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directly under the figures of the fame denomination in the greater number; that is, the units under units, tens under tens, and pounds, Dillings, pence, ounces, drams, &c. &c. directly under the same, as in addition. Then, beginning at the right hand, or the least denomination, take the value of each figure in the less number from that in the greater mumber, which stands directly over it; setting down the remainder underneath. Proceed in this manner till the work be finished. But if, as it frequently happens, any single figure in the less number he greater than that in the greater number, from which it is to be taken ; an unit is to be borrowed from the next figure towards the left hand of the greater number, and added to the uppermust figure, that the bottom figure may be taken therefrom; which borrowed unit must be paid, or added to the next figure of the less number on the left hand,

From 6954 Take 5443 Rem. 15 Proof 6954

Examples. 1729886

370290370 greater numbers, 1098907 128403492 less nunibers. 630979

241886878 1729886 370290370

To shew the use of this rule, I say-Suppose a merchant owed 6954), whereof he has paid $4437. : to know what re. mains to be paid, the sums are to be set orderly one under the other, according to the foregoing rule,, and as seen in the examples: then, beginning with the unit figures, in the first example, I say, take 3 from 4 and there remains i, which I set under the line; next take 4 from 5 and there remains

1, which is also fet under the line; again, take 4 from 9 and there remains 5, which must be set down as before; and, lastly, take 5 from 6 and there remains. I : thus there remains due 1511.

T Subtraction is proved by adding the remainder to the less of the two given numbers, and if the total of thefe two nyms1 bers amount to the exact fum of the greater pumber, the

work

work is right; otherwise not: thus, in this example, I add the remainder 1511. to the less number 5443l. and the amount is 69541.; the same as the greater number.

In the second example, I begin with the units, as before : saying, take 7 from 6 I cannot, but by borrowing 1 from the next figure 8, and which added to the 6 makes 16 (the 8 being the next fuperior number), I say, 7 from 16 and there remains 9; then, for the i that I borrowed, I carry 1 in return to the next figure of the less number, saying, i that I borrowed and o is but 1, therefore, i from 8 and there remains 7; again, 9 from 8 I cannot, but borrowing i as before from the next figure, I say, 9 from 18 and there re. mains 9; then for i that I borrowed, I must add 1 to the next figure 8, saying, 9 from 9 and there remains o (which may always be done when the two figures are the fame); again, 9 from 2 I cannot, but 9 from 12 (borrowing i from 7) and there remains 3 ; then i that I borrowed, added to the o, and taken from the 7, there remains 6: thus the work is finished. The proof demonstrates it right.

Proceeding in the same manner in the third example, I say, 2 from ol cannot, but 2 from 10 (borrowing one from the 7), there remains 8; then i that I borrowed and 9 is 10, 10 from 7 I cannot, but 10 from 17 and there remains 7; again, i that I borrowed and 4 is 5, 5 froin 3 I cannot, but 5 from 13 (borrowing 1) and there remains 8; then, 4 from o (or nothing) I cannot, but 4 from 10 and there remains 6; again, 1, that I borrowed, from 9 and there remains 8; and 4 from 12 and there remains 8; and i that I borrowed and 8 is 9, from

10, and there remains 1; and i that I borrowed and 2 is 3 from 7 and there remains 4; laftly, i from 3 and there remains 2.

More Examples. From 178927 22574390 13942583 greater numb. Take 147839 2107863 734098 less numbers. Rem. 31088 0466527 13208,85 Proof 178927 22574390

13942683

The number of years since any event happened, may be discovered by subtracting the date of the year the event happened from that of the present year. Thus :The present year 1808

1808

1808 The fire of London 1666

Location

Treason
Years since 142

203 Proof 1808

1808

1808

Gunpowder 1605

1588

paid

II

Proof 17

14

Subtraction of divers denominations is performed upon the same principle as subtraction of numbers of one denomination. Observing, that when an unit is borrowed of the next higher denomination, it must be considered according to its true intrinsic value, and must be repaid to the lower figure of the fame denomination; as will be seen in the following examples :20 124

20 1:4 k. s. d.

