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John Napier
John Napier

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1.7 Kepler and logarithms

Kepler was precisely the kind of practitioner for whom logarithms were of greatest benefit: a professional astronomer (and, of course, competent Latin scholar) driven at times to distraction – he tells us – by the magnitude and complexity of the calculations he needed to do. So when he was able to study a copy of Napier's Descriptio in about 1619 he welcomed it warmly, dedicating his next book to Napier (not realising he had been dead for two years); and he went further. Napier's book contained the definition of logarithms, and a set of logarithm tables, but not how to get from one to the other. (It is worth bearing in mind that although logarithms once calculated are a considerable saving of time, the calculation of the tables themselves was extraordinarily laborious – especially as one did not have the assistance of logarithm tables to help with the necessary computations.) So Kepler set to work recalculating their construction from first principles, basing his argument upon the classical theory of proportion of Euclid's Elements, Book V. This approach was consciously quite different from Napier's geometrical and kinematic considerations; Kepler wrote that logarithms were not associated, for him,

inherently with categories of trajectories, or lines of flow, or any other perceptible qualities, but (if one may say so) with categories of relationship and qualities of thought.

(Quoted in Yu. A. Belyi (1975) ‘Johannes Kepler and the development of mathematics’, in A. Beer and P. Beer (eds), Kepler: Four Hundred Years, Pergamon Press, p. 656.)

It is interesting to notice that Kepler felt that the appeal to kinematical intuition in Napier's work lacked rigour. He emphatically asserted that what he was supplying was rigorous proof, which was, for him, something cast in Euclidean mould. In the event, he had to approximate (just as Napier had done) to express potentially endless numbers to a suitable finite approximation, but Kepler's overall approach was influential nonetheless. You can see something of his style of argument in the extract linked below, which you should look at now.

Click the link below to open the extract.

Charles Hutton on Johannes Kepler’s construction of logarithms [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]

Figure 7
(Deutsches Museum, Munich) ©
Deutsches Museum, Munich
Figure 7 Frontispiece of Kepler's Rudolphine Tables (1627), the set of planetary tables on which he had been working for 26 years, latterly with the aid of logarithms. The design, by Kepler himself, includes amongst a host of symbolic detail the figure of the muse of arithmetic, on the roof, her halo consisting of the numbers 6931472 – the logarithm of 2 (or, which is numerically the same, log sin 30°)