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+\documentclass[oneside]{book}
+\usepackage[francais,english]{babel}
+\selectlanguage{english}
+\begin{document}
+
+\thispagestyle{empty}
+\small
+\begin{verbatim}
+Project Gutenberg's History of Modern Mathematics, by David Eugene Smith
+
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+**Welcome To The World of Free Plain Vanilla Electronic Texts**
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+*****These eBooks Were Prepared By Thousands of Volunteers!*****
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+
+Title: History of Modern Mathematics
+       Mathematical Monographs No. 1
+
+Author: David Eugene Smith
+
+Release Date: August, 2005 [EBook #8746]
+[Yes, we are more than one year ahead of schedule]
+[This file was first posted on August 9, 2003]
+
+Edition: 10
+
+Language: English
+
+Character set encoding: ASCII / TeX
+
+*** START OF THE PROJECT GUTENBERG EBOOK HISTORY OF MODERN MATHEMATICS ***
+
+
+Produced by David Starner, John Hagerson,
+and the Online Distributed Proofreading Team
+
+\end{verbatim}
+
+\normalsize
+\newpage
+
+
+
+\renewcommand{\chaptername}{Article}
+
+\frontmatter
+
+
+
+\begin{center}
+\noindent
+MATHEMATICAL MONOGRAPHS
+
+\bigskip
+EDITED BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD
+
+\bigskip\bigskip\huge
+No. 1
+
+\bigskip
+HISTORY OF MODERN MATHEMATICS.
+
+\bigskip\large
+BY
+
+\bigskip
+DAVID EUGENE SMITH,
+
+\bigskip\footnotesize\textsc{
+PROFESSOR OF MATHEMATICS IN TEACHERS COLLEGE, COLUMBIA UNIVERSITY.}
+
+\bigskip
+
+FOURTH EDITION, ENLARGED.
+
+1906
+
+\end{center}
+\newpage
+
+\fbox{\parbox{\columnwidth}{\textbf{MATHEMATICAL MONOGRAPHS.}\newline
+\small\textsc{edited by}\normalsize\newline
+\textbf{Mansfield Merriman and Robert S. Woodward.}
+
+\bigskip
+\textbf{No. 1. HISTORY OF MODERN MATHEMATICS.}\newline
+By \textsc{David Eugene Smith.}
+
+\bigskip
+\textbf{No. 2. SYNTHETIC PROJECTIVE GEOMETRY.}\newline
+By \textsc{George Bruce Halsted.}
+
+\bigskip
+\textbf{No. 3. DETERMINANTS.}\newline
+By \textsc{Laenas Gifford Weld.}
+
+\bigskip
+\textbf{No. 4. HYPERBOLIC FUNCTIONS.}\newline
+By \textsc{James McMahon.}
+
+\bigskip
+\textbf{No. 5. HARMONIC FUNCTIONS.}\newline
+By \textsc{William E. Byerly.}
+
+\bigskip
+\textbf{No. 6. GRASSMANN'S SPACE ANALYSIS.}\newline
+By \textsc{Edward W. Hyde.}
+
+\bigskip
+\textbf{No. 7. PROBABILITY AND THEORY OF ERRORS.}\newline
+By \textsc{Robert S. Woodward.}
+
+\bigskip
+\textbf{No. 8. VECTOR ANALYSIS AND QUATERNIONS.}\newline
+By \textsc{Alexander Macfarlane.}
+
+\bigskip
+\textbf{No. 9. DIFFERENTIAL EQUATIONS.}\newline
+By \textsc{William Woolsey Johnson.}
+
+\bigskip
+\textbf{No. 10. THE SOLUTION OF EQUATIONS.}\newline
+By \textsc{Mansfield Merriman.}
+
+\bigskip
+\textbf{No. 11. FUNCTIONS OF A COMPLEX VARIABLE.}\newline
+By \textsc{Thomas S. Fiske.}
+
+\normalsize
+} }
+\indent
+
+\newpage
+
+\chapter{EDITORS' PREFACE.}
+
+The volume called Higher Mathematics, the first edition of which was
+published in 1896, contained eleven chapters by eleven authors, each
+chapter being independent of the others, but all supposing the
+reader to have at least a mathematical training equivalent to that
+given in classical and engineering colleges. The publication of that
+volume is now discontinued and the chapters are issued in separate
+form. In these reissues it will generally be found that the
+monographs are enlarged by additional articles or appendices which
+either amplify the former presentation or record recent
+advances. This plan of publication has been arranged in order to
+meet the demand of teachers and the convenience of classes, but it
+is also thought that it may prove advantageous to readers in special
+lines of mathematical literature.
+
+It is the intention of the publishers and editors to add other
+monographs to the series from time to time, if the call for the same
+seems to warrant it. Among the topics which are under consideration
+are those of elliptic functions, the theory of numbers, the group
+theory, the calculus of variations, and non-Euclidean geometry;
+possibly also monographs on branches of astronomy, mechanics, and
+mathematical physics may be included. It is the hope of the editors
+that this form of publication may tend to promote mathematical study
+and research over a wider field than that which the former volume
+has occupied.
+
+\bigskip
+
+December, 1905.
+
+\normalsize
+
+\chapter{AUTHOR'S PREFACE.}
+
+This little work was published about ten years ago as a chapter in
+Merriman and Woodward's Higher Mathematics. It was written before
+the numerous surveys of the development of science in the past
+hundred years, which appeared at the close of the nineteenth
+century, and it therefore had more reason for being then than now,
+save as it can now call attention, to these later contributions. The
+conditions under which it was published limited it to such a small
+compass that it could do no more than present a list of the most
+prominent names in connection with a few important topics. Since it
+is necessary to use the same plates in this edition, simply adding a
+few new pages, the body of the work remains substantially as it
+first appeared. The book therefore makes no claim to being history,
+but stands simply as an outline of the prominent movements in
+mathematics, presenting a few of the leading names, and calling
+attention to some of the bibliography of the subject.
+
+It need hardly be said that the field of mathematics is now so
+extensive that no one can longer pretend to cover it, least of all
+the specialist in any one department. Furthermore it takes a century
+or more to weigh men and their discoveries, thus making the
+judgment of contemporaries often quite worthless. In spite of these
+facts, however, it is hoped that these pages will serve a good
+purpose by offering a point of departure to students desiring to
+investigate the movements of the past hundred years. The
+bibliography in the foot-notes and in Articles 19 and 20 will serve
+at least to open the door, and this in itself is a sufficient excuse
+for a work of this nature.
+
+\textsc{Teachers College, Columbia University,}
+
+December, 1905.
+
+\normalsize
+
+\tableofcontents
+
+%% CONTENTS.
+
+%% ART.
+
+%% 1. INTRODUCTION
+
+%% 2. THEORY OF NUMBERS
+
+%% 3. IRRATIONAL AND TRANSCENDENT NUMBERS
+
+%% 4. COMPLEX NUMBERS
+
+%% 5. QUATERNIONS AND AUSDEHNUNGSLEHRE
+
+%% 6. THEORY OF EQUATIONS
+
+%% 7. SUBSTITUTIONS AND GROUPS
+
+%% 8. DETERMINANTS
+
+%% 9. QUANTICS
+
+%% 10. CALCULUS
+
+%% 11. DIFFERENTIAL EQUATIONS
+
+%% 12. INFINITE SERIES
+
+%% 13. THEORY OF FUNCTIONS
+
+%% 14. PROBABILITIES AND LEAST SQUARES
+
+%% 15. ANALYTIC GEOMETRY
+
+%% 16. MODERN GEOMETRY
+
+%% 17. TRIGONOMETRY AND ELEMENTARY GEOMETRY
+
+%% 18. NON-EUCLIDEAN GEOMETRY
+
+%% 19. BIBLIOGRAPHY
+
+%% 20. GENERAL TENDENCIES
+
+%% INDEX
+
+\mainmatter
+
+\chapter{INTRODUCTION.}
+
+In considering the history of modern mathematics two questions at
+once arise: (1) what limitations shall be placed upon the term
+Mathematics; (2) what force shall be assigned to the word Modern? In
+other words, how shall Modern Mathematics be defined?
+
+In these pages the term Mathematics will be limited to the domain of
+pure science. Questions of the applications of the various branches
+will be considered only incidentally. Such great contributions as
+those of Newton in the realm of mathematical physics, of Laplace in
+celestial mechanics, of Lagrange and Cauchy in the wave theory, and
+of Poisson, Fourier, and Bessel in the theory of heat, belong rather
+to the field of applications.
+
+In particular, in the domain of numbers reference will be made to
+certain of the contributions to the general theory, to the men who
+have placed the study of irrational and transcendent numbers upon a
+scientific foundation, and to those who have developed the modern
+theory of complex numbers and its elaboration in the field of
+quaternions and Ausdehnungslehre. In the theory of equations the
+names of some of the leading investigators will be mentioned,
+together with a brief statement of the results which they
+secured. The impossibility of solving the quintic will lead to a
+consideration of the names of the founders of the group theory and
+of the doctrine of determinants. This phase of higher algebra will
+be followed by the theory of forms, or quantics. The later
+development of the calculus, leading to differential equations and
+the theory of functions, will complete the algebraic side, save for
+a brief reference to the theory of probabilities. In the domain of
+geometry some of the contributors to the later development of the
+analytic and synthetic fields will be mentioned, together with the
+most noteworthy results of their labors. Had the author's space not
+been so strictly limited he would have given lists of those who have
+worked in other important lines, but the topics considered have been
+thought to have the best right to prominent place under any
+reasonable definition of Mathematics.
+
+Modern Mathematics is a term by no means well defined. Algebra
+cannot be called modern, and yet the theory of equations has
+received some of its most important additions during the nineteenth
+century, while the theory of forms is a recent creation. Similarly
+with elementary geometry; the labors of Lobachevsky and Bolyai
+during the second quarter of the century threw a new light upon the
+whole subject, and more recently the study of the triangle has added
+another chapter to the theory. Thus the history of modern
+mathematics must also be the modern history of ancient branches,
+while subjects which seem the product of late generations have root
+in other centuries than the present.
+
+How unsatisfactory must be so brief a sketch may be inferred from a
+glance at the Index du Rep\'ertoire Bibliographique des Sciences
+Math\'ematiques (Paris, 1893), whose seventy-one pages contain the
+mere enumeration of subjects in large part modern, or from a
+consideration of the twenty-six volumes of the Jahrbuch \"uber die
+Fortschritte der Mathematik, which now devotes over a thousand pages
+a year to a record of the progress of the science.\footnote{The
+foot-notes give only a few of the authorities which might easily be
+cited. They are thought to include those from which considerable
+extracts have been made, the necessary condensation of these
+extracts making any other form of acknowledgment impossible.}
+
+The seventeenth and eighteenth centuries laid the foundations of
+much of the subject as known to-day. The discovery of the analytic
+geometry by Descartes, the contributions to the theory of numbers by
+Fermat, to algebra by Harriot, to geometry and to mathematical
+physics by Pascal, and the discovery of the differential calculus by
+Newton and Leibniz, all contributed to make the seventeenth century
+memorable. The eighteenth century was naturally one of great
+activity. Euler and the Bernoulli family in Switzerland,
+d'Alembert, Lagrange, and Laplace in Paris, and Lambert in Germany,
+popularized Newton's great discovery, and extended both its theory
+and its applications. Accompanying this activity, however, was a too
+implicit faith in the calculus and in the inherited principles of
+mathematics, which left the foundations insecure and necessitated
+their strengthening by the succeeding generation.
+
+The nineteenth century has been a period of intense study of first
+principles, of the recognition of necessary limitations of various
+branches, of a great spread of mathematical knowledge, and of the
+opening of extensive fields for applied mathematics. Especially
+influential has been the establishment of scientific schools and
+journals and university chairs. The great renaissance of geometry is
+not a little due to the foundation of the \'Ecole Polytechnique in
+Paris (1794-5), and the similar schools in Prague (1806), Vienna
+(1815), Berlin (1820), Karlsruhe (1825), and numerous other
+cities. About the middle of the century these schools began to exert
+a still a greater influence through the custom of calling to them
+mathematicians of high repute, thus making Z\"urich, Karlsruhe,
+Munich, Dresden, and other cities well known as mathematical centers.
+
+In 1796 appeared the first number of the Journal de l'\'Ecole
+Polytechnique. Crelle's Journal f\"ur die reine und angewandte
+Mathematik appeared in 1826, and ten years later Liouville began the
+publication of the Journal de Math\'ematiques pures et appliqu\'ees,
+which has been continued by Resal and Jordan. The Cambridge
+Mathematical Journal was established in 1839, and merged into the
+Cambridge and Dublin Mathematical Journal in 1846. Of the other
+periodicals which have contributed to the spread of mathematical
+knowledge, only a few can be mentioned: the Nouvelles Annales de
+Math\'ematiques (1842), Grunert's Archiv der Mathematik (1843),
+Tortolini's Annali di Scienze Matematiche e Fisiche (1850),
+Schl\"omilch's Zeitschrift f\"ur Mathematik und Physik (1856), the
+Quarterly Journal of Mathematics (1857), Battaglini's Giornale di
+Matematiche (1863), the Mathematische Annalen (1869), the Bulletin
+des Sciences Math\'ematiques (1870), the American Journal of
+Mathematics (1878), the Acta Mathematica (1882), and the Annals of
+Mathematics (1884).\footnote{For a list of current mathematical
+journals see the Jahrbuch \"uber die Fortschritte der Mathematik. A
+small but convenient list of standard periodicals is given in Carr's
+Synopsis of Pure Mathematics, p. 843; Mackay, J. S., Notice sur le
+journalisme math\'ematique en Angleterre, Association fran\c{c}aise
+pour l'Avancement des Sciences, 1893, II, 303; Cajori, F., Teaching
+and History of Mathematics in the United States, pp. 94, 277;
+Hart, D.~S., History of American Mathematical Periodicals, The Analyst,
+Vol. II, p. 131.} To this list should be added a recent venture,
+unique in its aims, namely, L'Interm\'ediaire des Math\'ematiciens
+(1894), and two annual publications of great value, the Jahrbuch
+already mentioned (1868), and the Jahresbericht der deutschen
+Math\-e\-ma\-tik\-er-Vereinigung (1892).
+%% Are those the correct hyphenation points?
+
+To the influence of the schools and the journals must be added that
+of the various learned societies\footnote{For a list of such
+societies consult any recent number of the Philosophical
+Transactions of Royal Society of London. Dyck, W., Einleitung zu dem
+f\"ur den mathematischen Teil der deutschen
+Universit\"atsausstellung ausgegebenen Specialkatalog, Mathematical
+Papers Chicago Congress (New York, 1896), p. 41.} whose published
+proceedings are widely known, together with the increasing
+liberality of such societies in the preparation of complete works of
+a monumental character.
+
+The study of first principles, already mentioned, was a natural
+consequence of the reckless application of the new calculus and the
+Cartesian geometry during the eighteenth century. This development
+is seen in theorems relating to infinite series, in the fundamental
+principles of number, rational, irrational, and complex, and in the
+concepts of limit, contiunity, function, the infinite, and the
+infinitesimal. But the nineteenth century has done more than
+this. It has created new and extensive branches of an importance
+which promises much for pure and applied mathematics. Foremost among
+these branches stands the theory of functions founded by Cauchy,
+Riemann, and Weierstrass, followed by the descriptive and
+projective geometries, and the theories of groups, of forms, and of
+determinants.
+
+The nineteenth century has naturally been one of specialization. At
+its opening one might have hoped to fairly compass the mathematical,
+physical, and astronomical sciences, as did Lagrange, Laplace, and
+Gauss. But the advent of the new generation, with Monge and Carnot,
+Poncelet and Steiner, Galois, Abel, and Jacobi, tended to split
+mathematics into branches between which the relations were long to
+remain obscure. In this respect recent years have seen a reaction,
+the unifying tendency again becoming prominent through the theories
+of functions and groups.\footnote{Klein, F., The Present State of
+Mathematics, Mathematical Papers of Chicago Congress (New York,
+1896), p. 133.}
+
+\chapter{THEORY OF NUMBERS.}
+
+The Theory of Numbers,\footnote{Cantor, M., Geschichte der
+Mathematik, Vol. III, p. 94; Smith, H.~J.~S., Report on the theory
+of numbers; Collected Papers, Vol. I; Stolz, O., Gr\"ossen und
+Zahien, Leipzig. 1891.} a favorite study among the Greeks, had its
+renaissance in the sixteenth and seventeenth centuries in the labors
+of Viete, Bachet de Meziriac, and especially Fermat. In the
+eighteenth century Euler and Lagrange contributed to the theory, and
+at its close the subject began to take scientific form through the
+great labors of Legendre (1798), and Gauss (1801). With the latter's
+Disquisitiones Arithmetic\ae (1801) may be said to begin the modern
+theory of numbers. This theory separates into two branches, the one
+dealing with integers, and concerning itself especially with (1) the
+study of primes, of congruences, and of residues, and in particular
+with the law of reciprocity, and (2) the theory of forms, and the
+other dealing with complex numbers.
+
+The Theory of Primes\footnote{Brocard, H., Sur la fr\'equence et la
+totalit\'e des nombres premiers; Nouvelle Correspondence de
+Math\'ematiques, Vols. V and VI; gives recent history to 1879.} has
+attracted many investigators during the nineteenth century, but
+the results have been detailed rather than general. Tch\'e\-bi\-chef
+(1850)
+% Another arbitary hyphenation point
+was the first to reach any valuable conclusions in the way of
+ascertaining the number of primes between two given limits. Riemann
+(1859) also gave a well-known formula for the limit of the number of
+primes not exceeding a given number.
+
+The Theory of Congruences may be said to start with Gauss's
+Disquisitiones. He introduced the symbolism $a \equiv b \pmod c$,
+and explored most of the field. Tch\'ebichef published in
+1847 a work in Russian upon the subject, and in France Serret has
+done much to make the theory known.
+
+Besides summarizing the labors of his predecessors in the theory of
+numbers, and adding many original and noteworthy contributions, to
+Legendre may be assigned the fundamental theorem which bears his
+name, the Law of Reciprocity of Quadratic Residues. This law,
+discovered by induction and enunciated by Euler, was first proved by
+Legendre in his Th\'eorie des Nombres (1798) for special
+cases. Independently of Euler and Legendre, Gauss discovered the law
+about 1795, and was the first to give a general proof. To the
+subject have also contributed Cauchy, perhaps the most versatile of
+French mathematicians of the century; Dirichlet, whose Vorlesungen
+\"uber Zahlentheorie, edited by Dedekind, is a classic; Jacobi,
+who introduced the generalized symbol which bears his name;
+Liouville, Zeller, Eisenstein, Kummer, and Kronecker. The theory has
+been extended to include cubic and biquadratic reciprocity, notably
+by Gauss, by Jacobi, who first proved the law of cubic reciprocity,
+and by Kummer.
