P Q 2c1 C3 1r2 C1

and he showed that small inaccuracies in the values of P and Q will lead to small errors in the heliocentric positions. Moreover, P and Q are approximated well by t2 - t1

respectively (the latter result being far from obvious). In determining the elements from P and Q , the same eighth-order algebraic equation arises as in Laplace's method, but Gauss transformed this into what is now sometimes referred to as 'Gauss' equation', a sin4 z = sin(z - f), where the unknown z is the angle between the heliocentric and geocentric position vectors at the middle observation, and a and f are known functions of P and Q. This equation has two or four real roots, and Gauss explained how to select the appropriate solution.

The Theoria motus is the work of a mathematical genius, fully aware that he is developing tools for practical use. As an example, we will consider Gauss' solution to what is often called 'Lambert's problem', i.e. the determination of an orbit from two points r1, r2, and the time, r, between them. It is a surprising fact that r depends only on the semi-major axis a, the sum of the distances r1 + r2 = s, for example, and the length of the chord \r2 - r1 \= d, say. Gauss described the history of this result, now known as Lambert's theorem, thus:

This formula appears to have been first discovered, for the parabola, by the illustrious Euler, who nevertheless subsequently neglected it, and did not extend it to the ellipse and hyperbola: they are mistaken, therefore, who attribute the formula to the illustrious Lambert, although the merit cannot be denied this geometer, of having independently obtained the expression when buried in oblivion, and of having extended it to the remaining conic sections.

In fact, the first analytic demonstration of the result was due to Lagrange in 1778, a year after Lambert's death, though Gauss failed to mention this. For an ellipse, Lagrange obtained r — = a - sin a - (f - sin f),

38 Gauss Theoria motus, Article 106 (Gauss (1963)).

where a and 3 are determined from 5 + d = 2a(l — cos a) and 5 — d = 2a (1 — cos 3), and from this Gauss deduced that r1 + r2 — 2 Jr1r2 cos f cos g a =--—2-'

2 sin2 g where f is half the angle between r1 and r2, and the unknown g is half the difference between the eccentric anomalies corresponding to the two positions.

This formula is, however, totally unsuitable for practical use since, if the positions are close together, the right-hand side is the difference of two almost equal quantities divided by a small quantity. So Gauss introduced a quantity l defined by

[77 [77 2(1 + 2l) cos f = — + / —, V 72 V 7l in terms of which, a can be represented in a suitable form. The problem manifest in the computation of a from the original formula now appears in the calculation of l, but Gauss got round this by defining a new quantity m through tan (1 n + &>) = ^

and then l cos f = sin2 2 f + tan2 2m, which is not sensitive to numerical error.

Gauss showed that g could be determined from the simultaneous solution of the two equations m2 3 2 22g — sin2g

0 0