Pythagorean Book II of the Elements Restored and Pythagorean Incommensurabilities Reconstructed

Unquestionably the greatest discovery of the PythagoreansPythagorean is the existence of incommensurable magnitudesIncommensurable magnitudes, most probably the incommensurability of the diameter to the side of a squareIncommensurability of the diameter to the side of a square,Incommensurability of the diameter to the side of a squarereconstruction of the original Pythagorean proof II.8/9 but there is no agreement among historians of Greek mathematics on their method of proof. In this chapter, we present novel arguments not only for an anthyphairetic reconstruction of the original PythagoreanPythagorean proof of incommensurabilityIncommensurability but also in favor of one that employs the PythagoreanPythagorean Application of Areas in ExcessApplication of Areasin ExcessApplication of Areasin Excess II.6/7 and, in fact, Geometric AlgebraGeometric Algebra. The main tool for this reconstruction is the restorationBook II of the Elementsrestoration of of Book II of the Elements to its original PythagoreanPythagorean form. Thus, we place the Pythagorean theoremPythagorean theorem II.4/5 at the position II.4/5 (namely immediately after Proposition II.4 of the Elements), the Application of Areas in DefectApplication of Areasin DefectApplication of Areasin Defect II.5/6 at II. 5/6, in ExcessApplication of Areasin ExcessApplication of Areasin Excess II.6/7 at II.6/7, theElegant theorem II.10/11.a Elegant TheoremElegant Theorem(s) at (II.9/10 and) II.10/11.a, and the proof of the PellPell property of the side and diameter numbers property of the side and diameter numbers by mathematical inductionInduction, mathematical at II.10/11.b. We then naturally expect that the original anthyphairetic PythagoreanPythagorean proof of incommensurability is placed before Propositions II.9 & 10 on the side and diameter numbers and after the tools needed for its proof, the Application of Areas II.6/7 (and II.8), hence, precisely at the position II.8/9.