Euclid and His Modern Rivals/Act II. Scene V.
ACT II.
Scene V.
Treatment of Parallels by revolving Lines.
Henrici.
'In order that an aggregate of elements may be called a spread, it is necessary that they follow continuously.'—Henrici's Art of Dining, p. 12.
Nie. I lay before you 'Elementary Geometry: Congruent Figures,' by Olaus Henrici, Ph.D., F.R.S., Professor of Pure Mathematics in University College, London, 1879.
Min. What is your Definition of a Line?
Nie. 'The boundary of a surface or of part of a surface is called a Line or a curve.' (p. 5.)
Min. Good—'Line,' I presume, meaning 'right Line.' But that throws us back upon 'surface.' Of course that is defined correctly?
Nie. I will tell you in a moment. (He turns over a few pages) Yes, here it is. 'A surface is the—' (He gives a perceptible start, stops reading, and turns a few pages back) Yes, it's all right. 'That which bounds a solid and separates it from other parts of Space is called its surface.' (p. 4.)
Min. (aside) There is more here than meets the eye! (Aloud) You will be good enough to read that other Definition of 'surface.'Nie. (innocently) What other Definition?
Min. No evasions, Sir! Read it at once! You know the one I mean.
Nie. (desperately) It's only this—'A surface is the path of a moving curve.' (p. 9.) Merely another way of looking at it, you know.
Min. (contemptuously) Oh! Merely another way of looking at it, is it? Of course the curve preserves its shape as it moves?
Nie. No doubt.
Min. Now look here. Take this Jargonelle pear—
Nie. Thank you very much. It is rather dry work—
Min. Stop! Don't eat it yet! Look at it. Would you call its curvature regular?
Nie. Certainly not: it bulges here and there, in all sorts of queer ways.
Min. Well, now take this bit of wire: bend it into any curve you like, and then move it so that its path may coincide with the surface of the pear.
Nie. (uneasily) I cannot do it.
Min. Well, eat it, then. That is possible, at all events. So! We start with a Definition which is simply ridiculous! Now for the distinction between 'right Line' and 'curve'—
Nie. Here my client's meaning is not very clear. The first Definition I can find is that of a curve. He says (p. 6) 'a point may be moved, and then it will describe a path. This path of a moving point is a curve.'
Min. Surely he does not mean that a point can never move straight? He must mean that there are two kinds of curves, 'curved curves,' and 'straight curves'—as the Irish talk of 'tay-tay' and 'coffee-tay.' But, if so, he makes 'Line' and 'curve' synonymous.
Nie. I have looked a little further on, and I find a description of a 'Line,' which seems to limit the word to bent Lines. He says (p. 7) 'The notion of a Line may be obtained directly by considering a wire bent into any shape and abstracting all thickness from it.'
Min. So then a 'Line' must be bent, though a 'curve' need not be so? Your client has clearly one merit—great originality of style!
Nie. Here is another definition of 'curve,' which may be more to your taste, 'A curve is a one-way spread, with points as elements.' (p. 10.)
Min. Too much like a dinner à la Russe. I don't like 'spread' at all.
Nie. He illustrates his use of 'spread' by applying it to other subjects. For instance, 'a musical tone allows of variations which form a two-way spread, with different degrees of intensity and of pitch as elements.' (p. 12.)
Min. That explains the phrase 'too-tooing on a flute.' How simple and intelligible all this must be to boys just beginning Geometry! But I am still waiting for a definition of 'right Line.'
Nie. (after turning over several pages) I have found it at last—after passing over a good deal about 'continuity' and 'space' and 'congruence.' We say (p. 17) 'If we suspend a weight by a string, the string becomes stretched; and we say it is straight.'Min. That will serve very well to give a notion of 'straight.' For a working definition we require of course some practical test, such as 'two straight Lines cannot enclose a space.'
Nie. We have that. At p. 20 we give you 'Axiom IV. Through two points always one, and only one, Line can be drawn.' And at p. 18 we at last distinguish 'Line' and 'curve.' 'A straight Line will in future be called a Line simply. All other Lines will be called curved Lines, or curves.'
Min. Better late than never: though it makes wild work of your former theory—in which you got the notion of 'Line' from a bent wire, and of 'curve' from the path of a moving point. Now for the Definition of 'angle.'
