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General Editor: William 1*. Milne, M.A., D.Sc 


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General Editor: WILLIAM ['. MILNE, M.A., D.Sc. 


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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. 





D. M. Y. SOMMERV1LLE, M.A., D.Sc. 






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. 



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 



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. 



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, 




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. * 






1 . Fundamental assumptions or axioms - 

2. Parallel linos ; the three geometries - 

3. Definition of parallel lines 

4. Plane GEOMRfair. Properties of parallelism 





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 


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. 







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 


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. 












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 



1. The prohliin ... 

2. Projective Representation 

3. The absolute 

4. Euclidean geometry 











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 






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 



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 



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 


Systems of circles -------- 

Types of pencils of circles 

Examples VII. -------- 

















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. 










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 - 




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 * 



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- 

1 12 vols., Leipzig, 1883-EM. 

! 3 vols., Cambridge, 1008. 



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 

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). 



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 

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). 



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 



[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] 


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). 


[,. 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 




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 




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 




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 

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 



[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] 



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 

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 

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.) 



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. 




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 

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 



[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] 



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 


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 . 





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 

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 



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 



[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 

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] 



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 

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). 



[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 



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 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/. 



(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. 



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. 


1. If the angle in a semicircle is constant, prove that it is a right 

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). 



[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, 


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. 




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 

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] 



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 

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. 



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 

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) 



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 

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. 


(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 

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 

[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] 



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' 


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, 


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 

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 


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, 


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] 


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 


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. 


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' 


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 


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 

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 

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- 

Draw MM' \\ BB\ and let the perpendicular C'N' to 
LX from any point C on BB' meet MM' in Q. Theu 

n. 13] 



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 

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. 


(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 




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 



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 




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. 


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] 



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 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. 


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 

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] 



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 


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 



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 


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- 


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- 


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. 



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 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 

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 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. 


"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. »] 



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 


=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, 


%m6 = 0-^ + ", -■ 


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. 


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). 


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] 



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, 

ft f* 

Hence sin II (p) - , and cos ll(p)=L\ 


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 



= e*. 




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 

We shall effect a simplification by taking in the following 
paragraphs (§§ 28-37) the constant k as the unit of length. 


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 ' ^ 


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 



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. 

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 



= 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, 


3. B , 

y o , 

X ft . 

4. X', 


ft a . 

5. fi, 

X B , 

a' y'. 

N.-E. O, 


Corresponding to these we get five associated rectilinear 
triangles : 



1. e, 


3. I , 

4. m, 

5. a', 



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 

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). 

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) 






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. 


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 



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), 

cosh (middle part) = product of hyp. sines of adjacent parts 
( B ) = product of hyp. cotangents of opposite 


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 ( 


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 


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). 


KlO. JL 



Then AD = *, BB = m , MOa = U lm - a) , DCi^U^l), md 

n{m-a] + Tl(l + b)^. 


Similarly mm+a t + uil -(,)=- _ ( g,. 

In the txi-rectangular quadrilateral with angle & and eide8 ffl order 


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<) 


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. 


Then, comparing (HI) and (3), we have 

Jl{m-v)= z-Ii(f+b) = Tl{m-a)i therefore v=a. 


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 



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. 



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. 


- 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 


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 


Let the centre go ta infinity, so that the circle becomes a horo- 
cvole. Then r-+ oo , 6~*0, and 


Comparing these with the formulae in 6; 28, we find S -h. 


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\ 


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 

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] 



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. 




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 

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 -. 


Hence the area of the sector = Jf tf/coah - - 1) 

= 2k i $mnh t —, 

and the area of the whole circle is 4irjt» sinh* -1 


^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 


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 


'(l- 2 cosh* .sinMJ 

1-0 2 

= | cos-' (1 - 2 coa?/f ) = fcir " 2B). 

Hence the area of the triangle 


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 


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). 


Now, by producing the sides of any triangle with angles ", /J, y, 
and drawing parallels, we have ( l-'ig. Si) 


Fltt. 54. 


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 +#«-**; 

which is independent of n. 


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] 


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. 


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). 


14. In euclidcan geometry prove that any convex quadrilateral 
can by repetition of itself bo made to cover the whole plane without 

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 




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 

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 



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 


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. 



fiu. a 

rn. 31 



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 





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 

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. 



fin. 5 

m. si 



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 


but in general the four points are projective in no other 


Properties which are unaltered by projection are called 
projective properties. Thus, points which are collinear. 
or lines which are concurrent, retain these properties after 


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 



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 

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 




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 

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. 



| in. <■. 

in. ol 


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'. 



[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 

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] 



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 


connected with the angle a between the lines by the relation 

a= —a, 


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 < ^. 


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 




III. fl 



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 

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). 



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. 



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. 



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 

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. 



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 



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 . 



