Niccolò Fontana Tartaglia
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Nicolo, known as Tartaglia (; 1499/1500 – 13 December 1557), was an Italian
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
,
engineer Engineers, as practitioners of engineering, are professionals who Invention, invent, design, build, maintain and test machines, complex systems, structures, gadgets and materials. They aim to fulfill functional objectives and requirements while ...
(designing fortifications), a surveyor (of
topography Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the landforms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary sci ...
, seeking the best means of defense or offense) and a bookkeeper from the then
Republic of Venice The Republic of Venice, officially the Most Serene Republic of Venice and traditionally known as La Serenissima, was a sovereign state and Maritime republics, maritime republic with its capital in Venice. Founded, according to tradition, in 697 ...
. He published many books, including the first Italian translations of
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
and
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
, and an acclaimed compilation of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs, known as
ballistics Ballistics is the field of mechanics concerned with the launching, flight behaviour and impact effects of projectiles, especially weapon munitions such as bullets, unguided bombs, rockets and the like; the science or art of designing and acceler ...
, in his ''Nova Scientia'' (''A New Science'', 1537); his work was later partially validated and partially superseded by
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...
's studies on falling bodies. He also published a treatise on retrieving sunken ships.


Personal life

Nicolo was born in
Brescia Brescia (, ; ; or ; ) is a city and (municipality) in the region of Lombardy, in Italy. It is situated at the foot of the Alps, a few kilometers from the lakes Lake Garda, Garda and Lake Iseo, Iseo. With a population of 199,949, it is the se ...
, the son of Michele, a dispatch rider who travelled to neighbouring towns to deliver mail. In 1506, Michele was murdered by robbers, and Nicolo, his two siblings, and his mother were left impoverished. Nicolo experienced further tragedy in 1512 when King Louis XII's troops invaded Brescia during the
War of the League of Cambrai The War of the League of Cambrai, sometimes known as the War of the Holy League and several other names, was fought from February 1508 to December 1516 as part of the Italian Wars of 1494–1559. The main participants of the war, who fough ...
against
Venice Venice ( ; ; , formerly ) is a city in northeastern Italy and the capital of the Veneto Regions of Italy, region. It is built on a group of 118 islands that are separated by expanses of open water and by canals; portions of the city are li ...
. 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, Nicolo and his family sought sanctuary in the local cathedral. But the French entered and a soldier sliced Nicolo'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. His surname at birth, if any, is disputed. Some sources have him as "Niccolò Fontana", but others claim that the only support for this is a will in which he named a brother, Zuampiero Fontana, as heir, and point out that this does not imply he had the same surname. Tartaglia's biographer Arnoldo Masotti writes that: Tartaglia moved to
Verona Verona ( ; ; or ) is a city on the Adige, River Adige in Veneto, Italy, with 255,131 inhabitants. It is one of the seven provincial capitals of the region, and is the largest city Comune, municipality in the region and in Northeast Italy, nor ...
around 1517, then to Venice in 1534, a major European commercial hub and one of the great centres of the Italian renaissance at this time. Also relevant is Venice's place at the forefront of European printing culture in the sixteenth century, making early printed texts available even to poor scholars if sufficiently motivated or well-connected — Tartaglia knew of Archimedes' work on the quadrature of the parabola, for example, from Guarico's Latin edition of 1503, which he had found "in the hands of a sausage-seller in Verona in 1531" (''in mano di un salzizaro in Verona, l'anno 1531'' in his words). Tartaglia's mathematics is also influenced by the works of medieval Islamic scholar
Muhammad ibn Musa Al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
from 12th Century Latin translations becoming available in Europe. Tartaglia eked out a living teaching practical mathematics in
abacus school Abacus school is a term applied to any Italian school or tutorial after the 13th century, whose commerce-directed curriculum placed special emphasis on mathematics, such as algebra, among other subjects. These schools sprang up after the publicatio ...
s and earned a penny where he could: He died in Venice.


