https://forum.wordreference.com/threads/the-abnormal-mind-is-quick-to-detect-and-attach-itself.2729976/

'The abnormal mind' stands for 'Those people with an abnormal mental outlook'.
'Is quick to detect' means 'rapidly discovers'.
'This quality' means 'the habit of reserving all judgements'.
'A politician' probably means 'a schemer or manipulative person'.
People accused him of being a schemer because he knew the secrets of the most strange undergraduates.
He had this information because his strange fellow-students would tell him their secrets.
They did this because they liked his non-judgemental attitude.

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