\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 



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 

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 


The points A lt A z are the centres of a similar elliptic 

involution on a. 



|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 

i4Ai'B l B l ')-x(A s A t 'B£ i ')-xiA 9 di'B i B 3 '). 


(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] 



than two points. Hence all the common perpendiculars 
to two paratactic lines cut the same two generators of the 


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 



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 


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 

Therefore ACAD + BAD > LCAB, i.e. > a right angle. 


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 




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 


- d. From the figure in § 15, it appears that Clifford's 

surface cuts the absolute in two generators of each 

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 



| 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, 

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. 


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). 


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 $, 




Again, the area ABB' A' =adr, and the sum of its exterior 
angles , * 

therefore adr= -2&?dQ, (§9) 



M dr~ k dr*' 
d?a a _ 

The solution of this is 

a=Csm(^ + <f>). 

Differentiating, -*- = 26 = v cos U + <f>) • 



[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 
Keep one part, say b, fixed. Let BAB' =dA, BB' =da, 

Fio. 73. 

AB'C=B+dB, N&^dc. 
Then cfc=ffacosB, 


h sin y rfj4 = NB = rfff sin S (2) 




The area of BAB' is obtained in two ways, (1) by inte- 
grating f, . c , . , 

a A ac 

= 1 fcshi 

= £ 3 <L4(l -cos jj ; 

(2) in terms of the angular excess 

= kHdA + 7r-B + B+dB-Tr) 
=&{dA +dB). 


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. 


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 




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. 



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. 


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] 



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- 

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. 



[Ex. in. 

If we write the parts 


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. 


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 — . 


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.] 



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 £ 


sin " = sin R sin £, sin p = tan cot -, 



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. 



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 

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 



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. 



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. 



[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 




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, 


y = k sin j_ = k sin , sin B, 



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 . 



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?; 


X»+tf-kH*=-^ = 0. 

Hence the ra&os of the coordinates of a point at infinity 
satisfy the equation 


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. 


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?' 

C- -k&ln i 

.v. 4] 


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' 

cos , = 

k j& +&+&**/** +#*+'!&* 

N.-E. O. 



[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. 


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. 


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] 



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 





-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] 



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- 

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 




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, 
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 

In euclidean geometry the distance between two points {x it #,), 
<*,.*) is zero if {Xl - Xs? + { y,. Vtr=0t ,,.. //,= 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- 


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 


£3 'la 



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- 

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. 



[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 

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 = & + *& + & 



(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 

iv. 13] 



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)% 



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 } • 




Xcos , +fi 

= cos r\ 


whence A 

sin, = sin-- and 


and the actual coordinates are 

- r. 

at, sin ^ + x s ain 


[rv. 14 



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] 



hyperbolic geometry the factor 2 sin gr, becomes 2* sinn ^ 

and the coordinates of the second middle point are all 


15. Properties of triangles. Centroid, in- and circum- 


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 



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 


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 

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.) 



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 

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 




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 


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 



[iv. 18 

opposite vertices, will be collmear in a line called the 
ortkaxis, o, which is the absolute polar of the ortho- 

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 



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. 




FIO. 83, 

19. Desmic system. 

In three dimensions we have similar interesting con- 
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] 



+ + + + 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 



[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 


= +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 ' 




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 


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 


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. 



fiv. 21 

Thus, the cross-ratio of a pencil is equal to that of any 
transversal, and cross-ratios are unaltered by projection. 


Further, it can be shown that 

(ABCD) . {ABDC) - 1 , (ABCD) + (ACBD) - 1. 

The harmonic property of the complete quadrilateral 
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.] 



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. 


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. 



[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 

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 

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 

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. 



(In the following questions, 18-22, the geometry is hyperbolic. 
The formulae are analogous to well-known formulae in spherical 

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 











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 



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 


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- 

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. 




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 


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*'__^ 

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 \ 


-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/ ""' ■ 


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 


i-t&ad a cos 0- sin cosfl+isinfl 

i + tan& icosfl+ainrJ cos 0-* sin W 
Therefore (« u', «»') = e s '<* ' *\ 


=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 


Let {PF)=d = k<j>, and 

X s + y* - £*z 2 = r*, x' a + y' 1 - AV 2 = r' 2 ; 



[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. 



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. 



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 

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 

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 



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 

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. 


S\-E. Q. 



[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 


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 



[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 




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. 



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 

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- 




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. 


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 | 



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 



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] 



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- 


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 * 


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 


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. 


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 

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, 



|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 

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. 



Fig. 03. 

n.-b. o 



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 


.' / 

FlO. 00. 




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 

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 



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) 



consider inversion in a circle which represents a straight 

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. 


31. Circular transformation, conformal and homo- 

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\ 



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 


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 = 



\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 

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 ' 


v. 331 



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 


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 

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 


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 

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 



, QY 3 fPX\ », . , RY 3 fPX\ 

1 k- ^ qx §£ \py) =f {Zl ' h) rx % \py)' 


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. 
(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 


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 ? 


