Biographical note: Eutocius (ca 480 CE–ca 530 CE) was born in Ascalon (now Ashqelon, in Israel). His chief writings are commentaries on several of Archimedes’ works (including On the Sphere and the Cylinder, which is the primary source for this article) and an edition with commentary of Apollonius’s Conics. He was likely a student of Ammonius, and a contemporary of Anthemius of Tralles. He dedicates his commentary on Apollonius to Anthemius, who was one of the architects of the Hagia Sophia in Constantinople. Read more about Eutocius at MacTutor.
Eutocius is engaging in two acts in his commentary. The first is mathematical. Since Archimedes’ proposition (SC II.1) depends on finding two mean proportionals, and Archimedes does not provide details as to how to find them, Eutocius is bridging the gap mathematically. While it may have been true that Archimedes and other research mathematicians of his day would have been sufficiently familiar with the result to not need this gap filled in, students or those in Eutocius’ day, who lived in a different era, may have needed it. The second act is historical. Since mathematically it would suffice to include only one method of finding two mean proportionals, Eutocius is clearly doing something different, beyond being mathematically thorough, by including twelve. Fortunately for us, Eutocius tells us exactly what he is doing in his own words:
(Heiberg 54.27) Having assumed the things related to the problem through the analysis of it, namely that it is necessary to find two mean proportionals in continuous proportion, but in the synthesis he says, "let them be found." But how to find them, being not written by him [Archimedes] at all, we have found in the writings on this problem by many famous men. Of these, we have excluded that of Eudoxus of Cnidus, since he says in his preface that he has found them by means of curved lines, but in the proof he does not make use of curved lines; and also because he passes off a discrete proportion as if it were continuous. This is impossible to imagine, I dare say for Eudoxus, but also for anyone even moderately engaged in geometry. And so, in order that the ideas of those men who have come to us might become well-known, the method of solution from each will be also (καί) written here.
We perhaps should pay special attention to the adverbial καί here, since in Eutocius’ day, it was clearly the case that these solutions were recorded elsewhere. Not so for us: most of Eutocius’ source texts are no longer extant. Thus, by including all of the solutions he could find, even ones that are nearly the same, he gives us a glimpse into the history of the problem and its solutions. It is worth noting that Eutocius will later engage in this exact sort of philological enterprise, with his commentary and new edition of the Conics.