Euclid Book 1 Definitions

Definition 1

Statement

A point is that which has no part.

Original statement

σημϵῖόν ἐστιν, οὗ μέρος οὐθέν.

Definition 2

Statement

A line is breadthless length.

Original statement

γραμμὴ δὲ μῆκος ἀπλατές.

Definition 3

Statement

The extremities of a line are points.

Original statement

γραμμῆς δὲ πέρατα σημϵῖα.

Definition 4

Statement

A straight line is a line which lies evenly with the points on itself.

Original statement

ϵὐθϵῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐϕ᾽ ἑαυτῆς σημϵίοις κϵῖται.

Definition 5

Statement

A surface is that which has length and breadth only.

Original statement

ἐπιϕάνϵια δέ ἐστιν, ὃ μῆκος καὶ πλάτος μόνον ἔχϵι.

Definition 6

Statement

The extremities of a surface are lines.

Original statement

ἐπιϕανϵίας δὲ πέρατα γραμμαί.

Definition 7

Statement

A plane surface is a surface which lies evenly with the straight lines on itself.

Original statement

ἐπίπϵδος ἐπιϕάνϵιά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐϕ᾽ ἑαυτῆς ϵὐθϵίαις κϵῖται.

Definition 8

Statement

A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

Original statement

ἐπίπϵδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπ᾽ ϵὐθϵίας κϵιμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις.

Definition 9

Statement

When the lines containing the angle are straight, the angle is called rectilinear.

Original statement

ὅταν δὲ αἱ πϵριέχουσαι τὴν γωνίαν γραμμαὶ ϵὐθϵῖαι ὦσιν, ϵὐθύγραμμος καλϵῖται ἡ γωνία.

Definition 10

Statement

When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.

Original statement

ὅταν δὲ ϵὐθϵῖα ἐπ᾽ ϵὐθϵῖαν σταθϵῖσα τὰς ἐϕϵξῆς γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν ἴσων γωνιῶν ἐστι, καὶ ἡ ἐϕϵστηκυῖα ϵὐθϵῖα κάθϵτος καλϵῖται, ἐϕ᾽ ἣν ἐϕέστηκϵν.

Definition 11

Statement

An obtuse angle is an angle greater than a right angle.

Original statement

ἀμβλϵῖα γωνία ἐστὶν ἡ μϵίζων ὀρθῆς.

Definition 12

Statement

An acute angle is an angle less than a right angle.

Original statement

ὀξϵῖα δὲ ἡ ἐλάσσων ὀρθῆς.

Definition 13

Statement

A boundary is that which is an extremity of anything.

Original statement

ὅρος ἐστίν, ὅ τινός ἐστι πέρας.

Definition 14

Statement

A figure is that which is contained by any boundary or boundaries.

Original statement

σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων πϵριϵχόμϵνον.

Definition 15

Statement

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.

Original statement

κύκλος ἐστὶ σχῆμα ἐπίπϵδον ὑπὸ μιᾶς γραμμῆς πϵριϵχόμϵνον ἣ καλϵῖται πϵριϕέρϵια, πρὸς ἣν ἀϕ᾽ ἑνὸς σημϵίου τῶν ἐντὸς τοῦ σχήματος κϵιμένων πᾶσαι αἱ προσπίπτουσαι ϵὐθϵῖαι πρὸς τὴν τοῦ κύκλου πϵριϕέρϵιαν ἴσαι ἀλλήλαις ϵἰσίν.

Definition 16

Statement

A point is called the center of a circle when from that point, all the straight lines falling upon the circle, among those lying within the circle, are equal to one another.

Original statement

κέντρον δὲ τοῦ κύκλου τὸ σημϵῖον καλϵῖται.

Definition 17

Statement

A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

Original statement

διάμϵτρος δὲ τοῦ κύκλου ἐστὶν ϵὐθϵῖά τις διὰ τοῦ κέντρου ἠγμένη καὶ πϵρατουμένη ἐϕ᾽ ἑκάτϵρα τὰ μέρη ὑπὸ τῆς τοῦ κύκλου πϵριϕϵρϵίας, ἥτις καὶ δίχα τέμνϵι τὸν κύκλον.

Definition 18

Statement

A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.

Original statement

ἡμικύκλιον δέ ἐστι τὸ πϵριϵχόμϵνον σχῆμα ὑπό τϵ τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπ᾽ αὐτῆς πϵριϕϵρϵίας. κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό, ὃ καὶ τοῦ κύκλου ἐστίν.

Definition 19

Statement

Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Original statement

σχήματα ϵὐθύγραμμά ἐστι τὰ ὑπὸ ϵὐθϵιῶν πϵριϵχόμϵνα, τρίπλϵυρα μὲν τὰ ὑπὸ τριῶν, τϵτράπλϵυρα δὲ τὰ ὑπὸ τϵσσάρων, πολύπλϵυρα δὲ τὰ ὑπὸ πλϵιόνων ἢ τϵσσάρων ϵὐθϵιῶν πϵριϵχόμϵνα.

Definition 20

Statement

Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Original statement

τῶν δὲ τριπλϵύρων σχημάτων ἰσόπλϵυρον μὲν τρίγωνόν ἐστι τὸ τὰς τρϵῖς ἴσας ἔχον πλϵυράς, ἰσοσκϵλὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλϵυράς, σκαληνὸν δὲ τὸ τὰς τρϵῖς ἀνίσους ἔχον πλϵυράς.

Definition 21

Statement

Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Original statement

ἔτι δὲ τῶν τριπλϵύρων σχημάτων ὀρθογώνιον μὲν τρίγωνόν ἐστι τὸ ἔχον ὀρθὴν γωνίαν, ἀμβλυγώνιον δὲ τὸ ἔχον ἀμβλϵῖαν γωνίαν, ὀξυγώνιον δὲ τὸ τὰς τρϵῖς ὀξϵίας ἔχον γωνίας.

Definition 22

Statement

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

Original statement

τῶν δὲ τϵτραπλϵύρων σχημάτων τϵτράγωνον μέν ἐστιν, ὃ ἰσόπλϵυρόν τέ ἐστι καὶ ὀρθογώνιον, ἑτϵρόμηκϵς δέ, ὃ ὀρθογώνιον μέν, οὐκ ἰσόπλϵυρον δέ, ῥόμβος δέ, ὃ ἰσόπλϵυρον μέν, οὐκ ὀρθογώνιον δέ, ῥομβοϵιδὲς δὲ τὸ τὰς ἀπϵναντίον πλϵυράς τϵ καὶ γωνίας ἴσας ἀλλήλαις ἔχον, ὃ οὔτϵ ἰσόπλϵυρόν ἐστιν οὔτϵ ὀρθογώνιον: τὰ δὲ παρὰ ταῦτα τϵτράπλϵυρα τραπέζια καλϵίσθω.

Definition 23

Statement

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Original statement

παράλληλοί ϵἰσιν ϵὐθϵῖαι, αἵτινϵς ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμϵναι ϵἰς ἄπϵιρον ἐϕ᾽ ἑκάτϵρα τὰ μέρη ἐπὶ μηδέτϵρα συμπίπτουσιν ἀλλήλαις.