The fact that Osiander, rather than Copernicus, was the author of the preface generally was not appreciated at the time. Kepler found out from a friend in Nuremberg and printed the information in 1609, though the authorship has been attributed wrongly to Copernicus on numerous occasions since then. Osiander's argument was not devised in response to Copernicus' revolutionary idea. The same beliefs concerning the nature of astronomical hypotheses had been held widely by ancient Greek philosophers and preserved through the works of, for example, Maimonides and St Thomas Aquinas (see Duhem (1969)).

15 See Gingerich (1992), Chapter 10.

essence, the same as Ptolemy's and, instead of devising new ones, he concentrated on producing a new physical interpretation with the Sun at the centre of the Universe, and on correcting the underlying geometrical parameters.16 Had Ptolemy been rather more forthcoming in explaining how he came up with his models, Copernicus might have been persuaded to spend time looking for new ways of fitting mathematical schemes to the known phenomena, but as it was, it was left to Kepler to discover, from a careful analysis of observational data, that a new mathematical model was required.

Copernicus did not make any original contributions to trigonometry, but the trigonometric theorems that he presented in Book I of On the Revolutions were well explained and quite wide-ranging, and Rheticus published them separately in 1542. Copernicus' trigonometry is based entirely on chords and sines, though he refers to the latter as half-chords subtending double arcs, and he notes that 'in demonstrations and calculations half-lines are used more frequently than whole lines'. Ptolemy's table of chords in the Almagest was based on a circle of radius 60 and the use of sexagesimal fractions, but by the sixteenth century the Hindu-Arabic numerals had been introduced into Europe and Copernicus used these since 'this numerical notation certainly surpasses every other, whether Greek or Latin, in lending itself to computations with exceptional speed'.18 However, decimal fractions did not come into common use until they were

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