4-52. Sophus Lie
Marius Sophus Lie (1842–1899) was born in Nordfjordeide, Norway, the son of Johan and Mette (Stabell) Lie. Johan was a former headmaster, who became a vicar in 1835. Tragedy struck the family in 1852 with the death of Mette, followed by that of Sophus’ younger brother Ludvig, two years later. In the fall of 1859 Lie matriculated at the Royal Fredrik’s University of Christiania (now Oslo), where he studied scientific subjects. Lie followed lectures by the mathematician Ludvig Sylow, which touched on the works of Abel and Galois; he also heard lectures by the mathematician and politician Ole Jacob Broch and the physicist C.A. Bjerknes. Upon graduation Lie applied unsuccessfully for an assistantship at the Christiania Observatory, and a lectureship at the university, while continuing to read mathematics on his own and attend lectures, including one held in July, 1868 by Zeuthen on the discoveries of Möbius, Plücker, and his former teacher Michel Chasles. The next fall, having succeeded in publishing a paper in Crelle’s Journal, Lie went to Berlin, where he befriended a former student of Plücker, Felix Klein, and made the acquaintance of Kummer, Kronecker and Weierstrass.
In early 1870, Lie left Berlin for Paris, where he met Gaston Darboux and Camille Jordan. Klein joined him in the summer, and the two men shared their ideas, which they soon published together. In the meantime, war broke out, and Lie was imprisoned in a French jail on suspicion of espionage. His condition soon became a cause célèbre in his native Norway, where the press regaled readers with items about the Norwegian scientist jailed as a German spy. After a month spent working out his new ideas and reading novels by Walter Scott, Lie was released, thanks to the intervention of Darboux, who was able to convince the authorities that Lie was not, in fact, a German spy, but a Norwegian mathematician.
Lie returned to Christiania a celebrity of sorts, and took up a teaching position in 1871 at the high school he had frequented. He defended his doctoral dissertation “on a class of geometric transformations” in June. The next year, a lectureship was created for Lie at the university. With Sylow, Lie took on the edition of Abel’s papers, and in 1876, he founded the journal Archiv for Mathematik og Naturvidenskab, where he published many of his own works. These writings did reach a large readership, and Lie found no promising students in Christiania. The latter situation improved when Lie succeeded Klein as professor of mathematics in Leipzig in 1886. One of his students from Leipzig, Georg Scheffers co-authored textbooks on differential equations, transformation groups, and contact transformations (Lie and Scheffers, 1891, 1893, 1896). With the help of Adolph Mayer’s doctoral student Friedrich Engel, who was also the son of a pastor, Lie published his theory of transformation groups (Lie and Engel, 1888, 1890, 1893).
In November 1889, Lie was admitted to a clinic in Hanover, where he stayed for the next seven months. He appears to have suffered from depression. Although his position in Leipzig was quite favorable, Lie made it known that he missed his home country. By the 1890s, Lie’s international renown was such that the a chair in Christiania was created for him in 1894. Lie did not resign his chair in Leipzig until 1898, when he moved to Christiania with his family and doctoral student E. O. Lovett. He was then suffering from an illness later diagnosed as pernicious anemia, for which there was no treatment at the time, and to which he succombed on 18 February, 1899. ^{1}^{1}endnote: ^{1} On Lie’s life and career, see Freudenthal (1973); Stubhaug (2002).
Lie and Poincaré
Lie and Poincaré first met in Paris in November, 1882. Lie gave a talk to the Mathematical Society of France on the third, and met with mathematicians at the Paris Academy of Sciences. Poincaré invited Lie for dinner at his flat on the eighteenth (§ 4-52-1). There ensued one of the more compelling mathematical exchanges in Poincaré’s Nachlass. In all, four letters from Poincaré to Lie have been preserved, and seven from Lie to Poincaré; all are transcribed here.
