I think other people have answered the question already but...
they were designed with a tall crown both to be visible in a crowd, and well made enough to offer protection if bashed on the head by a passing baddie, and was adopted by the Met, and other forces, in the 1860's.
It was originally a reinforced top hat. Which was designed to be strong enough to be stood on, should the need arise. The proper name for the traditional Bobbies headgear is the 'Custodian helmet'.
Of course with the introduction of police cars other flater types of headgear needed to be introduced...it was either that or have the cops driving around in Noddy cars.