4. d.
Money owing 17
14 92

123 14 74
10 10
61

III

19 95
Remains 7
4 23

14 104
14 9

123

7* In the first of these examples, beginning with the farthings, I say, 3 farthings from 2 I cannot, but borrowing an unit from the next denomination, or i penny from the 9 pence (and which added to the 2 farthings makes 6 farthings), I say, 3 farthings from 6 farthings and there remains 3 farthings, which I set under the farthings; then for the unit I borrowed of the pence, I add i to the 6, saying 6 and 1 is 7; now 7 from 9 and there remains 2 ; then 10 from 14 and there remains 4; lastly, 10 from 17 and there remains 7; thus the remains is 71. 45. 27d. which is proved in the example.

In the second example, I say, i farthing from 3 and there remains 2 farthings, or an halfpenny; which I set down in its proper place, viz. under the denomination of farthings ; then 9 from 7 I cannot, wherefore borrowing 1 Nhilling from the 14, and adding it to the 7 pence, which makes 19 pence, I say, 9 from 19 and there remains 10; then i that I bor

rowed

rowed and 19 is 20, 20 from 1+ I cannot, wherefore I borrow y pound fruin tlie pounds, whiclı, added to the 14, makes 34 Thillings; therefore, I say, 20 from 34 and there remains 14; then i sirat I borrowed and I is 2, 2 from 3 and there remainsit ; and i froin 2 and there remains 1 ; thus, the anfwer to this sum is 111. 145. Icd; and its proof, under the answer, shews that it is right.

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Due 145 12

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If the money paid be paid at several times, the sums so paid are to be added together, and the total subtracted from the sum firit due. 6. s. d.

£. so do 6 Received 1120 10

2 IO IO

Ο 10 6 Paid at several

30 9 8

Paid to several IO 8 8
times
0 10 6

persons I 2 1 2
3
9 4

20 19 10 12 O 6

9 O O Paid in all

Paid in all
59
Rests due

Rests due 1064 19 6

Proof I I20 IO

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S6 19

Proof 145 19

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4 28

12

20

2

Examples in Avoirdupois Weight.
4 28

16 16
C.
grs.
16.

C.
qrs.

lb. C. grs. Ib.
3
147 9

19 9 14

8 15 27 30 25

15 3

147
9

19 9 14
Examples

IO

2

2 1

116 3

12

From Take Remains

Proof

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12

20

20

16,4 Ej anples in Troy Weight.
24

24 oz. pwt. gr.

0%. pot. gr. : From 836 336. 10 20

3124

16 21 Tak 247 19

320

22

19 Remains 88

28 16 23 Proof 336

IS 20

311 21

18

II

22

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These examples, and all others of other denominations, are wrought in the same manner as subtraction of money ; having refpect to the table belonging to the denomination : thus, in the first example of avoirdupois weight, I fay, take z1 pounds, froin 20 I cannot, therefore I borrow i quarter of an hundred from the 3 quarters, and add it to the żó, which makes 48 pounds; then I say, 21 from 48 and there remains 27; next, that I borrowed and 2. is 3, 3 from 3 I cannot, wherefore I set down o, as before hinted; lastly, 10 from 12, and there remains 2.

The figures at the head of the columns fhew the number of units which each unit of the next higher dengmination contaijis.

SECT. IV.

OF MULTIPLICATION.

MULTIPŁACATION.is, perhaps, the most necessary rule in arithmetic for business, on account of its dispatch in refolving several long questions.

Multiplication teaches how, from two given numbers, to find a third, that shall contain pither of the two given num. bers as often as the other contains units : thus, 3 times 4 is 12; here 3 and 4 are the two given numbers, and 12 is the third number, or product, which contains 3 as often as 4 contains units, ráz. 4, cimes; or it contains 4 as often as 3 contains units, 3 times, Yol. I. T

There

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