arbitrary convention. If Geometry is to be expelled from Conic Sections, it must be expelled from Cosmical Dynamics also; and Newton must be replaced by Analytical Mechanics. But we are to recollect that, when this is done, besides the regret which we should feel in thrusting out Newton from our course of mathematics, what we have got in the place of our former system, is of little or no value as an instrument of Education. If Newton's Propositions are not worth proving in his own way, in our Educational Course, they are not worth proving in any way. If Conics is not worth retaining in a geometrical form, it is not worth retaining at all. It is true there are difficulties in Newton's reasonings; but it is by understanding these difficulties, and their solutions, that the reasoning of Newton becomes instructive. It is true, that the special properties of Conic Sections, in the geometrical method, require special reasonings; but these reasonings constitute what the mathematical world has always understood by Conics, as a province of mathematics. A person who is acquainted only with analytical treatises, knows nothing of Conics, in the sense in which the word has always heen used by all mathematicians.

236 I should, therefore, recommend that in our standard course of studies, the properties of the Conic Sections should be established by special geometrical reasonings; at least, so far as concerns the properties of the tangents, those of the circles of curvature, and the properties of oblique ordinates, which connect the other two sets of properties. The remaining properties, those regarding the asymptotes of the Hyperbola and the intercepts of its chords, for instance, may be proved by Algebra; for these algebraical proofs are instructive in themselves; and these properties are not directly connected with the others.

I shall, in the subsequent part of this volume, give a scheme of a standard course of Mathematical Studies;

and shall there endeavour to point out how Conics, among other subjects, may be read.

237 Again: that the subject of Mechanics has been rendered less valuable as a part of our Education, by the analytical character which has been given to its Elementary portions, I cannot but believe; although I fear I have had some share in bringing about the change. Dr Wood's Treatise on the subject might be considered as the standard work in the University, at the beginning of the present century. Among the peculiarities of this work, as we may now call them, were Newton's proof of the Composition of Forces, which goes upon the supposed identity of Statical and Dynamical Action; the Laws of the Collision of Bodies, also proved according to Newton; the Laws of Falling Bodies, Cycloidal Pendulums, and Projectiles, proved as Cotes had proved them, by elegant geometrical methods. The rest of the book, the properties of the Mechanical Powers and of the Centre of Gravity, had long had their places in elementary works on Mechanics. In this compilation, brief and simple as it was, there was no part which had not both a historical value and a geometrical rigour of proof. I do not think that any of the parts of the subject which I have mentioned deserved to be rejected out of our system, although it might be very proper to introduce other modes of dealing with these mechanical problems, as comments upon the standard proofs, and as preparations for the higher mathematical studies. The newer modes of treating mechanical questions employed in rival works, were more instructive when compared with those older and simpler reasonings; and it is to be regretted that Dr Wood's Mechanics has been allowed to vanish from among the books current in the University*.

* I am aware that a Volume was published in 1841, calling

238 I seem to be justified in speaking of my own book on Mechanics, as the successful rival of Dr Wood's, by the number of editions which it has gone through*; and by the so-entitled "New Edition" of Dr Wood's having adopted all the peculiarities in which it differed from Dr Wood's. The main features of difference, the distinction of Statics from Dynamics, and the statical proof of the Composition of Forces; features which give a new form to the subject of Elementary Mechanics, may, I think, be considered as having been fully accepted by the University. In the last edition of the work (the sixth) I endeavoured to make it approach more nearly than before to what a standard work ought to be; namely, as I there stated, "that it should consist of the principal, or classical propositions of the science, and of the other propositions which the proof of these renders indispensable." In one respect I fear that I have carried this principle too far; namely, by excluding "the Mechanical Powers" from the book, as being not an indispensable part of the subject, but mere examples;

itself a New Edition of Dr Wood's Mechanics; but this publication does not at all diminish the force of what I have said. There is, in this "New Edition" scarcely a vestige, either of Dr Wood's general arrangement, or of his treatment of particular questions; for the modes of teaching every subject appear to me to be taken from the rival works which had been published in the seventeen years elapsed since the last edition (the seventh) of Dr Wood's Mechanics. For instance, every one of the peculiarities of the work which I have above noticed in the text is obliterated: the Division of Statics and Dynamics, the statical proof of the Composition of Forces, the analytical investigation of the Resultant of Forces, and, I think, every noticeable feature in which Dr Wood's rivals differ from Dr Wood, are adopted. It is still to be hoped that some member of Dr Wood's College will give the Cambridge world a new Edition of his Mechanics.

* The work has also been extensively used in Educational Institutions in other parts of the British Empire and in America.

and further, as being examples not scientifically classified, but received by blind tradition from the later Greek mathematicians. On these grounds, I excluded the Mechanical Powers from the Elementary Treatise on Mechanics, giving them in a more complete form in the "Mechanics of Engineering." But I might more properly have said that these " Mechanical Powers" have so constantly had a place in Treatises of Mechanics that they are to be deemed Classical Propositions in the subject; and though they are not a complete classification of the simplest Mechanics, they serve very well to point out the mode of dealing with most statical problems. With the exception of the Toothed Wheel, the Wedge and the Screw, indeed, the Mechanical Powers are included in the University list of Propositions for the Polloi; but I should propose still to make them part of the standard Elementary Treatise of Mechanics, as they stood in the fifth edition of my own work.

239 There is one part of Dr Wood's Mechanics which it would be desirable to retain, namely, the doctrine of the Oscillations of Pendulums. This subject is so important, both historically, and in its application, that it should be made a portion of our standard Elementary Mechanics. The student of that science ought not to remain in ignorance of the laws of such Oscillations, till he falls in with them, if it so happens, as examples of the Differential Calculus. In all the Editions of my Mechanics, I proceeded upon this conviction, and proved the properties of Cycloidal Oscillations by the Method of Limits; modifying, however, Cotes's demonstration. But in the last edition I have, I think, rendered the proof more simple by restoring it nearly to the form in which Dr Wood gave it.

In the scheme of Mathematical Studies which I shall afterwards offer, I shall guide myself by the considerations now stated.

240 With regard to Newton's Principia, the text itself of the work is, of course, our standard; and our students ought to be able, not only to give Newton's proofs, but to show, by their explanation of the various steps, that they have fully entered into the train of reasoning of the great author. Several members of the University (myself among the rest*) have published books containing the Propositions of several Sections of the Principia, and especially the First Three Sections. But those books ought only to be considered as comments on the original. In the standard course of the University, Newton's Text should be the work adopted. Such books as I have mentioned would enable the student to understand, explain, and, when it is necessary, fill up the steps of Newton's proofs. 241 I have said that a portion of the Principia may be included in our Permanent Mathematical Studies; although the greater part of the book can be read only by our higher Mathematical Students. I think that both the general practice of the University, and the character of the subject itself, would direct us to take the First Three Sections as this Permanent Portion, to be studied by all candidates for our mathematical honours. The remainder of the work may employ, along with other capital mathematical works which I have mentioned, the labours of our more advanced students.

242 In the subject of Hydrostatics, Mr Vince's Hydrostatics might formerly be considered as the standard work in the University, though never very well suited to such a place. Mr Webster's Principles of Hydrostatics, from its form and mode of treating the subject, appears to me very fit to be adopted as our Standard Treatise on this subject.

*

First separately in 1832, and afterwards in The Doctrine of Limits in 1838.