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Second Discourse Upon a Phenomenology of Mathematical Progress Suitably Adapted for Interested Readers of Silicon Valley Executive Network (SVEN) – Silicon Valley, California, USA

Conveyed First to the Bulgarian Academy of Sciences, May 2019

 Approved for Publication by the Academic Senate of the Bulgarian Academy of Sciences, Bulgaria, Sofia, 1040
1 “15 Noemvri” Str.

Dr. Jonathan Kenigson, FRSA

Mathesis informs the contentions among classical philosophers regarding the differentiation of algebraic, as opposed to geometric, processes. This trend is absent from Plato and his forebears but debuts in Book 8 of Aristotle’s Topics (222) and Books 8-9 of Euclid’s Elements (127-190) in a discussion of incommensurable quantities and construction. In Proposition 3 of Measurement of a Circle (448-451), Archimedes references the Euclidean distinction between discrete and geometric quantity – a position taken in Conoids and Spheroids (455-456) in lemmata to the second proposition (456-457); in Proposition 10 of Spirals (488-489); and in the Sand Reckoner (520-526). Nicomachus’ Arithmetic, while more esoteric in its tone, speaks of the methods of arithmetic vs. geometric process in his taxonomy of the infinities in Book I (814-821). In Kepler’s refutation of Ptolemy and defense of Copernican astronomy, one finds a distinction between the algebraic and geometric processes most clearly in Book V of the Epitome (990), although the distinction is implicit throughout the work. Less directly, in Second Day of Two New Sciences, Galileo references the distinction between algebraic and geometric methods of computation. Descartes’ mathesis – described in Book I of Geometry (295-296); Part II of the Discourse of Method (46-47); and in Rules XVI (33-35) and XVIII (36-39) – effects the first full marriage of algebraic and geometric methods in calculation. Pascal further references the integration of algebraic and geometric methods in his number-theoretic Correspondence With Fermat (474-487) and Arithmetical Triangle (451-452; 458-459; 464-466). Locke and Berkeley treat this mathematically technical concept more cursorily; for Locke, the algebraic-geometric distinction appears only briefly in Section XI, Book IV of Human Understanding (378) and in sections 19 and 121-122 of Berkeley’s Human Knowledge (410; 436-437). The Critique of Pure Reason furnishes a more substantive discussion of the distinction as regards a-priori synthetic knowledge (17-18; 68-69; 212-213; 217). One effect of this Cartesian mathesis is the modern continuation of the attainment of generality in mathematical notation as well as in the disparate bodies of mathematical knowledge. This trend already appears in the 41st chapter of Book I of Aristotle’s Prior Analytics (68) and briefly in the Posterior Analytics (105). Nicomachus references symbolic generality as a goal of mathematical research in Book 11 of the Arithmetic (832), after which work is relatively sparse until the Cartesian mathesis makes symbolic generality a paramount task of algebro-geometric theoretical integration in Geometry (295; 298; 314; 353), Part II of the Discourse (46; 47), and in Rules (28; 40). Locke’s treatment is characteristically terse in the third chapter of Book IV of Human Understanding (318-319); Berkeley’s discussion is similarly cursory in Human Knowledge (409; 436-437). Kantian metaphysics explores the generalization of notation in light of the progress of reason in Pure Reason (68-69; 211-213), after which no other treatment in the putative Kantian corpus is present. Fourier cogently demonstrates the unification of algebraic and geometric notations in the infinitary representations of his Theory of Heat, fully realizing the methodology of the Cartesian (and Newtonian) trajectory toward high symbolic generality (172-173; 181; 233; 240).      

