On the notion of equal figures in Euclid
Beitrage zur Algebra und Geometrie
Euclid uses an undefined notion of “equal figures”, to which he applies the common notions about equals added to equals or subtracted from equals. This notion does not occur in modern geometrical theories such as those of Hilbert or Tarski. Therefore to account for Euclid in modern geometry, one must somehow replace Euclid’s “equal figures” with a defined notion. In this paper we present a new solution to this problem, and moreover we argue that “Euclid could have done it”. That is, it is based on mathematics that was available in Euclid’s time, including ideas related to Euclid’s Proposition I.44. The proof uses the theory of proportions. Hence we also discuss the “early theory of proportions”, which has a long history.
Area, Equal figures, Euclid, Euclidean geometry
Michael Beeson. "On the notion of equal figures in Euclid" Beitrage zur Algebra und Geometrie (2022). https://doi.org/10.1007/s13366-022-00649-9