Niccolò Fontana Tartaglia

Tartaglia (Niccoló Fontana)

Library of Congress

[b. Brescia (Italy), 1499, d. Venice, December 13, 1557]

Tartaglia is known mainly as the first to solve the general equation of a polynomial of degree 3 (the cubic). He also developed the rule that an elevation of 45° for a cannon provides the maximum-distance shot. He was the first to publish Pascal's triangle in Europe and translated Euclid into Italian and some of Archimedes into Latin.

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

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).

For the current Bishop of Paisley please see Philip Tartaglia.

Niccolò Fontana Tartaglia.

Niccolò Fontana Tartaglia (1499/1500, Brescia, Italy – December 13, 1557, Venice, Italy) was a mathematician, an engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then-Republic of Venice (now part of Italy). He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs; his work was later validated by Galileo's studies on falling bodies.

Niccolò Fontana was the son of Michele Fontana, a rider and deliverer. In 1505, Michele was murdered and Niccolò, his two siblings, and his mother were impoverished. Niccolò experienced further tragedy in 1512 when the French invaded Brescia during the War of the League of Cambrai. The militia of Brescia defended their city for seven days. When the French finally broke through, they took their revenge by massacring the inhabitants of Brescia. By the end of battle, over 45,000 residents were killed. During the massacre, a French soldier sliced Niccolò's jaw and palate. This made it impossible for Niccolò to speak normally, prompting the nickname "Tartaglia" (stammerer).

There is a story that Tartaglia learned only half the alphabet from a private tutor before funds ran out, and he had to learn the rest for himself. Be that as it may, he was essentially self-taught. He and his contemporaries, working outside the academies, were responsible for the spread of classic works in modern languages among the educated middle class.

His edition of Euclid in 1543, the first translation of the Elements into any modern European language, was especially significant. For two centuries Euclid had been taught from two Latin translations taken from an Arabic source; these contained errors in Book V, the Eudoxian theory of proportion, which rendered it unusable. Tartaglia's edition was based on Zamberti's Latin translation of an uncorrupted Greek text, and rendered Book V correctly. He also wrote the first modern and useful commentary on the theory. Later, the theory was an essential tool for Galileo, just as it had been for Archimedes.

Contents 1 Solution to cubic equations 2 Volume of a tetrahedron 3 Triangle 4 External links

Solution to cubic equations

Tartaglia is perhaps best known today for his conflicts with Gerolamo Cardano. Cardano nagged Tartaglia into revealing his solution to the cubic equations, by promising not to publish them. Several years later, Cardano happened to see unpublished work by Scipione del Ferro who independently came up with the same solution as Tartaglia. As the unpublished work was dated before Tartaglia's, Cardano decided his promise could be broken and included Tartaglia's solution in his next publication. Since Cardano credited his discovery, Tartaglia was extremely upset. He responded by publicly insulting Cardano.

Volume of a tetrahedron

Main article: Tetrahedron#Volume

Tartaglia is also known for having given an expression (Tartaglia's formula) for the volume of a tetrahedron (incl. any irregular tetrahedra) as the Cayley–Menger determinant of the distance values measured pairwise between its four corners:

where d _ij is the distance between vertices i and j. This is a generalization of Heron's formula for the area of a triangle.

Triangle

Tartaglia is known for having devised a method to obtain binomial coefficients called Tartaglia's Triangle (also called Pascal's Triangle).

External links

• O'Connor, John J.; Robertson, Edmund F., "Niccolò Fontana Tartaglia", MacTutor History of Mathematics archive .
• Tartaglia's work (and poetry) on the solution of the Cubic Equation at Convergence
•  "Nicolò Tartaglia". Catholic Encyclopedia. New York: Robert Appleton Company. 1913. http://en.wikisource.org/wiki/Catholic_Encyclopedia_(1913)/Nicol%C3%B2_Tartaglia. 
• The Galileo Project

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