Page 314 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

Αἱ ΑΒ, ΑΖ ἄρα ῥηταί εἰσι δυνάμει μόνον σύμμετροι, καὶ Thus, the (square) on AB does not have to the (square)
ἡ ΑΒ τῆς ΑΖ μεῖζον δύναται τῷ ἀπὸ τῆς ΖΒ ἀσυμμέτρου on BF the ratio which (some) square number has to
ἑαυτῇ μήκει· ὅπερ ἔδει δεῖξαι. (some) square number either. Thus, AB is incommensu-
rable in length with BF [Prop. 10.9]. And the square on
AB is greater than the (square on) AF by the (square) on
FB [Prop. 1.47], (which is) incommensurable (in length)
laþ are) commensurable in square only, and the square on †
with (AB).
Thus, AB and AF are rational (straight-lines which

AB is greater than (the square on) AF by the (square) on
FB, (which is) incommensurable (in length) with (AB).
(Which is) the very thing it was required to show.
√
† AB and AF have lengths 1 and 1/ 1 + k times that of AB, respectively, where k = p DE/CE.
2
Proposition 31
.
Εὑρεῖν δύο μέσας δυνάμει μόνον συμμέτρους ῥητὸν To find two medial (straight-lines), commensurable in
περιεχούσας, ὥστε τὴν μείζονα τῆς ἐλάσσονος μεῖζον square only, (and) containing a rational (area), such that
δύνασθαι τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει. the square on the greater is larger than the (square on
the) lesser by the (square) on (some straight-line) com-
mensurable in length with the greater.

Α Β Γ ∆ A B C D
᾿Εκκείσθωσαν δύο ῥηταὶ δυνάμει μόνον σύμμετροι αἱ Α, Let two rational (straight-lines), A and B, commensu-
Β, ὥστε τὴν Α μείζονα οὖσαν τῆς ἐλάσσονος τῆς Β μεῖζον rable in square only, be laid out, such that the square on
δύνασθαι τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει. καὶ τῷ ὑπὸ τῶν the greater A is larger than the (square on the) lesser B
Α, Β ἴσον ἔστω τὸ ἀπὸ τῆς Γ. μέσον δὲ τὸ ὑπὸ τῶν Α, Β· by the (square) on (some straight-line) commensurable
μέσον ἄρα καὶ τὸ ἀπὸ τῆς Γ· μέση ἄρα καὶ ἡ Γ. τῷ δὲ ἀπὸ in length with (A) [Prop. 10.29]. And let the (square)
τῆς Β ἴσον ἔστω τὸ ὑπὸ τῶν Γ, Δ· ῥητὸν δὲ τὸ ἀπὸ τῆς Β· on C be equal to the (rectangle contained) by A and B.
ῥητὸν ἄρα καὶ τὸ ὑπὸ τῶν Γ, Δ. καὶ ἐπεί ἐστιν ὡς ἡ Α πρὸς And the (rectangle contained by) A and B (is) medial
τὴν Β, οὕτως τὸ ὑπὸ τῶν Α, Β πρὸς τὸ ἀπὸ τῆς Β, ἀλλὰ τῷ [Prop. 10.21]. Thus, the (square) on C (is) also medial.
μὲν ὑπὸ τῶν Α, Β ἴσον ἐστὶ τὸ ἀπὸ τῆς Γ, τῷ δὲ ἀπὸ τῆς Thus, C (is) also medial [Prop. 10.21]. And let the (rect-
Β ἴσον τὸ ὑπὸ τῶν Γ, Δ, ὡς ἄρα ἡ Α πρὸς τὴν Β, οὕτως angle contained) by C and D be equal to the (square)
τὸ ἀπὸ τῆς Γ πρὸς τὸ ὑπὸ τῶν Γ, Δ. ὡς δὲ τὸ ἀπὸ τῆς Γ on B. And the (square) on B (is) rational. Thus, the
πρὸς τὸ ὑπὸ τῶν Γ, Δ, οὕτως ἡ Γ πρὸς τὴν Δ· καὶ ὡς ἄρα (rectangle contained) by C and D (is) also rational. And
ἡ Α πρὸς τὴν Β, οὕτως ἡ Γ πρὸς τὴν Δ. σύμμετρος δὲ ἡ Α since as A is to B, so the (rectangle contained) by A and
τῇ Β δυνάμει μόνον· σύμμετρος ἄρα καὶ ἡ Γ τῇ Δ δυνάμει B (is) to the (square) on B [Prop. 10.21 lem.], but the
μόνον. καί ἐστι μέση ἡ Γ· μέση ἄρα καὶ ἡ Δ. καὶ ἐπεί ἐστιν (square) on C is equal to the (rectangle contained) by
ὡς ἡ Α πρὸς τὴν Β, ἡ Γ πρὸς τὴν Δ, ἡ δὲ Α τῆς Β μεῖζον A and B, and the (rectangle contained) by C and D to
δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ, καὶ ἡ Γ ἄρα τῆς Δ μεῖζον the (square) on B, thus as A (is) to B, so the (square)
δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ. on C (is) to the (rectangle contained) by C and D. And
Εὕρηνται ἄρα δύο μέσαι δυνάμει μόνον σύμμετροι αἱ Γ, as the (square) on C (is) to the (rectangle contained) by

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