3 68 
A R I • T H M E T I C, 
er denomination ■; fubtraft the lower number, and fet 
down the remainder; then carry one, and add it to the low¬ 
er number of the next higher denomination, which will 
proceed as before. 
Money. Troy Weight. 
Ex. 1. 
1. 
2561 
876 
s. 
J 9 
14 
d. 
10 ! 
li 
lbs. 
Ex. 2. 354 
178 
oz. 
5 
7 
dwt. gr. 
1214 
18 6 
Difference 
1685 
5 
3 i 
1 75 
9 
14 8 
Proof 
2561 
'9 
IOJ 
354 
5 
12 14. 
The reafon of this rule and method of proof depend on 
the fame principle as in Simple Subtraction, and only dif¬ 
fer according to the value of different denominations. 
MULTIPLICATION 
Is the repeating a given number, or quantity, called the 
multiplicand, fo many times or parts of a time, as there are 
units in another given number, called the multiplier ; and 
the refult is called the prodidl. Both multiplicand and 
multiplier are called faEiors. Multiplication is a compen¬ 
dious method of addition, and is performed by the help of 
the following Table; 
2 
3 
4 
5 
6 | 7 
8 I 
9 | ioj 
”1 
I 2 
2 i 
4 
6 
8 
10 
12 114 
i 6 j 
i 8 | 20 } 
22 
24 
3 I 
6 
9 
12 
15 
1 8 ; 2 i 
24 ! 
2 7 i 3°1 
33 
36 
4 ! 
8 
I 2 
i 6|20 
24 ' 2 8 
32 ! 
36 | 4 °| 
44 ! 
48 
5 ! 
xo 
*5 
20 
25 
30(35 
40 ! 
4 s| 5 °l 
55 
60 
6 | 
12 
18 
24(30 
36 k 2 
481 
54 l 60 ] 
66 
72 
7 
14 
21 
28 
35 
42(49 
56 ! 
63 ! 70 : 
77 
84 
8 
16 
2 4 
32 
40 
48(56 
64 ! 
72 1 ; 80 
88 
96 
9 
18 
2 7 
36 
45 
54|63 
72 ! 
8 i| 90 : 
99 
108 
io| 
20 
30 
40 
5 ° 
60 70 
8 oj 
90 IOO 
I IO 
I 20 
II] 
22 
33 
44 j 55 
66 77 
88 ] 
99 I 10 
I 21 
13 ^ 
I 2 
24 
36 
48|60 
72 84 
96 .: 
ro 8 120 
132 
144 
The life of this table is this: Find one figure or faftor 
jn the top column of the table, the other in the left-hand 
column, and the fquare where thefe two columns meet is 
the produft. For infiance, fuppofe you would multiply 
8 by 9; find 9 in the top, and 8 in the fide, column, and in 
the fquare where they meet Hands 71; the product of 6 
times 7 is 42 ; of 8 by 8 is 64; of 6 by 12 is 72 ; of 3 by 
9 is 27 ; and 12 times 12 is 144, and fo of any other. 
The mark or character now ufed for multiplication, is 
Either the crofs (x )> or a (ingle point (.); the former be¬ 
ing introduced by Oughtred, and the latter by Leibnitz. 
Rule. Place the multiplier under the multiplicand, 
fo that units may Hand under units, &c. then draw a line. 
Multiply from the right hand to the left: thus, Begin with 
the units of the multiplier, by which multiply the units 
or firfi figure of the multiplicand, and fet down the over¬ 
plus above the tens, carrying the tens in your mind; then 
multiply the fecond figure of the multiplicand by the fame 
multiplier, adding fo many units as you had tens to carry; 
fet down the overplus, and carry the tens as before ; do 
thus till you come to the lafi figure in the multiplicand, 
whofe-produft fet down entire. Then take the fecond fi¬ 
gure of the multiplier, and multiply by it as you did by the 
firH, fetting the firH figure of the product under that you 
multiplied with; and in like manner take each figure in its 
order, till all the figures of the multiplicand are multi¬ 
plied into all the figures of the multiplier, always obfer- 
ving to fet the firH figure of each produft fo many places 
to the left hand as the multiplying figure is difiant from 
the units’ place. Lafily, add all thefe products together, 
.and their fum will be the produft of the two numbers given, 
s 
Ex. 1. 5864824134multiplicand. 
4 multiplier. 
2 3459 2 9 6 53 6 produft. 
When one or both faftors end with cyphers, multiply 
the other figures only as before direfted, and to the right 
of the prodiuft add as many cyphers as are on the right of 
both factors. 
Ex. Multiply 89764370 00 
By 28497600 
53S58622 
6283 ?oco 
80787933 
359 0 5748 
71811496 
1 795 2 874 __ 
255806911051200000 
When any number is to be multiplied by an unit, with 
cyphers to the right hand, as 10, 100, 1000, &c. annex fo 
many cyphers to the right of the number as there are in 
the multiplier. When the multiplier is a compofite num¬ 
ber, multiply continually by all the faftors infiead of it.. 
Ex. Multiply 8756 by 504 (=7x8X9) 
_ 7 
Produft of 7= 61292 
_8 
Do. of 56 = 490336 
__ 9 
Do. of 504 = 4413024 
When any figure in the multiplier is a multiple of ano= 
ther on its right hand, it is eafier and more concife to mul¬ 
tiply the produft of the lefs by its proper faftor for the 
produft of the greater. 
Multiply 80708615 
By 984^3 
242125845 = produft of 3, 
484251690 = do. of 3X 2 ' 
322834460 = do. of 4. 
645668920 = do. of 4X2. 
736377535 = do. of 3X3- 
7946812358745 the whole produft. 
Ex, 2. 549892994095' 
_ 3029 
4949036-946855 
549892994095 
16496789822850 
1660126949172805 
Sometimes the produft of two or more figures may be 
obtained at once, from the produft of a figure already found. 
Ex. 1. 14356 
64S 
114848 
918784 
9302688 
Ex. 2. 3462321 
96484 
13849284 
166x91408 
332382816 
334058579364 
In the fecond example, we multiply firH by 4; then, be- 
caufe 12 times 4 is 48, we multiply the firH line of the 
produft by 12, inHead of multiplying feparately by 8 and 
4; laHly, becaule twice 48 is 96, we multiply the fecond 
line of the produft by 2, inHead of multiplying feparately 
by 6 and 9. When we follow this method, we muH be 
careful to place the right hand figure of each produft un¬ 
der the right-hand figure of that part of the multiplier 
which it is derived from. 
To multiply by 5, which is the half of 10, annex a cy¬ 
pher to the multiplicand, and divide by 2. If the mul¬ 
tiplier be 25, which is the fourth of 100, annex two cy¬ 
phers and divide by 4. To multiply by 9, which is one 
lefs than ten, we may annex a cypher, and fubtraft the 
multiplicand from the number it compofes. And, if the 
multiplier be any number which is nearly equal to a cer- 
