94 MYSTICISM AND LOGIC 



axioms : (i) There is a class of entities called points. 

 (2) There is at least one point. (3) If a be a point, there 

 is at least one other point besides a. Then we bring in 

 the straight line joining two points, and begin again with 

 (4), namely, on the straight line joining a and b, there is 

 at least one other point besides a and b. (5) There is at 

 least one point not on the line ab. And so we go on, till 

 we have the means of obtaining as many points as we 

 require. But the word space, as Peano humorously 

 remarks, is one for which Geometry has no use at all. 



The rigid methods employed by modern geometers 

 have deposed Euclid from his pinnacle of correctness. It 

 was thought, until recent times, that, as Sir Henry Savile 

 remarked in 1621, there were only two blemishes in 

 Euclid, the theory of parallels and the theory of pro- 

 portion. It is now known that these are almost the only 

 points in which Euclid is free from blemish. Countless 

 errors are involved in his first eight propositions. That 

 is to say, not only is it doubtful whether his axioms are 

 true, which is a comparatively trivial matter, but it is 

 certain that his propositions do not follow from the 

 axioms which he enunciates. A vastly greater number 

 of axioms, which Euclid unconsciously employs, are re- 

 quired for the proof of his propositions. Even in the 

 first proposition of all, where he constructs an equilateral 

 triangle on a given base, he uses two circles which are 

 assumed to intersect. But no explicit axiom assures us 

 that they do so, and in some kinds of spaces they do not 

 always intersect. It is quite doubtful whether our space 

 belongs to one of these kinds or not. Thus Euclid fails 

 entirely to prove his point in the very first proposition. 

 As he is certainly not an easy author, and is terribly long- 

 winded, he has no longer any but an historical interest. 

 Under these circumstances, it is nothing less than a 



