Elements of the Differential Calculus
With Index (Classic Reprint)

Excerpt from Elements of the Differential Calculus: With Index<br><br>On first commencing to read the Differential Calculus, a subject which open a wide field of analytical research, the student enters upon an entirely new system of thought.<br><br>In his previous investigations he has always been accustomed to consider quantities, whether known or unknown, as having some fixed or determinate value; he has now to conceive the values of certain quantities to undergo continuous changes, and to operate upon these changes with new symbols and new processes, which in themselves have but a remote analogy to ordinary Algebra.<br><br>When two quantities, thus continuously variable, are connected by an analytical equation, and their values are therefore mutually dependent on each other, and they are supposed to be affected by simultaneous changes, it is evident that the increments will also be connected by some corresponding analytical relation. The primary object of the Calculus is to establish general methods of investigating the nature and properties of such relations when the changes or increments are supposed to be small. To effect this, it is first requisite to trace the successive values of the ratio subsisting between two increments, when the increments themselves are supposed to continuously decrease in magnitude, and to determine the limiting value of this ratio when they ultimately become infinitesimals. This ultimate or limiting valae is, in fact, that which represents the ratio when the increments are supposed absolutely to vanish, and it is completely defined and accurately determined by referring the successive values to the recognized law of continuity. The operation here described is the true foundation of the Calculus, and the condition of continuity, especially insisted upon in the present treatise, entirely removes from the limiting value that obscure and indeterminate character which otherwise forms an insuperable obstacle to a proper comprehension of the first principles.<br><br>We recommend the student to make himself familiar with the methods of limiting ratios and infinitesimals. The theory of Infinitesimals is literally that of the Differential Calculus, and the principal law which regulates this theory is directly inferred from the method of limiting ratios. The two methods are indeed virtually but modifications of the same idea.<br><br>About the Publisher<br><br>Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com<br><br>This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.