PROGRESS OF MATHEMATICS AND MECHANICS 239 



merchants, the struggles to improve arithmetic, are all set forth 

 here by Tartaglia in an extended but interesting fashion. 



Tartaglia, anticipating Galileo, taught that falling bodies of 

 different weight traverse equal distances in equal times, and that 

 a body swung in a circle if released flies off tangentially. 



GIROLAMO CARDAN (1501-1576) led a life of wild and more or 

 less disgraceful adventure, strangely combined with various forms 

 of scientific or semi-scientific activity, particularly the practice 

 of medicine. He studied at Pavia and Padua, travelled in France 

 and England, and became professor at Milan and Pavia. 



His Ars Magna (1545) contains the solution of the cubic equa- 

 tion fraudulently obtained from his rival Tartaglia. After its publi- 

 cation the aggrieved Tartaglia challenged Cardan to meet him in a 

 mathematical duel. This took place in Milan, August 10, 1548, 

 but Cardan sent his pupil Ferrari in his place. Tartaglia relates 

 that he was accompanied only by his brother, Ferrari by many 

 friends. Cardan had left for parts unknown. As Tartaglia began 

 to explain to the crowd the origin of the strife and to criticise 

 Ferrari's 31 solutions, he was interrupted by a demand that judges 

 be chosen. Knowing no one present he declines to choose; all 

 shall be judges. Being finally allowed to proceed he convicts his 

 opponent of an erroneous solution, but is then overwhelmed by 

 tumultuous clamor with demands that Ferrari must have the floor 

 to criticise his solution. In vain he insists that he be allowed to 

 finish, after which Ferrari may talk to his heart's content. Fer- 

 rari's friends are vehement ; he gains the floor and chatters about a 

 problem which he claims Tartaglia has not been able to solve till 

 the dinner hour arrives and Tartaglia, apprehending still worse 

 treatment, withdraws in disgust. 



Ferrari (1522-1565), this disciple of Cardan, even succeeded in 

 giving a general solution of the equation of the fourth degree, 

 beyond which, as has been shown only in quite recent times, the 

 solution can in general no longer be similarly expressed. 



Some idea of the difficulty of these sixteenth century achievements 

 may be conveyed by the corresponding modernized solutions. If 

 the given equation is ay? + by? + ex + d = the coefficient of the 



