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.
This is one of the few cases in which these two results usually connected to dyadic are combined.
The second part of the manuscript may appear less interesting, because we find here a study on the progression of the algorithm for the expansions of the squares, as it is already found in manuscripts like Periodus numerorum. In this manuscript however, the generalized progression of the squares is used to show that the second digit of a squared number written in binary form is always 0. While this result is present in other works, for the first time here it is demonstrated through calculations using binary algebra and not derived by a simple hypothetical reasoning.