MATHEMATICAL MONOGRAPHS
EDITED BY MANSFIELD MERRIMAN AND ROBERT S. WOODWARD
No. 1
HISTORY OF MODERN
MATHEMATICS.
BY
DAVID EUGENE SMITH,
PROFESSOR OF MATHEMATICS IN TEACHERS COLLEGE, COLUMBIA
UNIVERSITY.
FOURTH EDITION, ENLARGED.
1906
ii
MATHEMATICAL MONOGRAPHS.
edited by
Mansfield Merriman and Robert S. Woodward.
No. 1. HISTORY OF MODERN MATHEMATICS.
By David Eugene Smith.
No. 2. SYNTHETIC PROJECTIVE GEOMETRY.
By George Bruce Halsted.
No. 3. DETERMINANTS.
By Laenas Gifford Weld.
No. 4. HYPERBOLIC FUNCTIONS.
By James McMahon.
No. 5. HARMONIC FUNCTIONS.
By William E. Byerly.
No. 6. GRASSMANN’S SPACE ANALYSIS.
By Edward W. Hyde.
No. 7. PROBABILITY AND THEORY OF ERRORS.
By Robert S. Woodward.
No. 8. VECTOR ANALYSIS AND QUATERNIONS.
By Alexander Macfarlane.
No. 9. DIFFERENTIAL EQUATIONS.
By William Woolsey Johnson.
No. 10. THE SOLUTION OF EQUATIONS.
By Mansfield Merriman.
No. 11. FUNCTIONS OF A COMPLEX VARIABLE.
By Thomas S. Fiske.
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.
December, 1905.
iii
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.
Teachers College, Columbia University,
December, 1905.
iv
Contents
EDITORS’ PREFACE. iii
AUTHOR’S PREFACE. iv
1 INTRODUCTION. 1
2 THEORY OF NUMBERS. 4
3 IRRATIONAL AND TRANSCENDENT NUMBERS. 6
4 COMPLEX NUMBERS. 8
5 QUATERNIONS AND AUSDEHNUNGSLEHRE. 10
6 THEORY OF EQUATIONS. 12
7 SUBSTITUTIONS AND GROUPS. 16
8 DETERMINANTS. 18
9 QUANTICS. 20
10 CALCULUS. 23
11 DIFFERENTIAL EQUATIONS. 26
12 INFINITE SERIES. 30
13 THEORY OF FUNCTIONS. 33
14 PROBABILITIES AND LEAST SQUARES. 38
15 ANALYTIC GEOMETRY. 40
16 MODERN GEOMETRY. 45
17 ELEMENTARY GEOMETRY. 49
v
CONTENTS vi
18 NON-EUCLIDEAN GEOMETRY. 51
BIBLIOGRAPHY. 54
GENERAL TENDENCIES. 59

Article 1
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
1
ARTICLE 1. INTRODUCTION. 2
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.1
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),
1The 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.
ARTICLE 1. INTRODUCTION. 3
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).2 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 Mathematiker-
Vereinigung (1892).
To the influence of the schools and the journals must be added that of
the various learned societies3 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.4
2For 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¸caise 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.
3For 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.
4Klein, F., The Present State of Mathematics, Mathematical Papers of Chicago Congress
(New York, 1896), p. 133.
Article 2
THEORY OF NUMBERS.
The Theory of Numbers,1 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 Arithmeticæ(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 Primes2 has attracted many investigators during the nineteenth
century, but the results have been detailed rather than general. Tch´ebichef
(1850) 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  b (mod 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
1Cantor, 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.
2Brocard, 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.
4
ARTICLE 2. THEORY OF NUMBERS. 5
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 xn +yn = zn,
which Euler and Legendre had proved for n = 3, 4, Dirichlet showing that
x5 + y5 6= az5. 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.
Article 3
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 Cataldi,
1613),1 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
1But see Favaro, A., Notizie storiche sulle frazioni continue dal secolo decimoterzo al decimosettimo,
Boncompagni’s Bulletino, Vol. VII, 1874, pp. 451, 533.
