Page 411 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ΑΒ ἔστω σύμμετρος ἡ ΓΔ· λέγω, ὅτι καὶ ἡ ΓΔ μετὰ μέσου makes a medial whole, and let CD be commensurable (in
μέσον τὸ ὅλον ποιοῦσά ἐστιν. length) with AB. I say that CD is also a (straight-line)
῎Εστω γὰρ τῇ ΑΒ προσαρμόζουσα ἡ ΒΕ, καὶ τὰ αὐτὰ which with a medial (area) makes a medial whole.
κατεσκευάσθω· αἱ ΑΕ, ΕΒ ἄρα δυνάμει εἱσὶν ἀσύμμετροι For let BE be an attachment to AB. And let the same
ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπ᾿ αὐτῶν τετραγώνων construction have been made (as in the previous propo-
μέσον καὶ τὸ ὑπ᾿ αὐτῶν μέσον καὶ ἔτι ἀσύμμετρον τὸ sitions). Thus, AE and EB are (straight-lines which
συγκέιμενον ἐκ τῶν ἀπ᾿ αὐτῶν τετραγώνων τῷ ὑπ᾿ αὐτῶν. are) incommensurable in square, making the sum of the
καί εἰσιν, ὡς ἐδείχθη, αἱ ΑΕ, ΕΒ σύμμετροι ταῖς ΓΖ, ΖΔ, squares on them medial, and the (rectangle contained)
καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τετραγώνων by them medial, and, further, the sum of the squares on
τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ, τὸ δὲ ὑπὸ τῶν them incommensurable with the (rectangle contained) by
ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· καὶ αἱ ΓΖ, ΖΔ ἄρα δυνάμει them [Prop. 10.78]. And, as was shown (previously), AE
εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπ᾿ and EB are commensurable (in length) with CF and FD
αὐτῶν τετραγώνων μέσον καὶ τὸ ὑπ᾿ ἀὐτῶν μέσον καὶ (respectively), and the sum of the squares on AE and
ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπ᾿ αὐτῶν [τε- EB with the sum of the squares on CF and FD, and
τραγώνων] τῷ ὑπ᾿ αὐτῶν. the (rectangle contained) by AE and EB with the (rect-
῾Η ΓΔ ἄρα μετὰ μέσου μέσον τὸ ὅλον ποιοῦσά ἐστιν· angle contained) by CF and FD. Thus, CF and FD
rhþ ther, the sum of the [squares] on them incommensurable
ὅπερ ἔδει δεῖξαι. are also (straight-lines which are) incommensurable in
square, making the sum of the squares on them medial,
and the (rectangle contained) by them medial, and, fur-

with the (rectangle contained) by them.
Thus, CD is a (straight-line) which with a medial
(area) makes a medial whole [Prop. 10.78]. (Which is)
the very thing it was required to show.

Proposition 108
.
᾿Απὸ ῥητοῦ μέσου ἀφαιρουμένου ἡ τὸ λοιπὸν χωρίον A medial (area) being subtracted from a rational
δυναμένη μία δύο ἀλόγων γίνεται ἤτοι ἀποτομὴ ἢ ἐλάσσων. (area), one of two irrational (straight-lines) arise (as) the
square-root of the remaining area—either an apotome, or
a minor (straight-line).
Α Ε Β A E B

L G
Λ Η

Θ Κ Ζ H K F

Γ ∆ C D
᾿Απὸ γὰρ ῥητοῦ τοῦ ΒΓ μέσον ἀφῃρήσθω τὸ ΒΔ· λέγω, For let the medial (area) BD have been subtracted
ὅτι ἡ τὸ λοιπὸν δυναμένη τὸ ΕΓ μία δύο ἀλόγων γίνεται from the rational (area) BC. I say that one of two ir-
ἤτοι ἀποτομὴ ἢ ἐλάσσων. rational (straight-lines) arise (as) the square-root of the
᾿Εκκείσθω γὰρ ῥητὴ ἡ ΖΗ, καὶ τῷ μὲν ΒΓ ἴσον παρὰ τὴν remaining (area), EC—either an apotome, or a minor
ΖΗ παραβεβλήσθω ὀρθογώνιον παραλληλόγραμμον τὸ ΗΘ, (straight-line).
τῷ δὲ ΔΒ ἴσον ἀφῃρήσθω τὸ ΗΚ· λοιπὸν ἄρα τὸ ΕΓ ἴσον For let the rational (straight-line) FG have been laid
ἐστὶ τῷ ΛΘ. ἐπεὶ οὖν ῥητὸν μέν ἐστι τὸ ΒΓ, μέσον δὲ τὸ out, and let the right-angled parallelogram GH, equal to
ΒΔ, ἴσον δὲ τὸ μὲν ΒΓ τῷ ΗΘ, τὸ δὲ ΒΔ τῷ ΗΚ, ῥητὸν μὲν BC, have been applied to FG, and let GK, equal to DB,
ἄρα ἐστὶ τὸ ΗΘ, μέσον δὲ τὸ ΗΚ. καὶ παρὰ ῥητὴν τὴν ΖΗ have been subtracted (from GH). Thus, the remainder
παράκειται· ῥητὴ μὲν ἄρα ἡ ΖΘ καὶ σύμμετρος τῇ ΖΗ μήκει, EC is equal to LH. Therefore, since BC is a rational
ῥητὴ δὲ ἡ ΖΚ καὶ ἀσύμμετρος τῇ ΖΗ μήκει· ἀσύμμετρος ἄρα (area), and BD a medial (area), and BC (is) equal to

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