_Confirmation of this View._--It may be argued, as a confirmation of this view, that whenever mathematical reasoning comes to be applied to sciences involving facts either as the data, or as objects of investigation, where it is no longer possible to proceed entirely upon hypothesis, as, _e. g._, when you apply it to mechanics, physics, astronomy, practical geometry, etc., then it ceases to be demonstrative, and becomes merely probable reasoning. _Mathematical reasoning supposed by some to be identical._--It has been much discussed whether all mathematical reasoning is merely identical, asserting, in fact, nothing more than that _a=a_; that a given thing is equivalent to itself, capable of being resolved at last into merely this. This view has been maintained by Leibnitz, himself one of the greatest mathematicians, and by many others. It was for a long time the prevalent doctrine on the Continent. Condillac applies the same to all reasoning, and Hobbes seems to have had a similar view, _i. e._, that all reasoning is only so much addition or subtraction. Against this view Stewart contends that even if the propositions themselves might be represented by the formula _a=a_, it does not follow that the various steps of reasoning leading to the conclusion amount merely to that. A paper written in cipher may be said to be identical with the same paper as interpreted; but the evidence on which the act of deciphering proceeds, amounts to something more than the perception of identity. And further, he denies that the propositions are identical, _e. g._, even the simple proposition 2×2=4. 2×2 express one set of quantities, and 4 expresses another, and the proposition that asserts their equivalence is not identical; it is not saying that the same quantity is equal to itself, but that two different quantities are equivalent. II. PROBABLE REASONING. _Not opposed to Certainty._--It must be borne in mind, as already stated, that the probability now intended is not opposed to certainty. That Cæsar invaded Britain is _certain_, but the reasoning which goes to establish it, is only probable reasoning, because the thing to be proved is an event in history, contingent therefore, and not capable of demonstration. _Sources of Evidence._--Evidence of this kind of truths is derived from three sources: 1. Testimony; 2. Experience; 3. Analogy. 1. _Evidence of Testimony._