Page:Popular Science Monthly Volume 68.djvu/30

26 The above historical account has no doubt many important sins of omission besides those due to its brevity. But for the purpose of grasping the type of thought involved in these researches further general remarks would probably be less useful than a simple example.

In the academic year 1890-1 Professor C. Segre gave a course of lectures at the University of Turin in which he studied the analytic geometry of n dimensions. A point of n-dimensional space he defined as usual to be a set of n+1 homogeneous coordinates (x1, x2, . . . xn+1), a line as a set of points satisfying a set of linear equations, etc. To his students, however, he proposed the following problem:

To define a space Sn not by means of coordinates, but by a series of properties such that the representation with coordinates can be deduced as a consequence.

In other words, he asked for a set of axioms for n-dimensional space. The problem was taken up by one of the students, Gino Fano, now a professor at Turin, and the results published in the Giornale di Mathematiche. I wish to reproduce one of the many interesting constructions that Fano obtained and to illustrate by means of it certain concepts that have grown up since then.

Let us take the case of n-dimensional geometry where n=2 and proceed for a time as if to build the projective geometry of the plane. Let our undefined elements be called points and let us speak of certain undefined classes or sets of points which shall be called lines. Every one will recognize as valid of projective geometry the following propositions which are our axioms—our unproved propositions.

If we stop at this point and try to see how much we can prove on the strength of our assumptions, we are confronted at once by the fact that we can not prove the existence of even a single point. This must therefore be assumed by a further axiom. The assumption of one point is not enough either, but if we assume that there are two points, it follows from 4 that there must be at least three. There need not, however, be more than three, for if we suppose that the points referred to are A, B, C, and that the line AB consists merely of the points A and B, the line BC of the points B and C, and the line CA of the points C and A, then on rereading 1, 2, 3, 4 it is evident that they are all satisfied. Hence in order to get ahead we must assume:

But this does not postulate the existence of even a single point till we add