§5.1#2διὸ καὶ περιλαβεῖν ταύτας μιᾷ προτάσει ἐνδεχόμενον εὑρόντες οὕτως ἐγράψαμεν· ἐὰν ὑπτίου ἢ παρυπτίου τρία τὰ ἐπὶ μιᾶς σημεῖα ἢ παραλλήλου τῆς ἑτέρας τὰ δύο δεδομένα ᾖ, τὰ δὲ λοιπὰ πλὴν ἑνὸς ἅπτηται θέσει δεδομένης εὐθείας, καὶ τοῦθʼ ἅψεται θέσει δεδομένης εὐθείας.
For this reason, having found that it is possible to comprehend these in one proposition, we have written as follows: "If, in a concave or sub-concave (quadrilateral), three points on one straight line, or two points on the other if it is parallel, are given, and the rest except one lie on straight lines given in position, then this one also will lie on a straight line given in position." This is stated for four straight lines only, of which no more than two pass through the same point, but it is not known to be true when stated thus for any proposed number of straight lines.
τοῦτʼ ἐπὶ τεσσάρων μὲν εὐθειῶν εἴρηται μόνων, ὧν οὐ πλείονες ἢ δύο διὰ τοῦ αὐτοῦ σημείου εἰσίν, ἀγνοεῖται δὲ ἐπὶ παντὸς τοῦ προτεινομένου πλήθους ἀληθὲς ὑπάρχον οὕτως λεγόμενον τὸν δὲ Στοιχειωτὴν οὐκ εἰκὸς ἀγνοῆσαι τοῦτο, τὴν δʼ ἀρχὴν μόνην τάξαι.
But it is not likely that the writer of the Elements was ignorant of this, but he only set down the beginning.
καὶ ἐπὶ πάντων δὲ τῶν πορισμάτων φαίνεται ἀρχὰς καὶ σπέρματα μόνα πλήθει πολλῶν καὶ μεγάλων καταβεβλημένος, ὧν τὰ γένη οὐ κατὰ τὰς τῶν ὑποθέσεων διαφορὰς διαστέλλειν δεῖ, ἀλλὰ κατὰ τὰς τῶν συμβεβηκότων καὶ ζητουμένων.
And in all the Porisms he appears to have laid down only beginnings and seeds for a multitude of many and great things, of which the genera should be distinguished not by the differences of the hypotheses, but by those of the accidents and things sought.
αἱ μὲν γὰρ ὑποθέσεις ἅπασαι διαφέρουσιν ἀλλήλων εἰδικώταται οὖσαι, τῶν δὲ συμβαινόντων καὶ ζητουμένων ἕκαστον ἓν καὶ τὸ αὐτὸ ὂν πολλαῖς ὑποθέσεσι διαφόροις συμβέβηκε.
For all the hypotheses differ from one another, being most specific, but of the things happening and sought each, being one and the same, happens to many different hypotheses.
ποιητέον οὖν ἐν μὲν τῷ πρώτῳ βιβλίῳ ταῦτα τὰ γένη τῶν ἐν ταῖς προτάσεσι ζητουμένων· ἐν ἀρχῇ μὲν τοῦ βιβλίου διάγραμμα τοῦτο· ἐὰν ἀπὸ δύο δεδομένων σημείων πρὸς θέσει δεδομένην εὐθεῖαι κλασθῶσιν, ἀποτέμνῃ δὲ μία ἀπὸ θέσει
XV. ὅτι τόδε τὸ χωρίον ἢ τόδε μετὰ δοθέντος λόγον ἔχει πρὸς ἀποτομήν.
Therefore, we must set down in the first book these genera of things sought in the propositions; at the beginning of the book there is this diagram (description): "If from two given points straight lines are inflected to a straight line given in position, and one of them cuts off from a (straight line) given in position..." XV. That this area, or this area together with a given (area), has a ratio to the segment cut off.
XVI. ὅτι λόγος τοῦ ὑπὸ τῶνδε πρὸς ἀποτομήν.
XVI. That the ratio of the (rectangle) contained by these to the segment cut off (exists).
XVII. ὅτι λόγος τοῦ ὑπὸ συναμφοτέρων τῶνδε καὶ συναμφοτέρων τῶνδε πρὸς ἀποτομήν.
XVII. That the ratio of the (rectangle) contained by the sum of these and the sum of these to the segment cut off (exists).