6
ARTICLE 3. IRRATIONAL AND TRANSCENDENT NUMBERS. 7
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 Numbers2 were first distinguished from algebraic irrationals
by Kronecker. Lambert proved (1761) that  cannot be rational, and that en
(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  is
not the square root of a rational number. Liouville (1840) showed that neither
e nor e2 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 . Lindemann’s proof was much simplified by Weierstrass
(1885), still further by Hilbert (1893), and has finally been made elementary by
Hurwitz and Gordan.
2Klein, F., Vortr¨age ¨uber ausgew¨ahlte Fragen der Elementargeometrie, 1895, p. 38; Bachmann,
P., Vorlesungen ¨uber die Natur der Irrationalzahlen, 1892.
Article 4
COMPLEX NUMBERS.
The Theory of Complex Numbers1 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  + i sin )n = cos n + i sin n, and to Euler (1748)
the formula cos  + i sin  = ei.
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 ±p−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
1Riecke, 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.
8
ARTICLE 4. COMPLEX NUMBERS. 9
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  + i sin  the “direction factor”, and r = pa2 + b2 the “modulus”;
Cauchy (1828) called cos  + i sin  the “reduced form” (l’expression
r´eduite); Gauss used i for p−1, introduced the term “complex number” for
a + bi, and called a2 + b2 the “norm.” The expression “direction coefficient”,
often used for cos  + i sin , 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 types2 have been studied, besides the familiar a + bi, in which i is
the root of x2 + 1 = 0. Thus Eisenstein has studied the type a + bj, j being a
complex root of x3 − 1 = 0. Similarly, complex types have been derived from
xk − 1 = 0 (k prime). This generalization is largely due to Kummer, to whom
is also due the theory of Ideal Numbers,3 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)  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.
2Chapman, 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.
3Klein, F., Evanston Lectures, Lect. VIII.
Article 5
QUATERNIONS AND
AUSDEHNUNGSLEHRE.
Quaternions and Ausdehnungslehre1 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  + j sin ) such that while
r should be a real number and  a real angle, i, j, or k should be any directed
unit line such that i2 = j2 = k2 = −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.2
Following Hamilton, there have appeared in Great Britain numerous papers
and works by Tait (1867), Kelland and Tait (1873), Sylvester, and McAulay
1Tait, P. G., on Quaternions, Encyclopædia Britannica; Schlegel, V., Die Grassmann’sche
Ausdehnungslehre, Schl¨omilch’s Zeitschrift, Vol. XLI.
2These are set forth in a paper by J. W. Gibbs, Nature, Vol. XLIV, p. 79.
10
ARTICLE 5. QUATERNIONS AND AUSDEHNUNGSLEHRE. 11
(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 Grassmann3 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.4
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).
3For bibliography see Schlegel, V., Die Grassmann’sche Ausdehnungslehre, Schl¨omilch’s
Zeitschrift, Vol. XLI.
4For Macfarlane’s Digest of views of English and American writers, see Proceedings American
Association for Advancement of Science, 1891.
Article 6
THEORY OF
EQUATIONS.
The Theory of Numerical Equations1 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 Theorem2 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 nth 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
1Cayley, A., Equations, and Kelland, P., Algebra, in Encyclopædia Britannica; Favaro,
A., Notizie storico-critiche sulla costruzione delle equazioni. Modena, 1878; Cantor, M.,
Geschichte der Mathematik, Vol. III, p. 375.
2Loria, 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.
12
ARTICLE 6. THEORY OF EQUATIONS. 13
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 x5 +ax+b = 0 by the extraction of
square and cube roots only, was first shown to be possible by Bring (1786) and
independently by Jerrard3 (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.4 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.5 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
3Harley, R., A contribution of the history . . . of the general equation of the fifth degree,
Quarterly Journal of Mathematics, Vol. VI, p. 38.
4See Art. 7.
5Klein, F., Vorlesungen ¨uber das Ikosaeder, 1884.
ARTICLE 6. THEORY OF EQUATIONS. 14
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 xn − 1 = 0, admit of ready solution
by the familiar trigonometric formula x = cos 2k
n + i sin 2k
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
Pn−1
0 xm = 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.
ARTICLE 6. THEORY OF EQUATIONS. 15
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).
