1.1. Information from: Sophia Rare Books (2020).

First edition, the rare offprint issues, of the most original mathematical work of the fifteenth century, indeed the most important since Fibonacci's Liber Abaci almost three centuries earlier (a work which was also not published until the nineteenth century). Composed in 1484, but published here for the first time, this first French work on algebra introduced several ground-breaking innovations into European mathematics, notably the use of exponents to denote powers of a number and the use of negative numbers in the solution of equations. The first part of Le Triparty concerns the arithmetic operations on numbers, including an explanation of the Hindu-Arabic numerals. Chuquet gave a 'règle des nombres moyens' according to which a fraction could be found between any two given fractions by taking the sum of their numerators and dividing by the sum of their denominators. This rule could be used to find the solution of any problem soluble in rational numbers, once an upper and a lower bound for the solution had been found. The last, and most important part, concerns the 'règle des premiers', which is nothing less than what we would call 'algebra', 'premier' being Chuquet's name for the unknown. He also had specific names for the square, cube and fourth power of the unknown, but for higher powers he invented an exponential notation of great significance. In particular, it laid bare the laws of exponents, which played a crucial role in the subsequent invention of logarithms. Also in this part, Chuquet studied the solution of equations, making use for the first time of isolated negative numbers (and on one occasion of the number zero). Problèmes numériques contains Chuquet's statement and answers to 156 problems solved using the methods of Le Triparty. "Many of these problems have a long history going back at least as far as the Greek Anthology reputedly collected by Metrodorus some one thousand years earlier. In Chuquet they are solved by various methods, including the 'rule of three' and the 'rule of one and two positions', but the significant part of the section on problems is where Chuquet applies his 'rule of first terms,' that is, his algebra" (Flegg et al, page 25).

2.1. Review by: Joann Morse.
Isis 72 (1) (1981), 112-113.

Nicolas Chuquet's La géométrie, like other treatises on practical mathematics, is above all a collection of problems. It contains over two hundred, from the determination of a mountain's height to the calculation of the area of a regular heptagon. Chuquet drew his problems from an old and diverse tradition that encompassed Roman agrimensores, professors in medieval universities, and Italian masters of the abacus. We can determine the divisions and developments of this tradition only through the meticulous comparison of problem collections, tracing identical solutions or similar styles from one manuscript to another. Hervé l'Huillier's patient and scholarly edition provides this necessary background for understanding Chuquet's work.

La géométrie merits such detailed attention. Unlike most of his predecessors, Chuquet combined two strands in medieval practical geometry, a "Latin" geometry of instruments and a "vernacular" tradition of areas and volumes. The introduction and notes establish parallels between Chuquet's synthesis and that of contemporary maestri d'abaco, enhancing our understanding of these Italian sources and their dissemination north of the Alps. Moreover, many solutions rely on the algebra Chuquet had taught in the Triparty. Taken together, this treatise on algebra and La géométrie illustrates the connections between different aspects of practical mathematics. Finally, the editor presents variants from an earlier version along with the text of the 1484 manuscript. Thus we have a rare opportunity to study the development of one mathematician's practical geometry.

The edition, a readable photo-reduction of a typescript, is an exemplary presentation of a mathematical manuscript. L'Huillier places each carefully redrawn diagram near its problem, and frequently adds a translation of the proof into symbolic notation. The modern version never intrudes on Chuquet's own style, however, and l'Huillier conscientiously indicates where they differ. Two appendixes provide a helpful summary of Chuquet's algebraic notation and a glossary of mathematical terms-particularly useful for a text in fifteenth-century French. Those interested in mathematical education in the late Middle Ages will find a valuable source in l'Huillier's edition of La géométrie.

3.1. Preface by: Graham Flegg.

My attention was first drawn to Chuquet's mathematical manuscript whilst undertaking the necessary research for the preparation of the Open University's History of Mathematics course, presented initially in 1974. It was whilst editing the English edition of Mathématiques et Mathématiciens (P Dedron and J Itard, trans. J Field) that I noted that it was stated that "the whole manuscript ... comprises 324 folios, i.e. 648 pages", and that, in addition to the Triparty (by which the work is generally known) the manuscript includes sections on problems, on the application of algebraic methods to geometry, and on commercial arithmetic. It was clear from references to Chuquet in most of the secondary sources that even the existence of the last three parts of the manuscript was very largely unknown. It therefore became clear that a comprehensive presentation of the contents of the entire manuscript was a much needed resource for researches into the mathematics of the Renaissance period.

