644 POPULAR SCIENCE MONTHLY. 



In Gibeon the Lord appeared to Solomon in a dream by night: and God 

 said, Ask what I shall give thee. 



Then says Dante of his asking, 



'Twas not to know the number in which are 



Or if in semicircle can be made 

 Triangle so that it have no right angle. 



se del mezzo cerchio far si puote 

 Triangol si, ch'un retto non avesse. 



Par. C. XIII. 101-102. 



How startling this ! How strangely reinforced hy the fact that in 

 the fourth canto of the ' Divina Commedia,' with Caesar greatest of 

 the sons of men, Dante ranks, among exalted personages 



. . . who slow their eyes around 

 Majestically moved, and in their port 

 Bore eminent authority: 



Hippocrates of Chios, who found the quadrature of the lune 

 (nearest that ever man came to the quadrature of the circle until finally 

 John Bolyai squared it in non-Euclidean geometry and Lindemann 

 proved no man could square it in Euclidean geometry) ; Euclid, the 

 geometer, the elementist, preemptor, by his postulate, of the common 

 universe, Euclidean space; and then Ptolemy, first of the long line of 

 those who have tried by proof to answer the question Dante says 

 Solomon might have asked God and did not, a question crucial as to 

 whether Euclid or Bolyai owns the real world. 



Anyhow, the shock currents to scientific somnolence and compla- 

 cency breaking in to the entrenched thought camp from over the ram- 

 parts on the far frontiers almost simultaneously at Kazan on the 

 Volga and at Maros-Vasarhely in far Erdely, started an ever-deepen- 

 ing movement to sift, to revise the foundations of geometry, then of all 

 mathematics, then of all science, a movement of which the latest, as 

 the most charming and weightiest, outcome is that pair of wonder- 

 books, Poincare's ' Science and Hypothesis ' and ' The Value of 

 Science.' 



It was formerly customary to consider the assumption that space 

 is a triply extended continuous number-manifold as a self-evident 

 outcome of the continuity relations of space and the curves and sur- 

 faces in it. But since the rise of non-Euclidean geometry, it has come 

 more and more to be seen that such presupposition about space is only 

 admissible when one has already established and developed elementary 

 geometry synthetically. Lobachevski stressed this in 1836 in his intro- 

 duction to the ' New Elements,' where he says : 



One must necessarily make the beginning with synthesis, in order, finally, 

 after one has found the equations, therewith likewise to reach the limit beyond 

 which now all goes over into the science of numbers. 



For example, one demonstrates in geometry that two straights perpendicular 

 to a third never meet; that the equality of triangles follows from that of 

 certain of their parts. 