Article 7
SUBSTITUTIONS AND
GROUPS.
The Theories of Substitutions and Groups1 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 mth-degree equation having for roots m of the roots of a given nth-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 Soeur (1748)
and Waring (1762 to 1782) still further elaborated the idea.
Lagrange2 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.3 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.
1Netto, E., Theory of Substitutions, translated by Cole; Cayley, A., Equations, Encyclopædia
Britannica, 9th edition.
2Pierpont, James, Lagrange’s Place in the Theory of Substitutions, Bulletin of American
Mathematical Society, Vol. I, p. 196.
3Matthiessen, L. Grundz¨uge der antiken und modernen Algebra der litteralen Gleichungen,
Leipzig, 1878.
16
ARTICLE 7. SUBSTITUTIONS AND GROUPS. 17
The next great step was taken by Ruffini4 (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 r1, r2, . . . rn 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.
4Burkhardt, 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.
Article 8
DETERMINANTS.
The Theory of Determinants1 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,2
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
1Muir, 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.
2“Numerum bb − ac, cuius indole proprietates formæ(a, b, c) imprimis pendere in sequentibus
docebimus, determinantem huius uocabimus.”
18
ARTICLE 8. DETERMINANTS. 19
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.
Article 9
QUANTICS.
The Theory of Qualities or Forms1 appeared in embryo in the Berlin memoirs
of Lagrange (1773, 1775), who considered binary quadratic forms of the type
ax2 +bxy +cy2, and established the invariance of the discriminant of that type
when x + y is put for x. He classified forms of that type according to the sign
of b2−4ac, and introduced the ideas of transformation and equivalence. Gauss2
(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,
1Meyer, 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.
2See Art. 2.
20
ARTICLE 9. QUANTICS. 21
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  and order r
of a form of the mth order corresponds also a covariant or invariant of degree
m and of order r of a form of the 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.
Clebsch3 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 Christof-
3Klein’s Evanston Lectures, Lect. I.
ARTICLE 9. QUANTICS. 22
fel. 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.
Article 10
CALCULUS.
The Differential and Integral Calculus,1 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: R1
0 xn−1(1−x)n−1dx, R1
0 e−xxn−1dx, although these
were not the exact forms of Euler’s study. If n is integral, it follows that R1
0 e−xxn−1dx = n!, but if n is fractional it is a transcendent function. To
1Williamson, B., Infinitesimal Calculus, Encyclopædia 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.
23
ARTICLE 10. CALCULUS. 24
it Legendre assigned the symbol 􀀀, 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 􀀀x and log 􀀀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¸cois (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 Variations2 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’Hˆopital, 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
potential3 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.
2Carll, 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.
3Bacharach, M., Abriss der Geschichte der Potentialtheorie, 1883. This contains an extensive
bibliography.
ARTICLE 10. CALCULUS. 25
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.
Article 11
DIFFERENTIAL
EQUATIONS.
The Theory of Differential Equations1 has been called by Lie2 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, dny
dxn + A1
dn−1y
dxn−1 + · · · + Any = 0, depend on that
of the algebraic equation of the nth degree, F(z) = zn+A1zn−1+· · ·+An = 0,
in which zk takes the place of dky
dxk (k = 1, 2, · · · , 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
1Cantor, 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¸cons 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.
2“In der ganzen modernen Mathematik ist die Theorie der Differentialgleichungen die
wichtigste Disciplin.”
26
ARTICLE 11. DIFFERENTIAL EQUATIONS. 27
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). Imschenetsky3 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 eighteenthcentury
algebraists to find a method for solving the general equation of the nth
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,
3Grunert’s Archiv f¨ur Mathematik, Vol. LIV.
ARTICLE 11. DIFFERENTIAL EQUATIONS. 28
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. Clebsch4 (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’s5 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
4Klein’s Evanston Lectures, Lect. I.
5Klein’s Evanston Lectures, Lect. II, III.
ARTICLE 11. DIFFERENTIAL EQUATIONS. 29
question of the existence of integrals in any differential system and given a brief
summary of the history to 1895.6 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.
6Riquier, 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.
