ACT I.
Scene II.
Οὐκ άγαθὸν πολυκοιρανίη᾿ εἷς κΟίρανος ἔστω,
Εἷς βασιλεὺς
[Minos sleeping: to him enter the Phantasm of Euclid. Minos opens his eyes and regards him with a blank and stony gaze, without betraying the slightest surprise or even interest.]
§ 1. A priori reasons for retaining Euclid's Manual.
Euc. Now what is it you really require in a Manual of Geometry?
Min. Excuse me, but—with all respect to a shade whose name I have reverenced from early boyhood—is not that rather an abrupt way of starting a conversation? Remember, we are twenty centuries apart in history, and consequently have never had a personal interview till now. Surely a few preliminary remarks—Euc. Centuries are long, my good sir, but my time to-night is short: and I never was a man of many words. So kindly waive all ceremony and answer my question.
Min. Well, so far as I can answer a question that comes upon me so suddenly, I should say—a book that will exercise the learner in habits of clear definite conception, and enable him to test the logical value of a scientific argument.
Euc. You do not require, then, a complete repertory of Geometrical truth?
Min. Certainly not. It is the ὲνέργεια rather than the ἔργον that we need here.
Euc. And yet many of my Modern Rivals have thus attempted to improve upon me—by filling up what they took to be my omissions.
Min. I doubt if they are much nearer to completeness themselves.
Euc. I doubt it too. It is, I think, a friend of yours who has amused himself by tabulating the various Theorems which might be enunciated in the single subject of Pairs of Lines. How many did he make them out to he?
Min. About two hundred and fifty, I believe.
Euc. At that rate, there would probably be, within the limits of my First Book, about how many?
Min. A thousand, at least.
Euc. What a popular school-book it will be! How boys will bless the name of the writer who first brings out the complete thousand!
Min. I think your Manual is fully long enough already for all possible purposes of teaching. It is not in the region of new matter that you need fear your Modern Rivals: it is in quality, not in quantity, that they claim to supersede you. Your methods of proof, so they assert, are antiquated, and worthless as compared with the new lights.
Euc. It is to that very point that I now propose to address myself: and, as we are to discuss this matter mainly with reference to the wants of beginners, we may as well limit our discussion to the subject-matter of Books I and II.
Min. I am quite of that opinion.
Euc. The first point to settle is whether, for purposes of teaching and examining, you desire to have one fixed logical sequence of Propositions, or would allow the use of conflicting sequences, so that one candidate in an examination might use X to prove Y, and another use Y to prove X—or even that the same candidate might offer both proofs, thus 'arguing in a circle.'
Min. A very eminent Modern Rival of yours, Mr. Wilson, seems to think that no such fixed sequence is really necessary. He says (in his Preface, p. 10) 'Geometry when treated as a science, treated inartificially, falls into a certain order from which there can be no very wide departure; and the manuals of Geometry will not differ from one another nearly so widely as the manuals of algebra or chemistry; yet it is not difficult to examine in algebra and chemistry.'
Euc. Books may differ very 'widely' without differing in logical sequence—the only kind of difference which could bring two text-books into such hopeless collision that the one or the other would have to be abandoned. Let me give you a few instances of conflicting logical sequences in Geometry. Legendre proves my Prop. 5 by Prop. 8, 18 by 19, 19 by 20, 27 by 28, 29 by 32. Cuthbertson proves 37 by 41. Reynolds proves 5 by 20. When Mr. Wilson has produced similarly conflicting sequences in the manuals of algebra or chemistry, we may then compare the subjects: till then, his remark is quite irrelevant to the question.
Min. I do not think he will be able to do so: indeed there are very few logical chains at all in those subjects—most of the Propositions being proved from first principles. I think I may grant at once that it is essential to have one definite logical sequence, however many manuals we employ: to use the words of another of your Rivals, Mr. Cuthbertson (Pref. p. viii.), 'enormous inconvenience would arise in conducting examinations with no recognised sequence of Propositions.' This however applies to logical sequences only, such as your Props. 13, 15, 16, 18, 19, 20, 21, which form a continuous chain. There are many Propositions whose place in a manual would be partly arbitrary. Your Prop. 8, for instance, is not wanted till we come to Prop. 48, so that it might occupy any intermediate position, without involving risk of circular argument.
Euc. Now, in order to secure this uniform logical sequence, we should require to know, as to any particular Proposition, what other Propositions were its logical descendants, so that we might avoid using any of these in proving it?
Min. Exactly so.Euc. We might of course give this information by attaching to each enunciation references to its logical descendants: but this would be a very cumbrous plan. A better way would be to give them in the form of a genealogy, but this would be very bulky if the enunciations themselves were inserted: so that it would be desirable to have numbers to distinguish the enunciations. In that case (supposing my logical sequence to be adopted) the genealogy would stand thus:—(see Frontispiece).
Min. Would it not be enough to publish an arranged list (which would be all the better if numbered also), and to enact that no Proposition should be used to prove any of its predecessors?
Euc. That would hamper the writers of manuals very much more than the genealogy would. Suppose, for instance, that you adopted, in the list, the order of Theorems in my First Book, and that a writer wished to prove Prop. 8 by Prop. 47: this would not interfere with my logical sequence, and yet your list would bar him from doing so.
Min. But we might place 8 close before 48, and he would then be free to do as you suggest.
Euc. And suppose some other writer wished to prove 24 by 8?
Min. I see now that any single list must necessarily prevent many possible arrangements which would not conflict with the agreed-on logical sequence. And yet this is what the Committee of the Association for the Improvement of Geometrical Teaching have approved of, namely, 'a standard sequence for examination purposes,' and what the Association have published in their 'Syllabus of Plane Geometry.'
Euc. I think they have overlooked the fact that they are enacting many more sequences, as binding on writers, than the one logical sequence which they desire to secure. Their 'standard sequence' would be fitly replaced by a 'standard genealogy.' But in any case we are agreed that it is desirable to have, besides a standard logical sequence, a standard list of enunciations, numbered for reference?
Min. We are.
Euc. The next point to settle is, what sequence and numbering to adopt. You will allow, I think, that there are strong a priori reasons for retaining my numbers. The Propositions have been known by those numbers, for two thousand years; they have been referred to, probably, by hundreds of writers—in many cases by the numbers only, without the enunciations: and some of them, I. 5 and I. 47 for instance—'the Asses' Bridge' and 'the Windmill'—are now historical haracters, and their nicknames are 'familiar as household words.'
Min. Even if no better sequence than yours could be found, it might still be urged that a new set of numbers must be adopted, in order to introduce, in their proper places, some important Theorems which have been added to the subject since your time.
