§4#2Καὶ ἐπεὶ αἱ μὲν τῶν τρίτων ἀριθμῶν δέκα μυριάδες δυοκαιεικοστός ἐστιν ἀπὸ μονάδος ἀνάλογον, αἱ δὲ ρ μυριάδες ἕβδομος ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας, δῆλον ὡς ὁ γενόμενος ἐσσεῖται ὀκτωκαιεικοστὸς ἐκ τᾶς αὐτᾶς ἀναλογίας ἀπὸ μονάδος, Τῶν δὲ ὀκτὼ καὶ εἴκοσι τούτων ὀκτὼ μὲν οἱ πρῶτοι σὺν τᾷ μονάδι τῶν πρώτων καλουμένων ἐντί, οἱ δὲ μετὰ τούτους ἄλλοι ὀκτὼ τῶν δευτέρων, καὶ οἱ μετὰ τούτους ὀκτὼ τῶν τρίτων, οἱ δὲ λοιποὶ τέσσαρες τῶν τετάρτων καλουμένων, καὶ ὁ ἔσχατος αὐτῶν ἐστι χίλιαι μονάδες τῶν τετάρτων ἀριθμῶν.
And since on the one hand ten ten-thousands of the third numbers is the twenty-second from the unit in proportion, and on the other hand 100 ten-thousands is the seventh from the unit in the same proportion, it is clear that the product will be the twenty-eighth from the unit in the same proportion. And of these twenty-eight, the first eight, including the unit, belong to what are called the first numbers, and the other eight after these belong to the second numbers, and the eight after these belong to the third numbers, and the remaining four belong to what are called the fourth numbers, and the last of them is a thousand units of the fourth numbers.
Φανερὸν οὖν ὅτι τὸ τοῦ ψάμμου πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τᾷ σφαίρᾳ τᾷ τὰν διάμετρον ἐχούσᾳ σταδίων ἔλασσόν ἐστιν ἢ χίλιαι μονάδες τῶν τετάρτων ἀριθμῶν.
It is clear, then, that the quantity of sand having a size equal to the sphere having a diameter of 100 stadia is less than a thousand units of the fourth numbers.
Πάλιν δὴ ἁ σφαῖρα ἁ ἔχουσα τὰν διάμετρον μυρίων σταδίων πολλαπλασία ἐστὶ τᾶς σφαίρας τᾶς ἐχούσας τὰν διάμετρον σταδίων ρ ταῖς ρ μυριάδεσσιν, Εἰ οὖν γένοιτο ἐκ τοῦ ψάμμου σφαῖρα ταλικαύτα τὸ μέγεθος, ἁλίκα ἐστὶν ἁ σφαῖρα ἁ ἔχουσα τὰν διάμετρον σταδίων μυρίων, δῆλον ὅτι ἔλασσον ἐσσεῖται τὸ τοῦ ψάμμου πλῆθος τοῦ γενομένου ἀριθμοῦ πολλαπλασιασθεισᾶν τᾶν χιλιᾶν μονάδων τῶν τετάρτων ἀριθμῶν ταῖς ρ μυριάδεσσιν.
Again, also the sphere having a diameter of ten thousand stadia is as many times the sphere having a diameter of 100 stadia as 100 ten-thousands. If, then, there were a sphere of sand of such size as the sphere having a diameter of ten thousand stadia, it is clear that the quantity of sand will be less than the product of a thousand units of the fourth numbers multiplied by 100 ten-thousands.
Ἐπεὶ δʼ αἱ μὲν τῶν τετάρτων ἀριθμῶν χίλιαι μονάδες ὀκτωκαιεικοστός ἐστιν ἀπὸ μονάδος ἀνάλογον, αἱ δʼ ἑκατὸν μυριάδες ἕβδομος ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας, δῆλον ὅτι ὁ γενόμενος ἐσσεῖται ἐκ τᾶς αὐτᾶς ἀναλογίας τέταρτος καὶ τριακοστὸς ἀπὸ μονάδος.
And since a thousand units of the fourth numbers is the twenty-eighth 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 thirty-fourth from the unit in the same proportion.
