trans- 
posing,  however,  the  numbers  expressing  the  effects,  and 
using  x  to  express  the  number  sought.  There  will  then  be 
four  given  numbers  in  one  column  and  three  in  the  otJier. 
Divide  the  continued  product  of  the  former  by  the  con- 
tinued product  of  the  latter,  and  the  quotient  will  be  the 
number  required. 

Example  16.  —  If  a  troop  of  15  labourers,  working 
10  hours  a  day,  take  18  days  to  build  450  feet  of  wall, 
it  is  required  to  determine  how  many  workmen,  working 
for  12  hours  a  day,  would  build  480  feet  of  the  same 
wall  in  8  days  ? 

ANALYSIS.  —  Expressing  the  required  number  by 
x,  the  statement^  according  to  the  rule,  will  be  as  fol- 
lows :  — 

Agents         -         15         -         x 
Effects         -       480         -       450 
Days  18         -  8 

Hours  per  day      10         -         12 
COMPUTATION  :  — 

15x480x  18  x  10 


450x8  x  12 


=  30. 


Having  found  the  answer  to  such  a  question  as  this, 
the  teacher  can  frame  out  of  the  result  seven  other  ques- 
tions, by  taking  successively,  as  the  number  sought, 
each  of  the  numbers  which  are  given  in  the  above  ex- 
ample. Thus,  if  he  take  480  as  the  number  sought,, 
the  question  would  be  as  follows  :  — 

7/*15  labourers,  working  for  10  hours  a  day,  take  18 
days  to  build  450  feet  of  wall,  how  many  feet  of  wall 
will  30  labourers  build,  working  for  8  days  at  12  hours 
a  day  ?  Answer  480. 

Again,  if  12  be  taken  as  the  number  sought,  the 
question  would  be  as  follows  :  — 

If  15  labourers,  working  10  hours  a  day  for  18  days, 
build  450  feet  of  wall,  how  many  hours  a  day  must  30 
Y  3 


326  A    TREATISE    ON    ARITHMETIC.  BOOK  IV, 

labourers  work  in  order  to  build  480  feet  of  wall  in  8 
days? 

(409.)  In  the  last  two  examples  seven  numbers  are 
given  to  find  an  eighth.  Questions,  however,  of  still 
greater  complexity  may  be  proposed.  Although  such 
cases  rarely  happen,  either  in  commerce  or  science,  yet 
they  are  useful  as  arithmetical  exercises. 

Example  17.  —  500  men,  labouring  12  hours  a  day, 
are  engaged  for  57  days  in  cutting  a  canal,  which  mea- 
sures 1800  yards  in  length  by  7  in  breadth,  and  3  in 
depth :  it  is  required  to  know  how  many  days  will  be 
necessary  for  S60  men,  working  10  hours  a  day,  to 
cut  a  canal  measuring  2900  yards  in  length,  by  12  in 
breadth,  and  5  in  depth,  but  which  is  worked  through  a 
soil  which  is  3  times  more  difficult  than  the  former 


ANALYSIS.  —  This  question  is  resolved  into  the  fol- 
lowing statement :  — 

Men  500  860 

Length    -  1800  -        2900 

Breadth     -  7  12 

Depth  3  5 

Difficulty  1  3 

Days  57  x 

Hours  per  day    12  -  10 

In  this  case  the  effect  will  depend  conjointly  on  the 
length,  breadth,  and  depth  of  the  canal,  and  the  diffi- 
culty of  the  soil.  We  shall  find  the  number  of  cubic 
yards  to  be  cut,  by  each  set  of  men,  by  multiplying  the 
number  of  yards  in  the  length,  breadth,  and  depth 
(313.).  The  number,  therefore  cut  by  the  first 
troop,  will  be  1800  X  7  X  3,  and  by  the  second  troop 
2900  x  12  x  5.  But  the  second  troop  work  through 
a  soil  3  times  more  difficult,  which  will  cause  them, 
therefore,  3  times  as  much  labour,  and  the  effect 
will  be  the  same  as  if  they  had  cut  3  times  the 
quantity  of  soil  of  a  quality  like  that  through  which 
the  first  troop  worked.  We  shall,  therefore,  express 
the  labour  of  the  second  troop  by  estimating  it  at  3 


CHAP.  II.  COMPOUND    RULE    OF    THREE.  3%7 

times  as  many  cubic  yards  as  they  actually  cut :  their  la- 
bour, therefore,  will  be  expressed  by  2900  X  12  x  5  x  3. 
It  appears,  therefore,  that  the  effect,  in  this  case,  pro- 
duced by  each  troop  is  found  by  multiplying  together 
the  second,  third,  fourth,  and  fifth  numbers  in  their 
respective  columns;  so  that  the  statement  will  become  as 
follows :  — 

Men         -  500         -         860 

{Length      1800         -       2900 
Breadth     -7         -  12 

Depth      -       3         -  5 

Difficulty        1         -  3 

Days  57  x 

Hours  per  day     -         12         -  10 

But  by  rule  (408.)  we  must  transpose  the  effects : 
the  statement,  therefore,  becomes  — 

Men  500  860 

(Length  2900  -       1800 

Breadth  12  7 

Depth     -  5  -             3 

Difficulty  3  1 

Days  57  x 

Hours  per  day     -  12  -           10 

The  continued  product  of    all    the    numbers   in    the 
first  column,  divided  by  the  continued  product  of  those 
in  the  second  column,  will  give  a  quotient  which  will 
be  the  number  sought. 
COMPUTATION  :  — 

500  x  2900  x  12  x  5  x  3  x  57  x  12 

x  — _ =  549. 

860  x  1800  x  7  x  3  x  1  x  10 


328  A    TREATISE    ON    ARITHMETIC. 


CHAP.  III. 

INTEREST.  DISCOUNT.  —-PROFIT    AND    LOSS.  BROKERAGE.  

COMMISSION.  INSURANCE.  TARE    AND  TRET.  PARTNER- 
SHIP. 

(410.)  THERE  are  several  classes  of  arithmetical  pro- 
blems which  are  of  frequent  occurrence  in  commerce., 
and  which  fall  under  the  denominations  of  INTEREST, 
DISCOUNT,  PROFIT  AND  Loss,  BROKERAGE,  COMMISSION, 
INSURANCE,  TARE  AND  TRET,  &c.,  all  of  which  are  only 
appli cations  of  the  rule  of  three ;  and  the  principles  on 
which  they  are  solved,  are,  therefore,  fully  explained 
in  the  last  chapter.  Their  occurrence  in  the  common 
affairs  of  life  is,  however,  so  frequent,  and  the  calcula- 
tions depending  on  them  so  important  and  useful,  that 
they  are  generally  noticed  as  separate  commercial  rules. 
In  all  these  classes  of  problems,  the  object  is  generally 
to  compute  a  certain  small  fractional  part  of  a  gross 
amount,  whether  of  money  or  merchandise,  to  be  appro- 
priated to  some  specific  purpose,  and  this  part,  there- 
fore, always  increases  or  diminishes  in  proportion  to  the 
gross  amount  of  which  it  is  a  given  fraction. 

INTEREST. 

(411.)  When  a  sum  of  money  is  lent  by  one  indi- 
vidual to  another,  the  latter  pays  to  the  former  a  certain 
stipulated  sum  for  the  use  of  the  money  so  lent ;  this 
sum  is  called  INTEREST,  and  the  money  lent  is  called 
the  PRINCIPAL. 

The  amount  of  interest  is  computed  in  proportion  to 
the  amount  of  principal  lent,  and  to  the  time  it  continues 
in  the  hands  of  the  borrower.  Thus  the  interest  on 
a  given  sum  for  ten  years,  will  be  ten  times  the  interest 
on  the  same  sum  for  one  year,  and  the  interest  on  1 001. 
for  one  year  will  be  ten  times  the  interest  on  1 01.  for 
the  same  time.  - 


CHAP.  III.  INTEREST.  329 

Example  1. — If  the  interest  on  3571*  10s.  for  three 
years  be  531.  12s.  6d.}  what  will  be  the  interest  on 
681.  5s.  for  five  years  ? 

