Confequently bdb b) d d — b b ( d dd-db1, is the new Numerator. ( +db-bb And bd-bb) ddd-bbb (dd ddd-ddb+d+b the new Denominator. +ddb-bbb ddb-bbd +bbd-bbb Let both be multiplied with b, and then you will have d+b the Numerator, dd+bd+bb the Denominator, of the Fraction required. But if after all Means used (as above) there cannot be found one common Measure to both the Numerator and Denominator; then is that Fraction in it's leaft Terms already. Note, Thefe Operations will be understood by a Learner after he hath paffed thro' Multiplication, and Divifion of Fractions. Sect. 5. Addition and Subtraction of Fractional Quantities. THE 'HE given Fractions being of one Denomination, or if they are not, make them fo, per Sect. 4. Then, RULE. Add or fubtract their Numerators, as Occafion requires, and to their Sum or Difference, fubfcribe the common Denominators as in Vulgar Fractions. Examples in Subtraction. |bb+aa a+b13a+b+c I C 2 1-23 F b b - d+c 26 -a a+b-d 26 2a+c b. с Sect. 6. Multiplication of Fractional Quantities. IRST prepare mixed Quantities (if there be any) by making them improper Fractions, and whole Quantities by fubfcribing an Unit under them; as per Sect. 3. Then, RULE. Multiply the Numerators together for a new Numerator, and the Denominators together for a new Denominator; as in Vulgar Fractions. b C Suppose it were required to multiply 2a+25 with 36+4c. These prepared for the Work (per Sect. 3.) will 6 bac3bb75 b c + 8 ace + 4 b c — 100 cc C 46ba-71b+8ac-100c + N. B. Any Fraction is multiplied with it's Denominator by cafting off, or taking the Denominator away. Thus b a ха gives Sect. 7. Division of Fractional Quantities. THE Fractional Quantities being prepared, as directed in the laft Section. Then, RULE. Multiply the Numerator of the Dividend, into the Denominator of the Divifor, for a new Numerator; and multiply the other two toge ther for a new Denominator; as in Vulgar Fractions. When Fractions are of one Denomination, caft off the Deno minators, and divide the Numerators. Thus, if a 63 were to be it will be bb) a b3 (ab the Quotient required. a3 abb by Cafting off cd in both, it will be a a+ INvolve Nvolve the Number into itself for a new Numerator, and the Denominator into itself for a new Denominator; each as often as the Power requires. bb b 27 b b b c c c b b b + 3 b b d + 3 b d d + d d d aaa8aaaddd aaa заас + засс — ссс IF Sect. 9. Evolution of Fractional Quantities. F the Numerator and Denominator of the Fraction have each of them fuch a Root as is required (which very rarely happens) then evolve them; and their refpective Roots will be the Numerator and Denominator of the new Fraction required. Again! I 27 a a abb baaa+zaab +3 a b b + b b b 8 d d d ааа - 3aab+zabb — b b b 3ab 2 d a+b a-b Sometimes it fo falls out, that the Numerator may have such a Root as is required, when the Denominator hath not; or the Deno minator minator may have fuch a Root, when the Numerator hath not. In thofe Cafes the Operations may be fet down. But when neither the Numerator, nor the Denominator have juft such a Root as is required, prefix the radical Sign of the Root to the Fraction; and then it becomes a Surd; as in the laft Step, which brings me to the Bufinefs of managing Surds. CHAP IV. Of Surd Duantities. THE whole Doctrine of Surds (as they call it) were it fully handled, would require a very large Explanation (to render it but tolerably intelligible); even enough to fill a Treatife itfelf, if all the various Explanations that may be of Ufe to make it eafy should be inferted; without which it is very intricate and troublesome for a Learner to understand. But now these tedious Reductions of Surds, which were heretofore thought useful to fit Equations for fuch a Solution, as was then underfood, are wholly laid afide as useless: Since the new Methods of refolving all sorts of Equations render their Solutions equally eafy, although their Powers are never fo high. Nay, even fince the true Ufe of Decimal Arithmetick hath been well understood, the Butinefs of Surd Numbers has been managed that Way; as appears by feveral Inftances of that Kind in Dr Wallis's Hiftory of Algebra, from Page 23, to 29. I fhall therefore, for Brevity Sake, pafs over thofe tedious Re ductions, and only fhew the young Algebraift how to deal with. fuch Surd Quantities as may arife in the Solution of hard Questions. Se&t. 1. Addition and Subtraction of Surd Quantities. Cafe 1. WHEN the Surd Quantities are Homogeneal, (viz are alike) add, or fubtract the rational Part, if they Z 2 are |