## ‘Novelle Arithmetique binaire’. [With:] ‘Explication de l’Arithmetique binaire, qui se sert des seuls caracteres 0 & 1; avec des remarques sur son utilité, & sue ce qu’elle donne le sens des anciens figues Chinoises de Fohy’. Pages 58-63 (Histoires) and 85-9 (Mémoires) in Histoire de l’Académie Royale des Sciences Année MDCCIII. Avec les Mémoires de Mathématiques & de Physique, pour la même Année.

Paris: Boudot, 1705.

First edition, first issue, of Leibniz’s invention of binary arithmetic, the foundation of the electronic computer industry. This is the second of Leibniz’s great trilogy of works on mathematics and computation, following *Nova methodus pro maximis et minimis* (1684), his independent invention of calculus, and preceding *Brevis descriptio machinae arithmeticae* (1710), his (decimal) mechanical calculating machine. “A dated manuscript by Gottfried Wilhelm Leibniz, preserved in the Niedersachsische Landesbibliothek, Hannover, ‘includes a brief discussion of the possibility of designing a mechanical binary calculator which would use moving balls to represent binary digits.’ Though Leibniz thought of the application of binary arithmetic to computing in 1679, the machine he outlined was never built, and he published nothing on the subject until [the offered work]” (Norman). Leibniz viewed binary arithmetic less as a computational tool than as a means of discovering mathematical, philosophical and even theological truths. It was a candidate for the *characteristica generalis*, his long sought-for alphabet of human thought. With base 2 numeration Leibniz witnessed a confluence of several intellectual strands in his world view, including theological and mystical ideas of order, harmony and creation. ABPC/RBH list only one copy of this first issue (Zisska & Schauer, May 4, 2011, lot 461, €5,616). The copy of the extracted leaves sold at the Hans Merkle sale (Reiss, Auktion 85, October 15, 2002, lot 696) realized €6500.

“In the domain of mathematics, Leibniz regarded binary notation as intrinsically superior to decimal notation. Over and above this advantage, however, he believed that it contained the key to resolving both the problem of conceptual primitives and the problem of adequate characters. If it could be established, as Leibniz speculated from about 1679 onwards, that the only truly primitive concepts were those of God and Nothingness (or Being and Privation), then the symbols *1* and *0 *would form the basis for an adequate characteristic, whose simplest signs would stand in an immediate relation to the two conceptual primitives” (Jolley, pp. 236-237).

“About this time [1679] Leibniz also outlined a design for a calculating machine to operate the four rules in binary arithmetic, though he recognised that the development of such a machine would not be easy. Owing to the great number of wheels needed, the problems related to friction and smooth movement already encountered with the ordinary calculating machine would be more serious, while the greatest difficulty would be the mechanical conversion of ordinary numbers into binary and the binary answers into ordinary numbers. Perhaps it was on account of these seemingly insuperable obstacles that Leibniz failed to mention the binary calculating machine in his correspondence. Concerning the ‘binary progression’ itself, he remarked to Tschirnhaus in 1682 that he anticipated from its use discoveries in number theory that other progressions could not reveal” (Aiton, p. 104).

“… in April [1697] he [Leibniz] edited a collection of letters and essays by members of the Jesuit mission in China, entitled *Novissima Sinica* … One of the copies of the *Novissima Sinica* that Leibniz sent to to Verjus [Antoine Verjus, the leader of the mission] came into the hands of Joachim Bouvet, a member of the Mission who had just returned home to Paris on leave. Bouvet wrote to Leibniz on 18 October 1697 expressing his commendation of the *Novissima Sinica* and giving him more recent news from China … In the years that followed , the correspondence with Bouvet proved to be of great importance in relation to the dissemination of Leibniz’s binary arithmetic” (*ibid*., pp. 213-4).

