Page 313 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ἑαυτῇ. BF [Props. 5.19 corr., 3.31, 1.47]. And CD has to DE
Εὕρηνται ἄρα δύο ῥηταὶ δυνάμει μόνον σύμμετροι αἱ ΒΑ, the ratio which (some) square number (has) to (some)
ΑΖ, ὥστε τὴν μεῖζονα τὴν ΑΒ τῆς ἐλάσσονος τῆς ΑΖ μεῖζον square number. Thus, the (square) on AB also has to the
δύνασθαι τῷ ἀπὸ τῆς ΒΖ συμμέτρου ἑαυτῇ μήκει· ὅπερ ἔδει (square) on BF the ratio which (some) square number
δεῖξαι. has to (some) square number. AB is thus commensu-
rable in length with BF [Prop. 10.9]. And the (square)
on AB is equal to the (sum of the squares) on AF and
FB [Prop. 1.47]. Thus, the square on AB is greater than
(the square on) AF by (the square on) BF, (which is)
commensurable (in length) with (AB).
lþ square on the greater, AB, is larger than (the square on)
Thus, two rational (straight-lines), BA and AF, com-
mensurable in square only, have been found such that the

the lesser, AF, by the (square) on BF, (which is) com-
†
mensurable in length with (AB). (Which is) the very
thing it was required to show.

† BA and AF have lengths 1 and √ 1 − k times that of AB, respectively, where k = p DE/CD.
2
Proposition 30
.
Εὑρεῖν δύο ῥητὰς δυνάμει μόνον συμμέτρους, ὥστε τὴν To find two rational (straight-lines which are) com-
μείζονα τῆς ἐλάσσονος μεῖζον δύνασθαι τῷ ἀπὸ ἀσυμμέτρου mensurable in square only, such that the square on the
ἑαυτῇ μήκει. greater is larger than the (the square on) lesser by the
(square) on (some straight-line which is) incommensu-
rable in length with the greater.
Ζ F

Α Β A B

Γ Ε ∆ C E D
᾿Εκκείσθω ῥητὴ ἡ ΑΒ καὶ δύο τετράγωνοι ἀριθμοὶ οἱ Let the rational (straight-line) AB be laid out, and the
ΓΕ, ΕΔ, ὥστε τὸν συγκείμενον ἐξ αὐτῶν τὸν ΓΔ μὴ εἶναι two square numbers, CE and ED, such that the sum of
τετράγωνον, καὶ γεγράφθω ἐπὶ τῆς ΑΒ ἡμικύκλιον τὸ ΑΖΒ, them, CD, is not square [Prop. 10.28 lem. II]. And let the
καὶ πεποιήσθω ὡς ὁ ΔΓ πρὸς τὸν ΓΕ, οὕτως τὸ ἀπὸ τῆς semi-circle AFB have been drawn on AB. And let it be
ΒΑ πρὸς τὸ ἀπὸ τῆς ΑΖ, καὶ ἐπεζεύχθω ἡ ΖΒ. contrived that as DC (is) to CE, so the (square) on BA
῾Ομοίως δὴ δείξομεν τῷ πρὸ τούτου, ὅτι αἱ ΒΑ, ΑΖ (is) to the (square) on AF [Prop. 10.6 corr]. And let FB
ῥηταί εἰσι δυνάμει μόνον σύμμετροι. καὶ ἐπεί ἐστιν ὡς ὁ have been joined.
ΔΓ πρὸς τὸν ΓΕ, οὕτως τὸ ἀπὸ τῆς ΒΑ πρὸς τὸ ἀπὸ τῆς So, similarly to the (proposition) before this, we can
ΑΖ, ἀναστρέψαντι ἄρα ὡς ὁ ΓΔ πρὸς τὸν ΔΕ, οὕτως τὸ show that BA and AF are rational (straight-lines which
ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΖ. ὁ δὲ ΓΔ πρὸς τὸν ΔΕ are) commensurable in square only. And since as DC is
λόγον οὐκ ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον to CE, so the (square) on BA (is) to the (square) on AF,
ἀριθμόν· οὐδ᾿ ἄρα τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΖ λόγον thus, via conversion, as CD (is) to DE, so the (square) on
ἔχει, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν· AB (is) to the (square) on BF [Props. 5.19 corr., 3.31,
ἀσύμμετρος ἄρα ἐστὶν ἡ ΑΒ τῇ ΒΖ μήκει. καὶ δύναται ἡ 1.47]. And CD does not have to DE the ratio which
ΑΒ τῆς ΑΖ μεῖζον τῷ ἀπὸ τῆς ΖΒ ἀσυμμέτρου ἑαυτῇ. (some) square number (has) to (some) square number.

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