Euclid, Elements, 1.definitions (test1)
Christopher Blackwell / Euclid
- Created on 2018-03-15 17:03:44
- Modified on 2018-03-15 17:23:58
- Translated by Heath
- Aligned by Christopher Blackwell
Ἑλληνική Transliterate
English
urn:cts:greekLit:tlg1799.tlg001.heiberg:1.def
urn:cts:greekLit:tlg1799.tlg001.heath:1.def
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 , ὃ μῆκος καὶ πλάτος μόνον ἔχει .
ἐπιφανείας δὲ πέρατα γραμμαί .
ἐπίπεδος ἐπιφάνειά ἐστιν , ἥτις ἐξ ἴσου ταῖς ἐφ᾽ ἑαυτῆς εὐθείαις κεῖται .
ἐπίπεδος δὲ γωνία ἐστὶν ἡ 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 ἡ γωνία .
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 .
ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς .
ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς .
ὅρος ἐστίν , ὅ τινός ἐστι πέρας .
σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον .
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 .
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 , ὃ 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 ἢ 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 , 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 .
παράλληλοί εἰσιν εὐθεῖαι , αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ᾽ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις .
γραμμὴ δὲ μῆκος ἀπλατές .
γραμμῆς δὲ πέρατα σημεῖα .
εὐθεῖα γραμμή ἐστιν , ἥτις ἐξ ἴσου τοῖς ἐφ᾽ ἑαυτῆς σημείοις κεῖται .
ἐπιφάνεια δέ ἐστιν , ὃ μῆκος καὶ πλάτος μόνον ἔχει .
ἐπιφανείας δὲ πέρατα γραμμαί .
ἐπίπεδος ἐπιφάνειά ἐστιν , ἥτις ἐξ ἴσου ταῖς ἐφ᾽ ἑαυτῆς εὐθείαις κεῖται .
ἐπίπεδος δὲ γωνία ἐστὶν ἡ 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 ἡ γωνία .
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 .
ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς .
ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς .
ὅρος ἐστίν , ὅ τινός ἐστι πέρας .
σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον .
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 .
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 , ὃ 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 ἢ 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 , 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 .
παράλληλοί εἰσιν εὐθεῖαι , αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ᾽ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις .
A
point
is
that
which
has
no
part
.
A line is breadthless length .
The extremities of a line are points .
A straight line is a line which lies evenly with the points on itself .
A surface is that which has length and breadth only .
The extremities of a surface are lines .
A plane surface is a surface which lies evenly with the straight lines on itself .
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 .
And when the lines containing the angle are straight , the angle is called rectilineal .
urn : cts : greekLit : tlg1799 . tlg001 . heath : 1 . def . 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 .
An obtuse angle is an angle greater than a right angle .
An acute angle is an angle less than a right angle .
A boundary is that which is an extremity of anything .
A figure is that which is contained by any boundary or boundaries .
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 ;
And the point is called the centre of the circle .
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 .
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 .
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 .
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 .
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 .
urn : cts : greekLit : tlg1799 . tlg001 . heath : 1 . def . 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 .
urn : cts : greekLit : tlg1799 . tlg001 . heath : 1 . def . 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 .
A line is breadthless length .
The extremities of a line are points .
A straight line is a line which lies evenly with the points on itself .
A surface is that which has length and breadth only .
The extremities of a surface are lines .
A plane surface is a surface which lies evenly with the straight lines on itself .
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 .
And when the lines containing the angle are straight , the angle is called rectilineal .
urn : cts : greekLit : tlg1799 . tlg001 . heath : 1 . def . 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 .
An obtuse angle is an angle greater than a right angle .
An acute angle is an angle less than a right angle .
A boundary is that which is an extremity of anything .
A figure is that which is contained by any boundary or boundaries .
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 ;
And the point is called the centre of the circle .
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 .
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 .
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 .
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 .
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 .
urn : cts : greekLit : tlg1799 . tlg001 . heath : 1 . def . 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 .
urn : cts : greekLit : tlg1799 . tlg001 . heath : 1 . def . 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 .
Euclid, Elements, 1.definitions (test2)
Christopher Blackwell / Euclid
- Created on 2018-03-15 17:13:14
- Modified on 2018-03-19 13:52:53
- Translated by Heath
- Aligned by Christopher Blackwell
Euclid. Elements. Book 1. Defintions.
Ἑλληνική Transliterate
English
urn:cts:greekLit:tlg1799.tlg001.heiberg:1.def
urn:cts:greekLit:tlg1799.tlg001.heath:1.def
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 , ὃ μῆκος καὶ πλάτος μόνον ἔχει .
ἐπιφανείας δὲ πέρατα γραμμαί .
ἐπίπεδος ἐπιφάνειά ἐστιν , ἥτις ἐξ ἴσου ταῖς ἐφ᾽ ἑαυτῆς εὐθείαις κεῖται .
