Euclid and His Modern Rivals/Act IV.

by Charles L. Dodgson

 

 

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: 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 advocates—the 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. That substitute, also, I have seen reason to reject.

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.

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

 

§ 2. Euclid's Constructions.

I am told that you indulge too much in 'arbitrary restrictions.' Mr. Reynolds says (Pref. p. vi.) 'The arbitrary restrictions of Euclid involve him in various inconsistencies, and exclude his constructions from use. When, for instance, in order to mark off a length upon a straight Line, he requires us to describe five Circles, an equilateral Triangle, one straight line of limited, and two of unlimited length, he condemns his system to a divorce from practice at once and from sound reason.'

Euc. Mr. Reynolds has misunderstood me: I do not require all that construction in Prop. 3. To explain my meaning I must go back to Prop. 2, and I must ask your patience while I make a few general remarks on construction. The machinery I allow consists of a pencil, a ruler, and a pair of compasses to be used for drawing a Circle about a given centre and passing through a given point (that is what I mean by 'at any distance'), but not to be used for transferring distances from one part of a diagram to another until it has been shown that such transference can be effected by the machinery already allowed.

Min. But why not allow such transference without proving that possibility?

Euc. Because it would be introducing as a Postulate what is really a Problem. And I go on the general principle of never putting a Problem among my Postulates, nor a Theorem among my Axioms.

Min. I heartily agree in your general principle, though I need scarcely remind you that it has been frequently charged against you, as fault, that you state as an Axiom what is really a Theorem.

Euc. That charge has been met (see p. 40). To return to my subject. I merely prove, once for all, in Prop. 2, that a Line can be drawn, from a given point, and equal to a given Line, by the original machinery alone, and without transferring distances. After that, my reader is welcome to transfer a distance by any method that comes handy, such as a bit of string &c.: and of course he may now transfer his compasses to a new centre. And this is all I expect him to do in Prop. 3.

Min. Then you don't expect these five Circles &c. to be drawn whenever we have to cut off, from one Line, a part equal to another?

Euc. Pas si bête, mon ami.

Min. Some of your Modern Rivals are, however, a little discontented with the very scanty machinery you allow.

Euc. 'A bad workman always quarrels with his tools.'

Min. Their charge against you is 'the exclusion of hypothetical constructions.' Mr. Wilson says (Pref. p. i.) 'The exclusion of hypothetical constructions may bementioned as a self-imposed restriction which has made the confused order of his first book necessary, without any compensating advantage.'

Euc. In reply, I cannot do better than refer you to Mr. Todhunter's Essay on Elementary Geometry (p. 186). 'Confused order is rather a contradictory expression,' &c. (see p. 241).

Min. Your reply is satisfactory. Mr. Wilson himself is an instance of the danger of such a method. Three times at least (pp. 46, 70, 88) he produces Lines to meet without attempting to prove that they will meet.

§ 3. Euclid's Demonstrations.

Min. The next heading is 'Demonstration.' You are charged with an 'invariably syllogistic form of reasoning.' (Wilson, Pref. p. i.)

Euc. Do you know, I am vain enough to think that a merit rather than a defect? Let me quote what Mr. Cuthbertson says on this point (Pref. p. vii.). 'Euclid's mode of demonstration, in which the conclusion of each step is preceded by reasoning expressed with all the exactness of the minor premiss of a syllogism, of which some previous proposition is the major premiss, has been adopted as offering a good logical training, and also as being peculiarly adapted for teaching large classes, rendering it possible for the teacher to call first upon one, then upon another, and so on, to take up any link in the chain of argument.' Perhaps even Mr. Wilson's own book would not be the worse if the reasoning were a trifle more 'syllogistic'!

Min. A fair retort. You are also charged with 'too great length of demonstration.' Mr. Wilson says (Pref. p. i.) 'The real objections to Euclid as a text-book are … the length of his demonstrations.' And Mr. Cooley says (Pref. p. 1.) 'The important and fertile theorems, which crown the heights in this field of knowledge, are here all retained, and those only are omitted which seem to be but the steps of a needlessly protracted ascent. The short road thus opened will be found perfectly solid inconstruction, and at the same time far less tedious and fatiguing than the circuitous one hitherto in vogue.'

