NON-EUCLIDEAN
GEOMETRY
r
%'
BELL'S MATHEMATICAL SERIES
FOR SCHOOLS AND COLLEGES
General Editor: William 1*. Milne, M.A., D.Sc
NON-EUCLIDEAN GEOMETRY
Bell's Mathematical Series
FOR SCHOOLS AND COLLEGES.
General Editor: WILLIAM ['. MILNE, M.A., D.Sc.
FIRST LIST OF 'VOLUMES.
Crown %vo. is. bd.
Problem Papers in Arithmetic for Pre-
paratory Schools, By T. Cooi'kk Smith, M.A.
Lite Scholar of University College, Oxford; Mathe-
matical Master, St. Peter's Court, BroadaniTB,
Crown Bt>o. is. dd.
Arithmetic. By H, Frekman, M.A. Some-
time Scholar :tl Christ's College, Cambridge;
Mathematical Master nt the Haberdashers' Aske's
School.
Crown 8TW. 2s. (id.
Statics. Part I. By R. C. Fawort, M.A.,
B.Sc Sometime Scholar of Corpus Christi College,
Cambridge ; Head of the Military and Eogfoeeruig
Side, Clifton College.
Crown ivo. SJ.
The Elements of Non-Euclidean
Geometry. By D. M. V. .Sojijikkvii.i.k, M.A.,
D.Sc, Lecturer in Mathematics, University of
St. Andrews.
Others in active preparation.
LONDON: G. BELL AND SONS, LTD.
THE ELEMENTS OF
NON-EUCLIDEAN
GEOMETRY
BY
D. M. Y. SOMMERV1LLE, M.A., D.Sc.
LECTDRBK IK MATHEMATICS, UN1VBKSITV OK ST.^ANtlKKWS
LONDON
G. BELL AND SONS, LTD,
1914
PREFACE
The present work is an extension and elaboration of a
course of lectures on Non-Euclidean Geometry which I
delivered at the Colloquium held under the auspices of the
Edinburgh Mathematical Society in August, 1913.
Non-euclidean geometry is now a well-recognised branch
of mathematics. It is the general type of geometry of
homogeneous and continuous space, of which euclidean
geometry is a special form. The creation or discovery of
such types has destroyed the unique character of euclidean
geometry and given it a setting amongst geometrical
systems. There has arisen, so to speak, a science of Com-
parative Geometry.
Special care has, therefore, been taken throughout this
book to show the bearing of non-euclidean upon euclidean
geometry ; and by exhibiting euclidean geometry as a
really degenerate form — in the sense in which a pair of
straight lines is a degenerate come — to explain the apparent
want of symmetry and the occasional failure of the principle
of duality, which only a study of non-euclidean geometry
can fully elucidate.
VI
PREFACE
There are many ways of presenting the subject. In the
present work the primary exposition follows the lines of
elementary geometry in deduction from chosen postulates.
This was the method of Euclid, and it was also the method
of the discoverers of non-euclidean geometry. Restrictions
have, however, been made. It was felt that a rigorously
logical treatment, with a detailed examination of all the
axioms or assumptions, would both overload the boob and
tend to render it dry and repulsive to the average reader.
It is hoped, however, that the principles have been touched
upon sufficiently to indicate the nature of the problems
involved, especially in such cases where they throw light
upon ordinary geometry.
It is impossible thoroughly to appreciate non-euclidean
geometry without a knowledge of its history. I have there-
fore given in the first chapter a fairly full historical sketch
of the subject up to the epoch of its discovery. Chapters II.
and III. develop the principal results in hyperbolic and
elliptic geometries. Chapter IV. gives the basis of an
analytical treatment, the matter chosen for illustration here
being, for the most part, such as was not touched on in the
preceding chapters. This completes the rudiments of the
subject. The next two chapters exhibit non-euclidean
geometry in various lights, mathematical and philosophical,
and bring up the history to a later stage. In the last three
chapters some of the more interesting branches of geometry
are worked out for the non-euclidean case, with a view to
providing the serious student with a stimulus to pursue the
subject in its higher developments. The reader will find
PREFACE
vu
a list of text-books and references to all the existing litera-
ture up to 1910 in my Biblwgraphif of Non-EucMdeav
Geometry {London : Harrison, 1911).
Most of the chapters are furnished with exercises for
working. As no examination papers in the subject are yet
available, the examples have all been specially devised, or
culled from original memoirs. Many of them are theorems
of too special a character to be included in the text.
In preparing the treatise, the needs of the student reading
privately have been kept steadily in view. Hence it is
hoped that the work will prove useful to the " Scholarship
Candidate " in our Secondary Schools who wishes to widen
his geometrical horizon, to the Honours Student at our
Universities who chooses Geometry as his special subject,
and to the teacher of Geometry, in general, who desires to
see in how far strict logical rigour can be made compatible
with a treatment of the subject capable of comprehension
by schoolboys.
In acknowledging my indebtedness to previous writers
on the subject, special mention should be made of Bonola s
article in the collection, Qnestioni rignardanti la geamehio
demmtare, editct by Enriques (Bologna, 1900 ; German
translation, Leipzig, 1911); and Liemnann's N&kt-
eiiljidiwhe Geometrie (Leipzig, 2nd ed., 1912).
I take this opportunity of expressing my obligations to
Mr. Peter Fraser, M.A., B.Sc., Lecturer in Mathematics at
the University of Bristol, and Mr. E. K. Wakeford, Trinity
College, Cambridge, for kindly criticising the work while in
manuscript form and giving many valuable suggestions.
VIII
PREFACE
I am also greatly indebted to Mr. W. P. Milne, M.A., D.Sc.,
Clifton College, Bristol, for continued assistance, by criti-
cism and suggestion, all through the preparation of the
book. To Dr. A. E. Taylor, Professor of Moral Philosophy
at the University of St. Andrews, and Mr. C. D. Broad, B.A.,
Fellow of Trinity College, Cambridge, and Assistant to the
Professor of Logic at the University of St. Andrews, I have
also to express my thanks for reading and criticising
Chapter VI. In correcting the proofs I have profited by
the assistance of my wife and by the excellence of Messrs.
MacLehose's printing work.
D. M. y. s.
The University, St. Andrews,
April, 1914,
CONTENTS
CHAPTER I.
HISTORICAL.
1 . The origins of geometry
2. Euclid 'a Elements ; axioms and postulates
3. The Parallel-postulate ; attempts to prove it •
4. Playf air's Axiom
5. Thibaut'a rotation proof ; the direction fallacy
6. Bertrand's proof by infinite areas - - - -
7. Equidistant linos "
8. First glimpses of non-ouclidoan geometry ; Snccheri
9. Lambert
10 Gauss
11. Schweikarl, Taurinus and Wachter
12. Legondre „.-•--
13. The theory of parallels in Britain -
14. The discovery of non-euclidoan geometry ; Lobachevsky
16, Bolyai
16. The later development
Examples I. *
PACK
1
2
3
4
6
7
!)
11
18
14
14
in
is
20
21
24
25
CHAPTER n.
ELEMENTARY HYPERBOLIC GEOMETRY.
1 . Fundamental assumptions or axioms -
2. Parallel linos ; the three geometries -
3. Definition of parallel lines
4. Plane GEOMRfair. Properties of parallelism
27
20
30
31
x CONTENTS
PAfiE
5, Thoorera on the transversal - - 3a
0. Theorem of the exterior angle 34
7. The parallel-angle 35
8. Theorems on the quadrilateral 36
9. Intersecting lines are divergent ----- 37
10. Parallels are asymptotic in one direction and divergent in
the other - 3g
11. Parallel lines meet at infinity at a zero angle - ■ - 33
12. Non-intorscctors ; common perpendicular - - - 40
13. Classification of pairs of lines - - 41
14. Solid Geometry. Planes, dihedral angles, etc. - - 42
15. Systems of parallel lines, and lines poqjendicular to a
plane --------- 43
10. Pencils and bundles of lines 45
1 7. Points at infinity ; the absolute - - - - - 46
18. Idea! points 47
1ft. Parallel lines and planes 40
20. Principle of duality ...... -DO
21. The circle ; horocyele ; equidistant-curve - - - 81
22. The sphere ; horosphero ; equidistant-surface (52
23. Circles determined by three points or three tangents - 53
24. Geometry of a bundle of lines and planes 65
25. Trigonometry. Tho circular functions 66
20. Ratio of arcs of concentric horocyclee 67
27. The parallel-angle 68
28. Two formulae for the horocyelu - - - - - St
29. Tho right-angled trianglo ; notation ; complementary
angles and segments ------ 63
30. Correspondence between rectilinear and spherical triangles 03
31. Associated triangles ----- • - 65
32. Trigonometrical formulae for a right-angled triangle - 60
33. Engel-Napior rules ....... 67
34. Spherical trigonometry the same ns in euclidean apace - 00
35. Correspondence lx»twoen a right-angled triangle and a
tri-reetangular quadrilateral 70
30. Napier's rules for a tri-rectangular quadrilateral - - 74
37. Formulae for any triangle ..... 74
38. Euclidean geometry holds in the infinitesimal domain - 75
30. Circumference of a circle 76
40. Sum of the angles of a triangle ; defect and area - 77
CONTENTS
41. Relation between the unite of length and area ; area of
a sector and of a triangle by integration -
ii. The maximum triangle
):i. Gauss' proof of tlie defect-area theorem - - - -
14. Area of a polygon •
l.->. Another proof that the geometry on the horosphere is
euclidean -
Examples II.
XI
PAnK
70
81
82
83
84
84
CHAPTER III.
ELLIPTIC GEOMETRY.
1. Absolute polo and polar
2. Spherical and elliptic geometries - - - -
3. The elliptic plane is a one-sided surface - - - -
4. Absolute polar system - - -
5. Projective geometry ; summary of theorems ■
6. The absolute
7. Principle of duality - -
8. Relation betweon distance arid angle
0. Area of a triangle
10. The circle ; duality
1 1. Cylinder s rectilinear generators -
12. Common perpendiculars to two lines in space -
1 3. Para tactic linos or Clifford's parallels - - - -
14. Construction for common perpendiculars
15. Paratactio lines cut the same two generators of the
absolute
111. Comparison between pnrntAxy and euclidoiiii pnrnllclism •
17. Clifford's Surface - - - - - -
18. Trigonometrical formulae ; circumference of a circle
IB. The right-angled triangle ....
20. Associated triangles - • •
2 1 . Napier's Rules -
22. Spherical trigonometry the same as in euclidean space -
23. Tho trirectangnlar quadrilateral - • -
Examples HI.
ss
8fl
1)1
02
93
98
100
101
103
104
105
106
108
109
110
HI
112
114
no
117
no
120
121
122
xa
CONTENTS
CHAPTER IV.
ANALYTICAL GEOMETRY.
nun
1. Coordinates of a point ; Weierstrass' coordinates - - 125
2. The absolute 127
3. Equation of a straight -lino ; Weierstrass' line-coordinates 12S
4. Distance between two pointe 129
5. The absolute in elliptic geometry 130
6. Anglo between two tines • • - - . - 131
T. Distance of a point from a line ..... 132
8. Point of intersection of two lines ; imaginary points - 132
9. Line joining two points - - - - - - - 134
10. Minimal lines ........ 134
11. Concurrency and collincarity - - - - - - 135
12. The circle 13G
13. Coordinates of point dividing tho join of two points into
given parts ........ 137
14. Middle point of a segment .... - 138
15. Properties of triangles ; centroid, in- and eireum-centres 130
16. Explanation of apparent exception in euelidean geometry 13!)
17. Polar triangles ; orthocontre and orthaxis - - - 141
18. Desargues' Theorem ; configurations .... 142
1!). Dcsmic system ........ 144
20. Concurrency and collincarity 145
21. Position-ratio; croBS-ratio ; projection - - - 147
Examples TV. 149
CHAPTER V.
REPRESENTATIONS OF NOX-EUCLIDEAN
GEOMETRY IN EUCLIDEAN SPACE.
1. The prohliin ...
2. Projective Representation
3. The absolute
4. Euclidean geometry
153
154
154
155
CONTEXTS
XIII
8.
9.
10.
11.
12.
13.
1-1.
15.
16.
17.
18.
10.
20.
21.
22.
23.
24.
25.
28.
27.
23.
29.
30.
31.
32.
33.
34.
35.
36.
37.
:ts.
PAOI
The oiroular points - - - - - - -166
Expression of angle by logarithm of cross- ratio - • ISO
Projective expression for distance and angle in non-
euclidean geometry - - - - - 157
Metrical geometry reduced to projective ; Cay ley-Klein - 158
Example : construction of middle points of a segment - 1 B0
Classification of geometries with projective metric - - 160
Distance in euelidean geometry 161
Geometry in which the perimeter of a triangle is constant 162
Extension to three dimensions 103
Application to proof that geometry on the horospfoero is
ouclidcan - - - - - - - - 164
Geodesic Representation 165
Geometry upon a curved surface . - - - - 106
Measure of curvature - - - - - - -168
Surf acos of constant curvature ; Gauss' theorem - - 168
The Pseudosphero 1»8
The Cayley-Klein representation as a projection • - 170
Meaning of Weierstrass* coordinates - - - - 171
Conporkal Representation. Stereographic projection 172
The orthogonal circle or absolute I ' I
Conformal representation Wi
Point-pairs "6
Pencils of lines ; concentric eirclP3 . . - - 176
Distanco botween two points - - - - - -179
Motions ---------- 179
Reflexions - - - - - - - - -180
Complex numbers 181
Circular transformation ; conformal and nomographic - 182
Inversion ....----- 183
Types of motions - - - - - - - 185
The distance-function - - - - - - - 186
The line-element -------- 187
Simplification by taking fixed circle as a straight line - 188
Angle at which an equidistant-curve meets its axis - - 189
Extension to three dimensions - - - - - 191
XIV
CONTEXTS
CHAPTER VI.
" SPACE-CURVATURE " AND THE PHILOSOPHICAL
BEARING OF NON-ECCLIDEAN GEOMETRY.
PASS
1. Four periods in the history of non-euelideau geometry - 102
2. " Curved space " 193
3. Differential geometry ; Kienuiun 194
4. Free mobility of rigid bodies ; Hebnholtz - - -105
G. Continuous groups of transformations ; Lie - • - 1fl7
6. Assumption of coordinates ...... |gg
7. Space curvature and the fourth dimension - - - 199
8. Proof of the consistency of non-ouclidean geometry - 202
9. Which is tho true geometry T 203
10. Attempts to determine the space constant by astronomical
measurements 203
11. Philosophy of space - ..... 207
12. Tho inextricable entanglement of apace and matter - 209
CHAPTER VII.
RADICAL AXES, HOMOTHETIC CENTRES, AND
SYSTEMS OF CIRCLES.
1. Common points and tangents to two circles - - 211
2. Power of a- point with respect to a circle - - - 212
3. Power of a point with respect to an oquidistanl-eurve - 213
4. Reciprocal property - - - - - - -216
fi. Angles of intersection of two circles .... 218
0. Radical axes - - - - - - - - - 2 1 f!
", nomothetic centrea - -221
8. Radical centres and nomothetic axes - - - 221
9. Coaxal circles in elliptic geometry ----- 222
10. Uomocentric circles ....... 224
11. ('iiinpurison with nielidetm geomeay .... Jgfl
12. Linear equation of a circle - - - - - 227
CONTEXTS
xv
13. Systems of circles
1 1. Correspondence between circles and planes in hyperbolic
geometry ; marginal images
The margined images of two planes intereect in the marginal
image of the line of intersection of tho planes -
Two planes intersect ol the same angle as their marginal
images
Systems of circles --------
Types of pencils of circles
Examples VII. --------
13.
17.
IS.
rAtit
22S
229
231
231
232
233
235
CHAPTER VLU.
INVERSION AND ALLIED TRANSFORMATIONS
0.
7.
8.
9.
in.
II.
ia.
II
IV
in
California) and circular transformations •
A circular transformation is California]
Every congruent transformation of spoco gives a circular
trtiiisforinutioii of the plane -
Converse .......--
The general circular transformation is compounded of a
congruent transformation and a circular trans-
formation which leaves unaltered all the straight
lines through a fixed point -
Inversions and radiations ......
Formulae for inversion
Comparison with euelidcan inversion . - - -
Congruent transformations ; transformation of coordinates
Equations of transformation - - ...
Position of a point in terms of a complex parameter
Expression for congruent transformation by means of
nomographic transformation of tho complex para-
meter ... -
Groups of motions
Connection with quaternions ------
Equation of a circle in terms of complex parameters
EquaJ ion of inversion
Examples Vin.
230
237
238
239
240
241
244
246
247
248
248
249
250
251
252
253
XVI
CONTENTS
CHAPTER IX.
THE CONIC.
1. Equation of the second degree
2. Classification of conies -
3. Centres and axes; foei and Uiracl rices -
4. Focal distance property -
6. Focus-tangent properties
6. Focus-directrix property
7. Geometrical proof of the focal distance property
Examples XX. '
Index -
NON-EUCLTDEAN GEOMETRY
CHAPTER I.
HISTORICAL.
1, The origins of geometry.
Geometry, according to Herodotus, and the Greek deriva-
tion of the word, had its origin in Egypt in the mensuration
of land, and the fixing of boundaries necessitated by the
repeated inundations of the Nile. It consisted at first of
isolated facts of observation and rude rules for calculation,
until it came under the influence of Greek thought. The
honour of. having introduced the study of geometry from
Egypt falls to Thales of Miletus (640-546 B.O.), one of the
seven " wise men " of Greece. This marks the first step in
the raising of geometry from its lowly level ; geometric ele-
ments were abstracted from their material clothing, and
the geometry of lines emerged. With Pytiiaoohas (about
580-500 b.c.) geometry really began to be a metrical science,
and in the hands of his followers and the succeeding
Platom'sts the advance in geometrical knowledge was fairly
rapid. Already, also, attempts were made, by Hippocrates
of Chios (about 430 B.C.) and others, to give a connected and
logical presentation of the science in a series of propositions
based upon a few axioms and definitions. The most famous
of such attempts is, of course, that of Euclid (about 300
B.C.), and so great was his prestige that he acquired, like
N. -E. G. A *
HISTORICAL
[1.2
Aristotle, the reputation of infallibility, a fact which latterly
became a distinct bar to progress.
2. Euclid's Elements.
The structure of Euclid's Elements should be familiar to
every student of geometry, but owing to the multitude of
texts and school editions, especially in recent years, when
Euclid's order of the propositions has beeu freely departed
from, Euclid's actual scheme is apt to be forgotten. We
must turn to the standard text of Heiberg 1 in Greek and
Latin, or its English equivalent by Sir Thomas Heath. 2
Book L, which is the only one that immediately concerns
us, opens with a list of definitions of the geometrical figures,
followed by a number of postulates and common notions,
called also by other Greek geometers "axioms."
Objection may be taken to many of the definitions, as they appeal
simply to i h<* intuition. The definition of a straight line as " a
line which lies evenly with the points on itself " contains no state-
ment from which we can deduce any propositions. We now recog-
nise that wo must start with some terms totally uudefined, and rely
upon postulates to assign a more definite character to the objects.
A right angle and ft square are defined before it has been shown ihnt
objects corresponding to the definitions can exist.
An axiom or common notion was considered by Euclid as a pro-
position which is so self-evident that it needs no demonstration ;
a postulate as a proposition which, though it may not be self-evident,
cannot be proved by any simpler proposition. This distinction
has been frequently misunderstood —to such an extent that later
editors of Euclid have placed some of the postulates erroneously
among the axioms, A notable instance is the parallel-postulate,
No. o, which has figured for ages as Axiom 11 or 12.
The common notions of Euclid are five in number, and
deal exclusively with equalities and inequalities of magni-
tudes.
1 12 vols., Leipzig, 1883-EM.
! 3 vols., Cambridge, 1008.
1-S]
THE PARALLEL-POSTULATE
The postulates are also five in number and are exclusively
geometrical. The first three refer to the construction of
straight lines and circles. The fourth asserts the equality
of a! right angles, and the fifth is the famous Parallel-
Postulate : " If a straight line falling on two straight lines
make the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely,- meet
on that side on which are the angles less than two right
angles."
3. Attempts to prove the parallel-postulate.
It seems impossible to suppose that Euclid ever imagined
this to be self-evident, yet the history of the theory of
parallels is full of reproaches against the lack of self-evidence
of this " axiom." Sir Henry Savile 1 referred to it as one of
the great blemishes in the beautiful body of geometry ;
D'Aleinbert 1 called it " l'ecueil et le seandalc des elemens
de Geometrie,"
The universal converse of the statement. " if two straight lines
crossed by a transversal meet, they will make the interior angles
on that side less than two right angles," is proved, with the help
of another unexpressed assumption (that the straight line is of
unlimited length), in Prop. 17 ; while the eontrapositivc, " if the
interior angles on either side are not less than two right angle.* (<.-.,
bv Prop. 13, if they are equal to two right angles) the straight lines
will not meet," is proved, again with the same assumption, in
Prop. 28.
Such considerations induced geometers {and others), even
up to the present day, to attempt its demonstration. From
the invention of printing onwards a host of parallel-postu-
late demonstrators existed, rivalled only by the " circle-
squarers," the " flat- earth ers," and the candidates for the
1 Praetectionea, Oxford, 1621 (p. 140),
- Melange* de LiUeratwe, Amsterdam, 1759 (p. 180).
HISTORICAL
[1.4
Wotfskehl " Fcrmat " prize. Great ingenuity was expended,
but no advance was made towards a settlement of the
question, for each successive demonstrator showed the false-
ness of his predecessor's reasoning, or pointed out an un-
noticed assumption equivalent to the postulate which it
was desired to prove. Modern research has vindicated
Euclid, and justified his decision in putting this great
proposition among the independent assumptions which are
necessary for the development of euclidean geometry as a
logical system.
All this labour has not been fruitless, for it has led in
modern times to a rigorous examination of the principles,
not only of geometry, but of the whole of mathematics, and
even logic itself, the basis of mathematics. It has had a
marked effect upon philosophy, and has given us a freedom
of thought which in former times would have received the
award meted out to the most deadly heresies.
4. In a more restricted field the attempts of the postulate-
demonstrators have given us an interesting and varied
assortment of equivalents to Euclid's axiom. It would take
up too much of our space to examine the numerous demon-
strations, 1 but as some of the equivalent assumptions have
come into school text-books, and there appears still to
exist a belief that the Euclidean theory of parallels is a
necessity of thought, it will be useful to notice & few of
them.
One of the commonest of the equivalents used for Euclid's
axiom in school text- books is " Playfair's axiom " (really
due to Ludlain 2 ) : " Two intersecting straight lines cannot
1 A iLseftil account of these is Riven by W. B. Franldand in his Ttieoriex
of PamUdimu Cambridge. J £110.
• The Rudiments of Mathematics, Cam bridge, 1785 (p. W5).
L<]
THE ROTATION PROOF
both be parallel to the same straight line," which i3 equiva-
lent to the statement, " Through a given point not more
than one parallel can be drawn to a given straight line,"
and from this the properties of parallels follow very
Fio, I,
elegantly. The statement is simpler in form than Euclid's,
but it is none the less an assumption.
Another equivalent is : " The sum of the angles of a
triangle is equal to two right angles." I do not think that
fl
HISTORICAL
[i. 5
anyone has been bo bold as to assume this as an axiom,
but there have been many attempts to establish the theory
of parallels by obtaining first an intuitive proof of this
statement. A very neat proof, but particularly dangerous
unless it be regarded merely as an illustration, is the
" Rotation Proof," due to Thibaut. 1
5, Let a ruler (Fig. 1) be placed with its edge coinciding
with a side AC of a triangle, and let it be rotated succes-
sively about the three vertices A, B, C, in the direction
ABC, so that it comes to coincide in turn with AB. BC and
CA. When it returns to its original position it must have
rotated through four right angles. But this whole rotation
is made up of three rotations through angles equal to the
exterior angles of the triangle. The fault of this " proof"
is that the three successive rotations are not equivalent at
all to a single rotation through four right angles about a
definite point, but are equivalent to a translation, through
a distance equal to the perimeter of the triangle, along one
of the sides.
The construction may be performed equally well on the
surface of a sphere, with a ruler bent in the form of an arc
of a great circle ; and yet the sum of the exterior angles of
a spherical triangle is always less than four right angles.
A similar fallacy is contained in all proofs based upon the
idea of direction. Take the following : AB and CD (Fig. 2)
are two parallel roads which are intersected by another
road BC. A traveller goes along AB. and at B turns into
the road BC, altering his direction by the angle at B. At
C he turns into his original direction, and therefore must
haw turned back through the same angle. But this requires
1 Qrundrits dcr rtinen UathemalU; 2nd ed. Uottingcn, 1809.
t. 6]
THE DIRECTION FALLACY
a definition of sameness of direction, and this can only be
effected when the theory of parallels has been established.
The difficulty is made clear when we try to see what we
mean by the relative compass-bearing of two points on the
earth's surface. If we travelled due west from Plymouth
along a parallel of latitude, we should arrive at Newfound-
land', but the direct or shortest course would start IB a
" direction " WNW. and finish in the " direction " WSW.
A —
Fid. S
Other statements from which Euclid's postulate may be
deduced are
"Three points are either collinear or concyclic." (\\.
Bolyai. 1 )
" There is no upper limit to the area of a triangle.
(Gauss. 2 )
" Similar figures exist." (Wallis. 3 )
6 Another class of demonstrations is based upon con-
siderations of infinite areas. The following is " Beetrand's
Proof." *
1 Kvrzer OrunArm. 1851 (]'■ *<>)■
" Letter to W. Boiyai, Kith December, 17911.
3 Opera, Oxford, UI93 (t. ii. p. 670).
«L. Bertram), Devdojipement wm^mt dc la partie iiementaire da
nimhimaltgucs, Goneva, 1778 (t. ii. p. 19).
HISTORICAL
[,. G
Let a line AX (Fig. 3), proceeding to infinity in the direc-
tion of X, be divided into equal parts AB, BC, .... and let
the lines AA\ BB', ... each produced to infinity, make equal
angles with AX. Then the infinite strips A' ABB',
B'BCC, ... can all be superposed and have equal areas,
but it requires infinitely many of these strips to make up the
area A' AX, contained between the lines A A' and AX, each
produced to infinity. Again, let the angle A' AX be divided
into equal parts A'AP, PAQ, ... . Then all these sectors
can be superposed and have equal areas, but it requires
only a finite number of them to make up the area A' AX.
Fia. a.
Hence, however small the angle A'AP may be, the area
A'AP is greater than the area A' ABB', and cannot there-
fore be contained within it. AP must therefore cut BB' ;
and this result is easily recognised as Euclid's axiom.
The fallacy here consists in applying the principle of super-
position to infinite areas, as if they were finite magnitudes.
If we consider (Vig. i) two infinite rectangular strips A'ABH'
Bad A'PQli' with equal bases AB, PQ, and partially superposed,
then the two strips are manifestly unequal, or else tiie principle of
7]
INFINITE AREAS
9
superposition is at fault. Again, suppose we have two rectangular
strips A' ABB', C'CDD' (Fig. o). Mark ofT equal lengths A A,.
A t A t . ... along A A', each equal to CD. and equal lengths CC lt CiC t , ...
along t'C, each equal to AB, and divide the strips at these points
into rectangles. Then all the rectangles arc equal, and, if we
O 9 O'
A p **
FlO. 4.
number thein consecutively, then to every rectangle in the one
strip there corresponds the similarly numbered rectangle in the
other strip. Hence, if the ordinary theorems of congruence and
equality of areas are assumed, we must admit that the two strips
arc equal in area, and that therefore the area is independent of the
magnitude of A li. Such deductions are just as valid aa the de-
duction of Euclid's axiom from a consideration of infinite areas.
Pin. fi.
7. It will suffice to give one other example of the attempts
i" huso tin* theory of parallels on intuition. Suppose that,
instead of Euclid's definition of parallels as " straight lines,
which, being in the same plane, and being produced
indefinitely in both directions, do not meet one another in
10
HISTORICAL
[1.7
either direction,'" we define them as " straight lines which
an; everywhere equidistant," then the whole Euclidean
theory of parallels cornea out with beautiful simplicity. In
particular, the sum of the angles of any triangle ABC (Fig. 6)
is proved equal to two right angles by drawing through the
vertex A a parallel to the base BG. Then, if we draw per-
pendiculars from A, B, C on the opposite parallel, these
perpendiculars are all equal. The angle EAB = LB and
the angle CAF=LG.
It is scarcely necessary to point out, however, that this
definition contains the whole debatable assumption. We
have no warrant for assuming that a pair of straight lines
can exist with the property aseribed to them in the defini-
tion. To put it another way, if a perpendicular of constant
length move with one extremity on a fixed tine, is the locus
of its free extremity another straight line ? We shall find
reason later on to doubt this. In fact, non-euclidean
geometry has made it clear that the ideas of parallelism and
equidistance are quite distinct. The term " parallel "
Greek va/MXX^w~rttnnmg alongside) originally con-
noted equidistance, but the term is used by Euclid rather
in the sense " asymptotic " (Greek d-enJ/uirTwroy - non-inter-
secting), and this term has come to be used in the limiting
1.81
SACCHERI
II
case of curves which tend to coincidence, or the limiting
case between intersection and non-intersection. In non-
euclidean geometry parallel straight lines are asymptotic
in this sense, and equidistant straight lines in a plane do
not exist. This is just one instance of two distinct ideas
which are confused in euclidean geometry, but are quite
distinct in non-euclidean. Other instances will present
themselves.
8. First glimpses of Non-Euclidean geometry.
Among the early postulate-demonstrators there stands a
unique figure, that of a Jesuit, Gerolamo Saccheri (1667-
1738), contemporary and friend of Ceva. This man devised
an entirely different mode of attacking the problem, in an
attempt to institute a reductio ad absnrdtun. 1 At that time
the favourite starting-point was the conception of parallels
as equidistant 8'iaight lines, but Saccheri, like some of his
predecessors, saw that it would not do to assume this in
the definition. He starts with two equal perpendiculars AC
and BD to a line AB. When the ends 0, D are joined, it is
easily proved that the ang es at C and D are equal ; but
are they right angles ? Saccheri keeps an open mind, and
proposes three hypotheses :
(1) The Hypothesis of the Right Angle.
(2) The Hypothesis of the Obtuse Angle,
(3) The Hypothesis of the Acute Angle.
The object of his work is to demolish the last two hypo-
theses and leave the first, the Euclidean hypothesis, supreme;
1 Ettdide* ab omni iianvj vindicate*, Milan. 1733. KngBlh trims, by
Halslod. Amer. Math. MonUUg, vols. IS, 1894-08; German byStiickcl
and Engrf, [)it ThMrie d*r FaralUUinicn. Leipzig, 1895. (This book
by StHckel and Engcl contains a, valuable history of the theory of
parallels.)
L2
HISTORICAL
[i. 8
but the task- turns out to be more arduous than he expected.
He establishes a number of theorems, of which the most
important are the following :
If one of the three hypotheses is true in any one case,
the same hypothesis is true in every case.
On the hypothesis of the right angle, the obtuse angle,
or the acute angle, the sum of the angles of a triangle is
equal to, greater than, or less than two right angles.
On the hypothesis of the right angle two straight lines
intersect, except in the one case in which a transversal cuts
them at equal angles. On the hypothesis of the obtuse angle
two straight lines always intersect. On the hypothesis of
the acute angle there is a whole pencil of lines through a
given point which do not intersect a given straight liue, but
have a common perpendicular with it, and these are sepa-
rated from the pencil of lines which cut the given line by
two lines which approach the given line more and more
closely, and meet it at infinity.
The locus of the extremity of a perpendicular of constant
length which moves with its other end on a fixed line is
a straight line on the first hypothesis, but on the other
hypotheses it is curved ; on the hypothesis of the obtuse
angle it is convex to the fixed line, and on the hypothesis
of the acute angle it is concave.
^ Saccheri demolishes the hypothesis of the obtuse angle in
his Theorem 14 by showing that it contradicts Euclid I. 17
(that the sum of any two angles of a triangle is less Hum
two right angles) : but he requires nearly twenty more
theorems before he can demolish the hypothesis of the
acute angle, which he does by showing that two lines which
meet in a point at infinity can be perpendicular at that
t !l]
LAMBERT
13
point to the same straight line. In spite of all his efforts,
however, he does not seem to be quite satisfied with the
validity of his proof, and he offers another proof in which he
loses himself, like many another, in the quicksands of the
infinitesimal.
If Saccheri had had a little more imagination and been
less bound down by tradition, and a firmly implanted belief
that Euclid's hypothesis was the only true one, he would
have anticipated by a century the discovery of the two
non-euclidcan geometries which follow from his hypotheses
of the obtuse and the acute angle.
9. Another investigator, J. H. Lambert (1728-1777), 1
fifty years after Saccheri, also fell just short of this dis-
covery. His startiug-point is very similar to Saccheri's, and
he distinguishes the same three hypotheses ; but he went
further than Saccheri. He actually showed that on the
hypothesis of he obtuse angle the area of a triangle is
proportional to the excess of the sum of its angles over two
right angles, which is the case for the geometry on the
sphere, and he concluded that the hypothesis of the acute
angle would be verified on a sphere of imaginary radius.
He also luade the noteworthy remark that on the third
hypothesis there is an absolute unit of length which would
obviate the necessity of preserving a standard foot in the
Archives.
He dismisses the hypothesis of the obtuse angle, since it
requires that two straight lines should enclose a space, but
his argument against the hypothesis of the acute angle,
such as the non-existence of similar figures, he characterises
' Tlimrif. dtr ParalkUinitn, 178(1. (Rcprint«l in Stiicke] and Easel,
Th. tier Far., 1895.)
14
HISTORICAL
fi. 10
as arguments rib amore et inmdia ducla. Thus lie arrived
at no definite conclusion, and his researches were only
published some years after his death.
10. About this time (1799) the genius of Gauss (1777-
1855) was being attracted to the question, and, although
he published nothing on the subject except a few reviews,
it is clear from his correspondence and fragments of his
notes that he was deeply interested in it. He was a keen
critic of the attempts made by his contemporaries to
establish the theory of parallels ; and while at first lie
inclined to the orthodox belief, encouraged by Kant, that
Kuclidean' geometry was an example of a necessary troth,
he gradually came to see that it was impossible to demon-
strate it. He declares that he refrained from publishing
anything because he feared the clamour of the Boeotians,
or, as we should say, the Wise Men of Gotham ; indeed at
this time the problem of parallel lines was greatly dis-
credited, and anyone who occupied himself with it was
liable to be considered as a crank.
Gauss was probably the first to obtain a clear idea of the
possibility of a geometry other than that of Euclid, and we
owe the very name Non-Euclidean Geometry to him. 1 It is
clear that about the year 1820 he was in possession of many
theorems of non-euclidean geometry, and though he medi-
tated publishing his researches when he had sufficient
leisure to work them out in detail with his characteristic
elegance, he was finally forestalled by receiving in 1832,
from his friend W. Bolyai, a copy of the now famous
Appendix by his son, John Bolyai.
11. Among tbc contemporaries and pupils of Gauss tliere are throe
names which deserve mention. F. K. SchwkikaRt (17804859),
1 Letter to Taurinus, 8th November, 1834.
HJ
GAUSS
15
Professor of Law in Marburg, sent to Gauss in 1818 a pago of MR.
explaining a system of geometry which tie calls " Astral Geometry,"
in which the sum of the angles or a triangle is always less than
two right angles, and in which there is an absolute unit of length.
He did not publish any account of his researches, but he induced
his nephew, F. A. Taurinus (1794-1874}, to take up the question.
His uncle's ideas did not appeal to him, however, but a few years
later he attempted a treatment of the theory of parallels, and
having received some encouragement from Gauss, he published a
small book, Tkeorie der Pamlkllinim, in 1825. After its publication
he came across Camerer's new edition of Euclid in Greek and 1-atin.
wliiili, in an Excursus to Euclid I. 29. contains a very valuable
history of the theory of parallels, and there he found that his methods
had been anticipated by Saccheri and Lambert Nest year, accord-
ingly, he published another work, Gfiomelriae prima elentrnia, and
in the Appendix to this he works out some of tho most Important
trigonometrical formulae for non-euclidean geometry by using the
fundamental formulae of spherical geometry with an imaginary
radius. Instead of the notation of hyperbolic functions, which was
thin scarcely in use. be expresses his results in terms of logarithms
and exponentials, and calls his geometry the " Logarithmic Spherical
Geometry."
Though Taurinus must bo regarded as an Independent rJiscovcrer
of non-euclidean trigonometry, be always retained (be belief, unlike
Gauss and Schweikart, that Euclidean geometry was necessarily
the true one. Taurinus himself was aware, however, of the impor-
tance of liia contribution to the theory of parallels, and it was a
bitter disappointment to him when he found that his work attracted
no attention. In disgust he burned (he remainder of the edition of
his Elementa, which is now ono of the rarest of books.
The third to be mentioned as having arrived at the notion of a
geometry in which Euclid's postulate is denied is F. L WachteR
(1792-1817), a student under Gauss. It is remarkable that he
affirms that even if the postulate be denied, the geometry on a sphere
becomes identical with the geometry of Euclid when the radius is
indefinitely increased, though it is distinctly shown that the limiting
surface is not a plane. This was one of the greatest discoveries of
Lobachovsky and Bolyai. If Waehter had lived he might have
bean tJ» t Saeo v ewc of rmn-flrwtfcVwi geometry, far hi- tnabjbl into
the ouestion was far beyond that of the ordinary pa rail el -postulate
demonstrator.
16
HISTORICAL
[i. 12
12. While Gauss, Scliwetkart, Taurinus and others were
working in Germany, and had arrived independently at
some of the results of non-euclidean geometry, and were,
in fact, just on the threshold of its discovery, in France and
Britain the ideas were still at the old stage, though there
was a considerable interest iu the subject, inspired chie* 1 j
by A, M. Legendre (1752-1333). Legendre's researches
were published in the various editions of his Elements,
from 1794 to 1823, and collected in an extensive article
in the Memoirs of the Paris Academy in 1833.
Assuming all Euclid's definitions, axioms and postulates,
except the parallel-postulate and all that follows from it,
he proves some important theorems, two of which, Proposi-
tions A and B, are frequently referred to in later work as
Legendre's First and Second Theorems.
Prop. A. The mm of (lie three angles of a rectilinear tri-
angle cannot be greater than two right angles (v). (Elements,
3rd cd. 1800.)
In Fig. 7, A A 1 A 2 ... A„ is a straight line, and the tri-
angles AtfB^, A^BtA^ ... are all congruent, and the
vertices B^ ... B u are joined by a broken hue.
Suppose, if possible, that
sLAoBoAt + BtAoAt+AoAiB^v.
Now Z.B A A t =B J A 1 A s
and sLBtAyBi+BiAyAz+AoAiB^ir.
I. 12]
LEGENDRE
17
Therefore LA ( fi a A t >B n A l B u
and therefore A^A^B^.
Let A^A, -BoB,=d; then
A ll B \-B a B i +B 1 B 3 + ...+B n .,B„+BJ i ,=2A a B +nB Q B 1
=2AgB + nA A l -nd=A a A a + 2A B ~rtd,
/.- A 9 A»-A£ a + BoB l + ...+B n A* + (nd-2AtB ).
But, by increasing n, nil can be made to exceed the fixed
length 2.4 ,>B(,: and hence A^A,^ which is the length of the
straight line joining A a and A n , can be greater than the
sum of the parts of the broken line which joins the same
two points, which is absurd.
There are several points in this proof that require careful
examination.
In the first place, the assumption that nd can always exceed
2.-l W by takmp n sullieiently great lies at the basis of geometrical
eunlinttitjf, and is equivalent to the denial of the existence of infini-
tesimals. This is generally known as the Axiom- of Archimedes.
The question of continuity is fundamental in dealing with the
foundations of geometry, but it would be outside ttio scope of this
book to enter further into this extensive and difficult subject.
Twice in this proof we have assumed the " theorem of
the exterior angle " of a triangle (Euclid I. 16), first in the
statement that AbA^BqBi, and second in the assumption
that the straight line joining two points is the shortest
path (Euclid I. 20). This is equivalent to the rejection
of Saccheri's hypothesis of the obtuse angle. If this
hypothesis be followed to its logical conclusion, it can be
shown (see Chap. III.) that two straight lines in a plane
will ulways intersect, when produced in either direction.
The straight line is then re-entrant-, and there are at least
two straight paths connecting any two points. The straight
hne Aji u would not then of necesstfv be the shortest path
from A 9 to A n .
f.'K-G.
B
18
HISTORICAL
ft. 12
Prop. B. If (here exists a single triangle in which the mm
of the angles is equal to two right angles, then m every triangle
the sum of the augks must likewise be equal to two right
angles.
This proposition was already proved by Saccheri, along
with the corresponding theorem for the case in which the
sum of the angles is less than two right angles, and we need
not reproduce Legendre's proof, which proceeds by con-
structing successively larger and larger triangles, in each of
which the sum of the angles = sr.
Legendre makes an attempt to prove that the m» of
the angles of a triangle is equal to two rigid angles, as follows
{EUments, 12th ed. 1823) :
Let A^B i C l (Fig. 8) be a triangle, in whieh A& is the
greatest side and Bfi, the least. Join Ay to M lt the middle
point of lift,, and produce AJt t to C 2 so that A l C 2 = Aj; j .
On AyB, take AJi^KB^AyMy, and join C 2 K. Then
we get a second triangle A 2 B % V 2 , in which A 2 coincides
with A u and in which A 2 B„ is the greatest side and B$ t
the least. Denote the angles of the triangles AyByCy,
A^B.,0^ by single letters.
Then a Afi 2 K m AyB x My and a C z KB, m CyMyA t
Therefore LA 1 CJ(=B l! LKC S B 2 = C\, LMyAfi^B^.
Therefore A t ^A 2 +B, and B^C^C^.
i. 131
LECEXDBE
19
Hence Ay +By+C t =A 2 + B 2 + C\
sad area -4,^,(7, =area A 2 B 9 C 2 .
By repeating this construction we get a series of triangles
with the same area and angle-sum.
TSowA 2 <!,Ay, A^lA^lAy, ... , A nM < %n Ay,
B t <A ls B 3 <A 2 <^A lt ..., Bn+yK^-yA^
Hence the angles A n and B a both tend to zero, while the
vertex C„ ultimately lies on A H B a , The sum of the
angles thus reduces to the single angle C», which is ulti-
mately equal to two right angles.
In this proof there is a latent assumption and also a
fallacy. In the first place it is tacitly assumed that the
straight line is not re-entrant, for if it were re-entrant
the " theorem of the exterior angle," upon which the
proofs of the inequalities depend, could not be accepted,
and the whole proof is invalidated. Again, if we grant
the theorem of the exterior angle, B n and C„ both go to
infinity, and we cannot draw any conclusions as regards
the magnitude of the angle C„.
I-egendre's other attempts make use of infinite areas.
He makes reference to Bertrand's proof, and attempts to
prove the necessity of Playfair's axiom in this way : if it
be denied, then a straight line would be contained entirely
within the angle formed by two rays, but this is impossible
since the area enclosed by the angle is less than " half the
area of the whole plane."
13. In Britain the investigations of Legendre stimulated
such men as Playfaih and Leslie (Professors at Edin.-
tmigh), Ivory, Perrostet Thompson, and Henry Meikle.
°f these, however, none but Meikle had advanced beyond
20
HISTORICAL
[1. 14
the stage of Legendre. Meikle J actually proved in detail,
what had been foreshadowed fifty years before by Lambert,
that if the sum of the artglea of a triangle is less than two
right angles the defect is proportional to its area. He
rejected the hypothesis because he would not admit the
existence of a triangle with all its angles zero. He also
proved independently Saccheri's general form of Legendre's
second theorem.
But by this time the epoch-making works of Lobachevsky
and Bolyai had been published, and the discovery of a
logically consistent system of geometry in which the
parallel-postulate is denied proved once for all that all
attempts to deduce this postulate from the other axioms
are doomed to failure. It was not, however, in Germany
after all that non-euclidean geometry at last saw the light,
but simultaneously in remote districts of Russia and
Hungary.
14. The discovery of Non-Euclidean geometry.
Nikolai Ivanovich Lobachevsky (1793-1856), Professor
of Mathematics at Kazan, was interested in the theory of
parallels from at least 1815. Lecture notes of the period
1815-17 are extant, in which Lobachevsky attempts in
various ways to establish the Euclidean theory. He proves
Legendre's two propositions, and employs also the ideas
of direction and infinite areas. In 1823 he prepared a
treatise on geometry for use in the University, but it
obtained so unfavourable a report that it was not printed.
The MB. remained buried in the University Archives until
it was discovered and printed in 1909. In this book he
states that " a rigorous proof of the postulate of Euclid has
1 Edinburgh New Philos. Joum., 38 (1844 J, p. 313.
i 18]
LOBACHEVSKY AKD BOLYAI
2\
not hitherto been discovered ; those which have been
given may be called explanations, and do not deserve to
be considered as mathematical proofs in the full sense."
Just three years afterwards, he read to the physical and
Hiullicmatical section of the University of Kazan a paper
entitled " Exposition succinte des principes de la geometric
avec une demonstration rigoureuse du theoreme des
paralleles." In this paper, the manuscript of which has
unfortunately been lost, Lobachevsky explains the prin-
ciples of his " Imaginary Geometry," which is more general
than Euclid's, and in which two parallels can be drawn to
a given line through a given point, and in which the sum
of the angles of a triangle is always less than two right
angles.
In the course of a busy life Lobachevsky wrote some half
dozen extensive memoirs expounding the new geometry.
The first of these were in Russian, and therefore inaccessible.
In 1840 he tried to reach a wider circle with a small book
in German entitled Geonietrische Untersuchunffen zur Theorie
der Paiallctlinien, and just before his death he wrote a
summary of bis researches under the title " Pangeometry,"
which he put into French and contributed to the memorial
volume published at the jubilee of his own University. 1
IB. Bolyai Janos (John) (1802-1860) was the son of
Holv.m Farkas (Wolfgang) (1775-1856), a fellow-student
and friend of Gauss at Gottingen, The father was early
interested in the theory of parallels, and without doubt
discussed the subject with Gauss while at Gottingen. The
professor of mathematics at that time. A. G. Kaestner. had
' An Eniilisli translation of tho Geawrtiixtlit I'Memm-huni/fii was
PuMiahcd by Hoisted (Austin, Texas. ISiUJ. An extensive Life ot
Uroaohoveky was published, together with Oerman translations of two
°i the Russian papers, by Engel (Leipzig, 1SS18).
22
HISTORICAL
[i, 18
himself attacked the problem, and with hi help G. S.
KlLigel, one of his pupils, compiled in 17G3 the earliest
history of the theory of parallels.
In 1804 Wolfgang Bolyai, just after his appointment as
professor of mathematics in Maros-Vasarhely, sent to
Gauss a " Theory of Parallels," the elaboration of his
Gottingen studies. In this he gives a demonstration very
similar to that of Meikle and some of Perronet Thompson's,
in which he tries to prove that a series of equal segments
placed end to end at equal angles, like the sides of a regular
polygon, must make a complete circuit. Though Gauss
clearly revealed the fallacy. Botyai persevered and sent
Gauss, in 1808, a further elaboration of his proof. To
this Gauss did not reply, and Bolyai, wearied with his
ineffectual endeavours to solve the riddle of parallel lines,
took refuge in poetry and composed dramas. During the
next twenty years, amid various interruptions, he put
together his system of mathematics, and at length, in
1832-3. published in two volumes au elementary treatise l
on mathematical discipline which contains all his ideas with
regard to the first principles of geometry.
Meanwhile John Bolyai, while a student at the Royal
College for Engineers at Vienna, had been giving serious
attention to the theory of parallels, in spite of his father's
solemn adjuration to let the loathsome subject alone. At
first, like his predecessors) he attempted to find a proof
for the parallel-postulate, but gradually, as he foeussed
his attention more and more upon the results which would
follow from a denial of the axiom, there developed in his
mind the idea of a general or " Absolute Geometry " which
1 Tentamm jumntitkm /thidimam in elementa malkeseos . , . iniro-
ihimidi, Muro.-i.\"ttsarhdy. IS32-3.
,. I.V
BOLYAI
23
would contain ordinary or euclidean geometry as a special
or limiting case. Already, in 1823, lie had worked out the
main ideas of the non-euclidean geometry, and in a letter
of 3rd November he announces to his father his intention
of publishing a work on the theory of parallels, " for,"
he says, " I have made such wonderful discoveries that I
am myself lost in astonishment, and it would be an irre-
parable loss if they remained unknown. When you read
them, dear Father, you too will acknowledge it. I cannot
say more now except that out of nothing I have created a
new and another world. All that I have sent you hitherto
is as a house of cards compared to a tower." Wolfgang
advised his son, if his researches had really reached the
desired goal, to get them published as soon as possible,
for new ideas are apt to leak out, and further, it often
happens that a new discovery springs up spontaneously
in mam- places at once. " like the violets in springtime."
Bolyai's presentment was truer than he suspected, for
at this very moment Lobaehevsky at Kazan, Gauss at
Gotiinjren, Taurinus at Cologne, were all on the verge of
this great discovery It was not, however, till 1832 that
at length the work was published. It appeared in Vol. I.
of his father's Tenfamen, under the title "Appendix,
scientiam absolute veram exliibens."
W. Bolyai wrote one other book, 1 in German, in which
he refers to the subject, but the son, although he continued
to work at his theory of space, published nothing further.
Lobaehevsky 's Geome.fr inrhe Untersuckungen- came to his
1 ffwwr Omndrist einfet Versuelis, Maros-Vasarhely, lS.Tl. J. Bolyai's
'■ Appendix " has been translated into French, Italian, Herman, Kn^Mali
and Magyar; English by Halstcd (Austin, Texas, 18i)l}. A complete
life of the Bolyai. with Herman translations of the " Appendix," parts
of the Teiitam'en, etc., has been published by EUUckcl {Leipzig. HI13),
a* ■ companion book to Engel's Lo6o(«/w/»K/.
24
HISTORICAL
(I. IS
knowledge in 1848, and this spurred hini on to complete
the great work on " Rauinlehre," which he had already
planned at the time of the publication of his " Appendix,"
but he left this in large part as a rafts indigestaque moles,
and he never realised his hope of triumphing over his great
Russian rival.
On the other hand, Lobachevsky never seems to have
heard of Bolyai, though both were directly or indirectly
in communication with Gauss. Much has been written
on the relationship of these three discoverers, but it is
now generally recognised that John Boiyai and Lobachevsky
each arrived at their ideas independently of Gauss and of
each other ; and, since they possessed the convictions and
the courage to publish them which Gauss lacked, to them
alone is due the honour of the discovery,
16. The succeeding history of non-euclidean geometrv
will be passed over here very briefly. 1 The ideas inaugu-
rated by Lobachevsky and Bolyai did not for many years
attain any wide recognition, and it was only after Baltzer
had called attention to them in 1867, and at his request
Hoiiel had published French translations of the epoch-
making works, that the subject of non-euclidean geometry
began to be seriously studied.
It is remarkable that while Saccheri and Lambert both
considered the two hypotheses, it never occurred to
Lobachevsky or Bolyai or their predecessors, Gauss,
'Some of the later history will be Riven in Chap. VI. The best
history <>t the subject is !{. Brmnla : I M gcometria tum-fiieiidfa : etpo-
«iionr Htonm-rrthra ,1,1 «,to m-ilvppo (Holoimn. l!HH>) ; K na li.ili traiH i-
tion (based on (he German translation by Liebmann. Leipzig 190S)
by H. S. Cm-slaw (Chicago, 1<H2). A full' classified HbBoonipEy is lo
ito found in fvmttiiervillc s BMiixjrnphyofwm-t.uclid, ■,,/. inrhidino
the theory nfparallvb, the fumdalimw of gtomftn/ ,,»d */wrre ,,f n <Km«iuJoZ
(London, iSJl I).
Ex. i.
RtEMAXN AXD KLEIN
25
Schweikart, Taurinus and Waehter, to admit the hypo-
thesis that the sum of the angles of a triangle may be greater
than two right angles. This involves the conception of
a straight line as being unbounded but yet of finite length.
Somewhere " at the back of beyond " the two ends of the.
line meet and close it. We owe this conception first to
Bernhard Riemanh (1826-1866) in his Dissertation of
1854 J (published only in 1866 after the authors death),
but in his Spherical Geometry two straight lines intersect
twice like two great circles on a sphere. The conception
of a geometry in which the straight line is finite, and is,
without exception, uniquely determined by two distinct
points, is due to Felix Kletk. 2 Klein attached the now
usual nomenclature to the three geometries ; the geometry
of Lobachevsky he called Hyperbolic, that of Rieniann
Elliptic, and that of Euclid Parabolic.
EXAMPLES I.
1. If the angle in a semicircle is constant, prove that it is a right
angle.
2. A !i is a fixed line and P ft variable point such thai the angle
APH is constant. Show that the tangents at A and B to the
curve locus of P are equally inclined to AB.
3. If every chord in the locus of Question 2 has the property
that it subtends a constant angle at points on the curve, prove that
the sum of the angles of a, triangle must be equal to two right angles.
Examine the fallacies in the following proofs of Euclid's axiom :
4. If the side c and the angles A and B of a triangle arc given the
triangle is determined, and therefore the angle 0-J\A, B, c). But
aince this equation must be homogeneous, it cannot contain the
1 " fiber die Hvpoilirsen, welcho tier Geometric xn Grande liegen" ;
English translation by W, K. Clifford, Nature. S (1873).
: " CberdiesngenannteNicht-EiiklidischeGeoraetrie," Math. Anaalen,
4 US71), 8(1873).
26
HISTORICAL
[Ex. .
aide c. Hence C'-JIA, B). Let ABC be ii right-angled triangle,
and draw the (JcrpcurlitTuJar CD on tlio hypotenuse. Then the
two triangles ABC, ACD have two angles equal, and therefore tho
third angle ACD—B. Similarly BCD=A. Therefore A + B+C
=2 right angles. (A. M. Legendre, 1794.)
5. Let OB and NA be both perpendicular to ON, then OB does
not meet NA. Ijet 00, making a finite angle GOB, be tho last line
through which meets NA. Then A F .d can be produced beyond
its point of intersection with OG to if, ami OK still meets NA.
Hence 00 is not tho last, and therefore all lines through within
the angle NOB must meet NA. (J. D. Gergonnc, 1812.)
8. One altitude AD of an equilateral triangle ABC divides it into
two right-angled triangles, in which one acute unfile is double the
other. If the three altitudes meet in 0, each of the triangles .-I OK,
He, has one angle equal to half the angle of the equiliiti-ru! hinngle ;
hence tho angle OAE-\AOE Heme the sum of the angles of
the iriangle ABO is equal to half (lie .sum of the angles at 0, i.e.
equal to two right angles. (J. K. F. Hauff, 1819.)
7. A A' ±AB and ABB' is acute. From D, any point on J3.B',
is drawn the perpendicular BE to AB. O is any point on A A',
and BC cuts Di£ in F. O is the middle point of EF, and BO meets
AG in H. An isosceles triangle is drawn with base EF and sides
equal to ED, making the base angles-ra. Rotate the plane of the
ligure about AB through the angle «, Denote the point* in their
new positions by suffixes. Then DyO±EF, and B1I is the pro-
jection of WW,', // is t herefoiv the projection of a point on both
WW,' and AA t % and these lines therefore meet (K. Th. E. Gronau,
1902.)
CHAPTER n.
ELEMENTARY HYPERBOLIC GEOMETRY.
1. Fundamental assumptions.
In establishing any system of geometry we must start
by naming certain objects which we cannot define in terms
of anything more elementary, and make certain assump-
tions, from which by the laws of logic we can develop a
consistent system. These assumptions are the axioms of
the science." The axioms of geometry have been class) tied
by Hilbert ' under five groups :
1. Axioms of convection, or classification, connecting
point, line and plane.
