% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of The Evanston Colloquium Lectures on      %
% Mathematics, by Felix Klein                                             %
%                                                                         %
% 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 Evanston Colloquium Lectures on Mathematics                  %
%        Delivered From Aug. 28 to Sept. 9, 1893 Before Members of        %
%        the Congress of Mathematics Held in Connection with the          %
%        World's Fair in Chicago                                          %
%                                                                         %
% Author: Felix Klein                                                     %
%                                                                         %
% Release Date: May 18, 2011 [EBook #36154]                               %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK THE EVANSTON COLLOQUIUM ***   %
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The Project Gutenberg EBook of The Evanston Colloquium Lectures on
Mathematics, by Felix Klein

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 Evanston Colloquium Lectures on Mathematics
       Delivered From Aug. 28 to Sept. 9, 1893 Before Members of
       the Congress of Mathematics Held in Connection with the
       World's Fair in Chicago

Author: Felix Klein

Release Date: May 18, 2011 [EBook #36154]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK THE EVANSTON COLLOQUIUM ***
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Produced by Andrew D. Hwang, Brenda Lewis, and the Online
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file was produced from images from the Cornell University
Library: Historical Mathematics Monographs collection.)
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

\frontmatter
\pagenumbering{roman}
%% -----File: 001.png---Folio i-------
\null\vfill
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\Large LECTURES ON MATHEMATICS
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\underline{\large{\textit{THE EVANSTON COLLOQUIUM}}}
\vfill

\huge{\textsc{Lectures on Mathematics}}
\vfill

\footnotesize\scshape
delivered \\
From Aug.~28 to Sept.~9, 1893 \\[\TmpLen]
\itshape BEFORE MEMBERS OF THE CONGRESS OF MATHEMATICS \\
HELD IN CONNECTION WITH THE WORLD'S \\
FAIR IN CHICAGO \\[\TmpLen]
\upshape AT NORTHWESTERN UNIVERSITY \\
\scriptsize EVANSTON, ILL.
\vfill

BY \\
\large FELIX KLEIN
\vfill

\footnotesize \textit{REPORTED BY ALEXANDER ZIWET}
\vfill

PUBLISHED FOR H.~S. WHITE AND A.~ZIWET \\[\TmpLen]
\textgoth{New York} \\
\normalsize MACMILLAN AND CO. \\
\footnotesize AND LONDON \\
\small 1894 \\[\TmpLen]
\scriptsize \textit{All rights reserved}
\end{center}
\newpage
%% -----File: 004.png---Folio iv-------
\null\vfill
\begin{center}
\scriptsize\scshape Copyright, 1893, \\
By MACMILLAN AND CO.
\vfill

{\footnotesize\textgoth{Norwood Press:}} \\
\upshape J.~S. Cushing~\&~Co.---Berwick~\&~Smith. \\
Boston, Mass., U.S.A.
\end{center}
\newpage
%% -----File: 005.png---Folio v-------

\Preface

\First{The} Congress of Mathematics held under the auspices of
the World's Fair Auxiliary in Chicago, from the 21st to the
26th of August, 1893, was attended by Professor Felix Klein
of the University of Göttingen, as one of the commissioners of
the German university exhibit at the Columbian Exposition.
After the adjournment of the Congress, Professor Klein kindly
consented to hold a \textit{colloquium} on mathematics with such members
of the Congress as might wish to participate. The Northwestern
University at Evanston,~Ill., tendered the use of rooms
for this purpose and placed a collection of mathematical books
from its library at the disposal of the members of the colloquium.
The following is a list of the members attending the
colloquium:---
\begin{participants}
\Name{W.~W. Beman, A.M.}, professor of mathematics, University of Michigan.

\Name{E.~M. Blake, Ph.D.}, instructor in mathematics, Columbia College.

\Name{O.~Bolza, Ph.D.}, associate professor of mathematics, University of Chicago.

\Name{H.~T. Eddy, Ph.D.}, president of the Rose Polytechnic Institute.

\Name{A.~M. Ely, A.B.}, professor of mathematics, Vassar College.

\Name{F.~Franklin, Ph.D.}, professor of mathematics, Johns Hopkins University.

\Name{T.~F. Holgate, Ph.D.}, instructor in mathematics, Northwestern University.

\Name{L.~S. Hulburt, A.M.}, instructor in mathematics, Johns Hopkins University.

\Name{F.~H. Loud, A.B.}, professor of mathematics and astronomy, Colorado College.

\Name{J.~McMahon, A.M.}, assistant professor of mathematics, Cornell University.

\Name{H.~Maschke, Ph.D.}, assistant professor of mathematics, University of
Chicago.

\Name{E.~H. Moore, Ph.D.}, professor of mathematics, University of Chicago.
%% -----File: 006.png---Folio vi-------

\Name{J.~E. Oliver, A.M.}, professor of mathematics, Cornell University.

\Name{A.~M. Sawin, Sc.M.}, Evanston.

\Name{W.~E. Story, Ph.D.}, professor of mathematics, Clark University.

\Name{E.~Study, Ph.D.}, professor of mathematics, University of Marburg.

\Name{H.~Taber, Ph.D.}, assistant professor of mathematics, Clark University.

\Name{H.~W. Tyler, Ph.D.}, professor of mathematics, Massachusetts Institute of
Technology.

\Name{J.~M. Van~Vleck, A.M., LL.D.}, professor of mathematics and astronomy,
Wesleyan University.

\Name{E.~B. Van~Vleck, Ph.D.}, instructor in mathematics, University of Wisconsin.

\Name{C.~A. Waldo, A.M.}, professor of mathematics, De~Pauw University.

\Name{H.~S. White, Ph.D.}, associate professor of mathematics, Northwestern University.

\Name{M.~F. Winston, A.B.}, honorary fellow in mathematics, University of Chicago.

\Name{A.~Ziwet}, assistant professor of mathematics, University of Michigan.
\end{participants}

The meetings lasted from August~28th till September~9th;
and in the course of these two weeks Professor Klein gave a
daily lecture, besides devoting a large portion of his time to
personal intercourse and conferences with those attending the
meetings. The lectures were delivered freely, in the English
language, substantially in the form in which they are here
given to the public. The only change made consists in obliterating
the conversational form of the frequent questions and
discussions by means of which Professor Klein understands so
well to enliven his discourse. My notes, after being written
out each day, were carefully revised by Professor Klein himself,
both in manuscript and in the proofs.

As an appendix it has been thought proper to give a translation
of the interesting historical sketch contributed by Professor
Klein to the work \textit{Die deutschen Universitäten}. The translation
was prepared by Professor H.~W.~Tyler, of the Massachusetts
Institute of Technology.

It is to be hoped that the proceedings of the Chicago Congress
of Mathematics, in which Professor Klein took a leading
%% -----File: 007.png---Folio vii-------
part, will soon be published in full. The papers presented to
this Congress, and the discussions that followed their reading,
form an important complement to the Evanston colloquium.
Indeed, in reading the lectures here published, it should be kept
in mind that they followed immediately upon the adjournment
of the Chicago meeting, and were addressed to members of the
Congress. This circumstance, in addition to the limited time
and the informal character of the colloquium, must account
for the incompleteness with which the various subjects are
treated.

In concluding, the editor wishes to express his thanks to
Professors W.~W.~Beman and H.~S.~White for aid in preparing
the manuscript and correcting the proofs.

\hfill ALEXANDER ZIWET.\hspace{\parindent}

{\footnotesize\textsc{Ann Arbor, Mich.,} November, 1893.}
%% -----File: 008.png---Folio viii-------
%[Blank Page]
%% -----File: 009.png---Folio ix-------
\tableofcontents
\iffalse
CONTENTS.

Lecture       Page

I. Clebsch       1

II. Sophus Lie       9

III. Sophus Lie       18

IV. On the Real Shape of Algebraic Curves and Surfaces      25

V. Theory of Functions and Geometry       33

VI. On the Mathematical Character of Space-Intuition, and the
Relation of Pure Mathematics to the Applied Sciences       41

VII. The Transcendency of the Numbers $e$ and $\pi$       51

VIII. Ideal Numbers       58

IX. The Solution of Higher Algebraic Equations       67

X. On Some Recent Advances in Hyperelliptic and Abelian Functions       75

XI. The Most Recent Researches in Non-Euclidean Geometry        85

XII. The Study of Mathematics at Göttingen        94

The Development of Mathematics at the German Universities        99
\fi
%% -----File: 010.png---Folio x-------
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%% -----File: 011.png---Folio 1-------
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%[** TN: Text printed by the \Lecture command]
% LECTURES ON MATHEMATICS.
\Lecture{I.}{Clebsch.}

\Date{(August 28, 1893.)}

\First{It} will be the object of our \textit{Colloquia} to pass in review some
of the principal phases of the most recent development of mathematical
thought in Germany.

A brief sketch of the growth of mathematics in the German
universities in the course of the present century has been contributed
by me to the work \textit{Die deutschen Universitäten}, compiled
and edited by Professor \emph{Lexis} (Berlin, Asher, 1893), for
the exhibit of the German universities at the World's Fair.\footnote
  {A translation of this sketch will be found in the Appendix, \hyperref[addendum]{p.~\pageref{addendum}}.}
The strictly objective point of view that had to be adopted for
this sketch made it necessary to break off the account about
the year~1870. In the present more informal lectures these
restrictions both as to time and point of view are abandoned.
It is just the period since 1870 that I intend to deal with, and
I shall speak of it in a more subjective manner, insisting particularly
on those features of the development of mathematics
in which I have taken part myself either by personal work or
by direct observation.

The first week will be devoted largely to \emph{Geometry}, taking
this term in its broadest sense; and in this first lecture it will
surely be appropriate to select the celebrated geometer \emph{Clebsch}
%% -----File: 012.png---Folio 2-------
as the central figure, partly because he was one of my principal
teachers, and also for the reason that his work is so well known
in this country.

Among mathematicians in general, three main categories may
be distinguished; and perhaps the names \emph{logicians}, \emph{formalists},
and \emph{intuitionists} may serve to characterize them. (1)~The word
\emph{logician} is here used, of course, without reference to the mathematical
logic of Boole, Peirce,~etc.; it is only intended to indicate
that the main strength of the men belonging to this class
lies in their logical and critical power, in their ability to give
strict definitions, and to derive rigid deductions therefrom.
The great and wholesome influence exerted in Germany by
\emph{Weierstrass} in this direction is well known. (2)~The \emph{formalists}
among the mathematicians excel mainly in the skilful formal
treatment of a given question, in devising for it an ``algorithm.''
\emph{Gordan}, or let us say \emph{Cayley} and \emph{Sylvester}, must be ranged in
this group. (3)~To the \emph{intuitionists}, finally, belong those who
lay particular stress on geometrical intuition (\textit{Anschauung}), not
in pure geometry only, but in all branches of mathematics.
What Benjamin Peirce has called ``geometrizing a mathematical
question'' seems to express the same idea. Lord \emph{Kelvin} and
\emph{von~Staudt} may be mentioned as types of this category.

\emph{Clebsch} must be said to belong both to the second and third
of these categories, while I should class myself with the third,
and also the first. For this reason my account of Clebsch's
work will be incomplete; but this will hardly prove a serious
drawback, considering that the part of his work characterized
by the second of the above categories is already so fully appreciated
here in America. In general, it is my intention here,
not so much to give a complete account of any subject, as to
\emph{supplement} the mathematical views that I find prevalent in this
country.
%% -----File: 013.png---Folio 3-------

As the first achievement of Clebsch we must set down the
introduction into Germany of the work done previously by
Cayley and Sylvester in England. But he not only transplanted
to German soil their theory of invariants and the interpretation
of projective geometry by means of this theory; he
also brought this theory into live and fruitful correlation with
the fundamental ideas of Riemann's theory of functions. In
the former respect, it may be sufficient to refer to Clebsch's
\textit{Vorlesungen über Geometrie}, edited and continued by Lindemann;
to his \textit{Binäre algebraische Formen}, and in general to
what he did in co-operation with Gordan. A good historical
account of his work will be found in the biography of Clebsch
published in the \textit{Math.\ Annalen}, Vol.~7.

Riemann's celebrated memoir of 1857\footnote
  {\textit{Theorie der Abel'schen Functionen}, Journal für reine und angewandte Mathematik,
  Vol.~54 (1857), pp.~115--155; reprinted in Riemann's \textit{Werke}, 1876, pp.~81--135.}
presented the new
ideas on the theory of functions in a somewhat startling novel
form that prevented their immediate acceptance and recognition.
He based the theory of the Abelian integrals and their
inverse,\DPnote{** [sic], adjective?} the Abelian functions, on the idea of the surface now
so well known by his name, and on the corresponding fundamental
theorems of existence (\textit{Existenztheoreme}). Clebsch, by
taking as his starting-point an algebraic curve defined by its
equation, made the theory more accessible to the mathematicians
of his time, and added a more concrete interest to it
by the geometrical theorems that he deduced from the theory
of Abelian functions. Clebsch's paper, \textit{Ueber die Anwendung
der Abel'schen Functionen in der Geometrie},\footnote
  {Journal für reine und angewandte Mathematik, Vol.~63 (1864), pp.~189--243.}
and the work of
Clebsch and Gordan on Abelian functions,\footnote
  {\textit{Theorie der Abel'schen Functionen}, Leipzig, Teubner, 1866.}
are well known to
American mathematicians; and in accordance with my plan, I
proceed to give merely some critical remarks.
%% -----File: 014.png---Folio 4-------

However great the achievement of Clebsch's in making
the work of Riemann more easy of access to his contemporaries,
it is my opinion that at the present time the book of
Clebsch is no longer to be considered as the standard work
for an introduction to the study of Abelian functions. The
chief objections to Clebsch's presentation are twofold: they
can be briefly characterized as a lack of mathematical rigour
on the one hand, and a loss of intuitiveness, of geometrical
perspicuity, on the other. A few examples will explain my
meaning.

(\textit{a})~Clebsch bases his whole investigation on the consideration
of what he takes to be the most general type of an
algebraic curve, and this \emph{general} curve he assumes as having
only double points, but no other singularities. To obtain a
sure foundation for the theory, it must be proved that any
algebraic curve can be transformed rationally into a curve
having only double points. This proof was not given by
Clebsch; it has since been supplied by his pupils and followers,
but the demonstration is long and involved. See the
papers by Brill and Nöther in the \textit{Math.\ Annalen}, Vol.~7
(1874),\footnote
  {\textit{Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie},
  pp.~269--310.}
and by Nöther, \textit{ib}., Vol.~23 (1884).\footnote
  {\textit{Rationale Ausführung der Operationen in der Theorie der algebraischen Functionen},
  pp.~311--358.}

Another defect of the same kind occurs in connection with
the determinant of the periods of the Abelian integrals. This
determinant never vanishes as long as the curve is irreducible.
But Clebsch and Gordan neglect to prove this, and
however simple the proof may be, this must be regarded as
an inexactness.

The apparent lack of critical spirit which we find in the work
of Clebsch is characteristic of the geometrical epoch in which
%% -----File: 015.png---Folio 5-------
he lived, the epoch of Steiner, among others. It detracts in no-wise
from the merit of his work. But the influence of the
theory of functions has taught the present generation to be
more exacting.

(\textit{b})~The second objection to adopting Clebsch's presentation
lies in the fact that, from Riemann's point of view, many points
of the theory become far more simple and almost self-evident,
whereas in Clebsch's theory they are not brought out in all
their beauty. An example of this is presented by the idea of
the deficiency~$p$. In Riemann's theory, where $p$~represents the
order of connectivity of the surface, the invariability of~$p$ under
any rational transformation is self-evident, while from the point
of view of Clebsch this invariability must be proved by means
of a long elimination, without affording the true geometrical
insight into its meaning.

For these reasons it seems to me best to begin the theory
of Abelian functions with Riemann's ideas, without, however,
neglecting to give later the purely algebraical developments.
This method is adopted in my paper on Abelian functions;\footnote
  {\textit{Zur Theorie der Abel'schen Functionen}, Math.\ Annalen, Vol.~36 (1890), pp.~1--83.}
it is also followed in the work \textit{Die elliptischen Modulfunctionen},
Vols.\ I.~and~II., edited by Dr.~Fricke. A general account of the
historical development of the theory of algebraic curves in connection
with Riemann's ideas will be found in my (lithographed)
lectures on \textit{Riemann'sche Flächen}, delivered in 1891--92.\footnote
  {My lithographed lectures frequently give only an outline of the subject, omitting
  details and long demonstrations, which are supposed to be supplied by the
  student by private reading and a study of the literature of the subject.}

If this arrangement be adopted, it is interesting to follow
out the true relation that the algebraical developments bear
to Riemann's theory. Thus in Brill and Nöther's theory, the
so-called \emph{fundamental theorem} of Nöther is of primary importance.
%% -----File: 016.png---Folio 6-------
It gives a rule for deciding under what conditions an
algebraic rational integral function~$f$ of~$x$ and~$y$ can be put into
the form
\[
f = A \phi + B \psi,
\]
where~$\phi$ and~$\psi$ are likewise rational algebraic functions. Each
point of intersection of the curves $\phi = 0$ and $\psi = 0$ must of
course be a point of the curve $f = 0$. But there remains the
question of multiple and singular points; and this is disposed
of by Nöther's theorem. Now it is of great interest to investigate
how these relations present themselves when the
starting-point is taken from Riemann's ideas.

One of the best illustrations of the utility of adopting
Riemann's principles is presented by the very remarkable
advance made recently by Hurwitz, in the theory of algebraic
curves, in particular his extension of the theory of algebraic
correspondences, an account of which is given in the second
volume of the \textit{Elliptische Modulfunctionen}. Cayley had found
as a fundamental theorem in this theory a rule for determining
the number of self-corresponding points for algebraic correspondences
of a simple kind. A whole series of very valuable
papers by Brill, published in the \textit{Math.\ Annalen},\footnote
  {\textit{Ueber zwei Berührungsprobleme}, Vol.~4 (1871), pp.~527--549.---\textit{Ueber Entsprechen
  von Punktsystemen auf einer Curve}, Vol.~6 (1873), pp.~33--65.---\textit{Ueber die
  Correspondenzformel}, Vol.~7 (1874), pp.~607--622.---\textit{Ueber algebraische Correspondenzen},
  Vol.~31 (1888), pp.~374--409.---\textit{Ueber algebraische Correspondenzen. Zweite
  Abhandlung: Specialgruppen von Punkten einer algebraischen Curve}, Vol.~36 (1890),
  pp.~321--360.}
is devoted
to the further investigation and demonstration of this theorem.
Now Hurwitz, attacking the problem from the point of view
of Riemann's ideas, arrives not only at a more simple and
quite general demonstration of Cayley's rule, but proceeds to a
complete study of all possible algebraic correspondences. He
finds that while for \emph{general} curves the correspondences considered
%% -----File: 017.png---Folio 7-------
by Cayley and Brill are the only ones that exist, in the
case of \emph{singular} curves there are other correspondences which
also can be treated completely. These singular curves are
characterized by certain linear relations with integral coefficients,
connecting the periods of their Abelian integrals.

Let us now turn to that side of Clebsch's method which
appears to me to be the most important, and which certainly
must be recognized as being of great and permanent value;
I mean the generalization, obtained by Clebsch, of the whole
theory of Abelian integrals to the theory of algebraic functions
with several variables. By applying the methods he had
developed for functions of the form $f(x, y) = 0$, or in homogeneous
co-ordinates, $f(x_{1}, x_{2}, x_{3}) = 0$, to functions with four
homogeneous variables $f(x_{1}, x_{2}, x_{3}, x_{4}) = 0$, he found in~1868,
that there also exists a number~$p$ that remains invariant under
all rational transformations of the surface $f = 0$. Clebsch
arrives at this result by considering \emph{double integrals} belonging
to the surface.

It is evident that this theory could not have been found from
Riemann's point of view. There is no difficulty in conceiving a
four-dimensional Riemann space corresponding to an equation
$f(x, y, z) = 0$. But the difficulty would lie in proving the
``theorems of existence'' for such a space; and it may even be
doubted whether analogous theorems hold in such a space.

While to Clebsch is due the fundamental idea of this
grand generalization, the working out of this theory was
left to his pupils and followers. The work was mainly carried
on by Nöther, who showed, in the case of algebraic surfaces,
the existence of more than one invariant number~$p$ and of
corresponding moduli, \ie\ constants not changed by one-to-one
transformations. Italian and French mathematicians, in particular
Picard and Poincaré, have also contributed largely to the
further development of the theory.
%% -----File: 018.png---Folio 8-------

If the value of a man of science is to be gauged not by his
general activity in all directions, but solely by the fruitful new
ideas that he has first introduced into his science, then the
theory just considered must be regarded as the most valuable
work of Clebsch.

In close connection with the preceding are the general ideas
put forth by Clebsch in his last memoir,\footnote
  {\textit{Ueber ein neues Grundgebilde der analytischen Geometrie der Ebene}, Math.\
  Annalen, Vol.~6 (1873), pp.~203--215.}
ideas to which he
himself attached great importance. This memoir implies an
application, as it were, of the theory of Abelian functions to
the theory of differential equations. It is well known that the
central problem of the whole of modern mathematics is the
study of the transcendental functions defined by differential
equations. Now Clebsch, led by the analogy of his theory of
Abelian integrals, proceeds somewhat as follows. Let us consider,
for example, an ordinary differential equation of the first
order $f(x, y, y') = 0$, where $f$~represents an algebraic function.
Regarding~$y'$ as a third variable~$z$, we have the equation of an
algebraic surface. Just as the Abelian integrals can be classified
according to the properties of the fundamental curve that
remain unchanged under a rational transformation, so Clebsch
proposes to classify the transcendental functions defined by
the differential equations according to the invariant properties
of the corresponding surfaces $f = 0$ under rational one-to-one
transformations.

The theory of differential equations is just now being cultivated
very extensively by French mathematicians; and some
of them proceed precisely from this point of view first adopted
by Clebsch.
%% -----File: 019.png---Folio 9-------

\Lecture{II.}{Sophus Lie.}

\Date{(August 29, 1893.)}

\First{To} fully understand the mathematical genius of Sophus Lie,
one must not turn to the books recently published by him in
collaboration with Dr.~Engel, but to his earlier memoirs, written
during the first years of his scientific career. There Lie shows
himself the true geometer that he is, while in his later publications,
finding that he was but imperfectly understood by the
mathematicians accustomed to the analytical point of view, he
adopted a very general analytical form of treatment that is not
always easy to follow.

