and Astronomy in Columbia College. Tor his aid, in Riving to the work its present permanent form, 1 tender him my grateful acknowledgments. The edition of Legendre, referred to 1n the last paragraph, will not be altered in form or substance; and yet, Geometry must be made a more practical science. To attain this object, without deranging a system so long used, and so generally approved, an Appendix has been prepared and added to Legendre, embracing many Problems of Geometrical construction, and many applications of Algebra to Geometry. 1t would be unjust to those giving instruction, to add to their daily labors, the additional one, of finding appropriate solutions to so many difficult problems: hence, a Key has been made for the use of Teachers, in which the best methods of construction and solution are fully given. CHARLES DA V1ES. F1shk1ll-on-hudson, June, 1875. NOTE.—The edition of Leg-endre referred to in the foregoing preface was prepared by the late Professor Davies the year before his lamented death. The present edition is the result of a careful re-examination of the work, into which have been incorporated such emendations, in the way of greater clearness of expression or of proof, as could be made without altering it in form or substance. Practical exorcises have been placed at the end of the several books, and compriso additional theorems, problems, and numerical exercises upon the principles of the Book or Books preceding. They will, it 1s hoped, be found of service in accustoming students, early in and throughout their course, to make for themselves practical application of geometric principles, and constitute, in addition, a large body of review and test questions for the convenience of teachers. The Trigonometry has been carefully revised throughout, to simplify the discussions and to make the treatment conform in every particular to the latest and best methods. 1t is believed that in clearness and precision of de1inition, in general simplicity and rigor of demonstration, in orderly and logical development of the subject, and in compactness of form, Davies' Legendre is superior to any work of its grade for the general training of the logical powers of pupils, and for their instruction 1n the great body of elementary geometric truth. J. H. VAN AMR1NGE, Editor 0/ Davies' Course 0/ Mathematis4. Columb1a College, N. V., June, 1885. CONTENTS PACK Ratios and Proport1ons, 62 The CIrcle, and the Measurement of Anglos, 61 Problems relatIng to the First and ThIrd Books, 84 Proportions of Figures — Measurement of Areas, 97 Problems relating to the Fourth Book 183 Regular Polygons — Measurement of the C1rcle . 142 PACE Limiting Value of Circular Functions, 67 Relations of Circular Functions, 59—61 Functions of Negative Arcs 62—65 Particular Values of Certain Functions, 66 Formulas of Relation between Functions and Arcs 67—70 Spherical Trigonometry Defined 76 Formulas for Right-angled Triangles, 77—80 Solution of Right-angled Spherical Triangles 84—88 |