Sect. 4. Division 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 those 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. ababad + b dj-ad-bd Thus 1-2 -d + ala +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. 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. found in the Dividend; then fet them both down like a Vulgar Fraction, as in common Arithmetick. N. B. In Divifion one thing must be very carefully observed; 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 Division, in Vulgar Arithmetick. Examples of Divifion at large. |1|21 ba+ 15 da 35 bf 25 df (+ 3 a 2 7b+5d 2 x 3 a 321 ba+ 15 da 3 4 2x-5/15 4-5 6 2 734-5f the Quotient collected from the 3, and 5, Steps. Or Divifion of Quantities may stand as Numbers in common Arithmetick do; thus 3a6) 6 aaaa - 96 (ż aaa+4a a +8 a +16 6aaaa12aaa That is, 6 a aa-963a-6 gives za aa+4 aa +8a+ 16 for the Quotient, as may eafily be proved by Multiplication, viz. 2 aaa+4 aa+8a+ 16×3 a-6 will produce 6 a*—96; and fo for the rest. Sect. 5. Involution of whole Quantities. 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 ftill 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 ufually 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 thefe, 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 a+b were given to be involved to the fifth Power. Thus a+b called a Binomial Root. 14 aa+2ab+bb, the Square of a +6 a+b 4 x a 5 aaa+za a b + abb 4x66 + aab+ z a b b + b b b 1037aaa+za ab + 3 abboob, the Cube of a+b 7 xa 7aaa+3aab + 3 a b b + b b b 8a+ + 3 a3 b+3aab b + a b b b 7 × b 9 + a3b+3a a b b + z a b b b + b 4 1510a++ 4 a' b + 6 a a b b + 4 a b b b +b+ a+b 10 x a II I X b 12 as + 4a+b+6 a3 b b + 4 a a b3 +ab+ 1 @ 13a +5 a* b+10 a3 b b+10a a b3 +aba +63 &c. Again let ab, called a Refidual Root, be given. 7aaa aabzabb bbb 3aab3abb a a a — 3 a a b + 3 a b b — b b b, the Cube of a—¿ .b 4aaab6a abb -b 4 a + b + 6 a3 b b a+b+ 4a3 b b 5 a+b+10 a3 b b — 10 &c. a a b3 + 5 ab+_65 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 raised 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 decrease in Arithmetical Progreffion; viz. in the Square it is a a, a: In the Cube a a a, a a, a: In the Biquadrat a a a a, aaa, a a, a, &c. 3. The Indices of the other Quantity b do continually increase in Arithmetical Progreffion; viz. In the Square it is b, bb: In the Cube b, bb, bbb: In the Biquadrat b, b b, b b b, b b b b, &c. 4. The first and laft Terms, are always pure Powers of the fingle Quantities, and are both of the fame Height. 5. The Sum of the Indices of any two Letters joined together in the intermediate Terms, are always equal to the Index of the higheft Power, viz. of the firft or laft Term. Thefe Obfervations being duly confidered, it will be easy to conceive how the Terms of any propofed Power raised from a Binomial or Residual Root muft ftand, without their Uncia or Numeral Figures. For Inftance, fuppofe it were required to raise the Binomial Root a+b to the feventh Power; then the Terms of that Power will ftand without their Uncia in this Order. Viz. a2+ab+a3b2+a^ b3 + a3 b++ a2 b3+ab+b2. And because the Uncia (not only of any fingle Letter, but also) of every fingle Power, how high foever it be, is an Unit or i (which neither multiplies nor divides) and all the Powers of any Binomial or Refidual Root are naturally raised by multiplying of the precedent Power into it's original Root, which is done by only joining each Letter in the Root to the precedent Power, with it's Uncia, and then removing the faid Power, when it is fo joined to the fecond Letter, one place forward (either to the left or right Hand) it muft needs follow, That the Uncia of the fecond Terms (in any fuch Powet) will always be the Sum of fo many Units added together more one, as there have been Multiplications of the firft Root; which will always be determined by the Index of the firft Term in the Power. And because the Uncia of all the intermediate Terms, are only removed along with their Letters, it alfo follows; that if they are added together, their refpective Sums will produce the true Unciæ of the intermediate Terms in the new raised Power. As doth plainly appear from the following Numbers fo removed without their Letters; which both fhews and demonftrates an eafy Way of producing the Uncia of any ordinary Power (viz. of one not very high) raifed from either a Binomial or Refidual Root, Thus |