From the outset, Lie considered Poincaré as a skilled ally in his quest to promote the theory of transformation groups, along with Poincaré’s colleagues Émile Picard, Gaston Darboux, and Georges Halphen. In fact, in the 1880s Lie’s theory was appreciated more in Paris than it was in Berlin, as David Rowe (1989) has observed. The mathematicians in the geometry section of the Paris Academy were well-disposed toward Lie, and when the position of Correspondent became available, Lie was their favorite. It fell to Poincaré to write a report on Lie’s accomplishments, and to better prepare this, Poincaré wrote to Lie for help (§ 4-52-2). Lie’s reply has survived, but not the “out of scale” summary he wrote in longhand for the occasion. The report Poincaré delivered appears not to have relied on Lie’s response; it observes, in essence, that Lie’s work on continuous transformation groups was of fundamental importance to the theory of both partial and ordinary differential equations (§ 4-52-12). The concluding paragraphs of Poincaré’s report touch briefly on subjects less directly related to Lie’s theory of transformation groups, including that of minimal surfaces, with which Poincaré would later engage, and discuss with Lie (§ 4-52-11).^{2}^{2}endnote: ^{2} Poincaré (1895b, a, 1901b). Another such topic, which had already engaged Poincaré’s attention, was that of the foundations of geometry. Following Hermann von Helmholtz’s famous essay “On the facts underlying geometry”, which upheld the principle of free mobility of solids, Lie used his theory of continuous groups to give a more rigorous expression to Helmholtz’s axioms.^{3}^{3}endnote: ^{3} Helmholtz (1868); Lie (1890a, b). Lie recalled the latter result for Poincaré’s benefit by letter in March of 1892 (§ 4-52-7).^{4}^{4}endnote: ^{4} Poincaré was surely aware of Lie’s theorem, as Lie related his theory to that of H. von Helmholtz in a note presented to the Paris Academy on 29 February (Lie, 1892).
When writing his report on Lie’s candidacy, Poincaré probably knew that the geometry section’s ranking of the candidates would provoke no debate in the Academy, and as it happened, the election, which took place on 7 June, saw Lie easily outdistance his rivals Luigi Cremona and H.A. Schwarz (Comptes rendus 114, 1329).
Lie’s first visit to Paris after becoming Correspondent of the Academy of Sciences took place from 1–20 April, 1893, at the invitation of Darboux and Tannery.^{5}^{5}endnote: ^{5} Stubhaug (2002, 374). On this occasion, Lie met with the Norwegian physicist Kristian Birkeland, and his former student Ernest Vessiot. He also made the acquaintance of Élie Cartan, who was then writing his doctoral thesis on the structure of finite and continuous groups (Cartan, 1894).^{6}^{6}endnote: ^{6} Poincaré wrote a favorable report on Cartan’s candidacy for a chair at the Sorbonne in 1912, in which he commented that “group theory, in a manner of speaking, is all of mathematics, stripped of its substance and reduced to a pure form” (Poincaré, 1921). The chair of differential and integral calculus went to Cartan, who dominated the field of differential geometry for decades (Cogliati, 2018). While in Paris, Lie also met with Poincaré over dinner at Camille Jordan’s flat.^{7}^{7}endnote: ^{7} Stubhaug (2002, 390).
Poincaré may have been inspired by Lie to work on minimal surfaces, a subject they spoke about during Poincaré’s visit to Leipzig in June, 1895. As Poincaré recalled their conversation in his letter to Lie of early October, 1897 (§ 4-52-11), Lie spoke of a means of constructing all surfaces of double translation. At the time, Poincaré also sought a solution, not only for surfaces, but for manifolds of double translation, using an original method. When Lie read Poincaré’s memoir, he felt that the results were but special cases covered by his “general theorems”.^{8}^{8}endnote: ^{8} Poincaré (1895b, a); Lie and Scheffers (1896, 143). Lie was known for unbalanced attacks since 1893, when he published harsh criticism of his old friend Felix Klein for having neglected his priority (Tobies, 2019, 312). Lie continued to seek a solution for the general case, as did Poincaré and others, but a full solution to Lie’s problem remained out of reach until the Lie-Wirtinger theorem of 1938.^{9}^{9}endnote: ^{9} Lie (1897a); Poincaré (1901b); Wirtinger (1938). For details of the history of surfaces and manifolds of double translation, see Little (1983); Chern (1994).
