Euclid Book 3 Definitions

Definition 1

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.

Definition 2

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.

Definition 3

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.

Definition 4

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.

Definition 5

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.

Definition 6

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.

Definition 7

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.

Definition 8

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.

Definition 9

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.

Definition 10

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.

Definition 11

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.