logarithms had not yet been invented, and great inconvenience was therefore felt whenever it became necessary to multiply or divide trigonometrical quantities. To obviate this difficulty a method was invented, the so-called Prostaphæresis,[1] by which addition and subtraction were substituted for multiplication and division, and in the history of this invention, which was made independently by several mathematicians, the name of Tycho is also mentioned. The Arabs had had an idea of this method; at least, Ibn Tunis makes use of the formula[2]
cos A cos B = 12 [cos (A − B) + cos (A + B)]
but, like many other discoveries of the Arabs, this formula had to be deduced anew in Europe. It was found by Viète, as well as the corresponding formula:
sin A sin B = 12 [cos (A − B) − cos (A + B)]
but as Viète's Canon Mathematicus, which was published in 1579, seems only to have been printed in a few copies at his own expense, it is very possible that Tycho Brahe never saw it, or at least that he had not seen it in 1580, when, according to Longomontanus, he and Wittich invented Prostaphæresis.[3] This was among the inventions which Wittich a few years later brought to Cassel, where Bürgi soon developed the method further. It appears that Wittich merely had shown him the above formula for sin A sin B; but Bürgi applied the principle to the formulæ of spherical trigonometry, and ultimately was led to discover logarithms
- ↑ Astronomers need hardly be reminded that this word (formed from πρόσθεσις, addition, and ἄφαίρεσις, subtraction) had originally signified the equation of the centre, in which sense it was still used by Tycho.
- ↑ Delambre, Astr. du Moyen Age, pp. 112 and 164.
- ↑ Si autem de hujus compendii inventore quis quærat, nec Arabes aut Joannem Regiomontanum fuisse, scripta eorum analemmatica declarent; neminem certe habeo Tychone nostro & Vitichio Vratislaviensi antiquiorem: quorum scilicet mutua opera primum anno 1582 [should be 1580] in Huæna, sphærica quædam triangula tali pragmatiæ pro studiosis Vranicis sunt subjecta."—Longomontani Astr. Danica, p. 8.
