§10.prop3.114ἐὰν χωρίον περιέχηται ὑπὸ ἀποτομῆς καὶ τῆς ἐκ δύο ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά τέ ἐστι τοῖς τῆς ἀποτομῆς ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, ἡ τὸ χωρίον δυναμένη ῥητή ἐστιν.
If an area be contained by an apotome and a binomial straight line whose terms are commensurable with the terms of the apotome and in the same ratio, the straight line which is equal in square to the area is rational.
περιεχέσθω γὰρ χωρίον τὸ ὑπὸ τῶν ΑΒ, ΓΔ ὑπὸ ἀποτομῆς τῆς ΑΒ καὶ τῆς ἐκ δύο ὀνομάτων τῆς ΓΔ, ἧς μεῖζον ὄνομα ἔστω τὸ ΓΕ, καὶ ἔστω τὰ ὀνόματα τῆς ἐκ δύο ὀνομάτων τὰ ΓΕ, ΕΔ σύμμετρά τε τοῖς τῆς ἀποτομῆς ὀνόμασι τοῖς ΑΖ, ΖΒ καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ ἔστω ἡ τὸ ὑπὸ τῶν ΑΒ, ΓΔ δυναμένη ἡ Η· λέγω, ὅτι ῥητή ἐστιν ἡ Η.
Ἐκκείσθω γὰρ ῥητὴ ἡ Θ, καὶ τῷ ἀπὸ τῆς Θ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω πλάτος ποιοῦν τὴν ΚΛ·
For let the area contained by AB, ΓΔ be contained by the apotome AB and the binomial straight line ΓΔ, whose greater term let be ΓE, and let the terms ΓE, ED of the binomial straight line be commensurable with the terms AZ, ZB of the apotome and in the same ratio, and let H be the straight line which is equal in square to the area contained by AB, ΓΔ; I say that H is rational.
ἀποτομὴ ἄρα ἐστὶν ἡ ΚΛ, ἧς τὰ ὀνόματα ἔστω τὰ ΚΜ, ΜΛ σύμμετρα τοῖς τῆς ἐκ δύο ὀνομάτων ὀνόμασι τοῖς ΓΕ, ΕΔ καὶ ἐν τῷ αὐτῷ λόγῳ.
For let a rational straight line Θ be set out, and to ΓΔ let there be applied equal to the square on Θ a rectangle producing KΛ as breadth; therefore KΛ is an apotome, whose terms let be KM, MΛ, commensurable with the terms ΓE, ED of the binomial straight line and in the same ratio.
ἀλλὰ καὶ αἱ ΓΕ, ΕΔ σύμμετροί τέ εἰσι ταῖς ΑΖ, ΖΒ καὶ ἐν τῷ αὐτῷ λόγῳ· ἔστιν ἄρα ὡς ἡ ΑΖ πρὸς τὴν ΖΒ, οὕτως ἡ ΚΜ πρὸς ΜΛ. ἐναλλὰξ ἄρα ἐστὶν ὡς ἡ ΑΖ πρὸς τὴν ΚΜ, οὕτως ἡ ΒΖ πρὸς τὴν ΛΜ·
But ΓE, ED are also commensurable with AZ, ZB and in the same ratio; therefore, as AZ is to ZB, so is KM to MΛ. Alternately, therefore, as AZ is to KM, so is BZ to ΛM; therefore also the remainder AB is to the remainder KΛ as AZ is to KM.
καὶ λοιπὴ ἄρα ἡ ΑΒ πρὸς λοιπὴν τὴν ΚΛ ἐστιν ὡς ἡ ΑΖ πρὸς ΚΜ. σύμμετρος δὲ ἡ ΑΖ τῇ ΚΜ·
But AZ is commensurable with KM; therefore AB is also commensurable with KΛ.
σύμμετρος ἄρα ἐστὶ καὶ ἡ ΑΒ τῇ ΚΛ. καί ἐστιν ὡς ἡ ΑΒ πρὸς ΚΛ, οὕτως τὸ ὑπὸ τῶν ΓΔ, ΑΒ πρὸς τὸ ὑπὸ τῶν ΓΔ, ΚΛ·
And as AB is to KΛ, so is the rectangle contained by ΓΔ, AB to the rectangle contained by ΓΔ, KΛ; therefore the rectangle contained by ΓΔ, AB is also commensurable with the rectangle contained by ΓΔ, KΛ.
σύμμετρον ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΓΔ, ΑΒ τῷ ὑπὸ τῶν ΓΔ, ΚΛ. ἴσον δὲ τὸ ὑπὸ τῶν ΓΔ, ΚΛ τῷ ἀπὸ τῆς Θ·
But the rectangle contained by ΓΔ, KΛ is equal to the square on Θ; therefore the rectangle contained by ΓΔ, AB is commensurable with the square on Θ.
