Euclid and His Modern Rivals/Appendix I.
APPENDIX I.
Extract from Mr. Todhunter's essay on 'Elementary Geometry,' included in 'The Conflict of Studies, &c.'
It has been said by a distinguished philosopher that England is "usually the last to enter into the general movement of the European mind." The author of the remark probably meant to assert that a man or a system may have become famous on the continent, while we are almost ignorant of the name of the man and the claims of his system. Perhaps, however, a wider range might be given to the assertion. An exploded theory or a disadvantageous practice, like a rebel or a patriot in distress, seeks refuge on our shores to spend its last days in comfort if not in splendour. Just when those who originally set up an idol begin to suspect that they have been too extravagant in their devotions we receive the discredited image and commence our adorations. It is a less usual but more dangerous illustration of the principle, if just as foreigners are learning to admire one of our peculiarities we should grow weary of it.
In teaching elementary geometry in England we have for a long time been accustomed to use the well-known Elements of Euclid. At the present moment, when we learn from the best testimony, namely, the admission 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, we are urged by many persons to exchange our system for one which is falling out of favour on the continent.
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Many assertions have been made in discussion which rest entirely on the authority of the individual advocate, and thus it is necessary to be somewhat critical in our estimate of the value of the testimony. Two witnesses who are put prominently forward are MM. Demogeot and Montucci, who drew up a report on English education for the French Government. Now I have no doubt that these gentlemen were suited in some respects to report on English education, as they were selected for that purpose; but I have searched in vain for any evidence of their special mathematical qualifications. No list of mathematical publications that I have consulted has ever presented either of these names, and I am totally at a loss to conceive on what grounds an extravagant respect has been claimed for their opinions. The following sentence has been quoted with approbation from these writers: "Le trait distinctif de l'enseignement des mathématiques en Angleterre c'est qu'on y fait appel plutôt à la mémoire qu'à l'intelligence de l'élève." In the first place we ought to know on what evidence this wide generalisation is constructed. Did the writers visit some of the humbler schools in England in which the elements of arithmetic and mensuration were rudely taught, and draw from this narrow experience an inference as to the range of mathematical instruction throughout England? Or did they find on inspecting some of our larger public schools that the mathematical condition was unsatisfactory? In the latter case this might have arisen from exclusive devotion to classics, or from preference for some of the fashionable novelties of the day, or from want of attention and patience in the teachers. On the most unfavourable supposition the condemnation pronounced on the general mathematical training in England cannot be justified. But take some kind of experimental test. Let an inquirer carefully collect the mathematical examination papers issued throughout England in a single year, including those proposed at the Universities and the Colleges, and those set at the Military Examinations, the Civil Service Examinations, and the so-called Local Examinations. I say then, without fear of contradiction, that the original problems and examples contained in these papers will for interest, variety, and ingenuity surpass any similar set that could be found in any country of the world. Then any person practically conversant with teaching and examining can judge whether the teaching is likely to be the worst where the examining is the most excellent.
The sentence quoted from MM. Demogeot and Montucci, in order to have any value, ought to have proceeded from writers more nearly on a level with the distinguished mathematical teachers in England. So far as any foundation can be assigned for this statement, it will probably apply not to mathematics especially but to all our studies, and amount to this, that our incessant examinations lead to an over cultivation of the memory. Then as to the practical bearing of the remark on our present subject it is obvious that the charge, if true, is quite independent of the text-book used for instruction, and might remain equally valid if Euclid were exchanged for any modern author.
The French gentlemen further on contrast what they call Euclid's verbiage with the elegant conciseness of the French methods. It is surely more than an answer to these writers to oppose the high opinion of the merits of Euclid expressed by mathematicians of European fame like Duhamel and Hoüel. See the First Report of the Association for the Improvement of Geometrical Teaching, p. 10.
