As our excursion through Cardano’s solution to “cube equal to square and number” comes to a close, let us reflect on what we’ve just seen.
First, the abbaco ways of thinking provided an important element of the solution. Both Cardano and Tartaglia were trained in abbaco techniques, and their ability to draw on their stores of canonical problems helped them recognize a “problem of ten” in a geometric context.
Second, Cardano’s mastery of the decomposition of the cube and how the resulting solids could be arranged showed up in two places. The recognition that a solid of volume (ab)(bc)2 and one of volume (ab)2(bc) can be put together to give one solid of volume (ab)(bc)(ac) (where ac = ab + bc) was essential in creating the geometric situation in which he could see the “problem of ten.” But this mastery also showed itself in the geometric depression of the cubic, a tour-de-force of solid geometry. Recognizing that the cut at b in the side ac of the cube on ac can be specified to make two of the squared terms cancel each other out, and that this specification results in a problem having only a cubic term and a linear term, reveals an ability to move between geometric and abbaco thinking, a certain linking of geometry and rhetorical algebra.
As a teacher of mathematics and of history of mathematics, I find that this episode connects well with my students. They see a different method of depressing the cubic, one tied to solving a cubic equation rather than shifting the graph of a function. The geometry involved is solid geometry, neither the planar geometry so often emphasized in history of mathematics courses, nor the analytic geometry of calculus. This episode serves to illustrate the heights to which abbaco mathematics could rise, and the difficulty of doing algebra without symbolism, whetting students’ appetites for Viete, Thomas Harriot, and Descartes.
After all, this episode in the history of mathematics is both an ending and a beginning. The time of the abbaco master was coming to an end, as the discovery of the Arithmetica of Diophantus and the algebraic symbolism of Viete changed the ways of thinking about algebra. But the time of what we would call “research mathematics” was just beginning, and Cardano’s solutions of the cubics, solutions that had escaped the ancients, were among the first new discoveries about pure mathematics in Western Europe in a long time. The solutions of the cubics bolstered the confidence of other mathematicians that they could, indeed, go beyond the Greeks.
The Ars Magna, then, is both the glorious pinnacle of a dying tradition (the rhetorical algebra of the abbaco tradition) and a first baby step in the tradition then just beginning (the study of polynomial equations, and indeed that of modern mathematics). Cardano and his compeers Tartaglia and Lodovico Ferrari found these solutions using a mixture of both abbaco reasoning and geometric reasoning. Both were helpful; indeed, both were necessary.
I thank Kate Gill, Jeff Mullins, Judy Dorn and especially Steve Walk (all of St. Cloud State University) for their close readings of and insightful comments on various drafts of this paper. I greatly appreciate Janet Beery’s close reading of this manuscript, and the comments of an anonymous reviewer.
Bill Branson graduated from the University of California, Berkeley, with a B.A. in mathematics in 1991, and was awarded a Ph.D. in mathematics by the University of Illinois at Urbana-Champaign in 2000, with a thesis on complex differential equations. He has been teaching at St. Cloud State University in St. Cloud, Minnesota, since 2002, and during that time has shifted interest to history of mathematics, especially cultural aspects of Renaissance mathematics. When not ploughing through Cardano's collected works, he can be found out on one of the many disc golf courses in the St. Cloud area (yes, even in winter!) or in the kitchen, cooking up dishes with vegetables and herbs from the garden he maintains with his partner Kate.
Above: The author, Bill Branson, on a recent visit to Athens, Greece