ANALYSIS.  —  The  agents  are,  here,  the  two  principals, 
the  effects  are  the  interests  given  and  the  interest  sought, 
and  the  times  are  3  and  5  years.  We  have,  there- 
fore, in  conformity  with  what  has  been  explained  in 
the  preceding  chapter,  the  following  statement :  — 

Agents         -       ,£357   10  0      £68  5  0 
Effects          -  53  12  6        -       x 

Years  -  3-5 

Transposing  the  effects,  and  converting  the  shillings  into 
decimals  of  a  pound,  the  statement  is  as  follows : — 
Agents         -         357-5         -         68-25 
Effects         -  v         x  -         53-625 

Years  3  -  5 

COMPUTATION. —  Multiplying  the  three  numbers  in 
the  second  column,  we  obtain  the  product  18299*53125. 
and  multiplying  the  first  and  third  in  the  first  column, 
we  obtain  the  product  1072*5.  Dividing  the  former 
by  the  latter,,  we  obtain  the  quotient  17*0625,  which  by 
converting  the  decimals  into  shillings  and  pence,  gives 
17/.  1*.  3d.,  which  is  the  interest  of  6'8/.  5*.  for  five 
years,  on  the  supposition  that  the  interest  on  357^.  10*. 
for  three  years  would  be  53L  12s.  6d. 

In  this  example  the  use  made  of  the  principal  357/. 
10s.,  and  its  interest  for  three  years,  is  merely  to  fix 
the  rate  of  interest  which  should  be  paid  for  the  prin- 
cipal 68/.  5s. ;  but  it  is  usual  in  practice  to  express  the 
rate  of  interest,  not  by  an  uneven  principal  or  an  uneven 
time,  but  always  by  the  principal  of  100/.  invested  for 
one  year  ;  and  consequently  all  questions  in  interest  in- 
volve the  consideration  of  the  interest  of  1 001.  for  a  year, 
which  is  therefore  called  invariably  the  rate  of  interest. 
Thus,  if  for  every  1001.  principal,  51.  be  paid  for  every 
year  it  is  lent,  we  say  that  the  rate  of  interest  is  five  per 
cent,  per  annum,  or  simply  five  per  cent.,  the  time  being 
always  understood  to  be  one  year. 

Example  2.  —  Let  it  be  required  io  determine  the  rate 


330  A    TREATISE    ON   ARITHMETIC.  BOOK  IV. 

of  interest  when  the  interest  on  3511.  10*.  for  three 
years  amounts  to  531.  12*.  6d. 

ANALYSIS.  —  The  question  here  proposed  is  to  deter- 
mine what  the  interest  on  100/.  is  for  1  year,  if  the 
interest  on  357 1.  10*.  be  531.  12*.  6d.  for  3  years. 
The  statement  is  as  follows :  — 

Agents        -       £357  10  0     £100 
Effects         -  53  12  6  * 

Years  -  3  1. 

Converting  the  sums  of  money  into  decimals  of  a  pound, 
and  transposing  the  effects,  the  statement  becomes  as 
follows :  — 

Agents         -         357-5         -         100 
Effects         -  x  53-625 

Years  3  1. 

COMPUTATION.  —  Multiplying  the  numbers  in  the 
second  column,  we  obtain  the  product  5362*5,  by  merely 
moving  the  decimal  point  two  places  to  the  right.  By 
multiplying  the  first  and  third  terms  of  the  first  column, 
we  obtain  the  product  1072*5.  Dividing  the  former  by 
the  latter,  we  obtain  the  quotient  5  :  the  rate  of  interest 
is,  therefore,  5  per  cent,  per  annum. 

Example  3.  —  What  is  the  interest  of  45007.  for  2 
years  and  5  months  at  7  per  cent  ? 

ANALYSIS.  —  The  statement  of  this  question  is  as 
follows :  — 

Agents         -         £100         ~     £4500 
Effects  7  x 

Time  -  1  -      2/  5in. 

Converting  the  times  into  simple  numbers,  by  reducing 
both  to  months,  and  transposing  the  effects,  the  state- 
ment becomes  — 

Agents         -         £100          -     £4500 
Effects  x  7 

Months        -  12  29. 

COMPUTATION.  —  The  product  of  the  numbers  in  the 
second  column  is  913500,  and  the  product  of  the  first 
and  third  in  the  first  column  is  1200.  Dividing  the 


CHAP.  HI.  INTEREST.  331 

former  by  the  latter,  the  quotient  is  ?6l*25,  which, 
converted  into  pounds  and  shillings,  is  ?6l/.  5s.,  which 
is  therefore  the  interest  required. 

(412.)  In  all  cases  where,  after  the  columns  are 
arranged  for  computation,  numbers  are  found  in  each 
column  which  are  divisible  by  the  same  number,  the 
computation  may  be  abridged  by  previously  dividing 
them  by  that  number.  In  the  preceding  example,  the 
first  terms  in  each  column  may  be  divided  by  100,  by 
omitting  the  ciphers. 

The  most  common  class  of  questions  in  interest  is 
that  in  which  it  is  required  to  find  the  total  amount  of 
interest  for  a  given  time,  at  a  given  rate,  and  for  a  given 
principal.  It  will  be,  therefore,  advantageous  to  in- 
vestigate a  general  rule  for  the  solution  of  every  such 
question. 

Let  us  suppose  that  the  principal,  whatever  it  may 
be,  is  expressed  by  P,  the  rate  of  interest  by  R,  and  the 
time  expressed  in  years  by  Y.     The  analysis  of  the 
question  would  then  lead  to  the  following  statement :  — 
Agents  100  -  P 

Effects  R  -  x 

Time  -  1  Y. 

Transposing  the  effects  this  would  become,  — 

Agents        -          100  -  P 

Effects  x  -  R 

Time  1  Y. 

We   should,    accordingly,  by  what  has  been   already 

explained,  discover  the  amount  of  interest  sought,  which 

is  here  expressed  by  ae,  by  obtaining  the  continued  pro. 

duct  of  the  numbers  here  expressed  by  P,  R,  and  Y 

and  dividing  that  product  by  100  :  hence  the  following 

RULE  I. 

(413.)  To  find  the  interest  of  any  principal  for  any 
number  of  years  at  a  given  rate  of  interest,,  multiply  the 
principal  by  the  rate,  and  the  product  by  the  number  of 
years:  divide  the  product  thus  found  by  100,  and  the 
quotient  will  be  the  interest  sought. 


332  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

The  principal  and  rate,  if  they  are  not  whole  num- 
bers of  pounds,  should,  in  this  case,  be  expressed  as  deci- 
mals of  a  pound ;  and  if  the  time  be  not  a  complete  num- 
ber of  years,  it  should  be  expressed  in  decimals  of  a  year. 
The  method  of  reducing  sums  of  money  to  decimals  of 
a  pound  has  been  already  explained.  If  the  time  be 
not  a  round  number  of  years,  it  will  generally  be1  ex- 
pressed in  years,  months,  and  days.  In  such  cases  it  is 
sometimes  more  convenient  to  express  the  time  in  months 
or  days  only  than  in  decimals  of  a  year ;  but,  in  that 
case,  it  would  be  necessary  also  to  express  in  the  same 
denomination,  that  is,  in  months  or  days,  the  time  which 
is  expressed  by  1  in  the  first  column  :  the  statement 
would,  therefore,  be  modified  in  the  following  manner 
for  months: — 

Agents  -  100  P 

:  Effects  -  x  -         R 

Time  12          -         M. 

In  this  case  M  stands  for  the  number  of  months  for 
which  the  interest  is  to  be  calculated :  hence  we  would 
obtain  the  following 

RULE  II. 

