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301:. 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.
540:
36:
899:
422:, 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.
1768:
453:'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
105:
1596:
514:
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
328:
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
293:, 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
402:
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
315:
At the age of about fourteen, he went to a Master
Francesco to learn to write the alphabet; but by the time he reached âk,â he was no longer able to pay the teacher. âFrom that day,â he later wrote in a moving autobiographical sketch, âI never returned to a tutor, but continued to labour by myself
1780:
later in the sixteenth century, so the last figure would have been foreign to
Tartaglia, who always used fractions. All the same, his approach is in some ways a modern one, suggesting by example an algorithm for calculating the height of most or all irregular tetrahedra, but (as usual for him) he
881:
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
509:
the best treatise on arithmetic that appeared in Italy in his century, containing a very full discussion of the numerical operations and the commercial rules of the
Italian arithmeticians. The life of the people, the customs of the merchants, and the efforts at improving arithmetic in the 16th
386:
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 Aristotelian terms of projectile movement" in the words of Mary J.
395:, 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 (
501:), 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
390:
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
344:
This remarkable man was a self-educated mathematics teacher who sold mathematical advice to gunners and architects, ten pennies one question, and had to litigate with his customers when they gave him a worn-out cloak for his lectures on Euclid instead of the payment agreed
1763:{\displaystyle {\begin{aligned}V&=1/3\times {\text{ base }}\times {\text{ height }}\\&=1/3\times {\text{ Area }}(\triangle bcd)\times {\text{ height }}\\&=1/3\times 84\times {\sqrt {240{\frac {615}{3136}}}}\\&\approx 433.9513222\end{aligned}}}
886:. 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 "
329:
if sufficiently motivated or well-connected â 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" (
2210:
His student, Antonio Maria Fiore, knew the solution and attempted to gain a reputation by exploiting his master's discovery. He challenged
Tartaglia with thirty questions, all of which reduced to the solution of x + ax =
1585:
1463:
873:
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
415:
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,
378:... one of the most fundamental works on mechanics of the Renaissance, indeed, the first to transform aspects of practical knowledge accumulated by the early modern artillerists into a theoretical
2021:
See Malet, Antoni, "Euclidâs Swan Song: Euclidâs Elements in Early Modern Europe", where Tartaglia's work on Euclid is described as "mathematically cogent, innovative, and influential" (p. 207).
1601:
362:
878:
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.)
857:, 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.
308:", 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.
482:
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
1507:
2562:
1537:
1279:, 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
855:
787:
525:
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.
743:
2526:
2140:
594:
949:
1191:
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1329:
1001:
411:
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,
1127:
975:
1257:
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1021:
678:
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632:
1349:
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1277:
1211:
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1081:
1061:
1041:
698:
2606:
2269:
1545:
1357:
387:
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.
2626:
2551:
471:
2433:
417:
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57:
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357:
79:
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like Tartaglia taught not with the abacus but with paper-and-pen, inculcating algorithms of the type found in grade schools today.
2621:
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2523:
887:
160:
2601:
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294:
789:
as you go down the triangle. The symbols along the outside represent powers at this early stage of algebraic notation:
2596:
916:
he shows by example how to calculate the height of a pyramid on a triangular base, that is, an irregular tetrahedron.
539:
515:
with worked examples with much emphasis on methods and rules (that is, algorithms), all ready to use virtually as is.
50:
44:
898:
61:
2556:
2374:
Malet, Antoni (2012). "Euclid's Swan Song: Euclid's Elements in Early Modern Europe". In Olmos, Paula (ed.).
534:
519:
169:
2292:
Henninger-Voss, Mary J. (July 2002). "How the 'New Science' of Cannons Shook up the Aristotelian Cosmos".
261:. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs, known as
1482:
2329:
1512:
518:
Part II takes up more general arithmetic problems, including progressions, powers, binomial expansions,
2263:
792:
553:
Tartaglia was proficient with binomial expansions and included many worked examples in Part II of the
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748:
604:
597:
437:
2546:
2542:
412:
1990:
See Henninger-Voss, Mary J., "How the 'New Science' of Cannons Shook up the Aristotelian Cosmos",
316:
over the works of dead men, accompanied only by the daughter of poverty that is called industryâ (
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2238:
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1976:
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903:
875:
703:
246:
141:
123:
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2376:
Greek Science in the Long Run: Essays on the Greek Scientific Tradition (4th c. BCE-17th c. CE)
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1947:
Henninger-Voss, Mary J., "How the 'New Science' of Cannons Shook up the Aristotelian Cosmos",
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1961:
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229:
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1933:
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522:(also known as "Pascal's triangle"), calculations with roots, and proportions / fractions.
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274:
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403:
his works on ballistics as he set about solving the projectile problem once and for all.
17:
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Metallurgy, Ballistics and Epistemic Instruments: The Nova scientia of Nicolò Tartaglia
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Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia
1978:
Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia
1963:
Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia
1935:
Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia
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Tartaglia was a prodigious calculator and master of solid geometry. In Part IV of the
611:. His examples are numeric, but he thinks about it geometrically, the horizontal line
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The volume of the pyramid is easily gotten after that (not that Tartaglia gives it):
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into any modern European language. For two centuries Euclid had been taught from two
337:
234:
205:
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2222:
1773:
449:
theory of proportion, which rendered it unusable. Tartaglia's edition was based on
2483:
Zilsel, Edgar (2000), Raven, Diederick; Krohn, Wolfgang; Cohen, Robert S. (eds.),
2003:
Clagett, Marshall, "William of Moerbeke: Translator of Archimedes", pp. 356-366.
445:
translations taken from an Arabic source; these contained errors in Book V, the
374:(1537) was Tartaglia's first published work, described by Matteo Valleriani as:
258:
195:
191:
245:, seeking the best means of defense or offense) and a bookkeeper from the then
1580:{\displaystyle {\text{ height of pyramid }}={\sqrt {240{\frac {615}{3136}}}}.}
1458:{\displaystyle h^{2}=r^{2}-\left({{p^{2}+r^{2}-q^{2}} \over {2p}}\right)^{2},}
458:
262:
250:
242:
165:
2197:
273:, 1537); his work was later partially validated and partially superseded by
173:
2394:
Masotti, Arnoldo (1970). "Niccolò Tartaglia". In Gillispie, Charles (ed.).
2278:
Clagett, Marshall (1982). "William of Moerbeke: Translator of Archimedes".
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869:. In 1539, Cardano cajoled Tartaglia into revealing his solution to the
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454:
290:
249:. He published many books, including the first Italian translations of
119:
304:
His surname at birth, if any, is disputed. Some sources have him as "
1236:. He proceeds to erect a triangle in the plane perpendicular to line
426:
325:
298:
254:
137:
2435:
Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi
419:
Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi
2563:
Tartaglia's work (and poetry) on the solution of the Cubic Equation
897:
700:
is the apex of the triangle. Binomial expansions amount to taking
538:
470:
442:
356:
2398:. New York: Scribner & American Council of Learned Societies.
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for the calculation of the height of a 13-14-15-20-18-16 pyramid.
607:
one hundred years before Pascal, as shown in this image from the
557:, one a detailed explanation of how to calculate the summands of
336:
Tartaglia eked out a living teaching practical mathematics in
29:
2468:, Berlin: Edition Open Access / Max Planck Research Library,
865:
Tartaglia is perhaps best known today for his conflicts with
435:, was especially significant as the first translation of the
2012:
Henninger-Voss, Mary J., "'New Science' of Cannons", p. 392.
1472:(not that he cites any justification in this section of the
2273:. Vol. 26 (11th ed.). Cambridge University Press.
2158:
for discussion of the additive rule in "Pascal's triangle".
281:. He also published a treatise on retrieving sunken ships.
1539:, but his method is sound. The final (correct) answer is:
634:
at the top of the triangle being broken into two segments
1994:
63, 3 (July 2002), pp. 391-393 for discussion and quotes.
1479:
Tartaglia drops a digit early in the calculation, taking
2227:
Cardano v Tartaglia: The Great Feud Goes Supernatural.
