work has heen translated by Sir David Brewster. The book contains also a few classical propositions of Solid Geometry.

(4) CONIC SECTIONS.

[Dr Whewell's Conic Sections, or the Conic Sections in Mr Goodwin's Course.]

(5) MECHANICS.

[Dr Whewell's Elementary Treatise on Mechanics. 7th Edition. 1847.]

(6) NEWTON'S PRINCIPIA.

[Dr Whewell's Newton's Principia in the original Latin. 1846.]

I have already stated (240) that the standard book in the study of the Principia is Newton's text; and that other modes of presenting the subject are to be considered only as Commentaries upon that text. I have given such a Commentary in the Doctrine of Limits. The Doctrine of Limits, or in Newton's language, the Method of Prime and Ultimate Ratios, is the subject of the First Section of the Principia, and is the basis of all the succeeding portions of the work.

There are several Propositions concerning the Mensuration of the simplest figures, which are so familiarly assumed as known by Mathematicians, that it is not convenient to make them depend upon the Differential Calculus. Such are the Propositions which Archimedes proved concerning the Sphere and Cylinder. These Propositions may be conveniently proved by the reasoning of Limits. I will here state them.

(7) FAMILIAR RESULTS OF THE
METHOD OF LIMITS.

[Though this is not recognized as a distinct subject in the University Schedule, I retain it here, as being very instructive from the light which it throws both upon the Doctrine of Limits as upon the Principles of the Differential Calculus.]

1. To find the circumference of a circle, of given

radius.

2.

3.

4.

5.

the area of a circle.

the surface of a cylinder.

the solid content of a cylinder.
the surface of a cone.

6. The solid content of a cone is the circumscribing cylinder.

7. The solid content of a sphere is the circumscribing cylinder.

8. The surface of a spherical zone is equal the surface of the corresponding zone of the circumscribing cylinder.

9. The area of the parabola is the circumscribing parallelogram.

10. The solid content of the parabolic conoid is the circumscribing cylinder.

I have proved these propositions in the Doctrine of Limits*.

(8) DIFFERENTIAL CALCULUS.

[In the former edition I had inserted a Syllabus of those steps which appeared to me to be essential parts of a standard system of the Differential and Integral Calculus confined to the mere Elementary processes of

In this as well as in some other parts of the Doctrine of Limits were some material errours of the press, which were corrected by cancelling the pages.

the subject. But in drawing up the Schedule which contains the University standard, it was conceived, probably wisely, that the simplest way of limiting the scheme to Elementary Mathematics was to exclude the Differential Calculus altogether. I shall therefore omit the Syllabus of the subject.]

(9) INTEGRAL CALCULUS.

[This is of course excluded along with the Differential Calculus.]

(10) HYDROSTATICS.

As I have already stated (242), Mr Webster's Principles of Hydrostatics appears to me fitted for this part of our list. Of course when Hydrostatics is to be studied as a part of Progressive Mathematics, works which teach the subject by the aid of the Differential Calculus must be taken. [See also the "Hydrostatics" in Mr Goodwin's Course.]

(11) OPTICS.

I have stated in the last Chapter that I do not think the University can adopt any mode of presenting the Elements of Optics as a permanent subject better than is contained in the work of Dr Wood. Perhaps the calculation of the Aberration of Refracted Rays might be omitted at this stage of study. [See also the "Optics" in Mr Goodwin's Course.]

(12) ASTRONOMY.

I have already said that Dr Hymers' Astronomy appears to me fitted for general use in the University. But Astronomy in its methods of observing and calculating is a progressive science; and perhaps it might suffice to take, as the permanent part of it with which I am here concerned, Dr Hymers' First Chapter. [See also the "Astronomy" in Mr Goodwin's Course.]

SECT. 2.

Progressive Mathematical Subjects (for Wranglers and Senior Optimes).

The subjects of study which are to lead to the Higher Mathematical Honours ought to be the various branches of mathematics in their best form. The Systematic Treatises in which they are thus presented will necessarily vary from time to time; although it is desirable that the Treatises which we employ in our course should not be rapidly exchanged for others (79). I have already stated how, as seems to me, the selection of such Treatises might be made by the University. The Student's business would then be to master, in the form thus prescribed, a larger or smaller part of the subjects already mentioned; namely, Algebra, the Differential and Integral Calculus, with the solution of Differential Equations, Finite Integrals, the Calculus of Finite Differences, the Calculus of Variations; the properties of Curves and of Curve Surfaces, Analytical Mechanics and Hydrostatics; the Higher parts of Optics, both Formal and Physical, and Astronomy.

I have already mentioned certain works which may be considered as peculiarly the proper employment of our Highest Mathematical Students; namely, Newton's Principia, Lacroix's larger Traité; the Cambridge Examples to the Differential and Integral Calculus; Professor Airy's Tracts; and I add, as a Commentary on the Eleventh Section of the First Book of the Principia and on the Third Book, his Gravitation ; Lagrange's Théorie des Fonctions and Mécanique Analytique; Monge's Application de l'Algebre à la Géométrie; Laplace's Mécanique Céleste; to which I have added, on account of their practical application joined to their mathematical merit, M. Poncelet's Mécanique Industrielle, Professor Willis's Principles of Mechanism, and Count de Pambour's Theory of the Steam Engine.

SECT. 3. Other Progressive Sciences.

If the University were to carry into effect the suggestion which I have offered, and were to institute examinations in the Physical Sciences and Natural History, assigning honours and rewards to those who should distinguish themselves in such examinations, it would be proper, in some degree, to prescribe and define in each science the course of study which the examinations should require. If, as I have proposed, the Professors of the subjects included in these Examinations were to be the Examiners (the Professors of Chemistry, Anatomy, Botany, Geology, Mineralogy), these Professors would be the proper persons to mark out a University course of study in the progressive sciences; and might be invested with authority to draw up and prescribe such a course. In drawing up such a University course of study, the Lectures of the University Professors would naturally and properly be reckoned as necessary and important sources of knowledge. Attendance at those lectures, and the study of standard books in each science pointed out by the Professor, would constitute the University course of study in this department.

Among the subjects which, as I have mentioned in (284) are considered branches of the Higher Education, in addition to the Physical Sciences, there are several of which the University of Cambridge possesses Professorships, namely, Modern History and Languages, Moral Philosophy, and Political Economy, besides the Professorships belonging to the Faculties of Divinity, Laws, and Medicine, and the Professorships of Hebrew and Arabic. And on all these subjects Lectures are habitually given by the Professors. If it were thought right to give further weight to those studies, means might be devised for doing so, of the same kind as