Fortunately, I had the advantage of becoming intimately
acquainted with Lie's ideas at a very early period, when they
were still, as the chemists say, in the ``nascent state,'' and
thus most effective in producing a strong reaction. My lecture
to-day will therefore be devoted chiefly to his paper ``\textit{Ueber
Complexe, insbesondere Linien- und Kugel-Complexe, mit Anwendung
auf die Theorie partieller Differentialgleichungen}.''\footnote
  {Math.\ Annalen,\DPnote{** TN: Italicized in original} Vol.~5 (1872), pp.~145--256.}

To define the place of this paper in the historical development
of geometry, a word must be said of two eminent geometers
of an earlier period: Plücker (1801--68) and Monge (1746--1818).
Plücker's name is familiar to every mathematician,
through his formulæ relating to algebraic curves. But what is
of importance in the present connection is his generalized idea
%% -----File: 020.png---Folio 10-------
of the space-element. The ordinary geometry with the point as
element deals with space as three-dimensioned, conformably to
the three constants determining the position of a point. A dual
transformation gives the plane as element; space in this case
has also three dimensions, as there are three independent constants
in the equation of the plane. If, however, the straight
line be selected as space-element, space must be considered as
four-dimensional, since four independent constants determine
a straight line. Again, if a quadric surface~$F_{2}$ be taken as
element, space will have nine dimensions, because every such
element requires nine quantities for its determination, viz.\ the
nine independent constants of the surface~$F_{2}$; in other words,
space contains $\infty^{9}$~quadric surfaces. This conception of hyperspaces
must be clearly distinguished from that of Grassmann
and others. Plücker, indeed, rejected any other idea of a space
of more than three dimensions as too abstruse.---The work
of Monge that is here of importance, is his \textit{Application de
l'analyse à la géométrie}, 1809 (reprinted 1850), in which he
treats of ordinary and partial differential equations of the first
and second order, and applies these to geometrical questions
such as the curvature of surfaces, their lines of curvature,
geodesic lines,~etc. The treatment of geometrical problems by
means of the differential and integral calculus is one feature of
this work; the other, perhaps even more important, is the converse
of this, viz.\ the application of geometrical intuition to
questions of analysis.

Now this last feature is one of the most prominent characteristics
of Lie's work; he increases its power by adopting Plücker's
idea of a generalized space-element and extending this fundamental
conception. A few examples will best serve to give an
idea of the character of his work; as such an example I select
(as I have done elsewhere before) Lie's sphere-geometry (\textit{Kugelgeometrie}).
%% -----File: 021.png---Folio 11-------

Taking the equation of a sphere in the form
\[
x^{2} + y^{2} + z^{2} - 2Bx - 2Cy - 2Dz + E = 0,
\]
the coefficients, $B$, $C$, $D$, $E$, can be regarded as the co-ordinates
of the sphere, and ordinary space appears accordingly as a
manifoldness of four dimensions. For the radius,~$R$, of the
sphere we have
\[
R^{2} = B^{2} + C^{2} + D^{2} - E
\]
as a relation connecting the fifth quantity,~$R$, with the four co-ordinates,
$B$, $C$, $D$,~$E$.

To introduce homogeneous co-ordinates, put
\[
B = \frac{b}{a}, \quad C = \frac{c}{a},\quad  D =\frac{d}{a},\quad  E = \frac{e}{a}, \quad R = \frac{r}{a};
\]
then $a : b : c : d : e$ are the five homogeneous co-ordinates of the
sphere, and the sixth quantity~$r$ is related to them by means of
the homogeneous equation of the second degree,
\[
r^{2} = b^{2} + c^{2} + d^{2} - ae.
\Tag{(1)}
\]

Sphere-geometry has been treated in two ways that must be
carefully distinguished. In one method, which we may call \emph{the
elementary sphere-geometry}, only the five co-ordinates $a : b : c : d : e$
are used, while in the other, \emph{the higher}, or \emph{Lie's}, \emph{sphere-geometry},
the quantity~$r$ is introduced. In this latter system, a sphere
has six homogeneous co-ordinates, $a$,~$b$,~$c$, $d$,~$e$,~$r$, connected by
the equation~\Eq{(1)}.

From a higher point of view the distinction between these
two sphere-geometries, as well as their individual character, is
best brought out by considering the \emph{group} belonging to each.
Indeed, every system of geometry is characterized by its group,
in the meaning explained in my Erlangen \textit{Programm};\DPnote{** Semicolon ital. in orig.}\footnote
  {\textit{Vergleichende Betrachtungen über neuere geometrische Forschungen.\ Programm
  zum Eintritt in die philosophische Facultät und den Senat der K.~Friedrich-Alexanders-Universität
  zu Erlangen}. Erlangen, Deichert, 1872. For an English translation,
  by Haskell, see the Bulletin of the New York Mathematical Society, Vol.~2
  (1893), pp.~215--249.}
\ie\
%% -----File: 022.png---Folio 12-------
every system of geometry deals only with such relations of
space as remain unchanged by the transformations of its group.

In the elementary sphere-geometry the group is formed by
all the linear substitutions of the five quantities $a$,~$b$,~$c$, $d$,~$e$,
that leave unchanged the homogeneous equation of the second
degree
\[
b^{2} + c^{2} + d^{2} - ae = 0.
\Tag{(2)}
\]
This gives $\infty^{25-15} = \infty^{10}$ substitutions. By adopting this definition
we obtain point-transformations of a simple character.
The geometrical meaning of equation~\Eq{(2)} is that the radius is
zero. Every sphere of vanishing radius, \ie\ every point, is
therefore transformed into a point. Moreover, as the polar
\[
2bb' + 2cc' + 2dd' - ae' - a'e = 0
\]
remains likewise unchanged in the transformation, it follows
that orthogonal spheres are transformed into orthogonal spheres.
Thus the group of the elementary sphere-geometry is characterized
as the \emph{conformal group}, well known as that of the transformation
by inversion (or reciprocal radii) and through its
applications in mathematical physics.

Darboux has further developed this elementary sphere-geometry.
Any equation of the second degree
\[
F(a, b, c, d, e) = 0,
\]
taken in connection with the relation~\Eq{(2)} represents a point-surface
which Darboux has called \emph{cyclide}. From the point of
view of ordinary projective geometry, the cyclide is a surface of
the fourth order containing the imaginary circle common to all
spheres of space as a double curve. A careful investigation
%% -----File: 023.png---Folio 13-------
of these cyclides will be found in Darboux's \textit{Leçons sur la
théorie générale des surfaces et les applications géométriques du
calcul infinitésimal}, and elsewhere. As the ordinary surfaces of
the second degree can be regarded as special cases of cyclides,
we have here a method for generalizing the known properties
of quadric surfaces by extending them to cyclides. Thus Mr.\
M.~Bôcher, of Harvard University, in his dissertation,\footnote
  {\textit{Ueber die Reihenentwickelungen der Potentialtheorie}, gekrönte Preisschrift,
  Göttingen, Dieterich,~1891.}
has
treated the extension of a problem in the theory of the potential
from the known case of a body bounded by surfaces of
the second degree to a body bounded by cyclides. A more
extended publication on this subject by Mr.~Bôcher will appear
in a few months (Leipzig, Teubner).

In the higher sphere-geometry of Lie, the six homogeneous
co-ordinates $a : b : c : d : e : r$ are connected, as mentioned above,
by the homogeneous equation of the second degree,
\[
b^{2} + c^{2} + d^{2} - r^{2} - ae = 0.
\]

The corresponding group is selected as the group of the
linear substitutions transforming this equation into itself. We
have thus a group of $\infty^{36-21} = \infty^{15}$ substitutions. But this is
not a group of point-transformations; for a sphere of radius
zero becomes a sphere whose radius is in general different from
zero. Thus, putting for instance
\[
B' = B,\quad C' = C,\quad D' = D,\quad E' = E,\quad R' = R + \text{const.},
\]
it appears that the transformation consists in a mere dilatation
or expansion of each sphere, a point becoming a sphere of
given radius.

The meaning of the polar equation
\[
2bb' + 2cc' + 2dd' - 2rr' - ae' - a'e = 0
\]
%% -----File: 024.png---Folio 14-------
remaining invariant for any transformation of the group, is evidently
that the spheres originally in contact remain in contact.
The group belongs therefore to the important class of \emph{contact-transformations},
which will soon be considered more in detail.

In studying any particular geometry, such as Lie's sphere-geometry,
two methods present themselves.

(1)~We may consider equations of various degrees and inquire
what they represent. In devising names for the different configurations
so obtained, Lie used the names introduced by
Plücker in his line-geometry. Thus a single equation,
\[
F(a, b, c, d, e, r) = 0,
\]
is said to represent a \emph{complex} of the first, second,~etc., degree,
according to the degree of the equation; a complex contains,
therefore, $\infty^{3}$~spheres. Two such equations,
\[
F_{1} = 0,\quad F_{2} = 0,
\]
represent a \emph{congruency} containing $\infty^{2}$~spheres. Three equations,
\[
F_{1} = 0,\quad F_{2} = 0,\quad F_{3} = 0,
\]
may be said to represent a \emph{set} of spheres, the number being~$\infty^{1}$.
It is to be noticed that in each case the equation of the second
degree,
\[
b^{2} + c^{2} + d^{2} - r^{2} - ae = 0,
\]
is understood to be combined with the equation $F = 0$.

It may be well to mention expressly that the same names are
used by other authors in the elementary sphere-geometry, where
their meaning is, of course, different.

(2)~The other method of studying a new geometry consists
in inquiring how the ordinary configurations of point-geometry
can be treated by means of the new system. This line of
inquiry has led Lie to highly interesting results.
%% -----File: 025.png---Folio 15-------

In ordinary geometry a surface is conceived as a locus of
points; in Lie's geometry it appears as the totality of all the
spheres having contact with the surface. This gives a threefold
infinity of spheres, or a complex of spheres,
\[
F(a, b, c, d, e, r) = 0.
\]
But this, of course, is not a \emph{general} complex; for not every complex
will be such as to touch a surface. It has been shown
that the condition that must be fulfilled by a complex of
spheres, if all its spheres are to touch a surface, is the following:
\[
\left(\frac{\dd F}{\dd b}\right)^{2} +
\left(\frac{\dd F}{\dd c}\right)^{2} +
\left(\frac{\dd F}{\dd d}\right)^{2} -
\left(\frac{\dd F}{\dd r}\right)^{2} - \frac{\dd F}{\dd a}\, \frac{\dd F}{\dd e} = 0.
\]

To give at least one illustration of the further development of
this interesting theory, I will mention that among the infinite
number of spheres touching the surface at any point there are
two having stationary contact with the surface; they are called
the \emph{principal spheres}. The lines of curvature of the surface
can then be defined as curves along which the principal spheres
touch the surface in two successive points.

Plücker's line-geometry can be studied by the same two
methods just mentioned. In this geometry let $p_{12}$, $p_{13}$, $p_{14}$, $p_{34}$,
$p_{42}$, $p_{23}$ be the usual six homogeneous co-ordinates, where
$p_{ik} = -p_{ki}$. Then we have the identity
\[
p_{12}p_{34} + p_{13}p_{42} + p_{14}p_{23} = 0,
\]
and we take as group the $\infty^{15}$ linear substitutions transforming
this equation into itself. This group corresponds to the totality
of collineations and reciprocations, \ie\ to the projective group.
The reason for this lies in the fact that the polar equation
\[
p_{12}{p_{34}}' + p_{13}{p_{42}}' + p_{14}{p_{23}}' +
p_{34}{p_{12}}' + p_{42}{p_{13}}' + p_{23}{p_{14}}' = 0
\]
expresses the intersection of the two lines~$p$,~$p'$.
%% -----File: 026.png---Folio 16-------

Now Lie has instituted a comparison of the highest interest
between the line-geometry of Plücker and his own sphere-geometry.
In each of these geometries there occur six homogeneous
co-ordinates connected by a homogeneous equation of
the second degree. The discriminant of each equation is different
from zero. It follows that we can pass from either of these
geometries to the other by linear substitutions. Thus, to transform
\[
p_{12}p_{34} + p_{13}p_{42} + p_{14}p_{23} = 0
\]
into
\[
b^{2} + c^{2} + d^{2} - r^{2} - ae = 0,
\]
it is sufficient to assume, say,
\begin{alignat*}{3}
p_{12} &= b + ic,\quad & p_{13} &= d + r,\quad & p_{14} &= -a, \\
p_{34} &= b - ic,\quad & p_{42} &= d - r,\quad & p_{23} &= e.
\end{alignat*}
It follows from the linear character of the substitutions that
the polar equations are likewise transformed into each other.
Thus we have the remarkable result that \emph{two spheres that touch
correspond to two lines that intersect}.

It is worthy of notice that the equations of transformation
involve the imaginary unit~$i$; and the law of inertia of quadratic
forms shows at once that this introduction of the imaginary
cannot be avoided, but is essential.

To illustrate the value of this transformation of line-geometry
into sphere-geometry, and \textit{vice versa}, let us consider three
linear equations,
\[
F_{1} = 0,\quad F_{2} = 0,\quad F_{3} = 0,
\]
the variables being either line co-ordinates or sphere co-ordinates.
In the former case the three equations represent a \emph{set
of lines}; \ie\ one of the two sets of straight lines of a hyperboloid
of one sheet. It is well known that each line of either
set intersects all the lines of the other. Transforming to sphere-geometry,
%% -----File: 027.png---Folio 17-------
we obtain a \emph{set of spheres} corresponding to each
set of lines; and every sphere of either set must touch every
sphere of the other set. This gives a configuration well
known in geometry from other investigations; viz.\ all these
spheres envelop a surface known as Dupin's cyclide. We
have thus found a noteworthy correlation between the hyperboloid
of one sheet and Dupin's cyclide.

Perhaps the most striking example of the fruitfulness of this
work of Lie's is his discovery that by means of this transformation
the lines of curvature of a surface are transformed into
asymptotic lines of the transformed surface, and \textit{vice versa}.
This appears by taking the definition given above for the lines
of curvature and translating it word for word into the language
of line-geometry. Two problems in the infinitesimal geometry
of surfaces, that had long been regarded as entirely distinct,
are thus shown to be really identical. This must certainly be
regarded as one of the most elegant contributions to differential
geometry made in recent times.
%% -----File: 028.png---Folio 18-------

\Lecture{III.}{Sophus Lie.}

\Date{(August 30, 1893.)}

\First{The} distinction between analytic and algebraic functions,
so important in pure analysis, enters also into the treatment
of geometry.

\emph{Analytic} functions are those that can be represented by a
power series, convergent within a certain region bounded by
the so-called circle of convergence. Outside of this region
the analytic function is not regarded as given \textit{a~priori}; its
continuation into wider regions remains a matter of special
investigation and may give very different results, according to
the particular case considered.

On the other hand, an \emph{algebraic} function, $w = \text{Alg.}\,(z)$, is
supposed to be known for the whole complex plane, having a
finite number of values for every value of~$z$.

Similarly, in geometry, we may confine our attention to a
limited portion of an analytic curve or surface, as, for instance,
in constructing the tangent, investigating the curvature,~etc.;
or we may have to consider the whole extent of algebraic curves
and surfaces in space.

Almost the whole of the applications of the differential and
integral calculus to geometry belongs to the former branch of
geometry; and as this is what we are mainly concerned with in
the present lecture, we need not restrict ourselves to algebraic
functions, but may use the more general analytic functions
confining ourselves always to limited portions of space. I
%% -----File: 029.png---Folio 19-------
thought it advisable to state this here once for all, since here in
America the consideration of algebraic curves has perhaps been
too predominant.

The possibility of introducing new elements of space has been
pointed out in the preceding lecture. To-day we shall use again
a new space-element, consisting of an infinitesimal portion of a
surface (or rather of its tangent plane) with a definite point in
it. This is called, though not very properly, a \emph{surface-element}
(\emph{Flächenelement}), and may perhaps be likened to an infinitesimal
fish-scale. From a more abstract point of view it may be
defined as simply the combination of a plane with a point in it.

As the equation of a plane passing through a point~$(x, y, z)$
can be written in the form
\[
z' - z = p(x' - x) + q(y' - y),
\]
$x'$,~$y'$,~$z'$ being the current co-ordinates, we have $x$,~$y$,~$z$, $p$,~$q$ as the
co-ordinates of our surface-element, so that space becomes a
fivefold manifoldness. If homogeneous co-ordinates be used,
the point $(x_{1}, x_{2}, x_{3}, x_{4})$ and the plane $(u_{1}, u_{2}, u_{3}, u_{4})$ passing
through it are connected by the condition
\[
x_{1}u_{1} + x_{2}u_{2} + x_{3}u_{3} + x_{4}u_{4} = 0,
\]
expressing their united position; and the number of independent
constants is $3 + 3 - 1 = 5$, as before.

Let us now see how ordinary geometry appears in this
representation. A point, being the locus of all surface-elements
passing through it, is represented as a manifoldness of two
dimensions, let us say for shortness, an~$M_{2}$. A curve is represented
by the totality of all those surface-elements that have
their point on the curve and their plane passing through the
tangent; these elements form again an~$M_{2}$. Finally, a surface
is given by those surface-elements that have their point on the
%% -----File: 030.png---Folio 20-------
surface and their plane coincident with the tangent plane of the
surface; they, too, form an~$M_{2}$.

Moreover, all these~$M_{2}$'s have an important property in
common: any two consecutive surface-elements belonging to
the same point, curve, or surface always satisfy the condition
\[
dz - p\, dx - q\, dy = 0,
\]
which is a simple case of a Pfaffian relation; and conversely, if
two surface-elements satisfy this condition, they belong to the
same point, curve, or surface, as the case may be.

Thus we have the highly interesting result that in the geometry
of surface-elements points as well as curves and surfaces are
brought under one head, being all represented by twofold manifoldnesses
having the property just explained. This definition
is the more important as there are no other~$M_{2}$'s having the
same property.

We now proceed to consider the very general kind of transformations
called by Lie \emph{contact-transformations}. They are
transformations that change our element $(x, y, z, p, q)$ into
$(x', y', z', p', q')$ by such substitutions
\[
x' = \phi (x, y, z, p, q),\
y' = \psi (x, y, z, p, q),\
z' = \cdots,\
p' = \cdots,\
q' = \cdots,
\]
as will transform into itself the linear differential equation
\[
dz - p\, dx - q\, dy = 0.
\]
The geometrical meaning of the transformation is evidently that
any~$M_{2}$ having the given property is changed into an~$M_{2}$ having
the same property. Thus, for instance, a surface is transformed
generally into a surface, or in special cases into a point or a
curve. Moreover, let us consider two manifoldnesses~$M_{2}$ having
a contact, \ie\ having a surface-element in common; these~$M_{2}$'s
are changed by the transformation into two other~$M_{2}$'s having
%% -----File: 031.png---Folio 21-------
also a contact. From this characteristic the name given by
Lie to the transformation will be understood.

Contact-transformations are so important, and occur so frequently,
that particular cases attracted the attention of geometers
long ago, though not under this name and from this point
of view, \ie\ not as contact-transformations, so that the true
insight into their nature could not be obtained.

Numerous examples of contact-transformations are given
in my (lithographed) lectures on \textit{Höhere Geometrie}, delivered
during the winter-semester of 1892--93. Thus, an example
in two dimensions is found in the problem of wheel-gearing.
The outline of the tooth of one wheel being given, it is here
required to find the outline of the tooth of the other wheel,
as I explained to you in my lecture at the Chicago Exhibition,
with the aid of the models in the German university exhibit.

Another example is found in the theory of perturbations in
astronomy; Lagrange's method of variation of parameters as
applied to the problem of three bodies is equivalent to a
contact-transformation in a higher space.

The group of $\infty^{15}$~substitutions considered yesterday in
line-geometry is also a group of contact-transformations, both
the collineations and reciprocations having this character.
The reciprocations give the first well-known instance of the
transformation of a point into a plane (\ie\ a surface), and a
curve into a developable (\ie\ also a surface). These transformations
of curves will here be considered as transforming
the \emph{elements} of the points or curves into the \emph{elements} of the
surface.

Finally, we have examples of contact-transformations, not
only in the transformations of spheres discussed in the last
lecture, but even in the general transition from the line-geometry
of Plücker to the sphere-geometry of Lie. Let us
consider this last case somewhat more in detail.
%% -----File: 032.png---Folio 22-------

First of all, two lines that intersect have, of course, a
surface-element in common; and as the two corresponding
spheres must also have a surface-element in common, they
will be in contact, as is actually the case for our transformation.
It will be of interest to consider more closely the correlation
between the surface-elements of a line and those of a sphere,
although it is given by imaginary formulæ. Take, for instance,
the totality of the surface-elements belonging to a circle on
one of the spheres; we may call this a \emph{circular set} of elements.
In line-geometry there corresponds the set of surface-elements
along a generating line of a skew surface; and so on. The
theorem regarding the transformation of the curves of curvature
into asymptotic lines becomes now self-evident. Instead
of the curve of curvature of a surface we have here to consider
the corresponding elements of the surface which we may
call a \emph{curvature set}. Similarly, an asymptotic line is replaced
by the elements of the surface along this line; to this the name
\emph{osculating set} may be given. The correspondence between the
two sets is brought out immediately by considering that two
consecutive elements of a curvature set belong to the same
sphere, while two consecutive elements of an osculating set
belong to the same straight line.

One of the most important applications of contact-transformations
is found in the theory of partial differential equations;
I shall here confine myself to partial differential equations of
the first order. From our new point of view, this theory
assumes a much higher degree of perspicuity, and the true
meaning of the terms ``solution,'' ``general solution,'' ``complete
solution,'' ``singular solution,'' introduced by Lagrange
and Monge, is brought out with much greater clearness.

Let us consider the partial differential equation of the first
order
\[
f(x, y, z, p, q) = 0.
\]
%% -----File: 033.png---Folio 23-------
In the older theory, a distinction is made according to the way
in which $p$~and~$q$ enter into the equation. Thus, when $p$ and~$q$
enter only in the first degree, the equation is called linear.
If $p$ and~$q$ should happen to be both absent, the equation would
not be regarded as a differential equation at all. From the
higher point of view of Lie's new geometry, this distinction
disappears entirely, as will be seen in what follows.

The number of all surface elements in the whole of space is
of course~$\infty^{5}$. By writing down our equation we single out
from these a manifoldness of four dimensions,~$M_{4}$, of $\infty^{4}$~elements.
Now, to find a ``solution'' of the equation in Lie's
sense means to single out from this~$M_{4}$ a twofold manifoldness,~$M_{2}$,
of the characteristic property; whether this~$M_{2}$ be a point,
a curve, or a surface, is here regarded as indifferent. What
Lagrange calls finding a ``complete solution'' consists in
dividing the~$M_{4}$ into $\infty^{2}$~$M_{2}$'s. This can of course be done
in an infinite number of ways. Finally, if any singly infinite
set be taken out of the $\infty^{2}$~$M_{2}$'s, we have in the envelope of
this set what Lagrange calls a ``general solution.'' These
formulations hold quite generally for \emph{all} partial differential
equations of the first order, even for the most specialized forms.

