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 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 bi-quadratic equations. As in the case of cubics, so here, the first impulse was given by Colla, who, in 1540, proposed for solution the equation $$\scriptstyle{x^4+6x^2+36=60x}$$. To be sure. Cardan had studied particular cases as early as 1539. Thus he solved the equation $$\scriptstyle{13x^2=x^4+2x^3+2x+1}$$ by a process similar to that employed by Diophantus and the Hindoos; namely, by adding to both sides $$\scriptstyle{3x^2}$$ 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 bi-quadratic equations. Ferrari reduced Colla's equation to the form $$\scriptstyle{(x^2+6)^2=60x+6x^2}$$. In order to give also the right member the form of a complete square he added to both members the expression $$\scriptstyle{2(x^2+6)y+y^2}$$, containing a new unknown quantity y. This gave him $$\scriptstyle{(x^2+6+y)^2=(6+2y)x^2+60x+(12y+y^2)}$$. The condition that the right member be a complete square is expressed by the