Nie. (after turning the leaves backwards and forwards for some time, begins to read in an unsteady voice) 'The part of a pencil of half-rays, described by a half-ray on turning about its end point from one position to another, is called an angle.' (p. 47.)
Min. So you reject the notion of 'inclination' (or rather 'declination')? Well! This is an innovation! We must investigate it thoroughly. You mean by 'half-ray,' I presume, what Euclid calls 'a Line terminated in one direction but not in the other'?
Nie. Certainly.
Min. Now what is a 'pencil'?
Nie. 'The aggregate of all Lines in a plane which pass through a given point.' (p. 38.)
Min. Aha! And where will you get your angular magnitude, I should like to know? What kinds of magnitude is a Line capable of possessing?
Nie. Length only, of course.
Min. Two Lines?
Nie. (unasealy) Length only.
Min. A million?
Nie. (more unasealy) Length only.
Min. A pencil?
Nie. (faintly) Spare me!
Min. So much for the quality of your angular magnitude! Now for its quantity. What is the length of one of these half-rays?
Nie. Infinite, of course.
Min. And the aggregate length of all the half-rays in your 'angle' cannot well be less. Thus we may deduce a truly delightful definition of angular magnitude. 'As to quality, it is linear. As to quantity, it is infinite'!
Nie. (writhes, but says nothing).
Min. Will you not throw up your brief?
Nie. Not yet: I must fight it out.
Min. Then we must review this marvellous book 'to the bitter end.' What have you to say about 'right angles'?
Nie. We have 'angles of rotation' and 'angles of continuation' (p. 48); and the axiom 'all angles of rotation are equal' (p. 49) as a substitute for 'all right angles are equal.'
Min. It is a practicable method, but not so suitable for beginners as Euclid's. This matter I have already discussed (see p. 74). And now for the subject of Parallels.
Nie. We have Playfair's Axiom (or rather its equivalent) 'Through a given point only one Line can be drawn parallel to a given Line' (p. 68), but this we do not simply lay down as an Axiom. We lead up to it by two or three pages of reasoning.
Min. This is most interesting! Let us examine the argument minutely. A logical proof of that Axiom would be perhaps the greatest advance ever made in the subject since the days of Euclid.
Nie. 'Two indefinite Lines in a Plane may intersect, as we have seen. We shall now consider the possibility of there being such Lines which do not intersect.' (p. 65.)
Min. That, of course, you can easily prove, without appealing to any disputable Axiom. It is simply Euc. I. 27. Do you prove it in Euclid's way?
Nie. Not exactly. Our argument is quite different from Euclid's: and we come to two conclusions—one being the real existence of Parallels, the other the equivalent of Playfair's Axiom.
Min. I very much doubt your proving the first by any simpler method than Euclid's: and as to proving the second, by any method at all, without assuming some disputable Axiom, I defy you to do it! However, let us hear your argument.
Nie. We take a Line, and a point without it: and from the point we draw two 'half-rays' intersecting the line. These half-rays we then turn about the point, in opposite directions, until they cease to intersect the Line. And then we proceed to consider where their 'productions' have got to.
Min. Like 'little Bo-peep,' you are anxious about their 'tails' in fact; taking their 'heads' to be the ends which at first intersected the given Line.
Nie. We say that there are only three conceivable cases: one, where the tails fall next to
Min. I admit all that.
Nie. The first case we say is inadmissible because, if it were true, any Line through P, lying within the angle formed by the head of one ray and the tail of the other, would cut the given Line both ways.
Min. A reductio ad absurdum, no doubt; but it only holds good on the supposition that you can draw Lines through P, so as to lie within that angle. But this supposition requires a finite angle. If we suppose that, the moment one ray begins to revolve so as to bring its head nearer to the given line, it instantly coincides with the other ray, head with tail and tail with head, it will not then be possible to draw any such Line as you suggest: and then where is your reductio ad absurdum?
Nie. We do not seem to have noticed that case.
Min. In point of fact, your three cases are really five. Before going any further let us have them all clearly stated.
