Commentary & GEOMETOR Notes — PART IV — COMMENTARIES ON THE PHYSICS AND ON THE HEAVENS

Aristotelian Sources, Euclidean Counterparts, Hoenen’s Noetics, and Symbolic Constructive Geometry

Physics Book II, Lecture 3 — How the Mathematician Differs from the Natural Philosopher and Why There Is No Falsehood in Those Who Abstract

Primary Text: English (Phys. II, lect. 3) · Latin (Phys. II, lect. 3) · Summary

Aristotelian & Scholastic Foundations

Aquinas’s exposition comments on \ Aristotle‘s Physics II.2 (193b22–194a12), the locus classicus distinguishing natural philosophy (physike) from mathematics against Platonic Realism. Where Aristotle tersely remarks that the mathematician separates (chorizei) surfaces, volumes, lengths, and points in thought from motion without generating falsehood (oude ginetai pseudos chorizonton, 193b35—rendered in the Moerbeke-vetus tradition as neque fit mendacium abstrahentium), Aquinas supplies a full metaphysical justification drawn from Metaphysics VII.10–11 (1036a9–12) and \ Averroes‘s Long Commentary on the Physics (II, comm. 18–20). By establishing the ontological order in which accidents accrue to corporeal substance—dimensive quantity first, then sensible qualities, then active and passive powers and local or qualitative motions—Aquinas explains why mathematical abstraction is non-deceptive while Platonic separation of natural forms (flesh, bone, man) fails. Likewise, Aquinas’s analysis of the snub (to simon, 194a1–7) versus the curved (to koilon) and of the ta physikotera ton mathematon (194a7–12: optics, harmonics, astronomy) harmonizes Aristotle’s Physics II.2 with Posterior Analytics I.7–13 (75b14–17, 78b34–79a16) on subalternated sciences and De Caelo II.14 (297a8–298a20) on the physical versus astronomical proofs of the earth’s sphericity, echoing \ Simplicius (In Phys., ed. Diels, 290–293) who cites Geminus and Posidonius on the methodological boundary between causal physics and saving-the-phenomena mathematical astronomy.

Euclidean & Classical Geometric Counterparts

In \ Euclid‘s Elements, Book I Definitions 1–7 and Book XI Definitions 1–2 define point, line, straight line, surface, plane surface, and solid without a single reference to weight, temperature, optical medium, or physical resistance. Aquinas’s commentary on Physics II.3 provides the exact epistemological warrant for Euclid’s procedure: the Euclidean line is ontological reality—the very quantitative limit existing in natural bodies—considered apart from \ sensible matter (linea naturalis abstracte considerata). Moreover, Aquinas’s distinction between pure geometry and the sciences intermediate between mathematics and physics directly illuminates the Euclidean corpus itself, where pure planar and stereometric theorems in the Elements are subalternated in Euclid’s Optics and Catoptrics (applying abstract straight lines and equal angles to visual rays, lineae visuales) and in the Euclidean Sectio Canonis (applying arithmetic ratios of numbers to consonant musical intervals). Even the dual demonstration of terrestrial sphericity in n. 9 illustrates how Euclidean circular and spherical geometry furnishes formal propter quid middle terms (lunar shadow cross-sections and angular stellar altitudes) for phenomena grounded physically in elemental gravitation.

Hoenen’s De Noetica Geometriae & Judgment Theory

In De Noetica Geometriae (Chapter II, § 7, linking Summa Theologiae I, q. 85, a. 1, ad 2 directly with In II Phys., lect. 3, n. 5, and Chapter III, § 2), Peter \ Hoenen S.J. makes Aquinas’s ordering of quantity prior to sensible qualities the linchpin of geometric exactitude and necessity. Against John Stuart Mill’s empiricist reduction of geometric figures to inexact copies of sense impressions (minimum visibile) and J. Wellstein’s voluntarist claim that exact boundaries require an arbitrary act of the will, Hoenen argues that intellectual abstraction inspecting the \ phantasm grasps extended substance purely as extended—that is, clothed in intelligible matter while prescinding from all sensible qualities. Because there is no falsehood in those who abstract, the intellect sees with apodictic certainty that three-dimensional extension per se admits exact indivisible boundaries (surfaces, lines, and points) even if microscopic physical surfaces in sensible nature are violently wrinkled. Thus In II Phys., lect. 3 grounds Hoenen’s central thesis that mathematical existence is neither a Platonic fiction nor a formalist postulate, but an objective necessity apprehended in real extension.

