Max Easter
Furman Student
Euclid, Elements, Sections
Max Easter /
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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 , 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 .
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᾽ ἃ 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 ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ . δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστήσασθαι .
κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ὁ ΒΓΔ , καὶ πάλιν κέντρῳ μὲν τῷ Β διαστήματι δὲ τῷ ΒΑ κύκλος γεγράφθω ὁ ΑΓΕ , καὶ ἀπὸ τοῦ Γ σημείου , καθ᾽ ὃ 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 ἡ ΑΓ τῇ ΑΒ : πάλιν , ἐπεὶ τὸ Β σημεῖον κέντρον ἐστὶ τοῦ ΓΑΕ κύκλου , ἴση ἐστὶν ἡ ΒΓ τῇ ΒΑ . ἐδείχθη δὲ καὶ ἡ ΓΑ τῇ ΑΒ ἴση : ἑκατέρα ἄρα τῶν ΓΑ , ΓΒ τῇ ΑΒ ἐστὶν ἴση .
τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα : καὶ ἡ 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 .
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 .
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᾽ ἃ 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 ἡ δοθεῖσα εὐθεῖα πεπερασμένη ἡ ΑΒ . δεῖ δὴ ἐπὶ τῆς ΑΒ εὐθείας τρίγωνον ἰσόπλευρον συστήσασθαι .
κέντρῳ μὲν τῷ Α διαστήματι δὲ τῷ ΑΒ κύκλος γεγράφθω ὁ ΒΓΔ , καὶ πάλιν κέντρῳ μὲν τῷ Β διαστήματι δὲ τῷ ΒΑ κύκλος γεγράφθω ὁ ΑΓΕ , καὶ ἀπὸ τοῦ Γ σημείου , καθ᾽ ὃ 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 ἡ ΑΓ τῇ ΑΒ : πάλιν , ἐπεὶ τὸ Β σημεῖον κέντρον ἐστὶ τοῦ ΓΑΕ κύκλου , ἴση ἐστὶν ἡ ΒΓ τῇ ΒΑ . ἐδείχθη δὲ καὶ ἡ ΓΑ τῇ ΑΒ ἴση : ἑκατέρα ἄρα τῶν ΓΑ , ΓΒ τῇ ΑΒ ἐστὶν ἴση .
τὰ δὲ τῷ αὐτῷ ἴσα καὶ ἀλλήλοις ἐστὶν ἴσα : καὶ ἡ 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 .
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 .
John 1.43-51
Max Easter /
- Created on 2018-04-09 13:30:57
- Modified on 2018-04-17 14:27:54
- Aligned by Max Easter
Ἑλληνική Transliterate
English
[ 43 ] Τῇ ἐπαύριον ἠθέλησεν ἐξελθεῖν εἰς τὴν Γαλιλαίαν . καὶ εὑρίσκει Φίλιππον καὶ λέγει αὐτῷ ὁ Ἰησοῦς Ἀκολούθει μοι .
[ 44 ] ἦν δὲ ὁ Φίλιππος ἀπὸ Βηθσαιδά , ἐκ τῆς πόλεως Ἀνδρέου καὶ Πέτρου .
[ 45 ] εὑρίσκει Φίλιππος τὸν Ναθαναὴλ καὶ λέγει αὐτῷ Ὃν ἔγραψεν Μωυσῆς ἐν τῷ νόμῳ καὶ οἱ προφῆται εὑρήκαμεν , Ἰησοῦν υἱὸν τοῦ Ἰωσὴφ τὸν ἀπὸ Ναζαρέτ .
[ 46 ] καὶ εἶπεν αὐτῷ Ναθαναήλ Ἐκ Ναζαρὲτ δύναταί τι ἀγαθὸν εἶναι ; λέγει αὐτῷ ὁ Φίλιππος Ἔρχου καὶ ἴδε .
[ 47 ] εἶδεν Ἰησοῦς τὸν Ναθαναὴλ ἐρχόμενον πρὸς αὐτὸν καὶ λέγει περὶ αὐτοῦ Ἴδε ἀληθῶς Ἰσραηλείτης ἐν ᾧ δόλος οὐκ ἔστιν .
