Euclid and His Modern Rivals/Act III. Scene II. § 1.

by Charles L. Dodgson

 

 

ACT III.

 

Scene II.

 

§ 1. Syllabus of the Association for the Improvement of Geometrical Teaching. 1878.

 

'Nos numerus sumus.'

Nie. The last book to be examined is Mr. Wilson's new Manual, founded on the Syllabus of the Geometrical Association.

Min. We had better begin by examining the Syllabus itself. I own that I could have wished to do this in the presence of some member of the Committee, who might have supplied a few details for what is at present little more than a skeleton, but that I fear is out of the question.

Nie. Nay, you shall not have far to seek. I am a member of the Committee.

Min. (astonished) You! A German professor! No such member is included in the final list of the Committee, which a friend showed me the other day.

Nie. The final list, was it? Well, ask your friend whether, since the drawing up of that list, any addition has been made: he will say 'Nobody has been added.'

Min. Quite so.

Nie. You do not understand. Nobody—Niemand—see you not?

Min. What? You mean—

Nie. (solemnly) I do, my friend. I have been added to it!

Min. (bowing) The Committee are highly honoured, I am sure.

Nie. So they ought to be, considering that I am a more distinguished mathematician than Newton himself, and that my Manual is better known than Euclid's! Excuse my self-glorification, but any moralist will tell you that I—I alone among men—ought to praise myself.

Min. (thoughtfully) True, true. But all this is word-juggling—a most misleading analogy. However, as you now appear in a new character, you must at least have a new name!

Nie. (proudly) Call me Nostradamus!

[Even as he utters the mystic name, the air grows dense around him, and gradually crystallizes into living forms. Enter a phantasmic procession, grouped about a banner, on which is emblazoned in letters of gold the title 'Association for the Improvement of Things in General.' Foremost in the line marches Nero, carrying his unfinished 'Scheme for lighting and warming Rome'; while among the crowd which follow him may be noticed—Guy Fawkes, President of the 'Association for raising the position of Members of Parliament'—the Marchioness de Brinvilliers, Inventress of the 'Application of Alteratives to the Digestive Faculty'—and the Rev. F. Gustrell (the being who cut down Shakspeare's mulberry-tree), leader of the 'Association for the Refinement of Literary Taste.' Afterwards enter, on the other side, Sir Isaac Newton's little dog 'Diamond,' carrying in his mouth a half-burnt roll of manuscript. He pointedly avoids the procession and the banner, and marches past alone, serene in the consciousness that he, single-pawed, conceived and carried out his great 'Scheme for throwing fresh light on Mathematical Research,' without the aid of any Association whatever.]

Min. Nostra, the plural of nostrum, 'a quack remedy'; and damus, 'we give.' It is a suggestive name.

Nos. And, trust me, it is a suggestive book that I now lay before you. 'Syllabus—'.

Min. (interrupting) You mean 'a Syllabus', or 'the Syllabus'?

Nos. No, no! In this railroad-age, we have no time for superfluous words! 'Syllabus of Plane Geometry, prepared by the Association for the Improvement of Geometrical Teaching.' Fourth Edition, 1877.

Min. How do you define a Right Line?

Nostradamus reads.

P. 7. Def. 5. 'A straight line is such that any part will, however placed, be wholly on any other part, if its extremities are made to fall on that other part.'

Min. That looks more like a property of a Right Line than its essence. Euclid makes an Axiom of that property. Of course you omit his Axiom?

Nos. No. We have the Axiom (p. 10, Ax. 2) 'Two straight lines that have two points in common lie wholly in the same straight line.'

Min. Well! That is certainly the strangest Axiom I ever heard of! The idea of asserting, as an Axiom, that Right Lines answer to their Definition!

Nos. (bashfully) Well, you see there were several of us at work drawing up this Syllabus: and we've got it a little mixed: we don't quite know which are Definitions and which are Axioms.

Min. So it appears: not that it matters much: the practical test is the only thing of importance. Do you adopt Euc. I. 14?

Nos. Yes.