+
+To Gauss is also due the representation of numbers by binary
+quadratic forms. Cauchy, Poinsot (1845), Lebesque (1859, 1868), and
+notably Hermite have added to the subject. In the theory of ternary
+forms Eisenstein has been a leader, and to him and H.~J.~S.~Smith is
+also due a noteworthy advance in the theory of forms in
+general. Smith gave a complete classification of ternary quadratic
+forms, and extended Gauss's researches concerning real quadratic
+forms to complex forms. The investigations concerning the
+representation of numbers by the sum of 4, 5, 6, 7, 8 squares were
+advanced by Eisenstein and the theory was completed by Smith.
+
+In Germany, Dirichlet was one of the most zealous workers in the
+theory of numbers, and was the first to lecture upon the subject in
+a German university. Among his contributions is the extension of
+Fermat's theorem on $x^n+y^n=z^n$, which Euler and Legendre had proved
+for $n$ = 3, 4, Dirichlet showing that $x^5+y^5 \neq az^5$. Among
+the later French writers are Borel; Poincar\'e, whose memoirs are
+numerous and valuable; Tannery, and Stieltjes. Among the leading
+contributors in Germany are Kronecker, Kummer, Schering, Bachmann,
+and Dedekind. In Austria Stolz's Vorlesungen \"uber allgemeine
+Arithmetik (1885-86), and in England Mathews' Theory of Numbers
+(Part I, 1892) are among the most scholarly of general works.
+Genocchi, Sylvester, and J.~W.~L.~Glaisher have also added to the
+theory.
+
+\chapter{IRRATIONAL AND TRANSCENDENT NUMBERS.}
+
+The sixteenth century saw the final acceptance of negative numbers,
+integral and fractional. The seventeenth century saw decimal
+fractions with the modern notation quite generally used by
+mathematicians. The next hundred years saw the imaginary become a
+powerful tool in the hands of De Moivre, and especially of
+Euler. For the nineteenth century it remained to complete the
+theory of complex numbers, to separate irrationals into algebraic
+and transcendent, to prove the existence of transcendent numbers,
+and to make a scientific study of a subject which had remained
+almost dormant since Euclid, the theory of irrationals. The year
+1872 saw the publication of the theories of Weierstrass (by his
+pupil Kossak), Heine (Crelle, 74), G. Cantor (Annalen, 5), and
+Dedekind. M\'eray had taken in 1869 the same point of departure as
+Heine, but the theory is generally referred to the year
+1872. Weierstrass's method has been completely set forth by
+Pincherle (1880), and Dedekind's has received additional prominence
+through the author's later work (1888) and the recent indorsement by
+Tannery (1894). Weierstrass, Cantor, and Heine base their theories
+on infinite series, while Dedekind founds his on the idea of a cut
+(Schnitt) in the system of real numbers, separating all rational
+numbers into two groups having certain characteristic
+properties. The subject has received later contributions at the
+hands of Weierstrass, Kronecker (Crelle, 101), and M\'eray.
+
+Continued Fractions, closely related to irrational numbers {and due
+to Ca\-tal\-di, 1613),\footnote{But see Favaro, A., Notizie storiche
+sulle frazioni continue dal secolo decimoterzo al decimosettimo,
+Boncompagni's Bulletino, Vol. VII, 1874, pp. 451, 533.} received
+attention at the hands of Euler, and at the opening of the
+nineteenth century were brought into prominence through the writings
+of Lagrange. Other noteworthy contributions have been made by
+Druckenm\"uller (1837), Kunze (1857), Lemke (1870), and G\"unther
+(1872). Ramus (1855) first connected the subject with determinants,
+resulting, with the subsequent contributions of Heine, M\"obius, and
+G\"unther, in the theory of Kettenbruchdeterminanten. Dirichlet
+also added to the general theory, as have numerous contributors to
+the applications of the subject.
+
+Transcendent Numbers\footnote{Klein, F., Vortr\"age \"uber
+ausgew\"ahlte Fragen der Elementargeometrie, 1895, p. 38; Bachmann,
+P., Vorlesungen \"uber die Natur der Irrationalzahlen, 1892.} were
+first distinguished from algebraic irrationals by
+Kronecker. Lambert proved (1761) that $\pi$ cannot be rational, and
+that $e^n$ ($n$ being a rational number) is irrational, a proof,
+however, which left much to be desired. Legendre (1794) completed
+Lambert's proof, and showed that $\pi$ is not the square root of a
+rational number. Liouville (1840) showed that neither $e$ nor
+$e^2$ can be a root of an integral quadratic equation. But the
+existence of transcendent numbers was first established by Liouville
+(1844, 1851), the proof being subsequently displaced by G. Cantor's
+(1873). Hermite (1873) first proved $e$ transcendent, and Lindemann
+(1882), starting from Hermite's conclusions, showed the same for
+$\pi$. Lindemann's proof was much simplified by Weierstrass (1885),
+still further by Hilbert (1893), and has finally been made
+elementary by Hurwitz and Gordan.
+
+\chapter{COMPLEX NUMBERS.}
+
+The Theory of Complex Numbers\footnote{Riecke, F., Die Rechnung mit
+Richtungszahlen, 1856, p. 161; Hankel, H., Theorie der komplexen
+Zahlensysteme, Leipzig, 1867; Holzm\"uller, G., Theorie der
+isogonalen Verwandtschaften, 1882, p. 21; Macfarlane, A., The
+Imaginary of Algebra, Proceedings of American Association 1892,
+p. 33; Baltzer, R., Einf\"uhrung der komplexen Zahlen, Crelle, 1882;
+Stolz, O., Vorlesungen \"uber allgemeine Arithmetik, 2. Theil,
+Leipzig, 1886.} may be said to have attracted attention as early as
+the sixteenth century in the recognition, by the Italian
+algebraists, of imaginary or impossible roots. In the seventeenth
+century Descartes distinguished between real and imaginary roots,
+and the eighteenth saw the labors of De Moivre and Euler. To De
+Moivre is due (1730) the well-known formula which bears his name,
+$(\cos \theta + i \sin
+\theta)^{n} = \cos n \theta + i \sin n \theta$, and to Euler (1748)
+the formula $\cos \theta + i \sin \theta = e ^{\theta i}$.
+
+The geometric notion of complex quantity now arose, and as a result
+the theory of complex numbers received a notable expansion. The idea
+of the graphic representation of complex numbers had appeared,
+however, as early as 1685, in Wallis's De Algebra tractatus. In the
+eighteenth century K\"uhn (1750) and Wessel (about 1795) made
+decided advances towards the present theory. Wessel's memoir
+appeared in the Proceedings of the Copenhagen Academy for 1799, and
+is exceedingly clear and complete, even in comparison with modern
+works. He also considers the sphere, and gives a quaternion theory
+from which he develops a complete spherical trigonometry. In 1804
+the Abb\'e Bu\'ee independently came upon the same idea which Wallis
+had suggested, that $\pm\sqrt{-1}$ should represent a unit line, and
+its negative, perpendicular to the real axis. Bu\'ee's paper was
+not published until 1806, in which year Argand also issued a
+pamphlet on the same subject. It is to Argand's essay that the
+scientific foundation for the graphic representation of complex
+numbers is now generally referred. Nevertheless, in 1831 Gauss
+found the theory quite unknown, and in 1832 published his chief
+memoir on the subject, thus bringing it prominently before the
+mathematical world. Mention should also be made of an excellent
+little treatise by Mourey (1828), in which the foundations for the
+theory of directional numbers are scientifically laid. The general
+acceptance of the theory is not a little due to the labors of Cauchy
+and Abel, and especially the latter, who was the first to boldly use
+complex numbers with a success that is well known.
+
+The common terms used in the theory are chiefly due to the
+founders. Argand called $\cos \phi + i \sin \phi$ the ``direction
+factor'', and $r = \sqrt{a^2+b^2}$ the ``modulus''; Cauchy (1828)
+called $\cos \phi + i \sin \phi$ the ``reduced form'' (l'expression
+r\'eduite); Gauss used $i$ for $\sqrt{-1}$, introduced the term
+``complex number'' for $a+bi$, and called $a^2+b^2$ the ``norm.'' The
+expression ``direction coefficient'', often used for $\cos \phi + i
+\sin \phi$, is due to Hankel (1867), and ``absolute value,'' for
+``modulus,'' is due to Weierstrass.
+
+Following Cauchy and Gauss have come a number of contributors of
+high rank, of whom the following may be especially mentioned: Kummer
+(1844), Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis
+(1835, 1852), Peacock (1845), and De Morgan (1849). M\"obius must
+also be mentioned for his numerous memoirs on the geometric
+applications of complex numbers, and Dirichlet for the expansion of
+the theory to include primes, congruences, reciprocity, etc., as in
+the case of real numbers.
+
+Other types\footnote{Chapman, C. H., Weierstrass and Dedekind on
+General Complex Numbers, in Bulletin New York Mathematical Society,
+Vol. I, p. 150; Study, E., Aeltere und neuere Untersuchungen \"uber
+Systeme complexer Zahlen, Mathematical Papers Chicago Congress,
+p. 367; bibliography, p. 381.} have been studied, besides the
+familiar $a + bi$, in which $i$ is the root of $x^2 + 1 = 0$. Thus
+Eisenstein has studied the type $a + bj$, $j$ being a complex root
+of $x^3 - 1 = 0$. Similarly, complex types have been derived from
+$x^k - 1 = 0$ ($k$ prime). This generalization is largely due to
+Kummer, to whom is also due the theory of Ideal
+Numbers,\footnote{Klein, F., Evanston Lectures, Lect. VIII.} which
+has recently been simplified by Klein (1893) from the point of view
+of geometry. A further complex theory is due to Galois, the basis
+being the imaginary roots of an irreducible congruence, $F(x) \equiv 0$
+(mod $p$, a prime). The late writers (from 1884) on the general
+theory include Weierstrass, Schwarz, Dedekind, H\"older, Berloty,
+Poincar\'e, Study, and Macfarlane.
+
+\chapter{QUATERNIONS AND AUSDEHNUNGSLEHRE.}
+
+Quaternions and Ausdehnungslehre\footnote{Tait, P.~G., on
+Quaternions, Encyclop\ae{}dia Britannica; Schlegel, V., Die
+Grassmann'sche Ausdehnungslehre, Schl\"omilch's Zeitschrift,
+Vol. XLI.} are so closely related to complex quantity, and the
+latter to complex number, that the brief sketch of their development
+is introduced at this point. Caspar Wessel's contributions to the
+theory of complex quantity and quaternions remained unnoticed in
+the proceedings of the Copenhagen Academy. Argand's attempts to
+extend his method of complex numbers beyond the space of two
+dimensions failed. Servois (1813), however, almost trespassed on the
+quaternion field. Nevertheless there were fewer traces of the theory
+anterior to the labors of Hamilton than is usual in the case of
+great discoveries. Hamilton discovered the principle of quaternions
+in 1843, and the next year his first contribution to the theory
+
+
+appeared, thus extending the Argand idea to three-dimensional
+space. This step necessitated an expansion of the idea of $r(\cos
+\phi + j \sin \phi)$ such that while $r$ should be a real number and
+$\phi$ a real angle, $i$, $j$, or $k$ should be any directed unit
+line such that $i^2 = j^2 = k^2 = -1$. It also necessitated a
+withdrawal of the commutative law of multiplication, the adherence
+to which obstructed earlier discovery. It was not until 1853 that
+Hamilton's Lectures on Quarternions appeared, followed (1866) by his
+Elements of Quaternions.
+
+In the same year in which Hamilton published his discovery (1844),
+Grassmann gave to the world his famous work, Die lineale
+Ausdehnungslehre, although he seems to have been in possession of
+the theory as early as 1840. Differing from Hamilton's Quaternions
+in many features, there are several essential principles held in
+common which each writer discovered independently of the
+other.\footnote{These are set forth in a paper by J.~W.~Gibbs,
+Nature, Vol. XLIV, p. 79.}
+
+Following Hamilton, there have appeared in Great Britain numerous
+papers and works by Tait (1867), Kelland and Tait (1873), Sylvester,
+and McAulay (1893). On the Continent Hankel (1867), Ho\"uel (1874),
+and Laisant (1877, 1881) have written on the theory, but it has
+attracted relatively little attention. In America, Benjamin Peirce
+(1870) has been especially prominent in developing the quaternion
+theory, and Hardy (1881), Macfarlane, and Hathaway (1896) have
+contributed to the subject. The difficulties have been largely in
+the notation. In attempting to improve this symbolism Macfarlane has
+aimed at showing how a space analysis can be developed embracing
+algebra, trigonometry, complex numbers, Grassmann's method, and
+quaternions, and has considered the general principles of vector and
+versor analysis, the versor being circular, elliptic logarithmic, or
+hyperbolic. Other recent contributors to the algebra of vectors are
+Gibbs (from 1881) and Heaviside (from 1885).
+
+The followers of Grassmann\footnote{For bibliography see Schlegel,
+V., Die Grassmann'sche Ausdehnungslehre, Schl\"omilch's Zeitschrift,
+Vol. XLI.} have not been much more numerous than those of
+Hamilton. Schlegel has been one of the chief contributors in
+Germany, and Peano in Italy. In America, Hyde (Directional Calculus,
+1890) has made a plea for the Grassmann theory.\footnote{For
+Macfarlane's Digest of views of English and American writers, see
+Proceedings American Association for Advancement of Science, 1891.}
+
+Along lines analogous to those of Hamilton and Grassmann have been
+the contributions of Scheffler. While the two former sacrificed the
+commutative law, Scheffler (1846, 1851, 1880) sacrificed the
+distributive. This sacrifice of fundamental laws has led to an
+investigation of the field in which these laws are valid, an
+investigation to which Grassmann (1872), Cayley, Ellis, Boole,
+Schr\"oder (1890-91), and Kraft (1893) have contributed. Another
+great contribution of Cayley's along similar lines is the theory of
+matrices (1858).
+
+\chapter{THEORY OF EQUATIONS.}
+
+The Theory of Numerical Equations\footnote{Cayley, A., Equations,
+and Kelland, P., Algebra, in Encyclop\ae{}dia Britannica; Favaro, A.,
+Notizie storico-critiche sulla costruzione delle equazioni. Modena,
+1878; Cantor, M., Geschichte der Mathematik, Vol. III, p. 375.}
+concerns itself first with the location of the roots, and then with
+their approximation. Neither problem is new, but the first
+noteworthy contribution to the former in the nineteenth century was
+Budan's (1807). Fourier's work was undertaken at about the same
+time, but appeared posthumously in 1831. All processes were,
+however, exceedingly cumbersome until Sturm (1829) communicated to
+the French Academy the famous theorem which bears his name and which
+constitutes one of the most brilliant discoveries of algebraic
+analysis.
+
+The Approximation of the Roots, once they are located, can be made
+by several processes. Newton (1711), for example, gave a method
+which Fourier perfected; and Lagrange (1767) discovered an ingenious
+way of expressing the root as a continued fraction, a process which
+Vincent (1836) elaborated. It was, however, reserved for Horner
+(1819) to suggest the most practical method yet known, the one now
+commonly used. With Horner and Sturm this branch practically
+closes. The calculation of the imaginary roots by approximation is
+still an open field.
+
+The Fundamental Theorem\footnote{Loria, Gino, Esame di alcune
+ricerche concernenti l'esistenza di radici nelle equazioni
+algebriche; Bibliotheca Mathematica, 1891, p. 99; bibliography on
+p. 107. Pierpont, J., On the Ruffini-Abelian theorem, Bulletin of
+American Mathematical Society, Vol. II, p. 200.} that every
+numerical equation has a root was generally assumed until the latter
+part of the eighteenth century. D'Alembert (1746) gave a
+demonstration, as did Lagrange (1772), Laplace (1795), Gauss (1799)
+and Argand (1806). The general theorem that every algebraic equation
+of the $n$th degree has exactly $n$ roots and no more follows as a
+special case of Cauchy's proposition (1831) as to the number of
+roots within a given contour. Proofs are also due to Gauss, Serret,
+Clifford (1876), Malet (1878), and many others.
+
+The Impossibility of Expressing the Roots of an equation as
+algebraic functions of the coefficients when the degree exceeds 4
+was anticipated by Gauss and announced by Ruffini, and the belief in
+the fact became strengthened by the failure of Lagrange's methods
+for these cases. But the first strict proof is due to Abel, whose
+early death cut short his labors in this and other fields.
+
+The Quintic Equation has naturally been an object of special
+study. Lagrange showed that its solution depends on that of a
+sextic, ``Lagrange's resolvent sextic,'' and Malfatti and
+Vandermonde investigated the construction of resolvents. The
+resolvent sextic was somewhat simplified by Cockle and Harley
+(1858-59) and by Cayley (1861), but Kronecker (1858) was the first
+to establish a resolvent by which a real simplification was
+effected. The transformation of the general quintic into the
+trinomial form $x^5+ax+b=0$ by the extraction of square and cube
+roots only, was first shown to be possible by Bring (1786) and
+independently by Jerrard\footnote{Harley, R., A contribution of
+the history \ldots of the general equation of the fifth degree,
+Quarterly Journal of Mathematics, Vol. VI, p. 38.} (1834). Hermite
+(1858) actually effected this reduction, by means of Tschirnhausen's
+theorem, in connection with his solution by elliptic functions.
+
+The Modern Theory of Equations may be said to date from Abel and
+Galois. The latter's special memoir on the subject, not published
+until 1846, fifteen years after his death, placed the theory on a
+definite base. To him is due the discovery that to each equation
+corresponds a group of substitutions (the ``group of the equation'')
+in which are reflected its essential characteristics.\footnote{See
+Art. 7.} Galois's untimely death left without sufficient
+demonstration several important propositions, a gap which Betti
+(1852) has filled. Jordan, Hermite, and Kronecker were also among
+the earlier ones to add to the theory. Just prior to Galois's
+researches Abel (1824), proceeding from the fact that a rational
+function of five letters having less than five values cannot have
+more than two, showed that the roots of a general quintic equation
+cannot be expressed in terms of its coefficients by means of
+radicals. He then investigated special forms of quintic equations
+which admit of solution by the extraction of a finite number of
+roots. Hermite, Sylvester, and Brioschi have applied the invariant
+theory of binary forms to the same subject.
+
+From the point of view of the group the solution by radicals,
+formerly the goal of the algebraist, now appears as a single link in
+a long chain of questions relative to the transformation of
+irrationals and to their classification. Klein (1884) has handled
+the whole subject of the quintic equation in a simple manner by
+introducing the icosahedron equation as the normal form, and has
+shown that the method can be generalized so as to embrace the whole
+theory of higher equations.\footnote{Klein, F., Vorlesungen \"uber
+das Ikosaeder, 1884.} He and Gordan (from 1879) have attacked those
+equations of the sixth and seventh degrees which have a Galois group
+of 168 substitutions, Gordan performing the reduction of the
+equation of the seventh degree to the ternary problem. Klein (1888)
+has shown that the equation of the twenty-seventh degree occurring
+in the theory of cubic surfaces can be reduced to a normal problem
+in four variables, and Burkhardt (1893) has performed the reduction,
+the quaternary groups involved having been discussed by Maschke
+(from 1887).