Ballistics

] ''Nova Scientia'' (1537) was Tartaglia's first published work, described by Matteo Valleriani as: Then dominant Aristotelian physics preferred categories like "heavy" and "natural" and "violent" to describe motion, generally eschewing mathematical explanations. Tartaglia brought mathematical models to the fore, "eviscerat ngAristotelian terms of projectile movement" in the words of Mary J. Henninger-Voss. One of his findings was that the maximum range of a projectile was achieved by directing the cannon at a 45° angle to the horizon. Tartaglia's model for a cannonball's flight was that it proceeded from the cannon in a straight line, then after a while started to arc towards the earth along a circular path, then finally dropped in another straight line directly towards the earth. At the end of Book 2 of ''Nova Scientia'', Tartaglia proposes to find the length of that initial rectilinear path for a projectile fired at an elevation of 45°, engaging in a Euclidean-style argument, but one with numbers attached to line segments and areas, and eventually proceeds algebraically to find the desired quantity (''procederemo per algebra'' in his words). Mary J. Henninger-Voss notes that "Tartaglia's work on military science had an enormous circulation throughout Europe", being a reference for common gunners into the eighteenth century, sometimes through unattributed translations. He influenced Galileo as well, who owned "richly annotated" copies of his works on ballistics as he set about solving the projectile problem once and for all.


Translations

Archimedes' works began to be studied outside the universities in Tartaglia's day as exemplary of the notion that mathematics is the key to understanding physics, Federigo Commandino reflecting this notion when saying in 1558 that "with respect to geometry no one of sound mind could deny that Archimedes was some god". Tartaglia published a 71-page Latin edition of Archimedes in 1543
''Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi''
containing Archimedes' works on the parabola, the circle, centres of gravity, and floating bodies. Guarico had published Latin editions of the first two in 1503, but the works on centres of gravity and floating bodies had not been published before. Tartaglia published Italian versions of some Archimedean texts later in life, his executor continuing to publish his translations after his death. Galileo probably learned of Archimedes' work through these widely disseminated editions. Tartaglia's Italian edition of
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
in 1543
''Euclide Megarense philosopho''
was especially significant as the first translation of the '' Elements'' into any modern European language. For two centuries Euclid had been taught from two
Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...
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. This work went through many editions in the sixteenth century and helped diffuse knowledge of mathematics to a non-academic but increasingly well-informed literate and numerate public in Italy. The theory became an essential tool for
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642), commonly referred to as Galileo Galilei ( , , ) or mononymously as Galileo, was an Italian astronomer, physicist and engineer, sometimes described as a poly ...
, as it had been for
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
.


''General Trattato di Numeri et Misure''

Tartaglia exemplified and eventually transcended the abaco tradition that had flourished in Italy since the twelfth century, a tradition of concrete commercial mathematics taught at
abacus school Abacus school is a term applied to any Italian school or tutorial after the 13th century, whose commerce-directed curriculum placed special emphasis on mathematics, such as algebra, among other subjects. These schools sprang up after the publicatio ...
s maintained by communities of merchants. ''Maestros d'abaco'' like Tartaglia taught not with the abacus but with paper-and-pen, inculcating algorithms of the type found in grade schools today. Tartaglia's masterpiece was the ''General Trattato di Numeri et Misure'' (''General Treatise on Number and Measure''), a 1500-page encyclopedia in six parts written in the Venetian dialect, the first three coming out in 1556 about the time of Tartaglia's death and the last three published posthumously by his literary executor and publisher Curtio Troiano in 1560. David Eugene Smith wrote of the ''General Trattato'' that it was: Part I is 554 pages long and constitutes essentially commercial arithmetic, taking up such topics as basic operations with the complex currencies of the day (ducats, soldi, pizolli, and so on), exchanging currencies, calculating interest, and dividing profits into joint companies. The book is replete with worked examples with much emphasis on methods and rules (that is, algorithms), all ready to use virtually as is. Part II takes up more general arithmetic problems, including progressions, powers, binomial expansions, Tartaglia's triangle (also known as "Pascal's triangle"), calculations with roots, and proportions / fractions. Part IV concerns triangles, regular polygons, the Platonic solids, and Archimedean topics like the quadrature of the circle and circumscribing a cylinder around a sphere.


Tartaglia's triangle

] Tartaglia was proficient with binomial expansions and included many worked examples in Part II of the ''General Trattato'', one a detailed explanation of how to calculate the summands of (6 + 4)^7, including the appropriate
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s. Tartaglia knew of
Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...
one hundred years before Pascal, as shown in this image from the ''General Trattato''. His examples are numeric, but he thinks about it geometrically, the horizontal line ab at the top of the triangle being broken into two segments ac and cb, where point c is the apex of the triangle. Binomial expansions amount to taking (ac+cb)^n for exponents n = 2, 3, 4, \cdots as you go down the triangle. The symbols along the outside represent powers at this early stage of algebraic notation: ce = 2, cu = 3, ce.ce = 4, and so on. He writes explicitly about the additive formation rule, that (for example) the adjacent 15 and 20 in the fifth row add up to 35, which appears beneath them in the sixth row.


