Spherical Trigonometry

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. 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 Lardner, Lefebure 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 the 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. I have had the advantage of access to an unprinted 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. Considerable labor has been bestowed on the text in order to render it comprehensive and accurate, and the examples have all been carefully verified; and thus I venture to hope that the work will be found useful by Students and Teachers. I. TODHUNTER. CONTENTS I Great and Small Circles II Spherical Triangles III Spherical Geometry IV Relations between the Trigonometrical Functions of the Sides and the Angles of a Spherical Triangle V Solution of Right-angled Triangles VI Solution of Oblique-Angled Triangles VII Circumscribed and Inscribed Circles VIII Area of a Spherical Triangle. Spherical Excess IX On certain approximate Formula X Geodetical Operations XI On small variations in the parts of a Spherical Triangle XII On the connection of Formula in Plane and Spherical Trigonometry XIII Polyhedrons XIV Arcs drawn to fixed points on the Surface of a Sphere XV Miscellaneous Propositions XVI Numerical Solution of Spherical Triangles