SUBTRACTION, in Algebra, is performed by changing all the signs of the subtrahend (or conceiving them to be changed) and then connecting the quantities, as in addition.

In the second example, conceiving the signs of the subtrahend to be changed to their contrary, that of 3b becomes +; and so the signs of 36 and 56 being alike, the coefficients 3 and 5 are to be added together, by case 1 of addition. The same thing happens in the third example; since the sign of 3b, when changed, is, and therefore the same with that of 5b. But, in the fourth example, the signs of 3b and 5b, after that of 3b is changed, being unlike, the difference of the coefficients must be taken, conformable to case 2 in addition.

Other examples in subtraction, may be as follow:

In this last example the quantity a2 in the subtrahend, being without a coefficient, an unit is to be understood; for la2 and a2 mean the same thing. The like is to be observed in all other similar cases.

The grounds of the general rule for the subtraction of algebraic quantities may be explained thus: Let it be here required to subtract 5a-3b from 8a+ 5b (as in ex. 2.) It is plain, in the first place, that if the affirmative part 5a were alone to be subtracted, the remainder would then be 8a+ 5b-5a; but, as the quantity actually proposed to be subtracted is less than 5a by 3b, too much has been taken away by 3b; and therefore the true remainder will be greater than 8a+ 5b-5a by sb; and so will be truly expressed by 8a+5b-5a+3b: wherein the signs of the two last terms are both contrary to what they were given in the subtrahend; and where the whole, by uniting the like terms, is reduced to sa+ 8b, as in the example.

SECTION IV.

Of Multiplication.

BEFORE I proceed to lay down the necessary rules for multiplying quantities one by another, it may be proper to premise the following particulars, in order to give the learner a clear idea of the reason and certainty of such rules.

First, then, it is to be observed, that when several quantities are to be multiplied continually together, the result, or product, will come out exactly the same, multiply them according to what order you will. Thus a xbx c, a × c × b, bx c × a, &c. have all the same value, and may be used indifferently: to illustrate which we may suppose a = 2, b = 3, and c 4; then will axbx c 2 X 3 X 2 X 4 X 3 = 24; and bx c xa =

4

= 24; a xc x b 3 X 4 X 2 =

24.

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Secondly. If any number of quantities be multiplied continually together, and any other number of quantities be also multiplied continually together, and then the two products one into the other, the quantity thence arising will be equal to the quantity that arises by multiplying all the proposed quantities continually together. Thus will abc x de = axbx cx dxe; so that, if a was = 2, b = 3, c 4, d = 5, e 6, then would abc x de = 24 X 30 = 720, and a xbx c xdxe = 2 x 3 X4 X5 X6 = 720. The general demonstration of these observations is given be low, in the notes.

The following demonstrations depend on this principle, that if two quantities, whereof the one is n times as great as the other, (n being any number at pleasure,) be multiplied by one and the same quantity, the product, in the one case, will also be n times as great as in the other. The greater quantity may be conceived to be divided inton parts, equal, each, to the lesser quantity; and the product of each part (by the given multiplier) will

The multiplication of algebraic quantities may be considered in the seven following cases.

be equal to that of the said lesser quantity; therefore the sum of the products of all the parts, which make up the whole greater product, must necessarily be n times as great as the lesser product, or the product of one single part, alone.

This being premised, it will readily appear, in the first place, that ba and a × b are equal to each other: for, bx a being b times as great as 1 × a (because the multiplicand is b times as great) it must therefore be equal to 1x a (or a,) repeated b times, that is, equal to a x b, by the definition of multiplication.

In the same manner, the equality of all the variations, or products, abc, bac, acb, cab, bca, cba (where the number of factors is 3) may be inferred: for those that have the last factors the same (which I call of the same class) are manifestly equal, being produced of equal quantities multiplied by the same quantity: and, to be satisfied that those of different classes, as abc and acb, are likewise equal, we need only consider, that, since ac x b is c times as great as axb (because the multiplicand is c times as great) it must therefore be equal to a × b taken c times, that is, equal to a xbx c, by the definition of multiplica

tion.

Universally. If all the products, when the number of factors is n, be equal, all the products, when the number of factors is n+1, will likewise be equal: for those of the same class are equal, being produced of equal quantities multiplied by the same quantity: and to show that those of different classes are equal also, we need only take two products which differ in their two last factors, and have all the preceding ones according to the same order, and prove them to be equal. These two factors we will suppose to be represented by r and s, and the product of all the preceding ones by p; then the two products themselves will be represented by prs and psr, which are equal, by case 2.

1o. Simple quantities are multiplied together by multiplying the coefficients one into the other, and to the product annexing the quantity which, according to the method of notation, expresses the product of the species; prefixing the signor, according as the signs of the given quantities are like or unlike.

Thus, by way of illustration, abcde will appear to be = abced, &c. For, the former of these being equal to every other product of the class, or termination e (by hypothesis and equal multiplication,) and the latter equal to every other product of the class, or termination d'; it is evident, therefore, that all the products of different classes, as well as of the same class, are mutually equal to each other.

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So far relates to the first general observation: it remains to prove, that abcd x pqrst is a xbx cx dx p x q x r x sx t. In order to which, let abcd be denoted by x, then will abcd × pqrst be denoted by x × pqrst, or pqrst x x (by case 1,) that is, by p x q x rx sxtxx; which is equal to x x px q x rx sx t, or a x bx c x d x p × q × r x sxt, by the preceding demonstration.

The reason of Rule 1 depends on these two general observations: for it is evident from hence, that 2a x 3b (in the first example) is = 2 x ax 3 x b = 2 x 3 x a x b = 6 x a × b = 6ab: and, in the same manner, 11adfx 7ab (in the third example) appears to be = 11 xa x d x ƒ x 7 x à x b = 11 x 7 x a xa x b x dxf= 77 × aabdf=77 aabdf. But the grounds of the method of proceeding may be otherwise explained, thus: it has been observed that ab (according to the method of notation) defines the product of the species a, b (in the first example,) therefore the product of a by 3b, which must be three times as great (because the multiplier is here three