## Le Triparty en la Science des Nombres. Par Maistre Nicolas Chuquet Parisien, publié d’après le manuscript fonds français n°1346 de la Bibliothèque nationale de Paris et précédé d’une Notice par M. Aristide Marre. [Offered with:] Problèmes numériques faisant suite et servant d’application au Triparty en la Science des nombres de Nicolas Chuquet, Parisien. Extrait de la seconde partie du ms. n°1346 du fonds français de la Bibliothèque nationale, annoté et publié par Aristide Marre.

Rome: Imprimérie des Sciences Mathématiques, 1881; 1882.

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 groundbreaking 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, p. 25). These works are offprints from the Bullettino di bibliografa e di storia delle scienze matematiche e fisiche, tomo XIII & XIV. OCLC lists Harvard only in US for Le Triparty, no copies of the Problèmes numériques. ABPC/RBH lists one copy of Le Triparty (Bloomsbury Auctions, 18 July 2014, lot 445, £1240) and none of Problèmes numériques.

Provenance: Problèmes numériques signed and inscribed by Marre on upper wrapper to ‘Monsieur Ferdinand Denis.’ Jean-Ferdinand Denis (1798-1890), a historian specializing in Brazil, was director of the Bibliothèque Sainte-Geneviève (Paris) from 1865 to 1885.

“The ‘Triparty’ is a treatise on algebra, although the word appears nowhere in the manuscript. This algebra deals only with numbers, but in a very broad sense of the term. The first part concerns rational numbers. Chuquet’s originality in his rules for decimal numeration, both spoken and written, is immediately obvious. He introduced the practice of division into groups of six figures and used, besides the already familiar million, the words billion (1012), trillion (1018), quadrillion (1024), etc. …

“Chuquet’s study of the rules of three and of simple and double false position, clear but commonplace, served as pretext for a collection of remarkable linear problems, expounded in a chapter entitled ‘Seconde partie d’une position.’ Here he did not reveal his methods but reserved their exposition for a later part of the work, where he then said that after having solved a problem by the usual methods — double position or algebra (his ‘regle des premiers’) — one must vary the known numerical quantities and carefully analyze the sequence of computations in order to extract a canon (formula). This analysis generally led him to a correct formula, although at times he was mistaken and gave methods applicable only for particular values.

“Another original concept occurred in this group of problems. In a problem with five unknowns, Chuquet concluded: ‘I find 30, 20, 10, 0, and minus 10, which are the five numbers I wished to have.’ He then pointed out that zero added to or subtracted from a number does not change the number and reviewed the rules for addition and subtraction of negative numbers. In the thirteenth century Leonardo Fibonacci had made a similar statement but had not carried it as far as Chuquet in the remainder of his work.

“The first part of the ‘Triparty’ ends with the ‘regle des nombres moyens,’ the only discovery to which Chuquet laid claim. According to him — and he was right — this rule allows the solution of many problems that are unapproachable by the classic rule of three or the rules of simple or double false position. It consisted of establishing that between any two given fractions a third can always be interpolated that has for numerator the sum of the numerators of the other two fractions, and for denominator the sum of their denominators. It has been demonstrated in modern times that by repeating this procedure it is possible to arrive at all the rational numbers included between the two given fractions. It is obvious, therefore, that this rule, together with a lot of patience, makes it possible to solve any problem allowing of a rational solution. Further on, Chuquet utilizes it in order to approach indeterminately the square roots, cube roots, and so on, of numbers that do not have exact roots …

“The second part of the ‘Triparty’ deals with roots and ‘compound numbers.’ There is no trace of Euclidean nomenclature … The language has become simpler: there is no question of square roots or cube roots, but of second, third, fourth roots, “and so on, continuing endlessly.” The number itself is its own first root. Moreover, everything is called a ‘number’ — whole numbers, rational numbers, roots, sums, and differences of roots — which, in the fifteenth century, was audacious indeed …

“The third part is by far the most original. It deals with the ‘regle des premiers,’ a “truly excellent” rule that “does everything that other rules do and, in addition, solves a great many more difficult problems.” It is “the gateway and the threshold to the mysteries that are in the science of numbers.” Such were the enthusiastic terms in which Chuquet announced the algebraic method. First, he explained his notation and his computational rules. The unknown, called the ‘first number’ (nombre premier), is written as 11. Therefore, where Chuquet wrote 40, we should read 4; if he wrote 51, we should read 5x; and if he wrote 73, we should read 7x3

“In order to justify his rules of algebraic computation, and particularly those touching the product of the powers of a variable, he called upon analogy. He considered the sequence of the powers of 2 and showed, for example, that 22 x 23 = 25. He wished only to make clear, by an example that he considered commonplace and that goes back almost to Archimedes, the algebraic rule that if squares are multiplied by cubes the result is the fifth power.

