Reviews of ‘Euclid and his Modern Rivals,’ with the Author’s remarks thereon.
From the ‘English Mechanic and World of Science,’ May 2, 1879.
After serving for upwards of two thousand years as essentially the standard text-book, and as an unsurpassable and unsurpassed system of mathematical gymnastics to millions, it has been found or asserted in these latter days that the ‘Elements of Geometry’ of the mighty Alexandrian are deficient in some of the most rudimentary qualifications that they have been heretofore conceived to possess; and that the sooner they are remodelled or superseded altogether the better for the rising generation of mathematicians, and for the cause of education generally. Thus it has come to pass that Legendre among the French; Pierce, Chauvenet, and Loomis among the Americans; and Cooley, Cuthbertson, Morell, Reynolds, Willock, Wilson, and Wright among our own country-men, have each and everyone embodied his own idea of the way in which geometry should be taught, in treatises designed to supplant Euclid’s time-honoured work entirely. Nay, lest the authority of any individual should fail to prevail against that of the great Greek geometer, an Association for the Improvement of Geometrical Teaching has been formed, and, under Mr. Wilson’s editorship, has issued a ‘Syllabus Manual’ of its own.
Now it is to combat the views propounded by the authors whom we have just named, and to show what indifferent substitutes they have provided for the book they attack, that Mr. Dodgson’s remarkable work has been written—a work which takes the astonishing form of a drama in four acts and ten scenes! with six appendices, four of them bristling with formulæ. The argument of the play (if play it can be called) is briefly this: A college examiner, Minos, utterly wearied out by marking papers containing the most heterogeneous and illogical ‘proofs’ of the same theorem, falls asleep over his work; and to him appears the phantasm of Euclid himself. In the course of the conversation which ensues Euclid stoutly defends his own method of treatment and arrangement of propositions. The dramatic interest of this, by the way, is rather marred by the introduction of a somewhat repulsive symbolism (involving perpetual reference to the pages on which it is first explained), the notion of which is obviously taken from one of the common text-books on logic; but, putting this on one side, the ideas intended to be conveyed are clearly and definitely expressed, and Euclid quite holds his own. ‘Audi alteram partem,’ however, and it becomes necessary that Minos should hear what the modern rivals of the great Alexandrian have to say for themselves. But they being still in the flesh, the spirit of a certain German Professor, Niemand, is sent, who produces their works and acts as their advocate seriatim. We cannot, we confess, congratulate those whom we may provisionally call the heterodox party on the strength of this spiritual champion of theirs, whose advocacy strongly suggests to us the old story of the youthful and nervous barrister, who, after stammering out three or four times, ‘My lord, my unfortunate client; I say, my lord, my unfortunate client,’ was summarily arrested by the exclamation of the judge, ‘Go on, Mr. Smith; so far the Court is quite with you:’ Herr Niemand being in truth but a kind of straw giant set up by our author merely to be knocked down again. We should really prefer to hear the personal replies of Messrs. Cuthbertson and Wilson to Mr. Dodgson’s attack on their principles and methods, rather than those of their ghostly representative. It seems to us that they might one or both take much more formidable objections to certain pieces of hypercriticism on the part of ‘Minos’ than the spirit of the German professor does on their behalf.
On the whole, though, we regard our author as having triumphantly proved that, so far, no work has been produced which is comparable with Euclid’s immortal ‘Elements,’ as an introduction to geometry for beginners. Legendre’s proofs, for example, are very beautiful, but he treats of parallels by methods involving infinite series; a method presupposing a kind and amount of knowledge not ordinarily possessed by lads at the age at which they usually begin the study of Euclid. The chief points on which Mr. Dodgson is at issue with the innovators are the separation of problems and theorems; the substitution of Playfair’s Axiom for Euclid’s XIIth one; the treatment of parallels by (a) angles made with transversals; (b) by equidistances; and (c) by direction, as contradistinguished from the Euclidean method. He further assails the modern treatment of lines and angles; and the attacks on Euclid’s style, constructions, and demonstrations. He is particularly severe upon Mr. Wilson for the manner in which he handles the subjects of angles, the direction of lines, and parallels; and there can be no doubt that he does convict that writer both of logical inaccuracy and of loose writing, both defects fatal to a book claiming to supersede the time-honoured ‘Elements.’ Mr. Cuthbertson, too, by no means comes unscathed out of the fray; while our author points out how Mr. Morell and others have proved ‘theorems’ by illogical processes which furnish no proofs at all! His chief praise is given to the American books; but these he shows possess the cardinal fault of inapplicability to rudimentary instruction, however elegant and adapted to the advanced student their mode of treatment may be. In two out of his six appendices, he invokes the high authority of Mr. Todhunter and of the late Professor De Morgan, in favour of the views he has been advocating; and the testimony they give is as valuable as it is unhesitating. The remaining appendices call for no especial notice.
