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AN EDITION OF

Euclid's Elements

REVISED IN ACCORDANCE WITH THE REPORTS OF
THE CAMBRIDGE BOARD OF MATHEMATICAL
STUDIES, AND THE OXFORD BOARD OF THE
FACULTY OF NATURAL SCIENCE

BY

EDWARD It LANGLEY, M.A.

SENIOR MATHEMATICAL MASTER, BEDFORD MODERN SCHOOL

AND

W. SEYS PHILLIPS, M.A.

SENIOR MATHEMATICAL MASTER, BEDFORD GRAMMAR SCHOOL

RIVINGTONS

WATERLOO PLACE, LONDON

MDCCCXC

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I PREFACE

THIS work is intended to be strictly a School Edition of Euclid's Elements. While retaining his sequence of propositions, and basing their proofs entirely on his xxioms, the Editors have not scrupled to replace ome of his demonstrations by easier ones, and to discard much superfluous matter, irritating alike to student, teacher, and examiner, from those retained.

Symbols have been introduced at an early stage, but their use has been avoided in the first eight propositions, in order that beginners might not be confronted with two difficulties at once.

An attempt has been made in the Notes and Exercises to familiarise the student with such terms and ideas as he will be likely to meet with in his higher reading and in treatises on Elementary Geometry by other writers, and to indicate by difference of type important theorems and problems which should be well known to him although not given among Euclid's propositions. At the same time the work is not intended to be a substitute for such works as Casey's well-known Sequel to Euclid, but to serve as an introduction to them.

The Editors are of opinion that students of Geometry cannot attempt the solution of Geometrical Exercises too soon, and they have attached simple riders to most of the propositions which ought to be within the reach of all those who have mastered the book-work intelligently.

Much use has been made of the excellent Syllabus of Plane Geometry issued by the Association for the Improvement of Geometrical Teaching, and readers of Professor Henrici's remarkable little work on Congruent Figures will probably detect its influence throughout.

The short treatise 'On Quadrilaterals' will, it is hoped, be found interesting to those students who have mastered the previous exercises, and useful to teachers in supplying a large number of easy and instructive exercises in a short compass.

The final collection of Miscellaneous Exercises is purposely taken from widely different sources; some are—as far as it is safe to make such an assertion of any proposition in Elementary Geometry—original others have been taken from or suggested by various examination papers, and others are well known theo

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