Euclid and His Modern Rivals/Appendix II.
APPENDIX II.
Extract from Mr. De Morgan's review of Mr. Wilson's Geometry, in the 'Athenœum' for July 18, 1868.
The Schools' Inquiry Commission has raised the question whether Euclid be, as many suppose, the best elementary treatise on geometry, or whether it be a mockery, delusion, snare, hindrance, pitfall, shoal, shallow, and snake in the grass.
| · | · | · | · | · | · | · | · | · | · | · | · |
We pass on to a slight examination of Mr. 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. The consequences are ludicrous. On the one side we have, by confusion of words, the great logician Hamilton bringing forward two quantities which are 'one and the same quantity,' Breadth and Depth, while, within a few sentences, 'the greater the Breadth, the less the Depth.' On the other side, we have the great mathematician, Mr. Wilson, also by confusion of words, speaking of the 'invariably syllogistic form of his [Euclid's] reasoning,' and, to show that this is not a mere slip, he afterwards talks of the 'detailed syllogistic form' as a 'source of obscurity to beginners, and damaging to true geometrical freedom and power.'
Euclid a book of syllogistic form! We stared. We never heard of such a book, except the edition of Herlinus and Dasypodius (1566), who, quite ignorant that Euclid was syllogistic already, made him so, and reckoned up the syllogisms. Thus i. 47 has 'syllogismi novem' at the head. They did not get much thanks; the book was never reprinted, and was in oblivion-dust when Hamilton mentioned the zealous but thick-headed logicians, as he called them. Prof. Mansel, in our own day, has reprinted one of their propositions as a curiosity. In 1831, Mr. De Morgan, advocating the reduction of a few propositions to detailed syllogistic form as an exercise for students, gave i. 47 as a specimen, in the Library of Useful Knowledge; this was reprinted, we believe, in the preface to various editions of Lardner's Euclid. A look will show the difference between Euclid and syllogistic form. Had the Elements been syllogistic, it would have been quoted in all time, as a proof of the rapid diffusion of Aristotle's writings, that they had saturated his junior contemporary with their methods: with a controversy, most likely, raised by those who would have contended that Euclid invented syllogistic form for himself. Now it is well known that diffusion of Aristotle's writings commenced after his death, and that it was—not quite correctly—the common belief that evulgation did not take place until two hundred years after his death. Could this belief ever have existed if Euclid had invariably used 'syllogistic form'?
What could have been meant? Craving pardon if wrong, we suspect Mr. Wilson to mean that Euclid did not deal in arguments with suppressed premisses. 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. This is not a corollary from the theorem, but a matter taken for granted in proving it.
Leaving this, with a recommendation to the author to strengthen his armour by the study of logic, we pass on to the system. There is in it one great point, which brings down all the rest if it fall, and may perhaps—but we must see Part II. before we decide—support the rest if it stand. That point is the treatment of the angle, which amounts to this, that certain notions about direction, taken as self-evident, are permitted to make all about angles, parallels and all, immediate consequences. The notion of continuous change, and consequences derived from it, enters without even an express assumption: 'continually' is enough.
Mr. 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. On this we write, as Newton wrote on another matter, Falsa! Falsa! Non est Geometria!
What 'direction' is we are not told, except that 'straight lines which meet have different directions.' Is a direction a magnitude? Is one direction greater than another? We should suppose so; for an angle, a magnitude, a thing which is to be halved and quartered, is the 'difference of the direction' of 'two straight lines that meet one another.' A better definition follows; the 'quantity of turning' by which we pass from one direction to another. But hardly any use is made of this, and none at the commencement. And why two definitions? Is the difference of two directions the same thing as the rotation by which we pass from one to the other? Is the difference of position of London and Rugby a number of miles on the railroad? Yes, in a loosely-derived and popular and slip-slop sense: and in like manner we say that one man is a pigeon-pie, and another is a shoulder of lamb, when we describe their contributions to a pic-nic. But non est geometria! Metaphor and paronomasia can draw the car of poetry; but they tumble the waggon of geometry into the ditch.
Parallels, of course, are lines which have the same direction. It is stated, as an immediate consequence, that two lines which meet cannot make the same angle with a third line, on the same side, for they are in different directions. Parallels are knocked over in a trice. There is a covert notion of direction, which, though only defined with reference to lines which meet, is straightway transferred to lines which do not. According to the definition, direction is a relation of lines which do meet, and yet lines which have the same direction can be lines which never meet. There is a great quantity of turning wanted; turning of implied assumption into expressed. Mr. Wilson would, we have no doubt, immediately introduce and defend all we ask for; and we quite admit that his system has a right to it. How do you know, we ask, that lines which have the same direction never meet? Answer—lines which meet have different directions. We know they have; but how do we know that, under the definition given, the relation called direction has any application at all to lines which never meet? The use of the notion of limits may give an answer: but what is the system of geometry which introduces continuity and limits to the mind as yet untaught to think of space and of magnitude? Answer, a royal road. If the difficulty were met by expressed postulates, the very beginner himself would be frightened.
There is a possibility that Mr. Wilson may mean that lines which make the same angle with a third on the same side are in the same direction. If this be the case, either he assumes that lines equally inclined to one straight line are equally inclined to all,—and this we believe he does, under a play on the word 'direction'; or he makes a quibble only one degree above a pun on his own arbitrary assumption of his right to the word 'same': and this we do not believe he does. He should have been more explicit: he should have said, My system involves an assumption which has lain at the root of many attempts upon the question of parallels, and has always been scouted as soon as seen. He should have added, I assume Euclid's eleventh axiom: I have a notion of direction; I tell you that lines which meet have different directions; 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. The old geometry is a very English subject, and the heretics of this orthodoxy are the extreme of heretics: even Bishop Colenso has written a Euclid. And the reason is of the same kind as that by which the classics have held their ground in education. There is a mixture of good sense and of what, for want of a better name, people call prejudice: but to this mixture we owe our stability. The proper word is postjudice, a clinging to past experience, often longer than is held judicious by after times. We only desire to avail ourselves of this feeling until the book is produced which is to supplant Euclid; we regret the manner in which it has allowed the retention of the faults of Euclid; and we trust the fight against it will rage until it ends in an amended form of Euclid.