*Mathematics and Mathematicians from Abel to Zermelo*in

*Mathematics Magazine*

**26**(3) (1953), 127-146. He begins by giving a survey of 'Who Was Who in Mathematics in 1852'. Hille based his paper on two talks presented to the Mathematics Colloquium of Yale University in May 1952. He states that, "The material is culled from F Cajori's and G Loria's histories of mathematics and F Klein,

*Vorlesungen Über die Entwicklung der Mathematik im 19. Jahrhundert*."

We give below a version of this first part of Hille's paper:

**France.**

We shall try to reconstruct a Who's Who in Mathematics around 1850. We have to keep in mind that there were much fewer mathematicians in those days, fewer periodicals, little personal contact, no mathematical societies, no scientific meetings or congresses. Mathematics flourished in France, Germany, and Great Britain; outside of these countries research mathematicians were few and far between. There was interchange of ideas, however, mathematicians did write to each other, young students travelled abroad to study and so forth.In this survey it is reasonable to start with France where there is a splendid mathematical tradition going back to the seventeenth century. The Revolution, having abolished the old Académie des Sciences founded in 1666, and having liquidated most of its members, found it necessary for military reasons to revive the Academy as the Institut National in 1795 and to found the École Normale Supérieure and the École Polytechnique. The latter became the cradle of French mathematics, a role that it kept for about a century until this function was taken over by the École Normale. Organized and led by the geometer Gaspard Monge (1746-1818), the École Polytechnique gave an intense two year course concentrating on mathematics, descriptive geometry and drawing. Entrance was severely restricted, only 150 students could be accepted each year on the basis of searching examinations. Galois failed twice in this examination, during the first century of its existence J Hadamard held the highest score for admission with 1875 points out of a possible 2000. Only the best brains could secure admission, only hard work could carry the student through the intense training and the gruelling final examinations. Most French mathematicians of the nineteenth century belonged to this school in one capacity or another as student, instructor, professor or examiner.

The time of the revolution and the first empire saw the flourishing of mathematics in France. J L Lagrange (1736-1813) spent the last twenty years of his wandering life in France, men like J Fourier (1768-1830), P S Laplace (1749-1827), A M Legendre (1752-1833) and S D Poisson (1781-1840) brought French mathematics to new heights. A reaction is noticeable around 1830 and lasted for almost fifty years. The old heroes were dying off and the replacements are perhaps a shade less impressive.

The big name in Paris in 1850 was Augustin-Louis Cauchy (1789-1857). As an ardent royalist Cauchy had given up his professorship at the École Polytechnique after the July revolution in 1830 and had gone in exile with the Bourbons; they had trouble with loyalty oaths in those days too. He returned to Paris in 1838 and taught at a Jesuit college. In 1848 the February revolution opened the way to a professorship at the Sorbonne without any oath and Napoleon III did not interfere with him when the constitution was changed again in 1852. Cauchy was above all an analyst, the first rigorous analyst, the founder of function theory, of the theory of ordinary and partial differential equations, who also made important contributions to elasticity and optics as well as to algebra.

At the Collège de France in Paris we would have found Joseph Liouville (1809-1882), the man who started the mathematical theory of boundary value problems for linear second order differential equations, who produced the first integral equation and the first resolvent, but also the founder of the theory of transcendental numbers. He was the editor of the Journal de Mathématiques Pures et Appliquées, started in 1835 and still known as the Journal de Liouville. In connection with Liouville we often think of Jacques-Charles-François Sturm (1805-1855); he was Poisson's successor as professor of mechanics at the Sorbonne in 1850. Lesser lights, but famous in their days, were August-Albert Briot (1817-1882) and Jean-Claude Bouquet (1819-1885), great friends who collaborated in the theory of differential equations and elliptic functions. A young man by the name of Charles Hermite (1822-1901) was struggling for recognition, he still had to wait until 1869 for a professorship. Those mentioned so far were all analysts, but there were also geometers around: Jean-Victor Poncelet (1788-1867), the father of projective geometry, Charles Dupin (1784-1873) and Michel Chasles (1793-1880), were active. The algebraic geometer Chasles, who created the enumerative methods of geometry, devoted almost twenty-five years of his life to banking in his home town Chartres; in 1846 the Sorbonne created a professorship in higher geometry for him. The private fortune that he had amassed before returning to mathematics led him into collecting autographs and he was swindled into buying numerous forgeries. His most famous acquisition was a letter supposedly written by Mary Magdalene from Marseille to Saint Peter in Rome! Even Chasles ultimately had to admit that he had been swindled. Among French mathematicians we find few algebraists, the analysts attended to the theory of equations, and group theory was in its infancy. The publication of the work of Evariste Galois (1811-1832) did not take place until 1846.

