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 com- pressed 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 produc- tion 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, whic h 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 an d 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 prin- cipals 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 ad- vantage, but even to understand those m athematical 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 con- siderably 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 in- tended 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 bo dy of each Book are intended as applications of Euclid’s Prop ositions. 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´eor`emes et Probl`emes de G´eom´etrie El´ementaire, and the Trait´e de G´eom´etrie, by Rouch ´ e and De Comberousse. 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 escap ed 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., . . . . . . . 2 Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Propositions i.–xlviii., . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Questions for Examination, . . . . . . . . . . . . . . . . . . . . . . . . 45 Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 BOOK II. Theory of Rectangles, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Propositions i.–xiv., . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Questions for Examination, . . . . . . . . . . . . . . . . . . . . . . . . 65 Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 BOOK III. Theory of the Circle, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Propositions i.–xxxvii., . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Questions for Examination, . . . . . . . . . . . . . . . . . . . . . . . . 97 Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 BOOK IV. Inscription and Circumscription of Triangles and of Regular Polygons in and about Circles, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Propositions i.–xvi., . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Questions for Examination, . . . . . . . . . . . . . . . . . . . . . . . . 112 Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 BOOK V. Theory of Proportion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 iii Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Introduction, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Propositions i.–xxv., . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Questions for Examination, . . . . . . . . . . . . . . . . . . . . . . . . 133 Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 BOOK VI. Application of the Theory of Proportion, . . . . . . . . . . . . . . . . . . . . 135 Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Propositions i.–xxxiii., . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Questions for Examination, . . . . . . . . . . . . . . . . . . . . . . . . 163 BOOK XI. Theory of Planes, Coplanar Lines, and Solid Angles, . . . . . . . . . . . . . . 171 Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Propositions i.–xxi., . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 APPENDIX. Prism, Pyramid, Cylinder, Sphere, and Cone, . . . . . . . . . . . . . . . . . 183 Definitions, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Propositions i.–vii., . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Exercises, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 NOTES. A.—Modern theory of parallel lines, . . . . . . . . . . . . . . . . . . . . . 194 B.—Legendre’s pro of of Euclid, i., xxx ii., . . . . . . . . . . . . . . . . . . 194 ,, Hamilton’s ,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 C.—To inscribe a regular polygon of seventeen sides in a circle—Ampere’s solution simplified, . . . . . . . . . . . . . . . . . . . . . . . . . . 196 D.—To find two mean proportionals between two given lines—Philo’s so- lution, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 ,, Newton’s solution, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 E.—M c Cullagh’s proof of the minimum property of Philo’s line, . . . . . . 198 F.—On the trisection of an angle by the ruler and compass, . . . . . . . . 199 G.—On the quadrature of the circle, . . . . . . . . . . . . . . . . . . . . . 200 Conclusion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 iv [...]... the typical theorem, If X is Y, then Z is W, (i.) the hypothesis is that X is Y , and the conclusion is that Z is W Converse Theorems.—Two theorems are said to be converse, each of the other, when the hypothesis of either is the conclusion of the other Thus the converse of the theorem (i.) is— If Z is W, then X is Y (ii.) From the two theorems (i.) and (ii.) we may infer two others, called their contrapositives... GB, and the angle BF C equal to the angle CGB Hence the two triangles BF C, CGB have the two sides BF , F C in one equal to the two sides CG, GB in the other; and the angle BF C contained by the two sides of one equal to the angle CGB contained by the two sides of the other Therefore [iv.] these triangles have the angle F BC equal to the angle GCB, and these are the angles below the base Also the angle... angle ABD, the sum of the angles CBA, ABD is equal to the sum of the three angles CBE, EBA, ABD In like manner, the sum of the angles CBE, EBD is equal to the sum of the three angles CBE, EBA, ABD And things which are equal to the same are equal to one another Therefore the sum of the angles CBA, ABD is equal to the sum of the angles CBE, EBD; but CBE, EBD are right angles; therefore the sum of the angles... and we have the sum of BA, AC greater than the sum of BE, EC Again, the sum of the sides DE, EC of the triangle DEC is greater than DC: to each add BD, and we get the sum of BE, EC greater than the sum of BD, DC; but it has been proved that the sum of BA, AC is greater than the sum of BE, EC Therefore much more is the sum of BA, AC greater than the sum of BD, DC 2 The external angle BDC of the triangle... letters, as BAC, of which the middle one, A, is at the vertex, and the other two along the legs The angle is then read BAC xii The angle formed by joining two or more angles together is called their sum Thus the sum of the two angles ABC, P QR is the angle AB R, formed by applying the side QP to the side BC, so that the vertex Q shall fall on the vertex B, and the side QR on the opposite side of BC from... legs, and the point the vertex of the angle A light line drawn from the vertex and turning about it in the plane of the angle, from the position of coincidence with one leg to that of coincidence with the other, is said to turn through the angle, and the angle is the greater as the quantity of turning is the greater Again, since the line may turn from one position to the other in either of two ways, two... equal without producing the sides Also by producing the sides through the vertex 2 Prove that the line joining the point A to the intersection of the lines CF and BG is an axis of symmetry of the figure 3 If two isosceles triangles be on the same base, and be either at the same or at opposite sides of it, the line joining their vertices is an axis of symmetry of the figure formed by them 4 Show how to prove... XXIV.—Theorem If two triangles (ABC, DEF ) have two sides (AB, AC) of one respectively equal to two sides (DE, DF ) of the other, but the contained angle (BAC) of one greater than the contained angle (EDF ) of the other, the base of that which has the greater angle is greater than the base of the other Dem. Of the two sides AB, AC, let AB be the one which is not the greater, and with it make the angle... contained by the two sides of one equal to the angle D contained by the two sides of the other Hence [iv.] BC would be equal to EF ; but BC is, by hypothesis, greater than EF ; hence the angle A is not equal to the angle D 2 If A were less than D, then D would be greater than A, and the triangles DEF , ABC would have the two sides DE, DF of one respectively equal to the two sides AB, AC of the other, and the. .. perimeter of a quadrilateral is greater than the sum of its diagonals Def.—A line drawn from any angle of a triangle to the middle point of the opposite side is called a median of the triangle 9 The sum of the three medians of a triangle is less than its perimeter 10 The sum of the diagonals of a quadrilateral is less than the sum of the lines which can be drawn to its angular points from any point except the . hypothesis of either is the conclusion of the other. Thus the converse of the theorem (i.) is— If Z is W, then X is Y. (ii.) From the two theorems (i.) and (ii.) we may infer two others, called their contrapositives AT THE UNIVERSITY PRESS, BY PONSONBY AND WELDRICK PREFACE. This edition of the Elements of Euclid, undertaken at the request of the prin- cipals of some of the leading Colleges and Schools of. 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,