§1#1Ι. Οἴονταί τινες, βασιλεῦ Γέλων, τοῦ ψάμμου τὸν ἀριθμὸν ἄπειρον εἶμεν τῷ πλήθει· λέγω δὲ οὐ μόνον τοῦ περὶ Συρακούσας τε καὶ τὰν ἄλλαν Σικελίαν ὑπάρχοντος, ἀλλὰ καὶ τοῦ κατὰ πᾶσαν χώραν τάν τε οἰκημέναν καὶ τὰν ἀοίκητον.
I. There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean not only of the sand around Syracuse and the rest of Sicily, but also of that in every region, whether inhabited or uninhabited.
Ἐντί τινες δέ, οἳ αὐτὸν ἄπειρον μὲν εἶμεν οὐχ ὑπολαμβάνοντι, μηδένα μέντοι ταλικοῦτον κατωνομασμένον ὑπάρχειν ἀριθμόν, ὅστις ὑπερβάλλει τὸ πλῆθος αὐτοῦ.
And there are some who, while they do not believe it to be infinite, yet think that no number has been named of such a size as to exceed its multitude.
Οἱ δὲ οὕτως δοξάζοντες δῆλον ὡς, εἰ νοήσαιεν ἐκ τοῦ ψάμμου ταλικοῦτον ὄγκον συγκείμενον τὸ μέγεθος, ἁλίκος ὁ τᾶς γᾶς ὄγκος ἀναπεπληρωμένων ἐν αὐτῷ τῶν τε πελάγεων πάντων καὶ τῶν κοιλωμάτων τᾶς γᾶς εἰς ἴσον ὕψος τοῖς ὑψηλοτάτοις τῶν ὀρέων, πολλαπλασίως μὴ γνώσονται μηδένα κα ῥηθῆμεν ἀριθμὸν ὑπερβάλλοντα τὸ πλῆθος αὐτοῦ.
And it is clear that those who hold this view, if they were to imagine a mass made up of sand as large as the mass of the earth, with all the seas and hollows of the earth filled up to a height equal to the highest mountains, would be many times more far from knowing that any number can be spoken of which exceeds its multitude.
Ἐγὼ δὲ πειρασοῦμαί τοι δεικνύειν διʼ ἀποδείξιων γεωμετρικᾶν, αἷς παρακολουθήσεις, ὅτι τῶν ὑφʼ ἁμῶν κατωνομασμένων ἀριθμῶν καὶ ἐκδεδομένων ἐν τοῖς ποτὶ Ζεύξιππον γεγραμμένοις ὑπερβάλλοντί τινες οὐ μόνον τὸν ἀριθμὸν τοῦ ψάμμου τοῦ μέγεθος ἔχοντος ἴσον τᾷ γᾷ πεπληρωμένᾳ, καθάπερ εἴπαμες, ἀλλὰ καὶ τὸν τοῦ μέγεθος ἴσον ἔχοντος τῷ κόσμῳ, Κατέχεις δὲ διότι καλεῖται κόσμος ὑπὸ μὲν τῶν πλείστων ἀστρολόγων ἁ σφαῖρα, ἇς ἐστι κέντρον μὲν τὸ τᾶς γᾶς κέντρον, ἁ δὲ ἐκ τοῦ κέντρου ἴσα τᾷ εὐθείᾳ τᾷ μεταξὺ τοῦ κέντρου τοῦ ἁλίου καὶ τοῦ κέντρου τᾶς γᾶς· ταῦτα γὰρ ἐν ταῖς γραφομέναις παρὰ τῶν ἀστρολόγων δείξεσι διάκουσας.
But I will try to show you by geometric proofs, which you will be able to follow, that among the numbers named by us and published in our writings to Zeuxippus, some exceed not only the number of the sand of a size equal to the earth filled as we have said, but also of a size equal to the universe. You are aware that "universe" is the name given by most astronomers to the sphere of which the center is the center of the earth, while its radius is equal to the straight line between the center of the sun and the center of the earth; for these are the things you have heard in the proofs written by astronomers.
Ἀρίσταρχος δὲ ὁ Σάμιος ὑποθέσιών τινων ἐξέδωκεν γραφάς, ἐν αἷς ἐκ τῶν ὑποκειμένων συμβαίνει τὸν κόσμον πολλαπλάσιον εἶμεν τοῦ νῦν εἰρημένου.
But Aristarchus of Samos brought out books of certain hypotheses, in which it follows from the premises that the universe is many times larger than the one now mentioned.
