Unit 2 Notes: Gaultier, Steiner, Poncelet & Exact Circle Algebra¶

Historical Provenance & Architectural Notes on Coaxal Systems

1. Historical Synthesis: Gaultier (1813), Poncelet (1822) & Steiner (1826)¶

Three Continental memoirs converged to create the circle theory expounded in Casey’s Book III (pp. 41–49) and Book VI, Section V (pp. 113–126):

Louis Gaultier (Journal de l'École Polytechnique 16, 1813) coined axe radical (“radical axis”) and centre radical (“radical centre”), giving the three-circle secant construction for disjoint circles that Casey places in Book III, Props. 21 & 23.
Jean-Victor Poncelet (Traité des propriétés projectives des figures, 1822) introduced coaxal pencils and recognized the limiting points (points limites) $L_1, L_2$ of a non-intersecting pencil as real zero-radius circles dual to the ideal common chord—a breakthrough demonstrating how imaginary intersections in geometry manifest through real Euclidean invariants ($\delta = +64 \implies L_{1,2} = (\mp 8, 0)$).
Jakob Steiner (Crelle's Journal 1, 1826) named the Potenz eines Punktes (“power of a point”) $|PO|^2 - r^2$, unifying Euclid III.35 (interior chord product) and III.36 (exterior secant-tangent product) into a single signed quadratic functional.

Euclid & Elements: Generalizes III.17 (tangent construction), III.35–37 (intersecting chords and secants), and I.47 (orthogonality $d^2 = r_1^2 + r_2^2$).
Altshiller-Court: Corresponds directly to College Geometry Chapter VIII, §§ 358–376 (Inverse Points & Orthogonal Circles) and §§ 417–498 (Power of a Point, Radical Axis & Coaxal Circles).
Exact SymPy Gap Closure: Standard sympy.geometry.Circle lacks power, is_orthogonal, radical_axis, and limiting_points; geometor.casey.invariants and geometor.casey.coaxial supply all four with 100% exact rational/radical verification.