§4#3Θανερὸν οὖν ὅτι τὸ τοῦ ψάμμου πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τᾷ σφαίρᾳ τᾷ τὰν διάμετρον ἐχούσᾳ σταδίων μυριάδων μυριᾶν ἔλασσόν ἐστιν ἢ ῑ μυριάδες τῶν ἕκτων ἀριθμῶν.
It is clear, then, that the quantity of sand having a size equal to the sphere having a diameter of ten thousand ten-thousands of stadia is less than 10 ten-thousands of the sixth numbers.
Ἁ δὲ τὰν διάμετρον ἔχουσα σφαῖρα σταδίων μυριάκις μυριάδων ρ πολλαπλασία ἐστὶ τᾶς σφαίρας τᾶς ἐχούσας τὰν διάμετρον σταδίων μυριάδων μυριᾶν ταῖς ρ μυριάδεσσιν.
And the sphere having a diameter of a hundred ten-thousand ten-thousands of stadia is as many times the sphere having a diameter of ten thousand ten-thousands of stadia as 100 ten-thousands.
Εἰ οὖν γένοιτο ἐκ τοῦ ψάμμου σφαῖρα ταλικαύτα τὸ μέγεθος, ἁλίκα ἐστὶν ἁ σφαῖρα ἁ ἔχουσα τὰν διάμετρον σταδίων μυριάκις μυριάδων ρ, φανερὸν ὅτι τὸ τοῦ ψάμμου πλῆθος ἔλασσον ἐσσεῖται τοῦ γενομένου ἀριθμοῦ πολλαπλασιασθεισᾶν τᾶν ῑ μυριάδων τῶν ἕκτων ἀριθμῶν ταῖς ρ μυριάδεσσιν.
If, then, there were a sphere of sand of such size as the sphere having a diameter of a hundred ten-thousand ten-thousands of stadia, it is clear that the quantity of sand will be less than the product of 10 ten-thousands of the sixth numbers multiplied by 100 ten-thousands.
Ἐπεὶ δʼ αἱ μὲν τῶν ἕκτων ἀριθμῶν δέκα μυριάδες ἕκτος καὶ τετρωκοστός ἐστιν ἀπὸ μονάδος ἀνάλογον, αἱ δὲ ρ μυριάδες ἕβδομος ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας, δῆλον ὅτι ὁ γενόμενος ἐσσεῖται δυοκαιπεντακοστὸς ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας.
And since 10 ten-thousands of the sixth numbers is the forty-sixth from the unit in proportion, and 100 ten-thousands is the seventh from the unit in the same proportion, it is clear that the product will be the fifty-second from the unit in the same proportion.
Τῶν δὲ δύο καὶ πεντήκοντα τούτων οἱ μὲν ὀκτὼ καὶ τεσσαράκοντα σὺν τᾷ μονάδι οἵ τε πρῶτοι καλούμενοι ἐντὶ καὶ οἱ δεύτεροι καὶ τρίτοι καὶ τέταρτοι καὶ πέμπτοι καὶ ἕκτοι, οἱ δὲ λοιποὶ τέσσαρες τῶν ἑβδόμων καλουμένων ἐντί, καὶ ὁ ἔσχατος αὐτῶν ἐστι χίλιαι μονάδες τῶν ἑβδόμων ἀριθμῶν.
And of these fifty-two, the forty-eight, including the unit, belong to what are called the first, second, third, fourth, fifth, and sixth numbers, and the remaining four belong to what are called the seventh numbers, and the last of them is a thousand units of the seventh numbers.
Φανερὸν οὖν ὅτι τοῦ ψάμμου τὸ πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τᾷ σφαίρᾳ τᾷ τὰν διάμετρον ἐχούσᾳ σταδίων μυριάκις μυριάδων ρ ἔλασσόν ἐστιν ἢ,ᾱ μονάδες τῶν ἑβδόμων ἀριθμῶν.
