Lewis Carroll

“ 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. ”

“ 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 like to think of all the points in a circle really feeling a placid satisfaction in the thought that they are equidistant from the centre! They are infinite in number, and so can well afford to despise the arrogance of a point within, and to ignore the envious murmurs of a point without! ”

“ Wilson would not have ventured expressly to postulate that when a magnitude changes continuously, all magnitudes which change with it also change continuously. He knows that when a point moves on a line, an angle may undergo a sudden change of two right angles. He trusts to the beginner's perception of truth in the case before him: the whole truth would make that beginner feel that he is on a foundation of general principles made safe for him by selection, and only safe because the exceptions are not likely to occur to his mind. ”

“ 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. ”

“ Nevertheless it will, I trust, be found that I have permitted myself a glimpse of the comic side of things only at fitting seasons, when the tired reader might well crave a moment's breathing-space, and not on any occasion where it could endanger the continuity of a line of argument. ”

“ Wilson's book. We specially intend to separate the logician from the geometer. In the author's own interest, and that he may be as powerful a defender as can be of a cause we expect and desire to see fully argued, we recommend him to revise his notions of logic. We know that mathematicians care no more for logic than logicians for mathematics. The two eyes of exact science are mathematics and logic: the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye; each believing that it sees better with one eye than with two. ”

“ If, in a certain diagram, whose geometrical history we know, we want to test whether two Lines, passing through a common point, have, or have not, 'the same direction,' we have simply to take any other point on one of the Lines, and observe whether the other Line does, or does not, pass through it. ”

“ 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 ”

“ 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. ”

“ 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. ”

“ 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 imply that lines which make different angles with a third have also different directions; and I assume that lines of different directions will meet. Mr. Wilson is so concise that it is not easy to be very positive as to how much he will admit of the above, or how he will get over or round it. When put upon his defence he must be more explicit. Mr. Wilson gives four explicit axioms about the straight line: and not one about the angle.
We feel confidence that no such system as Mr. Wilson has put forward will replace Euclid in this country. ”

“ Euclid was quite right: the first reasonings presented to a beginner should be of full statement. He may be trained to suppression: but the true way to abbreviation is from the full length. Mr. Wilson does not use the phrases of reasoning consistently. He tells the student that a corollary is 'a geometrical truth easily deducible from a theorem': and then, to the theorem that only one perpendicular can be drawn to a straight line, he gives as a corollary that the external angle of a triangle is greater than the internal opposite. ”

“ Some amuse themselves by walking on one of the rails; some on another; others wander along the line, crossing and recrossing. Now as we are all bound for the same place, we may say, roughly speaking, that we are all moving 'in the same direction': but that is speaking very roughly indeed. We make our language more exact, if we exclude the wanderers, and say that those who are walking along the rails are so moving. But it seems to me that our phrase becomes still more exact, if we limit it to those who are walking on one and the same rail. ”

“ 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? ”

“ And how do you prove that the Lines will meet?
Nos. (smiling) We don't prove it: that is the reader's business: we only provide enunciations.
Min. You are like the gourmand who would eat so many oysters at supper that at last his friend could not help saying 'They are sure to disagree with you in the night.' 'That is their affair,' the other gaily replied. 'I shall be asleep!'
Your Syllabus has the same hiatus as the other writers who have rejected Euclid's 12th Axiom. If you will not have it as an Axiom, you ought to prove it as a Theorem. ”

“ You begin by dragging the unfortunate beginner straight into the most difficult part of the subject. Your first chapter positively bristles with difficulties about 'direction.' Then comes a long chapter on circles, including some very complicated figures, and a theory of tangents which depends upon moving lines and vanishing chords—all most disheartening to a beginner. What do you suppose he is likely to make of such a sentence as 'the direction of the motion of the generating point of any curve is that of the tangent to the curve at that point'? ”

“ 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. ”

“ A tremulous movement is seen amid the ghostly throng. They waver fitfully to and fro, and finally drift off in the direction of one corner of the ceiling. When the procession has got well under way, Niemand himself becomes hazy, and floats off to join them. ”

“ 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 object of this little book is to furnish evidence, first, that it is essential, for the purpose of teaching or examining in elementary Geometry, to employ one textbook only; secondly, that there are strong a priori reasons for retaining, in all its main features, and specially 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. ”

“ 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. ”

“ 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. ”