(414.)  To  compute  the  total  interest  on  any  given 
principal,  at  a  given  r ate }  for  a  given  number  of  months, 
multiply  the  principal  by  the  rate,  and  the  product  thus 
obtained  by  the  number  of  months,  and  divide  the  number 
thus  found  by  1200  :  the  quotient  will  be  the  amount  of 
interest  sought. 

If  the  time  be  expressed  in  years  and  months,  the 
interest  may  either  be  calculated  separately  for  the  years 
and  months,  by  Rules  I.  and  II.,  or  the  whole  time  may 
be  reduced  to  months,  and  the  calculation  made  by 
Rule  II.  alone.  If  the  time  be  expressed  in  days,  the 
statement  must  be  again  modified  thus  :  — 

Agents           -  100  -  P 

Effects  x  -  R 

Time              -  365  -  D. 
and  we  obtain  the  following 


CHAP.  III.  INTEREST. 

RULE  III. 

(41  5.)  To  find  the  interest  of  any  given  principal,  at 
any  given  rate,  for  any  given  number  of  days,  multiply 
the  principal  by  the  rate,  and  the  product  by  the  number 
of  days,  and  divide  the  number  thus  found  by  36500. 

If  the  time  be  expressed  in  years,  months,,  and  days, 
the  interest  for  the  years  may  be  calculated  by  Rule  I., 
for  months  by  Rule  II.,  and  for  the  days  by  Rule  III., 
and  the  sums  thus  obtained  added  together  ;  or  the  yeais 
and  months  may  be  reduced  to  months,  and  the  interest 
for  them  calculated  by  Rule  II.  ;  or  the  months  and 
days  may  be  reduced  to  days,  and  the  interest  for  them 
calculated  by  Rule  III.  In  the  selection  of  the  methods 
to  be  practised,  the  computist  must  be  guided  by  the 
peculiar  circumstances  of  the  question. 

Example  4  --  Find  the  interest  of  462  /.  for  85  days, 
at  5  per  cent,  by  Rule  III. 

462  x  5  x  85 


Example  5.  —  What  is  the  amount  of  interest  on 
751.  Ss.  6d.  at  4|  per  cent,  for  4  years,  7  months,  and 
27  days  ? 

ANALYSIS.  —  Since  the  calendar  months  have  unequal 
lengths,  it  would  be  necessary,  in  order  to  make  this 
computation,  to  know  the  time  of  the  year  at  which  the 
interest  begins  to  accrue  ;  but  we  shall,  in  the  present 
case,  consider  each  month  as  a  twelfth  part  of  the  year, 
and  make  the  computation  by  the  rules  already  given. 
The  principal,  reduced  to  decimals  of  apound,is75'425/.: 
the  rate  expressed  in  decimals  is  4*5  ;  we  have,  then.,  — 

£  £   s.    d. 

Interest  for   4")      75-425x4-5x4 
years        -        )  =        ~  [^  --  =  13-5765  =  13  H   6J 

Interest   for   7"!      75-425x4-5x7 

months      -    )=       --    --  =  7*97999  =  *   19 


£15  16 


Calculations  of  interest  are,  however,  made  with  much 


334  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

greater  expedition  and  accuracy  by  the  aid  of  tables  of 
interest^  which  tables  themselves,  however,  have  been 
previously  calculated  by  the  methods  just  explained. 
We  shall  not  here  enter  into  further  details  than  to  ex- 
plain the  method  of  using  such  tables.  The  following 
table  exhibits  the  interest  on  11.,  in  decimals  cf  a  pound, 
for  any  number  of  years  up  to  a  certain  limit,  at  the 
several  rates  of  3  per  cent.,  3 £  per  cent.,  4  per  cent., 
4£  per  cent.,  and  5  per  cent. 

TABLE  I.  —  A  TABLE  OF  SIMPLE  INTEREST. 

THE  INTEREST  OF  ONE  POUND  FOR  A  NUMBER  OF  YEARS. 


5 

s 

At  3  per 
cent. 

3i  per  cent 

4  per 
cent. 

4i  per  cent 

5  per  cent 

1 

i 

•03 

•035 

•04 

•045 

•06 

1 

2 

•06 

•07 

•08 

•09 

•1 

2 

3 

•09 

•105 

•12 

•135 

•15 

3 

4 

•12 

•14 

•16 

•18 

•2 

4 

5 

•15 

•175 

-2 

•225 

•25 

5 

6 

•18 

•21 

•24 

•27 

•3 

6 

7 

•21 

•245 

•28 

•315 

•35 

7 

8 

•24 

•28 

•32 

•36 

•4 

8 

9 

•27 

•315 

•36 

•405 

•45 

9 

10 

•3 

•35 

•4 

•45 

•5 

10 

11 

•33 

•385 

•44 

•495 

•55 

11 

12 

•36 

•42 

•48 

•54 

•6 

12 

13 

•39 

•455 

•52 

•585 

•65 

13 

14 

•42 

•49 

•56 

•63 

•7 

14 

15 

•45 

•525 

•6 

•675 

•75 

15 

16 

•48 

•56 

•64 

•72 

•8 

16 

17 

•51 

•595 

•68 

•765 

•85 

17 

18 

•54 

•63 

•72 

•81 

•9 

18 

19 

•57 

•665 

•76 

•855 

•95 

19 

20 

•6 

•7 

•8 

•9 

!• 

20 

21 

•63 

•735 

•84 

•945 

1-05 

21 

22 

•66 

•77 

•88 

•99 

1-1 

22 

23 

•69 

•805 

•92 

1  -035 

1-15 

23 

24 

•72 

•84 

•96 

1-08 

1-2 

24 

25 

•75 

•875 

1- 

1-125 

1-25 

25 

CHAP.  III.  INTEREST.  335 

The  following  is  part  of  a  table  which  gives  the 
interest  on  11.,  in  like  manner,  for  any  numher  of  days 
up  to  a  certain  limit. 

TABLE  II.  —  A  TABLE  OP  SIMPLE  INTEREST. 

THE    INTEREST    OF    ONE    POUND    FOR    ANY  NUMBER  OF  DAYS,  ETC. 


ri 

& 

3  per  cent. 

3£  per  cent. 

4  per  cent. 

4i  per  cent 

5  per  cent. 