We fear that the brief account which we have given of Mr. Dodgson’s curious book must convey but an imperfect idea of its contents. This, however, has its origin in the nature of the work itself, which is indescribable, save at a length to which our limited space prevents us from extending this notice. It is essentially a book to be carefully and deliberately read, as it doubtless will be by everyone interested in the teaching of geometry. Here and there the author has his adversaries unmistakably on the hip; in other places, we must repeat our impression that Herr Niemand has scarcely done so much for his clients as they might fairly have been expected to do for themselves. That the writers attacked will, some of them, reply to their assailant may only reasonably be expected; and we shall look forward with no little interest to any response which one or more of them may make. Meanwhile, we may reiterate our conviction that Mr. Dodgson has shown irrefragably that, whatever merits individual works may possess, there has not as yet appeared one destined to supplant Euclid’s ‘Elements of Geometry,’ as a means of instruction for the beginner; and we hold that pro tanto he has rendered good service to the cause of mathematical education, and to that of intellectual discipline generally.
From the ‘Saturday Review,’ May 10, 1879.
The movement for innovation in the teaching of elementary geometry has gone so far that the discussion is no longer a merely academical one, and Mr. Dodgson has thought it high time for a champion who is prepared to defend Euclid against all comers to arm himself and enter the lists. Mr. Dodgson has brought great knowledge and acuteness to his task, but we must regret the form in which he has cast his book—not on the score of his ‘abandoning the dignity of a scientific writer’ by putting his argument into dialogues between the ghost of Euclid, an examiner, and an imaginary German Professor—but, because to our mind the effect is to make the argument much harder reading than it would be otherwise. It is chopped up and frittered away, and what Mr. Dodgson has to say on any one point must be pieced out from half-a-dozen scraps of imaginary conversation. Mr. Dodgson’s own account of his method is this:—‘It is presented in a dramatic form, partly because it seemed a better way of exbibiting in alternation the arguments on the two sides of the question; partly that I might feel myself at liberty to treat it in a rather lighter style than would have suited an essay, and thus to make it a little less tedious and a little more acceptable to unscientific readers.’
We agree that the subject is one which might very fairly be treated by way of dialogue, provided the writer were impartial or versatile enough really to exhibit the arguments on both sides. But this Mr. Dodgson has hardly attempted. His phantasm of Herr Niemand, who brings up for discussion and judgment the various treatises of Euclid’s modern rivals, is a very poor ghost indeed—a mere ninepin of a ghost who stands up only just enough to be knocked down. The work is an argument for Euclid all through, and would have been more forcible and intelligible if presented consecutively. As to the ‘lighter style’ making the book ‘less tedious’ and ‘more acceptable to unscientific readers,’ that is a matter of taste; to our own taste few things are more tedious or less acceptable than to have the tenor of a closely reasoned discussion constantly interrupted by small jokes. [Against this remark of the Saturday Review my be set the following from the Spectator: ‘We only wish that Mr. Dodgson could have seen his way to putting more jokes into his pleadings than he has.’ Similar advice has reached me from private friends—some pleading earnestly for ‘more jokes,’ others, with equal earnestness, for ‘fewer jokes.’ It is not easy to express in words how practical and helpful I have found such advice, in preparing my second edition.] Certainly the little repartees exchanged between Euclid, Minos, and Niemand will not help any one to master the elaborate apparatus of tabular comparisons and symbolic vocabulary by which Mr. Dodgson brings into one view all hitherto imagined ways of treating the vexed doctrine of parallels. This last piece of work, on which Mr. Dodgson must have spent infinite pains and minute attention, is alone sufficient to make reference to his book almost indispensable for whoever treats the subject after him. Whether the symbolic notation really has a convenience in use proportionate to the author’s trouble in inventing it, and the reader’s in learning it, is a question on which we hesitate to offer a positive opinion. To some extent Mr. Dodgson seems to have been driven to the adoption of new terms by the ambiguous manner in which the word parallel is used by different writers. Of course we cannot follow him through his exhaustive account of the various plans which have been put forward as improvements on Euclid’s method in this point. But we must note one rather important omission. A review of Mr. Wilson’s book on Elementary Geometry by De Morgan is quoted in an appendix; but there is no reference either to De Morgan’s admirable article on Euclid in the Dictionary of Biography and Mythology, which is probably less known to mathematical readers than it ought to be, or to his article on ‘Parallels’ in the Penny Cyclopædia. What De Morgan says in the last-mentioned article is so much to the purpose that we make no scruple of repeating it here:—
‘Euclid obviously puts the whole difficulty into an assumption; which, though the most direct course, is not that which is best calculated to give the highest degree of evidence to geometrical truths. For it is a more obvious proposition that two lines which intersect one another cannot both be parallel to a third line, and, this being granted, Euclid’s axiom readily follows. If it should be objected that this is merely Euclid’s axiom in another form, it is replied that the form is a more easy one, and therefore preferable; just as it would be wiser to assume “Every A is B and every B is A,” than the identical but more complicated proposition, “Every A is B, and everything which is not A is not B.”’
