University of wichigan PLANE AND SOLID GEOMETRY BY C.A. HART INSTRUCTOR OF MATHEMATICS, WADLEIGH HIGH SCHOOL, CITY OF NEW YORK AND DANIEL D. FELDMAN PRINCIPAL OF CURTIS HIGH SCHOOL, NEW BRIGHTON, CITY OF NEW YORK WITH THE EDITORIAL COÖPERATION OF J. H. TANNER AND VIRGIL SNYDER PROFESSORS OF MATHEMATICS IN CORNELL UNIVERSITY NEW YORK: CINCINNATI CHICAGO From Math. Dept. COPYRIGHT, 1911, 1912, BY AMERICAN BOOK COMPANY. ENTERED AT STATIONERS' HALL, LONDON. H.-F. PLANE AND SOLID GEOMETRY. W. P. 7 PREFACE THIS book is the outgrowth of an experience of many years in the teaching of mathematics in secondary schools. The text has been used by many different teachers, in classes of all stages of development, and under varying conditions of secondary school teaching. The proofs have had the benefit of the criticisms of hundreds of experienced teachers of mathematics throughout the country. The book in its present form is therefore the combined product of experience, classroom test, and severe criticism. The following are some of the leading features of the book: The student is rapidly initiated into the subject. Definitions are given only as needed. The selection and arrangement of theorems is such as to meet the general demand of teachers, as expressed through the Mathematical Associations of the country. Most of the proofs have been given in full. In the Plane Geometry, proofs of some of the easier theorems and constructions are left as exercises for the student, or are given in an incomplete form. In the Solid Geometry, more proofs and parts of proofs are thus left to the student; but in every case in which the proof is not complete, the incompleteness is specifically stated. The indirect method of proof is consistently applied. The usual method of proving such propositions, for example, as Arts. 189 and 415, is confusing to the student. The method used here is convincing and clear. The exercises are carefully selected. In choosing exercises, each of the following groups has been given due importance: (a) Concrete exercises, including numerical problems and problems of construction. (b) So-called practical problems, such as indirect measurements of heights and distances by means of equal and similar triangles, drawing to scale as an application of similar figures, problems from physics, from design, etc. 426534 (c) Traditional exercises of a more or less abstract nature. The arrangement of the exercises is pedagogical. Comparatively easy exercises are placed immediately after the theorems of which they are applications, instead of being grouped together without regard to the principles involved in them. For the benefit of the brighter pupils, however, and for review classes, long lists of more or less difficult exercises are grouped at the end of each book. The student's The definitions of closed figures are unique. natural conception of a plane closed figure, for example, is not the boundary line only, nor the plane only, but the whole figure composed of the boundary line and the plane bounded. All definitions of closed figures involve this idea, which is entirely consistent with the higher mathematics. The numerical treatment of magnitudes is explicit, the fundamental principles being definitely assumed (Art. 336, proof in Appendix, Art. 595). This novel procedure furnishes a logical, as well as a teachable, method of dealing with incommensurables. The area of a rectangle is introduced by actually measuring it, thereby obtaining its measure-number. This method permits. the same order of theorems and corollaries as is used in the parallelogram and the triangle. The correlation with arithmetic in this connection is valuable. A similar method is employed for introducing the volume of a parallelopiped. Proofs of the superposition theorems and the concurrent line theorems will be found exceptionally accurate and complete. The many historical notes will add life and interest to the work. The carefully arranged summaries throughout the book, the collection of formulas of Plane Geometry, and the collection of formulas of Solid Geometry, it is hoped, will be found helpful to teacher and student alike. Argument and reasons are arranged in parallel form. This arrangement gives a definite model for proving exercises, renders the careless omission of the reasons in a demonstration impossible, leads to accurate thinking, and greatly lightens the labor of reading papers. Every construction figure contains all necessary construction lines. This method keeps constantly before the student a model for his construction work, and distinguishes between a figure for a construction and a figure for a theorem. The mechanical arrangement is such as to give the student every possible aid in comprehending the subject matter. The following are some of the special features of the Solid Geometry: The vital relation of the Solid Geometry to the Plane Geometry is emphasized at every point. (See Arts. 703, 786, 794, 813, 853, 924, 951, 955, 961, etc.) 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 knowledge of the things about him is constantly drawn upon. 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 same logical rigor that characterizes the demonstrations in the Plane Geometry is used throughout the Solid. 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 of the cone. This is in accordance with the recommendations of the leading Mathematical Associations throughout the country. 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. |