In 1897, Lie again suggested that Poincaré had neglected his results, this time, concerning Poincaré’s theory of integral invariants (Poincaré, 1890). As Lie understood it, Poincaré’s theory was a corollary of his own theory of differential invariants; likewise for a recent theorem by Gabriel Koenigs, later known as the Lie-Koenigs theorem, presented to the Paris Academy by Poincaré in 1895.^{10}^{10}endnote: ^{10} Koenigs (1895). Poincaré (1899a) developed the theory of integral invariants in without reference to group theory. On the Lie-Koenigs theorem (also known as the Lie-Königs theorem), see Whittaker (1917, 275). The latter theorem later found an interpretation in relativistic mechanics due to Hill and Kerner (1966). For Lie, there was only one mathematician – besides himself – to have “correctly grasped the state of affairs” with respect to integral invariants and his theory of groups: Élie Cartan.^{11}^{11}endnote: ^{11} (Lie, 1897b, 342). Cartan later explained that the essential difference between his approach to integral invariants and that of Poincaré as follows: instead of considering a multiple integral that is invariant with respect to a system of differential equations, one can consider its invariance with respect to a transformation group (Cartan, 1922, ix). Cartan’s work on the calculus of differential forms in this context is discussed by Hawkins (2000, 282).
Lie’s death in 1899 marks the beginning of Poincaré’s engagement with continuous groups. In his contribution to the Stokes Festschift of 1899, Poincaré proved a major theorem in what was later known as Lie algebra, referred to as the Poincaré-Birkhoff-Witt theorem.^{12}^{12}endnote: ^{12} Gray (2013, 492); Poincaré (1899b, c). Subsequent papers on continuous groups include Poincaré (1901a, 1908).
Along with the election of Lie as Correspondent of the geometry section of the Paris Academy of Sciences, a mark of the esteem in which Lie’s contributions were held in the 1890s is the invitation extended to honor Evariste Galois on the occasion of the centennial anniversary of the foundation of the École normale supérieure. Lie’s concluded his contribution with the prediction that transformation groups – that were then being called “Lie groups” – would find application in the natural world:
“What do natural phenomena represent, if not a succession of infinitesimal transformations, the invariants of which are the laws of the universe?”^{13}^{13}endnote: ^{13} “Que nous représentent en effet les phénomènes naturels, si ce n’est une succession de transformations infinitésimales, dont les lois de l’univers sont les invariants ?” (Lie, 1895, 489)
At the time of the centennial celebration in 1895, such an idea was somewhat far-fetched, although the related idea that geometric properties remain invariant under the transformations of a certain group gained significant recognition in the 1880s. The latter, geometric idea was expressed in Klein’s “Erlangen Program” of 1872, but had yet to find a significant echo in mathematical physics, at least beyond crystallography.^{14}^{14}endnote: ^{14} The Göttingen Privatdozent Arthur Schönflies employed group theory in the classification of crystallographic structures; see Schönflies (1891) and Scholz (1989a, b). Lie’s remark may have been inspired by the geometrization of dynamics, reflected in the work of Rudolf Lipschitz, Gaston Darboux, Wilhelm Killing, Paul Stäckel, and Heinrich Hertz; for an overview, see Lützen (1995). Traces of Klein’s program may be seen in modern physics, thanks to contributions from Poincaré, Minkowski, Weyl, Cartan and others, as observed by Goenner (2015).
A decade later, however, in May, 1905, Poincaré realized that the laws of the universe were invariant with respect to certain coordinate transformations, which formed a 10-parameter Lie group, known later as the inhomogeneous Lorentz group, or the “Poincaré” group.^{15}^{15}endnote: ^{15} See Poincaré to Lorentz, ca. May 1905 (§ 2-38-4). Newton’s law of universal gravitation was clearly not invariant with respect to these transformations, so Poincaré, following a method outlined by Lie and Scheffers in 1893, readily identified the principal invariants of the Lorentz group, from which he devised a Lorentz-invariant law of gravitation, expressed as a four-vector.^{16}^{16}endnote: ^{16} Walter (2007). This new mathematical view of the universe inspired – along with Albert Einstein’s theory of relativity – Hermann Minkowski’s discovery of spacetime.^{17}^{17}endnote: ^{17} Minkowski (1908). Unlike Einstein and Minkowski, Poincaré recommended that spacetime be considered not an empirical fact, but a convention, freely imposed by scientists in their search for understanding of the natural world.^{18}^{18}endnote: ^{18} In other words, Poincaré suggested Galilei spacetime as an alternative to Minkowski spacetime (Walter, 2009).