To illustrate, by an example, in what sense an equation of
the form $f(x, y, z) = 0$ may be regarded as a partial differential
equation and what is the meaning of its solutions, let
us consider the very special case $z = 0$. While in ordinary
co-ordinates this equation represents all the \emph{points} of the $xy$-plane,
in Lie's system it represents of course all the \emph{surface-elements}
whose points lie in the plane. Nothing is so simple
as to assign a ``complete solution'' in this case; we have only
to take the $\infty^{2}$~points of the plane themselves, each point being
an~$M_{2}$ of the equation. To derive from this the ``general solution,''
we must take all possible singly infinite sets of points
in the plane, \ie\ any curve whatever, and form the envelope
%% -----File: 034.png---Folio 24-------
of the surface-elements belonging to the points; in other words,
we must take the elements touching the curve. Finally, the
plane itself represents of course a ``singular solution.''

Now, the very high interest and importance of this simple
illustration lies in the fact that by a contact-transformation
every partial differential equation of the first order can be
changed into this particular form $z = 0$. Hence the whole disposition
of the solutions outlined above holds quite generally.

A new and deeper insight is thus gained through Lie's
theory into the meaning of problems that have long been
regarded as classical, while at the same time a full array of
new problems is brought to light and finds here its answer.

It can here only be briefly mentioned that Lie has done much
in applying similar principles to the theory of partial differential
equations of the second order.

At the present time Lie is best known through his theory of
continuous groups of transformations, and at first glance it
might appear as if there were but little connection between this
theory and the geometrical considerations that engaged our
attention in the last two lectures. I think it therefore desirable
to point out here this connection. \emph{It has been the final
aim of Lie from the beginning to make progress in the theory
of differential equations}; and as subsidiary to this end may be
regarded both the geometrical developments considered in these
lectures and the theory of continuous groups.

For further particulars concerning the subjects of the present
as well as the two preceding lectures, I may refer to my (lithographed)
lectures on \textit{Höhere Geometrie}, delivered at Göttingen,
in 1892--93. The theory of surface-elements is also fully developed
in the second volume of the \textit{Theorie der Transformationsgruppen},
by Lie and Engel (Leipzig, Teubner, 1890).
%% -----File: 035.png---Folio 25-------

\Lecture[Algebraic Curves and Surfaces.]
{IV.}{On the Real Shape of Algebraic
Curves and Surfaces.}

\Date{(August 31, 1893.)}

\First{We} turn now to \emph{algebraic} functions, and in particular to the
question of the actual geometric forms corresponding to such
functions. The question as to the reality of geometric forms
and the actual shape of algebraic curves and surfaces was somewhat
neglected for a long time. Otherwise it would be difficult
to explain, for instance, why the connection between Cayley's
theory of projective measurement and the non-Euclidean geometry
should not have been perceived at once. As these questions
are even now less well known than they deserve to be, I
proceed to give here an historical sketch of the subject, without,
however, attempting completeness.

It must be counted among the lasting merits of Sir~Isaac
Newton that he first investigated the shape of the plane curves
of the third order. His \textit{Enumeratio linearum tertii ordinis}\footnote
  {First published as an appendix to Newton's \textit{Opticks}, 1704.}
shows that he had a very clear conception of projective
geometry; for he says that all curves of the third order can
be derived by central projection from five fundamental types
(\Fig{1}). But I wish to direct your particular attention to the
paper by Möbius, \textit{Ueber die Grundformen der Linien der dritten
Ordnung},\footnote
  {Abhandlungen der Königl.\ Sächsischen Gesellschaft der Wissenschaften, math.-phys.\
  Klasse, Vol.~I (1852), pp.~1--82; reprinted in Möbius' \textit{Gesammelte Werke},
  Vol.~III (1886), pp.~89--176.}
where the forms of the cubic curves are derived by
%% -----File: 036.png---Folio 26-------
purely geometric considerations. Owing to its remarkable
elegance of treatment, this paper has given the impulse to
all the subsequent researches in this line that I shall have
to mention.

In 1872 we considered, in Göttingen, the question as to the
shape of surfaces of the third order. As a particular case,
Clebsch at this time constructed his beautiful model of the
%[Illustration: Fig.~1.]
\Figure{036}
\emph{diagonal surface}, with $27$~real lines, which I showed to you at
the Exhibition. The equation of this surface may be written
in the simple form
\[
\sum_{1}^{5}x_{i} = 0,\quad \sum_{1}^{5}x^{3}_{i} = 0,
\]
which shows that the surface can be transformed into itself by
the $120$~permutations of the~$x$'s.

It may here be mentioned as a general rule, that in selecting
a particular case for constructing a model the first prerequisite
is regularity. By selecting a symmetrical form for
the model, not only is the execution simplified, but what is of
more importance, the model will be of such a character as to
impress itself readily on the mind.

Instigated by this investigation of Clebsch, I turned to the
general problem of determining all possible forms of cubic surfaces.\footnote
  {See my paper \textit{Ueber Flächen dritter Ordnung}, Math.\ Annalen, Vol.~6 (1873),
  pp.~551--581.}
%% -----File: 037.png---Folio 27-------
I established the fact that by the principle of continuity
all forms of real surfaces of the third order can be derived
from the particular surface having four real conical points.
This surface, also, I exhibited to you at the World's Fair, and
pointed out how the diagonal surface can be derived from it.
But what is of primary importance is the completeness of
enumeration resulting from my point of view; it would be of
comparatively little value to derive any number of special forms
if it cannot be proved that the method used exhausts the
subject. Models of the typical cases of all the principal forms
of cubic surfaces have since been constructed by Rodenberg for
Brill's collection.

In the 7th~volume of the \textit{Math.\ Annalen} (1874) Zeuthen\footnote
  {\textit{Sur les différentes formes des courbes planes du quatrième ordre}, pp.~410--432.}
has
discussed the various forms of plane curves of the fourth order~$(C_{4})$.
He
%[Illustration: Fig.~2.]
\WFigure{1.25in}{037}
considers in particular the reality
of the double tangents on these curves. The
number of such tangents is~$28$, and they are
all real when the curve consists of four separate
closed portions (\Fig{2}). What is of particular
interest is the relation of Zeuthen's
researches on quartic curves to my own researches
on cubic surfaces, as explained by
Zeuthen himself.\footnote
  {\textit{Études des propriétés de situation des surfaces cubiques}, Math.\ Annalen, Vol.~8
  (1875), pp.~1--30.}
It had been observed before, by Geiser, that
if a cubic surface be projected on a plane from a point on the
surface, the contour of the projection is a quartic curve, and
that every quartic curve can be generated in this way. If a
surface with four conical points be chosen, the resulting quartic
has four double points; that is, it breaks up into two conics
%% -----File: 038.png---Folio 28-------
(\Fig{3}). By considering the shaded portions in the figure it
will readily be seen how, by the principle of continuity, the four
ovals of the quartic (\Fig{2}) are obtained. This corresponds
exactly to the derivation of the diagonal
surface from the cubic surface having four
conical points.

The attempts to extend this application
of the principle of continuity so as to gain
an insight into the shape of curves of the
$n$th~order have hitherto proved futile, as
far as a general classification and an enumeration
of all fundamental forms is concerned. Still, some
important results have been obtained. A paper by Harnack\footnote
  {\textit{Ueber die Vieltheiligkeit der ebenen algebraischen Curven}, Math.\ Annalen, Vol.~10
  (1876), pp.~189--198.}
and a more recent
%[Illustration: Fig.~3.]
\WFigure{1.5in}{038}
one by Hilbert\footnote
  {\textit{Ueber die reellen Züge algebraischer Curven}, Math.\ Annalen, Vol.~38 (1891),
  pp.~115--138.}
are here to be mentioned.
Harnack finds that, if $p$~be the deficiency of the curve, the
maximum number of separate branches the curve can have is~$p + 1$;
and a curve with $p + 1$~branches actually exists. Hilbert's
paper contains a large number of interesting special
results which from their nature cannot be included in the
present brief summary.

I myself have found a curious relation between the numbers
of real singularities.\footnote
  {\textit{Eine neue Relation zwischen den Singularitäten einer algebraischen Curve},
  Math.\ Annalen, Vol.~10 (1876), pp.~199--209.}
Denoting the order of the curve by~$n$,
the class by~$k$, and considering only simple singularities, we
may have three kinds of double points, say $d'$~ordinary and $d''$~isolated
real double points, besides imaginary double points;
then there may be $r'$~real cusps, besides imaginary cusps; and
similarly, by the principle of duality, $t'$~ordinary, $t''$~isolated
%% -----File: 039.png---Folio 29-------
real double tangents, besides imaginary double tangents; also
$w'$~real inflexions, besides imaginary inflexions. Then it can
be proved by means of the principle of continuity, that the
following relation must hold:
\[
n + w' + 2t'' = k + r' + 2d''.
\]

This general law contains everything that is known as to
curves of the third or fourth orders. It has been somewhat
extended in a more algebraic sense by several writers. Moreover,
Brill, in Vol.~16 of the \textit{Math.\ Annalen} (1880),\footnote
  {\textit{Ueber Singularitäten ebener algebraischer Curven und eine neue Curvenspecies},
  pp.~348--408.}
has shown
how the formula must be modified when higher singularities are
involved.

As regards quartic surfaces, Rohn has investigated an enormous
number of special cases; but a complete enumeration he
has not
%[Illustration: Fig.~4.]
\WFigure{1.5in}{039}
reached. Among the special
surfaces of the fourth order the Kummer
surface with $16$~conical points is
one of the most important. The
models constructed by Plücker in
connection with his theory of complexes
of lines all represent special
cases of the Kummer surface. Some
types of this surface are also included
in the Brill collection. But all these
models are now of less importance,
since Rohn found the following interesting
and comprehensive result.
Imagine a quadric surface with four generating lines of each set
(\Fig{4}). According to the character of the surface and the
reality, non-reality, or coincidence of these lines, a large number
of special cases is possible; all these cases, however, must be
%% -----File: 040.png---Folio 30-------
treated alike. We may here confine ourselves to the case of
an hyperboloid of one sheet with four distinct lines of each
set. These lines divide the surface into $16$~regions. Shading
the alternate regions as in the figure, and regarding the shaded
regions as double, the unshaded regions being disregarded, we
have a surface consisting of eight separate closed portions hanging
together only at the points of intersection of the lines; and
this is a Kummer surface with $16$~real double points. Rohn's
researches on the Kummer surface will be found in the \textit{Math.\
Annalen}, Vol.~18 (1881);\footnote
  {\textit{Die verschiedenen Gestalten der Kummer'schen Fläche}, pp.~99--159.}
his more general investigations on
quartic surfaces, \textit{ib}., Vol.~29 (1887).\footnote
  {\textit{Die Flächen vierter Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung},
  pp.~81--96.}

There is still another mode of dealing with the shape of
curves (not of surfaces), viz.\ by means of the theory of Riemann.
The first problem that here presents itself is to establish
the connection between a plane curve and a Riemann surface,
as I have done in Vol.~7 of the \textit{Math.\ Annalen} (1874).\footnote
  {\textit{Ueber eine neue Art der Riemann'schen Flächen}, pp.~558--566.}
Let us consider a cubic curve; its deficiency is $p = 1$. Now it
is well known that in Riemann's theory this deficiency is a
measure of the connectivity of the corresponding Riemann surface,
which, therefore, in the present case, must be that of a
\emph{tore}, or anchor-ring. The question then arises: what has the
anchor-ring to do with the cubic curve? The connection will
best be understood by considering the curve of the third \emph{class}
whose shape is represented in \Fig{5}. It is easy to see that of
%[Illustration: Fig.~5.] [** TN: Moved up one paragraph]
\Figure[1.75in]{041}
the three tangents that can be drawn to this curve from any
point in its plane, all three will be real if the point be selected
outside the oval branch, or inside the triangular branch; but that
only one of the three tangents will be real for any point in the
shaded region, while the other two tangents are imaginary. As
%% -----File: 041.png---Folio 31-------
there are thus two imaginary tangents corresponding to each
point of this region, let us imagine it covered with a double
leaf; along the curve the two leaves must, of course, be
regarded as joined. Thus we obtain a surface which can be
considered as a Riemann surface belonging to the curve, each
point of the surface corresponding to a single tangent of the
curve. Here, then, we have our anchor-ring. If on such a surface
we study integrals, they will be of double periodicity, and
the true reason is thus disclosed for the connection of elliptic
integrals with the curves of the third class, and hence, owing
to the relation of duality, with the curves of the third order.

To make a further advance, I passed to the general theory
of Riemann surfaces. To real curves will of course correspond
\emph{symmetrical} Riemann surfaces, \ie\ surfaces that reproduce
themselves by a conformal transformation of the second kind
(\ie\ a transformation that inverts the sense of the angles).
Now it is easy to enumerate the different symmetrical types
belonging to a given~$p$. The result is that there are altogether
$p+1$~``diasymmetric'' and $\left[\dfrac{p+1}{2}\right]$~``orthosymmetric'' cases.
If we denote as a line of symmetry any line whose points
%% -----File: 042.png---Folio 32-------
remain unchanged by the conformal transformation, the diasymmetric
cases contain respectively $p$, $p-1$,~$\dots$ $2$,~$1$,~$0$ lines
of symmetry, and the orthosymmetric cases contain $p+1$,~$p-1$,
$p-3$,~$\dots$ such lines. A surface is called diasymmetric or orthosymmetric
according as it does not or does break up into two
parts by cuts carried along all the lines of symmetry. This
enumeration, then, will contain a general classification of real
curves, as indicated first in my pamphlet on Riemann's theory.\footnote
  {\textit{Ueber Riemann's Theorie der algebraischen Functionen und ihrer Integrale},
  Leipzig, Teubner, 1882. An English translation by Frances Hardcastle (London,
  Macmillan) has just appeared.}
In the summer of 1892 I resumed the theory and developed
a large number of propositions concerning the reality of the
roots of those equations connected with our curves that can be
treated by means of the Abelian integrals. Compare the last
volume of the \textit{Math.\ Annalen}\footnote
  {\textit{Ueber Realitätsverhältnisse bei der einem beliebigen Geschlechte zugehörigen
  Normalcurve der~$\phi$}, Vol.~42 (1893), pp.~1--29.}
and my (lithographed) lectures
on \textit{Riemann'sche Flächen}, Part~II\@.

In the same manner in which we have to-day considered
ordinary algebraic curves and surfaces, it would be interesting
to investigate \emph{all} algebraic configurations so as to arrive at a
truly geometrical intuition of these objects.

In concluding, I wish to insist in particular on what I regard
as the principal characteristic of the geometrical methods that I
have discussed to-day: these methods give us an \emph{actual mental
image} of the configuration under discussion, and this I consider
as most essential in all true geometry. For this reason the
so-called synthetic methods, as usually developed, do not appear
to me very satisfactory. While giving elaborate constructions
for special cases and details they fail entirely to afford a general
view of the configurations as a whole.
%% -----File: 043.png---Folio 33-------

\Lecture{V.}{Theory of Functions and
Geometry.}

\Date{(September 1, 1893.)}

\First{A geometrical} representation of a function of a complex
variable $w = f(z)$, where $w = u + iv$ and $z = x + iy$, can be obtained
by constructing models of the two surfaces $u = \phi (x, y)$,
$v = \psi (x, y)$. This idea is realized in the models constructed
by Dyck, which I have shown to you at the Exhibition.

Another well-known method, proposed by Riemann, consists
in representing each of the two complex variables in the usual
way in a plane. To every point in the $z$-plane will correspond
one or more points in the $w$-plane; as $z$~moves in its plane, $w$~describes
a corresponding curve in the other plane. I may
refer to the work of Holzmüller\footnote
  {\textit{Einführung in die Theorie der isogonalen Verwandtschaften und der conformen
  Abbildungen, verbunden mit Anwendungen auf mathematische Physik}, Leipzig,
  Teubner, 1882.}
as a good elementary introduction
to this subject, especially on account of the large
number of special cases there worked out and illustrated by
drawings.

In higher investigations, what is of interest is not so much
the corresponding curves as corresponding areas or \emph{regions}
of the two planes. According to Riemann's fundamental
theorem concerning conformal representation, two simply connected
regions can always be made to correspond to each other
conformally, so that either is the conformal representation
%% -----File: 044.png---Folio 34-------
(\textit{Abbildung}) of the other. The three constants at our disposal
in this correspondence allow us to select three arbitrary points
on the boundary of one region as corresponding to three arbitrary
points on the boundary of the other region. Thus
Riemann's theory affords a geometrical definition for any function
whatever by means of its conformal representation.

This suggests the inquiry as to what conclusions can be
drawn from this method concerning the nature of transcendental
functions. Next to the elementary transcendental functions
the elliptic functions are usually regarded as the most
important. There is, however, another class for which at
least equal importance must be claimed on account of their
numerous applications in astronomy and mathematical physics;
these are the \emph{hypergeometric functions}, so called owing to their
connection with Gauss's hypergeometric series.

The hypergeometric functions can be defined as the integrals
of the following linear differential equation of the second order:
\begin{multline*}
\frac{d^{2}w}{dz^{2}}
  + \biggl[\frac{1 - \lambda' - \lambda''}{z - a} (a - b)(a - c)
         + \frac{1 - \mu' - \mu''}{z - b} (b - c)(b - a)  \\
  + \frac{1 - \nu' - \nu'' }{z - c } (c - a)(c - b)\biggr] \frac{dw}{dz}
  + \biggl[\frac{\lambda' \lambda'' (a - b)(a - c)}{z - a} \\
         + \frac{\mu' \mu'' (b - c)(b - a)}{z - b}
         + \frac{\nu' \nu'' (c - a)(c - b)}{z - c}\biggr]
    \frac{w}{(z - a)(z - b)(z - c)}= 0,
\end{multline*}
where~$z = a$, $b$,~$c$ are the three singular points and $\lambda'$,~$\lambda''$; $\mu'$,~$\mu''$;
$\nu'$,~$\nu''$ are the so-called exponents belonging respectively to
$a$,~$b$,~$c$.

If $w_{1}$~be a particular solution, $w_{2}$~another, the general solution
can be put in the form $\alpha w_{1} + \beta w_{2}$, where $\alpha$,~$\beta$ are arbitrary constants;
so that
\[
\alpha w_{1} + \beta w_{2}\quad \text{and}\quad \gamma w_{1} + \delta w_{2}
\]
represent a pair of general solutions.
%% -----File: 045.png---Folio 35-------

If we now introduce the quotient $\dfrac{w_{1}}{w_{2}} = \eta (z)$ as a new variable,
its most general value is
% [** TN: Inline equation in original]
\[
\frac{\alpha w_{1} + \beta w_{2}}{\gamma w_{1} + \delta w_{2}} =
\frac{\alpha\eta + \beta}{\gamma\eta + \delta}
\]
and contains therefore
three arbitrary constants. Hence $\eta$~satisfies a differential
equation of the third order which is readily found to be
\begin{multline*}
\frac{\eta'''}{\eta'} - \tfrac{3}{2} \left(\frac{\eta''}{\eta'}\right)^{2}\\
  = \frac{1}{(z - a)(z - b)(z - c)}
    \Biggl[\frac{\ \dfrac{1 - \lambda^{2}}{2}\ }{z - a} (a - b)(a - c)
         + \frac{\ \dfrac{1 - \mu^{2}}{2}\ }{z - b} (b - c)(b - a)\\
         + \frac{\ \dfrac{1 - \nu^{2}}{2}\ }{z - c} (c - a)(c - b)\Biggr],
\end{multline*}
in which the left-hand member has the property of not being
changed by a linear substitution, and is therefore called a differential
invariant. Cayley has named this function the Schwarzian
derivative; it has formed the starting-point for Sylvester's
investigations on reciprocants. In the right-hand member,
\[
±\lambda = \lambda' - \lambda'', \quad ±\mu = \mu' - \mu'', \quad ±\nu = \nu' - \nu''.
\]

As to the conformal representation (\Fig{6}), it can be shown
that the upper half of the $z$-plane, with the points $a$,~$b$,~$c$ on
%[Illustration: Fig.~6.]
\Figure{045}
the real axis and $\lambda$,~$\mu$,~$\nu$ assumed as real, is transformed for each
branch of the $\eta$-function into a triangular area~$abc$ bounded by
%% -----File: 046.png---Folio 36-------
three circular arcs; let us call such an area a \emph{circular triangle}
(\emph{Kreisbogendreieck}). The angles at the vertices of this triangle
are $\lambda\pi$,~$\mu\pi$,~$\nu\pi$.

This, then, is the geometrical representation we have to
take as our basis. In order to derive from it conclusions as
to the nature of the transcendental functions defined by the
differential equation, it will evidently be necessary to inquire
what are the forms of such circular triangles in the most
general case. For it is to be noticed that there is no restriction
laid upon the values of the constants $\lambda$,~$\mu$,~$\nu$, so that the
angles of our triangle are not necessarily acute, nor even
convex; in other words, in the general case the vertices will
be branch-points. The triangle itself is here to be regarded
as something like an extensible and flexible membrane spread
out between the circles forming the boundary.

I have investigated this question in a paper published in
the \textit{Math.\ Annalen}, Vol.~37.\footnote
  {\textit{Ueber die Nullstellen der hypergeometrischen Reihe}, pp.~573--590.}
It will be convenient to project
the plane containing the circular triangle stereographically on
a sphere. The question then is as to the most general form
of spherical triangles, taking this term in a generalized meaning
as denoting any triangle on the sphere bounded by the intersections
of three planes with the sphere, whether the planes
intersect at the centre or not.

This is really a question of elementary geometry; and it is
interesting to notice how often in recent times higher research
has led back to elementary problems not previously
settled.

The result in the present case is that there are two, and
only two, species of such generalized triangles. They are
obtained from the so-called elementary triangle by two distinct
operations: (\textit{a})~\emph{lateral}, (\textit{b})~\emph{polar attachment} of a circle.
%% -----File: 047.png---Folio 37-------

Let $abc$ (\Fig{7}) be the elementary spherical triangle. Then
the operation of lateral attachment consists in attaching to
the area~$abc$ the area enclosed by one of the sides, say~$bc$,
this side being produced so as to form a complete circle.
The process can, of course, be repeated any number of times
and applied to each side. If one circular area be attached at~$bc$,
the angles at $b$~and~$c$ are increased each by~$\pi$; if the
whole sphere be attached, by~$2\pi$,~etc. The vertices in this
way become branch-points. A triangle so obtained I call a\DPnote{** TN: italicized in original}
\emph{triangle of the first species}.

