Philoponus: On Aristotle's Prior Analytics

if they are equal to each other, the lines are parallel. For if not, let them meet each other at the point K and make a triangle HΘK, one side of the triangle, KH, having been extended to A. The exterior angle HAΘ is greater than the interior and opposite angle HΘK, as has been proven in the 16th in the sixteenth Z of the first book. But it was also given as equal, as alternate, therefore the same is both equal and greater, which is impossible. Therefore, the lines do not meet, but if the alternate angles are equal, the lines are also parallel. And x was forgotten by the rubricator that we may encompass the whole thought by recapitulating, if the lines are parallel, the interior angles on the same side are equal to two right angles. If the interior angles are equal to two right angles, the alternate angles are equal. If the alternate angles are equal, the lines are parallel. And it is gathered that if the lines are parallel, the lines are parallel the second "the lines are parallel" omitted in Y.

f. 160r (omitted Z) lemma (p. 66a 1): "It is evident, therefore, that the impossible not being related to the terms from the beginning:" two scholia follow, to one of which is written in the margin "Of Philoponus on the same."

Having established that the falsehood happens contrary to the premise when the impossible connects to the terms from the beginning, he says next that not even in this way will it always happen due to the hypothesis, that is, not even if the impossible connects to the terms from the beginning will the falsehood that happens always be contrary to the hypothesis. For let A be assumed for every B. And B for every Γ. And Γ for every Δ. For example, animal for every inanimate, inanimate for every stone, stone for every selenite a type of transparent stone or mineral. And let the impossible connect to the term from the beginning. And let there be a false conclusion: animal for every selenite. And this will happen entirely due to the hypothesis that says "animal for every inanimate." But if instead of the assumed B another term is set, for example "body," and it is said "animal for every body," and in this way the other remaining terms are connected, the stone and the selenite, there will again be the impossible connecting to the term from the beginning, but not because of the hypothesis that says "animal for every inanimate" will the falsehood happen. For that was removed by the change of the subject, and another was introduced instead. Even if the term predicated of it is still preserved. For it is possible to take the same term for many and different propositions.

f. 170r (omitted Z) lemma (p. 69a 35) "nor when BΓ is immediate;": of the two scholia, to one is ascribed in the margin "Of John Philoponus."

Geometers do not call a square simply a shape that has four angles, but when it is both equilateral and rectangular, the shape that has four angles. And a right angle is that which a perpendicular makes standing upon a straight line. For the geometer says that when a straight line standing upon a straight line makes the adjacent angles equal to each other, each of the equal angles is a right angle. Therefore, geometers comparing rectilinear shapes with rectilinear, for example a triangle with a square, for these are rectilinear as being contained by straight lines, sought a comparison or equalization, and of a square with a circle, that is, of a rectilinear with a curvilinear. This, then, many wise men sought but have not found up to now. Simplicius says somewhere that Iamblichus, in his commentaries on the Categories, says that Aristotle had not yet found the quadrature of the circle. But the Pythagoreans found it later. But also Archimedes and Hippocrates of Chios came close to finding it. Archimedes made the discovery more instrumental, but Hippocrates more logical and more geometrical. And the proof of Hippocrates is as follows: Let there be a straight line AB. And let a semicircle AΓB be described upon it. And let AB be cut in half at point Δ and let ΔΓ be joined. And from point Γ to points A and B let ΓA and ΓB be joined, the sides of the square inscribed within the circle. Then upon the square side ΓA let another semicircle be described. Since, therefore, the angle from AΓB is right, the square from AB is equal to the squares from AΓ and BΓ. But the square from AΓ is equal to that from BΓ. Therefore the square from AB is double that from AΓ. This, therefore, has been proven in the first elemental theorem called "the bride." Let this also be proposed now: Pythagoras in red. In right-angled triangles—which I have proven. (f. 171r, where in the margin it is written in red: "47 of the first book of Euclid's Elements.")