Ὑποτίθεται γὰρ τὰ μὲν ἀπλανέα τῶν ἄστρων καὶ τὸν ἅλιον μένειν ἀκίνητον, τὰν δὲ γᾶν περιφέρεσθαι περὶ τὸν ἅλιον κατὰ κύκλου περιφέρειαν, ὅς ἐστιν ἐν μέσῳ τῷ δρόμῳ κείμενος, τὰν δὲ τῶν ἀπλανέων ἄστρων σφαῖραν περὶ τὸ αὐτὸ κέντρον τῷ ἁλίῳ κειμέναν τῷ μεγέθει τηλικαύταν εἶμεν, ὥστε τὸν κύκλον, καθʼ ὃν τὰν γᾶν ὑποτίθεται περιφέρεσθαι, τοιαύταν ἔχειν ἀναλογίαν ποτὶ τὰν τῶν ἀπλανέων ἀποστασίαν, οἵαν ἔχει τὸ κέντρον τᾶς σφαίρας ποτὶ τὰν ἐπιφάνειαν.
For he hypothesizes that the fixed stars and the sun remain unmoved, while the earth revolves about the sun along the circumference of a circle, which is situated in the middle of the course, and that the sphere of the fixed stars, situated about the same center as the sun, is so great in size that the circle along which he hypothesizes the earth revolves has such a ratio to the distance of the fixed stars as the center of the sphere has to its surface.
Τοῦτό γʼ εὔδηλον ὡς ἀδύνατόν ἐστιν ἐπεὶ γὰρ τὸ τᾶς σφαίρας κέντρον οὐδὲν ἔχει μέγεθος, οὐδὲ λόγον ἔχειν οὐδένα ποτὶ τὰν ἐπιφάνειαν τᾶς σφαίρας ὑπολαπτέον αὐτό, Ἐκδεκτέον δὲ τὸν Ἀρίσταρχον διανοεῖσθαι τόδε·
This is indeed clearly impossible; for since the center of the sphere has no magnitude, it must be understood to have no ratio to the surface of the sphere.
ἐπειδὴ τὰν γᾶν ὑπολαμβάνομες ὥσπερ εἶμεν τὸ κέντρον τοῦ κόσμου, ὃν ἔχει λόγον ἁ γᾶ ποτὶ τὸν ὑφʼ ἁμῶν εἰρημένον κόσμον, τοῦτον ἔχειν τὸν λόγον τὰν σφαῖραν, ἐν ᾆ ἐστιν ὁ κύκλος, καθʼ ὃν τὰν γᾶν ὑποτίθεται περιφέρεσθαι, ποτὶ τὰν τῶν ἀπλανέων ἄστρων σφαῖραν τὰς γὰρ ἀποδείξιας τῶν φαινομένων οὕτως ὑποκειμένῳ ἐναρμόζει, καὶ μάλιστα φαίνεται τὸ μέγεθος τᾶς σφαίρας, ἐν ᾆ ποιεῖται τὰν γᾶν κινουμέναν, ἴσον ὑποτίθεσθαι τῷ ὑφʼ ἁμῶν εἰρημένῳ κόσμῳ, Φαμὲς δή, καὶ εἰ γένοιτο ἐκ τοῦ ψάμμου σφαῖρα τηλικαύτα τὸ μέγεθος, ἁλίκαν Ἀρίσταρχος ὑποτίθεται τὰν τῶν ἀπλανέων ἄστρων σφαῖραν εἶμεν, καὶ οὕτως τινὰς δειχθήσειν τῶν ἐν ἀρχᾷ ἀριθμῶν τῶν κατονομαξίαν ἐχόντων ὑπερβάλλοντας τῷ πλήθει τὸν ἀριθμὸν τὸν τοῦ ψάμμου τοῦ μέγεθος ἔχοντος ἴσον τᾷ εἰρημένᾳ σφαίρᾳ, ὑποκειμένων τῶνδε· πρῶτον μὲν τὰν περίμετρον τᾶς γᾶς εἶμεν ὡς τ μυριάδων σταδίων καὶ μὴ μείζω, καίπερ τινῶν πεπειραμένων ἀποδεικνύειν, καθὼς καὶ τὺ παρακολουθεῖς, ἐοῦσαν αὐτὰν ὡς λ μυριάδων σταδίων.
