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% The Project Gutenberg EBook of The Elements of non-Euclidean Geometry, by 
% Julian Lowell Coolidge                                                  %
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% This eBook is for the use of anyone anywhere at no cost and with        %
% almost no restrictions whatsoever.  You may copy it, give it away or    %
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% Title: The Elements of non-Euclidean Geometry                           %
%                                                                         %
% Author: Julian Lowell Coolidge                                          %
%                                                                         %
% Release Date: August 20, 2008 [EBook #26373]                            %
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% Language: English                                                       %
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% Character set encoding: ISO-8859-1                                      %
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% *** START OF THIS PROJECT GUTENBERG EBOOK NON-EUCLIDEAN GEOMETRY ***    %
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\begin{verbatim}
The Project Gutenberg EBook of The Elements of non-Euclidean Geometry, by 
Julian Lowell Coolidge

This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever.  You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.org


Title: The Elements of non-Euclidean Geometry

Author: Julian Lowell Coolidge

Release Date: August 20, 2008 [EBook #26373]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK NON-EUCLIDEAN GEOMETRY ***


Produced by Joshua Hutchinson, David Starner, Keith Edkins
and the Online Distributed Proofreading Team at
http://www.pgdp.net
\end{verbatim}

\pagestyle{empty}
\newpage

%[File: 001.png]

%[File: 002.png]
\begin{center}
{\Huge THE ELEMENTS OF}

\vspace{0.1in}

{\Huge NON-EUCLIDEAN GEOMETRY}

\vspace{0.5in}

{\small BY}

\vspace{0.3in}


{\Large \textsc{JULIAN LOWELL COOLIDGE Ph.D.}}

\vspace{0.1in}


{\small ASSISTANT PROFESSOR OF MATHEMATICS\\
IN HARVARD UNIVERSITY}

\vspace{4in}

OXFORD\\
AT THE CLARENDON PRESS\\
1909\\
\end{center}

%[File: 004.png]
\newpage
\mainmatter
\pagestyle{plain}


\begin{center}
{\Large PREFACE}
\end{center}

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 is now seldom undertaken, and those who
do undertake it, are considered in the class with
circle-squarers and searchers for perpetual 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 feeling any 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,%
\footnote{Detailed references given later}
%endfootnote
who
%[File: 005.png]
wish to build up certain clearly conceived geometrical
systems, and are careless of the details of the foundations
on which all is to rest. In the other category
are Hilbert, Vablen, 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
one about parallels'. A subject like ours must be
built up from explicitly stated assumptions, and nothing
else. The author would have preferred, in the first
chapters, to start from some system of axioms already
published, had he been familiar with any that seemed to
him suitable to establish simultaneously the euclidean
and the principal non-euclidean systems in the way that
he wished. The system of axioms here 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 elementary geometry of point, line,
and distance. This method is developed in the opening
chapters and is the most obvious. (2) Projective
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
%[File: 006.png]
because it seemed better not to interrupt the natural
course of the narrative by interpolating an alternative
beginning. (3) Differential geometry, with the concepts
of distance-element, extremal, and space constant.
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
feeling that, at any rate in a first study of the subject, the
gain in generality obtained by studying the geometry
of $n$-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 importance.
It is also evident that a system which gives
a simple and clear interpretation of ternary and quaternary
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 geometrical
importance than a non-desarguian plane that
cannot form part of a three-dimensional space.

The majority of material in the present work is,
naturally, old. A 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 geometry,
may find his greatest interest in Chapters X
and XVI, which contain the substance of a number of
%[File: 007.png]
recent papers on the extraordinary line geometry of
non-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
differential geometry in Chapter XV may prove to
be more comprehensive than that of Darboux, and
more comprehensible than that of Bianchi\index{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,
Professor Eduard Study of Bonn and Professor Corrado
Segre of Turin, and all others who have aided and
encouraged (or shall we say abetted?) him in the
present work.

%[File: 008.png]
\tableofcontents
%TABLE OF CONTENTS
%CHAPTER I
%FOUNDATION FOR METRICAL GEOMETRY IN A LIMITED REGION
%Fundamental assumptions and definitions
%Sums and differences of distances
%Serial arrangement of points on a line
%Simple descriptive properties of plane and space
%CHAPTER II
%CONGRUENT TRANSFORMATIONS
%Axiom of continuity
%Division of distances
%Measure of distance
%Axiom of congruent transformations
%Definition of angles, their properties
%Comparison of triangles
%Side of a triangle not greater than sum of other two
%Comparison and measurement of angles
%Nature of the congruent group
%Definition of dihedral angles, their properties
%CHAPTER III
%THE THREE HYPOTHESES
%A variable angle is a continuous function of a variable distance
%Saccheri's theorem for isosceles birectangular quadrilaterals
%The existence of one rectangle implies the existence of an infinite
%number
%Three assumptions as to the sum of the angles of a right triangle
%Three assumptions as to the sum of the angles of any triangle, their
%categorical nature
%Definition of the euclidean, hyperbolic, and elliptic hypotheses
%Geometry in the infinitesimal domain obeys the euclidean hypothesis
%CHAPTER IV
%TRIGONOMETRIC FORMULAE
%Limit of ratio of opposite sides of diminishing isosceles quadrilateral
%Continuity of the resulting function
%Its functional equation and solution
%Functional equation for the cosine of an angle
%Non-euclidean form for the pythagorean theorem
%Trigonometric formulae for right and oblique triangles
%%[File: 009.png]
%CHAPTER V
%ANALYTIC FORMULAE
%Directed distances
%Group of translations of a line
%Positive and negative directed distances
%Coordinates of a point on a line
%Coordinates of a point in a plane
%Finite and infinitesimal distance formulae, the non-euclidean plane
%as a surface of constant Gaussian curvature
%Equation connecting direction cosines of a line
%Coordinates of a point in space
%Congruent transformations and orthogonal substitutions
%Fundamental formulae for distance and angle
%CHAPTER VI
%CONSISTENCE AND SIGNIFICANCE OF THE AXIOMS
%Examples of geometries satisfying the assumptions made
%Relative independence of the axioms
%CHAPTER VII
%GEOMETRIC AND ANALYTIC EXTENSION OF SPACE
%Possibility of extending a segment by a definite amount in the
%euclidean and hyperbolic cases
%Euclidean and hyperbolic space
%Contradiction arising under the elliptic hypothesis
%New assumptions identical with the old for limited region, but permitting
%the extension of every segment by a definite amount
%Last axiom, free mobility of the whole system
%One to one correspondence of point and coordinate set in euclidean
%and hyperbolic cases
%Ambiguity in the elliptic case giving rise to elliptic and spherical
%geometry
%Ideal elements, extension of all spaces to be real continua
%Imaginary elements geometrically defined, extension of all spaces
%to be perfect continua in the complex domain
%Cayleyan Absolute, new form for the definition of distance
%Extension of the distance concept to the complex domain
%Case where a straight line gives a maximum distance
%CHAPTER VIII
%GROUPS OF CONGRUENT TRANSFORMATIONS
%Congruent transformations of the straight line
%   ''            ''        ''    hyperbolic plane
%%[File: 010.png]
%Congruent transformations of the elliptic plane
%   ''            ''         ''   euclidean plane
%   ''            ''         ''   hyperbolic space
%   ''            ''         ''   elliptic and spherical space
%Clifford parallels, or paratactic lines
%The groups of right and left translations
%Congruent transformations of euclidean space
%CHAPTER IX
%POINT, LINE, AND PLANE, TREATED ANALYTICALLY
%Notable points of a triangle in the non-euclidean plane
%Analoga of the theorems of Menelaus and Ceva
%Formulae of the parallel angle
%Equations of parallels to a given line
%Notable points of a tetrahedron, and resulting desmic configurations
%Invariant formulae for distance and angle of skew lines in line
%coordinates
%Criteria for parallelism and parataxy in line coordinates
%Relative moment of two directed lines
%CHAPTER X
%HIGHER LINE-GEOMETRY
%Linear complex in hyperbolic space
%The cross, its coordinates
%The use of the cross manifold to interpret the geometry of the
%complex plane
%Chain, and chain surface
%Hamilton's theorem
%Chain congruence, synectic and non-synectic congruences
%Dual coordinates of a cross in elliptic case
%Condition for parataxy
%Clifford angles
%Chain and strip
%Chain congruence
%CHAPTER XI
%THE CIRCLE AND THE SPHERE
%Simplest form for the equation of a circle
%Dual nature of the curve
%Curvature of a circle
%Radical axes, and centres of similitude
%Circles through two points, or tangent to two lines
%Spheres
%Poincar\'e's sphere to sphere transformation from euclidean to non-euclidean
%space
%%[File: 011.png]
%CHAPTER XII
%CONIC SECTIONS
%Classification of conics
%Equations of central conic and Absolute
%Centres, axes, foci, focal lines, directrices, and director points
%Relations connecting distances of a point from foci, directrices, \&c.,
%and their duals
%Conjugate and mutually perpendicular lines through a centre
%Auxiliary circles
%Normals
%Confocal and homothetic conics
%Elliptic coordinates
%CHAPTER XIII
%QUADRIC SURFACES
%Classification of quadrics
%Central quadrics
%Planes of circular section and parabolic section
%Conjugate and mutually perpendicular lines through a centre
%Confocal and homothetic quadrics
%Elliptic coordinates, various forms of the distance element
%String construction for the ellipsoid
%CHAPTER XIV
%AREAS AND VOLUMES
%Amplitude of a triangle
%Relation to other parts
%Limiting form when the triangle is infinitesimal
%Deficiency and area
%Area found by integration
%Area of circle
%Area of whole elliptic or spherical plane
%Amplitude of a tetrahedron
%Relation to other parts
%Simple form for the differential of volume of a tetrahedron
%Reduction to a single quadrature of the problem of finding the
%volume of a tetrahedron
%Volume of a cone of revolution
%Volume of a sphere
%Volume of the whole of elliptic or of spherical space
%%[File: 012.png]
%CHAPTER XV
%INTRODUCTION TO DIFFERENTIAL GEOMETRY
%Curvature of a space or plane curve
%Analoga of direction cosines of tangent, principal normal, and
%binormal
%Frenet's formulae for the non-euclidean case
%Sign of the torsion
%Evolutes of a space curve
%Two fundamental quadratic differential forms for a surface
%Conditions for mutually conjugate or perpendicular tangents
%Lines of curvature
%Dupin's theorem for triply orthogonal systems
%Curvature of a curve on a surface
%Dupin's indicatrix
%Torsion of asymptotic lines
%Total relative curvature, its relation to Gaussian curvature
%Surfaces of zero relative curvature
%Surfaces of zero Gaussian curvature
%Ruled surfaces of zero Gaussian curvature in elliptic or spherical
%space
%Geodesic curvature and geodesic lines
%Necessary conditions for a minimal surface
%Integration of the resulting differential equations
%CHAPTER XVI
%DIFFERENTIAL LINE-GEOMETRY
%Analoga of Kummer's coefficients
%Their fundamental relations
%Limiting points and focal points
%Necessary and sufficient conditions for a normal congruence
%Malus-Dupin theorem
%Isotropic congruences, and congruences of normals to surfaces of
%zero curvature
%Spherical representation of rays in elliptic space
%Representation of normal congruence
%Isotropic congruence represented by an arbitrary function of the
%complex variable
%Special examples of this representation
%Study's ray to ray transformation which interchanges parallelism
%and parataxy
%Resulting interchange among the three special types of congruence
%%[File: 013.png]
%CHAPTER XVII
%MULTIPLY CONNECTED SPACES
%Repudiation of the axiom of free mobility of space as a whole
%Resulting possibility of one to many correspondence of points and
%coordinate sets
%Multiply connected euclidean planes
%Multiply connected euclidean spaces, various types of line in them
%Hyperbolic case little known; relation to automorphic functions
%Non-existence of multiply connected elliptic planes
%Multiply connected elliptic spaces
%CHAPTER XVIII
%PROJECTIVE BASIS OF NON-EUCLIDEAN GEOMETRY
%Fundamental notions
%Axioms of connexion and separation
%Projective geometry of the plane
%Projective geometry of space
%Projective scale and cross ratios
%Projective coordinates of points in a line
%Linear transformations of the line
%Projective coordinates of points in a plane
%Equation of a line, its coordinates
%Projective coordinates of points in space
%Equation of a plane
%Collineations
%Imaginary elements
%Axioms of the congruent collineation group
%Reappearance of the Absolute and previous metrical formulae
%CHAPTER XIX
%DIFFERENTIAL BASIS FOR EUCLIDEAN AND NON-EUCLIDEAN
%GEOMETRY
%Fundamental assumptions
%Coordinate system and distance elements
%Geodesic curves, their differential equations
%Determination of a geodesic by a point and direction cosines of
%tangent thereat
%Determination of a geodesic by two near points
%Definition of angle
%Axiom of congruent transformations
%Simplified expression for distance element
%Constant curvature of geodesic surfaces
%Introduction of new coordinates; integration of equations of geodesic
%Reappearance of familiar distance formulae
%Recapitulation
%INDEX
%
%[File: 014.png]

%\mainmatter

%CHAPTER I
%\chapter{FOUNDATION FOR METRICAL GEOMETRY IN A LIMITED REGION}
\mychap{I}{FOUNDATION FOR METRICAL GEOMETRY IN A LIMITED REGION}

\mysect{Fundamental assumptions and definitions}

In any system of geometry we must begin by assuming
the existence of certain fundamental objects, the raw material
with which we are to work. What names we choose to
attach to these objects is obviously a question quite apart
from the nature of the logical connexions which arise from
the various relations assumed to exist among them, and in
choosing these names we are guided principally by tradition,
and by a desire to make our mathematical edifice as well
adapted as possible to the needs of practical life. In the
present work we shall assume the existence of two sorts
of objects, called respectively \emph{points}\index{Point} and \emph{distances}\index{Distance}.%
\footnote{There
is no logical or mathematical reason why the point should be taken
as undefined rather than the line or plane. This is, however, the invariable
custom in works on the foundations of geometry, and, considering the
weight of historical and psychological tradition in its favour, the point
will probably continue to stand among the fundamental indefinables. With
regard to the others, there is no such unanimity. Veronese, \emph{Fondamenti di
geometria}, Padua, 1891, takes the line, segment, and congruence of segments.
Schur\index{Schur}, `Ueber die Grundlagen der Geometrie,' \emph{Mathematische Annalen}, vol.\
lv, 1902, uses segment and motion. Hilbert\index{Hilbert}, \emph{Die Grundlagen der Geometrie},
Leipzig, 1899, uses practically the same indefinables as Veronese\index{Veronese}. Moore\index{Moore},
`The projective Axioms of Geometry,' \emph{Transactions of the American Mathematical
Society}, vol.~iii, 1902, and Veblen\index{Veblen}, `A System of Axioms for Geometry,' same
Journal, vol.~v, 1904, use segment and order. Pieri\index{Pieri}, `Della geometria
elementare come sistema ipotetico deduttivo,' \emph{Memorie della R. Accademia delle
Scienze di Torino}, Serie 2, vol.~xlix, 1899, introduces motion alone, as does
Padoa\index{Padoa}, `Un nuovo sistema di definizioni per la geometria euclidea,' \emph{Periodico
di matematica}, Serie 3, vol.~i, 1903. Vahlen\index{Vahlen}, \emph{Abstrakte Geometrie}, Leipzig, 1905,
uses line and separation. Peano\index{Peano}, `La geometria basata sulle idee di punto
e di distanza,' \emph{Atti della R. Accademia di Torino}, vol.~xxxviii, 1902-3, and
Levy\index{Levy}, `I fondamenti della geometria metrica-proiettiva,' \emph{Memorie Accad.
Torino}, Serie 2, vol.~liv, 1904, use distance. I have made the same choice as
the last-named authors, as it seemed to me to give the best approach to the
problem in hand. I cannot but feel that the choice of segment or order
would be a mistake for our present purpose, in spite of the very condensed
system of axioms which Veblen has set up therefor. For to reach congruence
and measurement by this means, one is obliged to introduce the
six-parameter group of motions (as in Ch. XVIII of this work), i.e.\ base
metrical geometry on projective. It is, on the other hand, an inelegance to
base projective geometry on a non-projective conception such as `between-ness',
whereas writers like Vahlen require both projective and `affine'
geometry, before reaching metrical geometry, a very roundabout way to
reach what is, after all, the fundamental part of the subject.}
%endfootnote
Our explicit assumptions shall be as follows:---

%[File: 015.png]
\medskip\textsc{Axiom}~I. \textbf{There exists a class of objects, containing at
least two members, called points.}

It will be convenient to indicate points by large Roman
letters as $A$, $B$, $C$.

\medskip\textsc{Axiom}~II\@. \textbf{The existence of any two points implies the
existence of a unique object called their distance.}

If the points be $A$ and $B$ it will be convenient to indicate
their distance by $\overline{AB}$ or $\overline{BA}$. We shall speak of this also
as the distance \emph{between} the two points, or from one to the
other.

We next assume that between two distances there may
exist a relation expressed by saying that the one is \emph{congruent}\index{Congruence01@Congruence of distances}
to the other. In place of the words `is congruent to' we
shall write the symbol $\equiv$. The following assumptions shall
be made with regard to the congruent relation:---

\medskip\textsc{Axiom}~III\@. \hspace{1in} $\overline{AB}\equiv\overline{AB}$.

\medskip\textsc{Axiom}~IV\@. \hspace{1in} $\overline{AA}\equiv\overline{BB}$.

\medskip\textsc{Axiom}~V. \textbf{If} $\overline{AB} \equiv \overline{CD}$ \textbf{and}
$\overline{CD} \equiv \overline{EF}$, \textbf{then}
$\overline{AB} \equiv \overline{EF}$.

\medskip These might have been put into purely logical form by
saying that we assumed that every distance was congruent
to itself, that the distances of any two pairs of identical
points are congruent, and that the congruent relation is
transitive.

\mysect{Sums and differences of distances}
\index{Sum03@Sum of distances|(}

Let us next assume that there may exist a triadic relation
connecting three distances which is expressed by a saying
that the first $\overline{AB}$ is congruent to the sum of the second
$\overline{CD}$
and the third $\overline{PQ}$. This shall be written
$\overline{AB}\equiv \overline{CD} + \overline{PQ}$.

\medskip\textsc{Axiom}~VI\@. \textbf{If}
$\overline{AB} \equiv \overline{CD} + \overline{PQ}$, \textbf{then}
$\overline{AB} \equiv \overline{PQ} + \overline{CD}$.

\medskip\textsc{Axiom}~VII\@. \textbf{If}
$\overline{AB} \equiv \overline{CD} + \overline{PQ}$ \textbf{and}
$\overline{PQ} \equiv \overline{RS}$, \textbf{then}
$\overline{AB} \equiv \overline{CD} + \overline{RS}$.

\medskip\textsc{Axiom}~VIII\@. \textbf{If}
$\overline{AB} \equiv \overline{CD} + \overline{PQ}$ \textbf{and}
$\overline{A'B'} \equiv \overline{AB}$, \textbf{then}
$\overline{A'B'} \equiv \overline{CD} + \overline{PQ}$.

\medskip\textsc{Axiom}~IX\@. \hspace{1in} $\overline{AB} \equiv \overline{AB} + \overline{CC}$.

\medskip\emph{Definition}.~The distance of two identical points shall be
called a \emph{null}\index{Null02@--- distance} distance.

%[File: 016.png]
\smallskip\emph{Definition}.~If $\overline{AB}$ and $\overline{CD}$
be two such distances that there
exists a not null distance $\overline{PQ}$ fulfilling the condition that
$\overline{AB}$
is congruent\index{Congruence01@Congruence of distances} to the sum of $\overline{CD}$ and $\overline{PQ}$, then
$\overline{AB}$ shall be said to be \emph{greater than}\index{Greater than}
$\overline{CD}$. This is written
$\overline{AB} > \overline{CD}$.

\smallskip\emph{Definition}.~If $\overline{AB} > \overline{CD}$, then $\overline{CD}$
shall be said to be \emph{less than}\index{Less than}
$\overline{AB}$. This is written $\overline{CD} < \overline{AB}$.

\medskip\textsc{Axiom}~X. \textbf{Between any two distances $\overline{AB}$ and $\overline{CD}$ there
exists one, and only one, of the three relations}
\[
  \overline{AB} \equiv \overline{CD}, \quad
  \overline{AB} > \overline{CD}, \quad
  \overline{AB} < \overline{CD}.
\]

\medskip\emph{Theorem}~1. If $\overline{AB} \equiv \overline{CD}$, then
$\overline{CD} \equiv \overline{AB}$.

For we could not have $\overline{AB} \equiv \overline{CD} + \overline{PQ}$
where $\overline{PQ}$ was
not null. Nor could we have
$\overline{CD} \equiv \overline{AB} + \overline{PQ}$ for then, by
VIII, $\overline{AB} \equiv \overline{AB} + \overline{PQ}$ contrary to X.

\smallskip\emph{Theorem}~2. If $\overline{AB} \equiv \overline{CD} + \overline{PQ}$ and $\overline{C'D'} \equiv \overline{CD}$, then
\[
  \overline{AB} \equiv \overline{C'D'} + \overline{PQ}.
\]

The proof is immediate.

\index{Extension01@Extension of segment}
\medskip\textsc{Axiom}~XI\@. \textbf{If $A$ and $C$ be any two points there exists
such a point $B$ distinct from either that}
\[
  \overline{AB} \equiv \overline{AC} + \overline{CB}.
\]

This axiom is highly significant. In the first place it
clearly involves the existence of an infinite number of points.
In the second it removes the possibility of a maximum distance.
In other words, there is no distance which may not
be extended in either direction. It is, however, fundamentally
important to notice that we have made no assumption as
to the magnitude of the amount by which a distance may
be so extended; we have merely premised the existence of
such extension. We shall make the concept of extension
more explicit by the following definitions.

\mysect{Serial arrangement of points on a line}

\smallskip\emph{Definition}.~The assemblage of all points $C$ possessing the
property that $\overline{AB} \equiv \overline{AC} + \overline{CB}$ shall be called the \emph{segment}\index{Segment}
of $A$ and $B$, or of $B$ and $A$, and written $(AB)$ or $(BA)$. The
points $A$ and $B$ shall be called the \emph{extremities}\index{Extremity of segment} of the segment,
all other points thereof shall be said to be \emph{within}\index{Within a segment} it.

\smallskip\emph{Definition}.~The assemblage of all points $B$ different from
$A$ and $C$ such that $\overline{AB} \equiv \overline{AC} + \overline{CB}$
shall be called the \emph{extension}
of $(AC)$ beyond $C$.

%[File: 017.png]
\begin{tabbing}
\medskip\hspace{\parindent}\textsc{Axiom}~XII\@. \textbf{If}\;
\=$\overline{AB} \equiv \overline{AC} + \overline{CB}$ \;\textbf{where}\; $  \overline{AC} \equiv \overline{AD} + \overline{DC}$, \\
\textbf{then}
\>$\overline{AB} \equiv \overline{AD} + \overline{DB}$ \;\textbf{where}\; $  \overline{DB} \equiv \overline{DC} + \overline{CB}$.
\end{tabbing}

The effect of this axiom is to establish a serial order among
the points of a segment and its extensions, as will be seen
from the following theorems. We shall also be able to show
that our distances are scalar magnitudes, and that addition of
distances is associative.

\medskip\textsc{Axiom}~XIII\@. \textbf{If}
$\overline{AB} \equiv \overline{PQ} + \overline{RS}$ \textbf{there is a single point
$C$ of $(AB)$ such that} $\overline{AC} \equiv \overline{PQ}$, $\overline{CB} \equiv \overline{RS}$.

\smallskip\emph{Theorem}~3. If $\overline{AB} > \overline{CD}$ and $\overline{CD} > \overline{EF}$, then $\overline{AB}>\overline{EF}$.

To begin with $\overline{AB} \equiv \overline{EF}$ is impossible. If then $\overline{EF} > \overline{AB}$,
let us put $\overline{EF} \equiv \overline{EG} + \overline{GF}$, where $\overline{EG} \equiv \overline{AB}$.
\begin{flalign*}
&\text{Then} \hfill &
  \overline{CD} &\equiv \overline{CH} + \overline{HD}; \quad
  \overline{CH} \equiv \overline{EF}.
\\
&\text{Then} \hfill &
  \overline{CD} &\equiv \overline{CK} + \overline{KD}; \quad
  \overline{CK} \equiv \overline{AB} & \hfill \phantom{\text{Then}}
\end{flalign*}
which is against our hypothesis.

We see as a corollary, to this, that if $C$ and $D$ be any two
points of $(AB)$, one at least being within it,
$\overline{AB} > \overline{CD}$.

It will follow from XIII that two distinct points of a
segment\index{Segment} cannot determine congruent distances from either end
thereof. We also see from XII that if $C$ be a point of $(AB)$,
and $D$ a point of $(AC)$, it is likewise a point of $(AB)$. Let
the reader show further that every point of a segment, whose
extremities belong to a given segment, is, itself, a point of
that segment.

\smallskip\emph{Theorem}~4. If $C$ be a point of $(AB)$, then every point $D$ of
$(AB)$ is either a point of $(AC)$ or of $(CB)$.

If $\overline{AC} \equiv \overline{AD}$ we have $C$ and $D$ identical.
If $\overline{AC} > \overline{AD}$\index{Greater than} we
may find a point of $(AC) \bigl[$and so of $(AB)\bigr]$ whose distance from
$A$ is congruent\index{Congruence01@Congruence of distances} to $\overline{AD}$,
and this will be identical with $D$. If
$\overline{AC} < \overline{AD}$\index{Less than} we find $C$ as a point of $(AD)$, and hence, by XII,
$D$ is a point of $(CB)$.

\smallskip\emph{Theorem}~5. If
$\overline{AB} \equiv \overline{AC} + \overline{CB}$ and
$\overline{AB} \equiv \overline{AD} + \overline{DB}$ while
$\overline{AC} > \overline{AD}$, then $\overline{CB} < \overline{DB}$.

\smallskip\emph{Theorem}~6. If
$\overline{AB} \equiv \overline{PQ} + \overline{RS}$ and
$\overline{A'B'} \equiv \overline{PQ} + \overline{RS}$, then
$\overline{A'B'} \equiv \overline{AB}$.

The proof is left to the reader.

\smallskip\emph{Theorem}~7. If
$\overline{AB} \equiv \overline{PQ} + \overline{RS}$ and
$\overline{AB} \equiv \overline{PQ} + \overline{LM}$, then
$\overline{RS} \equiv \overline{LM}$.\\
%[File: 018.png]
\index{Congruence01@Congruence of distances}For if
$\overline{AB} \equiv \overline{AC} + \overline{CB}$, and $\overline{AC} \equiv \overline{PQ}$, then
$\overline{CB} \equiv \overline{RS} \equiv \overline{LM}$.
\begin{flalign*}
&\text{If}
  &\overline{AB} &\equiv \overline{PQ} + \overline{RS} & \hspace{1.5in}\\
&\text{it will be convenient to write}
  &\overline{PQ} &\equiv (\overline{AB} - \overline{RS}),
\end{flalign*}
and say that $\overline{PQ}$ is the difference of the distances $\overline{AB}$ and $\overline{RS}$.
When we are uncertain as to whether $\overline{AB} > \overline{RS}$ or $\overline{RS} > \overline{AB}$,
we shall write their difference\index{Difference of distances}
$\left|\overline{AB} - \overline{RS}\right|$.

\smallskip\emph{Theorem}~8. If
$\overline{AB} \equiv \overline{PQ} + \overline{LM}$ and
$\overline{AB} \equiv \overline{P'Q'} + \overline{L'M'}$
\begin{flalign*}
&\text{while} \hfill&\overline{PQ} &\equiv \overline{P'Q'}, & \hfill \phantom{\text{while}} \\
&\text{then}  \hfill&\overline{LM} &\equiv \overline{L'M'}.
\end{flalign*}

\smallskip\emph{Theorem}~9. If\index{Greater than}\index{Less than}
$\overline{AB} \equiv \overline{PQ} + \overline{RS}$ and
$\overline{AB} \equiv \overline{P'Q'} + \overline{R'S'}$
\begin{flalign*}
&\text{while} \hfill&\overline{PQ} &> \overline{P'Q'}, & \hfill \phantom{\text{while}}  \\
&\text{then}  \hfill&\overline{RS} &< \overline{R'S'}.
\end{flalign*}\index{Sum03@Sum of distances|)}\vspace{-0.16in}

\emph{Definition}.~The assemblage of all points of a segment\index{Segment} and
its extensions shall be called a \emph{line}\index{Line}.

\smallskip\emph{Definition}.~Two lines having in common a single point are
said to \emph{cut} or \emph{intersect}\index{Intersection01@Intersection of lines} in that point.

Notice that we have not as yet assumed the existence of
two such lines. We shall soon, however, make this assumption
explicitly.


\medskip\textsc{Axiom}~XIV\@. \textbf{Two lines having two common distinct points
are identical.}

The line determined by two points $A$ and $B$ shall be written
$A B$ or $BA$.

\smallskip\emph{Theorem}~10. If $C$ be a point of the extension of $(AB)$
beyond $B$ and $D$ another point of this same extension, then $D$
is a point of $(BC)$ if $\overline{BC} \equiv \overline{BD}$ or
$\overline{BC} > \overline{BD}$; otherwise $C$ is
a point of $(BD)$.

\medskip\textsc{Axiom}~XV\@. \textbf{All points do not lie in one line.}

\medskip\textsc{Axiom}~XVI\@. \textbf{If $B$ be a point of $(CD)$ and $E$ a point of
$(AB)$ where $A$ is not a point of the line $BC$, then the line $DE$
contains a point $F$ of $(AC)$.}

The first of these axioms is clearly nothing but an existence
theorem. The second specifies certain conditions under which
two lines, not given by means of common points, must, nevertheless,
intersect. It is clear that some such assumption is
necessary in order to proceed beyond the geometry of a single
straight line.

%[File: 019.png]
\smallskip\emph{Theorem}~11. If two distinct points $A$ and $B$ be given, there
is an infinite number of distinct points which belong to their
segment\index{Segment}.

This theorem is an immediate consequence of the last two
axioms. It may be interpreted otherwise by saying that there
is no minimum distance, other than the null distance.

\smallskip\emph{Theorem}~12. The manifold\label{mainfold} of all points of a segment is
dense\index{Density of segment}\label{Density}.

\smallskip\emph{Theorem}~13. If $A$, $B$, $C$, $D$, $E$ form the configuration of
points described in Axiom XVI, the point $E$ is a point of $(DF)$.

Suppose that this were not the case. We should either
have $F$ as a point of $(DE)$ or $D$ as a point of $(EF)$. But then,
in the first case, $C$ would be a point of $(DB)$ and in the second
$D$ would be a point of $(BC)$, both of which are inconsistent
with our data.

\smallskip\emph{Definition}.~Points which belong to the same line shall be
said to be \emph{on} it or to be \emph{collinear}\index{Collinearity}. Lines which contain the
same point shall be said to pass through it, or to be \emph{concurrent}\index{Concurrence}.

\smallskip\emph{Theorem}~14. If $A$, $B$, $C$ be three non-collinear points, and $D$
a point within $(AB)$ while $E$ is a point of the extension of
$(BC)$ beyond $C$, then the line $DE$ will contain a point $F$
of $(AC)$.

Take $G$, a point of $(ED)$, different from $E$ and $D$. Then $AG$
will contain a point $L$ of $(BE)$, while $G$ belongs to $(AL)$. If $L$
and $C$ be identical, $G$ will be the point required. If $L$ be
a point of $(CE)$ then $EG$ goes through $F$ within $(AG)$ as
required. If $L$ be within $(BC)$, then $BG$ goes through $H$ of
$(AC)$ and $K$ of $(AE)$, so that, by 13, $G$ and $H$ are points
of $(BK)$. $H$ must then, by 4, either be a point of $(BG)$ or of
$(GK)$. But if $H$ be a point of $(BG)$, $C$ is a point of $(BL)$,
which is untrue. Hence $H$ is a point of $(GK)$, and $(AH)$
contains $F$ of $(EG)$. We see also that it is impossible that $C$
should belong to $(AF)$ or $A$ to $(FC)$. Hence $F$ belongs
to $(AC)$.

\smallskip\emph{Theorem}~15. If $A$, $B$, $C$ be three non-collinear points, no
three points, one within each of their three segments\index{Segment}, are
collinear.

The proof is left to the reader.

\smallskip\emph{Definition}.~If three non-collinear points be given, the locus
of all points of all segments determined by each of these, and
all points of the segment of the other two, shall be called
a \emph{Triangle}\index{Triangle}. The points originally chosen shall be called the
%[File: 020.png]
\emph{vertices}\index{Vertex05@--- of triangle}, their segments the \emph{sides}\index{Sides04@Sides of triangle}.
Any point of the triangle\index{Triangle},
not on one of its sides, shall be said to be
\emph{within}\index{Within a triangle} it. If the
three given points be $A$, $B$, $C$ their triangle shall be written
$\triangle ABC$. Let the reader show that this triangle is completely
determined by all points of all segments having $A$ as one
extremity, while the other belongs to $(BC)$.

It is interesting to notice that XVI, and 13 and 14, may be
summed up as follows\footnote{Some
writers, as Pasch\index{Pasch}\label{Pasch}, \emph{Neuere Geometrie}, Leipzig, 1882, p.~21, give Axiom
XVI in this form. I have followed Veblen\index{Veblen}, loc.\ cit., p.~351, in weakening the
axiom to the form given.}:---

\smallskip\emph{Theorem}~16. If a line contain a point of one side of a
triangle and one of either extension of a second side, it will
contain a point of the third side.

\mysect{Simple descriptive properties of plane and space}

\smallskip\emph{Definition}.~The assemblage of all points of all lines determined
by the vertices of a triangle and all points of the
opposite sides shall be called a \emph{plane}.

It should be noticed that in defining a plane in this manner,
the vertices of the triangle play a special rôle. It is our next
task to show that this specialization of function is only
apparent, and that any other three non-collinear points of the
plane might equally well have been chosen to define it.%
\footnote{The treatment of the plane and space which constitute the rest of this
chapter are taken largely from Schur, loc.\ cit. He in turn confesses his
indebtedness to Peano.}
%endfootnote

\smallskip\emph{Theorem}~17. If a plane be determined by the vertices of a
triangle, the following points lie therein:---

(\emph{a}) All points of every line determined by a vertex, and
a point of the line of the other two vertices.

(\emph{b}) All points of every line which contains a point of each
of two sides of the triangle.

(\emph{c}) All points of every line containing a point of one side
of the triangle and a point of the line of another side.

(\emph{d}) All points of every line which contains a point of the
line of each of two sides.

The proof will come at once from 16, and from the consideration
that if we know two points of a line, every other
point thereof is either a point of their segment, or of one of its
extensions. The plane determined by three points as $A$, $B$, $C$
shall be written the plane $ABC$. We are thus led to the
following theorem.

\smallskip\emph{Theorem}~18. The plane determined by three vertices of a
triangle is identical with that determined by two of their
number and any other point of the line of either of the
remaining sides.

%[File: 021.png]
\smallskip\emph{Theorem}~19. Any one of the three points determining a plane\index{Plane}
may be replaced by any other point of the plane, not collinear
with the two remaining determining points.

\smallskip\emph{Theorem}~20. A plane may be determined by any three of
its points which are not collinear.

\smallskip\emph{Theorem}~21. Two planes having three non-collinear points
in common are identical.

\smallskip\emph{Theorem}~22. If two points of a line lie in a plane, all points
thereof lie in that plane.

\medskip\textsc{Axiom}~XVII\@. \textbf{All points do not lie in one plane.}

\smallskip\emph{Definition}.~Points or lines which lie in the same plane shall
be called \emph{coplanar}. Planes which include the same line shall
be called \emph{coaxal}\index{Coaxality}. Planes, like lines, which include the same
point, shall be called \emph{concurrent}.

\smallskip\emph{Definition}.~If four non-coplanar points be given, the assemblage
of all points of all segments having for one extremity
one of these points, and for the other, a point of the triangle
of the other three, shall be called a \emph{tetrahedron}\index{Tetrahedron}. The four
given points shall be called its \emph{vertices},\index{Vertex04@--- of tetrahedron} their six segments its
\emph{edges},\index{Edge of tetrahedron} and the four triangles its
\emph{faces}\index{Face of tetrahedron}. Edges having no
common vertex shall be called \emph{opposite}\index{Opposite01@Opposite edges of tetrahedon}.
Let the reader show
that, as a matter of fact, the tetrahedron will be determined
completely by means of segments, all having a common
extremity at one vertex, while the other extremity is in the
face of the other three vertices. A vertex may also be said
to be opposite to a face, if it do not lie in that face.

\smallskip\emph{Definition}.~The assemblage of all points of all lines which
contain either a vertex of a tetrahedron, and a point of the
opposite face, or two points of two opposite edges, shall be
called a \emph{space}\index{Space}.

It will be seen that a space, as so defined, is made up of
fifteen regions, described as follows:---

(\emph{a})~The tetrahedron itself.

(\emph{b})~Four regions composed of the extensions beyond each
vertex of segments having one extremity there, and the other
extremity in the opposite face.

(\emph{c})~Four regions composed of the other extensions of the
segments mentioned in (\emph{b}).

(\emph{d})~Six regions composed of the extensions of segments
whose extremities are points of opposite edges.

\smallskip\emph{Theorem}~23. All points of each of the following figures
%[File: 022.png]
will lie in the space defined by the vertices of a given
tetrahedron.

(\emph{a})~A plane containing an edge, and a point of the opposite
edge.

(\emph{b})~A line containing a vertex, and a point of the plane
of the opposite face.


(\emph{c})~A line containing a point of one edge, and a point of the
line of the opposite edge.

(\emph{d})~A line containing a point of the line of each of two
opposite edges.

(\emph{e})~A line containing a point of one edge, and a point of the
plane of a face not containing that edge.

(\emph{f})~A line containing a point of the line of one edge, and
a point of the plane of a face not containing that edge.

The proof will come directly if we take the steps in the
order indicated, and hold fast to 16, and the definitions of
line, plane, and space\index{Plane}\index{Space}.

\smallskip\emph{Theorem} 24. In determining a space, any vertex of a tetrahedron\index{Tetrahedron}
may be replaced by any other point, not a vertex, on
the line of an edge through the given vertex.

\smallskip\emph{Theorem} 25. In determining a space, any vertex of a tetrahedron
may be replaced by any point of that space, not
coplanar with the other three vertices.

\smallskip\emph{Theorem} 26. A space may be determined by any four of its
points which are not coplanar.

\smallskip\emph{Theorem} 27. Two spaces which have four non-coplanar
points in common are identical.

\smallskip\emph{Theorem} 28. A space contains wholly every line whereof it
contains two distinct points.

\smallskip\emph{Theorem} 29. A space contains wholly every plane whereof
it contains three non-collinear points.

\smallskip\textsc{Practical limitation}. Points belonging to different spaces
shall not be considered simultaneously in the present work.%
\footnote{This means, of course, that we shall not consider geometry of more than
three dimensions. It would not, however, strictly speaking, be accurate to
say that we consider the geometry of a single space only, for we shall make
various mutually contradictory hypotheses about space.}
%endfootnote

Suppose that we have a plane containing the point $E$ of the
segment $(AB)$ but no point of the segment $(BC)$. Take $F$ and
$G$ two other points of the plane, not collinear with $E$, and
construct the including space by means of the tetrahedron
whose vertices are $A$, $B$, $F$, $G$. As $C$ lies in this space, it
must lie in one of the fifteen regions individualized by the
%[File: 023.png]
tetrahedron\index{Space}; or, more specifically, it must lie in a plane containing
one edge, and a point of the opposite edge. Every
such plane will contain a line of the plane $EFG$, as may be
immediately proved, and 16 will show that in every case this
plane must contain either a point of $(AC)$ or one of $(BC)$.

\smallskip\emph{Theorem}~30. If a plane contain a point of one side of a
triangle, but no point of a second side, it must contain a point
of the third.

\smallskip\emph{Theorem}~31. If a line in the plane of a triangle contain
a point of one side of the triangle and no point of a second
side, it must contain a point of the third side.

\smallskip\emph{Definition}.~If a point within the segment of two given
points be in a given plane, those points shall be said to be
on \emph{opposite sides}\index{Opposite04@--- sides of plane}
of the plane; otherwise, they shall be said to
be on the \emph{same side} of the plane. Similarly, we may define
opposite sides of a line.

\smallskip\emph{Theorem}~32. If two points be on the same side of a plane,
a point opposite to one is on the same side as the other; and
if two points be on the same side, a point opposite to one is
opposite to both.

The proof comes at once from 30.

\smallskip\emph{Theorem}~33. If two planes have a common point they have
a common line\index{Intersection02@--- of planes}.

Let $P$ be the common point. In the first plane take a line
through $P$. If this be also a line of the second plane, the
theorem is proved. If not, we may take two points of this
line on opposite sides of the second plane. Now any other
point of the first plane, not collinear with the three already
chosen, will be opposite to one of the last two points, and thus
determine another line of the first plane which intersects the
second one. We hereby reach a second point common to
the two planes, and the line connecting the two is common
to both.

It is immediately evident that all points common to the
two planes lie in this line\index{Plane}.

%[File: 024.png]
%CHAPTER II
\mychap{II}{CONGRUENT TRANSFORMATIONS}
\mysect{Axiom of continuity}

In Chapter I we laid the foundation for the present work.
We made a number of explicit assumptions, and, building
thereon, we constructed that three-dimensional type of
space wherewith we shall, from now on, be occupied. An
essential point in our system of axioms is this. We have
taken as a fundamental indefinable, distance, and this, being
subject to the categories greater and less, is a magnitude.
In other words, we have laid the basis for a metrical geometry.
Yet, the principal use that we have made of these metrical
assumptions, has been to prove a number of descriptive
theorems. In order to complete our metrical system properly
we shall need two more assumptions, the one to give us the
concept of continuity\index{Continuity01@Continuity, axiom of}, the other to establish the possibility of
congruent transformations.

\mysect{Division of distances}
\medskip\textsc{Axiom}~XVIII\@. \textbf{If all points of a segment\index{Segment}\index{Segment} $(AB)$ be
divided\index{Division of segment} into two such classes that no point of the first
shall be at a greater distance from $A$ than is any point
of the second; then there exists such a point $C$ of the
segment, that no point of the first class is within $(CB)$ and
none of the second within $(AC)$.}

It is manifest that $A$ will belong to the first class, and $B$ to
the second, while $C$ may be ascribed to either. It is the
presence of this point common to both, that makes it
advisable to describe the two classes in a negative, rather
than in a positive manner.

\smallskip\emph{Theorem}~1. If $\overline{AB}$ and $\overline{PQ}$ be any two distances whereof
the second is not null, there will exist in the segment $(AB)$
a finite or null number $n$ of points $P_k$ possessing the following
properties:
\[
\overline{PQ} \equiv \overline{AP_1} \equiv \overline{P_kP_{k+1}};\quad
 \overline{AP_{k+1}} \equiv \overline{AP_k} + \overline{P_kP_{k+1}};\quad
 \overline{P_nB}<\overline{PQ}.
\]

Suppose, firstly, that $\overline{AB} < \overline{PQ}$ then, clearly, $n = 0$. If,
however, $\overline{AB} \equiv \overline{PQ}$ then $n = 1$ and $P_1$ is identical with $B$.
There remains the third case where $\overline{AB} > \overline{PQ}$. Imagine the
theorem to be untrue. We shall arrive at a contradiction as
follows. Let us divide all points of the segment into two
%[File: 025.png]
classes. A point $H$ shall belong to the first class if we may
find such a positive integer $n$ that
\[
\overline{P_nH} < \overline{PQ},\quad
\overline{AH} \equiv \overline{AP_n} + \overline{P_nH},
\]
the succession of points $P_k$ being taken as above. All other
points of the segment shall be assigned to the second class. It
is clear that neither class will be empty. If $H$ be a point
of the first class, and $K$ one of the second, we cannot have
$K$ within $(AH)$, for then we should find
$\overline{AK} \equiv \overline{AP_n} + \overline{P_nK}$;
$\overline{P_nK} < \overline{PQ}$
contrary to the rule of dichotomy. We have
therefore a cut of the type demanded by Axiom XVIII, and
a point of division $C$. Let $D$ be such a point of $(AC)$ that
$\overline{DC} < \overline{PQ}$.
Then, as we may find $n$, so large that $\overline{P_nD} < \overline{PQ}$,
we shall either have $\overline{P_nC} < \overline{PQ}$
or else we shall be able to
insert a point $P_{n+1}$ within $(AC)$ making
$\overline{P_{n+1}C} < \overline{PQ}$. If,
then, in the first case we construct $P_{n+1}$, or in the second
$P_{n+2}$, it will be a point within $(CB)$, as
$\overline{P_nB}>\overline{PQ}$, and this
involves a contradiction, for it would require $P_{n+1}$ or $P_{n+2}$
to belong to both classes at once. The theorem is thus
proved.

It will be seen that this theorem is merely a variation of
the axiom of Archimedes\index{Archimedes},%
\footnote{A good deal of attention has been given in recent years to this axiom.
For an account of the connexion of Archimedes' axiom with the continuity
of the scale, see Stolz\index{Stolz}, `Ueber das Axiom des Archimedes,' \emph{Mathematische
Annalen}, vol.~xxxix, 1891. Halsted\index{Halsted}, \emph{Rational Geometry} (New York, 1904), has
shown that a good deal of the subject of elementary geometry can be built
up without the Archimedian assumption, which accounts for the otherwise
somewhat obscure title of his book. Hilbert\index{Hilbert}, loc.\ cit., Ch.~IV, was
the first writer to set up the theory of area independent of continuity,
and Vahlen\index{Vahlen} has shown, loc.\ cit., pp.~297--8, that volumes may be similarly
handled. These questions are of primary importance in any work that deals
principally with the significance and independence of the axioms. In our
present work we shall leave non-archimedian or discontinuous geometries
entirely aside, and that for the reason that their analytic treatment involves
either a mutilation of the number scale, or an adjunction of transfinite
elements thereto. We shall, in fact, make use of our axiom of
continuity\index{Continuity01@Continuity, axiom of}
XVIII wherever, and whenever, it is convenient to do so.}
%endfootnote
which says, in non-technical language,
that if a sufficient number of equal lengths be laid off on a
line, any point of that line may be surpassed. We are not
able to state the principle in exactly this form, however, for
we cannot be sure that our space shall include points of the
type $P_n$ in the extension of $(AB)$ beyond $B$.

\smallskip\emph{Theorem}~2. In any segment there is a single point whose
distances from the extremities are congruent.

The proof is left to the reader.

The point so found shall be called the \emph{middle point}\index{Middle point of segment} of the
%[File: 026.png]
segment. It will follow at once that if $k$ be any positive
integer, we may find a set of points $P_1P_2 \ldots P_{2^k-1}$ of the
segment $(AB)$ possessing the following properties
\[
\overline{AP_1} \equiv \overline{P_jP_{j+1}} \equiv \overline{P_{2^k-1}B}; \quad
\overline{AP_{j+1}} \equiv \overline{AP_j} + \overline{P_jP_{j+1}}.
\]

We may express the relation of any one of these congruent
distances to $\overline{AB}$ by writing $\overline{P_jP_{j+1}} \equiv \dfrac{1}{2^k}\overline{AB}$.

\smallskip\emph{Theorem}~3. If a not null distance $\overline{AB}$ be given and a
positive integer $m$, it is possible to find $m$ distinct points of
the segment $(AB)$ possessing the properties
\[
\overline{AP_1} \equiv \overline{P_jP_{j+1}}; \quad
\overline{AP_{j+1}} \equiv \overline{AP_j} + \overline{P_jP_{j+1}}.
\]

It is merely necessary to take $k$ so that $2^k > m + 1$ and
find $\overline{AP_1} \equiv \dfrac{1}{2^k}\overline{AB}$.

\smallskip\emph{Theorem}~4. When any segment $(AB)$ and a positive integer
$n$ are given, there exist $n-1$ points $D_1D_2\ldots D_{n-1}$ of the
segment $(AB)$ such that
\[
\overline{AD_1} \equiv \overline{D_jD_{j+1}} \equiv \overline{D_{n-1}B}; \quad
\overline{AD_{j+1}} \equiv \overline{AD_j} + \overline{D_jD_{j+1}}.
\]

If the distance $\overline{AB}$ be null, the theorem is trivial. Otherwise,
suppose it to be untrue. Let us divide\index{Division of segment}\index{Segment} the points of
$(AB)$ into two classes according to the following scheme.
A point $P_1$ shall belong to the first class if we may construct
$n$ congruent distances according to the method already
illustrated, reaching such a point $P_n$ of $(AB)$ that $\overline{P_nB} > \overline{AP_1}$;
all other points of $(AB)$ shall be assigned to the second class.
$B$ will clearly be a point of the second class, but every point
of $(AB)$ at a lesser distance from $A$ than a point of the first
class, will itself be a point of the first class. We have thus
once more a cut as demanded by Axiom XVIII, and a point
of division $D_1$; and this point is different from $A$.

Let us next assume that the number of successive distances
congruent to $\overline{AD_1}$ which, by 1, may be marked in $(AB)$, is $k$,
and let $D_k$ be the last extremity of the resulting segments,
so that $\overline{D_kB} < \overline{AD_1}$. Let $D_{k-1}$ be the other extremity of this
last segment. Suppose, first, that $k < n$. Let $\overline{PQ}$ be such
a distance that $\overline{AD_1} > \overline{PQ} > \overline{D_kB}$. Let $P_1$ be such a point of
$(AD_1)$ that $\overline{AP_1} > \overline{PQ}$, $k\overline{P_1D_1} < \overline{PQ} - \overline{D_kB}$. Then, by marking
$k$ successive distances by our previous device, we reach
%[File: 027.png]
$P_{k}$ such a point of $(AD_{k})$ that
\begin{equation*}
\overline{P_{k}B}<\overline{D_{k}B}+
(\overline{PQ}-\overline{D_{k}B})<\overline{PQ}<\overline{AP_{1}}.
\end{equation*}
But this is a contradiction, for $k$ is at most equal to $n-1$,
and as $P_{1}$ is a point of the first class, there should be at least
one more point of division\index{Division of segment}\index{Segment} $P_{k+1}$. Hence $k \geqq n$. But $k>n$
leads to a similar contradiction. For we might then find $Q_{I}$
of the second class so that $(k-2)\overline{D_{1}Q_{1}}<\frac{1}{2}\overline{AD_{1}}$. Then mark
$k-2$ successive congruent distances, reaching $Q_{k-2}$ such a
point of $(AD_{k-1})$ that $\overline{Q_{k-2}D_{k-1}}>\frac{1}{2}\overline{AD_{1}}$. Hence,
\begin{equation*}
\overline{Q_{k-2}D_{k}} > \tfrac{1}{2}\overline{AD_{1}}+\overline{AD_{1}} > \overline{AQ_{1}},
\end{equation*}
and we may find a $(k-1)$th point $Q_{k-1}$. But $k-1 \geqq n$ and
this leads us to a contradiction with the assumption that
$Q_{1}$ should be a point of the second class; i.e.\ $k=n$. Lastly,
we shall find that $D_{k}$ and $B$ are identical. For otherwise
we might find $Q_{1}$ of the second class so that $n\overline{D_{1}Q_{1}}<\overline{D_{n}B}$
and marking $n$ successive congruent distances reach $Q_{n}$ within
$(D_{n}B)$, impossible when $Q_{1}$ belongs to class two. Our theorem
is thus entirely proved, and $D_{1}$ is the point sought.

It will be convenient to write $\overline{AD_{1}} \equiv \dfrac{1}{n}\overline{AB}$.

\smallskip\emph{Theorem}~5. If $\overline{AB}$ and $\overline{PQ}$ be given, whereof the latter is
not null, we may find $n$ so great that $\dfrac{1}{n}\overline{AB}<\overline{PQ}$.

The proof is left to the reader.

\mysect{Measure of distance}\index{Measure01@Measure of distance}
We are at last in a position to introduce the concept of
number into our scale of distance magnitudes. Let $\overline{AB}$ and $\overline{PQ}$
be two distances, whereof the latter is not null. It may be
possible to find such a distance
$\overline{RS}$ that $q\overline{RS}\equiv\overline{PQ}$; $p\overline{RS}\equiv\overline{AB}$.
In this case the number $\dfrac{p}{q}$ shall be called the \emph{numerical
measure} of $\overline{AB}$ in terms of $\overline{PQ}$, or, more simply the \emph{measure}.
It is clear that this measure may be equally well written
$\dfrac{p}{q}$ or $\dfrac{np}{nq}$. There may, however, be no such distance as $\overline{RS}$.
Then, whatever positive integer $q$ may be, we may find $\overline{LM}$ so
that $q\overline{LM}\equiv\overline{PQ}$, and $p$ so that
$\overline{LM}>(\overline{AB}-p\overline{LM})$. By this
process we have defined a cut in our number system of such
a nature that $\dfrac{p}{q}$ and $\dfrac{p+1}{q}$ appear in the lower and upper
%[File: 028.png]
divisions respectively. If $\dfrac{p}{q}$ be a number of the lower, and
$\dfrac{p'+1}{q'}$ one of the upper division, we shall see at once by
reducing to a lowest common denominator that $\dfrac{p}{q}<\dfrac{p'+1}{q'}$.
Every rational number will fall into the one or the other
division. Lastly there is no largest number in the lower
division nor smallest in the upper. For suppose that $\dfrac{p}{q}$ is the
largest number of the lower division. Then if
$\overline{LM} > (\overline{AB}-p\overline{LM})$,
we may find $n$ so large that
$\dfrac{1}{n}\overline{LM} < (\overline{AB}-p\overline{LM})$. Let us
put $\overline{L_1M_1} \equiv \dfrac{1}{n}\overline{LM}$. At the same time as
$\overline{PQ} \equiv nq\overline{L_1M_1}$ we
may, by 1, find $k$ so large that
$\overline{L_1M_1} > (\overline{AB}-(np + k)\overline{L_1M_1})$.
Under these circumstances $\dfrac{np+k}{nq}$ is a number of the lower
division, yet larger than $\dfrac{p}{q}$. In the same way we may prove
that there is no smallest number in the upper. We have
therefore defined a unique irrational number, and this may be
taken as the measure of $\overline{AB}$ in terms of $\overline{PQ}$.

Suppose, conversely, that $\dfrac{p}{q}$ is any rational fraction, and
there exists such a distance $\overline{AB'}$ that
$q\overline{AB'}> p\overline{PQ}$. Then in
$(AB')$ we may find such a point $B$ that
$\overline{AB} \equiv \dfrac{p}{q}\overline{PQ}$, i.e.\ there
will exist a distance having the measure $\dfrac{p}{q}$ in terms of
$\overline{PQ}$. Next
let $r$ be any irrational number, and let there be such a number
$\dfrac{p+1}{q}$ in the corresponding upper division of the rational
number system that a distance
$q\overline{AB'}> ((p + 1)\overline{PQ})$ may be
found. Then the cut in the number system will give us a cut
in the segment $(AB')$, as demanded by XVIII, and a point of
division\index{Division of segment} $B$. The numerical measure of $\overline{AB}$ in terms of
$\overline{PQ}$
will clearly be $r$.

\smallskip\emph{Theorem}~6. If two distances, whereof the second is not null,
be given, there exists a unique numerical measure for the first
in terms of the second, and if a distance be given, and there
exist a distance having a given numerical measure in terms
%[File: 029.png]
thereof, there will exist a distance having any chosen smaller
numerical measure.

\smallskip\emph{Theorem}~7. If two distances be congruent, their measures
in terms of any third distance are equal.

It will occasionally be convenient to write the measure of $\overline{PQ}$
in the form $\Measure\overline{PQ}$.

\smallskip\emph{Theorem}~8. If $r > n$ and if distances $r\overline{PQ}$ and $n\overline{PQ}$ exist,
then $r\overline{PQ} > n\overline{PQ}$.

When $m$ and $n$ are both rational, this comes immediately by
reducing to a common denominator. When one or both of
these numbers is irrational, we may find a number in the
lower class of the larger which is larger than one in the upper
class of the smaller, and then apply I, 3.

\smallskip\emph{Theorem}~9. If $\overline{AB} > \overline{CD}$, the measure of $\overline{AB}$ in terms of
any chosen not null distance is greater than that of $\overline{CD}$ in
terms of the same distance.

This comes at once by reduction ad absurdum.

It will hereafter be convenient to apply the categories,
congruent greater and less, to segments\index{Congruence02@--- of segments}, when these apply
respectively to the distances of their extremities. We may
similarly speak of the measure\index{Measure01@Measure of distance} of a segment in terms of
another one. Let us notice that in combining segments or
distances, the associative, commutative, and distributive laws
of multiplication hold good; e.g.
\begin{equation*}
r \cdot n\overline{PQ} \equiv n \cdot r\overline{PQ} \equiv rn\overline{PQ}, \quad
  n(\overline{AB} + \overline{CD}) \equiv n\overline{AB} + n\overline{CD}.
\end{equation*}
Notice, in particular, that the measure of a sum is the sum of
the measures.

\index{Half-line|(}
\smallskip\emph{Definition}.~The assemblage of all points of a segment\index{Segment}, or of
all possible extensions beyond one extremity, shall be called
a \emph{half-line}\index{Bound of half-line}. The other extremity of the segment shall be
called the \emph{bound} of the half-line. A half-line bounded by $A$
and including a point $B$ shall be written $|AB$. Notice that
every point of a line is the bound of two half-lines thereof.

\smallskip\emph{Definition}.~A relation between two sets of points $(P)$ and
$(Q)$ such that there is a one to one correspondence of distinct
points, and the distances of corresponding pairs of points are
in every case congruent\index{Congruence01@Congruence of distances}, while the sum of two distances is
carried into a congruent sum, is called a \emph{congruent transformation}\index{Congruent figures}.
Notice that, by V, the assemblage of all congruent
transformations form a group. If, further, a congruent
transformation be possible $(P)$ to $(Q)$, and there be two sets
of points $(P')$ and $(Q')$ such that a congruent transformation
%[File: 030.png]
is possible from the set $(P)(P')$ to the set $(Q)(Q')$ then we
shall say that the congruent transformation\index{Congruent transformations}\index{Transformations, congruent} from $(P)$ to $(Q)$
has been \emph{enlarged\index{Enlargement of congruent transformation} to include the sets $(P')$ and $(Q')$}.

It is evident that a congruent transformation will carry
points of a segment, line, or half-line, into points of a segment,
line, or half-line respectively. It will also carry coplanar
points into coplanar points, and be, in fact, a collineation\index{Collineations},
or linear transformation as defined geometrically. In the
eighteenth chapter of the present work we shall see how the
properties of congruent figures may be reached by defining
congruent transformations as a certain six-parameter collineation
group.

\mysect{Axiom of congruent transformations}
\textsc{Axiom XIX.} \textbf{If a congruent transformation exist between
two sets of points, to each half-line bounded by a point
of one set may be made to correspond a half-line bounded
by the corresponding point of the other set, in such wise that
the transformation may be enlarged to include all points
of these two half-lines at congruent distances from their
respective bounds.}%
\footnote{The idea of enlarging a congruent transformation to include additional
points is due to Pasch\index{Pasch}, loc.\ cit. He merely assumes that if any point be
adjoined to the one set, a corresponding point may be adjoined to the other.
We have to make a much clumsier assumption, and proceed more circumspectly,
for fear of passing out of our limited region.}
%endfootnote

\smallskip\emph{Theorem} 10.~If a congruent transformation carry two chosen
points into two other chosen points, it may be enlarged to
include all points of their segments.

\smallskip\emph{Theorem} 11.~If a congruent transformation carry three
non-collinear points into three other such points, it may be
enlarged to include all points of their respective triangles.

\smallskip\emph{Theorem} 12.~If a congruent transformation carry four non-coplanar
points into four other such points, it may be enlarged
to include all points of their respective tetrahedra.

\smallskip\emph{Definition}.~Two figures which correspond in a congruent
transformation shall be said to be \emph{congruent}.

We shall assume hereafter that every congruent transformation
with which we deal has been enlarged to the greatest
possible extent. Under these circum\-stances:---

\smallskip\emph{Theorem} 13.~If two distinct points be invariant under a
congruent transformation, the same is true of all points of
their line.

\smallskip\emph{Theorem} 14.~If three non-collinear points be invariant
%[File: 031.png]
under a congruent transformation, the same is true of all
points of their plane.

\smallskip\emph{Theorem} 15.~If four non-coplanar points be invariant under
a congruent transformation the same is true of all points
of space.

\smallskip\emph{Definition}.~The assemblage of all points of a plane on one
side of a given line, or on that given line, shall be called
a \emph{half-plane}\index{Half-plane}. The given line shall be called the \emph{bound}\index{Bound of half-plane} of
the half-plane. Each line in a plane is thus the bound of two
half-planes thereof.

\mysect{Definition of angles, their properties}\index{Angle01@Angle, interior and exterior}
Suppose that we have two non-collinear half-lines with
a common bound $A$. Let $B$ and $C$ be two other points of
one half-line\label{onehalf}, and $B'$ and $C'$ two points of the other. Then
by Ch.~I,~16, a half-line bounded by $A$ which contains
a point of $(BB')$ will also contain a point of $(CC')$, and vice
versa. We may thus divide all half-lines of this plane,
bounded by this point, into two classes. The assemblage
of all half-lines which contain points of segments whose
extremities lie severally on the two given half-lines shall
be called the \emph{interior angle} of, or between, the given
half-lines. The half-lines themselves shall be called the \emph{sides}\index{Sides01@Sides of angle}
of the angle. If the half-lines be $|AB$, $|AC$, their interior
angle may be indicated $\measuredangle BAC$ or $\measuredangle
CAB$. The point $A$
shall be called the \emph{vertex}\index{Vertex01@Vertex of angle} of the angle.

\smallskip\emph{Definition}.~The assemblage of all half-lines coplanar with
two given non-collinear half-lines, and bounded by the
common bound of the latter, but not belonging to their
interior angle, shall be called the \emph{exterior angle} of the two
half-lines. The definitions for sides and vertex shall be as
before. If no mention be made of the words \emph{interior} or
\emph{exterior} we shall understand by the word \emph{angle}, \emph{interior
angle}. Notice that, by our definitions, the sides are a part of
the interior, but not of the exterior angle. Let the reader also
show that if a half-line of an interior angle be taken, the
other half-line, collinear therewith, and having the same bound
belongs to the exterior angle.

\smallskip\emph{Definition}.~The assemblage of all half-lines identical with
two identical half-lines, shall be called their \emph{interior angle}.
The given bound shall be the vertex, and the given half-lines
the sides of the angle. This angle shall also be called a \emph{null
angle}\index{Angle02@--- null}\index{Null01@Null angle}.
The assemblage of all half-lines with this bound, and
lying in any chosen plane through the identical half-lines,
shall be called their \emph{exterior angle} in this plane. The
definition
of sides and vertex shall be as before.
%[File: 032.png]

\smallskip\emph{Definition}.~Two collinear, but not identical, half-lines of
common bound shall be said to be \emph{opposite}.

\smallskip\emph{Definition}.~The assemblage of all half-lines having as bound
the common bound of two opposite half-lines, and lying in
any half-plane bounded by the line of the latter, shall be
called an \emph{angle} of the two half-lines in that plane. The
definitions of sides\index{Sides01@Sides of angle} and vertex\index{Vertex01@Vertex of angle} shall be as usual. We notice
that two opposite half-lines determine two angles in every
plane through their line.

We have thus defined the angles of any two half-lines of
common bound. The exterior angle of any two such half-lines,
when there is one, shall be called a \emph{re-entrant}\index{Angle05@--- re-entrant} angle.
Any angle determined by two opposite half-lines\index{Opposite02@--- half-lines} shall be
called a \emph{straight}\index{Angle04@--- straight} angle. As, by definition, two half-lines form
an angle when, and only when, they have a common bound,
we shall in future cease to mention this fact. Two angles
will be congruent\index{Congruence03@--- of angles}, by our definition of congruent figures,
if there exist a congruent transformation of the sides of one
into the sides of the other, in so far as corresponding distances
actually exist on the corresponding half-lines. Every half-line
of the interior or exterior angle will similarly be carried
into a corresponding half-line, or as much thereof as actually
exists and contains corresponding distances.

\smallskip\emph{Definition}.~The angles of a triangle\index{Angles01@Angles of a triangle}
shall be those non-re-entrant
angles whose vertices are the vertices of the triangle,
and whose sides include the sides of the triangle\index{Sides04@Sides of triangle}.

\smallskip\emph{Definition}.~The angle between a half-line including one
side of a triangle, and bounded at a chosen vertex, and the
opposite of the other half-line which goes to make the angle
of the triangle at that vertex, shall be called an
\emph{exterior angle}\index{Angles02@--- exterior of a triangle}
of the triangle. Notice that there are six of these, and that
they are not to be confused with the exterior angles of their
respective sides.

\mysect{Comparison of triangles}\index{Triangle|(}
\smallskip\emph{Theorem} 16.~If two triangles be so related that the sides of
one are congruent to those of the other, the same holds for the
angles.

This is an immediate result of 11.

The meanings of the words \emph{opposite} and \emph{adjacent} as applied
to sides and angles of a triangle are immediately evident, and
need not be defined. There can also be no ambiguity in
speaking of sides \emph{including} an angle.

\smallskip\emph{Theorem} 17.~Two triangles are congruent\index{Triangles, congruent}\index{Congruence04@--- of triangles}
if two sides and
the included angle of one be respectively congruent to two
sides and the included angle of the other.

%[File: 033.png]
The truth of this is at once evident when we recall the
definition of congruent angles, and 12\index{Congruence04@--- of triangles}.

\smallskip\emph{Theorem}~18. If two sides of a triangle\index{Sides04@Sides of triangle} be congruent, the
opposite angles are congruent.

Such a triangle shall, naturally, be called \emph{isosceles}\index{Isosceles02@--- triangle}.

\smallskip\emph{Theorem}~19. If three half-lines lie in the same half-plane
and have their common bound on the bound of this half-plane;
then one belongs to the interior angle of the other
two.

Let the half-lines be $|AB$, $|AC$, $|AD$. Connect $B$ with $H$
and $K$, points of the opposite half-lines bounding this half-plane.
If $|AC$, $|AD$ contain points of the same two sides
of the triangle $BHK$ the theorem is at once evident; if
one contain a point of $(BH)$ and the other a point of $(BK)$,
then $B$ belongs to $\measuredangle CAD$.

\smallskip\emph{Theorem}~20. If $|AB$ be a half-line of the interior $\measuredangle CAD$,
then $|AC$ does not belong to the interior $\measuredangle BAD$.

\smallskip\emph{Definition}.~Two non-re-entrant angles of the same plane
with a common side, but no other common half-lines, shall be
said to be \emph{adjacent}. The angle bounded by their remaining
sides, which includes the common side, shall be called their
\emph{sum}\index{Sum01@Sum of angles}. It is clear that this is, in fact, their logical sum,
containing all common points.

\smallskip\emph{Definition}.~An angle shall be said to be congruent to the
sum of two non-re-entrant angles, when it is congruent to the
sum of two adjacent angles, respectively congruent to them.

\smallskip\emph{Definition}.~Two angles congruent to two adjacent angles
whose sum is a straight angle shall be said to be \emph{supplementary}\index{Supplementary angles}.
Each shall be called the \emph{supplement} of the other.

\smallskip\emph{Definition}.~An angle which is congruent to its supplement
shall be called a \emph{right angle}\index{Angle03@--- right}\index{Right 01@Right angle}.

\smallskip\emph{Definition}.~A triangle, one of whose angles is a right angle,
shall be called a \emph{right triangle}\index{Right 02@--- triangle}.

\smallskip\emph{Definition}.~The interior angle formed by two half-lines,
opposite to the half-lines which are the sides of a given
interior angle, shall be called the \emph{vertical}\index{Vertical angles} of that angle. The
vertical of a straight angle will be the other half-plane,
coplanar therewith, and having the same bound.

\smallskip\emph{Theorem}~21. If two points be at congruent distances from
two points coplanar with them, all points of the line of the
first two are at congruent distances from the latter two.

%[File: 034.png]
For we may find a congruent transformation keeping the
former points invariant, while the latter are interchanged.

\smallskip\emph{Theorem}~22. If $|AA_1{}'$ be a half-line of the interior
$\measuredangle BAA_1$, then we cannot have a congruent transformation
keeping $|AB$ invariant and carrying $|AA_1$ into $|AA_1{}'$.

We may suppose that $A_1$ and $A_1{}'$ are at congruent distances
from $A$. Let $H$ be the point of the segment $(A_1A_1{}')$ equidistant
from $A_1$ and $A_1{}'$. We may find a congruent transformation
carrying $AA_1HA_1{}'$ into $AA_1{}'HA_1$. Let this take
the half-line $|AB$ into $|AC$ (in the same plane). Then if
$|\overline{AA_1}$ and $|\overline{AA_1{}'}$ be taken sufficiently
small, $A_1A_1{}'$ will
meet $AB$ or $AC$ as we see by I. 16. This will involve a
contradiction, however, for if $D$ be the intersection, it is easy
to see that we shall have simultaneously $\overline{DA_1} \equiv
\overline{DA_1{}'}$ and
$\overline{DA_1} > \overline{DA_1{}'}$ or $\overline{DA_1} <
\overline{DA_1{}'}$ for $D$ is unaltered by the congruent
transformation, while $A_1$ goes into $A_1{}'$.

There is one case where this reasoning has to be modified,
namely, when $|AC$ and $|AB$ are opposite half-lines, for here
I. 16 does not hold. Let us notice, however, that we may
enlarge our transformation to include the $\measuredangle BAA_1$ and
$\measuredangle BAA_1{}'$ respectively. If $|AB_1$ and $|AC_1$ be two
half-lines
of the first angle, $|AC_1$ being in the interior angle of
$\measuredangle BAB_1$,
to them will correspond $|AB_1{}'$ and $|AC_1{}'$, the latter being in
the interior angle of $\measuredangle BAB_1{}'$, while by definition,
corresponding
half-lines always determine congruent angles with
$|AB$. If, then, we choose any half-line $|AL$ of the interior
$\measuredangle BAA_1{}'$, it may be shown that we may find two
corresponding half-lines $|AL_1$ $|AL_1{}'$ so situated that $|AL_1$
belongs to the interior $\measuredangle BAL_1{}'$\label{bBAL} and $\measuredangle
L_1{}'AL$ is congruent\index{Congruence03@--- of angles}
to $\measuredangle LAL_1$. The proof is tedious, and depends on
showing that as a result of our Axiom XVIII, if in any
segment the points be paired in such a way that the
extremities correspond, and the greater of two distances from
an extremity correspond to the greater of the two corresponding
distances from the other extremity, then there is one
self-corresponding point.%
\footnote{Cf.\ Enriques\index{Enriques}, \emph{Geometria proiettiva}, Bologna, 1898, p.~80.}
%endfootnote
These corresponding half-lines
being found, we may apply the first part of our proof without
fear of mishap.
\index{Half-line|)}

\smallskip\emph{Theorem}~23. If $|AC$ be a half-line of the interior
$\measuredangle BAD$,
it is impossible to have $\measuredangle BAC$ and $\measuredangle BAD$
mutually
congruent.

%[File: 035.png]
\smallskip\emph{Theorem}~24. An angle is congruent to its vertical\index{Vertical angles}.

We have merely to look at the congruent transformation
interchanging a side of one with a side of the other.

We see as a result of 24 that if a half-line $|AB$ make right
angles\index{Right 01@Right angle} with the opposite half-lines $|AC$, $|AC'$, the verticals
obtained by extending $(AB)$ beyond $A$ will be right angles
congruent to the other two. We thus have four mutually
congruent right angles at the point $A$. Under these circumstances
we shall say that they are \emph{mutually perpendicular}\index{Perpendicularity}
there.

\smallskip\emph{Theorem}~25. If two angles of a triangle be congruent, the
triangle is isosceles\index{Isosceles02@--- triangle}.

This is an immediate result of 18.

Given two non-re-entrant angles. The first shall be said to
be \emph{greater than}\index{Greater than} the second, when it is congruent to the
sum\index{Sum01@Sum of angles}
of the second, and a not null angle. The second shall under
these circumstances, and these alone, be said to be \emph{less than}\index{Less than}
the first. As the assemblage of all congruent transformations
is a group, we see that the relations greater than, less than,
and congruent when applied to angles are mutually exclusive.
For if we had two angles whereof the first was both greater
than and less than the second, then we should have an angle
that would be both greater than and less than itself, an
absurd result, as we see from 23. We shall write $>$ in place
of \emph{greater than}, and $<$ for \emph{less than}, $\equiv$ means
\emph{congruence}\index{Congruence03@--- of angles}.
Two angles between which there exists one of these three
relations shall be said to be \emph{comparable}\index{Comparableness of angles}. We shall later see
that any two angles are comparable. The reason why we
cannot at once proceed to prove this fact, is that, so far,
we are not very clear as to just what can be done with our
congruent transformations. As for the \emph{a priori} question of
comparableness, we have perfectly clear definitions of greater
than, less than, and equal as applied to infinite assemblages,
but are entirely in the dark as to whether when two such
assemblages are given, one of these relations must necessarily
hold.%
\footnote{Cf.\ Borel\index{Borel}, \emph{Leçons sur la th\'eorie des fonctions}, Paris, 1898, pp.~102--8.}
%endfootnote

\smallskip\emph{Theorem}~26. An exterior angle of a triangle\index{Scalene triangles} is comparable
with either of the opposite interior angles.

Let us take the triangle $ABC$, while $D$ lies on the extension
of $(BC)$ beyond $C$. Let $E$ be the middle point of $(AC)$ and
let $DE$ meet $(AB)$ in $F$. If $\overline{DE}>\overline{EF}$ find $G$ of $(DE)$ so
that $\overline{FE}\equiv\overline{EG}$. Then we have $\measuredangle BAC$ congruent to $\measuredangle ECG$
%[File: 036.png]
and less than $\measuredangle ECD$. If $\overline{DE}<\overline{EF}$ we have $\measuredangle BAC$
greater than an angle congruent to $\measuredangle ECD$\index{Scalene triangles}.

\smallskip\emph{Theorem}~27. Two angles of a triangle are comparable\index{Comparableness of angles}.

For they are comparable to the same exterior angle.

\smallskip\emph{Theorem}~28. If in any triangle one angle be greater than
a second, the side\index{Sides04@Sides of triangle} opposite the first is greater than that
opposite the second.

Evidently these sides cannot be congruent. Let us then
have the triangle ABG where $\measuredangle BAG>\measuredangle BGA$. We may,
by the definition of congruence, find such a point $C_{1}$ of $(BG)$
that $\measuredangle C_{1}AG$ is congruent to $\measuredangle C_{1}GA$
and hence $\overline{C_{1}A}\equiv\overline{C_{1}G}$.
It thus remains to show that $\overline{AB}<(\overline{AC_{1}}+\overline{C_{1}B})$. Were such
not the case, we might find $D_{1}$ of $(AB)$ so that $\overline{AD_{1}}\equiv\overline{AC_{1}}$,
and the problem reduces to comparing $\overline{BC_{1}}$ and $\overline{BD_{1}}$. Now
in $\bigtriangleup BD_{1}C_{1}$ we have $\measuredangle BD_{1}C_{1}$ the supplement of
$\measuredangle AD_{1}C_{1}$
which is congruent to $\measuredangle AC_{1}D_{1}$ whose supplement is greater
than $\measuredangle BC_{1}D_{1}$. We have therefore returned to our original
problem, this time, however, with a smaller triangle. Now
this reduction process may be continued indefinitely, and if
our original assumption be false, the inequalities must always
lie the same way. Next notice that, by our axiom of continuity,
the points $C_{i}$ of $(BG)$ must tend to approach a point
$C$ of that segment as a limit, and similarly the points $D_{i}$ of
$(AB)$ tend to approach a limiting point, $D$. If two points of
$(AB)$ be taken indefinitely close to $D$ the angle which they
determine at any point of $(BG)$ other than $B$ will become
indefinitely small. On the other hand as $C_{i}$ approaches $C$,
$\measuredangle APC_{i}$ will tend to increase, where $P$
is any point of $(AB)$
other than $B$, in which case the angle is constant. This
shows that $C$, and by the same reasoning $D$, cannot be other
than $B$; so that the difference between $BC_{i}$ and $BD_{i}$ can be
made as small as we please. But, on the other hand
\[
\overline{C_{1}G}\equiv\overline{AC_{1}}\equiv\overline{AD_{1}};\quad
  (\overline{BA}-\overline{BG})\equiv (\overline{BD_{1}}-\overline{BC_{1}})
  \equiv (\overline{BD_{i}}-\overline{BC_{i}})
\]

Our theorem comes at once from this contradiction.

\smallskip\emph{Theorem}~29. If two sides of a triangle be not congruent,
the angle opposite the greater side is greater than that opposite
the lesser.

\mysect{Side of a triangle not greater than sum of other two}
\smallskip\emph{Theorem}~30. One side of a triangle cannot be greater than\index{Greater than}
the sum of the other two\index{Sum06@Sum of two sides of triangle}.

\smallskip\emph{Theorem}~31. The difference\index{Difference of distances} between two sides of a triangle
is less than\index{Less than} the third side.

The proofs of these theorems are left to the reader.\index{Triangle|)}


%[File: 037.png]
\smallskip\emph{Theorem}~32. Two distinct lines cannot be coplanar with
a third, and perpendicular\index{Perpendicularity} to it at the same point.

Suppose, in fact, that we have $AC$ and $AD$ perpendicular to
$BB'$ at $A$. We may assume $\overline{AB}\equiv\overline{AB'}$ so that by I.~31 $AD$
will contain a single point $E$ either of $(CB)$ or of $(CB')$. For
definiteness, let $E$ belong to $(CB')$. Then take $F$ on $(BC)$,
which is congruent\index{Congruence01@Congruence of distances} to $(B'C)$, so that $\overline{BF}\equiv\overline{B'E}$. Hence
$\measuredangle BB'F$ is congruent\index{Congruence03@--- of angles} to $\measuredangle B'BE$ and therefore congruent to
$\measuredangle BB'E$; which contradicts 23.%
\footnote{This is substantially Hilbert's\index{Hilbert} proof, loc.\ cit., p.~16. It is truly
astonishing how much geometers, ancient and modern, have worried over
this theorem. Euclid puts it as his eleventh axiom that all right angles
are equal. Many modern textbooks prove that all straight angles are equal,
hence right angles are equal, as halves of equal things. This is not usually
sound, for it is not clear by definition why a right angle is half a straight
angle. Others observe the angle of a fixed and a rotating line, and either
appeal explicitly to intuition, or to a vague continuity axiom.}
%endfootnote

\smallskip\emph{Theorem}~33. The locus of points in a plane at congruent
distances from two points thereof is the line through the middle
point of their segment perpendicular to their line.

\smallskip\emph{Theorem}~34. Two triangles are congruent if a side\index{Sides04@Sides of triangle} and two
adjacent angles of one be respectively congruent to a side and
two adjacent angles of the other.

\smallskip\emph{Theorem}~35. Through any point of a given line will pass
one line perpendicular to it lying in any given plane through
that line.

Let $A$ be the chosen point, and $C$ a point in the plane, not
on the chosen line. Let us take two such points $B$, $B'$ on the
given line, that $A$ is the middle point of $(BB')$ and $\overline{BB'}<\overline{CB}$,
$\overline{BB'}<\overline{CB'}$. If then $\overline{CB}\equiv\overline{CB'}$, $AC$ is the line required. If
not, let us suppose that $\overline{CB}>\overline{CB'}$. We may make a cut
in the points of $(CB)$ according to the following principle.
A point $P$ shall belong to the first class if no point of the
segment $(PB)$ is at a distance from $B$ greater than its distance
from $B'$, all other points of $(CB)$ shall belong to the second
class. It is clear that the requirements of Axiom XVIII are
fulfilled, and we have a point of division $D$. We could not
have $\overline{DB}<\overline{DB'}$, for then we might, by 31, take $E$ a point
of $(DC)$ so very near to $D$ that for all points $P$ of $DE$
$\overline{PB}<\overline{PB'}$, and this would be contrary to the law of the cut.
In the same way we could not have $\overline{DB}>\overline{DB'}$. Hence $AD$ is
the perpendicular required.

\smallskip\emph{Theorem}~36. If a line be perpendicular to two others at
%[File: 038.png]
their point of intersection, it is perpendicular to every line
in their plane through that point.

The proof given in the usual textbooks will hold.

\smallskip\emph{Theorem}~37. All lines perpendicular to a given line at
a given point are coplanar.

\smallskip\emph{Definition}.~The plane of all perpendiculars to a line at a
point, shall be said to be \emph{perpendicular}\index{Perpendicularity} to that line at that
point.

\smallskip\emph{Theorem}~38. A congruent transformation which keeps all
points of a line invariant, will transform into itself every plane
perpendicular to that line.

It is also clear that the locus of all points at congruent
distances from two points is a plane.

\smallskip\emph{Theorem}~39. If $P$ be a point within the triangle $ABC$ and
there exist a distance congruent to $\overline{AB}+\overline{AC}$, then\index{Greater than}\index{Less than}
\begin{equation*}
\overline{AB}+\overline{AC}>\overline{PB}+\overline{PC}.
\end{equation*}

To prove this let $BP$ pass through $D$ of $(AC)$. Then as
$\overline{AC}>\overline{AD}$ a distance exists congruent to $\overline{AB}+\overline{AD}$, and
$\overline{AB}+\overline{AD}>\overline{BP}+\overline{PD}$. As $\overline{AB}+\overline{AD}>\overline{PD}$
there exists a distance congruent to $\overline{PD}+\overline{DC}$,
and hence $\overline{PD}+\overline{DC}>\overline{PC}$,
\begin{equation*}
\overline{DC}>\overline{PC}-\overline{PD};\quad \overline{AB}+\overline{AC}>\overline{BP}+\overline{PC}.
\end{equation*}

\mysect{Comparison and measurement of angles}
\emph{Theorem}~40. Any two right angles are congruent.

Let these right angles be $\measuredangle AOC$ and $\measuredangle A'O'C'$. We
may assume $O$ to be the middle point of $(AB)$ and $O'$ the
middle point of $(A'B')$, where $\overline{OA}\equiv\overline{O'A'}$. We may also
suppose that distances exist congruent to $\overline{AC}+\overline{CB}$ and to
$\overline{A'C'}+\overline{C'B'}$. Then $\overline{AC}>\overline{AO}$ and
$\overline{A'C'}>\overline{A'O'}$. Lastly, we
may assume that $\overline{AC}\equiv\overline{A'C'}$. For if we had say,
$\overline{AC}>\overline{A'C'}$,
we might use our cut proceeding in $(OC)$. A point $P$ shall
belong to the first class, if no point of $(OP)$ determines with $A$
a distance greater then $\overline{A'C'}$, otherwise it shall belong to the
second class. We find a point of division $D$, and see at once
that $\overline{AD}\equiv\overline{A'C'}$. Replacing the letter $D$ by $C$, we have
$\overline{AC}\equiv\overline{A'C'}$, $\bigtriangleup ABC$ congruent to
$\bigtriangleup A'B'C'$, hence $\measuredangle AOC$
congruent to $\measuredangle A'O'C'$.

\smallskip\emph{Theorem}~41. There exists a congruent
transformation\index{Transformations, congruent}\index{Congruent transformations} carrying
any segment $(AB)$ into any congruent segment $(A'B')$ and
any half-plane\index{Half-plane} bounded by $AB$ into any half-plane bounded
by $A'B'$\label{theorem41}.

We have merely to find $O$ and $O'$ the middle points of $(AB)$
%[File: 039.png]
and $(A'B')$ respectively, and $C$ and $C'$ on the perpendiculars
to $AB$ and $A'B'$, at $O$ and $O'$ so that
$\overline{OC} \equiv \overline{O'C'}$.

\smallskip\emph{Theorem}~42. If $|OA$ be a given half-line\index{Half-line}, there will exist
in any chosen half-plane\index{Half-plane} bounded by $OA$ a unique half-line
$|OB$ making the $\measuredangle AOB$ congruent\index{Congruence03@--- of angles}
to any chosen angle.

The proof of this theorem depends immediately upon the
preceding one.

Several results follow from the last four theorems. To
begin with, any two angles are comparable, as we see at once
from 42. We see also that our Axioms III--XIII and XVIII,
may be at once translated into the geometry of the angle
if straight and re-entrant angles be excluded. We may then
apply to angles system of measurement entirely analogous
to that applied to distances\index{Angle09@--- measure of}. An angle may be represented
unequivocally by a single number, in terms of any chosen
not null angle. We may extend our system of comparison to
include straight and re-entrant angles as follows. A straight
angle shall be looked upon as greater than every non-re-entrant
angle, and less than every re-entrant one. Of two re-entrant
angles, that one shall be considered the less, whose corresponding
interior angle is the greater. A re-entrant angle
will be the logical sum of two non-re-entrant angles, and shall
have as a measure, the sum of their measures.

\mysect{Nature of the congruent group}
We have also found out a good deal about the congruent
group. The principal facts are as follows:---

(\emph{a})~A congruent transformation\index{Transformations, congruent}\index{Congruent transformations}
may be found to carry any
point into any other point.

(\emph{b})~A congruent transformation may be found to leave any
chosen point invariant, and carry any chosen line through
this point, into any other such line.

(\emph{c})~A congruent transformation may be found to leave
invariant any point, and any line through it, but to carry
any plane\index{Plane} through this line, into any other such plane.

(\emph{d})~If a point, a line through it, and a plane through the
line be invariant, no further infinitesimal congruent transformations are possible.

The last assertion has not been proved in full; let the
reader show that if a point and a line through it be invariant,
there is only one congruent transformation of the line possible,
besides the identical one, and so on. The essential thing
is this. We shall demonstrate at length in Ch.~XVIII that
the congruent group is completely determined by the requirement
that it shall be an analytic collineation\index{Collineations} group, satisfying
these four requirements.

%[File: 040.png]

Suppose that we have two half-planes on opposite sides
of a plane $a$ which contains their common bound $l$. Every
segment whose extremities are one in each of these half-planes\index{Half-plane}
will have a point in $a$, and, in fact, all such points will lie
in one half-plane of $a$ bounded by $l$, as may easily be shown
from the special case where two segments have a common
extremity.

\mysect{Definition of dihedral angles, their properties}
\smallskip\emph{Definition}.~Given two non-coplanar half-planes of common
bound. The assemblage of all half-planes with this bound,
containing points of segments whose extremities lie severally
in the two given half-planes, shall be called their \emph{interior
dihedral angle}, or, more simply, their \emph{dihedral angle}\index{Angle06@--- dihedral}. The
assemblage of all other half-planes with this bound shall be
called their \emph{exterior dihedral angle}. The two given half-planes
shall be called the \emph{faces}, and their bound the edge of the
dihedral angle.

We may, by following the analogy of the plane, define null,
straight, and re-entrant dihedral angles. The definition of the
dihedral angles of a tetrahedron will also be immediately
evident.

A plane perpendicular\index{Perpendicularity} to the edge of a dihedral angle will
cut the faces in two half-lines perpendicular to the edge.
The interior (exterior) angle of these two shall be called a
\emph{plane angle}\index{Angle07@--- plane, of dihedral} of the interior (exterior) dihedral angle.

\smallskip\emph{Theorem}~43. Two plane angles of a dihedral angle are
congruent.

We have merely to take the congruent transformation
which keeps invariant all points of the plane whose points
are equidistant from the vertices of the plane angles. Such
a transformation may properly be called a \emph{reflection}\index{Reflection01@Reflection in plane} in that
plane.

\smallskip\emph{Theorem}~44. If two dihedral angles be congruent\index{Congruence03@--- of angles}, any two
of their plane angles will be congruent, and conversely.

The proof is immediate. Let us next notice that we may
measure any dihedral angle in terms of any other not null one,
and that its measure is the measure of its plane angle in
terms of the plane angle of the latter.

\smallskip\emph{Definition}.~If the plane angle of a dihedral angle be a right
angle\index{Right 01@Right angle}, the dihedral angle itself shall be called \emph{right}, and the
planes shall be said to be \emph{mutually perpendicular}.

\smallskip\emph{Theorem}~45. If a plane be perpendicular to each of two
other planes, and the three be concurrent, then the first
plane is also perpendicular to the line of intersection of the
other two.
%[File: 041.png]

%CHAPTER III
\mychap{III}{THE THREE HYPOTHESES}
\mysect{A variable angle is a continuous function of a variable distance}

In the last chapter we discussed at some length the problem
of comparing distances and angles, and of giving them
numerical measures in terms of known units. We did not
take up the question of the sum of the angles of a triangle,
and that shall be our next task. The axioms so far set up
are insufficient to determine whether this sum shall, or shall
not, be congruent to the sum of two right angles, as we shall
amply see by elaborating consistent systems of geometry
where this sum is greater than, equal to, or less than two
right angles. We must first, however, give one or two
theorems concerning the continuous\index{Continuity02@--- in change of angles and sides of a triangle}
change of distances and
angles.

\smallskip\emph{Theorem}~1. If a point $P$ of a segment $(AB)$ may be taken
at as small a distance from $A$ as desired, and $C$ be any other
point, the $\measuredangle ACP$ may be made less than any given angle.

If $C$ be a point of $AB$ the theorem is trivial. If not, we
may, by III.~4, find $|CD$ in the half-plane bounded by $CA$
which contains $B$, so that $\measuredangle ACD$ is congruent to the
given
angle. If then $|AB$ belong to the internal $\measuredangle ACD$, we have
$\measuredangle ACB$ less than $\measuredangle ACD$, and, \emph{a
  fortiori}, $\measuredangle ACP < \measuredangle ACD$.
If $|AD$ belong to the internal $\measuredangle ACB$, $|AD$ must contain a
point $E$ of $CAB$, and if we take $P$ within $(AE)$, once more
\begin{equation*}
  \measuredangle ACP < \measuredangle ACD.
\end{equation*}

\smallskip\emph{Theorem}~2. If, in any triangle, one side and an adjacent
angle remain fixed, while the other side including this angle
may be diminished at will, then the external angle opposite
to the fixed side will take and retain a value differing from
that of the fixed angle by less than any assigned value.

Let the fixed side be $(AB)$, while $C$ is the variable vertex
within a fixed segment $(BD)$. We wish to show that if $\overline{BC}$
be taken sufficiently small, $\measuredangle ACD$ will necessarily
differ from
$\measuredangle ABD$ by less than any chosen angle.

Let $B_1$ be the middle point of $(AB)$, and $B_2$ the middle
point of $(B_1B)$, while $B_3$ is a point of the extension of $(AB)$
beyond $B$. Through each of the points $B_1, B_2, B_3$ construct
a half-line bounded thereby, and lying in that half-plane,
%[File: 042.png]
\index{Continuity02@--- in change of angles and sides of a triangle}bounded by $AB$ which contains $D$, and let the angles so
formed at $B_1$, $B_2$, $B_3$ all be congruent to $\measuredangle ABD$. We may
certainly take $\overline{BC}$ so small that $AC$ contains a point of each
of these half-lines, say $C_1$, $C_2$, $C_3$ respectively. We may moreover
take $\overline{BC}$ so tiny that it is possible to extend ($B_1C_1$)
beyond $C_1$ to $D_1$ so that $\overline{B_1C_1} \equiv \overline{C_1D_1}$. $AD_1$ will surely meet
$B_2C_2$ in a point $D_2$, when $\overline{B_1C_1}$ is very small, and as $\overline{AC_3}$
differs infinitesimally from $\overline{AB_3}$, and hence exceeds $\overline{AB}$ by
a finite amount, it is greater than $2\overline{AC_1}$ which differs infinitesimally
from $2\overline{AB_1}$, or $\overline{AB}$. We may thus find $C'$ on
the extension of ($AC_1$) beyond $C_1$ so that $\overline{AC_1} \equiv \overline{C_1C'}$. $C'$ will
be at a small distance from $C$, and hence on the other side of
$B_2D_2$ from $A$ and $D_1$. Let $D_1C'$ meet $B_2D_2$ at $H_2$. We now
see that, with regard to the $\triangle AB_1D_1$; the external angle at
$D_1$ (i.e.\ one of the mutually vertical external angles) is
$\measuredangle B_1D_1D_2$ congruent to ($\measuredangle B_1D_1C' + \measuredangle C'D_1D_2$), and $\measuredangle B_1D_1C'$
%[File: 043.png]
is congruent to $\measuredangle AB_1D_1$, and, hence congruent to $\measuredangle ABD$. The
$\measuredangle C'D_1D_2$ is the difference between $\measuredangle B_1D_1D_2$ and $\measuredangle B_1D_1H_2$,
and as $H_2$ and $D_2$ approach $B_2$ as a limiting position, the
angles determined by $B_2$, $D_2$ and $D_2$, $H_2$ at every point in
space decrease together towards a null angle as a limit.
Hence $\measuredangle C'D_1D_2$ becomes infinitesimal, and the difference
between $\measuredangle B_1D_1D_2$ and $\measuredangle ABD$ becomes and remains infinitesimal.
But as $\overline{AB_1} \equiv \overline{B_1B}$, and $\measuredangle AB_1D_1$ and $\measuredangle B_1BD$
are congruent, we see similarly that the difference between
$\measuredangle B_1CD$ and $\measuredangle ABD$ will become, and remain infinitesimal.
Lastly, the difference between $\measuredangle B_1CD$ and $\measuredangle ACD$ is $\measuredangle B_1CA$
which will, by our previous reasoning, become infinitesimal
with $\overline{B_1C_1}$. The difference between $\measuredangle ABD$ and $\measuredangle ACD$ will
therefore become and remain less than any assigned angle.\index{Continuity02@--- in change of angles and sides of a triangle}

%[Illustration: FIG. 1.]
\imgcent{200}{f001}{Fig. 1}

Several corollaries follow immediately from this theorem.

\smallskip\emph{Theorem}~3. If in any triangle one side and an adjacent
angle remain fixed, while the other side including this angle
becomes infinitesimal\index{Infinitesimal domain}, the sum of the angles of this triangle
will differ infinitesimally from a straight angle.

\smallskip\emph{Theorem}~4. If in any triangle one side and an adjacent
angle remain fixed, while the other side including this angle
varies, then the measures of the third side, and of the variable
angles will be continuous functions of the measure of the
variable side first mentioned.

Of course a constant is here included as a special case of
a continuous function.

\smallskip\emph{Theorem}~5. If two lines $AB$, $AC$ be perpendicular to $BC$, then
all lines which contain $A$ and points of $BC$ are perpendicular
to $BC$, and all points of $BC$ are at congruent distances from $A$.

To prove this let us first notice that our $\triangle ABC$ is isosceles,
and $\overline{AB}$ will be congruent to every other perpendicular
distance from $A$ to $BC$. Such a distance will be the distance
from $A$ to the middle point of ($BC$) and, in fact, to every
point of $BC$ whose distance from $B$ may be expressed in the
form $\dfrac{m}{2^n}\overline{BC}$ where $m$ and $n$ are integers. Now such points
will lie as close as we please to every point of $BC$, hence
by II.~31, no distance from $A$ can differ from $\overline{AB}$, and no
angle so formed can, by III.~2, differ from a right angle.

\smallskip\emph{Theorem}~6. If a set of lines perpendicular to a line $l$, meet
a line $m$, the distances of these points from a fixed point of $m$,
and the angles so formed with $m$, will vary continuously with
%[File: 044.png]
the distances from a fixed point of $l$ to the intersections with
these perpendiculars.

The proof comes easily from 2 and 5.

\smallskip\emph{Definition}.~Given four coplanar points $A$, $B$, $C$, $D$ so situated
that no segment may contain points within three of the
segments $(AB)$, $(BC)$, $(CD)$, $(DA)$. The assemblage of all points
of all segments whose extremities lie on these segments shall
be called a \emph{quadrilateral}\index{Quadrilateral01@Quadrilateral}. The given points shall be called
its vertices, and the given segments its sides\index{Sides03@--- of quadrilateral}. The four
internal angles $\measuredangle DAB$, $\measuredangle ABC$,
$\measuredangle BCD$, $\measuredangle CDA$ shall be
called its angles. The definitions of opposite sides and
opposite vertices are obvious, as are the definitions for
adjacent sides and vertices.

\smallskip\emph{Definition}.~A quadrilateral with right angles at two
adjacent vertices shall be called \emph{birectangular}\index{Birectangular quadrilateral}. If it have
three right angles it shall be called \emph{trirectangular}\index{Trirectangular quadrilateral}, and four
right angles it shall be called a \emph{rectangle}\index{Rectangle}. Let the reader
convince himself that, under our hypotheses, birectangular
and trirectangular quadrilaterals necessarily exist.

\smallskip\emph{Definition}.~A birectangular quadrilateral whose opposite
sides adjacent to the right angles are congruent, shall be said
to be \emph{isosceles}\index{Isosceles01@Isosceles quadrilateral}.

\mysect{Saccheri's theorem for isosceles birectangular quadrilaterals}
\smallskip\emph{Theorem}~7. Saccheri's\index{Saccheri}%
\footnote{Saccheri, \emph{Euclides ab omni
    naevo vindicatus}, Milan, 1732. Accessible in
  Engel\index{Engel} und Staeckel\index{Stackel@St\"ackel}, \emph{Theorie der Parallellinien von Euklid bis
    auf Gauss}, Leipzig,
  1895. The theorem given above covers Saccheri's theorems 1 and 2 on
    p.~50
  of the last-named work. Saccheri's is the first systematic attempt
    of which
  we have a record to prove Euclid's parallel postulate, and proceeds
  according
  to modern method of assuming the postulate untrue. He builded better
  than he knew, however, for the system so constructed is
    self-consistent, and
  not inconsistent, as he attempted to show.}.
In an isosceles birectangular quadrilateral
a line through the middle point of the side adjacent
to both right angles, which is perpendicular to the line of
that side, will be perpendicular to the line of the opposite
side and pass through its middle point. The other two angles
of the quadrilateral are mutually congruent.

Let the quadrilateral be $ABCD$, the right angles having
their vertices at $A$ and $B$. Then the perpendicular to $AB$
at $E$ the middle point of $(AB)$ will surely contain $F$ point of
$(CD)$. It will be easy to pass a plane through this line
perpendicular to the plane of the quadrilateral, and by taking
a reflection in this latter plane, the quadrilateral will be
transformed into itself, the opposite sides being interchanged.

This theorem may be more briefly stated by saying that
%[File: 045.png]
this line divides the quadrilateral\index{Quadrilateral01@Quadrilateral} into two mutually congruent
trirectangular ones.

\smallskip\emph{Theorem}~8. In a rectangle\index{Rectangle} the opposite sides are mutually
congruent, and any isosceles birectangular quadrilateral whose
opposite sides are mutually congruent is necessarily a rectangle.

\mysect{The existence of one rectangle implies the existence of an infinite number}
\smallskip\emph{Theorem}~9. If there exist a single rectangle, every isosceles
birectangular quadrilateral is a rectangle.

Let $ABCD$ be the rectangle. The line perpendicular to
$AB$ at the middle point of $(AB)$ will divide it into two
smaller rectangles. Continuing this process we see that we
can construct a rectangle whose adjacent sides may have any
measures that can be indicated in the form $\dfrac{m}{2^n}
\overline{AB}$, $\dfrac{p}{2^q} \overline{AC}$,
provided, of course, that the distances so called for exist
simultaneously on the sides of a birectangular isosceles
quadrilateral. Distances so indicated will be everywhere
dense on any line, hence, by 6 we may construct a rectangle
having as one of its sides one of the congruent sides of any
isosceles birectangular quadrilateral, and hence, by a repetition
of the same process, a rectangle which is identical with this
quadrilateral. All isosceles birectangular quadrilaterals, and
all trirectangular quadrilaterals are under the present circumstances
rectangles.

Be it noticed that, under the present hypothesis, Theorem~5
is superfluous.

\mysect{Three assumptions as to the sum of the angles of a right triangle}
\smallskip\emph{Theorem}~10. If there exist a single right triangle\index{Right 02@--- triangle} the sum
of whose angles is congruent to a straight angle, the same is
true of every right triangle.

Let $\triangle ABC$ be the given triangle, the right angle being
$\measuredangle ACB$ so that the sum of the other two angles is congruent
to a right angle. Let $\triangle A'B'C'$ be any other right triangle,
the right angle being $\measuredangle A'C'B'$. We have to prove that the
sum of its remaining angles also is congruent to a right angle.
We see that both $\measuredangle ABC$ and $\measuredangle BAC$ are
less than right
angles, hence there will exist such a point $E$ of $(AB)$ that
$\measuredangle EAC$ and $\measuredangle ECA$ are congruent. Then
$\measuredangle EBC \equiv \measuredangle ECB$
since $\measuredangle ACB$ is congruent to the sum of $\measuredangle
EAC$ and $\measuredangle EBC$.
If $D$ and $F$ be the middle points of $(BC)$ and $(AC)$ respectively,
as $\triangle EAC$ and $\triangle EBC$ are isosceles, we have, in the
quadrilateral $EDCF$ right angles at $D$, $C$, and $F$. The angle
at $E$ is also a right angle, for it is one half the straight angle,
$\measuredangle AEB$, hence $EDCF$ is a
rectangle\label{angleEDCF}. Passing now to the
$\triangle A'C'B'$ we see that the perpendicular to $A'C'$ at $F'$ the
%[File: 046.png]
middle point of $(A'C')$, will meet $(A'B')$ in $E'$, and the
perpendicular
to $E'F'$ at $E'$ will meet $(B'C')$ in $D'$. But, by
an easy modification of 9, as there exists one rectangle\index{Rectangle}, the
trirectangular quadrilateral $E'F'D'C'$ is also a rectangle. It
is clear that $\measuredangle D'E'B' \equiv \measuredangle D'E'C'$
since $\measuredangle F'E'D'$ is a right
angle and $\measuredangle F'E'A' \equiv \measuredangle F'E'C'$. Then
$\triangle C'E'B'$ is isosceles
like $\triangle A'E'C'$. From this comes immediately that the sum
of $\measuredangle E'B'C'$ and $\measuredangle E'A'C'$ is congruent to
a right angle, as
we wished to show.

\smallskip\emph{Theorem}~11. If there exist any right triangle\index{Right 02@--- triangle} where the
sum\index{Sum02@--- of angles of a triangle} of the angles is less than a straight angle, the same is
true of all right triangles.

We see the truth of this by continuity. For we may pass
from any right triangle to any other by means of a continuous
change of first the one, and then the other of the sides which
include the right angle. In this change, by 2, the sum of the
angles will either remain constant, or change continuously,
but may never become congruent to the sum of two right
angles, hence it must always remain less than that sum.

