Euclid · Elements §13.prop.3-13.prop.4

Fivefold and Threefold Square Relations in Golden Ratio

Passage 296 of 316 · Greek

Summary

In Book 13, Proposition 3, it is proved that the square on the sum of the lesser segment and half of the greater segment of a line cut in extreme and mean ratio is five times the square on the half of the greater segment, and Proposition 4 states that the sum of the squares on the whole line and on the lesser segment is three times the square on the greater segment.

§13.prop.3ἐὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ ἔλασσον τμῆμα προσλαβὸν τὴν ἡμίσειαν τοῦ μείζονος τμήματος πενταπλάσιον δύναται τοῦ ἀπὸ τῆς ἡμισείας τοῦ μείζονος τμήματος τετραγώνου.

If a straight line be cut in extreme and mean ratio, the lesser segment, if it have the half of the greater segment added to it, is five times the square on the half of the greater segment.

εὐθεῖα γάρ τις ἡ ΑΒ ἄκρον καὶ μέσον λόγον τετμήσθω κατὰ τὸ Γ σημεῖον, καὶ ἔστω μεῖζον τμῆμα τὸ ΑΓ, καὶ τετμήσθω ἡ ΑΓ δίχα κατὰ τὸ Δ· λέγω, ὅτι πενταπλάσιόν ἐστι τὸ ἀπὸ τῆς ΒΔ τοῦ ἀπὸ τῆς ΔΓ. Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΕ, καὶ καταγεγράφθω διπλοῦν τὸ σχῆμα.

¦5 For let some straight line AB be cut in extreme and mean ratio at the point C, and let AC be the greater segment, and let AC be bisected at D; I say that the square on BD is five times the square on DG. For let the square AE be described on AB, and let the double figure be drawn.

ἐπεὶ διπλῆ ἐστιν ἡ ΑΓ τῆς ΔΓ, τετραπλάσιον ἄρα τὸ ἀπὸ τῆς ΑΓ τοῦ ἀπὸ τῆς ΔΓ, τουτέστι τὸ ΡΣ τοῦ ΖΗ. καὶ ἐπεὶ τὸ ὑπὸ τῶν ΑΒΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ, καί ἐστι τὸ ὑπὸ τῶν ΑΒΓ τὸ ΓΕ, τὸ ἄρα ΓΕ ἴσον ἐστὶ τῷ ΡΣ. τετραπλάσιον δὲ τὸ ΡΣ τοῦ ΖΗ· τετραπλάσιον ἄρα καὶ τὸ ΓΕ τοῦ ΖΗ. πάλιν ἐπεὶ ἴση ἐστὶν ἡ ΑΔ τῇ ΔΓ, ἴση ἐστὶ καὶ ἡ ΘΚ τῇ ΚΖ. ὥστε καὶ τὸ ΗΖ τετράγωνον ἴσον ἐστὶ τῷ ΘΛ τετραγώνῳ.

Since AC is double DG, therefore the square on AC is four times the square on DG, that is, RS is four times ZH. And since the rectangle contained by AB, BC is equal to the square on AC, and GE is the rectangle contained by AB, BC, therefore GE is equal to RS. And RS is four times ZH; therefore GE is also four times ZH. Again, since AD is equal to DG, ThK is also equal to KZ. Therefore the square HZ is also equal to the square ThL.

ἴση ἄρα ἡ ΗΚ τῇ ΚΛ, τουτέστιν ἡ ΜΝ τῇ ΝΕ· ὥστε καὶ τὸ ΜΖ τῷ ΖΕ ἐστιν ἴσον.

Therefore HK is equal to KL, that is, MN to NE; so that MZ is also equal to ZE.

ἀλλὰ τὸ ΜΖ τῷ ΓΗ ἐστιν ἴσον· καὶ τὸ ΓΗ ἄρα τῷ ΖΕ ἐστιν ἴσον.

But MZ is equal to GH; therefore GH is also equal to ZE.