Descartes represents the first modern mathematician to undertake a mathesis of algebraic and geometric techniques, and Cartesian mathematics merits the close critical investigation that it receives in contemporary research literature. The Cartesian mathesis is a powerful unifying force in mathematical mechanics, in which the meta-language for the symbolic formulation of physical problems is taken to be analytic geometry. Books 1 and 2 of the Geometry (298-314; 322; 331) express this viewpoint powerfully, as does the Rules in sections IV, XIV, and XVI (5; 30; 33; 35-36). In his Principia, Newton adopts a Cartesian understanding of the mathesis as regards the search for a universal symbolic formulation of mathematical theory. He does not, however, believe that the development of analytical calculus makes the work of pure geometry irrelevant. This disposition can perhaps best be discerned in Lemma 19 of Book 1 of the Principia (57-58), which is a purely theoretical construction of the ratios of four plane lines. Newton opines that “…we have given in this corollary a solution of that famous problem of the ancients concerning four lines, begun by Euclid, and carried on by Appolonius [sic] and this not an analytical calculus, but a geometrical composition, such as the ancients required” (58). A universal language of symbolic mathematical discourse is necessary for the dialectic of theoretical physics with mathematics, but certain research in pure mathematics remains, in Newton, immune from such. Huygens’ On Light (610), Lavoisier’s Elements of Chemistry (1), and Fourier’s Theory of Heat (177-251) are similarly encomiastic of pure geometrical research. All, however, agree that a unification of algebraic and geometric formulation is necessary for the progress of the physical and mathematical sciences. Speculation regarding the nature of the infinitude of discrete quantities related in proportion is as old as Greek mathematics itself. Among the Platonic corpus, such inquiry is a problematization of the Heraclitean-Parmenidean debate about the nature of understandable change and appears most notably in the Phaedo (228-229), Parmenides (494; 500-502; 508; 510-511), and Theaetetus (518; 519). The Aristotelian corpus places the proportion of discrete quantities at the center of epistemology and ethics. Epistemological discussions include Book V, Chapters 13 and 15; Book X, Chapters 3 and V; and Book XIV, Chapter 1 of the Metaphysics (541-542; 581; 583-584; 620), and applications to Aristotle’s ethical doctrine of the mean occur throughout the Ethics but most notably in Book II, Chapter 6 (351-352) and Book V, Chapter 3 (378-381). A proportional theory of statehood and political virtue is found most paradigmatically in Book V, Chapter 1 of Politics (503). Following in this train, a proportional calculus of moral law adopting entelechy and Aristotelian Eudaimonism is to be found in the Summa of Aquinas, especially in Part 1 Q. 13; Q. 42, Rep. 1; Q. 47, A. 7, Rep. 2 and Part 1-2, Q. 113, A. 9 (68-70; 224-225; 257-258; 368-369). The theory of proportion as regards virtues and vices is developed organically without substantive interest in the underlying mathematical theory beyond that furnished by Aristotle in the citations on Ethics above. The Cartesian repudiation of scholasticism recuses proportion from moral law and casts proportion as a bridge between algebraic and geometric theories; this element of the mathesis figures more prominently in Books 1 and 3 of the Geometry (295-298 and especially 332-341) but alluded to in Part 2 of the Discourse of Method (47). Rousseau rejects the Aristotelian calculus of moral proportions in Book III of the Social Contract (407-408) but is uninterested in an explicitly mathematical repudiation. Curiously, Locke’s theory of politics neglects any mention of the proportion of discrete quantities and instead posits his repudiation of the apriority of proportion in a brief but obscure part of Sections 9-10, Chapter 2, Book 4 of Human Understanding (311). Berkeley and Hume have little to say on the topic as well, so Kant’s treatment is based principally from Locke and appears in Pure Reason (73-74; 211-213) and in an aesthetic guise, briefly and unassumingly, in Judgment (497-498). Among mathematicians, however, the theory of proportion of infinitary and discrete quantities is well appreciated. Borrowing from the pre-Socratics and Pythagoreans, Euclid’s treatment in the Elements is comparatively clear and comprehensive, and occurs in Book I, Propositions 4, 8, and 26 on the proportions and congruences of triangles (2; 4; 6-7; 16-17) and in Books V (81; 98) and Proposition 23 of Book VI on proportions of equiangular parallelograms (117); and in passing in Book X on proportions of commensurable and incommensurable magnitudes (191; 300). The Archimedean mathesis follows from the introduction of conics and quadrature to Euclidean analytical methods, as can be found in the Quadrature of the Parabola (527); On Spirals (484), and Book 1 of the Sphere and Cylinder (404). Nicomachus, in his passages entitled “Theory of Proportion” in Books I-II of the Arithmetic, proposes a purely number-theoretic treatment of quantity and proportion (821; 831; 841; 848), seeding the Neoplatonic interest in divine proportion advanced in Book III, Chapter 15 of the Sixth Ennead of Plotinus (289). The Galilean trajectory begins with Eudoxus and Archimedes in an analysis of proportions in the theory of infinite free-fall, among other topics; see, for instance, the First Day of Galileo’s Two New Sciences (142-145). Kepler’s Harmony of the World applies similar methods of proportion to planetary motion (1012-1014; 1078; 1080). The Newtonian Mathesis in the Principia – drawing on Kepler, Galileo, and the Archimedean corpus – is evident in Lemma 1 of Book 1, in which infinitude is perceived as a limit: “Quantities, and the ratio’s [sic] of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.” (25) Similar constructions are present in the scholium and Lemma 2 of Book 1 (31-32); Book 2, Lemma 1 (159); and Book 2, Lemma 2 (168-169), with the latter two lemmata relating the algebraic-geometric mathesis of proportions to the mechanics of bodies as advanced by Galileo and Kepler.       

