Excerpt from Elements of the Differential and Integral Calculus: With Examples and Applications

In this revision an attempt has been made to present in their unity the three methods commonly used in the Calculus. The concept of Rates is essential to a statement of the problems of the Calculus; the principles of Limits make possible general solutions of these problems, and the laws of Infinitesimals greatly abridge these solutions.

The Method of Rates, generalized and simplified, does not involve "the foreign element of time." For in measuring and comparing the rates of variables, the rate of any variable may be selected as the unit of rates, dy/dx is the x-rate of y, or the ratio of the rate of y to that of x, according as the rate of x is or is not the unit of rates.

The proofs of the principles of differentiation by the Method of Rates, and the numerous applications to geometry, mechanics, etc., found in Chapter II, render familiar the problem of rates before its solution by the Method of Limits or Infinitesimals is introduced.

In Chapter III, by proving that It (Δy/Δx) = dy/dx, the problem of rates is reduced to the problem of finding the limit of the ratio of infinitesimals.

The Theory of Infinitesimals is that part of the Theory of Limits which treats of variables having zero as their common limit. In approaching its limit an infinitesimal passes through a series of finitely small values before it reaches infinitely small values.

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