utilitarian study. Practical teachers and professors, however, are alone able to form a definite scheme; and while, if such a scheme were devised, it must in its general principles be regarded in a national system as imperative, the greatest amount of freedom that may be found possible should be allowed for modifications. Of all men, teachers should be men, not machines, and should be treated accordingly.

XII

TAKING it for granted-has it not been proved ?— that the Euclidean geometry should form a part of every system of general education, we now proceed to consider whether Euclid's method must be retained in all its detail. Innumerable efforts have been made to effect improvements, none, in our judgment, with any great measure of success. The only one that has obtained extensive acceptance is that of Legendre. Regarding Legendre's system not as anti-Euclidean, but as essentially Euclidean with important modifications, we must repeat the statement which we have already made in substance, that in our judgment the modifications are not improvements. Having tasted the new wine, we acknowledge it to be wine; and we do not say that it is not good wine, but we say that "the old is better." We brush aside multitudinous attempts to condense and simplify Euclid's demonstrations. These all proceed by deteriorating their rigidity, and so rob them of the main part of their value. Let us give an example of these simplifications. Rightly or wrongly, Euc. I. 5 has been regarded by many as presenting difficulties too great for the student at so early a stage in his course, whence it has derived the unenviable and opprobrious name by which it has long been designated. Now, such difficulty as there is might

be easily got over. If we bisected the vertical angle of the isosceles triangle, we should divide the triangle into two triangles, having one side in each equal to one side in the other, one side common to both, and an equal angle contained by these equal sides. It would at once follow, Prop. i. 4, that these two triangles are equal in every respect, and so that the angles at the base of an isosceles triangle are equal. Then the second part of the proposition relating to the angles at the other side of the base would follow from the first part, if we assumed as an axiom the almost axiomatic Prop. i. 13. Now all this Euclid knew just as well as his improvers. Why did he not adopt their method, which would certainly have resulted in a demonstration somewhat simpler than his? Manifestly because at that stage he had geometrically no cognisance of the half of a given angle. He knew that every angle has two halves. Had he not known that, he would not have sought to find the half of a given angle, as he does in Prop. I. 9. Till he had solved that problem, the half of a given angle was a thing unknown to him. It may be noticed that he proceeds on precisely the same principle with reference to straight lines. From his definition of a circle he knew that all radii of the same circle are equal. That was at the outset his only criterion of the equality of two lines, and upon this he proceeds in his first three propositions; whereas Legendre in his first proposition sets out by simply taking a line equal to a given line, that is, practically assuming as a postulate Euc. I. 2. While we are confident that Euclid's demonstration of I. 5 is preferable to that proposed to be substituted for it, we are of opinion that the Euclidean rigidity

might be united with more than the Euclidean simplicity by another method of proof. It will be sufficiently intelligible without a figure. Take an isosceles triangle whose base is AB and vertical angle C. To us it seems legitimate to compare the triangle ACB with the triangle BCA, and to infer from Prop. 4 that the angles A and B are equal. No doubt the triangles ACB and BCA are one and the same. But it does not seem to contravene the geometrical instinct to suppose them two pro hac vice. When the method of proof is put in the form of supraposing the triangle on itself with the sides reversed, Mr. Dodgson wittily compares it to the Irish feat of a man's jumping down his own throat! The wit is wholly commendable; its applicability to the matter in hand is questionable. At all events it is not applicable to the form in which we would put the demonstration.

Thus far, the matter stands thus. The systems of geometry, exclusive of the so-called systems which are not systems at all, are three: the Euclidean, the Cartesian, and the Legendrian. The first and second are essentially different, and are to be studied apart. We cannot afford, on utilitarian grounds, to discard the Cartesian; as little can we afford, on intellectual and therefore ultimately utilitarian grounds, to neglect the Euclidean. The Legendrian method, as a modification of the Euclidean, we regard as not an improvement, but the reverse. There is no reason why both should be studied by the mathematician, for in matter they are identical, while in method it may be freely admitted that Euclid has not always the advantage. But as an educational text-book, we hope that our own countrymen will never abandon Euclid; we can scarcely

hope that our French neighbours will ever abandon Legendre. Thus we must agree to differ. While we regard our own as "the more excellent way," we do not regard the difference as vital. When from time to time a great Legendrian geometer appears, we seek not to disparage his greatness. We rather argue that, great as he is, he might possibly have been greater still, if he had had the good fortune to be trained after the stricter fashion of the grand old man of Alexandria.

Attempts innumerable have been made to improve Euclid. These are handled with a degree of acuteness which is simply marvellous by Mr. Dodgson,1 who has shown that, with the exception of two or three emendations, and those of little or no consequence, Euclid still holds the field. While this is so, there is a very common-we might almost say a universal-impression that Euclid, with all his excellence and his superiority over all his rivals, is not absolutely perfect. Undeterred by the fate of so many who have failed, we have a lingering hope that success is possible, and that success might possibly fall even to our lot. Sustained above all by the consideration that "in great attempts 'tis glorious even to fail," 2 we venture to present ourselves, not as a rival, nor even as a humble editor, but as the offerer of some suggestions which may be worthy of consideration on the part of future editors.

If there be any defects in Euclid, they are in his definitions and axioms. In order to judge of these, we must have a clear apprehension of the real nature of definitions and axioms. This is all the more necessary

1 Euclid and his Rivals.

2 Magnis excidere ausis laudabile.