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 student's knowledge of mathematical geography has been appealed to in making clear the terms and the relations of figures connected with the sphere. The treatment of the Solid Geometry is logical. The same logical rigor that characterizes the demonstrations in the Plane Geometry is used consistently throughout the Solid. If a postulate is needed to make a proof complete, it is clearly stated, as in § 615. In the mensuration of the prism and the pyramid, the same general plan has been followed as that used in Book IV; in the mensuration of the cylinder, the cone, and the sphere, the method pursued is similar to that used in the mensuration of the circle. More proofs and parts of proofs are left to the student in the Solid, than in the Plane Geometry; but in every case in which the proof is not complete, the incompleteness is specifically stated. The treatment of the polyhedral angle (p. 336), of the prism (p. 345), and of the pyramid (p. 350), is similar to that of the cylinder and the cone. This is in accordance with the recommendations of the leading Mathematical Associations throughout the country. The complete collection of formulas of Solid Geometry at the end of the book, it is hoped, will be found helpful to teacher and student alike. The grateful acknowledgment of the authors is due to many friends for helpful suggestions; especially to Miss Grace A. Bruce, of the Wadleigh High School, New York; to Mr. Edward B. Parsons, of the Boys' High School, Brooklyn; and to Professor McMahon, of Cornell University. |