GEOMETOR Constructive & Symbolic Significance

For the GEOMETOR initiative, Aquinas’s principle that there is no falsehood in those who abstract defines the philosophical divide between exact symbolic geometry and numerical floating-point approximation. When \ geometor.model represents points, lines, and circles through exact algebraic expressions in SymPy—preserving quadratic and higher polynomial surds such as the square root of five rather than truncating coordinates to finite IEEE-754 floats or rasterized pixels—it enacts Aquinas’s intellectual abstraction from sensible matter (physical measurement noise and display resolution) while retaining intelligible matter (the continuous algebraic plane in which distinct congruent figures are individuated by coordinate position). Equally, Aquinas’s account of the sciences intermediate between mathematics and physics models the relationship within the GEOMETOR ecosystem between pure symbolic construction in \ geometor.model and \ geometor.elements on the one hand, and applied structural analysis in \ geometor.phyllotaxis or visual SVG rendering in \ geometor.explorer on the other, where pure geometric proportions and golden-ratio invariants are re-applied to botanical growth patterns and sensible visual diagrams.

Physics Book III, Lecture 5 — On Action and Passion and the Division of Being into Ten Categories Proportional to Modes of Predicating

Primary Text: English (Phys. III, lect. 5) · Latin (Phys. III, lect. 5) · Summary

Aristotelian & Scholastic Foundations

Commenting on Physics III.3 (202a13–202b29), Aquinas resolves \ Aristotle‘s famous aporia on the locus of motion—whether teaching (didaxis) and learning (mathesis), or action (poiesis) and passion (pathesis), are two motions or one—using Aristotle’s own topological analogy of a single road between Thebes and Athens (202b13–14) and a single spatial interval between two points (to metaxy, 202a18–20). Going far beyond a mere paraphrase, Aquinas inserts in nn. 14–17 one of the two great categorial deductions in his entire corpus (parallel to In V Metaphysicorum, lect. 9, n. 890), silently correcting \ Averroes (In Phys. III, comm. 20–23) and \ Avicenna on whether motion belongs to a category via ad rem (motus ad formam) or forma fluens. By distinguishing what motion is in rerum natura (an imperfect act reduced to the category of its terminus: quantity, quality, or where) from what reason apprehends in motion as a directed transition from agent to patient (grounding the extrinsic denominations of action and passion), Aquinas preserves both Categories 4 (1b25–2a4) and Physics V.1–2 (225b5–9) within a single analogical metaphysics of being.

Euclidean & Classical Geometric Counterparts

Two profound Euclidean insights emerge within Aquinas’s categorial deduction in In III Phys., lect. 5. First, in n. 15 Aquinas explicitly locates geometric figure (figura) in the fourth species of the category of quality (Categories 8, 10a11–24), observing that figure follows upon form yet is founded immediately upon dimensive quantity—specifically in lines and surfaces as quantitative boundaries (sicut color in superficie, et figura in lineis vel in superficiebus). This explains why Euclidean figures possess both metric magnitude (equality of area or length, governed by Common Notions 1–4) and qualitative shape (similarity of rectilinear figures in Elements Book VI Def. 1, governed by angular equality and proportional sides independent of scale). Second, Aquinas’s analysis in nn. 10–12 of the spatial distance (spatium distantium) between two points—identical in abstracto as a metric segment AB = BA, yet distinct in concreto when applied to ordered endpoints from A to B versus from B to A—captures the exact distinction in Euclidean construction between an undirected segment and a directed ray or circle construction (Postulate 3, where one endpoint serves as stationary center and the other determines radial extremity).

