Euclid and His Modern Rivals/Act II. Scene VI. § 3.
ACT II.
Scene VI.
§ 3. Willock.
'This work … no doubt, has its faults.’
Willock, Pref. p. 1.
Nie. I lay before you 'The Elementary Geometry of the Right Line and Circle' by W. A. Willock, D.D., formerly Fellow of Trinity College, Dublin, published in 1875.
Min. I have gone through the subject of 'direction' so minutely in reviewing Mr. Wilson's book, that I need not discuss with you any points in which your client essentially agrees with him. We may, I think, pass over the subject of the Right Line altogether?
Nie. Yes.
Min. And as to Angles and Right Angles, I see no novelty in Dr. Willock's book, except that he defines an Angle as 'the divergence of two directions,' which is virtually the same as Euclid's Definition.
Nie. That I think is all.Min. Then we can proceed at once to the subject of Parallels. Will you kindly give me your proof of Euc. I. 32 from the beginning?
Niemand reads.
P. 10. Th. 1. 'Two Directives can intersect in only one point.'
Min. By 'Directive' you mean an 'infinite Line'?
Nie. Yes.
Min. Well, I need hardly trouble you to prove it as a Theorem, being quite willing to grant it as an Axiom. What is the next Theorem?
Niemand reads.
P. 11. Th. 5. 'Parallel Directives cannot meet.'
Min. We will call them 'sepcodal,' if you please. I grant it, provisionally. If such Lines exist, they cannot meet.
Niemand reads.
P. 11. Th. 7. 'Only one Line, sepcodal to a Directive, can be drawn through a point.'
Min. Does that assert that one can be drawn? Or does it simply deny the possibility of drawing two?
Nie. The proof only applies to the denial: but the assertion is certainly involved in the enunciation. At all events, if not assumed here, it is assumed later on.
Min. Then I will at this point credit you with one unwarrantable Axiom, namely, that different Lines can have the same direction. The Theorem itself I grant.
Niemand reads.
P. 12. Th. 8. 'The angles of intersection of a Transversal with two sepcodal Directives are equal'.
Min. Do you prove that by Mr. Wilson's method?
Nie. Not quite. He does it by transferring an angle: we do it by divergence of directions.
Min. I prefer your method. All it needs to make it complete is the proof of the reality of such Lines: but that is unattainable, and its absence is fatal to the whole system. Nay, more: the fact, that the reality of such Lines leads by a logical necessity to the reality of Lines which make equal angles with any transversal, reacts upon that unfortunate Axiom, and destroys the little hope it ever had of being granted without proof. In point of fact, in asking to have the Axiom granted, you were virtually asking to have this other reality granted as axiomatic—but all this I have already explained (p. 125).
Niemand reads.
P. 13. Th. 10. 'If a Transversal cut two Directives and make the angles of intersection with them equal, the Directives are sepcodal'.
Min. The subject of your Proposition is indisputably real. If then you can prove this Theorem, you will thereby prove the reality of sepcodal Lines. But I fear you have assumed it already in Th. 7. There is still, however, a gleam of hope: perhaps you do not need Th. 7 in proving this?Nie. We do not: but I fear that will not mend matters, as we assume, in the course of this Theorem, that a Line can be drawn through a given point, so as to have the same direction as a given Line.
Min. Then we need not examine it further: it must perish with the faulty Axiom on which it rests. What is your next Theorem?
Nie. It answers to Euc. I, 16, 17, and is proved by the Theorem you have just rejected.
Min. Then I must reject its proof, but I will grant you the Theorem itself, if you like, as we know it can be proved from undisputed Axioms. What comes next?
Niemand reads.
P. 14. Th. 13. 'If a Transversal meet two Directives, and make angles with them, the External greater than the Internal, or the sum of the two Internal angles less than two right angles, the two directives must meet.'
Min. A proof for Euclid's Axiom? That is interesting.
Niemand reads.
'For, suppose they do not meet. Then, they should be sepcodal'
Min. (interrupting) 'Should be sepcodal'? Does that mean that they are sepcodal?
Nie. Yes, I think so.
Min. That is, you assume that separational Lines have the same direction?
Nie. We do.
Min. A fearful assumption! (A long silence) Well?Nie. I am waiting to know whether you grant it.
Min. Unquestionably not! I must mark it against you as an Axiom of the most monstrous character! Mr. Wilson himself does not assume this, though he does assume its Contranominal, that Lines having different directions will meet (see p. 115). And what I said then I say now—unaxiomatic! But supposing it granted, how would you prove the Theorem?
Niemand reads.
'Then, they should be sepcodal; and the external angle should be equal (Th. 8) to the internal; which is contrary to the supposition.'
Min. Quite so. But Th. 8, which you quote, itself depends on the reality of sepcodal Lines. Your Theorem rests on two legs, and both, I fear, are rotten!
Nie. The next Theorem is equivalent to Euc. I. 32. Do you wish to hear it?
Min. It is unnecessary: it follows easily from Th. 8.
