Humanitext Reader

Euclid · Elements §6.prop.16

Proportional Lines and Rectangles of Extremes and Means

Passage 95 of 316 · Greek

Summary

Proves that four straight lines are proportional if and only if the rectangle contained by the extremes is equal to the rectangle contained by the means.

§6.prop.16ἐὰν τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν ἄκρων περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν μέσων περιεχομένῳ ὀρθογωνίῳ· κἂν τὸ ὑπὸ τῶν ἄκρων περιεχόμενον ὀρθογώνιον ἴσον ᾖ τῷ ὑπὸ τῶν μέσων περιεχομένῳ ὀρθογωνίῳ, αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται.
If four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines will be proportional.
ἔστωσαν τέσσαρες εὐθεῖαι ἀνάλογον αἱ ΑΒ, ΓΔ, Ε, Ζ, ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ· λέγω, ὅτι τὸ ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ.
Let the four straight lines AB, ΓΔ, E, Z be proportional, as AB is to ΓΔ, so E to Z; I say that the rectangle contained by AB, Z is equal to the rectangle contained by ΓΔ, E.
ἤχθωσαν ἀπὸ τῶν Α, Γ σημείων ταῖς ΑΒ, ΓΔ εὐθείαις πρὸς ὀρθὰς αἱ ΑΗ, ΓΘ, καὶ κείσθω τῇ μὲν Ζ ἴση ἡ ΑΓ, τῇ δὲ Ε ἴση ἡ ΓΘ. καὶ συμπεπληρώσθω τὰ ΒΗ, ΔΘ παραλληλόγραμμα.
Let AH, ΓΘ be drawn from the points A, Γ at right angles to the straight lines AB, ΓΔ, and let AΓ be placed equal to Z, and ΓΘ equal to E. And let the parallelograms BH, ΔΘ be completed.
καὶ ἐπεί ἐστιν ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ, ἴση δὲ ἡ μὲν Ε τῇ ΓΘ, ἡ δὲ Ζ τῇ ΑΗ, ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΓΘ πρὸς τὴν ΑΗ. τῶν ΒΗ, ΔΘ ἄρα παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας.
And since as AB is to ΓΔ, so is E to Z, and E is equal to ΓΘ, and Z to AH, therefore, as AB is to ΓΔ, so is ΓΘ to AH. Therefore in the parallelograms BH, ΔΘ the sides about the equal angles are reciprocally proportional.
ὧν δὲ ἰσογωνίων παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας, ἴσα ἐστὶν ἐκεῖνα· ἴσον ἄρα ἐστὶ τὸ ΒΗ παραλληλόγραμμον τῷ ΔΘ παραλληλογράμμῳ.
And those equiangular parallelograms in which the sides about the equal angles are reciprocally proportional are equal; therefore the parallelogram BH is equal to the parallelogram ΔΘ.
καί ἐστι τὸ μὲν ΒΗ τὸ ὑπὸ τῶν ΑΒ, Ζ· ἴση γὰρ ἡ ΑΗ τῇ Ζ· τὸ δὲ ΔΘ τὸ ὑπὸ τῶν ΓΔ, Ε· ἴση γὰρ ἡ Ε τῇ ΓΘ· τὸ ἄρα ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ.
And BH is the rectangle contained by AB, Z; for AH is equal to Z; and ΔΘ is the rectangle contained by ΓΔ, E; for E is equal to ΓΘ; therefore the rectangle contained by AB, Z is equal to the rectangle contained by ΓΔ, E.
ἀλλὰ δὴ τὸ ὑπὸ τῶν ΑΒ, Ζ περιεχόμενον ὀρθογώνιον ἴσον ἔστω τῷ ὑπὸ τῶν ΓΔ, Ε περιεχομένῳ ὀρθογωνίῳ· λέγω, ὅτι αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται, ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ. τῶν γὰρ αὐτῶν κατασκευασθέντων, ἐπεὶ τὸ ὑπὸ τῶν ΑΒ, Ζ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΓΔ, Ε, καί ἐστι τὸ μὲν ὑπὸ τῶν ΑΒ, Ζ τὸ ΒΗ· ἴση γάρ ἐστιν ἡ ΑΗ τῇ Ζ· τὸ δὲ ὑπὸ τῶν ΓΔ, Ε τὸ ΔΘ· ἴση γὰρ ἡ ΓΘ τῇ Ε· τὸ ἄρα ΒΗ ἴσον ἐστὶ τῷ ΔΘ. καί ἐστιν ἰσογώνια.
But indeed let the rectangle contained by AB, Z be equal to the rectangle contained by ΓΔ, E; I say that the four straight lines will be proportional, as AB is to ΓΔ, so E to Z. For, with the same construction, since the rectangle contained by AB, Z is equal to the rectangle contained by ΓΔ, E, and the rectangle contained by AB, Z is BH; for AH is equal to Z; and the rectangle contained by ΓΔ, E is ΔΘ; for ΓΘ is equal to E; therefore BH is equal to ΔΘ. And they are equiangular.
τῶν δὲ ἴσων καὶ ἰσογωνίων παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας.
And in equal and equiangular parallelograms the sides about the equal angles are reciprocally proportional.
ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ ΓΘ πρὸς τὴν ΑΗ. ἴση δὲ ἡ μὲν ΓΘ τῇ Ε, ἡ δὲ ΑΗ τῇ Ζ· ἔστιν ἄρα ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ. ἐὰν ἄρα τέσσαρες εὐθεῖαι ἀνάλογον ὦσιν, τὸ ὑπὸ τῶν ἄκρων περιεχόμενον ὀρθογώνιον ἴσον ἐστὶ τῷ ὑπὸ τῶν μέσων περιεχομένῳ ὀρθογωνίῳ· κἂν τὸ ὑπὸ τῶν ἄκρων περιεχόμενον ὀρθογώνιον ἴσον ᾖ τῷ ὑπὸ τῶν μέσων περιεχομένῳ ὀρθογωνίῳ, αἱ τέσσαρες εὐθεῖαι ἀνάλογον ἔσονται· ὅπερ ἔδει δεῖξαι.
Therefore, as AB is to ΓΔ, so is ΓΘ to AH. And ΓΘ is equal to E, and AH to Z; therefore, as AB is to ΓΔ, so is E to Z. Therefore, if four straight lines be proportional, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines will be proportional; which was to be demonstrated.

Notes

  1. 3κἂν — A contraction of καὶ ἐάν. It expresses a concessive or conditional meaning ("even if", "and if"), here introducing the converse of the previous statement.
  2. 8ὡς ἡ ΑΒ πρὸς τὴν ΓΔ, οὕτως ἡ Ε πρὸς τὴν Ζ — A formulaic correlative structure expressing proportionality. The relative adverb ὡς ("as") and the demonstrative adverb οὕτως ("so") match to express the equality of the ratios AB:ΓΔ = E:Z.
  3. 12τῇ μὲν Ζ ἴση ἡ ΑΓ — The reading ΑΓ in the text is a scribal error for ΑΗ. As is clear from the geometrical construction and the subsequent sentence "ἴση γὰρ ἡ ΑΗ τῇ Ζ" (for AH is equal to Z), the segment to be set equal to Z must be AH.
  4. 31τῶν γὰρ αὐτῶν κατασκευασθέντων — A genitive absolute construction using the neuter plural perfect passive participle of κατασκευάζω without an explicit noun. A formulaic expression meaning "with the same construction having been made."

Cite this passage

Euclid, Elements §6.prop.16. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:6.prop.16

Please note the AI-draft status of the translation and the date accessed.

Translation, notes and summary are AI-generated drafts, revised through reader feedback.