Humanitext Reader

Euclid · Elements §4.def.1-4.def.7

Definitions of Inscribed and Circumscribed Figures

Passage 57 of 316 · Greek

Summary

This section contains Definitions 1-7 of Book IV, which define the terms 'inscribed' and 'circumscribed' for rectilineal figures and circles, as well as the concept of a straight line being 'fitted into' a circle as a chord.

§4.def.1σχῆμα εὐθύγραμμον εἰς σχῆμα εὐθύγραμμον ἐγγράφεσθαι λέγεται, ὅταν ἑκάστη τῶν τοῦ ἐγγραφομένου σχήματος γωνιῶν ἑκάστης πλευρᾶς τοῦ, εἰς ὃ ἐγγράφεται, ἅπτηται.
A rectilineal figure is said to be inscribed in a rectilineal figure when each of the angles of the inscribed figure touches each side of that in which it is inscribed.
§4.def.2σχῆμα δὲ ὁμοίως περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἑκάστης γωνίας τοῦ, περὶ ὃ περιγράφεται, ἅπτηται.
And similarly a figure is said to be circumscribed about a figure when each side of the circumscribed figure touches each angle of that about which it is circumscribed.
§4.def.3σχῆμα εὐθύγραμμον εἰς κύκλον ἐγγράφεσθαι λέγεται, ὅταν ἑκάστη γωνία τοῦ ἐγγραφομένου ἅπτηται τῆς τοῦ κύκλου περιφερείας.
A rectilineal figure is said to be inscribed in a circle when each angle of the inscribed figure touches the circumference of the circle.
§4.def.4σχῆμα δὲ εὐθύγραμμον περὶ κύκλον περιγράφεσθαι λέγεται, ὅταν ἑκάστη πλευρὰ τοῦ περιγραφομένου ἐφάπτηται τῆς τοῦ κύκλου περιφερείας.
And a rectilineal figure is said to be circumscribed about a circle when each side of the circumscribed figure touches the circumference of the circle.
§4.def.5κύκλος δὲ εἰς σχῆμα ὁμοίως ἐγγράφεσθαι λέγεται, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης πλευρᾶς τοῦ, εἰς ὃ ἐγγράφεται, ἅπτηται.
And similarly a circle is said to be inscribed in a figure when the circumference of the circle touches each side of that in which it is inscribed.
§4.def.6κύκλος δὲ περὶ σχῆμα περιγράφεσθαι λέγεται, ὅταν ἡ τοῦ κύκλου περιφέρεια ἑκάστης γωνίας τοῦ, περὶ ὃ περιγράφεται, ἅπτηται.
And a circle is said to be circumscribed about a figure when the circumference of the circle touches each angle of that about which it is circumscribed.
§4.def.7εὐθεῖα εἰς κύκλον ἐναρμόζεσθαι λέγεται, ὅταν τὰ πέρατα αὐτῆς ἐπὶ τῆς περιφερείας ᾖ τοῦ κύκλου.
A straight line is said to be fitted into a circle when its extremities are on the circumference of the circle.

Notes

  1. §4.def.1τοῦ, εἰς ὃ ἐγγράφεται — The neuter genitive article `τοῦ` functions substantively as the antecedent of the relative clause `εἰς ὃ ἐγγράφεται` ('into which it is inscribed'). The genitive `τοῦ` itself modifies the preceding noun `πλευρᾶς` ('each side of it').
  2. §4.def.1ἅπτηται — The verb `ἅπτομαι` ('touch') governs the genitive case, which is why its object `ἑκάστης πλευρᾶς` is in the genitive. Throughout the definitions of Book IV, the present subjunctive `ἅπτηται` is consistently used in the `ὅταν` clauses to denote contact between figures.
  3. §4.def.7ἐναρμόζεσθαι — The present passive infinitive of `ἐναρμόζω` ('to fit in' or 'to insert'). It is a mathematical term defining the action of a straight line being 'fitted' inside a circle such that its endpoints lie on the circumference (i.e., drawing a chord).

Cite this passage

Euclid, Elements §4.def.1-4.def.7. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:4.def.1-4.def.7

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.