CHAPTER I - CONTINUITY

Summary

Continuous Variation

In every problem of the Infinitesimal Calculus we have to deal with a number of magnitudes, or quantities, some of which may be constant, whilst others are regarded as variable, and (moreover) as admitting of continuous variation.

Thus in the applications to Geometry, the magnitudes in question may be lengths, angles, areas, volumes, &c; in Dynamics they may be masses, times, velocities, forces, &c.

Algebraically, any such magnitude is represented by a letter, such as a or x, denoting the ratio which it bears to some standard or ‘unit’ magnitude of its own kind. This ratio may be integral, or fractional, or it may be ‘incommensurable,’ i.e. it may not admit of being exactly represented by any fraction whose numerator and denominator are finite integers. Its symbol will in any case be subject to the ordinary rules of Algebra.

A ‘constant’ magnitude, in any given process, is one which does not change its value. A magnitude to which, in the course of any given process, different values are assigned, is said to be ‘variable.’ The earlier letters a, b, c, … of the alphabet are generally used to denote constant, and the later letters …u, v, w, x, y, z to denote variable magnitudes.

Some kinds of magnitude, as for instance lengths, masses, densities, do not admit of variety of sign. Others, such as altitudes, rotations, velocities, may be either positive or negative.