**“THE FIRST MATHEMATICIAN THOROUGHLY TO STUDY THE BINARY SYSTEM, UPON WHICH ALL MODERN DIGITAL COMPUTERS ARE BASED”: LEIBNIZ’ REVOLUTIONARY EXPLICATION DE L’ARITHMÉTIQUE**

LEIBNITZ, [G. W. von]. **Explication de l’arithmétique binaire, Que se sert seul caractéres 0 & 1; avec des Remarques sur son utilité, & sur ce qu’elle donne le sens des anciennes figures Chinoises de Fohy. IN: Histoire de L’Academie Royale des Sciences. Année M.DCC.III…. Seconde Edition, revûe, corrigée & augmentée. **Paris: Chez Charles-Estienne Hochereau, 1720. Thick quarto, period style full mottled brown calf, elaborately gilt decorated spine, raised bands, red morocco spine label, marbled endpapers.

*Second edition of Histoire de l’Academie Royale des Sciences, containing Leibniz’ groundbreaking work on binary arithmetic, which first appeared in the Royal Academy’s 1703 Memoires, this rare 1720 volume also with Bernouli’s Demonstration Generale, and seminal works by Cassini, de la Hire and other leading scientists of the age, with 12 engraved folding plates. Beautifully bound.*

Leibnitz (*alt*. Leibniz), who published his groundbreaking work on differential calculus in 1684, ultimately became “the leader of a small but very active group of mathematicians”—many embroiled in a controversy involving his work and that of Isaac Newton (DSB). Throughout his life Leibniz’ genius was nourished by a metaphysics evident in this seminal 1703 work, “*Explication de l’arithmétique binaire*… (An Explanation of Binary Arithmetic Using only the Characters 0 and 1…), which originally appeared in the journal *Memoires de l’Académie Royale des Sciences* [85-89]” (Lodder, *Binary Arithmetic*, 2), and includes a “Tables des nombres” with examples of numbers in both the binary and conventional system, and the four arithmetical operations in the binary system. “Leibniz was the first mathematician thoroughly to study the binary system, upon which all modern digital computers are based” (Heilbron, 172), and it was Leibniz who anticipated the main requirement of modern computer science by being the first to work out the properties of the binary system (Carpenter, *Notable Mathematicians*, 312). Leibniz “believed that he had found an historical precedent for his binary arithmetic in the ancient Chinese lineations or 64 hexagrams of the I-Ching. This, he thought, might be the origin of a universal symbolic language… He had been in possession of his ideas concerning binary arithmetic well before his 1703 publication. In 1679 Leibniz outlined plans for a binary digital calculating machine, and in 1697 he sent a congratulatory birthday letter to his patron Duke Rudolph August of Brunswick, in which he discusses binary numeration and the related creation theme with 0 denoting nothing and 1 denoting God” (Lodder, *Binary Arithmetic*).

Leibniz “sensed in the 0 and 1 bits the combination power to create the entire universe, which is exactly what happens in modern digital electronic computers and the rest of our digital information technology” (Chaitin, *Meta Math*). This volume’s introductory *Histoire* section (1-148), containing numerous essays on the works in *Memoires* (separately paginated; 149-467), also cites Leibniz in an informative essay (58-63). *Memoires* additionally features two major works by Jakob Bernoulli: *Demonstration General*, and *Extrait d’une lettre… de sa Regle du Centre de Balancement* (78-84; 272-83). With *Demonstration*, Bernouli “returned to the famous problem of the search for a center of oscillation, and gave a solution of it which contained the germ of d’Alembert’s principle [i.e. *Traite de dynamique*, 1743]” (Dugas, *History of Mechanics*, 243). *Memoires* futher contains key articles by mathematicians such as Varignon, Philippe de la Hire and Michel Rolle, astronomers such as James Cassini and other leading scientists. With 12 engraved folding plates, engraved title page vignette, engraved ornamental head- and tailpieces. Text in French. Armorial bookplate of English politician, mathematician and astronomer George Augustus William Shuckburgh-Evelyn, who was a Fellow of the Royal Society, and in 1798 was co-winner of the Copley Medal, the Society’s highest award.

Text and plates generally fresh. Beautifully bound.