§9.prop.19τριῶν ἀριθμῶν δοθέντων ἐπισκέψασθαι, πότε δυνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν.
Three numbers being given, to investigate when it is possible to find a fourth proportional to them.
ἔστωσαν οἱ δοθέντες τρεῖς ἀριθμοὶ οἱ Α, Β, Γ, καὶ δέον ἔστω ἐπισκέψασθαι, πότε δυνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν.
Let the three given numbers be A, B, Γ, and let it be required to investigate when it is possible to find a fourth proportional to them.
ἤτοι οὖν οὔκ εἰσιν ἑξῆς ἀνάλογον, καὶ οἱ ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους εἰσίν, ἢ ἑξῆς εἰσιν ἀνάλογον, καὶ οἱ ἄκροι αὐτῶν οὔκ εἰσι πρῶτοι πρὸς ἀλλήλους, ἢ οὔτε ἑξῆς εἰσιν ἀνάλογον, οὔτε οἱ ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους εἰσίν, ἢ καὶ ἑξῆς εἰσιν ἀνάλογον, καὶ οἱ ἄκροι αὐτῶν πρῶτοι πρὸς ἀλλήλους εἰσίν.
Now they either are not continuously proportional, and their extremes are prime to each other, or they are continuously proportional, and their extremes are not prime to each other, or they are neither continuously proportional nor are their extremes prime to each other, or they are both continuously proportional and their extremes are prime to each other.
εἰ μὲν οὖν οἱ Α, Β, Γ ἑξῆς εἰσιν ἀνάλογον, καὶ οἱ ἄκροι αὐτῶν οἱ Α, Γ πρῶτοι πρὸς ἀλλήλους εἰσίν, δέδεικται, ὅτι ἀδύνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν ἀριθμόν.
Now if A, B, Γ are continuously proportional, and their extremes A, Γ are prime to each other, it has been proved that it is impossible to find a fourth proportional number to them.
μὴ ἔστωσαν δὴ οἱ Α, Β, Γ ἑξῆς ἀνάλογον τῶν ἄκρων πάλιν ὄντων πρώτων πρὸς ἀλλήλους. λέγω, ὅτι καὶ οὕτως ἀδύνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν.
But indeed let A, B, Γ not be continuously proportional, their extremes being again prime to each other; I say that even so it is impossible to find a fourth proportional to them.
εἰ γὰρ δυνατόν, προσευρήσθω ὁ Δ, ὥστε εἶναι ὡς τὸν Α πρὸς τὸν Β, τὸν Γ πρὸς τὸν Δ, καὶ γεγονέτω ὡς ὁ Β πρὸς τὸν Γ, ὁ Δ πρὸς τὸν Ε. καὶ ἐπεί ἐστιν ὡς μὲν ὁ Α πρὸς τὸν Β, ὁ Γ πρὸς τὸν Δ, ὡς δὲ ὁ Β πρὸς τὸν Γ, ὁ Δ πρὸς τὸν Ε, διʼ ἴσου ἄρα ὡς ὁ Α πρὸς τὸν Γ, ὁ Γ πρὸς τὸν Ε. οἱ δὲ Α, Γ πρῶτοι, οἱ δὲ πρῶτοι καὶ ἐλάχιστοι, οἱ δὲ ἐλάχιστοι μετροῦσι τοὺς τὸν αὐτὸν λόγον ἔχοντας ὅ τε ἡγούμενος τὸν ἡγούμενον καὶ ὁ ἑπόμενος τὸν ἑπόμενον.
For, if possible, let Δ have been found, so that as A is to B, so is Γ to Δ, and let it be made as B is to Γ, so Δ to E. And since as A is to B, so is Γ to Δ, and as B is to Γ, so is Δ to E, therefore, ex aequali, as A is to Γ, so is Γ to E. And A, Γ are prime, and those which are prime are also least, and the least numbers measure those having the same ratio with them an equal number of times, the antecedent the antecedent and the consequent the consequent.
μετρεῖ ἄρα ὁ Α τὸν Γ ὡς ἡγούμενος ἡγούμενον.
Therefore A measures Γ as antecedent antecedent.
μετρεῖ δὲ καὶ ἑαυτόν· ὁ Α ἄρα τοὺς Α, Γ μετρεῖ πρώτους ὄντας πρὸς ἀλλήλους· ὅπερ ἐστὶν ἀδύνατον.
And it also measures itself; therefore A measures A, Γ which are prime to each other; which is impossible.
οὐκ ἄρα τοῖς Α, Β, Γ δυνατόν ἐστι τέταρτον ἀνάλογον προσευρεῖν.
Therefore it is not possible to find a fourth proportional to A, B, Γ.