%[Illustration: Fig.~7.]
%[Illustration: Fig.~8.]
\Figures{1.625in}{047a}{1.75in}{047b}
A \emph{triangle of the second species} is produced by the process
of polar attachment of a circle, say at~$bc$; the whole area
bounded by the circle~$bc$ is, in this case, connected with the
original triangle along a branch-cut reaching from the vertex~$a$
to some point on~$bc$. The point~$a$ becomes a branch-point,
its angle being increased by~$2\pi$. Moreover, lateral attachments
can be made at $ab$~and~$ac$.

The two species of triangles are now characterized as follows:
\emph{the first species may have any number of lateral attachments
at any or all of the three sides, while the second has a polar
attachment to one vertex and the opposite side, and may have
lateral attachments to the other two sides}.
%% -----File: 048.png---Folio 38-------

Analytically the two species are distinguished by inequalities
between the absolute values of the constants $\lambda$,~$\mu$,~$\nu$. For
the first species, none of the three constants is greater than
the sum of the other two, \ie\
\[
|\lambda| \leqq |\mu| + |\nu|, \quad
|\mu| \leqq |\nu| + |\lambda|, \quad
|\nu| \leqq |\lambda| + |\mu|;
\]
for the second species,
\[
|\lambda| \geqq |\mu| + |\nu|,
\]
where $\lambda$ refers to the pole.

For the application to the theory of functions, it is important
to determine, in the case of the second species, the
number of times the circle formed by the side opposite the
vertex is passed around. I have found this number to be
$E\left(\dfrac{|\lambda| - |\mu| - |\nu| + 1}{2}\right)$, where $E$~denotes the greatest positive
integer contained in the argument, and is therefore always zero
when this argument happens to be negative or fractional.

Let us now apply these geometrical ideas to the theory of
hypergeometric functions. I can here only point out one of
the results obtained. Considering only the real values that
$\eta = w_{1}/w_{2}$ can assume between $a$ and~$b$, the question presents
itself as to the shape of the $\eta$-curve between these limits.
Let us consider for a moment the curves $w_{1}$ and~$w_{2}$. It is
well known that, if $w_{1}$~oscillates between $a$ and~$b$ from one
side of the axis to the other, $w_{2}$~will also oscillate; their
quotient $\eta = w_{1}/w_{2}$ is represented by a curve that consists of
separate branches extending from $-\infty$ to~$+\infty$, somewhat like
the curve $y = \tan x$. Now it appears as the result of the
investigation that the number of these branches, and therefore
the number of the oscillations of $w_{1}$~and~$w_{2}$, is given precisely
by the number of circuits of the point~$c$; that is to say, it is
$E\left(\dfrac{|\nu| - |\lambda| - |\mu| + 1}{2}\right)$. This is a result of importance for all
%% -----File: 049.png---Folio 39-------
applications of hypergeometric functions which was derived
only later (by Hurwitz) by means of Sturm's methods.

I wish to call your particular attention not so much to the
result itself, however interesting it may be, as to the geometrical
method adopted in deriving it. More advanced researches on a
similar line of thought are now being carried on at Göttingen
by myself and others.

When a differential equation with a larger number of singular
points than three is the object of investigation, the triangles
must be replaced by quadrangles and other polygons. In my
lithographed lectures on \textit{Linear Differential Equations}, delivered
in 1890--91, I have thrown out some suggestions regarding
the treatment of such cases. The difficulty arising in these
generalizations is, strange to say, merely of a geometrical
nature, viz.\ the difficulty of obtaining a general view of the
possible forms of the polygons.

Meanwhile, Dr.~Schoenflies has published a paper on rectilinear
polygons of any number of sides\footnote
  {\textit{Ueber Kreisbogenpolygone}, Math.\ Annalen, Vol.~42, pp.~377--408.}
while Dr.\ Van~Vleck
has considered such rectilinear polygons together with the
functions they define, the polygons being defined in so general
a way as to admit branch-points even in the interior. Dr.~Schoenflies
has also treated the case of circular quadrangles,
the result being somewhat complicated.

In all these investigations the singular points of the $z$-plane
corresponding to the vertices of the polygons are of course
assumed to be real, as are also their exponents. There remains
the still more general question how to represent by conformal
correspondence the functions in the case when some of these
elements are complex. In this direction I have to mention the
name of Dr.~Schilling who has treated the case of the ordinary
hypergeometric function on the assumption of complex exponents.
%% -----File: 050.png---Folio 40-------

This treatment of the functions defined by linear differential
equations of the second order is of course only an example
of the general discussion of complex functions by means of
geometry. I hope that many more interesting results will be
obtained in the future by such geometrical methods.
%% -----File: 051.png---Folio 41-------

%[** TN: Added comma matches table of contents]
\Lecture[Mathematical Character of Space-Intuition]
{VI.}{On the Mathematical Character
of Space-Intuition\DPtypo{}{,} and the
Relation of Pure Mathematics to
the Applied Sciences.}

\Date{(September 2, 1893.)}

\First{In} the preceding lectures I have laid so much stress on
geometrical methods that the inquiry naturally presents itself
as to the real nature and limitations of geometrical intuition.

In my address before the Congress of Mathematics at Chicago
I referred to the distinction between what I called the
\emph{naïve} and the \emph{refined} intuition. It is the latter that we find in
Euclid; he carefully develops his system on the basis of well-formulated
axioms, is fully conscious of the necessity of exact
proofs, clearly distinguishes between the commensurable and
incommensurable, and so forth.

The naïve intuition, on the other hand, was especially active
during the period of the genesis of the differential and integral
calculus. Thus we see that Newton assumes without hesitation
the existence, in every case, of a velocity in a moving point,
without troubling himself with the inquiry whether there might
not be continuous functions having no derivative.

At the present time we are wont to build up the infinitesimal
calculus on a purely analytical basis, and this shows that
we are living in a \emph{critical} period similar to that of Euclid.
It is my private conviction, although I may perhaps not be
able to fully substantiate it with complete proofs, that Euclid's
%% -----File: 052.png---Folio 42-------
period also must have been preceded by a ``naïve'' stage of
development. Several facts that have become known only
quite recently point in this direction. Thus it is now known
that the books that have come down to us from the time of
Euclid constitute only a very small part of what was then
in existence; moreover, much of the teaching was done by
oral tradition. Not many of the books had that artistic finish
that we admire in Euclid's ``Elements''; the majority were
in the form of improvised lectures, written out for the use
of the students. The investigations of Zeuthen\footnote
  {\textit{Die Lehre von den Kegelschnitten im Altertum}, übersetzt von R.~v.~Fischer-Benzon,
  Kopenhagen, Höst, 1886.}
and Allman\footnote
  {\textit{Greek geometry from Thales to Euclid}, Dublin, Hodges, 1889.}
have done much to clear up these historical conditions.

If we now ask how we can account for this distinction
between the naïve and refined intuition, I must say that, in
my opinion, the root of the matter lies in the fact that \emph{the
naïve intuition is not exact, while the refined intuition is not
properly intuition at all, but arises through the logical development
from axioms considered as perfectly exact}.

To explain the meaning of the first half of this statement it
is my opinion that, in our naïve intuition, when thinking of
a point we do not picture to our mind an abstract mathematical
point, but substitute something concrete for it. In imagining
a line, we do not picture to
%[Illustration: Fig.~9.]
\WFigure{1.5in}{052}
ourselves ``length without
breadth,'' but a \emph{strip} of a certain width.
Now such a strip has of course \emph{always}
a tangent (\Fig{9}); \ie\ we can always
imagine a straight strip having a small
portion (element) in common with the curved strip; similarly
with respect to the osculating circle. The definitions in this
case are regarded as holding only approximately, or as far as
may be necessary.
%% -----File: 053.png---Folio 43-------

The ``exact'' mathematicians will of course say that such
definitions are not definitions at all. But I maintain that in
ordinary life we actually operate with such inexact definitions.
Thus we speak without hesitancy of the direction and curvature
of a river or a road, although the ``line'' in this case has certainly
considerable width.

As regards the second half of my proposition, there actually
are many cases where the conclusions derived by purely logical
reasoning from exact definitions can no more be verified by
intuition. To show this, I select examples from the theory of
automorphic functions, because in more common geometrical
illustrations our judgment is warped by the familiarity of the
ideas.

Let any number of non-intersecting circles $1$,~$2$,~$3$, $4$,~$\dots$, be
given (\Fig{10}), and let every circle be reflected (\ie\ transformed
%[Illustration: Fig.~10.]
\Figure[3in]{053}
by inversion, or reciprocal radii vectores) upon every other circle;
then repeat this operation again and again, \textit{ad~infinitum}. The
question is, what will be the configuration formed by the totality
%% -----File: 054.png---Folio 44-------
of all the circles, and in particular what will be the position of
the limiting points. There is no difficulty in answering these
questions by purely logical reasoning; but the imagination
seems to fail utterly when we try to form a mental image of
the result.

Again, let a series of circles be given, each circle touching the
following, while the last touches the first (\Fig{11}). Every circle
is now reflected upon every other just as in the preceding example,
and the process is repeated indefinitely. The special case
when the original points of contact happen to lie on a circle
%[Illustration: Fig.~11.]
\Figure[2.5in]{054}
being excluded, it can be shown analytically that the continuous
curve which is the locus of all the points of contact \emph{is not an
analytic curve}. The points of contact form a manifoldness that
is everywhere dense on the curve (in the sense of G.~Cantor),
although there are intermediate points between them. At
each of the former points there is a determinate tangent,
while there is none at the intermediate points. Second derivatives
do not exist at all. It is easy enough to imagine a \emph{strip}
covering all these points; but when the width of the strip is
reduced beyond a certain limit, we find undulations, and it seems
impossible to clearly picture to the mind the final outcome.
It is to be noticed that we have here an example of a curve
%% -----File: 055.png---Folio 45-------
with indeterminate derivatives arising out of purely geometrical
considerations, while it might be supposed from the usual
treatment of such curves that they can only be defined by
artificial analytical series.

Unfortunately, I am not in a position to give a full account
of the opinions of philosophers on this subject. As regards
the more recent mathematical literature, I have presented my
views as developed above in a paper published in~1873, and
since reprinted in the \textit{Math.\ Annalen}.\footnote
  {\textit{Ueber den allgemeinen Functionsbegriff und dessen Darstellung durch eine
  willkürliche Curve}, Math.\ Annalen, Vol.~22 (1883), pp.~249--259.}
Ideas agreeing in
general with mine have been expressed by Pasch, of Giessen,
in two works, one on the foundations of geometry,\footnote
  {\textit{Vorlesungen über neuere Geometrie}, Leipzig, Teubner, 1882.}
the other
on the principles of the infinitesimal calculus.\footnote
  {\textit{Einleitung in die Differential- und Integralrechnung}, Leipzig, Teubner, 1882.}
Another
author, Köpcke, of Hamburg, has advanced the idea that our
space-intuition is exact as far as it goes, but so limited as to
make it impossible for us to picture to ourselves curves without
tangents.\footnote
  {\textit{Ueber Differentiirbarkeit und Anschaulichkeit der stetigen Functionen}, Math.\
  Annalen, Vol.~29 (1887), pp.~123--140.}

On one point Pasch does not agree with me, and that is as to
the exact value of the axioms. He believes---and this is the
traditional view---that it is possible finally to discard intuition
entirely, basing the whole science on the axioms alone. I am
of the opinion that, certainly, for the purposes of research it is
always necessary to combine the intuition with the axioms. I
do not believe, for instance, that it would have been possible to
derive the results discussed in my former lectures, the splendid
researches of Lie, the continuity of the shape of algebraic curves
and surfaces, or the most general forms of triangles, without
the constant use of geometrical intuition.
%% -----File: 056.png---Folio 46-------

Pasch's idea of building up the science purely on the basis of
the axioms has since been carried still farther by Peano, in his
logical calculus.

Finally, it must be said that the degree of exactness of the
intuition of space may be different in different individuals, perhaps
even in different races.\DPnote{** Yikes} It would seem as if a strong
naïve space-intuition were an attribute pre-eminently of the
Teutonic race, while the critical, purely logical sense is more
fully developed in the Latin and Hebrew races. A full investigation
of this subject, somewhat on the lines suggested by
Francis Galton in his researches on heredity, might be interesting.

What has been said above with regard to geometry ranges
this science among the applied sciences. A few general
remarks on these sciences and their relation to pure mathematics
will here not be out of place. From the point of view
of pure mathematical science I should lay particular stress on
the \emph{heuristic value} of the applied sciences as an aid to discovering
new truths in mathematics. Thus I have shown (in my
little book on Riemann's theories) that the Abelian integrals
can best be understood and illustrated by considering electric
currents on closed surfaces. In an analogous way, theorems
concerning differential equations can be derived from the consideration
of sound-vibrations; and so on.

But just at present I desire to speak of more practical matters,
corresponding as it were to what I have said before about
the inexactness of geometrical intuition. I believe that the
more or less close relation of any applied science to mathematics
might be characterized by the degree of exactness attained,
or attainable, in its numerical results. Indeed, a rough classification
of these sciences could be based simply on the number
of significant figures averaged in each. Astronomy (and some
branches of physics) would here take the first rank; the number
%% -----File: 057.png---Folio 47-------
of significant figures attained may here be placed as high as
seven, and functions higher than the elementary transcendental
functions can be used to advantage. Chemistry would probably
be found at the other end of the scale, since in this science
rarely more than two or three significant figures can be relied
upon. Geometrical drawing, with perhaps $3$~to $4$~figures, would
rank between these extremes; and so we might go on.

The ordinary mathematical treatment of any applied science
substitutes exact axioms for the approximate results of experience,
and deduces from these axioms the rigid mathematical
conclusions. In applying this method it must not be forgotten
that mathematical developments transcending the limit of exactness
of the science are of no practical value. It follows that a
large portion of abstract mathematics remains without finding
any practical application, the amount of mathematics that can
be usefully employed in any science being in proportion to the
degree of accuracy attained in the science. Thus, while the
astronomer can put to good use a wide range of mathematical
theory, the chemist is only just beginning to apply the first
derivative, \ie\ the rate of change at which certain processes are
going on; for second derivatives he does not seem to have
found any use as yet.

As examples of extensive mathematical theories that do not
exist for applied science, I may mention the distinction between
the commensurable and incommensurable, the investigations on
the convergency of Fourier's series, the theory of non-analytical
functions,~etc. It seems to me, therefore, that Kirchhoff makes
a mistake when he says in his \textit{Spectral-Analyse} that absorption
takes place only when there is \emph{exact} coincidence between the
wave-lengths. I side with Stokes, who says that absorption
takes place \emph{in the vicinity} of such coincidence. Similarly, when
the astronomer says that the periods of two planets must be
exactly commensurable to admit the possibility of a collision,
%% -----File: 058.png---Folio 48-------
this holds only abstractly, for their mathematical centres; and it
must be remembered that such things as the period, the mass,
etc., of a planet cannot be exactly defined, and are changing all
the time. Indeed, we have no way of ascertaining whether
two astronomical magnitudes are incommensurable or not; we
can only inquire whether their ratio can be expressed approximately
by two \emph{small} integers. The statement sometimes made
that there exist only analytic functions in nature is in my
opinion absurd. All we can say is that we restrict ourselves
to analytic, and even only to simple analytic, functions because
they afford a sufficient degree of approximation. Indeed, we
have the theorem (of Weierstrass) that any continuous function
can be approximated to, with any required degree of accuracy,
by an analytic function. Thus if $\phi(x)$ be our continuous function,
and $\delta$~a small quantity representing the given limit of
exactness (the width of the strip that we substitute for the
curve), it is always possible to determine an \emph{analytic} function~$f(x)$
such that
\[
\phi(x) = f(x) + \epsilon, \quad\text{where}\quad |\epsilon| < |\delta|,
\]
within the given limits.

All this suggests the question whether it would not be possible
to create a, let us say, \emph{abridged} system of mathematics
adapted to the needs of the applied sciences, without passing
through the whole realm of abstract mathematics. Such a
system would have to include, for example, the researches of
Gauss on the accuracy of astronomical calculations, or the more
recent and highly interesting investigations of Tchebycheff on
interpolation. The problem, while perhaps not impossible, seems
difficult of solution, mainly on account of the somewhat vague
and indefinite character of the questions arising.

I hope that what I have here said concerning the use of
mathematics in the applied sciences will not be interpreted
%% -----File: 059.png---Folio 49-------
as in any way prejudicial to the cultivation of abstract mathematics
as a pure science. Apart from the fact that pure
mathematics cannot be supplanted by anything else as a means
for developing the purely logical powers of the mind, there
must be considered here as elsewhere the necessity of the
presence of a few individuals in each country developed in a
far higher degree than the rest, for the purpose of keeping
up and gradually raising the \emph{general} standard. Even a slight
raising of the general level can be accomplished only when
some few minds have progressed far ahead of the average.

Moreover, the ``abridged'' system of mathematics referred
to above is not yet in existence, and we must for the present
deal with the material at hand and try to make the best of it.

Now, just here a practical difficulty presents itself in the
teaching of mathematics, let us say of the elements of the
differential and integral calculus. The teacher is confronted
with the problem of harmonizing two opposite and almost contradictory
requirements. On the one hand, he has to consider
the limited and as yet undeveloped intellectual grasp of his
students and the fact that most of them study mathematics
mainly with a view to the practical applications; on the other,
his conscientiousness as a teacher and man of science would
seem to compel him to detract in nowise from perfect mathematical
rigour and therefore to introduce from the beginning
all the refinements and niceties of modern abstract mathematics.
In recent years the university instruction, at least in
Europe, has been tending more and more in the latter direction;
and the same tendencies will necessarily manifest themselves
in this country in the course of time. The second
edition of the \textit{Cours d'analyse} of Camille Jordan may be
regarded as an example of this extreme refinement in laying
the foundations of the infinitesimal calculus. To place a work
of this character in the hands of a beginner must necessarily
%% -----File: 060.png---Folio 50-------
have the effect that at the beginning a large part of the subject
will remain unintelligible, and that, at a later stage, the
student will not have gained the power of making use of
the principles in the simple cases occurring in the applied
sciences.

It is my opinion that in teaching it is not only admissible,
but absolutely necessary, to be less abstract at the start, to
have constant regard to the applications, and to refer to the
refinements only gradually as the student becomes able to
understand them. This is, of course, nothing but a universal
pedagogical principle to be observed in all mathematical
instruction.

Among recent German works I may recommend for the use
of beginners, for instance, Kiepert's new and revised edition of
Stegemann's text-book;\footnote
  {\textit{Grundriss der Differential- und Integral-Rechnung}, 6te~Auflage, herausgegeben
  von~Kiepert, Hannover, Helwing, 1892.}
this work seems to combine simplicity
and clearness with sufficient mathematical rigour. On
the other hand, it is a matter of course that for more advanced
students, especially for professional mathematicians, the study
of works like that of Jordan is quite indispensable.

I am led to these remarks by the consciousness of a growing
danger in the higher educational system of Germany,---the
danger of a separation between abstract mathematical science
and its scientific and technical applications. Such separation
could only be deplored; for it would necessarily be followed by
shallowness on the side of the applied sciences, and by isolation
on the part of pure mathematics.
%% -----File: 061.png---Folio 51-------

\Lecture[Transcendency of the Numbers $e$ and $\pi$.]
{VII.}{The Transcendency of the
Numbers $e$ and $\pi$.}

\Date{(September 4, 1893.)}

\First{Last} Saturday we discussed inexact mathematics; to-day we
shall speak of the most exact branch of mathematical science.

It has been shown by G.~Cantor that there are two kinds
of infinite manifoldnesses: (\textit{a})~\emph{countable} (\emph{abzählbare}) manifoldnesses,
whose quantities can be numbered or enumerated so that
to each quantity a definite place can be assigned in the system;
and (\textit{b})~\emph{non-countable} manifoldnesses, for which this is not possible.
To the former group belong not only the rational numbers,
but also the so-called \emph{algebraic} numbers, \ie\ all numbers defined
by an algebraic equation,
\[
a + a_{1}x + a_{2}x^{2} + \cdots + a_{n}x^{n} = 0
\]
with integral coefficients ($n$~being of course a positive integer).
As an example of a non-countable manifoldness I may mention
the totality of all numbers contained in a \emph{continuum}, such as
that formed by the points of the segment of a straight line.
Such a continuum contains not only the rational and algebraic
numbers, but also the so-called transcendental numbers. The
actual existence of transcendental numbers which thus naturally
follows from Cantor's theory of manifoldnesses had been proved
before, from considerations of a different order, by Liouville.
With this, however, is not yet given any means for deciding
whether any particular number is transcendental or not. But
%% -----File: 062.png---Folio 52-------
during the last twenty years it has been established that the
two fundamental numbers $e$ and~$\pi$ are really transcendental.
It is my object to-day to give you a clear idea of the very
simple proof recently given by Hilbert for the transcendency of
these two numbers.

%[** TN: Journal titles in next two footnotes (inconsistently) italicized in original]
The history of this problem is short. Twenty years ago,
Hermite\footnote
  {Comptes rendus, Vol.~77 (1873), p.~18,~etc.}
first established the transcendency of~$e$; \ie\ he
showed, by somewhat complicated methods, that the number~$e$
cannot be the root of an algebraic equation with integral
coefficients. Nine years later, Lindemann,\footnote
  {Math.\ Annalen, Vol.~20 (1882), p.~213.}
taking the developments
of Hermite as his point of departure, succeeded in
proving the transcendency of~$\pi$. Lindemann's work was
verified soon after by Weierstrass.

The proof that $\pi$~is a transcendental number will forever
mark an epoch in mathematical science. It gives the final
answer to the problem of squaring the circle and settles this
vexed question once for all. This problem requires to derive
the number~$\pi$ by a finite number of elementary geometrical
processes, \ie\ with the use of the ruler and compasses alone.
As a straight line and a circle, or two circles, have only two
intersections, these processes, or any finite combination of
them, can be expressed algebraically in a comparatively simple
form, so that a solution of the problem of squaring the circle
would mean that $\pi$~can be expressed as the root of an algebraic
equation of a comparatively simple kind, viz.\ one that is solvable
by square roots. Lindemann's proof shows that $\pi$~is not the
root of any algebraic equation.

The proof of the transcendency of~$\pi$ will hardly diminish the
number of circle-squarers, however; for this class of people has
always shown an absolute distrust of mathematicians and a
%% -----File: 063.png---Folio 53-------
contempt for mathematics that cannot be overcome by any
amount of demonstration. But Hilbert's simple proof will
surely be appreciated by all those who take interest in the
establishment of mathematical truths of fundamental importance.
This demonstration, which includes the case of the
number~$e$ as well as that of~$\pi$, was published quite recently
in the \textit{Göttinger Nachrichten}.\footnote
  {1893, No.~2, p.~113.}
Immediately after\footnote
  {\textit{Ib}., No.~4.}
Hurwitz
published a proof for the transcendency of~$e$ based on still
more elementary principles; and finally, Gordan\footnote
  {Comptes rendus,\DPnote{** TN: Ital. in original} 1893, p.~1040.}
gave a further
simplification. All three of these papers will be reprinted
in the next \textit{Heft} of the \textit{Math.\ Annalen}.\footnote
  {Vol.~43 (1894), pp.~216--224.}
The problem has
thus been reduced to such simple terms that the proofs for
the transcendency of $e$ and~$\pi$ should henceforth be introduced
into university teaching everywhere.

