> I learn a lot from David Fowler's postings to this list, and I
> particularly enjoyed his recent one about the non-existence of
> "rational numbers" in Greek arithmetic. I think his conclusion
>
>> Corollary. Our idea of the rational numbers was not part of the
>> Greek way of thinking.
>
> Is largely correct. I am curious, however, about how Diophantus fits
> into all this. From what I can tell, he does work with something
> quite close to rational numbers, for example when the solution to the
> problem of expressing 16 as a sum of two squares turns out to be
> 256/25 + 144/25.
>
> Judging from the selection of the Greek text reprinted in the Loeb
> library edition, these are even written as a sort of fraction (but
> "upside down" with respect to the way we do it). I don't have access
> to Heath's Diophantus right now, but the general histories I checked
> spend lots of time on Diophantus syncopated algebraic notation but
> not much (more often no time at all) on the issue of his use of (and
> notation for) fractions.
>
> David, could you comment? Does one know where Diophantus got his
> fractions?
>
> Thanks,
>
> Fernando
A very good point. I should have added, and usually do add, "Apart from
Diophantus"!
And here are a few further remarks, but I must confess at the outset that I
haven't looked at him for quite a time and have never studied him properly,
so what follows shouldn't be taken as being too reliable. I would welcome
corrections and supplements.
* Diophantus is remarkably unlike any other Greek writer and, if his
manuscript hadn't survived, nobody would ever have dreamed that there had
been anything like that. (So what other things might there have been but
not been preserved?!)
* The description of notations that everyone talks about only occurs in a
passage at the very beginning, and it is quite possible that this is a
later addition to the text.
* The newly discovered Arabic texts have a rather more basic, less
abbreviated style than the Greek ones, which may be a sign that the Arabic
texts come from an earlier exemplar, and the Greek ones have been more
edited and updated in transmission. There is and I feel there always will
be a tendency to modernise the ways of writing numerals and styles of
notation. (Some examples: I sometimes come across journals who give their
volume numbers in Roman, but I always change them to Arabic, and I freely
change the way dates are written into a variety of different ways. I have
read quite a bit of Euler on continued fractions, and write my notes using
index notation. Practically everyone describes Newton's calculus using
Leibniz' notation. And almost everyone writes Newton's fluxional notation
as 'x dot'. though he didn't introduce that until the 1690s.) So may some
of the special Diophantine features have been added in transmission?
* He shows stronger signs of Babylonian influence than any other Greek
mathematician. (Mathematician, not astronomer: Ptolemy's astronomy is a
blend of the Greek geometrical and qualitative and the Babylonian
arithmetical and quatantive approaches.)
* As to the the 'upside down' way of writing fractions, there is a lot
about that in my book, most of a rather turgid Chapter 7. In brief, I argue
that this is indeed a later scribal abbreviation of the early Greek way of
describing division, which I describe and illustrate at great length.
David Fowler