Hoenen’s De Noetica Geometriae & Judgment Theory

Peter \ Hoenen S.J. explicitly singles out In III Physicorum, lect. 5 in the Appendix (§ 5, p. 341) of De Noetica Geometriae alongside In V Metaphysicorum, lect. 9 as the master text demonstrating how Aquinas constructs the Aristotelian categories from modes of being (modi essendi) proportional to modes of predicating (modi praedicandi). For Hoenen, this text proves that human intellectual apprehension does not grasp naked static quiddities alone, but abstracts distinct existential and relational modes—inherence (esse in), substantiality (esse per se), and fluent actuality (esse fluens in motion). Moreover, in Chapter IV (§ 2 and § 4.2), Hoenen builds directly on the Thebes-to-Athens / hinc illuc directional analysis of Phys. III.3 and IV.11: while a static line segment AB is one single extension, actuating motion or passage through its endpoints reveals two really distinct, inverse directions (A to B versus B to A), showing that directional order (thesis) and quantitative measure are complementary intelligible aspects of the geometric \ continuum.

GEOMETOR Constructive & Symbolic Significance

Aquinas’s analysis of action as from this (ut ab hoc) and passion as in this (ut in hoc), together with his placement of figure as a form-like quality supervening on quantitative boundaries, resonates directly with constructive geometry as a directed acyclic graph (DAG) in \ geometor.model. Every compass-and-straightedge step in a GEOMETOR construction—joining two parent points to produce a line, or centering a circle at one point through another to cut an existing locus—embodies a single constructive operation that is simultaneously active with respect to its ancestral parent elements (ut ab hoc) and receptive in instantiating derived child intersections (ut in hoc). Furthermore, Aquinas’s categorial distinction between intrinsic quantitative measure (length, breadth, depth) and qualitative figure grounded on boundaries mirrors the separation between metric coordinate evaluation and projective or proportional invariants studied in \ geometor.divine and \ geometor.pappus, where golden sections and harmonic cross-ratios persist across scale and transformation as qualitative structural forms supervening upon continuous extension.

Physics Book IV, Lecture 7 — On the Definition of Immobile Place from Local Translation and Themistius’s Solution

Primary Text: English (Phys. IV, lect. 7) · Latin (Phys. IV, lect. 7) · Summary

Aristotelian & Scholastic Foundations

Aquinas expounds Physics IV.5 (212a31–212b29), where \ Aristotle applies his definition of place—the first immobile boundary of what contains (to tou periechontos peras akinēton prōton, 212a20)—to the cosmos as a whole. Because the aporia of the outermost sphere’s diurnal rotation became one of the fiercest battlegrounds of ancient and medieval natural philosophy, Aquinas stages in nn. 3–9 a magisterial disputation surveying John Philoponus’s and the Stoics’ three-dimensional void space (spatium dimensiones habens), Alexander of Aphrodisias, \ Avicenna (Sufficientia II.3, positing motion in situs), Avempace / Ibn Bajja (locating a sphere by its inner convex core), \ Averroes (In Phys. IV, comm. 43–45, positing per accidens location via the immobile earth at the center), and \ Themistius (In Phys. IV, ed. Schenkl, 120–122). Siding decisively with Themistius against Averroes (whom Aquinas even criticizes in n. 15 for forcing Aristotle’s text on water), Aquinas shows how Aristotle’s distinction between parts existing potentially in place within a homogeneous \ continuum (homoimeres) versus actually in place within a contiguous heap resolves the paradox without surrendering the relational surface-ontology of place.

Euclidean & Classical Geometric Counterparts

Aquinas’s critique of self-subsistent void space (n. 3), Avicenna’s indivisibility of situs (n. 4), Avempace’s inner-boundary theory (n. 5), and Themistius’s rotational invariance (nn. 7–14) turns throughout on Euclidean geometry. Against Avempace, Aquinas invokes the Euclidean axioms of containment and equality (Elements Common Notion 4: things coinciding with one another are equal to one another): the concave inner surface of a hollow shell is strictly smaller in area than its outer volume and fails to bound it externally. Even more strikingly, Aquinas’s analysis in n. 7 of a sphere rotating in place—where the whole sphere remains in the selfsame spatial locus in subject while its circularly ordered parts continuously exchange positions—corresponds directly to \ Euclid‘s genetic definition of the sphere in Elements Book XI Definition 14 (the figure comprehended when a semicircle revolves around its fixed diameter until it returns to the same position) and underpins the Euclidean method of superposition (epharmozein) in Elements I.4, I.8, and III.24, which presupposes that spatial translation and rotation neither stretch nor deform a figure’s metric determinations.

