54

L

L

42 39

170 5 4

70 4

2

15 3 9

175 4 9

66 9 8

18 6

42 57

73 86

12 13

7

163 7 4

45 3 2

15 3 9

148 5 8

18

6 8

9 6

5 10

5

2

1224 ΙΟ 1376 10 1419 17 I

Then we tranfcribe L 778, 16s. at the foot of the first and top of the fecond pages, L 1224: 10: 5 at the foot of the fecond and top of the third; and fo on.

CHAP. III. SUBTRACTION.

SUBTRACTION is the operation by which we take a leffer number from a greater, and find their differences. It is exactly oppofite to addition, and is performed by learners in a like manner, beginning at the greater and reckoning downwards the units of the leffer. The greater is called the minuend, and the leffer the fubtrabend.

If any figure of the fubtrahend be greater than the correfponding figure of the minuend, we add ten to that of the minuend, and, having found and marked the difference, we add one to the next place of the fubtrahend. This is called borrowing ten. The reafon will appear, if we confider that, when two numbers are equally increafed by adding the fame to both, their difference will not be altered. When we proceed as directed above, we add ten to the minuend, and we likewife add one to the higher place of the fubtrahend, which is equal to ten of the lower place.

RULE. "Subtract units from units, tens from tens, "and so on. If any figure of the fubtrahend be greater than the corresponding one of the minuend, "borrow ten."

Note 3. If the value of the higher denomination be not an even number of tens, fubtract the units and tens at once, borrowing according to the value of the higher denomination.

Note 4. Some chufe to fubtract the place in the fubtrahend, when it exceeds that of the minuend, from the value of the higher denomination, and add the minued to the difference. This is only a different order of proceeding, and gives the fame anfwer.

Note 5. As cuftom has established the method of placing the fubtrahend under the minuend, we follow it when there is no reafon for doing otherwife; but the minuend may be placed under the fubtrahend with equal propriety; and the learner thould be able to work it either way, with equal readiness, as this laft is fometimes more convenient; of which inftances will occur afterwards.

Note 6. The learner fhould alfo acquire the habit, when two numbers are marked down, of placing fuch a number under the leffer, that, when added together, the fum may be equal to the greater. The operation is the fame as fubtraction, though conceived in a different manner, and is useful in balancing accounts, and on other occafions.

It is often neceffary to place the fums in different columns, in order to exhibit a clear view of what is required. For inftance, if the values of feveral parcels of goods are to be added, and each parcel confifts of feveral articles, the particular articles fhould be placed in an inner column, and the fum of each parcel extended to the outer column, and the total added there.

If any perfon be owing an account, and has made fome partial payments, the payments must be placed in an inner column, and their fum extended under that of the account in the outer column, and fubtracted there. An example or two will make this plain. ift.] 30 yards linen at 2 s. L. 3 at I s. 6 d. 45 ditto 120 lb thread at 4 s. 40 ditto

To prove fubtraction, add the fubtrahend and remainder together; if their fum be equal to the minuend, the account is right.

E.

L 146 3 3

12 3 19

15 2 24 18

5. Received further In gold L. 10 10 In filver 13

12 236

Note 1. The reafon for borrowing is the fame as in fimple fubtraction. Thus, in fubtracting pence, we

2 3 28 II

CHAP. IV. MULTIPLICATION.

IN Multiplication, two numbers are given, and it is required to find how much the first amounts to, when reckoned as many times as there are units in the fecond. Thus, 8 multiplied by 5, or 5 times 8, is 40. The given numbers (8 and 5) are called factors; the first (8) the multiplicand; the fecond (5) the mul tiplier; and the amount (40) the product.

This operation is nothing elfe than addition of the fame number feveral times repeated. If we mark 8 five times under each other, and add them, the fum is 40 But, as this kind of addition is of frequent and extensive use, in order to shorten the operation, we mark down the number only once, and conceive it to be repeated as often as there are units in the multiplier.

For this purpose, the learner must be thoroughly acquainted with the following multiplication-table, which is compofed by adding each digit twelve times.

Twice Thrice (Four times Five times/Six times Seven times I is 2 1 is 3 I is 4 I is I is 6 I is

3456789

7

14

21 28

cation

and annex an equal number of cyphers to the product. Multipli Thus, if it be required to multiply by 50, we first multiply by 5, and then annex a cypher. It is the fame thing as to add the multiplicand fifty times; and this might be done by writing the account at large, dividing the column into 10 parts of 5 lines, finding the fum of each part, and adding thefe ten fums together.

