4

1. Cubic Equations

One of the first great advances beyond Arabic mathematics was

taken by Scipione del Ferro (14657-1526), who taught at the Univer-

sity of Bologna at the beginning of the sixteenth century. He was

the first, as reported some decades later by Cardano in his book Ars

Magna,2

to solve general cubic equations, namely those of the type

x3

+ px = q. Without making his method public, del Ferro revealed

it to his student Antonio Fior. At this time, Niccolo Fontana, alias

Tartaglia, was also working on solving cubic equations. Tartaglia,

who was working as a master calculator in Venice, was one of the

best mathematicians in Italy.3 And in fact, he figured out how to

solve cubic equations of the type

x3

+

px2

= q. However, his method

seems to have been less a general solution algorithm than a way of

setting up special equations whose solutions could be easily

found.4

At the competition mentioned at the beginning of the chapter,

Fior posed thirty problems of the type x3 + px = q for Tartaglia

to solve, while conversely, Tartaglia posed thirty somewhat atypical

problems, including cubic equations of the type x3 + px2 = q. At

first, neither contestant could solve any of the problems that he had

been posed. But shortly before the end of the competition, on 13

February 1535, Tartaglia figured out how to solve equations of the

form x3 -f px — q. Like del Ferro and Fior, he kept his method of

solution a secret.

And now there enters upon the stage the man whose name is as-

sociated today with the solution formula. Girolamo Cardano, known

today more for his discovery of what is called the Cardano wave and

Cardano suspension and by profession actually a physician, man-

aged to convince Tartaglia to reveal to him his formula, under the

guarantee—according to Tartaglia afterward—that he would keep it

2Girolamo Cardano, The Great Art or the Rules of Algebra, the English transla-

tion of the 1545 edition with additions from the editions of 1570 and 1663 (Cambridge,

Massachusetts, 1968); see the beginning of Chapter 1 and Chapter 11.

3An idea of the accomplishments of a master calculator, and in particular of the

person of Tartaglia, can be found in the historical novel Der Rechenmeister, by Dieter

Jorgensen, Berlin, 1999. A substantial part of the novel deals with the discovery of the

solution formula for cubic equations and the resulting conflict.

4See

the work cited by Renato Acampora, pp. 32-34. On the other hand, based

on the fact that Tartaglia is known to have studied the work of Archimedes, Phillip

Schultz speculates (Tartaglia, Archimedes and cubic equations, Australian Mathemat-

ical Society Gazette 11 (1984), pp. 81-84) that Tartaglia could have used a geometric

method in which he determined the intersection point of the parabola y =

x2

and the

hyperbola y — —q/(x + p).