Hi both... Hope you're still interested in finding out more about this thread. I'm from the 'Erasmus Cox' line in Evershot too, but not from THIS particular Erasmus. He is my '3rd cousin 5x removed', but there is a link, and I have various bits of information (which I appreciate you may already know by now, 6 years after your last posts). Erasmus tended to be the Christian name of the eldest son in the line, but not always so, and Evershot and the West Dorset area is FULL of related Coxes!
This Erasmus served in the Dorset Militia from when he was 22 (1816) so if you need these references, let me know. I don't have his death details yet (but haven't really looked, to be fair). Erasmus's parents were, I believe, Erasmus Cox (b.1761 and Christened in St Peter's Dorchester, but was living in Evershot in 1780, according to Land Tax Returns) and Judith Robert (b. 1767 in Dorchester). They married in May 1787 in Dorchester Holy Trinity.
Erasmus's parents were Robert Cox (b.1738, Christened Oct 19th at Dorchester Holy Trinity) and Mary Tilly (b.1738) and they married in West Stafford, just east of Dorchester in 1759. Robert was the second son, the eldest, Erasmus, having died as an infant less than one year old.
Their parents were also Erasmus Cox (b.1717 in Evershot) and Elizabeth Lester (b.1719). I am descended directly from this Erasmus's brother, Henry. Their parents were also Erasmus Cox (b. about 1676) and Eleanor Russell (b. about 1676), who married in May 1698 in Cattistock.
Although this is the earliest Cox record I can verify, two further generations of Erasmus Coxes can be found in nearby Toller Porcorum from as early as 1593. They are undoubtedly linked, but I can't find precise written evidence of the link.
I hope this helps, and please get back in contact if you'd like to continue the discussion. You may, by now, have some information that I don't have, which would be useful to know.