To understand the abbaco mathematics used by Cardano, we have to step back and look at the medieval tradition of abbaco schools and their masters. Though the subject is a fascinating and deep one, there is one particular aspect of this tradition that is crucial in the following account: abbaco masters thought in terms of canonical problems, and one particular canonical problem, the “Problem of Ten,” arises in the solution of the cubic that we will examine.
Throughout the Middle Ages and the Renaissance, city-states like Florence or Milan hired abbaco masters and established state-supported abbaco schools for children aged about nine to eleven. Students were introduced to Hindu-Arabic numerals; the basic arithmetic operations of addition, subtraction, multiplication, and division; proportions; some practical geometry; and often algebra up to the quadratic equation. The mathematical content was derived from Islamic texts, brought to Italy by writers such as Fibonacci. Abbaco masters often served their cities in capacities outside of the abbaco school itself, as engineers or accountants, roles in which their mastery of abbaco mathematics made them indispensable [Note 1].
Abbaco mathematics was rhetorical—in Cardano’s time, most of the algebraic symbols with which we are so familiar were either recently invented, concurrent with the Ars Magna, or were well in the future. For example, ‘\(+\)’ and ‘\(–\)’ were first recorded in the 1480s, and were not in common use in 1545, when the Ars Magna was published. Robert Recorde would not invent the equals sign until 1557, and the use of letters and exponential notation would have to await Francois Viete in the 1590s and the Geometrie of Rene Descartes of 1637 [Note 2]. What Descartes would write as \(x^3=ax+b,\) Cardano wrote as “cubus aequalis rebus & numero” [Cardano 1662, Chapter 12, p. 251].
Rhetorical formulas can be difficult to remember, so algebraic rules were presented with canonical examples, which encoded the rules as algorithms within the examples. Thus, the mind of the abbaco master was a storehouse of such canonical examples, to which he compared the new problems that he came across in his work. When he recognized a parallel structure between the new problem and a canonical problem, he could solve the new problem by making appropriate substitutions into the canonical example.
Such canonical examples occurred even in the foundational texts of abbaco mathematics, including the Algebra of al-Khwarizmi. An important example for us, one that occurs implicitly in Cardano’s solution to the cubic, is the “problem of ten” [Note 3]. Most abbaco texts had such problems, and one from Robert of Chester’s 1215 translation of al-Khwarizmi’s Algebra into Latin [al-Khwarizmi, p. 111] ran as follows:
Denarium numerum sic in duo diuido, vt vna parte cum altera multiplicata, productum multiplicationis in 21 terminetur. Iam ergo vnam partem, rem proponimus quam cum 10 sine re, quae alteram partem habent, multiplicamus...
In his translation of this passage into English, Louis Karpinski used \(x\) for ‘rem’ (thing), and so I offer my own translation, without symbols [Note 4]:
Ten numbers in two parts I divide in such a way, in order that one part with the other multiplied has the product of the multiplication conclude with 21. Now therefore one part we declare the thing, and then, with 10 without the thing, which the other part is, we multiply...
Al-Khwarizmi here first divided ten into two parts with product 21. He then named the two parts, one being 'thing' and the other '10 without the thing.' Continuing al-Khwarizmi’s solution symbolically, he distributed \(x\) over \(10-x\) to get \(10x-x^2=21.\) By applying ‘al-jabr’ or restoration [Note 5], he transformed the latter to \(x^2+21=10x\) or, in rhetorical style, to the problem of “square and number equal to thing.” Al-Khwarizmi had a solution for such a problem, complete with a geometric justification [al-Khwarizmi, p. 83]. Karpinski translated his solution as:
Take ½ of the unknown, that is 5, and multiply this by itself, giving 25. From this, subtract 21, giving 4. Take the root of this, 2, and subtract it from half of the root, leaving 3, which represents one of the parts.
The other solution is, of course, 7. There are neither symbols nor symbolic formulas in either al-Khwarizmi’s statement of the problem, or in his solution.
The structure of the “problem of ten” was that of a number \(a\) broken into two parts \(x\) and \(y,\) with a condition on the parts; symbolically: \[x+y=a\,\,{\rm and}\,\,f(x,y)=b\] for some function \(f(x,y)\) and number \(b.\) The usual method of solution was to express the two parts as “thing” and “number minus thing” and then to substitute into the condition, as al-Khwarizmi did above. The “problem of ten” was canonical for quadratic problems, and served as a way to remember the rules for solving such problems.
As we shall see, Cardano’s solution to the cubic rested upon a “problem of ten.”