Min. That is so.

Euc. Then, for anything we know to the contrary, class 'B' may be larger than class 'a.' Hence, if you assert anything of class 'B,' the logical effect is more extensive than if you assert it of class 'a': for you assert it, not only of that portion of class 'B' which is known to be included in class 'a,' but also of the unknown (but possibly existing) portion which is not so included.

Min. I see that now, and consider it a real and very strong reason for preferring your axiom.

But so far you have only answered Playfair. What do you say to the objection raised by Mr. Potts? A stronger objection appears to be that the converse of it forms Euc. I. 17; for both the assumed Axiom and its converse should be so obvious as not to require formal demonstration.'

Euc. Why, I say that I deny the general law which he lays down. (It is, of course, the technical converse that he means, not the logical one. All X is Y' has for its technical converse All Y is X'; for its logical, 'Some Y is X.) Let him try his law on the Axiom All right angles are equal,' and its technical converse All equal angles are right'!

Min. I withdraw the objection.

§ 6. The Principle of Superposition.

Min. The next subject is the principle of 'superposition.' You use it twice only (in Props. 4 and 8) in the First

Book: but the modern fancy is to use it on all possible occasions. The Syllabus indicates (to use the words of the Committee) 'the free use of this principle as desirable in many cases where Euclid prefers to keep it out of sight.'

Euc. Give me an instance of this modern method.

Min. It is proposed to prove I. 5 by taking up the isosceles Triangle, turning it over, and then laying it down again upon itself.

Euc. Surely that has too much of the Irish Bull about it, and reminds one a little too vividly of the man who walked down his own throat, to deserve a place in a strictly philosophical treatise?

Min. I suppose its defenders would say that it is conceived to leave a trace of itself behind, and that the reversed Triangle is laid down upon the trace so left.

Euc. That is, in fact, the same thing as conceiving that there are two coincident Triangles, and that one of them is taken up, turned over, and laid down upon the other. And what does their subsequent coincidence prove? Merely this: that the right-hand angle of the first is equal to the left-hand angle of the second, and vice versa. To make the proof complete, it is necessary to point out that, owing to the original coincidence of the Triangles, this same 'left-hand angle of the second' is also equal to the left-hand angle of the first: and then, and not till then, we may conclude that the base-angles of the first Triangle are equal. This is the full argument, strictly drawn out. The Modern books on Geometry often attain their much-vaunted brevity by the dangerous process of

Sc. II. § 6.] PRINCIPLE OF SUPERPOSITION.

49

omitting links in the chain; and some of the new proofs, which at first sight seem to be shorter than mine, are really longer when fully stated. In this particular case I think you will allow that I had good reason for not adopting the method of superposition?

Min. You had indeed.

Euc. Mind, I do not object to that proof, if appended to mine as an alternative. It will do very well for more advanced students. But, for beginners, I think it much clearer to have two non-isosceles Triangles to deal with.

Min. But your objection to laying a Triangle down upon itself does not apply to such a case as I. 24.

Euc. It does not. Let us discuss that case also. The Moderns would, I suppose, take up the Triangle ABC, and apply it to DEF so that AB should coincide with DE?

Min. Yes.

Euc. Well, that would oblige you to say 'and join C, in its new position, to E and F' The words 'in its new position' would be necessary, because you would now have two points in your diagram, both called 'C.' And you would also be obliged to give the points D and E additional names, namely 'A' and 'B.' All which would be very confusing for a beginner. You will allow, I think, that I was right here in constructing a new Triangle instead of transferring the old one?

Min. Cuthbertson evades that difficulty by re-naming the point C, and calling it 'Q.'

Euc. And do the points A and B take their names with them?

E

Min. No. They adopt the names 'D' and 'E.'
Euc. It is very like making a new Triangle!

Min. It is indeed. I think you have quite disposed of the claims of superposition.' The only remaining subject for discussion is the omission of the diagonals in Book II.

§ 7. The omission of diagonals in Euc. II.

Euc. Let us test it on my II. 4. We will go through my proof of it, and then the proof given by some writer who ignores the diagonal, supplying if necessary any of those gaps in argument which my Modern Rivals so often indulge in, and which give to their proofs a delusive air of neatness and brevity.

If a Line be divided into any two parts, the square of the Line is equal to the squares of the two parts with twice their rectangle.

H

B

团

D

Let AB be divided at C.

It is to be proved that square

of AB is equal to squares of AC, CB, with twice rectangle of AC, CB.

On AB describe Square ADEB; join BD; from C

Sc. II. § 7-] OMISSION OF DIAGONALS.

51

draw CF parallel to AD or BE, cutting BD at G; and through G draw HK parallel to AB or DE.

BD cuts Parallels AD, CF,

.. exterior angle CGB=interior opposite angle ADB.

Similarly HF is a Square and square of AC, for

HG=AC.

[I. 34

Also, AG, GE are equal, being complements,

[I. 43

.. AG and GE=twice AG;

twice rectangle of AC, CB.

But these four figures make up AE.

Therefore the square of AB &c.

Q. E. D.'

That is just 128 words, counting from 'On AB describe' down to the words 'rectangle of AC, CB.' What author shall we turn to for a rival proof?

Min. I think Wilson will be best.

Euc. Very well. Do the best you can for him. You may use all my references if you like, and if you can do so legitimately.

Min. 'Describe Square ADEB on AB. Through C draw CF parallel to AD, meeting