For the past six years, it has been my privilege to work with colleagues both within and outside the Open University in the preparation of a transcription of those parts of Chuquet's work never previously published and the translation of the whole work into English, and also in making our combined assessment of the role of the manuscript in the history of mathematics and Chuquet's importance as a Renaissance mathematician.

It has not been possible to publish here the complete translation; this would have involved a volume nearly twice the length of this present work. My colleagues and I have therefore had to be selective. The most important and interesting part of the manuscript, the Triparty, has been very extensively presented in English translation. The remaining parts have been presented sufficiently extensively to enable their nature and general content to be well appreciated. Our assessment has, however, been based on the total work.

In making the translation, we have attempted to retain the flavour and something of the structure of Chuquet's French. We have had as a first priority, however, the task of conveying the mathematical content of the work. Sometimes this has involved difficult decisions for us, when our various aims have come into conflict. Because of this, we have deliberately not always been strictly consistent in our translation of specific words, nor have we been by any means entirely faithful to Chuquet's punctuation. We hope, however, that what has resulted is a 'faithful' translation if not always an 'exact' one. We have, for example, often ignored Chuquet's tenses and also his use of singular and plural forms where an 'exact' translation would have made the English excessively awkward. On the other hand we have often retained his order of words, when this enabled us to keep the contemporary 'flavour' of his writing without destroying the sense of the English.

I would wish here to pay especial tribute to my two academic colleagues who have collaborated with me on this venture, Dr Cynthia Hay of the Open University and Ms Barbara Moss, formerly of Salford University but now a micro-computing consultant to the Economist group. We would wish to record our gratitude to Dr M S Sidarski and Mr J L Taylor who assisted with the initial transcription of Chuquet's manuscript. and to Mrs Doreen Tucker who undertook the somewhat difficult typing which preparation of this book has demanded. Our gratitude is also due to the Open University for providing funds to support this research, to the Bibliotheque Nationale for granting permission to photograph the entire Chuquet manuscript, to the President and Fellows of Queens College, Cambridge, for permission to reproduce an extract from de la Roche, and especially to our Publishers, who have been extremely patient and helpful when we encountered unavoidable delays in preparing this work for publication.

The book is presented to the public in the quincentenary year of the completion of Chuquet's manuscript in 1484. We offer it as a small tribute to that remarkable French Renaissance mathematician.

3.2. Review by: William Eamon.
Isis 77 (4) (1986), 690-691.

Nicolas Chuquet is known only through his mathematical manuscript (Bibliotheque Nationale, Paris, fonds français, 1346), completed in 1484. Although he was trained in medicine, Chuquet was apparently not a practicing physician. Instead, he set up shop as an escrivain and teacher of arithmetic in Lyon, a flourishing commercial and banking city, thus supplying skills urgently needed by a growing business community: copying legal and commercial documents and teaching the art of computation.

Chuquet's manuscript consists of four distinct parts. The first, entitled Triparty en la science des nombres, is a treatise on algebra and contains the only original discovery to which Chuquet laid claim, the rule of intermediate numbers, which allowed the solution of many problems that are unapproachable by the classic rule of three and the rules of one and two false positions. Appended to the Triparty is a set of mathematical problems of the sort that go back to the sixth-century anthology of Metrodorus. The problems are traditional, ranging from ways to compute profits and unknown quantities of goods to "think of a number" games; but Chuquet's solutions, which are presented in rudimentary algebraic terminology, represent a significant departure from conventional medieval treatises. The third part of the manuscript is a treatise on practical geometry, containing further applications of algebra to determine the length, area, or volume of bodies. The last part is a treatise on commercial arithmetic, again presented in the form of a series of problems, such as how to compute interest or determine how much profit a business transaction will earn.

