Hoenen–Aquinas–Aristotle–Euclid–GEOMETOR Concordance¶
Complete 78-Locus Cross-Reference Matrix Grounding Thomistic Geometric Noetics
Complete critical concordance mapping 100% of the 78 Thomistic and Pseudo-Thomistic citation keys in Peter Hoenen S.J.’s De Noetica Geometriae (1954)—together with Luca Pacioli’s De Divina Proportione (1509), Hoenen’s La théorie du jugement d’après St. Thomas d’Aquin (1946/1953), and Bernard Lonergan’s Verbum (1946/1967)—to their critical editions, underlying Aristotelian loci, Euclidean counterparts, and GEOMETOR symbolic architecture.
Total Hoenen Thomistic Citation Keys: 78 (76 authentic Thomas Aquinas keys + 2 Pseudo-Thomas scholastic keys)
Unabridged Full-Text Coverage in Pars I–V: 39 core geometric-noetic keys
Complementary Metaphysical & Logical Loci: 39 keys fully analyzed below
Sister Repository Bridges Across GEOMETOR¶
geometor-hoenen — P. H. J. Hoenen, S.J., De Noetica Geometriae: Origine Theoriae Cognitionis (Romae: Apud Aedes Universitatis Gregorianae, 1954)¶
Cites St. Thomas Aquinas across 78 distinct locus keys to establish that geometric axioms and theorems arise through intuitive formal abstraction (abstractio intuitiva formalis) within spatial phantasms (materia intelligibilis) prior to propositional judgment.
geometor-pacioli — Fra Luca Pacioli, O.F.M., De Divina Proportione (Venice: Paganino Paganini, 1509), Pars I, Capp. II–VII¶
Echoes the Aristotelian-Thomistic hierarchy of speculative sciences (Super Boetium De Trinitate qq. 5–6 and Posterior Analytics I.1): mathematical science stands in the primo grado di certezza (‘first degree of certainty’), abstracting continuous and discrete quantity to contemplate the Divine Proportion ($\phi$) and the five regular Platonic solids.
geometor-euclid / geometor-elements — Euclid, Elements (Books I–XIII; Byrne 1847 color diagrams & symbolic DAGs)¶
Supplies every concrete geometric paradigm analyzed by Aquinas: Definitions of point, line, surface, circle (Metaph. VII.10–11); Postulates & Common Notions (Anal. Post. I.5, I.18–19); equilateral construction I.1 (Anal. Post. I.2); congruence I.4 (Phys. IV.7); triangle angle-sum I.32 & semicircle right angle III.31 (Metaph. IX.10); and diagonal incommensurability (Metaph. V.9).
geometor-model / geometor-explorer / geometor-divine — GEOMETOR Symbolic Engine, Interactive Phantasm Canvas, and Golden Ratio Discovery¶
Computationally embodies Thomistic geometric noetics: geometor.model encodes pure materia intelligibilis in exact SymPy algebraic fields; geometor.explorer renders the determinate visual phantasma required by conversio ad phantasmata; and geometor.divine performs inventio per divisionem in actu to uncover emergent golden sections.
Master Summary Matrix (All 78 Citation Keys)¶
Hoenen Key |
Thomistic Work & Locus |
Aristotelian Source |
Hoenen Occurrence(s) |
Codex Link |
|---|---|---|---|---|
In Aristotelis Libros Posteriorum Analyticorum Expositio (passim) |
Posterior Analytics I–II (71a1–100b17) |
Caput V p.187 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.1.10) |
Posterior Analytics I.1, 71a1–11 |
Caput I p.11 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.10.7) |
Posterior Analytics I.4, 73b25–74a3 |
Caput II p.57 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.15) |
Posterior Analytics I.7, 75a38–b20 |
Caput VII p.223 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.17.4) |
Posterior Analytics I.9, 76a16–25 |
Caput I p.12 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.19.2) |
Posterior Analytics I.10, 76b23–34 |
Caput IV p.144; Caput IV p.145 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.19.4) |
Posterior Analytics I.10, 76b39–77a3 |
Caput IV p.145; Caput IV p.145 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.2.5) |
Posterior Analytics I.1, 71a11–17 |
Caput IV p.146; Caput VII p.223 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.42.7) |
Posterior Analytics I.27, 87a31–37 |
Caput III p.66 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.5.7) |
Posterior Analytics I.2, 72a14–24 |
Caput I p.12; Caput I p.12 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (I.6) |
Posterior Analytics I.3, 72b5–73a20 |
Caput I p.10 |
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In Aristotelis Libros Posteriorum Analyticorum Expositio (II.20) |
Posterior Analytics II.19, 99b15–100b17 |
Caput II p.51 |
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Sentencia Libri De Anima (2.16) |
De Anima II.8, 419b4–420b4 |
Appendix p.287 |
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Sentencia Libri De Anima (III.12.770-777) |
De Anima III.7, 431a14–b19 |
Caput I p.14 |
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Sentencia Libri De Anima (III.13.791) |
De Anima III.8, 431b20–432a14 |
Caput I p.14 |
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Sentencia Libri De Anima (III.8) |
De Anima III.4, 429b10–22 |
Caput II p.62 |
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Sentencia Libri De Anima (b1-l11) |
De Anima I.4, 409a3–7 |
Caput IV p.107 |
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In Libros Aristotelis De Caelo et Mundo Expositio (b1-l2) |
De Caelo I.1, 268a1–b10 |
Caput IV p.108 |
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In Libros Aristotelis De Caelo et Mundo Expositio (b2-l2) |
De Caelo II.2, 284b6–285b33 |
Caput IV p.108 |
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Quaestiones Disputatae de Malo (6.1.ad18) |
Metaphysics VI.4, 1027b25–27; De Anima III.6 |
Caput VII p.241 |
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Quaestiones Disputatae de Potentia Dei (7.2.ad9) |
Metaphysics V.7 & IX.8 |
Appendix p.279; Appendix p.279; Appendix p.283 (+1) |
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Quaestiones Disputatae de Potentia Dei (8.1) |
De Anima III.4–6 |
Appendix p.277; Appendix p.278 |
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Quaestiones Disputatae de Potentia Dei (9.5) |
De Anima III.6, 430a26–b6 |
Appendix p.277; Appendix p.278; Appendix p.279 (+1) |
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Expositio super Librum Boethii De Trinitate (5.1.ad3) |
Metaphysics VI.1, 1026a13–19; Ethics VI.7 |
Caput VII p.225 |
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Expositio super Librum Boethii De Trinitate (q5-a3) |
De Anima III.7, 431b12–17; Physics II.2, 193b22–194a12; Metaphysics VII.10, 1036a9–12 |
Caput IV p.101 |
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Expositio super Librum Boethii De Trinitate (q6.a1) |
Posterior Analytics I.2, 71b16–25; Ethics I.3, 1094b23–27 |
Caput V p.167 |
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Quaestiones Disputatae de Veritate (10.12.ad7) |
Posterior Analytics I.2, 72a25–b4 |
Appendix p.255 |
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Quaestiones Disputatae de Veritate (10.6.sc2) |
Posterior Analytics II.19, 99b35–100b17 |
Caput II p.38; Caput VII p.236 |
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Quaestiones Disputatae de Veritate (4.2) |
Peri Hermeneias I.1, 16a3–8 |
Appendix p.278; Appendix p.279 |
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Quaestiones Disputatae de Veritate (q10-a6) |
De Anima III.4–8; Posterior Analytics II.19 |
Caput IV p.97 |
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Quaestiones Disputatae de Veritate (q12-a3) |
Posterior Analytics I.18, 81a38–b9 |
Caput IV p.98 |
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Sententia Libri Ethicorum (IX.11) |
Nicomachean Ethics IX.9, 1170a25–b10 |
Appendix p.260 |
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Sententia super Metaphysicam (10.4.2017) |
Metaphysics X.3, 1054a20–b3 |
Appendix p.283; Appendix p.283 |
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Sententia super Metaphysicam (4.l4.n577) |
Metaphysics IV.3, 1005a19–b8 |
Caput V p.165 |
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Sententia super Metaphysicam (5.12.916) |
Metaphysics V.10, 1018a25–35 |
Appendix p.283 |
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Sententia super Metaphysicam (5.9) |
Metaphysics V.7, 1017a7–b9 |
Appendix p.284 |
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Sententia super Metaphysicam (5.9.890) |
Metaphysics V.7, 1017a22–30 |
Appendix p.284 |
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Sententia super Metaphysicam (9.10.1894) |
Metaphysics IX.9, 1051a21–33 |
Appendix p.273 |
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Sententia super Metaphysicam (9.3.1805) |
Metaphysics IX.3, 1047a30–b2 |
Appendix p.280 |
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Sententia super Metaphysicam (9.3.1806) |
Metaphysics IX.3, 1047a32–b2 |
Appendix p.280 |
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Sententia super Metaphysicam (III.7.416) |
Metaphysics III.2, 997b25–998a19 |
Caput III p.66 |
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Sententia super Metaphysicam (IX.3.1805) |
Metaphysics IX.3, 1047a30–b2 |
Appendix p.253 |
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Sententia super Metaphysicam (IX.7.1848) |
Metaphysics IX.8, 1049b12–17 |
Appendix p.265 |
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Sententia super Metaphysicam (IX.8.1857) |
Metaphysics IX.8, 1050a4–16 |
Appendix p.268 |
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Sententia super Metaphysicam (X.2.1938) |
Metaphysics X.1, 1052b20–1053a14 |
Caput VI p.216 |
