Euclid and His Modern Rivals/Act II. Scene VI. § 1.
ACT II.
Scene VI.
Treatment of Parallels by direction.
§ 1. Wilson.
'There is moreover a logic besides that of mere reasoning.'
Wilson, Pref. p. xiii.
Nie. You have made but short work of four of the five methods of treating Parallels.
Min. We shall have all the more time to give to the somewhat intricate subject of Direction.
Nie. I lay on the table 'Elementary Geometry,' by J. M. Wilson, M.A., late Fellow of St. John's College, Cambridge, late Mathematical Master of Rugby School, now Head Master of Clifton College. The second edition, 1869. And I warn you to be careful how you criticise it, as it is already adopted in several schools.
Min. Tant pis pour les écoles. So you and your client deliberately propose to supersede Euclid as a text-book?Nie. 'I am of opinion that the time is come for making an effort to supplant Euclid in our schools and universities.' (Pref. p. xiv.)
Min. It will be necessary, considering how great a change you are advocating, to examine your book very minutely and critically.
Nie. With all my heart. I hope you will show, in your review, 'the spirit without the prejudices of a geometrician.' (Pref. p. xv.)
Min. We will begin with the Right Line. And first, let me ask, how do you define it?
Nie. As 'a Line which has the same direction at all parts of its length.' (p. 3)
Min. You do not, I think, make any practical use of that as a test, any more than Euclid does of the property of lying evenly as to points on it?
Nie. No, we do not.
Min. You construct and test it as in Euclid, I believe? And you have his Axiom that 'two straight Lines cannot enclose a space?'
Nie. Yes, but we extend it. Euclid asserts, in effect, that two Lines, which coincide in two points, coincide between those points: we say they 'coincide wholly,' which includes coincidence beyond those points.
Min. Euclid tacitly assumes that.
Nie. Yes, but he has not expressed it.
Min. I think the addition a good one. Have you any other Axioms about it?
Nie. Yes, 'that a straight Line marks the shortest distance between any two of its points.' (p. 5. Ax. 1.)Min. That I have already fully discussed in reviewing M. Legendre's book (see p. 55).
Nie. We have also 'A Line may be conceived as transferred from any position to any other position, its magnitude being unaltered.' (p. 5. Ax. 3.)
Min. True of any geometrical magnitude: but hardly worth stating, I think. I have now to ask you how you define an Angle?
Nie. 'Two straight Lines that meet one another form an angle at the point where they meet.' (p. 5.)
Min. Do you mean that they form it 'at the point' and nowhere else?
Nie. I suppose so.
Min. I fear you allow your angle no magnitude, if you limit its existence to so small a locality!
Nie. Well, we don't mean 'nowhere else.'
Min. (meditatively) You mean 'at the point—and somewhere else.' Where else, if you please?
Nie. We mean—we don't quite know why we put in the words at all. Let us say 'Two straight Lines that meet one another form an angle.'
Min. Very well. It hardly tells us what an angle is, and, so far, it is inferior to Euclid's Definition: but it may pass. Do you put any limit to the size of an angle?
Nie. We have not named any, but the largest here treated of is what we call 'one revolution.'
Min. You admit reëntrant angles then?
Nie. Yes.
Min. Then your Definition only states half the truth: you should have said 'form two angles.'Nie. That would be true, no doubt.
Min. But this extension of limit will require several modifications in Euclid's language: for instance, what is your Definition of an obtuse angle?
Niemand reads.
P. 8. Def. 13. 'An obtuse angle is one which is greater than a right angle.'
Min. So you tumble headlong into the very first pitfall you come across! Why, that includes such angles as 180° and 360°. You would teach your pupil, I suppose, that one portion of a straight Line makes an obtuse angle with the other, and that every straight Line has an obtuse angle at each end of it!
Nie. It is an oversight—of course we ought to have added 'but less than two right angles.'
Min. A very palpable oversight. I fear we shall find more as we go on. What Axioms have you about angles?
Niemand reads.
P. 5. Ax. 4. 'An angle may be conceived as transferred to any other position, its magnitude being unaltered.'
Min. Hardly worth stating. Proceed.
Niemand reads.
P. 5. Ax. 5. 'Angles are equal when they could be placed on one another so that their vertices would coincide in position, and their arms in direction.'
Min. 'Placed on one another'! Did you ever see the child's game, where a pile of four hands is made on the table, and each player tries to have a hand at the top of the pile?
Nie. I know the game.
Min. Well, did you ever see both players succeed at once?
Nie. No.
Min. Whenever that feat is achieved, you may then expect to be able to place two angles 'on one another'! You have hardly, I think, grasped the physical fact that, when one of two things is on the other, the second is underneath the first. But perhaps I am hypercritical. Let us try an example of your Axiom: let us place an angle of 90° on one of 270°. I think I could get the vertices and arms to coincide in the way you describe.
Nie. But the one angle would not be on the other; one would extend round one-fourth of the circle, and the other round the remaining three-fourths.
Min. Then, after all, the angle is a mysterious entity, which extends from one of the Lines to the other? That is much the same as Euclid's Definition. Let us now take your definition of a Right Angle.
Nie. We first define 'one revolution,' which is the angle described by a Line revolving, about one extremity, round into its original position.
Min. That is clear enough.
Nie. We then say (p. 7. Def. 9) 'When it coincides with what was initially its continuation, it has described half a revolution, and the angle it has then described is called a straight angle.'Min. How do you know that it has described half a revolution?
Nie. Well, it is not difficult to prove. Let that portion of the Plane, through which it has revolved, be rolled over, using as an axis the arm (in its initial position) and its continuation, until it falls upon the other portion of the Plane. The two angular magnitudes will now together make up 'one revolution': therefore each is 'half a revolution.'
Min. A proof, I grant: but you are very sanguine if you expect beginners in the subject to supply it for themselves.
Nie. It is an omission, we admit.
Min. And then 'a straight angle'! 'Straight' is necessarily unbending: while 'angle' is from ἄγκος, 'a bend or hook': so that your phrase is exactly equivalent to 'an unbending bend'! In 'the Bairnslea Foaks' Almanack' I once read of 'a mad chap' who spent six weeks 'a-trying to maäk a straät hook': but he failed. He ought to have studied your book. Have you Euclid's Axiom 'all right angles are equal'?
Nie. We deduce it from 'all straight angles are equal': and that we prove by applying one straight angle to another.
Min. That is all very well, though I cannot think 'straight angles' a valuable contribution to the subject. I will now ask you to state your method of treating Pairs of Lines, as far as your proof of Euc. I. 32.
Nie. To do that we shall of course require parallel Lines: and, as our definition of them is 'Lines having the same direction,' we must begin by discussing direction.
Min. Undoubtedly. How do you define direction?
Nie. Well, we have not attempted that. The idea seemed to us to be too elementary for definition. But let me read you what we have said about it.
Reads.
P. 2. Def. 2. 'A geometrical Line has position, and length, and at every point of it it has direction…'
P. 3. Def. 4. ' A straight Line is a Line which has the same direction at all parts of its length. It has also the opposite direction… A straight Line may be conceived as generated by a point moving always in the same direction.'
