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(commonly known as projectile motion); 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.

Tartaglia was self taught in mathematics but, having an extraordinary ability, his mother was able to find him a patron. Ludovico Balbisonio took him to Padua to study there, but when he returned with his patron to Brescia he made himself unpopular by having an inflated opinion of himself. He left Brescia to earn his living teaching mathematics at Verona which he did between 1516 and 1518. Later, still in Verona, he taught at a school in the Palazzo Mizzanti but it is recorded that at that time he was married with a family, yet was very poor. He moved to Venice in 1534. As a lowly mathematics teacher in Venice, Tartaglia gradually acquired a reputation as a promising mathematician by participating successfully in a large number of debates.

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.

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.

The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement. On his deathbed, however, del Ferro passed on the secret to his (rather poor) student Fior. For mathematicians of this time there was more than one type of cubic equation and Fior had only been shown by del Ferro how to solve one type, namely 'unknowns and cubes equal to numbers' or (in modern notation) x3 + ax = b. As negative numbers were not used this led to a number of other cases, even for equations without a square term. Fior began to boast that he was able to solve cubics and a challenge between him and Tartaglia was arranged in 1535. In fact Tartaglia had also discovered how to solve one type of cubic equation since his friend Zuanne da Coi had set two problems which had led Tartaglia to a general solution of a different type from that which Fior could solve, namely 'squares and cubes equal to numbers' or (in modern notation) x3 + ax2 = b. For the contest between Tartaglia and Fior, each man was to submit thirty questions for the other to solve. Fior was supremely confident that his ability to solve cubics would be enough to defeat Tartaglia but Tartaglia submitted a variety of different questions, exposing Fior as an, at best, mediocre mathematician. Fior, on the other hand, offered Tartaglia thirty opportunities to solve the 'unknowns and cubes' problem since he believed that he would be unable to solve this type, as in fact had been the case when the contest was set up. However, in the early hours of 13 February 1535, inspiration came to Tartaglia and he discovered the method to solve 'squares and cubes equal to numbers'. Tartaglia was then able to solve all thirty of Fior's problems in less than two hours. As Fior had made little headway with Tartaglia's questions, it was obvious to all who was the winner. Tartaglia did not take his prize for winning from Fior, however, the honour of winning was enough.

Volume of a tetrahedron

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

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

1.A man dies leaving 17 horses to be divide amongst his heirs in the ratio 1/2:1/3:1/9.How can this be done.

Fontana's solution involved borrowing an extra horse to calculate the distribution, politely returning it after the fact one need to only multiply the solution by 18 to arrive at the solution 9,6,2.

2.How to put 3 quarts of liquid in two 10 quarts container ,using a five quart and four quart measures.

This problem is is similar to the "tower of hanoi type problems",which is originally said to be put forward by chienese around 100 b.c,these kind of problems are of utmost philosiphical value,and is often referred to the class of problem the describes the working of nature.Acyually satisfaction of human need and also depicts the principle of least action.

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