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Sententia super Metaphysicam (XI.11.2370) |
Metaphysics XI.11, 1067b25–1068a7 |
Appendix p.252 |
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Sententia super Metaphysicam (b7-l10) |
Metaphysics VII.10–11, 1035b31–1037a5 |
Caput IV p.101 |
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Expositio Libri Peryermeneias (1.5.17) |
Peri Hermeneias I.3, 16b19–25 |
Appendix p.279 |
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Expositio Libri Peryermeneias (1.5.5) |
Peri Hermeneias I.3, 16b6–25 |
Appendix p.279 |
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Commentaria in Octo Libros Physicorum Aristotelis (3.5) |
Physics III.3, 202b19–29 |
Appendix p.284; Appendix p.284; Appendix p.284 |
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Commentaria in Octo Libros Physicorum Aristotelis (4.17) |
Physics IV.11, 218b21–219b8 |
Appendix p.286 |
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Commentaria in Octo Libros Physicorum Aristotelis (II.3) |
Physics II.2, 193b22–194b15 |
Caput II p.62 |
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Commentaria in Octo Libros Physicorum Aristotelis (IV.3) |
Physics IV.2, 209b1–17 |
Appendix p.261 |
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Commentaria in Octo Libros Physicorum Aristotelis (IV.7) |
Physics IV.4, 211b5–212a21 |
Appendix p.262 |
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Commentaria in Octo Libros Physicorum Aristotelis (IV.7.7) |
Physics IV.4, 212a14–21 |
Appendix p.262 |
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Commentaria in Octo Libros Physicorum Aristotelis (V.2.8) |
Physics V.1, 225b5–9 |
Appendix p.251 |
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Commentaria in Octo Libros Physicorum Aristotelis (V.2.9) |
Physics V.1, 225b10–16 |
Appendix p.252 |
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Commentaria in Octo Libros Physicorum Aristotelis (VIII.1) |
Physics VIII.1, 250b11–252b6 (& Physics VI.1, 231a21–b18) |
Appendix p.252 |
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Commentaria in Octo Libros Physicorum Aristotelis (b4-l18) |
Physics IV.11, 219b9–33 |
Caput IV p.108 |
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Summa contra Gentiles (1.26.2) |
Posterior Analytics II.7, 92b13–14; Metaphysics III.3 |
Appendix p.284 |
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Summa contra Gentiles (I.36) |
Metaphysics V.7 & VI.4 |
Caput II p.57 |
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Scriptum super Libros Sententiarum (1.19.5.1.ad6) |
Metaphysics VI.4, 1027b25–27 |
Appendix p.280 |
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Scriptum super Libros Sententiarum (1.2.1.3) |
Metaphysics IV.2 & VII.1 |
Appendix p.281 |
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Scriptum super Libros Sententiarum (I.3.1.2) |
Posterior Analytics I.2; De Anima III.4 |
Caput II p.52 |
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Scriptum super Libros Sententiarum (I.4.2.1.ad1) |
Peri Hermeneias I.1–3 |
Caput II p.57 |
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Scriptum super Libros Sententiarum (III.11.1.4.ad6) |
Metaphysics V.7 |
Caput II p.58 |
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Summa Theologiae (I.12.4.ad3) |
De Anima III.4, 429a18–b22 |
Appendix p.280 |
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Summa Theologiae (I.13.12) |
Peri Hermeneias I.1–3, 16a1–16b25 |
Caput II p.57 |
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Summa Theologiae (I.7.1) |
Physics III.6, 207a7–15 |
Appendix p.258 |
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Summa Theologiae (I.75.6) |
De Anima I.4, 408b18–29; III.4, 429a24–b5 |
Appendix p.280 |
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Summa Theologiae (I.85.1.ad2) |
Metaphysics VII.10, 1036a9–12; Physics II.2, 193b31–35 |
Caput II p.62 |
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Summa Theologiae (I.85.5.ad3) |
De Anima III.6, 430a26–b6 |
Caput II p.57 |
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Summa Theologiae (I.q84.a6) |
De Anima III.4–5, 429a10–430a25 |
Caput III p.82 |
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Summa Theologiae (I.q84.a6.ad3) |
De Anima III.7, 431a16–17 |
Caput III p.82 |
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Summa Theologiae (II-II.24.4.ad3) |
Categories 8, 8b25–9a13; Physics VII.3 |
Appendix p.287 |
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Summa Theologiae (III.q11.a2.ad1) |
De Anima III.7, 431a14–17 |
Caput III p.83 |
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Summa Totius Logicae Aristotelis (passim) |
Posterior Analytics I.2 & I.10 |
Caput IV p.148; Caput V p.161 |
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Summa Totius Logicae Aristotelis (VII.2) |
Posterior Analytics I.10, 76b23–34 |
Caput IV p.148 |
Detailed Locus-by-Locus Analytical Concordance¶
In Aristotelis Libros Posteriorum Analyticorum Expositio¶
aquinas:anal-post — passim¶
Critical Edition: Leonine I* 2, Expositio libri Posteriorum Analyticorum, Lib. I–II
Aristotelian Foundation: Posterior Analytics I–II (71a1–100b17)
Euclidean Counterpart: Axiomatic architecture of Euclid’s Elements as paradigm of scientia demonstrativa
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput V p. 187 (
S. Th.)Hoenen’s Doctrinal Argument: Cites Aquinas’s Posterior Analytics commentary as the overarching scholastic synthesis of Aristotelian demonstrative science and Euclidean axiomatics against modern pure formalism.
GEOMETOR Architectural Significance: Provides the logical meta-theory for constructive DAGs and axiomatic derivation in geometor.elements.
aquinas:anal-post:I.1.10 — I.1.10¶
Critical Edition: Leonine I* 2, Lib. I, lect. 1, n. 10–12
Aristotelian Foundation: Posterior Analytics I.1, 71a1–11
Euclidean Counterpart: Euclid’s Elements as the canonical realization of certissimus modus demonstrationis
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput I p. 11 (
Anal. Post. I lect, 1 n. 10)Hoenen’s Doctrinal Argument: Establishes that mathematics alone among human sciences attains the ‘most certain mode of demonstration’ (certissimum modum) because it abstracts from sensible flux without transcending imaginative verification.
GEOMETOR Architectural Significance: Justifies exact symbolic computation (SymPy exact algebraic fields) over probabilistic or floating-point heuristics.
aquinas:anal-post:I.10.7 — I.10.7¶
Critical Edition: Leonine I* 2, Lib. I, lect. 10, n. 7
Aristotelian Foundation: Posterior Analytics I.4, 73b25–74a3
Euclidean Counterpart: Elements I.32 (interior angles equal to two right angles belonging per se and secundum quod ipsum to triangle qua triangle, not qua isosceles)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput II p. 57 (
S. Thomas Analyt. Post. I lect. 10 n. 7)Hoenen’s Doctrinal Argument: Defines universale / secundum quod ipsum (commensurate universal) predication: the geometer proves two right angles of triangle universally only when abstracting from equilateral, isosceles, or scalene particularities.
GEOMETOR Architectural Significance: Models structural invariance across parametric configurations in geometor.model and geometor.explorer.
aquinas:anal-post:I.15 — I.15¶
Critical Edition: Leonine I* 2, Lib. I, lect. 15
Aristotelian Foundation: Posterior Analytics I.7, 75a38–b20
Euclidean Counterpart: Prohibition of metabasis eis allo genos between arithmetic (Book VII–IX) and continuous magnitude (Book I–VI), bridged only by proportion (Book V)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput VII p. 223 (
Anal. Post. I l. 15)Hoenen’s Doctrinal Argument: Examines why demonstrations cannot cross from one genus subiectum to an unrelated genus except where a superior mathematical science supplies the propter quid to a subalternate science.
GEOMETOR Architectural Significance: Clarifies the algebraic bridge between discrete integer ratios and continuous quadratic/radical extensions in geometor.divine.
aquinas:anal-post:I.17.4 — I.17.4¶
Critical Edition: Leonine I* 2, Lib. I, lect. 17, n. 4–5
Aristotelian Foundation: Posterior Analytics I.9, 76a16–25
Euclidean Counterpart: Why geometry assumes its own Postulates and Common Notions while leaving their ultimate epistemological defense to First Philosophy
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput I p. 12 (
lect. 17 n. 4)Hoenen’s Doctrinal Argument: Shows why the geometer qua geometer does not dispute with one who denies geometric principles, whereas metaphysics (philosophia prima) sapientially defends the objective validity of first principles.
GEOMETOR Architectural Significance: Situates geometor-aquinas and geometor-hoenen as the foundational epistemological meta-layer of the GEOMETOR ecosystem.
aquinas:anal-post:I.19.2 — I.19.2¶
Critical Edition: Leonine I* 2, Lib. I, lect. 19, n. 2–3
Aristotelian Foundation: Posterior Analytics I.10, 76b23–34
Euclidean Counterpart: Euclid’s Postulates 1–5 (petitiones) vs. Hypotheses (suppositiones) vs. Common Notions (dignitates)
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 144 (
S. Th. lect.); Caput IV p. 145 (19 n. 2)Hoenen’s Doctrinal Argument: Distinguishes suppositiones (propositions to which the learner assents) from petitiones (postulates assumed even without prior opinion of the learner) and self-evident communes animi conceptiones.
