[1]. This is the proposition in proof of which nearly the whole of the demonstration of Euclid is spent.
(2.) It is false that there can be any whole which is not greater than its part (self evident).
(3.) Therefore it is false that there is any equiangular triangle which is not equilateral; or all equiangular triangles are equilateral.
When a proposition is established by proving the truth of the matters it contains, the demonstration is called direct; when by proving the falsehood of every contradictory proposition, it is called indirect. The latter species of demonstration is as logical as the former, but not of so simple a kind; whence it is desirable to use the former whenever it can be obtained.
The use of indirect demonstration in the Elements of Euclid is almost entirely confined to those propositions in which the converses of simple propositions are proved. It frequently happens that an established assertion of the form
| Every A is B | (1) |
may be easily made the means of deducing,
| Every (thing not A) is not B | (2) |
which last gives
| Every B is A | (3) |