We assume, in all three figures, that the ray-heads, as drawn, do not intersect the given line; but that either of them would, if it began to revolve towards the given Line, instantly intersect it. In other words we assume that any half-ray, drawn from P in the dotted angular space, would intersect the given Line: but that any half-ray, drawn from P in the undotted angular space, as well as the two ray-heads which limit that angular space, would not intersect it. And as to the ray-tails, it is obvious that in fig. 1 they do intersect the given Line, but in figs. 2, 3, they do not do so.
Nie. That is all clear enough.
Min. Then these are the five cases:—
(α) Figure 1. The head of each ray must revolve downwards through a finite angle before it can coincide with the tail of the other ray.
(β) Same figure. The head of each ray, on beginning to revolve downwards, instantly coincides with the tail of the other ray.
(γ) Figure 2. The head of each ray must revolve upwards through a finite angle before it can coincide with the tail of the other ray.
(δ) Same figure. The head of each ray, on beginning to revolve upwards, instantly coincides with the tail of the other ray.
(ε) Figure 3.
These five cases suggest a few observations.
In case (α) a number of Lines may be drawn through P, in the angular space contained between the head of one ray and the tail of the other: and all such Lines will intersect the given line both ways.
In case (β) this absurdity does not arise: all Lines, through P, intersect the given Line one way or the other: there is no instance of a Line intersecting it both ways, nor of one wholly separate from it.
In case (γ) a number of Lines may be drawn as in case (α): and all such Lines will be wholly separate from the given Line.
In case (δ) the two rays themselves, as drawn in the figure, are wholly separate from the given Line: but no other such Line can be drawn through P.
In case (ε) there is only one Line through P wholly separate from the given Line.
Now let us hear what you make of these five cases.
Nie. We exclude case (α), as I told you just now, by a reductio ad absurdum. Case (β) we have failed to notice.
Min. True: but it can be excluded by Euc. I. 27: so that if you can manage, by pure reasoning, from ordinary Axioms, and without assuming any disputable Axiom, to exclude cases (γ) and (δ), you will have achieved what geometricians have been vainly trying to do for the last two thousand years!
Nie. We go on thus. 'But our Axioms are not sufficient to decide which of the remaining two cases actually does occur.' (p. 67.)
Min. Or rather 'the remaining three cases.'
Nie. 'In looking at the figures the reader will at once feel that the third case' (we mean your 'case (ε)') 'is the true one.'
Min. An appeal to sentiment! What if the reader doesn't feel it?
Nie. 'But this cannot be considered decisive;'
Min. It cannot.
Nie. 'for the two Lines may include a very small angle—'
Min. Aye, or even a large one.
Nie. 'that is, they may very nearly coincide without actually doing so. Or it may be that sometimes the one, sometimes the other, happens, according as we take the point P at a smaller or greater distance from the Line.'
Min. That seems a fair statement of the difficulty. And now, how are you going to grapple with it?
Nie. 'The only way of settling this point is to make an assumption, and to see whether the consequences drawn from it do or do not agree with our experience.'
Min. If you find a consequence not agreeing with experience, you may of course conclude that your assumption was false; but, if it does agree, what then?
Nie. Nothing, I fear, unless you can prove that this is the case with one assumption only, and that all other possible assumptions lead to absurd results.
Min. Exactly so. If, then, you want to prove case (ε), your logical course is to assume case (γ) as true, and from that assumption to deduce some consequence which is evidently contrary to experience. And then to exclude case (δ) by a similar argument. Is that your method?
Nie. Well, hardly. We say 'The assumption to be made is, that the third case' (i.e. case (ε)) only happens, and this will give us a new axiom.' (p. 67.)
Min. You may assume it as an axiom, if you like. Then you will merely be in the same boat with Playfair. But if you are going to discuss the consequences of its being true, and get anything out of that, look to your feet! There are pitfalls about!
Nie. 'In the second case' (i.e. case (γ)) 'we should have to—'
Min. Oho! Then it is case (γ), after all, that you are provisionally assuming as true?
Nie. Apparently so.
Min. Well, go on. You are on the right track now.
Nie. In this case we should have to turn the ray 'through a finite angle' before its tail would cut the given Line: 'or there would be an indefinite number of Lines through P which do not cut' it. (p. 68.)
Min. What do you mean by 'or'? That one result would follow, or the other, but not both?
Nie. We mean that the two results are equivalent.
Min. Then you should say 'that is.' 'Or' is misleading. However, I grant you that this consequence would follow, if case (γ) were true. What then? Is there any obvious absurdity in such a consequence?
Nie. That we do not assert. We merely make the remark—and we now proceed to case (ε).
Min. A weak and pointless remark: but let that pass. Do you omit case (δ)?
Nie. We do. We proceed thus. 'But in the third case (i.e. in case (ε)) there would be only one Line through P which does not cut' the given Line. 'As soon as we turn this Line about P it would meet it to the right or to the left.'
Min. Certainly. And what then? Do you expect me to admit that, because case (ε) would lead to a consequence not obviously absurd, therefore it is the case which always happens, to the exclusion of cases (γ) and (α)?
Nie. (hesitatingly) Well, I think that is what we expect. But we first deduce the real existence of Parallels. 'Thus we are led to the conclusion that there exist Lines in a Plane which, though both be unlimited, do not meet. Such Lines are called parallel.'
Min. Oh most lame and impotent conclusion! After all these magnificent Catherine-wheels of revolving half-rays, to deduce Euc. I. 27! And even this wretched result you have no right to. Just consider what your argument has been. There are five conceivable cases, (α), (β), (γ), (δ), and (ε). If (α) or (β) were true, no Line could be drawn, through P, parallel to the given Line: if (γ), many such Lines could be drawn: if (δ), two such Lines: if (ε), one such Line. Now what have you proved? Positively nothing whatever but this—that case (α) would lead to an absurd result. You leave me perfectly free to range about among the other four cases, one of which, (β), denies the real existence of Parallels, which existence you tell me you have proved! And so, for the 'long course of logical reasoning' which you object to so much in Euclid, you substitute a short course of illogical reasoning! But you deduce another conclusion, do you not?Nie. Yes, one other. 'The assumption mentioned in § 113' (the assumption that case (ε) is the only true one) 'may now be stated thus:—Axiom VI. Through a given point only one Line can he drawn parallel to a given Line.'
Min. May it indeed? And why 'now' rather than three pages back? Is there a single word, in all this argument, which tends to show that case (ε) is—I will not say certainly true, but—even fairly probable?
Nie. (cautiously) I will not assert that there is.
Min. In point of fact the odds are exactly three to one against it—since you have only excluded one of the five cases, and the other four are, for anything we know to the contrary, equally probable.
Nie. I will not dispute it.
Min. Well! Then it only remains to say that your attempted proof of Playfair's Axiom is an utter failure. Anything more hopelessly illogical I have never met with, not even in Cooley—and that is saying a great deal!
Nie. I confess I do not see my way to defending this proof. But even if we abandon the whole of it, we are no worse off than any other writer who assumes Playfair's Axiom.
Min. That I quite admit.
Nie. And then, my client instructs me to plead, this Manual (handing it to Minos) being so distinctly better than Euclid's in every other particular—
Min. Gently, gently! You are anticipating. I have not yet had my general survey of the book.
Nie. (refiling his pipe) Well, let us have it then.
Min. I will begin with the general remark that the first 151 pages of this book (the rest of it going beyond the limits of Euc. I, II) contain (excluding 7 pages on Logic and 22 pages of Exercises) 122 pages of text, which I presume the learner is expected to master.
Nie. A great deal of that is merely explanatory.
Min. True: but even omitting all that, we have, of Definitions, 80: and of Theorems, 145. And when the unfortunate learner has mastered all these—more than there are in Euclid's first six Books—he finds he has learned no more Euclid than Props. 1 to 34!
Nie. But he will have learned a good deal that is not in Euclid.
Min. Undoubtedly: and it would have been easy to crowd in twice as many Theorems as Mr. Henrici has done, without passing Prop. 34. I believe the subject to be practically inexhaustible. But fancy having to master 145 Theorems before even hearing of so important a one as Prop. 47!
Nie. If all the new matter is good, it is a poor objection to raise that there is too much of it.
Min. You think the quantity unassailable? Well, let us test its quality a little, then.
The book begins with a page or two of very general considerations. Time and Force, Kinetics and Kinematics, Chemistry and Biology, cross the stage in a grand but shadowy procession. Then when the pupil has been sufficiently crushed by the spectacle of how much there is to know, we allow him, little by little, to contract his view: till at last we condescend to contemplate so trifling an entity as Infinite Space.And here I notice a singular mental process. 'Two material bodies,' we are told, 'cannot occupy the same space. We are thus led to recognise a third property common to all bodies: every body has position.' (p. 3.) The word 'thus' is what I want to call your special attention to: for I confess I can see no such sequence of thought as it would seem to imply. Suppose bodies could occupy the same space: wouldn't they have 'position' just as much as if they couldn't? Does an orange—to take the favourite logical entity—lose its position because another orange most uncivilly insists on permeating it and occupying the same portion of Space? But if not, what is the meaning of 'thus'? As Artemus Ward would say, 'why this thusness?'
Nie. I can't say.
Min. A little further on I find a 'therefore' which is equally shadowy. The writer's logical ideas—in spite of his actually introducing a 'Digression on Logic'—are, I fear, a little vague. He says 'If we bring different points together into the same position, they will never give us anything but a point; we never obtain any extension. We cannot, therefore, say that Space is made up of points' (p. 6). I venture to say that there is no such sequence as 'therefore' seems to imply: he has made the whole argument null and void by using the words 'into the same position.'
Nie. I do not understand you.
Min. I will put it in another way. The real reason why you cannot construct Space of points is that they have no size: if they had size you could do it. But, under the condition here laid down—of bringing them 'together into the same position'—you make the thing impossible, whether they have size or not.
I have often found it the best way for exhibiting the unsoundness of an argument, to make another exactly like it, but leading to an absurd conclusion. I will try it here. You grant that a cubic foot can be made up of cubic inches?
Nie. Certainly.
Min. Well, I will prove to you that it cannot; and I will do so by an argument just as good as Mr. Henrici's. 'If we bring different cubic inches together into the same position they will never give us anything but a cubic inch; we never obtain any extension—'
Nie. That won't do! You have the 'extension' of one cubic inch.
Min. Yes, but you had that to begin with. You don't 'obtain' any extension by squeezing in other cubic inches, do you?
Nie. No, I suppose not.
Min. Then the argument is sound so far. And now comes my triumphant conclusion, à la Henrici. 'We cannot, therefore, say that a cubic foot is made up of cubic inches.'
Nie. I see your meaning now. I give up the words 'into the same position.'
Min. I haven't quite done with points yet. I find an assertion that they never jump. Do you think that arises from their having 'position,' which they feel might be compromised by such conduct?Nie. I cannot tell without hearing the passage read.
Min. It is this:—'A point, in changing its position on a curve, passes, in moving from one position to another, through all intermediate positions. It does not move by jumps.' (p. 12.)
Nie. That is quite true.
Min. Tell me, then—is every centre of gravity a point?
Nie. Certainly.
Min. Let us now consider the centre of gravity of a flea. Does it—
Nie. (indignantly) Another word, and I shall vanish! I cannot waste a night on such trivialities.
Min. Forgive me. I drop the flea. My next remark shall be serious. I wish to point out to you the illogical tone of the book. I do not say that the instances I am going to give are crucial or fatal to the argument. But, however unimportant, and however easily corrected, they will, I think, justify me in asking 'Is a text-book, which contains such loosely reasoned arguments as these, to be trusted?'
My first selection is § 52, p. 23. For brevity's sake I shall omit superfluous words. The passages in parentheses are interpolations of my own.
(see Henrici, p. 23.)
'If we conceive a Plane (and a point A chosen anywhere in Space; then, either the Plane already passes through A, or) we may move it until a point on it comes to A, which has been chosen anywhere in Space. (If we now fix a second point B; then, either the Plane already passes through B, or) if we keep A fixed we may turn the Plane about it, until the Plane comes to pass also through B, likewise chosen arbitrarily in Space. (If we now fix a third point C; then, either the Plane already passes through C, or) we may still move the Plane, as only two points of it are fixed, by turning it about the Line joining them, until the Plane passes through C, chosen arbitrarily, like A and B. Thus it appears that we may place a Plane so as to pass through three points, A, B, C, chosen anywhere in Space.'
You accept that, interpolations and all?
Nie. Certainly.
Min. Omit the interpolations, and what do you say of it then?
Nie. It remains true. The three successive movings do no harm, but they are not always necessary.
Min. Would this statement be correct? 'Three "movings" are generally necessary: but there are three exceptions. If the Plane at first passes through A, the first "moving" is unnecessary; if, after being made to pass through A, it be found to pass through B also, the second "moving" is unnecessary; and if, after being made to pass through A and B, it be found to pass through C also, the third "moving" is unnecessary'?
Nie. Certainly.
Min. You would not, on finding some one 'moving' unnecessary, call it 'an open question' whether the result were attainable?
Nie. What? When it is already attained? By no means.Min. Now read this, at p. 23.
(hands the book)
'But if C happens to lie on the Line joining A and B, then a Plane through A and B, which did not pass through C, could never be made to pass through C by being rotated about A and B; for if it did contain C in one position, it would contain it in all positions, as this point would remain fixed during rotation.' What do you say to that?
Nie. Well, it is his way of discussing your third exception. Of course, when he talks of 'a Plane through A and B, which did not pass through C,' he is describing a nonentity: but it is all logical as an argument.
Min. What kind of argument?
Nie. (doubtfully) I should call it a—kind of—Reductio ad Absurdum.
Min. I don't wonder at your hesitation. A thoughtful boy might read it thus:—'then a Plane through A and B, which did not pass through C (but no such Plane can exist!), could never be made to pass through C by being rotated about A and B (why, it needs no 'making'!); for if it did contain C in one position (which it does!), it would contain it in all positions (which also it does!)'
You and I can recognise the Reductio ad Absurdum—though so abnormal and hideous—which the writer intends. But what do you think would be the effect, on a thoughtful boy, of a course of such arguments, where he is expected to accept as data what he knows to be absurd, and to recognise as an absurdity what he knows to be a necessary truth?
Nie. At first, Mania: ultimately. Dementia.
Min. Now read Mr. Henrici's deduction from this fearful argument, at p. 24.
'We ought, therefore, to limit the conclusion arrived at as follows:—Through three points which do not lie in a Line we may always pass a Plane. Whether a Plane may be drawn through three points which do lie in a Line, remains for the moment an open question.'
Are you prepared to back that statement? Is it an 'open question'?
Nie. I cannot say that it is.
Min. Now here is a most curious bit of bad Logic. (reads)
'If two Planes have two points, A and B, in common, they must necessarily have more points in common. For, since each extends continuously without limit, a point moving in the one Plane through A or B will cross the other Plane at this point;' (p. 25.)
I pause to ask—will it necessarily do so? How if it moved along their Line of intersection?
Nie. That is an exception, I grant.
Min. (reads) 'hence one Plane will lie partly on the one and partly on the other side of the second Plane. They must therefore intersect.'
Now the conclusion—that the Planes intersect—is undoubtedly true, so long as we assume that, by 'two planes,' the writer means 'two different Planes.' But does it follow from the premisses? Have the words 'hence' and 'therefore' any logical value?Nie. I fear not.
Min. At p. 74 I observe 'If two Lines be each perpendicular to a third, they will be parallel to one another.' This is not true. They might be coincidental. The same mistake is made in p. 75.
Now comes a wonderful specimen of slipshod writing. 'We understand by the angles of a Polygon those angles of which the part near the vertex lies within the Polygon.' Does not this oblige us to contemplate an angle as consisting of two parts—one 'near the vertex,' the other further off?
Nie. Undoubtedly.
Min. And if either part were gone, the angle would be less?
Nie. (uneasily) It would seem so.
Min. And this might be effected by shortening the Lines, so that they would not reach beyond the region 'near the vertex'?
Nie. I fear you have got us into a corner. Be merciful!
Min. You mean that I have driven you into 'that part of an angle which lies near the vertex.' Well, you may come out now. We will seek 'fresh fields and pastures new.'
At pages 91 to 96 I find no less than forty-six theorems on Symmetry, arranged in two columns—one headed 'Axial Symmetry,' the other 'Central Symmetry .' Here is a specimen pair, at p. 95.
| 'Corresponding Polygons are congruent but of opposite sense.' | 'Corresponding Polygons are congruent and of like sense.' |
I hardly know which to pity most—the master who has to teach these Theorems, or the boy who has to learn them!
But I have neither the 'one-way spread with moments as elements' nor the 'three-way spread with points as elements' to—
Nie. (gasping) What are you talking about?
