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I'm no great fan of Eton. It has long bred a feeling of superiority in its charges. Harrow, on the other hand, has long produced educated, louche, gentlemanly, eccentrics and reprobates who do have a sense of others. They don't lack self-confidence, they have plenty of that, but the arrogant edge of born superiority is not obvious. Eton has rather forgotten what Dr Arnold at Rugby, the creator of the modern Public School, held as a first principle, that every boy has to learn what it's like to be a servant and take orders before he can be trusted to employ anyone or give orders (hence the system of 'fagging' where the junior boys were the at the beck and call of six -formers)
If state schools all had the resources, the small classes, the personal monitoring of every pupil every day, the well-paid teachers, and the specialisation in type of pupil selected and in the general nature of the school; Westminster and St Paul's for very academic pupils, Gordonstoun and Blundells for outdoor types, Millfield for the sporting and so on; they would be as good
I think that Public Schools nowadays get the majority of their pupils from families where neither parent attended such a school and never boarded.