athematical 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.

In the sciences dealing with facts, all the best-established
experimental truths have passed through a conjectural stage. History
permits no doubt on this point. What makes it appear otherwise is the
fact that for centuries there has gradually come to be formed a body of
solid belief, making a whole, stored away in classic treatises from
which we learn from childhood, and in which they seem to be arranged of
themselves. We are not told of the series of checks and failures through
which[113] they have passed. Innumerable are the inventions that
remained for a long time in a state of conjecture, matters of pure
imagination, because various circumstances did not permit them to take
shape, to be demonstrated and verified. Thus, in the thirteenth century,
Roger Bacon had a very clear idea of a construction on rails similar to
our railroads; of optical instruments that would permit, as does the
telescope, to see very far, and to discover the invisible. It is even
claimed that he must have foreseen the phenomena of interferences, the
demonstration of which had to be awaited ten centuries.

On the other hand, there are guesses that have met success without much
delay, but in which the imaginative phase--that of the invention
preceding all demonstration--is easy to locate.