Reinventing or borrowing hot water? Early Latin and Tuscan algebraic operations with two unknowns. (English) Zbl 1476.01005

Høyrup, Jens

Gaṇita-Bhāratī 41, No. 1-2, 23-67 (2019).

Summary: In mature 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 infrequent that it has rarely been observed and never analyzed.
The present paper analyzes the five 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; the regular appearance in the treatises of Benedetto da Firenze; and finally what little we find in Pacioli’s Perugia manuscript and in his Summa. 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 – metaphorically, to which extent they represent reinvention of the hot water already available on the cooker in the neighbour’s kitchen; and it 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.

MSC:

01A35	History of mathematics in Late Antiquity and medieval Europe
01A32	History of Indian mathematics
11-03	History of number theory
11A05	Multiplicative structure; Euclidean algorithm; greatest common divisors
51-03	History of geometry

Keywords:

Indian mathematics; algebraic identities; algebra in several variables