In 1890, in fact, Henri Poincaré published a definitive paper on the three-body problem, making it clear not only that the problem has no closed-form solutions, but that it has another, even more disturbing quality: Its solutions are sometimes chaotic. That is, if you vary the initial conditions of the problem—the numbers equivalent to M and a in my two-body example—very slightly, the resulting orbits change drastically, beyond all recognition. Poincaré himself commented that one set of conditions produced “orbits so tangled that I cannot even begin to draw them.”
