Larismethique nouellement composee par maistre Estienne de La Roche dict Villefra[n]che natif de Lyo[n] sus le Rosne diuisee en deux parties dont la p[re]miere tracte des p[ro]prietes p[er]fectio[n]s et regles de la dicte scie[n]ce: come le no[m]bre entire, le no[m]bre rout, le regle de troys, la regle d’une faulse position, de deux faluses position[n]s, d’apposition et remotio[n], de la regle la chose, et de la qua[n]tite des p[ro]gressio[n]s et p[ro]portio[n]s. La seco[n]de tracte de la practique dicelle applicquee en fait des mo[n]oyes, en toutes marcha[n]dises comme drapperie, espicerie, mercerie et en toutes aultres marcha[n]dises qui se vendent a mesure au pois ou au nombre, en co[m]paignies et en tro[n]ques, es changes et merites, en fin dor et dargent et en lavaluer diceux. En arge[n]t le rey et en fin darge[n]t doze. Es deneraulx allyages et effaiz, tant de lot que de large[n]t. Et en geometrie aplicquee aux ars mecha[n]ique come aux masons charpe[n]tiers et a tous aultres besongna[n]s en art de mesure.

[Lyon]: Guillaume Huyon for Constantin Fradin, June 2, 1520.

First edition, extremely rare, of the first published work on algebra in French. This is a fine copy in a beautiful contemporary binding. Born in Lyon, then the principal commercial centre of France, La Roche was a student of Nicolas Chuquet and published for the first time in the present work large sections from Chuquet’s Le Triparty en la Science des Nombres, the most original mathematical work of the fifteenth century. Chuquet’s work, of which a single manuscript survives (BNF Fonds français 1346), remained unpublished until 1881. La Roche’s work thus printed for the first time several important innovations in arithmetic and algebra introduced by Chuquet: the use of exponents to denote powers of a number, often credited to Descartes who introduced them in his Géométrie more than a century later; the use of the ‘second unknown’ (see below) in the solution of systems of linear equations, which was an important step towards the invention of symbolic algebra by Viète; the use of negative numbers in the solution of equations; and the introduction of our terms ‘million’, ‘billion’ and ‘trillion’ for powers of 106. La Roche also includes Chuquet’s ‘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. La Roche intended his work to serve the mercantile class, and his account of commercial arithmetic goes considerably beyond Chuquet. “The second, and greater, part of La Roche’s work has, apart from some geometrical calculations at the end, a commercial character. The author states that as a basis he used ‘the flower of several masters, experts in the art’ of arithmetic, such as Luca Pacioli, supplemented by his own knowledge of business practice … [La Roche’s work] presented an outstanding view of contemporary methods of computation and their applications in trade” (DSB). OCLC lists copies at Columbia and Harvard only in North America. ABPC/RBH list only the Macclesfield copy (rebound in the 19th century) since Honeyman (Sotheby’s, April 14, 2005, lot 1204, £19,200 = $36,409). The present copy was offered by Librairie Thomas-Scheler in 1996 (Catalogue Nouvelle Série No. 15, n. 296, 120,000F).

“We do not know much about de la Roche (c. 1470-1530). Tax registers from Lyon reveal that his father lived in the Rue Neuve in the 1480s and that Estienne owned more than one property in Villefranche, from which he derived his nickname. De la Roche is described as a ‘master of argorisme’ as he taught merchant arithmetic for 25 years at Lyon. He owned the manuscript of the Triparty after the death of Chuquet (1488). It is therefore considered that de la Roche was on friendly terms with Chuquet and possibly learned mathematics from him.

“The importance of the Larismethique has been seriously underestimated. There are several reasons for this. Probably the most important one is Aristide Marre’s misrepresentation of the Larismethique as a grave case of plagiarism. Marre discovered that the printed work by Estienne de la Roche, contained large fragments that were literally copied from Chuquet’s manuscript (Marre, Le Triparty … par Maistre Nicolas Chuquet (1881), introduction)” (Heeffer, pp. 1-2). But Barbara Moss argues that “the charge of plagiarism against Estienne de la Roche is largely an anachronism … Before the spread of printing, academic knowledge had been disseminated through the copying of manuscripts, and Chuquet, like many of his contemporaries, must have written down for reference a large number of examples from the work of others, with or without a note of their source … De la Roche’s use of citations and sources is similar to that in a number of printed arithmetics of that period. Following the usual commendation of mathematics for its ‘great utility and necessity’, he continues: ‘I have collected and amassed the flowers of several masters expert in this art, such as master Nicolas Chuquet, Parisian, Philippe Frescobaldi, Florentine, and Brother Luke of Borgo [Pacioli], with some small addition of what I have been able to invent and test out in my time in its practice’ [first (unnumbered) page]” (Moss, pp. 117-9).