Axioms of order, explaining the idea of" between."
.-I.i/hm'n of i inij/mnHre.
Axiom of prmillete.
5. Axioms of continuity.
Without entering into these in more detail, 2 we shall
assume as deductions from them, the theorems relating
to the comparison and addition of segments and angles.
The method of superposition can be used as a /nfou rfe parler.
Strictly speaking, a geometrical figure is inoapablo of being moved ; *
' T). Hilbert, Qnuuttagen. der Gmmetrie, Leipzig, 1800, 4th ed, 1013 ;
English translation by Townsend. Chicago, 15)02.
"The reader who wishes In rtody r In- dsvakpSMBi of non-niclidean
geometry from u set of axioms may refer to .1. L. Coolidge, Element* of
Nttn.ftuclidean Geometry. Oxford, 1900
'Cf. Chap. VI. §4.
2.
:;,
I.
28 ELEMENTARY HYPERBOLIC GEOMETRY [». t
linos are not drawn, nor arc figures constructed. It is only the act
of the mind which fixes the attention on certain: geometrical figures
wliioli already exist, developing them out, so to speak, like the
picture on a photographic plate. And when we speak or applying
one figure to another by superposition, all (hut we mean is that a
comparison is made between the two figures and certain results
deduced by the axiom of congruence. VVhen a geometrical figure,
e.o. a line or a point, is spoken of as moving, we are really trans'
ferring our attention to a succession of lines or points in different
positions.
The measurement of angles is independent of the theory
of parallels. Vertically opposite angles are equal ; the sum
of the four angles made by two intersecting lines is an
absolute constant, and one quarter of this is a right angle.
An absolute unit of angle, therefore, exists. A " flat-
angle," which is equal to two right angles, is generally
denoted by the symbol tt. Through a given point only
one perpendicular can be drawn to a given straight line,
the usual construction for this being always possible.
The question of the numerical value of it, or, what is the same thing,
of the unit of angle, need not concern us until we come to consider
trigonometrical formulae (see g 39). VVe mav, however, state at
once tnat when r is treated as a number it has just the value
which wo are already accustomed to assign to it, viz, 3 J to a rough
approximation, or. accurately, 4 times the sum of the infinite series
- 3+Z-T+-- But it is necessary to warn the reader that ir does
not stand for the ratio of the circumference of a circle to its diameter
for in non-euclidean geometry this ratio is not constant s and the"
radian, or unit angle, in terms of which a flat-ande is represented
by the number tt, does not admit of the familiar construction by
means of a circle.
We shall assume, as deductions from the axioms of
congruence, the congruence-theorems for triangles (Euc. I.
4, 8, 26), and those on the base-angles of an isosceles
triangle (Euc. I. 5, G), which imply the symmetry of the
plane. The theorems relating to inequalities among the
II. 2]
FUN DAM EXTAL ASSUMPTIONS
2!1
sides and angles of a triangle (Euc. I. 16-20) are true within
a restricted region. In particular, the "theorem of the
exterior angle," upon which the others depend, is proved
in § 6 to be true without exception in hyperbolic geometry.
An important axiom of order which must be explicitly
mentioned is Pasch's Axiom. 1 If a straight line cuts one
side of a triangle and- does not pass through a vertex, it teill
also cut one of (he other sides (" side " being understood to
mean the segment subtended by the opposite interior angle
of the triangle).
A large part of geometry can be constructed without the
axioms of continuity, 2 but we shall in general assume
continuity.
The watershed, so to speak, between the euclidean and
the non-euclidean geometries which we are about to
develop, is the axiom of parallels.
2. Parallel lines.
Consider (Fig. 9) a straight line I and a point not on
the line. Let ON be ± I, and take any point P on I
The line OP cuts I in P. As the point P moves along /
away from N there are two possibilities to consider :
(1) P may return to its starting point after having
traversed a finite distance. This is the hypothesis of
Elliptic Geometry.
('2) P may continue moving, and the distance NP tend
to infinity. This hypothesis is true in ordinary geometry.
The ray OP then tends to a definite limiting 3 position OL,
1 M. Punch, VortemtHgm iibcr neuere Qeomttrit, Leipzig, 1882 ; 2nd cd.
'Of. 0. B. Hetrted, Rational Geometry, New Vtirk, 1904.
:> This assumes continuity. Wc might dispense with this assumption
by assuming a definite fine Oh which separates the interscctors of 8 A
from the non-intersectors.
30
ELEMENTARY HYPERBOLIC GEOMETRY fn. 3
and OL is said to be parallel to NA. If P moves along /
in the opposite sense, OP will tend to another limiting
position, OM, and OM || NB.
In Euclidean Geometry, the two rays OL and OM form
one line, and the angles NOL and NOM are right angles.
The hypothesis of Hyperbolic Geometry is that the
rays OL, OM are distinct, so that Playfair's axiom is
contradicted.
3. In this chapter we shall develop the fundamental
theorems of Hyperbolic Geometry.
Definition of Parallel Lines. AA' is said to be parallel
to BB in the sense thus indicated when
(') A A' and BB' lie in the same plane,
(2) AA' does not meet BB', both being produced
indefinitely, and
(3) every ray drawn through A within the angle
BAA' meets the ray BB'.
Through any point two parallels OL and OM can be
drawn to a given line AB, so that OL || NA and OM || NB.
The angles NOL and NOM are, by symmetry, equal, and
this angle depends only on the length <if the perpendicular
ON =p. It is called the angle of parallelism or the parallel-
angle, and is denoted by U{p). There are two distinct
senses of parallelism.
tl. 4)
P.YRALLEL LIXES
31
The two parallel lines separate all the lines through
into two classes, those which intersect AB and those which
are non-intersectors of AB.
Properties of Plane Figure3, Parallelism, etc.
4. Parallel lines possess in common with cuclidean
parallels the following properties :
( 1 ) The properly of parallelism is maintained, in the same
sense, throughout the ichole lenglh of (he Urn. (Property of
IraiifmissibilUg.}
Let AA' [| BB', and let P be any point in A A '. We have
FIO. 10.
to prove that within the angle BPA' every ray through
/' cuts BB'. and no other ray through P cuts BB'.
There are two cases to be considered, according as P is
on the side of A in the direction of parallelism or not.
In the first case draw any line PQ through P within
the angle BPA', and on it take a point K. Then the line
AK must cut BB' in some point L, and BP in M. Hence
PQ, which cannot cut. ag;tiu cither .1//, or BM, must cut
the third side BL of the triangle BML (Pasch's Axiom).
But PA' does not cut BB' ; therefore PA' || BB'.
In the second case it is only necessary to take K on QP
produced backwards.
32 ELEMENT ARY HYPERBOLIC GEOMETRY [m 4
(2) Parallelism, is reciprocal, i.e. if AA' || BB', then
BB'\\AA' (Fig. 11).
The bisectors of the angles BAA', ABB' meet in a point
M, since each meets the other parallel, Draw perpendi-
culars MP, MQ, ME from M on AA', BB', AB. By a
comparison of the triangles these perpendiculars are
equal. Draw MM' bisecting the angle PMQ. Then, if
PQ is Joined, PQ1MM', and makes equal angles with AA'
and BB'. The lines AA', BB' are therefore symmetrical
PIO. 11.
with respect to MM', and the reciprocity is therefore
established.
P, Q arc called corresponding points on the two parallels.
(3} Parallelism is transitive, i.e. if A A' II BB' and
BB' || CC, then 44' || CC. There are two cases to be
considered.
[a) Let BB' lie between ^^' and CC (Fig. 12). We
may suppose ABC to be collinear. Within the angle OAA'
draw any line AP. Since 44' || BB', AP cuts SB' in a
point Q. Then, since QB' \\ CC, PQ produced within the
angle CQB' must cut CC. Also A A' itself does not cut
CC ; therefore AA' || CC".
(ft) In the same figure let A A' and 2?S' be || CC. Then
It- 5]
PARALLELISM
33
any line within the angle CAA' must cut CC, and therefore
BB'. Also 44' itself cannot cut BB', for then we would
have two intersecting straight fines A A' and BB' both
parallel to CC in the given sense. Therefore A A' |j R6'.
Parallels in hyperbolic geometry are, however, sharply
distinguished from euclidean parallels by the following
property :
The distance between two parallels diminishes in the
direction of parallelism and tends to zero ; in the other
direction the distance increases and tends to infinity.
PlQ. 12.
Before we can prove this we shall require several pre-
liminary theorems.
5. If a transversal meets two lines making the sum of the
i hi u nor angles on the same side equal to ttca right angles,
the two lines cannot meet and are not parallel.
Let PQ be a transversa] cutting the two lines AA'
Bad BB' m p m & q (FJg . \s), a nd making the sum of the
angles APQ + PQB equal to two right angles. Then , si nee
the smn of the angles PQB+B'QP = W , therefore the
alternate angles APQ and B'QP are equal. .
*Wt PQ at M and draw MKlAA' and ML1.BB'
34 ELEMENTARY HYPERBOLIC GEOMETRY \n. 6
Then the triangles MKP and MLQ are congruent and
LKMF=LLMQ, Therefore KML is o straight line,
perpendicular to both AA' and BB'.
By symmetry, if A A? and #S' meet on one side, they will
also meet on the other. This is only possible in elliptic
geometry. Also if A A' and BB' are parallel in one sense,
they will be parallel also in the opposite sense, which is
only true in euclidean geometry. Hence, in hyperbolic
geometry they neither intersect nor are parallel.
Fro. is.
It follows that if a tramversal meets two parallel Urns it
am Irs the sum of the interior angles on tJie side of parallelism
less //torn two right angles.
6. An exterior angle of a triangle is greater than either
of the interior opposite angles.
Let ABC (Fig. 14) be a triangle with BC produced to D.
Then if the exterior angle AG1) is not greater than the
interior angle ABC it will be either equal to it or leas.
Suppose first that
tACD = ABC, then £ACB+ABC=t
and BA, CA cannot meet (except in elliptic geometry).
Second, if LACD<ABC._ draw BA' making LA'BC = AGD,
,i. 7] THEOREM OF TTIE EXTEKIOE ANGLE 35
Then BA' lies within the angle ABC and must meet
AC, while the sum of the angles A'BC \ A'CB = ir.
But this is impossible (except in eiUptic geometry).
FlO. 14,
Hence the " theorem of the exterior angle " is true,
except possibly in elliptic geometry.
7. The parallel angle II (p) diminishes as tlte distance p
increases.
Let AA' and BB' be || MM' (Fig. 15), and ABM ±MM' ;
and suppose AM >BM.
Fm. is.
Then £ MAA' +ABB'< it. (§ 5, Cor.)
Bl « LMBB'+ABB'=tt.
Therefore Z,MAA'< MBB'.
to avoid further prolixity we shall assume, or leave as
sxerdges to the reader, the theorems that II (p) is uniquely
36 ELEMENTARY HYPERBOLIC GEOMETRY [n. 8
defined for any value of p, and that there is a unique value
of p corresponding to any acute angle as parallel-angle.
Further II (p) is a continuous function of p. As y>-»-a>,
II (p) -+0, and as p -+0, II (p) -* 1 [. The analytical expression
for TI (p) will be found later in § 27. The range of p may
be extended into the negative region. If we suppose the
point A to move to the other side of the line MM', the
angle MAA' will become obtuse, and we have, in fact,
n(-y)+n(p)=Tr.
8. (a) Let ABNM be a quadrilateral with right angles
at the adjacent vertices M, N, and let MA=NB. If we
bisect MN perpendicularly by PQ we see from symmetry
that the angles M AH and NBA are equal.
Draw AA' and BB' \\ MN. Then, since MA =NB, the
angles MAA' and NBB' are equal.
But Z A' AB ■ + B'BA < w : therefore L B'BC >A'AB.
Hence LMAB<NBC, and the angles MAB and NBA
are acute. Thus, hyperbolic geometry implies Saccheri's
Hypothesis of the Acute Angle,
It follows, by considering the quadrilateral AMQP, that
□. 0]
SACCHERFS QUADRILATERAL
if a quadrilateral has three right angles tlie fourth angle tmtst
be acule.
(6) // AM, BN are perpendiculars to MN and AM>BN,
then the angle MAB<NBA.
Via. 17.
CatoSMA'=NB. Then
LNBA >NBA' - MA'B > MAB,
from the theorem of the exterior angle.
Conversely, if /-MAB<NBA, then MA>NB. {Proof
by r&luctio ad absurdum, using (a) and (&).)
9. The distance between two intersecting lines increases
vrithoitt limit:
Take two points P, P' on OA such that OP'>0P, and
drop perpendiculars PM> P'M' on ON. Then the angle
M'P'O and MPO are both acute.
38 ELEMENTARY HYPERBOLIC GEOMETRY [n. 10
Therefore L M'P'P < MPF, and hence M'P' > MP.
Take any kiigth 'V. Let O.V h<- the distance correspond-
ing to the parallel-angle AOJ, and draw NN' ION. Then
NN' || 04. Take A 7 // >G, and draw a line BK making the
acute angle N'HK. Then ///T, which lies within the angle
OBN', must meet OA in some point E. Draw KL±0N.
Then, since the angle tftftf is obtuse, LLKB<NHK\
therefore LK>NB>G. Hence the perpendicular PM
can exceed any 'ength.
10. (a) The distance between two -parallels diminishes in
the direction of parallelism and tends to zero.
Let AA' || MM', and let AM. BN be two. perpendiculars
dropped on MM' from points on AA'. such that B lies
<m the side of A in the direction of parallelism. Then
the angles MAA' and NBA' are both acute; therefore
/LMAB<NBA, therefore NB<MA {§8(6), converse).
Choose any length e, however small, and make MP<e.
Draw PBLMA. If PX || MAT', ZMPJC is acute ; there-
fore PB lies within the angle APX and will meet A A' in
some point B, since PA' ,| AA'.
Make ^tf/JP' =NBP, BP' ^BP, and draw P'M'±NM'.
Then #P' neither meets nor is parallel to AM/', and BA'
must lie within the angle M'BP', and therefore meets M'P'
n . nl DISTANCE BETW KEN PARALLELS 3d
in some point 4'. Then AT 'A' < M'P' < e . Hence the
distance between the parallels diminishes indefinitely.
Parallel lines are therefore asymptotic, and not equi-
distant as in euclidean geometry.
(/j) In the direction opposite to that of pamlklim the
iisiawe between two parallels increases without limit.
We have, in Fig. 19, AM>BN. Draw AL\\M'M
(Fig. 20). From P, any point on A' A, draw PN±M'M,
via. so.
cutting AL in fi, and draw PK± AL. Then PN >PR>PK,
and PK, the distance of P from Ah, can exceed any length.
Hence PN can exceed any length.
11. Two parallel lines can therefore be regarded, as meeting
'it >,ilir,it>f. and, pother, the angle of intersection must be
considered as being equal to zero.
Fio. 21.
Let AA'\\BB', and choose any small angle e. Draw
40 ELEMENTARY HYPERBOLIC GEOMETRY fn 12
AP making /_PAA'<e. Then AP cuts BB' in some
point P. Make PQ=PA, and join AQ. Then
£AQP = PAQ<PAA'<c.
Hence as BQ^oa , AQ tends to the position AA', and
/.AQB^O.
12. Non-intersectors.
If two lines are both perpendicular to a third, they cannot
meet and are not parallel ; and conversely, if two lines are
non-intersecting and not parallel, they will have a common
perpendicular.
Let AB* and LX be any two lines (Fig. 22). From any
point A on the one line draw a perpendicular AL to the
Flo. 22.
other. Then if AL is not perpendicular to both linea
it makes an acute angle with AB' at one side, say the angle
LAB' is acute. If BM is another perpendicular on the
side of AL next the acute angle, and such that the angle
MBB' is also acute, then MB< LA. The distance between
the two lines thus diminishes in this direction, but unless
the lines intersect or are parallel, it cannot diminish inde-
finitely.
Draw MM' \\ BB\ and let the perpendicular C'N' to
LX from any point C on BB' meet MM' in Q. Theu
n. 13]
NON I XTI-: US 1KTUKS
41
0'N'>QN'. But QN'->oo ; therefore CW->«. Thus
the distance between the two given lines AB' and LX at
first diminishes and finally tends to infinity. It must,
therefore, have a minimum value, and, if UV is that mini-
mum distance, UV must be perpendicular to both lines.
Hilherfs 1 construction for the common perpendicular.
Take A, B, any two points on the one line (Fig. 22), and
draw perpendiculars AL, BM on the other. If AL = BM,
the common perpendicular is found by bisecting LM
perpendicularly.
Suppose AL>BM. Make LP = MB, and the angle
LPP' = MBB'. Draw LL' || PP' and MM' || BB'. Then,
from the congruence of the figures XLPP' and XMBB',
the angles XLL' and XMM' are equal. Hence LL' is
not parallel to MM', and therefore is not parallel to BB' ;
nor docs it meet MM', therefore it must cut BB', There-
fore, since PP' || LL', it must meet BB' in some point C.
Make BC m PC. Draw the perpendiculars ON, C'N' to LX.
Then, comparing MBC'N' and LPCN, we find CN-CN\
and the common perpendicular is found by bisecting
NN' perpendicularly by UV.
13. If we make the common perpendicular to two non-
intersecting lines zero, the two lines will coincide, but if
the common perpendicular at the same time goes off to
infinity the two lines may become parallel.
Two straight lines maj' therefore be
(1) Intersecting, and have a real angle of intersection,
but no common perpendicular.
(2) Non-intersecting, and have a real shortest distance
or common perpendicular, but no real angle,
1 Gmndlatfen dtr (hointlH",, 2nd ed. (1803), Appendix III. § 1.
42 ELEMENTARY HYPERBOLIC GEOMETRY [n. W
(3) Parallel, with a zero angle and zero shortest
distance or common perpendicular at infinity.
Before the principles of non-euclidean geometry became known,
lines were Bomotimes classified as convergent, divergent and equi-
distant. In fact, from the assumption that two straight Uaec oanoof
first converge and then diverge wil (toot intersecting, Robert Simson
(1756) was enabled to prove Euclid's postulate. In non-euclidean
geometry equidistant straight lines cannot exist. Intersectors
are convergent or divergent in the same sense as in euciktean geo-
metry; parallels are convergent and asymptotic in one direction
and divergent in the other ; non-intersectors are ultimately divergent
in both directions.
Planes, Dihedral Angles, etc,
14. If two planes have a point in common they have a
line in common. 1 The diiiednd angle between two planes
is measured in the usual way by the angle between two
intersecting lines, one in each, plane, perpendicular to the
line of intersection. If this angle is a right angle the
planes are perpendicular.
The usual proof in euclidean geometry that the dihedral anglo
measured in this way is independent of the point chosen on the line
of intersection involves pamllsk and another proof is required.
Take P, I", any two points on the line of intersection of two
planes a, fi (Fig. 23). Draw PA -P'A'±PP' in (be plane a, und
PB=P'B'±PP' in the plane ji. Join PA' and P'A intersecting in
U, and PIT and P'B intersecting in F. Then, by comparing the
triangles PAP' and P'A'P, we find PA'^P'A Bad LPAV =P'A'U.
Hence PU =P'U. Similarly PB'=P'B and i'F = P'F. Hence,
by comparing triangles PUV, P'VV, we find lUPV = VP'V.
Then, by oomparing triangles PA'B' and P'AR, we find AB=A'B'.
Lastly, hy comparing triangles APB and A' P'B', w© obtain
I.APB =A'P'B'.
For the following theorems the usual proofs are valid.
1 This is mi wmmplion, explicitly excluding space of four or more
dimensions.
151
SOLID GEOMETRY
43
If a straight line p is perpendicular to each of two inter-
secting lines a, b at their point of intersection 0, it is
perpendicular to every line through in the plane ab, and
is said to be perpendicular to the plane ab. Every plane
through p is perpendicular to the plane ab. The line of
intersection of two planes which are both perpendicular
to a plane a is perpendicular to a. Two parade! lines tie
in the same plane (by definition). Two lines a, b, which
are both perpendicular to a plane y, lie in a plane, lor if a, h
cut y in A, B, then the planes aB and bA are both perpendi-
cular to y, and therefore coincide.
Three planes which have a point in common intersect in
paira in three concurrent- straight lines. Three lines which
intersect in pairs are either concurrent or coplanar.
15. {u) If two lities AA' and CC are both parallel to "
third line BB', then AA' j| CC (Fig. 24).
(The case in which all three lines lie in the same plane
rw
44 ELEMENTARY HYPERBOLIC GEOMETRY fn. 15
haa already been proved in §4.) Take three fixed points
A, B, C on the three lines, and any other point P on BB'.
Fro. St.
Join PA, PC. As P moves along BB', the plane PAC
rotates about ,10. In tlie limit, AP and CP become
parallel to BB', and coincide respectively with AA' and
CC ; therefore AA' and CC lie in the same plane.
Again, if CP is fixed while the plane PAC revolves, PA
tends to PB' and the plane CPA to CPB'. CA, the line
Fia. 2fi.
of intersection of the planes CPA and V'CAA', therefore
tends to CC, and CC \\ AA'.
This result may be stated also in the form : two planes
101
LINES IX SPACE
45
whk'h pffss respectively through tieo parallel lines intersect
in a line parallel to the two given lines.
(b) If three planes a, 0, y intersect m lines a, b, c, such
that a and b are neillier parallel nor intersect, then a, b, c
are all perpendicular to the same plane (Fig. 25).
Let AB be the common perpendicular to a, b. Then
the plane through Ala passes through B and is X the
plain; ab, and therefore ± b. Let this plane cut c in C.
Then the planes ac and be are both perpendicular to the
plane ABC, and therefore a, 6, c are all perpendicular to
this plane.
16. Pencils and bundles of lines.
A system of coplanar lines through a point is called
a pencil of lines with vertex 0. The whole system of lines
and planes through in space is called a bundle of lines
and planes.
FlO. S6.
If a system of lines is such that each is parallel in the same
sense to a given line, they are all parallel in pairs (§ 15 (a) ),
and form a pencil or bundle of parallel lines, or a parallel
handle. This is completely determined by one line with a
given direction.
46 ELEMENTARY HYPERBOLIC GEOMETRY [a, n
Denote a bundle of lines with vertex by 0, and a bundle
of lines parallel to fin a given sense by Q. Then these two
bundles uniquely determine a line Oil, which passes through
and is parallel to Tin the given sense.
Two parallel bundles Q, ff uniquely determine a line
mi', which is parallel to both I and V, The line QIV may
be constructed thus : Take any point A and determine
Ail and Ail' (pig. 26). Bisect the angle HAQ', and take
the distance AN corresponding to the parallel-angle kQAQ'.
The line through N ±AN in the plane QAQ' is i\AQ and
to Ail',
So also we can prove that any three bundles, ordinary
or parallel, uniquely determine a plane, for each pair
determines a straight line, and the three straight lines thus
determined are coplanar.
17, Points at infinity.
To an ordinary bundle corresponds a point 0, but to a
parallel bundle there is only a direction corresponding.
We shall extend the class of points by introducing a class
nl fictitious points called points at infinity. These points
function in exactly the same way as ordinary or, as we shall
nail them, actual points, and determine lines and planes
with each other or with actual points.
On every line there are two points at infinity, and the
assemblage of points at infinity in a plane is a curve of the
second degree or conic,' since it is met by any line in two
points. In three dimensions the assemblage is a surface
of the second degree or quadric. This figure, the assem-
blage of all the points at infinity, is called the Absolute.
' The definition of a eonie which m shall use is " a plane curve which is
cu by any straight fine in iu plane in two potato." W- UwwptaSSfan
of the case of " imaginary " intersection see Chap III § 5 a ^ mmma
n. 18]
POINTS AT INFINITY
47
When two points at infinity approach coincidence, the line
determined by them becomes a tangent to the absolute. As
Q, Q' approach, the angle QAQ' in Fig. 26 tends to zero
and AN-><x» . The line QiY therefore goes off to infinity.
Such a line is called a line at mfimiy. Similarly we obtain
dunes at infinity, which are tangent planes to the Absolute.
In euclidean geometry there ia just one parallel through a given
point to a given lino in a plane, and the two points at Infinity upon
a line coincide. The assemblage of points at infinity in a plane then
reduces to a double line, the line at infinity, which is a degenerate
case of a eonio. There ia in this caw; only one real line at infinity ;
but any line whose equation in rectangular coordinates is of the
form x±iy+c-0 is at an infinite distance from the origin, since
I ;• _o, tmd the assemblage of these lines consists of two imaginary
pencils. The equation of the line at infinity is asOr + 0y+ I =0,
and the equations of the two pencils are <u + Aa=0, u>' + Aa=0,
where w, b>' —x + iy.
The absolute in euclidean geometry thus consists, as a locus of
points, of the line at infinity it taken twice, and, as an envelope
of lines of two imaginary pencils <u+Aa = 0, «u' + Aa- 0. with their
verti.es on the line at infinity. These two imaginary vertices are
the points of intersection of the point-circle um>' ■ iP + lf -0 with the
line at infinity Since the equation of any eirele can be written
uu'+na=0, where u--0 represents a straight line, we see that
every circle passe.* through the two points (wiu'=0, a = 0), and for
this reason these twu imaginary points are called the circular poitO*.
In euclidean geometry of three dimensions the absolute consists,
aa a locus of poinK of the plane at infinity taken twice, and, as an
envelope of planes, of all the planes through tangents to an imaginary
eirele. the intersection of the point-sphere 3? + >f + zr=0 with the
plane at infinity.
The whole of metrical geometry is determined by the form of the
Ai -'"lute ; this will be more fully treated in Chap. V.
18. Ideal points.
// a system of lines is such that any kto are co-planar, while
tlic>i do not all lie in the same plane and are neither parallel
nor intersect, then they are all perpendicular to a fixed plane.
48 ELEMENTARY HYPERBOLIC GEOMETRY [,.. is
If a, b are any two of the lines they determine a plane T
which is perpendicular to both. If c is any third line'
which does not lie in the plane ab, it is the intersection of
two planes m and be, which are both 1 T , and therefore o
is X »(gl5(6)).
We shall call this system, which is completely determined
by two of the lines, or by a certain plane v, a bundle of
lines with an ideal vertex 0. The plane * is called the
axis of the bundle. All those lines of the system which lie
m a plane are perpendicular to a straight line I, the inter-
section of their plane with the fixed plane, and form a
pencil of lines with ideal vertex 0. The line I is called the
axis of the pencil.
The ideal paints thus introduced behave exactly like
actual points. They can be regarded as lying outside
the absolute, and are therefore ultra-spatial or ultra-infinite
points.
Two ideal points may determine a real or actual line
Considering only points in a plane, the two ideal points are
determined by two lines a, a'. If a, a' are non-intersecting
the common perpendicular to these lines belongs to both
pencils, and ,s therefore the line determined by the two
ideal points. If « || «', the line 00' is a line at infinity •
if a cute «', 00' is an ideal tine, which contains only ideal
points. An ideal line lies entirely outside the absolute
Similarly, m three-dimensional hyperbolic geometry we
lm-i! xl-nl planes,
It is left to the reader to show now that any two points
actual, at infinity or ideal, always determine uniquely
a line actual, at infinity or ideal ; and that any three points
actual, at infinity or ideal, always determine uniquely a
plane; actual, at infinity or ideal.
it. is]
IDEAL POINTS
49
These relations, in two dimensions, can be pictured more
dearly if we draw a conic to represent the absolute, or
assemblage of points at infinity (Fig. 26 bis). Actual
points are then represented by points in the interior of the
conic, ideal points by points outside the conic. Lines
which intersect on the conic represent parallel lines, those
Pointi at >* T
Fro. 2» bis.
winch intersect outside the conic represent non-inter-
sectors. If Oil and 012' are the tangents to the conic
from an ideal point 0, all the lines of the pencil with vertex
are perpendicular to ilQ'. For the present this may be
used as a mere graphical representation. Its full meaning
can only be understood in the light of projective geometry.
(See Chap. III. §§ 5, 6, and Chap. V. §§ 1-14.)
19. Extension to three dimensions.
If the point of intersection of a line with a plane is
at infinity, the line is said to be parallel to the plane. If
the point of intersection is ideal there is a unique line and
a unique plane perpendicular to both the given line and the
^IViill |>l;uie.
x. ■ K. a. D
50 ELEMENTARY HYPERBOLIC GEOMETRY [a 20
Two planes intersect in a line. If this line is at infinity
the planes are said to be parallel : if it is ideal the two
planes are non-intersecting and there is a unique line
perpendicular to both.
All planes parallel to a given lino in a given sense pass
through the same point at infinity and intersect in paira
in a parallel bundle of lines.
All planes perpendicular to a given plane pass through
the same ideal point and intersect in pairs in a bundle "of
hues with ideal vertex.
The following theorem is of great importance :
Through a line which is parallel to a plane passes just one
plane which is parallel to the given plane.
Let the line I out the plane « in the point at infinity 0.
Through D passes just one line at infinity », and this" line
determines with I a unique plane, which is parallel to a.
The actual construction may be obtained thus : Take any
point A on I and draw AN J_ «, Through A draw AB J. the
plane IN. Then Bl is the plane required.
Through a line which meets a plane a in an ideal point
pass two planes parallel to the plane «, for two tangents
can be drawn from to the section of the absolute made
by the plane a.
20. Principle of duality.
There is a correspondence between points and lines in
a plane, and between points and planes in space, which
gives rise to a sort of duality. To an actual plane a corre-
sponds uniquely an ideal point A, all the lines and planes
through which are perpendicular to the plane « ; and to
an actual point A corresponds an ideal plane «, which is
perpendicular to all the lines and planes through the
n. 211
THE CIRCLE
51
point A. Let B be any point on a ; then the plane R which
corresponds to B must pass through A, since every plane
perpendicular to « passes through A. If the plane it is
at infinity the corresponding point A is its point of contact
with the absolute. The points and planes are therefore
poles and polars with regard to the absolute. This reci-
procity will appear again in elliptic geometry, where the
elements are all real.
The Circle and the Sphere.
21. The circle.
In a plane the locus of a point which is at a constant
distance from a fixed point is a circle. The fixed point is
the centre, and the constant distance the radius.
A circle cuts aU its radii at right angles. This follows
in the limit when we consider a chord PQ, which forms an
isosceles triangle with the two radii CP, CQ. That is, a
circle is the orthogonal trajectory of a pencil of lines with a
real vertex.
Let the vertex go to infinity ; then the lines of the pencil
become parallel, and the circle takes a limiting form, which
is not, as in ordinary geometry, a straight line, but is a
Uniform curve. This curve, a circle with infinite radius, is
called a horocycle; it is the orthogonal trajectory of a pencil
of parallel lines. The parallel lines, normal to the horocycle,
are called its radii. AH horocycles are superposable.
To obtain the orthogonal trajectory of u pencil of lines
with ideal vertex we proceed thus :
Let AA' be the axis of the pencil (Fig. 27), and draw
perpendiculars to A A'. Cut off equal distances MP, NQ, ...
along these perpendiculars. Then the locus of P is again
a uniform curve, which is not, as in ordinary geometry, a
52 ELEMENTARY HYPERBOLIC GEOMETRY fn.22
straight line; and the curve cuts all the perpendiculars
to AA' at right angles. It is therefore the orthogonal
trajectory required. From the property that the curve is
e.|iiiclistaiit from the straight line A A' it is called an equi-
-7-
FIO. 27.
distant-curve. The complete curve consists of two branches,
symmetrical about the axis, and also symmetrical about any
line (radius) which is perpendicular to the axis.
As the axis tends to infinity, the perpendiculars tend to
become parallel, and the equidistant-curve becomes a
horocycle. We can thus pass continuously from an
iii|iiirlistatit-curve to a circle. When the axis goes to
infinity the centre also appears at infinity ; then the axis
becomes ideal and the centre becomes real.
There are therefore three sorts of circles :
(1) Proper circles, with real centre and ideal axis.
(2) Horocffcks, with centre and axis at infinity.
(3) Equidistant-curves, with ideal centre and real axis.
A straight line, or rather two coincident lines, is the limiting
case of an equidistant-curve when the distance vanishes.
22. The sphere.
In space of three dimensions the locus of a point which is
equidistant from a fixed point is a sphere. It is the ortho-
gonal trajectory of a bundle of fines with a real vertex.
When the centre is at infinity the surface is called a horo-
a. 23] THE CIRCLE AND THE SPHERE 53
sphere, and when the centre is ideal the surface is an
dji'idistanl-surface to a plane as axis.
A plane section of a sphere is always a circle ; the greatest
section, or the section of least curvature, is a diametral
section passing through the centre, that is, a great circle
on the sphere.
A plane section of a horosphere is a circle, except when
the section is normal to the surface, i.c. passes through a
normal, in which case the section i3 a horocycle.
A section of an equidistant surface by a plane which
does not cut the axial plane is a circle ; if the plane cuts
the axial plane the section is an equidistant-curve with
the line of intersection as axis ; if the plane is parallel to
the axial plane the section is a horocycle.
23. Circles determined by three points or three tangents.
Lot A, B, C be three given points I to ''"id the centre of a, circle
passing through A, B, C. Bisect the joins of the three points
perpendicularly. If two of tho perpendiculars meet, all three will
he concurrent in the centre required.
54
ELEMENTARY HYPERBOLIC GEOMETRY [n.
Suppose the perpendicular bisectors A' A - ' and LL' to AB and BC
are non- intersecting Let N'L' be their common iwpendtoafc*
Leu this line cut the perpendicular at M in if', and draw iho
popendicobn A A', BB', CC. Then, since NN> bisects AB
perpendicularly and is J. A'B'. AA'=BB\ Similarly BB' =CC.
In the quadrilaterals AA'M'M, CC'M'M, AA'=CC, AM=MG
Z ,' e ^"!f„ at . A i\f amI °' m ^ ht ' * llerafore « t«e quadri-
lateral «rjrjf be folded over i/jf, C will coincide with J, and
•moo only one perpendicular can he drawn from A upon 11/'
«v wl 1 cornc.de with ^ and the angles at W am right, Henee
ii , MM , NN are all perpendicular to A'C, and ABC he on an
equidistant-curve with axis A'B'C,
Suppose ' AW'HiZ,'; then VJ/' must be parallel to both. For,
,1 «!J« ' Cn by th ° first M " the fchreo lines »» concurrent ■
and fUW » a non-mtersector to LL', then by the second case Li/
and A A are nou-mtcrsecttog. Therefore J/Jf'||£i/ ABC
then lie on a horocyete. ' '
In addition to the circle, equidistant-curve or horocycle, which
ean he drawn through ABC in this way, there exist thr^ cqui-
chstant-ourves auch that one of the points lies on one branch while
the other two he on the other branch. Bisect AB, AC at if and ,V
Join MN and draw the perpendiculars A A', BB', CC" to MN
(See Fig. 50, p. 77.) Then AA'^BB'^CC, and an equlranT
curve with u. i/A passes through £, C and ,1. A triangle has
therefore four eiroumeireles, at least three of which are cquitlisUmt-
curvee. There cannot be more than one real circumcentro. This
point, which wo may call the cireumcentre, is the point of concur-
rence of the perpendicular bisectors of the sides, and may be real
at inlinity. or ideal. J ^
_ If L is the middle point of BC, the perpendicular from h on MN
is also ± BC, since it bisects the quadrilateral BB'C'C. Hence the
altitude. , of the triangle LMN are concurrent in the cireumcentre
of the triangle ABC. A triangle therefore possesses a unique
arttocnto, real at infinity, or ideal. If the orthocentro is Meal
there is a real orthaxu, which is perpendicular to the three
altitudes.
The construction for the circles touching the sides of a triangle
is, as usual, obtained by bisecting the angles. Three of the circles
may be equidistant-curves or horocyclea.
24]
GEOMETRY OF A BUNDLE
55
24. Geometry of a bundle of lines and planes.
In plane geometry we have points, lines, distances and
angles ; in a bundle of lines and planes through a point
we have lines, planes, plane angles and dihedral angles Let
us change the language to make it resemble the language of
plane geometry. Tn translating from one language to another
we require a dictionary. The following will suffice :
" Point " - - Line through 0.
" Line " - - - Plane through 0.
" Distance " between Angle between two lines
two ,: points " - through 0.
" Angle " between Dihedral angle between
two " lines " - two planes through 0.
" Parallel lines " - Parallel planes.
Then two " points " determine a " line " and two " lines "
determine a " point." " Parallel lines " only exist when
bat infinity or ideal.
When is at infinity, through a given " point " there
passes just one " tine " " parallel " to a given " line "
(§ 19), and when is ideal, two " parallels " can be drawn
through a given " point " to a given " line."
There are therefore three kinds of geometry of a bundle
according as the vertex ia actual, at infinity or ideal, and
these are exactly of the same form as elliptic, parabolic
{i.e. eticlidean) and hyperbolic plane geometry.
If a sphere be drawn with centre cutting the lines and
planes f the bundle, we can get a further correspondence.
When is an actual point we have a proper sphere. We
nave then the following dictionary :
" Point " - Pair of antipodal points on sphere.
" Line " - - Great circle.
56 ELEMENTARY HYPERBOLIC GEOMETRY [i,.a 6
"Distance" -
" Angle " between
" lines "
Length of are.
Angle between great circles.
Hence the geometry on a proper sphere, where great
circles represent lines, and pairs of antipodal points repre-
sent points is the same as elliptic geometry. (See farther
Chapter III.)
When is ideal, the sphere becomes an equidistant-
surface, and its geometry is hyperbolic ; when is at
infinity it becomes a horosphere, and ite geometry is
cuchdean : '< point " in each case being represented by a
point, and "lines" by normal sections, which are also
shortest lines or geodesies on the surface.
We have here the important and remarkable theorem
that the geometry on (lie liorosphere is eudidean.
Trigonometrical Formulae.
25. We shall now proceed to investigate the metrical
relations of figures, leading up to the trigonometric;,!
formulae for a triangle. The starting point is found in a
relation connecting the arcs of concentric horocycles and
this leads to the expression for the angle of parallelism
The great theorem which enables us to introduce the
circular functions, sines and cosines, etc., of an angle is
that the geometry of shortest lines (horocycles) traced on a
horosphem ra the same as plane ouclidean geometry.
Let- A, B, C be three points on a horosphere with centre Q
The planes ABQ, etc., cut the surface in horocycles, and
we have a triangle ABC formed of shortest lines or geodesies
which , are arcs of horocycles. The angles of this triangle
are the angles between the tangents to the arcs or the
n. »]
TRIGONOMETRY
r,7
dihedral angles between the planes QAB, 9.BC, Q.CA. If
the angle at C is a right angle, then the ratios of the arcs are
lie
AB
AC
=sinJ, 2b=cosj4, etc.
Hie circular functions could bo introduced independently of the
horosphere by defining them as analytical functions of the angle 0,
vte. i 0j ff,
3!
%m6 = 0-^ + ", -■
61
coa »= 1 — ijT + j i - *•• i
the unit of angle being such that the measure of a fiat-angle is
7T -3-14160... . Wo may call this "circular measure." Then it
could be shown that if ABC "m » rectilinear triangle, right-angled at
V, the ratios BC/AB and AC/AB tend to the limits sin A and cos A
as BC, AC and AB aU tend to zero, while the angle A is fixed
(Cf. Chap. HL § 18, footnote).
26. Ratio of arcs of concentric horocycles.
Let A SI, B t Q be two parallel lines, and let them be cut
by horocycles A t B lt A n B^ /i 3 B 3 with centre at infinity O.
A
Then the ratio of the arcs A& : A t B, depends only on the
distance A x A t =x. (See Ex. 26 and 26.)
AJ! 2
A x B t
U. *£-JWi then ffi-M — lfcr*>**
Therefore f(x + y) =/(«) - f(y).
68 ELEarENTARY HYPERBOLIC GEOMETRY |.,.»
This Is the fundamental law of indices, and the function
is therefore the exponential function :
f(x) »e»
c being an absolute constant greater than unity. f( x ) is
a pure ratio, and must be independent of the' arbitrary
umt of length which is selected ; therefore bg/(«) or x logfi
must be a pure ratio. Hence log c must be the reciprocal
of a length. We shall put log, c = 1 ft ; then
. , . /(*)=«*
where k is an absolute linear constant and e is the base of
the natural logarithms, k is called the space-constant ■
its actual value in numbers of course depends upon the*
arbitrary unit of length which is selected, but it forms
itself a natural unit of length, and it is often convenient
to make its value unity. This is one of the most remarkable
facts m non-euclidean geometry, that there is an absolute
umt for length as well as for angle. It can be proved (see
§39) that k M th length of the are of a horocyde which is
sucli thai the tangent at one extremity is parallel bo the radius
through the other extremity.
27. The parallel-angle.
We can apply this now to find the value of the parallel-
angle II {,,) m terms of p. This is the simplest case of the
determination of the relations between the sides and angles
of a triangle. The triangle in this case has two sides infinite,
one angle right and another angle zero.
Let AP H Bl\ and ABLBP (Kg. 30}. Erect a perpen-
d 1C ular at A to the plane of ABY. Draw the parallels BQ
and FU Draw the horosphere with centre at infinity Q
and passing through A, and let it cut Bll in B' and TO
in C. Let BB' = y, and the a res B'C, C'A ,AB'bea,b c
l! 87]
THE PARALLEL-ANGLE
m
Since BV±A£ the plane IIBV .L plane QAB, and since
the angles which the arcs AB', B'C, C'A make with the
lines QA, ilB, Off are all right,
LC'AB'=VAB=T\{p), LAB'C = \,
FIB. 30.
and since geometry on the horosphere is euclidean,
LAC'B'^-ll{p).
ft f*
Hence sin II (p) - , and cos ll(p)=L\
ff
therefore tan. , .n(p) = , ■*
' o + c,
The arc of the horocyelc b is a standard length, viz. the
length of the arc which is such that the tangent at one
extremity is parallel to the radius at the other extremity.
In Fig. 30 BK is such an arc, and =6. Hence
b
.(1)
= e*.
(2)
60 ELEMENTARY HYPERBOLIC GEOMETRY, ,, „ I ^ ^ the PARALLEL-ANGLE
61
Now fold the plane QAV about QA until it lies in the
plane ilAB (Fig. 31). Draw TB"±BQ. Then, if we draw
FIO. 31.
the horocyclic arc VCT with ceritref2i thisar{J = 6
■o/r -p ; therefore £'#" = j, _ y _
Therefore ^t_ c _/-r v ,«
Hence, multiplying (2) and (3) and using (1), we have
This relation may be put into other forms, e.g.
cot 11 (p)=sinh£;
other equivalent forms can be read off from the accom-
panymg figure (Fig. 32), treating the figure as a euclidean
triangle.
We shall effect a simplification by taking in the following
paragraphs (§§ 28-37) the constant k as the unit of length.
Utpi
Flo. 32.
The formulae may be restored in their general form by
dividing by it every letter which represents a length.
28. Two formulae for the horocycle.
Let AB=s be an arc of a horocyclc with centre £1, and
let S be the length of an arc of a horocycle such that the
tangent at one end is perpendicular to the radius at the
other end (Kg. 33). Let s<S.
Extend BA to M so that BM=8; then AM=S-s.
Take A, on the radius through A so that the perpendicular
at A. to A X A is parallel to MM* Then the arc A& -S.
Let BQ l cut Afil in D. Then DA^DB-t s*y. 1**
DA - %i. Then we have, comparing the arcs A^ and AM ,
S-s^Se-'-' 1 W
Extend AB to N so that BN «& Take 4, on the radius
through A so that the perpendicular at A % to AA % is parallel
to MJf. Then the arc A 2 M^S. And since BH || DA t ,
and 4 S S2 2 |! DB, Z)4, =-!>»-=*.
Then g+*«Sfi»-». ■ < 2 )
Adding these two equations, we get
or e" = i( e ' +e '') =cosl1 ' ^
62 ELEMENTARY HYPERBOLIC GEOMETRY t „J
Substituting in (1), wc get
V e'+e- t J- A ^r^ St&ahL - (a)
Then • -W-Stanh/.eoBh/.Sainln „/ B)
These two formulae (a) and (b) give the tangent and
ordmate at the extremity of an arc of a horo^Ctiz t
* is the arc AB of a horocycle, t the length of the talent
AT .ntercepted between the point of contact A an i TZ
s=-Stanh(=£sinb^.
,,.30] THE RIGHT-ANGLED TRIANGLE 63
29. The right-angled triangle; complementary
angles and segments.
Let ABC be a right-angled triangle with right angle at C.
Denote the sidea by a, b, the hypotenuse by c, the angles
opposite the sides by A, /jl.
Let a = II (a), etc. Then we have five segments and
five angles connected by the relations
a=II(a), $ = IL(b), y = II(c), X=H(0, M=U(m).
Fta. Si.
Let a' denote hw-a] then we have the complementary
segments and angles
«' = ll(a'), £'=11(0'), etc.
We have to deal with the circular functions of the angles,
and the complementary angles are of course connected by
the relations
sin a - cos a, tan «' = cot a, etc.
We have also to deal with the hyperbolic functions of
the segments, and we have the relations
sinli a = cot H {«) = cot a = tan a = tan 17 (o') = cosech a ' ,
cosh a = cosee II (a) = cosec n = sec a = sec II (a') = coth a',
30. Correspondence between rectilinear and spheri-
cal triangles.
Draw AQ ± the plane of the triangle (Fig. 35), and draw
Bil and CQ [| /IQ.
64 ELEMENTARY HYPERBOLIC GEOMETRY [„. m
Then BQ || 0ft, and BC ± plane 4CQ ; therefore BC±C9
the P«a*eUIM 1 plane ^C, and the angle between the
planes QfiCand ABV =11(1). AW since the planes 9AB
Fin. S5.
OK7 00* intersect in parallel lines, the sum of the angles
of mtaaeetioii-T; therefore the angle between theplaL
9ABmd9BC = ^-\.
Now draw a sphere with centre B, and we get a rieat-
aagted spherical triangle with hypotenuse f Jll(a) skies
M andy = n( C ), measured by the angles which they sub-
tend at the centre, and opposite angles X'«?-X and
.31]
ASSOCIATED TRIANGLES
= 11(6), i.e. to the rectilinear triangle (c, cA, b/x) corre-
sponds the spherical triangle (a, ft\\ y B).
31. Associated triangles.
To the spherical triangle (a, /u X', y B) we get four other
associated triangles by drawing the polars of the two
vertices (cf. Chapter III, § 20). This gives a star pentagon
(Fig. 36) whose outer angles are all right angles. The five
associated right-angled triangles have the parts indicated
in the figure. The inner simple pentagon has the measure
of each side equal to that of the opposite exterior angle.
If we write down in cyclic order the parts X', ft', «, y', B
as they occur on the sides of the simple pentagon, the parts
of the five associated spherical triangles can be written
down by cyclic permutation of these letters, thus :
I. a ,
ft X',
y B.
2. /,
a ft,
jS'X'.
3. B ,
y o ,
X ft .
4. X',
£'y\
ft a .
5. fi,
X B ,
a' y'.
N.-E. O,
66 ELEMENTARY HYPERBOLIC GEOMETRY [a. 32 „.33l
Corresponding to these we get five associated rectilinear
triangles :
NAPIER'S RULES
67
1. e,
2.6',
3. I ,
4. m,
5. a',
X,
c'
b a,
V y,
b p.
f «'
m* y
a 0'
c' X
Hence, if we establish a relationship between the sides
and angles of one triangle, we can obtain fonr other relation-
ships by applying the same result to the associated triangles,
or by a cyclic permutation of the letters (I'm'ac'b) (Ima'cb')
(xy« v 73)(x MU v/3')-
32. Trigonometrical formulae for a right-angled
triangle.
Produce the hypotenuse AB to D so that the per-
pendicular at D to AD is parallel to AC, Then AD=l,
BD-l-c. Draw the horocyclic arcs with centre fi passing
through D and B. Then (by § 28 (a), (b) and (8).)
g 1+ s s =S tanh?, s 2 =S tanh (l~e),
s, =se _ " =S sinh a/cosh (J - c).
Therefore
tanh I =tanh (I - c) + sinh a cosh (I - c),
sinh a cosh I - sinh I cosh {I - c) - cosh I sinh (I - c) = sinh c.
sinha = sinhc sinX.
From the associated triangles we get
sinh c' =sinh b' sin fx ; therefore sinh b =sinhc sin w .
sinh b = sinh I emu
sinh/' =siuh»isiny
sinhm' =sinho' sin ,6'
Prom (3), (4) and (5) we get
(1), (4) aud (5)
(1), (2) and (3)
(1), (2) and (5)
(2), (3) and (4)
sinh b = tanh « cot X.
cosh c = cot X cot ju.
sinh a = tanh 6 cot.u.
cosh c = cosh a cosh b.
cos X - tanh b coth c.
cosX =cosha sin/*,
cos/* = cosh b sin\.
cos ft. = tanh a coth c. ( 10)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
33. Engel-Napier rules,
These ten formulae, which connect all the five parte of
the triangle in set* of three, are of exactly the same form
as the formulae of spherical trigonometry, with hyperbolic
Fro. 37.
fin, 38.
68 ELEMENTARY HYPERBOLIC GEOMETRY [it 33
functions of the aides instead of circular functions, and can
be written down by Napier's rules.' If we write the five
parts b, X, e, ft, a in cyclic order as they occur in the
triangle (Pig. 38), then
sine (middle part) = product of cosines of opposite parts
| A ] = product of tangents of adjacent parts
it being understood that the circular functions of the angles'
and the hyperbolic functions of the sides, are taken and
each function of x is the " complement " of the correspond-
in.-; function of s, i.e. cosh i =sinh c, tan X = cot X, etc.
| Xotb.— A lias the same meaning as X', but c is not the
same as c'.]
Tim rale may be put in another form, which fa more homogeneous.
II. Hi
NAPIER'S RULES
i><)
Fro. 99.
tS !" th0 fo ™« la * i» t«nn« of the segments a', *, c, - V.
The formulae become : t ,,»•»» .
coah c =sinh I sinh m =eoth n' eotli 6'
£l rffl£ *?r h t° Sl '° mc lF was kid h - v Lo'^evskv, XZ Found..
- r 7 *£""£?' o!m P- * H ma! «* use of the dianriims in tinflM
Kl? • ?2£ J?-" n, r ifieti forraa of «N*rt Rales W ™«oN 8h od L
kngel ■■> on edition of Lobachevsky's , VeM » JPowdWftw* p s« *
and four otber pairs obtained by tbe cyclic permutation (hna'cb')
or {a'fcinb'}.
If we write the five parts a', I, c, m, b' in cyclic order (Fig. 39),
then
cosh (middle part) = product of hyp. sines of adjacent parts
( B ) = product of hyp. cotangents of opposite
parts.
It is easily verified that this rule holds, with circular functions
instead of hyperbolic functions, for a spherical triangle in euclidean
space with hypotenuse e, aides I' and m' and opposite angles a'
and 6'.
34. Expressing the formulae in terms of «', X, y, m. /3'.
we get, since
cosh x = eosec fr", sinh x = cot £, coth a; = sec £,
where x stands for any one of the letters a, b, c, I, m, and
£ for the corresponding Greek letter a, j3, y, X, ft,
sin y = tan X tan ft = cos a' cos /3',
Fig. 40.
with four other pairs obtained by the cyclic permutation
(«'XyM|8'). They may be read off from Fig. 40 by applying
Rule (b) with circular functions.
But these are the formulae for a right-angled spherical
triangle with hypotenuse y', sides «' and £' and opposite
angles ft' and X' ; or one with hypotenuse a, sides ft and y
L (
70 ELEMENTARY HYPERBOLIC GEOMETRY [n. 35 n. 88] TRI-RECTANGULAR QUADRILATERAL 71
and opposite angles X' and £ But this is just the spherical
triangle which we found to correspond to the rectilinear
triangle (§ 30), Hence (he formulae for a spherical triangle in
kyperbohe space are exactly the same as those for a. spherical
triangle in eudUean sjxice, when we take as the measure
of a side the angle which it subtends at the centre, and as
the measure of on angle the dihedral angle between the
planes passing through the sides and the centre (Cf
Chap. III. § 21.) It may be noted that the letters M ', a , y\
P, V in Fig. 40 are the same, and in the same order, as those
on the sides of the simple pentagon in Kg, 36.
35, Correspondence between a right-angled triangle
and a tn-rectangular quadrilateral.
c£ua!%£ ° A ' md mXBA and " CA (Fig " 4J) " Tl,en A D ■*
a-li=Il{c+l).... -( ij
P-X=Yl(c+m) (j/j
Similarly
Draw Bfl II AC, and DQ±BA and | AC (Fig. 42). Then AD ~l,
££8=", and a+/*=II(e-I) <*'
Similarly + A=n (C - m ) «
\_o- -\C
FlO. 42.
Wote.— If l>c, then ir-(cn-/i) = II(c-J);
ifl=c «+/*=II(0)=-;
which are both contained in (2) if we understand that
Draw PQ1C4 and || A*, and MJ-iKJ and H iL4 (Fig. 43).
o.
KlO. JL
f
72 ELEMENTARY HYPERBOLIC GEOMETRY [., 35
Then AD = *, BB = m , MOa = U lm - a) , DCi^U^l), md
n{m-a] + Tl(l + b)^.
(3)
Similarly mm+a t + uil -(,)=- _ ( g,.
In the txi-rectangular quadrilateral with angle & and eide8 ffl order
n{OT+a)+n(f-6)=!T.
u c^ J
Re. 44.
*. «, v, ta, draw SB fl ZW; and X ^5 (Fig. 44). Then fi £ = u% for
ii (!t ) + n ( «')= J ^fl^ntn ^fl=H(* + «',, and
n(«,)+n((+ W 'j = ^. ,. fI
Similarly IJ(f)+n<» + ir> ft .... .' '
.Draw AH || CD, and £fl^fl and B 67) (Fig. 45). Then «^
Mil = !!(«,), B/lii -r IT<< -«'), and
JI(w)+e=IT{(-«') m
si,nilarI y n«) * e=ij( W -yj (lr>
/a 1 '™- 2^/1 MfcJPiS' and ™ P%iT«-
r^-u , and d /UP=/, fl=I7(/), and we have
H(©'-t»)+n(w+/)«=|. (I11)
Similarly n (y '-u) + n(r ( /) = 5. (m<)
11.35] TRI-RECTANGULAR QUADRILATERAL 73
Now the quadrilateral is determined by l and «. L*t (=c and
a=»i'. Then
II(e + i») = 0-n{w)=/3- A, from (I) and (1').
n(c-H») = 0+n(u>)=j3+A ( from (I') and (2')-
Therefore 0=fi and ri(w>)=A ortc=J and/=6.
A
Then, comparing (HI) and (3), we have
Jl{m-v)= z-Ii(f+b) = Tl{m-a)i therefore v=a.
74 ELEMENTARY HYPERBOLIC GEOMETRY [u. 36
Hence, to a right-angled triangle {c, a A, b ft) there corresponds a
tri-rectangular quadrilateral with angle and sides, starting from
the angle, c, m' t a, I By reversing the order of tho sides, we get
the quadrilateral (/?, I am' a), to which corresponds a triangle
(I, m'y, b a'), or {I, b a', m'y). Fig. 47.
Flo, 47.
If we take a', I, c, m, b' as quantities determining the parts a, A,
c, ft, b of the triangle, then we get a triangle corresponding to the
quantities b', a', I, c, m, and similarly, by cyclic perm illation, we get
five associated triangles. This forms an independent proof of the
result deduced in ;<3I from spherical triangles.
