## R2 r r R p R p R2 p2 2pR e

The unknowns are p, p, p, e, and e. The last two of these can be approximated numerically at some given time, t , from a set of closely spaced observations, and then we have three scalar equations for three scalar unknowns p(t), p(T), and p(t). For a general elliptic orbit, this yields an eighth-order algebraic equation for p(t), which has to be solved numerically, though some reduction is possible if a parabolic orbit is assumed. We can then calculate r(T) and r(T) from which the elements are determined easily.

Laplace's method suffers from a number of weaknesses, not least of which is the need to determine the derivatives of observational data. If only three data points are available, this will probably lead to significant errors. Despite this and other problems, Laplace's procedure was understood easily and became popular quickly.

It was superseded by a method proposed by Olbers in 1797,37 which in turn was eclipsed by the brilliance of Gauss. Both Olbers and Gauss took as their starting point a fact that had been noted in 1733 by Pierre Bouguer. Since an orbiting body moves in a plane (neglecting perturbations), its heliocentric position vector ri at three different times ti, i = 1, 2, 3, are linearly related, i.e.

where c1 and c3 are some unknown constants. Crossing this equation with r3 and r1 shows that

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