ἐπίπεδος δὲ γωνία ἐστὶν ἡ 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 ἡ γωνία .
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 .
ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς .
ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς .
ὅρος ἐστίν , ὅ τινός ἐστι πέρας .
σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον .
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 .
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 , ὃ 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 ἢ 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 , 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 .
παράλληλοί εἰσιν εὐθεῖαι , αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ᾽ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις .
γραμμὴ δὲ μῆκος ἀπλατές .
γραμμῆς δὲ πέρατα σημεῖα .
εὐθεῖα γραμμή ἐστιν , ἥτις ἐξ ἴσου τοῖς ἐφ᾽ ἑαυτῆς σημείοις κεῖται .
ἐπιφάνεια δέ ἐστιν , ὃ μῆκος καὶ πλάτος μόνον ἔχει .
ἐπιφανείας δὲ πέρατα γραμμαί .
ἐπίπεδος ἐπιφάνειά ἐστιν , ἥτις ἐξ ἴσου ταῖς ἐφ᾽ ἑαυτῆς εὐθείαις κεῖται .
ἐπίπεδος δὲ γωνία ἐστὶν ἡ 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 ἡ γωνία .
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 .
ἀμβλεῖα γωνία ἐστὶν ἡ μείζων ὀρθῆς .
ὀξεῖα δὲ ἡ ἐλάσσων ὀρθῆς .
ὅρος ἐστίν , ὅ τινός ἐστι πέρας .
σχῆμά ἐστι τὸ ὑπό τινος ἤ τινων ὅρων περιεχόμενον .
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 .
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 , ὃ 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 ἢ 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 , 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 .
παράλληλοί εἰσιν εὐθεῖαι , αἵτινες ἐν τῷ αὐτῷ ἐπιπέδῳ οὖσαι καὶ ἐκβαλλόμεναι εἰς ἄπειρον ἐφ᾽ ἑκάτερα τὰ μέρη ἐπὶ μηδέτερα συμπίπτουσιν ἀλλήλαις .
A
point
is
that
which
has
no
part
.
A line is breadthless length .
The extremities of a line are points .
A straight line is a line which lies evenly with the points on itself .
A surface is that which has length and breadth only .
The extremities of a surface are lines .
A plane surface is a surface which lies evenly with the straight lines on itself .
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 .
And when the lines containing the angle are straight , the angle is called rectilineal .
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 .
An obtuse angle is an angle greater than a right angle .
An acute angle is an angle less than a right angle .
A boundary is that which is an extremity of anything .
A figure is that which is contained by any boundary or boundaries .
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 ;
And the point is called the centre of the circle .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
A line is breadthless length .
The extremities of a line are points .
A straight line is a line which lies evenly with the points on itself .
A surface is that which has length and breadth only .
The extremities of a surface are lines .
A plane surface is a surface which lies evenly with the straight lines on itself .
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 .
And when the lines containing the angle are straight , the angle is called rectilineal .
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 .
An obtuse angle is an angle greater than a right angle .
An acute angle is an angle less than a right angle .
A boundary is that which is an extremity of anything .
A figure is that which is contained by any boundary or boundaries .
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 ;
And the point is called the centre of the circle .
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 .
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 .
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 .
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 .
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 .
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 .
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 .
Euclid Elements, 9.prop.20
Christopher Blackwell / Euclid
- Created on 2018-03-29 14:12:14
- Translated by Heath
- Aligned by Christopher Blackwell
Euclid 9.20. The "infinitude of primes".
Ἑλληνική Transliterate
English
urn:cts:greekLit:tlg1799.tlg001.heiberg:9.prop.20
urn:cts:greekLit:tlg1799.tlg001.heath:9.prop.20
Οἱ πρῶτοι ἀριθμοὶ πλείους εἰσὶ παντὸς τοῦ προτεθέντος πλήθους πρώτων ἀριθμῶν .
Ἔστωσαν οἱ προτεθέντες πρῶτοι ἀριθμοὶ οἱ Α , Β , Γ : λέγω , ὅτι τῶν Α , Β , Γ πλείους εἰσὶ πρῶτοι ἀριθμοί .
Εἰλήφθω γὰρ ὁ ὑπὸ τῶν Α , Β , Γ ἐλάχιστος μετρούμενος καὶ ἔστω ὁ ΔΕ , καὶ προσκείσθω τῷ ΔΕ μονὰς ἡ ΔΖ .
ὁ δὴ ΕΖ ἤτοι πρῶτός ἐστιν ἢ οὔ .
ἔστω πρότερον πρῶτος : εὑρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ οἱ Α , Β , Γ , ΕΖ πλείους τῶν Α , Β , Γ .
Ἀλλὰ δὴ μὴ ἔστω ὁ ΕΖ πρῶτος : ὑπὸ πρώτου ἄρα τινὸς ἀριθμοῦ μετρεῖται .
μετρείσθω ὑπὸ πρώτου τοῦ Η :
λέγω , ὅτι ὁ Η οὐδενὶ τῶν Α , Β , Γ ἐστιν ὁ αὐτός .