Hoenen’s De Noetica Geometriae & Judgment Theory

In De Noetica Geometriae, Appendix § 2 (Hoc movetur, ergo locus est, pp. 115–138), Peter \ Hoenen S.J. explicitly cites and transcribes In IV Physicorum, lect. 7, n. 7 as a ‘beautiful text’ (pulchrum textum) establishing an intuitive existential nexus against Humean and Kantian skepticism: local translation immediately manifests the existence of place through the succession of bodies within an invariant surrounding extension. Moreover, in Chapter IV (§§ 1–3, pp. 61–158), Hoenen connects Aristotle’s and Aquinas’s doctrine of translation and rotation in Phys. IV with the foundation of geometric congruence against Bertrand \ Russell and Henri Poincaré. Russell objected that moving a triangle in Elements I.4 treats spatial points as material particles, while Poincaré claimed geometry depends empirically on physical rigid bodies. Hoenen replies via Aquinas’s materia intelligibilis and rotational continuity: just as the parts of Themistius’s homogeneous sphere continuously succeed one another in congruent positions without physical deformation, pure extension as the principle of geometric individuation guarantees a priori that any figure realized in one site can be identically realized across a continuous succession of sites.

GEOMETOR Constructive & Symbolic Significance

Aquinas’s twin doctrines in In IV Phys., lect. 7—that place is not an empty Cartesian pixel-box existing prior to extended bodies, and that circular rotation maps a homogeneous continuum onto itself in subject while permuting its parts—illuminate how \ geometor.model conceives spatial reference without arbitrary grid discretization. In \ geometor.model, the geometric plane is not a pre-allocated finite array of raster cells; rather, a relational coordinate frame is anchored by two initial foundational points that define the unit interval, from which every subsequent locus is generated by straightedge and compass incidence—an exact computational counterpart to Aristotle’s principle that local determinations arise only from extended bodies and their mutual boundaries. Likewise, Aquinas’s contrast between rectilinear translation (which shifts a whole figure into an extrinsic place) and circular rotation (which cyclically permutes consecutive parts within a self-contained locus) mirrors the interplay of linear extensions and circular compass sweeps in \ geometor.elements and \ geometor.explorer, where rotational symmetry around a center generates invariant regular polygons and golden-angle phyllotactic rings.

Physics Book VI, Lecture 1 — That No Continuum (Line, Time, or Motion) Can Be Composed of Indivisible Points

Primary Text: English (Phys. VI, lect. 1) · Latin (Phys. VI, lect. 1) · Summary

Aristotelian & Scholastic Foundations

Aquinas comments on the celebrated opening chapter of Physics VI (231a21–231b18), where \ Aristotle launches his refutation of spatial, temporal, and kinematic atomism (directed historically against \ Democritus, Xenocrates’ indivisible lines, and Zeno’s paradoxes, and elaborated by \ Simplicius, In Phys. 925–935, \ Themistius, and \ Averroes, In Phys. VI, comm. 1–6). Importing the three topological relations rigorously defined in Physics V.3 (226b18–227a32)—suneches (continuum, whose extremes are one), haptomenon (contactum / contiguum, whose extremes are together in the same immediate place), and ephexes (consequenter se habens, between which nothing of the same genus intervenes)—Aquinas constructs a tight syllogistic dilemma reinforced by the Book III.1 (200b18–20) definition of the \ continuum as infinitely divisible. Every escape route for point-atomism is sealed in nn. 7–8: two distinct points cannot touch (whole-to-whole contact collapses positional distinction), nor be consecutive without a gap, and any gap between two points must itself be infinitely divisible into further divisibles (in semper divisibilia), establishing strict isomorphism across spatial magnitude, motion, and time.

