Girolamo Cardano was a famous Italian physician, an avid gambler, and a prolific writer with a lifelong interest in mathematics. His widely read Ars Magna (1545; “Great Work”) contains the Renaissance era’s most systematic and comprehensive account of solving cubic and quartic equations. Cardano’s presentation followed the Islamic tradition of solving one instance of every possible case and then giving geometric justifications for his procedures, based on propositions from Euclid’s Elements. He also followed the Islamic tradition of expressing all coefficients as positive numbers, and his presentation was fully rhetorical, with no real symbolic manipulation. Nevertheless, he did expand the use of symbols as a kind of shorthand for stating problems and describing solutions. Thus, the Greek geometric perspective still dominated—for instance, the solution of an equation was always a line segment, and the cube was the cube built on such a segment. Still, Cardano could write a cubic equation to be solved as
cup p: 6 reb aequalis 20
(meaning: x3 + 6x = 20) and present the solution as
R.V: cu.R. 108 p: 10 m: R.V: cu. R. 108m: 10,
meaning
x = Cardano cubic solution.
Because Cardano refused to view negative numbers as possible coefficients in equations, he could not develop a notion of a general third-degree equation. This meant that he had to consider 13 “different” third-degree equations. Similarly, he considered 20 different cases for fourth-degree equations, following procedures developed by his student Ludovico Ferrari. However, Cardano was sometimes willing to consider the possibility of negative (or “false”) solutions. This allowed him to formulate some general rules, such as that in an equation with three real roots (including even negative roots), the sum of the roots must, except for sign, equal the coefficient of the square’s term.
In spite of his basic acceptance of traditional views on numbers, the solution of certain problems led Cardano to consider more radical ideas. For instance, he demonstrated that 10 could be divided into two parts whose product was 40. The answer, 5 + Square root of√−15 and 5 − Square root of√−15, however, required the use of imaginary, or complex numbers, that is, numbers involving the square root of a negative number. Such a solution made Cardano uneasy, but he finally accepted it, declaring it to be “as refined as it is useless.”
The first serious and systematic treatment of complex numbers had to await the Italian mathematician Rafael Bombelli, particularly the first three volumes of his unfinished L’Algebra (1572). Nevertheless, the notion of a number whose square is a negative number left most mathematicians uncomfortable. Where, exactly, in nature could one point to the existence of a negative or imaginary quantity? Thus the acceptance of numbers beyond the positive rational numbers was slow and reluctant