But we must assume that Aristarchus meant this: that, just as we conceive the earth to be as it were the center of the universe, the ratio which the earth has to the universe mentioned by us is the same as the ratio which the sphere containing the circle along which he hypothesizes the earth revolves has to the sphere of the fixed stars; for he adapts the proofs of the phenomena to such a hypothesis, and especially he appears to hypothesize the magnitude of the sphere in which he makes the earth move to be equal to the universe mentioned by us. We say then, even if there were to be a sphere of sand as great in magnitude as Aristarchus hypothesizes the sphere of the fixed stars to be, that even so some of the numbers named in the beginning would be shown to exceed in multitude the number of the sand having a magnitude equal to the said sphere, under the following hypotheses: first, that the perimeter of the earth is about three hundred thousand stades and not greater, although some have tried to prove, as you also know, that it is about three hundred thousand stades.
Ἐγὼ δʼ ὑπερβαλλόμενος καὶ θεὶς τὸ μέγεθος τᾶς γᾶς ὡς δεκαπλάσιον τοῦ ὑπὸ τῶν προτέρων δεδοξασμένου τὰν περίμετρον αὐτᾶς ὑποτίθεμαι εἶμεν ὡς μυριάδων σταδίων καὶ μὴ μείζω μετὰ δὲ τοῦτο τὰν διάμετρον τᾶς γᾶς μείζονα εἶμεν τᾶς διαμέτρου τᾶς σελήνας, καὶ τὰν διάμετρον τοῦ ἁλίου μείζονα εἶμεν τᾶς διαμέτρου τᾶς γᾶς, ὁμοίως τὰ αὐτὰ λαμβάνων τοῖς πλείστοις τῶν προτέρων ἀστρολόγων μετὰ δὲ ταῦτα τὰν διάμετρον τοῦ ἁλίου τᾶς διαμέτρου τᾶς σελήνας ὡς τριακονταπλασίαν εἶμεν καὶ μὴ μείζονα, καίπερ τῶν προτέρων ἀστρολόγων Εὐδόξου μὲν ὡς ἐννεαπλασίονα ἀποφαινομένου, Φειδία δὲ τοῦ ἁμοῦ πατρὸς ὡς δὴ δωδεκαπλασίαν, Ἀριστάρχου δὲ πεπειραμένου δεικνύειν ὅτι ἐστὶν ἁ διάμετρος τοῦ ἁλίου τᾶς διαμέτρου τᾶς σελήνας μείζων μὲν ἢ ὀκτωκαιδεκαπλασίων, ἐλάττων δὲ ἢ εἰκοσαπλασίων· ἐγὼ δὲ ὑπερβαλλόμενος καὶ τοῦτον, ὅπως τὸ προκείμενον ἀναμφιλόγως ᾖ δεδειγμένον, ὑποτίθεμαι τὰν διάμετρον τοῦ ἁλίου τᾶς διαμέτρου τᾶς σελήνας ὡς τριακονταπλασίαν εἶμεν καὶ μὴ μείζονα ποτὶ δὲ τούτοις τὰν διάμετρον τοῦ ἁλίου μείζονα εἶμεν τᾶς τοῦ χιλιαγώνου πλευρᾶς τοῦ εἰς τὸν μέγιστον κύκλον ἐγγραφομένου τῶν ἐν τῷ κόσμῳ.
But I, going far beyond and putting the magnitude of the earth as ten times that believed by my predecessors, hypothesize its perimeter to be about three million stades and not greater; and after this, that the diameter of the earth is greater than the diameter of the moon, and the diameter of the sun is greater than the diameter of the earth, taking the same views as most of the earlier astronomers; and after these, that the diameter of the sun is about thirty times the diameter of the moon and not greater, although of the earlier astronomers, Eudoxus declared it to be about nine times, and my father Phidias about twelve times, and Aristarchus tried to show that the diameter of the sun is greater than eighteen times and less than twenty times the diameter of the moon; but I, going beyond this also, so that the proposed matter may be proved indisputably, hypothesize the diameter of the sun to be about thirty times the diameter of the moon and not greater; and in addition to these, that the diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle of those in the universe.
Τοῦτο δὲ ὑποτίθεμαι Ἀριστάρχου μὲν εὑρηκότος τοῦ κύκλου τῶν ζῳδίων τὸν ἅλιον φαινόμενον ὡς τὸ εἰκοστὸν καὶ ἑπτακοσιοστόν, αὐτὸς δὲ ἐπισκεψάμενος τόνδε τὸν τρόπον ἐπειράθην ὀργανικῶς λαβεῖν τὰν γωνίαν, εἰς ἃν ὁ ἅλιος ἐναρμόζει τὰν κορυφὰν ἔχουσαν ποτὶ τᾷ ὄψει.
And I hypothesize this because, while Aristarchus found the apparent sun to be about the one-seven-hundred-and-twentieth part of the zodiac circle, I myself, having investigated in this manner, tried to obtain by instruments the angle which the sun fits, having its vertex at the eye.