Unit 1 Notes: Historical Provenance & Cross-Repository Commentary

Collation of the 1888 & 1892 Editions and Links to euclid, elements, altshiller-court, and etc

1. Why Casey Attaches Orthocentre & Mean Centres Directly to Book I

Unlike Euclid’s Elements—which postpones the circumcentre and incentre until Book IV (IV.4, IV.5) and never names the orthocentre or centroid—John Casey proves the concurrency of all four classical cevians directly in Book I (Props. 9–11, pp. 7–9) using only congruence (I.4, I.26) and parallelogram properties (I.31, I.34). Notice the elegance of Casey’s anticomplementary proof of the orthocentre (Prop. 10 Cor., f. 0034): instead of invoking cyclic quadrilaterals from Book III, he embeds $\triangle ABC$ as the medial triangle of $\triangle A'B'C'$ so that altitudes reduce immediately to perpendicular bisectors! In geometor/altshiller-court (College Geometry, §§ 93–96), Nathan Altshiller-Court adopts this exact anticomplementary reduction.

2. Centres of Mean Position & Barycentric Coordinates

Casey’s treatment of Centres of Mean Position (Book I, Props. 17–18, pp. 12–13; Book II, Props. 11–13) follows Simon Antoine Jean L’Huilier (1809) and August Ferdinand Möbius (Der barycentrische Calcul, 1827), translating static moments into pure Euclidean geometry so students reading “Mixed Mathematics” (mechanics and hydrostatics at Trinity College Dublin) could derive Leibniz’s identity $\sum m_i PA_i^2 = \sum m_i GA_i^2 + (\sum m_i) PG^2$ without analytic coordinates. In geometor/casey (barycentric_point()) and geometor/etc, every Kimberling triangle center is represented in exact homogeneous barycentric weights $(u : v : w)$, directly realizing Casey’s Mean Centre for $A, B, C$.

3. Lineage of the Nine-Points Circle (Brianchon–Poncelet 1821 & Feuerbach 1822)

Although Euler (1765) already knew that the orthocentre, centroid, and circumcentre are collinear and that the circle through the altitude feet passes through the side midpoints, the explicit nine-point concyclicity was published first by Charles Julien Brianchon and Jean-Victor Poncelet in Gergonne’s Annales de Mathématiques 11 (1820–1821) and independently a year later by Karl Wilhelm Feuerbach (Eigenschaften), who discovered its tangency with the four tritangent circles (proved in Unit 4).

Casey’s proof in Book IV, Prop. 5 (ff. 0083–0084, pp. 57–58) uses the three diameters $M_a E_a, M_b E_b, M_c E_c$ and Thales’s semicircle theorem (Euclid III.31). In Patrick A. E. Dowling's 1892 Sixth Edition, Book IV Prop. 5 and its corollaries on the orthocentric group $\{A, B, C, H\}$ (where any of the four points is the orthocentre of the remaining three, so all four triangles share the same Nine-Points Circle!) remain word-for-word aligned with the 1888 Fifth Edition.