SOLID GEOMETRY BY C. A. HART INSTRUCTOR OF MATHEMATICS, WADLEIGH HIGH SCHOOL, NEW YORK CITY AND DANIEL D. FELDMAN HEAD OF Department OF MATHEMATICS, ERASMUS HALL HIGH SCHOOL, BROOKLYN WITH THE EDITORIAL COÖPERATION OF J. H. TANNER AND VIRGIL SNYDER PROFESSORS OF MATHEMATICS IN CORNELL UNIVERSITY DEPARTMENT OF EDUCATION NEW YORK .:. CINCINNATI : CHICAGO 623176 COPYRIGHT, 1912, BY AMERICAN BOOK COMPANY. ENTERED AT STATIONERS' HALL, LONDON. H.-F. SOLID GEOMETRY. W. P. I PREFACE In addition to the features of the Plane Geometry, which are emphasized in the Solid as well, the chief characteristic of this book is the establishment, at every point, of the vital relation between the Solid and the Plane Geometry. Many theorems in Solid Geometry have been proved, and many problems have been solved, by reducing them to a plane, and simply applying the corresponding principle of Plane Geometry. Again, many proofs of Plane Geometry have been made to serve as proofs of corresponding theorems in Solid Geometry by merely making the proper changes in terms used. (See §§ 703, 786, 794, 813, 853, 924, 951, 955, 961, etc.) Other special features of the book may be summarized as follows: The student is given every possible aid in forming his early space concepts. In the early work in Solid Geometry, the average student experiences difficulty in fully comprehending space relations; that is, in seeing geometric figures in space. The student is aided in overcoming this difficulty by the introduction of many easy and practical questions and exercises, as well as by being encouraged to make his figures. (See § 605.) As a further aid in this direction, reproductions of models made by students themselves are shown in a group (p. 302), and at various points throughout Book VI. The student's fund of knowledge is constantly drawn upon. In the many questions, suggestions, and exercises, his knowledge of the things about him has been constantly appealed to. Especially is this true of the work on the sphere, where the |