Page 366 - Euclid's Elements of Geometry

ST EW iþ.

ELEMENTS BOOK 10

ἀλλα δὴ ἡ ΕΘ τῆς ΘΚ μεῖζον δυνάσθω τῷ ἀπὸ ἀσυμμέτρου on (some straight-line) commensurable (in length) with
ἑαυτῇ μήκει· καὶ ἀσύμμετρός ἐστιν ἑκατέρα τῶν ΕΘ, ΘΚ (EH), or by the (square) on (some straight-line) incom-
τῇ ΕΖ μήκει· ἡ ἄρα ΕΚ ἐκ δύο ὀνομάτων ἐστὶν ἕκτη. ἐὰν mensurable (in length with EH). Let it, first of all, be
δὲ χωρίον περιέχηται ὑπὸ ῥητῆς καὶ τῆς ἐκ δύο ὀνομάτων greater by the square on (some straight-line) commensu-
ἕκτης, ἡ τὸ χωρίον δυναμένη ἡ δύο μέσα δυναμένη ἐστίν· rable in length with (EH). And neither of EH or HK is
ὥστε καὶ ἡ τὸ ΑΔ χωρίον δυναμένη ἡ δύο μέσα δυναμένη commensurable in length with the (previously) laid down
ἐστίν. rational (straight-line) EF. Thus, EK is a third binomial
[῾Ομοίως δὴ δείξομεν, ὅτι κἂν ἔλαττον ᾖ τὸ ΑΒ τοῦ ΓΔ, (straight-line) [Def. 10.7]. And EF (is) rational. And if
ἡ τὸ ΑΔ χωρίον δυναμένη ἢ ἐκ δύο μέσων δευτέρα ἐστὶν an area is contained by a rational (straight-line) and a
ἤτοι δύο μέσα δυναμένη]. third binomial (straight-line) then the square-root of the
Δύο ἄρα μέσων ἀσυμμέτρων ἀλλήλοις συντιθεμένων αἱ area is a second bimedial (straight-line) [Prop. 10.56].
λοιπαὶ δύο ἄλογοι γίγνονται ἤτοι ἐκ δύο μέσων δευτέρα ἢ Thus, the square-root of EI—that is to say, of AD—
δύο μέσα δυναμένη. is a second bimedial. And so, let the square on EH
be greater than (the square) on HK by the (square)
᾿Η ἐκ δύο ὀνομάτων καὶ αἱ μετ᾿ αὐτὴν ἄλογοι οὔτε on (some straight-line) incommensurable in length with
τῇ μέσῃ οὔτε ἀλλήλαις εἰσὶν αἱ αὐταί. τὸ μὲν γὰρ ἀπὸ (EH). And EH and HK are each incommensurable in
μέσης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ῥητὴν καὶ length with EF. Thus, EK is a sixth binomial (straight-
ἀσύμμετρον τῇ παρ᾿ ἣν παράκειται μήκει. τὸ δὲ ἀπὸ τῆς line) [Def. 10.10]. And if an area is contained by a ra-
ἐκ δύο ὀνομάτων παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ tional (straight-line) and a sixth binomial (straight-line)
τὴν ἐκ δύο ὀνομάτων πρώτην. τὸ δὲ ἀπὸ τῆς ἐκ δύο then the square-root of the area is the square-root of (the
μέσων πρώτης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ sum of) two medial (areas) [Prop. 10.59]. Hence, the
τὴν ἐκ δύο ὀνομάτων δευτέραν. τὸ δὲ ἀπὸ τῆς ἐκ δύο square-root of area AD is also the square-root of (the
μέσων δευτέρας παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ sum of) two medial (areas).
τὴν ἐκ δύο ὀνομάτων τρίτην. τὸ δὲ ἀπὸ τῆς μείζονος παρὰ [So, similarly, we can show that, even if AB is less
ῥητὴν παραβαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων than CD, the square-root of area AD is either a second
τετάρτην. τὸ δὲ ἀπὸ τῆς ῥητὸν καὶ μέσον δυναμένης παρὰ bimedial or the square-root of (the sum of) two medial
ῥητὴν παραβαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων (areas).]
πέμπτην. τὸ δὲ ἀπὸ τῆς δύο μέσα δυναμένης παρὰ ῥητὴν Thus, when two medial (areas which are) incommen-
παραβαλλόμενον πλάτος ποιεῖ τὴν ἐκ δύο ὀνομάτων ἕκτην. surable with one another are added together, the remain-
τὰ δ᾿ εἰρημένα πλάτη διαφέρει τοῦ τε πρώτου καὶ ἀλλήλων, ing two irrational (straight-lines) arise (as the square-
τοῦ μὲν πρώτου, ὅτι ῥητή ἐστιν, ἀλλήλων δέ, ὅτι τῇ τάξει roots of the total area)—either a second bimedial, or the
οὐκ εἰσὶν αἱ αὐταί· ὥστε καὶ αὐταὶ αἱ ἄλογοι διαφέρουσιν square-root of (the sum of) two medial (areas).
ἀλλήλων.
A binomial (straight-line), and the (other) irrational
(straight-lines) after it, are neither the same as a medial
(straight-line) nor (the same) as one another. For the
(square) on a medial (straight-line), applied to a rational
(straight-line), produces as breadth a rational (straight-
line which is) also incommensurable in length with (the
straight-line) to which it is applied [Prop. 10.22]. And
the (square) on a binomial (straight-line), applied to a
rational (straight-line), produces as breadth a first bino-
mial [Prop. 10.60]. And the (square) on a first bimedial
(straight-line), applied to a rational (straight-line), pro-
duces as breadth a second binomial [Prop. 10.61]. And
the (square) on a second bimedial (straight-line), applied
to a rational (straight-line), produces as breadth a third
binomial [Prop. 10.62]. And the (square) on a major
(straight-line), applied to a rational (straight-line), pro-
duces as breadth a fourth binomial [Prop. 10.63]. And
the (square) on the square-root of a rational plus a medial

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