£ 

Q 

i 

c 

G 

4 
^ 

6 

8 
9 

10 

•0000,821 
•0001,641 
•0002,465 
•0003,287 
•0004,109 

•0000,958 
•0001,916 
•0002,876 
•0003,835 
•0004,794 

•0001,095 
•0002,191 
•0003,287 
•0004,383 
•0005,479 

•0001,232 
•0002,465 
•0003,698 
•0004,931 
•0006,164 

•0001,369 
•0002,739 
•0004,109 
•0005,479 
•0006,849 

1 
2 

g 

4 
£ 

•0004,931 
•0005,753 
•0006,575 
•0007,397 
•0008,219 

•0005,753 
•0006,712 
•0007,671 
•0008,630 
•0009,589 

•0006,575 
•0007,671 
•0008,767 
•0009,863 
•0010,958 

•0007,397 
•0008,630 
•0009,863 
•0011,095 
•0012,328 

•0008,219 
•0009,589 
•0010,958 
•0012,328 
•0013,698 

6 

7 
8 
9 

10 

11 
12 
13 
14 
15 

•0009,041 
•0009,863 
•0010,684 
•0011,506 
•0012,328 

•0010,547 
•0011,506 
•0012,465 
•0013,424 
•0014,383 

•0012,054 
•0013,150 
•0014,246 
•0015,342 
•0016,438 

•0013,561 
•0014,794 
•0016,027 
•0017,260 
•0018,493 

•0015,068 
•0016,438 
•0017,808 
•0019,178 
•0020,547 

11 
12 
13 
14 
15 

16 
17 
18 
19 

20 

•0013,150 
•0013,972 
•0014,794 
•0015,616 
•0016,438 

•0015,342 
•0016,301 
•0017,260 
•0018,219 
•0019,178 

•0017,530 
•0018,630 
•0019,726 
•0020,821 
•0021,917 

•0019,726 
•0020,958 
•0022,191 
•0023,424 
•0024,657 

•0021,917 
•0023,287 
•0024,657 
•0026,027 
•0027,397 

16 
17 
18 
19 
20 

21 

22 
23 
24 
25 

•0017,260 
•0018,082 
•0018,904 
•0019,726 
•0020,547 

•0020,137 
•0021,095 
•0022,054 
•0023,013 
•0023,972 

•0023,013 
•0024,109 
•0025,205 
•0026,301 
•0027,397 

•0025,890 
•0027,123 
•0028,356 
•0029,580 
•0030,821 

•0028,767 
•0030,137 
•0031,806 
•0032,876 
•0034,246 

21 

22 
23 
24 
25 

26 
27 
28 
29 
30 

•0021,369 
•0022,191 
•0023,013 
•0023,835 
•0024,657 

•0024,931 
•0025,890 
•0026,849 
•0027,808 
•0028,767 

•0028,493 
•0029,589 
•0030,684 
•0031,780 
•0032,876 

•0032,054 
•0033,287 
•0034,520 
•0035,753 
•0036,986 

•0035,616 
•0036,986 
•0038,556 
0039,726 
0041,095 

26 
27 
28 
29 
30 

336  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

When  the  interest  of  any  sum  of  money  is  required 
at  a  given  rate,  for  any  given  number  of  years,  look  in 
the  first  table  for  the  given  number  of  years  in  the  first 
or  last  column,,  and,  in  the  same  horizontal  line  with 
that  number,  you  will  find  under  the  given  rate  of 
interest  the  decimal  of  a  pound,  which  expresses  the 
interest  of  II.  for  the  required  number  of  years.  If 
this  decimal  be  multiplied  by  the  principal,  the  product 
will  be  the  interest  required. 

Example  6.  —  What  is  the  interest  on  381.,  at  3J  per 
cent.,  for  17  years? 

In  Table  I.,  opposite  17,  and  under  the  column  of  3£ 
per  cent.,  we  find  the  decimal  *595,  which  is  the  interest 
upon  II.  for  17  years:  we  multiply  this  by  38,  and  the 
product  is  22'6l/.  =  22/.  12*.  2$d. 

Example  7«  —  To  find  the  interest  on  51.  12s.  6d.,  at 
4  per  cent.,  for  23  years  and  6  months. 

In  Table  I.,  opposite  23  years,  and  in  the  column 
under  4  per  cent.,  we  find  the  decimal  '92,  which  is  the 
interest  on  11.  for  23  years:  opposite  1  year  we  find 
the  decimal  '04,  which  is  the  interest  upon  11.  for 
1  year ;  half  of  the  latter,  which  is  0'2,  is,  therefore, 
the  interest  on  II.  for  6  months ;  hence,  the  interest  on 
II.  for  23  years  and  6  months  is  '94.  Multiplying 
this  by  the  principal,  which  reduced  to  decimals,  is 
5-625,  we  obtain  the  product  5'3875/.  =  5/.  7*.  9^d., 
which  is  the  interest  required. 

Example  8.  —  Find  the  interest  on  75/.  8s.  6d.  for 
4  years,  7  months,  and  27  days,  at  4*%  per  cent. 

In  the  Table  I.  above,  opposite  4  years,  in  the  column 
under  4J  per  cent.,  we  find  the  decimal  -18,  which  is 
the  interest  upon  11.  for  4  years:  we  shall  find  the 
interest  on  11.  for  6  months  by  taking  half  the  interest 
on  11.  for  1  year ;  this  appears,  by  the  same  table,  to  be 
•045,  which,  divided  by  2,  gives  '0225,  which  is,  there- 
fore, the  interest  on  11.  for  6'  months  :  the  sixth  part  of 
this  is  -00375,  which  is  the  interest  for  1  month  ; 
adding  this  to  the  interest  for  6  months,  we  find  the 
interest  for  7  months  to  be  -02625.  To  find  the  in- 


CHAP.  III.  INTEREST.  337 

terest  for  27  days  we  refer  to  Table  II.,  and  opposite 
27  days,  and  under  the  column  headed  4J  per  cent.,  we 
find  the  decimal  -0033287.  Add  together  the  amounts 
of  interest  thus  found  for  4  years,  7  months,  and  27 
days. 

Interest  for  4  years       =  '18 
Interest  for  7  months    =  -02625 
Interest  for  27  days       =  -0033287 


Total  interest  on  £1     ~     -2095787 


Multiplying  this  interest  by  the  principal  75*425,  and 
omitting  all  the  digits  after  the  fourth  decimal  place  in 
the  product,  we  obtain  15'807/.  =  15J.  l6s.  2d.,  which 
is  the  interest  sought. 

(41 6.)  If  the  lender  instead  of  receiving  from  the 
borrower  the  interest  accruing  due  upon  his  principal 
from  year  to  year,  leaves  that  interest  in  the  hands  of 
the  borrower,  it  may  be  regarded  as  so  much  added  to 
Ihe  principal,  at  the  times  at  which  it  falls  due.  Thus, 
if  1001.  be  the  sum  lent,  and  the  interest  be  payable 
yearly  at  5  per  cent.,  then  at  the  end  of  the  first  year 
the  principal  will  become  105/.,  and  the  interest  at  the 
end  of  the  second  year  will  therefore  be,  not  the  in- 
terest on  100L,  but  the  interest  on  1051.  In  this  man- 
ner the  principal  which  produces  the  interest  each  year, 
is  increased  by  the  amount  of  the  interest  of  the  pre- 
ceding year. 

When  interest  is  thus  chargeable  upon  interest,  the 
increase  upon  the  original  principal  is  called  COMPOUND 
INTEREST. 

To  find  the  interest  on  any  principal  invested  at 
compound  interest  for  any  number  of  years,  it  is  neces- 
sary to  find  the  amount  of  the  principal  and  interest  at 
the  end  of  each  year,  and  considering  this  as  a  new 
principal,  compute  the  interest  upon  it  for  the  next 
year. 

Example  9.  —  To  find  the  compound  interest  .for  5 
years  on  1201.  at  5  per  cent.,  the  calculation  would  be  as 
follows :  — 

z 


338  A    TREATISE    ON    ARITHMETIC.  BOOK  IV, 


First  year's  principal 
First  year's  interest 

Second  year's  principal    - 
Second  year's  interest 

Third  year's  principal 
Third  year's  interest 

Fourth  year's  principal     - 
Fourth  year's  interest 

Fifth  year's  principal 
Fifth  year's  interest 

Amount  at  the  end  of  "1 
fifth  year           -          J 

£120    0     0 
600 

126     0     0 
660 

132     6     O 
6  12     3§ 

138   18     SJ 
6  18  10£ 

145  17     2 
7     5  10 

£153     3     0 

Example  10.  —  To  find  the  time  in  which  a  principal 
sum  will  be  doubled  at  Compound  Interest  at  5  per  cent. 

Let  us  suppose  the  principal  to  be  1001. ;  the  compu- 
tation must  be  continued  until  a  principal  be  obtained 
amounting  to  2001. 

First  year's  principal         -      £100    0     0 
First  year's  interest  -  500 

Second  year's  principal      • 
Second  year's  interest 

Third  year's  principal 
Third  year's  interest 

Fourth  year's  principal 
Fourth  year's  interest 

Fifth  year's  principal 
Fifth  year's  interest 

Sixth  year's  principal 
Sixth  year's  interest 

Seventh  year's  principal     - 
Seventh's  year's  interest    • 

Eighth  year's  principal 
Eighth  year's  interest 

Ninth  year's  principal 
Ninth  year's  interest 


CHAP.  III.  INTEREST. 