Euc. That want, if it were proved to exist, might, I think, be easily provided for without discarding my system of numbers. If you wished, for instance, to insert two new Propositions between I. 13 and I. 14, it would be far less inconvenient to call them 13 B and 13 C than to abandon the old numbers.
Min. I give up the objection.
Euc. You will allow then, I think, that my sequence and system of numbers should not be abandoned without good cause?
Min. Oh, certainly. And the onus probandi lies clearly on your Modern Rivals, and not on you.
Euc. Unless, then, it should appear that one of my Modern Rivals, whose logical sequence is incompatible with mine, is so decidedly better in his treatment of really important topics, as to make it worth while to suffer all the inconvenience of a change of numbers, you would not recognise his demand to supersede my Manual?
Min. On that point let me again quote Mr. Wilson. In his Preface, p. 15, he says, 'In a few years I hope that our leading mathematicians will have published, perhaps in concert, one or more text-books of Geometry, not inferior, to say the least, to those of France, and that they will supersede Euclid by the sheer force of superior merit.'
Euc. And I should be quite content to be so superseded. 'A fair field and no favour' is all I ask.
§ 2. Method of procedure in examining Modern Rivals.
Min. You wish me then to compare your book with those of your Modern Rivals?
Euc. Yes. But, in doing this, I must beg you to bear in mind that a Modern Rival will not have proved his case if he only succeeds in showing
(1) that certain Propositions might with advantage be omitted (for this a teacher would be free to do, so long as he left the logical sequence complete);
or (2) that certain proofs might with advantage be changed for others (for these might be interpolated as 'alternative proofs');
or (3) that certain new Propositions are desirable (for these also might be interpolated, without altering the numbering of the existing Propositions).
All these matters will need to be fully considered hereafter, if you should decide that my Manual ought to be retained: but they do not constitute the evidence on which that decision should be based.
Min. That, I think, you have satisfactorily proved. But what would you consider to be sufficient grounds for abandoning your Manual in favour of another?
Euc. Many grave charges have been brought against my Manual; but, of all these, there are only two which I regard as crucial in this matter. The first concerns my arrangement of Problems and Theorems: the second my treatment of Parallels.
If it be agreed that Problems and Theorems ought to be treated separately, my system of numbering must of course be abandoned, and no reason will remain why my Manual should then be retained as a whole; which is the only point I am concerned with. This question you can, of course, settle on its own merits, without examining any of the new Manuals.If, again, it be agreed that, in treating Parallels, some other method, essentially different from mine, ought to be adopted, I feel that, after so vital a change as that, involving (as no doubt it would) the abandonment of my sequence and system of numbering, the remainder of my Manual would not be worth fighting for, though portions of it might be embodied in the new Manual. To settle this question, you must, of course, examine one by one the new methods that have been proposed.
Min. You would not even ask to have your Manual retained as an alternative for the new one?
Euc. No. For I think it essential for purposes of teaching, that in treating this vital topic one uniform method should be adopted; and that this method should be the best possible (for it is almost inconceivable that two methods of treating it should be equally good). An alternative proof of a minor Proposition may fairly be inserted now and then as about equal in merit to the standard proof, and may make a desirable variety: but on this one vital point it seems essential that nothing but the best proof existing should be offered to the limited capacity of a learner. Vacuis committere venis nil nisi lene decet.
Min. I agree with you that we ought to have one system only, and that the best, for treating the subject of Parallels. But would you have me limit my examination of your 'Modern Rivals' to this single topic?
Euc. No. There are several other matters of so great importance, and admitting of so much variety of treatment, that it would be well to examine any method of dealing with them which differs much from mine—not with a view of substituting the new Manual for mine, but in order to make such changes in my proofs as may be thought desirable. There are other matters again, where changes have been suggested, which you ought to consider, but on general grounds, not by examining particular writers.
Let me enumerate what I conceive should be the subjects of your enquiry, arranged in order of importance.
- (1) The combination, or separation, of Problems and theorems.
- (2) The treatment of Pairs of Lines, especially Parallels, for which various new methods have been suggested. These may be classified as involving—
- (α) Infinite series: suggested by Legendre.
- (β) Angles made with transversals: Cooley.
- (γ) Equidistance: Cuthbertson.
- (δ) Revolving Lines: Henrici.
- (ε) Direction: Wilson, Pierce, Willock.
- (ζ) The substitution of 'Playfair's Axiom' for my Axiom 12.
If your decision, on these two crucial questions, be given in my favour, we may take it as settled, I think, that my Manual ought to be retained as a whole: how far it should be modified to suit modern requirements will be matter for further consideration.
- (3) The principle of superposition.
- (4) The use of diagonals in Book II. These two are general questions, and will not need the examination of particular authors.
Besides this, it will be well, in order that your enquiry into the claims of my Modern Rivals may be as complete as possible, to review them one by one, with reference to their treatment of matters not already discussed, especially:—
- (5) Right Lines.
- (6) Angles, including right angles.
- (7) Propositions of mine omitted.
- (8) Propositions of mine treated by a new method.
- (9) New Propositions.
- (10) And you may as well conclude, in each case, with a general survey of the book, as to style, &c.
The following may be taken as a fairly complete catalogue of the books to be examined:—
| 1. 2. 3. 4. 5. 6. 7. |
Legendre. Cooley. Cuthbertson. Henrici. Wilson Pierce. Willock. |
8. 9. 10. 11. 12. 13. |
Chauvenet. Loomis. Morell. Reynolds. Wright. Wilson's 'Syllabus'-Manual. |
You should also examine the Syllabus, published by the Association for the Improvement of Geometrical Teaching, on which the last-named Manual is based. Not that it can be considered as a 'Rival'— in fact, it is not a text-book at all, but a mere list of enunciations—but because, first, it comes with an array of imposing names to recommend it, and secondly, it discards my system of numbers, so that its adoption, as a standard for examinations, would seriously interfere with the retention of my Manual as the standard text-book.
Now, of these questions, I shall be most happy to discuss with you the general ones (I mean questions 1, 2 (ζ), 3, and 4) before we conclude this interview: but, when it comes to criticising particular authors, I must leave you to yourself, to deal with them as best you can.
Min. It will be weary work to do it all alone. And yet I suppose you cannot, even with your supernatural powers, fetch me the authors themselves?
Euc. I dare not. The living human race is so strangely prejudiced. There is nothing men object to so emphatically as being transferred by ghosts from place to place. I cannot say they are consistent in this matter: they are for ever 'raising' or 'laying' us poor ghosts—we cannot even haunt a garret without having the parish at our heels, bent on making us change our quarters: whereas if I were to venture to move one single small boy—say to lift him by the hair of his head over only two or three houses, and to set him down safe and sound in a neighbour's garden—why, I give you my word, it would be the talk of the town for the next month!
