CHAPTER IV.

PLAN OF A STANDARD CAMBRIDGE COURSE OF MATHEMATICS.

SECT. 1. Permanent Mathematical Subjects (for Junior Optimes.)

[Since the first edition of this Work was published, the University has established a standard scheme of the more elementary portions of Mathematics, namely, those portions which are required of Junior Optimes. See Part II. Sect. 5 of this book. Also Mr Harvey Goodwin has published An Elementary Course of Mathematics, the design of which is to include such portions of the science as belong to this scheme. Of this book I have spoken in the Second Part. I have moreover myself published Conic Sections, their principal properties proved geometrically. I have also published a new edition of my Elementary Treatise on Mechanics. I have in Part II. explained the reasons why I consider the course which this Work follows, more suitable to an Elementary Treatise than Mr Goodwin's. I have also published Newton's Principia, Book I. Sections I. II. III. in the original Latin, with explanatory Notes and References. As I published these works in order to embody the plan of a standard Course of Elementary Mathematical Subjects which I proposed in the former edition of this work, it will not be considered strange or presumptuous that I should introduce them here. 2nd Ed. Part I.]

(1) ARITHMETIC.

Mr Hind's Arithmetic, in the later editions, appears to me to be drawn up in such a manner as to be suited

for use in Schools for those who are intended to go to the University. It includes the use of Logarithms, and the Mensuration of various figures (Triangles, Circles, &c.), which I have spoken of as desirable appendages to the parts of Arithmetic usually learnt at school.

(2) ALGEBRA.

Dr Wood's Algebra may still be considered as marking the extent to which this subject should be read by the common student. In reading the First Part of the work the student will probably at first need additional explanations and examples, which he may obtain from many works in common use. In the Second and succeeding Parts the subject admits of developements much more extensive than Dr Wood has given; but still this work may be considered as the Standard of our Algebra, excluding its recent progress*.

(3) PLANE TRIGONOMETRY.

The work most worthy of being made our Standard work on this subject appears to me to be Legendre's Géométrie, which includes Trigonometry, both Plane and Spherical, and contains a few Notes which may be looked upon as classical in mathematical literature. There is an inconvenience, however, in his exclusive reference to the French graduation of the circle. The

Mr Lund, in his last edition of Dr Wood's Algebra (1845), has very properly kept his additions distinct from the original text by a difference of type. He has omitted the Second Part of the Treatise altogether, which I cannot but regret ; for that portion of Dr Wood's book represented very well the General Doctrine of Equations as a long established part of Mathematics; whereas Dr Hymers' Treatise, to which Mr Lund refers as replacing this Part, belongs to the Progressive Mathematical Studies of the University.

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.]