are joined to any, and to their Sum, or Difference, adjoin the irrational or Surd. 3 1+2 312 √ bc 10 b√ acl 4 b√ aa+ce 2 d 3 3√ aa—ccl3 Vaad vaard Cafe 2. When the Surd Quantities are Heterogeneal, (viz. their Indices are unlike) they are only to be added, or fubtracted And from thence will arife Surds by their Signs, viz. + or —. either Binomial, or Refidual, $+23│√be: +√ba|4d√a:+3b√/ac|3√ ac—ba: +√ ac+ba Sect. 2. Dultiplication of Surd Quantities. Cafe 1.HEN the Quant.ties are pure Surds of the fame Kind; W multiply them together, and to their Product prefix their radical Sign. EXAMPLES. √ ba+da Vaa + bb 1x23 √ ba√bcaa+dca a √ a à a a bbbb 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. Il dv bc | 5 c d √ ba + d a 236√a зайса 15 √ ab 5√d 1 x 2 3 3 db √ b c a 15 c d a √ b c aa + d ca al 75 √ a b d Sect. 6. Division 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. 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. Note, If any Square be divided by it's Root, the Quotient will be it's Root. a EXAMPLES. b b + 2 b c + c ca a aa—2bba a + b b b b va √bb+2bc+cc √ √ aa — 2bba a+b+ 1÷23│√ a│√ b b + 2 b c + c c √ √ a + — 2bba a+b+ 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. The Reafon of only taking away the radical Sign, as in Cafe 1. is eafily conceived, if you confider that any Root being involved into itfelf, produces a Square, &c. And from thence the Reafon of thofe Operations performed by the fecond Cafe may be thus ftated. Suppofe bax. Then va= x b 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 b baxx per Axiom 3. Which was to be proved. Again, Let 5d ca=x: Then ca= and ca 5 xx =25dd Allo from hence it will be eafy to deduce the Reason of multiplying Surd Quantities, according to both the Cafes. For 3bď from what is proved above, 5x3bd63bd ✔ b c a = zx, &c. for the reft. Divifion being the Converfe to Multiplication, needs no other Proof. CHA P. V. Concerning the Nature of Equations and bow to prepare them for a 3iution. WH HEN 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 difgulfed 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. Now Now the Knowledge of this difficult Part of the Work is only to be obtained by Practice, and a careful minding the Solution of fuch leading Questions as are in themselves very eafy. And for that Reafon I have inferted a Collection of feveral Queftions; wherein there is great Variety. Having got fo clear an Understanding of the Queftion proposed, as to place down all the Quantities concerned in their due Order, viz, all the fubftituted Letters, in fuch Order as their Nature requires; the next Thing must be to confider whether it be limited or not. That is, whether it admits of more Answers than one. And to discover that, obferve the two following Rules. RULE I. When the Number of the Quantities fought exceed the Number of the given Equations, the Queftion is capable of innumerable Answers. EXAMPLE. Suppofe a Question were propofed thus; there are three fuch Numbers, that if the firft be added to the fecond, their Sum will be 22. And if the second be added to the third, their Sum will be 46. What are thofe Numbers ? Let the three Numbers be reprefented by three Letters, thus, call the first a, the fecond e, and the third y. Then { ate=22 }according to the Question. Here the Number of Quantities fought are three; a, e, y, and the Number of the given Equations are but two. Therefore this Question is not limited, but admits of various Answers; because for any one of thofe three Letters you may take any Number at Pleasure, that is lefs than 22. Which with a little Confideration will be very easy to conceive. RULE 2. When the Number of the given Equations (not depending upon one another) are just as many as the Number of the Quantities fought; then is the Question truly limited, viz. each Quantity fought hath but one fingle Value. As for inftance, let the aforefaid Queftion be proposed thus. There are three Numbers (a, e, and y, as before) if the first be added to the second, their Sum will be 22; if the fecond be added to |