# Early two-author math papers

The middle of the twentieth-century featured several famous papers with two authors. For example, Eilenberg and Mac Lane's papers introducing categories and Eilenberg-MacLane spaces appeared in 1945. The Feit-Thompson Odd Order Theorem appeared in 1962. Atiyah and Singer published their index theorem in 1963.

I can't think of any important papers with two or more authors before the Eilenberg-Mac Lane collaboration, which could just be a lacuna in my historical knowledge. My question is: what are the first math papers with two or more authors? (A subsidiary question is: why were collaborations so rare before that?)

• Hardy-Littlewood? Feb 20, 2011 at 20:33
• Whitehead-Russell Feb 20, 2011 at 20:37
• The question is subjective - among all the multi-author papers, how do we determine which was the first important one? If we're just compiling a list of early multi-author papers, the question should be community wiki. Feb 20, 2011 at 22:47
• -1, because I don't like this kind of question. Feb 21, 2011 at 2:35
• This thread is long enough. I've started a meta discussion: tea.mathoverflow.net/discussion/965/oldest-joint-paper Please vote for this comment so that it appears above the fold. Feb 21, 2011 at 9:59

The famous paper of Dedekind and Weber:

R. Dedekind, H. Weber: Theorie der algebraischen Functionen einer Veränderlichen, J. Reine Angew. Math 92 (1882) 181-290.

is the first place where the points of a Riemann surface are described in terms of ideals of the ring of functions. To put this into context, Dedekind had only invented the notion of ideal a few years earlier. They also give an algebraic proof of the Riemann-Roch theorem.

I think the analogy between function fields and number fields started here.

• Already Gauss, in his unpublished part of the Disquisitiones [published later by Dedekind in Gauss's Werke), observed that the proof of unique factorization in $F_p[X]$ is very similar to that in ${\mathbb Z}$. This analogy was extended to the quadratic reciprocity law and forms over $F_p[x]$ by various writers such as Schoenemann, Serret, Heine, and Dedekind way before Dedekind-Weber. Feb 21, 2011 at 16:48
• Dedekind and Weber's paper developed the analogy in more depth than the earlier authors. Feb 22, 2011 at 2:48
• Maybe it can be said that Dedekind and Weber was the first paper to relate the geometry of function fields to the arithmetic of number fields, by describing the arithmetic analogue of points. And the idea of prime ideals as points leads to the modern notion of schemes... Sep 24, 2019 at 0:22

The Brill--Noether paper appeared in 1874: "Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie". Math. Annalen 7, 269–316.

• I never realized that the Noether of "Brill-Noether" referred to Max Noether, not Emmy Noether. Feb 20, 2011 at 21:51
• @Rex: 1874 is a bit early for Emmy Noether, gifted as she was. Feb 20, 2011 at 22:57
• Yes, of course. I had not been aware that Brill-Noether theory was that old. Feb 20, 2011 at 23:09
• the famous "brill noether" number occurs already in riemann for r=1. hence "brill noether" theory is even older than they are. Feb 21, 2011 at 3:15

Abraham Ecchellensis (Ibrahim Al-Haqilani) and Giovanni Alfonso Borelli published in 1661 a Latin translation of the 5th, 6th and 7th books of the Conics by Apollonius of Perga. Both Ecchellensis's and Borelli's names are on the title page, a highly unusual feature at the time.

For more details, see the paper Authorship and Teamwork Around the Cimento Academy by Domenico Bertoloni Meli, available here.

– user6976
Feb 21, 2011 at 14:43
• indiana.edu/~hpscdept/Fac-MeliPublications.shtml (go to Research articles and look for a paper published in 2001 in Early Science and Medicine).
– Did
Feb 21, 2011 at 15:24
• Strange, the link is working now. The paper seems interesting. It may be enough to answer the second question.
– user6976
Feb 21, 2011 at 17:43
• That's a really interesting paper. It directly addresses the question of how collaboration worked differently in the 1600s. Feb 21, 2011 at 22:24
• I don't think a jointly prepared translation of someone else's earlier work is in the spirit of the OP's question. Feb 22, 2011 at 2:52

Sturm and Liouville published joint papers in 1836-1837.