[ 48 ] λέγει αὐτῷ Ναθαναήλ Πόθεν με γινώσκεις ; ἀπεκρίθη Ἰησοῦς καὶ εἶπεν αὐτῷ Πρὸ τοῦ σε Φίλιππον φωνῆσαι ὄντα ὑπὸ τὴν συκῆν εἶδόν σε .
[ 49 ] ἀπεκρίθη αὐτῷ Ναθαναήλ Ῥαββεί , σὺ εἶ ὁ υἱὸς τοῦ θεοῦ , σὺ βασιλεὺς εἶ τοῦ Ἰσραήλ .
[ 50 ] ἀπεκρίθη Ἰησοῦς καὶ εἶπεν αὐτῷ Ὅτι εἶπόν σοι ὅτι εἶδόν σε ὑποκάτω τῆς συκῆς πιστεύεις ; μείζω τούτων ὄψῃ .
[ 51 ] καὶ λέγει αὐτῷ Ἀμὴν ἀμὴν λέγω ὑμῖν , ὄψεσθε " τὸν οὐρανὸν " ἀνεῳγότα καὶ " τοὺς ἀγγέλους τοῦ θεοῦ ἀναβαίνοντας καὶ καταβαίνοντας " ἐπὶ τὸν υἱὸν τοῦ ἀνθρώπου .
[ 44 ] ἦν δὲ ὁ Φίλιππος ἀπὸ Βηθσαιδά , ἐκ τῆς πόλεως Ἀνδρέου καὶ Πέτρου .
[ 45 ] εὑρίσκει Φίλιππος τὸν Ναθαναὴλ καὶ λέγει αὐτῷ Ὃν ἔγραψεν Μωυσῆς ἐν τῷ νόμῳ καὶ οἱ προφῆται εὑρήκαμεν , Ἰησοῦν υἱὸν τοῦ Ἰωσὴφ τὸν ἀπὸ Ναζαρέτ .
[ 46 ] καὶ εἶπεν αὐτῷ Ναθαναήλ Ἐκ Ναζαρὲτ δύναταί τι ἀγαθὸν εἶναι ; λέγει αὐτῷ ὁ Φίλιππος Ἔρχου καὶ ἴδε .
[ 47 ] εἶδεν Ἰησοῦς τὸν Ναθαναὴλ ἐρχόμενον πρὸς αὐτὸν καὶ λέγει περὶ αὐτοῦ Ἴδε ἀληθῶς Ἰσραηλείτης ἐν ᾧ δόλος οὐκ ἔστιν .
[ 48 ] λέγει αὐτῷ Ναθαναήλ Πόθεν με γινώσκεις ; ἀπεκρίθη Ἰησοῦς καὶ εἶπεν αὐτῷ Πρὸ τοῦ σε Φίλιππον φωνῆσαι ὄντα ὑπὸ τὴν συκῆν εἶδόν σε .
[ 49 ] ἀπεκρίθη αὐτῷ Ναθαναήλ Ῥαββεί , σὺ εἶ ὁ υἱὸς τοῦ θεοῦ , σὺ βασιλεὺς εἶ τοῦ Ἰσραήλ .
[ 50 ] ἀπεκρίθη Ἰησοῦς καὶ εἶπεν αὐτῷ Ὅτι εἶπόν σοι ὅτι εἶδόν σε ὑποκάτω τῆς συκῆς πιστεύεις ; μείζω τούτων ὄψῃ .
[ 51 ] καὶ λέγει αὐτῷ Ἀμὴν ἀμὴν λέγω ὑμῖν , ὄψεσθε " τὸν οὐρανὸν " ἀνεῳγότα καὶ " τοὺς ἀγγέλους τοῦ θεοῦ ἀναβαίνοντας καὶ καταβαίνοντας " ἐπὶ τὸν υἱὸν τοῦ ἀνθρώπου .
[
43
]
On
the
next
day
,
he
was
determined
to
go
out
into
Galilee
,
and
he
found
Philip
.
Jesus
said
to
him
,
"
Follow
me
.
"
[ 44 ] Now Philip was from Bethsaida , of the city of Andrew and Peter .
