Liber V.A — Mathematical Intermediates, Unity, Quantity & Theoretical Sciences — Structural Synopsis

Argumentative Architecture & Bekker Conspectus (Bekker 996a–998a, 1015b–1017a, 1020a, 1025b–1026a)

Architectural Thesis

Liber V.A gathers the architectonic chapters of Aristotle's Metaphysics on axioms, mathematical objects, unity, quantity, and the classification of the sciences: Book B.2 (996a18–998a19) raising the sharp aporiai of sensible vs. ideal geometry (Protagoras's circle-tangent objection, 998a2–4), Book $\Delta$.6 & 13 (1015b16–1017a6, 1020a7–32) defining the indivisible measure and quantity, and Book E.1 (1025b3–1026a32) establishing the canonical tripartition of theoretical philosophy into physics, mathematics, and theology/first philosophy.

Logical & Noetic Progression

Metaphysics B (III).2 (996a18–998a19) — Aporiai of Axioms & Mathematical Intermediates

Asks whether one science studies both substance and the common axioms used by all mathematicians (996b26–33), and confronts the dilemma of Plato's ‘intermediates’ (ta metaxy): sensible lines are not straight as the geometer defines them, nor does a physical circle touch a ruler at a single point as Protagoras objected (997b35–998a6), yet positing a separate duplicate heaven and duplicate mathematical solids generates absurdities (997b12–34).

Metaphysics Δ (V).6 (1015b16–1017a6) — Ontological Lexicon of Unity & the Indivisible Measure

Defines ‘the one’ (to hen) by continuity (1015b36), wholeness of form, and above all being the primary indivisible measure of each genus (τὸ γὰρ ἓν εἶναι ἀρχή τινί ἐστιν ἀριθμοῦ εἶναι· τὸ γὰρ πρῶτον μέτρον ἀρχή, 1016b17–21). Distinguishes indivisible in every respect and without position (monad) from indivisible in every respect with position (point, 1016b24–26), one-dimensionally divisible (line), two-dimensionally (plane), and three-dimensionally (body).

Metaphysics Δ (V).13 (1020a7–1020a32) — Ontological Lexicon of Quantity (Poson)

Gives the classic definition of quantity (1020a7–14): that which is divisible into two or more constituent parts each by nature a ‘one’ and a ‘this’—a multitude if countable (arithmēton, divisible into non-continuous parts) and a magnitude if measurable (metrēton, divisible into continuous parts: length, breadth, depth).

Metaphysics E (VI).1 (1025b3–1026a32) — Tripartition of Theoretical Science

Divides all rational knowledge into practical, productive, and theoretical (1025b25), and differentiates the three theoretical philosophies (1026a18–19): physics (objects inseparable and movable), mathematics (some branches studying objects immovable though presumably not separate, ὡς ἀκίνητα καὶ ὡς χωριστὰ θεωρεῖ, 1026a9–10), and first philosophy / theology (objects both separate and immovable).

Chapter & Bekker Span Conspectus

Treatise & Chapter	Bekker Span	Thematic Scope	Tracks
Metaphysica B (III).2	996a18–998a19	Aporiai on Axioms, Mathematical Intermediates & Protagoras’s Tangent (997b35–998a6)	EN · EL
Metaphysica Δ (V).6	1015b16–1017a6	Lexicon of Unity (To Hen): Continuity, Indivisible Measure, Point & Monad (1016b24–31)	EN · EL
Metaphysica Δ (V).13	1020a7–1020a32	Lexicon of Quantity (Poson): Numberable Multitude (Plethos) vs. Measurable Magnitude (Megethos)	EN · EL
Metaphysica E (VI).1	1025b3–1026a32	Tripartition of Theoretical Science: Physics, Mathematics & First Philosophy	EN · EL

Reception Snapshot in the GEOMETOR Canon

This unit encompasses 13 distinct Bekker loci cited across geometor/euclid, geometor/hoenen, and geometor/pacioli. See the companion Critical Notes & Dual Commentary and the master Bekker Concordance for full folio and page back-references.