I 7° 
ARITHMETIC. 
Ex. Divide 546^413214241246841 by 42 = 7X6. 
6)5468413214441246841(1 firff remainder 
7)911402202373541140(3 fecond remainder 
130200314634791591 quotient. 
Then 3X6 + 1 = 19, the true remainder. 
Compound Division, teacheth to find how often one 
number is contained in another, of different denominations. 
Rule. Place the divifor and dividend as in Simple Di- 
vifion. Begin at the higbeft denomination, and divide it 
by the divifor, as in Simple Divilion, and reduce the re¬ 
mainder (if any) to the next inferior, adding thereto what 
is in the dividend of the fame, denomination; divide this 
number as above, and fo proceed till the workisfinifhed. 
Ex. Divide 2o3lb. iooz. 7 dwts. 7grs. troy by 37. 
lb. oz. dwt. gr. 
37 ) 208 10 7 7 ( 51b. 
Rem. 23 
12 
37)286(7 oz. 
2.ft) 
Rem. 27 
f 20 
37 ) 54 - 7 ( 14 -Awts. 
37 
177 
148 
Rem. 29 
24 
37 ) 7°3 ( 1 9 S rs# 
37 
333 
333 
o Anfvver, 51b. 70Z. i4dwts. i9gr. 
The fame contractions, as well as method of proof, may 
be adopted here as in Simple Divilion. For in dividing a 
number of different denominations we only divide all the 
parts of which that number is compofed; and, though fome 
one of the parts lliould not be an exaCt multiple of the 
divifor, yet that number by which it exceeds the multiple 
has its proper value by being placed in a lower denomina¬ 
tion, and therefore the dividend will ftill be refolved into 
parts, and the true quotient found as before. 
Dr. Wallis in his Arithmetic, publifhed 1657, has pro¬ 
ved the four fundamental operations by calling out the 
nines, a method which depends on this principle, that any 
number divided by nine will leave the fame remainder as 
the furn of its digits divided by nine. To illuftrate this, 
let there be any number propofed, as 4329, which, ac¬ 
cording to the local value of each figure, becomes 4000+ 
300-1-20+9. But 4000=4X1060=4X999 + 1=4X999 
+4. In the fame manner,- 300=3X99+3 > and 20=2 X 
9+ 2. Th erefore, 4329 = 4 X 999 + 3 X 99 + 2 X 9 + 
But the firft part of this dividend, viz. 
4X999+3X99+-X9? is evidently divifible by 9 without 
a remainder; and'therefore the whole, or 4329, divided 
by 9, will leave the fame remainder as 4+3+2+9, or the 
fum of its digits, divided by 9. Hence, to prove Addi¬ 
tion, add the figures of each line of numbers together fe- 
verally, always calling out the nines as they arife, adding 
the overplus to the next figure, and letting at the end of 
each line what is over the nine or nines; then do the lame 
with'the fum total, as alfo with the former exceffes of 
ni ne, and the refults will agree if the work is right. 
Ex. 3 2 9 • • 5 l 
1562 . .5 I Excefs of nines in'each line refpeCt- 
20347 • • 7 j ively. 
7 1 2048 • • 4J 
734286 3 Ditto in the fum, the fame as in 5574.. 
In Subtraction, cad thfe nines out of the minuend, and 
alfo out of the fum of the fubducend and remainder ; if 
the operation is right, the excefs will be the fame. In 
Multiplication, cart the nines out of both the multiplicand 
and multiplier, and fet down the excefs in each ; multi¬ 
ply thefe exceffes together, and if the excefs of nines in 
their produCt equal the excefs of nines in the total pro¬ 
duct, the account may be prefumed right. To prove 
Divilion, cad the nines out of the divifor and quotient; 
multiply the exceii'es, (adding the remainder, if any,) and 
cad the nines out of their produCt; and if there is no 
error this lad excefs will agree with that obtained from the 
dividend. From the illudration given above, the reafon 
of this proof will appear evident; and, though the method 
is limple and eafy, yet it is liable to one objection. For 
if any two figures are tranfpofed, or reckoned in a wrong 
column, the refult arifi'ng from comparing the exceffes of 
nines will neverthelefs be the fame, and confequently the 
error remains undifcovered. 
REDUCTION. 
In computations relating to the value, weight, meafure, 
&rc. of articles, the folution is greatly facilitated by Re¬ 
duction, which is the converfion of numbers from one de¬ 
nomination to another of equal value ; and is of two kinds, 
viz. defending and afeending. 
Reduction descending, is when numbers of a greater name 
are to be reduced into thofe of a lefs, (as pounds into (hil¬ 
lings, pence, and farthings,) which is thus performed. 
Multiply continually all the denominations, from the given 
one to that fought, by fuch a number of the lefs as makes 
one of the greater; adding to each produCt, by the way, 
thofe of the fame denomination with itlelf, if fuch there be. 
Reduction afeending, is when integers of a lefs denomi¬ 
nation are to be reduced to thofe of a greater, (as farthings 
into pence, (hillings, and pounds,) which is thus performed. 
Divide continually all the denominations, from the given 
one to that fought, by fuch a number of each lefs as makes 
one of the greater; the lad quotient, with the feveral re¬ 
mainders, (if any,) will be the anfwer. Where note, Each 
remainder will always be of the fame denomination with 
its refpeCtive dividend. 
To perform Reduction, confider how many of the lefs 
denomination make one of the greater, as how many pence 
make a (lulling, or how many (hillings make a pound ; and 
multiply by that number when the reduction is defeending, 
but divide by it when it is afeending. So to reduce 23I. 
into pence, and converfely thofe pence into pounds, mul¬ 
tiply or divide by 12 and 20, as here below : 
23 pounds 12)5520 pence 
20 20)460 (hillings 
460 (hillings 23 pounds 
12 - 
55 2 ° P ence 
The method of proof is by working the converfe of the 
queftion ; thus the above examples prove each other. 
RULE of PROPORTION, 
Teacheth, from three numbers being given, to find a 
fourth, which may bear the fame proportion to the third 
as the fecond does to the firft; and, for this reafon, it is 
fometimes called the Rule of Three, and, from its great and 
extenfive ufe, it is often named the Golden Rule. In con- 
(idering the nature of any queftion, if it appears that a 
greater number requires a greater, or a lefs requires a lefs, 
it is called the Rule of Three DireB, or DircU Proportion. 
Rule. Place the three given terms fo that the firft and 
third may be of one name, the third being that which afks'' 
the queftion, and the fecond of the fame name with the. 
fourth term fought, and let thefe be reduced to the lowed 
denomination mentioned in eitherof them. Multiply the 
fecond and third terms together, and divide their produCt 
by the firft ; fo (hall the quotient be the anfwer in the fame 
denomination as the middle term. 
3 “Ex, 
