onstrations of mathematics. Axioms are certain self-ev But But axioms are by no means without their use, although their nature may have been misunderstood. They are properly and originally intuitive perceptions of the truth; and whether they be expressed in words, as we generally find them, or not, is of but little consequence, except as a matter of convenience to beginners, and in giving instruction. But those intuitive perceptions which are always implied in them are essential helps; and if by their aid alone we should be unable to complete a demonstration, we should be equally unable without them. We begin with definitions; we' compare together successively a number of propositions; and these intuitive perceptions of their agreement or disagreement, to which, when expressed in words, we give the name of axioms, attend us at every step. 187. The opposites of demonstrative reasonings absurd. In demonstrations we consider only one side of a question; it is not necessary to do anything more than this. The first principles in the reasoning are given; they are not only supposed to be certain, but they are assumed as such; these are followed by a number of propositions in succession, all of which are compared together; if the conclusion be a demonstrative one, then there has been a clear perception of certainty at every step in the train. Whatever may be urged against an argument thus conducted is of no consequence; the opposite of it will al ways imply some fallacy. Thus, the proposition that the three angles of a triangle are not equal to two right angles, and other propositions, which are the opposite of what has been demonstrated, will always be found to be false, and also to involve an absurdity; that is, are inconsistent with, and contradictory to, themselves. But it is not so in Moral Reasoning. And here, therefore, we find a marked distinction between the two great forms of ratiocination. We may arrive at a conclusion on a moral subject with a great degree of certainty; not a doubt may be left in the mind; and yet the opposite of that conclusion may be altogether within the limits of possibility. We have, for instance, the most satisfactory evidence that the sun rose to-day, but the opposite might have been true, without any inconsistency or contradiction, viz., That the sun did not rise. Again, we have no doubt of the great law in physics, that heavy bodies descend to the earth in a line directed towards its centre. But we can conceive of the opposite of this without involving any contradiction or absurdity. In other words, they might have been subjected, if the Creator had so determined, to the influence of a law requiring them to move in a different direction. But, on a thorough examination of a demonstrative process, we shall find ourselves unable to admit even the possibility of the opposite. § 188. Demonstrations do not admit of different degrees of belief. When our thoughts are employed upon subjects which come within the province of moral reasoning, we yield different degrees of assent; we form opinions more or less probable. Sometimes our belief is of the lowest kind; nothing more than mere presumption. New evidence gives it new strength; and it may go on, from cne degree of strength to another, till all doubt is excluded, and all possibility of mistake shut out.-It is different in demonstrations; the assent which we yield is at all times of the highest kind, and is never susceptible of being regarded as more or less. This results, as must be obvious on the slightest examination, from the nature of demonstrative reasoning. In demonstrative reasonings we always begin with certain first principles or truths, either known or taken for granted; and these hold the first place, or are the foundation of that series of propositions over which the mind successively passes until it rests in the conclusion. In mathematics, the first principles, of which we here speak, are the definitions. We begin, therefore, with what is acknowledged by all to be true or certain. At every step there is an intuitive perception of the agreement or disagreement of the propositions which are compared together. Consequently, however far we may advance in the comparison of them, there is no possibility of falling short of that degree of assent with which it is acknowledged that the series commenced. So that demonstrative certainty may be judged to amount to this. Whenever we arrive at the last step, or the conclusion of a series of propositions, the mind, in effect, intuitively perceives the relation, whether it be the agreement or disagreement, coincidence or want of coincidence, between the last step or the conclusion, and the conditions involved in the propositions at the commencement of the series; and, therefore, demonstrative certainty is virtually the same as the certainty of intuition. Although it arises on a different occasion, and is, therefore, entitled to a separate consideration, there is no difference in the degree of belief. § 189. Of the use of diagrams in demonstrations. In conducting a demonstrative process, it is frequently the case that we make use of various kinds of figures or diagrams. The proper use of diagrams, of a square, circle, triangle, or other figure which we delineate before us, is to assist the mind in keeping its ideas distinct, and to help in comparing them together with readiness and correctness. They are a sort of auxiliaries, brought in to the help of our intellectual infirmities, but are not absolutely necessary; since demonstrative reasoning, whereever it may be found, resembles any other kind of reasoning in this most important respect, viz., in being a comparison of our ideas. In proof that artificial diagrams are only auxiliaries, and are not essentially necessary in demonstrations it S may be remarked, that they are necessarily all of them imperfect. It is not within the capability of the wit and power of man to frame a perfect circle, or a perfect triangle, or any other figure which is perfect. We might argue this from our general knowledge of the imperfection of the senses; and we may almost regard it as a matter determined by experiments of the senses themselves, aided by optical instruments. "There never was," says Cudworth, "a straight line, triangle, or circle, that we saw in all our lives, that was mathematically exact; but even sense itself, at least by the help of microscopes, might plainly discover much unevenness, ruggedness, flexuosity, angulosity, irregularity, and deformity in them.”* Our reasonings, therefore, and our conclusions, will not apply to the figures before us, but merely to an imagined perfect figure. The mind can not only originate a figure internally and subjectively, but can ascribe to it the attribute of perfection. And a verbal statement of the properties of this imagined perfect figure is what we understand by a DEFINITION, the use of which, in this kind of reasoning in particular, has already been mentioned CHAPTER XI. MORAL REASONING. § 190. Of the subjects and importance of moral reasoning. MORAL REASONING, which is the second great division or kind of reasoning, concerns opinions, actions, and events; embracing, in general, those subjects which do not come within the province of demonstrative reasoning. The subjects to which it relates are often briefly expressed, by saying that they are matters of fact; nor would this definition, concise as it is, be likely to give an erroneous idea of them. Skill in this kind of reasoning is of great use in the formation of opinions concerning the duties and the gen * Treatise concerning Immutable Morality, bk. iv., ch iii. eral conduct of life. Some may be apt to think, that those who have been most practised in demonstrative reasoning can find no difficulty in adapting their intellectual habits to matters of mere probability. This opinion is not altogether well founded. Although that species of reasoning has a favourable result in giving persons a command over the attention, and in some other respects, whenever exclusively employed it has the effect, in some degree, to disqualify them for a correct judgment on those various subjects which properly belong to moral reason. ing. The last, therefore, which has its distinctive name from the primary signification of the Latin MORES, viz., manners, customs, &c., requires a separate consideration. 191. Of the nature of moral certainty. Moral reasoning causes in us different degrees of as sent, and in this respect differs from demonstrative. In demonstration there is not only an immediate perception of the relation of the propositions compared together; but, in consequence of their abstract and determinate nature, there is also a knowledge or absolute certainty of their agreement or disagreement. In moral reasoning the case is somewhat different.-In both kinds we begin with certain propositions, which are either known or regarded as such. In both there is a series of propositions successively compared. But in moral reasoning, in consequence of the propositions not being abstract and fixed, and, therefore, often uncertain, the agreement or disagreement among them is, in general, not said to be known, but presumed; and this presumption may be more or less, admitting a great variety of degrees. While, therefore, one mode of reasoning is attended with knowledge, the other can properly be said to produce, in most cases, only judgment or opinion.-But the probability of such judgment or opinion may sometimes arise so high as to exclude all reasonable doubt. And hence we then speak as if we possessed certainty in respect to subjects which admit merely of the application of moral reasoning. Al though it is possible that there may be some difference between the belief attendant on demonstration and that produced by the highest probability, the effect on our |