\smallskip\emph{Theorem}~12. If there exist a right triangle where the sum
of the angles is greater than two right angles, the same is
true of every right triangle.

This comes immediately by \emph{reductio ad absurdum}.

\mysect{Three assumptions as to the sum of the angles of any triangle, their categorical nature}
\smallskip\emph{Theorem}~13. If there exist any triangle where the sum of
the angles is less than (congruent to) a straight angle, then in
every triangle the sum of the angles is less than (congruent
to) a straight angle.

Let us notice, to begin with, that our given $\triangle ABC$
must have at least two angles, say $\measuredangle ABC$ and
$\measuredangle BAC$ which
are less than right angles. At each point of $(AB)$ there will
be a perpendicular to $AB$ (in the plane $BC$). If two of
these perpendiculars intersect, all will, by 5, pass through
this point, and a line hence to $C$ will surely be perpendicular
to $AB$. If no two of the perpendiculars intersect, then,
clearly, some will meet $(AC)$ and some $(BC)$. A cut will
thus be determined among the points of $(AB)$, and, by XVIII,
we shall find a point of division $D$. It is at once evident
that the perpendicular to $AB$ at $D$ will pass through $C$. In
every case we may, therefore, divide our triangle into two
right triangles. In one of these the sum of the angles must
surely be less than (congruent to) a straight angle, and the
same will hold for every right triangle. Next observe that
there can, under our present circumstances, exist no triangle
with two angles congruent to, or greater than right angles.
%[File: 047.png]
Hence every triangle can be divided into two right triangles
as we have just done. In each of these triangles, the sum of
the angles is less than (congruent to) a straight angle, hence
in the triangle chosen, the sum of the angles is less than
(congruent to) a straight angle.

\smallskip\emph{Theorem}~14. If there exist any triangle where the sum
of the angles is greater than a straight angle, the same will
be true of every triangle.

This comes at once by \emph{reductio ad absurdum}.

We have now reached the fundamental fact that the sum of
the angles of a single triangle will determine the nature
of the sum of the angles of every triangle. Let us set the
various possible assumptions in evidence.

\mysect{Definition of the euclidean, hyperbolic, and elliptic hypotheses}
The assumption that there exists a single triangle, the sum
of whose angles\index{Sum02@--- of angles of a triangle} is congruent to a straight angle is called the
\emph{Euclidean}\index{Euclidean01@Euclidean hypothesis}
or \emph{Parabolic}\index{Parabolic hypothesis} hypothesis.%
\footnote{There will
  exist, of course, numerous geometries, other than those which
  we give in the following pages, where the sum of the angles of a
  triangle is
  still congruent to a straight angle, e.g.\ those lacking our strong
  axiom of
  continuity. Cf.\ Dehn\index{Dehn}, `Die Legendre'schen S\"atze \"uber die Winkelsumme im
  Dreiecke', \emph{Mathematische Annalen}, vol.~liii, 1900, and
  R.~L. Moore\index{Moore}, `Geometry
  in which the sum of the angles of a triangle is two right angles',
  \emph{Transactions
  of the American Mathematical Society}, vol.~viii, 1907.}
%endfootnote

The assumption that there exists a triangle, the sum of
whose angles is less than a straight angle is called the
\emph{Lobatchewskian}\index{Lobatchewskian hypothesis}
or \emph{hyperbolic}\index{Hyperbolic01@Hyperbolic hypothesis}
hypothesis.%
\footnote{The
  three hypotheses were certainly familiar to Saccheri (loc.\ cit.),
  though
  the credit for discovering the hyperbolic system is generally given
  to Gauss,
  who speaks of it in a letter to Bolyai written in
  1799. Lobatchewsky's\index{Lobatchewsky} first
  work was published in Russian in Kasan, in 1829. This was followed
  by an
  article `G\'eom\'etrie imaginaire', \emph{Crelle's Journal}, vol.~xvii,
  1837. All spellings
  of Lobatchewsky's name in Latin or Germanic languages are
  phonetic. The
  author has seen eight or ten different ones.}
%endfootnote

The assumption that there exists a triangle, the sum of
whose angles is greater than a straight angle, is called the
\emph{Riemannian}\index{Riemannian hypothesis} or
\emph{elliptic hypothesis}\index{Elliptic02@--- hypothesis}.%
\footnote{Riemann\index{Riemann},
  \emph{Ueber die Hypothesen, welche der Geometrie zu Grunde liegen},
  first read
  in 1854; see p.~272 of the second edition of his \emph{Gesammelte
    Werke}, with
  explanations in the appendix by Weber\index{Weber}.}
%endfootnote

Only under the elliptic hypothesis can two intersecting
lines be perpendicular to a third line coplanar with them.

\smallskip\emph{Definition}.~The difference between the sum of the angles of
a triangle, and a straight angle shall be called the \emph{discrepancy}\index{Discrepancy of a triangle}
of the triangle.

\smallskip\emph{Theorem}~15. If in any triangle a line be drawn from one
vertex to a point of the opposite side, the sum of the discrepancies
of the resulting triangles is congruent to the
discrepancy of the given triangle.
%[File: 048.png]

The proof is immediate. Notice, hence, that if in any
triangle one angle remain constant, while one or both of the
other vertices tend to approach the vertex of the fixed angle,
along fixed lines, the discrepancy of the triangle, when not
zero, will diminish towards zero as a limit. We shall make
this more clear by saying---

\smallskip\emph{Theorem}~16. If, in any triangle, one vertex remain fixed,
the other vertices lying on fixed lines through it, and if a
second vertex may be made to come as near to the fixed vertex
as may be desired, while the third vertex does not tend to
recede indefinitely, then the discrepancy may be made less
than any assigned angle.

\smallskip\emph{Theorem}~17. If in any triangle one side may be made less
than any assigned segment, while neither of the other sides
becomes indefinitely large, the discrepancy may be made less
than any assigned angle.

If neither angle adjacent to the diminishing side tend to
approach a straight angle as a limit, it will remain less than
some non-re-entrant angle, and 16 will apply to all such
angles simultaneously. If it do tend to approach a straight
angle, let the diminishing side be $(AB)$, while $\measuredangle BAC$
tends
to approach a straight angle. Then, as neither $\overline{BC}$ nor
$\overline{AC}$
becomes indefinitely great, we see that $A$ must be very close
to some point of the extension of $(AB)$ beyond $A$, or to $A$
itself. If $C$ do not approach $A$, we may apply 1 to show that
$\measuredangle ACB$ becomes infinitesimal. If $C$ do approach $A$ we may
take $D$ the middle point of $(AC)$ and extend $(BD)$ to $E$ beyond
$D$ so that $\overline{DE} \equiv \overline{EB}$. Then we may apply
Euclid's own
proof%
\footnote{Euclid\index{Euclid}, Book I, Proposition 16.}
%endfootnote
that the exterior
angle of a triangle is greater than
either opposite interior one, so that the exterior angle at $A$
which is infinitesimal\index{Infinitesimal domain}, is yet greater than $\measuredangle ACB$.

\mysect{Geometry in the infinitesimal domain obeys the euclidean hypothesis}
\smallskip\emph{Theorem}~18. If, in any system of triangles, one side of each
may be made less than any assigned segment, all thus
diminishing together, while no side becomes indefinitely
great, the geometry of these triangles may be made to differ
from the geometry of the euclidean hypothesis by as little as
may be desired.

A specious, if loose, way of stating this theorem is to say
that in the infinitesimal domain, we have euclidean
geometry.%
\footnote{This theorem, loosely proved, is taken as the basis
  of a number of works
  on non-euclidean geometry, which start in the infinitesimal domain, and
  work to the finite by integration. Cf.\ e.g.\ Flye Ste-Marie\index{Marie, Ste-},
  \emph{\'Etudes analytiques
    sur la th\'eorie des parallèles}, Paris, 1871.}
%endfootnote

%[File: 049.png]
%CHAPTER IV
\mychap{IV}{THE INTRODUCTION OF TRIGONOMETRIC FORMULAE}
\mysect{Limit of ratio of opposite sides of diminishing isosceles quadrilateral}

The first fundamental question with which we shall have
to deal in this chapter is the following. Suppose that we
have an isosceles, birectangular quadrilateral $ABCD$, whose
right angles are at $A$ and $B$. Suppose, further, that $\overline{AB}$
becomes infinitesimally small, $\overline{AD}$ remaining constant; what
will be the limit of the fraction $\dfrac{\Measure
  \overline{CD}}{\Measure \overline{AB}}$ where
$\Measure \overline{XY}$ means the
measure of $\overline{XY}$ in terms of some convenient
unit.%
\footnote{The general treatment, and several of the actual proofs
  in this chapter
  are taken directly from G\'erard\index{Gerard@G\'erard}, \emph{La g\'eom\'etrie non-euclidienne},
  Paris, 1892. It has
  been possible to shorten some of his work by the consideration that we have
  euclidean geometry in the infinitesimal domain. On the other hand, several
  important points are omitted by him. There is no proof that the required
  limit does actually exist, and worse still, he gives no proof that
  the resulting
  function of $\Measure \overline{AD}$ is
  necessarily continuous, thereby rendering valueless his
  solution of its functional equation.}
%endfootnote
But, first
of all, we must convince ourselves, that, when $\overline{AD}$ is given
we may always construct a suitable quadrilateral; secondly,
and most important, we must show that a definite limit does
necessarily exist for this ratio, as $\overline{AB}$ decreases towards the
null distance.

\smallskip\emph{Theorem}~1. If $AD$ and $AX$ be two mutually perpendicular
lines we may find such a point $B$ on either half of $AX$ bounded
by $A$, that, a line being drawn perpendicular to $AB$ at any
point $P$ of $(AB)$ we may find on the half thereof bounded by
$P$, which lies in the same half-plane bounded by $AB$ as does $D$,
a point whose distance from $P$ is greater than $\overline{AD}$.

Let $E$ be a point of the extension of $(AD)$ beyond $D$. Draw
a line there perpendicular to $AD$. If $B$ be a point of $AX$
very close to $A$, and if a line perpendicular to $AB$ at $P$
of $(AB)$, meet the perpendicular at $E$ at a point $Q$,
$\overline{PQ}$ differs
but little from $\overline{AE}$, and, hence, is greater than
$\overline{AD}$.

%[File: 050.png]
The net result of theorem 1 is this. If $AD$ be given, and
the right $\measuredangle DAX$, any point of $AX$ very near to $A$ may
be
taken as the vertex of a second right angle of an isosceles
birectangular quadrilateral\index{Birectangular quadrilateral}\index{Quadrilateral01@Quadrilateral},
having $A$ as the vertex of one
right angle, and $(AD)$ as one of the congruent sides.

\smallskip\emph{Definition}.~We shall say that a distance may be made
infinitesimal compared with a second distance, if the ratio
of the measure of the first to that of the second may be made
less than any assigned value.

\smallskip\emph{Theorem}~2. If in a triangle whereof one angle is constant,
a second angle may be made as \emph{small} as desired, the side
opposite this angle will be infinitesimal compared to the other
sides of the triangle.

Suppose that we have, in fact, $\triangle PQR$ with $\measuredangle
PQR$ fixed,
while $\measuredangle PRQ$ becomes infinitesimal. It is clear that one
of the angles $\measuredangle PQR$ or $\measuredangle QPR$ must be
greater than a right
angle. Suppose it be $\measuredangle QPR$. Then, by hypothesis, no
matter how large a positive integer $n$ may be, I may find such
positions for $P$ and $R$, that $n$ points $Q_i$ may be found on $|PQ$
so that $\measuredangle PRQ \equiv \measuredangle QRQ_1 \equiv
\measuredangle Q_kRQ_{k+1}$, yet $\measuredangle QRQ_n$ is less
than any chosen angle. Now if $\overline{RQ}$ remain constantly greater
than a given not null distance, the theorem is perfectly
evident. If, on the other hand, $\overline{RQ}$ decrease indefinitely, we
may find $S$ on $|PQ$ but not in $(PQ)$, so that $\overline{QR} \equiv
\overline{QS}$. Then,
as geometry in the infinitesimal domain obeys the euclidean
hypothesis, $\measuredangle QRS$ will differ infinitesimally from one half
$\measuredangle PQR$. If, then, we require $\measuredangle QRQ_n$ to
be less than this last-named
amount, $Q_n$ will be within $(QS)$, and $\overline{PQ} <
\overline{Q_kQ_{k+1}}$
and $\overline{PQ} < \dfrac{1}{n} \overline{QR}$. A similar proof holds
when $\measuredangle PQR$ is
greater than a right angle.

It will follow, as a corollary, that if in any triangle, one
angle become infinitesimal, and neither of the other angles
approaches a straight angle as a limit, then the side opposite
the infinitesimal angle becomes infinitesimal as compared
with either of the other sides.

\smallskip\emph{Theorem}~3. If in an isosceles birectangular quadrilateral,
the congruent sides remain constant in value, while the side
adjacent to the two right angles decreases indefinitely, the
ratio\index{Ratio of opposite sides of quadrilateral}
of the measures of this and the opposite side approaches
a definite limit.

It will save circumlocution and involve no serious confusion
if, during the rest of this chapter, we speak of the ratio of two
%[File: 051.png]
distances, instead of the ratio of their measures, and write
such a ratio simply $\dfrac{\overline{PQ}}{\overline{XY}}$.  Let us then take the isosceles
birectangular quadrilateral\index{Isosceles01@Isosceles quadrilateral} $A'ABB'$, the right angles having
their vertices at $A$ and $B$. Let us imagine that $A$ and $A'$ are
fixed points, while $B$ is on a fixed line at a very small distance
from $A$. Let $C$ be the middle point of $(AB)$, and let the
perpendicular to $AB$ at $C$ meet $(A'B')$ at $C'$, which, by
Saccheri's\index{Saccheri} theorem, is the middle point of $(A'B')$. Now, by
III\@. 6, $\measuredangle C'A'A$ differs infinitesimally from a right angle,
as $\overline{AC}$ becomes infinitesimal, so that if $C_1$ be the point
of $(CC')$, or $(CC')$ extended beyond $C'$, for which $\overline{CC_1} \equiv \overline{AA'}$,
$\overline{C_1C'} < \dfrac{1}{n}\overline{A'C'}$. But $\dfrac{\overline{A'C'}}{\overline{AC}} \equiv
 \dfrac{\overline{A'B'}}{\overline{AB}}$. Hence $\dfrac{\overline{A'C_1}}{\overline{AC}} -
 \dfrac{\overline{A'B'}}{\overline{AB}} < \delta$
where $\delta$ may be made less than any assigned number. By a repeated use of this process we see that if $D$ be such a point of $(AB)$
that $\overline{AD} = \dfrac{k}{2^n}\overline{AB}$ and $D_1$ such a point of the perpendicular
at $D$ that $\overline{AA'} \equiv \overline{DD_1}$, then, however small $\epsilon$ may be,
$\dfrac{\overline{A'D_1}}{\overline{AD}} -
 \dfrac{\overline{A'B'}}{\overline{AB}} < \epsilon$, and what is more, we may take $\overline{AB}$ so
small that this inequality shall hold for all such points $D$
at once, for, as $\overline{AB}$ decreases, every ratio $\dfrac{\overline{A'D_1}}{\overline{AD}}$ gets nearer and
nearer to $\dfrac{\overline{A'B'}}{\overline{AB}}$.  Lastly, if $P$ be any point of $(AB)$, and $P_1$ lie
on the perpendicular at $P$ so that
$\overline{AA'} \equiv \overline{PP_1}$, we may find
one of our points recently called $D$ of such a nature that $\overline{DP_1}$
and $\overline{D_1P_1}$ are infinitesimal as compared with $\overline{AB}$. Hence
$\dfrac{\overline{A'P_1}}{\overline{AP}} -
 \dfrac{\overline{A'B'}}{\overline{AB}} < \epsilon$ where $\epsilon$ is infinitesimal with $\overline{AB}$. This
shows that $\dfrac{\overline{A'B'}}{\overline{AB}}$ approaches a definite limit, as $\overline{AB}$ approaches
the null distance.

This limit\index{Ratio of opposite sides of quadrilateral} is constantly equal to 1 in the euclidean case.
In the other cases it is a variable depending on the measure
of $\overline{AA'}$. If this measure be $x$, we may call our limit $\phi(x)$\index{Phi function}.

\mysect{Continuity of the resulting function}
Let us next show that the function $\phi$ is continuous. Take
$A'ABB'$ as before, while $A_1$ and $B_1$ are respectively on the
%[File: 052.png]
extensions of $(AA')$, beyond $A'$, and of $(BB')$ beyond $B'$. Let
the measure of $\overline{AA'}$ be $x$, while that of $\overline{A'A_1}$ is $\Delta x$,
\[
\frac{\overline{A'B'}}{\overline{AB}} = \phi(x) + \epsilon, \quad
  \frac{\overline{A_1B_1}}{\overline{AB}} = \phi(x+\Delta x) + \eta,
\]
\[
\left| \frac{\overline{A_1B_1}-\overline{A'B'}}{\overline{AB}}\right| =
  \Delta\phi(x)+\eta-\epsilon.
\]

\begin{flalign*}
&\text{\indent Now}
&\overline{A_1B_1} &< (\overline{A_1A'} + \overline{A'B'}) + \overline{B'B_1}, \quad
  2\overline{A_1A'} > |\overline{A_1B_1} - \overline{A'B'}|, &\phantom{\text{Now}}
\end{flalign*}
and, however great $m$ may be, we may take $\overline{A_1A'}$ so small
that
\begin{flalign*}
&&&\overline{A_1A'} < \frac{1}{2m}\overline{AB}, \\
&\text{then}& \hfill &\Delta\phi(x) < \frac{1}{m} + \delta,  \hfill & \phantom{\text{then}}
\end{flalign*}
and, hence, $\phi$ is a continuous function.

\mysect{Its functional equation and solution}
We shall find the actual form of $\phi$\index{Phi function}\index{Ratio of opposite sides of quadrilateral}
from its functional
equation. Let $x$ be the measure of $\overline{AC}$, $(x-y)$ that of $\overline{AC_1}$,
and $(x + y)$ that of $\overline{AC_2}$; where $C$ and $C_1$ are points within
$(AC_2)$. Take a corresponding set of distances upon a line near
by, $\overline{BD} \equiv \overline{AC}$;
$\overline{BD_1} \equiv \overline{AC_1}$;
$\overline{BD_2} \equiv \overline{AC_2}$ while $|AC$ and $|BD$
are in the same half-plane bounded by $AB$ and perpendicular
thereto. We know, by 1, that this construction is possible.
We shall presently suppose $\overline{AB}$ to be infinitesimal. The
perpendicular to $CD$ at $C$ will meet $C_2D_2$ and $C_1D_1$ in $P$ and $R$
respectively, while the perpendicular to $CD$ at $D$ will meet
these lines at $Q$ and $S$; the four last-named points will surely
exist, if $\overline{AB}$ be very tiny. $\measuredangle CC_2P$ and
$\measuredangle CC_1R$ will differ
infinitesimally from right angles, so that by 2

\[
\left| \frac{\overline{C_2P} - \overline{C_1R}}{\overline{CC_2}} \right| = \epsilon.
\]

This infinitesimal $\epsilon$ is, in fact, of the second order. For,
let us compare $\triangle CC_2P$ and $\triangle CC_1R$.
$\measuredangle C_1CR \equiv \measuredangle C_2CP$;
$\overline{CC_1} \equiv \overline{CC_2}$. Also $\measuredangle CC_2P$
and $\measuredangle CC_1R$ differ infinitesimally.
Hence, if, on $(CP)$ or $(CP)$ extended beyond $P$, we take
$\overline{CP'} \equiv \overline{CR}$ we have
$\overline{C_2P'}\equiv \overline{C_1R}$;
$\overline{C_2P}-\overline{C_1R} < \overline{PP'}$.  But
$\dfrac{\overline{PP'}}{\overline{C_2P}} < \delta$
as the angle opposite $(PP')$ is infinitesimal.
$\dfrac{2}{y}\overline{C_2P} = \dfrac{2}{y}\overline{C_1R}+2\epsilon$
where $\epsilon$ is infinitesimal, as compared with
$\Measure\overline{C_2P}$ meaning thereby the measure of $\overline{C_2P}$.
Lastly, let us
%[File: 053.png]
use letters of the type $\delta$, $\epsilon$, $\eta$ to indicate infinitesimals, and
remember that $\overline{AB}$ is an infinitesimal distance.

\begin{flalign*}
&&\overline{C_2P} & \equiv \overline{D_2Q},\quad \overline{C_1R} \equiv \overline{D_1S}, \\
&&2\overline{C_2P} & \equiv |\overline{C_2D_2} - \overline{PQ}|,\quad
  2\overline{C_1R} \equiv |\overline{C_1D_1} - \overline{RS}|, \\
&&\overline{CD} & \equiv \phi(x)\overline{AB} +
  \epsilon_1\overline{AB}, \\
&&\overline{C_1D_1} & \equiv \phi(x - y)\overline{AB} +
  \epsilon_2\overline{AB}, \\
&&\overline{C_2D_2} & \equiv \phi(x + y)\overline{AB} +
  \epsilon_3\overline{AB}, \\
&&\overline{PQ} & \equiv \phi(\Measure\overline{CP})\overline{CD} +
  \delta_1\overline{CD}, \\
&&\overline{RS} & \equiv \phi(\Measure\overline{CR})\overline{CD} +
  \delta_2\overline{CD}, \\
&\text{But } \hfill &\overline{C_2P} >\; & \overline{CC_2} - \overline{CP}
  \text{\ and } \overline{C_2P} \text{\ is infinitesimal.} \hfill &\phantom{\text{But }} \\
&&\overline{PQ} & \equiv \phi(y)\overline{CD} + \delta_3\overline{CD}, \\
&&\overline{RS} & \equiv \phi(y)\overline{CD} + \delta_4\overline{CD}.
\end{flalign*}

Substitute in the first equation connecting $\overline{C_2P}$ and
$\overline{C_1R}$
\begin{multline*}
[\phi(x+y) + \epsilon_3 -\phi(x)\phi(y) - \phi(x)\delta_3 -
  \phi(y)\epsilon_1 + \delta_3\epsilon_1]\;\Measure\overline{AB} = \\
 = [\phi(x)\phi(y) + \phi(x)\delta_4 + \phi(y)\epsilon_1 +
  \delta_4\epsilon_1 - \phi(x-y) - \epsilon_2]\;\Measure\overline{AB}
  + 2y\epsilon.
\end{multline*}

\index{Phi function}\index{Ratio of opposite sides of quadrilateral}Hence
$\phi(x+y) + \phi(x-y) - 2\phi(x)\phi(y) < \eta$ where $\eta$ may be
made less than any assigned value
\begin{equation*}
\phi(x+y) + \phi(x-y) = 2\phi(x)\phi(y)\tag{1}
\end{equation*}

This well-known equation may be easily solved. Let us
assume that the unit of measure of distance is well fixed
\[
\phi(0)=1, \quad \phi(2x)=2[\phi(x)]^2-1.
\]

Let $x_1$ be a value for $x$ in the interval to which the equation
applies, i.e.\ the measure of an actual distance. We may find
$k$ so that $\phi(x_1)=\cos\dfrac{x_1}{k}$\index{Cosine02@---
of distance}\index{Distance01@Distance of two points, cosine}. We have immediately
\[
\phi(2x_1) = \cos\frac{2x_1}{k}, \quad
  \phi\left(\frac{nx_1}{2^m}\right) =
  \cos\left(\frac{nx_1}{2^mk}\right).
\]
We also know that $\phi(x) - \cos\dfrac{x}{k}$ is a continuous function.
If, then, $x$ be any value of the argument, we may find $n$ and
$m$ such large integers that $x-\dfrac{nx_1}{2^m}$ is infinitesimal. Hence
$\phi(x)-\cos\dfrac{x}{k}$ will be less than any assigned quantity, or

\begin{equation*}
\phi(x)=\cos\frac{x}{k}.\tag{2}
\end{equation*}

%[File: 054.png]
The function cosine has, of course, a purely analytical
meaning, i.e.\ we write
\begin{equation*}
\phi (x) = 1 - \frac{x^2}{k^2\,.\,2!} + \frac{x^4}{k^4\,.\,4!} \cdots.
\end{equation*}


Of fundamental importance is the constant $k$. We shall
find that it gives the radius of a sphere (in our usual
euclidean geometry) upon which the non-euclidean plane
may be developed. We shall, therefore, define the constant
$\dfrac{1}{k^2}$ as the \emph{Measure of Curvature of
Space}\index{Measure02@--- of curvature of space}\index{Curvature06@--- of
space}\index{K@$\dfrac{1}{k^2}$ measure of curvature of space}\label{Curve}.%
\footnote{This fundamental concept is due to Riemann\index{Riemann}, loc.\ cit.
We shall consider it more fully in subsequent chapters, notably
XIX.}
%endfootnote
To find the
nature of the value of $k$, we see immediately that in the
parabolic case $\dfrac{1}{k^2} = 0$; in the elliptic $\phi$\index{Ratio of opposite sides of quadrilateral}
is, at most, equal
to $1$, hence $\dfrac{1}{k^2}$ is positive. In the hyperbolic case, $1$
constitutes a minimum value for $\phi$ and $\dfrac{1}{k^2}$ is negative, or $k$ a pure
imaginary. Under these circumstances, we may, if we choose,
remove all signs of imaginary values from (2) by writing
$k' = ik$,
\begin{equation*}
\phi (x) = \cosh \left(\frac{x}{k'}\right).
\end{equation*}
As a matter of fact, however, there is little or no gain in
doing this.

It is now necessary to calculate another limit, that of the
ratio of two simultaneously diminishing sides of a right
triangle. Let us, then, suppose that we have a right
$\triangle ABC$ whose right angle is $\measuredangle ABC$. We shall imagine that
$\overline{AB}$ becomes infinitesimal while $\measuredangle BAC$ is constant. We
seek the limit of $\dfrac{\overline{AB}}{\overline{AC}}$.%
\footnote{It is strange that G\'erard\index{Gerard@G\'erard}, loc.\ cit., assumes this ratio from
the euclidean case.}
%endfootnote
That such a limit will actually
exist may be proved by considerations similar to those which
established the existence of $\phi(x)$. We leave the details to
the reader. The limit is a function of the angle $\measuredangle BAC$, and
if $\theta$ be the measure of the latter, we may write our function
$f(\theta)$; including therein, of course, the possibility that this
function should be a constant.

First of all it is incumbent upon us to show that this
function is continuous. Take $C'$ on the extension of $(BC)$
beyond $C$, and let $\Delta \theta$ be the measure of $\measuredangle CAC'$.
If $\Delta \theta$ be
%[File: 055.png]
infinitesimal, then, by 2 $\overline{CC'}$ is infinitesimal as compared
with $\overline{AC}$. Hence
$\dfrac{\overline{AC'}}{\overline{AB}}-\dfrac{\overline{AC}}{\overline{AB}}$
will become and remain less than any assigned number, and
$f(\theta)$ is continuous.

Suppose, now, that we have two half-lines $|OY$, $|OZ$ lying
in a half-plane bounded by $|OX$. Let $\measuredangle XOY$
and $\measuredangle XOZ$
be each less than a right angle, and have the measures $\theta$,
$\theta + \phi; \phi < \theta$. Take $F$ on $|OZ$, and find $B$, so that
\[
\overline{OF} \equiv \overline{OB};\quad \measuredangle YOF \equiv
\measuredangle YOB,
\]
$|OB$ is within the interior angle $\measuredangle XOY$;
these points will certainly exist if $\overline{OF}$ be very
small. Connect $F$ and $B$ by a line meeting $|OY$ in $D$,
and through $F$, $D$, $B$ draw three lines perpendicular
to $|OX$, and meeting it in $E$, $C$, $A$ respectively, which
points also are sure to exist, if $\overline{OF}$ be small
enough. $C$ will
be separated from the middle point of $(EA)$ by a distance
infinitesimal compared with $\overline{EA}$, for the perpendicular
to $OX$
at such a point would meet $(BF)$ at a point whose distance
from $D$ was infinitesimal as compared with $\overline{OF}$.

\begin{flalign*}
\frac{\overline{OA}}{\overline{OB}} &= f(\theta - \phi) + \epsilon_{1},\\
%
\frac{\overline{OC}}{\overline{OB}} &= \frac{\overline{OC}}{\overline{OD}}\cdot
\frac{\overline{OD}}{\overline{OB}} = f(\theta)f(\phi)+\epsilon_{2},\\
%
\frac{\overline{CA}}{\overline{OB}} &= f(\theta - \phi) -
f(\theta)f(\phi)+\epsilon_{3}, \\
%
\frac{\overline{OE}}{\overline{OB}} &=
\frac{\overline{OE}}{\overline{OF}} = f(\theta +
\phi)+\epsilon_{4}\\
%
\frac{\overline{EC}}{\overline{OB}} &=  f(\theta)f(\phi)- f(\theta +\phi)+\epsilon_{5} \cdot
\frac{\overline{CA}}{\overline{OB}} -
\frac{\overline{EC}}{\overline{OB}} = \delta, \text{infinitesimal}.\\
%
&\hspace{4em} f(\theta + \phi)+ f(\theta - \phi)=2  f(\theta)f(\phi).
\end{flalign*}

\mysect{Functional equation for the cosine of an angle}
This is the functional equation that we had before, so that\index{Cosine01@Cosine of angle}
$f = \cos \dfrac{\theta}{l}$ and $l$ must be real. If, then we so
choose it that the measure of a right angle shall be
$\dfrac{\pi}{2}$,
\[f(\theta) = \cos \theta.
\]

%[File: 056.png]

Let us not fail to notice that since $\measuredangle ABC$ is a right angle
we have, by III\@. 17,

\begin{equation*}
\limdot\frac{\overline{BC}}{\overline{AC}} =
  \cos(\frac{\pi}{2} - \theta) = \sin\theta.\tag{3}
\end{equation*}

The extension of these functions to angles whose measures
are greater than $\dfrac{\pi}{2}$ will afford no difficulty, for, on the one
hand, the defining series remains convergent, and, on the
other, the geometric extension may be effected as in the
elementary books.

\mysect{Non-euclidean form for the pythagorean theorem}\index{Pythagorean theorem}
Our next task is a most serious and fundamental one, to
find the relations which connect the measures and sides and
angles of a right triangle\index{Right 02@--- triangle}. Let this be the $\triangle ABC$ with
$\measuredangle ABC$ as its right angle. Let the measure of $\measuredangle BAC$ be $\psi$
while that of $\measuredangle BCA$ is $\theta$. We shall assume that both $\psi$ and $\theta$
are less than $\dfrac{\pi}{2}$, an obvious necessity under the euclidean
or hyperbolic hypothesis, while under the elliptic, such will
still be the case if the sides of the triangle be not large, and
the case where the inequalities do not hold may be easily
treated from the cases where they do. Let us also call $a$, $b$, $c$
the measures of $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ respectively.

%[Illustration: FIG. 2.]
\imgcent{270}{f002}{Fig. 2}

We now make rather an elaborate construction.%
\footnote{See figure on next page.}
%endfootnote
Take $B_1$
in $(AB)$ as near to $B$ as desired, and $A_1$ on the extension
of $(AB)$ beyond $A$, so that $\overline{A_1A} \equiv \overline{B_1B}$, and construct
$\triangle A_1B_1C_1 \equiv \triangle ABC$, $C_1$ lying not far from $C$; a construction
which, by 1, is surely possible if $BB_1$ be small enough. Let
$B_1C_1$ meet $(AC)$ at $C_2$. $\measuredangle C_1C_2C$ will differ but little from
$\measuredangle BCA$, and we may draw $C_1C_3$ perpendicular to $CC_2$, where
$C_3$ is a point of $(CC_2)$. Let us next find $A_2$ on the extension
of $(AC)$ beyond $A$ so that $\overline{A_2A} \equiv \overline{C_2C}$ and $B_2$ on the extension
of $(C_1B_1)$ beyond $B_1$\label{beyondB2}
so that $\overline{B_1B_2} \equiv \overline{C_1C_2}$, which is certainly
possible as $\overline{C_1C_2}$ is very small. Draw $A_2B_2$. We saw that
$\measuredangle C_1C_2C$ will differ from $\measuredangle BCA$ by an infinitesimal (as $\overline{B_1B}$
decreases) and $\angle CC_1B_1$ will approach a right angle as a limit.
We thus get two approximate expressions for $\sin\theta$ whose
comparison yields
\[
\frac{\overline{C_1C_3}}{\overline{C_1C_2}} =
  \frac{\overline{CC_1}}{\overline{CC_2}} + \epsilon_1 =
  \frac{\cos\dfrac{a}{k}\overline{BB_1}}{\overline{CC_2}} + \epsilon_2,
\]
for $\overline{CC_1}-\cos\dfrac{a}{k}\overline{BB_1}$ is infinitesimal in comparison to $\overline{BB_1}$ or
%[File: 057.png]
$\overline{CC_1}$.
Again, we see that a line through the middle point,
of $(AA_1)$ perpendicular to $AA_2$ will also be perpendicular
to $A_1 C_1$, and the distance of the intersections will differ
infinitesimally from $\sin\psi\overline{AA_1}$.
We see that $\overline{C_1 C_3}$ differs by
a higher infinitesimal from
$\sin\psi\cos\dfrac{b}{k}\overline{AA_1}$, so that
\begin{equation*}
\cos \frac{b}{k} \sin \psi
   \frac{\overline{AA_1} }{\overline{CC_1}}
   + \epsilon_3 =
   \frac{\cos \dfrac{a}{k}\overline{BB_1} }
        {\overline{CC_2}} + \epsilon_2.
\end{equation*}
Next we see that
$\overline{A A_1} \equiv \overline{B B_1}$, and hence
\begin{equation*}
\cos \frac{b}{k} = \frac{1}{\sin \psi} \cos \frac{a}{k}
  \cdot \frac{\overline{C_1 C_2} }{\overline{CC_2} }
  + \epsilon_4.
\end{equation*}
Moreover, by construction
$ \overline{C_1 C_2} \equiv \overline{B_1 B_2}$,
$\overline{CC_2} \equiv \overline{AA_2}$.
A perpendicular to
$A A_1$ from the middle point of $(AA_2)$
will be
perpendicular to $A_2 B_2$, and the distance of the intersections
will differ infinitesimally from each of these expressions
\begin{flalign*}
&&& \sin \psi \overline{AA_2}, \quad
 \frac{1}{\cos \dfrac{c}{k} } \overline{B_1 B_2}. \\
%[File: 058.png]
&\text{Hence} \hfill &&   \cos \frac{b}{k} - \cos \frac{a}{k} \cos \frac{c}{k}
   < \epsilon, & \hfill \phantom{\text{Hence}}
\end{flalign*}
\begin{equation*}
   \cos \frac{b}{k} = \cos \frac{a}{k} \cos \frac{c}{k}.\tag{4}
\end{equation*}

\index{Cosines02@--- law of}To get the special formula for the euclidean case, we should
develop all cosines in power series, multiply through by $k^2$,
and then put $\dfrac{1}{k^2} = 0$, getting
\begin{equation*}
   b^2 = a^2 + c^2
\end{equation*}
the usual Pythagorean\index{Pythagorean theorem} formula.

\mysect{Trigonometric formulae for right and oblique triangles}
We have now a sufficient basis for trigonometry, the
development whereof merely requires a little analytic skill.
It may not perhaps be entirely a waste of time to work out
some of the fundamental formulae. Let $A$, $B$, $C$ be the
vertices of a triangle, and let us use these same letters, as
is usual in elementary work, to indicate the measures of the
corresponding angles, while the measures of the sides shall be
$a$, $b$, $c$ respectively. Begin by assuming that
$\measuredangle ABC$ is a right
angle so that $B = \dfrac{\pi}{2}$.
Let $D$ be such a point of $(AC)$ that $BD$
is perpendicular to $AC$; the measures of $ \overline{AD} $
and $ \overline{CD} $ being
$b_1$ and $b_2$, while the measure of $ \overline{BD}$ is $a_1$.
\begin{equation*}
   \cos \frac{b_1}{k} =
   \frac{\cos \dfrac{a}{k} }{\cos \dfrac{a_1}{k}}
    ,\quad
   \cos \frac{b_2}{k} =
   \frac{\cos \dfrac{c}{k} }{\cos \dfrac{a_1}{k}},
\end{equation*}
\begin{equation*}
   \cos \left( \frac{b_1 + b_2}{k} \right) =
   \cos \frac{b}{k} = \cos \frac{a}{k} \cos \frac{c}{k},
\end{equation*}
\begin{equation*}
   \cos \frac{a}{k} \cos \frac{c}{k} \left(1 - \cos^2 \frac{a_1}{k} \right)
    =
   \sqrt{\cos^2 \frac{a_1}{k} - \cos^2 \frac{a}{k}}
   \sqrt{\cos^2 \frac{a_1}{k} - \cos^2 \frac{c}{k}},
\end{equation*}
\begin{equation*}
  \cos^2 \frac{a}{k} - \cos^2 \frac{c}{k}
      \left( \cos^2 \frac{a_1}{k} -2 \right) =
  \cos^2 \frac{a_1}{k} - \cos^2 \frac{a}{k} -
      \cos^2 \frac{c}{k},
\end{equation*}
\begin{equation*}
  \left(1 - \cos^2 \frac{a_1}{k}\right)
    \left(1 - \cos^2 \frac{a}{k} \cos^2 \frac{c}{k}\right) =
  \left(1 - \cos^2 \frac{a}{k}\right)\left(1 - \cos^2 \frac{c}{k}\right),
\end{equation*}
\begin{equation*}
  \sin \frac{a_1}{k} \sin \frac{b}{k} =
     \sin \frac{a}{k} \sin \frac{c}{k},
\end{equation*}
\begin{equation*}
  \frac{\sin \dfrac{a}{k} }{\sin \dfrac{b}{k} } =
  \frac{\sin \dfrac{a_1}{k} }{\sin \dfrac{c}{k} }.
\end{equation*}

%[File: 059.png]
Now proceeding with the $\triangle ADB$ as we did with the $\triangle ABC$
we shall reach two more sines whose ratio is
\begin{equation*}
\frac{\sin\dfrac{a}{k}}{\sin\dfrac{b}{k}},
\end{equation*}
and so forth. Continuing thus we have in $(AB)$ and $(AC)$
two infinite series of points. Let the reader show that the
limit for each series cannot be other than the point $A$ itself.
Now we have just seen in (3) that the limit of this ratio
is $\sin A$, hence
\begin{equation*}
\sin\frac{a}{k}=\sin\frac{b}{k}\sin A.\tag{5}
\end{equation*}

Let the reader deduce from (4) and (5) that
\begin{align*}
\tan\frac{c}{k}&=\tan\frac{b}{k}\cos A.\tag{6} \\
\cos B&=\cos\frac{b}{k}\sin A.\tag{7}
\end{align*}

Let us next suppose that $\triangle ABC$ is any triangle. If none
of the angles be greater than a right angle, we may connect
any vertex with a point of the opposite side by a line
perpendicular to the line of that side, and we see at once that\index{Sines, law of}
\begin{equation*}
\sin\frac{a}{k} : \sin\frac{b}{k}:\sin\frac{c}{k}=\sin A:\sin B:\sin C.
\end{equation*}

Let us show that this formula holds universally, even when
this construction is not possible. Let us assume that $B >\dfrac{\pi}{2}$.
We may legitimately assume that $A$ and $C$ are less than $\dfrac{\pi}{2}$,
for the extreme case under the elliptic hypothesis where such
is not the fact may easily be treated after the simpler case
has been taken up. We shall still have
\begin{equation*}
\sin\frac{a}{k}:\sin\frac{c}{k}=\sin A:\sin C.
\end{equation*}
Let $E$ be that point of $(AC)$ which makes $BE$ perpendicular
to $AC$. Let the measures of $\overline{AE}$, $\overline{BE}$, and $\overline{CE}$ be $a'$, $b'$, $c'$,
while the measure of $\measuredangle ABE$ is $A'$ and that of $\measuredangle CBE$ is $C'$.

%[File: 060.png]
\[
  \cos A'=\frac{\tan\dfrac{b'}{k}}{\tan\dfrac{c}{k}},  \quad
  \cos C'=\frac{\tan\dfrac{b'}{k}}{\tan\dfrac{a}{k}},
\]
\[
  \sin A'=\frac{\sin\dfrac{a'}{k}}{\sin\dfrac{c}{k}},  \quad
  \sin C'=\frac{\sin\dfrac{c'}{k}}{\sin\dfrac{a}{k}},
\]
\[
\sin B=\sin(A'+C')
= \frac{\tan\dfrac{b'}{k}}{\sin\dfrac{a}{k} \sin\dfrac{c}{k}}
  \left( \cos\dfrac{c}{k}\sin\dfrac{c'}{k} + \cos\dfrac{a}{k}\sin\dfrac{a'}{k} \right),
\]
\[
  \cos\frac{c}{k}=\cos\frac{a'}{k}\cos\frac{b'}{k},  \quad
  \cos\frac{a}{k}=\cos\frac{c'}{k}\cos\frac{b'}{k},
\]
\[
  \sin B=\frac{\sin\dfrac{b'}{k}}{\sin\dfrac{a}{k}\sin\dfrac{c}{k}}
  \sin\left( \dfrac{a'}{k} + \dfrac{c'}{k} \right),
\]
\[
a'+c'=b;\quad
  \sin\frac{b'}{k} = \sin\frac{a}{k}\sin C = \sin\frac{c}{k}\sin A,
\]
\[
  \frac{\sin\dfrac{a}{k}}{\sin A} =
  \frac{\sin\dfrac{b}{k}}{\sin B} =
  \frac{\sin\dfrac{c}{k}}{\sin C}.
  \tag{8}
\]

\index{Sines, law of}Once more let us suppose that no angle of our triangle
is greater than a right angle, and let $D$ be such a point of
$(BC)$ that $AD$ is perpendicular to $BC$:
\begin{align*}
\cos\frac{b}{k} &=
  \frac{\cos\dfrac{\Measure\overline{DC}}{k} \cos\dfrac{c}{k}}
  {\cos\dfrac{\Measure\overline{BD}}{k}} \\
&= \frac{\cos\dfrac{c}{k}}{\cos\dfrac{\Measure\overline{BD}}{k}}
  \left[\cos\dfrac{a}{k}\cos\dfrac{\Measure\overline{BD}}{k} +
  \sin\dfrac{a}{k}\sin\dfrac{\Measure\overline{BD}}{k}\right] \\
&= \cos\frac{a}{k}\cos\frac{c}{k} +
    \sin\frac{a}{k}\sin\frac{c}{k}\cos B.
\end{align*}

%[File: 061.png]

If $B>\dfrac{\pi}{2}$ this proof is invalid. Here, however, following
our previous notation
\[
  \cos B = \cos(A'+C') = \frac{\tan^2\dfrac{b'}{k} \cos\dfrac{a}{k}
    \cos\dfrac{c}{k} - \sin\dfrac{a'}{k}
    \sin\dfrac{c'}{k}}{\sin\dfrac{a}{k} \sin\dfrac{c}{k}},
\]
\[
  \cos\frac{a}{k} = \cos\frac{b'}{k} \cos\frac{c'}{k},\quad
  \cos\frac{c}{k} = \cos\frac{b'}{k} \cos\frac{a'}{k},\quad b =
  a'+c',
\]
\begin{align*}
    \cos B &= \frac{\sin^2\dfrac{b'}{k} \cos\dfrac{a'}{k}
      \cos\dfrac{c'}{k} - \sin\dfrac{a'}{k}
      \sin\dfrac{c'}{k}}{\sin\dfrac{a}{k} \sin\dfrac{c}{k}}\\
    &= \frac{\cos\dfrac{b}{k} - \cos\dfrac{a}{k}
      \cos\dfrac{c}{k}}{\sin\dfrac{a}{k} \sin\dfrac{c}{k}},\\
    \cos\frac{b}{k} &= \cos\frac{a}{k} \cos\frac{c}{k} +
      \sin\frac{a}{k} \sin\frac{c}{k} \cos B.\tag{9}
\end{align*}

A correlative formula may be deduced as follows:%
\footnote{I owe this
  ingenious trigonometric analysis to my former pupil Dr. Otto
  Dunkel\index{Dunkel}.}
%endfootnote
\begin{flalign*}
& \text{\indent Let}
& \frac{\sin\dfrac{a}{k}}{\sin A}&= \frac{\sin\dfrac{b}{k}}{\sin B} =
  \frac{\sin\dfrac{c}{k}}{\sin C} = \lambda \neq 0, &\hfill \phantom{\text{\indent Let}} \\
\end{flalign*}\ungap
\begin{multline*}
  \cos^2\frac{b}{k} + \lambda^4\sin^2A\sin^2C\cos^2B -
    2\lambda^2\sin A\sin C\cos B\cos\frac{b}{k} =\\
  = \cos^2\frac{a}{k} \cos^2\frac{c}{k},
\end{multline*}\ungap
\begin{multline*}
  1 - \lambda^2\sin^2B + \lambda^4\sin^2A\sin^2C\cos^2B -
    2\lambda^2\sin A\sin C\cos B\cos\frac{b}{k} =\\
  = 1 - \lambda^2\sin^2A - \lambda^2\sin^2C +
    \lambda^4\sin^2A\sin^2C,
\end{multline*}\ungap
\begin{flalign*}
  &\quad\sin^2A + \sin^2C - \sin^2B
  &= \sin^2A\sin^2C\sin^2\frac{b}{k} + 2\sin A\sin C\cos B\cos\frac{b}{k},\quad
\end{flalign*}\ungap
\begin{multline*}
  1 - \sin^2A - \sin^2C + \sin^2A\sin^2C \\
  = \sin^2A\sin^2C\cos^2\frac{b}{k} - 2\sin A\sin C\cos\frac{b}{k}\cos
  B + \cos^2B,
\end{multline*}\ungap
%[File: 062.png]
\begin{align*}
  \cos A\cos C &= \cos\frac{b}{k}\sin A\sin C - \cos B,\\
  \cos B &= -\cos A\cos C + \sin A\sin C\cos\frac{b}{k}.\footnotemark\tag{10}
\end{align*}
\footnotetext{In finding this formula we have extracted a square
  root. To be sure
  that we have taken the right sign, we have but to consider the limiting
  case $A=0, B=\pi-C$.}

If $ABCD$ be an isosceles birectangular quadrilateral, the
right angles being at $A$ and $B$,
\begin{equation*}
  \cos\frac{\Measure\overline{CD}}{k} =
  \cos\frac{\Measure\overline{AC}}{k}
  \cos\frac{\Measure\overline{BD}}{k}
  \cos\frac{\Measure\overline{AB}}{k} +
  \sin\frac{\Measure\overline{AC}}{k}
  \sin\frac{\Measure\overline{BD}}{k}.\tag{11}
\end{equation*}

The proof of this is left to the reader, as well as the task of
showing that the formulae which we have here established
are identical with those for a euclidean sphere of radius $k$.
Let him also show that when $\dfrac{1}{k^2}=0$, our formulae pass over
into those for the euclidean plane.
%[File: 063.png]

%CHAPTER V
\mychap{V}{ANALYTIC FORMULAE}
\mysect{Directed distances}\index{Distance, directed}

At the beginning of Chapter I we posited the existence
of two undefined objects, points and distances. Between the
two existed the relation that the existence of two points
implied the existence of a single object, their distance. In
this relation the two points entered symmetrically.

These concepts may be further sharpened as follows.
Leaving aside the trivial case of the null distance, let us
imagine that a distinction is made between the two points,
the one being called the \emph{initial} and the other the \emph{terminal}
point. The concept distance, where this distinction is made
between the two points shall be called a \emph{directed distance},
or, more specifically, the directed distance from the initial\index{Initial point}
to the terminal point\index{Terminal point}. Any not null distance will, thus,
determine two directed distances. The directed distance from
$A$ to $B$ shall be written $\overrightarrow{AB}$. The relations congruent to,
greater than, and less than, when applied to directed distances,
shall mean that the corresponding distances have these
relations.

Suppose that we have two congruent segments ($AB$) and
($A'B'$) of the same line. It may be that a congruent transformation
which carries the line into itself, and transforms
$A$ and $B$ into $A'$ and $B'$, also transforms $A'$ into $A$. In this
case the middle point of ($AA'$) will remain invariant, the
extremities of every segment having this middle point will
be interchanged. Such a transformation shall be called a
\emph{reflection}\index{Reflection02@--- in point} in this middle point. Conversely, we easily see
that a congruent transformation whereby $A$ goes into $A'$,
and one other point of ($AA'$) also goes into a point of that
segment, is a reflection in the middle point of the segment.

\mysect{Group of translations of a line}
There are, however, other congruent transformations of the
line into itself besides reflections. For if $A$ go into $A'$, and
any point of ($AA'$) go into a point not of ($AA'$), then $A$ will
be the only point of ($AA'$) which goes into a point thereof,
there will be no invariant point on the line, and we have
a different form of congruent transformation called a \emph{translation}\index{Translations}.
It is at once evident that every congruent transformation
%[File: 064.png]
of the line into itself is either a reflection or a translation.
The inverse of a translation is another translation; the inverse
of a reflection is the reflection itself.

\smallskip\emph{Theorem}~1. The product of two translations is a translation\index{Translations}.
The assemblage of all translations is a group.

We see, to begin with, that every congruent transformation
has an inverse. This premised, suppose that we have a
translation whereby $A$ goes into $A'$, and a second whereby
$A'$ goes into $A''$. We wish to show that the product of
these two is not a reflection. Suppose, in fact, that it were.
A point $P_1$ of ($AA''$) close to $A$ must then go into another
point $P_3$ of ($AA''$) close to $A''$. If $A'$ be a point of ($AA''$), the
first translation will carry $P_1$ into $P_2$ a point of ($A'A''$), and
as $P_3$ is also a point of ($A'A''$) the second transformation
would be a reflection, and not a translation. If $A$ were
a point of ($A'A''$), $P_2$ would be a point of ($AA'$), and hence
of ($A'A''$), leading to the same fallacy. If $A''$ were a point of
($AA'$), $P_2$ would belong to the extension of ($A'A''$) beyond $A'$,
and $P_3$ would belong to ($A'A''$) and not to ($AA''$).

Let the reader show that the product of a reflection and
a translation is a reflection, and that the product of two
reflections is a translation.

\smallskip\emph{Definition}.~Two congruent directed distances of the same
line shall be said to have the same \emph{sense}\index{Sense01@Sense of directed distances}, if the congruent
transformation which carries the initial and terminal points
of the one into the initial and terminal points of the other be
a translation. They shall be said to have \emph{opposite senses}
if this transformation be a reflection. The following theorem
is obvious---

\smallskip\emph{Theorem}~2. The two directed distances determined by a
given distance have opposite senses.

Suppose, next, that we have two non-congruent directed
distances $\overrightarrow{AB}$, $\overrightarrow{A'C'}$ upon the same line,
so that $\overline{A'C'} > \overline{AB}$.
There will then (XIII) be a single such point $B'$ of ($A'C'$) that
$\overline{AB} \equiv \overline{A'B'}$. If then, $\overrightarrow{AB}$ and $\overrightarrow{A'B'}$ have the same sense, we
shall also say that $\overrightarrow{AB}$ and $\overrightarrow{A'C'}$ have the \emph{same sense}, or
\emph{like senses}. Otherwise, they shall be said to have \emph{opposite senses}\index{Opposite03@Opposite senses}.
The group theorem for translations gives at once---

\smallskip\emph{Theorem}~3. Two directed distances which have like or
opposite senses to a third, have like senses to one another,
and if two directed distances have like senses, a sense like
(opposite) to that of one is like (opposite) to that of the other,
%[File: 065.png]
while if they have opposite senses, a sense like (opposite)
to that of one is opposite (like) to that of the other.

\mysect{Positive and negative directed distances}\index{Sense01@Sense of directed distances}
Let us now make suitable conventions for the measurement
of directed distances. We shall take for the absolute value
of the measure of a directed distance, the measure of the
corresponding distance. Opposite directed distances of the
same line shall have measures with opposite algebraic signs.
If, then, we assign the measure for a single directed distance
of a line, that of every other directed distance thereof is
uniquely determined.\mysect{Coordinates of a point on a line}
If, further, we choose a fixed origin $D$
upon a line and a fixed unit for directed distances, every
point $P$ of the line will be completely determined by a single
coordinate
\[
x = \sin{\frac{\Measure\overrightarrow{OP}}{k}}.
\]

In an entirely similar spirit we may enlarge our concepts of
angle, and dihedral angle, to directed angle. We choose an
initial and a terminal side or face, and define as rotations
a certain one parameter, group of congruent transformation
which keep the vertex or edge invariant. We thus arrive
at the concept for sense of an angle, and set up a coordinate
system for half-lines or half-planes of common bound. If in
the $\measuredangle ABC$, $|AB$ be taken as initial side, the resulting directed
angle shall be written $\measuredangle \overrightarrow{AB}C$.

\mysect{Coordinates of a point in a plane}
We have at last elaborated all of the machinery necessary
to set up a coordinate system in the plane, and nearly all that
is necessary to set up coordinates\index{Coordinates02@Co-ordinates of a point} in space. Let us begin with
the plane, and choose two half-lines\index{Half-line} $|OX$, $|OY$ making a right
angle. Their lines shall naturally be called the \emph{coordinate
axes}\index{Axes, co-ordinate}, while $O$ is the \emph{origin}\index{Origin}. Let $P$ be any point of the plane,
the measure of $\overrightarrow{OP}$ being $\rho$, while those of $\measuredangle \overrightarrow{XOP}$
and $\measuredangle \overrightarrow{YOP}$ are $\alpha$ and $\beta$ respectively. We may then put
\[
\tag{1}
\begin{aligned}
\xi    &= k\sin{\frac{\rho}{k}}\cos{\alpha},\\
\eta   &= k\sin{\frac{\rho}{k}}\cos{\beta},\\
\omega &= \cos{\frac{\rho}{k}},
\end{aligned}
\]
with the further equation
\[
\xi^2 + \eta^2 + k^2\omega^2 = k^2.
\]
%[File: 066.png]

In practice it is better to use in place of $\xi$, $\eta$, $\omega$\label{zeta} homogeneous
coordinates defined as follows:---
\[
\tag{2}
\begin{aligned}
\omega &= \frac{x_0}{\sqrt{x_0{}^2 + x_1{}^2 + x_2{}^2}},\\
\xi    &= \frac{kx_1}{\sqrt{x_0{}^2 + x_1{}^2 + x_2{}^2}},\\
\eta   &= \frac{kx_2}{\sqrt{x_0{}^2 + x_1{}^2 + x_2{}^2}}.
\end{aligned}
\]

\mysect{Finite and infinitesimal distance formulae, the non-euclidean plane as a surface of constant Gaussian curvature}
What shall we say as to the signs to be attached to the
radicals appearing in these denominators? In the hyperbolic
case $\omega$ is essentially positive, so that the radical must have the
same sign as $x_0$. In the elliptic case it is not possible to have
two points, one with the coordinates $\xi$, $\eta$, $\omega$ and the other with
the coordinates $-\xi$, $-\eta$, $-\omega$, for their distance would be $k\pi$,
and the opposite angle of every triangle containing them both
would be straight, i.e.\ they might be connected by many
straight lines. On the other hand, it is not possible that
$\xi$, $\eta$, $\omega$ and $-\xi$, $-\eta$, $-\omega$ should refer to the same point, for
then that point would determine with itself two distinct
distances, which is contrary to Axiom II\@. Hence, in every
case, the radical must have a well-defined sign in order that
equations should give a point of our space.

In the limiting parabolic case
\[
\xi = \rho\cos{\alpha},\quad \eta = \rho\cos{\beta},\quad \omega = 1.
\]
The formula for the distance of two points $P$ and $P'$ with
coordinates $(x)$, $(x')$ is
\begin{align*}
\cos{\frac{\Measure\overline{PP'}}{k}} &=
\cos{\frac{\rho}{k}}\cos{\frac{\rho'}{k}} +
\sin{\frac{\rho}{k}}\sin{\frac{\rho'}{k}}\cos{(\alpha'-\alpha)}\\
&=\omega\omega' + \frac{\xi\xi' + \eta\eta'}{k^2}.
\\
\tag{3}
\cos{\frac{\Measure\overline{PP'}}{k}} &=
\frac{x_0 x_0{}' +  x_1 x_1{}' + x_2 x_2{}'}
{\sqrt{x_0{}^2 + x_1{}^2 + x_2{}^2}\;
 \sqrt{x_0{}^{'2} + x_1{}^{'2} + x_2{}^{'2}}}
\\
\tag{4}
\sin{\frac{\Measure\overline{PP'}}{k}} &=
\frac{\sqrt{\left\|
\begin{array}{ccc}
x_0 &x_1 &x_2	\\
x_0{}'& x_1{}' &x_2{}'
\end{array}
\right\|^2
}}
{\sqrt{x_0{}^2 + x_1{}^2 + x_2{}^2}\;
 \sqrt{x_0{}^{'2} + x_1{}^{'2} + x_2{}^{'2}}}
\end{align*}


The signs of the radicals in the denominators are, as we
have seen, well determined. The sign of the radical in the
numerator of (4), should be so taken as to give a positive
%[File: 067.png]
value to the whole. Should we seek the measures of directed
distances on the line $PP'$, then, after the adjunction of the
value of the sign of a single directed distance\index{Distance, directed}, that of every
other is completely determined. In the euclidean case
\[
\Measure\overline{PP'} =
\frac{1}{x_0 x_0{}'}\sqrt{(x_1 x_0{}' - x_0 x_1{}')^2 + (x_2 x_0{}' - x_0 x_1{}')^2}.
\]

Returning to (4) and putting $x_i{}' = x_i + dx_i$ we get for the
infinitesimal element\index{Distance04@Distance element} of arc
\[
\frac{ds^2}{k^2} =
\frac{
\left\|
\begin{array}{ccc}
x_0 &x_1 &x_2	\\
dx_0& dx_1 &dx_2
\end{array}
\right\|^2
}{(x_0{}^2 + x_1{}^2 + x_2{}^2)^2}.
\]
\begin{flalign*}
&\text{\indent Put}\hfill
&x  &= \frac{kx_1}{x_0},\quad
y  = \frac{kx_2}{x_0},\quad
x' = x + dx,\quad
y' = y + dy,\hfill&\phantom{\text{\indent Put}}
\end{flalign*}
\[
ds^2 = \frac{dx^2 + dy^2 + \dfrac{(ydx-xdy)^2}{k^2}}
{\Bigl[1 + \dfrac{x^2+y^2}{k^2}\Bigr]^2}. \tag{5}
\]


In the limiting euclidean case $\dfrac{1}{k^2} = 0$,
\[
ds^2 = dx^2 + dy^2.
\]

Returning to the general case, we may improve our formula
(5) as follows:---
\begin{flalign*}
&\text{let} \hfill & z  = &\sqrt{k^2 + x^2 + y^2},\quad
dz = \frac{xdx + ydy}
{\sqrt{k^2 + x^2 + y^2}}.  \\
&\text{\indent If}\hfill & dx^2 &+ dy^2 - dz^2 = d\sigma^2,
\quad ds = \frac{kd\sigma}{z}.  \\
&\text{\indent Put }\hfill && u = \frac{2kx}{k-z},\quad v = \frac{2ky}{k-z}. & \hfill\phantom{\text{}}
\end{flalign*}
\[
1 + \frac{u^2 + v^2}{4k^2} = \frac{-2z}{k-z},
\]
\[
\begin{aligned}
du^2 + dv^2 = & \frac{4k^2}{(k-z)^4}[(k-z)^2[dx^2 + dy^2]\\
& + 2(k - z)(xdx + ydy)dz + (x^2 + y^2)dz^2],
\end{aligned}
\]
\[
\begin{aligned}
\frac{(k - z)^2}{4k^2}(du^2 + dv)^2
&= \left[dx^2 + dy^2
+ \frac{2zdz^2}{k - z}
- \frac{k^2 - z^2}{(k - z)^2}dz^2\right]\\
&= d\sigma^2.\\
du^2 + dv^2 &= ds^2 \frac{4z^2}{(k -z)^2},
\end{aligned}
\]
%[File: 068.png]
\begin{equation*}
ds^2=\left[1+\frac{u^2+v^2}{4k^2}\right]^{-2}
  \left(du^2+dv^2\right).
  \tag{6}
\end{equation*}

Comparing this with the usual distance formula\index{Distance04@Distance element}
\[
ds^2=Edu^2+2Fdu\,dv+Gdv^2,
\]
\[
F=0,\quad E=G=\Bigl[1+\frac{u^2+v^2}{4k^2}\Bigr]^{-2}.
\]

Now if $K$ be the measure of curvature of the surface having
this distance formula
\begin{align*}
K & = -\frac{1}{2E}\Bigl(\frac{\partial^2\log E}{\partial u^2} +
                           \frac{\partial^2\log E}{\partial v^2}
                     \Bigr), \\
K & = \Bigl[1+\frac{u^2+v^2}{4k^2}\Bigr]^2
  \Biggl[
    \frac{\Bigl[ 1+\dfrac{u^2+v^2}{4k^2}        \Bigr]
           \Bigl( \dfrac{1}{2k^2}+\dfrac{1}{2k^2} \Bigr)
         - \dfrac{u^2}{4k^4} - \dfrac{v^2}{4k^4}          }
         {\Bigl[ 1+\dfrac{u^2+v^2}{4k^2} \Bigr]^2      }
  \Biggr], \\
K & = \frac{1}{k^2}.
\end{align*}

\smallskip\emph{Theorem}~4. The non-euclidean plane may be developed upon
a surface of constant curvature $\dfrac{1}{k^2}$ in euclidean space.

We shall return to questions of this sort in Chapters XV
and XIX%
\footnote{The idea of interpreting the non-euclidean plane as a surface of constant
curvature\index{Curvature02@--- Gaussian} in euclidean space must certainly have been present to Riemann's\index{Riemann}
mind, loc.\ cit. The credit for first setting the matter in a clear light is,
however, due to Beltrami\index{Beltrami}. See his `Teoria fondamentale degli spazii di
curvatura constante', \emph{Annali di Matematica}, Serie 2, vol.~ii, 1868, and
`Saggio
d'interpretazione della geometria non-euclidea', \emph{Giornale di Matematiche},vol.~vi, 1868.}
%endfootnote
of this work.

\mysect{Equation connecting direction cosines of a line}\index{Cosines01@Cosines, direction}
Let us now take up coordinates in three dimensions. We
must make some preliminary remarks about the direction
cosines of a half-line. Suppose, in fact, that we have three
mutually perpendicular half-lines, $|OX$, $|OY$, $|OZ$, and a
fourth half-line $|OP$. The angles $\measuredangle XOP$,
$\measuredangle YOP$, $\measuredangle ZOP$
whose measures shall be $\alpha$, $\beta$, $\gamma$ respectively, shall be called
the \emph{direction angles} of the half-line $|OP$. These angles are
not directed, but this will cause no inconvenience, as we shall
introduce them merely through the expressions $\cos \alpha$, $\cos \beta$.
$\cos \gamma$. These shall be called the \emph{direction cosines} of the half-line\index{Half-line},
$O$ shall be the origin, and $OX$, $OY$, $OZ$ the coordinate
axes\index{Axes, co-ordinate}, while the planes\index{Plane} determined by them are the coordinate
planes. Take a second half-line $|OP'$, with direction cosines
$\cos\alpha'$, $\cos\beta'$, $\cos\gamma'$. We shall imagine that
$\overline{OP}$ and $\overline{OP'}$ are
%[File: 069.png]
infinitesimal. Under these circumstances, we may find
$A$, $B$, $C$ where perpendiculars to the axes through $P$ meet
them, and $A'$, $B'$, $C'$ bearing the same relation to $P'$. Let $Q'$ be
that point of $|OP'$ which makes $\measuredangle PQ'O$ a right angle, and let
$\measuredangle POP'$ have a measure $\theta$. Now we know that geometry
in the infinitesimal domain\index{Infinitesimal domain} obeys the euclidean hypothesis,
hence we have
\[
  \Measure\overline{OQ'}
= \Measure\overline{OP}\cos\theta + \epsilon,
\]
the $\epsilon$ is infinitesimal as compared with $\Measure\overline{OP}$. In the same
spirit
\begin{flalign*}
&&  \Measure\overline{OQ'}
= \Measure&\overline{OA} \cos \alpha'
+ \Measure\overline{OB} \cos \beta'
+ \Measure\overline{OC} \cos \gamma' + \delta. \phantom{\text{clearly}} \\
&\text{\indent But clearly} &&\Measure\overline{OA}
= \Measure\overline{OP}\cos\alpha + \epsilon, \;\text{\&c.}
\end{flalign*}
Hence
\[
\Measure\overline{OP} \cos \theta
= \Measure\overline{OP} [\cos\alpha \cos\alpha'+
  \cos \beta \cos \beta' + \cos\gamma \cos \gamma']+\eta,
\]
or dividing out $\Measure\overline{OP}$,
\begin{equation*}
\tag{7}
\cos\theta = \cos\alpha \cos\alpha'+
   \cos\beta \cos\beta' +\cos\gamma \cos\gamma'.
\end{equation*}
In particular we shall have
\begin{equation*}
\tag{8}
1 = \cos^2 \alpha + \cos^2\beta + \cos^2\gamma.
\end{equation*}

\mysect{Coordinates of a point in space}
We now set up our coordinate system\index{Coordinates02@Co-ordinates of a point} as follows:---
\[
\tag{9}
\begin{aligned}
  \omega &=  \cos\frac{\Measure\overline{OP}}{k}, \\
  \xi    &= k\sin\frac{\Measure\overline{OP}}{k}\cos\alpha, \\
  \eta   &= k\sin\frac{\Measure\overline{OP}}{k}\cos\beta, \\
  \zeta  &= k\sin\frac{\Measure\overline{OP}}{k}\cos\gamma, \\
  k^2    &= \xi^2 + \eta^2 + \zeta^2 + k^2\omega^2.
\end{aligned}
\]

From these we pass, as before, to homogeneous coordinates
$x_0 : x_1 : x_2 : x_3$. But first we shall introduce a new symbol:
\begin{equation*}
\tag{10}
(xy) \equiv x_0y_0 + x_1y_1 + x_2y_2 + x_3y_3.
\end{equation*}

We then write
\[
  \omega = \frac{x_0}{\sqrt{(xx)}},\quad
  \eta   = \frac{kx_2}{\sqrt{(xx)}},
\]
\begin{equation*}
\tag{11}
  \xi  = \frac{kx_1}{\sqrt{(xx)}},\quad
  \zeta = \frac{kx_3}{\sqrt{(xx)}}.
\end{equation*}

%[File: 070.png]

Here, as in the case of the plane, there is no ambiguity arising
from the double sign of the radical. There is, however, one
modification which we shall occasionally make. We see,
in fact, that in the hyperbolic case, since $k^2 < 0$; $\xi$, $\eta$, $\zeta$, $\omega$ are
real, we must have $(xx) < 0$, and $x_0$ is a pure imaginary. To
remedy this let us write
$$
  k \dot x_0 = x_0,\  \dot x_1 = x_1,\  \dot x_2 = x_2,\  \dot x_3 = x_3.
$$
A point will now have real coordinates. This distinction
between coordinates $(x)$ and coordinates $(\dot x)$ shall be consistently maintained in the hyperbolic case.

The cosine of the measure of distance of two points\index{Distance01@Distance of two points, cosine} $(x)$ and
$(y)$ is easily found. We see at once that we shall have
\begin{equation*}
  \cos \frac{\Measure\overline{PP^\prime}}{k} = \frac{(xy)}{\sqrt{(xx)}\; \sqrt{(yy)}}. \tag{12}
\end{equation*}

\mysect{Congruent transformations and orthogonal substitutions}
Let us now see what effect a congruent
transformation\index{Congruent transformations}\index{Transformations, congruent} will
have upon our coordinates. First take a congruent transformation
keeping the origin invariant. We see at once that
the new direction cosines\index{Cosines01@Cosines, direction}, and so the new coordinates $(x^\prime)$, will
be linear functions of the old ones; for a plane through the
origin will be characterized by a linear relation\index{Collineations} connecting
the direction cosines of the half-lines with that bound. The
variables $\xi$, $\eta$, $\zeta$ are thus linearly transformed in such a way
that $\xi^2 + \eta^2 + \zeta^2$ has a constant value, while $\omega$ is unaltered.
Hence $x_0$, $x_1$, $x_2$, $x_3$ are linearly transformed so that $(xx)$
is an invariant (relative), i.e.\ they are subjected to an orthogonal
substitution\index{Orthogonal01@Orthogonal substitutions}.

Let us next suppose that we have a congruent transformation which
carries the planes $\xi = 0$ and $\eta = 0$ into themselves,
and every half-plane with this axis as bound into itself.
The assemblage of all such transformations will form a one-parameter
group, and this group may be represented by
  \begin{align*}
    \omega^\prime &= \omega \cos \dfrac{d}{k} + \zeta \sin \dfrac{d}{k}, \\
    \xi^\prime &= \xi, \\
    \eta^\prime &= \eta, \\
    \zeta^\prime &= -\omega \sin \dfrac{d}{k} + \zeta \cos \dfrac{d}{k}.
  \end{align*}

We see, in fact, that by this transformation every point
receives just the coordinates that it would obtain by a
translation of the axis $OZ$ into itself through a distance $d$,
so enlarged as to carry into itself every half-plane through
that axis. Once more we find that, in the coordinates $(x)$,
%[File: 071.png]
this will be an orthogonal substitution\index{Orthogonal01@Orthogonal substitutions}. Now, lastly, every
congruent transformation of space may be compounded out
of transformations of these two types. Hence:

\smallskip\emph{Theorem}~5. Every congruent
transformation\index{Congruent transformations}\index{Transformations, congruent} of space is
represented by an orthogonal substitution in the homogeneous
variables $x_0:x_l:x_2:x_3$.

In Chapter VIII we shall make a detailed study of these
congruent transformations. For the present, let us begin by
noticing that the coordinate planes have linear equations, and
as we may pass from one of these to any other plane by
linear transformations\index{Collineations}, so the equation of any plane\index{Plane} may
be written
\[
(ux) \equiv u_0x_0 + u_1x_1 + u_2x_2 + u_3x_3 = 0.
\]

\mysect{Fundamental formulae for distance and angle}
We see that $(xy)$, $(ux)$, $(uv)$ are concomitants of every
congruent transformation, and we shall use them to find
expressions for the distance from a point to a plane and the
angle between two planes\index{Angle10@--- of two planes, cosine}. The existence of the former of
these quantities is contingent upon the existence of a point
in the plane determining with the given point a line perpendicular
to the plane.

Let the plane $(u)$ be that which connects the axis $x_1 = x_2 = 0$
with the point $(y)$. Its equation is $y_2x_1 - y_1x_2 = 0$. The
cosines\index{Cosine01@Cosine of angle} of the angles which this makes with the plane $v_1x_1 = 0$
are the $x_2$ direction cosines of the two half-lines of $OP$. If
then, the measure of the angle be $\theta$, we have
\[
\cos \theta = \frac{y_2}{\sqrt{y_1{}^2 + y_2{}^2}} =
\frac{v_1y_2}{\sqrt{y_1{}^2 + y_2{}^2}\;\sqrt{v_1{}^2}} =
\frac{(uv)}{\sqrt{(uu)}\;\sqrt{(vv)}}.
\]
But both sides of this equation are absolute invariants for all
congruent transformations. Hence, we may write, in general:
\begin{equation*}
\cos \theta = \frac{(uv)}{\sqrt{(uu)}\;\sqrt{(vv)}}. \tag{13}
\end{equation*}

We find the distance from a point to a plane in the same
way. Let the point be $(x)$ and $d$ the distance thence to the
point where a perpendicular to the plane $u_1x_1=0$ meets it,
this being, by definition, the distance from the point to the
plane.\index{Sine of distance from point to plane}\index{Distance02@--- from point to plane}
\[
\sin \frac{d}{k} = \frac{\xi}{k} = \pm \frac{x_1}{\sqrt{(xx)}} =
\frac{u_1x_1}{\sqrt{(xx)}\;\sqrt{(uu)}}.
\]

Once more we have an invariant form, so that, in general:

\begin{equation*}
\sin \frac{d}{k} = \frac{(ux)}{\sqrt{(uu)}\;\sqrt{(xx)}}. \tag{14}
\end{equation*}

%[File: 072.png]
The sign of $\sqrt{(xx)}$ is determined. As for that of $\sqrt{(uu)}$, by
reversing it, we get opposite directed distances of the same line.

We have now reached the end of the first stage of our
journey. Our system of axioms has given us a large body
of elementary doctrine, a system of trigonometry, and a
system of analytic geometry wherein the fundamental metrical
invariants are easily expressed. All of these things will be of
use later. At present our task is different. We must show
that the system of axioms which has carried us safely so far,
will not break down later; i.e.\ that these axioms are essentially
compatible. We must also grapple with a disadvantage
which has weighed heavily upon us from the start, rendering
trebly difficult many a proof and definition. In Axiom XI we
assumed that any segment might be extended beyond either
extremity. Yes, but how far may it be so extended? This
question we have not attempted to answer, but have dealt
with the geometry of such a region as the inside of a sphere,
not including the surface. In fact, had we assumed that every
segment might be extended a given amount, we should have
run into a difficulty, for in elliptic space no distance may have
a measure $k\pi$ under our axioms.

The matter may be otherwise stated. Every point will
have a set of coordinates in our system. What is the extreme
limit of possibility for making points correspond to coordinate
sets, and what meaning shall we attach to coordinates to
which no point corresponds? We must also adjoin the complex
domain for coordinates, and give a new interpretation to
our fundamental formulae (12), (13), (14) covering the most
general case. Then only shall we be able to continue our
subject in the broadest and most scientific spirit.

%[File: 073.png]

%CHAPTER VI
\mychap{VI}{CONSISTENCY AND SIGNIFICANCE OF THE AXIOMS}\label{ChapVI}
\mysect{Examples of geometries satisfying the assumptions made}

The first fundamental question suggested at the close of
the last chapter was this. How shall we show that those
assumptions which we made at the outset are, in truth,
mutually consistent? We need not here go into that elusive
question which bothers the modern student of pure logic,
namely, whether any set of assumptions can ever be shown
to be consistent. All that we shall undertake to do is to
point to familiar sets of objects which do actually fulfil our
fundamental laws.

Let us begin with the geometry of the euclidean hypothesis\index{Euclidean01@Euclidean hypothesis},
and take as points any class of objects which may be put into
one to one correspondence with all triads of values of three
real independent variables $x$, $y$, $z$. By the distance\index{Distance} of two
points we shall mean the positive value of the expression
\begin{equation*}
\sqrt{(x'-x)^2+(y'-y)^2+(z'-z)^2}.
\end{equation*}
The sum of two distances shall be defined in the arithmetical
sense. It is a perfectly straightforward piece of algebra to
show that such a system of objects will obey all of our axioms
and the euclidean hypothesis; hence the consistency of our
axioms rests upon the consistency of the number system,
and that we may take as indubitable. Be it noticed that
we have another system of objects which obey all of our
axioms if we make the further assumption that
\begin{equation*}
x^2+y^2+z^2<|.
\end{equation*}

The net result, so far, is this. If we take our fundamental
assumptions and the euclidean hypothesis, points and distances
may be put into one to one correspondence with
expressions of the above types; and, conversely, any system
of geometry corresponding to these formulae will be of the
euclidean type. The elementary geometry of Euclid\index{Euclid} fulfils
these conditions. In what immediately follows we shall
assume this geometry as known, and employ its terminology.

Let us now exhibit the existence of a system of geometry
obeying the hyperbolic hypothesis\index{Hyperbolic01@Hyperbolic hypothesis}. We shall take as our
%[File: 074.png]
class of points the assemblage of all points in euclidean space
which lie within, but not upon, a sphere\index{Sphere} of radius unity.
We shall mean by the distance\index{Distance} of two points one half the
real logarithm of the numerically larger of the two cross ratios\index{Cross ratios}
which they make with the intersections of their line with
the sphere. The reader familiar with projective geometry
will see that the segment of two points in the non-euclidean
sense will be coextensive with their segment in the euclidean
sense, and the congruent group will be the group of
collineations\index{Transformations, congruent}\index{Congruent transformations}
which carry this sphere into itself. Lastly, we see that
we must be under the hyperbolic hypothesis\index{Hyperbolic01@Hyperbolic hypothesis}, for a line is
infinitely long, yet there is an infinite number of lines through
a given point, coplanar with a given line, which yet do not
meet it.

The elliptic\index{Elliptic02@--- hypothesis} case is treated similarly. We take as points
the assemblage of all points within a euclidean sphere of
small radius, and as the distance of two points $\dfrac{1}{2i}$ times, the
natural logarithm of a cross ratio which they determine with
the intersection of their line with the imaginary surface
\begin{equation*}
{x_0}^2+{x_1}^2+{x_2}^2+{x_3}^2=0.
\end{equation*}
By a proper choice of the cross ratio and logarithm, this
expression may be made positive, as before. The congruent
group will be so much of the orthogonal group\index{Orthogonal01@Orthogonal substitutions} as carries
at least one point within our sphere into another such point.
The elliptic hypothesis will prevail, for two coplanar lines
perpendicular to a third will tend to approach one another.

We may obtain a simultaneous bird's-eye view of our three
systems in two dimensions as follows. Let us take for our
class of points the assemblage of all points of a euclidean
sphere which are south of the equatorial circle. We shall
define the distance of two points in three successive different
ways:---

(\emph{a})~The distance of two points shall be defined as the
distance which the lines connecting them with the north pole
cut on the equatorial plane. A line will be a circle which
passes through the north pole. If we interpret the equatorial
plane as the Gauss plane, we see that the congruent group
will be
\begin{equation*}
z'=\alpha z + \beta,\quad \alpha\overline\alpha=1,
\end{equation*}
or rather so much of this group as will carry at least one
point of the southern hemisphere into another such point.
It is evident from the conformal nature of the transformation
from sphere to equatorial plane, that we are under the
euclidean hypothesis\index{Euclidean01@Euclidean hypothesis}.

%[File: 075.png]
(\emph{b})~The distance\index{Distance} of two points shall be defined as one half
the logarithm of the cross ratio on the circle through them
in a vertical plane which they determine with the two
intersections of this circle and the equator. A line here will
be the arc of such a circle. The congruent group will be
that group of (euclidean) collineations\index{Congruent transformations}\index{Transformations, congruent}
which carries into
itself the southern hemisphere. A line will be infinitely
long, yet there will be an infinite number of others through
any chosen point failing to meet it; i.e.\ we are under
the hyperbolic hypothesis.

(\emph{c})~The distance of two points shall be defined as the length
of the arc of their great circle. Non-euclidean lines will be
arcs of great circles.  Congruent transformations will be
rotations of the sphere\index{Sphere}, and it is easy to see that the sum
of the angles of a triangle is greater than a straight angle;
we are under the elliptic hypothesis\index{Elliptic02@--- hypothesis}.

\mysect{Relative independence of the axioms}
We have now shown that our system of axioms is sufficient,
for we have been able to introduce coordinates for our points,
and analytic expressions for distances and angles. The axioms
are also compatible, for we have found actual systems of
objects obeying them. Compared with these virtues, all other
qualities of a system of axioms are of small import. It will,
however, throw considerable light upon the significance of
these our axioms, if we examine in part, their mutual
independence, by examining the nature of those geometrical
systems where first one, and then another of our assumptions
is supposed not to hold.

Axiom XIX is popularly known as the axiom of free
mobility, or rather, it is the residue of that axiom when we
are confined to a limited space. It puts into precise shape
the statement that figures may be moved about freely without
suffering an alteration either in size or form. We have defined
congruent transformations by means of the relation \emph{congruent}
which is itself defined in the logical sense, but not descriptively.
We might, of course, have proceeded in the
reverse order.%
\footnote{Cf.\ Pieri\index{Pieri}, loc.\ cit.}
%endfootnote
The ordinary conception in the elementary
textbooks seems to be that two figures are congruent if they
may be superposed; superposed means that they may be
carried from place to place without losing size or shape, and
this in turn implies that throughout the transference, each
remains congruent to itself.%
\footnote{Cf.\ Veronese\index{Veronese}, loc.\ cit., p.~259, note 1,
and Russell\index{Russell}, \emph{The Principles of Mathematics},
vol.~i, Cambridge, 1903, p.~405.}
%endfootnote

With regard to the independence of this axiom, we have but
%[File: 076.png]
to look at any system where the measure of distance in one
plane is double that of all the rest of space. A triangle having
two vertices in this plane, and one elsewhere, could not be
congruently transformed into a triangle of a different sort.

Axiom XVIII is the axiom of continuity\index{Continuity01@Continuity, axiom of}. We have laid
special stress on it in the course of our work, although the
subject of elementary geometry may be pushed very far
without its aid.%
\footnote{Cf.\ Halsted\index{Halsted}, loc.\ cit.}
%endfootnote
We are not here concerned with the
question of the wisdom of such attempts, considered from
the didactic point of view. Systems of geometry where this
axiom does not hold will occur to every reader; e.g.\ the
Cartesian euclidean system where all points whose coordinates
are non-algebraic are omitted. It is interesting to note that
whereas the omission of XIX runs directly counter to our
sense experience, no amount of observation could tell us
whether or no our geometry were continuous.%
\footnote{Cf.\ R.~L. Moore\index{Moore}, loc.\ cit.}
%endfootnote

Axiom XVII is an existence theorem, not holding where
the geometry of the plane is alone considered. It is a very
curious fact that the projective geometry of the plane is not
entirely independent of that of space, for Desargues'\index{Desargues} theorem
that copolar triangles are also coaxal cannot be proved
without the aid either of a third dimension, or of the congruent
group.%
\footnote{Cf.\ Hilbert\index{Hilbert}, loc.\ cit., p.~70; Moulton, `A simple non-desarguesian plane
geometry,' \emph{Transactions of the American Mathematical Society}, vol.~iii, 1902;
Vahlen\index{Vahlen}, loc.\ cit., p.~67.}
%endfootnote

Axiom XVI gives a criterion for circumstances under which
two lines must necessarily intersect. It is evident that
without some such criterion we should have difficulty in
proceeding any distance at all among the descriptive properties
of a plane. It is difficult to show the independence
of this axiom. The only dense system of geometry known
to the writer where it is untrue is the following.%
\footnote{Cf.\ Levy\index{Levy}, loc.\ cit., p.~32.}
%endfootnote

Let us denote by R the class of all rational numbers whose
denominators are of the form
\begin{equation*}
({a_1}^2 + {b_1}^2)({a_2}^2 + {b_2}^2) \dots ({a_n}^2 + {b_n}^2)
\end{equation*}
where $a_i$ and $b_i$ are integers or one may be zero. Let us
take as points the assemblage of all points of the euclidean
plane whose Cartesian coordinates are rational numbers
of the class $R$. The whole field will be transported into
itself by a parallel translation from any one point to any
other. Moreover, let $x$, $y$ and $x'$, $y'$ be the coordinates of two
%[File: 077.png]
points of the class, where $x^2 + y^2 = {x'}^2 + {y'}^2$. We may imagine
in fact that
\begin{equation*}
x=\frac{p}{s}, \quad y=\frac{q}{s}, \quad x'=\frac{p'}{s}, \quad y'=\frac{q'}{s}, \quad \frac{p^2+q^2}{s^2}=\frac{{p'}^2+{q'}^2}{s^2}.
\end{equation*}
Then the cosine and sine of the angle which the two points
subtend at the origin will be respectively
\begin{equation*}
\frac{pp'+qq'}{p^2+q^2}, \quad \frac{pq'-p'q}{p^2+q^2},
\end{equation*}
and these are numbers of the class $R$. The whole field will
go into itself by a rotation about the origin. Our system
will, therefore, obey XIX\@. It is of course two-dimensional
and not continuous. Moreover XVI will not hold, as the
reader will see by easily devised numerical experiments.

There are, also, plenty of geometries of a finite number
of points where this axiom does not hold.%
\footnote{Veblen\index{Veblen}, loc.\ cit., pp.~350-51.}
%endfootnote

Axiom XV is, of course, an existence theorem, untrue in the
geometry of a single line.

Axiom XIV gives the fundamental property of straight
lines. As an example of a geometry where it does not hold,
let us consider the assemblage of all points within a sphere
of radius one, and define as the distance\index{Distance} of two points the
length of an arc of a circle of radius two which connects them.
The segment of two points is thus a cigar-shaped region
connecting them. We see that the extensions of such a segment
and the segment itself do not comprise the segment
of two points within the original, and the extensions of the
latter. Axioms XII and XIII are also in abeyance, and it
seems possible that these three axioms are not mutually
independent. The present writer is unable to answer this
question.

Axiom XI implies that space has no boundary, and will be
untrue of the geometry within and on a sphere.

The first ten axioms amount to saying that distances are
magnitudes among which subtraction is always possible, but
addition only under restriction.


%[File: 078.png]



%CHAPTER VII
\mychap{VII}{THE GEOMETRIC AND ANALYTIC EXTENSION OF SPACE}
\mysect{Possibility of extending a segment by a definite amount in the euclidean and hyperbolic cases}

We are now in a position to take up the second of those
fundamental questions which we proposed at the close of
Chapter V, namely, to determine what degree of precision
may be given to Axiom XI\@. This axiom tells us that,
popularly speaking, any segment may be extended\index{Extension02@--- of space} beyond
either end. How far may it be so extended? Are we able
to state that there exists a system of geometry, consistent
with our axioms, where any segment may be extended by
any chosen amount? Or, in more precise language, if $\overline{AB}$
and $\overline{PQ}$ be given, can we always find $C$ so that
\begin{equation*}
\overline{AC} \equiv \overline{AB}+\overline{BC}, \quad \overline{BC}\equiv \overline{PQ}.
\end{equation*}

\mysect{Euclidean and hyperbolic space}
We are already able to answer this question in the euclidean
case, and answer it affirmatively. We have seen that there
is no inconsistency in that system of geometry, where points
are in one to one correspondence with all triads of (real and
finite) values of three coordinates $x$, $y$, $z$, and where distances
are given by the positive values of expressions of the form
\begin{equation*}
\sqrt{(x'-x)^2+(y'-y)^2+(z'-z)^2}.
\end{equation*}
Here, if, as we have said, we restrict the values of $x$, $y$, $z$
merely to be real and finite, we have a space under the
euclidean hypothesis, where any segment may be extended
beyond either extremity by any desired amount. Such a
space shall be called \emph{euclidean space}\index{Euclidean02@--- space}.

The same result will hold in the hyperbolic case. We shall
have a consistent geometrical system if we assume that our
points are in one to one correspondence with values
\begin{equation*}
\dot x_0:\dot x_1:\dot x_2:\dot x_3, \quad k^2<0,
\end{equation*}
\begin{equation*}
k^2 {\dot x_0}^2+{\dot x_1}^2+{\dot x_2}^2+{\dot x_3}^2<0.
\end{equation*}
Here, also, there will exist on every line distances whose
measures will be as large as we please. The space under the
%[File: 079.png]
hyperbolic hypothesis\index{Hyperbolic01@Hyperbolic hypothesis}, where any segment may be extended
by any chosen amount shall be called \emph{hyperbolic space}\index{Hyperbolic02@--- space}\index{Space}. To
put the matter otherwise, we shall have euclidean or hyperbolic
geometry if we replace Axiom XII by:---

\medskip\textsc{Axiom}~XII$'$. \textbf{If the parabolic or hyperbolic hypothesis be
true, and if $\overline{AB}$ and $\overline{PQ}$ be any two distances, then there
will exist a single point $C$, such that}
\begin{equation*}
\overline{AC} \equiv \overline{AB} + \overline{BC},\quad \overline{BC} \equiv \overline{PQ}.
\end{equation*}

\mysect{Contradiction arising under the elliptic hypothesis}
When we turn to the elliptic case, we find a decidedly
different state of affairs. Suppose, in fact, that there is a one
to one correspondence between the assemblage of all points,
and all sets of real values $x_0:x_1:x_2:x_3$.
The distance\index{Distance}\index{Distance01@Distance of two points, cosine}
of two
points will depend upon the periodic function
\begin{equation*}
\cos^{-1} \frac{(xy)}{\sqrt{(xx)}\; \sqrt{(yy)}}.
\end{equation*}
If, to avoid ambiguity, we assume that the minimum positive
value should be taken for this expression, we should easily
find two not null distances, whose sum was a null distance,
which would be in disagreement with Axiom X.

\mysect{New assumptions identical with the old for limited region,
but permitting the extension of every segment by a definite amount}
The desideratum is this. To find a system of geometry
where each point belongs to a sub-class subject to Axioms I--XIX,
and the elliptic hypothesis, and where each segment may still
be extended\index{Extension02@--- of space} by any chosen amount, beyond either end.

\medskip\textsc{Axiom}~I. \textbf{There exists a class of objects, containing at
least two members, called points.}\index{Point}

\medskip\textsc{Axiom}~II$'$. \textbf{Every point belongs to a sub-class obeying
Axioms I--XIX.}

\smallskip\emph{Definition}.~Any such sub-class shall be called a \emph{consistent
region}\index{Consistent region}\index{Region consistent}.

\medskip\textsc{Axiom}~III$'$. \textbf{Any two consistent regions which have a
common point, have a common consistent region including
this point and all others determining therewith a sufficiently
small, not null, distance.}

\medskip\textsc{Axiom}~IV$'$. \textbf{If $P_0$ and $P_{n+1}$ be any two points there may
be found a finite number $n$ of points $P_1,P_2,P_3,\dots P_n$ possessing
the property that each set of three successive ones belong to
a consistent region, and $P_k$ is within the segment $(P_{k-1}P_{k+1})$.}

\smallskip\emph{Definition}.~The assemblage of all points of such segments,
and all possible successive extensions thereof shall be called
a \emph{line}\index{Line}.

%[File: 080.png]
An important implication of the last axiom is that any two
points may be connected (conceivably in many ways) by
a chain of consistent regions\index{Consistent region}\index{Region consistent},
where each successive pair
have a consistent sub-region in common. This shows that
if we set up a coordinate system like that of Chapter V in
any consistent region, we may, by a process of analytic extension\index{Extension02@--- of space},
reach a set of coordinates for every point in space.
We may also compare any two distances. We have merely
to take as unit of measure for one, a distance so small, that
a distance congruent\index{Congruence01@Congruence of distances} therewith shall exist in the first three
overlapping consistent regions; a distance congruent with
this in the second three and so on to the last region, and then
compare the measures of the two distances in terms of the
first unit of measure, and the unit obtained from this by the
series of congruent transformations. Let the reader show that
if once we find
%$\overline{AB} \equiv^{>}_{<} \overline{PQ}$
$\overline{AB}
\begin{array}{ccc}
> \\
\equiv \\
< \end{array}
\overline{PQ}$
the same relation will hold if we
proceed by any other string of overlapping regions. Having
thus defined the congruence of any two distances, we may
state our axiom for the extension of a segment\index{Extension01@Extension of segment}, as follows:---

\medskip\textsc{Axiom}~V$'$. \textbf{If $\overline{AB}$ and $\overline{PQ}$
be any two distances, there
exists a single point $C$ such that
$\overline{BC} \equiv \overline{PQ}$,
while $B$ is
within a segment whose extremities are $C$ and a point
of $(AB)$.}

An important corollary from this axiom is that there
must exist in the elliptic case a point having any chosen set
of homogeneous coordinates $(x)$ not all zero. For, let $(y)$ be
the coordinates of any known point. Consider the line
through it whose points have coordinates of the form
$\lambda(y) + \mu(x)$. As we proceed along this line, the ratio
$\dfrac{\lambda}{\mu}$ will
always change in the same sense, for such will be the case
in any particular consistent region. Moreover we may, by
our last axiom, find a number of successive points such that
the sum of the measures of their distances shall be $k\pi$.
Between the first and last of these points the value of
$\dfrac{\lambda}{\mu}$will
have run continuously through all values from $-\infty$ to $\infty$,
and hence have passed through the value $0$, giving a point
with the required coordinates.

The preceding paragraph suggests two interesting questions.
Is it possible that, by varying the method of analytic extension,
we might give to any point two different sets of
%[File: 081.png]
homogeneous coordinates in the same system?\index{Extension02@--- of space} Is it possible
that two different points should have the same homogeneous
coordinates? With regard to the first of these questions, it
is a fact that under our hypotheses a point may have several
different sets of coordinates, as we shall see at more length
in Chapter XVII\@. For the present it is, however, wiser to limit
ourselves to the classical non-euclidean systems, where a point
has a unique set of coordinates. We reach the desired
limitation by means of the following considerations.

A sufficiently small congruent
transformation\index{Congruent transformations}\index{Transformations, congruent}
of any consistent region\index{Consistent region}\index{Region consistent}
will effect a congruent transformation of any
chosen sub-region, and so of any consistent region including
this latter. It thus appears that if two consistent regions
have a common sub-region, a sufficiently small congruent
transformation of the one may be enlarged to be a congruent
transformation of the other. Proceeding thus, if we take any
two consistent regions of space, and connect them by a series
of overlapping consistent regions, then a small congruent
transformation of the one may be analytically extended to
operate a congruent transformation in the other. Will the
original transformation give rise to the same transformation
in the second space, if the connexion be made by means of
a different succession of overlapping consistent regions? It
is impossible to answer this question \emph{a priori}; we therefore
make the following explicit assumption:---

\mysect{Last axiom, free mobility of the whole system}
\medskip\textsc{Axiom}~VI$'$. \textbf{A congruent transformation of any consistent
region may be enlarged in a single way to be a congruent
transformation of every point.}

Evidently, as a result of this, a congruent transformation
of one consistent region can be enlarged in only one way
to be a congruent transformation of any other. Let us next
observe that it is impossible that two points of the same
consistent region should have the same coordinates in any
system. Suppose, on the contrary, that $P$ and $Q$ of a consistent
region have the coordinates $(x)$. There will be no
limitation involved in assuming that the coordinate axes were
set up in this consistent region, and the coordinates of $P$ found
directly as in Chapter V, while those of $Q$ are found by an analytic
extension through a chain of overlapping consistent regions.
Now it is not possible that every infinitesimal congruent
transformation which keeps $P$ invariant shall also keep $Q$
invariant, so that a transformation of this sort may be found
transforming each overlapping consistent region infinitesimally,
and carrying $Q$ to an infinitesimally near point $Q'$. But in
%[File: 082.png]
the analytic expression of this transformation, in the form
of an orthogonal substitution (in the non-euclidean cases)
the values $(x)$ will be invariant, so that $Q''$ will also have
the coordinates $(x)$, and by the same chain of extensions as
gave these coordinates to $Q$. Hence, reversing the order of
extensions, when we set up a coordinate system in the last
consistent region\index{Region consistent}, that which includes $Q$ and $Q'$, these two
points will have the same coordinates. But this is impossible
for the coordinate system explained in Chapter V, for a consistent
region gives distinct coordinates to distinct points.
This proof is independent of Axiom VI$'$.

\mysect{One to one correspondence of point and coordinate set in euclidean and hyperbolic cases}
Our desired uniqueness of coordinate sets will follow at
once from the foregoing. For, suppose that a point $P$ have
two sets of coordinate values $(x)$ and $(x')$, not proportional
to one another. Every infinitesimal transformation which
keeps the values $(x)$ invariant, will either keep $(x')$ invariant,
or transform them infinitesimally, let us say, to a set of
values $(x'')$. But there is a point distinct from $P$ and close to
it which has the coordinates $(x'')$, and this gives two points
of a consistent region with these coordinates, which we have
just seen to be impossible. Hence, the ratios of the coordinates
$({x_0}')$ must be unaltered by every infinitesimal orthogonal
substitution which leaves $(x)$ invariant, i.e.\ $x_0{}'=\rho x_i$. It is
evident, conversely, that if each point have but one set of
coordinates, Axiom VI$'$ must surely hold.

\mysect{Ambiguity in the elliptic case giving rise to elliptic and spherical geometry}
It is time to attack the other question proposed above,
by supposing that two distinct points shall have the same
homogeneous coordinates. They may not lie in the same
consistent region, and every congruent transformation which
leaves one invariant, will leave the other unmoved also.
Let us call two such points \emph{equivalent}\index{Equivalent points}. Every line through
one of these points will pass through the other. For let
a point $Q$ on a line through one of the points have coordinates
$(y)$. We may connect it with the other by a line,
and the two lines through $(Q)$ lie in part in a consistent
region, the coordinates of points on each being represented
in the form $\lambda y_i+\mu x_i$. The two lines are identical.

Let us consider the assemblage of all points whose coordinates
are linearly dependent on those of three non-collinear
points. This assemblage of points may properly be called
a \emph{plane}\index{Plane}, for those points thereof which lie in any consistent
region will lie in a plane as defined in Chapter II\@. It is
clearly a connex assemblage, and will contain every line
whereof it contains two non-equivalent points. Let $(y)$, $(z)$, $(t)$
be the coordinates of three points, no two of which are
%[File: 083.png]
equivalent. Let us consider the point $(x)$ whose coordinates
are
\begin{equation*}
(ux) = |uyzt|.
\end{equation*}
In the elliptic case, as we have seen, such a point surely
exists. In the hyperbolic or parabolic cases, there might not
be any such point. It is clear, however, that in these cases
there can be no equivalent points. Suppose, in fact, $P_0$ and
$P_{n+1}$ were equivalent\index{Equivalent points}. Connect them by a line whereon are
$P_1$, $P_2\dots P_n$. Move this line slightly so that the connecting
string of points are ${P_1}'$, ${P_2}' \dots {P_n}'$ very near to the former
points. We have constructed two triangles, and $(n-1)$
quadrilaterals, and as we are under the hyperbolic or euclidean
hypothesis, the sum of the measures of the angles of all the
triangles and quadrilaterals will be less than, or equal to
$\pi + (n-1)2 \pi + \pi$. But clearly the sum of the measures of
the angles at points $P_i$ and ${P_i}'$ is $2n\pi$, so that the sum of the
two angles which the two lines make at $P_0$ and $P_{n+1}$ is null
or negative; an absurd result. Equivalent points can then
occur only under the elliptic hypothesis, and there will surely
be a point $P$ with the coordinates $(x)$ above.

Let us next make a congruent transformation whereby $P$
goes into an equivalent point $P'$, the plane of $(y) (z) (t)$ goes
into itself congruently, for it constitutes the assemblage of all
points satisfying the condition $(xX) = 0$, and $(xX)$ is an
invariant under every orthogonal substitution. After $P$ has
been carried to $P'$, each point of the plane\index{Plane} may be returned
to its original position by means of a series of congruent
transformations\index{Congruent transformations}\index{Transformations, congruent},
each too small to change $P'$ to an equivalent
point, yet keeping the values $(x)$ invariant, coupled, at the
end, with a reflection in a plane\index{Reflection01@Reflection in plane} perpendicular to the given
one, in case the determinant of the original orthogonal
substitution is negative, and this too will leave $P'$ unchanged.
We may therefore pass from $P$ to any equivalent point by
a transformation which leaves in place every point of a plane.
But there is only one congruent transformation of space
which leaves every point of a plane invariant, besides, of
course, the identical one. Hence every point in space can
have but one equivalent at most.

Our results are, then, as follows. Under the euclidean and
hyperbolic hypotheses, there is but one point for each set
of coordinates, and our new Axioms I--VI$'$ will yield us
nothing more than euclidean or hyperbolic space. Under the
elliptic hypothesis there are two possibilities:---

\emph{Elliptic space}\index{Elliptic03@--- space}. This is a space obeying Axioms I--VI$'$, and
the elliptic hypothesis. If $n$ successive segments whose
%[File: 084.png]
measures are $\dfrac{k \pi}{n}$ be taken upon a line as indicated in V$'$, the
last extremity of the last segment will be identical with
the first extremity of the first. Two lines of the same plane
will have one and only one common point, so that no point
has an equivalent. We may take as a consistent region\index{Consistent region}\index{Region consistent} the
assemblage of all points whose distances from a given point
are of measure less than $\dfrac{k\pi}{4}$. If two points be of such a
nature that the expression for the cosine of the measure of
the $k$th part of their distance vanishes, we shall say that the
measure of their distance is $\dfrac{k\pi}{2}$. Two points will always
have a determinate distance and a single segment, unless the
measure of their distance is $\dfrac{k\pi}{2}$, in which case they determine
two segments with the same extremities. These last two
segments may also, with propriety, be called half-lines. The
definition of an interior angle given in Chapter II may be
retained, but the concept of half-plane is illusory, for a line
will not divide the plane. It may, however, be modified
much as we have modified the definition of a half-line, and
from it a definition built up for a dihedral angle. We leave
the details to the reader. An example of elliptic geometry\index{Elliptic03@--- space}
will be furnished by any set of points in one to one correspondence
with all sets of homogeneous values $x_0:x_1:x_2:x_3$
where also $\cos\dfrac{d}{k}=\dfrac{(xy)}{\sqrt{(xx)}\;\sqrt{(yy)}}$.
For instance, let us take as
points concurrent lines of a four dimensional space (euclidean,
for example) and mean by distance the measure of the angle
$\leqq \dfrac{\pi}{2}$ formed by two lines.

\emph{Spherical space}\index{Spherical space}. This is also a space obeying Axioms I--VI$'$
and the elliptic hypothesis. Each point will have one equivalent.
If $n$ successive congruent distances be taken upon
a line whose measures are $\dfrac{k\pi}{n}$, the last extremity of the last
will be equivalent to the first extremity of the first. We
may take as a consistent region the assemblage of all points
the measures of whose distances from a given point are less
than $\dfrac{k\pi}{2}$. The measure of the distance of two equivalent
points shall be defined as the number $k\pi$. Any two nonequivalent
%[File: 085.png]
points will have a well-defined segment. We may
find a definition for a half-line analogous to that given in
the elliptic case, and so for half-plane, internal angle, and
dihedral angle.

An example of spherical geometry will be furnished by the
geometry of a hypersphere in four dimensional euclidean
space, meaning by the distance of two points, the length
of the shorter arc of a great circle connecting them.

A simple example of a two dimensional elliptic geometry
is offered by the euclidean hemisphere, where opposite points
of the limiting great circle are considered as identical. A two
dimensional spherical geometry is clearly offered by the
euclidean sphere.