Min. Then we may go on to the next subject. Be good enough to define 'Angle.'

Nostradamus reads.

P. 8. Def. 11. 'When two straight lines are drawn from the same point, they are said to contain, or to make with each other, a plane angle.'

Min. Humph! You are very particular about drawing them from a point. Suppose they were drawn to the same point, what would they make then?

Nos. An angle, undoubtedly.

Min. Then why omit that case? However, it matters little. You say 'a plane angle,' I observe. You limit an angle, then, to a magnitude less than the sum of two right angles.

Nos. No, I can't say we do. A little further down we assert that 'two angles are formed by two straight lines drawn from a point.'

Min. Why, these are like Falstaff's 'rogues in buckram suits'! Are there more coming?

Nos. No, we do not go beyond the sum of four right angles. These two we call conjugate angles. 'The greater of the two is called the major conjugate, and the smaller the minor conjugate, angle.'

Min. These Definitions are wondrous! This is the first time I ever heard 'major' and 'minor' defined. One feels inclined to say, like that Judge in the story, when a certain barrister, talking against time, insisted on quoting authorities for the most elementary principles of law, 'Really, brother, there are some things the Court may be assumed to know!' Any more definitions?

Nos. We define 'a straight angle.'

Min. That I have discussed already (see p. 102).

Nos. But this, I think, is new:—

Reads.

P. 9. Def 12. 'When three straight Lines are drawn from a point, if one of them be regarded as lying between the other two, the angles which this one (the mean) makes with the other two (the extremes) are said to be adjacent angles.'

Min. That is new indeed. Let us try a figure:—

Now let us regard OA 'as lying between the other two.' Which are 'the angles which it makes with the other two'? For this line OA (which you rightly call 'the mean '—lying is always mean) makes, be pleased to observe, four angles altogether—two with OB and two with OC.

Nos. I cannot answer your question. You confuse me.

Min. I need not have troubled you. I see that I can obtain an answer from the Syllabus itself. It says (at the end of Def. 11) 'when the angle contained by two lines is spoken of without qualification, the minor conjugate angle is to be understood.' Here we have a case in point, as these angles are spoken of 'without qualification.' So that the angles alluded to are both of them 'minor conjugate' angles, and lie on the same side of OA. And these we are told to call 'adjacent' angles!

How do you define a Right Angle?

Nos. As in Euclid.

Min. Let me hear it, if you please. You know Euclid has no major or minor conjugate angles.

Nostradamus reads.

P. 9. Def. 14. 'When one straight line stands upon another straight line and makes the adjacent angles equal, each of the angles is called a right angle.'

Min. Allow me to present you with a figure, as I see the Syllabus does not supply one.

A

B

C

Here AB 'stands upon' BC and makes the adjacent angles equal. How do you like these 'right angles'?

Nos. Not at all.

Min. These same 'conjugate angles' will get you into many difficulties.

Have you Euclid's Axiom 'all right angles are equal'?

Nos. Yes; only we propose to prove it as a Theorem.

Min. I have no objection to that: nor do I think that your treatment of angles, as a whole, is actually illogical. What I chiefly object to is the general 'slipshoddity' (if I may coin a word) of the language of your Syllabus.

Does your proof of Euc. I. 32 differ from his?

Nos. No, except that we propose Playfair's Axiom, 'two straight Lines that intersect one another cannot both be parallel to the same straight Line,' as a substitute for Euc. Ax. 12.

Min. Is this your only test for the meeting of two Lines, or do you provide any other?

Nos. This is the only one.

Min. But there are cases where this is of no use. For instance, if you wish to make a Triangle, having, as data, a side and the two adjacent angles. Have you such a Problem?

Nos. Yes, it is Pr. 10, at p. 19.

Min. And how do you prove that the Lines will meet?

Nos. (smiling) We don't prove it: that is the reader's business: we only provide enunciations.

Min. You are like the gourmand who would eat so many oysters at supper that at last his friend could not help saying 'They are sure to disagree with you in the night.' 'That is their affair,' the other gaily replied. 'I shall be asleep!'

