Liber V.A — Mathematical Intermediates, Unity, Quantity & Theoretical Sciences — Critical Commentary & Notes

Historical-Mathematical (Heath) & Scholastic-Noetic (Hoenen) Apparatus · Bekker 996a–998a, 1015b–1017a, 1020a, 1025b–1026a

I. Mathematical-Historical Commentary (Secundum Heath & Euclidem)

In The Thirteen Books of Euclid's Elements (Vol. I, pp. 155–166; Vol. II, p. 113; Vol. III, p. 260), Metaphysics $\Delta$.6 and $\Delta$.13 are among Heath's most heavily cited Aristotelian texts (1016a2, 1016a13, 1016b24–27, 1020a11–13). Together they supply the complete philosophical lexicon behind Book I (points, lines, planes), Book V (magnitudes), Book VII (units and numbers), and Book XI (solids) of the Elements.

In Metaphysics B.2 (997b35–998a6), Aristotle preserves two vivid fourth-century mathematical illustrations against naive physicalism: (1) Protagoras's empirical attack claiming a bronze hoop touches a straightedge along a finite segment rather than at a single point (compare Elements III.2 & III.16), and (2) astronomy’s treatment of stars—visibly larger than points—as mathematical point-centers on celestial spheres (998a4–6). As Heath (Mathematics in Aristotle, pp. 201–203) observes, Book B stages the dialectical knot that Book M.1–3 will untie.

II. Scholastic-Noetic Commentary (Secundum Hoenen, Thomam & Paciolum)

In De Noetica Geometriae (Caput I–II & VI), Hoenen highlights Metaphysics B.2 (997b35, cited directly in Hoenen’s apparatus!) and E.1 (1026a7–19) as the historical fountainhead of the three degrees of formal abstraction codified by Aquinas (Super Boethium De Trinitate, q. 5, aa. 1 & 3). Protagoras’s objection proves conclusively that geometrical concepts are not passive copies of crude sensory edges; yet Aristotle’s refutation of Platonic intermediates proves they are not inhabitants of a separate third world either. Instead, mathematics studies real quantitative structures present in physical bodies, considered qua immovable and abstracted from sensible matter!

In De Divina Proportione (Compendio Cap. I–III, fols. 1r–3r), Fra Luca Pacioli builds his opening manifesto of Renaissance mathematical humanism directly upon Aristotle’s classification of theoretical sciences and Averroes's commentary (In Metaph. II, comm. 16): mathematical demonstrations stand in primo gradu certitudinis.

III. Locus-by-Locus Philological & Geometrical Notes

996b26–33 — Who Investigates the Common Axioms?

Aristotle poses the aporia solved in Book $\Gamma$ (IV).3: although geometers and arithmeticians use the common axioms restricted to their own genus (magnitudes or numbers), it belongs to First Philosophy to investigate them universally qua being.

997b35–998a6 — Protagoras’s Tangent to the Circle

Euclid III.16 proves rigorously that no straight line can enter the space between a circle’s tangent and its circumference, vindicating ideal geometric continuity against Protagoras’s sensory ruler.

1020a7–14 — Arithmēton (Countable) vs. Metrēton (Measurable)

This crisp binary—multitude is counted by indivisible units, magnitude is measured by a chosen continuous unit length without ever becoming discrete—explains why Greek mathematics treated ratio of magnitudes separately from fraction of integers.

1026a23–32 — Universal Mathematics (Katholou Mathēmatikē)

Echoing Posterior Analytics I.5 (74a17–25), Aristotle contrasts particular mathematical sciences (geometry, astronomy) with ‘universal mathematics common to all’ (ἡ δὲ καθόλου κοινὴ περὶ πάντων, 1026a27), i.e. Eudoxus’s general proportion theory.

IV. Sister-Corpus Citations Indexed in This Unit

Bekker Locus	Treatise & Chapter	Sister Repos	Cited Locations (Volume, Page / Folio & Display)
996b21	Metaphysica III.2	euclid	euclid (volume_I, II, p. 410): Metaph. 996 b 21
996b26–30	Metaphysica III.2	euclid	euclid (volume_I, p. 120): Metaph. 996 b 26—30
997a10	Metaphysica III.2	euclid	euclid (volume_I, p. 121): Metaph. 997 a 10
997a20–22	Metaphysica III.2	euclid	euclid (volume_I, p. 120): Metaph. 997 a 20—22
997b35	Metaphysica III.2	hoenen	hoenen (de_noetica_geometriae, Caput III, p. 66): Met. III 2, 997 b 35
998a25	Metaphysica III.2	euclid	euclid (volume_I, p. 116): Metaph. 998 a 25
1016a2	Metaphysica V.6	euclid	euclid (volume_I, I, p. 159): Metaph. 1016 a 2
1016a13	Metaphysica V.6	euclid	euclid (volume_I, I, p. 176): Metaph. 1016 a 13
1016b24	Metaphysica V.6	euclid	euclid (volume_I, I, p. 155): Metaph. 1016 b 24
1016b25–27	Metaphysica V.6	euclid	euclid (volume_II, VII, p. 279): Metaph. 1016 b 25; euclid (volume_I, I, p. 158): Metaph. 1016 b 25—27
1020a11	Metaphysica V.12	euclid	euclid (volume_I, I, p. 159): Metaph. 1020 a 11
1020a12	Metaphysica V.12	euclid	euclid (volume_I, I, p. 170): Metaph. 1020 a 12
1020a13	Metaphysica V.12	euclid	euclid (volume_II, VII, p. 280): Metaph. 1020 a 13; euclid (volume_III, XI, p. 262): Metaph. 1020 a 13, 11