Min. Excuse me. I fear I am getting demoralised. I meant to say—I have neither the time nor the space to criticise this book throughout.
I will, however, try to sum up its faults in a general description.
'Olla Podrida' is perhaps the best name for it, its contents are so hopelessly jumbled together. Most of the Axioms, and all the Theorems, are without numbers, and, as there is no index, the difficulty of finding them when wanted is obvious: and none the less that they are imbedded in oceans of 'padding.' Dip into the book anywhere, and you find yourself in the midst of some discursive talk, which perhaps culminates in an Axiom. Then perhaps comes a Definition. Then comes a little more talk, which, after appealing to sentiment, or probability, or some other motive degrading to Pure Mathematics, gradually becomes more and more logical, and at last warms into a regular proof—but of what? The reader has no warning as to what is to be proved. Unsuspectingly he glides on with the stream, till with a crash he is landed on an enunciation, and finds himself committed to an entire Theorem. This singular writer always reserves the enunciation for the end of the Proposition. It may be prejudice, but I cannot help thinking that Euclid's plan—of first clearly stating what he is going to prove and then proving it—is to be preferred to this conjurer's trick of 'forcing a card.'
The book is, I think, very hard for beginners to master: the majority of the new Theorems are much more fitted for 'exercises,' than to be embodied in a text-book: and, to crown all, the ambitious attempt to construct a proof of Playfair's Axiom is, as we have seen, a lamentable failure.
I think I cannot better conclude my review of this book than by giving you, in two parallel columns, Euclid's Props. I. 18, 19, and Mr. Henrici's proposed substitute for them, at p. 107.
| Euclid. | Henrici. |
The greater side of a Triangle is opposite to the greater angle.
 |
 |
| Let ABC be a triangle having AC > AB: then shall the angle ABC be > the angle C.
From AC cut off AD equal to AB; and draw BD. Then, ∵ AB = AD, ∴ the angle ABD = the angle ADB; but the angle ADB is exterior to the Triangle BCD, and ∴ < the angle C; ∴ the angle ABD also > the angle C; much more is the angle ABC > the angle C. Q. E. D. |
Let us now suppose a Triangle ABC, in which the bisector of the angle BAC is not an axis of symmetry. Then the contra-positive form of the theorem of § 162 tells us that AB is not equal to AC, that the angle B is not equal to the angle C, and that the bisector AD of the angle A is not perpendicular to BC, and hence, that the two angles ADB and ADC are unequal. Between these angles there exists the relation
'the angle ABC + the angle BDA = the angle C + the angle CDA,' for each sum makes with half the angle A an angle of continuation. Hence it follows that, if the angle ABC > the angle C, the angle BDA < the angle CDA. |
The greater angle of a Triangle is opposite to the greater side.
 Let ABC be a Triangle having the ∠ B > the angle C: then shall AC be > AB. For if not, it must be equal or less. It is not equal, for then the angle B would = the angle C. It is not less, for then the angle B would < the angle C. ∴ AC > AB. Q. E. D. |
If we now fold the figure along AD, then AB will fall along AC; and B will fall between A and C if we suppose that AB is the shorter of the two unequal Lines AB and AC. The line DB therefore takes the position DB' within the angle ADC. But the angle AB'D, which = the angle ABC, is exterior to the triangle DCB' and ∴ > the angle C.
Conversely, if the angle ADB < the angle ADC, the line DB will fall within the angle ADC, and ∴ B will fall between A and C; that is, AB will be less than AC. This always happens (see above) if the angle ABC > the angle C, for then the angle BDA < the angle ADC. Theorem. In every Triangle the greater side is opposite to the greater angle, and conversely, the greater angle is opposite to the greater side. |
Nie. (with a cunning smile) I don't think I could find such a schoolmaster.
Min. Ah, crafty man! You evade the question! I can't resist giving you just one more tit-bit—the definition of a Square, at p. 123.
'A quadrilateral which is a kite, a symmetrical trapezium, and a parallelogram is a Square'!
And now, farewell, Henrici! 'Euclid, with all thy faults, I love thee still!' Indeed I might say 'with twice thy faults,' or 'with thrice thy faults,' if the alternative be Henrici! (returns the book, which Niemand receives in solemn silence.)