A book on a somewhat loose and heterogeneous collection of concepts. Then, it was to combat the pop culture image of the Greeks as the perfect culture of rationality that the Enlightenment is so blameworthy for coming up with. (The middle ages get the equally and oppositely ridiculous image of the world of irrationality; for that I recommend C. S. Lewis's The Discarded Image.) To be sure, it uses the loose-goosey, pop culture notion of what's rationality and irrationality, but then, so does the image.
So it goes though monitions in Homer, whether the characters are said to be moved by gods, and the development of a guilt-culture from a shame culture and all the attendant development of pollution and catharsis, which originally meant ritual purification. An insane man might go through many ceremonies for many gods and goddesses known to cause insanity, and if it didn't work -- why, obviously, they had yet to propitiate the right god.
Inspiration as a form of madness, whether it caused prophecy, ritual dancing, or poetry.
Dreams. They did not think all dreams significant. (Unlike, say, Freud. I must say that it's a few decades and takes Freud rather more seriously than turned out to be wise.) But you have your premonitions and other abilities.
This took on a rather shamanistic slant -- he puts out a correlation to demonstrate that Orpheus was a shaman -- and the reason that the dreams can be prophetic is that the god-like soul is more god-like when semi-liberated by sleep. Logically, still more god-like when liberated by death. Which lead to Puritanism. One Pythagorean dictum was that pleasure was always bad, because souls were put in bodies to be punished, and they should be punished. (On the principle of taking your medicine as quickly as you could.)
The rationalistic culture and the dream of progress, rather like the Victorian. Also, like the Victorian, leading to a rapid backlash. Partly because questioning everything gave a good number of young men to believe in rights without responsibilities, Right Makes Might, and other beliefs that no society can possibly survive when they run wild. The persecution of which Socrates was perhaps the best known victim -- but there were other prominent ones. A great deal of discussion about society and whether it can survive such questioning. (I think his optimism in the last chapter is undercut by this one.)
Plato and his changing views on the irrational soul. Hellenism and the revival of magic and other irrationalities.
A fascinating grab-bag of information.