It has been the author's aim to treat the subject according to the latest and most approved methods. The book is designed for the use of colleges, technical schools, normal schools, secondary schools, and for those who take up the subject without the aid of a teacher. Special attention has been paid to the requirements of the College Entrance Board. The book contains more material than is required for some first courses in Trigonometry, but the matter has been so arranged that the teacher can make such omissions as will suit his particular needs. The trigonometric functions are defined as ratios; first for acute angles in right triangles, and then these definitions are extended to angles in general by means of coordinates. The student is first taught to use the natural functions of acute angles in the solution of simple problems involving right triangles. Attention is called to the methods shown in "23-29 for the reduction" of functions of angles outside of the first quadrant. In general, the first examples given under each topic are worked out, making use of the natural functions. A large number of carefully graded exercises are given, and the processes involved are summarized into working rules wherever practicable. Illustrative examples are worked out in detail under each topic. Logarithms are introduced as a separate topic, and attention is called to the fact that they serve to minimize the labor of computation. Granville's Four-Flace Tables of Logarithms is used. While no radical changes in the usual arrangement of logarithmic tables have been made, several improvements have been effected which greatly facilitate logarithmic computations. Particularly important is the fact that the degree of accuracy which may be expected in a result found by the aid of these tables is clearly indicated. Under each case in the solution of triangles are given two complete sets of examples, " one in which the angles are epxressed in degrees and minutes, and another in which the angles are expressed in degrees and the decimal part of a degree. This arrangement, which is characteristic of this book, should be of great advantage to those secondary schools in which college preparation involving both systems is necessary. To facilitate the drawing of figures and the graphical checking of results a combined ruler and protractor of celluloid is furnished with each copy of the book, and will be found on the inside of the back cover. In Spherical Trigonometry some simplifications have been idtroduced in the application of Napier's rule of circular parts to the solution of right spherical triangles. The treatment of oblique spherical triangles is unique. By making use of the Principle of Duality nearly one half of the work usually required in deriving the standard formulas is done away with, and the usual six cases in the solution of oblique spherical triangles have been reduced to three. An attempt has been made to treat the most important applications of Spherical Trigonometry to Geodesy, Astronomy, and Navigation with more clearness and simplicity than has been. the rule in elementary treatises.