εἰ γὰρ δυνατόν , ἔστω .
οἱ δὲ Α , Β , Γ τὸν ΔΕ μετροῦσιν :
καὶ ὁ Η ἄρα τὸν ΔΕ μετρήσει .
μετρεῖ δὲ καὶ τὸν ΕΖ :
καὶ λοιπὴν τὴν ΔΖ μονάδα μετρήσει ὁ Η ἀριθμὸς ὤν : ὅπερ ἄτοπον .
οὐκ ἄρα ὁ Η ἑνὶ τῶν Α , Β , Γ ἐστιν ὁ αὐτός .
καὶ ὑπόκειται πρῶτος .
εὑρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ πλείους τοῦ προτεθέντος πλήθους τῶν Α , Β , Γ οἱ Α , Β , Γ , Η :
ὅπερ ἔδει δεῖξαι .
Ἔστωσαν οἱ προτεθέντες πρῶτοι ἀριθμοὶ οἱ Α , Β , Γ : λέγω , ὅτι τῶν Α , Β , Γ πλείους εἰσὶ πρῶτοι ἀριθμοί .
Εἰλήφθω γὰρ ὁ ὑπὸ τῶν Α , Β , Γ ἐλάχιστος μετρούμενος καὶ ἔστω ὁ ΔΕ , καὶ προσκείσθω τῷ ΔΕ μονὰς ἡ ΔΖ .
ὁ δὴ ΕΖ ἤτοι πρῶτός ἐστιν ἢ οὔ .
ἔστω πρότερον πρῶτος : εὑρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ οἱ Α , Β , Γ , ΕΖ πλείους τῶν Α , Β , Γ .
Ἀλλὰ δὴ μὴ ἔστω ὁ ΕΖ πρῶτος : ὑπὸ πρώτου ἄρα τινὸς ἀριθμοῦ μετρεῖται .
μετρείσθω ὑπὸ πρώτου τοῦ Η :
λέγω , ὅτι ὁ Η οὐδενὶ τῶν Α , Β , Γ ἐστιν ὁ αὐτός .
εἰ γὰρ δυνατόν , ἔστω .
οἱ δὲ Α , Β , Γ τὸν ΔΕ μετροῦσιν :
καὶ ὁ Η ἄρα τὸν ΔΕ μετρήσει .
μετρεῖ δὲ καὶ τὸν ΕΖ :
καὶ λοιπὴν τὴν ΔΖ μονάδα μετρήσει ὁ Η ἀριθμὸς ὤν : ὅπερ ἄτοπον .
οὐκ ἄρα ὁ Η ἑνὶ τῶν Α , Β , Γ ἐστιν ὁ αὐτός .
καὶ ὑπόκειται πρῶτος .
εὑρημένοι ἄρα εἰσὶ πρῶτοι ἀριθμοὶ πλείους τοῦ προτεθέντος πλήθους τῶν Α , Β , Γ οἱ Α , Β , Γ , Η :
ὅπερ ἔδει δεῖξαι .
Prime
numbers
are
more
than
any
assigned
multitude
of
prime
numbers
.
Let A , B , C be the assigned prime numbers ; I say that there are more prime numbers than A , B , C .
For let the least number measured by A , B , C be taken , and let it be DE ; let the unit DF be added to DE .
Then EF is either prime or not .
First , let it be prime ; then the prime numbers A , B , C , EF have been found which are more than A , B , C .
Next , let EF not be prime ; therefore it is measured by some prime number .
Let it be measured by the prime number G .
I say that G is not the same with any of the numbers A , B , C .
For , if possible , let it be so .
Now A , B , C measure DE ; therefore G also will measure DE .
But it also measures EF .
Therefore G , being a number , will measure the remainder , the unit DF : which is absurd .
Therefore G is not the same with any one of the numbers A , B , C .
And by hypothesis it is prime .
Therefore the prime numbers A , B , C , G have been found which are more than the assigned multitude of A , B , C .
Q . E . D .
Let A , B , C be the assigned prime numbers ; I say that there are more prime numbers than A , B , C .
For let the least number measured by A , B , C be taken , and let it be DE ; let the unit DF be added to DE .
Then EF is either prime or not .
First , let it be prime ; then the prime numbers A , B , C , EF have been found which are more than A , B , C .
Next , let EF not be prime ; therefore it is measured by some prime number .
Let it be measured by the prime number G .
I say that G is not the same with any of the numbers A , B , C .
For , if possible , let it be so .
Now A , B , C measure DE ; therefore G also will measure DE .
But it also measures EF .
Therefore G , being a number , will measure the remainder , the unit DF : which is absurd .
Therefore G is not the same with any one of the numbers A , B , C .
And by hypothesis it is prime .
Therefore the prime numbers A , B , C , G have been found which are more than the assigned multitude of A , B , C .
Q . E . D .