in virtue of this insight that I know, that where ten are, there also are eight, six, four.
§ 39. Geometry.
The whole science of Geometry likewise rests upon the nexus of the position of the divisions of Space. It would, accordingly, be an insight into that nexus ; only such an insight being, as we have already said, impossible by means of mere conceptions, or indeed in any other way than by in tuition, every geometrical proposition would have to be brought back to sensuous intuition, and the proof would simply consist in making the particular nexus in question clear; nothing more could be done. Nevertheless we find Geometry treated quite differently. Euclid's Twelve Axioms are alone held to be based upon mere intuition, and even of these only the Ninth, Eleventh, and Twelfth are properly speaking admitted to be founded upon different, separate intuitions ; while the rest are supposed to be founded upon the knowledge that in science we do not, as in experience, deal with real things existing for themselves side by side, and susceptible of endless variety, but on the contrary with conceptions, and in Mathematics with normal intuitions, i.e. figures and numbers, whose laws are binding for all experience, and which therefore combine the comprehensiveness of the conception with the complete definiteness of the single representation. For although, as intuitive representations, they are throughout determined with complete precision—no room being left in this way by anything remaining undetermined—still they are general, because they are the bare forms of all phenomena, and, as such, applicable to all real objects to which such forms belong, What Plato says of his Ideas would therefore, even in Geometry, hold good of these normal intuitions, just as well as of conceptions, i.e. that two cannot be exactly