Hilbert's demonstration is based on two propositions. One
of these simply asserts the transcendency of~$e$, \emph{\ie\ the impossibility
of an equation of the form}
\[
a + a_{1}e + a_{2}e^{2} + \cdots + a_{n}e^{n} = 0,
\Tag{(1)}
\]
where $a$,~$a_{1}$, $a_{2}$,~$\dots$~$a_{n}$ are integral numbers. This is the original
proposition of Hermite. To prove the transcendency of~$\pi$,
another proposition (originally due to Lindemann) is required,
which asserts \emph{the impossibility of an equation of the form}
\[
a + e^{\beta_{1}} + e^{\beta_{2}} + \cdots + e^{\beta_{n}} = 0,
\Tag{(2)}
\]
where $a$~is an integer, and the exponents are algebraic numbers,
viz.\ the roots of an algebraic equation
\[
b\beta^{m} + b_{1}\beta^{m-1} + b_{2}\beta^{m-2} + \cdots + b_{m} = 0,
\]
$b$,~$b_{1}$, $b_{2}$,~$\dots~b_{m}$ being integers.
%% -----File: 064.png---Folio 54-------

It will be noticed that the latter proposition really includes
the former as a special case; for it is of course possible that
the~$\beta$'s are rational integral numbers, and whenever some of the
roots of the equation for~$\beta$ are equal, the corresponding terms
in the equation~\Eq{(2)} will combine into a single term of the form~$a_{k}e^{\beta_{k}}$.
The former proposition is therefore introduced only for
the sake of simplicity.

The central idea of the proof of the impossibility of equation~\Eq{(1)}
consists in introducing for the quantities $1 : e : e^{2} : \dots : e^{n}$, in
which the equation is homogeneous, proportional quantities
\[
I_{0} + \epsilon_{0} : I_{1} + \epsilon_{1} : I_{2} + \epsilon _{2} : \dots : I_{n} + \epsilon_{n},
\]
selected so that each consists of an integer~$I$ and a very small
fraction~$\epsilon$. The equation then assumes the form
\[
(aI_{0} + a_{1}I_{1} + \cdots + a_{n}I_{n}) + (a\epsilon_{0} + a_{1}\epsilon_{1} +\cdots + a_{n}\epsilon_{n}) = 0,
\Tag{(3)}
\]
and it can be shown that the $I$'s and~$\epsilon$'s can always be so
selected as to make the quantity in the first parenthesis, which
is of course integral, different from zero, while the quantity in
the second parenthesis becomes a proper fraction. Now, as
the sum of an integer and a proper fraction cannot be equal
to zero, the equation~\Eq{(1)} is proved to be impossible.

So much for the general idea of Hilbert's proof. It will be
seen that the main difficulty lies in the proper determination
of the integers~$I$ and the fractions~$\epsilon$. For this purpose Hilbert
makes use of a definite integral suggested by the investigations
of Hermite, viz.\ the integral
\[
J = \int_{0}^{\infty} z^\rho \bigl[(z - 1) \cdots (z - n)\bigr]^{\rho+1} e^{-z}\,dz,
\]
where $\rho$~is an integer to be determined afterwards. Multiplying
equation~\Eq{(1)} term for term by this integral and dividing
by~$\rho!$, this equation can evidently be put into the form
%% -----File: 065.png---Folio 55-------
\begin{multline*}
\left(a \frac{\int_{0}^{\infty}}{\rho!}
  + a_{1}e \frac{\int_{1}^{\infty}}{\rho!}
  + a_{2}e^{2}\frac{\int_{2}^{\infty}}{\rho!} + \cdots
  + a_{n}e^{n}\frac{\int_{n}^{\infty}}{\rho!}\right)\\
  + \left(a_{1}e \frac{\int_{0}^{1}}{\rho!}
     + a_{2}e^{2}\frac{\int_{0}^{2}}{\rho!} + \cdots
     + a_{n}e^{n}\frac{\int_{0}^{n}}{\rho!}\right) = 0,
\end{multline*}
or designating for shortness the quantities in the two parentheses
by $P_{1}$~and~$P_{2}$, respectively,
\[
P_{1} + P_{2} = 0.
\]

Now it can be proved that the coefficients of $a$,~$a_{1}$, $a_{2}$,~$\dots~a_{n}$
in~$P_{1}$ are all integers, that $\rho$~can be so selected as to make
$P_{1}$~different from zero, and that at the same time $\rho$~can be
taken so large as to make $P_{2}$ as small as we please. Thus,
equation~\Eq{(1)} will be reduced to the impossible form~\Eq{(3)}.

We proceed to prove these properties of $P_{1}$~and~$P_{2}$. The
integral~$J$ is readily seen to be an integer divisible by~$\rho!$,
owing to the well-known relation $\int_{0}^{\infty}z^{\rho}e^{-z}\,dz = \rho!$. Similarly,
by substituting $z = z' + 1$, $z = z' + 2$,~$\dots$ $z = z' + n$, it can be shown
that $e \int_{1}^{\infty}$, $e^{2} \int_{2}^{\infty}$,~$\dots \DPtypo{e}{e^{n}}\int_{n}^{\infty}$ are integers divisible by~$(\rho + 1)!$. It
follows that $P_{1}$~is an integer, viz.\
\[
P_{1}\equiv ±a(n!)^{\rho + 1} \pmod[sq]{\rho + 1}.
\]
If, therefore, $\rho$~be selected so as to make the right-hand member
of this congruence not divisible by~$\rho + 1$, the whole expression~$P_{1}$
is different from zero.

As regards the condition that $P_{2}$ should be made as small
as we please, it can evidently be fulfilled by selecting a sufficiently
large value for~$\rho$; this is of course consistent with
the condition of making $J$ not divisible by~$\rho + 1$. For by the
theorem of mean values (\textit{Mittelwertsatz}) the integrals can be
replaced by powers of constant quantities with $\rho$ in the exponent;
%% -----File: 066.png---Folio 56-------
and the rate of increase of a power is, for sufficiently
large values of~$\rho$, always smaller than that of the factorial which
occurs in the denominator.

The proof of the impossibility of equation~\Eq{(2)} proceeds on
precisely analogous lines. Instead of the integral~$J$ we have
now to use the integral
\[
J' = b^{m(\rho + 1)}\int_{0}^{\infty} z^{\rho}\bigl[(z - \beta_{1})(z - \beta_{2}) \cdots (z - \beta_{m})\bigr]^{\rho + 1}e^{-z}\,dz,
\]
the $\beta$'s being the roots of the algebraic equation
\[
b\beta^{m} + b_{1}\beta^{m-1} + \cdots + b_{m} = 0.
\]
This integral is decomposed as follows:
\[
\int_{0}^{\infty} = \int_{0}^{\beta} + \int_{\beta}^{\infty},
\]
where of course the path of integration must be properly
determined for complex values of~$\beta$. For the details I must
refer you to Hilbert's paper.

Assuming the impossibility of equation~\Eq{(2)}, the transcendency
of~$\pi$
%[Illustration: Fig.~12.]
\WFigure{2in}{066}
follows easily from the following considerations, originally
given by Lindemann. We notice
first, as a consequence of our theorem,
that, \emph{with the exception of
the point $x = 0$, $y = 1$, the exponential
curve $y = e^{x}$ has no algebraic
point}, \ie\ no point both of whose
co-ordinates are algebraic numbers.
In other words, however
densely the plane may be covered
with algebraic points, the exponential curve (\Fig{12}) manages
to pass along the plane without meeting them, the single point~$(0, 1)$
excepted. This curious result can be deduced as follows
from the impossibility of equation~\Eq{(2)}. Let~$y$ be any algebraic
%% -----File: 067.png---Folio 57-------
quantity, \ie\ a root of any algebraic equation, and let $y_{1}$,~$y_{2}$,~$\dots$
be the other roots of the same equation; let a similar notation
be used for~$x$. Then, if the exponential curve have any algebraic
point~$(x, y)$, (besides $x = 0$, $y = 1$), the equation
\[
\left.
\begin{array}{@{}l@{}l@{}l@{}l@{}}
  (y - e^{x})    & (y_{1} - e^{x})    & (y_{2} - e^{x})    &\cdots \\
  (y - e^{x_{1}}) & (y_{1} - e^{x_{1}}) & (y_{2} - e^{x_{1}}) &\cdots \\
  (y - e^{x_{2}}) & (y_{1} - e^{x_{2}}) & (y_{2} - e^{x_{2}}) &\cdots \\
\hdotsfor[3]{4}
\end{array}
\right\} = 0
\]
must evidently be fulfilled. But this equation, when multiplied
out, has the form of equation~\Eq{(2)}, which has been shown to be
impossible.

As second step we have only to apply the well-known identity
\[
\DPtypo{1}{-1} = e^{i\pi},
\]
which is a special case of $y = e^{x}$. Since in this identity $y = \DPtypo{1}{-1}$ is
algebraic, $x = i\pi$ must be transcendental.
%% -----File: 068.png---Folio 58-------

\Lecture{VIII.}{Ideal Numbers.}

\Date{(September 5, 1893.)}

\First{The} theory of numbers is commonly regarded as something
exceedingly difficult and abstruse, and as having hardly any
connection with the other branches of mathematical science.
This view is no doubt due largely to the method of treatment
adopted in such works as those of Kummer, Kronecker, Dedekind,
and others who have, in the past, most contributed to the
advancement of this science. Thus Kummer is reported as
having spoken of the theory of numbers as the only \emph{pure}
branch of mathematics not yet sullied by contact with the
applications.

Recent investigations, however, have made it clear that there
exists a very intimate correlation between the theory of numbers
and other departments of mathematics, not excluding
geometry.

As an example I may mention the theory of the reduction
of binary quadratic forms as treated in the \textit{Elliptische Modulfunctionen}.
An extension of this method to higher dimensions
is possible without serious difficulties. Another example you
will remember from the paper by Minkowski, \textit{Ueber Eigenschaften
von ganzen Zahlen, die durch räumliche Anschauung
erschlossen sind}, which I had the pleasure of presenting to
you in abstract at the Congress of Mathematics. Here geometry
is used directly for the development of new arithmetical
ideas.
%% -----File: 069.png---Folio 59-------

To-day I wish to speak on the \emph{composition of binary algebraic
forms}, a subject first discussed by Gauss in his \textit{Disquisitiones
arithmeticæ}\footnote
  {In the 5th~section; see Gauss's \textit{Werke}, Vol.~I, p.~239.}
and of Kummer's corresponding theory of \emph{ideal
numbers}. Both these subjects have always been considered as
very abstruse, although Dirichlet has somewhat simplified the
treatment of Gauss. I trust you will find that the geometrical
considerations by means of which I shall treat these questions
introduce so high a degree of simplicity and clearness that for
those not familiar with the older treatment it must be difficult
to realize why the subject should ever have been regarded as
so very intricate. These considerations were indicated by
myself in the \textit{Göttinger Nachrichten} for January,~1893; and
at the beginning of the summer semester of the present year
I treated them in more extended form in a course of lectures. I
have since learned that similar ideas were proposed by Poincaré
in~1881; but I have not yet had sufficient leisure to make a
comparison of his work with my own.

I write a binary quadratic form as follows:
\[
f = ax^{2} + bxy + cy^{2},
\]
\ie\ without the factor~$2$ in the second term; some advantages
of this notation were recently pointed out by H.~Weber, in
the \textit{Göttinger Nachrichten}, 1892--93. The quantities $a$,~$b$,~$c$, $x$,~$y$
are here of course all assumed to be integers.

It is to be noticed that in the theory of numbers a common
factor of the coefficients $a$,~$b$,~$c$ cannot be introduced or omitted
arbitrarily, as in projective geometry; in other words, we are
concerned with the form, not with an equation. Hence we
make the supposition that the coefficients $a$,~$b$,~$c$ have no
common factor; a form of this character is called a \emph{primitive
form}.
%% -----File: 070.png---Folio 60-------

As regards the discriminant
\[
D = b^{2} - 4ac,
\]
we shall assume that it has no quadratic divisor (and hence
cannot be itself a square), and that it is different from zero.
Thus $D$~is either $\equiv 0$ or $\equiv 1 \pmod{4}$. Of the two cases,
\[
D < 0\quad \text{and} \quad D > 0,
\]
which have to be considered separately, I select the former as
being more simple. Both cases were treated in my lectures
referred to before.

The following elementary geometrical interpretation of the
binary quadratic form was given by Gauss, who was much
inclined to using geometrical considerations in all branches of
mathematics. Construct a parallelogram (\Fig{13}) with two
%[Illustration: Fig.~13.]
\Figure[4in]{070}
adjacent sides equal to $\sqrt{a}$,~$\sqrt{c}$, respectively, and the included
angle~$\phi$ such that $\cos\phi = \dfrac{b}{2\sqrt{ac}}$. As $b^{2} - 4ac < 0$, $a$~and~$c$ have
necessarily the same sign; we here assume that $a$~and~$c$ are
%% -----File: 071.png---Folio 61-------
both positive; the case when they are both negative can
readily be treated by changing the signs throughout. Next
produce the sides of the parallelogram indefinitely, and draw
parallels so as to cover the whole plane by a network of
equal parallelograms. I shall call this a \emph{line-lattice} (\emph{Parallelgitter}).

We now select any one of the intersections, or \emph{vertices}, as
origin~$O$, and denote every other vertex by the symbol~$(x, y)$,
$x$~being the number of sides~$\sqrt{a}$, $y$~that of sides~$\sqrt{c}$, which
must be traversed in passing from~$O$ to~$(x, y)$. Then every
value that the form~$f$ takes for integral values of~$x$,~$y$ evidently
represents the square of the distance of the point~$(x, y)$ from~$O$.
Thus the lattice gives a complete geometrical representation
of the binary quadratic form. The discriminant~$D$ has
also a simple geometrical interpretation, the area of each parallelogram
being $= \frac{1}{2} \sqrt{-D}$.

Now, in the theory of numbers, two forms
\[
f = ax^{2} + bxy + cy^{2}\quad\text{and}\quad f' = a'x'^{2} + b'x'y' + c'y'^{2}
\]
are regarded as equivalent if one can be derived from the other
by a linear substitution whose determinant is~$1$, say
\[
x' = \alpha x + \beta y,\quad
y' = \gamma x + \delta y,
\]
where $\alpha \delta - \beta \gamma = 1$, $\alpha$, $\beta$, $\gamma$, $\delta$ being integers. All forms equivalent
to a given one are said to compose a \emph{class} of quadratic
forms; these forms have all the same discriminant. What
corresponds to this equivalence in our geometrical representation
will readily appear if we fix our attention on the vertices
only (\Fig{14}); we then obtain what I propose to call a \emph{point-lattice}
(\emph{Punktgitter}). Such a network of points can be connected
in various ways by two sets of parallel lines; \ie\ the
point-lattice represents an infinite number of line-lattices. Now
it results from an elementary investigation that the point-lattice
%% -----File: 072.png---Folio 62-------
is the geometrical image of the \emph{class} of binary quadratic
forms, the infinite number of line-lattices contained in
the point-lattice corresponding exactly to the infinite number
of binary forms contained in the class.

%[Illustration: Fig.~14.]
\Figure[4in]{072}
It is further known from the theory of numbers that to
every value of~$D$ belongs only a finite number of classes;
hence to every~$D$ will correspond a finite number of point-lattices,
which we shall afterwards consider together.

Among the different classes belonging to the same value of~$D$,
there is one class of particular importance, which I call the
\emph{principal class}. It is defined as containing the form
\[
x^{2} - \tfrac{1}{4} Dy^{2}
\]
when $D \equiv 0\pmod{4}$, and the form
\[
x^{2} + xy + \tfrac{1}{4}(1 - D)y^{2},
\]
when $D \equiv 1 \pmod{4}$. It is easy to see that the corresponding
lattices are very simple. When $D \equiv 0 \pmod{4}$, the principal
lattice is rectangular, the sides of the elementary parallelogram
%% -----File: 073.png---Folio 63-------
being~$1$ and~$\sqrt{-\frac{1}{4}D}$. For $D \equiv 1 \pmod{4}$, the parallelogram
becomes a rhombus. For the sake of simplicity, I shall here
consider only the former case.

Let us now define complex numbers in connection with the
principal lattice of the rectangular type (\Fig{15}). The point~$(x, y)$
%[Illustration: Fig.~15.]
\Figure[2.5in]{073}
of the lattice will represent simply the complex number
\[
x + \sqrt{-\tfrac{1}{4}D} · y;
\]
such numbers we shall call \emph{principal numbers}.

In any system of numbers the laws of multiplication are of
prime importance. For our principal numbers it is easy to
prove that the product of any two of them always gives a
principal number; \emph{\ie\ the system of principal numbers is, for
multiplication, complete in itself}.

We proceed next to the consideration of lattices of discriminant~$D$
that do not belong to the principal class; let us call
them \emph{secondary lattices} (\emph{Nebengitter}). Before investigating the
laws of multiplication of the corresponding numbers, I must
call attention to the fact that there is one feature of arbitrariness
in our representation that has not yet been taken into
account; this is the \emph{orientation} of the lattice, which may be
regarded as given by the angles, $\psi$~and~$\chi$, made by the sides
%% -----File: 074.png---Folio 64-------
$\sqrt{a}$,~$\sqrt{c}$, respectively, with some fixed initial line (\Fig{16}).
For the angle~$\phi$ of the parallelogram we have evidently $\phi = \chi - \psi$.
The point~$(x, y)$ of the lattice will thus give the complex number
\[
e^{i\psi} \left[\sqrt{a} · x + \frac{-b + \sqrt{D}}{2\sqrt{a}} · y\right]
  = e^{i\psi} · \sqrt{a} · x + e^{i\chi} · \sqrt{c} · y,
\]
which we call a \emph{secondary number}. The definition of a secondary
number is therefore indeterminate as long as $\psi$~or~$\chi$ is not
fixed.

Now, by determining~$\psi$ properly for every secondary point-lattice,
it is always possible to bring about the important result
%[Illustration: Fig.~16.]
\Figure[2.5in]{074}
that \emph{the product of any two complex numbers of all our lattices
taken together will again be a complex number of the system},
so that the totality of these complex numbers forms, likewise,
for multiplication, a complete system.

Moreover, the multiplication combines the lattices in a
definite way; thus, if any number belonging to the lattice~$L_{1}$
be multiplied into any number of the lattice~$L_{2}$, we always obtain
a number belonging to a definite lattice~$L_{3}$.

These properties will be seen to correspond exactly to the
characteristic properties of Gauss's \emph{composition of algebraic
forms}. For Gauss's law merely asserts that the product of
%% -----File: 075.png---Folio 65-------
two ordinary numbers that can be represented by two primitive
forms $f_{1}$,~$f_{2}$ of discriminant~$D$ is always representable by a
definite primitive form~$f_{3}$ of discriminant~$D$. This law is
included in the theorem just stated, inasmuch as the values of
$\sqrt{f_{1}}$,~$\sqrt{f_{2}}$,~$\sqrt{f_{3}}$ represent the distances of the points in the
lattices from the origin. At the same time we notice that
Gauss's law is not exactly equivalent to our theorem, since
in the multiplication of our complex numbers, not only the
distances are multiplied, but the angles~$\phi$ are added.

It is not impossible that Gauss himself made use of similar
considerations in deducing his law, which, taken apart from this
geometrical illustration, bears such an abstruse character.

It now remains to explain what relation these investigations
have to the ideal numbers of Kummer. This involves the
question as to the division of our complex numbers and their
resolution into primes.

In the ordinary theory of real numbers, every number can
be resolved into primes in only one way. Does this fundamental
law hold for our complex numbers? In answering this question
we must distinguish between the system formed by the totality
of all our complex numbers and the system of principal numbers
alone. For the former system the answer is: yes, every complex
number can be decomposed into complex primes in only
one way. We shall not stop to consider the proof which is
directly contained in the ordinary theory of binary quadratic
forms. But if we proceed to the consideration of the system
of principal numbers alone, the matter is different. There
are cases when a principal number can be decomposed in
more than one way into prime factors, \ie\ principal numbers
not decomposable into principal factors. Thus it may happen
that we have $m_{1}m_{2} = n_{1}n_{2}$; $m_{1}$,~$m_{2}$, $n_{1}$,~$n_{2}$ being principal primes.
The reason is,\DPnote{** [sic]} that these principal numbers are no longer primes
%% -----File: 076.png---Folio 66-------
if we adjoin the secondary numbers, but are decomposable as
follows:
\begin{alignat*}{2}
m_{1}& = \alpha · \beta,  \quad & m_{2} &= \gamma · \delta, \\
n_{1}& = \alpha · \gamma, \quad & n_{2} &= \beta  · \delta,
\end{alignat*}
$\alpha$,~$\beta$,~$\gamma$,~$\delta$ being primes in the enlarged system. \emph{In investigating
the laws of division it is therefore not convenient to consider the
principal system by itself; it is best to introduce the secondary
systems.} Kummer, in studying these questions, had originally
at his disposal only the principal system; and noticing the
imperfection of the resulting laws of division, he introduced
by definition his \emph{ideal} numbers so as to re-establish the ordinary
laws of division. These ideal numbers of Kummer are thus
seen to be nothing but abstract representatives of our secondary
numbers. The whole difficulty encountered by every one when
first attacking the study of Kummer's ideal numbers is therefore
merely a result of his mode of presentation. By introducing
from the beginning the secondary numbers by the side of
the principal numbers, no difficulty arises at all.

It is true that we have here spoken only of complex numbers
containing square roots, while the researches of Kummer himself
and of his followers, Kronecker and Dedekind, embrace all
possible algebraic numbers. But our methods are of universal
application; it is only necessary to construct lattices in spaces
of higher dimensions. It would carry us too far to enter into
details.
%% -----File: 077.png---Folio 67-------

\Lecture[Solution of Higher Algebraic Equations.]
{IX.}{The Solution of Higher Algebraic
Equations.}

\Date{(September 6, 1893.)}

\First{Formerly} the ``solution of an algebraic equation'' used to
mean its solution by radicals. All equations whose solutions
cannot be expressed by radicals were classed simply as \emph{insoluble},
although it is well known that the Galois groups belonging to
such equations may be very different in character. Even at
the present time such ideas are still sometimes found prevailing;
and yet, ever since the year~1858, a very different point of
view should have been adopted. This is the year in which
Hermite and Kronecker, together with Brioschi, found the
solution of the equation of the fifth degree, at least in its
fundamental ideas.

