A bit tangential, but this was one of the main ways that the Jesuits were able to gain access to the Imperial Court in Ming Era China. Predicting solar eclipses

A bit tangential, but this was one of the main ways that the Jesuits were able to gain access to the Imperial Court in Ming Era China. Predicting solar eclipses was one of the most important functions of the royal astronomers. However from the late Song Dynasty and into the Ming Dynasty there was some bureaucratic decay in the Astronomical Bureau and the quality of eclipse predictions declined. When the Jesuits arrived they were able to make superior eclipse predictions and leveraged that to gain appointments to the astronomical bureau. (And they were so successful at this that the Bureau was run by a Jesuit for a century and a half.) Incidentally, this didn't work everywhere Europeans went. When Guillaume Le Gentil arrived in southern India in the 18th century he found that Tamil eclipse prediction techniques were superior to the Western methods. (His own prediction was off by 68 seconds whereas the Tamil prediction was off by 41 seconds.)