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THIRD CLASS OF OBJECTS FOB THE SUBJECT. 157
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 G-eometry 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 diffe
rent, 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 compre
hensiveness of the conception with the complete definite-
ness of the single representation. For although, as intuitive
representations, they are throughout determined with com
plete 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