+
+Thus the attempt to solve the quintic equation by means of radicals
+has given place to their treatment by transcendents. Hermite (1858)
+has shown the possibility of the solution, by the use of elliptic
+functions, of any Bring quintic, and hence of any equation of the
+fifth degree. Kronecker (1858), working from a different standpoint,
+has reached the same results, and his method has since been
+simplified by Brioschi. More recently Kronecker, Gordan, Kiepert,
+and Klein, have contributed to the same subject, and the sextic
+equation has been attacked by Maschke and Brioschi through the
+medium of hyperelliptic functions.
+
+Binomial Equations, reducible to the form $x^n - 1 = 0$, admit of
+ready solution by the familiar trigonometric formula $x =
+\cos\frac{2k\pi}{n} + i\sin\frac{2k\pi}{n}$; but it was reserved for
+Gauss (1801) to show that an algebraic solution is
+possible. Lagrange (1808) extended the theory, and its application
+to geometry is one of the leading additions of the century. Abel,
+generalizing Gauss's results, contributed the important theorem that
+if two roots of an irreducible equation are so connected that the
+one can be expressed rationally in terms of the other, the equation
+yields to radicals if the degree is prime and otherwise depends on
+the solution of lower equations. The binomial equation, or rather
+the equation $\sum_0^{n-1} x^m = 0$, is one of this class
+considered by Abel, and hence called (by Kronecker) Abelian
+Equations. The binomial equation has been treated notably by
+Richelot (1832), Jacobi (1837), Eisenstein (1844, 1850), Cayley
+(1851), and Kronecker (1854), and is the subject of a treatise by
+Bachmann (1872). Among the most recent writers on Abelian equations
+is Pellet (1891).
+
+Certain special equations of importance in geometry have been the
+subject of study by Hesse, Steiner, Cayley, Clebsch, Salmon, and
+Kummer. Such are equations of the ninth degree determining the
+points of inflection of a curve of the third degree, and of the
+twenty-seventh degree determining the points in which a curve of the
+third degree can have contact of the fifth order with a conic.
+
+Symmetric Functions of the coefficients, and those which remain
+unchanged through some or all of the permutations of the roots, are
+subjects of great importance in the present theory. The first
+formulas for the computation of the symmetric functions of the
+roots of an equation seem to have been worked out by Newton,
+although Girard (1629) had given, without proof, a formula for the
+power sum. In the eighteenth century Lagrange (1768) and Waring
+(1770, 1782) contributed to the theory, but the first tables,
+reaching to the tenth degree, appeared in 1809 in the Meyer-Hirsch
+Aufgabensammlung. In Cauchy's celebrated memoir on determinants
+(1812) the subject began to assume new prominence, and both he and
+Gauss (1816) made numerous and valuable contributions to the
+theory. It is, however, since the discoveries by Galois that the
+subject has become one of great importance. Cayley (1857) has given
+simple rules for the degree and weight of symmetric functions, and
+he and Brioschi have simplified the computation of tables.
+
+Methods of Elimination and of finding the resultant (Bezout) or
+eliminant (De Morgan) occupied a number of eighteenth-century
+algebraists, prominent among them being Euler (1748), whose method,
+based on symmetric functions, was improved by Cramer (1750) and
+Bezout (1764). The leading steps in the development are represented
+by Lagrange (1770-71), Jacobi, Sylvester (1840), Cayley (1848,
+1857), Hesse (1843, 1859), Bruno (1859), and Katter
+(1876). Sylvester's dialytic method appeared in 1841, and to him is
+also due (1851) the name and a portion of the theory of the
+discriminant. Among recent writers on the general theory may be
+mentioned Burnside and Pellet (from 1887).
+
+\chapter{SUBSTITUTIONS AND GROUPS.}
+
+The Theories of Substitutions and Groups\footnote{Netto, E., Theory
+of Substitutions, translated by Cole; Cayley, A., Equations,
+Encyclop\ae{}dia Britannica, 9th edition.} are among the most important
+in the whole mathematical field, the study of groups and the search
+for invariants now occupying the attention of many
+mathematicians. The first recognition of the importance of the
+combinatory analysis occurs in the problem of forming an
+$m$th-degree equation having for roots $m$ of the roots of a given
+$n$th-degree equation ($m < n$). For simple cases the problem goes
+back to Hudde (1659). Saunderson (1740) noted that the determination
+of the quadratic factors of a biquadratic expression necessarily
+leads to a sextic equation, and Le S\oe{}ur (1748) and Waring (1762
+to 1782) still further elaborated the idea.
+
+Lagrange\footnote{Pierpont, James, Lagrange's Place in the Theory
+of Substitutions, Bulletin of American Mathematical Society, Vol. I, p.
+196.} first undertook a scientific treatment of the theory of
+substitutions. Prior to his time the various methods of solving
+lower equations had existed rather as isolated artifices than as
+unified theory.\footnote{Matthiessen, L. Grundz\"uge der antiken
+und modernen Algebra der litteralen Gleichungen, Leipzig, 1878.}
+Through the great power of analysis possessed by Lagrange (1770,
+1771) a common foundation was discovered, and on this was built the
+theory of substitutions. He undertook to examine the methods then
+known, and to show a priori why these succeeded below the quintic,
+but otherwise failed. In his investigation he discovered the
+important fact that the roots of all resolvents (r\'solvantes,
+r\'eduites) which he examined are rational functions of the roots
+of the respective equations. To study the properties of these
+functions he invented a ``Calcul des Combinaisons.'' the first
+important step towards a theory of substitutions. Mention should
+also be made of the contemporary labors of Vandermonde (1770) as
+foreshadowing the coming theory.
+
+The next great step was taken by Ruffini\footnote{Burkhardt, H.,
+Die Anf\"ange der Gruppentheorie und Paolo Ruffini, Abhandlungen zur
+Geschichte der Mathematik, VI, 1892, p. 119. Italian by E. Pascal,
+Brioschi's Annali di Matematica, 1894.} (1799). Beginning like
+Lagrange with a discussion of the methods of solving lower
+equations, he attempted the proof of the impossibility of solving
+the quintic and higher equations. While the attempt failed, it is
+noteworthy in that it opens with the classification of the various
+``permutations'' of the coefficients, using the word to mean what
+Cauchy calls a ``syst\`eme des substitutions conjugu\'ees,'' or
+simply a ``syst\`eme conjugu\'e,'' and Galois calls a ``group of
+substitutions.'' Ruffini distinguishes what are now called
+intransitive, transitive and imprimitive, and transitive and
+primitive groups, and (1801) freely uses the group of an equation
+under the name ``l'assieme della permutazioni.'' He also publishes a
+letter from Abbati to himself, in which the group idea is prominent.
+
+To Galois, however, the honor of establishing the theory of groups
+is generally awarded. He found that if $r_1, r_2, \ldots r_n$ are
+the $n$ roots of an equation, there is always a group of
+permutations of the $r$'s such that (1) every function of the roots
+invariable by the substitutions of the group is rationally known,
+and (2), reciprocally, every rationally determinable function of the
+roots is invariable by the substitutions of the group. Galois also
+contributed to the theory of modular equations and to that of
+elliptic functions. His first publication on the group theory was
+made at the age of eighteen (1829), but his contributions attracted
+little attention until the publication of his collected papers in
+1846 (Liouville, Vol. XI).
+
+Cayley and Cauchy were among the first to appreciate the importance
+of the theory, and to the latter especially are due a number of
+important theorems. The popularizing of the subject is largely due
+to Serret, who has devoted section IV of his algebra to the theory;
+to Camille Jordan, whose Trait\'e des Substitutions is a classic;
+and to Netto (1882), whose work has been translated into English by
+Cole (1892). Bertrand, Hermite, Frobenius, Kronecker, and Mathieu
+have added to the theory. The general problem to determine the
+number of groups of $n$ given letters still awaits solution.
+
+But overshadowing all others in recent years in carrying on the
+labors of Galois and his followers in the study of discontinuous
+groups stand Klein, Lie, Poincar\'e, and Picard. Besides these
+discontinuous groups there are other classes, one of which, that of
+finite continuous groups, is especially important in the theory of
+differential equations. It is this class which Lie (from 1884) has
+studied, creating the most important of the recent departments of
+mathematics, the theory of transformation groups. Of value, too,
+have been the labors of Killing on the structure of groups, Study's
+application of the group theory to complex numbers, and the work of
+Schur and Maurer.
+
+\chapter{DETERMINANTS.}
+
+The Theory of Determinants\footnote{Muir, T., Theory of Determinants
+in the Historical Order of its Development, Part I, 1890; Baltzer,
+R., Theorie und Anwendung der Determinanten. 1881. The writer is
+under obligations to Professor Weld, who contributes Chap. II, for
+valuable assistance in compiling this article.} may be said to take
+its origin with Leibniz (1693), following whom Cramer (1750) added
+slightly to the theory, treating the subject, as did his
+predecessor, wholly in relation to sets of equations. The recurrent
+law was first announced by Bezout (1764). But it was Vandermonde
+(1771) who first recognized determinants as independent
+functions. To him is due the first connected exposition of the
+theory, and he may be called its formal founder. Laplace (1772)
+gave the general method of expanding a determinant in terms of its
+complementary minors, although Vandermonde had already given a
+special case. Immediately following, Lagrange (1773) treated
+determinants of the second and third order, possibly stopping here
+because the idea of hyperspace was not then in vogue. Although
+contributing nothing to the general theory, Lagrange was the first
+to apply determinants to questions foreign to eliminations, and to
+him are due many special identities which have since been brought
+under well-known theorems. During the next quarter of a century
+little of importance was done. Hindenburg (1784) and Rothe (1800)
+kept the subject open, but Gauss (1801) made the next advance. Like
+Lagrange, he made much use of determinants in the theory of
+numbers. He introduced the word ``determinants'' (Laplace had used
+``resultant''), though not in the present
+signification,\footnote{``Numerum $bb-ac$, cuius indole
+proprietates form\ae $(a, b, c)$ imprimis pendere in sequentibus
+docebimus, determinantem huius uocabimus.''} but
+rather as applied to the discriminant of a
+quantic. Gauss also arrived at the notion of reciprocal
+determinants, and came very near the multiplication theorem. The
+next contributor of importance is Binet (1811, 1812), who formally
+stated the theorem relating to the product of two matrices of $m$
+columns and $n$ rows, which for the special case of $m = n$ reduces
+to the multiplication theorem. On the same day (Nov. 30, 1812) that
+Binet presented his paper to the Academy, Cauchy also presented one
+on the subject. In this he used the word ``determinant'' in its
+present sense, summarized and simplified what was then known on the
+subject, improved the notation, and gave the multiplication theorem
+with a proof more satisfactory than Binet's. He was the first to
+grasp the subject as a whole; before him there were determinants,
+with him begins their theory in its generality.
+
+The next great contributor, and the greatest save Cauchy, was Jacobi
+(from 1827). With him the word ``determinant'' received its final
+acceptance. He early used the functional determinant which Sylvester
+has called the ``Jacobian,'' and in his famous memoirs in Crelle for
+1841 he specially treats this subject, as well as that class of
+alternating functions which Sylvester has called ``Alternants.'' But
+about the time of Jacobi's closing memoirs, Sylvester (1839) and
+Cayley began their great work, a work which it is impossible to
+briefly summarize, but which represents the development of the
+theory to the present time.
+
+The study of special forms of determinants has been the natural
+result of the completion of the general theory. Axi-symmetric
+determinants have been studied by Lebesgue, Hesse, and Sylvester;
+per-symmetric determinants by Sylvester and Hankel; circulants by
+Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and
+Pfaffians, in connection with the theory of orthogonal
+transformation, by Cayley; continuants by Sylvester; Wronskians (so
+called by Muir) by Christoffel and Frobenius; compound determinants
+by Sylvester, Reiss, and Picquet; Jacobians and Hessians by
+Sylvester; and symmetric gauche determinants by Trudi. Of the
+text-books on the subject Spottiswoode's was the first. In America,
+Hanus (1886) and Weld (1893) have published treatises.
+
+\chapter{QUANTICS.}
+
+The Theory of Qualities or Forms\footnote{Meyer, W.~F., Bericht
+\"uber den gegenw\"artigen Stand der Invariantentheorie.
+Jahresbericht der deutschen Mathematiker-Vereinigung, Vol. I,
+1890-91; Berlin 1892, p. 97. See also the review by Franklin in
+Bulletin New York Mathematical Society, Vol. III, p. 187; Biography
+of Cayley, Collected Papers, VIII, p. ix, and Proceedings of Royal
+Society, 1895.} appeared in embryo in the Berlin memoirs of Lagrange
+(1773, 1775), who considered binary quadratic forms of the type
+$ax^2+bxy+cy^2$, and established the invariance of the discriminant
+of that type when $x+\lambda y$ is put for $x$. He classified forms
+of that type according to the sign of $b^2-4ac$, and introduced the
+ideas of transformation and equivalence. Gauss\footnote{See
+Art. 2.} (1801) next took up the subject, proved the invariance of
+the discriminants of binary and ternary quadratic forms, and
+systematized the theory of binary quadratic forms, a subject
+elaborated by H.~J.~S.~Smith, Eisenstein, Dirichlet, Lipschitz,
+Poincar\'e, and Cayley. Galois also entered the field, in his
+theory of groups (1829), and the first step towards the
+establishment of the distinct theory is sometimes attributed to
+Hesse in his investigations of the plane curve of the third order.
+
+It is, however, to Boole (1841) that the real foundation of the
+theory of invariants is generally ascribed. He first showed the
+generality of the invariant property of the discriminant, which
+Lagrange and Gauss had found for special forms. Inspired by Boole's
+discovery Cayley took up the study in a memoir ``On the Theory of
+Linear Transformations'' (1845), which was followed (1846) by
+investigations concerning covariants and by the discovery of the
+symbolic method of finding invariants. By reason of these
+discoveries concerning invariants and covariants (which at first he
+called ``hyperdeterminants'') he is regarded as the founder of what
+is variously called Modern Algebra, Theory of Forms, Theory of
+Quantics, and the Theory of Invariants and Covariants. His ten
+memoirs on the subject began in 1854, and rank among the greatest
+which have ever been produced upon a single theory. Sylvester soon
+joined Cayley in this work, and his originality and vigor in
+discovery soon made both himself and the subject prominent. To him
+are due (1851-54) the foundations of the general theory, upon which
+later writers have largely built, as well as most of the terminology
+of the subject.
+
+Meanwhile in Germany Eisenstein (1843) had become aware of the
+simplest invariants and covariants of a cubic and biquadratic form,
+and Hesse and Grassmann had both (1844) touched upon the
+subject. But it was Aronhold (1849) who first made the new theory
+known. He devised the symbolic method now common in Germany,
+discovered the invariants of a ternary cubic and their relations to
+the discriminant, and, with Cayley and Sylvester, studied those
+differential equations which are satisfied by invariants and
+covariants of binary quantics. His symbolic method has been carried
+on by Clebsch, Gordan, and more recently by Study (1889) and Stroh
+(1890), in lines quite different from those of the English school.
+
+In France Hermite early took up the work (1851). He discovered
+(1854) the law of reciprocity that to every covariant or invariant
+of degree $\rho$ and order $r$ of a form of the $m$th order
+corresponds also a covariant or invariant of degree $m$ and of order
+$r$ of a form of the $\rho$th order. At the same time (1854)
+Brioschi joined the movement, and his contributions have been among
+the most valuable. Salmon's Higher Plane Curves (1852) and Higher
+Algebra (1859) should also be mentioned as marking an epoch in the
+theory.
+
+Gordan entered the field, as a critic of Cayley, in 1868. He added
+greatly to the theory, especially by his theorem on the Endlichkeit
+des Formensystems, the proof for which has since been
+simplified. This theory of the finiteness of the number of
+invariants and covariants of a binary form has since been extended
+by Peano (1882), Hilbert (1884), and Mertens (1886). Hilbert (1890)
+succeeded in showing the finiteness of the complete systems for
+forms in $n$ variables, a proof which Story has simplified.
+
+Clebsch\footnote{Klein's Evanston Lectures, Lect. I.} did more than
+any other to introduce into Germany the work of Cayley and
+Sylvester, interpreting the projective geometry by their theory of
+invariants, and correlating it with Riemann's theory of
+functions. Especially since the publication of his work on forms
+(1871) the subject has attracted such scholars as Weierstrass,
+Kronecker, Mansion, Noether, Hilbert, Klein, Lie, Beltrami,
+Burkhardt, and many others. On binary forms Fa\`a di Bruno's work is
+well known, as is Study's (1889) on ternary forms. De Toledo (1889)
+and Elliott (1895) have published treatises on the subject.
+
+Dublin University has also furnished a considerable corps of
+contributors, among whom MacCullagh, Hamilton, Salmon, Michael and
+Ralph Roberts, and Burnside may be especially mentioned. Burnside,
+who wrote the latter part of Burnside and Panton's Theory of
+Equations, has set forth a method of transformation which is fertile
+in geometric interpretation and binds together binary and certain
+ternary forms.
+
+The equivalence problem of quadratic and bilinear forms has
+attracted the attention of Weierstrass, Kronecker, Christoffel,
+Frobenius, Lie, and more recently of Rosenow (Crelle, 108), Werner
+(1889), Killing (1890), and Scheffers (1891). The equivalence
+problem of non-quadratic forms has been studied by
+Christoffel. Schwarz (1872), Fuchs (1875-76), Klein (1877, 1884),
+Brioschi (1877), and Maschke (1887) have contributed to the theory
+of forms with linear transformations into themselves. Cayley
+(especially from 1870) and Sylvester (1877) have worked out the
+methods of denumeration by means of generating
+functions. Differential invariants have been studied by Sylvester,
+MacMahon, and Hammond. Starting from the differential invariant,
+which Cayley has termed the Schwarzian derivative, Sylvester (1885)
+has founded the theory of reciprocants, to which MacMahon, Hammond,
+Leudesdorf, Elliott, Forsyth, and Halphen have
+contributed. Canonical forms have been studied by Sylvester (1851),
+Cayley, and Hermite (to whom the term ``canonical form'' is due),
+and more recently by Rosanes (1873), Brill (1882), Gundelfinger
+(1883), and Hilbert (1886).
+
+The Geometric Theory of Binary Forms may be traced to Poncelet and
+his followers. But the modern treatment has its origin in connection
+with the theory of elliptic modular functions, and dates from
+Dedekind's letter to Borchardt (Crelle, 1877). The names of Klein
+and Hurwitz are prominent in this connection. On the method of nets
+(r\'eseaux), another geometric treatment of binary quadratic forms
+Gauss (1831), Dirichlet (1850), and Poincar\'e (1880) have written.
