Thank you - very interesting.
I deliberately didn't reveal all I know, as I wanted to discover if anything could be deduced from the hat alone, uninfluenced by extraneous information.
I found the picture online, with no source attributed.
The subject was my great great grandfather, a clergyman who lived from 1807-1879. I have no information about his character at all, so I wondered if something as personal as a hat might just possibly say something about the wearer.
It struck me that the hat was quite tall, perhaps pointing to early rather then later Victorian?
Also that it has a pronounced widening at the top, not straight-sided like a stovepipe or more modern shallower hat.
The brim has very little turn-up, unlike the Empire suggested, and what strikes me as quite a modest sweep front to back.
My guess would have been the choice of a would-be dandy, restrained as befitting a clergyman, but with a touch of man-about-town showing through? On the other hand, he is as pointed out mistreating his hat. This advice from the website of Oliver Browwn, hatters, of Chelsea:
"When wearing an antique top hat, special care should always be taken when setting it down on a flat surface. Always set the hat down upright, with the brim of the hat on the table. This maintains the condition of the crown, which is vulnerable to damage, and once worn can never be repaired, only blackened to minimise the effect of the damage."
Apart from basic stuff like family and catalogue of his livings, the only interesting thing I know about him is that as a young man he accidentally took his brother's eye out with a gun!
As with all family history, it's so frustrating that relatives didn't ask more questions of older members when they were alive. His daughter lived to 95 and my father knew her, his grandmother, well, so he could have asked her about her family and upbringing.