“In his reply of 2 December 1697 to Bouvet’s first letter, Leibniz described the nature of his own researches, in which he had shown by mathematics that the Cartesians did not have the true laws of nature. To arrive at these, he explained, it was necessary to suppose in nature not only matter but also force, and the forms or entelechies of the ancients were nothing other than forces. Bouvet, in his letter of 28 February 1698, written before his return to Peking, expressed the view that the ancient Chinese philosophy did not differ from that of Leibniz, for it supposed in nature only matter and movement, which was the same as form, or what Leibniz called force. The ancient Chinese philosophy, he added, was embodied in the hexagrams of the *I ching*, of which he had found the true meaning. In his view they represented in a very simple and natural manner the principles of all the sciences, or rather a complete system of a perfect metaphysics, of which the Chinese had lost the knowledge a long time before Confucius. It is in the ‘Great appendix’ of the *I ching* that the words ‘yin’ and ‘yang’ make their first appearance in philosophical terms, used to describe the fundamental forces of the universe, symbolising the broken and full lines of the trigrams and hexagrams” (*ibid*., p. 245).

“Early in 1700 Leibniz was elected a foreign member of the reconstituted Royal Academy of Sciences in Paris. This brought him into correspondence with Fontenelle … In return for his election to the Academy, he contributed papers on the binary system of arithmetic [offered here]” (*ibid*., p. 218).

“During his visit to Berlin in the summer of 1700, Leibniz evidently sought the collaboration of the Court mathematician Philippe Naudé in further researches on the binary system. For on his return from the conversations on Church reunion with Buchhaim in Vienna, he received a letter from Naudé containing tables of series of numbers in binary notation, including the natural numbers up to 1023. Thanking Naudé for the pains he had taken to compile these tables, Leibniz explained his intention to investigate the periods in the columns of the various series of numbers. For it was remarkable that series – such as the natural numbers, triples, squares, and figurate numbers generally – not only have periods in the columns but that in every case the intervals are the same, namely 2 in the units column, 4 in the twos column, 8 in the fours column, and so on. In the case of the triples, for example, the periods in the last three columns were 01, 0110 and 00101101. Already he had noticed a good theorem: that the periods consisted of two halves in which the 0s and 1s were interchanged; but the general rule for the periods in successive columns had thus far eluded him …

“The possession of Naudé’s tables enabled Leibniz to compose his *Essay d’une nouvelle science des nombres*, which he sent from Wolfenbüttel on 26 February 1701 to the Paris Academy of Sciences to mark his election as a foreign member. In his essay, and also in his letter to Fontenelle, he explained that the new system of arithmetic was not intended for practical calculation but rather for the development of number theory. To Fontenelle he remarked that, before publication, there was perhaps a need to add something more profound and he hoped that some young scholar might be stimulated to collaborate with him to this end … Concerning his decision to communicate his binary system, although the applications had not been achieved, Leibniz explained [in a latter to L’Hospital] that, in view of his many commitments that prevented him from bringing his researches to completion, he feared that his continued silence might lead to the loss of an idea which seemed worthy of conservation.

“Leibniz write to Bouvet on 15 February 1701, at the time he was compounding his essay for the Paris Academy, and it was therefore natural that he should describe for his correspondent the principles of his binary arithmetic, including the analogy of the formation of all the numbers from 0 to 1 with the creation of the world by God out of nothing. Bouvet immediately recognised the relationship between the hexagrams and the binary numbers and he communicated his discovery in a letter written in Peking on 4 November 1701. This reached Leibniz in Berlin, after a detour through England, on 1 April 1703. With the letter, Bouvet enclosed a woodcut of the arrangement of the hexagrams attributed to Fu-Hsi, the mythical founder of Chinese culture, which holds the key to the identification …

“Leibniz accepted Bouvet’s discovery with great enthusiasm. Having no reason to doubt the antiquity of the Fu-Hsi arrangement of the hexagrams that Bouvet had sent him, he was evidently delighted that this figure – ‘one of the most ancient monuments of science’, as he described it – should have been found to be in agreement with his own binary arithmetic” (*ibid*., pp. 245-7).