+
+\chapter{CALCULUS.}
+
+The Differential and Integral Calculus,\footnote{Williamson, B.,
+Infinitesimal Calculus, Encyclop\ae{}dia Britannica, 9th edition;
+Cantor, M., Geschichte der Mathematik, Vol. III, pp. 150-316;
+Vivanti, G., Note sur l'histoire de l'infiniment petit, Bibliotheca
+Mathematica, 1894, p. 1; Mansion, P., Esquisse de l'histoire du calcul
+infinit\'esimal, Ghent, 1887. Le deux centi\`eme anniversaire
+de l'invention du calcul diff\'erentiel; Mathesis, Vol. IV, p. 163.}
+dating from Newton and Leibniz, was quite complete in its general
+range at the close of the eighteenth century. Aside from the study
+of first principles, to which Gauss, Cauchy, Jordan, Picard, M\'eray,
+and those whose names are mentioned in connection with the theory of
+functions, have contributed, there must be mentioned the development
+of symbolic methods, the theory of definite integrals, the calculus
+of variations, the theory of differential equations, and the
+numerous applications of the Newtonian calculus to physical
+problems. Among those who have prepared noteworthy general treatises
+are Cauchy (1821), Raabe (1839-47), Duhamel (1856), Sturm (1857-59),
+Bertrand (1864), Serret (1868), Jordan (2d ed., 1893), and Picard
+(1891-93). A recent contribution to analysis which promises to be
+valuable is Oltramare's Calcul de G\'en\'eralization (1893).
+
+Abel seems to have been the first to consider in a general way the
+question as to what differential expressions can be integrated in a
+finite form by the aid of ordinary functions, an investigation
+extended by Liouville. Cauchy early undertook the general theory of
+determining definite integrals, and the subject has been prominent
+during the century. Frullani's theorem (1821), Bierens de Haan's
+work on the theory (1862) and his elaborate tables (1867),
+Dirichlet's lectures (1858) embodied in Meyer's treatise (1871), and
+numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke,
+Schl\"omilch, Elliott, Leudesdorf, and Kronecker are among the
+noteworthy contributions.
+
+Eulerian Integrals were first studied by Euler and afterwards
+investigated by Legendre, by whom they were classed as Eulerian
+integrals of the first and second species, as follows: $\int_0^1
+x^{n-1}(1 - x)^{n-1}dx$, $\int_0^\infty e^{-x} x^{n-1}dx$, although
+these were not the exact forms of Euler's study. If $n$ is
+integral, it follows that $\int_0^\infty e^{-x}x^{n-1}dx = n!$, but
+if $n$ is fractional it is a transcendent function. To it
+Legendre assigned the symbol $\Gamma$, and it is now called the
+gamma function. To the subject Dirichlet has contributed an
+important theorem (Liouville, 1839), which has been elaborated by
+Liouville, Catalan, Leslie Ellis, and others. On the evaluation of
+$\Gamma x$ and $\log \Gamma x$ Raabe (1843-44), Bauer (1859), and
+Gudermann (1845) have written. Legendre's great table appeared in
+1816.
+
+Symbolic Methods may be traced back to Taylor, and the analogy
+between successive differentiation and ordinary exponentials had
+been observed by numerous writers before the nineteenth
+century. Arbogast (1800) was the first, however, to separate the
+symbol of operation from that of quantity in a differential
+equation. Fran\c{c}ois (1812) and Servois (1814) seem to have been
+the first to give correct rules on the subject. Hargreave (1848)
+applied these methods in his memoir on differential equations, and
+Boole freely employed them. Grassmann and Hankel made great use of
+the theory, the former in studying equations, the latter in his
+theory of complex numbers.
+
+The Calculus of Variations\footnote{Carll, L. B., Calculus of
+Variations, New York, 1885, Chap. V; Todhunter, I., History of the
+Progress of the Calculus of Variations, London, 1861; Reiff, R., Die
+Anf\"ange der Variationsrechnung,
+Mathematisch-naturwissenschaftliche Mittheilungen, T\"ubingen,
+1887, p. 90.} may be said to begin with a problem of Johann
+Bernoulli's (1696). It immediately occupied the attention of Jakob
+Bernoulli and the Marquis de l'H\^opital, but Euler first elaborated
+the subject. His contributions began in 1733, and his Elementa
+Calculi Variationum gave to the science its name. Lagrange
+contributed extensively to the theory, and Legendre (1786) laid down
+a method, not entirely satisfactory, for the discrimination of
+maxima and minima. To this discrimination Brunacci (1810), Gauss
+(1829), Poisson (1831), Ostrogradsky (1834), and Jacobi (1837) have
+been among the contributors. An important general work is that of
+Sarrus (1842) which was condensed and improved by Cauchy
+(1844). Other valuable treatises and memoirs have been written by
+Strauch (1849), Jellett (1850), Hesse (1857), Clebsch (1858), and
+Carll (1885), but perhaps the most important work of the century is
+that of Weierstrass. His celebrated course on the theory is
+epoch-making, and it may be asserted that he was the first to place
+it on a firm and unquestionable foundation.
+
+The Application of the Infinitesimal Calculus to problems in physics
+and astronomy was contemporary with the origin of the science. All
+through the eighteenth century these applications were multiplied,
+until at its close Laplace and Lagrange had brought the whole range
+of the study of forces into the realm of analysis. To Lagrange
+(1773) we owe the introduction of the theory of the
+potential\footnote{Bacharach, M., Abriss der Geschichte der
+Potentialtheorie, 1883. This contains an extensive bibliography.}
+into dynamics, although the name ``potential function'' and the
+fundamental memoir of the subject are due to Green (1827, printed in
+1828). The name ``potential'' is due to Gauss (1840), and the
+distinction between potential and potential function to
+Clausius. With its development are connected the names of Dirichlet,
+Riemann, Neumann, Heine, Kronecker, Lipschitz, Christoffel,
+Kirchhoff, Beltrami, and many of the leading physicists of the
+century.
+
+It is impossible in this place to enter into the great variety of
+other applications of analysis to physical problems. Among them are
+the investigations of Euler on vibrating chords; Sophie Germain on
+elastic membranes; Poisson, Lam\'e, Saint-Venant, and Clebsch on
+the elasticity of three-dimensional bodies; Fourier on heat
+diffusion; Fresnel on light; Maxwell, Helmholtz, and Hertz on
+electricity; Hansen, Hill, and Gyld\'en on astronomy; Maxwell on
+spherical harmonics; Lord Rayleigh on acoustics; and the
+contributions of Dirichlet, Weber, Kirchhoff, F. Neumann, Lord
+Kelvin, Clausius, Bjerknes, MacCullagh, and Fuhrmann to physics in
+general. The labors of Helmholtz should be especially mentioned,
+since he contributed to the theories of dynamics, electricity, etc.,
+and brought his great analytical powers to bear on the fundamental
+axioms of mechanics as well as on those of pure mathematics.
+
+\chapter{DIFFERENTIAL EQUATIONS.}
+
+The Theory of Differential Equations\footnote{Cantor, M.,
+Geschichte der Mathematik, Vol. III, p. 429; Schlesinger, L.,
+Handbuch der
+Theorie der linearen Differentialgleichungen, Vol. I, 1895, an
+excellent historical view; review by Mathews in Nature, Vol. LII,
+p. 313; Lie, S., Zur allgemeinen Theorie der partiellen
+Differentialgleichungen, Berichte \"uber die Verhandlungen der
+Gesellschaft der Wissenschaften zu Leipzig, 1895; Mansion, P.,
+Theorie der partiellen Differentialgleichungen ter Ordnung, German
+by Maser, Leipzig, 1892, excellent on history; Craig, T., Some of
+the Developments in the Theory of Ordinary Differential Equations,
+1878-1893, Bulletin New York Mathematical Society, Vol. II, p. 119 ;
+Goursat, E., Le\c{c}ons sur l'int\'egration des \'equations aux
+d\'eriv\'ees partielles du premier ordre, Paris, 1895; Burkhardt,
+H., and Heffier, L., in Mathematical Papers of Chicago Congress,
+p.13 and p. 96.} has been called by Lie\footnote{``In der ganzen
+modernen Mathematik ist die Theorie der Differentialgleichungen die
+wichtigste Disciplin.''} the most important of modern
+mathematics. The influence of geometry, physics, and astronomy,
+starting with Newton and Leibniz, and further manifested through the
+Bernoullis, Riccati, and Clairaut, but chiefly through d'Alembert
+and Euler, has been very marked, and especially on the theory of
+linear partial differential equations with constant coefficients.
+The first method of integrating linear ordinary differential
+equations with constant coefficients is due to Euler, who made the
+solution of his type, $\frac {d^{n}y} {dx^{n}} + A_{1}\frac
+{d^{n-1}y} {dx^{n-1}} + \cdots + A_{n}y = 0$, depend on that of the
+algebraic equation of the
+$n$th degree, $F(z) = z^{n} + A_{1}z^{n-1} + \cdots + An = 0$, in
+which $z^{k}$ takes the place of $\frac {d^{k}y} {dx^{k}} (k = 1, 2,
+\cdots, n)$. This equation $F(z) = 0$, is the ``characteristic''
+equation considered later by Monge and Cauchy.
+
+The theory of linear partial differential equations may be said to
+begin with Lagrange (1779 to 1785). Monge (1809) treated ordinary
+and partial differential equations of the first and second order,
+uniting the theory to geometry, and introducing the notion of the
+``characteristic,'' the curve represented by $F(z) = 0$, which has
+recently been investigated by Darboux, Levy, and Lie. Pfaff (1814,
+1815) gave the first general method of integrating partial
+differential equations of the first order, a method of which Gauss
+(1815) at once recognized the value and of which he gave an
+analysis. Soon after, Cauchy (1819) gave a simpler method, attacking
+the subject from the analytical standpoint, but using the Monge
+characteristic. To him is also due the theorem, corresponding to the
+fundamental theorem of algebra, that every differential equation
+defines a function expressible by means of a convergent series, a
+proposition more simply proved by Briot and Bouquet, and also by
+Picard (1891). Jacobi (1827) also gave an analysis of Pfaff's
+method, besides developing an original one (1836) which Clebsch
+published (1862). Clebsch's own method appeared in 1866, and others
+are due to Boole (1859), Korkine (1869), and A. Mayer
+(1872). Pfaff's problem has been a prominent subject of
+investigation, and with it are connected the names of Natani (1859),
+Clebsch (1861, 1862), DuBois-Reymond (1869), Cayley, Baltzer,
+Frobenius, Morera, Darboux, and Lie. The next great improvement in
+the theory of partial differential equations of the first order is
+due to Lie (1872), by whom the whole subject has been placed on a
+rigid foundation. Since about 1870, Darboux, Kovalevsky, M\'eray,
+Mansion, Graindorge, and Imschenetsky have been prominent in this
+line. The theory of partial differential equations of the second
+and higher orders, beginning with Laplace and Monge, was notably
+advanced by Amp\`ere (1840). Imschenetsky\footnote{Grunert's Archiv
+f\"ur Mathematik, Vol. LIV.} has summarized the contributions to
+1873, but the theory remains in an imperfect state.
+
+The integration of partial differential equations with three or more
+variables was the object of elaborate investigations by Lagrange,
+and his name is still connected with certain subsidiary
+equations. To him and to Charpit, who did much to develop the
+theory, is due one of the methods for integrating the general
+equation with two variables, a method which now bears Charpit's name.
+
+The theory of singular solutions of ordinary and partial
+differential equations has been a subject of research from the time
+of Leibniz, but only since the middle of the present century has it
+received especial attention. A valuable but little-known work on the
+subject is that of Houtain (1854). Darboux (from 1873) has been a
+leader in the theory, and in the geometric interpretation of these
+solutions he has opened a field which has been worked by various
+writers, notably Casorati and Cayley. To the latter is due (1872)
+the theory of singular solutions of differential equations of the
+first order as at present accepted.
+
+The primitive attempt in dealing with differential equations had in
+
+
+view a reduction to quadratures. As it had been the hope of
+eighteenth-century algebraists to find a method for solving the
+general equation of the $n$th degree, so it was the hope of analysts
+to find a general method for integrating any differential
+equation. Gauss (1799) showed, however, that the differential
+equation meets its limitations very soon unless complex numbers are
+introduced. Hence analysts began to substitute the study of
+functions, thus opening a new and fertile field. Cauchy was the
+first to appreciate the importance of this view, and the modern
+theory may be said to begin with him. Thereafter the real question
+was to be, not whether a solution is possible by means of known
+functions or their integrals, but whether a given differential
+equation suffices for the definition of a function of the
+independent variable or variables, and if so, what are the
+characteristic properties of this function.
+
+Within a half-century the theory of ordinary differential equations
+has come to be one of the most important branches of analysis, the
+theory of partial differential equations remaining as one still to
+be perfected. The difficulties of the general problem of integration
+are so manifest that all classes of investigators have confined
+themselves to the properties of the integrals in the neighborhood of
+certain given points. The new departure took its greatest
+inspiration from two memoirs by Fuchs (Crelle, 1866, 1868), a work
+elaborated by Thom\'e and Frobenius. Collet has been a prominent
+contributor since 1869, although his method for integrating a
+non-linear system was communicated to Bertrand in 1868.
+Clebsch\footnote{Klein's Evanston Lectures, Lect. I.} (1873) attacked
+the theory along lines parallel to those followed in his theory of
+Abelian integrals. As the latter can be classified according to the
+properties of the fundamental curve which remains unchanged under a
+rational transformation, so Clebsch proposed to classify the
+transcendent functions defined by the differential equations
+according to the invariant properties of the corresponding surfaces
+$f = 0$ under rational one-to-one transformations.
+
+Since 1870 Lie's\footnote{Klein's Evanston Lectures, Lect. II,
+III.} labors have put the entire theory of differential equations
+on a more satisfactory foundation. He has shown that the integration
+theories of the older mathematicians, which had been looked upon as
+isolated, can by the introduction of the concept of continuous
+groups of transformations be referred to a common source, and that
+ordinary differential equations which admit the same infinitesimal
+transformations present like difficulties of integration. He has
+also emphasized the subject of transformations of contact
+(Ber\"uhrungstransformationen) which underlies so much of the recent
+theory. The modern school has also turned its attention to the
+theory of differential invariants, one of fundamental importance and
+one which Lie has made prominent. With this theory are associated
+the names of Cayley, Cockle, Sylvester, Forsyth, Laguerre, and
+Halphen. Recent writers have shown the same tendency noticeable in
+the work of Monge and Cauchy, the tendency to separate into two
+schools, the one inclining to use the geometric diagram, and
+represented by Schwarz, Klein, and Goursat, the other adhering to
+pure analysis, of which Weierstrass, Fuchs, and Frobenius are
+types. The work of Fuchs and the theory of elementary divisors have
+formed the basis of a late work by Sauvage (1895). Poincar\'e's
+recent contributions are also very notable. His theory of Fuchsian
+equations (also investigated by Klein) is connected with the general
+theory. He has also brought the whole subject into close relations
+with the theory of functions. Appell has recently contributed to the
+theory of linear differential equations transformable into
+themselves by change of the function and the variable. Helge von
+Koch has written on infinite determinants and linear differential
+equations. Picard has undertaken the generalization of the work of
+Fuchs and Poincar\'e in the case of differential equations of the
+second order. Fabry (1885) has generalized the normal integrals of
+Thom\'e, integrals which Poincar\'e has called ``int\'egrales
+anormales,'' and which Picard has recently studied. Riquier has
+treated the question of the existence of integrals in any
+differential system and given a brief summary of the history to
+1895.\footnote{Riquier, C., M\'emoire sur l'existence des
+int\'egrales dans un syst\`eme differentiel quelconque,
+etc. M\'emoires des Savants \'etrangers, Vol. XXXII, No. 3.} The
+number of contributors in recent times is very great, and includes,
+besides those already mentioned, the names of Brioschi,
+K\"onigsberger, Peano, Graf, Hamburger, Graindorge, Schl\"afli,
+Glaisher, Lommel, Gilbert, Fabry, Craig, and Autonne.
+
+\chapter{INFINITE SERIES.}
+
+The Theory of Infinite Series\footnote{Cantor, M., Geschichte der
+Mathematik, Vol. III, pp. 53, 71; Reiff, R., Geschichte der
+unendlichen Reihen, T\"ubingen, 1889; Cajori, F., Bulletin New York
+Mathematical Society, Vol. I, p. 184; History of Teaching of
+Mathematics in United States, p. 361.} in its historical
+development has been divided by Reiff into three periods: (1) the
+period of Newton and Leibniz, that of its introduction; (2) that of
+Euler, the formal period; (3) the modern, that of the scientific
+investigation of the validity of infinite series, a period beginning
+with Gauss. This critical period begins with the publication of
+Gauss's celebrated memoir on the series $1 +
+\frac{\alpha.\beta}{1.\gamma}x +
+\frac{\alpha.(\alpha+1).\beta.(\beta+1)}{1.2.\gamma.(\gamma+1)}x^2 +
+\cdots$, in 1812. Euler had already considered this series, but Gauss
+was the first to master it, and under the name ``hypergeometric
+series'' (due to Pfaff) it has since occupied the attention of
+Jacobi, Kummer, Schwarz, Cayley, Goursat, and numerous others. The
+particular series is not so important as is the standard of
+criticism which Gauss set up, embodying the simpler criteria of
+convergence and the questions of remainders and the range of
+convergence.
+
+Gauss's contributions were not at once appreciated, and the next to
+call attention to the subject was Cauchy (1821), who may be
+considered the founder of the theory of convergence and divergence
+of series. He was one of the first to insist on strict tests of
+convergence; he showed that if two series are convergent their
+product is not necessarily so; and with him begins the discovery of
+effective criteria of convergence and divergence. It should be
+mentioned, however, that these terms had been introduced long before
+by Gregory (1668), that Euler and Gauss had given various criteria,
+and that Maclaurin had anticipated a few of Cauchy's discoveries.
+Cauchy advanced the theory of power series by his expansion of a
+complex function in such a form. His test for convergence is still
+one of the most satisfactory when the integration involved is
+possible.
+
+Abel was the next important contributor. In his memoir (1826) on the
+series $1 + \frac{m}{1}x + \frac{m(m-1)}{2!}x^2 + \cdots$ he
+corrected certain of Cauchy's conclusions, and gave a completely
+scientific summation of the series for complex values of $m$ and $x$.
+He was emphatic against the reckless use of series, and showed the
+necessity of considering the subject of continuity in questions of
+convergence.
+
+Cauchy's methods led to special rather than general criteria, and
+the same may be said of Raabe (1832), who made the first elaborate
+investigation of the subject, of De Morgan (from 1842), whose
+logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have
+shown to fail within a certain region; of Bertrand (1842), Bonnet
+(1843), Malmsten (1846, 1847, the latter without integration);
+Stokes (1847), Paucker (1852), Tch\'ebichef (1852), and Arndt
+(1853). General criteria began with Kummer (1835), and have been
+studied by Eisenstein (1847), Weierstrass in his various
+contributions to the theory of functions, Dini (1867),
+DuBois-Reymond (1873), and many others. Pringsheim's (from 1889)
+memoirs present the most complete general theory.
