Euclid and His Modern Rivals/Act III. Scene I. § 4.

by Charles L. Dodgson

 

 

ACT III.

 

Scene I.

 

§ 4. Morell.

 

'Quis custodiet ipsos custodes?
Quis inspiciet ipsos inspectores?'

Nie. I lay before you 'Euclid Simplified, compiled from the most important French works, approved by the University of Paris and the Minister of Public Instruction,' by Mr. J. R. Morell, formerly H. M. Inspector of Schools, published in 1875.

Min. What have you about Lines, to begin with?

Nie. Here is a Definition. 'The place where two surfaces meet is called a Line.'

Min. Really! Let us take two touching spheres, for instance?

Nie. Ahem! We abandon the Definition.

Min. Perhaps we shall be more fortunate with the Definition of a straight Line.

Nie. It is 'an indefinite Line, which is the shortest between any two of its points.'

Min. An 'indefinite' Line! What in the world do you mean? Is a curved Line more definite than a straight Line?

Nie. I don't know.

Min. Nor I. The rest of the sentence is slightly elliptical. Of course you mean 'the shortest which can be drawn'?

Nie. (eagerly) Yes, yes!

Min. Well, we have discussed that matter already. Go on.

Nie. Next we have an Axiom, 'that from one point to another only one straight Line can be drawn, and that if two portions of a straight Line coincide, these Lines coincide throughout their whole extent.'

Min. You bewilder me. How can one portion of a straight Line coincide with another?

Nie. (after a pause) It can't, of course, in situ: but why not take up one portion and lay it on another?

Min. By all means, if you like. Let us take a certain straight Line, cut out an inch of it, and lay it along another inch of the Line. What follows?

Nie. Then 'these Lines coincide throughout their whole extent.'

Min. Do they indeed? And pray who are 'these Lines'? The two inches?

Nie. (gloomily) I suppose so.

Min. Then the Axiom is simple tautology.

Nie. Well then, we mean the whole straight Line and—and—

Min. And what else? You can't talk of 'one straight Line' as 'these Lines,' you know.

Nie. We abandon the Axiom.

Min. Better luck next time! Try another Definition.

Nie. 'A broken Line is a Line composed of straight Lines.'

Min. But a straight Line also is 'a Line composed of straight Lines,' isn't it?

Nie. Well, we abandon the Definition.

Min. This is quite a new process in our navigation. Instead of heaving the lead, we seem to be throwing overboard the whole of our cargo! Let us hear something about Angles.

Nie. 'The figure formed by two Lines that intersect is called an Angle.'

Min. What do you mean by 'figure'? Do you define it anywhere?

Nie. Yes. 'The name of figure is given to volumes, surfaces, and lines.'

Min. Under which category do you put 'Angle'?

Nie. I don't know.

Min. Anything new about the Definition, or equality, of right angles?

Nie. No, except that we prove that all right angles are equal.

Min. That we have discussed already (see p. 57). Let us go on to Pairs of Lines, and your proof of Euc. I. 29, 32.

Niemand reads.

'Th. 19. Two Lines perpendicular to the same Line are parallel.'

Min. Do you mean 'separational'?

Nie. Yes.

Min. Have you defined 'parallel' anywhere?

Nie. (after a search) I can't find it.

Min. A careless omission. Moreover, your assertion isn't always true. Suppose your two Lines were drawn from the same point?

Nie. We beg to correct the sentence. 'Two different Lines.'

Min. Very well. Then you assert Table I. 6. (See p. 29.) I grant it.

Niemand reads.

'Th. 20. Through a point situated outside a straight Line a Parallel, and only one, can be drawn to that Line.'

Min. 'A Parallel,' I grant at once: it is Table I. 9. But 'only one'! That takes us into Table II. What axiom do you assume?

Nie. 'It may be admitted that only one Parallel can be drawn to it.'

Min. That is Table II. 15 (b)—a contranominal of Playfair's Axiom. We need not pursue the subject: all is easy after that. Now hand me the book, if you please: I wish to make a general survey of style, &c.

At p. 4 I read:—'Two Theorems are reciprocal when the hypothesis and the conclusion of one are the conclusion and the hypothesis of the other.' (They are usually called 'converse'—the technical, not the logical, converse, as was mentioned some time ago (p. 47); but let that pass.) 'Thus the Theorem—if two angles are right angles, they are equal—has for its reciprocal—if two angles are equals, they are right angles.'

