A plurality of algebras, 1200–1600: Algebraic Europe from Fibonacci to Clavius¹

¹In a somewhat more detailed form, this paper was presented as a talk at the meeting ‘Mathematics Emerging: A Tribute to Jackie Stedall and Her Influence on the History of Mathematics’ held at The Queen's College, Oxford, 9–10 April, 2016. It draws from chapters 8 and 9 of Katz and Parshall (2014).

Abstract

In memory of Jackie Stedall, friend and colleague

As Jackie Stedall argued in her 2011 book, From Cardano's great art to Lagrange's reflections: filling a gap in the history of algebra, there was a ‘transition from the traditional algebra of equation-solving in the sixteenth and seventeenth centuries to the emergence of “modern” or “abstract” algebra in the mid nineteenth century’ (page vii). This paper traces the evolution from the thirteenth-century work of the Pisan mathematician, Leonardo Fibonacci, to the early seventeenth-century work of the German Jesuit Christoph Clavius of what came to be considered ‘traditional algebra’. It contends that rather than a single ‘traditional algebra’, in fact, a plurality of intimately related yet subtly different algebras emerged over the course of those four centuries in different yet interacting national settings.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ In a somewhat more detailed form, this paper was presented as a talk at the meeting ‘Mathematics Emerging: A Tribute to Jackie Stedall and Her Influence on the History of Mathematics’ held at The Queen's College, Oxford, 9–10 April, 2016. It draws from chapters 8 and 9 of Katz and Parshall (2014).

² See Rommevaux, Spiesser, and Massa Esteve 2012.

³ Lindberg 1978, 64.

⁴ Burnett 2001, 251–253. For more on the diffusion of al-Khwārizmī's text into the Latin West, see Katz and Parshall 2014, 178, note 6.

⁵ The thirteenth century witnessed much direct translation from Greek into Latin in southern Italy and on Sicily, two areas in which Greek, Latin, and Arabic were all actively in use. See Lindberg 1978, 70–75.

⁶ See Rashed 1994, 148, and Høyrup 2007, 41–44.

⁷ Some modern editions of this work use the spelling abaci in the title of this text, although Leonardo used abbaci. Here, I use the latter in order to avoid the common confusion between the book's title and the Chinese counting device, the abacus.

⁸ For more, see Sesiano 1999, 106–120.

⁹ Brown (1978, 190–197) gives an overview of Jordanus' work on the theory of statics.

¹⁰ See Hughes 1981, 3–4. Mahoney (1978, 159–162), briefly compared the work of Fibonacci and Jordanus.

¹¹ See Høyrup (1988, 334) for the example, and Høyrup (1988, 353) for the quotation.

¹² See Høyrup 2007, 27–44.

¹³ See the discussion in Høyrup 2007, 41–44.

¹⁴ See Høyrup 2007, 159–182. Franci (2002) provides a survey of fourteenth-century Italian algebra manuscripts.

¹⁵ See Van Egmond 1978, 156–162.

¹⁶ As Høyrup (2007, 321–323) pointed out, Jacopo also gave rules for dealing with various reducible cases of the quartic.

¹⁷ See, for example, Van Egmond 1978, 163; Franci and Toti Rigatelli 1983, 1985; Gilio of Siena 1983, xiv–xv; Van Egmond 1983; Dardi of Pisa 2001, 21–33.

¹⁸ The analysis of Dardi's work presented here follows that in Franci and Totigatelli (1985, 36–39), but compare Van Egmond (1983) and van der Waerden (1985, 47–52). A complete edition of Dardi's work may be found in Dardi of Pisa (2001, 37–297).

¹⁹ See Franci and Toti Rigatelli 1985, 39.

²⁰ On the different editions of Dardi's text, compare Van Egmond (1983, 419–420) and Dardi of Pisa (2001, 25). In Mantua, the Jewish mathematician, Mordecai Finzi, translated Dardi's text into Hebrew in 1473.

²¹ Compare Franci and Toti Rigatelli 1985, 43–44.

²² Sesiano (1985) treated the the appearance of negative solutions in medieval mathematics.

²³ See Franci and Toti-Rigatelli 1983, 299.

²⁴ Speziali (1973, 95–100), gives a brief overview of the mathematical contents of the Summa, while Heeffer (2010a) analyses Pacioli's sources.

²⁵ See Sarton 1938, 62 and Bühler 1948, 164. Slightly more than one tenth of the roughly 30,000 incunabula are scientific.

²⁶ See Cajori 1928, 90, and compare Parshall 1988, 140–141.

²⁷ As Flegg et al. (1985, 334) showed , , and R_x were becoming common in fifteenth-century Italian manuscripts before Pacioli's work was actually printed. Heeffer (2010a, 5–6) points out interesting differences between the printed version of the Summa, which employs abbreviations, and an earlier manuscript of a text on arithmetic and algebra written by Pacioli while he was teaching in Perugia, which employs a consistent symbolism.

²⁸ Katz (2009, 389–399) provided an overview of the algebraic work done elsewhere in Europe at this time, while Stedall (2012) discussed the notion of plurality of algebras as reflected in early printed European texts.

²⁹ Flegg et al. (1985) provided English translations of a significant part of Chuquet's Triparty, while Spiesser (2006) analysed his work in the context of French commercial arithmetic.

³⁰ See Heeffer (2012) for more on de la Roche's algebraic ideas.

³¹ See Recorde 1557, f. Ff1^r, and Katz 2009, 396.

³² Nuñes's Libro de algebra is available online in the Google Book archive at http://www.archive.org/details/librodealgebrae00nunegoog. Labarthe (2012) and Spiesser (2012) analysed Nuñes's sources, his use of a second unknown, and his sense of algebra as both a science superior to geometry and as an art for solving mathematical problems.

³³ Heeffer (2012) discusses Rudolff's sources.

³⁴ Also in Spain, Aurel's text was followed chronologically by the Arithmetica practica y speculativa (or Practical and Speculative Arithmetic) written by Juan Pérez de Moya (ca. 1513–ca. 1597) in 1562 and by the 1564 text, Arithmetica, by Antic Roca (ca. 1530–80). For more on the work of these Spanish algebraic practitioners, see, for example, Massa Esteve (2008) and Romero Vallhonesta (2012).

³⁵ For more on the use of algebraic symbolism in the sixteenth century, see Heeffer (2010b). Cifoletti (1992) analyses Peletier and his place in the history of sixteenth-century French algebra.

³⁶ Loget (2012) analyses all of these works.

³⁷ See Cifoletti (1996) for more on this theme.

³⁸ Cajori's classic (1928) treats this whole question of mathematical notation and its development, while Serafati (2005) is more targeted.

³⁹ Rommevaux (2012) explores Clavius's ideas on algebra.