Article 12
INFINITE SERIES.
The Theory of Infinite Series1 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 + .
1.
 x + .(+1)..(+1)
1.2.
.(
+1) x2 + · · ·, 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+ m
1 x+ m(m−1)
2! x2+· · · 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
1Cantor, 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.
30
ARTICLE 12. INFINITE SERIES. 31
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) = 1n + 2n + · · · + (x − 1)n. Genocchi (1852) has further contributed
to the theory.
Among the early writers was Wronski, whose “loi suprˆeme” (1815) was
hardly recognized until Cayley (1873) brought it into prominence. Transon
(1874), Ch. Lagrange (1884), Echols, and Dickstein2 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 Series3 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,
2Bibliotheca Mathematica, 1892-94; historical.
3Historical Summary by Bˆocher, Chap. IX of Byerly’s Fourier’s Series and Spherical Harmonics,
Boston, 1893; Sachse, A., Essai historique sur la repr´esentation d’une fonction . . . par
une s´erie trigonom´etrique. Bulletin des Sciences math´ematiques, Part I, 1880, pp. 43, 83.
ARTICLE 12. INFINITE SERIES. 32
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.
Article 13
THEORY OF
FUNCTIONS.
The Theory of Functions1 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 . Clairaut
(1734) used x, x, 
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, , F, · · ·, f0, 0, F0, · · · 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
1Brill, 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, Encyclopædia 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
. . . Bibliotheca Mathematica, 1891, p. 89.
33
ARTICLE 13. THEORY OF FUNCTIONS. 34
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 Functions2 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 ære 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
2Enneper, 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.
ARTICLE 13. THEORY OF FUNCTIONS. 35
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’s3
(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
3Klein, 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.
ARTICLE 13. THEORY OF FUNCTIONS. 36
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 snx, cnx, dnx. 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 Riemann4 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 }(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
4Klein, F., Riemann and Modern Mathematics, translated by Ziwet, Bulletin of American
Mathematical Society, Vol. I, p. 165.
ARTICLE 13. THEORY OF FUNCTIONS. 37
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 functions5 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).
5Bˆocher, M., A bit of mathematical history, Bulletin of New York Mathematical Society,
Vol. II, p. 107.
Article 14
PROBABILITIES AND
LEAST SQUARES.
The Theory of Probabilities and Errors1 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)2 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 = (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 1 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
1Merriman, 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.
2Enestr¨om, G., Review of Cantor, Bibliotheca Mathematica, 1896, p. 20.
38
ARTICLE 14. PROBABILITIES AND LEAST SQUARES. 39
ignorance of Legendre’s contribution, an Irish-American writer, Adrain, editor of
“The Analyst” (1808), first deduced the law of facility of error, (x) = ce−h2x2 ,
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.3
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.
3Bulletin of New York Mathematical Society, Vol. II, p. 57.
Article 15
ANALYTIC GEOMETRY.
The History of Geometry1 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 ordinis2 (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
1Loria, 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¸cu 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, Encyclopædia
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.
2Ball, W. W. R., On Newton’s classification of cubic curves. Transactions of London
Mathematical Society, 1891, p. 104.
40
ARTICLE 15. ANALYTIC GEOMETRY. 41
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.3
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 coordi-
3For details see Loria, Il passato e il presente, etc.
ARTICLE 15. ANALYTIC GEOMETRY. 42
nates, 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 Geometrie4
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’s5 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
4Loria, G., Notizie storiche sulla Geometria numerativa. Bibliotheca Mathematica, 1888,
pp. 39, 67; 1889, p. 23.
5Biographical Notice in Cayley’s Collected papers, Vol. VIII.
ARTICLE 15. ANALYTIC GEOMETRY. 43
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.
Clebsch6 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 nth 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,
6Klein, Evanston Lectures, Lect. I.
ARTICLE 15. ANALYTIC GEOMETRY. 44
Sch¨onflies, Kummer, and others.7
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.
7Dyck, W., Katalog mathematischer und mathematisch-physikalischer Modelle, M¨unchen,
1892; Deutsche Universit¨atsausstellung, Mathematical Papers of Chicago Congress, p. 49.