XVIII. ὅτι τὸ ὑπὸ τῆσδε καὶ συναμφοτέρου τῆσδε τε καὶ τῆς, πρὸς ἣν ἥδε λόγον ἔχει δοθέντα, καὶ τὸ ὑπὸ τῆσδε καὶ τῆς, πρὸς ἣν ἥδε λόγον ἔχει δοθέντα, λόγον ἔχει πρὸς ἀποτομήν.
XVIII. That the (sum of the rectangle) contained by this and the sum of this and that to which this has a given ratio, and the (rectangle) contained by this and that to which this has a given ratio, has a ratio to the segment cut off.
XIX. ὅτι λόγος συναμφοτέρου πρός τινα ἀπὸ τοῦδε ἕως δοθέντος.
XIX. That the ratio of the sum (of two segments) to some (segment) from this (point) to a given (point exists).
XX. ὅτι δοθὲν τὸ ὑπὸ τῶνδε.
XX. That the (rectangle) contained by these is given.
Ἐν δὲ τῷ τρίτῳ βιβλίῳ αἱ μὲν πλείονες ὑποθέσεις ἐπὶ ἡμικυκλίων εἰσίν, ὀλίγαι δὲ ἐπὶ κύκλου καὶ τμημάτων, τῶν δὲ ζητουμένων τὰ μὲν πολλὰ παραπλήσια τοῖς ἔμπροσθεν, περισσὰ δὲ ταῦτα· XXI. ὅτι λόγος τοῦ ὑπὸ τῶνδε πρὸς τὸ ὑπὸ τῶνδε.
And in the third book most of the hypotheses are on semicircles, and a few on a circle and segments, and of the things sought many are similar to the preceding, but these are additional: XXI. That the ratio of the (rectangle) contained by these to the (rectangle) contained by these (exists).
XXII. ὅτι λόγος τοῦ ἀπὸ τῆσδε πρὸς ἀποτομήν.
XXII. That the ratio of the square on this to the segment cut off (exists).
XXIII. ὅτι τὸ ὑπὸ τῶνδε τῷ ὑπὸ δοθείσης καὶ τῆς ἀπὸ τοῦδε ἕως δοθέντος.
XXIII. That the (rectangle) contained by these is equal to the (rectangle) contained by a given (straight line) and the (straight line) from this (point) to a given (point).
XXIV. ὅτι τὸ ἀπὸ τῆσδε τῷ ὑπὸ δοθείσης καὶ ἀπολαμβανομένης ὑπὸ καθέτου ἕως δοθέντος.
XXIV. That the square on this is equal to the (rectangle) contained by a given (straight line) and the (straight line) cut off by the perpendicular to a given (point).
XXV. ὅτι συναμφότερος ἥδε καὶ πρὸς ἣν ἥδε λόγον ἔχει δοθέντα λόγον ἔχει πρὸς ἀποτομήν.
XXV. That the sum of this and that to which this has a given ratio has a ratio to the segment cut off.
XXVI. ὅτι ἔστιν τι δοθὲν σημεῖον, ἀφʼ οὗ αἱ ἐπιζευγνύμεναι ἐπὶ τούσδε δοθὲν περιέξουσι τῷ εἴδει τρίγωνον.
XXVI. That there is some given point, from which the straight lines joined to these will contain a triangle given in species.
XXVII. ὅτι ἔστιν τι δοθὲν σημεῖον, ἀφʼ οὗ αἱ ἐπιζευγνύμεναι ἐπὶ τόνδε ἴσας ἀπολαμβάνουσι περιφερείας.
XXVII. That there is some given point, from which the straight lines joined to this cut off equal circumferences.
XXVIII. ὅτι ἥδε ἤτοι παρὰ θέσει ἐστὶν ἢ μετά τινος εὐθείας ἐπὶ δοθὲν νευούσης δοθεῖσαν περιέχει γωνίαν.
XXVIII. That this is either parallel in position or contains a given angle with some straight line verging to a given (point).
ἔχει δὲ τὰ τρία βιβλία τῶν Πορισμάτων λήμματα λη, αὐτὰ δὲ θεωρημάτων ἐστὶν ροα.
And the three books of the Porisms have 38 lemmas, and consist of 171 theorems.
De hoc loco cfr. Studien über Eukl. p. 64 sqq. , p. 72 sqq.
Concerning this passage see Studien über Eukl. p. 64 sqq., p. 72 sqq.