INTRODUCTION. Q UANTITY which can be measured, and is the object of Mathematics, is of two kinds, Number and Extenfion. The former is treated of in Arithmetic, the latter in Geometry. Numbers are ranged in a Scale, by the continued repetition of fome one number, which is called the Root; and, in consequence of this order, they are conveniently expreffed in words, and denoted by characters. The operations of arithmetic are easily derived from the established method of notation, and the moît fimple reafonings concerning the relations of magni tude. A Invef Investigations by the common arithmetic are greatly limited, from the want of characters to express the quantities that are unknown, and their different relations to one another, and to fuch as are known. Hence letters, and other convenient fymbols, have been introduced to fupply this defect; and thus gradually has arifen the science of Algebra, properly called UNIVERSAL A ope In the common arithmetic, too, the given numbers disappear in the course of the ration, fo that general rules can feldom be derived from it; but, in algebra, the known quantities, as well as the unknown, may be expreffed by letters, which, through the whole operation, retain their original form ; and hence may be deduced, not only general canons for like cafes, but the dependence of of the feveral quantities concerned, and likewife the determination of a problem, with out exhibiting which, it is not completely refolved. This general manner of expreffing quantities alfo, and the general reafonings concerning their connections, which may be founded on it, have rendered this fcience not lefs useful in the demonftration of theorems, than in the resolution of pro blems: If geometrical quantities be fuppofed to be divided into equal parts, their relations, in respect of magnitude, or their proportions, may be expreffed by numbers; one of these equal parts being denoted by the unit. Arithmetic, however, is used in expreffing only the conclufions of geometrical propofitions; and it is by algebra that the the bounds and application of geometry have been of late fo far extended. The proper objects of mathematical science are NUMBER and EXTENSION; but mathematical inquiries may be inftituted alfo concerning any physical quantities, that are capable of being measured or expressed by numbers and extended magnitudes : And, as the application of algebra may be equally univerfal, it has been called The Science of quantity in general. ELE O F ALGEBRA. I. PART I. DEFINITION S. Q UANTITIES which are known, 1 are generally represented by the first letters of the alphabet, as a, b, c, &c. and such as are unknown, by the last letters, as x, y, z, &c. II. The fign+ (plus) denotes, that the quantity before which it is placed is to be added. Thus a+b denotes the fum of a and b ; 3+5 denotes the fum of 3 and 5, or 8. When no fign is expreffed, is understood. III. |