Euclid is remembered as a geometer, when he is remembered at all. But Euclid's *Elements* was an omnibus compendium of all mathematical knowledge known to him at the time, beginning with the first mathematics, geometry, then addressing ratio and proportion--that is, *logos* and *analogos*--and ultimately arithmetic, irrationality, and other topics. “There is hardly anything in mathematics more beautiful than [Euclid's] wondrous fifth book,” wrote British mathematician Arthur Cayley. Indeed the definitions that begin book five of the treatise furnish a series of important concepts, first the mathematical ratio, then proportion, understood as an equality of ratios.

Definition 3: "A ratio is a sort of relation in respect of size between two magnitudes of the same kind" ["*Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πηλικότητα ποιὰ σχέσις*"].

Definition 6: "Let magnitudes which have the same ratio be called proportional" ["*Τὰ δὲ τὸν αὐτὸν ἔχοντα λόγον μεγέθη ἀνάλογον καλείσθω*"].

Digital and analog appear here on the same page, perhaps for the first time, at least so under the guise of *logos* and *analogos*. Of immediate interest is the expression "two magnitudes of the same kind" ("*δύο μεγεθῶν ὁμογενῶν*"), or, to mimic Euclid's terminology even more closely, two *homogenous *magnitudes. What does it take for two magnitudes to be homogeneous, to be "of the same genus"? They must contain a "part" or submultiple [*μέρος*] out of which each are measured without remainder. Hence 4 and 3 may form the ratio 4:3 because each is measurable by a shared, discrete submultiple, the simple arithmetical unit more commonly known as 1. But, apples and oranges are not comparable, as the old saying goes, and may form no discrete ratio, because they share no submultiple as a common basis for measurement. (This is one indication for why aesthetics and digitality belong to fundamentally different paradigms; perception easily accommodates qualitative difference while digitality constitutionally prohibits it.) The *logos* ratio is thus a strange beast, both multiple and homogenous. The digital begins with a differential cut, the cut of distinction. But beyond the initial cut all future differentiation is based on the same genus (the homogenous). Later in the treatise, Euclid expands this basic insight by stipulating that *logos* ratios are symmetric [*σύμμετρα*], literally “with measure” or commensurable through a shared, common part. Continue reading →