A characteristic feature of French mathematics, then as well as now, is the strong centralization. Paris, that is, the Sorbonne, Collège de France, École Polytechnique, École Normale Supérieure, and the many technical schools, has most of the desirable positions while the provincial universities vary in importance, but cannot hold their own against Paris. There were numerous facilities for publication in France. Joseph-Diaz Gergonne (1771-1859), who competed with Poncelet in discovering the principle of duality, published the first mathematical journal in the world, the

*Annales des mathématiques pures et appliquées*, in Nimes during 1810-31 where he was then professor at the local lycée. Liouville started his journal in 1835. Notes could be published quickly in the*Comptes Rendus de l'Académie des Sciences*at Paris which has appeared weekly since August 1835. The restriction to short notes (originally four pages, now two) was due to Cauchy flooding the Academy with his publications. The École Polytechnique and later also the École Normale had their own journals.**Germany.**

Turning now to Germany we find a totally different picture. German mathematics is unimportant during the eighteenth century and is essentially a one-man-show during the first quarter of the nineteenth. From then on German mathematics is in a steady upward march until the time of the first world war and even during the empire this movement is not centralized as in France but is attached to a number of local centres. The Germany of 1850 was a geographical and cultural unit but not a political one. It was split into autonomous kingdoms, grandduchies, duchies, principalities, and what have you, and this kaleidoscopic picture reflected itself in a large number of independent and thriving universities. Apparently the many political boundaries interfered very little with the students and the professors who moved freely around. Prussia had the largest number of universities of all German lands: Berlin was the largest and was outstanding in mathematics but there were also important schools in Bonn and Königsberg. In the kingdom of Hanover we find the Georgia Augusta University at Göttingen. Incidentally, it was founded two hundred years ago by the then elector of Hanover who as King George II of Great Britain and Ireland advanced the cause of learning in the then colonies by the founding of the College of New Jersey (Princeton), King's College (Columbia), and Queen's College (Rutgers). Leipzig was the intellectual centre of Saxony, in Bavaria Erlangen, in Baden Freiburg and Heidelberg, and in the Thuringian maze of principalities Jena kept the torch burning.The eighteenth century was the time when the Swiss Leonard Euler and the Frenchman Lagrange, born in Italy, lived and worked in Berlin and filled the memoirs of the Prussian Academy, but of native German mathematicians there were very few. The nineteenth century starts out with Carl Friedrich Gauss (1777-1855), a giant in mathematics, astronomy, geodesy and magnetism, who got his doctor's degree in Helmstadt in 1799 and from 1807 on spent his life in Göttingen as director of the astronomical observatory. His publications in number theory, algebra, and differential geometry were basic, but many later discoveries in function theory and geometry had been anticipated by Gauss who published only sparingly.