Min. I can well believe it. But what can you do for me? Are their Doppelgänger available?
Euc. I fear not. The best thing I can do is to send you the Phantasm of a German Professor, a great friend of mine. He has read all books, and is ready to defend any thesis, true or untrue.Min. A charming companion! And his name?
Euc. Phantasms have no names—only numbers. You may call him 'Herr Niemand,' or, if you prefer it, 'Number one-hundred-and-twenty-three-million-four-hundred-and-fifty-six-thousand-seven-hundred-and-eighty-nine.'
Min. For constant use, I prefer 'Herr Niemand.' Let us now consider the question of the separation of Problems and Theorems.
§ 3. The combination, or separation, of Problems and Theorems.
Euc. I shall be glad to hear, first, the reasons given for separating them, and will then tell you my reasons for mixing them.
Min. I understand that the Committee of the Association for the Improvement of Geometrical Teaching, in their Report on the Syllabus of the Association, consider the separation as 'equivalent to the assertion of the principle that, while Problems are from their very nature dependent for the form, and even the possibility, of their solution on the arbitrary limitation of the instruments allowed to be used, Theorems, being truths involving no arbitrary element, ought to be exhibited in a form and sequence independent of such limitations.' They add however that 'it is probable that most teachers would prefer to introduce Problems, not as a separate section of Geometry, but rather in connection with the Theorems with which they are essentially related.'
Euc. It seems rather a strange proposal, to print the Propositions in one order and read them in another. But a stronger objection to the proposal is that several of the Problems are Theorems as well—such as I. 46, for instance.
Min. How is that a Theorem?
Euc. It proves that there is such a thing as a Square. The definition, of course, does not assert real existence: it is merely provisional. Now, if you omit I. 46, what right would you have, in I. 47, to say 'draw a Square'? How would you know it to be possible?
Min. We could easily deduce that from I. 34.
Euc. No doubt a Theorem might be introduced for that purpose: but it would be very like the Problem: you would have to say 'if a figure were drawn under such and such conditions, it would be a Square.' Is it not quite as simple to draw it?
Then again take I. 31, where it is required to draw a Parallel. Although it has been proved in I. 27 that such things as parallel Lines exist, that does not tell us that, for every Line and for every point without that Line, there exists a real Line, parallel to the given Line and passing through the given point. And yet that is a fact essential to the proof of I. 32.
Min. I must allow that I. 31 and I. 46 have a good claim to be retained in their places: and if two are to be retained, we may as well retain all.
Euc. Another argument, for retaining the Problems where they are, is the importance of keeping the numbering unchanged—a matter we have already discussed.
But perhaps the strongest argument is that it saves you from 'hypothetical constructions,' the danger of which has been so clearly pointed out by Mr. Todhunter (see pp. 222, 241).
Min. I think you have proved your case very satisfactorily. The next subject is 'the treatment of Pairs of Lines.' Would it not be well, before entering on this enquiry, to tabulate the Propositions that have been enunciated, whether as Axioms or Theorems, respecting them?
Euc. That will be an excellent plan. It will both give you a clear view of the field of your enquiry, and enable you to recognise at once any doubtful Axioms which you may meet with.
Min. Will you then favour me with your views on this matter?
Euc. Willingly. It is a subject which I need hardly say I considered very carefully before deciding what Definitions and Axioms to adopt.
§ 4. Syllabus of propositions relating to Pairs of Lines.
Let us begin with the simplest possible case, a Pair of infinite Lines which have two common points, and which therefore coincide wholly, and let us consider how such a Pair may be defined, and what other properties it possesses.
After that we will take a Pair of Lines which have a common point and a separate point ('a separate point' being one that lies on one of the Lines but not on the other), and which therefore have no other common point, and treat it in the same way.
And in the third place we will take a Pair of Lines which have no common point.
And let us understand, by 'the distance between two points,' the length of the right Line joining them; and, by 'the distance of a point from a Line,' the length of the perpendicular drawn, from the point, to the Line.
Now the properties of a Pair of Lines may be ranged under four headings:—
- (1) as to common or separate points;
- (2) as to the angles made with transversals;
- (3) as to the equidistance, or otherwise, of points on the one from the other;
- (4) as to direction.
By 'coincidental Lines,' then, I shall mean Lines which either coincide or would do so if produced: and by 'intersectional Lines' I shall mean Lines which either intersect or would do so if produced; and, by 'separational Lines,' Lines which have no common point, however far produced.
In the same way, when I speak of 'Lines having a common point,' or of 'Lines having two common points,' I shall mean Lines which either have such points or would have them if produced.
It will also save time and trouble to agree on the use of a certain conventional phrase respecting transversals.
It admits of easy proof that, if a Pair of Lines make, with a certain transversal, either (a) a pair of alternate angles equal, or (b) an exterior angle equal to the interior opposite angle on the same side of the transversal, or (c) a pair of interior angles on the same side of the transversal supplementary; they will make, with the same transversal, (d) each pair of alternate angles equal, and (e) every exterior angle equal to the interior opposite angle on the same side of the transversal, and (f) each pair of interior angles on the same side of the transversal supplementary.
You will accept that as a simple Theorem, though with a somewhat lengthy enunciation?
Min. Certainly.
Euc. The phrase I propose is as follows. When I speak of a Pair of Lines as 'equally inclined to' a transversal, I wish it to be understood that they fulfil some one of the three conditions (a), (b), (c), and therefore all the three conditions (d), (e), (f).Min. A most convenient abridgment.
Euc. Similarly, it admits of easy proof that, if a Pair of Lines make, with a certain transversal, either (a) a pair of alternate angles unequal, or (b) an exterior angle unequal to the interior opposite angle on the same side of the transversal, or (c) a pair of interior angles on the same side of the transversal not supplementary; they will make, with the same transversal, (d) each pair of alternate angles unequal, and (e) every exterior angle unequal to the interior opposite angle on the same side of the transversal, and (f) each pair of interior angles on the same side of the transversal not supplementary.
And when I speak of a Pair of Lines as 'unequally inclined to' a transversal, I wish it to be understood that they fulfil some one of the three conditions (a), (b), (c), and therefore all the three conditions (d), (e), (f).
Min. Very well.
