Having caught up with some background reading http://mathworld.wolfram.com/IntegerRelation.html and discovered how far out-of-the-loop I was, it's time for third thoughts. The current jargon is apparently "Integer Relation Finding", and it turns out there's a pretty solid-looking but rather complex algorithm due to Ferguson et al from 1997 called PSLQ --- see Ferguson, H. R. P., Bailey, D. H., Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math. Comput. 68, 351-369, 1999. [online as cpslq.pdf] [This garnered some publicity at the time when Plouffe and others used it in a surprising algorithm to extract digits from \pi, recently mentioned in this list, I seem to recall --- see Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/paper.html] PSLQ is implemented in both Mathematica and Maple. I compared the Maple version with my home-grown geometrically-inspired "Umbrella" algorithm, only to discover that PSLQ detected the polynomial satisfied by a quadratic surd, using only half the precision required by Umbrella --- Umbrella retires, blown inside-out. All these algorithms have an inherent limitation: they detect only homogeneous relations (i.e. with RHS = 0) with integer coefficients, between a given bunch of reals; whereas the Lego orrery appears to demand a RHS equal to the gear ratio required to be approximated. I'm still thinking about this --- if only as a last-ditch attempt to salvage a little pride. Fred Lunnon