Article 16
MODERN GEOMETRY.
Descriptive1, 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.2
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
1Wiener, Chr., Lehrbuch der darstellenden Geometrie, Leipzig, 1884-87; Geschichte der
darstellenden Geometrie, 1884.
2On 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.
45
ARTICLE 16. MODERN GEOMETRY. 46
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
ARTICLE 16. MODERN GEOMETRY. 47
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¸cu 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,
3 the metrical properties being considered as projective relations to a
3Klein, F., Erlangen Programme of 1872, Haskell’s translation, Bulletin of New York Mathematical
Society, Vol. II, p. 215.
ARTICLE 16. MODERN GEOMETRY. 48
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.
Article 17
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.1 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,2 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
1Todhunter, I., History of certain formulas of spherical trigonometry, Philosophical Magazine,
1873.
2Gunther, S., Die Lehre von den gew¨ohnlichen und verallgemeinerten Hyperbelfunktionen,
Halle, 1881; Chrystal, G., Algebra, Vol. II, p. 288.
49
ARTICLE 17. ELEMENTARY GEOMETRY. 50
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. Lemoine3 (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.4
3Smith, 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¸caise
pour l’avancement des sciences.
4Klein, 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).
Article 18
NON-EUCLIDEAN
GEOMETRY.
The Non-Euclidean Geometry1 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’s2
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,
1St¨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 Encyclopædia Britannica, 9th edition; Vasiliev’s address
(German by Engel) also appears in the Abhandlungen zur Geschichte der Mathematik, 1895.
2Fink, E., Kant als Mathematiker, Leipzig, 1889.
51
ARTICLE 18. NON-EUCLIDEAN GEOMETRY. 52
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.3
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 view4 there have contributed to the
subject many of the leading mathematicians of the last quarter of a century,
3For 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.
4Klein. Evanston Lectures. Lect. IX.
ARTICLE 18. NON-EUCLIDEAN GEOMETRY. 53
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).
BIBLIOGRAPHY.
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.
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, IIi`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’oeil sur quelques progr`es des Sciences math´ematiques,
en France, 1830-35. Comptes Rendus, 1835.
Encyclopædia 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).
54
BIBLIOGRAPHY. 55
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 aoˆut 1889 `a l’inauguration de la nouvelle Sorbonne. Bulletin des Sciences
math´ematiques, 1890; also Nature, Vol. XLI, p. 597. (History of nineteenthcentury
mathematics in France.)
Hoefer, F., Histoire des math´ematiques (Paris, 1879).
Isely, L., Essai sur l’histoire des math´ematiques dans la Suisse fran¸caise
(Neuchˆatel, 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 Geschichte
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 XIXe 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.
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
BIBLIOGRAPHY. 56
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:
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’oeuvre math´ematique du XIXe 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).
Among the contributions to the history of applied mathematics in general
may be mentioned the following:
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).
The following should be mentioned as among the latest contributions to the
history of modern mathematics in particular countries:
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 XIXe si`ecle. (Co¨ımbre,
1900).
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:
BIBLIOGRAPHY. 57
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).
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:
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
BIBLIOGRAPHY. 58
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.
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).
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 Bˆocher in his St. Louis address
in 1904,5 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 rˆole of intuition
and of logic in mathematics,6 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 19007 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.8
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
5Bulletin of the American Mathematical Society (N. S.), Vol. XI, p. 115.
6Compte rendu du deuxi`eme congr`es international des math´ematiciens tenu `a Paris, 1900.
Paris, 1902, p. 115.
7G¨ottinger Nachrichten, 1900, p. 253; Archiv der Mathematik und Physik, 1901, pp. 44,
213; Bulletin of the American Mathematical Society, 1902, p. 437.
8Bulletin of the American Mathematical Society (N. S.), Vol. X, p. 443.
59
GENERAL TENDENCIES. 60
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 1894,9 has doubled its membership in
the past decade,10 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 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 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
9It was founded as the New York Mathematical Society six years earlier, in 1888.
10It is now, in 1905, approximately 500.
GENERAL TENDENCIES. 61
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).11
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.
11For 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.


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