+
+The Theory of Uniform Convergence was treated by Cauchy (1821), his
+limitations being pointed out by Abel, but the first to attack it
+successfully were Stokes and Seidel (1847-48). Cauchy took up the
+problem again (1853), acknowledging Abel's criticism, and reaching
+the same conclusions which Stokes had already found. Thom\'e used the
+doctrine (1866), but there was great delay in recognizing the
+importance of distinguishing between uniform and non-uniform
+convergence, in spite of the demands of the theory of functions.
+
+Semi-Convergent Series were studied by Poisson (1823), who also gave
+a general form for the remainder of the Maclaurin formula. The most
+important solution of the problem is due, however, to Jacobi (1834),
+who attacked the question of the remainder from a different
+standpoint and reached a different formula. This expression was
+also worked out, and another one given, by Malmsten (1847).
+Schl\"omilch (Zeitschrift, Vol.I, p. 192, 1856) also
+improved Jacobi's remainder, and showed the relation between the
+remainder and Bernoulli's function $F(x) = 1^n + 2^n + \cdots + (x -
+1)^n$. Genocchi (1852) has further contributed to the theory.
+
+Among the early writers was Wronski, whose ``loi supr\^eme'' (1815)
+was hardly recognized until Cayley (1873) brought it into
+prominence. Transon (1874), Ch. Lagrange (1884), Echols, and
+Dickstein\footnote{Bibliotheca Mathematica, 1892-94; historical.}
+have published of late various memoirs on the subject.
+
+Interpolation Formulas have been given by various writers from
+Newton to the present time. Lagrange's theorem is well known,
+although Euler had already given an analogous form, as are also
+Olivier's formula (1827), and those of Minding (1830), Cauchy
+(1837), Jacobi (1845), Grunert (1850, 1853), Christoffel (1858), and
+Mehler (1864).
+
+Fourier's Series\footnote{Historical Summary by B\^ocher, Chap. IX
+of Byerly's Fourier's Series and Spherical Harmonics, Boston, 1893;
+Sachse, A., Essai historique sur la repr\'esentation d'une fonction
+\ldots par une s\'erie trigonom\'etrique. Bulletin des Sciences
+math\'ematiques, Part I, 1880, pp. 43, 83.} were being investigated
+as the result of physical considerations at the same time that
+Gauss, Abel, and Cauchy were working out the theory of infinite
+series. Series for the expansion of sines and cosines, of multiple
+arcs in powers of the sine and cosine of the arc had been treated by
+Jakob Bernoulli (1702) and his brother Johann (1701) and still
+earlier by Vi\`ete. Euler and Lagrange had simplified the subject,
+as have, more recently, Poinsot, Schr\"oter, Glaisher, and
+Kummer. Fourier (1807) set for himself a different problem, to
+expand a given function of $x$ in terms of the sines or cosines of
+multiples of $x$, a problem which he embodied in his Th\'eorie
+analytique de la Chaleur (1822). Euler had already given the
+formulas for determining the coefficients in the series; and
+Lagrange had passed over them without recognizing their value, but
+Fourier was the first to assert and attempt to prove the general
+theorem. Poisson (1820-23) also attacked the problem from a
+different standpoint. Fourier did not, however, settle the question
+of convergence of his series, a matter left for Cauchy (1826) to
+attempt and for Dirichlet (1829) to handle in a thoroughly
+scientific manner. Dirichlet's treatment (Crelle, 1829), while
+bringing the theory of trigonometric series to a temporary
+conclusion, has been the subject of criticism and improvement by
+Riemann (1854), Heine, Lipschitz, Schl\"afli, and
+DuBois-Reymond. Among other prominent contributors to the theory of
+trigonometric and Fourier series have been Dini, Hermite, Halphen,
+Krause, Byerly and Appell.
+
+\chapter{THEORY OF FUNCTIONS.}
+
+The Theory of Functions\footnote{Brill, A., and Noether, M., Die
+Entwickelung der Theorie der algebraischen Functionen in alterer
+und neuerer Zeit, Bericht erstattet der Deutschen
+Mathematiker-Vereinigung, Jahresbericht, Vol. II, pp. 107-566,
+Berlin, 1894; K\"onigsberger, L., Zur Geschichte der Theorie der
+elliptischen Transcendenten in den Jahren 1826-29, Leipzig, 1879;
+Williamson, B., Infinitesimal Calculus, Encyclop\ae{}dia Britannica;
+Schlesinger, L., Differentialgleichungen, Vol. I, 1895; Casorati,
+F., Teorica delle funzioni di variabili complesse, Vol. I, 1868;
+Klein's Evanston Lectures. For bibliography and historical notes,
+see Harkness and Morley's Theory of Functions, 1893, and Forsyth's
+Theory of Functions, 1893; Enestr\"om, G., Note historique sur les
+symboles \ldots Bibliotheca Mathematica, 1891, p. 89.} may be said to
+have its first development in Newton's works, although algebraists
+had already become familiar with irrational functions in considering
+cubic and quartic equations. Newton seems first to have grasped the
+idea of such expressions in his consideration of symmetric functions
+of the roots of an equation. The word was employed by Leibniz
+(1694), but in connection with the Cartesian geometry. In its modern
+sense it seems to have been first used by Johann Bernoulli, who
+distinguished between algebraic and transcendent functions. He also
+used (1718) the function symbol $\phi$. Clairaut (1734) used $\Pi
+x$, $\Phi x$, $\Delta x$, for various functions of $x$, a symbolism
+substantially followed by d'Alembert (1747) and Euler
+(1753). Lagrange (1772, 1797, 1806) laid the foundations for the
+general theory, giving to the symbol a broader meaning, and to the
+symbols $f$, $\phi$, $F$, $\cdots$, $f^{\prime}$, $\phi^{\prime}$,
+$F^{\prime}$, $\cdots$ their modern signification. Gauss contributed
+to the theory, especially in his proofs of the fundamental theorem
+of algebra, and discussed and gave name to the theory of ``conforme
+Abbildung,'' the ``orthomorphosis'' of Cayley.
+
+Making Lagrange's work a point of departure, Cauchy so greatly
+developed the theory that he is justly considered one of its
+founders. His memoirs extend over the period 1814-1851, and cover
+subjects like those of integrals with imaginary limits, infinite
+series and questions of convergence, the application of the
+infinitesimal calculus to the theory of complex numbers, the
+investigation of the fundamental laws of mathematics, and numerous
+other lines which appear in the general theory of functions as
+considered to-day. Originally opposed to the movement started by
+Gauss, the free use of complex numbers, he finally became, like
+Abel, its advocate. To him is largely due the present orientation of
+mathematical research, making prominent the theory of functions,
+distinguishing between classes of functions, and placing the whole
+subject upon a rigid foundation. The historical development of the
+general theory now becomes so interwoven with that of special
+classes of functions, and notably the elliptic and Abelian, that
+economy of space requires their treatment together, and hence a
+digression at this point.
+
+The Theory of Elliptic Functions\footnote{Enneper, A., Elliptische
+Funktionen, Theorie und Geschichte, Halle, 1890; K\"onigsberger, L.,
+Zur Geschichte der Theorie der elliptischen Transcendenten in den
+Jahren 1826-29, Leipzig, 1879.} is usually referred for its origin
+to Landen's (1775) substitution of two elliptic arcs for a single
+hyperbolic arc. But Jakob Bernoulli (1691) had suggested the idea of
+comparing non-congruent arcs of the same curve, and Johann had
+followed up the investigation. Fagnano (1716) had made similar
+studies, and both Maclaurin (1742) and d'Alembert (1746) had come
+upon the borderland of elliptic functions. Euler (from 1761) had
+summarized and extended the rudimentary theory, showing the
+necessity for a convenient notation for elliptic arcs, and
+prophesying (1766) that ``such signs will afford a new sort of
+calculus of which I have here attempted the exposition of the first
+elements.'' Euler's investigations continued until about the time of
+his death (1783), and to him Legendre attributes the foundation of
+the theory. Euler was probably never aware of Landen's discovery.
+
+It is to Legendre, however, that the theory of elliptic functions is
+largely due, and on it his fame to a considerable degree
+depends. His earlier treatment (1786) almost entirely substitutes a
+strict analytic for the geometric method. For forty years he had the
+theory in hand, his labor culminating in his Trait\'e des Fonctions
+elliptiques et des Int\'egrales Eul\'eriennes (1825-28). A surprise
+now awaiting him is best told in his own words: ``Hardly had my work
+seen the light--its name could scarcely have become known to
+scientific foreigners,--when I learned with equal surprise and
+satisfaction that two young mathematicians, MM. Jacobi of
+K\"onigsberg and Abel of Christiania, had succeeded by their own
+studies in perfecting considerably the theory of elliptic functions
+in its highest parts.'' Abel began his contributions to the theory
+in 1825, and even then was in possession of his fundamental theorem
+which he communicated to the Paris Academy in 1826. This
+communication being so poorly transcribed was not published in full
+until 1841, although the theorem was sent to Crelle (1829) just
+before Abel's early death. Abel discovered the double periodicity of
+elliptic functions, and with him began the treatment of the elliptic
+integral as a function of the amplitude.
+
+Jacobi, as also Legendre and Gauss, was especially cordial in praise
+of the delayed theorem of the youthful Abel. He calls it a
+``monumentum \ae{}re perennius,'' and his name ``das Abel'sche
+Theorem'' has since attached to it. The functions of multiple
+periodicity to which it refers have been called Abelian
+Functions. Abel's work was early proved and elucidated by Liouville
+and Hermite. Serret and Chasles in the Comptes Rendus, Weierstrass
+(1853), Clebsch and Gordan in their Theorie der Abel'schen
+Functionen (1866), and Briot and Bouquet in their two treatises have
+greatly elaborated the theory. Riemann's\footnote{Klein, Evanston
+Lectures, p. 3; Riemann and Modern Mathematics, translated by
+Ziwet, Bulletin of American Mathematical Society, Vol. I, p. 165;
+Burkhardt, H., Vortrag uber Riemann, G\"ottingen, 1892.} (1857)
+celebrated memoir in Crelle presented the subject in such a novel
+form that his treatment was slow of acceptance. He based the theory
+of Abelian integrals and their inverse, the Abelian functions, on
+the idea of the surface now so well known by his name, and on the
+corresponding fundamental existence theorems. Clebsch, starting from
+an algebraic curve defined by its equation, made the subject more
+accessible, and generalized the theory of Abelian integrals to a
+theory of algebraic functions with several variables, thus creating
+a branch which has been developed by Noether, Picard, and
+Poincar\'e. The introduction of the theory of invariants and
+projective geometry into the domain of hyperelliptic and Abelian
+functions is an extension of Clebsch's scheme. In this extension, as
+in the general theory of Abelian functions, Klein has been a
+leader. With the development of the theory of Abelian functions is
+connected a long list of names, including those of Schottky,
+Humbert, C. Neumann, Fricke, K\"onigsberger, Prym, Schwarz,
+Painlev\'e, Hurwitz, Brioschi, Borchardt, Cayley, Forsyth, and
+Rosenhain, besides others already mentioned.
+
+Returning to the theory of elliptic functions, Jacobi (1827) began
+by adding greatly to Legendre's work. He created a new notation and
+gave name to the ``modular equations'' of which he made use. Among
+those who have written treatises upon the elliptic-function theory
+are Briot and Bouquet, Laurent, Halphen, K\"onigsberger, Hermite,
+Dur\`ege, and Cayley, The introduction of the subject into the
+Cambridge Tripos (1873), and the fact that Cayley's only book was
+devoted to it, have tended to popularize the theory in England.
+
+The Theory of Theta Functions was the simultaneous and independent
+creation of Jacobi and Abel (1828). Gauss's notes show that he was
+aware of the properties of the theta functions twenty years earlier,
+but he never published his investigations. Among the leading
+contributors to the theory are Rosenhain (1846, published in 1851)
+and G\"opel (1847), who connected the double theta functions with
+the theory of Abelian functions of two variables and established the
+theory of hyperelliptic functions in a manner corresponding to the
+Jacobian theory of elliptic functions. Weierstrass has also
+developed the theory of theta functions independently of the form of
+their space boundaries, researches elaborated by K\"onigsberger
+(1865) to give the addition theorem. Riemann has completed the
+investigation of the relation between the theory of the theta and
+the Abelian functions, and has raised theta functions to their
+present position by making them an essential part of his theory of
+Abelian integrals. H.~J.~S.~Smith has included among his
+contributions to this subject the theory of omega functions. Among
+the recent contributors are Krazer and Prym (1892), and Wirtinger
+(1895).
+
+Cayley was a prominent contributor to the theory of periodic
+functions. His memoir (1845) on doubly periodic functions extended
+Abel's investigations on doubly infinite products. Euler had given
+singly infinite products for $\sin x$, $\cos x$, and Abel had
+generalized these, obtaining for the elementary doubly periodic
+functions expressions for $\hbox{sn} x$, $\hbox{cn} x$, $\hbox{dn}
+x$. Starting from these expressions of Abel's Cayley laid a complete
+foundation for his theory of elliptic functions. Eisenstein (1847)
+followed, giving a discussion from the standpoint of pure analysis,
+of a general doubly infinite product, and his labors, as
+supplemented by Weierstrass, are classic.
+
+The General Theory of Functions has received its present form
+largely from the works of Cauchy, Riemann, and
+Weierstrass. Endeavoring to subject all natural laws to
+interpretation by mathematical formulas, Riemann borrowed his
+methods from the theory of the potential, and found his inspiration
+in the contemplation of mathematics from the standpoint of the
+concrete. Weierstrass, on the other hand, proceeded from the purely
+analytic point of view. To Riemann\footnote{Klein, F., Riemann and
+Modern Mathematics, translated by Ziwet, Bulletin of American
+Mathematical Society, Vol. I, p. 165.} is due the idea of making
+certain partial differential equations, which express the
+fundamental properties of all functions, the foundation of a general
+analytical theory, and of seeking criteria for the determination of
+an analytic function by its discontinuities and boundary
+conditions. His theory has been elaborated by Klein (1882, and
+frequent memoirs) who has materially extended the theory of
+Riemann's surfaces. Clebsch, L\"uroth, and later writers have based on
+this theory their researches on loops. Riemann's speculations were
+not without weak points, and these have been fortified in connection
+with the theory of the potential by C. Neumann, and from the
+analytic standpoint by Schwarz.
+
+In both the theory of general and of elliptic and other functions,
+Clebsch was prominent. He introduced the systematic consideration
+of algebraic curves of deficiency 1, bringing to bear on the theory
+of elliptic functions the ideas of modern projective geometry. This
+theory Klein has generalized in his Theorie der elliptischen
+Modulfunctionen, and has extended the method to the theory of
+hyperelliptic and Abelian functions.
+
+Following Riemann came the equally fundamental and original and more
+rigorously worked out theory of Weierstrass. His early lectures on
+functions are justly considered a landmark in modern mathematical
+development. In particular, his researches on Abelian transcendents
+are perhaps the most important since those of Abel and Jacobi. His
+contributions to the theory of elliptic functions, including the
+introduction of the function $\wp(u)$, are also of great
+importance. His contributions to the general function theory
+include much of the symbolism and nomenclature, and many
+theorems. He first announced (1866) the existence of natural limits
+for analytic functions, a subject further investigated by Schwarz,
+Klein, and Fricke. He developed the theory of functions of complex
+variables from its foundations, and his contributions to the theory
+of functions of real variables were no less marked.
+
+Fuchs has been a prominent contributor, in particular (1872) on the
+general form of a function with essential singularities. On
+functions with an infinite number of essential singularities
+Mittag-Leffler (from 1882) has written and contributed a fundamental
+theorem. On the classification of singularities of functions
+Guichard (1883) has summarized and extended the researches, and
+Mittag-Leffler and G. Cantor have contributed to the same
+result. Laguerre (from 1882) was the first to discuss the ``class''
+of transcendent functions, a subject to which Poincar\'e, Cesaro,
+Vivanti, and Hermite have also contributed. Automorphic functions,
+as named by Klein, have been investigated chiefly by Poincar\'e, who
+has established their general classification. The contributors to
+the theory include Schwarz, Fuchs, Cayley, Weber, Schlesinger, and
+Burnside.
+
+The Theory of Elliptic Modular Functions, proceeding from
+Eisenstein's memoir (1847) and the lectures of Weierstrass on
+elliptic functions, has of late assumed prominence through the
+influence of the Klein school. Schl\"afli (1870), and later Klein,
+Dyck, Gierster, and Hurwitz, have worked out the theory which Klein
+and Fricke have embodied in the recent Vorlesungen
+\"uber die Theorie der elliptischen Modulfunctionen
+(1890-92). In this theory the memoirs of Dedekind (1877), Klein
+(1878), and Poincar\'e (from 1881) have been among the most
+prominent.
+
+For the names of the leading contributors to the general and special
+theories, including among others Jordan, Hermite, H\"older, Picard,
+Biermann, Darboux, Pellet, Reichardt, Burkhardt, Krause, and
+Humbert, reference must be had to the Brill-Noether Bericht.
+
+Of the various special algebraic functions space allows mention of
+but one class, that bearing Bessel's name. Bessel's
+functions\footnote{B\^ocher, M., A bit of mathematical history,
+Bulletin of New York Mathematical Society, Vol. II, p. 107.} of
+the zero order arefound in memoirs of Daniel Bernoulli (1732) and
+Euler (1764), and before the end of the eighteenth century all the
+Bessel functions of the first kind and integral order had been used.
+Their prominence as special functions is due, however, to
+Bessel (1816-17), who put them in their present form in 1824. Lagrange's
+series (1770), with Laplace's extension (1777), had been regarded as the
+best method of solving Kepler's problem (to express the variable quantities
+in undisturbed planetary motion in terms of the time or mean anomaly),
+and to improve this method Bessel's functions were first prominently
+used. Hankel (1869), Lommel (from 1868), F.~Neumann, Heine, Graf
+(1893), Gray and Mathews (1895), and others have contributed to the
+theory. Lord Rayleigh (1878) has shown the relation between
+Bessel's and Laplace's functions, but they are nevertheless looked
+upon as a distinct system of transcendents. Tables of Bessel's
+functions were prepared by Bessel (1824), by Hansen (1843), and by
+Meissel (1888).
