I would add as an addendum to xlucid's contribution that reason that Eton kept a constant number of scholarships is that its mother school -
Winchester College - was founded to educate '70 poore and needie scholars' , according to Bishop
William of Wykham, who also founded
New College Oxford. Henry pledged when founding
Eton to remain constant to this.
Furthermore, xlucid does not quite explain the full paradox of the term 'public school' this side of the water. The term 'Public schools' is used to describe fee paying establishments catering for pupils between the ages of thirteen and eighteen, not including Grammar Schools, of which there now remain very few. Public schools are prohibitively expensive, hence the bizarrity of the word 'public': about 2% of the population are schooled in this way.
The term 'Private School' is usually used to refer to the fee-paying schools that handle pupils between the ages of seven and twelve, before they go to Public School. Again, these schools cater for a tiny percentage of children in that age group. Strong links exist between Private Schools and Public Schools owing to the symbiotic nature of their relationship.
Truly public schools are Comprehensives, which deal with the overwhelming majority of the populace. 'Public Schools' - Eton, Harrow et al - survive for a number of reasons. Hangovers from a time where education was regarded as a privilege and not a right, they purport to cater for the most intelligent (and richest, considering the fees) kids i the country. the truth is that whilst some do, the greater part educate sons and daughters of people who want their children to have been educated at their own cost. I imagine they feel there is kudos to be had from this. I don't know.
The Public School system as it stands in the UK is a body of about 250 fee-paying schools, most of whom offer scholarships, or endorsements, to greater or lesser extents. Winchester and Eton offer seventy academic scholarships and about 50 musical ones every five years, some places can afford fewer. It's a system riddled with prejudice and arrogance, elitism and snobbery. It's cliquey, and you can always spot a public school pupil in the street, before you even hear them talk. Public School pupils are one of the most stereotyped groups of young people, and unfortunately the stereotypes are accurate. The System itself aims to send its pupils to Russell Group universities (the top 13 in the country). It is under assault - or at least protracted hostility - from the current government because a party that is accused of drifting from its socialist roots needs an easy target. Such a bastion of privilege as the Public School System is that target. The Public School System will endure, because as long as there is conspicuous consumption, there will be fee-paying education. This is not a pretty truth: the system will not survive because it offers high quality education, but because of petty snobbery.