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ἧος ὃ ταῦθ᾽ ὥρμαινε κατὰ φρένα καὶ κατὰ θυμόν ,

ἕλκετο δ᾽ ἐκ κολεοῖο μέγα ξίφος , ἦλθε δ᾽ Ἀθήνη

οὐρανόθεν : πρὸ γὰρ ἧκε θεὰ λευκώλενος Ἥρη

ἄμφω ὁμῶς θυμῷ φιλέουσά τε κηδομένη τε :

στῆ δ᾽ ὄπιθεν , ξανθῆς δὲ κόμης ἕλε Πηλεΐωνα

οἴῳ φαινομένη : τῶν δ᾽ ἄλλων οὔ τις ὁρᾶτο :

θάμβησεν δ᾽ Ἀχιλεύς , μετὰ δ᾽ ἐτράπετ᾽ , αὐτίκα δ᾽ ἔγνω

Παλλάδ᾽ Ἀθηναίην : δεινὼ δέ οἱ ὄσσε φάανθεν :

ἕλκετο δ᾽ ἐκ κολεοῖο μέγα ξίφος , ἦλθε δ᾽ Ἀθήνη

οὐρανόθεν : πρὸ γὰρ ἧκε θεὰ λευκώλενος Ἥρη

ἄμφω ὁμῶς θυμῷ φιλέουσά τε κηδομένη τε :

στῆ δ᾽ ὄπιθεν , ξανθῆς δὲ κόμης ἕλε Πηλεΐωνα

οἴῳ φαινομένη : τῶν δ᾽ ἄλλων οὔ τις ὁρᾶτο :

θάμβησεν δ᾽ Ἀχιλεύς , μετὰ δ᾽ ἐτράπετ᾽ , αὐτίκα δ᾽ ἔγνω

Παλλάδ᾽ Ἀθηναίην : δεινὼ δέ οἱ ὄσσε φάανθεν :

While he pondered this in mind and heart , and was drawing from its sheath his great sword , Athene came from heaven . The white-armed goddess Hera had sent her forth , for in her heart she loved and cared for both men alike . She stood behind him , and seized the son of Peleus by his fair hair , appearing to him alone . No one of the others saw her . Achilles was seized with wonder , and turned around , and immediately recognized Pallas Athene . Terribly her eyes shone .

Now as he weighed in mind and spirit these two courses and was drawing from its scabbard the great sword , Athene descended from the sky . For Hera the goddess of the white arms sent her , who loved both men equally in her heart and cared for them . The goddess standing behind Peleus ' son caught him by the fair hair , appearing to him only , for no man of the others saw her . Achilleus in amazement turned about , and straightway knew Pallas Athene and the terrible eyes shining .

Ἑλληνική

English

English

ἧος ὃ ταῦθ᾽ ὥρμαινε κατὰ φρένα καὶ κατὰ θυμόν ,

ἕλκετο δ᾽ ἐκ κολεοῖο μέγα ξίφος , ἦλθε δ᾽ Ἀθήνη

οὐρανόθεν : πρὸ γὰρ ἧκε θεὰ λευκώλενος Ἥρη

ἄμφω ὁμῶς θυμῷ φιλέουσά τε κηδομένη τε :

στῆ δ᾽ ὄπιθεν , ξανθῆς δὲ κόμης ἕλε Πηλεΐωνα

οἴῳ φαινομένη : τῶν δ᾽ ἄλλων οὔ τις ὁρᾶτο :

θάμβησεν δ᾽ Ἀχιλεύς , μετὰ δ᾽ ἐτράπετ᾽ , αὐτίκα δ᾽ ἔγνω

Παλλάδ᾽ Ἀθηναίην : δεινὼ δέ οἱ ὄσσε φάανθεν :

ἕλκετο δ᾽ ἐκ κολεοῖο μέγα ξίφος , ἦλθε δ᾽ Ἀθήνη

οὐρανόθεν : πρὸ γὰρ ἧκε θεὰ λευκώλενος Ἥρη

ἄμφω ὁμῶς θυμῷ φιλέουσά τε κηδομένη τε :