The ‘more obvious proposition’ thus recommended by De Morgan is Playfair’s axiom; and it so happens that the point of his recommendation is not very successfully met by Mr. Dodgson. It is an easy victory to show that the axiom in this form is equivalent to Euclid’s; and then Mr. Dodgson proceeds to give reasons why Euclid’s should be preferred. He makes out that Euclid’s axiom is actually easier, since it puts before the learner ‘a Pair of Lines, a transversal, and two angles whose sum is less than two right angles—all clear positive conceptions’; while ‘Playfair requires him to realize a Pair of Lines which never meet, though produced to infinity—a negative conception which does not convey to the mind any clear notion of the relative position of the Lines.’ This appears to us to be little more than a play upon words. Positive conceptions are not necessarily easier to grasp than negative ones; and the picture of two parallels, whether on paper or in the mind, is a much simpler object of intuition than that of a pair of straight lines met by a transversal which makes two interior opposite angles less than two right angles. [When this reviewer talks of ‘the picture of two parallels,’ he cannot, surely, be using the word in Euclid’s sense? Perhaps he means ‘equidistant’: the picture of two equidistant lines, of (say) six inches long, would be a fairly simple object of apprehension (‘intuition’ is the apprehension of facts, not of things): but who ever succeeded in forming a picture, whether on paper or in the mind, of two lines produced to infinity? For that is what you must do, to realise Euclid’s parallels as ‘a picture.’ A pair of lines, a million miles long, would be of no use whatever: the fact that they had not met, so far, would be no evidence whatever that they would not meet afterwards—but would have the exactly opposite effect; it would leave a future meeting still possible, whereas, if they had met in the first million miles, they could not possibly do so again.]
But a more formidable objection is behind, which Mr. Dodgson fully brings out only when he comes to discuss Mr. Wilson’s treatment of the subject in detail. It may be most clearly seen by substituting for Playfair’s axiom the equivalent statement:—Through a given point outside a straight line only one parallel can be drawn to it. This at once raises the question, What business have you to assume that any parallel can be drawn? in other words, that parallels can and do exist in plane geometry, and that there is no external point through which a parallel cannot be drawn? The assumption, be it observed, is not made by Euclid. And we may further observe that it is not such a small one as it looks, especially in the light of modern geometrical speculations. [But who in the world wants to make this assumption?] For it results from the work of Lobatschewsky and others that our actual geometry is not an elucidation of eternal and immutable and unique relations, but is rather in the nature of a purely physical science. That is to say, it is the investigation of properties of space, or of things in so far as they occupy space, which might quite conceivably have been different. A consistent geometry (though of course inapplicable to our real experience) can be, and has been, founded on the categorical denial of Playfair’s axiom. Euclid’s geometry is the science, not of space absolutely, but of a particular kind of space; and in this view the doctrine of parallels lays down very characteristic and important properties of that kind of space. When we are investigating the properties of anything, our knowledge of them is not thoroughly scientific until it is connected, as far as possible, by proofs. We must know not only that the properties co-exist, but how far one implies the other. Now Euclid does not assume, but proves, the real existence of parallels to be a property of the space he is dealing with; and here he has a great advantage over most of the innovators. They commit precisely the same oversight of making a large tacit assumption which they are ready enough to charge Euclid with on other occasions. [This is news indeed! ‘Most of the innovators,’ as this reviewer seems to believe, assume Euc. I. 27!] The substance of the objection would be the same without appealing to imaginary geometry. But it appears to us (paradoxical as it may sound) that the considerations above suggested give it more reality. For one sees that it is a question, not of logical arrangement, but of real physical explanation. The assumption of Playfair’s axiom in the lump is objectionable in precisely the same way that it would be objectionable in physics to assume the conservation of energy as an axiom, and also to assume that a perpetual motion is impossible. Still the objection is not insuperable. [Oh joyful tidings! We breathe again.] It is possible to prove the general part of the alternative form of Playfair’s axiom—namely, that a parallel can always be drawn to a given straight line through an external point—before assuming the special part, on which depends the peculiar quality of the space we have the happiness to live in, namely that only one parallel can be so drawn. This was done by Mr. Hirst in a course of lectures on elementary geometry given by him several years ago. [This is perhaps the most extraordinary statement ever made by a writer professing to discuss a mathematical subject. It is as though one were to say ‘The all-important fact, that seven times eight is fifty-six, was pointed out by Colenso, in his treatise on Arithmetic’! Mr. Hirst, thus kindly immortalised, may perhaps modestly reply ‘but it was done 2000 years before I was born!’] Whether it is done in the published works of any of the ‘modern rivals’ whose claims are discussed by Mr. Dodgson is more than we can say. [Why, of course it is! By every one of them, except Pierce, Willock, and Wilson, who do not need it, as they use another definition of ‘parallel.’ Does the reviewer know of any writer on elementary geometry, who uses Euclid’s definition, and has omitted to prove this proposition?]
Mr. R. P. Wright’s auxiliary proposition (which Mr. Dodgson takes as a specimen of his treatise, in order to pass on it a rather supercilious criticism, apparently without seeing what it is meant to lead to, and what difficulties are being encountered) is that one, and only one, perpendicular can be drawn to a straight line from an external point. [My criticism (p. 153) was that the theorem is proved in a ‘wordy and unscientific style’—to which statement the questions ‘what it is meant to lead to,’ and ‘what difficulties are being encountered,’ are totally irrelevant. Yet I should like, very much, to know what difficulties are being encountered—in proving a proposition which is simply a deduction—of infantine simplicity—from Euc. I. 17! One is almost tempted to think that scientific books are occasionally reviewed by writers who do not entirely understand what they are talking about!] But he has to prove it by folding over the paper, which is a proceeding of doubtful fairness, and in fact involves assumptions about the nature of space of three dimensions. If such assumptions were made openly from the first; if surfaces and lines were defined and conceived as boundaries; if projections and other modern methods were freely introduced as soon as they could be made useful; if, in short, geometry were frankly treated as a physical science—then we should have before us a scheme of innovation really worth discussing.