Euc. Now the Propositions relating to Pairs of Lines may be divided into two classes, the first covering the ground occupied by my Axiom 10 ('two straight Lines cannot enclose a space') and my Propositions I. 16, 17, 27, 28, 31; the second that occupied by my Axiom 12 and Propositions I. 29, 30, 32. Those in the first class are logical deductions from Axioms which have never been disputed: the second class has furnished, through all ages, a battle-field for rival mathematicians. That some one of the Propositions in this class must be assumed as an Axiom is agreed on all hands, and each combatant in turn proclaims his own special favourite to be the one axiomatic truth of the series, insisting that all the rest ought to be proved as Theorems.Let us now consider the properties of Pairs of Lines.
Such pairs may be arranged in three distinct classes. I will take them separately, and enumerate, for each class, first the 'subjects,' and secondly the 'predicates,' of Propositions concerning it.
Min. Let us make sure that we understand each other as to those two words. I presume that a 'subject' will include just so much 'property' as is needed to indicate the Pair of Lines referred to, i.e. to serve as a sufficient Definition for them?
Euc. Exactly so. Now, if we are told that a certain Pair of Lines fulfil some one of the following conditions:—
- (1) they have two common points;
- or (2) they have a common point, and are equally inclined to a certain transversal;
- or (3) they have a common point, and one of them has two points on the same side of, and equidistant from, the other;
- or (4) they have a common point and identical directions;
we may conclude that they fulfil all the following conditions:—
- (1) they are coincidental;
- (2) they are equally inclined to any transversal;
- (3) they are 'equidistantial,' i. e. any two points on one are equidistant from the other;
- (4) they have identical directions.
Min. You mean, by 'conclude,' that we may prove our conclusion?
Euc. Yes, wherever proof is needed. Conclusions (1) and (4) need none, and are usually stated as Axioms.
Min. In subject (4), instead of 'identical directions,' why not say 'the same direction'?
Euc. Because I want to keep clearly in view that there are two Lines.
Min. In predicate (2), you speak of 'any transversal': a little while ago, you spoke of 'every exterior angle.' Do you make any distinction between 'any' and 'every'?
Euc. Where the things spoken of are limited in number, I use 'every'; where infinite, I use 'any' in order to bring the idea within the grasp of our finite intellects. For instance, you may talk of 'every grain of sand in the world': there are, no doubt, what country-folk would call 'a good few' of them, but still the number is limited, and the mind can just grasp the idea. But if you tell me that 'every cubic inch of Space contains eight cubic half-inches,' my mind is unable to form a distinct conception of the subject of your Proposition: you would convey the same truth, and in a form I could grasp, by saying 'any cubic inch.'
Min. The angles made with the transversal are a little bewildering when the Pair of Lines shrinks, as it does in this case, into one Line. For instance, what becomes of the pair of interior angles on the same side of the transversal?
Euc. A diagram will make it clear. 
By examining the second figure (in which, as you see, there are three points with double names) we find that the alternate angles AGF, EHD, have become vertical angles; that the exterior and interior opposite angles EGB, EHD, have become the same angle; and that the two interior angles BGF, DHE, have become adjacent angles.
Min. That is quite clear.
Euc. Let us do on to the second class of Pairs of Lines.
If we are told that a certain Pair of Lines fulfil some one of the following conditions:—
- (1) they have a common point and a separate point;
- or (2) they have a common point, and are unequally inclined to a certain transversal;
- or (3) they have a common point, and one of them has two points not-equidistant from the other;
- or (4) they have a common point and different directions;
we may conclude that they fulfil all the following conditions:—
- (1) they are separational;
- (2) they are unequally inclined to any transversal;
- (3) any two points on one, which are on the same side of the other, are not equidistant from it;
- (4) a point may be found on each, whose distance from the other shall exceed any assigned length;
- (5) they have different directions.
And thirdly, if we are told that a certain Pair of Lines fulfil some one of the following conditions:—
- (1) they have a separate point, and are equally inclined to a certain transversal;
- or (2) they have a separate point, and one of them has two points on the same side of, and equidistant from, the other;
we may conclude that they are separational.
Min. Why not use your own word 'parallel'?
Euc. Because that word is not uniformly employed, by modern writers, in one and the same sense. I would advise you, in discussing the works of my Modern Rivals, to disallow the use of the word 'parallel' altogether, and to oblige each writer to adopt a word which shall express his own definition.
Min. When you speak of two points on one Line, which are on the same side of the other, being 'equidistant from it,' do you include the case of their lying on the other Line?
Euc. Certainly. You may take them as lying on either side you like, and at zero-distances. The only case excluded is, where both points are outside the other Line, and on opposite sides of it.
Min. I understand you.Euc. We shall find the Table of Propositions, which I now lay before you, very convenient to refer to. I have placed contranominal Propositions (i.e. Propositions of the form 'All X is Y,' 'All not-Y is not-X') in the same section.
| Table I. |
| Containing twenty Propositions, of which some are undisputed Axioms, and the rest real and valid Theorems, deducible from undisputed Axioms. |
| [N.B. Those marked ∗ have been proposed as Axioms.] |
| ∗1. A Pair of Lines, which have two common points, are coincidental.
or ∗. Two Lines cannot enclose a space. [Euc. Ax.] |
|
2. (a) A Pair of Lines, which have a separate point, have not two common points. (b) A Pair of Lines, which have a common point and a separate point, are intersectional. |
| 3. If there be given a Line and a point, it is possible to draw a Line, through the given point, intersectional with the given Line. |
| 4. A Pair of intersectional Lines are unequally inclined to any transversal.