Chuquet's manuscript was therefore a comprehensive treatise covering both the principles of mathematics and their applications to the practical affairs of the business community. Unfortunately, Chuquet wrote for advanced students and teachers of arithmetic, and his manuscript was too difficult and abstract to find a readership among merchants. Chuquet's ideas, however, were transmitted by his pupil Estienne de la Roche, whose popular work Larismethique (1520) offered merchants a simplified version that better suited their needs. Although de la Roche is generally regarded in the history of mathematics as an incompetent plagiarist, ironically his handbook may have done more to advance mathematical knowledge than Chuquet's superior manuscript, which gathered dust in the Bibliotheque Nationale until Aristide Marre published the Triparty in 1880.

In the present work we are given an English translation of extended passages from all four parts of Chuquet's manuscript, an assessment of Chuquet's place in the history of mathematics, a running commentary clarifying Chuquet's terminology and methodology, and a detailed table of contents collating the pagination of the manuscript with the modern French editions and the editors' translation. The manuscript, which is more than three hundred folio pages long, is not translated in full; had the editors chosen to do so, they would have produced a volume nearly twice the size of the present work, with little gain for the non-specialist readers whom they wish to reach. Yet the translation is an accurate and faithful reproduction of Chuquet's exceptionally readable French. I would not recommend attempting the book on a drowsy late afternoon or evening, however; the endless repetition of mathematical problems, characteristic of treatises of this sort, requires more devotion to the subject than most readers will be able to muster.

Recent research in the history of Renaissance mathematics by Warren Van Egmond and others has demonstrated that it was not within the scholarly, Latin tradition that algebra was pursued, but within the vernacular tradition of practical, commercial mathematics. Algebra was a mathematical technique used to solve practical, commercial problems, and it was in the context of this tradition that Chuquet developed his ideas. This new translation will therefore be of interest not only to historians of mathematics but also to those investigating the social and economic context of scientific ideas.

3.3. Review by: Ralph Keen.
The Sixteenth Century Journal 17 (3) (1986), 364.

In the fifteenth century N Chuquet composed a French treatise called the "Triparty in the science of numbers," which remained in MS until the publication of the first of the four parts in 1880. Since that time its importance has become recognized, but the Triparty has never been fully edited. This volume makes his work available by presenting a large part of the text in an attractive English translation interspersed with clear running commentary and analysis.

The first part of the text is devoted to arithmetic and number theory. Because he is using arabic numerals he is able to be much freer and clearer with examples than those confined to the roman system, though in fact he does draw on Boethius, Nicomachus, and various other predecessors. The second part deals with roots and compounds, and is chiefly a practical manual of how various roots can be extracted and "simplified," and how they and compounds may be added, multiplied, and the like. The third section is Chuquet's exposition of algebra, the "rule of first terms," as he calls it, or "the manner of comparing and reducing one part composed of several diversities of number against another part, simple or compound." "Rule of first terms" is a name of Chuquet's own coining referring to his own notation rather than to algebra in general; and in fact he does not call this work of his "algebra" at any point.

Appended to the Triparty is a section of 166 mathematical problems, illustrating all three parts but mainly the third. After this is a small treatise in geometry, meant as a separate text for students and not an original text like the Triparty, but it is an interesting illustration of an attempt to relate geometry to algebra, both theoretically in its pure sections, and practically in the concrete problems with which, like the Triparty with its problem book, it tries to make itself useful in application. There follows a section of "Commercial Arithmetic," a set of problems much like the first appendix together with rules for their solution. These three parts, like the Triparty itself, are presented with Chuquet and the editors interspersed (italics/roman), with the result that one feels he is reading enough of Chuquet's own work to see the thread of the exposition, and with just enough of the expert assistance that he needs to understand it.

A final chapter discusses Chuquet's place in the history of mathematics. The editors seek to revise the ecstatic praise he was accorded in the nineteenth century by examining his relation to Fibonacci and to Pacioli, his studies, and the corruption of his ideas through de la Roche. In the end Chuquet's system of notation, which helped pave the way toward full symbolic algebra, remains undeniably his great achievement. And if Chuquet's work can be more fully appreciated as a result of this well-proportioned and helpful presentation, that will in turn be not a minor achievement of these editors.