Τῶν δὲ τεσσάρων καὶ τριάκοντα τούτων ὀκτὼ μὲν οἱ πρῶτοι σὺν τᾷ μονάδι τῶν πρώτων καλουμένων ἐντί, οἱ δὲ μετὰ τούτους ὀκτὼ τῶν δευτέρων, καὶ οἱ μετὰ τούτους ἄλλοι ὀκτὼ τῶν τρίτων, καὶ οἱ μετὰ τούτους ὀκτὼ τῶν τετάρτων, οἱ δὲ λοιποὶ δύο τῶν πέμπτων καλουμένων ἐσσοῦνται, καὶ ὁ ἔσχατος αὐτῶν ἐστι δέκα μονάδες τῶν πέμπτων ἀριθμῶν.
And of these thirty-four, the first eight, including the unit, belong to what are called the first numbers, and the eight after these belong to the second numbers, and the other eight after these belong to the third numbers, and the eight after these belong to the fourth numbers, and the remaining two will belong to what are called the fifth numbers, and the last of them is ten units of the fifth numbers.
Δῆλον οὖν ὅτι τὸ τοῦ ψάμμου πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τᾷ σφαίρᾳ τᾷ τὰν διάμετρον ἐχούσᾳ σταδίων μυρίων ἔλασσον ἐσσεῖται ἢ ῑ μονάδες τῶν πέμπτων ἀριθμῶν.
It is clear, then, that the quantity of sand having a size equal to the sphere having a diameter of ten thousand stadia will be less than 10 units of the fifth numbers.
Πάλιν δὴ ἁ σφαῖρα ἁ ἔχουσα τὰν διάμετρον σταδίων ρ μυριάδων πολλαπλασία ἐστὶ τᾶς σφαίρας τᾶς τὰν διάμετρον ἐχούσας σταδίων μυρίων ταῖς ρ μυριάδεσσιν.
Again, also the sphere having a diameter of 100 ten-thousands of stadia is as many times the sphere having a diameter of ten thousand stadia as 100 ten-thousands.
Εἰ οὖν γένοιτο ἐκ τοῦ ψάμμου σφαῖρα ταλικαύτα τὸ μέγεθος, ἁλίκα ἐστὶν ἁ σφαῖρα ἁ ἔχουσα τὰν διάμετρον σταδίων ρ μυριάδων, δῆλον ὡς ἐλάσσων ἐσσεῖται ὁ τοῦ ψάμμου ἀριθμὸς τοῦ γενομένου ἀριθμοῦ πολλαπλασιασθεισᾶν τᾶν δέκα μονάδων τῶν πέμπτων ἀριθμῶν ταῖς ρ μυριάδεσσιν.
If, then, there were a sphere of sand of such size as the sphere having a diameter of 100 ten-thousands of stadia, it is clear that the number of grains of sand will be less than the product of ten units of the fifth numbers multiplied by 100 ten-thousands.
Καὶ ἐπεὶ αἱ μὲν τῶν πέμπτων ἀριθμῶν δέκα μονάδες τέταρτός ἐστι καὶ τριακοστὸς ἀπὸ μονάδος ἀνάλογον, αἱ δὲ ρ μυριάδες ἕβδομος ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας, δῆλον ὅτι ὁ γενόμενος ἐκ τᾶς αὐτᾶς ἀναλογίας ἐσσεῖται τετρωκοστὸς ἀπὸ μονάδος, Τῶν δὲ τεσσαράκοντα τούτων ὀκτὼ μὲν οἱ πρῶτοι σὺν τᾷ μονάδι τῶν πρώτων καλουμένων ἐντί, οἱ δὲ μετὰ ταῦτα ἄλλοι ὀκτὼ τῶν δευτέρων, καὶ οἱ μετὰ τούτους ἄλλοι ὀκτὼ τῶν τρίτων, οἱ δὲ μετὰ τοὺς τρίτους ὀκτὼ τῶν τετάρτων, οἱ δὲ μετὰ τούτους ὀκτὼ τῶν πέμπτων καλουμένων, καὶ ὁ ἔσχατος αὐτῶν ἐστι χίλιαι μυριάδες τῶν πέμπτων ἀριθμῶν, Φανερὸν οὖν ὅτι τοῦ ψάμμου τὸ πλῆθος τοῦ μέγεθος ἔχοντος ἴσον τᾷ σφαίρᾳ τᾷ τὰν διάμετρον ἐχούσᾳ σταδίων ρ μυριάδων ἔλασσόν ἐστιν ἢ χίλιαι μυριάδες τῶν πέμπτων ἀριθμῶν.