Cor. 1. In either pair of alternate angles, that, which is on the side, of the transversal, remote from the point of intersection, is the greater. [I. 16.] Cor. 2. Every exterior angle, which is on the side, of the transversal, next to the point of intersection, is greater than the interior opposite angle on the same side.[I. 16.] Cor. 3. The pair of interior angles, which are on the side, of the transversal, next to the point of intersection, are together less than two right angles.[I. 17.] |
| 5. A Pair of Lines, which have a common point and are equally inclined to a certain transversal, are coincidental. |
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6. A Pair of Lines, which have a separate point and are equally inclined to a certain transversal, are separational.[I. 27, 28.] |
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7. If there be given a Line and a point without it, it is possible to draw a Line, through the given point, separational from the given Line.[I. 31.] |
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8. A Pair of intersectional Lines are such that any two points on one, which are on the same side of the other, are not equidistant from it. Cor. That which is the more remote from the point of intersection has the greater distance. |
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9. A Pair of Lines, which have a common point and of which one has two points on the same side of and equidistant from the other, are coincidental. |
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10. A Pair of Lines, which have a separate point and of which one has two points on the same side of and equidistant from the other, are separational. |
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11. Each of a Pair of intersectional Lines has, in each portion of it, a point whose distance from the other exceeds any given length. or A Pair of intersectional Lines diverge without limit. |
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12. A Pair of Lines, which have two common points, have identical directions. |
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∗13. (a) A Pair of Lines, which have different directions, have not two common points. (b) A Pair of Lines, which have a common point and different directions, are intersectional. |
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∗14. A Pair of intersectional Lines have different directions. |
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∗15. A Pair of Lines, which have a common point and identical directions, are coincidental. |
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∗16. If there be given a Line and a point without it: it is possible to draw a Line, through the given point, having a direction different from that of the given Line. |
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17. A Line, which has a point in common with one of two coincidental Lines has a point in common with the other also. |
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18. A Line, which has a point in common with one of two separational Lines, has a point separate from the other. |
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∗19. A Line, which has a point in common with one of two separational Lines and also a point in common with the other, is intersectional with both. |
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∗20. If there be three Lines; the first a right Line; the second, not assumed to be right, having a point separate from the first and being equidistantial from it; the third a right Line intersecting the first and diverging from it without limit on the side next to the second: the third is intersectional with the second. |
Min. I see that 2 (a) is the contranominal of 1. But where does 2 (b) come from?
Euc. It is got from 2 (a) by adding, to each term, the property 'having a common point'—just as if we were to deduce, from 'all men are mortal,' 'all fat men are fat mortals.'
Min. You mean 5 to be contranominal to 4, I suppose. But 'coincidental' is not equivalent to 'non-intersectional.'
Euc. True: but I have added a new condition, viz. 'which have a common point,' to the subject. Non-intersectional Lines, which have a common point, are coincidental, just as, in the next Proposition, non-intersectional Lines, which have a separate point, are separational. Min. 20 is rather a difficult enunciation to grasp.
Euc. A diagram will make it clear. As a matter of fact. No. 2 would be a right Line: but, as we have no right, at present, to assume this, I have drawn it as a wavy line.
Min. I can suggest two Contranominals which you have omitted: one, deducible from 13 (b), 'Two Lines, which are not intersectional and which have different directions, have no common point, i.e. are separational'; the other, deducible from 15, 'Two Lines, which have a separate point and identical directions, have no common point, i.e. are separational.'
Euc. They are valid deductions, but in neither case do we know the 'subject' to be real.
Min. The 'contranominality'—if such a fearful word be allowable—of 17, 18, 19, seems obscure.
Euc. I will do what I can to make it less so.
Let us name the three Lines 'A, B, C.'
Then 17 may be read 'A Line (C), which has a point in common with one (A) of two coincidental Lines (A, B), has a point in common with the other (B) also.'
From this we may deduce two Contranominals.
The first is 'If A, C, have a common point; and B, C, are separational: A, B have a separate point.' That is, 'a Line (A), which has a point in common with one (C) of two separational Lines (B, C), has a point separate from the other (B)': and thus we get 18.
The other Contranominal is 'If A, C, have a common point; and A, B, have a common point; and B, C, are separational: A, B, are intersectional.' That is, 'A Line (A), which has a point in common with one (C) of two separational Lines (B, C), and also a point in common with the other (B), is intersectional with that other (B).'
But we may evidently interchange B and C without interfering with the argument, and thus prove that A is also intersectional with C. Hence A is intersectional with both: and thus we get 19.
Min. That is quite clear.
Euc. We will now go a little further into the subject of separational Lines, as to which Table I. has furnished us with only three Propositions. There are, however, many other Propositions concerning them, which are fully admitted to be true, though no one of them has yet been proved from undisputed Axioms: and we shall find that they are so related to one another that, if any one be granted as an Axiom, all the rest may be proved; but, unless some one be so granted, none can be proved. Two thousand years of controversy have not yet settled the knotty question which of them, if any, can be taken as axiomatic.
If we are told that a certain Pair of Lines fulfil some one of the following conditions:—
- (1) they are separational;
- (2) they have a separate point and are equally inclined to a certain transversal;
- (3) they have a separate point, and one of them has two points on the same side of and equidistant from the other;
we may prove (though not without the help of some disputed Axiom) that they fulfil both the following conditions:—
- (1) they are equally inclined to any transversal;
- (2) they are equidistantial from each other.
These Propositions, with the addition of my own I. 30, I. 32, and certain others, I will now arrange in a tabular form, placing Contranominals in the same section.
| Table II. |
| Containing eighteen Propositions, of which no one is an undisputed Axiom, but all are real and valid Theorems, which, though not deducible from undisputed Axioms, are such that, if any one be admitted as an Axiom, the rest can he proved. |
| [N. B. Those marked ∗ have been, or parts of them have been, proposed as Axioms. |
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1. A Pair of separational Lines are equally inclined to any transversal. [I. 29.] |
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∗2. A Pair of Lines, which are unequally inclined to a certain transversal, are intersectional. [Euc. Ax.] |
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3. Through a given Point, without a given Line, a Line may be drawn such that the two Lines are equally inclined to any transversal. |
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4. A Pair of Lines, which are equally inclined to a certain transversal, are so to any transversal. |
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5. A Pair of Lines, which are unequally inclined to a certain transversal, are so to any transversal. |
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6. A Pair of separational Lines are equidistantial from each other. |
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∗7. A Pair of Lines, of which one has two points on the same side of, and not equidistant from, the other, are intersectional. |
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∗8. Through a given point, without a given Line, a Line may be drawn such that the two Lines are equidistantial from each other. |
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9. A Pair of Lines, of which one has two points on the same side of, and equidistant from, the other, are equally inclined to any transversal. |
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10. A Pair of Lines, which are unequally inclined to a certain transversal, are such that any two points on one, which are on the same side of the other, are not equidistant from it. |
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11. A Pair of Lines, which are equally inclined to a certain transversal, are equidistantial from each other. |
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12. A Pair of Lines, of which one has two points on the same side of, and not equidistant from, the other, are unequally inclined to any transversal. |
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13. A Pair of Lines, of which one has two points on the same side of, and equidistant from, the other, are equidistantial from each other. |
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14. A Pair of Lines, of which one has two points on the same side of, and not equidistant from, the other, are such that any two points on one, which are on the same side of the other, are not equidistant from it. |
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15. (a) A Pair of Lines, which are separational from a third Line, are not intersectional with each other. (b) A Pair of Lines, which have a common point and are separational from a third Line, are coincidental with each other. or, If there be given a Line and a point without it, only one Line can be drawn, through the given point, separational from the given Line. (c) A Pair of Lines, which have a separate point and are separational from a third Line, are separational from each other. [I. 30.] |
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∗16. (a) A Pair of intersectional Lines cannot both be separational from the same Line. (b) A Line, which is intersectional with one of two separational Lines, is intersectional with the other also. |
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∗17. A Line cannot recede from and then approach another; nor can one approach and then recede from another while on the same side of it. |
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18. (a) If a side of a Triangle be produced, the exterior angle is equal to each of the interior opposite angles. [I. 32.] (b) The angles of a Triangle are together equal to two right angles. [I. 32.] |
You will find it convenient to have the Propositions, that have been proposed as Axioms, repeated in a Table by themselves.
