This work is a development of the infinitesimal calculus as far as the first differentials of algebraic functions of one independent variable and their corresponding integrals. That is to say, it is restricted to the absolute rudiments of the science. Within these narrow limits however, the treatment is tolerably full, and suffices to show how far-reaching a mathematical instrument the Calculus is, even in its elementary steps. This seems to the author to be the best way of communicating a working knowledge of the science; namely, to teach a few elementary rules and then put them into immediate use, as far as they will go. The student then recognizes the necessity of the more advanced steps. Very little is said at the start about the logical basis of the science. The few remarks which it seemed necessary to make upon this topic are reserved for the final chapter. It seems wise to the author in any art or science to learn the practical process first, and the theory, or reason why, afterwards, taking every thing for granted in the mean time, more particularly when, as in the calculus, the theory is somewhat perplexing. The theory is much better approached after the student's confidence in the practical outcome of the thing has been fully established by actual work. Indeed in this way, he is liable to pick up a good deal of the doctrine as he goes along, and the knowledge of it that he thus acquires will guide him to what he still lacks. The recognized need will point to the best line. of investigation. A number of authorities have been consulted in the preparation of these few pages, while adopting a somewhat different treatment from any of them. Most of the examples are selected. Those who wish to pursue the study of the calculus in its more advanced developments, will find many admirable text books, both American and English, in which the subject is treated exhaustively. YONKERS, April, 1896. CHAPTER I. DIFFERENTIAL CALCULUS. The differential of the first power of a variable x, which may be considered as representing the varying length of a straight line, is an infinitely small increment, smaller than any quantity that can be expressed in figures, added to such variable. It is expressed thus: da. This expression is not to be regarded as the product of two factors, but simply as an infinitesimal linear quantity, expressed in terms of the variable. This quantity da, is a purely imaginary one, and has no numerical value. If we have, however, two such differentials dx, dy of two variable quantities, x and y for indicates an act example, the fraction dx dy ual ratio, which may receive a numerical value. Later on it will be seen what important practical results are derived from the ratio, dx dy of two differentials. But it |