The positions on the orbit indicated are F (Flamsteed's observation, 1690, 0 = -277°), B (point of maximum separation, 1736, 0 = -180°), U (discovery of Uranus, 1781, 0 = -86°), A (closest approach, 1822, 0 = 0°), and N (discovery of Neptune, 1846, 0 = 50°).

The ratio of the radii of the orbits of the two planets is about 1.57 : 1 and so in units chosen so that the minimum distance between them is unity, the radius of the orbit of Uranus is 1.76 (since 2.76/1.76 « 1.57). Of significance for the motion of Uranus is the gravitational attraction of Neptune, which is inversely proportional to the square of the distance between the two planets. Thus, if the maximum pull is scaled to 1, the pull when Uranus is at P is 1 /x2, where x2 = 1.762 + 2.762 - 2 x 1.76 x 2.76 cos 0. It follows that at F and U the effect of Neptune is about one-tenth of its maximum value, with a minimum value at B roughly half of this. When Neptune was discovered, its gravitational effect on Uranus was just over 20 per cent of its maximum value. 2

In the absence of any knowledge of the existence of Neptune then, the pre-discovery observations were the best on which to base an elliptic orbit for Uranus, as largely they were unaffected by the undiscovered planet. After correcting for the effects of Jupiter and Saturn, such an ellipse would, of course, still not fit well with the motion of Uranus while it was close to Neptune, but it would represent accurately the underlying orbit. Bouvard's decision to discard the 'old' observations in favour of the post-discovery ones was thus a rather unfortunate one! While he had the choice as to which data to use, he had no control over the relative positions of Uranus and Neptune when he was preparing his tables. As it happened, his timing was the worst possible, since the maximum interaction between the two planets occurred almost immediately after publication of the tables.

Perhaps the theoretical difficulty of the problem solved by Adams and Leverrier sometimes is overstated, and there is no doubt that, from a technical point of view, their work was not the most brilliant application of mathematics to a problem in astronomy; nevertheless, the discovery of Neptune ranks as one of the greatest achievements in the history of astronomy. Its psychological impact was immense, and it led to a new wave of optimism about what might

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