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**Johannes Regiomontanus: Problems and exercises in algebra. From the New York manuscript, Columbia University, Rare Book and Manuscript Library, Plimpton 188 (fol. 82v–96r) according to the preliminary work of Menso Folkerts. Tanslated and annotated by Martin Hellmann. On the occasion of Menso Folkerts’ 80th birthday.
(Johannes Regiomontanus: Aufgaben und Übungen zur Algebra. Aus der Handschrift New York, Columbia University, Rare Book and Manuscript Library, Plimpton 188 (fol. 82v–96r) nach den Vorarbeiten von Menso Folkerts. Übersetzt und kommentiert von Martin Hellmann. Menso Folkerts zum 80. Geburtstag.)**
*(German)*
Zbl 1515.01001

Algorismus 87. Augsburg: Dr. Erwin Rauner Verlag (ISBN 978-3-936905-78-6). XIV, 135 p. (2023).

In 1980, Menso Folkerts discovered that the first part of the manuscript Plimpton 188 contains material connected to Regiomontanus: A copy of Jean de Murs’ Quadripartitum numerorum with Regiomontanus’s commentaries, a copy of Gerard of Cremona’s translation of al-Khwārizmī’s algebra, and a collection of mostly algebraic problems.

The present volume contains a meticulous edition of the problem collection together with a close German translation, and a mathematical commentary. Hellmann shows that four scripts occur – almost certainly from Regiomontanus’s hand, changing because the collection has been produced over almost a full decade (1456–1465). This allows him to delineate the production process: a first part (problems #1–38 except #14, 15 and 25) was written immediately after the copying of the al-Khwārizmī-text – #16–31 drawn from a manuscript later also used by Friedrich Amann, the others from a different or sources; all are linear or of the second degree. Not much later he wrote commentaries to Jean, al-Khwārizmī and the first batch of problems, adding a few problems. At some later moment, he added rules and examples for reducible cubics and quartics, and two more problems (#47–54). Still later he wrote #55–60, dealing among other with the addition of square roots and with the iterated determination of approximate square roots; in all of these he seems to have worked independently. The last group (#61–64) is written while he was working on the De triangulis and mainly deals with related matters.

Noteworthy is, firstly, the use of symbolic first-degree algebra with unknowns \(a\), \(b\) and \(c\), a technique also known from Florentine manuscripts from the time (Regiomontanus’s likely source). And secondly the application of the letter technique created by Jordanus de Nemore in #55 and 60 (not symbolic algebra but rather a way to describe an algorism abstractly: each result is given a new letter name, \(a+c\) thus becomes \(c\).

The present volume contains a meticulous edition of the problem collection together with a close German translation, and a mathematical commentary. Hellmann shows that four scripts occur – almost certainly from Regiomontanus’s hand, changing because the collection has been produced over almost a full decade (1456–1465). This allows him to delineate the production process: a first part (problems #1–38 except #14, 15 and 25) was written immediately after the copying of the al-Khwārizmī-text – #16–31 drawn from a manuscript later also used by Friedrich Amann, the others from a different or sources; all are linear or of the second degree. Not much later he wrote commentaries to Jean, al-Khwārizmī and the first batch of problems, adding a few problems. At some later moment, he added rules and examples for reducible cubics and quartics, and two more problems (#47–54). Still later he wrote #55–60, dealing among other with the addition of square roots and with the iterated determination of approximate square roots; in all of these he seems to have worked independently. The last group (#61–64) is written while he was working on the De triangulis and mainly deals with related matters.

Noteworthy is, firstly, the use of symbolic first-degree algebra with unknowns \(a\), \(b\) and \(c\), a technique also known from Florentine manuscripts from the time (Regiomontanus’s likely source). And secondly the application of the letter technique created by Jordanus de Nemore in #55 and 60 (not symbolic algebra but rather a way to describe an algorism abstractly: each result is given a new letter name, \(a+c\) thus becomes \(c\).

Reviewer: Jens Høyrup (Roskilde)

### MSC:

01A75 | Collected or selected works; reprintings or translations of classics |

01A40 | History of mathematics in the 15th and 16th centuries, Renaissance |

00B30 | Festschriften |

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\textit{Regiomontanus} and \textit{M. Hellmann} (ed.), Johannes Regiomontanus: Aufgaben und Übungen zur Algebra. Aus der Handschrift New York, Columbia University, Rare Book and Manuscript Library, Plimpton 188 (fol. 82v--96r) nach den Vorarbeiten von Menso Folkerts. Übersetzt und kommentiert von Martin Hellmann. Menso Folkerts zum 80. Geburtstag. Augsburg: Dr. Erwin Rauner Verlag (2023; Zbl 1515.01001)