Elements of the Differential Calculus: With Index (Classic Reprint)

Differential Calculus, a subject which opens a wide field of analytical research, the student enters upon an entirely new system of thought. In his previous investigations he has always heen 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 hut a remote analogy to ordinary A lgebra. 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 value 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 f
(Typographical errors above are due to OCR software and don't occur in the book.)