§5.1#15.
5.
Pappus Συναγ. VII 3 p. 636, 18 sqq. (ed.
Pappus Collection VII 3 p. 636, 18 sqq. (ed.
Hultsch): Τῶν δὲ προειρημένων τοῦ ἀναλυομένου βιβλίων ἡ τάξις ἐστὶν τοιαύτη· Εὐκλείδου Δεδομένων βιβλίον α, Ἀπολλωνίου Λόγου ἀποτομῆς β, Χωρίου ἀποτομῆς β, Διωρισμένης τομῆς δύο, Ἐπαφῶν δύο, Εὐκλείδου Πορισμάτων τρία κτλ.
Hultsch): The order of the aforementioned books of the Treasury of Analysis is as follows: Euclid's Data, one book; Apollonius' Cutting-off of a Ratio, two books; Cutting-off of an Area, two books; Determinate Section, two books; Tangencies, two books; Euclid's Porisms, three books, etc.
Idem ibid. VII 13 p. 648, 18 sqq.
Idem ibid.
:
Μετὰ δὲ τὰς Ἐπαφὰς ἐν τρισὶ βιβλίοις Πορίσματά ἐστιν Εὐκλείδου, πολλοῖς ἄθροισμα φιλοτεχνότατον εἰς τὴν ἀνάλυσιν τῶν ἐμβριθεστέρων προβλημάτων, καὶ τῶν γενῶν ἀπερίληπτον τῆς φύσεως παρεχομένης πλῆθος οὐδὲν προστεθείκασι τοῖς ὑπὸ Εὐκλείδου γραφεῖσι πρώτου, χωρὶς εἰ μή τινες τῶν πρὸ ἡμῶν ἀπειρόκαλοι δευτέρας γραφὰς ὀλίγοις αὐτῶν παρατεθείκασιν ἑκάστου μὲν πλῆθος ὡρισμένον ἔχοντος ἀποδείξεων, ὡς ἐδείξαμεν, τοῦ δʼ Εὐκλείδου μίαν ἑκάστου θέντος τὴν μάλιστά πως ἐμφαίνουσαν.
VII 13 p. 648, 18 sqq.: After the Tangencies, in three books, are the Porisms of Euclid, a collection most highly artful for many in the analysis of more weighty problems, and of which the classes are incomprehensible, since nature provides an unlimited multitude of them. They have added nothing to what was first written by Euclid, except if some unrefined persons before us have set beside a few of them second proofs, while each of these had a defined number of proofs, as we have shown, but Euclid set down one proof for each, the one which most of all made it manifest.
ταῦτα δὲ λεπτὴν καὶ φυσικὴν ἔχει θεωρίαν καὶ ἀναγκαίαν καὶ καθολικωτέραν καὶ τοῖς δυναμένοις ὁρᾶν καὶ πορίζειν ἐπιτερπῆ.
And these have a subtle and natural theory, necessary and more universal, and delightful to those who are able to see and to find.
ἅπαντα δὲ αὐτῶν τὰ εἴδη οὔτε θεωρημάτων ἐστὶν οὔτε προβλημάτων, ἀλλὰ μέσον πως τούτων ἐχούσης ἰδέας, ὥστε τὰς προτάσεις αὐτῶν δύνασθαι σχηματίζεσθαι ἢ ὡς θεωρημάτων ἢ ὡς προβλημάτων, παῤ ὃ καὶ συμβέβηκε τῶν πολλῶν γεωμετρῶν τοὺς μὲν ὑπολαμβάνειν αὐτὰ εἶναι τῷ γένει θεωρήματα τοὺς δὲ προβλήμιατα ἀποβλέποντας εἰς τὸ σχῆμα μόνον τῆς προτάσεως.
And all their classes are neither of theorems nor of problems, but have a character intermediate between these, so that their propositions can be formulated either as theorems or as problems, owing to which it has also happened that among the many geometers, some suppose them to be theorems in genus, and others problems, looking only to the form of the proposition.
τὴν δὲ διαφορὰν τῶν τριῶν τούτων ὅτι βέλτιον ᾔδεσαν οἱ ἀρχαῖοι, δῆλον ἐκ τῶν ὅρων.
