Euclid

“ AS there are creeds in religion, so there are creeds in geometry. Most of us pin our faith upon Euclid, whose masterpiece of reasoning is still, after twenty-two centuries, the wonder of the world. The system that Euclid founded stands, as it were, four-square and solid; it meets every need in the only kind of space that we practically know. ”

“ For beginners we must admit that Euclid's method of treating this point is the best. But you will allow ours to be a legitimate and elegant method for the advanced student
Min. Most certainly. The whole of your beautiful treatise is admirably fitted for advanced students it is only from the beginner's point of view that I venture to criticise it at all.
Your treatment of angles and right angles does not, I think, differ much from Euclid's?
Nie. Not much. We prove, instead of assuming, that all right angles are equal, deducing it from the Axiom that two right Lines cannot enclose a space. ”

“ They would now study science, they would learn geometry (or the science of measurement) , and con the lessons of Euclid, such as boys still con at school and college. So eager were the young men in their new study that groups of them were to be seen in various rooms of the palace holding sticks in their hands, and scratching the figures of Euclid in the dust which was spread on the marble floors. ”

“ I consider the question, as to whether Euclid's system and numeration should be abandoned or retained, to be now set at rest: the subject of Parallels being disposed of, no minor points of difference can possibly justify the abandonment of our old friend in favour of any Modern Rival. Still it will be worth while to examine the other writers, whose works you have brought with you, as they may furnish some valuable suggestions for the improvement of Euclid's Manual. ”

“ The Propositions are pretty nearly in Euclid's order: with a few changes in order and numeration, the book would be a modernised Euclid, the only important differences being the adoption of Playfair's Axiom and the omission of the diagonals in Euc. II. I have no hostile criticisms to offer. Our American cousins set us an excellent example in the art of brief, and yet lucid, mathematical writing. ”

“ Euclid uses the reductio ad absurdum chiefly when he is demonstrating the converse of some former theorem; see I. 14, 19, 25, 40.
Some remarks on indirect demonstration by Professor Sylvester, Professor De Morgan, and Dr Adamson will be found in the volumes of the Philosophical Magazine for 1852 and 1853.
I. 6 is not required by Euclid before he reaches II. 4; so that I. 6 might be removed from its present place and demonstrated hereafter in other ways if we please. For example, I. 6 might be placed after I. 18 and demonstrated thus. ”

“ Do you prove it in Euclid's way?
Nie. Not exactly. Our argument is quite different from Euclid's: and we come to two conclusions—one being the real existence of Parallels, the other the equivalent of Playfair's Axiom.
Min. I very much doubt your proving the first by any simpler method than Euclid's: and as to proving the second, by any method at all, without assuming some disputable Axiom, I defy you to do it! However, let us hear your argument.
Nie. We take a Line, and a point without it: and from the point we draw two 'half-rays' intersecting the line. ”

“ Euclid should be taught in his own words,—just enough and none to spare: the employment of less must engender obscurity; and of more, a want of neatness and perspicacity. The best geometrician amongst us would have cut but a bad figure by the side of a lad of very average ability brought up to know Euclid by book.
Another twitch of the bell announced that the hour for playing at triangles had expired. In five minutes the slate was covered with bars of minims and crotchets, and the music lesson begun. ”

“ Look at Pr. (h) p. 69, where we are told to make, at the ends of a Line, two angles which are together less than two right angles, and where it is assumed that the Lines, so drawn, will meet. That is, you assume the truth of Euclid's 12th Axiom. And you do the same thing at pp. 70, 123, 143, and 185.
Nie. Euclid's 12th Axiom is easily proved from our Theorems.
Min. No doubt: but you have not done it, and the omission makes a very serious hiatus in your argument. It is not a thing that beginners are at all likely to be able to supply for themselves. ”

“ The Greek philosophers are said to have posted on the doors of their schools "Let no one enter here who does not know his Euclid." In the midst of the crumbling of old-fashioned methods of education in the introduction of the elective system, in the modern time, many of our best educators have insisted {74} that at least this portion of mathematics, Euclid's contribution to the science, should be a required study, and most educators feel, even when there is question of law or medical study, that one of the best preparations is to be found in a thorough knowledge of Euclid. ”

“ Manuals which reject Euclid's treatment of Parallels.
Scene I.
'E fumo dare lucem.'
[Minos sleeping. To him enter, first a cloud of tobacco-smoke; secondly the bowl, and thirdly the stem, of a gigantic meerschaum; fourthly the phantasm of Herr Niemand, carrying a pile of phantom-books, the works of Euclid's Modern Rivals, phantastically bound.]
Niemand. The first author we have to consider is M. Legendre, is it not?
Minos. (aside) Not a single word of greeting! He plunges in medias res with a more fearful suddenness than Euclid himself! (Aloud) It is so, mein lieber Herr. ”

“ The 'Elements' of Euclid, which have withstood the action of so many centuries, will have at least the honor before ending of arousing a long series of works admirably enchained which will contribute, in the most effective way, to the progress of mathematics, at the same time that they furnish to the philosophers the points of departure the most precise and the most solid for the study of the origin and of the formation of our cognitions. ”

