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HEAD OF THE DEPARTMENT OF MATHEMATICS, STATE NORMAL SCHOOL
ARITHMETIC, ALGEBRAS AND GEOMETRIES
JAMES F. MILLIS, A.M.
HEAD OF THE DEPARTMENT OF MATHEMATICS, FRANCIS W. PARKER SCHOOL
ου πόλλ' άλλα πολύ
BENJ. H. SANBORN & CO.
STONE-MILLIS MATHEMATICS SERIES
ELEMENTARY ARITHMETICS :
PLANE AND SOLID GEOMETRY
BENJ. H. SANBOHN: & Co., PUBLISHERS
BENJ, H. SANBORN & CO.
ang Lonis C. Karpmoki
THE STONE-MILLIS GEOMETRY, published in 1910, was a pioneer in its field, being the first of the series of American textbooks on geometry which of recent years have attempted in various ways to modernize the teaching of the subject. It marked, in fact, a wide departure from the traditional Greek geometry, after which textbooks for secondary schools had for generations been patterned. From the start, the text met with remarkable success. The educational ideals which it embodied are now national ideals, and are summarized in the REPORT OF THE NATIONAL COMMITTEE OF FIFTEEN on the teaching of geometry.
The present volume, by the same authors, is a thorough revision of that text. It has been prepared in the attempt to produce a text which shall preserve the distinctive features of the older book, but which, if possible, shall be more simple, practical, and teachable.
The main features of the older book, which have been preserved in the present volume, are :
(1) A concrete approach to the subject, developing a body of experience and imagery as a basis of formal geometry, the latter not being introduced until need for it is felt.
(2) A simplifying of the subject by the elimination of unnecessary material, and by the assumption of many of the simpler principles of geometry of which the student does not feel the need of proof.
(3) An attempt to motivate the teaching of geometry in the secondary school, and to make it function in the life of the student, by utilizing as exercises a large number of applied problems of geometry such as are actually encountered in real life.
(4) Correlation of geometry with algebra, and correlation of geometry with trigonometry by the introduction of simple work with the trigonometric ratios in connection with similar polygons.
(5) Elimination from elementary geometry of proofs of incommensurable cases of theorems by use of the theory of limits.
(6) A grading of geometry, through a rearrangement of the subject matter, which is made possible by abandoning the traditional Greek division of geometry into books and re-grouping the material in chapters.
(7) Use of the suggestive method in the treatment of theorems, the proofs of many of the simpler theorems being left, with suggestions, to the student.
In the revision, the sequence of the subject matter has been changed to some extent, and the material of the twelve chapters combined into eleven. The text has been simplified by reducing somewhat the number of formal theorems, corollaries, and constructions.
Complete proofs of more theorems are given than in the older edition, particularly at the beginning of the course, thus simplifying the subject and at the same time providing the student with more perfect models for his own proofs. Where proofs are provided, they are given in full. In the suggestive treatment of theorems, the character of the suggestions has been improved, the analysis which is extensively used affording the best training in original thinking on the part of the student.
The exercises have been thoroughly revised. Those which were too difficult or which demanded technical knowledge on the part of the student have been eliminated. Many new exercises, both theoretical and practical, have been introduced. The exercises have been placed at more frequent intervals, immediately following the theorems, etc., to which they are related.
New cuts have been made for all of the figures. Also much care has been given to making the page forms as attractive as possible and the entire volume in every way convenient to use.
Grateful acknowledgment of the authors is due to all those who by timely suggestions have aided in the preparation of this volume; especially to Professor H. E. Cobb and Professor A. W. Cavanaugh of Lewis Institute, Chicago; to Miss Alice M. Lord of the High School, Portland, Maine; and to Professor Guido H. Stempel of Indiana University. Special acknowledgment is due also to Mr. Charles McCauley, of Chicago, who has made the excellent illustrations.