Though Gauss had only few direct pupils, he had started something in Germany which was of lasting importance for mathematics. This development gathered momentum in the eighteen twenties and led to a flourishing of German mathematics which lasted until the days of Hitler. For the next phase of the development we have to turn to Berlin and Königsberg. An outward sign of the changing times was the founding in Berlin in 1826 of the

*Journal für die reine und angewandte Mathematik*by August Leopold Crelle (1780-1855), usually known as Crelle's Journal (1855-1880 Borchardt's Journal after the then editor). Crelle started out with a scoop: five memoirs by the young Norwegian mathematician Niels Henrik Abel (1802-1829) but there were also papers by Carl Gustav Jacob Jacobi (1804-1851) and by Jacob Steiner (1796-1863). In volume 3 appear the names of Peter Gustav Lejeune Dirichlet (1805-1859), August Ferdinand Möbius (1790-1868), and Julius Plucker (1801-1868). Crelle's Journal soon became known as the*Journal für die reine, und angewandte Mathematik*, much to the embarrassment of its editor who was much more prominent as an engineer than as a mathematician. In addition to Crelle, the German mathematicians had at their disposal a large number of academy publications. The*Mathematische Annalen*did not start until 1868.Jacobi got his degree in Berlin but came to Königsberg in 1826 where he did his fundamental work on elliptic functions in competition with Abel. He became ordinary professor in 1831, but retired owing to ill health in 1842, whereupon he moved back to Berlin where he got a research chair without teaching duties. Jacobi is the founder of the Königsberg school to which belonged men like L O Hesse (1811-1874) and R F A Clebsch (1833-1872), followed in later years by A Hurwitz (1859-1919), D Hilbert (1862-1942) and H Minkowski (1864-1909). In more recent years men like K Knopp (1882- ) and G Szego (1895- ) held professorships there. The last mathematician of note to get his training in Königsberg seems to have been Th Kaluza now in Göttingen. Perhaps there will also be a Kaliningrad mathematical school. Who knows?

Dirichlet was born in the Rhineland as the son of French emigrants. He spent the years 1822-27 as teacher in a private family in Paris where he came under the influence of the great French analysts of the period. He also read and reread Gauss's Disquisitiones Arithmeticae. Both facts are strongly reflected in his mathematical life. After a short time in Breslau he came to Berlin in 1829 as Privatdozent and became ordinary professor ten years later, the first of the great masters of the Berlin school. It is typical for German conditions, however, that he accepted a call to Göttingen to become Gauss's successor in 1855. He died four years later. Dirichlet worked in number theory, in particular analytic number theory (Dirichlet's series commemorates this fact), functions of a real variable, Fourier series, potential theory etc. Dirichlet was an excellent expositor and exercised a strong influence on the younger men who came to Berlin such as Leopold Kronecker (1823-1891), F G Eisenstein (1823-1852), Richard Dedekind (1831-1916), and Bernhard Riemann (1821-1866). At Berlin we also find the brilliant Swiss born geometer Jacob Steiner who, however, never got beyond the extraordinary professorship to which he was appointed in 1834. Kronecker, a man of wealth, came to Berlin in 1855 and became attached to the University in 1861. In 1856 Ernst Eduard Kummer (1810-1893) was called to Berlin as the successor of Dirichlet. He was the father of ideal theory and also a geometer of note. The same year Karl Weierstrass (1815-1897) was called to Berlin: he had twelve hours a week at the Gewerbeakademie (a college for trade and commerce) and an extraordinariat at the university. The ordinary professorship did not come until 1864.

Let us now turn to Göttingen and the year 1850. Gauss is still active. A young man by name of Riemann has just returned from a three year stay in Berlin and is now working on a dissertation which gave him the doctor's degree in 1851 and ultimate fame and glory. At this stage Riemann is scarcely influenced by Gauss, but he worked with Wilhelm Weber, a famous physicist and one of the discoverers of the electric telegraph. From Dirichlet and Weber he picked up an interest in mathematical physics which influenced both his research and his lectures, later published in book form as

