The Italian mathematician Niccolo Tartaglia (1500-1557) was the first person to apply mathematics to the solution of artillery problems.

Niccolo Tartaglia, born Niccolo Fontana in Brescia, was raised in poverty by his mother. His father was killed in the French occupation of the town in 1512, and it was then that Niccolo received a saber cut which was supposed to have been the cause of his stammering for the rest of his life. Because of this disability, he gave himself the nickname of Tartaglia, the "stutterer." He was a self-taught engineer, surveyor, and bookkeeper and is said to have used tombstones as slates because he was too poor to buy writing materials. As he grew to manhood, he demonstrated definite mathematical abilities, and he established himself as a teacher of mathematics in Venice in 1534.

## "New Science"

Tartaglia's pioneer work on ballistics and falling bodies, *Nova scientia* (1537; *New Science*) represents an original attempt to establish theories for knowledge which had previously been known empirically. Leonardo da Vinci had studied the science of ballistics earlier, but his work was not nearly so comprehensive. In his analysis of the dynamics of moving bodies, Tartaglia differentiated types of motion. Thus, a freely falling body possesses a natural motion if it is an "evenly heavy" body; by this phrase it was understood that the object was made of dense material and was of a form which would not develop much air resistance. Such bodies fall at an accelerated rate, and each has its maximum velocity at the moment of impact with the earth. The natural motion of descent varies with the distance traveled by the body.

The other case is that of the violent motion characteristic of a projectile. Tartaglia opposed the prevailing view that a projectile was subject to an initial acceleration and claimed that a violently propelled body starts to lose velocity as soon as it is detached from the propelling force. In his diagram of an evenly heavy body in violent motion, the first phase is a straight line upward at an angle, the second a curve, and the third a straight vertical line representing the body in a state of natural motion. He claimed that the curved part of the trajectory was the result of the body's own weight, but he recognized that this was theory inconsistent with his description of the first phase of violent motion. To save his theory, Tartaglia suggested that the whole path was actually curved but that the curvature was so slight as to be imperceptible.

In his discussions of violent motion, it is obvious that Tartaglia was still in harmony with the earlier "impetus" school of physics, which held that a quantity of force was impressed into a body when it was put in motion. Motion ceased when this force was exhausted, and a body in flight had its motion changed from violent to natural at that point.

## "Diverse Problems and Inventions"

In his second book on the subject, *Quesiti et inventioni diverse* (1546; *Diverse Problems and Inventions*), Tartaglia made some important modifications in the theories he had expounded in Nova scientia. He stated that a body could possess violent and natural motion at the same time and that the only motion which could occur as a straight line was purely vertical. Thus, in the case of a cannonball, unless the cannon was fired straight upward, the projectile was bound to have a curved path. Artillerymen, who based their conclusions on field observations, insisted that this was not so and that the force of propulsion of a shot guaranteed that it would move in a straight line for part of its flight. Some mathematicians agreed, but Tartaglia insisted that under the influences of violent and natural motion not even the smallest part of a missile's trajectory could be rectilinear.

In convincing his opponents, Tartaglia was less than successful, and they would accept only the triple-phase trajectory of his earlier work. Not until Galileo gave his mathematical proofs did scientists realize that all projectile motions are parabolic and hence trace a curved path.

## Later Years

Tartaglia's *Treatise on Numbers and Measurements* (3 vols., 1556-1560) was the best work on arithmetic written in Italy in his century. He also was responsible for the first translations of the works of Euclid into Italian and for the first Latin edition of Archimedes. Tartaglia died in Venice on Dec. 13, 1557.

## Further Reading on Niccolo Tartaglia

The reader who wishes to learn about Tartaglia and understand the Renaissance environment of science and mathematics should consult George Sarton, *Six Wings: Men of Science in the Renaissance* (1957). In addition, two books by Morris Kline are very helpful: *Mathematics in Western Culture* (1953) and *Mathematics and the Physical World* (1959).