# Niccolò Fontana Tartaglia

**Niccolò Fontana Tartaglia** (Italian: [nikkoˈlɔ ffonˈtaːna tarˈtaʎʎa]; 1499/1500, Brescia – 13 December 1557, Venice) was an Italian mathematician, 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, known as ballistics, in his *Nova Scientia* (*A New Science*); his work was later partially validated and partially superseded by Galileo's studies on falling bodies. He also published a treatise on retrieving sunken ships.

### Personal life

Niccolò Fontana was the son of Michele Fontana, a dispatch rider who travelled to neighboring towns to deliver mail. But in 1506, Michele was murdered by robbers, and Niccolò, his two siblings, and his mother were left impoverished. Niccolò experienced further tragedy in 1512 when the King Louis XII's troops invaded Brescia during the War of the League of Cambrai against Venice. 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, Niccolò and his family sought sanctuary in the local cathedral. But the French entered and a soldier sliced Niccolò's jaw and palate with a saber and left him for dead. His mother nursed him back to health but the young boy was left with a speech impediment, prompting the nickname "Tartaglia" ("stammerer"). After this he would never shave, and grew a beard to camouflage his scars.^{[1]}

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 by 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 classical works in modern languages among the educated middle class.

### Major works

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.

However, his best known work is his treatise * General Trattato di Numeri et Misure* published in Venice 1556–1560.^{[2]} This has been called the *best* treatise on arithmetic that appeared in the sixteenth century.^{[3]} Not only does Tartaglia have complete discussions of numerical operations and the commercial rules used by Italian arithmeticians in this work, but he also discusses the life of the people, the customs of merchants and the efforts made to improve arithmetic in the 16th century.

### Solution to cubic equations

Tartaglia is perhaps best known today for his conflicts with Gerolamo Cardano. Cardano cajoled Tartaglia into revealing his solution to the cubic equations by promising not to publish them. Tartaglia divulged the secrets of the solutions of three different forms of the cubic equation in verse.^{[4]} 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. Even though Cardano credited his discovery, Tartaglia was extremely upset and a famous public challenge match resulted between himself and Cardano's student, Ludovico Ferrari. Widespread stories that Tartaglia devoted the rest of his life to ruining Cardano, however, appear to be completely fabricated.^{[5]} Mathematical historians now credit both Cardano and Tartaglia with the formula to solve cubic equations, referring to it as the "Cardano–Tartaglia formula".

### Volume of a tetrahedron

Tartaglia is also known for having given an expression (**Tartaglia's formula**) for the volume of a tetrahedron (including 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.

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