+
+\chapter{PROBABILITIES AND LEAST SQUARES.}
+
+The Theory of Probabilities and Errors\footnote{Merriman, M., Method
+of Least Squares, New York, 1884, p. 182; Transactions of
+Connecticut Academy, 1877, Vol. IV, p. 151, with complete
+bibliography; Todhunter, I., History of the Mathematical Theory of
+Probability, 1865; Cantor, M., Geschichte der Mathematik, Vol. III,
+p. 316.} is, as applied to observations, largely a
+nineteenth-century development. The doctrine of probabilities dates,
+however, as far back as Fermat and Pascal (1654). Huygens (1657)
+gave the first scientific treatment of the subject, and Jakob
+Bernoulli's Ars Conjectandi (posthumous, 1713) and De Moivre's
+Doctrine of Chances (1718)\footnote{Enestr\"om, G., Review of
+Cantor, Bibliotheca Mathematica, 1896, p. 20.} raised the subject
+to the plane of a branch of mathematics. The theory of errors may
+be traced back to Cotes's Opera Miscellanea (posthumous, 1722), but
+a memoir prepared by Simpson in 1755 (printed 1756) first applied
+the theory to the discussion of errors of observation. The reprint
+(1757) of this memoir lays down the axioms that positive and
+negative errors are equally probable, and that there are certain
+assignable limits within which all errors may be supposed to fall;
+continuous errors are discussed and a probability curve is given.
+Laplace (1774) made the first attempt to deduce a rule for the
+combination of observations from the principles of the theory of
+probabilities. He represented the law of probability of errors by a
+curve $y = \phi(x)$, $x$ being any error and $y$ its probability,
+and laid down three properties of this curve: (1) It is symmetric as
+to the $y$-axis; (2) the $x$-axis is an asymptote, the probability
+of the error $\infty$ being $0$; (3) the area enclosed is $1$, it
+being certain that an error exists. He deduced a formula for the
+mean of three observations. He also gave (1781) a formula for the
+law of facility of error (a term due to Lagrange, 1774), but one
+which led to unmanageable equations. Daniel Bernoulli (1778)
+introduced the principle of the maximum product of the probabilities
+of a system of concurrent errors.
+
+The Method of Least Squares is due to Legendre (1805), who
+introduced it in his Nouvelles m\'ethodes pour la d\'etermination
+des orbites des com\`etes. In ignorance of Legendre's contribution,
+an Irish-American writer, Adrain, editor of ``The Analyst'' (1808),
+first deduced the law of facility of error, $\phi(x) = ce^{-h^2
+x^2}$, $c$ and $h$ being constants depending on precision of
+observation. He gave two proofs, the second being essentially the
+same as Herschel's (1850). Gauss gave the first proof which seems to
+have been known in Europe (the third after Adrain's) in 1809. To him
+is due much of the honor of placing the subject before the
+mathematical world, both as to the theory and its applications.
+
+Further proofs were given by Laplace (1810, 1812), Gauss (1823),
+Ivory (1825, 1826), Hagen (1837), Bessel (1838), Donkin (1844,
+1856), and Crofton (1870). Other contributors have been Ellis
+(1844), De Morgan (1864), Glaisher (1872), and Schiaparelli
+(1875). Peters's (1856) formula for $r$, the probable error of a
+single observation, is well known.\footnote{Bulletin of New York
+Mathematical Society, Vol. II, p. 57.}
+
+Among the contributors to the general theory of probabilities in
+the nineteenth century have been Laplace, Lacroix (1816), Littrow
+(1833), Quetelet (1853), Dedekind (1860), Helmert (1872), Laurent
+(1873), Liagre, Didion, and Pearson. De Morgan and Boole improved
+the theory, but added little that was fundamentally new. Czuber has
+done much both in his own contributions (1884, 1891), and in his
+translation (1879) of Meyer. On the geometric side the influence of
+Miller and The Educational Times has been marked, as also that of
+such contributors to this journal as Crofton, McColl, Wolstenholme,
+Watson, and Artemas Martin.
+
+\chapter{ANALYTIC GEOMETRY.}
+
+The History of Geometry\footnote{Loria, G., Il passato e il presente
+delle principali teorie geometriche. Memorie Accademia Torino,
+1887; translated into German by F. Schutte under the title Die
+haupts\"achlichsten Theorien der Geometrie in ihrer fr\"uheren und
+heutigen Entwickelung, Leipzig, 1888; Chasles, M., Aper\c{c}u
+historique sur l'origine et le d\'eveloppement des m\'ethodes en
+G\'eom\'etrie, 1889; Chasles, M., Rapport sur les Progr\`es de la
+G\'eom\'etrie, Paris, 1870; Cayley, A., Curves, Encyclop\ae{}dia
+Britannica; Klein, F., Evanston Lectures on Mathematics, New York,
+1894; A. V. Braunm\"uhl, Historische Studie \"uber die organische
+Erzeugung ebener Curven, Dyck's Katalog mathematischer Modelle,
+1892.} may be roughly divided into the four periods: (1) The
+synthetic geometry of the Greeks, practically closing with
+Archimedes; (2) The birth of analytic geometry, in which the
+synthetic geometry of Guldin, Desargues, Kepler, and Roberval merged
+into the coordinate geometry of Descartes and Fermat; (3) 1650 to
+1800, characterized by the application of the calculus to geometry,
+and including the names of Newton, Leibnitz, the Bernoullis,
+Clairaut, Maclaurin, Euler, and Lagrange, each an analyst rather
+than a geometer; (4) The nineteenth century, the renaissance of pure
+geometry, characterized by the descriptive geometry of Monge, the
+modern synthetic of Poncelet, Steiner, von Staudt, and Cremona, the
+modern analytic founded by Pl\"ucker, the non-Euclidean hypothesis
+of Lobachevsky and Bolyai, and the more elementary geometry of the
+triangle founded by Lemoine. It is quite impossible to draw the
+line between the analytic and the synthetic geometry of the
+nineteenth century, in their historical development, and Arts. 15
+and 16 should be read together.
+
+The Analytic Geometry which Descartes gave to the world in 1637 was
+confined to plane curves, and the various important properties
+common to all algebraic curves were soon discovered. To the theory
+Newton contributed three celebrated theorems on the Enumeratio
+linearum tertii ordinis\footnote{Ball, W.~W.~R., On Newton's
+classification of cubic curves. Transactions of London Mathematical
+Society, 1891, p. 104.} (1706), while others are due to Cotes
+(1722), Maclaurin, and Waring (1762, 1772, etc.). The scientific
+foundations of the theory of plane curves may be ascribed, however,
+to Euler (1748) and Cramer (1750). Euler distinguished between
+algebraic and transcendent curves, and attempted a classification of
+the former. Cramer is well known for the ``paradox'' which bears his
+name, an obstacle which Lam\'e (1818) finally removed from the
+theory. To Cramer is also due an attempt to put the theory of
+singularities of algebraic curves on a scientific foundation,
+although in a modern geometric sense the theory was first treated by
+Poncelet.
+
+Meanwhile the study of surfaces was becoming prominent. Descartes
+had suggested that his geometry could be extended to
+three-dimensional space, Wren (1669) had discovered the two systems
+of generating lines on the hyperboloid of one sheet, and Parent
+(1700) had referred a surface to three coordinate planes. The
+geometry of three dimensions began to assume definite shape,
+however, in a memoir of Clairaut's (1731), in which, at the age of
+sixteen, he solved with rare elegance many of the problems relating
+to curves of double curvature. Euler (1760) laid the foundations
+for the analytic theory of curvature of surfaces, attempting the
+classification of those of the second degree as the ancients had
+classified curves of the second order. Monge, Hachette, and other
+members of that school entered into the study of surfaces with great
+zeal. Monge introduced the notion of families of surfaces, and
+discovered the relation between the theory of surfaces and the
+integration of partial differential equations, enabling each to be
+advantageously viewed from the standpoint of the other. The theory
+of surfaces has attracted a long list of contributors in the
+nineteenth century, including most of the geometers whose names are
+mentioned in the present article.\footnote{For details see Loria,
+Il passato e il presente, etc.}
+
+M\"obius began his contributions to geometry in 1823, and four years
+later published his Barycentrische Calc\"ul. In this great work he
+introduced homogeneous coordinates with the attendant symmetry of
+geometric formulas, the scientific exposition of the principle of
+signs in geometry, and the establishment of the principle of
+geometric correspondence simple and multiple. He also (1852) summed
+up the classification of cubic curves, a service rendered by
+Zeuthen (1874) for quartics. To the period of M\"obius also belong
+Bobillier (1827), who first used trilinear coordinates, and
+Bellavitis, whose contributions to analytic geometry were
+extensive. Gergonne's labors are mentioned in the next article.
+
+Of all modern contributors to analytic geometry, Pl\"ucker stands
+foremost. In 1828 he published the first volume of his
+Analytisch-geometrische Entwickelungen, in which appeared
+the modern abridged notation, and which marks the beginning of a new
+era for analytic geometry. In the second volume (1831) he sets forth
+the present analytic form of the principle of duality. To him is due
+(1833) the general treatment of foci for curves of higher degree,
+and the complete classification of plane cubic curves (1835) which
+had been so frequently tried before him. He also gave (1839) an
+enumeration of plane curves of the fourth order, which Bragelogne
+and Euler had attempted. In 1842 he gave his celebrated ``six
+equations'' by which he showed that the characteristics of a curve
+(order, class, number of double points, number of cusps, number of
+double tangents, and number of inflections) are known when any three
+are given. To him is also due the first scientific dual definition
+of a curve, a system of tangential coordinates, and an
+investigation of the question of double tangents, a question further
+elaborated by Cayley (1847, 1858), Hesse (1847), Salmon (1858), and
+Dersch (1874). The theory of ruled surfaces, opened by Monge, was
+also extended by him. Possibly the greatest service rendered by
+Pl\"ucker was the introduction of the straight line as a space
+element, his first contribution (1865) being followed by his
+well-known treatise on the subject (1868-69). In this work he treats
+certain general properties of complexes, congruences, and ruled
+surfaces, as well as special properties of linear complexes and
+congruences, subjects also considered by Kummer and by Klein and
+others of the modern school. It is not a little due to Pl\"ucker that
+the concept of 4- and hence $n$-dimensional space, already suggested
+by Lagrange and Gauss, became the subject of later
+research. Riemann, Helmholtz, Lipschitz, Kronecker, Klein, Lie,
+Veronese, Cayley, d'Ovidio, and many others have elaborated the
+theory. The regular hypersolids in 4-dimensional space have been
+the subject of special study by Scheffler, Rudel, Hoppe, Schlegel,
+and Stringham.
+
+Among Jacobi's contributions is the consideration (1836) of curves
+and groups of points resulting from the intersection of algebraic
+surfaces, a subject carried forward by Reye (1869). To Jacobi is
+also due the conformal representation of the ellipsoid on a plane, a
+treatment completed by Schering (1858). The number of examples of
+conformal representation of surfaces on planes or on spheres has
+been increased by Schwarz (1869) and Amstein (1872).
+
+In 1844 Hesse, whose contributions to geometry in general are both
+numerous and valuable, gave the complete theory of inflections of a
+curve, and introduced the so-called Hessian curve as the first
+instance of a covariant of a ternary form. He also contributed to
+the theory of curves of the third order, and generalized the Pascal
+and Brianchon theorems on a spherical surface. Hesse's methods have
+recently been elaborated by Gundelfinger (1894).
+
+Besides contributing extensively to synthetic geometry, Chasles
+added to the theory of curves of the third and fourth degrees. In
+the method of characteristics which he worked out may be found the
+first trace of the Abz\"ahlende Geometrie\footnote{Loria, G.,
+Notizie storiche sulla Geometria numerativa. Bibliotheca Mathematica,
+1888, pp. 39, 67; 1889, p. 23.} which has been developed by Jonqui\`eres,
+Halphen (1875), and Schubert (1876, 1879), and to which Clebsch, Lindemann,
+and Hurwitz have also contributed. The general theory of correspondence starts
+with Geometry, and Chasles (1864) undertook the first special
+researches on the correspondence of algebraic curves, limiting his
+investigations, however, to curves of deficiency zero. Cayley (1866)
+carried this theory to curves of higher deficiency, and Brill (from
+1873) completed the theory.
+
+Cayley's\footnote{Biographical Notice in Cayley's Collected papers,
+Vol. VIII.} influence on geometry was very great. He early carried
+on Pl\"ucker's consideration of singularities of a curve, and showed
+(1864, 1866) that every singularity may be considered as compounded
+of ordinary singularities so that the ``six equations'' apply to a
+curve with any singularities whatsoever. He thus opened a field for
+the later investigations of Noether, Zeuthen, Halphen, and
+H.~J.~S.~Smith. Cayley's theorems on the intersection of curves
+(1843) and the determination of self-corresponding points for
+algebraic correspondences of a simple kind are fundamental in the
+present theory, subjects to which Bacharach, Brill, and Noether have
+also contributed extensively. Cayley added much to the theories of
+rational transformation and correspondence, showing the distinction
+between the theory of transformation of spaces and that of
+correspondence of loci. His investigations on the bitangents of
+plane curves, and in particular on the twenty-eight bitangents of a
+non-singular quartic, his developments of Pl\"ucker's conception of
+foci, his discussion of the osculating conics of curves and of the
+sextactic points on a plane curve, the geometric theory of the
+invariants and covariants of plane curves, are all noteworthy. He
+was the first to announce (1849) the twenty-seven lines which lie on
+a cubic surface, he extended Salmon's theory of reciprocal surfaces,
+and treated (1869) the classification of cubic surfaces, a subject
+already discussed by Schl\"afli. He also contributed to the theory
+of scrolls (skew-ruled surfaces), orthogonal systems of surfaces,
+the wave surface, etc., and was the first to reach (1845) any very
+general results in the theory of curves of double curvature, a
+theory in which the next great advance was made (1882) by Halphen
+and Noether. Among Cayley's other contributions to geometry is his
+theory of the Absolute, a figure in connection with which all
+metrical properties of a figure are considered.
+
+Clebsch\footnote{Klein, Evanston Lectures, Lect. I.} was also
+prominent in the study of curves and surfaces. He first applied the
+algebra of linear transformation to geometry. He emphasized the idea
+of deficiency (Geschlecht) of a curve, a notion which dates back to
+Abel, and applied the theory of elliptic and Abelian functions to
+geometry, using it for the study of curves. Clebsch (1872)
+investigated the shapes of surfaces of the third order. Following
+him, Klein attacked the problem of determining all possible forms of
+such surfaces, and 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. Zeuthen (1874) has discussed the various forms of plane
+curves of the fourth order, showing the relation between his results
+and those of Klein on cubic surfaces. Attempts have been made to
+extend the subject to curves of the $n$th order, but no general
+classification has been made. Quartic surfaces have been studied by
+Rohn (1887) but without a complete enumeration, and the same writer
+has contributed (1881) to the theory of Kummer surfaces.
+
+Lie has adopted Pl\"ucker's generalized space element and extended the
+theory. His sphere geometry treats the subject from the higher
+standpoint of six homogeneous coordinates, as distinguished from the
+elementary sphere geometry with but five and characterized by the
+conformal group, a geometry studied by Darboux. Lie's theory of
+contact transformations, with its application to differential
+equations, his line and sphere complexes, and his work on minimum
+surfaces are all prominent.
+
+Of great help in the study of curves and surfaces and of the theory
+of functions are the models prepared by Dyck, Brill, O. Henrici,
+Schwarz, Klein, Sch\"onflies, Kummer, and others.\footnote{Dyck,
+W., Katalog mathematischer und mathematisch-physikalischer Modelle,
+M\"unchen, 1892; Deutsche Universit\"atsausstellung, Mathematical
+Papers of Chicago Congress, p. 49.}
+
+The Theory of Minimum Surfaces has been developed along with the
+analytic geometry in general. Lagrange (1760-61) gave the equation
+of the minimum surface through a given contour, and Meusnier (1776,
+published in 1785) also studied the question. But from this time on
+for half a century little that is noteworthy was done, save by
+Poisson (1813) as to certain imaginary surfaces. Monge (1784) and
+Legendre (1787) connected the study of surfaces with that of
+differential equations, but this did not immediately affect this
+question. Scherk (1835) added a number of important results, and
+first applied the labors of Monge and Legendre to the
+theory. Catalan (1842), Bj\"orling (1844), and Dini (1865) have added
+to the subject. But the most prominent contributors have been
+Bonnet, Schwarz, Darboux, and Weierstrass. Bonnet (from 1853) has
+set forth a new system of formulas relative to the general theory of
+surfaces, and completely solved the problem of determining the
+minimum surface through any curve and admitting in each point of
+this curve a given tangent plane, Weierstrass (1866) has contributed
+several fundamental theorems, has shown how to find all of the real
+algebraic minimum surfaces, and has shown the connection between the
+theory of functions of an imaginary variable and the theory of
+minimum surfaces.
+
+\chapter{MODERN GEOMETRY.}
+
+Descriptive\footnote{Wiener, Chr., Lehrbuch der darstellenden
+Geometrie, Leipzig, 1884-87; Geschichte der darstellenden
+Geometrie, 1884.}, Projective, and Modern Synthetic Geometry are so
+interwoven in their historic development that it is even more
+difficult to separate them from one another than from the analytic
+geometry just mentioned. Monge had been in possession of his theory
+for over thirty years before the publication of his G\'eom\'etrie
+Descriptive (1800), a delay due to the jealous desire of the
+military authorities to keep the valuable secret. It is true that
+certain of its features can be traced back to Desargues, Taylor,
+Lambert, and Fr\'ezier, but it was Monge who worked it out in detail
+as a science, although Lacroix (1795), inspired by Monge's lectures
+in the \'Ecole Polytechnique, published the first work on the
+subject. After Monge's work appeared, Hachette (1812, 1818, 1821)
+added materially to its symmetry, subsequent French contributors
+being Leroy (1842), Olivier (from 1845), de la Gournerie (from
+1860), Vall\'ee, de Fourcy, Adh\'emar, and others. In Germany leading
+contributors have been Ziegler (1843), Anger (1858), and especially
+Fiedler (3d edn.~1883-88) and Wiener (1884-87). At this period
+Monge by no means confined himself to the descriptive geometry. So
+marked were his labors in the analytic geometry that he has been
+called the father of the modern theory. He also set forth the
+fundamental theorem of reciprocal polars, though not in modern
+language, gave some treatment of ruled surfaces, and extended the
+theory of polars to quadrics.\footnote{On recent development of
+graphic methods and the influence of Monge upon this branch of
+mathematics, see Eddy, H. T., Modern Graphical Developments,
+Mathematical Papers of Chicago Congress (New York, 1896), p 58.}
+
+Monge and his school concerned themselves especially with the
+relations of form, and particularly with those of surfaces and
+curves in a space of three dimensions. Inspired by the general
+activity of the period, but following rather the steps of Desargues
+and Pascal, Carnot treated chiefly the metrical relations of
+figures. In particular he investigated these relations as connected
+with the theory of transversals, a theory whose fundamental property
+of a four-rayed pencil goes back to Pappos, and which, though
+revived by Desargues, was set forth for the first time in its
+general form in Carnot's G\'eom\'etrie de Position (1803), and
+supplemented in his Th\'eorie des Transversales (1806). In these
+works he introduced negative magnitudes, the general quadrilateral
+and quadrangle, and numerous other generalizations of value to the
+elementary geometry of to-day. But although Carnot's work was
+important and many details are now commonplace, neither the name of
+the theory nor the method employed have endured. The present
+Geometry of Position (Geometrie der Lage) has little in common with
+Carnot's G\'eom\'etrie de Position.
