PREFACE.

The following words on line 13 make the error worse: τὸν δὲ αὐτόν, ὃν καὶ τὸ πολύγωνον the same [ratio], which also the polygon [has] (it should have been τὰ πολύγωνα the polygons), which should have been preceded by the words: διπλασίονα λόγον ἤπερ ἡ EK πρὸς AΛ a double ratio than that of EK to AΛ, since this is concluded from that. It appears, therefore, that these two passages are interpolated. But Eutocius seems to disagree, for he has these words: ἐπεὶ δέδεικται, ὅτι ἐστὶν ὡς τὸ πολύγωνον πρὸς τὸ πολύγωνον, οὕτως ὁ M κύκλος πρὸς τὸν N since it has been demonstrated that it is as the polygon is to the polygon, so is circle M to circle N. But I think he is speaking less precisely and is looking toward the ratio itself, which was demonstrated by Archimedes in the former part of the proposition: O : o = EK² : AΛ². For since O : o = M : N (by hypothesis) and

the ratio P : p = M : N follows so easily from this that Eutocius can correctly say that this ratio was demonstrated simultaneously with that one. In the same way Archimedes says: ἐδείχθη δὲ ὡς ἡ EK πρὸς AΛ, οὕτως ἡ ἐκ τοῦ κέντρου τοῦ M κύκλου πρὸς τὴν ἐκ τοῦ κέντρου τοῦ N κύκλου it was shown, however, that as EK is to AΛ, so is the radius of circle M to the radius of circle N (p. 174, 13), although he had only demonstrated this: O : o = EK² : AΛ², from which it is easily concluded that R : r = EK : AΛ. That those words are interpolated is also apparent from the fact that Archimedes in prop. 42 (p. 176, 25) proposes this very ratio O : o = P : p with these additional words: ἑκάτερος γὰρ τῶν λόγων διπλάσιός ἐστι τοῦ, ὃν ἔχει ἡ τοῦ περιγεγραμμένου πλευρὰ πρὸς τὴν τοῦ ἐγγεγραμμένου πλευράν for each of the ratios is double that which the side of the circumscribed [figure] has to the side of the inscribed [figure] (i.e., EK : AΛ). He would undoubtedly have added these words in prop. 41 if he had wished to use this ratio there as well; and furthermore, Eutocius explains the words "for each of the ratios, etc." at prop. 42, whereas it would have been enough to cite Archimedes' own words from prop. 41. Therefore, I think those words were inserted by a transcriber after the likeness of proposition 42.