Time-stamp: " 6.01.2023 19:22"
Notes
- ^{1} On Lie’s life and career, see Freudenthal (1973); Stubhaug (2002).
- ^{2} Poincaré (1895b, a, 1901b).
- ^{3} Helmholtz (1868); Lie (1890a, b).
- ^{4} Poincaré was surely aware of Lie’s theorem, as Lie related his theory to that of H. von Helmholtz in a note presented to the Paris Academy on 29 February (Lie, 1892).
- ^{5} Stubhaug (2002, 374).
- ^{6} Poincaré wrote a favorable report on Cartan’s candidacy for a chair at the Sorbonne in 1912, in which he commented that “group theory, in a manner of speaking, is all of mathematics, stripped of its substance and reduced to a pure form” (Poincaré, 1921). The chair of differential and integral calculus went to Cartan, who dominated the field of differential geometry for decades (Cogliati, 2018).
- ^{7} Stubhaug (2002, 390).
- ^{8} Poincaré (1895b, a); Lie and Scheffers (1896, 143). Lie was known for unbalanced attacks since 1893, when he published harsh criticism of his old friend Felix Klein for having neglected his priority (Tobies, 2019, 312).
- ^{9} Lie (1897a); Poincaré (1901b); Wirtinger (1938). For details of the history of surfaces and manifolds of double translation, see Little (1983); Chern (1994).
- ^{10} Koenigs (1895). Poincaré (1899a) developed the theory of integral invariants in without reference to group theory. On the Lie-Koenigs theorem (also known as the Lie-Königs theorem), see Whittaker (1917, 275). The latter theorem later found an interpretation in relativistic mechanics due to Hill and Kerner (1966).
- ^{11} (Lie, 1897b, 342). Cartan later explained that the essential difference between his approach to integral invariants and that of Poincaré as follows: instead of considering a multiple integral that is invariant with respect to a system of differential equations, one can consider its invariance with respect to a transformation group (Cartan, 1922, ix). Cartan’s work on the calculus of differential forms in this context is discussed by Hawkins (2000, 282).
- ^{12} Gray (2013, 492); Poincaré (1899b, c). Subsequent papers on continuous groups include Poincaré (1901a, 1908).
- ^{13} “Que nous représentent en effet les phénomènes naturels, si ce n’est une succession de transformations infinitésimales, dont les lois de l’univers sont les invariants ?” (Lie, 1895, 489)
- ^{14} The Göttingen Privatdozent Arthur Schönflies employed group theory in the classification of crystallographic structures; see Schönflies (1891) and Scholz (1989a, b). Lie’s remark may have been inspired by the geometrization of dynamics, reflected in the work of Rudolf Lipschitz, Gaston Darboux, Wilhelm Killing, Paul Stäckel, and Heinrich Hertz; for an overview, see Lützen (1995). Traces of Klein’s program may be seen in modern physics, thanks to contributions from Poincaré, Minkowski, Weyl, Cartan and others, as observed by Goenner (2015).
- ^{15} See Poincaré to Lorentz, ca. May 1905 (§ 2-38-4).
- ^{16} Walter (2007).
- ^{17} Minkowski (1908).
- ^{18} In other words, Poincaré suggested Galilei spacetime as an alternative to Minkowski spacetime (Walter, 2009).
References
- Sur la structure des groupes de transformations finis et continus. Ph.D. Thesis, Faculté des sciences de Paris, Paris. Cited by: Lie and Poincaré.
- Leçons sur les invariants intégraux. Hermann, Paris. Cited by: endnote 11.
