president's address. 



a column of figures correctly ; but it is a memory which fixes the order 

 in which the successive steps of reasoning follow each other without 

 necessarily retaining the details of the individual steps. This Poincar6 

 illustrates by contrasting the memory of a chess-player with that of a 

 mathematician. ' When I play chess,' he says, ' I reason out correctly 

 that if I were to make a certain move, I should expose myself to a 

 certain danger. I should, therefore, consider a number of other moves, 

 and, after rejecting each of them in turn, I should end by making the 

 one which I first contemplated and dismissed, having forgotten in the 

 meantime the ground on which I had abandoned it.' ' Why, then,' he 

 continues, ' does my memory not fail me in a difficult mathematical 

 reasoning in which the majority of chess-players would be entirely lost? 

 It is because a mathematical demonstration is not a juxtaposition of 

 syllogisms, but consists of syllogisms placed in a certain order; and 

 the order in which its elements are placed is much more important 

 than the elements themselves. If I have this intuition — so to speak — of 

 the order, so as to perceive at one glance the whole of the reasoning, 

 I need not fear to forget its elements : each of these will take its right 

 place of its own accord without making any call on my memory. ' ° 



Poincar^ next discusses the nature of the intellectual gift distinguish- 

 ing those who can enrich knowledge with new and fertile ideas of 

 discovery. Mathematical invention, according to him, does not consist 

 in forming new combinations of known mathematical entities, because 

 the number of combinations one could form are infinite, and most of 

 them would possess no interest whatever. Inventing consists, on the 

 contrary, in excluding useless combinations, and therefore : ' To invent 

 is to select — to choose.' . . . 'The expression "choose" perhaps 

 requires qualifying, because it recalls a buyer to whom one 

 offers a large number of samples which he examines before making his 

 choice. In our case the samples would be so numerous that a life- 

 time would not suffice to complete the examination. That is not the 

 way things are done. The sterile combinations do not even present them- 

 selves to the mind of the inventor, and those which may momentarily 

 enter his consciousness, only to be rejected, partake something of the 

 character of useful combinations. The inventor is therefore to be 

 compared with an examiner who has only to deal with candidates who 

 have already passed a previous test of competence. ' 



All those who have attempted to add something to knowledge must 

 recognize that there is a profound truth in these remarks. New ideas 

 may float across our consciousness, but, selecting the wrong ones for 

 more detailed study, we waste our time fruitlessly. We are bewildered 

 ,by the multitude of roads which open out before us, and, like Poincar6 



* Science e( Methode, pp. 46 and 47. 