[ 45 ] Philip found Nathanael , and said to him , " We have found him , of whom Moses in the law , and the prophets , wrote : Jesus of Nazareth , the son of Joseph . "
[ 46 ] Nathanael said to him , " Can any good thing come out of Nazareth ? " Philip said to him , " Come and see . "
[ 47 ] Jesus saw Nathanael coming to him , and said about him , " Behold , an Israelite indeed , in whom is no deceit ! "
[ 48 ] Nathanael said to him , " How do you know me ? " Jesus answered him , " Before Philip called you , when you were under the fig tree , I saw you . "
[ 49 ] Nathanael answered him , " Rabbi , you are the Son of God ! You are King of Israel ! "
[ 50 ] Jesus answered him , " Because I told you , ' I saw you underneath the fig tree , ' do you believe ? You will see greater things than these ! "
[ 51 ] He said to him , " Most assuredly , I tell you , hereafter you will see heaven opened , and the angels of God ascending and descending on the Son of Man . "
[ 44 ] Now Philip was from Bethsaida , of the city of Andrew and Peter .
[ 45 ] Philip found Nathanael , and said to him , " We have found him , of whom Moses in the law , and the prophets , wrote : Jesus of Nazareth , the son of Joseph . "
[ 46 ] Nathanael said to him , " Can any good thing come out of Nazareth ? " Philip said to him , " Come and see . "
[ 47 ] Jesus saw Nathanael coming to him , and said about him , " Behold , an Israelite indeed , in whom is no deceit ! "
[ 48 ] Nathanael said to him , " How do you know me ? " Jesus answered him , " Before Philip called you , when you were under the fig tree , I saw you . "
[ 49 ] Nathanael answered him , " Rabbi , you are the Son of God ! You are King of Israel ! "
[ 50 ] Jesus answered him , " Because I told you , ' I saw you underneath the fig tree , ' do you believe ? You will see greater things than these ! "
[ 51 ] He said to him , " Most assuredly , I tell you , hereafter you will see heaven opened , and the angels of God ascending and descending on the Son of Man . "
John 2:1-13
Max Easter /
- Created on 2018-04-17 14:32:26
- Modified on 2018-04-17 15:08:21
- Aligned by Max Easter
Ἑλληνική Transliterate
English
[ 1 ] Καὶ τῇ ἡμέρᾳ τῇ τρίτῃ γάμος ἐγένετο ἐν Κανὰ τῆς Γαλιλαίας , καὶ ἦν ἡ μήτηρ τοῦ Ἰησοῦ ἐκεῖ :
[ 2 ] ἐκλήθη δὲ καὶ ὁ Ἰησοῦς καὶ οἱ μαθηταὶ αὐτοῦ εἰς τὸν γάμον .
[ 3 ] καὶ ὑστερήσαντος οἴνου λέγει ἡ μήτηρ τοῦ Ἰησοῦ πρὸς αὐτόν Οἶνον οὐκ ἔχουσιν .
[ 4 ] καὶ λέγει αὐτῇ ὁ Ἰησοῦς Τί ἐμοὶ καὶ σοί , γύναι ; οὔπω ἥκει ἡ ὥρα μου .
[ 5 ] λέγει ἡ μήτηρ αὐτοῦ τοῖς διακόνοις Ὅτι ἂν λέγῃ ὑμῖν ποιήσατε .
[ 6 ] ἦσαν δὲ ἐκεῖ λίθιναι ὑδρίαι ἓξ κατὰ τὸν καθαρισμὸν τῶν Ἰουδαίων κείμεναι , χωροῦσαι ἀνὰ μετρητὰς δύο ἢ τρεῖς .
[ 7 ] λέγει αὐτοῖς ὁ Ἰησοῦς Γεμίσατε τὰς ὑδρίας ὕδατος : καὶ ἐγέμισαν αὐτὰς ἕως ἄνω .
[ 8 ] καὶ λέγει αὐτοῖς Ἀντλήσατε νῦν καὶ φέρετε τῷ ἀρχιτρικλίνῳ : οἱ δὲ ἤνεγκαν .
[ 9 ] ὡς δὲ ἐγεύσατο ὁ ἀρχιτρίκλινος τὸ ὕδωρ οἶνον γεγενημένον , καὶ οὐκ ᾔδει πόθεν ἐστίν , οἱ δὲ διάκονοι ᾔδεισαν οἱ ἠντληκότες τὸ ὕδωρ , φωνεῖ τὸν νυμφίον ὁ ἀρχιτρίκλινος
[ 10 ] καὶ λέγει αὐτῷ Πᾶς ἄνθρωπος πρῶτον τὸν καλὸν οἶνον τίθησιν , καὶ ὅταν μεθυσθῶσιν τὸν ἐλάσσω : σὺ τετήρηκας τὸν καλὸν οἶνον ἕως ἄρτι .