στῆ δ᾽ ὄπιθεν , ξανθῆς δὲ κόμης ἕλε Πηλεΐωνα

οἴῳ φαινομένη : τῶν δ᾽ ἄλλων οὔ τις ὁρᾶτο :

θάμβησεν δ᾽ Ἀχιλεύς , μετὰ δ᾽ ἐτράπετ᾽ , αὐτίκα δ᾽ ἔγνω

Παλλάδ᾽ Ἀθηναίην : δεινὼ δέ οἱ ὄσσε φάανθεν :

While he pondered this in mind and heart , and was drawing from its sheath his great sword , Athene came from heaven . The white-armed goddess Hera had sent her forth , for in her heart she loved and cared for both men alike . She stood behind him , and seized the son of Peleus by his fair hair , appearing to him alone . No one of the others saw her . Achilles was seized with wonder , and turned around , and immediately recognized Pallas Athene . Terribly her eyes shone .

Now as he weighed in mind and spirit these two courses and was drawing from its scabbard the great sword , Athene descended from the sky . For Hera the goddess of the white arms sent her , who loved both men equally in her heart and cared for them . The goddess standing behind Peleus ' son caught him by the fair hair , appearing to him only , for no man of the others saw her . Achilleus in amazement turned about , and straightway knew Pallas Athene and the terrible eyes shining .

Ἑλληνική

English

English

ἀλλ᾽ ὅτε δὴ τὸ τέταρτον ἐπὶ κρουνοὺς ἀφίκοντο ,

καὶ τότε δὴ χρύσεια πατὴρ ἐτίταινε τάλαντα ,

ἐν δ᾽ ἐτίθει δύο κῆρε τανηλεγέος θανάτοιο ,

τὴν μὲν Ἀχιλλῆος , τὴν δ᾽ Ἕκτορος ἱπποδάμοιο ,

ἕλκε δὲ μέσσα λαβών : ῥέπε δ᾽ Ἕκτορος αἴσιμον ἦμαρ ,

ᾤχετο δ᾽ εἰς Ἀΐδαο , λίπεν δέ ἑ Φοῖβος Ἀπόλλων .

καὶ τότε δὴ χρύσεια πατὴρ ἐτίταινε τάλαντα ,

ἐν δ᾽ ἐτίθει δύο κῆρε τανηλεγέος θανάτοιο ,

τὴν μὲν Ἀχιλλῆος , τὴν δ᾽ Ἕκτορος ἱπποδάμοιο ,

ἕλκε δὲ μέσσα λαβών : ῥέπε δ᾽ Ἕκτορος αἴσιμον ἦμαρ ,

ᾤχετο δ᾽ εἰς Ἀΐδαο , λίπεν δέ ἑ Φοῖβος Ἀπόλλων .

Then , at last , as they were nearing the fountains for the fourth time , the father of all balanced his golden scales and placed a doom in each of them , one for Achilles and the other for Hector , breaker of horses . As he held the scales by the middle , the doom of Hector fell down deep into the house of Hadēs - and then Phoebus Apollo left him .

But when for the fourth time they had come around to the well springs then the Father balanced his golden scales , and in them he set two fateful portions of death , which lays men prostrate , one for Achilleus , and one for Hektor , breaker of horses , and balanced it by the middle ; and Hektor ' s death-day was heavier and dragged downward toward death , and Phoibos Apollo forsook him .

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 , 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 .

English

Ἑλληνική Transliterate

A.S Kline

Gripped by terror we shrank back into a deep corner . He drove his well-fed flocks into the wide cave , the ones he milked , leaving the rams and he-goats outside in the broad courtyard . Then he lifted his door , a huge stone , and set it in place .

