Euclid · Elements §6.def.1-6.def.5

Definitions of Similar Figures, Mean Ratio, and Height

Passage 83 of 316 · Greek

Summary

Definitions 1 to 5 of Book 6, defining similar rectilineal figures, extreme and mean ratio, and the height of a figure (with definitions 2 and 5 being empty in this text).

§6.def.1ὅμοια σχήματα εὐθύγραμμά ἐστιν, ὅσα τάς τε γωνίας ἴσας ἔχει κατὰ μίαν καὶ τὰς περὶ τὰς ἴσας γωνίας πλευρὰς ἀνάλογον.

Similar rectilineal figures are such as have their angles severally equal and the sides about the equal angles proportional.

§6.def.3ἄκρον καὶ μέσον λόγον εὐθεῖα τετμῆσθαι λέγεται, ὅταν ᾖ ὡς ἡ ὅλη πρὸς τὸ μεῖζον τμῆμα, οὕτως τὸ μεῖζον πρὸς τὸ ἔλαττον.

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.

§6.def.4ὕψος ἐστὶ παντὸς σχήματος ἡ ἀπὸ τῆς κορυφῆς ἐπὶ τὴν βάσιν κάθετος ἀγομένη.

The height of any figure is the perpendicular drawn from the vertex to the base.

Notes

6.def.1κατὰ μίαν — An adverbial expression composed of the preposition κατά and the feminine accusative singular μίαν, meaning 'one by one' or 'severally'. It indicates that the corresponding angles between the two figures are equal.
6.def.3ἄκρον καὶ μέσον λόγον — The noun phrase 'in extreme and mean ratio' is in the accusative case (adverbial accusative or accusative of respect) modifying the infinitive τετμῆσθαι, meaning 'to have been cut in extreme and mean ratio'.
6.def.4κάθετος — Originally the feminine singular adjective κάθετος meaning 'let down' or 'perpendicular', here functioning as a noun meaning 'perpendicular line' with the feminine noun γραμμή ('line') understood.

Cite this passage

Euclid, Elements §6.def.1-6.def.5. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:6.def.1-6.def.5

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