Not content with inventing the world’s first computer, it seems that Archimedes (c. 287 BC – c. 212 BC) also invented calculus nearly two millennia before Newton and Liebniz worked out the mathematical details. This fact was only recently discovered after a Byzantine liturgical book was examined with modern techniques including x-ray photography. Underneath the thirteenth century Greek prayers were the partially erased traces of ancient Greek treatises by Archimedes on geometry and physics, three of which are the only known surviving copies today, and one of which is the earliest known text on calculus. The Method deals with the problem of measuring the area under a curve by slicing it into an infinite number of sections. What makes this treatise remarkable is that he chose to employ the concept of actual infinity, which was not thought to exist, instead of the more commonly used potential infinity. Newton and Leibniz followed this method as well, Leibniz with particular pride:
I am so in favour of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author.
Science News has a great illustrated article on the subject, and you can find out more at the Archimedes Palimpsest Project website. There is also a book on the subject, newly out in paperback: The Archimedes Codex: How a Medieval Prayer Book Is Revealing the True Genius of Antiquity’s Greatest Scientist. And stay tuned for news that Archimedes built an airplane, decoded DNA, and had a supercollider in his back yard…
Classical Bookworm 
Well, I'm impressed.
What amazes me even more is how these things are discovered. They were partially erased and beneath 13th century prayers? Really, the way historians work to reconstruct the past is astonishing!
It certainly is! It's a bit paradoxical that the more time passes the more we find out about the past.
I kept looking at the condition of the book and thinking: “They weren't raised in our house.” Anything which might mar a book was strictly forbidden in our home. No dog-ears; no face-down books; no highlighting; no marginal comments; no, no, no. I've decided that my mother must have grown up when books were still copied by hand.
Heh. Books certainly were more valuable before the printing press, but then the parchment was also valuable so the scribes didn't mind doing a bit of recycling. Either they didn't know the value of what they were destroying or there were plenty of other copies around at the time. Perhaps these palimpsests have lasted longer than other copies because the superimposed text was of more use in later times and so it was preserved. It may be a case of “use it or lose it.”