Page 457 - Euclid's Elements of Geometry

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Καὶ ἐπεὶ δύο αἱ ΤΡ, ΡΥ δυσὶ ταῖς ΑΛ, ΛΒ ἴσαι εἰσίν, And since the two (straight-lines) T R and RU are
καὶ γωνίας ἴσας περιέχουσιν, ἴσον ἄρα καὶ ὅμοιον τὸ ΡΧ equal to the two (straight-lines) AL and LB (respec-
παραλληλόγραμμον τῷ ΘΛ παραλληλογράμμῳ. καὶ ἐπεὶ tively), and they contain equal angles, parallelogram
πάλιν ἴση μὲν ἡ ΑΛ τῇ ΡΤ, ἡ δὲ ΛΜ τῇ ΡΣ, καὶ γωνίας RW is thus equal and similar to parallelogram HL
ὀρθὰς περιέχουσιν, ἴσον ἄρα καὶ ὅμοιόν ἐστι τὸ ΡΨ παραλ- [Prop. 6.14]. And, again, since AL is equal to RT ,
ληλόγραμμον τῷ ΑΜ παραλληλογράμμῳ. διὰ τὰ αὐτὰ δὴ and LM to RS, and they contain right-angles, paral-
καὶ τὸ ΛΕ τῷ ΣΥ ἴσον τέ ἐστι καὶ ὅμοιον· τρία ἄρα πα- lelogram RX is thus equal and similar to parallelogram
ραλληλόγραμμα τοῦ ΑΕ στερεοῦ τρισὶ παραλληλογράμμοις AM [Prop. 6.14]. So, for the same (reasons), LE is also
τοῦ ΨΥ στερεοῦ ἴσα τέ ἐστι καὶ ὅμοια. ἀλλὰ τὰ μὲν τρία equal and similar to SU. Thus, three parallelograms of
τρισὶ τοῖς ἀπεναντίον ἴσα τέ ἐστι καὶ ὅμοια, τὰ δὲ τρία solid AE are equal and similar to three parallelograms
τρισὶ τοῖς ἀπεναντίον· ὅλον ἄρα τὸ ΑΕ στερεὸν παραλλη- of solid XU. But, the three (faces of the former solid)
λεπίπεδον ὅλῳ τῷ ΨΥ στερεῷ παραλληλεπιπέδῳ ἴσον ἐστίν. are equal and similar to the three opposite (faces), and
διήχθωσαν αἱ ΔΡ, ΧΥ καὶ συμπιπτέτωσαν ἀλλήλαις κατὰ the three (faces of the latter solid) to the three opposite
τὸ Ω, καὶ διὰ τοῦ Τ τῇ ΔΩ παράλληλος ἤχθω ἡ αΤϡ, καὶ (faces) [Prop. 11.24]. Thus, the whole parallelepiped
ἐκβεβλήσθω ἡ ΟΔ κατὰ τὸ α, καὶ συμπεπληρώσθω τὰ ΩΨ, solid AE is equal to the whole parallelepiped solid XU
ΡΙ στερεά. ἴσον δή ἐστι τὸ ΨΩ στερεόν, οὗ βάσις μέν [Def. 11.10]. Let DR and WU have been drawn across,
ἐστι τὸ ΡΨ παραλληλόγραμμον, ἀπεναντίον δὲ τὸ Ωϟ, τῷ and let them have met one another at Y . And let aT b
ΨΥ στερεῷ, οὗ βάσις μὲν τὸ ΡΨ παραλληλόγραμμον, ἀπε- have been drawn through T parallel to DY . And let PD
ναντίον δὲ τὸ ΥΦ· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς εἰσι τῆς have been produced to a. And let the solids Y X and
ΡΨ καὶ ὑπὸ τὸ αὐτὸ ὕψος, ὧν αἱ ἐφεστῶσαι αἱ ΡΩ, ΡΥ, RI have been completed. So, solid XY , whose base is
Τϡ, ΤΧ, Σϛ, Σ˜ο, Ψϟ, ΨΦ ἐπὶ τῶν αὐτῶν εἰσιν εὐθειῶν τῶν parallelogram RX, and opposite (face) Y c, is equal to
ΩΧ, ϛΦ. ἀλλὰ τὸ ΨΥ στερεὸν τῷ ΑΕ ἐστιν ἴσον· καὶ τὸ ΨΩ solid XU, whose base (is) parallelogram RX, and oppo-
ἄρα στερεὸν τῷ ΑΕ στερεῷ ἐστιν ἴσον. καὶ ἐπεὶ ἴσον ἐστὶ site (face) UV . For they are on the same base RX, and
τὸ ΡΥΧΤ παραλληλόγραμμον τῷ ΩΤ παραλληλογράμμῳ· (have) the same height, and the (ends of the straight-
ἐπί τε γὰρ τῆς αὐτῆς βάσεώς εἰσι τῆς ΡΤ καὶ ἐν ταῖς αὐταῖς lines) standing up in them, RY , RU, T b, T W, Se, Sd,
παραλλήλοις ταῖς ΡΤ, ΩΧ· ἀλλὰ τὸ ΡΥΧΤ τῷ ΓΔ ἐστιν Xc and XV , are on the same straight-lines, Y W and
ἴσον, ἐπεὶ καὶ τῷ ΑΒ, καὶ τὸ ΩΤ ἄρα παραλληλόγραμμον eV [Prop. 11.29]. But, solid XU is equal to AE. Thus,

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