1951:
63, 3 (July 2002), pp. 371-397. "eviscerated": p. 376.
233:; 1499/1500 – 13 December 1557), was an Italian
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Tartaglia's biographer Arnoldo Masotti writes that:
2378:. Cambridge Scholars Publishing. pp. 205â234.
1193:triangles by dropping the perpendicular from point
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510:century are all set forth in this remarkable work.
2359:(2nd ed.), Reading: Addison Wesley Longman,
2280:Proceedings of the American Philosophical Society
507:
376:
342:
313:
331:in mano di un salzizaro in Verona, l'anno 1531
2407:, vol. I, New York: Dover Publications,
8:
2344:A philosophical and mathematical dictionary
241:(designing fortifications), a surveyor (of
104:
93:
2355:A History of Mathematics: An Introduction
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2108:
1874:
1863:Galileo at Work: His Scientific Biography
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80:Learn how and when to remove this message
486:maintained by communities of merchants.
43:This article includes a list of general
2552:MacTutor History of Mathematics Archive
1851:
2346:. Printed for the author. p. 482.
499:General Treatise on Number and Measure
361:Various projectile trajectories from
228:
7:
2485:The Social Origins of Modern Science
2456:General Trattato di Numeri et Misure
2333:. New York: Robert Appleton Company.
2240:General Trattato di Numeri et Misure
2167:
2154:General Trattato di Numeri et Misure
2099:General Trattato di Numeri et Misure
2083:General Trattato di Numeri et Misure
2067:General Trattato di Numeri et Misure
2051:General Trattato di Numeri et Misure
1921:The Social Origins of Modern Science
1906:General Trattato di Numeri et Misure
1838:General Trattato di Numeri et Misure
1829:General Trattato di Numeri et Misure
1820:General Trattato di Numeri et Misure
1811:General Trattato di Numeri et Misure
1802:General Trattato di Numeri et Misure
1793:General Trattato di Numeri et Misure
1299:of a triangle in terms of its sides
905:General Trattato di Numeri et Misure
546:General Trattato di Numeri et Misure
495:General Trattato di Numeri et Misure
476:General trattato di numeri et misure
466:General Trattato di Numeri et Misure
2607:16th-century Italian mathematicians
2340:"Tartaglia or Tartaglia (Nicholas)"
1502:{\displaystyle 305{\frac {31}{49}}}
902:13-14-15-20-18-16 pyramid from the
340:and earned a penny where he could:
2396:Dictionary of Scientific Biography
1892:Dictionary of Scientific Biography
1673:
1532:{\displaystyle 305{\frac {3}{49}}}
1259:through the pyramid's apex, point
257:, and an acclaimed compilation of
49:it lacks sufficient corresponding
25:
2321:Herbermann, Charles, ed. (1913).
850:{\displaystyle ce=2,cu=3,ce.ce=4}
27:Italian mathematician (1499â1557)
2454:Tartaglia, Niccolò (1556â1560),
493:Tartaglia's masterpiece was the
34:
2572:La Nova Scientia (Venice, 1550)
2294:Journal of the History of Ideas
2182:"The Cardano-Tartaglia dispute"
1992:Journal of the History of Ideas
1949:Journal of the History of Ideas
782:{\displaystyle n=2,3,4,\cdots }
425:Tartaglia's Italian edition of
2627:16th-century Italian engineers
2123:
2110:
1685:
1670:
726:
707:
577:
564:
1:
2426:, New York, NY: Pegasus Books
2180:Feldmann, Richard W. (1961).
2030:Tartaglia, Niccolò, 1556-1560
1551: height of pyramid
919:The base of the pyramid is a
2446:Euclide Megarense philosopho
1468:a formula deriving from the
1103:respectively. Base triangle
596:, including the appropriate
432:Euclide Megarense philosopho
295:War of the League of Cambrai
2464:Valleriani, Matteo (2013),
2443:Tartaglia, Niccolò (1543).
2432:Tartaglia, Niccolò (1543).