“In accordance with the custom of Chuquet’s time, all these rules of computation were simply set forth, illustrated by a few examples, and at times justified by analogy with more elementary arithmetic, but never ‘demonstrated’ in the modern sense of the term. Having set down these preliminaries, Chuquet dealt with the theory of equations, which he called the ‘method of equaling’” (DSB).

Chuquet’s original manuscript is in four parts, the first two being Le Triparty and the collection of Problèmes numériques published by Marre in the offered works. The two remaining parts of the manuscript, on geometry and on commercial arithmetic, remained unstudied until the late 20th century, and the last is still unpublished. Itard in DSB states that the manuscript is the work of a firm of copyists, but it has been shown more recently that it is in Chuquet’s hand.

The history of Chuquet’s manuscript, which survives in only a single copy, is described by Marre in the introduction to Le Triparty. After Chuquet’s death, probably in 1487, it fell into the possession of Étienne de la Roche (1470-1530), who may have been Chuquet’s pupil. According to a note written in Latin on a protecting sheet at the beginning, after the manuscript had been in de la Roche’s possession it was bought by an Italian, Leonardo de Villa. It then passed into the famous library of Jean Baptiste Colbert (1619-83), the finance minister of Louis XIV and founder of the French Academy of Sciences. In 1732, some 8000 volumes from Colbert’s library passed into the Royal Library of Louis XV (1710-74), among which was Chuquet’s manuscript. It then seems to have been forgotten for more than a century. In papers presented to the French Academy of Sciences in 1841 and 1842, Michel Chasles noted de la Roche’s reference to Chuquet’s ‘Treatise on algebra’ and expressed the hope that this treatise ‘has not been completely lost.’ “The publication of the first of the four sections of Chuquet’s work by Aristide Marre in 1880 therefore created something of a sensation for historians of mathematics” (Flegg et al, p. 18). Marre had located the manuscript in the Bibliothèque Nationale, where it is now no. 1346 of the Fonds français.

In his L’aristhmethique (1520), the first printed French work on algebra, de la Roche mentioned Le Triparty but then went on to plagiarise substantial portions of it without acknowledgement. Marre says of de la Roche, “without being accused of injustice or exaggeration one could say that he appropriated the work of Nicolas Chuquet, that he purely and simply copied the Triparty in a host of places, that he suppressed some of the most important passages, especially in the algebra, that he shortened or lengthened others, in order to compose his Arismethique, vastly inferior to the Triparty, and finally that, if for four centuries Nicolas Chuquet and his work have remained in the shadows, it is above all to him [de la Roche] that we must attribute the prime cause” (Flegg et al, p. 19).

Little is known of Chuquet’s life. He “called himself a Parisian. He spent his youth in that city, where he was probably born and where the name is yet known. There he pursued his extensive studies, up to the baccalaureate in medicine (which implies a master of arts as well). It is difficult to say more about his life. He was living in Lyons in 1484, perhaps practicing medicine but more probably teaching arithmetic there as ‘master of algorithms.’ The significant place given to questions of simple and compound interest, the repayment of debts, and such in his work leads one to suppose this. However, he used these questions only as pretexts for exercises in algebra.

“Chuquet’s mathematical learning was solid. He cites by name Boethius — whom everyone knew at that time — Euclid, and Campanus of Novara. He knew the propositions of Archimedes, Ptolemy, and Eutocius, which he stated without indicating his sources (referring to Archimedes only as ‘a certain wise man’). In geometry his language seems to be that of a translator, transposing terms taken from Greek or Latin into French. By contrast, in the parts devoted solely to arithmetic or algebra there is no borrowing of learned terminology. Everything is written in simple, direct language, with certain French neologisms that have not been preserved elsewhere” (DSB).

Flegg, Hay & Moss (eds.), Nicolas Chuquet, Renaissance Mathematician, 1985.

Large 4to. [Le Triparty:] (310 x 228 mm), pp. 229; [Problèmes numériques:] (310 x 228 mm) [3], 4-50. Original printed wrappers, edges with some chipping and wear, spine strips worn.

Item #4446

Price: \$2,800.00

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