Algebra Volume 1; An Elementary Text-Book, for the Higher Classes of Secondary Schools and for Colleges - George Chrystal

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1904 Excerpt: ...There are an infinite number of ways in which we may conceive one quantity y to depend upon, be calculable from, or, in technical mathematical language, be a function of, another quantity x. Thus we may have, for example, y=Zx, y=lA y = ax + b, y = az' + bz + c, and so on. For convenience x is called the independent variable, and y the dependent variable; because we imagine that any value we please is given to x, and the corresponding value of y derived from it by means of the functional relation. All the other symbols of quantity that occur in the above equations, such as 3, 17, a, b, c, VOL. I T 2, &c, are supposed to remain fixed, and are therefore called constants. Here wo attach meanings to the words variable and constant more in accordance with their use in popular language than those given above (chap. ii., 6). The justification of the double usage, if not already apparent, will be more fully understood when we come to discuss the theory of equations, and to consider more fully the variations of functions of various kinds (see chaps. xv.-xviii.) 18. In the meantime, we propose to discuss very briefly the simplest of all cases of the functional dependence of one quantity upon another, that, namely, which is characterised by the following property. Let tho following scheme denote any two corresponding pairs whatever of values of the independent and dependent variables, then the dependence is to be such that always y: y' = x: x' (1). It is obvious that this property completely determines the nature of the dependence of y upon x, as soon as any single corresponding pair of values are given. Suppose, in fact, that, when x has the value z, y has the value ya, then, by (1), V x Now we may keep x and y as a fixed standard..."