§1.def.1σημεῖόν ἐστιν, οὗ μέρος οὐθέν.
A point is that which has no part.
§1.def.2γραμμὴ δὲ μῆκος ἀπλατές.
A line is a breadthless length.
§1.def.3γραμμῆς δὲ πέρατα σημεῖα.
The extremities of a line are points.
§1.def.4εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφʼ ἑαυτῆς σημείοις κεῖται.
A straight line is that which lies equally with respect to the points on itself.
§1.def.5ἐπιφάνεια δέ ἐστιν, ὃ μῆκος καὶ πλάτος μόνον ἔχει.
A surface is that which has length and breadth only.
§1.def.6ἐπιφανείας δὲ πέρατα γραμμαί.
The extremities of a surface are lines.
§1.def.7ἐπίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφʼ ἑαυτῆς εὐθείαις κεῖται.
A plane surface is that which lies equally with respect to the straight lines on itself.
§1.def.8ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπʼ εὐθείας κειμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις.
A plane angle is the inclination to one another of two lines in a plane which meet each other and do not lie in a straight line.
§1.def.9ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν γραμμαὶ εὐθεῖαι ὦσιν, εὐθύγραμμος καλεῖται ἡ γωνία.
And when the lines containing the angle are straight, the angle is called rectilineal.
§1.def.10ὅταν δὲ εὐθεῖα ἐπʼ εὐθεῖαν σταθεῖσα τὰς ἐφεξῆς γωνίας ἴσας ἀλλήλαις ποιῇ, ὀρθὴ ἑκατέρα τῶν ἴσων γωνιῶν ἐστι, καὶ ἡ ἐφεστηκυῖα εὐθεῖα κάθετος καλεῖται, ἐφʼ ἣν ἐφέστηκεν.
And when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is a right angle, and the straight line standing on the other is called a perpendicular to that on which it stands.
§1.def.11ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς.
An obtuse angle is that which is greater than a right angle.
§1.def.12ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς.
And an acute angle is that which is less than a right angle.
§1.def.13ὅρος ἐστίν, ὅ τινός ἐστι πέρας.
A boundary is that which is an extremity of anything.
§1.def.14σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον.
A figure is that which is contained by any boundary or boundaries.
§1.def.15κύκλος ἐστὶ σχῆμα ἐπίπεδον ὑπὸ μιᾶς γραμμῆς περιεχόμενον, πρὸς ἣν ἀφʼ ἑνὸς σημείου τῶν ἐντὸς τοῦ σχήματος κειμένων πᾶσαι αἱ προσπίπτουσαι εὐθεῖαι ἴσαι ἀλλήλαις εἰσίν.
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.
§1.def.16κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται.
And the point is called the centre of the circle.
§1.def.17διάμετρος δὲ τοῦ κύκλου ἐστὶν εὐθεῖά τις διὰ τοῦ κέντρου ἠγμένη καὶ περατουμένη ἐφʼ ἑκάτερα τὰ μέρη ὑπὸ τῆς τοῦ κύκλου περιφερείας, ἥτις καὶ δίχα τέμνει τὸν κύκλον.
A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, which also bisects the circle.
§1.def.18ἡμικύκλιον δέ ἐστι τὸ περιεχόμενον σχῆμα ὑπό τε τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπʼ αὐτῆς περιφερείας.
A semicircle is the figure contained by the diameter and the circumference cut off by it.
κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό, ὃ καὶ τοῦ κύκλου ἐστίν.
And the centre of the semicircle is the same as that of the circle.
§1.def.19σχήματα 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.
Rectilineal figures are those contained by straight lines, trilateral figures being those contained by three, quadrilateral those by four, and multilateral those contained by more than four straight lines.
§1.def.20τῶν 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.
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.
§1.def.21ἔτι 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.
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.
§1.def.22τῶν 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·
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.
τὰ δὲ παρὰ ταῦτα τετράπλευρα τραπέζια καλείσθω.
And let quadrilaterals other than these be called trapezia.
§1.def.23παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφʼ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις.
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.