Galileo and Newton made many important advances in the study of motion (mechanics). They did not start from scratch. And they were not the first to use mathematics to describe this motion. There was a calculatory tradition from the late middle ages onward. The first group associated with this tradition is known as the Oxford Calculators. The most famous of the Calculatores was Thomas Bradwardine, a Roman Catholic Archbishop.
The advances of the Calculatores were impressive. They separated motion into categories such as uniform motion and uniformly accelerated motion, much as modern physicists do today. The "Mean Speed Theorem" was first published by William Heytesbury, a Calculator, in 1335. This tradition culminated in the Law of Free Fall, published by the Spanish Calculator, Father Domingo de Soto, a few years before Galileo's birth. Galileo is wrongly credited with originating the Law of Free Fall and the Mean Speed Theorem. The ideas of the Oxford Calculators spread through Europe, being incorporated into the work of the Parisian Doctors, and other schools of Calculators. It resulted in some surprisingly sophisticated mathematical descriptions of motion.
The Calculatores' Mean Speed Theorem was an innovative way of making calculations for constantly accelerated motion as simple as for constant motion. Calculating the distance travelled by a car at a fixed speed is simple; speed times time. Calculating the distance travelled by a car under constant acceleration is a little more complicated. We could calculate it by breaking the time into small sections then adding up the distance travelled in these small sections. The mathematics of the 1300's wasn't ready for this approach. Heytesbury suggested that if you take the mean speed during the acceleration and multiplied by time you would produce the same result. A few decades later, the Catholic Bishop, Nicole Oresme, and the Franciscan Friar, Giovanni di Casale, provided a geometric proof for the Theorem. Galileo provided a similar geometric proof a few centuries later.
Galileo knew of their work. He taught at a university that had been a hotbed for calculatory thought, the University of Padua.
How these figures were ignored for so long does raise some issues regarding historian bias (see Sarton:A Case for Bias). That is because key proponents of the Mertonian school were amongst the most honored academics during the middle ages and were also very widely taught throughout Europe. Bradwardine had even been mentioned in Geoffrey Chaucer's Canterbury Tales. The historians of science from the early twentieth century were positivists. The possibility of critical scientific advances occuring in a period largely dominated by the church conflicted with the positivist view of the history of science.