One can have a secret ballot even though it isn't a
government printed ballot. Alternatively, one can
have a government-printed ballot that isn't a secret
ballot. The terms shouldn't be used interchangeably.
The U.S. had party-printed ballots everywhere before
1888. But it was still possible to have a secret
ballot before 1888. Laws could mandate that all
party-printed ballots had to be of a certain color and
size, and so a voter who folded his ballot before he
approached the public polling place could still enjoy
a secret ballot.
And having a government-printed ballot didn't
necessarily guarantee a secret ballot, if the polling
place didn't have a curtained-off area for voters to
vote.
The idea of having a secret ballot, and the idea of
having a ballot printed by the government listing all
parties, were really two reforms which came in
simultaneously, for the most part.
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