cubits did compass it about." He would have seen that this statement, save in respect of the indefinite etc., was inaccurate as really, though not to the same extent, as the other, and that no statement would have been really accurate except "a line of 10 cubits did compass it about "! Would he have made either of these statements? Assuredly not. It is so probable as to be virtually certain that the Jews had not estimated the value of with any approach to the measure of accuracy of modern approximations. Without decimal notation and decimal arithmetic they could not. none the less is the inference that "the Jews had not paid much attention to geometry" illegitimately drawn from the fact that a historian makes use of somewhat broad approximations on a matter which even now could only be stated approximately, and where any closer approximation than that employed would even now be the perfection of finical pedantry.

But

XI

SUCH of our readers as accept the definition of a straight line as the shortest distance between two points, will, we fear, have difficulty in conceding to us that it has been by such a line that we have reached our present position. We admit that our course has been somewhat devious; yet we have pretty steadily held on our way, and have now to enter on the "last departure" of our allotted voyage. The question that lies before us is as to the future of Mathematical Study. Is Euclid to be deposed from the throne which he has occupied so long and so worthily? Or is the time come for his retiral from that position of distinguished honour and noble work?

In order to answer these questions aright, we must keep steadily in view the twofold use which we have regarded mathematical study as serving, according as it is viewed as a mental discipline, or as the means of acquiring important and useful knowledge. We frankly admit that, viewed in the latter aspect, the Euclidean geometry must take second place to the Cartesian in the technical or professional education of the mathematician. As an instrument of scientific investigation and discovery, the modern analysis is in ordinary hands far more potent than the ancient. While it is scarcely possible to conceive that Newton could have done more

by the former than he actually did by the latter, we are not entitled to say that what he so nobly did, he might not have done still better. This it were difficult to believe. We have his own strong testimony that he believed the contrary. But certainly what he did he would have done with much more ease to himself. And then it is to be considered that it is not certain that Newton did not make more or less use of the modern analysis in his investigations, while in his teaching he rigidly confines himself to the ancient. Some have supposed that he did so; but we are not aware that they have any ground for the supposition beyond the difficulty of conceiving the possibility of any man's accomplishing such an end by such means. But it is to be borne in mind that Newton was not any man. As there may have been "mute inglorious Miltons," so there may have been mute inglorious Newtons; but in all the ages there has been but one vocal and glorious Newton; and none can tell whether there shall ever be another.

In forecasting the future, we must bear in mind that in Newton's day the Euclidean geometry was mature, while the Cartesian was in its infancy-animosus infans, indeed, but quite immature. Since those days the former has indeed made progress, but, as might have been expected, the latter has made much more. The future successor of Newton, then, if he shall ever have a successor, will have but a slightly better instrument than Newton had; while the follower of Laplace will have ready to his hand a much better than was available to Laplace. The modern analysis will then-it cannot be doubted-be the staple of the modern mathematics as an instrument of scientific investigation. But it is to be earnestly hoped that the modern geo

meters will not cast aside the old love, however they may be under the paramount power of the new. There is no incompatibility between the two. Rather the new will gain in potency as the old is cultivated. It is not wholly with us a matter of theory, albeit our practice is little worth mention, that the use of the new is greatly enhanced when it is brought constantly into contact with the old, and translated into its language. Thus, and only thus, the student is able to estimate the progress which he has made, and to ascertain precisely the position which at any time he occupies. The modern analyst has as much need as any other of the Platonic caution against the neglect of geometry.

While it is freely admitted, then, that in the technical education of the mathematician of the future, and of all who are to be engaged professionally or otherwise in the Sciences of applied Mathematics, the non-Euclidean geometry must occupy a large place, we trust it shall never hold an exclusive one. The non-Euclidean is safe only in the hand of him who has drunk in the spirit of the Euclidean. The former is the motive power, potent and irresistible; the latter is the guiding principle, potent also and salutary. "Behold also the ships, which though they be so great, and are driven of fierce winds, yet are they turned about with a small helm, whithersoever the governor listeth."

Briefly, then, the matter stands thus. In all our primary schools we would have arithmetic, the only branch of mathematics that can be taught in these schools, made an indispensable subject of teaching; and we would take steps for having it well taught, much better taught than it usually was in the early days of men still living. We suppose that already there is a

great improvement; but in a matter of such importance every present improvement should be only an occasion for earnest inquiry whether further improvement be not possible.

In our secondary schools we would have the six books of the Elements thoroughly taught, and numerous exercises prescribed. We would also have algebra taught, but only as an extension of arithmetic. We would also treat plane trigonometry (geometrically).

In the non-technical, corresponding roughly with the undergraduate, department of our universities, the course should begin with a thorough revisal of the six books and plane trigonometry; then Books XI. and XII. of the Elements, or perhaps only parts of these books; then spherical trigonometry. This, with algebra treated somewhat more scientifically, would probably occupy the first session. The second we would devote to the conic sections, which we would treat both geometrically and analytically, and to analytical trigonometry.

In the technical schools of mathematics, and the technical departments of the universities, the modern analysis, as we have already hinted, must be paramount. In these schools the object of the teacher should not be so much to instruct his students as to lead them in the path of study, and initiate them in the lifelong work of instructing themselves.

It were not for us, comparatively inexperienced as we are, to dogmatise on the details of a course of mathematical study. But we believe that such a course as we have outlined would enable us to get all the good out of mathematical instruction that it is capable of yielding, both as a mental discipline and as a