WISM100: History from Ellipse to Elliptic Curve 2016/17
This page supplements the information in Blackboard and will be
updated with suggested reading, course notes, etc.
Important: change in hand in schedule
There will be 2 papers instead of 3, with the following deadlines:
Anything handed in after the deadline will suffer a grade deduction of
1 point per working day.
- Fri 2 June: deadline for Apollonius
- Fri 23 June: deadline for second paper with all the other stuff
Slides about Greek math and Apollonius
- a handout can be collected at my office HFG705 if you were not
- two tasks are suggested near the end of the slides; you have to
perform only one.
Elliptic integrals: intro and addition
forumlas NOTE: this is a jpeg image, open in an image viewer
and zoom to your preferred level.
Suggested reading for this part:
- Stillwell, Mathematics
and its History, par. 12.3, 12.4, 12.5, has
a easy-going introduction with nice excercises which will show you
the necessary techniques. Available online in the UU library.
- If you want more about the Bernoulli's and the paracentric
isochrone then have a look at
Bos, The lemiscate of Bernoulli,
in: Lectures in the History of Mathematics, 1993, or
Blåsjö, Transcendental Curves in the Leibnizian
Calculus, 2016, par. 7.3 and 8.3.
It is not my intention to lay too much stress on this topic, but it
may give you a clear idea of the much more geometric way of
thinking in the 17th century.
As an original source (Learn from the Masters!) Euler
(site) is highly
recommended. His work is indexed using Eneström index numbers,
beginning with the letter E. Search for the paper E251
"De integratione aequationis differentialis
mdx/√(1-x4)=ndy/√(1-y4)". There is an
English translation and you can also see a scan
of the original. Euler is a lucid writer so don't hesitate! You
could also look around the site for other articles of Euler's under
the subject "Elliptic functions".
Fagnano's article Teorema da cui si
deduce una nuova misura degli Archi Elittici... is available but
much harder to read.
- Excerpts of Euler's and Fagnano's texts are contained in Dirk
Struik's "Source Book in
Mathematics" here. Be warned: it
has a few typos.
From elliptic integrals to elliptic functions:
Gauss and Abel NOTE: this is a jpeg image again.
For down-to-earth stuff: continue reading in Stillwell, see link above.
Gauss: a very enjoyable article by David
Cox: "The arithmetic-geometric mean of Gauss"
in l'enseignement mathematiques, vol.30, 1984,
p.275-330. It covers more than only Gauss and agM. Be sure to
read section 3, Historical remarks. And
yes, it's in English! Download it from
free site but if you want to print it then
take this version with added
Reading Gauss himself is a bit steep because (a) mostly Latin
(b) not finished, publishable work and (c) leaves a lot of work
to the reader. But, if you like, follow any reference of your
liking from the
secondary literature (e.g., Cox) into the Gauss
Werke. Currently the University Library has free access: just
type "Gauss Werke" in
the catalog search bar.
Abel, first part of Recherches sur les fonctions
elliptiques available in original French in
and English translation at
Note: I WANT YOU TO STUDY THIS, IN particular paragraph
I up to the double periodicity. The functions phi, f and F are
defined in the introduction. Referring to the translations: page
6 is quite hard (but can perhaps be skipped), page 7-8 look
tough and unrewarding but on p.9 you get to the holy grail.
Slides (pdf) with suggested
reading at the end. There is also
a worksheet about Riemann surfaces.
Note: we made the decision that not all topics of the
lectures need to be covered in your paper.