GEOMETOR Architectural Significance: Directly illuminates Postulates 1–3 (line and circle constructors) and Postulate 5 (parallel postulate) in geometor.elements.
aquinas:anal-post:I.19.4 — I.19.4¶
Critical Edition: Leonine I* 2, Lib. I, lect. 19, n. 4–6
Aristotelian Foundation: Posterior Analytics I.10, 76b39–77a3
Euclidean Counterpart: Geometric diagrams drawn in dust/chalk vs. the intelligible lines signified by them
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 145 (
ibid. lect. 19 n. 4); Caput IV p. 145 (S. Th. ibid.)Hoenen’s Doctrinal Argument: Refutes the accusation that the geometer reasons from falsehoods when calling a rough line drawn in dust ‘one foot long’ or ‘straight’: the geometer demonstrates not of the sensible trace, but of the intelligible continuum signified through it.
GEOMETOR Architectural Significance: Core charter of GEOMETOR: an SVG rendering on a finite screen (geometor.explorer) is only the sensible/imaginative sign; the demonstration operates upon exact SymPy algebraic entities (geometor.model).
aquinas:anal-post:I.2.5 — I.2.5¶
Critical Edition: Leonine I* 2, Lib. I, lect. 2, n. 5
Aristotelian Foundation: Posterior Analytics I.1, 71a11–17
Euclidean Counterpart: Elements I.def.19–20 (nominal/quidditative definition of triangle) vs. Elements I.1 (demonstrative construction proving quia est)
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 146 (
Anal. Post. I lect. 2 n. 5); Caput VII p. 223 (Anal. Post. I lect. 2 n. 5)Hoenen’s Doctrinal Argument: Distinguishes the twofold foreknowledge (praecognitio): regarding principles one must foreknow quia verum est; regarding the subject-genus (unit, magnitude) both quid est and quia est; regarding derived figures and attributes (triangle, angle-sum) first quid nominis and then quia est via construction.
GEOMETOR Architectural Significance: Explains why Euclid I.1 constructs an equilateral triangle before assuming triangles exist: in geometor.model, geometric existence (quia est) is established by constructive intersection.
aquinas:anal-post:I.42.7 — I.42.7¶
Critical Edition: Leonine I* 2, Lib. I, lect. 42, n. 3–7
Aristotelian Foundation: Posterior Analytics I.27, 87a31–37
Euclidean Counterpart: Elements I.def.1 (point as indivisible having position in the continuum) vs. VII.def.1 (unit without position)
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput III p. 66 (
lect. 42 n. 7)Hoenen’s Doctrinal Argument: Demonstrates how geometry adds positio (spatial situs in materia intelligibilis) to the arithmetical unit: unitas est substantia non posita, punctus autem est substantia posita.
GEOMETOR Architectural Significance: Explains why every Point(x, y) in geometor.model is defined by exact algebraic coordinates encoding positio within the plane.
aquinas:anal-post:I.5.7 — I.5.7¶
Critical Edition: Leonine I* 2, Lib. I, lect. 5, n. 7
Aristotelian Foundation: Posterior Analytics I.2, 72a14–24
Euclidean Counterpart: Division of Euclidean first principles into Common Notions (dignitates), Definitions (positiones quid est), and Postulates/Hypotheses (suppositiones)
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput I p. 12 (
Anal. Post. I lect. 5 n. 7); Caput I p. 12 (ibid. lect. 5 n. 7)Hoenen’s Doctrinal Argument: Clarifies the rigorous epistemic hierarchy separating per se self-evident axioms (dignitates) known to every intellect upon grasping their terms from theses (positiones) divided into definitions and existential hypotheses.
GEOMETOR Architectural Significance: Maps directly onto the separation in geometor.elements between algebraic equivalence axioms (Common Notions) and generative construction primitives (Postulates 1–3).
aquinas:anal-post:I.6 — I.6¶
Critical Edition: Leonine I* 2, Lib. I, lect. 6
Aristotelian Foundation: Posterior Analytics I.3, 72b5–73a20
Euclidean Counterpart: Acyclicity of Euclidean proof DAGs grounded in indemonstrable first principles
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput I p. 10 (
Anal. Post. I lect. 6)Hoenen’s Doctrinal Argument: Refutes both radical skepticism (infinite regress) and coherentist circularity (demonstratio circularis): demonstration requires immediate, intrinsically evident principles grasped by intellectus.
GEOMETOR Architectural Significance: Guarantees that every construction in geometor.model and geometor.elements forms a strictly well-founded Directed Acyclic Graph (DAG) rooted in initial given points.
aquinas:anal-post:II.20 — II.20¶
Critical Edition: Leonine I* 2, Lib. II, lect. 20, n. 1–14
Aristotelian Foundation: Posterior Analytics II.19, 99b15–100b17
Euclidean Counterpart: Cognitive genesis of Euclid’s Common Notions and Postulates from sense, memory, and experimentum via intellectus
Codex Edition: Unabridged in English (pars-2) · Diplomatic Latin
Cited in Hoenen (1954): Caput II p. 51 (
Anal. Post. in ultima lectione)Hoenen’s Doctrinal Argument: Traces the genetic emergence of the habit of first principles (intellectus) from sense perception, memory, and experimentum—showing that inductio here is not statistical enumeration, but intellectual insight into necessary intelligible nexus within singular experience.
GEOMETOR Architectural Significance: Parallel to how visual exploration in geometor.explorer and pattern synthesis in geometor.seer uncover universal structural laws from concrete exemplars.
Sentencia Libri De Anima¶
aquinas:de-anima:2.16 — 2.16¶
Critical Edition: Leonine XLV.1, Lib. II, lect. 16; ed. Pirotta (Marietti), nos. 439–450
Aristotelian Foundation: De Anima II.8, 419b4–420b4
Euclidean Counterpart: Successive esse fluens of acoustic motion vs. permanent geometric extension (and harmonic ratios in music)
Codex Edition: Unabridged in English (pars-3) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 287 (
De Anim. II l. 16)Hoenen’s Doctrinal Argument: Cited in Hoenen’s Appendix on successive duration: contrasts permanent spatial continua (whose parts coexist simultaneously) with fluent realities like sound and motion whose esse consists in becoming.
GEOMETOR Architectural Significance: Bridges GEOMETOR’s spatial proportions (divine) with temporal/acoustic proportions (harmonic-resonance).
aquinas:de-anima:III.12.770-777 — III.12.770-777¶
Critical Edition: Leonine XLV.1, Lib. III, lect. 12; ed. Pirotta (Marietti), nos. 770–777
Aristotelian Foundation: De Anima III.7, 431a14–b19
Euclidean Counterpart: Demonstrating universal properties of an indeterminate triangle by placing a determinate finite triangle before the eyes
Codex Edition: Unabridged in English (pars-3) · Diplomatic Latin
Cited in Hoenen (1954): Caput I p. 14 (
S. Thomas lect. 12 ; Pirotta n. 770-772, n. 777)Hoenen’s Doctrinal Argument: Key psychological text of De Noetica Geometriae Caput I: nequaquam sine phantasmate intelligit anima—even in abstract geometry, the intellect beholds universal mathematical objects in phantasms, abstracting from determinate size while drawing a determinate magnitude.
GEOMETOR Architectural Significance: Explains why every abstract proof benefits from a concrete coordinate instantiation rendered in geometor.explorer.
aquinas:de-anima:III.13.791 — III.13.791¶
Critical Edition: Leonine XLV.1, Lib. III, lect. 13; ed. Pirotta (Marietti), nos. 787–794 (esp. n. 791)
Aristotelian Foundation: De Anima III.8, 431b20–432a14
Euclidean Counterpart: Intellect as species specierum and hand as organum organorum wielding compass and straightedge
Codex Edition: Unabridged in English (pars-3) · Diplomatic Latin
Cited in Hoenen (1954): Caput I p. 14 (
S. Thomas lect. 13 ; Pirotta n. 791)Hoenen’s Doctrinal Argument: Synthesizes the noetic hierarchy: intelligible objects exist inside sensible forms—even mathematical abstractions—so whoever lacked sense could neither learn nor use geometric science without turning to phantasms.
GEOMETOR Architectural Significance: Celebrates both the constructive instrument (the hand/compass/symbolic engine as organum organorum) and the intellect (species specierum).
aquinas:de-anima:III.8 — III.8¶
Critical Edition: Leonine XLV.1, Lib. III, lect. 8; ed. Pirotta (Marietti), nos. 705–719
Aristotelian Foundation: De Anima III.4, 429b10–22
Euclidean Counterpart: The straight (rectum, Euclid I.def.4) in the intelligible continuum vs. the snub (simum) in sensible flesh
Codex Edition: Unabridged in English (pars-3) · Diplomatic Latin
Cited in Hoenen (1954): Caput II p. 62 (
De anim. III lect. 8)Hoenen’s Doctrinal Argument: Expounds Aristotle’s proportion carnes et ossa : simum :: continuum : rectum: just as natural forms require sensible matter, geometric forms (rectum, curvum) require continuous extension (materia intelligibilis) apprehended by turning back upon phantasms.
GEOMETOR Architectural Significance: Shows why geometric forms cannot be reduced to non-spatial logicism (Russell): straightness intrinsically presupposes continuous extension.
aquinas:de-anima:b1-l11 — b1-l11¶
Critical Edition: Leonine XLV.1, Lib. I, lect. 11; ed. Pirotta (Marietti), nos. 168–177 (esp. n. 170)
Aristotelian Foundation: De Anima I.4, 409a3–7
Euclidean Counterpart: Kinematic generation of a line by a flowing point vs. Euclid I.def.2–4
Codex Edition: Unabridged in English (pars-3) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 107 (
lect. 11, ed. Pirotta n. 170)Hoenen’s Doctrinal Argument: Analyzes Democritus’s parallel with geometers who imagine a moving point describing a line: motion belongs not to abstract mathematical essence in itself, but to the constructive imaginatio tracing spatial continuity.
