not solve any of Tartaglia's. From now on, Tartaglia studied cubic equations with a will. In 1541 he discovered a general solution for the cubic , by transforming it into the form . The news of Tartaglia's victory spread all over Italy. Tartaglia was entreated to make known his method, but he declined to do so, saying that after his completion of the translation from the Greek of Euclid and
Archimedes, he would publish a large algebra containing his
method. But a scholar from Milan, named Hieronimo Cardano (1501–1576), after many solicitations, and after giving the most solemn and sacred promises of secrecy, succeeded in obtaining from Tartaglia a knowledge of his rules.
At this time Cardan was writing his Ars Magna, and he
knew no better way to crown his work than by inserting the much sought for rules for solving cubics. Thus Cardan broke his most solemn vows, and published in 1545 in his Ars Magna Tartaglia's solution of cubics. Tartaglia became desperate. His most cherished hope, of giving to the world an immortal work which should be the monument of his deep learning and power for original research, was suddenly destroyed; for the crown intended for his work had been snatched away. His first step was to write a history of his invention; but, to completely annihilate his enemies, he challenged Cardan and his pupil Lodovico Ferrari to a contest: each party should propose
thirty-one questions to be solved by the other within fifteen days. Tartaglia solved most questions in seven days, but the other party did not send in their solution before the expiration
of the fifth month; moreover, all their solutions except one were wrong. A replication and a rejoinder followed. Endless were the problems proposed and solved on both sides. The dispute produced much chagrin and heart-burnings to the parties, and to Tartaglia especially, who met with many other disappointments. After having recovered himself again, Tartaglia
began, in 1556, the publication of the work which he had had in his mind for so long; but he died before he reached the consideration of cubic equations. Thus the fondest wish
of his life remained unfulfilled; the man to whom we owe the greatest contribution to algebra made in the sixteenth century was forgotten, and his method came to be regarded as the discovery of Cardan and to be called Cardan's solution.
Remarkable is the great interest that the solution of cubics excited throughout Italy. It is but natural that after this great conquest mathematicians should attack biquadratic equations. As in the case of cubics, so here, the first impulse was given by Colla, who, in 1540, proposed for solution the equation
. To be sure, Cardan had studied particular cases as early as 1539. Thus he solved the equation by a process similar to that employed by Diophantus and the Hindoos; namely, by adding to both sides and thereby rendering both numbers complete squares. But Cardan failed to find a general solution; it remained for his pupil Ferrari to prop the reputation
of his master by the brilliant discovery of the general solution of biquadratic equations. Ferrari reduced Colla's equation to the form ( x 2 + 6 ) 2 = 60 x + 6 x 2 . In order to give also the right member the form of