§10.110.
10.
Apollonius Pergaeus Conic. I p. 4, 10 sqq. :
Τὸ δὲ τρίτον πολλὰ καὶ παράδοξα θεωρήματα χρήσιμα πρός τε τὰς συνθέσεις τῶν στερεῶν τόπων καὶ τοὺς διορισμούς, ὧν τὰ πλεῖστα καὶ κάλλιστα ξένα, ἃ καὶ κατανοήσαντες συνείδομεν μὴ συντιθέμενον ὑπὸ Εὐκλείδου τὸν ἐπὶ τρεῖς καὶ τέσσαρας γραμμὰς τόπον, ἀλλὰ μόριον τὸ τυχὸν αὐτοῦ καὶ τοῦτο οὐκ εὐτυχῶς·
Apollonius of Perga, Conics I, p. 4, 10 sqq.: The third book contains many remarkable theorems useful for the syntheses of solid loci and the determinations, of which the most and the most beautiful are new; having understood these, we realized that the locus on three and four lines had not been synthesized by Euclid, but only some chance part of it, and this not successfully.
οὐ γὰρ ἦν δυνατὸν ἄνευ τῶν προσευρημένων ἡμῖν τελειωθῆναι τὴν σύνθεσιν.
For it was not possible for the synthesis to be completed without the things newly discovered by us.
Cfr.
Cf.
Pappus VII 32 p. 676, 3 sqq.
Pappus VII 32, p. 676, 3 sqq.
Pappus VII 33 p. 676, 19 sqq. : Ἀπολλώνιος μὲν ταῦτα·
Pappus VII 33, p. 676, 19 sqq.: Apollonius says these things.
ὃν δέ φησιν ἐν τῷ τρίτῳ τόπον ἐπὶ γ καὶ δ γραμμὰς μὴ τελειωθῆναι ὑπὸ Εὐκλείδου, οὐδ᾿ ἂν αὐτὸς ἠδυνήθη οὐδ᾿ ἄλλος οὐδεὶς ἀλλ᾿ οὐδὲ μικρόν τι προσθεῖναι τοῖς ὑπὸ Εὐκλείδου γραφεῖσιν διά γε μόνων τῶν προδεδειγμένων ἤδη κωνικῶν ἄχρι τῶν κατ᾿ Εὐκλείδην, ὡς καὶ αὐτὸς μαρτυρεῖ λέγων ἀδύνατον εἶναι τελειωθῆναι, χωρὶς ὧν αὐτὸς προγράφειν ἠναγκάσθη.
But as for the locus on three and four lines which he says in the third book was not completed by Euclid, neither he himself nor anyone else would have been able to add even a little to what had been written by Euclid, through only the conics already demonstrated up to the time of Euclid, as indeed he himself testifies, saying that it is impossible to be completed without those things which he himself was forced to write beforehand.
ὁ δὲ Εὐκλείδης ἀποδεχόμενος τὸν Ἀρισταῖον ἄξιον ὄντα, ἐφ᾿ οἷς ἤδη παραδεδώκει κωνικοῖς, καὶ μὴ φθάσας ἢ μὴ θελήσας ἐπικαταβάλλεσθαι τούτων τὴν αὐτὴν πραγματείαν, ὅσον δυνατὸν ἦν τοῦ τόπου διὰ τῶν ἐκείνου κωνικῶν, ἔγραψεν οὐκ εἰπὼν τέλος ἔχειν τὸ δεικνύμενον.
But Euclid, accepting Aristaeus as being worthy on account of the conics which he had already handed down, and not having anticipated or not having wished to construct the same treatise of these, wrote as much of the locus as was possible through the conics of that man, without saying that the thing demonstrated was complete.
De loco ad tres uel quattuor lineas u. Zeuthen Die Lehre von den Kegelschn. p. 126 sq.
On the locus to three or four lines, see Zeuthen, Die Lehre von den Kegelschnitten, p. 126 sq.