See a transcription for the manuscript

Overview 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.

At the beginning of the first part, Leibniz presents a progression of numbers from 1 to 32, written both in the decimal numeral system and in the binary numeral system. He clearly identifies the connection between the position of a digit and its expression as a power of 2 in the binary system. This insight is particularly relevant because it is the premise for what in today’s terms, we would call a positional notation, where a single number is treated as a series of sums having a fixed order in which a change of position from right to left also represents an increase of the base index.

This general introduction is then followed by the description of a method to obtain the binary counterpart of a decimal number through multiple divisions by two:

[Fig. 1] Conversion between the decimal and the binary system.

Every time a number cannot be divided by two, we divide the closest divisible number and we write 1 as a carry, while every time a divisible number occurs, we write 0: in this way we progressively obtain the same number, expressed however in its binary form, as shown in [Fig. 1]. Leibniz also describes the four common arithmetical operations, showing the best way to implement them with binary numbers.

At the end of the first part Leibniz shifts from arithmetic to algebra. He starts to consider letters instead of numbers for his explorations, adding an important condition that is generally not followed in ordinary algebra: these letters can only represent either the number 1 or the number 0, in the context of the binary numeral system. Leibniz’s aim is to show that while there are some general algebraic properties that are both applicable in the context of the decimal and the binary numeral system, new unique relations between numbers could be found if we focus solely on the latter. This is the case of the expansion of a square in a sum of numbers, explored at the beginning of the second part of De Progressione Dyadica. While we can describe algebraically the expansion of a square in a form which is valid regardless of the numeral system adopted, Leibniz shows that there is an alternative version of this expansion which is valid only for the binary numeral system. A detailed explanation of why this is possible is present in Brancato (2021).

The last section of the second part is dedicated to the binary expression of fractions and their involvement in the expression of \frac{\pi}{4} as a series of sums, a topic Leibniz will frequently connect with dyadics. In the considerations on transcendental quantities present at the end of De Progressione Dyadica we witness here the premise to the great reflection on the possibility of expressing any type of number in the binary numeral system, which became one of Leibniz’ main focuses throughout his production on dyadics.

See a transcription for the manuscript