1781:gives no explicit formula.
861:Solution to cubic equations
738:{\displaystyle (ac+cb)^{n}}
2643:
2612:Italian military engineers
2264:"Tartaglia, Niccolò"
543:Tartaglia's triangle from
532:
2242:, Part IV, Book 2, p. 35r
2135:{\displaystyle (6+4)^{7}}
2101:, Part II, Book 2, p. 51v
1813:, Part III (Venice, 1556)
1351:to its opposite vertex):
907:, Part IV, Book 2, p. 35.
888:CardanoâTartaglia formula
589:{\displaystyle (6+4)^{7}}
548:, Part II, Book 2, p. 69.
211:
180:
161:CardanoâTartaglia formula
103:
18:Niccolò Fontana Tartaglia
2557:University of St Andrews
2487:, Springer Netherlands,
2458:, Venice: Curtio Troiano
2422:Strathern, Paul (2013),
2351:Katz, Victor J. (1998),
2324:"Nicolò Tartaglia"
2237:See Tartaglia, Niccolò.
2156:, Part II, Book 2, p. 72
2151:See Tartaglia, Niccolò.
2096:See Tartaglia, Niccolò.
1975:See Valleriani, Matteo,
1960:See Valleriani, Matteo,
1932:See Valleriani, Matteo,
1908:, Part IV, Book 3, p. 43
1903:See Tartaglia, Niccolò.
1840:, Part VI (Venice, 1560)
1822:, Part IV (Venice, 1560)
1804:, Part II (Venice, 1556)
944:{\displaystyle 13-14-15}
528:
2529:22 January 2012 at the
2338:Charles Hutton (1815).
2270:EncyclopĂŚdia Britannica
2186:The Mathematics Teacher
1910:for the sausage seller.
1831:, Part V (Venice, 1560)
1795:, Part I (Venice, 1556)
1186:{\displaystyle 9-12-15}
1154:{\displaystyle 5-12-13}
977:, with edges of length
894:Volume of a tetrahedron
397:procederemo per algebra
382:mathematical framework.
64:more precise citations.
2622:Italian mathematicians
2405:History of Mathematics
2136:
1764:
1581:
1533:
1503:
1459:
1345:
1331:(the height from side
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783:
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628:
590:
550:
512:
479:
384:
368:
347:
322:
320:, bk. VI, question 8).
2617:Engineers from Venice
2330:Catholic Encyclopedia
2306:10.1353/jhi.2002.0029
2137:
1765:
1582:
1534:
1504:
1460:
1346:
1326:
1324:{\displaystyle p,q,r}
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1274:
1254:
1231:
1208:
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1038:
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996:{\displaystyle 20,18}
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598:binomial coefficients
591:
542:
474:
457:, as it had been for
360:
2543:Robertson, Edmund F.
2508:Valleriani, Matteo,
2403:Smith, D.E. (1958),
2107:
2080:Tartaglia, Niccolò.
2064:Tartaglia, Niccolò.
2048:Tartaglia, Niccolò.
1980:, 2013, pp. 176-177.
1965:, 2013, pp. 169-181.
1865:, Dover, 1978, p. 3.
1835:Tartaglia, Niccolò,
1826:Tartaglia, Niccolò,
1817:Tartaglia, Niccolò,
1808:Tartaglia, Niccolò,
1799:Tartaglia, Niccolò,
1790:Tartaglia, Niccolò,
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1513:
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535:Tartaglia's triangle
529:Tartaglia's triangle
520:Tartaglia's triangle
170:Tartaglia's triangle
164:Early research into
2602:People from Brescia
2587:15th-century births
2541:O'Connor, John J.;
2536:The Galileo Project
2039:Smith 1985, p. 298.
1122:{\displaystyle bcd}
970:{\displaystyle bcd}
882:Cardano's student,
413:Federigo Commandino
349:He died in Venice.