\mysect{Ideal elements, extension of all spaces to be real continua}
The elliptic and spherical spaces which we have thus built
up are, in one respect, more complete than euclidean or
hyperbolic space, in that there is in the first two cases always
a point to correspond with every set of real values, not all
zero, that may be attached to our four homogeneous coordinates
$x$, while in the latter cases this is not so. We bring
our euclidean and hyperbolic geometries up to an equality
with the others by extending our concept \emph{point}\index{Point}. Let us begin
with the euclidean case where there is a point corresponding
to every real set of homogeneous values $x_0:x_1:x_2:x_3$, provided
that $x_0\ne 0$. Now a set of values $0: y_1: y_2: y_3$ will
determine at each real point $(x)$ a line, the coordinates of
whose points are of the form $\lambda y_i + \mu x_i$\label{y0}, and if $(x)$ be varied
off of this line, we get a second line coplanar with the first.
Our coordinates $0: y_1: y_2: y_3$ will thus serve to determine
a bundle of lines, and this will have exactly the same
descriptive properties as a bundle of concurrent lines. We
may therefore call the bundle an \emph{ideal point}\index{Ideal elements}, and assign to
it the coordinates $(y)$. Two ideal points will determine a
pencil of planes having the same descriptive properties as
a pencil of planes through a common line. We shall therefore
say that they determine, or have in common, an \emph{ideal
line}. Two lines whose intersection is ideal shall be said
to be \emph{parallel}, as also, two planes which meet in an ideal
line. These definitions of parallel are for euclidean space
only. The assemblage of all ideal points will be characterized
by the equation
\[
x_0=0.
\]
This we shall call the equation of the \emph{ideal plane} which is
supposed to consist of the assemblage of all ideal points.
Ideal points and lines shall also be called \emph{infinitely distant}\index{Infinitely distant elements},
while the ideal plane is called the \emph{plane at infinity}. We shall
%[File: 086.png]
in future use the words \emph{point}, \emph{line}, and \emph{plane} to cover both
ideal elements\index{Ideal elements}\index{Infinitely distant elements}
and those previously defined, which latter may
be called, in distinction, \emph{actual}\index{Actual elements}. Actual and ideal elements
stand on exactly the same footing with regard to purely
descriptive properties. No congruent transformation can
interchange actual and ideal elements. We shall later return
to the meaning of such words as \emph{distance} where ideal elements
enter.

In the hyperbolic case we may apply the same principles
with slight modification. There will be a real point corresponding
to each set of real homogeneous coordinates $(\dot{x})$ for
which
\[
k^2\dot{x}_0{}^2 + \dot{x}_1{}^2 + \dot{x}_2{}^2 + \dot{x}_3{}^2 < 0.
\]

\mysect{Imaginary elements geometrically defined, extension of all spaces
to be perfect continua in the complex domain}
A set of real homogeneous values for $(\dot{x})$, for which this
inequality does not hold, will determine a bundle of lines,
one through every actual point, any two of which are
coplanar; a bundle with the same descriptive properties as
a bundle of concurrent lines. We shall therefore say that
this bundle determines an ideal point having the coordinates
$(\dot{x})$. If
\[
k^2\dot{x}_0{}^2 + \dot{x}_1{}^2 + \dot{x}_2{}^2 + \dot{x}_3{}^2 = 0,
\]
the ideal point shall be said to be \emph{infinitely distant}. If
\[
k^2\dot{x}_0{}^2 + \dot{x}_1{}^2 + \dot{x}_2{}^2 + \dot{x}_3{}^2 > 0,
\]
the ideal point shall be said to be \emph{ultra-infinite}\index{Ultra-infinite elements}. Two lines
having an infinitely distant point in common shall be called
\emph{parallel}\index{Parallelism}. Through each actual point will pass two lines
parallel to a given line. An equation of the type
\[
(\dot{u}\dot{x}) = 0,\quad
  \frac{1}{k^2} \dot{u}_0{}^2 + \dot{u}_1{}^2
+ \dot{u}_2{}^2 + \dot{u}_3{}^2 > 0,
\]
will give a plane. If the inequality be not fulfilled, the assemblage
of all ideal points whose coordinates fulfil the equation
(and there can be no actual points which meet the requirement)
shall be called an \emph{ideal plane}, the coefficients $(\dot{u})$ being its
coordinates. There will thus be a plane corresponding to
each set of real homogeneous coordinates $(\dot{u})$ not all zero.
An ideal line shall be defined as in the euclidean case, and
the distinction between actual and ideal shall be the same
as there given. No congruent transformation, as defined so
far, can interchange actual and ideal elements.

Let us take account of stock. By the introduction of ideal
elements we have made each of our spaces a real analytic
continuum. In all but the spherical case there is a one to
one correspondence between points and sets of real homogeneous
values not all zero, in spherical space there is a one
%[File: 087.png]
to one correspondence of coordinate set and pair of equivalent
points. Each of our spaces will fulfil the fundamental
postulates of projective geometry, as we shall develop them
in Chapter XVIII, or as they have already been developed
elsewhere.%
\footnote{Cf.\ Pieri\index{Pieri}, `I principi della geometria di posizione.' \emph{Memorie della
R. Accademia delle Scienze di Torino}, vol.\ xlviii, 1899.}
%endfootnote
Let
us show hurriedly, how to find figures to
correspond to imaginary coordinate values. Four distinct
points\index{Point} will determine six numbers called their \emph{cross ratios}\index{Cross ratios},
which have a geometrical significance quite apart from all
concepts of distance or measurement.%
\footnote{Cf.
Pasch\index{Pasch}, loc.\ cit., p.\ 164, and Chapter XVIII of the present work.
The idea of assigning to four collinear points a projectively invariant
number originated with Von Staudt\index{Staudt@Von Staudt}, \emph{Beitr\"age zur Geometrie der Lage,} Part 2,
\S\S~19--22, Erlangen, 1858--66.}
%endfootnote
An involution\index{Involution} will
arise when the points of a line are paired in such a reciprocal
manner that the cross ratios of any four are equal to the
corresponding cross ratios of their four mates. If there be
no self-corresponding points, the involution is said to be
\emph{elliptic}. If the points of a line be located by means of
homogeneous coordinates $\lambda  : \mu$, it may be shown that every
involution may be expressed in the form
\[
  A\lambda\lambda' + B(\lambda\mu' + \lambda'\mu) + C\mu\mu'
= 0.
\]
In particular if $(y)$ and $(z)$ be the coordinates of two points,
there will exist an involution on their line determined by the
equations
\[
  (x) = \lambda(y) + \mu(z),\quad (x)'= \mu(y) - \lambda(z),
\]
and by a proper choice of running coordinates any elliptic
involution may be put into this form. Did we seek the
coordinates of self-corresponding points in this involution,
we should get
\[
  (x) = (y) \pm i(z).
\]
Conversely, every set of homogeneous complex values $(y) + i(z)$
will lead us in this way to a definite elliptic involution.
The involution may be taken to represent the two sets of
conjugate imaginary homogeneous values. We may separate
the conjugate values by the following device. It is not difficult
to show that if a directed distance be determined by two
points, it will have the same sense as the corresponding
directed distance determined by their mates in an elliptic
involution. To an elliptic involution may thus be assigned
either one of two \emph{senses of description}\index{Sense02@--- of description of involution},
and we shall define
as an imaginary point\index{Imaginary elements} an elliptic involution to which such
a sense has been attached. Had we taken the other sense\index{Opposite03@Opposite senses},
we should have said that we had the conjugate imaginary
%[File: 088.png]
point. An imaginary plane may similarly be defined as an
elliptic involution\index{Involution} among the planes of a pencil, with a
particular sense of description; an imaginary line as the
intersection of two imaginary planes. It may be shown
geometrically that by introducing imaginary elements\index{Imaginary elements} under
these definitions we have a system of points, lines, and planes,
obeying the same descriptive laws of combination as do the
real points of lines and planes of projective geometry, or
the assemblage of all real homogeneous coordinate sets, which
do not vanish simultaneously.%
\footnote{Cf.
Von Staudt\index{Staudt@Von Staudt}, loc.\ cit., \S~7, and L\"uroth\index{Luroth@L\"uroth},
`Das Imagin\"are in der Geometrie
und das Rechnen mit Wurfen,' \emph{Mathematische Annalen}, vol.~ix.}
%endfootnote
Introducing these imaginary
expressions, and the corresponding complex values for their
homogeneous coordinates, we extend our space to be a perfect
analytic continuum.

We must now see what extension must be given to the
concept distance, in order to fit the extended space with
which we are, henceforth, to deal. To begin with, we shall
from this time forth identify the two concepts \emph{distance}\index{Distance} and
\emph{measure of distance}\index{Measure01@Measure of distance}. In other words, as the concept distance
comes into our work effectively only in terms of its measure,
i.e.\ as a number, so we shall save circumlocution by replacing
the words \emph{measure of distance} by \emph{distance} throughout. The
distance of two points is thus dependent upon the two points,
and on the unit. In any particular investigation, however,
we assume that the unit is well known from the start, and
disregard its existence. We therefore give as the definition
of the distance of two points under the euclidean hypothesis
\[
  d = \frac{1}{x_0y_0} \sqrt{(x_1-y_1)^2 +(x_2-y_2)^2 +(x_3-y_3)^2}.  \tag{1}
\]
This is, at worst, a two valued function. When it takes
a real value, we give the positive root as the distance, when
it is imaginary we may make any one of several simple
conventions as to which root to take. If one or both of the
points considered be ideal, the expression for distance becomes
infinite, unless also the radical vanishes when no distance is
determined. Under these circumstances we shall leave the
concept of distance undefined, thus getting pairs of points
disobeying Axiom II$'$. Notice also that whenever the radical
vanishes for non-ideal points we have points which are
distinct, yet have a null distance, and when such points
are included, Axiom XIII may fail.

We shall in like manner identify the concepts \emph{angle}\index{Angle01@Angle, interior and exterior} and
%[File: 089.png]
\emph{measure of angle}\index{Angle09@--- measure of} in terms of the unit which gives to a right
angle the measure $\dfrac{\pi}{2}$.

We may proceed in a similar manner in the non-euclidean
cases. If $(x)$ and $(y)$ be the coordinates of two points, we
shall define as their distance $d$, the solution of
\[
  \cos\frac{d}{k} = \frac{(xy)}{\sqrt{(xx)}\;\sqrt{(yy)} }.  \tag{2}
\]

\mysect{Cayleyan Absolute, new form for the definition of distance}
This equation in $d$ has, of course, an infinite number of
solutions. Before taking up the question of which shall be
called the distance of the two points, let us approach the
matter in a different, and highly interesting fashion due to
Cayley.%
\footnote{Cayley\index{Cayley}, `A sixth memoir on Qualities,' \emph{Philosophical Transactions of the
Royal Society of London}, 1859.}
%endfootnote
This theory is of absolutely fundamental importance in all that follows.

The assemblage of points whose coordinates satisfy the
equation
\[
  (xx) = 0,    \tag{3}
\]
shall be called the \emph{Absolute}\index{Absolute}\index{Absolute}. This is a quadric surface, real
in the hyperbolic case, surrounding, so to speak, the actual
domain; imaginary in the elliptic and spherical cases; in the
last-named, it is the locus of points which coincide with their
equivalents. Every congruent transformation is an orthogonal
substitution, i.e.\ a linear transformation carrying the Absolute
into itself. Let us, by definition, enlarge our congruent group
so that every such transformation shall be called congruent;
certainly it carries a point into a point, and leaves distances
unaltered. In the euclidean case we take as \emph{Absolute} the
conic
\[
  x_0 = 0,\quad x_1{}^2 + x_2{}^2 + x_3{}^2 = 0,    \tag{4}
\]
and define as congruent transformations a certain six-parameter
sub-group of the seven-parameter collineation group which
carries it into itself. We shall return to the study of the
congruent group in the next chapter.

Returning to the non-euclidean cases, let us take two
points $P_1$, $P_2$ with coordinates $(x)$ and $(y)$, and let the line
connecting them meet the Absolute in two points $Q_1$, $Q_2$. We
obtain the coordinates of these by putting $\lambda(x) + \mu(y)$ into
the equation of the Absolute. The ratio of the roots of this
equation will give one of the two cross ratios\index{Cross ratios} formed by
the pair of points $P_1 P_2$ and the pair $Q_1 Q_2$; interchanging
%[File: 090.png]
the roots we get the other cross ratio\index{Cross ratios} of the two pairs of
points.%
\footnote{For the geometrical interpretation of a cross ratio when some of the
elements are imaginary, see Von Staudt\index{Staudt@Von Staudt}, loc.\ cit., \S 28, and L\"uroth\index{Luroth@L\"uroth}, loc.\ cit.}
%endfootnote
The value of such a cross ratio will thus be
\[
  \frac{(xy) + \sqrt{(xy)^2 - (xx)(yy)} }
       {(xy) - \sqrt{(xy)^2 - (xx)(yy)} }.
\]
By interchanging the signs of the radicals we change this
cross ratio into its reciprocal, and this amounts to interchanging
the members of one of the two point pairs. Let us
denote this expression by $e^{\dfrac{2id}{k}}$.
\begin{align*}
  e^{\dfrac{id}{k}}
&= \frac{(xy) + \sqrt{(xx)(yy)-(xy)^2} }
        {\sqrt{(xx)}\;\sqrt{(yy)} },
\\
  \cos\frac{d}{k} &= \frac{(xy)}{\sqrt{(xx)}\;\sqrt{(yy)} }.
\tag{5}
\end{align*}

\mysect{Extension of the distance concept to the complex domain}
If we write the cross ratios of the pair of points $P_1 P_2$ and
the pair $Q_1 Q_2$ as $(P_1P_2, Q_1Q_2)$, we may re-define our noneuclidean
distance\index{Distance} by the following theorem:---

\smallskip\emph{Theorem}. If $d$ be the distance of two points $P_1$ and $P_2$
whose line meets the Absolute in $Q_1$ and $Q_2$,
\[
  d = \frac{k}{2i}\log_e(P_1P_2, Q_1Q_2).   \tag{6}
\]

The great beauty of this definition is that it brings into
clear relief the connexion between distance and the congruent
group, for the cross ratio in question is, of course, invariant
under all linear transformation which carry the Absolute
into itself, i.e.\ under all congruent transformations. Let the
reader show that a corresponding projective definition may
be given for an angle.

Our distances, as so far defined, are infinitely multiple
valued functions. There is no great practical utility in
rendering them single valued by definition. It is, however,
perhaps worth while to carry it through in one case.

If we have two real points of the actual domain, the
expression $(P_1P_2, Q_1Q_2)$ will have two values, real in the
hyperbolic, pure imaginary in the elliptic and spherical case,
and these two are reciprocals, so that the resulting expressions
for $d$ will differ only in sign, for each determination of the
logarithm. We may therefore take the distance as positive.
%[File: 091.png]
Did we seek, not for a distance\index{Distance}, but a directed distance\index{Distance, directed}, then
it would be necessary to distinguish once for all between
$Q_1$ and $Q_2$ and in each particular case between the pair
$P_1 P_2$,
and the pair $P_2 P_1$, the directed distance will have a definite
value sometimes positive, sometimes negative.

Let us specialize by confining ourselves to the hyperbolic
case. We have defined the distance of two actual points.
Still restricting ourselves to the real domain, suppose that
we have an actual and an ultra-infinite point. Let us choose
such a unit of measure that $k^2 = -1$. Our cross ratio\index{Cross ratios} is here
negative, with an absolute value $r$ let us say, so that the distance
expression takes the form $\tfrac{1}{2}[\log r \pm (2m + 1)\pi i]$. Let us choose
in particular
\[
  d = \tfrac{1}{2}\log r + \frac{\pi i}{2}.
\]
Next consider two ultra-infinite points. If the line connecting
them meet the Absolute in real points, we shall have
a real cross ratio as before, and hence a real positive distance.
If, however, this real line meet the Absolute in conjugate
imaginary points, the expression for the cross ratio becomes
imaginary, and the simplest expression for their distance is
pure imaginary. The absolute value of this expression will
run between $0$ and $\dfrac{\pi}{2}$, for the roots of
$\tfrac{1}{2}\log A = X$ differ
by $\pi i$. We may, hence, represent all of these cross ratios in
the Gauss plane by points of the axis of pure imaginaries
between $0$ and $\dfrac{\pi}{2}$.

If the line connecting two ultra-infinite points be tangent
to the Absolute, the cross ratio is unity, and we may take
the distance as zero. The distance from a point of the
Absolute to a point not on its tangent will be infinite;
the distance to a point on the tangent is absolutely indeterminate,
for the cross ratio is indeterminate. We may,
in fact, consider the cross ratios of three coincident points
and a fourth, as the limiting case of any cross ratio which
we please.

Leaving aside the indeterminate case, we are thus able to
represent the distance of any two real points of hyperbolic
space in the Gauss plane by a point on the positive half
of the axis of reals, by a point of the segment of the origin
and $\dfrac{\pi}{2}i$, or by a point of the horizontal half-line $\left|\dfrac{\pi}{2}i\;\infty\right.$,
%[File: 092.png]
and as two points move continuously in the real domain of the
hyperbolic plane, the points which represent their distance
will move continuously on the lines described.

Let us now take two points of the hyperbolic plane, real or
imaginary. We see that the roots of $\frac{1}{2}\log A = X$ differ by
multiples of $\pi i$, so that we may assign to $d$ an imaginary
part whose Absolute value $\leqq\dfrac{\pi}{2}$. Moreover, by choosing
properly between the two reciprocal values of the cross ratio\index{Cross ratios},
we may ensure that the real part of $d$ shall not be negative.
If two points be conjugate imaginaries, while their line cuts
the Absolute in real points, the cross ratio is imaginary, and
the expression for distance is pure imaginary, which we may
represent by a point of the segment of the origin and
$-\dfrac{\pi}{2}i$. If both pairs of points be conjugate imaginaries, the
cross ratio is real and negative, so that the distance may
be represented in the form $X-\dfrac{\pi}{2}i$. We shall define as the
distance\index{Distance} of two points that value of the logarithm of a cross
ratio which they form with the intersection of their line and
the Absolute, which in the Gauss plane is represented by
a point of the infinite triangle whose vertices are
$\infty$, $0 + \dfrac{\pi}{2}i$,
$0 - \dfrac{\pi}{2}i$. The possible ambiguities for points on the sides of
this triangle have already been removed by definition.

We have already seen that when euclidean space\index{Euclidean02@--- space} has been
enlarged to be a perfect analytic continuum, imaginary points
and distances come in which do not obey all of our axioms.
In the hyperbolic case we shall find real, though ultra-infinite,
points which do not at all obey the principles laid down
for a consistent region.%
\footnote{The developments which follow are taken from Study\index{Study}, `Beitr\"age zur
nicht-euklidischen Geometrie.' \emph{American Journal of Mathematics}, vol.~xxix, 1907.}
%endfootnote
Let us take three points of the
ultra-infinite region of the actual hyperbolic plane $x_3 = 0$,
say $(x)$, $(y)$, $(z)$. As these points are supposed to be real we
may assume that $x_1$, $x_2$ are real, while $x_0$ is a pure imaginary,
and that a like state of affairs exists for $(y)$ and $(z)$. We
shall further assume that the lines connecting them shall
intersect the Absolute in real, distinct points. We have then
\[
\begin{aligned}
  (yz)^2-(yy)(zz) > 0, &\quad (xx) > 0, \\
  (zx)^2-(zz)(xx) > 0, &\quad (yy) > 0, \\
  (xy)^2-(xx)(yy) > 0, &\quad (zz) > 0.
\end{aligned}
\tag{7}
\]
%[File: 093.png]

Let us, for the moment, indicate the distance from $(x)$ to $(y)$
by $\overline{xy}$, and assume
\[
  \overline{yz} \geqq \overline{zx} \geqq \overline{xy}.
\]

\mysect{Case where a straight line gives a maximum distance}
We shall also take
\[
  h = i,\quad \cos \frac{d}{k} = \cosh d.
\]

Under what circumstances shall we have?\index{Greater than}\index{Less than}\index{Sum03@Sum of distances}
\[
  \overline{yz} \geqq \overline{zx} + \overline{xy},
\]
\[
  \cosh(\overline{yz} - \overline{zx}) \geqq \cosh \overline{xy},
\]
\begin{multline*}
  \sqrt{\frac{(yz)^2}{(yy)(zz)} }
  \sqrt{\frac{(zx)^2}{(zz)(xx)} }
- \sqrt{\frac{(xy)^2}{(xx)(yy)} }   \\
\geqq
  \sqrt{\frac{(yz)^2 - (yy)(zz)}{(yy)(zz)} }
  \sqrt{\frac{(zx)^2 - (zz)(xx)}{(zz)(xx)} }.
\end{multline*}
The terms on the left are essentially positive as they represent hyperbolic
cosines, those on the right are positive, being
hyperbolic sines; we may therefore square the inequality
\[
  (xx)(yy)(zz) + 2|(yz)(zx)(xy)| - (xx)(yz)^2
- (yy)(zx)^2 - (zz)(xy)^2 \leqq 0.  \tag{8}
\]

We see that if
\[
  (yz)(zx)(xy) > 0,    \tag{9}
\]
we are at liberty to drop the absolute value signs in the
second term, and the whole expression is the square of the
determinant $|xyz|$ which is zero or negative. We see, therefore, that under these circumstances,
\[
  |yz| \geqq |zx| + |xy|.
\]
To see what region of the ultra-infinite domain is determined
by (9), let us sketch the Absolute as a conic, and draw tangents
thereunto from $(y)$ and $(z)$. $X$ must lie within the quadrilateral
of these tangents or the vertical angle at $(y)$ or $(z)$.
The conic and tangents determine four quasi-triangles with
two rectilinear and one curvilinear side each. Since $(yy) > 0$
our inequality (9) will hold within the quasi-triangles whose
vertices are $(y)$ and $(z)$ and within the verticals of these
two angles.

%[Illustration: Fig. 3.]
\imgcent{340}{f003}{Fig. 3}

Let us now assume, on the contrary, that we are in the
other quasi-triangles
\[
  (yz)(zx)(xy) < 0.
\]
Our original inequality (8) will still hold if
\[
  |xyz|^2 - 4(yz)(zx)(xy) < 0,    \tag{10}
\]
%[File: 094.png]
and, conversely, this inequality certainly holds if (7) does.\index{Sum03@Sum of distances}
If we look on $(y)$ and $(z)$ as fixed, and $(x)$ as variable, the
curve
\begin{equation*}
|xyz|^2-4(yz)(zx)(xy)=0,
\end{equation*}
in so far as it lies in the two quasi-triangles we are now
considering, will play the part of the segment of $(y)$ and $(z)$.%
\footnote{For a complete discussion, see Study\index{Study}, loc.\ cit., pp.~103--8. Fig. 3 is taken
direct. }
%endfootnote
In a region where (8) holds, a rectilinear path is the longest
from $(y)$ to $(z)$.

%[File: 095.png]
%CHAPTER VIII
\mychap{VIII}{THE GROUPS OF CONGRUENT TRANSFORMATIONS}
\mysect{Congruent transformations of the straight line}
\index{Congruent transformations|(}
\index{Transformations, congruent|(}

The most significant idea introduced in the last chapter
was that of the Absolute\index{Absolute}, and its connexion with the concept
of distance. Every collineation of non-euclidean space which
keeps the Absolute in place was defined as a congruent
transformation; we had already seen in Chapter V that every
congruent transformation was such a collineation\index{Collineations}. We may
go one step further, and say that every analytic transformation
which carries the Absolute into itself alone is a congruent
transformation. Suppose that we have
\begin{multline*}
{x_0}'=f_0(x_0x_1x_2x_3), \quad x_1{}'=f_1(x_0x_1x_2x_3), \quad {x_2}'=
f_2(x_0x_1x_2x_3), \\ {x_3}'=f_3(x_0x_1x_2x_3),
\end{multline*}\label{x3}
\begin{equation*}
(x'x')=P(xx).
\end{equation*}
$P$ must be a constant, for were it a function of $(x)$ the
Absolute would be carried into itself, and into some other
surface $P = 0$, which is contrary to hypothesis. Replacing
$(x)$ by $\lambda (x) + \mu (y)$ we see that we shall also have
\begin{equation*}
(x'y') = P(xy),
\end{equation*}
whence we may easily show that the transformation is a
collineation.

%\mysect{Congruent transformations of the hyperbolic plane}
\mysect{\hspace{2em},,\hspace{5.9em},,\hspace{4em},,\hspace{1.5em}hyperbolic plane}
It is, of course, evident, that in the complex domain, the
congruent groups of elliptic and hyperbolic space are identical,
as they are merely the quaternary orthogonal group. In
the real domain, however, the structure of the two is quite
different, and our present task shall be the actual formation
of those groups, pointing out besides certain interesting subgroups.
We shall incidentally treat the euclidean group as
a limiting case where $\dfrac{1}{k^2} \equiv 0$.

The group of translations of the hyperbolic line will depend
on one parameter, and may be written, if $k^2=-1$,
\begin{equation*}\tag{1}
 \begin{split}
 {\dot x_0}'  =  \dot x_0 \cosh d + \dot x_1 \sinh d. \label{sin} \\
 {\dot x_1}'  =  \dot x_0 \sinh d + \dot x_1 \cosh d.
 \end{split}
\end{equation*}
We get a reflection by reversing the signs in the second
%[File: 096.png]
equation. In the elliptic or spherical case we shall have
similarly
\begin{equation*}\tag{2}
 \begin{split}
 {x_0}' = x_0 \cos d + x_1 \sin d, \phantom{-} \\
 {x_1}' = -x_0 \sin d+x_1 \cos d.
 \end{split}
\end{equation*}

To pass to the euclidean case, replace $x_0$, ${x_0}'$ by $kx_0$, $k{x_0}'$
and $d$ by $\dfrac{d}{k}$, divide out $k$, and then put $\dfrac{1}{k^2}=0$.
\begin{equation*}
\frac{{x_1}'}{{x_0}'}=x'=x-d.\tag{3}
\end{equation*}

The ternary domain is more interesting. Let us express
the Absolute\index{Absolute} in the hyperbolic plane in the following parametric
form
\begin{equation*}
\dot x_0={t_1}^2+{t_2}^2, \quad \dot x_1={t_1}^2-{t_2}^2, \dot x_2=2t_1t_2.
\end{equation*}
As the Absolute must be projectively transformed into itself,
we may put
\begin{equation*}
\begin{aligned}
\sigma {t_1}' & = \alpha_{11} t_1+\alpha_{12}t_2, \\
\sigma {t_2}' & = \alpha_{21} t_1+\alpha_{22}t_2,
\end{aligned} \quad
|\alpha_{ij}|=\Delta \neq 0,
\end{equation*}
and this will lead to the general ternary transformation
\begin{multline*}
\rho {\dot x_0}' = ({\alpha_{11}}^2+{\alpha_{12}}^2+{\alpha_{21}}^2+
{\alpha_{22}}^2) \dot x_0+ ({\alpha_{11}}^2+{\alpha_{21}}^2-{\alpha_{12}}^2-
{\alpha_{22}}^2) \dot x_1 \\
\hspace{2.8in}+2(\alpha_{11}\alpha_{12}+\alpha_{21}\alpha_{22}) \dot x_2, \phantom{(4)}
\end{multline*}\ungap
\begin{multline*}
\rho {\dot x_1}' = ({\alpha_{11}}^2-{\alpha_{21}}^2+{\alpha_{12}}^2-
{\alpha_{22}}^2) \dot x_0+ ({\alpha_{11}}^2-{\alpha_{21}}^2-{\alpha_{12}}^2+
{\alpha_{22}}^2) \dot x_1 \\
\hspace{2.8in}+2(\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22}) \dot x_2,  \tag{4}
\end{multline*}\ungap
\begin{multline*}
\rho {\dot x_2}' = 2(\alpha_{11}\alpha_{21}+\alpha_{12}\alpha_{22})
 \dot x_0+2(\alpha_{11}\alpha_{21}-\alpha_{12}\alpha_{22}) \dot x_1 \\
\hspace{2.8in}+2(\alpha_{11}\alpha_{22}+\alpha_{21}\alpha_{12}) \dot x_2. \phantom{(4)}
\end{multline*}

If we view the matter geometrically, we see that there are
three distinct possibilities. First the two fixed points of the
Absolute conic are conjugate imaginaries. The real line connecting
them is ultra-infinite, and has an actual pole with
regard to the Absolute. This will give a rotation about this
point, and we shall have
\begin{equation*}
(\alpha_{11}+\alpha_{22})^2-4\Delta=(\alpha_{11}-\alpha_{22})^2+4
\alpha_{12}\alpha_{21}<0.
\end{equation*}
If the fixed points of the Absolute conic be real, the transformation,
in the actual domain, will appear as a sliding along
a real line, if $\Delta > 0$, or a sliding combined with a reflection
in a perpendicular plane\index{Plane} through this line if $\Delta < 0$. In the
third case the two fixed points of the Absolute conic fall
together, and the third fixed point of the plane falls there
too. The transformation carries a pencil of parallel lines into
itself.

%[File: 097.png]
%\mysect{Congruent transformations of the elliptic plane}
\mysect{\hspace{2em},,\hspace{5.9em},,\hspace{4em},,\hspace{1.5em}elliptic plane}
The elliptic case is treated similarly, by a judicious introduction
of imaginaries. We may write the Absolute
\begin{align*}
x_0 & = i(t_1{}^2 + t_2{}^2), \\
x_1 & = t_1{}^2 - t_2{}^2, \\
x_2 & = 2t_1t_2.
\end{align*}
Let us now take the binary substitution
\begin{align*}
\sigma t_1{}' & = (\alpha + \beta i)t_1 - (\gamma + \delta i)t_2, \\
\sigma t_2{}' & = (\gamma - \delta i)t_1 + (\alpha - \beta i)t_2.
\end{align*}

We come thus to the general group of congruent transformations
\begin{equation*}\tag{5}
 \begin{split}
 \rho x_0{}' = (\alpha^2 - \beta^2 + \gamma^2 - \delta^2)x_0 + 2(\gamma \delta - \beta \alpha)x_1 + 2(\beta \gamma + \delta \alpha) x_2, \\
 \rho x_1{}' = 2(\gamma \delta + \beta \alpha)x_0 + (\alpha^2 - \beta^2 - \gamma^2 + \delta^2)x_1 + 2(\beta \delta - \gamma \alpha)x_2,  \\
 \rho x_2{}' = 2(\beta \gamma - \delta \alpha)x_0 + 2(\beta \delta + \gamma \alpha)x_1 + (\alpha^2 + \beta^2 - \gamma^2 - \delta^2)x_2. \\
 \Delta = (\alpha^2 + \beta^2 + \gamma^2 + \delta^2)^3. \hspace{1in}
 \end{split}
\end{equation*}

%\mysect{Congruent transformations of the euclidean plane}
\mysect{\hspace{2em},,\hspace{5.9em},,\hspace{4em},,\hspace{1.5em}euclidean plane}
These forms remind us at once of like forms occurring in
the theory of functions. Suppose, in fact, that we have the
euclidean sphere
\[X^2 + Y^2 + Z^2 = 1.
\]
The geometry thereof will be exactly our spherical geometry,
and we wish for the group of congruent transformations of
this sphere into itself. Let us project the sphere stereographically
from the north pole upon the equatorial plane,
and, considering this as the Gauss plane, take the linear
transformation

\[ \begin{array}{cc}
z' = \dfrac{(\alpha + \beta i)z - (\gamma + \delta i)}{(\gamma - \delta i)z + (\alpha - \beta i)}, &
\overline{z'} = \dfrac{(\alpha - \beta i)\overline{z} - (\gamma - \delta i)}{(\gamma + \delta i)\overline{z} + (\alpha + \beta i)}.
\end{array} \]

These equations are seen at once to be transformable into
the others by a simple change of variables.

To pass over to the euclidean case, put
\begin{equation*}\tag{6}
 \begin{split}
 x = \dfrac{x_1}{x_0}, \quad y = \dfrac{y_1}{y_0}, \hspace{0.75in} \\
 x' = C_1 + A_1x + B_1y, \hspace{0.6in} \\
 y' = C_2 + A_2x + B_2y, \hspace{0.6in} \\
 A_1B_2 - A_2B_1 = A_1{}^2 + B_1{}^2 = A_2{}^2 + B_2{}^2 = 1.
 \end{split}
\end{equation*}
Notice that here the group
\[ x' = c_1 + x, \quad y' = c_2 + y,
\]
is an invariant sub-group.

The congruent groups in three dimensions are of the same
general form as those in two, albeit the structure is a trifle
%[File: 098.png]
more complicated. We wish for the six-parameter groups
leaving invariant respectively a real, non-ruled quadric, an
imaginary quadric of real equation, and an imaginary conic
with two real equations. The solution has of course, long
been known.%
\footnote{The literature of this subject is large. The first writer to express the
general orthogonal substitution\index{Orthogonal01@Orthogonal substitutions} in terms of independent parameters was
Cayley\index{Cayley}, `Sur quelques propri\'et\'es des d\'eterminants gauches,' \emph{Crelle's Journal},
vol.~xxxii, 1846. The treatment here given follows broadly Chapters VI and
VII of Klein's\index{Klein} `Nicht-euklidische Geometric', lithographed notes, G\"ottingen,
1893. }
%endfootnote

%\mysect{Congruent transformations of the hyperbolic space}
\mysect{\hspace{2em},,\hspace{5.9em},,\hspace{4em},,\hspace{1.5em}hyperbolic space}
The Absolute\index{Absolute} of hyperbolic space may be interpreted as
a euclidean sphere of radius one, and the problem of finding
all congruent transformations of hyperbolic space, is the same
as that of finding all collineations carrying such a sphere into
itself. Let us represent this sphere parametrically in terms
of its rectilinear generators
\begin{align*}
\dot x_0 & = z \bar z+1, \\
\dot x_1 & = z \bar z-1, \\
\dot x_2 & = z+\bar z, \\
\dot x_3 & = -i(z-\bar z).
\end{align*}

Let us now take the linear transformation
\begin{equation*}
z'=\frac{\alpha z+\beta}{\gamma z+\delta}, \quad \bar z'=\frac{\bar \alpha
\bar z+\bar \beta}{\bar \gamma \bar z+\bar \delta}.
\end{equation*}

The six-parameter group of congruent transformations of
positive modulus will be
\begin{multline*}
\rho {\dot x_0}' = (\alpha \bar \alpha+\beta \bar \beta+\gamma \bar \gamma+\delta \bar \delta) \dot x_0
+(\alpha \bar \alpha-\beta \bar \beta+\gamma \bar \gamma-\delta \bar \delta) \dot x_1 \\
\hspace{1in}+(\alpha \bar \beta+\bar \alpha \beta+\gamma \bar \delta+\bar \gamma \delta) \dot x_2
+i(\alpha \bar \beta-\bar \alpha \beta+\gamma \bar \delta-\bar \gamma \delta) \dot x_3, \phantom{(7)}
\end{multline*}\ungap
\begin{multline*}
\rho {\dot x_1}' = (\alpha \bar \alpha+\beta \bar \beta+\gamma \bar \gamma-\delta \bar \delta) \dot x_0
+(\alpha \bar \alpha-\beta \bar \beta-\gamma \bar \gamma+\delta \bar \delta) \dot x_1 \\
\hspace{1in}+(\alpha \bar \beta+\bar \alpha \beta-\gamma \bar \delta-\bar \gamma \delta) \dot x_2
+i(\alpha \bar \beta-\bar \alpha \beta-\gamma \bar \delta+\bar \gamma \delta) \dot x_3, \tag{7}
\end{multline*}\ungap
\begin{multline*}
\rho {\dot x_2}' = (\alpha \bar \gamma+\bar \alpha \gamma+\beta \bar \delta+\bar \beta \delta)\dot x_0
+(\alpha \bar \gamma+\bar \alpha \gamma-\beta \bar \delta-\bar \beta \delta)\dot x_1 \\
\hspace{1in}+(\alpha \bar \delta+\bar \alpha \delta+\beta \bar \gamma+\bar \beta \gamma) \dot x_2
+i(\alpha \bar \delta-\bar \alpha \delta-\beta \bar \gamma+\bar \beta \gamma) \dot x_3, \phantom{(7)}
\end{multline*}\ungap
\begin{multline*}
-\rho {\dot x_3}' = i(\alpha \bar \gamma-\bar \alpha \gamma+\beta \bar \delta-\bar \beta \delta)\dot x_0
+i(\alpha \bar \gamma-\bar \alpha \gamma-\beta \bar \delta+\bar \beta \delta)\dot x_1 \\
\hspace{1in}+i(\alpha \bar \delta-\bar \alpha \delta+\beta \bar \gamma-\bar \beta \gamma) \dot x_2
-(\alpha \bar \delta+\bar \alpha \delta-\beta \bar \gamma-\bar \beta \gamma) \dot x_3. \phantom{(7)}
\end{multline*}
\begin{equation*}
\Delta=[(\alpha \delta-\beta \gamma)(\bar \alpha \bar \delta-\bar
\beta \bar \gamma)]^2.
\end{equation*}

This sub-group might properly be called the group of
motions\index{Motions}. The total group is made up of these and the
six-parameter assemblage of transformations of negative
%[File: 099.png]
discriminant called \emph{symmetry transformations}. We reach
these latter by writing
\begin{equation*}
z'=\frac{\alpha'\bar z+\beta'}{\gamma'\bar z+\delta'},\quad \bar z'=\frac{\bar \alpha' z+
\bar \beta'}{\bar \gamma' z+\bar \delta'}.
\end{equation*}

The distinction between motions and symmetry transformations
stands out in clear relief when we consider the effect
upon the Absolute\index{Absolute}. The sub-group of motions includes the
identical transformation, and any motion may be reached by
a continuous change in the six essential parameters from the
values which give the identical transformation, without ever
causing the modulus to vanish. This shows that as, under
the identical transformation, each generator of the Absolute
stays in place, so, under the most general motion, the generators
of each set are permuted among one another. On the contrary,
the most general symmetry transformation\index{Symmetry transformations} will arise
from the combination of the most general motion\index{Motions} with a
reflection, and it is easy to see that a reflection will interchange
the two sets of generators.

%\mysect{Congruent transformations of the elliptic and spherical space}
\mysect{\hspace{2em},,\hspace{5.9em},,\hspace{4em},,\hspace{1.5em}elliptic and spherical space}
In the elliptic case we shall have the group of all real
quaternary orthogonal substitutions\index{Orthogonal01@Orthogonal substitutions}. An extremely elegant
way of expressing these is offered by the calculus of
quaternions.

Let us, following the Hamiltonian\index{Hamilton} notation, assume three
new symbols $i$, $j$, $k$:
\begin{equation*}
i^2 = j^2 = k^2 = ijk =-1.
\end{equation*}
We assume that they obey the associative and commutative
laws of addition, the associative and distributive laws of
multiplication. An expression of the type
\begin{equation*}
p_0+p_1i+p_2j+p_3k
\end{equation*}
is called a \emph{quaternion}\index{Quaternions}, whereof
\begin{equation*}
\sqrt{(pp)}
\end{equation*}
is called the \emph{Tensor}\index{Tensor}. It is easy to show that the tensor of
the product of two quaternions is the product of their tensors.

Let us next write
\begin{equation*}
{x_0}'+{x_1}'i+{x_2}'j+{x_3}'k=P(x_0+x_1i+x_2j+x_3k)Q, \tag{8}
\end{equation*}
where $P$ and $Q$ are quaternions. Multiplying out the right-hand
side, and identifying the real parts and the coefficients
of $i$, $j$, $k$, we have ${x_0}'{x_1}'{x_2}'{x_3}'$
expressed as linear homogeneous
functions of $x_0x_1x_2x_3$. The modulus of the transformation
will be different from zero, and we shall have
\begin{equation*}
(x'x')=(xx) \cdot |P|^2 \cdot |Q|^2.
\end{equation*}

%[File: 100.png]
These equations will give the six-parameter group of
motions, the group of symmetry transformations will arise
from
\begin{equation*}
{x_0}'+{x_1}'i+{x_2}'j+{x_3}'k=P'(x_0-x_1i-x_2j-x_3k)Q',
\end{equation*}
the distinction between motions and symmetry transformations\index{Symmetry transformations}
being as in the hyperbolic case.

Our group of motions is half-simple, being made up of two
invariant sub-groups $G_3{G_3}'$ obtained severally by assuming
that $Q$ or $P$ reduces to a real number. We obtain their
geometrical significance as follows:---

The group of motions\index{Motions} $G_6$ can be divided into two invariant
three-parameter sub-groups $g_3{g_3}'$ by resolving it into
the two groups which keep invariant all generators of the
one or the other set on the Absolute\index{Absolute}. Now were it possible
to divide $G_6$ into invariant three-parameter sub-groups in
two different ways, the highest common factor of $g_3$ or ${g_3}'$
with $G_3$, would be an invariant sub-group, not only of $G_6$
but of $g_3$. This may not be, for $g_3$ is nothing but the binary
projective group which has no invariant sub-groups. Hence
the groups $g_3 {g_3}'$ are identical with $G_3 {G_3}'$, and the latter keep
the one or the other set of generators all in place.

\mysect{Clifford parallels, or paratactic lines}
\mysect{The groups of right and left translations}
It is well worth our while to look more deeply into the
properties of these sub-groups. Let us distinguish the two
sets of generators of the Absolute by calling the
one\index{Right and left generators of Absolute}\label{rlgen}\index{Left and right generators of Absolute} \emph{left},
and the other \emph{right}. This may be done analytically by
adjoining a number $i$ to our domain of rationals. Two lines
which cut the same left (right) generators of the Absolute
shall be called \emph{left (right) paratactic}\index{Left and right parataxy}\index{Parataxy}\index{Right and left parataxy}.%
\footnote{The more common name for such lines is `Clifford\index{Clifford} parallels'. The
word paratactic is taken from Study\index{Study}, `Zur Nicht-euklidischen und Liniengeometrie,'
\emph{Jahresbericht der deutschen Mathematikervereinigung}, xi, 1902. We
have already defined parallels as lines intersecting on the Absolute, and
although in the present case such lines cannot both be real, yet it is better to
be consistent in our terminology, especially since we shall find in Chapter XVI
a transformation carrying parallelism\index{Parallelism} into parataxy. Clifford's discussion
is in his `Preliminary Sketch of Biquaternions', \emph{Proceedings of the London
Mathematical Society}, vol.~iv, 1873}
%endfootnote
As the conjugate
imaginary to each generator of the Absolute belongs to the
same set as itself, we see that through each real point will
pass a real left and real right paratactic to each real line; and
the same will hold for each real plane. Of course there are
possible complications in the imaginary domain, but these
need not concern us here.

\index{Left and right translations}\index{Right and left translations}Let
us now look at a real congruent transformation which
keeps all right generators invariant. Two conjugate imaginary
left generators will also be invariant, and every line meeting these
%[File: 101.png]
two will be carried into itself, every other line will be carried
into a line right paratactic to itself. Such a transformation
shall be called a
\emph{left translation}\index{Left and right translations}\index{Right and left translations}\index{Translations},
since the path curves of all
points will be a congruence of left paratactic lines. In fact
this congruence will give the path curves for a whole one-parameter
family of left translations. Let the reader show
that under a translation, any two points will be transported
through congruent distances.

Before leaving the elliptic case, let us notice that in the
elliptic plane a reflection in a line is identical with a reflection
in a point, or a rotation through an angle $\pi$, in a spherical
plane they are different, and a reflection in a line is the same
as a rotation through an angle $\pi$ coupled with an interchange
of each point with its equivalent. In three dimensions, there
is never any identity between a rotation and a reflection, on
the other hand nothing new is brought in by interchanging
each point with its equivalent, for as each plane is hereby
transformed into self, we may split up the transformation
into a reflection in a plane, a reflection in a second plane
perpendicular to the first, and a rotation through an angle $\pi$
about a line perpendicular to both planes.

\mysect{Congruent transformations of euclidean space}
To pass to the limiting euclidean case
\[
\begin{aligned}
  x' &= A_0 + A_1 x + A_2 y + A_3 z,  \\
  y' &= B_0 + B_1 x + B_2 y + B_3 z,  \\
  z' &= C_0 + C_1 x + C_2 y + C_3 z,
\end{aligned}
\tag{9}
\]
where $\lVert A_1 B_2 C_3 \rVert$ is the matrix of a ternary orthogonal substitution.

There will be a three-parameter invariant sub-group; that
of all translations
\begin{align*}
  x' &= A_0 + x, \\
  y' &= B_0 + y, \\
  z' &= C_0 + z.
\end{align*}

In like manner we may find the six-parameter assemblage
of symmetry transformations.
\index{Congruent transformations|)}
\index{Transformations, congruent|)}

%[File: 102.png]
%CHAPTER IX
\mychap{IX}{POINT, LINE, AND PLANE TREATED ANALYTICALLY}
\mysect{Notable points of a triangle in the non-euclidean plane}

The object of the present chapter is to return, as promised
in Chapter VI, to the problems of elementary non-euclidean
geometry, from the higher point of view gained by extending
space to be a perfect analytic continuum. We shall find in
the Absolute a \emph{Deus ex Machina} to relieve us from many an
embarrassment. We shall leave aside the euclidean case,
and, for the most part, handle all of our non-euclidean cases
together, leaving to the reader the simple task of making
the distinction between the elliptic and the spherical cases.
Otherwise stated, our present task is to express the fundamental
metrical theorems of point, line, and plane, in terms
of the invariants of the congruent group.

Let us notice, at the outset, that the principle of duality
plays a fundamental rôle. The distance of two points is
$\dfrac{k}{2i} \times$~logarithm of the cross ratio that they form with the
points where their line meets the Absolute, the angle of two
planes is $\dfrac{1}{2i} \times$~logarithm of the cross ratio which they form
with two planes through their intersection, tangent to the
Absolute; the distance from a point to a plane is $\dfrac{\pi k}{2}$ minus its
distance to the pole of that plane with regard to the Absolute.
Two intersecting lines or planes which are conjugate with
regard to the Absolute are mutually perpendicular\index{Perpendicularity}. Two
points which are conjugate with regard to the Absolute shall
be said to be mutually \emph{orthogonal}\index{Orthogonal points}. In the real domain of
hyperbolic space, if one of two such points be actual, the other
must be ideal; the converse is not necessarily true.

Let us begin in the non-euclidean plane, say $x_3 = 0$. Let
us take two points $A$, $B$ with coordinates $(x)$ and $(y)$ respectively,
and find the two points of their line which are at
%[File: 103.png]
congruent distances from them. These shall be called the
\emph{centres of gravity}\index{Centre03@--- of gravity of points}
of the two points, and are, in fact, the two
points which divide harmonically the given points, and the
intersections of their line with the Absolute\index{Absolute}. We purposely
exclude the spherical case, where the centres of gravity will
be equivalent points.

The necessary and sufficient condition that the point
$\lambda(x) + \mu(y)$ should be at congruent distances from $(x)$ and
$(y)$ that
\[
  \lambda : \mu = \sqrt{(yy)} : \pm \sqrt{(xx)}.
\]
The coordinates of the centres of gravity will thus be
\[
  \Bigl( \frac{x}{\sqrt{(xx)}} \pm \frac{y}{\sqrt{(yy)}} \Bigr). \tag{1}
\]

Let the reader discover what complications may arise in the
ideal domain.

Let us next take three non-collinear points $A$, $B$, $C$ with
the coordinates $(x)$, $(y)$, $(z)$. A line connecting $(x)$ with a
centre of gravity of $(y)$ and $(z)$ will be
\[
  \sqrt{(yy)}\,|Xxz| + \sqrt{(zz)}\,|Xxy| = 0.
\]
It is clear that such lines are concurrent\index{Concurrence} by threes, in four
points which may be called the \emph{centres of gravity} of the three
given points. On the other hand the centres of gravity of
our pairs of points are collinear\index{Collinearity} in threes. Lastly, notice that
a dual theorem might be reached by interchanging the objects,
point and line, distance and angle; by taking, in fact, a polar
reciprocation in the Absolute:---

\medskip\twincol{\emph{Theorem}~1. The centres of
gravity of the pairs formed
from three given points are
collinear by threes on four
lines. The lines from the
given points to the centres
of gravity of their pairs are
concurrent by threes in four
points.}
{\emph{Theorem}~1$'$. The bisectors\index{Bisector of an angle} of
the angles formed by three
coplanar but not concurrent
lines are concurrent by threes
in four points. The points
where these bisectors meet
the given lines are collinear
by threes on four lines.}

The centres of gravity of the points $(x)$, $(y)$, $(z)$ are easily
seen to be
\[
  \Bigl( \frac{x}{\sqrt{(xx)}}
     \pm \frac{y}{\sqrt{(yy)}}
     \pm \frac{z}{\sqrt{(zz)}} \Bigr).  \tag{2}
\]

Returning to the line $BC$ we see that the coordinates of its
%[File: 104.png]
pole with regard to the Absolute\index{Absolute} will have the coordinates $(s)$,
where for every value of $(r)$
\[
(rs) \equiv |ryz|.
\]
The equation of the line connecting this point with $A$, i.e.\ the
line through $A$ perpendicular\index{Perpendicularity} to $BC$, will be
\[
(Xy)(zx)-(Xz)(xy) = 0.
\]
If we permute the letters $x$, $y$, $z$ cyclically twice, we get two
other equations of the same type, and the sum of the three
is identically zero, so that

\medskip\twincol{\emph{Theorem}~2. The lines
through each of three given
non-collinear points, perpendicular
to the line of the other
two, are concurrent\index{Concurrence}.}
{\emph{Theorem}~2$'$. The points on
each of three coplanar but not
concurrent lines, orthogonal
to the intersection of the other
two, are collinear\index{Collinearity}.}

Returning to a centre of gravity of the two points $BC$, we
see that a line through it perpendicular to the line $BC$ will
have the equation

\[
\begin{vmatrix}
  (xy) &  (xz)   \\
  \dfrac{(yy)}{\sqrt{(yy)}} + \dfrac{(yz)}{\sqrt{(zz)}}
& \dfrac{(yz)}{\sqrt{(yy)}} + \dfrac{(zz)}{\sqrt{(zz)}}
\end{vmatrix} = 0,
\]
\[
  \Bigl[ \frac{(yz)}{\sqrt{(yy)}\; \sqrt{(zz)}} - 1 \Bigr]
  \Bigl[ \frac{(xy)}{\sqrt{(yy)}} - \frac{(xz)}{\sqrt{(zz)}} \Bigr]
  = 0.
\]

The first factor will vanish (in the real domain) only when
$(y)$ and $(z)$ are identical, the equation will then be
\[
\frac{(xy)}{\sqrt{(yy)}} - \frac{(xz)}{\sqrt{(zz)}} = 0.
\]

We see immediately from the form of this equation, that
all points of this line are at congruent distances from $(y)$ and
$(z)$, thus confirming II.~33.

\medskip\twincol{\emph{Theorem}~3. If three non-collinear
points be given, the
perpendiculars to the lines of
their pairs at the centres of
gravity\index{Centre03@--- of gravity of points} of these pairs are
concurrent by threes in four
points, each at congruent distances
from all three of the
given points.}
{\emph{Theorem}~3$'$. If three coplanar
but not concurrent
lines be given, the points
orthogonal\index{Orthogonal points} to their intersections
on the bisectors\index{Bisector of an angle} of the
corresponding angles are collinear
by threes on four lines,
making congruent angles with
all three of the given lines.}\par\medskip
%end twincol

Let us now suppose that besides our three original points,
%[File: 105.png]
we have three others lying one on each of the lines of the
first set as follows
\begin{align*}
A' & = (ly + mz), \\
B' & = (pz + qx), \\
C' & = (rx + sy).
\end{align*}

\mysect{Analoga of the theorems of Menelaus and Ceva}
Let us, for the moment, suppose that we are restricted to
a consistent region of the plane. Then we shall easily see
from Axiom XVI that if $AA'$, $BB'$, $CC'$ be concurrent
\[
  \frac{\sin\dfrac{\overrightarrow{BA'}}{k} }
       {\sin\dfrac{\overrightarrow{CA'}}{k} }
\cdot
  \frac{\sin\dfrac{\overrightarrow{CB'}}{k} }
       {\sin\dfrac{\overrightarrow{AB'}}{k} }
\cdot
  \frac{\sin\dfrac{\overrightarrow{AC'}}{k} }
       {\sin\dfrac{\overrightarrow{BC'}}{k} } < 0.
\]
On the other hand, if $A'$, $B'$, $C'$ be collinear\index{Collinearity},
\[
  \frac{\sin\dfrac{\overrightarrow{BA'}}{k} }
       {\sin\dfrac{\overrightarrow{CA'}}{k} }
\cdot
  \frac{\sin\dfrac{\overrightarrow{CB'}}{k} }
       {\sin\dfrac{\overrightarrow{AB'}}{k} }
\cdot
  \frac{\sin\dfrac{\overrightarrow{AC'}}{k} }
       {\sin\dfrac{\overrightarrow{BC'}}{k} } > 0.
\]

Now, more specifically, we see that
\[
\sin^2\frac{\overline{BA'}}{k} =
  \frac{m^2\left[ (yy)(zz) - (yz)^2 \right]}
       {(yy)\left[ l^2(yy) + 2lm(yz) + m^2(zz) \right]},
\]
whence
\[
  \left[
  \frac{\sin\dfrac{\overrightarrow{BA'}}{k} }
       {\sin\dfrac{\overrightarrow{CA'}}{k} }
\cdot
  \frac{\sin\dfrac{\overrightarrow{CB'}}{k} }
       {\sin\dfrac{\overrightarrow{AB'}}{k} }
\cdot
  \frac{\sin\dfrac{\overrightarrow{AC'}}{k} }
       {\sin\dfrac{\overrightarrow{BC'}}{k} }
  \right]
= \left( \frac{mqs}{lpr} \right)^2.
\]
The equation of the line $AA'$ will be
\[
l\lvert Xxy \rvert + m\lvert Xzx \rvert = 0.
\]
And the condition for concurrence for the three lines
\[
(lpr + mqs) \cdot \lvert xyz \rvert^2 = 0,
\]
and this will give
\[
\frac{mqs}{lpr} = -1.
\]
On the other hand, we easily see that if $A'$, $B'$, $C'$ be collinear
\[
lpr-mqs = 0.
\]

\smallskip\emph{Theorem}~4. If $A'$, $B'$, $C'$ be three points lying respectively
%[File: 106.png]
on the lines $BC$, $CA$, $AB$, all six points being in a consistent
region, then the expression
\begin{equation*}
\frac{\sin \dfrac{\overrightarrow{BA'}}{k} }
     {\sin \dfrac{\overrightarrow{CA'}}{k} }
\cdot
\frac{\sin \dfrac{\overrightarrow{CB'}}{k} }
     {\sin \dfrac{\overrightarrow{AB'}}{k} }
\cdot
\frac{\sin \dfrac{\overrightarrow{BC'}}{k} }
     {\sin \dfrac{\overrightarrow{AC'}}{k} },
\end{equation*}
will be equal to $-1$ when, and only when, $AA'$, $BB'$, $CC'$
are concurrent, while it will be equal to 1, when, and only
when, $A'$, $B'$, $C'$ are collinear.

These are, of course, merely the analogs of the theorems
of Menelaus\index{Menelaus} and Ceva\index{Ceva}. It is worth noticing also, that they
will afford a sufficient ground for a metrical theory of cross
ratios.

Let us next suppose that $A'$ is a point where a bisector
of an angle formed by the lines $BA$, $CA$, meets $BC$. We
find $l$ and $m$ easily in this case, by noticing that $A'$ must be
at congruent distances from $AB$ and $AC$, thus getting
\begin{equation*}
(y\sqrt{(zz)(xx)-(xz)^2}+z\sqrt{(xx)(yy)-(xy)^2}),
\end{equation*}
\begin{equation*}
\sin \frac{\overline{BA'}}{k}:\sin \frac{\overline{CA'}}{k}
=\sin \frac{\overline{BA}}{k}:\sin \frac{\overline{CA}}{k}.
\end{equation*}

\medskip\twincol{\emph{Theorem}~5. If three noncollinear\index{Collinearity}
points be given, each
bisector of an angle formed by
the lines connecting two of
the points with the third will
meet the line of the two points
in such a point that the ratio
of the sines of the $k$th parts
of its distances from the two
points is equal to the corresponding
ratio for these two
with the third point.}
{\emph{Theorem}~5$'$. If three coplanar
but non-concurrent\index{Concurrence}
lines be given, each centre of
gravity of a pair of points
where two of the lines meet
a third determines with the
intersection of this pair of
lines such a line, that the ratio
of the sines of the angles which
it makes with these two lines,
is equal to the corresponding
ratio for the two lines with
the third.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~6. The locus of
a point which moves in a plane,
in such a way that the ratio
of the sines of the $k$th parts
of its distances from two points
is constant, is a curve of the
second order.}
{\emph{Theorem}~6$'$. The envelope of
a line which moves in such a
way in a plane, that the ratio
of the sines of its angles with
two fixed lines is constant, is
an envelope of the second class.}\par\medskip
%end twincol

%[File: 107.png]
It would be quite erroneous to suppose that either of these
curves would be, in general, a circle. Let the reader show
that if an angle inscribed in a semicircle be a right angle, the
euclidean hypothesis holds.

\mysect{Formulae of the parallel angle}
Our next investigation shall be connected with parallel\index{Parallelism}
lines. We suppose, for the moment, that we are in the
hyperbolic plane, and that $k=i$. We shall hunt for the
expression for the angle\index{Angle11@--- parallel}\index{Parallel angle}
which a parallel to a given line $l$
passing through a point $P$ makes with the perpendicular
to $l$ through $P$. This shall be called the \emph{parallel angle} of
the distance from the point to the line, and if the latter be $d$
the parallel angle shall be written%
\footnote{The concept \emph{parallel angle} and the notation $\Pi(d)$ are due to Lobatchewsky\index{Lobatchewsky}. }
%endfootnote
\begin{equation*}
\Pi(d).
\end{equation*}

Let us give to the point $P$ the coordinates $(y)$, while the
given line has the coordinates $(u)$. Let $(v)$ be the coordinates
of a parallel to $(u)$ through $(y)$. Let $D$ be the point where
the perpendicular to $(u)$ through $(y)$ meets $(u)$. We seek
$\cos \Pi(d)$.

Since $(u)$ and $(v)$ intersect on the Absolute\index{Absolute}
\begin{equation*}
(uu)(vv)-(uv)^2 = 0.
\end{equation*}
The equation of the line $PD$ will be
\begin{equation*}
|xyu|=0.
\end{equation*}
The cosine of the angle formed by $u$ and $PD$ will be
\begin{equation*}
\cos \Pi(d)=\frac{|yuv|}{\sqrt{\mystrut{6}vv}\;\sqrt{(uu)(yy)-(yu)^2}}
\end{equation*}
squaring, and remembering that
\begin{equation*}
(vy) = 0,
\end{equation*}
\begin{equation*}
\cos^2 \Pi(d)=\frac{\left| \begin{array}{ccc}
(yy) & (uy) & 0 \\
(uy) & (uu) & (uv) \\
0 & (uv) & (vv)
\end{array} \right| }
{(vv)[(uu)(yy)-(uy)^2]},
\end{equation*}
\begin{equation*}
\cos \Pi(d)=\frac{i(uy)}{\sqrt{(uu)(yy)-(uy)^2}},
\end{equation*}
\begin{equation*}
\cos \Pi(d) = \tanh d. \tag{3}
\end{equation*}
From these we easily see
\begin{equation*}
\sin \Pi(d) = \sech (d); \quad \tan \Pi(d) = \csch (d). \tag{4}
\end{equation*}
Furthermore, if $\measuredangle ACB$ be a right angle
\begin{equation*}
\cos \measuredangle ABC=\frac{\cos \Pi(\overline{BC})}{\cos \Pi(\overline{AB})}\quad
\sin \measuredangle ABC=\frac{\ctn \Pi(\overline{CA})}{\ctn \Pi(\overline{AB})}. \tag{5}
\end{equation*}

%[File: 108.png]
\begin{equation*}
\cos\measuredangle ABC =
  \frac{\sin\measuredangle CAB}{\sin\Pi (AC)}.          \tag{6}
\end{equation*}
\begin{equation*}
\sin \Pi (\overline{AB}) =
  \sin \Pi (\overline{BC})\sin\Pi(\overline{CA}) =
  \tan\measuredangle CAB\tan\measuredangle ABC.       \tag{7}
\end{equation*}\index{Angle11@--- parallel}\index{Parallel angle}

\mysect{Equations of parallels to a given line}
Let the reader prove the correctness of the following construction
for the parallels to $P$ through $l$:

Drop a perpendicular from $P$ on $l$ meeting it in $Q$. Take $S$
a convenient point on the perpendicular to $PQ$ at $P$, and let
the perpendicular to $PS$ at $S$ meet $l$ at $R$. Then with $P$ as
a centre, and a radius equal to $(QR)$, construct an arc meeting
$RS$ in $T$. $PT$ will be the parallel required.%
\footnote{The formulae given may be used as the basis for the whole trigonometric
structure. Cf.\ Manning\index{Manning}, \emph{Non-euclidean Geometry}, Boston, 1901. Manning's
reasoning is open to very grave question on the score of rigour. }
%endfootnote

Be it noticed that, as we should expect,
\[
\limit_{d \doteq 0}\frac{\cos \Pi (d)}{d} = 1.
\]

Let us now find the equations of the two parallels to the
line $(u)$ which pass through the point $(y)$. These two cannot,
naturally, be rationally separated one from the other, so that
we shall find the equations of both at once. Let the coordinates
of the line which connects the other intersections of the parallels
and the Absolute\index{Absolute} be $(w)$. The general form for an equation
of a curve of the second order through the intersections of
$(u)$ and $(w)$ with the Absolute will be
\[
  l(ux)(wx) - m(xx) - 0,
\]
and this will pass through $(y)$ if
\[
  l : m = (yy) : (uy)(wy).
\]

Since this curve is a pair of lines meeting in $(y)$ the polar
of $(y)$ with regard to it will be illusory, i.e.\ the coefficients of
$(x)$ will vanish in
\[
  (yy)(uy)(wx) + (yy)(wy)(ux) - 2(uy)(wy)(xy) = 0.
\]
This last equation may be written
\[
(uy) \begin{vmatrix}
       (wx) & (wy) \\ (yx) & (yy) \end{vmatrix} +
(wy) \begin{vmatrix}
       (ux) & (uy) \\ (yx) & (yy) \end{vmatrix} = 0.
\]

Now, by the harmonic theory of a quadrangle inscribed in
a curve of the second order, $w$ will pass through the intersection of $(u)$
with the polar of $y$ with regard to the Absolute,
so that we may write
\[
w_i = \lambda u_i + \mu y_i.
\]
Substituting
\[
[2\lambda(uy) + \mu(yy)]
  \begin{vmatrix} (ux) & (uy) \\ (yx) & (yy) \end{vmatrix} = 0.
\]
%[File: 109.png]
The coefficients of $x_0 x_1 x_2$ will vanish if
\[
  \lambda = -(yy),\quad \mu = 2(uy).
\]
Under these circumstances
\begin{align*}
  (wx) &= -(yy)(ux) + 2(uy)(xy),   \\
  (wy) &= (yy)(uy).
\end{align*}
Which leads to the required equation
\[
  (uy)^2(xx) + (ux)^2(yy) - 2(ux)(uy)(xy) = 0.   \tag{8}
\]

To get the euclidean formula, replace $x_0$ by $k_0 x_0$ and divide
by $k$. We get the square of the usual expression
\[
  [(uy)x_0 - (ux)y_0]^2 = 0.   \tag{9}
\]

The principles which we have followed in studying the
metrical invariants of the plane may be extended with ease
to three dimensions. We have merely to adjoin the fourth
homogeneous point or line coordinate.

\mysect{Notable points of a tetrahedron, and resulting desmic configurations}
Let us have four points, not in one plane, with the coordinates $(x)$, $(y)$, $(z)$, $(t)$ respectively. We easily see that the
eight points
\[
  \Bigl( \frac{x}{\sqrt{(xx)}}
     \pm \frac{y}{\sqrt{(yy)}}
     \pm \frac{z}{\sqrt{(zz)}}
     \pm \frac{t}{\sqrt{(tt)}} \Bigr),  \tag{10}
\]
will be points of concurrence, four by four, of lines from each
of the given points to the centres of gravity of the other three.
These eight may, in fact, be called the \emph{centres of gravity} of the
four points. The centres of gravity will form with the given
points a \emph{desmic configuration}\index{Desmic configuration}.%
\footnote{The desmic configuration was first studied by Stephanos\index{Stephanos}, `Sur la configuration
desmique de trois t\'etraèdres,' \emph{Bulletin des Sciences math\'ematiques},
s\'erie 2, vol.\ iii, 1878.
}
%endfootnote
The meaning of this phrase
is as follows. Let us indicate the centres of gravity by the
signs prefixed to their radicals, giving always to the first
radical a positive sign. We may then divide our twelve
points into three lots as follows:---
\[
\begin{array}{cccc}
  (x)   &  (y)   &  (z)   &  (t)   \\
 (++++) & (++--) & (+-+-) & (+--+) \\
 (+++-) & (++-+) & (+-++) & (+---)
\end{array}
\tag{11}
\]

We see that a line connecting a point of one lot, with any
point of a second, will pass through a point of the third. The
twelve points will thus lie by threes on sixteen lines, four
%[File: 110.png]
passing through each. In like manner we shall find that if
we take the twelve planes obtained by omitting in turn one
point of each lot, two planes\index{Plane} of different lots are always coaxal
with one of the third. Let the reader who is unfamiliar with
the desmic configuration\index{Desmic configuration}, study the particular case (in euclidean
space) of the vertices of a cube, its centre, and the ideal points
of concurrence of its parallel edges.

\medskip\twincol{\emph{Theorem}~7. If four non-coplanar\index{Coplanarity}
points be given, the
lines from each to the four
centres of gravity\index{Centre03@--- of gravity of points} of the other
three will pass by fours
through eight points which
form, with the original ones,
a desmic configuration.}
{\emph{Theorem}~7$'$. If four non-concurrent
planes be given,
the lines where each meets
the planes which severally are
coaxal with each of the three
remaining planes and a plane
bisecting\index{Bisector of an angle} a dihedral angle of
the two still left, lie by fours
in eight planes which, with
the original ones, form a
desmic configuration.}\par\medskip
%end twincol

Let the reader show that the centres of gravity of the six
pairs formed from the given points will determine a second
desmic configuration, and dually for the planes bisecting the
dihedral angles.

Let us seek for a point which is at congruent distances
from our four given points. It is easy to see that there cannot
be more than eight such points. Their coordinates are found
to be $(s)$ where, for all values of $r$,
\[
(rs) \equiv \sqrt{(xx)}\, \lvert ryzt \rvert
        \pm \sqrt{(yy)}\, \lvert rztx \rvert
        \pm \sqrt{(zz)}\, \lvert rtxy \rvert
        \pm \sqrt{(tt)}\, \lvert rxyz \rvert.
\tag{12}
\]

\medskip\twincol{\emph{Theorem}~8. If four non-coplanar
points be given, the
eight points which are severally
at congruent distances
from them form, with the
original four, a desmic configuration.}
{\emph{Theorem}~8$'$. If four non-concurrent
planes be given,
the eight planes which severally
meet them in congruent
dihedral angles, form, with the
original four, a desmic configuration.}\par\medskip
%end twincol

As there are eight points at congruent distances from the
four given points, so there will be eight planes at congruent
distances from them, we have but to take the polars of the
eight points with regard to the Absolute\index{Absolute}. In like manner,
if we consider not the points $(x)$, $(y)$, $(z)$, $(t)$ but their four
%[File: 111.png]
planes\index{Plane}, there will be eight points at congruent distances from
them. The coordinates of these latter eight will be
\begin{multline*}
\left|x\sqrt{\left\| \begin{array} {cccc}
y_0 & y_1 & y_2 & y_3 \\
z_0 & z_1 & z_2 & z_3 \\
t_0 & t_1 & t_2 & t_3 \end{array}\right\|^2} \pm
y \sqrt{\left\| \begin{array} {cccc}
z_0 & z_1 & z_2 & z_3 \\
t_0 & t_1 & t_2 & t_3 \\
x_0 & x_1 & x_2 & x_3 \end{array}\right\|^2}  \right. \\ \pm
\left. z \sqrt{\left\| \begin{array} {cccc}
t_0 & t_1 & t_2 & t_3 \\
x_0 & x_1 & x_2 & x_3 \\
y_0 & y_1 & y_2 & y_3 \end{array}\right\|^2} \pm
t \sqrt{\left\| \begin{array} {cccc}
x_0 & x_1 & x_2 & x_3 \\
y_0 & y_1 & y_2 & y_3 \\
z_0 & z_1 & z_2 & z_3\end{array}\right\|^2} \right|.
\end{multline*}

\medskip\twincol
{\emph{Theorem}~9. If four non-coplanar
points be given, the
eight points which, severally,
are at congruent distances
from the planes of the first
four, form, with the first four
points, a desmic configuration\index{Desmic configuration}.}
{\emph{Theorem}~9$'$. If four non-concurrent
planes be given, the
eight planes which, severally,
are at congruent distances
from the points of concurrence
of the first four, form,
with the first four planes, a
desmic configuration.}\par\medskip
%end twincol

\mysect{Invariant formulae for distance and angle of skew lines in line coordinates}
The parallel angle of a point with regard to a plane can be
defined as its parallel angle\index{Parallel angle} with regard to any line of the
plane through the foot of the perpendicular. If the distance
from the point to the plane be $x$, we shall have for the parallel
angle
\begin{equation*}
\cos \Pi(x) = k \tan \frac{x}{k}. \tag{13}
\end{equation*}

\emph{Definition}. A line shall be said to be parallel to a plane,
if the point common to the two be on the Absolute. The
cone of parallels to a plane $(u)$ through a point $(y)$ will have
the equation
\begin{equation*}
(uy)^2(xx) + (ux)^2 (yy) - 2(ux) (uy) (xy) = 0. \tag{14}
\end{equation*}

We now pass to certain metrical invariants of non-euclidean
space expressed in line coordinates\index{Coordinates01@Co-ordinates of a line}. We take as coordinates
for the line joining $(x)$ and $(y)$ the usual Plueckerian form
\begin{equation*}
p_{ij}=x_iy_j-x_jy_i.
\end{equation*}
The coordinates of the polar of this line with regard to the
Absolute, the Absolute polar let us say, will be
\begin{equation*}
q_{ij}=p_{kl}.
\end{equation*}
The condition for the intersection of two lines $(p)$ and $(p')$
will be, naturally
\begin{equation*}
(p \mid p') \equiv \smallsum p_{ij} p'_{kl} = 0. \tag{15}
\end{equation*}
%[File: 112.png]
Each will meet the Absolute\index{Absolute} of polar of the other if
\begin{equation*}
\smallsum p_{ij} {p'}_{ij}=0. \tag{16}
\end{equation*}
Notice that $(p \mid p')$ is an invariant under the general group of
collineations, while $\smallsum p_{ij} {p'}_{ij}$ is invariant under the congruent
group only.

We shall mean by the distance of two lines the distance
of their intersections with a third line perpendicular to them
both. It is easy to see that if two lines be not paratactic,
there will be two lines meeting both at right angles, and these
are indistinguishable in the rational domain, that is, in the
general case. If, thus, $d$ be taken to indicate the distance
of two lines, $\sin^2 \dfrac{d}{k}$ will be a root of an irreducible quadratic
equation, whose coefficients are rational invariants under the
congruent group. Let us seek for this equation.

Let one of our lines be $p$ given by the points $(x)$, $(y)$, while
the other is $(p')$ given by $(x')$ and $(y')$. For the sake of
simplifying our calculations we shall make the obviously
legitimate assumptions
\begin{equation*}
(xy) = (xy') = (x'y) = (x'y') = 0.
\end{equation*}
The distances\index{Distance03@--- of skew lines} which we wish to find are
\begin{equation*}
\sin \frac{d_1}{k}=\frac{\sqrt{(xx)(x'x')-(xx')^2}}{\sqrt{(xx)}\;\sqrt{(x'x')}}\quad
\sin \frac{d_2}{k}=\frac{\sqrt{(yy)(y'y')-(yy')^2}}{\sqrt{(yy)}\;\sqrt{(y'y')}}.
\end{equation*}
We have
\begin{equation*}
(xx)(yy)-(xy)^2=\smallsum {p_{ij}}^2,
\end{equation*}
and this will vanish only when $(p)$ is tangent to the Absolute,
a possibility which we now explicitly exclude both for $(p)$
and $(p')$.
\begin{equation*}
(xx) (yy) = \smallsum {p_{ij}}^2, \quad (x'x')(y'y')=\smallsum {{p'}_{ij}}^2,
\end{equation*}
\begin{align*}
(p \mid p')^2 & = |xyx'y'|^2 \\
& = \left| \begin{array}{cccc}
(xx) & 0 & (xx') & 0 \\
0 & (yy) & 0 & (yy') \\
(xx') & 0 & (x'x') & 0 \\
0 & (yy') & 0 & (y'y') \end{array} \right| \\
& = [(xx)(x'x')-(xx')^2][(yy)(y'y')-(yy')^2], \\
\sin^2 \frac{d_1}{k} \sin^2 \frac{d_2}{k} & = \frac{[(xx)(x'x')-(xx')^2]}{(xx)(x'x')} \frac{[(yy)(y'y')-(yy')^2]}{(yy)(y'y')}.
\tag{17}
\\
\sin^2 \frac{d_1}{k}\sin^2 \frac{d_2}{k} & = \frac{(p \mid p')^2}{\smallsum {p_{ij}}^2 \smallsum {{p_{ij}}'}^2}.
\tag{18} \\
%[File: 113.png]
\sin^2 \frac{d_1}{k} \sin^2 \frac{d_2}{k}
& = 1 - \cos^2 \frac{d_1}{k} - \cos^2 \frac{d_2}{k}
  + \frac{(xx')^2 (yy')^2}{\smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2}; \\
\smallsum p_{ij}p_{ij}{}'
& = \left|
   \begin{array}{cc}
    (xx') & (xy') \\
    (yx') & (yy')
   \end{array}
  \right|
= (xx')(yy'),
\end{align*}
%
\begin{equation*}
\cos^2 \frac{d_1}{k} + \cos^2 \frac{d_2}{k}
= 1 + \frac{(\smallsum p_{ij}p_{ij}{}')^2 - (p \mid p')^2}
           {\smallsum p_{ij} \smallsum p_{ij}{}'^2},
\end{equation*}
%
\begin{equation*}
\sin^2 \frac{d_1}{k} + \sin^2 \frac{d_2}{k}
= 1 - \frac{(\smallsum p_{ij}p_{ij}{}')^2 - (p \mid p')^2}
           {\smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2},
\end{equation*}
%
\begin{multline*}
\smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2 \sin^4 \frac{d}{k}
   + \left[(\smallsum p_{ij}p_{ij}{}')^2 - (p \mid p')^2
    - \smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2 \right]
   \sin^2 \frac{d}{k} \\
   + (p \mid p')^2 = 0. \tag{19}
\end{multline*}\ungap
%
\begin{multline*}
\smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2 \cos^4 \frac{d}{k}
   + \left[(p \mid p')^2 - (\smallsum p_{ij}p_{ij}{}')^2
    - \smallsum p_{ij}{}^2 p_{ij}{}'^2 \right]
   \cos^2 \frac{d}{k} \\
   + (\smallsum p_{ij}p_{ij}{}')^2 = 0. \tag{20}
\end{multline*}

The square roots of the products of the roots of these two
equations are well-known metrical invariants, and have been
studied under the names of \emph{moment}\index{Moment01@Moment, relative of two lines}
and \emph{commoment} of the
two lines.%
\footnote{See D'Ovidio\index{Ovidio@d'Ovidio},
`Studio sulla geometria proiettiva,' \emph{Annali di Matematica},
vi, 1873, and `Le funzioni metriche fondamentali negli spazii di quantesivogliono
dimensioni', \emph{Memorie dei Lincei}, i, 1877.}
%endfootnote
We shall return to the moment presently, attaching
a particular value to the signs of the radicals in the
denominator. If two lines intersect the moment must be zero,
and if each intersect the absolute polar of the other, the
commoment must vanish, thus bringing us back to equations
(15), (16).

To reach the limiting euclidean case we replace, as usual,
$x_0$ by $kx_0$, divide out $k^2$, and put $\dfrac{1}{k^2} = 0$. Then, since
\begin{equation*}
\lim_{k \doteq \infty} k \sin \frac{d}{k} = d.
\end{equation*}
\begin{multline*}
\text{We have } d^2 =\\
\frac{(p \mid p')^2}{(p_{01}{}^2 + p_{02}{}^2 + p_{03}{}^2)(p_{01}{}'^2 + p_{02}{}'^2 + p_{03}{}'^2)
    - (p_{01}p_{01}{}' + p_{02}p_{02}{}' + p_{03}p_{03}{}')^2},
\tag{21}
\end{multline*}
the usual formula\index{Distance03@--- of skew lines}.

With regard to the signs of the roots in (19) we see that in
the hyperbolic case, where the two lines are actual, one of
%[File: 114.png]
the points chosen to determine each line will be actual and the
other ideal, so that
\[
\smallsum p_{ij}{}^2 < 0,\quad
\smallsum p_{ij}{}'^2 < 0,
\]
\[
(p \mid p')^2 < 0,
\]
\[
\sin^2{\frac{d_1}{k}}
\sin^2{\frac{d_2}{k}} < 0.
\]
The square of the moment of the two lines is negative, so that
one distance will be real and the other pure imaginary. In
the elliptic case the two distances will be real.

\mysect{Criteria for parallelism and parataxy in line coordinates}
We shall mean by the angle of two non-intersecting lines
the angles of the plane, one through each, which contain the
same common perpendicular. This will be $k$ times the corresponding
distance of the absolute polars of the lines. We
thus get for the angles\index{Angle08@--- of skew lines} $\theta$ of the two lines ($p$), ($p'$)
\begin{multline*}
\smallsum p_{ij}{}^2\smallsum p_{ij}{}'^2\sin^4{\theta} +
[(\smallsum p_{ij}p_{ij}{}')^2 -
(p \mid p')^2 -
\smallsum p_{ij}{}^2\smallsum p_{ij}{}'^2]
\sin^2{\theta} \\
+ (p \mid p')^2 = 0.
\end{multline*}
To get the euclidean formula we make the usual substitutions
and divisions, and put $\dfrac{1}{k} = 0 $, thus getting the well-known
formula
\[
\sin^2{\theta} =
\frac{
(x_1{}^2 + x_2{}^2 + x_3{}^2)
(x_1{}'^2 + x_2{}'^2 + x_3{}'^2) -
(x_1x_1{}' + x_2x_2{}' + x_3x_3{}')^2
}
{
(x_1{}^2 + x_2{}^2 + x_3{}^2)
(x_1{}'^2 + x_2{}'^2 + x_3{}'^2)
}.
\tag{22}
\]
The coordinates of the line $q$ cutting $p$ and $p'$ at right angles
will be given by
\[
(p \mid q) =
(p' \mid q) =
\smallsum p_{ij} q_{ij} =
\smallsum p_{ij}{}' q_{ij} =
(q \mid q) = 0.
\]
We have defined as a parallel\index{Parallelism}, two lines whose intersection
is on the Absolute\index{Absolute}; let us now give the name
\emph{pseudoparallel}\index{Pseudo-parallelism of lines}
to two coplanar lines whose plane touches the Absolute. The
necessary and sufficient condition that two lines should be
either parallel or pseudoparallel is that they should intersect,
and that there should be but a single line of their pencil
tangent to the Absolute. These conditions will be expressed
by the equations
\[
(p \mid p') =
[\smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2 -
(\smallsum p_{ij}p_{ij}{}')^2] = 0
\tag{23}
\]

Let the reader notice that when we pass to the limit in the
usual way for the euclidean case, our equations (23) become
\[
(p \mid p') = \sin{\theta} = 0.
\tag{24}
\]

Let us now look at paratactic lines, i.e.\ lines which meet
the same two generators of one set of the Absolute. Of course
%[File: 115.png]
it is in the elliptic case only that two such lines can be real.
It is immediately evident that two paratactic lines have an
infinite number of common perpendiculars whereon they
always determine congruent distances, we have, in fact,
merely to look at the one-parameter group of translations
of space which carry these two lines into themselves. Conversely,
suppose that the distances of two lines be congruent.
Besides our previous equations connecting ($x$) ($y$) ($x'$) ($y'$), we
have
\[
\frac{(xx')^2}{(xx)(x'x')} =
\frac{(yy')^2}{(yy)(y'y')}.
\]

The lines $p$, $p'$ meet the Absolute respectively in the points
\[
\left
(x\sqrt{(yy)} \pm
iy\sqrt{(xx)}
\right)
\left
(x'\sqrt{(y'y')} \pm
iy'\sqrt{(x'x')}
\right).
\]
It is clear, however, that every point of the line
\[
\left(
x\sqrt{(yy)} +
iy\sqrt{(xx)}
\right)
\left
(x'\sqrt{(y'y')} +
iy'\sqrt{(x'x')}
\right),
\]
and of the line
\[
\left(
x\sqrt{(yy)} -
iy\sqrt{(xx)}
\right)
\left
(x'\sqrt{(y'y')} -
iy'\sqrt{(x'x')}
\right),
\]
\label{andoftheline}
belongs to the Absolute; the lines are paratactic. Lastly,
the absolute polars of paratactic lines are, themselves, paratactic\index{Parataxy}.
Hence

\smallskip\emph{Theorem}~10. The necessary and sufficient condition that
two lines should be paratactic is that their distances or angles
should be congruent.

This condition may be expressed analytically by equating
to zero the discriminant of either of our equations (19), (20).
\begin{multline*}
\{[(p \mid p') + (\smallsum p_{ij}p_{ij}{}')]^2 -
\smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2 \}
\{[(p \mid p') + (\smallsum p_{ij}p_{ij}{}')]^2 \\
- \smallsum p_{ij}{}^2 \smallsum p_{ij}{}'^2\} = 0. \tag{25}
\end{multline*}

This puts in evidence that intersecting lines cannot be
paratactic unless they be parallel, or pseudoparallel.

\mysect{Relative moment of two directed lines}
In conclusion, let us return for an instant to the moment\index{Distance03@--- of skew lines} of
two real lines.
\[
\sin{\frac{d_1}{k}}
\sin{\frac{d_2}{k}} =
\frac{(p \mid p')}
{\sqrt{\smallsum p_{ij}{}^2}\;
\sqrt{\smallsum p_{ij}{}'^2}}.
\]

We shall assume that the radicals in the denominator
are taken positively, so that the sign of the moment is
that of ($p \mid p'$). We now proceed to replace our concept of
a line by the sharper concept of a \emph{ray}\index{Ray} as follows. Let us,
%[File: 116.png]
in the hyperbolic case assume always $\dot x_0 > 0$, and in the
elliptic case $x_0 > 0$. The coordinates
\[
p_{ij'} = \left| \begin{matrix}
y_i{}' & y_j{}' \\
z_i{}' & z_j{}' \end{matrix} \right|,\quad
p_{ij} = \left| \begin{matrix}
y_i & y_j \\
z_i & z_j \end{matrix} \right|,
\]
shall be called the coordinates of the ray from $(y)$ to $(z)$, and
this shall be considered equivalent to any other ray\index{Ray} whose
coordinates differ therefrom by a positive factor. Interchanging
$(y)$ and $(z)$ will give a second ray, said to be \emph{opposite}
to this. The relative moment\index{Moment02@--- relative of two rays} of two rays is thus determined,
both in magnitude and sign. We shall later see various
applications of this concept.

%[File: 117.png]
%CHAPTER X
\mychap{X}{THE HIGHER LINE GEOMETRY}
\mysect{Linear complex in hyperbolic space}

In Chapter IX we took some first steps in non-euclidean
line-geometry. The object of the present chapter is to
continue the subject in the special direction where the
fundamental element is not, in general, a line, but a pair
of lines invariantly connected.%
\footnote{Practically
the whole of this chapter is sketched, without proofs, by
Study\index{Study} in his article, `Zur nicht-euklidischen etc.,' loc.\ cit. The elliptic case
is developed at length in the author's\index{Author} dissertation, `The dual projective
geometry of elliptic and spherical space,' Greifswald, 1904. For the hyperbolic
case, see the dissertation of Beck\index{Beck}, `Die Strahlenketten im hyperbolischen
Raume,' Hannover, 1905.}
%endfootnote

Let us start in the real domain of hyperbolic space and
consider a linear complex\index{Complex of lines} whose equation is
\[
(\dot{a} \mid \dot{p}) = 0.
\]
The dots indicate that the coordinates of a point are
$\dot{x}_0$, $\dot{x}_1$, $\dot{x}_2$, $\dot{x}_3$,  and choosing such a unit of measure that
$k^2 = -1$, we have for the Absolute\index{Absolute}
\[
- \dot{x}_0{}^2 + \dot{x}_1{}^2 + \dot{x}_2{}^2 + \dot{x}_3{}^2 = 0.
\]
The polar of the given complex will have the coordinates
\[
\dot{a}_{0i} = r\dot{b}_{jk},\quad
\dot{a}_{jk} =  - r\dot{b}_{0i},\quad
i, j, k = 1, 2, 3,
\]
and the congruence, whose equations are
\[
(\dot{a} \mid \dot{p}) =
\smallsum \dot{a}_{0i} \dot{p}_{0i} -
\smallsum \dot{a}_{jk} \dot{p}_{jk} = 0,
\]
will be composed of all lines of our complex and its absolute
polar, or common to all complexes of the pencil
\[
(l\dot{a}_{01} - m\dot{a}_{23})\ldots
(l\dot{a}_{23} + m\dot{a}_{01}).
\]

These complexes shall be said to form a \emph{coaxal
pencil}\index{Coaxal pencil of complexes}\index{Pencil01@Pencil of complexes}, and
the two mutually absolute polar lines, which are the directrices
of the congruence, shall be called \emph{axes}\index{Axis05@--- of a pencil of complexes}
of the pencil. We get
their pl\"uckerian coordinates by giving to $l:m$ such values
that the complex shall be special. Let us now write
\[\tag{1}
\begin{aligned}
& \dot{a}_{01} + i\dot{a}_{23} = \rho X_1, \\
& \dot{a}_{02} + i\dot{a}_{31} = \rho X_2, \\
& \dot{a}_{03} + i\dot{a}_{12} = \rho X_3.
\end{aligned}
\]
%[File: 118.png]
A complex coaxal with the given line will be obtained by
multiplying the numbers $(X)$ by $(l + mi)$.

\mysect{The cross, its coordinates}
A pair of real lines which are mutually absolute polar,
neither of which is tangent to the Absolute\index{Absolute}, shall be called
a \emph{proper cross}\index{Cross}. They will determine a pencil of coaxal
complexes. If either of the lines have the pl\"uckerian coordinates
$(a)$, then the three numbers $(X)$ given by equations (1)
may be taken to represent the cross. These coordinates $(X)$
are homogeneous in the complex (i.e.\ imaginary) domain, for
the result of multiplying them through by $(l + mi)$ is to
replace the complex $(\dot{a})$ by a coaxal complex, and therefore
to leave the axes of the pencil unaltered.

Conversely, suppose that we have a triad of coordinates $(X)$
which are homogeneous in the imaginary domain. The coordinates
of the lines of the corresponding cross will be found
from (1) by assigning to $\rho$ such a value that the coordinates
$(\dot{a})$ shall satisfy the fundamental pl\"uckerian identity. For
this it is necessary and sufficient that the imaginary part of
$\rho^2(XX)$ should vanish, i.e.
\[\tag{2}
\begin{aligned}
  \sigma\dot{a}_{0\,1} &= \Bigl( \frac{X_i}{\sqrt{\mystrut{9}(XX)}} +
  \frac{\overline{X_i}}{\sqrt{\mystrut{9}(\bar{X}\bar{X})}} \Bigr),
\\
  \sigma\dot{a}_{j\,k} &= -i\Bigl(\frac{X_i}{\sqrt{\mystrut{9}(XX)}} -
  \frac{\overline{X_i}}{\sqrt{\mystrut{9}(\bar{X}\bar{X})}} \Bigr).
\end{aligned}
\]
To get the other line of the cross, i.e.\ the Absolute polar
of the line $(\dot{a})$, we merely have to reverse the sign of one
of our radicals.

There is one, and only one case, where our equations (2)
become illusory, namely where
\[
(XX) = 0.
\]

This will arise when
\[
(\dot{a} \mid \dot{a})
= \smallsum\dot{a}_{0i}{}^2 - \smallsum\dot{a}_{ik}{}^2 = 0,
\]
i.e.\ when the directrices of the congruence are tangent to the
Absolute. All complexes of the pencil will here be special,
and will be determined severally by lines intersecting the
various tangents to the Absolute at this point. Any mutually
polar lines of the pencil of tangents, will, conversely, serve to
determine the coaxal system. We may then represent such
a pencil of tangent lines by a set of homogeneous values $(X)$
where $(XX) = 0$, and, conversely, every such set of homogeneous
values will determine a pencil of tangents to the
Absolute. We shall therefore define such a pencil of tangents
as an \emph{improper cross}\index{Improper01@Improper cross}.

%[File: 119.png]
\smallskip\emph{Theorem}~1. There exists a perfect one to one correspondence
between the assemblage of all crosses in hyperbolic space, and
the assemblage of all points of the complex plane of elliptic
space. Improper crosses will correspond to points of the
elliptic Absolute\index{Absolute}.

We shall say that two crosses intersect if their lines intersect.
The N.~S. condition for this in the case of two proper
crosses will be
\begin{equation*}
  \frac{(XY)}{\sqrt{\mystrut{9}(XX)}\;\sqrt{\mystrut{9}(YY)}} =
  \frac{\pm(\bar{X}\bar{Y})}{\sqrt{\mystrut{9}(\bar{X}\bar{X})}\;
  \sqrt{\mystrut{9}(\bar{Y}\bar{Y})}}.
\end{equation*}
Geometrically a line may intersect either member of a cross\index{Cross}.
This ambiguity disappears in the case of perpendicular\index{Perpendicularity} intersection.

\smallskip\emph{Theorem}~2. Two intersecting crosses will correspond to
points, the cosine of whose distance is real, or pure imaginary;
crosses intersecting orthogonally will correspond to orthogonal
points\index{Orthogonal points} of the elliptic plane.

\mysect{The use of the cross manifold to interpret the geometry of the complex plane}
The assemblage of crosses which intersect a given cross
orthogonally will be given by means of a linear equation.
A linear equation will be transformed linearly into another
linear equation, if the variables and coefficients be treated
contragrediently. Geometrically we shall imagine that our
assemblage of crosses, \emph{cross space}\index{Cross space} let us say, is doubly
overlaid,
the crosses of one layer\index{Layer of cross space} being represented by points and
those of the other by lines in the complex plane\index{Plane}, we have then

\smallskip\emph{Theorem}~3. The necessary and sufficient condition that two
crosses of different layers should intersect orthogonally is that
the corresponding line and point of the complex plane should
be in united position.

If a cross be improper\index{Improper01@Improper cross}, the assemblage of all crosses cutting
it orthogonally will be made up of all lines through the point
of contact, and all lines in the plane of contact. This assemblage,
reducible in point space, is irreducible in cross space.

The collineation group of cross space, is the general group
depending on eight complex, or sixteen real parameters
\begin{equation*}
  \rho X_i{}' = \sum_j^{1..3} a_{ij}X_i,\quad |a_{ij}| \neq 0.\tag{3}
\end{equation*}

When will this indicate a transformation of point space?
It is certainly necessary that improper crosses should go into
improper crosses, hence the substitution must be of the orthogonal
%[File: 120.png]
type. Moreover, the Absolute\index{Absolute} of hyperbolic space will
be transformed into itself, so that our transformation of point
space must be a congruent one. Conversely, it is immediately
evident that a congruent transformation will transform cross
space linearly into itself. Also, an orthogonal substitution in
cross coordinates will carry an improper cross\index{Cross} into an improper
cross, and will carry intersecting crosses into other
intersecting crosses. The corresponding transformation in
point space is not completely determined, for a polar reciprocation
in the Absolute of point space appears as the identical
transformation of cross space. A transformation which
carries intersecting crosses into intersecting crosses may thus
be interpreted either as a collineation\index{Collineations}, or a correlation of
point space.

\smallskip\emph{Theorem}~4. Every collineation or correlation of hyperbolic
space which leaves the Absolute invariant will be equivalent
to an orthogonal substitution in cross space, and every such
orthogonal substitution may be interpreted either as a congruent
transformation of hyperbolic space, or a congruent
transformation coupled with a polar reciprocation in the
Absolute.

\mysect{Chain, and chain surface}
Let us now inquire as to what are the simplest figures of
cross space. The simplest one dimensional figure is the \emph{chain}\index{Chain02@--- of crosses}
composed of all crosses whose coordinates are linearly dependent,
by means of real coefficients, on those of two given
crosses,
\begin{equation*}
\rho X_i = aY_i + bZ_i, \quad i = 1,\: 2,\: 3.
\tag{4}
\end{equation*}

Interpreting these equations in the complex plane we see
that we have $\infty^1$ points of a line so related that the cross ratio
of any four is real. If this line be represented in the Gauss
plane, the chain will be represented by a circle. If the line
be imaginary, the real lines, one through each point of the
chain, will generate a linear pencil or a regulus.%
\footnote{The concept `chain of imaginary points' is due to Von Staudt. See his
`Beitr\"age', loc.\ cit., pp.~137--42. For an extension, see Segre\index{Segre}, `Su un nuovo
campo di ricerche geometriche,' \emph{Atti della R. Accademia delle Scienze di Torino},
vol.~xxv, 1890.}
%endfootnote

The crosses of the chain will cut orthogonally another cross
(of the other layer\index{Layer of cross space})
called the \emph{axis}\index{Axis04@--- of a chain} of the chain. The axis
being proper, the chain will contain two improper crosses,
namely, the pencils of tangents to the Absolute where it
meets the actual line of the chain.

There is a theorem of very great generality connected with
chains, which we shall now give. Suppose that we have a
%[File: 121.png]
congruence of lines of such a nature that the corresponding
cross coordinates $(U)$ are analytic functions of two real
parameters $u, v$. The cross of common perpendiculars to the
cross $(U)$ and the adjacent cross $(U+dU)$ will be given by
\begin{equation*}
X_i=\left|\begin{array}{cc}
U_j & U_k \smallskip\\
\dfrac{\partial U_j}{\partial u} & \dfrac{\partial U_k}{\partial u}
\end{array} \right| du +
\left|\begin{array}{cc}
U_j & U_k \smallskip\\
\dfrac{\partial U_j}{\partial v} & \dfrac{\partial U_k}{\partial v}
\end{array} \right| dv. \tag{5}
\end{equation*}
There are two sharply distinct sub-cases, (\emph{a})
\begin{equation*}
\left| U \frac{\partial U}{\partial u} \frac{\partial U}{\partial v} \right|
\equiv 0. \tag{6}
\end{equation*}

\mysect{Hamilton's theorem}
Here there is but one common perpendicular to $(U)$ and all
adjacent crosses. Such a congruence shall be called
\emph{synectic}\index{Congruence05@--- synectic}\index{Synectic congruence}.
Let us exclude this case for the moment and pass to the other,
where, (\emph{b})
\begin{equation*}
\left| U \frac{\partial U}{\partial u} \frac{\partial U}{\partial v} \right|
\not\equiv 0. \tag{7}
\end{equation*}
We shall mean by the \emph{general position} of a line in such a congruence,
one where this determinant does not vanish. We
have then the theorem:%
\footnote{The analogous theorem for euclidean space is due to Hamilton\index{Hamilton}, see
his paper on `Systems of Rays', \emph{Transactions of the Royal Irish Academy}, vol.~xv,
1829.}
%endfootnote

\smallskip\emph{Theorem}~5. The common perpendiculars to a line, in the
general position, of a non-synectic congruence, and each
adjacent line will generate a chain.

Let us find, in point coordinates, the equation of the surface
obtained by splitting off from a chain\index{Chain02@--- of crosses} its improper crosses.
We easily see that there will be two crosses of the chain
which intersect orthogonally; taking these and the axes to
determine the coordinate system, we may express our chain
in the simple form
\begin{equation*}
X_1=a(p+qi), \quad X_2=b(r+si), \quad X_3=0.
\end{equation*}

Eliminating $a/b$ we get
\begin{equation*}
(p \dot a_{31}+q \dot a_{02})(r \dot a_{10}-s \dot a_{23})=(p \dot a_{02}-q
\dot a_{31})(r \dot a_{23}+s \dot a_{01}).
\end{equation*}
This gives the equation of the chain surface in point coordinates
\begin{equation*}
(ps-qr)(-\dot x_0{}^2+\dot x_3{}^2) \dot x_1 \dot x_2 + (pr + qs) (\dot x_1{}^2 +
\dot x_2{}^2 ) \dot x_0 \dot x_3 = 0. \tag{8}
\end{equation*}
%[File: 122.png]
\begin{flalign*}
&\text{If} & (ps-qr) &= 0 \quad \mbox{or} \quad (pr+qs)=0, & \phantom{\text{If}}
\end{flalign*}
we have two real and two imaginary linear pencils; the conditions
for this in cross coordinates will be invariant under
the orthogonal, but not under the general group. The general
form of our surface is a ruled quartic, having a strong similarity
to the euclidean cylindroid.

\mysect{Chain congruence, synectic and non-synectic congruences}
The simplest two dimensional system of crosses is the \emph{chain
congruence}\index{Chain01@Chain congruence}\index{Congruence06@--- chain}.
This is made up of all crosses which have coordinates
linearly dependent with real coefficients on those of
three given crosses which do not cut a fourth orthogonally
\begin{equation*}
X_i=a Y_i+b Z_i+c T_0,
\end{equation*}
\begin{equation*}
| XYZ | \neq 0,  \quad i = 1,\: 2,\: 3. \tag{9}
\end{equation*}

\smallskip\emph{Theorem}~6. The crosses which correspond to the assemblage
of all points of the real domain of a plane will generate
a chain congruence.

\smallskip\emph{Theorem}~7. The common perpendiculars to pairs of crosses
of a chain congruence will generate a second chain congruence
in the other layer. Each congruence is the locus of the axes
of the $\infty^2$ chains of the other; the two are said to be reciprocal
to one another.

The reciprocal to the chain congruence (9) will have equations
\begin{equation*}
U_i=p \left| \begin{array}{cc}
Y_i & Y_k \\
Z_j & Z_k
\end{array} \right|+
q \left| \begin{array}{cc}
Z_j & Z_k \\
T_j & T_k
\end{array} \right|+
r \left| \begin{array}{cc}
T_j & T_k \\
Y_i & Y_k
\end{array} \right|. \tag{10}
\end{equation*}

Let the reader show that the chain congruence may be
reduced to the canonical form
\begin{equation*}
X_1=a(p+qi), \quad X_2=b(r+si), \quad X_3=c(t+ri),
\end{equation*}
where $a$, $b$, $c$ are real homogeneous variables.

There are various sub-cases under the congruent group. If
\begin{equation*}
(ps-qr) = 0,
\end{equation*}
the congruence will be transformed into itself by a one-parameter
group of rotations.

Again, let
\begin{equation*}
(ps-qr) = 0, \quad (pr-qt) = 0.
\end{equation*}
Here we see that
\begin{equation*}
\frac{(XX')}{\sqrt{(XX)}\;\sqrt{(X'X')}}
\end{equation*}
is real for any two crosses of the congruence, i.e.\ the congruence
consists in all crosses through the point $(1,\:0,\:0,\:0)$.

Leaving aside the special cases the following theorems may
be proved for the general case.

\smallskip\emph{Theorem}~8. The chain congruence, considered as an assemblage
%[File: 123.png]
of lines in point space, is of the third order and class.
It is generated by common perpendiculars to the pairs of lines
of a regulus. Those lines of the congruence which meet a line
of the reciprocal congruence, orthogonally generate a quartic
surface, those which meet such a line obliquely generate a
regulus whose conjugate belongs to the reciprocal congruence.
The two congruences have the same focal surface of order and
class eight.

Another simple two-parameter system of crosses is the
following
\begin{equation*}
\rho X_i=a Y_i+b Z_i+c T_i,
\end{equation*}
\begin{equation*}
p Y_i+q Z_i+s T_i \neq 0, \quad |YZT|=0,\; (a\:b\:c\:p\:q\:r) \text{\ real}.
\end{equation*}
All these crosses cut orthogonally the cross
\begin{equation*}
U_i=\left|\begin{array}{cc}
Y_j & Y_k \\
Z_j & Z_k
\end{array} \right|.
\end{equation*}

Conversely, let us show that every cross orthogonally intersecting
$(U)$ may be expressed in this form. As such a form
as this is invariant for all linear transformations, we may
suppose
\begin{equation*}
Y_3 = Z_3 = T_3 = 0.
\end{equation*}

We have then the equations
\begin{align*}
a Y_1+b Z_1+c T_1&=(r+ir')X_1, \\
a Y_2+b Z_2+c T_2&=(r+ir')X_2,
\end{align*}
which amount to four linear homogeneous equations in five
unknowns $a$, $b$, $c$, $r$, $r'$ and these may always be solved. There
will be found to be one singular case where the same cross
has $\infty^1$\label{inftyp} determinations.

The assemblage of crosses cutting a cross orthogonally is
but a special case of what we have already defined as a
synectic congruence\index{Congruence05@--- synectic}\index{Synectic congruence}. If
\begin{equation*}
X = X(uv), \quad \left| X \frac{\partial X}{\partial u} \frac{\partial X}{\partial v} \right| \equiv 0,
\end{equation*}
there will be but one common perpendicular to a cross and its
adjacent crosses. This corresponds to the fact that there will
exist an equation
\begin{equation*}
f(X_1,X_2,X_3)=0,
\end{equation*}
so that our congruence is represented by a curve, the tangent
at any point representing the common perpendicular just
mentioned (in the other layer), and, conversely, every curve
will be represented by a synectic congruence. The points and
tangents will be represented by two synectic congruences so
%[File: 124.png]
related that each cross of one is a cross of striction of a cross
of the other, and all its adjacent crosses. We may reach
a still clearer idea of these congruences by anticipating some
of the results of differential geometry to be proved in later
chapters. For, if we look upon the congruence of lines
generated by our crosses, we see that the two focal points
on each are orthogonal and the two focal planes mutually
perpendicular. From this we shall conclude that our line-congruence
is one of normals\index{Congruence10@--- of normals, to surfaces of Gaussian curvature zero},
and the characteristics of the
developable surfaces of the congruence will be geodesics of
the focal surface, to which the lines of the other congruence
are binormals. We shall, moreover, show in a later chapter
that if $r_1$ and $r_2$ be the radii of curvature of normal sections of
a surface in planes of curvature, then the Gaussian\index{Curvature02@--- Gaussian} expression
for the curvature of the surface at that point will be
\[
  \frac{1}{k\tan\dfrac{r_1}{k}} \cdot
  \frac{1}{k\tan\dfrac{r_2}{k}} +
  \frac{1}{k^2}.
\]
In the present instance as the two focal points are orthogonal
\[
  \frac{r_2}{k} = \frac{\pi}{2} + \frac{r_1}{k}, \quad
  \frac{1}{k\tan\dfrac{r_1}{k}} \cdot
  \frac{1}{k\tan\dfrac{r_2}{k}} +
  \frac{1}{k^2} = 0.
\]
Our congruence is made up of normals to surfaces of Gaussian
curvature zero, i.e.\ to surfaces whose distance element may
be written
\[
ds^2 = du^2 + dv^2.
\]

\smallskip\emph{Theorem}~9.%
\footnote{Cf.\ Study\index{Study}, `Zur nicht-euklidischen etc.,' cit., p.~328.}
%endfootnote
A synectic congruence will represent the points
of a curve of the complex plane. It will be made up of crosses
whose lines are normals to a series of surfaces of Gaussian
curvature zero\index{Curvature08@--- surfaces of zero}. The characteristics of the developable surfaces
are geodesics of the focal surfaces. Their orthogonal trajectories
are a second set of geodesics whose tangents will
generate a like congruence.

