The Elements of Non-Euclidean Geometry

316 Pages·2009·15.576 MB·English

by  J. Coolidge

THE ELEMENTS OF
NON-EUCLIDEAN GEOMETRY
BY
JULIAN LOWELL COOLIDGE PH.D.
ASSISTANT PROFESSOR OF MATHEMATICS
IN HARVARD UNIVERSITY
OXFORD
AT THE CLARENDON PRESS
1909
HENRY FROWDE, M.A.
PUBLISHER TO THEUNIVERSITY OF OXFORD
LONDON, EDINBURGH,NEWYORK
TORONTOANDMELBOURNE
PREFACE
THE heroic age of non-euclidean geometry is passed.
It is long since the days when Lobatchewsky timidly
referred to his system as an * imaginary geometry',
and the new subject appeared as a dangerous lapse
from the orthodox doctrine of Euclid. The attempt to
prove the parallel axiom by means of the other usual
assumptions isnow seldom undertaken, and those who
do undertake are considered in the class with
it,
circle-squarersand searchersforperpetual motion sad
by-products of the creative activity of modern science.
In this, as in all other changes, there is subject both
for rejoicing and regret. It is a satisfaction to a writer
on non-euclidean geometry that he may proceed at
once to his subject, without feelingany need to justify
himself, or, at least, any more need than any other
who adds to our supply of books. On the other hand,
he will miss the stimulus that comes to one who feels
that he is bringing out something entirely new and
strange. The subject of non-euclidean geometry is, to
the mathematician, quite as well established as any
other branch of mathematical science ; and, in fact, it
may lay claim to a decidedly more solid basis than
some branches, such as the theory of assemblages, or
the analysis situs.
Recent books dealing with non-euclidean geometry
fall naturally into two classes. In the one we find
the works of Killing, Liebmann, and Manning,* who
* Detailedreferencesgiven later.
A2
4 PREFACE
wish to build up certain clearly conceived geometrical
systems, and are careless of the details of the founda-
tions on which all is to rest. In the other category
are Hilbert, Vahlen, Veronese, and the authors of
a goodly number of articles on the foundations of
geometry. These writers deal at length with the
consistency, significance, and logical independence of
their assumptions, but do not go very far towards
raising a superstructure on any one of the foundations
suggested.
The present work is, in a measure, an attempt to
unite the two tendencies. The author's own interest,
be it stated at the outset, lies mainly in the fruits,
rather than in the roots but the day is past when the
;
matter of axioms may be dismissed with the remark
that we ' make all of Euclid's assumptions except the
A
one about parallels'. subject like ours must be
builtupfromexplicitlystated assumptions, andnothing
else. The author would have preferred, in the first
chapters, to start from some system of axioms already
published, hadhebeenfamiliar with anythatseemed to
him suitable to establish simultaneously the euclidean
and the principalnoii-euclideaii systems in thewaythat
hewished. Thesystemofaxiomshere used is decidedly
more cumbersome than some others, but leads to the
desired
goal.
There are three natural approaches to non-euclidean
geometry. (1) The elementarygeometry ofpoint, line,
and distance. This method is developed in the open-
ing chapters and is the most obvious. (2) Projcctive
geometry, and the theory of transformation groups.
This method is not taken up until Chapter XVIII, not
because it is one whit less important than the first, but
PREFACE
5
because it seemed better not to interrupt the natural
course of the narrative by interpolating an alternative
beginning. (3) Differential geometry, with the con-
ceptsofdistance-element, extremal, and spaceconstant.
This method is explained in the last chapter, XIX.
The author has imposed upon himself one or two
very definite limitations. To begin with, he has not
gone beyond three dimensions. This is because of his
feelingthat, atanyrateinafirststudyofthesubject, the
gain in generality obtained by studying the geometry
of ^-dimensions is more than offset by the loss of
clearness and naturalness. Secondly, he has confined
himself, almost exclusively, to what may be called the
'classical ' non-euclidean systems. These are much
more closely allied to the euclidean system than are
any others, and have by far the most historical impor-
tance. It is also evident that a system which gives
a simple and clear interpretation of ternary and qua-
ternary orthogonal substitutions, has a totally different
sort of mathematical significance from, let us say, one
whose points are determined by numerical values in
a non-archimedian number system. Or again, a non-
euclidean plane which may be interpreted as a surface
of constant total curvature, has a more lasting geo-
metrical importance than a non-desarguian plane that
cannot form part of a three-dimensional space.