36. Wo can deduce from this correspondence that the relations
between the parts of a tri-rectangular quadrilateral can be written
down by rules exactly analogous to Napier's rules. If the angle is
O andthe sidesja order are o, m, I, b, write down in a circle tho
parts C, a, m, I, b. Then
sine (middle part) = product of cosines of opposite parts
= product of tangents of adjacent parts,
wiili the same understanding as in the ease of the triangle (§ 33 (a)).
If wo write the parts in the cyclic order C, I, a, b, m, we get rules
analogous to (b) at the end of § 33, viz. :
eos (middle part) = product of sines of adjacent parts
=product of cotangents of opposite parts.
37. The formulae for a general triangle can be obtained
from those for a right-angled triangle by dividing the
triangle into two right-angled trianglea (Pig. 48).
Thus, sinh p =sinh a sin B = sinh b sin A.
Hence sinh a = sinh ft _ sin lw
sin A sin B ™ sin C
a. 381
THE GENERAL TRIANGLE
75
Again, cosh c - cosh c, cosh c s + sinh c, sinh r a ,
cos C - cos C'j cos C. - sin C\ sin C« .
cosh a ■ cosh c, cosh p. cosh 6 - cosh c 2 cosh p,
sinh c t - sinh a sin C, , sinh c a = sinh b sin C t ,
cos (7, = coth a tanb p, cos C 2 =coth b tanh p.
c
Also
F10. 48.
Therefore cosh c =cosh a cosh b sech 3 ?)
+sinh a sinh & sin C x sin C' a
= cosh (t cosh b sech V + sinh a sinh 6
x (coth « coth b tanh 2 j) - cos C)
= cosh a cosh b - sinh « sinh b cos C.
Similarly
- cos C =cos A cos B - sin A sin B cosh c.
It is needless to write down other formulae, which may
be obtained from the corresponding formulae of spherical
trigonometry by putting cosh for cos and tsinh for sin,
when operating upon the sides, leaving the functions of the
angles unaltered.
38. The formulae of hyperbolic trigonometry become
those of euclidean plane trigonometry when the constant
.
76 ELEMENTARY HYPERBOLIC GEOMETRY j „. aa
To a first approximation
The formula
cosh I =cosh l coah I ~ -«nh ^sinh * cos
becomes 1 +1* J\ + I ?Vi + U ^ « & „
or <^=« !! +6 a -2a6co3C.
This shows that when we are dealing with a small region
M. small in comparison with k, the geometry is sensibly
the same as that of Euclid.
39. Circumference of a circle.
Let <fe be the length of the are PQ of a, circle of ^^ wU h
aubtends an angle rf£? at the centre. Then
ainJl 2j = ainh^8inl^ I
Honce the length of the whole circumference is faft sm |, *
Here, for the fiat time, we require to consider the actual value of r,
for the formula to, ^ 1, which is here warned, fa true only
nRTSKirSS th0 tangent at p - *- «- •» «-*
* = £#sinh r ,
1/ ^
sink • - sinh - sin ft
ft k '
sinli = tanh 7 cot ft
„.4i>] CIRCUMFERENCES OF A CIRCLE 77
Let the centre go ta infinity, so that the circle becomes a horo-
cvole. Then r-+ oo , 6~*0, and
tan/'
Comparing these with the formulae in 6; 28, we find S -h.
P
Jbsiiih? = *.^--»-* and fctanh 7 = .s. —£—-*«.
ho *
!"!•• r.'.
40. Sum of the angles and area of a triangle.
Join MN 7 the middle points of AB, AC, and construct
the equidistant-curve with MN as axis, which passes through
FIO. 50.
B, C and A. Then the perpendiculars AA', BB', CC to
MN ate all equal, and <LB'BM =MAA', LC'CN =NAA\
78 ELEMENTARY HYPERBOLIC GEOMETRY [ir. 40
Denote by ABE the angle which AB makes with the
tangent to the equidistant-curve at B ; the angle B'BE is
a right angle. Then
ABAC + ABE + ACE =B'BM + MBE + C'CN + NCE = r .
Hence the sum of the angles of the triangle ABC
= - - 2CBE. The difference r-{A+B + C)n called the
defect of the triangle.
Again, the area of the triangle ABC
- BMNC + MA A' +NAA' =B'BCC.
Hence all triangles with base BC and vertex on the other
branch of the equidistant-curve which passes through
B, C and A have the same area and the same angle-sum
or defect.
Now, if we are given any two triangles, we can transform
one of them into another of the aame area and defect, and
having one of its sides equal to one of the sides of the other
triangle.
Let ABC, DEF be, the two triangles, and let DF be the
greatest of the six sides. Construct an equidistant-curve
passing through B, C and A. With centre C and radius
equal to DF, draw a circle cutting the branch of the equi-
distant-curve on which A ties in A'. Then the triangle
A'BC has the same area and defect as the triangle ABC,
and has the side A'C equal to DF.
Again, if the perpendicular bisector of the base BC of
a triangle ABC meets the other branch of the equidistant-
curve in A', the isosceles triangle A'BC has the same area
and defect as the triangle ABC.
Hence, if two triangles have the same area they can be
transformed into the same isosceles triangle, and have
therefore the same defect, and conversely.
n. 41]
AREA AND ANGLE-SUM
79
Now, let a triangle ABC with area A and defect S be
divided into two triangles ABD, ADC with areas ^ and A 2
and defects Si and S t ■
Then S^tt-BAD-B-ADB,
& % = Tr-DAC-C-ADC.
Therefore &i + & t =%*--A-B-C -■* = ■* -A-B-C = &,
and A! + A a = A.
If Ai = A a , then S t =S, and A=2A 15 5 = 2^.
Hence the defect is proportional to the area, or
± = \{t-A-B-C).
The value of this constant \ depends upon the unite of
angle and area which are employed ; but when these have
been chosen it is given absolutely.
41, Relation between the units of length and area.
In euclidean geometry the units of length and area are
immediately connected by taking as the unit of area the
area of a square whose side is the unit of length. In fact
the relationship is so obvious that there is constant con-
fusion, though we are not always aware of it, between the
area of a rectangle and the product of two numbers. Thus
modern treatment has tended to confuse the theorems of
the second book of Euclid, which are purely geometrical
theorems relating to areas of squares and rectangles, with
algebraic theorems relating to "squares" and products
of numbers. The expression " product of two lines " has
no meaning until we frame a suitable definition consistent
with the rest of the subj ect-mattcr. The area of a rectangle
is not equal to the product of its sides, but the number of
units of area in the area of a rectangle is equal to the product
of the numbers of units of length in its sides.
80 ELEMENTARY HYPERBOLIC GEOMETRY [ir. 41 n.42]
AREAS
81
It would take us too far out of our way to examine
completely the notion of area. We shall simply take
advantage of the fact, that when we are dealing with a very
small region of the plane we can apply euclidean geometry.
Thus, while there exists no such thing as a euclidean square
in non-euclidean geometry, if we take a regular quadri-
lateral 1 with all its sides very small we may take as its
area the square of the number of units of length in its
sides ; or, more accurately, the units of length and area are
so adjusted that the ratio of the area of a regular quadri-
lateral to the square of the number of units of length in its
side tends to the limit unity as the sides are indefinitely
diminished.
I*t us apply tliis to find the area of a sector of a circle POQ, the
angle POQ --- $ being very small.
Fig. si.
Produce OP, OQ to P' and Q\ lMOP = OQ = r PP'-QO'- d*
Then * ~ w ~ ar
area of PQQ'P> = dr .PQ = k$drwDh -.
k
Hence the area of the sector = Jf tf/coah - - 1)
= 2k i $mnh t —,
al-
and the area of the whole circle is 4irjt» sinh* -1
w
^A regular polygon k one which has all ita sides equal and all ita angles
We can apply this now to find the area of a triangle by another
method. It is sufficient to take a triangle ABC with a right angle
at C. Divide it into small sectors by lines drawn through A. Then
the area ifl given by
fW4fcoah|-l\
Express c in terms of A and the constant 6, write tanh ^=(, and
put cosM =y, and we get, after some reductions,
u c ia ~ d y ~ d,J
cosh , 4A = ■ - — _ = — ... , - '
The integral of this term, from y= 1 to y =cosM, is
1 ,2y-(l+C) 1
BOS'
'(l- 2 cosh* .sinMJ
1-0 2
= | cos-' (1 - 2 coa?/f ) = fcir " 2B).
Hence the area of the triangle
■^B-4)**(*-f^-Jl)
42. It appears then that, when the angles are measured
in " circular " measure, the constant \ =F, and the formula
for the area of a triangle becomes
/± = k*(ir-A-B-C).
As the area of a triangle increases the sum of the angles
diminishes, but, so long as the vertices are real, the angles
are positive quantities ; the area cannot therefore exceed
v&. This is therefore the maximum limit to the area of a
triangle whcu its angles all tend to zero. A triangle of
maximum area has all its vertices at infinity and its sides
are parallel in pairs.
n.-b. a. P
82 ELEMENTARY HYPERBOLIC GEOMETRY [n. 48
43. On account of its neatness, wo add the proof that Causa
gave of the formula for the area of a triangle, in a letter' to
W. Botyai acknowledging the receipt of the " Appendix."
Gauss starts by assuming that the area enclosed by a straight line
and two tinea through a [joint parallel to it is finite, and a certain
function /(jt-.£) or the angle between the two parallels ; and
further, that the area of a triangle whoso vertices are alt at iiiliiiily
is a certain finite quantity t
Then wo have, from Fig, 52, /(jr-jp)+/(a>)=f.
i'lti, 62.
Again, from Pig. 53, f(<f>) +/( $ +/( 5 r - <p - $ =(,
Flo. 53.
Whence /('/') = A&
where k is a constant, and therefore (=Air.
1 Cth March, 1832 (Gauss' Werke, viii. 221).
n.441 GAUSS* PROOF Of ANGLE-SUM 83
Now, by producing the sides of any triangle with angles ", /J, y,
and drawing parallels, we have ( l-'ig. Si)
Therefore
Fltt. 54.
*=/(«)+/(#+/(y)+A-
A = A(7T-U-/3-7).
44. Area of a polygon.
The area of a polygon can be found by breaking it up into
triangles. By joining one vertex to each of the others,
we divide an R-gon into n-2 triangles. The sum of the
angles of the rt-gon is equal to the sum of the angles of &£
n-2 triangles.
Let Aj, A s ,... be the areas, and S lt S 2 , ... the defects
of these triangles ; then, if S is the sum of the angles and
A the area of the polygon,
A = £A -EW =h*(n~2 .tt-S).
If S' is the sum of the exterior angles, flf +#«-**;
therefore
which is independent of n.
84 ELEMENTARY HYPERBOLIC GEOMETRY [a 45
45. Wo add here another proof of the result that tlte geometry of
ionoydea on the horosphere w the same as the geometry of Mraiijht
lines on the eudidean plane.
Let the three parallel lines iu apace Ail, J5fi, Cfl be cut by a
Ex. iu]
EXAMPLES
FI9. 65.
horosphere with centre fl in A, B, 0, and make AA'=BB'=CC,
so that A'B'C lie again on a horosphere with centre il. [See Ex. 8.)
Let the dihedral angles between the planes BCil, CAB, ABU
be a, B, y, and let the angles of the rectilinear triangle A'B'C be
<*', 8', y', and its area A.
Then, &S ,1.1' increase, (be angles ilA'B', RA'C, etc., all tend to
right angles ; hence a', B', y" tend to the values a, B, y. Also
A -0.
NowA-P(r-«'-j3'-/); hence a + Bi 7=3-, i.e. when tone
planes intersect in pail* in three parallel lines the sum of the dihedral
angles ia equal to two right angles. Heme 1 In- sum of the atigli>« nf
a geodesic triangle on the horosphere ia equal to two right angles.
EXAMPLES n.
1. Prove that the four axes of the circuineircles of a triangle form
a complete quadrilateral whose diagonal triangle is the given triangle ;
ami state the reciprocal theorem.
2. Tf a simple quadrilateral is inscribed in a circle, horoeycle or
one branch of an equidistant-curve, prove that the sum of one pair
of opposite angles is equal to the sum of the other pair of opposite
angles. Show that this holds also for a crossed quadrilateral if
the angles are measured always in the same sense, and for a quadri-
lateral whose vertices are distributed between the two branches of
an equidistant-curve if the angles on opposite branches are reckoned
of opposite sign.
3 If a simple quadrilateral is circumscribed about a circle, prove
that the sum of one pair of opiwsite sides is equal to the sum of the
other pair of opposite aides. Examine the case of a crossed quadri-
lateral circumscribed about a circle, equidistant-curve or horoeycle.
4. If a is the chord of an arc a of a horoeycle, prove that
o = 2fc sinh $a/k.
5. If 6 is the angle which the chord of a horoeycle makes with the
tangent at either end, and a is the arc, prove that a = 2k tan ft
6 If * is the angle which the tangent at one extremity of an arc a
of a horoeycle makes with the radius through the other extremity,
prove that a -k cos ft
7. Prove that the arc of an equidistant- curve of distance a,
corresponding to a segment* on its axis, is x eosh ajk.
8 HA B are corresponding points on the parallels A A', BB',
and A, C are corresponding points on the parallels AA',CC\ prove
that B, C are corresponding points on the parallels BS , CO .
9 Prove the following construction for the parallel from to
AM. Draw ON±NA. Take any point A on AM, draw OB J.OA
and ABJ.OB. With centre and radius equal to AM, draw a
circle cutting AB in P. Then OP |l AM.
10. Prove that the radius of the inscribed circle of a triangle of
maximum area is U'foff.3.
11. In a quadrilateral of maximum area, if 2a, 26 are the lengths
of the common perpondiculars of opposite sides, prove that
sinh -sinh 7 = 1.
k k
12. A regular quadrilateral is symmetrically inscribed in a regular
maximum quadrilateral ; prove that each of its angles is cos «fc
13. If the three escribed circles of a triangle are all horocyctcs,
prove that each side of the triangle is cosh""?, and that the radius
of the inscribed circle is tanh"M. «»" «» radiua o£ the crcumcircle
is tanh _1 J (fc being unity).
86 ELEMENTARY HYPERBOLIC GEOMETRY [Ex. u.
14. In euclidcan geometry prove that any convex quadrilateral
can by repetition of itself bo made to cover the whole plane without
overlapping.
15. In hyperbolic geometry prove that any convex polygon with
an even number of sides can by repetition of itself be made to cover
the whole plane without overlapping, provided the sum of its angles
is equal to or a submultiple of four right, singles. Show that the
same is possible if the number of sides is odd, provided the sum of the
angles is equal to or a submultiple of two right angles.
16. If a is the side and a the angle of a regular «-gon, prove that
_ v .a a
cos -= sm ^ cosh — .
n 2 21:
17. If r is the radius of the inscribed circle, if that of the circum-
scribed circle of a regular n-gon with side a and angle a, prove that
sin li ■ a cot - tanh ~, and cosh = - = cot - cot -.
fr n. 2k k n 2
18. A regular network is formed of regular a-gons, p at each point.
Show that the area of each polygon is l?ir{2nlp - n 4 2),
18. A semiregular network is formed of triangles and hexagons
with the same length of side, three of each being a t each point.
Prove that the length of the side is 2k cosh" 1 Vi(4+V3).
20. A -wiiii regular network consists of regular polygons all with
the sa-rae length of side. At each vertex there arc p, %-gons,
Pt »a-gon s . p, n a -gons, etc If each %-gon has area A,, prove that
Kx. n.i
EXAMPLES
87
^*[<>**)-%}
21. If a ring of « equal circles can be placed round an equal
circle, each one touching the central circle and two adjacent ones,
prove that the radius of each eirele is given by 2 cosh r sin - = 1
k n
22. Prove that the area included between an arc of an cquidistant-
eurve of distance a, its axis, and two ordinates at distance z is
23. AA' ;; BR' and they make equal angles with AB, ACX. BB'
awl AD LAG. If the angle A'AD=z, prove that the area of the
circle whose radius is AB is equal to ?rA? tan's. (J. Bolyai.)
24. Prove that the volume of a sphere of radius r is
/ 2r r\
25. These parallel lines AQ, BB, CQ are cut by two horocycles
with centre SI in A, B, C and A', B', C. Prove that the arcs
AB:BC=A'B' BC.
23. AtB r , A t B„ AtB s arc arcs of concentric horocycles as in
Fig. 29, and AiA t =A,A^ Prove that j4,B, : A ,B 1 =j4, J B t : A t B t ,
Hence show that the ratio A t Bi : A t B t depends only on the length
27. Prove that the sides of a pentagon whose angles are all right
angles are connected by the relations
cosh (middle side) = product of hyp. cots, of adjacent sides
= product of hyp. sines of opposite sides.
28. A simple spherical pentagon, each of whoso vertices is the pole
of the opposite side, is projected from the centre upon any plane.
IYove that the projection is a pentagon whose altitudes are con-
current ; and that the product of the hyp. tangents of the segments
into which each altitude is divided is the same.
fat) sinh
J>
CHAPTER III.
ELLIPTIC GEOMETRY.
1. The hypothesis of elliptic geometry is that the straight
line, instead of being of infinite length, is closed and of
finite length. Two straight lines in the same plane will
always meet, even when they are both perpendicular to
a third straight. line.
Let I, m, n, be three straight lines drawn perpendicular to
another straight line a at the points L, M, N. Let m, n
meet in A ; n, I in B ; and l t m in 0,
"When LB is produced it will meet a again either in L or
in some other point. Let V be the first point in which
it again meets a.
Then, from isosceles triangles, we have BL = BN =BL'
CL=CM=CU. Hence B and C are both the middle
m .2l ABSOLUTE POLE AXD POLAR 89
point of the segment LBV, and must therefore coincide.
In the same way A, B and all coincide.
Hence all the perpendiculars to a given line a on one side
of it meet in a point A, and A is equidistant from all points
on the line. The point A is called the absolute pok of the
line a, and a is called the absolute pohr of A. If P is any
point on a, the distance AP is called a quadrant, and A is
said to be orthogonal to P, or A and P are called absolute
conjugate points.
2. The perpendiculars drawn in the other sense will
similarly meet in a point A'.
The question arises : are A and A' distinct points ?
On the hypothesis that A and A' are distinct points, two
straight lines have two points in common. It could be
proved that in this case any two straight lines would
intersect in a pair of points distant from one another two
quadrants. A consistent system of geometry results, which
is exactly like the geometry on a sphere, straight lines being
represented by great circles, and is therefore called Spheri-
cal Geometry. The two points of intersection of two lines
:iig called antipodal points. Two points determine a line
uniquely except when they are antipodal points ; a pair
of antipodal points determine a whole pencil of lines.
On the hypothesis that A and A' are one and the same
point, two straight lines always cut in just one point, and
two distinct points uniquely determine a line. This gives
again a consistent system of geometry, which is called
Elliptic Geometry. 1
1 Somotimea both of these systems are called Elliptic geometry, and
ttey are distinguished as the Antipodal or Double form aud (ho Polar
or Single form. We shall, however, keep the term Elliptic geometry
for the' latter form.
90
ELLIPTIC GEOMETRY
fiu. a
rn. 31
GEOMETRY OF A BUNDLE
01
While spherical geometry admits more readily of being
realised by means of tbe sphere, elliptic geometry is by
far the more symmetrical, and our attention will be confined
entirely to this type. Elliptic geometry has also the advan-
tage that it more nearly resembles euclidean geometry,
since in euclidean geometry all the perpendiculars to a
straight line in a plane have to be regarded as passing
through one point (at infinity).
Another mode of representation of these two geometries
exists, which exhibits them both with equal clearness.
Consider a bundle of straight lines and planes through a
point 0. If we call a straight line of the bundle a " point,"
and a plane of the bundle a " line," we have the following
theorems with their translations. (Cf. Chap. II. § 24.)
Two lines through Two "points" uniquely
uniquely determine a - plaue determine a " line."
through 0.
Two planes through
intersect always in a single
line through 0.
All the planes through
perpendicular to a given
plane a through pass
through a fixed line a through
Two " lines " intersect
always in a single " point."
All the " lines " perpendi-
cular to a given " line " a
pass through a fixed " point"
A, which is orthogonal to
every " point " lying in a.
0, which is orthogonal to
every line through lyiug
in a.
Hence elliptic geometry can be represented by the
geometry of a bundle of lines and planes. In the same way
spherical geometry can be represented by the geometry
of a bundle of rays (or half-lines) and half-planes. Two
rays which together form one and the same straight line
represent a pair of antipodal points.
In elliptic geometry all straight lines are of the same
finite length 2q, equal to two quadrants.
If we extend these considerations to three dimensions,
all tbe perpendiculars to a plane « pass through a point A,
the absolute pole of a, and the locus of points a quadrant
distant from a point A is a plane a, the absolute polar of A.
3. The plane in elliptic geometry, or, as we may call it,
the elliptic, plane, differs in an important particular from
the euclidean ot hyperbolic plane. It is not divided by a
straight line into two distinct regions.
Imagine a set of three rectangular lines Oxyz with Oif
on the°line AM and Oz always cutting the fixed line AP.
Fig. 57.
As moves along AM it will return to A, but now Oz is
turned downwards and Ox points to the left instead of to
the right. The point z has thus moved in the plane PAM
and come to the other side of the line AM without actually
crossing it.
A concrete illustration of this peculiarity is afforded by
what i» called Menus' sheet, which consists of a band of
paper half twisted and with its ends joined. A line traced
92
ELLIPTIC GEOMETRY
[in.41 m . 5 ] ABSOLUTE POLAR SYSTEM
93
along the centre of the band will return to its starting
point, but on the opposite surface of the sheet. The two
The elliptic
Fra, 68.
sides of the sheet are continuously connected,
plane is therefore a one-sided surface.
If we carry out the same procedure for tlio euclideau plane, we
shall obtain exactly similar results, with the exception that a point
passes through infinity in going from one side of the Line to the other.
This ia well illustrated by the ease of a curve which runs along an
asymptote. Ordinarily the curve lies on opposite sides of the asymp-
tote at the two ends, and thus appears to cross the asymptote.
When it does actually cross the asymptote at infinity it has a point
of inflexion there and lies on the same side of the asymptote at each
end.
4. Absolute polar system.
To every point in space corresponds a plane, and vice
versa, which are absolute pole and polar.
If the polar of a point A passes through B, the polar of B
will pass through A, because the distance AR is a quadrant.
Let A, B be two points on a line I ; the potars of A and B
intersect in a line I'. Let A' and B' be any two points
on l' ■ then the polar of A' passes through both A and B.
Hence the polars of all points on the line V pass through the
Hue I If P is any point on I, its polar will pass through
A' and B'. Therefore the polars of all points on the line I
pass through the line I'.
To every line I, therefore, there corresopnds a line I , the
absolute polar of I, such that the polar of any point on I
passes through V and vice versa. All points of l' are a
quadrant distant from all points of I, and every line which
meets both I and I' cuts them at. right angles.
If we confine ourselves to a plane, to every point m the
plane corresponds a line in the plane and vice versa.
These relations arc exactly the same as those that we
get in ordinary geometry by taking poles and polars with
regard to a conic in a plane, or a surface of the second
degree in space. The points on the conic or quadnc
surface have the property that they lie on their polars ; the
polar is a tangent to the conic or quadric and the pole is the
point of contact.
5 Projective geometry.
These relations of polarity with regard to a conic belong
to pure projective geometry, and have nothing whatever to
do with actual measurement, distances or angles. All the
theorems of projective geometry can be at once transferred
to non-cuclidean geometry, for, so long as we are not
dealing with actual metrical relations, non-euchdean
geometry is in no way whatever distinguished from
euclideau. Pure projective geometry tabes no notice of
points at infinity, for infinity bare implies infinite distance,
and is therefore irrelevant to the subject. It has there-
fore nothing to do with parallel lines.
94
ELLIPTIC GEOMETRY
fin. 5
m. si
PROJECTIVE GEOMETRY
95
l.'siinriinwtclv most Ba^irii best-hooka on projective
geometry start by assuming euclidean metrical geometry.
A harmonic range is defined in terms of tlic ratios of seg-
ments, and a conic is obtained as the " projection " of a
circle. This treatment unnecessarily limits the generality
of projective geometry, and attaches a quite unmerited
importance to euclidean metric.
Tho use of analytical geometry might [» thought to supply a
means for a general treatment, for tho algebraic relations brtwoa)
numbers which express the relations of projective geometry are just
theorems of arithmetic, and these may be applied to any subject
matter which cou be subjected to numerical treatment, whether that
subject matter is euclidean or non-euclidean geometry. But ihc
difficulty in applying this procedure is that the subject matter must
first be prepared for numerical treatment. This mean* either
postponing the introduction of projective geometry until metrical
geometry, with a system of coordinates, has been established. 1
which is just the fault we wish to avoid, or the establishment of a
system of projective coordinates independent of distance. In
cither case we have to assume much more than is really necessary.
For convenience of reference we shall give a summary nf
the theorems of projective geometry which we shall require,
assuming that proofs of these are available which do not
involve metrical geometry. (Reference may be made to
Reye, Geometry of Position, Part I., translated by Holgate,
New York, 1898, or Veblen and Young, Projective Geometry,
Vol. L, Boston, 1910.)
If two ranges of points arc made to correspond in such a
way that to every point P on the one range corresponds
uniquely a point P' on the other, and vice verm, the ranges
are said to be homograph r c
Notation. {P)-k{P'\.
The simplest way of obtaining a range which is homo-
1 Of. Chap. IV. §21.
graphic with a given range is as follows. Take any point 0,
not on the axis of the range : join to the points of the
range, and cut these rays by any transversal. The range
on this transversal is called the projection of the first range
and is nomographic with it. In this special position, in
which the lines joining pairs of corresponding points are
concurrent, the ranges arc said to be in perspective, with
centre 0.
Notation. {P}*o{P'} or {P}a(P'}.
It can be proved that two homographic ranges can
always be connected by a finite number of projections,
and in fact this number can in general be reduced to two.
It can be proved that
(ABCD)-x(BADC)-x(CDAB)-{I)CBA),
but in general the four points are projective in no other
order.
Properties which are unaltered by projection are called
projective properties. Thus, points which are collinear.
or lines which are concurrent, retain these properties after
projection.
A harmonic range is projected into a harmonic range.
We cannot define a harmonic range in terms of the ratios
of segments, because a segment is not projective. We
define a harmonic range thus ; Let X, Y be two given
points on a line, and P a third point. (See Fig. 85, Chap.
IV.) Through P draw any line PST t and on it take any
two poiuts 8, T. Join S and T to X and F ; let SX cut
TY in F, and SY cut TX in V. Join UY, and let it cut
XY in Q. Q is called the harmonic conjugate of P with
regard to X and Y. This construction can be proved to
be unique ; P, Q are distinct, and are separated by and
96
ELLIPTIC GEOMETRY
pit. 5
separate X and Y. If (XY, PQ) is a harmonic range
(XY, PQ)-x(XY, QP).
If we start with three points on a line, we can derive an
indefinite number of other points by the above quadrilateral
construction, and in fact we can find a new point lying
between any two given points. AH the points derived in
this way form a net of rationality. They do not give all the
points on the line. To secure this we would require an
assumption of continuity.
If three points A, B, C of one line are projected on to
three points A', B', C" of another line, the correspondence
between all the points of the two ranges is determined.
This is the fundamental, theorem of projective geometry.
Two homographic ranges can exist on the same line.
If three points A, B, C are self corresponding, it follows by
the fundamental theorem that all the points are self-
corresponding. Hence two homographic ranges oa the
same line cannot have more than two self-corresponding
points.
That it is possible in certain cases to have two self-
corresponding points is shown in Fig. 59. I is the given
line, l t an intermediate line on which a range of points {P}
is projected from centre 5, , and £„ is a second centre of
projection from which the projected range {P,} is projected
on to I.
In this way P' corresponds to P. Let Z, cut I in Y, and
let SjSg cut Z in X. Then X and Y are self-corresponding
points. If Z, passes through X, the two self-corresponding
points will coincide.
If \P) and {P'\ are two homographic ranges on the same
line, such that to P corresponds P\ in general to P' will
correspond another point P". If P" coincides with P, the
»]
INVOLUTIONS
97
points of the line are connected in pairs and are said to
form an involution. If D x and D., are the double or self-
corresponding points of an involution, and X, X' axe a pair
of corresponding points, (D^JCX^^iD^JC'X), so that
(DjZ>aAX) is a harmonic range. If two real self-corre-
sponding points do not exist, we introduce by definition
conjugate pairs of " imaginary " points, much in the
same way as ideal points were introduced into hyperbolic
geometry.
When the double points are real the involution is said to
be hyperbolic, and when they arc imaginary it is said to be
elliptic. If the double points coincide, the conjugate of
any point P coincides with D, and the involution is said to
be parabolic.
If ( p} and I p' | are two homographic pencils with different
vertices, the locus of the points of intersection of corre-
sponding lines is a curve with the property that any line
cuts it in two points, real, coincident or imaginary. A line /
N.-E. G.
98
ELLIPTIC GEOMETRY
| in. <■.
in. ol
THE ABSOLUTE
cuts the two pencils in homograpiiic ranges, and the self-
corresponding points of these ranges are points on the locus.
FlO. 60.
This curve is called a point-conic ; it is the general curve
of the second degree, characterised by the property that
any line cuts it in two points.
Similarly the envelope of the lines joining pairs of
corresponding points on two homograpiiic ranges is a curve
of the second class, or line-conic, characterised by the
property that from any pomt two tangents can be drawn
to it.
It can be proved that a point-conic is also a line-conic, and
vice versa. The term conic can then be applied to either.
6. The absolute.
Let us return now to the absolute polar system in a plane.
We shall prove the theorem : In every polar system in a
plane which has Ike reciprocal properly (km " if the polar of
a point A passes through B, (he polar of B passes through A,"
tliere is afmtl conic, the locus of points or ike envelope of lines
which an inchlent with their pohirs.
Consider a line I, The polar of a point P on I cuts I in
;i point P' t and the polar of P' passes through P. Hence
the points of I are connected in pairs and form an involution
whose double points arc incident with their polars. Every
line therefore cuts the locus in two points, and the locus
ia a point-conic. Similarly the envelope is a line-conic.
If I cuts the locus in P and Q, the polars of P and Q are lines
of the One-conic. Further, the polar of P does not cut the
locus in any second point, since the polar of any point
upon it passes through P ; hence the polar of P is a tangent
to the point-conic, and the point- and line-couics form one
and the same conic.
Similarly, in three dimensions a polar system determines
a surface of the second degree or quadric surface.
Applying this theorem to the absolute polar system, we
mid a conic in a plane or a quadric surface in space which is
given absolutely. But as a real point cannot lie on its
polar, since it is at the fixed distance of a quadrant from
any point of it, this conic or quadric can have no real points.
This imaginary conic, or in space the imaginary quadric
surface, is called the Absolute.
Let P, P' and Q. Q' be two pairs of conjugate points
on a line g, so that PP' = QQ' = a quadrant. Therefore
Fro. 61.
PQ ~ P'Q'. Let g cut the absolute in X and Y ; then
P, P' and Q, Q' are harmonic conjugates with regard to
X and Y. Let Q coincide with X ; then Q' will also
coincide with X, and the equation PQ^P'Q' becomes
PX=FX = PX-PP'.
100
ELLIPTIC GEOMETRY
[in. 7
Therefore 11- PP'/PX, Therefore PX must be infinite.
Every point on the absolute is therefore at an infinite
distance from any real point, and the absolute is, like the
real conic in hyperbolic geometry, the locus of points at
infinity.
7. Principle of duality.
The polar system with regard to the absolute conic
establishes the principle of duality, In euclidean geometry
t he principle of duality holds so long as we are dealing with,
purely descriptive properties, i.e. it holds in projective
geometry, which is independent of any hypothesis regarding
parallel lines, but it has only a very limited range in metrical
geometry, and is often applied more as a principle of analogy
than as a scientific principle with a logical foundation.
Thus, four circles can be drawn to touch three given
lines, but only one circle can be drawn to pass through three
points. A circle is the locus of a point which is always
at a fixed distance from a given point, but we cannot con-
sider it also as the envelope of a line which makes a constant
ainilc with a fixed straight line.
In hyperbolic geometry, when we consider equidistant
curves as circles, we find it true that four circles are deter-
mined by three points ; and if we introduce freely points
at infinity and ideal points, we can make the principle of
duality fit fairly well.
In elliptic geometry, however, the principle of duality
has its widest field of validity, and extends to the whole
of metrical geometry. The reason for this is found in t lie
nature of the absolute and the measure of distance and
angle. In a pencil of lines with vertex there are always
two absolute hues, the tangents from to the absolute,
in. 8]
PRINCIPLE OF DUALITY
101
and in all three geometries these two lines are conjugate
imaginaries. They form the double lines of the elliptic
involution of pairs of conjugate or rectangular lines through
0. In a range of points on a line I there are similarly two
absolute points, the points of intersection of I with the
absolute. They form the double points of the involution
of pairs of conjugate points with regard to the absolute.
But in hyperbolic, elliptic and euclidean geometry this
involution is respectively hyperbolic, elliptic and parabolic.
Thus it is only in elliptic geometry that the involution on a
line is of the same nature as that in a pencil.
8. As a consequence of this, in elliptic geometry the
distance between heo points is proportional to the ajujle
between their absolute polars.
Consider two lines OP, OQ. Let P', Q' be the poles of
OP and OQ. Then P'Q' is the polar of 0. PP' = QQ'=q.
Now distances measured along PQ are proportional to the
angles at 0.
Therefore ^ = / - />0 ^ a „d P'Q' = PQ.
Therefore the distance d between the poles of the lines is
102 ELLIPTIC GEOMETRY [m.a
connected with the angle a between the lines by the relation
a= —a,
7T
and if the unit of distance is such that ? = T, then d = a.
Here we must observe that two points have two distances,
viz. d and 2q - d ; two lines have two angles, q and «■ - «.
In the above relation we have made the smaller distance
correspond to the smaller angle.
Consider, however, a triangle ABC, in which we shall
suppose each of the sides < q, and each of the angles < ^.
33
FlO. 6S.
The absolute polar figure is another triangle A'B'C, in
which B'C is the polar of A, etc. Let AB, AC meet B'C
in M and N, Then
and
B'C
IT IT
III. fl
AREA OF A TRIANGLE
103
To the angle j4, <\, corresponds therefore the segment
To a segment d corresponds an angle '(2q-d), and to
an angle a corresponds a segment — (it - «). if the segment
d>d', the corresponding angle a<a'.
Tn applying the principle of duality, therefore, we must
interchange point and line, segment and angle, greater and
less.
9. Area of a triangle.
Two lines enclose an area proportional to the angle
between them, = 2#vi, say, where k is a linear constant,
MB fit
and the area of the whole plane is 2A 2 tt. In Fig. 64 the
areas enclosed by the angles of the triangle are shaded, and
these areas cover the area of the triangle three times, and
the rest of the plane only once.
We have, therefore, 2k a -{A +B+C) -2ftV +2A,
whence A=k*(A+B + C -x).
104
ELLIPTIC GEOMETRY
firt. 10
The area of a triangle is therefore proportional to the
excess of the sum of its angles over two right angles. If
A u B u C l are the exterior angles,
A=&(2tt-A 1 -Bi-C 1 ).
The absolute polar of the triangle ABC is a triangle
A'B'V with sides a', b', c' = ?$■ {-w - A), etc.
Hence
A=k*l(4q-a'-b'-c%
at
or the perimeter of a triangle falls sftort of Aq by m amount
■proportional to lite area of (lie polar triangle.
These results hold also for the sum of the exterior angles
and the perimeter of any simple polygon.
10. The circle.
A circle is the locus of points equidistant from a fixed
point, the centre, and by the principle of duality it is also
i in I-..-.,
the envelope of lines which make a constant angle with a
fixed line, the axis.
in
ill THE CIRCLE AND THE SPHERE 105
Let C be the centre and c the polar of C. Let P be any
point on the circle, and draw the tangent PT. Then
CMXTM and also J. TP. Therefore T is the pole of CP.
PM =q-r=-u; therefore u is a constant angle = | - 1 r.
Further, since PM = q -f, the circle is an equidistant-curve
with c as axis. Just as in hyperbolic geometry, the circle
or equidistant-curve lies symmetrically on both sides of
the axis, but the two branches are continuously connected.
In elliptic geometry, therefore, equidistant-curves arc
proper circles. When the radius of a circle is a quadrant
the circle becomes a double straight line, the axis taken
twice.
11. In three dimensions the surface equidistant from a
plane is a proper sphere.
A remarkable surface exists which is equidistant from a
line. With this property it resembles a cylinder in ordinary
space. A section by a plane perpendicular to the axis is
a circle. A section by a plane through the axis is an
equidistant-curve to the axis, but this is also a circle, and
the surface can be generated by revolving a circle about its
axis. It thus also resembles an anchor ring (Fig. 66). ^ But
sections perpendicular to the axis do not cut it in pairs of
circles, but only in single circles, and so it also resembles a
hyperboloid of' one sheet. Every point is at a distance d
from the axis I, and is therefore at a distance q - d from the
line P, the absolute conjugate of L The surface has there-
fore two conjugate axes, and can be generated by the
revolution of a circle about either of these. It is therefore
a double surface of revolution. It is a surface of the second
degree, since a straight line cuts it in two points.
106
ELLIPTIC GEOMETRY
frrr. 12
From its resemblance to a hyperboloid, the existence of
rectilinear generators is suggested. If it does possess
rectilinear generators, these lines must be everywhere
Fig. SO.'
equidistant from either axis. We shall therefore investigate
tlie existence of such lines, and return in § 17 to a description
of this surface {Cliffords surface).
12. Common perpendicular to two lines in space.
Consider two lines a, b not in the same plane. Let a'
and b' be their absolute polars. Any line which cute both
a and a' is perpendicular to both; hence any line which
,i! TU t P io . turc of Clifford"* ™rfacc will bo best understood after readme
Limp, V. In the conformal representation of nnn-cuelidcan Konnu-iry
in euciidean space, planes and spheres ore all represented by spheres
straight lines and circles by circles. Clifford's surface is represented by
an anchor-ring, one axis Wing represented by the axis of the ring, the
other axis being represented by a line at infinity. The circular sections
of the surface by plane* through an axis (which arc lines of oun
are represented by the meridians and parallels of the anchor-ring (which
arc also lines of curvature). The rectilinear generators are represented
by the l.itariLTiir oircular sections of the ring. The two systems of these
lnst-mentioncd eireles arc depicted in the figure. Thev inierseet at a-
constant angle.
m. 121
LINES IN SPACE
107
meets the four lines a, b, a\ b' cuts them all at right angles.
Now, three of these lines a, 6, a' determine a ruled surface
of the second degree, and the fourth line b' cuts this surface
in two points P, Q. The two generators p, q of the opposite
system through P and Q are common transversals of the
four lines a, a', b, ¥, and therefore cut all four at right
iWgks, The two lines a, b have therefore two common
perpendiculars. The two common perpendiculars p, q are
absolute polars, For. since O, a' and 6, b' cut p and q at
right angles, they also cut the polars p' and q', but they
have only two common transversals; therefore p' must
coincide with q, and q' with p.
Of the two common perpendiculars one is a minimum
and the other a maximum perpendicular from one line on
PlO. 67.
the other. Take any point P t on a and draw P,Q 2 1&,
Q^P.La, and so on. Then P t Q»>Q s Pt>P*Q a > - > s0
that" the perpendiculars form a decreasing sequence which
must tend to a finite limit A&. So if we continue
the sequence in the other direction, drawing PA ±o,
Q,P„ _!.&,..., we have an increasing sequence which tends
to the other common perpendicular A 2 B Z .
108
ELLIPTIC GEOMETRY
\m. 13
13. Paratactic lines.
If A l B 1 = A f B 2 , all the intermediate perpendiculars must
also be equal ; the two lines have then an infinity of common
perpendiculars, and therefore the four lines a, b, a', b' all
belong to the same regulus of a ruled surface of the second
degree. The two lines are equidistant, though not coplanar ;
they are analogous to parallel lines in ordinary geometry
and possesses many of their properties. They were dis-
covered by W. K. Clifford, and have therefore been called
Clifford's parallels. A more distinctive name, suggested
by Study, is paratactic lines.
Through any point two lines can be drawn paratactic
to a given straight line, one right-handed and the other left-
handed. Each is obtained from the original line by screwing
it along the perpendicular NO either right-handedly or left-
handedly. The angle through which it has to be turned is pro-
portional to the distance through which it has to be moved.
■ In the plane ONM draw OMlON, cutting the given
line in M, the pole of ON in this plane. Draw MP the
P FIO. 6&
polar of ON, which is therefore perpendicular to NM, and
along it cut off MP - MP' =d. Then OP and OP' are the
two lines through paratactic to NM. Also d - 2(f Q.
(ii. 141
PARATAXY
109
14. The above construction for a common perpendicular
to two skew lines can only be carried out in elliptic geo-
metry, for in hyperbolic geometry the polar of a real point
is ideal, and in euelidean geometry it is at infinity. Con-
sider a svstem of pairs of planes at right angles to each other
Hi.. 69.
drawn through the line a. This forms an elliptic involution,
the double elements of which are the imaginary planes
through a which touch the absolute. These planes are cut
by the line 6 in a range of points forming an elliptic involu-
tion. Although the double points of this involution are
imaginary, the centres B lt B % of the segments determined by
the double points are always real. These form a pair of
elements of the involution a quadrant apart. In elliptic
geometry there are two real centres, in euelidean geometry
one is at infinity, and in hyperbolic geometry one is ideal.
The perpendiculars to b at B lt B s are the two common
perpendiculars.
Since B, and B z are conjugate points and A&l B^^
A,B t is the polar of B t in the plane Ajb ; therefore
A , B 8 _L AyB v But the plane aB t A. the plane aB s ; therefore
AJlt is 1 the plane aB«; therefore A^Xa. Similarly
A^B^La.
The points A lt A z are the centres of a similar elliptic
involution on a.
no
ELLIPTIC GEOMETRY
|rn, fa
15. Two paratactic lines cut the same two generators, of the
nam system of the absolute.
Let three common transversals fc, l t , l 3 cut the four
lines a, a', b, V in A,, A,' r B., B.', and the absolute in
Xi, Y, (*-l, 2, 3). Then, since Aj, A x ' and B lt B,' are
FIO. 70.
harmonic conjugates with regard to X,, Y u (A^', B l B 1 i )
is an involution with double points X„ Y\. Also, by a
fundamental property of a ruled surface of the second
degree,
i4Ai'B l B l ')-x(A s A t 'B£ i ')-xiA 9 di'B i B 3 ').
Therefore
(A 1 A 1 'B i B l 'X,Y l )7,(A 2 A i 'B i B.,'X 2 Y s )
^{A 3 A 3 'B 3 B 3 'X,Y A ).
Therefore X x X t X 3 and Y s F,y s are two generators x, y of
the same regulus as A,A S A 3 , etc. ; they are also generators
of the same regulus of the absolute, since they cut it in more
ni. IB]
PAEATAXY
111
than two points. Hence all the common perpendiculars
to two paratactic lines cut the same two generators of the
absolute.
Let a cut the absolute in a lt u 2 . Through each of these
points passes one generator (g t and g t ) of the absolute of
the opposite system to a;, y, and therefore cutting x, y in
&&» nm- Si and St mu8t als0 Delon 8 to the same Te ^ m
as l u l t , k, since they cut the surface in more than two
points. Therefore they cut a\ b and b' also. Hence a and
b cut the same two generators of the absolute, q.e.d,
Conversely, if a and b cut two generators g z , g 2 of the
absolute in ai 8 lt u*#z, let A,, h t be the two generators of
the other system through a l ,u 2 , then the polar of a is the
intersection of the planes (gjij, {^J, and therefore cuts
both g, and (jr 2 . Hence gr, and g % are common transversals
of a, a\ b, 6\ " But, by § 12, if a, a', b, b' do not all belong
to the same regulus, they have only two common trans-
versals, which are absolute polars. Now g t and g t are not
absolute polars (each being its own polar), hence a, a\ b, b'
belong to the same regulus, and have an infinity of common
transversals. Therefore a,, b are paratactic.
The two seta of generators of the absolute may be called
right-handed and left-handed. Two lines which cut the same
two left (right) generators of the absolute are called left
(right) paratactic lines. We see, therefore, that all the
common transversals of two right paratactic lines are left
paratactic lines. Further, if a and b are both right (left)
paratactic to c, then a is right (left) paratactic to b ; for
a, b, c all cut the same two generators of the absolute.
16. Paratactic lines have many of the properties of
ordinary euclidean parallels. In particular they have the
112
ELLIPTIC GEOMETRY
tin. 17
characteristic property of being equidistant. They are
not, however, coptanar. We shall use the symbol fl for
right parataxy, and U for left parataxy.
If AB n CD and if AC and BD are both 1 CD, they are
also ± AB; AC=BD and Li BD, and AB=CD. Also
AD cuts both pairs of lines at equal angles. The figure
ABDC is a skew rectangle ; its opposite sides are equal
and paratactic.
If ABUCD and = CD, then joining AC, BD and AD,
LADC = LDAB, and we find two congruent triangles
ACD and DBA ; therefore
AC=BD and LACD=LDBA.
Conversely, if AB = CD and LABD = LDCA, or if
CAB + ACD = 2 right angles, then AB is paratactic to CD.
Hence, if ^3 = and n CD, it follows that AC = and U BD.
ABDC is analogous to a parallelogram.
Real parataxy can only exist in elliptic space. For if
ABDC is a skew rectangle, the lines AB, AC, AD are not
co-planar.
Therefore ACAD + BAD > LCAB, i.e. > a right angle.
But LBAD = LADQ;
therefore L_ CAD + ADC +ACD>2 right angles ;
therefore the geometry is elliptic,
17. Clifford's surface.
If «0 c and 6 P c, so that a U b, the common transversals
of a, b, c are all LI , and form one regulus of a ruled surface
of the second degree ; the lines of the other regulus are all
11 a, b and c. If a' is a generator of the opposite system to
a, b, c, then any line which cuts a and is u a' cuts 6 and c.
The surface is therefore generated by a line which cuts a
17]
CLIFFORD'S SURFACE
113
fixed line and is paratactic to another fixed line. By § 16
it cuts the fixed line at a constant angle, W.
Let OP be the fixed line, which is cut by the variable
line OP' (Fig. 68). Draw OJVLthe plane POP' and
=d=2$ql-!r. Draw OM bisecting the angle POP' {=2$),
and draw NMLON in the plane MON. NM is then para-
tactic to both OP and OP' , and ON is supposed to be drawn
in the direction such that NM (1 OP and U OP'. Any other
tine which cuts OP and is U OP', i.e. any generator of the
left-handed system, is also U NM. and is at the same
distance d from NM. Hence NM is an axis of revolution
of the surface, and similarly the polar of NM is also an
axis of revolution.
This surface is, therefore, just the surface of revolution
of a circle about its axis which we considered in §11. In
fact, through any point P of this surface there pass two
lines paratactic to the axis, and since these lines arc equi-
distant from the axis, they lie entirely in the surface. This
surface, which is called Clifford's Surface, is therefore
a ruled surface. All the generators of one set are n to the
axis I, and ail the generators of the other set are U I. Two
generators of opposite systems cut at a constant angle
IT
- d. From the figure in § 15, it appears that Clifford's
surface cuts the absolute in two generators of each
system.
Suppose the surface is cut along two generators. The
whole surface is covered with a network of lines inter-
secting at a fixed angle W, and can be conformly repre-
sented upon a euclidean rhombus with this angle. The
geometry on this surface is therefore exactly the same as
that upon a finite portion of the euclidean plane bounded
N.-E. Q. H
114
ELLIPTIC GEOMETRY
| in. 13
by a rhombus whose opposite sides are to be regarded
as coincident. As an immediate consequence, the area of
the surface is found to be 4tp . sin 20, since the side of the
rhombus = 2j. We have therefore the remarkable result
Fia. 7i.
that both in hyperbolic and in elliptic space there exist
surfaces {viz. horospberes and Clifford's surfaces respec-
tively) upon which euchdean geometry holds.
Circumference of a
18. Trigonometrical formulae,
circle.
In investigating the trigonometrical formulae we shall
use a method which might equally well have been employed
in hyperbolic geometry. The starting point is the assump-
tion that euclidean geometry holds in the infinitesimal
domain. 1
1 The truth of this assumption in indicated by the fact that when the
sides (if ;i t rtuneje tend to iiero, the Hum of the angles tend* In tin- value jr.
The steps of the proof are as follows. Let A BC be a triangle with right
angle H 0. We have to prove (1) that the nitio AC: AB tends to a
limit. This limit is a function of the angle A, say f(A). We liuvo to
prove (2) that/(j-IJ is continuous, and (3) that its vnluc is cos A. The
last step is best obtained bv the formation of a functional equation
fi» + 4>) -> f{0~<t>)=2fW ■/{•/>)■ See Coolidge, Chap. IV.
m. 18] CIRCUMFERENCE OF A CIRCLE 115
Let AOB be a smaU angle a, OA =OB=r, AA'=BB' =oV,
AB = a, A'B' = a + da. The angle OAB is nearly = £ , Let
OAB^OBA^-0, OA'B =OB'A' = ~-(6+d6).
altfa
Flo. 72,
Draw BM, making the angle ABM - ABO - ^ - $.
Then, neglecting higher inlinitesimals, we have A'M = AB;
therefore MB' =da, and BM=dr.
Therefore da = 2dr sin A MBB' m 2dr sin $,
da
or
dr
=20.
Again, the area ABB' A' =adr, and the sum of its exterior
angles , *
therefore adr= -2&?dQ, (§9)
or
i.e.
M dr~ k dr*'
d?a a _
The solution of this is
a=Csm(^ + <f>).
Differentiating, -*- = 26 = v cos U + <f>) •
116
ELLIPTIC GEOMETRY
[ra, 19
When r=0, «=0 and 2(^-#J =■*--«, therefore 20= a,
so that = C sin q4, a = , cos <p ; whence 0=0 and C=ka.
Hence we have, finally,
j ■ 7
a = tea sin v.
Since a and a are small, we can take a as the arc of a
circle of radius r. The whole circumference of a circle is
therefore 2irk sin j-
When r = |7rjt, the circumference is 2-&, which is twice
the length of a complete line, and therefore q = l-xk.
19. Trigonometrical formulae for a right-angled
triangle.
Keep one part, say b, fixed. Let BAB' =dA, BB' =da,
Fio. 73.
AB'C=B+dB, N&^dc.
Then cfc=ffacosB,
•0)
h sin y rfj4 = NB = rfff sin S (2)
•
in. 201 THE EIGHT- ANGLED TRTANGLE
117
The area of BAB' is obtained in two ways, (1) by inte-
grating f, . c , . ,
a A ac
= 1 fcshi
Jo
= £ 3 <L4(l -cos jj ;
(2) in terms of the angular excess
= kHdA + 7r-B + B+dB-Tr)
=&{dA +dB).
e
(3)
Equating these, dB= -cos, dA
Eliminate (fa and dA between these three equations,
and we get
kdBb&a C ,= -ks\n ,dA = - da sin B = - dc tan B,
giving a differential equation in B and c. The integral of
this is c
sin j sin B =f(b),
since b is the only constant part.
Putting B = ™, c=6, and we find /(6) = sin g
ft f*
Hence we have sin , =sin 'sin B.
20. Associated triangles.
Id order to obtain the other relations between the sides
and angles, we shall establish a sequence of associated
triangles which form the basis for Napier's rules in spherical
trigonometry. This sequence has already been referred
to, and a similar sequence was found in hyperbolic geometry.
118 ELLIPTIC GEOMETRY fin. 20
We shall first introduce the following notation. Let
u=a/k, a' = 2 -«; then the angles «, £, y correspond to
the sides a, b, c of the triangle.
Draw the absolute polars of the vertices A and B. These
form, with the sides produced of the given triangle, a star
Fro. 74.
pentagon. Mark on each side the angle which corresponds
to it : and we get the figure (Fig. 74). Each of the five outer
angles is a right angle. Each vertex of the simple pentagon
is the pole of the opposite side. We obtain then five
associated right-angled triangles. If we write down the five
quantities A, «', y, /3', B which correspond to the parts of
the first triangle A, «, y, 8, B t the corresponding quantities
in the same order for the second triangle are «'. y, B', B, A,
but these are the same as the five quantities corresponding
to the first triangle permuted cyclically ; and they are
represented in proper order by the sides of the simple
pentagon.
Ui.21]
NAPIER'S RULES
119
21. Napier's rules.
Now we have proved for the first triangle that
.6 . c . n
sin ,,=sih .sniff.
Writing this in terms of A, «', y, &\ B, we have
cos j8' = sin y sin B,
and since this equation can be applied to each of the five
triangles, and therefore transformed by cyclic permutation,
we can state a general rule as follows :
Write the five angles A, «', y, fi', B in order on the sides
of a simple pentagon. Then, calling any one part the
middle part and the other two pairs the adjacent parts and
the opposite parts, we have
cos (middle part) = product of sines of adjacent parts, (a)
Taking in succession y, A, B as middle parts, we get
cos y =sin a sin B',
cos A= sin a sin S,
cosB = sin/3'sin^.
Hence cos y =cot A cot B,
i..e. cos (middle part) = product of cotangents of opposite
parts (b)
There is a relation of one of these forms between any three
parts of the triangle. For convenience we write down the
ten relations in terms of a, b,c, A, B.
cos t = cos ,-' cos j - cot A cot B,
k k K
Bin t = sin v sin A = tan y cot ff ,
a . n . c b
cos A = cos t sin ff - cot % tan t .
and two other pairs formed by interchanging a, 6 and A, B.
120
ELLIPTIC GEOMETRY
im. 22
These are exactly the same as the relations which exist
between the parts of a spherical triangle. Tlte trigonometry
of the elliptic plane is tlierefore exactly the same as ordinary
s /i//i'r >'>;,/ lrifjo7ioinetry
If we write the parts «, B', y', A\ 8 in the order in which
they occur in the triangle, we get the more familiar rules of
Napier :
sine (middle part) = product of cosines of opposite parts
= product of tangents of adjacent parts.
22. Id elliptic space the formulae for spherical trigonometry
are die same as in euclidean space, when we take as the
measure of a side of a spherical triangle the angle which it
subtends at the centre, and as the measure of an angle the
dihedral angle between the planes passing through the
sides and the centre.
Fro.
Let be the centre of the sphere, and let 0A, OB, OC
cut the polar plane of in A', B', C. Then we get a
hi. as]
SPHERICAL TRIGONOMETRY
121
rectilinear triangle A'B'O with sides a', 6', c'. The angles
which the radii 0A', etc., make with the sides are right
angles ; hence A' = the dihedral angle between the planes
OAB and OAC, is. A' =A. Also a' =ka. Hence the rela-
tions between a. j8, y, A, B, C are the same as those between
^i >iri A\ R, C, which are the same as those of
ordinary spherical trigonometry.
The measurement of angle, plane or dihedral, is the same
in all three kinds of space, and spherical trigonometry
involves only angular measurement. This explains why
spherical trigonometry is the same in all three geome-
tries.
23. The trirectangular quadrilateral.
As in hyperbolic geometry, there is a correspondence between a
right-angled triangle and a trirectangular quadrilateral In fact
Fro. 70.
wo see that, by producing two opposite sides of the quadrilateral
to meet, we get, corresponding to the trirectangular quadrilateral
CamBi, a right-angled triangle with hypotenuse ?,jrfc-&=6, sides
a and {irk -m- m. and tho opposite angles I and r - C.
122
ELLIPTIC GEOMETRY
[Ex. in.
If we write the parts
IT
r + O.
/ b
a IT nt. ir
2 Tl " F 2~ fc' I
in oyclio order, then we have tbe rules :
sine (middle part) = product of cosines of opposite parts
-product of tangents of adjacent parts.
EXAMPLES m.
1. Prove that the bisectors of the vertical angle of a triangle
divide the base into segments whose nines are in the ratio of tbe
sines of the sides.