Euc. I think Mr. Wilson's Th. 17 (p. 27), with its five figures (all necessary, though he only draws one), and still more his marvellous Problem, 'approached by four stages,' which fills pages 69 to 72, are pretty good instances of lengthy demonstration. And Mr. Cooley's 'short and solid road' contains, if I remember right, a rather breakneck crevasse!

Min. The next charge against you is 'too great brevity of demonstration.' Mr. Leslie (a writer whom I have not thought it necessary to review as a 'Modern Rival,' as his book is nearly seventy years old) says (Pref. p. vi.) 'In adapting it' (the Elements of Euclid) 'to the actual state of the science, I have … sought to enlarge the basis … The numerous additions which are incorporated in the text, so far from retarding will rather facilitate progress, by rendering more continuous the chain of demonstration. To multiply the steps of ascent, is in general the most expeditious mode of gaining a summit.'

Euc. I think you had better refer him to Mr. Wilson and Mr. Cooley: they will answer him, and he in his turn will confute them!

Min. The last charge relating to demonstration is, in Mr. Wilson's words (Pref. p. viii.) 'the constant reference to general Axioms and general Propositions, which are no clearer in the general statement than they are in the particular instance,' which practice, he says, makes the study of Geometry 'unnecessarily stiff, obscure, tedious and barren.'

Euc. One advantage of making a general statement, and afterwards referring to it instead of repeating it, is that you have to go through the mental process of affirming or proving the truth once for all: apparently Mr. Wilson would have you begin de novo and think out the truth every time you need it! But the great reason for always referring back to your universal, instead of affirming the particular (Mr. Wilson is merely starting the old logical hare 'Is the syllogism a Petitio Principii?'), is that the truth of the particular does not rest on any data peculiar to itself, but on general principles applicable to all similar cases; and that, unless those general principles prove the conclusion for all cases, they cannot he warranted to prove it for any one selected case. If, for instance, I see a hundred men, and am told that some assertion is true of ninety-nine of them, but am not told that it is true of all, I am not justified in affirming it of any selected man; for he might chance to be the excepted one. Now the assertion, that the truth of the particular case under notice depends on general principles, and not on peculiar circumstances, is neither more nor less than the assertion of the universal affirmative which Mr. Wilson deprecates.

§ 4. Euclid's Style.

Min. Quite satisfactory. I will now take the third heading, namely 'Style.'

You are charged with Artificiality, Unsuggestiveness, and Want of Simplicity. Mr. Wilson says (Pref. p. i.) 'The real objections to Euclid as a text-book are his artificiality … and his unsuggestiveness,' and again, 'he has sacrificed, to a great extent, simplicity and naturalness in his demonstrations, without any corresponding gain in grasp or cogency.'

Euc. Well, really I cannot deal with general charges like these. I prefer to abide by the verdict of my readers during these two thousand years. As to 'unsuggestiveness,' that is a charge which cannot, I admit, be retorted on Mr. Wilson: his book is very suggestive—of remarks which, perhaps, would not be wholly 'music to his ear'!

§ 5. Euclid's treatment of Lines and Angles.

Min. Let us now take the subjects of Right Lines and Angles; and first, the 'Right Line.'

I see, by reference to the original, that you define it as a Line 'which lies evenly as to points on it.' That of course is only an attempt to give the mind a grasp of the idea. It leads to no geometrical results, I think?

Euc. No: nor does any definition of it, that I have yet seen.

Min. I have no rival Definitions to propose. Mr. Wilson's 'which has the same direction at all parts of its length' has perished in the collapse of the 'direction' theory: and M. Legendre's 'the shortest course from one point to another' is not adapted for the use of a beginner. And I do not know that any change has been suggested in your test of a right Line in Prop. 14.

The next subject is 'Angles.'

Your definition would perhaps be improved, if for 'inclination to' we were to read 'declination from,' for, the greater the angle the greater the declination, and the less (as it seems to me) the inclination.

Euc. I agree with you.

Min. The next point is that you limit the size of an angle to something less than the sum of two right angles.

Euc. What advantage is claimed for the extension of the Definition?

Min. It is a prospective rather than an immediate one. It must be granted you that the larger angles are not needed in the first four Books—

Euc. In the first six Books.

Min. Nay, surely you need them in the Sixth Book?

Euc. Where?

