Philoponus: On Aristotle's Prior Analytics

The explanation of Psellus on this theorem follows, then another scholion (omitted Z), to which is ascribed in the margin "Of Philoponus."

Since this also has been proven in the twelfth book of the Elements, that as the squares from the diameters have, so also the circles and semicircles have. Therefore the semicircle ABΓ is double the semicircle AΕΓ. Therefore the quadrant of the circle AΖΓΔ is equal to the semicircle AΕΓ. Therefore let the common segment of the circle AΖΓ, which is less than a semicircle, be subtracted. Therefore the remaining triangle AΔΓ is equal to the remaining lune a crescent-shaped geometric figure AΕΓ. Hippocrates, having thus shown a lune equal to a triangle, next attempts to square the circle as follows: Let there be a straight line AB and let a semicircle be described upon it. And let ΓΔ be double of AB. And let the semicircle ΓΗΔ be described upon it, and let the hexagonal sides be joined within it, ΓΕ, ΕΖ, ΖΔ. And upon these hexagonal sides let semicircles be described, the ΓΗΕ, ΕΘΖ, ΖΚΔ. And since each of the hexagonal sides is half of the diameter ΓΔ, but the diameter AB was also given as half of the diameter ΓΔ, therefore each semicircle ΓΗΕ, ΕΘΖ, ΖΚΔ is equal to the semicircle AB. Therefore the four are equal to each other. And since the square from the double diameter is four times the square from the half diameter. And as the squares from the diameters have, so also have the circles and the semicircles. Therefore the semicircle ΓΗΔ is four times the semicircle AB. Therefore the semicircle ΓΗΔ is equal to the four semicircles AB, ΓΗΕ, ΕΘΖ, ΖΚΔ. Let the common segments of the circles ΓΛΕ, ΕΜΖ, ΖΝΔ be subtracted, therefore the remaining trapezoid ΓΕΖΔ is equal to the remaining lunes ΓΗΕΛ, ΕΘΖΜ, ΖΚΔΝ, and the semicircle AB. But since it is possible to analyze every rectilinear shape into triangles, if the trapezoid were analyzed into triangles and three triangles were given equal to the three lunes. For this was proven above. The remaining triangle will be equal to the semicircle AB. But the AB, when doubled, will make a square shape, and there will be a square equal to the circle. And this circle will be squared. The attempt is indeed technical and proven from geometrical principles. Yet it errs to the extent that the previously proven lune was equal to a triangle consisting of a square side, whereas these present lunes consist of hexagonal sides. Yet it is possible to say, as Simplicius also says, that even in this way it is possible for the circle to be squared, even if it is laborious to state it now due to the refinement of the theorem figures are drawn in the margin.

f. 173r (f. 350v Z) lemma (p. 69 b 38) "One must also examine concerning the other objections (of etc. omitted Z):" in the margin "Of Philoponus (omitted Z):" Since objections are expressed not only categorically but also hypothetically, he says one must also examine those expressed hypothetically, for example, those from the contrary. As when someone claims pleasure is good, we object by saying: "If pleasure is good, then pain is evil. For if the contrary is for the contrary, then the contrary is for the contrary. But surely not every pain is evil. Therefore, not every pleasure is good." And if someone were to say justice is a science, we object that it is not at all. Since injustice is not ignorance. And from the similar, as when someone says the surface is a part of the body, we object by saying it is not. Since the line is not a part of the surface. Nor the point a part of the line. And from the conventional, as if someone should say that every soul is mortal, we would respond that it is not. For to Aristotle and Plato, being wise men, it seemed that human souls are immortal. And concerning the negative, that it is impossible to take it from the middle figure, it has been said that one must not turn the objection toward other figures, that is, toward different ones, but the same ones. And that it is not possible to take the particular negative from the first [figure], the same argument will fit. For here too, there is need of conversion. For if we object to the one saying "one science exists for all contraries," wishing to establish "not for all," we shall say that the known and the unknown are contraries. And inverted things are known and unknown contraries. But of the known and the unknown, there is not one science.

f. 173 (f. 351r Z) lemma (p. 70 a 7): "And a sign wishes to be a demonstrative or necessary or reputable proposition (proposition etc. omitted Z)" with two scholia, to one of which is prefixed "Of Philoponus (in margin Z):" The sign is predicated as a genus of the probable and the proofs. And a proof tekmērion is the convertible proposition. As fire and smoke.