an enquiry "whether it be expedient to lay down more exact rules respecting the Elementary part of the Examination, and especially the selection of the simpler parts of Natural Philosophy." This implies, what we know to have been the case, in the opinion of several members of the University, that the Studies of the University had not become fixed and definite; that the questions were not such as the students anticipated; nor all the subjects duly introduced. The language of this Grace appears to imply a suggestion of an authoritative selection of the simpler parts of Natural Philosophy. There had already been an example of such a selection on the part of the University. The Grace of 1837, modifying the Examinations for Ordinary Degrees, gave a Syllabus of the Elementary Parts of Mechanics and Hydrostatics, which became from that time the authoritative guide of a portion of the University Examinations*.

232 It is believed, by many persons in the University, that the defects which have been referred to still exist; that the want of fixity and definiteness in the studies of the University, and the frequent frustration of the anticipations of the candidates as to the questions which will be asked, are still felt, in a sufficient degree to make it worth while to consider whether some remedy cannot be found. It is probable, too, that these defects prevent the mathematical knowledge

* I believe this mode of defining the portion of Mechanics and Hydrostatics required for the common degrees has been found satisfactory. Such a system may lead to the examinee giving propositions by rote, but not more than the Examination in Euclid may do so. The effectual remedy for this would be a few questions vivâ voce in each subject. It is said that the Algebra which is required in this Examination is very imperfectly given. I believe it will continue to be so, as long as young men come from classical schools destitute, as they commonly are, of all familiarity with Arithmetic.

of our students from being sound and coherent; for their attention during their studies is directed, not to overcome the real difficulties of the subject, but to seize some such portion of it, or to put it in some such form, as may fall in with the turn that the Examination may take.

These evils are included among those which I have pointed out, as likely to result from a System of mere paper Examinations, especially if the distinction of Permanent and Progressive Educational Studies be neglected, and if the branches of Mathematics be treated in an Analytical manner. And I must now say a few words respecting the remedies for these evils, as they are suggested by the train of considerations in which we have already been engaged.

SECT. 2. Suggestions of Improvements in the Educational System of Cambridge.

233 We have already seen that it is important, in an Educational course, to attend to the Permanent, before proceeding to the Progressive portions of Mathematics: that these Permanent Mathematical Studies ought to be laid before the student in standard works: and that such works must be geometrical rather than analytical in their treatment of the subject. I shall proceed to follow out these remarks into further detail, in their application to the present state of the studies at Cambridge.

234 A Standard System of the more Elementary portions of Mathematics which constitute our Permanent Studies in that department, has been always, in some degree, assumed in the University legislation on this subject. Nor does it appear likely that we shall avoid the evils which have been and are complained of, till such a Standard is in some way established.

I shall offer a few remarks containing views which seem to me fitted to direct those who may be entrusted by the University with the office of selecting a course of standard works, and thus fixing the Permanent Mathematical Studies of the University.

235 I have already shewn that the use of analytical methods has rendered the branches of Elementary Mathematics less suited for Permanent Educational Studies; taking in order Geometry, Trigonometry, Conic Sections, Statics, Dynamics, and Astronomy (46-51). This might be illustrated still further by instances of the changes which have taken place in the books in common use in the University of Cambridge. For instance, by adopting books which employ the analytical method of establishing the properties of the Parabola, Ellipse, and Hyperbola, the subject of Conic Sections has lost all its value, both as an example of geometrical reasoning extended into a new region, as an historical province of mathematics, and as an introduction to the reading of Newton's Principia. Looking particularly at the matter in the latter point of view, we may venture to say that a student who knows no other methods of drawing tangents and circles of curvature to the Conic Sections, than are given in the analytical treatises, as now current among us, is not likely to see the meaning of Newton's reasonings concerning the Conic Sections, and curves in general, in the first book of the Principia. If, in our standard mode of treating the subject, we draw tangents and circles of curvature by the Method of Limits, or some equivalent geometrical reasoning, the reference in Newton's proofs to the properties so established, is consistent and intelligible. But if we acquire our knowledge of the properties of Conic Sections by general analytical formulæ, the introduction of a few geometrical steps of Newton's reasoning, as a sequel to such formulæ, is incongruous; and must appear to the student a mere

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