When the given Surds are Incommenfurable, neither their Sum nor Difference can be expreft by any Simple Root; but they are to be added by +, and Subftracted by ; whence arife Surds, Binomial and Refidual. Thus the Sum of √6 and 7 is 6+7, and the difference of 6 and 7 is 7 -6. 7—61. c. I But Univerfally the Sum of land is (manifeftly). Also b3 — c3 is = b + c — 2 b = " The Arithmetick of Compound and Univerfal Surds, depends upon the Rules above given about Simple Surds. + because the Square of b2 + c2 or V8+ Vc is b+2 √bel+c, or because the Squares of b = + c = and b+c+2bc) are e= qual # because the Cubes of b+c+383 ( ++385 (3) 3 and ·squal. PART VII. Concerning the Nature of Equations, and how to prepare them for a Solution. AN Equation is the mutual comparing of two equal Quantities differently expreft, (which are call'd the two Parts the eof and is ufually denoted with the Sign between them, and the fingle Quantities of both Parts are call'd the Terms of the Equation; Thus ac+d+eg is an Equation, In which the angle Quantity a, one Part thereof, is equal to the Compound Quantity c+d+eg, the other Part; and the Terms of both Parts, are the fingle Quantities a, c, d3 and eg. An Equation is two-fold, viz. 1. Pure or Simple, or 2. Affected or Compound. A Pure or Simple Equation, is that wherein the Quantity fought (as fuppose a) is expreft by one Power only, as by the ift, 2d, 3d, &c. Power; thus 24+c=d+f, and ca3 = db, are Simple Equations. An Affected or Compound Equation is that in which there are two or more different Degrees or Powers of the Quantity fought (4), As in this Equation a + 2a2 — ca3 cd, there are three different Powers of a, viz. à3, a2 and a3s. When any Question or Problem is propos'd to be Analytically refolved; it is, very requifite that the true Defign, or meaning thereof, be fully and clearly comprehended (in all its Parts) that fo it may be truly abftracted from fuch ambiguous Words, as Questions of this kind are often difguis'd with; otherwise it will be very difficult, if not impoffible, to ftate the Question Right in its fubftituted Letters, and ever to bring it to an Equation, by fuch various Methods of ordering thofe Letters, as the Nature of the Queftion may require. 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 Queftions, as are in themselves very easy. Having got fo clear a Notion of the Queftion propos'd, as to place down all the Quantities concern'd in their due Order ; viz. ; all all the Subftituted Letters in fuch Order, as the Nature of the Queftion 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 following Rules.. Kule 1. When the Number of Quantities fought, exceed the Number of the given Equations, the Question is capable of innumerable Answers. Example. Suppofe a Queftion were propos'd thus; There are three fuch Numbers, that if the First be added to the Second, 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 Second e, and the Third y. Then ate=223 according to the Question, 46. Here the Number of Quantities fought are Three, viz. a, e, and y, and the Number of the given Equations are but two; therefore this Question is not Limited, but admits of various Anfwers; because for any one of those three Letters, you may take any Number at Pleafure that is less than 22, which with a little Confideration, will be very eafy to conceive. Rule 2. When the Number of the given Equations (not depending upon one another) are juft 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 Question be propos'd thus. J There are three Numbers (a, e, and y, as before) if the First be added to the Second, their Sun will be 22; if the Second be added to the Third, their Sum will be 46; and if the First be added to the Third, their Sum will be 36. What are the Numbers ? That is, a te= 22; e + y = 46; and 4+y=36. Now Now the Question is perfectly Limited, each fingle Quantity having but one fingle Value, to wit a=6; c=16; and y = 30. N. B. If the Number of the given Equations exceeds the Number of Quantities fought; they not only limit the Queftion, but oftentimes render it impoffible, by being propos'd inconsistent one to another. Having truly ftated the Queftion in its fubftituted Letters, and found it Limited to an Antwer (or at least so bounded as to have a certain determinate Number of Anfwers) then let all thofe fubftituted Letters be fo ordered or compared together, either by Adding, Subftracting, Multiplying or Dividing them, &c. according as the Nature of the Question requires, until all the unknown Quantities except One, are caft off or vanished; bur therein great Care must be taken to keep them to an exact Equality; and when that unknown Quantity, or fome Power of it (as Square, Cube, &c.) is found equal to thole that are known; then the Question is faid to be brought to an Equation, and confequently to a Solution, viz, fitted for an Anfwer. But no particular Rules can be prefcribed for the cafting off, or getting away Quantities out of an Equation; that part of the Art is only to be obtained by Care and Practice. And when that is done, it generally happens fo, that the unknown Quantity which is retained in the Equation, is fo mix'd and entangled with those that are known; that it often requires fome Trouble and Skill to bring it (or its Powers, &c.) to one Side of the quazion, and thole that are known to the other Side; (ftill keeping them to a juft Equality) which the ingenious Van Schooten in his Principia Mathefeos Univerfalis calls Reduction of Equations, The Bufinefs of reducing Equations (as of moft, if not all Algebraick Operations) is grounded and depends upon a right Application of the five Axioms propofed in Page 6. and therefore, if thofe Axioms be well underftood, the Reafon of fuch Operations must needs appear very plain, and the Work be easily perform'd; as in the following Sections. a Sect. 1. Of Reduction by Addition. Reduction by Addition is grounded upon Axiom 1, and is only the Tranfpofing (viz. the removing) of any Negative Quantity from either Side of an Equation to the other Side, with the Sign+before it; as in thefe Examples Suppole 244 -- dd + b + dc dcc3baa — aaa - TCC звал ++ aaa 2 aaa +2da - d Sect. 2. Heduction by Subftracion. Reduction by Subftraction is grounded upon Axiom 2. and perform'd by Tranfpofing (or removing) any Affirmative Quantity from either fide of the Equation, to the other, fide, with the Sign before it. As in thele Let |1|a2 +d=c2 + 3ba2 + 2da 1—3b4 — 2da — d [2] a2 — 3ba2 — 2da — c2 — d Sect |