How in
certain cases transference is possible I will explain later.
Arithmetical demonstration and the other sciences likewise
possess, each of them, their own genera; so that if the
demonstration is to pass from one sphere to another, the genus must be
either absolutely or to some extent the same. If this is not so,
transference is clearly impossible, because the extreme and the middle
terms must be drawn from the same genus: otherwise, as predicated,
they will not be essential and will thus be accidents. That is why
it cannot be proved by geometry that opposites fall under one science,
nor even that the product of two cubes is a cube. Nor can the
theorem of any one science be demonstrated by means of another
science, unless these theorems are related as subordinate to
superior (e.g. as optical theorems to geometry or harmonic theorems to
arithmetic). Geometry again cannot prove of lines any property which
they do not possess qua lines, i.e. in virtue of the fundamental
truths of their peculiar genus: it cannot show, for example, that
the straight line is the most beautiful of lines or the contrary of
the circle; for these qualities do not belong to lines in virtue of
their peculiar genus, but through some property which it shares with
other genera.
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