Are Book 1 “things” and Book 5 “magnitudes” the same objects in Euclid’s Elements?

Determine whether the objects referred to as “things” under the common notions in Book 1 of Euclid’s Elements are the same as the “magnitudes” employed in Book 5’s theory of proportions; specifically, ascertain whether line segments, angles, and plane figures are treated as one unified class of magnitudes across both books or whether distinct categories are intended.

Background

The note highlights that Euclid uses the term “common notions” in Book 1 when discussing equality and comparison, and a different term—translated as “magnitudes”—in Book 5 when developing proportion theory. The author observes that the Greek words differ and points out the ambiguity regarding whether these refer to the same kind of mathematical objects.

Clarifying this relationship is central to understanding how Euclid compares and operates on line segments, angles, and figures across the Elements, and whether the framework for proportions consistently applies to the objects introduced earlier.