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108 he had been informed in a dream of all that was to happen to him. His costume, his bearing, corresponded with his strange character. He appeared sometimes in rags, sometimes splendidly dressed; ran through the streets at night, and the next day was drawn in a three-wheeled carriage. Yet he published a treatise on mathematics, Ars magna, which was remarkable for the age. Pertinently to the publication of this work he had controversies with Tartaglia, of which something should be said, for the curious picture they offer of the manners of the learned world in the sixteenth century.

Tartaglia, as we have said, discovered the solution of cubic equations. Cardan employed toward him all the persuasions in his power to obtain a communication to himself of the famous discovery. "I swear to you on the holy gospels," he promised, "that if you teach me your discoveries I will never publish them, and will, besides, record them for myself in cipher, so that no one shall be able to understand them after my death." Tartaglia, trusting in Cardan's good faith, communicated to him his rules summarized in twenty-seven mnemotechnic verses, in three strophes of nine verses each. Cardan, assisted by his pupil Ferrari, succeeded in extending the rules, solved equations of the fourth degree, and published the whole in the Ars magna. Tartaglia, irritated at the algebraist astrologer's violation of his word, fell into a violent rage. He sent to his enemy, according to the fashion of the time, several challenges, and in one of them went so far as to threaten Cardan and his pupil that he would wash their heads together and at the same time, "a thing which no barber in Italy could do." Cardan finally agreed to attend a disputation, which was to be held in a church in Milan on the 10th of August, 1548. He did not appear, but sent his pupil Ferrari. Ferrari bore his part in the contest alone, and the affair would have resulted in favor of Tartaglia if the hostile attitude of Cardan's friends had not caused him to leave Milan by a byroad. "These mathematical jousts," says M. Victorien Sardou, "these challenges proclaimed by heralds and trumpets, with great parade of pompous words and swelling eulogies, were more becoming to charlatans than to really learned men; but charlatanism was then in fashion; a discovery was the finder's secret, and a method of calculating was speculated upon as if it was a new medicinal powder." We do not wholly agree with M. Sardou. We see an example of intellectual activity and find a proof of the importance that was attached to algebraic discoveries in these scientific tournaments in which all classes of society are interested as formerly, in ancient Greece, they applauded the challenges of poets and the contests of athletes.

Leaving the Italian mathematicians and crossing the Alps, we