§8.prop.23ἐὰν τέσσαρες ἀριθμοὶ ἑξῆς ἀνάλογον ὦσιν, ὁ δὲ πρῶτος κύβος ᾖ, καὶ ὁ τέταρτος κύβος ἔσται.
If four numbers are continuously proportional, and the first is a cube, the fourth will also be a cube.
ἔστωσαν τέσσαρες ἀριθμοὶ ἑξῆς ἀνάλογον οἱ Α, Β, Γ, Δ, ὁ δὲ Α κύβος ἔστω· λέγω, ὅτι καὶ ὁ Δ κύβος ἐστίν.
Let four numbers A, B, Γ, Δ be continuously proportional, and let A be a cube; I say that Δ is also a cube.
ἐπεὶ γὰρ τῶν Α, Δ δύο μέσοι ἀνάλογόν εἰσιν ἀριθμοὶ οἱ Β, Γ, οἱ Α, Δ ἄρα ὅμοιοί εἰσι στερεοὶ ἀριθμοί.
For since two numbers B, Γ are in continued proportion between A, Δ, therefore A, Δ are similar solid numbers.
κύβος δὲ ὁ Α· κύβος ἄρα καὶ ὁ Δ· ὅπερ ἔδει δεῖξαι.
But A is a cube; therefore Δ is also a cube; which it was required to prove.
§8.prop.24ἐὰν δύο ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, ὁ δὲ πρῶτος τετράγωνος ᾖ, καὶ ὁ δεύτερος τετράγωνος ἔσται.
If two numbers have to one another the ratio which a square number has to a square number, and the first is a square, the second will also be a square.
δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς ἀλλήλους λόγον ἐχέτωσαν, ὃν τετράγωνος ἀριθμὸς ὁ Γ πρὸς τετράγωνον ἀριθμὸν τὸν Δ, ὁ δὲ Α τετράγωνος ἔστω· λέγω, ὅτι καὶ ὁ Β τετράγωνός ἐστιν.
For let two numbers A, B have to one another the ratio which a square number Γ has to a square number Δ, and let A be a square; I say that B is also a square.
ἐπεὶ γὰρ οἱ Γ, Δ τετράγωνοί εἰσιν, οἱ Γ, Δ ἄρα ὅμοιοι ἐπίπεδοί εἰσιν.
For since Γ, Δ are squares, therefore Γ, Δ are similar plane numbers.
τῶν Γ, Δ ἄρα εἷς μέσος ἀνάλογον ἐμπίπτει ἀριθμός.
Therefore one number falls between Γ, Δ in continued proportion.
καί ἐστιν ὡς ὁ Γ πρὸς τὸν Δ, ὁ Α πρὸς τὸν Β· καὶ τῶν Α, Β ἄρα εἷς μέσος ἀνάλογον ἐμπίπτει ἀριθμός.
And, as Γ is to Δ, so is A to B; therefore also one number falls between A, B in continued proportion.
καί ἐστιν ὁ Α τετράγωνος· καὶ ὁ Β ἄρα τετράγωνός ἐστιν· ὅπερ ἔδει δεῖξαι.
And A is a square; therefore B is also a square; which it was required to prove.
§8.prop.25ἐὰν δύο ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχωσιν, ὃν κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν, ὁ δὲ πρῶτος κύβος ᾖ, καὶ ὁ δεύτερος κύβος ἔσται.
If two numbers have to one another the ratio which a cube number has to a cube number, and the first is a cube, the second will also be a cube.
δύο γὰρ ἀριθμοὶ οἱ Α, Β πρὸς ἀλλήλους λόγον ἐχέτωσαν, ὃν κύβος ἀριθμὸς ὁ Γ πρὸς κύβον ἀριθμὸν τὸν Δ, κύβος δὲ ἔστω ὁ Α· λέγω, ὅτι καὶ ὁ Β κύβος ἐστίν.
For let two numbers A, B have to one another the ratio which a cube number Γ has to a cube number Δ, and let A be a cube; I say that B is also a cube.
ἐπεὶ γὰρ οἱ Γ, Δ κύβοι εἰσίν, οἱ Γ, Δ ὅμοιοι στερεοί εἰσιν· τῶν Γ, Δ ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί.
For since Γ, Δ are cubes, Γ, Δ are similar solid numbers; therefore two numbers fall between Γ, Δ in continued proportion.
