Overview of Diophantea seu Arithmetica figurata absoluta methodo dyadica
Despite being a small fragment, manuscript LH 35 3 A 16 Bl. 27 represents an important part of Leibniz’s corpus on dyadics. It was first published by Couturat in Opuscules et fragments inédits de Leibniz (1907, 571) with a fairly precise transcription, albeit small mistakes.
Its importance comes first from the fact that Leibniz himself highlights here the result he achieved by writing before the text multiple times the word “NB”, i.e. “Nota Bene”. At the same time, we don’t witness in the manuscript a major breakthrough in the evolution of the binary system: here, as much as in many other places, the superiority of dyadics is connected with the possibility of dealing with a number’s power in an easier way than in the decimal system, provided we express numbers in an indefinite dyadic form (a possible reference to what I defined as binary algebra). Since this approach is present in Leibniz’s production since the very beginning, connecting these two observations (the enthusiasm showed by Leibniz about his results and the fact that they are among the first he obtained) we could cautiously infer that this might be one of the first manuscripts in which Leibniz realizes dyadics’ great potential on the matter, suggesting a possible dating between 1676 and 1679. A later dating is suggested by Alcantara, following Couturat (J. Alcantara, « cette caractéristique secrète et sacrée… » : Leibniz et Bouvet, lecteurs du yijing, Intervention au séminaire du Centre d’Études et de Recherches sur l’Humanisme et l’Âge Classique, UPRES-A CNRS 5037, équipe de recherche du Centre International Blaise Pascal, Chiffres et secrets, organisé par M. Dominique Descotes, 21 janvier 2006, Université de Lyon III), but this is probably due to the fact that evidences of Leibniz’s work on dyadics before 1679 were found only recently.
Another important element present in the manuscript is the explicit connection between dyadic and what Leibniz calls Diophantea or Arithmetica Figurata, a connection which is already evident mathematically throughout Leibniz’s whole production, but rarely made this explicit.
The fact that Leibniz was interested in Diophantine equations is largely proven by his works and his knowledge of Fermat’s works. The connection with dyadic becomes particularly important if we compare Leibniz’s use of the binary system with the contemporary notions of Diophantine geometry, algebraic geometry, arithmetic geometry, and number theory. There is in fact here more than a simple reference to arithmetic, as Couturat suggested in his La logique de Leibniz (1901, 293), because the equivalence between Diophantine problems and Figurative Arithmetic problems highlighted by Leibniz is not only a reference to the fact that in such topics we use and study geometric progressions and geometric series: Figurative Arithmetic is also involved in the study of figurative numbers, as intended by the Pythagorean tradition. In Leibniz’s mind then, the work on Diophantine equations is also a way to prove the relevance ofthis tradition and, above all, a way to promote his own, more sophisticated version of an arithmetic connected to geometrical representation and yet founded on the binary system: as much as Pythagoras’ figurative numbers were highlighting mathematical relations through the study of their geometrical representation and position, so Leibniz’s Arithmetica Figurata based on dyadics highlights in the best way possible mathematical relations that can be expressed by Diophantine equations. This is consistent on one side with Weigel’s early Pythagorean influence on Leibniz and on the other side with Leibniz’s superior mathematical knowledge with respect to his former teacher, paving the way to his own evolution of these traditional topics.