## Lettres de A[mos] Dettonville contenant Quelques-unes de ses Inventions de Géométrie. Sçavoir – La Résolution de tous les Problèmes touchant la Roulette qu’il avoit proposez publiquement au mois de Juin 1658. L’Égalité entre les Lignes courbes de toutes sortes de Roulettes, & des Lignes Eliptiques. L’Égalité entre les Lignes Spirale, & Parabolique, demonstrée à la manière des Anciens. La Dimension d’un Solide formé par le moyen d’une Spirale autour d’un Cone. La Dimension & le Centre de gravité des Triangles Cylindriques. La Dimension et le Centre de gravité de l’Escalier. Un Traitté des Trilignes & de leurs Onglets. Un Traitté des Sinus, et des Arcs de Cercle, Un Traitté des Solides Circulaires.

Paris: Guillaume Desprez, 1659.

First edition, extremely rare (one of about 120 copies printed), of one of Pascal’s most brilliant works. “Édition originale, extrêmement rare, de la dernière œuvre de Pascal, l’une des plus éclatantes de son génie” (Lucien Scheler in Tchemerzine, V, pp. 54-55). “Pascal devoted himself during 1658 and the first part of 1659 to the perfection of the “theory of indivisibles,” an ancestor of the integral calculus. In 1658, after using the method of indivisibles to solve several infinitesimal problems relating to the cycloid, he proposed the problems he had solved as a challenge to other mathematicians, then announced his own superior solutions in four letters published in December 1658 and January 1659 under the pseudonym A. Dettonville. These pamphlets were collected in February 1659 under the above title. The structure of this work is very complex, with the first letter (*Lettre de A. Dettonville à Monsieur de Carcavy*) containing five separately paginated sections and the remaining three letters (*Lettre de A. Dettonville à Monsieur A. D. D. S.* …, *Lettre de A. Dettonville à Monsieur De Sluze* …, and *Lettre de A. Dettonville à Monsieur Huggyens* ([i.e., Huygens]) appearing in inverse order of their composition” (Norman). In a diagram (fig. 26) in the treatise *Traité des Sinus du quart de Cercle*, Pascal introduced what Leibniz later called the ‘characteristic triangle’ and used to establish the differential calculus. Pascal made us of it “to determine the sum of the sines (ordinates) of a portion of the curve, that is, the area under this portion. If Pascal had at this point only been more interested in arithmetic considerations and in the problem of tangents, he might have anticipated the important concept of the limit of a quotient and have discovered the significance of this for the determination of both tangents and quadratures. Had he done this, he would have hit upon the crucial point in the calculus some seven years before Newton and about fourteen years before Leibniz” (Boyer, p. 153). “Cette édition originale ne fut donc tirée qu’à 120 exemplaires, le tirage classique de l’époque étant d’environ 3000” (‘Mémoires sur la vie de M. Pascal par Marguerite Périer, sa nièce,’ p. 40 in Pascal, *Œuvres complètes*, Bibliothèques de la Pléiade, N.R.F., 1957 and Tchémerzine, V, 55). ABPC/RBH list six copies, those of Pierre Berge, Macclesfield, Norman, Bute, Honeyman and that of Jean Jacques Amelot, Seigneur de Chaillou (1689-1749). The Berge copy made $118,262 in 2015 (“contemporary vellum, loss to lower board, plates reinforced at folds, second plate detached”). Apart from these copies, we know of two that were sold in the French trade since 2000, a copy in contemporary vellum in 2005 for €295,000 and another in 2008 for €250,000. Our copy is complete with all four part-titles and the two-page *Lettre de Monsieur de Carcavy à Monsieur Dettonville*, some of which are often lacking (e.g., the Bute copy had only the first of the four part-titles). OCLC lists three copies in US (Harvard, NYPL, Yale).

According to the testimony of Gilberte Périer, his sister, it was to forget very painful toothache that in 1657 Pascal suddenly resumed his mathematical research, interrupted since his religious conversion late in 1654. According to Pascal himself, the solution to the problem of finding the area of a cycloid (the path travelled by a point on a circle as it rolls along a plane) came to him in his sleep. Initially he wrote nothing of this discovery as he regarded it as a distraction from his work on religion, but his friend the Duke of Roannez pointed out that God may have provided this vision to give more strength to his work against atheists and libertines, because by showing them the depth of his genius they would be less likely to challenge his proofs of religious doctrine.

“During 1658 and the first months of 1659 Pascal devoted most of his time to perfecting the “theory of indivisibles,” a forerunner of the methods of integral calculus. This new theory enabled him to study problems involving infinitesimals: calculations of areas and volumes, determinations of centers of gravity, and rectifications of curves.

