(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) The conversion of the second proposition into the third is usually made by an indirect demonstration, in the following manner. If possible, let there be one B which is not A, (2) being true. Then there is one thing which is not A and is B; but every thing not A is not B; therefore there is one thing which is B and is not B: which is absurd. It is then absurd that there should be one single B which is not A; or, Every B is A. The following proposition contains a method which is of frequent use. HYPOTHESIS.—Let there be any number of propositions or assertions,—three for instance, A, B, and C,—of which it is the property that one or the other must be true, _and one only_. Let there be three other propositions, P, Q, and R, of which it is also the property that one, and one only, must be true. Let it also be a connexion of those assertions, that When A is true, P is true When B is true, Q is true When C is true, R is true CONSEQUENCE: then it follows that When P is true, A is true When Q is true, B is true When R is true, C is true For, when P is true, then Q and R must be false; consequently, neither B nor C can be true, for then Q or R would be true. But either A, B, or C must be true, therefore A must be true; or, when P is true, A is true. In a similar way the remaining assertions may be proved. Case 1. If │When P is Q, A is B │When P is not Q, A is not B It follows that│When A is B, P is Q │When A is not B, P is not Q Case 2. If │When A is greater than B, P is greater than Q „ │When A is equal to B, P is equal to Q „ │When A is less than B, P is less than Q