Tenth  year's  principal        -         155     2     9 
Tenth  year's  interest  -  7  15     2 

Eleventh  year's  principal   -         162  17  11 
Eleventh  year's  interest     -  8211 

Twelfth  year's  principal     -         171     0  10 
Twelfth  year's  interest        -  8  ]  1     0 

Thirteenth  year's  principal 
Thirteenth  year's  interest 

Fourteenth  year's  principal 
Fourteenth  year's  interest 

Fifteenth  year's  principal 
Fifteenth  year's  interest 

Amount  at  the  end  of  \     ^207  18   10 
15  years  J 

From  this  calculation  it  appears  that  the  principal 
will  he  more  than  douhled  in  15  years.  If  it  he  re- 
quired to  find  the  period  in  which  it  will  be  exactly 
doubled,  we  have  only  to  find  the  number  of  days  which 
197^.  19$«  lOrf.  must  be  placed  at  interest  to  produce 
2/.  Os.  %d.,  or,  what  will  be  nearly  the  same,  how  many 
days  198/.  must  be  placed  at  interest  to  produce  2/. 
To  solve  this  question  we  would  have  the  following 
statement,  transposing  the  effects  as  usual,  — 

Agents         -         100         -         198 
Effects        -  2-5 

Days  -          365         -  x. 

Multiplying  the  numbers  in  the  first  column  we  obtain 
the  product  73000,  and  dividing  this  by  the  product  of 
the  first  two  numbers  in  the  second  column  we  obtain 
the  quotient  74.  The  principal  would,  therefore,  be 
doubled  in  14  years  and  74  days,  at  compound  interest. 
In  the  above  calculation  small  fractions  have  been 
omitted. 

When  the  principal  is  large  and  the  time  considerable, 

the  computations  at  compound  interest  are   generally 

complex  and  embarrassing.     By  the  aid  of  the  higher 

mathematics  and  logarithmic  tables  more  compendious 

z  2 


340  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

methods  may  be  obtained;  but,  the  most  convenient 
and  expeditious  means  of  calculation  are  tables  of  com- 
pound interest,  by  which  all  such  problems  may  be 
solved  with  very  little  or  no  calculation.  , 

DISCOUNT. 

(417-)  When  a  payment  is  made  at  any  time  previous 
to  the  date  at  which  it  falls  due,  the  party  thus  antici- 
pating the  payment  is  entitled  to  deduct  from  the  amount 
to  be  paid  a  certain  sum,  in  consideration  of  the  interest 
which  would  accrue  on  the  money  paid,  between  the 
time  of  the  actual  payment  and  the  time  at  which  the 
amount  would  fall  due.  This  deduction  or  abatement 
is  called  Discount. 

Thus,  if  it  be  required  to  ascertain  the  present  value 
of  a  bill  of  exchange  for  a  certain  amount,  which  will 
fall  due  at  a  future  period,  it  will  be  necessary  to  deduct 
from  the  actual  amount  for  which  the  bill  is  drawn  the 
amount  of  discount. 

Again,  if  it  be  proposed  to  pay  a  tradesman  in  cash, 
for  goods  for  which  it  is  the  custom  of  the  trade  to  give 
a  certain  length  of  credit,  the  buyer  is  entitled  to  dis- 
count, proportionate  to  the  amount  of  the  account  and 
the  length  of  credit. 

The  method  of  computing  discount  will  be  easily  de- 
duced from  an  example. 

Example  1. — A  merchant  presents  a  bill  of  exchange 
for  30001.  payable  at  the  end  of  one  year  to  his  banker 
to  be  discounted.  It  is  required  to  know  what  discount 
the  banker  should  deduct  from  the  amount  of  the  bill? 

ANALYSIS.  —  It  is  evident  that  the  question  here  to 
be  determined  is,  what  is  that  sum  of  money  which 
being  now  placed  at  interest  would,  at  the  end  of  one 
year,  be  worth  3000/.  ?  for  such  is  the  sum  which  the 
banker  ought  to  pay  to  the  merchant ;  and  the  difference 
between  this  sum,  whatever  it  be,  and  3000/.  is  the 
amount  of  the  discount.  To  solve  this  question,  it  is 
therefore  necessary  that  the  rate  of  interest  should  be 


CHAP.   III.  DISCOUNT.  341 

previously  settled:  suppose  this  to  be  6  per  cent.  A 
present  sum  of  1001.  would,  at  this  rate,  increase  to  the 
amount  of  I06l.  at  the  end  of  one  year:  the  question, 
therefore,  is,  if  1001.  at  the  end  of  one  year  become  106/. 
what  sum,  at  the  end  of  one  year,  would  become  SOOOL  ? 
The  statement  would  be  thus : — 

£106  :  £3000  =  £100  :  x. 

COMPUTATION.  —  To  find  the  amount  represented  by 
x,  multiply  the  amount  of  the  bill  by  100,  and  divide 
the  product  by  106 :  the  quotient  will,  in  this  case,  be 
28301.  3s.  9^d.,  which  is  the  sum  which  the  banker 
should  pay  to  the  merchant  for  the  bill ;  the  difference 
between  this  and  30001.,  which  is  l6$l.  l6s.  2±d., 
gives  the  amount  of  the  discount. 

The  above  analysis  has  led  us  to  compute  the  sum  to 
be  paid  by  the  merchant  to  the  banker  ;  but  it  is  more 
convenient  and  usual  to  compute,  in  the  first  instance 
the  amount  of  the  discount,  which  is  easily  done. 

ANALYSIS.  —  If  a  bill  for  106/.  was  presented  to  the 
banker,  it  is  clear  that  the  sum  which  he  would  be  en- 
titled to  deduct  from  it  for  discount  would  be  6l.,  in- 
asmuch as  1001.  paid  at  present,  would,  at  the  end  of  a 
year,  be  worth  106/.  The  question  therefore  is,  if  106L 
gives  a  discount  of  6l.  for  one  year,  what  discount  will  be 
given  for  one  year  by  3000/.  ?  which  leads  to  the  fol- 
lowing statement :  — 

£106  :  6  =  £3000  :  x. 
where  x  expresses  the  discount  required. 

COMPUTATION.  —  To  find  x,  multiply  3000  by  6, 
and  divide  the  product  by  106  :  the  quotient  will  be 
l69/.  l6s.  2^d.,  which  is  the  discount  sought. 

In  the  preceding  example,  the  time  for  which  the 
discount  is  computed  is  one  year,  a  circumstance  which 
renders  the  analysis  more  simple  than  if  any  other  time 
had  been  supposed.  We  shall  now  take  a  more  general 
case. 

Example  2.  —  To  find  the  discount  to  be  deducted 
z  3 


342  A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 

for  the  present  payment  of  48 50/.,  payable  in  13  years 
6'  months,  the  rate  of  interest  being  5  per  cent,  per  annum. 
ANALYSIS.  —  In  this  case,  the  sum  which  would  pro- 
duce a  discount  of  5  per  cent,  in  one  year  would  be 
105/. :  the  statement  will  then  be  as  follows  :  — 

Agents    -      -          105         -         4850 
Effects  5  -  x 

Time  -  1*         -         13y  6m. 

Transposing  the  effects  and  reducing  the  amounts  to 
decimals  of  a  year,  the  statement  becomes  — 

Agents          -          105         -         4850 
Effects  x  -  5 

Time  -  V         -       13-5y. 

Multiply  the  three  numbers  in  the  second  column,  and 
find  their  continued  product ;  divide  this  by  105,  and 
the  quotient  will  be  3 1 1 7 '9521.  =  3 1 1 H  1 9s.  0%d.  which 
is  the  discount  sought.  In  this  process  we  have  mul- 
tiplied the  principal  by  the  rate  of  interest  and  mul- 
tiplied the  product  thus  obtained  by  the  time  in  years, 
and  decimals  of  a  year.  The  product  thus  found  is 
then  divided  by  105,  which  is  the  number  found  by 
adding  the  rate  of  interest  to  100 :  hence  we  obtain  the 
following 

RULE. 

