The Student is recommended to omit the articles marked † at the first reading. Some of the more difficult examples, such as those in Art. 41 and Ex. 31, should be employed as occasional exercises ELEMENTS OF ALGEBRA. CHAPTER I. INTRODUCTION DEFINITIONS, WITH ILLUSTRATIVE EXAMPLES. 1. In a great number of arithmetical operations, it is impossible to carry on a continuous train of reasoning by means of which the various data are successively introduced in their proper places, and the conclusions to which they respectively lead, combined and worked into the final result. It may be that the data themselves depend on the result, or on some property which can only be expressed at a period subsequent to their introduction into the calculation; it may be that the required conclusion is only a particular case of more general considerations, contemplated in, and necessarily introduced by, the hypothesis. From whatever cause it may arise, it will and does frequently happen, that we must of necessity have in view the conclusion itself, or something involving it, even at the very outset of the solution. Should we attempt to proceed by arithmetic simply, it would be requisite to employ some specific artifice suggested by the nature of the particular problem, whereby the introduction of the thing sought might be rendered obvious in the process, and thus the mind might be relieved from the complex considerations incident to calculations applied to the thing yet to be found. B The science of Algebra has for its primary object the exhibition to the eye, of all the operations which in this case would have to be represented only to the mind. Whereas, in arithmetic, nothing can be represented but that which is known either by the conditions of the problem, or by the calculations which have resulted from them; in algebra no operations are suffered to remain without representation. In the former case, then, the mind has, without assistance, to pursue the track, and to keep in view both the previous operations and the results which are to be obtained from them to the full extent in both directions; in the latter, the mind is relieved from all retrospection, and almost all prospective action, and is concentrated on the point immediately before it; the eye being made the guide to what is to follow, as well as the depositary of that which has been effected. 2. An example will illustrate this: A's money is four times B's; if we add £15 to what each has, A's money will be only three times B's. Required what each respectively possesses. Here the operation consists in altering the relative magnitude of the two required numbers, by adding the same number to each of them; it is, therefore, strictly an operation on quantities which are unknown. Our solution, if direct, must then be something like the following:-A's present money is four times B's present money; A's increased money is therefore four times B's present money, together with £15. Now B's increased money being B's present money, together with £15, the triple of this sum is equal to three times B's present money, together with £45. But by the question, A's increased money is equal to three times B's increased money: hence, by our shewing, four times B's present money, together with £15, is the same thing as three times B's present money, together with £45; whence it is evident, that B's present money is £30, and consequently A's is £120. This solution is obvious enough, yet its prolixity in statement is calculated to create confusion, even in an example so |