Oaths in ancient Hellas were, quite literally, sacred. The ancient Hellenes tended to swear oaths by the Gods. In different places different deities were more important than others and were more likely to sworn by. Women were more likely than men to swear by female deities, and vise versa. Part of the choice most likely also had to do with the amount of kharis they'd built up with that particular God. Perhaps the situation that made you swear had an effect on the choice of God they swore by as well. I found a very lovely podcast about oaths yesterday, which describes the value and practice of oaths and oath breaking.

Melvyn Bragg, Alan Sommerstein (Professor of Greek at the University of Nottingham), Paul Cartledge (Professor of Greek History at the University of Cambridge), and Mary Beard (Professor in Classics at the University of Cambridge) feature in this podcast which asks questions like: how did the Classical world come to understand the oath? Why did oaths come to occupy such a central place in the political, social and legal life of the Athenian State? And what role did oath-making play in the expanding Roman Empire?

When you want to swear an oath to the Theoi, hold up your right hand, bend your pinkie and ringfinger down to your palm and extend all other fingers up. An example of an oath sworn is the following from the 'Argonautica' by Apollonius Rhodes:

"I swear by Leto's Son [Apollon], who of His own accord taught me prophetic lore; by my own ill-starred fate; by the dark cloud that veils my eyes; by the Powers below [presumably Haides, Persephone and the Erinyes] – and may They blast me if I die forsworn – that you will not incur the wrath of Heaven by helping me." [2.259]