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. (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!

АСТ II.

SCENE III.

Treatment of Parallels by angles made with transversals.

COOLEY.

The verbal solemnity of a hollow logic.'

COOLEY, Pref. p. 20.

• The

Nie. I have now the honour to lay before you Elements of Geometry, simplified and explained,' by W. D. COOLEY, A.B., published in 1860.

Min. Please to hand me the book for a moment. I wish to read you a few passages from the Preface. It is always satisfactory-is it not?-to know that a writer, who attempts to simplify' Euclid, begins his task in a becoming spirit of humility, and with some reverence for a name that the world has accepted as an authority for two thousand years.

Nie. Truly.

MINOS reads.

'The Elements of Plane Geometry are here presented in the reduced compass of 36 Propositions, perfectly coherent, fully demonstrated, and reaching quite as far as the 173 Propositions contained in the first six books of Euclid.' Modest, is it not?

Nie. A little high-flown, perhaps. Still, you know, if they really are 'fully demonstrated

6

Min. If! In page 4 of the Preface he talks of Euclid's circumlocutory shifts': in the same page he tells us that 'the doctrine of proportion, as propounded by Euclid, runs into prolixity though wanting in clearness': and again, in the same page, he states that most of Euclid's ex absurdo proofs though containing little,' yet 'generally puzzle the young student, who can hardly comprehend why gratuitous absurdities should be so formally and solemnly dealt with. These Propositions therefore are omitted from our Book of Elements, and the Problems also, for the science of Geometry lies wholly in the Theorems. Thus simplified and freed from obstructions, the truths of Geometry may, it is hoped, be easily learned, even by the youngest.' But perhaps the grandest sentence is at the end of the Preface. Then as to those Propositions (the first and last of the 6th Book), in which, according to the same authority' (he is alluding to the Manual of Euclid by Galbraith and Haughton), 'Euclid so beautifully illustrates his celebrated Definition, they appear to our eyes to exhibit only the verbal solemnity of a hollow logic, and to exemplify nothing but the formal application of