| spherical trigonometry (ocr) |
| 1 | rjL. is. i h 6 SPHERICAL TRIGONOMETRY MACMILLAN AND CO., Limited LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO SPHERICAL TEIGONOMETRT FOB THE USE OF COLLEGES AND SCHOOLS BY THE LATE I. TODHUNTEE, M.A., F.B.S. HONORARY FEW.OW O* - ST. JOHN'S COLLEGE, CAMBRIDGE REVISED BV J. G. LEATHEM, M.A., I). So. |
| 2 | FELLOW A.VD LECTURER OJf ST. JOHK'S COLLEGE, CAMBRIDGE MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 1914 COPYRIGHT. First Revised Edition 1901. Reprinted 1903, 1907, 1911, 1914. GLASGOW : PRINTED AT THB UNIVERSITY PRESS BV ROBERT MACLEHOSE AND CO. LTD. PREFACE. The present work is constructed on the same plan as my treatise on Plane Trigonometry, to which it is intended as a sequel ; it contains all the propositions usually included under the head of Spherical Trigonometry, together with a large collection of examples for exercise. |
| 3 | In the course of the work reference is made to preceding writers from whom assistance has been obtained ; besides these writers I have consulted the treatises on Trigonometry by Lard nee, Lcfebure de Fourcy, and Snowball, and the Treatise on Geometry published in the Library of Useful Knowledge. The examples have been chiefly selected from the University and College Examination Papers. In tbe account of Napier's Rules of Circular Parts an explanation has been given of a method of proof devised by Napier, which seems to have been overlooked by most modern writers on the subject. |
| 4 | I have had the advantage of access to an imprinted Memoir on this point by the late R. L. Ellis, of Trinity College ; Mr, Ellis had in fact rediscovered for himself Napier's own method. For the use of this Memoir and for some valuable references on the subject I am indebted to the Dean of Ely. v i PREFACE. Considerable labour has boon bestowed on the text in order to render it comprehensive and accurate, and the ex- amples have all been carefully verified ; and thus I venture to hope that the work will be found useful by Students and Teachers. |
| 5 | I. TODHUNTEK. St, John's College, August 15, 1859. REVISER'S PREFACE. In the present revision of Dr. Todhuntkr's Spherical Trigonometry so many changes have been made that only a comparatively small portion of the last edition remains in its original form. The introductory chapter, and the chapters on Gcodetical Operations and on Polyhedrons, are almost un- touched, and in the chapter on Arcs Drawn to Fixed Points only one paragraph has T>cen altered. |
| 6 | But that part of the book which deals with the Formulae of the Triangle and the Solution of Triangles has been re-written, and the remain- ing chapters include extensive alterations and additions. I have followed the example of the late Dr. Casey in intro- ducing chapters on Spherical Geometry, and I am indebted to his Spluriml Trigonometry, and to Baltzer's Elemerde tier Mathematik, for references to the important writings on the subject. |
| 7 | Passing over, however, a number of geometrical methods of considerable interest bub of restricted application, I have given the central place in the present edition to the Principle of Duality as exemplified in theorems relating to circles on the sphere. Though the principle and some of its applications to Spherical G-eometry have been known for viii PREFACE. a long time, I have not found any connected account of the subject, such as is contained in Chapter X, Coaxal circles have been discussed in such a way as to shew their analogy with coaxal circles on a plane ; and the coaxal sys- tem and the reciprocal of a coaxal system, to which I have given the name eohmar, are selected as examples of Duality, partly because the properties of the latter afford a new treatment of Hart's Theorem, but chiefly because, on tran- sition to the plane, they present an interesting relation between systems of circles on the plane, possessed in the one case of a common radical axis, in the other of a common centre of similitude * A chapter has been devoted to the generalisation of the Spherical Triangle, based on a recent memoir by Dr E. |
| 8 | Study ; and another gives a brief account of Prof Frobenius's application of determinants to the geometry of the sphere. *In this connexion a remark, which it is now too late to insert in its natural place in the text, may be made here. Just as the constant of Art. 169 ia called the Spherical Power of the point with respect to the small circle, so the constant of Art. 171 may be called the Spherical Power of the great circle with respect to the small circle. |
| 9 | (Tf the great and the small circle intersect at an angle (p, the spherical power is equal to tan 2 0. ) Then, as the radical circle of two small circles is the lorn* of points whose spherical pow-ers with respect to them are equal, the centre of similitude of two small circlos is the envelope of great circles whose spherical powers with respect to them are equal. Of course by the centre of similitude of two circles is meant the external or the internal centre of similitude, according as the circles have the same or opposite senses of rotation assigned to them. This view of centres of similitude completes the analogy between coaxal and colunar circles, whether on a sphere or on a plane. |
| 10 | PREFACE. ix In both these chapters special attention has been paid to the conventions used for the purpose of avoiding ambiguity. It is hoped that a sufficient emphasis has thus been laid on the importance of assigning to every circle a certain direction and a unique pole, a method whose utility has been exemplified also in Chapter X. Some examples have been added, taken, for the most part, from the papers of the Science and Art Examinations and of the Royal University of Ireland - } a few are from Rkidt'S collection. |
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