Page 185 - Euclid's Elements of Geometry

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ELEMENTS BOOK 6

ΑΔ. παραβεβλήσθω γὰρ παρὰ τὴν ΑΒ εὐθεῖαν τὸ ΑΖ πα- [parallelogrammic] figures similar, and similarly laid out,
ραλληλόγραμμον ἐλλεῖπον εἴδει παραλληλογράμμῳ τῷ ΖΒ to DB, the greatest is AD. For let the parallelogram AF
ὁμοίῳ τε καὶ ὁμοίως κειμένῳ τῷ ΔΒ· λέγω, ὅτι μεῖζόν ἐστι have been applied to the straight-line AB, falling short by
τὸ ΑΔ τοῦ ΑΖ. the parallelogrammic figure FB (which is) similar, and
similarly laid out, to DB. I say that AD is greater than
AF.
∆ Ε D E

Ν N
Η Λ Ζ Μ Θ G L F M H

Α Γ Κ Β A C K B
᾿Επεὶ γὰρ ὅμοιόν ἐστι τὸ ΔΒ παραλληλόγραμμον τῷ ΖΒ For since parallelogram DB is similar to parallelo-
παραλληλογράμμῳ, περὶ τὴν αὐτήν εἰσι διάμετρον. ἤχθω gram FB, they are about the same diagonal [Prop. 6.26].
αὐτῶν διάμετρος ἡ ΔΒ, καὶ καταγεγράφθω τὸ σχῆμα. Let their (common) diagonal DB have been drawn, and
᾿Επεὶ οὖν ἴσον ἐστὶ τὸ ΓΖ τῷ ΖΕ, κοινὸν δὲ τὸ ΖΒ, let the (rest of the) figure have been described.
ὅλον ἄρα τὸ ΓΘ ὅλῳ τῷ ΚΕ ἐστιν ἴσον. ἀλλὰ τὸ ΓΘ τῷ Therefore, since (complement) CF is equal to (com-
ΓΗ ἐστιν ἴσον, ἐπεὶ καὶ ἡ ΑΓ τῇ ΓΒ. καὶ τὸ ΗΓ ἄρα τῷ ΕΚ plement) FE [Prop. 1.43], and (parallelogram) FB is
ἐστιν ἴσον. κοινὸν προσκείσθω τὸ ΓΖ· ὅλον ἄρα τὸ ΑΖ τῷ common, the whole (parallelogram) CH is thus equal
ΛΜΝ γνώμονί ἐστιν ἴσον· ὥστε τὸ ΔΒ παραλληλόγραμμον, to the whole (parallelogram) KE. But, (parallelogram)
τουτέστι τὸ ΑΔ, τοῦ ΑΖ παραλληλογράμμου μεῖζόν ἐστιν. CH is equal to CG, since AC (is) also (equal) to CB
Πάντων ἄρα τῶν παρὰ τὴν αὐτὴν εὐθεῖαν παραβαλ- [Prop. 6.1]. Thus, (parallelogram) GC is also equal
λομένων παραλληλογράμμων καὶ ἐλλειπόντων εἴδεσι παραλ- to EK. Let (parallelogram) CF have been added to
ληλογράμμοις ὁμοίοις τε καὶ ὁμοίως κειμένοις τῷ ἀπὸ τῆς both. Thus, the whole (parallelogram) AF is equal to
khþ straight-line, and falling short by a parallelogrammic
ἡμισείας ἀναγραφομένῳ μέγιστόν ἐστι τὸ ἀπὸ τῆς ἡμισείας the gnomon LMN. Hence, parallelogram DB—that is to
παραβληθέν· ὅπερ ἔδει δεῖξαι. say, AD—is greater than parallelogram AF.
Thus, for all parallelograms applied to the same

figure similar, and similarly laid out, to the (parallelo-
gram) described on half (the straight-line), the greatest
is the [parallelogram] applied to half (the straight-line).
(Which is) the very thing it was required to show.
Proposition 28
.
†
Παρὰ τὴν δοθεῖσαν εὐθεῖαν τῷ δοθέντι εὐθυγράμμῳ To apply a parallelogram, equal to a given rectilin-
ἴσον παραλληλόγραμμον παραβαλεῖν ἐλλεῖπον εἴδει πα- ear figure, to a given straight-line, (the applied parallel-
ραλληλογράμμῳ ὁμοίῳ τῷ δοθέντι· δεῖ δὲ τὸ διδόμενον ogram) falling short by a parallelogrammic figure similar
εὐθύγραμμον [ᾧ δεῖ ἴσον παραβαλεῖν] μὴ μεῖζον εἶναι τοῦ to a given (parallelogram). It is necessary for the given
ἀπὸ τῆς ἡμισείας ἀναγραφομένου ὁμοίου τῷ ἐλλείμματι [τοῦ rectilinear figure [to which it is required to apply an equal
τε ἀπὸ τῆς ἡμισείας καὶ ᾧ δεῖ ὅμοιον ἐλλείπειν]. (parallelogram)] not to be greater than the (parallelo-
῎Εστω ἡ μὲν δοθεῖσα εὐθεῖα ἡ ΑΒ, τὸ δὲ δοθὲν gram) described on half (of the straight-line) and similar
εὐθύγραμμον, ᾧ δεῖ ἴσον παρὰ τὴν ΑΒ παραβαλεῖν, τὸ Γ μὴ to the deficit.
μεῖζον [ὂν] τοῦ ἀπὸ τῆς ἡμισείας τῆς ΑΒ ἀναγραφομένου Let AB be the given straight-line, and C the given
ὁμοίου τῷ ἐλλείμματι, ᾧ δὲ δεῖ ὅμοιον ἐλλείπειν, τὸ Δ· δεῖ δὴ rectilinear figure to which the (parallelogram) applied to

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