Page 399 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ποιοῦν τὴν ΓΖ· λέγω, ὅτι ἡ ΓΖ ἀποτομή ἐστι δευτέρα. and CD a rational (straight-line). And let CE, equal to
῎Εστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΗ· αἱ ἄρα ΑΗ, the (square) on AB, have been applied to CD, producing
ΗΒ μέσαι εἰσὶ δυνάμει μόνον σύμμετροι ῥητὸν περιέχουσαι. CF as breadth. I say that CF is a second apotome.
καὶ τῷ μὲν ἀπὸ τῆς ΑΗ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω For let BG be an attachment to AB. Thus, AG and
τὸ ΓΘ πλάτος ποιοῦν τὴν ΓΚ, τῷ δὲ ἀπὸ τῆς ΗΒ ἴσον τὸ GB are medial (straight-lines which are) commensurable
ΚΛ πλάτος ποιοῦν τὴν ΚΜ· ὅλον ἄρα τὸ ΓΛ ἴσον ἐστὶ τοῖς in square only, containing a rational (area) [Prop. 10.74].
ἀπὸ τῶν ΑΗ, ΗΒ· μέσον ἄρα καὶ τὸ ΓΛ. καὶ παρὰ ῥητὴν And let CH, equal to the (square) on AG, have been ap-
τὴν ΓΔ παράκειται πλάτος ποιοῦν τὴν ΓΜ· ῥητὴ ἄρα ἐστὶν plied to CD, producing CK as breadth, and KL, equal
ἡ ΓΜ καὶ ἀσύμμετρος τῇ ΓΔ μήκει. καὶ ἐπεὶ τὸ ΓΛ ἴσον to the (square) on GB, producing KM as breadth. Thus,
ἐστὶ τοῖς ἀπὸ τῶν ΑΗ, ΗΒ, ὧν τὸ ἀπὸ τῆς ΑΒ ἴσον ἐστὶ the whole of CL is equal to the (sum of the squares)
τῷ ΓΕ, λοιπὸν ἄρα τὸ δὶς ὑπὸ τῶν ΑΗ, ΗΒ ἴσον ἐστὶ τῷ on AG and GB. Thus, CL (is) also a medial (area)
ΖΛ. ῥητὸν δέ [ἐστι] τὸ δὶς ὑπὸ τῶν ΑΗ, ΗΒ· ῥητὸν ἄρα τὸ [Props. 10.15, 10.23 corr.]. And it is applied to the ratio-
ΖΛ. καὶ παρὰ ῥητὴν τὴν ΖΕ παράκειται πλάτος ποιοῦν τὴν nal (straight-line) CD, producing CM as breadth. CM
ΖΜ· ῥητὴ ἄρα ἐστὶ καὶ ἡ ΖΜ καὶ σύμμετρος τῇ ΓΔ μήκει. is thus rational, and incommensurable in length with CD
ἐπεὶ οὖν τὰ μὲν ἀπὸ τῶν ΑΗ, ΗΒ, τουτέστι τὸ ΓΛ, μέσον [Prop. 10.22]. And since CL is equal to the (sum of the
ἐστίν, τὸ δὲ δὶς ὑπὸ τῶν ΑΗ, ΗΒ, τουτέστι τὸ ΖΛ, ῥητόν squares) on AG and GB, of which the (square) on AB
ἀσύμμετρον ἄρα ἐστὶ τὸ ΓΛ τῷ ΖΛ. ὡς δὲ τὸ ΓΛ πρὸς τὸ is equal to CE, the remainder, twice the (rectangle con-
ΖΛ, οὕτως ἐστὶν ἡ ΓΜ πρὸς τὴν ΖΜ· ἀσύμμετρος ἄρα ἡ tained) by AG and GB, is thus equal to FL [Prop. 2.7].
ΓΜ τῇ ΖΜ μήκει. καί εἰσιν ἀμφότεραι ῥηταί· αἱ ἄρα ΓΜ, And twice the (rectangle contained) by AG and GB [is]
ΜΖ ῥηταί εἰσι δυνάμει μόνον σύμμετροι· ἡ ΓΖ ἄρα ἀποτομή rational. Thus, FL (is) rational. And it is applied to the
ἐστιν. λέγω δή, ὅτι καὶ δευτέρα. rational (straight-line) FE, producing FM as breadth.
FM is thus also rational, and commensurable in length
with CD [Prop. 10.20]. Therefore, since the (sum of the
squares) on AG and GB—that is to say, CL—is medial,
and twice the (rectangle contained) by AG and GB—
that is to say, FL—(is) rational, CL is thus incommen-
surable with FL. And as CL (is) to FL, so CM is to
FM [Prop. 6.1]. Thus, CM (is) incommensurable in
length with FM [Prop. 10.11]. And they are both ra-
tional (straight-lines). Thus, CM and MF are rational
(straight-lines which are) commensurable in square only.
G A Z B H CF is thus an apotome [Prop. 10.73]. So, I say that (it

is) also a second (apotome).
G
B
A
C
F
N K
M
D E X

H
O
E
D
Τετμήσθω γὰρ ἡ ΖΜ δίχα κατὰ τὸ Ν, καὶ ἤχθω διὰ τοῦ For let FM have been cut in half at N. L And let
Ν τῇ ΓΔ παράλληλος ἡ ΝΞ· ἑκάτερον ἄρα τῶν ΖΞ, ΝΛ ἴσον NO have been drawn through (point) N, parallel to
ἐστὶ τῷ ὑπὸ τῶν ΑΗ, ΗΒ. καὶ ἐπεὶ τῶν ἀπὸ τῶν ΑΗ, ΗΒ CD. Thus, FO and NL are each equal to the (rectan-
τετραγώνων μέσον ἀνάλογόν ἐστι τὸ ὑπὸ τῶν ΑΗ, ΗΒ, καί gle contained) by AG and GB. And since the (rectan-
ἐστιν ἴσον τὸ μὲν ἀπὸ τῆς ΑΗ τῷ ΓΘ, τὸ δὲ ὑπὸ τῶν ΑΗ, gle contained) by AG and GB is the mean proportional
ΗΒ τῷ ΝΛ, τὸ δὲ ἀπὸ τῆς ΒΗ τῷ ΚΛ, καὶ τῶν ΓΘ, ΚΛ to the squares on AG and GB [Prop. 10.21 lem.], and
ἄρα μέσον ἀνάλογόν ἐστι τὸ ΝΛ· ἔστιν ἄρα ὡς τὸ ΓΘ πρὸς the (square) on AG is equal to CH, and the (rectangle
τὸ ΝΛ, οὕτως τὸ ΝΛ πρὸς τὸ ΚΛ. ἀλλ᾿ ὡς μὲν τὸ ΓΘ πρὸς contained) by AG and GB to NL, and the (square) on

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