(This, by the way, is a capital instance of the distinction between 'technical' and 'logical.' Here the technical converse is wild nonsense, while the logical converse is of course as true as the Theorem itself: it is 'some cases of two angles being equal are cases of their being right.')

'All Propositions are direct, reciprocal, or contrary—allso closely connected that either of the two latter' (I presume he means 'the latter two') 'is a consequence of the other two.'

A 'consequence'! Can he mean a logical consequence? Would he let us make a syllogism of the three, using the 'direct' and 'reciprocal' (for instance) as premisses, and the 'contrary' as the conclusion?

However, let us first see what he means by a 'contrary' Proposition.

'It is a direct Proposition to prove that all points in a circle enjoy a certain property, e.g. the same distance from the centre.'

(This notion of sentient points, by the way, is very charming. I like to think of all the points in a circle really feeling a placid satisfaction in the thought that they are equidistant from the centre! They are infinite in number, and so can well afford to despise the arrogance of a point within, and to ignore the envious murmurs of a point without!)

'The contrary Proposition shows that all points taken outside or inside the figure do not enjoy this property.'

So then this is his trio:—

1. Direct. 'All X are Y.'
2. Reciprocal. 'All Y are X.'
3. Contrary. 'All not-X are not-Y.'

Here of course No. 2 and No. 3, being Contranominals, are logically deducible from each other, No. 1 having no logical connection with either of them.

And yet he calls the three 'so closely connected that either of the two latter is a consequence of the other two'! Shade of Aldrich! Have we come to this? You say nothing, mein Herr?

Nie. I say that, if you grant what you call the 'premisses,' you cannot deny the conclusion.

Min. True. It reminds me of an answer given some years ago in the Schools at Oxford, when the Examiner asked for an example of a syllogism. After much patientthought, the candidate handed in

'All men are dogs;
All dogs are men:
Therefore, All men are dogs.'

This certainly has the form of a syllogism. Also it avoids, with marked success, the dangerous fallacy of 'four terms.' And it has the great merit of Mr. Morell's syllogism, that, if you grant the premisses, you cannot deny the conclusion. Nevertheless I feel bound to add that it was not commended by the Examiner.

Nie. I can well believe it.

Min. I proceed. 'The direct and the reciprocal proofs are generally the simpler, and do not require a fresh construction.' Why 'fresh'? The 'direct' comes first, apparently; so that, if it requires a construction at all, it must be a 'fresh' one.

Nie. Be not hypercritical.

Min. Well, it is rather 'small deer,' I confess: let us change the subject.

Here is a pretty proof in Th. 4.

'Then m + o = m + x.
But m = m.
Therefore o = x.'

Isn't that 'but m = m' a delightfully cautious parenthesis? Your client seems to be nearly as much at home in Algebra as in Logic, which is saying a great deal!

At p. 9, I read 'The base of an isosceles Triangle is the unequal side.'

'The unequal side'! Is an equilateral Triangle isosceles, or is it not? Answer, mein Herr!

Nie. Proceed.

Min. At p. 17, I read 'From one and the same point three equal straight Lines cannot be drawn to another straight Line; for if that were the case, there would be on the same side of a perpendicular two equal obliques, which is impossible.'

Kindly prove the italicised assertion on this diagram, in which I assume FD, FC, FE, to be equal Lines, and

have made the middle one of the three a perpendicular to the 'other straight Line.'

Nie. (furiously) I will not!

Min. Look at p. 36. 'A circumference is generally described in language by one of its radii.' Let us hope that the language is complimentary—at least if the circumference is within hearing! Can't you imagine the radius gracefully rising to his feet, rubbing his lips with his table-napkin? 'Gentlemen! The toast I have the honour to propose is &c. &c. Gentlemen, I give you the Circumference!' And then the chorus of excited Lines, 'For he's a jolly good felloe!'

Nie. (rapturously) Ha, ha! (checking himself) You are insulting my client.

Min. Only filling in his suggestive outlines. Try p. 48. 'Th. 13. If two circumferences are interior,' &c. Can your imagination, or mine, grasp the idea of two circumferences, each of them inside the other? No! We are mere prosaic mortals: it is beyond us!

In p. 49 I see some strange remarks about ratios. First look at Def. 44. 'When a magnitude is contained an exact number of times in two magnitudes of its kind, it is said to be their common measure.' (The wording is awkward, and suggests the idea of their having only one 'common measure'; but let that pass.) 'The ratio of two magnitudes of the same kind is the number which would express the measure of the first, if the second were taken as unity.'

