Commentary on Proposition IV.15

Back to Proposition IV.15.

Heiberg, I think with good reason, considers the Porism to this proposition to be referred to in the instance which Proclus (p. 304, 2) gives of a porism following a problem. As the text of Proclus stands, “the (porism) found in the second Book (τὸ δὲ ἐν τῷ δευτέρῳ βιβλίῳ κείμενον) is a porism to a problem”; but this is not true of the only porism that we find in the second Book, namely the porism to II. 4. Hence Heiberg thinks that for τῷ δευτέρῳ βιβλίῳ should be read τῷ δʹ βιβλίῳ, i.e. the fourth Book. Moreover Proclus speaks of the porism in the particular Book, from which we gather that there was only one porism in Book IV. as he knew it, and therefore that he did not regard as a porism the addition to IV. 5. Cf. note on that proposition.

It appears that Theon substituted for the first words of the Porism to

  1. 15 “And in like manner as in the case of the pentagon” (ὁμοίως δὲ

τοῖς ἐπὶ τοῦ πενταγώνου) the simple word “and” or “also” (καί), apparently thinking that the words had the same meaning as the similar words lower down. This is however not the case, the meaning being that “if, as in the case of the pentagon, we draw tangents, we can prove, also as was done in the case of the pentagon, that the figure so formed is a circumscribed regular hexagon.”