 The issues addressed in Dedekind’s Stetigkeit und irrationale Zahlen (Continuity and Irrational Numbers) grow out of the “rigorization” and “arithmetization” of analysis (the mathematical theory) in the first half of the nineteenth century. Yet their roots go deeper, all the way down, or back, to the discovery of incommensurable magnitudes in Ancient Greek geometry, e.g., diagonals in unit squares that have length \(\sqrt{2}\) (Jahnke 2003, ch. 1). The Greeks’ response to this startling discovery culminated in Eudoxos’ theory of ratios and proportionality, presented in Chapter V of Euclid’s Elements (Mueller 1981, ch. 3, Corry, ch. 3). This theory brought with it a sharp distinction between discrete multitudes (numbers) and continuous magnitudes, which led to the long dominant view of mathematics as the science of number, on the one hand, and of magnitude, on the other hand. Dedekind’s first foundational work concerns, at bottom, the relationship between the two sides of this dichotomy.