Suppose it were required to divide aa3bba by af bi a +46 The Work (when prepar'd as before directed) will ftand thus When Fractions are of one Denomination, Caft off the Denominators, and divide one Numerator by the other. abbb Thus, If were to be divided by C (ab the Quotient required. bb it will be bb) abbb By Cafting off cd from both, it will ftand thus PART III. אן CHA P. I.. Involution of whole Quantities, Nvolution is the Raifing or Producing of Powers, from any propos'd Root; and is perform'd in all respects like Multiplication; fave only in this; Multiplication admits of any different Factors, but Involution ftill Retains the fame. 1 & 33 aaa 144aaaa +-aaaa | Biquadrat, or 4th Power. $¦s|aaaaa |—aaaaa | Surfolid, or 5th Power, &c. Note, The Figures plac'd in the Margent, 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 Coefficients, the Coefficients must be Involved along with the Quantities. As in thefe. Thus +8гaaaa 25bbcc! 125bbbccc 625bbbbcece 1044 16aaaa 1055 32aaaaa-243aaaaa 3125bs cs. &c. Involution of Compound Quantities is performed in the fame manner, due regard being had to their Signs and Coefficients. As for Inftance, Suppole a + b were given to be Multiplyed to the 5th Power. 23, or 124aa2ab+bb the Square of a b a+b 5+6, or 1 3 7 aaa+zaab+3abbbbb the Cube of +b, a+b 8 +9, or 1410a++4a2b+6a2b2+4ab3b4 the Biquadrat a+b 10 × a 11 a3 + 4a+b+6a3b2 +4a2b3 + ab↑ 10x12 a+b+4a3b2+6a2b3+4ab4+bs (of a+b the |a3-+5a+b+i0a3b2+1©a2b3+5ab4+bs Surfolid or 5th Powr of ab required. &c. 1 Again, Letaba Residual Root, be given to be Involved. 523—2ab+abb 4x—b 6 — a2b÷zab2—b3 3742-34b+sab-b the Cube of a-b 7x a 8a+ —3a3b+3a2b2. ab3 7x-b9 a3b+3a2b2—3ab3 + b1 1410a1-4a3b +6a2b2-4ab3+b1 the 4th Power of a- b 10x a 1145-444b+6a3b2_4a2b3 + abs Tox-biz a+b+4a3b2—6a2b3 †4ab4_b3 1513 a3 — 5a+b+10a3l·2 —10ab35ab4—bs the 5th Power of a-b, &c. By comparing these two Examples together, you may make the following Obfervations. 1. That the Powers rais'd from a Refidual Root, (viz. the Dif ference of the two Quantities) are the fame with their like Powers rais'd 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; in the Cube a3, a, a; in the Biquadrat 4", a3, a2, a'; &c. f 3. The Indices of the other Quantity b, do continually increase in Arithmetical Progreffion, viz. in the Square it is B, b; in the Cube b, b3, b3; in the Biquadrat b, b2, b3, b1 ; &c. 4. The firft 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 easy 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 Coefficients. For Inftance, fuppofe it were required to raise the Binomial Root ab to the 7th Power; then the Terms of that Power will ftand without their Uncie in this Order, Viz. a? + ab + a2b2 + aa b3 -+ a? b2 + a2 b3 + ab2 + b7 And 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 (which neither Multiplies nor Divides) and all the Powers of any Binomial of Refidual Root are naturally raised 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 Uncia of the Second 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 removed along with their Letters, it also follows; that if they are Added together, their respective Sums will produce the true Uncia 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 easie way of producing the Uncia of any ordinary Power (viz. of one not very high) raised from either a Binomial, or Refidual Root. I : 3 5.10.10 The Uncia of the Square. I I . 4 S I. 5.10.10 The Uncie of the Cube. The Untie of the 4th Power. Uncia of the 5th Power. 6.15.20 .15 .6 6 .15 .20 .15 Uncia of the 6th Power. 7.21 .35 .35 .21 .7.1 Uncie of the 7th Power. And so on in this manner ad infinitum: Now |
