Page 172 - Euclid's Elements of Geometry

ST EW þ.

ELEMENTS BOOK 6

περιεχόμενον ὀρθογώνιον ἴσον ᾖ τῷ ὑπὸ τῶν μέσων περιε- angle contained by the (two) outermost is equal to the
χομένῳ ὀρθογωνίῳ, αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται. rectangle contained by the middle (two) then the four
straight-lines will be proportional.
Θ H
Η G

Α Β Γ ∆ A B C D

Ε Ζ E F
῎Εστωσαν τέσσαρες εὐθεῖαι ἀνάλογον αἱ ΑΒ, ΓΔ, Ε, Ζ, Let AB, CD, E, and F be four proportional straight-
ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ· λέγω, ὅτι lines, (such that) as AB (is) to CD, so E (is) to F. I say
τὸ ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ that the rectangle contained by AB and F is equal to the
ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ. rectangle contained by CD and E.
῎Ηχθωσαν [γὰρ] ἀπὸ τῶν Α, Γ σημείων ταῖς ΑΒ, ΓΔ [For] let AG and CH have been drawn from points
εὐθείαις πρὸς ὀρθὰς αἱ ΑΗ, ΓΘ, καὶ κείσθω τῇ μὲν Ζ ἴση A and C at right-angles to the straight-lines AB and CD
ἡ ΑΗ, τῇ δὲ Ε ἴση ἡ ΓΘ. καὶ συμπεπληρώσθω τὰ ΒΗ, ΔΘ (respectively) [Prop. 1.11]. And let AG be made equal to
παραλληλόγραμμα. F, and CH to E [Prop. 1.3]. And let the parallelograms
Καὶ ἐπεί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς BG and DH have been completed.
τὴν Ζ, ἴση δὲ ἡ μὲν Ε τῇ ΓΘ, ἡ δὲ Ζ τῇ ΑΗ, ἔστιν ἄρα ὡς And since as AB is to CD, so E (is) to F, and E
ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΓΘ πρὸς τὴν ΑΗ. τῶν ΒΗ, (is) equal CH, and F to AG, thus as AB is to CD, so
ΔΘ ἄρα παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ CH (is) to AG. Thus, in the parallelograms BG and DH
περὶ τὰς ἴσας γωνίας. ὧν δὲ ἰσογωνίων παραλληλογράμμων the sides about the equal angles are reciprocally propor-
ἀντιπεπόνθασιν αἱ πλευραί αἱ περὶ τὰς ἴσας γωνάις, ἴσα ἐστὶν tional. And those equiangular parallelograms in which
ἐκεῖνα· ἴσον ἄρα ἐστὶ τὸ ΒΗ παραλληλόγραμμον τῷ ΔΘ the sides about the equal angles are reciprocally propor-
παραλληλογράμμῳ. καί ἐστι τὸ μὲν ΒΗ τὸ ὑπὸ τῶν ΑΒ, Ζ· tional are equal [Prop. 6.14]. Thus, parallelogram BG
ἴση γὰρ ἡ ΑΗ τῇ Ζ· τὸ δὲ ΔΘ τὸ ὑπὸ τῶν ΓΔ, Ε· ἴση γὰρ ἡ is equal to parallelogram DH. And BG is the (rectangle
Ε τῇ ΓΘ· τὸ ἄρα ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον contained) by AB and F. For AG (is) equal to F. And
ἴσον ἐστὶ τῷ ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογώνιῳ. DH (is) the (rectangle contained) by CD and E. For E
᾿Αλλὰ δὴ τὸ ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον (is) equal to CH. Thus, the rectangle contained by AB
ἴσον ἔστω τῷ ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ. and F is equal to the rectangle contained by CD and E.
λέγω, ὅτι αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται, ὡς ἡ ΑΒ And so, let the rectangle contained by AB and F be
πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ. equal to the rectangle contained by CD and E. I say that
Τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ τὸ ὑπὸ τῶν the four straight-lines will be proportional, (so that) as
ΑΒ, Ζ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΓΔ, Ε, καί ἐστι τὸ μὲν ὑπὸ AB (is) to CD, so E (is) to F.
τῶν ΑΒ, Ζ τὸ ΒΗ· ἴση γάρ ἐστιν ἡ ΑΗ τῇ Ζ· τὸ δὲ ὑπὸ τῶν For, with the same construction, since the (rectangle
ΓΔ, Ε τὸ ΔΘ· ἴση γὰρ ἡ ΓΘ τῇ Ε· τὸ ἄρα ΒΗ ἴσον ἐστὶ contained) by AB and F is equal to the (rectangle con-
τῷ ΔΘ. καί ἐστιν ἰσογώνια. τῶν δὲ ἴσων καὶ ἰσογωνίων tained) by CD and E. And BG is the (rectangle con-
παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς tained) by AB and F. For AG is equal to F. And DH
ἴσας γωνίας. ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΓΘ (is) the (rectangle contained) by CD and E. For CH
πρὸς τὴν ΑΗ. ἴση δὲ ἡ μὲν ΓΘ τῇ Ε, ἡ δὲ ΑΗ τῇ Ζ· ἔστιν (is) equal to E. BG is thus equal to DH. And they
ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ. are equiangular. And in equal and equiangular parallel-
᾿Εὰν ἄρα τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν ograms the sides about the equal angles are reciprocally
ἄκρων περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν proportional [Prop. 6.14]. Thus, as AB is to CD, so CH
μέσων περιεχομένῳ ὀρθογωνίῳ· κἂν τὸ ὑπὸ τῶν ἄκρων (is) to AG. And CH (is) equal to E, and AG to F. Thus,
περιεχόμενον ὀρθογώνιον ἴσον ᾖ τῷ ὑπὸ τῶν μέσων περιε- as AB is to CD, so E (is) to F.
χομένῳ ὀρθογωνίῳ, αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται· Thus, if four straight-lines are proportional then the
ὅπερ ἔδει δεῖξαι. rectangle contained by the (two) outermost is equal to
the rectangle contained by the middle (two). And if the
rectangle contained by the (two) outermost is equal to

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