Euclidean & Classical Geometric Counterparts

Aquinas’s commentary on Physics VI.1 is the indispensable philosophical key to reading the opening definitions of \ Euclid‘s Elements Book I without falling into point-set anachronism. Euclid defines a point as that which has no part (semeion estin hou meros outhen, Def. 1), a line as breadthless length (Def. 2), and—crucially—the extremities of a line are points (grammes de perata semeia, Def. 3), parallel to Book I Def. 6 (the extremities of a surface are lines) and Book XI Def. 2 (the extremity of a solid is a surface). Far from building lines out of points like beads on a string, Euclid treats the continuous line as ontologically prior to its endpoint markers and internal section points! Furthermore, Aquinas’s theorem that every continuum is divisible in semper divisibilia directly underwrites Elements Book I Proposition 10 (bisecting any given finite straight line), Book X Proposition 1 (the exhaustion principle of endless bisection), Book X Proposition 2 (incommensurable magnitudes whose Euclidean antenairesis never terminates in a common atomic unit), and Book II Proposition 11 / Book VI Proposition 30 (cutting a line in extreme and mean ratio, whose golden-section anthyphairesis continues in infinitum).

Hoenen’s De Noetica Geometriae & Judgment Theory

In De Noetica Geometriae (Chapter III, §§ 1–2, pp. 88–178), Peter \ Hoenen S.J. deploys the Aristotelian-Thomistic continuum doctrine of Physics VI.1 to diagnose the foundational crisis of nineteenth- and twentieth-century arithmetized analysis (Cantor, Dedekind, Wellstein, \ Russell). Modern rigorists sought to build the geometric continuum upward from an uncountable point-set because they mistakenly began with the point as an isolated corpuscle shrunk to zero (Grenzprozess), rendering its mathematical existence unintelligible. Reversing that order with Aristotle and Aquinas, Hoenen begins from the primary intellective datum: three-dimensional continuous extension, whose first propria passio is infinite divisibility. When an extended magnitude is divided without separating its parts, the boundary between the parts—through which transition occurs in an indivisible—emerges in act: first the surface, then the line, and finally the point, an indivisible having position. Because points exist in potency within the undivided continuum and arise in act only as indivisible cuts or intersections, no collection of points—finite or transfinite—can ever constitute extension by contact or consecutiveness; rather, between any two constructible points an infinitely divisible line segment necessarily intervenes.

GEOMETOR Constructive & Symbolic Significance

Nowhere does Aquinas’s philosophy of mathematics speak more directly to GEOMETOR than in In VI Physicorum, lect. 1. Digital graphics engines approximate geometry on discrete pixel or floating-point lattices—a computational atomism in which lines are literally consecutive arrays of indivisible raster points, causing false misses or rounded drift whenever two loci intersect at irrational coordinates. By contrast, \ geometor.model implements the exact Thomistic ontology of the continuum: lines and circles are represented as continuous algebraic loci (linear and quadratic polynomial equations), and points arise solely as exact symbolic boundaries and intersections (termini) where two continuous loci cut one another. Moreover, Aquinas’s proof that between any two distinct points lies a continuum divisible in semper divisibilia is the exact mathematical condition enabling the golden ratio (phi) investigations of \ geometor.divine: because the golden section is quadratic and irrational, cutting a segment in extreme and mean ratio induces a self-similar cascade of nested golden sections without ever hitting an indivisible atomic quantum.

On the Heavens Book I, Lecture 2 — On the Perfection of Three-Dimensional Body and the Geometers’ Mode of Speaking of a Moving Point

Primary Text: English (De Caelo I, lect. 2) · Latin (De Caelo I, lect. 2) · Summary

Aristotelian & Scholastic Foundations

Commenting on the opening argument of De Caelo I.1 (268a1–268b10), Aquinas harmonizes the formal definition of the \ continuum in Categories 6 (5a1–2: parts joined at a common boundary, pros hena koinon horon) with the material definition in Physics III.1 and VI.1 (eis aei diaireta diaireton) to establish corporeal three-dimensionality. Particularly noteworthy is Aquinas’s critical methodological restraint in n. 7, drawn directly from \ Simplicius‘s Commentary on De Caelo (ed. Heiberg, 6–9; translated into Latin by William of Moerbeke in 1271, immediately utilized by Aquinas): recognizing that \ Aristotle‘s appeal to Pythagorean triads, religious libations, and linguistic usage (ambo vs. omnes) is dialectically probable rather than apodictic, Aquinas cites Claudius \ Ptolemy‘s lost treatise On Dimensionality (Peri diastaseos, preserved in Simplicius), which proves geometrically that at most three mutually perpendicular straight lines can intersect at a single point. Equally incisive is n. 9, explaining Aristotle’s remark on impossibility of metabasis eis allo genos beyond solid body (268b1–5) by identifying the fluxion of point into line, line into surface, and surface into body as an imaginative idiom of geometers (cf. De Anima I.4, 409a3–5).

