CONTENTS. On the Methods of Combining and Incorporating Algebra- Page CHAP. I. DEFINITIONS AND FIRST PRINCIPLES OF THE SCIENCE. 1. ALGEBRA may be defined to be, the science of Definition general reasoning by symbolical language. of Algebra. It is impossible however, by any simple definition, to express fully its objects and applications, which can only be clearly comprehended by a person acquainted with the science: it has been termed Universal Arithmetic; but this definition is defective, inasmuch as it assigns for the general object of the science, what can only be considered as one of its applications. 2. The symbols of Algebra may be made the representatives of every species of quantity, whether abstract or concrete the operations to which they are subject are perfectly general, and are in no respect affected by the nature of the quantities which the symbols denote, being determined solely by the definitions and assumptions which constitute the first principles of the science. Of the Sym bols of Al gebra. 3. The symbols most commonly used are the letters Symbols of the alphabet great or small: as the choice of them is most commonly used. perfectly arbitrary, those are most commonly adopted, which are most easily written: in some cases, in order to exhibit to the eye the connection between the symbol and the thing signified, we make use of the initial letter of the term which designates the quantity represented in other cases, when different quantities of the same kind are involved in an algebraical operation, we denote them, in order to indicate their connection with each other, by the same letter, with different accents, as a', a", a", a", &c.; or by the same letter with numbers written underneath to the right hand, as a1, α2, az, α, &c.; or by the same letter in different alphabets, as a, A, a, a, &c. A Symbols of 4. In many of the operations of Algebra, it is necesquantities considered sary to distinguish such quantities as are known and deteras known or minate, from such as are unknown, or whose values are unknown. found by means of algebraical operations: it is usual to Determi terminate. make use of the first letters of the alphabet, such as a, b, c, &c. to denote the former, and of the last letters of the alphabet, such as v, u, x, y, &c. to denote the latter. It is also convenient sometimes to distinguish indeternate or inde- minate quantities, whose values are arbitrary and assignable at the pleasure of the operator, from such as are determinate, whether known, or unknown: they are commonly represented by the middle letters of the alphabet, such as l, m, n, p, q, r, &c. Variable and invariable. Addition and Sub Variable quantities, admitting of every value between given limits, when the variation is continuous, or of a certain number of such intermediate values, when the change is discontinuous, are commonly distinguished from such as are invariable, whether assigned or not, in the same manner as unknown quantities are distinguished from such as are known, by representing them respectively by the last and first letters of the alphabet. All quantities of the same kind admit of being traction de- added to or subtracted from, each other: and the operations noted by the of addition and subtraction, which are of all others the most signs+and used in considering the relations of quantity, are denoted by the signs and the first denoting Addition, and called plus or the positive sign: and the second denoting Subtraction, and called minus or the negative sign. Further use and mean 6. In symbols of concrete quantities of the same kind, other relations besides those of greater or less, may be consigns+and sidered: thus, if the symbols represented lines, some may represent lines drawn in one direction, and others lines drawn in the direction opposite: if they represented portions of time, some may represent time past, others time to come: if they denoted forces in the same direction, one symbol may designate a force which pushes, another a force which pulls; and similarly, in other cases: in order that |