« PreviousContinue »
That a positive Quantity, Multiplied by a Negative ; or a new gative Quantity, Multiplyed by a pofitive, produces a Negative may be thus prov'd.
If + b, be Multiplied by — 5, I say the Product =p is - bc.
= pt ba
Demonstration. supposes !!! cta Ixb21bb==-box
4 xb/s1bbt be=ba 5 - bc 61bb =- bc + ba 3=6712 +ba=-bctuba; that is p=-bc.Q.E.D.
= That a Negative Multiplyed by' a Negative Quantity, produces e Positive one, may be prov'd thus.
If-bbc Multiplyed by - c; I say the Product p is = + bc.
cat-cx b = C. the above Demon. that is 3 ca +p=
41 =bfc 48&ls s fcb16
=cb. & E. D. Note, that sometimes Products are expressid only by the Quantities to be Multiplyed with the Sign x between them, thus che Product of a and b, is a xb, and the Product of a + %, and 6 zis =a +xx03, and the Product of = •
a + x, moty, and e trzisait xxmantyxot 27. zz
cb - CC
That a xb is = b xa, may be thus prov'd. Let a consist of any number of Units, as 1,1,1,1,1,1, c. and bbe equal to any
I, I, I, I, I, I, &c. otber number of Units, as 1,1,1,1,&c.
I, I, I, I, I, I, c. = Now if you suppose the number of Vuits. I, 1, 1, 1, 1, 1, &c.
1 equivalent to a, written over one an
I, I, 1,'i, 1, 1, $6.
GQGQ other, as often as there are Onits in b, (as you see in this Scheme to the Right li fi Hand) their. Sum will be (by tbe Defi. nition of Multiplication) = 4 xb. I, I, I, 1, 8c. =
And if you suppose the number of Units eI, I, I, i, c.
b qual to b written one over the other, as often as 1, 1, 1, 1,
b i here are Onits in a, (as you see in this Scheme 1, I, I, I,
&c. b to the Left Hand) their Sum will be = 1,1, !,. 1, &c. Eb b. x a. But the number of Units in the I, 1, 1, ‘I, c. = b
one Scheme, must be equal to the number of QQ
Units in the other; altho' a and b were equal to any two Numbers you please, as is manifeft, by viewing each Scheme. And
consequently the sum of the one, or a xk is = to the Sum of the other, ir bx a.
See this Proposition otherwise Demonstrated in the 16th Propofition of the jth Book of Euclid.
CH A P. V.
Division of whole Quantities. Divifion in species is the converfe, or dire& contrary to that of
Multiplication, and consequently perform’d by converse Operations (as in common Arithmetick.) The general Rule of Division is this.
fiule. Place the Divifor under the Dividend, with a Line between them (as in Vulgar Arithmetick.)
Or place this Sign + between the Dividend and Divisor, and let the former be to the Left Hand of it,
Thus a being Divided by b, will give - or a Ċ b, for a Quotient. And abeg + dg, Divided by 6gf +
394, is =
zbog + dę, or = zbog + dg = 6fg 6fg - 394
. Bur if any Quantity be found to be a common Multiplyer in both the Dividend and Divisor ; place that Quantity, which being Multiplyed by char common Multiplyer, will produce the Dividend, over that Quantity which, being Multiplyed by the said common Multiplyer, will produce the Divisor, with a Linc between them
ab + cdb
56 a tica
50 tad Note, when the Quantities in the Divisor and Dividend are all ebe fame, and have the Jame Signs, the Quotient will be an Unit or i.
Examples. 8. b; b=1
N. B. In Division of whole Quantities one thing must be careful, obferved, viz. That like Signs give t, and unlike Signs give which holds bere as well as in Multiplication; as may be proved thus,
Every Dividend being a Produ&t made by Multiplying the Quotient into the Divifor, the Sign of each Factor must be such, 4s according to the former Rules in Multiplication, can produce the Dividend; wherefore, if the Dividend be Positive, the Divisor
' and Quotient muff bave like Signs, that is, if the Divifor be Negative, the Quotient must Be fo too, and of consequence a positive Dividend, divided by a negaSive Divifor, gives a negative Quotient, and if the Dividendb3 Negarlythe Divisor and Quotient must have unlike Signs ; (as has
been Demonstrated in Multiplication) and consequently a negatide Dividend, divided by a positive Divisor, gives a negative Quotient ; but a negative Dividend, divided by a negative Divifor, gives an affirmative or positive Quotient.
In Division of compound Quantities, the Terms as well of the Dividend as Divisor, must be plac'd in order ; according to the Dimensions of some Letter in both of them; that is to say, the first, second, third, c. Terms of each must be they which have the first, second, third, &c. greatest power of the said Letter respectively.
Then you must place such a Quantity in the Quotient, as being Multiplied by the first Term of the Divisor shall produce the first Term of the Dividend, which Quantity Multiply into the whole Divisor, then Subftract the Product from the respective Terms of the Dividend; and to the Remainder adjòyn, with their proper Signs, as many more of the next following Terms of the Dividend, as is requisitė, and caļl this Sum your new Dividend. Then find another Term of the Quotient, such as being Multiplyed by the first Term of your Divisor, will produce the first Term of your new Dividend, which Term, when found, place in your Quotient, then Multiply and Subftract as aforesaid; and fo proceed.
Thus, If ana t 3a9e + zace + ece, be given to be Divided by a fe; I place them in this manner daa+3aae + zace +
a + c then I seek such a Quantity as being Multiplyed by a (the first Term of the Divisor) will produce daa, (the first Term of the Dividend) and find it to be a4 ; and therefore I place. ar in the Quotient, then | Multiply aa by ste, and the Product aaa t aae, 1 Subftract from ada + zaae, (the refa 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 2aae + 3aee is my new Dividend. Again, I seek a Quantity, which being Multiplyed by a, (the first Term of the Divisor) will produce zade, (the firft Term of the new Dividend) and find it to be + 2ac, which + 2ae, I place in the Quotient, then I Multiply it by (or into) a te, and the Product zdaě 28cc, 1 Substract from 24ae + zace, and the Remainder is aée, to which I adjoin the next and lait Term of the Dividend, viz. + eee, and the Sum 41c + eee 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 + ce in the Quotient, and Multiply a to by it, and the Product des feed, 1 Sub
stract from my last foregoing Dividend, and the Remainder is nothing. See the following Operation,
Quotient 1aaa + gaae + 3ace + eee (aa+2act-
4 + 34ee s 124ae + 3ace
7! ot nee
8aee teee 2 x ee 9aee + eee 8 9. Io
Or DiviGon of Quantities may be better perform'd thus,
Suppose it was required to divide aaa + 4aac + dai + dda +4dac + ddd, by a + d.
If aaa be made the first Term of the Dividend, + 4can + daa, + 4dca + dda, + ddd, must be the second, third, and fourth Terms of it respectively, and a, + d, the first and second Terms of the Divisor respectively; and then the Division will ftand thus,