| Table III. |
| Containing five Propositions, taken from Table II, which have been proposed as Axioms. |
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Euclid's Axiom. A Pair of Lines, which have a separate point and make, with a certain transversal, two interior angles on one side of it together less than two right angles, are intersectional on that side. [This is one case of II. 2, with an additional statement as to the side of the transversal on which the Lines will meet.] |
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T. Simpson's Axiom. A Pair of Lines, which have a separate point and of which one has two points on the same side of, and not equidistant from, the other, are intersectional. [This is II. 7.] |
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Clavius' Axiom. Through a given Point, without a given Line, a Line may be drawn equidistantial from the given Line. [This is part of II, 8.] |
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Playfair's Axiom. A pair of intersectional Lines cannot both be separational from the same Line. [This is II. 16 (a).] |
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R. Simpson's Axiom. A Line cannot recede from and then approach another: nor can one approach and then recede from another while on the same side of it. [This is II. 17.] |
Min. In the predicate of 2, what right have you to say 'are intersectional'? The true contradictory of 'separational' would be 'have a common point.'
Euc. True: but we may assume as an Axiom 'A Pair of coincidental Lines are equally inclined to any transversal.' This, combined with 1, gives 'A Pair of not-intersectional Lines are equally inclined to any transversal,' whose Contranominal is 2.
Similarly, we may combine, with 6, the Axiom 'A Pair of coincidental Lines are equidistantial from each other,' and thus get a Theorem whose Contranominal is 7.
Min. In classing 15 (a), (b), and (c) under one number, you mean, I suppose, that they are so related that, if any one of them be granted, the others may be deduced ?
Euc. Certainly.
Min. I see that if (a) be given, (b) may be deduced by simply adding 'having a common point' to subject and predicate. And I see that (b) and (c) are Contranominals, so that, if either be given, the other follows. But I don't see how, if (b) only were given, you would prove (a).
Euc. You can prove (c) from it, as you say: and then, from (b) and (c) combined, you can prove (a) thus:—
Any Pair of Lines, which are separational from a third Line, must belong to one or both of the two classes, 'having a common point,' 'having a separate point.' Hence if we take these two classes together, we include any Pair that can be proposed. Thus we get the Theorem 'Any Pair of Lines, which are separational from a third Line, are either coincidental or separational'; the predicate of which is equivalent to 'are not intersectional.'
Min. I see. And how are 16 (c) and 16 (b) related to 15?
Euc. Each of them is a Contranominal of 15 (a); and they are also contranominal to each other.
Min. I should like to see that drawn out.
Euc. Let 'A, B, C' be three Lines. Then 16 (a) may be written 'A Pair of Lines (A, B), which are separational from a third Line (C), are not intersectional with each other.'
This yields three Contranominals. The first is 'If A, B, are intersectional; it cannot be true that B, C, are separational, and also A, C.' i.e. 'A Pair of intersectional Lines (A, B) cannot both be separational from a third Line (C)': the second is 'If B, C are separational, and A, B intersectional; then A, C are not separational.' i.e. 'A Line (A) which is intersectional with one (B) of two separational Lines (B, C), is not separational from the other (C)': and the third proves a similar Theorem for B.
Min. Yes, but your conclusion now is 'A is not separational from C': whereas 16 (b) says 'is intersectional.'
Euc. That is so: but since A is intersectional with a Line (B) which is separational from C, it is axiomatic that it has a point separate from C, and so cannot be coincidental with it. Hence, its being 'not separational from C' proves that it must be intersectional with it.Min. I suppose I must take it on trust that any one ofthese 18 is sufficient logical basis for the other 17: I can hardly ask you to go through 306 demonstrations!
Euc. I can do it with 11. You will grant me that, when two Propositions are contranominal, so that each can be proved from the other, I may select either of the two for my series of proofs, but need not include both?
Min. Certainly.
Euc. Here are the proofs, which you can read afterwards at your leisure. (See Appendix III.)
§ 5. Playfair's Axiom.
Euc. The next general question to be discussed is the proposed substitution of Playfair's Axiom for mine. With regard to mine, I am quite ready to admit that it is not axiomatic until Prop. 17 has been proved. What is an Axiom at one stage of our knowledge is often anything but an Axiom at an earlier stage.
Min. The great question is whether it is axiomatic then.
Euc. I am quite aware of that: and it is because this is not only the great question of the whole First Book, but also the crucial test by which my method, as compared with those of my 'Modern Rivals,' must stand or fall, that I entreat your patience in speaking of a matter which cannot possibly be dismissed in a few words.
Min. Pray speak at whatever length you think necessary to so vital a point.Euc. Let me remark in the first place—it is a minor matter, but yet one that must come in somewhere, and I do not want to break the thread of my argument—that we need, in any complete geometrical treatise, some practical geometrical test by which we can prove that two given finite Lines will meet if produced. My Axiom serves this purpose—a secondary purpose it is true—but it is incumbent on any one, who proposes to do away with it, to provide some sufficient substitute.
Min. I admit all that.
Euc. Now, if the test I propose—that the two Lines make with a certain transversal two interior angles on the same side of it together less than two right angles—be objected to as not sufficiently simple, the question arises, what simpler test can be proposed?
Min. The supporters of Playfair's Axiom would of course reply 'that one of the two Lines should cut a Line known to be parallel to the other.'
Euc. Assuming that what is needed is a distinct conception of the geometrical relationship of the two Lines, whose future meeting we are asked to believe in, which picture, think you, is the more likely to yield us such a conception—two finite Lines, both intersected by a transversal, and having a known angular relation to that transversal and so to each other—or two Lines 'known to be parallel,' that is two Lines of whose geometrical relationship, so far as our field of vision extends, we know absolutely nothing, but can only say that, in the far-away region of infinity, they do not meet?
Min. In clearness of conception, your picture seems to have the advantage. In fact, I could not form any mental picture at all of the relative position of two finite Lines, if all I knew about them was their never meeting however far produced: and it would be equally impossible to form any mental picture of the position which a Line, crossing one of them, would have relatively to the other. But, though your picture may be more easy to conceive, I doubt if it is enough so to constitute an axiom.
