Note, The Figures placed in the Margin after the Sign (w) of Evolution, denote the Index of the Root to be extracted. If the given Powers have Co-efficients, (viz. Numbers prefixed to them ;) then you must extract their refpective Roots, as in Vulgar Arithmetick. But if the Root required cannot be truly extracted out of both the Co-efficients and Indices of the given Power; then it is a Surd, and must have the Sign of the Root required prefixed to it, as 67 at Thus f I w 333a5367 at 2 1 6 b b b d d d Evolution of compound Quantities or Powers, is a little more troublesome than that of fingle Powers; and would require a great many Words to explain the Manner and Reafon of forming the feveral Canons, that are commonly used in extracting the Roots of compound Quantities; especially if the Powers be very high, &c. I fhall therefore for Brevity's Sake omit them, and inftead thereof propose an eafy Method of difcovering the Roots of all compound Powers in general. And in order to that it will be neceffary to premife, that if either the Sum or Difference of feveral Quantities be involved to any Power, there will arife fo many fingle Powers of the fame Height, as there are different Quantities. As for Inftance, if a+b+d be fquared, that is, be involved to the second Power, it will be an+2ab+2 ad+bb+2bd+dd, here you have a a, bb, and dd. Again, if a+b+d'were cubed, viz. involved to the third Power, then you will have aa a, b bb, ddd, in it, &c. Υ Whense Whence it follows, that in extracting the Roots of all compound Quantities, there must be confidered 1. How many different Letters (or Quantities) there are in the given Power. 2. Whether the fingle Powers of each of thofe Letters be of an equal Height, and have in them fuch a fingle Root as is required: which if they have, extract it as before. 3. Connect thofe fingle Roots together with the Sign +, and involve them to the fame Height with the given Power; that being done, compare the new raised Power with the given Power; and if they are alike in all their respective Terms, then you have the Root required; or if they differ only in their Signs, the Root may be eafily corrected with the Sign as occafion requires. -- Example 1. Let it be required to extract the Square Root of cc + 2 cb - 2 c d + b b → 2 b d + dd. In this Compound Square, there are three diftinct Powers, viz. bb, cc, dd, whofe fingle Roots are b, c, d, wherefore I fuppofe the Root fought to be b+c+d, or rathered, because in the given Power, there is 2 cd, and 2 bd, therefore I conclude it is; then b+c-d, being fquared, produces bb + 2 b c -2bd+cc-2cd+dd, which I find to be the fame in all it's Terms with the given Power, although they stand in a different Position; confequently b+c-d is the true Root required. Example 2. It is required to extract the Square Root of a -2 aabb + b2. Here are but two fingle Powers, viz. at and +, whofe Square Roots are a a, and bb. And because in the given Power there is 2 a abb, therefore I conclude it muft either be aa-bbor bb-aa. Both which, being involved, will produce "a+2 aabb +b+; confequently the Root fought may either be aabb, or bba a, according to the Nature or Defign of the Queftion from whence the given Power was produced. - Example 3. Let it be required to extract the Square Root of 36 aaaa + 108 aa+81. Here the two fingle Powers are 36aaaa, and 81, whofe Roots are 6 a a and 9. And because the Signs are all therefore I fuppofe the Root to be 6 aa +9, the which being involved doth produce 36 at 108 aa+81; confequently 6 aa + 9 is the true Root required. Example 4. Suppofe it were required to extract the Cube Root of 125 aaa + 300 a ae 450 aa + 250 ae e -720 a·8 +64 eee +540a-288 ee+432e-216. In this Example there are three diftinct Powers, viz. 125 a a a, 64 eee, and — 216. And the Cube Root of 125 aaa is 5a; of 64 eee is 4 e; of 216 is 6. Wherefore I fuppofe the Root fought to be 5 a +4-6, which being involved to the third Fower, does pro duce duce the fame with the