what i have sought to explain in the preceding pages is how the scientist should guide himself in choosing among the innumerable facts offered to his curiosity, since indeed the natural limitations of his mind compel him to make a choice, even though a choice be always a sacrifice. i have expounded it first by general considerations, recalling on the one hand the nature of the problem to be solved and on the other hand seeking to better comprehend that of the human mind, which is the principal instrument of the solution. i then have explained it by examples; i have not multiplied them indefinitely; i also have had to make a choice, and i have chosen naturally the questions i had studied most. others would doubtless have made a different choice; but what difference, because i believe they would have reached the same conclusions.

there is a hierarchy of facts; some have no reach; they teach us nothing but themselves. the scientist who has ascertained them has learned nothing but a fact, and has not become more capable of foreseeing new facts. such facts, it seems, come once, but are not destined to reappear.

there are, on the other hand, facts of great yield; each of them teaches us a new law. and since a choice must be made, it is to these that the scientist should devote himself.

doubtless this classification is relative and depends upon the weakness of our mind. the facts of slight outcome are the complex facts, upon which various circumstances may exercise a sensible influence, circumstances too numerous and too diverse for us to discern them all. but i should rather say that these are the facts we think complex, since the intricacy of these circumstances surpasses the range of our mind. doubtless a mind vaster and finer than ours would think differently of them. but what matter; we can not use that superior mind, but only our own.

the facts of great outcome are those we think simple; may be they really are so, because they are influenced only by a small number of well-defined circumstances, may be they take on an appearance of simplicity because the various circumstances upon which they depend obey the laws of chance and so come to mutually compensate. and this is what happens most often. and so we have been obliged to examine somewhat more closely what chance is.

facts where the laws of chance apply become easy of access to the scientist who would be discouraged before the extraordinary complication of the problems where these laws are not applicable. we have seen that these considerations apply not only to the physical sciences, but to the mathematical sciences. the method of demonstration is not the same for the physicist and the mathematician. but the methods of invention are very much alike. in both cases they consist in passing up from the fact to the law, and in finding the facts capable of leading to a law.

to bring out this point, i have shown the mind of the mathematician at work, and under three forms: the mind of the mathematical inventor and creator; that of the unconscious geometer who among our far distant ancestors, or in the misty years of our infancy, has constructed for us our instinctive notion of space; that of the adolescent to whom the teachers of secondary education unveil the first principles of the science, seeking to give understanding of the fundamental definitions. everywhere we have seen the r?le of intuition and of the spirit of generalization without which these three stages of mathematicians, if i may so express myself, would be reduced to an equal impotence.

and in the demonstration itself, the logic is not all; the true mathematical reasoning is a veritable induction, different in many regards from the induction of physics, but proceeding like it from the particular to the general. all the efforts that have been made to reverse this order and to carry back mathematical induction to the rules of logic have eventuated only in failures, illy concealed by the employment of a language inaccessible to the uninitiated. the examples i have taken from the physical sciences have shown us very different cases of facts of great outcome. an experiment of kaufmann on radium rays revolutionizes at the same time mechanics, optics and astronomy. why? because in proportion as these sciences have developed, we have the better recognized the bonds uniting them, and then we have perceived a species of general design of the chart of universal science. there are facts common to several sciences, which seem the common source of streams diverging in all directions and which are comparable to that knoll of saint gothard whence spring waters which fertilize four different valleys.

and then we can make choice of facts with more discernment than our predecessors who regarded these valleys as distinct and separated by impassable barriers.

it is always simple facts which must be chosen, but among these simple facts we must prefer those which are situated upon these sorts of knolls of saint gothard of which i have just spoken.

and when sciences have no direct bond, they still mutually throw light upon one another by analogy. when we studied the laws obeyed by gases we knew we had attacked a fact of great outcome; and yet this outcome was still estimated beneath its value, since gases are, from a certain point of view, the image of the milky way, and those facts which seemed of interest only for the physicist, ere long opened new vistas to astronomy quite unexpected.

and finally when the geodesist sees it is necessary to move his telescope some seconds to see a signal he has set up with great pains, this is a very small fact; but this is a fact of great outcome, not only because this reveals to him the existence of a small protuberance upon the terrestrial globe, that little hump would be by itself of no great interest, but because this protuberance gives him information about the distribution of matter in the interior of the globe, and through that about the past of our planet, about its future, about the laws of its development.