Page 353 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ἀπὸ συμμέτρου ἑαυτῇ. καί ἐστιν ἡ ΜΗ σύμμετρος τῇ ΔΕ rable in square only. DG is thus a binomial (straight-line)
μήκει. [Prop. 10.36]. So, we must show that (it is) also a second
῾Η ΔΗ ἄρα ἐκ δύο ὀνομάτων ἐστὶ δευτέρα. (binomial straight-line).
For since (the sum of) the squares on AC and CB is
greater than twice the (rectangle contained) by AC and
CB [Prop. 10.59], DL (is) thus also greater than MF.
Hence, DM (is) also (greater) than MG [Prop. 6.1].
And since the (square) on AC is commensurable with
the (square) on CB, DH is also commensurable with
KL. Hence, DK is also commensurable (in length) with
KM [Props. 6.1, 10.11]. And the (rectangle contained)
by DKM is equal to the (square) on MN. Thus, the
xbþ length) with (DM) [Prop. 10.17]. And MG is commen-
square on DM is greater than (the square on) MG by
the (square) on (some straight-line) commensurable (in

surable in length with DE.
Thus, DG is a second binomial (straight-line) [Def.
10.6].

† In other words, the square of a first bimedial is a second binomial. See Prop. 10.55.

.
Proposition 62
Τὸ ἀπὸ τῆς ἐκ δύο μέσων δευτέρας παρὰ ῥητὴν παρα- The square on a second bimedial (straight-line) ap-
βαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων τρίτην. plied to a rational (straight-line) produces as breadth a
third binomial (straight-line). †
∆ Κ Μ Ν Η D K M N G

Ε Θ Λ Ξ Ζ E H L O F

Α Γ Β A C B
῎Εστω ἐκ δύο μέσων δευτέρα ἡ ΑΒ διῃρημένη εἰς τὰς Let AB be a second bimedial (straight-line) having
μέσας κατὰ τὸ Γ, ὥστε τὸ μεῖζον τμῆμα εἶναι τὸ ΑΓ, ῥητὴ been divided into its (component) medial (straight-lines)
δέ τις ἔστω ἡ ΔΕ, καὶ παρὰ τὴν ΔΕ τῷ ἀπὸ τῆς ΑΒ ἴσον at C, such that AC is the greater segment. And let DE be
παραλληλόγραμμον παραβεβλήσθω τὸ ΔΖ πλάτος ποιοῦν some rational (straight-line). And let the parallelogram
τὴν ΔΗ· λέγω, ὅτι ἡ ΔΗ ἐκ δύο ὀνομάτων ἐστὶ τρίτη. DF, equal to the (square) on AB, have been applied to
Κατεσκευάσθω τὰ αὐτὰ τοῖς προδεδειγμένοις. καὶ ἐπεὶ DE, producing DG as breadth. I say that DG is a third
ἐκ δύο μέσων δευτέρα ἐστὶν ἡ ΑΒ διῃρημένη κατὰ τὸ Γ, binomial (straight-line).
αἱ ΑΓ, ΓΒ ἄρα μέσαι εἰσὶ δυνάμει μόνον σύμμετροι μέσον Let the same construction be made as that shown pre-
περιέχουσαι· ὥστε καὶ τὸ συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, viously. And since AB is a second bimedial (straight-
ΓΒ μέσον ἐστίν. καί ἐστιν ἴσον τῷ ΔΛ· μέσον ἄρα καὶ τὸ line), having been divided at C, AC and CB are thus
ΔΛ. καὶ παράκειται παρὰ ῥητὴν τὴν ΔΕ· ῥητὴ ἄρα ἐστὶ καὶ ἡ medial (straight-lines) commensurable in square only,
ΜΔ καὶ ἀσύμμετρος τῇ ΔΕ μήκει. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΜΗ and containing a medial (area) [Prop. 10.38]. Hence,
ῥητή ἐστι καὶ ἀσύμμετρος τῇ ΜΛ, τουτέστι τῇ ΔΕ, μήκει· the sum of the (squares) on AC and CB is also medial
ῥητὴ ἄρα ἐστὶν ἑκατέρα τῶν ΔΜ, ΜΗ καὶ ἀσύμμετρος τῇ [Props. 10.15, 10.23 corr.]. And it is equal to DL. Thus,
ΔΕ μήκει. καὶ ἐπεὶ ἀσύμμετρός ἐστιν ἡ ΑΓ τῇ ΓΒ μήκει, DL (is) also medial. And it is applied to the rational
ὡς δὲ ἡ ΑΓ πρὸς τὴν ΓΒ, οὕτως τὸ ἀπὸ τῆς ΑΓ πρὸς τὸ (straight-line) DE. MD is thus also rational, and in-

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