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304, "-305, 3. 13 ibid. p. 426, 6-912 ibid. pp. 428, 7-429,9, 3 CH. IV] PROCLUS AND HIS SOURCES 37 so far as the story of the sacrifice is concerned, the same thing is told of Thales in connexion with his discovery that the angle in a semicircle is a right angle l , and Plutarch is not certain whether the ox was sacrificed on the discovery of I. 47 or of the problem about application of areas". Plutarch's doubt suggests that he knew of no evidence for the story beyond the vague allusion in the distich of Apollodorus "Logisticus" (the "calculator") cited by Diogenes Laertius also s ; and Proclus may have had in mind this couplet with the passages of Plutarch.

Ov)7 when reproducing what he found in the ancient writers; sometimes it is clear that he left out altogether proofs or constructions of things by no means easy8. Geminus. The discussions about the date and birthplace of Geminus form a whole literature, for an account of which I must refer the reader to the recent edition by Manitius of Gemini elementa astronomiae (Teubner, 1898)9. It must suffice here to state the general conclusion arrived at by Manitius 10• Though the name looks like a Latin name (GemYnus), 1 Diog.

II. Porphyry. D. Whether he really wrote a systematic commentary on the Elemmts is uncertain. The passages in Proclus which seem to make this probable are two in which he mentions him (I) as having demonstrated the necessity of the words "not on the same side" in the enunciation of I. I4 \ 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 error 2• These passages, showing that Porphyry carefully analysed Euclid's enunciations in these cases, certainly suggest that his remarks were part of a systematic commentary.

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Thirteen Books of Euclid's Elements. Books I-II by Heath


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