Xerxes invaded Greece, &c., are inquiries belonging to moral reasoning.

§ 282. 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 mathematical 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.

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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 other treatises involving geometrical principles; 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, ir olved in propositions susceptible of demonstration, would have been beyond our reach. (See § 205.)

But axioms are by no means without their use, alhough 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.

283. 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 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. Again, we have ne

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.

$284. 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 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. 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

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.

285. 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, 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 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

* Treatise concerning Immutable Morality book iv., ch. iii.

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.

§ 286. 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 perceptions. 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 learned 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 likely, under the same circumstances, to cause the same sensation. When the infant sees before him some glittering toy, he reaches his hand towards it, and is evidently induced to do so by a thought of this kind, that the acquisition of the object will soon follow the effort of the hand, as it has a similar effort previously made.-Here is reasoning without words; it is purely internal; nevertheless, no one will presume to say that words are not great helps in reasoning. And thus in demonstrative reasoning, although diagrams, and numerical and algebraic signs are assistances, they do not constitute the process; nor can it be even said that they are indispensably essential to it.

"Some geometricians," says Buffier (First Truths, pt. i., ch. 6)," are led into a palpable error in imagining that things demonstrated by Geometry exist, out of their thought, exactly similar to the demonstration formed of them in their mind. They must be quickly sensible of their mistake, if they will but reflect a moment on the VOL L-H H