Reinventing or Borrowing Hot Water? Early Latin and Tuscan algebraic operations with two unknowns

In developed symbolic algebra, from Viète onward, the handling of several algebraic unknowns was routine. Before Luca Pacioli, on the other hand, the
simultaneous manipulation of three algebraic unknowns was absent from European algebra and the use of two unknowns so rare that it has rarely been observed and never analyzed.
The present paper analyzes the three occurrences of two algebraic unknowns in Fibonacci’s writings; the gradual unfolding of the idea in Antonio de’ Mazzinghi’s Fioretti; the distorted use in an anonymous Florentine algebra from ca 1400; and finally the regular appearance in the treatises of Benedetto da Firenze. It asks which of these appearances of the technique can be counted as independent rediscoveries of an idea present since long in Sanskrit and Arabic mathematics, and raises the question why the technique once it had been discovered was not cultivated – pointing to the line diagrams used by Fibonacci as a technique that was as efficient as rhetorical algebra handling two unknowns and much less cumbersome, at least until symbolic algebra developed, and as long as the most demanding problems with which algebra was confronted remained the traditional recreational challenges.