  Ivory's Lemma is a geometrical statement in the heart of J. Ivory's
calculation of the gravitational potential of a homeoidal shell. In the
simplest planar case, it claims that the diagonals of a curvilinear
quadrilateral made by arcs of confocal ellipses and hyperbolas are equal.
  In the first part of this paper, we deduce Ivory's Lemma and its numerous
generalizations from complete integrability of billiards on conics and
quadrics. In the second part, we study analogs of Ivory's Lemma in Liouville
and St\"ackel metrics. Our main focus is on the results of the German school of
differential geometry obtained in the late 19 -- early 20th centuries that
might be lesser know today.
  In the third part, we generalize Newton's, Laplace's, and Ivory's theorems on
gravitational and Coulomb potential of spheres and ellipsoids to the spherical
and hyperbolic spaces. V. Arnold extended the results of Newton, Laplace, and
Ivory to algebraic hypersurfaces in Euclidean space; we generalize Arnold's
theorem to the spaces of constant curvature.