And since on the one hand ten units of the fifth numbers is the thirty-fourth from the unit in proportion, and on the other hand 100 ten-thousands is the seventh from the unit in the same proportion, it is clear that the product will be the fortieth from the unit in the same proportion. And of these forty, the first eight, including the unit, belong to what are called the first numbers, and the other eight after these belong to the second numbers, and the other eight after these belong to the third numbers, and the eight after the third belong to the fourth numbers, and the eight after these belong to what are called the fifth numbers, and the last of them is a thousand ten-thousands of the fifth numbers. It is clear, then, that the quantity of sand having a size equal to the sphere having a diameter of 100 ten-thousands of stadia is less than a thousand ten-thousands of the fifth numbers.
Ἁ δὲ τὰν διάμετρον ἔχουσα σφαῖρα σταδίων μυριᾶν μυριάδων πολλαπλασίων ἐστὶ τᾶς σφαίρας τᾶς ἐχούσας τὰν διάμετρον σταδίων ρ μυριάδων ταῖς ρ μυριάδεσσιν.
And the sphere having a diameter of ten thousand ten-thousands of stadia is as many times the sphere having a diameter of 100 ten-thousands of stadia as 100 ten-thousands.
Εἰ δὴ γένοιτο ἐκ τοῦ ψάμμου σφαῖρα ταλικαύτα τὸ μέγεθος, ἁλίκα ἐστὶν ἁ σφαῖρα ἁ ἔχουσα τὰν διάμετρον σταδίων μυριᾶν μυριάδων, φανερὸν ὅτι ἔλασσον ἐσσεῖται τὸ τοῦ ψάμμου πλῆθος τοῦ γενομένου ἀριθμοῦ πολλαπλασιασθεισᾶν τᾶν χιλιᾶν μυριάδων τῶν πέμπτων ἀριθμῶν ταῖς ρ μυριάδεσσιν.
If, indeed, there were a sphere of sand of such size as the sphere having a diameter of ten thousand ten-thousands of stadia, it is clear that the quantity of sand will be less than the product of a thousand ten-thousands of the fifth numbers multiplied by 100 ten-thousands.
Ἐπεὶ δʼ αἱ μὲν τῶν πέμπτων ἀριθμῶν χίλιαι μυριάδες τετρωκοστός ἐστιν ἀπὸ μονάδος ἀνάλογον, αἱ δὲ ρ μυριάδες ἕβδομος ἀπὸ μονάδος ἐκ τᾶς αὐτᾶς ἀναλογίας, δῆλον ὡς ὁ γενόμενος ἐσσεῖται ἕκτος καὶ τετρωκοστὸς ἀπὸ μονάδος, Τῶν δὲ τεσσαράκοντα καὶ ἓξ τούτων ὀκτὼ μὲν οἱ πρῶτοι σὺν τᾷ μονάδι τῶν πρώτων καλουμένων ἐντί, ὀκτὼ δὲ οἱ μετὰ τούτους τῶν δευτέρων, καὶ οἱ μετὰ τούτους ἄλλοι ὀκτὼ τῶν τρίτων, οἱ δὲ μετὰ τοὺς τρίτους ἄλλοι ὀκτὼ τῶν τετάρτων, καὶ οἱ μετὰ τοὺς τετάρτους ὀκτὼ τῶν πέμπτων, οἱ δὲ λοιποὶ ἓξ τῶν ἕκτων καλουμένων ἐντί, καὶ ὁ ἔσχατος αὐτῶν ἐστι ῑ μυριάδες τῶν ἕκτων ἀριθμῶν.
And since a thousand ten-thousands of the fifth numbers is the fortieth 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 forty-sixth from the unit in the same proportion. And of these forty-six, the first eight, including the unit, belong to what are called the first numbers, and the eight after these to the second numbers, and the other eight after these to the third numbers, and the other eight after the third to the fourth numbers, and the eight after the fourth to the fifth numbers, and the remaining six belong to what are called the sixth numbers, and the last of them is 10 ten-thousands of the sixth numbers.