§3#1ΙΙΙ. Ἃ μὲν οὖν ὑποτίθεμαι, ταῦτα·
III. These, then, are the things which I assume.
χρήσιμον δὲ εἶμεν ὑπολαμβάνω τὰν κατονόμαξιν τῶν ἀριθμῶν ῥηθῆμεν, ὅπως καὶ τῶν ἄλλων οἱ τῷ βιβλίῳ μὴ περιτετευχότες τῷ ποτὶ Ζεύξιππον γεγραμμένῳ μὴ πλανῶνται διὰ τὸ μηδὲν εἶμεν ὑπὲρ αὐτᾶς ἐν τῷδε τῷ βιβλίῳ προειρημένον.
But I conceive it useful to explain the naming of the numbers, so that those of the others who have not met with the book written to Zeuxippus may not be misled by there being nothing previously said concerning it in this book.
Συμβαίνει δὴ τὰ ὀνόματα τῶν ἀριθμῶν ἐς τὸ μὲν τῶν μυρίων ὑπάρχειν ἁμῖν παραδεδομένα, καὶ ὑπὲρ τὸ τῶν μυρίων ἀποχρεόντως γιγνώσκομες μυριάδων ἀριθμὸν λέγοντες ἔστε ποτὶ τὰς μυρίας μυριάδας.
It indeed happens that the names of the numbers have been handed down to us as far as ten thousand, and beyond ten thousand we know them sufficiently by speaking of a number of ten thousands, up to ten thousand ten-thousands.
Ἔστων οὖν ἁμῖν οἱ μὲν νῦν εἰρημένοι ἀριθμοὶ ἐς τὰς μυρίας μυριάδας πρῶτοι καλούμενοι, τῶν δὲ πρώτων ἀριθμῶν αἱ μύριαι μυριάδες μονὰς καλείσθω δευτέρων ἀριθμῶν, καὶ ἀριθμείσθων τῶν δευτέρων μονάδες καὶ ἐκ τᾶν μονάδων δεκάδες καὶ ἑκατοντάδες καὶ χιλιάδες καὶ μυριάδες ἐς τὰς μυρίας μυριάδας.
Let, therefore, the numbers now mentioned up to ten thousand ten-thousands be called by us the "first numbers", and of the first numbers let the ten thousand ten-thousands be called the "unit of the second numbers", and let the units of the second numbers be counted, and from the units, the tens and hundreds and thousands and ten-thousands, up to ten thousand ten-thousands.
Πάλιν δὲ καὶ αἱ μύριαι μυριάδες τῶν δευτέρων ἀριθμῶν μονὰς καλείσθω τρίτων ἀριθμῶν, καὶ ἀριθμείσθων τῶν τρίτων ἀριθμῶν μονάδες καὶ ἀπὸ τᾶν μονάδων δεκάδες καὶ ἑκατοντάδες καὶ χιλιάδες καὶ μυριάδες ἐς τὰς μυρίας μυριάδας.
Again, let also the ten thousand ten-thousands of the second numbers be called the "unit of the third numbers", and let the units of the third numbers be counted, and from the units, the tens and hundreds and thousands and ten-thousands, up to ten thousand ten-thousands.
Τὸν αὐτὸν δὲ τρόπον καὶ τῶν τρίτων ἀριθμῶν μύριαι μυριάδες μονὰς καλείσθω τετάρτων ἀριθμῶν, καὶ αἱ τῶν τετάρτων ἀριθμῶν μύριαι μυριάδες μονὰς καλείσθω πέμπτων ἀριθμῶν, καὶ ἀεὶ οὕτως προάγοντες οἱ ἀριθμοὶ τὰ ὀνόματα ἐχόντων ἐς τὰς μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας.
In the same manner, let also the ten thousand ten-thousands of the third numbers be called the "unit of the fourth numbers", and let the ten thousand ten-thousands of the fourth numbers be called the "unit of the fifth numbers", and let the numbers, advancing always in this way, have their names up to the ten thousand ten-thousands of the ten-thousand-myriadth numbers.
Ἀποχρέοντι μὲν οὖν καὶ ἐπὶ τοσοῦτον οἱ ἀριθμοὶ γιγνωσκόμενοι, ἔξεστι δὲ καὶ ἐπὶ πλέον προάγειν.
Now, the numbers known up to this point are indeed sufficient, but it is possible to advance even further.
Ἔστων γὰρ οἱ μὲν νῦν εἰρημένοι ἀριθμοὶ πρώτας περιόδου καλούμενοι, ὁ δὲ ἔσχατος ἀριθμὸς τᾶς πρώτας περιόδου μονὰς καλείσθω δευτέρας περιόδου πρώτων ἀριθμῶν.
For let the numbers now mentioned be called those of the "first period", and let the last number of the first period be called the "unit of the first numbers of the second period".
Πάλιν δὲ καὶ αἱ μύριαι μυριάδες τᾶς δευτέρας περιόδου πρώτων ἀριθμῶν μονὰς καλείσθω τᾶς δευτέρας περιόδου δευτέρων ἀριθμῶν.
Again, let also the ten thousand ten-thousands of the first numbers of the second period be called the "unit of the second numbers of the second period".
Ὁμοίως δὲ καὶ τούτων ὁ ἔσχατος μονὰς καλείσθω δευτέρας περιόδου τρίτων ἀριθμῶν, καὶ ἀεὶ οὕτως οἱ ἀριθμοὶ προάγοντες τὰ ὀνόματα ἐχόντων τᾶς δευτέρας περιόδου ἐς τὰς μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας.
Similarly, let also the last of these be called the "unit of the third numbers of the second period", and let the numbers, advancing always in this way, have their names of the second period up to the ten thousand ten-thousands of the ten-thousand-myriadth numbers.
Πάλιν δὲ καὶ ὁ ἔσχατος ἀριθμὸς τᾶς δευτέρας περιόδου μονὰς καλείσθω τρίτας περιόδου πρώτων ἀριθμῶν, καὶ ἀεὶ οὕτως προαγόντων ἐς τὰς μυριακισμυριοστᾶς περιόδου μυριακισμυριοστῶν ἀριθμῶν μυρίας μυριάδας.
Again, let also the last number of the second period be called the "unit of the first numbers of the third period", and so let them always advance up to the ten thousand ten-thousands of the ten-thousand-myriadth numbers of the ten-thousand-myriadth period.
Τούτων δὲ οὕτως κατωνομασμένων, εἴ κα ἔωντι ἀριθμοὶ ἀπὸ μονάδος ἀνάλογον ἑξῆς κείμενοι, ὁ δὲ παρὰ τὰν μονάδα δεκὰς ᾖ, ὀκτὼ μὲν αὐτῶν οἱ πρῶτοι σὺν τᾷ μονάδι τῶν πρώτων ἀριθμῶν καλουμένων ἐσσοῦνται, οἱ δὲ μετʼ αὐτοὺς ἄλλοι ὀκτὼ τῶν δευτέρων καλουμένων, καὶ οἱ ἄλλοι τὸν αὐτὸν τρόπον τούτοις τῶν συνωνύμων καλουμένων ἐσσοῦνται τᾷ ἀποστάσει τᾶς ὀκτάδος τῶν
These being thus named, if there are numbers starting from a unit in continuous proportion, and the one next to the unit is ten, the first eight of them, including the unit, will belong to the so-called "first numbers", and the other eight after them to the so-called "second numbers", and the others in the same manner will belong to those called by the same names, according to the distance of the octad of...