LECTURE III.

ON THE DEFINITIONS, POSTULATES, AND AXIOMS OF

GEOMETRY.

Object of definitions different from that of axioms-without reference to the nature of the thing no interpretation or extension possibleLocke's views need no refutation.-Definitions in Euclid's first Element-proof of the coincidence of planes of the equality of right angles-postulates-axioms-theory of parallels, dependent on our idea of straightness-connection between the two axioms of a straight line.

THE first thing to be done in entering on the discussion of a subject, is to lay down clearly what it is about which the arguments are to be adduced. To effect this is the proper duty of definitions of the first class. A science framed without reference to the right conception of the things with which it is conversant, must terminate in its own conclusions. It is not that definitions are requisite for demonstration, for they are not. Conceive yourselves to have no notion of the form of what is called a straight line, or, if you please, assign to it an incorrect notion. Suppose this the representation of a straight line, agreeably to your ideas of it. To exhibit to you the relations of such lines, I put you in possession of a property or affection of them; as, for instance, that "two straight lines cannot coincide in two points without coinciding altogether." You accept it as the characteristic of that which you have figured to yourself, if indeed you have procured any conception of the

thing at all; or abstractedly as the mark or quality of a certain thing if you have not. This being premised, I am in a position to demonstrate to you the earlier theorems of Euclid's first Element. I can compel your assent to the truth of the proposition, that "any two sides of a triangle are together greater than the third side;" and this without requiring from you any notion of a straight line at all. But I must stop here. I cannot require you to admit that the line in which a stone descends to the earth is that which I have called a straight line. It is true you may deduce from my defining property such other affections as to convey the idea to your mind. You may prove from it that "a straight line is the shortest distance between two points."* But this is to effect nothing worth the labour. The mind can certainly pass more easily from conceptions to properties than from properties to conceptions. But the question is, which can be done most securely? Now, I am of opinion that the former has the advantage here too. And if this be true, whatever merits a science without axioms may possess as an independent system, it has no recommendation as a component part of the body of knowledge. It moves not in unison with the other members, and has no affections in common with them.

But further, in defining by means of a property, we can never be sure that we express all the affections which our conceptions of the thing taught us to expect from it. When we designate as a straight line that which possesses a quality in common with another, we bar ourselves from considering what it is in itself. This, as far as the sci* As it is done in Colonel Thompson's Theory of Parallels, Note 6.

ence of geometry is concerned, is of no importance. Comparison is all that this science effects. It is only when we want to pass from this to another subject that we are met by a difficulty. But what is far worse, the property which we select may be an exclusive property. It may not only fail to exhibit what the thing is in itself, but may be of such a nature as to bar out other species of comparison. And this is what the defining property of a straight line actually does. Its characteristic feature is mutual coincidence. Relations which belong not to coincidence may assuredly flow from it, but relations which forbid the conception of coincidence never can. Now, parallelism, view it as you will, is of the latter nature; and I do not think I speak too strongly when I say, that the problems of the quadrature of the circle and the duplication of the cube are as promising as the doctrine of parallels. The straight line is endued with two properties; the one relative to a line with which it does and must come in contact, the other relative to a line with which it does not and must not come in contact. Both these properties flow naturally enough from the same fundamental idea of a straight line, but the one does not and cannot (as I think) flow from the other. Here, then, we are presented with another reason for not defining, by means of properties,—the thing defined may possess two affections not mutually connected with one another. From what has been said, it appears that a science framed on abstract definitions will not only be useless in its results, but imperfect in its construction. It will not be interpretable, will not be applicable to quantity or to any thing in our conceptions, but must be a mere creation of the intellect, in

capable of being transferred to existences or our ideas of them. It will either be confined within narrow limits, each branch being distinct from and unconnected with the other, or will owe its extension to the assigning of qualities to defined things which may be inconsistent with their hypothetical nature; and, if otherwise, inasmuch as they are purely empirical, their fitness can never be evidenced by the mind.

All our science, then, is based on axioms. It may seem incumbent on me, in making this statement, to reply to the arguments of Locke,* who asserts that " axioms are not, nor have been the foundations on which any science hath been built." Those, however, who examine Locke's arguments will, I believe, excuse me for omitting to refute him. The axioms which he alludes to are of two kinds, general axioms and hypotheses, and although he does occasionally† quote one of Euclid's, as, for instance, that which Aristotlet says is common to all quantities, he never once reflects on it as applied to reasoning on diverse forms, for the purpose of transferring a property belonging to a quality of one thing to the like quality of another. Neither does he allude to the more important axioms of particular parts of space, as of the straight line and the circle. But my limits compel me to abstain from further remarks. Let us therefore proceed to a detailed examination of the

DEFINITIONS OF GEOMETRY.
1.

A point has no parts nor magnitude, but is position merely.
Euclid gives the former clause alone. To this mode of

* On the Human Understanding, B. 4, C. 7, § 11.

defining, there are obviously two objections, 1st, that it contains only a negative; and, 2dly, that it is not convertible. Proclus* defends the definition against the first objection, on the ground that an element must commence with negatives; inasmuch as it is the terminus or first limit of our conceptions. The ancients generally do not appear to have required any more accurate statement. Cicerot quotes the definition in Euclid's own words, "punctum esse, quod magnitudinem nullam habet." It is true the Pythagoreans‡ expressed it as "an unit (or monad) having position," which, by a slight change, makes the definition we have adopted. They did so, however, I believe, not for the purpose of annexing more definite properties to that which is without magnitude, but to distinguish between geometrical and abstract unity. This was to relieve the definition from the second charge, the want of convertibility. Proclus himself anticipates this second objection, when he asks, “Is a point, then, the only impartible? or may we not affirm the same of the now in time and of unity in numbers ?" The addition of the second clause, which assigns to it position, obviates this difficulty, and, at the same time, gives the real characteristic of a point. It is rather as position than as want of magnitude that geometers make use of it. The word onusov, as also its Latin punctum, does appear, indeed, to refer to a thing, but it must be remembered that the language has reference rather to that which fixes the idea, than to the idea itself. It is well remarked

* Com. on Def. 1. See also Savile Prælectiones, p. 51. Scarburgh English Euclid, p. 1. Compare Arist. Top. vi.

† Acad. Quest. iv. 36.

Proclus in Def.