324:Tartaglia moved to
289:Nicolo was born in
230:[tarËtaĘĘa]
2597:Ballistics experts
2547:"Nicolo Tartaglia"
2132:
1886:Masotti, Arnoldo,
1760:
1758:
1693: height
1638: height
1577:
1529:
1499:
1455:
1341:
1321:
1289:
1269:
1252:{\displaystyle bc}
1249:
1229:{\displaystyle bc}
1226:
1203:
1183:
1151:
1119:
1093:
1073:
1053:
1033:
1016:{\displaystyle 16}
1013:
993:
967:
941:
910:
876:Scipione del Ferro
847:
779:
735:
690:
673:{\displaystyle cb}
670:
650:{\displaystyle ac}
647:
627:{\displaystyle ab}
624:
603:Tartaglia knew of
586:
551:
480:
369:
247:Republic of Venice
142:Republic of Venice
124:Republic of Venice
2475:978-3-8442-5258-3
2385:978-1-4438-3775-0
1888:Niccolò Tartaglia
1778:decimal fractions
1741:
1739:
1694:
1668:
1639:
1631:
1572:
1570:
1552:
1527:
1497:
1440:
1344:{\displaystyle p}
1292:{\displaystyle h}
1272:{\displaystyle a}
1206:{\displaystyle d}
1096:{\displaystyle d}
1076:{\displaystyle c}
1056:{\displaystyle b}
1036:{\displaystyle a}
693:{\displaystyle c}
605:Pascal's triangle
215:
214:
182:Scientific career
90:
89:
82:
16:(Redirected from
2634:
2559:
2497:
2478:
2459:
2450:
2439:
2427:
2417:
2399:
2389:
2369:
2358:
2347:
2334:
2326:
2317:
2287:
2274:
2266:
2245:
2235:
2229:
2220:
2214:
2213:
2177:
2171:
2165:
2159:
2149:
2143:
2141:
2139:
2138:
2133:
2131:
2130:
2094:
2088:
2078:
2072:
2062:
2056:
2046:
2040:
2037:
2031:
2028:
2022:
2019:
2013:
2010:
2004:
2001:
1995:
1988:
1982:
1973:
1967:
1958:
1952:
1945:
1939:
1930:
1924:
1917:
1911:
1901:
1895:
1884:
1878:
1872:
1866:
1856:
1769:
1767:
1766:
1761:
1759:
1746:
1742:
1740:
1732:
1727:
1713:
1699:
1695:
1692:
1669:
1667: Area
1666:
1658:
1644:
1640:
1637:
1632:
1630: base
1629:
1621:
1586:
1584:
1583:
1578:
1573:
1571:
1563:
1558:
1553:
1550:
1538:
1536:
1535:
1530:
1528:
1520:
1508:
1506:
1505:
1500:
1498:
1490:
1474:General Trattato
1464:
1462:
1461:
1456:
1451:
1450:
1445:
1441:
1439:
1431:
1430:
1429:
1417:
1416:
1404:
1403:
1393:
1383:
1382:
1370:
1369:
1350:
1348:
1347:
1342:
1330:
1328:
1327:
1322:
1298:
1296:
1295:
1290:
1278:
1276:
1275:
1270:
1258:
1256:
1255:
1250:
1235:
1233:
1232:
1227:
1212:
1210:
1209:
1204:
1192:
1190:
1189:
1184:
1160:
1158:
1157:
1152:
1129:partitions into
1128:
1126:
1125:
1120:
1102:
1100:
1099:
1094:
1082:
1080:
1079:
1074:
1062:
1060:
1059:
1054:
1042:
1040:
1039:
1034:
1022:
1020:
1019:
1014:
1002:
1000:
999:
994:
976:
974:
973:
968:
950:
948:
947:
942:
914:General Trattato
884:Ludovico Ferrari
867:Gerolamo Cardano
856:
854:
853:
848:
788:
786:
785:
780:
744:
742:
741:
736:
734:
733:
699:
697:
696:
691:
679:
677:
676:
671:
656:
654:
653:
648:
633:
631:
630:
625:
609:General Trattato
595:
593:
592:
587:
585:
584:
555:General Trattato
503:General Trattato
488:Maestros d'abaco
232:
227:
202:Notable students
134:13 December 1557
108:
94:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
2642:
