This necessity, that if we wish to have abstract knowledge of space-relations (_i.e._, rational knowledge, not mere intuition or perception), space with its three dimensions must be translated into time which has only one dimension, this necessity it is, which makes mathematics so difficult. This becomes very clear if we compare the perception of curves with their analytical calculation, or the table of logarithms of the trigonometrical functions with the perception of the changing relations of the parts of a triangle, which are expressed by them. What vast mazes of figures, what laborious calculations it would require to express in the abstract what perception here apprehends at a glance completely and with perfect accuracy, namely, how the co-sine diminishes as the sine increases, how the co-sine of one angle is the sine of another, the inverse relation of the increase and decrease of the two angles, and so forth. How time, we might say, must complain, that with its one dimension it should be compelled to express the three dimensions of space! Yet this is necessary if we wish to possess, for application, an expression, in abstract concepts, of space-relations. They could not be translated directly into abstract concepts, but only through the medium of the pure temporal quantity, number, which alone is directly related to abstract knowledge. Yet it is worthy of remark, that as space adapts itself so well to perception, and by means of its three dimensions, even its complicated relations are easily apprehended, while it eludes the grasp of abstract knowledge; time, on the contrary, passes easily into abstract knowledge, but gives very little to perception. Our perceptions of numbers in their proper element, mere time, without the help of space, scarcely extends as far as ten, and beyond that we have only abstract concepts of numbers, no knowledge of them which can be presented in perception. On the other hand, we connect with every numeral, and with all algebraical symbols, accurately defined abstract concepts. We may further remark here that some minds only find full satisfaction in what is known through perception. What they seek is the reason and consequent of being in space, sensuously expressed; a demonstration after the manner of Euclid, or an arithmetical solution of spacial problems, does not please them. Other minds, on the contrary, seek merely the abstract concepts which are needful for applying and communicating knowledge. They have patience and memory for abstract principles, formulas, demonstrations in long trains of reasoning, and calculations, in which the symbols represent the most complicated abstractions. The latter seek preciseness, the former sensible perception. The difference is characteristic.