It is clear, then, that the quantity of sand having a size equal to the sphere having a diameter of a hundred ten-thousand ten-thousands of stadia is less than 1,000 units of the seventh numbers.
Ἐπεὶ οὖν ἐδείχθη ἁ τοῦ κόσμου διάμετρος ἐλάσσων ἐοῦσα σταδίων μυριάκις μυριάδων ρ, δῆλον ὅτι καὶ τοῦ ψάμμου τὸ πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τῷ κόσμῳ ἔλασσόν ἐστιν ἢ,ᾱ μονάδες τῶν ἑβδόμων ἀριθμῶν.
Since, then, the diameter of the world was shown to be less than a hundred ten-thousand ten-thousands of stadia, it is clear that also the quantity of sand having a size equal to the world is less than 1,000 units of the seventh numbers.
Ὅτι μὲν οὖν τὸ τοῦ ψάμμου πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τῷ ὑπὸ τῶν πλείστων ἀστρολόγων καλουμένῳ κόσμῳ ἔλασσόν ἐστιν ἢ,ᾱ μονάδες τῶν ἑβδόμων ἀριθμῶν δέδεικται· ὅτι δὲ καὶ τὸ πλῆθος τοῦ ψάμμου τοῦ μέγεθος ἔχοντος ἴσον τᾷ σφαίρᾳ ταλικαύτᾳ, ἁλίκαν Ἀρίσταρχος ὑποτίθεται τὰν τῶν ἀπλανέων ἄστρων σφαῖραν εἶμεν, ἔλασσόν ἐστιν ἢ,ᾱ μυριάδες τῶν ὀγδόων ἀριθμῶν δειχθήσεται.
That, then, the quantity of sand having a size equal to the world called by most astronomers is less than 1,000 units of the seventh numbers has been shown; and that also the quantity of sand having a size equal to such a sphere as Aristarchus supposes the sphere of the fixed stars to be is less than 1,000 ten-thousands of the eighth numbers will be shown.
Ἐπεὶ γὰρ ὑπόκειται τὰν γᾶν τὸν αὐτὸν ἔχειν λόγον ποτὶ τὸν ὑφʼ ἁμῶν εἰρημένον κόσμον, ὃν ἔχει λόγον ὁ εἰρημένος κόσμος ποτὶ τὰν τῶν ἀπλανέων ἄστρων σφαῖραν, ἃν Ἀρίσταρχος ὑποτίθεται, καὶ αἱ διάμετροι τᾶν σφαιρᾶν τὸν αὐτὸν ἔχοντι λόγον ποτʼ ἀλλάλας.
For since it is assumed that the earth has the same ratio to the world mentioned by us as the mentioned world has to the sphere of the fixed stars which Aristarchus supposes, also the diameters of the spheres have the same ratio to each other.
Ἁ δὲ τοῦ κόσμου διάμετρος τᾶς διαμέτρου τᾶς γᾶς δέδεικται ἐλάσσων ἐοῦσα ἢ μυριοπλασίων δῆλον οὖν ὅτι καὶ ἁ διάμετρος τᾶς τῶν ἀπλανέων ἄστρων σφαίρας ἐλάσσων ἐστὶν ἢ μυριοπλασίων τᾶς διαμέτρου τοῦ κόσμου.
And the diameter of the world has been shown to be less than ten thousand times the diameter of the earth; it is clear, then, that also the diameter of the sphere of the fixed stars is less than ten thousand times the diameter of the world.
Ἐπεὶ δὲ αἱ σφαῖραι τριπλάσιον λόγον ἔχοντι ποτʼ ἀλλάλας τᾶν διαμέτρων, φανερὸν ὅτι ἁ τῶν ἀπλανέων ἄστρων σφαῖρα, ἃν Ἀρίσταρχος ὑποτίθεται, ἐλάττων ἐστὶν ἢ μυριάκις μυρίαις μυριάδεσσι πολλαπλασίων τοῦ κόσμου.
And since spheres have to each other the triplicate ratio of their diameters, it is clear that the sphere of the fixed stars which Aristarchus supposes is less than ten thousand ten-thousand ten-thousands of times the world.