This solution of the quintic equation is often referred to as
a ``solution by elliptic functions''; but this expression is not
accurate, at least not as a counterpart to the ``solution by
radicals.'' Indeed, the elliptic functions enter into the solution
of the equation of the fifth degree, as logarithms might be said
to enter into the solution of an equation by radicals, because
the radicals can be computed by means of logarithms. \emph{The
solution of an equation will, \emph{in the present lecture}, be regarded
as consisting in its reduction to certain algebraic normal equations.}
That the irrationalities involved in the latter can, in
the case of the quintic equation, be computed by means of
tables of elliptic functions (provided that the proper tables of
%% -----File: 078.png---Folio 68-------
the corresponding class of elliptic functions were available)
is an additional point interesting enough in itself, but not to
be considered by us to-day.

I have simplified the solution of the quintic, and think that
I have reduced it to the simplest form, by introducing the
\emph{icosahedron equation} as the proper normal equation.\footnote
  {See my work \textit{Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen
  vom fünften Grade}, Leipzig, Teubner, 1884.}
In other
words, the icosahedron equation determines the typical irrationality
to which the solution of the equation of the fifth
degree can be reduced. This method is capable of being so
generalized as to embrace a whole theory of the solution of
higher algebraic equations; and to this I wish to devote the
present lecture.

It may be well to state that I speak here of equations with
coefficients that are not fixed numerically; the equations are
considered from the point of view of the theory of functions,
the coefficients corresponding to the independent variables.

In saying that an equation is solvable by radicals we mean
that it is reducible by algebraic processes to so-called pure
equations,
\[
\eta^{n} = z,
\]
where $z$~is a known quantity; then only the new question
arises, how $\eta = \sqrt[n]{z}$ can be computed. Let us compare from
this point of view the icosahedron equation with the pure
equation.

The icosahedron equation is the following equation of the
$60$th~degree:
\[
\frac{H^{3}(\eta)}{1728f^{5}(\eta)} = z,
\]
where $H$~is a numerical expression of the~$20$th, $f$~one of the
$12$th~degree, while $z$~is a known quantity. For the actual
%% -----File: 079.png---Folio 69-------
forms of $H$ and~$f$ as well as other details I refer you to the
\textit{Vorlesungen über das Ikosaeder}; I wish here only to point
out the characteristic properties of this equation.

(1)~Let $\eta$ be any one of the roots; then the $60$~roots can
all be expressed as linear functions of~$\eta$, with known coefficients,
such as for instance,
\[
\eta,\quad \frac{1}{\eta},\quad \epsilon \eta,\quad
\frac{(\epsilon - \epsilon^{4})\eta - (\epsilon^{2} - \epsilon^{3})}
     {(\epsilon^{2} - \epsilon^{3})\eta + (\epsilon - \epsilon^{4})},\quad \text{etc.},
\]
where $\epsilon = e^{\frac{2i\pi}{5}}$. These $60$~quantities, then, form a group of $60$~linear
substitutions.
%[Illustration: Fig.~17.]
\Figure{079a}

(2)~Let us next illustrate geometrically the dependence of~$\eta$
on~$z$ by establishing the conformal representation of the $z$-plane
on the $\eta$-plane, or rather (by stereographic projection) on a
sphere (\Fig{17}).
%[Illustration: Fig.~18.]
\WFigure{1.625in}{079b}
The triangles corresponding
to the upper (shaded) half of
the $z$-plane are the alternate (shaded)
triangles on the sphere determined by
inscribing a regular icosahedron and
dividing each of the $20$~triangles so
obtained into six equal and symmetrical
triangles by drawing the altitudes (\Fig{18}).
This conformal representation on the sphere assigns to
every root a definite region, and is therefore equivalent to a
%% -----File: 080.png---Folio 70-------
perfect separation of the $60$~roots. On the other hand, it corresponds
in its regular shape to the $60$~linear substitutions
indicated above.

(3)~If, by putting $\eta = y_{1}/y_{2}$, we make the $60$~expressions
of the roots homogeneous, the different values of the quantities~$y$
will all be of the form
\[
\alpha y_{1} + \beta y_{2},\quad  \gamma y_{1} + \delta y_{2},
\]
and therefore satisfy a linear differential equation of the
second order
\[
y'' + py' + q\DPtypo{}{y} = 0,
\]
$p$~and~$q$ being definite rational functions of~$z$. It is, of course,
always possible to express every root of an equation by means
of a power series. In our case we reduce the calculation of~$\eta$
to that of $y_{1}$ and~$y_{2}$, and try to find series for these quantities.
Since these series must satisfy our differential equation
of the second order, the law of the series is comparatively
simple, any term being expressible by means of the two
preceding terms.

(4)~Finally, as mentioned before, the calculation of the
roots may be abbreviated by the use of elliptic functions,
provided tables of such elliptic functions be computed beforehand.

Let us now see what corresponds to each of these four
points in the case of the \emph{pure} equation $\eta^{n} = z$. The results are
well known:

(1)~All the $n$~roots can be expressed as linear functions
of any one of them,~$\eta$:
\[
\eta,\quad \epsilon \eta,\quad \epsilon^{2} \eta, \quad\dots\quad \epsilon^{n-1} \eta,
\]
$\epsilon$~being a primitive $n$th~root of unity.
%% -----File: 081.png---Folio 71-------

(2)~The conformal representation (\Fig{19}) gives the division
of the sphere into $2n$~equal lunes whose great circles all pass
through the same two points.

%[Illustration: Fig.~19.]
\Figure{081}
(3)~There is a differential equation of the first order in~$\eta$,
viz.,
\[
nz · \eta' - \eta = 0,
\]
from which simple series can be derived for the purposes of
actual calculation of the roots.

(4)~If these series should be inconvenient, logarithms can be
used for computation.

The analogy, you will perceive, is complete. The principal
difference between the two cases lies in the fact that, for the
pure equation, the linear substitutions involve but one quantity,
while for the quintic equation we have a group of \emph{binary} linear
substitutions. The same distinction finds expression in the
differential equations, the one for the pure equation being of
the first order, while that for the quintic is of the second order.

Some remarks may be added concerning the reduction of the
general equation of the fifth degree,
\[
f_{5}(x) = 0,
\]
to the icosahedron equation. This reduction is possible because
the Galois group of our quintic equation (the square root of the
discriminant having been adjoined) is isomorphic with the group
%% -----File: 082.png---Folio 72-------
of the $60$~linear substitutions of the icosahedron equation. This
possibility of the reduction does not, of course, imply an answer
to the question, what operations are needed to effect the reduction.
The second part of my \textit{Vorlesungen über das Ikosaeder} is
devoted to the latter question. It is found that the reduction
cannot be performed rationally, but requires the introduction of
a square root. The irrationality thus introduced is, however, an
irrationality of a particular kind (a so-called \emph{accessory} irrationality);
for it must be such as not to reduce the Galois group of
the equation.

I proceed now to consider the general problem of an analogous
treatment of higher equations as first given by me in the
\textit{Math.\ Annalen}, Vol.~15 (1879).\footnote
  {\textit{Ueber die Auflösung gewisser Gleichungen vom siebenten und achten Grade},
  pp.~251--282.}
I must remark, first of all,
that for an accurate exposition it would be necessary to distinguish
throughout between the homogeneous and projective
formulations (in the latter case, only the ratios of the homogeneous
variables are considered). Here it may be allowed to
disregard this distinction.

%[** TN: Variables inside italics are upright in the original]
Let us consider the very general problem: \emph{a finite group of
homogeneous linear substitutions of $n$~variables being given, to
calculate the values of the $n$~variables from the invariants of the
group.}

This problem evidently contains the problem of solving an
algebraic equation of any Galois group. For in this case all
rational functions of the roots are known that remain unchanged
by certain \emph{permutations} of the roots, and permutation is, of
course, a simple case of \emph{homogeneous linear transformation}.

Now I propose a general formulation for the treatment of
these different problems as follows: \emph{among the problems having
isomorphic groups we consider as the simplest the one that has the}
%% -----File: 083.png---Folio 73-------
\emph{least number of variables, and call this the normal problem. This
%[** TN: Wording below from 1911 reprint]
problem must be considered as solvable by series of \DPtypo{any}{some} kind.
The question is to reduce the other isomorphic problems to the
normal problem.}

This formulation, then, contains what I propose as a general
solution of algebraic equations, \ie\ a reduction of the equations
to the isomorphic problem with a minimum number of
variables.

The reduction of the equation of the fifth degree to the
icosahedron problem is evidently contained in this as a special
case, the minimum number of variables being two.

In conclusion I add a brief account showing how far the general
problem has been treated for equations of higher degrees.

In the first place, I must here refer to the discussion by
myself\footnote
  {Math.\ Annalen, Vol.~15 (1879), pp.~251--282.}
and Gordan\footnote
  {\textit{Ueber Gleichungen siebenten Grades mit einer Gruppe von $168$~Substitutionen},
  Math.\ Annalen, Vol.~20 (1882), pp.~515--530, and Vol.~25 (1885), pp.~459--521.}
of those equations of the seventh degree
that have a Galois group of $168$~substitutions. The minimum
number of variables is here equal to three, the ternary group
being the same group of $168$~linear substitutions that has since
been discussed with full details in Vol.~I. of the \textit{Elliptische
Modulfunctionen}. While I have confined myself to an exposition
of the general idea, Gordan has actually performed the
reduction of the equation of the seventh degree to the ternary
problem. This is no doubt a splendid piece of work; it is
only to be deplored that Gordan here, as elsewhere, has disdained
to give his leading ideas apart from the complicated
array of formulæ.

Next, I must mention a paper published in Vol.~28 (1887) of
the \textit{Math.\ Annalen},\footnote
  {\textit{Zur Theorie der allgemeinen Gleichungen sechsten und siebenten Grades}, pp.~499--532.}
where I have shown that for the \emph{general}
%% -----File: 084.png---Folio 74-------
equations of the sixth and seventh degrees the minimum number
of the normal problem is four, and how the reduction can
be effected.

Finally, in a letter addressed to Camille Jordan\footnote
 {Journal de mathématiques, année 1888, p.~169.}
I pointed
out the possibility of reducing the equation of the $27$th~degree,
which occurs in the theory of cubic surfaces, to a normal problem
containing likewise four variables. This reduction has
ultimately been performed in a very simple way by Burkhardt\footnote
  {\textit{Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen. Dritter
  Theil}, Math.\ Annalen, Vol.~41 (1893), pp.~313--343.}
while all quaternary groups here mentioned have been considered
more closely by Maschke.\footnote
  {\textit{Ueber die quaternäre, endliche, lineare Substitutionsgruppe der Borchardt'schen
  Moduln}, Math.\ Annalen, Vol.~30 (1887), pp.~496--515; \textit{Aufstellung des vollen Formensystems
  einer quaternären Gruppe von $51840$~linearen Substitutionen}, ib., Vol.~33
  (1889), pp.~317--344; \textit{Ueber eine merkwürdige Configuration gerader Linien im
  Raume}, ib., Vol.~36 (1890), pp.~190--215.}

This is the whole account of what has been accomplished;
but it is clear that further progress can be made on the same
lines without serious difficulty.

A first problem I wish to propose is as follows. In recent
years many groups of permutations of $6, 7, 8, 9, \dots$ letters have
been made known. The problem would be to determine in
each case the minimum number of variables with which isomorphic
groups of linear substitutions can be formed.

Secondly, I want to call your particular attention to the case
of the general equation of the eighth degree. I have not been
able in this case to find a material simplification, so that it
would seem as if the equation of the eighth degree were its
own normal problem. It would no doubt be interesting to
obtain certainty on this point.
%% -----File: 085.png---Folio 75-------

\Lecture[Hyperelliptic and Abelian Functions.]
{X.}{On Some Recent Advances in
Hyperelliptic and Abelian Functions.}

\Date{(September 7, 1893.)}

\First{The} subject of hyperelliptic and Abelian functions is of such
vast dimensions that it would be impossible to embrace it in
its whole extent in one lecture. I wish to speak only of the
mutual correlation that has been established between this
subject on the one hand, and the theory of invariants, projective
geometry, and the theory of groups, on the other. Thus in
particular I must omit all mention of the recent attempts to
bring arithmetic to bear on these questions. As regards the
theory of invariants and projective geometry, their introduction
in this domain must be considered as a realization and farther
extension of the programme of Clebsch. But the additional
idea of groups was necessary for achieving this extension.
What I mean by establishing a mutual correlation between
these various branches will be best understood if I explain it
on the more familiar example of the \emph{elliptic functions}.

To begin with the older method, we have the fundamental
elliptic functions in the Jacobian form
\[
\sin\am\left(v, \frac{K'}{K}\right),\quad
\cos\am\left(v, \frac{K'}{K}\right),\quad
\Delta\am\left(v, \frac{K'}{K}\right),
\]
as depending on two arguments. These are treated in many
works, sometimes more from the geometrical point of view of
Riemann, sometimes more from the analytical standpoint of
%% -----File: 086.png---Folio 76-------
Weierstrass. I may here mention the first edition of the work
of Briot and Bouquet, and of German works those by Königsberger
and by Thomae.

The impulse for a new treatment is due to Weierstrass. He
introduced, as is well known, three homogeneous arguments,
$u$,~$\omega_{1}$,~$\omega_{2}$, instead of the two Jacobian arguments. This was
a necessary preliminary to establishing the connection with
the theory of linear substitutions. Let us consider the discontinuous
ternary group of linear substitutions,
\begin{alignat*}{3}
u'          &= u + & m_{1} \omega_{1} &+&  m_{2} \omega_{2}&, \\
\omega_{1}' &=     &\alpha \omega_{1} &+&  \beta \omega_{2}&, \\
\omega_{2}' &=     &\gamma \omega_{1} &+& \delta \omega_{2}&,
\end{alignat*}
where $\alpha$,~$\beta$,~$\gamma$,~$\delta$ are integers whose determinant $\alpha \delta - \beta \gamma = 1$,
while $m_{1}$,~$m_{2}$ are any integers whatever. The fundamental
functions of Weierstrass's theory,
\[
p (u, \omega_{1}, \omega_{2}),\quad
p'(u, \omega_{1}, \omega_{2}),\quad
g_{2}(\omega_{1}, \omega_{2}),\quad
g_{3}(\omega_{1}, \omega_{2}),
\]
are nothing but the complete system of invariants of that group.
It appears, moreover, that $g_{2}$,~$g_{3}$ are also the ordinary (Cayleyan)
invariants of the binary biquadratic form $f_{4}(x_{1}, x_{2})$, on
which depends the integral of the first kind
\[
\int\frac{x_{1}\,dx_{2} - x_{2}\,dx_{1}}{\sqrt{f_{4}(x_{1}, x_{2})}}.
\]
This significant feature that the transcendental invariants turn
out to be at the same time invariants of the algebraic irrationality
corresponding to the transcendental theory will hold in
all higher cases.

As a next step in the theory of elliptic functions we have to
mention the introduction by Clebsch of the systematic consideration
of algebraic curves of deficiency~$1$. He considered
in particular the plane curve of the third order~($C_{3}$) and the
%% -----File: 087.png---Folio 77-------
first species of quartic curves~($C_{4}^{1}$) in space, and showed how
convenient it is for the derivation of numerous geometrical
propositions to regard the elliptic integrals as taken along these
curves. The theory of elliptic functions is thus broadened by
bringing to bear upon it the ideas of modern projective geometry.

By combining and generalizing these considerations, I was
led to the formulation of a very general programme which may
be stated as follows (see \textit{Vorlesungen über die Theorie der elliptischen
Modulfunctionen}, Vol.~II.).

Beginning with the discontinuous group mentioned before
\begin{alignat*}{3}
u'         &= u + & m_{1} \omega_{1} &+&  m_{2} \omega_{2}&, \\
\omega_{1}'&=     &\alpha \omega_{1} &+& \beta  \omega_{2}&, \\
\omega_{2}'&=     &\gamma \omega_{1} &+& \delta \omega_{2}&,
\end{alignat*}
our first task is to construct all its sub-groups. Among these
the simplest and most useful are those that I have called
\emph{congruence sub-groups}; they are obtained by putting
\[
\left.
\begin{alignedat}{2}
m_{1}  &\equiv 0,\quad & m_{2} &\equiv 0, \\
\alpha &\equiv 1,\quad & \beta &\equiv 0, \\
\gamma &\equiv 0,\quad &\delta &\equiv 1,
\end{alignedat}
\right\} \pmod{n}.
\]

The second problem is to construct the invariants of all
these groups and the relations between them. Leaving out
of consideration all sub-groups except these congruence sub-groups,
we have still attained a very considerable enlargement
of the theory of elliptic functions. According to the value
assigned to the number~$n$, I distinguish different \emph{stages} (\emph{Stufen})
of the problem. It will be noticed that Weierstrass's theory
corresponds to the first stage ($n = 1$), while Jacobi's answers,
generally speaking, to the second ($n = 2$); the higher stages
have not been considered before in a systematic way.

Thirdly, for the purpose of geometrical illustration, I apply
Clebsch's idea of the algebraic curve. I begin by introducing
%% -----File: 088.png---Folio 78-------
the ordinary square root of the binary form which requires the
axis of~$x$ to be covered twice; \ie\ we have to use a~$C_{2}$ in an~$S_{1}$.
I next proceed to the general cubic curve of the plane
($C_{3}$ in an~$S_{2}$), to the quartic curve in space of three dimensions
($C_{4}$ in an~$S_{3}$), and generally to the elliptic curve~$C_{n+1}$ in an~$S_{n}$.
These are what I call the normal elliptic curves; they serve best
to illustrate any algebraic relations between elliptic functions.

I may notice, by the way, that the treatment here proposed
is strictly followed in the \textit{Elliptische Modulfunctionen}, except
that there the quantity~$u$ is of course assumed to be zero, since
this is precisely what characterizes the modular functions. I
hope some time to be able to treat the whole theory of elliptic
functions (\ie\ with $u$~different from zero) according to this
programme.

The successful extension of this programme to the theory of
hyperelliptic and Abelian functions is the best proof of its
being a real step in advance. I have therefore devoted my
efforts for many years to this extension; and in laying before
you an account of what has been accomplished in this rather
special field, I hope to attract your attention to various lines of
research along which new work can be spent to advantage.

As regards the \emph{hyperelliptic functions}, we may premise as a
general definition that they are functions of \emph{two} variables $u_{1}$,~$u_{2}$,
with \emph{four} periods (while the elliptic functions have \emph{one} variable~$u$,
and \emph{two} periods). Without attempting to give an
historical account of the development of the theory of hyperelliptic
functions, I turn at once to the researches that mark
a progress along the lines specified above, beginning with the
geometric application of these functions to surfaces in a space
of any number of dimensions.

Here we have first the investigation by Rohn of Kummer's
surface, the well-known surface of the fourth order, with $16$~conical
%% -----File: 089.png---Folio 79-------
points. I have myself given a report on this work in
the \textit{Math.\ Annalen}, Vol.~27 (1886).\footnote
  {\textit{Ueber Configurationen, welche der Kummer'schen Fläche zugleich eingeschrieben
  und umgeschrieben sind}, pp.~106--142.}
If every mathematician is
struck by the beauty and simplicity of the relations developed
in the corresponding cases of the elliptic functions (the~$C_{3}$ in
the plane,~etc.), the remarkable configurations inscribed and
circumscribed to the Kummer surface that have here been
developed by Rohn and myself, should not fail to elicit interest.

Further, I have to mention an extensive memoir by Reichardt,
published in~1886, in the \textit{Acta Leopoldina}, where the connection
between hyperelliptic functions and Kummer's surface is
summarized in a convenient and comprehensive form, as an
introduction to this branch. The starting-point of the investigation
is taken in the theory of line-complexes of the second
degree.

Quite recently the French mathematicians have turned their
attention to the general question of the representation of surfaces
by means of hyperelliptic functions, and a long memoir by
Humbert on this subject will be found in the last volume of the
\textit{Journal de Mathématiques.}\footnote
  {\textit{Théorie générale des surfaces hyperelliptiques}, année~1893, pp.~29--170.}

I turn now to the abstract theory of hyperelliptic functions.
It is well known that Göpel and Rosenhain established that
theory in~1847 in a manner closely corresponding to the Jacobian
theory of elliptic functions, the integrals
\[
u_{1} = \int \frac{dx}{\sqrt{f_{6}(x)}},\quad
u_{2} = \int \frac{x\,dx}{\sqrt{f_{6}(x)}}
\]
taking the place of the single elliptic integral~$u$. Here, then,
the question arises: what is the relation of the hyperelliptic
functions to the invariants of the binary form of the sixth order
$f_{6}(x_{1}, x_{2})$? In the investigation of this question by myself and
%% -----File: 090.png---Folio 80-------
Burkhardt, published in Vol.~27 (1886)\footnote
  {\textit{Ueber hyperelliptische Sigmafunctionen}, pp.~431--464.}
and Vol.~32 (1888)\footnote
  {pp.~351--380 and 381--442.}
of the \textit{Math.\ Annalen}, we found that the decompositions of
the form~$f_{6}$ into two factors of lower order, $f_{6} = \phi_{1} \psi_{5} = \phi_{3} \psi_{3}$,
had to be considered. These being, of course, irrational decompositions,
the corresponding invariants are irrational; and a
study of the theory of such invariants became necessary.

