INTRODUCTION. Q UANTITY which can be measured, and is the object of Mathematics, is of two kinds, Number and Extenson. The former is treated of in Arithmetic, the latter in Geometry. Numbers are ranged in a Scale, by the continued repetition of some one number, which is called the Root; and, in confequence of this order, they are conveniently expressed in words, and denoted by characters. The operations of arithmetic are easily derived from the established method of notation, and the most simple reasonings concerning the relations of magni tude, A Invefs. ! 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 such as are known. Hence letters, and other convenient fymbols, have been introduced to supply this defect; and thus gradually has arifen the science of Algebra, properly called UNIVERSAL A RITHMETIC. In the common arithmetic, too, the given numbers disappear in the course of the ope ration, so that general rules can seldom be derived from it; but, in algebra, the known quantities, as well as the unknown, may be expressed by letters, which, through the whole operation, retain their original form; and hence may be deduced, not only general canons for like cases, but the dependence of of the several quantities concerned, and likewise the determination of a problem, with out exhibiting which, it is not completely resolved. This general manner of exprefsing quantities also, 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 prow blems. If geometrical quantities be supposed to be divided into equal parts; their relations, in respect of magnitude, or their proportions, may be expressed by numbers; one of these equal parts being denoted by the unit. Arithmetic, however, is used in expressing only the conclusions of geometrical propositions ; and it is by algebra that the the bounds and application of geometry have been of late so far extended. The proper objects of mathematical science are NUMBER and EXTENSION; but mathematical inquiries may be instituted also 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 universal, it has been called The science of quantity in general. OF A L G E B R A. PAR T I. DEFINITIONS. I. Q UANTITIES which are known, 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 sign + (plus) denotes, that the quantity before which it is placed is to be added. Thus a+b denotes the sum of a and b; 3+5 denotes the sum of 3 and 59 or 8. . When no sign is expressed, + is understood. III. |