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From the PREFACE. The present treatise, as its title points out, has been based on the principles of the infinitesimal method. I think that this method, besides being > the simplest of all for teaching Calculus, as all admit, is also the most consistent and philosophical. Some modern writers, indeed, have tried to discredit it; but, when we examine their reasonings, we soon discover that they have done so because they have failed to grasp the true nature of infinitesimal quantities. This I have endeavored to show in my introduction to this work, where ihe reader will find not only what I believe to be the exact notion of the infinitesimal, but also a hint at the grounds (elsewhere developed) upon which the infinitesimal method rests its claim to be preferred to its fashionable rival, the method of limits. This work being intended for young men who are supposed to devote a considerable part of their time to the study of mental and of natural philosophy, it has been necessary to limit its developments by giving less prominence to the analytical than to the practical portion of it. But, while anxious not to overtax our young people, I have nevertheless collected and condensed all that seemed to be of practical use in this branch of study; and, though I have made it a point to be concise, I have constantly endeavored to make all things as plain as the subject matter permitted. I hope that the average student will need only a moderate effort to understand the object and the processes of differentiation and of integration as laid down in this treatise; though, as to integration, his success will often depend also on his endurance of analytical work. To assist him in the performance of this task, I have developed a sufficient number of geometrical and of mechanical problems, together with some fundamental notions of rational mechanics, which were indispensable, and which will serve as an introduction to the regular study of this latter science. -J. B.

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