In conclusion, let us emphasize the distinction between
these congruences and the non-synectic ones, where the
common perpendiculars to a cross and its adjacent ones
generate a chain.

Did we wish to represent the imaginary as well as the
real members of a synectic or non-synectic congruence, we
should be obliged to introduce into our representing plane,
points with hypercomplex coordinates. We shall not enter
into this extension, for, after all, the real point of interest of
%[File: 125.png]
the subject lies merely in this, namely, to give a real interpretation
for the geometry of the complex plane.

\mysect{Dual coordinates of a cross in elliptic case}
As we identify the geometry of the cross\index{Cross} in hyperbolic
space with that of a point of the complex plane, so we may
relate a cross of elliptic (or spherical) space to a pair of real
points of two plane. The modus operandi is as follows:---

We start, as before, with a pencil\index{Coaxal pencil of complexes} of coaxal linear complexes
defined by

\begin{align*}
a_{01}+a_{23}=\rho\,_lX_1, \quad &a_{01}-a_{23}=\sigma\,_rX_1,\\
a_{02}+a_{31}=\rho\,_lX_2, \quad &a_{02}-a_{31}=\sigma\,_rX_2,\tag{11}\\
a_{03}+a_{12}=\rho\,_lX_3, \quad &a_{03}-a_{12}=\sigma\,_rX_3,
\end{align*}

If we replace our complex by another coaxal therewith, we
shall merely multiply $({}_lX) ({}_rX)$ by two different constants.
Conversely, when we wish to move back from the independently
homogeneous sets of coordinates $({}_lX) ({}_rX)$ to the
degenerate complexes of the pencil, i.e.\ to the lines of the
cross defined thereby, we have to take for $\rho$ and $\sigma$ such values
that the fundamental pl\"uckerian identity is satisfied,
\[\tag{12}
\begin{aligned}
\tau a_{0i}&={}_lX_i\sqrt{(_rX_rX)}+ {}_rX_i\sqrt{({}_lX_lX)},\\
\tau a_{jk}&={}_lX_i\sqrt{(_rX_rX)}- {}_rX_i\sqrt{({}_lX_lX)}.
\end{aligned}
\]

The two separately homogeneous coordinate triads $( _lX)( _rX)$
may be taken to represent this proper cross, and, conversely,
as all quantities involved so far are supposed to be real, every
real pair of triads will correspond to a single cross.

\smallskip\emph{Theorem}~10. The assemblage of all real crosses of elliptic
or spherical space may be put into one to one correspondence
with the assemblage of all pairs of points one in each of two
real planes.

Our doubly homogeneous coordinates have a second interpretation
which is of the highest interest. Let us write the
coordinates of a point of the Absolute\index{Absolute} in terms of two independent
parameters, i.e.\ of the parameters determining the
one and the other set of linear generators
\begin{equation*}
x_0:ix_1:x_2:ix_3\\
=(\lambda_1\mu_1-\lambda_2\mu_2):(\lambda_1\mu_1+\lambda_2\mu_2):(\lambda_1\mu_2+\lambda_2\mu_1):(\lambda_1\mu_2-\lambda_2\mu_1).
\end{equation*}

The pl\"uckerian coordinates of a generator of the left or right
system\index{Left and right generators of Absolute}\index{Right and left generators of Absolute}
will thus be
\begin{align*}
&p_{01}=p_{23}=2\lambda_1\lambda_2, &q_{01}&=-q_{23}=2\mu_1\mu_2,\\
&p_{02}=p_{31}=i(\lambda_1{}^2+\lambda_2{}^2), &q_{02}&=-q_{31}=i(\mu_1{}^2+\mu_2{}^2),\\
&p_{03}=p_{12}=(\lambda_1{}^2-\lambda_2{}^2), &q_{03}&=-q_{12}=-(\mu_1{}^2+\mu_2{}^2).
\end{align*}
%[File: 126.png]
The parameter $(\lambda)$ of a left generator which meets a given
line $(a)$ will satisfy
\begin{equation*}
2\lambda_1\lambda_2(a_{01}+a_{23})+i(\lambda_1{}^2+\lambda_2{}^2)(a_{02}+a_{31})+(\lambda_1{}^2-\lambda_2{}^2)(a_{03}+a_{12})=0.
\end{equation*}
Similarly, for a right generator we have
\begin{equation*}
2\mu_1\mu_2(a_{01}-a_{23})+i(\mu_1{}^2+\mu_2{}^2)(a_{02}-a_{31})-(\mu_1{}^2-\mu_2{}^2)(a_{03}-a_{12})=0.
\end{equation*}\index{Left and right generators of Absolute}\index{Right and left generators of Absolute}

\mysect{Condition for parataxy}
We thus get as a necessary and sufficient condition that
two lines should be right (left) paratactic\index{Parataxy}, that the differences
(sums) of complementary pairs of pl\"uckerian coordinates in
the one shall be proportional to the corresponding differences
(sums) in the other. If the lines be $(p)$ and $(p')$, the first of
these conditions will be
\begin{equation*}
[(p \mid p')+\Sigma p_{ij}p{ij}']^2-\Sigma p_{ij}{}^2\Sigma p_{ij}{}'^2=0,
\end{equation*}
while the second is
\begin{equation*}
[(p \mid p')-\Sigma p_{ij}p{ij}']^2-\Sigma p_{ij}{}^2\Sigma p_{ij}{}'^2=0.
\end{equation*}

If these equations be multiplied together, we get (25) of
Chapter IX.

If a line pass through the point $(1, 0, 0, 0)$ its last three
pl\"uckerian coordinates will vanish, while the first three are
proportional to those of its intersections with $x_0 = 0$. It thus
appears that in (11) the coordinates $( _lX)$ and $( _rX)$ are nothing
more nor less than the coordinates of the points, where the
plane $x_0 = 0$ is met respectively by the left and the right
paratactic through the point $(1, 0, 0, 0)$ to the two lines of the
cross, for a line paratactic to the one is also paratactic to
the other. It will, however, be more convenient to consider
$( _lX)$ and $( _rX)$ as standing for points in two different planes,
called, respectively, \emph{the left and right representing planes}.
We shall speak of two crosses as being paratactic, when their
lines are so, and the necessary and sufficient condition therefore,
invariant under the group of cross space, is that they
should be represented by identical points in the one or the
other plane.%
\footnote{The whole question of left and right is considered most carefully in
Study's\index{Study} `Beitr\"age', cit., pp.~126, 156.}
%endfootnote

As in the hyperbolic case, so here, we shall look upon cross
space as doubly overlaid, and assign a cross to the upper layer\index{Layer of cross space}
if it be determined by two points in the representing planes,
while it shall be assigned to the lower layer if it be determined
by two lines. Under these circumstances we may say:---

\smallskip\emph{Theorem}~11. In order that two crosses\index{Cross} of different layers
should intersect orthogonally, it is necessary and sufficient
%[File: 127.png]
that they should be represented by line elements in the two
planes.

\mysect{Clifford angles}
We may go still further in this same direction. We shall
mean by the right and left \emph{Clifford angles}\index{Angles03@--- Clifford}\index{Clifford}
of two crosses, the
angles of right and left paratactics to them through any chosen
point. Let the reader show that the magnitude of these angles
is independent of the choice of the last-named point. If, thus,
we choose the point $(1, 0, 0, 0)$, the cosines of the Clifford
angles will be
\begin{equation*}
\frac{( _lX_lY)}{\sqrt{( _lX_lX)}\; \sqrt{( _lY_lY)}}, \quad
\frac{( _rX_rY)}{\sqrt{( _rX_rX)}\; \sqrt{( _rY_rY)}}.
\end{equation*}

Now, from equations (19) and (20) of Chapter IX, we see that
\begin{align*}
&\sin\frac{d}{k} \sin\frac{d'}{k}=\sin\theta \sin\theta'=\frac{(p \mid p')}{\sqrt{\smallsum p_{ij}{}^2}\; \sqrt{\smallsum p_{ij}{}'^2}}, \\
&\cos\frac{d}{k} \cos\frac{d'}{k}=\cos\theta \cos\theta'=\frac{\smallsum p_{ij}p_{ij}{}'}{\sqrt{\smallsum p_{ij}{}^2}\; \sqrt{\smallsum p_{ij}{}'^2}};
\end{align*}
hence, we easily find
\begin{multline*}
\quad\text(1)\quad\cos\Bigl(\frac{d}{k}+\frac{d'}{k}\Bigr)=\frac{(_rX_rY)}{\sqrt{(_rX_rX)}\; \sqrt{(_rY_rY)}},
\\
\cos\Bigl(\frac{d}{k}-\frac{d'}{k}\Bigr)=\frac{(_lX_lY)}{\sqrt{(_lX_lX)}\; \sqrt{(_lY_lY)}}, \tag{13}
\end{multline*}
or else\label{rXrX}
\begin{multline*}
\quad\text(2)\quad\cos\Bigl(\frac{d}{k}-\frac{d'}{k}\Bigr)=\frac{(_rX_rY)}{\sqrt{(_rX_rX)}\; \sqrt{(_rY_rY)}},
\\
\cos\Bigl(\frac{d}{k}+\frac{d'}{k}\Bigr)=\frac{(_lX_lY)}{\sqrt{(_lX_lX)}\; \sqrt{(_lY_lY)}}. \phantom{(13)}
\end{multline*}

The ambiguity can be removed by establishing certain conventions
with regard to the signs of the radicals, into which
we shall not enter.%
\footnote{For an elaborate discussion, see Study\index{Study},
`Beitr\"age,' cit., especially p.~130.}
%endfootnote
We may, however, state the following
theorem:---

\smallskip\emph{Theorem}~12. The Clifford angles of two lines have the same
measures as the sums and differences of the $k$th parts of their
distances, or the sums and differences of their angles. The
necessary and sufficient condition that two lines should intersect
is that their Clifford angles should be equal or supplementary.

%[File: 128.png]
When we adjoin the imaginary domain to the real one,
serious complications will arise which can only be removed
by careful definition. Without going into a complete discussion,
we merely give the facts.%
\footnote{Cf.\ the author's\index{Author} `Dual projective Geometry', loc.\ cit., \S~3.}
%endfootnote

If $( _lX_lX) = 0$, $( _rX_rX) \not= 0$, we shall say that we have
a \emph{left improper cross}\index{Improper01@Improper cross}, and denote thereby a left generator
of the Absolute\index{Absolute}, conjoined to a non-parabolic involution
among the right generators. There will be $\infty^3$ such improper
crosses, and $\infty^3$ right improper crosses, whose definition is
obvious. Left and right improper crosses together will constitute
what shall be called \emph{improper crosses of the first sort}.
Improper crosses of the second sort shall be defined, as in
hyperbolic space, as pencils of tangents to the Absolute,
corresponding to sets of values for which $( _lX_lX) = ( _rX_rX) = 0$.
The definitions of parataxy and orthogonal intersection may
be extended to all cases, their analytic expression being as
in the real domain.

The general group of linear transformations of cross space
will depend upon sixteen essential parameters. It will be
made up of the sixteen-parameter sub-group $G_{16}$ of all
transformations of the type
%
\begin{equation*}
\rho \, _lX_i{}' = \sum_j a_{ij} \,_lX_j, \quad
\sigma \, _rX_i{}' = \sum_j b_{ij} \, _rX_j, \quad
|a_{ij}| \times |b_{ij}| \not=0,
\tag{14}
\end{equation*}
%
and the sixteen-parameter assemblage $H_{16}$ of all
transformations of the type
%
\begin{equation*}
\rho \, _lX_i{}' = \sum_j a_{ij} \, _rX_j, \quad
\sigma \, _rX_i{}' = \sum_j b_{ij} \, _lX_j, \quad
|a_{ij}| \times |b_{ij}| \not=0.
\tag{15}
\end{equation*}
%
Notice that under $G_{16}$ left and right parataxy of crosses of
the same layer are invariant, while under $H_{16}$ the two sorts
of parataxy are interchanged.

The group $G_{16}$ will contain, as a sub-group, the group of all
motions, while $H_{16}$ includes the assemblage of all symmetry
transformations\index{Symmetry transformations}. Let the reader show that there can be no
collineations\index{Collineations} of point space under $G_{16}$, except congruent transformations,
and that the necessary and sufficient condition
that (14) should represent a motion of point space is that
the transformations of the two representing planes should
be of the orthogonal type.

The group $G_{16}$ is half-simple, being composed entirely of
two invariant sub-groups $_lG_8$, $_rG_8$, of which the former is
made up of the general linear transformation for $(_lX)$ with
%[File: 129.png]
the identical transformation for $(_rX)$, while in the latter, the
r\^oles of $(_lX)$ and $(_rX)$ are interchanged. The highest common
factors of the group of motions with $_lG_8$ and $_rG_8$ respectively,
will be the groups of left and right translations\index{Translations} (cf.
Chapter IX).

The simplest assemblages of crosses in elliptic space bear
a close analogy to those of hyperbolic space, although possessing
more variety in the real domain. Let
%
\[
_lX_i = a\, _lY_i + b\, _lZ_i, \quad
_rX_i = a\, _rY_i + b\, _rZ_i,
\]
\[
|_lX\, _lY\, _lS| \times
|_rX\, _rY\, _rT| \not\equiv 0.
\]

\mysect{Chain and strip}
The assemblage of crosses so defined shall be called a \emph{chain}\index{Chain02@--- of crosses}.
The properties of these chains are entirely analogous to those
in the hyperbolic case. For instance, take a congruence of
crosses whose coordinates are analytic functions of two essential
parameters $(u)$, $(v)$. Let us further assume that $( _lY)$ $( _rY)$
being crosses of the system
%
\begin{equation*}
\left|
_lY \frac{\partial}{\partial u}\, _lY\frac{\partial}{\partial v}\, _lY
\right|
\times
\left|
_rY \frac{\partial}{\partial u}\, _rY\frac{\partial}{\partial v}\, _rY
\right|
\not\equiv 0.
\end{equation*}
%
The meaning of this restriction is that neither $(_lY)$ nor $(_rY)$
can be expressed as functions of a single parameter, so that
the crosses of the congruence cannot be assembled into the
generators of $\infty^1$ surfaces, those of each surface being paratactic.
Let the reader then show that for every such
congruence, the common perpendiculars to a line in the
general position, and its immediate neighbours, will generate
a chain.

The chains of elliptic cross space will have the same subclassifications
under the congruent group, as in the hyperbolic
plane. Let the reader show that the general chain may be
represented by means of a homographic relation between the
points of two linear ranges in the representing planes, and
that the special chain, composed of two pencils, arises, when
the relation is a congruent one.

Suppose, next, that we have
%
\[
\rho\, {}_lX_i = a\, {}_lY_i + b\, {}_lZ_i, \quad
\sigma\, {}_rX_i = a\, {}_rY_i + b\, {}_rZ_i,
\]
\[
{}_lY_i = r\, {}_lZ_i, \quad
|{}_rY {}_rZ {}_rT| \not\equiv 0.
\]
%
This is a new one-parameter family of crosses called a \emph{strip}\index{Strip},
or, more exactly, a \emph{left strip}. The common perpendiculars
to pairs of crosses of the left strip will generate a right strip
(whereof the definition is obvious), and each strip shall be said
to be reciprocal to the other. A left strip of the upper layer
%[File: 130.png]
will be represented by a point of the left plane, and a linear
range of the right plane. The reciprocal strip in the lower
layer will be represented by the pencil through the point
in the left plane, and the line of the range in the right.

In point space, the lines of a strip\index{Strip} are generators of a
quadric, whose other generators belong to the reciprocal strip.
Owing to the parataxy of the generators of such a quadric,
it will intersect the Absolute\index{Absolute} in two generators of each set.
We shall call our quadric a \emph{Clifford surface}\index{Clifford}, when we wish
to refer to it as a figure of point space. We shall show in
Chapter XV, that these surfaces have Gaussian curvature zero,
since they are generated by paratactic\index{Parataxy} lines, and are minimal
surfaces\index{Minimal surfaces}, since their asymptotic lines form an orthogonal
system.%
\footnote{Cf.\ Klein\index{Klein}, `Zur nicht-euklidischen Geometrie,' \emph{Mathematische Annalen},
vol.~xxxvii, 1890.}
%endfootnote

\mysect{Chain congruence}
The simplest two dimensional system of crosses will be, as
before, the chain congruence\index{Chain01@Chain congruence}\index{Congruence06@--- chain}
\[
_lX_i=a\, _lY_i+b\, _lZ_i+c\, _lT_i, \quad _rX_i=a\, _lX_i+b\, _lY_i+c\,  _lZ_i
\]
\[
| {}_lY {}_lZ {}_lT| \times | {}_rY {}_rZ{} _rT| \neq 0.
\]
We may solve the first three equations for $a,b,c$, and substitute
in the last
\begin{equation*}
_rX_i=\sum_j a_{ij}\, _lX_j, \quad |a_{ij}| \neq 0.
\end{equation*}
This, again, may easily be reduced to the canonical form
\begin{equation*}
_rX_i=a_i\, _lX_i \tag{16}
\end{equation*}
The reciprocal congruence will be given by
\begin{equation*}
_lU_i = a_i\, _rU_i. \nonumber
\end{equation*}

There are various sub-classes under the congruent group.
If the squares of no two of our quantities $a_i$ in (16) be equal,
we have the general congruence, if we have one such equality,
the congruence will be transformed into itself by a one-parameter
group of rotations. If all three squares be equal,
we have a bundle of crosses through a point. The general
congruence will have all of the properties mentioned in (8).

A different sort of congruence will arise in the case where
\begin{equation*}
|{}_lY {}_lZ {}_lT|=0, \quad |{}_rY {}_rZ {}_rT| \neq 0. \tag{17}
\end{equation*}
This congruence will contain $\infty^1$ strips, whose reciprocals
generate the reciprocal congruence. The common perpendiculars
to all non-paratactic crosses of the congruence will
generate a bundle, those to paratactic crosses, the reciprocal
%[File: 131.png]
congruence. Such a congruence will be generated by the
common perpendiculars to the paratactic lines of two pencils
which have different centres and planes, but a common line
and paratactic axes. In point space the line congruence will
be of order and class two. The canonical form will be%
\footnote{Apparently
nothing has ever been published concerning this type of
congruence. The theorems here given are taken from an unpublished section
of the author's\index{Author} dissertation, cit.}
%endfootnote
\begin{align*}
_lX_1&=a_1\, _rX_1, \\
_lX_2&=a_2\, _rX_2, \\
_lX_3&=0.
\end{align*}

If, in addition to (17), we require the first minors of $|_lY\, _lZ\, _lT|$
all to vanish, we shall have a bundle of paratactic crosses.
If, on the other hand, we have
\begin{equation*}
|_lY{}_lZ{}_lT|=|_rY{}_rZ{}_rT|=0,
\end{equation*}
without the vanishing of the first minors of either determinant,
we have $\infty^2$ crosses cutting a given cross orthogonally. The
equations of the congruence may be reduced to the canonical
form
\begin{align*}
\rho\, _lX_1=a, & \quad\sigma\, _rX_1=b,\\
\rho\, _lX_2=b, & \quad\sigma\, _rX_2=c,\tag{18} \\
\rho\, _lX_3=0, & \quad\sigma\, _rX_3=0.
\end{align*}
The cross $(1, 0, 0)$ $(0, 1, 0)$ will be singular, having $\infty^1$ determinations.

In general, if we have
\begin{equation*}
F( _lX_1\, _lX_2\, _lX_3)=0,
\quad\phi( _rX_1\, _rX _2\, _rX_3)=0,
\end{equation*}
the line-congruence can be assembled into $\infty^1$ surfaces with
left, and $\infty^1$ surfaces with right paratactic generators. Such
surfaces will have Gaussian curvature\index{Curvature02@--- Gaussian} zero. We shall show
also in Chapter XVI that the lines of such a congruence are
normals to a series of surfaces of Gaussian curvature zero.

%[File: 132.png]



%CHAPTER XI
\mychap{XI}{THE CIRCLE AND THE SPHERE}
\mysect{Simplest form for the equation of a circle}
\index{Circle01@Circle|(}

The simplest curvilinear figures in non-euclidean geometry
are circles, and it is now time to study their properties.%
\footnote{For a very simple treatment of this subject by means of pure Geometry,
see Riccordi\index{Riccordi}, `I cercoli nella geometria non-euclidea,' \emph{Giornale di Matematica},
xviii, 1880. Riccordi's results had previously been reached analytically by
Battaglini\index{Battaglini}, `Sul rapporto anarmonico sezionale e tangenziale delle coniche,'
ibid., xii, 1874.}
%endfootnote

\smallskip\emph{Definition}.~The locus of all points of a plane at a constant
distance from a given point which is not on the Absolute
is called a \emph{circle}. The given point shall be called the \emph{centre}
of the circle, its absolute polar, which will also turn out to
be its polar with regard to the circle, shall be called the \emph{axis}\index{Axis01@Axis of a circle}
of the circle. A line through the centre of the circle shall be
called a \emph{diameter}. Let the reader show that all points of
a circle are at constant distances from the axis, a distance
whose measure becomes infinite in the limiting euclidean case.

\mysect{Dual nature of the curve}
To get the equation of the circle whose centre is $(a)$ and
whose radius is $r$, i.e.\ this shall be the measure of the distance
of all points from the centre, we have
\[
  \frac{(ax)}{\sqrt{(aa)}\;\sqrt{(xx)} } = \cos\frac{r}{k},
\]
\[
  \cos^2\frac{r}{k}(aa)(xx)-(ax)^2 = 0.     \tag{1}
\]
It is evident that when $\cos^2 \dfrac{r}{k} \neq 0$, this curve has double
contact with the Absolute, the secant of contact being the
axis, and, conversely, every such curve of the second order
will be a circle. The absolute polar of a circle will, hence,
be another circle, so that the circle is self-dual:---

\medskip\twincol{\emph{Theorem} 1. \emph{Definition}. The
locus of all points of a plane
at a constant distance from
a given point thereof is a
circle whose centre is the given
point.}
{\emph{Theorem}~1$'$. The envelope
of all lines of a plane which
make a constant angle with
a given line is a circle having
the given line as axis.}\par\medskip
%end twincol

Note that a circle of radius $\dfrac{\pi k}{2}$ is a line, and that circle of
radius $0$ is two lines.

%[File: 133.png]
Restricting ourselves, for the moment, to the real domain
of the hyperbolic plane, we see that if the centre be ideal,
the axis will be actual, and the curve will appear in the actual
domain as the locus of points at a constant distance from the
axis, an actual line. In this case the circle is sometimes
called an \emph{equidistant curve}\index{Equidistant01@Equidistant curves}. If the centre be actual we shall
have what may be more properly called a \emph{proper} circle.
Notice that to a dweller in a small region of the hyperbolic
space, a proper circle would appear much as does a euclidean
circle to a euclidean dweller, while an equidistant curve would
appear like two parallel lines. These distinctions will,
naturally, disappear in the elliptic case; in the spherical, the
circle will have two centres, which are equivalent points.

If the point $(a)$ tend to approach the Absolute\index{Absolute} (analytically
speaking) the equation (1) will tend to approach an indeterminate
form. The limiting form for the curve will be
a conic having four-point contact with the Absolute. Such a
curve shall be called a \emph{horocycle}\index{Horocycle}, the point of contact being
called the centre, and the common tangent the axis. If $(u)$ be
the coordinates of the axis, we have
\[
(uu) = 0,
\]
and the equation of the horocycle takes the form
\[
(u_1{}^2 + u_2{}^2) (xx) + C (ux)^2 = 0.
\]

\medskip\twincol{\emph{Theorem}~2. A tangent to
a circle is perpendicular\index{Perpendicularity} to the
diameter through the point of
contact.}
{\emph{Theorem}~2$'$. A point on a
circle is orthogonal\index{Orthogonal points} to the
point where the tangent thereat
meets the axis.}\par\medskip
%end twincol

These simple theorems may be proved in a variety of ways.
For instance every circle will be transformed into itself by
a reflection in any diameter, hence the tangent where the
diameter meets the curve must be perpendicular to the diameter.
Or, again, if $\overline{AB} \equiv \overline{AC}$, a line from $A$ to one centre of gravity
of $B$, $C$ will be perpendicular to $BC$; then let $B$ and $C$ close
up on this centre of gravity. Or, lastly, the equation of the
tangent to the circle (1) at a point $(y)$ will be
\[
(xy) (aa)-N(ax) (ay) = 0.
\]

The diameter through $(y)$ will have the equation
\[
  \lvert xya \rvert = 0.
\]

If we indicate these two lines by $(u)$ and $(v)$, then
\[
(uv) = (aa)\, \lvert yay \rvert - N(ay)\, \lvert aya \rvert.
\]

Let the reader show that these theorems hold also in the
case of the horocycle.

%[File: 134.png]
\medskip\twincol{\emph{Theorem}~3. The locus of
the centres of gravity\index{Centre03@--- of gravity of points} of pairs
of points of a circle whose
lines are concurrent on the
axis, is the point of concurrence,
and the diameter perpendicular
to these lines.}
{\emph{Theorem}~3$'$. The envelope
of the bisectors\index{Bisector of an angle} of the angles
of tangents to a circle from
points of a diameter, is this
diameter, and its absolute
pole.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~4. If two tangents
to a circle (horocycle) make
a constant angle, the locus of
their point of intersection is
a concentric circle (horocycle).}
{\emph{Theorem}~4$'$. If two points of
a circle (horocycle) are at a
constant distance, the envelope
of their line is a coaxal circle
(horocycle).}\par\medskip
%end twincol

The element of arc of a circle of radius $(r)$ will be, by
Chapter IV (5),
\[
ds = k\sin\frac{r}{k}d\theta.
\]

The circumference of the circle is thus
\[
k\sin\frac{r}{k} \int_0^{2\pi} d\theta = 2\pi k\sin\frac{r}{k}.
\]

\mysect{Curvature of a circle}
\index{Curvature01@Curvature of a curve}Let the tangents at $P$ and $P'$ meet at $Q$, the centre of the
circle being $A$. Let $\Delta \phi$ be the angle between the tangents,
and let $P''$ be the point on the tangent at $P$ whose distance
from $P$ equals $PP'$, or, in the infinitesimal, equals $ds$. The
$\Delta PAP'$ and $\Delta P'PP''$ are isosceles, hence
\begin{flalign*}
&&\Delta\phi &= 2\measuredangle P'PP'',
\\
&&\tan \frac{\overline{P'P''}}{2k} &=
  \sin\frac{\overline{PP'}}{k} \tan\frac{\Delta\phi}{4}
\\
&&\limit\frac{\Delta\phi}{ds} &=
  \limit\frac{4\tan\dfrac{\overline{P'P''}}{2k}}
  {\overline{PP'}\sin\dfrac{\overline{PP'}}{k}} =
  \limit\frac{\overline{2P'P''}}{\overline{PP'^2}}.
\\
&\text{But} \hfill &
\tan\frac{\overline{PQ}}{k} &=
  \sin\frac{r}{k}\tan\tfrac{1}{2}d\theta =
  \frac{ds}{2k}, & \hfill &\phantom{\text{But}}
\\
&&\tan\frac{\overline{PQ}}{k} &=
  \tan\frac{\overline{AQ}}{k} \cos\tfrac{1}{2}(\pi-\Delta\phi)
 \text{\ by IV (6),}
\\
&&\limit \tan \frac{\overline{PQ}}{k} &=
  \tfrac{1}{2}\tan\frac{r}{k}\Delta\phi.\\
%[File: 135.png]
&\text{Hence}
&  \limit\frac{\Delta\phi}{ds}
&= \limit\frac{\overline{2P'P''}}{\overline{PP'^2}}
= \frac{1}{k\tan\dfrac{r}{k}}.  \tag{2}
\end{flalign*}

We shall subsequently define this expression as the curvature of the
circle at the point $(P)$. We see that, as we should
expect, it is constant.

We shall next take up simple systems of circles. We leave
to the reader the task of making the slight modifications in
what follows necessary to adapt it to the case of spherical
geometry. In the general case two circles, neither of which
is a line, will intersect in four points, real, or imaginary, in
pairs. If two circles lie completely without one another they
will have four real common tangents, the absolute polars of
such circles will intersect in four real points. The difficulty
of visualization disappears in the hyperbolic case where we
take one at least of the circles as an equidistant curve. If we
identify the euclidean hemisphere, where opposite points of
the equator are considered identical, with the elliptic plane,
we see how two circles there also can intersect in four real
points. In the spherical case, by Chapter VIII, the Absolute\index{Absolute} is
the locus of all points which are identical with their equivalents.
A point will have one absolute polar, a line two
equivalent absolute poles. The absolute polar of a circle is
two equivalent circles, which are also the absolute polars of
the equivalent circle. Two real circles cannot intersect in
more than two real points.

\mysect{Radical axes, and centres of similitude}
Two circles which intersect in four points will have three
pairs of common secants. The problem of finding the common
secants of two conics will, in general, lead to an irreducible
equation of the third degree. When, however, the two conics
have double contact with a third, the equation is reducible,
and one pair of secants appears which intersect on the chords
of contact, and are harmonically separated by them.%
\footnote{This theorem is, of course, well known. Cf.\ Salmon\index{Salmon}, \emph{Conic Sections},
sixth edition. London, 1879, p.~242.}
%endfootnote
In the case of two circles these secants shall be called the \emph{radical
axes}\index{Axis02@--- radical of two circles}.\label{theywill}

\medskip\twincol{\emph{Theorem}~5. If two circles
intersect in four points, two
common secants called radical
axes are concurrent\index{Concurrence} with the
axes\index{Axis01@Axis of a circle} of the circles and
harmonically
%[File: 136.png]
separated by them,
They are perpendicular to one
another and to the line of centres. The centres of gravity
of the intersections of the
circles with a radical axis are
the intersections with the
other radical axis and with
the line of centres.}
{\emph{Theorem}~5$'$. If two circles
have four common tangents,
two intersections of these,
called centres of similitude\index{Centre05@--- of similitude}\index{Similitude, centres of}, lie
on the line\index{Collinearity} of centres, are
%[File: 136.png bis]
harmonically separated by the
centres and are mutually
orthogonal\index{Orthogonal points}. The bisectors\index{Bisector of an angle} of
angles of the tangents at a
centre of similitude are the
line of centres and the line to
the other centre of similitude\index{Centre05@--- of similitude}\index{Similitude, centres of}.}\par\medskip
%end twincol

If the equations of the two circles be
\[
  \cos^2 \frac{r_1}{k}(aa)(xx) - (ax)^2 = 0,\quad
  \cos^2 \frac{r_2}{k}(bb)(xx) - (bx)^2 = 0,
\]
the equations of the radical axes will be
\begin{multline*}
  \left( \cos\frac{r_2}{k}\sqrt{(bb)}(ax)
       + \cos\frac{r_1}{k}\sqrt{(aa)}(bx) \right)
\\
  \left( \cos\frac{r_2}{k}\sqrt{(bb)}(ax)
       - \cos\frac{r_1}{k}\sqrt{(aa)}(bx) \right) = 0. \tag{3}
\end{multline*}
The last factor equated to zero will give
\[
  \frac{\dfrac{(ax)}{\sqrt{(aa)}\;\sqrt{(xx)}} }
       {\cos \dfrac{r_1}{k}                  }
\\
= \frac{\dfrac{(bx)}{\sqrt{(bb)}\;\sqrt{(xx)}} }
       {\cos \dfrac{r_2}{k}                  },
\]
and the two sides of this equation will, by Ch.~IV~(4), be the
cosines of the $k$th parts of the distances from $(x)$ to the points
of contact of tangents, thence to the two circles.

\medskip

\mysect{Circles through two points, or tangent to two lines}
\twincol{\emph{Theorem}~6. If a set of circles
through two points have the
line of these points as a radical
axis\index{Axis02@--- radical of two circles}, the points of contact of
tangents to all of them from
a point of the line lie on a
circle whose centre\index{Centre01@Centre of a circle} is this
point.}
{\emph{Theorem}~6$'$. If a set of circles
tangent to two given lines
have the intersection of the
lines as a centre of similitude,
the envelope of tangents to
them at the points where they
meet a line through this
centre of similitude will be
a circle with this line as axis\index{Axis01@Axis of a circle}.}\par\medskip
%end twincol

Consider the assemblage of all circles through two given
points. If the line connecting the two points be a radical
axis for two of these circles it will be perpendicular\index{Perpendicularity} to their
line of centres at one centre of gravity\index{Centre03@--- of gravity of points} of the two points, and
in every case a perpendicular from the centre of a circle on
%[File: 137.png]
a secant will meet it at a centre of gravity\index{Centre03@--- of gravity of points} of the two points
of the circle on that line. We thus see---

\medskip\twincol{\emph{Theorem}~7. The assemblage
of all circles through two
common points will fall into
two families according as the
perpendicular\index{Perpendicularity} from the centre
on the line of these points
passes through the one or the
other of their centres of
gravity. Two circles of the
same family, and they only, will
have the line as a radical axis\index{Axis02@--- radical of two circles}.}
{\emph{Theorem}~7$'$. The assemblage
of all circles tangent to two
lines will fall into two families
according as the centres\index{Centre01@Centre of a circle} lie on
the one or the other bisector\index{Bisector of an angle}
of the angles of the lines.
Two circles of the same
family, and they only, will
have the intersection of the
lines as a centre of similitude\index{Centre05@--- of similitude}\index{Similitude, centres of}.}\par\medskip
%end twincol

Let us now take a third point, and consider the circles that
pass through all three.

\medskip\twincol{\emph{Theorem}~8. Four circles will
pass through three given
points. Each line connecting
two of the given points will
be a radical axis for two pairs
of circles.}
{\emph{Theorem}~8$'$. Four circles will
touch three given lines. Each
intersection of two lines will
be a centre of similitude for
two pairs of circles.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~9. The radical axes
of three circles pass by threes
through four points.}
{\emph{Theorem}~9$'$. The centres of
similitude of three circles lie
by threes on four lines.}\par\medskip
%end twincol

Of course when two circles touch one another, their common
tangent replaces one radical axis, and the point of contact one
centre of similitude. Two circles will have double contact
when, and only when, they are concentric. We get at once
from (6) and (9)

\medskip\twincol{\emph{Theorem}~10. Four circles
may be constructed to cut
each of three circles at right
angles twice.}
{\emph{Theorem}~10$'$. Four circles
may be constructed so that
the points of contact of tangents
common to them and to
each of three given circles
form two pairs of orthogonal\index{Orthogonal points}
points.}\par\medskip
%end twincol

It is here assumed that no two of the given circles are concentric.
There is no reason to expect that because two circles
intersect at right angles in two points they will in the other
two. Let the circles be
\[
  \cos^2 \frac{r_1}{k} (aa) (xx)-(ax)^2 = 0, \quad
  \cos^2 \frac{r_2}{k} (bb) (xx)-(bx)^2 = 0.
\]

%[File: 138.png]
Let $(y)$ be a point of intersection; the lines thence to the
centres are
\[
  \lvert xya \rvert = 0, \quad \lvert xyb \rvert = 0.
\]

The cosine of the angle formed by them will be
\[
\begin{split}
\cos\theta &=
  \frac{\begin{vmatrix} (yy) & (ay) \\ (by) & (ab) \end{vmatrix} }
       {\sqrt{(yy)(aa)-(ay)^2}\; \sqrt{(yy)(bb)-(by)^2}}
\\
&=\frac{(ab) - \cos\dfrac{r_1}{k} \cos\dfrac{r_2}{k}
                \sqrt{(aa)}\; \sqrt{(bb)}            }
       {\sin\dfrac{r_1}{k} \sin\dfrac{r_2}{k}
         \sqrt{(aa)}\; \sqrt{(bb)}                   }.
\end{split}
\tag{4}
\]

This gives two values for the angle which will be equal when,
and only when
\[
(ab) = 0.
\]

The condition of contact will be
\begin{equation*}
\tag{5}
  \cos\theta = \pm 1, \quad
  \cos \left( \frac{r_1}{k} \pm \frac{r_2}{k} \right) =
  \frac{(ab)}{\sqrt{(aa)}\; \sqrt{(bb)}};
\end{equation*}
and of orthogonal intersection
\begin{equation*}
\tag{6}
\cos\frac{r_1}{k} \cos\frac{r_2}{k} = \frac{(ab)}{\sqrt{(aa)}\;\sqrt{(bb)}},
\end{equation*}
these last two facts being, also, geometrically evident. We
see that two circles cannot have four rectangular intersections,
for if
\begin{equation*}
\tag{7}
(ab) = 0, \quad \cos \frac{r_2}{k} = 0,
\end{equation*}
the circle is a line.

\medskip\twincol{\emph{Theorem}~11. The necessary
and sufficient condition that
two circles should cut at the
same angle at all points is
that their centres\index{Centre01@Centre of a circle} should be
mutually orthogonal\index{Orthogonal points}.}
{\emph{Theorem}~11$'$. The necessary
and sufficient condition that
two circles should determine
by their points of contact,
congruent distances on all four
common tangents, is that their
axes should be mutually perpendicular\index{Perpendicularity}.}\par\medskip
%end twincol

Notice that these two conditions are really identical.
\index{Circle01@Circle|)}

%[File: 139.png]
\mysect{Spheres}\index{Sphere|(}
We shall define as a sphere that surface which is the locus
of all points of space at congruent distances from a point not
on the Absolute.

\medskip\twincol{\emph{Theorem}~12. A sphere is
the locus of all points at a
constant distance from a given
point not on the Absolute\index{Absolute}.
It is, when not a plane, a
quadric with conical contact
with the Absolute.}
{\emph{Theorem}~12$'$. A sphere is
the envelope of planes meeting
at a constant angle a plane
which is not tangent to the
Absolute. It is, when not a
point, a quadric with conical
contact with the Absolute.}\par\medskip
%end twincol

Note that a plane and point are special cases of the sphere.

The fixed point shall be called the centre, the plane of conical
contact the \emph{axial plane}\index{Axial plane of sphere} of the sphere. A line connecting any
point with the centre of a sphere is perpendicular\index{Perpendicularity} to the polar
plane of the point, a tangent plane is perpendicular to the line
from the point of contact to the centre, to the diameter through
the point of contact let us say.

\medskip\twincol{\emph{Theorem}~13. Two spheres
will intersect in two circles
whose planes are perpendicular to the line of centres
and to one another, and are
harmonically separated by the
axial planes.}
{\emph{Theorem}~13$'$. The common
tangent planes to two spheres
envelop two cones of revolution whose vertices are
mutually orthogonal\index{Orthogonal points} and
harmonically separated by
the centres.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~14. Three spheres
not containing a common
circle will meet in three pairs
of circles whose planes are
collinear by threes in four
lines.}
{\emph{Theorem}~14$'$. Three spheres
not tangent to a cone of revolution have three such
pairs of common tangent cones
whose vertices are collinear in
threes on four lines.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~15. Four spheres
whose centres are not coplanar\index{Coplanarity} intersect in twelve
circles whose planes pass by
sixes through eight points
which, with the centres of the
spheres, form a desmic configuration.}
{\emph{Theorem}~15$'$. Four spheres
whose axial planes are not
concurrent are enveloped in
pairs by twelve cones of revolution whose vertices lie
by sixes in eight planes which,
with the axial planes, determine a desmic configuration\index{Desmic configuration}.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~16. The necessary
and sufficient condition that
two spheres should cut at the
same angle along their two
%[File: 140.png]
circles is that their centres
should be mutually orthogonal\index{Orthogonal points}.}
{\emph{Theorem}~16$'$. The necessary
and sufficient condition that
two spheres should, by their
contact, determine congruent
%[File: 140.png bis]
distances on the generators of
the two circumscribed cones,
is that their axial planes
should be mutually perpendicular\index{Perpendicularity}.}\par\medskip
%end twincol

\mysect{Poincar\'e's sphere to sphere transformation from euclidean to non-euclid\-ean space}
We shall terminate this chapter by giving an unusually
elegant transformation from euclidean to non-euclidean space.%
\footnote{This transformation seems to have been first given in the second edition
of \emph{Wissenschaft und Hypothese}, by Poincar\'e\index{Poincar\'e}, translated by F. and L. Lindemann\index{Lindemann},
Leipzig, 1906, p.~258. This is fruitfully used in the dissertation of M\"unich\index{Munich@M\"unich},
`Nicht-euklidische Cykliden,' Munich, 1906. We have adapted the notation
to conform to our own usage.}
%endfootnote
Let us assume that we have a euclidean space where a point
has the homogeneous coordinates $x$, $y$, $z$, $t$ and a hyperbolic
space for which $k^2=-1$, a point being given by our usual $(\dot{x})$
coordinates. Let us then write
\begin{equation*}
  \rho x = \dot{x}_1,\quad \rho y=\dot{x}_2,\quad
  \rho z = \sqrt{\dot{x}_0{}^2 - \dot{x}_1{}^2
                - \dot{x}_2{}^2 - \dot{x}_3{}^2 },\quad
  \rho t = \dot{x}_0-\dot{x}_3.
\tag{8}
\end{equation*}

To each point of hyperbolic space will correspond two
points of euclidean space. Let us choose that for which the
real part of
$\sqrt{\dot{x}_0{}^2 - \dot{x}_1{}^2 - \dot{x}_2{}^2 - \dot{x}_3{}^2}$
is greater than zero. When
the real part vanishes, we may, by adjoining to our domain of
rationality a square root of minus one, distinguish between the
imaginary roots, and so choose one in particular. We may
thus say that to every point of hyperbolic space, not on the
Absolute, will correspond a point of euclidean space above
the plane $z = 0$, and to each points of the Absolute will
correspond points of this plane. The transformation is real,
so that real and actual points will correspond to real ones.
Conversely, we get from (8)
\begin{multline*}
  \sigma \dot{x}_0 = x^2 + y^2 + z^2 + t^2,\quad
  \sigma \dot{x}_1 = 2xt,\quad \sigma \dot{x}_2 = 2yt,
\\
  \sigma \dot{x}_3 = x^2 + y^2 + z^2 - t^2,
\tag{9}
\end{multline*}
and to each point of euclidean space, above, or on the $z$ plane,
will correspond a point of hyperbolic space, not on, or on the
Absolute.

Suppose that we have a euclidean sphere of centre $(a, b, c, d)$
and radius $r$. If we write for short
\[
(a^2 + b^2 + c^2 - d^2r^2) = p^2,
\]
the equation of this sphere may be written
\begin{align*}
  &(dx-at)^2 + (dy-bt)^2 + (dz-ct)^2 = d^2 r^2 t^2,
\\
  &d^2(x^2+y^2+z^2) - 2dt(ax+by+cz) + p^2 t^2 = 0.
\tag{10}
\end{align*}

%[File: 141.png]
Transforming we get, after splitting off a factor
$\dot{x}_3 - \dot{x}_0$ which
corresponds to the euclidean plane at infinity,
\[
d^2 (\dot{x}_0 + \dot{x}_3) - 2d (a\dot{x}_1 + b\dot{x}_2
  + c\sqrt{\dot{x}_0{}^2 - \dot{x}_1{}^2 - \dot{x}_2{}^2 - \dot{x}_3{}^2})
  + p^2(\dot{x}_0 - \dot{x}_3) = 0,
\]
\begin{multline*}
\tag{11}
[ (d^2+p^2)\dot{x}_0 - 2ad\dot{x}_1 - 2bd\dot{x}_2
+ (d^2-p^2)\dot{x}_3 ]^2   \\
= 4c^2d^2 (\dot{x}_0{}^2 - \dot{x}_1{}^2 - \dot{x}_2{}^2 - \dot{x}_3{}^2).
\end{multline*}

This is a sphere of hyperbolic space whose centre is
\[
(d^2+p^2,\; 2ad,\; 2bd,\; p^2-d^2),
\]
and whose radius $r_1$ is given by
\[
\cosh r_1 = \frac{c}{\sqrt{p^2 - a^2 - b^2}}.
\]

Conversely, if we have the hyperbolic sphere
\begin{multline*}
\tag{12}
( \dot{a}_0\dot{x}_0 - \dot{a}_1\dot{x}_1
- \dot{a}_2\dot{x}_2 - \dot{a}_3 \dot{x}_3 )^2   \\
= \cosh^2 r_1(\dot{a}_0{}^2 - \dot{a}_1{}^2 - \dot{a}_2{}^2 - \dot{a}_3{}^2)
             (\dot{x}_0{}^2 - \dot{x}_1{}^2 - \dot{x}_2{}^2 - \dot{x}_3{}^2),
\end{multline*}
we get from (9)
\begin{multline*}
\tag{13}
[(\dot{a}_0 - \dot{a}_3) (x^2 + y^2 + z^2)
- 2\dot{a}_1 xt - 2\dot{a}_2yt + (\dot{a}_0 + \dot{a}_3)t^2]   \\
= \pm 2\cosh r_1
  \sqrt{\dot{a}_0{}^2 - \dot{a}_1{}^2 - \dot{a}_2{}^2 - \dot{a}_3{}^2} zt.
\end{multline*}

We have here two spheres which differ merely in the $z$ coordinate
of their centre, i.e.\ two spheres which are reflections
of one another in the $z$ plane. If the hyperbolic sphere were
real and actual, one of the euclidean spheres would lie wholly
above the $z$ plane, and the other wholly below it. We may
say that (leaving aside special cases) a hyperbolic sphere will
correspond to so much of a euclidean sphere as is above or
in the $z$ plane, and to the reflection in the $z$ plane of so much
of the sphere as is below it.

A euclidean sphere for which $c = 0$, that is, one whose
centre is in the $z$ plane will correspond to a plane in hyperbolic
space, a hyperbolic sphere for which
\[
\dot{a}_0 - \dot{a}_3 = 0,
\]
that is, one whose centre is in the plane which corresponds
to the euclidean plane at infinity, will correspond to a plane
in euclidean space. A euclidean circle perpendicular to the
$z$ plane will correspond to a hyperbolic line, a hyperbolic circle
which is perpendicular to the plane $\dot{a}_0-\dot{a}_3 = 0$, will correspond
to a euclidean line.

We may go a step further in this direction. Suppose that
we have two euclidean spheres given by an equation of the
%[File: 142.png]
type (13), and the condition that they shall be mutually orthogonal is that
\begin{multline*}
  -\dot{a}_0\dot{a}_0{}' + \dot{a}_1\dot{a}_1{}' + \dot{a}_2\dot{a}_2{}'
\\
  \pm \cosh r_1 \cosh r_1{}'
  \sqrt{\dot{a}_0{}^2 - \dot{a}_1{}^2 - \dot{a}_2{}^2 - \dot{a}_3{}^2}\;
  \sqrt{\dot{a}_0{}^{'2} - \dot{a}_1{}^{'2} - \dot{a}_2{}^{'2} - \dot{a}_3{}^{'2}}
\\
+ \dot{a}_3\dot{a}'_3 = 0,
\end{multline*}
\[
  \cosh r_1 \cosh r_1{}'
= \pm \frac{-\dot{a}_0\dot{a}_0{}' + \dot{a}_1\dot{a}_1{}'
           + \dot{a}_2\dot{a}_2{}' + \dot{a}_3\dot{a}'_3 }
           {\sqrt{-\dot{a}_0{}^2 + \dot{a}_1{}^2
                  + \dot{a}_2{}^2 + \dot{a}_3{}^2  }\;
             \sqrt{-\dot{a}_0{}^{'2} + \dot{a}_1{}^{'2}
                  + \dot{a}_2{}^{'2} + \dot{a}_3{}^{'2}}       }.
\]

But this gives immediately that the corresponding hyperbolic
spheres are also mutually orthogonal, and conversely. We
thus have a correspondence of orthogonal spheres to orthogonal
spheres. We see next that the lines of curvature of any
surface will go into any lines of curvature of the corresponding
surface, and hence the Darboux-Dupin\index{Darboux}\index{Dupin} theorem must hold in
hyperbolic space, namely, in any triply orthogonal system
of surfaces, the intersections are lines of curvature.

Were we willing to sacrifice the real domain, we might in
a similar manner establish a correspondence between spheres
of euclidean and of elliptic space.\index{Sphere|)}

%[File: 143.png]


%CHAPTER XII
\mychap{XII}{CONIC SECTIONS}
\mysect{Classification of conics}
\index{Conic|(}

The study of the metrical properties of conics in the non-euclidean
plane, is, in the last analysis, nothing more nor less
than a study of the invariants and covariants of two conics.
We shall not, however, go into general questions of invariant
theory here, but rather try to pick out those metrical properties
of non-euclidean conics which bear the closest analogy
to the corresponding euclidean properties.%
\footnote{The treatment of conics in the present chapter is in close accord with
three articles by D'Ovidio\index{Ovidio@d'Ovidio}, `Le propriet\`a focali delle coniche,' `Sulle coniche
confocali,' and `Teoremi sulle coniche', all in the \emph{Atti della R. Accademia delle
Scienze di Torino}, vol.~xxvi, 1891. These articles suffer from the curious
blemish, not uncommon in Italian mathematical publications, that the
theorems are not given in distinctive type. See also Story\index{Story}, `On the
non-euclidean Properties of Conics,' \emph{American Journal of Mathematics}, vol.~v, 1882;
Killing\index{Killing}, `Die nicht-euklidische Geometrie in analytischer Behandlung,'
Leipzig, 1885, and Liebmann\index{Liebmann}, `Nicht-euklidische Geometrie,' in the \emph{Sammlung
Schubert}, xlix, Leipzig, 1904.}
%endfootnote

First of all, let us classify our conics under the real congruent group;
that is, in relation to their intersections with
the Absolute\index{Absolute}. This may be done analytically by means of
Weierstrass's\index{Weierstrass} elementary divisors, but the geometric question
is so easy that we give the results merely. We shall begin
with the real conics in the actual domain of hyperbolic space.

(1) Convex hyperbolas\index{Hyperbola}. Four real absolute points, no real
absolute tangents.

(2) Concave hyperbolas. Four real absolute points, four
real absolute tangents.

(3) Semi-hyperbolas\index{Semi-hyperbola}. Two real and two imaginary absolute
points and tangents.

(4) Ellipses\index{Ellipse}. Four imaginary absolute points and tangents.

(5) Concave hyperbolic parabolas\index{Parabola}. Two coincident, and
two real and distinct absolute points and tangents.

(6) Convex hyperbolic parabolas. Two coincident, and two
real and distinct absolute points. Two coincident, and two
conjugate imaginary absolute tangents.

(7) Elliptic parabolas. Two coincident, and two conjugate
imaginary absolute points and tangents.

(8) Osculating parabolas. Three real coincident, and one
real distinct absolute point, and the same for absolute tangents.

%[File: 144.png]
(9) Equidistant curves\index{Equidistant01@Equidistant curves}.

(10) Proper circles\index{Circle01@Circle}.

(11) Horocycles\index{Horocycle}.

In the real elliptic, or spherical, plane, we shall have
merely---

(1) Ellipses\index{Ellipse};

(2) Circles.

\mysect{Equations of central conic and Absolute}\index{Central conic|(}
In what follows we shall limit ourselves to \emph{central conics},
i.e.\ to those which cut the Absolute\index{Absolute} in four distinct points.
A real central conic in the actual domain of the hyperbolic
plane will have a common self-conjugate triangle with the
Absolute which is real, except in the case of the semi-hyperbola.
In the elliptic case it will surely be real. Taking this
as the coordinate triangle we may write the equation of the
Absolute in typical form, while that of the conic is

\[
\tag{1}
\sum_i^{0..2}c_ix_i{}^2=0.
\]

We assume that no two of our $c$'s are equal, and that none
of them are equal to zero.

\mysect{Centres, axes, foci, focal lines, directrices, and director points}
Our plane being $x_3 = 0$, we shall use the letters $h$, $k$, $l$ as
a circular permutation of the numbers $0$, $1$, $2$, and define the
vertices of the common self-conjugate triangles as \emph{centres}\index{Centre02@--- of a conic} of
the conic, while its sides are called the \emph{axes}\index{Axis03@--- of a conic}. Be it noticed
that in speaking of \emph{triangle} in this sense we are using the
terminology of projective geometry where a triangle is a figure
of three coplanar, but not concurrent lines, and not the exact
definition of Chapter I, which is meaningless except in a restricted
domain. There will, however, arise no confusion
from this.

\medskip\twincol{\emph{Theorem}~1. Each centre of
a central conic is a centre
of gravity\index{Centre03@--- of gravity of points} for every pair of
points of the conic collinear
therewith.}
{\emph{Theorem}~1$'$. Each axis of
a central conic is a bisector\index{Bisector of an angle}
of an angle of each pair of
tangents to the conic concurrent
thereon.}\par\medskip
%end twincol


The three pairs of lines which connect the pairs of intersections
of a central conic with the Absolute shall be called
its pairs of \emph{focal lines}. The three pairs of intersections
of its absolute tangents shall be called its pairs of \emph{foci}.

\medskip\twincol{\emph{Theorem}~2. Conjugate points
of a focal line of a conic are
mutually orthogonal\index{Orthogonal points}.}
{\emph{Theorem}~2$'$. Conjugate lines
through a focus of a conic are
mutually perpendicular\index{Perpendicularity}.}
%end twincol


%[File: 145.png]

\twincol{\emph{Theorem}~3. Two focal lines\index{Focal03@--- lines}
of a central conic pass through
each vertex, and are perpendicular to the opposite axis.}
{\emph{Theorem}~3$'$. Two foci\index{Foci} of a
central quadric lie on each
axis, and are orthogonal to
the opposite centre.}\par\medskip
%end twincol

The coordinates of the focal lines $f_h$, $f_h{}'$, through the centre
$u_h = 0$, will be
\[
  u_h : u_k : u_l
= 0 : \sqrt{c_h-c_k} : \pm \sqrt{c_l-c_h}.   \tag{2}
\]

The coordinates of the foci $F_h$, $F_h{}'$ on the opposite axis
will be
\[
  x_h : x_k : x_l
= 0 : \sqrt{c_l(c_h-c_k)} : \pm \sqrt{c_k(c_l-c_h)}. \tag{3}
\]

\mysect{Relations connecting distances of a point from foci, directrices, \&c., and their duals}
The polars of the foci with regard to the conic shall be called
\emph{directrices}\index{Director points and directrices},
the poles of the focal lines its \emph{director points}.
A directrix $d_h$ perpendicular to the axis $x_h$ will have the
equation
\[
  \sqrt{c_k(c_h-c_k)}x_k + \sqrt{c_l(c_l-c_h)}x_l = 0. \tag{4}
\]

Let $(x)$ be a point of the conic. Eliminating $x_h$ by means
of (1) we get
\[
  (xx) = \frac{(c_h-c_k)}{c_h}x_k{}^2 - \frac{(c_l-c_h)}{c_h}x_l{}^2.
\]

We then have
\begin{align*}
  \cos\frac{\overline{PF_h}}{k}
&= \frac{\sqrt{c_l(c_h-c_k)}x_k + \sqrt{c_k(c_l-c_h)}x_l        }
        {\sqrt{(c_h-c_k)x_k{}^2 - (c_l-c_h)x_l{}^2}\;\sqrt{(c_l-c_k)} }. \tag{5}
\\
  \sin\frac{\overline{PF_h}}{k}
&= \frac{\sqrt{c_k(c_h-c_k)}x_k + \sqrt{c_l(c_l-c_h)}x_l        }
        {\sqrt{(c_h-c_k)x_k{}^2 - (c_l-c_h)x_l{}^2}\;\sqrt{(c_k-c_l)} }. \tag{6}
\end{align*}

If $d_h$ be the corresponding directrix
\[
  \sin\frac{\overline{Pd_h}}{k}
= \frac{\sqrt{c_k(c_h-c_k)}x_k + \sqrt{c_l(c_l-c_h)}x_l            }
       {\sqrt{(c_h-c_k)x_k{}^2 - (c_l-c_h)x_l{}^2}\;
         \sqrt{\dfrac{c_k}{c_h}(c_h-c_k) + \dfrac{c_l}{c_h}(c_l-c_h)} },
\tag{7}
\]
the signs of the radicals in the numerators of the two expressions being the same
\begin{align*}
  \frac{\sin\dfrac{\overline{PF_h}}{k}}
       {\sin\dfrac{\overline{Pd_h}}{k}}
&= \frac{\sqrt{c_k(c_h-c_k) + c_l(c_l-c_h)} }
        {\sqrt{c_h(c_k-c_l)}                }
\\
&= \sqrt{\frac{c_h - (c_k+c_l)}{c_k}}.  \tag{8}
\end{align*}

%[File: 146.png]

\medskip\twincol{\emph{Theorem}~4. The ratio\index{Ratios, constant connected with conics} of the
sines of the $k$th parts of the
distances from a point of a
central conic to a focus\index{Foci} and
to the corresponding directrix\index{Director points and directrices}
is constant.}
{\emph{Theorem}~4$'$. The ratio of
the sines of the angles which
a tangent to a central conic
makes with a focal line\index{Focal03@--- lines} and
the absolute polar of the
corresponding director point
is constant.}\par\medskip
%end twincol


\[
  \cos\frac{\overline{F_hF_h{}'}}{k}
= \frac{c_l(c_h-c_k) - c_k(c_l-c_h)}
       {c_l(c_h-c_k) + c_k(c_l-c_h)},\quad
  \tan^2\tfrac{1}{2}\frac{\overline{F_hF_h{}'}}{k}
= \frac{c_k(c_l-c_h)}{c_l(c_h-c_k)},
\]
\begin{multline*}
  \tan^2\tfrac{1}{2}\frac{\overline{F_hF_h{}'}}{k}
  \tan^2\tfrac{1}{2}\frac{\overline{F_kF_k{}'}}{k}
  \tan^2\tfrac{1}{2}\frac{\overline{F_lF_l{}'}}{k}
\\
= \tan^2\tfrac{1}{2} \measuredangle f_h f_h{}'
  \tan^2\tfrac{1}{2} \measuredangle f_k f_k{}'
  \tan^2\tfrac{1}{2} \measuredangle f_l f_l{}' = 1.
\tag{9}
\end{multline*}\ungap
\begin{multline*}
  \sin\frac{\overline{PF_h}}{k}
  \sin\frac{\overline{PF_h{}'}}{k}
= \frac{c_k(c_h-c_k)x_k{}^2 - c_l(c_l-c_h)x_l{}^2}
       {\bigl[(c_l-c_h)x_l{}^2 - (c_h-c_k)x_k{}^2\bigr](c_k-c_l)}
\\
= \frac{\smallsum c_h{}^2 x_h{}^2}{c_h(c_k-c_l)(xx)}, \phantom{(9)}
\end{multline*}\ungap
\begin{multline*}
  \sin\frac{\overline{PF_h}}{k} \sin\frac{\overline{PF_h{}'}}{k}\ :
  \sin\frac{\overline{PF_k}}{k} \sin\frac{\overline{PF_k{}'}}{k}\ :
  \sin\frac{\overline{PF_l}}{k} \sin\frac{\overline{PF_l{}'}}{k}
\\
= \frac{1}{c_h(c_k-c_l)}\ :
  \frac{1}{c_k(c_l-c_h)}\ :
  \frac{1}{c_l(c_h-c_k)}.
\tag{10}
\end{multline*}
\[
  \csc\frac{\overline{PF_h}}{k} \csc\frac{\overline{PF_h{}'}}{k}
+ \csc\frac{\overline{PF_k}}{k} \csc\frac{\overline{PF_k{}'}}{k}
+ \csc\frac{\overline{PF_l}}{k} \csc\frac{\overline{PF_l{}'}}{k} = 0.
\tag{11}
\]
\[
 \cos\frac{\overline{PF_h}}{k} \cos\frac{\overline{PF_h{}'}}{k}
= \frac{c_l(c_h-c_k)x_k{}^2 - c_k(c_l-c_h)x_l{}^2 }
       {\bigl[(c_h-c_k)x_k{}^2 - (c_l-c_h)x_l{}^2\bigr](c_l-c_k) },
\]
\label{xhxl}
\[
\cos\Bigl[ \frac{\overline{PF_h}}{k}
         \pm\frac{\overline{PF_h{}'}}{k} \Bigr]
= \frac{c_k+c_l}{c_k-c_l}.
\tag{12}
\]
\[
 \tan\tfrac{1}{2}\Bigl[ \frac{\overline{PF_h }}{k}
                     \pm\frac{\overline{PF_h{}'}}{k} \Bigr]
 \tan\tfrac{1}{2}\Bigl[ \frac{\overline{PF_k }}{k}
                     \pm\frac{\overline{PF_k{}'}}{k} \Bigr]
 \tan\tfrac{1}{2}\Bigl[ \frac{\overline{PF_l }}{k}
                     \pm\frac{\overline{PF_l{}'}}{k} \Bigr]
= \pm 1.
\tag{13}
\]

With regard to the ambiguity of signs: the upper sign in
(12) will go with the upper sign throughout in (13), and so
for the lower sign. It is also geometrically evident that
in the case of an ellipse we must take the upper, and in
the case of a hyperbola the lower sign (when in the real
domain).

%[File: 147.png]
\medskip\twincol{\emph{Theorem}~5. The sum of the
distances\index{Sum04@Sum of distances connected with a conic} from real points of
an ellipse and the difference
of the distances from real
points of a hyperbola or semi-hyperbola
to two real foci on
the same axis is constant.}
{\emph{Theorem}~5$'$. The sum of the
angles which the real tangents
to an ellipse\index{Ellipse} or convex hyperbola\index{Hyperbola},
or the difference of the
angles which the real tangents
to a concave hyperbola or a
semi-hyperbola make with
two real focal lines through
a centre is constant.}\par\medskip
%end twincol

Reverting to our point $(x)$ we see

\[
\sin\frac{\overline{Pf_h}}{k} =
  \frac{\sqrt{c_h-c_k}x_k + \sqrt{c_l-c_h}x_l}
  {\sqrt{(c_l-c_h)x_l{}^2 - (c_h-c_k)x_k{}^2}\;
  \sqrt{\dfrac{-(c_k-c_l)}{c_h}}},
\]
\[
\sin\frac{\overline{Pf_h}}{k}
  \sin\frac{\overline{Pf_h{}'}}{k} = \pm \frac{c_h}{c_k-c_l}.
\]

\medskip\twincol{\emph{Theorem}~6. The product\index{Products connected with a conic} of
the sines of the $k$th parts
of the distances from a point
of a central conic to two focal
lines through the same centre
is constant.}
{\emph{Theorem}~6$'$. The product of
the sines of the $k$th parts
of the distances to a tangent
from two foci of a central
conic on the same axis is
constant.}\par\medskip
%end twincol

Let us now recall Desargues'\index{Desargues} theorem, whereby a transversal
meets the conics of a pencil in pairs of points of an involution.
This will apply to a central conic, the Absolute\index{Absolute}, and the pairs
of focal lines\index{Focal03@--- lines}. A dual theorem will of course hold for a central
conic, the Absolute, and the pairs of foci\index{Foci}.

\medskip\twincol{\emph{Theorem}~7. The intersections
of a line with a central
conic, and with its pairs of
corresponding focal lines, all
have the same centres of
gravity\index{Centre03@--- of gravity of points}.}
{\emph{Theorem}~7$'$. The tangents
from a point to a central
conic, and the pairs of lines
thence to its pairs of corresponding
foci, form angles
with the same two bisectors\index{Bisector of an angle}.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~8. The polar of a
point with regard to a central
conic passes through one
centre of gravity of the intersections
of each focal line with
the tangents from the point to
the conic.}
{\emph{Theorem}~8$'$. The pole of a
line with regard to a central
conic lies on one bisector of
the angle determined at each
focus by the lines thence to
the intersections of the given
line with the conic.}\par\medskip
%end twincol

%[File: 148.png]
A variable point of a conic will determine projective pencils
at any two fixed points thereof, and these will meet any line
in projective ranges, hence

\medskip\twincol{\emph{Theorem}~9. If a variable point of a central conic be connected with
two fixed points thereof, the distance which these lines cut on any focal line\index{Focal03@--- lines}
is constant.}
{\emph{Theorem}~9$'$. If a variable tangent to a central conic be brought to intersect two fixed
tangents thereof, the angle of the lines from a chosen focus\index{Foci}
to the two intersections is constant.}\par\medskip
%end twincol

Recalling the properties of the eleven-point conic\index{Conic, eleven-point or line} of two
given conics and a line:

\medskip\twincol{\emph{Theorem}~10. If a line and a central conic be given, the two
mutually conjugate and orthogonal points of the line, the points of the
focal lines orthogonal to their intersections with the line, and the
three centres lie on a conic.}
{\emph{Theorem}~10$'$. If a point and a central conic be given,
the two lines through the point which are mutually conjugate and
perpendicular, the perpendiculars on the line from the foci,
and the three axes all touch a conic.}\par\medskip
%end twincol

It is a well-known theorem that the locus of points, whence
tangents to two conics form a harmonic set, is a conic passing
through the points of contact with the common tangents.

\medskip\twincol{\emph{Theorem}~11. The locus of points whence tangents to a central conic are
mutually perpendicular is a conic meeting the given conic where it meets its
directrices\index{Director points and directrices}.}
{\emph{Theorem}~11$'$. The envelope of lines which meet a central conic in pairs of mutually
orthogonal points is a conic touching the tangents to the given circle from its director points.}

It is clear that neither of these conics will, in general, be
a circle, as in the euclidean case. If the mutually perpendicular
tangents from the point $(y)$ be

\[
(ux) = 0,\quad (vx) = 0.\\
\]
\[
\sum_i^{0..2} \frac{u_i{}^2}{c_i} = \sum_i^{0..2} \frac{v_i{}^2}{c_i} = \sum_i^{0..2} u_i v_i=0,\\
\]
\[
\sum_i^{0..2} \frac{u_i{}^2}{c_i} (vv) + \sum_i^{0..2} \frac{v_i{}^2}{c_i} (uu)-2\sum_i^{0..2} \frac{u_i v_i}{c_i} (uv)=0,
\]
%[File: 149.png]
\[
  \sum_h^{0..2} c_h(c_k+c_l)y_i{}^2 = 0.   \tag{14}
\]
Let the reader show that the equation of the other conic
will be
\[
  \sum_h^{0..2} (c_k+c_l)u_i{}^2 = 0.
\]
We may extend the usual euclidean proof to the first of the
following theorems---

\medskip\twincol{\emph{Theorem}~12. The locus of
the reflection of a real focus
of an ellipse in a variable
tangent, is a circle whose
centre is the corresponding
focus.}
{\emph{Theorem}~12$'$. The envelope
of the reflection in a variable
point of an ellipse, of a real
focal line, is a circle whose
axis is the corresponding focal
line.}\par\medskip
%end twincol

\mysect{Conjugate and mutually perpendicular lines through a centre}
Let $(y)$ be the coordinates of a point $P$ of our conic. The
equation of a line through the centre\index{Centre02@--- of a conic} $O_h$ conjugate to the line
$O_hP$ will be
\[
  c_k y_k x_k + c_l y_l x_l = 0.
\]
This will meet the conic in two points $P'$ having the coordinates
\[
  x_h  : x_k  : x_l
= \pm\sqrt{c_k c_l}y_h  : c_l y_l  : - c_k y_k,
\]
\[
  \tan^2 \frac{\overline{OP }}{k}
+ \tan^2 \frac{\overline{OP'}}{k}
= \frac{-c_h(c_k+c_l)}{c_k c_l}   \tag{15}
\]

\medskip\twincol{\emph{Theorem}~13. The sum\index{Sum04@Sum of distances connected with a conic} of
the squares of the tangents
of the $k$th parts of the distances from a centre of a
central conic to any pair of
intersections with two conjugate lines through this
centre is constant.}
{\emph{Theorem}~13$'$. The sum of
the squares of the tangents
of the angles which an axis of
a central conic makes with a
pair of tangents to the curve
from two conjugate points of
this axis is constant.}\par\medskip
%end twincol

We shall call two such diameters\index{Diameters01@Diameters of conic} as $O_hP$, $O_hP'$ \emph{conjugate
diameters}\index{Conjugate01@Conjugate diameters of a conic}.
\begin{align*}
  \sin \measuredangle PO_hP'
&= \frac{(c_k y_k{}^2 + c_l y_l{}^2)}
        {\sqrt{y_k{}^2 + y_l{}^2}\;\sqrt{c_k{}^2y_k{}^2 + c_l{}^2y_l{}^2}}
\\
&= \frac{-c_h y_h{}^2 }
        {\sqrt{y_k{}^2 + y_l{}^2}\;\sqrt{c_k{}^2y_k{}^2 + c_l{}^2y_l{}^2}} ;
\end{align*}
\[
  \tan \frac{\overline{O_hP }}{k}
  \tan \frac{\overline{O_hP'}}{k}
  \sin \measuredangle PO_hP'
= \pm \frac{c_h}{\sqrt{c_k c_l}}.
\]

%[File: 150.png]
\medskip\twincol{\emph{Theorem}~14. The product
of the tangents of the $k$th
parts of the distances from a
centre\index{Centre02@--- of a conic} of a central conic to
two intersections with a pair
of conjugate diameters\index{Diameters01@Diameters of conic} through
that centre, multiplied by the
sine of the angle of these
diameters is constant.}
{\emph{Theorem}~14$'$. The product\index{Products connected with a conic}
of the tangents of the angles
which an axis of a central
conic makes with two tangents
to it from a pair of conjugate
points of this axis, multiplied
by the sine of the $k$th part of
the distance of these points is
constant.}\par\medskip
%end twincol

The equation of a line through the centre $O_h$ perpendicular
to $O_hP$ will be
\[
  y_k x_k + y_l x_l = 0.
\]
This will meet the conic in points $P''$ having coordinates
\begin{align*}
  x_h : x_k : x_l
&=\pm\sqrt{\frac{-(c_l y_k{}^2 + c_k y_l{}^2)}{c_h}} : y_l : -y_k,
\\
  \cos\frac{\overline{OP''}}{k}
&= \frac{\sqrt{-(c_l y_k{}^2 + c_k y_l{}^2)       } }
        {\sqrt{(c_h-c_l)y_k{}^2 + (c_h-c_k)y_l{}^2} },
\\
  \ctn^2 \frac{\overline{OP''}}{k}
&= \frac{-(c_l y_k{}^2 + c_k y_l{}^2)}{c_h(y_k{}^2 + y_l{}^2)},
\\
  \ctn^2 &\frac{\overline{OP  }}{k}
+ \ctn^2 \frac{\overline{OP''}}{k}
= -\frac{c_k+c_l}{c_h}.     \tag{16}
\end{align*}%[** Overlines assumed]

\medskip\twincol{\emph{Theorem}~15. The sum\index{Sum04@Sum of distances connected with a conic} of
the squares of the cotangents
of the $k$th parts of the distances
from a centre of a
central conic to two intersections
of the curve with
mutually perpendicular diameters
through this centre is
constant.}
{\emph{Theorem}~15$'$. The sum of
the squares of the cotangents
of the angles which an axis of
a central conic makes with
two tangents from a pair of
orthogonal points of this axis
is constant.}\par\medskip
%end twincol

The equation of the tangent $t'$ at the point $P'$ is
\[
  c_h y_h x_h + \sqrt{c_k c_l}(x_k y_l - x_l y_k) = 0.
\]

From this we get
\[
  \sin^2 \frac{\overline{O_h t'}}{k}
= \frac{c_h{}^2 y_h{}^2}{(c_l-c_h)c_k y_k{}^2 + (c_k-c_h)c_l y_l{}^2},
\]
\[
  \tan \frac{\overline{O_h P }}{k}
  \tan \frac{\overline{O_h t'}}{k}
= \frac{c_h}{\sqrt{c_k c_l}}.      \tag{17}
\]

%[File: 151.png]
\medskip\twincol{\emph{Theorem}~16. The product\index{Products connected with a conic}
of the tangents of the $k$th
parts of the distances from
a centre of a central conic\index{Centre02@--- of a conic}
to a point of the curve and
to the tangent where the curve
meets a diameter\index{Diameters01@Diameters of conic} conjugate to
that from the centre to the
point of the curve, is constant.}
{\emph{Theorem}~16$'$. The product
of the tangents of the angles
which an axis of a central
conic makes with a tangent
and with the absolute polar
of a point of contact with
a tangent from a point of this
axis conjugate to the intersection
with the given tangent,
is constant.}\par\medskip
%end twincol

The equations of two conjugate diameters through $O_h$ have
already been written
\[
y_l x_k - y_k x_l = 0,\quad c_k y_k x_k + c_l y_l x_l = 0.
\]

The product of the tangents of the angles which they make
with the $x_k$ axis is
\[
\frac{y_k c_l y_l}{y_l c_k y_k} = \frac{c_l}{c_k}.
\]

\medskip\twincol{\emph{Theorem}~17. The product
of the tangents of the angles
which two conjugate diameters
through a centre make
with either axis through this
centre is constant.}
{\emph{Theorem}~17$'$. The product
of the tangents of the $k$th
parts of the distances of two
conjugate points of an axis
from either centre on this
axis is constant.}\par\medskip
%end twincol

Let $P_h$, $P'_h$ be the intersections of the $x_h$ axis with the conic
\[
\cos\frac{\overline{P_h P'_h}}{k} =
  \frac{c_h + c_l}{c_k - c_l},
\]
%[**c_k + c_l?]
\[
\tag{18}
  \tan^2 \tfrac{1}{2} \frac{\overline{P_h P'_h}}{k} \cdot
  \tan^2 \tfrac{1}{2} \frac{\overline{P_k P'_k}}{k} \cdot
  \tan^2 \tfrac{1}{2} \frac{\overline{P_l P'_l}}{k} = -1.
\]

\medskip\twincol{\emph{Theorem}~18. The product
of the squares of the tangents
of the $2k$th parts of the
distances determined by a
central conic on the axes is
equal to $-1$.}
{\emph{Theorem}~18$'$. The product
of the squares of the tangents
of the half-angles of the
pairs of tangents to a central
conic from its centres is constant.}\par\medskip
%end twincol

If a circle have double contact with a conic, we have, with
the Absolute, the figure of two conics having double contact
with a third, already studied in the last chapter.

\medskip\twincol{\emph{Theorem}~19. If a circle have
double contact with a conic,
its axis\index{Axis01@Axis of a circle} and the lines connecting
%[File: 152.png]
the points of contact are
harmonically separated by a
pair of focal lines\index{Focal03@--- lines}.}
{\emph{Theorem}~19$'$. If a circle have
double contact with a conic,
its centre and the intersections
%[File: 152.png bis]
of the common tangents are
harmonically separated by a
pair of foci\index{Foci}.}\par\medskip
%end twincol

Of course we mean by foci and focal lines of any conic what
we mean in the special case of the central conic.

\mysect{Auxiliary circles}
A circle\index{Circle01@Circle} which has double contact with a central conic
where the latter meets an axis is called an \emph{auxiliary circle}.
There will clearly be six such circles, their centres being the
centres of the conic. Consider the circle having its centre
at $O_k$ while it has double contact with our central conic at
the intersections with $x_h = 0$.
\[
  p\sum_i^{0..2} c_i x_i{}^2 + q x_h{}^2 \equiv
  r(xx) + s x_k{}^2 = 0,
\]
\[
  \sum_i^{0..2} c_i x_i{}^2 + (c_l-c_h) x_h{}^2 =
  c_h x_h{}^2 + c_k x_k{}^2 + c_l x_l{}^2 = 0.
\]

This will meet the line $(u)$ through $O_h$ in points $Q$, $Q'$, having
coordinates
\[
  x_h  : x_k  : x_l =
  \sqrt{-\Bigl( \frac{c_l u_k{}^2}{c_l}
              + \frac{c_k u_l{}^2}{c_l} \Bigr)}  : u_l  : -u_k.
\]

The same line will meet the conic in points $P$, $P'$, having
coordinates
\[
  x_h  : x_k  : x_l =
  \sqrt{-\frac{c_l u_k{}^2 + c_k u_l{}^2}{c_h}}  : u_l  : -u_k,
\]
\[
  \tan^2\frac{\overline{O_h Q}}{k} =
  \frac{-c_l(u_k{}^2 + u_l{}^2)}{c_l u_k{}^2 + c_k u_l{}^2},
\quad
  \tan^2\frac{\overline{O_h P}}{k} =
  \frac{-c_h(u_k{}^2 + u_l{}^2)}{c_l u_k{}^2 + c_k u_l{}^2},
\]
\[
\tag{19}
  \tan\frac{\overline{O_h Q}}{k}  : \tan\frac{\overline{O_h P}}{k}
= \sqrt{c_l}  : \sqrt{c_h}.
\]

Let us remark, finally, that the tangent of the $k$th part of the
distance from a point to a line, is the cotangent of the $k$th part
of its distance to the pole of the line, and that if the tangents
of two distances bear a constant ratio\index{Ratios, constant connected with conics}, so do their cotangents:

\medskip\twincol{\emph{Theorem}~20. If the tangents
of the $k$th parts of the distances
from the points of a
circle to any diameter\index{Diameters01@Diameters of conic} be
%[File: 153.png]
altered in a constant ratio, the
locus of the resulting points
will be a conic having the
given circle as an auxiliary\index{Circle02@Circle auxiliary to conic}.}
{\emph{Theorem}~20$'$. If the tangents
of the angles which the tangents
to a circle make with a
diameter be altered in a constant
%[File: 153.png bis]
ratio, the envelope of the
resulting lines will be a conic
having the given circle as an
auxiliary circle.}\par\medskip
%end twincol

\mysect{Normals}
The normal at any point of a conic is the line connecting
it with the absolute pole of its tangent. This line is also
perpendicular to the absolute polar of the given point, so that
the conic and its absolute polar conic are geodesically parallel
curves. The equation of the normal to our conic (1) will be
\begin{equation*}
\tag{20}
\sum_i^{0..2}\frac{c_k - c_l}{y_h} x_h = 0.
\end{equation*}

The tangents to a central conic from a centre shall be called
\emph{asymptotes}\index{Asymptotes}. The equation of the pair of asymptotes through
the centre $(O_h)$ will evidently be
\begin{equation*}
  c_k x_k{}^2 + c_l x_l{}^2 = 0.
\tag{21}
\end{equation*}

The tangent at the point $P$ with coordinates $(y)$ will meet
them in two points $R$, $R'$, whose coordinates are
\begin{equation*}
x_h  : x_k  : x_l =
  \sqrt{-c_k c_l} (\sqrt{-c_l} y_l \pm \sqrt{c_k} y_k)  :
  \mp c_h y_h \sqrt{-c_l}  :
  -c_h y_h \sqrt{c_k},
\end{equation*}\label{whatsce}
\begin{equation*}
\tag{22}
\tan\frac{\overline{O_h R}}{k} \tan\frac{\overline{O_h R'}}{k} =
  \frac{(c_l-c_k) c_h{}^2 y_h{}^2}{c_k c_l (c_ky_k{}^2+c_ly_l{}^2)} =
  \frac{c_h (c_k-c_l)}{c_k c_l}.
\end{equation*}

\medskip\twincol{\emph{Theorem}~21. The product of
the tangents of the $k$th parts
of the distances from a centre
of a central conic to the intersection
with the asymptotes
through that centre of a tangent
is constant.}
{\emph{Theorem}~21$'$. The product
of the tangents of the angles
which an axis of a central
conic makes with the lines
from a point of the curve to
the intersections of the curve
with this axis is constant.}\par\medskip
%end twincol

\mysect{Confocal and homothetic conics}
A set of conics which meet the Absolute\index{Absolute} in the same four
points shall be said to be \emph{homothetic}\index{Homothetic01@Homothetic conics}. If they have the same
four absolute tangents they shall be called \emph{confocal}. We get
at once from Desargues' involution theorem:---

\medskip\twincol{\emph{Theorem}~22. One conic
homothetic to a given conic
will pass through every point
of space, and two will touch
every line, not through a point
%[File: 154.png]
common to all the conics, in
the centres of gravity\index{Centre03@--- of gravity of points} of all
pairs of intersections of the
homothetic conics\index{Homothetic01@Homothetic conics} with this
line.}
{\emph{Theorem}~22$'$. One conic confocal
with a given conic will
touch every line, and two
will pass through every point
not on the common tangents
%[File: 154.png bis]
to all. The tangents to these
two will bisect\index{Bisector of an angle} the angles of
the pairs of tangents from that
point to all of the confocal
conics\index{Confocal conics}.}\par\medskip
%end twincol

Concentric circles are a special case both of homothetic and
of confocal conics. The general form for the equations of conics
homothetic and confocal respectively to our conic (1) will be
\[
  \sum_i^{0..2} (c_i + m)x_0{}^2 = 0.       \quad(23);
\qquad
  \sum_i^{0..2} \frac{c_i}{l+c_i}x_i = 0. \quad(24)
\]
\index{Conic|)}\index{Central conic|)}

\mysect{Elliptic coordinates}
It is sometimes useful to modify the second of these
equations, in order to introduce the elliptic coordinates\index{Elliptic01@Elliptic co-ordinates} of
a point, i.e.\ the two parameters giving the conics of the
confocal system which pass through it. Let us write $\dfrac{1}{c_i}$ in
place of $c_i$.
\[
  \frac{x_i}{\sqrt{(xx)}} = X_i.
\]

Our confocal conics have, then, the general equation
\[
  \sum_i^{0..2} \frac{X_i}{c_i-\lambda} = 0. \tag{25}
\]

If $\lambda_1$ and $\lambda_2$ be the parameter values of the conic through $(X)$
we have
\[
  X_h
= \sqrt{\frac{(c_k-c_l) (c_h-\lambda_1) (c_h-\lambda_2) }
             {\sum\limits_h^{0..2} c_h{}^2(c_h-c_l)           }}. \tag{26}
\]
\[
ds^2 = \sum_i^{0..2} dX_i{}^2
= \frac{1}{4}
  \left[ \frac{(\lambda_1 - \lambda_2) d\lambda_1{}^2}
              {\prod\limits_i^{0..2} (c_i-\lambda_1) }
       + \frac{(\lambda_2 - \lambda_1) d\lambda_2{}^2}
              {\prod\limits_i^{0..2} (c_i-\lambda_2) }
 \right]. \tag{27}
\]

With the aid of these coordinates, we may easily prove for
the non-euclidean case Graves'\index{Graves} theorem, namely, if a loop
of thread be cast about an extremely thin elliptic disk, and
pulled taut at a point, that point will trace a confocal ellipse\index{Ellipse}.
We shall not give the details here, however, for in the next
chapter we shall work at length the more interesting corresponding
problem in three dimensions, and the calculations
are too fatiguing to make it advisable to carry them through
twice.

%[File: 155.png]



%CHAPTER XIII
\mychap{XIII}{QUADRIC SURFACES}
\mysect{Classification of quadrics}

The discussion of non-euclidean quadric surfaces may be
carried on in the same spirit as that of conic sections in the
preceding chapter. There is not, however, the same wealth
of easy and interesting theorems, owing to the greater complication
in the formation of the simultaneous covariants
of two quadrics.

Let us begin by classifying non-euclidean quadrics under
the group of real congruent transformations.%
\footnote{The classification here given is that which appears in the author's\index{Author}
article `Quadric Surfaces in Hyperbolic Space', \emph{Transactions of the American
Mathematical Society}, vol.~iv, 1903. This classification was simplified and
put into better shape by Bromwich\index{Bromwich}, `The Classification of Quadratic Loci,'
ibid., vol.~vi, 1905. The latter, however, makes use of Weierstrassian
Elementary Divisors, and it seemed wiser to avoid the introduction of these
into the present work. Both Professor Bromwich and the author wrote in
ignorance of the fact that they had been preceded by rather a crude article
by Barbarin\index{Barbarin}, `\'Etude de g\'eom\'etrie non-euclidienne,' \emph{M\'emoires couronn\'es par
l'Acad\'emie de Belgique}, vol.~vi, 1900.}
%endfootnote
We begin
in the actual domain of hyperbolic space, giving only those
surfaces which have a real part in that domain and a non-vanishing
discriminant. The names adopted are intended to
give a certain idea of the shape of the surface. We shall
mean by \emph{curve}, the curve of intersection of the surface and
Absolute\index{Absolute}, while \emph{developable} is the developable of common
tangent planes.

\medskip A. Central Quadrics.

(1) Ellipsoid\index{Ellipsoid}. Imaginary quartic curve and developable.

(2) Concave, non-ruled hyperboloid. Real quartic curve
and developable.

(3) Convex non-ruled hyperboloid.  Real quartic curve,
imaginary developable.

(4) Two-sheeted ruled hyperboloid. Real quartic curve
and developable.

(5) One-sheeted ruled hyperboloid.  Real quartic curve,
imaginary developable.

(6) Non-ruled semi-hyperboloid\index{Semi-hyperboloid}. Real quartic curve and
developable.

(7) Ruled semi-hyperboloid. Real quartic curve and developable.

The last two surfaces differ from the preceding ones in that
%[File: 156.png]
here two vertices of the common self-conjugate tetrahedron
(in the sense of projective geometry) of the surface and
Absolute\index{Absolute} are conjugate imaginaries, while in the first five
cases all four are real.

\medskip B.

(8) Elliptic paraboloid\index{Paraboloid}. Imaginary quartic curve with real
acnode, imaginary developable.

(9) Tubular non-ruled hyperbolic paraboloid. Real quartic
with acnode, real developable.

(10) Cup-shaped non-ruled hyperbolic paraboloid.  Real
quartic with acnode, imaginary developable.

(11) Open ruled hyperbolic paraboloid. Real acnodal
quartic, real developable.

(12) Gathered ruled hyperbolic paraboloid. Real crunodal
quartic, imaginary developable.

(13) Cuspidal non-ruled hyperbolic paraboloid. Real cuspidal quartic curve, real developable.

(14) Cuspidal ruled hyperbolic paraboloid. Real cuspidal
quartic curve, real developable.

(15) Horocyclic non-ruled hyperbolic paraboloid. The curve
is two mutually tangent conics, developable real.

(16) Horocyclic elliptic paraboloid. Curve is two mutually
tangent imaginary conics, developable imaginary.

(17) Horocyclic ruled hyperbolic paraboloid. Curve is two
real mutually tangent conics, developable imaginary.

(18) Non-ruled osculating semi-hyperbolic paraboloid. The
curve is a real conic and two conjugate imaginary generators
meeting on it. The developable is a real cone, and two
imaginary lines.

\medskip C. Surfaces of Revolution.\index{Revolution, surfaces of}

(19) Prolate spheroid\index{Spheroid}. Curve is two imaginary conics in
real ultra-infinite planes, imaginary developable.

(20) Oblate spheroid. Curve is two imaginary conics in
conjugate imaginary planes meeting in an ultra-infinite line,
imaginary developable.

(21) Concave non-ruled hyperboloid\index{Hyperboloid} of revolution. Curve
is two real conics whose planes meet in an ideal line, real
developable.

(22) Convex non-ruled hyperboloid of revolution. Absolute
curve two real conics whose planes meet in an ideal line,
imaginary developable.

(23) Ruled hyperboloid of revolution.  Curve two real
conics whose planes meet in an ideal line, imaginary developable.

%[File: 157.png]
(24) Semi-hyperboloid of revolution. The curve is a real
conic, and an imaginary one in a real plane, the developable
is a real cone and an imaginary one.

(25) Elliptic paraboloid of revolution. The absolute curve
is an imaginary conic in an ultra-infinite plane, and two
imaginary generators not intersecting on the conic. The developable
is an imaginary cone, and the same two generators.

(26) Tubular semi-hyperbolic paraboloid of revolution.
The curve is a real conic and two imaginary generators not
intersecting on it; the developable is the same two lines and
a real cone.

(27) Cup-shaped semi-hyperbolic paraboloid of revolution\index{Revolution, surfaces of}.
Real conic and two imaginary lines not meeting on it. Developable
same two lines and imaginary cone.

(28) Clifford\index{Clifford} surface. Curve and developable two generators
of each set.

\medskip D. Canal Surfaces\index{Canal surface}.%
\footnote{Called \emph{Surfaces of Translation} in the author's\index{Author} article `Quadric Surfaces',
loc.\ cit.}
%endfootnote

(29) Elliptic canal surface. Curve is two imaginary conics
whose planes meet in an actual line, developable imaginary.

(30) Non-ruled hyperbolic canal surface. Two real conics
whose planes meet in an actual line, developable two real
cones.

(31) Ruled hyperbolic canal surface. Curve two real conics
whose planes meet in an actual line, imaginary developable.

\medskip E. Spheres.\index{Sphere}

(32) Proper sphere.  Curve is two coincident imaginary
conics, developable imaginary.

(33) Equidistant surface\index{Equidistant02@--- surfaces}. Curve two real coincident conics,
developable two real coincident cones.

(34) Horocyclic surface\index{Horocyclic surface}. Curve and developable
two conjugate imaginary intersecting generators, each counted twice.

\medskip In elliptic or spherical space the number of real varieties
will, of course, be much smaller. We have

(1) Non-ruled ellipsoid\index{Ellipsoid}.

(2) Ruled ellipsoid.

(3) Prolate spheroid\index{Spheroid}.

(4) Oblate spheroid.

(5) Ruled ellipsoid of revolution.

(6) Clifford surface.

(7) Sphere.

%[File: 158.png]

It is worth mentioning that the Clifford\index{Clifford} surface of elliptic
space has real linear generators, while that in hyperbolic space
has not.

\mysect{Central quadrics}
\index{Central quadric|(}
Let us next turn our attention to that class of quadrics
which we have termed central, and which are distinguished
by the existence of a non-degenerate tetrahedron (in the
projective sense) self-conjugate with regard both to the surface
and the Absolute\index{Absolute}. The vertices of this tetrahedron shall be
called the \emph{centres}\index{Centre04@--- of quadric} of the surface,
and its planes the \emph{axial planes}.
When this tetrahedron is chosen as the basis of the coordinate
system, the Absolute may be written in the typical form
while the equation of the surface involves none but squared
terms.

\medskip\twincol{\emph{Theorem}~1. A centre of a
central quadric is equidistant
from the intersections with
the surface of every line
through this centre.}
{\emph{Theorem}~1$'$. An axial plane
of a central quadric bisects\index{Bisector of an angle}
a dihedral angle of every two
tangent planes to the surface
which meet in a line of this
axial plane.}\par\medskip
%end twincol

We obtain a good deal of information about our central
quadrics by enumerating the Cayleyan\index{Cayley} characteristics of their
curves of intersection with the Absolute, and the corresponding
developables. The curve is a twisted quartic of deficiency
one. Its osculating developable is of order eight and class
twelve. It has sixteen stationary tangent planes, thirty-eight
lines in every plane lie in two osculating planes, two secants,
i.e.\ two lines meeting the curve twice, pass every point not on
the curve, sixteen points in every plane are the intersection of
two tangents, eight double tangent planes pass through every
point. The developable will, of course, possess the dual
characteristics.

\medskip\twincol{\emph{Theorem}~2. Through an
arbitrary point in space will
pass twelve planes cutting a
central quadric in osculating
parabolas, eight planes of
parabolic\index{Paraboloid} section will pass
through an arbitrary line. An
arbitrary point will be the
centre of one section. Sixteen
planes cut the surface in horocycles, sixteen points in an
%[File: 159.png]
arbitrary plane are the centres
of circular sections, eight
planes of circular section pass
through an arbitrary point.}
{\emph{Theorem}~2$'$. In an arbitrary
plane there will be twelve
points, vertices of cones circumscribed to a central
quadric which have stationary
contact with the cone of tangents
to the Absolute, eight
points on an arbitrary line
are vertices of circumscribed
cones which touch the Absolute.
An arbitrary plane will
%[File: 159.png bis]
be a plane of symmetry for
one circumscribed cone. Sixteen
points are vertices of
circumscribed cones which
have four-plane contact with
the Absolute. Sixteen planes
through an arbitrary point are
perpendicular to the axes of
revolution of circumscribed
cones of revolution.}\par\medskip
%end twincol

\mysect{Planes of circular section and parabolic section}
The planes of circular section are those which touch the
cones whose vertices are the centres of the quadric, and which
pass through the Absolute curve. It may be shown that not
more than six real planes of circular section will pass through
an actual point, and that only two of these will cut the surface
in proper circles.%
\footnote{See the author's\index{Author} `Quadric Surfaces', loc.\ cit., p.~164.}
%endfootnote

Let us write as the equation of a typical quadric
\[
  \sum_i^{0..3} c_i x_i{}^2 = 0.  \tag{1}
\]

No two of the $c$'s shall be equal, and none shall equal
zero.

The cones whose vertices are the centres and which pass
through the Absolute curves shall be called the \emph{focal cones}\index{Focal01@Focal cones}.
In like manner there will be four \emph{focal conics}\index{Focal02@--- conics} in the axial
planes. The equation of the focal cone whose vertex is $O_h$
will be
\[
  \sum_i^{0..3} (c_i - c_h)x_i{}^2 = 0.  \tag{2}
\]

The focal conic in the corresponding axial plane will be
\[
  x_h = 0,\quad \sum_i^{0..3} \frac{c_h-c_i}{c_h c_i} w_i{}^2 = 0.  \tag{3}
\]

Let the reader show that each of these conics passes through
two foci of each other one.

We next seek the locus of points whence three mutually
tangent planes may be drawn to the surface. Let these be the
planes $(v)$, $(w)$, $(\omega)$, and let the equation of the surface and
%[File: 160.png]
the Absolute in plane coordinates be, in the Clebsch-Aronhold\index{Clebsch}\index{Aronhold}
notation
\[
{u_\gamma}^2 = 0, \quad {u_\alpha}^2 \equiv u_\alpha{}'^2 = 0,
\]
\[
{v_\gamma}^2 = {w_\gamma}^2 = {\omega_\gamma}^2 = 0,
\]
\[
v_\alpha w_\alpha = w_\alpha \omega_\alpha
= \omega_\alpha v_\alpha = 0,
\]
\[
\begin{vmatrix}
  v_\gamma  & w_\gamma  & \omega_\gamma \\
  v_\alpha  & w_\alpha  & \omega_\alpha \\
  v_\alpha{}' & w_\alpha{}' & \omega_\alpha{}'
\end{vmatrix}^2
= \lvert \gamma\alpha\alpha' x \rvert^2 = 0,
\]
where $(x)$ is the point of concurrence of the planes $(v)$, $(w)$, $(\omega)$.
Returning to actual coefficients, the coefficients of $x_i x_j$ will
vanish, for they involve $\gamma_i \gamma_j$ or $\alpha_k \alpha_l$ which are zero. We
shall find eventually
\begin{equation*}
\tag{4}
\sum_h^{0..3} c_h (c_k c_l + c_l c_m + c_m c_k) x_h{}^2 = 0.
\end{equation*}

This quadric is also the locus of points whence triads of
tangents to the Absolute are conjugate with regard to the
given quadric, hence interchanging $\gamma$ and $\alpha$, we get the locus
of points whence triads of mutually perpendicular tangents
may be drawn to the quadric (1)
\begin{equation*}
\tag{5}
\sum_h^{0..3} c_h (c_k + c_l + c_m) x_h{}^2 = 0.
\end{equation*}

If the quadric be ruled, the former of these loci will intersect
it along a curve where generators of different sets intersect
at right angles.

\medskip\twincol{\emph{Theorem}~3. A line will meet
a central quadric and its focal
cones\index{Focal01@Focal cones} in five pairs of points
with the same centres of
gravity\index{Centre03@--- of gravity of points}.}
{\emph{Theorem}~3$'$. The tangent
planes to a central quadric
and its focal conics\index{Focal02@--- conics} through
a line form five sets of dihedral
angles with the same bisectors\index{Bisector of an angle}.}\par\medskip
%end twincol

The proof of these two theorems is immediate.

\mysect{Conjugate and mutually perpendicular lines through a centre}
If we mean by a diameter of a quadric\index{Diameters02@--- of quadric}, a line through
a centre, we see that we may pass from any set of three
concurrent conjugate diameters to any other such set through
that same centre by changing two diameters at a time, and
keeping the third one fixed. We may thus continually apply
Theorem 14, of Chapter XII\@. In the same way we may pass
from any set of three mutually perpendicular diameters to
%[File: 161.png]
any other such set, and apply Theorem 15 of the same
chapter.

\medskip\twincol{\emph{Theorem}~4. The sum of the
squares of the tangents of the
$k$th parts of the distances
from a centre of a central
quadric to three intersections
of the surface with three conjugate diameters\index{Diameters02@--- of quadric} through that
centre is constant.}
{\emph{Theorem}~4$'$. The sum of the
squares\index{Sum05@Sum of distances connected with a quadric} of the tangents of the
angles which an axial plane
of a central quadric makes
with three tangent planes
through three conjugate lines
in that axial plane is constant.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~5. The sum of the
squares of the cotangents of
the $k$th parts of the distances
from a centre of a central
quadric to three intersections
with the surface of three
mutually perpendicular lines
through that centre is constant.}
{\emph{Theorem}~5$'$. The sum of the
squares of the cotangents of
the angles which an axial
plane of a central quadric
makes with three tangent
planes through three mutually
perpendicular lines in that
axial plane is constant.}\par\medskip
%end twincol

To find the values of the constants referred to in Theorems 4
and 5, we have but to choose a particular set of diameters,
say the intersections of the axial planes through $O_h$. We
thus get
\[
  \tan^2 \frac{\overline{O_hP}}{k}
+ \tan^2 \frac{\overline{O_hP'}}{k}
+ \tan^2 \frac{\overline{O_hP''}}{k}
= -c_h\left( \frac{1}{c_k} + \frac{1}{c_l} + \frac{1}{c_m} \right). \tag{6}
\]
\[
  \ctn^2 \frac{\overline{O_hQ}}{k}
+ \ctn^2 \frac{\overline{O_hQ'}}{k}
+ \ctn^2 \frac{\overline{O_hQ''}}{k}
= -\frac{(c_k + c_l + c_m)}{c_h}.
\tag{7}
\]
\index{Central quadric|)}

A set of quadrics having the same absolute focal curve, and,
hence, the same focal cones, shall be called \emph{homothetic}\index{Homothetic02@--- quadrics}. A set
inscribed in the same absolute developable, and possessing,
in consequence the same focal conics shall be called \emph{confocal}\index{Confocal quadrics}.

\medskip

\mysect{Confocal and homothetic quadrics}
\twincol{\emph{Theorem}~6. An arbitrary
line will meet a set of confocal quadrics in pairs of
points with the same centres
of gravity.}
{\emph{Theorem}~6$'$. The tangent
planes to a set of confocal
quadrics through an arbitrary
line, form dihedral angles with
the same bisectors.}\par\medskip
%end twincol

\twincol{\emph{Theorem}~7. Three homothetic quadrics will touch an
arbitrary plane in three
mutually orthogonal points.}
{\emph{Theorem}~7$'$. Three confocal
quadrics will pass through an
arbitrary point, and intersect
orthogonally.}\par\medskip
%end twincol

%[File: 162.png]
\mysect{Elliptic coordinates, various forms of the distance element}
Let us now set up our system of elliptic coordinates as we
did in the plane
\begin{equation*}
\tag{8}
X_i=\frac{x_i}{\sqrt{(xx)}},\quad (XX) = 1.
\end{equation*}

These coordinates $(X)$ are inapplicable to points of the
Absolute\index{Absolute}; we imagine that all such points are excluded from
consideration. The general equation for the system of quadrics
confocal with that given by (1) will be,%
\footnote{The residue of the present chapter is closely analogous to the treatment
of the corresponding euclidean problem given by Klein\index{Klein} in his `Einleitung in
die h\"ohere Geometrie', lithographed notes, G\"ottingen, 1893, pp.~38--73, and
Staude\index{Staude}, `Fadenconstruktion des Ellipsoids,' \emph{Mathematische Annalen}, vol.~xx,
1882. Staude returns to the subject in his \emph{Die Fokaleigenschaften der Fl\"achen
zweiter Ordnung}, Leipzig, 1896. This book is intended as a supplement to the
usual textbooks on analytic geometry, and is somewhat prolix in its attempts
at simplicity.}
%endfootnote
if we replace $c_i$ by $\dfrac{1}{c_i}$,
\begin{equation*}
\tag{9}
\sum_i^{0..3}\frac{X_i{}^2}{c_i-\lambda}=0.
\end{equation*}
If the roots be $\lambda_1$, $\lambda_2$, $\lambda_3$, we have
\begin{equation*}
\tag{10}
X_h = \sqrt{\frac{(c_h-\lambda_1)(c_h-\lambda_2)(c_h-\lambda_3)}
  {(c_h-c_k)(c_h-c_l)(c_h-c_m)}}.
\end{equation*}
For the differential of distance we have
\begin{equation*}
\tag{11}
\frac{ds^2}{k^2} =
  \frac{(xx)(dx\,dx)-(x\,dx)^2}{(xx)^2} = (dX\,dX).
\end{equation*}
We wish to express this in terms of our elliptic coordinates\index{Elliptic01@Elliptic co-ordinates}.
It will be found that the coefficients of $d\lambda_p\; d\lambda_q$ will vanish,
and, indeed, this is \emph{a priori} evident if we have in mind that
our coordinate system is a triply orthogonal one, and the
general formulae for orthogonal curves, as will be shown in
Chapter XV, are the same for euclidean as for non-euclidean
We thus get
\[
\frac{ds^2}{k^2} =
  \frac{1}{4}\sum_h^{0..3}\sum_p^{1..3}
  \frac{(c_h-\lambda_q)(c_h-\lambda_r)d\lambda_p{}^2}
  {(c_h-c_k)(c_h-c_l)(c_h-c_m)(c_h-\lambda_p)}.
\]
If we give to $c_h$ each of its four values, divide the terms into
partial fractions and recombine, we get
\begin{equation*}
\tag{12}
\frac{ds^2}{k^2} =
  \frac{1}{4}\sum_p^{1..3}
  \frac{(\lambda_p-\lambda_q)(\lambda_p-\lambda_r)d\lambda_p{}^2}
  {\prod\limits_i^{0..3}(c_i-\lambda_p)}.
\end{equation*}
The analogy to the corresponding formula in euclidean space
is striking.

