The topics in this book are arranged for primary courses in calculus in which the formal division into differential calculus and integral calculus is deemed necessary. The book is mainly made up of matter from my Infinitesimal Calculus, Changes, however, have been made in the treatment of several topics, and some additional matter has been introduced, in particular that relating to indeterminate forms, solid geometry, and motion. The articles on motion have been written in the belief that familiarity with the notions of velocity and acceleration, as treated by the calculus, is a great advantage to students who have to take mechanics. Part of the preface of my Infinitesimal Calculus applies equally well to this book. Its purpose is to provide an introductory course for those who are entering upon the study of calculus either to prepare themselves for elementary work in applied science or to gratify and develop their interest in mathematics. Little more has been discussed than what may be regarded as the essentials of a primary course. An attempt is made to describe and emphasise the fundamental principles of the subject in such a way that, as much as may reasonably be expected, they may be clearly understood, firmly grasped, and intelligently applied by young students. There has also been kept in view the development in them of the ability to read mathematics and to prosecute its study by themselves. With regard to simplicity and clearness in the exposition of the subject, it may be said that the aim has been to write a book that will be found helpful by those who begin the study of calculus without the guidance and aid of a teacher. For these students more especially, throughout the work suggestions and remarks are made concerning the order in which the various topics may be studied, the relative importance of the various topics in a first study of calculus, the articles that must be thoroughly mastered, and the articles that may advantageously be omitted or lightlypassed over at its first reading, and so on. The notion of anti-differentiation is presented simultaneously with the notion of differentiation, and exercises thereon appear earlyin the text; but when integrationis formally taken up the idea of integrationas a process of summation is considered before the idea of integrationas a process which is the inverse of differentiation. There is considerable difference of opinionas to the propriety or the advantage of this order. The decision to follow it here has been made mainly for the reason that students appear " at least so it seems to me, but other teachers may have a different experience" to understand more clearlyand vividly the relation of integration to many practicalproblems when the summation idea is put in the forefront. In teaching, the one order can be taken as readily as the other.