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BY

HERBERT E. HAWKES, Ph.D.

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PROFESSOR OF MATHEMATICS IN COLUMBIA UNIVERSITY

WILLIAM A. LUBY, M.A.

HEAD OF THE DEPARTMENT OF MATHEMATICS IN THE

JUNIOR COLLEGE OF KANSAS CITY

AND

FRANK C. TOUTON, Ph.D.

LECTURER IN EDUCATION, DEPARTMENT OF EDUCATION

UNIVERSITY OF CALIFORNIA

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GINN AND COMPANY
BOSTON NEW YORK CHICAGO LONDON
ATLANTA DALLAS COLUMBUS SAN FRANCISCO

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.

H 34

COPYRIGHT, 1922, BY
HERBERT E, HAWKES, WILLIAM A. LUBY

AND FRANK C. TOUTON

ALL RIGHTS RESERVED

322.

JORI

The Athenæum Press
GINN AND COMPANY · PRO.
PRIETORS. BOSTON • U.S.A.

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PREFACE

During recent years the study of solid geometry has occupied a somewhat less commanding position in the mathematical curriculum than formerly. Important and essential as its subject matter is admitted to be, it has been little more than an appendage to plane geometry, both in the methods of its presentation and in its scientific results.

The authors of this text feel that the subject is much more vital than such a tendency would indicate. Not only are the bare truths gained from a study of solid geometry essential to the student of science, but through its medium a multitude of mathematical ideas can be presented and elucidated in a natural and convincing manner. In fact, no subject of elementary mathematics can be compared to solid geometry as a climax and capstone of mathematical study for the student who pursues the subject no farther. It not only utilizes and applies much that he has learned in other courses, but serves as a point of vantage from which may be gained many glimpses of scientific fields which he is not to enter.

In this text the authors have presented the subject in such form that a minimal course as prescribed by the colleges and the various examining boards may be covered. At the same time it affords at every turn a richness of suggestion and development for those who have the time and the inclination to do more than that minimum.

One of the important opportunities afforded by the study of solid geometry is that of using and developing the scientific

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imagination. This text, through its hundreds of queries, aims to encourage the student to regard the subject not merely as a logical sequence of theorems but as a subject inviting reflection and the play of speculation. These queries should be used in class as a basis for discussion and will be found to render the more formal work not only more interesting but more intelligible. If time does not permit any attention to the queries, they may be omitted from the class assignments without disturbing the continuity of the subject. Their use, however, is strongly urged by the authors.

Exercises illustrating or dependent upon the various theorems are scattered throughout the text and afford as much drill of this kind as many teachers can profitably use. The collections at the end of each book may be regarded as supplementary. Great care has been exercised to provide a collection of originals that is fresh, interesting, not too difficult, but illustrating all parts of the subject.

The assumption of Cavalieri's Theorem as a basis for the theorems on measurement is the result of many years of classroom experience. The simplicity and power of this procedure should commend it both to teachers and to students.

The geometry of the sphere and its relation to plane geometry is also elaborated with care and in such a manner as to give the student an insight into the meaning of geometrical science.

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