This question is kind of difficult to explain but I'll give it a try Don't judge a book by its cover. ( by here means through the means of; so in other words 'its cover' is directly responsible for judging a book; this is also very idiomatic) It was unfair to judge her on such a brief acquaintanceship. ( on here means 'as a result of' so the brief acquaintanceship itself isn't directly responsible for judging her; the results of the brief acquaintanceship are) Judging by her last letter, they are having a wonderful time.
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