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ARITHMETIC and the elementary parts of ALGEBRA; namely, the Rules for the fundamental Operations upon Algebraical 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 on 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 Oscillations.
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 non-elastic Fluids, 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; Telescopes.
The elementary parts of ASTRONOMY; so far as they are necessary for the explanation of the more simple phenomena, without calculation.
4. That in all these subjects, Examples, and Questions arising directly out of the Propositions, shall be introduced into the Examination, in addition to the Propositions themselves.
5. (This article refers merely to the days and hours of Examination, and is therefore omitted.)
6. That the Moderators and Examiners shall be authorized to declare Candidates, though they have not deserved Mathematical Honours, to have deserved to pass for an Ordinary Degree, so far as the Mathematical part of the Examination for such degree is concerned; and such persons shall accordingly be excused the Mathematical part of the Examination for an Ordinary Degree, and shall only be required to pass in the other subjects, namely, in the parts of the Examination assigned in the Schedule to the last two days: but such excuse shall be available to such persons only for the Examination then in progress.
When the preceding regulations had passed into law, it struck me very forcibly that, in order to carry out the expressed wishes of the University, it would be desirable, if not necessary, that a short course of mathematics should be compiled, of which the Schedule agreed upon by the Senate should be, as it were, the table of contents. appeared to me that, with regard to several of the subject, there were no books in use, which would put before the student the portions to which it would be necessary for him to devote his attention, without an accumulation of other matter which would be likely to confuse and perplex. I mentioned the necessity of such a book several times in the course of conversation, and found that others agreed as to the want, but I did not hear of any one who seemed disposed to undertake the labour requisite for its supply. Under these circumstances I determined to attempt the task myself, trusting that the intention would be appreciated, however much the execution of the design might fall short of my own hopes, or the requirements of the case.
For indeed it is a task of no ordinary degree of diffi
culty, to write an elementary work upon an abstruse subject; points, which appear to the writer plain and intelligible without explanation, sometimes assume a very dif ferent aspect to the beginner, and difficulties of which the author is scarcely aware, may be of huge dimensions to a mind not already familiarized with the mode of thinking which belongs to each particular subject. Hence it has come to pass that so few elementary works have long retained their ground; and hence also, an author may conclude the expediency of endeavouring, so far as he may, to follow in the steps of those few who have shewn an aptitude for this kind of writing. Time has, I think, proved that, of all works which Cambridge has produced, that which the most nearly fulfils the conditions of a perfect elementary treatise, is Wood's Algebra, a work which it is impossible too much to admire for its simplicity and admirable perspicuity. In writing on Algebra, therefore, I have endeavoured, as far as possible, to take Dr Wood's treatise as a model: and, indeed, in all other parts of my work, where the nature of the case allowed, I have endeavoured, though I fear not always successfully, to keep the same example in view.
I may remark also, in general, that it appeared to me that the worst fault into which I could fall, in such a work as the present, would be an affectation of originality. Originality belongs to the progression of science, but not to the exposition of those portions which may be regarded as permanent. I have, therefore, endeavoured to deviate as little as possible from the methods pursued in those books which appeared to me, on the whole, to be the best and the most generally acknowledged.
In each treatise I have included all propositions which, according to my judgment, can fairly be included
in the intention of the Grace of the Senate, omitting however some which are usually given, but which are only applications of, or deductions from, the fundamental propositions. To take an example, in the treatise on Statics I have not given the investigations for pullies, when their own weight is taken into account, nor when the strings of the pullies are not parallel, because these are only deductions from the simplest case; and it would seem to be unadvisable to load a treatise, intended to be of the simplest description, with deductions and applications which may be indefinitely multiplied. Nevertheless my judgment may have led me into error; and, indeed, the only authorised comment on the meaning of the Grace, will be the Examination Papers of 1848 and the few succeeding years: the character of those papers may perhaps render it necessary to modify some portions of my work in a succeeding edition, should that be called for.
I will, however, be bold to give my opinion, that the success of the new scheme of examination depends, to a considerable extent, upon the Grace of the Senate being interpreted in the narrowest manner possible. The great number of the subjects of which a knowledge is required, renders it impossible for students of no great ability, or whose attention is principally devoted to other academical pursuits, to obtain a very extended knowledge of each; a very real and useful knowledge may doubtless be acquired, but it can scarcely be expected to be adequate to the task of answering questions proposed, unless those questions be almost confined to that small number of propositions which may be considered to be classical in each subject.
A few remarks may be made respecting certain of the following treatises, and the plan upon which they have been written.
The treatise on Algebra has, as I have before observed, been formed as much as possible, on the model of that by the late Dr Wood. I have given Euler's proof of the Binomial Theorem for fractional and negative indices, as being at once the most elegant and the most useful as a mental exercise.
In the treatise on the Conic Sections, I have principally followed the demonstrations given by Mr Hustler, partly because they appeared to me as elegant as could be desired, and partly because that work having been long published, and being usually counted the text book in this subject, I thought it well to deviate as little as possible from the beaten track. I have, however, abbreviated to some extent by omission, making it in general a necessary condition of the admission of any proposition that it should be necessary to the understanding of the first three sections of Newton's Principia; by this and other means, I have endeavoured to render as little formidable as might be a subject confessedly difficult and unpalatable.
In the subject of Mechanics, it is more difficult than in either of the preceding to determine precisely the limits to which questions may, according to the Grace of the Senate, extend. On principles to which I have before referred, I have made the treatises on both Statics and Dynamics as brief as possible.
In giving an English version of the first three sections of the Principia, I have endeavoured to adhere, as nearly as circumstances would allow, to the original, only giving the demonstrations a form more convenient for the purposes of the student; and any interpolations of my own have been enclosed in brackets. In one instance (Lemma VI.) I have, after the example of a version much used in the University, substituted a different mode of demonstration for the very short method given by Newton; but