GEOMETOR Architectural Significance: Mirrors animated compass-and-straightedge sweeps in geometor.explorer tracing static algebraic loci.
In Libros Aristotelis De Caelo et Mundo Expositio¶
aquinas:de-caelo:b1-l2 — b1-l2¶
Critical Edition: Leonine III, Lib. I, lect. 2, n. 9
Aristotelian Foundation: De Caelo I.1, 268a1–b10
Euclidean Counterpart: Perfection of three-dimensional solid geometry (Books XI–XIII) and kinematic generation of line/surface/solid
Codex Edition: Unabridged in English (pars-4) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 108 (
De Caelo I lect. 2 n. 9)Hoenen’s Doctrinal Argument: Establishes that three dimensions exhaust continuous spatial extension (line -> surface -> body) and clarifies that geometers speak of a moving point generating a line imaginando for pedagogical manifestation, not physical causation.
GEOMETOR Architectural Significance: Connects planar geometry (model) to the five regular Platonic solids of Euclid Book XIII and Luca Pacioli (pacioli).
aquinas:de-caelo:b2-l2 — b2-l2¶
Critical Edition: Leonine III, Lib. II, lect. 2, n. 11
Aristotelian Foundation: De Caelo II.2, 284b6–285b33
Euclidean Counterpart: Orientation, chirality, and positional directions (above/below, right/left) in geometric figures
Codex Edition: Unabridged in English (pars-4) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 108 (
ibid. II lect. 2 n. 11)Hoenen’s Doctrinal Argument: Shows that while abstract mathematical figures lack intrinsic natural powers of ‘up’ or ‘right’, they acquire positional orientation (secundum positionem ad nos) through our imaginative perspective.
GEOMETOR Architectural Significance: Directly models oriented cycles, signed angles, and viewport orientation in geometor.model and geometor.explorer.
Quaestiones Disputatae de Malo¶
aquinas:de-malo:6.1.ad18 — 6.1.ad18¶
Critical Edition: Leonine XXIII, De Malo, q. 6, a. 1, ad 18
Aristotelian Foundation: Metaphysics VI.4, 1027b25–27; De Anima III.6
Euclidean Counterpart: Necessity of assenting to demonstrated Euclidean theorems once resolved back to first principles
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput VII p. 241 (
Q. D. de Malo q. 6 a. 1 ad 18)Hoenen’s Doctrinal Argument: Explains how the intellect is moved of necessity by objective evidence both in first indemonstrable principles and in demonstrated geometric conclusions rigorously reduced to them.
GEOMETOR Architectural Significance: Corresponds to deterministic symbolic verification in geometor.model and geometor.elements.
Quaestiones Disputatae de Potentia Dei¶
aquinas:de-potentia:7.2.ad9 — 7.2.ad9¶
Critical Edition: ed. Bazzi (Marietti), De Potentia, q. 7, a. 2, co. & ad 9
Aristotelian Foundation: Metaphysics V.7 & IX.8
Euclidean Counterpart: Existential actuality (esse) grounding the reality of mathematical subjects and the truth of geometric judgments
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 279 (
de Pot. q. 7, a. 2, ad 9); Appendix p. 279 (loc. cit.); Appendix p. 283 (De Pot. q. 7 a. 2 ad 9); Appendix p. 286 (Pot. 7, 2 ad 9)Hoenen’s Doctrinal Argument: Classic Thomistic milestone: hoc quod dico esse est inter omnia perfectissimum… actualitas omnium actuum, et propter hoc est perfectio omnium perfectionum—form relates to esse as potency to act.
GEOMETOR Architectural Significance: Grounds Hoenen’s existential theory of judgment: even mathematical necessity is rooted in intelligible compatibility with esse.
aquinas:de-potentia:8.1 — 8.1¶
Critical Edition: ed. Bazzi (Marietti), De Potentia, q. 8, a. 1, co.
Aristotelian Foundation: De Anima III.4–6
Euclidean Counterpart: Fourfold distinction: intelligible geometric object (res intellecta), species, act of understanding, and conceptio / verbum mentis (Euclidean definition or proposition)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 277 (
de Pot. q. 8 a. 1); Appendix p. 278 (Pot. 8, 1)Hoenen’s Doctrinal Argument: Cited with Lonergan’s Verbum: distinguishes the impressed intelligible species (principle of understanding) from the expressed conceptio intellectus (the definition or theorem formed by the mind).
GEOMETOR Architectural Significance: Maps cleanly onto input geometric data -> solver reasoning -> expressed canonical symbolic theorem in geometor.model and geometor.seer.
aquinas:de-potentia:9.5 — 9.5¶
Critical Edition: ed. Bazzi (Marietti), De Potentia, q. 9, a. 5, co.
Aristotelian Foundation: De Anima III.6, 430a26–b6
Euclidean Counterpart: Twofold verbum mentis: the definition (incomplexum, Euclid Defs.) and the proposition/judgment (complexum, Euclid Props.)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 277 (
ibid. q. 9); Appendix p. 278 (a. 5); Appendix p. 279 (9, 5); Appendix p. 279 (Pot. 9, 5)Hoenen’s Doctrinal Argument: Shows that the inner word (verbum) arises in both operations of the intellect: first as definition of essence, second as affirmative/negative propositional judgment.
GEOMETOR Architectural Significance: Corresponds to Element constructors (definitions) vs. Invariant / theorem assertions (propositions) in GEOMETOR.
Expositio super Librum Boethii De Trinitate¶
aquinas:de-trin:5.1.ad3 — 5.1.ad3¶
Critical Edition: Leonine L, q. 5 a. 1 ad 3; ed. Decker, p. 170
Aristotelian Foundation: Metaphysics VI.1, 1026a13–19; Ethics VI.7
Euclidean Counterpart: Elements Books I–XIII as pure speculative geometry vs. applied mensuration
Codex Edition: Unabridged in English (pars-1) · Diplomatic Latin
Cited in Hoenen (1954): Caput VII p. 225 (
Boet. de Trin. q. 5 a. 1 ad 3)Hoenen’s Doctrinal Argument: Distinguishes pure speculative mathematics from practical operational rules: mathematical objects cannot be fabricated by human action yet serve as canonical measures for art and nature.
GEOMETOR Architectural Significance: Grounds GEOMETOR’s distinction between exact algebraic invariants in geometor.model and physical or pixel approximations.
aquinas:de-trin:q5-a3 — q5-a3¶
Critical Edition: Leonine L, q. 5 a. 3 co. & ad 3; ed. Decker, pp. 181–189
Aristotelian Foundation: De Anima III.7, 431b12–17; Physics II.2, 193b22–194a12; Metaphysics VII.10, 1036a9–12
Euclidean Counterpart: Elements I.def.1–23 (continuous magnitudes abstracted from sensible qualities while retaining spatial extension)
Codex Edition: Unabridged in English (pars-1) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 101 (
In Boet. de Trin. q. 5 a. 3 ad 3)Hoenen’s Doctrinal Argument: Foundational locus for the twofold operation of the intellect: separation (negative judgment of esse in metaphysics) vs. abstraction of the whole (physics) vs. abstraction of form from sensible matter leaving materia intelligibilis (substance as subject of continuous quantity).
GEOMETOR Architectural Significance: Defines the exact ontological domain of geometor.model: pure continuous spatial extension stripped of sensible noise yet individuated by distinct algebraic positions.
aquinas:de-trin:q6.a1 — q6.a1¶
Critical Edition: Leonine L, q. 6 a. 1 (q. 2) & q. 6 a. 2; ed. Decker, pp. 205–218
Aristotelian Foundation: Posterior Analytics I.2, 71b16–25; Ethics I.3, 1094b23–27
Euclidean Counterpart: Deductive disciplinalis order of Euclid’s Definitions, Postulates, Common Notions, and Propositions
Codex Edition: Unabridged in English (pars-1) · Diplomatic Latin
Cited in Hoenen (1954): Caput V p. 167 (
In Boet. de Trin. q. 6 a. 1)Hoenen’s Doctrinal Argument: Explains why mathematics proceeds disciplinaliter (with supreme pedagogical certainty between sense-bound physics and purely intellectual metaphysics) and why mathematical judgment terminates at the imagination (terminari ad imaginationem, q. 6 a. 2).
GEOMETOR Architectural Significance: Illuminates the synergy between deductive symbolic proof (geometor.elements) and imaginative visual verification (geometor.explorer).
Quaestiones Disputatae de Veritate¶
aquinas:de-veritate:10.12.ad7 — 10.12.ad7¶
Critical Edition: Leonine XXII.2, De Veritate, q. 10, a. 12, ad 7
Aristotelian Foundation: Posterior Analytics I.2, 72a25–b4
Euclidean Counterpart: Self-evidence (per se notum) of geometric and metaphysical axioms upon understanding their terms
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 255 (
Q. D. de Ver. q. 10 a. 12 ad 7)Hoenen’s Doctrinal Argument: Distinguishes propositions per se notae secundum se from those per se notae quoad nos whose predicate is immediately grasped inside the ratio of the subject.
GEOMETOR Architectural Significance: Clarifies why elementary axioms require zero proof steps whereas deep properties (like golden ratio emergences) require constructive mediation.
aquinas:de-veritate:10.6.sc2 — 10.6.sc2¶
Critical Edition: Leonine XXII.2, De Veritate, q. 10, a. 6, sed contra 2 & co.