Euc. Taken by itself, it may be, as you say, not entirely axiomatic. But I think I can put before you a few considerations which will make it more acceptable.
Min. They will be well worth having. An absolute proof of it, from first principles, would be received, I can assure you, with absolute rapture, being an ignis fatuus that mathematicians have been chasing from your age down to our own.
Euc. I know it. But I cannot help you. Some mysterious flaw lies at the root of the subject. Probabilities are all I have to offer you.
Now suppose you were assured, with regard to two finite Lines placed before you, that, when produced in a certain direction, one of them approached the other, that is, contained two points, of which the second was nearer, to the other Line, than the first, would you not think it probable—if not absolutely certain—that they would meet at last?
Min. Utilising—as I suppose you will allow me to do—my knowledge of the properties of asymptotes, I should say 'No. The mere fact of approach, granted as to two Lines, does not secure a future meeting.'Euc. But, if you look into the depths of your own consciousness—assuming such depths to exist—you will find, I believe, an eternal distinction maintained, in this respect, between straight and curved Lines: so that Lines of the one kind must, if they approach, ultimately meet, whereas those of the other kind need not.
Min. I will grant it, provisionally, if only to know what you are going to deduce from it.
Euc. I will now ask you to consider this diagram.
Suppose it given that the Lines BD, CE, make with BC two angles together less than two right angles. My object is to show that probably—if not certainly—they will meet, if produced towards D, E.
Let BF be so drawn that the angles FBC, BCE, may be together equal to two right angles.
Now, if any point in BD be nearer to CE than B is, what is required is proved, since BD approaches CE.
But, if this be not so, then F (which is obviously further from CE than some point in BD is) must also be further from CE than B is; i.e. FB must approach EC; i.e. FB and EC must ultimately meet, below BC, and so form a Triangle, whose angles at B and C will be (by my Prop. 17) less than two right angles. Hence the angles FBC, BCE, must be greater than two right angles, since the four angles are (by my Prop. 13) equal to four right angles. But this is absurd, since they were made equal to two right angles.
Hence D is nearer to CE than B is; i.e. BD approaches CE, and so will meet it if produced.
Min. You certainly have made your Axiom a little more axiomatic. It is, I presume, an afterthought of yours: otherwise you would have made your Axiom deal with approaching Lines, and would then have proved your present Axiom as a Theorem.
Euc. Excuse me. Whatever the habits of modern geometricians may be, in our day we always investigated a subject down to the very roots. No 'afterthought' was possible. You of the nineteenth century may 'look before and after,' if it so please you, so long as we have liberty to look at what is at our feet: you may 'sigh for what is not,' and welcome, so long as we may chuckle at what is!
Min. Flippancy will not serve your turn. If you have no better reason than that—
Euc. I have a better reason. How could I have dealt with approaching Lines without first strictly defining 'the distance of a point from a Line'?
Min. Nohow, I grant you.
Euc. Which would have entailed a definition of 'the distance of a point from a point,' i.e. the length of the shortest path by which the one can pass to the other—which again would have entailed the comparison of all possible paths—which again would have entailed the estimation of the lengths of curved Lines—which again—
Min. This is uncanny! It is whichcraft!
Euc. (preserves a disgusted silence).Min. I beg your pardon. I grant that you have made out a very good case for your own Axiom, and but a bad one for Playfair's.
Euc. I will make it worse yet, before I have done. My next remark will be best explained with the help of a diagram.
Let AB and CD make, with EF, the two interior angles BEF, EFD, together less than two right angles. Now if through E we draw the Line GH such that the angles HEF, EFD may be equal to two right angles, it is easy to show (by Prop. 28) that GH and CD are 'separational.'
Min. Certainly.
Euc. We see, then, that any Lines which have the property (let us call it 'α') of making, with a certain transversal, two interior angles together less than two right angles, have also the property (let us call it 'β') that one of them intersects a Line which is separational from the other.
Min. I grant it.
Euc. Now suppose you decline to grant my 12th Axiom, but are ready to grant Playfair's Axiom, that two intersectional Lines cannot both be separational from the same Line: then you have in fact granted my Axiom.
Min. Be good enough to prove that.
Euc. Lines, which have property 'α,' have property 'β.' Lines, which have property 'β,' meet if produced; for, if not, there would be two Lines both separational from the same Line, which is absurd. Hence Lines, which have property 'α,' meet if produced.
Min. I see now that those who grant Playfair's Axiom have no right to object to yours: and yours is certainly the more simple one.
Euc. To make assurance doubly sure, let me give you two additional reasons for preferring my Axiom.
In the first place, Playfair's Axiom (or rather the Contranominal of it which I have been using, that 'a Line which intersects one of two separational Lines will also meet the other') does not tell us which way we are to expect the Lines to meet. But this is a very important matter in constructing a diagram.
Min. We might obviate that objection by re-wording it thus:—'If a Line intersect one of two separational Lines, that portion of it which falls between them will, if produced, meet the other.'
Euc. We might: and therefore I lay little stress on that objection.
Euc. In the second place, Playfair's Axiom asserts more than mine does: and all the additional assertion is superfluous, and a needless strain on the faith of the learner.
Min. I do not see that in the least.
Euc. It is rather an obscure point, but I think I can make it clear. We know that all Pairs of Lines, which have property 'α,' have also property 'β'; but we do not know as yet (till we have proved I. 29) that all, which have property 'β,’ have also property 'α.'Min. That is so.
Euc. Then, for anything we know to the contrary, class 'β' may be larger than class 'α.' Hence, if you assert anything of class 'β,' the logical effect is more extensive than if you assert it of class 'α': for you assert it, not only of that portion of class 'β' whitch is known to be included in class 'α,' but also of the unknown (but possibly existing) portion which is not so included.
Min. I see that now, and consider it a real and very strong reason for preferring your axiom.
But so far you have only answered Playfair. What do you say to the objection raised by Mr. Potts? 'A stronger objection appears to be that the converse of it forms Euc. I. 17; for both the assumed Axiom and its converse should be so obvious as not to require formal demonstration.'
Euc. Why, I say that I deny the general law which he lays down. (It is, of course, the technical converse that he means, not the logical one. 'All X is Y' has for its technical converse 'All Y is X'; for its logical, 'Some Y is X.') Let him try his law on the Axiom 'All right angles are equal,' and its technical converse 'All equal angles are right'!
Min. I withdraw the objection.
§ 6. The Principle of Superposition.