(418.)  To  find  the  discount  to  be  allowed  for  the  pre- 
sent payment  of  a  given  sum,  due  at  a  future  time,  mul- 
tiply the  given  sum  by  the  rate  of  interest,  and  multiply 
the  product  thus  obtained  by  the  time  from  the  present 
date  until  the  date  at  which  the  sum  falls  due,  this  time 
being  expressed  in  years  and  decimals  of  a  year :  divide 
the  continued  product  thus  obtained  by  the  number  which 
you  will  find  by  adding  the  rate  of  interest  to  100;  the 
quotient  will  be  the  discount  sought. 

It  is  sometimes  more  convenient  to  express  the  time 
in  months  or  days,  than  in  decimals  of  a  year. 

Example  3.  —  To  find  the  discount  on  a  bill  for 
251.  12s.  6d.  due  in  1  year  and  5  months  from  the 
present  time,  the  rate  of  interest  being  5  per  cent. 


CHAP.  III.  DISCOUNT.  543 

ANALYSIS.  —  The  statement  will  be  as  follows,  reduc- 
ing the  time  to  months  :  — 

Agents       .       105       -      £25  12     6 
Effects  5  x 

Months      -       12        -  17. 

Transposing  the  effects,  and  converting  the  shillings  and 
pence  into  decimals  of  a  pound,  the  statement  becomes — 

Agents         -         105        -         25-625 
Effects  x  5 

Months       -          12         -  17. 

Multiplying  the  numbers  in  the  second  column,  and  di- 
viding their  continued  product  by  the  product  of  105 
and  12  in  the  first  column,  we  obtain  the  quotient 
=£1 '728  =  17.  14s.  6|oL,  which  is  the  discount  sought. 
From  this  example  we  derive  the  following 

RULE. 

(419.)  If  the  time  be  expressed  in  months,  multiply 
the  amount  to  be  paid  by  the  rate  of  interest,  and  mul- 
tiply the  product  by  the  number  of  months  from  the  pre- 
sent time  to  the  time  it  will  be  due :  divide  the  product 
thus  found  by  the  product  which  is  found  by  multiplying 
by  12  the  number  obtained  by  adding  the  rate  of  interest 
to  100. 

If  the  time  be  expressed  in  days,  there  will  be  no 
other  difference  in  the  statement,  except  that  the  third 
term  in  the  first  column  will  then  be  3^5,  and  we  shall 
have  the  following 

RULE. 

(420.)  When  the  time  is  expressed  in  days,  multiply 
as  before,  the  amount  to  be  discounted  by  the  rate  of  in- 
terest, and  the  product  by  the  number  of  days  ;  the  pro- 
duct thus  found  must  then  be  divided  by  another  product, 
which  will  be  found  by  multiplying  365  by  The  number 
obtained  by  adding  the  rate  of  interest  to  100. 

The  above  are  the  fair  principles  upon  which  all  dis- 
z  4 


344  A    TREATISE    ON    ARITHMB    1C.  BOOK  IV. 

count  should  be  computed.  It  has  been,  however,  cus- 
tomary in  this  country,  in  commercial  business,  instead 
of  considering  discount  under  this  point  of  view,  to  com- 
pute it  merely  as  the  interest  which  the  whole  amount 
to  be  discounted  would  produce,  from  the  present  time 
to  the  time  at  which  the  amount  falls  due.  Thus  the 
discount  on  a  bill  for  100/.,  payable  in  a  year,  at  5 
per  cent,  interest,  would  be  considered  to  be  51. ,  whereas 
51.  is,  in  truth,  the  discount  on  a  bill  for  1051.  payable 
in  a  year. 

According  to  this  estabh'shed  custom,  discount  would 
be  calculated  by  the  following 

RULE. 

(421 .)  Find  the  interest  on  the  sum  to  be  discounted, 
from  the  day  on  which  it  is  discounted  to  the  day  on 
which  it  becomes  payable,  and  this  interest  will  be  the 
discount. 

PROFIT  AND  Loss. 

(422.)  When  money  is  placed  at  interest  it  is  always 
supposed  that  the  principal  is  secured  to  its  owner,  and 
that  he  enters  into  no  speculation  upon  it.  When  a  sum 
of  money  is  invested  in  any  speculation,  it  is  called 
CAPITAL,  and  the  increase  which  it  receives  from  year 
to  year  is  called  PROFIT.  Profit,  like  interest,  is  calcu- 
lated at  so  much  on  every  100/.  in  a  year,  or  so  much 
per  cent,  per  annum,  supposing  the  profit  from  year  to 
year  to  be  uniform.  The  same  methods  of  calculation 
will  be  applicable  to  profit  on  capital  as  have  been  already 
explained  for  computing  interest  on  principal. 

To  compute  the  rate  of  profit,  a  merchant  will  esti- 
mate his  capital  and  stock  at  the  beginning  and  end  of 
the  year,  and  he  will  subtract  the  amount  at  the  begin- 
ning from  the  amount  at  the  end;  the  remainder  will  be 
the  total  profit.  To  find  the  profit  per  cent,  it  will  be 
only  necessary  to  annex  two  ciphers  to  the  total  profit, 
and  divide  the  number  thus  obtained  by  the  capital  at 
the  beginning  of  the  year. 


CHAP.  III.      PROFIT BROKERAGE INSURANCE.       345 

BROKERAGE,  COMMISSION,  &c. 

(423.)  BROKERAGE,  COMMISSION,  &c.  are  allowances 
of  so  much  per  cent,  which  are  usually  made  to  factors 
or  agents,  employed  by  merchants  to  effect  sales.  In 
such  computations  time  does  not  enter  as  an  element, 
and  the  problem  is  merely  confined  to  determine  a  given 
fraction  of  a  sum  of  money. 

Example  1.  —  To  find  the  brokerage  or  commission  on 
sales  amounting  to  53QL  14*.  at  2J  per  cent. 

The  question  here  to  be  solved  is,  if  WOL  produce  a 
commission  of  2/.  10*.,  what  commission  will  539^.  14*. 
uroduce  ?  We  have  hence  the  following  proportion  : — 

£100  :  £539  14  0  =  £2  10:  x. 

where  x  expresses  the  commission  sought ;  we  there- 
fore must  multiply  539^-  14*.  by  2£.  10*.,  and  divide 
the  product  by  100.  The  quotient  will  in  this  case  be 
1 31. 9*.  10|d.  From  this  analysis  we  derive  the  following 

RULE. 

(424.)  To  find  the  total  commission,  at  a  given  rate 
per  cent.,  on  a  given  amount ,  multiply  the  amount  by  the 
rate,  and  divide  the  product  by  100  :  the  quotient  will  be 
the  total  commission  sought. 

It  frequently  happens  that  it  is  more  convenient  to 
express  the  rate  of  commission  by  so  much  in  the  pound, 
than  by  so  much  in  tLe  1001.  Thus  a  commission  of 
5  per  cent,  will  be  a  shilling  in  the  pound,  and  in  such 
case  the  total  commission  will  always  be  found  by  taking 
the  number  of  pounds  in  the  amount  to  express  shillings. 
Again,  if  the  commission  be  2^  per  cent.,  it  will  be 
equivalent  to  6d.  in  the  pound,  and  we  shall  therefore 
find  the  commission  in  shillings,  by  dividing  the  num- 
ber of  pounds  by  2. 

INSURANCE. 

(425.)  INSURANCE  is  an  indemnity  given  to  make 
good  a  future  contingent  loss,  in  consideration  of  a  per 
centage  paid  down  on  the  extent  of  the  value  insured. 
The  party  insuring  thus  undertakes  to  make  good  any 


346 


A    TREATISE    ON    ARITHMETIC.  BOOK  IV. 


loss  of  property  incurred  by  fire,  by  storms,,  or  other 
accidents  at  sea,  by  loss  of  life,  £c.  &c. 

