§1#2Τὸ μὲν οὖν ἀκριβὲς λαβεῖν οὐκ εὐχερές ἐστι διὰ τὸ μήτε τὰν ὄψιν μήτε τὰς χεῖρας μήτε τὰ ὄργανα, διʼ ὧν δεῖ λαβεῖν, ἀξιόπιστα εἶμεν τὸ ἀκριβὲς ἀποφαίνεσθαι·
To obtain then the exact measurement is not easy, because neither the eye, nor the hands, nor the instruments required to obtain it are trustworthy for declaring the exact truth.
περὶ δὲ τούτων ἐπὶ τοῦ παρόντος οὐκ εὔκαιρον μακύνειν ἄλλως τε καὶ πλεονάκις τοιούτων ἐμπεφανισμένων ἀποχρὴ δέ μοι ἐς τὰν ἀπόδειξιν τοῦ προκειμένου γωνίαν λαβεῖν, ἅτις ἐστὶ μὴ μείζων τᾶς γωνίας, εἰς ἃν ὁ ἃλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει, καὶ πάλιν ἄλλαν γωνίαν λαβεῖν, ἅτις ἐστὶν οὐκ ἐλάττων τᾶς γωνίας, εἰς ἃν ὁ ἃλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει.
But it is not opportune at present to prolong the discussion on these matters, especially as such things have been shown many times before. But it is sufficient for me, for the proof of the proposed thesis, to take an angle which is not greater than the angle which the sun fits, having its vertex at the eye, and again to take another angle which is not less than the angle which the sun fits, having its vertex at the eye.
Τεθέντος οὖν μακροῦ κανόνος ἐπὶ πόδα ὀρθὸν ἐν τόπῳ κείμενον, ὅθεν ἤμελλεν ἀνατέλλων ὁ ἅλιος ὁρᾶσθαι, καὶ κυλίνδρου μικροῦ τορνευθέντος καὶ τεθέντος ἐπὶ τὸν κανόνα ὀρθοῦ εὐθέως μετὰ τὰν ἀνατολὰν τοῦ ἁλίου, ἔπειτʼ ἐόντος αὐτοῦ ποτὶ τῷ ὁρίζοντι καὶ δυναμένου ἀντιβλέπεσθαι ἐπεστράφη ὁ κανὼν εἰς τὸν ἅλιον, καὶ ἁ ὄψις κατεστάθη ἐπὶ τὸ ἄκρον τοῦ κανόνος· ὁ δὲ κύλινδρος ἐν μέσῳ κείμενος τοῦ τε ἁλίου καὶ τᾶς ὄψιος ἐπεσκότει τῷ ἁλίῳ.
Therefore, a long ruler having been placed on a straight pedestal situated in a place from which the rising sun was about to be seen, and a small cylinder having been turned and placed upright on the ruler immediately after the rising of the sun, then, when the sun was near the horizon and could be looked at directly, the ruler was turned towards the sun, and the eye was placed at the end of the ruler; and the cylinder, being situated in the middle between the sun and the eye, obscured the sun.
Ἀποχωριζόμενος οὖν ἀπὸ τᾶς ὄψιος, ἐν ᾧ ἄρξατο παραφαίνεσθαι τοῦ ἁλίου μικρὸν ἐφʼ ἑκάτερα τοῦ κυλίνδρου, κατεστάθη ὁ κύλινδρος.
Then, as the cylinder was moved away from the eye, it was fixed at the position where a little of the sun began to appear on either side of the cylinder.
Εἰ μὲν οὖν συνέβαινεν τὰν ὄψιν ἀφʼ ἑνὸς σαμείου βλέπειν, εὐθειᾶν ἀχθεισᾶν ἀπʼ ἄκρου τοῦ κανόνος, ἐν ᾧ τόπῳ ἁ ὄψις κατεστάθη, ἐπιψαυουσᾶν τοῦ κυλίνδρου ἁ περιεχομένα γωνία ὑπὸ τᾶν ἀχθεισᾶν ἐλάσσων κα ἦς τᾶς γωνίας, εἰς ἃν ὁ ἅλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει, διὰ τὸ παραβλέπεσθαί τι τοῦ ἁλίου ἐφʼ ἑκάτερα τοῦ κυλίνδρου ἐπεὶ δʼ αἱ ὄψιες οὐκ ἀφʼ ἑνὸς σαμείου βλέποντι, ἀλλὰ ἀπό τινος μεγέθεος, ἐλάφθη τι μέγεθος στρογγύλον οὐκ ἔλαττον ὄψιος, καὶ τεθέντος τοῦ μεγέθεος ἐπὶ τὸ ἄκρον τοῦ κανόνος, ἐν ᾧ τόπῳ ἁ ὄψις κατεστάθη, ἀχθεισᾶν εὐθειᾶν ἐπιψαυουσᾶν τοῦ τε μεγέθεος καὶ τοῦ κυλίνδρου ἁ οὖν περιεχομένα γωνία ὑπὸ τᾶν ἀχθεισᾶν ἐλάττων ἦς τᾶς γωνίας, εἰς ἃν ὁ ἅλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει.
Now, if the eye were to see from a single point, then if straight lines were drawn from the end of the ruler where the eye was placed, touching the cylinder, the angle contained by the drawn lines would have been less than the angle which the sun fits, having its vertex at the eye, because of a part of the sun being seen on either side of the cylinder; but since the eye does not see from a single point, but from a certain magnitude, a round magnitude not less than the pupil was taken, and this magnitude being placed at the end of the ruler where the eye was situated, straight lines were drawn touching both the magnitude and the cylinder; the angle contained by the drawn lines was therefore less than the angle which the sun fits, having its vertex at the eye.
Τὸ δὲ μέγεθος τὸ οὐκ ἔλαττον τᾶς ὄψιος τόνδε τὸν τρόπον εὑρίσκεται δύο κυλίνδρια λαμβάνεται λεπτὰ ἰσοπαχέα ἀλλάλοις, τὸ μὲν λευκόν, τὸ δὲ οὔ, καὶ προτίθενται πρὸ τᾶς ὄψιος, τὸ μὲν λευκὸν ἀφεστακὸς ἀπʼ αὐτᾶς, τὸ δὲ οὐ λευκὸν ὡς ἔστιν ἐγγυτάτω τᾶς ὄψιος, ὥστε καὶ θιγγάνειν τοῦ προσώπου.
The magnitude which is not less than the pupil is found in the following manner: two thin cylinders of equal thickness are taken, one white and the other not, and are placed before the eye, the white one at a distance from it, and the non-white one as close as possible to the eye, so as even to touch the face.
Εἰ μὲν οὖν κα τὰ λαφθέντα κυλίνδρια λεπτότερα ἔωντι τᾶς ὄψιος, περιλαμβάνεται ὑπὸ τᾶς ὄψιος τὸ ἐγγὺς κυλίνδριον καὶ ὁρῆται ὑπὸ αὐτᾶς τὸ λευκόν, εἰ μέν κα παρὰ πολὺ λεπτότερα ἔωντι, πᾶν, εἰ δέ κα μὴ παρὰ πολύ, μέρεά τινα τοῦ λευκοῦ ὁρῶνται ἐφʼ ἑκάτερα τοῦ ἐγγὺς τᾶς ὄψιος, λαφθέντων δὲ τῶνδε τῶν κυλινδρίων ἐπιταδείων πως τῷ πάχει ἐπισκοτεῖ τὸ ἕτερον αὐτῶν τῷ ἑτέρῳ καὶ οὐ πλείονι τόπῳ τὸ δὴ ταλικοῦτον μέγεθος, ἁλίκον ἐστὶ τὸ πάχος τῶν κυλινδρίων τῶν τοῦτο ποιούντων, μάλιστά πώς ἐστιν οὐκ ἔλαττον τᾶς ὄψιος.
Now, if the cylinders taken are thinner than the pupil, the nearer cylinder is enveloped by the sight, and the white one is seen by the eye; if they are much thinner, the whole of it is seen, but if they are not much thinner, certain parts of the white one are seen on either side of the cylinder near the eye; but when these cylinders are taken of a certain suitable thickness, the one of them obscures the other without covering a larger space; a magnitude of this size, equal to the thickness of the cylinders producing this effect, is most nearly not less than the pupil.
