The First Six Books of the Elements of Euclid

Johneuclid Casey, John Casey, Euclid

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• Publié par: shauh
• Publié le: 08 décembre 2010
• Nombre de lectures: 38
• Langue: English
• Poids de l'ouvrage: 1 Mo

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Project Gutenberg’s First Six Books of the Elements of Euclid, by John Casey This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net Title: The First Six Books of the Elements of Euclid Subtitle: And Propositions I.-XXI. of Book XI., and an Appendix on the Cylinder, Sphere, Cone, etc., Author: John Casey Author: Euclid Release Date: April 14, 2007 [EBook #21076] Language: English Character set encoding: TeX *** START OF THE PROJECT GUTENBERG EBOOK ELEMENTS OF EUCLID *** Produced by Joshua Hutchinson, Keith Edkins and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images from the Cornell University Library: Historical Mathematics Monographs collection.) Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell’s replacement volume on paper that meets the ANSI Standard Z39.48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. Digital file copyright by Cornell University Library 1991. Transcriber’s Note: The Index has been regenerated to fit the pagination of this edition. Despite the author’s stated hope that “few misprints have escaped detection” there were several, which have here been corrected and noted at the end of the text. CORNELL UNIVERSITY LIBRARY THE EVAN WILHELM EVANS MATHEMATICAL SEMINARY LIBRARY THE GIFT OF LUCIEN AUGUSTUS WAIT ....................................................................... THE ELEMENTS OF EUCLID. NOW READY Price 3s. A TREATISE ON ELEMENTARY TRIGONOMETRY, With Numerous Examples and Questions for Examination. Third Edition, Revised and Enlarged, Price 3s. 6d., Cloth. A SEQUEL TO THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID, Containing an Easy Introduction to Modern Geometry: With numerous Examples. Third Edition, Price 4s. 6d.; or in two parts, each 2s. 6d. THE ELEMENTS OF EUCLID, BOOKS I.—VI., AND PROPOSITIONS I.—XXI., OF BOOK XI.; Together with an Appendix on the Cylinder, Sphere, Cone, &c.: with Copious Annotations & numerous Exercises. Price 6s. A KEY TO THE EXERCISES IN THE FIRST SIX BOOKS OF CASEY’S ELEMENTS OF EUCLID. Price 7s. 6d. A TREATISE ON THE ANALYTICAL GEOMETRY OF THE POINT, LINE, CIRCLE, & CONIC SECTIONS, Containing an Account of its most recent Extensions, With numerous Examples. DUBLIN: HODGES, FIGGIS, & CO. LONDON: LONGMANS & CO. THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID, AND PROPOSITIONS I.-XXI. OF BOOK XI., AND AN APPENDIX ON THE CYLINDER, SPHERE, CONE, ETC., WITH COPIOUS ANNOTATIONS AND NUMEROUS EXERCISES. BY J O H N C A S E Y, LL. D., F. R. S., FELLOW OF THE ROYAL UNIVERSITY OF IRELAND; MEMBER OF COUNCIL, ROYAL IRISH ACADEMY; MEMBER OF THE MATHEMATICAL SOCIETIES OF LONDON AND FRANCE; AND PROFESSOR OF THE HIGHER MATHEMATICS AND OF MATHEMATICAL PHYSICS IN THE CATHOLIC UNIVERSITY OF IRELAND. THIRD EDITION, REVISED AND ENLARGED. DUBLIN: HODGES, FIGGIS, & CO., GRAFTON-ST. LONDON: LONGMANS, GREEN, & CO. 1885. DUBLIN PRINTED AT THE UNIVERSITY PRESS, BY PONSONBY AND WELDRICK PREFACE. This edition of the Elements of Euclid, undertaken at the request of the principals of some of the leading Colleges and Schools of Ireland, is intended to supply a want much felt by teachers at the present day—the production of a work which, while giving the unrivalled original in all its integrity, would also contain the modern conceptions and developments of the portion of Geometry over which the Elements extend. A cursory examination of the work will show that the Editor has gone much further in this latter direction than any of his predecessors, for it will be found to contain, not only more actual matter than is given in any of theirs with which he is acquainted, but also much of a special character, which is not given, so far as he is aware, in any former work on the subject. The great extension of geometrical methods in recent times has made such a work a necessity for the student, to enable him not only to read with advantage, but even to understand those mathematical writings of modern