Overview 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.
The use of binary algebra, even if at first might seen pointless, has here in Leibniz’s mind two advantages: it shows the same property belonging to $bx=by$, restricting however the possible value of $b$ to two, one for which the property is satisfied and another one for which it is false. This brings a kind of elegance or simplicity to the demonstration, because we are encouraged to prove the same universal truth resorting however to fewer elements, since $p$, instead of $b$, cannot be any possible number, but only 1 or 0.
The second advantage is a direct consequence of this approach: since the value assigned to the letters in the whole equation can only be ones and zeroes, substituting the various letters with their different values becomes easier, hence we can easily imagine all the possible cases. This hypothetical approach that analyses an equation through a series of possibilities is very typical of Leibniz’s binary algebra, as it is present also in manuscripts like De Progressione Dyadica and Summum Calculi Analytici Fastigium. It is particularly interesting because it merges combinatorial approach with the use of binary letters almost like truth-values, closing the gap between binary mathematics and logic.