+
+Projective Geometry had its origin somewhat later than the period of
+Monge and Carnot. Newton had discovered that all curves of the third
+order can be derived by central projection from five fundamental
+types. But in spite of this fact the theory attracted so little
+attention for over a century that its origin is generally ascribed
+to Poncelet. A prisoner in the Russian campaign, confined at
+Saratoff on the Volga (1812-14), ``priv\'e,'' as he says, ``de toute
+esp\`ece de livres et de secours, surtout distrait par les
+malheurs de ma patrie et les miens propres,'' he still had the vigor
+of spirit and the leisure to conceive the great work which he
+published (1822) eight years later. In this work was first made
+prominent the power of central projection in demonstration and the
+power of the principle of continuity in research. His leading idea
+was the study of projective properties, and as a foundation
+principle he introduced the anharmonic ratio, a concept, however,
+which dates back to Pappos and which Desargues (1639) had also
+used. M\"obius, following Poncelet, made much use of the anharmonic
+ratio in his Barycentrische Calc\"ul (1827), but under the name
+``Doppelschnitt-Verh\"altniss'' (ratio bisectionalis), a term now in
+common use under Steiner's abbreviated form ``Doppelverh\"altniss.''
+The name ``anharmonic ratio'' or ``function'' (rapport anharmonique,
+or fonction anharmonique) is due to Chasles, and ``cross-ratio'' was
+coined by Clifford. The anharmonic point and line properties of
+conics have been further elaborated by Brianchon, Chasles, Steiner,
+and von Staudt. To Poncelet is also due the theory of ``figures
+homologiques,'' the perspective axis and perspective center (called
+by Chasles the axis and center of homology), an extension of
+Carnot's theory of transversals, and the ``cordes id\'eales'' of
+conics which Pl\"ucker applied to curves of all orders, He also
+discovered what Salmon has called ``the circular points at
+infinity,'' thus completing and establishing the first great
+principle of modern geometry, the principle of continuity. Brianchon
+(1806), through his application of Desargues's theory of polars,
+completed the foundation which Monge had begun for Poncelet's (1829)
+theory of reciprocal polars.
+
+Among the most prominent geometers contemporary with Poncelet was
+Gergonne, who with more propriety might be ranked as an analytic
+geometer. He first (1813) used the term ``polar'' in its modern
+geometric sense, although Servois (1811) had used the expression
+``pole.'' He was also the first (1825-26) to grasp the idea that
+the parallelism which Maurolycus, Snell, and Viete had noticed is a
+fundamental principle. This principle he stated and to it he gave
+the name which it now bears, the Principle of Duality, the most
+important, after that of continuity, in modern geometry. This
+principle of geometric reciprocation, the discovery of which was
+also claimed by Poncelet, has been greatly elaborated and has found
+its way into modern algebra and elementary geometry, and has
+recently been extended to mechanics by Genese. Gergonne was the
+first to use the word ``class'' in describing a curve, explicitly
+defining class and degree (order) and showing the duality between
+the two. He and Chasles were among the first to study scientifically
+surfaces of higher order.
+
+Steiner (1832) gave the first complete discussion of the projective
+relations between rows, pencils, etc., and laid the foundation for
+the subsequent development of pure geometry. He practically closed
+the theory of conic sections, of the corresponding figures in
+three-dimensional space and of surfaces of the second order, and
+hence with him opens the period of special study of curves and
+surfaces of higher order. His treatment of duality and his
+application of the theory of projective pencils to the generation of
+conics are masterpieces. The theory of polars of a point in regard
+to a curve had been studied by Bobillier and by Grassmann, but
+Steiner (1848) showed that this theory can serve as the foundation
+for the study of plane curves independently of the use of
+coordinates, and introduced those noteworthy curves covariant to a
+given curve which now bear the names of himself, Hesse, and Cayley.
+This whole subject has been extended by Grassmann, Chasles,
+Cremona, and Jonqui\`eres. Steiner was the first to make prominent
+(1832) an example of correspondence of a more complicated nature
+than that of Poncelet, M\"obius, Magnus, and Chasles. His
+contributions, and those of Gudermann, to the geometry of the sphere
+were also noteworthy.
+
+While M\"obius, Pl\"ucker, and Steiner were at work in Germany, Chasles
+was closing the geometric era opened in France by Monge. His Aper\c{c}u
+Historique (1837) is a classic, and did for France what Salmon's
+works did for algebra and geometry in England, popularizing the
+researches of earlier writers and contributing both to the theory
+and the nomenclature of the subject. To him is due the name
+``homographic'' and the complete exposition of the principle as
+applied to plane and solid figures, a subject which has received
+attention in England at the hands of Salmon, Townsend, and
+H.~J.~S.~Smith.
+
+Von Staudt began his labors after Pl\"ucker, Steiner, and Chasles had
+made their greatest contributions, but in spite of this seeming
+disadvantage he surpassed them all. Joining the Steiner school, as
+opposed to that of Pl\"ucker, he became the greatest exponent of pure
+synthetic geometry of modern times. He set forth (1847, 1856-60) a
+complete, pure geometric system in which metrical geometry finds no
+place. Projective properties foreign to measurements are
+established independently of number relations, number being drawn
+from geometry instead of conversely, and imaginary elements being
+systematically introduced from the geometric side. A projective
+geometry based on the group containing all the real projective and
+dualistic transformations, is developed, imaginary transformations
+being also introduced. Largely through his influence pure geometry
+again became a fruitful field. Since his time the distinction
+between the metrical and projective theories has been to a great
+extent obliterated,\footnote{Klein, F., Erlangen Programme of
+1872, Haskell's translation, Bulletin of New York Mathematical
+Society, Vol. II, p. 215.} the metrical properties being considered
+as projective relations to a fundamental configuration, the circle
+at infinity common for all spheres. Unfortunately von Staudt wrote
+in an unattractive style, and to Reye is due much of the popularity
+which now attends the subject.
+
+Cremona began his publications in 1862. His elementary work on
+projective geometry (1875) in Leudesdorf's translation is familiar
+to English readers. His contributions to the theory of geometric
+transformations are valuable, as also his works on plane curves,
+surfaces, etc.
+
+In England Mulcahy, but especially Townsend (1863), and Hirst, a
+pupil of Steiner's, opened the subject of modern geometry. Clifford
+did much to make known the German theories, besides himself
+contributing to the study of polars and the general theory of curves.
+
+\chapter{ELEMENTARY GEOMETRY.}
+
+Trigonometry and Elementary Geometry have also been affected by the
+general mathematical spirit of the century. In trigonometry the
+general substitution of ratios for lines in the definitions of
+functions has simplified the treatment, and certain formulas have
+been improved and others added.\footnote{Todhunter, I., History of
+certain formulas of spherical trigonometry, Philosophical Magazine,
+1873.} The convergence of trigonometric series, the introduction of
+the Fourier series, and the free use of the imaginary have already
+been mentioned. The definition of the sine and cosine by series, and
+the systematic development of the theory on this basis, have been
+set forth by Cauchy (1821), Lobachevsky (1833), and others. The
+hyperbolic trigonometry,\footnote{Gunther, S., Die Lehre von den
+gew\"ohnlichen und verallgemeinerten Hyperbelfunktionen, Halle, 1881;
+Chrystal, G., Algebra, Vol. II, p. 288.} already founded by Mayer and
+Lambert, has been popularized and further developed by Gudermann
+(1830), Ho\"uel, and Laisant (1871), and projective formulas and
+generalized figures have been introduced, notably by Gudermann,
+M\"obius, Poncelet, and Steiner. Recently Study has investigated the
+formulas of spherical trigonometry from the standpoint of the modern
+theory of functions and theory of groups, and Macfarlane has
+generalized the fundamental theorem of trigonometry for
+three-dimensional space.
+
+Elementary Geometry has been even more affected. Among the many
+contributions to the theory may be mentioned the following: That of
+M\"obius on the opposite senses of lines, angles, surfaces, and
+solids; the principle of duality as given by Gergonne and Poncelet;
+the contributions of De Morgan to the logic of the subject; the
+theory of transversals as worked out by Monge, Brianchon, Servois,
+Carnot, Chasles, and others; the theory of the radical axis, a
+property discovered by the Arabs, but introduced as a definite
+concept by Gaultier (1813) and used by Steiner under the name of
+``line of equal power''; the researches of Gauss concerning
+inscriptible polygons, adding the 17- and 257-gon to the list below
+the 1000-gon; the theory of stellar polyhedra as worked out by
+Cauchy, Jacobi, Bertrand, Cayley, M\"obius, Wiener, Hess, Hersel,
+and others, so that a whole series of bodies have been added to the
+four Kepler-Poinsot regular solids; and the researches of Muir on
+stellar polygons. These and many other improvements now find more or
+less place in the text-books of the day.
+
+To these must be added the recent Geometry of the Triangle, now a
+prominent chapter in elementary mathematics. Crelle (1816) made
+some investigations in this line, Feuerbach (1822) soon after
+discovered the properties of the Nine-Point Circle, and Steiner also
+came across some of the properties of the triangle, but none of
+these followed up the investigation. Lemoine\footnote{Smith,
+D. E., Biography of Lemoine, American Mathematical Monthly,
+Vol. III, p. 29; Mackay, J. S., various articles on modern geometry
+in Proceedings Edinburgh Mathematical Society, various years;
+Vigari\'e, \'E., G\'eom\'etrie du triangle. Articles in recent
+numbers of Journal de Math\'ematiques sp\'eciales, Mathesis, and
+Proceedings of the Association fran\c{c}aise pour l'avancement des
+sciences.} (1873) was the first to take up the subject in a
+systematic way, and he has contributed extensively to its
+development. His theory of ``transformation continue'' and his
+``g\'eom\'etrographie'' should also be mentioned. Brocard's
+contributions to the geometry of the triangle began in 1877. Other
+prominent writers have been Tucker, Neuberg, Vigari\'e, Emmerich,
+M'Cay, Longchamps, and H. M. Taylor. The theory is also greatly
+indebted to Miller's work in The Educational Times, and to
+Hoffmann's Zeitschrift.
+
+The study of linkages was opened by Peaucellier (1864), who gave the
+first theoretically exact method for drawing a straight line. Kempe
+and Sylvester have elaborated the subject.
+
+In recent years the ancient problems of trisecting an angle,
+doubling the cube, and squaring the circle have all been settled by
+the proof of their insolubility through the use of compasses and
+straight edge.\footnote{Klein, F., Vortr\"age \"uber ausgew\"ahlten
+Fragen; Rudio, F., Das Problem von der Quadratur des Zirkels.
+Naturforschende Gesellschaft Vierteljahrschrift, 1890; Archimedes,
+Huygens, Lambert, Legendre (Leipzig, 1892).}
+
+\chapter{NON-EUCLIDEAN GEOMETRY.}
+
+The Non-Euclidean Geometry\footnote{St\"ackel and Engel, Die
+Theorie der Parallellinien von Euklid bis auf Gauss, Leipzig, 1895;
+Halsted, G. B., various contributions: Bibliography of Hyperspace
+and Non-Euclidean Geometry, American Journal of Mathematics,
+Vols. I, II; The American Mathematical Monthly, Vol. I; translations
+of Lobachevsky's Geometry, Vasiliev's address on Lobachevsky,
+Saccheri's Geometry, Bolyai's work and his life; Non-Euclidean and
+Hyperspaces, Mathematical Papers of Chicago Congress, p. 92. Loria,
+G., Die haupts\"achlichsten Theorien der Geometrie, p. 106;
+Karagiannides, A., Die Nichteuklidische Geometrie vom Alterthum bis
+zur Gegenwart, Berlin, 1893; McClintock, E., On the early history of
+Non-Euclidean Geometry, Bulletin of New York Mathematical Society,
+Vol. II, p. 144; Poincar\'e, Non-Euclidean Geom., Nature, 45:404;
+Articles on Parallels and Measurement in Encyclop\ae{}dia Britannica,
+9th edition; Vasiliev's address (German by Engel) also appears in
+the Abhandlungen zur Geschichte der Mathematik, 1895.} is a natural
+result of the futile attempts which had been made from the time of
+Proklos to the opening of the nineteenth century to prove the fifth
+postulate (also called the twelfth axiom, and sometimes the eleventh
+or thirteenth) of Euclid. The first scientific investigation of
+this part of the foundation of geometry was made by Saccheri (1733),
+a work which was not looked upon as a precursor of Lobachevsky,
+however, until Beltrami (1889) called attention to the fact. Lambert
+was the next to question the validity of Euclid's postulate, in his
+Theorie der Parallellinien (posthumous, 1786), the most important of
+many treatises on the subject between the publication of Saccheri's
+work and those of Lobachevsky and Bolyai. Legendre also worked in
+the field, but failed to bring himself to view the matter outside
+the Euclidean limitations.
+
+During the closing years of the eighteenth century
+Kant's\footnote{Fink, E., Kant als Mathematiker, Leipzig, 1889.}
+doctrine of absolute space, and his assertion of the necessary
+postulates of geometry, were the object of much scrutiny and
+attack. At the same time Gauss was giving attention to the fifth
+postulate, though on the side of proving it. It was at one time
+surmised that Gauss was the real founder of the non-Euclidean
+geometry, his influence being exerted on Lobachevsky through his
+friend Bartels, and on Johann Bolyai through the father Wolfgang,
+who was a fellow student of Gauss's. But it is now certain that
+Gauss can lay no claim to priority of discovery, although the
+influence of himself and of Kant, in a general way, must have had
+its effect.
+
+Bartels went to Kasan in 1807, and Lobachevsky was his pupil. The
+latter's lecture notes show that Bartels never mentioned the subject
+of the fifth postulate to him, so that his investigations, begun
+even before 1823, were made on his own motion and his results were
+wholly original. Early in 1826 he sent forth the principles of his
+famous doctrine of parallels, based on the assumption that through a
+given point more than one line can be drawn which shall never meet a
+given line coplanar with it. The theory was published in full in
+1829-30, and he contributed to the subject, as well as to other
+branches of mathematics, until his death.
+
+Johann Bolyai received through his father, Wolfgang, some of the
+inspiration to original research which the latter had received from
+Gauss. When only twenty-one he discovered, at about the same time as
+Lobachevsky, the principles of non-Euclidean geometry, and refers to
+them in a letter of November, 1823. They were committed to writing
+in 1825 and published in 1832. Gauss asserts in his correspondence
+with Schumacher (1831-32) that he had brought out a theory along the
+same lines as Lobachevsky and Bolyai, but the publication of their
+works seems to have put an end to his investigations. Schweikart
+was also an independent discoverer of the non-Euclidean geometry, as
+his recently recovered letters show, but he never published anything
+on the subject, his work on the theory of parallels (1807), like
+that of his nephew Taurinus (1825), showing no trace of the
+Lobachevsky-Bolyai idea.
+
+The hypothesis was slowly accepted by the mathematical world. Indeed
+it was about forty years after its publication that it began to
+attract any considerable attention. Ho\"uel (1866) and Flye
+St. Marie (1871) in France, Riemann (1868), Helmholtz (1868),
+Frischauf (1872), and Baltzer (1877) in Germany, Beltrami (1872) in
+Italy, de Tilly (1879) in Belgium, Clifford in England, and Halsted
+(1878) in America, have been among the most active in making the
+subject popular. Since 1880 the theory may be said to have become
+generally understood and accepted as legitimate.\footnote{For an
+excellent summary of the results of the hypothesis, see an article
+by McClintock, The Non-Euclidian Geometry, Bulletin of New York
+Mathematical Society, Vol. II, p. 1.}
+
+Of all these contributions the most noteworthy from the scientific
+standpoint is that of Riemann. In his Habilitationsschrift (1854)
+he applied the methods of analytic geometry to the theory, and
+suggested a surface of negative curvature, which Beltrami calls
+``pseudo-spherical,'' thus leaving Euclid's geometry on a surface of
+zero curvature midway between his own and Lobachevsky's. He thus set
+forth three kinds of geometry, Bolyai having noted only two. These
+Klein (1871) has called the elliptic (Riemann's), parabolic
+(Euclid's), and hyperbolic (Lobachevsky's).
+
+Starting from this broader point of view\footnote{Klein. Evanston
+Lectures. Lect. IX.} there have contributed to the subject many of
+the leading mathematicians of the last quarter of a century,
+including, besides those already named, Cayley, Lie, Klein, Newcomb,
+Pasch, C.~S.~Peirce, Killing, Fiedler, Mansion, and
+McClintock. Cayley's contribution of his ``metrical geometry'' was
+not at once seen to be identical with that of Lobachevsky and
+Bolyai. It remained for Klein (1871) to show this, thus simplifying
+Cayley's treatment and adding one of the most important results of
+the entire theory. Cayley's metrical formulas are, when the Absolute
+is real, identical with those of the hyperbolic geometry; when it
+is imaginary, with the elliptic; the limiting case between the two
+gives the parabolic (Euclidean) geometry. The question raised by
+Cayley's memoir as to how far projective geometry can be defined in
+terms of space without the introduction of distance had already been
+discussed by von Staudt (1857) and has since been treated by Klein
+(1873) and by Lindemann (1876).
+
+\backmatter
+
+\chapter{BIBLIOGRAPHY.}
+
+%% In the book, the titles are slightly smaller then the rest
+%% of the text; should we follow that here?
+
+The following are a few of the general works on the history of
+mathematics in the nineteenth century, not already mentioned in the
+foot-notes. For a complete bibliography of recent works the reader
+should consult the Jahrbuch \"uber die Fortschritte der Mathematik,
+the Bibliotheca Mathematica, or the Revue Semestrielle, mentioned
+below.
+
+\bigskip
+Abhandlungen zur Geschichte der Mathematik (Leipzig).
+
+Ball, W.~W.~R., A short account of the history of mathematics
+(London, 1893).
+
+Ball, W.~W.~R., History of the study of mathematics at Cambridge
+(London, 1889).
+
+Ball, W.~W.~R., Primer of the history of mathematics (London, 1895).
+
+Bibliotheca Mathematica, G. Enestr\"om, Stockholm. Quarterly.
+Should be consulted for bibliography of current articles and works
+on history of mathematics.
+
+Bulletin des Sciences Math\'ematiques (Paris, II\up{i\`eme} Partie).
+
+Cajori, F., History of Mathematics (New York, 1894).
+
+Cayley, A., Inaugural address before the British Association,
+1883. Nature, Vol. XXVIII, p. 491.
+
+Dictionary of National Biography. London, not completed. Valuable
+on biographies of British mathematicians.
+
+D'Ovidio, Enrico, Uno sguardo alle origini ed allo sviluppo della
+Matematica Pura (Torino, 1889).