ἡμεῖς
δὲ
δείσαντες
ἀπεσσύμεθ̓
ἐς
μυχὸν
ἄντρου
.
αὐτὰρ
ὅ
γ̓
εἰς
εὐρὺ
σπέος
ἤλασε
πίονα
μῆλα
πάντα
μάλ̓
ὅσσ̓
ἤμελγε
,
τὰ
δ̓
ἄρσενα
λεῖπε
θύρηφιν
,
ἀρνειούς
τε
τράγους
τε
,
βαθείης
ἔκτοθεν
αὐλῆς
.
αὐτὰρ
ἔπειτ̓
ἐπέθηκε
θυρεὸν
μέγαν
ὑψόσ̓
ἀείρας
,
ὄβριμον
·
οὐκ
ἂν
τόν
γε
δύω
καὶ
εἴκοσ̓
ἄμαξαι
ἐσθλαὶ
τετράκυκλοι
ἀπ̓
οὔδεος
ὀχλίσσειαν
·
τόσσην
ἠλίβατον
πέτρην
ἐπέθηκε
θύρῃσιν
.
ἑζόμενος
δ̓
ἤμελγεν
ὄις
καὶ
μηκάδας
αἶγας
,
πάντα
κατὰ
μοῖραν
,
καὶ
ὑπ̓
ἔμβρυον
ἧκεν
ἑκάστῃ
.

Ἑλληνική Transliterate

English

χωόμενος δ᾽ ὁ γέρων πάλιν ᾤχετο : τοῖο δ᾽ Ἀπόλλων

εὐξαμένου ἤκουσεν , ἐπεὶ μάλα οἱ φίλος ἦεν ,

ἧκε δ᾽ ἐπ᾽ Ἀργείοισι κακὸν βέλος : οἳ δέ νυ λαοὶ

θνῇσκον ἐπασσύτεροι , τὰ δ᾽ ἐπῴχετο κῆλα θεοῖο

πάντῃ ἀνὰ στρατὸν εὐρὺν Ἀχαιῶν : ἄμμι δὲ μάντις

εὖ εἰδὼς ἀγόρευε θεοπροπίας ἑκάτοιο . ’’’

εὐξαμένου ἤκουσεν , ἐπεὶ μάλα οἱ φίλος ἦεν ,

ἧκε δ᾽ ἐπ᾽ Ἀργείοισι κακὸν βέλος : οἳ δέ νυ λαοὶ

θνῇσκον ἐπασσύτεροι , τὰ δ᾽ ἐπῴχετο κῆλα θεοῖο

πάντῃ ἀνὰ στρατὸν εὐρὺν Ἀχαιῶν : ἄμμι δὲ μάντις

εὖ εἰδὼς ἀγόρευε θεοπροπίας ἑκάτοιο . ’’’

So
the
old
man
went
back
again
in
anger
;
and
Apollo
heard
his
prayer
,
for
he
was
very
dear
to
him
,
and
sent
against
the
Argives
an
evil
shaft
.
Then
the
people
began
to
die
thick
and
fast
,
and
the
shafts
of
the
god
ranged
everywhere
throughout
the
wide
camp
of
the
Achaeans
.
But
to
us
the
prophet
with
sure
knowledge
declared
the
oracles
of
the
god
who
strikes
from
afar
.

Ἑλληνική Transliterate

English

γνώσει , τέχνης σημεῖα τῆς ἐμῆς κλύων .

εἰς γὰρ παλαιὸν θᾶκον ὀρνιθοσκόπον

ἵζων , ἵν᾽ ἦν μοι παντὸς οἰωνοῦ λιμήν ,

ἀγνῶτ᾽ ἀκούω φθόγγον ὀρνίθων , κακῷ

κλάζοντας οἴστρῳ καὶ βεβαρβαρωμένῳ .

καὶ σπῶντας ἐν χηλαῖσιν ἀλλήλους φοναῖς

ἔγνων : πτερῶν γὰρ ῥοῖβδος οὐκ ἄσημος ἦν .

εὐθὺς δὲ δείσας ἐμπύρων ἐγευόμην

βωμοῖσι παμφλέκτοισιν :

εἰς γὰρ παλαιὸν θᾶκον ὀρνιθοσκόπον

ἵζων , ἵν᾽ ἦν μοι παντὸς οἰωνοῦ λιμήν ,

ἀγνῶτ᾽ ἀκούω φθόγγον ὀρνίθων , κακῷ

κλάζοντας οἴστρῳ καὶ βεβαρβαρωμένῳ .

καὶ σπῶντας ἐν χηλαῖσιν ἀλλήλους φοναῖς

ἔγνων : πτερῶν γὰρ ῥοῖβδος οὐκ ἄσημος ἦν .