From the ‘Scotsman,’ May 15, 1879.
Mr. Dodgson’s book on Euclid and His Modern Rivals will, or at least should, command a good deal of attention in academic circles. It is an attempt to demonstrate the inutility and positive mischief of having a large number of text-books for the teaching of elementary geometry. Mr. Dodgson does not take his stand on any narrow principle, or oppose modern attempts to supersede Euclid merely from a dislike to innovation. What he tries, and with very considerable success, to show is, that none of the recent text-books contain anything of importance that is not in Euclid; that the axioms and definitions which have been proposed in addition to, or in substitution for, those laid down by him will not, for the most part, stand the test of close examination; and that in simplicity and logical sequence of method he has the advantage of all his rivals. In order to establish these points, Mr. Dodgson subjects the dis tinctive features of the modern text-books of geometry to a very close and strongly reasoned criticism and comparison with Euclid; but nevertheless his book has not the solemnity and dryness of tone which might be thought inseparable from the treatment of such a subject. It is cast in dialogue form, and is full of a quaint satiric humour, which does not in the least diminish its scientific value, but which certainly makes it much pleasanter reading than most books on mathematical topics.
From the ‘British Quarterly Review,’ July, 1880.
The object of the present work, to use the words of the author, ‘is to furnish evidence, first, that it is essential, for the purpose of teaching or examining in elementary geometry, to employ one text-book only; secondly, that there are strong a priori reasons for retaining, in all its main features, and especially in its sequence and numbering of propositions, and in its treatment of parallels, the manual of Euclid; and, thirdly, that no sufficient reasons have yet been shown for abandoning it in favour of any one of the modern manuals which have been offered as substitutes.’ The evidence is presented in a dramatic form, as affording a better opportunity of marshalling the arguments on both sides of the question, and of a more lively and interesting treatment than the usual essay-form. The main object of the work is the vindication of Euclid’s masterpiece, with certain modifications, against its modern rivals. The dramatis personæ who uphold this view are borrowed from ancient history, e. g., Minos, Rhadamanthus, Plato, Aristotle, and, of course, Euclid himself, and the views of the modern rivals are represented by Herr Niemand, who is introduced as carrying a pile of books, the works of the following authors: Legendre, Cooley, Cuthbertson, Wilson, Pierce, Willock, Chauvenet, Loomis, Morell, Reynolds, and Wright. The arguments brought forward in favour of geometrical text-books are discussed with great clearness and acuteness, and, what is more, with a good amount of sound common-sense, while ample justice is done to the advantage of substituting different methods of proof from those employed by Euclid himself. The perplexing difficulties which harass examiners and examined alike are vividly described, and very satisfactory reasons are given why the text-book which is adopted as the standard should follow more closely the line of argument pursued in the original masterpiece. For details we must refer our mathematical readers to the work itself, the reading of which will amply repay them on whichever side of the question they may have taken their position. We ourselves agree with Mr. Dodgson, that no modern text-book has as yet made good its claim to the place of the ancient geometrician. A text-book constructed somewhat after the plan suggested by the author would be an unmistakable boon both to teachers and taught.
From the ‘Journal of Science,’ July, 1879.
The chief points of attack on Euclid’s Modern Rivals are Mr. Wilson’s two works—‘Elementary Geometry’ and the ‘Manual founded on the Association Syllabus.’ The author makes comparatively short work of Legendre’s book as unsuited to beginners, though doubtless valuable to advanced students; and of Cooley’s, in which a certain theorem breaks down through a faulty definition of parallel lines. [‘A certain theorem’! This is as if a builder were to say ‘They actually presumed to condemn the bridge I had built, because a certain stone had given way!’ (Said ‘certain stone’ being the key-stone.)] But to Mr. Wilson’s two Manuals he devotes nearly a third of his volume. Much of the criticism on these, however, is mere cavilling: for instance, at page 177, Minos says, speaking of Wilson’s ‘Syllabus’ Manual:—‘At p. 57 I see an Exercise (No. 5). “Show that the angles of an equiangular triangle are equal to two-thirds of a right angle.” In this attempt I feel sure I should fail. In early life I was taught to believe them equal to two right angles—an antiquated prejudice no doubt; but it is difficult to eradicate these childish instincts.’ This is mere straw-splitting; strictest accuracy would of course require the insertion of ‘each’ before ‘equal,’ but if the sum of the interior angles had been intended to be understood ‘together’ before ‘equal’ would have been absolutely necessary. [Thus we never say ‘2 and 2 are 4.’]
At page 160 there is a criticism on the definition of a right angle as given by the Association for the Improvement of Geometrical Teaching in their Syllabus. This is—‘When one straight line stands upon another straight line, and makes the adjacent angles equal, each of the angles is called a right angle.’ Since the Association Syllabus admits of angles equal to or greater than two right angles, this is open to the objection that it does not debar the case in which one line stands on the end of the other, making the adjacent angles equal to one another, and to two right angles as right angles are generally considered. That is certainly a grave objection, but the same applies equally to Euclid’s definition, or else a proof must be supplied, which is not that in the case mentioned the two lines are in one and the same straight line; and so this interpretation is debarred by Euclid’s limiting clause.’ [This is beyond my comprehension.]