The  per  centage  on  the  amount  insured  is  payable 
yearly,  and  is  called  the  premium.  Its  amount  is  cal- 
culated by  the  rule  already  explained  for  computing 
commission,  &c. 

TARE  AND  TRET. 

(426.)  The  GROSS  WEIGHT  of  any  sort  of  merchandise 
is  the  weight  which  is  found  by  weighing  it  with  the 
box,  sack,  barrel,  or  whatever  be  the  envelope  in  which 
it  shall  be  contained. 

TARE  is  an  allowance  made  to  the  buyer,  for  the 
weight  of  the  envelope  containing  the  merchandise. 

TRET  is  an  allowance  made  to  the  buyer  for  waste. 

When  these,  or  other  similar  allowances  are  deducted 
from  the  gross  weight,  the  remainder  is  called  net  weight. 
Generally,  such  allowances  are  either  a  fraction  of  the 
total  weight,  or  a  given  fraction  of  the  value  of  the 
merchandise.  In  any  case,  the  methods  of  computation 
consist  in  finding  the  required  fraction  of  the  whole 
weight  or  value  of  the  merchandise. 

When  such  allowances  are  expressed  at  so  much  per 
parcel,  their  total  amount  will  be  found  by  multiplying 
the  allowance  by  the  number  of  parcels. 

When  they  are  expressed  at  so  much  per  hundred 
weight,  we  have  only  to  find  what  fractional  part  of  a 
hundred  weight  the  allowance  is,  and  multiply  the  total 
weight  by  the  fraction  thus  obtained :  the  product  will 
be  the  total  amount  of  the  allowance. 

Example  1.  —  A  broker  is  employed  to  sell  goods  to  the 
amount  of  627^.  10*.  on  an  allowance  of  2s.  8d.  per 
cent.  :  what  is  his  total  brokerage  ? 

ANALYSIS. — Divide  the  total  amount  by  100,  and 
multiply  the  quotient  by  2s.  8d. 

COMPUTATION. —  Dividing  627/.  10s.  by  100,  we 
get  the  quotient  6'275,  and  multiplying  2*.  Sd.  by 
6*275,  we  obtain  the  product  16s.  8|d.,  which  is  the 
amount  of  the  brokerage  sought. 


CHAP.  III.   TARE  AND  TRET PARTNERSHIP.       347 

Example  2.  —  A  factor  purchases  goods  to  the  amount 
of  5001.  14s.  at  a  commission  of  2J  per  cent.,  what  is 
the  total  amount  of  his  commission  ? 

COMPUTATION.  —  Divide  500/.  14s.  by  100,  and  we 
obtain  the  quotient  51.  Os.  lT^d:  multiply  this  by 
2/.  10*.  and  we  obtain  the  product  12/.  10*.  4^d. 

Example  3.  —  A  factor  sells  goods  to  the  amount  of 
2301.  12s.  at  a  commission  of  3  per  cent. :  what  is  the 
total  amount  of  his  commission  ? 

COMPUTATION. — Divide  2SO/.  12s.  by  100,  and  the 
quotient  is  2/.  6s.  l^tffrd.:  multiply  this  by  3,  and  the 
product  is  6l.  18s.  4-^%^.,  which  is  the  commission 
sought. 

Example  4.  —  What  premium  must  be  paid  for  insur- 
ing goods  to  the  amount  of  31JI.  18s.  6d.  at  if  per  cent. 

COMPUTATION.  —  Divide  317 1*  18s.  6d.  by  100,  and 
the  quotient  is  31.  3s.  l-^-^d. :  multiply  this  by  II.  1 5s. 
and  the  product  will  be  51.  11s.  3-ffQd. 

PARTNERSHIP. 

(427.)  When  two  or  more  persons  subscribe  to  a  com- 
mon-stock as  capital  in  trade,  and  carry  on  a  joint  busi- 
ness, the  profits  which  accrue  from  year  to  year  on  the 
common  capital  should  be  distributed  between  the  part- 
ners, in  the  proportion  of  the  capital  they  respectively 
subscribe,  provided  that  the  capital  of  each  partner  re- 
mains for  the  same  time  invested  in  the  business. 

Let  us  suppose  that  the  whole  joint  stock  in  trade 
consists  of  a  number  of  equal  shares,  represented  each 
by  11.  of  the  original  capital.  It  is  clear  that,  in  the 
distribution  of  profits,  the  total  profits  should  be  divided 
into  as  many  equal  parts  as  there  are  pounds  in  the 
common  capital  subscribed  ;  and  that,  in  the  distribu- 
tion of  profits,  each  partner  should  receive  as  many  of 
those  parts  as  there  were  pounds  in  his  subscribed  ca- 
pital. This  will  be  easily  understood  by  an  example. 

Example  1. — Let  A,  B,  and  C,  invest  respectively 
10551.  7561.  and  5571.  in  trade,  and  at  the  end  of  the 
first  year  suppose  that  a  profit  of  468  J.  15s.  has  been  06- 


348  A    TREATISE    ON    ARITHMETIC.  BOOK  IV, 

tained.       It  is  required  to  know  how  this  profit  must  be 
distributed  between  the  partners  ? 

Adding  together  the  portions  of  capital  subscribed, 
we  find  the  total  capital  invested  to  be  23681.  Let  us 
consider  this,  the  whole  stock  in  trade,  to  consist  of 
2368  shares  of  II.  each.  It  is  evident,  therefore,  that 
if  the  whole  profit  be  divided  by  2368,  we  shall  obtain 
the  profit  on  each  share  of  1L  To  find  therefore  the 
portion  of  profit  to  which  each  partner  is  entitled,  we 
have  only  to  multiply  the  profit  on  11.  by  the  number 
of  pounds  in  his  subscribed  capital :  the  process  would 
then  be  as  follows  :  — 

Divide  the  total  profit  by  the  number  of  pounds  in 
the  total  capital,  and  multiply  the  quotient  by  the  number 
of  pounds  in  the  subscribed  capital :  the  product  will  be 
the  portion  of  the  total  profit  to  which  each  partner  is 
entitled.  We  should  obtain  the  same  result  by  reversing 
the  order  of  these  two  operations,  viz.  by  first  multiplying 
the  total  profit  by  the  number  of  pounds  in  the  subscribed 
capital  of  each  partner,  and  then  dividing  the  product  by 
the  number  of  pounds  in  the  total  subscribed  capital. 
This  method  would,  in  practice,  generally  be  the  most 
expeditious.  In  the  present  example  the  process  of  cal- 
culation would  be  as  follows :  — 

£    s.  d. 

Total  profit  468   15  0 

x  A's  capital        -  1055     0  0 

Divide  by  total  capital       -     2368)494231     5  0 


A's  share  of  profit  -  208  16  9J 


Total  profit  468   15  0 

x  B's  capital        -  -  756     0  0 


Divide  by  total  capital        -     2368)354375     0  0 


B's  share  of  profit  149  13  0| 

Total  profit  -  468   15  0~ 

x   C's  capital         -  -  557     0  0 


Divide  by  total  capital        -     2368)261093150 


C's  shaie  of  profit  -.  £  110     5 


CHAP.  III.   TARE  AND  TRET  —  PARTNERSHIP.     34$ 

To  verify  this  computation,  we  need  only  add  together 
the  three  shares  into  which  the  whole  profit  is  to  be 
divided ;  and  if  the  total  obtained  by  this  addition 
amount  to  468/.  15s.  the  computation  has  been  correctly 
performed. 

£    5.  d. 

A's  share  of  profit  208  16  9f|f 

B's  share  of  profit  -  149  13  0§§| 

C's  share  of  profit  «  110     5  2if£ 

^468  15  0 


INDEX. 


ADDITION,  50.  Explained  by  coun- 
ters, 51.  By  figures,  54.  Signs 
used  to  express  the  operation  of, 
56,  Table  of,  57.  Mode  of  testing 
questions  in,  61.  General  rule 
for,  6G.  Compound  addition, 
242.  Examples  in,  243. 