ἀλλὰ δὴ πάλιν ἔστωσαν οἱ Α, Β, Γ ἑξῆς ἀνάλογον, οἱ δὲ Α, Γ μὴ ἔστωσαν πρῶτοι πρὸς ἀλλήλους. λέγω, ὅτι δυνατόν ἐστιν αὐτοῖς τέταρτον ἀνάλογον προσευρεῖν.
But indeed let A, B, Γ again be continuously proportional, and let A, Γ not be prime to each other; I say that it is possible to find a fourth proportional to them.
ὁ γὰρ Β τὸν Γ πολλαπλασιάσας τὸν Δ ποιείτω· ὁ Α ἄρα τὸν Δ ἤτοι μετρεῖ ἢ οὐ μετρεῖ.
For let B by multiplying Γ make Δ; then A either measures Δ or does not measure it.
μετρείτω αὐτὸν πρότερον κατὰ τὸν Ε· ὁ Α ἄρα τὸν Ε πολλαπλασιάσας τὸν Δ πεποίηκεν.
Let it first measure it according to E; therefore A by multiplying E has made Δ.
ἀλλὰ μὴν καὶ ὁ Β τὸν Γ πολλαπλασιάσας τὸν Δ πεποίηκεν· ὁ ἄρα ἐκ τῶν Α, Ε ἴσος ἐστὶ τῷ ἐκ τῶν Β, Γ. ἀνάλογον ἄρα ὡς ὁ Α πρὸς τὸν Β, ὁ Γ πρὸς τὸν Ε· τοῖς Α, Β, Γ ἄρα τέταρτος ἀνάλογον προσηύρηται ὁ Ε.
ἀλλὰ δὴ μὴ μετρείτω ὁ Α τὸν Δ·
But indeed B also by multiplying Γ has made Δ; therefore the product of A, E is equal to the product of B, Γ. Therefore proportionally, as A is to B, so is Γ to E; therefore a fourth proportional to A, B, Γ, namely E, has been found.
λέγω, ὅτι ἀδύνατόν ἐστι τοῖς Α, Β, Γ τέταρτον ἀνάλογον προσευρεῖν ἀριθμόν.
But indeed let A not measure Δ; I say that it is impossible to find a fourth proportional number to A, B, Γ.
εἰ γὰρ δυνατόν, προσευρήσθω ὁ Ε· ὁ ἄρα ἐκ τῶν Α, Ε ἴσος ἐστὶ τῷ ἐκ τῶν Β, Γ. ἀλλὰ ὁ ἐκ τῶν Β, Γ ἐστιν ὁ Δ· καὶ ὁ ἐκ τῶν Α, Ε ἄρα ἴσος ἐστὶ τῷ Δ. ὁ Α ἄρα τὸν Ε πολλαπλασιάσας τὸν Δ πεποίηκεν· ὁ Α ἄρα τὸν Δ μετρεῖ κατὰ τὸν Ε· ὥστε μετρεῖ ὁ Α τὸν Δ. ἀλλὰ καὶ οὐ μετρεῖ· ὅπερ ἄτοπον.
For, if possible, let E have been found; therefore the product of A, E is equal to the product of B, Γ. But the product of B, Γ is Δ; therefore the product of A, E is also equal to Δ. So that A by multiplying E has made Δ; therefore A measures Δ according to E; so that A measures Δ. But indeed it is also assumed not to measure it; which is absurd.
οὐκ ἄρα δυνατόν ἐστι τοῖς Α, Β, Γ τέταρτον ἀνάλογον προσευρεῖν ἀριθμόν, ὅταν ὁ Α τὸν Δ μὴ μετρῇ.
Therefore it is not possible to find a fourth proportional number to A, B, Γ, when A does not measure Δ.
ἀλλὰ δὴ οἱ Α, Β, Γ μήτε ἑξῆς ἔστωσαν ἀνάλογον μήτε οἱ ἄκροι πρῶτοι πρὸς ἀλλήλους.
But indeed let A, B, Γ neither be continuously proportional nor let their extremes be prime to each other.
καὶ ὁ Β τὸν Γ πολλαπλασιάσας τὸν Δ ποιείτω.
And let B by multiplying Γ make Δ.
ὁμοίως δὴ δειχθήσεται, ὅτι εἰ μὲν μετρεῖ ὁ Α τὸν Δ, δυνατόν ἐστιν αὐτοῖς ἀνάλογον προσευρεῖν, εἰ δὲ οὐ μετρεῖ, ἀδύνατον· ὅπερ ἔδει δεῖξαι.
Then in like manner it will be proved that, if A measures Δ, it is possible to find a proportional to them, but if it does not measure it, impossible; which it was required to prove.