That is, when did astronomy figure out how to predict when and where a solar eclipse will be visible?

It seems people noticed fairly early on that the Sun, Moon and Earth return to the same approximate relative positions periodically (~18 years and 11 days and 8 hours). This tells them how long it takes for an eclipse to (almost) repeat itself. However, the shadow of the moon on Earth is only ~150km wide during a solar eclipse, and so only a small part of the planet will actually see it happening. Thus:

... since the Saros cycle does not include an integral number of days ... the next eclipse in the cycle will occur one third of the way around the world from the previous eclipse in the cycle. So, the Saros cycle allowed ancient astronomers to predict when a solar eclipse could be expected, but it did not help them figure out where the eclipse would occur.

_{- http://ds9.ssl.berkeley.edu/solarweek/THURSDAY/history.html}

So when did astronomers understood the entire mechanics well enough to be able to predict both the time and place of a solar eclipse?

Inspired by this question on History.SE. Note that I'm not just asking for when one eclipse was correctly predicted (by chance or otherwise), but rather when the physics of it is understood well enough for reliable predictions.