The Mesopotamian and Babylonian civilisations were probably the first to develop mathematical astronomy, primarily during the Seleucid Kingdom (ca. 320BC to ca. 620AD). These early mathematical astronomers developed techniques for predicting eclipses and positions of the heavenly bodies. They created tables for reference. In Egypt, methods developed for land surveying were also being applied to astronomy. Eventually, the Greeks adopted the mathematical approach taken by the Babylonians.
These early methods were used with few changes for centuries. In the 1700s, Caroline Herschel helped her brother, William, develop the modern mathematical approach to astronomy. William Herschel used many of these methods to help discover Uranus.
In the 1800s, Urbain Jean Joseph Le Verrier used mathematical equations to calculate the existence of Neptune. He gave his calculations to astronomer Johann Gottried Galle at the Berlin Observatory. Using Le Verrier's calculations, Galle was able to observe the planet within one hour of starting. Le Verrier expected to be declared the sole discoverer of Neptune, but months prior to his calculations being completed John Couch Adams, an English mathematician, had accomplished the same feat. As a consequence, Le Verrier and Adams share the honor as Neptune's discoverers.
Le Verrier also theorized that there was a second asteroid belt in our solar system. He believed the second belt was between the Sun and Mercury. We now know that this second belt does not exist.