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# Archimedes Codex – The Lost Palimpsest

In recent years I have become interested in the history of mathematics. What is interesting is that a lot of what I learned about the history of maths, usually through asides from my lecturers, is actually wrong. And often what I hear on the radio and TV makes me cringe as it is usually innaccurate.

One area I am developing expertise in is ancient Greek mathematics. (The name is misleading – most of the important maths happened outside of what we think of Greece although it was certainly influenced by Hellenistic culture.) Following my interest my favourite mathematical theorem is Archimedes Quadrature of the Parabola. Also, there are many great stories – see for example the Antikythera Mechanism. Another interesting story to emerge in recent years is that of a copy of work by Archimedes discovered written in a medieval prayer book. In the 12th century books were rare, precious and difficult to make so people would disassemble old books, scrape off the ink and write something new on the cleaned paper. This fate befell a text book on the work of Archimedes which was turned into a prayer book.

Fortunately, some of the mathematical writing was still visible and the scholar Heiberg found it in a religious establishment, publishing a translation of some of it in around 1906. The book then mysteriously disappeared, resurfacing in the 90s in an auction where it was sold for $2 million. The project to find more secrets in the book was detailed in Archimedes Codex by Reviel Netz and William Noel. (Be warned the book is told in the first person by two people so can be a bit confusing!). The following video of the first author tells the story and discusses the work of Archimedes:

Very interesting (though some of the lecturer’s mannerisms were a bit distracting and some of the sound a little distorted), particularly the way the Greeks apparently used diagrams, and the way diagrams seemed to be essential to their way of reasoning. For instance, the way in which it is shown diagrammatically that a straight line can only be tangent to a spiral at one point. A sort of visual proof by contradiction, I guess. I was fascinated also by the ratio of that pyramidal cylindrical section to the enclosing block and I wish it had been explained just a little more. As I recall, reference was made to getting this ratio from the ratios of two infinitely thin shapes (2 sections of a triangle) which are then stacked together. It reminded me a little of what I’ve done with my calculus mini-project – you’ll perhaps see what I mean when you’re able to look through it. I also had no idea whatsoever that Archimedes or anyone else at that time interested themselves in combinatorics. I wonder if they ever got on to Pascal’s triangle? That, I admit, seems far-fetched.

Best of luck anyway with ancient Greek mathematics – all I got of that was the lower slopes of Euclid at a small private day-school in Hampstead, north London. Perhaps you might investigate why there was no subsequent scientific or industrial revolution as happened nearly 2,000 years later in Europe – and also where the mathematics came from (you say above, ‘most of the important maths happened outside of what we think of as Greece…’. Did it come from Babylon – or even the Far East? Did the Minoans produce anything before being snuffed out by the consequences of the Santorini eruption in 1470 BCE? I’ll look forward to anything you publish, though maybe that won’t happen for quite a while.

If you are interested in the discussion of why there was no scientific or industrial revolutions, then you might like to read The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn by Lucio Russo. It’s a bit pricey and a bit academic in places but does discuss some interesting ideas on the topic.

The reason I say the important work was done outside Greece is because a lot happened in places such as Alexandria, Egypt. For example, Euclid worked there and so did Eratosthenes. Less well known Egyptians were Pappus and Diophantus. A number of other important mathematicians came from places now in Turkey, eg Thales and and Apollonius. And the most important mathematician of the era, Archimedes, was from Sicily. Of course, these people were working in the Hellenistic empire and Alexandria flourished under Ptolemy who deliberately imitated Greek civilization but the fact remains that these mathematicians were not swanning around the centre of Athens.

Many thanks for the tip re Lucio Russo – yes, it is a bit pricey I have to admit, may give it to myself as a Christmas present! Meanwhile I hope all’s well with the Leeds Mathematics Dept & that the academic year’s off to a good start. I see now what you mean by ‘not Greek’ – yes of course, mainly Alexandria and Sicily. I certainly remember Euclid being Alexandrian. I got a lot of this from a book called ‘Mathematics and Western Culture’ by Morris Kline – a bit dated now but when I was at university in the early 70s was quite a revelation about the place of mathematics in broader culture.

I still wonder about the Minoans, though – I’ve been a fan of volcanoes since childhood and in about 1971 or 72 my parents pointed me in the direction of a book called ‘The End of Atlantis’ about excavations on Crete which at that time had just uncovered what seemed to be a highly sophisticated civilisation brought down by a Krakatau-scale or bigger eruption on Thera Santorini. A Google search on ‘Minoans mathematics’ throws up some stuff but it looks a bit sensationalist & not scholarly so I don’t know.

Russo might be a bit heavy for Christmas reading, but hey, I once go VI Arnol’d’s book on Ordinary Differential Equations for Christmas so I can’t really make any comment.

I’ve got a copy of Kline somewhere. What’s interesting about his book (see also Dan Pedoe, W. Sawyer or Lancelot Hogben) is that it contained quite a lot of complicated maths and yet was considered a “popular” account of mathematics. I’m not sure a publisher would attempt that sort of book these days.

Oops! I gave the wrong date for the Santorini eruption – my apologies. Correct date 1600 BCE approximately as far as I can see from an authoritative source: http://www.volcanolive.com/santorini.html (the site owner’s been active in the volcano business since 1988). Other estimates are for about 1630 BCE. At any rate 1470 BCE seems to be right out of the running.