Min. In Prop. 33, where you treat of 'any equimultiples whatever' of an angle, of an arc, and of a sector. You cannot possibly assume the multiple angle to be always less than two right angles.

Euc. You think, then, that a multiple of an angle must itself be an angle?

Min. Surely.

Euc. Then a multiple of a man must itself be a man. If I contemplate a man as multiplied by the number ten thousand, I must realise the idea of a man ten thousand times the size of the first?

Min. No, you need not do that.

Euc. Thanks: it is rather a strain on the imaginative faculty.

Min. You mean, then, that the multiple of an angle may be conceived of as so many separate angles, not in contact, nor added together into one?

Euc. Certainly.

Min. But you have to contemplate the case where two such angular magnitudes are equal, and to infer from that, by III. 26, that the subtending arcs are equal. How can you infer this when your angular magnitude is not one angle but many?

Euc. Why, the sum total of the first set of angles is equal to the sum total of the second set. Hence the second set can clearly be broken up and put together again in such amounts as to make a set equal, each to each, to the first set: and then the sum total of the arcs, and likewise of the sectors, will evidently be equal also.

But if you contemplate the multiples of the angles as single angular magnitudes, I do not see how you prove the equality of the subtending arcs: for my proof applies only to cases where the angle is less than the sum of two right angles.

Min. That is very true, and you have quite convinced me that we ought to observe that limit, and not contemplate 'angles of rotation' till we enter on the subject of Trigonometry.

As to right angles, it has been suggested that your Axiom 'all right angles are equal to one another' is capable of proof as a Theorem.

Euc. I do not object to the interpolation of such a Theorem, though there is very little to distinguish so simple a Theorem from an Axiom.

Min. Let us now consider the omissions, alterations, and additions, which have been proposed by your Modern Rivals.

§ 6. Omissions, alterations, and additions, suggested by Modern Rivals.

Euc. Which of my Theorems have my Modern Rivals proposed to omit?

Min. Without dwelling on such extreme cases as that of Mr. Pierce, who omits no less than 19 of the 35 Theorems in your First Book, I may say that the only two, as to which I have found anything like unanimity, are I. 7 and II. 8.

Euc. As to I. 7, I have several reasons to urge in favour of retaining it.

First, it is useful in proving I. 8, which, without it, is necessarily much lengthened, as it then has to include three cases: so that its omission effects little or no saving of space.

Secondly, the modern method of proving I. 8 independently leaves I. 7 still unproved.

Min. That reason has no weight unless you can prove I. 7 to be valuable for itself.

Euc. True, but I think I can prove it; for, thirdly, it shows that, of all plane Figures that can be made by hingeing rods together, the three-sided ones (and these only) are rigid (which is another way of stating the fact that there cannot be two such figures on the same base). This is analogous to the fact, in relation to solids contained by plane surfaces hinged together, that any such solid is rigid, there being no maximum number of sides.

And fourthly, there is a close analogy between I. 7, 8 and III. 23, 24. These analogies give to Geometry much of its beauty, and I think that they ought not to be lost sight of.

Min. You have made out a good case. Allow me to contribute a 'fifthly.' It is one of the very few Propositions that have a direct bearing on practical science. I have often found pupils much interested in learning that the principle of the rigidity of Triangles is of constant use in architecture, and even in so homely a matter as the making of a gate.

The other Theorem which I mentioned, II. 8, is now so constantly ignored in examinations that it is very often omitted, as a matter of course, by students. It is believed to be extremely difficult and entirely useless.

Euc. Its difficulty has, I think, been exaggerated. Have you tried to teach it?

Min. I have occasionally found pupils amiable enough to listen to what they felt sure would be of no service in examinations. My experience has been wholly among undergraduates, any one of whom, if of average ability, would, I think, master it in from five to ten minutes.

Euc. No very exorbitant demand on your pupil's time. As to its being 'entirely useless,' I grant you it is of no immediate service, but you will find it eminently useful when you come to treat the Parabola geometrically.

Min. That is true.

Euc. Let us now consider the new methods of proof suggested by my Rivals.

Min. Prop. 5 has been much attacked—I may say trampled on—by your Modern Rivals.

Euc. Good. So that is why you call it 'The Asses' Bridge'? Well, how many new methods do they suggest for crossing it?

