This Chapter deals with the early physico-mathematics of Descartes in 1618–1620, which he pursued at first in conjunction with his mentor, Isaac Beeckman, who also had conveyed to him his first inkling of corpuscular-mechanism as an approach to natural philosophy. In the physico-mathematical program, the traditional view of the mixed mathematical sciences as subordinate to natural philosophy, and devoid of explanatory power, was challenged. The mixed mathematical disciplines were intended to become more integrally linked to questions of matter and cause; in other words, to questions of a natural philosophical type. In the case of Descartes and Beeckman, this meant an unsystematized, but firmly held, corpuscular-mechanism. The Chapter deals with three case studies of Descartes' physico-mathematics: his manuscript on hydrostatics and the hydrostatic paradox; his work with Beeckman on the nature of accelerated fall, which is treated here in a new way as an exercise in physico-mathematics; and a widely overlooked, but extremely important, geometrical and physical optical fragment on refraction of light, adapted from bits of the work of Kepler.