Spinoza and Euclidean Arithmetic: The Example of the Fourth Proportional

Abstract

Spinoza’s example of the fourth proportional has often been mistreated: it has generally been considered either trivial or ill-chosen. My aim in this article is to show that it is perfectly germane. To do so it is enough to take seriouly his reference to Euclid’s Elements on the assumption that Spinoza was both aware of its implications, and addressed himself to readers whom he expected to be equally aware; we shall then see that the example illustrates with extreme precision the distinction he establishes in the Tractatus de Intellectus Emendatione between the last two ‘modes’ of knowledge, if we grant that these are identical with the last two ‘genres’ of knowledge in the Ethics. Since the texts of the Tractatus de Intellectus Emendatione are, at one and the same time, those which are most in need of elucidation as well as those which best elucidate that example, we shall concentrate principally on that work, saving the Ethics for the end.