κοινὸν προσκείσθω τὸ ΓΝ· ὁ ἄρα ΞΟΠ γνώμων ἴσος ἐστὶ τῷ ΓΕ. ἀλλὰ τὸ ΓΕ τετραπλάσιον ἐδείχθη τοῦ ΗΖ· καὶ ὁ ΞΟΠ ἄρα γνώμων τετραπλάσιός ἐστι τοῦ ΖΗ τετραγώνου.

Let the common GN be added; therefore the gnomon XOP is equal to GE. But GE was shown to be four times HZ; therefore the gnomon XOP is also four times the square ZH.

ὁ ΞΟΠ ἄρα γνώμων καὶ τὸ ΖΗ τετράγωνον πενταπλάσιός ἐστι τοῦ ΖΗ. ἀλλὰ ὁ ΞΟΠ γνώμων καὶ τὸ ΖΗ τετράγωνόν ἐστι τὸ ΔΝ. καί ἐστι τὸ μὲν ΔΝ τὸ ἀπὸ τῆς ΔΒ, τὸ δὲ ΗΖ τὸ ἀπὸ τῆς ΔΓ. τὸ ἄρα ἀπὸ τῆς ΔΒ πενταπλάσιόν ἐστι τοῦ ἀπὸ τῆς ΔΓ· ὅπερ ἔδει δεῖξαι.

Therefore the gnomon XOP and the square ZH is five times ZH. But the gnomon XOP and the square ZH is DN. And DN is the square on DB, and HZ is the square on DG. Therefore the square on DB is five times the square on DG; which it was required to prove.

§13.prop.4ἐὰν εὐθεῖα γραμμὴ ἄκρον καὶ μέσον λόγον τμηθῇ, τὸ ἀπὸ τῆς ὅλης καὶ τοῦ ἐλάσσονος τμήματος, τὰ συναμφότερα τετράγωνα, τριπλάσιά ἐστι τοῦ ἀπὸ τοῦ μείζονος τμήματος τετραγώνου.

If a straight line be cut in extreme and mean ratio, the square on the whole and the square on the lesser segment, both together, are three times the square on the greater segment.

Notes

13.prop.3προσλαβὸν ... πενταπλάσιον δύναται — The participle `προσλαβὸν` agrees with the subject `τὸ ἔλασσον τμῆμα` (the lesser segment), expressing an attendant circumstance or condition. The verb `δύναται` is used here in the mathematical sense 'is equal in square to' or 'has the power of the square of', meaning that the square on the subject segment is equal to the genitive object, here 'five times the square on the half of the greater segment'.
13.prop.3διπλοῦν τὸ σχῆμα — The phrase 'double figure' is an idiomatic expression in Euclidean geometry referring to the construction of a composite figure or a complete parallelogram system drawn over a single base figure (here, the square AE) by adding auxiliary lines.
13.prop.3τὸ ὑπὸ τῶν ΑΒΓ — A formulaic expression meaning 'the rectangle contained by AB and BC'. Although three letters `ΑΒΓ` are listed, it denotes the rectangle formed by the straight line AB and the segment BC (which lies on the same line).
13.prop.3ὁ ΞΟΠ ἄρα γνώμων καὶ τὸ ΖΗ τετράγωνον πενταπλάσιός ἐστι — With a compound subject (the masculine noun `γνώμων`, gnomon, and the neuter noun `τετράγωνον`, square), the verb `ἐστι` is in the third person singular, and the predicate adjective `πενταπλάσιός` is masculine singular. This occurs either because the sum of these two figures is conceived as a single mathematical entity, or because of grammatical attraction to the closer or leading masculine subject.
13.prop.4τὸ ἀπὸ τῆς ὅλης καὶ τοῦ ἐλάσσονος τμήματος, τὰ συναμφότερα τετράγωνα — The article `τὸ` is neuter singular, but it stands in apposition with the subsequent phrase `τὰ συναμφότερα τετράγωνα` (both squares together). Literally, it means 'the [square] on the whole and [that] on the lesser segment, the two squares together', indicating the sum of the square on the whole and the square on the lesser segment.

Cite this passage

Euclid, Elements §13.prop.3-13.prop.4. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:13.prop.3-13.prop.4

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