*Differentialgleichungen der Physik*, first edited by Hattendorf, from the fourth edition by Heinrich Weber, and the seventh edition of 1925-26 by Philipp Frank and Richard von Mises. Riemann qualified for the venia legendi in 1854, that is, for the right to give lectures as a Privatdozent. This required writing another dissertation (Habilitationsschrift) and giving a lecture (Habilitationsvortrag) before the faculty on one out of three pre-assigned topics. The former was his paper on trigonometric series, the latter dealt with the basic hypotheses of geometry. Riemann became ordinary dealt with the basic hypotheses of geometry. Riemann became ordinary professor in 1859. With Gauss, Dirichlet and Riemann as fathers of the Göttingen mathematical school a star was born which has never set.There are few names to be added from the other German universities. Möbius (with the strip!) was professor of astronomy in Leipzig. He was a pupil of Gauss who turned him into an astronomer, but most of his production was in mathematics: number theory, combinatorics, and the barycentric calculus, his magnum opus of 1827, while the non-orientable manifolds was work done in his old age appearing in 1858. In Bonn we find another famous geometer, Julius Plücker, professor of physics and mathematics from 1838 until his death in 1868. As a physicist he was an experimental one and did pioneering work in spectral analysis which took up most of his time. But his contributions to geometry such as the Plücker coordinates in line geometry, a subject created by him, and the Plücker formulas in algebraic geometry, are fundamental. Plücker had several famous pupils, the physicist Hittorf was outstanding in spectral analysis, the mathematician Felix Klein (1849-1925) published his collected works. Finally we have to mention Christian von Staudt (1798-1867), professor in Erlangen since 1835 until his death in 1867. Among his main achievements are his making projective geometry independent of metric considerations and the introduction of imaginary elements in geometry. His ideas ripened slowly, his Geometrie der Lage appeared in 1847, followed by three further Beiträge in 1856, 1857 and 1860.

**Great Britain and Ireland.**

Due to the priority fights between Newton's pupils and those of Leibnitz, mathematics in the British isles had enjoyed a splendid but unhealthy isolation for over a century, the after-effects of which were not overcome until after 1900. One of the consequences of this was an almost complete lack of feeling for analysis per se. What there was of analysis went into mathematical physics. As an example we could take George Green (1793-1841) who applied analysis to electricity and magnetism. Drink carried him off to an early death, but his name lives (Green's theorem, Green's function etc.). George Gabriel Stokes (1819-1903), another Cambridge don, made fundamental contributions to optics and hydrodynamics. There is a theorem of Stokes in the Calculus, there is also a phenomenon of Stokes in the theory of differential equations. To the same tradition belongs the work of the famous physicists James Clerk Maxwell (1831-1879) and William Thomson, Lord Kelvin (1824-1907). Sir William Rowan Hamilton (1805-1865) was professor of astronomy at Trinity College in Dublin, but is more famous for his work in optics and mechanics (the Hamiltonian equations) and as creator of vector analysis and the theory of quaternions.But there is also another direction in British mathematics which breaks through around 1850, represented by the three names Arthur Cayley (1821-1895), George Salmon (1819-1904), and James Joseph Sylvester (1814-1897). They were personal friends. Cayley and Sylvester read for the bar together in London. Cayley and Salmon had a joint paper on the twenty-seven lines on the cubic surface in 1849 and Cayley on the twenty-seven lines on the cubic surface in 1849 and Cayley helped Salmon to revise his Higher Plane Curves. Salmon stayed all his life at Trinity, he was a fellow from 1840 on, became professor of divinity in 1866, and provost in 1888. He is best known for his books on geometry and algebra which were much read and translated into French and German, but he also wrote books an theology.

Cayley was one of the most prolific mathematicians of all time: starting production in 1841 he produced a total of 887 papers. And this intense activity was kept up while he was reading for the bar and during the fourteen years he practiced law - he was a conveyancer - as a matter of fact, most of his basic ideas stem from this period. In 1863 he retired from practice and accepted the Sadlerian professorship of pure mathematics in Cambridge which he held until his death. Cayley's nine memoirs on quantics (= forms) are famous, he started the theory of matrices and, together with Sylvester, the theory of invariants. The very name of invariant is due to Sylvester who also introduced covariant, contragrediant, discriminant, etc. Sylvester used to refer to himself as the new Adam since he had named so many things. Sylvester held a multitude of positions, including a brief stay at the University of Virginia in the early forties. He had been an actuary, he was called to the bar in 1850, from 1855 to 1870 he taught at the Military Academy at Woolwich, 1876-1883 he spent at The Johns Hopkins University where he started the

