HENRI POINCAEE NOEDMANN. 757 



Mathematical truths are derived from a few self-evident axioms by an unimpeach- 

 able chain of reasoning. They rule not only over us but over nature herself. They 

 limit in a way even the Creator. He can choose only between a few possible solu- 

 tions. We need, then, only a few trials to know what choice he made. From every 

 experiment many consequences may follow through a series of mathematical deduc- 

 tions, each one leading into knowledge of some new corner of the universe. 



Note the significonce of these facts for the good of the people; the importance to 

 those colleges which first discover the physical basis of some srientific truth. But 

 note how they have misunderstood the relation of experiments and mathematics; for 

 hundreds of years philosophers have made worlds of dreams based as little as possible 

 upon facts. 



Poincare first undertook to show the weakness of that creed which 

 refers all phenomena to time, number, and space, and which was left 

 to us by the traditions of the seventeenth and eighteenth centuries. 

 The "mathematical universe," that dream sketched by Descartes and 

 elaborated b}^ the great encyclopaedists, expressed the very essence 

 of everything in an absolute, definite geometrical form. According 

 to the Cartesian conception, all the properties of matter are reducible 

 to extension and movement; matter hid nothing further. This 

 ambitious dream was indulged in not only by the people and the 

 colleges, as Poincare has stated, but even in our days by scientists of 

 considerable repute, notably in the work of the celebrated German 

 naturalist ITaeckel, who developed such a system and with naive 

 arrogance believed lie had solved the "riddle of the universe." 



There has been quite a little doubt since Kant whether these no- 

 tions of time and space upon which this metaphysical structure is 

 based, this absolute pragmatism, if I may use that term, are not a 

 little subjective. That at once renders the very foundations of their 

 structure insecure. But it was Poincar^'s task to show in a not easily 

 refutable, scientific manner what was to be thought of these funda- 

 mental ideas. For that he examined in turn the various sciences 

 based upon geometrical form; first, geometry itself, then mechanics, 

 and finally physics. 



Mathematics was first tried. Complete rationalism after ha^^ng 

 first pursued dogma and the absolute into their ancient fortress, by 

 a strange and somewhat paradoxical turn restored them to mathe- 

 matics. He believed that mathematics could not be what it seemed. 

 There seemed to be something of fatality, necessity, wliich could 

 not be got away from about it. Wlien all our ideas melted away it 

 alone remained solid like a rock in the ocean, under cover of the con- 

 tingencies and the relative. 



Now, if with Pioncare we examine the sciences of number and exten- 

 sion, especially the first principles, which are the most frail parts just 

 because of their apparent and undcmonstrable tnitlis, we find this: 

 The postulate of Euclid, upon which all geometry is based, states 

 that "through a point we can pass but one line parallel to a given 

 85360°— SM 1912 49 