I will next quote what we have said about two Lines having 'the same direction' and 'different directions.'
Min. We will take that presently: I have a good deal to say first as to what you have read. I gather that you consider direction to be a property of a geometrical entity, but not itself an entity?
Nie. Just so.
Min. And you ascribe this property to a Line, and also to the motion of a point?
Nie. We do.
Min. For simplicity's sake, we will omit all notice of curved Lines, etc., and will confine ourselves to straight Lines and rectilinear motion, so that in future, when I use the word 'Line,' I shall mean 'straight Line.' Now may we not give a notion of 'direction' by saying—that a moving point must move in a certain 'direction'—that, if two points, starting from a state of coincidence, move along two equal straight Lines which do not coincide (so that their movements are alike in point of departure, and in magnitude), that quality of each movement, which makes it differ from the other, is its 'direction'—and similarly that, if two equal straight Lines are terminated at the same point, but do not coincide, that quality of each which makes it differ from the other, is its 'direction' from the common point?
Nie. It is all very true: but you are using 'straight Line' to help you in defining 'direction.' We, on the contrary, consider 'direction' as the more elementary idea of the two, and use it in defining 'straight Line.' But we clearly agree as to the meanings of both expressions.
Min. I am satisfied with that admission. Now as to the phrase 'the same direction,' which you have used in reference to a single Line and the motion of a single point. May we not say that portions of the same Line have 'the same direction' as one another? And that, if a point moves along a Line without turning back, its motion at one instant is in 'the same direction' as its motion at another instant?
Nie. Yes. That expresses our meaning in other language.
Min. I have altered the language in order to bring out clearly the fact that, in using the phrase 'the same direction,' we are really contemplating two Lines, or two motions. We have now got (considering 'straight Line' as an understood phrase) accurate geometrical Definitions of at least two uses of the phrase. And to these we may add a third, viz. that two coincident Lines have 'the same direction.'
Nie. Certainly, for they are one and the same Line.
Min. And you intend, I suppose, to use the word 'different' as equivalent to 'not-same.'
Nie. Yes.
Min. So that if we have, for instance, two equal Lines terminated at the same point, but not coinciding, we say that they have 'different directions'?
Nie. Yes, with one exception. If they are portions of one and the same infinite Line, we say that they have 'opposite directions.' Remember that we said, of a Line, 'it has also the opposite direction.'
Min. You did so: but, since 'same' and 'different' are contradictory epithets, they must together comprise the whole genus of 'pairs of directions.' Under which heading will you put 'opposite directions'?
Nie. No doubt, strictly speaking, 'opposite directions' are a particular kind of 'different directions.' But we shall have endless confusion if we include them in that class. We wish to avoid the use of the word 'opposite' altogether, and to mean, by 'different directions,' all kinds of directions that are not the same, with the exception of 'opposite.'
Min. It is a most desirable arrangement: but you have not clearly stated it in your book. Tell me whether you agree in this statement of the matter. Every Line has a pair of directions, opposite to each other. And if two Lines be said to have 'the same direction,' we must understand 'the same pair of directions'; and if they be said to have 'different directions,' we must understand 'different pairs of directions.' And even this is not enough: for suppose I draw, on the map of England, a straight Line joining London and York; I may say 'This Line has a pair of directions, the first being "London-to-York" and the second "York-to-London."' I will now place another Line upon this, and its pair of directions shall be, first "York-to- London" and second "London-to-York." Then it has a different first-direction from the former Line, and also a different second-direction: that is, it has a 'different pair of directions.' Clearly this is not intended: but, in order to exclude such a possibility, we must extend yet further the meaning of the phrase, and, if two Lines be said to have 'the same direction,' we must understand 'pairs of directions which can be arranged so as to be the same'; and if they be said to have 'different directions,' we must understand 'pairs of directions which cannot be arranged so as to be the same.'
Nie. Yes, that expresses our meaning.
Min. You must admit, I think, that your theory of direction involves a good deal of obscurity at the very outset. However, we have cleared it up, and will not use the word 'opposite' again. Tell me now whether you accept this as a correct Definition of the phrases 'the same direction' and 'different directions,' when used of a Pair of infinite Lines which have a common point:—
If two infinite Lines, having a common point, coincide, they have 'the same direction'; if not, they have 'different directions.'Nie. We accept it.
Min. And, since a finite Line has the same direction as the infinite Line of which it is a portion, we may generalise thus:—'Coincidental Lines have the same direction. Non-coincidental Lines, which have a common point, have different directions.'
But it must be carefully borne in mind that we have as yet no geometrical meaning for these phrases, unless when applied to two Lines which have a common point.
Nie. Allow me to remark that what you call 'coincidental Lines' we call 'the same Line' or 'parts of the same Line,' and that what you call 'non-coincidental Lines' we call 'different Lines.'
Min. I understand you: but I cannot employ these terms, for two reasons: first, that your phrase 'the same Line' loses sight of a fact I wish to keep in view, that we are considering a Pair of Lines; secondly, that your phrase 'different Lines' might be used, with strict truth, of two different portions of the same infinite Line, so that it is not definite enough for my purpose.
Let us now proceed 'to consider the relations of two or more straight Lines in one Plane in respect of direction.'
And first let me ask which of the propositions of Table II you wish me to grant you as an axiom?
Nie. (proudly) Not one of them! We have got a new patent process, the 'direction' theory, which will dispense with them all.
Min. I am very curious to hear how you do it.
Niemand reads.
P. 11. Ax. 6. 'Two different Lines may have either the same or different directions.'
Min. That contains two assertions, which we will consider separately. First, you say that 'two different Lines (i.e. 'non-coincidental Lines,' or 'Lines having a separate point') 'may have the same direction'. Now let us understand each other quite clearly. We will take a fixed Line to begin with, and a certain point on it: there is no doubt that we can draw, through that point, a second Line coinciding with the first: the direction of this Line will of course be 'the same' as the direction of the first Line; and it is equally obvious that if we draw the second Line in any other direction, so as not to coincide with the first, its direction will not be 'the same' as that of the first: that is, they will have 'different' directions. If we want a geometrical definition of the assertion that this second Line has 'the same direction' as the first Line, we may take the following:—'having such a direction as will cause the Lines to be the same Line.' If we want a geometrical construction for it, we may say 'take any other point on the fixed Line; join the two points, and produce the Line, so drawn, at both ends': this construction we know will produce a Line which will be 'the same' as the first Line, and whose direction will therefore be 'the same' as that of the first Line. If, in a certain diagram, whose geometrical history we know, we want to test whether two Lines, passing through a common point, have, or have not, 'the same direction,' we have simply to take any other point on one of the Lines, and observe whether the other Line does, or does not, pass through it. This relationship of direction, which you call 'having the same direction,' and I 'having identical directions,' we may express by the word 'co-directional.'
Nie. All very true. My only puzzle is, why you have explained it at such enormous length: my meerschaum has gone out while I have been listening to you!