ὅσοι δὲ εἰς τοὺς Γ, Δ μεταξὺ κατὰ τὸ συνεχὲς ἀνάλογον ἐμπίπτουσιν, τοσοῦτοι καὶ εἰς τοὺς τὸν αὐτὸν λόγον ἔχοντας αὐτοῖς· ὥστε καὶ τῶν Α, Β δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί.
But as many as fall between Γ, Δ in continuous proportion, so many also fall between those having the same ratio with them; so that two numbers fall between A, B in continued proportion.
ἐμπιπτέτωσαν οἱ Ε, Ζ. ἐπεὶ οὖν τέσσαρες ἀριθμοὶ οἱ Α, Ε, Ζ, Β ἑξῆς ἀνάλογόν εἰσιν, καί ἐστι κύβος ὁ Α, κύβος ἄρα καὶ ὁ Β· ὅπερ ἔδει δεῖξαι.
Let them fall as E, Z. Since, therefore, four numbers A, E, Z, B are continuously proportional, and A is a cube, therefore B is also a cube; which it was required to prove.
§8.prop.26οἱ ὅμοιοι ἐπίπεδοι ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσιν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
Similar plane numbers have to one another the ratio which a square number has to a square number.
ἔστωσαν ὅμοιοι ἐπίπεδοι ἀριθμοὶ οἱ Α, Β· λέγω, ὅτι ὁ Α πρὸς τὸν Β λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν.
Let A, B be similar plane numbers; I say that A has to B the ratio which a square number has to a square number.
ἐπεὶ γὰρ οἱ Α, Β ὅμοιοι ἐπίπεδοί εἰσιν, τῶν Α, Β ἄρα εἷς μέσος ἀνάλογον ἐμπίπτει ἀριθμός.
For since A, B are similar plane numbers, therefore one number falls between A, B in continued proportion.
ἐμπιπτέτω καὶ ἔστω ὁ Γ, καὶ εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Γ, Β οἱ Δ, Ε, Ζ· οἱ ἄρα ἄκροι αὐτῶν οἱ Δ, Ζ τετράγωνοί εἰσιν.
Let it fall and let it be Γ, and let the least numbers of those having the same ratio as A, Γ, B be taken, namely Δ, E, Z; therefore their extremes Δ, Z are squares.
καὶ ἐπεί ἐστιν ὡς ὁ Δ πρὸς τὸν Ζ, οὕτως ὁ Α πρὸς τὸν Β, καί εἰσιν οἱ Δ, Ζ τετράγωνοι, ὁ Α ἄρα πρὸς τὸν Β λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· ὅπερ ἔδει δεῖξαι.
And since, as Δ is to Z, so is A to B, and Δ, Z are squares, therefore A has to B the ratio which a square number has to a square number; which it was required to prove.
§8.prop.27οἱ ὅμοιοι στερεοὶ ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχουσιν, ὃν κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν.
Similar solid numbers have to one another the ratio which a cube number has to a cube number.
ἔστωσαν ὅμοιοι στερεοὶ ἀριθμοὶ οἱ Α, Β· λέγω, ὅτι ὁ Α πρὸς τὸν Β λόγον ἔχει, ὃν κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν.
Let A, B be similar solid numbers; I say that A has to B the ratio which a cube number has to a cube number.
ἐπεὶ γὰρ οἱ Α, Β ὅμοιοι στερεοί εἰσιν, τῶν Α, Β ἄρα δύο μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί.
For since A, B are similar solid numbers, therefore two numbers fall between A, B in continued proportion.
ἐμπιπτέτωσαν οἱ Γ, Δ, καὶ εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον ἐχόντων τοῖς Α, Γ, Δ, Β ἴσοι αὐτοῖς τὸ πλῆθος οἱ ε, Ζ, Η, Θ· οἱ ἄρα ἄκροι αὐτῶν οἱ Ε, Θ κύβοι εἰσίν.
Let them fall as Γ, Δ, and let the least numbers of those having the same ratio as A, Γ, Δ, B, equal to them in multitude, be taken, namely E, Z, H, Θ; therefore their extremes E, Θ are cubes.
καί ἐστιν ὡς ὁ Ε πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Β· καὶ ὁ Α ἄρα πρὸς τὸν Β λόγον ἔχει, ὃν κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν· ὅπερ ἔδει δεῖξαι.
And as E is to Θ, so is A to B; therefore A also has to B the ratio which a cube number has to a cube number; which it was required to prove.