The problem of squaring the circle: Given a circle, construct (with straightedge and compass) a square with the same area. While browsing the web for more information on this problem (for the blog entry on problems that might be similar to P vs NP: here) I came across the following:
In the Gilbert and Sullivan comic opera Princess Ida, in the song Gently, Gently is the line:
... and the circle they will square it one fine day.
(To hear the song see here. The line is towards the end.)
They lyrics are here. That website begins gsarchive.net which made me wonder Did I at one time set up a website of math refs in Gilbert and Sullivan plays (gsarch is very close to gasarch) ? which IS the kind of thing I would do. The answer is no: gsarch stands for Gilbert and Sullivan archive. They could have called it gasarch if they used the and in Gilbert and Sullivan but abbreviated archive as arch. Then I would have been far more confused.
In 1884 Princess Ida opened in 1884. For more on this comic opera see here.
In 1882 pi was proven transcendental and hence one cannot square the circle. For more on pi being transcendental see here.
Kolmogorov Random Thoughts on all of this
0) The song is sung my three men who are making fun of the notion of a women's college. The song is about all the things the women are trying to do that are absurd such as squaring the circle. They also mention perpetual motion machines.
1) Did G and S know that the squaring the circle had been proven impossible, or just that it was thought to be impossible, or just that it was thought to be hard?
2) Was it known that perpetual motion machines were impossible? Or just very hard?
3) G and S used Mathematics in at least one other song: I am the very model of a modern major general, from The Pirates of Penzance has the lines:
I'm very well acquainted too with matters mathematical
I understand equations, both the simple and quadratical,
About binomial theorems I'm teeming with the a lot o' news---
With many cheerful facts about the square of the hypotenuse
I'm very good at integral and differential calculus
See here for all the lyrics. The website mentioned in the next point has a pointer to a YouTube video of people singing it.
4) There are many parodies of Modern Major General. The earliest ones I know of is Tom Lehrer's The Elements. Since making a website of them IS the kind of thing I would do, while writing this post I did it (Are we compelled to do things that fit our image of ourselves? Yup.) The website is here. It has 36 parodies (as of Oct 17, 2021 when I wrote this blog--- it may have more if you read this later.) That may seem like a lot, but it pales in comparison to the most satirized song of all time: The 12 days of Christmas which I did an ugly lyrics-only website for back before html had nice tools, see here. It has 143 songs on it but I am sure there are many more. (Note to self: redo that website when you have time. Maybe when I retire.)
4) I suspect that G and S knew more math, or perhaps knew of more math, than Broadway composers know now. I suspect this is a more general trend: people are more specialized now. Having said that, I need to mention the off-Broadway musical Fermat's last Tango which I liked more than Lance (see his post on it here).
5) How much math would you need to know in order to insert some into your play or movie? With Wikipedia and other web sources you could find out some things, but you would have to have some idea what you are looking for. And perhaps you would need some math background in order to even want to insert some math into your work in the first place.