In fact, though Mr. Dodgson’s book is interesting and often witty, he fails to prove his point, because he takes a one-sided view of the question, and merely exhibits the blunders of Euclid’s Rivals without balancing them against Euclid’s own. [The wonderful calmness of this assumption is worthy of all praise. Perhaps the writer will kindly furnish me with a list of ‘Euclid’s blunders’?] Besides which—though perhaps it is more readable than an essay—a dialogue does not seem to be the clearest form for setting forth arguments and facts. [Compare with this the remark of the Saturday Review (see p. 335): ‘We agree that the subject is one which might very fairly be treated by way of dialogue.’ So hard is it to please everybody.]
From the ‘Nature,’ July 10, 1879.
By a curious chance these two works reached our hands nearly on the same day, and as Mr. Dodgson devotes a great portion of his space (62 pp.) to the consideration of Mr. Wilson’s Geometries, we have thought it well to notice the two authors at the same time. As however it is patent from the fact of Mr. Wilson’s work having reached a fourth edition, that his method is not unknown to, and, may we add, not unappreciated by, a large section of mathematical teachers, we shall at once pass on to a consideration of Mr. Dodgson’s book, only noticing Mr. Wilson’s book in connection with the criticisms put forward in ‘Euclid and His Modern Rivals.’
A few words by way of introduction. Mr. Dodgson has been a teacher of geometry at Oxford, we believe, for nearly five-and-twenty years, and during that time has had frequent occasion to examine candidates in that subject. For a great part of the above-stated period things went pretty smoothly, and King Euclid held undisputed sway in the ‘Schools;’ but eleven years ago a troubler of the geometrical Israel came upon the scene, and read a paper before the Mathematical Society, entitled ‘Euclid as a Text-Book of Elementary Geometry.’ The agitation thus commenced acquired strength, and at length, in consequence of a correspondence carried on in these columns, the Geometrical Association was formed. A prime mover in this matter was that Mr. Wilson who wrote the paper, and subsequently brought out the geometry cited. Mr. Dodgson is one of the gentlemen opposed to this change, and the moving cause of the present Iliad is the ‘vindication of Euclid’s master-piece.’ Another consequence of the agitation is that many have tried their prentice hands on the production of new geometries—‘rivals,’ our author calls them—‘forty-five were left in my rooms to-day.’ Can we wonder then, that, his soul being stirred within him, he should overhaul a selection of them to see what blots he could ‘spot’ in them? He might well have taken for his motto one once familiar to us—
‘If there’s a hole in a’ your coats,
I rede ye tent it;
A chiel’s amang ye takin’ notes,
An’ faith he’ll prent it!’
Our author’s criticism takes a peculiar form, but we shall not blame him for this, for he has afforded us much amusement, and we quite hold with the Horatian line he cites in extenuation of his mode of procedure: ‘Ridentem dicere verum quid vetat?’ We believe he has made a good many hits, but at times his wit, we think, has led him too far. We shall not, however, here give any account of his plot—we prefer to refer our readers to the work itself—but confine our notice to the remarks upon Mr. Wilson’s books, and upon Mr. Morell’s ‘Euclid Simplified.’
Mr. Dodgson devotes forty-eight pages to Mr. Wilson’s ‘Elementary Geometry’ (second edition, 1869). We can hardly see why so much space should be devoted to a work which seems tacitly to have been withdrawn by the author, or, at any rate, to have been considered inferior to the work under review. [I have given my reasons at the foot of p. 171]. Is it that the ‘scene’ was written some time since, and was considered to be too good to be sacrificed? Happily it is not our business to defend Mr. Wilson’s views on ‘direction’; he is perfectly competent to defend his own views, and no doubt, should he see fit, will do so at the right time.
We pass over many passages we had marked, with saying that in many cases the objections are sound but trivial. Objection is taken to Mr. Wilson’s remark, ‘Every theorem may be shown to be a means of indirectly measuring some magnitude,’ and Niemand abandons ‘every’. We think, however, that Niemand might have made a better fight of it and suggested that what is intended is that, for instance, all the theorems of the first book are directly or indirectly required for the proof of the 47th Proposition, which is surely a proposition concerned with the measurement of magnitude.
A word or two on Morell’s (J. R.) ‘Euclid Simplified.’ It is very easy work to pick this little book to pieces, but we cannot understand a statement of Mr. Dodgson’s on p . 148. Of the proposition ‘Every convex closed line enveloped by any other closed line is less than it,’ he says the method used fails, ‘as of course all methods must, the thing not being capable of proof.’ We cannot call to mind any English text-book in which the proposition is proved, but there is what we have thought was a proof in Sannia and D’Ovidio’s ‘Elementi di Geometria,’ p. 32. [The sentence, from which the above words are quoted, is ‘all depends on our proving the perimeter less than the perimeter , which this method has failed to do—as of course all methods must, the thing not being capable of proof.’ Surely, according to the ordinary laws of English grammar, the antecedent to ‘which’ is the clause immediately preceding it. Why should the reviewer go a whole page back in search of an antecedent?]
From ‘The Examiner,’ Oct. 25, 1879.
