Joined: Oct. 2005
I just finished reading this. I doubt it is the one you are after but it covers the same subject.
The core of the book is about how astronomers proceeded to measure the age of the Universe.
However it starts with a brief overview of why biology and geology went to demand an ancient Earth.
Then covers the physics version of aging the sun and solar system.
Then the main subject of trying to date the Universe and the slightly embarrassing period where the most up to date science of the time had the oldest stars being dated as older than the age of the Universe they inhabit.
Probably not the one you wanted, but the subject does cover your requiremnt.