When we compare the lustre of the mathematical reputation of these latter names with the obscurity of the two former, it seems that there is a great want of accuracy in the statement made in a recent circular: 'The opinion of French mathematicians on this question, is plainly expressed in the Report of MM. Demogeot and Montucci…'
I should have to quote very largely indeed if I wished to draw attention to every hazardous statement which has been advanced; I must therefore severely restrain myself. Consider the following: 'Unquestionably the best teachers depart largely from his words, and even from his methods. That is, they use the work of Euclid, but they would teach better without it. And this is especially true of the application to problems. Everybody recollects, even if he have not the daily experience, how unavailable for problems a boy's knowledge of Euclid generally is.' The value of such a statement depends entirely on the range of the experience from which it has been derived. Suppose for instance that the writer had been for many years an examiner in a large University in which against each candidate's name the school was recorded from which he came; suppose that the writer had also been much engaged in the numerous examinations connected with the military institutions; suppose that he had also been for a quarter of a century in residence at one of the largest colleges at Cambridge, and actively employed in the tuition; suppose also that it had been his duty to classify the new students for lecture purposes by examining them in Euclid and other parts of elementary mathematics; and finally suppose that he was in constant communication with the teachers in many of the large schools: then his opinion would have enjoyed an authority which in the absence of these circumstances cannot be claimed for it.
If I may venture to refer to my own experience, which I fear commenced when the writer whom I have just quoted was in his cradle, I may say that I have taught geometry both Euclidean and non-Euclidean, that my own early studies and prepossessions were towards the latter, but that my testimony would now be entirely in favour of the former.
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I admit that to teach Euclid requires patience both from the tutor and the pupil; but I can affirm that I have known many teachers who have succeeded admirably, and have sent a large number of pupils to the University well skilled in solving deductions and examples; nor have I ever known a really able and zealous teacher to fail. I am happy to supplement my own testimony by an extract from the very interesting lecture on Geometrical Teaching by Dr. Lees, of St. Andrews. 'Whatever may be the cause of failure in England, it is clear as any demonstration can be that the failure cannot be ascribed to Euclid. Because in Scotland we do employ Euclid as the text-book for our students, and in Scotland we have the teaching of Geometry attended with the most complete success; and this not only in the colleges, but in all the higher and more important schools and academies of the country, and in many of the parish schools even, where the attention of the teacher is necessarily so much divided.' See also the remarkable Narrative-Essay on a Liberal Education, by the Rev. S. Hawtrey, A.M., Assistant-Master, Eton.
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During the existence of the East India Company's military college at Addiscombe, it is well known that the cadets were instructed in mathematics by the aid of a course drawn up by the late Professor Cape. The geometry in this course was of the kind which our modern reformers recommend, being founded on Legendre, and adopting the principle of hypothetical constructions which is now so emphatically praised. In certain large schools where youths were trained for the military colleges it was usual to instruct a class of candidates for Woolwich, in Euclid, and a class of candidates for Addiscombe in Cape's adaptation of Legendre. Fairness in the procedure was secured by giving the same number of hours by the same masters to each class; and the honour and rewards which attended success supplied an effectual stimulus both to teachers and pupils. Now consider the result. I was assured by a teacher who was for many years distinguished for the number and the success of his pupils, that the training acquired by the Euclid class was far superior to that acquired by the Legendre class. The Euclid was not more difficult to teach and was more potent and more beneficial in its influence. The testimony made the stronger impression on me because at the time I was disposed from theoretical considerations to hold an opposite opinion; I was inclined for example to support the use of hypothetical constructions. Such experience as I afterwards gained shewed the soundness of the judgment at which the practical teacher had arrived; and I have also received the emphatic evidence of others who had good opportunities of considering the question, and had come to the same conclusion. I have myself examined at Woolwich and at Addiscombe, and am confident that the teaching in both institutions was sound and zealous; but I have no hesitation in saying that the foundation obtained from Euclid was sounder than that from Legendre.