σύμμετρον ἄρα ἐστὶ τὸ ὑπὸ τῶν ΓΔ, ΑΒ τῷ ἀπὸ τῆς Θ. τῷ δὲ ὑπὸ τῶν ΓΔ, ΑΒ ἴσον ἐστὶ τὸ ἀπὸ τῆς Η·
And to the rectangle contained by ΓΔ, AB the square on H is equal; therefore the square on H is commensurable with the square on Θ.
σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς Η τῷ ἀπὸ τῆς Θ. ῥητὸν δὲ τὸ ἀπὸ τῆς Θ· ῥητὸν ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς Η·
But the square on Θ is rational; therefore the square on H is also rational; therefore H is rational.
ῥητὴ ἄρα ἐστὶν ἡ Η. καὶ δύναται τὸ ὑπὸ τῶν ΓΔ, ΑΒ.
ἐὰν ἄρα χωρίον περιέχηται ὑπὸ ἀποτομῆς καὶ τῆς ἐκ δύο ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά ἐστι τοῖς τῆς ἀποτομῆς ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, ἡ τὸ χωρίον δυναμένη ῥητή ἐστιν.
And it is equal in square to the area contained by ΓΔ, AB. If therefore an area be contained by an apotome and a binomial straight line whose terms are commensurable with the terms of the apotome and in the same ratio, the straight line which is equal in square to the area is rational.
Πόρισμα
καὶ γέγονεν ἡμῖν καὶ διὰ τούτου φανερόν, ὅτι δυνατόν ἐστι ῥητὸν χωρίον ὑπὸ ἀλόγων εὐθειῶν περιέχεσθαι.
Porism: And it has been made manifest to us by this also that it is possible for a rational area to be contained by irrational straight lines.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.
§10.prop3.115ἀπὸ μέσης ἄπειροι ἄλογοι γίνονται, καὶ οὐδεμία οὐδεμιᾷ τῶν πρότερον ἡ αὐτή.
From a medial straight line infinitely many irrational straight lines arise, and none of them is the same as any of the preceding.
ἔστω μέση ἡ Α· λέγω, ὅτι ἀπὸ τῆς Α ἄπειροι ἄλογοι γίνονται, καὶ οὐδεμία οὐδεμιᾷ τῶν πρότερον ἡ αὐτή.
Let A be a medial straight line; I say that from A infinitely many irrational straight lines arise, and none of them is the same as any of the preceding.
Ἐκκείσθω ῥητὴ ἡ Β, καὶ τῷ ὑπὸ τῶν Β, Α ἴσον ἔστω τὸ ἀπὸ τῆς Γ· ἄλογος ἄρα ἐστὶν ἡ Γ· τὸ γὰρ ὑπὸ ἀλόγου καὶ ῥητῆς ἄλογόν ἐστιν.
For let a rational straight line B be set out, and let the square on Γ be equal to the rectangle contained by B, A; therefore Γ is irrational; for the area contained by an irrational and a rational straight line is irrational.
καὶ οὐδεμιᾷ τῶν πρότερον ἡ αὐτή· τὸ γὰρ ἀπʼ οὐδεμιᾶς τῶν πρότερον παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ μέσην.
And it is not the same as any of the preceding; for the square on none of the preceding, applied to a rational straight line, produces a medial as breadth.
πάλιν δὴ τῷ ὑπὸ τῶν Β, Γ ἴσον ἔστω τὸ ἀπὸ τῆς Δ·
Again, let the square on Δ be equal to the rectangle contained by B, Γ; therefore the square on Δ is irrational.
ἄλογον ἄρα ἐστὶ τὸ ἀπὸ τῆς Δ. ἄλογος ἄρα ἐστὶν ἡ Δ· καὶ οὐδεμιᾷ τῶν πρότερον ἡ αὐτή·
Therefore Δ is irrational; and it is not the same as any of the preceding; for the square on none of the preceding, applied to a rational straight line, produces Γ as breadth.
τὸ γὰρ ἀπʼ οὐδεμιᾶς τῶν πρότερον παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ τὴν Γ. ὁμοίως δὴ τῆς τοιαύτης τάξεως ἐπʼ ἄπειρον προβαινούσης φανερόν, ὅτι ἀπὸ τῆς μέσης ἄπειροι ἄλογοι γίνονται, καὶ οὐδεμία οὐδεμιᾷ τῶν πρότερον ἡ αὐτή· ὅπερ ἔδει δεῖξαι].
Likewise, since this order proceeds to infinity, it is manifest that from the medial straight line infinitely many irrational straight lines arise, and none of them is the same as any of the preceding; which was to be proved.