[ 11 ] Ταύτην ἐποίησεν ἀρχὴν τῶν σημείων ὁ Ἰησοῦς ἐν Κανὰ τῆς Γαλιλαίας καὶ ἐφανέρωσεν τὴν δόξαν αὐτοῦ , καὶ ἐπίστευσαν εἰς αὐτὸν οἱ μαθηταὶ αὐτοῦ .
[ 12 ] ΜΕΤΑ ΤΟΥΤΟ κατέβη εἰς Καφαρναοὺμ αὐτὸς καὶ ἡ μήτηρ αὐτοῦ καὶ οἱ ἀδελφοὶ καὶ οἱ μαθηταὶ αὐτοῦ , καὶ ἐκεῖ ἔμειναν οὐ πολλὰς ἡμέρας .
[ 13 ] Καὶ ἐγγὺς ἦν τὸ πάσχα τῶν Ἰουδαίων , καὶ ἀνέβη εἰς Ἰεροσόλυμα ὁ Ἰησοῦς .
[ 2 ] ἐκλήθη δὲ καὶ ὁ Ἰησοῦς καὶ οἱ μαθηταὶ αὐτοῦ εἰς τὸν γάμον .
[ 3 ] καὶ ὑστερήσαντος οἴνου λέγει ἡ μήτηρ τοῦ Ἰησοῦ πρὸς αὐτόν Οἶνον οὐκ ἔχουσιν .
[ 4 ] καὶ λέγει αὐτῇ ὁ Ἰησοῦς Τί ἐμοὶ καὶ σοί , γύναι ; οὔπω ἥκει ἡ ὥρα μου .
[ 5 ] λέγει ἡ μήτηρ αὐτοῦ τοῖς διακόνοις Ὅτι ἂν λέγῃ ὑμῖν ποιήσατε .
[ 6 ] ἦσαν δὲ ἐκεῖ λίθιναι ὑδρίαι ἓξ κατὰ τὸν καθαρισμὸν τῶν Ἰουδαίων κείμεναι , χωροῦσαι ἀνὰ μετρητὰς δύο ἢ τρεῖς .
[ 7 ] λέγει αὐτοῖς ὁ Ἰησοῦς Γεμίσατε τὰς ὑδρίας ὕδατος : καὶ ἐγέμισαν αὐτὰς ἕως ἄνω .
[ 8 ] καὶ λέγει αὐτοῖς Ἀντλήσατε νῦν καὶ φέρετε τῷ ἀρχιτρικλίνῳ : οἱ δὲ ἤνεγκαν .
[ 9 ] ὡς δὲ ἐγεύσατο ὁ ἀρχιτρίκλινος τὸ ὕδωρ οἶνον γεγενημένον , καὶ οὐκ ᾔδει πόθεν ἐστίν , οἱ δὲ διάκονοι ᾔδεισαν οἱ ἠντληκότες τὸ ὕδωρ , φωνεῖ τὸν νυμφίον ὁ ἀρχιτρίκλινος
[ 10 ] καὶ λέγει αὐτῷ Πᾶς ἄνθρωπος πρῶτον τὸν καλὸν οἶνον τίθησιν , καὶ ὅταν μεθυσθῶσιν τὸν ἐλάσσω : σὺ τετήρηκας τὸν καλὸν οἶνον ἕως ἄρτι .
[ 11 ] Ταύτην ἐποίησεν ἀρχὴν τῶν σημείων ὁ Ἰησοῦς ἐν Κανὰ τῆς Γαλιλαίας καὶ ἐφανέρωσεν τὴν δόξαν αὐτοῦ , καὶ ἐπίστευσαν εἰς αὐτὸν οἱ μαθηταὶ αὐτοῦ .