Your Syllabus has the same hiatus as the other writers who have rejected Euclid's 12th Axiom. If you will not have it as an Axiom, you ought to prove it as a Theorem. Your treatise is incomplete without it.

13—15

14, 5

26 α

16

16

18—24

18

25

26 β

17

The Theorems contained in the first 26 Propositions of Euclid are thus rearranged in the Syllabus. The only advantage that I can see in the new arrangement is that it places first the three which relate to Lines, thus getting all those which relate to Triangles into a consecutive series. All the other changes seem to be for the worse, and specially the separation of Theorems from their converses, e.g. Props. 5, 6, and 24, 25.

The third part of Prop. 29 is put after Prop. 32: and Props. 33, 34 are transposed. I can see no reason for either change.

Prop. 47 is put next before Prop. 12 in Book II. This would be a good arrangement (if it were ever proved to be worth while to abandon Euclid's order), as the Theorems are so similar; and the placing Prop. 48 next after II. 13 is a necessary result.

In Book II, Props. 9, 10 are placed after Props. 12, 13. I see no reason for it.

It does not appear to me that the new arrangements, for the sake of which it is proposed to abandon the numeration of Euclid, have anything worth mentioning to offer as an advantage.

I will now go through a few pages of 'this many-headed monster,' and make some general remarks on its style.

P. 4. 'A Theorem is the formal statement of a Proposition that may be demonstrated from known Propositions.These known Propositions may themselves be Theorems or Axioms.'

This is a truly delightful jumble. Clearly, 'a Proposition that may be demonstrated from known Propositions' is itself a Theorem. Hence a Theorem is 'the formal statement' of a Theorem. The question now arises—of itself, or of some other Theorem? That a Theorem should be 'the formal statement' of itself, has a comfortable domestic sound, something like 'every man his own washerwoman,' but at the same time it involves a fearful metaphysical subtlety. That one Theorem should be 'the formal statement' of another Theorem, is, I think, degrading to the former, unless the second will consent to act on the 'claw me, claw thee' principle, and to be 'the formal statement' of the first.

Nos. You bewilder me.

Min. Perhaps, however, it is intended that the teacher who uses this Manual should, on reaching the words 'a Proposition that may be demonstrated,' recognise the fact that this is itself 'a Theorem,' and at once go back to the beginning of the sentence. He will thus obtain a Definition closely resembling a Continued Fraction, and may go on repeating, as long as his breath holds out, or until his pupil declares himself satisfied, 'a Theorem is the formal statement of the formal statement of the formal statement of the——'

Nos. (widly) Say no more! My brain reels!

Min. I spare you. Let us go on to p. 5, where I find the following:—

'Rule of Conversion. If of the hypotheses of a group of demonstrated Theorems it can be said that one must be true, and of the conclusions that no two can be true at the same time, then the converse of every Theorem of the group will necessarily be true.'

Let us take an instance:—

If 5 > 4, then 5 > 3.
If 5 < 2, then 5 < 3.

Those will do for 'demonstrated Theorems,' I suppose?

Nos. I suppose so.

Min. And the 'hypothesis' of the first 'must be true,' simply because it is true.

Nos. It would seem so.

Min. And it is quite clear that 'of the conclusions no two can be true at the same time,' for they contradict each other.

Nos. Clearly.

Min. Then it ought to follow that 'the converse of every Theorem of the group will necessarily be true.' Take the converse of the second, i.e.

If 5 < 3, then 5 < 2.

Is this 'necessarily true'? Is every thing which is less than 3 necessarily less than 2?

Nos. Certainly not. I think you have misinterpreted the phrase 'it can be said that one must be true,' when used of the hypotheses. It does not mean 'it can be said, from a knowledge of the subject-matter of some one hypothesis, that it is, and therefore must be, true,' but 'it can be said, from a knowledge of the mutual logical relation of all the hypotheses, as a question of form alone, and without any knowledge of their subject-matter, that one must be true, though we do not know which it is.'

Min. Your power of uttering long sentences is one that does equal honour to your head and your—lungs. And most sincerely do I pity the unfortunate learner who has to make out all that for himself! Let us proceed.

