Scientist of the Day - Diophantus of Alexandria
Title page, Arithmeticorvm libri sex, by Diophantus of Alexandia, ed. and transl. by Claude Bachet, 1621 (Linda Hall Library)
Diophantus of Alexandria was a Greek mathematician of, we believe, the 3rd-century C.E., although he could have lived 300 years earlier and we could not tell. We know nothing about his life, and needless to say, we don't know his birthday or death date, so this is not an anniversary notice. It is just a celebration of someone you ought to know more about.
Assuming he did live in the mid-3rd century C.E., which is still the best guess, Diophantus lived about 125 years after Ptolemy of Alexandria, 460 years after Archimedes, 550 years after Euclid, and 800 years after Pythagoras, so he was heir to a considerable Greek tradition in mathematics. But most of his predecessors were geometers. Diophantus, by contrast, was an arithmetician, with an interest in solving equations, what we would now call algebra. He was the first of his kind.
There is a puzzle in the Greek Anthology of about 500 C.E. that presents very nicely the kind of problem that Diophantus liked to solve. It is, in fact, a puzzle about the lifespan of Diophantus. To get the answer, you need to write and solve an equation. It goes more or less like this (I paraphrase for brevity): “Diophantus was a boy for 1/6 of his lifespan, married after another 1/7 of his lifespan, grew a beard after 1/12 more, had a son 5 years later, who lived only half as long as his father, who died 4 years after his son died. If we let X be the lifespan of Diophantus, we can write an equation that looks like this:
X = X/6 + X/7 + X/12 + 5 + X/2 + 4
It's a little messy, but it is not a great problem to solve the equation for X, revealing that Diophantus lived to be 84, his son 42. The trick was to realize that you could write an equation with one unknown incorporating all that information.
Diophantus was hindered by the fact that he lacked the symbolic apparatus of modern algebra, but he wasn't interested in general equations anyway. He liked to solve particular problems. The Pythagorean theorem was his kind of equation, a2 +b2 = c2. What sets of whole numbers satisfy that equation? Everyone knows about 3,4,5, but there are lots of other Pythagorean triples, such as 5,12,13, and 28,45,53. An equation of the form a2 +b2 = c2, where you seek integer solutions only, is now known as a Diophantine equation.
Diophantus wrote a book about Diophantine equations, called Arithmetica. Originally composed in 13 books, six books survive in their original Greek, and several more (discovered only recently) in Arabic. The first Latin translation of the Arithmetica was published in Paris in 1621, a copy of which we have in our collections (first image). An oft-repeated story is associated with this edition. Pierre de Fermat was reading it in the 1630s when he ran across Diophantus’s discussion of the Pythagorean equation, where Diophantus speculated that a similar equation to the third power, a3 + b3 = c3, has no whole number solutions. Fermat wrote in the margin of his copy that the conjecture was correct, for all exponents greater than 2, and that he had a proof, but the margin was too small to contain it. This was Fermat's famous Last Theorem, which he never did write down or prove.
Title page, Arithmeticorvm libri sex, by Diophantus of Alexandia, ed. and transl. by Claude Bachet, with notes by Samuel de Fermat, 1670 (Linda Hall Library)
Another edition of the Arithmetica of Diophantus was published in 1670, edited by Fermat's son Samuel (Fermat senior had died in 1665), which we also have in our collections (second image). When he got to book 2, problem 8, containing the discussion of Pythagorean-like equations to the second, third, and higher powers, the son inserted an editor's paragraph relating the story of the proof that would not fit in the margin (third image). It is a good thing he did so, as Pierre's copy of the 1621 edition, with his notes, has not survived, and without the printed comment in the 1670 edition, we would not know about Fermat's Last Theorem.
Editorial comment on Fermat’s Last Theorem, Book 2, Question 8, Arithmeticorvm libri sex, by Diophantus of Alexandia, ed. and transl. by Claude Bachet, with notes by Samuel de Fermat, detail of p. 61, 1670 (Linda Hall Library)
Many modern short bios of Diophantus, such as the one on Wikipedia, insist on providing a modern attempt at a portrait of Diophantus. That might be appropriate if the portrait had some historical interest of its own – if Fermat had drawn one, for example, or Leonard Euler. But to just make up a portrait and insert it in an article that is supposedly based on fact, seems presumptive at best. We would like to discourage the practice.
We conclude by offering for your solution one of the 130 problems that Diophantus discussed in his book. Consider the two expressions, 10X + 9, and 5X + 4. Find a value for X such that both expressions are squares. There is no general method for solving such a problem, just intuition and a little brute force, and the assurance that X is not huge (less than 50). You can check your answer in the excellent entry on Diophantus on the MacTutor website, which should always be your first stop when pursuing historical information on matters mathematical.
William B. Ashworth, Jr., Consultant for the History of Science, Linda Hall Library and Associate Professor emeritus, Department of History, University of Missouri-Kansas City. Comments or corrections are welcome; please direct to ashworthw@umkc.edu.
