Selections

Nik Verbeke / Euclids Elements
  • Created on 2018-03-20 14:06:50
  • Modified on 2018-03-20 14:17:00
  • Translated by Heath
  • Aligned by Nik Verbeke
Ἑλληνική Transliterate
English
σημεῖόν ἐστιν , οὗ μέρος οὐθέν .



γραμμὴ δὲ μῆκος ἀπλατές .



γραμμῆς δὲ πέρατα σημεῖα .



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



ἐπιφάνεια δέ ἐστιν , μῆκος καὶ πλάτος μόνον ἔχει .



ἐπιφανείας δὲ πέρατα γραμμαί .



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



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



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



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



ἀμβλεῖα γωνία ἐστὶν μείζων ὀρθῆς .



ὀξεῖα δὲ ἐλάσσων ὀρθῆς .



ὅρος ἐστίν , τινός ἐστι πέρας .



σχῆμά ἐστι τὸ ὑπό τινος τινων ὅρων περιεχόμενον .



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



κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται .



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 .



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



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



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 .



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 .



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



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



POSTULATES .



Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν .



καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖν .



καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι .



καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι .



καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ , ἐκβαλλομένας τὰς δύο εὐθείας ἐπ᾽ ἄπειρον συμπίπτειν , ἐφ᾽ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες .





COMMON NOTIONS .



τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα .



καὶ ἐὰν ἴσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἴσα .



καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀφαιρεθῇ , τὰ καταλειπόμενά ἐστιν ἴσα .



καὶ ἐὰν ἀνίσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἄνισα .



καὶ τὰ τοῦ αὐτοῦ διπλάσια ἴσα ἀλλήλοις ἐστίν .



καὶ τὰ τοῦ αὐτοῦ ἡμίση ἴσα ἀλλήλοις ἐστίν .



καὶ τὰ ἐφαρμόζοντα ἐπ᾽ ἄλληλα ἴσα ἀλλήλοις ἐστίν .



καὶ τὸ ὅλον τοῦ μέρους μεῖζον ἐστιν .



καὶ δύο εὐθεῖαι χωρίον οὐ περιέχουσιν .



PROPOSITION 1



ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συστήσασθαι .



ἔστω δοθεῖσα εὐθεῖα πεπερασμένη ΑΒ . δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστήσασθαι .



κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ΒΓΔ , καὶ πάλιν κέντρῳ μὲν τῷ Β διαστήματι δὲ τῷ ΒΑ κύκλος γεγράφθω ΑΓΕ , καὶ ἀπὸ τοῦ Γ σημείου , καθ᾽ τέμνουσιν ἀλλήλους οἱ κύκλοι , ἐπὶ τὰ Α , Β σημεῖα ἐπεζεύχθωσαν εὐθεῖαι αἱ ΓΑ , ΓΒ .



καὶ ἐπεὶ τὸ Α σημεῖον κέντρον ἐστὶ τοῦ ΓΔΒ κύκλου , ἴση ἐστὶν ΑΓ τῇ ΑΒ : πάλιν , ἐπεὶ τὸ Β σημεῖον κέντρον ἐστὶ τοῦ ΓΑΕ κύκλου , ἴση ἐστὶν ΒΓ τῇ ΒΑ . ἐδείχθη δὲ καὶ ΓΑ τῇ ΑΒ ἴση : ἑκατέρα ἄρα τῶν ΓΑ , ΓΒ τῇ ΑΒ ἐστὶν ἴση .



τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα : καὶ ΓΑ ἄρα τῇ ΓΒ ἐστὶν ἴση : αἱ τρεῖς ἄρα αἱ ΓΑ , ΑΒ , ΒΓ ἴσαι ἀλλήλαις εἰσίν .



ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον , καὶ συνέσταται ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τῆς ΑΒ .



Ἐπὶ τῆς δοθείσης ἄρα εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συνέσταται : ὅπερ ἔδει ποιῆσαι .
1 A point is that which has no part .



2 A line is breadthless length .



3 The extremities of a line are points .



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



5 A surface is that which has length and breadth only .



6 The extremities of a surface are lines .



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



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 .



9 And when the lines containing the angle are straight , the angle is called rectilineal .



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 .



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



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



13 A boundary is that which is an extremity of anything .



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



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 ;



16 And the point is called the centre of the circle .



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 .