“Within a week of receiving Bouvet’s letter, Leibniz had communicated the discovery to his friend Carlo Mauritio Vota, the Confessor of the King of Poland, and sent to Abbe Bignoin for publication in the Memoires of the Paris Academy his *Explication de l’Arithmetique binaire, qui se sert des seuls caracteres 0 & 1*; *avec des remarques sur son utilité, & sue ce qu’elle donne le sens des anciens figues Chinoises de Fohy*. Ten days later he sent a brief account to Hans Sloane, the Secretary of the Royal Society” (*ibid*., p. 247).

“Owing to his separation from the real scholars of the time, who for political reasons shunned the Court circles on which he had to rely for his information, Bouvet had been mistaken in his belief in the antiquity of the Fu-Hsi arrangement of the hexagrams. For this order was the creation of Shao Yung, who lived in the eleventh century … In the *I ching* the hexagrams are arranged in a different order, attributed to King Wen (ca. 1050 BC) … This order lacks even a superficial resemblance to a number system.

“Bouvet’s great discovery, to which Leibniz gave hia enthisuasitc support, was therefore a misinterpretation based on bad Sinology. Generously but mistakenly, Leibniz had been willing to follow Bouvet in attributing his own invention to Fu-Hsi, thereby giving support to the myth that the ancient Chinese possessed advanced scientific knowledge which later generations had lost” (*ibid*., 247-8).

Nevertheless, combinatorial aspects susceptible to binary interpretations do exist in the Figures of Fu-Hsi, as has been demonstrated by F. van der Blij of the Mathematical Institute at Utrecht.

The article ‘Nouvelle Arithmetique binaire’ in the *Histoire* part of this volume is unsigned, but is actually by Bernard Le Bovier de Fontenelle. His article constituted an editorial comment on the ‘Explication’ of Leibniz.

“Fontenelle pointed out that ten need not be the base of our arithmetic, and that indeed certain other bases would have advantages over it. Base 12, for example, would simplify dealings with certain fractions such as 1/3 and 1/4. He also noted that numbers have two sorts of properties, essential ones and those dependent on the manner of expressing them. As an example of the former he cited the property that the sum of the first *n* odd numbers equals *n*^{2}, and of the latter that a number divisible by 9 has a digit sum also divisible by 9. This same property would hold for 11 in the case of base 12. He reported that Leibniz had worked with the simplest of all possible bases, base two. This base was not recommended for common use because of the excessive length of its number representations, but Leibniz considered it particularly suitable for difficult research and as possessing advantages absent from other bases. Fontenelle reported further that Leibniz had communicated this binary arithmetic in 1702, but had asked that no mention of it be made in the *Histoire* until he could supply an application. This application eventually came forth in the binary interpretation of the Figures of Fohy. The rest of Fontenelle’s article is devoted to reporting that binary arithmetic was invented not only by Leibniz, but also by Professor Lagny at about the same time [i.e., Tomas Fantet de Lagny (1660-1734)]” (Glaser, p. 44).

Lagny “attempted to establish trigonometric tables through the use of transcription into binary arithmetic, which he termed ‘natural logarithm’ and the properties of which he discovered independently of Leibniz” (DSB). He “used binary arithmetic in his text *Trigonométrie française ou reformée* published in Rochefort in 1703” (MacTutor).

This volume of the *Histoire de l’Académie Royale des Sciences* was reissued at Paris in 1710 (this is the edition reproduced on Gallica), and later in octavo format at Amsterdam.

Aiton. *Leibniz: A Biography*, 1985. Glaser, *A History of Binary and Other Non-Decimal Numeration*, 1971. Jolley (ed.), *Cambridge Companion to Leibniz*, 1995. Van der Blij, ‘Combinatorial Aspects of the Hexagrams in the Chinese Book of Changes,’ *Scripta Mathematica* 28 (1967), pp. 37-49.

4to (243 x 182 mm), pp. [x], 148, 467, [2], with engraved frontispiece and 12 engraved plated (10 folding). Contemporary calf with richly gilt spine, hinges and capitals with some very well done leather restoration.

Item #4824

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Price:
$19,500.00
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