The Hero-Apollonius Method of Cube Duplication

Abstract

The method of cube duplication presented by Hero survives in at least six different versions. It appears in the Belopoeica of Hero, still extant in Greek (HB); it appears also, in a somewhat different form, in Hero’s Mechanics, accessible to us now only in the medieval Arabic translation by Qustâ ibn Lûqâ (HM). Furthermore, Pappus presents a version of Hero’s text in Collection III (HP), while Eutocius includes yet another version among the 12 methods surveyed in his commentary on Archimedes’ Sphere and Cylinder (HE). To these we may add one of the three methods given by John Philoponus in his commentary on Aristotle’s Posterior Analytics; although the procedure there described is identical with that of the Heronian group, Philoponus assigns it to Apollonius (hence our designation of it as AJ). Support for this attribution comes from Eutocius, who likewise assigns to Apollonius an alternative method (AE) that technically, if not textually, is equivalent to the Heronian method.¹ We may infer from Eutocius and Philoponus that this manner of solution was known through a textual tradition stemming from Apollonius, whence the procedure itself was discovered no later than the time of Apollonius. The attribution to Hero must have arisen merely from his having chosen it for inclusion in his mechanical writings, rather than from his being its discoverer. A closer look at these texts will reveal the nature of their interdependence, as well as the manner of editorial intervention by the later writers.