A SEQUEL TO THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID

Containing an Easy Introduction to Modern Geometry, with Numerous Examples (1888 / 1892)

JOHN CASEY, LL.D., F.R.S. (1820–1891)
Fellow of the Royal University of Ireland; Member of the Council of the Royal Irish Academy
Fifth Edition, Revised and Enlarged — Hodges, Figgis, & Co., Dublin; Longmans, Green, & Co., London, 1888 (pp. i–xviii, 1–251; 276 folios)
Sixth Edition, Revised and Enlarged, edited by PATRICK A. E. DOWLING, B.A. — Dublin / London, 1892 ([casey:sequel:1892])

“I have endeavoured to collect and arrange in a systematic form all those elementary geometrical propositions not given in Euclid which a student will require in reading Mixed Mathematics, or before entering on the study of the Higher Pure Mathematics.”

—John Casey, Preface to A Sequel to Euclid, p. v (f. 0007)

Conspectus of the 3-Track Digital Edition

Each unit below is published as a synchronized 3-Track Book Codex (English | Summary | Notes) with track-preserving navigation, multimodal scan-verified marginal folio milestones (PDF Folio = Printed Page + 26), calibrated interactive .euclid-graphic SVGs, and hoverable :person:, :cite:, and :term: apparatus cards.

Unit 1: Books I, II & IV Extensions — Triangles, Euler Line, Medians & the Nine-Points Circle

Concurrency of Altitudes (Orthocentre), Centres of Mean Position, Maxima & Minima, Euler Line, Apollonius’s & Stewart’s Theorems, Heron’s Formula, and the Nine-Points Circle

English · Summary · Notes — Folios 0027–0089 (pp. 1–63)

Unit 2: Book III & Book VI Sec. V Extensions — Power of a Point, Orthogonal & Coaxal Circles

Inverse Points, Power of a Point, Orthogonal Circles, Radical Axis, Radical Centre, Coaxal Systems of Circles, and Poncelet’s Limiting Points

English · Summary · Notes — Folios 0067–0075, 0139–0152 (pp. 41–49, 113–126)

Unit 3: Book IV & VI Extensions, Part A — Transversals, Similitude, Harmonic & Anharmonic Ratios

Ceva’s & Menelaus’s Theorems, Centres of Similitude & Monge’s Theorem, Harmonic Section, Complete Quadrilaterals, Anharmonic (Cross) Ratios, Desargues’s & Pascal’s Theorems

English · Summary · Notes — Folios 0093–0120, 0153–0166 (pp. 67–94, 127–140)

Unit 4: Book VI Extensions, Part B & Book III Props. 25, 27 — Inversion, Casey’s & Feuerbach’s Theorems, Poles & Polars

Circular Inversion, Pappus’s Arbelos Chain, Casey’s Generalized Ptolemy Theorem (1866), Feuerbach’s Nine-Point Tangency Theorem, Hart’s Circle, Poles & Polars, La Hire’s Reciprocity, and Brianchon’s Theorem

English · Summary · Notes — Folios 0070–0073, 0121–0138, 0167–0177 (pp. 44–47, 95–112, 141–151)

Unit 5: Supplementary Chapter, Secs. I & III (Part 1) — Symmedians, Lemoine’s Point & Lemoine Circles

Isogonal & Isotomic Conjugates, Antiparallels, Symmedians, Lemoine’s Point $K = X(6)$, Lagrange’s Minimum Property, Lemoine’s First (Triplicate-Ratio) Circle, and the Cosine Circle

English · Summary · Notes — Folios 0191–0200, 0205–0209 (pp. 165–174, 179–183)

Unit 6: Supplementary Chapter, Secs. II–VII & Misc. — Brocard Geometry, Tucker/Taylor Circles, Harmonic Polygons & Associated Points

Figures Directly Similar, Brocard Points $\Omega, \Omega'$ & Brocard Angle $\omega$, Tucker & Taylor Circles, Brocard’s Circle & First Triangle, M’Clelland’s Second Proof of Feuerbach, Harmonic Polygons, Steiner $X(99)$, Tarry $X(98)$, Neuberg & Longchamps $X(20)$ Circles

English · Summary · Notes — Folios 0201–0274 (pp. 175–248)

Apparatus Criticus & Cross-Repository Indices