The majority of material in the present work is,
A
naturally, old. reader, new to the subject, may find
it wiser at the first reading to omit Chapters X, XV,
XVI, XVIII, and XIX. On the other hand, a reader
already somewhat familiar with non-euclidean geo-
X
metry, may find his greatest interest in Chapters
and XVI, which contain the substance of a number of
PREFACE
6
recent papers on the extraordinary line geometry of
rion-euclidean space. Mention may also be made
of Chapter XIV which contains a number of neat
formulae relative to areas and volumes published
many years ago by Professor d'Ovidio, which are not,
perhaps, very familiar to English-speaking readers,
and Chapter XIII, where Staude's string construction
of the ellipsoid is extended to non-euclidean space.
It is hoped that the introduction to non-euclidean
XV
differential geometry in Chapter may prove to
be more comprehensive than that of Darboux, and
more comprehensible than that of Bianchi.
The author takes this opportunity to thank his
colleague, Assistant-Professor Whittemore, who has
read in manuscript Chapters XV and XIX. He would
also offer affectionate thanks to his former teachers,
ProfessorEduardStudy ofBonn and Professor Corrado
Segre of Turin, and all others who have aided and
encouraged (or shall we say abetted?) him in the
work.
present
TABLE OF CONTENTS
CHAPTER
I
FOUNDATION FORMETRICAL GEOMETRY IN A LIMITED REGION
PAGE
Fundamentalassumptions and definitions IS
Sums and differences ofdistances 14
Serialarrangementofpoints on aline 15
.17
Simple descriptive propertiesofplaneand space . . .
CHAPTER
II
CONGRUENT TRANSFORMATIONS
Axiom ofcontinuity 23
Division ofdistances 23
Measureofdistance 26
......
Axiom ofcongruent transformations 21)
Definition ofangles, theirproperties 30
Comparison oftriangles 32
Side ofatriangle notgreaterthan sum ofothertwo . . . 35
Comparison and measurementofangles 37
....
Nature ofthe congruent group 38
Definition ofdihedral angles, theirproperties . 3d
CHAPTER IH
THE THREE HYPOTHESES
A variable angle is acontinuousfunction ofa variable distance . 40
Saecheri'stheorem forisoscelesbirectangular quadrilaterals . . 43
The existence of onerectangleimplies the existence of an infinite
number 44
Three assumptionsasto the sum ofthe angles ofarighttriangle . 45
Three assumptionsastothe sum oftheanglesofanytriangle,their
categorical nature 4fr
Definition ofthe euclidean, hyperbolic, and elliptic hypotheses . 46
Geometryintheinfinitesimaldomainobeystheeuclideanhypothesis 47
CHAPTER IV
TRIGONOMETRIC FORMULAE
Limit of ratio of opposite sides of diminishing isosceles quadri-
lateral 4H
Continuityofthe resultingfunction 50
....
Itsfunctional equation and solution 51
....
Functional equation forthe cosine ofanangle 54
Non-euclidean form forthepythagoreantheorem 55
Trigonometric formulaeforright and oblique triangles . . . 57
8 CONTENTS
CHAPTER V
ANALYTIC FORMULAE
PAGE
Directed distances 62
Groupoftranslations ofaline 62
Positive and negative directed distances 64
.64
Coordinates ofa point on aline
Coordinates ofa pointin aplane .... 65
Finite andinfinitesimal distanceformulae,thenon-e.ucl.idea.npl.ane
asasurface ofconstant Gaussian curvature 65
Equation connecting direction cosines ofaline 67
Coordinates ofapoint in space 68
Congruenttransformationsand orthogonal substitutions . . 69
Fundamental formulae for distance and angle 70
CHAPTER VI
CONSISTENCE AND SIGNIFICANCE OF THE AXIOMS
......
Examplesofgeometries satisfying the assumptions made . . 72
Relative independence ofthe axioms 74
CHAPTER VII
GEOMETRIC AND ANALYTIC EXTENSION OF SPACE
Possibility of extending a segment by a definite amount in the
euclidean andhyperbolic cases 77
....
