THE ELEMENTS OF EUCLID.

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THE ELEMENTS OF EUCLID, BOOKS I.—VI., AND

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Cone, &c.: with

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A TREATISE ON THE ANALYTICAL GEOMETRY OF

THE POINT, LINE, CIRCLE, & CONIC SECTIONS,

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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.

PREFACE.

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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 Théorèmes et Problèmes de Géométrie Elémentaire, and the Traité de Géométrie, by Rouché 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 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 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. |

## Contents

THE ELEMENTS OF EUCLID.

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INTRODUCTION.

Geometry is the Science of figured Space. Figured Space is of one, two, or three dimensions, according as it consists of lines, surfaces, or solids. The boundaries of solids are surfaces; of surfaces, lines; and of lines, points. Thus it is the province of Geometry to investigate the properties of solids, of surfaces, and of the figures described on surfaces. The simplest of all surfaces is the plane, and that department of Geometry which is occupied with the lines and curves drawn on a plane is called Plane Geometry; that which demonstrates the properties of solids, of curved surfaces, and the figures described on curved surfaces, is Geometry of Three Dimensions. The simplest lines that can be drawn on a plane are the right line and circle, and the study of the properties of the point, the right line, and the circle, is the introduction to Geometry, of which it forms an extensive and important department. This is the part of Geometry on which the oldest Mathematical Book in existence, namely, Euclid’s Elements, is written, and is the subject of the present volume. The conic sections and other curves that can be described on a plane form special branches, and complete the divisions of this, the most comprehensive of all the Sciences. The student will find in Chasles’ Aperçu Historique a valuable history of the origin and the development of the methods of Geometry.

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In the following work, when figures are not drawn, the student should construct them from the given directions. The Propositions of Euclid will be printed in larger type, and will be referred to by Roman numerals enclosed in brackets. Thus [III. xxxii.] will denote the 32nd Proposition of the 3rd Book. The number of the Book will be given only when different from that under which the reference occurs. The general and the particular enunciation of every Proposition will be given in one. By omitting the letters enclosed in parentheses we have the general enunciation, and by reading them, the particular. The annotations will be printed in smaller type. The following symbols will be used in them:—

Circle | will be denoted by | ⊙ |

Triangle | ,, | △ |

Parallelogram | ,, | |

Parallel lines | ,, | ∥ |

Perpendicular | ,, | ⊥ |

In addition to these we shall employ the usual symbols +, −, &c. of Algebra, and also the sign of congruence, namely ≡. This symbol has been introduced by the illustrious Gauss.

BOOK I.

THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND PARALLELOGRAMS.

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DEFINITIONS.

i. A point is that which has position but not dimensions.

A geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line. But a point is neither a solid, nor a surface, nor a line; hence it has no dimensions—that is, it has neither length, breadth, nor thickness.

ii. A line is length without breadth.

A line is space of one dimension. If it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness.

iii. The intersections of lines and their extremities are points.

iv. A line which lies evenly between its extreme points is called a straight or right line, such as AB.

If a point move without changing its direction it will describe a right line. The direction in which a point moves in called its “sense.” If the moving point continually changes its direction it will describe a curve; hence it follows that only one right line can be drawn between two points. The following Illustration is due to Professor Henrici:—“If we suspend a weight by a string, the string becomes stretched, and we say it is straight, by which we mean to express that it has assumed a peculiar definite shape. If we mentally abstract from this string all thickness, we obtain the notion of the simplest of all lines, which we call a straight line.”

The Plane.

v. A surface is that which has length and breadth.

A surface is space of two dimensions. It has no thickness, for if it had any, however small, it would be space of three dimensions.

vi. When a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane.

A plane is perfectly flat and even, like the surface of still water, or of a smooth floor.—Newcomb.

vii. Any combination of points, of lines, or of points and lines in a plane, is called a plane figure. If a figure be formed of points only it is called a stigmatic figure; and if of right lines only, a rectilineal figure.

viii. Points which lie on the same right line are called collinear points. A figure formed of collinear points is called a row of points.

ix. The inclination of two right lines extending out from one point in different directions is called a rectilineal angle.

x. The two lines are called the 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 angles are formed by two lines drawn from a point.

Thus if AB, AC be the legs, a line may turn from the position AB to the position AC in the two ways indicated by the arrows. The smaller of the angles thus formed is to be understood as the angle contained by the lines. The larger, called a re-entrant angle, seldom occurs in the “Elements.”

xi. Designation of Angles.—A particular angle in a figure is denoted by three 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.