investigator should not include some course in design in his work, in painting, architecture, music, poetry or sculpture. Courses in the appreciation of art, rather than the criticism of art, might also be very serviceable indirectly. The constructive philosophers, like Plato or Bergson, might furnish valuable indirect training. Reading that leads to an appreciation of the beauty and sublimity of the universe is of the same value. In any case whatever would intensify the esthetic sensitiveness would be worth while.
When the intuition does not favor us, the golden butterfly fails to emerge from its chrysalis, what is to be done? Here is his answer for whom time did not count, taken from one of his most recent papers.[1] There is a note of pathos in it as well as a hint of premonition. He presents some incomplete results of a new and very important theorem in geometrical transformation, which he is convinced is true, yet the proof of it encounters great difficulties. Every particular case he has been able to settle is favorable to the theorem. After explaining why he is publishing an incomplete paper for the first time, he says:
It would seem that in this situation I should abstain from all publication be long as I have not solved the problem, but after fruitless efforts for many months it seems to me wisest to let the whole problem ripen during several years. That would indeed be well, were I sure of some day being able to take it up again, but at my age I can not go bail for this. On the other hand, the importance of the subject is great. . . and the totality of results so far obtained is too considerable for me to resign myself to definitively allowing them to become unfruitful. I may hope that the mathematicians who interest themselves in the problem and who will be more fortunate than I without doubt will find some means to resolve it.
Again, Poincaré points out that these flashes from below the horizon of consciousness must be preceded by periods of prolonged attentive work. It is like setting Pegasus to plowing corn, but this conscious effort is necessary. This discouraging wandering over the hills and rocks, examining the promising paths and the fragments that point to a nearby mine, day after day, is indispensable to success. It is the weary search over the face of the mountain and the driving of many fruitless drifts that eventually lead the prospector to his mine of gold. On this kind of drudgery Poincaré spent two periods of two hours each daily. The unconscious action of his mind did the rest of his work.
Neither does the discovery of the mine develop it. After the unconscious power has led us to our eldorado, it has done all it can. The deductions, the demonstrations, the applications, must be carried out at the expense of prolonged effort again. The intuition can not do this kind of work. Its region is the nebulous part of thought where the mental ions unite, dissolve, and whirl away,—or we may say that
- ↑ Rend. Circ. Mat. Palermo, 33 (1912), p. 375.