- Sophus Lie and differential geometry. See Proceedings of the Sophus Lie Memorial Conference, Oslo 1992, Laudal and Jahren, pp. 129–152. Cited by: endnote 9.
- Writing Small Omegas: Elie Cartan’s Contributions to the Theory of Continuous Groups 1894–1926. Academic Press, London. Cited by: endnote 6.
- Lie, Marius Sophus. See Dictionary of Scientific Biography, Volume 8: Jonathon Homer Lane–Pierre Joseph Macquer, Gillispie, pp. 323–327. Cited by: endnote 1.
- Dictionary of Scientific Biography, Volume 8: Jonathon Homer Lane–Pierre Joseph Macquer. Charles Scribner’s Sons, New York. Cited by: H. Freudenthal (1973).
- Klein’s ‘Erlanger Programme’: do traces of it exist in physical theories?. See Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics, Ji and Papadopoulos, pp. 77–90. Cited by: endnote 14.
- Henri Poincaré: A Scientific Biography. Princeton University Press, Princeton. link1 Cited by: endnote 12.
- Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics. Springer, Berlin. Cited by: endnote 11.
- The Significance of the Hypothetical in the Natural Sciences. Walter de Gruyter, Berlin. Cited by: S. A. Walter (2009).
- Über die Thatsachen, die der Geometrie zum Grunde liegen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen 15, pp. 193–221. Cited by: endnote 3.
- Unique canonical representation of the inhomogeneous Lorentz group in relativistic particle dynamics. Physical Review Letters 17, pp. 1156–1158. link1, link2 Cited by: endnote 10.
- Sophus Lie and Felix Klein: The Erlangen Program and Its Impact in Mathematics and Physics. European Mathematical Society, Zürich. Cited by: H. Goenner (2015).
- Application des invariants intégraux à la réduction au type canonique d’un système quelconque d’équations différentielles. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 121 (24), pp. 875–878. link1 Cited by: endnote 10.
- Proceedings of the Sophus Lie Memorial Conference, Oslo 1992. Scandanavian University Press, Oslo. Cited by: S. Chern (1994).
- Theorie der Transformationsgruppen, Volume 1. Teubner, Leipzig. link1 Cited by: 4-52. Sophus Lie.
- Theorie der Transformationsgruppen, Volume 2. Teubner, Leipzig. link1 Cited by: 4-52. Sophus Lie.
- Theorie der Transformationsgruppen, Volume 3. Teubner, Leipzig. link1 Cited by: 4-52. Sophus Lie.
- Vorlesungen über Differentialgleuchungen mit bekannten infinitesimalen Transformationen. Teubner, Leipzig. link1 Cited by: 4-52. Sophus Lie.
- Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen. Teubner, Leipzig. link1 Cited by: 4-52. Sophus Lie.
- Geometrie der Berührungstransformationen. Teubner, Leipzig. Cited by: 4-52. Sophus Lie, endnote 8.
- Über die Grundlagen der Geometrie (I). Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physische Klasse 42, pp. 284–321. link1 Cited by: endnote 3.
- Über die Grundlagen der Geometrie (II). Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physische Classe 42, pp. 355–418. link1 Cited by: endnote 3.
- Sur les fondements de la géométrie. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 114 (9), pp. 461–463. link1 Cited by: endnote 4.
- Influence de Galois sur le développement des mathématiques. See Le centenaire de l’École normale 1795–1895, Perrot, pp. 481–489. link1 Cited by: endnote 13.
- Das Abel’sche Theorem und die Translationsmannigfaltigkeiten. Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physische Klasse 49, pp. 181–248. link1 Cited by: endnote 9.
- Die Theorie der Integralinvarianten ist ein Corallar der Theorie der Differentialinvarianten. Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physische Klasse 49, pp. 342–357. link1 Cited by: endnote 11.
- Translation manifolds and the converse of Abel’s theorem. Compositio Mathematica 40 (2), pp. 147–171. link1 Cited by: endnote 9.
- Interactions between mechanics and differential geometry in the 19th century. Archive for History of Exact Sciences 49, pp. 1–72. Cited by: endnote 14.
