I But Univerfally the Sum of b and c is manifestly I fo bc is =√:b+c= 2√bc: And b+c3is= The Operations of Compound and Univerfal Surds are fo eafily deduc'd from what hath been said of Simple ones, that I do not think it worth while to infert them. Add to this, that 'tis very difficult to print them, which was one Motive I had for not publishing them in the firft Edition of this Book, 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 exprefs'd, which are call'd the Parts or Sides thereof, and is ufually denoted with the Sign = between them; and the fingle Quantities of both Parts are called the Terms of the Equation. Thus a=c3+eg+d is an Equation, in which the fingle Quantity a, one Part thereof, is equal to the Compound Quantity c3+eg+d the other Part, and the Terms of both Parts are the fingle Quantities a, c3, eg and d. Equations are to be confider'd chiefly after two ways; viz. either as the laft Conclufions to which you come in the Refolution of Questions; or, as means by the help whereof you are to obtain final or fingle Equations. An Equation of the former Kind, is compos'd only of one unknown Quantity, Involv'd with known ones, if the Question be determin'd, and propofes fomething certain to be found out. But thofe of the latter Kind, Involve two or more unknown Quantities, which, for that Reafon, must be compar'd among one another, and fo connected, that out of all these may emerge a new Equation, in which there is only one unknown Quantity, which we feek, mix'd with known Quantities; which unknown Quantity, that it may be the more easily discover'd, that Equation must be transform'd, moft commonly various Ways, until it becomes the moft Simple that it can: And when that is done, the Equation is faid to be reduc'd, or fitted for an Answer. A fingle Equation is two-fold, viz. Pure or Simple, or Adfected or Compound. A Pure or Simple fingle Equation is that wherein the Quanrity fought (as fuppofe a) is exprefs'd by one Power only, as by the 1ft, ad, or 3d, &c. Power: Thus 2a+c=d-ƒ, and cadbare Simple Equations. An An Adfected, or Compound Equation is that in which there are two or more different Degrees, or Powers of the Quantity fought (a), as in this Equation, a32a - ca=cd, there are three different Powers of a, viz. a3, a' and a1. When any Question is propos'd to be refolv'd, it is requisite that the true Design and Meaning thereof be fully and clearly comprehended, fo as you may be able to place down all the Quantities concern'd in their due order; viz. all the fubftituted Letters in fuch order as the Nature of the Question requires. The next Thing to be done, is to confider whether the Question be limitted or not, that is, whether it admits of a determinate Number of Anfwers, or not; and to discover that, obferve the following Rules. Kule 1. When the Number of the Quantities fought, exceeds the Number of the given Equations; moft Questions of this Kind are capable of innumerable Answers. Example. Suppofe a Question were propos'd thus. There are three fuch Numbers, that if the first 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 fought be represented by three Let ters, thus ; Call the firft a, the fecond e, and the third y. Then, ate=22, and e+y=46, according to the State of the Question. Here the Number of the Quantities fought is three, viz. a, e and y, and the Number of the given Equations is but two; therefore this Question is not limitted, but admits of innumerable Answers; because, for any one of those three Letters, you take any Number at Pleasure that is lefs than 22, which, with a little Confideration, will be easy to conceive. may Kule 2. When the Number of the given Equations (not depending upon one another) are just as many as the Number of the Quan tities fought; then is the Queftion truly limitted; viz. each Quantity fought hath a determinate Number of Values. As for Inftance, Let the aforefaid Question (with one Condition added to it) be propos'd thus: There are three Numbers (a, e and y, as before). If the firft be added to the fecond, their Sum will be 22; if the fe cond 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 thofe Numbers? That is, ate=22; e+y=46; and a+y=36. = Now the Question is perfectly limitted, each fingle Quantity having but one fingle Value; to wit a 6, e16, and y= 30. N.B. If the Number of the given Equations exceeds the Number of the Quantities fought, they not only limit the Queftion, but oftentimes render it impoffible, by being propos d inconfiftent one to another. After you deduce, from the feveral Equations concern'd in the Queftion, a fingle Equation, it often happens that the Powers of the unknown Quantity therein are fo mix'd and intangled with known ones, that it requires fome Trouble and Skill to bring them to one Side of the Equation, and those that are known to the other Side (ftill keeping them to a juft Equality), which, by the Writers of Algebra, is call'd the Reduction of Equations. The Methods of doing which I fhall comprize in the following Sect. referring you for more Examples to Part XI. CHA P. I. The Reduction of Single Equations. Sect. 1. Of Reduction by Addition. R Eduction by Addition is grounded upon Axiom 1. and is only the Tranfpofing (viz. the Removing) of one or more Negative Quantities from either Side of the Equation to the c ther Side, with the Sign+before each of them. As in these Examples. Examples. Suppofelia-b-4d 2+b; a=d+4+ & diftinguish it from the other Numbers: As the fecond step of this Example. Let Ifaa 7bcdd2ba = 1+7bc2ba2aa + 2ba dd +7bc A Sect. 2. Reduction by Subtraction; Reduction by Subtraction is grounded upon Axiom 2. and is perform'd by Tranfpofing (or Removing) one or more Affirmative Quantities from either Side of the Equation to the other Side, with the Sign before each of them. As in these Examples. Sect. 3. Reduction by Multiplication. When any of the Terms of an Equation containing one or more Powers of the unknown Quantity, is, or are Fractions; Multiply the Equation by the Denominator of any of these Fractions; and with the Equation thus produc'd, if need be, proceed as above directed: And fo on, 'till all the Terms, con taining the Power, or Powers of the unknown Quantity, are reduc'd into Integers. Or Multiply the first Equation, by the Product of all the Denominators therein, and you'll reduce it into an Equation, where in all the Terms will be Integers. Examples |