Min. The next subject is the principle of 'superposition.' You use it twice only (in Props. 4 and 8) in the First Book: but the modern fancy is to use it on all possible occasions. The Syllabus indicates (to use the words of the Committee) 'the free use of this principle as desirable in many cases where Euclid prefers to keep it out of sight.'
Euc. Give me an instance of this modern method.
Min. It is proposed to prove I. 5 by taking up the isosceles Triangle, turning it over, and then laying it down again upon itself.
Euc. Surely that has too much of the Irish Bull about it, and reminds one a little too vividly of the man who walked down his own throat, to deserve a place in a strictly philosophical treatise?
Min. I suppose its defenders would say that it is conceived to leave a trace of itself behind, and that the reversed Triangle is laid down upon the trace so left.
Euc. That is, in fact, the same thing as conceiving that there are two coincident Triangles, and that one of them is taken up, turned over, and laid down upon the other. And what does their subsequent coincidence prove? Merely this: that the right-hand angle of the first is equal to the left-hand angle of the second, and vice versâ. To make the proof complete, it is necessary to point out that, owing to the original coincidence of the Triangles, this same 'left-hand angle of the second' is also equal to the left-hand angle of the first: and then, and not till then, we may conclude that the base-angles of the first Triangle are equal. This is the full argument, strictly drawn out. The Modern books on Geometry often attain their much-vaunted brevity by the dangerous process of omitting links in the chain; and some of the new proofs, which at first sight seem to be shorter than mine, are really longer when fully stated. In this particular case I think you will allow that I had good reason for not adopting the method of superposition?
Min. You had indeed.
Euc. Mind, I do not object to that proof, if appended to mine as an alternative. It will do very well for more advanced students. But, for beginners, I think it much clearer to have two non-isosceles Triangles to deal with.
Min. But your objection to laying a Triangle down upon itself does not apply to such a case as I. 24.
Euc. It does not. Let us discuss that case also. The Moderns would, I suppose, take up the Triangle ABC, and apply it to DEF so that AB should coincide with DE?
Min. Yes.
Euc. Well, that would oblige you to say 'and join C, in its new position, to E and F.' The words 'in its new position' would be necessary, because you would now have two points in your diagram, both called 'C.' And you would also be obliged to give the points B and E additional names, namely 'A' and 'B.' All which would be very confusing for a beginner. You will allow, I think, that I was right here in constructing a new Triangle instead of transferring the old one?
Min. Cuthbertson evades that difficulty by re-naming the point C, and calling it 'Q.'
Euc. And do the points A and B take their names with them?Min. No. They adopt the names 'D' and 'E.'
Euc. It is very like making a new Triangle!
Min. It is indeed. I think you have quite disposed of the claims of 'superposition.' The only remaining subject for discussion is the omission of the diagonals in Book II.
§ 7. The omission of diagonals in Euc. II..
Euc. Let us test it on my II. 4. We will go through my proof of it, and then the proof given by some writer who ignores the diagonal, supplying if necessary any of those gaps in argument which my Modern Rivals so often indulge in, and which give to their proofs a delusive air of neatness and brevity.
'If a Line be divided into any two parts, the square of the Line is equal to the squares of the two parts with twice their rectangle.
Let AB be divided at C. It is to be proved that square of AB is equal to squares of AC, CB, with twice rectangle of AC, CB.
On AB describe Square ADEB; join BD; from C draw CF parallel to AD or BE, cutting BD at G; and through G draw HK parallel to AB or BE.
∵ BD cuts Parallels AD, CF,
∴ exterior angle CGB = interior opposite angle ADB. [I. 29
also ∵ AD = AB,
∴ angle ADB = angle ABD; [I. 5
∴ angle CGB = angle ABD;
∴ CG = CB; [I. 6
but BK = CG, and GK = CB; [I. 34
∴ CK is equilateral.
also, ∵ angle CBK is right,
∵ CK is rectangular; [I. 46. Cor.
∴ CK is a Square.
Similarly HF is a Square and = square of AC, for HG = AC. [I. 34
Also, ∵ AG, GE are equal, being complements, [I. 43
∴ AG and GE = twice AG;
= twice rectangle of AC, CB.
But these four figures make up AE.
Therefore the square of AB &c. Q. E. D.'
That is just 128 words, counting from 'On AB describe' down to the words 'rectangle of AC, CB.' What author shall we turn to for a rival proof?
Min. I think Wilson will be best.
Euc. Very well. Do the best you can for him. You may use all my references if you like, and if you can do so legitimately.
Min. 'Describe Square ADEB on AB. Through C draw CF parallel to AB, meeting——'Euc. You must insert 'or BE,' to make the comparison fair.
Min. Certainly. I will mark the necessary insertions by parentheses. 'Through C draw CF parallel to AD (or BE), meeting DE in F.'
Euc. You may omit those four words, as they do not occur in my proof.
Min. Very well. 'Cut off (from CF) CG = CB. Through G draw HK parallel to AB (or DE). It is easily shewn that CK, HF are squares of CB, AC; and that AG, GE, are each of them rectangle of AC, CB.'
Euc. We can't admit 'it is easily shewn'! He is bound to give the proof.
Min. I will do it for him as briefly as I can. '∵ CG = CB, and BK = CG, and GK = CB, ∴ CK is equilateral. It is also rectangular, since angle CBK is right. ∴ CK is a Square.' I'm afraid I mustn't say 'Similarly HF is a Square'?
Euc. Certainly not: it requires a different proof.
Min. 'Because CE = AD = AB, and CG, CB, parts of them, are equal, ∴ remainder GF = remainder AC, = HG. But HD = GF, and DF = HG; ∴ HF is equilateral. It is also rectangular, since angle HDF is right. ∴ HF is a Square, and = square of AC. Also AG is rectangle of AC, CB.' I fear I can't assume GE to be equal to AG?
Euc. I fear I cannot permit you to assume the truth of my I. 43.
Min. 'Also GE is rectangle of AC, CB, since GF = AC, and GK = CB. ∴ AG and GE = twice rectangle of AC, CB.'Euc. That will do. How many words do you make it?
Min. 145.
Euc. Then the omission of the diagonal, instead of shortening the proof, has really lengthened it by seventeen words! Well! Has it any advantage in the way of neatness to atone for its greater length?
Min. Certainly not. It is quite unsymmetrical. I very much prefer your method of appealing to the beautiful Theorem of the equality of complements.
Euc. Then that concludes our present interview: we will meet again when you have reviewed my Modern Rivals one by one. If you had any slow music handy, I would vanish to it: as it is
(Vanishes without slow music.)