But another new step had to be taken. The hyperelliptic
integrals involve the form~$f_{6}$ under the square root,~$\sqrt{f_{6}(x_{1}, x_{2})}$.
The corresponding Riemann surface has, therefore, two leaves
connected at six points; and the problem arises of considering
binary forms of $x_{1}$,~$x_{2}$ on such a Riemann surface, just as ordinarily
functions of $x$~alone are considered thereon. It can be
shown that there exists a particular kind of forms called \emph{primeforms},
strictly analogous to the determinant $x_{1}y_{2} - x_{2}y_{1}$ in the
ordinary complex plane. The primeform on the two-leaved
Riemann surface, like this determinant in the ordinary theory,
has the property of vanishing only when the points $(x_{1}, x_{2})$ and
$(y_{1}, y_{2})$ co-incide (on the same leaf). Moreover, the primeform
does not become infinite anywhere. The analogy to the determinant
$x_{1}y_{2} - x_{2}y_{1}$ fails only in so far as the primeform is no
longer an algebraic but a transcendental form. Still, all algebraic
forms on the surface can be decomposed into prime
factors. Moreover, these primeforms give the natural means
for the construction of the $\theta$-functions. As an intermediate
step we have here functions called by me $\sigma$-functions in analogy
to the $\sigma$-functions of Weierstrass's elliptic theory. In the
papers referred to (\textit{Math.\ Annalen}, Vols.~27,~32) all these considerations
are, of course, given for the general case of hyperelliptic
functions, the irrationality being $\sqrt{f_{2p+2}(x_{1}, x_{2})}$, where
$f_{2p+2}$ is a binary form of the order~$2p+2$.
%% -----File: 091.png---Folio 81-------

Having thus established the connection between the ordinary
theory of hyperelliptic functions of $p = 2$ and the invariants of
the binary sextic, I undertook the systematic development of
what I have called, in the case of elliptic functions, the \emph{Stufentheorie}.
The lectures I gave on this subject in~1887--88
have been developed very fully by Burkhardt in the \textit{Math.\
Annalen}, Vol.~35 (1890).\footnote
  {\textit{Grundzüge einer allgemeinen Systematik der hyperelliptischen Functionen~I.
  Ordnung}, pp.~198--296.}

As regards the first stage, which, owing to the connection
with the theory of \emph{rational} invariants and covariants, requires
very complicated calculations, the Italian mathematician, Pascal,
has made much progress (\textit{Annali di matematica}). In this
connection I must refer to the paper by Bolza\footnote
  {\textit{Darstellung der rationalen ganzen Invarianten der Binärform sechsten Grades
  durch die Nullwerthe der zugehörigen $\theta$-Functionen}, pp.~478--495.}
in \textit{Math.\
Annalen}, Vol.~30 (1887), where the question is discussed in
how far it is possible to represent the rational invariants of
the sextic by means of the zero values of the $\theta$-functions.

For higher stages, in particular stage three, Burkhardt has
given very valuable developments in the \textit{Math.\ Annalen}, Vol.~36
(1890), p.~371; Vol.~38 (1891), p.~161; Vol.~41 (1893), p.~313.
He considers, however, only the hyperelliptic modular functions
($u_{1}$~and~$u_{2}$ being assumed to be zero). The final aim, which
Burkhardt seems to have attained, although a large amount
of numerical calculation remains to be filled in, consists here
in establishing the so-called \emph{multiplier-equation} for transformations
of the third order. The equation is of the $40$th~degree;
and Burkhardt has given the general law for the formation
of the coefficients.

I invite you to compare his treatment with that of Krause
in his book \textit{Die Transformation der hyperelliptischen Functionen
erster Ordnung}, Leipzig, Teubner, 1886. His investigations,
%% -----File: 092.png---Folio 82-------
based on the general relations between $\theta$-functions, may
go farther; but they are carried out from purely formal
point of view, without reference to the theories of invariants,
of groups, or other allied topics.

So much as regards hyperelliptic functions. I now proceed
to report briefly on the corresponding advances made in the
theory of Abelian functions. I give merely a list of papers;
they may be classed under three heads:

(1)~A \emph{preliminary} question relates to the invariant representation
of the integral of the third kind on algebraic curves of
higher deficiency. Pick\footnote
  {\textit{Zur Theorie der Abel'schen Functionen}, Math.\ Annalen, Vol.~29 (1887), pp.~259--271.}
has considered this problem for plane
curves having no singular points. On the other hand, White,
in his dissertation,\footnote
  {\textit{Abel'sche Integrale auf singularitätenfreien, einfach überdeckten, vollständigen
  Schnittcurven eines beliebig ausgedehnten Raumes}, Halle, 1891, pp.~43--128.}
briefly reported in \textit{Math.\ Annalen}, Vol.~36
(1890), p.~597, and printed in full in the \textit{Acta Leopoldina}, has
treated such curves in space as are the complete intersection
of two surfaces and have no singular point. We may here
also notice the researches of Pick and Osgood\footnote
  {Osgood, \textit{Zur Theorie der zum algebraischen Gebilde $y^{m} = R(x)$ gehörigen
  Abel'schen Functionen}, Göttingen, 1890, 8vo, 61~pp.}
on the so-called
binomial integrals.

(2)~An exposition of the general theory of forms on Riemann
surfaces of any kind, in particular a definition of the
primeform belonging to each surface, was given by myself
in Vol.~36 (1890) of the \textit{Math.\ Annalen}.\footnote
  {\textit{Zur Theorie der Abel'schen Functionen}, pp.~1--83.}
I may add that
during the last year this subject was taken up anew and
farther developed by Dr.~Ritter; see \textit{Göttinger Nachrichten}
%[** TN: Correct volume number from 1911 reprint]
for~1893, and \textit{Math.\ Annalen}, Vol.~\DPtypo{43}{44}. Dr.~Ritter considers
the algebraic forms as special cases of more general forms, the
\emph{multiplicative forms}, and thus takes a real step in advance.
%% -----File: 093.png---Folio 83-------

(3)~Finally, the particular case $p = 3$ has been studied on the
basis of our programme in various directions. The normal
curve for this case is well known to be the plane quartic~$C_{4}$
whose geometric properties have been investigated by Hesse
and others. I found (\textit{Math.\ Annalen}, Vol.~36) that these
geometrical results, though obtained from an entirely different
point of view, corresponded exactly to the needs of the Abelian
problem, and actually enabled me to define clearly the $64$
$\theta$-functions with the aid of the~$C_{4}$. Here, as elsewhere, there
seems to reign a certain pre-established harmony in the development
of mathematics, what is required in one line of research
being supplied by another line, so that there appears to be
a logical necessity in this, independent of our individual
disposition.

In this case, also, I have introduced $\sigma$-functions in the place
of the $\theta$-functions. The coefficients are irrational covariants
just as in the case $p = 2$. These $\sigma$-series have been studied at
great length by Pascal in the \textit{Annali di Matematica}. These
investigations bear, of course, a close relation to those of
Frobenius and Schottky, which only the lack of time prevents
me from quoting in detail.

Finally, the recent investigations of an Austrian mathematician,
\emph{Wirtinger}, must here be mentioned. First, Wirtinger has
established for $p = 3$ the analogue to the Kummer surface; this
is a manifoldness of three dimensions and the $24$th~order in an~$S_{7}$;
see \textit{Göttinger Nachrichten} for~1889, and \textit{Wiener Monatshefte},
1890. Though apparently rather complicated, this manifoldness
has some very elegant properties; thus it is transformed into
itself by $64$~collineations and $64$~reciprocations. Next, in
Vol.~40 (1892), of the \textit{Math.\ Annalen},\footnote
  {\textit{Untersuchungen über Abel'sche Functionen vom Geschlechte}~3, pp.~261--312.}
Wirtinger has discussed
the Abelian functions on the assumption that only
%% -----File: 094.png---Folio 84-------
\emph{rational} invariants and covariants of the curve of the fourth
order are to be considered; this corresponds to the ``first
stage'' with $p = 3$. The investigation is full of new and
fruitful ideas.

In concluding, I wish to say that, for the cases $p = 2$ and
$p = 3$, while much still remains to be done, the fundamental
difficulties have been overcome. The great problem to be
attacked next is that of $p = 4$, where the normal curve is of the
sixth order in space. It is to be hoped that renewed efforts
will result in overcoming all remaining difficulties. Another
promising problem presents itself in the field of $\theta$-functions,
when the general $\theta$-series are taken as starting-point, and not
the algebraic curve. An enormous number of formulæ have
there been developed by analysts, and the problem would be
to connect these formulæ with clear geometrical conceptions
of the various algebraic configurations. I emphasize these
special problems because the Abelian functions have always
been regarded as one of the most interesting achievements
of modern mathematics, so that every advance we make in
this theory gives a standard by which we can measure our
own efficiency.
%% -----File: 095.png---Folio 85-------

\Lecture{XI.}{The Most Recent Researches
in Non-Euclidean Geometry.}

\Date{(September 8, 1893.)}

\First{My} remarks to-day will be confined to the progress of non-Euclidean
geometry during the last few years. Before reporting
on these latest developments, however, I must briefly
summarize what may be regarded as the general state of
opinion among mathematicians in this field. There are three
points of view from which non-Euclidean geometry has been
considered.

(1)~First we have the point of view of elementary geometry, of
which Lobachevsky and Bolyai themselves are representatives.
Both begin with simple geometrical constructions, proceeding
just like Euclid, except that they substitute another axiom for
the axiom of parallels. Thus they build up a system of non-Euclidean
geometry in which the length of the line is infinite,
and the ``measure of curvature'' (to anticipate a term not used
by them) is negative. It is, of course, possible by a similar
process to obtain the geometry with a positive measure of
curvature, first suggested by Riemann; it is only necessary
to formulate the axioms so as to make the length of a line
finite, whereby the existence of parallels is made impossible.

(2)~From the point of view of projective geometry, we begin
by establishing the system of projective geometry in the sense
of von~Staudt, introducing projective co-ordinates, so that
straight lines and planes are given by \emph{linear} equations. Cayley's
%% -----File: 096.png---Folio 86-------
theory of projective measurement leads then directly to
the three possible cases of non-Euclidean geometry: hyperbolic,
parabolic, and elliptic, according as the measure of
curvature~$k$ is $< 0$,~$= 0$, or~$> 0$. It is here, of course, essential
to adopt the system of von~Staudt and not that of
Steiner, since the latter defines the anharmonic ratio by
means of distances of points, and not by pure projective
constructions.

(3)~Finally, we have the point of view of Riemann and Helmholtz.
Riemann starts with the idea of the element of distance~$ds$,
which he assumes to be expressible in the form
\[
ds = \sqrt{\sum a_{ik}\,dx_{i}\,dx_{k}}.
\]
Helmholtz, in trying to find a reason for this assumption, considers
the motions of a rigid body in space, and derives from
these the necessity of giving to~$ds$ the form indicated. On the
other hand, Riemann introduces the fundamental notion of the
\emph{measure of curvature of space}.

The idea of a measure of curvature for the case of two
variables, \ie\ for a surface in a three-dimensional space, is due
to Gauss, who showed that this is an intrinsic characteristic of
the surface quite independent of the higher space in which the
surface happens to be situated. This point has given rise to a
misunderstanding on the part of many non-Euclidean writers.
When Riemann attributes to his space of three dimensions a
measure of curvature~$k$, he only wants to say that there exists
an invariant of the ``form'' $\sum{a_{ik}\,dx_{i}\,dx_{k}}$; he does not mean to
imply that the three-dimensional space necessarily exists as a
curved space in a space of four dimensions. Similarly, the
illustration of a space of constant positive measure of curvature
by the familiar example of the sphere is somewhat misleading.
Owing to the fact that on the sphere the geodesic lines (great
circles) issuing from any point all meet again in another definite
%% -----File: 097.png---Folio 87-------
point, antipodal, so to speak, to the original point, the existence
of such an antipodal point has sometimes been regarded as a
necessary consequence of the assumption of a constant positive
curvature. The projective theory of non-Euclidean space shows
immediately that the existence of an antipodal point, though
compatible with the nature of an elliptic space, is not necessary,
but that two geodesic lines in such a space may intersect in
one point if at all.\footnote
  {This theory has also been developed by Newcomb, in the \textit{Journal für reine
  und angewandte Mathematik}, Vol.~83 (1877), pp.~293--299.}

I call attention to these details in order to show that there
is some advantage in adopting the second of the three points of
view characterized above, although the third is at least equally
important. Indeed, our ideas of space come to us through the
senses of vision and motion, the ``optical properties'' of space
forming one source, while the ``mechanical properties'' form
another; the former corresponds in a general way to the projective
properties, the latter to those discussed by Helmholtz.

As mentioned before, from the point of view of projective
geometry, von~Staudt's system should be adopted as the basis.
It might be argued that von~Staudt practically assumes the
axiom of parallels (in postulating a one-to-one correspondence
between a pencil of lines and a row of points). But I have
shown in the \textit{Math.\ Annalen}\footnote
  {\textit{Ueber die sogenannte Nicht-Euklidische Geometrie}, Math.\ Annalen, Vol.~6
  (1873), pp.~112--145.}
how this apparent difficulty can
be overcome by restricting all constructions of von~Staudt to a
limited portion of space.

I now proceed to give an account of the most recent researches
in non-Euclidean geometry made by Lie and myself.
Lie published a brief paper on the subject in the \textit{Berichte} of
the Saxon Academy~(1886), and a more extensive exposition
of his views in the same \textit{Berichte} for 1890 and~1891. These
%% -----File: 098.png---Folio 88-------
papers contain an application of Lie's theory of continuous
groups to the problem formulated by Helmholtz. I have the
more pleasure in placing before you the results of Lie's investigations
as they are not taken into due account in my paper
on the foundations of projective geometry in Vol.~37 of the
\textit{Math.\ Annalen} (1890),\footnote
  {\textit{Zur Nicht-Euklidischen Geometrie}, pp.~544--572.}
nor in my (lithographed) lectures on
non-Euclidean geometry delivered at Göttingen in~1889--90; the
last two papers of Lie appeared too late to be considered, while
the first had somehow escaped my memory.

I must begin by stating the problem of Helmholtz in modern
terminology. The motions of three-dimensional space are~$\infty^{6}$,
and form a group, say~$G_{6}$. This group is known to have an
invariant for any two points $p$,~$p'$, viz.\ the distance $\Omega (p, p')$
of these points. But the form of this invariant (and generally
the form of the group) in terms of the co-ordinates $x_{1}$,~$x_{2}$,~$x_{3}$,
$y_{1}$,~$y_{2}$,~$y_{3}$ of the points is not known \textit{a~priori}. The question
arises whether the group of motions is fully characterized by
these two properties so that none but the Euclidean and the
two non-Euclidean systems of geometry are possible.

For illustration Helmholtz made use of the analogous case
in two dimensions. Here we have a group of $\infty^{3}$~motions;
the distance is again an invariant; and yet it is possible to
construct a group not belonging to any one of our three
systems, as follows.

Let $z$ be a complex variable; the substitution characterizing
the group of Euclidean geometry can be written in the well-known
form
\[
z' = e^{i\phi}z + m + in = (\cos\phi + i \sin\phi)z + m + in.
\]
Now modifying this expression by introducing a complex
number in the exponent,
\[
z' = e^{(a+i)\phi}z + m + in = e^{a\phi} (\cos\phi + i \sin\phi)z + m + in,
\]
%% -----File: 099.png---Folio 89-------
we obtain a group of transformations by which a point (in
the simple case $m = 0$, $n = 0$) would not move about the origin
in a circle, but in a logarithmic spiral; and yet this is a group~$G_{3}$
with three variable parameters $m$,~$n$,~$\phi$, having an invariant
for every two points, just like the original group. Helmholtz
concludes, therefore, that a new condition, that of \emph{monodromy},
must be added to determine our group completely.

I now proceed to the work of Lie. First as to the results:
Lie has confirmed those of Helmholtz with the single exception
that in space of three dimensions the axiom of monodromy is
not needed, but that the groups to be considered are fully
determined by the other axioms. As regards the proofs, however,
Lie has shown that the considerations of Helmholtz must
be supplemented. The matter is this. In keeping one point of
space fixed, our $G_{6}$ will be reduced to a~$G_{3}$. Now Helmholtz
inquires how the differentials of the lines issuing from the fixed
point are transformed by this~$G_{3}$. For this purpose he writes
down the formulæ
\begin{align*}
dx_{1}' &= a_{11}\, dx_{1} + a_{12}\, dx_{2} + a_{13}\, dx_{3}, \\
dx_{2}' &= a_{21}\, dx_{1} + a_{22}\, dx_{2} + a_{23}\, dx_{3}, \\
dx_{3}' &= a_{31}\, dx_{1} + a_{32}\, dx_{2} + a_{33}\, dx_{3},
\end{align*}
and considers the coefficients $a_{11}$, $a_{12}$,~$\dots$ $a_{33}$ as depending on
three variable parameters. But Lie remarks that this is not
sufficiently general. The linear equations given above represent
only the first terms of power series, and the possibility
must be considered that the three parameters of the group may
not all be involved in the linear terms. In order to treat all
possible cases, the general developments of Lie's theory of
groups must be applied, and this is just what Lie does.

Let me now say a few words on my own recent researches in
non-Euclidean geometry which will be found in a paper published
in the \textit{Math.\ Annalen}, Vol.~37 (1890), p.~544. Their
%% -----File: 100.png---Folio 90-------
result is that our ideas as to non-Euclidean space are still very
incomplete. Indeed, all the researches of Riemann, Helmholtz,
Lie, consider only a portion of space surrounding the origin;
they establish the existence of analytic laws in the vicinity of
that point. Now this space can of course be continued, and
the question is to see what kind of connection of space may
result from this continuation. It is found that there are different
possibilities, each of the three geometries giving rise
to a series of subdivisions.

To understand better what is meant by these varieties of
connection, let us compare the geometry on a sphere with that
in the sheaf of lines formed by the diameters of the sphere.
Considering each diameter as an infinite line or ray passing
through the centre (not a half-ray issuing from the centre), to
each line of the sheaf there will correspond two points on the
sphere, viz.\ the two points of intersection of the line with the
sphere. We have, therefore, a one-to-two correspondence
between the lines of the sheaf and the points of the sphere.
Let us now take a small area on the sphere; it is clear that
the distance of two points contained in this area is equal to
the angle of the corresponding lines of the sheaf. Thus the
geometry of points on the sphere and the geometry of lines in
the sheaf are identical as far as small regions are concerned, both
corresponding to the assumption of a constant positive measure
of curvature. A difference appears, however, as soon as we
consider the whole closed sphere on the one hand and the complete
sheaf on the other. Let us take, for instance, two geodesic
lines of the sphere, \ie\ two great circles, which evidently intersect
in two (diametral) points. The corresponding pencils of
the sheaf have only \emph{one} straight line in common.

A second example for this distinction occurs in comparing
the geometry of the Euclidean plane with the geometry on a
closed cylindrical surface. The latter can be developed in the
%% -----File: 101.png---Folio 91-------
usual way into a strip of the plane bounded by two parallel
lines, as will appear from \Fig{20}, the arrows indicating that
the opposite points of the edges are coincident on the cylindrical
surface. We notice at once the difference: while in the
plane all geodesic lines are infinite, on the cylinder there is
%[Illustration: Fig.~20.]
\Figure[2.5in]{101a}
one geodesic line that is of finite length, and while in the plane
two geodesic lines always intersect in one point, if at all, on
the cylinder there may be $\infty$~points of intersection.

This second example was generalized by Clifford in an
address before the Bradford meeting of the British Association~(1873).
%[Illustration: Fig. 21.]
\Figure[2in]{101b}
In accordance with Clifford's general idea, we
may define a closed surface by taking a parallelogram out of
an ordinary plane and making the opposite edges correspond
point to point as indicated in \Fig{21}. It is not to be
understood that the opposite edges should be brought to
%% -----File: 102.png---Folio 92-------
coincidence by bending the parallelogram (which evidently
would be impossible without stretching); but only the logical
convention is made that the opposite points should be considered
as identical. Here, then, we have a closed manifoldness
of the connectivity of an anchor-ring, and every one
will see the great differences that exist here in comparison
with the Euclidean plane in everything concerning the lengths
and the intersections of geodesic lines, etc.

It is interesting to consider the $G_{3}$ of Euclidean motions on
this surface. There is no longer any possibility of moving the
surface on itself in $\infty^{3}$~ways, the closed surface being considered
in its totality. But there is no difficulty in moving any
small area over the closed surface in $\infty^{3}$~ways.

We have thus found, in addition to the Euclidean plane,
two other forms of surfaces: the strip between parallels and
Clifford's parallelogram. Similarly we have by the side of
ordinary Euclidean space three other types with the Euclidean
element of arc; one of these results from considering a
parallelepiped.

Here I introduce the axiomatic element. There is no way
of proving that the whole of space can be moved in itself in
$\infty^{6}$~ways; all we know is that small portions of space can be
moved in space in $\infty^{6}$~ways. Hence there exists the possibility
that our actual space, the measure of curvature being taken as
zero, may correspond to any one of the four cases.

Carrying out the same considerations for the spaces of constant
positive measure of curvature, we are led back to the two
cases of elliptic and spherical geometry mentioned before. If,
however, the measure of curvature be assumed as a negative
constant, we obtain an infinite number of cases, corresponding
exactly to the configurations considered by Poincaré and myself
in the theory of automorphic functions. This I shall not stop
to develop here.
%% -----File: 103.png---Folio 93-------

I may add that Killing has verified this whole theory.\footnote
  {\textit{Ueber die Clifford-Klein'schen Raumformen}, Math.\ Annalen, Vol.~39 (1891),
  pp.~257--278.}
It
is evident that from this point of view many assertions concerning
space made by previous writers are no longer correct
(\textit{e.g.}\ that infinity of space is a consequence of zero curvature),
so that we are forced to the opinion that our geometrical
demonstrations have no absolute objective truth, but are true
only for the present state of our knowledge. These demonstrations
are always confined within the range of the space-conceptions
that are familiar to us; and we can never tell
whether an enlarged conception may not lead to further
possibilities that would have to be taken into account.
From this point of view we are led in geometry to a certain
modesty, such as is always in place in the physical sciences.
%% -----File: 104.png---Folio 94-------

\Lecture{XII.}{The Study of Mathematics
at Göttingen.}

\Date{(September 9, 1893.)}

\First{In} this last lecture I should like to make some general
remarks on the way in which the study of mathematics is
organized at the university of Göttingen, with particular reference
to what may be of interest to American students. At the
same time I desire to give you an opportunity to ask any questions
that may occur to you as to the broader subject of mathematical
study at German universities in general. I shall be
glad to answer such inquiries to the extent of my ability.

It is perhaps inexact to speak of an \emph{organization} of the
mathematical teaching at Göttingen; you know that \textit{Lern- und
Lehr-Freiheit} prevail at a German university, so that the organization
I have in mind consists merely in a voluntary agreement
among the mathematical professors and instructors. We distinguish
at Göttingen between a general and a higher course
in mathematics. The general course is intended for that large
majority of our students whose intention it is to devote themselves
to the teaching of mathematics and physics in the higher
schools (\textit{Gymnasien}, \textit{Realgymnasien}, \textit{Realschulen}), while the
higher course is designed specially for those whose final aim
is original investigation.

As regards the former class of students, it is my opinion that
in Germany (here in America, I presume, the conditions are
very different) the abstractly theoretical instruction given to
%% -----File: 105.png---Folio 95-------
them has been carried too far. It is no doubt true that what
the university should give the student above all other things
is the scientific ideal. For this reason even these students
should push their mathematical studies far beyond the elementary
branches they may have to teach in the future. But the
ideal set before them should not be chosen so far distant, and
so out of connection with their more immediate wants, as to
make it difficult or impossible for them to perceive the bearing
that this ideal has on their future work in practical life.
In other words, the ideal should be such as to fill the future
teacher with enthusiasm for his life-work, not such as to make
him look upon this work with contempt as an unworthy
drudgery.

