XI. Of the Rules for finding the number of impoffible roots in an equation XII. Containing a general Demonftration of Sir Ifaac Newton's rule for finding A TREA- TREATISE OF ALGEBRA. PART I. A CHAP. I. Definitions and Illuftrations. LGEBRA is a general method of com putation by certain figns and fymbols which have been contrived for this purpose, and found convenient. It is called an UNIVERSAL ARITHMETICK, and proceeds by operations and rules fimilar to thofe in common arithmetick, founded upon the fame principles. This, however, is no argument against its usefulness or evidence; fince arithmetick is not to be the lefs valued that it is common, and is allowed to be one of the moft clear and evident of the fciBut as a number of fymbols are admitted into this fcience, being neceffary for giving it that extent and generality which is its greatest excellence; the import of thofe fymbols is to be clearly ftated, that no obfcurity or error may arife from the frequent ufe and complication of them. $2. In GEOMETRY, lines are reprefented by a line, triangles by a triangle, and other figures by a figure of the fame kind; but, in ALGEBRA, quantities are reprefented by the fame letters of the alphabet; and various figns have been imagined for reprefenting their affections, relations, and dependencies. In Geometry the representations are more natural, in Algebra more arbitrary: the former are like the first attempts towards the expreffion of objects, which was by drawing their refemblances; the latter correfpond more to the prefent ufe of languages and writing. Thus the evidence of Geometry is fometimes more fimple and obvious; but the ufe of Algebra more extenfive, and often more ready: efpecially fince the mathematical fciences have acquired fo vaft an extent, and have been applied to fo many enquiries. §3. In thofe fciences, it is not barely magnitude that is the object of contemplation: but there quan there are many affections and properties of tities, and operations to be performed upon them, that are neceffarily to be confidered. In eftimating the ratio or proportion of quantities, magnitude only is confidered. (Elem. 5. Def. 3.) But the nature and properties of figures depend on the pofition of the lines that bound them, as well as on their magnitude. In treating of motion, the direction of motion as well as its velocity; and the direction of powers that generate or deftroy motion, as well as their forces, muft be regarded. In OPTICS, the pofition, brightnefs, and diftinctnefs of images, are of no lefs importance than their bignefs; and the like is to be faid of other fciences. It is neceffary therefore that other fymbols be admitted into Algebra befide the letters and numbers which reprefent the magnitude of quantities. §4. The relation of equality is expreffed, by the fign; thus to exprefs that the quantity represented by a is equal to that which is reprefented by b, we write a = b. But if we would exprefs that a is greater than b, we write a b; and if we would exprefs algebraically that a is lefs than b, we write a b. of parts, § 5. QUANTITY is what is made up or is capable of being greater or lefs. It is in. creafed by Addition, and diminished by Subtracwhich are therefore the two primary ope sion; - rations that relate to quantity. Hence it is, that any quantity may be fuppofed to enter into algebraic computations two different ways which have contrary effects; either as an increment or as a decrement; that is, as a quantity to be added or as a quantity to be fubtracted. The fign + (plus) is the mark of Addition, and the fign (minus) of Subtraction. Thus the quantity being represented by a,+ a imports that a is to be added, or represents an increment; but a imports that a is to be subtracted, and represents a decrement. When feveral fuch quantities are joined, the figns ferve to fhew which are to be added, and which are to be fubtracted. Thus + a + b denotes the quantity that arises when a and are both confidered as increments, and therefore expreffes the fum of a and b. But +ab denotes the quantity that arifes when from the quantity a the quantity b is fubtracted; and expreffes the excess of a above b. When a is greater than b, then ab is itself an increment; when ab, then abo; and when a is less than b, then a-b is itself a de crement. § 6. As addition and fubtraction are oppofite, or an increment is oppofite to a decrement, there is an analogous oppofition between the affections of quantities that are confidered in the mathematical fciences. As between excefs and defect; between the value of effects or |