Min. Allow me to hand you a light. As to the 'enormous length' of my explanation, we are in troubled waters, my friend! There are breakers ahead, and we cannot 'heave the lead' too often.
Nie. It is 'lead' indeed!
Min. Let us now return to our fixed Line: and this time we will take a point not on it, and through this point we will draw a second Line. You say that we can, if we choose, draw it in 'the same direction' as that of the first Line?
Nie. We do.
Min. In that case let me remind you of the warning I gave you a few minutes ago, that we have no geometrical meaning for the phrase 'the same direction,' unless when used of Lines having a common point. What geometrical meaning do you attach to the phrase when used of other Lines?
Nie. (after a pause) I fear we cannot give you a geometrical definition of it at present.
Min. No? Can you construct such Lines?
Nie. No, but really that is not necessary. We allow of 'hypothetical constructions' now-a-days.Min. Well then, can you test whether a given Pair of Lines have this property? I mean, if I give you a certain diagram, and tell you its geometrical history, can you pronounce, on a certain Pair of finite Lines, which have no visible common point, as to whether they have this property?
Nie. We cannot undertake it.
Min. You ask me, then, to believe in the reality of a class of 'Pairs of Lines' possessing a property which you can neither define, nor construct, nor test?
Nie. We can do none of these things, we admit: but yet the class is not quite so indefinite as you think. We can give you a geometrical description of it.
Min. I shall be delighted to hear it.
Nie. We have agreed that a Pair of coincidental finite Lines have a certain relationship of direction, which we call 'the same direction,' and which you allow to be an intelligible geometrical relation?
Min. Certainly.
Nie. Well, all we assert of this new class is that their relationship of direction is identical with that which belongs to coincidental Lines.
Min. It cannot be identical in all respects, for it certainly differs in this, that we cannot reach the conception of it by the same route. I can form a conception of 'the same direction,' when the phrase is used of two Lines which have a common point, but it is only by considering that one 'falls on' the other—that they have all other points common—that they coincide. When you ask me to form a conception of this relationship of direction, when asserted of other Lines, you know that none of these considerations will help me, and you do not furnish me with any substitutes for them. To me the relationship does not seem to be identical: I should prefer saying that separational Lines have 'collateral,' or 'corresponding,' or 'separational' directions, to using the phrase 'the same direction' over again. It is, of course, true that 'collateral' directions produce the same results, as to angles made with a transversal, as 'identical' directions; but this seems to me to be a Theorem, not an Axiom.
Nie. You say that the relationship does not seem to you to be identical. I should like to know where you think you perceive any difference?
Min. I will try to make my meaning clearer by an illustration.
Suppose that I and several companions are walking along a railway, which will take us to a place we wish to visit. Some amuse themselves by walking on one of the rails; some on another; others wander along the line, crossing and recrossing. Now as we are all bound for the same place, we may say, roughly speaking, that we are all moving 'in the same direction': but that is speaking very roughly indeed. We make our language more exact, if we exclude the wanderers, and say that those who are walking along the rails are so moving. But it seems to me that our phrase becomes still more exact, if we limit it to those who are walking on one and the same rail.
As a second illustration, suppose two forces, acting on a certain body; and let them be equal in amount and opposite in direction. Now, if they are acting along the same Line, we know that they neutralise each other, and that the body remains at rest. But if one be shifted ever so little to one side, so that they act along parallel Lines, then, though still equal in amount and (according to the 'direction' theory) opposite in direction, they no longer neutralise each other, but form a 'couple'.
As a third illustration, take two points on a certain Plane. We may, first, draw a Line through them and cause them to move along that Line: they are then undoubtedly moving 'in the same direction.' We may, secondly, draw two Lines through them, which meet or at least would meet if produced, and cause them to move along those Lines: they are then undoubtedly moving 'in different directions.' We may, thirdly, draw two parallel Lines through them, and cause them to move along those Lines. Surely this is a new relationship of motion, not absolutely identical with either of the former two? But if this new relationship be not absolutely identical with that named 'in the same direction,' it must belong to the class named 'in different directions.'
Still, though this new relationship of direction is not identical with the former in all respects, it is in some: only, to prove this, we must use some disputed Axiom, as it will take us into Table II. For instance, they are identical as to angles made with transversals: this fact is embodied in Tab. II. 4. (See p. 34). Would you like to adopt that as your Axiom?
Nie. No. We are trying to dispense with Table II altogether.
Min. It is a vain attempt.There is another remark I wish to make, before considering your second assertion. In asserting that there is a real class of non-coincidental Lines that have 'the same direction,' are you not also asserting that there is a real class of Lines that have no common point? For, if they had a common point, they must have 'different directions.'
Nie. I suppose we are.
Min. We will then, if you please, credit you with an Axiom you have not expressed, viz. 'It is possible for two Lines to have no common point.' And here I must express an opinion that this ought to be proved, not assumed. Euclid has proved it in I. 27, which rests on no disputed Axiom; and I think it may be recorded as a distinct defect in your treatise, that you have assumed, as axiomatic, a truth which Euclid has proved.
My conclusion, as to this first assertion of yours, is that it is most decidedly not axiomatic.
Let us now consider your second assertion, that some non-coincidental Lines have 'different directions.' Here I must ask, as before, are you speaking of Lines which have a common point? If so, I am quite ready to grant the assertion.
Nie. Not exactly. It is rather a difficult matter to explain. The Lines we refer to would, as a matter of fact, meet if produced, and yet we do not suppose that fact known in speaking of them. What we ask you to believe is that there is a real class of non-coincidental finite Lines, which we do not yet know to have a common point, but which have 'different directions.' We shall assert presently, in another Axiom, that such Lines will meet if produced; but we ask you to believe their reality independently of that fact.
Min. But the only geometrical meaning I know of, as yet, for the phrase 'different directions,' refers to Lines known to have a common point. What geometrical meaning do you attach to the phrase when used of other Lines?
Nie. We cannot define it.
Min. Nor construct it? Nor test it?
Nie. No.
Min. You ask me, then, to believe in the reality of two classes of 'pairs of Lines,' each possessing a property that you can neither define, nor construct, nor test?
Nie. That is true. But surely you admit the reality of the second class? Why, intersectional Lines are a case in point.
Min. Certainly. And so much I am willing to grant you. I allow that some non-coincidental Lines, viz. intersectional Lines, have 'different directions.' But as to 'the same direction,' you have given me no reason whatever for believing that there are any non-coincidental Lines which possess that property.
Nie. But surely there are two real distinct classes of non-coincidental Lines, 'intersectional' and 'separational'?
Min. Yes. Thanks to Euc. I. 27, you may now assume the reality of both.
Nie. And you will hardly assert that the relationship of direction, which belongs to a Pair of intersectional Lines, is identical with that which belongs to a Pair of separational Lines?
Min. I do not assert it.Nie. And you allow that intersectional Lines have 'different directions'?
Min. Yes. Are you going to argue, from that, that separational Lines must have 'the same direction'? Why may I not say that intersectional Lines have one kind of 'different directions' and that separational Lines have another kind?
Nie. But do you say it?
