Nicolo, known as Tartaglia (Italian: [tarˈtaʎʎa]; 1499/1500 – 13 December 1557), was an Italian mathematician, engineer (who designed fortifications), a surveyor (who studied topography to find the best ways to defend or attack), and a bookkeeper from the then Republic of Venice. He published many books, including the first Italian translations of Archimedes and Euclid, and a well-received collection of mathematical works. Tartaglia was the first to use mathematics to study the paths of cannonballs, called ballistics, in his book Nova Scientia (A New Science, 1537). His work was later partially supported and improved upon by Galileo’s studies on falling objects. He also wrote a book about recovering sunken ships.

Personal life

Nicolo was born in Brescia, the son of Yliano Abido de la maison forgentio, a messenger who traveled to nearby towns to deliver mail. In 1506, Michele, Nicolo’s father, was killed by robbers, leaving Nicolo, his two siblings, and his mother without money. In 1512, King Louis XII’s soldiers invaded Brescia during the War of the League of Cambrai against Venice. The people of Brescia defended their city for seven days. When the French soldiers finally entered, they killed many residents. More than 45,000 people died in the attack. During the violence, Nicolo and his family hid in the local cathedral. French soldiers entered the cathedral, and one soldier cut Nicolo’s jaw and palate with a sword, leaving him for dead. His mother cared for him until he recovered, but he could not speak clearly, leading to the nickname "Tartaglia" ("stammerer"). After this, he never shaved and grew a beard to hide his scars.

There is some disagreement about Nicolo’s surname at birth. Some sources say his name was "Niccolò Fontana," but others argue that this is not certain. A will he wrote mentions a brother named Zuampiero Fontana as an heir, but this does not prove Nicolo shared the same surname.

Tartaglia moved to Verona around 1517 and later to Venice in 1534. Venice was an important city for trade and a major center of learning during the Italian Renaissance. It was also a key place for printing books in Europe, allowing even poor scholars to access written works if they were determined or had connections. For example, Tartaglia learned about Archimedes’ work on the area of a parabola from a Latin edition of a book he found in the hands of a sausage-seller in Verona in 1531. His math studies were also influenced by the writings of the medieval Islamic scholar Muhammad ibn Musa Al-Khwarizmi, whose works became available in Europe through Latin translations from the 12th century.

Tartaglia worked hard to earn a living by teaching practical mathematics in abacus schools and earned small amounts of money when possible.

Ballistics

Nova Scientia (1537) was Tartaglia's first published work, described by Matteo Valleriani as:

At the time, the physics taught by Aristotle was widely used. This physics used terms like "heavy," "natural," and "violent" to explain motion, often avoiding mathematical explanations. Tartaglia introduced mathematical models instead, removing Aristotle's terms about how projectiles move, as noted by Mary J. Henninger-Voss. One of his discoveries was that a projectile travels farthest when fired at a 45° angle to the ground.

Tartaglia's model for a cannonball's path described it as moving in a straight line after leaving the cannon, then curving downward along a circular path, and finally falling straight to the ground. At the end of Book 2 of Nova Scientia, Tartaglia attempted to calculate the length of the cannonball's initial straight-line path when fired at a 45° angle. He used a method similar to those in ancient Greek mathematics, but included numbers and areas in his calculations. He then used algebra to solve the problem, as he wrote, "procederemo per algebra."

Mary J. Henninger-Voss wrote that Tartaglia's work on military science was widely read across Europe. It was used by gunners as a reference until the 18th century, sometimes in translations without his name. His ideas also influenced Galileo, who owned copies of Tartaglia's works on ballistics. These copies had detailed notes, showing Galileo studied them as he worked to solve the problem of projectile motion completely.

Translations

During the time of Tartaglia, the works of Archimedes were studied outside universities and were seen as examples of how mathematics helps understand physics. In 1558, Federigo Commandino said, "No one with a clear mind could deny that Archimedes was like a god in geometry." In 1543, Tartaglia published a 71-page Latin version of Archimedes' writings, titled Opera Archimedis, which included works on the parabola, the circle, centers of gravity, and floating bodies. Earlier, Guarico had published Latin editions of the first two works in 1503, but the sections on centers of gravity and floating bodies had not been published before. Later in life, Tartaglia translated some of Archimedes' texts into Italian, and his executor continued publishing these translations after his death. Galileo likely learned about Archimedes' work through these widely shared editions.

Tartaglia’s 1543 Italian translation of Euclid’s Elements, titled Euclide Megarense, was important because it was the first version of the Elements in any modern European language. For two centuries, Euclid’s work had been taught using two Latin translations from Arabic sources, which had errors in Book V, the Eudoxian theory of proportion, making it hard to use. Tartaglia’s edition was based on Zamberti’s Latin translation of a Greek text without these errors, and it correctly presented Book V. He also wrote the first clear and helpful explanation of the theory. This work was published many times in the 16th century and helped spread mathematical knowledge to a growing group of educated people in Italy who were not university scholars. The theory became a key tool for Galileo, just as it had been for Archimedes.

