The author does not give sufficient credit for a particular difficulty in calculating the location of a solar eclipse. He mentions in pathing that "the rotation of the Earth is gradually slowing down" and that "exactly when a new leap second will be needed is unpredictable". More precisely, he should have said that the Earth's rotation has a tendency to slow down - in contrast to today, it accelerated somewhat in the last third of the 19th century, for example. But more significant in this context, however, is the aspect that the estimation of the degree of change for a given period of time becomes quickly more and more inaccurate when we go forward or backward in time. In the case of ancient eclipses we cannot calculate their exact location from a prior knowledge of the Earth's rotational speed. It is the other way round: since we have historical reports from a particular region about an eclipse we can deduce the change in the Earth's rotational speed between now and then.

For more information about the change of Earth's rotational speed see the Wikipedia article on "ΔT (timekeeping)": https://en.wikipedia.org/wiki/%CE%94T_(timekeeping)