“ Nie. We give Euclid's 12th Axiom, which we prove from Prop. 32, using the principle of Euc. X. 1 (second part) , that 'if the greater of two unequal magnitudes be bisected, and if its half be bisected, and so on; a magnitude will at length be reached less than the lesser of the two magnitudes.'
Min. That again is a mode of proof entirely unsuited to beginners. ”

“ 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. ”

“ Proclus preserves a reply made by Euclid to King Ptolemy, who asked whether he could not learn geometry more easily than by studying the Elements—“There is no royal road to geometry.” Pappus of Alexandria, in his Mathematical Collection, says that Euclid was a man of mild and inoffensive temperament, unpretending, and kind to all genuine students of mathematics. This being all that is known of the life and character of Euclid, it only remains therefore to speak of his works.
Among those which have come down to us the most remarkable is the Elements (Στοιχεῖα) (see Geometry) . ”

“ The trouble about old Euclid was that he had no soul, and few of his books have that emotional appeal for which we look in these days. My aim would be to bring home his discoveries to the young by clothing them with human interest; and I should at the same time demonstrate to the adult how often they might be made practically useful in everyday life. When one thinks of the times one draws a straight line at right angles to another straight line, and how seldom one does it Euclid's way ... every time one writes a T.... ”

“ The question has often been mooted, to what extent Euclid, in his Elements, is a discoverer or a compiler. To this question no entirely satisfactory answer can be given, for scarcely any of the writings of earlier geometers have come down to our times. We are mainly dependent on Pappus and Proclus for the scanty notices we have of Euclid’s predecessors, and of the problems which engaged their attention; for the solution of problems, and not the discovery of theorems, would seem to have been their principal object. ”

“ 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." ”

“ As to 'Parallels,' there is a good deal to be said, and that not very flattering, I fear.
In Ax. 6, you assert the reality of different Lines having the same direction—a property you can neither define, nor construct, nor test.
You also assert (by implication) the reality of separational Lines, which Euclid proves.
You also assert the reality of Lines, not known to have a common point, but having different directions—a property you can neither define, nor construct, nor test. ”

“ The Jesus of the four gospels is also alleged to have fed large multitudes of people under circumstances of a most ultra-thaumaturgic character. To the first book of Euclid is, prefixed an axiom that "the whole is greater than its part." John Wesley is alleged to have eschewed mathematics lest it should lead him to Infidelity. John Wesley was wise, for if any man be foolish enough to accept Euclid's axiom, he will be compelled to reject the miraculous feeding of 5,000 people with five loaves and two small fishes. It is difficult under any circumstances to perform a miracle. ”

“ Min. By no means. You are too impatient. The one great merit, as I was saying, of Mr. Wilson's new book (and a most blessed change it is!) is that it ignores the whole theory of 'direction.' That he has finally abandoned that night-mare of Elementary Geometry, I dare not hope: so all I have said about it had better stand, lest in some future fit of inspiration he should bring out a yet more agonising version of it.
But it has the usual hiatus of a system which replaces Euclid's Axiom by Playfair's: it provides no means of proving that the Lines contemplated by Euclid will meet if produced. ”

“ 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. ”

“ Euclid.—What a popular school-book it will be! How boys will bless the name of the writer who first brings out the complete thousand!
With a view to discussing and criticising his various modern rivals, Euclid promises to send to Minos the ghost of a German Professor (Herr Niemand) who "has read all books, and is ready to defend any thesis, true or untrue."
"A charming companion!" as Minos drily remarks.
This brings us to Act II., in which the Manuals which reject Euclid's treatment of Parallels are dealt with one by one. Those Manuals which adopt it are reserved for Act III., Scene i. ”

“ Accordingly in his own demonstration Euclid confines himself to the right-hand figure; and he shews that this case cannot exist, because the straight line BD would be a diameter of both circles, and would therefore be bisected at two different points; which is absurd.
Euclid might have used a similar method for the second part of the proposition; for as there cannot be a point of contact out of the straight line joining the centres, it is obviously impossible that there can be a second point of contact when the circles touch externally. It is easy to see this ”

“ A distinction should be drawn between Euclid's definition and his manner of applying it. Every one who understands it must see that it is an application of arithmetic, and that the defective and unwieldy forms of arithmetical expression which never were banished from Greek science, need not be the necessary accompaniments of the modern use of the fifth book. For ourselves, we are satisfied that the only rigorous road to propor- tion is either through the fifth book, or else through something much more difficult than the fifth book need be. ”

“ 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. ”

“ And so, for the 'long course of logical reasoning' which you object to so much in Euclid, you substitute a short course of illogical reasoning! But you deduce another conclusion, do you not?
Nie. Yes, one other. 'The assumption mentioned in § 113' (the assumption that case (ε) is the only true one) 'may now be stated thus:—Axiom VI. Through a given point only one Line can he drawn parallel to a given Line.'
Min. May it indeed? ”