Euclid and His Modern Rivals/Act II. Scene II.
ACT II.
Scene II.
Treatment of Parallels by methods involving infinite series.
Legengre.
'Fine by degrees, and beautifully less.'
Nie. I lay before you 'Éléments de Géométrie' by Mons. A. M. Legendre, the 14th edition, 1860.
Min. Let me begin by asking you (since I consider you and your client as one in this matter) how you define a straight Line.
Nie. As 'the shortest path from one point to another.'
Min. This does not seem to me to embody the primary idea which the word 'straight' raises in the mind. Is not the natural process of thought to realise first the notion of 'a straight Line,' and then to grasp the fact that it is the shortest path between two points?Nie. That may be the natural process: but surely you will allow our Definition to be a legitimate one?
Min. I think not: and I have the great authority of Kant to support me. In his 'Critique of Pure Reason,' he says (I quote from Meiklejohn's translation, in Bohn's Philosophical Library, pp. 9, 10), 'Mathematical judgments are always synthetical … "A straight Line between two points is the shortest" is a synthetical Proposition. For my conception of straight contains no notion of quantity, but is merely qualitative. The conception of the shortest is therefore wholly an addition, and by no analysis can it be extracted from our conception of a straight Line.'
This may fairly be taken as a denial of the fitness of the Axiom to stand as a Definition. For all Definitions ought to be the expressions of analytical, not of synthetical, judgments: their predicates ought not to introduce anything which is not already included in the idea corresponding to the subject. Thus, if the idea of 'shortest distance' cannot be obtained by a mere analysis of the conception represented by 'straight Line,' the Axiom ought not to be used as a Definition.
Nie. We are not particular as to whether it be taken as a Definition or Axiom: either will answer our purpose.
Min. Let us then at least banish it from the Definitions. And now for its claim to be regarded as an Axiom. It involves the assertion that a straight Line is shorter than any curved Line between the two points. Now the length of a curved Line is altogether too difficult a subject for a beginner to have to consider: it is moreover unnecessary that he should consider it at all, at least in the earlier parts of Geometry all he really needs is to grasp the fact that it is shorter than any broken Line made up of straight Lines.
Nie. That is true.
Min. And all cases of broken Lines may be deduced from their simplest case, which is Euclid's I. 20.
Nie. Well, we will abate our claim and simply ask to have I. 20 granted us as an Axiom.
Min. But it can be proved from your own Axioms and it is a generally admitted principle that, at least in dealing with beginners, we ought not to take as axiomatic any Theorem which can be proved by the Axioms we already possess.
Nie. For beginners we must admit that Euclid's method of treating this point is the best. But you will allow ours to be a legitimate and elegant method for the advanced student
Min. Most certainly. The whole of your beautiful treatise is admirably fitted for advanced students it is only from the beginner's point of view that I venture to criticise it at all.
Your treatment of angles and right angles does not, I think, differ much from Euclid's?
Nie. Not much. We prove, instead of assuming, that all right angles are equal, deducing it from the Axiom that two right Lines cannot enclose a space.
Min. I think some such proof a desirable interpolation.
I will now ask you how you prove Euc. I. 29.
Nie. What preliminary Propositions will you grant us as proved?Min. Euclid's series consists of Ax. 12, Props. 4, 5, 7, 8, 13, 15, 16, 27, 28. I will grant you as much of that series as you have proved by methods not radically differing from his.
Nie. That is, you grant us Props. 4, 13, and 15. Prop. 16 is not in our treatise. The next we require is Prop. 6.
Min. That you may take as proved.
Nie. And, next to that, Prop. 20: that we assume as an Axiom, and from it, with the help of Prop. 6, we deduce Prop. 19.
Min. For our present purpose you may take Prop. 19 as proved.
Nie. From Props. 13 and 19 we deduce Prop. 32; and from that, Ax. 12; from which Prop. 29 follows at once.
Min. Your proof of Prop. 32 is long, but beautiful. I need not, however, enter on a discussion of its merits. It is enough to say that what we require is a proof suited to the capacities of beginners, and that this Theorem of yours (Prop, xix, at p. 20) contains an infinite series of Triangles, an infinite series of angles, the terms of which continually decrease so as to be ultimately less than any assigned angle, and magnitudes which vanish simultaneously. These considerations seem to me to settle the question. I fear that your proof of this Theorem, though a model of elegance and perspicuity as a study for the advanced student, is wholly unsuited to the requirements of a beginner.
Nie. That we are not prepared to dispute.
Min. It seems superfluous, after saying this, to ask what test for the meeting of Lines you have provided: but we may as well have that stated, to complete the enquiry.
Nie. We give Euclid's 12th Axiom, which we prove from Prop. 32, using the principle of Euc. X. 1 (second part), that 'if the greater of two unequal magnitudes be bisected, and if its half be bisected, and so on; a magnitude will at length be reached less than the lesser of the two magnitudes.'
Min. That again is a mode of proof entirely unsuited to beginners.
The general style of your admirable treatise I shall not attempt to discuss: it is one I would far rather take as a model to imitate than as a subject to criticise.
I can only repeat, in conclusion, what I have already said, that your book, though well suited for advanced students, is not so for beginners.
Nie. At this rate we shall make short work of the twelve Modern Rivals!