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Although I have admitted that the study of Euclid is one that really demands patient attention from the beginner, yet I cannot admit that the tax is unreasonable. My own experience has been gained in the following manner. Some years since on being appointed principal mathematical lecturer in my college, more systematic arrangements were introduced for the lectures of the freshmen than had been previously adopted; and as the Euclid seemed to be one of the less popular subjects I undertook it myself. Thus for a long period the way in which this has been taught in schools, and the results of such teaching, have been brought under my notice. It need scarcely be said that while many of the students who have thus presented themselves to me have been distinguished for mathematical taste and power, yet the majority have been of other kinds; namely, either persons of ability whose attention was fully occupied with studies different from mathematics, or persons of scanty attainments and feeble power who could do little more than pass the ordinary examination. I can distinctly affirm that the cases of hopeless failure in Euclid were very few; and the advantages derived from the study, even by men of feeble ability, were most decided. In comparing the performance in Euclid with that in Arithmetic and Algebra there could be no doubt that the Euclid had made the deepest and most beneficial impression: in fact it might be asserted that this constituted by far the most valuable part of the whole training to which such persons were subjected. Even the modes of expression in Euclid, which have been theoretically condemned as long and wearisome, seemed to be in practice well adapted to the position of beginners. As I have already stated there appears to me a decided improvement gradually taking place in the knowledge of the subject exhibited by youths on entering the University. My deliberate judgment is that our ordinary students would suffer very considerably if instead of the well-reasoned system of Euclid any of the more popular but less rigid manuals were allowed to be taken as a substitute.
Let me now make a few remarks on the demand which has been made to allow other books instead of Euclid in examinations. It has been said: "We demand that we should not be,—as we are now, by the fact of Euclid being set as a text-book for so many examinations,—practically obliged to adhere to one book. Surely such a request, made by men who know what they want, and are competent to form an opinion on the subject,—and made in earnest,—should induce the Universities and other examining bodies to yield their consent. The grounds of the demand then are three; that it is made in earnest, that it is made by those who know what they want, and that it is made by those who are competent to form an opinion on the subject. I need not delay on two of the grounds; the experience of every day shews that claimants may know what they want, and be terribly in earnest in their solicitations, and yet it may be the duty of those to whom the appeal is made to resist it. Moreover it is obvious that the adoption of Euclid as a text book is prescribed by those who are equally in earnest and know what they recommend. In short if no institution is to be defended when it is attacked knowingly and earnestly, it is plain that no institution is safe.
I turn then to the other ground, namely that the demand is made by men who are competent to form an opinion on the subject. Now it is not for me to affect to speak in the name of the University of Cambridge; mine is the opinion of only a private unofficial resident. But I have little doubt that many persons here will maintain, without questioning the competence of the claimants to form an opinion, that we ourselves are still much more competent to form an opinion.
For it will not be denied that in all which relates to mathematical knowledge we have an aggregate of eminence which far surpasses what has yet been collected together to press the demand on the University. Moreover as inspectors and judges we occupy a central position as it were, and thus enjoy opportunities which do not fall to isolated teachers however eminent and experienced. The incessant demands made upon the University to furnish examiners for schools and for the local examinations keep us as a body practically familiar with the standard of excellence attained in various places of instruction. Then as college lecturers and private tutors we have the strongest motives for keenly discriminating the state of mathematical knowledge in different schools, as shewn by the performance of the candidates when brought under our notice. Moreover some of the residents in the University by continued intercourse with old pupils, now themselves occupying important positions as teachers, are enabled to prolong and enlarge the experience which they may have already obtained directly or indirectly. If it is obvious that certain teachers by ability and devotion have for many years sent up well-trained pupils, the University may well consider that it would be neither right nor wise to deprive its best friends of their justly earned distinction, by relaxing in any way the rigour of the examinations. Instead then of urging an instant acquiescence with demands on the ground that those who make them are well qualified to judge, the claimants should endeavour by argument to convince others who are still better qualified to judge.
Here let me invite attention to the following remark which has been made in support of the claim: "In every other subject this is freely accorded; we are not obliged to use certain grammars or dictionaries, or one fixed treatise on arithmetic, algebra, trigonometry, chemistry, or any other branch of science. Why are we to be tied to one book in geometry alone?" Now in the first place it may be said that there are great advantages in the general use of one common book; and that when one book has long been used almost exclusively it would be rash to throw away certain good in order to grasp at phantasmal benefits. So well is this principle established that we have seen in recent times a vigorous, and it would seem successful, effort to secure the use of a common Latin Grammar in the eminent public schools. In the second place the analogy which is adduced in the remark quoted above would be rejected by many persons as involving an obvious fallacy, namely that the word geometry denotes the same thing by all who use it. By the admirers of Euclid it means a system of demonstrated propositions valued more for the process of reasoning involved than for the results obtained. Whereas with some of the modern reformers the rigour of the method is of small account compared with the facts themselves. We have only to consult the modern books named in a certain list, beginning with the Essentials of Geometry, to see that practically the object of some of our reformers is not to teach the same subject with the aid of a different text-book, but to teach something very different from what is found in Euclid, under the common name of geometry.
