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Euclid · Elements §2.prop.5

Rectangle from Unequal Segments and Square on the Half

Passage 28 of 316 · Greek

Summary

Euclid proves that if a straight line is cut into equal and unequal segments, the rectangle contained by the unequal segments together with the square on the segment between the points of section is equal to the square on the half.

§2.prop.5ἐὰν εὐθεῖα γραμμὴ τμηθῇ εἰς ἴσα καὶ ἄνισα, τὸ ὑπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς μεταξὺ τῶν τομῶν τετραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ἡμισείας τετραγώνῳ.
If a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half.
εὐθεῖα γάρ τις ἡ ΑΒ τετμήσθω εἰς μὲν ἴσα κατὰ τὸ Γ, εἰς δὲ ἄνισα κατὰ τὸ Δ· λέγω, ὅτι τὸ ὑπὸ τῶν ΑΔ, ΔΒ περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΓΔ τετραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΒ τετραγώνῳ.
For let a straight line AB be cut into equal segments at C, and into unequal segments at D; I say that the rectangle contained by AD, DB together with the square on CD is equal to the square on CB.
Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΓΒ τετράγωνον τὸ ΓΕΖΒ, καὶ ἐπεζεύχθω ἡ ΒΕ, καὶ διὰ μὲν τοῦ Δ ὁποτέρᾳ τῶν ΓΕ, ΒΖ παράλληλος ἤχθω ἡ ΔΗ, διὰ δὲ τοῦ Θ ὁποτέρᾳ τῶν ΑΒ, ΕΖ παράλληλος πάλιν ἤχθω ἡ ΚΜ, καὶ πάλιν διὰ τοῦ Α ὁποτέρᾳ τῶν ΓΛ, ΒΜ παράλληλος ἤχθω ἡ ΑΚ. καὶ ἐπεὶ ἴσον ἐστὶ τὸ ΓΘ παραπλήρωμα τῷ ΘΖ παραπληρώματι, κοινὸν προσκείσθω τὸ ΔΜ· ὅλον ἄρα τὸ ΓΜ ὅλῳ τῷ ΔΖ ἴσον ἐστίν.
For let the square CEZB be described on CB, and let BE be joined, and through D let DH be drawn parallel to either CE or BZ, and through Q let KM again be drawn parallel to either AB or EZ, and again through A let AK be drawn parallel to either CL or BM. And since the complement CQ is equal to the complement QZ, let DM be added to each; therefore the whole CM is equal to the whole DZ.
ἀλλὰ τὸ ΓΜ τῷ ΑΛ ἴσον ἐστίν, ἐπεὶ καὶ ἡ ΑΓ τῇ ΓΒ ἐστιν ἴση· καὶ τὸ ΑΛ ἄρα τῷ ΔΖ ἴσον ἐστίν.
But CM is equal to AL, since AC is also equal to GB; therefore AL is also equal to DZ.
κοινὸν προσκείσθω τὸ ΓΘ· ὅλον ἄρα τὸ ΑΘ τῷ ΜΝΞ γνώμονι ἴσον ἐστίν.
Let CQ be added to each; therefore the whole AQ is equal to the gnomon MNX.
ἀλλὰ τὸ ΑΘ τὸ ὑπὸ τῶν ΑΔ, ΔΒ ἐστιν· ἴση γὰρ ἡ ΔΘ τῇ ΔΒ· καὶ ὁ ΜΝΞ ἄρα γνώμων ἴσος ἐστὶ τῷ ὑπὸ ΑΔ, ΔΒ. κοινὸν προσκείσθω τὸ ΛΗ, ὅ ἐστιν ἴσον τῷ ἀπὸ τῆς ΓΔ· ὁ ἄρα ΜΝΞ γνώμων καὶ τὸ ΛΗ ἴσα ἐστὶ τῷ ὑπὸ τῶν ΑΔ, ΔΒ περιεχομένῳ ὀρθογωνίῳ καὶ τῷ ἀπὸ τῆς ΓΔ τετραγώνῳ.
But AQ is the rectangle contained by AD, DB; for DQ is equal to DB; therefore the gnomon MNX is also equal to the rectangle contained by AD, DB. Let LH, which is equal to the square on CD, be added to each; therefore the gnomon MNX and LH are equal to the rectangle contained by AD, DB and the square on CD.
ἀλλὰ ὁ ΜΝΞ γνώμων καὶ τὸ ΛΗ ὅλον ἐστὶ τὸ ΓΕΖΒ τετράγωνον, ὅ ἐστιν ἀπὸ τῆς ΓΒ· τὸ ἄρα ὑπὸ τῶν ΑΔ, ΔΒ περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς ΓΔ τετραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΒ τετραγώνῳ.
But the gnomon MNX and LH are the whole square CEZB, which is on CB; therefore the rectangle contained by AD, DB together with the square on CD is equal to the square on CB.
ἐὰν ἄρα εὐθεῖα γραμμὴ τμηθῇ εἰς ἴσα καὶ ἄνισα, τὸ ὑπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων περιεχόμενον ὀρθογώνιον μετὰ τοῦ ἀπὸ τῆς μεταξὺ τῶν τομῶν τετραγώνου ἴσον ἐστὶ τῷ ἀπὸ τῆς ἡμισείας τετραγώνῳ· ὅπερ ἔδει δεῖξαι.
If therefore a straight line be cut into equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half; which was to be proved.

Notes

  1. §2.prop.5τὸ ὑπὸ τῶν ἀνίσων τῆς ὅλης τμημάτων περιεχόμενον ὀρθογώνιον — Meaning 'the rectangle contained by the unequal segments of the whole.' The phrase 'τὸ ὑπό...' is a standard formula in Greek geometry denoting the rectangle contained by specified lines, with the genitive phrase 'τῶν ἀνίσων... τμημάτων' depending on the preposition 'ὑπό'.
  2. §2.prop.5τετμήσθω — Third-person singular perfect passive imperative of τέμνω ('to cut'). It functions to postulate a geometric setting ('let it have been cut'), characteristically used in mathematical texts to introduce a construction.
  3. §2.prop.5ὁποτέρᾳ τῶν ΓΕ, ΒΖ παράλληλος — The adjective 'παράλληλος' (parallel) governs the dative pronoun 'ὁποτέρᾳ' (to either of the two), which in turn takes the genitive 'τῶν ΓΕ, ΒΖ' (of CE, BZ) to specify the objects of choice.
  4. §2.prop.5γνώμονι — Dative of 'γνώμων' (gnomon). In Euclidean geometry, it refers to the L-shaped figure remaining after a smaller, similar parallelogram is removed from a corner of a larger parallelogram.

Cite this passage

Euclid, Elements §2.prop.5. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:2.prop.5

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