Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/61

does not of itself provide a readily applicable criterion by the use of which the question may be answered in respect of a particular number.

The scope of Euclidean determinations

Before proceeding to describe the manner in which it was finally shewn that the number $\pi$ is a transcendental number, it is desirable to explain in what way this result is connected with the problems of the quadrature and rectification of the circle by means of Euclidean determinations.

The development of Analytical Geometry has made it possible to replace every geometrical problem by a corresponding analytical one which involves only numbers and their relations. As we have already remarked, every Euclidean problem of what is called construction consists essentially in the determination of one or more points which shall satisfy certain prescribed relations with regard to a certain finite number of assigned points, the data of the problem. Such a problem has as its analytical counterpart the determination of a number, or a finite set of numbers, which shall satisfy certain prescribed relations relatively to a given set of numbers. The determination of the required numbers is always made by means of a set of algebraical equations.

The development of the theory of algebraical equations, especially that due to Abel, Gauss, and Galois, led the Mathematicians of the last century to scrutinize with care the limits of the possibility of solving geometrical problems subject to prescribed limitations as to the nature of the geometrical operations regarded as admissible. In particular, it has been ascertained what classes of geometrical problems are capable of solution when operations equivalent in practical geometry to the use of certain instruments are admitted^[1]. The investigations have led to the discovery of cases such as that of inscribing a regular polygon of 17 sides in a circle, in which a problem, not previously known to be capable of solution by Euclidean means, has been shewn to be so.

We shall here give an account of as much of the theory of this subject as is necessary for the purpose of application to the theory of the quadrature and rectification of the circle.

In the first place we observe that, having given two or more points in a plane, a Cartesian set of axes can be constructed by means of a

1. An interesting detailed account of investigations of this kind will be found in Euriques' Questions of Elementary Geometry, German Edition, 1907.