2641:
2637:
2636:
2635:
2633:
2632:
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2577:
2576:
2540:
2531:Wayback Machine
2520:
2505:
2503:Further reading
2495:
2482:
2476:
2463:
2453:
2442:
2431:
2421:
2415:
2402:
2393:
2386:
2373:
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2350:
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2320:
2291:
2277:
2257:
2254:
2249:
2248:
2236:
2232:
2221:
2217:
2179:
2178:
2174:
2166:
2162:
2150:
2146:
2122:
2105:
2104:
2095:
2091:
2079:
2075:
2063:
2059:
2047:
2043:
2038:
2034:
2029:
2025:
2020:
2016:
2011:
2007:
2002:
1998:
1989:
1985:
1974:
1970:
1959:
1955:
1946:
1942:
1931:
1927:
1919:Zilsel, Edgar,
1918:
1914:
1902:
1898:
1885:
1881:
1873:
1869:
1857:
1853:
1848:
1787:
1757:
1756:
1744:
1743:
1697:
1696:
1642:
1641:
1607:
1595:
1594:
1544:
1543:
1511:
1510:
1481:
1480:
1421:
1408:
1395:
1388:
1387:
1374:
1361:
1356:
1355:
1333:
1332:
1301:
1300:
1281:
1280:
1261:
1260:
1238:
1237:
1215:
1214:
1195:
1194:
1163:
1162:
1131:
1130:
1105:
1104:
1085:
1084:
1065:
1064:
1045:
1044:
1025:
1024:
1005:
1004:
979:
978:
953:
952:
921:
920:
896:
871:cubic equations
863:
791:
790:
747:
746:
725:
702:
701:
682:
681:
659:
658:
636:
635:
613:
612:
576:
559:
558:
537:
531:
469:
409:
399:in his words).
355:
333:in his words).
306:Niccolò Fontana
287:
225:
172:
168:
163:
144:
135:
126:
117:
99:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
2640:
2638:
2630:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2579:
2578:
2575:
2574:
2569:
2560:
2538:
2533:
2519:
2518:External links
2516:
2515:
2514:
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2500:
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2440:
2429:
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2413:
2400:
2391:
2384:
2371:
2365:
2348:
2335:
2318:
2300:(3): 371â397.
2289:
2275:
2261:, ed. (1911).
2259:Chisholm, Hugh
2253:
2250:
2247:
2246:
2230:
2215:
2192:(3): 160â163.
2172:
2160:
2144:
2129:
2125:
2121:
2118:
2115:
2112:
2103:for expanding
2089:
2073:
2057:
2041:
2032:
2023:
2014:
2005:
1996:
1983:
1968:
1953:
1940:
1925:
1912:
1896:
1879:
1875:Strathern 2013
1867:
1859:Stillman Drake
1850:
1849:
1847:
1844:
1843:
1842:
1833:
1824:
1815:
1806:
1797:
1786:
1783:
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1698:
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1678:
1675:
1672:
1664:
1661:
1657:
1653:
1650:
1647:
1645:
1643:
1635:
1627:
1624:
1620:
1616:
1613:
1610:
1608:
1606:
1603:
1602:
1588:
1587:
1576:
1569:
1566:
1561:
1556:
1526:
1523:
1518:
1496:
1493:
1488:
1470:Law of Cosines
1466:
1465:
1454:
1449:
1444:
1438:
1435:
1428:
1424:
1420:
1415:
1411:
1407:
1402:
1398:
1391:
1386:
1381:
1377:
1373:
1368:
1364:
1340:
1320:
1317:
1314:
1311:
1308:
1288:
1268:
1248:
1245:
1225:
1222:
1202:
1182:
1179:
1176:
1173:
1170:
1150:
1147:
1144:
1141:
1138:
1118:
1115:
1112:
1092:
