I have endeavoured in the present work to exhibit a comprehensive view of the Differential Calculus on the method of Limits. In the more elementary portions I have entered into considerable detail in the explanations with the hope that a reader who is without the assistance of a tutor may be enabled to acquire a competent acquaintance with the subject. To. the different chapters will be found appended Examples -sufficiently numerous to render another book unnecessary. These examples have been selected almost exclusively from the College and University Examination Papers; the greater part of them will be found to present no very serious difficulty to the student, although a few may require peculiar analytical skill.