It is generally known that for some time past dissatisfaction has been felt with Euclid’s Elements as a text-book on Geometry. Professor Sylvester was among the first to suggest that Euclid ought to be banished from our schools; and the cry has been taken up by a small host of modern rivals, who have gone so far as to publish books intended to supersede the time-honoured work of the Alexandrian geometer. The ‘unsuggestiveness,’ the ‘tediousness,’ and the circuitous and cumbrous methods’ of Euclid had been tacitly admitted long before any serious attempt had been made to displace the Elements by any other text-book. The vis inertiæ, or difficulty of change, seemed to render Euclid’s position practically secure, and consequently the various attacks to which from time to time his work was exposed remained unchallenged, even by his supporters. The fear of being thought old-fashioned, prejudiced, or indifferent to progress, prevented many, who honestly believed that they owed to the discipline of Euclid their skill and accuracy in reasoning, from undertaking his defence. Moreover, the example of foreign teachers was not without its influence on Englishmen, who found themselves alone in using Euclid as a school text-book.
Thus it came about that attempts were made to introduce into our schools systems of geometry, diverging more or less widely from Euclid’s Elements; and such well-known teachers as Wright and Wilson ventured to publish works that were avowedly intended as substitutes for Euclid. The cry for reform became so loud that an Association was formed for the express purpose of improving geometrical teaching, and a ‘Syllabus’ was issued setting forth the new lines on which geometrical teaching ought to proceed. Euclid’s supporters had now, at length, the opportunity of declaring themselves. Their enemies had written books. Mr. Wilson’s work was subjected to the cutting criticism of the late Professor De Morgan. Mr. Todhunter, the well-known writer of many text-books, and author of an edition of Euclid free from any modern improvements, published an essay in defence of his master; and now, whilst the battle is still raging hotly, Mr. Dodgson writes a book in which, single-handed, he fights with the most conspicuous of Euclid’s rivals, and professes to have slain them all.
Whilst these changes have been under contemplation, and partly adopted, (for some schools have introduced modern methods of teaching this subject,) professional examiners have been sorely troubled. Fear and trembling have taken hold of them. For if candidates are to be permitted to use other works than Euclid’s in preparation for their examination, how is it possible for the examiner to test the validity of a proof presented to him? With Euclid for a text-book, examination was rendered easy and satisfactory. Half-a-dozen propositions were set, three or four deductions were added, which were seldom answered, and the candidate’s knowledge of geometry was infallibly ascertained. No mechanical appliance for examination could be more satisfactory. But if the order of Euclid’s propositions be once changed, if other proofs be substituted for the time-honoured demonstrations of past generations, examinations in this subject must become hopelessly confusing. Examiners would be expected to know every system of geometry which candidates might have studied; and candidates, by disregarding Euclid’s established sequence of proofs, would be free to indulge in the fallacy of petitio principii without fear of detection. These considerations have helped to rouse Euclid’s defenders from their apathy, by showing that the requirements of examinations necessitate the retention of Euclid as a school text-book. The Universities of Oxford and Cambridge have realised this fact; and as they exert, through their local examinations, considerable influence on school-teaching, Euclid’s rivals have fared but badly. So decided a corporate feeling exists on this question, that we find in the Report of the Oxford and Cambridge Schools Examination Board for last year the following remark on the subject:—‘At one or two schools there is some departure from Euclid’s methods. This practice has generally led to confusion and inaccuracy without any compensating advantage in power of working riders.’ This observation cannot be without effect on those schools which have ventured to tread the new paths, and will doubtless drive them back into the beaten track, in which ‘the propositions are generally written out clearly and accurately’ (Report, p. 9). Rebellion against the supremacy of Euclid must be crushed out at all costs! There can be no doubt that Mr. Dodgson deserves well of University examiners; but it is open to question whether the advocates of improved geometrical teaching will give that weight to this form of argument which Mr. Dodgson and his friends attach to it. [It is in the interests of teachers, rather than of examiners, that I have written.]
On first opening Mr. Dodgson’s book we must acknowledge to having felt somewhat uncertain whether the defence of Euclid against his rivals was intended as a piece of serious reasoning, or was a mere attempt on the part of the author to give us ‘a glimpse of the comic side of things,’ even in the study of Geometry, as an instance of his versatility of humour. But a glance through the contents of the book showed us that Mr. Dodgson was in grim earnest; and we venture to think that no one who has made himself master of the pages of symbols, of which may be taken as a type, will be inclined to regard the book as a ‘mere jeu d’esprit.’ No, Mr. Dodgson has come forward to show that the Universities of Oxford and Cambridge have other reasons than the convenience of examiners for advising, if not insisting on, the retention of Euclid in our schools. He desires to silence Euclid’s rivals by force of argument; and to this end he lays down the chief points of difference between Euclid and those who wish to supersede him, and discusses them seriatim, showing, it is needless to remark, that in nearly every instance, the advantages are strongly in favour of retaining Euclid.
The plan of the book is original. It is cast into a dramatic mould. In the first Act the ghost of Euclid appears and defends himself with respect to certain contested points, which he agrees to consider as crucial questions between him and his opponents. These having been satisfactorily and rather summarily disposed of, Euclid is made to say: ‘This then concludes our present interview; we will meet again when you have reviewed my modern rivals one by one. If you had any slow music handy, I would vanish to it, as it is ‘—vanishes without slow music. In the second and third Acts, an omniscient and ubiquitous German, Herr Niemand, undertakes the introduction and defence of Euclid’s several rivals, including Legendre, Cooley, Wilson, Willock, Wright, the Society’s Syllabus, and others. In Act IV, Euclid comes back triumphant; but before finally returning to his shady home, he generously confesses that so long as the order and numbering of bis propositions (presumably for the sake of examiners) are retained, he is content that his proofs should be ‘abridged and improved,’ and that new problems and new theorems should be interpolated—which is rather a wholesale concession. The last words of the book are those in which scientific teachers will perhaps most cordially agree: ‘In all these matters my Manual (Euclid) is capable of almost unlimited improvement.’
