My 'construction' of the reals, referred to by Ross and Mike, was suggested by an observation of Tate about bilinear maps many years earlier, but essentially goes back to Eudoxus, who noted that to measure the ratio of a line segment to another with any desired accuracy, it isn't necessary to cut either of them up--instead multiply the 'numerator' segment by a large integer--thus pointing the way to a direct construction of the reals as ring from the integers as mere additive group. (Incidentally, I take the task to be relating the continuous to the discrete, rather than constructing the former from the latter, but never mind.) R is the ring of endomorphisms E(T(L))of the group T(L) of translations (rigid maps at finite distance from the identity) of the line L. If we start instead with a discrete line D (a line of dots), then T(D) is isomorphic to Z (additive group without preferred generator). E(T(D) is the ring Z, but instead take A(T(D)), 'almost homomorphisms' (as in Mike's description) from Z to Z. A(T(D)) is an additive goup with a 'multiplication' by composition, but not a ring, since one distributive law and commutativity of multiplication fail; but A(T(D)) modulo bounded maps is R. The maps in both directions are easy: send the real r to the map 'multiply by r and round down', and send the almost homomorphism f to the limit of the Cauchy sequence f(n)/n. Steve Schanuel