[ 12 ] ΜΕΤΑ ΤΟΥΤΟ κατέβη εἰς Καφαρναοὺμ αὐτὸς καὶ ἡ μήτηρ αὐτοῦ καὶ οἱ ἀδελφοὶ καὶ οἱ μαθηταὶ αὐτοῦ , καὶ ἐκεῖ ἔμειναν οὐ πολλὰς ἡμέρας .
[ 13 ] Καὶ ἐγγὺς ἦν τὸ πάσχα τῶν Ἰουδαίων , καὶ ἀνέβη εἰς Ἰεροσόλυμα ὁ Ἰησοῦς .
[
1
]
The
third
day
,
there
was
a
marriage
in
Cana
of
Galilee
.
Jesus
'
mother
was
there
.
[ 2 ] Jesus also was invited , with his disciples , to the marriage .
[ 3 ] When the wine ran out , Jesus ' mother said to him , " They have no wine . "
[ 4 ] Jesus said to her , " Woman , what does that have to do with you and me ? My hour has not yet come . "
[ 5 ] His mother said to the servants , " Whatever he says to you , do it . "
[ 6 ] Now there were six water pots of stone set there after the Jews ' manner of purifying , containing two or three metretes apiece .
[ 7 ] Jesus said to them , " Fill the water pots with water . " They filled them up to the brim .
[ 8 ] He said to them , " Now draw some out , and take it to the ruler of the feast . " So they took it .
[ 9 ] When the ruler of the feast tasted the water now become wine , and didn ' t know where it came from ( but the servants who had drawn the water knew ) , the ruler of the feast called the bridegroom ,
[ 10 ] and said to him , " Everyone serves the good wine first , and when the guests have drunk freely , then that which is worse . You have kept the good wine until now ! "
[ 11 ] This beginning of his signs Jesus did in Cana of Galilee , and revealed his glory ; and his disciples believed in him .
[ 12 ] After this , he went down to Capernaum , he , and his mother , his brothers , and his disciples ; and they stayed there a few days .
[ 13 ] The Passover of the Jews was at hand , and Jesus went up to Jerusalem .
[ 2 ] Jesus also was invited , with his disciples , to the marriage .
[ 3 ] When the wine ran out , Jesus ' mother said to him , " They have no wine . "
[ 4 ] Jesus said to her , " Woman , what does that have to do with you and me ? My hour has not yet come . "
[ 5 ] His mother said to the servants , " Whatever he says to you , do it . "
[ 6 ] Now there were six water pots of stone set there after the Jews ' manner of purifying , containing two or three metretes apiece .
[ 7 ] Jesus said to them , " Fill the water pots with water . " They filled them up to the brim .
[ 8 ] He said to them , " Now draw some out , and take it to the ruler of the feast . " So they took it .
[ 9 ] When the ruler of the feast tasted the water now become wine , and didn ' t know where it came from ( but the servants who had drawn the water knew ) , the ruler of the feast called the bridegroom ,
[ 10 ] and said to him , " Everyone serves the good wine first , and when the guests have drunk freely , then that which is worse . You have kept the good wine until now ! "
[ 11 ] This beginning of his signs Jesus did in Cana of Galilee , and revealed his glory ; and his disciples believed in him .
[ 12 ] After this , he went down to Capernaum , he , and his mother , his brothers , and his disciples ; and they stayed there a few days .
[ 13 ] The Passover of the Jews was at hand , and Jesus went up to Jerusalem .