In his treatment of the rival systems, Mr. Dodgson cannot be said to reason without a strong bias in favour of Euclid. His objections in most cases amount to little more than verbal quibbles, and we doubt sometimes whether in his client’s cause he has not purposely criticised unimportant differences and passed over those of greatermoment. He is very great on the definitions and axioms of the rival systems. The various definitions of that undefinable thing, ‘a straight line,’ are successively shown to be inadequate or redundant, or not in strict accordance with the ordinary canons of formal logic. At the same time not a word is said in defence of Euclid’s definition, for which these are intended as substitutes. If our author before writing this book had read Henrici’s Geometry, which appeared in the early part of this year, he would have seen what is probably the only scientific way of treating this part of the subject, and would have learned how comparatively unimportant to the study of geometry are these verbal definitions on the accuracy of which he so vainly insists. [I now present to the reader a review of ‘what is probably the only scientific way of treating this part of the subject’!] In the matter of axioms, he defends Euclid’s 12th axiom, which no schoolboy ever yet regarded as a selfevident proposition, by pointing out that ‘it is not axiomatic till Prop. 28, Bk. I. has been proved,’ without apparently seeing that this statement reduces the proposition to a corollary to Prop. 28, and takes away from it its axiomatic character altogether. [The Reviewer is no doubt aware of the essential distinction between an ‘axiom’ and a ‘corollary’: but he has got the two ideas so hopelessly mixed in this sentence, that it may be worth while to explain it. An ‘axiom,’ then, always requests the voluntary assent of the reader to some truth, for which no proof is offered, and which he is not logically compelled to grant: a ‘corollary’ is logically deduced from what has been already granted, and the reader must accept it, however unwilling he may be to do so. But though he has free choice as to whether he will, or will not, accept any axiom that may be proposed, his willingness to do so largely depends on the amount of truth he has already grasped in connection with it.]
In answer to the charge that Euclid’s proofs are in many cases unnecessarily tedious, Mr. Dodgson admits that alternative proofs might be allowed , although he is very unwilling to grant that the proposed alterations are real improvements. In the same way he suggests that one or two propositions, omitted by Euclid, might be interpolated into the text, without interfering with the order and numbering of the propositions. Mr. Dodgson endeavours to reduce to a minimum the changes and interpolations which are required to make Euclid a serviceable text-book of geometry; and amongst various omissions we may notice that he does not refer to Euclid’s improper [!] treatment of the several cases in which two triangles can be proved to coincide. [I cannot fairly be expected to ‘refer to’ what I deny to exist!] To point out the various reasons not mentioned by our author for superseding Euclid by some other text-book, would occupy us far too long. Suffice it to say, that if Euclid were amended according to the notions of modern teachers of geometry, or even according to the suggestions of our author, there would be little left of the original text; and under such circumstances it is difficult to see any reason, save the convenience of examiners, for retaining the order and numbers of the propositions. [If amended ‘according to the suggestions of our author,’ there would be, I should think, at least 99-100ths of the original text left without any material change.]
Mr. Dodgson admits some few imperfections in Euclid; but those of his modern rivals are far more numerous and unpardonable. Of Mr. Wilson’s Manual, which he justly considers a formidable rival, he speaks in unmeasured terms of disapproval. ‘The abundant specimens of logical inaccuracy, and of loose writing generally, which I have here collected would, I feel sure, in a more 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.’ Strong language! But Mr. Dodgson is well versed in logic and his statements are not loose utterances. Let us test one of them—the most ‘monstrous’ of the charges brought against Mr. Wilson. In his book we are told there is found one instance of ‘Illicit process of the Minor’ (p. 117). We will examine it.
Mr. Dodgson says (p. 108):—
‘At p. 9 we have a deduction from a definition, and an axiom which involves the fallacy “Illicit process of the Minor.” The passage is as follows:—Def. II. “A straight line is said to be perpendicular to another straight line when it makes a right angle with it. Hence there can be only one perpendicular to a given line at a given point, on one side of that line, because only one line can make a right angle with the given line at that point.”
‘Thrown into syllogistic form the argument may be stated thus:—
‘All lines drawn at right angles to a given line at a given point, on one side of it, are coincident; all lines drawn at right angles to a given line, &c., are perpendiculars to that line, &c.; therefore, all perpendiculars to a given line, &c., are coincident.’ That is, “All X is Y; all X is Z; therefore all Z is Y.”’
Not so at all. The syllogism expressed formally should be stated thus:—
All X is Y,
All X is all Z,
∴ All Z is Y;
which is logically sound, involving no illicit process at all.