1072:
1052:
1032:
1012:
992:
989:
986:
966:
963:
960:
940:
937:
934:
931:
928:
895:
892:
862:
859:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
778:
775:
772:
769:
766:
763:
760:
757:
754:
745:for exponents
732:
728:
724:
721:
718:
715:
712:
709:
689:
680:, where point
669:
666:
646:
643:
623:
620:
583:
579:
575:
572:
569:
566:
533:Main article:
530:
527:
484:abacus schools
468:
463:
408:
405:
354:
351:
338:abacus schools
286:
283:
279:falling bodies
277:'s studies on
213:
212:
209:
208:
203:
199:
198:
189:
185:
184:
178:
177:
158:
157:Known for
154:
153:
150:
146:
145:
136:
132:
128:
127:
118:
114:
110:
109:
101:
100:
97:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2639:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2584:
2582:
2573:
2570:
2568:
2564:
2561:
2558:
2554:
2553:
2548:
2544:
2539:
2537:
2534:
2532:
2528:
2525:
2524:History Today
2522:
2521:
2517:
2513:
2512:
2507:
2506:
2502:
2496:
2494:0-7923-6457-0
2490:
2486:
2481:
2477:
2471:
2467:
2462:
2457:
2452:
2448:
2447:
2441:
2437:
2436:
2430:
2425:
2420:
2416:
2414:0-486-20429-4
2410:
2406:
2401:
2397:
2392:
2387:
2381:
2377:
2372:
2368:
2366:0-321-01618-1
2362:
2357:
2356:
2349:
2345:
2341:
2336:
2332:
2331:
2325:
2319:
2315:
2311:
2307:
2303:
2299:
2295:
2290:
2286:(5): 356â366.
2285:
2281:
2276:
2272:
2271:
2265:
2260:
2256:
2255:
2251:
2243:
2241:
2234:
2231:
2228:
2224:
2219:
2216:
2212:
2207:
2203:
2199:
2195:
2191:
2187:
2183:
2176:
2173:
2169:
2164:
2161:
2157:
2155:
2148:
2145:
2127:
2119:
2116:
2113:
2102:
2100:
2093:
2090:
2086:
2084:
2077:
2074:
2070:
2068:
2061:
2058:
2054:
2052:
2045:
2042:
2036:
2033:
2027:
2024:
2018:
2015:
2009:
2006:
2000:
1997:
1993:
1987:
1984:
1981:
1979:
1972:
1969:
1966:
1964:
1957:
1954:
1950:
1944:
1941:
1938:
1937:, 2013, p. 1.
1936:
1929:
1926:
1922:
1916:
1913:
1909:
1907:
1900:
1897:
1893:
1889:
1883:
1880:
1876:
1871:
1868:
1864:
1860:
1855:
1852:
1845:
1841:
1839:
1834:
1832:
1830:
1825:
1823:
1821:
1816:
1814:
1812:
1807:
1805:
1803:
1798:
1796:
1794:
1789:
1788:
1784:
1782:
1779:
1775:
1753:
1750:
1748:
1736:
1733:
1728:
1723:
1720:
1717:
1714:
1710:
1706:
1703:
1701:
1688:
1682:
1679:
1676:
1662:
1659:
1655:
1651:
1648:
1646:
1633:
1625:
1622:
1618:
1614:
1611:
1609:
1604:
1593:
1592:
1591:
1574:
1567:
1564:
1559:
1554:
1542:
1541:
1540:
1524:
1521:
1516:
1494:
1491:
1486:
1477:
1475:
1471:
1452:
1447:
1442:
1436:
1433:
1426:
1422:
1418:
1413:
1409:
1405:
1400:
1396:
1389:
1384:
1379:
1375:
1371:
1366:
1362:
1354:
1353:
1352:
1338:
1318:
1315:
1312:
1309:
1306:
1286:
1266:
1246:
1243:
1223:
1220:
1200:
1180:
1177:
1174:
1171:
1168:
1148:
1145:
1142:
1139:
1136:
1116:
1113:
1110:
1090:
1070:
1050:
1030:
1010:
990:
987:
984:
964:
961:
958:
938:
935:
932:
929:
926:
917:
915:
908:
906:
900:
893:
891:
889:
885:
879:
877:
872:
868:
860:
858:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
