HENRI POINCARE AS AN INVESTIGATOR 219 



them any more than that a collection of beams and stones will make a 

 cathedral. Mere haphazard construction does not produce the cathedral 

 either. To reach the end it is necessary to have the end in view from 

 the beginning. It is not only necessary to choose a route, but we must 

 see that it is the route to be chosen. This implies a power of the mind 

 which Poincare calls intuition. It is that power which enables us to 

 perceive the plan of the whole, to seize the unity in the matter at hand. 

 This power is necessary not only to the investigator, but it is also neces- 

 sary — in less degree, perhaps — to him who desires to follow the in- 

 vestigation. Why is it, he asks, that any one can ever fail to understand 

 mathematics? Here is a subject constructed step by step with infallible 

 logic, yet many do not comprehend it at all. Not on account of poor 

 memory — that may lead to errors in calculation, but has little to do with 

 comprehension of the subject. Sylvester, for example, was notorious for 

 his inability to remember even what he himself had proved. It is not 

 due to lack of the power of attention, for while concentration is neces- 

 sary in the development of a demonstration, or in following a piece of 

 logic, it does riot give this appreciation of mathematics. A mathematical 

 demonstration is a series of inferences, but it is above all a series of 

 inferences in a certain order. The important thing is the order, just 

 as in chess the mere moving according to the rules is not enough, it is 

 the plan of the game that counts. If one appreciates it, this order, this 

 plan, this unity, this harmony, he need have no fear of a poor memory, 

 nor need he weary his concentration. The student deficient in this 

 power may learn demonstrations by heart, he may assent to each step as 

 logically proved, yet he will know little of the theorem itself. Those 

 who possess this kind of insight which reveals hidden relations, this 

 divining power for the discovery of mines of gold, may hope to become 

 investigators, creators. Those who do not have it must find it or give up 

 the task. The great educational question of the day is the problem of 

 the development of the intuition. If we learn to cultivate this spirituelle 

 flower it will open all doors of invention and discovery of Jaws. It is an 

 interesting problem for even the grade teacher. If it be true, as Boris 

 Sidis and others have claimed, that there are superior methods of edu- 

 cation (which seem really to lie along this line) then they must become 

 the methods of future education. We will begin to educate for genius. 

 One thing seems evident, that too prolonged adherence to the methods 

 of rigid reasoning leads to sterility. In mathematics at least both logic 

 and intuition are indispensable, one furnishes the architect's plan of the 

 structure, the other bolts it and cements it together. Logic, says Poin- 

 care, is the sole instrument of certitude, intuition of creation. Yet even 

 the steps of a logical deduction are planned in their entirety by the 

 intuition. In discussing the partial differential equations of physics 9 



• Amer. Jour. Math., Vol. 12. 