Euclidean andhyperbolic space 77
Contradiction arising underthe elliptic hypothesis 78
New assumptions identical with the old forlimited region,but per-
mittingthe extension ofevery segmentby adefinite amount . 78
Last axiom,free mobility ofthe whole system 80
One to one correspondence ofpointand coordinatesetin euclidean
andhyperbolic cases 81
Ambiguity inthe elliptic case givingrise to elliptic and spherical
geometry 81
Ideal elements, extension ofallspacestobe real con.tinu.a .. .. 84
Imaginaryelements geometrically defined, extension of all spaces
to be perfect continuain the complexdomain 85
Cayleyan Absolute, newform forthe definition ofdistance . . 88
CEaxsteenwshieorneofatshteradiigshttalnicneecgoinvceespatmtoaxthiemucommpdilsetxandcoemai.n .. .. 9189
CHAPTER VIII
GROUPS OF CONGRUENT TRANSFORMATIONS
.....94
Congruenttransformations ofthe straightline . .
,, hyperbolicplane 94
CONTENTS 9
PAGE
.....96
Congruent transformations ofthe elliptic plane ..... .
,, ,, euclidean plane 96
hyperbolic space 97
elliptic and spherical space . . 98
Clifford parallels, orparatacticlines 99
The groups ofright and lefttranslations 99
Congruenttransformationsofeuclidean space 100
CHAPTER IX
POINT, LINE, AND PLANE, TREATED ANALYTICALLY
Notable points ofa triangle in the non-euclidean plane . . . 101
Analogaofthe theorems ofMenelaus and Ceva . . . .104
Formulae ofthe parallel angle 106
Equations ofparallels to agivenline 107
Notablepointsofatetrahedron,and resultingdesmicconfigurations 108
Invariant formulae for distance and angle of skew lines in line
coordinates' 110
Criteria forparallelism and parataxy in line coordinates. . . 113
Relative moment oftwo directed lines 114
CHAPTER X
HIGHER LINE-GEOMETRY
Linear complex in hyperbolic space 11ti
The cross, itscoordinates 117
The use of the cross manifold to interpret the geometry of the
complex plane 118
Chain, and chain surface 119
Hamilton'stheorem 120
Chain congruence, synectic and non-synectic congruences . .121
Dual coordinates ofa crossin elliptic case 124
Condition for parataxy 125
Clifford angles 126
Chainand strip . . . 128
Chain congruence 129
CHAPTER XI
THE CIRCLE AND THE SPHERE
Simplestform for the equation of a circle 131
Dualnatureofthe curve 131
Curvature ofacircle 133
Radicalaxes, and centresofsimilitude 134
Circlesthroughtwo points, ortangent totwolines . . . .135
Spheres 138
Poincare's sphere to sphere transformation from euclidean to non-
euclideanspace 139
CONTENTS
10
CHAPTER XII
CONIC SECTIONS
PAGE
Classification ofconies 142
Equations ofcentral conicand Absolute 148
Centres, axes,foci, focal lines, directrices, and directorpoints . 14B
Relations connectingdistances ofa pointfromfoci,directrices,&c.,
andtheir duals 144
Conjugate and mutually perpendicular linesthroughacentre . 148
Auxiliary circles 150
Normals 152
Confocal and homothetic conies 152
Elliptic coordinates 152
CHAPTER XIII
QUADRIC SURFACES
.154
Classification ofquadrics .
Central quadrics 157
Planes ofcircularsection and parabolic section . - . .158
Conjugate and mutuallyperpendicularlinesthrough a centre . 159
Confocalandhomothetic quadrics 160
Elliptic coordinates, variousforms of thedistance element . .161
Stringconstruction forthe ellipsoid 166
CHAPTER XIV
AREAS AND VOLUMES
Amplitude ofatriangle 170
Relationtootherparts .171
Limitingformwhen the triangle is infinitesimal . . . .174
Deficiency and area 175
Area foundbyintegration . . . . . . . .176
Area ofcircle 178
Area ofwholeelliptic orspherical plane . . . . . .178
Amplitude ofatetrahedron 178
Relationtootherparts 179
Simple formfor the differential ofvolume ofatetrahedron . .181
Reduction to a single quadrature of the problem of finding the
volumeofatetrahedron 184
Volume ofaconeofrevolution 185
Volume ofasphere 186
Volumeofthewhole ofelliptic orofsphericalspace . . . 186