When a criminal is examined, a _procès-verbal_ is made of his statement in order that we may judge of its truth from its consistency. But this is only a makeshift, and we are not satisfied with it if it is possible to investigate the truth of each of his answers for itself; especially as he might lie consistently from the beginning. But Euclid investigated space according to this first method. He set about it, indeed, under the correct assumption that nature must everywhere be consistent, and that therefore it must also be so in space, its fundamental form. Since then the parts of space stand to each other in a relation of reason and consequent, no single property of space can be different from what it is without being in contradiction with all the others. But this is a very troublesome, unsatisfactory, and roundabout way to follow. It prefers indirect knowledge to direct, which is just as certain, and it separates the knowledge that a thing is from the knowledge why it is, to the great disadvantage of the science; and lastly, it entirely withholds from the beginner insight into the laws of space, and indeed renders him unaccustomed to the special investigation of the ground and inner connection of things, inclining him to be satisfied with a mere historical knowledge that a thing is as it is. The exercise of acuteness which this method is unceasingly extolled as affording consists merely in this, that the pupil practises drawing conclusions, _i.e._, he practises applying the principle of contradiction, but specially he exerts his memory to retain all those data whose agreement is to be tested. Moreover, it is worth noticing that this method of proof was applied only to geometry and not to arithmetic. In arithmetic the truth is really allowed to come home to us through perception alone, which in it consists simply in counting. As the perception of numbers is in _time alone_, and therefore cannot be represented by a sensuous schema like the geometrical figure, the suspicion that perception is merely empirical, and possibly illusive, disappeared in arithmetic, and the introduction of the logical method of proof into geometry was entirely due to this suspicion. As time has only one dimension, counting is the only arithmetical operation, to which all others may be reduced; and yet counting is just intuition or perception _a priori_, to which there is no hesitation in appealing here, and through which alone everything else, every sum and every equation, is ultimately proved. We prove, for example, not that (7 + 9 × 8 - 2)/3 = 42; but we refer to the pure perception in time, counting thus makes each individual problem an axiom. Instead of the demonstrations that fill geometry, the whole content of arithmetic and algebra is thus simply a method of abbreviating counting. We mentioned above that our immediate perception of numbers in time extends only to about ten. Beyond this an abstract concept