Moreover, “giving a transcription of the problem text only, Marre withholds that for many of the solutions to Chuquet’s problems de la Roche uses different methods and an improved symbolism. In general, the Larismethique is a much better structured text than the Triparty and one intended for a specific audience. Chuquet was a bachelor in medicine educated in Paris within the scholarly tradition and well acquainted with Boethius and Euclid. On the other hand, de la Roche was a reckoning master operating within the abbaco tradition. It becomes clear from the structure of the book that de la Roche had his own didactic program in mind. He produced a book for teaching and learning arithmetic and geometry which met the needs of the mercantile class. He rearranges Chuquet’s manuscript using Pacioli’s Summa as a model. He even adopts Pacioli’s classification in books, distinctions and chapters. He moves problems from Chuquet’s Appendice to relevant sections within the new structure. He adds introductory explanations to each section of the book, such as for the second unknown, discussed below. With the judgment of an experienced teacher, he omits sections and problems from the Triparty which are of less use to merchants and craftsmen and adds others which were not treated by Chuquet such as problems on exchange and barter” (Heeffer, p. 2).

“As far as the first book of the Triparty is concerned, de la Roche is reasonably faithful to his teacher. He does include an extra chapter, on the connotations of the numbers 1 to 12, which he took from Pacioli, and Pacioli from St Augustine’s Civita Dei. He also prefers some of the more conventional names, like ‘rule of false position’ rather than ‘rule of one position’, and disagrees with Chuquet’s assessment that the rule of apposition and remotion for solving indeterminate equations in integers ‘is a science of little recommendation’. However, he includes two of the distinctive contributions of the Triparty: nomenclature in terms of the powers of 106 up to the nonillion [(106)9], and the rule of intermediate terms, or ‘rule of mediation between the greater and the less’ [règle des nombres moyens]” (Moss, p. 121).

“[Chuquet] employs the words byllion, trillion, quadrillion, quyllion, sixlion, septillion, octillion, nonillion, ‘et ainsi des aultres se plus oultre ou voulait proceder’ to denote the second, third, etc. powers of a million. Evidently Chuquet had solved the difficult question of numeration. The new words used by him appear in 1520 in the printed work of La Roche. Thus, the great honour of having simplified numeration of large numbers appears to belong to the French. In England and Germany the new nomenclature was not introduced until about a century and a half later. In England the words billion, trillion, etc. were new when Locke wrote, about 1687 [Human Understanding, Chap. XVI]. In Germany these new terms appear for the first time in 1681 … but they did not come into general use until the eighteenth century” (Cajori, p. 144).

“De la Roche’s algebra contains several topics that are not found in the Triparty … he has chapters on algebraic fractions, on the rule of quantity (a mixture of algebra and the rule of false position used for solving equations in more than one unknown), and on equations with more than one solution. He also gives methods for ‘proving’, or checking, results on algebra analogous to Chuquet’s proofs in arithmetic.

“The four canons of the rule of the first terms (one for solving generalized linear equations, and three for the three acceptable forms of the generalized quadratic, with all coefficients positive) are stated in Chuquet’s terms, although de la Roche gives more elementary examples, and fewer which require solution by means of compound roots or roots of high order. However, special cases of cubic equations, with no constant term, are included among the quadratics.

“The fourth canon, for solving equations of the form x2 + b = ax, which may have two positive roots, gave rise to a minor controversy. In 1559, Jean Buteon [in his Logistica] attacked de la Roche’s rule, and claimed that it is impossible for an equation to have more than one solution. De la Roche’s example to the contrary should be disallowed because he gives 1 as one of the roots, and 1, according to Buteon, should not be considered as a number (!) …

“The weaknesses [in de la Roche’s algebra] are essentially those of conservatism, though his work, like Chuquet’s, is not free from careless errors. As in the case of radicals, he presents the concepts of algebra both in the terminology and notation of the Triparty and in a more traditional, restricted, and qualitative system involving symbols not involving numbers for powers of the unknown, and he prefers to use the latter … However, the index notation is presented. It attracted the attention of Michel Chasles in the nineteenth century … Chasles saw no anticipation of Viete’s ideas among the Italians; but he instanced the German Stifel [Arithmetica integra, 1544] and the Frenchmen Peletier [L'Algèbre, 1554] and Buteon, because they used letters for unknowns and a crude form of index notation. The development of an adequate notation for exponents had hitherto been accredited to Descartes; but Chasles claimed that such a notation is already present in de la Roche’s Larismethique nouellement composee” (Moss, pp. 121-4).

“That de la Roche made an important contribution to the emergence of symbolic algebra during the sixteenth century can best be argued by his treatment of the second unknown, sometimes called ‘Regula quantitatis’ or ‘Rule of Quantity’ … The importance of the use of letters to represent several unknowns goes much further than the introduction of a useful system of notation. It contributed to the development of the modern concept of unknown and that of a symbolic equation. These developments formed the basis on which Viète [In artem analyticum isagoge, 1591] could build his theory of equations … De la Roche first mentions ‘la regle de la quantite’ in the beginning of distinction six together with ‘la regle de la chose’. He properly introduces the second unknown in a separate chapter titled Le neufiesme chapitre de la regle de la quantite annexee avec le dict primier canon, et de leur application, in the sixth distinction of the first part [f. 42v] …

“In the Rule of Quantity, de la Roche sees a perfection of algebra itself. The use of several unknowns allows for an easy solution to several problems which might otherwise be more difficult or even impossible to solve … After this introduction, de la Roche gives six examples of the rule of quantity applied to the typical linear problems, though he removes the practical context. Then he presents five indeterminate problems under the heading ‘questions which have multiple responses’, without use of the second unknown. Finally he solves five problems using the second unknown under the heading ‘other inventions on numbers’. At the end of the book there is a chapter on applications in which four more problems are given (ff. 149v-150r). In total, there are twenty problems solved by the regle de la quantite” (Heeffer, pp. 3-7).

