admits of the contrary supposition without absurdity, the other does not; the one is contingent, the other necessary Now these two classes of truths, differing as they do, in this important particular, admit of, and require, very different methods of reasoning. The one class is susceptible of demon stration, the other admits only that species of reasoning called probable or moral. It must be remembered, however that when we thus speak we do not mean that this latter class of truths is deficient in proof; the word probable is not, as thus used, opposed to certainty, but only to demonstr tion. That there is such a city as Rome, or London, is just as certain as that the several angles of a triangle are equal to two right-angles; but the evidence which substantiates the one is of a very different nature from that of the other. The one can be demonstrated, the other cannot. The one is an eternal and necessary truth, subject to no contingence, no possibility of the opposite. The other is of the nature of an event taking place in time, and dependent on the will of man, and might, without any absurdity, be supposed not to be as it is.

I. DEMONSTRATIVE REASONING.

Field of Demonstrative Reasoning. Its field, as we have seen, is necessary truth. It is limited, therefore, in its range, takes in only things abstract, conceptions rather than reali ties, the relations of things rather than things themselves, as existences. It is confined principally, if not entirely, to mathematical truths.

No degrees of Evidence. There are no degrees of evi dence or certainty in truths of this nature. Every step follows irresistibly from the preceding. Every conclusion is inevitable. One demonstration is as good as another, so far as regards the certainty of the conclusion, and one is as good as a thousand. It is quite otherwise in probable rea soning.

Two Modes of Procedure. In demonstration, we may

proceed directly, or indirectly; as, e. g., in case of two trian gles to be proved equal. I may, by super-position, prove this directly; or I may suppose them unequal, and proceed to show the absurdity of such a supposition; or I may make a number of suppositions, one or the other of which must be true, and then show that all but the one which I wish to establish are false.

Force of Mathematical reasoning. The question arises whence the peculiar force of mathematical, in distinction from other reasoning? a fact observed by every one, but not easily explained: how happens this, and on what does it depend, this irresistible cogency which compels our assent? Is it owing to the pains taken to define the terms employed, and the strict adherence to those definitions? I think not; for other sciences approximate to mathematics in this, but not to the cogency of its reasoning. The explanation given by Stewart is certainly plausible. He ascribes the peculiar force of demonstrative reasoning to the fact, that the first principles from which it sets out, i. e., its definitions, are purely hypothetical, involving no basis or admixture of facts, and that by simply reasoning strictly upon these assumed hypotheses the conclusions follow irresistibly. The same thing would happen in any other science, could we (as we cannot) construct our definitions to suit ourselves, instead of proceeding upon facts as our data. The same view is ably maintained by other writers.

If this be so, the superior certainty of mathematical, over all other modes of reasoning, if it does not quite vanish, becomes of much less consequence than is generally supposed. Its truths are necessary in no other sense than that certain definitions being assumed, certain suppositions made, then the certain other things follow, which is no more than may be said of any science.

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 be comes merely probable reasoning.

Mathematical reasoning supposed by some to be iden tical. — 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 con tends 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 x2=4. 2 × 2 express one set of quantities, and 4 expresses another, and the proposi tion that asserts their equivalence is not identical; it is not saying that the same quantity is equal to itself, but that two lifferent quantities are equivalent.

II. PROBABLE REASONING.

Not opposed to Certainty. It must be borne in mir.d, 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 capabie 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.

In itself probable. - This is, à priori, probable. We ar so constituted as to be inclined to believe testimony, and it is only when the incredibility of the witness has been ascertained by sufficient evidence, that we refuse our assent. The child believes whatever is told him. The man, long conversant with human affairs, becomes wary, cautious, suspicious, 'ncredulous. It is remarked by Reid that the evidence of testimony does not depend altogether on the character of the witness. If there be no motive for deception, especially if there be weighty reasons why he should speak truth, or if the narrative be in itself probable and consistent, and tallies with circumstances, it is in such cases to be received even from those not of unimpeachable integrity.

Limits of Belief. What are the limits of belief in testimony? Suppose the character of witnesses to be good, the narrative self-consistent, the testimony concurrent of various witnesses, explicit, positive, full, no motive for deception; are we to believe in that case whatever may be testified? One thing is certain, we do in fact believe in such cases; we are so constituted. Such is the law of our nature. Nor can it be shown irrational to yield such assent. It has been shown by an eminent mathematician that it is always possible to assign a number of independent witnesses, sc great that the falsity of their concurrent testimony shall be mathematically more improbable, and so more incredible, than the truth of their statement, be it what it may.

Case supposed. Suppose a considerable number of men of undoubted veracity, should, without concert, and agree ing in the main as to particulars, all testify, one by one, that

they witnessed, on a given day and hour, some very strange occurrence, as, e. y., a ball of fire, or a form of angelic brightness, hovering in the air, over this building, or any like unwonted and inexplicable phenomenon. Are we to withhold or yield our assent? I reply, if the number of witnesses is large, and the testimony concurrent, and without concert, and no motive exists for deception, and they are men of known integrity, especially if they are sane and sober men, not easily imposed upon, I see not how we can reasonably withhold assent. Their testimony is to be taken as true testimony, i. e., they did really witness the phenomenon described. The proof becomes stronger or weaker in proportion as the circumstances now mentioned coëxist to a greater or less extent, i.e., in proportion as there are more or fewer of these concurring and corroborating circumstances. If there was but a single witness, or if a number of the witnesses were not of the best character, or if there were some possible motive for deception, or if they were not altogether agreed as to important features of the case, so far the testimony would of course be weakened. But we may always suppose a case so strong that the falsity of the witnesses would be a greater miracle than the truth of the story. This is the case with the testimony of the witnesses to our Saviour's miracles.

Distinction to be made. An important distinction is here to be noticed between the falsity, and the incorrectness, of the witness, between his intention to deceive, and his being himself deceived. He may have seen precisely what he describes; he may be mistaken in thinking it to have been an angel, or a spirit, or a ball of fire. Just as in the case of certain illusions of sense an oar in the water-the eye correctly reports what it sees, but the judgment is in error, in thinking the oar to be crooked. So the witness may be true, and the testimony true in the case of a supposed miracle or other strange phenomenon; the appearance may have been just as stated, but the question may stil' be raised,