Aristotelian Foundation: Posterior Analytics II.19, 99b35–100b17
Euclidean Counterpart: Euclid Common Notion 5 (omne totum est maius sua parte) known through singular sensible experience illuminated by agent intellect
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Caput II p. 38 (
(Q. D. de Ver. q. 10 a. 6 sed contra 2)); Caput VII p. 236 (Q. D. de Ver. q. 10 a. 6 « sed contra » 2)Hoenen’s Doctrinal Argument: Twice cited by Hoenen (Caput II & Caput VII): omnis nostra cognitio originaliter consistit in notitia primorum principiorum indemonstrabilium; horum autem cognitio in nobis a sensu oritur.
GEOMETOR Architectural Significance: Explains how Euclid’s Common Notion 5 (‘the whole is greater than the part’) is immediately intuited in any concrete segment division (including the golden section!).
aquinas:de-veritate:4.2 — 4.2¶
Critical Edition: Leonine XXII.1, De Veritate, q. 4, a. 2
Aristotelian Foundation: Peri Hermeneias I.1, 16a3–8
Euclidean Counterpart: Interior mental formulation of a geometric theorem prior to vocal or symbolic notation
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 278 (
de Ver. q. 4 a. 2); Appendix p. 279 (Ver. 4, 2)Hoenen’s Doctrinal Argument: Analyzes the interior verbum cordis as the luminous expression of intellectual insight into truth.
GEOMETOR Architectural Significance: Distinguishes the invariant mathematical insight from its external Python, SymPy, RST, or SVG serialization.
aquinas:de-veritate:q10-a6 — q10-a6¶
Critical Edition: Leonine XXII.2, De Veritate, q. 10, a. 6 (entire article)
Aristotelian Foundation: De Anima III.4–8; Posterior Analytics II.19
Euclidean Counterpart: Cooperation of phantasms and lumen intellectus agentis in generating Euclidean axioms and theorems
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 97 (
Q. D. de Ver. q. 10 a. 6 « sed contra » 2)Hoenen’s Doctrinal Argument: Demonstrates that sensible phantasms do not infuse purely spiritual truth on their own, but determine the specific intelligible content rendered actual by the light of the agent intellect.
GEOMETOR Architectural Significance: Models the interplay between concrete geometric configurations and universal symbolic deduction.
aquinas:de-veritate:q12-a3 — q12-a3¶
Critical Edition: Leonine XXII.2, De Veritate, q. 12, a. 3, ad 2
Aristotelian Foundation: Posterior Analytics I.18, 81a38–b9
Euclidean Counterpart: Indispensability of spatial sense/imagination for acquiring geometric science (deficientibus sensibus deficit scientia)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput IV p. 98 (
Q. D. de Ver. q. 12 a. 3 ad 2)Hoenen’s Doctrinal Argument: Shows that intellectual light without imaginative data yields no determinate natural or mathematical science: imagination supplies the determinate figure in which universal ratios are read.
GEOMETOR Architectural Significance: Affirms the essential place of visual rendering (geometor.explorer) in mathematical discovery.
Sententia Libri Ethicorum¶
aquinas:eth-nic:IX.11 — IX.11¶
Critical Edition: Leonine XLVII.2, Sententia libri Ethicorum, Lib. IX, lect. 11
Aristotelian Foundation: Nicomachean Ethics IX.9, 1170a25–b10
Euclidean Counterpart: Reflexive intellectual awareness (percipimus nos intelligere) during geometric contemplation
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 260 (
i. h. l. (lect. 11))Hoenen’s Doctrinal Argument: Examines luminous intellectual self-presence in the act of sensing and understanding: perceiving that we perceive and understand.
GEOMETOR Architectural Significance: Connects contemplative geometric insight with reflexive critical verification (reflection in Hoenen and Lonergan).
Sententia super Metaphysicam¶
aquinas:met:10.4.2017 — 10.4.2017¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. X, lect. 4, nos. 2017 sqq.
Aristotelian Foundation: Metaphysics X.3, 1054a20–b3
Euclidean Counterpart: Identity (eadem), equality (aequalia, Common Notions 1–4), and similarity (similia, Book VI) in geometry
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 283 (
S. Th. l. 4 n. 2017 sqq.); Appendix p. 283 (l. c. n. 2017)Hoenen’s Doctrinal Argument: Systematizes the transcendental and quantitative modes of unity: unity in substance is identitas, unity in quantity is aequalitas, and unity in quality/figure is similitudo (congruent angles with proportional sides).
GEOMETOR Architectural Significance: Provides the exact conceptual triad governing geometor.elements and geometor.divine: identical points, equal segments (:eq:), and similar golden figures.
aquinas:met:4.l4.n577 — 4.l4.n577¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. IV, lect. 4, n. 577
Aristotelian Foundation: Metaphysics IV.3, 1005a19–b8
Euclidean Counterpart: Mathematicians using Euclid’s Common Notions (aequalia ab aequalibus) within continuous magnitude
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput V p. 165 (
Met. IV lect. 4 n. 577)Hoenen’s Doctrinal Argument: Explains why mathematicians employ common ontological axioms (dignitates) restricted proportionately to quantitative magnitude, while metaphysics treats them universally of ens inquantum ens.
GEOMETOR Architectural Significance: Shows how general algebraic equality laws specialize cleanly to segment, angle, and area equalities in geometor.elements.
aquinas:met:5.12.916 — 5.12.916¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. V, lect. 12, n. 916
Aristotelian Foundation: Metaphysics V.10, 1018a25–35
Euclidean Counterpart: Geometric possibility/impossibility (e.g., commensurability vs. incommensurability of diagonal)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 283 (
S. Th. l. 12 n. 916)Hoenen’s Doctrinal Argument: Distinguishes possibile/impossibile in the order of real active/passive potency from necessary/impossible truth in mathematical propositions (where ‘impossible’ means its contrary is necessarily true, as with a commensurable diagonal).
GEOMETOR Architectural Significance: Clarifies algebraic constructibility vs. impossibility (e.g., degree-of-extension criteria in compass-and-straightedge constructions).
aquinas:met:5.9 — 5.9¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. V, lect. 9, nos. 885–897
Aristotelian Foundation: Metaphysics V.7, 1017a7–b9
Euclidean Counterpart: Truth of geometric propositions (e.g., incommensurability of the square’s diagonal, Euclid X.117) signified by the copula est
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 284 (
Met. V l. 9)Hoenen’s Doctrinal Argument: Distinguishes the fourfold sense of ens: per accidens, per se in the ten categories, ens ut verum signified by the judgmental copula (using the diagonal’s incommensurability with the side as prime example!), and actus/potentia.
GEOMETOR Architectural Significance: Connects quadratic irrationality ($\sqrt{2}$ in the square diagonal, $\sqrt{5}$ in the golden ratio) directly to Thomistic propositional truth!
aquinas:met:5.9.890 — 5.9.890¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. V, lect. 9, n. 890
Aristotelian Foundation: Metaphysics V.7, 1017a22–30
Euclidean Counterpart: Categorical grounding of quantum (magnitude), quale (figure), and ad aliquid (ratio/proportion)
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 284 (
Met. l. c. n. 290)Hoenen’s Doctrinal Argument: Classic formulation that being contracts to the ten categories according to diverse modes of predicating consequential upon diverse modes of being (quoties ens dicitur, toties esse significatur).
GEOMETOR Architectural Significance: Establishes the ontological distinction between magnitudes (segments), qualities of quantity (polygonal figures), and relations (golden ratios).
aquinas:met:9.10.1894 — 9.10.1894¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. IX, lect. 10, nos. 1888–1894 (esp. n. 1894)
Aristotelian Foundation: Metaphysics IX.9, 1051a21–33
Euclidean Counterpart: Euclid I.32 (triangle angle-sum via parallel auxiliary line) and Euclid III.31 (Thales’s right angle in a semicircle via auxiliary radius)
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 273 (
S. Th. lect. 10 n. 1894)Hoenen’s Doctrinal Argument: Crowning heuristic text of the Codex: geometrae inveniunt veritatem dividendo! Geometric theorems latent in potency become immediately manifest (statim manifestum est videnti) the moment auxiliary divisions are drawn in act.
GEOMETOR Architectural Significance: The exact philosophical charter of GEOMETOR: adding auxiliary lines and circles in geometor.model actualizes latent intersections so geometor.divine reveals emergent golden sections!
aquinas:met:9.3.1805 — 9.3.1805¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. IX, lect. 3, n. 1805
Aristotelian Foundation: Metaphysics IX.3, 1047a30–b2
Euclidean Counterpart: Derivation of actus (energeia) from kinematic construction before extending to static geometric actuality
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 280 (
Met. IX l. 3 n. 1805)Hoenen’s Doctrinal Argument: Explains how the name and concept of actus were first grasped in sensible local motion and then analogically extended to immobile mathematical and metaphysical actualities.
GEOMETOR Architectural Significance: Illuminates the transition from dynamic construction steps to timeless algebraic truth.
aquinas:met:9.3.1806 — 9.3.1806¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. IX, lect. 3, n. 1806
Aristotelian Foundation: Metaphysics IX.3, 1047a32–b2
Euclidean Counterpart: Distinction between non-existent objects of imagination and really possible geometric constructions
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 280 (
1806)Hoenen’s Doctrinal Argument: Clarifies why non-beings may be thought or imagined without yet possessing esse in actu, linking potency and act to existential judgment.