Mathesis informs the vacillation between Platonistic and Aristotelian perspectives in the debate regarding how mathematics is and should be applied in mechanics. Aristotle discusses the relation of mathematics to mechanics In Posterior Analytics, Book 1, Chapters 9 (104) (regarding reasoning from primary premises and middle terms) and 13 (108) (predicability and syllogism, facts and reasoned facts); Physics, Book 2 Chapter 2 (270) (the distinction between mathematical and physical knowledge and the sufficiency of physical knowledge for scientific progress); and Book 7 Chapter 5 (333) (on the nature of single and combined forces acting over variable distances). In the Metaphysics Book 13, Chapter 3, Aristotle considers the mathematical sciences as neither separable from, nor strictly a part of, the science of mechanics (609). The Archimedean mathesis is present in Περὶ ἐπιπέδων ἱσορροπιῶν (Equilibrium of Planes, 502-519) among computations of the centers of gravity of plane figures and theories of proportion applied to levers. Such results admit of strong generalization to parabolic segments, heralding a non-physical application of the results from the first book. Περὶ τῶν ἐπιπλεόντων σωμάτων (On Floating Bodies, 538-560) culminates with a discussion of the displacement of paraboloids in fluids of varying specific gravities; Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος (On the Method of Mechanical Theorems) admits a full generalization of the theory of infinitesimals to curvilinear shapes with rational volumes (569-592) – a far more abstract task, and one which illustrates the indispensability of mathematics in Archimedean mechanics. In Marcellus (252-255), Plutarch speaks of the dialectic between mathematics and mechanics, attributes the birth of mechanics to Eudoxus and Archytas, and relates the dissatisfaction of Plato with the pedestrian nature of mechanics, which consists “of abstract things descended…to the things of sense” (253). In the Summa, Aquinas very briefly espouses the Aristotelian view, returning to the descriptive phenomenology of the Physics (see especially Part 1-2, Q. 35, A. 8, 779-780). Book 5 of Kepler’s Epitome makes mathematics indispensable to mechanics (964-965), and indeed, to all of natural science – a position clearly held by Galileo (First Day of Two New Sciences, 131-133; Second-Fourth Day, 178-260). Repudiating the scholastics and the Summa, the Cartesian mathesis defends an essentialist view of mechanics as a science of discovered natural laws to be presented in mathematical form (see Rules 4 (7) and 14 (31); Parts 1 and 3 of the Discourse of Method (43; 50); Book 2 of Geometry (322-331); and Objections and Replies, 169-170). The Newtonian mathesis is entirely congruous in its philosophical aims (see, for instance, Principia Book 1, Propositions 1-17 (32-50); Propositions 30-98 (76-157); the Scholium (130-131); and Book 3, 169-170 for a concise account). In the Theory of Heat, Fourier adopts the Newtonian mathesis and argues that mathematics is the foundation of mechanics as a discipline (169; 172; 173; 175-177; 182-184; 249). Faraday’s Researches in Electricity is devoid of this foundationalism and casts mechanics as an experimental (as opposed to a mathematical) science. This methodology is congruous with Bacon’s inductive empiricism as presented in Advancement of Learning (46) and of Hume’s comments on mechanics in Section 4, Section 27 of Human Understanding (460).