It may be said that I am assuming the point in question, namely, that Euclid is the best book in geometry; but this is not the case. I am not an advocate for finality in this matter; though I do go so far as to say that a book should be decidedly better than Euclid before we give up the advantages of uniformity which it will be almost impossible to secure if the present system is abandoned. But, as it has been well observed by one of the most distinguished mathematicians in Cambridge, "The demand is unreasonable to throw aside Euclid in favour of any compendium however meagre and however unsound; and this is really the demand which is made: it will be time enough to consider about the discontinuance of Euclid when a better book is deliberately offered." It may be added that the superiority to Euclid must be established by indisputable evidence; not by the author's own estimation, the natural but partial testimony of parental fondness; not by the hasty prediction of some anonymous and irresponsible reviewer; not by the authority of eminent men, unless the eminence is founded on mathematical attainments; not even by the verdict of teachers who are not conspicuous for the success of their pupils. The decision must rest with students, teachers, and examiners, of considerable reputation in the range of the mathematical sciences.
It must be allowed that there is diversity of opinion among the opponents of Euclid, for while the majority seem to claim freedom in the use of any text-books they please, others rather advocate the construction and general adoption of a new text-book. The former class on the whole seem to want something easier and more popular than Euclid; among the latter class there are a few whose notion seems to be that the text-book should be more rigorous and more extensive than Euclid. There are various considerations which seem to me to indicate that if a change be made it will not be in the direction of greater rigour; the origin of the movement, the character of the text-books which have hitherto been issued, and the pressure of more modern and more attractive studies, combine to warn us that if the traditional authority which belongs to Euclid be abandoned geometry will be compelled to occupy a position in general education much inferior to that which it now holds.
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There is one very obvious mode of advancing the cause of the anti-Euclidean party, which I believe will do far more for them than the most confident assertions and predictions of the merits of the course which they advocate: let them train youths on their system to gain the highest places in the Cambridge Mathematical Tripos, and then other teachers will readily follow in the path thus opened to distinction. But it may naturally be said that as long as Euclid is prescribed for the text-book, the conditions of competition are unfair towards those who adopt some modern substitute; I will examine this point. In the Cambridge Examination for Mathematical Honours there are at present sixteen papers; a quarter of the first paper is devoted to book-work questions on Euclid. Now suppose that 1000 marks are assigned to the whole examination, and that about five of these fall to the book-work in Euclid. A student of any modern system would surely be able to secure some of these five marks, even from a stern Euclidean partisan. But to take the worst case, suppose the candidate deliberately rejects all chance of these five marks, and turns to the other matter on the paper, especially to the problems; here the advantage will be irresistibly on his side owing to the "superiority of the modern to the ancient methods of geometry" which is confidently asserted. It must be remembered that in spite of all warning and commands to the contrary, examiners will persist in making their papers longer than can be treated fully in the assigned time, so that the sacrifice of the book-work will be in itself trifling and will be abundantly compensated by the greater facility at the solution of problems which is claimed for the modern teaching, as compared with the "unsuggestiveness" of Euclid, and by the greater accuracy of reasoning, since we are told that "the logical training to be got from Euclid is very imperfect and in some respects bad." Thus on the whole the disciple of the modern school will even in the first paper of the Cambridge Tripos Examination be more favourably situated than the student of Euclid; and of course in the other papers the advantages in his favour become largely increased. For we must remember that we are expressly told that Euclid is "an unsuitable preparation for the higher mathematical training of the present day;" and that "those who continue their mathematical reading with a view of obtaining honours at the University … will gain much through economy of time and the advantage of modern lights."
