Euclid and His Modern Rivals/Act II. Scene VI. § 2.

by Charles L. Dodgson

 

 

ACT II.

 

Scene VI.

 

§ 2. Pierce.

 

'dum brevis esse laboro,

Obscurus fio.’

Nie. I lay before you 'An Elementary Treatise on Plane and Solid Geometry' by Benjamin Pierce, A.M., Perkins Professor of Astronomy and Mathematics in Harvard University, published in 1872.

Min. As I have already considered, at great length, the subject of direction as treated by Mr. Wilson, I need not trouble you as to any matters where Mr. Pierce's treatment does not materially differ from his. Is there any material difference in the treatment of a straight line?

Nie. He has a Definition of direction which will, I think, be new to you:—

Reads.

P. 5. § 11, Def. 'The Direction of a Line in any part is the direction of a point at that part from the next preceding point of the Line.'

Min. That sounds mysterious. Which way along a Line are 'preceding' points to be found?

Nie. Both ways. He adds, directly afterwards, 'a Line has two different directions,' etc.

Min. So your Definition needs a postscript? That is rather clumsy writing. But there is yet another difficulty. How far from a point is the 'next' point?

Nie. At an infinitely small distance, of course. You will find the matter fully discussed in any work on the Infinitesimal Calculus.

Min. A most satisfactory answer for a teacher to make to a pupil just beginning Geometry! I see nothing else to remark on in your treatment of the Line, except that you state, as an Axiom, that 'a straight Line is the shortest way from one point to another.' I have already given, in my review of M. Legendre, my reasons for thinking that this is not a fair Axiom, and ought to be a Theorem (see p. 55).

There is nothing particular to notice in your treatment of angles and right angles. Let us go on to Parallels. How do you prove Euc. I. 32?

Niemand reads.

P. 9. § 27, Def. 'Parallel Lines are straight Lines which have the same Direction.'

Min. I presume you do not mean to include coincidental Lines?

Nie. Certainly not. We see the omission. Allow us to insert the word 'different.'

Min. Very well. Then your Definition combines the two properties 'different' and 'having the same direction.' Bear in mind that you have yet to prove the reality of such Lines. And may I request you in future to call such Lines 'sepcodal'? But if you wish to assert any thing of them which is also true of coincidental Lines, you had better drop the 'sep-' and simply call them 'Lines which have the same direction,' so as to include both classes.

Nie. Very well.

Niemand reads.

P. 9. § 28. Th. 'Sepcodal Lines cannot meet, however far they are produced.'

Min. Or rather 'could not meet, if they existed.' Proceed.

Niemand reads.

P. 9. § 29. Th. 'Two angles are equal, when their sides have the same direction.'

Min. How do you define 'same direction' for different Lines?

Nie. We cannot define it.

Min. Then I cannot admit that such Lines exist. But even if I did admit their reality, why should the angles be equal?

Nie. Because 'the difference of direction' is the same in each case.

Min. But how would that prove the angles equal? Do you define 'angle' as the 'difference of direction' of two lines?

Nie. Not exactly. We have stated (p. 6, § 19) 'The magnitude of the angle depends solely upon the difference of direction of its sides at the vertex.'

Min. But the difference of direction also possesses 'magnitude.' Is that magnitude a wholly independent entity? Or does it, in its turn, depend to some extent upon the angle? Seriously, all these subtleties must be very trying to a beginner. But we had better proceed to the next Theorem. I am anxious to see where, in this system, these creatures of the imagination, these sepcodal Lines, are to appear as actually existent.

Nie. We next prove (p. 9. § 30) that Lines, which have the same direction, make equal angles with all transversals.

Min. That is merely a particular case of your last Theorem.

Nie. And then that two Lines, which make equal angles with a transversal, have the same direction.

Min. Ah, that would bring them into existence at once! Let us hear the proof of that.

Nie. The proof is that if, through the point where the first Line is cut by the transversal, a Line be drawn having the same direction as the second, it makes equal angles with the transversal, and therefore coincides with the first Line.

Min. You assume, then, that a Line can be drawn through that point, having the same direction as the second Line?

Nie. Yes.

Min. That is, you assume, without proof, that different Lines can have the same direction. On the whole, then, though Mr. Pierce's system differs slightly from Mr. Wilson's, both rest on the same vicious Axiom, that different Lines can exist, which possess a property called 'the same direction'—a phrase which is intelligible enough when used of two Lines which have a common point, but which, when applied to two Lines not known to have a common point, can neither be defined, nor constructed. We need not pursue the subject further. Have you provided any test for knowing whether two given finite Lines will meet if produced?

Nie. We have not thought it necessary.

Min. Then the only other remark I have to make on this singularly compendious treatise is that, of the 35 Theorems which Euclid gives us in his First Book, it reproduces just sixteen: the omissions being 16, 17, 25, 26 (2), 27 and 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 47, and 48.

Nie. Most of those are in the book. For example, § 30 answers to Euc. I. 29.

Min. Only by proving that separational Lines have the same direction: which you have not done.

Nie. At any rate we have Euc. I. 47 in our § 256.

Min. Oh, no doubt! Long after going through ratios, which necessarily include incommensurables; and long after the Axiom (§ 99) 'Infinitely small quantities may be neglected'! No, no: so far as beginners are concerned, there is no Euc. I. 47 in this book!

My conclusion is that, however useful this Manual may be to an advanced student, it is not adapted to the wants of a beginner.