Δέδεικται δὲ ὅτι τὸ τοῦ ψάμμου πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τῷ κόσμῳ ἔλασσόν ἐστιν ἢ,ᾱ μονάδες τῶν ἑβδόμων ἀριθμῶν· δῆλον οὖν ὅτι, εἰ γένοιτο ἐκ τοῦ ψάμμου σφαῖρα ταλικαύτα τὸ μέγεθος, ἁλίκαν ὁ Ἀρίσταρχος ὑποτίθεται τὰν τῶν ἀπλανέων ἄστρων σφαῖραν εἶμεν, ἐλάσσων ἐσσεῖται ὁ τοῦ ψάμμου ἀριθμὸς τοῦ γενομένου ἀριθμοῦ πολλαπλασιασθεισᾶν τᾶν χιλιᾶν μονάδων ταῖς μυριάκις μυρίαις μυριάδεσσιν, Καὶ ἐπεὶ αἱ μὲν τῶν ἑβδόμων ᾱ μονάδες δυοκαιπεντακοστός ἐστιν ἀπὸ μονάδος ἀνάλογον, αἱ δὲ μυριάκις μύριαι μυριάδες τρισκαιδέκατος ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας, δῆλον ὅτι ὁ γενόμενος ἐσσεῖται τέταρτος καὶ ἑξηκοστὸς ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας·
And it has been shown that the quantity of sand having a size equal to the world is less than 1,000 units of the seventh numbers; it is clear, then, that if there were a sphere of sand of such size as Aristarchus supposes the sphere of the fixed stars to be, the number of grains of sand will be less than the product of a thousand units multiplied by ten thousand ten-thousand ten-thousands. And since on the one hand 1,000 units of the seventh numbers is the fifty-second from the unit in proportion, and on the other hand ten thousand ten-thousand ten-thousands is the thirteenth from the unit in the same proportion, it is clear that the product will be the sixty-fourth from the same proportion.
οὗτος δὲ ἐστι τῶν ὀγδόων ὄγδοος, ὅς κα εἴη χίλιαι μυριάδες τῶν ὀγδόων ἀριθμῶν.
And this is the eighth of the eighth numbers, which would be a thousand ten-thousands of the eighth numbers.
Φανερὸν τοίνυν ὅτι τοῦ ψάμμου τὸ πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τᾷ τῶν ἀπλανέων ἄστρων σφαίρᾳ, ἃν Ἀρίσταρχος ὑποτίθεται, ἔλασσόν ἐστιν ἢ,ᾱ μυριάδες τῶν ὀγδόων ἀριθμῶν.
It is manifest, therefore, that the quantity of sand having a size equal to the sphere of the fixed stars which Aristarchus supposes is less than 1,000 ten-thousands of the eighth numbers.
Ταῦτα δέ, βασιλεῦ Γέλων, τοῖς μὲν πολλοῖς καὶ μὴ κεκοινωνηκότεσσι τῶν μαθημάτων οὐκ εὔπιστα φανήσειν ὑπολαμβάνω, τοῖς δὲ μεταλελαβηκότεσσιν καὶ περὶ τῶν ἀποστημάτων καὶ τῶν μεγεθέων τᾶς τε γᾶς καὶ τοῦ ἁλίου καὶ τᾶς σελήνας καὶ τοῦ ὅλου κόσμου πεφροντικότεσσιν πιστὰ διὰ τὰν ἀπόδειξιν ἐσσεῖσθαι διόπερ ᾠήθην καὶ τὶν οὐκ ἀνάρμοστον εἶμεν ἐπιθεωρῆσαι ταῦτα.
And these things, King Gelon, I suppose will appear not easy to believe to the majority of people who have not shared in mathematics, but to those who have acquired experience and have reflected upon the distances and sizes both of the earth and of the sun and of the moon and of the whole world, they will be credible because of the demonstration; for which reason I thought that it would not be unbefitting also for you to contemplate these things.