The final result is this; according to the promises of the geometrical reformers, one of their pupils might sacrifice five marks out of a thousand, while for all the remaining 995 his chance would be superior to that of a Euclid-trained student. It may be added that in future the Cambridge Mathematical Examinations are to be rather longer than they have been up to the date of my writing; so that the advantage of the anti-Euclidean school will be increased. Moreover we must remember that in the Smith's Prizes Examination the elementary geometry of Euclid scarcely appears, so that the modern reformers would not have here any obstacle to the triumphant vindication of their superiority as teachers of the higher mathematics. The marvellous thing is that in these days of competition for educacational prizes those who believe themselves to possess such a vast superiority of methods do not keep the secret to themselves, instead of offering it to all, and pressing it on the reluctant and incredulous. Surely instead of mere assertion of the benefits to be secured by the modern treatment, it will be far more dignified and far more conclusive to demonstrate the proposition by brilliant success in the Cambridge Mathematical Tripos. Suppose we were to read in the ordinary channels of information some such notice as this next January: "The first six wranglers are considered to owe much of their success to the fact that in their training the fossil geometry of Alexandria was thrown aside and recent specimens substituted;" then opposition would be vanquished, and teachers would wonder, praise, and imitate. But until the promises of success are followed by a performance as yet never witnessed we are reminded of the case of a bald hairdresser who presses on his customers his infallible specific for producing redundant locks.
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To those who object to Euclid as an inadequate course of plane geometry it may then be replied briefly that it is easy, if thought convenient or necessary, to supply any additional matter. But for my part I think there are grave objections to any large increase in the extent of the course of synthetical geometry which is to be prepared for examination purposes. One great drawback to our present system of mathematical instruction and examination is the monotony which prevails in many parts. When a mathematical subject has been studied so far as to master the essential principles, little more is gained by pursuing these principles into almost endless applications. On this account we may be disposed to regard with slender satisfaction the expenditure of much time on geometrical conic sections; the student seems to gain only new facts, but no fresh ideas or principles. Thus after a moderate course of synthetical geometry such as Euclid supplies, it may be most advantageous for the student to pass on to other subjects like analytical geometry and trigonometry which present him with ideas of another kind, and not mere repetitions of those with which he is already familiar.
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It has been said, and apparently with great justice, that examination in elementry geometry under a system of unrestricted text-books will be a very troublesome process; for it is obvious that in different systems the demonstration of a particular proposition may be more or less laborious, and so may be entitled to more or fewer marks. This perplexity is certainly felt by examiners as regards geometrical conic sections; and by teachers also who may be uncertain as to the particular system which the examiners may prefer or favour. It has been asserted that the objection thus raised is imaginary, and that "the manuals of geometry will not differ from one another nearly so widely as the manuals of algebra or chemistry: yet it is not difficult to examine in algebra and chemistry." But I am unable to feel the confidence thus expressed. It seems to me that much more variety may be expected in treatises on geometry than on algebra; certainly if we may judge from the experience of the examiners at Cambridge the subject of geometrical conics is the most embarrassing which occurs at present, and this fact suggests a conclusion very different from that which is laid down in the preceding quotation. Of course there will be no trouble in examining a single school, because the system there adopted will be known and followed by the examiner.
I have no wish to exaggerate the difficulty; but I consider it to be real and serious, more especially as it presents itself at the outset of a youth's career, and so may cause disappointment just when discriminating encouragement is most valuable. But I think the matter must be left almost entirely to the discretion of examiners; the attempts which have been made to settle it by regulation do not seem to me very happy. For example, I read: "As the existing text-books are not very numerous, it would not be too much to require examiners to be acquainted with them sufficiently for the purpose of testing the accuracy of written, or even, if necessary, of oral answers." The language seems to me truly extraordinary. Surely examiners are in general men of more mathematical attainments than this implies; for it would appear that all we can expect them to do is to turn to some text-book and see if the student has correctly reproduced it. The process in a viva voce examination would be rather ignominious if when an answer had been returned by a candidate some indifferent manual had to be consulted to see if the answer was correct.
I have heard that an examining board has recently issued instructions to its executive officers to make themselves acquainted with the various text-books. This does not enjoin distinctly, what the above quotation implies, that the examiner is to accept all demonstrations which are in print as of nearly equal value; but it seems rather to suggest such a course. The point is important and should be settled. Suppose a candidate offered something taken from the Essentials of Geometry, and the examiner was convinced that the treatment was inadequate or unsound; then is the candidate nevertheless to obtain full marks? Again, it may be asked, why printed books alone are to be accepted; and why a student who has gone through a manuscript course of geometry should be precluded from following it? The regulation might be made that he should submit a copy of his manuscript course to the examiner in order that it might be ascertained whether he had reproduced it accurately. As I have already intimated, the only plan which can be adopted is to choose able and impartial men for examiners, and trust them to appreciate the merits of the papers submitted by the candidate to them.
