"In addition to that branch of geometry which is concerned with magnitudes and which has always received the greatest attention, there is another branch, previously almost unknown, which Leibniz first mentioned, calling it the geometry of position [geometria situs]. This branch is concerned only with the determination of position and its properties; it does not involve measurements, nor calculations made with them [...] Hence, when a problem was recently mentioned, which seemed geometrical but was so constructed that it did not require the measurement of distances, nor did calculation help at all, I had no doubt that it was concerned with the geometry of position.” - L. Euler (1736) ‘Solutio problematis ad geometriam situs pertinentis’, Commentarii Academiae Scientiarum Imperialis Petropolitanae, 8: 28-140.
The problem Euler is referring to is, of course, the Bridges of Königsberg problem. The allusion to the “the geometry of position” concerns Leibniz’s attempt to build a new foundation for, and devise a new formalism for, geometry. In a series of manuscripts, papers, and notes (unpublished in his lifetime) Leibniz tried to reduce geometry to a system of basic relations between points — the latter is connected to his theory of relational space. Leibniz’s analysis situs (or sometimes called geometria situs) affords a way of expressing various “situations” of points relative to one another, and promised a calculus so that all geometrical theorems could be deduced from propositions concerning the points and relations (see especially volumes 5 and 7 of Gerhardt’s edition of the Mathematische Schriften). But although Euler was clearly inspired by Leibniz's analysis situs, it is a misattribution to credit Leibniz with the key graph-theoretical ideas that Euler goes on to develop in the paper. (See discussion in Benson Mates (1986) The Philosophy of Leibniz: Metaphysics and Language, Oxford University Press, p. 235-240.)