Karl Gustav Jacob Jacobi
This page is not being actively maintained, but I
will try to make small changes as I find time.
The translations on this site date to my graduate
student days, when I was first learning about elliptic functions.
Incidentally, the translations came directly from the versions
published in Crelle's Journal, and not from Jacobi's Collected
(The Collected Works weren't readily available to me at the
It shouldn't be difficult to find errors in these translations
-- I would certainly appreciate hearing about any that you find.
This page was last modified on:
Fri Aug 1 11:51:33 EDT 2003
by Eric Conrad.
Mail comments to (firstname.lastname@example.org).
Please let me know if there is some Jacobi or elliptic functions material
on the net which I should add to this site.
[REVISED 27 Apr 97] The table of contents (postscript
formats) for Jacobi's Fundamenta Nova Theoriae Functionum Ellipticarum
translated from Latin into English. Pages numbers refer to the version
printed in volume 1 of his collected works, available from the American
Mathematical Society. (1 page in postscript format. Keywords: Jacobi elliptic
functions, theta functions, elliptic integrals, algebraic combinatorics.)
[MINOR REVISIONS 28 Jun 98] C. Jacobi, A
new application of elliptic functions to algebra, originally in
Crelle's Journal 7(1832), 41-43, translated from French into English.
(3 pages in postscript format. Keywords: Jacobi elliptic functions, continued
[REVISED 28 Jun 98] C. Jacobi, Note
sur les fonctions elliptiques, originally published in Crelle's
Journal 3(1828), 251-254, translated from French into English. (4
pages in postscript format. Keywords: Jacobi elliptic functions, theta
[Corrected Jan 2004] C. Jacobi, Suite
des notices sur les fonctions elliptiques, originally published
in Crelle's Journal 3(1828), 255-263, translated from French into
English. (9 pages in postscript format. Keywords: Jacobi elliptic functions,
theta functions, sums of squares, modular transformations.) (Note:
The source file was corrupt. The corrections were applied to a backup
Also available as a
A biography of Carl Gustav Jacob Jacobi (1804-1851). (MacTutor web site) Also on the MacTutor site:
- Niels Abel (1802-1829). Abel discovered inversion of elliptic integrals shortly before Jacobi.
- Adrien-Marie Legendre (1752-1833). Legendre was mentor to both Abel and Jacobi.
- August Leopold Crelle (1780-1855). Crelle was a civil engineer who published much of Abel's and Jacobi's early work, including work on elliptic functions in his new mathematical journal. His journal, officially titled Journal fur die reine und angewante Mathematik ("Journal of pure and applied mathematics"), has been informally called Crelle's Journal since its inception. (A short exception occurs for a brief period after Crelle's death when, under the stewardship of Carl Wilhelm Borchardt (1817-1880), the journal was referred to as Borchardt's Journal.)
[May 21, 2000] Search the catalog
of the Biblioteque Nationale de France (BnF). With
a small amount of searching -- in French of course --
you should have no trouble in finding
an online Acrobat (.pdf) photocopy version
of Jacobi's collected works.
gives some help in the use of the BnF site. As I find time, I may be able
to provide Jacobi-specific help here. (I thank Mr. Bifet
for providing me these links.)
Math Forum message by Emile Bifet
Jacobi communicated work in progress to others in seminars. Among
those influenced by Jacobi's seminars was Bernhard Riemann. Riemann, a
student of Gauss, was reportedly frustrated by Gauss's secretiveness about
his research and attended Jacobi's seminars to see mathematics research
in progress. (Among Riemann's collected works is an appendix to section
40 of Jacobi's Fundamenta Nova.)
Four Square Theorem. (Also available in postscript
format [11 pages].) [CONSTRUCTION IN PROGRESS]
[April 25, 2000] A simple program to generate four-square representations
and some sample output to test Jacobi's version of the Four Square Theorem.
program foursq.C and sample
output foursq.txt). The program finds all representations by brute
force (in this case brute force is essentially a depth first search). The
search is optimized somewhat by using congruences pare the searches for
two-square and three-square representations. (It is easy to verify that
a two-square representation does not exist for integers congruent to 3
modulo 4, and that a three-square representation does not exist for integers
congruent to 7 modulo 8.)
[April 28, 2000] A simple program to generate r-square representations.
rsq.C and sample
output rsq.txt). Essentially the same algorithm is used, except that
negatives and positives are handled separately. A few additional
trivial cases are eliminated using congruences. The number of representations
goes up rapidly as the number of squares r increases. The program
will calculate all representations of 100 as a sum of 10 squares, but expect
to wait a while.
[April 28, 2000] A Mathematica program to count the number of representations
of (small) integers as sums of r squares. For example, the program results
show that 36 has more than 100,000,000 representations as 11 squares. (See
the very last entry in the sample run for details.) This number was computed
exactly without producing any representations of 36 as a sum of squares.
program nsq.m and sample
output are available for your amusement. In addition, I used the program
to show that there are exactly 1,282,320,348 representations of 100 as
a sus of ten squares. The computation (nine convolutions) took about two
seconds. With a little thought, you should see that I could have done this
using only four convolutions. (Mathematica input
and a transcript
of the program run).
[REVISED 28 Jun 98] A short exercise from section 39 in the Fundamenta
Nova. In this exercise, you are asked to formally transform an infinite
product (equation 1 section 39) into a Fourier series expansion (equation
6 section 39). The techniques are straightforward -- most are are taught
in a typical first year calculus sequence. (I suggest you not try
this on your first year calculus students without their consent!) Analysts
will probably want to assume that q<1 and fill in details about convergence.
Elliptic functions and integrals
(MacTutor) Elliptic functions and integrals.
A brief early survey from Wallis to Jacob Bernoulli.
Conrad's home page.