For accidents are not necessary: and, further, since
accidents are not necessary one does not necessarily have reasoned
knowledge of a conclusion drawn from them (this is so even if the
accidental premisses are invariable but not essential, as in proofs
through signs; for though the conclusion be actually essential, one
will not know it as essential nor know its reason); but to have
reasoned knowledge of a conclusion is to know it through its cause. We
may conclude that the middle must be consequentially connected with
the minor, and the major with the middle.
7
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: (1) what is
proved, the conclusion-an attribute inhering essentially in a genus;
(2) the axioms, i.e. axioms which are premisses of demonstration;
(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.
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