εὐθὺς δὲ δείσας ἐμπύρων ἐγευόμην

βωμοῖσι παμφλέκτοισιν :

Ἑλληνική Transliterate

English

χωόμενος δ᾽ ὁ γέρων πάλιν ᾤχετο : τοῖο δ᾽ Ἀπόλλων

εὐξαμένου ἤκουσεν , ἐπεὶ μάλα οἱ φίλος ἦεν ,

ἧκε δ᾽ ἐπ᾽ Ἀργείοισι κακὸν βέλος : οἳ δέ νυ λαοὶ

θνῇσκον ἐπασσύτεροι , τὰ δ᾽ ἐπῴχετο κῆλα θεοῖο

πάντῃ ἀνὰ στρατὸν εὐρὺν Ἀχαιῶν : ἄμμι δὲ μάντις

εὖ εἰδὼς ἀγόρευε θεοπροπίας ἑκάτοιο . ’’’

εὐξαμένου ἤκουσεν , ἐπεὶ μάλα οἱ φίλος ἦεν ,

ἧκε δ᾽ ἐπ᾽ Ἀργείοισι κακὸν βέλος : οἳ δέ νυ λαοὶ

θνῇσκον ἐπασσύτεροι , τὰ δ᾽ ἐπῴχετο κῆλα θεοῖο

πάντῃ ἀνὰ στρατὸν εὐρὺν Ἀχαιῶν : ἄμμι δὲ μάντις

εὖ εἰδὼς ἀγόρευε θεοπροπίας ἑκάτοιο . ’’’

So
the
old
man
went
back
again
in
anger
;
and
Apollo
heard
his
prayer
,
for
he
was
very
dear
to
him
,
and
sent
against
the
Argives
an
evil
shaft
.
Then
the
people
began
to
die
thick
and
fast
,
and
the
shafts
of
the
god
ranged
everywhere
throughout
the
wide
camp
of
the
Achaeans
.
But
to
us
the
prophet
with
sure
knowledge
declared
the
oracles
of
the
god
who
strikes
from
afar
.

English

Ἑλληνική Transliterate

The wrath sing , goddess , of Peleus ' son , Achilles , that destructive wrath which brought countless woes upon the Achaeans , and sent forth to Hades many valiant souls of heroes , and made them themselves spoil for dogs and every bird ; thus the plan of Zeus came to fulfillment , from the time when first they parted in strife Atreus ' son , king of men , and brilliant Achilles . Who then of the gods was it that brought these two together to contend ? The son of Leto and Zeus ; for he in anger against the king roused throughout the host an evil pestilence , and the people began to perish , because upon the priest Chryses the son of Atreus had wrought dishonour .

μῆνιν
ἄειδε
θεὰ
Πηληϊάδεω
Ἀχιλῆος
οὐλομένην
,
ἣ
μυρί᾽
Ἀχαιοῖς
ἄλγε᾽
ἔθηκε
,
πολλὰς
δ᾽
ἰφθίμους
ψυχὰς
Ἄϊδι
προΐαψεν
ἡρώων
,
αὐτοὺς
δὲ
ἑλώρια
τεῦχε
κύνεσσιν
οἰωνοῖσί
τε
πᾶσι
,
Διὸς
δ᾽
ἐτελείετο
βουλή
,
ἐξ
οὗ
δὴ
τὰ
πρῶτα
διαστήτην
ἐρίσαντε
Ἀτρεΐδης
τε
ἄναξ
ἀνδρῶν
καὶ
δῖος
Ἀχιλλεύς
.
τίς
τ᾽
ἄρ
σφωε
θεῶν
ἔριδι
ξυνέηκε
μάχεσθαι
;
Λητοῦς
καὶ
Διὸς
υἱός
:
ὃ
γὰρ
βασιλῆϊ
χολωθεὶς
νοῦσον
ἀνὰ
στρατὸν
ὄρσε
κακήν
,
ὀλέκοντο
δὲ
λαοί
,
οὕνεκα
τὸν
Χρύσην
ἠτίμασεν
ἀρητῆρα
Ἀτρεΐδης