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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:05:12 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:05:12 -0700 |
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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/36154-pdf.pdf b/36154-pdf.pdf Binary files differnew file mode 100644 index 0000000..99ce15b --- /dev/null +++ b/36154-pdf.pdf diff --git a/36154-pdf.zip b/36154-pdf.zip Binary files differnew file mode 100644 index 0000000..50e3fe4 --- /dev/null +++ b/36154-pdf.zip diff --git a/36154-t.zip b/36154-t.zip Binary files differnew file mode 100644 index 0000000..ec77486 --- /dev/null +++ b/36154-t.zip diff --git a/36154-t/36154-t.tex b/36154-t/36154-t.tex new file mode 100644 index 0000000..49e6a02 --- /dev/null +++ b/36154-t/36154-t.tex @@ -0,0 +1,5600 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% 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] % +% Most recently updated: June 11, 2021 % +% % +% Language: English % +% % +% Character set encoding: UTF-8 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK THE EVANSTON COLLOQUIUM *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{36154} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. 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+\end{minipage}\hfil +\refstepcounter{figno} +\begin{minipage}{0.4\textwidth} +\Input[#3]{#4} +\caption{Fig.~\thefigno.} +\label{fig:\thefigno} +\end{minipage} +\end{figure} +} + +\newcommand{\WFigure}[2]{% +\begin{wrapfigure}{o}{#1} + \refstepcounter{figno} + \centering + \Input[#1]{#2} + \caption{Fig.~\thefigno} + \label{fig:\thefigno} +\end{wrapfigure} +} + +\newcommand{\Fig}[1]{\hyperref[fig:#1]{Fig.~#1}} + +% Equation anchors and links +\newcommand{\Tag}[1]{% + \phantomsection + \label{eqn:\LectureNo#1} + \tag*{\normalsize\ensuremath{#1}} +} + +\newcommand{\Eq}[1]{\hyperref[eqn:\LectureNo#1]{\ensuremath{#1}}} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} + +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +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] +Most recently updated: June 11, 2021 + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK THE EVANSTON COLLOQUIUM *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Brenda Lewis, and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images from the Cornell University +Library: Historical Mathematics Monographs collection.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\phantomsection +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter +\pagenumbering{roman} +%% -----File: 001.png---Folio i------- +\null\vfill +\begin{center} +\Large LECTURES ON MATHEMATICS +\end{center} +\vfill +\newpage +%% -----File: 002.png---Folio ii------- +\null\vfill +\begin{center} +%[MacMillan Publisher's device] +\Input[1.5in]{002} +\end{center} +\vfill +\newpage +%% -----File: 003.png---Folio iii------- +\begin{center} +\linestretch{1.2}% +\setlength{\TmpLen}{16pt} +\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------- +%[Blank Page] +%% -----File: 011.png---Folio 1------- +\mainmatter +\pdfbookmark[-1]{Main Matter.}{Main Matter.} + +%[** 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------- +%[Blank Page] +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\cleardoublepage + +\backmatter +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\phantomsection +\pdfbookmark[0]{PG License}{Project Gutenberg License} +\fancyhead[C]{\textsc{LICENSING}} + +\begin{PGtext} +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-pdf.pdf or 36154-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/1/5/36154/ + +Produced by Andrew D. 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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 *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{36154} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Standard DP encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. 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+\label{fig:\thefigno} +\end{minipage} +\end{figure} +} + +\newcommand{\WFigure}[2]{% +\begin{wrapfigure}{o}{#1} + \refstepcounter{figno} + \centering + \Input[#1]{#2} + \caption{Fig.~\thefigno} + \label{fig:\thefigno} +\end{wrapfigure} +} + +\newcommand{\Fig}[1]{\hyperref[fig:#1]{Fig.~#1}} + +% Equation anchors and links +\newcommand{\Tag}[1]{% + \phantomsection + \label{eqn:\LectureNo#1} + \tag*{\normalsize\ensuremath{#1}} +} + +\newcommand{\Eq}[1]{\hyperref[eqn:\LectureNo#1]{\ensuremath{#1}}} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{document} + +\pagestyle{empty} +\pagenumbering{Alph} + +\phantomsection +\pdfbookmark[-1]{Front Matter}{Front Matter} + +%%%% PG BOILERPLATE %%%% +\phantomsection +\pdfbookmark[0]{PG Boilerplate}{Project Gutenberg Boilerplate} + +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +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 *** +\end{PGtext} +\end{minipage} +\end{center} + +\clearpage + + +%%%% Credits and transcriber's note %%%% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang, Brenda Lewis, and the Online +Distributed Proofreading Team at http://www.pgdp.net (This +file was produced from images from the Cornell University +Library: Historical Mathematics Monographs collection.) +\end{PGtext} +\end{minipage} +\end{center} +\vfill + +\begin{minipage}{0.85\textwidth} +\small +\phantomsection +\pdfbookmark[0]{Transcriber's Note}{Transcriber's Note} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% + +\frontmatter +\pagenumbering{roman} +%% -----File: 001.png---Folio i------- +\null\vfill +\begin{center} +\Large LECTURES ON MATHEMATICS +\end{center} +\vfill +\newpage +%% -----File: 002.png---Folio ii------- +\null\vfill +\begin{center} +%[MacMillan Publisher's device] +\Input[1.5in]{002} +\end{center} +\vfill +\newpage +%% -----File: 003.png---Folio iii------- +\begin{center} +\linestretch{1.2}% +\setlength{\TmpLen}{16pt} +\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------- +%[Blank Page] +%% -----File: 011.png---Folio 1------- +\mainmatter +\pdfbookmark[-1]{Main Matter.}{Main Matter.} + +%[** 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------- +%[Blank Page] +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% + +\cleardoublepage + +\backmatter +\phantomsection +\pdfbookmark[-1]{Back Matter}{Back Matter} +\phantomsection +\pdfbookmark[0]{PG License}{Project Gutenberg License} +\fancyhead[C]{\textsc{LICENSING}} + +\begin{PGtext} +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-pdf.pdf or 36154-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/1/5/36154/ + +Produced by Andrew D. 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THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..cc4e9ea --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #36154 (https://www.gutenberg.org/ebooks/36154) |