“From the end of the sixteenth century many authors, including Stevin (1586), L. Valerio (1604), and Kepler (1609 and 1615), had tried to solve these fundamental problems by using simpler and more intuitive methods than that of Archimedes, which was considered a model of virtually unattainable rigor. The publication in 1635 of Cavalieri’s *Geometria* marked the debut of the method of indivisibles; its principles, presentation, and applications were discussed and elaborated in the later writings of Cavalieri (1647 and 1653) and in those of Galileo (1638), Torricelli (1644), Guldin (1635–1641), Gregory of Saint-Vincent (1647), and A. Tacquet (1651). (The research of Fermat and Roberval on this topic remained unpublished.) The method, which assumed various forms, constituted the initial phase of development of the basic procedures of integral calculus, with the exception of the algorithm.

“Pascal first referred to the method of indivisibles in a work on arithmetic of 1654, “Potestatum numericarum summa.” He observed that the results concerning the summation of numerical powers made possible the solution of certain quadrature problems. As an example he stated a known result concerning the integral of *x ^{n}* for whole number

*n*, in modern notation. This arithmetical interpretation of the theory of indivisibles permitted Pascal to give a sufficiently precise idea of the order of infinitude and to establish the natural relationship between “la mesure d’une grandeur continue” and “la sommation des puissances numériques.” In the fragment “De l’esprit géométrique”, composed in 1657, he returned to the notion of the indivisible in order to specify its relationship to the notions of the infinitely small and of the infinitely large and to refute the most widespread errors concerning it.

“At the beginning of 1658 Pascal believed that he had perfected the calculus of indivisibles by refining his method and broadening its field of application. Persuaded that in this manner he had discovered the solution to several infinitesimal problems relating to the cycloid or *roulette,* he decided to challenge other mathematicians to solve these problems. Although rather complicated, the history of this contest is worth a brief recounting because of its important repercussions during a crucial phase in the birth of infinitesimal calculus. In an unsigned circular distributed in June 1658, Pascal stated the conditions of the contest and set its closing date at 1 October. In further unsigned circulars and pamphlets, issued between July 1658 and January 1659, he modified or specified certain of the conditions and announced the results. He also responded to the criticism of some participants and sought to demonstrate the importance and the originality of his own solutions.

“Most of the leading mathematicians of the time followed the contest with interest, either as participants (A. de Lalouvère and J. Wallis) or as spectators working on one or several of the questions proposed by Pascal or on related problems—as did R. F. de Sluse, M. Ricci, Huygens, and Wren. Their solutions having been judged incomplete and marred by errors, Lalouvére and Wallis were eliminated. Their heated reactions to this decision were partially justified by the bias it displayed and the commentaries that accompanied it. This bias was the source of intense polemics concerning, in particular, the importance of Torricelli’s original contribution. At the end of the contest Pascal published his own solutions to some of the original problems and to certain problems that had been added in the meantime. In December 1658 and January 1659 he brought out, under the pseudonym A. Dettonville, four letters setting forth the principles of his method and its applications to various problems concerning the cycloid, as well as to such questions as the quadrature of surfaces, cubature of volumes, determination of centers of gravity, and rectification of curved lines. In February 1659 these four pamphlets were collected in *Lettres de A. Dettonville contenant quelques-unes de ses inventions de géométrie* ….

“This publication of some 120 pages has a very complex structure. The first of the *Lettres* consists of five sections with independent paginations, and the three others appear in inverse order of their composition. Thus only by returning to the original order is it possible to understand the logical sequence of the whole, follow the development of Pascal’s method, and appreciate the influence on it of the new information he received and of his progress in mastering infinitesimal problems.

“When he began the contest, Pascal knew of the methods and the chief results of Stevin, Cavalieri, Torricelli, Gregory of Saint-Vincent, and Tacquet; but he was not familiar with the bulk of the unpublished research of Roberval and Fermat. Apart from this information, and in addition to the arithmetical procedures that he applied, starting in 1654, to the solution of problems of the calculus of indivisibles, Pascal possessed a new method inspired by Archimedes. It was elaborated on a geometric foundation, its point of departure being the principle of the balance and the concepts of static moment and center of gravity. Pascal learned of the importance of the results obtained by Fermat and Roberval—notably in the study of the cycloid—at the time he issued his first circular. This information led him to modify the subject of the contest and to develop his own method further. Similarly, in August 1658, when he was informed of the result of the rectification of the cycloid, Pascal extended rectification to other arcs of curves and then undertook to determine the center of gravity of these arcs, as well as the area and center of gravity of the surfaces of revolution generated by their revolution about an axis. Consequently the *Lettres* present a method that is in continual development, appearing increasingly complex as it becomes more precise and more firmly based. The most notable characteristics of this work, which remained unfinished, are the importance accorded to the determination of centers of gravity, the crucial role of triangular sums and statical considerations, its stylistic rigor and elegance, and the use of a clear and precise geometric language that partially compensates for the absence of algebraic symbolism. Among outstanding contributions of the work are the discovery of the equality of curvature of the generalized cycloid and the ellipse; the deepening of the concept of the indivisible; a first step toward the concept of the definite integral and the determination of its fundamental properties; and the indirect recourse to certain methods of calculation, such as integration by parts.

