Page 333 - Euclid's Elements of Geometry

ST EW iþ. ELEMENTS BOOK 10

Vroi deÔteroi its component terms) at different points. Thus, it can be

(so) divided at one [point] only.

q q q
† In other words, k ′1/4 [1 + k/(1 + k ) ]/2 + k ′1/4 [1 − k/(1 + k ) ]/2 = k ′′′1/4 [1 + k /(1 + k ′′2 1/2 ]/2
2 1/2
′′
2 1/2
)
q
′
′′
)
′′
+k ′′′1/4 [1 − k /(1 + k ′′2 1/2 ]/2 has only one solution: i.e., k = k and k ′′′ = k .
Definitions II
.
εʹ. ῾Υποκειμένης ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων 5. Given a rational (straight-line), and a binomial
διῃρημένης εἰς τὰ ὀνόματα, ἧς τὸ μεῖζον ὄνομα τοῦ (straight-line) which has been divided into its (compo-
ἐλάσσονος μεῖζον δύναται τῷ ἀπὸ συμμέτρου ἑαυτῇ μήκει, nent) terms, of which the square on the greater term is
ἐὰν μὲν τὸ μεῖζον ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ larger than (the square on) the lesser by the (square)
ῥητῇ, καλείσθω [ἡ ὅλη] ἐκ δύο ὀνομάτων πρώτη. on (some straight-line) commensurable in length with
ϛʹ. ᾿Εὰν δὲ τὸ ἐλάσσον ὄνομα σύμμετρον ᾖ μήκει τῇ (the greater) then, if the greater term is commensurable
ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων δευτέρα. in length with the rational (straight-line previously) laid
ζʹ. ᾿Εὰν δὲ μηδέτερον τῶν ὀνομάτων σύμμετρον ᾖ μήκει out, let [the whole] (straight-line) be called a first bino-
τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ δύο ὀνομάτων τρίτη. mial (straight-line).
ηʹ. Πάλιν δὴ ἐὰν τὸ μεῖζον ὄνομα [τοῦ ἐλάσσονος] μεῖζον 6. And if the lesser term is commensurable in length
δύνηται τῷ ἀπὸ ἀσυμμέτρου ἑαυτῇ μήκει, ἐὰν μὲν τὸ μεῖζον with the rational (straight-line previously) laid out then
ὄνομα σύμμετρον ᾖ μήκει τῇ ἐκκειμένῃ ῥητῇ, καλείσθω ἐκ let (the whole straight-line) be called a second binomial
δύο ὀνομάτων τετάρτη. (straight-line).
θʹ. ᾿Εὰν δὲ τὸ ἔλασσον, πέμπτη. 7. And if neither of the terms is commensurable in
ιʹ. ᾿Εὰν δὲ μηδέτερον, ἕκτη. length with the rational (straight-line previously) laid out
then let (the whole straight-line) be called a third bino-
mial (straight-line).
8. So, again, if the square on the greater term is
larger than (the square on) [the lesser] by the (square)
on (some straight-line) incommensurable in length with
mhþ out, let (the whole straight-line) be called a fourth bino-
(the greater) then, if the greater term is commensurable
in length with the rational (straight-line previously) laid

mial (straight-line).
9. And if the lesser (term is commensurable), a fifth
(binomial straight-line).
10. And if neither (term is commensurable), a sixth
(binomial straight-line).

.
Proposition 48
Εὑρεῖν τὴν ἐκ δύο ὀνομάτων πρώτην. To find a first binomial (straight-line).
᾿Εκκείσθωσαν δύο ἀριθμοὶ οἱ ΑΓ, ΓΒ, ὥστε τὸν Let two numbers AC and CB be laid down such that
συγκείμενον ἐξ αὐτῶν τὸν ΑΒ πρὸς μὲν τὸν ΒΓ λόγον their sum AB has to BC the ratio which (some) square
ἔχειν, ὃν τετράγωνος ἀριθμὸς πρὸς τετράγωνον ἀριθμόν, number (has) to (some) square number, and does not
πρὸς δὲ τὸν ΓΑ λόγον μὴ ἔχειν, ὃν τετράγωνος ἀριθμὸς have to CA the ratio which (some) square number (has)
πρὸς τετράγωνον ἀριθμόν, καὶ ἐκκείσθω τις ῥητὴ ἡ Δ, καὶ to (some) square number [Prop. 10.28 lem. I]. And let
τῇ Δ σύμμετρος ἔστω μήκει ἡ ΕΖ. ῥητὴ ἄρα ἐστὶ καὶ ἡ some rational (straight-line) D be laid down. And let EF
ΕΖ. καὶ γεγονέτω ὡς ὁ ΒΑ ἀριθμὸς πρὸς τὸν ΑΓ, οὕτως be commensurable in length with D. EF is thus also ra-
τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΖΗ. ὁ δὲ ΑΒ πρὸς τὸν tional [Def. 10.3]. And let it have been contrived that as
ΑΓ λόγον ἔχει, ὃν ἀριθμὸς πρὸς ἀριθμόν· καὶ τὸ ἀπὸ τῆς the number BA (is) to AC, so the (square) on EF (is)
ΕΖ ἄρα πρὸς τὸ ἀπὸ τῆς ΖΗ λόγον ἔχει, ὃν ἀριθμὸς πρὸς to the (square) on FG [Prop. 10.6 corr.]. And AB has to
ἀριθμόν· ὥστε σύμμετρόν ἐστι τὸ ἀπὸ τῆς ΕΖ τῷ ἀπὸ τῆς AC the ratio which (some) number (has) to (some) num-

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