Notes — Books VI & VII: Spectral Matrix Invariants & the Lost Book VIII¶
1. Linear-Algebraic Meaning of Apollonius’s Two Theorems¶
From the vantage point of modern linear algebra and Rational Trigonometry (Wildberger 2005), Propositions 129 and 136 (`VII.12–13, 31`) express the two fundamental orthogonal invariants (trace and determinant) of a matrix!
Let and be any pair of conjugate semi-diameter vectors of the ellipse , where . Conjugacy means maps the unit circle (or unit hyperbola ) onto the conic, so satisfies :
Trace Invariance (`VII.12`): !
Determinant Invariance (`VII.31`): !
Apollonius discovered and proved both spectral invariants synthetically in 200 BCE using solely Euclidean proportions and auxiliary segments MHMH on the axis!
2. How Book VII Reconstructs the Lost Book VIII¶
In the Preface to Book VII (Heath p. lxxv), Apollonius explicitly tells Attalus why he forged the intricate inequality theorems VII.21–51: “Several examples of these occur in the determinate conic problems solved and demonstrated by me in the eighth book, which is by way of an appendix.” Both Ibn al-Haytham (c. 1024 CE) and Edmond Halley (1710) used this clue—together with Pappus's preserved lemmas for Books VII and VIII intertwined in Synagoge VII—to reconstruct Book VIII as the constructive solution of conics given prescribed conjugate-diameter sums, differences, ratios, or parameters.