• The paper springerlink.com/content/k28765tu63333677/fulltext.pdf on the development of Sturm--Liouville theory says they published only one joint paper on S--L theory, in 1837. The paper they published together in 1836, also mentioned in that article, was about counting complex roots of a polynomial inside a contour. The bibliography lists a paper by Colladon and Sturm from 1834 about compression of liquids. Feb 22, 2011 at 4:03

Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen by N.H. Abel, Th. Clausen, J. Steiner, published in 1827, Journal für die reine und angewandte Mathematik. Volume 1827, Issue 2, Pages 286–292. I think every paper by Abel must be important.

Also:

Nouvelles formules analogues aux séries de Taylor et de Maclaurin. by B. Clapeyron G. Lamé, in Journal für die reine und angewandte Mathematik. Volume 1830, Issue 6, Pages 40–44.

I do not know who was Clapeyron, but Lamé is well known. These two papers may be the first collaborative journal papers because Crelle was the first math journal (or one of the first journals). But this site: http://www.daviddarling.info/encyclopedia/B/Bernoulli.html talks about a joint paper in astronomy produced by Bernoulli family. At that time (~1730) astronomy and mathematics were not that far apart.

• Clapeyron is probably the Physicist: en.wikipedia.org/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron In France, he seems mostly remembered for work in Thermodynamics. Feb 20, 2011 at 22:43
• @Thierry: Of course, I only just realized who he was. Before now, I only saw Russian transcription of his name. Thanks!
– user6976
Feb 20, 2011 at 22:54
• The "Aufgaben und Lehrsätze" is not a joint paper, it is a collection of problems to be solved by the readers, of which the first four were submitted by Abel, the next few by Clausen etc. Feb 21, 2011 at 17:32
• As for the Bernoulli paper, wikipedia (mathematik.ch/mathematiker/daniel_bernoulli.php) has this to say: "Daniel Bernoulli submitted an entry for the Grand Prize of the Paris Academy for 1734 giving an application of his ideas to astronomy. This had unfortunate consequences since Daniel's father, Johann Bernoulli, also entered for the prize and their entries were declared joint winners of the Grand Prize." Feb 21, 2011 at 17:37
• @Franz: you are right about Abel's paper, but was it a joint venture or the journal combined three papers together? That is why I included the second paper in my answer. That one seems really collaborative.
– user6976
Feb 21, 2011 at 17:42

Here is one from the eighteenth century. There must be others.

Theoria motuum lunae, nova methodo pertractata una cum tabulis astronomicis, unde ad quodvis tempus loca lunae expedite computari possunt incredibili studio atque indefesso labore trium academicorum: Johannis Alberti Euler, Wolffgangi Ludovici Krafft, Johannis Andreae Lexell. Opus dirigente Leonhardo Eulero acad. scient. Borussicae directore vicennali et socio acad. Petrop. Parisin. et Lond. Petropoli, typis academiae imperialis scientiarum. 1772.

• Translation, anyone? "Theory of the motion of the moon...something" Feb 21, 2011 at 8:09
• Here's my clumsy translation: 'A theory of the movements of the moon, treated by means of a single new method with astronomical tables, by which the position of the moon at any given time may be easily calculated thanks to the marvellous studies and indefatigable efforts of three academicians.' Feb 21, 2011 at 15:11
• Euler's son, Krafft and Lexell were among the "scribes" responsible for writing up Euler's work after he had become blind. Feb 21, 2011 at 17:40
• I don't think a jointly prepared write-up of another (blind) person's work is in the spirit of the OP's question. Feb 22, 2011 at 2:54
• Lexell published 66 papers, only 4 of them were coauthored with Euler. He was eventually appointed as Euler's successor. I don't think he should be labelled a "scribe." Wikipedia has an article about Lexell: en.wikipedia.org/wiki/Anders_Johan_Lexell. Feb 22, 2011 at 13:15

The first explicit construction $E_8$ lattice (positive definite even unimodular quadratic form in $8$ variables) was given by A. Korkin and G. Zolotarev in 1873, see "Sur les formes quadratiques". Mathematische Annalen. 6: 366–389.