Has Mr. Dodgson never heard that it is a logical postulate to state explicitly what is thought implicitly, or does he suppose that the logic of the Stagirite is sufficient for all reasoning processes? Surely, Sir Wm. Hamilton and the Archbishop of York might have taught him otherwise, not to speak of Mill, Boole, and Jevons! We have quoted this—the most formidable of Mr. Dodgson’s charges—as an instance of the style of reasoning adopted throughout the work. [I can only say, again, that the Definition in question is of the form ‘all X is Z,’ and is not of the form ‘all X is all Z.’ Let us try it on new subject-matter. ‘A man is said to be honest, when he pays his friend a debt which that friend had forgotten.’ I say this is equivalent to ‘All who pay forgotten debts are honest men.’ My reviewer identifies it with ‘All who pay forgotten debts are all honest men’ (i. e. they are the only honest men alive). Let the reader judge between us. And why should this be called ‘the most formidable’ of my charges? Surely it is not quite so important as (say) the treatment of Parallels, which so entirely collapses in Mr. Wilson’s hands?]
From the ‘Educational Times,’ March 1, 1880.
[Extract from a Paper read by Mr. Philip Magnus at an Evening Meeting of the College of Preceptors.]
To all foreigners it appears very strange that Geometry should be taught to English school children by aid of a treatise not designed for school purposes, and written more than two thousand years ago. In Germany and France, modern methods of teaching Geometry have for many years been adopted; and, although it is not a good thing that Englishmen should imitate too readily the educational methods of foreigners, still the fact that we stand alone among European nations in using Euclid’s Elements as our school text-book of Geometry requires explanation, and suggests the reflection that we are not wise in doing so. [‘At the present moment, we learn from the best authority, namely, the testimony of anti-Euclideans, that both in France and Italy dissatisfaction is felt with the system hitherto used, accompanied with more or less desire to adopt ours.’ Todhunter’s Essay on Elementary Geometry.] The explanation probably consists in the difficulty and inconvenience of change. There is a manifest advantage in having a fixed and recognised text-book, to which both teachers and examiners can refer; and there is, moreover, the very natural apprehension that, if Euclid is abandoned, a number of different text-books may come into use, which , however much they might improve the teaching of Geometry, would terribly perplex the examiner.
The way of Euclid is not always smooth and easy. The difficulties are not slight which the beginner has to surmount; and those who have practice in teaching well know that the artificiality of Euclid proves a stumbling-block which many pupils never succeed in overcoming. When this is the case, the pupil draws upon his memory for the assistance which his reason ought to afford, and endeavours to deceive his teacher and examiner by learning his propositions by heart. Unfortunately this utter waste of time is not so uncommon as it might be thought. The difficulty of mastering the proposition, together with the necessity of adhering to the exact words of the text, tempt the unhappy pupil to this assumption of knowledge. [This ‘necessity of adhering to the exact words of the text’ is, I venture to say, one that could only arise with a thoroughly incompetent examiner. The reasoning may be given correctly, with an almost unlimited variety of language.] An oral teacher cannot be deceived; he may vary the letters, alter the figure, and adopt a number of other simple devices to test the real knowledge of his pupil. But in written examinations deception is less easily detected. The examiner who finds the propositions ‘written out clearly and accurately,’ cannot withhold marks from his candidates, although no single rider may have been attempted, and although the work presented to him may be the result of a mere exercise of memory.
I have said that the Conservatives admit that Euclid’s proofs may be advantageously modified, and that other changes may be introduced into the ‘Elements’ without interfering with the Euclidian method. This view is maintained with considerable ability by Mr. Dodgson, Mathematical Lecturer of Christ Church, Oxford, in a work entitled ‘Euclid and his Modern Rivals,’ which should be read by all persons interested in the question we are now considering. Mr. Dodgson’s discussion of the various systems of Geometry that have been put forth as rivals to Euclid, is enlivened by occasional flashes of humour which make the work as amusing as it is profound. Mr. Dodgson’s ultimate conclusions seem to be hardly worthy of the efforts he has made to establish them. For, whilst he subjects the definitions and demonstrations of Wilson, Wright, and others to the most searching criticism, and establishes in each case the superiority of Euclid to any modern writer, he ultimately concedes that, so long as Euclid’s sequence and numbering of propositions is retained, together with his doctrine of parallels, his proofs may be altered, abridged, and indefinitely improved. ‘Leave me [Euclid] these untouched, and I shall look on with great contentment while other changes are made—while my proofs are abridged and improved—while alternative proofs are appended to mine—and while new problems and theorems are interpolated. In all these matters my Manual is capable of almost unlimited improvement’ (p. 199).
The reason for insisting so strongly on the retention of the numbering and sequence of Euclid’s propositions can be none other than the convenience of external examiners; for the precautions against assuming propositions subsequent to the one to be proved, cannot be necessary in examinations conducted by the teacher himself, since he would of course know very well how much his own pupils might legitimately take for granted. But it is easily understood that an external examiner who is familiar with no other than Euclid’s method of teaching Geometry, would be somewhat bewildered if, for instance, Euclid I. 24 were assumed in the proof of Euclid I. 8. But, surely, it must be admitted that methods of teaching a subject ought not to be considered from the standpoint of examinational requirements. [If examinations were done away with altogether, there would still be abundant reason for adopting the same logical sequence in all sciences where some logical sequence is essential: otherwise any attempt at communication between one mathematical student and another is merely a revival of the scenes enacted around the Tower of Babel.]