CONTENTS. On the determination of the highest common divisors, and the On the general principles of the interpretation of the signs of On the determination of the sides and angles of triangles, or On the limits of the values of series proceeding according to ascending or descending powers of a symbol, which is capable CHAPTER XI. ON TAE OPERATIONS OF ADDITION AND SUBTRACTION IN SYMBOLICAL ALGEBRA. : a 543. The symbols in Arithmetical Algebra represent num- Distinction between bers, whether abstract or concrete, whole or fractional, and the Arithmetic operations to which they are subject are assumed to be identical and Arith metical in meaning and extent with the operations of the same name Algebra in common Arithmetic: the only distinction between the two sciences consisting in the substitution of general symbols for digital numbers. Thus, if a be added to b, as in the expression a+b, it is assumed that a and b are either abstract numbers or concrete numbers of the same kind: if b be subtracted from a, as in the expression a – b, it is assumed that a is greater than b, which implies likewise that they are numbers of the same kind: if a be multiplied by b, as in the expression ab (Art. 34), or if a be divided by b, as in the expression g (Art. 71), it is assumed that 6 is an abstract number. In all these cases, the operation required to be performed, whether it be addition or subtraction, multiplication or division, is clearly defined and understood, and the result which is obtained, is a necessary consequence of the definition: the same observation applies to all the results of Arithmetical Algebra. 544. But the symbols, which are thus employed, do not The asconvey, either to the eye or to the mind, in the same manner as sumption of the undigital numbers and geometrical lines, the limitations of value to limited values of which they are subject in Arithmetical Algebra: for they are the symbols equally competent to represent quantities of all kinds, and of all employed involves the relations of magnitude. But if we venture to ascribe to them necessary a perfect generality of value, (upon which a conventional limi- of the indetation was imposed in Arithmetical Algebra), it will be found pendent use of the signs to involve, as an immediate and necessary consequence, the + and VOL. II. A |