Alkindi,  the  celebrated  Arabian 
writer,  his  work  on  arithmetic 
entitled,  "  Hisabu  1'  Hindi," 
36. 

American  nomenclature  of  num. 
bers,  15. 

Arabic  notation,  history  of,  35. 

Arithmetic,antiquity  of  the  science 
of,  36. 

Asiatic  and  African  nomenclature 
of  numbers,  19. 

Avoirdupois  weight,  table  of,  226. 
Divisions  of,  227. 


B. 

Binary  and  duodecimal  systems, 

21. 

Bissextile  year,  212. 
Brokerage,  345. 


Chinese  nomenclature  of  numbers, 
14.  Method  of  reckoning  on  the 
fingers,  24. 

Cipher,  uses  and  origin  of,  47. 

Commission,  345. 

Complex  numbers  in  general,  209. 
Their  reduction  and  simplifica- 
tion, 210.  Rule  for  the  addition 
of,  244.  Examples  in,  245.  Sub- 
traction of,  246.  Multiplication 
of,  250.  Examples  in  the  divi- 
sion of,  272. 

Compound  addition,  examples  in, 
242.  Subtraction,  examples  in, 


246.  Multiplication,  examples 
in,  250.  Division,  examples  in. 
272.  Rule  of  three,  311.  Ana- 
lysis and  computation  of  ques- 
tions in,  312.  Interest,  examples 
in,  338. 

Computation  by  counters,  as  prac- 
tised throughout  Europe  during 
the  middle  ages,  and  continued 
until  it  was  superseded  by  the 
improved  arithmetic  now  in  use, 
43.  Arrangement  in  periods, 
45. 

Cubic  measure,  220. 

D. 

Decimal  nomenclature,  7. 

Decimals,  189.  Reducible  to  equi- 
valent vulgar  fractions,  194. 
Methods  by  which  the  elemen- 
tary arithmetical  operations  may 
be  performed  on,  196.  Rule  for 
the  multiplication  of,  198.  Gen- 
eral rule  for  the  division  of,  200. 
General  rule  for  converting  a 
vulgar  fraction  into  an  equiva- 
lent decimal,  203. 

Direct  rule  of  three,  296.  Analysis 
and  computation  of  questions  in, 
297. 

Discount,  340.  Method  of  comput- 
ing, 341. 

Division,  118.  Process  of,  119. 
Examples  in,  123.  Rule  for,  125. 
Practical  process  by  which  the 
dividend  is  resolved  into  mul- 
tiples of  the  divisor,  127.  Ex- 
amples in  which  the  divisor 
consists  of  three  figures,  132. 
Examples  in  long  division,  135. 
General  rules  for  the  solution  of 
questions  in,  140.  Compound 
division,  272. 

Duodecimal  and  binary  system*, 


352 


INDEX. 


Egyptian  hieroglyphics  expressive 
of  numbers,  30. 


F. 

Fractions,  explanation  of,  147. 
Notation  of,  149.  Examples  in 
the  division  of,  151.  General 
method  for  reducing  a  fraction 
to  its  lowest  terms,  155.  Addi- 
tion and  subtraction  of,  161. 
Rule  for,  162.  General  rule  to 
convert  a  mixed  number  into  an 
improper  fraction,  163.  Multi- 

?lication  of,  171.     Division  of, 
81.    Rule  for,  184.    Method  of 
proving     compound    •  fractions, 

French  metrical  system,  228. 


G. 

Greeks,  their  method  of  notation, 
26.  Greek  characters  expressing 
numbers,  30.  Their  method  of 
teaching  the  elements  of  arith. 
metic,  39. 

H. 

Hebrews,  their  method  of  notation, 
25. 


Imperial  liquid  and  dry  measure, 
table  of,  221. 

Insurance,  345. 

Integers,  or  whole  numbers,  147. 

Interest,  328.  Analysis  and  com. 
putation  of  questions  in,  329. 
General  rule  for  the  solution  of 
questions  in,  331.  Tables  of 
simple  interest,  334.  Compound 
interest,  337. 

Inverse  rule  of  three,  304.  Ana- 
lysis and  computation  of  ques- 
tions in,  305.-, 


Length,  imperial  measure  of,  216. 
Leonardo  Pisano,  his  treatise  on 

arithmetic,  38. 
Leslie,  professor,  40. 


M. 

Magnitude  of  solids,  219. 


Measure,  reduction  of,  238. 

Measures  of  time,  210.  Of  space, 
214.  Imperial  measures  of 
length,  216.  Square  measure, 
218.  Cubic  measure,  220.  Im- 
perial liquid  and  dry  measure, 
,221.  Old  wine  measure,  222. 
Old  dry  or  Winchester  measure, 
223.  Of  weight,  225. 

Mohammed  ben  Muza,  35. 

Money,  table  of,  213.  Reduction 
of,  233.  Examples  in,  236. 

Multiplication,  86.     Processes  of, 

90.  Examples  in,  by  counters, 

91.  Table  of,  94.    Simple  ques- 
tions in,  97.    Rule  for,  98.    More 
complex      questions     in,      101. 
General  rule  for,  108.     Mode  of 
proving  questions  in,  113.    Table 
of,  116.    Tables  of  squares  and 
cubes,  117. 


Numbers,  1.  Method  of  forming 
ideas  of  large  numbers,  4.  A 
curious  example  illustrative  of, 
as  given  in  the  history  of  the 
island  of  Madagascar,  5.  Their 
expression  by  words,  6.  Method 
of  expressing  numbers  by  sym- 
bols or  figures,  23. 

Numerical  notation,  system  of,  31. 
Invention  of,  35.  Antiquity  of, 


O. 

Oriental  nomenclature  of  num- 
bers, 13. 

P. 

Partnership,  347. 

Phenicians,  their  method  of  nota- 
tion, 25. 

Profit  and  loss,  344. 

Proportion,  280.  Its  practical  ap- 
plication, 287.  Examples  in, 
290. 

R. 

Ratio,  explanation  of,  288. 
Reduction,  160.     Examples  in,  233. 

Of  money,  236.    Of  weight,  239. 
Roman     numeral    notation,     27. 

Table  for  explaining  the  way  in 

which  their  symbols  are   com. 

bined  to  express  numbers,  28. 
Roman  abacus,  39. 
Rule  for  obtaining    the    greatest 

common  measure  of  two  num. 

bers,  157. 


INDEX. 


353 


Rule  of  three,  295.  Analysis  and 
computation  of  questions  in  the 
direct  rule  of  three,  296.  Ana- 
lysis and  computation  of  ques- 
tions in  the  inverse  rule  of  three, 
304.  Analysis  and  computation 
of  questions  in  the  compound 
rule  of  three,  311. 


Sagard,  G.,  a  Franciscan  monk,  his 

work  entitled  "  Le  GrandVoyage 

des  Hurons,"  15. 
Sanskrit    nomenclature    of  num. 

bers,  13. 

Solids,  magnitude  of,  219. 
Space,  measures  of,  214. 
Square  measure,  218. 
Subtraction,  68.     Process  of,   by 

counters,  69.    Examples  in,  71. 

General  rule  for,  78.    Examples 

in  compound  subtraction,  246. 
Swan-pan,    a   calculating    instru. 

ment  used  by  the  Chinese,  42. 


T. 

Tare  and  tret,  346. 

Thibet,  nomenclature  of  numbers. 

12. 
Time,  measure  of,  210.    Table  of, 

212. 
Troy  weight,  table  of,  226. 


W. 

Weight,  measures  of,  225.  Tables 
of  Troy  and  avoirdupois,  226 
Reduction  of,  239.  Examples  in, 
240. 

Welsh,  Erse,  and  Gaelic  names  for 
numbers,  17. 


Y. 

Young,  Dr.,  his  investigations  re- 
specting the  Egyptian  hierogly- 
phics expressive  of  numbers, 
explanation  of,  30. 


A  A 


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