About John Casey & the Digital Edition¶
Life of John Casey (1820–1891), Editorial Lineage of the Sequel (1881–1892), and Digital Scriptorium Methodology
John Casey, LL.D., F.R.S. (1820–1891)¶
John Casey (born May 12, 1820, Kilbehenny, County Limerick; died January 3, 1891, Dublin) stands alongside George Salmon, Richard Townsend, and Robert Stawell Ball at the summit of the nineteenth-century Dublin school of pure geometry. Educated first in local Munster schools and largely self-taught in higher mathematics while serving as headmaster of the Kilkenny Central Model Schools, Casey entered Trinity College Dublin as a pensioner in 1858 at the age of thirty-eight—having already communicated original discoveries on the geometry of the triangle and circle to Richard Townsend and George Salmon. Electing Scholarship in 1861 and graduating B.A. in 1862, he subsequently served as Professor of Higher Mathematics and Mathematical Physics at the Catholic University of Ireland (from 1873) and Fellow of the Royal University of Ireland (from 1882), elected Fellow of the Royal Society in 1875.
In 1866 (Proceedings of the Royal Irish Academy, vol. 9, pp. 396–423), Casey published his celebrated generalization of Ptolemy’s Theorem (Book VI, Section IV, Proposition 10): four circles $O_1, O_2, O_3, O_4$ are tangent to a fifth circle (or straight line) if and only if the six pairwise common tangents $\overline{ij}$ satisfy a vanishing Cayley–Menger / Ptolemy relation
$$ \overline{12}\cdot\overline{34} \pm \overline{14}\cdot\overline{23} \pm \overline{13}\cdot\overline{24} = 0. $$
Already as an undergraduate in Quarterly Journal of Pure and Applied Mathematics 4 (1861), pp. 245–252, Casey had discovered the three-line circular-inversion proof of Feuerbach’s Theorem—inverting the incircle and the $A$-excircle into themselves from the midpoint $M_a$ of $BC$ with squared radius $\frac{1}{4}(b - c)^2$ so that the fourth common tangent maps directly onto the nine-points circle!
Editions of the Sequel (1881–1892)¶
Published in the Dublin University Press Series (Dublin: Hodges, Figgis, & Co.; London: Longmans, Green, & Co.):
1st–3rd Editions (1881–1884): Books I–IV and Book VI Sections I–VII, culminating in inversion, Casey’s theorem, Feuerbach’s theorem, Hart’s circle, poles and polars, and coaxal circles.
4th Edition (1886): Major expansion adding the Supplementary Chapter on the Recent Elementary Geometry of the Triangle (pp. 165–222), introducing English-speaking students systematically to Lemoine’s symmedian point $K = X(6)$, Brocard’s points $\Omega, \Omega'$ and Brocard angle $\omega$, Tucker’s and Taylor’s circles, Neuberg’s and Tarry’s theorems, and harmonic polygons.
5th Edition (1888): Revised and enlarged by Casey himself (pp. i–xviii, 1–251; 276 archival folios), expanding Theory of Harmonic Polygons and Associated Figures. Inspected directly via 300-DPI multimodal scan vision as the primary witness of this codex.
6th Edition (1892): Posthumously seen through the press and revised by Casey’s pupil and son-in-law Patrick A. E. Dowling, B.A. (pp. i–xviii, 1–248), preserving the exact proposition and section numbering of the 5th edition; cited canonically as
[casey:sequel:1892]in Nathan Altshiller-Court’s College Geometry (1952).
Digital Scriptorium Verification Protocol¶
Every unit in docsrc/sequel_to_euclid/ obeys the GEOMETOR Multimodal Scan Vision protocol:
Archival page images rendered at 300 DPI (PDF Folio = Printed Page + 26 in the 1888 5th-edition witness) were visually inspected directly via multimodal scan vision rather than OCR layers.
Each unit pairs synchronized English (historical enunciations, demonstrations, corollaries, and calibrated
.euclid-graphicSVGs), Summary (mathematical synthesis and invariants ledger), and Notes (historical provenance, Dowling 1892 collation, and cross-references toeuclid,elements,altshiller-court,etc, andpappus).