776:
773:
770:
767:
764:
761:
758:
755:
752:
730:
722:
719:
716:
713:
710:
687:
667:
664:
644:
641:
621:
618:
610:
606:
601:
599:
581:
573:
570:
567:
556:
549:
547:
541:
536:
526:
523:
521:
516:
511:
506:
505:that it was:
504:
500:
496:
491:
489:
485:
477:
473:
467:
464:
462:
460:
456:
452:
448:
444:
440:
439:
434:
433:
428:
423:
421:
420:
414:
406:
404:
400:
398:
394:
393:Nova Scientia
388:
383:
381:
375:
373:
372:Nova Scientia
367:
365:
364:Nova Scientia
359:
352:
350:
346:
341:
339:
334:
332:
327:
321:
319:
312:
309:
307:
302:
300:
296:
292:
285:Personal life
284:
282:
280:
276:
272:
271:A New Science
268:
267:Nova Scientia
264:
260:
256:
252:
248:
244:
240:
236:
235:mathematician
231:
223:
219:
210:
207:
206:Ostilio Ricci
204:
200:
197:
193:
190:
186:
183:
179:
175:
171:
167:
162:
159:
155:
151:
147:
143:
139:
133:
129:
125:
121:
115:
111:
107:
102:
95:
92:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
2550:
2510:
2484:
2465:
2455:
2445:
2434:
2423:
2404:
2395:
2375:
2354:
2343:
2328:
2297:
2293:
2283:
2279:
2268:
2239:
2233:
2223:Tony Rothman
2218:
2209:
2189:
2185:
2175:
2163:
2153:
2147:
2098:
2092:
2082:
2076:
2066:
2060:
2050:
2044:
2035:
2026:
2017:
2008:
1999:
1991:
1986:
1977:
1971:
1962:
1956:
1948:
1943:
1934:
1928:
1920:
1915:
1905:
1899:
1891:
1887:
1882:
1870:
1862:
1854:
1837:
1828:
1819:
1810:
1801:
1792:
1774:Simon Stevin
1772:
1589:
1478:
1473:
1467:
1043:from points
918:
913:
911:
904:
880:
864:
608:
602:
554:
552:
545:
524:
517:
513:
508:
502:
498:
494:
492:
487:
481:
475:
465:
436:
431:
424:
418:
410:
407:Translations
401:
396:
392:
389:
385:
379:
377:
371:
370:
363:
348:
343:
335:
330:
323:
317:
314:
310:
305:
303:
288:
270:
266:
221:
217:
216:
181:
91:
76:
67:
48:
2592:1557 deaths
2567:Convergence
1754:433.9513222
259:mathematics
220:, known as
196:engineering
192:Mathematics
149:Nationality
62:introducing
2581:Categories
2252:References
459:Archimedes
353:Ballistics
263:ballistics
251:Archimedes
243:topography
166:ballistics
70:April 2014
45:references
2449:. Venice.
2438:. Venice.
2424:Venetians
2314:170464547
2198:0025-5769
2168:Katz 1998
2085:, Part IV
2069:, Part II
1776:invented
1751:≈
1724:×
1718:×
1689:×
1674:△
1663:×
1634:×
1626:×
1419:−
1385:−
1178:−
1172:−
1146:−
1140:−
951:triangle
936:−
930:−
777:⋯
429:in 1543,
265:, in his
222:Tartaglia
174:Artillery
116:1499/1500
98:Tartaglia
2527:Archived
2206:27956338
2170:, p. 359
2053:, Part I
1923:, p. 35.
1877:, p. 189
1213:to side
451:Zamberti
447:Eudoxian
438:Elements
297:against
239:engineer
226:Italian:
1890:in the
455:Galileo
318:Quesiti
291:Brescia
275:Galileo
152:Italian
120:Brescia
58:improve
2491:
2472:
2411:
2382:
2363:
2312:
2204:
2196:
1083:, and
1003:, and
478:, 1556
427:Euclid
326:Verona
299:Venice
255:Euclid
218:Nicolo
188:Fields
176:theory
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