Nicolo Tartaglia and the Cubic Equation

John Hood, Dickson College, ACT, 2005, With assistance from Jan Bentley, Mathematics Department, Dickson College

  • John Hood is currently a Master Teacher at Dickson College, ACT, and teaches Medieval and Modern History. Jan Bentley is Senior Teacher in Mathematics at Dickson College.

Portrait of Tartaglia, illustrated on the cover of the 1546. edition of 'Quesiti et Inventioni Diverse'
Nicolo Tartaglia was born in Brescia, Lombardy, at the very beginning of the sixteenth century. He was by tradition self-educated, but went on to become a teacher of mathematics. He is regarded as one of the more important contributors to the progress of mathematical knowledge in the sixteenth century and, in particular, the development of algebra. He died in Venice in 1557, at the age of 57, alone and poor in spite of his considerable mathematical accomplishments. Our knowledge of these achievements is derived principally from his published works. He is credited above all with solving the complexities of 3rd degree equations (cubic equations). He also translated Euclid into contemporary Italian. Perhaps from the perspective of his time, his most important contribution would have been the application of mathematical knowledge to the study of ballistics and fortifications.

The details of his life are few and obscure. According to Abbot (1985), he was born in Brescia in c.1500 and left there in 1516 to become a teacher of the abacus in Verona. He was then 16 years old. Williams (1969) gives a later date of 1521 for his move to Verona, though he implies that Tartaglia may have arrived earlier than this. Both agree that he was teaching mathematics at this early stage. He had survived the French seige of Brescia in 1511, though he was wounded in the mouth by a sword cut which caused him to stammer thereafter. This physical defect apparently became his professional name, though whether he adopted it first or was given the name by others is not clear. At any rate, he seems to have remained in Verona for some 18 years. According to Abbot, he was in charge of a school in the city from c.1529 to 1533. After this, he moved to Venice, where he seems to have remained for the rest of his life. Williams places him back in Brescia in 1548, teaching geometry, but says that ‘soon after’ he went back to Venice and remained there till his death in 1557.

It was during this period of 23 years in Venice that his books were published. They represent the principal source of our knowledge about him. His first publication was Nuova Scienza, or ‘New Science” in which he
Cover illustration to Nuova Scienza, published in Venice1537
outlined his theory of ballistics. His work is perhaps the first application of mathematics to gunnery. In 1543 Tartaglia published a translation of the ancient Greek mathematician Euclid, from Latin into Italian, making the work accessible to a much greater readership. Euclid was one of those ‘lost’ books of the ancient world which had been re-discovered through Arabic sources, appearing in Latin translations initially, and now, thanks to Tartaglia, in contemporary Italian. According to Williams, Tartaglia followed his translation of Euclid with a Latin edition of another ancient Greek mathematician, Archimedes, in 1543. These achievements link Tartaglia very much with the spirit of the 16th century ‘renaissance’. Tartaglia’s next publication, which appeared in 1546, was entitled Quesiti et Inventioni Diverse. Amongst other topics it, too, dealt with the mathematics of gunnery. It is interesting not only for the fact that it was dedicated to King Henry VIII of England, but also because it contains Tartaglia’s claim that he had discovered the solution to algebraic equations of the third degree, that is, of cubic equations. Both Abbot and Williams describe Tartaglia’s victory in a mathematical contest in 1535 with Antonio Fiorido, in response to a challenge to solve the problem of cubic equations. Williams also discusses Tartaglia’s claim that his method was stolen by Antonio Cardano, who published the solution in 1545, a year before the publication of Tartaglia’s Quesiti et Inventioni Diverse. His largest publication, according to Williams, was the Trattato Generale, the “best mathematical compilation of the time.” It covered arithmetic, mensuration, geometry and algebra and is a unique summary of the mathematical knowledge of the sixteenth century.

Tartaglia’s contribution to the science of gunnery and warfare is very important. Elton asserts that ‘the works of Biringuccio and Tartaglia underlay nearly everything written on artillery for two hundred years.’ (Elton, 1958, p.482) Tartaglia’s theories on the design of fortifications are discussed in the Quesiti, in the form of a dialogue with the Prior of Barletta. When the Prior asks Tartaglia if he does not think the new walls of Turin are the finest possible, “Tartaglia answers no. They are good as regards materia (brute strength) but poor as regards forma (design). Walls should not be straight, but curved.” (Elton, 1958, p.493). Tartaglia’s principles of design were in fact put into effect by King Henry VIII of England, in the construction of a series of coastal fortresses. Henry, facing possible invasion from the French and Spanish, defended his southern coasts with fortresses such as Deal Castle and Southsea Castle, both of which are still intact. As Elton points out (p.493), these ‘castles’ are entirely new in English fortification. In place of high walls and towers, the new forts are low in profile and protected by very wide moats. Their walls, curved in mathematical precision, function essentially as emplacements for batteries of cannon.
Southsea Castle, England
The guns are placed in mutual support, on five levels in the case of Deal Castle, and even the moats are defended by casements built into their base. These forts are not defensive in concept. They function primarily as static offence, using firepower as their means of defence instead of thick walls. They are designed to destroy the enemy before the walls can be attacked. In this, they demonstrate the impact of gunpowder technology on medieval war. As Sancha points out, the French had cut a swathe of destruction through the cities of the Italian peninsula in 1494, thanks to their possession of a new artillery. Not only were the new designs and enhanced mobility of cannon significant, but also the development of hand-held weapons such as the arquebus symbolised a new era of battlefield conflict. Tartaglia’s significance is symbolised by the theorem outlined in his publications. He formulated what is still today called Tartaglia’s Theorem – that the trajectory of a projectile is a curved line everywhere, and the maximum range is obtained
Deal Castle, England. Picture from English Heritage
with an elevation of 45 degrees. Given the impossibility of observing the flight path of projectiles such as cannon balls with the human eye, It is clear that his theorem is based on mathematical calculation rather than observation or practical experience.

The events of Tartaglia’s life and his mathematical achievements can be traced in broad outline from the two secondary sources cited. However, neither source gives much specific detail. One of the problems in reconstructing the life and achivements of Tartaglia is the lack of primary sources. Abbot gives no bibliography whatsoever. Williams lists two references – a journal article by A.Favaro which appeared in a 1913 edition of Isis, and a 1962 publication by Ateneo di Brescia entitled Quatro Centenario della Morte de N.Tartaglia – ‘Four Hundredth Anniversary of the Death of N. Tartaglia.’ The next step is to investigate Tartaglia’s work in gunnery and sixteenth century warfare and to find out more detail of his work concerning cubic equations.

Abbot, David, (ed), 1985, Biographical Dictionary of Scientists: Mathematicians, Blond Educational Press, London.
Williams, Trevor (ed) 1969, Biographical Dictionary of Scientists, Adam&Charles Black,London.
Elton, G.R, (ed), 1958, The New Cambridge History, Vol II The Renaissance. CUP Cambridge.
Sancha, Shiela, The Castle Story.