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 :

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 .

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 :

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 .

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 .

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 .

English

Ἑλληνική Transliterate

For he first devised unseemly deeds . She conceived a cunning , evil trick . Around them a white foam from the immortals kin began to arise .

English

Ἑλληνική Transliterate

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 .

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 . ’’’

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 :

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 . ’’’

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 .