Posted by: **Dave Richeson** | May 31, 2011

## Hankel on Diophantus

Diophantus of Alexandria was one of the last (c. 250 AD) great mathematicians of the Hellenistic period. He is often called the “father of algebra.” An entire branch of mathematics is named for him. It was in the margin of his book *Arithmetica* that Fermat penned his famous note.

Today, while looking up some information on Diophantus, I came across this wonderfully descriptive quote by Hermann Hankel (1874) on Diophantus and his mathematics (the English translation is by Heath).

Of more general comprehensive methods there is in our author no trace discoverable: every question requires a quite special method, which often will not serve even for the most closely allied problems. It is on that account difficult for a modern mathematician even after studying 100 Diophantine solutions to solve the 101st problem; and if we have made the attempt, and after some vain endeavors read Diophantus’ own solution we shall be astonished to see how suddenly he leaves the broad high-road, dashes into a side-path and with a quick turn reaches the goal with reaching which we should not be content; we expect to have to climb a toilsome path, but to be rewarded at the end by an extensive view; instead of which our guide leads by narrow, strange, but smooth ways to a small eminence; he has finished! He lacks the calm and concentrated energy for a deep plunge into a single important problem; and in this way the reader also hurries with inward unrest from problem to problem as in a game of riddles, without being able to enjoy the individual one. Diophantus dazzles more than he delights. He is in a wonderful measure shrewd, clever, quick-sighted, indefatigable, but does not penetrate thoroughly or deeply into the root of the matter. As his problems seem framed in obedience to no obvious scientific necessity, but often only for the sake of the solution, the solution itself also lacks completeness and deeper signification. He is a brilliant performer in the *art* of indeterminate analysis invented by him, but the *science* has nevertheless been indebted, at least directly, to his brilliant genius for few methods, because he was deficient in the speculative thought which sees in the True more than the Correct. That is the general impression which I have derived from a thorough and repeated study of Diophantus’ arithmetic.

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It’s certainly expressed very well, but I suspect that he is wrong, i.e., that there actually is an underlying method, though it’s not the kind of method a modern mathematician wants.

By:

Fernandoon June 1, 2011at 9:24 am

You’re probably right. I don’t know much about (today’s) Diophantine analysis or Diophantus’s work—so I can’t offer an opinion. I just found this quote to be an interesting mixture of respect, awe, frustration, and dismissiveness.

By:

Dave Richesonon June 1, 2011at 9:32 am