2. Prove that the arc of an equidistant- curve of distance a, corre-
sponding to a segment x 011 its axis, is x cos «//,'.
3. Prove that the area of a circle of radius r is iirfcsm 2 — .
2k
4. Prove that the area included between an arc of an equidistant-
ourve of distance o, its axis, and two ordinate's m distance x, is
■ . a
kxmn -.
5. Prove that the area of tke whole plane is 2wl.*, and the volume
of the whole of spaco is ir^fc 5 .
8. Prove that the volume of a sphere of radius r is
.,/2r . 2A
(In the following examples it is unity.)
7. If R is i In- radius of the cironmsphere of a regular tetrahedrou
whose side is a, show that
sin in - \/| sin H,
8. If 2d is the distance between opposite edges of a cube of edge 2a,
8ft 1 be distance between opposite faces, and ft the radius of the
circumaphere, prove that
sin x d - 2 tan s ffl, sin'ft = sin'a/oos 2o, sin ! fl = 3sin*a,
9. A semiregukr network is formed of triangles and quadrilaterals,
two of each at each node. Prove that this can only exist in elliptic
spaco, and that the length of the aide ia .'. -.
Kx. in.]
EXAMPLES
12^
10. In elliptic geometry show that there can exist six equal
circles, each touching each of the others, and of radius given by
2eoBrsin- = l; three equal circles each having double contact
with the other two, and of radius % ; and (with overlapping) four
4 1
circles each touching the other three, and of radius cos- 1 — ,
11. Prove that five spheres, each of radius g, can be placed each
touching the other four ; eight spheres, each of radius -, and each
having double contact with four others ; and four spheres of radius
- each having double contact with the other throe,
4 *
12. For a regular polyhedron :
a = length of edge.
a = angle subtended by edge at centre,
8 - angle of each polygon.
S = dihedral angle between faces,
w = number of sides of each face.
p = number of edges at each vertex.
R = radius of circumscribed sphere.
r = radios of inscribed sphere.
p = radius of sphere touching the edges.
R a = radius of circum circle of each face.
r = radius of incirclc of each face.
Prove the relations :
7T . ir a
cos ■ sin cos i,
n p 2
ir . 8 a
cos - = sm - cos -,
n I £
a
sin " = sin R sin £, sin p = tan cot -,
a
V
sin - =sin fl„sin -, sin r a = tan ; oot -,
cos R = cos r cos /?„, cosp=cosr coar,,, sin r„= tan root-,
sin r - sin p sin 3.
n .it / / ,0 ,t\
,^ 2 co a y^-cos.-j.
124
ELLIPTIC GEOMETRY
I Ex. hi.
13. For a regular tetrahedron prove that cos 8=cosa/(l +2cos«>.
hexahedron „ co9S=(cosa-l}/2 coso
„ octahedron „ cosS= -i/(l + 2cosa>.
is •■ dodeoahedron prove that
f08 6={2coso-(l+V5)}/{4cosa + (l-^5)}.
For a regular icosijiedroti prove that
cos 8= {( 1 - V5) cos a - { ] + Vo) }/2(l + 2 coao).
14. Prove that elliptic space can be filled twice over by 5 regular
tetrahedra of side cwT't - }), with 3 at each edge and 4 at each
vertex.
15. Prove that elliptic space can be filled in tJio following ways :
(1) 4 cubes, of edge |, 3 at oaoh edge and 4 at each vortex.
(2) 8 tefcrahedra, of edge I, 4 at each edge and 8 at each vertex.
(3) 12 octahedra, of edge ~ , 3 at each edge and 6 at each vertex.
(4) 60 dodecahedrn, of edge cos" 1 — S z H t 3 at each edge and
4 at each vertex. 8
(5) 300 tetrahodra, of edge cos \ 5 at each edge and 20 at each
vertex.
CHAPTER TV.
ANALYTICAL GEOMETRY.
1. Coordinates.
We shall assume elliptic geometry as the standard case,
and construct a system of coordinates. The formulae
can be adapted immediately to hyperbolic geometry by
changing the sign of fc 8 .
Take two rectangular axes Ox, Ojf. Let P be any point,
and draw the perpendiculars PM=u and PN =v. Let
0P=r, xOP = 6.
M
71
FTC. 77.
r, $ are the polar coordinates of the point, it, v might
be taken as rectangular coordinates, but we shall find it
more convenient to take certain functions of these.
We have
. u , r a
sin ■,•= sin , cosy,
sin , =sin . sinO.
126
ANALYTICAL GEOMETRY
[IV. I
For any point on OP, therefore, sin V = sin "tan B.
This is the equation of OP in terms of the coordinates u
and v.
Consider any line. Draw the perpendicular ON=p,
and let xON =«. -p and a are always real, and completely
fio, ?a
determine the line. If P is any point on the line with
coordinates u, «,
7) <f
tan ^ cot , =cos (8 - a).
Therefore tan |cos| = sin|cos«+ain %in rt .
This equation is linear and homogeneous in
■ u , v r
m l' sm r C0S F
We shall effect a great simplification, therefore, if we
take as coordinates certain multiples of these functions.
The equation of a straight line being now of the first degree,
the degree of any homogeneous equation in these coordi-
nates gives the number of points in which a straight line
B]
COORDINATES
127
meets the curve, i.e. the degree of the equation is the same
as the degree of the curve.
In order that the coordinates of a real point may be real
numbers, both in elliptic and in hyperbolic geometry, we
shall define the coordinates as follows :
a; = « sin -r = « sm v cos tf,
v
y = k sin j_ = k sin , sin B,
r
2=COSt.
These are called Wekrstmss' point-coordinates.
The three homogeneous coordinates are connected by a
fixed relationship. We have
i.e. :c* + ^ + ifcV=A a .
As any equation in x, y, z may be made homogeneous
by using this identical relation, we need only, in general,
use the ratios of the coordinates.
2. The absolute.
In hyperbolic geometry, putting ik instead of k, we find
the coordinates
a;=Asiuli . ,
j/=Asinh v,
z=cosh r*
and x, y, z are connected by the relationship
x* + y*-k*z i =-k t .
128
ANALYTICAL GEOMETRY
Iiv. 3
tf / is infinite, x, y, z are all infinite, but they have
definite limiting ratios. Let a, 8, y be the actual values,
*, y, z the ratios, so that a = \x, 8 =Xy, y = \z, and \-h» . '
Then a* + 8 a -k*y*=-li?;
therefore
X»+tf-kH*=-^ = 0.
Hence the ra&os of the coordinates of a point at infinity
satisfy the equation
x*+y*-]fr?=0.
This is the equation of the absolute, which is therefore
a curve of the second degree or a conic. In hyperbolic
geometry it is a real curve ; in elliptic geometry the equation
is x* +y s +A?« s =0, which represents an imaginary conic.
3. Normal form of the equation of a straight line.
Line-coordinates.
We found the equation of a straight line in terms of the
perpendicular p and the angle «, which this perpendicular
makes with the z-axis, in the form
x cos a +y sin a = kz tan ?.
which may be written
The ratios ( : n : f determine the line, and can be taken
as its line-coordinates. It is convenient to take certain
multiples of these as actual homogeneous coordinates, viz.
£=cosa cost*
F/=Sin a cos?'
k
C- -k&ln i
ft
.v. 4]
THE STRAIGHT LINE
which are connected by the identical relation
These are called Weierstrass' line-coordinates.
In hyperbolic geometry
£ = cos a cosh*?. »j=sina cosh?. f=-&sinh^.
and the identical relation is
If p-»-co, £, i), f all ->oo. Let the actual values be
a, 8, y, and let a =\£ 8=\r}, y = Xf ; then
Hence the coordinates of a fine at infinity satisfy the
equation it^ + *V - f* = 0,
A homogeneous equation in line-coordinates £, q, f repre-
sents an envelope of lines. This equation represents an
envelope of class 2, {.e. a conic. This is the same conic
as we had before and represents the absolute, since it
expresses the condition that the line (£ v, f) should be a
tangent to x* +y z - khP =0.
4. Distance between two points.
Let P{x y y, z) and P'(x', y', z') be the two points, PP' =d.
Then, if the polar coordinates are (r, 6) and (/, $'),
d t t' . t . t'
cos t ■ cos r cos -, + sin y sm -r cos (9 - &'}
, xx 1 yi/
or, in terms of the ratios of the coordinates,
jRc'+yy' + k?zz'
d
cos , =
k j& +&+&**/** +#*+'!&*
N.-E. O.
1*1
ANALYTICAL GEOMETRY
[tv. fl
It is convenient to introduce here a brief notation. If
(x, y, z), (as*, y', z') are the coordinates of two points, we shall
define xx 1 +yy' +l^zzf = (xx') t
and we shall speak of the points {x) and {if).
Then the distance between the points (x) and (a;') is
g' ven b y d (xx'\
COS , = i ■! ,
k J(xx)J(x'x')
5 In elliptic geometry the distance-function is periodic.
d
Suppose d = hvk ; then cos , =0, and
xx" + y>/ + k % zz' =0,
i.e. all points on this hne are at the distance £tt& or a
quadrant from (x\ y 1 , z'). This is therefore the equation
of the absolute polar of (x', y\ z'). It is the polar with
respect to the conic
a?+^ + itV=0.
This is therefore the equation of the absolute.
d
Suppose d = irifc ; then cos t = -1, and, with actual values
of the coordinates,
xx' + yy" +k?zz' ■ -k-.
but x*+y 2 +kH 2 = **,
and x'*+y'* +kH'*=k*;
therefore, multiplying the first equation by 2 and adding
to the others,
(x + x'f + (y + y'f + k*{z + z'f =0,
which requires that x' = - x, y' = - y, z' = -z.
In spherical geometry these would represent antipodal
points. In elliptic geometry antipodal points coincide,
LV. o]
DISTANCE AND ANGLE
131
and therefore in every case, if two points have their co-
ordinates in the same ratios, they must coincide.
6. Angle between two lines.
From the figure (Fig. 79) we have
sin v 1 =sin -, sin 0,, cos t =cot t tan j",
sin ^ = sin [sin^j, coB/3 t =coti tan y,
no. 7u.
cos </>i = sin $ 1 cos ~J, 0, + a = ir - <f>,
cos 2 = sin /S,; cos jf, ft + fi z = « 3 - n t ,
cos (0! + 2 ) =sin /3, sin 0„ cos V} cos 7 , 2
- coscc 2 , sin ¥ sin ,"■
k k k
cos (/?, + S ) = cot 2 T tan y" tan y - sin 8i si n /3 2
^CGSfas-a,).
,n
132 ANALYTICAL GEOMETRY jiv.7
Therefore
-f . p, . p s
cos ^ =coaec 8 T sm V sin v
fC A' tC
+ (cos «j - «! - cot* v tan y ten y ) cos y 1 cos y 2
= sin 41 sin ', a + cos jJ cos ^f cos (a a - n^)
= l^ + i^t + mnz,
or, in terms of the ratios,
COS (/>
J Pfr + &** + & n'^? + *V + &
If (££) =0 is the line-equation of the absolute.
9 </&&M&£,)
7. Distance of a point from a line.
If tl is the distance of a point from a line, kirk-d is the
distance of the point from the pole of the line. Let the
coordinates of the point be (x, y, z) and of the line (f, >;, f).
The pole of the line is (&*f , khi, 0. Therefore
&+ny+& m £* + mi 1 cj
■ + £ a + W Jg* + >? + &I& J(asc) V( I
. d
sin . = -==
8. Point of intersection of two lines (&, i; lt £,),
The coordinates of the point of intersection are pro-
portional to Uf s - nsfi » &&-£*&» &J»-&m)-
iv. 8]
INTERSECTION OF LINES
133
Tf the actual values are a, ft, y, so that « =X#, etc., then
+# s (£'J2-£tfl) 2 ]
= A !! ifc a [l-COS 3 f];
therefore X = cosec ^,
where is the angle between the lines.
I f the coordinates of the point of intersection satisfy the
equation K a + ii/ 2 + A-V=0, i.e. if the lines intersect on the
absolute, X is infinite and tj> is zero. The two lines in this
case are parallel.
If the ratios of the coordinates make x 2 + tf + k i z t <0,
X is imaginary. The two lines have then no real point of
intersection, and the angle is imaginary, The lines may
be said to intersect outside the absolute. (These two cases
can, of course, only happen in hyperbolic geometry.)
In the latter case the two lines have a common perpendi-
cular.
Let gx + iftf + £z =0 be perpendicular to both ; then
Ut+vh + &/** =0- a* + w> +&VA 2 =0 ;
therefore f : n : i=tt^2-ldi '■ 6& " £•£ : ^(&9»-&Ji) 5
but this line is just the polar of their point of intersection.
The length p of the common perpendicular is equal to k<f>,
and we have a ^
COS <j> = COS I = i ,f 2 + W 2 + U ^ 2 .
The actual Weierstrass coordinates of an ideal point are therefore
purely imaginary numbers of the form (ix, ly, iz), and their ratios
are real. If we let the coordinates {z, //. s) be any complex numbers,
we get points bolonging to the whole " complex domain." This
134
ANALYTICAL GEOM ETR V
[iv.
includes (1) real actual points, for which the ratios x : // : s are real
and i- i if - && has the same sign as L~ (2) real ideal points, for
which the ratios x:y :i are real, and (** + j*-i-JPstyJP is negative,
(8) imaginary points, for which at least one of the ratios x;y:B
is imaginary. The line joining a pair of conjugate imaginary
points is a real lino, actual, at infinity or ideal. The distance
between a pair of conjugate imaginary points is real only if their
join is ideal.
9. Line joining two points.
Similarly the line-coordinates of the line joining two
points (xj, y, , Zj), (x s , y t , z t ) are proportional to #,2 2 -y.fr,
ZjOSg-ZjSj, Xjifi - sr-ji/, . The actual values of the line-
coordinates are found by multiplying by the factor cosec .,
where d is the distance between the two points.
If the ratios of the line-coordinates satisfy the equation
£* + «* + £*//?=(), the line is at infinity, and the distance d
is zero.
If the ratios make g* + ir + ^/i 2 <0, the line is wholly
ideal, and the distance d is imaginary.
10. Minimal lines,
When the join of two points is a tangent to the absolute.
the distance between the two points is zero. For this
reason the tangents to the absolute are called int'itininl
lines.
In euclidean geometry the distance between two points {x it #,),
<*,.*) is zero if {Xl - Xs? + { y,. Vtr=0t
Li.it ,,.. //,= d ){*i-Xi},
i.e. if the join of the two points passes through one of the circular
points (Chap, II. §17). The line at infinity itself passes through
both of the circular points, ant! it is the only real line which passes
through them. The distance between two [joints at infinity should
thus be BMO. But again, any point on the line at infinity is in-
ly, li] CONCUBBENCY AND COLLINEARITY 135
finitely distant from any other point. Hence the distance between
two points, both of which are at infinity, becomes indeterminate.
In relation to the rest of the plane wc must consider such distances
m infinite, and the geometry of points at infinity becomes quite
unmanageable. The geometry upon the line at infinity by itself,
however, is really elliptic, since the absolute upon this line consists
of a pair of imaginary points ; the " distance " between two points
at infinity could then be represented hy the angle which they subtend
at any finite point.
11. Concurrency and collinearity.
The condition that the lines (£, 17, , &)> etc., be con-
current is
£1 >h
&
£2 >h
u
£3 'la
&
=0.
The condition that the points (a^, y t , «,), etc., be collinear is
a% tfx % ,=0.
aj £ y 2 z 2
#3 Vs z 'i
These conditions arc, of course, the same as those in
ordinary analytical geometry, with homogeneous co-
ordinates.
Since the equation of a straight line is homogeneous and
of the first degree in the coordinates, all theorems of
ordinary geometry which do not involve the actual values
of the coordinates, or the distance-formulae, will be true
also in non-euclidean geometry. These theorems are
those of projective geometry. The difference between
euclidean and non-euclidean geometry only appears in
the form of the identical relation which connects the point
and line coordinates, i.e. in the form of the absolute.
136
ANALYTICAL GEOMETRY
[iv. 12
12. The circle.
A circle is the locus of points equidistant from a. fixed
point. Let {x 1 , jfe «,) be the centre and r the radius ; then
the equation of the circle is
or, when rationalised,
(aacftxiXj) cos 2 £ = (xar,) a .
This equation is of the second degree, and from its form we
see that it represents a conic touching the absolute (xx) =0
at the points where it is cut by the line (xx l ) =0. (axe,) =0
ia the polar of the centre, and is therefore equidistant from
the circle, i.e. it is the axis of the circle. Hence A circle is
a conk having double contact with the absolute ; ite axis is
the common- chord and its centre is the pole of the common
chord.
The equidistant-curve. Let (£, i hl f,) be the coordinates
trf the axis, and d the constant distance; then the equation
of the curve is
s \ a d = & + *& + &
or
J
(xx) ($i) sin^ = {£sr + .,# + &f.
This again represents a conic having double contact with
the absolute, the common chord being the axis. The pole
of the axis is equidistant from the curve, and so the equi-
distant-curve is a circle. In elliptic geometry both centre
and axis are real, in hyperbolic geometry the centre atone
is real for a proper circle, and the axis alone is real for an
equidistant-curve.
iv. 13]
THE CIRCLE
137
The horocycle. In hyperbolic geometry, the equation
of the absolute being x* + if - k*z 2 = 0, the equation of a
horocycle is of the form
2? + y & - kH z = \(ax + by + cz)%
where
*+*-£
13. Coordinates of a point dividing the join of two
points into given parts.
If (x 1 , Jf t ,?j), (aij, 3/g, z 2 ) are any two points, the coordinates
of any point on the line joining them are
(Kx^+fixt, Ay, + /*?/«. Az,+,u2 2 ),
for if ax + by+cz = is the equation of the line, so that
it is satisfied by the coordinates of the two given points, it
will he satisfied also by the coordinates of any point with
coordinates of this form. Similarly, if we consider these
as the line-coordinates of two lines, the coordinates of any
line through their point of intersection are of this form.
In fact the line
X(«jx + 6,y + Cjz) + tt(a& + b$ + c#) =0,
whose coordinates are (A<ri + /ia*, ...)• passes through the
intersection of the two given lines a 1 x + b i y+e t z = Q and
a& + o a y + CjZ = 0.
To find the coordinates of a point dividing the join of two
points whose actual coordinates are (34 , y t , z,) and (a! 2 , y 2 , z 2 )
into two parts r, and r s , where r l +r i -r.
Let(\Xj+ux s ,...) be the actual coordinates of the re-
quired point. Then
g, (Xx! + nx t )+ y, (Ay, + /xy a ) + *fej (As, + pxj - k* cos J ;
therefore A + /* cos ,- % = cos } •
138
ANALYTICAL GEOMETRY
Similarly
Xcos , +fi
= cos r\
k
whence A
sin, = sin-- and
Msin|=sin^
and the actual coordinates are
- r.
at, sin ^ + x s ain
k
[rv. 14
sm
I
If (^i. fh, %). etc., are only the ratios of the coordinates, we
must first find their actual values by dividing by the
factor (zx)/k.
If the line is divided externally into two parts r, and
r t , we have only to observe the proper signs off, r t and ■/■„.
14, Middle point of a segment.
In particular, if r, -r, we get the ratios of the coordinates
of the middle point of the segment (a:, +x it g, +y t , 3, + Zg ) t
or, if at,, etc., are only proportional to the coordinates, the
ratios of the coordinates of the middle point are
% t
¥»
the actual values being obtained by dividing by 2 cos X.
The join of too points has a second middle point with
coordinates -J!JL=- ,% n : . : . . the actual values being
obtained by dividing by 2ain^. In elliptic geometry
these points are both real and a quadrant apart ; ' in
1 In spherical geometry the tmo middle points of a segment are
antipodal, and are not (as in elliptic geometry) harmonic: conjaentcs
with respect to the given points.
rv. 16]
CEXTROTDS
139
hyperbolic geometry the factor 2 sin gr, becomes 2* sinn ^
and the coordinates of the second middle point are all
imaginary.
15. Properties of triangles. Centroid, in- and circum-
centres.
Through each vertex of a triangle (a;,), (sj s ), (a%) pass
two medium, and the medians are concurrent in sets of
three in four cenlroids, denoted, in the notation of §4, by
JU±-
*)■
Vtatjai) V(ij*s) V(!KjXs; J
The same combination of signs is taken for all three co-
ordinates, and theTe are four different combinations of
signs, one corresponding to each of the centroids.
Similarly, the middle points of the aides are collinear in
sets of three in four lines, the axes of the circumscribed
circles.
The bisectors of the angles are concurrent in sets of three
in four points, the centres of the inscribed circles ; and their
points of intersection with the opposite sides are collinear
in sets of three in four lines.
16. Explanation of apparent exception in euclidean
geometry.
In euclidean geometry four cycles can be drawn to
touch the sides of a triangle, but apparently only one can
be circumscribed. Of the four circumcireles of a triangle
in hyperbolic geometry, three are equidistant-curves. In
euclidean geometry the equidistant-curve through B, C
and A reduces to the line EC and the line through A \\ BC.
(Ci. Chap. II. f 23.)
140
ANALYTICAL GEOMETRY
flv. 16
The conception of a pair of parallel straight lines as
forming a circle iu euclidean geometry is consistent with
the definition of a circle as a conic having double contact
with the absolute, for the absolute in this case is a pair of
coincident straight lines, and this is cut by a pair of parallel
lines in two pairs of coincident points. A single straight
line is not, of course, a tangent to the absolute, though it
cuts it in two coincident points : this case is just the same
as that of a line which passes through a double point on a
curve, but which is not considered as being a tangent.
But when we have a pair of parallel lines cutting the
absolute S> in four points all coincident, we can regard O
as being a tangent to the curve consisting of this pair of
lines. Fig. 80 represents the case approximately when the
absolute is still a proper conic and the pair of straight lines
is also a proper conic, having double contact with the
absolute.
Fia. so.
The axis of the circle consisting of a pair of parallel
lines is the line lying midway between them ; the absolute
pole of this (a point at infinity) is the centre. When the
171
ORTBOCENTRE
141
axis passes through the centre, i.e. when it coincides with
the line at infinity, the circle becomes a horocycle, which
is thus represented in euclidean geometry by a straight
line together with the line at infinity.
Two equidistant-enrves, with parallel axes, have the
same centre at infinity. In hyperbolic geometry two
equidistant-curves, with parallel axes intersecting at
infinity at 0, have their centres on the tangent at 0, and
therefore at a zero distance apart though not coincident.
17. Polar triangles. Orthoeentre and orthaxis.
If A, B, C is a triangle and A', B', C the absolute poles
of the sides o, 6, c, then the sides a', b', c' of the second
triangle are the absolute polars of the vertices A, B, C of
the given triangle. Two such triangles are called polar
triangles.
Tf the coordinates of A, B, are (a^, ft, Zj), etc., the
equations of their polars are (2X1) =0, etc.
The point-coordinates of the vertices A', B\ C are
&&-?&' *x**-*A. (Siya-Ssffi)/* 2 . etc -
The equation of AA', which joins (x,,ft, »i) to the point
of intersection of (axe 9 ) - and {xx 3 ) =0, is
(xXi) (x&) - {xx 3 ) (x^) =0.
Writing down two other equations by a cyclic permuta-
tion of the suffixes, we get the equations of BB' and CC,
and the sum of these vanishes identically. Hence AA',
BB', CC are concurrent. AA'LBC and B'C ; hence the
point of concurrence is the common ortkomUre of the
triangles ABC, A' B'C.
The absolute poles of A A'. BB', CC, i.e. the points on
the sides of the triangles distant a quadrant from the
142
ANALYTICAL GEOMETRY
[iv. 18
opposite vertices, will be collmear in a line called the
ortkaxis, o, which is the absolute polar of the ortho-
centre.
The two triangles ABC, A' BO' are in perspective with
centre and axis o.
18. Desargues' theorem. Configurations.
The last result is a particular case of Desargues' theorem
for perspective triangles, which, since it expresses a pro-
jective property, is true in non-enclideau geometry, and
can be proved (using space of three dimensions) in a purely
projective manner.
In the figure for Desargues' theorem (Fig. 81) we have
two triangles with their corresponding vertices lying on
three concurrent lines, and their corresponding sides inter-
secting in three collinear points. There are thus 10 points
and 10 lines : through each point pass 3 lines, and on each
r*. 181
CONFIGURATIONS
143
line lie 3 points. A figure of points and lines with this
property, that through every point pass the same number
of lines and on every line lie the same number of points, is
called a configuration. If p m denotes the number of lines
through a point, p i0 the number of points on a line, p 0Q
the whole number of points, and p u the whole number of
lines, the configuration may be denoted by the symbol
7>ao Poi
3>io Pn
Desargues' configuration is represented by
and is reciprocal. A convenient notation for the points is
by pairs of the numbers from 1 to 5. The three points
which lie on one line are denoted by the combinations
with the same three numbers.
The configuration formed by the six middle points 8f«,
M & , etc., of the sides of a triangle ABV and the four points
of concurrency G , <?„ G%, G 3 of the medians is a Desargues
configuration of a special kind (Fig. 82). The points G
form a complete quadrangle, and the points H are the
vertices of a complete quadrilateral, both having ABC as
diagonal triangle. This is called, therefore, the quadrangle-
quadrilateral configuration. Each vertex M ra of the quadri-
lateral lies on a side G r G. of the quadrangle.
Similarly, the eix bisectors of the angles and the four
lines of collinearity of the points in which they meet the
sides of the triangle form the same configuration.
144
ANALYTICAL GEOMETRY
I
FIO. 83,
19. Desmic system.
In three dimensions we have similar interesting con-
figurations.
If fa), (x t ), (x 3 ), (x t ) are four points in space,
v(*a) A*&i) Az&z) -J(z& 4 y
represent ^ the eight centroids of the four points. Each
centroid is on a line joining one of the points to the cen-
troid of the other three.
If the four given points be denoted by A lt A t , A s , A t .
and the other points corresponding to the different com-
binations of signs be represented as follows :
it. 20]
DESMIC SYSTEM
145
+ + + + B u
+ + - - B 2 ,
+ - + - B 3 ,
+ - - + B t ,
+ c„
+ - + + O a ,
+ + - + c a ,
+ + + - c t ,
then the join of any B with any C passes through an A,
e.g. B a - C t gives A 3 . So the 12 points lie in sets of 3 on
10 lines. They form three tetrahedra, any two of which
are in perspective in four different ways, the centres of
perspective being the vertices of the third tetrahedron.
Corresponding planes of two perspective tetrahedra inter-
sect in four lines which are coplanar, and these planes are
the faces of the third tetrahedron. A system of tetrahedra
of this kind is called a desmic system.
In a similar way it may be proved that the centres or
axial planes of the 8 circum- or in-scribed spheres form with
the given tetrahedron a desmic system.
A simple example of a desmic system in ordinary space is
afforded by the corners of a cube, its centre and the points
of concurrency (at infinity) of its edges.
20. Concurrency and collinearity.
In euclidean geometry we have the two useful theorems
of Menelaus and Ceva as tests for collinearity and con-
currency. Theorems corresponding to these hold also in
non- euclidean geometry.
I. If a transversal meets the sides of a triangle ABC in
XYZ, and a, j8, y are the angles of intersection, taken
positively, we have (Fig. 83)
sinBX_ siny sinC'Ysinq sin^Z = _sin£
sin BZ~ ~ sirTa ' sin CX ~sm8' sin A Y sin y '
N.-e. a. K
146
ANALYTICAL GEOMETRY
[rv. 20
the positive directions on the sides being in the cyclic
order ABC. Hence
sin BX si n CY sin AZ
sin CX sin AY sin HZ
A
= +1.
Fio. 83.
II. If three concurrent lines tlirough the vertices meet
the opposite sides of a triangle ABC in XYZ, and a, ft, y
are the angles between the lines (Fig, 84),
siuiJA'siiiy , sin C<£ _ sinj3
sin OB "sin X sin 00 "sin X'
sin BX _ sin OB sin y
sin CX sin OC ' sin ft'
singy _ sin 00 sin a
sin>4F~ sinOZ siny"
sin AZ sin 0,4 s in ft
sin BZ ~ sin OB sin a '
therefore
Similarly
and
Therefore
sin .a* sinC'F sin^Z
--].
sin OX' sin AY smBZ'
Conversely, the points X, Y, Z are col linear, or AX, BY,
CZ are concurrent, according as
siiiBX sinCY sin AZ _
.sin OX ' sin At ' ^mBZ " + l ot ~ L
w.sil CONCURRENCY AND OOLLINEARITY 147
This condition may be put in another form. Since
siaBX sin AB sin BAX ... .... , .
• . - ,__. = -. ... ■ . i-nri the condition reduces to
sin CX sin AC sin CMa
nn.K4£ sin Cay sin ACZ
sin 321 ' sin ABY ' sin BCZ~ ~ '
in which form it is the same as the condition in cuclidean
geometrv.
From this it follows at once that if AX, BY, CZ are three
concurrent lines through 0, their isogonal conjugates with
respect to the sides of the triangle are concurrent in the
iBogonal conjugate of 0.
21. Position-ratio. Cross-ratio.
If X, Y, P are colli near, the ratio . ., „ is called the
position-ratio of P with respect to X and Y, and the double
ta tio EJL_- _^?ML_* is called the cross-ratio of the range
sin YP sin YQ
( ,Y Y, PQ).
Similar definitions can be given for pencils of rays, and
the whole theory of cross-ratio can be developed on the
same lines as in ordinary geometry.
148
ANALYTICAL GEOMETRY
fiv. 21
Thus, the cross-ratio of a pencil is equal to that of any
transversal, and cross-ratios are unaltered by projection.
T
Further, it can be shown that
(ABCD)=(BADC) - (CDAB)=(DCBA),
(ABCD) . {ABDC) - 1 , (ABCD) + (ACBD) - 1.
The harmonic property of the complete quadrilateral
follows.
For (Fig. 85), (XYPQ)t? t (UYMQ), and also^FtfMQ).
Therefore ( VV, MQ) = ( VU, MQ) = - 1.
If A (a^ , y t , s,) and B(x 2 , y 2 , z s ) are two fixed points, and
P a variable point with coordinates
fo + \x t , y t + \y« , s, + As a ),
then, if AP=r t , PB = r 2 , AB = r,
we found A=sin -/ sin , 2 = the position -ratio of P with
respect to A and B. If Q is the point corresponding to
the parameter p, the cross-ratio (AB, PQ)=-. The cross-
Ex. IV.]
CROSS-RATIO
149
ratio of the two pairs of points corresponding to the para-
meters A, A' and n, m' is
These results are the same as in euclidean geometry.
EXAMPLES IV.
1. Prove that the actual Weierstrass line -coordinates of the
absolute polar of {%, y, z) are (x/k, y/k, kz), and the actual point-
coordinates of the absolute pole of {£, »;, f ) are (k£, to), (/k).
2. If the distance between the points (-r 1( y u Zj), (*,, y„ z.) vanishes,
prove that their join touches the absolute,
3. If to+ir,, ifi + iy t , «! + «i) * rc the actual Weierstrass coordi-
nates of a point (£,, y, . etc., being real numbers), prove that (z, , y, , Zt)
and (a:,, y,, n t ) are conjugate with regard to the absolute,
4. If («i + «Xj,...) (a, i t'<ju....)are the actual Weierstrass coordinates
of two points at a real distance (a-, , ?/, , etc., being real numbers),
prove that, for all values of X, (*, + Au, , ...) and {x,+ Aa,,..,) are
conjugate with regard to the absolute.
5. If ds is the element of arc of a curve and dx, dy, «fe the differen-
tials of the Weieratrass coordinates, prove that di?=dx t +d?f ! +k?dz l .
li" r, ff are the polar coordinates, prove that (&*=(?»* -t-fr* sin* f/fl*.
6. ABCD i» a skew quadrilateral, PQIiS are points on the four
sides AB, BC, CD, DA. Prove thai if
sin AP sin BQ sin Oil sin Z>S=sin BP sin CQ sin DR sin A3,
the four points PQKS lie in one plane.
7. 1, 2, 3, 4 are the vertices of a tetrahedron. A plane cuts
each of the six edges. If the edge 12 is out at A, and the ratio
mnlAfwi2A is denoted by (12), prove that (12)(23}(34)(41) = 1.
Conversely, ir {12)(23)(:U){ 11}- 1, prove that the points 12, 23,
34, 41 (i.e. the corresponding points on these edges) are coplanar.
8. If (12)(23)(34)(41)-I=(12)(24){43)(31) = (13K32){24)(41),
prove that either (i) the sets of points 12, 23, 31, etc., are collinear,
or (ii) the lines (12, 34), (13, 24), (14, 23) are concurrent.
150
ANALYTICAL CEOMETRY
[Ex. iy.
9. Four circles (ouch in succession, each raw touching two others
(the number of external contacts being even) ; show that the four
points of contact lie on a circle, and that the four tangents at the
points of contact touch a circle.
10. Four spheres touch in succession, each one touching two
others (the number of external contacts being even) ; show that the
Four points of contact lie on a circle, and that the Four tangent planes
at the points of contact touch a sphere. .Show Further that, whatever
the nature of the contacts, the four tangent planes pass through one
point.
11. Five spheres touch in succession, each one touching two
others (the number of externa) contacts being even) ; show that the
live points of contact lie on a sphere, and that the five tangent planes
at the points of contact touch a sphere. (Edue. Times (n.s.), xi
p. 57.)
12. D, E, F are the feet of the pcr]>endiculars from a point on
the sides of the triangle ABC. Prove that
cos BD cos CE cob A F = cos CD cos A E cos BF.
13. A BC is a given triangle, anil I is any line. P, Q, R are the
feet of the perpendiculars fr .1 . B, C on I. PP J ±BC, QQ'LCA,
RR'IAB. Prove that PP', QQ\ BR' meet in a point {the arik&pole
oil).
14. Prove that the locus of a point such that the ratio of the
cosines of its distances from two fixed points is const-ant is a straight
line.
15. If L, M, N ; £n, All, A\ ; etc., are the points of contact of
the in- and e-scribed circles of the triangle ABC with the sides
a, b, e. and 2s=a + b + e, prove the relations:
A M, ■ AJS t = BN t = BL t = CL 3 =CM 3 =s,
AM=AN = BN s =BLs=CL t = OM l =&-a, etc.
16. Establish the reciprocal relations to those in Question 15 for
the circumcircles.
17. Prove that the envelope of a line which makes with two fixed
lines a triangle of constant perimeter is a circle. Prove also that
the envelope is a circle if the excess of the sum of two sides over the
third side is constant. What is the reciprocal theorem ?
Ex, rv.
EXAMPLES
151
(In the following questions, 18-22, the geometry is hyperbolic.
The formulae are analogous to well-known formulae in spherical
trigonometry.)
18. If /.■«. kb. kc are the sides, and A, B, C the angles of a triangle,
prove that
A _ /sinh j) sinh (s - a) gm A _ /sinh (« - b) sinh (s -ej
2 t sinh b sinh c 2 \ sinh b sinh c
19. If r, r,, r t , r, are the radii of the in- and e-scribed circles of a
triangle ABC, prove that
tanh r sinh /s = tanh r, sinh {# tt} = tanh r, sinh (a - b)
=tanhr, sinh (s-c)
=v'sinh (s -a) sinh (a - b) sinh (a - c).
20. Prove that
tanh r, tanh r* tanhrj =\/sinh (a - a) sinh (a-b) sinh (» - c),
21. If R is the radius of the circumcircle of the triangle ABC,
prove that
. a , b c sinh a audi 6 sinh o
2 cosh -cosh _ cosh -tanh /? = — — r = . - = - ; „.
2 2 2 sin A am B sin
If D u D t , D 3 are the distances of the circumscribed equi-
distant-curves, prove that
. a , , b . , c - _ sinh a
2 cosh- sinh -sinh-oolh D,- , ■•> ete .
2 2 2 sin A
22. Prove that
eothif+tanh /J, -i fanh /) 3 I tanh JJ a ='2 cosh# sin A, 'sinh a,
col h R I tanh O, - tanh /J, - tanh l) 3 -2 cosh (n - a) sin i4/sinh a, etc
23. Prove that, in the desmic configuration in § 1 9, the following
sets of points are coplanar: AiA.BtB^Ct, AtAJBiBJSiCt. and
those obtained from these by cyclic permutation of ABC or of 234.
Deduce that the configuration has the symbol
12 4
6
3
16
S
6
4
12
152
ANALYTICAL GEOMETRY
[Ex.
24. If ono pair of altitudes of a tetrahedron ABCl) intersect,
prove that the other pair will also intersect; and if one altitude
intersects two others, all fonr arc concurrent. If these conditions
are satisfied, prove that
cos AB cosCO^cos AC cos BD=coa AD ma BO.
25. Prove that there is a circle which touches the in- and the
e-scribed circles of a triangle. [In spherical geometry this is Hart's
circle, and corresponds to the nine-point circle in ordinary geometry.
See M'Clelland and Preston's Spherical Trigonometry, Chap. VI
Art. 88.]
^ 26. Prove that there is a circle which touches the four circum-
ctrcles of a triangle, [In euclidean geometry the circumscribed
equidistant-curves are three pairs of parallel lines and form a triangle
A'B'C, of which A, B, C are the middle points of the sides, The
circumcircle of ABC m the nine-point circle of A'B'C. and touches
the inscribed circle of A 'B'C. That is. the last-named circle touches
the four " circumcireles ■ of the triangle A BC.J
CHAPTER V
REPRESENTATIONS OF NON-EUCLIDEAN GEOMETRY
IN EUCLIDEAN SPACE.
1. The problem of Representation is one that faces us
whenever we try to realise the figures of non-euclidean
geometry. There already exists in the mind, whether
intuitively or as the result of experience, a more or less clear
idea of euclidean geometry. This geometry has from time
immemorial been applied to the space in which we live;
and now, when it is pointed out to us that there are other
conceivable systems of geometry, each as self-consistent
as Euclid's, it is a matter of the greatest difficulty to
conjure up a picture of space endowed with non-cucltdunn
properties. The image of euclidean space constantly
presents itself and suggests as the easiest solution of the
difficulty a representation of non-euclidean geometry by
the figures of euclidean geometry. Thus, upon a sheet of
paper, which is for us the rough model of a euclidean plane,
we draw figures to represent the entities of non-euclidean
geometry. Sometimes we represent the non-euclidean
straight lines by straight lines and sometimes by curves,
according as the idea of straightness or that of shape happens
to be uppermost in the mind. But we must never forget
that the figures that we are constructing are only repre-
sentations, and that the non-euclidean straight line is
154 CONCRETE REPRESENTATIONS [v. 2
every bit as straight as its euclidean counterpart. The
problem of representing non-euclidean geometry cm the
euclidean plane is exactly analogous to that of map-
projection.
Projective Representation.
2. The fact that a straight line can be represented by an
equation of the first degree enables us to represent non-
euclidean straight lines by straight lines on the euclidean
plane. Distances and angles will not, however, be truly J
represented, and we must find the functions of the euclidean
distances and angles which give the actual distances and
angles of non-euclidean geometry.
3. The absolute is represented by a conic. In hyperbolic
geometry this conic is real, in elliptic geometry it is wholly
imaginary, but in every case the polar of a real point is
a real line. The conic always has a real equation. In
the case in which the absolute is a real conic, we could
if we like, represent it by a circle, but except in special
cases this does not give any gain in simplicity.
Two lines whose point of intersection is on the absolute
are parallel ; two lines whose point of intersection lies
outside the absolute are non-intersectors. The points
outside the absolute have to be regarded as ultra-infinite,
and are called ideal points. They are distinguished from
other imaginary points by the fact that, while their actual
coordinates are all imaginary, the ratios of their coordinates
are real. In the present representation they are repre-
1 In the sense of map-projections ; i.e. angles which are equal in the
euclidean representation, when measured hy euclidean standards,
do not in general represent oipin] angles in the non-euclidean geometry,
but, again in the sense of map- project ions, figures are distorted.
•■1
THE ABSOLUTE
155
sented by real points ; other imaginary points are repre-
sented by imaginary points. (Cf. Chap. IV. § 8.)
A real line has two points at infinity, and part of the line
lies in the ideal region. A line which touches the absolute
Idtrtt point!
Fro. SB.
has one point at infinity, and all the rest of the line is ideal.
A line which lies outside the absolute is wholly ideal.
Through any point two parallels can be drawn to a given
line, viz. the points joining the given point to the two
points at infinity on the given line. A triangle which has
its three vertices on the absolute has a constant area.
In elliptic geometry the absolute is imaginary, and there
are no ideal points.
4. Euclidean geometry.
Euclidean geometry is a limiting case, where the space-
constant k-><x> . The coordinates of a point become Hie
usual rectangular coordinates x and y with z = l. The
equation of the absolute becomes in point-coordi nates
z = 0, and in line-coordinates £« + »j a =0, ie. the absolute
degenerates as a locus to a straight line counted twice—
the straight line at infinity, and as an envelope to two
imaginary pencils of lines, i + i», = and £-iij=0, whose
156 CONCRETE REPRESENTATIONS fv.s
vertices lie on the line at infinity since the line-coordinates
of their join are f = 0, ,=0, f«£ and its equation is
therefore z=0. The equations of the lines of these ima-
ginary penci Is are of the forms x + iy + at = 0,x-iy + cz=Q.
The formula for the distance between two points,
ja£+0£+* s a*'__^
r
cos 7= ._
■Jx 2 + -f + Ktyyfx^+y'* + jfcVi'
becomes 1 -\ • jj.^ + yy' + k *) ■ £ . (l _ J . ^/ )
V 2 JF" /
~( ii ^y | 6- 1 ■ ^ + ^. +i ' a tii s \
or
-i- 1 (^T + to-yT
r 2 = (j;-^) 3 + (y-y')2.
5. The circular points.
The equation of a circle becomes of the general form
x* +y* +z(ax+by + cz)=Q,
and this represents a conic passing through the points of
intersection of the line s=0 with the pair of imaginary
lines x + iy =0 and x - iy = 0, i.e. every circle passes tliiou-h
the vertices of the imaginary pencils. For this-; reason
these two points are called the circular points. This pro-
perty of the circle is the equivalent of the property that
it has double contact with the absolute. (Chap. IV. § Hi.)
6. Now, in ordinary geometry the angle between two lines
can be expressed in terms of the two lines joining their
point of intersection to the circular points. 1
BteE uffsssT " mtC 8Ur b a "" ;oriC deS foyc,rs '" AW Ann - J/ ""' ■
v. 71 ANGLE AND DISTANCE FUNCTIONS 157
Let the equations of the two lines w, «' through the
origin be y=x tan 6, y =x tan 0', and denote the two lines
joining to the circular points by m, w \ their equa-
tions are y = ix, y^ -kk. The cross-ratio of the pencil
(ttu',<m') is tan (J -i^ tan 0' -i
tan 6 + i ' tan 0' +i
Now
i-t&ad a cos 0- sin cosfl+isinfl
i + tan& icosfl+ainrJ cos 0-* sin W
Therefore (« u', «»') = e s '<* ' *\
and
=c 2,s .
= 0' - Q = | log (u»'. mw').
i.e. fte aflfffe between two tines is a certain multiple of the
logarithm of ilw cross-ratio of the pencil formed by the too
lines and the lines joining their point of intersection to the
circular points.
7. Now let us return to the case where the absolute is
a real conic a£ + y* - Bz z = 0. Consider two points P(x,y,z),
P'{x', y', z'). The point (x + \x r , y+\t/, s+Xz') lies on
their join. If this point is on the absolute,
(x + \x') 2 + (y+\yy-&(z + te'?=0,
i.e. X* (x' a + y'* - A-V s ) + 2X (xx 1 + yy' - 1M)
+ (x % + f- -&*z*)=0.
Let X,, A j be the roots of this quadratic. The line PP'
cuts the absolute in the two points X, Y, corresponding to
these parameters, and the cross-ratio of the range
{PF,XY)=h
Let {PF)=d = k<j>, and
X s + y* - £*z 2 = r*, x' a + y' 1 - AV 2 = r' 2 ;
158
CONCRETE REPRESENTATIONS
[v, 8
then the quadratic for \ becomes
A V 3 + 2\rr' cos ih + r 3 = ;
whence X lf X 2 = ( - cos oi:t J - sin 3 0)r/r' = - e +f *r/r'.
Therefore XJK^e-*'* and £ = A* log (W, XT).
Therefore d = Hi log (PP', 1 V ) ,
i.e. f/i« distance between two points is a certain multiple of
the logarithm of tJte cross- ratio of live range formed by the two
points and the two points in which their join cuts the absolute.
In a similar way it can be shown that the angle between
two straight lines is a certain multiple of the logarithm of tJte
cross-ratio of the pencil formed by the two lines and the two
tangents from their point of intersection to the absolute.
If the unit angle is such that the auglu between two lines
which are conjugate with regard to the absolute is |tt, then
<f> = \i\og{pp', xg).
8. By this representation the whole of metrical geometry
is reduced to projective geometry, for cross-ratios are
unaltered by projection. Any projective transformation
which leaves the absolute unaltered will therefore leave
distances and angles unaltered. Such transformations are
called congruent transformations and form the most general
motions of rigid bodies.
This projective metric is associated with the name of
Cayley. 1 who invented the term Absolute, lie was the
1 "A sixth memoir upon quantics," London Phil. Trans. R. Sor... 149
(1859). Cayley wrote <i number of papers dealing specially with non-
euolidean geometry, but although he imisl he regarded us une of the
epoch-makers, he never quite arrived at a just appreciation of the science,
lit liis mind nou-cuclidean geometry scarcely attained to an indejieiidcnt
existence, hot was always either the geometry upon a certain claas of
curved surfaces, like spherical geometry, or a mode of representation
of certain projective rotations in euolidean geometry.
9]
PROJECTIVE METRIC
first to develop the theory of the absolute, though only as
a geometrical representation of the algebra of qualities.
Klein ' introduced the logarithmic expressions and showed
the connection between Cayley's theory and Lobaehevsky s
geometry. 2
9. As an example of a projective solution of a metrical problem,
let us find the middle points of a, segment PQ. Let PQ cat the
absolute in X, Y, and let 31,, M s he the double points of the involu-
tion {PQ, XY). Then (X YPM,)7:{ YXQMJ7=:(X YM k Qh therefore
tli9t.(/*jl/,)=dist(Jf 1 Q), jtf„ ju% are therefore the middb points
of the segment (PQ).
Since M u M t are harmonic conjugates with respect to X, Y and
also with respect to P, Q, the construction is therefore as follows.
1 " Uber die sogenannte Nicht-Euklidischo Geometrie," Math Ami.,
4(1871), 6(1873).
1 Since the definition of the cross-ratio of a range is the same in non*
euolidean geometry, the logarithmic expressions for distance and angle
hold not only in the eucb'dean representation of the geometry, but also
in Hie actual non-ouclidean geometry itself.
ICO
CONCRETE REPRESENTATIONS [v. 10
Join 0, the pole of PQ, to P and Q, cutting the absolute in A A', BB\
AB\ A'B intersect in M, and AB, A'fi' in M 3 . For by this con-
struction OM l M s is a self-conjugate triangle and M,, M 3 are harmonic
conjugates with respect to A', Y, and also with respect to P, Q.
10. Classification of geometries with projective
metric.
Having arrived at the result that metrical plane geometry
is projective geometry in relation to an absolute conic,
distances and angles being determined by the projective
expressions
dist. (PQ) = K log (X Y, PQ), angle (pq) = h log (xy, pq),
we may reverse the process, and define distances and angles
by these expressions. We thus get a general system of
geometry which will include euclidean, hyperbolic and
elliptic geometries as special cases. The nature of the
geometry will be determined when the absolute conic is
fixed, and the values of the constants K and k have been
determined. Generally speaking, the values of these
constants depend only on the units of distance and angle
which are selected, but there is an essential distinction
according as the constants are real, imaginary or infinite.
There is no distinction, for example, between the cases
corresponding to different real values of K. This simply
corresponds to a different choice of the arbitrary unit of
length ; just as in angular measurement the constant k may
be chosen so that the measure of a right angle may be
- or 180 or any other number. As each of the constants
may conceivably be real, infinite or imaginary, there are
nine species of plane geometry.
The points of the absohite are at an infinite distance
v. llj
THE EUCLIDEAN CASE
161
from all other points, and the rangente to the absolute
make an infinite angle with all other lines.
If the measure of angle is to be the same as in ordinary
geometry, the tangents from a real point to the absolute
must be imaginary ; the cross-ratio (pq, xy) will be imagi-
nary, and k must be purely imaginary. When p, q are
conjugate with regard to the absolute they are at right
angles, and if the unit angle is such that the angle in this
case is ^ , then k = \i.
Then there are three cases according as the absolute is a
real proper conic (hyperbolic geometry, K real), an ima-
ginary conic (elliptic geometry. K imaginary), or degenerate
to two coincident lines and two imaginary points (parabolic
or euclidean geometry. K infinite).
11. In the last case there is a difficulty in determin-
ing the distance. Since X. Y coincide, the cross-ratio
(PQ, XY) is zero and K must be infinite; but the dis-
tance becomes now indeterminate.
Suppose PY=PX + e, where e is small.
Then (PQ^y^fxVA^Q^i^A)' 1
QXi
x+6 \QX"Pxr
neglecting squares and higher powers of e.
and (PQ) = K\og(PQ,XY) = K*Q x -±>).
Let K—~kx> and «— >0 in such a way that Ke- >a finite
limit A.
Then
(pQ)-*px-qx-
S\-E. Q.
162
CONCRETE R E PRESENTATIONS
[v. 12
Now, to fix X we must choose a point E such tbat
(PE) =1, the unit distance.
PX.EX PQ XB.XQ , yp ™.
PE ' P~X7QX = PE • PQ-\ At ' B ^ h
Then (PQ) =
If we measure distances from P = as origin,
<GQ)=(AD, EQ) = (0«>, £1)=^^,
which agrees with the expression in euclideau geometry,
since ;=1, and 01 = 1.
oo 1
This case differs in one marked respect from the case of
elliptic geometry. In that system there is a natural unit
of length, which may be token as the length of the complete
straight line — the period, in fact, of linear measurement ;
just as in ordinary angular measurement there is a natural
unit of angle, the complete revolution. In euclideau
geometry, however, the unit of length has to be chosen
conventionally, the natural unit having become infinite.
12. The other geometries, in which the measure of angle is either
hyperbolic or parabolic, are of a somewhat bizarre nature.
For example, if the absolute degenerates to two imaginary lines
a, <i)', and two coincident points ii, the case is just the reciprocal
of the cuclidean case; Sinoar measurement in elliptic, K being
imaginary, and angular measurement is parabolic, k being infinite.
In this geometry the straight lino is of finite length =!rA'i". If the
positive direction along any one line is defined, the positive directions
along alt other lines in a plane are determined, for this is determined
by the sense of rotation about tiie point Ii. The sides of a triangle
are defined as the segments which subtend the opposite angles
which do not contain ii, just as in ouclidean geometry the angles
of a triangle are defined as the angles which arc subtended by the
opposite segments which do not cut the line at infinity.
Thus the sides of the triangle A BC in (he figure ( Fig, 88) are
represented by the heavy lines. If the positive direction on each
v. 13] RECIPROCAL OF EUCLIDEAN CASE 103
line is then defined as the direction corresponding to clockwise
rotation about il. then
a + ti-t e=the length of the complete line,
i.e. the pciimeier of a triangk is constant and = 5rJsT»»
Fio, SB.
13. Extension to three dimensions.
In three dimensions the absolute is a quadric surface.
If the measures of angle between lines and between planes
are to be elliptic, the tangent planes through an actual line
must be imaginary, and the tangents through an actual
point in an actual plane must be imaginary.
(1) Let the quadric be real. If the quadric has real
generators, i.e. if it is a ruled quadric, every plane cuts it
in a real conic, for it cuts all the generators in real points.
Actual points must lie within the section and actual lines
must cut the surface. But the tangent planes passing
through a line which cuts a ruled quadric are real, and so
the measure of dihedral angles would be hyperbolic. The
quadric cannot therefore be ruled.
Through a hue which does not cut a non-ruled quadric
164
CONCRETE REPRESENTATIONS
[v. 1 4
two real tangent planes pass; hence actual lines, and there-
fore planes, must cut the surface, and actual points are
within the surface, " This gives hyperbolic geometry. The
absolute could be represented by a real sphere. All points
outside the sphere are ideal points.
(2) Let the quadric be imaginary. The measure of
distance is also elliptic, and the geometry is elUftic.
(3) Let the quadric degenerate. If the quadric degene-
rates to a cone, necessarily with real vertex, the measure
of dihedral angle must be parabolic. If the quadric
degenerates to two planes, unless the planes coincide they
will have a real line of intersection and the measure of
plane angle must be parabolic. Hence the quadric must
degenerate to two coincident planes.
A quadric which reduces, as a locus, to two coincident
planes, reduces as an envelope to a conic lying in this plane.
If the measure of angle is elliptic this conic must be ima-
ginary. This is the case of euclidean geometry. The
absolute consists of an imaginary conic in the plane at
infinity. Any quadric which passes through this conic is
cut by every piano in a conic which passes through the
two absolute, or circular, points in this plane, i.e. every
plane section is a circle, and the quadric is a sphere. The
imaginary conic itself must be regarded as a circle since
it is the plane section of a sphere. This is the imaginary
circle at infinity.
14. Other three-dimensional geometries can bo constructed in
which the measure of plane or dihedral angle is hyperbolic or para-
bolic, but they are not of much interest, as they resemble ordinary
geometry too slightly.
One application of these bwarro geometries may bo given. It is
obvious that in euclidean space the geometry on the plane at infinity
is elliptic, since the absolute consists of the imaginary circle in this
i.->|
CKODKSIC RKi'lIKSKNTATluX
165
plane, and it follows, as we have already seen, that the geometry
of complete straight lines through a point is elliptic, the geometry
of rays, or of points on a sphere, being of course the spherical or
antipodal variety.
Now consider three-dimensional hyperbolic space. A tangent
plane to the absolute rata the absolute in a degenerate conic con-
sisting of two imaginary straight lines and two coincident points fl ;
hence the geometry on such a plane is the reciprocal of euclidean,
i.f. the measure of distance is elliptic, while angular measurement is
parabolic. Now, the polar of a |>oiiit (or line) on this plane is a
plane (or line) passing through H Hence, by this second reciproca-
tion, we find that the geometry of a bundle of parallel lines and planes
is euclidean, and if we cut the system by the surface (horosphere)
which cuts each line and plane orthogonally, wo find that the geometry
on the horosphere ts euclidean.
Geodesic Representation.
15. It has been seen that the trigonometrical formulae
of elliptic geometry with constant k are exactly the same
as those of spherical trigonometry on a sphere of radius k,
and therefore elliptic geometry can be truly represented on
a sphere, straight lines being represented by great circles,
and antipodal points being regarded as identical. Within
a limited region of the sphere which contains no pair of
antipodal points, the geometry is exactly the same as
elliptic geometry. We do not require, as in Cayley's
representation, to obtain a distance- or angle-function ;
distances and angles are represented by the actual distances
and angles on the sphere.
The corresponding representation for hyperbolic geometry
appears at first sight to be imaginary, since hyperbolic
geometry is the same as the geometry upon a sphere of
purely imaginary radius. It is possible, however, to obtain
a real representation of this kind, though confined to a
limited portion of the hyperbolic plane.
166
CONCRETE REPRESENTATIONS [v. 10
16. Geometry upon a curved surface.
We must first understand what we mean by the geometry
upon a surface which is not, like the sphere, uniform. The
straight line joining two given points has the property that
the distance measured along it is less than that measured
along any other Hue joining the same two points. This
is the property which we shall retain upon the surface.
