Unit 5 Notes: Lemoine’s 1873 Revival & the English Reception via Casey

Historical Provenance & Cross-Canon Concordance (etc, altshiller-court)

1. From Lhuilier & Grebe to Lemoine (1873) and d’Ocagne’s “Symmedian”

Although the minimum-square property of $K = X(6)$ had been noticed in passing by Simon Lhuilier (1809) and Ernst Wilhelm Grebe (1847—hence the German name Grebescher Punkt), modern triangle geometry began in earnest at the 1873 Lyon Congress of the Association Française pour l’Avancement des Sciences, where Émile Lemoine presented Note sur un point remarquable du plan d'un triangle, discovering both six-point circles ((Ml D1) and (K Cb2)). Ten years later, Maurice d’Ocagne (Nouvelles Annales de Mathématiques, 1883) coined the enduring French term symédiane (“symmedian”), while Robert Tucker (QJM 19, 1883) rediscovered Lemoine’s First Circle in England under the name Triplicate-Ratio Circle before recognizing it as a member of the full one-parameter Tucker family (treated in Unit 6).

2. Casey’s Pivotal Role in the English-Speaking World

When John Casey added his 58-page Supplementary Chapter on the Recent Elementary Geometry of the Triangle to the 4th edition of the Sequel in 1886 (expanded further in the 1888 5th and 1892 6th editions), it was the very first textbook in any language to organize the discoveries of Lemoine, Brocard, Tucker, Taylor, Neuberg, and Tarry into systematic Euclidean propositions! Nathan Altshiller-Court's Chapter X (Recent Geometry of the Triangle, §§ 557–590, 616–628, 661–673 in geometor/altshiller-court) descends directly from Casey’s architecture, and geometor/etc indexes Lemoine’s point as X(6) and the First Lemoine / Brocard centre as X(182).