§4.14.
4.
Proclus Comm. in Encl. p. 212,12sqq. : Τὸ δὲ πόρισμα λέγεται μὲν καὶ ἐπὶ προβλημάτων τινῶν, οἷον τὰ Εὐκλείδῃ γεγραμμένα πορίσματα, λέγεται δὲ ἰδίως, ὅταν ἐκ τῶν ἀποδεδειγμένων ἄλλο τι συναναφανῇ θεώρημα μὴ προθεμένων ἡμῶν, ὃ καὶ διὰ τοῦτο πόρισμα κεκλήκασιν ὥσπερ τι κέρδος ὂν τῆς ἐπιστημονικῆς ἀποδείξεως πάρεργον.
Proclus Comm. in Encl. p. 212,12sqq.: The term "porism" is used on the one hand for certain problems, such as the Porisms written by Euclid, and on the other hand, in a special sense, when from what has been demonstrated another theorem appears at the same time, without our proposing it, which for this very reason they have called "porism", being as it were a certain windfall, a byproduct of the scientific demonstration.
Idem ibid. p. 301,21 sqq. : Ἕν τι τῶν γεωμετρικῶν ἐστιν ὀνομάτων τὸ πόρισμα.
Idem ibid. p. 301,21 sqq.: "Porism" is one of the geometrical terms.
τοῦτο δὲ σημαίνει διττόν· καλοῦσι γὰρ πορίσματα, καὶ ὅσα θεωρήματα συγκατασκευάζεται ταῖς ἄλλων ἀποδείξεσιν οἷον ἕρμαια καὶ κέρδη τῶν ζητούντων ὑπάρχοντα, καὶ ὅσα ζητεῖται μέν, εὑρέσεως δὲ χρῄζει καὶ οὔτε γενέσεως μόνης οὔτε θεωρίας ἁπλῆς.
And this signifies two things: for they call "porisms" both all those theorems which are constructed along with the demonstrations of other things, being like windfalls and windfalls of those who seek, and all those things which are sought indeed, but require discovery, and belong neither to mere production nor to simple contemplation.
ὅτι μὲν γὰρ τῶν ἰσοσκελῶν αἱ πρὸς τῇ βάσει ἴσαι, θεωρῆσαι δεῖ, καὶ ὄντων δή τινων1) πραγμάτων ἐστὶν ἡ τοιαύτη γνῶσις, τὴν δὲ γωνίαν δίχα τεμεῖν ἢ τρίγωνον συστήσασθαι ἢ ἀφελεῖν ἢ προσθέσθαι2), ταῦτα πάντα ποίησίν τινος ἀπαιτεῖ· τοῦ δὲ δοθέντος κύκλου τὸ κέντρον εὑρεῖν ἢ δύο δοθέντων συμμέτρων μεγεθῶν τὸ μέγιστον καὶ κοινὸν μέτρον εὑρεῖν ἢ ὅσα τοιάδε μεταξύ πώς ἐστι προβλημάτων καὶ θεωρημάτων.
For that the angles at the base of isosceles triangles are equal must be contemplated, and such knowledge is of things that already exist, but to bisect an angle, or to construct a triangle, or to cut off, or to add, all these require the production of something; but to find the center of a given circle, or to find the greatest common measure of two given commensurable magnitudes, or whatever other things are of this kind, lie somehow between problems and theorems.
οὔτε γὰρ γενέσεις εἰσὶν ἐν τούτοις τῶν ζητουμένων, ἀλλʼ εὑρέσεις, οὔτε θεωρία ψιλή· δεῖ γὰρ ὑπʼ ὄψιν ἀγαγεῖν καὶ πρὸ ὀμμάτων ποιήσασθαι τὸ ζητούμενον.
For in these there are no productions of the things sought, but discoveries, nor is there mere contemplation; for it is necessary to bring the thing sought under observation and to make it before one's eyes.
τοιαῦτα ἄρα ἐστὶν καί, ὅσα Εὐκλείδης πορίσματα γέγραφε γ βιβλία Πορισμάτων συντάξας.
Such then are also the things which Euclid wrote in his porisms, having composed three books of Porisms.
ἀλλὰ περὶ μὲν τῶν τοιούτων πορισμάτων παρείσθω λέγειν.
But let us pass over speaking about such porisms.
Cfr. scholia in Eucl. l nr. 61 et 62 (V p. 144, 9sqq. , p. 146, 19 sqq. ). 1)
Cf. scholia on Euclid I, nos. 61 and 62 (V p. 144, 9sqq., p. 146, 19 sqq.). 1)