Complete Extant Works of Hero of Alexandria, Vol. 3

for since the preamble of the dioptric commentary was used by that anonymous writer in composing the preface (chapter I), it is credible that he also took from the same arsenal those things which he could not but propose concerning the structure of the dioptra itself. Since these things are so, with the hope of filling that gap from more complete codices abandoned, it is necessary to restore the form of Hero's dioptra with the help of those indications that are found scattered throughout the latter part of the commentary.

Since I have explained what resources protect the edition of the dioptric commentary, it must be spoken of concerning the interpolations.

And first, Fr. Hultsch1) Preface to the Reliquiae of Hero of Alexandria, p. XVII. opined that the very significant theorem by which the measure of a triangular area is effected from three sides (chapter XXX) was inserted into the middle of Hero’s book by a certain interpolator. If this were true, that most memorable chapter could seem to have been taken from the first book of the Rationes Dimetiendi Principles of Measurement; for in that work, that demonstration is proposed in almost the same words (p. 20, 6 ff.). But an unassailable argument is at hand to refute Hultsch’s opinion. For Hero himself says in chapter XXVII: "It is possible to measure the H-K-Λ triangle, since I have its sides; for this we shall demonstrate next." Since he points his finger, as it were, at chapter XXX with these words placed in chapter XXVII, it appears clearly that either neither of these chapters or both were written by him. This reasoning is confirmed by two clearly similar examples. For what is written in chapter XXIV: "It will be necessary to know how to subtract from a given trapezoid a trapezoid equal to the given one; but this we shall demonstrate next," by these we are relegated to chapter XXVIII; likewise, what is read in chapter XXVI: "how it is necessary to subtract a triangle or to add, we shall demonstrate next," these point to the things that are demonstrated in chapter XXIX. He who has weighed these things, I think, will easily understand that chapters XXVIII, XXIX, and XXX are not only not alien to the plan of the dioptric commentary, but are even its necessary supplements, since they contain some difficult mathematical demonstrations, the mention of which has been made in the previous chapters.