Dr Turner slipped in this speculation almost unnoticed and certainly without applying it.

Turner's associated end note 21 at p. 660  advises: "For the manner in which Newman's personality  in the 1820s conforms to that of persons believing in magic, consult Stuart A. Vyse, Believing in Magic: The Psychology of Superstition (New York: Oxford University Press, 1997), pp. 24-58, italics, underlining mine)."

Now pp. 24-58 represent the whole of Vyse's Chapter 2, "The Superstitious Person: Group and Individual Differences in Magical Belief." Both the book and the chapter are good reads with much material drawn from recent tests of college sophomores taking introductory psychology courses and other sources. Superstition and religious belief are similar but acknowledged by Vyse also to be different. "...both involve an act of faith. The believer must place trust in forces beyond understanding." (Vyse, op. cit., p.40 f)

The odd manner, however, in which Dr Turner presents the theme of Newman and magic makes the reader expect that Vyse's book will show us Newman as a Victorian Dr Faustus. What shall we learn about Newman and magic? Nothing, I submit. 

Meanwhile a non-specialist like me finds more light in this passage from Avery Dulles's NEWMAN (2002, p. 56f ) on what type evidence supports Christian faith: In the last chapter of the GRAMMAR OF ASSENT
 

he (Newman) points to what he calls coincidences -- events which, 'though not in themselves miraculous, do irresistibly force upon us, almost by the law of our nature, the presence and extraordinary action of Him whose being we already acknowledge. An example might be the reception of a greatly needed and unexpected gift soon after the prayer of a saintly person.  [UNDERLINING MINE]