Table Of ContentTHE FIRST SIX BOOKS
OF THE
ELEMENTS OF EUCLID,
AND
PROPOSITIONS I.-XXI. OF BOOK XI.,
AND AN
APPENDIX ON THE CYLINDER, SPHERE,
CONE, E ,
TC.
WITH
COPIOUS ANNOTATIONS AND NUMEROUS EXERCISES.
BY
JOHN CASEY, LL.D., F.R.S.,
FELLOW OF THE ROYAL UNIVERSITY OF IRELAND;
MEMBER OF COUNCIL, ROYAL IRISH ACADEMY;
MEMBEROFTHEMATHEMATICALSOCIETIESOFLONDONANDFRANCE;
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
suchaworkanecessityforthestudent, toenablehimnotonlytoreadwithad-
vantage, 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.
IncompilinghisworktheEditorhasreceivedinvaluableassistancefromthe
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-
tendedasspecimens,whichtheteacheroughttofollowthroughtheentirework.
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
withgreatcare. ThoseinthebodyofeachBookareintendedasapplicationsof
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´eor`emes et Probl`emes de G´eom´etrie El´ementaire, and
the Trait´e de G´eom´etrie, by Rouche´ and De Comberousse.
The second edition has been thoroughly revised and greatly enlarged. The
newmatterincludesseveralalternativeproofs,importantexaminationquestions
oneachofthebooks,anexplanationoftheratioofincommensurablequantities,
thefirsttwenty-onepropositionsofBookXI.,andanAppendixontheproperties
of the Prism, Pyramids, Cylinder, Sphere, and Cone.
ThepresentEditionhasbeenverycarefullyreadthroughout,anditishoped
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., . . . . . . . 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 proof of Euclid, i., xxxii., . . . . . . . . . . . . . . . . . . 194
,, Hamilton’s ,, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
C.—Toinscribearegularpolygonofseventeensidesinacircle—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.—McCullagh’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 ELEMENTS OF EUCLID.
INTRODUCTION.
GeometryistheScienceoffiguredSpace. FiguredSpaceisofone, two, orthree
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 prop-
erties 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
anextensiveandimportantdepartment. ThisisthepartofGeometryonwhich
the oldest Mathematical Book in existence, namely, Euclid’s Elements, is writ-
ten, 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’ Aper¸cu Historique a valuable history of the origin and the
development of the methods of Geometry.
In the following work, when figures are not drawn, the student should con-
structthemfromthegivendirections. ThePropositionsofEuclidwillbeprinted
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 num-
ber of the Book will be given only when different from that under which the
reference occurs. The general and the particular enunciation of every Propo-
sition will be given in one. By omitting the letters enclosed in parentheses we
have the general enunciation, and by reading them, the particular. The anno-
tations will be printed in smaller type. The following symbols will be used in
them:—
J
Circle will be denoted by
Triangle ,, 4
Parallelogram ,,
Parallel lines ,, k
Perpendicular ,, ⊥
Inadditiontotheseweshallemploytheusualsymbols+,−,&c.ofAlgebra,
and also the sign of congruence, namely ≡. This symbol has been introduced
by the illustrious Gauss.
1
BOOK I.
THEORY OF ANGLES, TRIANGLES, PARALLEL LINES, AND
PARALLELOGRAMS.
DEFINITIONS.
The Point.
i. A point is that which has position but not dimensions.
Ageometricalmagnitudewhichhasthreedimensions,thatis,length,breadth,andthick-
ness,isasolid;thatwhichhastwodimensions,suchaslengthandbreadth,isasurface;and
thatwhichhasbutonedimensionisaline. Butapointisneitherasolid,norasurface,nor
aline;henceithasnodimensions—thatis,ithasneitherlength,breadth,northickness.
The Line.
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
bespaceoftwodimensions;andifinadditionithadanythicknessitwouldbespaceofthree
dimensions;hencealinehasneitherbreadthnorthickness.
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.
Ifapointmovewithoutchangingitsdirectionitwilldescribearightline. Thedirectionin
whichapointmovesincalledits“sense.” Ifthemovingpointcontinuallychangesitsdirection
it will describe a curve; hence it follows that only one right line can be drawn between two
points. ThefollowingIllustrationisduetoProfessorHenrici:—“Ifwesuspendaweightbya
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,weobtainthenotionofthesimplestofalllines,whichwecallastraightline.”
The Plane.
v. A surface is that which has length and breadth.
Asurfaceisspaceoftwodimensions. Ithasnothickness,forifithadany,howeversmall,
itwouldbespaceofthreedimensions.
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.
2
Figures.
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.
The Angle.
ix. Theinclinationoftworightlinesextendingoutfromonepointindifferent
directions is called a rectilineal angle.
x. The two lines are called the legs, and the point the vertex of the angle.
Alightlinedrawnfromthevertexandturningaboutit
in the plane of the angle, from the position of coincidence
withonelegtothatofcoincidencewiththeother,issaidto
turn through the angle, and the angle is the greater as the
quantityofturningisthegreater. Again,sincethelinemay
turn from one position to the other in either of two ways,
twoanglesareformedbytwolinesdrawnfromapoint.
Thus if AB, AC be the legs, a line may turn from the
position AB to the position AC in the two ways indicated
bythearrows. Thesmalleroftheanglesthusformedistobe
understoodastheanglecontainedbythelines. Thelarger,
calledare-entrant angle,seldomoccursinthe“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.
xii. The angle formed by joining two or more angles together is called
their sum. Thus the sum of the two angles ABC, PQR is the angle AB0R,
formedbyapplyingthesideQP tothesideBC,
so that the vertex Q shall fall on the vertex B,
and the side QR on the opposite side of BC
from BA.
xiii. When the sum of two angles BAC,
CAD is such that the legs BA, AD form one
right line, they are called supplements of each
other.
Hence,whenonelinestandsonanother,thetwoangleswhichitmakesonthesameside
ofthatonwhichitstandsaresupplementsofeachother.
3
xiv. When one line stands on another, and
makes the adjacent angles at both sides of itself
equal, each of the angles is called a right angle,
and the line which stands on the other is called a
perpendicular to it.
Hencearightangleisequaltoitssupplement.
xv. An acute angle is one which is less than
a right angle, as A.
xvi. An obtuse angle is one which is greater than a right angle, as BAC.
Thesupplementofanacuteangleisobtuse,andconversely,thesupplementofanobtuse
angleisacute.
xvii. When the sum of two angles is a right angle,
each is called the complement of the other. Thus, if
the angle BAC be right, the angles BAD, DAC are
complements of each other.
Concurrent Lines.
xviii. Three or more right lines passing through
the same point are called concurrent lines.
xix. Asystem ofmore thanthree concurrent lines is calleda pencil oflines.
Each line of a pencil is called a ray, and the common point through which the
rays pass is called the vertex.
The Triangle.
xx. Atriangle isafigureformedbythreerightlinesjoinedendtoend. The
three lines are called its sides.
xxi. A triangle whose three sides are unequal is said to be scalene, as A;
a triangle having two sides equal, to be isosceles, as B; and and having all its
sides equal, to be equilateral, as C.
xxii. A right-angled triangle is one that has one of its angles a right angle,
as D. The side which subtends the right angle is called the hypotenuse.
4
