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In the present revision of Dr. 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 Geodetical Operations and on Polyhedrons, are almost untouched, and in the chapter on Arcs Drawn to Fixed Points only one paragraph has been altered. 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 remaining chapters include extensive alterations and additions.

I have followed the example of the late Dr. in introducing chapters on Spherical Geometry, and I am indebted to his Spherical Trigonometry, and to Elemente der Muathematik, for references to the important writings on the subject. Passing over, however, a number of geometrical methods of considerable interest but 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 Geometry have been known for