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 .



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 .



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 .



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 .



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 .



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 .



POSTULATES .



1 Let the following be postulated : To draw a straight line from any point to any point .



2 To produce a finite straight line continuously in a straight line .



3 To describe a circle with any centre and distance .



4 That all right angles are equal to one another .



5 That , if a straight line falling on two straight lines make the interior angles on the same side less than two right angles , the two straight lines , if produced indefinitely , meet on that side on which are the angles less than the two right angles .



COMMON NOTIONS .



1 Things which are equal to the same thing are also equal to one another .



2 If equals be added to equals , the wholes are equal .



3 If equals be subtracted from equals , the remainders are equal .



7 Things which coincide with one another are equal to one another .



8 The whole is greater than the part .



PROPOSITION 1



On a given finite straight line to construct an equilateral triangle .



Let AB be the given finite straight line . Thus it is required to construct an equilateral triangle on the straight line AB .



With centre A and distance AB let the circle BCD be described ; [ Post . 3 ] again , with centre B and distance BA let the circle ACE be described ; [ Post . 3 ] and from the point C , in which the circles cut one another , to the points A , B let the straight lines CA , CB be joined . [ Post . 1 ]



Now , since the point A is the centre of the circle CDB , AC is equal to AB . [ Def . 15 ] Again , since the point B is the centre of the circle CAE , BC is equal to BA . [ Def . 15 ] But CA was also proved equal to AB ; therefore each of the straight lines CA , CB is equal to AB .



And things which are equal to the same thing are also equal to one another ; [ C . N . 1 ] therefore CA is also equal to CB . Therefore the three straight lines CA , AB , BC are equal to one another .



Therefore the triangle ABC is equilateral ; and it has been constructed on the given finite straight line AB .



Being what it was required to do .

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Euclid, Elements, Selections

Nathaniel Bilodeau / Euclid, Elements, Selections
  • Created on 2018-03-20 13:53:55
  • Modified on 2018-03-20 14:21:53
  • Aligned by Nathaniel Bilodeau
Ἑλληνική Transliterate
English
DEFINITIONS .



σημεῖόν ἐστιν , οὗ μέρος οὐθέν .



γραμμὴ δὲ μῆκος ἀπλατές .



γραμμῆς δὲ πέρατα σημεῖα .



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



ἐπιφάνεια δέ ἐστιν , μῆκος καὶ πλάτος μόνον ἔχει .



ἐπιφανείας δὲ πέρατα γραμμαί .



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



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



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



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



ἀμβλεῖα γωνία ἐστὶν μείζων ὀρθῆς .



ὀξεῖα δὲ ἐλάσσων ὀρθῆς .



ὅρος ἐστίν , τινός ἐστι πέρας .



σχῆμά ἐστι τὸ ὑπό τινος τινων ὅρων περιεχόμενον .



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



κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται .



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 .



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



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



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 .



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 .



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



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



POSTULATES .



Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν .



καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖν .



καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι .



καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι .



καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ , ἐκβαλλομένας τὰς δύο εὐθείας ἐπ᾽ ἄπειρον συμπίπτειν , ἐφ᾽ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες .





COMMON NOTIONS .



τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα .



καὶ ἐὰν ἴσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἴσα .



καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀφαιρεθῇ , τὰ καταλειπόμενά ἐστιν ἴσα .



καὶ ἐὰν ἀνίσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἄνισα .



καὶ τὰ τοῦ αὐτοῦ διπλάσια ἴσα ἀλλήλοις ἐστίν .



καὶ τὰ τοῦ αὐτοῦ ἡμίση ἴσα ἀλλήλοις ἐστίν .



καὶ τὰ ἐφαρμόζοντα ἐπ᾽ ἄλληλα ἴσα ἀλλήλοις ἐστίν .



καὶ τὸ ὅλον τοῦ μέρους μεῖζον ἐστιν .



καὶ δύο εὐθεῖαι χωρίον οὐ περιέχουσιν .



PROPOSITION 1



ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συστήσασθαι .



ἔστω δοθεῖσα εὐθεῖα πεπερασμένη ΑΒ . δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστήσασθαι .



κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ΒΓΔ , καὶ πάλιν κέντρῳ μὲν τῷ Β διαστήματι δὲ τῷ ΒΑ κύκλος γεγράφθω ΑΓΕ , καὶ ἀπὸ τοῦ Γ σημείου , καθ᾽ τέμνουσιν ἀλλήλους οἱ κύκλοι , ἐπὶ τὰ Α , Β σημεῖα ἐπεζεύχθωσαν εὐθεῖαι αἱ ΓΑ , ΓΒ .



καὶ ἐπεὶ τὸ Α σημεῖον κέντρον ἐστὶ τοῦ ΓΔΒ κύκλου , ἴση ἐστὶν ΑΓ τῇ ΑΒ : πάλιν , ἐπεὶ τὸ Β σημεῖον κέντρον ἐστὶ τοῦ ΓΑΕ κύκλου , ἴση ἐστὶν ΒΓ τῇ ΒΑ . ἐδείχθη δὲ καὶ ΓΑ τῇ ΑΒ ἴση : ἑκατέρα ἄρα τῶν ΓΑ , ΓΒ τῇ ΑΒ ἐστὶν ἴση .



τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα : καὶ ΓΑ ἄρα τῇ ΓΒ ἐστὶν ἴση : αἱ τρεῖς ἄρα αἱ ΓΑ , ΑΒ , ΒΓ ἴσαι ἀλλήλαις εἰσίν .



ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον , καὶ συνέσταται ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τῆς ΑΒ .



Ἐπὶ τῆς δοθείσης ἄρα εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συνέσταται : ὅπερ ἔδει ποιῆσαι .
DEFINITIONS .



1 A point is that which has no part .



2 A line is breadthless length .



3 The extremities of a line are points .



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



5 A surface is that which has length and breadth only .



6 The extremities of a surface are lines .



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



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 .



9 And when the lines containing the angle are straight , the angle is called rectilineal .



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 .



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



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



13 A boundary is that which is an extremity of anything .



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



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 ;



16 And the point is called the centre of the circle .



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 .



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 .



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 .



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 .



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 .



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 .



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 .



POSTULATES .



1 Let the following be postulated : To draw a straight line from any point to any point .



2 To produce a finite straight line continuously in a straight line .



3 To describe a circle with any centre and distance .



4 That all right angles are equal to one another .



5 That , if a straight line falling on two straight lines make the interior angles on the same side less than two right angles , the two straight lines , if produced indefinitely , meet on that side on which are the angles less than the two right angles .



COMMON NOTIONS .



1 Things which are equal to the same thing are also equal to one another .



2 If equals be added to equals , the wholes are equal .



3 If equals be subtracted from equals , the remainders are equal .



7 Things which coincide with one another are equal to one another .



8 The whole is greater than the part .



PROPOSITION 1



On a given finite straight line to construct an equilateral triangle .



Let AB be the given finite straight line . Thus it is required to construct an equilateral triangle on the straight line AB .



With centre A and distance AB let the circle BCD be described ; [ Post . 3 ] again , with centre B and distance BA let the circle ACE be described ; [ Post . 3 ] and from the point C , in which the circles cut one another , to the points A , B let the straight lines CA , CB be joined . [ Post . 1 ]



Now , since the point A is the centre of the circle CDB , AC is equal to AB . [ Def . 15 ] Again , since the point B is the centre of the circle CAE , BC is equal to BA . [ Def . 15 ] But CA was also proved equal to AB ; therefore each of the straight lines CA , CB is equal to AB .



And things which are equal to the same thing are also equal to one another ; [ C . N . 1 ] therefore CA is also equal to CB . Therefore the three straight lines CA , AB , BC are equal to one another .



Therefore the triangle ABC is equilateral ; and it has been constructed on the given finite straight line AB .



Being what it was required to do .

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Euclid, Elements, 1.postulates

Janey Capers Newland / Euclid
Euclid, Elements, book 1, definitions

Euclid, Elements, Selections

Neal Yates / Geometry Greek Drawing
  • Created on 2018-03-20 08:41:28
  • Aligned by Neal Yates
Ἑλληνική Transliterate
English
Euclid, Elements, Selections
σημεῖόν ἐστιν , οὗ μέρος οὐθέν .



γραμμὴ δὲ μῆκος ἀπλατές .



γραμμῆς δὲ πέρατα σημεῖα .



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



ἐπιφάνεια δέ ἐστιν , μῆκος καὶ πλάτος μόνον ἔχει .



ἐπιφανείας δὲ πέρατα γραμμαί .



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



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



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



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



ἀμβλεῖα γωνία ἐστὶν μείζων ὀρθῆς .



ὀξεῖα δὲ ἐλάσσων ὀρθῆς .



ὅρος ἐστίν , τινός ἐστι πέρας .



σχῆμά ἐστι τὸ ὑπό τινος τινων ὅρων περιεχόμενον .



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



κέντρον δὲ τοῦ κύκλου τὸ σημεῖον καλεῖται .



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 .



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



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



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 .



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 .



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



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



POSTULATES .



Ἠιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖν .



καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖν .



καὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράφεσθαι .



καὶ πάσας τὰς ὀρθὰς γωνίας ἴσας ἀλλήλαις εἶναι .



καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ , ἐκβαλλομένας τὰς δύο εὐθείας ἐπ᾽ ἄπειρον συμπίπτειν , ἐφ᾽ μέρη εἰσὶν αἱ τῶν δύο ὀρθῶν ἐλάσσονες .





COMMON NOTIONS .



τὰ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα .



καὶ ἐὰν ἴσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἴσα .



καὶ ἐὰν ἀπὸ ἴσων ἴσα ἀφαιρεθῇ , τὰ καταλειπόμενά ἐστιν ἴσα .



καὶ ἐὰν ἀνίσοις ἴσα προστεθῇ , τὰ ὅλα ἐστὶν ἄνισα .



καὶ τὰ τοῦ αὐτοῦ διπλάσια ἴσα ἀλλήλοις ἐστίν .



καὶ τὰ τοῦ αὐτοῦ ἡμίση ἴσα ἀλλήλοις ἐστίν .



καὶ τὰ ἐφαρμόζοντα ἐπ᾽ ἄλληλα ἴσα ἀλλήλοις ἐστίν .



καὶ τὸ ὅλον τοῦ μέρους μεῖζον ἐστιν .



καὶ δύο εὐθεῖαι χωρίον οὐ περιέχουσιν .



PROPOSITION 1



ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συστήσασθαι .



ἔστω δοθεῖσα εὐθεῖα πεπερασμένη ΑΒ . δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστήσασθαι .



κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ΒΓΔ , καὶ πάλιν κέντρῳ μὲν τῷ Β διαστήματι δὲ τῷ ΒΑ κύκλος γεγράφθω ΑΓΕ , καὶ ἀπὸ τοῦ Γ σημείου , καθ᾽ τέμνουσιν ἀλλήλους οἱ κύκλοι , ἐπὶ τὰ Α , Β σημεῖα ἐπεζεύχθωσαν εὐθεῖαι αἱ ΓΑ , ΓΒ .



καὶ ἐπεὶ τὸ Α σημεῖον κέντρον ἐστὶ τοῦ ΓΔΒ κύκλου , ἴση ἐστὶν ΑΓ τῇ ΑΒ : πάλιν , ἐπεὶ τὸ Β σημεῖον κέντρον ἐστὶ τοῦ ΓΑΕ κύκλου , ἴση ἐστὶν ΒΓ τῇ ΒΑ . ἐδείχθη δὲ καὶ ΓΑ τῇ ΑΒ ἴση : ἑκατέρα ἄρα τῶν ΓΑ , ΓΒ τῇ ΑΒ ἐστὶν ἴση .



τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα : καὶ ΓΑ ἄρα τῇ ΓΒ ἐστὶν ἴση : αἱ τρεῖς ἄρα αἱ ΓΑ , ΑΒ , ΒΓ ἴσαι ἀλλήλαις εἰσίν .



ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον , καὶ συνέσταται ἐπὶ τῆς δοθείσης εὐθείας πεπερασμένης τῆς ΑΒ .



Ἐπὶ τῆς δοθείσης ἄρα εὐθείας πεπερασμένης τρίγωνον ἰσόπλευρον συνέσταται : ὅπερ ἔδει ποιῆσαι .

1 A point is that which has no part .



2 A line is breadthless length .



3 The extremities of a line are points .



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



5 A surface is that which has length and breadth only .



6 The extremities of a surface are lines .



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



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 .



9 And when the lines containing the angle are straight , the angle is called rectilineal .



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 .



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



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



13 A boundary is that which is an extremity of anything .



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



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 ;



16 And the point is called the centre of the circle .



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 .



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 .



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 .



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 .



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 .



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 .