%[File: 163.png]
The cones whose vertices are all at an arbitrary point, and
which are circumscribed to a set of confocal quadrics, will
themselves be confocal, i.e.\ they will have four common
tangent planes which touch the Absolute\index{Absolute}. Any two of these
cones will intersect orthogonally. This shows that the congruence\index{Congruence09@--- of normals}
of lines tangent to two confocal quadrics will be
a normal\index{Normals02@--- to surface} one, the edges of regression of their developable
surfaces being geodesics of the quadrics. These facts, well
known in the euclidean case, will be proved for the noneuclidean
one in Chapter XVI\@. Notice that we get the
system of geodesics of a quadric by means of its $\infty^3$ common
tangents with confocal quadrics. The difficulties which arise
for special positions, as umbilical points\index{Umbilical points}, need not concern
us here.

The equation of the cone whose vertex is $(Y)$ and which
circumscribes the quadric (1) will be

\begin{equation*}
\sum_i^{0..3} \frac{Y_i{}^2}{c_i-\lambda} \sum_i^{0..3} \frac{X_i{}^2}{c_i-\lambda} - \Bigl[\sum_i^{0..3} \frac{Y_iX_i}{c_i-\lambda}\Bigr]^2 =0.
\end{equation*}
\begin{equation*}
\sum_i^{0..3} \sum_j^{0..3} \frac{(X_iY_j-X_jY_i)^2}{(c_i-\lambda)(c_j-\lambda)}=0.
\end{equation*}

Putting $X = Y+dY$ we get the differential form

\begin{equation*}
\sum_i^{0..3} \sum_j^{0..3} \frac{(Y_idY_j-Y_jdY_i)^2}{(c_i-\lambda)(c_j-\lambda)}=0.
\end{equation*}

Let us change this also to the elliptic form. We notice
that the coefficients of the expressions $d\lambda_p$ $d\lambda_q$ will be 0,
for the axial planes of the cones will be given by tangents to

\begin{equation*}
\lambda_p=0,\quad \lambda_q=0,\quad \lambda_r=0.
\end{equation*}

The $\infty^1$ confocal cones form a one-parameter family all
touching the same tangent planes to the cone $ds^2=0$. The
%[File: 164.png]
equation of one cone of the family may be thrown into
the form
\[
\sum_p^{1..3}
  \frac{(\lambda_p-\lambda_q)(\lambda_p-\lambda_r)}
  {\Bigl[\prod\limits_i^{0..3}(c_i-\lambda_p)\Bigr]L_p}
  d\lambda_p{}^2=0,
\]
where $L_p$ is a function of $\lambda$. Hence the general form will be
\[
\sum_p^{1..3}
  \frac{(\lambda_p-\lambda_q)(\lambda_p-\lambda_r)}
  {\Bigl[\prod\limits_i^{0..3}(c_i-\lambda_p)\Bigr](L_p-\mu)}
  d\lambda_p{}^2=0.
\]
It remains to find the value of $L_p-\mu$. It is clearly a polynomial
in powers of $\lambda$, which vanishes only when
$\lambda = \lambda_p$,
for then only shall we have $d\lambda_p{}^2 = 0$. We thus get
\[
L_p-\mu=A_p(\lambda_p-\lambda)^m,
\]
where $A_p$ is a constant. Again, as two of these confocal
quadrics contain every line through the vertex, we must
have $m = 1$. Lastly, our expression is symmetrical in $p$, $q$, $r$,
hence
\[
A_p=A_q=A_r.
\]
We finally get for our cone
\begin{equation*}
\tag{13}
\sum_p^{1..3}
  \frac{(\lambda_p-\lambda_q)(\lambda_p-\lambda_r)d\lambda_p{}^2}
       {\Bigl[\prod\limits_i^{0..3}
        (c_i-\lambda_p)\Bigr](\lambda_p-\lambda)}=0.
\end{equation*}
For progress along an arc of a geodesic\index{Geodesic02@--- lines} of
$\lambda_r = $ const., we
have
\[
d\lambda_p\sqrt{\frac{\lambda_p-\lambda_r}
  {(\lambda_p-\lambda)\prod\limits_i^{0..3}(c_i-\lambda_p)}}
  \pm d\lambda_q\sqrt{\frac{\lambda_q-\lambda_r}
  {(\lambda_q-\lambda)\prod\limits_i^{0..3}(c_i-\lambda_q)}} = 0.
\]
so that the problem of finding the geodesics of a quadric
depends merely upon elliptic integrals. If we take
$\lambda_r = \lambda$,
%[File: 165.png]
we have double tangents to the surface, i.e.\ rectilinear
generators,
\[
    \frac{d\lambda_p}{\sqrt{\prod\limits_i^{0..3}(c_i-\lambda_p)}}
\pm \frac{d\lambda_q}{\sqrt{\prod\limits_i^{0..3}(c_i-\lambda_q)}} = 0.
\]
The general differential of arc on a surface
$\lambda_r = \text{\ const.}$ is
\[
\frac{ds^2}{k^2}
= \frac{1}{4}
  \left[ \frac{(\lambda_p-\lambda_q)(\lambda_p-\lambda_r)}
              {\prod\limits_i^{0..3}(c_i-\lambda_p) } d\lambda_p{}^2
       + \frac{(\lambda_q-\lambda_p)(\lambda_q-\lambda_r)}
              {\prod\limits_i^{0..3}(c_i-\lambda_q) } d\lambda_q{}^2
 \right],
\]
we have, then, for a distance along a generator
\[
  s = \frac{k}{2}
      \int\frac{(\lambda_p-\lambda_q) d\lambda_p}
               {\sqrt{\prod\limits_i^{0..3}(c_i-\lambda_p)}}.
\]
This expression is independent of $\lambda_r$, whence

\medskip\twincol{\emph{Theorem}~8. If from a set of
confocal central quadrics\index{Confocal quadrics} a one-parameter
set of linear generators be so chosen that all
intersect the same $\infty^1$ lines of
curvature of $\infty^1$ confocal quadrics of the system, then any
two of these lines of curvature
will cut congruent distances
on all of these linear
generators.}
{\emph{Theorem}~8$'$. If from a set of
homothetic central quadrics
a one-parameter set of linear
generators be so chosen that
all touch $\infty^1$ developables
circumscribed to pairs of
quadrics of the homothetic
system, then the tangent
planes to any two of these
developables will determine
congruent dihedral angles
whose edges are the given
linear generators.}\par\medskip
%end twincol

Theorem 8 may also be easily proved by showing that the
generators of a set of confocal quadrics form an isotropic
congruence, whereof much more later.%
\footnote{The general theorem concerning isotropic
congruences\index{Congruence11@--- isotropic}\index{Isotropic02@--- congruence} upon which this
depends will be proved in Chapter XVI, where also will be found a bibliography of the subject.}
%endfootnote

%[File: 166.png]
We now seek for the expression for the element of distance
upon a common tangent to two confocal quadrics $\lambda$, $\lambda'$.
\[
  \sum_p^{1..3}
  \frac{(\lambda_q-\lambda_p) (\lambda_r-\lambda_p)d\lambda_p{}^2}
       {\Bigl[ \prod\limits_i^{0..3} (c_i-\lambda_p) \Bigr]
        (\lambda-\lambda_p)}
= 0,
\]
\[
  \sum_p^{1..3}
  \frac{(\lambda_q-\lambda_p) (\lambda_r-\lambda_p)d\lambda_p{}^2}
       {\Bigl[ \prod\limits_i^{0..3} (c_i-\lambda_p) \Bigr]
        [\lambda'-\lambda_p]}
= 0.
\]
\[
    \frac{(\lambda_p - \lambda_q) d\lambda_q                      }
         {\sqrt{\Bigl[ \prod\limits_i^{0..3} (c_i-\lambda_q) \Bigr]
                 (\lambda-\lambda_q) (\lambda'-\lambda_q)        } }
\\
=\pm\frac{(\lambda_p - \lambda_r) d\lambda_r                      }
         {\sqrt{\Bigl[ \prod\limits_i^{0..3} (c_i-\lambda_r) \Bigr]
                 (\lambda-\lambda_r) (\lambda'-\lambda_r)        } }.
\tag{14}
\]
\[
    \frac{(\lambda_q - \lambda_p) d\lambda_p                      }
         {\sqrt{\Bigl[ \prod\limits_i^{0..3} (c_i-\lambda_p) \Bigr]
                 (\lambda-\lambda_p) (\lambda'-\lambda_p)        } }
\\
=\pm\frac{(\lambda_q - \lambda_r) d\lambda_r                      }
         {\sqrt{\Bigl[ \prod\limits_i^{0..3} (c_i-\lambda_r) \Bigr]
                 (\lambda-\lambda_r) (\lambda'-\lambda_r)        } }.
\]
\[
    \frac{ds}{k}
=\pm\frac{d\lambda_p  }
         {2\sqrt{\Bigl[ \prod\limits_i^{0..3} (c_i-\lambda_p) \Bigr]} }
 \times\frac{\sqrt{(\lambda_q - \lambda_p) (\lambda_r - \lambda_p)}}
            {\sqrt{(\lambda   - \lambda_p) (\lambda'  - \lambda_p)}}
 \times \\
 \frac{\sqrt{\begin{vmatrix}
                \lambda_p{}^2 & \lambda_q{}^2 & \lambda_r{}^2 \\
                \lambda_p   & \lambda_q   & \lambda_r   \\
                1           & 1           & 1
              \end{vmatrix}}                 }
            {\sqrt{\lambda_q - \lambda_r } }
\]
%[File: 167.png]
\[
(\lambda_q-\lambda_r)\frac{ds}{k}=\frac{d\lambda_p}{2\sqrt{\prod\limits_i^{0..3} (c_i-\lambda_p)}}
\times \frac{(\lambda_p-\lambda_q)(\lambda_q-\lambda_r)(\lambda_r-\lambda_p)}
{\sqrt{(\lambda-\lambda_p)(\lambda'-\lambda_p)}}.
\]

Multiplying through by $(\lambda-\lambda_p)$, $(\lambda'-\lambda_p)$, and summing for $p=1,\; 2,\; 3$

\begin{align*}
\frac{ds}{k}&=\frac{1}{2} \sqrt{\frac{(\lambda-\lambda_1)(\lambda'-\lambda_1)}
{\prod\limits_i^{0..3} (c_i-\lambda_1)}}d\lambda_1\\
&\hspace{0.5in}+\frac{1}{2} \sqrt{\frac{(\lambda-\lambda_2)(\lambda'-\lambda_2)}{\prod\limits_i^{0..3} (c_i-\lambda_2)}}d\lambda_2\\
&\hspace{1in}+\frac{1}{2} \sqrt{\frac{(\lambda-\lambda_3)(\lambda'-\lambda_3)}{\prod\limits_i^{0..3} (c_i-\lambda_3)}}d\lambda_3. \tag{15}
\end{align*}


For a geodesic on $\lambda=\lambda_1$ whose tangent touches $\lambda'$ we have

\[
\frac{ds}{k}=\frac{1}{2} \sqrt{\frac{(\lambda-\lambda_2)(\lambda'-\lambda_2)}
{\prod\limits_i^{0..3} (c_i-\lambda_2)}}d\lambda_2
+\frac{1}{2} \sqrt{\frac{(\lambda-\lambda_3)(\lambda'-\lambda_3)}{\prod\limits_i^{0..3} (c_i-\lambda_3)}}d\lambda_3.
\tag{16}
\]

For a line of curvature common to $\lambda=\lambda_1$, $\lambda'=\lambda_2$

\begin{equation*}
\frac{ds}{k}=\frac{1}{2}
\sqrt{\frac{(\lambda_1-\lambda_3)(\lambda_2-\lambda_3)}{\prod\limits_i^{0..3} (c_i-\lambda_3)}}d\lambda_3. \tag{17}
\end{equation*}

\mysect{String construction for the ellipsoid}
It is now necessary to look more closely into the signs
of the radicals in (15). We know that, at least in a restricted
domain, three confocal quadrics will pass through each point.
In elliptic space one of these will be ruled, and the other two
not ruled; assuming, of course, that we are dealing with the
case of central quadrics. In hyperbolic space, two possible
cases can arise in the actual domain. If the developable be
%[File: 168.png]
real, two ruled, and one non-ruled hyperboloid will pass
through each point. If it be imaginary we shall have an
ellipsoid, a ruled, and a not-ruled hyperboloid.%
\footnote{See the Author's\index{Author} `Quadric Surfaces', p.~165.}
%endfootnote
Let us
confine ourselves to this case, taking $\lambda_3$ as the parameter of
the non-ruled hyperboloid, $\lambda_2$ as that of the ruled one, while $\lambda_1$
gives the ellipsoid\index{Ellipsoid}. The elliptic case will follow immediately
if we suppress the word \emph{hyperboloid}\index{Hyperboloid} substituting \emph{ellipsoid}.
In (15) let us assume that $\lambda$ refers to an ellipsoid, and $\lambda'$
to a ruled hyperboloid. In two of the three actual axial
planes we shall have real focal conics\index{Focal02@--- conics}. There will be a real
focal ellipse\index{Ellipse} which, looked upon as an envelope, constitutes
the transition between the ellipsoid and the ruled hyperboloid.
It will be surrounded by all ellipsoids, and surround all ruled
hyperboloids. If we take a point in this axial plane, without
the focal ellipse, the ellipsoid and non-ruled hyperboloid will
subsist, the ruled hyperboloid, looked upon as a point locus,
will shrink into the plane counted doubly. The other real
focal conic will be a hyperbola, and will serve as a transition
between the two sorts of hyperboloids, looked upon as
envelopes. It will surround the non-ruled hyperboloids, but
be surrounded by the ruled ones. The plane counted doubly,
will replace a non-ruled hyperboloid for each point without
the hyperbola\index{Hyperbola}. If a point be taken in the remaining axial
plane, this plane, counted doubly, will replace a non-ruled
hyperboloid for each of its points. Similar considerations
will hold in the elliptic case.

Once more, let us look at the signs of the terms in (15).
$d\lambda_i$ will change sign as a point passes through an axial plane
that counts doubly in the $\lambda_i$ family, or when passing along
a tangent to one of these surfaces, the point of contact is
traversed. On the other hand we see from (14) that when
$d\lambda_i$ changes sign, the radical associated with it in (15) changes
sign also, and vice versa. The radical associated with $d\lambda_3$
will change sign as we pass through a point of the axial plane
with an imaginary focal conic (which we shall call $\pi_3$), and
for a point of the axial plane $\pi_2$ of the focal hyperbola, which
is without this hyperbola. The radical with $d\lambda_2$ will change
sign for points of $\pi_1$, the plane of the focal ellipse without this
curve, or points of $\pi_2$ within the focal hyperbola. The radical
with $d\lambda_1$ will change sign for points of $\pi_1$ within the focal
ellipse.

We next suppose that a loop of inextensible thread is slung
about an ellipsoid $\lambda$, and a confocal, ruled, one-sheeted
hyperboloid $\lambda'$, and pulled taut at a point $P$. The loop is supposed
%[File: 169.png]
to surround the ellipsoid\index{Ellipsoid}, so that it winds partly on each of
the portions of the hyperboloid, which, in a restricted domain,
are separated by the ellipsoid. The form for the element of
length throughout the whole string will be that given by (15).
For when we pass from the ellipsoid to the hyperboloid\index{Hyperboloid} we
pass along a geodesic whose tangent touches both surfaces,
and this will be true throughout the continuation of that
geodesic, for a geodesic is traced by a line rolling on a quadric,
and touching a confocal one. The same form of distance
element will hold for the rectilinear parts of the loop. We
see, moreover, that two, and only two surfaces, of a confocal
system will touch any line; hence $\lambda$ and $\lambda'$ are the only two
which will touch the rectilinear parts of the loop. Lastly,
let us limit ourselves to those regions of the plane where the
various portions of the loop may be named in order: straight,
hyperboloidal, ellipsoidal, hyperboloidal, ellipsoidal, straight.
The constant length of the thread may be written
\[
C=\int_{\lambda_1}^{\lambda_1}F_1d\lambda_1 +
  \int_{\lambda_2}^{\lambda_2}F_2d\lambda_2 +
  \int_{\lambda_3}^{\lambda_3}F_3d\lambda_3.
\]

We see that $F_3$ can never vanish, for $\lambda$ and $\lambda'$ are the parameters
of an ellipsoid and ruled hyperboloid respectively,
while $\lambda_3$ refers to a non-ruled hyperboloid. It will become
infinite four times, twice when the loop passes $\pi_2$ the plane
of the focal\index{Focal02@--- conics} hyperbola\index{Hyperbola}, and twice when it passes $\pi_3$. We may,
however, integrate right up to these limits, and, as we have
seen, $d\lambda_3$ changes sign with the radical. We thus have
\begin{align*}
  \int_{\lambda_3}^{\lambda_3}F_3d\lambda_3 &=
  \int_{\lambda_3}^{c_3}F_3d\lambda_3 -
  \int_{c_3}^{c_2}F_3d\lambda_3 +
  \int_{c_2}^{c_3}F_3d\lambda_3 -
  \int_{c_3}^{c_2}F_3d\lambda_3 +
  \int_{c_2}^{\lambda_3}F_3d\lambda_3 \\
  &=4 \int_{c_2}^{c_3}F_3d\lambda_3 =\text{const}.
\end{align*}

We may approach the second integral in the same spirit.
$F_2$ will become infinite twice when the loop passes the plane
of the focal ellipse\index{Ellipse} $\pi_1$. It will vanish throughout those two
portions of the loop that lie on the ruled hyperboloid $\lambda_2 = \lambda'$,
and these two are separated by an intersection with $\pi_1$ We
have then
\begin{align*}
\int_{\lambda_2}^{\lambda_2}F_2d\lambda_2 &=
  \int_{\lambda_2}^{\lambda'}F_2d\lambda_2 -
  \int_{\lambda'}^{c_1}F_2d\lambda_2 +
  \int_{c_1}^{\lambda'}F_2d\lambda_2 -
  \int_{\lambda'}^{c_1}F_2d\lambda_2 +
  \int_{c_1}^{\lambda_2}F_2d\lambda_2 \\
 & =4 \int_{c_1}^{\lambda'}F_2d\lambda_2 =\text{const}.
\end{align*}

%[File: 170.png]
We must, in conclusion, consider the first integral. It will
never become infinite, but will vanish along those two portions
of the loop which lie on the ellipsoid\index{Ellipsoid} $\lambda = \lambda_1$. We have
therefore:
\[
  \int_{\lambda_1}^{\lambda_1} F_1 d\lambda_1
= \int_{\lambda_1}^{\lambda  } F_1 d\lambda_1
- \int_{\lambda  }^{\lambda_1} F_1 d\lambda_1
=2\int_{\lambda_1}^{\lambda  } F_1 d\lambda_1 = \phi(\lambda_1).
\]
We have therefore, since the first two integrals and the
sum are constant,
\[
  \phi(\lambda_1) = \text{\ const.,}
\]
and the locus of the moving point is an ellipsoid. Lastly, let
the ellipsoid and hyperboloid\index{Hyperboloid} shrink down to the focal\index{Focal02@--- conics} ellipse
and focal hyperbola\index{Hyperbola} respectively, we have in the limiting case:

\smallskip\emph{Theorem}~9. If an ellipse and hyperbola in mutually
perpendicular planes pass each through two foci of the other, and
if a loop of inextensible thread\index{Thread construction} be slung around the ellipse
and pulled taut at a point $P$ in such a way that it meets
the two curves alternately, then the locus of $P$ will be an
ellipsoid confocal with the given ellipse and hyperbola.

%[File: 171.png]
%CHAPTER XIV
\mychap{XIV}{AREAS AND VOLUMES}
\mysect{Amplitude of a triangle}

The subjects \emph{area}\index{Area01@Area} and \emph{volume} offer some of the most
striking points of disparity between euclidean and non-euclidean
geometry.%
\footnote{For a bibliographical account of the subject-matter of the present chapter
see the dissertation of Dannmeyer\index{Dannmeyer}, \emph{Die Oberfl\"achen- und Volumenberechnung f\"ur
Lobatschefskijsche R\"aume}, G\"ottingen, 1904.}
%endfootnote
A first notable difference arises from
the fact that, in the non-euclidean cases, two different functions
of a triangle\index{Triangle} appear to play the r\^ole of the euclidean
area. The first is present in the analoga of those formulae
which give the area in terms of the sides and angles; the
second appears when the area is defined as the limit of a
sum, i.e.\ as a definite integral. We shall reserve the name
\emph{area} for the second of these, giving to the first the name
\emph{amplitude}\index{Amplitude of triangle}.%
\footnote{The concept \emph{amplitude} of a triangle, and the various trigonometric identities
connected with it, are taken directly from an admirable paper by
D'Ovidio\index{Ovidio@d'Ovidio}, `Su varie questioni di metrica proiettiva,' \emph{Atti della R. Accademia
delle Scienze di Torino}, vol.~xxviii, 1893. Unfortunately the author gives, p.~20,
an incorrect formula for the volume of a tetrahedron.}
%endfootnote

Let us, as in elementary geometry, use the letters $A$, $B$, $C$
to indicate, either the vertices of a triangle, or the measures
of its angles. We assume that these points are real, and,
in the hyperbolic case, situated in the actual domain. We
shall define triangle as in Chapter II\@. We might carry
through the same sort of work for any three points, but,
as we saw in the closing pages of Chapter VII, we should
thereby be compelled, in the hyperbolic case at least, to
introduce certain very delicate considerations as to algebraic;
sign, not only in our analytic expressions, but even in the
trigonometric formulae first introduced in Chapter IV.

We begin by rewriting IV\@. 9
\[
 -\sin \frac{b}{k} \sin\frac{c}{k} \cos A =
  \cos \frac{b}{k} \cos\frac{c}{k}-\cos\frac{a}{k}.
\]

This formula, established for one region, is seen at once to
hold for all the others.
%[File: 172.png]
\begin{align*}
\sin \frac{b}{k} \sin\frac{c}{k} &\sin A \\
& = \Bigl[ \sin^2\frac{b}{k} \sin^2\frac{c}{k} -
           \cos^2\frac{b}{k} \cos^2\frac{c}{k} +
          2\cos\frac{a}{k} \cos\frac{b}{k} \cos\frac{c}{k} -
           \cos^2\frac{a}{k} \Bigr]^{\frac{1}{2}} \\
& = \Bigl[ 1 - \cos^2\frac{a}{k} - \cos^2\frac{b}{k} -
           \cos^2\frac{c}{k} +
          2\cos\frac{a}{k} \cos\frac{b}{k} \cos\frac{c}{k}
    \Bigr]^{\frac{1}{2}}.
\end{align*}

The right-hand side is symmetrical in the three letters $a$, $b$, $c$,
so that we may write
\begin{multline*}
  \sin \frac{b}{k} \sin \frac{c}{k} \sin A =
  \sin \frac{c}{k} \sin \frac{a}{k} \sin B =
  \sin \frac{a}{k} \sin \frac{b}{k} \sin C
\\
\tag{1}
= \begin{vmatrix}
    1               & \cos\dfrac{c}{k} & \cos\dfrac{b}{k} \\
    \mystrut{18}\cos\dfrac{c}{k} & 1               & \cos\dfrac{a}{k} \\
    \mystrut{18}\cos\dfrac{b}{k} & \cos\dfrac{a}{k} & 1
  \end{vmatrix}^{\frac{1}{2}}.
\end{multline*}

In the real domain, if the measures of sides and angles be
taken positively, the left side is essentially negative in the
hyperbolic case, and positive in the elliptic, so that the
radical on the right must be chosen accordingly. It will
vanish only when the three points are collinear (under the
restrictions made at the outset of this chapter), and shall be
called the \emph{Sine Amplitude}\index{Amplitude of triangle} of the triangle, written
$\sin (ABC)$.

Let the reader show that if the coordinates of $A$, $B$, $C$ be
$(x)$, $(y)$, $(z)$ respectively
\begin{equation*}
\tag{2}
\sin (ABC) =
  \frac{\begin{vmatrix}
  (xx) & (xy) & (xz) \\
  (yx) & (yy) & (yz) \\
  (zx) & (zy) & (zz) \end{vmatrix}^{\frac{1}{2}}}
  {\sqrt{(xx)}\;\sqrt{(yy)}\;\sqrt{(zz)}} =
  \frac{\lvert x y z \rvert}
  {\sqrt{(xx)}\;\sqrt{(yy)}\;\sqrt{(zz)}}.
\end{equation*}

\mysect{Relation to other parts}
We may rewrite (1) in the form
\begin{equation*}
\tag{3}
  \frac{\sin A}{\mystrut{13.5}\sin\dfrac{a}{k}} =
  \frac{\sin B}{\sin\dfrac{b}{k}} =
  \frac{\sin C}{\mystrut{13.5}\sin\dfrac{c}{k}} \equiv
  \frac{\sin(ABC)}{\sin\dfrac{a}{k}\sin\dfrac{b}{k}\sin\dfrac{c}{k}}.
\end{equation*}

If $A'$, $B'$, $C'$ be the points where the sides of the triangle
meet the perpendiculars from the vertices, we have
\begin{equation*}
\tag{4}
  \sin\frac{a}{k} \sin\frac{\overline{AA'}}{k} =
  \sin\frac{b}{k} \sin\frac{\overline{BB'}}{k} =
  \sin\frac{c}{k} \sin\frac{\overline{CC'}}{k} =
  \sin(ABC).
\end{equation*}
%[File: 173.png]
We see at once the close analogy of the sine amplitude\index{Amplitude of triangle} of a
non-euclidean triangle\index{Triangle} to double area of a euclidean triangle.
Let the reader show that
\[
  \Limdot\frac{1}{k^2} = 0,\quad k^2
  \sin (ABC) = 2\text{\ Area }\triangle ABC.
\]

A function correlative to the sine amplitude may be
obtained from the correlative formula
\begin{align*}
  \sin B \sin C \cos \frac{a}{k} = \cos B &\cos C + \cos A.
\\
  \sin B \sin C \sin \frac{a}{k}
= \sin C \sin A \sin \frac{b}{k}
&= \sin A \sin B \sin \frac{c}{k}
\\
\hfill &= \begin{vmatrix}   1   & \cos C & \cos B  \\
                         \cos C &    1   & \cos A  \\
                         \cos B & \cos A &    1
          \end{vmatrix}^{\frac{1}{2}}
\\
       &= \sin (abc).                 \tag{5}
\end{align*}
This $> 0$ in the elliptic case, pure imaginary in the hyperbolic
\[
  \frac{\sin\dfrac{a}{k} }{\sin A}
= \frac{\sin\dfrac{b}{k} }{\sin B}
= \frac{\sin\dfrac{c}{k} }{\sin C}
= \frac{\sin(abc) }{\sin A\sin B\sin C}.   \tag{6}
\]
\[
\sin A \sin\frac{\overline{AA'}}{k}
= \sin B \sin\frac{\overline{BB'}}{k}
= \sin C \sin\frac{\overline{CC'}}{k} = \sin(abc).   \tag{7}
\]
\[
  \frac{\sin\dfrac{a}{k}}{\sin A}
= \frac{\sin\dfrac{b}{k}}{\sin B}
= \frac{\sin\dfrac{c}{k}}{\sin C}
= \frac{\sin(ABC)}{\sin(abc)}.   \tag{8}
\]
\[
  \sin(abc)
= \frac{\sin^2(ABC)}
       {\sin\dfrac{a}{k} \sin\dfrac{b}{k} \sin\dfrac{c}{k}}, \quad
  \sin(ABC)
= \frac{\sin^2(abc)}
       {\sin A \sin B \sin C}.   \tag{9}
\]


\begin{flalign*}
&\text{\indent If}  &&a + b + c = 2s, &\phantom{\text{\indent If}}
\end{flalign*}
\begin{align*}
  \cos A &= \frac{\cos\dfrac{a}{k} - \cos\dfrac{b}{k} \cos\dfrac{c}{k} }
                {\sin\dfrac{b}{k} \sin\dfrac{c}{k}                   },
\\
  \sin\tfrac{1}{2}A
&= \left[ \frac{\sin\dfrac{s-b}{k} \sin\dfrac{s-c}{k} }
              {\sin\dfrac{b  }{k} \sin\dfrac{c  }{k} }
  \right]^{\frac{1}{2}},
\\
%[File: 174.png]
\cos \tfrac{1}{2} A &=
\left[ \frac{\sin \dfrac{s}{k} \sin \dfrac{s-a}{k}}
{\sin \dfrac{b}{k} \sin \dfrac{c}{k}} \right]^\frac{1}{2}, \\
\ctn \tfrac{1}{2} A &=
\left[ \frac{\sin \dfrac{s}{k} \sin \dfrac{s-a}{k}}
{\sin \dfrac{s-b}{k} \sin \dfrac{s-c}{k}} \right]^\frac{1}{2},
\end{align*}
\[
\sin (ABC)=2 \sqrt{\sin \frac{s}{k} \sin \frac{s-a}{k} \sin \frac{s-b}{k} \sin \frac{s-c}{k}}. \tag{10}
\]

In like manner, let us put\index{Amplitude of triangle}
\begin{align*}
&A+B+C = 2\sigma. \\
\sin \tfrac{1}{2} \frac{a}{k} &= \left[
\frac{-\cos \sigma \cos (\sigma-A)}{\sin B \sin C}
\right]^\frac{1}{2}, \\
\cos \tfrac{1}{2} \frac{a}{k} &= \left[
\frac{\cos (\sigma-B) \cos (\sigma-C)}{\sin B \sin C}
\right]^\frac{1}{2}, \\
\ctn \tfrac{1}{2} \frac{a}{k} &= \left[
\frac{\cos (\sigma-B) \cos (\sigma-c)}{-\cos \sigma \cos (\sigma-a)}
\right]^\frac{1}{2},
\end{align*}
\[
\sin (abc)=2 \sqrt{-\cos \sigma \cos (\sigma-A) \cos (\sigma-B) \cos (\sigma-C)}.\tag{11}
\]
\begin{equation*}
\sin \tfrac{1}{2}A \sin \tfrac{1}{2}B \sin \tfrac{1}{2}C=
\frac{\Bigl(\sin \dfrac{s-a}{k} \sin \dfrac{s-b}{k} \sin \dfrac{s-c}{k}\Bigr)}
{\sin \dfrac{a}{k} \sin \dfrac{b}{k} \sin \dfrac{c}{k}},
\end{equation*}
\begin{align*}
\sin \frac{s}{k} &= \frac{\sin (abc)}{4 \sin \tfrac{1}{2}A \sin \tfrac{1}{2}B \sin \tfrac{1}{2}C}. \tag{12} \\
-\cos \sigma &= \frac{\sin (ABC)}{4 \cos \tfrac{1}{2} \dfrac{a}{k} \cos \tfrac{1}{2}
\dfrac{b}{k} \cos \tfrac{1}{2} \dfrac{c}{k}}. \tag{13}
\end{align*}

It should be noticed that the denominator on the right of
equation (13) is essentially positive. The numerator is
negative in the hyperbolic case, as we have already seen,
%[File: 175.png]
but here also $\sigma < \dfrac{\pi}{2}$ and $\cos \sigma > 0$.
In the elliptic case the
numerator is positive but $\sigma > \dfrac{\pi}{2}$, $\cos \sigma < 0$.

In Chapter III we defined as the \emph{discrepancy}\index{Discrepancy of a triangle}
of a triangle\index{Triangle},
the absolute value of the difference between the sum of the
measures of the angles and $\pi$. Let us now define as the
\emph{excess}\index{Excess of a triangle} of our triangle the expression
\[
  e = A + B + C - \pi.
\]

This will have the same sign as $\dfrac{1}{k^2}$ the measure of curvature
of space. We have
\[
  \sin\frac{e}{2} = -\cos\sigma
= \frac{\sin(ABC)}{4\cos\tfrac{1}{2}\dfrac{a}{k}
                    \cos\tfrac{1}{2}\dfrac{b}{k}
                    \cos\tfrac{1}{2}\dfrac{c}{k} }.   \tag{14}
\]

\mysect{Limiting form when the triangle is infinitesimal}
Passing to the limiting case where the triangle becomes
infinitesimal\index{Infinitesimal domain}, we have
\begin{align*}
\Limdot \frac{\sin(ABC)}{\sin\dfrac{e}{2}}
&= 4\limdot \left( \cos\tfrac{1}{2}\dfrac{a}{k}
                      \cos\tfrac{1}{2}\dfrac{b}{k}
                      \cos\tfrac{1}{2}\dfrac{c}{k} \right)
\\
&= 4
\\
  \limdot\, e &= \tfrac{1}{2}\limdot\, (ABC)
\\
&= \frac{1}{2k^2}\limdot bc \sin A
\\
&= \frac{1}{2k^2}\limdot\, a\, \overline{AA'}.
\end{align*}

\smallskip\emph{Theorem}~1. In an infinitesimal triangle the limit of the
ratio of the excess to the product of the euclidean area and
the measure of curvature of space is unity.

Let us next examine the infinitesimal quadrilateral\index{Quadrilateral01@Quadrilateral}, whose
vertices are $A$, $B$, $C$, $D$. $AB$ and $CD$ shall intersect in $H$ (actual
or ideal) while $AC$ and $BD$ intersect in $K$; the latter two
points remaining at a finite distance from $A$, $B$, $C$, $D$.
\[
  \frac{\sin\dfrac{\overline{AB}}{k}}
       {\sin\dfrac{\overline{BK}}{k}}
= \frac{\sin K}{\sin A},
\quad
  \frac{\sin\dfrac{\overline{CD}}{k}}
       {\sin\dfrac{\overline{DK}}{k}}
= \frac{\sin K}{\sin C},
\]
\[
  \limdot \frac{\sin A}{\sin C} = 1,\quad
  \limdot \frac{\overline{AB}}{\overline{CD}} = 1.
\tag{Ch.~III.~2.}
\]
%[File: 176.png]
\begin{eqnarray*}
    \limdot \frac{\sin(CAB)}{\sin(DAB)}
&=& \limdot \frac{\sin\dfrac{\overline{AB}}{k}\sin\dfrac{\overline{AC}}{k}\sin A}
             {\sin\dfrac{\overline{DB}}{k}\sin\dfrac{\overline{DC}}{k}\sin D} \\
&=& \limdot \frac{\overline{AB}\,.\,\overline{AC}\,.\,\sin A}
             {\overline{DB}\,.\,\overline{DC}\,.\,\sin D} \\
&=& 1.
\end{eqnarray*}

\mysect{Deficiency and area}
We shall define as the \emph{area}\index{Area01@Area}\index{Area05@--- of a triangle}
of an infinitesimal\index{Infinitesimal domain} triangle\index{Triangle}
the common value of $k^2$ times its excess\index{Excess of a triangle}, its half-amplitude,
and the euclidean expression for its area.

\smallskip\emph{Theorem}~2. If the opposite sides of an infinitesimal
quadrilateral do not intersect in points infinitesimally near
the vertices, the limit of the ratio of the areas of the triangles
into which it is divided by a diagonal is unity.

The sum of these two infinitesimal areas shall be called
the \emph{area} of the infinitesimal quadrilateral; it will be equal
(always neglecting infinitesimals of higher order) to the
product of two adjacent sides multiplied into the sine of the
included angle.

Suppose now that we have a region of the plane, connex
right up to the boundary, which is limited by one or more
closed curves, and let this be covered by a network of infinitesimal
quadrilaterals of the sort just described. Let the
area of each of these be multiplied by the value for a point
therein of a continuous function of the coordinates of the
point. The limit of this sum as the individual areas tend
uniformly toward zero shall be called the \emph{surface integral}\index{Surface integral} of
the given function for the given area. The proof of the
existence of such a limit, and its independence of network
employed will be identical with that used in the corresponding
euclidean case, and need not detain us here.%
\footnote{Conf.\ e.g. Picard\index{Picard}, \emph{Trait\'e d'Analyse}, first ed., Paris, 1891, vol.~i, pp.~83--102.}
%endfootnote

\smallskip\emph{Definition}.~When the surface integral of the function $1$
exists over a region of the plane, that integral shall be defined
as the \emph{area} of the region.

\smallskip\emph{Theorem}~3. The area of a region of a plane is the sum of
the areas of any two regions into which it may be divided
provided that these two have no common area.

This follows immediately from the definition given above.
As an application of these principles let us determine the
%[File: 177.png]
area of a triangle\index{Area05@--- of a triangle}\index{Triangle}.
It is the limit of the sum of the areas of
a network of infinitesimal triangles, or by (1) the limit
of the sum of $k^2$ times their excesses\index{Excess of a triangle}. Now it is perfectly
clear that if a triangle be divided in two by a segment whose
extremities are a vertex and a point of the opposite side, the
excess of the original triangle is the sum of the excesses of
the parts, and we may establish our network by a repetition
of this process of division\label{processor}, hence%
\footnote{It is surprising to see how unsatisfactory are the proofs usually given for
this, the best-known theorem of non-euclidean geometry. In Frischauf\index{Frischauf},
\emph{Elemente der absoluten Geometrie}, Leipzig, 1876, will be found a geometrical proof
applicable to the hyperbolic case but not, so far as I can see, to the elliptic,
and the same remark will apply to the book of Liebmann\index{Lindemann}, cit. Manning\index{Manning},
loc.\ cit., makes an attempt at a general proof, but the use of intuition is
scarcely disguised. In Clebsch-Lendemann\index{Clebsch}, \emph{Vorlesungen \"uber Geometrie}, Leipzig,
1891, vol.~ii, p.~49, is a proof by integration, but the analysis is unnecessarily
complicated owing to the fact that, apparently, the author overlooked the
consideration that it is sufficient to prove the theorem for a right triangle.}
%endfootnote

\smallskip\emph{Theorem}~4. The area of a triangle is the quotient of the
excess divided by the measure of curvature of space\index{Measure02@--- of curvature of
space}\index{K@$\dfrac{1}{k^2}$ measure of curvature of space}\index{Curvature06@--- of space}.

\mysect{Area found by integration}
Let us give a second demonstration of this fundamental
theorem with the aid of integration. It will be sufficient to
do so in the case of a right triangle, and we shall take a right
triangle with one angle at $C$ the intersection of $x_1 = 0$, $x_2 = 0$,
the right angle being at $B$ a point of the axis $x_2 = 0$. We
may introduce polar coordinates\index{Coordinates02@Co-ordinates of a point}
%
\begin{equation*}
\frac{x_1}{x_0} = k\tan \frac{r}{k}\cos\phi, \quad
\frac{x_2}{x_0} = k\tan \frac{r}{k}\sin\phi,
\end{equation*}
%
the elements of arc along $\phi = $ const.\  and $r = $ const.\  will be
$dr$ and $k\sin\dfrac{r}{k}d\phi$ respectively. The element of area will be
%
\[
df = k\sin\frac{r}{k}dr\,d\phi. \tag{15}
\]
\[
k \int^R_0 \sin\frac{r}{k}dr = k^2\left(1-\cos\frac{R}{k}\right),
\]
\[
\tan\frac{R}{k} = \tan\frac{\overline{BC}}{k}\sec\phi. \tag{Ch. IV\@. (6).}
\]
\[
\cos\frac{R}{k} = \frac{\cos\phi}{\sqrt{\cos^2\phi + \tan^2\dfrac{\overline{BC}}{k}}}.
\]

Remembering that the limits for $\phi$ are $0$ and $C$
%[File: 178.png]
\begin{equation*}
\text{Area} = k^2\int^C_0 d\phi
   - k^2\int^C_0 \frac{\cos\phi\,d\phi}{\sqrt{\cos^2\phi + \tan^2\dfrac{\overline{BC}}{k}}}.
\end{equation*}\index{Area05@--- of a triangle}\index{Triangle}
%
The first integral is $k^2C$. If, further, we put $\sin\phi = x$,
%
\begin{equation*}
  \int\frac{dx}{\sqrt{\sec^2\dfrac{\overline{BC}}{k} - x^2}}
= \sin^{-1}\Bigl[x\cos\frac{\overline{BC}}{k}\Bigr] + \text{const.}
\end{equation*}
%
Hence our second integral will be
%
\begin{equation*}
-k^2\left\{\sin^{-1}\left[\sin\phi\cos\frac{\overline{BC}}{k}\right]\right\}^C_0.
\end{equation*}
%
This vanishes at the lower limit. On the other hand by
Chapter IV\@. (7)
%
\begin{equation*}
\cos A = \sin C \cos \frac{\overline{BC}}{k},
\end{equation*}
%
our second integral becomes
%
\[
-k^2\left[\frac{\pi}{2} - A\right] = k^2\left[A + B - \pi\right],
\]
\[
\text{Area} = k^2(A + B + C - \pi). \tag{16}
\]


Two regions with the same area may, naturally, have very
different shapes. There are, however, three simple cases
where the equivalence of area is immediately evident. First,
where the two figures are congruent; second, when they are
composed of the same number of non-overlapping sub-regions
(i.e.\ sub-regions no two of which have in common a region
which has an area) congruent in pairs; third, where by the
adjunction of pairs of mutually congruent non-overlapping
sub-regions to them, they may be transformed into congruent
regions. In this latter case they may be said to be \emph{equivalent
by completion.}%
\footnote{The term \emph{equivalent by completion} is borrowed from
Halsted\index{Halsted}, loc.\ cit., p.~109.
The distinction between \emph{equivalent} and \emph{equivalent by completion} is, I believe, due
to Hilbert\index{Hilbert}, loc.\ cit., p.~40. For an admirable discussion of the question of
area see Amaldi\index{Amaldi}, in the fifth article in Enriques\index{Enriques},
\emph{Questioni riguardanti la geometria elementare}, Bologna, 1900.}
%endfootnote

\smallskip\emph{Definition}.~Given $n$ successive coplanar segments $(A_1A_2)$,
$(A_kA_{k+1})$, $(A_{n-1}$ $A_1)$ so situated that no line other than one
through a point $A_i$ can contain points of more than two of the
segments; the assemblage of all points of all segments whose
%[File: 179.png]
extremities are points of the given segments shall be called a
\emph{convex polygon} or, more simply, a polygon\index{Polygon}. The definition
of sides, vertices, and angles is immediate. If one vertex, say
$A_1$ be connected with all the others, the polygon will be
divided into $n-2$ triangles, no two of which have in common
any area. The area\index{Area01@Area} of the polygon\index{Area04@--- of a polygon} will thus be the sum of
the areas of these triangles. We may convince ourselves of
the compatibility of these statements as follows. A triangle
is certainly a polygon, and if a polygon of $n-1$ sides exist,
we may easily enlarge it to have $n$ sides by taking an additional
vertex near one side. On the other hand, if a polygon
of $n-1$ sides may be divided up in the manner suggested,
it is immediately evident that one of $n$ sides may be so
divided also.

\smallskip\emph{Theorem}~5. The area of a convex polygon is the quotient of
the excess\index{Excess of a triangle} of the sum of its angles over $(n-2)\pi$ divided by
the measure of curvature of Space.

\mysect{Area of circle}
Let the reader show that the area of a proper circle\index{Circle01@Circle}\index{Area02@--- of a circle} is
\begin{equation*}
\tag{17}
2\pi k^2\left( 1 - \cos\frac{r}{k} \right).
\end{equation*}
\mysect{Area of whole elliptic or spherical plane}
The total areas of the elliptic and the spherical planes\index{Area03@--- of a plane} will be
respectively
\[
2\pi k^2, \quad 4\pi k^2.
\]
In the hyperbolic plane regions may be found having any
desired area.

\mysect{Amplitude of a tetrahedron}
Our next undertaking shall be to see how far the methods
which we have established for studying areas are applicable
in three dimensions. We shall begin, as before, with amplitudes,
following, however, an analytical rather than a trigonometric
method.

Let the vertices of a tetrahedron, as defined in Chapter II,
be $A$, $B$, $C$, $D$ with the coordinates $(x)$, $(y)$, $(z)$, $(t)$ respectively.
The opposite faces shall be $\alpha$, $\beta$, $\gamma$, $\delta$ with coordinates $(u)$, $(v)$,
$(w)$, $(\omega)$, so that, e.g.
\[
r(\omega X) \equiv (X\,xyz).
\]
We shall define as sine amplitude of the tetrahedron\index{Amplitude of tetrahedron}
\begin{align*}
\sin (ABCD)
&= \left\lvert \cos\frac{\overline{AA}}{k}
               \cos\frac{\overline{BB}}{k}
               \cos\frac{\overline{CC}}{k}
               \cos\frac{\overline{DD}}{k} \right\rvert^{\frac{1}{2}} \\
&= \frac{\lvert (xx)(yy)(zz)(tt) \rvert^{\frac{1}{2}}    }
        {\sqrt{(xx)} \sqrt{(yy)} \sqrt{(zz)} \sqrt{(tt)} } \\
%[File: 180.png]
&= \frac{\lvert xyzt \rvert}
       {\sqrt{(xx)} \sqrt{(yy)} \sqrt{(zz)} \sqrt{(tt)}}. \tag{18}
\end{align*}

\mysect{Relation to other parts}
We shall give to the radicals involved such signs that
$k$ sine amplitude shall have the sign of $k^2$. Recalling the
concept of the moment of two lines introduced in Chapter IX,
we get
\[
\tag{19}
\sin\frac{\overline{AB}}{k} \sin\frac{\overline{CD}}{k}
  (\Moment AB,CD)=\sin(ABCD).
\]
\[
\sin(ABC)
= \frac{\lvert (xx)(yy)(zz) \rvert^{\frac{1}{2}}}
       {\sqrt{(xx)}\sqrt{(yy)}\sqrt{(zz)}}.
\]

Let $A'$, $B'$, $C'$, $D'$ be the points where perpendiculars from
the vertices of a tetrahedron meet the opposite faces. Then
\begin{align*}
\sin\frac{\overline{DD'}}{k} &=
  \frac{\lvert xyzt \rvert}
  {\sqrt{(tt)}\,\lvert (xx)(yy)(zz) \rvert^{\frac{1}{2}}},
\\
  \sin(BCD)\sin\frac{\overline{AA'}}{k} =&
  \sin(CDA)\sin\frac{\overline{BB'}}{k} =
  \sin(DBA)\sin\frac{\overline{CC'}}{k}
\\
\tag{20}
=& \sin(ABC)\sin\frac{\overline{DD'}}{k} = \sin(ABCD).
\end{align*}

If we mean by $\measuredangle \alpha\beta$ the dihedral angle of these two faces
\begin{align*}
\cos\measuredangle \alpha\beta &= \frac{(uv)}{\sqrt{(uu)}\,\sqrt{(vv)}}
\\
&= \frac{\begin{vmatrix}
  (xy) & (xz) & (xt) \\
  (zy) & (zz) & (zt) \\
  (ty) & (tz) & (tt) \end{vmatrix} }
  {\sqrt{\dfrac{\partial\lvert (xx)(yy)(zz)(tt) \rvert}{\partial(xx)}}\;
   \sqrt{\dfrac{\partial\lvert (xx)(yy)(zz)(tt) \rvert}{\partial(yy)}}
  },
\end{align*}
\[
\sin\measuredangle\alpha\beta =
  \frac{\sin(ABCD)\sin\dfrac{\overline{AB}}{k}}
       {\sin(BCD)\sin(ACD)},
\]
\[
\tag{21}
  \sin(BCD) \sin(ACD)
  \frac{\sin\measuredangle\alpha\beta}
       {\sin\dfrac{\overline{AB}}{k}} = \sin(ABCD).
\]

The geometry of lines through a point is an example of the
%[File: 181.png]
geometry of the elliptic plane, where $k^2 = 1$. We may thus
speak of the sine amplitude of a trihedral angle
\begin{align*}
\sin(AB, AC, AD) &=
  \frac{\begin{vmatrix}
          \begin{vmatrix} (tt) & (tx) \\ (xt) & (xx) \end{vmatrix}
        & \begin{vmatrix} (tt) & (ty) \\ (xt) & (xy) \end{vmatrix}
        & \begin{vmatrix} (tt) & (tz) \\ (xt) & (xz) \end{vmatrix}
        \smallskip \\
          \begin{vmatrix} (tt) & (ty) \\ (xt) & (xy) \end{vmatrix}
        & \begin{vmatrix} (tt) & (ty) \\ (yt) & (yy) \end{vmatrix}
        & \begin{vmatrix} (tt) & (tz) \\ (yt) & (yz) \end{vmatrix}
        \smallskip \\
          \begin{vmatrix} (tt) & (tx) \\ (zt) & (zx) \end{vmatrix}
        & \begin{vmatrix} (tt) & (ty) \\ (zt) & (zy) \end{vmatrix}
        & \begin{vmatrix} (tt) & (tz) \\ (zt) & (zz) \end{vmatrix}
        \end{vmatrix}^{\frac{1}{2}}                                 }
    {\sqrt{\begin{vmatrix} (tt) & (tx) \\ (xt) & (xx) \end{vmatrix}}\;
     \sqrt{\begin{vmatrix} (tt) & (ty) \\ (yt) & (yy) \end{vmatrix}}\;
     \sqrt{\begin{vmatrix} (tt) & (tz) \\ (zt) & (zz) \end{vmatrix}}} \\
&= \frac{(tt)\, \lvert xyzt \rvert}
    {\sqrt{\begin{vmatrix} (tt) & (tx) \\ (xt) & (xx) \end{vmatrix}}\;
     \sqrt{\begin{vmatrix} (tt) & (ty) \\ (yt) & (yy) \end{vmatrix}}\;
     \sqrt{\begin{vmatrix} (tt) & (tz) \\ (zt) & (zz) \end{vmatrix}}}.
\end{align*}
\[
\tag{22}
  \sin\frac{\overline{DA}}{k}
  \sin\frac{\overline{DB}}{k}
  \sin\frac{\overline{DC}}{k}
  \sin(AB,AC,AD) = \sin(ABCD).
\]

The reader will not fail to notice in formulae (19), (20), and
(22) the striking analogy between the sine amplitude\index{Amplitude of tetrahedron} and six
times the euclidean volume. There will be a function correlative
to $\sin (ABCD)$ which we shall call $\sin (\alpha\beta\gamma\delta)$.
\[
\tag{23}
  \sin\measuredangle\alpha\beta
  \sin\measuredangle(\gamma\delta) (\Moment AB, CD) =
  \sin(\alpha\beta\gamma\delta).
\]
\[
\tag{24}
  \sin(\alpha\beta\gamma)\sin\frac{\overline{DD'}}{k}
= \sin(\alpha\beta\gamma\delta).
\]
\[
\tag{25}
  \sin(\alpha\gamma\delta)\sin(\beta\gamma\delta)
  \frac{\sin\dfrac{\overline{AB}}{k}}
       {\sin\measuredangle\alpha\beta} =
  \sin(\alpha\beta\gamma\delta).
\]
\[
\tag{26}
  \sin\measuredangle\alpha\delta
  \sin\measuredangle\beta\delta
  \sin\measuredangle\gamma\delta
  \sin(\alpha\delta,\beta\delta,\gamma\delta) =
  \sin(\alpha\beta\gamma\delta).
\]
\[
\tag{27}
  \frac{\sin(BCD )}{\sin(\beta \gamma\delta)} =
  \frac{\sin(CDA )}{\sin(\gamma\delta\alpha)} =
  \frac{\sin(DBA )}{\sin(\delta\beta \alpha)} =
  \frac{\sin(ABC )}{\sin(\alpha\beta \gamma)} =
  \frac{\sin(ABCD)}{\sin(\alpha\beta\gamma\delta)}.
\]

Our two tetrahedral functions are connected by the relations
\begin{align*}
\sin(\alpha\beta\gamma\delta) &=
  \frac{\sin^3(ABCD)}
       {\sin(BCD)\sin(CDA)\sin(DBA)\sin(ABC)},
\\
\sin(ABCD) &=
  \frac{\sin^3(\alpha\beta\gamma\delta)}
       {\sin(\beta\gamma\delta) \sin(\gamma\delta\alpha)
        \sin(\delta\beta\alpha) \sin(\alpha\beta\gamma)}.
\end{align*}

%[File: 182.png]
\mysect{Simple form for the differential of volume of a tetrahedron} % ??
The analogy between the sine amplitude\index{Amplitude of tetrahedron} and the sextuple
of the euclidean expression for the volume appears even more
distinctly in the infinitesimal domain.
\begin{align*}
   \Limdot\sin (ABC)
&= \frac{1}{k^2}\overline{AB}\,.\, \overline{AC}\,.\, \sin \measuredangle BAC
\\
&= \frac{2}{k^2}\Area \triangle ABC.
\\
   \Limdot\sin (ABCD)
&= \limdot\, (ABCD)
\\
&= \frac{6}{k^3} \Voltet ABCD.
\tag{28}
\end{align*}

Following our previous analogy, suppose that we have six
planes, no three coaxal, passing by fours through four actual
or ideal, but not collinear points. Let the remaining intersections
be at a finite distance from the three chosen points,
but infinitesimally near one another. An infinitesimal
region will thus be formed, on the analogy of a euclidean
parallelepiped, which may be divided into six tetrahedra\index{Tetrahedron} of
such sort that the limit of the ratio of the sine amplitudes, or
of the euclidean volumes\index{Volume01@Volume}, of any two is unity. Six times the
euclidean volume of any one of these tetrahedra may be defined
as the euclidean volume of the region.

So far the analogy between two and three dimensions has
been sufficiently good. Each time we have had a function
called \emph{sine amplitude} corresponding in many particulars to
a simple multiple of the euclidean area or volume, and approaching
a multiple of the area or volume as a limit,
when the figure becomes infinitesimal. In the plane there
appeared, besides half the sine amplitude and the euclidean
area, a third expression, namely, the discrepancy or excess.
In three dimensions this function is, sad to relate, entirely
lacking; that is to say, there is no simple function of the
measures of a tetrahedron which possesses the property that
when one tetrahedron is the logical sum of two others, the
function of the sum is the sum of the functions. It is the
lack of this function that renders the problem of non-euclidean
volumes difficult.%
\footnote{It is highly interesting that in four dimensions a function playing the
role of the discrepancy appears once more. See Dehn\index{Dehn}, `Die eulersche Formel
in Zusammenhang mit dem Inhalt in der nicht-euklidischen Geometrie,'
\emph{Mathematische Annalen}, vol.~lxi, 1906.}
%endfootnote

Suppose, in general, that we have a three dimensional
region connex up to the boundary, and that we divide it
%[File: 183.png]
into a number of extremely tiny tetrahedra\index{Tetrahedron}. The limit of
the sum of the euclidean volume\index{Volume05@--- of tetrahedron} of each, multiplied by the
value for a point therein of a continuous function of the
coordinates of that point, as all the volumes approach zero
uniformly, shall be called the \emph{volume integral}\index{Volume02@--- integral} for that region
of that function. The proofs for the existence of that volume
integral, and its independence of the method of subdivision,
are analogous to those already referred to for the surface
integral. In particular, the volume integral of the function
unity shall be called the \emph{volume}\index{Volume01@Volume} of the region. Two regions
will have the same volume if they be congruent, made up
of the same number of parts, mutually congruent in pairs,
or if by the adjunction of such pairs they may be completed
to be congruent.

If the limiting surface of a region be made up of a series of
plane surfaces, and if no line, not lying in a plane of the
surface, can contain more than two points of the surface, then
it is easy to show that the region may be divided up into a
number of tetrahedra, and the problem of finding the volume of
any such region reduces to the problem of finding the volume
of a tetrahedron. This problem may, in turn, be reduced
to that of finding the volume of a tetrahedron of particularly
simple structure. To begin with, we may assume that there
is one face which makes with the three others dihedral angles
whose measures are less than $\dfrac{\pi}{2}$, for the bisectors of the
dihedral angles of the original tetrahedron will always divide
it into smaller tetrahedra possessing this property. The perpendicular\index{Perpendicularity}
on the plane of this face, from the opposite vertex,
will, then, pass through a point within the face, and, with the
help of this perpendicular, we may subdivide into three
smaller tetrahedra, for each of which the line of one edge is
perpendicular to the plane of one face.

Consider, next, a tetrahedron where the line of one edge
is indeed perpendicular to the plane of a face. There are
two possibilities. First, in the plane of this face neither of
the face angles whose vertex is not at the foot of the
perpendicular is obtuse; secondly, one of these angles is
obtuse. (The case where both were obtuse could not occur
in a small region.) In the first case we might draw a line
from the foot of the perpendicular to a point of the opposite
edge in this particular face, perpendicular to the line of
that edge, and thus, by a familiar theorem in elementary
geometry, which holds equally in the non-euclidean case,
divide the tetrahedron into two others, each of which possesses
%[File: 184.png]
the property that the lines of two opposite edges are perpendicular
to two of the faces. These we shall for the moment
call \emph{simplest type}. In the second case, from the vertex of the
obtuse angle mentioned, draw a line perpendicular to the line
of the opposite edge in this particular face (and passing
through a point within this edge), and connect the intersection
with the vertex opposite this face. The tetrahedron\index{Tetrahedron} will be
divided up into a tetrahedron of the simplest type, and one
of the sort considered in case 1. We have, then, merely to
consider the volume of a tetrahedron\index{Volume05@--- of tetrahedron} of the simplest type.

Let the vertices of the tetrahedron be $A$, $B$, $C$, $D$, where $AB$
is perpendicular to $BCD$ and $DC$ perpendicular to $ABC$. Let a
plane perpendicular to $AB$ contain a point $B_1$ of $(AB)$ whose
distance from $A$ shall have the measure $x$; while this plane
meets $(AC)$ and $(AD)$ in $C_1$ and $D_1$ respectively. The volume
of the region bounded by this plane, and an adjacent one of
the same type and the three faces through $A$, will be $dx$, multiplied
by the surface integral over the $\triangle B_1 C_1 D_1$ of the cosine
of the $k^{\text{th}}$ part of the distance of a point from $B_1$ (Cf.\ Ch.
IV\@. (2).) This integral takes a striking form.%
\footnote{The integration which follows is a very special case of a much more
general one for \emph{n} dimensions given by Schl\"afli\index{Schl\"afli}, \emph{Theorie der vielfachen Kontinuit\"at},
Zurich, 1901, p.~646. This paper of Schl\"afli's is posthumous; it was originally
written in 1855, when the science of non-euclidean geometry had not reached
its present recognition. It is very general, extremely difficult reading, and
hampered by a fearful and wonderful terminology, e.g.\ our tetrahedron of the
simplest type is a special case of an \emph{Artiothoscheme}. It is, however, a striking
piece of geometrical work. Schl\"afli gives a shorter account of his work in
his `R\'eduction d'une int\'egrale multiple qui comprend l'arc d'un cercle et
l'aire d'un triangle sph\'erique comme cas partieuliers', \emph{Lionville's Journal}, vol.\
xxii, 1855.}
%endfootnote

Let the distance from $B_1$ to a variable point $P$ of the
triangle be $r$, while $\phi$ is the measure of $\measuredangle C_1 B_1 P$. We wish
to find
$$
k\iint\sin\frac{r}{k}\cos\frac{r}{k}dr\,d\phi.
$$
Let $B_1P$ meet $(C_1 D_1)$ in $E_1$. The limits of integration for $r$
are 0 and $\overline{B_1E_1}$; hence we have merely to find
$$
\frac{k^2}{2}\int_0^{\measuredangle C_1B_1D_1}\sin^2\frac{\overline{B_1E_1}}{k} d\phi.
$$
Now $C_1D_1$ is perpendicular\index{Perpendicularity} to $B_1C_1$, hence
$$
\tan\phi\sin\frac{\overline{B_1C_1}}{k} = \tan\frac{\overline{E_1C_1}}{k},
$$

%[File: 185.png]
\[
  \cos\frac{\overline{B_1 E_1}}{k}
= \cos\frac{\overline{B_1 C_1}}{k}
  \cos\frac{\overline{E_1 C_1}}{k}
  \tan\frac{\overline{B_1 E_1}}{k}
= \tan\frac{\overline{B_1 C_1}}{k} \sec\phi. \tag{Ch.~IV.~(5), (6).}
\]
\[
  \sin^2\frac{\overline{B_1 E_1}}{k} d\phi
= \frac{1}{k} \sin\frac{\overline{B_1 C_1}}{k} d\overline{E_1 C_1}.
\]
Our required integral is then
\[
  \frac{k}{2} \int \sin\frac{\overline{B_1 C_1}}{k}
  d\overline{E_1 C_1}
= \frac{k}{2} \sin\frac{\overline{B_1 C_1}}{k}
  \cdot \overline{C_1 D_1}
\]

Let the reader note the astonishing feature of this result,
namely, that it involves one side of a triangle directly, and
another trigonometrically.

Let the measure of the dihedral angle whose edge is $(C_1 D_1)$
be $\theta$, this will also be the measure of
$\measuredangle A C_1 B_1$ which is the
plane angle of the dihedral one.
\begin{align*}
  \cos\theta
&= \cos\frac{\overline{A B_1}}{k} \sin\measuredangle BAC,
\\
  \sin\theta d\theta
&= \frac{1}{k} \sin\frac{\overline{A B_1}}{k}
   \sin\measuredangle BAC dx
\\
&= \frac{1}{k}  \sin\frac{\overline{A   B_1}}{k}
   \cdot \frac{\sin\dfrac{\overline{B_1 C_1}}{k} }
              {\sin\dfrac{\overline{A   C_1}}{k} } dx
\\
&= \frac{1}{k} \sin\theta \sin\frac{\overline{B_1 C_1}}{k} dx.
\end{align*}

\mysect{Reduction to a single quadrature of the problem of finding the volume of a tetrahedron}
We thus get for our volume\index{Volume05@--- of tetrahedron} the strange formula%
\footnote{See Schl\"afli\index{Schl\"afli}, \emph{R\'eduction}, p.~381,
where it is stated that this integral cannot
be evaluated by integration by parts. This same integral was discovered,
apparently independently, by Richmond\index{Richmond}, `The Volume of a Tetrahedron in
Elliptic Space,' \emph{Quarterly Journal of Mathematics}, vol.~xxxiv, 1902, p.~175.}
%endfootnote
\[
  \text{Vol.} = \frac{k^2}{2} \int \overline{C_1 D_1} d\theta. \tag{29}
\]

We can easily express this integral in terms of $\theta$,
\begin{align*}
   \tan\frac{\overline{C_1 D_1}}{k}
&= \sin\frac{\overline{A   C_1}}{k}
   \tan\measuredangle DAC
=a\sin\frac{\overline{A   C_1}}{k},
\\
   \cos\frac{\overline{A   C_1}}{k}
&= \ctn\measuredangle BAC \ctn\theta
= b\ctn\theta.
\\
  \text{Vol.}
&= \frac{k^3}{2} \int \tan^{-1}
  [ a\sqrt{1-b^2\ctn^2\theta} ] d\theta.  \tag{30}
\end{align*}

%[File: 186.png]
This formula apparently represents about as close an
approach as can be made towards \emph{finding the volume} of this
tetrahedron\index{Volume05@--- of tetrahedron}, for, in the general case,%
\footnote{Schl\"afli\index{Schl\"afli}, \emph{Vielfache Kontinuit\"at}, p.~95,
gives a formula for the special case
where the sum of the squares of the cosines of the dihedral angles is equal
to unity. The proof is highly intricate, and not suitable to reproduce here.}
%endfootnote
it does not seem possible
to effect the quadrature in terms of elementary functions.

\mysect{Volume of a cone of revolution}
If a right triangle be rotated completely about one of the
sides adjacent to the right angle, the figure so generated shall
be called a \emph{cone of revolution}\index{Cone of revolution}. The volume\index{Volume03@--- of cone}
within the surface
may be found as follows. Let the vertex of the cone be $A$
and the centre of the base $O$, while $P$ is a point within the
cone. Let $Q$ be the intersection of $(AO)$ with a perpendicular
from $P$, while the base circle meets the plane $A OP$ in $B$.
$(AB)$ shall meet $PQ$ in $R$. Let us also write
\begin{align*}
  \overline{AB} =& s,\quad \overline{AR} = r,\quad
  \overline{AO} = h,\quad \measuredangle OAB = \theta.
\\
  \text{Vol.}
&= k\int_0^{\overline{QR}} \int_0^h \int_0^{2\pi}
  \sin\frac{\overline{QP}}{k} \cos\frac{\overline{QP}}{k}
  d\overline{AQ}\, d\overline{QP}\, d\theta
\\
&= 2\pi k\int_0^h \int_0^{\overline{QR}}
  \sin\frac{\overline{QP}}{k} \cos\frac{\overline{QP}}{k}
  d\overline{AQ}.\, d\overline{QP}
\\
&= \pi k^2 \int_0^h \sin^2 \frac{\overline{QR}}{k} d\overline{AQ}.
\end{align*}
\[
  \tan \frac{\overline{AQ}}{k}
= \tan\frac{r}{k} \cos \theta. \tag{Ch.~V.~(6).}
\]
\[
  d\overline{AQ}
= \frac{\cos\theta \sec^2\dfrac{r}{k} dr}
        {1 + \cos^2\theta \tan^2\dfrac{r}{k}}.
\]
\[
  \sin\frac{\overline{QR}}{k}
= \sin\frac{r}{k} \sin\theta.
\]
\[
  \text{Vol.}
= \pi k^2\sin^2\theta \cos\theta
  \int_0^h \frac{\tan^2\dfrac{r}{k}                 }
                {1 + \cos^2\theta \tan^2\dfrac{r}{k}} dr.
\]
\[
  \text{Put}\quad\tan\frac{r}{k} = x.
\]
%[File: 187.png]
\begin{align*}
\text{Vol. } &= \pi k^3\cos\theta\sin^2\theta
  \int_0^{\tan\frac{h}{k}}
  \frac{x^2dx}{(1+x^2)(1+x^2\cos^2\theta)} \\
&= \pi k^3\cos\theta \Bigl[
  \int_0^{\tan\frac{h}{k}}
  \frac{dx}{1+x^2\cos\theta} -
  \int_0^{\tan\frac{h}{k}}
  \frac{dx}{1+x^2} \Bigr] \\
&= \pi k^3\cos\theta \Bigl[
  \frac{1}{\cos\theta}\tan^{-1}(x\cos\theta)-\tan^{-1}x
  \Bigr]_0^{\tan\frac{h}{k}} \\
\tag{31}
&= \pi k^2[h-s\cos\theta].\footnotemark
\end{align*}
\footnotetext{This formula is given without sufficiently detailed proof by Frischauf\index{Frischauf},
loc.\ cit., p.~99. A tedious demonstration was subsequently worked out by
Von Frank\index{Frank@Von Frank}, `Der K\"orperinhalt des senkrechten Cylinders und Kegels in der
absoluten Geometrie,' \emph{Grunerts Archiven}, vol.~lix, 1876.}
%endfootnote

\mysect{Volume of a sphere}
To find the volume within a proper sphere\index{Volume04@--- of sphere}, where the
distance from the centre to every point of the surface has the
constant value $R$,
\begin{align*}
\text{Vol.} &= k^2 \int_0^R \int_0^\pi \int_0^{2\pi}
  \sin^2\frac{2}{k}\sin\theta\, dr\, d\theta\, d\phi \\
&= 4\pi k^2\int_0^R \sin^2\frac{r}{k}dr \\
\tag{32}
&= \pi k^3\left(\frac{2R}{k}-\sin\frac{2R}{k}\right).
\end{align*}

\mysect{Volume of the whole of elliptic or of spherical space}
Let the reader show that the total volumes of elliptic and
of spherical space, where $k = 1$ will be, respectively,
\[
\pi^2,\quad 2\pi^2.
\]

%[File: 188.png]



%CHAPTER XV
\mychap{XV}{INTRODUCTION TO DIFFERENTIAL GEOMETRY}
\mysect{Curvature of a space or plane curve}

The task which we shall undertake in the present chapter
is to develop the differential geometry of curves and surfaces
in non-euclidean space.%
\footnote{The developments of this chapter follow the general scheme worked out
for the euclidean case in Bianchi-Lukat\index{Bianchi}, \emph{Vorlesungen \"uber Differentialgeometrie},
Leipzig, 1899, Chapters I, III, IV, and VI\@. In Chapters XXI and XXII of the
same work will be found a different development of the non-euclidean case.
It is, however, so general, yet so concise, as to be scarcely suitable to serve
as an introduction to the subject.}
%endfootnote
We shall introduce a notable simplification in our work by abandoning
homogeneous coordinates\index{Coordinates02@Co-ordinates of a point}, and assuming that
\[
(xx) = k^2.   \tag{1}
\]
In the elliptic case we shall take $x_0 \geqq 0$; in the hyperbolic,
$\dot{x}_0 = \dfrac{1}{k}x_0 \geqq 0$ for all real points.

Of course in exceptional cases, where we wish to include
points of the Absolute\index{Absolute} or beyond, this proceeding is not
legitimate; we shall therefore assume, unless we specifically
state the contrary, that we are limiting ourselves to a real
region, where no absolute or ultra infinite\index{Ultra-infinite elements} points are included
in the hyperbolic case. We shall, further, have for the
distance\index{Distance04@Distance element} of two points $(x)$, $(x')$.
\[
  \cos\frac{d}{k} = \frac{(xx')}{k^2},\quad
  \sin^2\frac{d}{k}
= \frac{\begin{Vmatrix}
           x_0 & x_1 & x_2 & x_3 \\
           x_0{}'& x_1{}'& x_2{}'& x_3{}'
         \end{Vmatrix}^2           }
       {k^4                        }.  \tag{2}
\]
When $x_i{}' = x_i + dx_i$ we have for the square of the differential
of distance
\[
  k^2 \frac{ds^2}{k^2} = ds^2 = \frac{(xx)(dx\,dx) - (x\,dx)^2}{k^2}.
\]
\[
  (x+dx,\;x+dx) = k^2,\quad (x\,dx) = -\tfrac{1}{2}(dx\,dx),
\]
\[
  ds^2 = (dx\,dx).   \tag{3}
\]

We shall mean by an analytic curve, such a curve that the
coordinates of its points are analytic functions of a single
variable. The formulae developed in this chapter will hold
%[File: 189.png]
equally well under the supposition that the functions and
their first three partial derivatives exist and are finite in our
region, but the gain in generality is of little interest to the
geometer, and we shall assume from here on that when we
speak of curve we mean analytic curve.

Let us imagine that at a chosen point of a curve, say $P$, a
tangent is drawn. We shall take two near points $P'$ and $P''$
on the curve and tangent respectively, so situated near $P$ and
on the same side of the normal plane that
$\overline{PP'} = \overline{PP''}$. Then
we shall define%
\footnote{This definition is taken from Bianchi\index{Bianchi}, loc.\ cit., p.~603. It is there
ascribed to Voss\index{Voss}.}
%endfootnote
\[
\limdot \frac{2\overline{P'P''}}{\overline{PP'^2}},
\]
as the curvature of the given curve at that point. If we
compare with Chapter XI.~(2), and define as the osculating
circle\index{Circle01@Circle} to a curve at a point, the limit of the circle through
that and two adjacent points, we shall have

\smallskip\emph{Theorem}~1. The curvature of a curve\index{Curvature01@Curvature of a curve} at any point is equal
to that of its osculating circle, and is equal to the absolute
value of the product of the square root of the curvature of
space and the cotangent of the $k^{\text{th}}$ part of the distance of each
point of the circle from its centre.

Let us now suppose that the equations of our curve are
written in the form
\begin{align*}
x_i &= x_i(t_0)+(t-t_0)x_i{}'(t_0)
+ \frac{(t-t_0)^2}{2}x_i{}''(t_0) + \ldots
\\
x_i{}^{(n)} &= \frac{d^n}{dt^n}x_i(t).
\end{align*}
Then for a point on the tangent we shall have coordinates\index{Coordinates02@Co-ordinates of a point}
\[
X_i=\lambda[x_i(t_0)+(t-t_0)x_i{}'(t_0)].
\]
To get the value of $\lambda$.
\[
(XX) = xx = k^2, \quad (xx') = 0,
\]
\[
X_i = \frac{x_i(t_0) + (t-t_0)x_i{}'(t_0)}
           {\sqrt{1 + \dfrac{(t-t_0)^2}{k^2}(x_i{}' x_i{}')}}.
\]

Developing by the binomial theorem, and rejecting powers
of $(t-t_0)$ above the second
\[
X_i = x_i(t_0) + (t-t_0)x_i{}'(t_0)
    - \frac{(t-t_0)^2}{2k^2}(x'x')x_i(t_0).
\]

%[File: 190.png]
Subtracting from the series development of $x_i$ we get for
our curvature $\dfrac{1}{\rho}$.
\[
\frac{1}{\rho^2} =
  \frac{\left[ (x''x'')+\dfrac{2}{k^2}(xx'')(x'x')
             + \dfrac{1}{k^4}(x'x')^2(xx) \right]}
       {(x'x')^2},
\]
\[
\tag{4}
\frac{1}{\rho^2} = \frac{(x''x'')}{(x'x')^2} - \frac{1}{k^2}.
\]

\smallskip\emph{Theorem}~2. The square of the curvature of a curve\index{Curvature01@Curvature of a curve} is the
square of its curvature treated as a curve in a four-dimensional
euclidean space, minus the measure of
curvature\index{Measure02@--- of curvature of space}\index{K@$\dfrac{1}{k^2}$ measure of curvature of space}\index{Curvature06@--- of space}
of the non-euclidean
space.

\mysect{Analoga of direction cosines of tangent, principal normal, and binormal}
It will be convenient to consider, besides our point $(x)$,
three other points allied to it. $(t)$ shall be orthogonal\index{Orthogonal points} to $(x)$
and on the tangent, $(z)$ orthogonal to $(x)$ on the principal
normal, and $(\xi)$ orthogonal to $(x)$ on the binormal. These
three will replace the direction cosines of tangent, principal
normal, and binormal, which figure so prominently in the
euclidean theory. In hyperbolic space these points lie without
the actual domain to which we suppose $(x)$ confined.
\[
(xt) = (xz) = (x\xi) = (tz) = (t\xi) = (z\xi) = 0.
\]

If a point trace an infinitesimal arc $ds$, the angle of the
corresponding absolute polar planes is
$\left|\sqrt{\dfrac{ds^2}{k^2}}\right|$.

We shall, hereafter, take as our parameter on the given
curve $s$, the length of arc, so that
\[
x_i{}'=\frac{dx_i}{ds}, \quad (x'x')=1.
\]

As $(t)$ lies on the tangent, its coordinates will be of the form
\[
t_i = lx_i+mx_i{}',
\]
\[
(tt) = (xx) = k^2, \quad (tx) = 0, \quad (xx') = 0,
\]
\[
\tag{5}
t_i = kx_i{}'.
\]

For the point $(z)$ we shall have
\[
z_i = \lambda x_i + \mu x_i{}' + \nu x_i{}'',
\]
\[
(zx) = (zx') = (xx') = (x'x') + (xx'') = 0,
\]
\[
(zz) = (xx) =k^2, \quad (x'x') = 1.
\]
%[File: 191.png]
\begin{align*}
  z_i &= \frac{x_i + k^2x_i{}''}{\sqrt{k^2(x''x'')-1}},  \\
  z_i &= \frac{\rho}{k}(x_i + k^2x_i{}'').  \tag{6}
\end{align*}
To determine $\xi$ we shall have the conditions
\[
  (\xi x) = (\xi z) =  (\xi t) = 0,\quad  (\xi\xi) = k^2,
\]
\[
  \xi_i = \rho\frac{\partial}{\partial y_i} \lvert y x x' x''\rvert. \tag{7}
\]

We shall define the \emph{torsion}\index{Torsion} of our curve as the limit of the
ratio of the angle of two successive osculating planes to the
differential of arc. We thus get
\[
  \frac{1}{T} = \frac{1}{k} \frac{\sqrt{(d\xi\,d\xi)}}{ds}.  \tag{8}
\]

\mysect{Frenet's formulae for the non-euclidean case}
Reverting to our formulae $(5)$ and $(6)$
\[
  \frac{dt_i}{ds} = \frac{z_i}{\rho} - \frac{x_i}{k}.   \tag{9}
\]
\[
  (x\xi) = (x'\xi) = (x''\xi) = (x\xi') = (x'\xi') = (\xi\xi') = 0.
\]
\begin{flalign*}
&\text{Hence}
&\frac{d\xi_i}{ds} &= lz_i,
\\
&\text{or, more specifically}
&  \frac{d\xi_i}{ds} &= \frac{z_i}{T}.   \tag{10}
\\
&\text{We have also}
&  (xz) = (xz') = &(x'z) = (zz') = 0, & \phantom{\text{We have also}}\\
&&  z_i{}' = \lambda &t_i + \mu\xi_i,   \\
&&  \frac{dz_i}{ds} = -&\frac{t_i}{\rho} - \frac{\xi_i}{T}.   \tag{11}
\end{flalign*}

The reader will see at once that $(9)$, $(10)$, $(11)$ are the
analoga of Frenet's\index{Frenet} formulae for euclidean curves.

We have, so far, overlooked the question of the sign of the
torsion, but that is well determined from the above formulae,
and it is important now to find the geometric difference
between the case where the torsion is negative, and that
where it is positive. We shall carry through the work for
the elliptic case only, the hyperbolic may be treated in the
same way, but it is wiser there to replace the coordinates
$(x)$ by $(\dot{x})$.

As before we shall choose $s$ as the independent variable,
so that
\[
  (xx) = k^2,\quad (xx') = (x'x'') = 0,\quad (x'x') = -(xx'')= 1.
\]

%[File: 192.png]
The sign of $t_i$ (which may be ideal) will be found from (5),
that of $z_i$ from (6), and that of $\xi_i$ from (7), while the sign
of T\index{Torsion} will be given by (10).

The equation of the plane of the tangent and binormal
will be
\begin{equation*}
|Xxt\xi| = (Xx) + k^2(Xx'') = 0.
\end{equation*}
Putting in the coordinates of a near-by point of the curve,
\begin{equation*}
x_i + x_i{}'\Delta s + x_i{}''\frac{(\Delta s)^2}{2},
\end{equation*}
\begin{equation*}
k^2 - \frac{(\Delta s)^2}{2} - k^2 + k^2(x''x'')\frac{\Delta s^2}{2} = \frac{k^2}{\rho^2}\frac{(\Delta s)^2}{2},
\end{equation*}\label{no2}
and this is essentially positive, so that, in general, the curve
will not cross this plane here. Again, we see by (6) that
we may give to a point on the principal normal close to $(x)$
the coordinates
\begin{equation*}
x_i + \epsilon x_i{}''.
\end{equation*}
Substituting in the equation of the plane we get
\begin{equation*}
\frac{k^2}{\rho^2}\epsilon,
\end{equation*}
so that this will lie on the same side as the curve if $\epsilon > 0$.

Let us call \emph{positive} that part of the curve near our point for
which $\Delta s > 0$. The positive part of the tangent shall be that
which lies on the same side of the normal plane as the
positive part of the curve, while that part of the principal
normal shall be called positive which lies on the same side of
the plane of tangent and binormal as does the curve. Let us
find the Plueckerian coordinates of a ray from $x_i + x_i{}'\Delta s$ on
the positive part of the tangent to $x_i + \epsilon x_i{}''$ on the positive
part of the principal normal. We get
\begin{equation*}
p_{ij} = \epsilon\left|\begin{array}{cc}x_i&x_j\\x_i{}''&x''_j\end{array}\right|
+ \Delta s\left|\begin{array}{cc}x_i{}'&x'_j\\x_i&x_j\end{array}\right|
+ \epsilon\Delta s\left|\begin{array}{cc}x_i{}'&x'_j\\x_i{}''&x''_j\end{array}\right|.
\end{equation*}

In like manner for a ray\index{Ray} from $(x)$ to a point on the positive
part of the curve
\begin{equation*}
x_i + x_i{}'\Delta_1 s + x_i{}''\frac{(\Delta_1 s)^2}{2} + x_i{}'''\frac{(\Delta_1 s)^3}{3!},
\end{equation*}
we get
\begin{equation*}
q_{kl} = \Delta_1 s\left|\begin{array}{cc}x_k&x_l\\x_k{}'&x_l{}'\end{array}\right|
+ \frac{(\Delta_1 s)^2}{2}\left|\begin{array}{cc}x_k&x_l\\x_k{}''&x_l{}''\end{array}\right|
+ \frac{(\Delta_1 s)^3}{3!}\left|\begin{array}{cc}x_k&x_l\\x_k{}'''&x_l{}'''\end{array}\right|.
\end{equation*}
The relative moment of these two rays, as defined at the
close of Chapter IX, will be
\begin{equation*}
\smallsum p_{ij}q_{kl} \equiv \epsilon\frac{\Delta s (\Delta_1 s)^3}{6}|xx'x''x'''|.
\end{equation*}

%[File: 193.png]
The factors outside of the determinant are all, by hypothesis,
positive, so that the sign depends merely upon that of the
determinant, and this by (7) is equal to $\dfrac{\rho(\xi x''')}{k}$.

Now
\[
(\xi x'') = 0, \quad (\xi x''') = -(\xi' x'').
\]
Hence the relative moment will have the sign of
\[
  \frac{-\rho}{kT}(zx'')
= \frac{-\rho^2}{k^2T} [(xx'')+k^2(x''x'')]
= -\frac{1}{T}.
\]

\mysect{Sign of the torsion}
\smallskip\emph{Theorem}~3. The torsion\index{Torsion} at a general point of a curve is
positive when the relative moment\index{Moment02@--- relative of two rays}
of a ray\index{Ray} thence to a point
on the positive part of the curve, and a ray from a point on
the positive part of the tangent to one on the positive part of
the principal normal\index{Normals01@Normals to curve} is negative; when the latter product is
positive, the torsion is negative.

Intuitively stated this means that the torsion is positive
when the curve resembles a left-hand screw, otherwise
negative.

\mysect{Evolutes of a space curve}
We shall next take up the evolutes\index{Evolutes} of a curve. Let $(\overline{x})$
be a point of an evolute. Then
\begin{align*}
x_i &= \cos\frac{\overline{s}}{k}\overline{x_i}
     - \sin\frac{\overline{s}}{k}\overline{t_i}, \\
\frac{dx_i}{d\overline{s}}
&= -\sin\frac{\overline{s}}{k} \frac{\overline{z_i}}{\overline{\rho}}.
\end{align*}
Remembering that $\dfrac{dx_i}{d\overline{s}} = kt_i$, while $\overline{z_i}$ is on the principal
normal of the evolute.

\smallskip\emph{Theorem}~4. A tangent to an analytic curve at a general
point will be in the osculating plane at the corresponding
point of any evolute.

Since $(\overline{x})$ lies in the normal plane at $(x)$, we may write
\[
w\overline{x_i} = x_i + u\xi_i + vz_i,
\]
\[
w\frac{d\overline{x_i}}{ds} =
  -\frac{1}{w} (x_i+u\xi_i+vz_i)\frac{dw}{ds} +
  \frac{t_i}{k} + u\frac{z_i}{T} -
  v\left( \frac{t_i}{\rho} + \frac{\xi_i}{T} \right)
  +\xi_i\frac{du}{ds} + z_i\frac{dx}{ds}.
\]

Now $\dfrac{d\overline{x_i}}{ds}$ is linearly dependent on $(x)$ and
$(\overline{x})$,
\[
\left( t\frac{d\overline{x}}{ds} \right) = 0, \quad
v = \frac{\rho}{k},
\]
%[File: 194.png]
and, for the same reason, the assemblage of all terms in $(\xi)$
and $(z)$ must be a linear combination of $(x)$ and $(\overline{x})$, and so
proportional to $w\overline{x_i} - x_i = u\xi_i + vz_i$
\[
  \Bigl[ \frac{du}{ds} - \frac{\rho}{kT} \Bigr]\xi_i +
  \Bigl[ \frac{u}{T} + \frac{d\rho}{kds} \Bigr]z_i =
  \lambda\left[ u\xi_i + \frac{\rho}{k}z_i \right],
\]
\[
  \frac{\dfrac{du}{ds}-\dfrac{\rho}{kT} }
       {\dfrac{u}{T} + \dfrac{1}{k}\dfrac{d\rho}{ds}} =
  \frac{u}{\dfrac{\rho}{k} },
\]
\[
  \tan^{-1}\left( \frac{u}{\dfrac{\rho}{k}} \right) =
  \int\frac{ds}{T} + C =
  (\sigma + C).
\]

To get $(w)$ we have
\[
(\overline{x}\,\overline{x}) = k^2,
\]
\[
w=\sqrt{1+u^2+v^2} =
  \sqrt{1+\frac{\rho^2}{k^2}\sec^2(\sigma+C)}.
\]
\[
\tag{12}
\overline{x_i} =
  \frac{\left( x_i+z_i\dfrac{\rho}{k} \right)\cos(\sigma+C)
       + \dfrac{\rho}{k}\xi_i\sin(\sigma+C) }
       {\sqrt{\dfrac{\rho^2}{k^2} + \cos^2(\sigma+C)} }.
\]

The coordinates of the point of the line $(x)(\overline{x})$
orthogonal\index{Evolutes}\index{Normals01@Normals to curve}\index{Perpendicularity}
to $(x)$ will be
\[
\lambda x_i + \mu \overline{x_i},
\]
\begin{align*}
\lambda k^2 + \frac{\mu k^2\cos(\sigma+C)}
  {\sqrt{\dfrac{\rho^2}{k^2} + \cos^2(\sigma+C)}} &= 0,
\\
(\lambda^2+\mu^2)k^2 +
  \frac{2\lambda\mu k^2\cos(\sigma+C)}
  {\sqrt{\dfrac{\rho^2}{k^2} + \cos(\sigma+C)}} &= k^2,
\\
\mu = \frac{-\sqrt{\dfrac{\rho^2}{k^2} + \cos(\sigma+C)}}
           {\dfrac{\rho}{k} },   \quad
  \lambda &= \frac{\cos(\sigma+C)}
                 {\dfrac{\rho}{k} }.
\end{align*}

The point in question will therefore have the coordinates\index{Coordinates02@Co-ordinates of a point}
\[
\xi_i\sin(\sigma+C) + z_i\cos(\sigma+C).
\]

%[File: 195.png]
This gives us the significance of $\sigma$, namely $(\sigma + C)$ is the $k^{\text{th}}$
part of the distance from this point to $(z)$, i.e.\ $(\sigma + C)$ represents
the angle which this normal\index{Normals01@Normals to curve} makes with the principal
normal. If, then, we take two evolutes\index{Evolutes} of our curve the
angle between their corresponding tangents, i.e.\ those which
meet on the involute, is
\[
  (\sigma + C_1) - (\sigma + C_2) = C_1 - C_2.
\]

\smallskip\emph{Theorem}~5. Corresponding tangents to two evolutes of
a curve meet at a constant angle.

\smallskip\emph{Theorem}~6. If the generators of a developable surface be
turned through a constant angle about the tangents to one
of their orthogonal trajectories, the resulting surface is
developable.

\smallskip\emph{Theorem}~7. The tangents to an evolute of a plane curve
make a constant angle with the plane of the curve.

\mysect{Two fundamental quadratic differential forms for a surface}
The foregoing theorems and formulae exhibit sufficiently
the close analogy between the differential theory of curves
in euclidean and in non-euclidean space. It is our next task
to take up the theory of surfaces, and we shall find a no
less striking analogy there. We shall mean by an \emph{analytic
surface} the locus of a point whose coordinates are analytic
functions of two independent parameters. We shall exclude
from consideration all singular points of such surfaces. If
the parameters be $(u)$ and $(v)$, we shall have for the squared
distance element\index{Distance04@Distance element}
\[
  ds^2 = E\,du^2 + 2 F\,du\,dv + G\,dv^2,
\]
\[
  E = \left( \frac{\partial x}{\partial u}
             \frac{\partial x}{\partial u} \right),\quad
  F = \left( \frac{\partial x}{\partial u}
             \frac{\partial x}{\partial v} \right),\quad
  G = \left( \frac{\partial x}{\partial v}
             \frac{\partial x}{\partial v} \right),
\]
\[
  EG - F^2
= \begin{Vmatrix}
    \dfrac{\partial x_0}{\partial u}
  & \dfrac{\partial x_1}{\partial u}
  & \dfrac{\partial x_2}{\partial u}
  & \dfrac{\partial x_3}{\partial u}
  \smallskip\\
    \dfrac{\partial x_0}{\partial v}
  & \dfrac{\partial x_1}{\partial v}
  & \dfrac{\partial x_2}{\partial v}
  & \dfrac{\partial x_3}{\partial v}
  \end{Vmatrix}^2.                \tag{13}
\]

This is a positive definite form in the elliptic case, and in
the actual domain of hyperbolic space, to which we shall
restrict ourselves. The discriminant, under this same restriction,
will always be greater than zero, for it will vanish only
when the tangent plane to the surface\index{Tangent plane to surface} is also tangent to the
Absolute.

The equation of the tangent plane at $(x)$ will be
\[
  \left |  Xx
  \frac{\partial x}{\partial u}
  \frac{\partial x}{\partial v}
   \right|
= 0.
\]

%[File: 196.png]
The Absolute pole of this plane\index{Tangent plane to surface} will be
\[
  y_i = \frac{\dfrac{\partial}{r_i}
               \left|
                 rx
               \dfrac{\partial x}{\partial u}
               \dfrac{\partial x}{\partial v}
               \right|  }
             {\sqrt{EG-F^2}  }.   \tag{14}
\]

We shall consistently use the letter $(y)$ throughout the
present chapter to indicate this point. The equation of the
plane through the normal, and the point $(x + dx)$, will be
\[
  \begin{vmatrix}
    (Xx) & (xx)
  & \left( x\dfrac{\partial x}{\partial u} \right)
\\
    \left( X\dfrac{\partial x}{\partial u} \right)
  & \left( x\dfrac{\partial x}{\partial u} \right) & E
\\
    \left( X\dfrac{\partial x}{\partial v} \right)
  & \left( x\dfrac{\partial x}{\partial v} \right) & F
  \end{vmatrix} du
+
  \begin{vmatrix}
    (Xx) & (xx)
  & \left( x\dfrac{\partial x}{\partial u} \right)
\\
    \left( X\dfrac{\partial x}{\partial u} \right)
  & \left( x\dfrac{\partial x}{\partial u} \right) & F
\\
    \left( X\dfrac{\partial x}{\partial v} \right)
  & \left( x\dfrac{\partial x}{\partial v} \right) & G
  \end{vmatrix} dv
= 0.
\]
\[
  \sum_i^{0.. 3} \left[
  \left( F\frac{\partial x_i}{\partial u}
       - E\frac{\partial x_i}{\partial v} \right) du
+
  \left( G\frac{\partial x_i}{\partial u}
       - F\frac{\partial x_i}{\partial v} \right) dv
 \right] X_i = 0.
\]

The cosine of the angle which this plane makes with that
through the normal and the point $(x + \delta x)$, or the cosine of
the angle of the two arcs from $(x)$ to $(x + dx)$ and
$(x + \delta x)$,
will be
\[
  \frac{E\,du\,\delta u
       + F (du\,\delta v + \delta u\,dv)
       + G\,dv\,\delta v }
       {ds\,\delta s }.             \tag{15}
\]

\mysect{Conditions for mutually conjugate or perpendicular tangents}
The two will be mutually perpendicular\index{Conjugate02@--- directions on a surface} if
\[
  E\,du\,\delta u
+ F (du\,\delta v + \delta u\,dv)
+ G\,dv\,\delta v = 0.
\]

The condition for perpendicularity between the parameter
curves will be
\[
  F = 0.      \tag{16}
\]

The equation of the tangent plane at $(x + dx)$ is
\begin{multline*}
  \left| X
  \Bigl( x
       + \frac{\partial x}{\partial u}du
       + \frac{\partial x}{\partial v}dv \Bigr)
  \Bigl( \frac{\partial x}{\partial u}
       + \frac{\partial^2 x}{\partial u^2}du
       + \frac{\partial^2 x}{\partial u\,\partial v}dv \Bigr)
  \right.
\\
  \left.
  \Bigl( \frac{\partial x}{\partial v}
       + \frac{\partial^2 x}{\partial u\,\partial v}du
       + \frac{\partial^2 x}{\partial v^2}dv           \Bigr)
  \right|
= 0.
\end{multline*}
Neglecting differentials of higher order than the first, we
have
\begin{multline*}
  \left| Xx \frac{\partial x}{\partial u}
            \frac{\partial x}{\partial v} \right|
+ \left[
  \left| Xx \frac{\partial^2 x}{\partial u^2}
            \frac{\partial x}{\partial v} \right|
+
  \left| Xx \frac{\partial x}{\partial u}
            \frac{\partial^2 x}{\partial u\,\partial v} \right|
  \right] du
\\
+ \left[
  \left| Xx \frac{\partial^2 x}{\partial u\,\partial v}
            \frac{\partial x}{\partial v} \right|
+
  \left| Xx \frac{\partial x}{\partial u}
            \frac{\partial^2 x}{\partial v^2} \right|
  \right] dv
= 0.
\end{multline*}
%[File: 197.png]
The line of intersection with the tangent plane at $(x)$ will be
found by equating to zero separately the first and the last
four terms. This line will contain the point $(x + \delta x)$ if
\[
  D\,du\,\delta u + D'(du\,\delta v + dv\,\delta u)
+ D''dv\,\delta v = 0.
\]
\[
  D  = \frac{\left| x \dfrac{\partial x}{\partial u}
                      \dfrac{\partial x}{\partial v}
                      \dfrac{\partial^2 x}{\partial u^2} \right|}
            {\sqrt{EG-F^2}},
\quad
  D' = \frac{\left| x \dfrac{\partial x}{\partial u}
                      \dfrac{\partial x}{\partial v}
                      \dfrac{\partial^2 x}{\partial u\,\partial v}
             \right|}
            {\sqrt{EG-F^2}},
\quad
  D''= \frac{\left| x \dfrac{\partial x}{\partial u}
                      \dfrac{\partial x}{\partial v}
                      \dfrac{\partial^2 x}{\partial v^2} \right|}
            {\sqrt{EG-F^2}}.
\tag{17}
\]

The signs of $D$, $D'$, $D''$ to be determined presently.

These are the equations for tangents to conjugate systems
of curves, or, briefly put, the equations determining differentials
in conjugate directions\index{Conjugate02@--- directions on a surface}. The parameter curves will
be mutually conjugate if
\[
  D' = 0.     \tag{18}
\]

The differential equation for self-conjugate, or asymptotic\index{Asymptotic lines}
lines, will be
\[
  D\,du^2 + 2D'\,du\,dv + D''\,dv^2 = 0.    \tag{19}
\]

\mysect{Lines of curvature}
Returning to the point $(y)$, the pole of the tangent plane,
we have
\[
  \Bigl( y\frac{\partial^2 x}{\partial u^2} \Bigr) = D,\quad
  \Bigl( y\frac{\partial^2 x}{\partial u\,\partial v} \Bigr) =D',\quad
  \Bigl( y\frac{\partial^2 x}{\partial v^2} \Bigr) = D'';
\]
\[
  (xy) = (y\,dx) = (x\,dy) = 0,
\]
\[
\begin{split}
  \Bigl( \frac{\partial x}{\partial u}
         \frac{\partial y}{\partial u} \Bigr)
=-\Bigl(y\frac{\partial^2 x}{\partial u^2} \Bigr)
\quad
  \Bigl( \frac{\partial x}{\partial u}
         \frac{\partial y}{\partial v} \Bigr)
= \Bigl( \frac{\partial x}{\partial v}
         \frac{\partial y}{\partial u} \Bigr)
=-\Bigl(y\frac{\partial^2 x}{\partial u\,\partial v} \Bigr)
\hfill \\ \hfill
  \Bigl( \frac{\partial x}{\partial v}
         \frac{\partial y}{\partial v} \Bigr)
=-\Bigl( y\frac{\partial^2 x}{\partial v^2} \Bigr),
\end{split}
\]
\[
  -(dy\,dx) = D\,du^2 + 2D'\,du\,dv + D''\,dv^2.    \tag{20}
\]

These equations will determine the signs of $D$, $D'$, $D''$.

Under what circumstances will the normals at two adjacent
points intersect, i.e.\ when will their minimum distance be
an infinitesimal of higher order than the element of arc?
Geometrically we see that the characteristic of the two
adjacent tangent planes must be perpendicular to its conjugate.
Conversely, when we do progress along such an infinitesimal
arc, the tangent plane may be said to rotate about a line
%[File: 198.png]
perpendicular\index{Perpendicularity} to the element of progression, and adjacent
normals\index{Normals02@--- to surface} are coplanar. At any general point of the surface,
except at an umbilical point where the involution of conjugate
tangents is made up of mutually perpendicular tangents,
there will be just two tangents which are mutually conjugate
and mutually perpendicular, and these give the elements
desired.

This fairly plausible geometrical reasoning may easily be
put on a sound analytical basis. The necessary and sufficient
condition that the four points $(x)$, $(y)$, $(x + dx)$, $(y + dy)$ should
be coplanar is

\[
 | yx\,dx\,dy | = 0,
\]
\[
 \left|\begin{array}{ccc}
(xx) & (x\,dx) & (x\,dy) \\
\Bigl(x\dfrac{\partial x}{\partial u}\Bigr) & \Bigl(\dfrac{\partial x}{\partial u}dx\Bigr) & \Bigl(\dfrac{\partial x}{\partial u}dy \Bigr) \smallskip\\
 \Bigl(x\dfrac{\partial x}{\partial v}\Bigr) & \Bigl(\dfrac{\partial x}{\partial  v}dx\Bigr) &
 \Bigl(\dfrac{\partial x}{\partial v}dy \Bigr)
 \end{array}\right|  = 0. \tag*{by (14)}
\]
\[
\begin{array}{|cc|}
Edu + Fdv & Ddu + D'dv\\
Fdu + Gdv & D'du + D''dv
\end{array} = 0. \tag{21}
\]

This is the Jacobian of the binary homogeneous forms (13)
and (20), and gives the two tangents which are both mutually
perpendicular and mutually conjugate; the indetermination
mentioned above occurs in the case where
\[
E:F:G = D:D':D''.
\]

\smallskip\emph{Theorem}~8. The normals to a surface may be assembled
into two families of developable surfaces. Each normal, with
the exception of those at umbilical points, lies in one surface
of each family.

The integral curves of the differential equation (20) are
called \emph{lines of curvature}. We see at once that

\smallskip\emph{Theorem}~9. If two surfaces intersect along a line which is
a line of curvature for each, they intersect at a constant
angle, and if two surfaces intersect at a constant angle along
a curve which is a line of curvature for one it is a line of
curvature for the other.

\mysect{Dupin's theorem for triply orthogonal systems}
This is the theorem of Joachimsthal\index{Joachimsthal}, well known in the
euclidean case. No less celebrated is the beautiful theorem
of Dupin\index{Dupin}.

\smallskip\emph{Theorem}~10. In any triply orthogonal system of surfaces\index{Orthogonal02@--- system of surfaces},
the curves of intersection are lines of curvature.

%[File: 199.png]
Let the three families of surfaces be given by the equations
\[
x_i=f_i(uv),\quad x_i=\phi_i(vw),\quad x_i=\psi_i(wu),
\]
\[
(xx)=k^2,\quad
  \Bigl(x\frac{\partial x}{\partial u}\Bigr) =
  \Bigl(x\frac{\partial x}{\partial v}\Bigr) =
  \Bigl(x\frac{\partial x}{\partial w}\Bigr) = 0.
\]

As the parameter lines are, in every case, mutually perpendicular
\[
\Bigl(\frac{\partial x}{\partial v} \frac{\partial x}{\partial w} \Bigr) =
  \Bigl(\frac{\partial x}{\partial w} \frac{\partial x}{\partial u} \Bigr) =
  \Bigl(\frac{\partial x}{\partial u} \frac{\partial x}{\partial v} \Bigr) =0,
\]
\begin{multline*}
\Bigl(\frac{\partial x}{\partial u} \frac{\partial^2 x}{\partial v\,\partial w}
  \Bigr) +
\Bigl(\frac{\partial x}{\partial w} \frac{\partial^2 x}{\partial u\,\partial v}
  \Bigr) =
\Bigl(\frac{\partial x}{\partial u} \frac{\partial^2 x}{\partial v\,\partial w}
  \Bigr) +
\Bigl(\frac{\partial x}{\partial v} \frac{\partial^2 x}{\partial w\,\partial u}
  \Bigr)
\\ = \Bigl(\frac{\partial x}{\partial v} \frac{\partial^2 x}{\partial w\,\partial u}
  \Bigr) +
\Bigl(\frac{\partial x}{\partial w} \frac{\partial^2 x}{\partial u\,\partial v}
  \Bigr) = 0,
\end{multline*}
\[
\Bigl(x\frac{\partial x}{\partial w}\Bigr) =
  \Bigl(\frac{\partial x}{\partial u} \frac{\partial x}{\partial w} \Bigr) =
  \Bigl(\frac{\partial x}{\partial v} \frac{\partial x}{\partial w} \Bigr) =
  \Bigl(\frac{\partial^2 x}{\partial u\,\partial v}
    \frac{\partial x}{\partial w}\Bigr) = 0,
\]
\[
\left| x \frac{\partial x}{\partial u}
  \frac{\partial x}{\partial v}
  \frac{\partial^2 x}{\partial u\,\partial v} \right| =
  D^{\prime}\sqrt{EG-F^2} = 0,
\]
\[
D^{\prime} = 0.
\]

The vanishing of $D^{\prime}$ and $F$ proves our theorem. Our statement
in Chapter XIII that confocal quadrics intersect in lines
of curvature\index{Curvature07@--- lines of} is hereby justified.

A surface all of whose curves are lines of curvature must
be a sphere. The normal at any point $P$ will determine, with
any other point $Q$ of the surface, a plane. The normals to
the surface along this curve, will, by hypothesis, generate an
evolute, and hence, by (7) make a fixed angle with the plane;
and this angle must be null, since, by hypothesis, one normal
lies in the plane. Hence the normals at $P$ and $Q$ intersect, or
all normals must pass through one point. Evidently the
orthogonal surface to a bundle of concurrent lines is a sphere.

Let us suppose that we have a conformal transformation\index{Conformal transformations} of
space. It will carry a triply orthogonal system of surfaces\index{Orthogonal02@--- system of surfaces}
into another such system, hence a line of curvature into a line
of curvature. It will, therefore, carry any surface all of
whose curves are lines of curvature into another such surface,
hence

\smallskip\emph{Theorem} 11.~Every conformal transformation of space
carries a sphere into a sphere.

Of course a plane is here regarded as a special case of a
sphere.