The examiners will find many perplexing cases I have no doubt; one great source of trouble seems to me to consist in the fact that what may be a sound demonstration to one person with adequate preliminary study is not a demonstration to another person who has not gone through the discipline. To take a very simple example: let the proposition be, The angles at the base of an isosceles triangle are equal. Suppose a candidate dismisses this briefly with the words, this is evident from symmetry; the question will be, what amount of credit is to be assigned to him. It is quite possible that a well-trained mathematician may hold himself convinced of the truth of the proposition by the consideration of symmetry, but it does not follow that the statement would really be a demonstration for an early student. Or suppose that another imbued with "the doctrine of the imaginary and inconceivable" says as briefly "the proposition is true, for the inequality of the angles is inconceivable and therefore false;" then is the examiner to award full marks, even if he himself belongs to the school of metaphysics which denies that the inconceivable is necessarily the false?
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It has been urged as an objection against Euclid that the number of his propositions is too great. Thus it has been said that the 173 propositions of the six books might be reduced to 120, and taught in very little more than half the time required to go through the same matter in Euclid. So far as the half time is concerned this seems to be only an expression of belief as to the result of an untried experiment; it is based on the comparison of a few other books with Euclid, one of these being the Course of the late Professor Cape; as I have already stated, actual experience suggests a conclusion directly contrary to the present prediction. As to the number of propositions we readily admit that a reduction might be made, for it is obvious that we may in many cases either combine or separate according to our taste. But the difficulty of a subject does not vary directly as the number of propositions in which it is contained; a single proposition will in some cases require more time and attention than half a dozen others. I have no doubt that the mixture of easy propositions with the more difficult is a great encouragement to beginners in Euclid; and instead of diminishing the number of propositions I should prefer to see some increase: for example I should like to have Euclid i. 26 divided into two parts, and Euclid i. 28 into two parts.
Again, it has been said that Euclid is artificial, and that he "has sacrificed to a great extent simplicity and naturalness in his demonstrations;" it is a curious instance of the difference of opinion which we may find on the same subject, for, with a much wider experience than the writer whom I quote, I believe that Euclid maintains, and does not sacrifice, simplicity and naturalness in his system, assuming that we wish to have strictness above all things.
The exclusion of hypothetical constructions has been represented as a great defect in Euclid; and it has been said that this has made the confused order of his first book necessary. Confused order is rather a contradictory expression; but it may be presumed that the charge is intended to be one of confusion: I venture to deny the confusion. I admit that Euclid wished to make the subject depend on as few axioms and postulates as possible; and this I regard as one of his great merits; it has been shown by one of the most distinguished mathematicians of our time how the history of science teaches in the clearest language that the struggle against self-imposed restrictions has been of the most signal service in the advancement of knowledge.
The use of hypothetical constructions will not present itself often enough to produce any very great saving in the demonstrations; while the difficulty which they produce to many beginners, as shown by the experience to which I have already referred, is a fatal objection to them. Why should a beginner not assume that he can draw a circle through four given points if he finds it convenient?
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Finally, I hold that Euclid, in his solution of the problems he requires, supplies matter which is simple and attractive to beginners, and which therefore adds practically nothing to their labours, while it has the advantage of rendering his treatise far more rigorous and convincing to them.
The objections against Euclid's order seem to me to spring mainly from an intrusion of natural history into the region of mathematics; I am not the first to print this remark, though it occurred to me independently. It is to the influence of the classificatory sciences that we probably owe this notion that it is desirable or essential in our geometrical course to have all the properties of triangles thrown together, then all the properties of rectangles, then perhaps all the properties of circles; and so on. Let me quote authority in favour of Euclid, far more impressive than any which on this point has been brought against him: "Euclid … fortunately for us, never dreamed of a geometry of triangles as distinguished from a geometry of circles, … but made one help out the other as he best could."
