By breaking up certain of the Propositions of Euc. I, II, and including some of the Corollaries, we get 73 Propositions in all-57 Theorems and 16 Problems. Of these 73, this Manual omits 14 (10 Theorems and 4 Problems); it proves 43 (32 Theorems and 11 Problems) by methods almost identical with Euclid's; for 10 of them (9 Theorems and a Problem) it offers new proofs, against which I have recorded my protest, one being illogical, 2 (needlessly) employing superposition,' 2 deserting Geometry for Algebra, and the remaining 4 omitting the diagonals in Euc. II; and finally it offers 6 new proofs, which I think may fairly be introduced as alternatives for those of Euclid.

In all this, and in all the matters previously discussed, I fail to see one atom of reason for abandoning Euclid. Have you any yet-unconsidered objections to urge against my proposal that the sequence and numeration of Euclid be kept unaltered'?

[Dead silence is the only reply.]

Carried, nemine contragemente! And now, Prisoner at the Bar (I beg your pardon, I should say 'Professor on the Sofa'), have you, and your attendant phantoms, any other reasons to urge for regarding this Manual as in any sense a substitute for Euclid's- -as in any sense anything else than a revised edition of Euclid?

Nie. We have nothing more to say.

Min. Then I can but repeat with regard to this newborn follower' of the Syllabus, what I said of the

Syllabus itself. Restore the Problems (which are also Theorems) to their proper places; keep to Euclid's numbering (interpolating your new Propositions where you please); and your new book may yet prove a valuable addition to the literature of Elementary Geometry.

[A tremulous movement is seen amid the ghostly throng. They waver fitfully to and fro, and finally drift off in the direction of one corner of the ceiling. When the procession has got well under way, NIEMAND himself becomes hazy, and floats off to join them. The whole procession gradually melts away into vacancy, DIAMOND going last, nibbling at the heels of NERO, for which a pair of gorgeous Roman sandals seem to afford but scanty protection.]

ACT IV.

'Old friends are best."

[Scene as before. Time, the early dawn. MINOS slumbering uneasily, having fallen forwards upon the table, his forehead resting on the inkstand. To him enter EUCLID on tip-toe, followed by the phantasms of ARCHIMEDES, PYTHAGORAS, ARISTOTLE, PLATO, &c., who have come to see fair play.]

§ 1. Treatment of Pairs of Lines.

Euc. Are all gone?

Min.

'Be cheerful, sir:

Our revels now are ended: these our actors,

As I foretold you, were all spirits, and

Are melted into air, into thin air!'

Euc. Good. Let us to business. And first, have you found any method of treating Parallels to supersede mine?

Min. No! A thousand times, no! The infinitesimal method, so gracefully employed by M. Legendre, is unsuited to beginners: the method by transversals, and the method by revolving Lines, have not yet been offered in a logical form: the 'equidistant' method is too cumbrous:

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and as for the method of direction,' it is simply a rope of sand-it breaks to pieces wherever you touch it!

Euc. We may take it as a settled thing, then, that you have found no sufficient cause for abandoning either my sequence of Propositions or their numbering, and that all that now remains to be considered is whether any important modifications of my Manual are desirable?

Min. Most certainly.

Euc. Have you met with any striking novelty on the subject of a practical test for the meeting of Lines?

Min. There is one rival to your 12th Axiom which is formidable on account of the number of its advocatesthe one usually called 'Playfair's Axiom.'

Euc. We have discussed that matter already (p 40). Min. But what have you to say to those who reject Playfair's Axiom as well as yours?

Euc. I simply ask them what practical test, as to the meeting of two given finite Lines, they propose to employ. Not only will they find it necessary to prove, in certain Theorems, that two given finite Lines will meet if produced, but they will even find themselves sometimes obliged to prove it of two Lines, of which the only geometrical fact known is that they possess the very property which forms the subject of my Axiom. I ask them, in short, this question:- Given two Lines making, with a certain transversal, two interior angles together less than two right angles, how do you propose to prove, without my Axiom, that they will meet if produced?'

Min. The advocates of the direction' theory would of course reply, 'We can prove, from the given property,

that they have different directions: and then we bring in the Axiom that Lines having different directions will meet if produced.'

Euc. All that you have satisfactorily disposed of in your review of Mr. Wilson's Manual.

Min. The only other substitute, that I know of, belongs to the equidistant' theory, which replaces your Axiom by three or four new Axioms and six new Theorems. substitute, also, I have seen reason to reject.

That

My general conclusion is that your method of treatment of all these subjects is the best that has yet been suggested.

Euc. Any noticeable innovations in the treatment of Right Lines and Angles?

Min. Those subjects I should be glad to talk over with

you.

Euc. With all my heart. And now how do you propose to conduct this our final interview?

Min. I should wish, in the first place, to lay before you the general charges which have been brought against you: then to discuss your treatment of Lines and Angles, as contrasted with that of your Rivals'; and lastly the omissions, alterations, and additions proposed by them. Euc. Good. Let us begin.

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Min. I will take the general charges under three headings:-Construction, Demonstration, and Style. And first as to Construction :