The study of higher-order equations in Italy before Pacioli (Q2883425)

Luca Pacioli (um 1445--1517) ist bekannt durch: Summa di arithmetica geometria proportioni et proportionalita, Venedig 1494; sein Wirken: ``the definitive exposition of the state of mathematical knowledge at the end of the Renaissance.\dots One of the few areas of mathematics in which Pacioli appeared to break new ground was in his treatment of algebra.\dots Pacioli went beyond the standard treatment and presented several cases of higher-order equations, specifically, eight cases containing terms raised to the fourth power.'' (p. 303) Der Verf. führt entsprechende Beispiele an, bezieht al-Hwārizmī\ (um 780 -- nach 847) und Leonardo Fibonacci (um 1170 -- nach 1240) ein, benennt Gründe für das Auftreten von Gleichungen höheren Grades. -- ``Pacioli's discussion of algebra was the first printed treatment of this subject in Europe, so that the inclusion of these higher-order equations marks the first time, the possibility of extending the rules of algebra to higher degrees was publicly presented to the European world.'' (p. 303) Nach einigen Bemerkungen hinsichtlich der fachlichen Sichtweise zwischen Girolamo Cardano (1501--1576) und Evariste Galois (1811--1832) bekennt er aber: ``\dots yet during all of this time the sources of Pacioli's higher-order equations has remained a mystery. Where did they come from? Did Pacioli personally invent the idea of formulating and solving cubic and quartic equations? And if he didn't, who did? -- It is now possible to answer this question. Recent research into the hand-written books of Italian mathematics from the `hidden centuries' between Leonardo Pisano and Luca Pacioli has revealed that Pacioli did not pioneer the study of higher-order equations. Instead, he drew upon a deep, rich, and complex tradition of investigation into the solution of trans-quadratic equations that was already more than 150 years old at the time he wrote and had already succeeded in giving correct solutions for most of the simple and reducible cases up to the eighth degree and some limited solutions for a few cases of the irreducible cubits and quartics.\dots The sources we draw upon are the Italian abbacus books, the comparatively short, vernacular treatments of practical mathematics that were the dominant genre of mathematical writing during the two centuries from 1300 to 1500. These `trattati o libri d'abbaco', as they were called, first began to appear at the end of the thirteenth century, clearly modeled after Leonardo Pisano's Liber abbaci of 1202, another landmark of European mathematics that had brought the new symbols, terms, and methods of Arabic mathematics into Italy.\dots'' (p. 304) -- Mehr als 300 dieser ``abbacus books'' sind noch zugänglich, womit der Verf. auch auf eigene frühere Arbeiten hindeutet, wobei: ``Yet although the focus of these books was on presenting a form of mathematics that was useful for merchants and geometers, it is notable that more than a third of the surviving books contain at least some treatment of algebra, and that, of those that do, about 80\% include some discussion of higher-order equations. -- Indeed, the study of higher-order equations in Italy begins with the very oldest vernacular presentation of algebra that survives, the Libro di ragioni of Paolo Gerardi (fl. 1327), a Florentine abbacist who was living in the South of France in January of 1328\dots At the end of a somewhat rambling collection of 193 practical number and measure problems that were solved by various rules of proportion, Gerardi suddenly begins, without any explanation, a list of 15 equations or `Regolle delle cose'. For each case he sets out the equation, presents an algorithmic rule that solves it, and then illustrates it with a numerical problem.'' (p. 304 sq.); originale Beispiele Gerardi's werden angedeutet, auch auf Fehler verwiesen. ``But this did not keep Gerardi's work from being taken up and repeated in countless abbacus texts that followed over the next century and a half, for even though some of his solutions were wrong, his work was seminal. It began the tradition of vernacular algebra in Europe\dots'' (p. 305). Der Verf. führt im folgenden Werk auf, die von Gerardi abhängen, einschließlich Standort: zunächst ``a short time later'' (p. 305), anonym: Libro di molte ragioni d'abaco, 16 Gleichungen; ca. 20 Jahre später, anonym: Trattato dell'alcibra amuchabile, 24 Gleichungen davon 15 wie bei Gerardi; Libro merchatantesche und Regole dell'algebre, weisen je 27 Gleichungen auf. -- ``By the middle of the next century it is possible to find simple equations containing powers as high as the 8th degree and compound equations of the 5th and 6th degrees.'' (p. 306) -- Der Verf. gliedert und teilt auch mit, wie er an sein Wissen gelangte: [I] Group I: The Standard Texts and the `Standard Series': ``\dots most of the texts that contain higher-order equations seem to draw upon a common corpus of 68 equations that frequently appear in the same order.\dots the enumeration of this `Standard Series' provides a useful framework for comparing the texts and identifying relationships between them.'' (p. 306); relevante Exempel: [I.1] The `Standard Series' of Higher-Order Equations: 68 Nachweise; gegliedert, z.B.: No.: 60, Equation: Sextic series \(ax^6= bx\), Reduced form: \(ax^5= b\), First use: Vat.Lat.460614 (p. 309); [I.2] The Standard Families: ``\dots it is possible to identify a number of text families that give a common set of equations in a common order, and probably reflect a common textual tradition.'' (p. 310), im Einzelnen: [I.2.1] The Gerardi family -- 23 texts (1328--1505): ``Paolo Gerardi's Libro di ragioni is the first in a long series of texts\dots'' (p. 310) enthält 24 Nachweise; anschließend wird detailgetreu fortgesetzt, d.h. Standort, Signatur, Besonderheiten: [I.2.2] The Lucca family -- 4 texts (1330--1435); The Quadri family -- 9 texts (1390--1501); The Benedetto family -- 8 texts (1390--1470); The Antonio family -- 3 texts (1400--1480); The Nicheletto family -- 2 texts (c. 1400); The Baldovinetti family -- 3 texts (1414--1470); The Vallicelliana family -- 3 texts (1445--1495); The Nardi family -- 4 texts (1450--1514); bis [I.2.10] Additional Standard Texts (13 Werke; 2 Gruppen; 14./15. Jh.). -- [II] Group II: The Non-Standard Texts: ``\dots This `non-standard' group includes some of the most important works of the period.\dots'' (p. 316): [II1.1.1] The Dardi family -- 11 texts (1344--1495); [II.1.2] The Gori family -- 2 texts (1485--1544). -- [II.2] Additional Non-Standard Texts: ``Of the six\dots two are particularly notable\dots'' (p. 317); 15. Jh. -- [III] Group III: Algebra Texts that contain no higher-order Equations (p. 318): Texts listing only the six linear and quadratic equations, 18 Werke, 1390 bis 1564; Texts containing no equations, 7 Werke, 1370 bis um 1570; ferner vier weitere Werke, 16./17. Jh. -- Conclusion: ``So, where did Pacioli's eight higher-order equations come from?\dots from the very beginning of the Italian algebra tradition in 1328, the Italian algebraists of the Renaissance struck out on a bold new research program\dots'' (p. 320). -- Anerkennung gebührt dem Verf. für den detailgetreuen, inhaltsreichen und verdienstvollen Beitrag voller neuer Erkenntnisse; der von ihm hierfür benötigte Arbeits- und Zeitaufwand steht freilich in keinem Verhältnis zu dem knappen, sehr aussagekräftigen Ergebnis.NEWLINENEWLINEFor the entire collection see [Zbl 1236.01002].