Acts 27:1-10
Max Easter /
- Created on 2018-04-19 14:16:40
- Modified on 2018-04-19 16:58:09
- Aligned by Max Easter
Ἑλληνική Transliterate
English
Ὡς δὲ ἐκρίθη τοῦ ἀποπλεῖν ἡμᾶς εἰς τὴν Ἰταλίαν , παρεδίδουν τόν τε Παῦλον καί τινας ἑτέρους δεσμώτας ἑκατοντάρχῃ ὀνόματι Ἰουλίῳ σπείρης Σεβαστῆς . [ 2 ] ἐπιβάντες δὲ πλοίῳ Ἁδραμυντηνῷ μέλλοντι πλεῖν εἰς τοὺς κατὰ τὴν Ἀσίαν τόπους ἀνήχθημεν , ὄντος σὺν ἡμῖν Ἀριστάρχου Μακεδόνος Θεσσαλονικέως : [ 3 ] τῇ τε ἑτέρᾳ κατήχθημεν εἰς Σιδῶνα , φιλανθρώπως τε ὁ Ἰούλιος τῷ Παύλῳ χρησάμενος ἐπέτρεψεν πρὸς τοὺς φίλους πορευθέντι ἐπιμελείας τυχεῖν . [ 4 ] κἀκεῖθεν ἀναχθέντες ὑπεπλεύσαμεν τὴν Κύπρον διὰ τὸ τοὺς ἀνέμους εἶναι ἐναντίους , [ 5 ] τό τε πέλαγος τὸ κατὰ τὴν Κιλικίαν καὶ Παμφυλίαν διαπλεύσαντες κατήλθαμεν εἰς Μύρρα τῆς Λυκίας . [ 6 ] Κἀκεῖ εὑρὼν ὁ ἑκατοντάρχης πλοῖον Ἀλεξανδρινὸν πλέον εἰς τὴν Ἰταλίαν ἐνεβίβασεν ἡμᾶς εἰς αὐτό . [ 7 ] ἐν ἱκαναῖς δὲ ἡμέραις βραδυπλοοῦντες καὶ μόλις γενόμενοι κατὰ τὴν Κνίδον , μὴ προσεῶντος ἡμᾶς τοῦ ἀνέμου , ὑπεπλεύσαμεν τὴν Κρήτην κατὰ Σαλμώνην , [ 8 ] μόλις τε παραλεγόμενοι αὐτὴν ἤλθομεν εἰς τόπον τινὰ καλούμενον Καλοὺς Λιμένας , ᾧ ἐγγὺς ἦν πόλις Λασέα . [ 9 ] Ἱκανοῦ δὲ χρόνου διαγενομένου καὶ ὄντος ἤδη ἐπισφαλοῦς τοῦ πλοὸς διὰ τὸ καὶ τὴν νηστείαν ἤδη παρεληλυθέναι , παρῄνει ὁ Παῦλος λέγων αὐτοῖς [ 10 ] Ἄνδρες , θεωρῶ ὅτι μετὰ ὕβρεως καὶ πολλῆς ζημίας οὐ μόνον τοῦ φορτίου καὶ τοῦ πλοίου ἀλλὰ καὶ τῶν ψυχῶν ἡμῶν μέλλειν ἔσεσθαι τὸν πλοῦν .
When
it
was
determined
that
we
should
sail
for
Italy
,
they
delivered
Paul
and
certain
other
prisoners
to
a
centurion
named
Julius
,
of
the
Augustan
band
.
[
2
]
Embarking
in
a
ship
of
Adramyttium
,
which
was
about
to
sail
to
places
on
the
coast
of
Asia
,
we
put
to
sea
;
Aristarchus
,
a
Macedonian
of
Thessalonica
,
being
with
us
.
[
3
]
The
next
day
,
we
touched
at
Sidon
.
Julius
treated
Paul
kindly
,
and
gave
him
permission
to
go
to
his
friends
and
refresh
himself
.
[
4
]
Putting
to
sea
from
there
,
we
sailed
under
the
lee
of
Cyprus
,
because
the
winds
were
contrary
.
[
5
]
When
we
had
sailed
across
the
sea
which
is
off
Cilicia
and
Pamphylia
,
we
came
to
Myra
,
a
city
of
Lycia
.
[
6
]
There
the
centurion
found
a
ship
of
Alexandria
sailing
for
Italy
,
and
he
put
us
on
board
.
[
7
]
When
we
had
sailed
slowly
many
days
,
and
had
come
with
difficulty
opposite
Cnidus
,
the
wind
not
allowing
us
further
,
we
sailed
under
the
lee
of
Crete
,
opposite
Salmone
.
[
8
]
With
difficulty
sailing
along
it
we
came
to
a
certain
place
called
Fair
Havens
,
near
the
city
of
Lasea
.
[
9
]
When
much
time
was
spent
,
and
the
voyage
was
now
dangerous
,
because
the
Fast
had
now
already
gone
by
,
Paul
admonished
them
,
[
10
]
and
said
to
them
,
"
Sirs
,
I
perceive
that
the
voyage
will
be
with
injury
and
much
loss
,
not
only
of
the
cargo
and
the
ship
,
but
also
of
our
lives
.
"