A NEW ASTRONOMY 193 



investigations. He showed independence of thinking, but his 

 astronomical theories were too little developed and too specula- 

 tive to constitute real progress in an age not yet quite ripe for 

 their reception. 



Peurbach (1423-1461), who had as a youth met Nicholas of 

 Cusa in Rome, became professor of astronomy and mathematics 

 in Vienna and has been called " the founder of observational and 

 mathematical astronomy in the West/' Recognizing the imper- 

 fections of the Alfonsine tables he published a new edition of 

 the Almagest with tables of natural sines instead of chords 

 computed for every ten minutes. He depended mainly, however, 

 on imperfect Arabic translations. 



His more eminent pupil and successor, Johann Miiller, of 

 Konigsberg, better known as Regiomontanus (1436-1476), was 

 the most distinguished scientific man of his time. After the 

 fall of Constantinople he was among the first to avail himself of 

 the opportunities for more direct acquaintance with the works 

 of Archimedes, Apollonius, and Diophantus. For the defective 

 version of the Almagest which had come through Arabic 

 channels he substituted the Greek original, while his tables, pub- 

 lished in 1475, were important both for astronomy and for the 

 voyages of discovery of Vasco da Gama, Vespucci, and Colum- 

 bus. These tables covered the period 1473 to 1560, giving sines 

 for each minute of arc, longitudes for sun and moon, latitude for 

 the moon, and a list of predicted eclipses from 1475 to 1530. An- 

 other work on astrology includes a table of natural tangents for 

 each degree. A wealthy merchant of Nuremberg erected an 

 elaborately equipped observatory for Regiomontanus, and the 

 printing-press recently established there became the most impor- 

 tant in Germany. Accepting, however, a summons to Rome to 

 reform the calendar, he was murdered at the age of 40. 



His De Triangulis (1464) is the earliest modern trigonometry. 

 Four of its five books are devoted to plane trigonometry, the 

 other to spherical. He determines triangles from three given 

 conditions, using sines and cosines, and employs quadratic equa- 

 tions successfully in some of his solutions. One of his problems 



