Unfortunately, however, the verification only seemed to be successful. Aristotle chose to exemplify his theory of scientific proof from the mathematical sciences. His choice was natural enough, because they were the only sciences which had reached any considerable development in his day, and they had, moreover, an apparent necessity and universality and a fascinating appearance of exactness. But he had unwittingly chosen the most difficult and deceptive exemplification of scientific procedure. Because the mathematical sciences were in a relatively advanced condition they seemed to lend themselves to his design. He could there find terms whose meaning, and principles whose truth, was no longer in dispute. They could in consequence be argued from with as much assurance as debaters could assume the recognized meanings of words. And the fact that results seemed to follow from mathematical definitions and premisses which were not merely verbal, shed a delusive glory on the forms of dialectical proof by which they had been reached. Hence it easily escaped notice that the logical superiority of mathematics was an achievement, not a datum. Just because the mathematical sciences were very ancient, their origins had been forgotten, and with them the tentative gropings which had first selected, and subsequently confirmed, their principles. They had become immediately certain and ‘self-evident’, and no one was disposed to dispute them. On this psychological fact the whole theory of logical proof was erected. Again, it was natural to suppose that the true nature of scientific knowing must be revealed in its most perfect specimens: no one stopped to reflect that even so the real difficulties of making a science are more keenly felt and more easily seen in the nascent stage than in one which has victoriously overcome them, and has rewritten its history in the assurance of its prosperous issue. Lastly, the subtle ambiguity which pervades all mathematical reasoning, according as its terms are taken as _pure_ or as applied, was overlooked entirely--with the disastrous result that the universality, certainty, and exactness pertaining (hypothetically) to the ideal creations of ‘pure’ mathematics were erroneously transferred to their ‘applied’ counterparts. To this day logicians are found to argue that real space is homogeneous because it is convenient in Euclidean geometry to abstract from the multitudinous deformations to which bodies moving through it are subjected, and to leave them to be treated by physics;[382] nor are they aware of any lack of ‘exactness’ and discrimination when they identify the ideal triangle with the figures they draw on the blackboard.