Pre-Eudoxean Geometric Algebra

By : Stelios Negrepontis, Vasiliki Farmaki, Demetra Kalisperi

Page No: 107-152

Abstract
In the light of our re-interpretation of Plato’s philosophy and of our reconstruction of the proofs of quadratic incommensurabilities by the Pythagoreans, Theodorus, and Theaetetus, in terms of periodic anthyphairesis, we re-examine the Geometric Algebra hypothesis in Greek Mathematics, originally enunciated by Zeuthen and Tannery and supported by van der Waerden and Weil, but challenged by Unguru and several modern historians. Our reconstruction of these proofs employs, for the computation of the anthyphairetic quotient at every step, the solution of a Pythagorean Application of Areas, either in excess or in defect, and is thus qualified as “school algebra” in the spirit of van der Waerden. For the Application of Areas in defect in the Theaetetean Books X and XIII of the Elements, by which the alogoi lines are characterized, the periodic nature of their anthyphairesis is revealed by the Scholia in Eucliden X.135 and 185 and by our re-interpretation of the ill-understood Meno 86e-87b passage. In conclusion, the pre-Eudoxean uses of Applications of Areas fall under the description of “school algebra” solutions of quadratic equations. It is interesting that these early uses stand in sharp contrast to the later uses of more general versions of Application of Areas by Appolonius in his Conic Sections, and which, according to Zeuthen, qualify as Geometric Algebra too, but in the form of pre-Analytic Geometry.
 

Authors:
Stelios Negrepontis : Department of Mathematics, Athens University, Athens 157 84, Greece
Vasiliki Farmaki : Department of Mathematics, Athens University, Athens 157 84, Greece
Demetra Kalisperi : Department of Mathematics, Athens University, Athens 157 84, Greece
 

DOI: https://doi.org/10.32381/GB.2022.44.2.1