It is Emperor John's turn, in 1379,…

Niccolo Fontana, the Italian mathematician and professional military engineer better known (to historians of science) as Tartaglia, serves as a consultant on scientific problems to the rulers of several principalities.

When the master of ordnance at the castle of Verona suggests that Tartaglia consider what angular elevation of a gun barrel would yield the greatest range for a shot, he calculates the requisite elevation of one-half of a right angle—in traditional measure, forty-five degrees.

Tartaglia establishes the science of ballistics in 1537 with his Nuova scienzia, the first systematic treatment of the ballistics of gunnery.

Niccolò Tartaglia is the first to apply mathematics to the investigation of the paths of cannonballs; Galileo's studies on falling bodies will validate his work.

Gerolamo Cardano reveals Tartaglia's solution of the cubic equation—a betrayal—in his Ars magna (“The Great Art”), the first great Latin treatise on algebra.

Nicccolò Fontana Tartaglia writes a popular arithmetic text, and is, in 1543, the first Italian translator and publisher of Euclid's Elements.

Tartaglia also publishes Latin editions of Archimedes.

Also an engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper of the Republic of Venice, Tartaglia is the first to apply mathematics to the investigation of the paths of cannonballs; his work will later be validated by Galileo's studies on falling bodies.

Gerolamo Cardano, two years after accepting a professorship in medicine at Pavia in 1543, reveals Niccolo Tartaglia's solution of the cubic equation—which he had promised not to do—in his Ars magna (“The Great Art”), the first great Latin treatise on algebra.

He also reveals the solution of the quartic equation found by Cardano's former servant, Ferrari.

Ferrari, the discoverer of the solution, or root, of the quartic equation (fourth-degree polynomial equation), defends his teacher Cardano in a debate in Milan with Tartaglia in 1548.

The debate was provoked by Cardano’s publication three years earlier of Tartaglia's solution of the cubic equation (third-degree equation), which Tartaglia, having solved it in 1535, had confided to Cardano on the condition that it not be published.

This debate brings Ferrari public attention and gains for him a position as tax assessor in Mantua.