Ἁ δὲ γωνία ἁ οὐκ ἐλάττων τᾶς γωνίας, εἰς ἃν ὁ ἅλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει, οὕτως ἐλάφθη ἀποσταθέντος ἐπὶ τοῦ κανονίου τοῦ κυλίνδρου ἀπὸ τᾶς ὄψιος οὕτως, ὡς ἐπισκοτεῖν τὸν κύλινδρον ὅλῳ τῷ ἁλίῳ, καὶ ἀχθεισᾶν εὐθειᾶν ἀπʼ ἄκρου τοῦ κανόνος, ἐν ᾧ τόπῳ ἁ ὄψις κατεστάθη, ἐπιψαυουσᾶν τοῦ κυλίνδρου ἁ περιεχομένα γωνία ὑπὸ τᾶν ἀχθεισᾶν εὐθειᾶν οὐκ ἐλάττων γίνεται τᾶς γωνίας, εἰς ἃν ὁ ἅλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει.
The angle which is not less than the angle which the sun fits, having its vertex at the eye, was obtained in this way: the cylinder on the ruler having been moved away from the eye to such a distance that the cylinder obscured the whole of the sun, and straight lines having been drawn from the end of the ruler where the eye was placed, touching the cylinder; the angle contained by the drawn straight lines becomes not less than the angle which the sun fits, having its vertex at the eye.
Ταῖς δὴ γωνίαις ταῖς οὕτως λαφθείσαις καταμετρηθείσας ὀρθᾶς γωνίας ἐγένετο ἁ ἐν στίγῳ διαιρεθείσας τᾶς ὀρθᾶς εἰς ρξδ ἐλάττων ἢ ἓν μέρος τούτων, ἁ δὲ ἐλάττων διαιρεθείσας τᾶς ὀρθᾶς εἰς σ μείζων ἢ ἓν μέρος τούτων δῆλον οὖν ὅτι καὶ ἁ γωνία, εἰς ἃν ὁ ἅλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει, ἐλάττων μὲν ἐστιν ἢ διαιρεθείσας τᾶς ὀρθᾶς εἰς ρξδ τούτων ἓν μέρος, μείζων δὲ ἢ διαιρεθείσας τᾶς ὀρθᾶς εἰς σ τούτων ἓν μέρος.
When right angles were measured by the angles thus obtained, the greater angle was less than one part of a right angle divided into 164, and the lesser angle was greater than one part of a right angle divided into 200. It is therefore clear that the angle which the sun fits, having its vertex at the eye, is also less than one part of a right angle divided into 164, and greater than one part of a right angle divided into 200.
Πεπιστευμένων δὲ τούτων δείκνυται ἁ διάμετρος τοῦ ἁλίου μείζων ἐοῦσα τᾶς τοῦ χιλιαγώνου πλευρᾶς τοῦ εἰς τὸν μέγιστον κύκλον ἐγγραφομένου τῶν ἐν τῷ κόσμῳ, Νοείσθω γὰρ ἐπίπεδον ἐκβεβλημένον διά τε τοῦ κέντρου τοῦ ἁλίου καὶ τοῦ κέντρου τᾶς γᾶς καὶ διὰ τᾶς ὄψιος, μικρὸν ὑπὲρ τὸν ὁρίζοντα ἐόντος τοῦ ἁλίου, τεμνέτω δὲ τὸ ἐκβληθὲν ἐπίπεδον τὸν μὲν κόσμον κατὰ τὸν ΑΒΓ κύκλον, τὰν δὲ γᾶν κατὰ τὸν △ΕΖ, τὸν δὲ ἅλιον κατὰ τὸν ΣΗ κύκλον, κέντρον δὲ ἔστω τᾶς
These things being assumed, it is shown that the diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle of those in the universe. For let a plane be conceived to be passed through the center of the sun, the center of the earth, and the eye, when the sun is a little above the horizon; and let the passed plane cut the universe along the circle ΑΒΓ, the earth along the circle △ΕΖ, and the sun along the circle ΣΗ, and let the center of the...