&c., so far as they are susceptible of being accurately expressed by numbers, or other mathematical signs. But the subjects of moral reasoning, upon which we are to remark hereafter more particularly, are matters of fact, including their connection with other facts, whether constant or variable, and all attendant circumstances.-That the exterior angle of a triangle is equal to both the interior and opposite angles, is a truth, which comes within the province of demonstration. That Homer was the author of the Iliad, that Xerxes invaded Greece, &c. are inquiries, belonging to moral reasoning.

§. 321. Use of definitions and axioms in demonstrative reasoning.

In every process of reasoning there must be at the commencement of it something to be proved; there must also be some things either known, or taken for granted as such, with which the comparison of the propositions begins. The preliminary truths in demonstrative reasonings are involved in such definitions as are found in all mathemati cal treatises. It is impossible to give a demonstration of the properties of a circle, parabola, ellipse, or other mathematical figure, without first having given a definition of them. DEFINITIONS, therefore, are the facts assumed, the FIRST PRINCIPLES in demonstrative reasoning, from which by means of the subsequent steps the conclusion is derived.We find something entirely similar in respect to subjects, which admit of the application of a different form of reasoning. Thus in Natural Philosophy, the general facts in relation to the gravity and elasticity of the air may be considered as first principles. From these principles in Physics are deduced, as consequences, the suspension of the mercury in the barometer, and its fall, when carried up to an eminence.

We must not forget here the use of axioms in the demonstrations of mathematics. Axioms are certain self-evident propositions, or propositions, the truth of which is discovered by intuition, such as the following; "Things, equal to the same, are equal to one another;" From equals take away equals, and equals remain." We generally find

a number of them prefixed to treatises of geometry, and it has been a mistaken supposition, which has long prevailed, that they are at the foundation of geometrical, and of all other demonstrative reasoning. But axioms, taken by themselves, lead to no conclusions. With their assistance alone, it cannot be denied, that the truth involved in propositions susceptible of demonstration, would have been beyond our reach. (See §. 279.)

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.

§. 322. The opposites of demonstrative reasoning absurd.

In demonstrations we consider only one side of a question; it is not necessary to do any thing 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 always 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. But on a thorough examination of a demonstrative process, we shall find ourselves unable to admit even the possibility of the opposite.

§. 323. Demonstrative reasonings 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 one 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. In short, all demonstrations are certain.But a question first arises, What is certainty? (See §. 64.) And again, What in particular do we understand by that certainty, which is ascribed to the conclusions, to which we are conducted in any process of demonstrative reasoning?

§. 324. Of the nature of demonstrative certainty.

In proceeding to answer this inquiry, it is again to be observed, that in demonstrative reasonings we always begin with certain first principles or truths, cither 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 the belief.

§. 325. Of the use of diagrams in demonstrations.

Mr. Locke has advanced the opinion, that moral subjects are no less susceptible of demonstration, than mathematical. However this may be, we are certainly more frequently required to practice this species of reasoning in the mathematics, than any where else; and in conducting the process, nothing is more common, than to 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, wherever 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 may be remarked, that they are necessarily all of them imperfect. It is not within the capability of the wit and the 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 experiment of the senses themselves, aided by optical instruments. "There never was (says Cudworth,) a strait 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 uneveness, 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.

§. 326. Of signs in general as connected with reasoning.

The statements in the last section will appear the less exceptionable, when it is recollected, that in all cases reasoning is purely a mental process. From beginning to end, it is a succession of feelings. Neither mathematical signs, nor words constitute the process, but are only its attendants and auxiliaries. We can reason without diagrams or other signs employed in mathematics, the same as an infant reasons, before it has learnt artificial language.

When the infant has once put his finger in the fire, he avoids the repetition of the experiment, reasoning in this way, that there is a resemblance between one flame and another, and that what has once caused him pain, will be

*Treatise concerning Immutable Morality, Bk. IV, Ch. 3.