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. The arrangement of the exercises is pedagogical. Exercises 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 fundamental 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. BOOK V. REGULAR POLYGONS. MEASUREMENT OF THE Measurement of the Circumference and of the Circle To Every Straight Line there belongs a Measure-number. viii right. straight. exterior. interior. alternate. definition. axiom. postulate. hypothesis. proposition. prop. th. cor. corollary. cons. construction. ex. fig. iden. exercise: figure. identity. comp. complementary. sup. supplementary. adj. adjacent. homol. homologous. Q.E.D. Quod erat demonstrandum, which was to be proved. Q.E. F. The signs +, —, ×, ÷ have the same meanings as in algebra. |