  The Tait-Kneser theorem, first demonstrated by Peter G. Tait in 1896, states
that the osculating circles along a plane curve with monotone non-vanishing
curvature are pairwise disjoint and nested. This note contains a proof of this
theorem using the Lorentzian geometry of the space of circles. We show how a
similar proof applies to two (apparently) new variations on the theorem,
concerning the osculating Hooke and Kepler conics along a plane curve.
