As noted in the comment on I.01.091, the meaning of the verb eukhesthai as ‘declare’ has to do with speaking for the record in the form of ‘boasting’ or ‘praying’ or ‘juridically declaring’ (Muellner 1976). According to the speaker here, who is the god Poseidon himself in the guise of Kalkhas, Hector aspires to be immortal by way of ‘boasting’, as expressed by the verb eukhesthai, that he is the son of the god Zeus.