### Aristotle - The Organon ANALYTICA POSTERIORA Book 1 Part 7

## No demonstration may pass from one genus to another

1.
It follows that we cannot in demonstrating pass from one genus to
another.
We cannot, for instance, prove geometrical truths by
arithmetic. For there are three elements in demonstration:

2.
(1) what is
proved, the conclusion - an attribute inhering essentially in a genus;

3.
(2) the axioms, i.e. axioms which are premisses of demonstration;

4.
(3) the subject-genus whose attributes, i.e. essential
properties, are
revealed by the demonstration.
The axioms which are premisses of
demonstration may be identical in two or more sciences: but in the
case of two different genera such as arithmetic and geometry you
cannot apply arithmetical demonstration to the properties of
magnitudes unless the magnitudes in question are numbers. How in
certain cases transference is possible I will explain later.

5.
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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