any Examination holding the same place with reference to the University Honours, which the Examination for Bachelor of Civil Law does. The Examination for the Degree of Bachelor of Medicine is of a kind not general, but professional; although undoubtedly attendance at the Lectures and Examinations of several of the above Professors of the Natural Sciences are introduced as conditions for that Degree. But there does not appear to be, at present, any necessity for any change in the requirement for Medical Degrees. The professional Education now given at Cambridge to medical students, is a sound and extensive one: and the Degrees are not sought except for professional objects. We do not want at present a Board of Natural Studies which shall connect them with the Medical Faculty. But we may observe that for the purposes of the Examination for the Natural Sciences Tripos, the Professors who conduct the Examination, with the additional Examiner, would constitute a Board of Natural Studies, and would be able to act as such for all needful purposes.

SECT. 5. The Mathematical Tripos.

352 I have already stated (286) that the more elementary parts of Mathematics should be defined by means of some standard, and that a satisfactory proficiency in them according to this standard should be made a condition of competition to Higher Mathematical Honours. This arrangement has since been made, and came into operation in January 1848.

The standard of the more Elementary Mathematics, thus established, was in the following terms: Euclid, Book 1. to v1.; Book x1., Prop. 1. to XXI.; Book XII., Prop. I. and 11.

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"Arithmetic and the Elementary parts of Algebra; namely, the Rules for the fundamental operations upon

Algebraic Symbols, with their proofs, the solution of simple and quadratic Equations, Arithmetical and Geometrical Progression, Permutations and Combinations, the Binomial Theorem, and the principles of Logarithms.

"The Elementary parts of Plane Trigonometry so far as to include the Solution of Triangles.

"The Elementary parts of Conic Sections, treated geometrically, together with the values of the Radius of Curvature and of the Chords of Curvature passing through the Focus and Centre.

"The Elementary parts of Statics, treated without the Differential Calculus; namely, the Composition and Resolution of Forces acting in one plane at a point, the Mechanical Powers and the Properties of the Centre of Gravity.

"The Elementary parts of Dynamics, treated without the Differential Calculus, namely, the Doctrine of Uniform and Uniformly Accelerated Motion, of Falling Bodies, Projectiles, Collision and Cycloidal Oscilla

tions.

"The 1st, 2nd, and 3rd Sections of Newton's Principia, the propositions to be proved in Newton's

manner.

"The Elementary parts of Hydrostatics, treated without the Differential Calculus; namely, the Pressure of Elastic Fluid, Specific Gravities, Floating Bodies, the Pressure of the Air, and the construction and use of the more simple Instruments and Machines.

"The Elementary parts of Optics, [treated geometrically,] namely, the laws of Reflection and Refraction of Rays at Plane and Spherical Surfaces, not including Aberrations; the Eye, Telescope, &c.

"The Elementary parts of Astronomy, so far as they are necessary for the explanation of the more simple phenomena, without calculations."

The First Report of the Mathematical Board re

commended that in the subject of Optics the words "treated geometrically" should be omitted; and a Grace was accordingly passed to that effect.

353 It was to be expected that this Regulation of the Elementary portions of Mathematics by the University would lead to the publication of one or more works combining those portions treated in accordance with the Regulation; so as to provide students with the means of preparing for this part of the Examination. And it was desirable that there should be some one work of this kind which might meet with general approval. If this were the case, we should have not only a standard List of subjects, but a standard Book; a condition under which I conceive that students are able to work with much more benefit to themselves than when they have to search through many books with only the chance of finding what is asked for, or to depend upon the guidance of Private Tutors. I should therefore be most glad to accept a work containing an Elementary Course of Mathematics, conformable to the Grace of the Senate, as the standard for our studies and Examinations.

354 Mr Harvey Goodwin has published a book of this kind: "An Elementary Course of Mathematics;" 1st Edition 1846; 2nd Edition 1847, and has stated that his object in compiling the work was to conform to the Regulations of the University above mentioned. Mr Goodwin made, in his Second Edition, some changes, which were intended to remove some want of full conformity of the work with the University Regulations which I, and perhaps other persons, had conceived to exist in the First Edition. I hope I shall not be deemed unreasonably critical if I point out some passages which still appear to me unsatisfactory. It is to be recollected that we are considering the work as one which we would wish to have fit to be a standard work, permanently used in the University by

all Mathematical students; and permanently used for what I have termed Permanent Mathematical studies in opposition to Progressive. All the Mathematical improvements of modern times are proper subjects of attention for our students; but they come before us in the Progressive Portion of the subject: the Permanent Portion retaining its original form; as we retain our Geometry, for instance, in the form which Euclid has given to it.

355 I cannot but regret therefore that Mr Goodwin in his mode of presenting Trigonometry, has adopted a mode of defining the Trigonometrical lines and proving their properties, which is a Cambridge novelty of a few years standing only, and familiar to none but Cambridge Mathematicians. I mean that he defines. sines, chords, tangents, &c., not as lines, but as ratios. By this means, the meaning of the terms becomes quite obscure, and has no obvious connection with the construction, as in the old and ordinary method it has. If our students learn this Trigonometry only, they will be unable to understand any works on the subject except Cambridge works of the last few years; a kind of learning which deprives the study of a great part of its value. To this it may be added, that these definitions of the sine, tangent, &c. are really not applicable when we come to deal with Trigonometry in a form adapted to logarithmic computation: the only form in which Trigonometry is of very extensive use. Mr Goodwin, indeed, afterwards gives the old and ordinary mode of defining the Trigonometrical lines, from whence, as he says, the meaning of their names will be more distinctly seen. But in all his reasonings he takes the new Cambridge definition. Of course I am quite aware that this novelty was introduced in order to get rid of some changes which used to perplex young students in Trigonometry, and which occurred as sines, tangents, &c. were taken "to radius r" or "to

radius 1." But novelties introduced for such purposes should be employed as illustrations of the old form of presenting the subject, and as elucidations of its difficulties, not as substitutes for the old definitions. As I have already repeatedly said, the value of a certain portion of our Mathematics ought to depend upon its permanent form, as for instance, Geometry. I have heard of an Inspector of a School, in which Trigonometry was one of the subjects studied, blaming the managers of the school because they had not got the new Cambridge Trigonometry. It would have been just as wise to blame them because instead of the Euclidean demonstration of the square on the hypothenuse they had not given some one of the many pretty substitutes for it which have been produced since Euclid's time. It is very well that the students should know those novelties; but the way to make them valuable is to have, as a ground work, the old proposition to compare them with. I should be much better satisfied therefore to see, in our standard Course of Elementary Mathematics, Trigonometry presented in the old and classical form.

356 I am obliged to make an objection partly of the same kind to Mr Goodwin's mode of treating the subject of Statics. He has first proved the properties of the Composition of Forces at a point, by a method which is ingenious, but not, I think, likely to appear clear or satisfactory to a beginner; and he has then deduced the properties of the Lever from those of the Parallelogram of Forces. Now there is an independent proof of the properties of the Lever which has always been current in elementary works on Mechanics, which is eminently simple and satisfactory; and which is remarkable for being as old as Archimedes, who invented it, and from being thus the first monument of a demonstrative Science of Mechanics. These appear to me to be strong reasons for making it the basis of