Euclid has been blamed for his adherence to the syllogistic method; but it is not necessary to say much on this point, because the reformers are not agreed concerning it: those who are against the syllogism may pair off with those who are for the syllogism. We are told in this connexion that, "the result is, as every one knows, that boys may have worked at Euclid for years, and may yet know next to nothing of Geometry." We may readily admit that such may be the case with boys exceptionally stupid or indolent; but if any teacher records this as the average result of his experience, it must I think be singularly to his own discredit.
There is, I see, a notion that the syllogistic form, since it makes the demonstrations a little longer, makes them more difficult; this I cannot admit. The number of words employed is not a test of the difficulty of a demonstration. An examiner, especially if he is examining viva voce, can readily find out where the difficulties of the demonstrations really lie; my own experience leads me to the conclusion that the syllogistic form instead of being an impediment is really a great assistance, especially to early students.
"Unsuggestiveness" has been urged as a fault in Euclid; which is interpreted to mean that it does not produce ability to solve problems. We are told: "Everybody recollects, even if he have not the daily experience, how unavailable for problems a boy's knowledge of Euclid generally is. Yet this is the true test of geometrical knowledge; and problems and original work ought to occupy a much larger share of a boy's time than they do at present." I need not repeat what I have already said, that English mathematicians, hitherto trained in Euclid, are unrivalled for their ingenuity and fertility in the construction and solution of problems. But I will remark that in the important mathematical examinations which are conducted at Cambridge the rapid and correct solution of problems is of paramount value, so that any teacher who can develop that power in his pupils will need no other evidence of the merits of his system.
Euclid's treatment of proportion has been especially marked out for condemnation; indeed, with the boldness which attaches to many assertions on the subject of elementary geometry, it has been pronounced already dead. In my own college it has never been laid aside; only a few months since one of our most influential tutors stated that he was accustomed to give a proposition out of the fifth book of Euclid to some candidates for emoluments, and he considered it a very satisfactory constituent of the whole process of testing them.
I should exceedingly regret the omission of the fifth book of Euclid, which I hold to be one of the most important parts of the training supplied by Elementary Geometry. I do not consider it necessary for beginners to go through the entire book; but the leading propositions might be mastered, and the student led to see how they can be developed if necessary. I may refer here to some valuable remarks which have been made on the subject by the writer of a Syllabus of Elementary Geometry… who himself I believe counts with the reformers. He sums up thus "… any easy and unsatisfactory short cuts (and I have sometimes seen an inclination for such) should be scouted, as a simple deception of inexperienced students."
However, I must remark that I see with great satisfaction the following Resolution which was adopted at a recent meeting of the Association for the Improvement of Geometrical Teaching: "That no treatise on Geometry can be regarded as complete without a rigorous treatment of ratio and proportion either by Euclid's method, or by some equally rigorous method of limits." It would be injudicious to lay much stress on resolutions carried by a majority of votes; but at least we have a striking contradiction to the confident statement that Euclid's theory of proportion is dead. We shall very likely see here, what has been noticed before, that a course may be proposed which differs widely from Euclid's, and then, under the guidance of superior knowledge and experience, the wanderers are brought back to the old path. Legendre's return to Euclid's treatment of parallels is a conspicuous example; see the valuable paper by Professor Kelland on Superposition in the Edinburgh Transactions, Vol. xxi.
I cannot avoid noticing one objection which has been urged against Euclid in relation to his doctrine of proportion; namely, that it leaves "the half-defined impression that all profound reasoning is something far fetched and artificial, and differing altogether from good clear common sense." It appears to me that if a person imagines that "good clear common sense" will be sufficient for mastering pure and mixed mathematics, to say nothing of contributing to their progress,—the sooner he is undeceived the better. Mathematical science consists of a rich collection of investigations accumulated by the incessant labour of many years, by which results are obtained far beyond the range of unassisted common sense; of these processes Euclid's theory of proportion is a good type, and it may well be said that from the degree of reverent attention which the student devotes to it, we may in most cases form a safe estimate of his future progress in these important subjects.
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In conclusion I will say that no person can be a warmer advocate than I am for the improvement of Geometrical Teaching: but I think that this may be obtained without the hazardous experiment of rejecting methods, the efficacy of which a long experience has abundantly demonstrated.