%[File: 200.png]
Let us now examine the normals along a line of curvature\index{Curvature07@--- lines of}.
Let $r$ be the distance from the point $(x)$ to the intersection
of the normal there with the adjacent normal, a point whose
coordinates shall be called $(\bar{x})$.

\[ \bar{x}_i = x_i \cos\frac{r}{k} - y_i \sin\frac{r}{k},
\]

\[ \frac{d\bar{x}_i}{ds} = \frac{dx_i}{ds}\cos\frac{r}{k} - \frac{dy_i}{ds}\sin\frac{r}{k} - \left [x_i\sin\frac{r}{k} - y_i\cos\frac{r}{k}\right ]\frac{dr}{ds}.
\]

Now, by hypothesis, $\left (\dfrac{d\bar{x}}{ds}\right )$ is linearly dependent on $(x)$
and $(y)$.
\[ dx_i\cos\frac{r}{k} - dy_i\sin\frac{r}{k} = \lambda(x_i + \mu y_i).
\]
\begin{flalign*}
&\text{\indent But} &(x\,dx) &= (x\,dy) = (y\,dx) = (y\,dy) = (xy) = 0, & \phantom{\text{\indent But}}
\end{flalign*}
\[ \lambda = \mu = 0,
\]
\[ dx_i \equiv dy_i\tan\frac{r}{k},
\]
\[ \frac{\partial x_i}{\partial u}du + \frac{\partial x_i}{\partial v} dv = \tan\frac{r}{k}\Bigl[ \frac{\partial y_i}{\partial u}du + \frac{\partial y_i}{\partial v}dv \Bigr].
\]
\label{partial}

\mysect{Curvature of a curve on a surface}
In particular, let us take as parameter lines the lines of
curvature

\[ \frac{\partial x_i}{\partial u} = \tan\frac{r_1}{k} \frac{\partial y_i}{\partial u}, \quad \frac{\partial x_i}{\partial v} = \tan\frac{r_2}{k} \frac{\partial y_i}{\partial v},
\]

\[ (dx\,dy) = \frac{E}{\tan\dfrac{r_1}{k}}du^2 + \frac{G}{\tan\dfrac{r_2}{k}}dv^2,
\]

\[ (dy\,dy) = \frac{E}{\tan^2\dfrac{r_1}{k}}du^2 + \frac{G}{\tan^2\dfrac{r_2}{k}}dv^2. \tag{22}
\]

In the general case,

\[ Edu + Fdv = -\tan\frac{r}{k} [Ddu + D'dv],
\]

\[ Fdu + Gdv = -\tan\frac{r}{k} [D'du + D''dv].
\]

Eliminating $\tan\dfrac{r}{k}$ we get our previous differential equation
for the lines of curvature. On the other hand, if we eliminate
$du$, $dv$ we get
%[File: 201.png]
\[
  (DD'' - {D'}^2)\tan^2 \frac{r}{k}
+ [ED'' + GD - 2FD']\tan\frac{r}{k} + (EG - F^2) = 0.   \tag{23}
\]
\[
  \frac{1}{k\tan\dfrac{r_1}{k}} + \frac{1}{k\tan\dfrac{r_2}{k}}
= -\frac{ED'' + GD - 2FD'}{k[EG - F^2]};
\]
\[
  \frac{1}{k^2\tan\dfrac{r_1}{k} \tan\dfrac{r_2}{k}}
= \frac{DD'' - {D'}^2}{k^2(EG - F^2)}.                 \tag{24}
\]

These last two expressions shall be called the \emph{mean relative
curvature}\index{Curvature04@--- mean relative}
and the \emph{total relative curvature}\index{Curvature05@--- total relative}, respectively.
They are, by XI.~(2), the sum and the product of the curvatures of normal sections through the tangents to the lines of
curvature. Notice that they are absolute simultaneous
invariants of the two binary forms (13), (20).

Let us now look at the more general question of the
curvature of a curve\index{Curvature01@Curvature of a curve} on our surface. As, by (4), this does not
involve derivatives of higher order than the second, the
curvature at any point of a curve of the surface is identical
with that of the curve of intersection of the osculating plane
with the surface. Along our curve $u$ and $v$ will be functions
of $s$ the parameter of length of arc, so that, using our previous
notation,
\[
  t_i = k\Bigl[ \frac{\partial x_i}{\partial u}\frac{du}{ds}
              + \frac{\partial x_i}{\partial v}\frac{dv}{ds} \Bigr].
\]

The cosine of the angle which the principal normal to this
curve makes with the normal to the surface may be written
\[
  \cos \sigma = \pm \frac{(yz)}{k^2},
\]
\[
  \frac{z_i}{\rho} = \frac{dt_i}{ds} + \frac{x_i}{k},\quad
  \frac{\cos\sigma}{\rho}
= \pm\left( \frac{y\dfrac{dt}{ds}}{k^2} \right),
\]
\label{spotthediff}\begin{multline*}
  \frac{dt_i}{ds}
= k\left[
    \frac{\partial^2 x_i}{\partial u^2}\Bigl(\frac{du}{ds}\Bigr)^2
 + 2\frac{\partial^2 x_i}{\partial u\,\partial v}
    \frac{du}{ds}\frac{dv}{ds}
 + \frac{\partial^2 x_i}{\partial v^2}\Bigl(\frac{dv}{ds}\Bigr)^2
   \right]
\\
+ k\Bigl[ \frac{\partial x_i}{\partial u}\frac{d^2u}{ds^2}
        + \frac{\partial x_i}{\partial v}\frac{d^2v}{ds^2} \Bigr],
\end{multline*}
\[
  \frac{\cos\sigma}{\rho}
= \pm\frac{Ddu^2 + 2D'du\,dv + D''dv^2}
          {k[Edu^2 + 2Fdu\,dv + Gdv^2]}.
\]

The indetermination of sign may be used to make the
curvature essentially positive.

%[File: 202.png]
\smallskip\emph{Theorem}~12. Meunier's\index{Meunier}. The curvature of a
curve\index{Curvature01@Curvature of a curve} on a
surface at any point is equal to the curvature of the normal
section with the same tangent divided by the cosine of the
angle which the principal normal makes with the normal to
the surface.

Reverting to our previous expressions $r_1$, $r_2$ and taking the
lines of curvature as parameter lines, the curvature of the
normal sections through the tangents to the lines of curvature
are
\[
\frac {1} {k \tan \dfrac {r_1} {k} },\quad
\frac {1} {k \tan \dfrac {r_2} {k} },
\]
\[
dx_i = \tan \frac {r_1} {k} dy_i,\quad
\delta x_i = \tan \frac {r_2} {k} \delta y_i,
\]
\[
E = \tan \frac {r_1} {k} D,\quad
G = \tan \frac {r_2} {k} D'',
\]
\[
\frac {1} {\rho} = \pm \Biggl[\frac {E} {k \tan \dfrac {r_1} {k}}
\Bigl( \frac {du} {ds} \Bigr)^2 +
\frac {G} {k \tan \dfrac {r_2} {k}} \Bigl(\frac {dv} {ds} \Bigr)^2\Biggr];
\]
or, if $\theta$ be the angle which the chosen tangent makes with
that to $v = $ cons.

\[
\frac {1} {\rho} =
\frac {\cos^2 \theta} {k \tan \dfrac {r_1} {k} } +
\frac {\sin^2 \theta} {k \tan \dfrac {r_2} {k} }.
\]

\smallskip\emph{Theorem}~13. The normal sections of a surface at any point
having the greatest and the least curvature are those determined
by the tangents to the lines of curvature.

\smallskip\emph{Theorem}~14. If on each tangent to a surface at a point
a distance be laid off equal to the square root of the reciprocal
of the measure of curvature of the normal section with that
tangent, the locus of the points so formed will be a central
conic.

\mysect{Dupin's indicatrix}
We leave to the reader the task of filling in the details of
the proof of the last theorem, they will come very easily from
considering the equation of a central conic as given in
Chapter XII\@. Of course the theorem is untrue at a point
where the tangents to the two lines of curvature coincide.
This central conic is called \emph{Dupin's Indicatrix}\index{Indicatrix of Dupin}\index{Dupin} in the
euclidean case, and we may well use the same name in
the non-euclidean case also.

%[File: 203.png]
The curvature of a surface bears a close relation to the
element of arc of the point $(y)$.
\[
-(dx\,dy)=Ddu^2+2D'du\,dv+D''dv^2,
\]
\[
(dy\,dy)=e\,du^2+2f\,du\,dv+g\,dv^2,
\]
\[
\Bigl(y\frac{\partial y}{\partial u}\Bigr)=\Bigl(x\frac{\partial y}{\partial u}\Bigr)=
D'\Bigl(\frac{\partial x}{\partial u} \frac{\partial y}{\partial u}\Bigr)-
D\Bigl(\frac{\partial x}{\partial v} \frac{\partial y}{\partial u}\Bigr)=0,
\]
\[
\lambda \frac{\partial y_i}{\partial u} =
D' \frac{\partial}{\partial s_i}\left\vert sxy\frac{\partial x}{\partial u}
\right\vert -D\frac{\partial}{\partial s_i}\left\vert sxy\frac{\partial x}{\partial v}\right\vert,
\]
\[
\lambda\Bigl(\frac{\partial x}{\partial u} \frac{\partial y}{\partial u}\Bigr)=-\lambda D=-D\sqrt{EG-F^2},
\]
\[
\Bigl(\frac{\partial y}{\partial u}\frac{\partial y}{\partial u}\Bigr)=\frac{D'^2E+D^2G-2DD'F}{EG-F^2},
\]
\[
\Bigl(\frac{\partial y}{\partial u}\frac{\partial y}{\partial v}\Bigr)=\frac{D'D''E-(DD''+D'^2)F+DD'G}{EG-F^2},
\]
\[
\Bigl(\frac{\partial y}{\partial v}\frac{\partial y}{\partial v}\Bigr)=\frac{D''^2E-2D'D''F+D'^2G}{EG-F^2},
\]
\begin{flalign*}
-(e\,du^2 + 2f\,du\,dv &+ g\,dv^2)\\
&=\frac{1}{\tan\dfrac{r_1}{k}\tan\dfrac{r_2}{k}} (Edu^2 + 2Fdu\,dv + Gdv^2)+\\
&+\left(\frac{1}{\tan\dfrac{r_1}{k}}+\frac{1}{\tan\dfrac{r_2}{k}}\right)(Ddu^2 + 2D'du\,dv + D''dv^2).
\tag{25}
\end{flalign*}

\mysect{Torsion of asymptotic lines}
An asymptotic curve\index{Asymptotic lines} has the property that as a point moves
along it, the tangent plane to the surface tends to rotate
about the tangent to this curve, i.e.\ the tangent plane to the
surface is the osculating plane to the curve, and the normal
to the surface is the binormal to the curve. In dealing with
such a curve the point $(y)$ on the normal will replace the
point we previously called $(\xi)$. The torsion of any asymptotic
line will be, by (8),

\begin{equation*}
\frac{1}{T}=\frac{\sqrt{(dy\,dy)}}{kds}.
\end{equation*}

But, in the case of an asymptotic curve, the second part
of the right-hand side of (25) will be zero, while the parenthesis
%[File: 204.png]
in the first part is equal to $ds^2$, hence, for an asymptotic
line
\[ \frac{(dy\,dy)}{k^2ds^2} = \frac{1}{T^2} = \frac{-1}
{k^2\tan\dfrac{r_1}{k}\tan\dfrac{r_2}{k}}. \]

It is not difficult to see that the two asymptotic lines\index{Asymptotic lines} at a
point, when real, have torsion\index{Torsion} with opposite signs, we have
but to look at the special case of a ruled quadric, hence:

\smallskip\emph{Theorem}~15. The two asymptotic lines at a point, when
real, have torsions equal to the two square roots of the
negative of the total relative curvature of the surface.

\smallskip\emph{Theorem}~16. In any surface of constant total relative
curvature, the torsion of every asymptotic line is constant
and equal to a square root of the total relative curvature, and
the necessary and sufficient condition that a surface should
have constant total relative curvature is that the asymptotic
lines of one set should have constant torsion. Under these
circumstances the asymptotic lines of the other set will have a
constant torsion equal to the negative of that already given,
and the square of either torsion will be the total relative
curvature.

\mysect{Total relative curvature, its relation to Gaussian curvature}
In speaking of the total curvature of a surface we have
used the word \emph{relative}\index{Curvature05@--- total relative}. It is now time to explain why that
adjective is chosen. Let us try to express our total relative
curvature in terms of $E$, $F$, $G$ and their derivatives. We have
\[ \frac{1}{k^2\tan\dfrac{r_1}{k}\tan\dfrac{r_2}{k}} =
\frac{DD''-D'^2}{k^2(EG - F^2)}. \tag{24}\]

For the sake of simplicity we shall take as parameter lines
$u$, $v$ the isotropic curves\index{Isotropic01@Isotropic curves} of the surface, i.e.\ those whose
tangents also touch the Absolute. We assume that our
surface is not a developable circumscribed to the Absolute,
and that in the region considered no tangent plane to the
surface touches the Absolute. The isotropic curves at every
point will therefore be distinct. We shall have

\[ E= G = 0,\quad (xx) = k^2,
\]

\[
\Bigl ( x \frac{\partial x}{\partial u} \Bigr ) =
\Bigl ( x \frac{\partial x}{\partial v} \Bigr ) =
\Bigl ( x \frac{\partial^2 x}{\partial u^2} \Bigr ) =
\Bigl ( x \frac{\partial^2 x}{\partial v^2} \Bigr ) =
0,
\]

\[ 2Fdu\,dv = ds^2,
\]

\[ \Bigl ( \frac{\partial^2 x}{\partial u^2} \frac{\partial x}{\partial v} \Bigr ) = \frac{\partial F}{\partial u},
\]

%[File: 205.png]
\[ \Bigl( \frac{\partial x}{\partial u} \frac{\partial^2 x}{\partial v^2} \Bigr) = \frac{\partial F}{\partial v},
\]

\[ \Bigl(\frac{\partial^2 x}{\partial u^2}\frac{\partial^2 x}{\partial v^2}\Bigr)
- \Bigl(\frac{\partial^2 x}{\partial u\,\partial v}\frac{\partial^2 x}{\partial u\,\partial v}\Bigr) = \frac{\partial^2 F}{\partial u\,\partial v}; \]

\[
D'^2 = \frac{-1}{F^2}
\begin{vmatrix}
k^2 & 0 & 0 & -F\\
0 & 0 & F & 0\\
0 & F & 0 & 0\\
-F & 0 & 0 & \Bigl(\dfrac{\partial^2 x}{\partial u\,\partial v}\dfrac{\partial^2 x}{\partial u\,\partial v}\Bigr)
\end{vmatrix},
\]

\[
DD''=\frac{-1}{F^2}
\begin{vmatrix}
k^2 & 0 & 0 & 0\\
0 & 0 & F & \dfrac{\partial F}{\partial v}\smallskip\\
0 & F & 0 & 0\\
0 & 0 & \dfrac{\partial F}{\partial u} & \Bigl(\dfrac{\partial^2 x}{\partial u^2}\dfrac{\partial^2 x}{\partial v^2}\Bigr)
\end{vmatrix},
\]

\[
\frac{DD''-D'^2}{k^2(EG-F^2)}
=\frac{1}{F^2}\Bigl[\frac{1}{F} \frac{\partial F}{\partial u}\frac{\partial F}{\partial v}
- \frac{\partial^2 F}{\partial u\,\partial v}\Bigr] - \frac{1}{k^2}. \tag{26}
\]

The first expression on the right is the Gaussian curvature\index{Curvature02@--- Gaussian}
of a two-dimens\-ional manifold whose squared distance element
is $2Fdu\,dv$%
\footnote{Cf.\ Bianchi\index{Bianchi}, loc.\ cit., p.~68.}.

\smallskip\emph{Theorem}~17%
\footnote{Cf.\ Bianchi, loc.\ cit., p.~609.}.
The total relative curvature\index{Curvature05@--- total relative} of a surface is
equal to the difference between its total Gaussian curvature
and the measure of curvature of
space\index{K@$\dfrac{1}{k^2}$ measure of curvature of space}\index{Measure02@--- of curvature of space}\index{Curvature06@--- of space}.

The Gaussian curvature may also be called the \emph{total
absolute curvature}. Notice that this theorem remains true
in euclidean space where the measure of curvature is $0$.


\mysect{Surfaces of zero relative curvature}
The problem of finding all surfaces of total relative curvature
zero\index{Curvature08@--- surfaces of zero} is quickly solved. Let us assume that

\[
\tan\frac {r_2} k= \infty.
\]

Then, by an equation just preceding (22), as

\[
\frac{\partial x_i}{\partial v} \neq 0, \quad \frac{\partial y_i}{\partial v}=0, \]
and there will be the same tangent plane all along $u = \text{const}$.

\smallskip\emph{Theorem}~18. A surface of total relative curvature zero is
a developable.

%[File: 206.png]
Clearly every developable has total relative curvature zero.

Much more interest attaches to the surfaces of total Gaussian
curvature\index{Curvature02@--- Gaussian} zero, i.e.\ those which are developable upon the
euclidean plane. The total relative curvature will be $-\dfrac{1}{k^2}$.
There is an advantage in considering the hyperbolic and
elliptic cases separately.

In the hyperbolic case let $(y)$ be the centre of a sphere, the
constant distance thence to points of the surface being $r$

\[ \cos\frac r k = \frac {(xy)}{k^2}, \quad k^2\tan^2\frac r k = k^2\left[\frac{(xx)(yy)-(xy)^2}{(xy)^2}\right].
\]

If the surface is to be actual $(xx) = k^2$. If the sphere be
a proper one $(yy) = k^2$, the total relative curvature will be
$> \dfrac{-1}{k^2}$. In the case of a horocyclic surface\index{Horocyclic surface} we may not
assume $(yy) = k^2$, but must treat $(y)$ as homogeneous coordinates
where $(yy)=0$. We get then

\[ \frac{1}{k^2 \tan^2\dfrac{r}{k}} = - \frac{1}{k^2}.
\]

\smallskip\emph{Theorem}~19.%
\footnote{Cf.\ Manning\index{Manning}, loc.\ cit., p.~52; Killing, \emph{Die Grundlagen der Geometrie},
Paderborn, 1898, p.~33.}
%endfootnote
The horocyclic surface of hyperbolic space is developable on the euclidean plane.

\mysect{Surfaces of zero Gaussian curvature}
In elliptic space there is a peculiarly notable class of
surfaces of Gaussian curvature zero, ruled surfaces. We have
already seen one example, the Clifford\index{Clifford} Surface of Chapter X.
This quadric, be it remembered, cuts the Absolute\index{Absolute} in two
generators of each set, and its own generators form an orthogonal
system. Now Dupin's\index{Dupin} indicatrix\index{Indicatrix of Dupin} shows that the
normal sections of greatest and of least curvature will be
determined by tangents bisecting the angles of the two
generators, and the planes of these normal sections will cut
the surface in two circles whose axes are the axes of revolution
of the surface, and whose centres lie on these axes. The
centres are thus mutually orthogonal points\index{Orthogonal points}, hence the total
relative curvature is $- \dfrac {1} {k^2},$ and the Gaussian curvature is zero.
This statement was given without proof in Chapter X. We
notice also that the generators of either set are paratactic,
and the question arises, will not this fact alone constitute
a sufficient condition that a surface should have Gaussian
curvature zero?

%[File: 207.png]
Let us imagine that we have a surface generated by $\infty^1$
paratactic\index{Parataxy} lines.%
\footnote{For an interesting treatment of these surfaces see Bianchi\index{Bianchi}, `Le superficie
a curvatura nulla nella geometria ellitica,' \emph{Annali di Matematica}, Serie 2,
Tomo 24, 1896.}
%endfootnote
The parameter $v$ shall give the actual distance measured on each line from an orthogonal trajectory
$v =$ const. We have for our distance element
\[
  ds^2 = Edu^2 + dv^2.
\]

We know, moreover, by Chapter IX that if two lines be
paratactic they have an infinite number of common perpendiculars
on which they determine congruent distances.
Hence $E$ is a function of $u$ alone, and we may choose $u$ so
that it shall be equal to unity
\[
  ds^2 = du^2 + dv^2,     \tag{27}
\]
and the Gaussian curvature\index{Curvature02@--- Gaussian}\index{Curvature08@--- surfaces of zero} is zero.

Conversely, suppose that we have a ruled surface of
Gaussian curvature zero. The square of the element of arc
may be written
\[
  ds^2 = Edu^2 + dv^2.
\]

Since the Gaussian curvature is zero
\[
  \frac{\partial^2 \sqrt{E}}{\partial v^2} = 0,\quad
  \sqrt{E} = \theta(u)v + \psi(u).
\]
On the other hand we may write our surface parametrically
in the form
\[
  x_i = f_i(u)\cos \frac{v_i}{k} + \phi_i(u)\sin \frac{v}{k},
\]
with the additional conditions
\[
  (f\!f) = (\phi\phi) = k^2,\quad
  (f\!f') = (\phi\phi') = (f\phi) = (f\phi') + (\phi f') = 0;
\]
\[
  E = (f'f')\cos^2 \frac{v}{k} + (\phi'\phi')\sin^2 \frac{v}{k}
    + 2(f'\phi')\sin \frac{v}{k} \cos \frac{v}{k},
\]
\[
  kF
= (\phi f')\cos^2 \frac{v}{k} - (f\phi')\sin^2 \frac{v}{k} = 0,\quad
  (f\phi') = (\phi f') = 0.
\]

These are identical with previous
\[
  E = [\theta(u)]^2 v^2 + 2\theta(u)\psi(u)v + [\psi(u)]^2,
\]
only when
\[
  \theta(u) \equiv 0.
\]

We may, then, take
\[
  E = 1,\quad ds^2 = du^2 + dv^2;
\]
%[File: 208.png]
and this shows that two adjacent generators determine equal
distances on all their orthogonal trajectories, and so are
paratactic\index{Parataxy}.

\mysect{Ruled surfaces of zero Gaussian curvature in elliptic or spherical space}
\smallskip\emph{Theorem}~20. The necessary and sufficient condition that
a ruled surface in elliptic space should have Gaussian curvature
zero\index{Curvature02@--- Gaussian}\index{Curvature08@--- surfaces of zero}
is that its generators should be paratactic.

Another highly interesting criterion for a surface of constant
Gaussian curvature zero is obtained as follows:
\[
E=G=1, \quad F=0;
\]
\[
\Bigl( \frac{\partial   x}{\partial u  }
       \frac{\partial^2 x}{\partial u \,\partial v} \Bigr) =
\Bigl( \frac{\partial^2 x}{\partial u^2}
       \frac{\partial   x}{\partial v  } \Bigr) =
\Bigl( \frac{\partial   x}{\partial u  }
       \frac{\partial^2 x}{\partial v^2} \Bigr) = 0, \quad
\Bigl(x\frac{\partial^2 x}{\partial u^2}\Bigr) = -1.
\]

The coordinates of the absolute pole of the tangent plane
are
\[
y_i = \frac{\partial}{\partial s_i}
  \left\lvert sx \frac{\partial x}{\partial u}
                 \frac{\partial x}{\partial v} \right\rvert.
\]

The coordinates of the absolute pole of the osculating plane
to the orthogonal trajectory of the generators, i.e.\ to a curve
$v = \text{const}$, are
\[
\lambda\xi_i =   \frac{\partial    }{\partial r_i}
  \left\lvert rx \frac{\partial   x}{\partial u  }
                 \frac{\partial^2 x}{\partial u^2} \right\rvert,
\]
\[
(y\xi) = 0.
\]

This shows that the generators are binormals to their
orthogonal trajectories. Our given surface may be written
in the form
\[
x_i = x_i(u) \cos\frac{v}{k} + \xi_i(u) \sin\frac{v}{k},
\]
\[
ds^2 = dv^2 +
  \Bigl[                 \cos^2\frac{v}{k}
       + \frac{k^2}{T^2} \sin^2\frac{v}{k} \Bigr] du^2.
\]

This reduces to
\[
du^2+dv^2,
\]
when, and only when
\[
\frac{1}{T^2} = \frac{1}{k^2}.
\]

\smallskip\emph{Theorem}~21. The necessary and sufficient condition that a
ruled surface should have Gaussian curvature zero is that it
should be generated by the binormals to a curve whose
squared torsion\index{Torsion} is equal to the measure of curvature of
space\index{Measure02@--- of curvature of space}.

The proof given holds equally in hyperbolic space; the
surface is, however, in that case imaginary. If we compare
theorems 16 and 21, we get

%[File: 209.png]
\smallskip\emph{Theorem}~22. The necessary and sufficient condition that
it should be possible to assemble the
normals\index{Congruence09@--- of normals}\index{Congruence10@--- of normals, to surfaces of Gaussian curvature zero}
to a surface\index{Normals02@--- to surface} into
one parameter families of left (right)
paratactics\index{Left and right parataxy}\index{Parataxy}\index{Right and left parataxy}, is that the
given surface should have
Gaussian curvature\index{Curvature02@--- Gaussian}
zero\index{Curvature08@--- surfaces of zero}. It will,
then, be possible to assemble the normals into families of
right (left) paratactics also. The intersections of the given
surface with the various families of paratactics will be the
asymptotic lines of the former.

\mysect{Geodesic curvature and geodesic lines}
We shall, as in euclidean space, define as the geodesic
curvature at any point of a curve on our surface, the curvature
of its orthogonal projection on the tangent plane at that point.
Let us denote this by $\dfrac{1}{\rho_g}$, while $\sigma$ is the angle which the
osculating plane makes with the tangent plane to the surface.
Then, applying Meunier's\index{Meunier} theorem to the projecting cone
\[
  \frac{1}{\rho_g} = \frac{\cos\sigma}{\rho}.     \tag{28}
\]

As a first exercise, assuming $F = 0$, let us find the geodesic
curvature\index{Curvature03@--- geodesic}\index{Geodesic01@Geodesic curvature} of one of our parameter lines
\begin{align*}
  ds_v &= \sqrt{G}\,dv,
\\
  t_i  &= \frac{k}{\sqrt{G}} \frac{\partial x_i}{\partial v},
\end{align*}
\[
  \frac{z_i}{\rho} = \frac{dt_i}{ds} + \frac{x_i}{k}
= \frac{k}{\sqrt{G}}
  \left[ \frac{\partial}{\partial v}
         \Bigl(\frac{1}{\sqrt{G}}
         \frac{\partial x_i}{\partial v}\Bigr)
  \right] + \frac{x_i}{k}.
\]

To find $\cos\sigma$ must determine the distance of $(z)$ from
the point orthogonal to $(x)$ on the curve $v =$ const., i.e.\ to the
point
$\dfrac{k}{\sqrt{E}}\Bigl( \dfrac{\partial x}{\partial u} \Bigr)$.
\[
  \cos\frac{\sigma}{\rho}
= \frac{1}{\sqrt{EG}}
  \Bigl( \frac{\partial x}{\partial u}
         \frac{\partial}{\partial v}
         \bigl( \frac{1}{\sqrt{G}}
                \frac{\partial x}{\partial v} \bigr)
  \Bigr),
\]
\[
  \frac{1}{\rho_g}
= \frac{-1}{\sqrt{EG}} \frac{\partial \sqrt{G}}{\partial u}.  \tag{29}
\]

For the other parameter line
\[
  \frac{1}{\rho_g}
= \frac{-1}{\sqrt{EG}} \frac{\partial \sqrt{E}}{\partial v}.
\]

Let us now, more generally, find the geodesic curvature of
the curve
\[
  v \cdot {} = v \cdot(u).
\]

%[File: 210.png]
Once more we shall make use of the isotropic\index{Isotropic01@Isotropic curves} parameters,
so that

\[ E = G = 0,
\]

\[ ds = \sqrt{2Fv'}du, \quad v'= \frac{dv}{du},
\]
\ungap

\begin{multline*}
\quad t_i = \frac{k}{\sqrt{2Fv'}}\Bigl[\frac{\partial x_i}{\partial u} + v'\frac{\partial x_i}{\partial v}\Bigr],
\\
\shoveleft{\frac{z_i}{\rho} = \frac{k}{2Fv'}\Bigl[\frac{\partial^2 x_i}
{\partial u^2} + 2\frac{\partial^2 x_i}{\partial u\,\partial v}v' +
\frac{\partial^2 x_i}{\partial v^2}v'^2 + \frac{\partial x_i}{\partial v}v''\Bigr] +}
\\
+ \frac{k}{\sqrt{2Fv'}}\Bigl[\frac{\partial x_i}{\partial u} +
\frac{\partial x_i}{\partial v}v'\Bigr]\frac{d}{du}\frac{1}{\sqrt{2Fv'}} + \frac{x_i}{k}.
\end{multline*}
For an orthogonal trajectory to this curve
\begin{align*}
\frac{\delta v}{\delta u} &= - \frac{dv}{du} = - v', \\
\delta s &= \delta u \sqrt{-2Fv'}, \\
\bar{t}_i &= \frac{-k}{\sqrt{-2Fv'}}\left[ \frac{\partial x_i}{\partial u} -
v'\frac{\partial x_i}{\partial v}\right], \\
\cos\frac{\sigma}{\rho} &= \frac{1}{\rho k^2}(z\bar{t}), \\
\frac{1}{\rho_g} &= \frac{-1}{\sqrt{-2Fv'}}\left[\frac{\dfrac
{\partial F}{\partial v}v' - \dfrac{\partial F}{\partial u}}{2F} +
\frac{v''}{2v'}\right] \\
&\qquad = \frac1{\sqrt{-F^2}}\Bigl[\frac d{du}\frac{\sqrt{F}}{\sqrt{2v'}} -
\frac{\partial}{\partial v}\sqrt{2Fv'}\Bigr].\tag{30}
\end{align*}

What will be the nature of those curves whose geodesic
curvature\index{Curvature03@--- geodesic}\index{Geodesic01@Geodesic curvature}
vanishes, i.e.\ those curves whose osculating planes
pass through the normal? These shall be called geodesic
lines\index{Geodesic02@--- lines}, and, evidently, we shall have

\[ \frac{dv}{du}\frac{\sqrt{F}}{\sqrt{2v'}} =
\frac{\partial}{\partial v}\sqrt{2Fv'}. \]

This merely tells us that our given curve is an extremal,
i.e.\ the first variation of the length between two fixed points
is zero. If we assume that two sufficiently near points can
always be connected by a curve of minimum length%
\footnote{For a proof of the existence of this curve, see Bolza\index{Bolza},
\emph{Lectures on the Calculus of Variations}, Chicago, 1904, Ch. VIII.}
%endfootnote
we shall get

%[File: 211.png]
\smallskip\emph{Theorem}~23.  The curve of shortest length between two
points of a surface is a geodesic line\index{Geodesic02@--- lines}.

Remembering 21, we have further


\smallskip\emph{Theorem}~24. The orthogonal trajectories of a family of
paratactic lines are geodesics of the surface generated by
these lines.

If we consider the two planes through the normal\index{Normals02@--- to surface} to a
surface and the two tangents to the lines of curvature, we
see that they are mutually perpendicular, and that each
touches the focal surface of the congruence of normals at the
point of intersection of the two adjacent normals in the other
plane.%
\footnote{For a simple proof of this general theorem see Picard\index{Picard}, loc.\ cit., vol.~i,
pp 307, 308.}
%endfootnote

\smallskip\emph{Theorem}~25. In any congruence of normals\index{Congruence09@--- of normals}, the edges of
regression of the developable surfaces are geodesics of the
focal surfaces\index{Focal05@--- surfaces} of the congruence.

The osculating plane to any straight line is indeterminate;
the line is, therefore, a geodesic for all space; a result also
evident from Chapter II.~30. It is also clear that as the
expressions for the geodesic curvature of a parameter line in
terms of $E$, $F$, $G$ and their derivatives are the same in euclidean
and in non-euclidean space, and the formula for the distance
element is written in the same shape, so will the formula for
the geodesic curvature of any curve be the same. We might,
for instance, have given this formula in terms of the Beltrami\index{Beltrami}
invariants. We have, however, purposely avoided the introduction
of these into the present work, and will therefore
merely refer the reader to the current textbooks in differential
geometry,%
\footnote{e.g.\ Bianchi\index{Bianchi}, \emph{Differentialgeometrie}, cit.\ p.~258.}
%endfootnote

\mysect{Necessary conditions for a minimal surface}
\index{Minimal surfaces|(}

As a last problem in the differential geometry of surfaces
let us take up that of minimal surfaces. To begin with, what
will be the element of area? It is perfectly clear that the
expression for this will be the same as that in the euclidean
case. The sine of the angle formed by the parameter lines
will be, by (15)
\[
  \frac{\sqrt{EG-F^2}}{\sqrt{EG}},
\]
and the area of the elementary quadrilateral
\[
  \sqrt{EG-F^2}du\,dv.
\]

%[File: 212.png]
Let us, in particular, take the lines of curvature as parameter
lines. The formula for the area enclosed by a given
curve will be
\[
\iint\sqrt{EG}du \,dv.
\]

Let us compare this with the area\index{Area01@Area} enclosed by this curve
upon a surface reached by laying off on each normal an
extremely small distance $w(uv)$.
\begin{align*}
\overline{x_i} &= x_i\cos\frac{w}{k} + y_i\sin\frac{w}{k},
\\
d\overline{x_i}&=dx_i\cos\frac{w}{k} +dy_i\sin\frac{w}{k} -
\frac{1}{k}\left[ x_i\cos\frac{w}{k} - y_i\sin\frac{w}{k} \right]dw.
\end{align*}

The squared element of arc for this surface will be by (22)
\[
d\overline{s}^2 =
  E\left[ \cos\frac{w}{k} +
    \frac{\sin\dfrac{w}{k}}{\tan\dfrac{r_1}{k}} \right]^2 du^2 +
  G\left[ \cos\frac{w}{k} +
    \frac{\sin\dfrac{w}{k}}{\tan\dfrac{r_2}{k}} \right]   dv^2 +
  \frac{dw^2}{k^2}.
\]

This becomes, when we neglect powers of $w$ above the first,
\[
d\overline{s}^2 =
  E\left[1 + \frac{2\dfrac{w}{k}}{\tan\dfrac{r_1}{k}}\right] du^2 +
  G\left[1 + \frac{2\dfrac{w}{k}}{\tan\dfrac{r_2}{k}}\right] dv^2.
\]

For the surface element we have
\[
\sqrt{EG}\left[1+
  2\frac{w}{k}
  \left( \frac{\tan\dfrac{r_1}{k} + \tan\dfrac{r_2}{k}}
              {\tan\dfrac{r_1}{k}   \tan\dfrac{r_2}{k}} \right) +
  4\frac{\dfrac{w^2}{k^2}}
              {\tan\dfrac{r_1}{k}   \tan\dfrac{r_2}{k}}
  \right]^\frac{1}{2} du \,dv.
\]

Developing by the binomial theorem, and neglecting higher
powers of $w$ we have
\[
\iint\sqrt{EG}
  \left[ 1
       + \frac{w}{k}
         \left( \frac{\tan\dfrac{r_1}{k} + \tan\dfrac{r_2}{k}}
                     {\tan\dfrac{r_1}{k}   \tan\dfrac{r_2}{k}} \right)
  \right] du \,dv.
\]

%[File: 213.png]
If we define as a minimal surface one where the first
variation of the area is zero, certainly a necessary condition,
we have

\smallskip\emph{Theorem}~26. The necessary and sufficient condition that a
surface should be minimal is that the mean relative curvature
should be zero.

We see from (23) that the numerator of the expression for
the relative mean curvature\index{Curvature04@--- mean relative} is the simultaneous invariant
of (13) and (20), and vanishes when, and only when, the
tangents to the asymptotic lines\index{Asymptotic lines} are harmonically separated
by those to the isotropic ones, hence

\smallskip\emph{Theorem}~27. The necessary and sufficient condition that a
surface should be minimal is that the asymptotic lines should
form an orthogonal system.

This theorem justifies our statement in Chapter X that a
Clifford\index{Clifford} surface is a minimal surface. It is very interesting
that in non-euclidean space we should have an algebraic
minimal surface (other than the plane) whose order is as low
as two.

We may go one long step further towards the solution of
the problem of minimal surfaces, namely, exhibit the differential
equations on which they depend.%
\footnote{Cf.\ Darboux\index{Darboux}, \emph{Leçons
sur la th\'eorie g\'en\'erale des surfaces}, vol.~iii, ch.~xiv, Paris,
1894. The reader is strongly urged to read this interesting chapter in connection with the present work.}
%endfootnote

\mysect{Integration of the resulting differential equations}
We shall once more take as parameter lines the isotropic
ones. These will form a conjugate system, since they are
harmonically separated by the asymptotic lines, hence
\[
  E = G = D' = 0,
\]
\[
  \frac{\partial^2 x_i}{\partial u\,\partial v}
= Ax_i + B\frac{\partial x_i}{\partial u}
       + C\frac{\partial x_i}{\partial v},
\]
\[
  BF = \frac{1}{2}\frac{\partial G}{\partial u} = 0, \quad
  CF = \frac{1}{2}\frac{\partial E}{\partial u} = 0, \quad
  F = -Ak^2,
\]
\[
  \frac{\partial^2 x_i}{\partial u\,\partial v} + \frac{1}{k^2}Fx_i = 0.
\tag{31}
\]

It is merely necessary to find $F$ and take for $(x)$ four
solutions of (3) subject to the restriction $(xx) = k^2$.
Let us put
\[
  \frac{\partial^2 x_i}{\partial u^2}
= P\frac{\partial x_i}{\partial u}
+ Q\frac{\partial x_i}{\partial v} + Rx_i + Sy_i,
\]
%[File: 214.png]
which is certainly possible, since
\begin{equation*}
   \left| x y \frac{\partial x_i}{\partial u} \frac{\partial x}{\partial v} \right| \not\equiv 0.
\end{equation*}

We easily find
\begin{equation*}
   R = Q = 0, \quad FP = \frac{\partial F}{\partial u} ,
\end{equation*}
\begin{equation*}
   \frac{\partial^2 x_i}{\partial u^2} = \frac{1}{F} \frac{\partial F}{\partial w} \frac{\partial x_i}{\partial u} + S y_i.
\end{equation*}

\label{d3x}Now
\begin{equation*}
   \frac{\partial}{\partial v} \Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial^2 x}{\partial u^2} \Bigr) = 2 \Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial^3 x}{\partial u^2 \, \partial v} \Bigr) = -2 \Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial}{\partial u} (Fx) \Bigr) = 0,
\text{\ by (31)}.
\end{equation*}
\begin{flalign*}
&\text{\indent Hence}
   &&\Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial^2 x}{\partial u^2} \Bigr) = \phi (u).
&\phantom{\text{\indent Hence}}
\end{flalign*}
\begin{flalign*}
&\text{\indent If}
   &&\phi (u) \equiv 0, \quad D = 0.
&\phantom{\text{\indent If}}
\end{flalign*}

The total relative curvature is zero, and the surface is
developable. In a developable surface the asymptotic lines\index{Asymptotic lines}
fall together, by (24); hence a minimal developable must be
circumscribed to the Absolute, and cannot be real in the
actual domain. Conversely it is clear that every developable
circumscribed to the Absolute is a minimal surface in that its
asymptotic lines are mutually perpendicular, even though it
lie in a region of our space where the concept area has not
been defined.

In the second case let us suppose $\phi (u) \not\equiv 0$.

Let us replace $u$ by $\bar{u} (u)$ so that $\Bigl( \dfrac{\partial^2 x}{\partial \bar{u}}
\dfrac{\partial^2 x}{\partial \bar{u}} \Bigr)
= \dfrac{1}{k^2}$. Then
replace the letter $\bar{u}$ by the letter $u$ once more.

\begin{flalign*}
&\text{\indent Then}
&& S = \frac{1}{k^2}, \quad \frac{\partial^2 x_i}{\partial u^2} =
   \frac{1}{F} \frac{\partial F}{\partial u} \frac{\partial x_i}{\partial u} + \frac{y_i}{k^2}.
&\phantom{\text{\indent Then}}
\end{flalign*}
\begin{flalign*}
&\text{\indent In like manner}
&& \frac{\partial^2 x_i}{\partial v^2} =
   \frac{1}{F} \frac{\partial F}{\partial v} \frac{\partial x_i}{\partial v} + \frac{y_i}{k^2}.
&\phantom{\text{\indent In like manner}}
\end{flalign*}

Multiplying through by $\dfrac{\partial^2 x_i}{\partial u^2}$ and adding
\begin{equation*}
   \Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial^2 x}{\partial v^2} \Bigr) =
   \frac{1}{k^2} + \frac{1}{F} \frac{\partial F}{\partial u} \frac{\partial F}{\partial v}.
\end{equation*}

%[File: 215.png]
On the other hand

\begin{align*}
\frac{\partial^2 F}{\partial u \,\partial v}=&
  \Bigl(\frac{\partial^2 x}{\partial u^2}
  \frac{\partial^2 x}{\partial v^2}\Bigr) +
  \Bigl(\frac{\partial x}{\partial u}\frac{\partial^3 x}{\partial u \,\partial v^2}\Bigr),\\
=&\Bigl(\frac{\partial^2 x}{\partial u^2}
  \frac{\partial^2 x}{\partial v^2}\Bigr)-\frac{1}{k^2}\Bigl(\frac{\partial x}{\partial u}
  \frac{\partial}{\partial v} (xF)\Bigr),\\
=&\Bigl(\frac{\partial^2 x}{\partial u^2} \frac{\partial^2 x}{\partial v^2}\Bigr)-\frac{1}{k^2}F^2,\\
\frac{\partial^2 F}{\partial u \,\partial v} =&
  \frac{1-F^2}{k^2}+\frac{1}{F} \frac{\partial F}{\partial u} \frac{\partial F}{\partial v},
\end{align*}
\[
k^2 \frac{\partial^2\log F}{\partial u \,\partial v}=\frac{1}{F}-F.\tag{32}
\]
\begin{flalign*}
&\text{\indent Lastly, let us put}
&&F=e^{2iw},
&\phantom{\text{\indent Lastly, let us put}}
\end{flalign*}
\begin{equation*}
k^2 \frac{\partial^2 w}{\partial u \,\partial v}+\sin2w=0.\tag{33}
\end{equation*}

When $F$ has been found we may, as already noted, find $(x)$
from (31).
\index{Minimal surfaces|)}

%[File: 216.png]
%CHAPTER XVI
\mychap{XVI}{DIFFERENTIAL LINE-GEOMETRY}
\mysect{Analoga of Kummer's coefficients}
\index{Congruence07@--- of lines, analytic|(}

In Chapter IX we gave the foundations of the Pl\"uckerian
line-geometry, and the fundamental invariants of a metrical
character; in Chapter X we saw what advantages arose from
taking the cross instead of the line as element, and introducing
suitable coordinates. Chapter XV was given to the
differential geometry of curves and surfaces. It is the object
of the present chapter to draw all of these threads together
into a theory of differential line-geometry, and, in particular,
a theory of two-parameter line systems or congruences.%
\footnote{The first part of the present chapter follows, with slight modifications,
a rather inaccessible memoir by Fibbi\index{Fibbi}, `I sistemi doppiamente infiniti di
raggi negli spazii di curvatura costante,' \emph{Annali della R. Scuola Normale Superiore},
Pisa, 1891.}
%endfootnote

We shall define as an analytic line-congruence a system
whose Pl\"uckerian coordinates are analytic functions of two
independent parameters, say $u$ and $v$. This is equivalent to
supposing that our lines are determined by two points, which
we may assume mutually orthogonal\index{Orthogonal points}, whose coordinates are
analytic functions of the two independent parameters in
question.
\begin{equation*}
\tag{1}
x_i = x_i(uv), \quad y_i = y_i(uv), \quad
(xx) = (yy) = k^2, \quad (xy) = 0.
\end{equation*}

Following Kummer's\index{Kummer} classical method, we shall write the
following fundamental quadratic expression:
\begin{multline*}
k^2(dx\,dx)-(y\,dx)^2 =
  \begin{Vmatrix}
     y_0 &  y_1 &  y_2 &  y_3 \\
    dx_0 & dx_1 & dx_2 & dx_3
  \end{Vmatrix}^2
\\= Edu^2 + 2Fdu\,dv + Gdv^2, \phantom{(2)'''}
\end{multline*}\ungap
\begin{multline*}
\tag{2}
k^2(dy\,dy)-(x\,dy)^2 =
  \begin{Vmatrix}
     x_0 &  x_1 &  x_2 &  x_3 \\
    dy_0 & dy_1 & dy_2 & dy_3
  \end{Vmatrix}^2
\\= E'du^2 + 2F'du\,dv + G'dv^2.
\end{multline*}\ungap\label{dudv}
\begin{align*}
k^2(dx\,&dy) = e\,du^2 + (f + f')du\,dv + g\,dv^2,
\\
&k^2\Bigl( \frac{\partial x}{\partial u}
           \frac{\partial x}{\partial u} \Bigr) -
    \Bigl(y\frac{\partial x}{\partial u} \Bigr)^2 = E,
\\
&k^2\Bigl( \frac{\partial x}{\partial u}
           \frac{\partial x}{\partial v} \Bigr) -
    \Bigl(y\frac{\partial x}{\partial u} \Bigr)
    \Bigl(y\frac{\partial x}{\partial v} \Bigr) = F,
\\
%[File: 217.png]
& k^2\Bigl( \frac{\partial x}{\partial v}
            \frac{\partial x}{\partial v} \Bigr)
-    \Bigl(y\frac{\partial x}{\partial v} \Bigr)^2 = G.   \tag{3}
\\
& k^2\Bigl( \frac{\partial y}{\partial u}
            \frac{\partial y}{\partial u} \Bigr)
-    \Bigl(x\frac{\partial y}{\partial u} \Bigr)^2 = E',
\\
& k^2\Bigl( \frac{\partial y}{\partial u}
            \frac{\partial y}{\partial v} \Bigr)
-    \Bigl(x\frac{\partial y}{\partial u} \Bigr)
     \Bigl(x\frac{\partial y}{\partial v} \Bigr) = F',
\\
& k^2\Bigl( \frac{\partial y}{\partial v}
            \frac{\partial y}{\partial v} \Bigr)
-    \Bigl(x\frac{\partial y}{\partial v} \Bigr)^2 = G'.   \tag{4}
\\
& k^2\Bigl( \frac{\partial x}{\partial u}
     \frac{\partial y}{\partial u} \Bigr) = e,\phantom{f'}
\quad
  k^2\Bigl( \frac{\partial x}{\partial v}
     \frac{\partial y}{\partial u} \Bigr) = f,
\\
& k^2\Bigl( \frac{\partial x}{\partial u}
     \frac{\partial y}{\partial v} \Bigr) = f',\phantom{e}
\quad
  k^2\Bigl( \frac{\partial y}{\partial v}
     \frac{\partial y}{\partial v} \Bigr) = g.   \tag{5}
\\
  EG - F^2
&= \left| yx \frac{\partial x}{\partial u}
            \frac{\partial x}{\partial v} \right|^2 \equiv \Delta^2,
\quad
  E'G' - F'^2
= \left| xy \frac{\partial y}{\partial u}
            \frac{\partial y}{\partial v} \right|^2 \equiv \Delta'^2.
\tag{6}
\end{align*}

\mysect{Their fundamental relations}
The following relations will subsist between these various
expressions:
\[
  \Delta x_i = k^2\frac{\partial \Delta}{\partial x_i},\quad
  \Delta' y_i = k^2\frac{\partial \Delta'}{\partial y_i},
\]
\begin{align*}
&\text{since}
&(xy)
& = \Bigl( x \frac{\partial x}{\partial u} \Bigr)
= \Bigl( x \frac{\partial x}{\partial v} \Bigr)
= \Bigl( y \frac{\partial y}{\partial u} \Bigr)
= \Bigl( y \frac{\partial y}{\partial v} \Bigr) = 0,
&\phantom{\text{since}}
\end{align*}
\begin{align*}
& E' = k^2 \Bigl( \frac{\partial y}{\partial u}
                  \frac{\partial y}{\partial u} \Bigr)
     - \frac{k^4}{\Delta^2}
       \left| y \frac{\partial y}{\partial u}
                \frac{\partial x}{\partial u}
                \frac{\partial x}{\partial v} \right|^2,
\\
& E' = \frac{1}{\Delta^2} [Ge^2 - 2Fef + Ef^2],
\\
& F' = \frac{1}{\Delta^2} [Gef' - F(eg + f\!f') + Efg].   \tag{7}
\\
& G' = \frac{1}{\Delta^2} [Gf'^2 - 2F(f'g) + Eg^2],
\\
& E = \frac{1}{\Delta'^2} [G'e^2 - 2F'ef' + E'f'^2].    \tag{8}
\\
& F = \frac{1}{\Delta'^2} [G'ef - F'(eg + f\!f') + E'f'g],
\\
& G = \frac{1}{\Delta'^2} [G'f^2 - 2F'(fg) + E'g^2],
\end{align*}\label{gprime}
\[
\Delta\Delta' = (eg - f\!f').                           \tag{9}
\]

%[File: 218.png]
Notice that $\Delta$ and $\Delta'$ being square roots of positive definite
forms cannot vanish in the real domain.

We remember from Chapter IX, that two lines which are
not paratactic have two common perpendiculars\index{Perpendicularity} meeting them
in pairs of mutually orthogonal points\index{Orthogonal points}. Let us, as a first
problem, find where the common perpendicular to a line of
our congruence and an adjacent line meets the given line.
The coordinates of an arbitrary point of our line may be
written $\left( x\cos\dfrac{r}{k} + y\sin\dfrac{r}{k} \right)$ while an arbitrary point of an
adjacent line will be $\lambda(x + dx) + \mu(y + dy)$.

Let us begin by writing that the second of these points is
orthogonal to $\left( x\sin\dfrac{r}{k} - y\cos\dfrac{r}{k} \right)$ the point of the first line
orthogonal to the first point, while, on the other hand, the first
point lies in the absolute polar plane of
$\mu(x + dx) - \lambda(y + dy)$.
There will result two linear homogeneous equations in $\lambda$ and $\mu$
whose determinant must be equated to zero. When this is
simplified in view of the identities
\[
  (x\,dx) = -\tfrac{1}{2}(dx\,dx), \quad
  (y\,dy) = -\tfrac{1}{2}(dy\,dy),
\]
\[
  (x\,dy) + (y\,dx) = -(dx\,dy),
\]
we shall have
\begin{multline*}
  \left|
  \begin{aligned}
   &[k^2 - \tfrac{1}{2}(dx\,dx)]\sin\frac{r}{k}
  - (y\,dx)\cos\frac{r}{k}   \\
  -&[k^2 - \tfrac{1}{2}(dy\,dy)]\sin\frac{r}{k}
  - (x\,dy)\cos\frac{r}{k}
  \end{aligned}
  \right.
\\
  \left.
  \begin{aligned}
    (x\,dy)\sin\frac{r}{k}
  -&[k^2 - \tfrac{1}{2}(dy\,dy)]\cos\frac{r}{k}\\
    (y\,dx)\sin\frac{r}{k}
  +&[k^2 - \tfrac{1}{2}(dx\,dx)]\cos\frac{r}{k}
  \end{aligned}
  \right| = 0.   \tag{10}
\end{multline*}
Casting aside infinitesimals above the second order
\begin{multline*}
  k^2(dx\,dy) \left( \cos^2\frac{r}{k} - \sin^2\frac{r}{k} \right)
\\
- \left[k^2(dx\,dx) - (y\,dx)^2 - k^2(dy\,dy) + (x\,dy)^2\right]
   \sin\frac{r}{k}\cos\frac{r}{k} = 0, \phantom{(11)}
\end{multline*}\ungap
\begin{multline*}
  (e\,du^2 + (f+f')du\,dv + g\,dv^2)
   \left( \cos^2\frac{r}{k} - \sin^2\frac{r}{k} \right)
\\
+ \left[(E-E')du^2 + 2(F-F')du\,dv + (G-G')dv^2\right]
   \sin\frac{r}{k}\cos\frac{r}{k} = 0.         \tag{11}
\end{multline*}

%[File: 219.png]
This will give $\infty^1$ determinations for $r$ in the general case
where
\begin{equation*}
e : \left( \frac{f+f'}{2} \right) : g \not\equiv (E-E') : (F-F') : (G-G'), \tag{12}
\end{equation*}
and, as we saw in Chapter X, Theorem 5, with the corresponding
elliptic case, these common perpendiculars will
generate a surface of the fourth order, analogous to the
euclidean cylindroid. We shall call a congruence where
inequality (12) holds a `general' congruence\index{Congruence08@--- of lines, general}.

Let us now ask what are the maximum and minimum
values for $r$ in (11). Equating to zero the partial derivatives
to $du$ and $dv$ we get
\[
\Bigl[ e\,du + \frac{f+f'}{2}dv \Bigr]
  \left(\tan^2\frac{r}{k}-1 \right)
  +[(E-E')du + (F-F')dv]\tan\frac{r}{k} = 0,
\]
\[
\Bigl[\frac{(f+f')}{2}du + g\,dv \Bigr]
  \left(\tan^2\frac{r}{k}-1 \right)
  +[(F-F')du + (G-G')dv]\tan\frac{r}{k} = 0.
\]
\label{noplus}

Eliminating $r$ we have
\begin{multline*}
\Bigl[e(F-F') - \frac{(f+f')}{2}(E-E')\Bigr]du^2 \\
  +[e(G-G')-g(E-E')]du\,dv \hspace{0.9in} \\
  +\Bigl[\frac{(f+f')}{2}(G-G')-g(E-E')\Bigr]dv^2 = 0. \tag{13}
\end{multline*}

Each root of this will give two values to $\tan\dfrac{r}{k}$ corresponding
to two mutually orthogonal points. On the other hand, if we
eliminate $du : dv$ we get
\begin{multline*}
\left(eg-\tfrac{1}{4}(f+f')^2\right)
  \left(\tan^2\frac{r}{k}-1\right)^2 +
  [e(G-G') \\
  \quad -(F-F')(f+f') + g(E-E')]
  \left(\tan^2\frac{r}{k}-1\right)\tan\frac{r}{k} \\
  + [(E-E')(G-G')-(F-F')^2]\tan^2\frac{r}{k} = 0.\tag{14}
\end{multline*}

\mysect{Limiting points and focal points}
The left-hand side of this equation is the discriminant
of (11) looked upon as an equation in $du: dv$. It gives,
therefore, those points of the given line where the two perpendiculars
coalesce. Such points shall be called `limiting
%[File: 220.png]
points'\index{Limiting points and planes}. They will determine two regions (when real) point
by point mutually orthogonal\index{Orthogonal points}, which contain the intersections
of the line with the real common perpendiculars\index{Perpendicularity}. In the
same way we might find limiting planes through the line
determining two dihedral angles whose faces are, in pairs,
mutually perpendicular, and which when real, with their
verticals, determine all planes wherein lie all real common
perpendiculars to the given line and its immediate neighbours.

\medskip\twincol{\emph{Theorem}~1. A line of a
general analytic congruence
contains four limiting points,
mutually orthogonal in pairs,
and these, when real, determine
two real regions of the line
where it meets the real common perpendiculars
with adjacent lines of the congruence.
They are also the points where
the two perpendiculars coincide.}
{\emph{Theorem}~1$'$. Through a
line of a general analytic congruence will pass four limiting
planes, mutually perpendicular in pairs, and these, when
real, determine two real regions of the axial pencil
through the line which contain all planes wherein are
real common perpendiculars
to the line and adjacent lines
of the congruence. They are
also the planes in which the
two perpendiculars coincide.}\par\medskip
%end twincol

We shall now look more closely into the question of the
reality of limiting points and places. We may so choose our
coordinate system that the equations of the line in question
shall be $x_1 = x_2 = 0$. Reverting to equation (8) of Chapter
X the equation of the ruled quartic surface will be, in the
hyperbolic case\label{hyberbolic}
\[
  a(-\dot{x}_0{}^2 + \dot{x}_3{}^2 )\dot{x}_1\dot{x}_2
+ b( \dot{x}_1{}^2 + \dot{x}_2{}^2 )x_0 x_3 = 0.   \tag{$15$}
\]

Let the reader show%
\footnote{See the author's \emph{Dual Projective Geometry}, cit., p.~26.}
%endfootnote
that in the elliptic case we shall have
\[
  (a_1 - a_2)(x_0{}^2 + x_3{}^2)x_1x_2
+ (a_1 + a_2)(x_1{}^2 + x_2{}^2)x_0x_3 = 0.   \tag{15$'$}
\]

To find the limiting points on the line $x_1 = x_2 = 0$, equate
to zero the discriminant of this looked upon as an equation in
\[
  \dot{x}_1  : \dot{x}_2 \quad \text{\ or } \quad x_1  : x_2.
\]
\[
  a^2(-\dot{x}_0{}^2 + \dot{x}_3{}^2) - 4b^2\dot{x}_0{}^2\dot{x}_3{}^2 = 0.
\tag{$16$}
\]
\[
  (a_1 - a_2)^2(x_0{}^2 + x_3{}^2)^2 - 4(a_1 + a_2)^2 x_0{}^2 x_3{}^2 = 0.  \tag{16$'$}
\]

In like manner for the limiting planes we shall have
\[
  b^2(\dot{x}_1{}^2 + \dot{x}_2{}^2)^2 + 4a^2\dot{x}_1{}^2\dot{x}_2{}^2 = 0.      \tag{$17$}
\]
\[
  (a_1 + a_2)^2(x_1{}^2 + x_2{}^2) - 4(a_1 - a_2)^2x_1{}^2 x_2{}^2 = 0.
\tag{17$'$}
\]

%[File: 221.png]
Notice that the centres of gravity\index{Centre03@--- of gravity of points} of the limiting points
are $(1, 0, 0, 0)$ $(0, 0,$ $0, 1)$; while the bisectors\index{Bisector of an angle} of the dihedral
angles of the limiting planes\index{Limiting points and planes} are $(0, 1,$ $0, 0)$ $(0, 0, 1, 0)$.

If we look more closely into the roots of the last four
equations we see that the roots of (16) are all real, those of
(17) all imaginary. As for the two equations (16$'$) and (17$'$)
the one will have real roots, the other imaginary ones, whence

\smallskip\emph{Theorem}~2. In hyperbolic space the limiting points of an
actual line are real, and the limiting planes imaginary. In
elliptic space this may occur, or the planes may be all real
and the points all imaginary.

Giving to $x_0:x_3$ one of the values from (16$'$) we see that
\begin{equation*}
\frac{x_0{}^2 + x_3{}^2}{x_0x_3}= \pm\frac{2(a_1 +a_2)}{a_1 - a_2}.
\end{equation*}

Substituting in (15$'$) we have

\begin{equation*}
x_1+x_2=0 \quad \text{\ or } \quad x_1-x_2=0
\end{equation*}

The four limiting points will yield but these two planes,
hence

\medskip\twincol{\emph{Theorem}~3. The perpendiculars\index{Perpendicularity}
at the limiting points
line in two planes called
`principal planes'\index{Principal points and planes} whose dihedral
angles have the same
bisectors as pairs of limiting
planes.}
{\emph{Theorem}~3$'$. The perpendiculars
in the limiting planes
meet the line in two points
called `principal points' whose
centres of gravity are those of
two pairs of limiting points.}\par\medskip
%end twincol

Reverting to (16$'$) we see that we may also write

\begin{equation*}
x_0:x_3= \pm(\sqrt{a_1}\pm\sqrt{a_2}):(\sqrt{a_1}\mp\sqrt{a_2}).
\end{equation*}

Let us pick out a pair of limiting points which are not
mutually orthogonal, say\label{pointpair}
\begin{equation*}
(\sqrt{a_1}+\sqrt{a_2}, 0, 0, \sqrt{a_1}-\sqrt{a_2})\;
(-(\sqrt{a_1}+\sqrt{a_2}), 0, 0, \sqrt{a_1}-\sqrt{a_2}).
\end{equation*}

The perpendicular from the point $(x)$ to the line $x_1=x_2=0$
meets it in the point $(x_0, 0, 0, x_3)$. Calling $d_1$, $d_2$ the distances
thence to the limiting points just chosen we have
\begin{align*}
\tan{\frac{d_1}{k}}&=\frac{(\sqrt{a_1}-\sqrt{a_2})x_0-
(\sqrt{a_1}+\sqrt{a_2})x_3}{(\sqrt{a_1}+\sqrt{a_2})x_0+(\sqrt{a_1}-\sqrt{a_2})x_3},\\
\tan{\frac{d_2}{k}}&=\frac{(\sqrt{a_1}-\sqrt{a_2})x_0+
(\sqrt{a_1}+\sqrt{a_2})x_3}{-(\sqrt{a_1}+\sqrt{a_2})x_0+(\sqrt{a_1}-\sqrt{a_2})x_3}.
\end{align*}

%[File: 222.png]
Further, let $(\omega)$ be the angle which the plane through
$x_1 = x_2 = 0$ and $(x)$ makes with the principal plane
\[
  x_1 + x_2 = 0.
\]
\[
  \cos^2\omega = \frac{(x_1 - x_2)^2}{2(x_1{}^2 + x_2{}^2)},\quad
  \sin^2\omega = \frac{(x_1 + x_2)^2}{2(x_1{}^2 + x_2{}^2)},
\]
\[
  \tan\frac{d_1}{k}\cos^2\omega + \tan\frac{d_2}{k}\sin^2\omega = 0.
\tag{18}
\]

This is, of course, the direct analog of Hamilton's\index{Hamilton} well-known formula for the cylindroid.%
\footnote{For the Hamiltonian equation see Bianchi, \emph{Differentialgeometrie}, cit., p.~261.
For the non-euclidean form here given, cf.\ Fibbi\index{Fibbi}, loc.\ cit., p.~57. Fibbi's work
is burdened with many long formulae; one cannot help admiring his skill
in handling such cumbersome expressions at all.}
%endfootnote

Returning to the notations wherewith we opened the
present chapter, let us find the focal points\index{Focal04@--- points and planes} of our line, i.e.\ the
points where it intersects adjacent lines of the congruence,
or rather, the points where the distance becomes infinitesimal
to a higher order. Here, if the focal point be
\[
  \Bigl( x\cos\frac{r}{k} + y\sin\frac{r}{k} \Bigr),
\]
we shall have
\begin{multline*}
  x_i\cos\frac{r}{k} + y_i\sin\frac{r}{k}
= (x_i + dx_i)\cos\frac{r+dr}{k} + (y_i + dy_i)\sin\frac{r+dr}{k}
\\
  dx_i\cos\frac{r}{k} + dy_i\sin\frac{r}{k}
- \frac{1}{k} \Bigl( x_i\cos\frac{r}{k} - y_i\sin\frac{r}{k} \Bigr)dr = 0.
\end{multline*}
\[
  k\,dr = (x\,dy),
\]
\begin{multline*}
  \Bigl[
    k^2 \Bigl( \frac{\partial x_i}{\partial u} \cos\frac{r}{k}
             + \frac{\partial y_i}{\partial u} \sin\frac{r}{k} \Bigr)
    - \Bigl( x_i\cos\frac{r}{k} - y_i\sin\frac{r}{k} \Bigr)
    \Bigl( x \frac{\partial y}{\partial u} \Bigr)
  \Bigr] du +
\\
+ \Bigl[
    k^2 \Bigl( \frac{\partial x_i}{\partial v} \cos\frac{r}{k}
             + \frac{\partial y_i}{\partial v} \sin\frac{r}{k} \Bigr)
    - \Bigl( x_i\cos\frac{r}{k} - y_i\sin\frac{r}{k} \Bigr)
    \Bigl( x \frac{\partial y}{\partial v} \Bigr)
  \Bigr] dv = 0.
\end{multline*}

Multiplying through by $\dfrac{\partial y_i}{\partial u}$ and adding, then multiplying
through by $\dfrac{\partial y_i}{\partial v}$ and adding again
\begin{align*}
  [e\,du + f\,dv]\cos\frac{r}{k} + [E'du + F'dv]\sin\frac{r}{k} = 0,\\
  [f'du + g\,dv]\cos\frac{r}{k} + [F'du + G'dv]\sin\frac{r}{k} = 0.
\end{align*}

%[File: 223.png]
Replacing $(y\,dx)$ by $(-x\,dy)$ we have, similarly

\[\tag{19}
\begin{split}
\left[e\,du+f'dv\right]\sin{\frac{r}{k}}+[Edu+Fdv]\cos{\frac{r}{k}}=0,
\\
\left[f\,du+g\,dv\right]\sin{\frac{r}{k}}+[Fdu+Gdv]\cos{\frac{r}{k}}=0.
\end{split}
\]

Eliminating $r$
\[\tag{20}
\begin{split}
(E'f'-F'e)du^2+[E'g-F'(f-f')-G'e]du\,dv+(F'g-G'f)dv^2=0&,
\\
(Ef-Fe)du^2+[Eg-F(f'-f)-Ge]du\,dv+(Fg-Gf)dv^2=0&.
\end{split}
\]

Eliminating $du:dv$
\[\tag{21}
\begin{split}
(E'G'-F')^2\tan^2{\frac{r}{k}}+[E'g-F'(f+f')+G'e]\tan{\frac{r}{k}}+
(eg-f\!f')=0&,
\\
(eg-f\!f')\tan^2{\frac{r}{k}}+[Eg-F(f+f')+Ge]
\tan{\frac{r}{k}}+(EG-F^2)=0&.
\end{split}
\]

Subtracting one of these equations from the other
\begin{multline*}
[(eg-f\!f')-(E'G'-F'^2)]\tan^2{\frac{r}{k}}+[(E-E')g-(F-F')(f+f') \\
+(G-G')e]\tan{\frac{r}{k}}+[(EG-F^2)-(eg-f\!f')]=0.\tag{22}
\end{multline*}

We see at once that the middle coefficients are identical in
(14) and (22), and these will vanish when, and only when, we
are measuring from a centre of gravity\index{Centre03@--- of gravity of points} of the roots.

\medskip\twincol{\emph{Theorem}~4. The centres of gravity of the focal points
are identical with those of two pairs of limiting points\index{Limiting points and planes}.}
{\emph{Theorem}~4$'$. The bisectors\index{Bisector of an angle} of the dihedral angles of
two focal planes\index{Focal04@--- points and planes} are identical with those of
two pairs of limiting planes.}

\mysect{Necessary and sufficient conditions for a normal congruence}
The focal properties of a congruence of
normals\index{Congruence09@--- of normals}\index{Normals02@--- to surface}
are especially
interesting. Here we may suppose that $(y)$ is the
Absolute pole of the tangent plane to the surface described
by $(x)$. We have then
\begin{equation*}
\Bigl(y\frac{\partial x}{\partial u}\Bigr)=\Bigl(y\frac{\partial x}{\partial v}\Bigr)=\Bigl(x\frac{\partial y}{\partial u}\Bigr)=\Bigl(x\frac{\partial y}{\partial v}\Bigr)=0,
\end{equation*}

\begin{equation*}
-\Bigl(x\frac{\partial^2 y}{\partial u \,\partial v}\Bigr)=f=f'.
\end{equation*}

%[File: 224.png]
Suppose, conversely, that
\[
  f = f'.
\]

Let us put $\bar{x}_i = x_i\cos\dfrac{r}{k} + y_i\sin\dfrac{r}{k}$ and show that we may
find $r$ so that our line is normal\index{Normals02@--- to surface} to the surface traced by $(\bar{x})$.
For this it is necessary and sufficient that the point of the
line orthogonal to $(\bar{x})$ should be orthogonal to every displacement of $(\bar{x})$. This point being
$\left( x\sin\dfrac{r}{k} - y\cos\dfrac{r}{k} \right)$, we must
have
\[
  \sin\frac{r}{k}(x\,d\bar{x}) - \cos\frac{r}{k}(y\,d\bar{x}) = 0,
\]
\[
  (y\,dx) = -k\,dr,
\]
and $(y\,dx)$ must be an exact differential, i.e.
\[
  \frac{\partial}{\partial u}
  \Bigl( y\frac{\partial x}{\partial v} \Bigr)
= \frac{\partial}{\partial v}
  \Bigl( y\frac{\partial x}{\partial u} \Bigr), \quad
  f = f'.   \tag{23}
\]

This condition can be put into a more geometrical form.
Let us, in fact, find the necessary and sufficient condition that
the focal planes\index{Focal04@--- points and planes} should be mutually perpendicular. Writing
their equations in the form
\[
  \lvert Xxy\,dx \rvert = 0, \quad \lvert Xxy\,\delta x \rvert = 0,
\]
the numerator of the expression for the cosine of their angle
will be
\[
  \begin{vmatrix}
    k^2     & 0      & (y\,\delta x)   \smallskip \\
    0       & k^2    & -\tfrac{1}{2}(\delta x\,\delta x)   \smallskip \\
    (y\,dx) & -\tfrac{1}{2}(dx\,dx) & (dx\,\delta x)
  \end{vmatrix}
= k^2 [k^2(dx\,\delta x) - (y\,dx)(y\,\delta x)].
\]

For perpendicularity,
\[
  Edu\,\delta u + F(du\,\delta v + \delta u\,dv) + Gdv\,\delta v = 0.
\]

Now, by (20),
\[
  \frac{du\,\delta u}{dv\,\delta v} = \frac{Fg-Gf'}{Ef-Fe},
\quad
  \Bigl[\frac{du}{dv} + \frac{\delta u}{\delta v}\Bigr]
= \frac{Ge + F(f'-f)-Eg}{Ef-Fe}.
\]

Hence
\[
  (EG - F^2)(f - f') = 0.
\]

Let us give the name \emph{pseudo-normal} to the absolute polar
of a normal congruence\index{Congruence09@--- of normals}. We thus get

%[File: 225.png]
\medskip\twincol{\emph{Theorem}~5. The necessary
and sufficient condition that
a congruence should be normal
is that the focal planes\index{Focal04@--- points and planes} through
each line should be mutually
perpendicular\index{Perpendicularity}.}
{\emph{Theorem}~5$'$. The necessary
and sufficient condition that a
congruence should be pseudo-normal\index{Pseudo-normal congruence}
is that the focal points
on each line should be mutually
orthogonal\index{Orthogonal points}.}\par\medskip
%end twincol

If we subtract one of the equivalent equations (20) from the
other, we get an equation which reduces to (13) when, and
only when
\[
f=f'.
\]

\smallskip\emph{Theorem}~6. The necessary and sufficient condition that
a general congruence should be composed of
normals\index{Normals02@--- to surface}\index{Congruence09@--- of normals}
is that
the focal points should coincide with a pair of limiting points.

In a normal congruence let us suppose that $(x)$ traces a
surface to which the given lines are normal so that
\[
(y\,dx) =-(x\,dy) = 0.
\]

Let us then put
\[
  \overline{x_i} = x_i\cos\frac{r}{k} + y_i\sin\frac{r}{k},\quad
  \overline{y_i} = x_i\cos\frac{r}{k} - y_i\sin\frac{r}{k},
\]
where $y$ is constant. We see at once that
\[
(\overline{y}\,d\overline{x}) = -(\overline{x}\,d\overline{y}) = 0.
\]

\smallskip\emph{Theorem}~7. If a constant distance be laid off on each normal
to a surface from the foot, in such a way that the points
on adjacent normals are on the same side of the tangent
plane\index{Plane} corresponding to either, the locus of the points so found
is a surface with the same normals as the original one.

Let us suppose that we have a normal congruence determined by mutually
orthogonal points $(x)$ and $(y)$, where
$x_i = x_i(uv)$ traces a surface, not one of the orthogonal trajectories
of the congruence. We shall choose as parameter
lines in this surface the isotropic curves, so that
\[
  \Bigl( \frac{\partial x}{\partial u}
         \frac{\partial x}{\partial u} \Bigr) =
  \Bigl( \frac{\partial x}{\partial v}
         \frac{\partial x}{\partial v} \Bigr) = 0.
\]

The sine of the angle which our given line makes with the
normal to this surface is
\[
\sin\theta = \sqrt{
  \frac{2\Bigl(y\dfrac{\partial x}{\partial u} \Bigr)
         \Bigl(y\dfrac{\partial x}{\partial v} \Bigr)   }
       {k^2\Bigl( \dfrac{\partial x}{\partial u}
                  \dfrac{\partial x}{\partial v} \Bigr)}}
\]

%[File: 226.png]
Let all the lines of our congruence\index{Congruence09@--- of normals} be reflected or refracted
in this surface in such a way that
\[
\sin \overline{\theta} = n \sin \theta.
\]

We must replace $y$ by $\overline{y}$ where
\[
\overline{y_i} = ny_i +
  \lambda\frac{\partial}{\partial t_i}
  \left|
    tx
   \dfrac{\partial x}{\partial u}
   \dfrac{\partial x}{\partial v}
  \right|.
\]

It is easily seen that for the new congruence also
\[
f=f'.
\]

\mysect{Malus-Dupin theorem}
\smallskip\emph{Theorem}~8. If a normal congruence be subjected to any
finite number of reflections or refractions, the resulting congruence
is normal.

We shall now abandon the general congruence and assume
that, contrary to (12)
\begin{equation*}
\tag{24}
  e  : \frac{f+f'}{2}  : g \equiv
  (E-E')  : (F-F')  : (G-G').
\end{equation*}

There are two sharply distinct sub-cases which must not
be confused:
\[
\text{(a) }f\equiv f', \quad \text{(b) } f \not\equiv f'.
\]

In either case, as we readily see, (11) is illusory, and there
is no ruled quartic determined by the common perpendiculars
to a line and its neighbours; these perpendiculars will either
all meet the given line at one of two mutually orthogonal
points, or two adjacent lines will be paratactic\index{Parataxy}, and have $\infty^1$
common perpendiculars.

\mysect{Isotropic congruences, and congruences of normals to surfaces of zero curvature}
Our condition for focal points expressed in (23) was independent
of (12), and this shows that our two sub-cases just
mentioned differ in this, that the first is a normal\index{Normals02@--- to surface} congruence,
while the second is not. Let $(x)$ be a point where our line
meets a set of perpendiculars, $(y)$ being thus the other such
point. Then under our first hypothesis, we shall have
\[
e=f = f'=g = 0.
\]

We see that the focal points will fall into $(x)$ and $(y)$ likewise.
These are mutually orthogonal, and so by equation (26) of the
last Chapter, that the total relative curvature of the surface
will be $-\dfrac{1}{k^2}$ or the Gaussian curvature zero. We see also by
theorem (22) of that chapter that it is possible to assemble the
lines of our congruence into families of left or right
paratactics\index{Left and right parataxy}\index{Right and left parataxy}
according as we assemble them by means of the one or the
other set of asymptotic lines of the given surface. Conversely,
%[File: 227.png]
if we have given a congruence of
normals\index{Congruence09@--- of normals}\index{Congruence10@--- of normals,
to surfaces of Gaussian curvature zero} to a surface of
Gaussian curvature zero\index{Curvature08@--- surfaces of zero}, two
normals\index{Normals02@--- to surface} adjacent to a given one
are paratactic thereunto. There must be, then, two values of
$du: dv$ for which (11), looked upon as an equation in $r$, becomes
entirely illusory. Hence (24) must hold, and as we have
normal congruence (23) is also true.

We now make the second assumption
\[
f \not\equiv f'.
\]

We shall still take $(x)$ as a point where the line meets the
various common perpendiculars, so that we may put
\[
e = \frac{f+f'}{2} = g = 0.
\]

We may take as coordinates of a focal plane\index{Focal05@--- surfaces}
\[
u_i = \frac{\partial}{\partial t_i} |t\,xy\,dx|,
\]
\[
(uu) = k^2 [Edu^2 +2Fdu\,dv +Gdv^2].
\]

But by (20) this expression vanishes. Hence the focal
planes all touch the Absolute\index{Absolute}, and the focal surface must be
a developable circumscribed thereunto. It is clear that the
lines of such a congruence cannot be assembled into paratactic
families.

This type of congruence shall be called
`isotropic'\index{Congruence11@--- isotropic}\index{Isotropic02@--- congruence}.%
\footnote{The earliest discussion of these interesting congruences in non-euclidean
space will be found in the author's\index{Author} article `Les congruences isotropes qui
servent \`a repr\'esenter les fonctions d'une variable complexe', \emph{Atti della R.
Accademia delle Scienze di Torino}, xxxix, 1903, and xl, 1904. In the same
number of the same journal as the first of these will be found an article
by Bianchi\index{Bianchi}, `Sulla rappresentazione di Clifford delle congruenze rettilinee
nello spazio ellitico.' Professor Bianchi uses the word `isotropic' to cover
both what we have here defined as isotropic congruences, and also congruences
of normals to surfaces of Gaussian curvature zero, distinguishing the latter
by the name of `normal'. The author, on the other hand, included in his
definition of isotropic congruences those which, later, we shall define as
`pseudo-isotropic'. A discussion of these definitions will be found in a note
at the beginning of the second of the author's articles.}
%endfootnote

Let us take an isotropic congruence, or congruence of
normals to a surface of Gaussian curvature zero, and choose
$(x)$ and $(y)$ so that
\[
e = \tfrac{1}{2}(f+f')=g=0,
\]
\[
\overline{x_i} = x\cos\frac{r}{k} + y\sin\frac{r}{k},
\]
\[
(d\overline{x}\,d\overline{x}) =
  \cos^2\frac{r}{k}(dx\,dx) + \sin^2\frac{r}{k}(dy\,dy).
\]

%[File: 228.png]
This expression will be unaltered if we change $r$ into $-r$.
Conversely, when such is the case, we must have $(dx\,dy) = 0$,
and the congruence will be either
isotropic\index{Congruence11@--- isotropic}\index{Isotropic02@--- congruence}, or composed of
normals\index{Normals02@--- to surface}\index{Congruence09@--- of normals}\index{Normals02@--- to surface}
to a surface of Gaussian curvature
zero\index{Congruence10@--- of normals, to surfaces of Gaussian curvature zero}\index{Curvature08@--- surfaces of zero}.

\smallskip\emph{Theorem}~9. The necessary and sufficient condition that a
congruence should be either isotropic, or composed of normals
to a surface of Gaussian curvature zero, is that it should consist
of lines connecting corresponding points of two mutually
applicable surfaces, which pairs of points determine always
the same distance. The centres of gravity of these pairs of
points will be the points where the various lines meet the
common perpendiculars to themselves and the adjacent lines.

In elliptic (or spherical) space, there is advantage in studying our last
two types of congruence from a different point
of view, suggested by the developments of Chapter X.

Let us rewrite the equations (11) there given.
\begin{align*}
  (x_0 y_i - x_i y_0) + (x_j y_k - x_k y_j) &= {}_lX_i,   \\
  (x_0 y_i - x_i y_0) - (x_j y_k - x_k y_j) &= {}_rX_i.   \tag{25}
\end{align*}

These equations were originally written under the supposition that $(x)$ and $(y)$
were homogeneous. At present if we so
choose the unit of measure that $k = 1$ we have
\[
  ({}_lX\,{}_lX) = ({}_rX\,{}_rX) = 1.      \tag{26}
\]

\mysect{Spherical representation of rays in elliptic space}
These coordinates $({}_lX)$, $({}_rX)$ were formerly looked upon
as giving the lines through the origin $(1, 0, 0, 0)$ respectively
left and right paratactic to the given line. They may now be
looked upon as coordinates of two points of two unit spheres\index{Sphere}
of euclidean space, called, respectively, the left and right
\emph{representing spheres}\index{Spheres, representing}%
\footnote{This representation was first published independently by Study, `Zur
nichteuklidischen etc.,' and Fubini\index{Fubini}, `Il parallelismo di Clifford negli spazii
ellitici,' \emph{Annali della R. Scuola Normale di Pisa}, Vol.~ix, 1900. The latter writer
does not, however, distinguish with sufficient clearness between rays\index{Ray} and
lines.}
%endfootnote
The representation is not, however,
unique. On the one hand the two lines of a cross will be
represented by the same points, on the other, we get the same
line if we replace either representing point by its diametrical
opposite. We shall avoid ambiguity by assuming that each
line is doubly overlaid with two opposite `rays', meaning
thereby a line with a sense or sequence attached to its points,
as indicated in the beginning of Chapter V or end of Chapter
IX\@. We shall assume that by reversing the signs in one triad
of coordinates we replace our ray by a ray on the absolute
%[File: 229.png]
polar of its line, while by reversing both sets of signs, we
replace the ray\index{Ray} by its opposite.

\smallskip\emph{Theorem}~10. There is a perfect one to one correspondence
between the assemblage of all real rays of elliptic or spherical
space, and that of pairs of real points of two euclidean spheres.
Opposite rays of the same line will be represented by diametrically
opposite pairs of points, rays on mutually absolute
polar lines by identical points on one sphere\index{Sphere} and opposite
points of the other. Rays on left (right) paratactic lines will
be represented by identical or opposite points of the left (right)
sphere\index{Spheres, representing}.

Two rays shall be said to be paratactic when their lines are.
Reverting to Theorem 12 of Chapter X.

\smallskip\emph{Theorem}~11. The perpendicular distances of the lines of two
rays or the angles of these rays are half the difference and
half the sum of the pairs of spherical distances of their representing points.

\smallskip\emph{Theorem}~12. The necessary and sufficient condition that the
lines of two rays should intersect is that the spherical distances
of the pairs of representing points should be equal; each will
intersect the absolute polar of the other if these spherical
distances be supplementary.

\smallskip\emph{Theorem}~13. Each ray of a common perpendicular to the
lines of two rays will be represented by a pair of poles of two
great circles which connect the pairs of representing points.

It is clear that an analytic congruence may be represented
in the form
\begin{equation*}
_lX_i={}_lX_i(uv),\quad
_rX_i={}_rX_i(uv),
\end{equation*}

or else, in general,
\begin{equation*}
_lX_i={}_lX_i(_rX_1\,_rX_2\,_rX_3).
\end{equation*}

Two adjacent rays will intersect, or intersect one another's
polars if
\begin{equation*}
(d_lX\,d_lX)=(d_rX\,d_rX).
\end{equation*}

The common perpendicular to two adjacent rays will have
coordinates
\begin{equation*}
\lambda_lY_i=\frac{\partial}{\partial_lZ_i}\left|_lZ\,_lX\,d_lX\right|,\quad
\mu_rX_i=\frac{\partial}{\partial_rZ_i}\left|_rZ\,_rX\,d_rX\right|.
\end{equation*}

\mysect{Representation of normal congruence}
The condition that a congruence should be either normal or
pseudo-normal is
\begin{equation*}
(d_lXd_lX)=(d_rXd_rX),
\end{equation*}
\begin{equation*}
(\delta_lX\delta_lX)=(\delta_rX\delta_rX),
\end{equation*}
%[File: 230.png]
\[
  \begin{vmatrix}
  ( _lX \,_lX)( _lX\, \delta \,_lX) \smallskip \\
  ( _lX \,d \,_lX)(d \,_lX \,\delta \,_lX) \end{vmatrix} = \pm
  \begin{vmatrix}
  ( _rX \,_rX)( _rX\, \delta \,_rX) \smallskip \\
  ( _rX \,d \,_rX)(d \,_rX \,\delta \,_rX) \end{vmatrix} ,
\]
from these
\begin{equation*}
\tag{27}
(d \,_l X\, \delta \,_lX) = \pm (d\,_rX\,\delta\,_rX).
\end{equation*}

Let us determine the significance of the double sign. If, in
particular, we take the congruence of
normals\index{Congruence09@--- of normals}\index{Normals02@--- to surface} to a sphere
whose centre is $(1, 0, 0, 0)$ we shall get the equations
\[
  _lX_i = {}_rX_i,
\]
and this transformation keeps areas invariant in value and
sign. On the other hand, the congruence of rays in the
absolute polar of this plane will be
\[
  {}_lX_i = - {}_rX_i,
\]
a transformation which changes the signs of all areas. Lastly,
we may pass from one normal congruence to another by a
continuous change, wherein the sign in equation (27) will not
be changed, hence%
\footnote{Cf.\ Study\index{Study}, loc.\ cit., p.~321; Fubini\index{Fubini}, p.~46.}
%endfootnote

\medskip\twincol{\emph{Theorem}~14. A normal congruence
will be represented
by a relation between the two
spheres which keeps areas invariant
in actual value and
sign, and every such relation
will give a normal congruence.}
{\emph{Theorem}~14$'$. A pseudo-normal\index{Pseudo-normal congruence}
congruence will be
represented by a relation between
the two spheres where
the sum of corresponding areas
on the two is zero, and every
such relation will give a
pseudo-normal congruence.}\par\medskip
%end twincol

\mysect{Isotropic congruence represented by an arbitrary function of the complex variable}
Let us next take an isotropic congruence. Here two
common perpendiculars to two adjacent lines necessarily
intersect, or each intersects the absolute polar of the other.
The same will hold for the absolute polar of an isotropic
congruence, a `pseudo-isotropic' congruence\index{Pseudo-isotropic congruence}, let us say. Such
a congruence will not have a focal surface at all, but a focal
curve, which lies on the Absolute. On the representing
spheres, in the case of either of these congruences, two intersecting
arcs of one will make the same angle, in absolute
value, as the corresponding arcs on the other. In the particular
case of the isotropic congruence of all lines through
the point $(1, 0, 0, 0)$ the relation between the two representing
spheres is a directly conformal one, while in the case of the
pseudo-isotropic congruence of all lines in the plane $(1, 0, 0, 0)$
we have an inversely conformal relation. We may now repeat
%[File: 231.png]
the reasoning by continuity used in the case of the normal
congruence, and get%
\footnote{First given in the Author's\index{Author} first article on isotropic congruences, recently cited.}
%endfootnote

\medskip\twincol{\emph{Theorem}~15. The necessary and sufficient condition that a congruence
should be isotropic\index{Congruence11@--- isotropic}\index{Isotropic02@--- congruence}
is that the corresponding relation between the representing
spheres should be directly conformal.}
{\emph{Theorem}~15$'$. The necessary and sufficient condition that a congruence should be
pseudo-isotropic\index{Pseudo-isotropic congruence}
is that the corresponding relation between the representing spheres
should be inversely conformal.}\par\medskip
%end twincol

Let us take up the isotropic case more fully. Any directly
conformal relation between the real domains of two euclidean
spheres of radius unity may be represented by an analytic
function of the complex variable. Let us give the coordinates
of points of our representing spheres in the following parametric form:
\begin{align*}
_lX_1&=\frac{u_1u_2-1}{u_1u_2+1},\quad &
_rX_1&=\frac{z_1z_2-1}{z_1z_2+1},\\
_lX_2&=\frac{i(u_1-u_2)}{u_1u_2+1},\quad &
_rX_2&=\frac{i(z_1-z_2)}{z_1z_2+1}, & \tag{28} \\
_lX_3&=\frac{u_1+u_2}{u_1u_2+1},\quad &
_rX_3&=\frac{z_1+z_2}{z_1z_2+1}.
\end{align*}

We shall get a real ray when
\begin{equation*}
u_2=\bar u_1,\quad z_2=\bar z_1.
\end{equation*}

In order to have a real directly conformal relation between
the two spheres, our transformation must be such as to carry
a rectilinear generator into another generator, i.e.
\begin{equation*}
u_1=u_1(z_1),\quad u_2=\bar u_1(z_2).\tag{29}
\end{equation*}

For an inversely conformal transformation
\begin{equation*}
u_1=u_1(z_2),\quad u_2=\bar u_1(z_1).\tag{30}
\end{equation*}

All will thus depend on the single analytic function $u_1(z)$.

The opposite of the ray $(u)$ $(z)$ will be
\begin{align*}
u_1{}'&=-\frac{1}{u_2}, \qquad z_1{}' =-\frac{1}{z_2}, \\
u_2{}'&=-\frac{1}{u_1}, \qquad z_2{}' =-\frac{1}{z_1}.
\end{align*}

%[File: 232.png]
Let us now inquire under what circumstances the following equation will hold:
\[
  \bar{u}_1\Bigl(-\frac{1}{z_1}\Bigr) = \frac{-1}{u_1(z_1)}. \tag{31}
\]

If this hold identically, the opposite of every ray of the
congruence will belong thereto. If not, there will still be
certain rays of the congruence for which it is true. To begin
with it will be satisfied by all rays of the congruence for
which
\[
  u_1 u_2 + 1 = 0, \quad z_1 z_2 + 1 = 0.
\]

This amounts to putting
\[
  (_lX\,_lX) = (_rX\,_rX) = 0.
\]

We saw in Chapter X that, interpreted in cross\index{Cross} coordinates,
these are the equations which characterize an improper cross\index{Improper01@Improper cross}
of the second sort, which is made up of a pencil of tangents
to the Absolute\index{Absolute}. Such a pencil we may also call an improper
ray\index{Improper02@--- ray} of the second sort. Let us see under what circumstances
such a ray $(uz)$ will intersect a proper ray $(u'z')$ orthogonally.
Geometrically, we see that either the proper ray must pass
through the vertex of the pencil, or lie in the plane thereof,
and analytically we shall have
\[
  (u_1 - u_1{}')(u_2 - u_2{}') = (z_1 - z_1{}')(z_2 - z_2{}') = 0,
\]
\[
  u_1 u_2 + 1 = z_1 z_2 + 1 = 0.
\]

There are four solutions to these equations. By considering
a special case we are able to pick out those two where the ray
lies in the plane of the pencil\label{z1prime}
\[
\begin{aligned}
  u_1 &= u_1{}', \qquad &  z_1 &= z_1{}',   \\
  u_2 &= -\frac{1}{u_1{}'}, \qquad &  z_2 &= -\frac{1}{z_1{}'},
\end{aligned}
\]
or else
\[
\begin{aligned}
  u_1 &= -\frac{1}{u_2{}'}, \qquad &  z_1 &= -\frac{1}{z_2{}'}, \\
  u_2 &= u_2{}', \qquad &  z_2 &= z_2{}',
\end{aligned}
\]

The proper ray $(u') (z')$ was supposed to belong to our
congruence. The condition that the improper one $(u) (z)$ shall
also belong thereto will be
\[
  \bar{u}_1\Bigl(-\frac{1}{z_1{}'}\Bigr) = -\frac{1}{u_1(z_1{}')}.
\]

%[File: 233.png]
\smallskip\emph{Theorem}~16.%
\footnote{See the Author's\index{Author} second note on isotropic congruences, p.~13.}
%endfootnote
The necessary and sufficient condition that
the opposite of a real ray of an isotropic congruence should
also belong thereunto is that the ray should be coplanar with
an improper ray\index{Improper02@--- ray} of the second sort belonging to the congruence.
When the latter are present in infinite number in an
irreducible congruence, the congruence contains the opposite
of each of its rays.

The two cases here given may be still more sharply distinguished
by geometrical considerations. The focal surface\index{Focal05@--- surfaces}
of an isotropic congruence\index{Isotropic02@--- congruence}\index{Congruence11@--- isotropic}
is a developable circumscribed to
the Absolute\index{Absolute}, and will have a real equation when the congruence
is real. There are two distinct possibilities; first, the
equation of this surface is reducible in the rational domain;
second, it is not. In the first case the surface is made up of
two conjugate imaginary portions; in the second there is one
portion which is its own conjugate imaginary. In the first
case there will be a finite number of planes which touch the
Absolute and also each of the two portions of the focal surface
at the same point, namely, those which touch the Absolute
at the points of intersection of the two curves of contact with
the two portions of the focal surface. In these planes only
shall we have improper rays of the second sort belonging to
the congruence. If, on the other hand, the focal surface be
irreducible, every point of the curve of contact may be looked
upon as being in the intersection of two adjacent planes
tangent to the Absolute, and the focal surface which is its
own conjugate imaginary. The tangents at each of these
points will be improper rays of the second sort of the congruence.
Theorem 17 may now be given in a better form.

\smallskip\emph{Theorem}~17. The necessary and sufficient condition that
an isotropic congruence should contain the opposite of each
of its rays is that the focal surface should be irreducible.

\mysect{Special examples of this representation}
It is very easy to observe the distinction between the two
cases in the case of the linear function
\begin{equation*}
u_1=\frac{\alpha z_1+\beta}{\gamma z_1+\delta}.
\end{equation*}

If $\beta=-\bar\gamma$, $\delta=\bar\alpha$, (29) is identically satisfied. But here
it will be seen that if we write
\begin{equation*}
\alpha=a+bi,\quad \gamma=-c+di,
\end{equation*}
our congruence is nought else than the assemblage of all rays
through the point $(a, b, c, d)$. The focal surface is the cone of
%[File: 234.png]
tangents thence to the Absolute\index{Isotropic02@--- congruence}, clearly its own conjugate
imaginary. On the other hand, when $\alpha$, $\beta$, $\gamma$, $\delta$ are not connected
by these relations, we shall have a line congruence of
the fourth order, and second class, as is easily verified. It is
well known%
\footnote{Cf.\ Sturm\index{Sturm}, \emph{Gebilde erster und zweiter Ordnung der Liniengeometrie},
Leipzig, 1892-96, Vol.~ii, p.~320.}
%endfootnote
that a congruence of the second order and fourth
class has no focal surface, but a focal curve composed of two
conics, so our present congruence has as focal surface two
conjugate imaginary quadric cones which are circumscribed
to the Absolute. When their conjugate imaginary centres fall
together in a real point, we revert to the previous case.

When $(u)$ and $(z)$ are connected by the vanishing of a
polynomial of order $m$ in $u_1$ and order $n$ in $z_1$, in the general
case where (31) does not hold identically, we shall have
a line-congruence of order $(m+n)^2$. When, however, (31) does
hold, we must subtract from this the order of the curve of
contact of the focal surface and Absolute, and then divide by
2 to allow for the fact that there are two opposite rays on
each line.

If $u_1$ be a function of $z_1$ that possesses an essential singularity
corresponding to a certain value of $z_1$ we see that as $u_1$
takes all possible values (except at most two) in the immediate
neighbourhood, there will be a whole bundle of right paratactic
lines in the congruence. If $u_1$ be periodic, there will be an
infinite number of lines of the congruence left paratactic\index{Parataxy} to
each line thereof. If $u_1$ be one of the functions of the regular
bodies, we have a congruence which is transformed into itself
by a group of orthogonal substitutions in $(_rX)$, i.e.\ by a group
of left translations.

We have still to consider the congruence of normals to a
surface of Gaussian curvature zero in ray coordinates. Here
there will be $\infty^1$ paratactics of each sort to each line. We
may therefore express $(_lX)$ and $(_rX)$ each as functions of one
independent variable, or merely write
\begin{equation*}
\phi(_lX_1\,_lX_2\,_lX_3)=\psi(_rX_1\,_rX_2\,_rX_3)=0.\tag{32}
\end{equation*}

\mysect{Study's ray to ray transformation which interchanges parallelism and para\-taxy}
All our work here developed for the elliptic case may be
brought into immediate relation with the hyperbolic case, and
in so doing we shall get to the inmost kernel of the whole
matter. The parameters $u_1u_2$ will determine generators of
the left representing sphere. They have, however, a more
direct significance. For if $u_2$ remain constant while $u_1$ varies,
the left paratactics to the ray in question passing through the
point $(1, 0, 0, 0)$ will trace a pencil, and this pencil will lie in
%[File: 235.png]
a plane tangent to the Absolute\index{Absolute}, for there is only one value
for $u_1$, namely, $-\dfrac{1}{u_2}$, which will make the moving ray\index{Ray} tangent
to the Absolute. When, therefore, $u_2$ is fixed, one of the left
generators of the Absolute met by the ray in question is fixed,
and this shows that $u_1u_2$ are the parameters determining the
left generators which the ray intersects, while $z_1z_2$ in like
manner determine the right
generators\index{Left and right generators of Absolute}\index{Right and left generators of Absolute}.

If two rays meet the same two generators of one set they
are paratactic\index{Parataxy}, i.e.\ their lines are. If they meet the same two
generators of different sets, they are either parallel\index{Parallelism} or
pseudoparallel\index{Pseudo-parallelism of lines}.
The conditions for parallelism or pseudo-parallelism
will be that two rays shall have the same value for one $(u)$
and for one $(z)$. Let us, in fact, assume that the subscripts
are assigned to the letters $u_1u_2$, $z_1z_2$ in such a way that a
direct conformal transformation, or isotropic
congruence\index{Congruence11@--- isotropic}\index{Isotropic02@--- congruence}, is
given by equations (29). Such a congruence will contain $\infty^1$
rays pseudo-parallel to a given ray, but only a finite number
parallel to it. The conditions for pseudo-parallelism will
thus be
\begin{equation*}
u_1{}'=u_1,\; z_1{}'=z_1, \text{\ or }u_2{}'=u_2,\; z_2{}'=z_2.\tag{33}
\end{equation*}

On the other hand a pseudo-isotropic congruence\index{Pseudo-isotropic congruence} will be
given by (30), and the conditions for parallelism will be
\begin{equation*}
u_1{}'=u_1,\; z_2{}'=z_2, \text{\ or }u_2{}'=u_2,\; z_1{}'=z_1.\tag{34}
\end{equation*}

To pass to the hyperbolic case, let us now assume that
$(_lX) (_rX)$ are two points of the hyperbolic Absolute, and that,
taken in order, they give a ray from $(_lX)$ to $(_rX)$. Two rays
will be parallel if
\begin{equation*}
(_lX)=(_lX')\text{\ or }(_rX)=(_rX').
\end{equation*}

\mysect{Resulting interchange among the three special types of congruence}
Equations (33) will give the conditions for parataxy, while
(34) give those for pseudo-parallelism. We might push the
matter still further by distinguishing between syntaxy and
anti-taxy, synparallelism and anti-parallelism, but we shall
not enter into such questions here. Equations (29) will give
a congruence whose rays can be assembled into surfaces with
paratactic generators, i.e.\ a congruence of normals to a surface
of Gaussian curvature zero; (30) will give an isotropic congruence,
while (32) will give a pseudo-isotropic congruence.
We may tabulate our results as follows.%
\footnote{The Author's\index{Author} attention was first called to this remarkable
correspondence by Professor Study\index{Study} in a letter in the summer of 1905.
It is developed, without proof, but in detail, in his second memoir,
`Ueber nichteuklidische und Liniengeometrie,' \emph{Jahresbericht
der deutschen Mathematikervereinigung}, xv, 1906.}
%endfootnote

\medskip
%[File: 236.png]
\twincol{\hspace{0.5in}\emph{Hyperbolic Space.}}{\hspace{0.5in}\emph{Elliptic Space.}}\par
\twincol{Ray\index{Ray}.}{Ray.}\par
\twincol{Real ray in actual domain or pencil of tangents to Absolute.}{Real Ray}\par\smallskip
\twincol{Real parallelism\index{Parallelism}.}{Real parataxy\index{Parataxy}.}\par
\twincol{Imaginary pseudo-parallelism\index{Pseudo-parallelism of lines}.}{Imaginary parallelism.}\par
\twincol{Imaginary parataxy.}{Imaginary pseudo-parallelism.}\par
\twincol{Real congruence of normals\index{Congruence09@--- of normals}\index{Congruence10@--- of normals, to surfaces of Gaussian curvature zero}
to surface of Gaussian curvature zero\index{Curvature08@--- surfaces of zero}.}{Real isotropic congruence.}\par\smallskip
\twincol{Real isotropic congruence\index{Congruence11@--- isotropic}\index{Isotropic02@--- congruence}.}
{Real pseudo-isotropic congruence\index{Pseudo-isotropic congruence}.}\par
\twincol{Real pseudo-isotropic congruence.}{Real congruence of normals to a surface of Gaussian curvature zero.}
\index{Congruence07@--- of lines, analytic|)}


%[File: 237.png]
%CHAPTER XVII
\mychap{XVII}{MULTIPLY CONNECTED SPACES}
\mysect{Repudiation of the axiom of free mobility of space as a whole}

In Chapters I and II we laid down a system of axioms
for our fundamental objects \emph{points} and \emph{distances}, and showed
how, thereby, we might build up the geometry of a restricted
region. We also saw that with the addition of an assumption
concerning the sum of the angles of a single triangle, we
were in a position to develop fully the elliptic, hyperbolic\index{Hyperbolic02@--- space},
or euclidean geometry of the restricted region in question.
Our spaces so defined were not, however, perfect analytic
continua, even in the real domain. To reach such continua
it was necessary to assume that any chosen segment might
be extended beyond either extremity by a chosen amount.
We saw in the beginning of Chapter VII that this assumption,
though allowable in the euclidean and hyperbolic cases,
will involve a contradiction when added to the assumptions
already made for elliptic space. The difficulty was overcome
by assuming the existence of a space which contained as
sub-regions (called \emph{consistent regions}\index{Consistent region}\index{Region consistent})
spaces where our
previous axioms held good. For this new type of space we
set up our Axioms I$'$--VI$'$.

Our next task was to show that under Axioms I$'$--V$'$ each
point will surely have one set of homogeneous coordinates\index{Coordinates02@Co-ordinates of a point} $(x)$,
and conversely, to each set of real coordinates subject to the
restriction that in hyperbolic space
\begin{flalign*}
 && k^2 \dot{x}_0{}^2 + \dot{x}_1{}^2 + &\dot{x}_2{}^2 + \dot{x}_3{}^2 < 0,  \\
  &\text{in elliptic space}       &(xx) &> 0, &\phantom{\text{in elliptic space}} \\
  &\text{and in euclidean space}  &x_0 &\neq 0,
\end{flalign*}
there will surely correspond one real point. Under the
euclidean or hyperbolic hypotheses each set of real coordinates
can correspond to one real point, at most; under the
elliptic hypothesis, on the contrary, we found it necessary
to distinguish between elliptic space where but one point
goes with each coordinate set, and the spherical case where
two equivalent points\index{Equivalent points} necessarily have the same coordinates.

One further point was established in connexion with these
developments; to each point there will correspond but a single
set of homogeneous coordinates $(x)$. The proof of this depended
%[File: 238.png]
upon Axiom VI$'$, which required that a congruent transformation
of one consistent region\index{Consistent region}\index{Region consistent} should produce one definite
transformation of space as a whole. Of course such an
assumption, when applied to our space of experience, can
neither be proved nor disproved empirically. In the present
chapter we shall set ourselves the task of examining whether,
under Axioms I$'$--V$'$ of Chapter VII, it be possible to have
a space where each point shall correspond to several sets of
coordinate\index{Coordinates02@Co-ordinates of a point} values.%
\footnote{The present chapter is in close accord
with Killing\index{Killing}, \emph{Die Grundlagen der Geometrie}, Paderborn, 1893,
Part iv. Another account will be found in Woods'\index{Woods} `Forms of
Non-Euclidean-Space', published in \emph{Lectures on Mathematics},
Woods, Van Vleck, and White, New York, 1905.}
%endfootnote
For simplicity we shall assume that no
two different points can have the same coordinates.

\mysect{Resulting possibility of one to many correspondence of points and coordinate sets}
What will be the meaning of the statement that under our
set of axioms two sets of coordinate values $(x)$, $(x')$ belong
to the same point? Let a coordinate system be set up, as
in Chapter V, in some consistent region; let this region be
connected with the given point by two different sets of overlapping
consistent regions; then $(x)$ and $(x')$ shall be two
different sets of coordinate values for this point, obtained by
two different sets of analytic extension of the original coordinate
system.