Min. Certainly not. There is no evidence, at present, one way or the other. For anything we know, Pairs of separational Lines may always have 'the same direction,' or they may always have 'different directions,' or there may be Pairs of each kind. I fear I must decline to grant the first part of your Axiom altogether, and the second part in the sense of referring to Lines not known to have a common point. You may now proceed.
Niemand reads.
P. 11. Ax. 7. 'Two different straight Lines which meet one another have different directions.'
Min. That I grant you, heartily. It is, in fact, a Definition for the phrase 'different directions,' when used of Lines which have a common point.
Niemand reads.
P. 11. Ax. 8. 'Two straight Lines which have different directions would meet if prolonged indefinitely.'
Min. Am I to understand that, if we have before us a Pair of finite Lines which are not known to have a common point, but of which we do know that they have different directions, you ask me to believe that they will meet if produced?
Nie. That is our meaning.
Min. We had better heave the lead once more, and return to our fixed Line, and a point not on it, through which we wish to draw a second Line. You ask me to grant that, if it be drawn so as to have a direction 'different' from that of the first Line, it will meet it if prolonged indefinitely?
Nie. That is our humble petition.
Min. Will you be satisfied if I grant you that some Lines, so drawn, will meet the first Line? That I would grant you with pleasure. I could draw millions of Lines which would fulfil the conditions, by simply taking points at random on the given Line, and joining them to the given point. Every Line, so constructed, would have a direction 'different' from that of the given Line, and would also meet it.
Nie. We will not be satisfied, even with millions! We ask you to grant that every Line, drawn through the given point with a direction 'different' from that of the given Line, will meet the given Line: and we ask you to grant this independently of, and antecedently to, any other information about the Lines except the fact that they have 'different' directions.
Min. But what meaning am I to attach to the phrase 'different directions,' independently of, and antecedently to, the fact that they have a common point?
Nie. (after a long silence) I fear we can suggest none.
Min. Then I must decline to accept the Axiom.Nie. And yet this Axiom is the converse of the preceding, which you granted so readily.
Min. The technical converse, my good sir, not the logical! I will not suspect you of so gross a logical blunder as the attempt to convert a universal affirmative simpliciter instead of per accidens. The only converse, as you are no doubt aware, to which you have any logical right, is 'Some Lines, which have "different directions," would meet if produced'; and that I grant you. It is true of intersectional Lines, and I would limit the Proposition to such Lines, so that it would be equivalent to 'Lines, which would meet if produced, would meet if produced'—an indisputable truth, but not remarkable for novelty! You may proceed.
Nie. I beg to hand in this diagram, and will read you our explanation of it:—
'Thus A and B in the figure have the same direction; and C and D, which meet, have different directions; and E and F, which have different directions, would meet if produced far enough.'
Min. I grant the assertion about C and D; but I am wholly unable to guess on what grounds you expect me to grant that A and B 'have the same direction,' and that E and F 'have different directions.' Do you expect me to judge by eye? How if the lines were several yards apart? Is this what Geometry is coming to? Proceed.
Niemand reads.
Def. 19. 'Straight Lines, which are not parts of the same straight Line, but have the same direction, are called Parallels.'
Min. A Definition is of course unobjectionable, since it does not assert the existence of the thing defined: in fact, it asserts nothing except the meaning which you intend to attach to the word 'parallel.' But, as this word is used in different senses, I will thank you to substitute for it, in what you have yet to say about this matter, the phrase 'having a separate point, but the same direction,' which you may condense into one compound word, if you like:—'sepuncto-codirectional.'
Nie. (sighing) A terrible word! And I shall have to use it so often!
Min. I will try to abridge it for you. Let us take 'sep-' and 'cod-' from the beginnings of the two words, and '-al' for a termination. That will give us 'sepcodal.'
Nie. That sounds a little harsh.
Min. 'What? Is it harder. Sirs, than Gordon,
Colkitto, or Macdonald, or Galasp?'
Nie. (doubtfully) I think I prefer it to Colkitto.' But it is from you Moderns I have learned to be so sensitive about long words. How I would have liked to take you to an Egyptian restaurant I used to frequent, centuries ago, in a phantasmic sort of way, if only to hear the names of some of the dishes! Why, one thought nothing of seeing a gentleman rush in, carpet-bag in hand, and shout out 'ὐᾷτερ!' (that was the way we addressed the attendant in those days) 'A plate of λεπαδοτεμαχοσελαχογαλεοκρανιολειψανοδριμυποτριμματοσιλφιοπαραομελιτοκατακεχυμενοκιχλεπικοσσυφοφαττοπεριστεραλεκτρυονοπτεγκεφαλοκιγκλοπελειολαγωοσιραιοβαφητραγανοπτερὐγων, and look sharp about it! I'm in a hurry!'
Min. If the gentleman wanted to catch his train—by the way, had they trains in Egypt in ancient days?
Nie. Certainly. Read your 'Antony and Cleopatra,' Act I, Scene 1. 'Exeunt Antony and Cleopatra with their train.'
Min. In that case, wouldn't it be enough to say 'A plate of λεπαδο'?
Nie. Most certainly not—at least not in a fashionable restaurant. But this is a digression. I am willing to adopt the word 'sepcodal.'
Min. Now, before you read any more, let us get a clear idea of your Definition. We know of two real classes of Pairs of Lines, 'coincidental' and 'intersectional'; and to these we may (if we credit you with a Corollary to Euc. I. 27, 'It is possible for two Lines to have no common point') add a third class, which we may call 'separational.'
We also know that if a Pair of Lines has a common point, and no separate point, it belongs to the first class; if a common point, and a separate point, to the second. Hence all Pairs of Lines, having a common point, must belong to one or other of these classes. And since a Pair, which has no common point, belongs to the third class, we see that every conceivable Pair of Lines must belong to one of these three classes.
We also know thatNie. (sighing deeply) You are heaving the lead again!
Min. I am: but we shall be in calmer water soon.
We also know that the 'Coincidental' class possesses two properties—they are coincidental and have identical directions; and that the 'Intersectional' class also possesses two properties—they are intersectional and have different directions.
Now if you choose to frame a Definition by denying one property of each of these two classes, any Pair of Lines, so defined, is excluded from both of these classes, and must, if it exist at all, belong to the 'Separational' class. Remember, however, that you may have so framed your Definition as to exclude your Pair of Lines from existence. For instance, if you choose to combine two contradictory conditions of direction, and to say that Lines, which have identical and intersectional directions, are to be called so-and-so, you are simply describing a nonentity.
Nie. That is all quite clear.
Min. Your Definition, then, amounts to this:—Lines, which are not coincidental, but which have identical directions, are said to be 'sepcodal.'
Nie. It does.
Min. Well, here is another Definition for Parallels, which will answer your purpose just as well:—'Lines, which are not intersectional, but which have different directions.'
Nie. But I think I can prove to you that you have now done the very thing you cautioned me against: you have annihilated your Pair of Lines.
Min. That is a matter which we need not consider at present. Proceed.
Niemand reads.