“Assimilated and exploited by Pascal’s successors, these innovations contributed to the elaboration of infinitesimal methods. His most productive contribution, however, appears to have been his implicit use of the characteristic triangle. Indeed, Leibniz stated that Pascal’s writings on the characteristic triangle were an especially fruitful stimulus for him. This testimony from one of the creators of infinitesimal calculus indicates that Pascal’s work marked an important stage in the transition from the calculus of indivisibles to integral calculus” (DSB).

As noted in Sotheby’s Macclesfield sale catalogue, the four works contained in this volume were separately printed, and the first item, *Lettres … a Monsieur Carcavy*, was printed in two shops. The *Lettre de M. de Carcavy*, *Lettre de M.D. a M. de C.* and *Traité générale de la roulette* were printed in shop A. The *Traité des trilignes rectangles*, *Traité des sinus*, and *Petit traité des solides circulaires* were printed in shop B. Desprez was, as it were, the official publisher for Port Royal, and published the *Pensées* in 1670, the second edition, and what became the *textus receptus* in 1678 (see H.J. Martin, ‘Guillaume Desprez, libraire de Pascal & de Port-Royal’, *Le Livre français sous l'ancien régime* (Paris: Promodis, 1987), pp. 65-78).

Amos Dettonville is an anagram of Louis de Montalte, the pen name used by Pascal in *Les Provinciales* (1657).

Macclesfield 1599; Norman 1649; Sotheran, 1, 3475; Tchemerzine, V, pp. 54-55. Boyer, *The History of the Calculus and its Conceptual Development*, 1949.

Nine parts in one volume, 4to, pp. [2] (general title dated 1659); [2] (Lettre de Monsieur de Carcavy a Monsieur Dettonville); [2, part title dated 1658 – Lettre de A. Dettonville à Monsieur de Carcavy, en lui envoyant, une Méthode générale pour trouver les Centres de gravité de toutes sortes de grandeurs. Un Traitté des Trilignes et de leurs Onglets. Un Traitté des Sinus du quart de Cercle. Un Traitté des Arcs de Cercle. Un Traitté des Solides circulaires. Et enfin un Traitté générale de la Roulette, contenant la Solution de tous les Problemes touchant la Roulette qu’il auoit proposez publiquement au mois de juin 1658]; 26 (Lettre de Monsieur Dettonville à Monsieur de Carcavy); 25 (Traité des Trilignes & de leurs Onglets); 8 (Propriétez des Sommes Simples, Triangulaires & Pyramidales); 24 (Traité des Sinus du quart de Cercle; Traite des Arcs de Cercle); 7 (Petit Traite des Solides); 10 (Traitte general de la roulette ou, Problèmes touchant la Roulette proposez publiquement & résolue par A. Dettonville); [2, part title dated 1659 – Lettre de A. Dettonville à Monsieur Huggyens de Zulichem, en lui envoyant La Dimension des Lignes de toutes sortes de Roulettes, lesquelles il montre être égales à des Lignes Elliptiques], 7 (Dimension des Lignes Courbes de toutes les Roulettes); [2, part title dated 1658 – Lettre de A. Dettonville à Monsieur De Sluze Chanoine de la Cathédrale du Liège, en lui envoyant la Dimension & le Centre de gravité de l’Escalier. La Dimension & le Centre de gravité des Triangles Cylindriques. La Dimension d’un Solide formé par le moyen d’une Spirale autour d’un Cone], 8 (De l’Escalier, des Triangles Cylindriques, & de la Spirale au tour d’un Cone; Pour la Dimension et le Centre de gravite de l’Escalier; Pour la Dimension et le Centre de gravite des Triangles Cylindriques; Dimension d’un Solide Forme par le moyen d’une Spirale autour d’un Cone); [2, part title dated 1658 – Lettre de A. Dettonville a Monsieur A. D. D. S. en lui envoyant La Démonstration à la maniere des Anciens de l’Égalité des Lignes Spirale & Parabolique], 16 (Égalité des Lignes Spirale & Parabolique). With 4 folding engraved plates of geometrical diagrams.

Item #4403

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Price:
$75,000.00
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