MATHEMATICAL CURIOSITIES. 107 



were used. The sign was not required, for the fruitful theory of 

 negative quantities was not as yet known. In equations the co- 

 efficients of the unknown quantities were always figures, which 

 became combined with the other factors during the operations, 

 and of which no trace appeared in the final result. "We may 

 conceive/' says M. Chasles, in his History of Geometrical Meth- 

 ods, " that this cramped condition of imperfection did not consti- 

 tute an algebraic science like that of our days, the power of which 

 resides in those combinations of the signs themselves which as- 

 sist the reasonings of intuition and lead by a mysterious way to 

 the results sought/' 



Tartaglia Nicolo was an illustrious figure among the mathe- 

 maticians of Italy. Born at Brescia in 1500, he was terribly mu- 

 tilated at an early age, when his native city was captured by 

 Gaston de Foix. His skull was broken in three places and his 

 brain exposed, his jaws were split by a wound across his face, and 

 he could not speak or eat. He nevertheless recovered, but always 

 stammered, whence his name (tartagliare, to stammer). He was 

 his own schoolmaster, and, after he had learned to read and 

 write, devoted himself to the study of the ancient geometricians. 

 At thirty-five years of age he taught mathematics in Venice. 

 There he accepted a challenge which Fiori sent him, to solve 

 twenty problems, all of which depended upon a particular case of 

 cubic equations. Tartaglia solved them in less than two hours, 

 and to commemorate his triumph composed mnemotechnic verses 

 containing the solution. He was also the author of the ingenious 

 formula for finding directly the area of a triangle of which all 

 three of the sides are known. 



Cardan Jerome, who was born in Paris, of Italian parents, 

 September 24, 1501, was one of the most extraordinary men of his 

 time. At twenty-two years of age, when he had just terminated 

 his studies at the University of Pavia, he taught Euclid publicly. 

 He also taught medicine, traveled in Scotland, Germany, and the 

 Low Countries, and returning established himself in Rome as a 

 pensioner of Pope Gregory XIII, and died there in 1576. Sca- 

 liger and De Thou assert that he had calcuiated the day of his 

 death by astrology, and then starved himself to secure the fulfill- 

 ment of his predictions.* 



Such was the final eccentricity of this mathematician, who be- 

 lieved firmly in astrology and had visions, and he professed that 



* In one of his excursions to England he cast the horoscope of Edward VI, for whom 

 he predicted a long life. Unfortunately, the king died in the next year. Having become 

 used to such accidents, he was not disconcerted, but revised his calculations, rectified some 

 of the figures, and found that the king had died in full accordance with the rules of 

 astrology. 



