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 fuch 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 THE CHA P. IV. of Surd Duantities. 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 Treatife itfelf, if all the various Explanations that may be of Ufe to make it eafy fhould be inferted; without which it is very intricate and troublesome for a Learner to understand. But now thefe tedious. Reductions of Surds, which were heretofore thought ufeful to fit Equations for fuch a Solution, as was then understood, are wholly laid afide as ufelefs: Since the new Methods of refolving all Sorts of Equations render their Solutions equally eafy, although their Powers are ever fo high. Nay, even fince the true Ule of Decimal Arithmetick hath been well underflood, the Bufinefs 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 Reductions, and only fhew the young Algebraift how to deal with fuch Surd Quantities as may arife in the Solution of difficult Questions. Sect. 1. Addition and Subtraction of Surd Quantities. Cafe 1. WHEN the Surd Quantities are Homoreneal, (viz. are alike) add, or fubtract the rational Part, if they Z 2 are are joined to any, and to their Sum, or Difference, adjoin the irrational or Surd. Cafe 2. When the Surd Quantities are Heterogeneal, (viz. their Indices are unlike) they are only to be added, or fubtracted by their Signs, viz. + or —. And from thence will arife Surds either Binomial, or Residual. 1+23 4d√ a 36 √ ac √bc:+ √ba | 4d√a:+3b√ac | 3√ ac—ba:+√ ac+ba Examples in Subtraction. 2 3 √bc - ✔ balb-d↓ aaa+ca:—d-\-2 a ↓ bd + dd Se&t. Sect. 2. Multiplication of Surd Quantities. Cafe 1.WHEN the Quantities are pure Surds of the fame Kind; multiply them together, and to their Product prefix EXAMPLES. their radical Sign. Cafe 2. If Surd Quantities of the fame Kind (as before) are joined to rational Quantities, then multiply the rational into the rational, and the Surd into the Surd, and join their Products together. EXAMPLES. d√ b c 5cd ✓ ba+da 236a 3a√ca 15 √ ab 5√d 1 x 2 | 3 | 3 db √ b c a | 15 c d aybe a a‡dcaa 175 √ abd Sect. 6. Divifion of Surd Quantities. Cafe 1.WHEN the Quantities are pure Surds of the fame Kind, and can be divided off, (viz. without leaving a Remainder) divide them, and to their Quotient prefix their radical Sign. EXAMPLES. √ bcaa + d ca a aa aa √ba + da Jaa bb 1-2 3 Cafe 2. If Surd Quantities, of the fame Kind, are joined to rational Quantities; then divide the rational by the rational, if it can be, and to their Quotient join the Quotient of the Surd divided by the Surd with it's firft radical Sign. EXAMPLES. 1 1 3 db √ bca | 15 c d a √ b c a a + d ca a 75 √ a b d 3a√ ca 5√d 236√a 1-2131 d√bc 5 c d √ ba + da 15 √ ab Note Note, If any Square be divided by it's Root, the Quotient will be it's Root. a HA 2 1÷213 EXAMPLES. bb+2bc+cc|aaaa-2bba a + b b b b a √bb +2bc+ccl√ a4 -2bba a + b4 | √ al √ b b + 2 b c + c c | √ at −2 b b a a+b4 Sect. 4. Involution of Surd Quantities. Cafe 1.WHEN the Surds are not joined to rational Quantities; they are involved to the fame Height as their Index denotes, by only taking away their radical Sign. Cafe 2. When the Surds are joined to rational Quantities; involve the rational Quantities to the fame Height as the Index of the Surd denotes; then multiply thofe involved Quantities into the Surd Quantities, after their radical Sign is taken away, as before. 2 2 Ib√ a 5d √↓ ca | 3 baa — d d [1]a: 3 b c | 3 d : 3 Ja a + b b 132 aaabc | + 27 dddaa 27 dddbb da: 3/b dddaaab The Reafon of only taking away the radical Sign, as in Cafe 1. is easily conceived, if you confider that any Root being involved into it elf, produces a Square, &c. And from thence the Reafon of thofe Operations performed by the fecond Cafe may be thus ftated. X b Suppose bax. Then a = per Axiom 4. and both Sides of the Equation being equally involved, it will be a = x x Then multiplying both Sides of the Equation into bb, it bb becomes bbaxx per Axiom 3. Which was to be proved. Again, Let 5 d✔ca=x: Then ✔ca = xx 25 dd Alfo from hence it will be eafy to deduce the Reafon of multiplying Surd Quantities, according to both the Cafes. For 6 √ba = xx. which was to be proved. c ==% } Example 1. Cafe 2. d ✓ b c = z 23b√ a = x Z 3√ √ b c = 2/4 ÷ 364 √a = x 36 Divifion being the Converse to Multiplication, needs no other Proof. CHA P. V. Concerning the Nature of Equations, and how to prepare them for a Solution. WHEN any Problem or Queftion is propofed to be analytically refolved; it is very requifite that the true Design or Meaning thereof be fully and clearly comprehended (in all it's Parts) that fo it may be truly abftracted from fuch ambiguous Words as Queftions of this Kind are often difguised with; otherwife it will be very difficult, if not impoffible, to ftate the Queftion right in it's fubftituted Letters, and ever to bring it to an Equation by fuch various Methods of ordering thofe Letters as the Nature of the Queftions may require. 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