412 



SCIENCE. 



[N. S. Vol. XIX. No. 480. 



the great Cayley: 'There is hardly any- 

 thing in mathematics more beautiful than 

 his wondrous fifth book.' 



For my own part, nothing ever better 

 repaid study. But the contest is over, for 

 now, at last, without sacrificing a whit of 

 rigor, we are able to give the whole matter 

 by an algebra as simple as if only approxi- 

 mate, like Euclid, including ineommen- 

 surables without even mentioning them. 



Again, we shall regain the pristine 

 purity of Euclid in the matter of what 

 Jules Andrade calls 'eette malheureuse et 

 illogique definition' (Phillips and Fisher, 

 §7) : 'A straight line is a line which is the 

 shortest path between any two of its 

 points. ' 



As to this hopeless muddle, which has 

 been condemned ad nauseam, notice that it 

 is senseless without a definition for the 

 length of a curve. Yet, Professor A. 

 Lodge, in a discussion on reform, says: 

 "I believe we could not do better than 

 adopt some French text-book as our model. 

 Also I., 24, 25, being obviously related to 

 I., 4, are made to immediately follow it 

 in such of the French books as define a 

 straight line to be the shortest distance 

 between two points." Professor Lodge, 

 then, does not know that the French 

 themselves have repudiated this nauseous 

 pseudo-definition. Of it Laisant says (p. 

 223): 



This definition, almost unanimously abandoned, 

 represents one of the most remarkable examples 

 of . the persistence with which an absurdity can 

 propagate itself throughout the centuries. 



In the first place, the idea expressed is incom- 

 prehensible to beginners, since it presupposes the 

 notion of the length of a curve; and further, it is 

 a vicious circle, since the length of a curve can 

 only be understood as the limit of a sum of recti- 

 linear lengths; moreover, it is not a definition at 

 all, since, on the contrary, it is a demonstrable 

 proposition. 



As to what a tremendous affair this 

 proposition really is, consult Georg Hamel 



in Mathematische Annalen for this very 

 year (p. 242), who employs to adequately 

 express its content the refinements of the 

 integral calculus and the modern theory of 

 functions. 



Moreover, underneath all this even is 

 the assumption of the theorem, Euclid, I., 

 20: 'Any two sides of a triangle are to- 

 gether greater than the third side'; upon 

 which proposition, which the Sophists said 

 even donkeys knew, Hilbert has thrown 

 brilliant new light in the Proceedings of 

 the London Mathematical Society, 1902, 

 pp. 50-68, where he creates a geometry in 

 which the donkeys are mistaken, a geom- 

 etry in which two sides of a triangle may 

 be together less than the third side, exhib- 

 iting as a specific and defiinite example a 

 right triangle in which the sum of the two 

 sides is less than the hypothenuse. 



Any respectably educated person knows 

 that in general the length of a curve is 

 defined by the aggregate formed by the 

 lengths of a proper sequence of inscribed 

 polygons. 



The curve of itself has no length. This 

 definition in ordinary cases creates for the 

 curve a length; but in case the aggregate 

 is not convergent, the curve is regarded as 

 not rectifiable. It had no length, and even 

 our creative definition has failed to endow 

 it with length; so it has no length, and 

 lengthless it must remain. 



If, hoAvever, it can be shown that the 

 lengths of these inscribed polygons form a 

 convergent aggregate which is independent 

 of the particular choice of the polygons of 

 the sequence, the curve is rectifiable, its 

 length being defined by the number given 

 by the aggregate. 



21. GEOMETRY WITHOUT ANT CONTINUITY 

 ASSUMPTION. 



Euclid in his very first proposition and 

 again in I., 22, 'to make a triangle from 

 given sides,' uses unannounced a contin- 



