§3#2ἀριθμῶν ἀπὸ τᾶς πρώτας ὀκτάδος τῶν ἀριθμῶν.
...of the numbers from the first octad of numbers.
Τᾶς μὲν οὖν πρώτας ὀκτάδος τῶν ἀριθμῶν ὁ ὄγδοός ἐστιν ἀριθμὸς χίλιαι μυριάδες, τᾶς δὲ δευτέρας ὀκτάδος ὁ πρῶτος, ἐπεὶ δεκαπλασίων ἐστὶν τοῦ πρὸ αὐτοῦ, μύριαι μυριάδες ἐσσεῖται οὗτος δέ ἐστι μονὰς τῶν δευτέρων ἀριθμῶν.
Of the first octad of numbers, then, the eighth number is a thousand ten-thousands, and the first of the second octad, since it is ten times the one before it, will be ten thousand ten-thousands; and this is the unit of the second numbers.
Ὁ δὲ ὄγδοος τᾶς δευτέρας ὀκτάδος ἐστὶ χίλιαι μυριάδες τῶν δευτέρων ἀριθμῶν, Πάλιν δὲ καὶ τᾶς τρίτας ὀκτάδος ὁ πρῶτος, ἐπεὶ δεκαπλασίων ἐστὶ τοῦ πρὸ αὐτοῦ, μύριαι μυριάδες ἐσσεῖται τῶν δευτέρων ἀριθμῶν· οὗτος δέ ἐστιν μονὰς τῶν τρίτων ἀριθμῶν, Φανερὸν δὲ ὅτι καὶ πολλοσταὶ ὀκτάδες ἑξοῦντι ὡς εἴρηται.
And the eighth of the second octad is a thousand ten-thousands of the second numbers. Again, also the first of the third octad, since it is ten times the one before it, will be ten thousand ten-thousands of the second numbers; and this is the unit of the third numbers. It is clear that any number of octads will proceed as has been said.
Χρήσιμον δὲ ἐστι καὶ τόδε γιγνωσκόμενον.
And this also is useful to be known.
Εἴ κα ἀριθμῶν ἀπὸ τᾶς μονάδος ἀνάλογον ἐόντων πολλαπλασιάζωντί τινες ἀλλάλους τῶν ἐκ τᾶς αὐτᾶς ἀναλογίας, ὁ γενόμενος ἐσσεῖται ἐκ τᾶς αὐτᾶς ἀναλογίας ἀπέχων ἀπὸ μὲν τοῦ μείζονος τῶν πολλαπλασιαξάντων ἀλλάλους, ὅσους ὁ ἐλάττων τῶν πολλαπλασιαξάντων ἀπὸ μονάδος ἀνάλογον ἀπέχει, ἀπὸ δὲ τᾶς μονάδος ἀφέξει ἑνὶ ἐλάττονας ἢ ὅσος ἐστὶν ὁ ἀριθμὸς συναμφοτέρων, οὓς ἀπέχοντι ἀπὸ μονάδος οἱ πολλαπλασιάξαντες ἀλλάλους.
If, while numbers are in proportion starting from a unit, some of those from the same proportion multiply one another, the product will be of the same proportion, being distant from the greater of those multiplying one another by as many (terms) as the smaller of those multiplying one another is distant from the unit in the proportion, and from the unit it will be distant by one fewer than the sum of both numbers of terms by which the multipliers are distant from the unit.
Ἔστων γὰρ ἀριθμοί τινες ἀνάλογον ἀπὸ μονάδος οἱ Α, Β, Γ, △, Ε, Ζ, Η, Θ, Ι, Κ, Λ, μονὰς δὲ ἔστω ὁ Α, καὶ πεπολλαπλασιάσθω ὁ △ τῷ Θ, ὁ δὲ γενόμενος ἔστω ὁ Χ. Λελάφθω δὴ ἐκ τῆς ἀναλογίας ὁ Λ ἀπέχων ἀπὸ τοῦ Θ τοσούτους, ὅσους ὁ △ ἀπὸ μονάδος ἀπέχει·
For let some numbers in proportion starting from a unit be A, B, G, D, E, Z, H, Th, I, K, L, and let A be the unit, and let D be multiplied by Th, and let the product be X. Let, then, L be taken from the proportion, being distant from Th by as many (terms) as D is distant from the unit; it must be shown that X is equal to L.
δεικτέον ὅτι ἴσος ἐστὶν ὁ Χ τῷ Λ. Ἐπεὶ οὖν ἀνάλογον ἐόντων ἀριθμῶν ἴσους ἀπέχει ὅ τε △ ἀπὸ τοῦ Α καὶ ὁ Λ ἀπὸ τοῦ Θ, τὸν αὐτὸν ἔχει λόγον ὁ △ ποτὶ τὸν Α, ὃν ὁ Λ ποτὶ τὸν Θ. Πολλαπλασίων δὲ ἐστιν ὁ △ τοῦ Α τῷ △· πολλαπλασίων ἄρα ἐστὶν καὶ ὁ Λ τοῦ Θ τῷ △ ὥστε ἴσος ἐστὶν ὁ Λ τῷ Χ. Δῆλον οὖν ὅτι ὁ γενόμενος ἐκ τᾶς ἀναλογίας τέ ἐστιν καὶ ἀπὸ τοῦ μείζονος τῶν πολλαπλασιαξάντων ἀλλάλους ἴσους ἀπέχων, ὅσους ὁ ἐλάττων ἀπὸ τᾶς μονάδος ἀπέχει.
Since, then, when numbers are in proportion, D is as distant from A as L is from Th, D has the same ratio to A as L has to Th. And D is D times A; therefore L is also D times Th, so that L is equal to X. It is clear, then, that the product is of the proportion and is distant from the greater of those multiplying one another by as many (terms) as the smaller is distant from the unit.
Θανερὸν δὲ ὅτι καὶ ἀπὸ μονάδος ἀπέχει ἑνὶ ἐλάττονας ἢ ὅσος ἐστὶν ὁ ἀριθμὸς συναμφοτέρων, οὓς ἀπέχοντι ἀπὸ τᾶς μονάδος οἱ △, Θ οἱ μὲν γὰρ Α, Β, Γ, △, Ε, Ζ, Η, Θ τοσοῦτοί ἐντι, ὅσους ὁ Θ ἀπὸ μονάδος ἀπέχει, οἱ δὲ Ι, Κ, Λ ἐνὶ ἐλάττονες ἢ ὅσους ὁ △ ἀπὸ μονάδος ἀπέχει· σὺν γὰρ τῷ Θ τοσοῦτοί ἐντι.
And it is manifest that it is also distant from the unit by one fewer than the sum of both numbers of terms by which D, Th are distant from the unit; for A, B, G, D, E, Z, H, Th are as many as Th is distant from the unit, while I, K, L are one fewer than D is distant from the unit; for with Th they are that many.