*American Journal of Mathematics*as well as the high traditions of Johns Hopkins in mathematics. At seventy he accepted a call to Oxford as successor of the number theory man H J S Smith (1826-1883) as Savilian professor of geometry.If to this list we add the logicians Augustus de Morgan (1806-1871) and George Boole (1815-1864) we have a fairly good picture of British mathematics at the middle of the last century. The

*Cambridge Journal of Mathematics*started in 1837, after four volumes it became the*Cambridge and Dublin Mathematical Journal*, later (in 1855) the*Quarterly Journal of Pure and Applied Mathematics*. The London Mathematical Society, the first of its kind, was organized in 1865.**Other countries.**

The compiler of our Who's Who would have found a single research mathematician in this country, namely Benjamin Peirce (1809-1880), professor at Harvard since 1833, whose work on linear associative algebra was really fundamental. In the seventies things are beginning to stir elsewhere also; we have already referred to Sylvester's stay at Johns Hopkins. In 1871 Josiah Willard Gibbs (1839- 1903) became professor of mathematical physics at Yale; his main work on thermodynamics, equilibra and the phase rule date from 1873-78. His work on vector analysis was of some importance and one or two of his students made mathematical history. The real awakening did not come until the eighteen nineties, however.Back in Europe again, let us look briefly at the situation in Italy where the mathematical tradition goes back to the year 1200. The beginning of the nineteenth century is meagre, however. After the death of Paolo Ruffini (1765-1822), there is not much mathematics in Italy, but the awakening in mathematics came with the general national awakening and shortly before the unification. As fathers of modern Italian mathematics it is customary to acknowledge Francesco Brioschi (1824-1897), Enrico Betti (1823-1892), Felice Casorati (1835-1890) and Luigi Cremona (1830-1903). Brioschi became professor of applied mathematics in Pavia in 1852; in 1858 he made a journey to France and Germany together with Betti and Casorati and it is from this journey that the revival is dated. Brioschi organized the technological institute of Milan from 1862 on, Crenmona played a similar part in Rome, and Betti became the director of the Scuola Normale Superiore of Pisa. To these names should be added that of Eugenio Beltrami (1835-1900). Most of the Italian mathematicians turned towards algebraic geometry.

Euler spent over twenty years in Russia but, beyond filling the memoirs of the Saint Petersburg Academy (for which purpose he had been imported), there is little trace of his activities. Around 1850 we would have found at least three prominent mathematicians in Russia: Victor Yakovlevich Bunyakovskii (1804-1889), Pafnuty Lvovich Chebyshev (1821-1894), both in St Petersburg (= Petrograd = Leningrad), and Nikolai Ivanovich Lobachevsky (1792-1856) in Kazan. Bunyakovskii's inequality was the customary name in Russia for what we know as Schwarz's inequality (usually misspelt Schwartz in this country). Chebyshev started the investigation of extremal problems in function theory and problems of best approximation. He also worked with prime numbers. Lobachevsky had been pondering over the parallel postulate since 1815; around 1826 he arrived at the alternate postulate of two parallels and his investigations were published in seven memoirs 1829-1856.

Turning to Hungary we find the other discoverer of non-euclidean geometry Janos (= John or Johann) Bolyai (1802-1860) whose discovery was published in an appendix to a mathematical memoir of his father's, Farkas (= Wolfgang) Bolyai (1775-1856), in 1832-1835. It is well known that Gauss had been thinking along similar lines; it is perhaps more than a coincidence that both Wolfgang Bolyai and Lobachevsky's teacher Bartels in Kazan were personal friends of Gauss. There is little activity elsewhere in Europe. Switzerland had over- exerted itself during the eighteenth century with all the Bernoullis and with Euler and was recovering from the strain. Scandinavia had produced Abel and new talent was in the making both in Norway and in Sweden, but there is nothing to report until after 1860 to 1870.