times which require an accurate knowledge of Elementary Geometry, and to which it is in reality the best introduction. In compiling his work the Editor has received invaluable assistance from the late Rev. Professor Townsend, s.f.t.c.d. The book was rewritten and considerably altered in accordance with his suggestions, and to that distinguished Geometer it is largely indebted for whatever merit it possesses. The Questions for Examination in the early part of the First Book are intended as specimens, which the teacher ought to follow through the entire work. Every person who has had experience in tuition knows well the importance of such examinations in teaching Elementary Geometry. The Exercises, of which there are over eight hundred, have been all selected with great care. Those in the body of each Book are intended as applications of Euclid’s Propositions. They are for the most part of an elementary character, and may be regarded as common property, nearly every one of them having appeared already in previous collections. The Exercises at the end of each Book are more advanced; several are due to the late Professor Townsend, some are original, and a large number have been taken from two important French works—Catalan’s Th´or`mes et Probl`mes de G´om´trie El´mentaire, and e e e e e e ´ the Trait´ de G´om´trie, by Rouche and De Comberousse. e e e The second edition has been thoroughly revised and greatly enlarged. The new matter includes several alternative proofs, important examination questions on each of the books, an explanation of the ratio of incommensurable quantities, the first twenty-one propositions of Book XI., and an Appendix on the properties of the Prism, Pyramids, Cylinder, Sphere, and Cone. The present Edition has been very carefully read throughout, and it is hoped that few misprints have escaped detection. The Editor is glad to find from the rapid sale of former editions (each 3000 copies) of his Book, and its general adoption in schools, that it is likely to i accomplish the double object with which it was written, viz. to supply students with a Manual that will impart a thorough knowledge of the immortal work of the great Greek Geometer, and introduce them, at the same time, to some of the most important conceptions and developments of the Geometry of the present day. JOHN CASEY. 86, South Circular-road, Dublin. November, 1885. ii Contents Introduction, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 BOOK I. Theory of Angles, Triangles, Parallel Lines, and parallelograms., Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . Propositions i.–xlviii., . . . . . . . . . . . . . . . . . . . Questions for Examination, . . . . . . . . . . . . . . . . . Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 8 45 46 BOOK II. Theory of Rectangles, . . . . . . Definitions, . . . . . . . . . Propositions i.–xiv., . . . . Questions for Examination, Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 49 50 65 66 BOOK III. Theory of the Circle, . . . . . . Definitions, . . . . . . . . . Propositions i.–xxxvii., . . Questions for Examination, Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 68 69 97 98 BOOK IV. Inscription and Circumscription of Triangles and of Regular Polygons in about Circles, . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propositions i.–xvi., . . . . . . . . . . . . . . . . . . . . . . . . . Questions for Examination, . . . . . . . . . . . . . . . . . . . . . Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and . . . . . . . . . . . . . . . 101 101 101 112 112 BOOK V. Theory of Proportion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 iii Definitions, . . . . . . . . . Introduction, . . . . . . . . Propositions i.–xxv., . . . Questions for Examination, Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 116 121 133 134 BOOK VI. Application of the Theory of Proportion, Definitions, . . . . . . . . . . . . . Propositions i.–xxxiii., . . . . . . Questions for Examination, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 135 135 163 BOOK XI. Theory of Planes, Coplanar Definitions, . . . . . . Propositions i.–xxi., . Exercises, . . . . . . . Lines, and Solid Angles, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .