Humanitext Reader

Euclid · Elements §3.def.1-3.def.11

Definitions of Circles, Tangents, and Segments

Passage 35 of 316 · Greek

Summary

The first eleven definitions of Book 3 concerning circles. They define equal circles, tangent lines, touching circles, distance from the center, segments of circles, angles of/in segments, angles standing on circumferences, sectors, and similar segments.

§3.def.1ἴσοι κύκλοι εἰσίν, ὧν αἱ διάμετροι ἴσαι εἰσίν, ἢ ὧν αἱ ἐκ τῶν κέντρων ἴσαι εἰσίν.
Equal circles are those whose diameters are equal, or whose radii (the straight lines from the centers) are equal.
§3.def.2εὐθεῖα κύκλου ἐφάπτεσθαι λέγεται, ἥτις ἁπτομένη τοῦ κύκλου καὶ ἐκβαλλομένη οὐ τέμνει τὸν κύκλον.
A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
§3.def.3κύκλοι ἐφάπτεσθαι ἀλλήλων λέγονται οἵτινες ἁπτόμενοι ἀλλήλων οὐ τέμνουσιν ἀλλήλους.
Circles are said to touch one another which, meeting one another, do not cut one another.
§3.def.4ἐν κύκλῳ ἴσον ἀπέχειν ἀπὸ τοῦ κέντρου εὐθεῖαι λέγονται, ὅταν αἱ ἀπὸ τοῦ κέντρου ἐπʼ αὐτὰς κάθετοι ἀγόμεναι ἴσαι ὦσιν.
Straight lines in a circle are said to be equally distant from the center when the perpendiculars drawn from the center to them are equal.
§3.def.5μεῖζον δὲ ἀπέχειν λέγεται, ἐφʼ ἣν ἡ μείζων κάθετος πίπτει.
And that is said to be at a greater distance on which the greater perpendicular falls.
§3.def.6τμῆμα κύκλου ἐστὶ τὸ περιεχόμενον σχῆμα ὑπό τε εὐθείας καὶ κύκλου περιφερείας.
A segment of a circle is the figure contained by a straight line and a circumference of a circle.
§3.def.7τμήματος δὲ γωνία ἐστὶν ἡ περιεχομένη ὑπό τε εὐθείας καὶ κύκλου περιφερείας.
And an angle of a segment is that contained by a straight line and a circumference of a circle.
§3.def.8ἐν τμήματι δὲ γωνία ἐστίν, ὅταν ἐπὶ τῆς περιφερείας τοῦ τμήματος ληφθῇ τι σημεῖον καὶ ἀπʼ αὐτοῦ ἐπὶ τὰ πέρατα τῆς εὐθείας, ἥ ἐστι βάσις τοῦ τμήματος, ἐπιζευχθῶσιν εὐθεῖαι, ἡ περιεχομένη γωνία ὑπὸ τῶν ἐπιζευχθεισῶν εὐθειῶν.
And an angle in a segment is, when some point is taken on the circumference of the segment and straight lines are joined from it to the extremities of the straight line which is the base of the segment, the angle contained by the joined straight lines.
§3.def.9ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν εὐθεῖαι ἀπολαμβάνωσί τινα περιφέρειαν, ἐπʼ ἐκείνης λέγεται βεβηκέναι ἡ γωνία.
And when the straight lines containing the angle cut off any circumference, the angle is said to stand on that circumference.
§3.def.10Τομεὺς δὲ κύκλου ἐστίν, ὅταν πρὸς τῷ κέντρῳ τοῦ κύκλου συσταθῇ γωνία, τὸ περιεχόμενον σχῆμα ὑπό τε τῶν τὴν γωνίαν περιεχουσῶν εὐθειῶν καὶ τῆς ἀπολαμβανομένης ὑπʼ αὐτῶν περιφερείας.
And a sector of a circle is, when an angle is constructed at the center of the circle, the figure contained by the straight lines containing the angle and the circumference cut off by them.
§3.def.11ὅμοια τμήματα κύκλων ἐστὶ τὰ δεχόμενα γωνίας ἴσας, ἢ ἐν οἷς αἱ γωνίαι ἴσαι ἀλλήλαις εἰσίν.
Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.

Notes

  1. §3.def.1αἱ ἐκ τῶν κέντρων — The feminine noun εὐθεῖαι (straight lines) is omitted; the phrase means "the straight lines drawn from the centers," i.e., the radii. This is the standard Greek mathematical expression for radius.
  2. §3.def.8ἡ περιεχομένη γωνία ὑπὸ τῶν ἐπιζευχθεισῶν εὐθειῶν — This noun phrase is placed appositively after the long conditional clause ὅταν ... ἐπιζευχθῶσιν εὐθεῖαι to define and specify the actual "angle" introduced in the main clause "ἐν τμήματι δὲ γωνία ἐστίν" (an angle in a segment is...).
  3. §3.def.11τὰ δεχόμενα — The verb δέχομαι (to receive/admit) is used in a geometrical sense, meaning to "admit" or contain a specific angle as an inscribed angle.

Cite this passage

Euclid, Elements §3.def.1-3.def.11. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:3.def.1-3.def.11

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