The last reference to Heron made by an-Nairīzī occurs in the note on VIII. 27, so that the commentary of the former must at least have reached that point.
II. Porphyry.
The Porphyry here mentioned is of course the Neo-Platonist who lived about 232-304 A.D. Whether he really wrote a systematic commentary on the Elements is uncertain. The passages in Proclus which seem to make this probable are two in which he mentions him (1) as having demonstrated the necessity of the words “not on the same side”
in the enunciation of I. 141, and (2) as having pointed out the necessity of understanding correctly the enunciation of I. 26, since, if the particular injunctions as to the sides of the triangles to be taken as equal are not regarded, the student may easily fall into error2. These passages, showing that Porphyry carefully anaiysed Euclid's enunciations in these cases, certainly suggest that his remarks were part of a systematic commentary. Further, the list of mathematicians in the Fihrist gives Porphyry as having written “a book on the Elements.”
It is true that Wenrich takes this book to have been a work by Porphyry mentioned by Suidas and Proclus (Theolog. Platon.), περὶ ἀρχῶν libri II.3.
There is nothing of importance in the notes attributed to Porphyry by Proclus.
(1) Three alternative proofs of I. 20, which avoid producing a side of the triangle, are assigned to Heron and Porphyry without saying which belonged to which. If the first of the three was Heron's, I agree with van Pesch that it is more probable that the two others were both Porphyry's than that the second was Heron's and only the third Porphyry's. For they are similar in character, and the third uses a result obtained in the second4.
(2) Porphyry gave an alternative proof of I. 18 to meet a childish objection which is supposed to require the part of AC equal to AB to be cut off from CA and not from AC.
Proclus gives a precisely similar alternative proof of I. 6 to meet a similar supposed objection; and it may well be that, though Proclus mentions no name, this proof was also Porphyry's, as van Pesch suggests5.
Two other references to Porphyry found in Proclus cannot have anything to do with commentaries on the Elements. In the first a work called the Συμμικτά is quoted, while in the second a philosophical question is raised.
III. Pappus.
The references to Pappus in Proclus are not numerous; but we have other evidence that he wrote a commentary on the Elements. Thus a scholiast on the definitions of the Data uses the phrase “as ”
1 Proclus, pp. 297, 1-298, 10.
2 ibid. p. 352, 13, 14 and the pages preceding,
3 Fihrist (tr. Suter), p. 9, 10 and p. 45 (note 5).
4 Van Pesch, De Procli fontibus, pp. 129, 130. Heiberg assigned them as above in his Euklid-Studien (p. 160), but seems to have changed his view later. (See Besthorn-Heiberg. Codex Leidensis, p. 93, note 2.)
5 Van Pesch, op. cit. pp. 130-1.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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