- Die Grundgleichungen für die electromagnetischen Vorgänge in bewegten Körpern. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pp. 53–111. link1 Cited by: endnote 17.
- Le centenaire de l’École normale 1795–1895. Hachette, Paris. link1 Cited by: S. Lie (1895).
- Sur le problème des trois corps et les équations de la dynamique. Acta mathematica 13, pp. 1–270. link1 Cited by: Lie and Poincaré.
- Remarques diverses sur les fonctions abéliennes. Journal de mathématiques pures et appliquées 1 (3), pp. 219–314. link1 Cited by: endnote 2, endnote 8.
- Sur les fonctions abéliennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 120 (5), pp. 239–243. link1 Cited by: endnote 2, endnote 8.
- Les méthodes nouvelles de la mécanique céleste, Volume 3. Gauthier-Villars, Paris. link1 Cited by: endnote 10.
- Sur les groupes continus. Transactions of the Cambridge Philosophical Society 18, pp. 220–255. link1 Cited by: endnote 12.
- Sur les groupes continus. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 128, pp. 1065–1069. link1 Cited by: endnote 12.
- Quelques remarques sur les groupes continus. Rendiconti del Circolo matematico di Palermo 15, pp. 321–366. link1 Cited by: endnote 12.
- Sur les surfaces de translation et les fonctions abéliennes. Bulletin de la Société mathématique de France 29, pp. 61–86. link1 Cited by: endnote 2, endnote 9.
- Nouvelles remarques sur les groupes continus. Rendiconti del Circolo matematico di Palermo 25, pp. 81–130. link1 Cited by: endnote 12.
- Rapport sur les travaux de M. Cartan. Acta mathematica 38, pp. 137–145. link1 Cited by: endnote 6.
- The Genesis of General Relativity, Volume 3: Theories of Gravitation in the Twilight of Classical Physics, Between Mechanics, Field Theory, and Astronomy. Springer, Berlin. link2 Cited by: S. A. Walter (2007).
- The History of Modern Mathematics, Volume 1: Ideas and Their Reception. Academic Press, New York. Cited by: D. E. Rowe (1989).
- The History of Modern Mathematics, Volume 2, Institutions and Applications. Academic Press, New York. link1 Cited by: E. Scholz (1989a).
- The early geometrical works of Felix Klein and Sophus Lie. See The History of Modern Mathematics, Volume 1: Ideas and Their Reception, Rowe and McCleary, pp. 209–273. Cited by: Lie and Poincaré.
- Crystallographic symmetry concepts and group theory (1850–1880). See The History of Modern Mathematics, Volume 2, Institutions and Applications, Rowe and McCleary, pp. 3–27. Cited by: endnote 14.
- Symmetrie–Gruppe–Dualität: Zu Beziehung zwischen theoretischer Mathematik und Anwendungen in Kristallographie und Baustatik des 19. Jahrhunderts. Birkhäuser, Basel. Cited by: endnote 14.
- Krystallsysteme und Krystallstruktur. Teubner, Leipzig. link1 Cited by: endnote 14.
- The Mathematician Sophus Lie: It Was the Audacity of My Thinking. Springer, Berlin. Cited by: endnote 1, endnote 5, endnote 7.
- Felix Klein: Visionen für Mathematik, Anwendungen und Unterricht. Springer, Berlin. link2 Cited by: endnote 8.
- Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910. See The Genesis of General Relativity, Volume 3: Theories of Gravitation in the Twilight of Classical Physics, Between Mechanics, Field Theory, and Astronomy, Renn and Schemmel, pp. 193–252. link1, link2 Cited by: endnote 16.
- Hypothesis and convention in Poincaré’s defense of Galilei spacetime. See The Significance of the Hypothetical in the Natural Sciences, Heidelberger and Schiemann, pp. 193–219. link1 Cited by: endnote 18.
- A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge. link1 Cited by: endnote 10.
- Lie’s Translationsmannigfaltigkeiten und Abel’sche Integrale. Monatshefte für Mathematik und Physik 46 (1), pp. 384–431. link1, link2 Cited by: endnote 9.