Should we assign as belonging solely to the imagination every invention or discovery--in a word, whatever is new--in the well-organized sciences that form a body of solid, constantly-broadening doctrine? It is a hard question. That which raises scientific knowledge above popular knowledge is the use of an experimental method and rigorous reasoning processes; but, is not induction and deduction going from the known to the unknown? Without desiring to depreciate the method and its value, it must nevertheless be admitted that it is preventive, not inventive. It resembles, says Condillac, the parapets of a bridge, which do not help the traveler to walk, but keep him from falling over. It is of value especially as a habit of mind. People have wisely discoursed on the "methods" of invention. There are none; but for which fact we could manufacture inventors just as we make mechanics and watchmakers. It is the imagination that invents, that provides the rational faculties with their materials, with the position, and even the solution of their problems. Reasoning is only a means for control and proof; it transforms the work of the imagination into acceptable, logical results. If one has not imagined beforehand, the logical method is aimless and useless, for we cannot reason concerning the completely unknown. Even when a problem seems to advance towards solution wholly through the reason, the imagination ceaselessly intervenes in the form of a succession of groupings, trials, guesses, and possibilities that it proposes. The function of method is to determine its value, to accept or reject it.[111] Let us show by a few examples that conjecture, the work of the combining imagination, is at the root of the most diverse scientific inventions.[112] Every mathematical invention is at first only an hypothesis that must be demonstrated, i.e., must be brought under previously established general principles: prior to the decisive moment of rational verification it is only a thing imagined. "In a conversation concerning the place of imagination in scientific work," says Liebig, "a great French mathematician expressed the opinion to me that the greater part of mathematical truth is acquired not through deduction, but through the imagination. He might have said 'all the mathematical truths,' without being wrong." We know that Pascal discovered the thirty-second proposition of Euclid all by himself. It is true that it has been concluded, wrongly perhaps, that he had also discovered all the earlier ones, the order followed by the Greek geometrician not being necessary, and not excluding other arrangements. However it be, reasoning alone was not enough for that discovery. "Many people," says Naville, "of whom I am one, might have thought hard all their lives without finding out the thirty-two propositions of Euclid." This fact alone shows clearly the difference between invention and demonstration, imagination and reason.