Jo Ellen Barnett has written two books here, hidden between one set of covers. The first part goes over the history of how we have measured (and, truly, how we have defined) the time of day, starting with the "temporary hours" of a sundial (longer in summer, shorter in winter, and not even counted during the night), and transitioning to equal (but still based on the sun) hours (local apparent time), then to local mean time, to standard (time-zone) time, and ending up with the current Coordinated Universal Time, based on atomic phenomena. The story is absorbing and well written, and would be an enjoyable book all by itself.
Then she has a second part, concerned with the way we have determined the age of the earth. This could be said to start with the speculations of the Babylonians and Greeks, but really took off in earnest with medievals' attempts to build everything on a Biblical basis, reading into the Biblical account whatever they needed to build their chronology. When the geologists tried to account for their own observations, however, it was clear that the few thousand years the Bible literalists derived for the age of the earth was far too small, but physicists like Lord Kelvin (while arriving at a longer time than the Bible provided) still reached an age of the earth too brief to mesh with the geological evidence. Only with the discovery of radioasctivity and the refinement of the techniques of deriving chronological data from radioactivity measurements could the physicists and geologists be reconciled.
Both parts make it clear that ultimately time has become defined in terms of atomic phenomena (though different parts of the atom) and only through our measurement of these can accuracy be attained (whether in the case of the time of day or the time the earth has taken to evolve since its origin).
Unlike some other two-part books I have reviewed, this one puts them both together successfully. It is a very interesting book.