From line-systems to sphere-systems — Schläfli’s double six, Lie’s line-sphere transformation, and Grace’s theorem

If each four spheres in a set of five unit spheres in R3R3 have nonempty intersection, then all five spheres have nonempty intersection. This result is proved using Grace’s theorem: the circumsphere of a tetrahedron encloses none of its escribed spheres. This paper provides self-contained proofs of these results; including Schläfli’s double six theorem and modified version of Lie’s line-sphere transformation. Some related problems are also posed.