A curve lying on a surface and having this minimum
property is called a geodesic. The geodesies of a sphere are
all great circles. Now, if a surface can be bent in any way,
without stretching, creasing or tearing, geodesies will
remain geodesies, lengths of lines and magnitudes of angles
will remain unaltered, and the geometry on the surface
remains precisely the same. Two surfaces which can be
transformed into one another in this way are called ap)>li-
cable surfaces.
If, for example, a plane is bent into the form of a cylinder,
the geometry, at least of a limited region, will be unaltered.
The same holds for any surface which can be laid flat or
developed on the plane.
The sphere is a surface which cannot be developed on the
plane, and it possesses a geometry of its own, A complete
Sphere cannot in fact be bent- at all without cither si i-ptching
or kinking, but a limited portion of it can be bent into
different shapes without altering the character of the
geometry.
17. Measure of curvature.
Now the sphere and the plane possess this property in
common, that congruent figures, e.g. triangles with equal
corresponding sides and angles, can be constructed in any
positions on the surface, or, to use the language of kinc-
17]
MEASURE OF CURVATURE
167
matics, a rigid figure is freely movable on the surface. It
follows that the surface is applicable to itself at all its
points. This property is expressed analytically by saying
that there is a certain quantity, called the measure of
curvature, which is the same at all points of the surface and
is not altered by bending.
To see what this invariant quantity is, consider any plane
section of a surface passing through a tangent hue OT
at ; the section is a curve having this line as tangent
at 0. The more obliquely the plane cuts the surface the
sharper is the curvature of the section, until, when the plane
touches the surface at 0, the section is just a point.' The
section of least curvature occurs when the plane is per-
pendicular to the tangent plane, or passes through the
normal to the surface.
Again, if we revolve the cutting plane about the normal,
the curvature of the section will vary continuously and
have a maximum and a minimum value. These occur for
sections at right angles, and are called the principal curva-
tures of the surface at 0. The curvature of a curve at a
point being defined as the reciprocal of the radius of the
circle of closest fit to the curve at 0, the product of
the principal u uiv a fa tt es at 0, denoted by M, is called the
measure of curvature of the surface at 0. If the two curva-
tures are in the same sense M is positive, if in opposite
senses M is negative ; if one is zero, as in the case of a
cylinder or any developable surface, M is zero. For a
sphere of radius it, M is the same at all points and = 1/ft*.
1 This holds for a convex surface like a sphere. In the general case
the section of a surf.vr l.v a tangent plane is a curve which has a node
ai Hie pnim <>r .outset, with real or imaginary tangents. In the cn»c ot
a surface of the second degree the section consists of two straight lines,
real or imaginary. In the text we arc considering tho case of a node
with imaginary tanjsenU, which apjicars just as a point.
168 CONCRETE REPRESENTATIONS [v. 18
18. Surfaces of constant curvature.
C4auss, who founded the differential geometry of surfaces,
as well as being almost the discoverer of non-euclidean
geometry, discovered the beautiful theorem * that wlmi
a surface is bent in any way without stretching or kin/ting,
the measure of curvature at every point remains vnaUered.
It follows, then, that the only surfaces upon which, free
mobility is possible are those which are applicable, upon
themselves in all positions, and therefore for which M has
the same value at all points.
There are three kinds of surfaces of constant curvature,
(1) those of constant positive curvature, of which the
sphere is a type ; (2) those of constant negative curvature,
saddle- backed at all points tike a " diabolo " ; (3) those of
zero curvature, the plane and all developables.
19. The pseudosphere.
Fortunately, we do not require to take an imaginary
sphere as the type of surfaces of constant negative curva-
ture. There are different forms of such surfaces, even of
revolution, but the simplest is the surface called the
Pseudosphere, which is formed by revolving a tractrix about
its asymptote.
The tractrix is connected with the simpler curve, the
catenary, which is the form in which a uniform chain
hangs under gravity. The equation of the catenary referred
to the axes Ox, Oy is y=Acosh| It has the property
that the distance of the foot of the ordinate N from the
tangent at Q is coustant and equal to k, while QP= the
im£ 1 2™**' Disquisaio,m S^nmUts circa superficies cumu, Oiittingcn,
v. in |
THE PSEUDOSPHERE
\m
arc AQ. It follows then that if a string AQ is unwound
from the curve, its extremity will describe a curve AP with
the property that the length of the tangent PN is constant
and equal to h This curve is the Tractrix. Ox is an
asymptote. Now, if the tractrix is revolved about the
asymptote we get a surface of revolution whose principal
sections at P are the meridian section in which the tractrix
lies, and a section through the normal PT at right angles
to the plane of the curve. The radii of curvature of these
sections are respectively PQ and PT, and we have
PQ . PT = PN 2 = t?, but since the curvatures are in opposite
senses, the measure of curvature = - 1/k 1 .
The pseudosphere, therefore, gives a real surface upon
which hyperbolic geometry is verified— within a limited
region. The surface does not, of course, represent the
whole of the hyperbolic plane, for it has only a single point
170
CONCRETE REPRESENTATIONS fv.M
at infinity. The meridian curves are geodesies passing
through this point at infinity, and therefore represent
a system of parallel lines. So the surface only corresponds
Mb. bo,
to a port-ion of the hyperbolic plane bounded by two
lines and an arc of a horocvcle. 1
20. The Cay ley- Klein representation as a projection.
Through the medium of the geodesic representation we
can now get a geometrical interpretation of the Cayley-
Klein representation. If we project a sphere centrally,
great circles are projected into straight lines, since their
planes pass through the centre of projection. Hence the
Cayley- Klein representation of elliptic geometry can be
regarded as the central or gnomonic projection of the
geometry on a sphere.
The equations of transformation are easily found.
'The intrinsic equation of the Iraetrix is a=fctog ensce ifi. and since
y = isiu^ we have <j—ke ■■*. The ratio of the corresponding urcs of
two horocycles (sections _|_ to tin axis) is therefore «"-"*, which agrees
with the expression wc have already found (Chap. II. § 28).
v. 21]
CENTRAL PROJECTION
171
Let the plane of projection be chosen for convenience
as the tangent plane at 0, and take rectangular axes, Oz
through the centre of the sphere 8, and Ox, Oy in the plane
no. «i.
of projection. Let the coordinates of P, any point on the
sphere, be (x, y, z) and the coordinates of its projection F
be(/,/,0). Tbm#+tf+(* -if-*,
af_,/ OF OS k
x~y~ON OS-NP h-z
~J®-x*-tf~ *
21. Meaning of Weierstrass' coordinates.
Let it, v, w be the angles which SP makes with the planes
yz, zx and xy ; u, v, w can he regarded as coordinates on the
sphere, and
z=&sini&, y=h.&mv, z-k = k$mw.
Then k sin u, jfcsinw and sin ware Weierstrass' coordi-
172 CONCRETE REPRESENTATIONS \v. 22
nates, denoted by X, Y, Z, and connected by the relation
3P+r*+l*E*-JP, In terms of the spatial coordinates
ofP, X=x, Y=y,kZ=z-k.
We see thus that Weierstrass 1 coordinates are pro-
portional to the sines of the distances of P from the sides
of a self-polar triangle, and are therefore analogous to
trilinear coordinates.
Dually, the line-coordinates of a line are defined as pro-
portional to the sines of the distances of the line from the
vertices of the fundamental triangle. We may also define
the point-coordinates as proportional to the cosines of
the distances from the vertices, and the line-coordinates
as proportional to the eosiues of the angles which the
line makes with the sides of the triangle. In the three-
dimensional representation the point-coordinates are the
direction-sines of the point referred to rectangular axes.
Conformal Representation.
22. Stereographic projection.
There is another very useful projection of a sphere, the
stereographic projection. In this case the centre of pro-
jection is taken on the surface.
Let S be the centre of projection, and the centre of the
sphere of radius k. Take the plane of projection perpendi-
cular to SC, and at distance SO=d. Choose rectangular
axes with OS as axis of z. Let the coordinates of P, any
point on the sphere, be (x, y, z), and the coordinates of its
projection P' be (x', y', 0). Then, since SPA is a right
angle - SOP', SP . SP' = SA . SO = 2H.
The formulae of transformation are :
x' = y' ^OP' ^SP' _ d m x^+y'*+d*
x y ON SP 4-z x* + if + (z-df a83 *
v. 221 STEREOGRAPHIC PROJECTION 173
If the plane of projection is chosen to pass through C,
and the xy plane of P is the tangent plane at S,d = k, and
the formulae become :
x' _ y' I 2fc* = x'^+ y' & + #
x~y~z 85*+/ + ^ 2& £
FIO. 62.
A plane ax + by + cz + d=Q becomes
2fc* (atf +by'+ek)+ d{a;' B + y Ji + k*) = 0,
which represents a circle. Hence all circles on the
are represented by circles.
Consider two planes
h+my + nz=nh l'x+m'y+n'z = n'k
through the centre, and cutting the sphere in great
174 CONCRETE REPRESENTATIONS [v. 23
circles. The angle between the great circles is equal to
the angle between the planes, and is given by
cos = W + mm' + nn'.
The projections of the great circles are the circles
2A- 2 (h + my + nk) - nk (s* + y 2 + &*) = 0,
The angle <f> at which the circles cut is given by
therefore cos = H' + mm' + nn\
i.e. the projections cut at the same angle as the great circles.
Stereographic projection is three-dimensional inversion,
for SP.SP' = const., circles are changed into circles, and
angles are unaltered. A small figure on the sphere will
therefore be projected into a similarly shaped small figure
on the plane, with corresponding angles all equal. For this
reason the representation is called nmformul.
23. The orthogonal circle or absolute.
A circle in the projection, which represents a straight line
in the non-euclidean geometry, has for its equation
* & + f) ~2k(h + my) - n& = 0,
and this cuts at right angles the fixed circle
This circle, imaginary in elliptic geometry, real in hyper-
bolic, is the projection of the absolute, which cuts all
straight lines at right angles.
v. 25] CONFORMAL REPRESENTATION 175
24. Conforms! representation.
We shall now consider generally the problem of the
conformal representation of the non-euclidean plane upon
the euclidean plane, straight lines being represented by
circles. We shall set aside stereographic projection
entirely, as this assumes the geodesic representation on a
sphere, and treat the problem purely as a problem in
correspondence.
It may be shown directly that the circles which represent
straight lines all cut a fixed circle orthogonally. For, if
the circles
x* + 1/* + 2#x h Ify + c =
represent straight lines, they must have the property of
being determined uniquely by two points. Hence the
three coefficients g, /, c must be connected by a fixed
equation of the first degree, say
-'/.'/' +2#' = c + c',
but this is just the condition that the circle should cut
orthogonally the fixed circle
x i +y t +2g'z + 2fy + e'=0.
In elliptic geometry this circle is imaginary, in hyperbolic
geometry it is real. If the fixed circle reduces to a point
(the transition between a real and an imaginary circle), all
the circles which represent straight lines pass through this
fixed point. Now, if we invert the system with this point
as centre, the circles become straight lines. Hence the
straight lines of euclidean geometry can be represented by
a system of circles passing through a fixed point.
25. Point-pairs.
This representation has a fault which we must try to
correct. In non-euclidean geometry, hyperbolic or elliptic,
17(1
CONCRETE REPRESENTATIONS
|v. 28
two straight lines intersect in only one point ; but the
circles which represent tlieni intersect in a pair of points.
In the representation of hyperbolic geometry the fixed
circle is real, and two orthogonal circles may intersect in
real, imaginary or coincident points, according as the
straight lines which they represent are interseetors, non-
intorsectors or parallel. The pair of points which corre-
spond to a single point are inverses with respect to the
fixed circle. We must therefore consider pairs of points
which are inverses with respect to the fixed circle as forming
just one point.
In the representation of spherical geometry, as distinct
from elliptic geometry, the points of a pair will be con-
sidered as distinct and constituting a pair of antipodal
points.
In the representation of eucMean geometry, one of the
points of a pair is always the fixed point itself.
26. Pencils of lines. Concentric circles.
To a pencil of lines through a point P corresponds a
pencil of circles through the two points P u P t which
correspond to P. The radical axis of this system is itself
a circle of the system, and is in no way distinguished from
any other circle of the system.
The representation of straight lines by circles is not
necessarily a conformal one, nor is it by any means the only
possible conformal representation. If the representation
is conformal we can show that when straight lines are
represented by circles, circles also are represented by circles. 1
For a system of concentric circles cut all the lines of a pencil
with vertex P at right angles. They will therefore be
1 For the converse of this theorem, see Ch»j>. VIII. 8 2.
PEN-OILS OF LINES
177
Fig. 03.
n.-b. o
178
CONCRETE REPRESENTATIONS \v. 26
represented by a system of curves catting orthogonally a
system of coaxal circles ; but. this is also a system of coaxal
circles, and has P lt P 3 as limiting points (Fig. 93).
A circle is represented actually by a pair of circles ;
these are inverses with respect to the fixed circle, and are
coaxal with the fixed circle. The two limiting points form
its centre.
Corresponding to a pencil of lines through an ideal
point F, i.e. a system of lines cutting a fixed line I at right
angles, we have a system of circles cutting at right angles
the fixed circle and the circle /' which represents the fixed
line (Fig. 94). But this is a coaxal system whose radical
axis is the common chord of the fixed circle and the circle I' ;
its limiting points are the common points Q x and Q* of V
and the fixed circle. The circles with centre P are equi-
distant-curves, and are represented by a system of circles
passing through Q„ Q 2 ,
Corresponding to a pencil of lines through a point at
9)
.' /
FlO. 00.
28]
CONCENTRIC CIRCLES
179
infinity P, i.e. a system of parallel lines, we have a system
of circles cutting the fixed circle orthogonally at a fixed point
on it (Fig. 95). The horocycles, circles with centre P, are
represented by circles touching the fixed circle at thia
point.
27. The distance between two points.
If ABC is a circle which represents a circle in non-
euclidean geometry, and is the point which represents
its centre, the radii are represented by arcs of circles through
cutting the given circle and the fixed circle orthogonally.
The arcs OA, OB. OC, .,. represent equal distances in non-
euclidean geometry. We require then to find what
function of the positions of the points and A represents
the distance between their corresponding points.
28. motions.
In order to investigate this function we shall make use of
the idea of motion. By a motion or displacement in the
general sense is meant not a change in position of a single
point or of any bounded figure, but a displacement of the
ISO
CONCRETE REPRESENTATIONS [v. 29
whole space, or, if we are dealing only with two dimensions,
of the whole plane. A motion is a transformation which
changes each point P uniquely into another point P' in
such a way that distances and angles are unchanged. It
follows that straight lines remain straight lines, and the
displacement is a particular case of a collineation (the general
one-one point- transformation which changes straight lines
into straight lines). Further, it will change circles into
circles, and the fixed circle must remain fixed as a whole.
We require therefore to find what is the sort of transforma-
tion of the euclidean plane which will change circles into
circles and leave a fixed circle unaltered.
29. Reflexions.
The process of inversion with respect to a circle at once
suggests itself, since this transformation leaves angles
unaltered and changes rircles into circle*. KuHher, since
rq. or.
the fundamental circle must be unaltered as a whole, the
circle of inversion must cut it orthogonally. Let us then
v. :w)
MOTIONS AST) REFLEXIONS
181
consider inversion in a circle which represents a straight
line.
In euclidean geometry, when the circle of inversion
becomes a straight line, inversion reduces to reflexion in
this line. Now any motion or displacement in euclidean
geometry can be reduced to a pair of reflexions in two suitably
chosen lines.
If AB is displaced to A'B' (Fig. 97), first take MO the
perpendicular bisector of A A' ; the reflexion of AB in MO
is A'B X . Then take M'O the perpendicular bisector of
Bfl, which passes through A', and the reflexion of A'B Y
is A'B', Since OB = OB 1 = OB', lies also on the perpen-
dicular bisector of BB', and is in fact the centre of
rotation for the given displacement.
30. Complex numbers.
We shall find now what is the most general transformation
which changes circles into circles and the fundamental
circle into itself.
The equation of any circle is
a* -m/ 2 + 2gx + 2fy+c = 0.
The procedure is greatly simplified by the introduction
of complex numbers and the use of Argand's diagram.
Let z^x+iy, p=g+if, and write the conjugate complex
numbers z=x-iy, p-g-tf- Then the equation of the
circle becomes
zz + pz+pz+c=0, (1)
a lineo-linear expression in z, z. Its centre is z= -p 7 and
the square of its radius is pp - c.
182 CONCRETE REPRESENTATIONS [v. 31
31. Circular transformation, conformal and homo-
graphic.
Now it is proved in the theory of functions that any
transformation of the form
*-/</), z=f(z')
is conformal, leaving angles unchanged. A real trans-
formation of this form which leaves the form of the equation
(1) unaltered, i,e. which changes circles into circles, is one
in which z, z' both occur only to the first power, 1 or
az' +8 -_aZ+J3
where «, 8, y, S are any complex numbers such that uS=j=8y.
By this transformation any complex number z is trans-
formed into a complex number z', and the point [x, y)
corresponding to z is transformed into the point (x', i/')
corresponding to z'.
If z,, z a , 23,34 are any four complex numbers which are
transformed into z,', z 2 ', 2*,', a/, and if we define the cross-
ratio (zjZsj, SjZj)
Zi -Z t " z« - z t
then (z,z„ , Zgz 4 ) = (%'«,', 23'?/).
The transformation is therefore said to be tomographic.
Let % be the modulus, and 13 the amplitude of the
complex number z, -23, so that z t -z 3 =r n e m <=, then the
cross- ratio (2,2,, z&,) has modulus -S /[** and amplitude
1 Xba only other type of real tranaformation having this proporty is
Mint which diffore only from thi* one by interchnngine; z and i. Bat
tln.» oniy differs from the former by a. reflexion in the z-nxis, z— z\
'
v.MJ HOMOCRA1MIH' TRANSFORMATIONS 183
The cross-ratio is real only when its amplitude is a
multiple of it, is. when the points corresponding to the
four numbers z,, z t , Zs, z t are eoncyclic, and its value is
then
PiPa ' *A
(Fig. 98).
Flo. (IS.
The transformation has to satisfy the further condition
that it transforms the fundamental circle into itself.
It can be proved that if the fundamental circle is
a? +?/* + /£=(), or 22 + A"=0, the general form of the
transformation is
tit - Kj$
~ frf+a "
K the. fundamental circle is Jf =--0. or 2 = s, it can be proved
that the general transformation is
az' +b
~cz' ±d'
where <*, 6, c, d are real and ad+bc.
32. Inversion.
Consider now the equations of transformation of inversion
in a circle cutting the fundamental circle orthogonally.
s = -
z =
184
CONCRETE REPRESENTATIONS
\v. :t-2
The equation of a circle cutting zz + K = orthogonally
ia zz+pz+pz-K = 0.
Let G( -p) be the centre, P(z) and P'(z'} a pair of inverse
points.
Fra. as.
Let the complex numbers represented by CP and CP'
be u, u'. Then
z=~p + u, z'=-p+ u '.
Also, since u, «' have the same amplitude, and the
product of their moduli is equal to the square of the radius
of the circle of inversion,
uu'=pp + K.
Therefore (z + p) (z' +p)-pp-g^Q i
or &'+pz+pz'~K=Q,
._ -pi' + K
z'+P '
%.e.
v. 331
IXVKltSIuNS AXD MOTION'S
185
A second inversion in the circle zz + qz +qz -K-0 gives
(K + ]xj)z"-K{p-q)
(p-q)2r + (K + pg)-
This will not hold when the circle of inversion is a straight
Hue — (j>. Here inversion becomes reflexion, and we have
z=re if >, s'=rc (:! *- e) , ?-l**-*»;
therefore z = z'e-'* ,
This combined with an inversion gives
-pz" + K ....
r +p
Then,
Let tf> - I - ^, /? =6* ; then e 2 '+ - -«-»* « -|.
if p,8=«, the transformation becomes
iisf-K jS
% ~ f-izT+n ■
Hence in either case the transformation is of this form.
Hence the general <l is placement of a plane figure is eqaivalent
to a pair of inversions in two circles which cut the fundamental
circle orthogonally.
33. Types of motions.
In the general displacement there are always two points
which are unaltered, for if z' =z we have the quadratic
equation
0z z + {a-a)z + K/3=O.
If we substitute z = - Kf%, the equation becomes
%' a + (« < -«)z'+^/9 = 0;
therefore z' is also a root. The two points are therefore
inverses with regard to the fundamental circle. This
point-pair corresponds to the centre of rotation in the
general displacement. In hyperbolic geometry there are
186 CONCRETE REPRESENTATIONS [v. 34
three distinct types of displacement according as the
centre of rotation ia real, ideal, or at infinity. The first
case is similar to ordinary rotation ; the second case is
motion of translation along a fixed line, and points not on
this line describe equidistant-curves ; in the third case
all points describe arcs of horocycles.
34. The distance-function.
We have now to find the expression for the distance
between two points P, Q, i.e. the function of their co-
ordinates or complex numbers (zj, z 2 ). which remains
invariant during a motion.
The two points determine uniquely a circle cutting the
fundamental circle orthogonally hi A', F, This circle
represents the straight line jo inin g PQ, and X, Y represent
the points at infinity on this line. If the motion is one
of translation along this line, the straight line as well as
the fundamental circle are unaltered, and X, Y arc fixed
points. Let x, y be the complex numbers corresponding
to X, Y; then the cross-ratio (z^, xy) Temains constant.
If we suppose, therefore, that for points on this line the
distance (PQ) is a function of this cross-ratio, wc can write
(PQ)=f(h> z 2 ). If P> 6- R are three points on the line,
corresponding to the numbers %, % %, this function has
to satisfy the relation (PQ) + (QR) = (PR), or
Ah,^)+/(s2,^)=f(h-^)-
This is a functional equation by which the form of the
function is determined. Consider z as a parameter deter-
mining the position of a point, and differentiate with regard
to ?j. Then, since
v«i 2. m Zi _ y z ^_ x pY g X ,
v. 35]
we have
THE DISTANCE-FUNCTION
187
, QY 3 fPX\ », . , RY 3 fPX\
1 k- ^ qx §£ \py) =f {Zl ' h) rx % \py)'
Hence
f'(z u z 2 )_QX RYfPX RY\_(PX QY\JW a ,W)
J (z, ,z s )~QY' RX " VP Y ' RXJ -\PY' QXJ (z jZz , xy)'
i.e. (%, s 2 )/'(^, z 2 )=( z i, ts)f'(*i, %)= const. =^.
Integrating, we find
and substituting in the functional equation we find C-0.
Hence
(PQ) = M log ( 2l z 3J xg) = M log (py ■ §£) =M l0g {PQ - XY) >
(PQ, X Y) being the cross-ratio of the four points P, Q, X, Y
on the circle, i.e. the cross-ratio of the pencil Q(PQ, XY),
where is any point on the circle.
In hyperbolic geometry K= -&, and the fundamental
circle is real. The distance between two conjugate points
is kivk, and the cross- ratio (PQ, XY)= -1. Then
{PQ)=^l-
Therefore & = h.
35. The line-element.
If, returning to the stereograph ic projection, we take the formulae
in § 22, we can find an expression for the line-element (Is. We
have, r, y, z heing the coordinates of a point on the sphere,
Expressing this in terms of x' and >/, we get
4JftP (<&'* + (ft/'")
rf« 3 =-
(x's+y'+d 1 ?
188 CONCRETE REPRESENTATIONS [t.3D
In particular, if rf=3£,so that the plane of projection k the
tangent plane at A (Fig. 92), we get
where n — [/$ft
36. There is a gain in simplicity when the fundamental
circle is taken as a straight line, say the axis of x. Then
straight lines are represented by circles with their centra?
on the axis of x. Pairs of points equidistant from the axis
«&•**"
B V
f .A
\
■r ;G
Pis. 100.
of x represent the same point, and we may avoid dealing
with pairs of points by considering ouly those points above
v. 37]
AN APPARENT PARADOX
189
the a:-axis. A proper circle is represented by a circle lying
entirely above the ar-axis ; a horoeyele by a circle touching
the x-axis ; an equidistant-curve by the upper part of a
circle cutting the z-axis together with the reflexion of the
part which lies below the axis.
Through three points A, B, pass four circles. If
A', B', C are the reflexions of A, B, C, the four circles are
represented by ABC, A'BC, AB'G, ABC. The last three
are certainly equidistant-curves ; the first may be a proper
circle, a horoeyele or an equidistant- curve.
37. Angle at which an equidistant-curve meets its
axis.
Fig, 100 shows that the two branches of an equidistant- curve oat
at inlinity at a Unite angle, a fact that is not apparent in the Cay ley -
Klein representation. Let APBQA (Fig. 101) be the equidistant-
curve, AMB its axis, represented by the circle on A B as diameter,
190 CONCRETE REPRESENT ATIOKS (V. 37
and let C be the centre of the circle APB. Draw CX ± A U meeting
the two branches of the equidistant- curve and its axis in P, Q, M.
Let PAQ = 2a; then CAX=a, tan a=^-. Let </ be the dia-
XA
tan ce of the equidistant-curve from its axis.
The hue PX being ± AB represents a straight line; it cute ,1 S
in X, and the second point at infinity on the line is represented by
Y at infinity.
Henee *«ft \og(PM, XY)=klog~, also d = k]og¥£,
,!/ A QX
Now CX={{PX-QX);
therefore
PX-QX l/'' - J \ d
2 V
We can get a geometrical meaning for this result. Draw PLJ.PN
and PE\\NE (Fig. 102). Then the equidistant-curve and the
parallel and the axis all meet at infinity at E.
Fro. 102.
The angle LPE is then = a.
Consider a chord PQ of the equidistant-curve : like a circle, the
curve cute the chord at equal angles. Keeping P fixed, let Q go to
infinity. PQ becomes parallel to NX, and makes a zero angle with
it ; hence the angle between the curve and the axis is equal to the
angle LPE.
The explanation of the apparent contradiction shown in the
Caytey-Klein representation, where the two branches of the eani-
distnnt-curve form one continuous curve, lies in the fact that the
angle between two lines becomes indeterminate when their point of
intersection is on the absolute and at the same time one of the lines
touches the absolute. If the first alone happens the angle is zero,
if the second the angle is infinite.
v.sa] EXTENSION TO THREE DIMENSIONS 191
38. Extension to three dimensions.
The conforinal representation of non-euclidean geometry
can be extended to three dimensions, planes being repre-
sented by spheres cutting a fundamental sphere orthogon-
ally. A propel sphere is represented by a sphere which does
not cut the fundamental sphere, a horosphere by a sphere
touching the fundamental sphere, and an equidistant-
snx&Qfi by a sphere cutting the fundamental sphere.
A horocycle is represented by a circle touching the funda-
mental sphere. The horocycles which lie on a horosphere
all pass through the same point on the sphere, viz. the
point of contact. This is exactly similar to the system of
circles on a plane representing tin* straight lines of euclidean
geometry, and thus we have another verification that
the geometry on the horosphere is euclidean,
This suggests that the three geometries can be repre-
sented on the plane of any one of them by systems of
circles cutting a fixed circle orthogonally.
CHAPTER VI.
"SPACE CURVATURE" AND THE PHILOSOPHICAL
BEARINU OF XON-EUCLIDEAN GEOMETRY.
1. Pour periods in the history of non-euclidean
geometry.
The projective and the geodesic representations of non-
euclidean geometry have an important bearing on the
history of the subject, for it was through these that Cayley
and Rtcmaiiu arrived independently at non-euclidean
geometry.
Klein has divided the history of non-euclidean geometry
into three periods. The first period, which contains Gauss,
Lobachevsky and Bolyai, is characterised by the syn-
thetic method, and applies the methods of elementary
geometry. The second period is related to the geodesic
representation, and employs the methods of differential
geometry. It begins with Hihmanx's classical dissertation,
and includes also the work of Helmholtz, Lie and
Beltrami on the formula for the line-element. The
Surd period is related to the projective representation, and
applies the principles of pure projective geometry. It
begins with Cayley, whose ideas were developed and put
into relationship with non-euclidean geometry by Ki.kis\
To these a fourth period, has now to be added, which is
connected with no representation, but is concerned with the
vi. 2]
CURVED SPACE"
193
logical grounding of geometry upon sets of axioms. It is
inaugurated by Pasch, though we must go back to von
Stactxt for the true beginnings. This period contains
Hilbert and an Italian school represented by Peano and
Pieri • in America its chief representative is Veblen.
It has led to the severe logical examination of the founda-
tions of mathematics represented by the Principia Mathe-
matica of Russell and Whitehead.
2. "Curved space."
If we attempt to extend the geodesic representation
of non-euclidean geometry to space of three dimensions,
we find ourselves at a loss, for the representation of plane
geometry already requires tliree dimensions. It is quite
a legitimate mathematical conception, however, to extend
space to four dimensions. A limited portion of elliptic
space of three dimensions could be represented on a portion
of a " hypersphcre " in space of four dimensions, or the
whole of elliptic space of three dimensions could be repre-
sented completely on a hyperspbere, with the understanding
that a point in elliptic space is represented by a pair of
antipodal points on the hypersphere.
A hypersphere is a locus of constant curvature, just as
a sphere is a surface of constant curvature. Analog)' with
the geometry of surfaces leads to the conception of the
curvature of a three-dimensional locus in space of four
dimensions, and just as the curvature of a surface can be
determined at any point by intrinsic considerations, such
as by measuring the angles of a geodesic triangle, so by
similar measurements in the three-dimensional locus we
could, without going outside that locus, obtain a notion of
its curvature.
n.-e. a. N
194
PHILOSOPHICAL
[vi. 3
3, Application of differential geometry.
This was the path traversed by Riemann iu his cele-
brated Dissertation. Space, he teaches us, is an example
of a " manifold " of three dimensions, distinguished from
other manifolds by nature of its homogeneity and the
possibility of measurement. Space is unbounded, but not
necessarily infinite. Thereby he expresses the possibility
that the straight line may be of finite length, though without
end— a conception that was absent from the minds of any
of his predecessors. The position of a point P can be
determined by three numbers or coordinates, .r, y, z ; and
if x + dx, y +dy t z + dz are the values of the coordinates for
a neighbouring point Q, then the length of the small element
of length PQ, =ds, must be expressed in terms of the
increments dx, dy, dz. If the increments are all increased
in the same ratio, ds will be increased in the same ratio,
and if all the increments are changed in sign the value of ds
will be unaltered. Hence ds must be an even root, square,
fourth, etc., of a positive homogeneous function of dx, dy, dz
of the second, fourth, etc., degree. The simplest hypothesis
is that & 2 is a homogeneous function of dx, dy, dz of the
second degree, or by proper choice of coordinates d$*=&
homogeneous linear expression in dx-, dy-, dz*. For
example, with rectangular coordinates in ordinary space,
ds* = dx*+dy z +dz % .
By taking the analogy of Gauss' formulae for the curva-
ture of a surface, Ricmann defines a certain function of the
differentials as the measure of curvature of the manifold.
In order that congruence of figures may be possible, it is
necessary that the measure of curvature should be every-
where the same ; but it may be positive or zero. (Riemann
had no conception of Lobachevsky's geometry, for which
VI. 4|
RIEMAXX AND HELMHOLTZ
IflG
the measure of curvature is negative.) He gives without
proof the following expression for the line-element. If
a denotes the measure of curvature, then
ds = ^fa*/{\+$a2x*).
(Cf. Chap. V. § 35.) If k is what has already been called
the space-constant, « ■ I//.: 2 .
4, Free mobility of rigid bodies.
About the same time that Niemann's Dissertation was
being published, Hermann von Helmholtz (1821-189-1)
was conducting very similar investigations from the point
of view of the general intuition of space, being incited
thereto by his interest in the physiological problem of the
localisation of objects in the field of vision.
Helmholtz ' starts Irani the idea of congruence, and, by
assuming certain principles such as Hint of free mobility of
rigid bodies, and nmnodromy, i.e. that a body returns
unchanged to its original position after rotation about an
axis, he proves — what is arbitrary in Riemanu's investiga-
tion—that the square of the line-element is a homogeneous
function of Hie second decree in the differentials.
That the form of the function which expresses the
distance between two points is limited by the possibility
of the existence of congruent figures in different positions
is shown as follows. Suppose we have five points in space,
A, B, C, D, E. The position of each point is determined by
three coordinates, and connecting each pair of points there
is a certain expression involving the coordinates, which
corresponds to the distance between the two points. Let
1 " Uober die Tbatsachen, die d(>r Geometric zuni (jmncle beget),'*
Q Winger KachricJiten, 1868. An abstract of this paper "' ;,y published
in 18b*G.
196
PHILOSOPHICAL
fvi. 4
us try to construct a figure A'D'C'D'E' with exactly the
same distances between pairs of corresponding points as
the figure ABODE. A' may be taken arbitrarily. Than
B' must lie on a certain surface, since its coordinates are
connected by one equation. C" has to satisfy two condi-
tions, and therefore lies on some curve, and then D' is
completely determined by its distances from A', B' and C".
Similarly E' will be completely determined by its distances
from A', B' and C, but we cannot now guarantee that
the distance D'E' will be equal to DE. The distance-
function is thus limited by one condition. And with more
than five points a still greater number of conditions must
be satisfied. 1
It is customary to speak, as Helmholtss does, of the
transformation of a figure into another congruent figure
as a dwpfammenl of a single rigid figure from one position
to another. This language often enables us to abbreviate
our statements.
Titus, employing this language, we may argue fur the general case
as Follows. If there are u points, the figure lias 3» degrees of freedom,
and there are £«(«- 1) equations connecting the distances of pairo
cif points. But a rigid body has only 6 degrees of freedom j therefore
the number of equations determining the distance- function is
!«(jT.-l)-3M + 6=£{M-3Hn-'i>.
But it is necessary to avoid here a dangerous confusion.
Points in space are fixed objects and cannot be conceived
as altering their positions. When we speak of a motion
of a rigid figure we are thinking of material bodies. The
assumption which llelmholtz makes, which is expressed
by the phrase, the " free mobility of rigid bodies," is thus
1 This method was employed by J. M. do Tilly, Brnxdlts, Mem, Acad,
Rmj. (8vo collection), 47 (1893), to find the expression for the distance-
function without using iniinitesuimls.
vi. 5]
HELMIIOLTZ AND LIE
107
simply an assumption that there is such a thing as absolute
space.
While, psychologically, the idea of congruence may be
based on the idea of rigid bodies, if it were really dependent
upon the actual existence of rigid bodies it would have a
very insecure foundation. Not only are the most solid
bodies within out experience elastic and deformable, but
modern researches in physics have given a high degree of
probability to the conception that all bodies suffer a change
in their dimensions when they are in motion relative to
the aether. As all bodies, including our measuring rods,
suffer equally in this distortion, however, we can never be
conscious of it.
5. Continuous groups of transformations.
Helmholtz's researches, though of great importance in
the history of the foundations of geometry, lacked the
thoroughness which we would have expected had the author
been a mathematician by profession.
The whole question was considered over again from a
severely mathematical point of view by Sophus Lie 1
(1842-1899), who reduced the idea of motions to trans-
formations between systems of coordinates, and congruence
to invariance under such transformations. The underlying
idea is that of a group of transformations.
Suppose we have a set of operations R, S } T, ... such that
(1) the operation R followed by the operation S is again
an operation (denoted by the product RS) of the set, and
(2) (RS)T =R(ST), then the set of operations is said to form
a group. The operation, if it exists, which leaves the operand
1 S. lie, Theorie der TrajuiforiinUitnuigruppen, vol. iii. {Leipzig, 18fKJ),
Abt V. Kap. 20-24 ; and " Obcrdio Grundlagen der Geometric," Leipager
Berkhtc, 42 (1890).
PHILOSOPHICAL
[vr. 5
unaltered, is called the identical transformation, and is
denoted by I.
Tims, if R, S, T are the operations of rotation about a
fixed point through J, 2 and 3 right angles, the operations
1, R, 8, T form a group, and this is a sub-group of the
group consisting of the 8 operations of rotation through
every multiple of j-
The transformations which Lie considers are infinitesimal
transformations, and the groups are continuous groups,
auch as the group of nil the rotations about a fixed point.
All tbe transformations which change points into points,
straight lines into straight lines, and planes into planes
form a continuous group which is called the general pro-
jective group.
The assumption from which Lie starts in his geometrical
investigation is the " axiom of free mobility in the infini-
tesimal " :
" If, at least within a certain region, a point P and a line-
element through P are fixed, continuous motion is still
possible, but if, in addition, a plane-clement through P is
fixed, no motion is possible."
Starting then with the group of projective transforma-
tions, he determines the character of the transformations
so that this assumption may be verified, and he proves that
they form a group which leaves unaltered eii her a non-ruled
surface of the second degree (real or imaginary ellipsoid,
hyperboloid of two sheets or elliptic paraboloid), or a plane
and an imaginary conic lying on this plane. This invariant
figure is just the Absolute. The motions of space, therefore,
form a sub-group of the general projective group of point-
transformations which leave the Absolute invariant. And
vi. 7]
COORDINATES
199
so, without Helmholtz's axiom of monodromy, but using
a definite assumption of free mobility. Lie establishes that
the only possible types of metrical geometry are the three
m which the absolute is a real non-ruled quadric (hyperbolic,
geometry), an imaginary quadric (elliptic geometry), and
a plane with an imaginary conic (euclidean geometry).
6, Assumption of coordinates.
There are several points on which the investigations of
Riemann, Helmholtz and Lie admit of criticism. The
outstanding difficulty which strikes one at once lies in the
use of coordinates. How can we define the coordinates of
a point before we have fixed the idea of congruence ? This
question has been settled by an appeal to the famous
procedure of von Statidt (1798-1867), the founder of
projective geometry. He has shown 1 how, by means of
repeated application of the quadrilateral-construction for
a harmonic range (see Chap. HI. § 5), numbers may be
assigned to all the points of a line. This, and other
questions involved, have now been solved by the modern
procedure of Paseh, Hilbert and the Italian school repre-
sented by Pieri. This procedure, which marks a return to
the classical method of Euclid, consists in developing
geometry as a purely logical system deduced from an
appropriately chosen system of axioms or assumptions.
7. Space-curvature and the fourth dimension.
A misunderstanding, which is especially common among
philosophers, has grown around Riemanivs use of the
term " curvature." Helmholtz, whose philosophical
1 C K. Cli. v. Standi, Qmmeirizdtr Lage.. Xiirnterg, 1847, and Beitriige
zurQwmvtriederLage, Niiraberg, 1850-57-60.
200
PHILOSOPHICAL
fvi. 7
writings 1 are much better known than his mathematical
researches, has unfortunately contributed largely to this
error. The use of the terra " space- curvature " has led
to the idea that non-euclidean geometry of three dimensions
necessarily implies space of four dimensions, for curvature
of space has no meaning except to relation to a fourth
dimension. But when we assert that space has only three
dimensions, we thereby deny that space has four dimensions.
The geometry of this space of three dimensions, whether
it is euclidean or non-euclidean, follows logically from
certain assumed premises, one of which will certainly be
equivalent to the statement that space has not more than
three dimensions (cf. Chap. n. § 14, footnote). The origin
of the fallacy lies in the failure to recognise that the
geometry on a curved surface is nothing but a representa-
tion of the non-euclidean geometry.
This is brought out still more clearly by the fact that,
as non-euclidean geometry, elliptic or hyperbolic, can be
represented on certain curved surfaces in euclidean space,
the converse is also true, that euclidean geometry can be
represented t>n certain curved surfaces in elliptic or hyper-
bolic space ; and, of course, we do not consider the euclidean
plane as being a curved surface.
While, therefore, the conception of non-euclidean space
of three dimensions in no way implies necessarily space-
curvature or a fourth dimension, it is still an interesting
speculation to suppose that we exist really in a space of
four dimensions, but with our experience confined to a
certain curved locus in this space, just as Helmholtz's
"two-dimensional beings" were confined to the surface
1 EL v. Hclmlinliz. -'The origin and meaning of geometrical axioms,"
Mind, 1 (1.S7C), 3 (1878) s also in Papular Scientific Lactam (London,
18S1), vol. ii.
vi. 7]
THE FOURTH DIMENSION
201
of a sphere in space of three dimensions, and acquired in
this way the idea that their geometry is non-euclidean.
W. K. Clifford ' has gone further than this and imagined
that the phenomena of electricity, etc., might be explained
by periodic variations in the curvature of space. But we
cannot now say that this three-dimensional universe in
which we have our experience is space in the old sense, for
space, as distinct from matter, consists of a changeless set
of terms in changeless relations. There are two alternatives.
We must either conceive that space is really of four dimen-
sions and our universe is an extended sheet of matter
existing in this space, the aether % if we like ; and then,
just as a plane surface is to our three-dimensional intelli-
gence a pure abstraction, so our whole universe will become
an ideal abstraction existing only in a mind that perceives
space of four dimensions— an argument which has been
brought to the support of Bishop Berkeley ! 3 Or, we must
resist our innate tendencies to separate out space and
bodies as distinct entities, and attempt to build up a
monistic theory of the physical world in terms of a single
set of entities, material points, conceived as altering their
relations with time.* In either case it is not space that is
altering its qualities, but matter which is changing its form
or relations with time.
1 The Common Swwe of tin timet Sciences (London, 1 883). chap. iv. § tU.
1 Cf. W. W. Rouse Bull, " A hypothesis relating to the nature of the
ether and gravity," Messenger of Math., 21 (ISitl).
s See C H. Hinton, Scientific Romance, First Series, p. 31 (London,
188S). For other four-dtoettriona] theories of physical phenomena
see Hinton. The Fourth Dimension (l-oudon. ISKM).
l ('f A N. Whitehead. "On mathematical concepts of the material
world,'" hit. Trans., A 205 (1900).
202 PHILOSOPHICAL [vi . 8
8. Proof of the consistency of non-euclidean geometry.
The characteristic feature of the second period in the
history of non-euclidean geometry is brought out for the
first time by Beltrami' (1835-1900), who showed that
Lobachevsky's geometry is represented upon a surface of
constant curvature. This is historically the first euclidean
representation of non-euclidean geometry, and is of import-
ance in providing a proof of the consistency of the non-
euclidean systems. While the development of hyperbolic
geometry in the hands of Lobachevsky and Bolyai led to
no apparent internal contradiction, a doubt remained that
inconsistencies might yet be discovered if the investigations
were pushed far enough. This doubt was removed by
Beltrami's concrete representation by means of the pseudo-
sphere, which reduced the consistency of non-euclidean
geometry to depend upon that of euclidean geometry,
which everyone admits to be self-consistent.
Any concrete representation of non-euclidean geometry
in euclidean space can be applied with the same object.
In fact, the Cayley representation is more suitable for this
purpose, since it affords an equally good representation of
three-dimensional geometry. The advantage of Beltrami's
representation is that, distances and angles are truly repre-
sented, and the arbitrariness which may perhaps be felt
in the logarithmic expressions for distances and angles
is eliminated.
At the present time no absolute test of consistency is
v'f' ^o lm l; f m '°. d J '""rpn*"'"™* ^"a gcowttria non-eurii.hn.
.Naples imn, JS.-ltrnuu n!»o showetl Mint, since the equation of a geodesic
in * :...!-.«■ coordinates ia liomr, tin- surface eon t>o represented on a
plane, geodesies being represented by straight lines, and real points bcin K
r "l ,r ''~';"" 11 ,V " l " , "' h "-' lu:1,m;l fi.vn! Hrckr. il. , -A , , be < ran!
aiUon from the geodesic to the projective representation of On lev
vi. 10 1
TEST OF CONSISTENCY
203
known to exist, and the only test which we can apply is to
construct a concrete representation by means of a body
of propositions whose consistency is universally granted.
In the case of non-euclidean geometry the test which has
just been applied suffices to prove the impossibility of
demonstrating Euclid's postulate. For, if Euclid's postu-
late could be mathematically or logically proved, this
would establish an inconsistency in the non-euclidean
systems ; but any such inconsistency would appear again
in the concrete representation. The mathematical truth
of the euclidean and the non-euclidean geometries is equally
si rong.
9. Which is the true geometry ?
There being no <i priori means of deciding from the
mathematical or logical side which of the three forms of
geometry does in actual fact represent the true relations
of things, three questions arise :
(1) Can the question of the true geometry be decided
a posterior*, or experimentally ?
(2) Can it be decided on philosophical grounds '?
(3) Is it, after all, a proper question to ask, one to which
an answer can be expected '.
10. Attempts to determine the space-constant by
astronomical measurements.
Let us consider what form of experiment we can contrive
to determine, if possible, the geometrical character of
space. Essentially it must consist in the measurements
of distances and angles, the sort of triangulation which is
employed to determine the figure of the earth, but on a
prodigiously larger scale. If we could measure the angles
204
PHILOSOPHICAL
fvt. 10
of some very large triangle, the difference between their
sum and two right angles migM give us the necessary data
for determining the value of the space-constant. We do
not say that such an experiment will give us the necessary
data, for, as we shall see presently, the whole argument is
destroyed by a vicious circle {§ 12) ; but let us assume, for
the sake of illustrating the argument, that the experiment
can be made, and see to what conclusions it leads.
The largest triangles, whose vertices are all accessible
and whose angles we can measure directly, are far too small
to allow of any discrepancy being observed. We must
turn to astronomy to provide us with triangles of a suitable
size. The largest triangles, of which two vertices are
accessible, are those determined by a star and the observer
in two different positions.
Let S be a star and E t , E % two positions of the earth at
opposite ends of a diameter of its orbit, the sun ; and
Fig. 103.
let CSLE t E». The angle E t SC, subtended by the earth's
radius R at the star, is called the poraflms of the star :
blowing this angle and applying euclidean geometry, we
can find the-star's distance.
There are two methods of determining the angle EfiC.
The first, or direct method, is to measure the angle SEfi by
vi. 10] ASTRONOMICAL MEASUREMENTS 205
the transit circle. Then the parallax is, by assumption
of euclidean geometry, the angle ?-SEfi. The second
method, that of Bessel, is to compare the position of the
star <S with those of neighbouring stars which, from their
faintness and other considerations, are believed to be much
farther away than S. Considering S' as at infinity, and
again assuming euclidean geometry, Eft' || OS and the
parallactic angle E t SC =S'E 1 S.
But on the hypothesis that geometry is hyperbolic, these
two methods will give different results, and the angle.
SEfi t-S'^iS is in fact not equal to ^, but is the parallel-
angle corresponding to the distance R. Let 2$ be the small
difference (^-SE t c) -S'E V S ; then
2e=J-n(ff).
Also
e » : = tan \\\{R) =tan(j - f)) -(1 -tan 0)/(l +tan $) ;
therefore
fl/fr=log,{(l +tan 0)1(1 -tan 6)} =2 tan B, approx.
■ Now we have records of the determination of the star
« Centauri by both methods. An early measurement by
the direct method yielded the value I'M", while Bessel's
method gives the value Q-76" ±0-01". Taking 20 therefore
equal to 0-38", we have tan = 92 x 10" 8 , and kffi = 530000
approx. The direct method is not susceptible of very
great accuracy, and the value P14* for the parallax is
probably much too large, but even from these data, if we
admit the soundness of our argument, we should be
206
PHILOSOPHICAL
fvi, 10
warranted in stating that the space-constant must be at
least half a million times the radius of the earth's orbit.
The data, so far as they go, seem to favour the hypothesis
of hyperbolic geometry rather than that of elliptic, since
the calculation leads to a real value, for k.
Thp hypothesis of elliptic geometry, however, leads to
the result that a star would be visible in opposite directions
unless there is some absorption of light in space, 1 If we
assume that the light from a. .star which is at a distance of
h-n-k (i.e. half the total length of the straight line in elliptic
space) is so diminished by absorption that the star becomes
invisible, then the parallax of the farthest, visible stars,
measured by the direct method, would, as on the euclidean
hypothesis without absorption, be zero. And if the light
is totally absorbed in a distance of say \irk, the case would
be similar to that on the hyperbolic hypothesis, or on the
assumption of absorption in euclidean space. Thus, if
we admit the hypothesis of absorption of light in free space,
it becomes impossible to draw any definite conclusion as
to the nature of actual space, except perhaps that the
space-constant is very large.
The diivci appeal to experiment therefore leads only to
the conclusion that the space-constant, if not infinite, must
be very large compared with any of (he usual units of
length, and is very large in comparison with the. distances
which we have ordinarily to deal wiHi. These experiments
do not contradict euclidean geometry, but they only verify
it within the limits of experimental error. No amount of
1 A complication, however, arista owing to tin- Unite rate nf propagation
of light. The two images «l tho star Been in opposite directions will
represent, the star tit clinVrettt times, and in "pnenil therefore in different
positions, so that, even if then- Merc no adsorption of light, the appearance
Of in., sky would nol necessarily be symmetrical. (C'f. W. IJ. Franklund.
Math. Gazette, July 1013.)
I
'I]
INTUITION
207
experimental evidence of this kind can ever prove that the
geometry of 3pace is strictly euclidean, for ( here will always
be a margin of error. On the other band, so far as we have
gone, it remains conceivable that further refinements in our
instruments and more accurate information regarding the
laws of absorption of light might enable us to establish an
vpjicr limit to the value, of the space-constant, and thus
demonstrate that the. geometry of actual space is non-
euchdean.
11, Philosophy of space.
This way of regarding experience as the source of our
spatial ideas is in striking contrast to Kant's attitude
towards space, which is expressed by his dicta : that space
is not an empirical concept derived from external experience,
but a framework already exist ing in the mind without which
no external phenomena would be possible. 1 From the new
point of view, geometry applied to actual space has become
an experimental science, or a branch of applied mathe-
matics. We are not forced to accept ite axioms, but shall
onlv do so when we find them convenient and in sufficiently
close agreement with the facts of experience. Since Kant's
time the intuitive has become discredited. We now know
that there are things which formerly appeared to be intui ti ve
which are in fact false. Thus, it was formerly believed that
every continuous function possessed a differential co-
efficient ; the proposition appeared, indeed, to be intuitive.
But Weierstrass gave an example which showed that the
belief was false. In the extreme empiricist view the
parallel-postulate has to be ranked with the law of gravita-
tion as a law of observation, which is verified within the
limits of experimental error.
1 1. Kant, Critiqm of Pure Reason, ohap, i.
208
PHILOSOPHICAL
|>I. 11
As regards the second question, therefore, the powers
of philosophy have been narrowly circumscribed by the
stricture kid upon intuition. Obviously the fact that a
coherent mental picture can be formed of euclidean space
does not constitute a proof that this is the form of actual
space, since the same thing applies to the non-euclideaii
systems. But the philosopher may say he has an intuition
of euclidean space. What does this mean ? Has he an
intuition that the sum of the angles of a triangle is equal
to two right angles ? Does he perceive intuitively that
two straight lines which are both perpendicular to a third
remain equidistant ! What intuitions or beliefs would
the philosopher have had if he had been deprived of powers
of locomotion and the sense of touch, and been provided
with only one eye ? He would believe, because his eye
told him bo, that two railway lines converge to a point, that
objects change their shapes when they are moved about :
and he would perhaps demonstrate that the sum of the
angles of a triangle is greater than two right angles. His
intuitions are merely beliefs, and perhaps not even true
ones.
We have really to distinguish between different kinds of
apace. The space of experience is brought to our knowledge
through the senses principally of sight and touch, and is a
composite of two spaces, " visual space " and " tactual
space." Pure visual space, which is the limited field of
our imaginary one-eyed sessile philosopher, is a crude
elliptic two-dimensional space ; l the three-dimensional
form of tactual space is conditioned probably in part by
the semi-circular canals of the ear. From this composite
1 Cf, Thomas Rcid. ,4n Inquiry into the Human Mind. Edinburgh, 1704,
chap. vi. " On Seeing," § " Of the geometry of visibles."
vi. 12]
VISUAL AND TACTUAL SPACE
20!)
space, which is far front being the beautiful mathematical
continuum which we have arrived at after generations of
thought, we get by abstraction a conceptual space which is
conditioned only by the laws of logic, but to which we find
it convenient to ascribe the particular form which we call
euclidean space, for the reason that this is the simplest of
the logically possible forms which correspond with sufficient
closeness to the space of experience. Whether there is,
besides these, an intuitional space, we shall leave to philo-
sophy to settle if it can. We may, perhaps, leave Kant in
possession of an a priori space as the framework of his
external intuitions, but this space is amorphous, and only
experience can lead us to a conception of its geometrical
properties.
12. The inextricable entanglement of space and
matter.
A further point--and this is the " vicious circle : ' of which
we spoke above— arises in connection with the astronomical
attempts to determine the nature of space. These experi-
ments are based upon the received laws of astronomy and
optics, which are themselves based upon the euclidean
assumption. It might well happen, then, that a discre-
pancy observed in the sum of the angles of a triangle could
admit of an explanation by some modification of these laws
or that even the absence of any such discrepancy might
still be compatible with the assumptions of non-euclidean
geometry.
"' All measurement involves both physical and geometrical
assuniptiun.s and ihi> two things, apace and matter, ore not gtVflB
separately, but analysed out of a common experience. Subject
to the general condition (bat space is to be changeless and matter
to move about in space, we can explain the same observed results
N.-K. O. O
210
PHILOSOPHICAL
\vi. 12
in many different ways by making compensatory chanpr* in
the qualities that we assign to space and the qualities we assign
to matter. Hence it seems theoretically impossible to decide by
any cxporiment what are the qualities of one of them in distinction
from the other." '
It was on such grounds that Poincare.* maintained the
essential impropriety of the question, " Which is the true
geometry ? " In Ms view it is merely a matter of con-
venience. Facts are and always will be most simply
described on the euclidean hypothesis, but they can still
be described on the non-euclidean hypothesis, with suitable
modifications of the physical laws. To ask which is the
true geometry is then just as unmeaning as to ask whether
the old or the metric system is the true one. The con-
clusion thus arrived at by Poincare is quite akin to the
modern doctrine in physics expressed by the Principle of
Relativity. Just as, according to this doctrine, it is
impossible by any means to obtain a knowledge of absolute
motion, so, according to Poincare. it is beyond our power
to obtain a knowledge of absolute space.
1 Mr. C, I). Broad, with whom I have discussed thif chapter, has put
this point of view so well that I quote his words.
5 H. Poincare, 1st science el rhypatklse (Pans, !ft03), chap. v. ; Etagfci
translation by W. .). Grcenstreet, London, !U€5.
CHAPTER Vn.
RADICAL AXES, HOMOTrTRTTC CENTRES AND
SYSTEMS OF CIRCLES.
1. Common points and tangents to two circles.
Two circles intersect in four points and have four common
tangents. Various cases arise according as these points
uih.I lines are coincident or imaghttfj in pain,
In hyperbolic geometry two equidistant-curves whose
axes intersect have their common points and tangents
all real. A proper circle which cuts both branches of ao
equidistant-enrve has four real common tangents with it.
If it cuts only one branch, two of the common points and
two of the common tangents are imaginary. Two proper
circles cannot have more than two of their common points
real ; their common tangents are then two real and two
imaginary. If two proper circles do not intersect, their
common tangents are all real or all imaginary. The case
of four real common points and four imaginary common
tangents cannot occur in hyperbolic geometry ; two real
and two imaginary common points can only occur along
with two real and two imaginary common tangents.
In elliptic geometry, if two circles intersect in two real
and two imaginary points, they have two real and two
imaginary common tangents. If each lies entirely outside
the other, their common points are all imaginary and their
212
SYSTEMS OF CIRCLES
fvn. 2
common tangents are all real. The absolute polars of two
such circles have four real common points and their common
tangents all imaginary. If one lies entirely within the other,
their common points and tangents are all imaginary. The
case of four real common points and Four real common
tangents cannot occur in elliptic geometry,
2. The power of a point with respect to a circle.
Let. be a fixed point in the plane of a proper circle with
centre C and radius a. Through draw any secant cutting
Fio. 104.
the circle in P, Q. Draw CN 1.0PQ. Let OC=d and
COP = B, OP=r, OQ=r', so that
Otf = l(r'4-r),Ptf = Hr'-r).