'The measure of the first'! Do you understand that? Is it a 'measure' such as you have just defined? or some other kind?

Nie. Some other kind, I think. But there is a slight obscurity somewhere.

Min. Perhaps this next enunciation will clear it up. 'If two magnitudes of the same kind, A and B, are mutually commensurable' (by the way, 'mutually' is tautology), 'their ratio is a whole or fractional number, which is obtained by dividing the two numbers one by the other, and which expresses how many times these magnitudes contain their common measure M.' Do you understand that?

Nie. Well, no!

Min. Let us take an instance—£3 and 10s. A shilling is a common measure of these two sums: will you accept it as 'their common measure'?

Nie. We will do it, provisionally.

Min. Now the number, 'obtained by dividing the two numbers' (I presume you mean 'the two magnitudes') 'one by the other,' is '6,' is it not?

Nie. It would seem so.

Min. Well, does this number 'express how many times these magnitudes contain their common measure,' viz. a shilling?

Nie. Hardly.

Min. Did you ever meet with any one number that could 'express' two distinct facts?

Nie. We would rather change the subject.

Min. Very well, though there is plenty more about it, and the obscurity deepens as you go on. We will 'vary the verse' with a little bit of classical criticism. Look at p. 81. 'Homologous, from the Greek ὁμοῖος, like or similar, λὀγος, word or reason.' Do you think this school-inspector ever heard of the great Church controversy, where all turned on the difference between ὃμος and ὁμοῖος?

Nie. (uneasily) I think not. But this is not a mathematical slip, you know.

Min. You are right. Revenons à nos moutons. Turn to p. 145, art. 65. 'To measure areas, it is usual to take a square as unity.' To me, who have always been accustomed to regard 'a square' as a concrete magnitude and 'unity' as a pure number, the assertion comes rather as a shock. But I acquit the author of any intentional roughness. Nothing could surpass the delicacy of the next few words:—'It has been already stated that surfaces are measured indirectly'! Lines, of course, may be measured anyhow: they have no sensibilities to wound: but there is an open-handedness—a breadth of feeling—about a surface, which tells of noble birth—'every (square) inch a King!'—and so we measure it with averted eyes, and whisper its area with bated breath!

Nie. Return to other muttons.

Min. Well, take p. 156. Here is a 'scholium' on a theorem about the area of a sector of a circle. The 'scholium ' begins thus:—'If α is the number of degrees in the arc of a sector, we shall have to find the length of this arc ——.' I pause to ask 'If β were the number, should we have to find it then?'

Nie. (solemnly) We should!

Min. 'For the two Lines which are multiplied in all rules for the measuring of areas must be referred to the same linear unity.' That, I take it, is fairly obscure: but it is luminous when compared with the note which follows it. 'If the linear unit and angular unit are left arbitrary, any angle has for measure the ratio of the numbers of linear units contained in the arcs which the angle in question and the irregular unit intercept in any circumference described from their summit as common centres.' Is not that a useful note? 'The irregular unit'! Linear, or angular, I wonder? And then 'common centres'! How many centres does a circumference usually require? I will only trouble you with one more extract, as a bonne bouche to wind up with.

'Th. 9. (P. 126.) Every convex closed Line ABCD enveloped by any other closed Line PQRST is less than it.

'All the infinite Lines ABCD, PQRST, &c.'——by the way, these are curious instances of 'infinite Lines'?

Nie. (hastily) We mean 'infinite' in number, not in length.

Min. Well, you express yourself oddly, at any rate '——which enclose the plane surface ABCD, cannot be equal. For drawing the straight Line MD, which does not cut ABCD, MD will be less than MPQD; and adding to both members the part MTSRQD, the result will be MDQRSTM less than MPQRSTM.' Is that result proved?

Nie. No.

Min. Is it true?

Nie. Not necessarily so.

Min. Perhaps it is a lapsus pennœ. Try to amend it.

Nie. If we add to MD the part MTSRQD, we do get MDQRSTM, it is true: but, if we add it to MPQD, we get QD twice over; that is, we get MPQRSTM together with twice QD.

Min. How does that addition suit the rest of the proof?

Nie. It ruins it: all depends on our proving the perimeter MDQRSTM less than the perimeter MPQRSTM, which this method has failed to do—as of course all methods must, the thing not being capable of proof.

Min. Then the whole proof breaks down entirely?

Nie. We cannot deny it.

Min. Let us turn to the next author.