Number theory: Book 9 Proposition 18
Translations
Δύο ἀριθμῶν δοθέντων ἐπισκέψασθαι, εἰ δυνατόν ἐστιν αὐτοῖς τρίτον ἀνάλογον προσευρεῖν. Ἔστωσαν οἱ δοθέντες δύο ἀριθμοὶ οἱ Α, Β, καὶ δέον ἔστω ἐπισκέψασθαι, εἰ δυνατόν ἐστιν αὐτοῖς τρίτον ἀνάλογον προσευρεῖν. Οἱ δὴ Α, Β ἤτοι πρῶτοι πρὸς ἀλλήλους εἰσὶν ἢ οὔ. καὶ εἰ πρῶτοι πρὸς ἀλλήλους εἰσίν, δέδεικται, ὅτι ἀδύνατόν ἐστιν αὐτοῖς τρίτον ἀνάλογον προσευρεῖν. Ἀλλὰ δὴ μὴ ἔστωσαν οἱ Α, Β πρῶτοι πρὸς ἀλλήλους, καὶ ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Γ ποιείτω: ὁ Α δὴ τὸν Γ ἤτοι μετρεῖ ἢ οὐ μετρεῖ. μετρείτω πρότερον κατὰ τὸν Δ: ὁ Α ἄρα τὸν Δ πολλαπλασιάσας τὸν Γ πεποίηκεν. ἀλλὰ μὴν καὶ ὁ Β ἑαυτὸν πολλαπλασιάσας τὸν Γ πεποίηκεν: ὁ ἄρα ἐκ τῶν Α, Δ ἴσος ἐστὶ τῷ ἀπὸ τοῦ Β. ἔστιν ἄρα ὡς ὁ Α πρὸς τὸν Β, ὁ Β πρὸς τὸν Δ: τοῖς Α, Β ἄρα τρίτος ἀριθμὸς ἀνάλογον προσηύρηται ὁ Δ. Ἀλλὰ δὴ μὴ μετρείτω ὁ Α τὸν Γ: λέγω, ὅτι τοῖς Α, Β ἀδύνατόν ἐστι τρίτον ἀνάλογον προσευρεῖν ἀριθμόν. εἰ γὰρ δυνατόν, προσηυρήσθω ὁ Δ. ὁ ἄρα ἐκ τῶν Α, Δ ἴσος ἐστὶ τῷ ἀπὸ τοῦ Β. ὁ δὲ ἀπὸ τοῦ Β ἐστιν ὁ Γ: ὁ ἄρα ἐκ τῶν Α, Δ ἴσος ἐστὶ τῷ Γ. ὥστε ὁ Α τὸν Δ πολλαπλασιάσας τὸν Γ πεποίηκεν: ὁ Α ἄρα τὸν Γ μετρεῖ κατὰ τὸν Δ. ἀλλὰ μὴν ὑπόκειται καὶ μὴ μετρῶν: ὅπερ ἄτοπον. οὐκ ἄρα δυνατόν ἐστι τοῖς Α, Β τρίτον ἀνάλογον προσευρεῖν ἀριθμόν, ὅταν ὁ Α τὸν Γ μὴ μετρῇ: ὅπερ ἔδει δεῖξαι.
Given two numbers, to investigate whether it is possible to find a third proportional to them. Let A, B be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them. Now A, B are either prime to one another or not. And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [IX. 16] Next, let A, B not be prime to one another, and let B by multiplying itself make C. Then A either measures C or does not measure it. First, let it measure it according to D; therefore A by multiplying D has made C. But, further, B has also by multiplying itself made C; therefore the product of A, D is equal to the square on B. Therefore, as A is to B, so is B to D; [VII. 19] therefore a third proportional number D has been found to A, B. Next, let A not measure C; I say that it is impossible to find a third proportional number to A, B. For, if possible, let D, such third proportional, have been found. Therefore the product of A, D is equal to the square on B. But the square on B is C; therefore the product of A, D is equal to C. Hence A by multiplying D has made C; therefore A measures C according to D. But, by hypothesis, it also does not measure it: which is absurd.