THE WORLD AS IDEA. 69
enduring, arranged, and planned activity must therefore proceed from principles, — that is, from abstract knowledge, and it must be conducted in accordance with them. Thus, for example, the knowledge of the relation of cause and effect arrived at by the understanding, is in itself far completer, deeper and more exhaustive than anything that can be thought about it in the abstract; the understanding alone knows in perception directly and completely the nature of the effect of a lever, of a pulley, or a cog-wheel, the stability of an arch, and so forth. But on account of the peculiarity of the knowledge of perception just referred to, that it only extends to what is immediately present, the mere understanding can never enable us to construct machines and buildings. Here reason must come in; it must substitute abstract concepts for ideas of perception, and take them as the guide of action; and if they are right, the anticipated result will happen. In the same way we have perfect knowledge in pure perception of the nature and constitution of the parabola, hyperbola, and spiral; but if we are to make trustworthy application of this knowledge to the real, it must first become abstract knowledge, and by this it certainly loses its character of intuition or perception, but on the other hand it gains the certainty and preciseness of abstract knowledge. The differential calculus does not really extend our knowledge of the curve, it contains nothing that was not already in the mere pure perception of the curve; but it alters the kind of knowledge, it changes the intuitive into an abstract knowledge, which is so valuable for application. But here we must refer to another peculiarity of our faculty of knowledge, which could not be observed until the distinction between the knowledge of the senses and understanding and abstract knowledge had been made quite clear. It is this, that relations of space cannot as such be directly translated into abstract knowledge, but only temporal quantities, — that is, numbers, are suitable for this.
