HENRI POINCARE AS AN INVESTIGATOR 213 



abruptly; and integro-difference equations would mean that they de- 

 pend on all preceding states discontinuously. Each is able to account 

 for certain relations in the states. In the same sense the word atom is 

 the name for a set of relations, and though it may change and the atom 

 itself become a solar system, yet what we really mean by the atom is 

 permanent and represents an objective reality. We are witnesses too of 

 an evolution in science and mathematics from the continuous to the dis- 

 continuous. In mathematics it has produced the function defined over 

 a range rather than a line — a chaos, as it were, of elements — and the 

 calculable numbers of Borel. In physics it has produced the electron, 

 the magneton, and the theory of quanta, 5 about which Poincare said 

 shortly before his death : 



A physical system is capable of only a finite number of distinct states; it 

 abruptly jumps from one state to another without passing through the inter- 

 mediate states. 



In biology we have the corresponding theory of mutations. Yet 

 despite this apparent reduction of old ideas into dust, contradictory to 

 our hopes of its permanence; as Poincare put it: this is right and the 

 other is not wrong. They are in harmony, only the language varies; 

 both set forth certain true relations. 



Just as Maxwell and Kelvin were able to invent mechanical models 

 of the ether, so Poincare is perhaps the most profound genius the world 

 has ever known in devising the more subtle machinery of thought to 

 represent the relations he found not only between numbers and geo- 

 metric figures, but between the phenomena of physics. His mind seemed 

 to create new structures of this kind continually, finding expression for 

 the most intricate relations. Nowadays this is the same as saying that 

 he was a mathematician, for this ideal world of relations is the one with 

 whose structure mathematics is concerned, and which mathematics 

 claims sovereignty over, verifying Gauss's assertion : " Mathematics is 

 Queen of all the Sciences." 



In the address of Masson when Poincare was made one of the forty 

 immortals, he said : 



You were born, you have lived, you will live, and you will die a mathe- 

 matician; the vital function of your brain is to invent and to resolve more cases 

 in mathematics; everything about you relates to that. Even when you seem to 

 desert mathematics for metaphysics, the former furnishes the examples, the 

 reasoning, the paradoxes. It is in you, possesses you, harries you, dominates 

 you; in repose, your brain automatically pursues its work, without your being 

 aware of it — the fruit forms, grows, ripens, and falls, and you have yourself 

 told us of your wonder at finding it in your hand so perfect. You furnish an 

 admirable example of the mathematical type. Since Archimedes it is classical 

 but legendary. Earely will historian have found an occasion so fit to note in life 

 its external characters, and in place of relating your works, rather is not this 



6 See Jour, de Physique, 1912 (5), 2; pp. 5-34; 347-360. 



