Browse books
Euclid Book 3 Definitions
Statement
Equal circles are those the diameters of which are equal, or the radii of which are equal.
Original statement
ἴσοι κύκλοι ϵἰσίν, ὧν αἱ διάμϵτροι ἴσαι ϵἰσίν, ἢ ὧν αἱ ἐκ τῶν κέντρων ἴσαι ϵἰσίν.
Statement
A straight line is said to touch a circle which, meeting the circle and being produced, does not cut the circle.
Original statement
ϵὐθϵῖα κύκλου ἐϕάπτϵσθαι λέγϵται, ἥτις ἁπτομένη τοῦ κύκλου καὶ ἐκβαλλομένη οὐ τέμνϵι τὸν κύκλον.
Statement
Circles are said to touch one another which, meeting one another, do not cut one another.
Original statement
κύκλοι ἐϕάπτϵσθαι ἀλλήλων λέγονται οἵτινϵς ἁπτόμϵνοι ἀλλήλων οὐ τέμνουσιν ἀλλήλους.
Statement
In a circle straight lines are said to be equally distant from the center when the perpendiculars drawn to them from the center are equal.
Original statement
ἐν κύκλῳ ἴσον ἀπέχϵιν ἀπὸ τοῦ κέντρου ϵὐθϵῖαι λέγονται, ὅταν αἱ ἀπὸ τοῦ κέντρου ἐπ᾽ αὐτὰς κάθϵτοι ἀγόμϵναι ἴσαι ὦσιν.
Statement
Among straight lines in a circle, that straight line is said to be at a greater distance from the center on which the greater perpendicular from the center falls.
Original statement
μϵῖζον δὲ ἀπέχϵιν λέγϵται, ἐϕ᾽ ἣν ἡ μϵίζων κάθϵτος πίπτϵι.
Statement
A segment of a circle is the figure contained by a straight line and a circumference of a circle.
Original statement
τμῆμα κύκλου ἐστὶ τὸ πϵριϵχόμϵνον σχῆμα ὑπό τϵ ϵὐθϵίας καὶ κύκλου πϵριϕϵρϵίας.
Statement
An angle of a segment is that contained by a straight line and a circumference of a circle.
Original statement
τμήματος δὲ γωνία ἐστὶν ἡ πϵριϵχομένη ὑπό τϵ ϵὐθϵίας καὶ κύκλου πϵριϕϵρϵίας.
Statement
An angle in a segment is the angle which, when a 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, is contained by the straight lines so joined.
Original statement
ἐν τμήματι δὲ γωνία ἐστίν, ὅταν ἐπὶ τῆς πϵριϕϵρϵίας τοῦ τμήματος ληϕθῇ τι σημϵῖον καὶ ἀπ᾽ αὐτοῦ ἐπὶ τὰ πέρατα τῆς ϵὐθϵίας, ἥ ἐστι βάσις τοῦ τμήματος, ἐπιζϵυχθῶσιν ϵὐθϵῖαι, ἡ πϵριϵχομένη γωνία ὑπὸ τῶν ἐπιζϵυχθϵισῶν ϵὐθϵιῶν.
Statement
An angle is said to stand upon a circumference when the straight lines containing the angle cut off the circumference.
Original statement
ὅταν δὲ αἱ πϵριέχουσαι τὴν γωνίαν ϵὐθϵῖαι ἀπολαμβάνωσί τινα πϵριϕέρϵιαν, ἐπ᾽ ἐκϵίνης λέγϵται βϵβηκέναι ἡ γωνία.
Statement
A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.
Original statement
Τομϵὺς δὲ κύκλου ἐστίν, ὅταν πρὸς τῷ κέντρῳ τοῦ κύκλου συσταθῇ γωνία, τὸ πϵριϵχόμϵνον σχῆμα ὑπό τϵ τῶν τὴν γωνίαν πϵριϵχουσῶν ϵὐθϵιῶν καὶ τῆς ἀπολαμβανομένης ὑπ᾽ αὐτῶν πϵριϕϵρϵίας.
Statement
Similar segments of circles are those which admit equal angles, or in which the angles are equal to one another.
Original statement
ὅμοια τμήματα κύκλων ἐστὶ τὰ δϵχόμϵνα γωνίας ἴσας, ἢ ἐν οἷς αἱ γωνίαι ἴσαι ἀλλήλαις ϵἰσίν.