Now 1 cmd = co&CN cos i(r' +r),
cos a = cos CN cos £ (/ - r).
_. . cosd cosi(/+r) 1 - tan \r tan lr' .
Therefore CM a = C03 I ^j = i+tau £r tan jf '
therefore tan \r tan £/ = const, = tan it(d +a) ton i(rf - a).
1 Elliptic geometry is token as the standard case, and the space-con-
stant it w taken as the unit of length.
vn, 3]
POWER OF A POINT
2LJ
In hyperbolic geometry tan is replaced by i tanh. This
product may be called the power of the point with respect
to the circle. It is positive if is outside, negative if is
inside the circle. In the former case, if I is the length of
the tangent from to the circle, the power of is equal
to tan*Jfc
3. Power of a point with respect to an equidistant-
curve.
(1) Let the secant cut one branch of the curve in P, Q,
i.e. in hyperbolic geometry the secant does not cut the
o
M
PlO. 105.
axis of the curve, in elliptic geometry neither of the
finite segments OP, OQ cuts the axis.
Let M be the middle point of PQ, and draw MN JL the
axis ; then MN is also 1 PQ, Draw OH 1 the axis. Let
OH^d, MN-x,0P=-r, OQ=r>, so that OJtf=- 4 {/+*),
PJlf = A (/-/).
Then, from the trirectangular quadrilaterals OHNM,
PKNM,
sin d = cos A (/ +r) sin x, sin a =cos A, (/ - r) sin a; ;
sin d __ cos A (/ + r) 1 -tan A.rtanA/ .
sin r(.~cos !(/ -r) I +tan Artan \r' '
therefore tan \r tan lr' = const. =tan A {a -rf)/tan | (d +o).
therefore
214
SYSTEMS OF CIRCLES
fvn. 3
(2) Let the secant cut both branches of the curve, i.e.
the point of intersection A with the axis is real and one
of the segments OP, OQ cuts the axis.
no. ioa.
Let OAN = 6, ON =rf, OP =r, OQ =r', so that
0A = k{r'+r), PA = !.<r'-r).
sintf = sin ,1 (r' + *■ ) sin f),
sina=sin .](/ -r) sin 6.
sind = sinj ( / +r ) = ta n lr' -r tan \r
sia a = sin £(/ - r) ~ tan \r' - tan \r
tanjr_ _tani(d-n)
i t — const. — z * » i —
taiUr tani{d + a)
Then
Therefore "M:
Therefore
Note. Figs. 105 and 106 have been drawn for the ease of
hyperbolic geometry. In elliptic geometry the equidistant- curve
is convex towards the axis. In Fig. 105. in this case, either
OR < PK or lies between P and 0. If is the same point in the
VII. A
POWER 01-' A POINT
215
two figures, the values of tan {r tan i/ and tanArtau*/, respec-
tively for the secant which cuts and the secant which does not cut
t be axis, are equal.
Hence, if a variable line tJirough a fixed point cuts « circle
in P, Q and its axis in A, either the ratio or the product of
the tangents of half the segments OP, OQ is constant, according
as (1) one, or (2) both or neither of the segments contains the
point A. If OT is a tangent- to the curve, the constant is
equal to tatPlOT, and is catted the power of the point with
respect to the circle.
Tho two cases are simply explained in elliptic geometry. Let
the clotted circle A A' represent the axis of the circle, which is
no. 107,
represented in the diagram by a pair of circles. The secant cuts
the two circles in P, P' ; Q,Q'; and the axis in A, A'. These pairs
of course represent single points.
AA' = PP'=QQ' = ir;
therefore
OCi'=v-OQ.
tan hOP
Therefore tan \OP tan \0Q = tan \ OP cot \OQ' = ^rgjg>
216
■SYSTEMS OK ('.] lU'LKS
F\n. 4
4. Reciprocally, if P is a variable point on a fixed line PN,
p
Fia, 108.
and the tangents FT, PT from P to a fixed circle make
angles 0, & frith PN, we have in Fig. 108,
sind=sin /-sin \($' + 6),
sin a = sin r sin \(ff - Q) t
sin d _ 3i n|(e'+fl) _ tan iff + tan W
whence
sin a sin !, [ff - $) tan h $' - tan 1 '
vu. 4j
POWKK OF A LINK
217
This result is true also in euclidean geometry, the constant
reducing to {d - a)/(d +<?).
For an equidistant-curve, let the line cut the axis in N
at an angle « ( Kg. 109).
Then. 8 and ff being taken positively,
TPL<*TPLml(6+&%
NPL = \(8-ff),
cos a - cos x sin 1 (0 - W\
cos a = cos x sin J, (# -i- 8') ;
whence, as before. . — ^a, ia constant,
tan An
If the angles 8. $' are measured in the same sense, then
for & we must put - -8', and we have
tan if) tan W = const.
If, the angles 8, 8' being measured in the same sense,
both or neither of them contains the line joining / J to the
centre, then we have (Fig. 110)
LPC = i(t)' +*■ +8) =| + l(ff + 0),
NPC = ^-!,(ff+e),
TPC=wpr=!A-r-ff+Q)-*-W -fl-
Fiu. 110.
218
Then
SYSTEMS OF CIRCLES
|yn. 8
sin d =sin r cos A ($' + $),
sin a = sin r cos A (f)' - 0),
and tan IB ten ifl' =const.
Hence, if from a mriable paint on a fixed line I the
tangents to a circle are p, q, and (fie line to lite centre is a,
either Ike ratio or the product of the tangents of half the angles
Ity), ity) is constant, according as (I) one, or (2) both or
neither of the angles contains the line a.
5. Angles of intersection of two circles.
Since two circles may intersect in four points, there are
four angles of intersection to consider.
It is easy to show geometrically that if two circles have
only two real points of intersection, the two angles of
intersection are equal.
Suppose a circle cuts an equidistant-curve in four points,
P, P' on one branch, Q, Q 1 on the other branch. Then,
no. in.
drawing PM, P'M' _L the axis and joining P, P' to C, the
centre of the circle,
L CPP' = CP'P, Z MPP' - M'P'P i
therefore /_ MPC = M'P'C,
and the angles of intersection at P, P' are equal, and simi-
larly the angles of intersection at Q, Q' are equal.
vn. 0]
INTERSECTION OP CIRCLES
219
But LCPM + CQN =2CPQ, since LMPQ=NQP ;
therefore OPM and CQN are not in genera! equal. If
<1>M=CQN, then each =CPQ=CQP; M and N then
coincide, and lies on the axis of the equidistant-curve.
Similarly, it may be shown that if two equidistant-curves
intersect in four- points, the angles at the points of inter-
section which are on different branches are equal, but all
four angles cannot be equal unless the axes are at right
angles.
When two of the angles of intersection are right,
the circles are said to cut orthogonally, All four angles
cannot be right, for then the centre C of the one circle must
lie on the axis of the other, and if CT, CT' are the tangents
to the second circle, CT is a radius of the first circle.
But CT is a quadrant ; therefore the first circle must reduce
to two coincident straight lines.
6. Radical axes.
Let P, P', Q, Q' be the points of intersection of two
circles, with axes «=0 and /3=0. Then, if 5=0 is the
equation of the absolute, the equations of the circles can
be written S - <t 2 - 0, 8 - j3* = 0.
The equation [S -«*) -{S- ) S 1! )=0 represents a conic
passing through their common points, but this breaks up
into the two straight lines a±fi=Q, and these represent
a pair of common chords which pass through the point of
intersection of the axes. Tliey form with a and /3 a har-
monic pencil.
If y is the polar of the intersection of the axes, i.e. the line
of centres, the other pairs of common chords pass through
«y and j3y.
If we take any point on one of the first pair of common
220
SYSTEMS OF ni:''l.KN
j \ n, 'i
chords, say PP', the power of with respect to either circle
is tan |OP/tan \()I y . These two lines are therefore the locus
of points from which equal tangents can be drawn to the
two circles.
But if we take, a point on PQ, the power with respect to
one circle is the product, and with respect to the other
circle the ratio of tan WP and tan \0Q, and this chord does
not possess the property of equal tangents.
Fro. n±
Hence, of the three pairs of common chords of two circles,
one paii* pass through the intersection of the axes and are
harmonically separated by them, and possess the property
that the tangents from any point on either to the two
circles are equal.
These two lines are called the radical axes of the two circles.
1
YTI. 8]
RADICAL AXES
221
7. Homothetic centres.
Rcciproeallv, two circles have four common tangents,
which intersect in three pairs of points. One pair lie on
the line joining the centres, and are harmonically separated
by them, the other pairs he on the lines joining the centres
to the pole of the line of centres. The first pair possess
the property that any line drawn through one of them cuts
the two circles a t eq ual a ngles. These two poi nts are called
the homothetic centres of the two circles.
8. Radical centres and homothetic axes.
The three pairs of radical axes of thee circles taken in pairs
pass thmwjhfonr points, the radical centres of the three circles.
Let ABC be the triangle formed by the axes «, 6, c of
the three circles ; a pair of radical axes eii« 2 , $$%, y&z
passes through each of these points.
Fu:. IIS.
If one radical axis y, of the circles A, B, and one radical
axis a of the circles A, C intersect in P, then the tangents
from P to the three circles are all equal. Therefore P lies
SYSTEMS OF CIRCLES
[vn. :;
on a radicnl axis «, of the circles B, C. Wc have then
«i. B t , ji concurrent in P. Let y t cut Ul in S, and join BS.
Then, since (ab, y,yj is harmonic, B(AC, SP) is harmonic ,
therefore BS is ft , i.e. a% ,B l ,y 1 pass through 5. Similarly
«2' A, Vz are concurrent in Q, and « 2 , B 3 , y^ in /?. The
quadrangle PQRS has ^£6' as harmonic triangle.
Reciprocally, the three -pairs of homolketie centres of three
circles taken in pairs lie in sets of three on four limes, the
nomothetic axes of the three circles. They form a complete
quadrilateral, whose harmonic triangle is the triangle formed
by the centres of the circles.
9, Coaxal circles in elliptic geometry.
The locus of the centre of a circle which passes through
two fixed points D u D z on a line I consists of the two
perpendicular bisectm-s OL, O'L of the segments D,/^ and
0»A (Kg- H4). All the circles through /),, Z>, therefore
fall into two groups; any two circles belonging to the
same group have I as a radical axis. Each group is there-
fore called a system of coaxal circles with common points
/>,. D q . When the centre is at 0, the circle is a
minimum, and it increases up to a maximum, which is
just the straight line / itself, when the centre is at L.
Let C ls C\ on OL = V be the centres of two circles of the
one system, and take two points R tl A, on I Draw the
tangents KJJ U KJU* A' 2 F„ K s f s to the circles C„ C % .
Then K,0 1 =K l V t and I.F,-*^,. Hence the points
V lie on a circle with centre A"„ and V on a circle with
centre K t . Also, since K t U t is a tangent, to the circle 6 T ,
and a radius of the circle K u C,*/, is a tangent to the
circle E t ; and since O t U l m(J i y i , C\ lies on a radical axis
of the circles K„ K t . Hence the circles K have V as a
vn. I>]
COAXAL CIRCLES
223
radical axis. We get then a system of coaxal circles K
associated with the system C, and every circle of the one
system cuts orthogonally every circle of the other system.
As KD, diminishes the circle tends to vanish. Z),, I) s
are called the Hmiliitg points of the A' system. If K lies
*.
\\y. _— — — i»&-j
(' 1
t /
3 y
fig. 114.
in the segment D t OD zt the circle is imaginary. As K
approaches 0', the circle becomes the straight line ¥.
The A' system is a non-intersecting system, i.e. it has
imaginary common points. The system has imaginary
limiting points.
If the segment D 1 D i vanishes so that the common points
224
SYSTEMS OF CIRCLES
[vH. 10
D u 7> 2 coincide at 0. the circles C all touch the line I at 0,
and the circles K all touch the line (' at 0.
If the segment D } D 2 becomes », so that the common
points coincide at 0'. the circles all reduce to straight
lines passing through 0', while the circles K become con-
centric circles with centre 0'.
10. Homocentric circles.
The locus of the centre of a circle which touches two
fixed lines d u & % through a point L consists of the two
bisectors o, o' of the angles between rf,, d s (Fig. 115). All
the circles touching t^, d» therefore fall into two groups ;
any two circles belonging to the same group have L as a
hoinothetic centre. Each group is therefore called a system
of homocentric circles with common tangents rf, , d%. When
the centre is at L, the circle is a minimum and reduces to
the point L itself; as the centre moves along o', the
circle increases up to a maximum when the centre is at
0, the pole of o.
Let fli, c 2 through It, the intersection of o and /, be the
axes of two circles of the one system. Take two lines k, , fc,
through L, and let « n , « 2 , v, , v z be the tangents to the circles
C,, C a at their points of intersection with jfc,. k„. Then the
angles (£,*',) = (/f- t K 2 ) and {k i v i ) = (k i r i ). Hence the lines u
are tangents to a circle with axis Jc u and v are tangents to
a circle with axis k 2 . Also, since (i^) lies on the circle ( ,
and on the axis of the circle A',, and since the angle (<?,«,)
= (<V'i). fi passes through a hoinothetic centre of the circles
A',, A* 2 . Hence the two circles K have // as a hoinothetic
centre. We get then a system of homocentric circles K
associated with the system C, and every circle of the one
system is tangentially distant a quadrant from every circle
to. io| HOMOCENTRIC CIRCLES
of the other system. As k approaches d, the circle K
becomes the straight line rf, . d t , d 2 are called the limiting
lines of the K system. If h lies outside the angle djd.^,
Fig. 115.
the eircle becomes imaginary. As k approaches ©', the
circle reduces to the point L". The A' system has imaginary
common tangents; the C system has imaginary limiting
lines.
If the angle d^^ vanishes, so that the common tangents
N.-E. Q. F
220
SYSTEMS OF CIRCLES
tvn. 11
d t , t? a coincide with o', the circles V all reduce to points
on o\ while the circles K become concentric circles with
axis a".
If the angle d t d 2 becomes w, so that the common tangents
d t! f/ a coincide with o, the circles all touch o at L, and the
circles K all touch o at //.
11, In euclidean and hyperbolic geometry this duality
does not hold, since in euclidean geometry the envelope
of a system of lines cutting a fixed line at a constant angle
is a point at infinity, and in hyperbolic geometry it is an
ideal circle. In hyperbolic geometry, as K goes to infinity
the circle becomes a horocyelc. Between the horocyele
and the straight line lies a system of branches of equidistant-
curves. The other branches complicate the figure as they
intersect the other circles of the system. The same thing,
of course, occurs in the other coaxal system passing through
In euclidean geometry a system of coaxal circles is a linear
system, i.e. through a given point only one circle of the
system passes. In non-euclidcan geometry, through three
given points four circles pass, -i.e. four circles can be drawn
through any point P to pass through two fixed points X, Y.
Denote these circles by PX F, / J 'A' Y, etc. ; then of the four
circles, PXY, P'XY have their centres on the one per-
pendicular bisector of XY, and belong to the one coaxal
system, while PX'Y, PXY' belong to the other. Hence,
through a given point there pass only two circles of a given
coaxal system.
In euclidean geometry a system of coaxal circles is equivalent to
a system of conies through four points ; in non-euciidean geometry
it is equivalent to a system of conies through two points and having
vtr. I2|
LINEAR SYSTEMS
double contact with a fixed conic. 1 The reciprocal system, i.e. a
system of circles touching two lines, is equivalent in non-euclidcan
geometry, to a system of conies touching two lines and having
doable contact with a fixed conic ; and in euclidean geometry to
a system of conies passing through two points and touching two
huos. Hence the complexity of the latter system compared with
a system of coaxal circles.
The analytical treatment of systems of coaxal circles in non-
euclidean geometry can he reduced to the consideration of linear
systems in the following way,
12. Linear equation of a circle.
If {x, y, z) are the actual Weierstrass coordinates, the equation
of a circle, with centre (.r, , #, , s,) and radius r, is
**« + Wi + ^^i - ** coa 7 •
Let n=Aa",, lj=ky u c = A£%, rf= - AJPeos-, so that
k
W -h jW + e 1 = A'A-^a:, 1 + y» + L% 3 ) = XH-* =jj>, say.
Then the equation reduces to
ax + by + cz+tl II.
The non-homogeneous linear equation, willi real cnefheicnts, in
actual Weierstrasis coordinates, therefore represents a circle with
centre (a, b, c/A*), axis (& > by I «=0, and radius r, such that
on . -the positive value of -,
k p
where f- = IrtC- \ W+A
In elliptic geometry fr 5 is positive and p 2 is always positive. The
centre is always real, and the radius is real if iP~k*a s +&!£+<?. If
d=0 the circle becomes a straight line, and if d=p it reduces to a
[joint.
In hyperbolic geometry, changing the sign of A*, we have
r d
jj»=cS-A% a -i 5 i 2 , aud cosh ,=tho positive value of -.
K p
1 It is therefore exactly equivalent to a svstem of circles in eticlidenn
geometry having double contact with n fixed conic. The limiting points
are represented by the foci of the conic.
228
SYSTEMS OF CIRCLES
fvn. 13
The centre is real, ideal or at inlinity, according as jf>, = or <0.
The curve is therefore
A real circle if tP>& -IftP-tW^-Q, reducing to a point if
An imaginary circle if <?■ - Ifia* - Ifi&xP^O.
An equidistant-curve if c 1 <i(- a (!i 2 -! -Ir), reducing to a straight line
if<J=0.
A ho recycle if ^ = jt s (a' + 6»).
The two equations ax + bi/ + cz+d=Q represent the same circle.
In dliptio geometry this is verified, since (a:, y,z) and { -x, -y, 2)
represent the same point. In hyperbolic geometry, for a proper
circle or a horocyclc only one of these equations can be satisfied,
since z must be positive ; for an equidistant-curve the two
equations represent the two branches.
The points of intersection of two circles «,a; + 6, y + cfi ±d l —0 and
a& + b&+c&+d r =0 are found by solving these equations simul-
taneously with the equation x i +y I +&& = li*. These give four sets
of values of &, y*, #, and therefore four points of intersection.
13. Systems of circles.
If S, =0 and S,-0 are equations of circles in this form, 5, + AS--0
represents a circle, and for all values of A represents a pencil of circles
passing through two fixed points. If <l t =d t , the circle S, -S,=0 of
the system reduces to a straight line, and if d t +d a =0 the circle
S,+8 t =0 is another straight line. These are the radical axes of
tho two circles.
S i + AjS, + /kiS»=0 represents a linear two-parameter system or
1 iiiudlc of circles. Ha circle of the system passes through the point V,
Sj' + AS.'+iiiS.'^O,
and we have (SfiJ - 5,'5 3 ) + A [5*8,' - 5,'S a } = 0,
which represents a linear one parameter system or pencil of circles.
Hcneo all circles of a bundle which pass through one fixed point
form a coaxal system and pass through another fixed point.
If rf, + Ad- + /wtf, =0, we get a pencil of straight lines If the vertex
of this pencil is real, choose it as origin ; then the linear system can
be reduced to the form x + ky+piz 1 e)=0 Then one radical axis
of every pair of circles of the system passes through the origin.
i.e. the circles have a common radical centre at the origin. If
tangents are drawn from this point to the circles of the system they
are all equal, and hence all circles of a bundle eut orthogonally a
TO. Hi
MARGINAL IMAGES
229
fixed circle. In hyperbolic geometry the orthogonal circle may 1k>
a proper circle, real or imaginary, an equidistant -curve or a horo-
cyclo. If tho orthogonal circle reduces to a point, all the circles pass
through this point.
All these results admit, in the case of elliptic geometry,
of a simple interpretation by means of the central projection
of the sphere. To a plane corresponds a circle, to an axial
pencil of planes corresponds a pencil of circles, and to a
bundle of planes through a fixed point corresponds a
bundle of circles, the orthogonal circle of which corresponds
to the polar plane of with respect to the sphere.
This representation fails in hyperbolic geometry, since
the sphere becomes imaginary, but there is a correspondence
in tween the circles of the hyperbolic plane and the planes
of hyperbolic space.
14, Correspondence between circles and planes in
hyperbolic geometry. Marginal images, 1
Consider a fixed plane F and a plane E. From any
point P on E drop a perpendicular PQ on F. The assem-
Pra. 110.
■This theory is due, analytically, in F. Hnusdorff, " Anftlytische
Beitrage zur niehtcuklidieeheii (Jcumutric," tteiptiger BtrkMa, 61
(189!)}, p. 177, and geometrically to H. Liebmann, " Synthetische
Ableitung der Kreisi'erwnndtachnflen in tier LnbaUoiicfslrijschen
Geometric," Mpzitjer Berichu, 54 ( I !X>2}. p. 250. Cf. also Liebmann,
Niehlettktidische Qeomttrie, 2nd ed., Leipzig, 1912, p, 63.
230
SYSTEMS OF CIRCLES
,
[vn. II
blage of all points Q lie within a curve called the marginal
linage of the plane E on the plane F.
(1) Let E and F he non-intersci 'tin». and have a common
perpendicular AB (Fig. 116). Through A in the plane
B draw any line AP, and in the pbae P4JJ, which cuts F
in 5Q, draw <?PJ_£<3 and ||/(P. Then <? lies on the
marginal image of E.
If AB =p and BQ =p\ then sinh p sinh j/ = 1. Hence
// is constant, and the marginal image is a circle with centre
B and radius p' given by sinh p sinh p' = 1.
(2) Let E cut J at an angle <, in the line MN (Kg. 117).
Draw a plane 1 MA*, cutting E in MP and F in MQ.
Kin. 117.
Draw QP 1 MQ and || M P. Then „ - IT (MQ). Hence MQ
is constant, and the locus of Q is an equidistant-curve
with axis MN and distance a such that !!(«)=«.
(3) Let E be parallel to F. Then the line MN goes to
infinity, and the equidistant-curve becomes a homcyclc.
Hence there is a (1, 1) correspondence between the
circles in a plane and -the planes in space.
vn. 181 MARGINAL IMAGES 231
15. When the planes E, E' intersect in a straigfd line I,
their marginal images intersect in. two points which form the
■marginal image of the line l.
Let the planes E, E' cut the absolute in conies C, C,
and let be the absolute pole of the plane F, Then the
marginal images /, I' of B, E' are the projections of C, C
on the plane F with ceutre of projection 0. The conies
C, C intersect only in two points P, Q, the points of inter-
section of I with the absolute. The cones OC, OC cut the
absolute each in a second conic C,, C,'.
Now C, C cut in P, Q ; l\ . t 7 "it is P, , Q, ;
G, Ci „ R, S ', C, C\ „ Ri, Si,
and the points P, P, «ive the same projection on F, and so
also do the other pairs. Hence the marginal images cut
in four points, two of which form the marginal image of the
line of intersection I,
16. The angle between two planes is equal to the angk of
intersection of (heir marginal images.
Let Ei, E t be two planes cutting in TT 1 . Let P,Qj be
i'lii. 118.
232
SYSTEMS OF CIRCLES
|vn. 17
the common perpendicular of E x and F, P t Q 2 that of g,
and P. The plane P&Qft X P cute TT in £. Draw
PA'i JV*' in the, planes ^.fi.nrr, and SS' ±F
and 1 1 TT'. The planes P,ft.S fl, and P 2 <^S = P, are ± F.
The marginal images r u r s of ff t , £ 2 are circles with centres
Q,, Q s and intersecting in 5.
We have then four planes E u B a , R 2 , R, whoso lines of
intersection 7T', P a /> 2 ', SS', jyy are parallel.
Therefore z (g,g J + (g^jj + (R 2 R t) + (JW = 2 _
But ftft) -£-(*£■) an d pyy. f -(^
Therefore 0M-fV£
17. Systems of circles.
A pencil of planes through a line I is represented on F
by a system of circles through two fixed points, the marginal
image of I. The planes perpendicular to I form a pencil
of planes with ideal axis f , the absolute polar of I These
are represented on F by a system of circles through two
imaginary fixed points, the marginal image of I', and every
circle of the first system cuts orthogonally every circle of
the second system. These form therefore conjugate systems
of coaxal circles.
A bundle of planes thxongh a point P is represented by a
system of circles any two of which intersect in a pair of
points which are the marginal image of a line through P.
If is the foot of the perpendicular from P on F, the two
points of each pair lie on a line through and are equi-
distant from 0. O is the radieal centre of the system, and
all the circles cut orthogonally the circle which is the
marginal image of the polar plane of P.
vu. IE
PENCILS OP CIRCLES
233
18. Types of pencils of circles.
(I) Lot the axis I of the pencil of planes he a non-mtersector of
the piano F, and let ffl be the common perpendicular of I and F.
Lot. .-l f B be the marginal image of I ; O is the middle point of AB.
Then the marginal images of the plant's through I are circles through
the real pointeil, B. One of these is the line A B ; then, as the plane 8
is tilted, we get branches of equidistant-curves, then a horaeycJe, and
lastly circles, onding with the state on AB as diameter, the circle
of least diameter.
(I') In the conjugate system the planes E arc all perpendicular
to a Used line I, and the axis V is ideal. The marginal images are
first a straight line through 0±AB, then a series of equidisfunt-
curves with axes 1 A B and increasing distances, then a horocyele,
and lastly a series of circles with diminishing radii tending to the
limiting point A ; and a similar series tending to the other limiting
point B.
(2) Let the axis ( cut F in 0. The marginal images are first the
straight line AB. then a scries of equidistant-curvrs with axes
through 0, one branch passing- through each of the points A, B,
ending with the equidistaut-eurve of greatest distance whose axis
J.AB.
(2a) lf l±F, A and B coincide, and the morginal images are
concurrent straight lines through O.
(26) If Hies in F. A and B are at infinity in opposite senses. Tho
marginal images are equidistant- curves with common axis AB.
(2') The conjugate system to (2) is similar to (1'), hut instead of
starting with a straight line we ha%'e first an equidistant-curve with
a minimum distance
(2'a) When l±F the limiting points coincide and the marginal
images become concentric circles
(2'b) When ( lies in F the limiting points are at infinity La opposite
senses, and the marginal images are straight lines 1. AB.
(3) Let 1 1! F, then one of the points, B say, is at infinity. The
marginal images are equidistant-curves through A with axes parallel
to AB. one being the straight line AB and one the horocyele ± AB.
(3') In the conjugate system the limiting point B is at infinity.
We have in <>l oqutilisiant-eurvea with increasing distances.
234
SYSTEMS OF CIRCLES
[vn, 18
then a horoeycle, and lastly a series of circles with diminishing radii
ending with the limiting point A.
(4) Let I be at infinity with P as point at infinity, and BOppose
P is not on F. The planes E are all parallel The marginal image
of I is a point A, i.e. A, B coincide. A is the orthogonal projec-
tion of P on F. There ia a real plane through l±F cutting F in
a line *. Wo have then, as the marginal images of the planes E,
first the straight lino *, then a series of equidistant curves, then
a horoeycle, and finally a series of circles, all touching I at A,
which is both a limiting point and a common point.
( ■(') The conjugate system is of exactly the same form, since the
absolute polar of the line I at infinity touching the absolute at / J
is also a line touching the absolute at P. The marginal images all
touch a line ± t, A being the point of contact.
(5) In (4) let P be on F. so that A coincides with P at infinity.
The parallel planes E make a constant angle a with if. We have
then, as marginal images, a series of equidistant-curves with axes
parallel to the direction through A, and constant distance a. such
that il(o)=o.
(5a) If the conjugate axis t lies in F, a= £jt, and the equidistant-
curves reduce to a system of parallel straight lines.
{5b) It I lies in F , u=0, and the marginal images are a system or
concentric horoeycles.
{6') The conjugate system to (5) is a system of exactly the same
form with the angle a'=- - a.
(5'a) is the same as (06) and (a'b) the same as {;V«),
Note— In f>} we appear to have a pencil of circles with coincident
common points, but we must consider this actually as a pencil with
one real common point, and an ideal common point which is the
ravened o «ri*4 rasped to fee absolute, Smttarfy En $ m should
regard the two branches of the equidistant-curves separately, and
regard the whole system as consisting of two pencils, each with one
actual and one ideal common point. (Cf, IJx. VIII. 19. 20.) This
is rendered clearer if. in finding the marginal images.' we confine
our attention to the parts of the planes and lines which lie on one
side of the plane F.
Ex. vi i.J
EXAMPLES
EXAMPLES Vn.
1. Prove that if the common tangents to two circles are all real,
the distances between the points of coated are equal in pairs, and
that all four distances will be equal only if the axes of the circles an
orthogonal.
2. Prove that the second radical axis of two circles which pass
through A, I! pa**"* through the middle point of one of the segments
All. HA.
3. Prove thatx i ky-kz=0 represents, for parameter A, a pencil
of lines parallel to the positive axis of x.
4. In elliptic geometry, prove that the circles a,*-i hjij + c 1 z±d, = *>
and ap + etc. - will cut orthogonally i f IP (<t s a t + bfa) + e,c, ± d^d. a 0.
5. If a bundle of circles contains a pencil of lines parallel to the
positive axis of x, show thai the equation of the bundle can bo
written in the form {x+plt)+ ky-rti(x+p)=0.
6. If a bundle of eircles contains a pencil of lines perpendicular
to the axis of x. show that the equation of the bundle can be written
7 Prove that tho orthogonal circle of the bundle of circles
x-i Ay <- /*(z -4 c)=0 is es=L
8. Prove thiit every circle of the system {x- pfc)+Xp+ft(* -p)=Q
cuts orthogonally the horooycle p{x~kz)--k.
9. If the orthogonal circle of the bundle of circles
aj+Aj/+fi(2 + c)=0
is imaginary, prove that every circle of the system passes through
the ends of a diameter of the fixed circle z +e-0.
10. Prove that tho locus of the centres of point-circles of the
bundle z + A(0 + 6)i-/is=O is the equidistant-curve by- +K
11. If sinhpsinh p'=l. prove that
ll(,,)t-n(ji.') = ^, and p'«lag0Oth|.
12. Given a circle, equidistant- curve or horocyclc in a plane F,
show how to construct the plane A* of which it is the marginal image
on the plane f '.
CHAPTER Vm.
INVERSION - AND ALLIED TRANSFORMATIONS.'
1. In euclidcan geometry, Inversion, or the transformation
by reciprocal radii, is a transformation which changes any
point P into a point P', and P' into P ; the line PP' passes
through a fixed point 0. the centre of inversion, and the
segments OP, OP" are connected by the relation OP .OP
= constant. This transformation lias the properties that it
changes circles into circles and transforms angles unaltered
in magnitude. It is a special case of a conformal trans-
formation which preserves angles, and of the more special
type of conformal transformation, the rimekr transforniutiim
which changes circles into circles.
We shall consider in this chapter the circular trans-
formations in the non-euclidcan plane, and first we shall
prove the following theorem.
2. Any print-transformation, which changes circles into
circles is conformal.
Two circles which intersect at equal angles at A, B are
transformed into two circles which intersect at equal angles
at A', B', i.e. certain pairs of equal angles are transformed
into pairs of equal angles. We shall show that this holds
for all pairs of equal angles.
1 See the references to Hiuistloril and Liobiwuin in chap, vii. § JU.
vm. 81 CIRCULAR TRANSFORMATIONS 237
Let the lines o„ 6, through S, and a it 6 a through S. z
make equal angles in tke same sense, Let flj, « 2 meet in 0,
FIG. no.
Draw
and let S,S S make equal angles with a 1 and %.
6 through >? 3 making u t b t -6h», then ba i =aj} > .
Two circles A £ can be drawn with their centres on the
bisector of the angle at 0, passing through 8, and S 3 and
having a,, a 2 and fc, , b as tangents ; and similarly a circle C
can he drawn with its centre on the perpendicular bisector
of Sj£, and having 6, b 3 as tangents.
The equal angles 0,6, and ba s are transformed into equal
A A
angles, and the equal angles ba t and aJ> a are also trans-
farmed into equal angles, is. a pair of equal angles in any
position are transformed into a pair of equal angles.
Hence two adjacent right angles are transformed into two
adjacent right angles, half a right angle is transformed into
half a right angle, and so on. Hence angles are unchanged.
3. Consider two planes J, , F*. All the planes in space
are represented on each of these by circles, and we have a
1
238 INVERSION AND TRANSFORMATIONS [vm. 4
correspondence between the circles of F t and the circles
of F* through the medium of the planes of space. Then, if
F a is made to coincide with F u we have established a corre-
spondence between the circles of F^ itself, i.e. we have
effected a transformation of F t , changing circles into circles.
Instead of supposing the plane F t to move, we may suppose
F to be a fixed plane and let the whole of space move rigidly,
with the exception of F. To a circle C corresponds a plane
E. This plane is moved to E' and gives another circle C.
To a pencil of circles corresponds an axial pencil of planes,
and this gives again a pencil of circles. To a bundle of
circles with common radical centre corresponds a bundle
of planes through a point P ; P is moved to P', and we get
another bundle of circles with common radical centre 0'.
Hence this effects a transformation of the plane F, changing
a point into a point, and a circle into a circle. It does
not change a straight line into a straight line, but in general
into a ctrola
The motion of space which has just been considered is
a kind of congruent transformation, i.e. it does not alter
distances or angles. Rut a congruent transformation
considered more generally may reverse the order of objects,
changing, for example, a right-hand ^Iovi> into a left-hand
glove. Such a transformation is produced by a reflexion in
a plane. A motion is equivalent to two reflexions.
We may extend the above result, therefore, and say :
Every congruent transformation of sjmce gives rise to a circular
transformation of a plane.
4. Conversely : Every paint -transformation of the plane
which duinges circles into circles can be represented by a
congruent transfornuttion of space.
vm.5] CIRCULAR TRANSFORMATIONS 239
To a circle C corresponds a plane E, and to the corre-
sponding circle C" corresponds a plane E'. Hence a plane
is transformed iuto a plane, and the angle between two
planes is equal to the angle between the corresponding
planes. Further, a pencil of circles is transformed into a
pencil of circles (since the transformation is a point-trans-
formation) ; hence a straight line, the axis of a pencil of
planes, is transformed into a straight line. Also a bundle
of circles is tmnsformed into a similar system ; hence a
point, the vertex of a bundle of planes, is transformed into
a point. The transformation of space therefore changes
points, lines and planes iuto points, lines and planes,
and leaves angles unaltered, ie. it is a congruent trans-
formation. 1
6. The general circular transformation which we have
been considering is more general than inversion, for in-
version leaves unaltered a point 0, the centre of inversion,
and also all straight lines through 0.
In general a system of lines through a point is trans-
formed iuto a pencil of circles. In a pencil of circles through
two points A, B there is always one straight line, the
straight line AB ; and if a pencil of circles contains two
straight lines it must consist entirely of straight lines;
for the planes corresponding to the two lines are both
perpendicular to F, and any plane through their line of
intersection is also perpendicular to F.
Now a pencil of lines through a point A is transformed
into a pencil of circles through A', B' . Hence one line of
the pencil is transformed into the straight line A' ft. Hence
J Tn ouclidran space these conditions would specify only a similar
transformation. In non-euelidean geometry, when the angles of a
triangle are given, its aides are also determined.
240 INVERSION AND TRANSFORMATIONS [vm. 8
tlirounh any point A there is one straight line g which is
transformed into a straight line g'. Let h be another line
which is transformed into a straight line h'. Then the
pencil igh), which consists entirely of straight lines, is trans-
formed into a pencil (g'h% which consists entirely of straight
lines. lfg t h intersect in 0, then^'. k' intersect in 0\ and
FIO. 120,
the corresponding angles at and 0' are equal. Let
lie moved into coincidence with 0' and g with g' . Then
either h and h' coincide or can be brought into coincidence
by flapping the whole plane over about g', i.e. by a reflexion
i n //. Then all the other lines of the first pencil will coincide
with their correspondents, since angles are unaltered.
Hence, Uie gemral circular transformation is compotmded
of a congruent transformation of tlie plane and a circular
transformatio, t which leaves unaltered all the straight lines
through a fixed point.
6. Of this simpler form of circular transformation, which
keeps one point fixed, there are three types, according as
the fixed point is real, ideal or at infinity. These are called
the hyperbolic, elliptic and parabolic types.
And further, there are two forms of each, according as
vm.7] INVERSIONS AND RADIATIONS 241
the corresponding congruent transformation of space is
a reflexion or a motion.
In the first case points are connected in pairs, since the
relation between a point and its image is symmetrical.
If P is transformed into P', then by the same transformation
P' is transformed into P. A repetition of the transforma-
tion will reproduce the slahts quo. The transformation is
therefore periodic with period 2. or, as it is called, involu-
tory. This form of transformation is called an inversion.
In the other case, by repeated transformation the trans-
formed points on a fixed line always go in the same direction.
This form is called a radiation.
7. We shall now determine the metrical relations which
*
define inversion.
(1) Hyperbolic Inversion, with real centre 0. Draw a
line 0D=d perpendicular to the plane F, and through I)
.A'
draw a plane K±OD. We shall obtain a hyperbolic
inversion by a reflexion of space in the plane K. Take
any point P in F and draw PAA.F, and QA±0D
anil H PA. Let Q' be the reflexion of Q, so that DQ' =QD,
and in the plane POQ draw Q'A' LOD, and P'A'LF
n,-e. a. H
i,
242 INVERSION AND TRANSFORMATIONS [vm. 7
and || Q'A'. Then P' is the point which corresponds to P.
Construct the point C which corresponds to D. Let
OC = c,QD=DQ' =y, OP =z, OP' -as'.
Then
ainhe srrnW = l = sinh as sinh (d +?/)= sinh as' sinh (d-y);
therefore sinha:=cosech{rf+;y) and cosha; = coth(<£+y).
tanh ^x = coth x - cosech x
= cosh (d + y ) - sinh (d + y) = e - W+rt,
Similarly tanh Jx'=e- {d -5'> and tanh §c = e-''.
Hence tauh Jx tanh J35' =e~- <( =tanlvHc.
This is the formula for inversion in a circle of radius c.
(2) Elliptic Inversion, with ideal centre 0. The fixed
lines are all perpendicular to a fixed line I. Draw a plane
K through I making an angle u with F, and take this as the
plane of reflexion. Then in Fig. 122, where
AOD = DO A' = $,
PA ±0P and || OA, P'A' ±0P' and || OA', OP -z, OP' =x>,
OC=c, we have
u(x)=u+e, u(x')= <t -e.
vm. 71
FORMULAE FOR INVERSION
243
Therefore 1 1 {x) + IT (as') = 2a = 2T7 (c).
This is the formula for inversion in an equidistant-curve
of distance c.
If a = I, this gives x' = -x, a reflexion in a straight
line.
If a = j, we have II (x) + II (x') «», or sinh x sinli x' = 1 ,
a form of transformation which was frequently used by
Lobacbevsky in establishing the trigonometrical formulae.
(3) Parabolic Inversion, with centre £2 at infinity. The
corresponding congruent transformation of space consists
of a reflexion in a plane E [| F.
In Fig. 123 XD is the trace of the fixed plane K, C the
marginal image of D ; VA is the trace of a plane || F, V'A'
FIO. 123.
the trace of the reflexion of VA in K t and P, P' the marginal
images of A and A'.
Draw the horocyclic arcs PU t CV, P'U'. Let CP=x,
CP'=x', x being positive and x' negative.
Then PV =0X =P'V =i = l, since each is the arc of a
244 INVERSION AND TRANSFORMATIONS fm s
horocycle having the tangent at one end parallel to the
radius at the other end.
XV=XV, CY=PU.e*, CV'~P'U'.<*;
also CF + CT" = 2£X
Therefore e r + e* = 2.
This is the formula for inversion in a horocycle.
8. There is one property in which non-euclidean inversion
differs from euclidean. In euclidean inversion the inverse
P' of a point P with respect to a circle of radius OA is the
Fig. 12*.
point of intersection of the radius OP with the polar of P.
This does not hold in non-euclidean geometry.
If P'T is a tangent to the circle, and OP=r, 0P' = /,
we have
cos TOP' = coth OP' tanh a = tanh OP coth a.
Hence tanh r tanh / = tanh%,
whereas the distances of the inverse points are connected
by the relation ^ y ^ ^ = ^^
In euclidean geometry these both reduce to the same,
m > = a a .
The transformation which is called in euclidean geometiy
" quadric inversion," and which is obtained by the above
vnt.9] CONGRUENT TRANSFORMATIONS 245
construction with the circle replaced by any conic, is, there-
fore, in non-euclidean geometry, not a generalization of
inversion.
9. Congruent transformations. Transformation of
coordinates.
The equations which determine a congruent transforma-
tion of the plane are given at once by the equations for
transformation of coordinates.
Let the rectangular axes Ox, Oy be moved into the
position O'x', O'y', still remaining rectangular. Let the
Fro. 125.
coordinates of 0' be (a , b e , c ), and the equations of
O'x', O'y',
a^ + i^y +0^=0.
Then, the geometry being hyperbolic,
-{D
t
346 INVERSION AND TRANSFORMATIONS [vm. 8
the new axes are rectangular and pass
Also, since
through 0',
a,a +6,6 +BiC =0, (2)
whence
Oq : b : c : 1 =6jC s - 6^ : c,a a - c^ : afi a - a^ : R.
To determine the factor R, we have
- k*R? = (V* - & A) 2 + («v»i - c^o,) 1 - ^{fflj&j - a A) a
= -#>(V + 6,2 - Cl «/JP) («*• + 6," - Cl »/P)
+ &{<i i a 2 + b l b t -c 1 c 2 fk z y
= -&.
Therefore R = ±l.
There are two cases in the transformation, according as
the new axes follow the same order as the old, or are
reversed. The second case is obtained from the first by
interchanging x and y. If the axes are supposed to be
fixed while the whole plane moves, we call these two cases
respectively a motion and a reflexion of the plane. A
reflexion may be produced by flapping the plane over about
a line in it.
If the axes are unchanged we have a 8 =0, 6 t = l, 0,-1,
&i=0, e = l, an d therefore aj> t - a t b l = \. If the axes
are interchanged a 2 = l, 6 a =0, ^=0, 6, = 1, c fl = l, and
therefore Oi& a -«.&= -I. Hence for a motion R= +1,
for a reflexion R = - 1.
We have then
b ] e z -b 2 c 1 =Ra , Cia 2 -c 8 a 1 =E6 ,
&iCo + Vi/** = -R<t 2 , a l c +<t l p t /k s =Rb,,,
(ijb 2 -a 3 b 1 =Rc e ,
fesC + &0C2/P = P% , Ojc + fcoCa/A 2 = - Bfr, , a 2 b ~a i fi t =Rc 1 .
vu>. 10] CONGRUENT TRANSFORMATIONS 247
10. Let P be any point, whose coordinates referred to
the old and the new axes are (x, y t z) and (a;', y', z'). Then,
expressing the distances of P from 0'x\ O'y' and O', we
have x'=fcsinhM'P/fc, y 1 =k sinh N'Pfk, z* = cosh O'P/k;
hence a;' = e^a; + bji/ + <\z, \
t/=a t x + b i y+c 2 z, V ... (a)
- JfcV =aoa; + &(,!/ - fc%s.J
where the nine coefficients are connected by the six relations
(l)and(2).
Further, since x'* +r/ i -kh' s = -h 2 , we have
a* + a a a - «,»/# = 1 , Vi + b&t + 6«Po = °>
V + 6 a 2 - 6 a /fc* = 1 , 0,0! + cja s + c^ = 0,
Multiplying the equations (a) respectively by %, a v
- ajk* and adding, we get
a; = a 1 a:'+<* s / +«(£'. 1
Similarly ?/ =&!»' + &$' +W- \ •• ( A ')
-A*z=c,!b'h-Cj1/' -IPc&', J
from which we see that the coordinates of 0/, Ox',
referred to the new axes are
(a u «,, a a ), (6 t , b it b a ), (c,, c 2 , c^.
(a') is the inverse transformation to (a). Both can be
represented by the scheme
a; y ikz
x'
0] Oj Cjlik
y'
a t b 2 cjik
ikz'
a-oM bfjik c
which may be read either horizontally or vertically.
i
248 INVERSION AND TRANSFORMATIONS [v.n. n
The determinant of the substitution = + 1 f or a motion,
- 1 for a reflexion.
11. The transformation admits of a very simple repre-
sentation.
Since x i +y i -kh i = -&, we can write
{x + iy) {% - iy) = k*(z - 1) (z + 1 ).
Let
Then
and
x = x+iy = k(z- 1) c- j8- ?^_ _ t(g-l)
ft(s + I) s-iy' A(z + 1)~ s+iy '
1-XX 1-XX 1-XX
The coordinates of a point are then expressed in terms of
the complex parameter X, just as in Argand's diagram.
12. Let X' be the parameter of P referred to the new
axes. We have to express X' in terms of X.
x' + iy' = (a-, + iajx + (6, + ib J y + (e, + w t ) z,
W + I ) = (p - a & - b$ + k*c£)lh
Multiplying these by I -XX, we get
N =k{a l + ta-^(X+\) + ik(b t +?o g )(X -X)
+ («i+m 2 )(1+XX)
=&X[(% + ia z ) -{(6, +?6 2 )]
+fcX[(a 1 +«,) +*<«, +{& a )] + (i 4-XA)to +«;»).
D-i(l -XX) -a (X +X) -i6 (X -X) +fe (l +XX).
Now («.„ - t6 )[(a, + w 2 ) + i(fc, + $,)]
= - c oC] + Re! + i( - Co c a + JfcJ - - ( Co _ #) (c, + ic^,
and K+*o)l(«i+Wt)-ifo+#J]= -(Co + fljfcj+tc,).
vm. 13J THE COMPLEX PARAMETER 249
Let X be the parameter of 0', so that
Then, for a motion, K= +1, and
iVX = (c, + M! £ )(X -X)(l -XoX),
Z) = i(c + l)(l-XX ){l-XoX).
Now c^+c^=P(c Q z -l);
therefore c x + ic 2 = - As/c * - 1 e*, where # = tt + tan - ' J
and a a + 6 8 = f(c 2 -1);
therefore X (c + 1) = Je a ~lef i , where i/r = tan - 1 -*.
"0
Therefore A T - A(c + 1)(X - X )(l - XtX)e iW " ♦>•
Hence
\' = ^ =
N X-X
<L «.«»->»
^e
"# 1-XX
Let e**-*-itfS, and X a = -)3, X 5= -,3; then
mXh-j8
X' =
£X+5'
i.e. the general transformation of coordinates, or the general
motion in the plane, can be represented by a type of homo-
graphic transformation of a complex parameter.
13. If S denotes the operation wbich changes X into X'
by tbe above equation, then the product of two such
operations S x aud S A leads to
yi « a X +p 3
"^aX'+aj
_ u 2 {ai\ +A) ^-ffgt&X+ci) = aX +0
where a = ajct s + /8i/3g, /3 = u iA + ui j3* ;
\
250 INVERSION AND TRANSFORMATIONS [m U
therefore $ } S 2 is an operation of the same form. The
operations 8 have therefore the property that the product
of any two of them is again an operation S.
Further, it can be proved that (ftS a )S 3 =ft(S 2 ft,), and
the operations are associative, A set of operations satisfy-
ing these two conditions is called a group. This group "is
called the group of non-eiwlidean motions.
The nomographic transformation which represents a
motion is a particular case of the general homographic
transformation
y _ «X+j fl
>X + <T
where «, ft y, § are any complex numbers. This trans-
formation belongs to a more general group, the group of
hmtiogmphic tmnsformutions, and the group of motions is
a sub-group of this larger group.
14. In elliptic geometry a motion is represented by the
transformation aK-B
\' =■= — ".
8\+~a
If S is the product of two operations ft, S t , we have
a = am t - ft ft,, 8 = a ,ft + ^8,.
Put_a=d + ia, 8=b-tc, where a, b, o, d are real and
i =•%/ - 1 ; then we have
o= «irf 2 +6 1 c 2 -Cj6 2 + (i 1 a a ,
6= -a 1 c 3 +6 1 rf 2 + c 1 <? a -i-d 1 6 2 ,
C= ffii&s-ftjfflg + c^ + ^Cj,
Now these relations are exactly the same as those which
we obtain from the equation
™+fy + ck + d = (a 1 i+b t j + c 1 k+d l )(a s i + b 2 j+c z k+dJ,
where i? = j a _ $■ _ _ j
jk=i= - kj, U =j =-ik, ij - & = _#,
vm. is] GROUTS AND QUATERNIONS 251
Here ai + bj+ck + d (=q) is a quaternion. Hence the
rule for compounding operations of the group
\'={a\-|8)/(8\+a)
is exactly the same as that for quaternions. The meaning
of this can be explained as follows. The operation q( )q l
performed upon a vector ( ) has the effect of a rotation
about a definite line. The product of two such operations
h(Si< kf l }H~ l -Ubl )?rV lta ?( )?"'■
where q = q->qi, and is therefore another operation of the
same form. These operations form the group of rotations
about a fixed point, or the group of motions on the sphere.
15. If we take polar coordinates (r, 0),
•Y T t
x +vy =& sinh , (cos + isin 0) =& sinh j_e i9 ,
z+l = cosh t + 1 =2 cosh* sr-
Therefore
A = tanh 9 , e
. ,r 2JW
sinh ,- = -.
& 1-XX
XX = tanh 3 ^, >/xXe» = X, cosh £ =^xx'
The equation of a straight line aa; + &iy + C2=0 becomes,
when expressed in terms of X,
ai(\+X)+»&*(X-X) + c(l+\A)-0,
i.c e(AA + l)+A(a+i&)X-i-&(a-*)A=0,
which is of the form
XX-dA-aX+l=0.
If the line passes through the origin, c=0, and the
equation reduces to
5X+aX=0.
i
232 INVERSION AND TRANSFORMATIONS [m 10
The equation of a circle,
cosh ^ = coah ^ cosh t - sinh f sinh t cos (0 - a),
with centre (c, a) and radius a, becomes
Xx(cosh ^ +coah |) - sinh r(Xe"*' + Ac" 1 )
+ cosh j- cosh "=0.
In general, therefore, the equation
XX -SiX-aX +6=0
represents a circle (equidistant-curve or horocycle) which
reduces to a straight line when 6 = 1.
16. The general homographic transformation of A leaves
the form of the equation of a circle unaltered, and is there-
fore a circular transformation. The transformation of
inversion is included in this. Inversion is characterised
by connecting points in pairs. The parameters A, A' of
a pair of inverse points must therefore be connected by a
lineo-Iinear equation of one of the forms
y _aX+8
yX+6'
y_a\+8
yX+V
■(I)
y\+S' yX+S'
4*
The first form characterises motions, the second re-
flexions, when y = 8 and <5 = 5. Inversion belongs to the
second form and is a symmetrical transformation, i.e.
X' is expressed in terms of X by exactly the same equation
as that which expresses A in terms of A'.
If y=/=0, we can take X' = »*+#
X+S
Ex.vm.] INVOLUTORY TRANSFORMATION 253
Then X =-<?"— ® and A =
-$x'+8
X' -a
Hence S = -u and 8 = 8, i.e. 8 is a real number = -6.
The transformation for inversion is therefore of the form
., aX-b
X — <i
•(I)
If the points X, A' coincide, so that A' =X,
AA -aX -«X +6=0 (r)
which is the equation of the circle of inversion. If X, A'
are a pair of corresponding points, equation (i) gives
A'X-aX'-aX+6=G.
If y =0, we can take X' =«A +8. Proceeding as before
we find aa = l and 8= -a/3. The transformation then
reduces to the form
oX'+aA = 6.
In this case the circle of inversion is the inverse with
regard to the absolute of a circle which passes through the
origin.
These results should be compared with the corresponding
formulae for euclidean geometry in Chapter V. §§ 31, 32.
EXAMPLES VEX
1. In elliptic geometry, show that the general transformation of
coordinates is expressed by A'=} ; ^k+ta-E f ™ °'
are real.
~{c-id)k+{a-ib}'
2. Prove that the general homographic transformation X'= ■ . ,
changes circles into circles.
ak * li
3. Show that the transformations A'=—r— -. form a group.
yA + o
i
254 INVERSION AND TRANSFORMATIONS [Ex. vm.
4. Show that the general reflexion nf the plane in hyperbolic
geometry is represented by \' = ~-!-.
tfA, + a.
5. Show that the reflexions of (lie plane tlo not form a group, but
that the product of two reflexions is a motion.
6. Show that the operations of the group A'=^— ^ leave un-
Ex. vm.]
EXAMPLES
255
altered the equation A A = 1
/3A + a
7. Show that the equation »/=0 is unaltered by the operations of
the group A'= - — -, where a, b, c, d are real.
CA +«
8. Show (lint the equation x=0 is unaltered by the ojjerations of
the group A'= — — where a, b, c, d are real.
tCA + tt
9. If the points A,, A,, A 3 are collincar, prove that
1 + A,a, A, A, =0.
1 + AjAj Aj Aj
1 + A..A, A, A,
10. Verify that if
A, A^cosh |+eo8h *\ - sinh |{ A,e - '»+ A>«) + cosh C - - cosh "=0,
the points A,, A, are collinear with the point whose polar coordi-
nates are (c, a).
11. Prove that the formula for a hyperbolic radiation, correspond-
ing to a translation in space through distance rf, is sinh x'=e a sinh x.
12. Prove that the transformation tanb rtanh /= const, changes
a straight hue into a curve of the second degree.
13. Prove that the inverse of the absolute in a circle of radius c
is a circle of radius equal to k log cosh c/k. (This circle is called the
vanishing circle ; cf. vanishing plane in the theory of perspective.)
14. Prove that the inverse of a straight line is a circle cutting the
vanishing circle orthogonally.
15. Prove that the inverse of a horocyele is a circle touching the
vanishing circle.
16. Prove that any circle which cuts the circle of inversion ortho-
gonally is unaltered by inversion.
17. Prove that the inverse of a system of parallel lines is a system
of circles all tombing at the sumo point.
18. Prove that a horosphere which cuts the vanishing sphere
orthogonally is inverted into a plane touching the vanishing sphere ;
and that a horocyele traced on the horosphere is inverted into a
circle lying in this plane and passing through the point of contact.'
Hence deduce that the geometry on the horosphere is euclidean.
19. Show that the equations x +}iy= U («- 1) and x+fa/ = kt{s + \),
where ( -tanh a/it, represent two pencils of branches of equidistant -
curves, the first passing through the origin, the second through the
point on the axis of x at distance 2a from the origin. Prove that the
inverses of these systems with respect to a circle with centre and
radius 2a are respectively x+fiy -kt 3 (z+ 1) and t(x + /ty)=k{s - 1 ).
20. Prove that the inverse of the pencil of straight lines x + p.y = ktz,
where f - tanh a/k, wit h respect to a circle with centre and radius a,
is the pencil of circles 2p*(»+/»gfl -kt [(p*+ l)z+(p*- 1)1, where
p at tanh fafk. Show that the common points of this pencil are on
the axis of k at distances from the origin equal to a and 6, where
tanh \bjk =p*.
21. Prove that inversion with regard to the absolute is represented
by X'A=1. .Show that this transformation leaves every straight
line unaltered, and changes the circle a*+%+« + rf=0 into
ax+by+cz-d^O, i.e. interchanges the two branches of an equi-
distant-curve.
22. 1'rove that two successive inversions in the two branches of
mi equidistant-curve of distance k sinh" l l, followed by a reflexion
in its axis, are equivalent to an inversion in the absolute.
i
CHAPTER IX.
THE CONIC.
1. A conic is a curve of the second degree, i.e. one which
is cut by any straight line in two points. Since the equation
of a straight line in Weicrstrass' coordinates is homogeneous
and of the first degree, the equation of the conic will be a
homogeneous equation of the second degree. In Cayley's
representation a conic will be represented by a conic. This
is the chief beauty of Cayley's representation, that the
degree of a curve is kept unaltered.
