perception” is a function of distance. Moreover, when we reflect upon anything, we do not exactly remember the impressions that we received at the time, but “represent to ourselves” the picture of a space abstracted from them. But this representation may and does deceive us regarding the living actuality. Kant let himself be misled; he should certainly not have permitted himself to distinguish between forms of perception and forms of ratiocination, for _his_ notion of Space in principle embraced both.[182] Just as Kant marred the Time-problem by bringing it into relation with an essentially misunderstood arithmetic and—on that basis—dealing with a phantom sort of time that lacks the life-quality of direction and is therefore a mere spatial scheme, so also he marred the Space-problem by relating it to a common-place geometry. It befell that a few years after the completion of Kant’s main work Gauss discovered the first of the Non-Euclidean geometries. These, irreproachably demonstrated as regards their own internal validity, enable it to be proved that there are _several_ strictly mathematical kinds of three-dimensional extension, all of which are _a priori_ certain, and none of which can be singled out to rank as the genuine “form of perception.” It was a grave, and in a contemporary of Euler and Lagrange an unpardonable, error to postulate that the Classical school-geometry (for it was that which Kant always had in mind) was to be found reproduced in the forms of Nature around us. In moments of attentive observation at very short range, and in cases in which the relations considered are sufficiently small, the living impressions and the rules of customary geometry are certainly in approximate agreement. But the _exact_ conformity asserted by philosophy can be demonstrated neither by the eye nor by measuring-instruments. Both these must always stop short at a certain limit of accuracy which is very far indeed below that which would be necessary, say, for determining which of the Non-Euclidean geometries is the geometry of “empirical” Space.[183] On the large scales and for great distances, where the experience of depth completely dominates the perception-picture (for example, looking on a broad landscape as against a drawing) the form of perception is in fundamental contradiction with mathematics. A glance down any avenue shows us that parallels meet at the horizon. Western perspective and the otherwise quite different perspective of Chinese painting are both alike based on this fact, and the connexion of these perspectives with the root- problems of their respective mathematics is unmistakable.