Min. One is 'hypothetical construction,' M. Legendre bisecting the base, and Mr. Pierce the vertical angle, but without any proof that the thing can be done.

Euc. So long as we agree that beginners in Geometry shall be limited to the use of Lines and Circles, so long will it be unsafe to assume a point as found, or a Line as drawn, merely because we are sure it exists. For example, it is axiomatic, of course, that every angle has a bisector: but it is equally obvious that it has two trisectors: and if I may assume the one as drawn, why not the others also? However we have discussed this matter already (p. 20).

Min. A second method is 'superposition,' adopted by Mr. Wilson and Mr. Cuthbertson—a method which here involves the reversing of the triangle, before applying it to its former position.

Euc. That also we have discussed (p. 47). What is the method adopted in the new Manual founded on the Syllabus of the Association?

Min. The same as Mr. Pierce's. Mr. Reynolds has a curious method: he treats the sides as obliques 'equally remote from the perpendicular.'

Euc. Curious, indeed.

Min. But perhaps the most curious of all is Mr. Willock's method: he treats the sides as radii of a circle, and the base as a chord.

Euc. He had better have made them asymptotes of a hyperbola at once! C'est magnifque, mais ce n'est pas la—Géometrie.

Min. Two of your Rivals prove Prop. 8 from Prop. 24.

Euc. 'Putting the cart before the horse,' in my humble opinion.

Min. For a brief proof of Prop. 13, let me commend to your notice Mr. Reynolds'—consisting of the seven words 'For they fill exactly the same space.'

Euc. Why so lengthy? The word 'exactly' is superfluous.

Min. Instead of your chain of Theorems, 18, 19, 20, several writers suggest 20, 19, 18, making 20 axiomatic.

Euc. That has been discussed already (p. 56).

Min. Mr. Cuthbertson's proof of Prop. 24 is, if I may venture to say so, more complete than yours. He constructs his diagram without considering the lengths of the sides, and then proves the 3 possible cases separately.

Euc. I think it an improvement.

Min. There are no other noticeable innovations, that have not been already discussed, except that Mr. Cuthbertson proves a good deal of Book II by a quasi-algebraical method, without exhibiting to the eye the actual Squares and Rectangle: while Mr. Reynolds does it by pure algebra.

Euc. I think the actual Squares, &c. most useful for beginners, making the Theorems more easy to understand and to remember. Algebraical proofs of course introduce the difficulty of 'incommensurables.'

Min. We will now take the new Propositions, &c. which have been suggested.

Here is an Axiom:—'Two lines cannot have a common segment.'

Euc. Good. I have tacitly assumed it, but it may as well be stated.

Min. Several new Theorems have been suggested, but only two of them seem to me worth mentioning. They are:—

'All right angles are equal.'

Euc. I have already approved of that (p. 219).

Min. The other is one that is popular with most of your Rivals:—

'Of all the Lines which can be drawn to a Line from a point without it, the perpendicular is least; and, of the rest, that which is nearer to the perpendicular is less than one more remote; and the lesser is nearer than the greater; and from the same point only two equal Lines can he drawn to the other Line, one on each side of the perpendicular.'

Euc. I like it on the whole, though so long an enunciation will be alarming to beginners. But it is strictly analogous to III. 7. Introduce it by all means in the revised edition of my Manual. It will be well, however, to lay it down as a general rule, that no Proposition shall be so interpolated, unless it be of such importance and value as to be thought worthy of being quoted as proved, in the same way in which candidates in examinations are now allowed to quote Propositions of mine.

Min. (with a fearful yawn) Well! I have no more to say.

 

§ 7. The summing-up.

Euc. 'The cock doth craw, the day doth daw,' and all respectable ghosts ought to be going home. Let me carry with me the hope that I have convinced you of the importance, if not the necessity, of retaining my order and numbering, and my method of treating straight Lines, angles, right angles, and (most especially) Parallels. Leave me these untouched, and I shall look on with great contentment while other changes are made—while my proofs are abridged and improved—while alternative proofs are appended to mine—and while new Problems and Theorems are interpolated.

In all these matters my Manual is capable of almost unlimited improvement.

[To the sound of slow music, Euclid and the other ghosts 'heavily vanish,' according to Shakespeare's approved stage-direction. Minos wakes with a start, and betakes himself to bed, 'a sadder and a wiser man.']