Manuscripts on Dyadica
This section contains Leibniz’s unpublished manuscripts concerning his studies on the binary numeral system or dyadics, as he called it. While Leibniz is rightfully considered the discoverer of the binary numeral system, or at least of its importance in a computational context and is thus the father of today’s computer science, of the over 170 manuscript sheets concerning dyadics less than 10% of them have been transcribed at the time of writing, oftentimes without a proper critical edition. Leibniz started working on dyadics in around 1679, as is believed from one of the earliest manuscripts, De Progressione Dyadica, dated 15 March 1679. Even though Leibniz’ production on dyadics spans his life, he never really showed publicly his most complex results on this topic, probably because he thought they were too far from what was usually debated during those times in mathematical circles. Despite this reluctance, he was, on a more general level, a keen advocate of the importance of the new numeral system, as it is shown for example by his discussions with the Bernoulli brothers (A III, 8, N. 233) or his correspondence with Joachim Bouvet on the connection between dyadics and the Chinese I Ching (A I, 19, N. 202, A I, 20, N. 318-319), which culminated with the publication of the Explication de l'arithmétique binaire for the Académie royale des sciences in 1703 (LBr. 68, 21r). As the way in which Leibniz develops dyadics is a unique case of a direct relationship between metaphysics and mathematics, he often discusses the topic from a philosophical standpoint: the correspondence between Leibniz, Rudolf Augustus, Duke of Braunschweig-Wolfenbüttel, and Müller on the possibility of producing a coin representing the connection between 1 and 0 and unity and nothingness is the most famous example of this relationship (LBr. F 15). The collection presents manuscripts relating mostly to Leibniz’s mathematical achievements: the foundation and development of binary arithmetic, studies on the binary expression of numbers and on the binary expansion of their squares.
Overviews
Overview of Appropinquatio Circuli Per Radices Dyadice Expressas
The first part of this manuscript, dated between 1683 and 1685, is a reflection on the formula that gives the perimeter of a regular polygon inscribed in a circle. Since Leibniz shows that the formula depends on powers of 2, we could use dyadics to help in the calculations.
In this reflection Leibniz combines two strategies already utilized in other manuscripts: the notion that powers of 2 simplify the calculations, i.e. in this case the fact that the product between a number and a power of 2 entails always the addition of 0, and the notion that 10.00000... is equal to 1.11111... . Leibniz states then that the extraction of 1.11111 ... is easier than the extraction of 10, meaning that we are allowed to substitute it in the $\sqrt{2}$ progression created.
viewOverview of De Progressione Dyadica (1679)
De Progressione Dyadica is probably the most famous manuscript concerning the development of the binary numeral system. It is dated by Leibniz himself (15 March 1679) and is thus considered the earliest reliable document on this topic, but at the same time, no one has yet offered a complete transcription of it: it is in fact only mentioned by Couturat [Leibniz 1903, 574] as early as 1903, while in 1966 a facsimile of the manuscript with an incomplete German translation appeared in Leibniz [1966].
The manuscript is divided into two parts: the first focuses on the binary numeral system in general, its properties and its binary arithmetic, while the second introduces an innovative method of discovering mathematical truths through a unique combination of algebra and binary numbers, defined by some scholars as binary algebra.
viewOverview 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.
viewOverview of Notae Ad Arithmeticam Et Dyadicam [1676 (?)]
This manuscript is a reflection on the mathematical truth stating that from the identity $bx=by$ doesn’t follow necessarily $x=y$, unless we establish that $b$ is different from $0$. What makes the fragment interesting is the fact that Leibniz, in order to show this, appeals to binary algebra: he considers a similar algebraic identity, stressing that in this particular case the utilized letters stand for binary numbers, more precisely either the number 1 or the number 0. Just like in the general case, from the identity $p,1−ϕ=p.qu(ϕ−p)$ doesn’t follow necessarily that $1−ϕ=qu(ϕ−p)$, unless we take $p$ different from 0, i. e. equal to 1.
viewOverview of Periodus numerorum (1705?)
Periodus Numerorum is a manuscript written by Overbeck for Leibniz in around 1705 and represents a collection of results obtained by Leibniz on the topic of dyadics. The content of the manuscript strongly suggests that it was dictated word by word by Leibniz, as that level of knowledge of the binary system it includes was available only to him at that time.
The manuscript starts by introducing a new notation to express what had already been defined by Leibniz in other manuscripts as the period of natural numbers:
[latex]0_{1024}1_{1024}|0_{512}1_{512}|0_{256}1_{256}|0_{128}1_{128}|0_{64}1_{64}|0_{32}1_{32}|0_{16}1_{16}|0_{8}1_{8}|0_{4}1_{4}|0_{2}1_{2}|0_{1}1_{1}=N[/latex]
viewOverview of Summa seriei binariae [1676 (?)]
viewOverview of Tentata Expressio Circuli Per Progressionem Dyadicam
This manuscript, dated around 1680, is particularly important in the context of the evolution of Leibniz’s work on dyadics, because it shows once again that the major developments in his approach happened after 1679 are connected with its study on the expression of a circle having diameter 1 (π/4) in the form of Gregory’s series.
This study is a comparison between a dyadic progression, expressed in the form :
and called dyadic because it has powers of 2 at the denominator, and the standard Gregory series expressed in the form
view