An explanation to “Rolling simplexes and their commensurability” (field equations in accordance with Tycho Brahe)

Various Cartesian models of central power fields with quadratic dynamics are studied. These examples lead the reader to comprehension of basic aspects of the differential algebraic-geometrical Brahe–Descartes–Wotton theory, which embraces central power fields whose dynamics is composed of flat affine algebraic curves of degree at most $N$ ($N=1,2,3,\dots$). When $N=2$, a quadratic rolling simplex law is proved by purely algebraic means.