The idea of a function of an infinite number of variables, like so many other mathematical ideas, has its origin in the study of problems of applied mathematics. Such problems, as is usual, preceded by some considerable time the explicit formulation of the idea. As an introduction to this discussion it will be sufficient to consider briefly two of them.

The Problem of the Brachistochrone.—Given two points P, Q in a vertical plane, what is the shape of a smooth curve through P, Q for which the time required to descend from P to Q under gravity is a mimimum.

It is worthy of note that the imaginary was introduced into mathematics in the course of strictly non-utilitarian researches on the theory of quadratic equations, being for a long time rejected by the mathematicians as an entirely useless conception. This formal condemnation of the imaginary seems less extraordinary when we recollect that at the time of its inception, and indeed for long afterwards, serious scruples were experienced by some mathematicians even regarding the admission of negative quantities in their results. The last writers who objected to the use of negative quantities were Baron Maseres, a Fellow of Clare College, Cambridge, and William Frend, a second wrangler. Both these men had written extensively in condemnation of the use of negative quantities; their attitude in this matter is clearly evinced by the following extract from the preface to Frend’s Principles of Algebra, published in 1796, at which period, strangely enough, the notion of the imaginary was firmly eatablished.

One of the charms of elementary geometry is that there is no theorem so well known but that one may hope to discover improvements or at least interesting variations in the proof; and the same thing is true of the constructions.

In order to decide between variations in these it is necessary to have some way of measuring the complexity of a construction. This is done by counting the number of settings of the ruler or compass and the number of lines or arcs drawn.