GEOMETOR Architectural Significance: Distinguishes arbitrary unconstrained sketches from algebraically consistent constructions in geometor.model.
aquinas:met:III.7.416 — III.7.416¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. III, lect. 7, n. 416
Aristotelian Foundation: Metaphysics III.2, 997b25–998a19
Euclidean Counterpart: Tangent line touching a circle at a single indivisible point (Euclid III.16) vs. Protagoras’s empiricist objection
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput III p. 66 (
S. Th. lect. 7 n. 416)Hoenen’s Doctrinal Argument: Refutes Protagoras’s claim that because a physical ruler touches a bronze hoop along a segment rather than at a single point, geometry is false: mathematical tangency holds rigorously in materia intelligibilis.
GEOMETOR Architectural Significance: In geometor.model, a tangent line and circle yield a polynomial system with discriminant zero—an exact double root at a single symbolic point!
aquinas:met:IX.3.1805 — IX.3.1805¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. IX, lect. 3, nos. 1805–1806
Aristotelian Foundation: Metaphysics IX.3, 1047a30–b2
Euclidean Counterpart: Analogical extension of energeia from motion to geometric actualization
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 253 (
i. h. l. Cath. n. 1805)Hoenen’s Doctrinal Argument: Cross-reference in Hoenen’s Appendix reinforcing that motion is the primary sensible manifestation of actus (energeia) for human cognition.
GEOMETOR Architectural Significance: Links interactive user gestures in geometor.explorer to underlying symbolic actualization.
aquinas:met:IX.7.1848 — IX.7.1848¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. IX, lect. 7, n. 1848
Aristotelian Foundation: Metaphysics IX.8, 1049b12–17
Euclidean Counterpart: Cognitive priority of actual geometric construction (actus) over definitional potency (potentia)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 265 (
i. h. l. (n. 1848))Hoenen’s Doctrinal Argument: Demonstrates that actus is prior to potentia in definition (secundum rationem): we can define the power to construct or divide a figure only through the actualized construction itself.
GEOMETOR Architectural Significance: Explains why constructive operators in geometor.model are specified by the exact loci they actualize.
aquinas:met:IX.8.1857 — IX.8.1857¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. IX, lect. 8, n. 1857
Aristotelian Foundation: Metaphysics IX.8, 1050a4–16
Euclidean Counterpart: Pedagogical practice of teaching geometry by making the student construct figures in act (operando)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 268 (
i. h. l. ex Met. IX (l. 8 n. 1857))Hoenen’s Doctrinal Argument: Highlights Aristotle’s and Aquinas’s pedagogical insight: teachers judge that students have attained the end of geometric science when they show them constructing and demonstrating in actu.
GEOMETOR Architectural Significance: Embodies GEOMETOR’s constructive pedagogy: understanding geometry by building exact constructions step by step.
aquinas:met:X.2.1938 — X.2.1938¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. X, lect. 2, n. 1938
Aristotelian Foundation: Metaphysics X.1, 1052b20–1053a14
Euclidean Counterpart: Commensurable unit measure vs. incommensurable magnitudes (Euclid Book X) requiring two measures
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput VI p. 216 (
l. 2 n. 1938)Hoenen’s Doctrinal Argument: Analyzes the nature of quantitative measure (mensura): while arithmetic measures by an absolutely indivisible unit, continuous geometry posits a conventional unit foot and discovers magnitudes (like the diagonal) measurable only across two incommensurable dimensions.
GEOMETOR Architectural Significance: Directly explains why GEOMETOR begins from two initial points (0, 0) and (1, 0) establishing the unit segment, from which $\sqrt{2}$, $\sqrt{3}$, and $\phi$ emerge.
aquinas:met:XI.11.2370 — XI.11.2370¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. XI, lect. 11, n. 2370
Aristotelian Foundation: Metaphysics XI.11, 1067b25–1068a7
Euclidean Counterpart: Immobility of mathematical abstractions vs. physical categories of motion
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 252 (
n. 2370)Hoenen’s Doctrinal Argument: Reaffirms in the Metaphysics summary of Physics V that per se motion belongs only to sensible realizations of quantity, quality, and place.
GEOMETOR Architectural Significance: Reinforces the timeless algebraic stability of symbolic constructions in geometor.model.
aquinas:met:b7-l10 — b7-l10¶
Critical Edition: ed. Cathala–Spiazzi (Marietti), Lib. VII, lect. 10, nos. 1494–1500 (esp. n. 1496) & lect. 11, nos. 1501–1536
Aristotelian Foundation: Metaphysics VII.10–11, 1035b31–1037a5
Euclidean Counterpart: Elements I.def.15 (universal definition of the circle) vs. individual circles in materia intelligibilis and semicircles as material parts
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Caput IV p. 101 (
Met. VII lect. 10 (Cath. n. 1496))Hoenen’s Doctrinal Argument: Shows that just as bronze individuates sensible circles, materia intelligibilis individualis (designated continuous dimensions) individuates multiple mathematical circles of the same species, while materia intelligibilis communis enters mathematical definitions.
GEOMETOR Architectural Significance: Explains how geometor.model instantiates many distinct Circle objects sharing one universal algebraic form yet individuated by distinct centers and radii.
Expositio Libri Peryermeneias¶
aquinas:periherm:1.5.17 — 1.5.17¶
Critical Edition: Leonine I* 1, Expositio libri Peryermeneias, Lib. I, lect. 5, n. 17–22
Aristotelian Foundation: Peri Hermeneias I.3, 16b19–25
Euclidean Counterpart: Consignification of composition by est in timeless geometric truths
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 279 (
ibid. n. 17)Hoenen’s Doctrinal Argument: Crucial text for Hoenen’s judgment theory: est principally signifies actuality (actualitas) in the mode of verb and consignifies propositional composition uniting predicate with subject.
GEOMETOR Architectural Significance: Links ontological actuality with formal equality assertions.
aquinas:periherm:1.5.5 — 1.5.5¶
Critical Edition: Leonine I* 1, Expositio libri Peryermeneias, Lib. I, lect. 5, n. 5
Aristotelian Foundation: Peri Hermeneias I.3, 16b6–25
Euclidean Counterpart: Propositional synthesis (compositio) in geometric enunciation (protasis)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 279 (
Peri Herm. I l. 5, n. 5)Hoenen’s Doctrinal Argument: Analyzes why verbs signify with time and serve as the formal copula of predication in every scientific assertion.
GEOMETOR Architectural Significance: Relates natural-language geometric enunciations (euclid) to formal relational assertions.
Commentaria in Octo Libros Physicorum Aristotelis¶
aquinas:phys:3.5 — 3.5¶
Critical Edition: Leonine II, Lib. III, lect. 5, nos. 14–16; ed. Maggiòlo, nos. 321–323
Aristotelian Foundation: Physics III.3, 202b19–29
Euclidean Counterpart: Category of Quantity (quantum, consequential upon matter) vs. Quality (figura, modification of quantity) and Situs
Codex Edition: Unabridged in English (pars-4) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 284 (
Phys. III l. 5); Appendix p. 284 (Phys. l. c.); Appendix p. 284 (ibid.)Hoenen’s Doctrinal Argument: Demonstrates the isomorphism between predication and ontology (modi essendi proportionales sunt modis praedicandi), deriving Quantity as the intrinsic accident following upon matter.
GEOMETOR Architectural Significance: Grounds the type hierarchy of geometor.model (substance-substrate -> continuous plane -> bounded segments, circles, polygons, ratios).
aquinas:phys:4.17 — 4.17¶
Critical Edition: Leonine II, Lib. IV, lect. 17
Aristotelian Foundation: Physics IV.11, 218b21–219b8
Euclidean Counterpart: Continuity of time derived from continuity of spatial magnitude (magnitudo -> motus -> tempus)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 286 (
S. Thomas l. 17)Hoenen’s Doctrinal Argument: Traces the causal and cognitive order magnitudo -> motus -> tempus: temporal succession is continuous only because the underlying spatial magnitude is continuous.
GEOMETOR Architectural Significance: Grounds animation keyframes and step sequencing in geometor.explorer upon underlying geometric magnitude.
aquinas:phys:II.3 — II.3¶
Critical Edition: Leonine II, Lib. II, lect. 3, nos. 1–9; ed. Maggiòlo, nos. 157–165
Aristotelian Foundation: Physics II.2, 193b22–194b15
Euclidean Counterpart: Surfaces, lines, points, and curves abstracted from physical bodies without mendacity (abstrahentium non est mendacium)
Codex Edition: Unabridged in English (pars-4) · Diplomatic Latin
Cited in Hoenen (1954): Caput II p. 62 (
In Phys. II lect. 3)Hoenen’s Doctrinal Argument: Defends the objective realism of geometry: because quantitative extension inheres in corporeal substance prior to sensible qualities (heat, weight, color), considering figure apart from sensible motion involves zero falsehood (abstrahentium non est mendacium).
GEOMETOR Architectural Significance: Validates studying natural phyllotaxis (geometor.phyllotaxis) and physical symmetries via pure Euclidean/projective geometry.
aquinas:phys:IV.3 — IV.3¶
Critical Edition: Leonine II, Lib. IV, lect. 3, in fine
Aristotelian Foundation: Physics IV.2, 209b1–17
Euclidean Counterpart: Three-dimensional corporeal extension vs. surrounding locus
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 261 (
S. Th. l. 3 in fine)Hoenen’s Doctrinal Argument: Critiques Plato’s identification of matter and place/extension while showing why bounded spatial intervals suggest a distinct receptive dimension.