P. 9. Def. 13. 'The bisector of an angle is the straight Line that divides it into two equal angles.'

This assumes that 'an angle has one and only one bisector,' which appears as Ax. 4, at the foot of p. 10.

P. 10. Def. 21. 'The opposite angles made by two straight Lines that intersect &c.'

This seems to imply that 'two Lines that intersect' always do make 'opposite angles.'

Nos. Surely they do?

Min. By no means. Look at p. 12, Def. 32, where, in speaking of a Triangle, you say 'the intersection of the other two sides is called the vertex.'

Nos. A slip, I confess.

Min. One of many.

P. 12, Def. 31. 'All other Triangles are called acute-angled Triangles.' What? If a Triangle had two right angles, for instance?

Nos. But there is no such Triangle.

Min. That is a point you do not prove till we come to Th. 18, Cor. 1, two pages further on. The same remark applies to your Def. 33, in the same page. 'The side… which is opposite to the right angle,' where you clearly assume that it cannot have more than one.

P. 12, Def. 32. 'When two of the sides have been mentioned, the remaining side is often called the base.' Well, but how if two of the sides have not been mentioned?

Nos. In that case we do not use the word.

Min. Do you not? Turn to p. 22, Th. 2, Cor. 1, 'Triangles on the same or equal bases and of equal altitude are equal.'

Nos. We abandon the point.

Min. You had better abandon the Definition.

P. 12, Def. 34. Is not 'identically equal' tautology? Things that are 'identical' must surely be 'equal' also. Again, 'every part of one being equal,' &c. What do you mean by 'every part' of a rectilineal figure?

Nos. Its sides and angles, of course.

Min. Then what do you mean by Ax (b) in p. 3. 'The whole is equal to the sum of its parts'? This time, I think I need not 'pause for a reply'!

P. 15, Def. 38. 'When a straight Line intersects two other straight Lines it makes with them eight angles etc.'

Let us count the angles at G. They are, the 'major' and 'minor' angles which bear the name EGA; do. for EGB; do. for AGH; and do. for BGH. That is, eight angles at G alone. There are sixteen altogether.

P. 17, Th. 30. 'If a quadrilateral has two opposite sides equal and parallel, it is a Parallelogram.'

This re-asserts part of its own data.

P. 17, Th. 31. 'Straight Lines that are equal and parallel have equal projections on any other straight Line; conversely, parallel straight Lines that have equal projections on another straight Line are equal.'

The first clause omits the case of Lines that are equaland in one and the same straight Line. The second clause

is not true: if the parallel Lines are at right angles to the other Line, their projections are equal, both being zero, whether the Lines are equal or not.

P. 18, Th. 32. 'If there are three parallel straight Lines, and the intercepts made by them on any straight Line that cuts them are equal, then etc.'

The subject of this Proposition is inconceivable: there are three intercepts, and by no possibility can these three be equal.

P. 25, Prob. 5. 'To construct a rectilineal Figure equal to a given rectilineal Figure and having the number of its sides one less than that of the given Figure.'

May I ask you to furnish me with the solution of this Problem, taking, as your 'given rectilineal Figure,' a Triangle?

Nos. (indignantly) I decline to attempt it!

Min. I will now sum up the conclusions I have come to with respect to your Syllabus.

In the subjects of Lines, Angles, and Parallels, the changes you propose are as follows:—

You give a very unsatisfactory Definition of a 'Right Line,' and then most illogically re-state it as an Axiom.

You extend the Definition of Angle—a most disastrous innovation.

Your Definition of 'Right Angle' is a failure.

You substitute Playfair's axiom for Euclid's 12th.

All these things are very poor compensation indeed for the vital changes you propose—the separation of Problems and Theorems, and the abandonment of Euclid's order and numeration. Restore the Problems (which are also Theorems) to their proper places, keep to Euclid's numbering (interpolating your new Propositions where you please), and your Syllabus may yet prove to be a valuable addition to the literature of Elementary Geometry.