1 x 23154-30=24 54-27=271-54+27=-27 That a Pofitive Quantity Multiplyed by a Negative, or a Negative Quantity Multiplyed by a Pofitive, produces a Negative one, may be thus prov'd. Ifb be Multiplyed by -c, I fay the Product P is= -bc. 2 x b3bb=ba+bx-e, by the Nature of Multiplication. that is 4 bbba + p Ixb5bbbc = ba 5-bc6bbba - b c 4=67│ba+p=ba-bc; that is p―b c. Q. E. D. That a Negative Multiplied by a Negative Quantity, produces 4 Pofitive one, may be prov'd thus. Ifb be Multiplyed by -c, I fay the Product P +bc. Putab + c I 1 -b2a- b=c Demonftration. is= 2 x -c13 ca + bx-c-cc, by the above Demonftra ca+p==cc c15 I can 5+cb6 cacb= cc 4=617 (tion. ca+p=ca+cb pcb. Q. E. D. Note, that fometimes Products are exprefs'd only by the Quantities to be Multiplyed with the Sign x between them, and each Compound Factor being connexed by Colons, or a Line over it: Thus the Product of a and b, is a xb, and the Product of a+x, and e- is : a-t-x:x:e - : or axxe СНАР: CHAP. V. Division of whole Quantities. DIvifion in Species is the Converfe, or direct contrary to that of Multiplication, and confequently perform'd by converse 0perations (as in common Arithmetick.) The general Role of Divifion is this; viz. Place the Divifor under the Dividend, with a Line between them, and this Fraction is the Quotient (as in common Arithmetick.) Or place the Sign between the Dividend and Divifor, and let the latter be to the Right-hand of it; Or place the Sign) between the Dividend and Divifor, and let the latter be to the Left-hand of it; Thus a being Divided by b, will give, or ab, or b) a for a Quotient: And 2 beg dg Divided by 6 g f − 3 g a, is = 2 b c g + d g or : 2bcgdg:68f-38ai, 6gf-384 &c. But if any Quantity be found to be a common Multiplyer in both the Dividend and Divifor; Place the Quantity which, being Multiplyed by that common Multiplyer, will produce the Dividend over that Quantity which, being Multiplyed by the common Multiplyer will produce the Divifor, with a Line between them. Note, When the Quantities in the Divifor and Dividend are all the fame, and have the fame Signs, the Quotient will be an Unit or 1; because every thing contains it felf once: So b÷b= 1; also: a+b:: a b = 1, and : a a b FI. N.B. In Division of Quantities one thing must be carefully obServed, viz. That like Signs give, and unlike Signs give —, which holds here as well as in Multiplication, as may be prov'd thus. Every Dividend being a Product made by Multiplying the Quotient into the Divifor, the Sign of each Factor must be fuch, as according to the former Rules of Multiplycation, can produce the Dividend: Wherefore if the Dividend be pofitive, the Divifor and Quotient must have like Signs; that is, if the Divifor be Negative, the Quotient must be fo too; and of confequence a pofitive Dividend divided by a negative Divifor, gives a negative Quotient. And if the Dividend be Negative, the Divifor and Quotient must have unlike Signs (as has been Demonftrated in Multiplication); and confequently a negative Dividend divided by a pofitive Divifor, gives a negative Quotient; but a negative Dividend divided by a negative Divifor, gives an affirmative or pofitive Quotient. In Divifion of Compound Quantities, the Terms of, as well the Dividend as Divifor, muft be plac'd in order, according to the Dimensions of fome Letter in both of them, which will be judg'd the most Commodious for the Purpofe; that is to fay, the first, fecond, third, &c. Terms of the Dividend, and of the Divifor, must be thofe which contain the greateft, greatest but one, greateft but two, &c. power of the faid Letter refpectively. Then feek fuch a Quantity, as being Multiplyed by the first Term of the Divifor, thall produce the firft Term of the Dividend, which Quantity, when found, place in the Quotient, and then Multiply it into the whole Divifor, the Product fubtract from the refpective Terms of the Dividend; and to the Remainder adjoin, with their proper Signs, as many more of the next following Terms of the Dividend as are requifite, and call this Sum your Dividual. Again, feek another Term of the Quotient, viz. Such as being Multiplyed by the firft Term of the Divifor, will produce the first Terin of your Dividual, which Term, when found, place in the Quotient; then Multiply, and Subtract as aforefaid; and fo proceed. Thus, If a aa3aae + 3 aeeeee be given to be Divided by a e, I place them in this Manner, aaa3aae + 3 a ce e e e a t e Then Part I: Then I feek fuch a Quantity as, being Multiplyed by a (the firft Term of the Divifor) will produce a a a (the first Term of the Dividend) and finding it to be a a, I place a a in the Quotient, then I Multiply ae by a a, and the Product a a a aae I fubtract from a 4 a + 3a a e (the respective Terms of the Dividend) and the Remainder is 2 a a e; to which I adjoin the next Term of the Dividend, viz. + 3a e e, and the Sum 2 a ac3 a ee is my Dividual. Again I feek a Quantity, which, being Multiplyed by a _(the firft Term of the Divifor) will produce 2 a a e (the firft Term of my Dividual) and find it to be + 2 a e, which - 2 a e I place in the Quotient, then I Multiply it by (or into) a † e, and the Product 2.a a e + 2 a e e, I fubtract from 2aae+3ace, and the Remainder is a e e; to which I adjoin the next and last Term of the Dividend, viz. + e e e, and the Sum aeeeee is my Dividual. Then I feek a Quantity which, being Multiplyed by a, will produce a ee (or ask how ofta is contained in aee) and finding it to be ee, I place e e in the Quotient, and Multiply ae by it, and the Product aeeeee I Subtract from my last Dividual, and the Remainder is nothing. See the following Operation. 2x443 Example 11. Quotient I aaa +3aae +3aee +eee (aa † 2ae+ e e 314 aaa + zaaе a a a + a ae 。 +2aae 2aae +3 ace 2aae2aee 。 + aee aee teee 2xee 9 aee teee 8 910 Or Divifion of compound Quantities may be better perform'd thus. Ex. 12. Suppose it were required to divide aaa +4caa + daa + 4cda +dda + ddd by a + d. If aaa be made the firft Term of the Dividend, +4caa +daa, +4cda + dda, ddd must be the fecond, third, and fourth Terms of it refpectively; and a, +d the first and second Terms of the Divifor refpectively: And then the Divifion will ftand thus, +4caa 4cda a+d) +da+da+ddd (aa+4ca+dd Here the Members of the fecond Term, as alfo thofe of the third are united by adding in each Term the Factors of the Powers of the Letter a, in refpect of which the Terms of the Divident were plac'd. D But 1 |
