--- title: "Elements · Στοιχεῖα" author: "Euclid" translator: "Thomas Little Heath" languages: [grc, en] shelf: "Classical Greek" section: 1 of 49 url: https://scriptura.si/works/euclid-elements/1/ cite: "Euclid, tr. Thomas Little Heath, Elements · Στοιχεῖα, section 1. Scriptura (scriptura.si), https://scriptura.si/works/euclid-elements/1/" --- # Elements · Στοιχεῖα *Euclid, tr. Thomas Little Heath* The thirteen books of geometry and number theory that taught mathematics for two thousand years, from the definition of a point to the five regular solids. > Greek: ed. Johan Ludvig Heiberg, from the Perseus Digital Library (canonical-greekLit, tlg1799.tlg001, CC BY-SA 4.0). > English: Thomas Little Heath, from the Perseus Digital Library (tlg1799.tlg001.perseus-eng2). Notes are not included. > Rows follow the references the two texts share (books, chapters and sections), cut where both reach the end of a sentence. ## 1. Book I · Definitions Source: https://scriptura.si/works/euclid-elements/1/ · cite a paragraph as https://scriptura.si/works/euclid-elements/1/#p-N [¶0] 1. A point is that which has no part. > Greek: 1. σημεῖόν ἐστιν, οὗ μέρος οὐθέν. [¶1] 2. A line is breadthless length. > Greek: 2. γραμμὴ δὲ μῆκος ἀπλατές. [¶2] 3. The extremities of a line are points. > Greek: 3. γραμμῆς δὲ πέρατα σημεῖα. [¶3] 4. A straight line is a line which lies evenly with the points on itself. > Greek: 4. εὐθεῖα γραμμή ἐστιν, ἥτις ἐξ ἴσου τοῖς ἐφʼ ἑαυτῆς σημείοις κεῖται. [¶4] 5. A surface is that which has length and breadth only. > Greek: 5. ἐπιφάνεια δέ ἐστιν, ὃ μῆκος καὶ πλάτος μόνον ἔχει. [¶5] 6. The extremities of a surface are lines. > Greek: 6. ἐπιφανείας δὲ πέρατα γραμμαί. [¶6] 7. A plane surface is a surface which lies evenly with the straight lines on itself. > Greek: 7. ἐπίπεδος ἐπιφάνειά ἐστιν, ἥτις ἐξ ἴσου ταῖς ἐφʼ ἑαυτῆς εὐθείαις κεῖται. [¶7] 8. 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. > Greek: 8. ἐπίπεδος δὲ γωνία ἐστὶν ἡ ἐν ἐπιπέδῳ δύο γραμμῶν ἁπτομένων ἀλλήλων καὶ μὴ ἐπʼ εὐθείας κειμένων πρὸς ἀλλήλας τῶν γραμμῶν κλίσις. [¶8] 9. And when the lines containing the angle are straight, the angle is called rectilineal. > Greek: 9. ὅταν δὲ αἱ περιέχουσαι τὴν γωνίαν γραμμαὶ εὐθεῖαι ὦσιν, εὐθύγραμμος καλεῖται ἡ γωνία. [¶9] 10. 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. > Greek: 10. ὅταν 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. [¶10] 11. An obtuse angle is an angle greater than a right angle. > Greek: 11. ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς. [¶11] 12. An acute angle is an angle less than a right angle. > Greek: 12. ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς. [¶12] 13. A boundary is that which is an extremity of anything. > Greek: 13. ὅρος ἐστίν, ὅ τινός ἐστι πέρας. [¶13] 14. A figure is that which is contained by any boundary or boundaries. > Greek: 14. σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον. [¶14] 15. 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; > Greek: 15. κύκλος ἐστὶ σχῆμα ἐπίπεδον ὑπὸ μιᾶς γραμμῆς περιεχόμενον [ἣ καλεῖται περιφέρεια], πρὸς ἣν ἀφʼ ἑνὸς σημείου τῶν ἐντὸς τοῦ σχήματος κειμένων πᾶσαι αἱ προσπίπτουσαι εὐθεῖαι [πρὸς τὴν τοῦ κύκλου περιφέρειαν] ἴσαι ἀλλήλαις εἰσίν. [¶15] 16. And the point is called the centre of the circle. > Greek: 16. κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται. [¶16] 17. A diameter of the circle is any straight line drawn through the centre and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle. > Greek: 17. διάμετρος 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. [¶17] 18. A semicircle is the figure contained by the diameter and the circumference cut off by it. And the centre of the semicircle is the same as that of the circle. > Greek: 18. ἡμικύκλιον δέ ἐστι τὸ περιεχόμενον σχῆμα ὑπό τε τῆς διαμέτρου καὶ τῆς ἀπολαμβανομένης ὑπʼ αὐτῆς περιφερείας. κέντρον δὲ τοῦ ἡμικυκλίου τὸ αὐτό, ὃ καὶ τοῦ κύκλου ἐστίν. [¶18] 19. Rectilineal 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. > Greek: 19. 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. [¶19] 20. 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. > Greek: 20. 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. [¶20] 21. 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 acuteangled triangle that which has its three angles acute. > Greek: 21. 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. [¶21] 22. 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. > Greek: 22. 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. [¶22] 23. 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. > Greek: 23. παράλληλοί εἰσιν εὐθεῖαι, αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφʼ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις.