A year or so after China established diplomatic ties with America in 1979,
Chinese officials started asking for a new agreement.
They wanted then-President Ronald Reagan to commit to ending arms sales to Taiwan,
the self-governed island that China claims.
His secretary of state, Alexander Haig, backed the idea,
arguing that China would soon be the world's "most important country".
Reagan sacked Haig
but agreed to a fudge.
A new communiqué with China in 1982 promised "gradually to reduce" arms sales to Taiwan,
provided Chinese intentions were peaceful.
He then sent secret assurances to Taiwan,
reaffirming American support.
In the years since, peace in the western Pacific has often depended on such ambiguities
in the liturgy of relations between China, America and Taiwan.
Chinese officials have made many more attempts to dilute America's commitment
to help Taiwan defend itself.
And successive American presidents have rebuffed them.
But as Donald Trump prepares to visit Beijing,
China is making a new push to win concessions on Taiwan.
And fears are growing in Taipei and Washington that Mr Trump might at some point give ground