were used. The sign—was not required, for the fruitful theory of negative quantities was not as yet known. In equations the coefficients 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 Methods, "that this cramped condition of imperfection did not constitute an algebraic science like that of our days, the power of which resides in those combinations of the signs themselves which assist the reasonings of intuition and lead by a mysterious way to the results sought."
Tartaglia Nicolo was an illustrious figure among the mathematicians of Italy. Born at Brescia in 1500, he was terribly mutilated 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 (tariagliare, 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. Scaliger and De Thou assert that he had calculated the day of his death by astrology, and then starved himself to secure the fulfillment of his predictions.
Such was the final eccentricity of this mathematician, who believed 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.