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+* text=auto
+*.txt text
+*.md text
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% 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 ***    %
+%                                                                         %
+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+
+\def\ebook{26373}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% The Elements of non-Euclidean Geometry, by Julian Lowell Coolidge     %%
+%%                                                                       %%
+%% Packages and substitutions:                                           %%
+%%                                                                       %%
+%% book   :   Document class.                                            %%
+%% amsmath:   Basic AMS math package.                                    %%
+%% amssymb:   Basic AMS symbols                                          %%
+%% babel:     Hyphenation                                                %%
+%% inputenc:  Encoding                                                   %%
+%% graphicx   Basic graphics for images.                                 %%
+%% makeidx    Support \makeindex \printindex                             %%
+%% wasasym    Additional symbols from wasy fonts                         %%
+%% verbatim   Preformated text                                           %%
+%%                                                                       %%
+%% PDF Pages: 282                                                        %%
+%%                                                                       %%
+%% 4 overfull hboxes (in the publisher's adverts)                        %%
+%%                                                                       %%
+%% 4 illustrations (f001.png through f004.png)                           %%
+%% are included as .png files in the images directory.                   %%
+%%                                                                       %%
+%% Compile sequence:                                                     %%
+%%         pdflatex x2                                                   %%
+%%         makeindex                                                     %%
+%%         pdflatex                                                      %%
+%%                                                                       %%
+%% Compile History:                                                      %%
+%%                                                                       %%
+%% Aug 08: Laverock. Compiled with pdflatex:                             %%
+%%         [pdfeTeX, Version 3.141592-1.30.6-2.2 (MiKTeX 2.5)]           %%
+%%                                                                       %%
+%%                                                                       %%
+%% August 2008: pglatex.                                                 %%
+%%   Compile this project with:                                          %%
+%%   pdflatex 26373-t.tex ..... TWO times                                %%
+%%   makeindex 26373-t.idx                                               %%
+%%   pdflatex 26373-t.tex                                                %%
+%%                                                                       %%
+%%   pdfeTeX, Version 3.141592-1.30.5-2.2 (Web2C 7.5.5)                  %%
+%%                                                                       %%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\documentclass[oneside]{book}
+\usepackage[reqno]{amsmath}[2000/07/18]		% basic math support
+\usepackage{amssymb}[2002/01/22]		% math symbols
+\usepackage[latin1]{inputenc}[2006/05/05]	% encoding system
+\usepackage{makeidx}[2000/03/29]		% support \makeindex \printindex
+\usepackage{verbatim}[2003/08/22]		% preformated text
+\usepackage{graphicx}[1999/02/16]		% basic graphics for images.
+\usepackage{wasysym}[2003/10/30]		% Additional symbols from wasy fonts
+\makeindex
+\allowdisplaybreaks[4]
+
+%separation operator based on upright integral from wasysym
+\newcommand\sep{\hspace{-0.2em}\displaystyle\varint\hspace{-0.2em}}
+\newcommand\nsep{\hspace{0.2em}\diagdown\hspace{-1.1em}\displaystyle\varint}
+
+%simulated chapter heading & toc command
+\newcommand\mychap[2]{
+\newpage
+\medskip\begin{center}{\Large CHAPTER #1}\par\bigskip{\large #2}\end{center}
+\addtocontents{toc}{\bigskip\protect\goodbreak \protect\centering CHAPTER #1\\ #2 \\*[6pt]}
+}
+
+%simulated section entry - toc command only
+\newcommand\mysect[1]{\addcontentsline{toc}{section}{\hspace{-1.5em}#1}}
+
+% centred image - point width, file name (without the .png), caption
+\newcommand\imgcent[3]{
+\begin{figure}[htb]
+\begin{center}
+\emph{\includegraphics*[width=#1pt]{images/#2.png} \\ \textsc{#3}}
+\end{center}
+\end{figure}
+}
+
+% twin columns for parallel theorems
+\newcommand\twincol[2]{
+\noindent\parbox[t]{.47\linewidth}{\hspace{1.5em}\stretchspace#1}%
+\hspace{.06\linewidth}%
+\parbox[t]{.47\linewidth}{\hspace{1.5em}\stretchspace#2}\medskip}
+\newcommand{\stretchspace}{\spaceskip0.5em plus 0.5em minus 0.25em}
+
+% small text M
+\newcommand\Measure{\mbox{\scriptsize M}\,}
+
+% remove excess gap between 2 multlines
+\newcommand\ungap{\vspace{-0.33in}}
+
+% strut to adjust surd signs, also matrix row spacing, fraction denominators ...
