26 POPULAR SCIENCE MONTHLY 



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 gen- 

 eral 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 -f- 1 homogeneous coordinates (x 1} x 2 , • • • x ), 

 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 prop- 

 erties such that the representation with coordinates can be deduced as a con- 

 sequence. 



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 con- 

 structions that Fano obtained and to illustrate by means of it cer- 

 tain 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 cer- 

 tain 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. 



1. If A and B are points there is one line which contains them both; and 



2. There is not more than one such line. 



3. Any two lines have in common at least one point. 



4. Not all the points are on the same line. 



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 : 



5. In each line there are at least three points. 



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

 we add 



