reductio ad ahsurdum is obtained by assuming an absurdity in the process of proof. The conception of unequal infinites is absurd, but so is that of equal infinites. Neither equality nor inequality can be predicated of infinites. But the author evidently thought that infinites must be equal, and it is by assuming this in the course of proof that he lands us in an absurdity. He proceeds on the axiom, that if equals be multiplied by unequals, the products must be unequal. The unequals are the respective units or periods of revolution, the equals are the infinites by which they are multiplied; and consequently the resulting infinites are unequal. Here he assumes that infinites must be equal; which is absurd, for equality can be predicated only of finite qualities. The remark, that the idea of matter may be entirely dropped from the formula without in the least affecting the argument, is applicable to both of the above arguments; so that, if the reasoning have any force, it tells equally against all existence, mind as well as matter.
The antinomies of Kant involve the same fallacy. He assumes one definition in the thesis, and adopts another in the antithesis. The essence of the fallacy lies in tacitly assuming that infinity is a definite whole, of which equality and inequality may be predicated. For example, he proves the contradiction involved in a past eternity in this manner. If we reckon the past eternity from to-day, we have the following
