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ELEMENTARY GEOMETRY

PLANE AND SOLID

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PLANE AND SOLID

FOR USE IN HIGH SCHOOLS AND ACADEMIES

BY

THOMAS F. HOLGATE

PROFESSOR OF APPLIED MATHEMATICS IN NORTHWESTERN

UNIVERSITY

New York
THE MACMILLAN COMPANY

LONDON: MACMILLAN & CO., LTD.

1901

All rights reserved

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PREFACE

Elementary Geometry deals only with forms whose determining parts are points, straight lines, and circles. Its method is that employed by the ancient writers. The amount of such study to be included in the high school or academy course has been fixed by tradition, as, in plane geometry, the equivalent of Euclid's first six books together with some additional material on mensuration; and, in solid geometry, the equivalent of Euclid's eleventh and twelfth books, to which is also added the mensuration of solids. Modern concepts and modern methods have given to this old material a correlation and a symmetry which it did not at one time present, and have opened fields of investigation which were entirely beyond the range of the early geometers. From the modern point of view, many isolated and apparently independent theorems have proved to be but special cases of broader and more general ones, or to be related to each other in an easily defined way. The so-called modern geometry possesses great beauty and strength, but how much of it can be wisely woven into a first course is a matter about which there is no general concensus of opinion. Some recent writers have deemed it wise to introduce general principles very early, while others have held rigorously to the old methods and old materials. My own belief is that the pupil must become quite familiar with the incidental and particular facts of geometry before he is capable of much generalization, and while I have written with the modern notions distinctly in mind, I have preferred not to depart far from the well-beaten path.

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