Page 251 - Euclid's Elements of Geometry

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τετράγωνον ἀριθμόν. a(nother) square number. ELEMENTS BOOK 8
᾿Επεὶ γὰρ οἱ Α, Β ὅμοιοι ἐπίπεδοί εἰσιν, τῶν Α, Β ἄρα For since A and B are similar plane numbers, one
εἷς μέσος ἀνάλογον ἐμπίπτει ἀριθμός. ἐμπιπτέτω καὶ ἔστω ὁ number thus falls (between) A and B in mean propor-
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Γ, καὶ εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον tion [Prop. 8.18]. Let it (so) fall, and let it be C. And
ἐχόντων τοῖς Α, Γ, Β οἱ Δ, Ε, Ζ· οἱ ἄρα ἄκροι αὐτῶν οἱ let the least numbers, D, E, F, having the same ratio
Δ, Ζ τετράγωνοί εἰσιν. καὶ ἐπεί ἐστιν ὡς ὁ Δ πρὸς τὸν Ζ, as A, C, B have been taken [Prop. 8.2]. The outermost
οὕτως ὁ Α πρὸς τὸν Β, καί εἰσιν οἱ Δ, Ζ τετράγωνοι, ὁ Α of them, D and F, are thus square [Prop. 8.2 corr.]. And
ἄρα πρὸς τὸν Β λόγον ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς since as D is to F, so A (is) to B, and D and F are square,
τετράγωνον ἀριθμόν· ὅπερ ἔδει δεῖξαι. A thus has to B the ratio which (some) square number
(has) to a(nother) square number. (Which is) the very
thing it was required to show.

Proposition 27
.
Οἱ ὅμοιοι στερεοὶ ἀριθμοὶ πρὸς ἀλλήλους λόγον ἔχου- Similar solid numbers have to one another the ratio
σιν, ὃν κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν. which (some) cube number (has) to a(nother) cube num-
ber.
Α Ε A E

Γ Ζ C F
∆ Η D G

Β Θ B H
῎Εστωσαν ὅμοιοι στερεοὶ ἀριθμοὶ οἱ Α, Β· λέγω, ὅτι ὁ Let A and B be similar solid numbers. I say that A
Α πρὸς τὸν Β λόγον ἔχει, ὃν κύβος ἀριθμὸς πρὸς κύβον has to B the ratio which (some) cube number (has) to
ἀριθμόν. a(nother) cube number.
᾿Επεὶ γὰρ οἱ Α, Β ὅμοιοι στερεοί εἰσιν, τῶν Α, Β ἄρα δύο For since A and B are similar solid (numbers), two
μέσοι ἀνάλογον ἐμπίπτουσιν ἀριθμοί. ἐμπιπτέτωσαν οἱ Γ, numbers thus fall (between) A and B in mean proportion
Δ, καὶ εἰλήφθωσαν ἐλάχιστοι ἀριθμοὶ τῶν τὸν αὐτὸν λόγον [Prop. 8.19]. Let C and D have (so) fallen. And let the
ἐχόντων τοῖς Α, Γ, Δ, Β ἴσοι αὐτοῖς τὸ πλῆθος οἱ Ε, Ζ, Η, least numbers, E, F, G, H, having the same ratio as A,
Θ· οἱ ἄρα ἄκροι αὐτῶν οἱ Ε, Θ κύβοι εἰσίν. καί ἐστιν ὡς ὁ C, D, B, (and) equal in multitude to them, have been
Ε πρὸς τὸν Θ, οὕτως ὁ Α πρὸς τὸν Β· καὶ ὁ Α ἄρα πρὸς taken [Prop. 8.2]. Thus, the outermost of them, E and
τὸν Β λόγον ἔχει, ὃν κύβος ἀριθμὸς πρὸς κύβον ἀριθμόν· H, are cube [Prop. 8.2 corr.]. And as E is to H, so A (is)
ὅπερ ἔδει δεῖξαι. to B. And thus A has to B the ratio which (some) cube
number (has) to a(nother) cube number. (Which is) the
very thing it was required to show.

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