Very important joint work one of whose authors is a somewhat well-known mathematician, but not a mathematics paper was the construction of the first electromagnetic telegraph by C. F. Gauss and W. E. Weber (1833), and then:

Gauss, Carl Friedrich; Weber, Wilhelm Eduard (1840). Atlas Des Erdmagnetismus: Nach Den Elementen Der Theorie Entworfen. Leipzig: Weidmann'sche Buchhandlung.

Don't know if that counts, but I don't know of any joint work of Gauss in pure mathematics...

In differential geometry a fundamental collaborative paper is: Ricci, Gregorio, Levi-Civita, Tullio (March 1900). "Méthodes de calcul différentiel absolu et leurs applications". Mathematische Annalen (Springer) 54 (1–2): 125–201. doi:10.1007/BF01454201.

Its digitalization is freely available here.

Edit: I replaced the link to a pay-walled copy with another to the GDZ repository.

Obvious answer to subsidiary question: communication was vastly more difficult then, and travel similarly. For that matter, in the U.S., prior to the advent of the internet, it was crazily slow to communicate with anyone far away (long-distance phone calls cost an absurd amount, too much to "discuss" anything, even if paid-for by an NSF grant... and physical mail would take a week each direction, due to the additional slow-down of campus mail interface).

Further, "in the old days", people up for tenure or promotion would vastly have preferred solo papers, because there was more presumption (I guess due to the predominance of solo papers...) that a joint paper was surely much less a product of the junior author... or something.

Even the now-mundane issues of revision would have been grossly complicated in those days: no TeX, so one usually gave a hand-written ms. to a technical-typing-secretary, who would have a bit of technology ("IBM selectric"?) unavailable to most mortals, and produce a presentable mostly-typed document. But introduce errors. To photocopy and physical-mail to co-author far away would introduce an extra two-week delay...

The practical difficulties certainly explain the relative ease of contemporary collaboration, and, thus, the relative plausibility, etc.

• What, authors in the same department or university couldn't collaborate? Dec 14, 2019 at 13:24
• @MarkL.Stone, of course co-located authors could collaborate, but the hiring and tenure patterns tended to make it unlikely that there'd be two people of comparable seniority with similar research programs in the same dept. (E.g., there would have been a risk that they'd have been viewed as competing for tenure...) Dec 14, 2019 at 14:28

As far as the number of joint authors is concerned, this one is hard to beat: Malmsten, C.J., Almgren, T.A., Camitz, G., Danelius, D., Moder, D.H., Selander, E., Grenander, J.M.A., Themptander, S., Trozelli, L.M., Föräldrar, Ä., Ossbahr, G.E., Föräldrar, D.H., Ossbahr, C.O., Lindhagen, C.A., Moder, D.H., Syskon, Ä., Lemke, O.V., Fries, C., Laurenius, L., Leijer, E., Gyllenberg, G., Morfader, M.V., Linderoth, A.: Specimen analyticum, theoremata quædam nova de integralibus definitis, summatione serierum earumque in alias series transformatione exhibens (Eng. trans.: "Some new theorems about the definite integral, summation of the series and their transformation into other series") [Dissertation, in 8 parts]. Upsaliæ, excudebant Regiæ academiæ typographi. Uppsala, Sweden (April—June 1842)

Edit (December 20, 2016): t turns out the information in Blagouchine's citation is imprecise and the number of Malmsten's co-authors has to be reduced. More detailed information which I received from Lennart Börjeson (thanks!) indicates first of all that Moder, Föräldrar, Syskon and Morfader are not last names but rather parts of respective dedications (translation: mother, parents, siblings and maternal grandfather). Indeed, in the scanned copy of the paper available here, goo.gl/bZIWZZ one such dedication can be seen on p. 2. Another point: the remaining names are those of students and the parts of the paper are probably their theses. This makes the co-authorship questionable, since before 1852 in Sweden it was common that the professor wrote the thesis bearing the student's name rather than the student himself. Still, the names appear jointly in print (only some of them can be seen here, but the paper version from which the scan was made might have been a part of a bigger series of publications).

