2o6 POPULAR SCIENCE MONTHLY 



the two kinds of intuition; they have not the same object and seem to 

 call into play two different faculties of our soul; one would think of 

 two search-lights directed upon two worlds strangers to one another. 



It is the intuition of pure number, that of pure logical forms, which 

 illumines and directs those we have called analysts. This it is which 

 enables them not alone to demonstrate, but also to invent. By it 

 they perceive at a glance the general plan of a logical edifice, and that 

 too without the senses appearing to intervene. In rejecting the aid 

 of the imagination, which, as we have seen, is not always infallible, 

 they can advance without fear of deceiving themselves. Happy, there- 

 fore, are those who can do without this aid! We must admire them; 

 but how rare they are ! 



Among the analysts there will then be inventors, but they will be 

 few. The majority of us, if we wished to see afar by pure intuition 

 alone, would soon feel ourselves seized with vertigo. Our weakness 

 has need of a staff more solid, and, despite the exceptions of which we 

 have just spoken, it is none the less true that sensible intuition is in 

 mathematics the most usual instrument of invention. 



Apropos of these reflections, a question comes up that I have not 

 the time either to solve or even to enunciate with the developments it 

 would admit of. Is there room for a new distinction, for distinguish- 

 ing among the analysts those who above all use this pure intuition and 

 those who are first of all preoccupied with formal logic? 



M. Hermite, for example, whom I have just cited, can not be classed 

 among the geometers who make use of the sensible intuition; but 

 neither is he a logician, properly so called. He does not conceal his 

 aversion to purely deductive procedures which start from the general 

 and end in the particular. 



(To be continued.) 



