Throughout the Middle Ages, motion and change were seen as the fundamental and immediate expressions of the innate natures of physical things. To understand their causes and effects was to grasp the nature of physical reality and to approach an understanding of higher realities such as spiritual substances and God. Philosophical treatments of motion and change in the Middle Ages arose out of teaching and commenting on Aristotle's natural works (the libri naturales) by masters in the medieval universities. Although banned several times at Paris early in the thirteenth century, by mid-century these works came to constitute almost the entire program in natural philosophy (physica or philosophia naturalis), which, along with logic, moral philosophy, and metaphysics, made up the core of the medieval arts curriculum. Beyond the faculty of arts, theories of motion and change came to be applied to theological problems such as grace and the sacraments, and they were in turn influenced by theological and other considerations. Motion and change were matters of intense interest to medieval natural philosophers, logicians, mathematicians, and theologians alike, and the scope of their studies extended far beyond the narrow limits of what in the seventeenth century would become dynamics and kinematics. Within medieval treatments of motion and change, there emerged a number of scientific and mathematical advances of great interest and lasting importance in the history of science, including a rule on the relation between force and speed, impetus theory, the quantification of qualities, the mean-speed theorem, and the graphical representation of qualities and motion.