The Complete Works of Archimedes, Vol. 2

Hermann in Orph. p. 769 conjectured: τοῖσδε σύ γ’ ἐν. Otherwise, Hiller believes the whole letter to be spurious, without having brought forward any sufficient cause. Cf. Cantor: Vorlesungen p. 284 note 2. Not even the solutions of Menaechmus or Archytas are to be called into question; rather, only the words themselves were changed by Eutocius himself or his source (who, in Archytas alone, is Eudemus) and adapted to the speech of later Greek mathematicians. On the other hand, Blass appears to me as well to have justly condemned Plato's solution in De Platone mathemat. p. 27 sq. Cf. Cantor: Vorlesungen p. 200 sq. Concerning these solutions, cf. Proclus in Timaeum p. 353 ed. Schneider: "How, therefore, given two straight lines, it is possible to take two mean proportionals—we, having found the Archytean demonstration at the end of the treatise [cf. p. 384], shall write it down, having chosen this rather than that of Menaechmus, because he uses conic lines, and likewise that of Eratosthenes, because he uses the placement of a ruler."

Regarding Quadr. Parabol. prop. 23, L. Oppermann, a man especially skilled in ancient mathematics, astutely observed that it contains nothing but a certain peculiar case of the proposition which is in Euclid IX, 35 (if any number of integers are in continued proportion, and from the second and the last are subtracted amounts equal to the first, it will be as the excess of the second to the first, so is the excess of the last to all those before it, i.e., if a : b = b : c = c : d, then b ÷ a : a = d ÷ a : a + b + c). For from Quadr. Parabol. 23, it will be d + c + b + a + 1/3 a = 4/3 d, if d = 4c, c = 4b, b = 4a; or a + b + c = 1/3 (d ÷ a),