P. 11. 'From this Definition, and the Axioms above given, the following results are immediately deduced:
(1) That parallel—I beg your pardon—that 'sepcodal' Lines would not meet however far they were produced. For if they met'
Min. You need not trouble yourself to prove it. I grant that, if such Lines existed, they would not meet. Your assertion is simply the Contranominal of Ax. 7 (p. 115), and therefore is necessarily true if the subject be real.
But remember that, though I have granted to you that, if we are given a Line and a point not on it, we can draw, through the point, a certain Line separational from the given Line, we do not yet know that it is the only such Line. That would take us into Table II. With our present knowledge, we must allow for the possibility of drawing any number of Lines through the given point, all separational from the given Line: and all I grant you is, that your ideal 'sepcodal' Line will, if it exist at all, be one of this group.
Niemand reads.
(2) 'That Lines which are sepcodal with the same Line are sepcodal with each other. For'
Min. Wait a moment. I observe that you say that such Lines are sepcodal with each other. Might they not be 'compuncto-codirectional'?
Nie. Certainly they might: but we do not wish to include that case in our predicate.Min. Then you must limit your subject, and say 'different Lines.'
Nie. Very well.
Reads.
'That different Lines, which are sepcodal with the same Line, are sepcodal with each other. For they each have the same direction as that Line, and therefore the same direction as the other.'
Min. I am willing to grant you, without any proof, that, if such Lines existed, they would have the same direction with regard to each other. The phrase 'they each have' is not remarkably good English. However, you may proceed.
Niemand reads.
P. 12. Ax. 9. 'An angle may be conceived as transferred from one position to another, the direction of its arms remaining the same.'
Min. Let us first consider the right arm by itself. You assert that it may be transferred to a new position, its direction remaining the same?
Nie. We do.
Min. You might, in fact, have here inserted an Axiom 'A Line may be conceived as transferred from one position to another, its direction remaining the same'?
Nie. That would express our meaning.
Min. And this is virtually identical with your Axiom 'Two different Lines may have the same direction'?
Nie. Certainly. They embody the same truth. But the one contemplates a single Line in two positions, and the other contemplates two Lines: the difference is very slight.
Min. Exactly so. Now let me ask you, do you mean, by the word 'angle,' a constant or a variable angle?
Nie. I do not quite understand your question.
Min. I will put it more fully. Do you mean that the arms of the angle are rigidly connected, so that it cannot change its magnitude, or that they are merely hinged loosely together, as it were, so that it depends entirely on the relative motions of the two arms whether the angle changes its magnitude or not?
Nie. Why are we bound to settle the question at all?
Min. I will tell you why. Suppose we say that the arms are merely hinged together: in that case all you assert is that each arm may be transferred, its direction remaining the same; that is, you merely assert your 6th Axiom twice over, once for the right arm and once for the left arm; and you do not assert that the angle will retain its magnitude. But in the Theorem which follows, you clearly regard it as a constant angle, for you say 'the angle AOD would coincide with the angle EKH. Therefore the angle AOD = EKH.' But the 'therefore' would have no force if AOD could change its magnitude. Thus you would be deducing, from an Axiom where 'angle' is used in a peculiar sense, a conclusion in which it bears its ordinary sense. You have heard of the fallacy 'A dicto secundum Quid ad dictum Simpliciter'?
Nie. (hastily) We are not going to commit ourselves to that. You may assume that we mean, by 'angle,' a rigid angle, which cannot change its magnitude.Min. In that case you assert that, when a pair of Lines, terminated at a point, is transferred so that its vertex has a new position, these three conditions can be simultaneously fulfilled:—
- (1) the right arm has 'the same direction' as before;
- (2) the left arm has 'the same direction' as before;
- (3) the magnitude of the angle is unchanged.
Nie. We do not dispute it.
Min. But any two of these conditions are sufficient, without the third, to determine the new state of things. For instance, taking (1) and (3), if we fix the position of the right arm, by giving it 'the same direction' as before, and also keep the magnitude of the angle unchanged, is not that enough to fix the position of the left arm, without mentioning (2)?
Nie. It certainly is.
Min. Your Axiom asserts, then, that any two of these conditions lead to the third as a necessary result?
Nie. It does.
Min. Your Axiom then contains two distinct assertions: the data of the first being (1) and (3) [or (2) and (3), which lead to a similar result], the data of the second being (1) and (2). These I will state as two separate Axioms:—
9 (α). If a Pair of Lines, terminated at a point, be transferred to a new position, so that the direction of one of the Lines, and the magnitude of the included angle, remain the same; the direction of the other Line will remain the same.
9 (β). If a Pair of Lines, terminated at a point, be transferred to a new position, so that their directions remain the same; the magnitude of the included angle will remain the same.
Have I represented your meaning correctly?
Nie. We have no objection to make.
Min. We will return to this subject directly. I must now ask you to read the enunciation of Th. 4, omitting, for simplicity's sake, all about supplementary angles, and assuming the Lines to be taken 'the same way.'
Niemand reads.
P. 12. Th. 4. 'If two Lines are respectively sepcodal with two other Lines, the angle made by the first Pair will be equal to the angle made by the second Pair.'
Min. The 'sep' is of course superfluous, for if the Lines are 'compuncto-codirectional,' it is equally true. May I re-word it thus?—
'If two Pairs of Lines, each terminated at a point, be such that the directions of one Pair are respectively the same as those of the other; the included angles are equal.'
Nie. Yes, if you like.
Min. But surely the only difference, between Ax. 9 (β) and this, is that in the Axiom we contemplated a single Pair of Lines transferred, while here we contemplate two Pairs?
Nie. That is the only difference, we admit.
Min. Then I must say that it is anything but good logic to take two Propositions, distinguished only by a trivial difference in form, and to call one an Axiom and the other a Theorem deduced from it! A very gross case of 'Petitio Principii,' I fear!Nie. (after a long pause) Well! We admit that it is not exactly a Theorem: it is only a new form of the Axiom.
Min. Quite so: and as it is a more convenient form for my purpose, I will with your permission adopt it as a substitute for the Axiom. Now as to the corollary of this Theorem: that, I think, is merely a particular case of Ax. 9 (β), one of the arms being slid along the infinite Line of which it forms a part, and thus of course having 'the same direction' as before?
Nie. It is so.
Min. And, as this is a more convenient form still, I will restate your assertions, limiting them to this particular case:—
Ax. 9 (α). Lines, which make equal corresponding angles with a certain transversal, have the same direction.
Ax. 9 (β). Lines, which have the same direction, make equal corresponding angles with any transversal.
Am I right in saying that these two assertions are virtually involved in your Axiom?
Nie. We cannot deny it.
Min. Now in 9 (α) you ask me to believe that Lines possessing a certain geometrical property, which can be defined, constructed, and tested, possess also a property which, in the case of different Lines, we can neither define, nor construct, nor test. There is nothing axiomatic in this. It is much more like a Definition of 'codirectional' when asserted of different Lines, for which we have as yet no Definition at all. Will you not permit me to insert it, as a Definition, before Ax. 6 (p. 108)? We might word it thus:—'If two different Lines make equal angles with a certain transversal, they are said to have the same direction: if unequal, different directions.'
