"Bürgi not only used this method, but... improved it. He found the second formula, for Brahe and Wittich only knew the first. In addition, he improved the computation of the ...\cos c = \cos a \cos b + \sin a \sin b \cos CUsing the method of prosthaphæresis... cos a cos b and sin a sin b... could be computed but two new multiplications were... left... Bürgi realized that... prosthaphæresis could be used a second time, and... all multiplications could be replaced by additions or subtractions."
Jost Bürgi, active primarily at the courts in Kassel and Prague, was a Swiss clockmaker, mathematician, and writer. Burgi was the brother-in-law and adoptive father of Benjamin Bramer.
"... had been devised by ... and was likely brought to by... ... in 1580... This method was based on...\sin A \sin B = \frac{1}{2}[ \cos (A-B)- \cos (A+B)], \cos A \cos B = \frac{1}{2}[ \cos (A-B)+ \cos (A+B)].With... a table of sines, these... could... replace multiplications by additions and subtractions, something...Wittich found out, but apparently Werner didn’t realize."
Jost Bürgi"It would seem that J. Bürgi, independently of Napier, had constructed before 1611 a table of antilogarithms of a series of natural numbers: this was published in 1620. ...Bürgi also employed decimal franctions ..."
Jost Bürgi"In such a way, with much trouble and labor, the whole Canon has been established. For many hundreds of years, up to now, our ancestors have been using this method because they were not able to invent a better one. However, this method is uncertain and dilapidated as well as cumbersome and laborious. Therefore we want to perform this in a different, better, more correct, easier and more cheerful way. And we want to point out now how all sines can be found without the troublesome inscription [of polygons], namely by dividing a right angle into as many parts as one desires."
Jost Bürgi"Joost Bürgi... a Swiss watch and instrument maker and an assisitant to Kepler in Prague was... interested in facilitating astronomical calculations; he invented logarithms independently of Napier about 1600 but did not publish his work, Progress Tabulen, until 1620. Bürgi too was stimulated by Stifels remarks that multiplication and division of terms in a geometric progression can be performed by adding and subtracting the exponents. His arithmetical work was similar to Napiers."
Jost Bürgi"I do not have to explain to which level of comprehensibility this extremely deep and nebulous theory has been corrected and improved by the tireless study of my dear teacher, Justus Bürgi... by assiduous considerations and daily thought. ...Therefore neither I nor my dear teacher, the inventor and innovator of this hidden science, will ever regret the trouble and the labor which we have spent."
Jost Bürgi"Divide a right angle in as many parts as you want and construct herefrom the sine table. (Einen rechten Winckell in also viel theile theilenn alß man will, vnnd aus demselben den Canonen Sinuum vermachenn.)"
Jost Bürgi