Solution to cubic equations

Tartaglia is perhaps best known today for his conflicts with
Gerolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; ; ; 24 September 1501– 21 September 1576) was an Italian polymath whose interests and proficiencies ranged through those of mathematician, physician, biologist, physicist, chemist, astrologer, as ...
. In 1539, Cardano cajoled Tartaglia into revealing his solution to the
cubic equation In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
s by promising not to publish them. Tartaglia divulged the secrets of the solutions of three different forms of the cubic equation in verse. Several years later, Cardano happened to see unpublished work by Scipione del Ferro who independently came up with the same solution as Tartaglia. (Tartaglia had previously been challenged by del Ferro's student Fiore, which made Tartaglia aware that a solution existed.) 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. 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 was a prodigious calculator and master of solid geometry. In Part IV of the ''General Trattato'' he shows by example how to calculate the height of a pyramid on a triangular base, that is, an irregular tetrahedron.See Tartaglia, Niccolò
''General Trattato di Numeri et Misure'', Part IV, Book 2, p. 35r
for the calculation of the height of a 13-14-15-20-18-16 pyramid.
The base of the pyramid is a 13-14-15 triangle ''bcd'', and the edges rising to the apex ''a'' from points ''b'', ''c'', and ''d'' have respective lengths 20, 18, and 16. The base triangle ''bcd'' partitions into 5-12-13 and 9-12-15 triangles by dropping the perpendicular from point ''d'' to side ''bc''. He proceeds to erect a triangle in the plane perpendicular to line ''bc'' through the pyramid's apex, point ''a'', calculating all three sides of this triangle and noting that its height is the height of the pyramid. At the last step, he applies what amounts to this formula for the height ''h'' of a triangle in terms of its sides ''p'', ''q'', ''r'' (the height from side ''p'' to its opposite vertex): :h^2 = r^2 - \left(\frac\right)^2, a formula deriving from the
law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...
(not that he cites any justification in this section of the ''General Trattato''). Tartaglia drops a digit early in the calculation, taking as , but his method is sound. The final (correct) answer is: :\text = \sqrt. The volume of the pyramid is easily obtained from this, though Tartaglia does not give it: :\begin V &= \tfrac13 \times \text \times \text \\ &= \tfrac13 \times \text (\triangle bcd) \times \text \\ &= \tfrac13 \times 84 \times \sqrt \\ &\approx 433.9513222 \end
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a County_of_Flanders, Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He a ...
invented
decimal fractions The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of the ...
later in the sixteenth century, so the approximation would have been foreign to Tartaglia, who always used fractions. His approach is in some ways a modern one, suggesting by example an algorithm for calculating the height of irregular tetrahedra, but (as usual) he gives no explicit general formula.


Works

* Tartaglia, Niccolò
''General Trattato di Numeri et Misure'', Part I (Venice, 1556)
* Tartaglia, Niccolò
''General Trattato di Numeri et Misure'', Part II (Venice, 1556)
* Tartaglia, Niccolò
''General Trattato di Numeri et Misure'', Part III (Venice, 1556)
* Tartaglia, Niccolò
''General Trattato di Numeri et Misure'', Part IV (Venice, 1560)
* Tartaglia, Niccolò
''General Trattato di Numeri et Misure'', Part V (Venice, 1560)
* Tartaglia, Niccolò
''General Trattato di Numeri et Misure'', Part VI (Venice, 1560)


Notes


References

* * . * * * * . * . * * . * . * * * . * . * .


Further reading

* Valleriani, Matteo


External links


History Today


*
Tartaglia's work (and poetry) on the solution of the Cubic Equation
a
Convergence

La Nova Scientia (Venice, 1550)
{{DEFAULTSORT:Tartaglia, Niccolo 15th-century births 1557 deaths Republic of Venice people Ballistics experts People from Brescia 16th-century Italian mathematicians Italian military engineers Engineers from Venice Italian mathematicians 16th-century Italian engineers