Euclidean & Classical Geometric Counterparts

Aquinas’s synthesis in In I De Caelo, lect. 2 bridges the planar geometry of Elements Books I–VI and the solid stereometry of Books XI–XIII. Ptolemy’s three-perpendiculars proof cited by Aquinas in n. 7 rests directly on Elements Book XI Propositions 4–6 and Definition 3–4 (a straight line perpendicular to a plane is at right angles to all straight lines meeting it in that plane; and through a given point on a plane only one perpendicular straight line can be erected, exhausting orthogonal dimensionality at three). Simultaneously, Aquinas’s careful gloss on the modus loquendi geometrarum—that a moving point makes a line, a moving line a surface, and a moving surface a solid—illuminates both the classical Greek tradition reported by Proclus (In Eucl. 97.6–13) and Heron of Alexandria, and \ Euclid‘s own kinematic definitions of the sphere, cone, and cylinder in Elements Book XI Definitions 14, 18, and 21 (generated by rotating a semicircle, right triangle, and rectangle around a fixed axis), clarifying that constructive generation expresses dimensional dependency without violating the non-atomistic continuum of Physics VI.1.

Hoenen’s De Noetica Geometriae & Judgment Theory

Peter \ Hoenen S.J. cites In I De Caelo, lect. 2, n. 9 verbatim in De Noetica Geometriae (Chapter IV, § 2, p. 128 and fn. 9, paired with In II De Caelo, lect. 2, n. 11 and In IV Phys., lect. 18, n. 4) when evaluating kinematic descriptions in pure geometry. Hoenen shows that Aquinas strikes the exact balance between static ontological rigor and dynamic constructive fruitfulness: strictly speaking, a line cannot be defined quidditatively through a moving point without circularity (since motion already presupposes an extended path along which the point moves, as Hoenen demonstrated in Chapter III, § 1), which is why Aquinas attributes this idiom to the imagination of geometers (imaginantes quod punctus motus facit lineam). Yet as a genetic description for tracing special curves (such as Archimedes’ spiral or Hippias’s quadratrix, where time enters as an auxiliary parameter and is subsequently eliminated to leave a pure geometric locus), kinematic imagination is completely legitimate because it rests on the intelligible continuity and directionality of three-dimensional extension—and in Chapter III, § 2.3, Hoenen likewise echoes Aquinas and Ptolemy on the intuitive clarity of three dimensions against uncritical four-dimensional spatial speculation.

GEOMETOR Constructive & Symbolic Significance

Aquinas’s dual insight in In I De Caelo, lect. 2—the completeness of three-dimensional orthogonal extension and the distinction between ontological continuum structure and the constructive mode of speaking of geometers—maps directly onto the architecture of GEOMETOR. In \ geometor.explorer and \ geometor.model, constructions unfold sequentially in time as animated sweeps of straightedges and compasses across the canvas, appealing to the geometer’s constructive imagination (modus loquendi geometrarum) to reveal how complex harmonic configurations emerge step by step from simple seeds. Yet beneath that kinematic animation lies the timeless algebraic continuum of SymPy equations, in which every locus exists complete in its formal definition rather than being assembled out of discrete motion-steps. Furthermore, the culmination of three-dimensional completeness studied in De Caelo I.2 points directly toward \ geometor.pacioli and Elements Book XIII, where planar golden-ratio sections ascend into the five regular Platonic solids—above all the dodecahedron and icosahedron—inscribed within the complete three-dimensional sphere.