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] 



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 


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, 


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- 


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); 


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.] 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. 



1. Pour periods in the history of non-euclidean 

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 

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] 



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 



[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| 



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. 



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] 



simply an assumption that there is such a thing as absolute 

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). 


[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] 



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. 



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] 



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 
.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! Hrckr. il. , -A , , be < ran! 

aiUon from the geodesic to the projective representation of On lev 

vi. 10 1 



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 



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 


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 


e » : = tan \\\{R) =tan(j - f)) -(1 -tan 0)/(l +tan $) ; 


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 



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.) 





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- 

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. 



|>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 

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] 



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 

12. The inextricable entanglement of space and 

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 

"' 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 



\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. 



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 



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] 



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- 
(1) Let the secant cut one branch of the curve in P, Q, 

i.e. in hyperbolic geometry the secant does not cut the 



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, 

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). 




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) 


Therefore "M: 


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 




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; 



tan hOP 

Therefore tan \OP tan \0Q = tan \ OP cot \OQ' = ^rgjg> 



F\n. 4 

4. Reciprocally, if P is a variable point on a fixed line PN, 

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 


sin a sin !, [ff - $) tan h $' - tan 1 ' 

vu. 4j 



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, 
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. 



|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] 



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 

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 



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. 


YTI. 8] 



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 


[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>] 



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 



[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 


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 
If the angle d^^ vanishes, so that the common tangents 

N.-E. Q. F 



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| 


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 


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 

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. 



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 

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 



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. 




[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. 



|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 



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 

(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. 



[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 


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 

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 '. 


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. 


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. 


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 



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. 


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. 


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 


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) 


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 



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'. 


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 



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 

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 


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 


construction with the circle replaced by any conic, is, there- 
fore, in non-euclidean geometry, not a generalization of 

9. Congruent transformations. Transformation of 

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, 



the new axes are rectangular and pass 

Also, since 
through 0', 

a,a +6,6 +BiC =0, (2) 


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 . 


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 
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 


0] Oj Cjlik 


a t b 2 cjik 


a-oM bfjik c 

which may be read either horizontally or vertically. 



The determinant of the substitution = + 1 f or a motion, 
- 1 for a reflexion. 

11. The transformation admits of a very simple repre- 

Since x i +y i -kh i = -&, we can write 

{x + iy) {% - iy) = k*(z - 1) (z + 1 ). 



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,). 


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 -*. 


Therefore A T - A(c + 1)(X - X )(l - XtX)e iW " ♦>• 


\' = ^ = 

N X-X 

<L «.«»->» 


"# 1-XX 

Let e**-*-itfS, and X a = -)3, X 5= -,3; then 


X' = 


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 
_ u 2 {ai\ +A) ^-ffgt&X+ci) = aX +0 

where a = ajct s + /8i/3g, /3 = u iA + ui j3* ; 



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 


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 

\' =■= — ". 
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 - & = _#, 


Here ai + bj+ck + d (=q) is a quaternion. Hence the 
rule for compounding operations of the group 

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- 


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 

If the line passes through the origin, c=0, and the 
equation reduces to 




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 




y\+S' yX+S' 


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' = »*+# 



Then X =-<?"— ® and A = 


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 


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 


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 

These results should be compared with the corresponding 
formulae for euclidean geometry in Chapter V. §§ 31, 32. 

1. In elliptic geometry, show that the general transformation of 

coordinates is expressed by A'=} ; ^k+ta-E f ™ °' 
are real. 


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 



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.] 



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- 

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. 



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 



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 



ix. 3 


(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 

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 


(10) Absolute points and tangents all imaginary and coincident 

in pairs. 
Proper circle. 

(11) Absolute points and tangents all coincident. 

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 



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. 


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 
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 



r tx. I 

ix. 15] 




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 


_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'. 


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 

smr = 


Va? + sf~+3?jfb~- c)(a -b-c) ' 
yjb(a - b) +z*/c(c - a) 
■J-j^ + if-i&Jtiib-c) 

siiir_ la 
mid \ 


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. 



(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 



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). 



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). 


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. 



[Ex. ix. 

Ex, jx.J 




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 

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 


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. 

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. 






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, 

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. 
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. 

Configurations, 143. 
Confocal conies, Ex. ix. IB. 
Conforrual representation, 172- 
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, 
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, 

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, 

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. 


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, 

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. 






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., 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, 

Lublam, V\'., 4, 

Manifold, 194. 
Marginal images, 229. 
Maximum triangle, 7, SI ; Ex ii 
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, 

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, 

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- 


volume, Ex. ii, 24 ; iii. 6. 

Spherical geometry, 25, 89, 130, 
138 » . 
triangles, Q3. 
trigonometry, 09, 70, 120, 



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. 

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- 

associated, 05. 74, 117. 
exterior angle, If, 111, 2B, 34- 

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. 


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