GEOMETOR Architectural Significance: Clarifies the distinction between intrinsic figure dimensions and ambient coordinate space.
aquinas:phys:IV.7 — IV.7¶
Critical Edition: Leonine II, Lib. IV, lect. 7, nos. 1–12
Aristotelian Foundation: Physics IV.4, 211b5–212a21
Euclidean Counterpart: Elements Common Notion 4 & Proposition I.4 (congruence/superposition of figures across immobile spatial positions)
Codex Edition: Unabridged in English (pars-4) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 262 (
lectionem 7)Hoenen’s Doctrinal Argument: Explains how the mind abstracts immobile place (terminus continentis immobilis primus) by observing successive bodies replacing one another through local translation.
GEOMETOR Architectural Significance: Provides the Thomistic justification for Euclidean congruence, rigid motions, and superposition (:coincide: role in photon_platform.publish).
aquinas:phys:IV.7.7 — IV.7.7¶
Critical Edition: Leonine II, Lib. IV, lect. 7, n. 7
Aristotelian Foundation: Physics IV.4, 212a14–21
Euclidean Counterpart: Invariance of geometric relations under spatial translation (fixed ordo ad totum caelum)
Codex Edition: Unabridged in English (pars-4) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 262 (
lect. cit. n. 7)Hoenen’s Doctrinal Argument: Highlights Aquinas’s refinement of Themistius: although a vessel or river current moves materially, the ratio loci remains immobile by preserving a fixed positional relation (situs) to the whole cosmos.
GEOMETOR Architectural Significance: Corresponds to global coordinate frame invariance in geometor.model.
aquinas:phys:V.2.8 — V.2.8¶
Critical Edition: Leonine II, Lib. V, lect. 2, n. 8
Aristotelian Foundation: Physics V.1, 225b5–9
Euclidean Counterpart: Immobility of geometric essences and relations per se
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 251 (
i. h. l. lect. 2 n. 8)Hoenen’s Doctrinal Argument: Shows that there is no per se motion in the category of Substance or Relation (nor in abstract mathematical forms as such), but only in Quantity, Quality, and Ubi.
GEOMETOR Architectural Significance: Confirms that proportional relations (such as $\phi$) are invariant relational truths rather than mutable processes.
aquinas:phys:V.2.9 — V.2.9¶
Critical Edition: Leonine II, Lib. V, lect. 2, n. 9
Aristotelian Foundation: Physics V.1, 225b10–16
Euclidean Counterpart: Invariance of geometric equality and proportion unless an underlying magnitude changes
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 252 (
l. c. n. 9)Hoenen’s Doctrinal Argument: Explains why a new relation arises without intrinsic change in a term when another term moves—crucial for relational geometry and spatial situs.
GEOMETOR Architectural Significance: Supports relational collineation and harmonic cross-ratio tracking in geometor.pappus and geometor.divine.
aquinas:phys:VIII.1 — VIII.1¶
Critical Edition: Leonine II, Lib. VIII, lect. 1–2 (coupled with Lib. VI, lect. 1 in Pars IV s4)
Aristotelian Foundation: Physics VIII.1, 250b11–252b6 (& Physics VI.1, 231a21–b18)
Euclidean Counterpart: Infinite divisibility of the continuum (Pars IV, s4: In VI Phys. lect. 1) vs. impossibility of composition from indivisibles
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 252 (
lect. 1 et 2)Hoenen’s Doctrinal Argument: Analyzes potential vs. actual infinity in continuous division and motion, building upon In VI Phys. lect. 1’s proof that no line is composed of indivisible points.
GEOMETOR Architectural Significance: Explains why recursive geometric constructions (such as nested pentagrams and Fibonacci spirals) are endlessly iterable without hitting atomic ‘pixels’.
aquinas:phys:b4-l18 — b4-l18¶
Critical Edition: Leonine II, Lib. IV, lect. 18, n. 4
Aristotelian Foundation: Physics IV.11, 219b9–33
Euclidean Counterpart: Analogy between the moving point (punctum fluens) giving unity to a line and the flowing nunc in time
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput IV p. 108 (
Phys. IV lect. 18 n. 4)Hoenen’s Doctrinal Argument: Examines how the geometer’s punctum both continues a line and divides it in potency, analogous to the nunc in time.
GEOMETOR Architectural Significance: Illuminates how intersection points divide segments into golden-ratio sub-segments in geometor.divine.
Summa contra Gentiles¶
aquinas:scg:1.26.2 — 1.26.2¶
Critical Edition: Leonine XIII, Summa contra Gentiles, Lib. I, cap. 26, n. 2
Aristotelian Foundation: Posterior Analytics II.7, 92b13–14; Metaphysics III.3
Euclidean Counterpart: Distinction of esse from generic/specific geometric essence
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 284 (
S. c. G. I 26 n. 2)Hoenen’s Doctrinal Argument: Proves that esse cannot be a genus because every formal nature contracts esse by distinct modes of being.
GEOMETOR Architectural Significance: Clarifies why mathematical objects form a specific quantitative domain within universal being.
aquinas:scg:I.36 — I.36¶
Critical Edition: Leonine XIII, Summa contra Gentiles, Lib. I, cap. 36
Aristotelian Foundation: Metaphysics V.7 & VI.4
Euclidean Counterpart: Synthesis of diverse conceptual ratios (e.g., ‘equilateral’ and ‘equiangular’) into one geometric subject via judgment
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput II p. 57 (
S. c. G. I 36)Hoenen’s Doctrinal Argument: Shows how our intellect forms affirmative propositions to unite diverse conceptual aspects signifying a single reality.
GEOMETOR Architectural Significance: Models multi-property equivalence on a single geometric figure in geometor.elements (e.g., Euclid I.5–I.6).
Scriptum super Libros Sententiarum¶
aquinas:sent:1.19.5.1.ad6 — 1.19.5.1.ad6¶
Critical Edition: ed. Mandonnet, In I Sent., d. 19, q. 5, a. 1 (co. & ad 6–7)
Aristotelian Foundation: Metaphysics VI.4, 1027b25–27
Euclidean Counterpart: Formal ratio veritatis realized in demonstrated geometric judgment grounded in esse rei
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 280 (
Sent. I D. 19 q. 5 a. 1 ad 6)Hoenen’s Doctrinal Argument: Famous early Thomistic thesis: truth has its fundamentum in the esse of the thing (rather than bare quiddity alone) and its formal completion in the judging intellect.
GEOMETOR Architectural Significance: Connects symbolic verification (model) to ontological proportion in nature (phyllotaxis).
aquinas:sent:1.2.1.3 — 1.2.1.3¶
Critical Edition: ed. Mandonnet, In I Sent., d. 2, q. 1, a. 3
Aristotelian Foundation: Metaphysics IV.2 & VII.1
Euclidean Counterpart: Objective foundation (fundamentum in re) of distinct mathematical ratios in one continuous figure
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 281 (
Sent. I D. 2 q. 1 a. 3)Hoenen’s Doctrinal Argument: Establishes that distinct conceptual rationes are not empty fictions when reality itself offers an objective foundation (aliquid in re correspondens).
GEOMETOR Architectural Significance: Validates uncovering multiple distinct harmonic and golden proportions inside one geometric construction (geometor.divine).
aquinas:sent:I.3.1.2 — I.3.1.2¶
Critical Edition: ed. Mandonnet, In I Sent., d. 3, q. 1, a. 2
Aristotelian Foundation: Posterior Analytics I.2; De Anima III.4
Euclidean Counterpart: Immediate per se evidence of Euclidean Common Notions from their terms
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput II p. 52 (
I Sent. Dist. III q. 1 a. 2)Hoenen’s Doctrinal Argument: Analyzes what makes a proposition per se nota in itself and to us through the intelligible content of its terms.
GEOMETOR Architectural Significance: Grounds axiomatic base cases in geometor.elements.
aquinas:sent:I.4.2.1.ad1 — I.4.2.1.ad1¶
Critical Edition: ed. Mandonnet, In I Sent., d. 4, q. 2, a. 1, ad 1
Aristotelian Foundation: Peri Hermeneias I.1–3
Euclidean Counterpart: Subject taken materially/suppositionally vs. predicate taken formally in geometric predication
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput II p. 57 (
I Sent. Dist. 4 q. 2 a. 1 ad 1)Hoenen’s Doctrinal Argument: Explains the asymmetry of predication: the subject stands for the supposit (suppositum) while the predicate signifies the formal nature (forma) attributed to it.
GEOMETOR Architectural Significance: Matches object-attribute modeling in geometor.model (Segment instance vs. algebraic length / golden predicate).
aquinas:sent:III.11.1.4.ad6 — III.11.1.4.ad6¶
Critical Edition: ed. Moos, In III Sent., d. 11, q. 1, a. 4, ad 6
Aristotelian Foundation: Metaphysics V.7
Euclidean Counterpart: Precision of formal predication in rigorous mathematical definitions and theorems
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput II p. 58 (
III Sent. Dist. 11 q. 1 a. 4 ad 6)Hoenen’s Doctrinal Argument: Analyzes rules of supposition and formal predication so no equivocation enters demonstrative syllogisms.
GEOMETOR Architectural Significance: Ensures strict type safety between points, lines, segments, circles, and polygons in geometor.model.
