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By I. TODHUNTER, M. A.
FELLOW AND ASSISTANT TUTOR OF ST JOHN'S COLLEGE,
MACMILLAN AND CO.
IN writing the present treatise on the INTEGRAL CALCULUS, the object has been to produce a work at once elementary and complete-adapted for the use of beginners, and sufficient for the wants of advanced students. In the selection of the propositions, and in the mode of establishing them, I have endeavoured to exhibit fully and clearly the principles of the subject, and to illustrate all their most important results. The process of summation has been repeatedly brought forward, with the view of securing the attention of the student to the notions which form the true foundation of the Integral Calculus itself, as well as of its most valuable applications. Considerable space has been devoted to the investigations of the lengths and areas of curves and of the volumes of solids, and an attempt has been made to explain those difficulties which usually perplex beginners-especially with reference to the limits of integrations.
The transformation of multiple integrals is one of the most interesting parts of the subject, and the experience of teachers shews that the usual modes of treating it are not free from obscurity. I have therefore adopted a method different from those of previous elementary writers, and have endeavoured to