This interpolation would have the advantage of making Ax. 6 (which I have hitherto declined to grant) indisputably true.
Nie. (after a pause) No. We cannot adopt it as a Definition so early in the subject.
Min. You are right. You probably saw the pitfall which I had ready for you, that this same Definition would make your 8th Axiom (p. 115) exactly equivalent to Euclid's 12th! From this catastrophe you have hitherto been saved solely by the absence of geometrical meaning in your phrase 'the same direction,' when applied to different Lines. Once define it, and you are lost!
Nie. We are aware of that, and prefer all the inconvenience which results from the absence of a Definition.
Min. The 'inconvenience,' so far, has consisted of the ruin of Ax. 6 and Ax. 8. Let us now return to Ax. 9.
As to 9 (β), it is of course obviously true with regard to coincidental Lines: with regard to different 'Lines, which have the same direction,' I grant you that, if such Lines existed, they would make equal corresponding angles with any transversal; for they would then have a relationship of direction identical with that which belongs to coincidental Lines. But all this rests on an 'if'—if they existed.
Now let us combine 9 (β) with Axiom 6, and see what it is you ask me to grant. It is as follows:—
'There can be a Pair of different Lines that make equal angles with any transversal.'I am not misrepresenting you, I think, if I say that you propound this as axiomatic truth—which, I need hardly remark, is a corollary deducible from the fourth Proposition in Table II. (see p. 34).
Nie. We accept the responsibility of the two Axioms separately, but not of a logical deduction from the two.
Min. There are certainly some logical deductions from Axioms (Contranominals for instance) that are not so axiomatic as the Axioms from which they come: but surely if you tell me 'it is axiomatic that X is Y' and 'it is axiomatic that Y is Z,' it is much the same as saying 'it is axiomatic that X is Z'?
Nie. It is very like it, we admit.
Min. Now take one more combination. Take 9 (α) and 9 (β). We thus eliminate the mysterious property altogether, and get a Proposition whose subject and predicate are perfectly definite geometrical conceptions—a Proposition which you assert to be, if not perfectly axiomatic, yet so nearly so as to be easily deducible from two Axioms—a Proposition which again lands us in Table II, and which, I will venture to say, is less axiomatic than any Proposition in that Table that has yet been proposed as an Axiom. We get this:—
'Lines which make equal corresponding angles with a certain transversal do so with any transversal,' which is Tab. II. 4 (see p. 34).
Here we have, condensed into one appalling sentence, the whole substance of Euclid I. 27, 28, and 29 (for the fact that the lines are 'separational' may be regarded as merely a go-between). Here we have the whole difficulty of Parallels swallowed at one gulp. Why, Euclid's much-abused 12th Axiom is nothing to it! If we had (what I fear has yet to be discovered) a unit of 'axiomaticity,' I should expect to find that Euclid's 12th Axiom (which you call in your Preface, at p. xiii, 'not axiomatic') was twenty or thirty times as axiomatic as this! I need not ask you for any further proof of Euc. I. 32. This wondrous Axiom, or quasi-Axiom, is quite sufficient machinery for your purpose, along with Euc. I. 13, which of course we grant you. Have you thought it necessary to provide any other machinery?
Nie. No.
Min. Euclid requires, besides I. 13, the following machinery:—Props. 4, 5, 7, 8, 15, 16, Ax. 12, Props. 27, 28, and 29. And for all this you offer, as a sufficient substitute, one single Axiom!
Nie. Two, if you please. You are forgetting Ax. 6.
Min. No, I repeat it—one single Axiom. Ax. 6 is contained in Ax. 9 (α): when the subject is known to be real, the Proposition necessarily asserts the reality of the predicate.
Nie. That we must admit to be true.
Min. I need hardly say that I must decline to grant this so-called 'Axiom,' even though its collapse should involve that of your entire system of 'Parallels.' And now that we have fully discussed the subject of direction, I wish to ask you one question which will, I think, sum up the whole difficulty in a few words. It is, in fact, the crucial test as to whether 'direction' is, or is not, a logical method of proving the properties of Parallels.You assert, as axiomatic, that different Lines exist, whose relationship of direction is identical with that which exists between coincidental Lines.
Nie. Yes.
Min. Now, does the phrase 'the same direction,' when used of two Lines not known to have a common point, convey to your mind a clear geometrical conception?
Nie. Yes, we can form a clear idea of it, though we cannot define it.
Min. And is that idea (this is the crucial question) independent of all subsequent knowledge of the properties of Parallels?
Nie. We believe so.
Min. Let us make sure that there is no self-deception in this. You feel certain you are not unconsciously picturing the Lines to yourself as being equidistant, for instance?
Nie. No, they suggest no such idea to us. We introduce the idea of equidistance later on in the book, but we do not feel that our first conception of 'the same direction' includes it at all.
Min. I think you are right, though Mr. Cuthbertson, in his 'Euclidian Geometry,' says (Pref. p. vi.) 'the conception of a parallelogram is not that of a figure whose opposite sides will never meet…, but rather that of a figure whose opposite sides are equidistant.' But do you feel equally certain that you are not unconsciously using your subsequent knowledge that Lines exist which make equal angles with all transversals?
Nie. We are not so clear about that. It is, of course, extremely difficult to divest one's mind of all later knowledge, and to place oneself in the mental attitude of one who is totally ignorant of the subject.
Min. Very difficult, no doubt, but absolutely essential, if you mean to write a book adapted to the use of beginners. My own belief as to the course of thought needed to grasp the theory of 'direction' is this:—first you grasp the idea of 'the same direction' as regards Lines which have a common point; next, you convince yourself, by some other means, that different Lines exist which make equal angles with all transversals; thirdly, you go back, armed with this new piece of knowledge, and use it unconsciously, in forming an idea of 'the same direction' as regards different Lines. And I believe that the course of thought in the mind of a beginner is simply this:—he grasps, easily enough, the idea of 'the same direction' as regards Lines which have a common point; but when you put before him the idea of different Lines, and ask him to realise the meaning of the phrase, when applied to such Lines, he, finding that the former geometrical conception of 'coincidence' is not applicable in this case, and knowing nothing of the idea, which is latent in your mind, of Lines which make equal angles with all transversals, simply fails to attach any idea at all to the phrase, and accepts it blindly, from faith in his teacher, and is from that moment, until he reaches the Theorem about transversals, walking in the dark.
Nie. If this be true, of course the theory of 'direction,' however beautiful in itself, is not adapted for purposes of teaching.
Min. That is my own firm conviction. But I fear I may have wearied you by discussing this matter at such great length. Let us turn to another subject. What is your practical test for knowing whether two finite Lines will meet if produced?
Nie. You have already heard our 8th Axiom (p. 11). 'Two straight lines which have different directions would meet if produced.'
Min. But, even if that were axiomatic (which I deny), it would be no practical test, for you have admitted that you have no means of knowing whether two Lines, not known to have a common point, have or have not different directions.
Nie. We must refer you to p. 14. Th. 5. Cor. 2, where we prove that Lines, which make equal angles with a certain transversal, have the same direction.
