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 UNIVFRS/TV NEW YORK .:. CINCINNATI .:. CHICAGO ? 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. Proofs of some of the easier theorems and constructions are left as exercises for the student, or are given in an incomplete form; but in every case in which the proof is not complete, the incompleteness is specifically stated. The authors believe that the proofs of most of the propositions should be complete, first, in order to serve as models for the handling of exercises; second, to prevent the serious error of making the student feel contented with loose and slipshod reasoning which defeats the main purpose of instruction in geometry; and third, as an excellent means of reviewing the previous theorems on which they depend. The indirect method of proof is consistently applied. The usual method of proving such propositions as Arts. 189 and 459957 415, e.g., is confusing to the student. The method used here is convincing and clear. The exercises are carefully selected. In choosing the exercises, each of the following groups has been given due impor tance: (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. (c) The traditional exercises given in a more or less abstract setting. Exercises The arrangement of the exercises is pedagogical. of a rather easy nature are placed immediately after the theorems of which they are applications, instead of being grouped together without regard to the principles involved in them. In many instances the exercises are so arranged as to constitute a careful line of development, leading gradually from a very simple construction or exercise to others that are more difficult. For the benefit of the brighter pupils, however, and for review classes, large lists of more or less difficult exercises are grouped at the end of each book. The definitions of plane closed figures are unique. The student's natural conception of a plane closed figure 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 funda mental principles being definitely assumed (Art. 336, proof in Appendix, Art. 595). This procedure is novel and is believed to be the only logical, and at the same time teachable, method of dealing with incommensurables. Teachers who find these subjects too difficult, however, can easily omit them without interruption of sequence. 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 triangle. The correlation with arithmetic in this connection is valuable. The number concepts already found so useful and practical in the modern treatment of ratio and proportion have been developed in connection with areas, as well as in other portions of the book Proofs of the superposition theorems and the concurrent line theorems will be found exceptionally accurate and complete. The many historical notes are such as will add life and interest to the work. The carefully arranged summaries throughout the book, and the collection of formulas of plane geometry at the end of the book, 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 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 City; to Mr. Edward B. Parsons of the Boys' High School, Brooklyn; and to Professor McMahon of Cornell University. |