But that the ancients knew better the difference between these three is clear from the definitions.
ἔφασαν γὰρ θεώρημα μὲν εἶναι τὸ προτεινόμενον εἰς ἀπόδειξιν αὐτοῦ τοῦ προτεινομένου, πρόβλημα δὲ τὸ προβαλλόμενον εἰς κατασκευὴναὐτοῦ τοῦ προτεινομένου, πόρισμα δὲ τὸ προτεινόμενον εἰς πορισμὸν αὐτοῦ τοῦ προτεινομένου.
For they said that a theorem is what is proposed for the demonstration of the very thing proposed, a problem is what is proposed for the construction of the very thing proposed, and a porism is what is proposed for the finding of the very thing proposed.
μετεγράφη δὲ οὗτος ὁ τοῦ πορίσματος ὅρος ὑπὸ τῶν νεωτέρων μὴ δυναμένων ἅπαντα πορίζειν, ἀλλὰ συγχρωμένων τοῖς στοιχείοις τούτοις καὶ δεικνύντων αὐτὸ μόνον τοῦθʼ, ὅτι ἔστι τὸ ζητούμενον, μὴ ποριζόντων δὲ τοῦτο καὶ ἐλεγχομένων ὑπὸ τοῦ ὄρου καὶ τῶν διδασκομένων.
But this definition of the porism was rewritten by the moderns who were not able to find everything, but made use of these Elements and showed only this very thing, that the thing sought exists, but did not find it, and were refuted by the definition and by those who are taught.
ἔγραψαν δὲ ἀπὸ συμβεβηκότος οὕτως· πόρισμά ἐστιν τὸ λεῖπον ὑποθέσει τοπικοῦ θεωρήματος. τούτου δὲ τοῦ γένους τῶν πορισμάτων εἶδός ἐστιν οἱ τόποι καὶ πλεονάζουσιν ἐν τῷ ἀναλυομένῳ· κεχωρισμένον δὲ τῶν πορισμάτων ἤθροισται καὶ ἐπιγράφεται καὶ παραδίδοται διὰ τὸ πολύχυτον εἶναι μᾶλλον τῶν ἄλλων εἰδῶν.
And they wrote from an accidental property as follows: 'A porism is that which is deficient in the hypothesis of a locus theorem.' And of this genus of porisms, loci are a species, and they abound in the Treasury of Analysis; but having been separated from the porisms, they have been collected and inscribed and handed down because they are more abundant than the other species.
συμβέβηκε δὲ καὶ τοῦτο τοῖς πορίσμασιν τὰς προτάσεις ἔχειν ἐπιτετμημένας διὰ τὴν σκολιότητα πολλῶν συνήθως συνυπακουομένων· ὥστε πολλοὺς τῶν γεωμετρῶν ἐπὶ μέρους ἐκδέχεσθαι, τὰ δὲ ἀναγκαιότερα ἀγνοεῖν τῶν σημαινομένων.
And it has also happened to porisms that they have their propositions shortened because of the complexity of many things that are commonly understood together; so that many of the geometers understand them only in part, but are ignorant of the more necessary of the things signified.
περιλαβεῖν δὲ πολλὰ μιᾷ προτάσει ἥκιστα δυνατὸν ἐν τούτοις διὰ τὸ καὶ αὐτὸν Εὐκλείδην οὐ πολλὰ ἐξ ἑκάστου εἴδους τεθεικέναι, ἀλλὰ δείγματος ἕνεκα ἐκ τῆς πολυπληθείας ἓν ἢ ὀλίγα.
But to comprehend many things in one proposition is least of all possible in these, because Euclid himself did not set down many from each species, but for the sake of an example, one or a few out of the great multitude.
πρὸς ἀρχῇ δὲ ὅμως τοῦ πρώτου βιβλίου τέθεικεν ὁμοειδῆ τινα ἐκείνου τοῦ δαψιλεστέρου εἴδους τῶν τόπων ὡς ι τὸ πλῆθος.
Nevertheless, near the beginning of the first book he has set down some of the same class from that more abundant species of loci, about ten in number.