given Power; confequently 5a+4e- — 6 is the Cube Root required. But if the new Power, raised from the supposed Root (being involved to it's due Height) fhould not prove the fame with the given Power, viz. if it hath either more or fewer Terms in it, &c. then you may conclude the given Power to be a Surd, which muft have it's proper Sign prefixed to it, and cannot be otherwise expreffed, until it come to be involved in Numbers. Example 5. Suppofe it were required to extract the Cube Root of 27 a aa +54 baa+8bbb. Here are two diftinct and perfect Cubes, viz. 27 a aa, and 8 bbb, whofe Cube Roots are 3 a and 2 b. Wherefore one may fuppofe the Root fought to be 3 a +2b, which being involved to the Third Power, is 27 a aa +54baa +36bba +8bb. Now this new raised Power hath one Term (viz. 36 bba) more in it than the given Power hath; but this being a perfect Cube, one may therefore conclude the given Power is not fo, viz. it is a Surd, and hath not such a Root as was required, but must be expreffed, or fet down, Thus 3/27 aaa+54baa +8bbb. If thefe Examples be well understood, the Learner will find it very easy, by this Method of proceeding, to discover the true Root of any given Power whatsoever. CHA P. III. Of Algebjaick Fractions, or Broken Duantities. Sect. 1. Notation of Fractional Quantities. FRational Quantities are expreffed or fet down like Vulgar Fractions in common Arithmetick. • Thus {/ 2 b c 5b-4 a Numerators. d 4d+7b Denominators. How they come to be fo, fee Cafe 4, in the laft Chapter of Divifion. Thefe Fractional Quantities are managed in all refpects like Vulgar Fractions in common Arithmetick. Y 2 Scot. Sect. 2. To Alter or Change different Fractions into one Denomination, retaining the fame Value. RULE. MULTIPLY all the Denominators into each other for a new Denominator, and each Numerater into all the Deno minators but it's own for new Numerators. EXAMPLES. Let it be required to bring and into one Denomination, b C Firft axe, and dx b, will be the Numerators, and bx c will ca b d be the common Denominator, viz. and are the two b.c bc bd d = and = Τ ** be brought into one Denomination, bb + b c bd de ad-acbd-bc &c. Sect. 3. To Bang whole Quantities into Fractions of a given Denomination. MUL RULE. ULTIPLY the whole Quantities into the given Denonominator for a Numerator, under which fubfcribe the given Denominator, and you will have the Fraction required. EXAMPLES. Let it be required to bring a + b into a Fraction, whofe Denominator is d-a. First a+bxda is dab d-aa-ba: da+bd-aaba is the Fraction required. Then d-a ab And When whole Quantities are to be fet down Fraction-wife, fubfcribe an Unit for the Denominator. Thus ab is Sect. 4. To Abbyviate, or Reduce Fractional Quantities into their lowest Denomination. RULE. Ivide both the Numerator and Denominator by their greatest common Divifor, viz. by fuch Quantities as are found in both and their Quotients will be the Fraction in it's lowest Term. is And a+bdc=a+d. b.c In fuch fingle Fractions as thefe, the common Divifors (if there be any) are easily discovered by Inspection only; but in compound Fractions it often proves very troublesome, and must be done either by dividing the Numerator by the Denominator, until no thing remains, when that can be done: or elfe finding their common Measure, by dividing the Denominator by the Numerator, and the Numerator by the Remainder, and so on, as in Vulgar Fractions (Sect. 4. Page 51.) aac-aad Suppose cd-dd EXAMPLES. were to be reduced lower. Then e d—dd) a a c—a ad (aa the Fraction aac aad first by an and 88 the quotient is c_d and then a ac -. required gives a, and Scd -dd: - ed by c-a the goo gives d in the In this Example it fo happens that the Numerator is divided juft inte off by the Denominator; but in the next it is otherwise, and requires a double Divifion to find out the common Measure, viz. Let it be required to reduce aaa- abb aa+2ab+bb. Firfta a +2ab+b b) a aa—a bb (a to it's lowest Terms. qaa+2aab tab b |