Traité du triangle arithmétique

4 parts in 1 quarto volume of (2) leaves, 12 pp., 8 pp., 16 pp., 48 pp.; 1 large folding diagram, complete.

Olive green calf, double gilt fillet framing the covers, floral cornerpieces, spine with raised bands and gilt decoration, marbled edges. English binding circa 1830.

221 x 163 mm.

Pascal, Blaise. Treatise on the Arithmetical Triangle with some other short treatises on the same subject.

Paris, Guillaume Desprez, 1665 (in fact, 1654).

"Very rare first edition, illustrated with a folding plate" (D. Courvoisier).

Published after the author's death, the work paved the way for research in the field of probability calculus.

It contains eleven parts including the Treatise on Numerical Orders and two treatises in Latin.

"The principle of the arithmetical triangle was current in Pascal’s lifetime, but it was he who used it fully and derived the greatest number of applications. The text is in both French and Latin and is divided into eleven inter-related sections, numbers I-IV in French, and the rest in Latin".

"Between July and October 1654, Pascal corresponded with Fermat about two problems in gaming: one about the probability of a player getting a certain face of a dice in a given number of throws, and the other about the proportion of the stakes to be returned to the several players in an interrupted game. Fermat solved these problems by combinatorial analysis. Pascal however used reasoning by recursion".

The Treatise on the Arithmetical Triangle is closely related to Pascal's research on the "problem of points" and probability. This triangle consists of a series of whole numbers arranged in columns in the shape of an endless triangle, and in which each number is obtained by adding the numbers above it. While this figure was not entirely new, Pascal discovered innumerable applications for it. He would go on to develop them in a series of short treatises. The Treatise on Numerical Orders as well as two treatises in Latin—one on the Character of Divisibility of Numbers and the other on the Summation of Numerical Powers—are also related to this same line of research. Pascal established one of the essential bases of integral calculus. These mathematical considerations can be related to the metaphysical developments in the Pensées on the "three orders".

"The Treatise on the Arithmetical Triangle was published with the date 1665, three years after the author's death, by the care of his family, in the form of a thin quarto volume. But an examination of the surviving copies shows that only a few initial leaves were then printed, including the title page and the figure of the triangle. The body of the work had been put to press during Pascal's lifetime, and all copies were available as early as summer 1654, at which time Fermat received one. He was likely the only informed recipient. Why the subsequent silence? No doubt because of the powerful spiritual conversion Pascal underwent at the time of the Memorial (23 November 1654). The edition printed in 1654 presents some curious disparities. Not only does the Treatise on the Arithmetical Triangle, with some applications, constitute only a first part, with a second formed by several other short mathematical treatises, but while the first part and the start of the second are in French, the rest is in Latin. Among the books gathered by Pascal's heirs there is another printed version of the work, all in Latin, beginning with a 'Triangulus arithmeticus' (without the applications). There were thus two versions and two successive printings of the work, which appear to have followed just weeks apart." Jean Mesnard, member of the Institut, professor emeritus at Université de Paris IV – Sorbonne.

Among the 4 copies recorded on the international market for over 40 years, the Phillips copy had renewed endpapers (Sotheby’s, 29 November 1976); the Norman copy was washed and rebound in later vellum (Christie’s New York, sold in that state for €36,000, 22 years ago, on 16 June 1998); the copy from the library of Shirburn Castle sold for €85,000 on 5 May 2006 was bound in 17th-century calf, and the copy rebacked in 18th-century morocco was sold for €59,000 11 years ago.

The theory of the arithmetical triangle is closely linked with Pascal's research on the problem of points and probabilities. The triangle allows the determination of numerical orders, serves in the calculation of combinations, to find the successive sums of like powers, to calculate the powers of binomials; it is also applicable to the theory of probability and the problem of points. Finally, this discovery contains all the fundamental principles of the science of statistics (Philippe Zoummeroff, En Français dans le texte, no. 101).

Among the short treatises annexed to that of the Arithmetical Triangle are the De numeris multiplicibus, dealing with the criteria for the divisibility of numbers, and the Potestatum numericarum summa, where Pascal teaches how to calculate the sum of the like powers of the terms in an arithmetic progression.

Famous copy from my Robert Honeyman library, the only one without flaws along with the Shirburn Castle copy.