## Fourier, Hamilton and Ptolemy

As we stagger into the last week of term I find myself with just two lectures to give in my second-year module on Vector Calculus and Fourier Series. I didn’t want to present the two topics mentioned in the title as disconnected, so I linked them in a lecture in which I used the divergence theorem of vector calculus to derive the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

Fourier’s work was so influential and widely admired that it inspired a famous Irish mathematician William Rowan Hamilton to write the following poem:

The serious thing that strikes me is not the quality of the verse, but how many scientists of the 19th Century, Hamilton included, saw their scientific interrogation of Nature as a manifestation of the human condition just as the romantic poets saw their artistic contemplation and how many poets of the time were also interested in science.

Anyway I was looking for nice demonstrations of Fourier series to help my class get to grips with them when I remembered this little video recommended to me some time ago by esteemed Professor George Ellis. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

This reminds me of a point I’ve made a few times in popular talks about Astronomy. It’s a common view that Kepler’s laws of planetary motion according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as the elements used in the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed-form expression for such an orbit is considerably more compact and elegant than a Fourier representation, and also encapsulates a deeper level of physical understanding.

### 6 Responses to “Fourier, Hamilton and Ptolemy”

1. I see it the other way – once it is realised that almost any type of periodic motion can be accurately modelled if you use enough epicycles and deferents, it’s no longer clear that there is a meaningful physical basis for the Ptolemy model!

2. Anton Garrett Says:

The serious thing that strikes me is… how many scientists of the 19th Century, Hamilton included, saw their scientific interrogation of Nature as a manifestation of the human condition

They didn’t necessarily see it as an interrogation. It is possible to see it as a romantic wooing of nature, persuading her to give up her secrets. In ancient Hebrew culture – one of the formative influences on the culture of Western Europe in which the scientific revolution took place – there is no ascription of femininity to nature, but *wisdom* is portrayed as feminine; certainly scientists see beauty in the workings of the creation. Francis Bacon and other Renaissance men openly used sexual metaphors (not excluding rape, as I recall) for the then new project of probing nature’s inmost secrets. Certainly there is a passion in the doing of science.

3. “The serious thing that strikes me is not the quality of the verse, but how many scientists of the 19th Century, Hamilton included, saw their scientific interrogation of Nature as a manifestation of the human condition just as the romantic poets saw their artistic contemplation and how many poets of the time were also interested in science.”

It’s been mentioned here before, but Richard Holmes’s book The Age of Wonder is a good read if you are interested in those times.

4. “Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series.”

Perhaps as two-dimensional projections on the sky. But the two systems make different predictions regarding the radial distance as a function of time.

5. Bryn Jones Says:

Coincidentally, I came across Hamilton’s poem about Fourier while writing next month’s article on the history of astronomy in the Vale of Glamorgan. It appeared in a three volume biography of Hamilton from the 1880s (R. P. Graves, “Life of Sir William Rowan Hamilton”, volume 1, publ. Hodges, Figgis & Co., Dublin, 1882). I’m using a scan in the Internet Archive: https://archive.org/details/lifeofsirwilliam01gravuoft/page/596

(My article is about Lord Adare, later the Third Earl of Dunraven, of Dunraven Castle in the Vale of Glamorgan, who was a private student of Hamilton at Dunsink Observatory in the 1830s.)

6. Joe Whitbourn Says:

The ‘joy born of awe’ line reminded me of Rebecca Elson’s poem ‘We Astronomers’: https://www.poemhunter.com/poem/we-astronomers/

She’s an interesting 20th century counterpart as a poetry writing scientist. I’d recommend her posthumous collection ‘A Responsibility for Awe’