THE ELEMENTS OF EUCLID. - Project Gutenberg

1 Project Gutenberg s First Six Books of the ELEMENTS of Euclid,by John CaseyThis eBook is for the use of anyone anywhere in the United States andmost other parts of the world at no cost and with almost no restrictionswhatsoever. You may copy it, give it away or re-use it under the termsof the Project Gutenberg License included with this eBook or online If you are not located in the United States, youwill have to check the laws of the country where you are located beforeusing this : The First Six Books of the ELEMENTS of EuclidSubtitle: And Propositions of Book XI., and anAppendix on the Cylinder, Sphere, Cone, etc.,Author: John CaseyAuthor: EuclidRelease Date: April 14, 2007 [eBook #21076][Most recently updated: December 29, 2021]Language: EnglishCharacter set encoding: UTF-8 Produced by Joshua Hutchinson, Keith Edkins and theOnline Distributed Proofreading Team at (This file was produced from images from the Cornell UniversityLibrary: Historical Mathematics Monographs collection.)

2 Revised by Richard Tonsing.** START OF THE Project Gutenberg EBOOK ELEMENTS OF EUCLID **Production NoteCornell University Library produced this volume to replace theirreparably deteriorated original. It was scanned using Xeroxsoftware and equipment at 600 dots per inch resolution and com-pressed prior to storage using CCITT Group 4 compression. Thedigital data were used to create Cornell s replacement volumeon paper that meets the ANSI Standard The produc-tion of this volume was supported in part by the Commission onPreservation and Access and the Xerox Corporation. Digitalfile copyright by Cornell University Library s Note:The Index has been regenerated to fit the pagination of this edition.

3 Despitethe author s stated hope that few misprints have escaped detection there were several, whichhave here been corrected and noted at the end of the UNIVERSITY LIBRARYTHE EVAN WILHELM EVANSMATHEMATICAL SEMINARY LIBRARYTHE GIFT OFLUCIEN AUGUSTUS WAIT..THE ELEMENTS OF READYP rice TREATISE ON ELEMENTARY TRIGONOMETRY,With Numerous Examples and Questions for Edition, Revised and Enlarged, Price 3s. 6d., SEQUEL TO THE FIRST SIX BOOKS OF THEELEMENTS OF EUCLID,Containing an Easy Introduction to Modern Geometry:With numerous Edition, Price 4s. 6d.; or in two parts, each 2s. ELEMENTS OF EUCLID, BOOKS I. VI., ANDPROPOSITIONS I.

4 XXI., OF BOOK XI.;Together with an Appendix on the Cylinder, Sphere,Cone, &c.: withCopious Annotations & numerous KEY TO THE EXERCISES IN THE FIRST SIXBOOKS OF CASEY S ELEMENTS OF 7s. TREATISE ON THE ANALYTICAL GEOMETRY OFTHE POINT, LINE, CIRCLE, & CONIC SECTIONS,Containing an Account of its most recent Extensions,With numerous : HODGES, FIGGIS, & : LONGMANS & FIRST SIX BOOKSOF THEELEMENTS OF EUCLID,ANDPROPOSITIONS OF BOOK XI.,AND ANAPPENDIX ON THE CYLINDER, SPHERE,CONE, ETC.,WITHCOPIOUS ANNOTATIONS AND NUMEROUS 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 OFMATHEMATICAL PHYSICS IN THE CATHOLIC UNIVERSITY OF EDITION, REVISED AND : HODGES, FIGGIS, & CO.

5 , : LONGMANS, GREEN, & AT THE UNIVERSITY PRESS,BY PONSONBY AND edition of the ELEMENTS of Euclid, undertaken at the request of the prin-cipals of some of the leading Colleges and Schools of Ireland, is intended tosupply a want much felt by teachers at the present day the production of awork which, while giving the unrivalled original in all its integrity, would alsocontain the modern conceptions and developments of the portion of Geometryover which the ELEMENTS extend. A cursory examination of the work will showthat the Editor has gone much further in this latter direction than any of hispredecessors, for it will be found to contain, not only more actual matter thanis given in any of theirs with which he is acquainted, but also much of a specialcharacter, which is not given, so far as he is aware, in any former work on thesubject.

6 The great extension of geometrical methods in recent times has madesuch a work a necessity for the student, to enable him not only to read with ad-vantage, but even to understand those mathematical writings of modern timeswhich require an accurate knowledge of Elementary Geometry, and to which itis in reality the best compiling his work the Editor has received invaluable assistance from thelate Rev. Professor Townsend, book was rewritten and con-siderably altered in accordance with his suggestions, and to that distinguishedGeometer it is largely indebted for whatever merit it Questions for Examination in the early part of the First Book are in-tended as specimens, which the teacher ought to follow through the entire person who has had experience in tuition knows well the importance ofsuch examinations in teaching Elementary Exercises, of which there are over eight hundred, have been all selectedwith great care.

7 Those in the body of each Book are intended as applications ofEuclid s Propositions. They are for the most part of an elementary character,and may be regarded as common property, nearly every one of them havingappeared already in previous collections. The Exercises at the end of eachBook are more advanced; several are due to the late Professor Townsend, someare original, and a large number have been taken from two important Frenchworks Catalan sTh eor`emes et Probl`emes de G eom etrie El ementaire, andtheTrait e de G eom etrie, byRouch eandDe second edition has been thoroughly revised and greatly enlarged.

8 Thenew matter includes several alternative proofs, important examination questionson 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 propertiesof the Prism, Pyramids, Cylinder, Sphere, and present Edition has been very carefully read throughout, and it is hopedthat few misprints have escaped Editor is glad to find from the rapid sale of former editions (each 3000copies) of his Book, and its general adoption in schools, that it is likely toiaccomplish the double object with which it was written, viz. to supply studentswith a Manual that will impart a thorough knowledge of the immortal workof the great Greek Geometer, and introduce them, at the same time, to someof the most important conceptions and developments of the Geometry of thepresent , South Circular-road.

9 1 BOOK of Angles, Triangles, Parallel Lines, and parallelograms.,..2 Definitions, ..2 Propositionsi. xlviii.,..8 Questions for Examination, .. 46 Exercises, .. 47 BOOK of Rectangles,.. 50 Definitions, .. 50 Propositionsi. xiv.,.. 51 Questions for Examination, .. 66 Exercises, .. 67 BOOK of the Circle,.. 69 Definitions, .. 69 Propositionsi. xxxvii.,.. 70 Questions for Examination, .. 98 Exercises, .. 99 BOOK and Circumscription of Triangles and of Regular Polygons in andabout Circles,.. 102 Definitions, .. 102 Propositionsi. xvi.,.. 102 Questions for Examination, .. 113 Exercises.

10 113 BOOK of Proportion,.. 117iiiDefinitions, .. 117 Introduction, .. 117 Propositionsi. xxv.,.. 122 Questions for Examination, .. 134 Exercises, .. 135 BOOK of the Theory of Proportion,.. 136 Definitions, .. 136 Propositionsi. xxxiii.,.. 136 Questions for Examination, .. 164 BOOK of Planes, Coplanar Lines, and Solid Angles,.. 172 Definitions, .. 172 Propositionsi. xxi.,.. 172 Exercises, .. , Pyramid, Cylinder, Sphere, and Cone,.. 184 Definitions, .. 184 Propositionsi. vii.,.. 185 Exercises, .. Modern theory of parallel lines, .. 196B. Legendre s proof of Euclid,i., xxxii.,.. 196,,Hamilton s.