If the multiplier confift of feveral fignificant figures, we multiply feparately by each, and add the products. It is the fame as if we divided a long account of addition into parts correfponding to the figures of the multiplier.

Example. To multiply 7329 by 365. 7329 7329 7329 60

5

300

36645= 439740 = 60 times. 2198700 300 times.

5 times.

36645 439740 2198700

2675085 365 times. It is obvious that 5 times the multiplicand added to 60 times, and to 300 times, the fame must amount to the product required. In practice, we place the products at once under each other; and, as the cyphers arifing from the higher places of the multiplier, are loft in the addition, we omit them. Hence may be inferred the following

RULE." Place the multiplier under the multipli"cand, and multiply the latter fucceffively by the fignificant figures of the former; placing the right"hand figure of each product under the figure of the "multiplier from which it arifes; then add the pro. "duct."

5

35

12 6

18 6

24 6

30 6 36 6

42

40 8

48 8

56

63

50 10

60 10

5511 6611

77

48/12

6012 72/12 84

Eight times Nine times(Ten times] Eleven times | Twelve times

365

8704

24 3

27 3

30 3

33 3

36

43974

385065

113538

657692

32 4

364

40 4

44 4

48

21987

75692

751648

40 5

45 5

3893435

60 6

661

72

2675085

8893810 817793024

A number which cannot be produced by the multiplication of two others is called a prime number; as 3, 5, 7, II, and many others.

A number which may be produced by the multiplication of two or more fmaller ones, is called a compofite number. For example, 27, which arifes from the multiplication of 9 by 3; and thefe numbers (9 and 3) are called the component parts of 27.

Contractions and Varieties in Multiplication.

First, If the multiplier be a compofite number, we. may multiply fucceffively by the component parts. Ex. 7638 by 45 or 5 times 9 7638 1ft,

by 67=64+3 Here becaufe 8 times 8

L. 287 2

8

=8 times.

by 8, which gives L.2296, 16s. equal to 64 times the multiplicand; then we find the amount of 3 times the L. 2296 1664 times. multiplicand, which is 107 13 3 3 times. L. 107: 13:3; and it is evident that thefe added, L. 2404 9 3=67 times. amount to 67, the multiplicand.

RULE IV. "If there be a compofite number a lit"tle above the multiplier, we may multiply by that "number, and by the difference, and fubtract the "fecond product from the first."

17 4 5 by 106-108-2 Here we multiply 12 108 9×12 by 12 and 9, the component parts of 108, and obtain a product of L. 1860, 6 s. equal to 1c8 times the multiplicand; and, as this is twice oftener than was required, we subtract the multiplicand doubled, and the remainder is the number fought.

17 =108 times. 8 10=

L. 1859 34

L. 1825 8

2 times.

2 106 times.

Example. L. 34: 8: 2+ by 3465. RULE V. "If the multiplier be large, multiply by 10, and multiply the product again by 10; by "which means you obtain an hundred times the given number. If the multiplier exceed 1000, multiply by ten again; and continue it farther if the "multiplier require it; then multiply the given "number by the unit-place of the multiplier; the "first product by the ten place, the fecond product "by the hundred-place; and fo on. Add the pro"ducts thus obtained together." 34 8 2 by SL. 172 1 =

L.

5 times

1000 times L. 34410 8

divifion, two numbers are given; and it is required to find how often the former contains the latter. Thus, it may be asked how often 21 contains 7, and the answer is exactly 3 times. The former given number (21) is called the Dividend; the latter (7) the Divifor; and the number required (3) the Quotient. It frequently happens that the divifion cannot be completed exactly without fractions. Thus it may be asked, how often 8 is continued in 19 ? the anfwer is twice, and a remainder of 3.

This operation confifts in fubtracting the divifor from the dividend, and again from the remainder, as often as it can be done, and reckoning the number

ΤΟ

4 by 3 103231 5 3000 times L. 119232 1 1043465 times The ufe of multiplication is to compute the amount of any number of equal articles, either in respect of measure, weight, value, or any other confideration. The nrultiplicand expreffes how much is to be reckoned for each article; and the multiplier expreffes how many times that is to be reckoned. As the multiplier points out the number of articles to be added, it is always an abstract number, and has no reference to any value or measure whatever. It is therefore quite improper to attempt the multiplication of fhillings by fhillings, or to confider the multiplier as expreffive of any denomination. The most common inftances in which the practice of this ope. ration is required, are, to find the amount of any number of parcels, to find the value of any number of articles, to find the weight or measure of a number of articles, &c.

N° 28.

As this operation, performed at large, would be very tedious, when the quotient is a high number, it is proper to fhorten it by every convenient method; and, for this purpose, we may multiply the divifor by any number whofe product is not greater than the dividend, and fo fubtract it twice or thrice, or oftener, at the fame time. The best way is to multiply it by the greatest number, that does not raise the product too high, and that number is also the quotient. For example, to divide 45 by 7, we inquire what is the greatest multiplier for 7, that does

not

II

Divifion. not give a product above 45; and we shall find that it is 6; and 6 times 7 is 42, which, fubtracted from 45, leaves a remainder of 3. Therefore 7 may be fubtracted 6 times from 45; or, which is the fame thing, 45, divided by 7, gives a quotient of 6, and a remainder of 3.

If the divifor do not exceed 12, we readily find the highest multiplier that can be ufed from the multiplication table. If it exceed 12, we may try any multiplier that we think will answer. If the product be greater than the dividend, the multiplier is too great; and, if the remainder, after the product is fubtracted from the dividend, be greater than the divifor, the multiplier is too fmall. In either of these cafes, we must try another. But the attentive learner, after some practice, will generally hit on the right multiplier at firft.

If the divifor be contained oftener than ten times in. the dividend, the operation requires as many fteps as there are figures in the quotient. For inftance, if the quotient be greater than 100, but lefs than 1000, it requires 3 fteps. We first inquire how many hundred times the divifor is contained in the dividend, and fubtract the amount of thefe hundreds. Then we inquire how often it is contained ten times in the remainder, and subtract the amount of these tens. Laftly, we inquire how many fingle times it is contained in the remainder. The method of proceeding will appear from the following example: To divide 5936 by 8.

From

5936

It is obvious, that as often as 8 is contained in 59, fo many hundred times it will be contained in 5900, or in 5936; and, as often as it is contained in 33, fo many ten times it will be contained in 330, or in 336; and thus the higher places of the quotient will be obtained with equal eafe as the lower. The operation might be performed by fubtracting 8 continually from the dividend, which will lead to the fame conclufion by a very tedious procefs. After 700 fubtractions, the remainder would be 336; after 40 more, it would be 16; and after 2 more, the dividend would be entirely exhaufted. In practice, we omit the cyphers, and proceed by the following

RULE ft, "Affume as many figures on the left "hand of the multiplier as contain the divifor once "or oftener find how many times they contain it, "and place the anfwer as the higheft figure of the "" quotient.

:

have

2d, "Multiply the divifor by the figure you "found, and place the product under the part of "the dividend from which it is obtained.

3d, "Subtract the product from the figures above it. 4th, " Bring down the next figure of the dividend to the remainder, and divide the number it makes up, as before."

VOL. II. Part I.

17

504

The numbers which we divide, as 59, 33, and 16, in the first example, are called dividuals.

It is ufual to mark a point under the figures of the dividend, as they are brought down, to prevent miftakes.

If there be a remainder, the divifion is completed by a vulgar fraction, whofe numerator is the remainder, and its denominator the divifor. Thus, in Ex. 3. the quotient is 2671, and remainder 17; and the quo tient completed is 2671.

A number which divides another without a remainder is faid to measure it; and the feveral numbers which measure another, are called its aliquot parts. Thus, 2, 4, 6, 8, and 12, are aliquot parts of 24. As it is often ufeful to difcover numbers which meafure others, we may obferve,

ift, Every number ending with an even figure, that is, with 2, 4, 6, 8, or o, is measured by 2. 2d. Every number ending with 5, or o, is meafured by 5.

3d, Every number, whofe figures, when added, amount to an even number of 3's or 9's, is meatured by 3 or 9, refpectively.

Contractions and Varieties in Divifion.

First, When the divifor does not exceed 12, the whole computation may be performed without fetting down any figures except the quotient.

Ex. 7)35868(5124

or

7)35868 5124