For this reason we insist that our students of this class, in
addition to their lectures on pure mathematics, should pursue
a thorough course in physics, this subject forming an integral
part of the curriculum of the higher schools. Astronomy is
also recommended as showing an important application of
mathematics; and I believe that the technical branches, such
as applied mechanics, resistance of materials,~etc., would form
a valuable aid in showing the practical bearing of mathematical
science. Geometrical drawing and descriptive geometry form
also a portion of the course. Special exercises in the solution
of problems, in lecturing,~etc., are arranged in connection with
the mathematical lectures, so as to bring the students into
personal contact with the instructors.

I wish, however, to speak here more particularly on the
higher courses, as these are of more special interest to American
students. Here specialization is of course necessary.
Each professor and docent delivers certain lectures specially
designed for advanced students, in particular for those studying
for the doctor's degree. Owing to the wide extent of modern
mathematics, it would be out of the question to cover the whole
%% -----File: 106.png---Folio 96-------
field. These lectures are therefore not regularly repeated every
year; they depend largely on the special line of research that
happens at the time to engage the attention of the professor.
In addition to the lectures we have the higher seminaries, whose
principal object is to guide the student in original investigation
and give him an opportunity for individual work.

As regards my own higher lectures, I have pursued a certain
plan in selecting the subjects for different years, my general
aim being \emph{to gain, in the course of time, a complete view of the
whole field of modern mathematics, with particular regard to the
intuitional or} (in the highest sense of the term) \emph{geometrical
standpoint}. This general tendency you will, I trust, also find
expressed in this colloquium, in which I have tried to present,
within certain limits, a general programme of my individual
work. To carry out this plan in Göttingen, and to bring it to
the notice of my students, I have, for many years, adopted the
method of having my higher lectures carefully written out, and,
in recent years, of having them lithographed, so as to make
them more readily accessible. These former lectures are at the
disposal of my hearers for consultation at the mathematical
reading-room of the university; those that are lithographed can
be acquired by anybody, and I am much pleased to find them
so well known here in America.

As another important point, I wish to say that I have always
regarded my students not merely as hearers or pupils, but as
collaborators. I want them to take an active part in my own
researches; and they are therefore particularly welcome if they
bring with them special knowledge and new ideas, whether
these be original with them, or derived from some other source,
from the teachings of other mathematicians. Such men will
spend their time at Göttingen most profitably to themselves.

I have had the pleasure of seeing many Americans among
my students, and gladly bear testimony to their great enthusiasm
%% -----File: 107.png---Folio 97-------
and energy. Indeed, I do not hesitate to say that, for
some years, my higher lectures were mainly sustained by students
whose home is in this country. But I deem it my duty
to refer here to some difficulties that have occasionally arisen
in connection with the coming of American students to Göttingen.
Perhaps a frank statement on my part, at this opportunity,
will contribute to remove these difficulties in part. What I wish
to speak of is this. It frequently happens at Göttingen, and
probably at other German universities as well, that American
students desire to take the higher courses when their preparation
is entirely inadequate for such work. A student having
nothing but an elementary knowledge of the differential and
integral calculus, usually coupled with hardly a moderate familiarity
with the German language, makes a decided mistake in
attempting to attend my advanced lectures. If he comes to Göttingen
with such a preparation (or, rather, the lack of it), he
may, of course, enter the more elementary courses offered at our
university; but this is generally not the object of his coming.
Would he not do better to spend first a year or two in one of
the larger American universities? Here he would find more
readily the transition to specialized studies, and might, at the
same time, arrive at a clearer judgment of his own mathematical
ability; this would save him from the severe disappointment
that might result from his going to Germany.

I trust that these remarks will not be misunderstood. My
presence here among you is proof enough of the value I attach
to the coming of American students to Göttingen. It is in
the interest of those wishing to go there that I speak; and
for this reason I should be glad to have the widest publicity
given to what I have said on this point.

Another difficulty lies in the fact that my higher lectures
have frequently an encyclopedic character, conformably to the
general tendency of my programme. This is not always just
%% -----File: 108.png---Folio 98-------
what is most needful to the American student, whose work
is naturally directed to gaining the doctor's degree. He will
need, in addition to what he may derive from my lectures, the
concentration on a particular subject; and this he will often
find best with other instructors, at Göttingen or elsewhere.
I wish to state distinctly that I do not regard it as at all desirable
that all students should confine their mathematical studies
to my courses or even to Göttingen. On the contrary, it
seems to me far preferable that the majority of the students
should attach themselves to other mathematicians for certain
special lines of work. My lectures may then serve to form
the wider background on which these special studies are projected.
It is in this way, I believe, that my lectures will
prove of the greatest benefit.

In concluding I wish to thank you for your kind attention,
and to give expression to the pleasure I have found in meeting
here at Evanston, so near to Chicago, the great metropolis of
this commonwealth, a number of enthusiastic devotees of my
chosen science.
%% -----File: 109.png---Folio 99-------

\Addendum{The Development of Mathematics}{at the
German Universities.\protect\footnotemark}
{By F.~Klein.}

\footnotetext{Translation, with a few slight modifications by the author, of the section \textit{Mathematik}
  in the work \textit{Die deutschen Universitäten}, Berlin, A.~Asher \&~Co., 1893,
  prepared by Professor Lexis for the World's Columbian Exposition at Chicago.}

\First{The} eighteenth century laid the firm foundation for the
development of mathematics in all directions. The universities
as such, however, did not take a prominent part in this
work; the \emph{academies} must here be considered of prime importance.
Nor can any fixed limits of nationality be recognized.
At the beginning of the period there appears in Germany no
less a man than \emph{Leibniz}; then follow, among the kindred
Swiss, the dynasty of the \emph{Bernoullis} and the incomparable
\emph{Euler}. But the activity of these men, even in its outward
manifestation, was not confined within narrow geographical
bounds; to encompass it we must include the Netherlands,
and in particular Russia, with Germany and Switzerland. On
the other hand, under Frederick the Great, the most eminent
French mathematicians, Lagrange, d'Alembert, Maupertuis,
formed side by side with Euler and Lambert the glory of
the Berlin Academy. The impulse toward a complete change
in these conditions came from the French Revolution.

The influence of this great historical event on the development
of science has manifested itself in two directions.
On the one hand it has effected a wider separation of nations
%% -----File: 110.png---Folio 100-------
with a distinct development of characteristic national qualities.
Scientific ideas preserve, of course, their universality;
indeed, international intercourse between scientific men has
become particularly important for the progress of science;
but the cultivation and development of scientific thought now
progress on national bases. The other effect of the French
Revolution is in the direction of educational methods. The
decisive event is the foundation of the École polytechnique at
Paris in~1794. That scientific research and active instruction
can be directly combined, that lectures alone are not sufficient,
and must be supplemented by direct personal intercourse
between the lecturer and his students, that above all it is of
prime importance to arouse the student's own activity,---these
are the great principles that owe to this source their recognition
and acceptance. The example of Paris has been the more
effective in this direction as it became customary to publish in
systematic form the lectures delivered at this institution; thus
arose a series of admirable text-books which remain even now
the foundation of mathematical study everywhere in Germany.
Nevertheless, the principal idea kept in view by the founders
of the Polytechnic School has never taken proper root in the
German universities. This is the combination of the technical
with the higher mathematical training. It is true that, primarily,
this has been a distinct advantage for the unrestricted
development of theoretical investigation. Our professors, finding
themselves limited to a small number of students who, as
future teachers and investigators, would naturally take great
interest in matters of pure theory, were able to follow the bent
of their individual predilections with much greater freedom
than would have been possible otherwise.

But we anticipate our historical account. First of all we
must characterize the position that Gauss holds in the science
of this age. Gauss stands in the very front of the new development:
%% -----File: 111.png---Folio 101-------
first, by the time of his activity, his publications reaching
back to the year~1799, and extending throughout the entire
first half of the nineteenth century; then again, by the wealth of
new ideas and discoveries that he has brought forward in almost
every branch of pure and applied mathematics, and which still
preserve their fruitfulness; finally, by his methods, for Gauss
was the first to restore that \emph{rigour} of demonstration which we
admire in the ancients, and which had been forced unduly into
the background by the exclusive interest of the preceding period
in \emph{new} developments. And yet I prefer to rank Gauss with
the great investigators of the eighteenth century, with Euler,
Lagrange,~etc. He belongs to them by the universality of his
work, in which no trace as yet appears of that specialization
which has become the characteristic of our times. He belongs
to them by his exclusively academic interest, by the absence of
the modern teaching activity just characterized. We shall have
a picture of the development of mathematics if we imagine a
chain of lofty mountains as representative of the men of the
eighteenth century, terminating in a mighty outlying summit,---\emph{Gauss},---and
then a broader, hilly country of lower elevation;
but teeming with new elements of life. More immediately connected
with Gauss we find in the following period only the
astronomers and geodesists under the dominating influence of
\emph{Bessel}; while in theoretical mathematics, as it begins henceforth
to be independently cultivated in our universities, a new
epoch begins with the second quarter of the present century,
marked by the illustrious names of \emph{Jacobi} and \emph{Dirichlet}.

\emph{Jacobi} came originally from Berlin and returned there for
the closing years of his life (died~1851). But it is the period
from 1826 to~1843, when he worked at Königsberg with \emph{Bessel}
and \emph{Franz Neumann}, that must be regarded as the culmination
of his activity. There he published in~1829 his \textit{Fundamenta
nova theoriæ functionum ellipticarum}, in which he gave, in
%% -----File: 112.png---Folio 102-------
analytic form, a systematic exposition of his own discoveries
and those of Abel in this field. Then followed a prolonged residence
in Paris, and finally that remarkable activity as a teacher,
which still remains without a parallel in stimulating power as
well as in direct results in the field of pure mathematics. An
idea of this work can be derived from the lectures on dynamics,
edited by Clebsch in~1866, and from the complete list of his
Königsberg lectures as compiled by Kronecker in the seventh
volume of the \textit{Gesammelte Werke}. The new feature is that
Jacobi lectured exclusively on those problems on which he was
working himself, and made it his sole object to introduce his
students into his own circle of ideas. With this end in view
he founded, for instance, the first mathematical seminary. And
so great was his enthusiasm that often he not only gave the
most important new results of his researches in these lectures,
but did not even take the time to publish them elsewhere.

\emph{Dirichlet} worked first in Breslau, then for a long period
(1831--1855) in Berlin, and finally for four years in Göttingen.
Following Gauss, but at the same time in close connection
with the contemporary French scholars, he chose mathematical
physics and the theory of numbers as the central points
of his scientific activity. It is to be noticed that his interest is
directed less towards comprehensive developments than towards
simplicity of conception and questions of principle; these are
also the considerations on which he insists particularly in his
lectures. These lectures are characterized by perfect lucidity
and a certain refined objectivity; they are at the same time
particularly accessible to the beginner and suggestive in a high
degree to the more advanced reader. It may be sufficient to
refer here to his lectures on the theory of numbers, edited by
Dedekind; they still form the standard text-book on this subject.

With Gauss, Jacobi, Dirichlet, we have named the men who
have determined the direction of the subsequent development.
%% -----File: 113.png---Folio 103-------
We shall now continue our account in a different manner,
arranging it according to the universities that have been most
prominent from a mathematical standpoint. For henceforth,
besides the special achievements of individual workers, the
principle of co-operation, with its dependence on local conditions,
comes to have more and more influence on the advancement
of our science. Setting the upper limit of our account
about the year~1870, we may name the universities of \emph{Königsberg},
\emph{Berlin}, \emph{Göttingen}, and \emph{Heidelberg}.

Of Jacobi's activity at Königsberg enough has already been
said. It may now be added that even after his departure the
university remained a centre of mathematical instruction.
\emph{Richelot} and \emph{Hesse} knew how to maintain the high tradition of
Jacobi, the former on the analytical, the latter on the geometrical
side. At the same time \emph{Franz Neumann's} lectures on
mathematical physics began to attract more and more attention
A stately procession of mathematicians has come from
Königsberg; there is scarcely a university in Germany to
which Königsberg has not sent a professor.

Of Berlin, too, we have already anticipated something in our
account. The years from 1845 to~1851, during which \emph{Jacobi}
and \emph{Dirichlet} worked together, form the culminating period of
the first Berlin school. Besides these men the most prominent
figure is that of \emph{Steiner} (connected with the university
from 1835 to~1864), the founder of the German synthetic
geometry. An altogether original character, he was a highly
effective teacher, owing to the one-sidedness with which he
developed his geometrical conceptions.---As an event of no
mean importance, we must here record the foundation (in~1826)
of \emph{Crelle's} \textit{Journal für reine und angewandte Mathematik}. This,
for decades the only German mathematical periodical, contained
in its pages the fundamental memoirs of nearly all the eminent
representatives of the rapidly growing science in Germany.
%% -----File: 114.png---Folio 104-------
Among foreign contributions the very first volumes presented
Abel's pioneer researches. \emph{Crelle} himself conducted this periodical
for thirty years; then followed \emph{Borchardt}, 1856--1880;
now the Journal has reached its 110th~volume.---We must
also mention the formation (in~1844) of the \textit{Berliner physikalische
Gesellschaft}. Men like \emph{Helmholtz}, \emph{Kirchhoff}, and
\emph{Clausius} have grown up here; and while these men cannot
be assigned to mathematics in the narrower sense, their work
has been productive of important results for our science in
various ways. During the same period, \emph{Encke} exercised, as
director of the Berlin astronomical observatory (1825--1862),
a far-reaching influence by elaborating the methods of astronomical
calculation on the lines first laid down by Gauss.---We
leave Berlin at this point, reserving for the present the
account of the more recent development of mathematics at
this university.

The discussion of the \emph{Göttingen school} will here find its
appropriate place. The permanent foundation on which the
mathematical importance of Göttingen rests is necessarily the
Gauss tradition. This found, indeed, its direct continuation
on the physical side when \emph{Wilhelm Weber} returned from
Leipsic to Göttingen~(1849) and for the first time established
systematic exercises in those methods of exact electro-magnetic
measurement that owed their origin to Gauss and himself.
On the mathematical side several eminent names follow in
rapid succession. After Gauss's death, Dirichlet was called
as his successor and transferred his great activity as a teacher
to Göttingen, for only too brief a period (1855--59). By his
side grew up \emph{Riemann} (1854--66), to be followed later by
\emph{Clebsch} (1868--72).

Riemann takes root in Gauss and Dirichlet; on the other
hand he fully assimilated Cauchy's ideas as to the use of
complex variables. Thus arose his profound creations in the
%% -----File: 115.png---Folio 105-------
theory of functions which ever since have proved a rich and
permanent source of the most suggestive material. Clebsch
sustains, so to speak, a complementary relation to Riemann.
Coming originally from Königsberg, and occupied with mathematical
physics, he had found during the period of his work
at Giessen (1863--68) the particular direction which he afterwards
followed so successfully at Göttingen. Well acquainted
with the work of Jacobi and with modern geometry, he introduced
into these fields the results of the algebraic researches of
the English mathematicians Cayley and Sylvester, and on the
double foundation thus constructed, proceeded to build up new
approaches to the problems of the entire theory of functions,
and in particular to Riemann's own developments. But with
this the significance of Clebsch for the development of our
science is not completely characterized. A man of vivid imagination
who readily entered into the ideas of others, he influenced
his students far beyond the limits of direct instruction;
of an active and enterprising character, he founded, together
with C.~Neumann in Leipsic, a new periodical, the \textit{Mathematische
Annalen}, which has since been regularly continued,
and is just concluding its 41st~volume.

We recall further those memorable years of Heidelberg, from
1855 to perhaps~1870. Here were delivered Hesse's elegant
and widely read lectures on analytic geometry. Here Kirchhoff
produced his lectures on mathematical physics. Here,
above all, Helmholtz completed his great papers on mathematical
physics, which in their turn served as basis for Kirchhoff's
elegant later researches.

It remains now to speak of the \emph{second Berlin school}, beginning
also about the middle of the century, but still operating upon
the present age. \emph{Kummer}, \emph{Kronecker}, \emph{Weierstrass}, have been
its leaders, the first two, as students of Dirichlet, pre-eminently
engaged in developing the theory of numbers, while the last,
%% -----File: 116.png---Folio 106-------
leaning more on Jacobi and Cauchy, became, together with
Riemann, the creator of the modern theory of functions.
Kummer's lectures can here merely be named in passing;
with their clear arrangement and exposition they have always
proved especially useful to the majority of students, without
being particularly notable for their specific contents. Quite
different is the case of Kronecker and Weierstrass, whose
lectures became in the course of time more and more the
expression of their scientific individuality. To a certain extent
both have thrust intuitional methods into the background
and, on the other hand, have in a measure avoided
the long formal developments of our science, applying themselves
with so much the keener criticism to the fundamental
analytical ideas. In this direction Kronecker has gone even
farther than Weierstrass in trying to banish altogether the
idea of the irrational number, and to reduce all developments
to relations between integers alone. The tendencies thus
characterized have exerted a wide-felt influence, and give a
distinctive character to a large part of our present mathematical
investigations.

We have thus sketched in general outlines the state reached
by our science about the year~1870. It is impossible to carry
our account beyond this date in a similar form. For the developments
that now arise are not yet finished; the persons whom
we should have to name are still in the midst of their creative
activity. All we can do is to add a few remarks of a more
general nature on the present aspect of mathematical science
in Germany. Before doing this, however, we must supplement
the preceding account in two directions.

Let it above all be emphasized that even within the limits
here chosen, we have by no means exhausted the subject. It
is, indeed, characteristic of the German universities that their
life is not wholly centralized,---that wherever a leader appears,
%% -----File: 117.png---Folio 107-------
he will find a sphere of activity. We may name here, from an
earlier period, the acute analyst \textit{J.~Fr.~Pfaff}, who worked in
Helmstädt and Halle from 1788 to~1825, and, at one time, had
Gauss among his students. Pfaff was the first representative
of the \emph{combinatory} school, which, for a time, played a great rôle
in different German universities, but was finally pushed aside in
the manifold development of the advancing science. We must
further mention the three great geometers, \emph{Möbius} in Leipsic,
\emph{Plücker} in Bonn, \emph{von~Staudt} in Erlangen. Möbius was, at the
same time, an astronomer, and conducted the Leipsic observatory
from 1816 till~1868. Plücker, again, devoted only the first
half of his productive period (1826--46) to mathematics, turning
his attention later to experimental physics (where his researches
are well known), and only returning to geometrical investigation
towards the close of his life (1864--68). The accidental circumstance
that each of these three men worked as teacher only in
a narrow circle has kept the development of modern geometry
unduly in the background in our sketch. Passing beyond
university circles, we may be allowed to add the name of
\emph{Grassmann}, of Stettin, who, in his \textit{Ausdehnungslehre} (1844 and~1862),
conceived a system embracing the results of modern
geometrical speculation, and, from a very different field, that of
\emph{Hansen}, of Gotha, the celebrated representative of theoretical
astronomy.

We must also mention, in a few words, the \emph{development of
technical education}. About the middle of the century, it became
the custom to call mathematicians of scientific eminence to the
polytechnic schools. Foremost in this respect stands Zürich,
which, in spite of the political boundaries, may here be counted
as our own; indeed, quite a number of professors have taught
in the Zürich polytechnic school who are to-day ornaments of
the German universities. Thus the ideal of the Paris school,
the combination of mathematical with technical education,
%% -----File: 118.png---Folio 108-------
became again more prominent. A considerable influence in
this direction was exercised by \emph{Redtenbacher's} lectures on the
theory of machines which attracted to Carlsruhe an ever-increasing
number of enthusiastic students. Descriptive geometry and
kinematics were scientifically elaborated. \emph{Culmann} of Zürich,
in creating graphical statics, introduced the principles of modern
geometry, in the happiest manner, into mechanics. In connection
with the scientific advance thus outlined, numerous new
polytechnic schools were founded in Germany about 1870 and
during the following years, and some of the older schools were
reorganized. At Munich and Dresden, in particular, in accordance
with the example of Zürich, special departments for the
training of teachers and professors were established. The
polytechnic schools have thus attained great importance for
mathematical education as well as for the advancement of the
science. We must forbear to pursue more closely the many
interesting questions that present themselves in this connection.

If we survey the entire field of development described above,
this, at any rate, appears as the obvious conclusion, in Germany
as elsewhere, that the number of those who have an earnest
interest in mathematics has increased very rapidly and that, as a
consequence, the amount of mathematical production has grown
to enormous proportions. In this respect an imperative need
was supplied when \emph{Ohrtmann} and \emph{Müller} established in Berlin
(1869) an annual bibliographical review, \textit{Die Fortschritte der
Mathematik}, of which the 21st~volume has just appeared.

In conclusion a few words should here be said concerning the
modern development of university instruction. The principal
effort has been to reduce the difficulty of mathematical study
by improving the seminary arrangements and equipments.
Not only have special seminary libraries been formed, but
study rooms have been set aside in which these libraries
are immediately accessible to the students. Collections of
%% -----File: 119.png---Folio 109-------
mathematical models and courses in drawing are calculated
to disarm, in part at least, the hostility directed against the
excessive abstractness of the university instruction. And
while the students find everywhere inducements to specialized
study, as is indeed necessary if our science is to flourish, yet
the tendency has at the same time gained ground to emphasize
more and more the mutual interdependence of the different
special branches. Here the individual can accomplish but
little; it seems necessary that many co-operate for the same
purpose. Such considerations have led in recent years to the
formation of a German mathematical association (\textit{Deutsche
Mathematiker-Vereinigung}). The first annual report just issued
(which contains a detailed report on the development of the
theory of invariants) and a comprehensive catalogue of mathematical
models and apparatus published at the same time indicate
the direction that is here to be followed. With the
present means of publication and the continually increasing
number of new memoirs, it has become almost impossible to
survey comprehensively the different branches of mathematics.
Hence it is the object of the association to collect, systematize,
maintain communication, in order that the work and
progress of the science may not be hampered by material
difficulties. Progress itself, however, remains---in mathematics
even more than in other sciences---always the right
and the achievement of the individual.

{\footnotesize\textsc{Göttingen}, January, 1893.}
%% -----File: 120.png---Folio 110-------
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% End of the Project Gutenberg EBook of The Evanston Colloquium Lectures on
% Mathematics, by Felix Klein                                             %
%                                                                         %
% *** END OF THIS PROJECT GUTENBERG EBOOK THE EVANSTON COLLOQUIUM ***     %
%                                                                         %
% ***** This file should be named 36154-t.tex or 36154-t.zip *****        %
% This and all associated files of various formats will be found in:      %
%         http://www.gutenberg.org/3/6/1/5/36154/                         %
%                                                                         %
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