On the Heavens Book II, Lecture 2 — On Positional Directions in Natural Bodies and How They Are Considered by Position in Mathematical Figures

Primary Text: English (De Caelo II, lect. 2) · Latin (De Caelo II, lect. 2) · Summary

Aristotelian & Scholastic Foundations

Aquinas comments here on the first half of De Caelo II.2 (284b6–285b8), where \ Aristotle critiques the Pythagorean attribution of right and left to the cosmos (cf. Metaphysics I.5, 986a22–26, the ten-column systoichia of principles) by importing the biological mechanics of De Incessu Animalium (On the Progression of Animals, chs. 2–5, 704b12–706b16) and De Anima II.4 / III.9–10. Following \ Simplicius (In De Caelo, ed. Heiberg, 383–393), Aquinas distinguishes three senses of dimensional priority—by dimensional magnitude (length, breadth, depth), by genetic construction (line, surface, body), and by causal origin of vital motion (nutritive above, sensory before, locomotive right)—and highlights the pivotal philosophical contrast in nn. 6–7 (285a1–10): whereas organisms possess intrinsic functional chirality and polarity independent of external orientation, inanimate bodies and geometric objects lack internal directional principles and receive above and below, right and left, and before and behind solely quoad nos—including enantiomorphic inversion in specular reflection (in speculo).

Euclidean & Classical Geometric Counterparts

In Aristotle’s Greek text at De Caelo II.2, 285a1–4 (reflected in Aquinas’s analysis of objects lacking an intrinsic principle of motion in nn. 6–7 and geometric generation in n. 11), triangles and squares are the classic exemplars of inanimate/mathematical objects whose above, below, right, and left are assigned purely by positional convention (thesei / per comparationem ad nos) and reverse whenever the figure is rotated. This principle is fundamental to \ Euclid‘s Elements: in Book I Definitions 20–22 and Propositions 4, 8, and 26, a triangle’s ‘base’ (basis), ‘vertex’ (koruphe), ‘left side,’ and ‘right side’ are never intrinsic properties of the figure itself, as Euclid explicitly signals in I.4 and I.5 by choosing whichever side serves the proof as base. Even more remarkably, Aquinas’s third relational mode in n. 7—contraria positio in a mirror image (imago quae resultat in speculo), where right and left interchange while preserving figure (conveniunt in figura, n. 9)—identifies exact enantiomorphic (chiral) congruence, the very relation tested when Euclid applies inversely oriented congruent triangles in Elements I.5 (Pons Asinorum) without altering metric equality.

Hoenen’s De Noetica Geometriae & Judgment Theory

In De Noetica Geometriae (Chapter IV, § 2, fn. 9, citing In II De Caelo, lect. 2, n. 11 on how length precedes breadth because according to geometers a moving line generates a surface, and Chapter IV, § 4.2, pp. 183–205 on the notion of direction and orientation), Peter \ Hoenen S.J. develops Aquinas’s distinction between intrinsic vital polarity and relational geometric position into a rigorous theory of spatial isotropy and relative orientation. Pure geometric extension (materia intelligibilis) is completely homogeneous and isotropic: no point or axis possesses an intrinsic above or right prior to a chosen reference line. Yet once a single directed line a passing through a point P in a plane is posited, two opposite senses of angular divergence (‘on the right’ and ‘on the left’ relative to a, p. 194) immediately become objectively distinguishable as relational differences in site alone. Thus Hoenen shows how Aquinas’s per comparationem ad nos in De Caelo II.2 translates mathematically into frame-relative orientation and reflection symmetry, preserving Euclidean invariance across every rotation of the coordinate frame.

GEOMETOR Constructive & Symbolic Significance

Aquinas’s demonstration in In II De Caelo, lect. 2 that geometric figures possess no absolute up, down, left, or right in themselves—acquiring directional labels solely relative to an adopted reference frame and inverting chirality under mirror reflection—captures a foundational principle of symbolic modeling in \ geometor.model and \ geometor.divine. Although \ geometor.model typically seeds a canonical construction with two initial points along a horizontal axis at (0, 0) and (1, 0) to fix a reference orientation (quoad nos), the emergent golden sections, harmonic ranges, and equilateral or pentagonal symmetries detected by \ geometor.divine are completely invariant under rigid rotation, translation, and mirror reflection (in speculo). Moreover, Aquinas’s observation in nn. 9 and 13 that right and left agree in figure (conveniunt in figura) while reversing orientation anticipates bilateral and chiral symmetry analysis in \ geometor.phyllotaxis and \ geometor.seer, where parastichy spirals (clockwise versus counterclockwise Fibonacci numbers) and visual grid transformations exhibit exact mirror duality.