Let us first assume that there is a consistent region which
is reached by each chain of overlapping consistent regions,
a statement which will always hold true when there is a single
point so reached. We may set up a coordinate system in
this region, and then make successive analytic extensions for
the change of axes from one to another of the overlapping
consistent regions, until we have run through the whole
circuit, and come back to the region in which we started.
If, then, one point of the region have different values for its
coordinates from what it had at the start, the same will be
true of all, or all but a finite number of points of the region,
and the new coordinate values will be obtained from the old
ones (in the non-euclidean cases) by means of an orthogonal
substitution. If $(x)$ and $(x')$ be two sets of coordinates for
one point we shall have

\begin{equation*}
x_i{}'=\sum_j^{0..3}a_{ij}x_j,\quad |a_{ij}|\neq 0.\tag{1}
\end{equation*}

Conversely, if these equations hold for any point, they will
represent an identical transformation of the region, and give
two sets of coordinate values for every point of the region.
%[File: 239.png]
We see also by analytic extension that these equations will
give two sets of coordinate values for every point in space.

There is one possible variation in our axioms which should
be mentioned at this point. It is entirely possible to build
up a geometrical system where IV$'$ holds \emph{in general} only,
and there are special points, called \emph{singular points}\index{Singular region}, which
can lie in two consistent regions\index{Consistent region}\index{Region consistent}
which have no sub-region
in common. In two dimensions we have a simple example
in the case of the geometry of the euclidean cone with
a singular line. We shall, however, exclude this possibility
by sticking closely to our axioms.

\index{Multiply connected space|(}
\index{Space|(}
Let us suppose that we have two overlapping systems of
consistent regions going from the one wherein our coordinate
axes were set up to a chosen point $P$. We may connect $P$
with a chosen point $A$ of the original region by two continuous
curves, thus making, in all, a continuous loop. If
now, $P_1$ be a point which will have two different sets of
coordinate values, according as we arrive at it by the one
or the other set of extensions, we see that our loop is of a sort
which cannot be reduced in size beyond a definite amount
without losing its characteristic property. This shows that,
in the sense of analysis situs, our space is multiply connected\index{Connectivity of space}.
In speaking of spaces which obey Axioms I$'$--V$'$, but where
each point can have several sets of coordinate values, we
shall use the term \emph{multiply connected spaces}.

Suppose that we have a third set of coordinate values for
a point of our consistent region. These will be connected
with the second set by a relation

\begin{equation*}
x_i{}''=\sum_j^{0..3}b_{ij}x_j{}',\quad |b_{ij}|\neq 0.
\end{equation*}

We see that $(x'')$ and $(x)$ are also connected by a relation
of this type, hence

\smallskip\emph{Theorem}~1. The assemblage of all coordinate transformations
which represent the identical transformation of a multiply
connected space form a group.

If $(x)$ and $(x')$ be two sets of coordinates for the same point
the expression
\begin{equation*}
\left|\cos^{-1}\frac{(xx')}{\sqrt{(xx)}\;\sqrt{(x'x')}}\right|
\end{equation*}
cannot sink below a definite minimum value greater than
zero, for then we should have two different points of the same
%[File: 240.png]
consistent region with the same coordinate values, which we
have seen is impossible (Chapter VII).

\index{Fundamental01@Fundamental region|(}
\index{Region01@Region, fundamental|(}
For the sake of clearness in our subsequent work let us
introduce, besides our multiply connected space $S$, a space $\Sigma$,
having the same value for the constant $k$ as our space $S$.
and giving to each point one set of coordinate values only.
The group of identical transformations of $S$ will appear
in $\Sigma$ as a group of congruent
transformations\index{Congruent transformations}\index{Transformations, congruent}, a group which
has the property that none of its transformations can leave
a real point of the actual domain invariant, nor produce an
infinitesimal transformation of that domain. We lay stress
upon the actual domain of $\Sigma$, for in $S$ we are interested in
actual points only. Let us further define as \emph{fundamental}
such a region of $\Sigma$, that every point of $\Sigma$ has an equivalent
in this region under the congruent sub-group which we are
now considering, yet no two points of a fundamental region
are equivalent to one another. The points of $S$ may be
put into one to one correspondence with those of a fundamental
region of this sort or of a portion thereof, and,
conversely, such a fundamental region will furnish an example
of a multiply connected space obeying Axioms I$'$--V$'$.

\smallskip\emph{Theorem}~2. Every real group of congruent transformations
of euclidean\index{Euclidean02@--- space}, hyperbolic, or elliptic space, which carries the
actual domain into itself, and none of whose members leave
an actual point invariant, nor transport such a point an
infinitesimal amount, may be taken as the group of identical
transformations of a multiply connected space whose points
may be put into one to one correspondence with the points
of a portion of any fundamental domain of the given space
for that group.

Our interest will, from now on, centre in the space $\Sigma$. We
shall also find it advisable to treat the euclidean and the two
non-euclidean cases separately.

\mysect{Multiply connected euclidean planes}
We shall begin by asking what groups of congruent transformations
of the euclidean plane fulfil the requirements of
Theorem 2. Every congruent transformation\index{Collineations} of the euclidean
plane is either a translation\index{Translations} or a rotation, but the latter type
is inadmissible for our present purpose. What then are the
groups of translations of the euclidean plane? The simplest
is evidently composed of the repetitions of a single translation.
If the amplitude of the translation be $l$, while $n$ is an integer,
positive or negative, this group may be expressed in the form
\[
  x' = x + nl,\quad y' = y.
\]
The fundamental regions will be strips bounded by lines
%[File: 241.png]
parallel to the $y$ axis, each strip including one of the bounding
lines. A corresponding space $S$ will be furnished by a euclidean\index{Euclidean02@--- space}
cylinder of circumference $l$.

What translation groups can be compounded from two given
translations? It is clear that the lines of motion of the two
should not be parallel. For if, in that case, their amplitudes
were commensurable, we should fall back upon the preceding
system; but if the amplitudes were incommensurable, the
group would contain infinitesimal transformations; and these
we must exclude. On the other hand, the group compounded
from repetitions of two non-parallel translations will suit our
purpose very well. If the amplitudes of the two be $l$ and $\lambda$,
while $m$ and $n$ are integers, we may write our group in the
form
\[
x'=x + nl,\quad y' = y + m\lambda.
\]

The fundamental regions are parallelograms, each including
two adjacent sides, excepting two extremities. The Clifford\index{Clifford}
surface discussed in Chapters X and XV offers an excellent
example of a multiply connected surface of this type.

It is interesting to notice that with these two examples
we exhaust the possibilities of the euclidean plane. Suppose,
in fact, that $P$ is any point of this plane, that is to say,
any point in the finite domain. The points equivalent to it
under the congruent group in question may not cluster anywhere,
hence there is one equivalent, or a finite number of
such, nearer to it than any other. If these nearest equivalents
do not all lie on a line with $P$, we may pick out two of them,
non-collinear with $P$, thus determining one-half of a fundamental
parallelogram. If the nearest equivalents are collinear
with $P$ (and, hence, two only in number), we may pick out
one of them and one of the next nearest (which will be off
that line, unless we are under our previous first case), and
thus construct a parallelogram within which there is no
equivalent to $P$, for every point within such a parallelogram
is nearer to one vertex than any two vertices are to one
another. This parallelogram, including two adjacent sides,
except the vertices which are not common, will constitute
a fundamental region, and we are back on the second previous
case. Let the reader notice an exactly similar line of reasoning
will show that there cannot exist any single valued continuous
function of the complex variable which possesses more than
two independent periods.

\mysect{Multiply connected euclidean spaces, various types of line in them}
In a three-dimensional euclidean space we shall find suitable
groups compounded of one, two, or three independent translations\index{Translations}.
The fundamental regions will be respectively layers
%[File: 242.png]
between parallel planes, four-faced prismatic spaces, and
parallelepipeds. It is easy to determine how much of the
bounding surface should be included in each case. It is also
evident that there can be no other groups composed of
translations only, which fulfil the requirements.

Let us glance for a moment at the various forms of straight
line which will exist in a multiply connected euclidean space $S$,
which corresponds to a euclidean parallelepiped in $\Sigma$. The
corresponding lines in $\Sigma$ shall all pass through one vertex
of the fundamental parallelepiped. If the line in $\Sigma$ be one
edge of the parallelepiped, the line in $S$ will be a simple loop
of length equal to one period. If the line in $\Sigma$ connect the
vertex with any other equivalent point, the line in $S$ will still
be a loop, but of greater length. If, lastly, the line in $\Sigma$ do
not contain any other point equivalent to the vertex, the line
in $S$ will be open, but, if followed sufficiently far, will pass
again as close as desired to the chosen point.

There are other groups of motions of euclidean space\index{Euclidean02@--- space},
besides translations which give rise to multiply connected
spaces. An obvious example is furnished by the repetitions
of a single screw motion. This may be expressed, $n$ being
an integer, in the form
\[
x'= x\cos n\theta-y\sin n\theta,\quad
  y' = x\sin n\theta + y\cos n\theta,\quad
  z'=z + nd.
\]

The fundamental regions in $\Sigma$ will be layers bounded by
parallel planes. In $S$ we shall have various types of straight
lines. The $Z$ axis will be a simple closed loop of length $d$.
Will there be any other closed lines in $S$? The corresponding
lines in $\Sigma$ must be parallel to the axis, there being an infinite
number of points of each at the same distance from that axis.
When $\theta$ and $2\pi$ are commensurable, we see that every parallel
to the $Z$ axis will go into a closed line of the type required,
when $\theta$ and $2\pi$ are incommensurable, the $Z$ axis is the only
closed line.

Let us now take two points of $\Sigma$ separated by a distance $r$
\begin{align*}
\xi & = x + r \cos \alpha, \\
\eta & = y + r \cos \beta, \\
\zeta & = z + r \cos \gamma.
\end{align*}

The necessary and sufficient condition that they should be
equivalent is
\begin{align*}
x\cos n\theta-y\sin n\theta & = x + r \cos \alpha, \\
x \sin n\theta + y \cos n \theta & = y + r \cos \beta, \\
nd & = r \cos \gamma.
\end{align*}

The last of these equations shows that a line in $\Sigma$ perpendicular
%[File: 243.png]
to the $Z$ axis (i.e.\ parallel to a line meeting it
perpendicularly) cannot return to itself. On the other hand, if
\[
  \cos \alpha = \cos \beta = 0:\quad n\theta = 2 m\pi,
\]
and we have a closed loop of the type just discussed. If
$\alpha$, $\beta$, $\gamma$, $n$ be given, $r$ may be determined by the last equation,
and $x$, $y$ from the two preceding, since the determinant of
the coefficients will not, in general, vanish. We thus see that
in $S$ the lines with direction angles $\alpha$, $\beta$, $\gamma$, and possessing
double points, will form an infinite discontinuous assemblage.
If, on the other hand, $x$, $y$, $z$, $n$ be given, $\alpha$, $\beta$, $\gamma$, $r$ may be
determined from the given equations, coupled with the fact
that the sum of the squares of the direction cosines is unity;
through each point in $S$, not on the $Z$ axis, will pass an infinite
number of straight lines, having this as a double point.

The planes\index{Plane} in $S$ will be of three sorts. Those which are
perpendicular to the $Z$ axis will contain open lines only, those
whose equations lack the $Z$ term will contain all sorts of lines.
Other planes will contain no lines which are simple loops.

Another type of multiply connected space will be determined by
\begin{align*}
  x' &= (-1)^l x + ma,  \\
  y' &= (-1)^l y + nb,  \\
  z' &= z + lc.
\end{align*}
$l$, $m$, $n$ being integers.

The fundamental regions in $\Sigma$\index{Euclidean02@--- space} will be triangular right
prisms. Lines in $\Sigma$ parallel to the $Z$ axis will appear in $S$
as simple closed loops of length $2c$. To find lines which cross
themselves, let us write
\begin{align*}
  x + r\cos\alpha &= (-1)^l x + ma,   \\
  y + r\cos\beta  &= (-1)^l y + nb,   \\
  z + r\cos\gamma &= z + lc.
\end{align*}

For each even integral value of $l$, and each integral value
of $m$ and $n$, we get a bundle of loop lines in $S$ with direction
cosines
\[
  \cos\alpha = \frac{ma}{\sqrt{m^2a^2 + n^2b^2 + l^2c^2}}\text{, \&c.}
\]

When $l$ is odd, we shall have through each point an infinite
number of lines which have a double point there, the direction
cosines being
\[
  \cos\alpha
= \frac{-2x + ma}{\sqrt{(-2x+ma)^2 + (-2y+nb)^2 + l^2c^2}}
  \text{, \&c.}
\]

Such lines will, in general, be open. We see, however, that
%[File: 244.png]
whereas the length of a loop perpendicular to the $x$, $y$ plane\index{Plane}
is $2c$, if the point $\dfrac{ma}{2}$, $\dfrac{nb}{2}$ happen to be on such a loop, this
point is reached again after a distance $C$.  This loop has
therefore, the general form of a lemniscate.%
\footnote{These and the preceding example are taken from Killing, \emph{Grundlagen},
loc.\ cit. The last is not, however, worked out.}
%endfootnote

\mysect{Hyperbolic case little known; relation to automorphic functions}
When we turn from the euclidean to the hyperbolic
hypothesis, we find a less satisfactory state of affairs. The
real congruent group of the hyperbolic plane was shown in
Chapter VIII to depend upon the real binary group
\begin{align*}
\sigma t_1{}' & = \alpha_{11}t_1 + \alpha_{12}t_2, \\
\sigma t_2{}' & = \alpha_{21}t_1 + \alpha_{12}t_2,
\end{align*}
the homogeneous coordinates $(t)$ being supposed to define
a point of the absolute conic. The two fixed points must
be real, in order that the line joining them shall be actual,
and its pole, the fixed point, ideal. In other words, we wish
for groups of binary linear substitutions which contain
members of the hyperbolic type exclusively. Apparently
such groups have not, as yet, been found. It might seem,
at first, that parabolic transformations where the two fixed
points of the conic fall together, would also answer, but
such is not the case. We may show, in fact, that in such
a substitution there will be points of the plane which are
transformed by as small a distance as we please. The path
curves are horocycles\index{Horocycle} touching the absolute conic at the fixed
point: having in fact, four-point contact with it. It is merely
necessary to show that a horocycle of the family may be found
which cuts two lines through the fixed point in two points
as near together as we please. Let this fixed point be $(0, 0, 1)$
while the absolute conic has an equation of the form
\[
x_0{}^2 + x_1x_2 = 0.
\]

The general type for the equation of a horocycle tangent
at $(0, 0, 1)$ will be
\[
(x_0{}^2+x_1x_2) + px_1{}^2 = 0.
\]
This will intersect the two lines
\[
x_0-lx_1= 0,\quad x_0-mx_1 = 0,
\]
in the points $(l, 1,-(l^2+p))$ $(m, 1,-(m^2+p))$. The cosine of
the $k$th part of their distance will be
\[
\frac{(l-m)^2+2p}{2p},
\]
%[File: 245.png]
an expression which will approach unity as a limit, as $\dfrac{1}{p}$
approaches zero.

The group of hyperbolic motions in three dimensions will,
as we saw in Chapter VIII, depend upon the linear function
of the complex variable
\begin{equation*}
z'=\frac{\alpha z+\beta}{\gamma z+\delta}.
\end{equation*}

The group which we require must not contain rotations
about a line tangent to the Absolute\index{Absolute}, for the reason which
we have just seen, hence the complex substitution must not
be parabolic. Again, we may not have rotations about actual
lines, hence the path curves on the Absolute may not be conics
in planes through an ideal line (the absolute polar of the axis
of rotation); the substitutions may not be elliptic. The only
allowable motions of hyperbolic space are rotations about
ideal lines, which give hyperbolic substitutions, and screw
motions, which give loxodromic ones. There does not seem
to be any general theory of groups of linear transformations
of the complex variable, which include merely hyperbolic
and loxodromic members only.%
\footnote{For the general theory of discontinuous
groups of linear substitutions, see Fricke-Klein\index{Fricke}\index{Klein},
\emph{Vorlesungen \"uber die Theorie der automorphen Funktionen}, vol.~i, Leipzig, 1897.}
%endfootnote

The group of repetitions of a single rotation about an ideal
line may be put into the form $(k^2=-1)$,
\begin{align*}
\dot{x}_0{}'&=\dot{x}_0 \cosh{n\theta}-\dot{x}_3\sin{n\theta},\\
\dot{x}_1{}'&=\dot{x}_1,\\
\dot{x}_2{}'&=\dot{x}_2,\\
\dot{x}_3{}'&=\dot{x}_0 \sinh{\theta}+\dot{x}_3\cosh{\theta}.
\end{align*}

The fundamental regions in $\Sigma$ will be bounded by pairs
of planes through the line
\begin{equation*}
x_0=x_3=0.
\end{equation*}

The orthogonal trajectories of planes through this line will
be equidistant curves whose centres lie thereon. A line in $\Sigma$
connecting two points which are equivalent under the group
will appear in $S$ as a line crossing itself once.

We may, in like manner, write the group of repetitions
of a single screw motion
\begin{align*}
\dot{x}_0{}'&=\dot{x}_0\cosh{n\theta}-\dot{x}_3\sinh{n\theta},\\
\dot{x}_1{}'&=\dot{x}_1\cos{n\phi}-\dot{x}_2\sin{n\phi},\\
\dot{x}_2{}'&=\dot{x}_1\sin{n\phi}+\dot{x}_2\cos{n\phi},\\
\dot{x}_3{}'&=\dot{x}_0\sinh{n\theta}+\dot{x}_3\cosh{n\theta}.
\end{align*}

%[File: 246.png]
\mysect{Non-existence of multiply connected elliptic planes}
In elliptic space\index{Elliptic03@--- space} we obtain rather more satisfactory results.
Every congruent transformation of the real elliptic plane is
a rotation about an actual point, there being no ideal points.
Hence, there are no two-dimensional multiply connected
elliptic spaces. In three dimensions the case is different. Let
us assume that $k=1$, and consider the group of repetitions
of a single screw motion. The angle of rotation about one
axis is equal to the distance of translation along the other,
and the two distances or angles of rotation must be of the
form $\dfrac{\lambda\pi}{\nu}$, $\dfrac{\mu\pi}{\nu'}$
in order that there shall be no infinitesimal
transformations in the group. Moreover, these two fractions
must have the same denominator, for otherwise the group
would contain rotations. We may therefore write the general
equations
\begin{align*}
x_0{}'&=x_0\cos{n\frac{\lambda\pi}{\nu}}-x_1\sin{n\frac{\lambda\pi}{\nu}},\\
x_1{}'&=x_0\sin{n\frac{\lambda\pi}{\nu}}+x_1\cos{n\frac{\lambda\pi}{\nu}},\\
x_2{}'&=x_2\cos{n\frac{\mu\pi}{\nu}}-x_3\sin{n\frac{\mu\pi}{\nu}},\\
x_3{}'&=x_2\sin{n\frac{\mu\pi}{\nu}}+x_3\cos{n\frac{\mu\pi}{\nu}},
\end{align*}
where $\lambda$, $\mu$, $\nu$ are constant integers, and $n$ a variable integer.
It will be found that the cosine of the distance of the points
$(x)$, $(x')$ will be equal to unity only when $n$ is divisible by $\nu$,
i.e.\ we have the identical transformation, so that there are no
real fixed points nor points moved an infinitesimal distance.
If $\lambda=\mu$ we have a translation (cf.\ Chapter VIII), for our
transformation may be written in the quaternion\index{Quaternions} form:%
\footnote{Killing\index{Killing}, \emph{Grundlagen},
cit.\ p.~342, erroneously states that these translations\index{Translations}
are the only motions along one fixed line yielding a group of the desired
type. The mistake is corrected by Woods\index{Woods}, loc.\ cit., p.~68.}
%endfootnote
\begin{equation*}
(x_0{}'+x_1{}'i+x_2{}'j+x_3{}'k)=(\cos{n\frac{\lambda\pi}{\nu}}+\sin{n\frac{\lambda\pi}{\nu}i}) (x_0+x_1i+x_2j+x_3k).
\end{equation*}

\mysect{Multiply connected elliptic spaces}
The path-curves in $\Sigma$ will be lines paratactic to either axis
of rotation, and they will appear in $S$ as simple closed loops
of length $\dfrac{\pi}{\nu}$. Notice the close analogy of this case to the
simplest case in euclidean space.

%[File: 247.png]
There is another translation group of elliptic space giving
rise to a multiply connected space of a simple and interesting
description. Let $\lambda_1:\lambda_2$ be homogeneous parameters, locating
the generators of one set on the Absolute. Each linear transformation
of these will determine a translation. In particular,
if we put
\[
x_0+ix_1 = \lambda_1,\quad x_2-ix_3=\lambda_2,
\]
then the translation
\[
(x_0{}' + x_1{}'i + x_2{}'j + x_3{}'k) = (a+bi+cj+dk)(x_0+x_1i+x_2j+x_3k),
\]
may also be written
\begin{align*}
\lambda_1{}'&=(a+bi)\lambda_1-(c+di)\lambda_2,\\
\lambda_2{}'&=(c-di)\lambda_1+(a-bi)\lambda_2.
\end{align*}

Now this is precisely the formula for the rotation of the
euclidean sphere. The cosine of the distance traversed by
the point ($x$) will be
\[
\frac{a}{\sqrt{a^2+b^2+c^2+d^2}},
\]
which becomes equal to unity only when $b = c = d = 0$, i.e.\ when
we have the identical transformation. The groups of
elliptic translations which contain no infinitesimal transformations,
are therefore identical with those of euclidean
rotations about a fixed point which contain no infinitesimal
members, whence

\smallskip\emph{Theorem}~3.%
\footnote{Cf.\ Woods\index{Woods}, loc.\ cit., p.~68.}
%endfootnote
If a multiply connected elliptic space be
transformed identically by a group of translations\index{Translations}, that group
is isomorphic with one of the groups of the regular solids.
Conversely each group of the regular solids gives rise to a
group of right or left elliptic
translations\index{Left and right translations}\index{Right and left translations},
suitable to define
a multiply connected space of elliptic type.

Of course the inner reason for this identity is that a real
line meets the elliptic Absolute\index{Absolute} in conjugate imaginary points,
corresponding to diametral imaginary values of the parameter
for either set of generators, and a real point of a euclidean
sphere is given by the value of its coordinate as a point of
the Gauss sphere, while diametrically opposite points will be
given by diametral values of the complex variable. The
problem of finding elliptic translations, or euclidean rotations,
depend therefore, merely on the problem of finding linear
transformations of the complex variable which transport
diametral values into diametral values.
\index{Multiply connected space|)}
\index{Space|)}
\index{Fundamental01@Fundamental region|)}
\index{Region01@Region, fundamental|)}

%[File: 248.png]
%CHAPTER XVIII
\mychap{XVIII}{THE PROJECTIVE BASIS OF NON-EUCLIDEAN GEOMETRY}
\mysect{Fundamental notions}

Our non-euclidean system of metrics, as developed in
Chapter VII and subsequently, rests in the last analysis,
upon a projective concept, namely, the cross ratio\index{Cross ratios}. The group
of congruent transformations appeared in Chapter VII as
a six-parameter collineation group, which left invariant a
certain quadric called the Absolute. An exception must be
made in the euclidean case where the congruent group was
a six-parameter sub-group of the seven-parameter group which
left a conic in place. We thus come naturally to the idea
that a basis for our whole edifice may be found in projective
geometry, and that non-euclidean metrical geometry may be
built up by positing the Absolute, and defining distance as
in Chapter VII\@. It is the object of the present chapter to
show precisely how this may be done, starting once more
at the very beginning.%
\footnote{The first writer to set up a suitable set of axioms for projective geometry
was Pieri\index{Pieri}, in his \emph{Principii della geometria di posizione}, cit. He has had many
successors, as Enriques\index{Enriques},
\emph{Lezioni di geometria proiettiva}, Bologna, 1898, or Vahlen\index{Vahlen},
\emph{Abstrakte Geometric}, cit., Parts II and III\@.
Veblen\index{Veblen} and Young\index{Young}, `A system of
axioms for projective geometry,' \emph{American Journal of Mathematics}, Vol.~xxx,
1908.}
%endfootnote

\mysect{Axioms of connexion and separation}
\medskip\textsc{Axiom}~I. \textbf{There exists a class of objects, containing at
least two distinct members, called points\index{Point}.}

\medskip\textsc{Axiom}~II\@. \textbf{Each pair of distinct points belongs to a single
sub-class called a line.}

The points shall also be said to be on the line, the line
to pass through the points. A point common to two lines
shall be called their intersection. It is evident from Axiom II
that two lines with two common points are identical. We
have thus ruled out the possibility of building up spherical
geometry upon the present basis.

\medskip\textsc{Axiom}~III\@. \textbf{Two distinct points determine among the
remaining points of their line two mutually exclusive sub-classes\index{Separation02@--- classes},
neither of which is empty.}

If the given points be $A$ and $B$, two points belonging to
%[File: 249.png]
different classes according to Axiom III shall be said to be
\emph{separated} by them, two belonging to the same class \emph{not
separated}.%
\footnote{The axioms of separation\index{Separation01@Separation} were first given by Vailati\index{Vailati},
`Sulle propriet\`a
caratteristiche delle variet\`a a una dimensione,' \emph{Rivista di Matematica}, v, 1895.}
%endfootnote
We shall call such classes \emph{separation classes}.

\medskip\textsc{Axiom}~IV\@. \textbf{If $P$ and $Q$ be separated by $A$ and $B$, then
$Q$ and $P$ are separated by $A$ and $B$.}

\medskip\textsc{Axiom}~V. \textbf{If $P$ and $Q$ be separated by $A$ and $B$, then
$A$ and $B$ are separated by $P$ and $Q$.}

\label{sep}We shall write this relation $PQ\sep AB$ or $AB\sep PQ$. If $PQ$
be not separated by $A$ and $B$, though on a line\index{Line}, or collinear,
with them, we shall write $PQ\nsep AB$.

\medskip\textsc{Axiom}~VI\@. \textbf{If four distinct collinear points be given there
is a single way in which they may be divided into two
mutually separating pairs.}

\smallskip\emph{Theorem}~1. $AB \sep CD$ and $AE\sep CD$, then $EB\nsep CD$.

For $C$ and $D$ determine but two separation classes on the
line, and both $B$ and $E$ belong to that class which does not
include $A$.

\smallskip\emph{Theorem}~2. If five collinear points be given, a chosen pair
of them will either separate two of the pairs formed by the
other three or none of them.

Let the five points be $A$, $B$, $C$, $D$, $E$. Let $AC\sep DE$. Then, if
$BC\sep DE$, $AB\hspace{-0.5pt}\nsep\hspace{-0.5pt} DE$, and if $AB\hspace{-0.5pt} \sep\hspace{-0.5pt} DE$,
$BC\hspace{-0.5pt}\nsep\hspace{-0.5pt} DE$. But if we had
$BC\hspace{-0.5pt}\nsep\hspace{-0.5pt} DE$ and $AB\hspace{-0.5pt}\nsep\hspace{-0.5pt} DE$, $ABC$ would belong to the same
separation class with regard to $DE$, and hence $AC\nsep DE$.

\smallskip\emph{Theorem}~3. If $AC\sep BD$ and $AE\sep CD$, then $AE\sep BD$.

To begin with $BC\nsep AD$, $EC\nsep AD$; hence $BE\nsep AD$. Again,
if we had $AB\sep ED$, we should have $AB\sep EC$, i.e.\ $AE\nsep BC$.
But we have $AE\sep CD$, hence $AE\sep BD$ a contradiction with
%[File: 250.png]
$AB \sep ED$. As a result, since
$BE \nsep AD$ and
$AB \nsep ED$, we must
have $AE \sep BD$.

It will be clear that this theorem includes as a special case
Theorem 3 of Chapter I. We have but to take $A$ at a great
distance.

\smallskip\emph{Theorem}~4. If $PA \sep CD$, $PB \sep CD$, $PQ \sep AB$, then $PQ \sep CD$.
The proof is left to the reader.

It will follow from the fact that neither of our separation\index{Separation01@Separation}
classes\index{Separation02@--- classes} is empty that the assemblage of all points of a line
is infinite and dense. We have but to choose one point of
the line\index{Line}, and say that a point is between two others when
it be separated thereby from the chosen point.

\medskip\textsc{Axiom}~VII\@. \textbf{If all points of either separation class determined
by two points $A$, $B$, be so divided into two sub-classes
that no point of the first is separated from $A$ by $B$ and
a point of the second, there will exist a single point $C$ of
this separation class of such a nature that no point of the
first sub-class is separated from $A$ by $B$ and $C$, and none
of the second is separated from $B$ by $A$ and $C$.}

It is clear that $C$ may be reckoned as belonging to either
sub-class, but that no other point enjoys this property.
This axiom is one of continuity\index{Continuity01@Continuity, axiom of}, let the reader make a careful
comparison with XVIII of Chapter II.

\medskip\textsc{Axiom}~VIII\@. \textbf{All points do not belong to one line.}

\mysect{Projective geometry of the plane}
\smallskip\emph{Definition}.~The assemblage of all points of all lines determined
by a given point and all points of a line not containing
the first shall be called a \emph{plane}\index{Plane}. Points or lines in the same
plane shall be called \emph{coplanar}.

\medskip\textsc{Axiom}~IX\@. \textbf{A line
intersecting\index{Intersection01@Intersection of lines} in distinct points two of
the three lines determined by three non-collinear points,
intersects the third line.}

Let the reader compare this with the weaker Axiom XVI
of Chapter I.

\smallskip\emph{Theorem}~5. A plane will contain completely every line
whereof it contains two points.

Let the plane be determined by the point $A$ and the line
$BC$. If the two given points of the given line belong to $BC$
or be $A$ and a point of $BC$, the theorem is immediate. If not,
%[File: 251.png]
let the line contain the points $B'$ and $C'$ of $AB$ and $AC$
respectively. Let $P$ be any other point of the given line.
Then $BP$ will intersect $AC$, hence $AP$ will intersect $BC$ or
will lie in the given plane\index{Plane}.

\smallskip\emph{Theorem}~6. If $A$, $B$, $C$ be three non-collinear points, then
the planes determined by $A$ and $BC$, by $B$ and $CA$, and by
$C$ and $AB$ are identical.

We have but to notice that the lines generating each plane
lie wholly in each of the others.

\smallskip\emph{Theorem}~7. If $A'$, $B'$, $C'$ be three non-collinear points of the
plane determined by $ABC$, then the planes determined by
$A'B'C'$ and $ABC$ are identical.

This will come immediately from the two preceding.

\smallskip\emph{Theorem}~8. Two lines in the same plane always intersect.

Let $B$ and $C$ be two points of the one line, and $A$ a point
of the other, If $A$ be also a point of $BC$ the theorem is proved.
If not, we may use the point $A$ and the line $BC$ to determine
the plane, and our second line must be identical with a line
through $A$ meeting $BC$.

\mysect{Projective geometry of space}
\medskip\textsc{Axiom}~X. \textbf{All points do not lie in one plane.}

\smallskip\emph{Definition}.~The assemblage of all points of all lines which
arc determined by a chosen point, and all points of a plane
not containing the first point shall be called a \emph{space}\index{Space}.

We leave to the reader the proofs of the following very
simple theorems.

\smallskip\emph{Theorem}~9. A space contains completely every line whereof
it contains two points.

\smallskip\emph{Theorem}~10. A space contains completely every plane
whereof it contains three non-collinear points.

\smallskip\emph{Theorem}~11. The space determined by a point $A$ and the
plane $BCD$ is identical with that determined by $B$ and the
plane $CDA$.

\smallskip\emph{Theorem}~12. If $A'$, $B'$, $C'$, $D'$ be four non-coplanar points of
the space determined by $A$, $B$, $C$, $D$, then the two spaces determined
by the two sets of four points are identical.

With regard to the last theorem it is clear that all points of
the space determined by $A'$, $B'$, $C'$, $D'$ lie in that determined by
$A$, $B$, $C$, $D$. Let us assume that $B'$, $C'$, $D'$ are points of
$AB$, $AC$,
$AD$ respectively. The planes $BCD$ and $B'C'D'$ have a common
%[File: 252.png]
line $l$, which naturally belongs to both spaces\index{Space}. Let us first
assume that $AA'$ does not intersect this line. Let $A''$ be the
intersection of $AA'$ with $BCD$. Then $A''B$ meets both $A'B'$
and $l$, hence, has two points in each space, or lies in each.
Then the plane $BCD$ lies in both spaces, as do the line $A'A''$
and the point $A$; the two spaces are identical. If, on the
other hand, $AA'$ meet $l$ in $A''$, then $A$ lies in both spaces.
Furthermore $A'B$ will meet $A''B'$ in a point of both spaces,
so that $B$ will lie in both, and, by similar reasoning, $C$ and $D$
lie in both.

\smallskip\emph{Theorem}~13. Two planes\index{Plane} in the same space have a common
line\index{Intersection02@--- of planes}.

\smallskip\emph{Theorem}~14. Three planes in the same space have a common
line or a common point.

\smallskip\emph{Practical limitation}. All points, lines, and planes hereinafter
considered are supposed to belong to one space.

\smallskip\emph{Theorem}~15. If three lines $AA'$, $BB'$, $CC'$ be concurrent\index{Concurrence},
then the intersections of $AB$ and $A'B'$, of $BC$ and $B'C'$, of $CA$
and $C'A'$ are collinear\index{Collinearity}, and conversely.

This is Desargues'\index{Desargues} theorem of two triangles. The following
is the usual proof. To begin with, let us suppose that the
planes $ABC$ and $A'B'C$ are distinct. The lines $AA'$, $BB'$,
and $CC'$ will be concurrent in $O$ outside of both planes. Then
as $AB$ and $A'B'$ are coplanar, they intersect in a point which
must lie on the line $l$ of intersection of the two planes $ABC$
and $A'B'C'$, and a similar remark applies to the intersections
of $BC$ and $B'C'$, of $CA$ and $C'A'$. Conversely, when these
last-named three pairs of lines intersect, the intersections
must be on $l$. Considering the lines $AA'$, $BB'$, and $CC'$, we
see that each two are coplanar, and must intersect, but all
three are not coplanar. Hence the three are concurrent.
The second case occurs where $A'B'C'$ are three non-collinear
points of the plane determined by $ABC$. Let $V$ and $V'$ be
two points without this plane collinear with $O$ the point of
concurrence of $AA'$, $BB'$, $CC'$. Then $VA$ will meet $V'A'$ in
$A''$, $VB$ will meet $V'B'$ in $B''$, and $VC$ will meet $V'C'$ in $C''$.
The planes $ABC$ and $A''B''C''$ will meet in a line $l$, and
$B''C''$ will meet both $BC$ and $B'C'$ in a point of $l$. In the
same way $CA$ will meet $C'A'$ on $l$, and $AB$ will meet $A'B'$
on $l$. Conversely, if the last-named three pairs of lines meet
in points of a line $l$ in their plane, we may find $A''B''C''$
non-collinear points in another plane through $l$, so that $B''C''$
meets $BC$ and $B'C'$ in a point of $l$, and similarly for $C''A''$,
%[File: 253.png]
$CA$, $C'A'$ and for $A''B''$, $AB$, $A'B'$. Then by the converse
of the first part of our theorem $AA''$, $BB''$, $CC''$ will be
concurrent in $V$, and $A'A''$, $B'B''$, $C'C''$ concurrent in $V'$.
Lastly, the three coaxal planes $VV'A''$, $VV'B''$, $VV'C''$ will
meet the plane $ABC$ in three concurrent lines $AA'$, $BB'$, $CC'$.

We have already remarked in Chapter VI on the dependence
of this theorem for the plane either on the assumption of the
existence of a third dimension, or of a congruent group.

\smallskip\emph{Definition}.~If four coplanar points, no three of which are
collinear, be given, the figure formed by the three pairs of
lines determined by them is called a \emph{complete quadrangle}\index{Quadrangle, complete}.
The original points are called the \emph{vertices}\index{Vertex02@--- of quadrangle}, the pairs of lines
the \emph{sides}\index{Sides02@--- of quadrangle}. Two sides which do not contain a common vertex
shall be said to be \emph{opposite}. The intersections of pairs of
opposite sides shall be called \emph{diagonal points}\index{Diagonal points of quadrangle}.

\smallskip\emph{Theorem}~16. If two complete quadrangles be so situated
that five sides of one meet five sides of the other in points
of a line, the sixth side of the first meets the sixth side of the
second in a point of that line.

The figure formed by four coplanar lines, no three of which
are concurrent, shall be called a \emph{complete quadrilateral}\index{Quadrilateral02@--- complete}.
Their six intersections shall be called the \emph{vertices}\index{Vertex03@--- of quadrilateral}; two vertices
being said to be opposite when they are not on the same side\index{Sides03@--- of quadrilateral}.
The three lines which connect opposite pairs of vertices shall
be called \emph{diagonals}\index{Diagonals of quadrilateral}.

\smallskip\emph{Definition}.~If $A$ and $C$ be two opposite vertices of a complete
quadrilateral, while the diagonal which connects them
meets the other two in $B$ and $D$, then $A$ and $B$ shall be said
to be \emph{harmonically separated}\index{Harmonic02@--- separation}\index{Separation03@--- harmonic}
by $C$ and $D$.

\smallskip\emph{Theorem}~17. If $A$ and $C$ be harmonically separated by
$B$ and $D$, then $B$ and $D$ are harmonically separated by $A$
and $C$.

The proof will come immediately from 15, after drawing
two or three lines; we leave the details to the reader.

\smallskip\emph{Definition}.~If $A$ and $C$ be harmonically separated by
$B$ and $D$, each is said to be the
\emph{harmonic conjugate}\index{Conjugate03@--- harmonic}\index{Harmonic01@Harmonic conjugate} of the
other with regard to these two points; the four points may
also be said to form a \emph{harmonic set}\index{Harmonic03@--- set}.

\smallskip\emph{Theorem}~18. A given point has a unique harmonic conjugate
with regard to any two points collinear with it.

This is an immediate result of 16.

%[File: 254.png]
\smallskip\emph{Theorem}~19. If a point $O$ be connected with four points
$A$, $B$, $C$, $D$ not collinear with it by lines $OA$, $OB$, $OC$, $OD$, and
if these lines meet another line in $A'$, $B'$. $C'$, $D'$ respectively,
and, lastly, if $A$ and $C$ be harmonic
conjugates\index{Conjugate03@--- harmonic}\index{Harmonic01@Harmonic conjugate} with regard
to $B$ and $D$, then $A'$ and $C'$ are harmonic conjugates with
regard to $B'$ and $D'$.

We may legitimately assume that the quadrilateral\index{Quadrilateral02@--- complete} construction
which yield\-ed $A$, $B$, $C$, $D$ was in a plane which did
not contain $O$, for this construction may be effected in any
plane\index{Plane} which contains $AD$. Then radiating lines through $O$
will transfer this quadrilateral construction into another
giving $A'$, $B'$, $C'$, $D'$.

\smallskip\emph{Definition}.~If $a$, $b$, $c$, $d$ be four concurrent lines which pass
through $A$, $B$, $C$, $D$ respectively, and if $A$ and $C$ be harmonically
separated by $B$ and $D$, then $a$ and $c$ may properly be said
to be harmonically separated\index{Harmonic02@--- separation}\index{Separation03@--- harmonic}
by $b$ and $d$, and $b$ and $d$
harmonically separated by $a$ and $c$. We may also speak of
$a$ and $c$ as harmonic conjugates with regard to $b$ and $d$, or
say that the four lines form a harmonic set\index{Harmonic03@--- set}.

\smallskip\emph{Theorem}~20. If four planes $\alpha$, $\beta$, $\gamma$, $\delta$ determined by a line $l$
and four points $A$, $B$, $C$, $D$ meet another line in four points
$A'$, $B'$, $C'$, $D'$ respectively, and if $A$ and $C$ be harmonically
separated by $B$ and $D$, then $A'$ and $C'$ are harmonically
separated by $B'$ and $D'$.

It is sufficient to draw the line $AD'$ and apply 19.

\smallskip\emph{Definition}.~If four coaxal planes $\alpha$, $\beta$, $\gamma$, $\delta$
pass respectively
through four points $A$, $B$, $C$, $D$ where $A$ and $C$ are harmonically
separated by $B$ and $D$; then we may speak of $\alpha$ and $\gamma$ as
harmonically separated by $\beta$ and $\delta$, or $\beta$ and $\delta$ as harmonically
separated by $\alpha$ and $\gamma$. We shall also say that $\alpha$ and $\gamma$ are
harmonic conjugates with regard to $\beta$ and $\delta$, or that the four
planes form a harmonic set.

We shall understand by \emph{projection}\index{Projection} the transformation
(recently used) where\-by coplanar points and lines are carried,
by means of concurrent lines, into other coplanar points and
lines. With this in mind, we have the theorem.

\smallskip\emph{Theorem}~21. Any finite number of projections and intersections
will carry a harmonic set into a harmonic set.

\medskip\textsc{Axiom}~XI\@. \textbf{If four coaxal planes meet two lines respectively
in $A$, $B$, $C$, $D$ and $A'$, $B'$, $C'$, $D'$ distinct points, and
if $AC \sep BD$ then $A'C' \sep B'D'$.}

%[File: 255.png]
\smallskip\emph{Definition}.~If $AC\sep BD$ and $l$ be any line not intersecting
$AD$, we shall say that the planes $lA$ and $lC$ separate the
planes $lB$ and $lD$.

\smallskip\emph{Definition}.~If the planes $\alpha$ and $\gamma$ separate the planes $\beta$
and $\delta$, and if a fifth plane meet the four in $a$, $b$, $c$, $d$
respectively,
then we shall say that $a$ and $c$ separate $b$ and $d$.
A complete justification for this terminology will be found
in Axiom XI and in the two theorems which now follow.

\smallskip\emph{Theorem}~22. The laws of separation laid down for points
in Axioms III--VII hold equally for coplanar concurrent lines,
and coaxal planes.

We have merely to bring the four lines or planes to intersect
another line in distinct points, and apply XI.

\smallskip\emph{Theorem}~23. The relation of separation is unaltered by any
finite number of projections and intersections.

\smallskip\emph{Theorem}~24. If $A$, $B$, $C$, $D$ be four collinear points, and $A$
and $C$ be harmonically separated\index{Harmonic02@--- separation}\index{Separation03@--- harmonic}
by $B$ and $D$, then $AC\sep BD$.

We have merely to observe that our quadrilateral construction
for harmonic separation permits us to pass by
two projections from $A$, $B$, $C$, $D$ to $C$, $B$, $A$, $D$ respectively, so
that if we had $AB\sep CD$ we should also have $CB\sep AD$, and
vice versa. Hence our theorem.

Before proceeding further, let us glance for a moment at the
question of the independence of our axioms.

The author is not familiar with any system of projective
geometry where XI is lacking. X naturally fails in plane
geometry. Here IX must be suitably modified, and Desargues'
theorem, our 15, must be assumed as an axiom. IX is lacking
in the projective euclidean geometry where the ideal
plane is excluded. VIII fails in the geometry of the single
line, while VII is untrue in the system of all points with
rational Cartesian coordinates. III, IV, V, VI may be shown
to be serially independent.%
\footnote{Vailati\index{Vailati}, loc.\ cit., note quoting Padoa\index{Padoa}.}
%endfootnote
II is lacking in the geometry
of four points.

Besides being independent, our axioms possess the far more
important characteristic of being consistent. They will be
satisfied by any class of objects in one to one correspondence
%[File: 256.png]
with all sets of real homogeneous coordinate values
$x_0:x_1:x_2:x_3$ not all simultaneously zero. A line may be
defined as the assemblage of all objects whose coordinates
are linearly dependent on those of two. If $A$ and $C$ have
the coordinates ($x$) and ($y$)\label{xanyy} respectively, while $B$ and $D$ have
the coordinates $\lambda(x) + \mu(y)$ and $\lambda'(x) + \mu'(y)$, then $A$ and $C$
shall be said to be separated\index{Separation01@Separation} by $B$ and $D$ if
\[
\frac{\lambda\mu'}{\lambda'\mu}<0.
\]
When this is not the case, they shall be said to be not
separated by $B$ and $D$.

\mysect{Projective scale and cross ratios}
As a next step in our development of the science of projective
geometry, let us take up the concept of cross ratio.
Suppose that we have three distinct collinear points $P_\infty$, $P_0$, $P_1$.
Construct the harmonic
conjugate\index{Conjugate03@--- harmonic}\index{Harmonic01@Harmonic conjugate}
of $P_0$ with regard to $P_1$
and $P_\infty$, and call it $P_2$, that of $P_1$ with regard to $P_2$ and $P_\infty$,
and call it $P_3$, that of $P_1$ with regard to $P_0$ and $P_\infty$, and
call it $P_{-1}$, and so, in general, construct $P_{n+1}$ and $P_{n-1}$
harmonic conjugates with regard to $P_n$ and $P_\infty$. The construction
is very rapidly performed as follows. Take $O$ and $V$
collinear with $P_\infty$, while our given points lie on the line $l_0$.
Let $l_1$ be the line from the intersection of $OP_1$ and $VP_0$ to $P_\infty$.
Then $OP_{n+1}$ and $VP_n$ will always intersect on $l_1$, the generic
name for such a point being $Q_{n+1}$.%
\footnote{See Fig. 4 on page following.}
%endfootnote

\begin{flalign*}
&\text{\indent \emph{Theorem}~25.} &  P_0P_{n+1}&\sep P_nP_\infty \text{\ if }  n>0. &
\phantom{\text{\indent \emph{Theorem}~25.}}
\end{flalign*}

The theorem certainly holds when $n=1$. Suppose that
$P_0P_n\sep P_{n-1}P_\infty$. We also know that $P_{n-1}P_{n+1}\sep P_nP_\infty$. Hence,
clearly $P_0P_{n+1}\sep P_nP_\infty$. We notice also that $P_0P_{n+2}\sep P_nP_\infty$,
and, in general $P_0P_{n+k}\sep P_nP_\infty$.  A similar proof may be found
for the case where negative subscripts are involved.

\smallskip\emph{Theorem}~26. If $P$ be any point which satisfies the condition
$P_0P\sep P_1P_\infty$, then such a positive integer $n$ may be found
that $P_0P\sep P_nP_\infty$, $P_0P_{n+1}\sep PP_\infty$.

%[Illustration: FIG. 4]
\imgcent{300}{f004}{Fig. 4}

Let us divide all points of the separation class\index{Separation02@--- classes} determined
by $P_0P_\infty$ which include $P_1$ and $P$ the positive separation class
let us say, into two sub-classes as follows. A point $A$ shall
be assigned to the first class if we may find such a positive
%[File: 257.png]
integer $n$ that $P_0P_{n+1} \sep AP_\infty$, otherwise it shall be assigned
to the second class, i.e.\ for every point of the second class
and every positive integral value of $n$, $P_0B \sep P_{n+1}P_\infty$. Then,
by 3, as long as $A$ and $B$ are distinct we shall have
$P_0B\sep AP_\infty$, giving a dichotomy of the sort demanded by
Axiom VII, and a point of division $D$. Let us further assume
that $OD$ meets $l_1$ in $\overline{D}$, and $V\overline{D}$ meets $l_0$ in $C$. We know that
$\overline{D}Q_1 \nsep Q_0Q_\infty$. Hence lines from $P_0$ to $V$ and $\overline{D}$ are not
separated by those to $O$ and $P_\infty$. Hence lines from $\overline{D}$ to $P_0$
and $V$, are not separated by those to $O$ and $P_\infty$, so that
$P_0C \nsep DP_\infty$ or $C$ is a point of the first sub-class. We may,
then, find $n$ so great that $P_0P_n \sep CP_\infty$, hence $Q_1Q_{n+1} \sep DP_\infty$
and $P_1P_{n+1} \sep DP_\infty$. But $P_0P_1 \nsep DP_\infty$; hence $P_0P_{n+1} \sep DP_\infty$. This,
however, is absurd, for a point separated\index{Separation01@Separation} from $P_0$ by $D$ and
$P_{n+1}$ would have to belong to both classes. Our theorem
results from this contradiction.

We might treat the case where $P_0P \sep P_{-1}P_\infty$ in exactly the
same way. Our net result is that if $P$ be any point of the
line $l_0$, it is either a point of the system we have constructed,
%[File: 258.png]
or else we may find two such successive integers (calling ${}_0$ an
integer) $n$, $n + 1$ that $P_n P_{n+1} \sep PP_\infty$.

Our next care shall be to find points of the line to which we
may properly assign fractional subscripts. Let $l_k$ be the line
from $P_\infty$, to the intersection of $OP_k$ with $VP_0$. Then I say
that $VP_m$ and $OP_{m+k}$ meet on $l_k$. This is certainly true when
$k = 1$ Let us assume it to be true in the case of $l_{k-1}$ so
that $VP_1$ and $OP_k$ meet on $l_{k-1}$. Then $l_k$ is constructed with
regard to $l_{k-1}$ as was $l_1$ with regard to $l_0$, for we take a point
of $l_{k-1}$, connect it with $O$ and find where that line meets
$VP_0$. In like manner $VP_2$ meets $OP_{k+1}$ on $l_{k-1}$ and $OP_{k+2}$
on $l_k$ and so on; $VP_m$ meets $OP_{m+k}$ on $l_k$, which was to be
proved.

As an application of this we observe that $l_n$ meets $VP_n$
on the line $OP_{2n}$, hence we easily see that $P_n$ and $P_\infty$ are
harmonically
separated\index{Harmonic02@--- separation}\index{Separation01@Separation}\index{Separation03@--- harmonic}\index{Conjugate03@--- harmonic}
by $P_0$ and $P_{2n}$. Secondly, find the
points into which the points $P_h$, $P_k$, $P_l$ are projected from $O$ on
the line $VP_m$. These points lie on the lines $l_{h-m}$, $l_{k-m}$, $l_{l-m}$.
Find the intersections of the latter with $VP_n$ and project back
from $O$ on $l_0$; we get the points $P_{n+h-m}$, $P_{n+k-m}$, $P_{n+l-m}$.
A particular result of this will be that $P_kP_{k+n}P_{k+2n}P_\infty$ form
a harmonic set\index{Separation02@--- classes}.

Let us now draw a line from $P_1$ to the intersection of $VP_0$
and $l_n$, and let this meet $P_\infty V$ in $V_\frac{1}{n}$ Then if $P_0$, $P_k$, $P_l$, $P_\infty$ be
projected from $O$ upon $P_0V$ and then projected back from $V_\frac{1}{n}$
upon $l_0$, we get points which we may call $P_0$, $P_\frac{k}{n}$, $P_\frac{l}{n}$, $P_\infty$ where
$P_\frac{n}{n} = P_1$. Connect $P_\infty$ with the intersection of $VP_0$ and $OP_\frac{1}{n}$ by
a line $l_\frac{1}{n}$. We may use this line to find $P_\frac{k}{n}$ as formerly we
used $l_1$ to find $P_k$. We shall thus find that $P_0$ and $P_\frac{2n}{n}$ are
harmonically separated by $P_\frac{n}{n}$ and $P_\infty$, or $P_\frac{2n}{n}$ is identical
with $P_2$, and similarly $P_\frac{rn}{n}$ is identical with $P_r$. Subdividing
still further we shall find that $P_\frac{r}{rn}$ is identical with $P_\frac{1}{n}$ or $P_\frac{rm}{rn}$
identical with $P_\frac{m}{n}$. We have thus found a single definite
point to correspond to each positive rational subscript.
Negative rational subscripts might be treated in the same
way, and eventually we shall find a single point whose subscript
%[File: 259.png]
is any chosen rational number. We shall also find
by reducing to a common denominator, that if
\begin{equation*}
q > p > 0,\quad P_0P_q\sep P_pP_\infty,
\end{equation*}
with a similar rule for negative numbers.

It remains to take up the irrational case. Let $P$ be any
point of the positive separation class determined by $P_0$ and $P_\infty$.
Then either it is a point with a rational subscript, according
to our scheme, or else, however great soever $n$ may be, we
may find $m$ so that $P_0P\sep P_\frac{m}{n}P_\infty$, $P_0P_\frac{m+1}{n}\sep PP_\infty$.
We thus have a dichotomy of the positive rational number system
of such a nature that a number of the lower class\index{Separation02@--- classes} will
correspond to a point separated\index{Separation01@Separation} from $P_\infty$ by $P_0$ and $P$ while
one of the upper class will correspond to a point separated
from $P_0$ by $P$ and $P_\infty$. There will be no largest number in
the lower class. We know, in fact, that wherever $R$ may
be in the positive separation class of $P_0P_\infty$ we may find $n'$
so great that $P_0P_{n'}\sep RP_\infty$. We may express this by saying
that $P_{n'}$ approaches $P_\infty$ as a limit as $n'$ increases. Hence,
as separation is invariant under projection, $l_{n'}$ approaches $P_\infty O$
as a limit and $P_\frac{1}{n'}$ approaches $P_0$ as a limit, or $P_{\frac{m}{n}+\frac{1}{n'}}$
approaches $P_\frac{m}{n}$ as a limit. We can thus find $n'$ so large that
$P_{\frac{m}{n}+\frac{1}{n'}}$ is also a number of the first class, and surely
$\dfrac{m}{n}+\dfrac{1}{n'}>\dfrac{m}{n}$. In the
same way we show that there can be no smallest number
in the upper class. Finally each number of the upper is
greater than each of the lower. Hence a perfect dichotomy
is effected in the system of positive rationals defining a precise
irrational number, and this may be assigned as a subscript
to $P$. A similar proceeding will assign a definite subscript to
each point of the other negative separation class of $P_0P_\infty$.

Conversely, suppose that we have given a positive irrational
number. This will be given by a dichotomy in the system of
positive rationals, and corresponding thereto we may establish
a classification among the points of the positive separation
class of $P_0P_\infty$ according to the requirement of Axiom VII\@.
We shall, in fact, assign a point $A$ of this separation class
to the lower sub-class if we may find such a number in the
lower number class that the point with the corresponding
%[File: 260.png]
subscript is separated\index{Separation01@Separation} from $P_0$ by $P_\infty$ and $A$; otherwise a point
shall be assigned to the upper sub-class. If thus $A$ and $B$
be any two points of the lower and upper sub-classes respectively,
we can find $\dfrac{m}{n}$ in the lower number class so that
$P_{0}P_{\frac{m}{n}}\sep AP_\infty$ whereas
$P_0B\sep P_{\frac{m}{n}}P_\infty$, and, hence, by 3, $P_0B\sep AP_\infty$.

This shows that all of the requirements of Axiom VII are
fulfilled, we may assign as subscript to the resulting point
of division the irrational in question. In the same way we
may assign a definite point to any negative irrational. The
one to one correspondence between points of a line and the real
number system including $\infty$ is thus complete.

\smallskip\emph{Definition}.~If $A$, $B$, $C$, $D$ be four collinear points, whereof
the first three are necessarily distinct, the subscript which
should be attached to $D$, when $A$, $B$, $C$ are made to play
respectively the r\^oles of $P_\infty$, $P_0$, $P_1$ in the preceding discussion,
shall be called a \emph{cross ratio}\index{Cross ratios} of the four given points, and
indicated by the symbol $(AB, CD)$. Four points which are
distinct would thus seem to have twenty-four different cross
ratios, as a matter of fact they have but six.

We know that the harmonic relation\index{Conjugate03@--- harmonic} is unaltered by any
finite number of projections and intersections. We may therefore
define the cross ratios of four concurrent coplanar lines,
or four coaxal planes\index{Plane}, by the corresponding cross ratios of
the points where they meet any other line.

\smallskip\emph{Theorem}~27. Cross ratios are unaltered by any finite
number of projections and intersections.

\smallskip\emph{Definition}.~The range of all collinear points, the pencil
of all concurrent coplanar lines, and the pencil of coaxal
planes shall be called \emph{fundamental one-dimensional
forms}\index{Forms, fundamental one-dimensional}\index{Fundamental02@--- one-dimensional forms}.

\smallskip\emph{Definition}.~Two fundamental one-dimensional forms shall
be said to be \emph{projective}\index{Projectivity} if they may be put into such a one to one
correspondence that corresponding cross ratios are equal.

\smallskip\emph{Theorem}~28. If in two projective one-dimensional forms
three elements of one lie in the corresponding elements of
the other, then every element of the first lies in the corresponding
element of the second.

For we may use these three elements in each case as $\infty$, $0$, $1$,
and then, remembering the definition of cross ratio, make use
of the fact that the construction of the harmonic conjugate
%[File: 261.png]
of a point with regard to two others is unique. This theorem
is known as the fundamental one of projective geometry.%
\footnote{For an interesting historical note concerning this theorem, see Vahlen\index{Vahlen},
loc.\ cit., p.~161.}
%endfootnote

\smallskip\emph{Theorem}~29. If two fundamental one-dimensional
forms\index{Forms, fundamental one-dimensional}\index{Fundamental02@--- one-dimensional forms} be
connected by a finite number of projections\index{Projection} and intersections
they are projective.

This comes immediately from 27.

\smallskip\emph{Theorem}~30. If two fundamental one-dimensional forms be
projective\index{Projectivity}, they may be connected by a finite number of
projections and intersections.

It is, in fact, easy to connect them with two other projective
forms whereof one contains three, and hence all corresponding
members of the other.

Let us now turn back for a moment to our cross ratio\index{Cross ratios} scale.
We have already seen that in the case of integers, and, hence,
by reducing to least common denominator, in the case of
all rational numbers $k$, $l$, $m$, $n$.
\[
(P_\infty P_m, P_l P_n)=(P_\infty P_{m+k}, P_{l+k}P_{n+k}).
\]

By letting $k$, $l$, $m$, $n$ become irrational, one at a time, and
applying a limiting process, we see that this equation is
always true.

In like manner we see that $P_0$, $P_q$, $P_{2q}$, $P_\infty$ form a harmonic
set, as do $P_k$, $P_{q+k}$, $P_{2q+k}$, $P_\infty$. In general, therefore,
\begin{flalign*}
&&(P_\infty P_0, P_1P_\nu)&= (P_\infty P_0, P_n P_{n\nu})
\\
&&&= (P_\infty P_\alpha, P_{n+\alpha}P_{n\nu+\alpha})
\\
&&&= \nu.
\\
&\text{\indent Putting}          &n+\alpha=\beta&,\quad  n\nu+\alpha=\gamma, &\phantom{\text{\indent Putting}}
\\
&&(P_\infty P_\alpha&,\, P_\beta P_\gamma)=\frac{\gamma-\alpha}{\beta-\alpha}.
\end{flalign*}

We next remark that the cross ratio of four points is that
of their harmonic conjugates with regard to two fixed points.
Reverting to our previous construction for $P_{\frac{1}{n}}$ we see that it is
collinear with $V_{\frac{1}{n}}$ and $Q_1$. $VQ_0P_{-1}$ are also on a line. If,
then, we compare the triads of points $VP_0Q_0$, $V_{\frac{1}{n}}P_1Q_1$, since
lines connecting corresponding points are concurrent in $P_\infty$,
the intersections of corresponding lines are collinear. But
%[File: 262.png]
the line from $O$ to the intersection of
$V_{\frac{1}{n}} P_1$ with $VP_0$ (or $VQ_1$)
is, by construction, the line $OP_n$. Hence $VP_{-1}$, which is
identical with $VQ_0$, meets $V_{\frac{1}{n}}P_{\frac{1}{n}}$ on $OP_n$. Furthermore $O$ and
$Q_1$ are harmonically separated by the intersections of their
line with $VP_{-1}$ and $V_{\frac{1}{n}}P_1$; i.e.\ by $P_1$ and the intersection with
$VQ_0$. Project these four upon $l_0$ from the intersection of $OP_n$
and $VP_{-1}$. We shall find $P_n$ and $P_{\frac{1}{n}}$ are harmonic
conjugates\index{Conjugate03@--- harmonic}\index{Harmonic01@Harmonic conjugate}
with regard to $P_1$ and $P_{-1}$. Let the reader show that this last
relation holds equally when $n$ is a rational fraction, and,
hence, when it takes any real value.

The preceding considerations will enable us to find the
cross ratio\index{Cross ratios} of four points which do not include $P_\infty$ in their
number. To begin with\label{PorB}
\begin{align*}
  (P_0 P_\beta, P_\gamma P_\delta)
&= (P_\infty P_{\frac{1}{\beta}},
    P_{\frac{1}{\gamma}} P_{\frac{1}{\delta}})
\\
&= \frac{\gamma}{\beta} \times \frac{\beta-\delta}{\beta-\gamma}.
\end{align*}

Let us project our four points from $V$ upon $l_\alpha$, then back
upon $l_0$ from $O$. This will add $\alpha$ to each subscript. Then
replace $\gamma + \alpha$ by $\gamma$, \&c.
\[
  (P_\alpha B_\beta, P_\gamma P_\delta)
= \frac{\alpha-\gamma}{\alpha-\delta} \times
  \frac{\beta-\delta}{\beta-\gamma}.        \tag{1}
\]

\smallskip\emph{Theorem}~31. Four elements of a fundamental one-dimensional
form\index{Forms, fundamental one-dimensional}\index{Fundamental02@--- one-dimensional forms}
determine six cross ratios which bear to one
another the relations of the six numbers
\[
  \lambda,\quad \frac{1}{\lambda},\quad 1-\lambda,\quad
  \frac{1}{1-\lambda},\quad \frac{\lambda-1}{\lambda},\quad
  \frac{\lambda}{\lambda-1}.
\]

The proof is perfectly straightforward, and is left to the
reader.

\mysect{Projective coordinates of points in a line}
If three points be taken as fundamental upon a straight
line, any other point thereon may be located by a pair of
homogeneous coordinates whose ratio is a definite cross ratio
of the four points. We shall assign to the fundamental points
the coordinates $(1,0)$, $(0,1)$, $(1,1)$. A cross ratio of four points
$(x)$, $(y)$, $(z)$, $(t)$ will then be
\[
  \frac{\begin{vmatrix}
          x_0 & z_0  \\
          x_1 & z_1
        \end{vmatrix}
        \centerdot
        \begin{vmatrix}
          y_0 & t_0  \\
          y_1 & t_1
        \end{vmatrix}}
       {\begin{vmatrix}
          y_0 & z_0  \\
          y_1 & z_1
        \end{vmatrix}
        \centerdot
        \begin{vmatrix}
          x_0 & t_0  \\
          x_1 & t_1
        \end{vmatrix}}.   \tag{2}
\]

%[File: 263.png]
\mysect{Linear transformations of the line}
Any projective transformation of the line into itself, i.e.\ any
point to point transformation which leaves cross ratios\index{Cross ratios} unaltered,
will thus take the form
\begin{equation*}
\begin{aligned}
Px_0{}'=a_{00}x_0+a_{01}x_1,\\
Px_2{}'=a_{10}x_0+a_{11}x_1,
\end{aligned}
\qquad
|a_{ij}|\neq 0.\tag{3}
\end{equation*}

To demonstrate this we have merely to point out that surely
this transformation is a projective one, and that we may so
dispose of our arbitrary constants as to carry any three distinct
points into any other three, the maximum amount of freedom
for any projective transformation of a fundamental one-dimensional
form. Let the reader show that the necessary
and sufficient condition that there should be two real self-corresponding
points which separate\index{Separation01@Separation} each pair of corresponding
points is
\[
|a_{ij}|<0.
\]

Two projective\index{Projectivity} sets on the same fundamental one-dimensional
form whose elements correspond interchangeably, are
said to form an \emph{involution}. By this is meant that each
element of the form has the same corresponding element
whether it be assigned to the first or to the second set.
It will be found that the necessary and sufficient condition
for an involution in the case of equation (3) will be
\begin{equation*}
a_{01}=a_{10}.
\tag{4}
\end{equation*}

When the determinant $|a_{ij}|>0$, there will be no self-corresponding
points, and the involution is said to be \emph{elliptic}.
Let the reader show that under these circumstances each pair
of the involution separates each other pair.

\mysect{Projective coordinates of points in a plane}
Our next task shall be to set up a suitable coordinate
system for the plane and for space. Let us take in the plane
four points $A$, $B$, $C$, $D$, no three being collinear. We shall
assign to these respectively the coordinates $(1, 0, 0)$, $(0, 1, 0)$,
$(0, 0, 1)$, $(1, 1, 1)$. Let $AD$ meet $BC$ in $A_1$, $BD$ meet $CA$
in $B_1$, and $CD$ meet $AB$ in $C_1$. The intersections of $AB$, $A_1B_1$,
of $BC$, $B_1C_1$, and of $CA$, $C_1A_1$, are, by 15, on a line $d$. Now
let $P$ be any other point in the plane
\begin{align*}
(AB\,AC, AD\,AP) &= (PC_1PC, PD\,PA) = (PC\,PC_1, PA\,PD)\\
(BC\,BA, BD\,BP) &= (PC\,PC_1, PD\,PB)\\
(CA\,CB, CD\,CP) &= (PC_1PC, PA\,PB) = \frac{1}{(PC\,PC_1, PA\,PB)}.
\end{align*}

From this it is clear that the product of the three is equal
to unity, and we may represent them by three numbers of the
%[File: 264.png]
type $\dfrac{x_1}{x_0}$, $\dfrac{x_2}{x_1}$, $\dfrac{x_0}{x_2}$.
We may therefore take $x_0: x_1: x_2$ as three
homogeneous coordinates\index{Coordinates02@Co-ordinates of a point} for the point $P$. One coordinate
will vanish for a point lying on one of the lines $AB$, $BC$, $CA$.
Let the reader convince himself that the usual cartesian
system is but a special case of this homogeneous coordinate
system where two of the four given points are ideal, and
\[
\frac{x_1}{x_0}=x,\quad \frac{x_2}{x_0}=y.
\]

\mysect{Equation of a line, its coordinates}
The equations of the lines connecting two of the points
$A$, $B$, $C$ are of the form
\[
x_i=0.
\]

Those which connect each of these with the point $D$ are
similarly
\[
x_i-x_j=0.
\]

If ($y$) and ($z$) be two points, not collinear with $A$, $B$, or $C$,
while $P$ is a variable point with coordinates $\lambda(y) + \mu(z)$, the
lines connecting it with $A$ and $B$ will meet $BC$ and ($CA$)
respectively in the points
\[
(0,\; \lambda y_1+\mu z_1,\; \lambda y_2+\mu z_2)\quad
(\lambda y_0+\mu z_0,\; 0,\; \lambda y_2+\mu z_2).
\]

It is easy to see that the expressions for corresponding cross
ratios in these two ranges are identical, hence the ranges are
projective. The pencils which they determine at $A$ and $B$
are therefore projective, and have the line $AB$ self-corresponding,
for this will correspond to the parameter value
\[
\lambda:\mu=z_2:-y_2.
\]

But it will follow immediately from 28, that if two pencils
be coplanar and projective, with a self-corresponding line,
the locus of the intersection of their corresponding members
is also a line. Hence the locus of the point $P$ with the
coordinates $\lambda(y)+\mu(z)$ is the line connecting ($y$) and ($z$).
Conversely, it is evident that every point of the line from
($y$) to ($z$) will have coordinates linearly dependent on those
of ($y$) and ($z$). If, then, we put
\[
x_i=\lambda y_i + \mu z_i,
\]
and eliminate $\lambda:\mu$, we have as equation of the line
\[
|xyz|=(ux)=0.
\]

Conversely, it is evident that such an equation will always
represent a line, except, of course, in the trivial case where
the $u$'s are all zero. Let the reader show that the coefficients
%[File: 265.png]
$u_i$ have a geometrical interpretation dual to that of the
coordinates $x_i$; for this purpose the line which we have above
called $d$ will be found useful.

\mysect{Projective coordinates of points in space}
Our system of homogeneous coordinates may be extended
with great ease to space. Suppose that we have given five
points $A$, $B$, $C$, $D$, $O$ no four being coplanar. Let $P$ be any
other point in space. We may write

\begin{equation*}
(ABC\, ABD, ABO\, ABP) = \frac{x_3}{x_2}, \quad
(ACD\, ACB, ACO\, ACP) =\frac{x_1}{x_3},
\end{equation*}
\begin{equation*}
(ADB\, ADC, ADO\, ADP) =\frac{x_2}{x_1}.
\end{equation*}

We shall then be able to write also

\begin{equation*}
(CDA\, CDB, CDO\, CDP) = \frac{x_1}{x_0},\quad
(DBA\, DBC, DBO\, DBP) = \frac{x_2}{x_0},
\end{equation*}
\begin{equation*}
(BCD\, BCA, BCO\, BCP) = \frac{x_0}{x_3}.
\end{equation*}

\mysect{Equation of a plane}
In other words, we may give to a point four homogeneous
coordinates $x_0:x_1:x_2:x_3$. Two points collinear with $A$, $B$,
$C$, or $D$ will differ (or may be made to differ) in one coordinate
only. An equation of the first degree in three coordinates
will represent a plane through one of these four points.
Every line will be the intersection of two such planes, and
will be represented by the combination of two linear equations
one of which lacks $x_i$ while the other lacks $x_j$. The coordinates
of all points of a line\index{Coordinates01@Co-ordinates of a line} may therefore be expressed as
a linear combination of the coordinates of any two points\index{Coordinates02@Co-ordinates of a point}
thereof. A plane\index{Plane}\index{Coordinates03@--- of a plane}
may be represented as the assemblage of
all points whose coordinates are linearly dependent on those
of three non-collinear points. Eliminating the variable parameters
from the four equations for the coordinates of a point
in a plane, we see that a plane may also be given by an
equation of the type

\begin{equation*}
(ux) = 0.\tag{5}
\end{equation*}

\index{Cross ratios}Conversely, the assemblage of all points whose coordinates
satisfy an equation such as (5) will be of such a nature that
it will contain all points of a line whereof it contains two
distinct points, yet will meet a chosen line, not in it, but
once. Let the reader show that such an assemblage must
be a plane. The homogeneous parameters $(u)$ which, naturally,
may not all vanish together, may be called the coordinates
%[File: 266.png]
of the plane. They will have a significance dual to that
of the coordinates of a point.%
\footnote{The treatment of cross ratios\index{Cross ratios} in the present chapter is based on that of
Pasch\index{Pasch}, loc.\ cit. The development of the coordinate system is also taken from
the same source, though it has been possible to introduce notable simplification,
especially in three dimensions. This method of procedure seemed to
the author more direct and natural than the more modern method of
`Streckenrechnung' of Hilbert\index{Hilbert} or Vahlen\index{Vahlen}, loc.\ cit.}
%endfootnote

If we have four collinear points
\[
(y),\quad (z),\quad \lambda(y)+\mu(z),\quad \lambda'(y)+\mu'(z),
\]
\begin{flalign*}
&\text{one cross ratio will be}
&&\frac{\lambda\mu'}{\lambda'\mu}.
&\phantom{\text{one cross ratio will be}}
\end{flalign*}

The proof will consist in finding the points where four
coaxal planes\index{Plane} through these four points meet the line
\[
x_2=x_3=0
\]
and then applying (2).

Suppose that we have a transformation of the type
\begin{equation*}
\rho x_i{}'=\sum_{j}^{0.. 3}a_{ij}x_j.
\tag{6}
\end{equation*}

\mysect{Collineations}
This shall be called a \emph{collineation}\index{Collineations}.  We shall restrict
ourselves to those collineations for which
\[
|a_{ij}|\neq 0.
\]

The transformation is, clearly, one to one, with no exceptional
points. It will carry a plane into a plane, a line
into a line, a complete quadrilateral\index{Quadrilateral02@--- complete} into a complete quadrilateral,
and a harmonic set into a harmonic set. It will
therefore leave cross ratios invariant. Moreover, every point
to point and plane to plane transformation will be a
collineation. For every such transformation will enjoy all
of the properties which we have mentioned with regard to
a collineation, and will, therefore, be completely determined
when once we know the fate of five points, no four of which
are coplanar. But we easily see that we may dispose of the
arbitrary constants in (6), to carry any such five points into
any other five.

\mysect{Imaginary elements}
It is worth while to pause for a moment at this point in
order to see what geometrical meaning may be attached to
coordinate sets which have imaginary values. This question
%[File: 267.png]
has already been discussed in Chapter VII\@. Every set of
complex coordinates
\[
(y)+i(z),
\]
may be taken to define the elliptic involution\index{Involution}
\begin{equation*}
(x)=\lambda(y)+\mu(z),\quad x'=\lambda'(y)+\mu'(z),\quad \lambda\lambda'+\mu\mu'=0.
\tag{7}
\end{equation*}

To verify this statement we have merely to notice that an
involution will, by definition, be carried into an involution
by any number of projections and intersections, and that
equations such as (7) will go into other such equations. But
in the case of the line
\[
x_2=x_3=0,
\]
these equations will give an involution, for the relation
between ($x$) and ($x'$) may readily be reduced to the type of (3)
and (4). Did we seek the analytic expression for the coordinates
of a self-corresponding point\index{Point} in (7) we should get
the values
\[
(y)+i(z).
\]

Conversely, it is easy to show that any elliptic involution
may be reduced to the type of (7). There is, therefore, a one
to one correspondence between the assemblage of all elliptic
point involutions, and all sets of pairs of conjugate imaginary
coordinate values\index{Imaginary elements}.

The correspondence between coordinate sets and elliptic
involutions may be made more precise in the following fashion.
Two triads of collinear points $ABC$, $A'B'C'$ shall be said to
have the \emph{same sense}\index{Sense02@--- of description of involution}
when the projective transformation which
carries the one set, taken in order, into the other, has a positive
determinant; when the determinant is negative they shall be
said to have \emph{opposite senses}\index{Opposite03@Opposite senses}. In this latter case alone, as we
have already seen, will there be two real self-corresponding
points which separate each distinct pair of corresponding
points. Two triads which have like or opposite senses to
a third, have like senses to one another, for the determinant
of the product of two projective transformations of the line
into itself is the product of the determinants. We shall also
find that the triads $ABC$, $BCA$, $CAB$ have like senses, while
each has the sense opposite to that of either of the triads
$ACB$, $CBA$, $BAC$. We may thus say that three points given
in order will determine a sense of description for the whole
range of points on the line, in that the cyclic order of any
other three points which are to have the same sense as the
first three is completely determined. It is immediately
evident that any triad of points and their mates in an
elliptic involution have the same sense. We may therefore
%[File: 268.png]
attach to such an elliptic involution\index{Involution} either the one or the
other sense of description for the whole range of points.

\smallskip\emph{Definition}.~An elliptic involution of points to which is
attached a particular sense of description of the line on which
they are situated shall be defined as an \emph{imaginary point}\index{Imaginary elements}.
The same involution considered in connexion with the other
sense shall be called the \emph{conjugate imaginary point}.

Starting with this, we may define an imaginary plane as
an elliptic involution in an axial pencil, in connexion with
a sense of description for the pencil; when the other sense
is taken in connexion with this involution we shall say that
we have the conjugate imaginary plane. An imaginary point
shall be said to be in an imaginary plane if the pairs of the
involution which determine the point lie in pairs of planes
of the involution determining the plane, and if the sense of
description of the line associated with the point engenders
among the planes the same sense as is associated with the
imaginary plane. Analytically let us assume that besides
the involution of points given by (7) we have the following
involution of planes.
\begin{equation*}
(u)=l(v)+m(w),\quad (u')=l'(v)+m'(w),\quad ll'+mm'=0,
\end{equation*}
\begin{equation*}
(vy)=(wz)=0.\tag{8}
\end{equation*}

The plane ($u$) will contain the point $l(vz) (y)-m (wy) (z)$
while its mate in the involution contains the point
\[
m(vz)(y)+l(wy)(z).
\]
These points will be mates in the point involution, if
\[
[(vz) + (wy)]\,[(vz)-(wy)] = 0,
\]
and these equations tell us that the imaginary plane $(v) + i (w)$
will contain either the point $(y) +i (z)$, or the point $(y)-i (z)$.
An imaginary line may be defined as the assemblage of all
points common to two imaginary planes. Imaginary points,
lines, and planes obey the same laws of connexion as do
real ones. A geometric proof may be found based upon the
definitions given, but it is immediately evident analytically.%
\footnote{See von Staudt\index{Staudt@Von Staudt}, loc.\ cit.,
and L\"uroth\index{Luroth@L\"uroth}, loc.\ cit. It is to be noted that in
these works the idea of sense of description is taken intuitively, and not given
by precise definitions.}
%endfootnote

\smallskip\emph{Theorem}~32. If a fundamental one-dimensional
form\index{Forms, fundamental one-dimensional}\index{Fundamental02@--- one-dimensional forms} be
projectively\index{Projectivity} transformed into itself there will be two distinct
or coincident self-corresponding elements.

We have merely to put ($\rho x$) for ($x'$) in (3), and solve the
%[File: 269.png]
quadratic equation in $\rho$ obtained by equating to zero the determinant
of the two linear homogeneous equations in $x_0$, $x_1$.

The assemblage of all points whose coordinates satisfy an
equation of the type
\[
\smallsum a_{ij}x_i x_j=0,\quad |a_{ij}|\neq 0,
\]
shall be called a \emph{quadric}. We should find no difficulty in
proving all of the well-known theorems of a descriptive sort
connected with quadrics in terms of our present coordinates.

We have now, at length, reached the point where we may
profitably introduce metrical concepts. Let us recall that the
group of congruent transformations which we considered in
Chapter II, and, more fully, in Chapter VIII, is a group of
collineations which leaves invariant either a quadric or a
conic, and depends upon six parameters. We also saw in
Chapter II, that the congruent group may be characterized
as follows (cf.\ p.~38):---

(\emph{a})~Any real point of a certain domain may be carried into
any other such point.

(\emph{b})~Any chosen real point may be left invariant, and any
chosen real line through it carried into any other such line.

(\emph{c})~Any real point and line through it may be left invariant,
and any real plane\index{Plane} through this line may be carried into any
other such plane.

(\emph{d})~If a real point, a line through it, and a plane through
the line be invariant, no further infinitesimal congruent
transformations are possible.

It shall be our present task to show that these assumptions,
or rather the last three, joined to the ones already made in
the present chapter, will serve to define hyperbolic elliptic
and euclidean geometry.

\mysect{Axioms of the congruent collineation group}
It is assumed that there exists an assemblage of transformations,
called congruent transformations, obeying the following
laws:---

\medskip\textsc{Axiom}~XII\@. \textbf{The assemblage of all congruent transformations
is a group of collineations, including the inverse of
each member.}%
\footnote{It is highly remarkable that this axiom is superfluous. Cf.\ Lie-Engel\index{Lie},
\emph{Theorie der Transformationsgruppen}, Leipzig, 1888--93, vol.~iii, Ch.~XXII, \S{}~98.
The assumption that our congruent
transformations\index{Congruent transformations}\index{Transformations, congruent}
are collineations\index{Collineations}, does,
however, save an incredible amount of labour, and, for that reason, is included
here.}
%endfootnote

%[File: 270.png]
\medskip\textsc{Axiom}~XIII\@. \textbf{The group of congruent
transformations\index{Congruent transformations}\index{Transformations, congruent} may
be expressed by means of analytic relations among the
parameters of the general collineation group.}

\smallskip\emph{Definition}.~The assemblage of all real points whose coordinates
satisfy three inequalities of the type
\begin{equation*}
\xi_i<\begin{array}{cc}x_i \\ x_0\end{array}<X_i,\quad i=1,2,3,
\end{equation*}
shall be called a \emph{restricted region}\index{Region02@--- restricted}.

\medskip\textsc{Axiom}~XIV\@. \textbf{A congruent transformation may be found
leaving invariant any point of a restricted region, and
transforming any real line through that point into any other
such line.}

\medskip\textsc{Axiom}~XV\@. \textbf{A congruent transformation may be found
leaving invariant any point of a restricted region, and any
real line through that point; yet carrying any real plane
through that line into any other such plane.}

\medskip\textsc{Axiom}~XVI\@. \textbf{There exists no continuous assemblage of
congruent transformations which leave invariant a point of
a restricted region, a real line through that point, and a real
plane\index{Plane} through that line.}

\smallskip\emph{Theorem}~33. The congruent group is transitive for a sufficiently
small restricted region.

This comes at once by \emph{reductio ad absurdum}. For the
tangents to all possible paths which a chosen point might
follow would, if 33 were untrue, generate a surface or set
of surfaces, or a line or set of lines, and this assemblage of
surfaces or lines would be carried into itself by every congruent
transformation which left this point invariant. The
tangent planes to the surfaces, or the lines in question, could
not, then, be freely interchanged with other planes or lines
through the point.

\smallskip\emph{Theorem}~34. The congruent group depends on six essential
parameters.

The number of parameters is certainly finite since the
congruent group arises from analytic relations among the
fifteen essential parameters of the general collineation group.
The transference from a point to a point imposes three
restrictions, necessarily distinct, as three independent parameters
are needed to determine a point. A fixed point being
chosen, two more independent restrictions are imposed by
%[File: 271.png]
determining the fate of any chosen real line through it.
When a point and line through it are chosen, one more
restriction is imposed by determining what shall become of
any assigned plane through the line. When, however, a real
plane, a real line therein, and a real point in the line are
fixed, there can be no independent parameter remaining, as no
further infinitesimal transformations are possible.

Let us now look more closely at the one-parameter family
of projective transformations of the axial pencil through
a fixed line of the chosen restricted region.%
\footnote{Cf.\ Lie-Scheffers\index{Lie}, \emph{Vorlesungen \"uber continuierliche Gruppen}. Leipzig, 1893,
p. 125.}
%endfootnote
Let us determine
any plane through this line by two homogeneous
parameters $\lambda_1:\lambda_2$, and take an infinitesimal transformation
of the group
\[
\Delta\Bigl(\begin{array}{cc}\lambda_1 \\ \lambda_2\end{array}\Bigr)
=f\Bigl(\begin{array}{cc}\lambda_1 \\ \lambda_2\end{array}\Bigr)dt.
\]

The product of two such infinitesimal transformations will
belong to our group, hence also, as none but analytic functions
are involved, the limit of the product of an infinite number
of such transformations as $dt$ approaches zero; that is to say,
the transformation obtained by integrating this equation
belongs to the group. Now this integral will involve one
arbitrary constant, which may be used to make the transformation
transitive, and for all transformations obtained by
this integration, that pair of planes will be invariant which
was invariant for the infinitesimal transformation. Our
one-parameter group has thus a transitive one-parameter sub-group
with a single pair of planes invariant. These planes
are surely conjugate imaginary, for otherwise there would
be infinitesimal congruent
transformations\index{Congruent transformations}\index{Transformations, congruent} which left a point,
line, and real plane invariant; contrary to our last axiom.
The question of whether our whole one-parameter group is
generated by this integration or not, need not detain us here.
What is essential is that this pair of planes will be invariant
for the whole group. For suppose that $S_i$ indicate a generic
transformation of the sub-group which leaves invariant the
two planes $\alpha$, $\alpha'$, and the transformation $T$ carries the two
planes $\alpha$, $\alpha'$ into two planes $\beta$, $\beta'$. Then all transformations
of the type
\[
TS_iT^{-1}
\]
will belong to our group, and leave the planes $\beta$, $\beta'$ invariant,
and combining these with the transformations $S_i$ we have
a two-parameter sub-group of our one-parameter group; an
absurd result.

%[File: 272.png]
Let us next consider the three-parameter congruent group
composed of all transformations which have a fixed point.
If a real line $l$ be carried into a real line $l'$, then the two
planes which were invariant with $l$ will go into those which
are invariant with $l'$. To prove this we have but to repeat
the reasoning which lately showed that the two planes which
were invariant for a sub-group, are invariant for the total
one-parameter group. The envelope of all these invariant
planes which pass through a point will thus depend upon
one parameter, for if it depended on two it would include
real planes, and this is not the case. It is well known that
this system of planes must envelope lines or a quadric cone.%
\footnote{Cf.\ Lie-Scheffers\index{Lie}, loc.\ cit., p.~289.}
%endfootnote
The first case is surely excluded for such lines would have
to appear in conjugate imaginary pairs, giving rise to invariant
real planes through this point, and there are no such
in the three-parameter group. The envelope is therefore
a cone with no real tangent planes. Each pair of conjugate
imaginary tangent planes must touch it along two conjugate
imaginary lines; the plane connecting these is real, and
invariant for the one-parameter congruent group associated
with the line of intersection of the two imaginary planes.
Let us fix our attention upon one such one-parameter group
and choose our coordinate system in such a way that the
non-homogeneous coordinates $u, v, 1$ of our three fixed planes
are proportional respectively to
\begin{equation*}
(0,\, 0,\, 1),\quad (1,\, i,\, 0),\quad (1,\,-i,\, 0).
\end{equation*}

The general linear transformation keeping these three
invariant is
\begin{equation*}
u'=r\cos{\theta u}-r\sin{\theta v},\quad  v'=r\sin{\theta u}+r\cos{\theta v}.
\end{equation*}

Here $r$ must be a constant, as otherwise we should have
congruent transformations\index{Congruent transformations}\index{Transformations, congruent} of the type
\begin{equation*}
u'=ru,\quad v'=rv,
\end{equation*}
which kept a point, a line, and all planes through that line
invariant, yet depended on an arbitrary parameter. In order
to see what sort of cones are carried into themselves by this
group, the cone we are seeking for being necessarily of the
number, let us take an infinitesimal transformation
\begin{flalign*}
&&\Delta u=&-vd\theta,\quad \Delta v=ud\theta.
\\
&\text{\indent Integrating}
&&u^2+v^2=C.
&\phantom{\text{\indent Integrating}}
\end{flalign*}
The cone we seek is therefore a quadric cone.

%[File: 273.png]
We see by a repetition of the sort of reasoning given above
that if we take a congruent transformation that carries
a point $P$ into a point $P'$, it will carry the invariant quadric
cone whose vertex is $P$ into that whose vertex is $P'$. The
envelope of these quadric cones is, thus, invariant under the
whole congruent group. The envelope of these cones must
be a quadric or conic\index{Conic}. This theorem is simpler when put
into the dual form, i.e.\ a surface which meets every plane
in a conic is a quadric or quadric cone. For it has just the
same points in every plane as the quadric or cone through
two of its conics and one other of its points. In our present
case our quadric must have a real equation, since it touches
the conjugate to each imaginary plane tangent thereto. There
are, hence, three possibilities:

(\emph{a}) The quadric is real, but the restricted region in question
is within it.

(\emph{b}) The quadric is imaginary.

(\emph{c}) The quadric is an imaginary conic in a real plane.

\smallskip\emph{Theorem}~35. The congruent group is a six-parameter collineation
group which leaves invariant a quadric or a conic.

\mysect{Reappearance of the Absolute and previous metrical formulae}
It remains for us to find the expression for distance. We
make the following assumptions.

\medskip\textsc{Axiom}~XVII\@. \textbf{The distance\index{Distance} of two points of a restricted
region is a real value of an analytic function of their
coordinates.}

\medskip\textsc{Axiom}~XVIII\@. \textbf{If $ABC$ be three collinear real points, and
if $B$ be separated by $A$ and $C$ from a point of their line not
belonging to this restricted region; then the distance from
$A$ to $C$ is the sum of the distance from $A$ to $B$ and the
distance from $B$ to $C$.}

Let the reader show that this definition is legitimate as all
points separated from $A$ by $B$ and $C$, or from $C$ by $A$ and $B$
will belong to the restricted region.

Let us first take cases (\emph{a}) and (\emph{b}) together. The distance
must be a continuous function of each cross ratio determined by
the two points and the intersections of their line with the
quadric. If we call a distance $d$, and the corresponding cross
ratio of this type $c$, we must have
\begin{equation*}
c =f(d).
\end{equation*}

Moreover, from equation (1) and Axiom XIII,
\begin{equation*}
f(d)\times f(d')=f(d + d').
\end{equation*}

%[File: 274.png]
Now this functional equation is well known, and the only
continuous solution is%
\footnote{Cf.\ e.g. Tannery\index{Tannery}, \emph{Th\'eorie des fonctions d'une variable},
second edition, Paris, 1904, p.~275.}
%endfootnote
\begin{eqnarray*}
c&=&e^{2i\frac{d}{k}}.\\
\frac{d}{k}&=&\frac{1}{2i}\log_ec.
\end{eqnarray*}

If, in particular, the two points be $P_1P_2$ while their line
meets the quadric in $Q_1Q_2$, we shall have for our distance,
equation (5) of Chapter VII
\begin{equation*}
\frac{d}{k}=\frac{1}{2i}\log_e(P_1P_2,Q_1Q_2).
\end{equation*}

From this we may easily work back to the familiar expressions
for the cosine of the $k$th part of the distance\index{Distance}.

The case of an invariant conic is handled somewhat
differently. Let the equations of the invariant conic be
\begin{equation*}
x_0=0,\quad x_1{}^2+x_2{}^2+x_3{}^2=0.
\end{equation*}

These are unaltered by a seven-parameter group
\begin{align*}
x_0{}'&=a_{00}x_0,\\
x_1{}'&=a_{10}x_0+a_{11}x_1+a_{12}x_2+a_{13}x_3,\\
x_2{}'&=a_{20}x_0+a_{21}x_1+a_{22}x_2+a_{23}x_3,\\
x_3{}'&=a_{30}x_0+a_{31}x_1+a_{32}x_2+a_{33}x_3,
\end{align*}
where $\parallel a_{11}\,a_{22}\,a_{33}\parallel$ is the matrix of a ternary orthogonal
substitution. For our congruent group we must have the
six-parameter sub-group where the determinant of this orthogonal
substitution has the value $a_{00}{}^3$, for then only will there
be no further infinitesimal transformations possible when
a point, a line through it, and a plane through the line are
fixed. We shall find that, under the present circumstances
the expression
\begin{equation*}
D=\left|\sqrt{\Bigl(\frac{x_1}{x_0}-\frac{y_1}{y_0}\Bigr)^2+
\Bigl(\frac{x_2}{x_0}-\frac{y_2}{y_0}\Bigr)^2+\Bigl(\frac{x_3}{x_0}-
\frac{y_3}{y_0}\Bigr)^2}\right|
\end{equation*}
is an absolute invariant. If the distance of two points $(x)$, $(y)$
be $d$, we shall have
\begin{equation*}
d=f(D).
\end{equation*}

This function is continuous and real, and satisfies the
functional equation

\begin{equation*}
f(D)+f(D')=f(D+D').
\end{equation*}

%[File: 275.png]
The solution of this equation is easily thrown back upon
the preceding one. Let us put
\begin{align*}
f(x)&=\log\phi(x),\\
\phi(x)\phi(y)&=\phi(x+y),\\
\phi(x)&=e^{rx}.
\end{align*}

We thus get finally
\begin{equation*}
d=r\left|\sqrt{\Bigl(\frac{x_1}{x_0}-\frac{y_1}{y_0}\Bigr)^2+
\Bigl(\frac{x_2}{x_0}-\frac{y_2}{y_0}\Bigr)^2+
\Bigl(\frac{x_3}{x_0}-\frac{y_3}{y_0}\Bigr)^2}\right|.
\end{equation*}

\smallskip\emph{Theorem}~36. Axioms I--XVIII are compatible with the
hyperbolic\index{Hyperbolic01@Hyperbolic hypothesis}, elliptic\index{Elliptic02@--- hypothesis},
or euclidean hypotheses\index{Euclidean01@Euclidean hypothesis}, and with these
only.

%[File: 276.png]
%CHAPTER XIX
\mychap{XIX}{THE DIFFERENTIAL BASIS FOR EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY}
\mysect{Fundamental assumptions}
\index{Geodesic02@--- lines|(}

We saw in Chapter XV, Theorem 17, that the Gaussian
curvature of a surface is equal to the sum of the total relative
curvature, and the measure of curvature of space. A noneuclidean
plane is thus a surface of Gaussian curvature equal
to $\dfrac{1}{k^2}$\index{K@$\dfrac{1}{k^2}$ measure of curvature of space}\index{Measure02@--- of curvature of space}.
This fact was also brought out in Chapter V, Theorem 3,
and we there promised to return in the present chapter to
a more extensive examination of this aspect of our noneuclidean
geometry.

\mysect{Coordinate system and distance elements}
In Chapter II, Theorem 30, we saw that the sum of the
distances from a point to any other two, not collinear with
it, when such a sum exists, is greater than the distance of
these latter. We thus come naturally to look upon a straight
line as a geodesic, or curve of minimum length between two
points. A plane may be generated by a pencil of geodesics
through a point; the geometrical simplicity of the plane may
be said to arise from the fact that it is capable of $\infty^2$ such
generations. The task which we now undertake is as
follows:---to determine the nature of a three-dimensional
point-manifold which possesses the property that every surface
generated by a pencil of geodesics has constant Gaussian
curvature\index{Curvature02@--- Gaussian}. We must begin, as in previous chapters, with
a sufficient set of axioms.%
\footnote{The first writer to
approach the subject from this point of view was
Riemann\index{Riemann}, loc.\ cit. The best presentation of the problem in its general form,
and in a space of $n$-dimensions, will be found in Schur\index{Schur},
`Ueber den Zusammenhang der R\"aume constanten Riemannschen
Kr\"ummungsmasses mit den
projectiven R\"aumen,' \emph{Mathematische Annalen}, vol.~27, 1886.}
%endfootnote

\smallskip\emph{Definition}.~Any set of objects which may be put into one
to one correspondence with sets of real values of three independent
coordinates\index{Coordinates02@Co-ordinates of a point}
$z_1$, $z_2$, $z_3$ shall be called \emph{points}\index{Point}.

\smallskip\emph{Definition}.~An assemblage of points shall be said to form
a \emph{restricted region}, when their coordinates are limited merely
by inequalities of the type
\begin{equation*}
\zeta_i<z_i<Z_i,\quad i=1,2,3.
\end{equation*}

%[File: 277.png]
\medskip\textsc{Axiom}~I. \textbf{There exists a restricted region\index{Region02@--- restricted}.}

\medskip\textsc{Axiom}~II\@. \textbf{There exist nine functions} $a_{ij},~i,j=1,2,3$
\textbf{of} $z_1,z_2,z_3$ \textbf{real and analytic throughout the restricted region,
and possessing the following properties}
\begin{equation*}
a_{ij}=a_{ji},\quad \lvert a_{ij} \rvert \not\equiv 0.
\end{equation*}
\begin{equation*}
\sum_{ij}^{1,2,3}a_{ij}dz_idz_j
\end{equation*}
\textbf{is a positive definite form for all real values of} $dz_1$, $dz_2$, $dz_3$
\textbf{and all values of} $z_1$, $z_2$, $z_3$ \textbf{corresponding to points of the given
restricted region.}

\smallskip\emph{Limitation}. We shall restrict ourselves to such a portion
of the original restricted region that for no point thereof shall
the discriminant of our quadratic form be zero. This amounts
to confining ourselves to the original region, or to a smaller
restricted region within the original one.

\index{Distance04@Distance element|(}
\smallskip\emph{Definition}.~The expression
\begin{equation*}
ds=+\sqrt{\sum_{i,j}^{1,2,3}a_{ij}dz_idz_j}
\end{equation*}
shall be called the \emph{distance element}.

\smallskip\emph{Definition}.~The assemblage of all points whose coordinates
are analytic functions of a single parameter shall be called an
\emph{analytic curve}, or, more simply, a curve. As we have defined
only those points whose coordinates are real, it is evident that
the functions involved in the definition of a curve must be
real also. The definite integral of the distance element
between two chosen points along a curve shall be called the
\emph{length} of the corresponding portion or \emph{arc}\index{Length of arc} of the curve. If
the curve pass many times through the chosen points, the
expression \emph{length} must be applied to that portion along which
the integration was performed.

\smallskip\emph{Definition}.~An arc of a curve between two fixed points
which possesses the property that the first variation of its
length is zero, shall be called \emph{geodesic arc}. The curve whereon
this arc lies shall be called a \emph{geodesic} connecting the two
points.

%[File: 278.png]
\mysect{Geodesic curves, their differential equations}
Let us begin by setting up the differential equations for
a geodesic. Let us write
\begin{equation*}
ds=\sqrt{\sum_{ij}^{1,2,3} a_{ij} \frac{dz_i}{dt} \frac{dz_j}{dt}} dt.
\end{equation*}

It is clear that $s$ is an analytic function of $t$ with no
singularities in our region, hence $t$ is an analytic function of $s$.
We may, then, by taking our restricted region\index{Region02@--- restricted} sufficiently
small, express $a_{ij}$ as functions of $s$, and write
\begin{equation*}
\sum_{i,j}^{1,2,3} a_{ij} \frac{dz_i}{ds} \frac{dz_j}{ds} = 1.\tag{1}
\end{equation*}

Replacing $\dfrac{dz_i}{ds}$ temporarily by $z_i{}'$, we have
\begin{equation*}
s = \int_0^s \sqrt{\sum_{i,j}^{1,2,3} a_{ij} z_i{}' z_j{}'}ds.
\end{equation*}

We have now a simple problem in the calculus of
variations.
\begin{equation*}
2\delta s=\int_0^s \sum_{i,j}^{1,2,3} \sum_k^{1,2,3}
\Bigl(\frac{\partial a_{ij}}{\partial z_k} z_i{}' z_j{}' \delta z_k + 2a_{ij}z_i{}' \delta z_j{}'\Bigr) ds.
\end{equation*}
\begin{equation*}
\frac{d}{ds} \sum_{i,j}^{1,2,3} a_{ij} z_i{}' \delta z_j =
\sum_{ij}^{1,2,3} \frac{d(a_{ij}z_i{}')}{ds} \delta z_j +
\sum_{ij}^{1,2,3} a_{ij} z_i{}' \delta z_j{}',
\end{equation*}
hence, since $\delta z_j$ vanishes at the extremities of the interval
\begin{equation*}
2\delta s=\int_0^s \sum_{j}^{1,2,3}\left[\sum_{ik}^{1,2,3}
\frac{\partial a_{ik}}{\partial z_j} z_i{}' z_k{}' -2\sum_i \frac{d}{ds}(a_{ij} z_i{}')\right] \delta z_j ds,
\end{equation*}
the increments $\delta z_j$ are arbitrary, hence the coefficients of each
must vanish, or
\begin{equation*}
\frac{d}{ds} \sum_i^{1,2,3} a_{ij} \frac{dz_i}{ds} =
\frac{1}{2} \sum_{ik}^{1,2,3} \frac{\partial a_{ik}}{\partial z_j}
\frac{\partial z_i}{\partial s} \frac{\partial z_k}{\partial s}.\tag{2}
\end{equation*}

These three equations are of the second order. There will
exist a single set of solutions corresponding to a single set
of initial values for $(z)$ and $(z')$.%
\footnote{Cf.\ e.g. Jordan\index{Jordan},
\emph{Cours d'Analyse}, Paris, 1893-6, vol.~iii, p.~88.}
%endfootnote
Let these be $(z^0)$ and $(\zeta)$
%[File: 279.png]
respectively. Any point of such a geodesic will be determined
by $\zeta_1\zeta_2\zeta_3$ and $r$ the length of the arc connecting it with ($z^0$).
We have thus
\begin{equation*}
z_i=z_i{}^0+r\zeta_i+r^2\sum_{j,k}^{1,2,3}\alpha_{jk}\zeta_j\zeta_k.
\tag{3}
\end{equation*}

Now the expression $\dfrac{D(z_1z_2z_3)}{D(r\zeta_1\,r\zeta_2\,r\zeta_3)}$ has the value unity when
$r = 0$. We may therefore revert our series, and write
\begin{equation*}
  r\zeta_i=z_i-z_i{}^0+\sum_{jk}^{1,2,3}\beta_{jk}(z_j-z_j{}^0)(z_k-z_k{}^0)+\ldots.
\tag{4}
\end{equation*}

We shall take our restricted region\index{Region02@--- restricted} so small that (4) shall
be uniformly convergent therein, for all values for ($z$) and ($z^0$)
in the region. Hence two points of the region may be connected
by a single geodesic arc lying entirely therein.%
\footnote{Cf.\ Darboux\index{Darboux}, loc.\ cit., vol.~ii, p.~408.}
%endfootnote

\mysect{Determination of a geodesic by two near points}
\smallskip\emph{Theorem}~1. Two points of a restricted region whose coordinates
differ by a sufficiently small amount may be connected
by a single geodesic arc lying wholly in a sufficiently small
restricted region which includes the two points.

We shall from now on, suppose that we have limited
ourselves to such a small restricted region that any two
points may be so connected by a single geodesic arc.

\smallskip\emph{Definition}.~A real analytic transformation of a restricted
region which leaves the distance element absolutely invariant
shall be called a congruent
transformation\index{Congruent transformations}\index{Transformations, congruent}.

\mysect{Determination of a geodesic by a point and direction cosines of tangent thereat}
\smallskip\emph{Definition}.~Given a geodesic through a point ($z^0$). The
three expressions
\[
\frac{dz_i}{ds}\!
\begin{array}{cc}
{}\\{}_{z_i=z_i{}^0}
\end{array}\!=\zeta_i,\quad i=1,2,3
\]
shall be called the \emph{direction cosines}\index{Cosines01@Cosines, direction} of the geodesic at that
point. Notice that
\begin{align*}
1-\sum_{i,j}^{1,2,3} a_{ij}\zeta_i\dot\zeta_j=
\sum_{i,j}^{1,2,3}a_{ij}\zeta_i&\zeta_j\sum_{ij}^{1,2,3}a_{ij}
\dot\zeta_i\dot\zeta_j-\Biggl(\sum_{i,j}^{1,2,3}a_{ij}\zeta_i\dot\zeta_j\Biggr)^2\\
&\,=\sum_{ij}^{1,2,3}(a_{ii}a_{jj}-a_{ij}{}^2)(\zeta_i\dot\zeta_j-\zeta_j\dot\zeta_i)^2.
\end{align*}

%[File: 280.png]
This is a positive definite form, for the coefficients are the
minors of a positive definite form. Hence
\[
\sum_{i,j}^{1,2,3}a_{ij}\zeta_i\dot\zeta_j\leqq 1.
\]

\mysect{Definition of angle}
This expression shall be defined as the \emph{cosine\index{Cosine01@Cosine of angle} of the
angle}\index{Angle01@Angle, interior and exterior}
formed by the two geodesics. When it vanishes, the geodesics
shall be said to be \emph{mutually perpendicular}\index{Perpendicularity} or to cut at right
angles\index{Right 01@Right angle}.

\smallskip\emph{Theorem}~2. The angle of two intersecting geodesics is an
absolute invariant for all congruent
transformations\index{Congruent transformations}\index{Transformations, congruent}.

This comes at once from the fact that
\[
\frac{\sum\limits_{ij}^{1,2,3}a_{ij}dz_i\delta z_j}{ds\delta s}
\]
is obviously an absolute invariant for all congruent transformations.

