I Examples in Subtraction. dta 20 to atbd d+c d dta b atd I-23 d to d dta u 2 a to Sect. 6. gultiplication of Fractional Quantities. FIRST prepare mixed Quantities (if there be any) by making them improper Fractions, and whole Quantities by subscribing an Unit under them; as per Sect. 3. Then, RUL E. I Multiply the Numerators together for a new Numerator, and the Denominators together for a new Denominator ; as in Vulgar Fractions. Thus ab 132 — 26 2d+c 40 + 26 d 2 a 6 - 466 1 x 2 3 cf 2 ddt de с 25 with b Suppose it were required to multiply 20+ 36+46. These prepared for the Work (per Sect. 3.) will itand | 2 actb-25 I c Thus 36+40 6bac+365-7500+80cc+46c-100CC 2 x 2 3 2 N. B. Any Fraction is multiplied with it’s Denominator by cafting off, or taking the Denominator away. Thus xa gives b Sect. 7. Division of Fractional Quantities. THE HE Fractional Quantities being prepared, as directed in the lait Section. Then, R U L E. Multiply the Numerator of the Dividend, into the Denominator of the Divifor, for a new Numerator ; and muliiply the other two together for a new Denominator ; as in Vulgar Fractions. EXAMPLES. abd ab Let be divided by the Work may stand thusg of ab'd rabdo d per Sect. 4. of labe f labd bbb Or thus i of d abt66 a 46 atb аа- а 3 abb Suppose it were required to divide a at by a + b. a + 4b The Work will ffand thus, aaa + 4 a ab+3abb paaa+4 a a 2a a +4 a abt 3abb But a +46 aatsbat 466 aa a + 4 a ab +3abb a a +36 a á + 5 bat 406 (per Sect. 4.) a +46 When Fractions are of one Denomination, caft off the Denominators, and divide the Numerators. Thus, if were to be a h divided by by bol it will be ! b) a b) (ab the Quotient required. z For 83 za b c For But =ab(per Sect. 4.) a3 abo Again, suppose it were required to divide by 6-8 aa + 2ab +66 Casting off c d in both, it will be eat cd ba 2 ab +66) a aarabb &c. atb Sect. 8. Involution of Fractional Quantities. À 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. 3bc 6td Thus I 2 4 Q add 66 19bbcc 166+2bd + dd 2 ar too 66 27 bbbccc1868 +366d+36dd todd 3 a a a18 aa add dla a a заас+3асс — ссс I 3 Sect. 9. Evolution of Fractional Quantities. IF F the Numerator and Denominator of the Fraction have each of them such a Root as is required (which very rarely happens) then evolve them; and their respective Roots will be the Nume rator and Denominator of the new Fraction required. 9 a abblaa + 2ab +66 Thus 2 abtbb a I 6 27 a a abbbja a a +32ab +3abb+bbb Again I 8 d d d la a a — 3 aab +3 abb-bob I 4dd +6 2 Sometimes it so 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 such a Root, when the Numerator hath not, In those Cases the Operations may be set down, But when neither the Numerator, nor the Denominator have just 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 last Step, which brings me to the Business of managing Surds. CHA P. IV.. Of Surd Duantities. THE HE 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 Treatise itfell, if all the various Explanations that may be of Use to make it easy thould be inserted; 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 such a Solution, as was then understood, are wholly laid afide as useless : Since the new Methods of resolving all Sorts of Equations render their Solutions equally easy, although their Powers are never so high. Nay, even fince the true Use of · Decimal Arithmetick hath been well understood, the Business of Surd Numbers has been managed that Way; as appears by several Inftances of that Kind in Dr Wallis's History of Algebra, from Page 23, to 29. I shall therefore, for Brevity Sake, pass over those tedious Rex ductions, and only thew the young Ålgebraift how to deal with fuch Surd Quantities as may arise in the Solution of hard Questions. Sect. 1. Addition and Subtraction of Surd Quantities. Case1. WHEN the Surd Quantities are Homogeneal, (virgo are alike) add, or subtract the rational Part, if they arc are joined to any, and to their Sum, or Difference, adjoin the Examples in addition. 7 , bcl abraczba aatio 1+213112 V bclicóva0|4hvaa toc 2 Cafe 2. When the Surd Quantities are Heterogeneal, (viz, their Indices are unlike) they are only to be added, or subtracted by their Sigos, viz. + or -- And from thence will arise Surds cither Binomial, or Refidual. 1 Examples ire additioii. a caba Vact ba Examples in Subtraction. b-dvaaatta d-2avba+dd 1---213 1vbe-bab-dvāa a toca ;-d+2avbd + dd Seet. |