[HM] rational points on conics

Jim Propp (propp@math.mit.edu)
Wed, 16 Dec 1998 07:40:51 -0500 (EST)

Who should be credited with the insight that rational points P on a (rational)
conic (with at least one rational point O) are parametrized by rationals, AND
that this parametrization can be interpreted geometrically, via the slope of
line OP (or equivalently by the intersection of line OP with a fixed rational
line)?

I'm under the impression that Diophantus knew about rational points on conics,
but that he only gave algebraic formulas, and that part of the programme of
17th century mathematics was to reconstruct the underlying geometry. So it
wouldn't surprise me if Fermat or Descartes had figured out the geometry
underlying the study of rational points on conics. (Though of course they
wouldn't have phrased their insight in modern language.)

I am particularly interested in the case of circles (which may well have
been studied first, given Diophantus' interest in writing numbers as sums
of squares).

Can someone knowledgeable about early analytic geometry comment?

Jim Propp
Department of Mathematics
University of Wisconsin