\smallskip\emph{Definition}.~A set of geodesics through a chosen point whose
direction cosines there, are linearly dependent upon those
of two of their number, shall be said to form a \emph{pencil}. The
surface which they trace shall be called a \emph{geodesic surface}\index{Pencil02@--- of geodesics}.
We shall later show that the choice of the name \emph{geodesic
surface}\index{Geodesic03@--- surfaces} is entirely justified, for each surface of this sort
may be generated in $\infty^2$ ways by means of pencils of
geodesics.

\mysect{Axiom of congruent transformations}
\medskip\textsc{Axiom}~III\@. \textbf{There exists a congruent transformation
which carries two sufficiently small arcs of two intersecting
geodesics whose lengths are measured from the common
point, into two arcs of equal length on any two intersecting
geodesics whose angle is equal to the angle of the
original two.}%
\footnote{Our Axioms I--III, are, with slight verbal alterations, those used by
Woods\index{Woods}, loc.\ cit. His article, though vitiated by a certain haziness of definition,
leaves nothing to be desired from the point of view of simplicity. In
the present chapter we shall use a different coordinate system from his, in
order to avoid too close plagiarism. It is also noteworthy that he uses $k$
where we conformably to our previous practice use $\dfrac{1}{k}$.}
%endfootnote

It is clear that a congruent transformation will carry an
arc whose variation is zero into another such, hence a geodesic
%[File: 281.png]
into a geodesic. It will also transform a geodesic surface
into a geodesic surface, for it is immediately evident that
we might have defined a geodesic surface\index{Geodesic03@--- surfaces} as generated by
those geodesics through a point which are perpendicular to
a chosen geodesic through that point.

It is now necessary to choose a particular coordinate system,
and we shall make use of one which will turn out to be
identical with the polar coordinate system of elementary
geometry. Let us choose a fixed point ($z^0$), and a fixed
geodesic through it with direction cosines ($\zeta^0$). Finally, we
choose a geodesic surface determined by our given geodesic,
and another through ($z^0$). Let $\phi$ be the angle which a geodesic
through ($z^0$) makes with the geodesic ($\zeta^0$), while $\theta$ is the angle
which a geodesic perpendicular to the last chosen geodesic
and to ($\zeta^0$) makes with a geodesic perpendicular to the given
geodesic surface, i.e.\ perpendicular to the geodesics of the
generating pencil. Let $r$ be the length of the geodesic arc of
($\zeta$) from ($z^0$) to a chosen point. We may take $\phi$, $\theta$, $r$ as coordinates
of this point. The square of the distance element
will take the form
\begin{equation*}
ds^2 = dr^2 + Ed\theta^2 + 2 Fd\theta\, d\phi + Gd\phi^2.
\tag{5}
\end{equation*}

We see, in fact, that there will be no term in $dr\,d\phi$ or $dr\,d\theta$.
For if we take $\theta = $ const.\  we have a geodesic surface, and
the geodesic lines of space radiating from ($z^0$) and lying in
this surface will be geodesics of the surface. The curves
$r = $ const.\  will be orthogonal to these radiating geodesics.%
\footnote{Bianchi\index{Bianchi}, \emph{Differentialgeometrie}, cit., p.~160.}
%endfootnote
The surfaces $\phi=$ const.\ are not geodesic surfaces, but the
curves $\theta = $ const.\ and $r = $ const.\ form an orthogonal system for
the same reason as before. The coefficients $E$, $F$, $G$ are independent
of $\theta$, for, by Axiom III, we may transform
congruently\index{Congruent transformations}\index{Transformations, congruent}
from one surface $\theta = $ const.\ into another such. The coefficient
$G$ is independent of $\phi$ also, for in any surface $\theta = $ const.\ we
may transform congruently from any two geodesics
through ($z^0$) into any other two making the same angle.
\mysect{Simplified expression for distance element}
We may, in fact, write
\[
E=G(r)E'(\phi),\quad F=G(r)F'(\phi),
\]
for the square of any distance element can be put into the
form
\[
ds^2=dr^2+Gd\phi_1{}^2,
\]
where $\phi_1$ is a function of $\phi$ and $\theta$.


%[File: 282.png]
Let us at this point rewrite our differential equations (2)
in terms of our present coordinates
\begin{align*}
&\frac{d}{ds} \Bigl[\frac{dr}{ds}\Bigr]=
\frac{1}{2}\Bigl[\frac{\partial E}{\partial r}\Bigl(\frac{d\theta}{ds}\Bigr)^2+
2\frac{\partial F}{\partial r}\Bigl(\frac{d\theta}{ds}\Bigr)
\Bigl(\frac{d\phi}{ds}\Bigr)+\frac{\partial G}{\partial r}
\Bigl(\frac{d\phi}{ds}\Bigr)^2\Bigr],
\\
&\frac{d}{ds}\Bigl[E\frac{d\theta}{ds}+F\frac{d\phi}{ds}\Bigr]=0,\tag{6}
\\
&\frac{d}{ds}\Bigl[F\frac{d\theta}{ds}+G\frac{d\phi}{ds}\Bigr]=
\frac{1}{2}\Bigl[\frac{\partial E}{\partial \phi}\Bigl(\frac{d\theta}{ds}\Bigr)^2 +
2\frac{\partial F}{\partial\phi}\Bigl(\frac{d\theta}{ds}\Bigr)
\Bigl(\frac{d\phi}{ds}\Bigr)\Bigr].
\end{align*}
\index{Geodesic02@--- lines|)}

Consider the geodesic surface $\phi=\dfrac{\pi}{2}$ which may, indeed,
be taken to stand for any geodesic surface\index{Geodesic03@--- surfaces}. Here we must
have
\begin{equation*}
E=cG,
\end{equation*}
where $c$ is constant. The differential equations for a geodesic
curve on this surface will be%
\footnote{Bianchi\index{Bianchi}, ibid., p.~153.}
%endfootnote
\begin{align*}
&\frac{d}{ds}\Bigl[\frac{dr}{ds}\Bigr]=
\frac{1}{2}\Bigl[\frac{\partial E}{\partial r}\Bigl(\frac{d\theta}{ds}\Bigr)^2\Bigr],
\\
&\frac{d}{ds}\Bigl[G\frac{d\theta}{ds}\Bigr]=0.
\end{align*}

These are exactly equivalent to the combination of (6) and
$\phi=$const. Lastly, if we remember that two near points of
a surface can be connected by a single geodesic arc lying
therein.

\smallskip\emph{Theorem}~2. The geodesic connecting two near points of
a geodesic surface lies wholly in that surface, and is identical
with the geodesic of the surface which connects those two
points.

\smallskip\emph{Theorem}~3. There is a group of $\infty^3$ congruent transformations
which carry a geodesic surface transitively into itself.

\mysect{Constant curvature of geodesic surfaces}
\smallskip\emph{Theorem}~4. All geodesic surfaces have the same constant
Gaussian curvature.

These theorems enable us to solve completely our differential
equations (6). The Gaussian\index{Curvature02@--- Gaussian} curvature\index{Curvature06@--- of space}
of each geodesic
surface is an invariant of space which we may call its
\emph{measure of curvature}\index{Measure02@--- of curvature of space}. We shall denote this constant by
$\dfrac{1}{k^2}$,\index{K@$\dfrac{1}{k^2}$ measure of curvature of space}
and distinguish with care the two following cases
\begin{equation*}
\frac{1}{k^2} \neq 0,\quad \frac{1}{k^2}=0.
\end{equation*}

%[File: 283.png]
The determination of our coefficients $E$, $F$, $G$ is now an
easy task. The square of the distance element for a geodesic
surface $\theta = $ const., will be
\[
ds^2 = dr^2 + G(r)d\phi^2.
\]

Writing that this shall have Gaussian curvature\index{Curvature02@--- Gaussian} $\dfrac{1}{k^2}$, we get
\[
\frac{-1}{\sqrt{G}} \frac{\partial^2\sqrt{G}}{\partial r^2} = \frac{1}{k^2},
\]
\[
\sqrt{G} = A\sin\frac{r}{k} + B\cos\frac{r}{k}.
\]

The determination of the constants $A$, $B$ requires a little
care. It is clear to begin with that when
\[
r = 0,\quad G=0.
\]
\begin{flalign*}
&\text{\indent Hence} &B &= 0.
\\
&\text{\indent Again}
&G =
  \sum_{i,j}^{1,2,3}a_{ij}\frac{\partial z_i}{\partial \phi}
    \frac{\partial z_j}{\partial \phi} &=
  r^2\sum_{ij}^{1,2,3}a_{ij}\frac{\partial\zeta_i}{\partial\phi}
    \frac{\partial\zeta_j}{\partial\phi} +
  r^3 R(r\phi),
&\phantom{\text{\indent Again}}
\\
&&\Bigl(\frac{\partial\sqrt{G}}{\partial r}\Bigr)_{r=0} =
  &\sqrt{\sum_{i,j}^{1,2,3}a_{ij}\frac{\partial\zeta_i}{\partial\phi}
    \frac{\partial\zeta_j}{\partial\phi}}.
\end{flalign*}

But, from (1)
\[
1 = \sum_{i,j}^{1,2,3}a_{ij}\zeta_i\zeta_j =
  \sum_{i,j}^{1,2,3}a_{ij}
  \Bigl(\zeta_i + \frac{\partial\zeta_i}{\partial\phi}d\phi\Bigr)
  \Bigl(\zeta_j + \frac{\partial\zeta_j}{\partial\phi}d\phi\Bigr),
\]
\[
\cos d\phi =
  \sum_{i,j}^{1,2,3}a_{ij}\zeta_i
  \Bigl(\zeta_j + \frac{\partial\zeta_i}{\partial\phi}d\phi\Bigr),
\]
\[
\cos\frac{d\phi}{2} =
  1 - \frac{1}{2}\sum_{i,j}^{1,2,3}a_{ij}
  \frac{\partial\zeta_i}{\partial\phi}
  \frac{\partial\zeta_j}{\partial\phi}d\phi,
\]
\[
\sin\frac{d\phi}{2} =  \frac{d\phi}{2} =
  \frac{d\phi}{2}\sqrt{\sum_{i,j}^{1,2,3}a_{ij}
    \frac{\partial\zeta_i}{\partial\phi}
    \frac{\partial\zeta_j}{\partial\phi}},
\]
giving eventually
\[
\Bigl(\frac{\partial\sqrt{G}}{\partial r}\Bigr)_{r=0} = 1;\quad A=k.
\]

Hence, by the equations preceding (6)\label{dphidphi}
\[
ds^2 = dr^2 + k^2\sin^2\frac{r}{k}
  [E'd\theta^2 + 2F'd\theta\, d\phi + d\phi^2].
\]

%[File: 284.png]
We proceed to calculate $F'$. The differential equations for
a geodesic curve of the surface $\theta = $ const., will be
\[
\frac{d}{ds} \Bigl( \frac{dr}{ds} \Bigr) =
  \frac{1}{2}\frac{\partial \theta}{\partial r}
    \Bigl(\frac{d\phi}{ds}\Bigr)^2,
\]
\[
\frac{d}{ds} \Bigl( G\frac{d\phi}{ds} \Bigr) = 0.
\]

These must be equivalent to those obtained from (6), when
$\theta = $ const., i.e.\ we must have
\[
\frac{d}{ds} \Bigl( F'G\frac{d\phi}{ds} \Bigr) = 0,
\]
\[
F' = \text{const.},
\]
and as $F'$ is not a function of $\theta$ it is a constant everywhere.
Now when $\phi = 0$, there is no $d\theta$ term in $ds^2$, so that $E = 0$;
but $\dfrac{E}{\sqrt{FG}}$, which is the cosine of the angle which curves
$\theta =$ const.\ and $\phi = $ const., make on the surface $r = $ const.,
is surely less than unity. Hence
\[
F' = 0.
\]

Lastly, we must find $E'$. The surfaces $r = $ const.\  have
constant Gaussian curvature\index{Curvature02@--- Gaussian}, for each is capable of $\infty^3$ congruent
transformations into itself. Hence
\[
ds^2 = k^2\sin^2\frac{r}{k}[E'd\theta^2+d\phi^2],
\]
\[
\frac{1}{\sqrt{E'}} \frac{d^2\sqrt{E'}}{d\phi^2} = \text{const.},
\]
\[
\sqrt{E} = A \sin l\phi + B \cos l\phi.
\]

As we saw a moment ago $B = 0$, for $E$ vanishes with $\phi$.
On the other hand, when
\begin{flalign*}
&&\phi=\frac{\pi}{2}, \quad &\sqrt{E'}=1,\quad A\sin\frac{l\pi}{2} = 1.
\\
&\text{\indent But also}
&&A \sin l\pi = 0.
&\phantom{\text{\indent But also}}
\end{flalign*}

Hence $l$ is an odd\label{oldint} integer, and
\[
A^2 = 1.
\]
\[
\tag{7}
ds^2 = dr^2 +
  k^2\sin^2\frac{r}{k} [\sin^2\phi\,d\theta^2 + d\phi^2].
\]

%[File: 285.png]
This is our ultimate form for the square of the distance
element. Let the reader show that under the second case
$\dfrac{1}{k^2} = 0$, we have
\[
  ds^2 = dr^2 + r^2\Bigl[\sin^2\phi\,d\theta^2 + d\phi^2\Bigr].  \tag{7$'$}
\]
\index{Distance04@Distance element|)}

\mysect{Introduction of new coordinates; integration of equations of geodesic}
It is now time to return to coordinates of a more familiar
sort. Let us write
\[
\begin{aligned}
  x_0 &= k\cos\frac{r}{k},  \\
  x_1 &= k\sin\frac{r}{k} \cos\theta \cos\phi,  \\
  x_2 &= k\sin\frac{r}{k} \sin\theta \cos\phi,  \\
  x_3 &= k\sin\frac{r}{k} \sin\phi,  \\
  (xx) &= k^2,   \\
  (dx\,dx) &= ds^2.
\end{aligned}
\tag{8}
\]

To find the differential equation of a geodesic\index{Geodesic02@--- lines}, we have
a problem in relative minima
\[
  \int_0^s \Bigl( \sqrt{\Bigl( \frac{dx}{ds}\frac{dx}{ds} \Bigr)}
                + \lambda[(xx)-k^2] \Bigr) ds = 0,
\]
\[
  \frac{d}{ds}\Bigl(\frac{dx_i}{ds}\Bigr) = 2\lambda x_i,\quad
  i = 0, 1, 2, 3.
\]

To determine $\lambda$
\[
  (xx) = k^2,\quad (x\,dx) = -\tfrac{1}{2}ds^2,
\]
\[
  (x\,d^2x) + ds^2 = d(-\tfrac{1}{2}ds^2) = 0.
\]

But from our equations
\[
  (x\,d^2x) + ds^2 = 2\lambda k^2 ds^2,
\]
\[
  2\lambda = -\frac{1}{k^2}.
\]

\mysect{Reappearance of familiar distance formulae}
We thus get for the final form for our differential equation
\[
  \frac{d^2x_i}{ds^2} + \frac{x_i}{k^2} = 0.    \tag{9}
\]

Let the reader show that in the other case we have
\[
  \frac{d^2x}{ds^2} = \frac{d^2y}{ds^2} = \frac{d^2z}{ds^2} = 0. \tag{9$'$}
\]

%[File: 286.png]

Integrating
\[
x_i = y_i\cos\frac{s}{k} + z_i\sin\frac{s}{k},
\]
\[
k^2 = (xx) = (yy) =(zz),
\]
\[
(yz) = 0.
\]

\index{Cosine02@--- of distance}\index{Distance01@Distance of two points, cosine}
We have then for the length of the geodesic arc from $(y)$
to $(x)$
 \[
k^2\cos\frac{d}{k} = (xy),
\]
or, if we replace our coordinates by homogeneous ones proportional to them
\begin{equation*}
\tag{10}
\cos\frac{d}{k} = \frac{(xy)}{\sqrt{(xx)}\;\sqrt{(yy)}}.
\end{equation*}

Let the reader show that when $\dfrac{1}{k^2} = 0$,
\[
d = \sqrt{(x-x_0)^2 + (y-y_0)^2+(z-z_0)^2}.
\]

\smallskip\emph{Theorem}~5. Axioms I, II, III are compatible with the
euclidean\index{Euclidean01@Euclidean hypothesis}
hyperbolic\index{Hyperbolic01@Hyperbolic hypothesis}
and elliptic\index{Elliptic02@--- hypothesis} hypotheses, and with these
alone.

\midrule

\mysect{Recapitulation}
Our task is now completed. At bottom, the essential
feature of a geometrical system where the elements are points
is the expression for distance\index{Distance}, for the projective theory is
the same for a limited domain in all restricted regions. We
have established our distance formulae three several times,
each time approaching the subject from a new point of view.
In Chapters I--IV we took as fundamental the concepts point,
distance, and sum of distances. We reached our analytic
formulae by proceeding from elementary geometry to trigonometry,
and then introducing a simple coordinate system, such
as we do when we first take up the study of elementary
analytic geometry. The Chapters VI--XVII were devoted to
erecting a superstructure upon the foundation which we had
established. In Chapter XVIII we took a fresh start, laid
down point line and separation as fundamental, constructed
the common projective geometry for all of our systems (except
the spherical, which would involve slight modifications), and
established the system of projective coordinates. We then
introduced certain collineations called \emph{congruent transformations},
and worked around to our previous distance formulae
through group-theory. In the present chapter we took as
fundamental the concepts point and correspondence of point
and coordinate set. The essentials in our development were
the distance element, the geodesic curve, and the space constant,
%[File: 287.png]
or measure of curvature. We reached our familiar
formulae by means of surface theory, integration, and the
calculus of variations.

Which of the three methods of approach is the best? To
this question no definite answer may be given, for that method
which is best for one purpose is not, necessarily, best for
another. The first method depended upon the simplest and
most natural fundamental conceptions, and presupposed a
minimum of mathematical knowledge. It also corresponded
most closely to the line of historical development. On the
other hand it is the longest, even after cutting out a number
of theorems, interesting in themselves, but not essential as
steps towards the ultimate goal. The second method possessed
the advantage of beginning with the assumptions which serve
as a basis for the important subject of projective geometry;
metrical ideas were grafted upon this stem as a natural
development. Moreover, the fundamental importance of the
six-parameter collineation group which keeps a conic or
quadric invariant was brought into the clearest light. On
the other hand, we were obliged to develop a coordinate
system, which to some readers might seem a trifle unnatural
or forced, and exposed ourselves to being put down among
those whom the late Professor Tait\index{Tait} has stigmatized as `That
section of mathematicians for whom transversals and anharmonic
pencils have a, to us, incomprehensible charm'.%
\footnote{Tait, \emph{An Elementary Treatise on Quaternions}, third edition, Cambridge, 1890,
p.~309.}
%endfootnote
third and last method is, beyond a peradventure, the quickest
and most direct; and has the advantage of bringing out the
full significance of the space constant. It may, however,
be urged with some justice, that too high a price has been
paid for this directness, by assuming at the outset that space
is something whose elements depend in a definite manner on
three independent parameters. The modern tendency is to
take a more abstract view, to look upon space, in the last
analysis, as a set of objects which can be arranged in multiple
series.%
\footnote{Cf.\ Russell, loc.\ cit., p.~372.}
%endfootnote
The battle is more than half over when the coordinate
system has been set up.

No, there is no answer to the question which method of
approach is the best. The determining choice among the
three, will, in the end, be a matter of personal aesthetic
preference. And this is well. Let us not forget that, in
large measure, we study pure mathematics to satisfy an
aesthetic need. We are fortunate when, as in the present case,
we are free at the outset to choose our line of approach.

%[File: 288.png]

\newpage
\mysect{\textsc{Index}}
\begingroup
\stretchspace
\printindex
\endgroup
%{\centering \Large\textbf{INDEX} }

%Absolute, 88, 94, 95, 97, 98, 99, 101,
%102, 103, 106, 107, 110, 111, 113,
%116, 117, 118, 119, 124, 127, 129,
%132, 134, 138, 142, 143, 146, 152,
%154, 155, 157, 161, 162, 187, 205,
%226, 231, 232, 233, 234, 244, 246.
%Actual elements, 85.
%Amaldi, 177.
%Amplitude of tetrahedron, 179, 180, 181.
%Amplitude of triangle, 170, 171, 172, 173.
%Angle, interior and exterior, 30, 87, 88, 279.
%--null, 30.
%--right, 32.
%--straight, 31.
%--re-entrant, 31.
%--dihedral, 39.
%--plane, of dihedral, 39.
%--of skew lines, 113.
%--measure of, 38, 87.
%--of two planes, cosine, 70.
%--parallel, 106, 107.
%Angles of a triangle, 31.
%--exterior of a triangle, 31.
%--Clifford, 126.
%Archimedes, 24.
%Area, 170, 175, 178, 211.
%--of a circle, 178.
%--of a plane, 178.
%--of a polygon, 178.
%--of a triangle, 175, 176, 177.
%Aronhold, 159.
%Asymptotes, 152.
%Asymptotic lines, 196, 202, 203, 212, 213.
%Author, 116. 127, 130, 154, 156,
%158, 167, 226, 230, 232, 234.
%Axes, co-ordinate, 64, 67.
%Axial plane of sphere, 138.
%Axis of a circle, 131, 134, 135, 150.
%--radical of two circles, 134, 135, 136.
%--of a conic, 143.
%Axis of a chain, 119.
%--of a pencil of complexes, 116.
%Barbarin, 154.
%Battaglini, 131.
%Beck, 116.
%Beltrami, 67, 210.
%Bianchi, 6, 187, 188, 204, 206, 210,
%226, 280, 281.
%Birectangular quadrilateral, 43. 44,
%49.
%Bisector of an angle, 102, 103, 109,
%133, 135, 136, 143, 146, 153, 157,
%159, 220, 222.
%Bolza, 209.
%Borel, 34.
%Bound of half-line, 28.
%Bound of half-plane, 30.
%Bromwich, 154.
%Canal surface, 156.
%Cayley, 88, 97, 157.
%Central conic, 143-153.
%Central quadric, 157-60.
%Centre of a circle, 135, 136. 137.
%--of a conic, 143, 148, 149, 150.
%--of gravity of points, 102, 103,
%109, 133, 135, 136, 143, 146, 153,
%159, 220, 222.
%Centre of quadric, 157.
%--of similitude, 134, 135, 136.
%Ceva, 105.
%Chain congruence, 121, 129.
%--of crosses, 119, 120, 128.
%Circle, 131-137, 143, 151, 178, 188.
%--auxiliary to conic, 152.
%Clebsch, 159, 176.
%Clifford, 99, 126, 129, 156, 157, 205,
%212, 240.
%Coaxal pencil of complexes, 116, 124.
%Coaxality, 20.
%Collinearity, 18, 102, 103, 104, 105,
%134, 136, 251.
%%%[File: 289.png]
%Collineations, 29, 38, 69, 70, 94, 119,
%127, 239, 265, 266, 268.
%Comparableness of angles, 34, 35.
%Complex of lines, 116.
%Concurrence, 18, 102, 103, 105, 134,
%136, 251.
%Cone of revolution, 185.
%Confocal conics, 153.
%Confocal quadrics, 160, 164.
%Conformal transformations, 198.
%Congruence of distances, 14, 15, 16,
%17, 28, 36, 79.
%--of segments, 28.
%--of angles, 31, 33, 34, 36, 38, 39.
%--of triangles, 31, 32.
%--synectic, 120, 122.
%--chain, 121, 129.
%--of lines, analytic, 215-235.
%--of lines, general, 218.
%--of normals, 162, 208, 210, 222,
%223, 224, 225, 226, 227, 229, 235.
%--of normals, to surfaces of Gaussian
%curvature zero, 123, 208, 226,
%227, 235.
%isotropic, 164, 226, 227, 230, 232,
%234, 235.
%Congruent figures, 28.
%Congruent transformations, 29, 37,
%38, 69, 70, 73, 74, 80, 82, 92-100,
%239, 268, 269, 270, 271, 278, 279,
%280.
%Conic, 142-53, 272.
%Conic, eleven-point or line, 147.
%Conjugate diameters of a conic, 148.
%--directions on a surface, 195, 196.
%--harmonic, 252, 253, 254, 257,
%259, 261.
%Connectivity of space, 238.
%Consistent region, 78, 79, 80, 83,
%236, 237. 238.
%Continuity, axiom of, 23, 24, 75, 249.
%--in change of angles and sides of
%a triangle, 40, 41, 42.
%Co-ordinates of a line, 110, 264.
%--of a point, 64, 68, 176, 187, 188,
%194, 236, 237, 263, 264, 275.
%--of a plane, 264.
%Coplanarity, 109, 138.
%Cosine of angle, 54, 70, 279.
%--of distance, 52, 285.
%Cosines, direction, 67, 69, 278.
%--law of, 57.
%Cross, 117, 118, 119, 124, 125, 231.
%Cross ratios, 73, 86, 88, 89, 90, 91,
%247, 259, 260, 261, 262, 264, 265.
%Cross space, 118.
%Curvature of a curve, 133, 188, 189,
%200, 201.
%--Gaussian, 67, 123, 130, 204, 205,
%206, 207, 208, 275, 281, 282, 283.
%--geodesic, 208, 209.
%--mean relative, 200, 212.
%--total relative, 200, 203, 204.
%--of space, 53, 176, 189, 204,
%281.
%--lines of, 198, 199.
%--surfaces of zero, 123, 204,
%206, 207, 208, 226, 227, 235.
%Dannmeyer, 170.
%Darboux, 141, 212, 278.
%Dehn, 46, 181.
%Density of segment, 16.
%Desargues, 75, 146, 251.
%Desmic configuration, 108, 109, 110,
%138.
%Diagonal points of quadrangle, 252.
%Diagonals of quadrilateral, 252.
%Diameters of conic, 148, 149, 150,
%151.
%--of quadric, 159, 160.
%Difference of distances, 17, 35.
%Director points and directrices, 144,
%145, 146.
%Discrepancy of a triangle, 46, 174.
%Distance, 13, 72. 73, 74, 76, 78, 87,
%89, 90, 91, 272, 273, 285.
%Distance, directed, 62, 66, 90.
%Distance of two points, cosine, 52,
%69, 78, 285.
%--from point to plane, 70.
%--of skew lines, 111, 112, 114.
%--element, 66, 67, 187, 194, 276-84.
%Division of segment, 24, 25, 26, 27.
%Dunkel, 60.
%Dupin, 141, 197, 201, 205.
%Edge of tetrahedron, 20.
%Ellipse, 142, 143, 146, 153, 167, 168.
%Ellipsoid, 154, 156, 167, 168, 169.
%Elliptic co-ordinates, 153, 161.
%--hypothesis, 46, 73, 74, 274. 285.
%--space, 82, 83, 245.
%Engel, 43.
%Enlargement of congruent transformation, 29.
%Enriques, 33, 177, 247.
%Equidistant curves, 132, 143.
%--surfaces, 156.
%%%[File: 290.png]
%Equivalent points, 81, 82, 236.
%Euclid, 47, 72.
%Euclidean hypothesis, 46, 72, 73,
%274, 285.
%--space, 77, 91, 239, 240, 241, 242.
%Evolutes, 192, 193, 194.
%Excess of a triangle, 174, 175, 176,
%177.
%Extension of segment, 15, 79.
%--of space, 77, 78, 79, 80.
%Extremity of segment, 15.
%Face of tetrahedron, 20.
%Fibbi, 215, 221.
%Focal cones, 158, 159.
%--conics, 158, 159, 167, 168. 169.
%--lines, 144, 145, 146, 147, 151.
%--points and planes, 221, 222, 223,
%224.
%--surfaces, 210. 226, 232.
%Foci, 144, 145, 146, 147, 151.
%Forms, fundamental one-dimensional,
%259, 260, 261, 267.
%Von Frank, 186.
%Frenet, 190.
%Fricke, 244.
%Frischauf, 176, 186.
%Fubini, 227, 229.
%Fundamental region, 239-46.
%--one-dimensional forms, 259, 260,
%261, 267.
%Geodesic curvature, 208, 209.
%--lines, 163, 209, 210, 274-81, 284,
%285.
%--surfaces, 279, 280, 281.
%G\'erard, 48. 53.
%Graves, 153.
%Greater than, 15, 16, 17, 34, 35,
%37, 92.
%Half-line, 28-33, 38, 64, 67.
%Half-plane, 30, 37, 38, 39.
%Halsted, 24, 75, 177.
%Hamilton, 98, 120, 221.
%Harmonic conjugate, 252, 253, 255,
%261.
%--separation, 252, 253, 254, 257.
%--set, 252, 253.
%Hilbert, 13, 24, 36, 75, 177, 265.
%Homothetic conics, 152, 153.
%--quadrics, 160.
%Horocycle, 132, 143, 243.
%Horocyclic surface, 156, 205.
%Hyperbola, 142, 146, 167, 168, 169.
%Hyperbolic hypothesis, 46, 72, 73,
%78, 274, 285.
%--space, 78, 236.
%Hyperboloid, 155, 167, 168, 169.
%Ideal elements, 84, 85.
%Imaginary elements, 86, 87, 266,
%267.
%Improper cross, 117, 118, 127, 231.
%--ray, 231, 232.
%Indicatrix of Dupin, 201, 205.
%Infinitely distant elements, 84, 85.
%Infinitesimal domain, 42, 47, 68,
%174, 175.
%Initial point, 62.
%Intersection of lines, 17, 249.
%--of planes, 22, 251.
%Involution, 86, 87, 266, 267.
%Isosceles quadrilateral, 43, 50.
%--triangle, 32, 34.
%Isotropic curves, 203, 209.
%--congruence, 164, 226, 227, 230,
%232, 234, 235.
%Joachimsthal, 197.
%Jordan, 277.
%$\frac{1}{k^2}$ measure of curvature of space,
%53, 176, 189, 204, 275, 281.
%Killing, 142, 237, 245.
%Klein, 97, 129, 161, 244.
%Kummer, 215.
%Layer of cross space, 118, 119, 125.
%Left and right generators of Absolute,
%99, 124, 125, 234.
%Left and right translations, 99, 100,
%245.
%Left and right parataxy, 99, 208, 225.
%Length of arc, 276.
%Less than, 15, 16, 17, 34, 35, 37, 92.
%Levy, 13, 75.
%Lie, 268, 270, 271.
%Liebmann, 142.
%Limiting points and planes, 219,
%220, 222.
%Lindemann, 139, 176.
%Line, 17, 78, 248, 249.
%Lobatchewsky, 46, 106.
%Lobatchewskian hypothesis, 46.
%L\"uroth, 87, 89, 267.
%Manning, 107, 176, 205.
%Marie, Ste-, 47.
%%[File: 291.png]
%Measure of distance, 27, 28, 87.
%--of curvature of space, 33, 176,
%189, 204, 207, 275, 281.
%Menelaus, 105.
%Meunier, 201, 208.
%Middle point of segment, 24.
%Minimal surfaces, 129, 210-14.
%Moment, relative of two lines, 112.
%--relative of two rays, 115, 192.
%Moore, 13, 46, 75.
%Motions, 97, 98, 99.
%Multiply connected space, 238-46.
%M\"unich, 139.
%Normals to curve, 192, 193, 194.
%--to surface, 162, 197, 208, 210, 222,
%223, 224, 225, 226, 227, 229, 235.
%Null angle, 30.
%--distance, 14.
%Opposite edges of tetrahedon, 20.
%--half-lines, 31.
%--senses, 63, 86, 266.
%--sides of plane, 22.
%Origin, 64.
%Orthogonal points, 101, 103, 118,
%132, 135, 136, 137, 138, 139, 143,
%189, 205, 215, 217, 219, 224.
%Orthogonal substitutions, 69, 70, 73,
%97, 98.
%--system of surfaces, 197, 198.
%d'Ovidio, 112, 142, 170.
%Padoa, 13, 254.
%Parabola, 142, 143.
%Parabolic hypothesis, 46.
%Paraboloid, 155, 157.
%Parallel angle, 106, 107, 110.
%Parallelism, 85, 99, 106, 113, 234,
%235.
%Parataxy, 99, 114, 125, 129, 206,
%207, 208, 225, 233, 234, 235.
%Pasch, 13, 29, 86, 265.
%Peano, 13.
%Pencil of complexes, 116.
%--of geodesics, 279.
%Perpendicularity, 34, 36, 37, 39,
%101, 103, 118, 132, 135, 136, 137,
%138, 139, 143, 182, 183, 193, 197,
%217, 219, 220, 224, 279.
%Phi function, 50, 51, 52.
%Picard, 175, 210.
%Pieri, 13, 74, 86, 247.
%Plane, 20, 21, 22, 38, 67, 70, 81, 82,
%95, 109, 110, 118, 224, 242, 243,
%249, 250, 251, 253, 259, 264 265,
%268, 269.
%Poincar\'e, 139.
%Point, 13, 78, 84, 86, 247, 266,
%275.
%Polygon, 178.
%Principal points and planes, 220.
%Products connected with a conic,
%145, 149, 150.
%Projection, 253, 260.
%Projectivity, 259, 260, 262, 267.
%Pseudo-isotropic congruence, 229,
%230, 234, 235.
%Pseudo-normal congruence, 224,
%229.
%Pseudo-parallelism of lines, 113,
%234, 235.
%Pythagorean theorem, 55, 57.
%Quadrangle, complete, 252.
%Quadrilateral, 43, 44, 49, 174.
%--complete, 252, 253, 266.
%Quaternions, 98, 245.
%Ratio of opposite sides of quadrilateral,
%49, 50, 51, 52, 53.
%Ratios, constant connected with
%conics, 144, 151.
%Ray, 114, 115, 191, 192, 227, 228,
%234, 235.
%Rectangle, 43, 44, 45, 46.
%Reflection in plane, 39, 82.
%--in point, 62.
%Region consistent, 78, 79, 80, 81,
%83, 236, 237, 238.
%Region, fundamental, 239-46.
%--restricted, 269, 276, 277, 278.
%Revolution, surfaces of, 155, 156.
%Riccordi, 131.
%Richmond, 184.
%Riemann, 46, 53, 67, 275.
%Riemannian hypothesis, 46.
%Right angle, 32, 34, 39, 279.
%--triangle, 32, 44, 45, 55.
%Right and left generators of Absolute,
%94, 124, 125, 234.
%Right and left parataxy, 99, 208,
%225.
%Right and left translations, 99, 100,
%245.
%Russell, 74.
%Saccheri, 43, 50.
%Salmon, 134.
%Scalene triangles, 34, 35.
%%[File: 292.png]
%Schl\"afli, 183, 184, 185.
%Schur, 13, 275.
%Segment, 15, 16, 17, 18, 23, 24, 25,
%26, 28.
%Segre, 119.
%Semi-hyperbola, 142.
%Semi-hyperboloid, 154.
%Sense of directed distances, 63, 64.
%--of description of involution, 86,
%266.
%Separation, 248, 249, 255, 256, 257,
%258, 259, 262.
%--classes, 247, 249, 255, 257, 258.
%--harmonic, 252, 253, 254, 257.
%Sides of angle, 30, 31.
%--of quadrangle, 252.
%--of quadrilateral, 43, 252.
%--of triangle, 19, 31, 32, 35, 36.
%Similitude, centres of, 134, 135,
%136.
%Sine of distance from point to plane,
%70.
%Sines, law of, 58, 59.
%Singular region, 238.
%Space, 20, 21, 22, 78, 238-46, 250,
%251.
%Sphere, 73, 74, 138-41, 156, 227,
%228.
%Spheres, representing, 227, 228.
%Spherical space, 83.
%Spheroid, 155, 156.
%St\"ackel, 43.
%Staude, 162.
%Von Staudt, 86, 87, 89, 267.
%Stephanos, 108.
%Stolz, 24.
%Story, 142.
%Strip, 128, 129.
%Study, 91, 93, 99, 116, 123, 125, 126,
%229, 234.
%Sturm, 233.
%Sum of angles, 32, 34.
%--of angles of a triangle, 45, 46.
%Sum of distances, 14-17, 92, 93.
%Sum of distances connected with a
%conic, 145, 148, 149.
%Sum of distances connected with a
%quadric, 160.
%Sum of two sides of triangle, 35.
%Supplementary angles, 32.
%Surface integral, 175.
%Symmetry transformations, 98, 99,
%127.
%Synectic congruence, 120, 122.
%Tait, 286.
%Tangent plane to surface, 194, 195.
%Tannery, 273.
%Terminal point, 62.
%Tetrahedron, 20, 21, 181, 182, 183.
%Tensor, 98.
%Thread construction, 169.
%Torsion, 190, 191, 192, 203, 207.
%Transformations, congruent, 29, 37,
%38, 69, 70, 73, 74, 80, 82, 92-100,
%239, 268, 269, 270, 271, 278, 279,
%280.
%Translations, 62, 63, 100, 128, 239,
%240, 245, 246.
%Triangle, 18, 19, 31-5, 170, 172,
%174, 175, 176, 177.
%Triangles, congruent, 31.
%Trirectangular quadrilateral, 43.
%Ultra-infinite elements, 85, 187.
%Umbilical points, 162.
%Vablen, 13, 24, 75, 247, 260, 265.
%Vailati, 248, 254.
%Veblen, 13, 19, 76, 247.
%Veronese, 13, 74.
%Vertex of angle, 30, 31.
%--of quadrangle, 252.
%--of quadrilateral, 252.
%--of tetrahedron, 20.
%--of triangle, 19.
%Vertical angles, 32, 34.
%Volume, 181, 182.
%--integral, 182.
%--of cone, 185.
%--of sphere, 186.
%--of tetrahedron, 182, 183, 184,
%185.
%Voss, 188.
%Weber, 46.
%Weierstrass, 142.
%Within a segment, 15, 18.
%Within a triangle, 19.
%Woods, 237, 245, 246, 279.
%Young, 247.
%[File: 293.png]
\pagestyle{empty}
\vbox{}
\vfill\vfill
\begin{center}
OXFORD\\
{\scshape
printed at the clarendon press\\
by horace hart, m.a.\\
printer to the university
}
\end{center}
\vfill\vfill\vfill

%[File: 294.png]
\newpage

\pagestyle{plain}
\pagenumbering{arabic}
\begin{center}
{\Huge CLARENDON PRESS BOOKS}

\medskip{\huge HISTORY}

\medskip{\huge Greece, Italy, Egypt, etc}
\end{center}

\adless\begin{large}Clinton's Fasti Hellenici,\end{large} from the LVIth to the CXXIIIrd Olympiad.
Third edition. 4to. £1 14s.\ 6d.\ net. From the CXXIVth Olympiad to the Death
of Augustus. Second edition. 4to. £1 12s.\ net. Epitome. 8vo. 6s.\ 6d.\ net.

\adnorm\begin{large}Clinton's Fasti Romani,\end{large} from the death of Augustus to the death of Heraclius. Two volumes. 4to. £2 2s.\ net. Epitome. 8vo. 7s.\ net.

\adnorm\begin{large}Greswell's Fasti Temporis Catholici.\end{large} 4 vols. 8vo. £2 10s.\ net. Tables and Introduction to Tables. 8vo. 15s.\ net. Origines Kalendariae Italicae.
4 vols. 8vo. £2 2s.\ net. Origines Kalendariae Hellenicae. 6vols. 8vo. £4 4s.\ net.

\adnorm\begin{large}A Manual of Greek Historical Inscriptions.\end{large} By \textsc{E.~L. Hicks}. New edition, revised by \textsc{G.~F. Hill}. 8vo. 10s.\ 6d.\ net.

\adnorm\begin{large}Latin Historical Inscriptions,\end{large} illustrating the history of the Early
Empire. By \textsc{G.~M\textsuperscript{c}N. Rushforth}. 8vo. 10s.\ net.

\adnorm\begin{large}Sources for Greek History\end{large} between the Persian and Peloponnesian Wars. By \textsc{G.~F. Hill}. 8vo. Reissue, revised. 10s.\ 6d.\ net.

\adnorm\begin{large}Sources for Roman History,\end{large} \textsc{b.c.} 133-70. By \textsc{A.~H.~J. Greenidge}
and \textsc{A.~M. Clay}. Crown 8vo. 5s.\ 6d.\ net.

\adnorm\begin{large}A Manual of Ancient History.\end{large} By \textsc{G. Rawlinson}. 2nd ed. 8vo. 14s.

\adnorm\begin{large}Finlay's History of Greece\end{large} from its Conquest by the Romans (\textsc{b.c.}146)
to \textsc{a.d.} 1864. A new edition, revised, and in part re-written, with many
additions, by the Author, and edited by \textsc{H.~F. Tozer}. 7 vols. 8vo. 63s.\ net.

\adnorm\begin{large}The History of Sicily\end{large} from the earliest times. By \textsc{E.~A. Freeman}. 8vo.\\
Vols.\ I and II\@. The Native Nations: The Phoenician and Greek Settlements
to the beginning of Athenian Intervention. £2 2s.\ net.\\
Vol. III\@. The Athenian and Carthaginian Invasions. £1 4s.\ net.\\
Vol. IV\@. From the Tyranny of Dionysios to the Death of Agathokl\^es.
Edited from posthumous MSS, by \textsc{A.~J. Evans}. £1 1s.\ net.

\adnorm\begin{large}Italy and her Invaders\end{large} (\textsc{a.d.} 376-814).
With plates and maps. Eight volumes. 8vo. By \textsc{T. Hodgkin}. Vols. I--IV in the second edition.\\
I--II\@. The Visigothic, Hunnish, and Vandal Invasions, and the Herulian
Mutiny. £2 2s.\\
III--IV\@. The Ostrogothic Invasion. The Imperial Restoration. £1 16s.\\
V--VI\@. The Lombard Invasion, and the Lombard Kingdom. £1 16s.\\
VII--VIII\@. Frankish Invasions, and the Frankish Empire. £1 4s.

\adnorm\begin{large}The Dynasty of Theodosius;\end{large} or, Seventy Years' Struggle with the Barbarians. By the same author. Crown 8vo. 6s.

\adnorm\begin{large}Aetolia; its Geography, Topography, and Antiquities.\end{large} By \textsc{W.~J. Woodhouse}. With maps and illustrations. Royal 8vo. £1 1s.\ net.

\adnorm\begin{large}The Islands of the Aegean.\end{large} By \textsc{H.~F. Tozer}. Crown 8vo. 8s.\ 6d.

\adnorm\begin{large}Dalmatia, the Quarnero, and Istria;\end{large} with Cettigne and Grado. By \textsc{T.~G. Jackson}. Three volumes. With plates and illustrations. 8vo. 31s.\ 6d.\ net.

\adnorm\begin{large}Cramer's Description of Asia Minor.\end{large} Two volumes. 8vo. 11s.

\adnorm\begin{large}\phantom{Cramer's }Description of Ancient Greece.\end{large} 3 vols. 8vo. 16s.\ 6d.

%[File: 295.png]
\adnorm\begin{large}The Cities and Bishoprics of Phrygia.\end{large} By \textsc{W.~M. Ramsay}.
Royal 8vo. Vol. I, Part I\@. The Lycos Valley and South-Western Phrygia.
18s.\ net. Vol. I, Par. II\@. West and West Central Phrygia. £1 1s.\ net.

\adnorm\begin{large}Stories of the High Priests of Memphis,\end{large} the Sethon of
Herodotus, and the Demotic Tales of Khamnas. By \textsc{F.~Ll.\ Griffith}. With
Portfolio containing seven facsimiles. Royal 8vo. £2 7s.\ 6d.\ net.

\adnorm\begin{large}The Arab Conquest of Egypt.\end{large} By \textsc{A.~J. Butler}. With maps and plans. 8vo. 16s.\ net.

\adnorm\begin{large}Baghdad during the Abbasid Caliphate,\end{large} from contemporary
sources. By \textsc{G. Le Strange}. With eight plans. 8vo. 10s.\ net.

\newpage

\medskip\begin{center}
{\huge Archaeology}
\end{center}

\adless\begin{large}Ancient Khotan.\end{large}  Detailed report of Archaeological explorations
in Chinese Turkestan carried out and described under the orders of H.M.
Indian Government by \textsc{M. Aurel Stein}. Vol.\ I\@. Text, with descriptive list
of antiques, seventy-two illustrations in the text, and appendices. Vol. II.\
One hundred and nineteen collotype and other illustrations and a map.
2 vols. 4to. £5 5s.\ net.

\adnorm\begin{large}Catalogue Of the Coins\end{large} in the Indian Museum, Calcutta, including the Cabinet of the Asiatic Society of Bengal. (Published for the Trustees of
the Indian Museum.) Royal 8vo, with numerous collotype plates. Vol. I,
by \textsc{V.~A. Smith}, 30s.\ net; or Part I (Early Foreign Dynasties and Guptas),
15s.\ net, Part II (Ancient Coins of Indian Types), 6s.\ net. Part III (Persian,
Mediaeval, South Indian, Miscellaneous), 10s.\ 6d.\ net. Vol. II, by \textsc{H.~N.\
Wright} (the first section of Part II by Sir \textsc{J. Bourdillon}), 30s.\ net (Sultáns
of Delhí, Contemporary Dynasties in India). Vol. III, by \textsc{H.~N. Wright},
40s.\ net (Mughal Emperors).

\adnorm\begin{large}Ancient Coptic Churches of Egypt.\end{large} By \textsc{A.~J. Butler}. 2 vols.
8vo. 30s.

\adnorm\begin{large}A Catalogue of the Cyprus Museum.\end{large} By \textsc{J.~L. Myres} and
\textsc{Max Ohnefalsch-Richter}. 8vo. With eight plates, 7s.\ 6d.\ net.

\adnorm\begin{large}A Catalogue of the Sparta Museum.\end{large}  By \textsc{M.~N. Tod} and
\textsc{A.~J.~B. Wace}. 8vo. 10s.\ 6d.\ net.

\adnorm\begin{large}Catalogue of the Greek Vases in the Ashmolean Museum.\end{large} By \textsc{P. Gardner}. Small folio, linen, with 26 plates. £3 3s.\ net.

\adnorm\begin{large}The Cults of the Greek States.\end{large} By \textsc{L.~R. Farnell}. 8vo.
Vols.\ I and II, with 61 plates and over 100 illustrations. £1 12s.\ net:
Vols.\ III and IV, with 86 plates. £1 12s.\ net.

\adnorm\begin{large}Classical Archaeology in Schools.\end{large} By \textsc{P. Gardner} and \textsc{J.~L. Myres}. 8vo. Second edition. Paper covers, 1s.\ net.

\adnorm\begin{large}Introduction to Greek Sculpture.\end{large} By \textsc{L.~E. Upcott}. Second
edition. Crown 8vo. 4s.\ 6d.

\adnorm\begin{large}Marmora Oxoniensia,\end{large} inscriptiones Graecae ad Chandleri exempla editae, cur.\ \textsc{Gul. Roberts}, 1791. Crown 8vo. 3s.\ 6d.

\adnorm\begin{large}De Antiquis Marmoribus,\end{large} Blasii Caryophili. 1828. 7s.\ 6d.

\adnorm\begin{large}Fragmenta Herculanensia.\end{large} A Catalogue of the Oxford copies of the Herculanean Rolls, with texts of several papyri. By \textsc{W. Scott}. Royal 8vo. £1 1s.
Thirty-six Engravings of Texts and Alphabets from the Herculanean Fragments. Folio. Small paper, 10s.\ 6d., large paper, £1 1s.

\adnorm\begin{large}Herculanensium Voluminum\end{large} Partes II\@. 1824. 8vo. 10s.

\newpage

%[File: 296.png]
\medskip\begin{center}
{\huge English History: Sources}
\end{center}

\adnorm\begin{large}Baedae Opera Historica,\end{large} edited by \textsc{C. Plummer}.  Two volumes. Crown 8vo, leather back. £1 1s.\ net.

\adnorm\begin{large}Asser's Life of Alfred, with the Annals of St.\ Neot,\end{large}
edited by \textsc{W.~H. Stevenson}. Crown 8vo. 12s.\ net.

\adnorm\begin{large}The Alfred Jewel,\end{large} an historical essay. With illustrations and a map, by \textsc{J. Earle}. Small 4to, buckram. 12s.\ 6d.\ net.

\adnorm\begin{large}Two of the Saxon Chronicles Parallel;\end{large}
with supplementary
extracts from the others. A Revised Text, edited, with introduction, notes,
appendices, and glossary, by \textsc{C. Plummer} and \textsc{J. Earle}. Two volumes.
Crown 8vo, leather back. Vol.\ I\@. Text, appendices, and glossary. 10s.\ 6d.
Vol.\ II\@. Introduction, notes, and index. 12s.\ 6d.

\adnorm\begin{large}The Saxon Chronicles\end{large} (787-1001 \textsc{a.d.}). Crown 8vo, stiff covers. 3s.

\adnorm\begin{large}Handbook to the Land-Charters,\end{large} and other Saxonic Documents, by \textsc{J. Earle}. Crown 8vo. 16s.

\adnorm\begin{large}The Crawford Collection\end{large} of early Charters and Documents, now in the Bodleian Library. Edited by \textsc{A.~S. Napier} and \textsc{W.~H. Stevenson}.
Small 4to, cloth. 12s.\ net.

\adnorm\begin{large}The Chronicle of John of Worcester,\end{large} 1118-1140. Edited by
\textsc{J.~R.~H. Weaver}. Crown 4to. 7s.\ 6d.\ net.

\adnorm\begin{large}Dialogus de Scaccario.\end{large} Edited by \textsc{A. Hughes, C.~G. Crump}, and \textsc{C. Johnson}, with introduction and notes. 8vo. 12s.\ 6d.\ net.

\adnorm\begin{large}Passio et Miracula Beati Olaui.\end{large} Edited from the Twelfth-century
MS by \textsc{F. Metcalfe}. Small 4to. 6s.

\adnorm\begin{large}The Song of Lewes.\end{large} Edited from the MS, with introduction and
notes, by \textsc{C.~L. Kingsford}. Extra fcap 8vo. 5s.

\adnorm\begin{large}Chronicon Galfridi le Baker de Swynebroke,\end{large} edited by Sir
\textsc{E. Maunde Thompson, K.C.B.} Small 4to, 18s.; cloth, gilt top, £1 1s.

\adnorm\begin{large}Chronicles of London.\end{large} Edited, with introduction and notes, by \textsc{C.~L. Kingsford}. 8vo. 10s.\ 6d.\ net.

\adnorm\begin{large}Gascoigne's Theological Dictionary\end{large} (`Liber Veritatum'): selected passages, illustrating the condition of Church and State, 1403-1458. With
an introduction by \textsc{J.~E. Thorold Rogers}. Small 4to. 10s.\ 6d.

\adnorm\begin{large}Fortescue's Governance of England.\end{large} A revised text, edited,
with introduction, etc, by \textsc{C. Plummer}. 8vo, leather back. 12s.\ 6d.

\adnorm\begin{large}Stow's Survey of London.\end{large} Edited by \textsc{C.~L. Kingsford}. 8vo, 2 vols., with a folding map of London in 1600 (by \textsc{Emery Walker} and \textsc{H.~W. Cribb}) and other illustrations. 30s.\ net.

\adnorm\begin{large}The Protests of the Lords,\end{large} from 1624 to 1874; with introductions. By \textsc{J.~E. Thorold Rogers}. In three volumes. 8vo. £2 2s.

%[File: 297.png]
\begin{center}
\adnorm\begin{large}The Clarendon Press Series of Charters,
Statutes, etc\end{large}
\medskip\\
From the earliest times to 1307. By Bishop \textsc{Stubbs}.
\end{center}\vspace{-0.12in}

\adless\begin{large}Select Charters\end{large} and other illustrations of English Constitutional History. Eighth edition. Crown 8vo. 8s.\ 6d.

\begin{center}From 1558 to 1625. By \textsc{G.~W. Prothero}.\end{center}\vspace{-0.12in}

\adless\begin{large}Select Statutes and other Constitutional Documents of the Reigns of Elizabeth and James I.\end{large} Third edition. Crown 8vo. 10s.\ 6d.

\begin{center}From 1625 to 1660. By \textsc{S.~R. Gardiner}.\end{center}\vspace{-0.12in}

\adless\begin{large}The Constitutional Documents of the Puritan Revolution.\end{large} Third edition. Crown 8vo. 10s.\ 6d.

\smallskip\begin{center}\begin{large}Calendars, etc\end{large}\end{center}

\adnorm\begin{large}Calendar of Charters and Rolls\end{large} preserved in the Bodleian Library. 8vo. £1 11s.\ 6d.

\adnorm\begin{large}Calendar of the Clarendon State Papers\end{large} preserved in the
Bodleian Library. In three volumes. 1869-76. Vol.\ I\@. From 1523 to January 1649. 8vo. 18s. Vol. II\@. From 1649 to 1654. 8vo. 16s. Vol. III\@. From 1655 to 1657. 8vo. 14s.

\midrule

\adnorm\begin{large}Hakluyt's Principal Navigations,\end{large} being narratives of the Voyages of the Elizabethan Seamen to America. Selection edited by \textsc{E.~J. Payne}.
Crown 8vo, with portraits. Second edition. Two volumes. 5s.\ each.\\
\indent Also abridged, in one volume, with additional notes, maps, \&c., by \textsc{C. Raymond Beazley}. Crown 8vo, with illustrations.  4s.\ 6d.\ Also, separately, The Voyages of Hawkins, Frobisher, and Drake; The Voyages of Drake and Gilbert, each 2s.\ 6d.

\adnorm\begin{large}Aubrey's `Brief Lives,'\end{large} set down between the Years 1669 and 1696. Edited from the Author's MSS by \textsc{A. Clark}. Two volumes. 8vo. £1 5s.

\adnorm\begin{large}Whitelock's Memorials\end{large} of English Affairs from 1625 to 1660. 4 vols, 8vo. £1 10s.

\adnorm\begin{large}Ludlow's Memoirs,\end{large} 1625-1672. Edited, with Appendices of Letters and illustrative documents, by \textsc{C.~H. Firth}. Two volumes. 8vo. £1 16s.

\adnorm\begin{large}Luttrell's Diary.\end{large} A brief Historical Relation of State Affairs, 1678-1714. Six volumes. 8vo. £1 10s.\ net.

\adnorm\begin{large}Burnet's History of James II.\end{large} 8vo. 9s.\ 6d.

\adnorm\begin{large}\phantom{Burnet's }Life of Sir M. Hale,\end{large} with \begin{large}Fell's Life of Dr.\ Hammond.\end{large} Small 8vo. 2s.\ 6d.

\adnorm\begin{large}\phantom{Burnet's }Memoirs of James and William, Dukes of
Hamilton.\end{large} 8vo. 7s.\ 6d.

%[File: 298.png]
\adnorm\begin{large}Burnet's History of My Own Time\end{large} A new edition based on that of \textsc{M.~J. Routh}. Edited by \textsc{Osmund Airy}. Vol.\ I\@. 12s.\ 6d.\ net.
Vol. II\@. (Completing Charles the Second, with Index to Vols.\ I and II.)
12s.\ 6d.\ net.

\smallskip\begin{large}Supplement,\end{large} derived from Burnet's Memoirs, Autobiography, etc, all hitherto unpublished. Edited by \textsc{H.~C. Foxcroft}, 1902. 8vo. 16s.\ net.

\adnorm\begin{large}The Whitefoord Papers,\end{large} 1739 to 1810.
Ed.\ by \textsc{W.~A.~S. Hewins}. 8vo. 12s.\ 6d.

\medskip\begin{center}\begin{large}History of Oxford\end{large}\end{center}

\begin{center}A complete list of the Publications of the Oxford Historical Society
can be obtained from Mr.\ Frowde.\end{center}

\adnorm\begin{large}Manuscript Materials relating to the History of Oxford;\end{large}
contained in the printed catalogues of the Bodleian and College Libraries.
By \textsc{F. Madan}. 8vo. 7s.\ 6d.

\adnorm\begin{large}The Early Oxford Press.\end{large} A Bibliography of Printing and Publishing at Oxford, `1468'-1640. With notes, appendices, and illustrations. By
\textsc{F. Madan}. 8vo. 18s.

\begin{center}\begin{large}Bibliography\end{large}\end{center}

\adnorm\begin{large}Cotton's Typographical Gazetteer.\end{large} First Series. 8vo. 12s.\ 6d.

\adnorm\begin{large}Ebert's Bibliographical Dictionary.\end{large} 4 vols. 8vo. £3 3s.\ net.

\bigskip\begin{center}\begin{large}Bishop Stubbs's and Professor Freeman's Books\end{large}\end{center}

\adnorm\begin{large}The Constitutional History of England,\end{large} in its Origin and
Development. By \textsc{W. Stubbs}. Library edition. Three volumes. Demy
8vo. £2 8s. Also in three volumes, crown 8vo, price 12s.\ each.

\adnorm\begin{large}Seventeen Lectures\end{large} on the Study of Mediaeval and Modern History and kindred subjects, 1867-1884. By the same. Third edition, revised and
enlarged, 1900. Crown 8vo, half-roan. 8s.\ 6d.

\adnorm\begin{large}History of the Norman Conquest of England;\end{large} its Causes
and Results. By \textsc{E.~A. Freeman}. Vols. I, II and V (English edition) are
out of print.

Vols. III and IV\@. £1 1s.\ each. Vol.\ VI (Index). 10s.\ 6d.

\adnorm\begin{large}A Short History of the Norman Conquest of England.\end{large}
Third edition. By the same. Extra fcap 8vo. 2s.\ 6d.

\adnorm\begin{large}The Reign of William Rufus\end{large} and the Accession of Henry the First. By the same. Two volumes. 8vo. £1 16s.

%[File: 299.png]

\newpage\begin{center}\begin{large}Special Periods and
Biographies\end{large}\end{center}

\adnorm\begin{large}Ancient Britain and the Invasions of Julius Caesar.\end{large} By
\textsc{T. Rice Holmes.} 8vo. 21s.\ net.

\adnorm\begin{large}Life and Times of Alfred the Great,\end{large}
being the Ford Lectures
for 1901. By \textsc{C. Plummer}. 8vo. 5s.\ net.

\adnorm\begin{large}The Domesday Boroughs.\end{large} By \textsc{Adolphus Ballard.} 8vo. 6s.\ 6d.\ net.

\adnorm\begin{large}Villainage in England.\end{large}  Essays in English Mediaeval History. By
\textsc{P. Vinogradoff.} 8vo. 16s.\ net.

\adnorm\begin{large}English Society in the Eleventh Century.\end{large} Essays in
English Mediaeval History. By \textsc{P. Vinogradoff.} 8vo. 16s.\ net.

\adnorm\begin{large}The Gild Merchant:\end{large} a contribution to British municipal history. By
\textsc{C. Gross.} Two volumes. 8vo, leather back, £1 4s.

\adnorm\begin{large}The Welsh Wars Of Edward I;\end{large} a contribution to mediaeval
military history. By \textsc{J.~E. Morris.} 8vo. 9s.\ 6d.\ net.

\adnorm\begin{large}The Great Revolt of 1381.\end{large} By \textsc{C. Oman.} With two maps. 8vo.
8s.\ 6d.\ net.

\adnorm\begin{large}Lancaster and York.\end{large} (\textsc{a.d.} 1399-1485.) By Sir \textsc{J.~H. Ramsay.} Two
volumes. 8vo, with Index, £1 17s.\ 6d.\ Index separately, 1s.\ 6d.

\adnorm\begin{large}Life and Letters of Thomas Cromwell.\end{large} By \textsc{R.~B. Merriman.}
In two volumes. [Vol.\ I, Life and Letters, 1523-1535, etc. Vol. II, Letters,
1536-1540, notes, index, etc.] 8vo. 18s.\ net.

\adnorm\begin{large}Edward Hyde, Earl of Clarendon,\end{large} A lecture by \textsc{C.~H. Firth.}
8vo. 1s.\ net.

\adnorm\begin{large}A History of England,\end{large} principally in the Seventeenth Century. By
\textsc{L.~von Ranke.} Translated under the superintendence of \textsc{G.~W. Kitchin}
and \textsc{C.~W. Boase.} Six volumes. 8vo. £3 3s.\ net. Index separately, 1s.

\adnorm\begin{large}Sir Walter Ralegh,\end{large} a Biography, by \textsc{W. Stebbing.} Post 8vo. 6s.\ net.

\adnorm\begin{large}Henry Birkhead\end{large} and the foundation of the Oxford Chair of Poetry.
By \textsc{J.~W. Mackail.} 8vo. 1s.\ net.

\adnorm\begin{large}Biographical Memoir of Dr.\ William Markham,\end{large} Archbishop
of York, by Sir \textsc{Clements Markham,} K.C.B\@. 8vo. 5s.\ net.

\adnorm\begin{large}The Life and Works of John Arbuthnot.\end{large} By \textsc{G.~A. Aitken.}
8vo, cloth extra, with Portrait. 16s.

%[File: 300.png]
\adnorm\begin{large}Life and Letters of Sir Henry Wotton.\end{large} By \textsc{L. Pearsall
Smith}. 8vo. Two volumes. 25s.\ net.


\adnorm\begin{large}Great Britain and Hanover.\end{large} By \textsc{A.~W. Ward}. Crown 8vo. 5s.


\adnorm\begin{large}History of the Peninsular War.\end{large} By \textsc{C. Oman}. To be completed
in six volumes, 8vo, with many maps, plans, and portraits.

Already published: Vol.\ I\@. 1807-1809, to Corunna. Vol. II\@. 1809, to
Talavera. Vol. III\@. 1809-10, to Torres Vedras. 11s.\ net each.


\adnorm\begin{large}Anglo-Chinese Commerce and Diplomacy:\end{large} mainly in the
nineteenth century. By \textsc{A.~J. Sargent}. 12s.\ 6d.\ net.


\adnorm\begin{large}Frederick York Powell.\end{large}  A Life and a selection from his Letters
and Occasional Writings. By \textsc{Oliver Elton}. Two volumes. 8vo. With
photogravure portraits, facsimiles, etc. 21s.\ net.


\adnorm\begin{large}David Binning Monro:\end{large} a Short Memoir.  By \textsc{J. Cook Wilson}.
8vo, stiff boards, with portrait. 2s.\ net.


\adnorm\begin{large}F.~W. Maitland.\end{large}  Two lectures by \textsc{A.~L. Smith}.  8vo. 2s.\ 6d.\ net.




\bigskip\begin{center}\begin{large}School Books\end{large}\end{center}


\adnorm\begin{large}Companion to English History\end{large} (Middle Ages).  Edited by \textsc{F.~P. Barn\-ard}.
With 97 illustrations. Crown 8vo. 8s.\ 6d.\ net.


\adnorm\begin{large}School History of England\end{large} to the death of Victoria. With maps,
plans, etc. By \textsc{O.~M. Edwards}, \textsc{R.~S. Rait}, and others. Crown 8vo, 3s.\ 6d.




\begin{center}\begin{large}Oxford School Histories\end{large}\end{center}

\begin{center}Crown 8vo, with many illustrations, each 1s.\ 6d.\ net.\end{center}


\adnorm\begin{large}Berkshire,\end{large} by \textsc{E.~A.~G. Lamborn}.


\adnorm\begin{large}Oxfordshire,\end{large} by \textsc{H.~A. Liddell}.

\begin{center}Others in preparation.\end{center}


\begin{center}Also, for junior pupils, illustrated, each 1s.\end{center}

\adnorm\begin{large}Stories from the History of Berkshire.\end{large}  By \textsc{E.~A.~G. Lamborn}.


\adnorm\begin{large}Stories from the History of Oxfordshire.\end{large} By \textsc{John Irving}.


\newpage

%[File: 301.png]
\bigskip\begin{center}\begin{huge}History and Geography of America
and the British Colonies\end{huge}\end{center}

\begin{center}For other Geographical books, see pages 59, 60.\end{center}


\adnorm\begin{large}History of the New World called America.\end{large} By \textsc{E.~J. Payne}.\\
Vol.\ I\@. 8vo. 18s. Bk.\ I\@. The Discovery. Bk.\ II, Part I\@. Aboriginal America.\\
Vol.\ II\@. 8vo. 14s. Bk.\ II, Part II\@. Aboriginal America (concluded).



\adnorm\begin{large}A History of Canada, 1763-1812.\end{large} By Sir \textsc{C.~P. Lucas, K.C.M.G.}
8vo. With eight maps. 12s.\ 6d.\ net.


\adnorm\begin{large}The Canadian War of 1812.\end{large} By Sir \textsc{C.~P. Lucas, K.C.M.G.} 8vo.
With eight maps. 12s.\ 6d.\ net.


\adnorm\begin{large}Historical Geography of the British Colonies.\end{large} By Sir \textsc{C.~P. Lucas, K.C.M.G.}
Crown 8vo.

\smallskip\begin{large}Introduction.\end{large} New edition by \textsc{H.~E. Egerton.} 1903. (Origin and
growth of the Colonies.) With eight maps. 3s.\ 6d.\ In cheaper binding,
2s.\ 6d.

\smallskip\begin{large}Vol.\ I\@. The Mediterranean and Eastern Colonies.\end{large}
With 13 maps. Second edition, revised and brought up to date, by
\textsc{R.~E. Stubbs.} 1906. 5s.

\smallskip\begin{large}Vol.\ II\@.  The West Indian Colonies.\end{large}  With twelve
maps. Second edition, revised and brought up to date, by \textsc{C. Atchley},
I.S.O\@. 1905. 7s.\ 6d.

\smallskip\begin{large}Vol.\ III\@.  West Africa.\end{large}  Second Edition.  Revised to the
end of 1899 by \textsc{H.~E. Egerton.} With five maps. 7s.\ 6d.

\smallskip\begin{large}Vol.\ IV\@. South and East Africa.\end{large} Historical and Geographical.
With eleven maps. 9s.\ 6d.\\
Also Part I\@. Historical. 1898. 6s.\ 6d.\ Part II\@. 1903. Geographical.
3s.\ 6d.

\smallskip\begin{large}Vol.\ V\@. Canada,\end{large} Part I\@. 1901. 6s. Part II, by \textsc{H.~E. Egerton.}
4s.\ 6d.\ Part III (Geographical) in preparation.

\smallskip\begin{large}Vol.\ VI\@. Australasia.\end{large} By \textsc{J.~D. Rogers.} 1907. With 22 maps.
7s.\ 6d.\ Also Part I, Historical, 4s.\ 6d.\ Part II, Geographical, 3s.\ 6d.

\adless History of the Dominion of Canada. By \textsc{W.~P. Greswell.} Crown 8vo. 7s.\ 6d,

\adless Geography of the Dominion of Canada and Newfoundland. By the same author.
With ten maps. 1891. Crown 8vo. 6s.

\adless Geography of Africa South of the Zambesi. By the same author. With maps.
1892. Crown 8vo. 7s.\ 6d.


\adnorm\begin{large}The Claims of the Study of Colonial History upon the
attention of the University of Oxford.\end{large} An inaugural lecture
delivered on April 28, 1906, by \textsc{H.~E. Egerton.} 8vo, paper covers, 1s.\ net.


\adnorm\begin{large}Historical Atlas.\end{large} Europe and her Colonies, 27 maps. 35s.\ net.


\adnorm\begin{large}Cornewall-Lewis's Essay on the Government of Dependencies.\end{large}
Edited by Sir \textsc{C.~P. Lucas, K.C.M.G.} 8vo, quarter-bound, 14s.

\newpage

%[File: 302.png]
\medskip\begin{center}\begin{large}Rulers of India\end{large}\end{center}

Edited by Sir \textsc{W.~W. Hunter.} Crown 8vo. 2s.\ 6d.\ each.

\adnorm\begin{large}Asoka.\end{large}  By \textsc{V.~A. Smith.}

\adnorm\begin{large}Bábar.\end{large}  By \textsc{S. Lane-Poole.}

\adnorm\begin{large}Albuquerque.\end{large}  By \textsc{H. Morse Stephens.}

\adnorm\begin{large}Akbar.\end{large}  By Colonel \textsc{Malleson.}

\adnorm\begin{large}Aurangzíb.\end{large}  By \textsc{S. Lane-Poole.}

\adnorm\begin{large}Dupleix.\end{large}  By Colonel \textsc{Malleson.}

\adnorm\begin{large}Lord Clive.\end{large}  By Colonel \textsc{Malleson.}

\adnorm\begin{large}Warren Hastings.\end{large}  By Captain \textsc{L.~J. Trotter.}

\adnorm\begin{large}Mádhava Ráo Sindhia.\end{large}  By \textsc{H.~G. Keene.}

\adnorm\begin{large}The Marquis of Cornwallis.\end{large} By \textsc{W.~S. Seton-Karr.}

\adnorm\begin{large}Haidar Alí and Tipú Sultán.\end{large} By \textsc{L.~B. Bowring.}

\adnorm\begin{large}The Marquis Wellesley, K.G.\end{large} By \textsc{W.~H. Hutton.}

\adnorm\begin{large}Marquess of Hastings.\end{large}  By Major \textsc{Ross-of-Bladensburg.}

\adnorm\begin{large}Mountstuart Elphinstone.\end{large} By \textsc{J.~S. Cotton.}

\adnorm\begin{large}Sir Thomas Munro.\end{large} By \textsc{J. Bradshaw.}

\adnorm\begin{large}Earl Amherst.\end{large}  By \textsc{Anne T. Ritchie} and \textsc{R. Evans.}

\adnorm\begin{large}Lord William Bentinck.\end{large} By \textsc{D.~C. Boulger.}

\adnorm\begin{large}The Earl of Auckland.\end{large}  By Captain \textsc{L.~J. Trotter.}

\adnorm\begin{large}Viscount Hardinge.\end{large}  By his son, Viscount \textsc{Hardinge.}

\adnorm\begin{large}Ranjit Singh.\end{large}  By Sir \textsc{L. Griffin.}

\adnorm\begin{large}The Marquess of Dalhousie.\end{large} By Sir \textsc{W.~W. Hunter.}

\adnorm\begin{large}James Thomason.\end{large}  By Sir \textsc{R. Temple.}

\adnorm\begin{large}John Russell Colvin.\end{large} By Sir \textsc{A. Colvin.}

\adnorm\begin{large}Sir Henry Lawrence.\end{large} By Lieut.-General \textsc{J.~J.~M\textsuperscript{c}Leod Innes.}

\adnorm\begin{large}Clyde and Strathnairn.\end{large} By Major-General Sir \textsc{O.~T. Burne.}

\adnorm\begin{large}Earl Canning.\end{large}  By Sir \textsc{H.~S. Cunningham.}

\adnorm\begin{large}Lord Lawrence.\end{large} By Sir \textsc{C. Aitchison.}

\adnorm\begin{large}The Earl of Mayo.\end{large}  By Sir \textsc{W.~W. Hunter.}


\adnorm\begin{large}Sketches of Rulers of India.\end{large} Abridged from the \emph{Rulers of India}
by \textsc{G.~D. Oswell.} Vol.\ I, The Mutiny and After; Vol.\ II, The Company's
Governors; Vol.\ III, The Governors-General; Vol.\ IV, The Princes of India.
Crown 8vo. 2s.\ net each.

%[File: 303.png]
\adnorm\begin{large}The Imperial Gazeteer of India.\end{large} New Edition. To be completed
in twenty-six volumes. 8vo. Subscription price, cloth, £5 net;
morocco back, £6 6s.\ net. The four volumes of `The Indian Empire'
separately 6s.\ net each, in cloth, or 7s.\ 6d.\ net with morocco back; the
Atlas separately 15s.\ net in cloth, or 17s.\ 6d.\ net with morocco back.
Subscriptions may be sent through any bookseller.

Vol.\ I\@. Descriptive.

Vol.\ II\@. Historical.

Vol.\ III\@. Economic.

Vol.\ IV\@. Administrative.

Vol.\ V-XXIV\@. Alphabetical Gazetteer.

Vol.\ XXV\@. Index.

Vol.\ XXVI\@. Atlas.\\
Each volume contains a map of India specially prepared for this Edition.


\begin{center}Reprints from the Imperial Gazetteer.\end{center}

\adless A sketch of the Flora of British India. By Sir \textsc{Joseph Hooker.} 8vo. Paper
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\adless The Indian Army. A sketch of its History and Organization. 8vo. Paper
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\adnorm\begin{large}A Brief History of the Indian Peoples.\end{large} By Sir \textsc{W.~W. Hunter.}
Revised up to 1903 by \textsc{W.~H. Hutton.} Eighty-ninth thousand. 3s.\ 6d.


\adnorm\begin{large}The Government of India,\end{large} being a digest of the Statute Law relating
thereto; with historical introduction and illustrative documents. By Sir
\textsc{C.~P. Ilbert.} Second edition, 1907. 10s.\ 6d.\ net.


\adnorm\begin{large}The Early History of India from 600 \textsc{b.c.} to the Muhammadan
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Second edition, revised and enlarged. 14s.\ net.


\adnorm\begin{large}The Oxford Student's History of India.\end{large} By \textsc{V.~A. Smith.}
Crown 8vo. With 7 maps and 10 other illustrations. 2s.\ 6d.


\adnorm\begin{large}The English Factories in India:\end{large} By \textsc{W. Foster.} 8vo. (Published
under the patronage of His Majesty's Secretary of State for India in Council.)

Vol.\ I\@. \begin{large}1618-1621.\end{large} 12s.\ 6d.\ n.\ Vol.\ II\@. \begin{large}1622-1623.\end{large} 12s.\ 6d.\ n.

(The six previous volumes of Letters received by the East India Company
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\adnorm\begin{large}Court Minutes of the East India Company, 1635-1639.\end{large}
By \textsc{E.~B. Sainsbury.} Introduction by \textsc{W. Foster.} 8vo. 12s.\ 6d.\ net.

The Court Minutes of the Company previous to 1635 have been calendared
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Record Office.


\adnorm\begin{large}Wellesley's Despatches, Treaties,\end{large} and other Papers relating to his
Government of India. Selection edited by \textsc{S.~J. Owen.} 8vo. £1 4s.


\adnorm\begin{large}Wellington's Despatches, Treaties,\end{large} and other Papers relating to
India. Selection edited by \textsc{S.~J. Owen.} 8vo. £1 4s.


\adnorm\begin{large}Hastings and the Rohilla War.\end{large} By Sir \textsc{J. Strachey.} 8vo. 10s.\ 6d.

\newpage

%[File: 304.png]
\medskip\begin{center}\begin{large}European History\end{large}\end{center}


\adnorm\begin{large}Historical Atlas of Modern Europe,\end{large} from the Decline of the
Roman Empire. 90 maps, with letterpress to each: the maps printed by
\textsc{W.~\&~A.~K. Johnston, Ltd.}, and the whole edited by \textsc{R.~L. Poole.}

In one volume, imperial 4to, half-persian, £5 15s.\ 6d.\ net; or in selected
sets--British Empire, etc, at various prices from 30s.\ to 35s.\ net each;
or in single maps, 1s.\ 6d.\ net each. Prospectus on application.


\adnorm\begin{large}Genealogical Tables\end{large} illustrative of Modern History.  By \textsc{H.~B. George.}
Fourth (1904) edition. Oblong 4to, boards. 7s.\ 6d.


\adnorm\begin{large}The Life and Times of James the First of Aragon.\end{large} By
\textsc{F.~D. Swift.} 8vo. 12s.\ 6d.


\adnorm\begin{large}The Renaissance and the Reformation.\end{large} A textbook of European
History, 1494-1610. By \textsc{E.~M. Tanner.} Crown 8vo, with 8 maps. 3s.\ 6d.


\adnorm\begin{large}A History of France,\end{large} with numerous maps, plans, and tables, by
\textsc{G.~W. Kitchin.} Crown 8vo; Vol.\ I (to 1453), revised by \textsc{F.~F. Urquhart};
Vols.\ II (1624), III (1795), revised by \textsc{A. Hassall.} 10s.\ 6d.\ each volume.


\adnorm\begin{large}De Tocqueville's L'Ancien R\'egime et la R\'evolution.\end{large}
Edited, with introductions and notes, by \textsc{G.~W. Headlam.} Crown 8vo. 6s.


\adnorm\begin{large}The Principal Speeches of the Statesmen and Orators
of the French Revolution,\end{large} 1789-1795. Ed.\ \textsc{H. Morse Stephens.} Two vols.
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\adnorm\begin{large}Documents of the French Revolution, 1789-1791.\end{large} By
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\adnorm\begin{large}Napoleonic Statesmanship: Germany.\end{large} By \textsc{H.~A.~L. Fisher.}
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\adnorm\begin{large}Bonapartism.\end{large}  Six lectures by \textsc{H.~A.~L. Fisher.}  8vo. 3s.\ 6d.\ net.


\adnorm\begin{large}Thiers' Moscow Expedition,\end{large} edited by \textsc{H.~B. George.}  Cr. 8vo,
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\adnorm\begin{large}Great Britain and Hanover.\end{large} By \textsc{A.~W. Ward.} Crown 8vo. 5s.


\adnorm\begin{large}History of the Peninsular War.\end{large} By \textsc{C. Oman.} To be completed
in six volumes, 8vo, with many maps, plans, and portraits.

Already published: Vol.\ I\@. 1807-1809, to Corunna. Vol.\ II\@. 1809, to
Talavera. Vol.\ III\@. 1809-10, to Torres Vedras.  14s.\ net each.



\newpage


\medskip\begin{center}\begin{huge}The Oxford Geographies\end{huge}\end{center}


\adnorm\begin{large}Relations of Geography and History.\end{large} By \textsc{H.~B. George.}
With two maps. Crown 8vo. Third edition. 4s.\ 6d.


\adnorm\begin{large}Geography for Schools,\end{large} by \textsc{A. Hughes.} Crown 8vo. 2s.\ 6d.


\adnorm\begin{large}The Oxford Geographies.\end{large}  By \textsc{A.~J. Herbertson.}  Crown 8vo.\\
Vol.\ I\@. The Preliminary Geography, Ed.\ 2, 72 maps and diagrams, 1s.\ 6d.\ \\
Vol.\ II\@. The Junior Geography, Ed.\ 2, 166 maps and diagrams, 2s. \\
Vol.\ III\@. The Senior Geography, Ed.\ 2, 117 maps and diagrams, 2s.\ 6d.



\adnorm\begin{large}Practical Geography.\end{large} By \textsc{J.~F. Unstead.} Crown 8vo. Part I,
27 maps and diagrams, Part II, 21 maps and diagrams, each 1s.\ 6d.;
together 2s.\ 6d.

\newpage

%[File: 305.png]
\medskip\begin{center}\begin{large}Geography and Anthropology\end{large}\end{center}


\adnorm\begin{large}The Dawn of Modern Geography.\end{large} By \textsc{C.~R. Beazley.} in three
volumes. £2 10s. Vol.\ I (to \textsc{a.d.} 900). Not sold separately. Vol.\ II (\textsc{a.d.}
900-1260). 15s.\ net. Vol.\ III\@. 20s.\ net.


\adnorm\begin{large}Regions of the World.\end{large} Geographical Memoirs under the general
editorship of \textsc{H.~J. Mackinder.} Medium 8vo. 7s.\ 6d.\ net per volume.

Britain and the British Seas. Second edition. By \textsc{H.~J. Mackinder.}---Central
Europe. By \textsc{John Partsch.}---The Nearer East. By \textsc{D.~G. Hogarth.}---North
America. By \textsc{J. Russell.}---India. By Sir
\textsc{Thomas Holdich.}---The Far East. By \textsc{Archibald Little.}


\adnorm\begin{large}Frontiers:\end{large} Romanes Lecture for 1907. By Lord \textsc{Curzon of Kedleston.} 8vo.
2s.\ net.


\adnorm\begin{large}The Face of the Earth.\end{large}  By \textsc{Eduard Suess.} See p.~92.

\midrule

\adnorm\begin{large}Transactions of the Third (1908) International Congress
for the History of Religions.\end{large} Royal 8vo. 2 vols. 21s.\ net.


\adnorm\begin{large}Anthropological Essays\end{large} presented to \textsc{Edward Burnett Tylor} in
honour of his seventy-fifth birthday; by \textsc{H. Balfour,} \textsc{A.~E. Crawley,}
\textsc{D.~J. Cunningham,} \textsc{L.~R. Farnell,} \textsc{J.~G. Frazer,} \textsc{A.~C. Haddon,} \textsc{E.~S. Hartland,}
\textsc{A. Lang,} \textsc{R.~R. Marett,} \textsc{C.~S. Myers,} \textsc{J.~L. Myres,} \textsc{C.~H. Read,}
Sir \textsc{J. Rh\^ys}, \textsc{W. Ridgeway,} \textsc{W.~H.~R. Rivers,} \textsc{C.~G. Seligmann}, \textsc{T.~A. Joyce},
\textsc{N.~W. Thomas}, \textsc{A. Thomson}, \textsc{E. Westermarck}; with a bibliography by
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\adnorm\begin{large}The Evolution of Culture, and other Essays,\end{large} by the late
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Introduction by \textsc{H. Balfour}. 8vo, with 21 plates, 7s.\ 6d.\ net.


\adnorm\begin{large}Anthropology and the Classics.\end{large} Six lectures by \textsc{A. Evans},
\textsc{A. Lang}, \textsc{G.~G.~A. Murray}, \textsc{F.~B. Jevons}, \textsc{J.~L. Myres}, \textsc{W.~W. Fowler}.
Edited by \textsc{R.~R. Marett}. 8vo. Illustrated. 6s.\ net.


\adnorm\begin{large}Folk-Memory.\end{large} By \textsc{Walter Johnson}. 8vo. Illustrated. 12s.\ 6d.\ net.


\adnorm\begin{large}Celtic Folklore: Welsh and Manx.\end{large} By \textsc{J. Rh\^ys}, 2 vols. 8vo. £1 1s.


\adnorm\begin{large}Studies in the Arthurian Legend.\end{large} By \textsc{J. Rh\^ys}. 8vo. 12s.\ 6d.


\adnorm\begin{large}Iceland and the Faroes.\end{large} By \textsc{N. Annandale}. With an appendix
on the Celtic Pony, by \textsc{F.~H.~A. Marshall}. Crown 8vo. 4s.\ 6d.\ net.


\adnorm\begin{large}Dubois' Hindu Manners.\end{large} Translated and edited by \textsc{H.~K. Beauchamp}.
Third edition. Crown 8vo. 6s.\ net. On India Paper, 7s.\ 6d.\ net.


\adnorm\begin{large}The Melanesians,\end{large} studies in their Anthropology and Folk-Lore. By
\textsc{R.~H. Codrington}. 8vo. 16s.\ net.


\adnorm\begin{large}The Masai, their Language and Folk-lore.\end{large} By \textsc{A.~C. Hollis}.
With introduction by Sir \textsc{Charles Eliot}. 8vo. Illustrated. 14s.\ net.


\adnorm\begin{large}The Nandi, their Language and Folk-lore.\end{large} By \textsc{A.~C. Hollis}.
With introduction by Sir \textsc{Charles Eliot}. 8vo. Illustrated. 16s.\ net.


\adnorm\begin{large}The Ancient Races Of the Thebaid:\end{large} an anthropometrical study.
By \textsc{Arthur Thomson} and \textsc{D. Randall-MacIver}.  Imperial 4to, with 6
collo-types, 6 lithographic charts, and many other illustrations. 42s.\ net.


\adnorm\begin{large}The Earliest Inhabitants of Abydos.\end{large} (A craniological study.)
By \textsc{D. Randall-MacIver}. Portfolio. 10s.\ 6d.\ net.

\newpage

%[File: 306.png]
\medskip\begin{center}\begin{huge}LAW\end{huge}\end{center}



\begin{center}\begin{large}Jurisprudence\end{large}\end{center}


\adnorm\begin{large}Bentham's Fragment on Government.\end{large}  Edited by \textsc{F.~C.
Montague}. 8vo. 7s.\ 6d.


\adnorm\begin{large}Bentham's Introduction to the Principles of Morals and
Legislation.\end{large}  Second edition. Crown 8vo. 6s.\ 6d.


\adnorm\begin{large}Studies in History and Jurisprudence.\end{large} By the Right Hon.
\textsc{James Bryce}. 1901. Two volumes. 8vo. £1 5s.\ net.


\adnorm\begin{large}The Elements of Jurisprudence.\end{large} By \textsc{T.~E. Holland}. Tenth
edition. 1906. 8vo. 10s.\ 6d.


\adnorm\begin{large}Elements of Law,\end{large} considered with reference to Principles of General
Jurisprudence. By Sir \textsc{W. Markby, K.C.I.E.} Sixth edition revised, 1905.
8vo. 12s.\ 6d.



\begin{center}\begin{large}Roman Law\end{large}\end{center}


\adnorm\begin{large}Imperatoris Iustiniani Institutionum Libri Quattuor;\end{large}
with introductions, commentary, and translation, by \textsc{J.~B. Moyle}. Two
volumes. 8vo. Vol.\ I (fourth edition, 1903), 16s.; Vol.\ II, Translation
(fourth edition, 1906), 6s.


\adnorm\begin{large}The Institutes Of Justinian,\end{large} edited as a recension of the Institutes
of Gaius. By \textsc{T.~E. Holland}. Second edition. Extra fcap 8vo. 5s.


\adnorm\begin{large}Select Titles from the Digest of Justinian.\end{large} By \textsc{T.~E. Holland}
and \textsc{C.~L. Shadwell}. 8vo. 14s.

Also, sold in parts, in paper covers: Part I\@. Introductory Titles. 2s.\ 6d.
Part II\@. Family Law. 1s. Part III\@. Property Law. 2s.\ 6d.\ Part IV.\
Law of Obligations. No. 1. 3s.\ 6d. No. 2. 4s.\ 6d.


\adnorm\begin{large}Gai Institutionum Iuris Civilis Commentarii Quattuor:\end{large}
with a translation and commentary by the late \textsc{E. Poste}. Fourth edition.
Revised and enlarged by \textsc{E.~A. Whittuck}, with an historical introduction
by \textsc{A.~H.~J. Greenidge}. 8vo. 16s.\ net.


\adnorm\begin{large}Institutes of Roman Law,\end{large} by \textsc{R. Sohm}. Translated by \textsc{J.~C.
Ledlie}: with an introductory essay by \textsc{E. Grueber}. Third edition.
8vo. 16s.\ net.


\adnorm\begin{large}Infamia;\end{large} its place in Roman Public and Private Law.  By \textsc{A.~H.~J.\
Greenidge}. 8vo. 10s.\ 6d.


\adnorm\begin{large}Legal Procedure in Cicero's Time.\end{large} By \textsc{A.~H.~J. Greenidge}.
8vo. 25s.\ net.


\adnorm\begin{large}The Roman Law of Damage to Property:\end{large} being a commentary
on the title of the Digest `Ad Legem Aquiliam' (ix.\ 2), with an introduction
to the study of the Corpus Iuris Civilis. By \textsc{E. Grueber}. 8vo. 10s.\ 6d.


\adnorm\begin{large}Contract of Sale in the Civil Law.\end{large} By \textsc{J.~B. Moyle}. 8vo. 10s.\ 6d.

\midrule

\adnorm\begin{large}The Principles of German Civil Law.\end{large} By \textsc{Ernest J. Schuster},
8vo. 12s.\ 6d.\ net.

%[File: 307.png]
\medskip\begin{center}\begin{large}English Law\end{large}\end{center}


\adless\begin{large}Principles of the English Law of Contract,\end{large} and of Agency in
its relation to Contract. By Sir \textsc{W.~R. Anson}. Eleventh edition. 1906. 8vo.
10s.\ 6d.


\adnorm\begin{large}Law and Custom of the Constitution.\end{large} By the same. In two
volumes. 8vo.

Vol.\ I\@. Parliament. (Out of print. New edition in the press.)

Vol.\ II\@. The Crown. Third edition. Part I, 10s.\ 6d.\ net. Part II,
8s.\ 6d.\ net.



\adnorm\begin{large}Introduction to the History of the Law of Real Property.\end{large}
By Sir \textsc{K.~E. Digby}. Fifth edition. 8vo. 12s.\ 6d.


\adnorm\begin{large}Legislative Methods and Forms.\end{large} By Sir \textsc{C.~P. Ilbert, K.C.S.I.}
1901. 8vo, leather back, 16s.


\adnorm\begin{large}Modern Land Law.\end{large} By \textsc{E. Jenks}. 8vo. 15s.


\adnorm\begin{large}Essay on Possession in the Common Law.\end{large} By Sir \textsc{F.
Pollock} and Sir \textsc{R.~S. Wright}. 8vo. 8s.\ 6d.


\adnorm\begin{large}Outline of the Law of Property.\end{large} By \textsc{T. Raleigh}. 8vo. 7s.\ 6d.


\adnorm\begin{large}Law in Daily Life.\end{large}  By \textsc{Rud.\ von Jhering}.  Translated with Notes
and Additions by \textsc{H. Goudy}. Crown 8vo. 3s.\ 6d.\ net.


\adnorm\begin{large}Cases illustrating the Principles of the Law of Torts,\end{large}
with table of all Cases cited. By \textsc{F.~R.~Y. Radcliffe} and \textsc{J.~C. Miles}. 8vo.
1904. 12s.\ 6d.\ net.


\adnorm\begin{large}The Management of Private Affairs.\end{large} By \textsc{Joseph King}, \textsc{F.~T.~R.
Bigham}, \textsc{M.~L. Gwyer}, \textsc{Edwin Cannan}, \textsc{J.~S.~C. Bridge}, \textsc{A.~M. Latter}.
Crown 8vo. 2s.\ 6d.\ net.


\midrule

\adless\begin{large}Calendar of Charters and Rolls,\end{large} containing those preserved in the
Bodleian Library. 8vo. £1 11s.\ 6d.


\adnorm\begin{large}Handbook to the Land-Charters,\end{large} and other Saxonic Documents.
By \textsc{J. Earle}. Crown 8vo. 16s.


\adnorm\begin{large}Fortescue's Difference between an Absolute and a Limited
Monarchy.\end{large}  Text revised and edited, with introduction, etc, by \textsc{C.
Plummer}. 8vo, leather back, 12s.\ 6d.


\adnorm\begin{large}Villainage in England.\end{large} By \textsc{P. Vinogradoff}. 8vo.  16s.\ net.


\adnorm\begin{large}Welsh Mediaeval Law:\end{large} the Laws of Howel the Good.  Text,
translation, etc., by \textsc{A.~W. Wade Evans}. Crown 8vo. 8s.\ 6d.\ net.



\newpage\begin{center}\begin{large}Constitutional Documents\end{large}\end{center}


\adnorm\begin{large}Select Charters\end{large} and other Illustrations of English Constitutional History,
from the earliest times to Edward I\@. Arranged and edited by \textsc{W. Stubbs}.
Eighth edition. 1900. Crown 8vo. 8s.\ 6d.


\adnorm\begin{large}Select Statutes and other Constitutional Documents,\end{large}
illustrative of the reigns of Elizabeth and James I\@.  Edited by \textsc{G.~W.\
Prothero}. Third edition. Crown 8vo. 10s.\ 6d.


\adnorm\begin{large}Constitutional Documents\end{large} of the Puritan Revolution, selected and
edited by \textsc{S.~R. Gardiner}. Third edition. Crown 8vo. 10s.\ 6d.

%[File: 308.png]
\medskip\begin{center}\begin{large}International Law\end{large}\end{center}


\adnorm\begin{large}International Law.\end{large}  By \textsc{W.~E. Hall}.
Fifth edition by \textsc{J.~B. Atlay}.
1904. 8vo. £1 1s.\ net.


\adnorm\begin{large}Treatise on the Foreign Powers and Jurisdiction of the
British Crown.\end{large}  By \textsc{W.~E. Hall}. 8vo. 10s.\ 6d.


\adnorm\begin{large}The European Concert in the Eastern Question,\end{large} a collection
of treaties and other public acts. Edited, with introductions and notes, by
\textsc{T.~E. Holland}. 8vo. 12s.\ 6d.


\adnorm\begin{large}Studies in International Law.\end{large} By \textsc{T.~E. Holland}. 8vo. 10s.\ 6d.


\adnorm\begin{large}The Laws of War on Land.\end{large}  By \textsc{T.~E. Holland}. 8vo. 6s.\ net.


\adnorm\begin{large}Gentilis Alberici de Iure Belli Libri Tres\end{large} edidit \textsc{T.~E.
Holland}. Small quarto, half-morocco. £1 1s.


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%[File: 310.png]

\newpage
\pagenumbering{roman}
\vspace*{1in} % make it different from the ads, which all start at the top of the page
\begin{center}\Large % make the heading a bit more noticeable
\textsc{Typographical Errors corrected\\in Project Gutenberg
edition}\end{center}

p.~\pageref{mainfold}.~`Theorem~12. The mainfold of all points' in text,
read `manifold'.\

p.~\pageref{onehalf}.~`two other points of one-half-line', the first hyphen removed as apparently spurious.\

p.~\pageref{bBAL}.~`$|AL_1$ belongs to the interior $\measuredangle b_1BAL_1{}'$' in text,
the last group is meaningless and was amended to `$\measuredangle BAL_1{}'$'.\

p.~\pageref{theorem41}.~Theorem 41, statement has `any half-plane bounded by $AB$ into any half-plane bounded
by $A'B$', the rest of this section requires `bounded by $A'B'$'. The overline on the final $\overline{O'C'}$
of the explanation is assumed, it is not visible on the scan.\

p.~\pageref{angleEDCF}.~`hence $\measuredangle EDCF$ is a rectangle' in text,
the $\measuredangle$ is evidently spurious, read `$EDCF$ is a rectangle'.\

p.~\pageref{beyondB2}.~`$B_2$ on the extension of $(C_1B_1)$ beyond $B_2$' in text,
obviously garbled, read `beyond $B_1$' (and see Fig. 2).\

p.~\pageref{zeta}.~`in place of $\xi$, $\eta$, $\zeta$' in text,
context requires `$\xi$, $\eta$, $\omega$'.\

p.~\pageref{ChapVI}.~Chapter VI heading `CONSISTENCY A SIGNIFICANCE OF THE AXIOMS' in text,
amended to `CONSISTENCY AND SIGNIFICANCE OF THE AXIOMS' as per the original ToC.\

p.~\pageref{y0}.~`the coordinates of whose points are of the form $\lambda y_0 + \mu x_i$' in text,
there is no $y_0$ in this discussion, read `$\lambda y_i + \mu x_i$'.\

p.~\pageref{x3}.~The first term of the transformation is given as `${x_0}'=f_0(x_0x_1x_2){x_3}'$',
this makes no sense so I have made it `${x_0}'=f_0(x_0x_1x_2x_3)$' in line with the other terms.\

p.~\pageref{sin}.~`${\dot x_0}'  =  \dot x_0 \cosh d + \dot x_1 \sin d$' in text,
but the context is hyperbolic, so read `$\cdots + \dot x_1 \sinh d$'.\

p.~\pageref{andoftheline}.~`and of the line $(x\sqrt{(yy)} - iy\sqrt{(xx)})(x'\sqrt{(y'y')} -iy'\sqrt{(xx)})$' in text,
read `$\cdots -iy'\sqrt{(x'x')})$' to conform with 2 previous expressions.\

p.~\pageref{inftyp}.~`the same cross has $\infty'$\ determinations.' in text,
taken to mean `$\infty^1$'.\

p.~\pageref{rXrX}.~Equation (2) first fraction numerator is `$(_rX_rX)$' in text,
evidently should be `$(_rX_rY)$'.\

p.~\pageref{theywill}.~The paragraph before Theorem 5 ends with the dangling words `They will'.
These have been removed.\

p.~\pageref{xhxl}.~In the equation after (11), the RHS was printed
\[
\frac{c_l(c_h-c_k)x_k{}^2 - c_k(c_l-c_h)x_l{}^2 }
       {\bigl[(c_h-c_k)x_k{}^2 - (c_l-c_h)x_h{}^2\bigr](c_l-c_k) }.\
\]
For homogeneity with the rest of this discussion the second denominator term needs
to be $(c_l-c_h)x_l{}^2$. Compare, in particular, the expression after (9), which must be
added to this one to give the simplification (12).\

p.~\pageref{whatsce}.~The equation before (22) was
\[
x_h  : x_k  : x_l =
  \sqrt{-c_k} c_e (\sqrt{-c_l} y_l \pm \sqrt{c_k} y_k)  :
  \mp c_h y_h \sqrt{-c_l}  :
  -c_h y_h \sqrt{c_k},
\]
there is no $c_e$ here, I have rather tentatively amended to $\sqrt{-c_k c_l}$ - note the
extended root sign. The overlines in (22) were not visible in the scan.\

p.~\pageref{processor}.~`a repetition of this process or division' in text,
taken to mean `process of division'.\

p.~\pageref{no2}.~Text has
\[
k^2 - \frac{(\Delta s)^2}{2} - k^2 + k^2(x''x'')\frac{\Delta s^2}{2} = \frac{k^2}{\rho^2}\frac{(\Delta s)^2}{}
\]
from the argument (back to equation (4)), last term reconstructed to $\dfrac{k^2}{\rho^2}\dfrac{(\Delta s)^2}{2}$.

p.~\pageref{partial}.~The equation before `In particular' was
\[ \frac{\partial x_i}{\partial u}du + \frac{\partial x_i}{\partial v} \partial v = \tan\frac{r}{k}\Bigl[ \frac{\partial y_i}{\partial u}du + \frac{\partial y_i}{\partial v}dv \Bigr].
\]
obviously the second term on the left is meant to be $\dfrac{\partial x_i}{\partial v} dv$.

p.~\pageref{spotthediff}. Spot the 4 differences! Text has
\begin{multline*}
  \frac{dt_i}{ds}
= k\left[
    \frac{\partial^2 x_i}{\partial u^2}\Bigl(\frac{du}{ds}\right)^2
 + 2\frac{\partial^2 x_i}{\partial u\,\partial v}
    \frac{du}{ds}\frac{dv}{ds}
 + \frac{\partial x_i}{\partial v^2}\left(\frac{dv}{ds}\Bigr)^2
   \right]
\\
+ k\Bigl[ \frac{\partial x_i}{du}\frac{d^2u}{ds^2}
        + \frac{\partial x_i}{dv}\frac{dv^2}{ds^2} \Bigr]
\end{multline*}
Corrected to
\begin{multline*}
  \frac{dt_i}{ds}
= k\left[
    \frac{\partial^2 x_i}{\partial u^2}\Bigl(\frac{du}{ds}\right)^2
 + 2\frac{\partial^2 x_i}{\partial u\,\partial v}
    \frac{du}{ds}\frac{dv}{ds}
 + \frac{\partial^2 x_i}{\partial v^2}\left(\frac{dv}{ds}\Bigr)^2
   \right]
\\
+ k\Bigl[ \frac{\partial x_i}{\partial u}\frac{d^2u}{ds^2}
        + \frac{\partial x_i}{\partial v}\frac{d^2v}{ds^2} \Bigr]
\end{multline*}

p.~\pageref{d3x}.~`Now \ldots by (31)' equation was printed:
\begin{equation*}
   \frac{\partial}{\partial v} \Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial^2 x}{\partial u^2} \Bigr) = 2 \Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial^3 x}{\partial u \, \partial v} \Bigr) = -2 \Bigl( \frac{\partial^2 x}{\partial u^2} \frac{\partial}{\partial u} (Fx) \Bigr) = 0,
\end{equation*}
the second term should be $2 \Bigl( \dfrac{\partial^2 x}{\partial u^2} \dfrac{\partial^3 x}{\partial u^2 \, \partial v} \Bigr)$.

\smallskip p.~\pageref{dudv}.~The equation before (4) was:
\[k^2\Bigl( \frac{\partial y}{\partial u}
            \frac{\partial y}{\partial v} \Bigr)
-    \Bigl(x\frac{\partial y}{\partial u} \Bigr)
            \Bigl(\frac{\partial y}{\partial v} \Bigr) = F',
\]
for consistency I take it to be $\Bigl(x\dfrac{\partial y}{\partial u} \Bigr)
            \Bigl(x\dfrac{\partial y}{\partial v} \Bigr)$.

\smallskip p.~\pageref{gprime}.~The equation before (9) was printed
\[
G' = \frac{1}{\Delta'^2} [G'f^2 - 2F'(fg) + E'g^2],
\]
to fit the pattern this should be $G = \ldots$

p.~\pageref{noplus}.~The plus sign in the second equation after `Let us now ask' was not printed.\

p.~\pageref{hyberbolic}.~Before (15) `in the hyberbolic case', read `hyperbolic'.

p.~\pageref{pointpair}.~The pair of limiting points are \emph{both} given as
$(\sqrt{a_1}+\sqrt{a_2}, 0, 0, \sqrt{a_1}-\sqrt{a_2})$. From the following discussion
it appears the intention is to negate the first element of the second one.\

p.~\pageref{z1prime}.~The first solution is given as
\[
\begin{aligned}
  u_1 &= u'_1, \qquad &  z_1 &= z'_1,   \\
  u_2 &= -\frac{1}{u'_1}, \qquad &  z_2 &= -\frac{1}{z'_2},
\end{aligned}
\]
but following the equations the last equality should be $z_2 = -\dfrac{1}{z'_1}$.\

%\newpage
p.~\pageref{xanyy}.~`If $A$ and $C$ have the coordinates ($x$) any ($y$) respectively' in text,
taken to read `($x$) and ($y$)'.\

p.~\pageref{PorB}.~Equation is printed
\[
  (P_0 P_\beta, P_\gamma B_\delta)
= (P_\infty P_{\frac{1}{\beta}},
   P_{\frac{1}{\gamma}} P_{\frac{1}{\delta}})
\]
- there are no $B$s in this discussion so switched to $(P_0 P_\beta, P_\gamma P_\delta)$. Similarly
just below in equation (1) $(P_\alpha B_\beta, P_\gamma P_\delta)$.\

p.~\pageref{dphidphi}.~$ds^2 = dr^2 + k^2\sin^2\dfrac{r}{k}
  [E'd\theta^2 + 2F'd\phi\, d\phi + d\phi^2]$ in text, corrected to $\ldots+2F'd\theta\, d\phi+\ldots$.\

p.~\pageref{oldint}.~`Hence $l$ is an old integer' corrected to `odd'.\

%Index:

%`Density of segment, 16' corrected to `18' (then repaginated as \pageref{Density}).\

%`Measure of curvature of space, 33' corrected to `53' (then repaginated as \pageref{Curve}).\

%`Pasch, 13' corrected to `19' (then repaginated as \pageref{Pasch}).\

%`Right and left generators of Absolute, 94' corrected to `99' (then repaginated as \pageref{rlgen}).\

\newpage
\small
\pagenumbering{Roman}
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% End of the Project Gutenberg EBook of The Elements of non-Euclidean Geometry, by 
% Julian Lowell Coolidge                                                  %
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% *** END OF THIS PROJECT GUTENBERG EBOOK NON-EUCLIDEAN GEOMETRY ***      %
%                                                                         %
% ***** This file should be named 26373-t.tex or 26373-t.zip *****        %
% This and all associated files of various formats will be found in:      %
%         http://www.gutenberg.org/2/6/3/7/26373/                         %
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