• Moder, D.H. gets listed twice? Oct 24, 2016 at 22:11
• @GerryMyerson: The misprint is not mine. This particular list of authors appears in MR3258600 Blagouchine, Iaroslav V. Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results. Ramanujan J. 35 (2014), no. 1, 21–110 There is no reproduction of the title page on Malmsten et al.'s thesis, but Blagouchine warns about misprints, so this might be one of them. Oct 25, 2016 at 16:01

Joint mathematical papers were pioneered by Polish School after WWI. Offhand, let me mention popular Knaster-Kuratowski-Mazurkiewicz paper on the fixed point theorem or Borsuk-Ulam paper on antipodes or Banach-Tarski paradox. There were many more.

• Gergonne’s Annales contains many nominally “multi-authored” articles. Most are just collations of independent parts (e.g. “solutions of problems”) by the editor, but the paper

Recherches sur la détermination d'une hyperbole équilatère, au moyen de quatre conditions données

Par MM. BRIANCHON , capitaine d’artillerie , professeur de mathématiques à l’école d’artillerie de la garde royale , et PONCELET , capitaine du génie , employé à Metz.

Annales de mathématiques pures et appliquées XI, nº VII (1821), pp. 205-220

seems to be both joint work and older than the (imho) earliest genuinely collaborative papers yet mentioned on this page (viz., Sturm and Liouville 1836-1837).

• Earlier (but could be regarded as teaching rather than original research): Monge & Hachette, Application d’algèbre à la géométrie; Hachette & Poisson, Addition au mémoire précédent, both in Journal de l’école polytechnique IV, nº XI, (1802).

• Earlier (but maybe subject to the same objections as Theoria motuum lunae): Théorie de la lune (1770 prize essay, labeled “par MM. Léonard Euler & J. A. son fils, conjointement”; Johann Albrecht Euler has 31 papers of his own.)

• Earlier (and original, but not a “paper”): Le Seur & Jacquier, Elémens du calcul intégral (1768).

There was a known paper by Alexandrov-Urysohn on compact spaces (y. 1929) -- they called them bicompact spaces in the Soviet Union for a long time.

Though it was initially likely more intended (and perceived) as physics, the paper which introduced what has since then become known as the Korteweg-De Vries equation [Korteweg, D. J.; De Vries, G., On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves., Phil. Mag. (5) XXXIX, 422-443 (1895). JFM 26.0881.02] may be counted here due to its lasting influence on the analysis of PDE.

Felix Klein and Sophus Lie published at least a couple papers together:

• Ueber diejenigen ebenen Curven, welche durch ein geschlossenes System von einfach unendlich vielen vertauschbaren linearen Transformationen in sich übergehen. Mathematische Annalen 4 (1871): 50-84. http://eudml.org/doc/156514.

• Ueber die Haupttangentencurven der Kummer'schen Fläche vierten Grades mit 16 Knotenpunkten. Mathematische Annalen 23 (1884): 579-586. https://doi.org/10.1007/BF01446604

(These are the joint articles listed at https://de.wikisource.org/wiki/Sophus_Lie, I do not know if there are others.)

The work introducing the Phragmén–Lindelöf principle should likely also be mentioned here:

Phragmén, E.; Lindelöf, E., Sur une extension d’un principe classique de l’analyse et sur quelques propriétés de fonctions monogènes dans le voisinage d’un point singulier., Acta Math. 31, 381-406 (1908). Zbl 39.0465.01.

The Lotka-Sharpe population model was introduced in their joint 1911 paper Sharpe, F. R.; Lotka, A. J., A problem in age-distribution., Phil. Mag. (6) 21, 435-437 (1911). JFM 42.1030.02. and has been much in use in demography since then.

The classification of finite groups with only abelian subgroups by _G. A. Miller and H. C.Moreno Non-Abelian groups in which every subgroup is Abelian., American M. S. Trans. 4, 398-404 (1903). ZBL 34.0173.01., which is still frequently used, might also be worth mentioning.

Another example worth to be mentioned might be the Bernstein-Doetsch theorem on convex functions [Bernstein, Felix; Doetsch, Gustav, On the theory of convex functions, Math. Ann. 76, 514-526 (1915). JFM 45.0627.02].

The Liénard–Chipart stability criterion [Liénard; Chipart, Sur le signe de la partie réelle des racines d’une équation algébrique., Journ. de Math. (6) 10, 291-346 (1914). JFM 45.1226.03.] is still used in control theory.