Summa Theologiae¶
aquinas:st:I.12.4.ad3 — I.12.4.ad3¶
Critical Edition: Leonine IV, ST I, q. 12, a. 4, co. & ad 3
Aristotelian Foundation: De Anima III.4, 429a18–b22
Euclidean Counterpart: Apprehension of incorporeal mathematical ratios vs. connatural dependence on corporeal phantasms
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 280 (
S. Th. I q. 12 a. 4 ad 3)Hoenen’s Doctrinal Argument: Explains how the human intellect, connaturally proportioned to natures existing in corporeal matter, knows individualized natures through sense and universal essences by abstracting from sense—grounding the existential import of judgment.
GEOMETOR Architectural Significance: Links concrete sensory diagrams with universal algebraic forms.
aquinas:st:I.13.12 — I.13.12¶
Critical Edition: Leonine IV, ST I, q. 13, a. 12
Aristotelian Foundation: Peri Hermeneias I.1–3, 16a1–16b25
Euclidean Counterpart: Affirmative propositions expressing identity in construction vs. conceptual distinction of subject and predicate
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput II p. 57 (
S. Th. I q. 13 a. 12)Hoenen’s Doctrinal Argument: Shows that in every true affirmative proposition, subject and predicate signify what is identical in reality (secundum rem) while differing in intelligible aspect (secundum rationem)—precisely how ‘triangle’ and ‘having angles equal to two right angles’ unite in Euclidean theorems.
GEOMETOR Architectural Significance: Explains why algebraic simplification (sympy.simplify(a - b) == 0) proves non-trivial geometric theorems: distinct construction paths converge on the identical algebraic entity.
aquinas:st:I.7.1 — I.7.1¶
Critical Edition: Leonine IV, ST I, q. 7, a. 1 (& q. 7, a. 3)
Aristotelian Foundation: Physics III.6, 207a7–15
Euclidean Counterpart: Euclid Postulate 2 (indefinite extendibility of a straight line) vs. determinate species of geometric figures (ST I, q. 7, a. 3)
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Appendix p. 258 (
S. Th. I q. 7 a. 1)Hoenen’s Doctrinal Argument: Establishes that ipsum esse is most formal of all (illud quod est maxime formale omnium est ipsum esse), whereas quantitative infinity comes from the side of matter lacking formal boundary—and in q. 7 a. 3 shows that the geometer never needs an actually infinite line, only a finite line extendable in potency as far as needed.
GEOMETOR Architectural Significance: Models unbounded symbolic lines (Line through two points in geometor.model) instantiated on any bounded canvas in geometor.explorer.
aquinas:st:I.75.6 — I.75.6¶
Critical Edition: Leonine V, ST I, q. 75, a. 6
Aristotelian Foundation: De Anima I.4, 408b18–29; III.4, 429a24–b5
Euclidean Counterpart: Necessary connection between form and per se property (sicut triangulo habere tres angulos aequales duobus rectis, Euclid I.32)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 280 (
S. Th. I q. 75 a. 6)Hoenen’s Doctrinal Argument: Uses Euclid I.32 as the paradigm of per se ontological necessity: just as a triangle cannot lack three angles equal to two right angles, so esse inseparably follows subsistent intellectual form.
GEOMETOR Architectural Significance: Illustrates how geometric necessity serves Aquinas as the clearest analogate for metaphysical necessity.
aquinas:st:I.85.1.ad2 — I.85.1.ad2¶
Critical Edition: Leonine V, ST I, q. 85, a. 1, co. & ad 2
Aristotelian Foundation: Metaphysics VII.10, 1036a9–12; Physics II.2, 193b31–35
Euclidean Counterpart: Threefold abstraction: sensible qualities stripped away while continuous materia intelligibilis remains in geometric definitions
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Caput II p. 62 (
S. Th. I q. 85 a. 1 ad 2)Hoenen’s Doctrinal Argument: Canonical mature synthesis of materia sensibilis vs. materia intelligibilis (both signata and communis): quantities such as lines, numbers, and figures can be understood without sensible qualities, though never without the substance underlying intelligible quantity.
GEOMETOR Architectural Significance: Defines the exact abstraction boundary of geometor.model.
aquinas:st:I.85.5.ad3 — I.85.5.ad3¶
Critical Edition: Leonine V, ST I, q. 85, a. 5, ad 3
Aristotelian Foundation: De Anima III.6, 430a26–b6
Euclidean Counterpart: Apprehending a geometric property within its subject prior to composing/dividing in explicit propositional judgment
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput II p. 57 (
q. 85 a. 5 ad 3)Hoenen’s Doctrinal Argument: Pivotal text for Hoenen’s La théorie du jugement: before composing or dividing, the intellect apprehends the nexus of parts and properties in the concrete whole through simple apprehension.
GEOMETOR Architectural Significance: Explains pre-propositional structural recognition of geometric symmetries in geometor.explorer and geometor.divine.
aquinas:st:I.q84.a6 — I.q84.a6¶
Critical Edition: Leonine V, ST I, q. 84, a. 6, co.
Aristotelian Foundation: De Anima III.4–5, 429a10–430a25
Euclidean Counterpart: Origin of geometric concepts neither from Platonic innate ideas nor from Democritean sense impressions alone, but via intellectus agens illuminating phantasms
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Caput III p. 82 (
S. Th. I. q. 84 a. 6 in fine)Hoenen’s Doctrinal Argument: Mediates between Platonic apriorism and Democritean empiricism: phantasms serve as instrumental material causes illuminated by the agent intellect to impress intelligible species.
GEOMETOR Architectural Significance: Rejects both blind pixel empiricism and ungrounded formalism in favor of constructive insight (geometor.seer + geometor.model).
aquinas:st:I.q84.a6.ad3 — I.q84.a6.ad3¶
Critical Edition: Leonine V, ST I, q. 84, a. 6, ad 3 (& q. 84, a. 7)
Aristotelian Foundation: De Anima III.7, 431a16–17
Euclidean Counterpart: Constructing new auxiliary diagrams (formare alia phantasmata, q. 84 a. 7) to grasp geometric theorems
Codex Edition: Unabridged in English (pars-5) · Diplomatic Latin
Cited in Hoenen (1954): Caput III p. 82 (
resp. 3 eiusdem articuli)Hoenen’s Doctrinal Argument: Explains how the higher intellect requires sensitive phantasms not as total efficient cause, but as indispensable materia causae, and in q. 84 a. 7 proves conversio ad phantasmata from the everyday experience of forming diagrams when trying to understand geometry!
GEOMETOR Architectural Significance: Directly describes the human user forming diagrams in geometor.explorer to behold universal truths.
aquinas:st:II-II.24.4.ad3 — II-II.24.4.ad3¶
Critical Edition: Leonine VIII, ST II-II, q. 24, a. 4, ad 3
Aristotelian Foundation: Categories 8, 8b25–9a13; Physics VII.3
Euclidean Counterpart: Intensive perfection of scientific habit (growing by extending to more theorems or deeper certitude) vs. extensive addition of continuous quantity
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Appendix p. 287 (
S. Th. II, II q. 24 a. 4 ad 3)Hoenen’s Doctrinal Argument: Contrasts quantitative addition of homogeneous parts in geometry with intensive growth in scientific habitus.
GEOMETOR Architectural Significance: Distinguishes geometric measure addition from deductive depth in geometor.elements.
aquinas:st:III.q11.a2.ad1 — III.q11.a2.ad1¶
Critical Edition: Leonine XI, ST III, q. 11, a. 2, ad 1
Aristotelian Foundation: De Anima III.7, 431a14–17
Euclidean Counterpart: Beholding universal conclusions inside singular geometric diagrams (inspicere universale in singulari)
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput III p. 83 (
S. Th. III q. 11 a. 2 ad 1)Hoenen’s Doctrinal Argument: Confirms that human intellective habits always turn to phantasms in act to behold universal intelligible species within particular representations.
GEOMETOR Architectural Significance: Reinforces the permanent cognitive role of visual diagrams alongside symbolic algebra.
Summa Totius Logicae Aristotelis¶
pseudo-thomas:summa-totius-logicae — passim¶
Critical Edition: Opuscula spuria (ed. Mandonnet, Opuscula Omnia, t. V, tract. VII, pp. 102–111)
Aristotelian Foundation: Posterior Analytics I.2 & I.10
Euclidean Counterpart: Scholastic reception of Euclidean petitiones and constructive definitions
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput IV p. 148 (
Summa totius logicae); Caput V p. 161 (Summae totius logicae)Hoenen’s Doctrinal Argument: Philological note by Hoenen distinguishing the authentic Thomistic Posterior Analytics commentary from the anonymous scholastic Summa totius logicae Aristotelis printed among the Opuscula.
GEOMETOR Architectural Significance: Exemplifies critical philological provenance separating authentic Thomistic texts from later scholastic compilations.
pseudo-thomas:summa-totius-logicae:VII.2 — VII.2¶
Critical Edition: ed. Mandonnet, Opuscula Omnia, t. V, tract. VII, cap. 2 & 6, pp. 102–104, 111
Aristotelian Foundation: Posterior Analytics I.10, 76b23–34
Euclidean Counterpart: Classification of geometric postulates (petitiones) alongside Albert the Great’s logical commentary
Codex Status: Complementary Metaphysical / Logical Locus
Cited in Hoenen (1954): Caput IV p. 148 (
op. cit. tract. VII cap. 2 et 6 (in editione Mandonnet pagg. 102-104, pag. 111))Hoenen’s Doctrinal Argument: Examines Tractatus VII cap. 2 & 6 alongside St. Albert the Great on how constructive geometric postulates function in Euclidean demonstration.
GEOMETOR Architectural Significance: Reinforces the constructive interpretation of Euclid’s Postulates 1–3.