+
+Dupin, Ch., Coup d'\oe{}il sur quelques progr\`es des Sciences
+math\'ematiques, en France, 1830-35. Comptes Rendus, 1835.
+
+Encyclop\ae{}dia Britannica. Valuable biographical articles by Cayley,
+Chrystal, Clerke, and others.
+
+Fink, K., Geschichte der Mathematik (T\"ubingen, 1890). Bibliography
+on p. 255.
+
+Gerhardt, C.~J., Geschichte der Mathematik in Deutschland (Munich,
+1877).
+
+Graf, J.~H., Geschichte der Mathematik und der Naturwissenschaften
+in bernischen Landen (Bern, 1890). Also numerous biographical
+articles.
+
+G\"unther, S., Vermischte Untersuchungen zur Geschichte der
+mathematischen Wissenschaften (Leipzig, 1876).
+
+G\"unther, S., Ziele und Resultate der neueren
+mathematisch-historischen Forschung (Erlangen, 1876).
+
+Hagen, J.~G., Synopsis der h\"oheren Mathematik. Two volumes
+(Berlin, 1891-93).
+
+Hankel, H., Die Entwickelung der Mathematik in dem letzten
+Jahrhundert (T\"ubingen, 1884).
+
+Hermite, Ch., Discours prononc\'e devant le pr\'esident de la
+r\'epublique le 5 ao\^ut 1889 \`a l'inauguration de la nouvelle
+Sorbonne. Bulletin des Sciences math\'ematiques, 1890; also Nature,
+Vol. XLI, p. 597. (History of nineteenth-century mathematics in
+France.)
+
+Hoefer, F., Histoire des math\'ematiques (Paris, 1879).
+
+Isely, L., Essai sur l'histoire des math\'ematiques dans la Suisse
+fran\c{c}aise (Neuch\^atel, 1884).
+
+Jahrbuch \"uber die Fortschritte der Mathematik (Berlin, annually,
+1868 to date).
+
+Marie, M., Histoire des sciences math\'ematiques et physiques.
+Vols. X, XI, XII (Paris, 1887-88).
+
+Matthiessen, L., Grundz\"uge der antiken und modernen Algebra der
+litteralen Gleichungen (Leipzig, 1878).
+
+Newcomb, S., Modern mathematical thought. Bulletin New York
+Mathematical Society, Vol. III, p. 95; Nature, Vol. XLIX, p. 325.
+
+Poggendorff, J.~C., Biographisch-literarisches Handw\"orterbuch
+zur Ge\-schi\-chte der exacten Wissenschaften. Two volumes (Leipzig,
+1863), and two later supplementary volumes.
+
+Quetelet, A., Sciences math\'ematiques et physiques chez les Belges
+au commencement du XIX\up{e} si\`ecle (Brussels, 1866).
+
+Revue semestrielle des publications math\'ematiques r\'edig\'ee sous
+les auspices de la Soci\'et\'e math\'ematique d'Amsterdam. 1893 to
+date. (Current periodical literature.)
+
+Roberts, R.~A., Modern mathematics. Proceedings of the Irish
+Academy, 1888.
+
+Smith, H.~J.~S., On the present state and prospects of some branches
+of pure mathematics. Proceedings of London Mathematical Society,
+1876; Nature, Vol. XV, p. 79.
+
+Sylvester, J.~J., Address before the British Association. Nature,
+Vol. I, pp. 237, 261.
+
+Wolf, R., Handbuch der Mathematik. Two volumes (Zurich, 1872).
+
+Zeitschrift f\"ur Mathematik und Physik. Historisch-literarische
+Abtheilung. Leipzig. The Abhandlungen zur Geschichte der Mathematik
+are supplements.
+
+\bigskip
+
+For a biographical table of mathematicians see Fink's Geschichte der
+Mathematik, p. 240. For the names and positions of living
+mathematicians see the Jahrbuch der gelehrten Welt, published at
+Strassburg.
+
+Since the above bibliography was prepared the nineteenth century has
+closed. With its termination there would naturally be expected a
+series of retrospective views of the development of a hundred years
+in all lines of human progress. This expectation was duly
+fulfilled, and numerous addresses and memoirs testify to the
+interest recently awakened in the subject. Among the contributions
+to the general history of modern mathematics may be cited the
+following:
+
+\bigskip
+Pierpont, J., St. Louis address, 1904. Bulletin of the American
+Mathematical Society (N. S.), Vol. IX, p. 136. An excellent survey
+of the century's progress in pure mathematics.
+
+G\"unther, S., Die Mathematik im neunzehnten Jahrhundert. Hoffmann's
+Zeitschrift, Vol. XXXII, p. 227.
+
+Adh\'emar, R. d', L'\oe{}uvre math\'ematique du XIX\up{e} si\`ecle. Revue
+des questions scientifiques, Louvain Vol. XX (2), p. 177 (1901).
+
+Picard, E., Sur le d\'eveloppement, depuis un si\`ecle, de quelques
+th\'eories fondamentales dans l'analyse
+math\'ematique. Conf\'erences faite \`a Clark University (Paris,
+1900).
+
+Lampe, E., Die reine Mathematik in den Jahren 1884-1899 (Berlin,
+1900).
+
+\bigskip
+
+Among the contributions to the history of applied mathematics in
+general may be mentioned the following:
+
+\bigskip
+
+Woodward, R.~S., Presidential address before the American
+Mathematical Society in December, 1899. Bulletin of the American
+Mathematical Society (N. S.), Vol. VI, p. 133. (German, in the
+Naturwiss. Rundschau, Vol. XV; Polish, in the Wiadomo\'sci
+Matematyczne, Warsaw, Vol. V (1901).). This considers the century's
+progress in applied mathematics.
+
+Mangoldt, H. von, Bilder aus der Entwickelung der reinen und
+angewandten Mathematik w\"ahrend des neunzehnten Jahrhunderts mit
+besonderer Ber\"ucksichtigung des Einflusses von Carl Friedrich
+Gauss. Festrede (Aachen, 1900).
+
+Van t' Hoff, J.~H., Ueber die Entwickelung der exakten
+Naturwissenschaften im 19. Jahrhundert. Vortrag gehalten in Aachen,
+1900. Naturwiss. Rundschau, Vol. XV, p. 557 (1900).
+
+\bigskip
+
+The following should be mentioned as among the latest contributions
+to the history of modern mathematics in particular countries:
+
+\bigskip
+
+Fiske, T.~S., Presidential address before the American Mathematical
+Society in December, 1904. Bulletin of the American Mathematical
+Society (N. S.), Vol. IX, p. 238. This traces the development of
+mathematics in the United States.
+
+Purser, J., The Irish school of mathematicians and physicists from
+the beginning of the nineteenth century. Nature, Vol. LXVI, p. 478
+(1902).
+
+Guimar\~aes, R. Les math\'ematiques en Portugal au XIX\up{e} si\`ecle.
+(Co\"{\i}mbre, 1900).
+
+\bigskip
+
+A large number of articles upon the history of special branches of
+mathematics have recently appeared, not to mention the custom of
+inserting historical notes in the recent treatises upon the subjects
+themselves. Of the contributions to the history of particular
+branches, the following may be mentioned as types:
+
+\bigskip
+
+Miller, G.~A., Reports on the progress in the theory of groups of a
+finite order. Bulletin of the American Mathematical Society (N. S.),
+Vol. V, p. 227; Vol. IX, p. 106. Supplemental report by Dickson,
+L. E., Vol. VI, p. 13, whose treatise on Linear Groups (1901) is a
+history in itself. Steinitz and Easton have also contributed to this
+subject.
+
+Hancock, H., On the historical development of the Abelian functions
+to the time of Riemann. British Association Report for 1897.
+
+Brocard, H., Notes de bibliographie des courbes g\'eom\'etriques.
+Bar-le-Duc, 2 vols., lithog., 1897, 1899.
+
+Hagen, J.~G., On the history of the extensions of the calculus.
+Bulletin of the American Mathematical Society (N. S.), Vol. VI,
+p. 381.
+
+Hill, J.~E., Bibliography of surfaces and twisted curves. Ib., Vol.
+III, p. 133 (1897).
+
+Aubry, A., Historia del problema de las tangentes. El Progresso
+matematico, Vol. I (2), pp. 129, 164.
+
+Comp\`ere, C., Le probl\`eme des brachistochrones. Essai historique.
+M\'emoires de la Soci\'et\'e d. Sciences, Li\`ege, Vol. I (3),
+p. 128 (1899).
+
+St\"ackel, P., Beitr\"age zur Geschichte der Funktionentheorie im
+achtzehnten Jahrhundert. Bibliotheca Mathematica, Vol. II (3),
+p. 111 (1901).
+
+Obenrauch, F.~J., Geschichte der darstellenden und projektiven
+Geometrie mit besonderer Ber\"ucksichtigung ihrer Begr\"undung in
+Frankreich und Deutschiand und ihrer wissenschaftlichen Pflege in
+Oesterreich (Br\"unn, 1897).
+
+Muir, Th., The theory of alternants in the historical order of its
+development up to 1841. Proceedings of the Royal Society of
+Edinburgh, Vol. XXIII (2), p. 93 (1899). The theory of screw
+determinants and Pfaffians in the historical order of its
+development up to 1857. Ib., p. 181.
+
+Papperwitz, E., Ueber die wissenschaftliche Bedeutung der
+darstellenden Geometrie und ihre Entwickelung bis zur
+systematischen Begr\"undung durch Gaspard Monge. Rede (Freiberg
+i./S., 1901).
+
+\bigskip
+
+Mention should also be made of the fact that the Bibliotheca
+Mathematica, a journal devoted to the history of the mathematical
+sciences, began its third series in 1900. It remains under the able
+editorship of G. Enestr\"om, and in its new series it appears in
+much enlarged form. It contains numerous articles on the history of
+modern mathematics, with a complete current bibliography of this
+field.
+
+Besides direct contributions to the history of the subject, and
+historical and bibliographical notes, several important works have
+recently appeared which are historical in the best sense, although
+written from the mathematical standpoint. Of these there are three
+that deserve special mention:
+
+\bigskip
+
+Encyklop\"adie der mathematischen Wissenschaften mit Einschluss
+ihrer Anwendungen. The publication of this monumental work was begun
+in 1898, and the several volumes are being carried on
+simultaneously. The first volume (Arithmetik and Algebra) was
+completed in 1904. This publication is under the patronage of the
+academies of sciences of G\"ottingen, Leipzig, Munich, and Vienna. A
+French translation, with numerous additions, is in progress.
+
+Pascal, E., Repertorium der h\"oheren Mathematik, translated from
+the Italian by A. Schepp. Two volumes (Leipzig, 1900, 1902). It
+contains an excellent bibliography, and is itself a history of
+modern mathematics.
+
+Hagen, J. G., Synopsis der h\"oheren Mathematik. This has been for
+some years in course of publication, and has now completed Vol. III.
+
+\bigskip
+
+In the line of biography of mathematicians, with lists of published
+works, Poggendorff's Biographisch-literarisches Handw\"orterbuch zur
+Geschichte der exacten Wissenschaften has reached its fourth volume
+(Leipzig, 1903), this volume covering the period from 1883 to
+1902. A new biographical table has been added to the English
+translation of Fink's History of Mathematics (Chicago, 1900).
+
+\chapter{GENERAL TENDENCIES.}
+
+The opening of the nineteenth century was, as we have seen, a period
+of profound introspection following a period of somewhat careless
+use of the material accumulated in the seventeenth century. The
+mathematical world sought to orientate itself, to examine the
+foundations of its knowledge, and to critically examine every step
+in its several theories. It then took up the line of discovery once
+more, less recklessly than before, but still with thoughts directed
+primarily in the direction of invention. At the close of the
+century there came again a period of introspection, and one of the
+recent tendencies is towards a renewed study of foundation
+principles. In England one of the leaders in this movement is
+Russell, who has studied the foundations of geometry (1897) and of
+mathematics in general (1903). In America the fundamental
+conceptions and methods of mathematics have been considered by
+B\^ocher in his St. Louis address in 1904,\footnote{Bulletin of the
+American Mathematical Society (N. S.), Vol. XI, p. 115.} and the
+question of a series of irreducible postulates has been studied by
+Huntington. In Italy, Padoa and Bureli-Forti have studied the
+fundamental postulates of algebra, and Pieri those of geometry. In
+Germany, Hilbert has probably given the most attention to the
+foundation principles of geometry (1899), and more recently he has
+investigated the compatibility of the arithmetical axioms (1900). In
+France, Poincar\'e has considered the r\^ole of intuition and of
+logic in mathematics,\footnote{Compte rendu du deuxi\`eme congr\`es
+international des math\'ematiciens tenu \`a Paris, 1900. Paris,
+1902, p. 115.} and in every country the foundation principles have
+been made the object of careful investigation.
+
+As an instance of the orientation already mentioned, the noteworthy
+address of Hilbert at Paris in 1900\footnote{G\"ottinger
+Nachrichten, 1900, p. 253; Archiv der Mathematik und Physik, 1901,
+pp. 44, 213; Bulletin of the American Mathematical Society, 1902,
+p. 437.} stands out prominently. This address reviews the field of
+pure mathematics and sets forth several of the greatest questions
+demanding investigation at the present time. In the particular line
+of geometry the memoir which Segr\'e wrote in 1891, on the
+tendencies in geometric investigation, has recently been revised and
+brought up to date.\footnote{Bulletin of the American Mathematical
+Society (N. S.), Vol. X, p. 443.}
+
+There is also seen at the present time, as never before, a
+tendency to co\"operate, to exchange views, and to internationalize
+mathematics. The first international congress of mathematicians
+was held at Zurich in 1897, the second one at Paris in 1900, and
+the third at Heidelberg in 1904. The first international congress
+of philosophy was held at Paris in 1900, the third section
+being devoted to logic and the history of the sciences (on this
+occasion chiefly mathematics), and the second congress was
+held at Geneva in 1904. There was also held an international
+congress of historic sciences at Rome in 1903, an international
+committee on the organization of a congress on the history of
+sciences being at that time formed. The result of such gatherings
+has been an exchange of views in a manner never before
+possible, supplementing in an inspiring way the older form of
+international communication through published papers.
+
+In the United States there has been shown a similar tendency
+to exchange opinions and to impart verbal information
+as to recent discoveries. The American Mathematical Society,
+founded in 1894,\footnote{It was founded as the New York
+Mathematical Society six years earlier, in 1888.} has doubled
+its membership in the past decade,\footnote{It is now, in 1905,
+approximately 500.}
+and has increased its average of annual papers from 30 to 150.
+It has also established two sections, one at Chicago (1897) and
+one at San Francisco (1902). The activity of its members and
+the quality of papers prepared has led to the publication of the
+\emph{Transactions}, beginning with 1900. In order that its members
+may be conversant with the lines of investigation in the various
+mathematical centers, the society publishes in its \emph{Bulletin} the
+courses in advanced mathematics offered in many of the leading
+universities of the world. Partly as a result of this activity,
+and partly because of the large number of American students
+who have recently studied abroad, a remarkable change is at
+present passing over the mathematical work done in the universities
+and colleges of this country. Courses that a short time ago
+were offered in only a few of our leading universities are now
+not uncommon in institutions of college rank. They are often
+given by men who have taken advanced degrees in mathematics,
+at G\"ottingen, Berlin, Paris, or other leading universities abroad,
+and they are awakening a great interest in the modern field.
+A recent investigation (1903) showed that 67 students in ten
+American institutions were taking courses in the theory of functions,
+11 in the theory of elliptic functions, 94 in projective geometry,
+26 in the theory of invariants, 45 in the theory of groups,
+and 46 in the modern advanced theory of equations, courses
+which only a few years ago were rarely given in this country.
+A similar change is seen in other countries, notably in England
+and Italy, where courses that a few years ago were offered only
+in Paris or in Germany are now within the reach of university
+students at home. The interest at present manifested by American
+scholars is illustrated by the fact that only four countries (Germany,
+Russia, Austria, and France) had more representatives
+than the United States, among the 336 regular members at the
+third international mathematical congress at Heidelberg in 1904.
+
+The activity displayed at the present time in putting the
+work of the masters into usable form, so as to define clear points
+of departure along the several lines of research, is seen in the
+large number of collected works published or in course of publication
+in the last decade. These works have usually been
+published under governmental patronage, often by some learned
+society, and always under the editorship of some recognized
+authority. They include the works of Galileo, Fermat, Descartes,
+Huygens, Laplace, Gauss, Galois, Cauchy, Hesse, Pl\"ucker,
+Grassmann, Dirichlet, Laguerre, Kronecker, Fuchs, Weierstrass,
+Stokes, Tait, and various other leaders in mathematics. It is
+only natural to expect a number of other sets of collected works
+in the near future, for not only is there the remote past to draw
+upon, but the death roll of the last decade has been a large one.
+The following is only a partial list of eminent mathematicians
+who have recently died, and whose collected works have been
+or are in the course of being published, or may be deemed worthy
+of publication in the future: Cayley (1895), Neumann (1895),
+Tisserand (1896), Brioschi (1897), Sylvester (1897), Weierstrass
+(1897), Lie (1899), Beltrami (1900), Bertrand (1900), Tait (1901),
+Hermite (1901), Fuchs (1902), Gibbs (1903), Cremona (1903),
+and Salmon (1904), besides such writers as Frost (1898), Hoppe
+(1900), Craig (1900), Schl\"omilch (1901), Everett on the side of
+mathematical physics (1904), and Paul Tannery, the best of
+the modern French historians of mathematics (1904).\footnote{For
+students wishing to investigate the work of mathematicians who died
+in the last two decades of the nineteenth century, Enestr\"om's "Bio-bibliographie
+der 1881-1900 verstorbenen Mathematiker," in the Bibliotheca Mathematica
+Vol. II (3), p. 326 (1901), will be found valuable.}
+
+It is, of course, impossible to detect with any certainty the
+present tendencies in mathematics. Judging, however, by the
+number and nature of the published papers and works of the
+past few years, it is reasonable to expect a great development in
+all lines, especially in such modern branches as the theory of
+groups, theory of functions, theory of invariants, higher geometry,
+and differential equations. If we may judge from the works in
+applied mathematics which have recently appeared, we are
+entering upon an era similar to that in which Laplace labored,
+an era in which all these modern theories of mathematics shall
+find application in the study of physical problems, including
+those that relate to the latest discoveries. The profound study
+of applied mathematics in France and England, the advanced
+work in discovery in pure mathematics in Germany and France,
+and the search for the logical bases for the science that has distinguished
+Italy as well as Germany, are all destined to affect the
+character of the international mathematics of the immediate
+future. Probably no single influence will be more prominent
+in the internationalizing process than the tendency of the younger
+generation of American mathematicians to study in England,
+France, Germany, and Italy, and to assimilate the best that each
+of these countries has to offer to the world.
+
+\newpage
+
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