Page 362 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ΖΔ ῥητόν. EB (is) commensurable with the sum of the (squares)
oþ dial, and the (rectangle contained) by CF and FD (is)
῾Ρητὸν ἄρα καὶ μέσον δυναμένη ἐστὶν ἡ ΓΔ· ὅπερ ἔδει on CF and FD, and the (rectangle contained) by AE
δεῖξαι. and EB with the (rectangle contained) by CF and FD.
And hence the sum of the squares on CF and FD is me-

rational.
Thus, CD is the square-root of a rational plus a medial
(area) [Prop. 10.40]. (Which is) the very thing it was
required to show.
Proposition 70
.
῾Η τῇ δύο μέσα δυναμένῃ σύμμετρος δύο μέσα δυναμένη A (straight-line) commensurable (in length) with the
ἐστίν. square-root of (the sum of) two medial (areas) is (itself
also) the square-root of (the sum of) two medial (areas).
Α Ε Β A E B

Γ Ζ ∆ C F D
῎Εστω δύο μέσα δυναμένη ἡ ΑΒ, καὶ τῇ ΑΒ σύμμετρος Let AB be the square-root of (the sum of) two medial
ἡ ΓΔ· δεικτέον, ὅτι καὶ ἡ ΓΔ δύο μέσα δυναμένη ἐστίν. (areas), and (let) CD (be) commensurable (in length)
᾿Επεὶ γὰρ δύο μέσα δυναμένη ἐστὶν ἡ ΑΒ, διῃρήσθω with AB. We must show that CD is also the square-root
εἰς τὰς εὐθείας κατὰ τὸ Ε· αἱ ΑΕ, ΕΒ ἄρα δυνάμει of (the sum of) two medial (areas).
εἰσὶν ἀσύμμετροι ποιοῦσαι τό τε συγκείμενον ἐκ τῶν ἀπ᾿ For since AB is the square-root of (the sum of) two
αὐτῶν [τετραγώνων] μέσον καὶ τὸ ὑπ᾿ αὐτῶν μέσον καὶ medial (areas), let it have been divided into its (compo-
ἔτι ἀσύμμετρον τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ nent) straight-lines at E. Thus, AE and EB are incom-
τετραγώνων τῷ ὑπὸ τῶν ΑΕ, ΕΒ· καὶ κατεσκευάσθω τὰ mensurable in square, making the sum of the [squares]
αὐτὰ τοῖς πρότερον. ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ΓΖ, ΖΔ on them medial, and the (rectangle contained) by them
δυνάμει εἰσὶν ἀσύμμετροι καὶ σύμμετρον τὸ μὲν συγκείμενον medial, and, moreover, the sum of the (squares) on AE
ἐκ τῶν ἀπὸ τῶν ΑΕ, ΕΒ τῷ συγκειμένῳ ἐκ τῶν ἀπὸ τῶν ΓΖ, and EB incommensurable with the (rectangle) contained
ΖΔ, τὸ δὲ ὑπὸ τῶν ΑΕ, ΕΒ τῷ ὑπὸ τῶν ΓΖ, ΖΔ· ὥστε καὶ by AE and EB [Prop. 10.41]. And let the same construc-
τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων μέσον tion have been made as in the previous (propositions).
ἐστὶ καὶ τὸ ὑπὸ τῶν ΓΖ, ΖΔ μέσον καὶ ἔτι ἀσύμμετρον τὸ So, similarly, we can show that CF and FD are also
συγκείμενον ἐκ τῶν ἀπὸ τῶν ΓΖ, ΖΔ τετραγώνων τῷ ὑπὸ incommensurable in square, and (that) the sum of the
τῶν ΓΖ, ΖΔ. (squares) on AE and EB (is) commensurable with the
῾Η ἄρα ΓΔ δύο μέσα δυναμένη ἐστίν· ὅπερ ἔδει δεῖξαι. sum of the (squares) on CF and FD, and the (rectangle
contained) by AE and EB with the (rectangle contained)
oaþ CF and FD (is) medial, and, moreover, the sum of the
by CF and FD. Hence, the sum of the squares on CF
and FD is also medial, and the (rectangle contained) by

squares on CF and FD (is) incommensurable with the
(rectangle contained) by CF and FD.
Thus, CD is the square-root of (the sum of) two me-
dial (areas) [Prop. 10.41]. (Which is) the very thing it
was required to show.

Proposition 71
.
῾Ρητοῦ καὶ μέσου συντιθεμένου τέσσαρες ἄλογοι γίγνον- When a rational and a medial (area) are added to-
ται ἤτοι ἐκ δύο ὀνομάτων ἢ ἐκ δύο μέσων πρώτη ἢ μείζων gether, four irrational (straight-lines) arise (as the square-
ἢ ῥητὸν καὶ μέσον δυναμένη. roots of the total area)—either a binomial, or a first bi-

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