The projective properties of a conic are the same as in
ordinary geometry, and it is only in metrical properties
that there is any distinction. Since metrical geometry is
reduced to projective geometry in relation to the absolute
conic, the metrical geometry of a conic in non-euclidean
space reduces to the projective geometry of a pair of conies.
The metrical properties are those which are not altered by
any projective transformation which transforms the
absolute into itself The metrical geometry of a conic
therefore reduces to a study of the invariants and co-
variants of a pair of conies.
We shall confine ourselves here to au enumeration of the
different types of conies, and a few theorems relating to
the focal properties of the central conies which bear the
closest resemblance to those in ordinary geometry.
is. -J
CLASSIFICATION OF CONICS
257
2. Classification of conies.
Tn euclidean geometry, leaving out degenerate forms,
there are three species of conies, according as they cut the
line at infinity in real, coincident or imaginary points.
These are the hyperbola, the parabola and the ellipse.
Also, as a special case of the ellipse, we have the circle,
whose imaginary intersections with the line at infinity
are the two circular points.
In non-euclidean geometry conies are classified similarly
with reference to their intersections with the absolute.
Two conies cut in four points, and reciprocally they have
four common tangents. The points and lines which a
conic has in common with the absolute are called the
absolute pout* and tangents. These elements may be all
real, or imaginary or coincident in pairs. When two
absolute points are coincident, two absolute tangents are
also coincident. When two points are real and two ima-
ginary, the same is true for the tangents. When the
points are all real, the tangents may be all real or all
imaginary. When the points are all imaginary, the conic
must be within the absolute (for we need not notice a conic
which is wholly ideal), and the tangents are all imaginary.
Conies are therefore oteflrittod im> follows :
(J) Absolute points and tangents all real.
team hypabdht, with two real branches concave towards
a point between thorn.
(2) Absolute points nil real, absolute tangents sill imaginary.
liiiptrM'i. with two real branches, resembling an
ordinary hyperbola.
(3) Absolute points and tangent* all imaginary.
fillips, n rinsed curve.
(4) Absolute points and tangents two real and two imaginary.
Semi-hifperhofti. with one real branch.
N.-E. O, R
258
THE C0X1C
ix. 3
I
(5) Absolute points and tangents imi coincident and two real.
Concave hyperbolic parabola, two real branches touching
the absolute at the same point.
(6) Absolute points two coincident and two real, absolute
tangents two coincident and two imaginary.
Convex hyperbolic parabola, one real branch and an ideal
branch touching the absolute.
(7) Absolute points and tangents two coincident and two
imaginary.
Elliptic parabola, resembling nu ordinary parabola.
(8) Absolute points and tangents three coincident and one real.
(habiting parabola, one real branch osculating the absolute
at one end.
(9) Absolute points and tangents, two pairs of each real and
coincident.
EquidiKtanl-mrve.
(10) Absolute points and tangents all imaginary and coincident
in pairs.
Proper circle.
(11) Absolute points and tangents all coincident.
Iloroeijck.
In elliptic geometry the absolute points and tangents are all
imaginary, and we have only ellipses and proper circles.
3. The four absolute points form a complete quadrangle.
The diagonal points forai a triangle C\C Z C' S which is self-
con j ugate with regard to the conic and the absolute. Every
chord through any of these points is bisected at the point.
The points C,C' 2 C 3 are therefore centres of the conic, and
their joins are the axes.
The four absolute tangents form a complete quadrilateral.
Its diagonal triangle is formed by the three axes. In
euciidean geometry the foci of a conic are the intersection*
of the tangents from the circular points. Those are the
absolute tangents, and we call therefore the three pairs of
•*■«
CENTRES, AXES, FOCI, ETC.
intersections of the absolute tangents the/oe* of the conic.
Similarly the three pairs of joints of the absolute points are
called focal lines.
The polars of the foci with regard to the conic are called
tlttYctrux's. Tv. m pass fhruujih each centre and are per-
pendicular to the opposite axis.
Fill.
The poles of the focal lines with regard to the conic are
called director point.?. Two lie on each axis.
In euciidean geometry the focal lines degenerate in two
pairs to the linn at infinity. The third pair become the
asymptotes. Four of the director points coincide with
the centre, and the other pair coincide with the points at
infinity on the conic. In euciidean geometry the a.si/mptntcs
are the tangents to the conic at the points where it cuts the
absolute ; but in non-euclidean geometry the lines which
N.E. O. Illi
200
THE CONIC
r tx. I
ix. 15]
FOf'M, PROPERTIES
261
i
most closely resemble the euelidean asymptotes are the
taDgents to the conic from a centre, and arc therefore six
in number.
4. By taking the triangle formed by the centres as triangle of
reference, the equations of the absolute and the conic can be taken
in the form . , .
or in line- coordinates
a b e
The coordinates of the common (mints are given by
a? : >?: z'—b-o-.c-aia-b,
and the coordinates of the common tangents
£»: */» : f==a(6- e) : b{c-a) : c(a-b).
The focus F t is the intersection of two absolute tangents
■Ja (6 -c)x+\/b(c -a)y - <Jc[a -b}z =0,
Ja (b - c)x - Jb (e - a)y+ s /c(a^b)z=0 ;
therefore its coordinates are
0, \/c{a b), ,/i(c-a).
F L ' is the intersection of the other pair of absolute tangents, and
Its coordinates are „ , — r ,-
Ifi/. d' arc tho distances of a point P (x, y, z\ from F lr F,',
y s/e(a - b) + z -Jb (c - a)
coa if- T -
d -i y^( g -b)+zjc{c-a)
+ sin
Hence ~»^ i ^- (^-^-^-^Hftla-W-^-tf I
&+tf+z*)a(c-b\
_o+b (a-b)tj t -(c~ a)z* e +6
- c-6* o(a?T!/'+#)^ = c~& ,
i.e. e#/*er //w SUin or the difference of Ike distances of any
point on a conic from a pair of foci is constant.
Reciprocally, either the sum or (he difference of the angles
which, any tangent to a conic makes with a pair of focal lines
is constant.
A tangent makes a triangle with a pair of focal lines.
In the case in which the sum of the interior angles is con-
stant the sum of the angles of the triangle is constant, and
hence the area is constant. This result may be compared
with the property of a hyperbola in euelidean geometry, a
tangent to which makes with the asymptotes a triangle of
constant area,
6. The conic, the absolute, and a pair of focal lines form
three conies passing through the same four points. Any
lino is cat bv these three conies in involution. Let the line
cut the conic in P, Q. the absolute in X, Y, and the local
lines in M, N. Then (XY, PQ, MN) is an involution.
Ijet G, G' be the middle points of PQ. so that
(XY, PG)~(XY, GQ) and (XY, PG')7:(XY t G'Q).
Then G, G' are the double points of the involution, and
(AT. MQ)*(YX, NG)-*(XY, GN);
therefore G, G" are also the middle points of MN, i.e. the
segments determined by the points of intersection of any line
I with a conic and the points in which I cuts a pair of focal
faies have the same two middle pcmits.
Reciprocally, the tangents from any point P to a conic and
the lines joining P to a pair of foci have the same two bisectors.
If P lies on the conic, the tangent and normal to the conic
at P are the bisectors ofPF, PF'.
I
THE CONIC [nc.a
6. Take a focus F with coon) iiuites
The equation of the corresponding directrix is
y-Jb(a -b)+zJc[c-a)-0.
Let d bo the distance of any point P on the conic from the
directrix and r its distance from the focus; then
sin.tf-
smr =
Therefore
Va? + sf~+3?jfb~- c)(a -b-c) '
yjb(a - b) +z*/c(c - a)
■J-j^ + if-i&Jtiib-c)
siiir_ la
mid \
■b-e
i.e. the ratio of the sines of the dtstamxs of a point- on a conic
from a focus and the c»firs/it»ttlin<f directrix is constant.
Reciprocally, the ratio of the sine of th& angle which a
tangent to a conic makes with a focal line to the sine of Us
distance from die corresponding director point w constant.
7. It is interesting to obtain a geometrical proof of the focal
distance property. 1
Lot Si be the absolute and any conic Inning four real common
tangents with 12. Let two jmirs of the common iuiigen(« intersect
in the pair of foci F, F'. Let P be any point on C. .loin PF and
PF', cutting 12 in X, 7 and A", Y'. Then we have to prove that
dist, (/\F)±dist. (W)=const„
or, in terms of cross-ratios,
log (PF, XY) ± log {PF', A'}")=conat.,
i.e. either the product or the quotient of the cross- ratios is constant.
Let A' A"', YY' cut FF' in A and /J. A" J' ami A" J" cut FF' in
A' and B'.
Then (PF, XY)~K x -( F'F, A A')
and (FJ", ZTITCtCW, A'BpZ{ F'F, BA'}.
1 For part of thin proof I am indebted to Dr. VV. P. Milne.
a. 71 FOCAL DISTANCE PROPERTY 2611
Therefore
(PF, XY)HPF\ X'Y')={F'F, AA')MF'F, BA')=(F'F,AB).
Similarly {PF, XY) . (PF', X'Y')=(FF\ A'B').
Wo have therefore to prove one of these cross-ratios constant.
Four eonics through the points X, X', Y, Y' are Si ; X Y, A" i ;
XX', YY'; X Y', X'Y. Let 12 cut FF' in V, V. We have then an
Fie. 1S7.
involution determined by (UV, FF% and this contains also the pairs
A, B and A', B". If therefore {FF', AB) is a given cross-ratio,
A, B must be fixed points. •
Now, supposing that A, B are fixed points, the point P is con-
structed thus : Si is a fixed conic and F, F' two fixed points. FF
cuts Si in fixed point* U, V, and .4, B are a pair of fixed points m
the involution determined by (FF', UV).
Through F anv line u is drawn cutting Si in X, Y. XA cuts 11
again in X', and'wc get the line X'F'=n' corresponding to u. P is
264
THE CONIC
fix. 7
the point of intersection of u, ■*'. If F'X' cuts again in Y% then
iU"r% m B ' tU ° point ^responding to A m the involution
{ft , UV).
Since « cuts in two points, there are two lines u' correspond in K
to u, and similarly there are two lines u corresponding to u' The
rays « «' are therefore connected by a (2, 2) correspondence. The
locus of P is therefore a curve of the fourth dc^ve. liuf v.lun
/(coincides with H", so also do both the corresponding lines V,and
MM Bfw: (h-^n-rr ill,- |.»:iis eonUins ilu, | m ,. W uvirr It
therefore consists of this line doubled and a conic.
Also, if u is a tangent to the two lines «' coincide, and P is a
double point on «, Therefore m is a tangent to the locus of P
Hence the conic which is the locus of P touches the four taneents
drawn from F. F' to fl. »s »
Further, if P is taken on fi, X and X' coincide with V>; hence the
tiiii-I'THs I,, fi „ | tt-s points of teittBBOtiOOa ui:h MBS. l!ir..iM>
Cither .4 or >S. re
4, .ff arc therefore the fixed points in which the tangents to 11 at
its intersections with C cut FF', and therefore
(PF, XY)+(PF, X'Y') is constant.
Tho fooi f, F' are real only when the absolute tangents are
nil aginary.
In the case of the conver hyperbola the order of tho points
P. F X, Y and P, f, X'. Y' is the same, and the difference of the
focal dtstanr&s is constant (Fig. 128).
Pw, X2S.
Ex.de.] FOCAL DISTANCE PROPERTY
265
In the case of the ellipse the points P, F, X, Y and P, F', Y', X'
have the same order, and the mm of the focal distances is constant
(Fig. 129).
EXAMPLES IX.
1, If tho equation of the absolute is aP + tf+HW^O, prove that
the coordinates of the three pairs of foci of the conic
x t la+f/b + l^z'/c=(i
are («, rf^g /:?), (**^J, 0. ^}
where £».=&- e, f3=c-a, y—a-b.
2. In hyperbolic geometry, where the otpiation of the absolute
is ^+2^-^=0, show that the equation x*/a +,</*/& -iV/e=0
represents (1) an imaginary conic if a>0, 6>0, e<0, (2) a real ellipse
if a, b, c are positive and c docs not lie between a and 6, (3) an ideal
ellipse if «<0, 6>e>0 or 6<0. «>c>0, (4) a concave hyperbola
if a, b, c are all positive and e lies between a and b, (5) a convex
hyperbola if «<0, e>6>0 or 6<0, c>«>0.
•>m
THK OONIC
[Ex. ix.
Ex, jx.J
EXAMPLES
267
i
a. In elliptic geometry, prove that an ellipse, real or imaginary,
has always one pair of real and two pairs of imaginary foci.
4. In hyperbolic geometry, prove that the three pairs of Foci are
(1) one real and two imaginary for a real or imaginary ellipse or a
convex hyperbola, (2) all ideal for an ideal ellipse or a concave
hyperbola.
5. A,B are tixed points and APB is a right angle ; show that tho
locus of P is an ellipse. If AB=2a, prove that the real foci are on
AR at a distance from 0, the middle point of AB, such that
Unhz/£=tanh s «/fc, or a5=H*logcosh2a/J(:.
Hence prove the following construction for the iboi • Draw OR
making the angle AOR=Tl(a) and cutting the circle on AB as
diameter in R. If. Then F, F are the feet of the perpendiculars
on AB from R. K.
6. A, B are fixed points and P is a variable point, such that I he
angle APR is constant ; prove that the loans of P is a curve of the
fourth degree,
7. A, B are fixed points and j° is a variable point, such that
cosh ^i. cosh —
A. K
is constant ; prove (hat the locus of P is an ellipse.
8. Prove that the locus of a point, such that the ratio of the sines
of its distances from two fixed points is constant, is a conic.
9. A, B are fixed points and P is a variable point, such that the
sum or the difference of tho angles ARP, BAP is constant; prove
that in each case the locus of P is a conic passing through A and B.
10. A variable lino cuts off on two fixed axes intercepts whose
sum or difference is constant ; prove that in each case the envelope
of tho line is a conic touching the axes.
11. Prove that the product, of the sines of the distances from a
pair of foci to a tangent is constant State the reciprocal theorem.
12. Prove that the locus of points from which tangents to a central
conic are at right angles is a conic meeting the given conic where
it meets its directrices. State the reciprocal theorem.
13. Prove that the locus of a point which makes with two given
points a trianplc, whose perimeter is constant, is a conic with the
two given points as foci. Show that the locus is also a conic if the
excess of the sum of two sides over the third side is constant.
14. Prove that the envelope of a line which makes with two given
lines' a triangle of constant area is a conic. Show that the envelope
is also a conic if the excess of the sum of two angles of tho triangle
over the third is constant.
15. Prove that -^r + r^t + — y =« represents, for all values
0+ A 0+ A C+ A
of A., & system of confocal conies.
16. Show that in the conformal representation, in which straight
lines are represented by circles, a conic is represented by a qnnrtic
curve having two nodes at the circular points, i.e. a bicircular
quartic.
INDEX
Tho numbore rcfrr to tin? tsiifls, except itwso iirccwtetl by Ex., which
nta to ',.■ Bxampiu; n suriiiites fuomote.
Absolute, the, 40, 08, 1 54, JOS.
ks equation, 127. 129, 174.
in Euo. (Jeom,. 47, 155, 104.
Absolute (leometry, 22.
polar system. H,
space, 107, 210.
Absolute unit, of angle, 88.
of longlh, 13, IS, 5S. Mil
Absorption of ligbt, 309.
Actual points, 40.
Author, 197, 201.
Al.ESlBKKT, d', 3.
AhiLudes of a triangle. 54, 141.
of a, t.tialii'dii.ii, Ex. iv. 24.
A n eh or- ring, 106 n.
Angle, 28, 121
nhranlt 42.
flat, 28.
formula, 131.
logarithmic expression, 157.
of parallelism, 30, 35, 58.
right. 28.
in a semi ci rcli-. Ex, i I ; ix. 5.
Anglo-smn of a triangle, 5. 10, 12,
IB, 10, 18,21.
and area, 13, 30, 77. 82, 104.
Antipodal points, 55, 80, 130, 178.
Ap p heebie surfaces , Wo.
Archimedes, axiom of, 17.
Area, of circle. 80 ; Ex. ii. 23 j
iii, 3.
of equidistant-curve, Ex. ii.
22 ; iii. 4.
of plane, 10 ; Ex. iii. 5.
of polygon, 83, 104.
of triangle, 13, 20, 77-70, 81,
82, 103-104.
Area, of triangle, maximum, 7,81.
infinite, 7-0', 19, 20.
unit of, 79.
Argand's diagram, 181,248.
AlIISTOTLE, 2.
Astral geometry, 15.
Astronomy, 203-207.
Asymptote.'!, 259.
Asymptotic linos, 10, 30, 42.
Axioms, 2, 27.
of Archimedes, 17.
of Pnscii, 20.
Axis of a circle, 52, 104, 136.
of a conic, 258.
of a pencil of lines, 48.
radical, 219,228.
Halt., W. W. Rouse, 201 u.
IUi.tzer, H. B., 24.
15ki.tha.mi, E., 202.
Berkeley, Q., 201.
Bkuthakd, L., 7.
Hieircular (piartic, Ex. ix. Iff.
Bisectors of angles, Ex. iii. 1 ; 139.
Bolyai.J., 14, 15,21-24.
Bot.yai, W., 7, 14, 21-24.
li.'Not.A, K., 24 n.
I Broab, C. D., 210n.
Bundle of circles, 228.
of lines, 45, 55, DO.
Camerer, J. W-, 15.
Carsiaw.H. S., 24 n.
Cayley, A., 158, 102.
i Centre of circle, 51, 104.
of conic. 2ss.
nomothetic, 221.
270
INDEX
INDEX
271
'%
Centra, radical, 221,
Controid, 139.
Ceva's theorem, 145.
Cirolo, 51, 104.
circumference, 7(4, 1 14.
equal ton, 1 3ft, 227. 808,
in relation to Absolute, 130,
258.
in Hue. (3enm., 47, 140.
of infinite radius, 51 [see oho
!! "ii n-.\ (■]<■).
through three, points, 53, 189.
Circle at infinity, 104.
Circular functions, 57, I14ji.
measure of angle, 57, 81.
points, 47, 156.
transformations, 181 ; Chap.
viii.
Cireumcentre of triunglo, 64.
Circumeireles of triangle, 53, 189 ;
Ex. ii. I, 13 ; iv. 10, 21, 22.
of regular polygon, Ex. ii. 17.
Circumscribed quadrilateral, Ex.
ii. 3.
sphere, Ex. iii. 7, 12,
Clifford, W. K., 25 «, 201.
Clifford's parallels. 108.
surface, 10(1. 112.
Coaxal oircles, 222, 232.
Collinaarity, 135, 145 ; Ex. viii. fl.
Collinoation, 180.
CftmpWW hearings, 7.
Complementary scgmonts, 03.
Complex numbers, 181, 248.
Concurrency, 135, 145 ; Ex. iv.
12.
Configurations, 143.
Confocal conies, Ex. ix. IB.
Conforrual representation, 172-
191.
i i wisformation, 182: Chap, viii,
Congruenee, 28, 194-197.
■ if infinite areas, 8.
Congruent transformation, 158,
238, 245.
Conies, 40 n, 08 ; Chap. ix. ; Ex.
viii. 12.
Conjugate coaxal circles, 232.
harmonic, 05, 148.
isogonal, 147.
points, 80.
Conaifltency of N.-E, G., 202.
Continuity, 17, 29, 90.
(.'(Divergent lines, 42.
Cotit.tinii-:, .1 I. ., Z~ ...
Coordinates, 125, 199.
homogeneous, 135,
lino, 128, 172.
polar, 125.
tri linear, 172,
Weieretrasa', 127, 129, 171,
227,
Corresponding points, 32 ; Ex. ii.
Cross-ratio, 147,
Cube, Ex, iii. 8.
Curvature, measure of, 160.
of space, 193, 199.
surfaces of constant, 168,
Cyclic quadrilateral, Ex. ii. 2.
Cylinder, 105.
Defect of triangle, 20, 78.
Definitions, Euclid's, 2.
I )■■;.'. derate eases in Eue. Geom.,
11,47, 75, 82, 139-141. 166-
150, 161-162, 217, 220, 244,
259, 201.
Desargues' theorem, 142.
Deamio system, 144 ; Ex. iv. 23.
Dovolopabte surfaces, 160.
Dihedral angles, 42,
Dimensions of space, 208.
Direction fallacy, 0, 20.
Director points, 259.
Directrix, 259,
Displacement, 179, 190.
Distance, absolute unit of, 13, 15,
in fine. Ili-nm., 75, IfiH. Mil.
formula, 120. 132, 158, 186.
Divergent lines, 12.
Duality. 69, I nil. 226.
Egypt, 1,
Element of length, 187, 104 ; Ex,
iv. 5.
Ellipse, 267,
Elliptic geometry, 25, 20, 55 ;
Chap. iii. ; 200, 208 («M also
Spherical geometry),
inversion or radiation, 240.
involution, 97.
Empiricism, 207.
EnsHXs F., 1 1 it, 13 n, 21 n, 08 n.
Engol-Napi'i' nth-*, 07.
Envelopes, 104, 120; Ex. iv. 17;
ix. 10, 14.
Equidistance, 10, 42.
Equidistant- curve, 12, 52, 105,
258.
equation, 130, 228.
length of arc, Ex. ii. 7. iii. 2.
Eimidislant-surfaco, 53, 105.
Escribed circle*, Ex. ii. 13.
ICCl'I-lU, 1, 2.
Euclidean geometry, 30, 47, 76,
711. 00,92, 134, 139-141, 155-
157, 101, 176, 220, 259, 201.
Kvcc-s ni a trinnule. Hit
Exterior angle, theorem, 17, 19,
29, 31-35.
Focal distance properly, 200, 202.
Focal lines, 25!).
Foci, 259 ; Ex. ix. 1, 3, 4, 5
K. .ens-directrix property, 202.
Wam dimensions of space, 42 ?t,
193, 199.
FRANKi-ASn, W. B.. 4 n, 200 >i-
Froe. mobility, 107, 108, 195-100.
Fundamental theorem of projec-
tive geometry, '.Iii
Gauss, C. F,, on area of triangle,
7, 82-83.
on curved surfaces, 108.
on parallels, 14, 22, 24.
pentagram, 08 n.
C-dili-sir.,, 166
Geometry, Absolute, 22.
Analytical, Chap, iv.
Astral, 15.
Bizarre, 162, 104.
I H I'ferontial, 194.
Elliptic, Chap. iii.
Euclidean, g.v.
Hyperbolic, Chap. ii.
Imaginary, 21.
in the infinitesimal, 70, 1 14.
Log. -spherical, 15.
Non-Euclidean, 14, 20.
of a bundle. 55, W.
on Clifford's surface, 1 13.
(leomotry on curved surface, 160.
on equidistant-surface, 50.
on horosphore, 15, 50, 84, 165 ;
Ex. vih. 18.
on imaginary sphere, 13, 15,
105.
on plane at infinity, 135, 105.
on sphere, 50, 105,
origins, 1.
Parabolic, 25.
Projective, 93-98.
Spherical, 25, 89, 130. 138 ».
with hyperbolic or parabolic
measure of angle. 102, 104,
with projective metric. 100.
QaaaaKKB, ■' l> . Bx- ■ ■ *».
Greek geometry. 1.
GllKENSTBKET, W. ■!., 210 «,
Chonai:. K '(' !■'. . Ex. i. 7.
Groups, 197, 250.
Halsted, G, B., 1 1 ii, 21 n. 23 n.
Harmonic range, 95.
Hart's circle. Ex iv. 25.
Haitst, J. K. F., Ex i. 0.
Havsdorfk. F, 229 ».
HK\TIt. 'I'. L. 2.
Ih'iHiiKii, J- L., 2.
Hnst,MH(M.T2, H. von, 105- 1 '.IT.
I'.l'.l
Heboootus, 1.
Hii.bert, D, 27, 41, 11&.
HlNTON, C. H.. 201 II.
IlllTDnltATF-S, I.
I [0 HiATK, T. F„ 94.
lloiiuHviitric circles, 221
Homojjmphic transformation,
182, 240.
Homography, 94.
Hornochetic centres and axes,
221.
Horocyele, 51, 258,
equation, 137, 228.
leu-lb .if arc, 57 ; E\ n I
Horosphero, 52.
geometry on, 16, 56, 84, 160 1
Ex. viii. 18.
Ui.Tki.. -I .21
Hyperbolas, 257.
Hyperbolic functions, 15, 03.
geometry, Chap. ii. ; 25, 30.
272
INDEX
1KDEX
273
I
Hyperbolic inversion or radia-
Won, 240.
involution, 97.
Ideal elements, 47, 154.
Imaginary points, 97, 133, 134.
Indujinablea, 2, 27,
Inequalities, 28.
Infinite us. unbounded, 194.
Iiilimt,- areas. 7-9. |ii. 2u.
Infinitesimal domaio, 7ii, I 1 4,
transformation, 198.
Infinity, points at, 4fi.
Inscribed circles of n triangle, 54
139; Ex. ii. 10, 13; iv. 15,
19, .'11
of a regular polygon, Ex. ii. 17.
Inscribed quadrilateral. En. ii. 2
Intersection of lines, 132.
of circles, 211, 228.
angle of, of circles, 218.
Iniiiition, 2, 207.
Inversion, 241, 252
in Eire, tieotn . ISO, 183. 244.
quadrie, 244; Ex. viii. 12.
Involut '.17.
Involulory transformation, 241.
Isogonol conjugates, 147.
Isosceles triangle, 28.
Ivory, J., 19.
Kak.st.vkb, A. O., 21.
Kant, I., 1 4, 207.
Ki.kin, IV. 2.7. 159, 192.
Kluokj,, G. S„ 22.
Laoukkkk, E., )i5ii ,1.
Lamukrt, J. H., 13-14, 15, 20.
Lbhbn-dke, A. M., 10-18,20; Ex.
i. 4.
Lehi.ii-:, J., 19.
Lit:, S., 197.
I.ikiimann. H., 24 n, 229 n.
Limiting points, 223.
tines, ii:'.".
Lin e-c oordinates, 128,
Linear systems of eii-ctes. 220.
Line-element. 187, 194 : Ex t>, B.
LoBAfHEVSKY, N. I., 15 "II - ■> ]
23,24, ISSn, 243.
Loci, Ex. iv, 14; i.\. 5-fl, 12, IS,
Logarithmic expression for dis-
tanbe anil angto, 157.
LogariUmiie-spherieal geometry,
15
Lublam, V\'., 4,
Manifold, 194.
Marginal images, 229.
Maximum triangle, 7, SI ; Ex ii
19.
quadrilateral, Ex. ii. II, 12.
Medians. 139.
Meiklk, H., 19, 22.
Menelaus" theorem, 145.
Middle point of segment. 138, 159.
Milne, \V. I'., 202 ft.
Minimal lines, 134 ; Ex. iv. 2.
Mo bins' sheet, 91.
Monodromy, 195.
Motions. 28, 17!), IS5, 190, 248.
Naj?ier, J., 08 n.
rnli*, 08, 74, 119, 122.
Net of rationality, 9li.
Networks, Ex. ii. 11, IT), 18-20;
iii. 9.
Nine-point circle, Ex. iv. 25, 20.
-Non-euclidcan geometry, 14, 20.
Non-intorsectors, 12, 31, -10.
Normal to conic, 201.
One-sided sru i,ei<. u2.
Order, 27.
Orfhaxis, 54, 141.
Otthoecntre, 54, III.
Orthogonal points, 89.
circles, 219 ; Ex. vii. 4.
trajectory, 51,
Onhopolo, Ex. iv, 1.!.
.. 9g 75
UAp). 90, 38, 59; Ex. vii. II.
Parabolas, 258.
Parebolic geometry. 25.
inversion of radiation, 249.
involution, 97.
Parallax, 204.
Parallel angle, 30, 35, 5H.
lines, 29, 30 ; Ex. ii. ; 133.
planes, 50.
postulate, Chap, i. ; 27, 203.
Farotaxy, 103.
PaBOH, M., 193.
axiom, 29.
Peano, G„ 193.
Pencil of circles, 228, 233.
of hoes, 45, 170.
Pentagmmmo. mirifieum, 68,
08 n, 118.
Perimeter of triangle, 104, 103.
Perpendicular to a line, 28.
to plane, 43.
to eoplannr lines, 12, 40.
ro two lines in apace, 100, 109.
Perspective triangles, 142.
PerspcctiviiY, 95
PrERi, ML, 193.
Planes, 42.
Pinion ists, 1.
Playfair, J., 19.
axiom, 4, 30.
POINCAKK, H„ 210.
Polar system, 92, 98.
triangles, 102, 141.
Pole and polar, 51, 89, 130 ; Ex.
tv. 1.
Polygon, area, 83, 104.
regular, 80 m ; Ex. ii. 19-20.
Polyhedrn, Ex. iii 7, 8, 12-15.
Position-ratio, 147.
Postulates, 2, 3, 27.
Power of a point, 212.
of 11 line, 210
Projection, 95, 148.
gnomon io, 170.
stereographic, 172.
!Yejer-ii\-e geometry. 93-98, 135.
group, 198.
metric, 158.
Pseudoaphere, 198.
Pythagoras, 1.
Quadrangle-quadrilateral con-
figuration. 143.
Quadrant, 89, 110.
Quadrilateral, circumscribed, Ex.
». 3.
complete, '48.
inscribed, Ex. ii. 2.
of maximum area, Ex. ii 11.
1 -
tr-irectangukr, 70, 121.
Quadrilateral -ei instruction, 96,
199.
Quaternions, 251.
Radian, 28.
Radiation and inversion, 241 ;
Ex. viii, 11,
Radical axes, 219, 228.
centres, 221.
Radius of circle, 51.
Kaiitmality, net of, 00.
Reciprocity of parallelism, 32,
Rectangle, 9. 79, 112.
Reflexions. 238, 240.
in Krrc, Geom., 180.
Reid, T., 208 n.
Relativity-principle, 197, 210.
Reye, T., 94.
Ribuakn, B., 26, 194.
Rigid figure, 107, 195.
Rotation proof of parallel -post u
hue, 6.
Rotations, 185.
Russell, B., 193.
Saccheri, G.. 11-13, 15, 18,24.
Swit.E, H., 3.
Sc-hweikaiit, F. K., 14.
Sections of sphere, cic., 63.
Segments, congruence of, 27.
Sell-corresponding elements, 00.
Semi-circular eonals of the ear,
20H.
Semi-hyperbola, 257.
Similar figure*, 7, 13,
1 runsformation, 239 n.
Simson, R., 42.
Sommbbviixe, D. M. Y., 24«,
Space-constant, 58,
construction for, 77.
physical measurement of, 203.
Space-fillings, Ex. ii. 14, 15, 18-
21 ; iii: 14. 15.
Sphere, 62, 105.
.if infinite radius, 15 <«er Horo-
spliere).
volume, Ex. ii, 24 ; iii. 6.
Spherical geometry, 25, 89, 130,
138 » .
triangles, Q3.
trigonometry, 09, 70, 120,
274
INDEX
Square, 2.
St.ukbl, P., Jin, 13 », 23 n.
Staudt, G. K. CIi. von, 103. IBB.
Stereogruphie projection, 172.
Straight lino, Euclid's dot,, 2.
as shortest path, 17.
ro-eutrant, 17. 19, 11)1.
Superposition, 8, 27.
Synimolry of space, 28.
Systems of circles, 228, 232.
Tactual space, 208.
Tangents to two circles, 21 1.
to a conic, 2U1.
Tauki.vus, F. A., Iff.
IVlriilH'tlniii, li.\. iv. 7, 8.
dl Miiio, I 1 1
radius of eircunmpherc, Ex, iii.
7.
with concurrent altitudes. F.\
iv. 24.
Tiialks, I.
Tii i iiaut, B. F., 6.
TH.I.IMI-WIV. T IV'rronnt, 19, 22
Tilly, .1. 11. de, ]im«.
Townsbno, E. J., 27 re.
Transformations, circular, 181 ;
CI lap. viii.
conformal, 182; Chap, viii.
congruent, 138, 238. 245.
hom "graphic, 182, 24B.
infinitesimal, IDS.
of coordinates, 245.
<>r inversion, ISO, is:!. 21 1. 2H2
Transitivity of paralleli-tn, 32.
Transmiwibuity of parallelism, i
Trims vo real theorem, 33.
Triangle, angle-sum, 5, 10, 12, 13,
15, Hi, 18, 2«, 21. 77. 104.
area, 13, 20, 77-79, 81, 82, 103-
104.
associated, 05. 74, 117.
exterior angle, If, 111, 2B, 34-
35.
of maximum area, 7, 81.
periinotcr, 104, 163.
perspective, 142.
points connected with, 139.
rectilinear und spitoriciil, n:i.
right-angled, (13, 1 Hi.
side, 2D.
trigonometry of. lili. 1 Mi.
Tiiiingulation, 203.
Trigonometry, 1 5, 5<i, I Hi.
Trirect angular quadrilateral, 3<i,
71), 121.
t'h m-.-iptii i:il sjatnents, 48.
Unit of ongle, 28, 100.
of area, 79.
of l.-n K ili, 13, 15,58, IfiO. 162,
Vanishing circle. En. viii. 13.
Vkblen, O., 94, 198
Visual ■puce, 2<w,
Volumes, Ex. ii. 24; iii. 5, fi.
Wachter, F. L., 15.
U'allts, J., 7.
Wcici-siriij^ inordinate*, 127
129, 171.
WmsOBUB, A X . 193, 201re.
Yocno, J. W., 94.
A TREATISE ON HYDROMECHANICS
Part I.— HYDROSTATICS, By W. H. Buurr, Sc.D.,
F.R.S., Fellow and lace Lecturer of St. John's College,
Cambridge, and A, S. Ramsey, M.A., Fellow and Lecturer
of Magdalene College, Cambridge. Seventh F.dition, revised.
Demy Svo. 7 s. 6d. net.
Part II.— HYDRODYNAMICS. By A. S. Ra.usky, M.A.
Demy Svo, 10s. 6d. net.
AN ELEMENTARY TREATISE ON HYDRO-
DYNAMICS AND SOUND
By A. B. Basset, M.A., F.R.S., Trinity College, Cambridge.
Second Edition, Revised and Enlarged. 8vo. 8s.
NOTES ON ROULETTES AND GLISSETTES
By W. H. Bksant, Sc.D., F.R.S., Fellow and late Lecturer,
St. John's College, Cambridge, Second Edition, Enlarged.
Crown 8vo. }».
ELEMENTARY GEOMETRICAL OPTICS
»
OUUWOW : PRINTS AT TUX DKlvKlHITV 1>M»H RV HOBKHT MACLniOS AMD CO. LTD.
By A, S. Ra.msiy. I>emy Jsvo. $s. net.
G, BELL it SONS, LTD., LONDON, VV.C.
A LIST OF BOOKS
SELECTED FROM
Bell's Educational
Catalogue
CONTENTS
MSI
Latin and Greek .......... 1
Mathematics 5
English a
Modern Lanc.uac.es ra
Science and Technology 14
Mental and Mom At. Science 15
History , 15
Horn's Pctulak Library tfj
MESSRS. BELL are at all limes glad to
receive visits from members of the teaching
profession, and to avail themselves of the oppor-
limiiv to discttSi matters of iimlunl interest, to
submit their latest publications, and to talk
over new methods and ideas.
i
LONDON: G. BELL AND SONS, LTD.
PORTUGAL STREET, KINGS WAV, W.C.
CAMBRIDGE - - DEIGHTON, HELL &. CO.
NEW YORK - - THE MACMILLAX COMPANY
BOMBAY •.. A. H, WHEELER & CO.
G. Bell
c-
Son/
Select Educational Catalogue
\
LATIN AND GREEK
FnU Catatogut of Classical Books sent mi application
Bell's Illustrated Classics
Edited by E. C, March ant, M.A.
Edited with Introductions, Notes and Vocabularies. Wild ilhtst rations, maps and
plans, is. Oil. each; except the Grelk I'LAVS, which are 2J. each.
Ctonar. Hook I. By A. C. Tjimitr i„ M.A.
Book 1 [. B? A. C. LM.mii.i., M.A.
Book 111. By F, H. CtusOK, M.A,, and
I*. M. GwVTttBK, M.A.
Hook IV. By Rev. A. W. Uitiht, D.D.
Beat V. ^ By A. Ricraoms, M.A.
- — Books IV*. and V., in ope volume, is. 6ti,
Book VI. By [. T. Pmr.Lo-so.v, M.A.
Books, V. anil VI.. in one volume, ar. <*/.
Book VII. By h: ]■;. Wis holt, M.A.
Cwsar'B Invasions of Britain (He Hello
Ufa Lib. IV. XX. -V. XXIH.Jl By
Kov. A. W. UitoTT, D.D., anil A. Rsv-
HOt Dfi, M.A.
GlCeTO, Speeches against Catiline. I. utlij
II. (i vol,), ByF, Hkkkfni;, M.A.
* — ■ Selections. Bv J. l\ CitAUI.KH, B.A.
DeAmidtla. By ILL L..1. MawiS.M.A.
Di; Sa a ac tm o. By A- S, Wanman, b.A.
Cornelius Nepos, Epamjuonde*, Hanr.lkd,
("alo. By H. L. Eakl, It, A.
Eutroplna. Books 1. and II. (i vol.). By
J, G. Si>:schr, B.A,
Homer : Iliad. Book L By L. D. Wain-
WHiuur, M.A.
Horace I Odes. Book I. By C. C. Bot.
tiko, HA.
Book It By C. C. Bottino, B.A,
Boot III. By H. Lattkk, M.A.
- — Lkiuk IV, l!v 11. La: :■]■::;, M.A.
Livy. Book IX , ue, i-xi*. By W. a
Flamstkau Waltehs, M.A.
Livy. Hannibal's First t'ainpatG;n hi Italy,
(SfllocWd iVom Book XXL) By F. E, A.
Tkaves. M.A.
Lucian: Vera Historin. By R. K. Yatks,
1.1. 'I
Ovid : Metamorphoses. Book I. By G. h.
Wells. M.A,
Selection from the Metamorphoses.
BvJ. W. E, PllAHC*. M.A.
Elegiac Selections Bv F. CdVBaLSV
Smith. B.A,
Tristia. Book!. By A. E. Rogers, M.A.
■ Tristia. Book 111. By H. R. Wont,-
Rvcil, M.A.
PhaedruB: A Selection. By Rev. R. H.
CsAJtfffSKB, M.A,
Stories of Great Men. By Rev, F. Com.
wav, M.A.
Virgil : AeoefcL Hook 1. By Rev. E. II. S.
EsctiTT, SLA.
Book 11. ByL. I), Waishi-right, ALA.
»w,k III. By L. Ii. Waiskhight, M.A.
Book IV. By A. S. Wabnax, B.A.
Book V. Bv j. T. Pnn-LIPSON. M.A.
Book VI. Bv J. T. Piiili.ipsok, M.A.
Books VII., Y'llL. IS., X., XL, XII.
By L, D. WaIKWKKSH r, M.A. 6 vols.
■ Sefactifto team Books VII. to XII, By
W, G. Coast, B.A.
(I corgi oh. Book IV. By L. D. Wain.
WKNitrr, M.A.
Xenophon . Anabasis. Books I., TL f III.
liv K. C. Mauoiaxt, M.A, 3 vols.
GREEK FLAYS (21. each)
Aeschylus: Prometheus Vinclus- ByC. E.
Lai-klxch, M-A,
Euripides : Akejuis. By E. H Blakbney,
M.A.
Euripides : Baccttae, By G. M. Gu-vthe*,
JI.A.
Heeilha.
Medea.
By Rev. A. W. UrcoTT, M.A.
By Rev. T. KKKUH, SLA.
Bell's Illustrated Classics— Intermediate Series
Edited for higher forms, without VoeabaUries
Cfesar: Seventh C^uip/ii-.rn in ii.i.l, 11. c. 57,
Do Bello Galileo. Lib. VII. By too Rev.
W. Con K worthy Com prox , M.A. aj. fit/.
he Bello Civili. Book I. By tlie Rev.
W. J. BfitffiuRV, M.A. b£,b<£ net,
livy.' Hook XXL Edited by F. E. A.Tkaves,
M.A. as. 61. net.
Tacitiw: Areola, By J. W. E, Pbarce,
M.A. «,
With Illustrations and Maps.
By G. H. Wells,
Sophocles: An 1 igoo e.
.M.A. is, Srf. t.et.
Homer :
GHAtf'l',
Odyssey.
M.A. !U
Book I. By E. C Mar.
Athenians in Sicily. Being ponions of
■J'hucydidci, Books VI, and VI J. Bv the
Rev. W. C0.1KK0KTHV Cojbton, to. A.
St. bl. tWt.
Bell's Simplified Latin Classics
Edited, with Notes, Exercises, and Vocabulary, by S. E, Wjmioi.t, M.A., Qirist's
llnspital. Morsham. ('rimnSvo. With iiiimertnit tllustraiimis. Is. fit «h-!i.
PiSST /-«/■ OF VOLVMKS, XOIf HEADY
Cesar's Invasions of Britain.
Iivy : s Kings of Rome.
Vergil's TaMng o( Troy,
Vergil's Attilatx Sports.
Simple Selections from Cicero's Letters. 1
Uniform with ' Bell's Siiitfiliftiul Latin. Classics.'
Dialogues of Roman Life. By S. E. U'i-jumlt, M.A, Ftiup, 3™. Ulustrated. Willi or
without v oo a h o l aTy, as t
Latin and Greek Class Books
Caisar'fl Firth Campaign (Imm De Bello
Callica B00J1 V.j.
Tacitus' Agricola.
Sallust's Catiline.
Bell's ninstrated Latin Readers.
I.ditedby E. t". Mahciia;;t. M.A.
I'oti bv,>. Will, brief Xntes, Vocabylaries,
and n nieerous IfttaitnUlDltl, i J. each.
Scalae Primae. A Selection of Simple
Stories for Tttinsl.ition into Ivn^lis^l.
Scalae Mediae, bhort Extraos from
Eutrojjius and CssuDT.
Scalae Tertlae. Seleei ions In Prose and Verse
front I'liaeitnts, Gvid, Nupos and Ctcero,
Latin Picture Cards. Edited by Prof,
Pka&k S. Gmam,i:«, M.A, Sixteen cards
printed in eo!otir>, Willi Vocabuiarics and
Exercises, u, yt. tlel (ier yl.
Bell's Dlttatrated Latin Course, for the
First > ear. In tlnoe Bans. By K. (.".
M utcHAHT, M,v, and J. Q. Smtcea, B.A,
WiLk numerous LlluMnilloits, rr. &/. each.
Latin Unseens, Selei:ted and nrmnged by
E. C MAkUHANT, M.A.
Latin Reader I Verse and Prose), Bv W,
, M.A.,
M.A.
KlSi. CilLl.Il£S,
, and II. J. AK11.CKS0X.,
Latin Of the Empire (Prose arid Verses,
By W. Kim: Cll.l.llts. M.A,, and A. R.
1 "1 ,:m:m;. M-A. 4*, 6ft.
First Exercises in Latin Prose Com-
position. By K. A. Waitf, M-A. Wiih
Voeabnlarv. w.
Materials for Latin Prose Composition.
Bvilie Rt\. P. MKMT, M.A. _*j. Key. +J. :iet.
Passages for Translation into Latin
PTOSe, By PTO£ II, NKrrLKSJill", M-A,
3*. Key, -j-r. b*t. net.
Easy Translations from Nepos, Ca-sir,
Ctcero, Livy, &c, for Reiratlslalion into
I^itin. By ■ i'. Collins. M.A. 2J.
MemorahiUaLatlna. By 1'. W. Levakbee,
K.R.A.S. 11.
Test Questions on the Latin Language,
Bv F. W, Lkvamihii, F.R.A.S. is. W.
Latin Syntax ExercisoB. By L. l.\
Wainvv^icilt, M.A. I''ivc Parts, Srf. each,
A Latin Verse Book. By tiie Rev, p.
FROST, M-A- «» Kiy, sr. net.
Latin Elegiac Verse, E*w Eierases fa
ByiheRev. J. PliSKo^l". .'J. Key. y. 6J. net.
Bell's Concise Latin Course. Pan 1.
B; K, C Mabciiast- M.A., and J. C
SI-EM.I--H, B.A. 25.
Bell's Concise Latfn Course. Part It.
By E. C MAkcn-\>."T. M.A., and S. E
U'imi.u. 1. M.A. bc, o-f.
CothUrUUlUS. Tliree Short T-atin Historical
I'lajs. By Prof, F.. V. Aunoi.ii, l.i'rr.l).
With 01 without Vocabulary, i*. Vucaba.
jury snjiniMLLely, 4Y.
Easy Latin Play a. By M. L Kswhar. 6/.
EclOeTM Latinsa; or. First Latin Ke-iding
Hook. With Xoles and Voi:ahiilaiy by the
laii. Rev. P. Pkos-t. M.A, u. fiv/.
Latin Exercises and Grammar Papers.
Bv T. CllLLIN!-. M.-V. 2J. 6,1.
Unseen Papers in Latin Pro?* and Verse.
B) I. (j o.l.i vs. M.A. zj. 6it.
FOliorum SilVUla. F'art L PasKmes for
I'ran.l.iEioti into Laiin Kle^iac tuid lieroic
Vur.se. By H. A. Hoijdes, LL. 1>. 7s. 6,1.
How to Pronounce Latin. By J. P.
FOftTGATS, LlrT.LK is. ntU
Res Romanae, bcintj brief Aids to the llis-
Lmry, G eeg t aj my, LtteiatLtie and Antiii'.ih ii",
of Aindeiii Rome. By E, P. Col.SiMi";,r: ;
M.A. Willi 3 maps. ai. Off.
Climax Prote, A F rst Greek Reader.
With Hints and Vocabukwy, Dy K. il.
Makliia.vt, M.A. Willi 30 illustrations.
is. &/.
Greek Verbs. By J. S. Baiko.T.GD. h.&l
Analecta Graven Minora. W i t h Notes and
Dictionary. By the Rev. P. Fkost, M.A,
tlnseen Papers m Greek Proao and Verse,
ByT. t.'m.Lixs, M.A. 3.1.
Notes on Greek Accents. By the Rt. Rev.
X. Bahuv, D.I), tr.
Res Graecae, Being Aids to the study oi
tlie History, Gography, Arcllienlogy, and
Literature of Ancient Athens. By F.. P.
t'oLEWOOH, -M.A. With 3 Maps, 7 Plain,
and 17 uthcr til list rat ions. 51,
Hotahilia Quaedam, 11,
G. Bell & Sons'
Select Educational Catalogue
LATIN AND GREEK- wntinud
Other Editions, Texts, &c.
Bell's Classical Translations— continued
I
Anthologia LatLna. A Ssbctfow tf Cboica
l^attjs Poetry, with Notes. By Rev. F, St.
John" Thacjckkay, M.A. i^aio, 4*. 6d+
Anthologia Graeca* A Selection From the
Greek Poets. By Rev* F. Sr. Jon::
ThackekaVt M-A- itimo. 4*. &/.
ArtatophaniB Comotdfam Baited by II. .V
Ho&BRN., LL.B. Utmyflvo, iS*.
The Plays separately: Aeharnenses, aj. ;
l'jt| u 1 tts, is. 6d, ; Vefjfme, ax. ; Pax, sj, ;
Lysbtrata, etTli^mopboriaiiisae, *tJ* ; Aves.
at, ■ R&ttaeu ?s, ; Plums* or,
CatUllUB. Edited by J. P. Postdate, M. A.,
Lirr.D. Fcag. 8 vol y>
Corpus Paetaruin Latmorum. Kditedby
Hat-kf.3. 1 thick vhj|. 6w f Cloth, 181.
Mundus Alter et Idem. Kdhed as a
School Reader by H* J, Amjehsqs, M-A
HorftCO. The l*atin Text, with Conington s
Translation on opposite pagafc Pocket Edi-
tion* 4 J. (iet; or in leather, gj. net. Also
in a vols., limp leather. The Odes, £?. net ;
Satires acid Eptatfas, a/, &i, net.
Livy, The first five Rooks. Phenbevili.e's
edition revised by J* If* &R8SS&, M*A*
Books I-, II., Ill-* iV„ V. it. &£ each.
Lucau. Tie PharaaJia, By C. E. Has-
Kiffs, M-A* Wi:h an fmruducUon by
W. K, Hritlanii, M.A. Demy Svo, 141.
Lucretius. Titl Lnoratt Cart de re-
rum natura llbii sex. Edited with
Notes, Introduction, ;t:ut Translation, by
the late IL A, j. Munro. 3 vols* ftvo.
Vols, 1. and II, IntvodncLicn., Test, and
Notes, i&s. VoL TIT. TratislatEon, 6>.
OVict Ths Me taiuorp hoses. BookXlll.
With Introduction and Notes, by Prof. C. 11*
Kki;k, M.A. as. 6V/.
Ovid. The MetamorphcaeB. Boole XJV.
U ith ItttrocfaiCtfOQ and Notes by Prof*
t:. H. Kiiese. M.A. at. 6d.
. ■ * Books XI 1 1 , and X 1 V. together. 31. $£
Porslus. A Perali Flacci Bflttxarum
Liber* Edited* with Introduction aijj •
Notet by A. F'BhTfUK, M.A. 3*. &/- net.
Plato. The Ft-oem to the Kepttblic of
PlAtO, (Book Land Book H. chaps, j-io.)
Sdnetf, with introduction,, Critical Note*,
and Commentary, by Prof T, G. Tuck sit,.
Ltrt.D. dr.
PetTonli Ceiia Trlmalchlonls. Edited
and Ti-iiJiw lated by W. I >. Lo w h , M. A*
71. V. net.
Propertiua, Sexti Properti Canniita
rc-ttKtiovit J. P, PosTr.ATi:, Litt.D. 4(0.
3J. net.
Rut lit us : Eutllii Cluudii Namatianl tie
Reditu Sua Libri Duo, With in:. >■ ..
lion and .Votes by Pruf, C'. II. Keenk, M.A.,
niid Kn^i tih Verse U'j.nv-il^iii.U] by *J, F.
Savage ahmstroho, M.a. yjt, &??, net,
Theocritus. Editod widi Introduccba and
Notes, by R, j» CiiOLMELifv. M.A, Crown
Theoffnis. The Elejfiea of Theognls;
ana other Elegies included rn the
Theognidean Sylloge* Wkh Introdoc-
tion. CtSMBSatitaityt and Apj.wndices, by
J. IIvdson' Weluams, M.A, Crov^n Bvo.
7j h 6£ net.
Thuoydidea, The History of the Pelo-
ponneatan War. With Notes and a
relation of the MSS. By the bte
K- SutLLBTO, M.A. Book I. Svo, 6j, 6^*
Book 11. §ft &/.
Bells Classical Translations
Crown Svo. Paper Covers, n. each
JEsChylUS: Translated bv Walter Hear.
I,AM, LtTT.D., Mid C. E, S, HftAtJLAM, M.A*
Apntrntmnon— The Suppliants— Cboephoroo
—Eumenides — PfOmewflUS Bodnd — Per-
sians—Seven against '] )■.'.■'. .■::.„
Aristophanes : The AcharQinh.^ Trans.
Luedly W. H. CovjKfiTOS% H.A.
The PLmns. Translated by M. T.
Qlins* M.A,
Csssar'B Gallic War* Tfanslawd by W. A.
MTJisvtTiK. P,A. 1 Vols, (Books 1. -IV.,
aad Rooks V.-VIU
Cicero. Friendship and Old 'Age, Trans-
bi t dby(;*H, V?KU*S, M.'A.
Orations. Tr^u^lfsicd hy Pmf r C I J.
?09MSO| M-A. fi vol. 1 ;- (Jntiiine, Murcua,
Si.lla and Aycnifis (in one vt-U), ManUjAii
Demosthenes oti the Crown. Translaled
Eimpldea-. Trajisbiml by E, P-CoLERttMrijE,
M.A. T4v;^U. Medcst— Alcwtifj ■HroMlst.
dai— Hippolytus- ■ S'.iiTjiliu.eis Tw^deS— Ion
— Andromache — Baccaec — Hecuba — Her-
Oilex Furens— Orestes — Ipbigenbi in Tauris.
Homers Ulati, Boofel t. and IL, Hookx
IN. .IV., Books V.-VI. t Root* VIL-VIII.,
Hooks IX. -X., BouksXL-Xn.. JiooksXlI I-
XIV,. Books XV, and XVJ., Boohs XVII*
and XVHL, Books XIX, and XX. Trans-
laiad hv K. H. BlaKSMBV, M.A* ro vol*.
Book XXIV. Translated by E. H,
Bl.AKRNEV, M.A.
Hoi a ce . Tj'aai slated b y A. Ha m 1 i,ti j m
BavCS, l.L.D* 4 vols. OdflSj Books L a)id
]|. Oilc-=. H"nL> HI. aji,: iV.* Cajnicn
Seeulare and lipodn-- S*tiiea- Epistles Jt£d
Ar» PociiiCt
LlVy. Ruoks 1 . , [I . , U I. , I V. Trtnslatcdby
J. H. FsftESB, M.A, With Slaps. 4 vols.
Books V. anil VI. Translated by K. S.
Wevmouth, M. A. Lund. WitbMapi. avuls.
Hook IX. TraJnkl«I by FbasciS
Stohk, MA. Wilh Map.
. B«As XXI., XXII., XXIJI. Train.
lated by J. IIkksahh Baker, M.A. 3 volt
Luca& ; The Pbarsallj. llook I. Tron^
lated by J'Knmtwt:.: Conwav, M.A.
OV.d'B I'nsli. Ti;sii-.l.il';il by IIltNKy T.
RttEV, M.A. 3 vuls. Books I. and IL—
liooks HI. raid IV.— Bonks V, and VI,
Ovid's TrUiia. Translated by Hkmbv T.
Rii_et, M.A.
Plato ; ApoioEV of SocmtEs and Crito (t vol.),
Pbardo, and Protagoras, Ttanslatcd by 1 1 .
Cahv, M.A. 3 vofi.
Piautua : TrtttLummis, AidniLu-iit. MeitLCL-hiui,
Rwdena, and Captiv'i. Translated by riSNRV
T. Kn.BV, Sl,.\, 4 VLils.
Sophocles. Translated by E. P, < 1 it 1
kiDCE, M.A. 7 vols, Antigone— I'ldLx;.
tetcs— iEdiuiis ^ Res — (Edtpiis CbkxHRts—
Klectra — Tracbinite Ajajc
Tillieydidea* Book V 1. Translated by
E. C. II ARCH AST, M.A.
. Book VII. Translated by E. C. Ma It.
ctlAST, M.A.
Virgil, Trutssbwd by A. Ramtltom Hkyck,
X.T.. I). 6 volfi, Bucolics — Gears ii;s *
-Eneid, 1*3— ^nuid, 4.6— .Entiitl, 7^—
.Entdd. iu-r.-.
XeilOpJlOn'S Au:ilinsis. TranslalKtl by the
Rev. J, s, Watkjn, M.A. With Map. j
voE. Books 1, and II. -Books III., IV.,
and V.— Books VI. and VII.
llellcniis. Books I. and II. Trans-
lated by the Rev. II. n a LB, M.A.
Far ether Translations from tht Classics, ?tt the Catalogue ef Bokiis Libraries,
■which -mill he fbrwkrdtd en application
MATHEMATICS
FuU Catalogttc of Naihtmathal Books post fret en npptktslhn
Cambridge Mathematical Series
Public School Arithmetic. By W, M.
Baker, M.A,, imd A* A. BoUB»«, M.A,
js, Hd. Or with Aiiihwers, 4J. 6*/*
THe Student's Aritbmeiiie. By w M«
Uakf-R, M.A., and A. A* Bouhnb, M.A.
