CHARLES HUTTON, LL.D. F.R.S. FORMERLY PROFESSOR OF MATHEMATICS IN THAT CONTINUED AND AMENDED BY OLINTHUS GREGORY, LL.D. F.R.A.S. VOL. II. TWELFTH EDITION, WITH CONSIDERABLE ALTERATIONS AND ADDITIONS, BY THOMAS STEPHENS DAVIES, F.R.S. L. & E. F.S.A. ROYAL MILITARY ACADEMY. LONDON: LONGMAN, BROWN, & CO.; J. M. RICHARDSON; J. G. F. & J. RIVINGTON; HAMILTON THE EDITOR'S PREFACE. THE public approbation of my efforts to render the first volume of Dr. Hutton's Course of Mathematics such as the present state of Mathematics requires, has been too undisguisedly manifested by the extensive sale of that volume, to be other than very gratifying to my feelings and I hope in the next edition of it, to be able to fully carry out those contemplated improvements, which, from various circumstances, I was only able to partially develope in the last. The second volume, here laid before the public, will, I trust, be considered more completely in accordance with the views under which I have officiated as editor, than was the case in the preceding one: and I anticipate with confidence, that the patronage with which the former volume was honoured, will be extended to the present one. This, at least, I can assure my readers, that no exertion or labour has been spared to render it worthy of the continuance of general approbation: and I cannot offer stronger evidence of this, than the fact, that there is not a single line of the original work which has not been recomposed. It may, likewise, be added, that I undertook only the ordinary duties of an editor-those of correcting the work for press, and reading the proofs-on corresponding pecuniary conditions. I may, hence, fairly disclaim every mercenary motive in the labours here brought to a close. Hutton's Course was, in fact, the first mathematical work I studied. I may, hence, be allowed to entertain a desire to render it as complete as possible, and, perhaps, to feel a deeper interest in its improvement, than in any work exclusively of my own production. I now proceed to give a short account of the contents of the present volume. Prefatory to the Spherical Trigonometry there is given a chapter on the Geometry of the Sphere, comprising, besides the propositions usually to be found in other treatises, several curious and important ones used in the course of this work for the completion of discussions of trigonometrical properties, which have either not been noticed, or but very imperfectly developed. In the Trigonometry itself, it will be seen that I have taken a view of Napier's rules as a scientific method, altogether the reverse of that usually adopted. The deduction of the general properties of spherical triangles from the right-angled triangle is probably new and the entire series of formulæ of solution is obtained by successive transformations without the aid of the polar triangle; as I consider the exercise in trigonometrical reduction thus obtained, to be infinitely more valuable than the time saved by the use of the polar triangle, elegant as that method is admitted to be. For the convenience, however, of those teachers who think otherwise, the method of deduction by the polar triangle has been added in the note on page 242. I have given numerous properties of the system of associated triangles and their polar ones, which are interesting for their remarkable symmetry and manifold applications in research; and likewise all the known formulæ (some of them, indeed, but little known) for the areas of spherical triangles. In the solution of spherical triangles, I have introduced a new classification of the cases, which, from their analogy to the three cases of plane triangles, will render the rules of spherical trigonometry equally simple and easy to recollect with those in plane trigonometry. In all the examples it has been my aim to give the best working formularies, as well as the most elegant algebraical formulæ. The doctrine of solid angles is curious in itself, and useful in its applications; and it is interesting to the readers of Hutton's Course from its being the re-invention of my late venerated colleague, Dr. Gregory,the little that Stevin had written on the subject never having excited the least attention. I have, however, remodelled the chapter to adapt it to the general view which I entertain of the units of geometrical magnitude. The series of properties of spherical figures contains a considerable number of theorems of much elegance in themselves, and still more for their analogy to, and indeed almost identity with, the most admired series of properties of plane figures investigated in the modern geometry. I had published some of them previously, though in a less completed state, in Leybourn's Repository, and in the Ladies' and Gentleman's Diaries. The chapter on Spherical Astronomy, brief as it is, contains all that is required for the purpose. If practical astronomers, instead of tabulating an immense number of formulæ, would only take the trouble to familiarise themselves with the best solutions of spherical triangles, and use the two astronomical triangles, they would find the claims upon their memory and attention greatly abridged. The idea of the two astronomical triangles was suggested by a remark of Dr. Pearson, who, in his invaluable work on Practical Astronomy, speaks of the astronomical triangle which is our second one. The investigation of the Lunar problem is from the Trigonometry of Professor Young, who is undoubtedly amongst the most able, and by far the most useful, elementary mathematical writer of our age. The treatise on the Conic Sections in the preceding edition was drawn up (as far as it was altered) by me; and, except a few changes in language and arrangement, it remains as it was then given. Those of greatest moment in this edition are in the parabola; and I have introduced, amongst other theorems, Lambert's very remarkable property, which has been attributed to several more recent authors. (Vid. prop. xviii. p. 198, and Phil. Mag. Jan. 1843.) The chapter entitled general properties of the conic sections is new in this work; and being drawn up on the principles of the great authors of antiquity, embodying their most general properties and modes of investigation, it cannot but be a useful contribution to the young geometer's means of study. It also includes the doctrine of poles and polars originating in the earliest researches of the great Pascal, and terminating with the Mystic Hexagram of that extraordinary man, and the correlative property discovered by the very able French geometer, Brianchon. There are some theorems interspersed from the writings of Simson and Maclaurin-men not less gifted than the greatest of their predecessors of antiquity. In many respects I am much indebted to the Geometry of Curve Lines, of the late, and very imperfectly estimated, Sir John Leslie*, in the The scientific character of Leslie has been greatly under-rated. As a geometer, he was, not only in learning, but invention, equalled by none of his cotemporaries, except Lowry and Swale; whilst in the literature of science he had no English competitor. His language, indeed, sometimes has too much eloquent hyperbole, and in, perhaps, a few of his demonstrations he has committed oversights: yet with these trivial objections, I can mention no work so improving to the taste of the young student as Leslie's Geometrical Analysis and Doctrine of Curve Lines. My present opinion, too, was that of a no less eminent judge of geometrical merit than the author of the Life of Simson; and Dr. Traill urged with much earnestness my close attention to this work at an early period of my own studies. No man who reads a life of Leslie in vol. vi. of Leybourn's Repository, can help regretting that so distinguished a patron of mathematical science should have admitted into his work such a coarse, ignorant, and illiterate piece of biography; and no one who is familiar with Leslie's mathematical writings can rise from the reading of that paper without feelings of the deepest disgust. |