Or these may be otherwife proved by Numbers. Thus, fuppofe {20} and {12}{or any other Num Then bers. a=b=6 c―d=4 per Axiom 2. Confequently, a-bxc-d=6x4=24, per Axiom 3. but a-bxc-d, according to the precedent Rules, will be, ac-cb+bd-da, which if true must be equal to 24. Proof {&d=14x ac 20 x 12 = 240 8=112 cb= 12 x 14 = 168 160 Hence ac+bd=352 per Axiom 1. And Note, If the Multiplier confifts of feveral Terms, then every one of those Terms must be multiplied into all the Terms of the Multiplicand; and the Sum of those particular Products, will be the Product required, as in Common Arithmetick. 35bf-25 df 3+45 aa-da−bb+db 21 ba+15 da—35 b f→ 25 df Sect. 4. Divifion of whole Duantities. Divifion of Species, is the converfe or direct contrary to that of Multiplication, and confequently is performed by converse Operations, (as in common Arithmetick) and admits of four Cafes. Cafe 1. When the Quantities in the Dividend, have like Signs to thofe in the Divifor, and no Co-efficients in either; caft off or expunge all the Quantities in the Dividend, that are like thofe in the Divifor; and fet down the other Quantities with the Sign + for the Quotient required. 1-2 3 a + aa +b la+b Cafe 2. When the Quantities in the Dividend have unlike Signs to thofe in the Divifor; then fet down the Quotient Quantities found as before, with the Sign before them. a b c + b c d + b c f bc -adf Cafe 3. If the Quantities in the Dividend and Divifor, have Co-efficients; divide the Numbers (as in common Arithmetick) and to their Quotients adjoin the Quotient Quantities. Note, When the Quantities and Co-efficients in the Divifor and Dividend are all the fame, the Quotient will be an Unit, or 1. Cafe 4. When the Quantities in the Divifor cannot be exactly found in the Dividend; then fet them both down like a Vulgar Fration, as in common Arithmetick. N. B. In Divifion one thing must be very carefully obferved; viz. that like Signs give + and unlike Signs give in the Quotient; which needs no other Proof than that already laid down in the laft Section, if duly compared with what hath been faid concerning Multiplication and Divifion, in Vulgar Arithmetick. Examples of Division at large. |1|21 ba+ 15 da- 35 bf-25 df (+3 a 2 76+5d 4-5 6 O I - 2 734-5f the Quotient collected from the 3, and 5, Steps, Or Division of Quantities may ftand as Numbers in common 3 @~6) 6 aaaa-96 (z aaa+4aa + 8a+ 16 бааая +o + -12a a a 12 a da - 96 That is, 6 a aa—963 a—6 gives a aaa+4aa+8a+ 16 for the Quotient, as may eafily be proved by Multiplication, viz. 2 aaa+ 4aa+8a+ 16 x 3 a 6 will produce 6 a+—96; and fo for the rest. Sect. 5. Involution of whole Duantities. Nvolution is the raifing or producing of Powers, from any propofed Root, and is performed in all refpects like Multiplication, fave only in this: Multiplication admits of any different Factors, but Involution Rill retains the fame. EXAMPLES. Note, The Figures placed in the Margin, after the Sign () of Involution, fhew to what Height the Root is involved; and are called Indices of the Power; and are usually placed over the involved Quantities, in order to contract the Work, especially when the Powers are any thing high. If the Quantities have Co-efficients, the Co-efficients must be involved along with the Quantities, as in these, Involution of Compound Quantities is performed in the fame manner, due regard being had to their Signs and Co-efficients, if there be any. As for inftance, fuppofe ab were given to be involved to the fifth Power. Thus ab called a Binomial Root. 124 aa+2ab+bb, the Square of a+b 7aaa + za ab+3aba+000, the Cube of a+b 7aaa+za a b + 3 a b b + bbb 8a+ + 3 a3 b + 3a a b b + 9 10 II 12 a a2 abbb + a3b+3a a b b + z a b b b + b+ + b + 4a+b+ 6 a 3 b b + 4 a a 63 13 as +5 a+ b + 10 a3 b b + 10 aab3 +ab++b2 &c. Again let ab, called a Refidual Root, be given. 4 x a 4 x →→→ 123 86 7aaa-zaab+3abbbbb, the Cube of a-b аааа- 3aaabza abb b abb b +za abb z a b b b + b b b b 4 a b b b + b b b b 5 a+b+10a3 bb10a a b3 +5 a ba—65 &c. By comparing these two Examples together, you may make the following Obfervations. 1. That the Powers raised from a Refidual Root (viz. the Difference of two Quantities) are the fame with their like Powers raifed from a Binomial Root (or the Sum of two Quantities) fave only in their Signs; viz. the Binomial Powers have the Sign + to every Term, but the Refidual Powers have the Signs and -interchangeably to every other Term. 2. The Indices of the Powers of the leading Quantity (a) continually decreafe in Arithmetical Progreffion; viz. in the Square |