Unit 4 Notes: Casey’s Dublin Memoirs (1861 & 1866), Hart’s Circle & Polar Duality

Historical Context of Casey’s Original Discoveries & Cross-Canon Links

1. John Casey’s Two Famous Memoirs (1861 & 1866)

Unit 4 preserves John Casey’s own most celebrated contributions to pure mathematics, integrated directly into a student textbook:

  • 1861 Feuerbach Inversion Proof (:cite:`QJM 4, 1861 <casey:1861:feuerbach>`): When Karl Wilhelm Feuerbach proved his nine-point tangency theorem in 1822, his proof relied on heavy trigonometric computation. While still an undergraduate pensioner at Trinity College Dublin in 1860–1861, Casey noticed that the circle centered at $M_a$ of radius $\frac{1}{2}|b-c|$ cuts $(I)$ and $(I_a)$ orthogonally and inverts their fourth common tangent into the Nine-Points Circle! Immediately upon seeing Casey’s proof, Trinity Vice-Provost Andrew Searle Hart extended it to three arbitrary circles (QJM 4, 1861), discovering Hart’s Circle (Book VI, Sec. IV, Prop. 11).

  • 1866 Generalized Ptolemy Theorem (:cite:`Proc. R.I.A. 9, 1866 <casey:1866:circles>`): Five years later, Casey discovered the bitangent criterion $\overline{12}\cdot\overline{34} \pm \overline{14}\cdot\overline{23} \pm \overline{13}\cdot\overline{24} = 0$ while investigating systems of circles touching three fixed circles on the plane and sphere—turning Feuerbach’s theorem into a three-line algebraic corollary! Note that for sufficiency, modern expositions (Zacharias, Altshiller-Court § 502) specify compatible contact orientations (signed radii), which prove_casey_theorem_symbolic() verifies explicitly for both internal incircle and external excircle cases.