With or without AflBsratlt, 2J. oV.
New School Arithmetic. ByC ^rnui.h-
wmy* M.A., and f. E. Romnstnt. M.A,
W r jih Of wittujut Answers. 4J. eV/^ In
Two Parts. 7t. fyt. ftflwtfe*
Key W Pari 1 1., St. fu/. net.
New school Examples in a wrpartittf
volume, }*. Or in Two Pam. u* 6d, ft rid 9*.
Arithmetic, wfeh Sooo Examp'ts. Ity C*
PENin.raiurtv, M.A. 4J. &d. In Two Parts.
uj, 6<l. each,. Key to Fart J L h jw, 6^^ Wt
Examples in Arithmetic fcrtracted from
the above. 31. Or in Two Parts u. &/*
and as,
Commercial Arithmetic. Hv C Pi-.n-dlr-
BWRy, M.A., nnd W. iv Beard. fr'.R.G.S.
^r. &/. I'bji 1* scpamtoly, u. Pan It., 1/. W.
Arithmetic for Iniliaa Schopls. Hy C
PfittttLBBURY, M-A., Wid T. S, TaIT» 3J.
Exp-mple b in Ar ithm«tic By C* O* I H m . 1 v
M.A. Willi dt wiLliuut An^wwa. jf.
Junior Practical Mathematics. By W,
J. Staiskr, bJL 5i..wlth.'\ttsw.'ers 1 2f. S<£
Part I*, 1 J. *d, t wilh Answws, u. 6^.
Part 1!., ij, ^rf.
Elementary Algehra. By w. M. F{akeh.
M.A., antl A. A. BOUWW, M.A- New and
U..:v[>i:lI KditiLJi,, ^.r, ftrf* Also Part [.,
■2s. 6tf,, or with AossvBJSj 3J* Part II., with
ur whhfi'.u Answrs's. m: &/. Key toj, net ;
or In 2 PortKj ^j. nrt each.
A Shorter Algebra* By W H M. Rakki*,
M-V, au.l A. A. BrkUltMi". M.A, vs. &/,
Examples tn. Algebra, Extracted
above. Wiih nr without Answcra, 31. Of
in Two Pan*. Pari I., u, &/., or wiih
Answers, *>*f. Part H. B with or without
Examplrs in Algebrp, By a p. Tulkkv,
M.A. W nil or wUhout Answers, J&
— Supplemesitary Examplee, 6./. tict,
Elementary Algebra for uee in Indian
Scb.OOl3. IK' In T, llArHORMIIWAtTE-
M.A- 7x.
Choice and Chance- tiy W. a* Whit-
wok-mi, M.A. 7 j; 6d.
■■ DCC EXPrciaee, triLluding Hints fbt ibn
Solatluti of ill the ijticstions to "Choice
and Change. " df.
Euclid Boohs 1.— VI. v and part yfPuok XJ.
By Horace DEMitiTON, M.A. 41, Srf., ^r
in eepsiatfl bookti.
Introduction to Euclid. By Han a etc
Desgjitox, M.A., and O. Emtat.e, P. A,
i j. for".
d. Bell & Sous'
Select r^du-caiional Catalogue
, Cambridge Mathematical Series - eoutfoutd
Other Mathematical Works
Euclid Exercise* on ICuclid nut] in Modem
Geometry. By J. McUtiWKI .]., M.A. tit
Elementary Graphs, By W. M, Baki-h,
JI.A,, and A. A. HouKhVK, M.A, 6rf. net.
A New Geometry- lii-w. m. lum-n. ji.a..
anil A. A. Boukxb. JJ' A. Own Svu. at erf.
Also Bucks 1--1 []. separately, It 6rf.
Elementary Geometry. Bg w. ji. Bakkk.
M.A., and A. A. BtWICNB, M.A. AX. 6,f. Or
in Pans. Answer?., 64. net. Key! 6t ti*-T,
Examples in Practical Geometry and
Mensuration. By |.W. Marshal!, M. A.,
and C. O. Ti-cki.^, M.A. it, fit,
Geometry (or Schools, Hy W. G, Bok-
CIIAJiiHj M.A_.nnd the Rev. A. 11. Pekhdtt,
M.A. Complete, 4t Grf, ; abo Vol. I., is ■
Vol. II., it &/. ; Vol, 111., hc; Vols. I,.
III., it. 6rf. ; Vol. IV., it; Vols. 1.-1 V„
v. r Vol. V., t*. : Vol. vi.. ,t erf. : Vols. !.-
V., $s. &f. : Vols. IV,. V., .:,.
A Now Trigonometry for Schools. By
W. (.;. BtiKiHAKbr, J|,A., anil Rev. A. 1).
I'liKiturr, JI.A. 4t erf. Or 'IVo I'liiis,
at erf. cadi. Key, toft tier ; or i Parts, t£,
net each.
Numerical Trigonometry, Be- w. c,
BriKtmKijr, M.A., ami lite Bev. A. II,
1'KkHtJIT, MA, _'.-. bit.
Junior Trigonometry. lit w, <:, Bu K .
I N un/r, M.. V., and tht Rev. A. It. I'FRRorr
M.A. v. 6/.
Elementary Trigonometry. ByCnAsuca
I' hN mi' i,i- kv, M. a„ I •'. H. A.S. ai. e t.
Short Course of Elementary Plane Tri-
gonometry. By I.: It. Mi I. IIS PBKDtETOKV,
Elementary Trigonometry. By r, M.
I)h:k. M.A-, ami the Km-. K. H. Wlin.
i DMBB, M.A, 4J. erf.
Algebraic Geometry. By W. U. Bakes.
M.A. 6r. Pari 1. (The Straight line anil
Circle), it &/. Key, 71. orf, net.
Practical Solid Geometry. By the Rev.
I'|,.kl.v CJmwik, M.A. nt. b*.
Analytical Geometry for Beginners
Uy Kev. T. tk Vyvvax, M.A. Parti. The
Ssltalghl I, hit; and Chile. u,t, erf.
Conic Sections, i rcitcil Cc ru-hall-.-. Bi-
ff, II. IVi*AM-,.St:.n., I-'. U.S. A s.U Kev
5X. nee.
Elementary ConicG, being the gnu 3 chap-
ters tit the above 3J, erf.
Conies, the Elementary Geometry of
Uy kev. C Taylor, I). It. 5 ,,.
Calculus for Beginners. Hv JV \i
BaKHK, M.A. jr. ' '
Differential Calculus for Beginners.
By A. LOSCB, M.A. Willi Introduction by
bit Oliver Louuk, 4j. erf.
Integral Calculus for Beginners By
A. LOBGB, M.A.
Hoiilettes and Glis-ettes. u y w. h.
Bosart, ScBt, KRA 5s.
Geometrical Optics. An Elementary
I twain h y \v. s. Arms, M.A. *«,
Practical Mathematics, I w H. a. Stfrn
\l. V. and \\. H. Tiii-iiAM. dr. ; or Part I
is. 6,1. : Part II.. jt M.
Elementary Hydroa'atics. By \v. 11.
BBUKT, Stl). tt tuf. SobUjoOj, 5,. net
Elements of Hydrostatics. By c. \f.
Jbssot, M.A., o«d o. w. Cabot, m.'a!
af, erf.
Elementary Mecha.nics. Uy c. Jl. Iksshi>
JI.A., and .(. H. H.M'iii.ocK", M.A., D.Sc
t-i'. 6 r.
Experimental Mechanica for Schools
•."itAi.i.i:,, M.A.. and \V if
llmurr. M.A.. Jl.Sc;. i>. 6C
Tho Student's OyDoinics. Cornpristng
SiKitaand KmeticK ByG. M. Minxhi.v.
M.A, r r.R.S. ^r. fvf.
Elementary Dynajnlcs. By W, jr.
JiAK Irn, JI.A. New Uevixejl KJiiion, . j. t,,{.
Kiiy, 107, erf. net.
Elementary Dynamics, By W. (;j, miT
JI.A,, Ij.CIj. er,
tynamlca, A Trratisn no. By W tr
BlKANT, |c. ».. R U.S. km. 6rf.
Heat, An Elcmeiiiarv Trtatiie on. Bv W
«**xirrr, MjA., D.CI. ttU
Elementary Physics, Ks a mules and Ei-
atntnation Pa;iers in. Hy \V. a.n lati v
M.A. 4J.
Mechanics, A Cojaotion of ProMenn in
hltnticlltnry. By W. WAt.'ltJS', M.A. 6s.
Cui/itr/n Vol ii me
Geometa-ical Drawing, tor Army and
MM) Kt.-nitnniions, By R. I Iaiikis. ]jt6rf.
The Mathematical Gazette. Bolted by
W, |. I1n!:];m.tki:i:t. M.A. (Jan., Mardi,
Jlay, July, Uot. and Dec.) it td. net.
The Teaching of Elementary Mathe-
matics, berag llieRepans t.i llicCt-'inliiinee
of the Matlleinalii^Ll Assotjatko, 6rf. net.
Tha Toaohmg of Elementary Algebra
and Numerical Trigonometry. Being
the Rqmil of the JlathL ,-Vssot, COBaaaUB,
inn, "irf. net.
A New Shilling Arithmetic. Uy C,
Pkndleiiuhv, M.A., and F. B. Konissnx,
JI.A. it. 1 or wkh Answers, t^. ,ip/,
A Shilling Arithmetic. By CuAtuats
1't.vin.ltHL nv. SI. A., and W. S. HuAltn,
r.H.tl.S. u. With Atiswets, iJ. 4rf,
Elementary Arithmetic By Chakus
Pkhouiii RV, JI.A. With or without
Answet 1 -. it. cirf.
A Preparatory Arithmetic. 1 : y C . i iu -
l'Ksui.Kiiuhv, M.A. With or wtehout
Answers, is. erf.
Problem Papers for Preparatory
Schools (Arithmetic). By T. Cooiek
.Smith, ll.A. it 6rf.
Bell's Indoor and Outdoor Experi-
mental At ithme-tic- Hy U. B. t5ot»>-
ackr, K.R.ti.S. I';irts I. -II I., iiata;r. tc£
each, cloth, 4rf. each j Pans IV. and V.,
Stper, ^rf. VBfibi cloth, 5rf. each. Teachers'
B >k, jg, oVA net.
Graduated Arithmetic, Hn Junior and
I'ltvnlt' Soliooltt By the same Authors.
I 'a ris I., II., and 111,, irf. each ; Parts IV.,
V.. and VI,, ,,.». each ; Pan VII., firf.
Answers e.j PWts I- and I J., *rf. net;
Pam 1 1 1. VI I., jrf, net each.
Aritthmetlc for the standards (Scheme
B). M.-iitdard I.. sewed, srf. , vl. ,th, arf. 1
11., 111., IV.. an'.l V-- .-ewe-.?, -rf. f*n:h, c] ittl,
4rf, ench J VI, tutd VII., sewed, jrf, each,
cioth, orf. each. Answers to each Standard,
4rf, MA W&k
ExercUos and Examination Papers in
Arithmetic, Logarithms and Monsura-
tOU, I'l f", I'iLS-|>i.Kar_"HY, JI.A, - af , erf.
New B dlSon .
Test Cards in Arithmetic (Scheme: it)
ByC. I'lt.Mii.itiiuttv, JI.A. rorStanilan.sl!
III., IV., V,, VI. and VII. it nel oath.
Public School Examination Papers in
Mathematics, Compiled hy P. A, QtttK'
shaw, JI.A, it 6rf.
Bell's Hew Practical Arithmetic, By
W, .1. Svainsk. JI.A. jst, and, jrd, 4th,
5th and 6lh Ye:u- t paper, yt. each, cloth,
4rf. eadl I ?lh Vear, pallet, 4't 1 ., cloth, 6,/.
Teachers' hii'iihs. c-rf. tiet each \"i:iir.
Bell's Hew practical Arithmetic Test
Cards, (of the and, 3rd, 4th, .sill, 6lh, and
71)1 ycttrn. is, ijrf. net each.
Graduated Exercises in Addition (simple
and Compound). By W, S. I;i:m:i.. m.
Algebra for Elementary Schools. By
W. M. Bakbe, JI.A., and A. A. BotJJWit,
JI.A. Thtee stnses, erf, each. Cluth, 81I.
each. Answers, at net each.
A First Year's Coarse in Geometry
and Physici. By Ernest youwi, M.A.,
B.Sc at erf. Parts 1. and II. I J. 6rf. J or
P.ilIJ. it
Trigonometry, Ksaminatlon Papers in.
By ti. II. W Attn, M-A. ai. firf. Key, Jt act.
Euclid, The Elements Of. The Knunda-
tiens and I'iKttres. By the L-iie J. Brassb,
I). I*, tt \Vithout the Figure-,, 6rf.
HydromechanicB. By W. H. Hksamt,
So, I)., and A. S. RAMSEY, .M.A. Pan 1.,
Hydrinlalici, 7tSrf.nct. J'att J !., Hydro-
dyrinndcs. By A. S. Kamkf.y, M.A. icu. bd,
n-i.
Hydrodynamics and Sound, An lilcmcn-
tory Treatise on. By A. B. Hassht, J1..V,
F.K..S. S».
The Geometry of Surfaces. By A. b.
BaSSBT, M.A., F.K-S. tor. erf.
Elementary Treatise on Cubic and
Quartlc Curves. By A. Ii. Basshi-, M.A..
F.H.S. tot, M.
Analytical Geometry. By Rev. T. G.
Vvvvan', JL.V. 4-r. 6rf.
I
The Junior Cambridge Mathematical Series.
A Junior Arithmetic, Bj i . pg -,-;,: E bl-rv,
M.A., and F. E. RontxdoN, M.A, tt erf,
With Ans-A-crs. is.
Exannlaj from a Junior Arithmet:c.
l-jittactecl ftom the above. jj. Willi
Answers, u, erf.
A First Algebra. % W. M. n A!;rEi j r A
and A. A. BwftxK, M.A, u. of. ; or will
Aoswetft, fe£.
A First Geometry. By W. M. Bakkr.
M.A., and A. A. BtJCltaB, M.A, \v;
v,t:h,^ii Answer*, is. &/
Elementary Mensuratton. Bv w. jj.
Bakkk, \I.A.,andA..\.llt,v«si- \( 1 ,,4,
Book-keeping
Booli-lcefping by Double Entry, Theo-
retical, I J r,n tii.al. and (or KxitmlltAtiott
l l urposcs, H> J. T- Mt-;oiu;i,s'J\ A.K.C,
F.S.S. 1 J. firf.
BoolE^Kseping, Exanuint^ii P.ijj,-;s in.
(.'oaipiled hy jtaiiN T. MVfillURKT, A.IC.C,
F.S.S. jr. Key, 2s. &.'. net.
BOOlC-lCeeplng, Gi-adnateLl EMrCtHC* and
I'.v.i'.iia.ihnii Papers in. CoinpUw) hy 1'.
Mi-uitAy. F.S.S.S., F.Sc.S.(I.und.), m, 6rf,
Text Book of the Principle a and Prac-
tice of Book-keeping and Eatate-
Ofilce Work. By Prof. A. W. Thomson,
B.^c. st
G. Bell & Sons'
SeAcl Educational Catalogue
\
ENGLISH
Full Catalogue of English Bwfopcstjret en affh't-aticn.
Bell's English Texts for Secondary Schools
Mason's New English Grammars. Re-
vised by a. J. Anrootf, M.A.
A Juni or Eagfiih (irammar. n.
Intermediate Knstlish Grainmnr. vs.
Senior English Grammar, jr. 6;f.
Works by C. P, MaMW, B.A., F.IT.P.
First Motions of Grammar for Young
Learners, u.
First Btepa in English Grammar, for
Junior Classes, m.
Outlines of English Grammar, for the
Use of Junior Classes, at.
English Grammar; including the principles
of Grammatical Analysis, -jj. 6d.
A Shorter English Grammar. 31. fd.
Practice and Help in the Analysis of
Sentences, =1,
English Grammar Practice. «.
Elementary English Grammar through
Composition. [Jy .1. I). Kr.su, M.A. »,
Advanced English Grammar through
Composition. By [<-hk ji. Ri.sh, m.a.
31. &/.
Aids to the Writing of English Compo-
sition. By V. W. IWWStlSJt, B.A. is, lies.
Preparatory English Grammar. By
W, Smtsas, M.A. New Edition, u. mm.
Rudiments of English Grammar and
AnalysiB. By J&msxr Adams, Pm.D. is.
Examples for Analysis in Verse and
P^cas* Selected by F. KowAkbs. if.
The Paraphrase of Poetry. By Enwsws
Cawdler. ii.
Essays and Essay- Writing, for Public
Eliminations. By A. wi Rsiur, ELA.
jr. 6t£.
Frocis and Precis-Writing. B>- A. w.
Rkauy, B.A. 31. 6,/. Or without Key, 2j, &/,
Hatriculattou Precis. ByS, E. WrseoLT,
M.A. is- net. Key, &/. net.
Elements of the English Language. By
KnNHST Aimks, I'u.l). Revised By J. F,
Davis, M.A.., D.Llr. 4,1, 6.Y.
History of the English Language. By
Prof. T. R. Iajunswry. 51, net.
The Teaching of English literature in
the Secondary School. By R. S. Bate
M.A. M. M. net.
An OrrtUne History of English Litera-
ture. By W. II, Ill'csQN. 31. 6*f. act.
Rapresentative Extract* from English
Llteraturs. By w. ft. Hwso* «. sw.
Ten Brink's Early EngUsh Literature
■} vjjs. 31. &£ each.
Introduction to English Literature.
By Hkshv S. I'axcoast. 51. net.
A First View of English Literature. By
Hiss'kv S. Pancoaot and Peulv Vas Dyke
SHSJJ.V. Crown S to. u, net.
Introduction to American Literature.
By H. S. PaBOOAST. it. 6.1, [ieL
The Foreign Debtof English Literature.
By T. G. ll'CKKR, I,i rr. I). Post Svo. Si.
net.
Handbooks of English Literature
toiled by Prof. H.st.KS. 31. 6/, net each.
The Age of Alfred. (640-1154). By F. J,
The Age of Chaucer. (1346.1400.) ByF. J.
Smell, M.A,
The Age of Transition, (uco-jjto.) By
I, j. Snei.i., M.A. 3 vols.
The Age of Shakespeare. (n;g.i6-,t.) By
fHOMAS Seccosihe and j. W. Aiiev
j vol* Vol. I. Poetry ^.d pr^
>oi. II. Dtaina.
The Age of Milton. (1631-1680.) By the
Rev. I. II. B. MA4TEHMAW, M.A.,
with Introduction, etc., by J. Bass
Mci.LTStlER, M.A.
The Age or Dryden. (1660-1700.) Bv
K. liAHKEir, LI..I)., C.U.
The Age of Pope. (1700-1744,) By John
Dennis.
The Age of Johnson. (1744-1703 1 By
Thomas Skccosuih.
The Age of Wordsworth. fiToS-iSira \
By Prof, a H. Hbehmu,. L,tTd! 3 '
T; ^ -l s f,° r T<:nn yson. (iS^o-iSjo.) By
Prof. HucJii Wai.kkb.
Kotes on Shakespeare's Plays
JKl'F BAa.-iEIT, B.A. it. ea^h.
By T.
Mi. ..^imrner N: K hts l>reoin.- fniitis C^sar.
-The lemijest.— .Madieih.- Henry V. .
Ilamlet. — Merchant of Venice. — Kins
Rteha«UI.- ; Kl W John.-King Rirh^rd
H 1 - - ^^ng Lear.— C-onobnos. - Twelf-h
Nwnt.-As You I.ilte It.- Much Ado
About Nothing.
Principles of English Verse
I.trtvis -- —
5J. Mer.
Introduction to Poetry.
ALUKK. Jr.
By C. M.
By Ravjhoxc. M.
Edited by A. Gl'thkelcii, M.A.
Browning/i The Pied Piper, and other
Poems. Kdited by A. Guiukki-ch. &rf.
Fairy Poetry. Selected and edited by
R. S. Bate, M.A. u.
Hawthorne'a Wonder Book and Tangle-
wood Tales. Selected and Kdiied by H.
H.iMh-.i..i M.A, it.
Kingsley" s Heroes, lid i ted hy L. H. Po n r>,
R A. With 3 itiaj^, u,
Lamb's Tales from Bhakospeare. Be>
lected and edited b y R . S, Bate, M . A. lorf.
Lamh'a Adventures of TJlyssea, Se>o
lions. Ediieri by A. C. DoMSTAtl, Pit. D. 2^.
Stor.es of King Arthur, from Malory
and Tennyson. Edited by R. s. Baib,
M.A. u.
The Story of Enid, from Tennyson and
The MablD&gion. By II. A. Tkkiilb,
M.A. iocf.
Scott's A Legend Of Montrose. Abridged
and edited by F. C. Lucehlhst. it.
Charles Reade's The Cloister and the
Hearth. Abridged and edited by the Rev.
A. E. Haue, I'. A. 11.
Coleridge's The Ancient Mariner ; »nd
Selected Old English Ballads, td.te.1
by A. Lj^thkeiaih, M.A. u.
HailttytS Voyages, A Selection edited by
the Hev. A. K. HA1.I,, B.A. tr.
Selections from Boswell's Life of
Johnson. EJned by K, A. j. Maksh. u.
Selecti ona from Ruskin. Edited by H.
HamI'SHLRe, M.A. j^.
Lockhaxt's Life Of Scott Selections edited
by A, Baftke, LI..A. ir.
Charles Lamb's Selected Essays and
Letters. Edited by A. UeTBiMuat, M.A.
With Map of London. 11. ~-
Selectaons from Carlyle. Edited by
Ei.t7-Aiie-ni Lee. ir.
English Odes. Iv.litcd by e, a. J. Maww,
M.A. u.
Bell's English Classics
General Intelligence Papers With
Bswcises in Rngliih Compoiitioo. Bv u
BWttZ ai. M. '
Bacon's Essays. (Selected.) Edited by
A. E, Roheets, M.A. 1 j.
Browning Selections from. Edited by
F. Kclamo, M.A. 11. 6rf.
-—Strafford. Kdited by E. H. Hickkv.
11. 6d.
Burke's Conoillation with Amorioa.
By Pinf. J. Mokhisos. u. W.
Enrke's Letters on a Regicide Peace.
I. and II. Edited by H. Q. K.EENB, M.A.,
CUE. 11. trf.
Byron's Siege of Corinth. Edited by P.
HoKHERf*. ri.
Byron's Chllde Harold. Edited by
H. U. Keknk, M.A., C I.E. 3/. Also
Cantos I. and II., sewed, a. Comos III.
aMd IV., lewcrl, 11.
Carlyle' s Hero aB Man of Letters.
Edited by MahK IIv/ntkk, M.A. is. bi/,
Hero as Divinity. By Makk H'.nteh,
M.A. n. &/.
Chaucer's Minor Poems, Selections
from. Edited by I. B. Bildekiieck, M.A.
II. f>i.
De Quincey's Revolt of the Tartars
and the English Mail-Coach. Edited
by Cecil- M. Bakrow, M.A., and Mark
tlUNTKR, M.A. W.
\* Revuit ^ the 1 ;irtais, separately, ti,
Opium Eater. Edited by Mark
Huntek, M,A. sj. 6rf.
Goldsmith's Good-Matured Man a n d
She Stoops to Connaer. Edited by K.
DEt&lITOK. l%ach 11.
* * n,e two plays together, n. &/.
— Traveller and Deserted Village.
Edited by the Bev. A. E. Woobwaeo, M.A.
Cloth. 11. 61*.. of leptirately, sewed, ic«i. each.
Irvlng's Sketch Book. Edited by R. G.
Oxenham, M.A. Sewed, is. dd.
Johnson's Life of Addieon. Edited by F.
Rvt-ANn, M.A. 11.
Life Of Pope. Edited by F. RvtA.Nti,
M-A. ffj.
V The Lives of Swift and Pope, together,
sewed- ai. bit.
Johnson's Life of Milton. Edited by T.
Rvt-AKD, M.A. 11. 6rf.
Life of Dry den. Edited by F. Rvla n d,
M.A. 11. &>'.
*,* The Lives of Milton iind Drydcn, together,
sewed, 31. Off 1 .
Life of Bwift. lidited by F. Ryi.ani',
M.A. 11.
Lives of Prior and Congreve.
Edited by F. RVLAKO, M.A» «.
Kingsley's Heroes. Edited by A. K.
Roeerts, M.A. Illus. u. 6d. Sewed, n.
Lamb's Essays. Selected and Edited by
K- Deighton, jj. 61/.
Longfellow, Selections from, includ-
ing Evangeline. Ed i 1 ed by -M. T. Qu 1 » k ,
M.A. u. hi,
V Evangeline, stsnarati-ly, si-weJ, torf.
10
G. Bell & Sons'
Sclerf Educational Catalogue
It
Bell's English Classijs-^«*«tf
Readers continued
Macatilay'B lays of Ancient Rome.
— Essay on Cliire. KUitcd by Qien.
— War of tlie Spanish Succession,
■ Milled by A, W, Ready, r.t. &£
Massingers A New Way to Fay Old
Debts. Edited by K. DejOHTtw. it 6i.
Milton's Paradise Lost Booln lit and 1 v.
Edited by K. C. Oxhxiiam, M.A. ir. ; or
mpaancaly, oa»e4t io ^- eath.
Milton's Paradise Regained Kdfced by
£. LiiLEtirri'o.v. u
Kdileitt by F.
by K.
Fope's Essay on Man.
kvi_v\-ij. .Yf.a. ij.
Pope, Selections from. t,ii !L il
Dt::i:i[ [■■>,. i
Scott's Lady of tlie Laie. Edited by the
»ev. A. E. Woowwako, M.A. »f.«rf. Ti.c
&I3C Unites separately, sewtrl, 6cr\ each
SlialtespejLro's Julius Csssar. Edited bv-
' & JFr - B **-« gr t. B-A. (to»dA „. 6 *
— Merchant of Venice. Ediiud by
J. Ll: . i BASnrrr, B.A. (I end.), ,.,, &f.
Tempest. EdfcetlbyT. IXrrrlUtxvrr,
I...V (Lond.>. t.i, erf.
Wordsworth's Excursion BootL Kdited
by M, 1. QutKM, M.A. Sewed, u.
Bell's Sixpenny English Texts
Poems by John Milton.
Spenser's ' Faerie Queene." Buuk 1,
Poems by Tennyson.
Selections from Byron,
Macaulay'a 'History of England.'
Gibbon's 'Decline and Fall.' Chapter „
1, loUI. H
Hound in limp cloth, 6d. each.
Selections from Pope,
Poems by Gray and Cowper,
Plutarch's Lives orcresar and Cicero.
English Elegiacs.
'Selections from Chancer.
Kingsley's Heroes.
m i-ohtmes marked with a„ attend a,;: supplied irttrUmsd sad in,,,,,/
m cloth botirify, \s.
English Readings. i6mo.
Burke : Selections. Edited bj Buss Pebrv.
Byron : Selections, BcHwd by f. j. Cak.
p&ktks, sr, int.
Colerictpro: Prose Selections. Edited ■■•■,
Iik\.;v A. I !!-:]. ;:is. si.
Dry den: Essays on the Drama, Kdiio]
ay Whjjjih grapsg. is,
Jobnaon : Prose selections. Editd by
c ij, Oscooix v.
Milton i Minor English Poems. Kdhed
by Maetlv W. Si vmttt. at. tW,
Swift : Prose Selections, Edited by
EHKIlliHICK C. PitBSCOTT. is. Ctf,
Tennyson : The Princess. Edited by L. a.
Thackeray: English Humourists. Edited
by William 1 ,v. . n Pueuts. it, 6,1,
Readers
The Story of Peter Pan (...- tofd in " The
Perer Pan HcMHe Bfx.t;. J. With 16 lUas-
initio™ .-md StiiTgs from the Play in Tonic
Solfa mid Old NoMiion, grf,
Alice in Wonderland. By Lewis Cak-
!-.(■],!,. lllns.1 ruled by A.jcb II. Wm.hmvabu
i)tf.
T 4 rlf ! t - » A Gammon Statue Bosk for GM».
li> !■, FttvT. ;./ net,
York Readerc. A now toiai of Literary
Readem, uitb Coloured and other lllus.
bulno^
Primer I, yf. Primer 1!. 11/.
York Headers-, rJ«,v W Hfff.
Infant Reader, 6t£.
Introductory Header. &£
Reader. Book I., «: Book 11., i<n/. Book
in., u, Bool iv., a jrf. lijok v.,
York Poetry Books. 3 Books. Paiwrcove-,
8 '. each ; cloth, :w. em*.
Bell's Poetry Books. InSureaPartJ. Pnee
j^, cadi I'art, papa covin J or jrf, eloth
rovers.
Poetry for Upper Classes. Selected b.
fc. A, Hi.l.k. TI , o,,-, '
Books for Young Readers. tttasfe*H*£
6r/. «:it:1i.
.-Ksvp'n t'abUrv ! TW and the Cat, etc
"I'he Old IJffiit-HMii>.*r 1 ctu-
Thc CaL ami tlvc >kn. OCC*
( be Two hfflU. I The Losi Pira-
*rhe Swry of Ttma .%ionkcyr>
The S&oIlT of .. I '..I
Ijii-een Ucc ntut Uit^y ttre. | fiijll's Crui;.
Bell's Continuous Readers. Bound fri
t'luth. *pL esoht
Suitable fer Simmtmrd ill.
I be story of Foter Past
liic Advefltuivi of a Duukej
Tlit l.sft: uf CnhlnibflS.
Ttn: TSth.^ Miil-,liiviii[i:n.
$uha J tefor Sttimhirti / K.
Alice in Wonderland.
The Water Babfe&
Tbe Fit^ifole* from N^turit
l. 'tide Tunis Calrin.
Rw'nmson Cruwoc*
SuitaM* fov .St/tntM'--' I ".
Twin Un.iwn'f, Sclii )«1 days.
The l^Kt of irim Muh Scans,
Fesits on the Fiord,
The LJulc Dukb
Htrewzird the Wnke.
Suitable for ±t*iKd*iM-(ts t'Ljtrni i*U,
Tim l^isi Days of PowpcH,
Oliver Twist.
The Tale of Two (,'IiEcj.
Ivnnhoe*
Jjimbs Tfk^e> frnin Shakespenirt,
Bell's Heading Books and Literature
Readers. Strongly famuxl iaClotb* liu&-
iniltxl. u. awbi
Suitable for Standard t/f.
Adventures of a Donkey,
Great D»oda bl EBngtiah USstory*
(Ifitnm's <»fiTTHi'.ii 'I ales.
Ai'.ileisen's TJ.m.Uh 'I'lles.
(ireat EngB^iBWO.
(iiraT lr^hitif-11.
] ,ife of t'nlwnihiiS.
*flte Three Midshipmen.
Suifaf'fcfi'' Standard IV*
Great Suits men,
Uncle Tom'K (.'aliln.
Hwiss i-ninlly Robinson.
4.*rent Kltglishwfinifcii.
rhdiiren of the N«tv Forest.
Seultrs in Camuin.
K-dgewonh'-. i'ales*
The Wstfor Rabies.
Parables fn.ni. Nature,
Su finite fy* Sttutdar,' I'.
l.yrj^d Poetiy.
The Story of I-lt de Sett
MnilcrlliiLn Re.^dy.
liuUivcrs Travels.
Kobinsoji C: iiv,r.
Poor Jack.
Arabian Nighl5,
Bell^ Reading Books, &c. iaft/him,t,
"I he l<a."it of the Mul.iiu.~uns.
FcmtHon i he KtottI.
I h- L.htli: lJukr 4
Suitti&fejf'r Sfuttdttrds 1'f. and f"/A
*l he Talisman. | IvunJuae.
Woodstock. I Oliver Twftt.
Tliff Vicar of Wakefield.
I . 'nil's T"a1es fix-m Sliakc^ponre,
.Southcy's Life of XcImiij.
Sir Ruger d^ Covtflcy.
Itaedft that Won tlie- Jjujiire.
Six tv Sixteen. ; I'inbts for the Wing,
Bell's Supplementary Readers. Gown
S vo. t II i ist rated. LlffipCletlfc fjd. -ml vlid U
Suitable far Standards HI, and I P.
Ander*jt>n's Thinisli Tales.
<"jrc;it Ueedsm Eii^i^h Hifftory
< it iimni's Tales.
Adventures of a Tlonkej**
(Intnl ICn^livhiiLca.
life of CaLumhus.
Sftitrthltfttr Standards ll\ and fc
F: 1 rubles Emm Nature,
Uncle Tow's Cabin.
Swiss Family Robiusnn*
tlrcat Kn^lishwoiiifin.
SeitlerK in Canada.
Suitable for Standards V. a:ti*f l"I
Mitiittrmao Ready*
Rmhiusoft Cntsoe.
LbNr.lrr'ii. of i he Ninv Forest.
Sutitildf far Standards I 'I. and Fit,
'\ ll'' 'I'..'!-: -11.-il.--l. i I V Mil. (;..
Oliver Turlst, | WoodttiOjclc,
BeD's Geographical Readers* Ky M. J.
liAKRI.Vl'.TPiiN-WAHl*. ALA.
Ilie ChfloTs Geography. Mlustrated. 6V.
Mi- Roond Worlti (Standard II. J u.
Ahum F:t.;bn.. : .. CSiarnl. l[!,)ll]ns. m. +d.
The Care Of Babies. A Readine m, ,,:
for (ilrfa* Sdnoofe. Iflwtn 1. Qautj tt*
Bell's History Readers on the Con-
centric Method. K:i(ly IIWriLted,
Finn Lwsons in KuuMhIi Bntor^. io*/.
A Junior Hktory of Kn^laud. if. 6j£
A Scfitor Histjory of Kogbrock aj.
Abbey History Readers* Revised by the
Rl &*Vh P. a. <;as* i h-i: r, D.n. HhettvAed.
ECnriy K:ir;l[sh Kiatory (to iu6oX «i.
StonM front Englfatti HHrtory (1060% 1485).
ThrTr.iior Period CH^ *Ooj)i Tf - ^£
I ho Kttrart Period (iLVt-i?**)- •* <»'*
TJie llant^-eri:tfi IVriud {jji^t^^j),
ts. fd,
BelVs History Readers, Diu^tr.M ■].
Kiirly ICng&Ji rii.iri.'i) ha 106^). t*.
Stones from KnglEih Hnioty itufi^-iji/i.
The 'I udor Pentwl ( .1 48 ,• 10*03). u. yh
The Stuart Period (i6oi-t;u)v ur. fivi'.
The Han^TertanFcriodX "7 14-1637). u. -t",
12
G Belt & Si
ons
Select F. durational Cata/opie
MODERN LANGUAGES
French and German Class Books
Bell'a French Course. By R. P. athrkt™,
M,A- Illustrated- t Pans, is. &r", each,
key to die Exercises. Part 1., &/. net ;
Part 11., tt. net,
BolTa First French Reader. DyK. p.
AriiF.BTOfj, M,A. JllnstrMerl. «,
The Direct Method of Teaching: French.
By D. M ACKAY, M, A,,and F,J,< "uktis, Ph. D.
Flnat French Book, it net.
Becona French Book, ts w. net.
Teacher's Handbook uv net.
Subject Wall Picture iCototMsdi 71. a,/.
net.
Bell'a French Picture Cards. Edited by
H. N. AnAIk, NLA. Two Nc:s of Sixteen
Cards. Printed in Colours, with question-
naire nil the back of each. ti. yf. net eaclt.
Bell's Illustrated French Headers.
Pott.8vo, Fully Illustrated.
*»■ fuH List .m t&plicatfen.
French Historical Reader, By h. N
Adair, M.A. New Comjjosition Supple,
mail, is. ; ar without .Supplement, i*". 6J.
Supplement separately, &/ net.
Simple French Stories. Bv Maec Csw 1,
1'cap. Svo. With or without Vocabulary rind
Notes, i.T,
Coutes Franjftis. Edited, with Introduc-
tion and \nw, by Mak Cbiii. With or
without Vocabulary, u. f^i. Handbcxik or*
Exercise* arid IJuestioitiinirca, fcrf,
Taies from Moli<>re. Hy Marc Cam.
F'cap, Ivu. Wiih W^ibularyund Notes, a$.
Text only, ir, &/".
A French Dramatic Reader, By Mark
Cevii. With Sotcs. reap. Svo. it.
Contes d'Hler et d'AuJourd'bal First
Series. Hy I, S. NoKMAB, M.A., nod
I rtAHlB»RoUKHT-Du»tAS. I HuatdUtsd. tfctfrf.
Second Series, ar.
Le Francois de Prance. Bv Madame
V alette Vkknkt. With Illustrations, if.
Granunaire Pratique. I\ur "1* Fran.
jiL.-i de Prance." By Ma.l.ime Valette
Veknet. to/.
Stortesand Anecdotes for Translation
IntoFreuOh. By Carl Hbatm, ii.
1
French Composition. II,- M. K/ananv,
M..\. l,kit:i, W.
Vocabulaire Franr-als. French VsaUr-
■'•' ll " : ' : " Kejjutm ||y J. p. R . ...
tlLSL., I J, &J,
_. .. .Case's French Course
First French Book. i».
Second French Book. 1.. erf
^tLS K " rat al " 1 8 econd French
IMOko. ij. rW. net.
French Fables for Beginners. .1.
Eistoires Amuaantoset Instruct ves u
^nX^T,,* M ° dem F « n ^
Frencii Poetry f or the Ycnur. Wkh
"^Hr 18 for Frai"* Prose Com-
POBitioU. J». Key, 3,. nct . wu "*
Prosateurs Contenipora'ns. at
t Pe .« Compagnon; > Frasdi jaicBook
tor Luclc (. hlldrciL u.
^^ABie Rev. A, C. Clapin
French Grammar for Pubiic Schools
A French Primer, n
Primer ofFreuchPhilology. ,r,
^f, nl Pa3ca i*ea for Translation into
tTeucll. 2t. E«/, Key, ji, net.
A ^^ iaM GrKmmu f< "* Public Schools.
A Spanish Primer, u,
Bell's First German Course Bv L n r
t-liArniv, M,A. ;.i,
BeU'a First German Reader. By L. n. T,
'- "Ai rt:v. M.A. tlHestnuod -.j "
German Historical Reader, By J. E.
B |f| e Hknlan.ff' ,3 ^^T
Materials for Gorman Prose com.
position, its- Dr. C a. j" ci ST. Jrg
mi ke 'V"v ,ftni '• ™ d "• ■>*■ nct - J ^"
First Book of German Prose, B
K ,•■ ■»'i 11- S the above, iviElT
VocjAukw, t* 6,/.
Kurser Leitraden dor Deutschen Dich-
TUH£T. liy A. E. Lai', at, 6uJ.
Gasc's French Dictionaries
hffiSWSftgE BSWg DiCTIOHARY. Ne, ,d f!ii; „ , i:ll Sup .
OOKCISE FRENCH DICTIONARY. M«li um -tao. ,, &i. Qc'mt^V^ „ «,
POCKET DICTIONARY OF THE FRENCH AND ENGLISH IANGD AGES !0 -no. '-/L'
LITTLE GEM FRENCH DICTIONARY. N 3mw Svo. ,, net. Lin.p L 5a ., lfr , J, ntt '
French and German Annotated Editions
Bell'a French Flays. (Kascd ou Gumbcrt s
Frciicii Dnuno.) Kilited by MaBC CSW1.
Paper, 6rf. ; doth, 8<f.
First r r 4temet%
MolK-re. Le Tartulle. — L*Avarc. — Le
Misanthrope,
Racine. Lcs 1'Uiidures.
Voltaire. 'Zaire.
CornelHe. Le CM,
Oombert's French Drama. Re-edited,
with Notes, by I". E. A Gasc. Sewed,
cW. each.
Mollfere. Le Misanthrope.— L'Avnre.— Le
BOBreeoU tieutilhomme.—Le TaiturTe. —
Le Slalade Iniaeiiiaire. — Les Femmcs
Savantes.~l^s Fourberies de Seapin, —
Ijs Prtcieuses Ridicules.— L'F,colc des
FemmB- - L' EcoledesMirU.— Le Medecin
Mal^re Lui,
RaOlne. L* Tliiholde.— Ijm Plaideurs.—
Iisiiigenie. — Hrilannicuil. — rhidre. —
Esther,— A thnlie,
CornelHe. Le Cid.— Horace.— Oniia.—
Polyeucse.
Voltaire. Zaire,
FtiltelOn. Avcntnres tie Teiemaque, By
C. J. tlKC.tl.s.tt. aJ. &/.
La. Fontaine. Betsoi Fables. By F, E. A.
Gas-z. u, rW.
Lamaxtine. T.r TuUeor de Pitrres de Sainl-
Point, Hy .1. IloiELLE, l'..-is-J. u. of.
Balntlne. I'iccioln. By Dr. DtJOBft
is. 6./.
Voltaire. Chotlei XU. By I- DmY.
German Ballads from HWand. Goethe,
and Schiller, By (J, L. Btrasraui.
If, :'.>.','■.
Goethe. Hetinann untl Dorothea, By B
Bell, M.A., and E. Woui n. is. 6J.
LessineT, Minna von llanihelrj:. By Prof.
A. 11. Nichols. 3J. 6d.
BcttlHer. WaUcnstcin. By TV. Bl-chiikim,
V. Or the LaHer and Ptrcjlumini, yr, {id.
VVallenstein's Tod, 3.'. 6il.
Maid of Orleans. By Dt. W. WagMBH.
I.-. W.
| . Maria Stuart. By V. KASTriEiL U. id.
Bell's Modern Translations
A Series of Translations ftom Modem r^nguages, with Memoirs, Introductions, etc
Crown 8vo, I J. each.
by
the Rev,
Rev.
Dante. Inferno, Translated
H. I". Cakv, M.A.
Poreatorio. Translated by the
H. F. Cakv, M.A.
Paiadiso. Translated by the Res*. H. F.
Cary, M.A.
Goethe, Kemeni. Ttanslaied by Ansa
SiWANSVlCK.
Iphigenia in Tauris. Translaled by ANNA
SWAirwiCK,
— - Goeti von Berlidjinsen. Translated by
Sir WA1.TB" Scott.
Hermann and Dorothea. Translated by
E. A BoWRtKG, C.B.
Banff. The Caravan. Translaled by S.
Mendei»
The Inn in tbo Spesuut. Translaled by
S. Mendbl.
Lessing, Laekoon. Translated by E. C
BbaslRV. ,., , , ,
m ^ Minna von B^rnhcliu. J r;tnsJated by
Ernest BCU, M.A.
Lessirig. Kaflsaa the Wise. Traps'ated by
K. Dll.LOM BoYLAN-
Moliere. Translated, by C. Hf.rcim Wali-
H vols. The Misanthrope. — The Doctor in
Spite or Himself.— Tartune. -The Miser,—
Itie Shopkeeper tumetl Gentleman,— The
Affected l.ailies.— The Learned Women.—
The Impostures ol Scapin.
Racine. Trnnstated by R, Bbltce Boswiilij,
M.A. S vols. Albftlie. - Flsther. — Iphi-
genia. — Andromache. — Britai miens,
ScMHer. William Telb Translated by Sif
Theodcikk Makhs, K.Cti., LL.D. .Vod
Edtt&n, tntirely rei'tied.
• The Maid of Orleans. Translated by
Ansa SWAHWICX.
Mary Stuart. Translated by J, Mei,I.ISII.
— — WallcnMcin's Camp and the Piccolomim.
Transtaied by J. Churchill and S, T
CotBBCTCHfc
llie Death or Walleiwleiti. Translated
by S, T, COLIIKIOGS.
%• Fertfher Translation* frew Modem Languages, see the CaJule^ue of Baku's
Libraries, which itritl be forwarded bu nf-;r:o;t">n.
»4
G. Bell & S t
CV.f
Seket Edntatioml GrfafoQite
>5
SCIENCE AND TECHNOLOGY
thtOiltd Cii/al.^'rie sent m i!fi/://r,!/,\,it
Elementary Botany, iw Pekcv 6m
M.A., U.St.. KL.S. Will, 2/ - ; UluNtrjttior.^.
^^fntMT Botany, r.y <;. f. a ntttoox,
Botany for Schools and Colleges jw
<-. I. Al'KIMtiiK. lili„trau:.f. .. h.t „,.■ '
Practical Plant Physiology. By 'i\,',',.
■RICK KitiriiLK, M.A. I'm,,,,!,,!, " 3 , 6lj
A ££! 0T S U !F, c ? aTee m Plant Physio-
logy. HyW. 1-. Casom;, Ph.1). ^ Slid
The Botanist's Pocket-Book. '% w. R
ilAVWAKi.. Revii«l b] <;. I .:. [imriT. +6 M.
An Introduction to the Study of the
comparative Anatomy of Animals
«!y <.. C BofBKS, M.A. U..X-. WtA
ntiincrmis illiisiramni^ v Vols,
W.i I. Anted Organization, -| i„. |>,„.
loan uiii! (.icleiiwrai.i. Kcvtaed Edition, g .
* ul J r. I be < tmaamut, 6s.
A Manual of Zoology. Ity Kioiak ,, ; ,.
wtc, Ir™fet«l l, y Profcj.S. Kdkmmv.
luastrated. raj. 6d. iip-t
Injurious and Useful Inaecta. An Intro.
daeSaa io ike Study of Economic Knto.
m-.w. By Pr. ; f. I_ r. Miai i„ i--.n. s
«1!|| IMl lIltLSlrnUOIIS. 3 Arf
Civil Service Examination Papers :
Chemistry Papera, Theoretical and
Practical. liv A. p. x,. :lv , u .,. , fp u
A First Year's Course of Chemistry Hv
JAMIlH SINCLAIR. ]'.:■.,/. J
An Introduction to Chemistry liv D s.
Mackaik, Ph.D., i;.,s l . JJ, j r**»
Elementary Inorganic Chemistry n v
Ii-.i. Iamj.sUai.ki..,,-, D.Sc «. «* y
Introduction to Inorganic CnemiBfcry
l-S Hi. Al.BXA.M.KIf SMITH. -J. &/ .,,.,"
Oensrai Chemistry for Colleges ; ..'
A Phv-fe, _*?!*.•"■? in Practical
fc - / \, ' * -!■""« Mixer. a i,i, i ySJj*
A Coilego '.Text-Book of Physics. By
;n,,,';„,, K ' M, ' 1M - ''"■"- ta**n»Z
The Principles of Physics. B. w r
„ ' V ' ' i:i::-;:..:-.l - . .... ,„.,
^?' \v 1 r !a c * ricit3 |, ana "^etism.
ano luilnrgert. 7j.6f.11c1.
Electrons. B^OuvuUn,. & . neL
i. run 11 ,,\,i, Numerous tttus, m SJ ,,,,.
Exorcises in Metal Work. Is,' \ ' ■ I
Technological Handbooks
Edited by Sir II. Tkusman Woon
SpeciaHy ada^d fa "-J^fc-Jtata of * Gt y and «.„,
Woollen and Worsted Cloth Manufac
mm. Ily Prof. Roubkts ItKAujins-r,
o„.» -u , [A i r " Jl '"' / ''''"' '" trefaratim.
Soap Manufacture, ity W. i, lMnr i,
Gaoo, F.J.C, f.c.S. v
Plnnitiing : Its Principles and Practice
By S. Sthven-s IIkli.veh. 51
Silk Dyeing and Finishing. By a u
rliiK-ii, RC.S. }*. til.
Printing. A Pra« ; cul Trcallse, By C T
Jacoiii. 7J. t5rf.
C ^S^ir^^XT^
HOWlHIU PkiE5-v.ua.-:. ft. B y
Music
Music, A Complete Text-Book of, 11, I Masic, A Concise Hiatorv nf 1- , ■•
HISTORY
Catalogue of Historical Banks sent fost free on application
Liugartl'B History of England. Alnn S «l ,
^^ Continued by Dom H. K. BiBJ «nh 1
n !'rrf:i.i= byAun.li (ia-iji-bt. II. Jl, New
Kdiliiiii. Wiih Sfinu.. V. rW- : or 111 : vob.
V,il. I. fw t 4 Sj), jf. ' Vol. II. (1435-ioi-V. JJ.
An Introduction to English Industrial I
History. By Hbkbv Am .soit, H.a. si.
English History Source Books. Edited
IA- S. ft VBM.T, Mi., and Kr.NSBTII
Bfll, M.A. i>. t&
«9-tci«6. TheWcldincofthcRacc. F.-litcd
hy Rev. Jons Wai.i.is, M.A.
tot*, int. The Normans in Kn£l.iiid.
KilitMl by A. ft lii.ANi', M.A.
t 1 ^ i -1 1 1 6, Toft A»b»vh» and die C bAt Ler.
EditedAyS. M. Tovkb, M.A.
I3i»5 1307. Thr. lir-vith ol I'arluraent.
Edited by W- I). RiililsoM.
1 307-1- 00 WW and Morale l.dilcti by
A. A. Loots. ., ,. ,
two-was- Ite io»» of FowJafiBB. l-.il tt «1
6} W. r.ABMON |HK1W. M.A.
nSf-.lU7, The Uoforinatioil anil llie Rc-
rabsroe. Bdtod by ft « : - Bitwawaii
iS4 7 -ifo3- The \iie of Bmbettk tdit*d
"by Akusoeu- Ksiiaii.i-- M.-A.
too^tftstio. PiiTilani.in and Liljcny. b.dilcd
ted by KeKMBm Rell, M.A.
]6&^i7H. A Coii5iitution 111 Muking.
Kdited by C>. B, I'limiKTr, M.A.
1711-1760 WaVi'.r- and llliiiilinni, l:Ali!ed
by K. A. Esn*iu'. . , , ,
1760-1801. Amenoin Indepi:ndini--e ana
the French Reyoltllion. Edited by
S. E. WtHBOLT, M.A
1801-1815. Eniclnnd and Kapoleon. hililci)
by S. K. Wtsiiui-T, M.A.
iBi;-i8jt. Peaw and Rctorm, I'.dneri by
A. C. w. Edwajuw,
18(6-1876. Flora Palmeraton M Itlsttieli.
Ediied by Kwim II MUMSO. B-A-
iBj6-lS57. Imperialism and Mr. Gbidsloae,
Hy R. H. DutSTTOK, M.A.
.,,),,. Caoiata. By James Mijsbo.
Medlsival England : 105S-148S. A Pomte.
I«»l< of History, liy S ; . M. InvKK, M.A,
Cnmn S™. u mw.
First Lessone In English History.
IlUisiiated- li.
A Junior History of England. By E.
HtWiC. llluitraletl. If. 6s.
A Senior History of England. Hy A.
MiKlixiAii. M.A. Crown 8v„. lllns. m.
Highways of the World, to A. E,
MtKit.MAM, M.A. Second lidiii.ni. Ke-
vineil. Tron-n Bvo, tVith U*pn and
|Bo9tmt£oB& \$. 6i/.
A Social History of England. By
Okih.i.k tiuivsi-. Crown Svo. Willi natny
I lluKt alionli, ijf. 6.1'*
EngUsh Church HiRtory to AD. 1000-
iiv w. H. rutaam, M.A.. D.c.i u. s-/.
Civil Service Examination Papera:
I Btory Questions. By A P*wtr*«.
NbVTTORt M.A. IJ,
Ancient History for Schools, liy K.
Ni-a.s and H. R. Sri' el. m.
Strickland's Lives of the Queens of
England. Swdt. 55. cath.
'■, ,, Abridged olili'.n for Schools, 6j. 611'.
Landmarks in the History of Europe.
Ily K. M. RtOiAKOSOX, l'..A. (.rownBvu. vs.
BuildlnK of the British Empire. By
K, M. RixatAjroBO*, 11. A. u. W.
An Atlas of European History. Ky
Km;i,k W. lXnv. 6>. net.
The Foundations of Modern Eurooe.
hl i nv. i-:-,in. Kiatii. s 1 - iK1 -
Dyer'a History of Modern Europe.
B«5od throunlnml by ARriitnl 1 1 ASSAM.,
M.A. 6 villi. Willi Majw. 31. tW. cailh.
Life of Mapoleon L By Jom» Holuiho
RosH.Ln-r.il. ivoh. 101.net.
Carlyle'a French Kevoluton. Bdited
liv J. H.ii.i.A-ii' Bjosb, I-m.». J •••"■-
\Vith nomerout IBwHrrtfema. «. netcach-
Mtgnet's History of the French Revo-
lution, froai 1 7 Hi) lu 1814. K i»'l-
Select Historical Documents of the
Middle Ages. It .ii-.l.-.led and willed by
Bjworrft Hkmikhs<^. I'h-I'- 5*-
Mensel'a History of Germany, j vols.
V. inf. catb.
Eanke'a History Of the Popes. Trans,
lined by ft Fostbb. New Edition. Ke-
virted. 3 vols. I*, nut each.
Banke's History of the Latin and
Teutonic Nations. Revised I riiiulation
by 1.1. R- DWSMB, U.A Willi an Inlroduc
liun by Edwaki) Akmstkosg, .M.A. is. net.
Bohn's
Popular library
THE PIONEEB SERIES OF CHEAP REPRINTS IN NSW AND
DISTINCTIVE FORMAT.
First List of 40 Volumes.
Strongly bound, in Cloth.
One Shilling Net.
12
1. SWIFT (J.) Gutu van's Travels.
2-4. MOTLEY (J. L.) Risk of tiih Dutch Republic. 3 voU.
6-6. EMERSON (R. W.) Works i Vol. I.— Essays and Representative
Men. Vol. II. — English Trails, Nature, and Conduct of Life.
BURTON (Sir R.) Pilgrimage to Al-.Wadi.vah and Mecca.
2 Vols.
LAMB (O.J Essays of Ei.ia and Last Essays of Eli a.
HOOPER (G.) Waterloo : The Downfall of the First Napoleon.
FIELD! vg (H.) Joseph Andrews.
CERVANTES. Don Quixotic 2 vols.
CALVERLEY (0. s.) Thk Idylls of Theocritus with Tits
Eclot.dbs of Virgil.
BUHNEY IF.) Evelina.
16. COLERIDGE (S. T.) An.'S TO Reflection.
18. GOETHE. Poetry and Tri'tii from My Own Life. 2 vols.
EBER1 (Gsory). An Egyptian Princess.
YOTJNG (Arthur), Travels in France.
BURNE? (F.l The Early Diary ov Frances Eurnby (Madame
D'Arblay), 1768-1778. 2 vols.
CARLYLE'S History of the French Revolution. Introduc-
tion ond Votes by J. Holland Rose, Litl.D. 3 vols.
EMERSON (R. W.) Works. Vol. III.— Society and Solitude ;
Letters and Social Aims; Addresses. Vol. IV. —Miscellaneous Pieces.
FIELDING (H.) Tom Jones. 2 vols.
JAMESON (Mrs.) Shakespeare's Heroines.
MARCUS AURELIUS ANTONINUS, The TuotJCUTs or.
Translated by George Long, M.A.
MIGNET'S i I [story of the French Revolction, from 1789 to 1814.
MONTAIGNE. Essays. Cotton*! Translation. 3 vols.
RANKE. History of the Popes. Mrs. Foster's Translation. 3 vols.
TROLLOPE [Anthony). The Warden. Introduction by Frederic
Harrison.
TROLLOPE (Anthony). Barchkstf.r Towers.
Others in active preparation.
WRITE TO-DAY far a copy ol the protpactus containing a history at
the famous Bobo'« Libraries from tbclr Inauguration to the prttent day.
G. BELL AND SONS, LTD., PORTUGAL STREET, LONDON, W.C.
7-8.
9.
10.
11.
-13.
14.
IB.
17-
19.
20.
21-22.
23 25.
36-27.
28-29.
30.
81.
32.
33-35.
86-38.
39.
40.
B
whjwpi