+\newcommand\mystrut[1]{\rule{0pt}{#1pt}}
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+\AtBeginDocument{\def\contentsname{TABLE OF CONTENTS}
+  \def\indexname{INDEX}}%
+
+\begin{document}
+\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.
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+
+\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
+covers. 1s.\ net.
+
+\adless The Indian Army. A sketch of its History and Organization. 8vo. Paper
+covers. 1s.\ net.
+
+
+\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
+Conquest,\end{large} including the invasion of Alexander the
+Great. By \textsc{V.~A. Smith.} 8vo. With maps, plans, and other illustrations.
+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
+from its Servants in the East (1602-1617) may also be obtained, price
+15s.\ each volume.)
+
+
+\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
+in the Calendars of State Papers, East Indies, published by the Public
+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.
+Crown 8vo. £1 1s.
+
+
+\adnorm\begin{large}Documents of the French Revolution, 1789-1791.\end{large} By
+\textsc{L.~G. Wickham Legg.} Crown 8vo. Two volumes. 12s.\ net.
+
+
+\adnorm\begin{large}Napoleonic Statesmanship: Germany.\end{large} By \textsc{H.~A.~L. Fisher.}
+8vo, with maps. 12s.\ 6d.\ net.
+
+
+\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,
+6 maps. 5s.
+
+
+\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
+\textsc{Barbara W. Freire-Marreco}. Imperial 8vo. 21s.\ net.
+
+
+\adnorm\begin{large}The Evolution of Culture, and other Essays,\end{large} by the late
+Lieut.-Gen.\ \textsc{A. Lane-Fox Pitt-Rivers}; edited by \textsc{J.~L. Myres}, with an
+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.
+
+
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+New edition, revised and enlarged. 8vo. 15s.
+
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+
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+
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+Holderness, C.S.I.} (1907.) Crown 8vo. 5s.\ net.
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+
+%[File: 309.png]
+\medskip\begin{center}\begin{huge}Political Science and Economy\end{huge}\end{center}
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+\begin{center}For Bryce's \emph{Studies} and other books on general jurisprudence and political
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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}
+\begin{verbatim}
+End of the Project Gutenberg EBook of The Elements of non-Euclidean Geometry, by 
+Julian Lowell Coolidge
+
+*** END OF THIS PROJECT GUTENBERG EBOOK NON-EUCLIDEAN GEOMETRY ***
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+% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
+%                                                                         %
+% End of the Project Gutenberg EBook of The Elements of non-Euclidean Geometry, by 
+% Julian Lowell Coolidge                                                  %
+%                                                                         %
+% *** END OF THIS PROJECT GUTENBERG EBOOK NON-EUCLIDEAN GEOMETRY ***      %
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diff --git a/LICENSE.txt b/LICENSE.txt
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+This eBook, including all associated images, markup, improvements,
+metadata, and any other content or labor, has been confirmed to be
+in the PUBLIC DOMAIN IN THE UNITED STATES.
+
+Procedures for determining public domain status are described in
+the "Copyright How-To" at https://www.gutenberg.org.
+
+No investigation has been made concerning possible copyrights in
+jurisdictions other than the United States. Anyone seeking to utilize
+this eBook outside of the United States should confirm copyright
+status under the laws that apply to them.
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+Project Gutenberg (https://www.gutenberg.org) public repository for
+eBook #26373 (https://www.gutenberg.org/ebooks/26373)