Page 184 - Euclid's Elements of Geometry

ST EW þ.

ELEMENTS BOOK 6

ὅτι περὶ τὴν αὐτὴν διάμετρόν ἐστι τὸ ΑΒΓΔ τῷ ΑΖ. AF have been subtracted (which is) similar, and similarly
laid out, to ABCD, having the common angle DAB with
it. I say that ABCD is about the same diagonal as AF.
Α Η ∆ A G D

Ε Ζ E F

Κ K
Θ H

Β Γ B C
Μὴ γάρ, ἀλλ᾿ εἰ δυνατόν, ἔστω [αὐτῶν] διάμετρος ἡ For (if) not, then, if possible, let AHC be [ABCD’s]
ΑΘΓ, καὶ ἐκβληθεῖσα ἡ ΗΖ διήχθω ἐπὶ τὸ Θ, καὶ ἤχθω diagonal. And producing GF, let it have been drawn
διὰ τοῦ Θ ὁπορέρᾳ τῶν ΑΔ, ΒΓ παράλληλος ἡ ΘΚ. through to (point) H. And let HK have been drawn
᾿Επεὶ οὖν περὶ τὴν αὐτὴν διάμετρόν ἐστι τὸ ΑΒΓΔ τῷ through (point) H, parallel to either of AD or BC
ΚΗ, ἔστιν ἄρα ὡς ἡ ΔΑ πρὸς τὴν ΑΒ, οὕτως ἡ ΗΑ πρὸς [Prop. 1.31].
τὴν ΑΚ. ἔστι δὲ καὶ διὰ τὴν ὁμοιότητα τῶν ΑΒΓΔ, ΕΗ καὶ Therefore, since ABCD is about the same diagonal as
ὡς ἡ ΔΑ πρὸς τὴν ΑΒ, οὕτως ἡ ΗΑ πρὸς τὴν ΑΕ· καὶ ὡς KG, thus as DA is to AB, so GA (is) to AK [Prop. 6.24].
ἄρα ἡ ΗΑ πρὸς τὴν ΑΚ, οὕτως ἡ ΗΑ πρὸς τὴν ΑΕ. ἡ ΗΑ And, on account of the similarity of ABCD and EG, also,
ἄρα πρὸς ἑκατέραν τῶν ΑΚ, ΑΕ τὸν αὐτὸν ἔχει λόγον. ἴση as DA (is) to AB, so GA (is) to AE. Thus, also, as GA
ἄρα ἐστὶν ἡ ΑΕ τῇ ΑΚ ἡ ἐλάττων τῇ μείζονι· ὅπερ ἐστὶν (is) to AK, so GA (is) to AE. Thus, GA has the same
ἀδύνατον. οὐκ ἄρα οὔκ ἐστι περὶ τὴν αὐτὴν διάμετρον τὸ ratio to each of AK and AE. Thus, AE is equal to AK
ΑΒΓΔ τῷ ΑΖ· περὶ τὴν αὐτὴν ἄρα ἐστὶ διάμετρον τὸ ΑΒΓΔ [Prop. 5.9], the lesser to the greater. The very thing is
παραλληλόγραμμον τῷ ΑΖ παραλληλογράμμῳ. impossible. Thus, ABCD is not not about the same di-
kzþ to the whole, having a common angle with it, then (the
᾿Εὰν ἄρα ἀπὸ παραλληλογράμμου παραλληλόγραμμον agonal as AF. Thus, parallelogram ABCD is about the
ἀφαιρεθῇ ὅμοιόν τε τῷ ὅλῳ καὶ ὁμοίως κείμενον κοινὴν same diagonal as parallelogram AF.
γωνίαν ἔχον αὐτῷ, περὶ τὴν αὐτὴν διάμετρόν ἐστι τῷ ὅλῳ· Thus, if from a parallelogram a(nother) parallelogram
ὅπερ ἔδει δεῖξαι. is subtracted (which is) similar, and similarly laid out,

subtracted parallelogram) is about the same diagonal as
the whole. (Which is) the very thing it was required to
show.

.
Proposition 27
Πάντων τῶν παρὰ τὴν αὐτὴν εὐθεῖαν παραβαλλομένων Of all the parallelograms applied to the same straight-
παραλληλογράμμων καὶ ἐλλειπόντων εἴδεσι παραλληλογράμ- line, and falling short by parallelogrammic figures similar,
μοις ὁμοίοις τε καὶ ὁμοίως κειμένοις τῷ ἀπὸ τῆς ἡμισείας and similarly laid out, to the (parallelogram) described
ἀναγραφομένῳ μέγιστόν ἐστι τὸ ἀπὸ τῆς ἡμισείας παρα- on half (the straight-line), the greatest is the [parallelo-
βαλλόμενον [παραλληλόγραμμον] ὅμοιον ὂν τῷ ἐλλείμμαντι. gram] applied to half (the straight-line) which (is) similar
῎Εστω εὐθεῖα ἡ ΑΒ καὶ τετμήσθω δίχα κατὰ τὸ Γ, to (that parallelogram) by which it falls short.
καὶ παραβεβλήσθω παρὰ τὴν ΑΒ εὐθεῖαν τὸ ΑΔ παραλ- Let AB be a straight-line, and let it have been cut in
ληλόγραμμον ἐλλεῖπον εἴδει παραλληλογράμμῳ τῷ ΔΒ ἀνα- half at (point) C [Prop. 1.10]. And let the parallelogram
γραφέντι ἀπὸ τῆς ἡμισείας τῆς ΑΒ, τουτέστι τῆς ΓΒ· λέγω, AD have been applied to the straight-line AB, falling
ὅτι πάντων τῶν παρὰ τὴν ΑΒ παραβαλλομένων παραλλη- short by the parallelogrammic figure DB (which is) ap-
λογράμμων καὶ ἐλλειπόντων εἴδεσι [παραλληλογράμμοις] plied to half of AB—that is to say, CB. I say that of all
ὁμοίοις τε καὶ ὁμοίως κειμένοις τῷ ΔΒ μέγιστόν ἐστι τὸ the parallelograms applied to AB, and falling short by

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