I'm not sure how helpful it will be, but if you click
http://www.hmc.org.uk/
you will be taken to the website of the Headmasters' and Headmistresses' Conference. That is the body which controls all major British independent schools. Once there, you can click on the 'History' heading to find out how they started - just after the Clarendon Commission reported in the 1860s, actually - and how many of them there are now. Their e-mail address is
[email protected] so you could perhaps simply put your question directly to
them...the horse's mouth, as it were. Good luck.