Abel on Elliptic Integrals: A Translation

Editor’s note: This article was published in June 2007.

Niels Henrik Abel was one of Norway’s great mathematicians. He was born on August 5, 1802,in Frindoe (near Stavanger), Norway, and died at the young age of 26,on April 6, 1829 in Froland, Norway, a victim of tuberculosis.

It was only around the time of his death that Abel’s mathematical talents were beginning to be recognized around Europe. Sadly, he never got to realize in how high esteem others would eventually consider his works. There are some good existing accounts of Abel’s life and mathematics: Arlid Stubhaug has written a masterful biography of Abel’s life; Christian Houzel surveys at length Abel’s mathematics; Henrik Kragh Sørensen has written a useful and solid Ph.D. thesis on Abel’s mathematics, which he has generously made available Online.

Abel’s “Recherches sur les fonctions elliptiques” (1827) was the first published account that made significant inroads on the theory of elliptic integrals, that is, integrals where the integrand is a quotient with a  rational function for a numerator and a square root of a polynomial of degree 3 or 4 for the denominator. Abel’s key insight was to invert these integrals, that is, consider the function u(x), where x is the upper limit of integration of the elliptic integral. He discovered that these “elliptic functions” were unlike other functions considered up to that time; for instance, they were found to be doubly periodic. Gauss had made similar discoveries in the late 1790s, but failed to publish them. It only became widely known that Gauss had undertaken such studies after his collected works were published in the later half of the nineteenth century.

 The translation made available here is only the first part of Abel’s “Recherches sur les fonctions elliptiques” and represents a small portion of his work in this field. Abel’s work highly influenced later mathematicians, in particular, Bernhard Riemann  (1826 - 1866), who would introduce bold new ideas into mathematics, including Riemann surfaces.  Riemann introduced these in order to get a better handle on elliptic functions and what are known as Abelian functions, generalizations of elliptic functions where the denominator in the integrand is no longer just a square root of a polynomial of degree 3 or 4, but of higher degree.

Pedagogical Perspective: Abel, using only elementary methods (nothing more complex than an intuitive version of the inverse function theorem) and crafty manipulation, is able to establish basic properties of elliptic functions such as their double periodicity. He is also able to determine specific values and derive an addition theorem, among other things. This is all done in a more concrete way than is typically found in all contemporary treatments of elliptic functions. As well, unlike Newton’s Principia, for example, what is written is easily understood by the modern reader. So students with only a basic grasp of calculus can profitably go through this text and get an intuitive idea of some basic properties of elliptic functions, something that would be quite difficult for students at that level to do using modern treatments.

Translation Notes: A translator necessarily has to make certain decisions about language and notation. I have followed Abel’s writing quite closely, though I have tried to insure that the English flows as well as possible given his writing style. Also, I have used in places more standard notation than what Abel sometimes uses in the original. All in all, I hope that you find the translation to be of some use.

Acknowledgments: I would like to thank Prof. Tom Archibald for various forms of assistance during the course of this translation project.

Click here  to read the translation of "Recherches sur les fonctions elliptiques."

References

Abel, N. H., 1827. Recherches sur les fonctions elliptiques. Journal für die reine und angevandte Mathematik, 2. 101-181.

Houzel, Christian, 2004. The Work of Niels Henrik Abel. In: Laudal, Olav Arnfinn and Piene, Ragni (Ed.), 2004. The Legacy of Niels Henrik Abel, The Abel Bicentennial, Oslo, 2002. Springer-Verlag, Berlin, 21-177.

O'Connor, J J and Robertson, E F. Niels Henrik Abel. From: The MacTutor History of Mathematics archive. http://turnbull.mcs.st-and.ac.uk/history/Biographies/Abel.html

Ore, Oystein, 1974/1957. Niels Henrik Abel, mathematician extraordinary. Chelsea Publishing Company, New York, N.Y.

Sørenson, Henrik Kragh, 2002. The Mathematics of Niels Henrik Abel - Continuation and New Approaches in Mathematics During the 1820s. Ph.D. dissertation, History of Science Department, Faculty of Science, University of Aarhus, Denmark. Available Online at http://www.henrikkragh.dk/phd/

Stubhaug, Arlid, 2000. Niels Henrik Abel and his times: Called too soon by flames afar. Springer-Verlag, Berlin.

Stubhaug, Arlid, 2004. The Life of Niels Henrik Abel. In: Laudal, Olav Arnfinn and Piene, Ragni (Ed.), 2004. The Legacy of Niels Henrik Abel, The Abel Bicentennial, Oslo, 2002. Springer-Verlag, Berlin, 17-20.