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Overview of De curvis similibus et similiter positis et parallelis

The text discusses the notion of parallelism in relation to curves. This is the concluding text in a series of studies by Leibniz on parallelism, in which Leibniz had defined parallelism as equidistance. The problem Leibniz saw with this notion is that in general a curve equidistant to a given curve is not similar (in the sense of Euclidean similarity) to the original curve. This had been noted by Leibniz about conic sections, and had to do with the transformations by evolution that Leibniz (following Huygens) was studying in the 1680s. The present essay offers a full discussion of this topic.

The fundamental problem underlying these Leibnizian investigations, however, was the provability of the famous Parallel Postulate. Many attempts to prove this postulate, in fact, were based on a definition of parallelism through equidistance, and then in the assumption that the line equidistant to a straight line is also a straight line. This last assumption is false in non-Euclidean geometry, and if one assumes it, it actually becomes possible to prove the Parallel Postulate. Giovanni Alfonso Borelli, in his Euclides Restitutus (1658) had noted that therefore several demonstrations of the Parallel Postulate (such as that of Clavius, 1589) were in fact petitiones principii.

Leibniz treats the argument from a more general point of view, and showing that in curved lines the equidistant curve is not similar to the original curve, he exposes an argument to doubt that the equidistant from a straight line is also a straight line—thus concurring with Borelli on the need to reject as incomplete those demonstrations of the Parallel Postulate.

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