Min. Which you had already asserted, if you remember, in Ax. 9.
Nie. Well then, we refer you to Ax. 9 as containing the same truth.
Min. And having got that truth, whether lawfully or not, what do you do with it?
Nie. Why, surely it is almost the same as saying that, if they make unequal angles, they have different directions.
Min. And what then?
Nie. Then, combining this with the Axiom you refused to grant, namely, that Lines having different directions will meet, we get a practical test, such as you were asking for.
Min. (dreamily) I see! You get rid of the 'different directions' altogether, and the result is that 'Lines, which make unequal angles with a certain transversal, will meet if produced,' which is Tab. II. 2 (see p. 34). And this you assert as axiomatic truth?Nie. (uneasily) Yes.
Min. Surely I have read something like it before? Could it have been Euclid's 12th Axiom? And have I not somewhere read words like these:—'Euclid's treatment of parallels distinctly breaks down in Logic. It rests on an Axiom which is not axiomatic'?
Nie. We have nowhere stated this Axiom which you put into our mouth.
Min. No? Then how, may I ask, do you prove that particular Lines will meet? You must have to prove it sometimes, you know.
Nie. We have not had to prove it anywhere, that we are aware of.
Min. Then there must be some gaps in your arguments. Let us see. Please to turn to p. 46. Prob. 7. Here you make, at the ends of a Line CD, angles equal to two given angles (which, as you tell us below, 'must be together less than two right angles'), and you then say 'let their sides meet in O.' How do you know that they will meet?
Nie. You have found one hiatus, we grant. Can you point out another in the whole book?
Min. I can. At p. 70 I find the words 'Join QG, and produce it to meet FH produced in S.' And again at p. 88. 'Hence the centre must be at O, the point of intersection of these perpendiculars.' In both these cases I would ask, as before, how do you know that the Lines in question will meet?
Nie. We had not observed the omissions before, and we must admit that they constitute a serious hiatus.
Min. A most serious one. A student, who had been taught such proofs as these, would be almost sure to try the plan in cases where the Lines would not really meet, and his assumption would lead him to results more remarkable for novelty than truth.
Let us now take a general survey of your book. And First, as to the Propositions of Euclid which you omit—
Nie. You are alluding to Prop. 7, I suppose. Surely its only use is to prove Prop. 8, which we have done very well without it.
Min. That is quite a venial omission. The others that I miss are 27, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 40, 41, and 43: rather a formidable list.
Nie. You are much mistaken! Nearly all of those are in our book, or could be deduced in a moment from theorems in it.
Min. Let us take I. 34 as an instance.
Nie. That we give you, almost in the words of Euclid, at p. 37.
Reads.
Th. 22. 'The opposite angles and sides of a Parallelogram will be equal, and the diagonal, or the Line which joins its opposite angles, will bisect it.'
Min. Well, but your Parallelogram is not what Euclid contemplates. He means by the word that the opposite sides are separational—a property whose reality he has demonstrated in I. 27; whereas you mean that they have the same direction—a property whose reality, when asserted of different Lines, has nowhere been satisfactorily proved.
Nie. We have proved it at p. 14. Th. 5. Cor. 2.Min. Which, if traced back, will be seen to depend ultimately on your 6th Axiom, where you assume the reality of such Lines. But, even if your Theorem had been shown to refer to a real figure, how would that prove Euc. I. 34?
Nie. You only need the link that separational Lines have the same direction.
Min. Have you supplied that link?
Nie. No: but the reader can easily make it for himself. It is the 'Contranominal' (as you call it) of our 8th Axiom, 'two straight Lines which have different directions would meet if prolonged indefinitely.'
Min. Your pupils must be remarkably clever at drawing deductions and filling up gaps in an argument, if they usually supply that link, as well as the proof that separational Lines exist at all, for themselves. But, as you do not supply these things, it seems fair to say that your book omits all the Propositions which I have enumerated.
I will now take a general survey of your book, and select a few points which seem to call for remark.
Minos reads.
P. 14. Th. 5. Cor. 1. 'Hence if two straight Lines which are not parallel are intersected by a third, the alternate angles will be not equal, and the interior angles on the same side of the intersecting Line will be not supplementary.' Excuse the apparent incivility of the remark, but this Corollary is false.
Nie. You amaze me!Min. You have simply to take, as an instance, a Pair of coincidental Lines, which most certainly answer to your description of 'not parallel.'
Nie. It is an oversight.
Min. So I suppose: it is a species of literary phenomenon in which your Manual is rich.
Your proof of Cor. 2. is a delicious collection of negatives.
Reads.
'Cor. 2. Hence also if the corresponding angles are equal, or the alternate angles equal, or the interior angles supplementary, the Lines will he parallel.
'For they cannot be not parallel, for then the corresponding and alternate angles would be unequal by Cor. 1.'
Should I be justified in calling this a somewhat knotty passage?
Nie. You have no right to make such a remark. It is a mere jest!
Min. Well, we will be serious again.
At p. 9, you stated more than the data authorised: we now come to a set-off against this, since we shall find you asserting less than you ought to do. I will read the passage:—
P. 26. Th. 15. 'If two Triangles are equiangular to one another and have a side of the one equal to the corresponding side of the other, the Triangles will be equal in all respects.'
This contains a superfluous datum: it would have been enough to say 'if two Triangles have two angles of the one equal to two angles of the other &c.'
Nie. Well, it is at worst a superfluity: the enunciation is really identical with Euclid's.
Min. By no means. The logical effect of a superfluous datum is to limit the extent of a Proposition: and, if the Proposition be 'universal,' it reduces it to 'particular'; i.e. it changes 'all A is B' into 'some A is B.' For suppose we take the Proposition 'all A is B,' and substitute for it 'all that is both A and X is B,' we may be accidentally making an assertion of the same extent as before, for it may happen that the whole class 'A' possesses the property 'X'; but, so far as logical form is concerned, we have reduced the Proposition to 'some things that are A (viz. those which are also X) are B.'
I turn now to p. 27, where I observe a new proof for Euc. I. 24.
Nie. New and, we hope, neat and short.
Min. Charmingly neat and short, as it stands: but this method really requires the discussion of five cases, each with its own figure.
Nie. How do you make that out?
Min. The five cases are:—
(1) Vertical angles together less than two right angles, and adjacent base angles acute (the case you give).
(2) Adjacent base-angles right.
(3) Adjacent base-angles obtuse.
These two cases are proved along with the first.
(4) Vertical angles together equal to two right angles.
This requires a new proof, as we must substitute for the words 'the bisector of the angle FAC,' the words 'the perpendicular to FC drawn through A.'
(5) Vertical angles together greater than two right angles.
This also requires a new proof, as we must insert, after the words 'the bisector of the angle FAC,' the words 'produced through A,' and must then prove (by your Th. 1) that the angles OAC, OAF, are equal.
On the whole, I take this to be the most cumbrous proof yet suggested for this Theorem.
We now come to what is probably the most extraordinary Corollary ever yet propounded in a geometrical treatise. Turn to pages 30 and 31.
