Suppose it were required to divide aa -† a 3bba + 4b by a + b. 46 The Work (when prepar'd as before directed) will stand thus, a+b aaa + 4aab + zabb aaa + 4aab + zabb a + 46 sab + 466 +466 When Fractions are of one Denomination, reject the Denominators, and divide one Numerator by the other. abbb Thus, If were to be divided by C ab the Quotient required. Again, fuppofe it was required to divide 444 — abb aa+2ab+bb C -- by PART III. Involution. I CHA P. I. Involution of whole Quantities. Nvolution is the Raifing or Producing of Powers, from any propos'd Root, and is perform'd by Multiplication. I a 1 - 2 2 aa +aa Examples. | the Root, or fingle Power. | Square, or fecond Power. 133aaa | -aaa | Cube, or third Power. 144 aaaa |+aaaa | Biquadrat, or 4th Power. 155aaaaa| —aaaaa| Surfelid, or 5th Power. &c. Note, The Figures plac'd in the Margin after the Sign () of Involution, thew to what Height the Root is Involved; and are call'd Indices of the Power; and are ufually plac'd over the Involved Quantities, in order to contract the Work, especially when the Powers are of high Dimensions. If the Quantities have Co-efficients, the Co-efficients must be Involved along with the Quantities. As in these, Thus Involution of Compound Quantities is performed in the fame Manner, due Regard being had to their Signs and Co-efficients. As for Inftance, Suppofe a+b were given to be Involved to the 5th Power. Thus Ι a+b called a Binomial Root. IX 4 a+b 2aa+ab 3 +ab + bb 5+6 7 aaa+3aab + zabb + bbb the Cube of a+b. a+b 7×4 8 aaaa +3aaab+3aabb+ abbb 7xb 9 +aaab+3aabb+3abbb+bb b b 8 +9 10 aaaa+4aaab+6aabb+4abbb+bbbb a + b 10x a II as† 4a4b + 6a3bb + 4aab3 † ab* 10×12 a4b + 4a3bb + 6aab3 + qaba-|- bs II + 12/13 as+5aab + 10a3bb +10aab3 + 5ab4+bs the Surfolid, or the 5th Power of a+b required. Again, Let ab a Refidual Root be given to be Involved. ba By comparing these two Examples together, you may make the following Obfervations. 1. That the Powers rais'd 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 Quantitics) 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 a2, a1; In the Cube a3, a3, a1; In the Biquadrat a*, a3, a2, a'; &c. 3. The Indices of the other Quantity b, do continually increafe in Arithmetical Progreffion; viz. in the Square it is b', b2; In the Cube b, b, b3; In the Biquadrat b, ba, b3, b4 ; &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 join'd together in the intermediate Terms, are always equal to the Index of the higheft Power, viz. of the first or laft Term.. Thefe Obfervations being duly confider'd, it will be eafy to conceive how the Terms of any propos'd Power rais'd from a Binomial or Refidual Root, muft ftand without their Uncia, or Numeral Figures, or Co-efficients. For Infiance, Suppofe it were required to raife the Binomial Root ab to the 7th Power; then the Terms of that Pow er will stand without their Uncia in this Order; 2 Viz. a7 -- ab -- a3 U2 − 1 − a+ L3 + a3 ba I a2 bs + abε -|- b7. And And because the Uncie (not only of any fingle Letter, but also) of every fingle Power, how high foever it be, is an Unit or í (which neither Multiplies nor Divides) and all the Powers of any Binomial or Residual are naturally rais'd by Multiplying of the precedent Power into its original Root, which is done by only joining each Letter in the Root to the precedent Power with its Uncia, and then removing the faid Power, when it is fo join'd, to the fecond Letter, one place forwards (either to the Left or Right-hand) it muft needs follow, That the Uncie of the fecond Term (in any fuch Power) will always be the Sum of fo many Units added together more one, as there hath been Multiplications of the firft Root, which will always be determined by the Index of the firft Term in the Power. And, because the Uncie of all the intermediate Terms are only remov'd along with their Letters, it alfo follows, that if they are added together, their respective Sums will produce the true Uncia of the intermediate Terms in the new rais'd Power; As doth plainly appear from the following Numbers so remov'd without their Letters; which both fhews and Demonftrates an eafy Way of producing the Uncie of any ordinary Power (viz. of one not very high) raised from either a Binomial, or Refidual Root. Add { Add { Add { Add { Add { Add Thus I I I I the Uncia of the Root, I 4 6 5 .10 .10 the Uncia of the 5th Power. SI6.15.20 .15. 6. 1 the Uncie of the 6th Power. I . 7 .21 .35 .35 .21 7 I the Uncia of the 7th Power. And fo on in this Manner ad infinitum Now |