As far as I know, we don't know of any work like that of Euclid from those
earlier times, but Euclid's work came at the end of a long development
which predates classical Greece.
There's been much debate and study for many years now on the extent of
so-called oriental ("Near Eastern")influences on developments in classical
Greece. As far as mathematics and astronomy are concerned, the work of
Otto Neugebauer and his students in this connection was preceded in the
later 19th century by that of others, e.g. Father Kugler. I feel that it
has been shown that there was considerable scope and depth to certain kinds
of mathematics before the time of classical Greece, e.g. the approximation
techniques of some Babylonian astronomers, the "binary" numerical
algorithms of some Egyptians, various kinds of work in applied geometry,
etc.
Of course, what we know depends on what documents or other evidence has
survived. I suggest that it is a dubious procedure to base too much weight
on such documents as the Rhind and Berlin papyri. For all one knows, these
could have been something on the order of students' notes, and should not
be evaluated as if they were treatises, or parts of treatises, in the way
Euclid's work is, or the now lost earlier Greek "elements" of geometry may
have been.
On the other hand, the classical Greeks appear to have been the first to
introduce notions of proof of the sort that we have become accustomed to.
Related to this is the discover of the connectivity between mathematical
results which goes with axiomatic mathematical techiques. I suppose we
must assume these were Greek innovations until someone uncovers some
remains proving otherwise, if such exist or have ever existed. This notion
of proof may be especially what Hardy (and maybe Littlewood) had in mind.
At any rate, I vote for a better appreciation of earlier mathematical work
which most likely influenced classical Greek developments that is evidenced
by the (I would say) condescending remark by Littlewood (reported by Hardy)
to the effect that the mathematics of earlier times was all at a level
which could be attained by a clever schoolboy -- unless maybe the clever
schoolboy was Gauss?
Gordon Fisher gfisher@shentel.net