IN these fifty pages the authors have brought together most of the chief geometrical properties of the sphere, intending the book to be used as a companion to their larger one, on mensuration of the simpler figures, by students preparing for the intermediate examinations in Arts and in Science of the University of London. The three chapters into which the subject is divided lead the reader from the elementary definitions relating to great and small circles, poles, lunes, &c., through the numerous geometrical properties of spherical triangles and their antipodal triangles, polar triangles, supplemental triangles, and finally to the determination of the area of lunes, spherical triangles, spherical polygons, and the spherical excess. The definitions and theorems are expressed quite clearly throughout, while the figures leave nothing to be desired. As an introduction to works on spherical trigonometry, students will find this book a most helpful guide. Two minor slips in construction will be found: one on page 6, line 6, where for CT read TC; and the other on page 18, line 9, where for oa and ob read ao and bo.