How do modern scientists predict eclipses?

Astronomers first have to work out the geometry and mechanics of how the Earth and the Moon orbit the Sun under the influence of the gravitational fields of these three bodies. Using Newton’s laws of motion, they mathematically calculate the motion of these bodies in three-dimensional space, taking into account the fact that these bodies have a finite size and are not perfect spheres. Scientists then plug the current positions and velocities of the Earth and the Moon into these complex equations, and programme a computer to ‘integrate’ these equations forwards or backwards in time in order to calculate the relative positions of the Moon and the Sun as seen from the Earth’s perspective.
Eclipses are specific configurations of these bodies that can be identified by computers. Current eclipse predictions have a timing accuracy of less than one minute over a period of hundreds of years.