That a pofitive Quantity, Multiplied by a Negative; or a negative Quantity, Multiplyed by a pofitive, produces a Negative may be thus prov'd. If+b, be Multiplied by c, I say the Product -bc. -c, Productp, is Demonftration. fuppofeli b 1 x b2bb = -- bx c + bas by the Nature of Multi that is 3 bbp+ba · 1+c4b+c=a 4xbs bb + be ba 5 bc 6 bbbc ba plication. that is p――be.Q.E.D. 3=67p+ba-bcba; that is p= That a Negative Multiplyed by a Negative Quantity, produces a Pofitive one, may be prov'd thus. If-b. be Multiplyed by-c; I fay the Product p is = +bc. a Note, that fometimes Products are exprefs'd only by the Quantities to be Multiplyed with the Sign x between them, thus the Product of a and b, is a xb, and the Product of 4 + x, and zis=a+xxe むっ and the Product of a +x, m-nty, ande+zz is a + xxmn+yxo+z ‡xxm. e That That a xbis = b xa, may be thus prov'd. Let a confift of any number of Units,as 1,1,1,1,1,1, &c. and b be equal to any other number of Units, as 1,1,1,1,&c. Now if you fuppofe the number of Uuits equivalent to a, written over one another, as often as there are Units in b, I, I, I, I, I, I, I, J, (as you fee in this Scheme to the Right I I, I, I, I, &c.—b I, I, I, I, =b 1, I, I, &c. = b -&c. &c.. And if you fuppofe the number of Units equal to b written one over the other, as often as bibere are Units in a, (as you fee in this Scheme to the Left Hand) their Sum will be = b.x a. But the number of Units in the one Scheme, must be equal to the number of Units in the other; altho' a and b were equal to any two Numbers you please, as is manifeft, by viewing each Scheme. And confequently the Sum of the one, or a xb, b is = to the Sum of the other, or b xa. See this Propofition otherwife Demonftrated in the 16th Propofi tion of the 7th Book of Euclid. CHAP. V. Divifion of whole Quantities. Divifion in Species is the converfe, or direct contrary to that of Multiplication, and confequently perform'd by converfe Operations (as in common Arithmetick.) The general Rule of Divifion is this, Aule. Place the Divifor under the Dividend, with a Line between them (as in Vulgar Arithmetick.) Or place this Sign between the Dividend and Divifor, and let the former be to the Left-Hand of it, or ab, for a Thus a being Divided by b, will give 7, Quotient. And abeg + dg, Divided by 6gf t 2bcg + dg, or=2bcg + dg ÷ 6fg — 39a. 6fg-39a But if any Quantity be found to be a common Multiplyer in both the Dividend and Divifor; place that Quantity, which being Multiplyed by that common Multiplyer, will produce the Dividend, over that Quantity which, being Multiplyed by the faid 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 Jame Signs, the Quotient will be an Unit or 1. N. B. In Divifion of whole Quantities one thing must be carefully obferved, viz. That like Signs givet, and unlike Signs givewhich holds here as well as in Multiplication; as may be proved 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 in Multiplication, can produce the Dividend'; wherefore, if the Dividend be Pofitive, the Divifor and Quotient must bave like Signs, that is, if the Divifor be Negative, the Quotient must be fo too, and of confequence a pofitive Dividend, divided by a negaFive Divifor, gives a negative Quotient; and if the Dividend be Negative, the Divifor and Quotient must have unlike Signs; (as has been been Demonftrated in Multiplication) and confequently a negative Dividend, divided by a pofitive Divifor, gives a negative Quotient 3 but a negative Dividend, divided by a negative Divifor, gives an affirmative or pofitive Quotient. In Divifion of compound Quantities, the Terms as well of the Dividend as Divifor, must be plac'd in order; according to the Dimensions of fome Letter in both of them; that is to fay, the firft, fecond, third, &c. Terms of each must be they which have the firft, fecond, third, &c. greateft power of the faidLetter respectively. Then you must place fuch a Quantity in the Quotient, as being Multiplied by the firft Term of the Divifor fhall produce the firft Term of the Dividend, which Quantity Multiply into the whole Divifor, then Subftract the Product from the refpective Terms of the Dividend; and to the Remainder adjoyn, with their proper Signs, as many more of the next following Terms of the Dividend, as is requifite, and call this Sum your new Dividend. Then find another Term of the Quotient, fuch as being Multiplyed by the first Term of your Divifor, will produce the firft Term of your new Dividend, which Term, when found, place in your Quotient, then Multiply and Substract as aforefaid; and fo pro ceed. Thus, If aaa3aae +3ace + ece, be given to be Divided by ate; I place them in this manner aaa +3aae + zace + cee a + c then I feek fuch a Quantity as being Multiplyed by a (the firft Term of the Divifor) will produce aaa, (the firft Term of the Dividend) and find it to be aa; and therefore I place at in the Quotient, then I Multiply aa by ae, and the Product aaa aae, 1 Subftract from aaa+ 3aae, (the ref pective Terms of the Dividend) and the Remainder is 2aae, to which I adjoin the next Term of the Dividend, viz. 3ace, and the Sum 2aae3aee is my new Dividend. Again, I feek a Quantity, which being Multiplyed by 4, (the firft Term of the Divifor) will produce zaae, (the firft Term of the new Dividend) and find it to be 2ae, which+2ae, I place in the Quotient, then I Multiply it by (or into), and the Product zaae2ace, 1 Subftract from 24ae3ace, and the Remainder is ace, to which I adjoin the next and laft Term of the Dividend, viz.eee, and the Sum aaeeee is my new Dividend. Then I feck a Quantity, which being Multiplyed by a, will produce aee, and finding it to be ee, I place +ee in the Quotient, and Multiply ae by it, and the Product aeeeee, I Sub Atract ftract from my laft foregoing Dividend, and the Remainder is nothing. See the following Operation. Or Division of Quantities may be better perform'd thus, Ex. 12. 34—6) 6aaaa-96 (2aaa+4aa+84†iö Suppofe it was required to divide aaa † 4aac + dan † dda +4dac + ddd, by a + d. If aaa be made the firft Term of the Dividend, + 4can +daa, +4dca+dda, ddd, must be the fecond, third, and fourth Terms of it relpectively; and a, + d, the first and second Terms of the Divifor refpectively; and then the Divifion will ftand thus, |