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PREFACE

The mathematical series of which this book is the first to be published is founded on the works of the late Professor Elias Loomis. In the present instance, however, the work can scarcely be called a revision. We have utilized many of the terse and accurate statements and definitions of the Loomis Geometry, and have aimed to maintain the high standard of that work for rigorous demonstrations, but, aside from these similarities, the arrangement and method of presentation are essentially new.

While the book speaks for itself, we would call attention to some of its most important features.

The Introduction presents in the shortest possible compass the general outlines of the science to be studied, and leads at once to the actual study itself.

The definitions are distributed through the book as they are needed, instead of being grouped in long lists many pages in advance of the propositions to which they apply. An alphabetical index is added for easy reference.

. The constructions in the Plane Geometry are also distributed, so that the student is taught how to make a figure at the same time that he is required to use it in demonstration.

In the Geometry of Space, the figures consist of half-tone engravings from the photographs of actual models recently constructed for use in the class-rooms of Yale University. By the side of these models are skeleton diagrams for the student to copy.

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Extensive use has been made of natural and symmetrical methods of demonstration. Such methods are used for deducing the formula for the sum of the angles of a triangle, for the sum of the exterior and interior angles of a polygon, for parallel lines, for the theorems on regular polygons, and for similar figures in both Plane and Solid Geometry.

The theory of limits is treated with rigor, and not passed over as self-evident.

Attention is also called to the theorems of proportion, the use of corollaries as exercises to supply the need of “inventional geometry,” and the Introduction to Modern Geometry.

We would here express our grateful acknowledgments to all who have aided in the preparation of this book; to Miss Elizabeth H. Richards, whose successful experience in fitting students for college in Plane Geometry has rendered her criticisms and suggestions most valuable, to Mr. E. H. Lockwood, of the Sheffield Scientific School, whose skill as a draughtsman has been of essential service in the preparation of the figures, and to our colleagues, Messrs. W. M. Strong and Joseph Bowden, Jr. Mr. Strong has selected, for the most part, the exercises at the end of the book, and has written a large part of the Introduction to Modern Geometry.

Mr. Bowden, whose large experience in teaching successive Freshman classes has given him an unusual equipment, has written a considerable portion of the Solid Geometry, and has examined critically the references and proof-sheets of the book.

ANDREW W. PHILLIPS,

IRVING FISHER. YALE UNIVERSITY, June, 1896.

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154

158

160

PROBLEMS FOR COMPUTATION

BOOK IV

AREAS OF POLYGONS

170

193

PROBLEMS OF DEMONSTRATION

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