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

Statical reasoning, as it has generally been made. And I much fear that students who see the property of the Lever proved in the complex and indirect way in which Mr Goodwin proves it, will never see what a simple and cardinal truth it is. It is true that Mr Goodwin has given in a note the independent proof of the property of the Lever; but this is far, I fear, from being likely to remove the confusion which is likely to arise from giving to a Science so old, a new form in an Elementary Course. Many of the questions which have usually been discussed in books of Mechanics become unintelligible in this way of treating the subject*.

357 I should be disposed to object, although in a less degree, to the mode in which the subject of Cycloidal Motion is treated. Mr Goodwin has, in a note in the second edition, introduced a method of proving the properties of Cycloidal Oscillations which is conformable to the University Regulation; the method introduced in the text being certainly one which the University could not have contemplated in passing the Grace. I have every temptation to be pleased with Mr Goodwin's adoption of this mode of proof, for it is, in substance, one which I invented, and inserted in my Mechanics several years ago. It was invented for the sake of introducing ratios, instead of an arbitrary radius; the ground, as I have already said, on which our Cambridge Mathematicians made the innovations

* I may remark that my objection to this mode of dealing with the subject of Statics arises from no want of interest in the proofs of the composition of forces at a point. Several such proofs have been devised in recent times, one of which is that which Mr Goodwin gives. I have myself invented one such proof, which was published in a book entitled Analytical Statics, in 1833; and which coincides with a proof published about the same time by M. Poisson, though invented quite independently. And I believe I was the first person who published, for the use of Cambridge Students, a Treatise on Mechanics in which the Composition of Forces at a Point was statically demonstrated.

in Trigonometry, of which I have spoken. But I confess I still prefer the older method invented by Cotes, and adopted by Wood; and I think the student would derive from it a clearer notion of the proof; as certainly it has a far better claim to be considered as a classical proposition than mine has.

358 I do not think that I entertain any other objections to Mr Goodwin's book which are so important that I need mention them here. Some parts of his work are well suited to their purpose.

The Conic Sections, as treated by Mr Goodwin, though brief, are sufficiently rigorous. The Newton will not give a very exact notion of Newton's reasoning to those who do not consult the original text: but it is to be hoped that all the more active minded students will do so. The Optics would, I think, have been clearer for elementary students if it had been more Geometrical; but the propositions are very properly illustrated by several good figures, which in a great measure remedy the defect. The Astronomy is such as meets the occasion; and if there be introduced more concerning astronomical instruments than is absolutely needed, this part may serve to turn the attention of the better students to an interesting subject. And the same may be said of the mode of constructing Solar Eclipses, which makes an acceptable addition to the Astronomical part of the Course. I cannot but rejoice that we have the whole of the Elementary Course of Mathematics to which we direct the candidates for Mathematical Honours thus collected in a single moderate octavo volume.

359 The separation of the higher Mathematics from the elementary in the Examination of our candidates for Mathematical Honours was in operation in the Examinations of January 1848, and January 1849; and so far as can yet be judged, has tended much to produce, or to ascertain, the state of things which was