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Russell (1964).

44 One might think that Kepler would have used his third law to compute the mean solar distances of the planets from their sidereal periods, which were determined very accurately, but he did not. There are at least two possible reasons for this. First, he may have anticipated criticism for the use of a law that was grounded in his speculations about harmony rather than derived from systematic observation, and second, he may well have believed that, as with other essentially correct laws, the third law was subject to small variations.

From the preface to the Rudolphine Tables, which is translated in Gingerich (1972).

Fig. 7.6. Napier's geometrical definition of logarithms.

Fig. 7.6. Napier's geometrical definition of logarithms.

Another source of delay was Kepler's discovery of logarithms, which he described as a 'happy calamity', and his subsequent investigations of them. The invention of logarithms simplified greatly many of the tedious arithmetical calculations required by astronomers and, unlike many other mathematical advances, was an almost instant success. Laplace later remarked that:

it is an admirable contrivance ... which, by reducing to a few days the labour of many months, we may almost say doubles the life of astronomers, and spares them the errors and disgusts inseparable from long calculations.

Logarithms were developed in the 1590s by the Scotsman, John Napier, and

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