And I need not ask you what practical test you provide for the meeting of two Lines, seeing that you have Euclid's 12th Axiom itself.
Nie. Proved as a Theorem.
Min. Attempted to be proved as a Theorem. I will now take a hasty general survey of your client's book.
The first point calling for remark is the arrangement. You begin by dragging the unfortunate beginner straight into the most difficult part of the subject. Your first chapter positively bristles with difficulties about 'direction.' Then comes a long chapter on circles, including some very complicated figures, and a theory of tangents which depends upon moving lines and vanishing chords—all most disheartening to a beginner. What do you suppose he is likely to make of such a sentence as 'the direction of the motion of the generating point of any curve is that of the tangent to the curve at that point'? (p. 29.) Or this again, 'it is also evident that, the circle being a simple curve, there can be only one tangent to it at any point'? (p. 29.) What is 'a simple curve'?
Nie. I do not know.
Min. Then comes a chapter of Problems, and then—when your pupil has succeeded in mastering thirty-four pages of your book, and has become tolerably familiar with tangents and segments, with diametral lines and reëntrant angles, with 'oval forms' and 'forms semi-convex, semi-concave,'—you at last confront him with that abstruse and much dreaded Theorem, Euc. I. 4! True, he has the 'Asses' Bridge' to help him in proving it, that in its turn being proved, apparently, by properties of the circle; but, even with all these assistances, it is an arduous task!
Nie. You are hard on my client.
Min. Well, jesting apart, let me say in all seriousness that I think it would require very great ingenuity to make a worse arrangement of the subject of Geometry, for purposes of teaching, than is to be found in this little book.
I do not think it necessary to criticise the book throughout: but I will mention one or two passages which have caught my eye in glancing through it.Here, for instance, is something about 'Directives,' which seem to be a curious kind of Loci—quite different from Right Lines, I should say.
Nie. Oh no! They are exactly the same thing!
Min. Well, I find, at p. 4, 'Directives are either divergent or parallel': and again, at p. 11, 'Parallel Directives cannot meet.' Clearly, then, Directives can never by any possibility coincide: but ordinary Right Lines occasionally do so, do they not?
Nie. It is a curious lapsus pennae.
Min. At p. 7, I observe an article headed 'The principle of double conversion,' which I will quote entire.
Reads.
'If four magnitudes, a, b, A, B, are so related, that when a is greater than b, A is greater than B; and when a is equal to b, A is equal to B: then, conversely, when A is greater than B, a is greater than b; and, when A is equal to B, a is equal to b.
'The truth of this principle, which extends to every kind of magnitude, is thus made evident:—If, when A is greater than B, a is not greater than b, it must be either less than or equal to b. But it cannot be less; for, if it were, A should, by the antecedent part of the proposition, be less than B, which is contrary to the supposition made. Nor can it be equal to b; for, in that case, A should be equal to B, also contrary to supposition. Since, therefore, a is neither less than nor equal to b, it remains that it must be greater than b.'
Now let a and A be variables and represent the ordinates to two curves, mnr and MNR, for the same abscissa; and let b and B be constants and represent their intercepts on the Y-axis; i.e. let On = b, and ON = B.
Does not this diagram fairly represent the data of the proposition? You see, when we take a negative abscissa, so as to make a greater than b, we are on the left-hand branch of the curve, and A is also greater than B; and again, when a is equal to b, we are crossing the Y-axis, where A is also equal to B.
Nie. It seems fair enough.
Min. But the conclusion does not follow? With a positive abscissa, A is greater than B, but a less than b.
Nie. We cannot deny it.
Min. What then do you suppose would be the effect on a simple-minded student who should wrestle with this terrible theorem, firm in the conviction that, being in a printed book, it must somehow be true?
Nie. (gravely) Insomnia, certainly; followed by acute Cephalalgia; and, in all probability, Epistaxis.
Min. Ah, those terrible names! Who would suppose that a man could have all those three maladies, and survive? And yet the thing is possible!
Let me now read you a statement (at p. 112) about incommensurables:—
'When one of the magnitudes can be represented only by an interminable decimal, while the other is a finite whole number, or finite decimal, no finite common submultiple can exist; for, though a unit be selected in the last place of the whole number or finite decimal, yet the decimal represented by all the figures which follow the corresponding place in the interminable decimal, being less than that unit in that place and unknown in quantity, cannot be a common measure of the two magnitudes, and is only a remainder.'
Now can you lay your hand upon your heart and declare, on the word of an honest man, that you understand this sentence—beginning at the words 'yet the decimal'?
Nie. (vehemently) I cannot!
Min. Of the two reasons which are mentioned, to explain why it 'cannot be a common measure of the two magnitudes,' does the first—that it is 'less than that unit in that place'—carry conviction to your mind? And does the second—that it is 'unknown in quantity' ripen that conviction into certainty?
Nie. (wildly) Not in the least!
Min. Well, I will not 'slay the slain' any longer. You may consider Dr. Willock's book as rejected. And I think we may say that the whole theory of 'direction' has collapsed under our examination.
Nie. I greatly fear so.