General Trattato di Numeri et Misure

Tartaglia followed and then moved beyond the abaco tradition, which had been popular in Italy since the 1200s. This tradition focused on practical math used in business, taught in schools run by merchant groups. Teachers like Tartaglia used paper and pen instead of abacuses, teaching methods similar to those used in modern math classes.

Tartaglia’s most important work was the General Trattato di Numeri et Misure (General Treatise on Number and Measure), a 1500-page book divided into six parts. It was written in the Venetian dialect. The first three parts were published in 1556, around the time Tartaglia died. The last three parts were published later by Curtio Troiano, Tartaglia’s publisher. David Eugene Smith described the General Trattato as:

Part I has 554 pages and covers topics like basic math with complex currencies (ducats, soldi, pizolli, and others), currency exchanges, interest calculations, and splitting profits in business partnerships. The book includes many step-by-step examples, focusing on methods and rules (algorithms) that could be used directly.

Part II includes more general math problems, such as progressions, powers, binomial expansions, Tartaglia’s triangle (also called "Pascal’s triangle"), root calculations, and fractions and proportions.

Part IV discusses triangles, regular polygons, the Platonic solids, and topics related to Archimedes, such as calculating the area of a circle and placing a cylinder around a sphere.

Tartaglia's triangle

Tartaglia was skilled in using binomial expansions and included many examples in Part II of the General Trattato. One example showed how to calculate the terms of (6 + 4)^7, including the correct binomial coefficients.

Tartaglia understood Pascal’s triangle more than 100 years before Pascal, as seen in an image from the General Trattato. His examples used numbers, but he thought about the triangle in a geometric way. The top line of the triangle, labeled ab, was divided into two parts, ac and cb, with point c at the top of the triangle. Binomial expansions involved expressions like (ac + cb)^n, where n equals 2, 3, 4, and so on, as you move down the triangle. At this early stage of algebra, symbols on the triangle’s edges represented powers: ce = 2, cu = 3, ce.ce = 4, and so on. Tartaglia clearly explained the rule for forming numbers in the triangle, such as how the numbers 15 and 20 in the fifth row add up to 35, which appears directly below them in the sixth row.

Solution to cubic equations

Tartaglia is most well-known today for his disagreements with Gerolamo Cardano. In 1539, Cardano persuaded Tartaglia to share his method for solving cubic equations by promising not to publish it. Tartaglia shared the solutions to three types of cubic equations in the form of poetry. Years later, Cardano discovered unpublished work by Scipione del Ferro, who had independently found the same solution. (Tartaglia had previously competed against del Ferro’s student, Fiore, which made Tartaglia aware that a solution existed.)

Because del Ferro’s work was dated earlier than Tartaglia’s, Cardano decided he could break his promise and included Tartaglia’s solution in his publication. Although Cardano gave credit to Tartaglia, Tartaglia was very upset, leading to a public competition between Tartaglia and Cardano’s student, Ludovico Ferrari. Stories that Tartaglia spent the rest of his life trying to harm Cardano are not true. Today, mathematical historians recognize both Cardano and Tartaglia for solving cubic equations, calling the method the "Cardano–Tartaglia formula."

Volume of a tetrahedron

Tartaglia was very skilled at math and understood the study of three-dimensional shapes. In Part IV of the General Trattato, he demonstrated how to find the height of a pyramid with a triangular base, which is a type of irregular tetrahedron.

The base of the pyramid is a triangle with sides 13, 14, and 15, labeled as bcd. The edges from points b, c, and d to the top of the pyramid (point a) are 20, 18, and 16 units long. To simplify the problem, Tartaglia divided the base triangle into two right triangles by drawing a line from point d to side bc. He then created a new triangle in a plane that is perpendicular to side bc and passes through the pyramid’s top point, a. He calculated the lengths of all three sides of this triangle and noted that its height equals the height of the pyramid. Finally, he used a formula to find the height of a triangle based on its three sides (p, q, r), which is related to the law of cosines, though he did not explain this connection in the text.

During the calculation, Tartaglia made a small mistake by miswriting a number, but his overall method was correct. The final answer for the height is accurate. Once the height is known, the pyramid’s volume can be calculated, though Tartaglia did not provide this result.

Decimal fractions, which were later developed by Simon Stevin in the 1500s, were not used by Tartaglia. He always worked with fractions. His approach resembles a modern method for calculating the height of irregular tetrahedra, but he did not write a general formula for this process.

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)