The section on commercial arithmetic provides information about the economic life of fifteenth-century Lyon, which had become one of the most important commercial centres of the Western world after Louis XI gave royal protection to the fairs established there in 1464. De la Roche discusses the currencies in use, the weights and measures, and the financial arrangements between partners in business enterprises. The various currencies mentioned come from different parts of France, and from Germany and Italy, probably all to be found in the market at Lyon. From some examples one can infer that exchange rates fluctuated considerably. Other examples may relate to actual methods of counting used by money-changers.

“La Roche begins the commercial section independently of Chuquet by stating that numbers can be considered in three ways according to the Bible’s Book of Wisdom, that God the creator ordered all in measure, number and weight. And thus, all the affairs of the world are governed and managed by these three things, which la Roche continues to examine from the monetary perspective … Dealing with numbers, la Roche returns to the basic arithmetic operations in relation to the monetary and coinage system, then in relation to measure, and third as weight. La Roche divides the entire second part into ten chapters, each divided into a number of subsections …

“La Roche first states that all countries use the £ (Lire, Pound), the solz (sous, shilling), and deniers (penny). The Lire is always worth 20s (solz), and the solz is 12d (denier, penny). But the £ of one country has a different value from that of another country … The section ‘numbers as measure’ is divided into three kinds according to the three dimensions in geometry: length (ell, toyse, cane, brasse, palme, etc.), which is used for cloth measures. The second is the square (toyse, pye, etc.) used for tapestries, mural work, fields. The third way number can be considered is as weight of silver, copper, lead, saffron, ginger, pepper, etc. The weights systems are similar but the values differ. After the introduction follow a number of examples of calculation of different monetary values by addition and subtraction. The examples la Roche mainly took from Chuquet.

“La Roche proceeds to explain multiplication by taking a fraction of a fraction of a higher unit … The method is very simple and easy to explain: if you buy 1200 items at 1 Euro per item, the total price will be 1200 Euro; if the price is ½ Euro (50 cents), the price is half the number of items, which is 600 Euro, etc. … The second chapter is about merchandise sold by length or size, such as linen … Similar counting methods are followed in calculating the price of merchandise sold by weight in the third chapter, and in the fourth chapter, which describes selling merchandise by number (dozen, gross, hundred, thousand). Subsequent chapters deal with liquid measures (5), corporations (6), barter (7), exchange of money and banking (8), profit and discount (9), and gold and silver (10)” (Ulff-Møller).

The final section of the book, on the application of the science of numbers to geometry, includes the measurement of areas and volumes. There are problems on circles, triangles and polygons, including many examples about inscribed figures, the representation of square and cube roots, various simple constructions, and notes on the quadrature of the circle. This last presented some results about lunes, the triangulation of the circle, and series of inscribed and circumscribed circles and squares. Algebra is applied to problems of volumes leading to simple cubic equations. Thus, geometry is viewed as an area to which algebra can be applied, rather than a means of justifying algebraic rules.

“The last few pages are devoted to gauging and show the same instrument as was described by La Court (La fabrique et usage de la jauge, 1588)” (Tomash). This gauge or compass (so called because of its shape) was used in measuring barrels. The results achieved with this instrument were at best approximate as only the bung diameter and a diagonal distance were measured – the shape of the individual barrels was not taken into account.

A second edition of la Roche’s book was published at Lyon in 1538. There is no modern edition.

Bechtel L-47; Brunet III 842; Hoock & Jeannin L5.1; USTC 30158; von Gültlingen, Huyon 12. Cajori, History of Elementary Mathematics, 2007. Smith, Rara Arithmetica, p. 128. Ulff-Møller, ‘Estienne de la Roche. Larismethique nouellement compose. Lyon 1520, and second edition 1538,’ in Rechenmeister und Mathematiker der frühen Neuzeit, Gebhardt (ed.), 2017. For Chuquet, see Flegg, Graham, Hay, C., Moss, B. Nicolas Chuquet, Renaissance Mathematician. A study with extensive translation of Chuquet’s mathematical manuscript completed in 1484, 1985.

Folio (255 x 178 mm), ff. [iv], 230, title printed in red and black within a beautiful woodcut border, woodcut printer’s device on title-page, woodcut initials and diagrams (small tear to head of title-page, ink stains on n4v, lacking front free endpaper). Contemporary binding using a fifteenth-century vellum manuscript leaf (slightly soiled).

Item #4636

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