Th. 20. 'If two triangles have two sides of the one equal to two sides of the other, and the angle opposite that which is not the less of the two sides of the one equal to the corresponding angle of the other, the triangles shall be equal in all respects.
'Cor. 1. If the side opposite the given angle were less than the side adjacent, there would be two triangles, as in the figure; and the proof given above is inapplicable.
'This is called the ambiguous case.'
The whole Proposition is a grand specimen of obscure writing and bad English, 'is' and 'are,' 'could,' and 'would,' alternating throughout with the most charming impartiality: but what impresses me most is the probable effect of this wondrous Corollary on the brain of a simple reader, coming breathless and exhausted from a death-struggle with the preceding theorem. I can imagine him saying wildly to himself 'If two Triangles fulfil such and such conditions, such and such things follow: but, if one of the conditions were to fail, there would be two Triangles! I must be dreaming! Let me dip my head in cold water, and read it all again. If two Triangles… there would be two Triangles. Oh, my poor brains!'
Nie. You are pleased to be satirical: it is rather obscure writing, we confess.
Min. It is indeed! You do well in calling it the ambiguous case.
At p. 33, I see the heading 'Theorems of equality': but you only give two of them, the second being 'the bisectors of the three angles of a triangle meet in one point,' which, as a specimen of 'Theorems of equality,' is probably unique in the literature of Geometry. I cannot wonder at your not attempting to extend the collection.
At p. 40 I read, 'It is assumed here that if a circle has one point inside another circle, the circumferences will intersect one another.' This I believe to be the boldest assumption yet made in Modern Geometry.
At pp. 40, 42 you assume a length 'greater than half a given Line, without having shewn how to bisect Lines. Two cases of 'Petitio Principii.' (See p. 58.)
P. 69. Here we have a Problem (which you call 'the quadrature of a rectilineal area') occupying three pages and a half. It is 'approached' by four 'stages,' which is a euphemism for saying that this fearful Proposition contains four of Euclid's Problems, viz. I. 42, 44, 45, and II. 14.
P. 73. 2. 'Find a point equally distant from three given straight lines.' Is it fair to give this without any limitation? What if the given lines were parallel?
P. 84. 'If A, B, C … as conditions involve D as a result, and the failure of C involves a failure of D; then A, B, D … as conditions involve C as a result.' If not-C proves not-D, then D proves C. A and B are irrelevant and obscure the statement. I observe, in passing, the subtle distinction which you suggest between 'the failure of C' and 'a failure of D.' D is a habitual bankrupt, who has often passed through the court, and is well used to failures: but, when C fails, his collapse is final, and 'leaves not a wrack behind'!
P. 90. 'Given a curve, to ascertain whether it is an arc of a circle or not.' What does 'given a curve' mean? If it means a line drawn with ink on paper, we may safely say at once 'it is not a circle.'
P. 96. Def. 15. 'When one of the points in which a secant cuts a circle is made to move up to, and ultimately coincide with, the other, the ultimate position of the secant is called the tangent at that point.' (The idea of the position of a Line being itself a Line is queer enough: I suppose you would say 'the ultimate position of Whittington was the Lord Mayor of London.' But this is by the way: of course you mean 'the secant in its ultimate position.') Now let us take three points on a circle, the middle one fixed, the others movable; and through the middle one let us draw two secants, each passing through one of the other points; and then let us make the other points 'move up to, and ultimately coincide with,' the middle one. We have no ground for saying that these two secants, in their ultimate positions, will coincide. Hence the phrase 'the tangent' assumes, without proof, Th. 7. Cor. 1, viz. 'there can be only one tangent to a circle at a given point.' This is a 'Petitio Principii.'
P. 97. Th. 6. The secant consists of two portions, each terminated at the fixed point. All that you prove here is that the portion which has hitherto cut the circle is ultimately outside: and you jump, without a shadow of proof, to the conclusion that the same thing is true of the other portion! Why should not the second portion begin to cut the circle at the precise moment when the first ceases to do so? This is another 'Petitio Principii.'
P. 129, line 3 from end. 'Abstract quantities are the means that we use to express the concrete.' Excluding such physical 'means' as pen and ink or the human voice (to which you do not seem to allude), I presume that the 'means' referred to in this mysterious sentence are 'pure numbers.' At any rate the only instances given are 'seven, five, three.' Now take P. 130, l. 5, 'Abstract quantities and ratios are precisely the same things.' Hence all ratios are numbers. But in the middle of the same page we read that 'all numbers are ratios, but all ratios are not numbers.' I leave this without further remark.
I will now sum up the conclusions I have come to with respect to your Manual.
(1) As to 'straight Lines' you suggest a useful extension of Euclid's Axiom.
(2) As to angles and right angles, your extension of the limit of size is, in my opinion, objectionable. In other respects your language, though hazy, agrees on the whole with Euclid.
(3) As to 'Parallels,' there is a good deal to be said, and that not very flattering, I fear.
In Ax. 6, you assert the reality of different Lines having the same direction—a property you can neither define, nor construct, nor test.
You also assert (by implication) the reality of separational Lines, which Euclid proves.
You also assert the reality of Lines, not known to have a common point, but having different directions—a property you can neither define, nor construct, nor test.
In Ax. 8, you assert that the undefined Lines last mentioned would meet if produced.
These Axioms, therefore, are not axiomatic.
In proving result (2), you are guilty of the fallacy 'Petitio Principii.'
In Ax. 9 and Th. 4 taken together, if the word 'angle' in Ax. 9 means 'variable angle,' you are guilty of the fallacy 'A dicto secundum Quid ad dictum Simpliciter'; if 'constant angle,' of the fallacy 'Petitio Principii.’
In Ax. 9 (α), you assert that Lines possessing a certain real geometrical property, viz. making equal angles with a certain transversal, possess also the before-mentioned undefined property. This is not axiomatic.
In Ax. 9 (β) combined with Ax. 6, you assert the reality of Lines which make equal angles with all transversals. This is not more axiomatic than Euc. Ax. 12.In Ax. 9 (α) combined with Ax. 9 (β), you assert that Lines, which make equal angles with a certain transversal, do so with all transversals. This I believe to be the most unaxiomatic Axiom ever yet proposed.
(4) You furnish no practical test for the meeting of finite Lines, and consequently you never prove (however necessary for the matter in hand) that any particular Lines will meet. And when we come to examine what practical test can possibly be extracted from your Axioms, the only result is an imperfect edition of Euclid's 12th Axiom!
The sum total of the chief defects which I have noticed is as follows:—
- fourteen of Euclid's Theorems in Book I. omitted;
- seven unaxiomatic Axioms;
- six instances of 'Petitio Principii.'
The abundant specimens of logical inaccuracy, and of loose writing generally, which I have here collected would, I feel sure, in a mere popular treatise be discreditable—in a scientific treatise, however modestly put forth, deplorable—but in a treatise avowedly put forth as a model of logical precision, and intended to supersede Euclid, they are simply monstrous.
My ultimate conclusion on your Manual is that it has no claim whatever to be adopted as the Manual for purposes of teaching and examination.