The dialectic between Platonic and Aristotelian modes of ontology is also attributable to Platonic cosmology, and is the beginning of another perpetual coincidentia oppositorum that represents the progress of mathesis in the philosophy of science. The Timaeus (particularly salient are 447; 450-451; 451-452; 460) is the source of most Platonic cosmology. Book 10 of the Republic (438-439) and Book 7 of Laws also treat the periodicity of celestial motion; the eternity of celestial motion is broached in the Timaeus alone. In the Timaeus, the δημιουργός (demiurge) attributes ensoulment to the celestial bodies; their motions are eternal and spherical, for eternity is more perfect than temporality and the sphere is perfect among all shapes (451-452). Apparent time is the number of the motion of the spheres; time is motion itself, rather than a measure of motion. The celestial bodies are themselves responsible for the modulation of time – this is there raison de etre (451-460). According to the Timaeus, the composition of celestial and terrestrial matter is, as the Presocratics held, the four fundamental elements (450-451). These are comprised of atoms that assume the forms of perfect Polyhedra, which are in turn comprised of right triangles conjoined in a predictable manner (450-451). Books 7 and 10 of the Republic (see especially 394-396; 438-439); Book 7 of the Laws (especially 730); and the Timaeus (451-452) comprise the bulk of the Platonic discussion of the corresponding forms of the celestial motions (circles, ellipses, and their equants). The Aristotelian treatment of celestial motion is broader and more subtle. The infinite periodicity of celestial motion is discussed in Books I-II of the Heavens (359-389) and in Book 12, Chapter 8 of the Metaphysics (603-605); the corresponding forms of celestial motions are addressed in Book 8, Chapters 8-9 of the Physics (348-353); Book 1, Chapters 2-5 (359-364) and Book 2, Chapters 4, 5, 8, and 12 of the Heavens (378-379; 381-382; 384); and Book 12 of the Metaphysics (601-602; 604), wherein crystalline spheres and their perfect motions are imputed to underlie cosmological regularities. In Aristotle, the primum movens replaces the demiurge as the origin of understandable change; the universe, like all things, has an entelechy and is the subject of cause. In Books 5 and 6 of De Rerum Natura, Lucretius renounces the idea that cosmological bodies have either telos or soul; rather, these bodies are comprised of indestructible atoms and subject to the naturalistic force of chance (fortuna) that governs the interactions of such (61; 62; 64-65; 67-69; 76-77; 88). Aurelius resurrects the Parmenidean stasis in his Meditations through the repudiation of Lucretius’ fortuna and asserts instead the eternal, static, and immutable nature (pneuma) of the entire cosmos (see, for instance, Book 6 Section 13 (271) and Book 11 Section 27 (306) for a discussion of the periodicity of celestial motion; see Book 9, Section 28 (293-294) for a discussion of the eternity of celestial motion). The Ptolemaic mathesis of the eternity of celestial motion presents itself in Books 1 and 4 of the Almagest (7-8; 12-14; 109; 112) via a prolegomena to Aristotelian cosmology and the properties of lunar motion and apparent lunar apogee. Book 8 of the Almagest deals with the eternity of celestial motion in terms of the variabilities of the apparent regresses of stellar positions (492) The mathematicization of the Aristotelian universe present in Book 1 is drawn both from Euclid and Nicomachus and forms a full, predictive mathematical cosmology when paired with the theory of Deferents, Equants, and Epicycles present in Book 3 (on the precession of Equinoxes, 83; 86), Book 5 (lunar Parallax and Apogee; 148-157), and Book 9 (models of the visible planets and the motions of Mercury; 270; 291-296). Plotinus’ refutation of Aristotelian and Ptolemaic empiricism is present early in the Enneads, which discuss the ensoulment of the celestial spheres, their perfect motions, and their eternal nature in terms of the One and its transcendental emanations. On the periodicity of celestial motions, one finds the Second Ennead, TR 2, Chapter 1 (40-41) and Fourth Ennead, TR 4, Chapter 8 (162) particularly salient. On the eternity of celestial motions one finds the Second Ennead, TR 9, Chapters 3-8 (67-70) and TR 8, Chapters 5-13 of the Third Ennead (122; 124; 126; 129) on the ensoulment of planetary spheres and the perpetuity of their motions. In Book 5 of Confessions and Book 2, Chapter 29 of Doctrina Christiana (651), it seems that Augustine is strongly indebted to the Neoplatonic cosmology, both of the immaterialism of the soul and the perfection of the eternal motions of the spheres – a position diffusely advanced by Aquinas in the Summa,  Part 1, Q. 32; A. 1, Rep. 2 on the application of reason to the Ptolemaic cosmos (175-178) and Q. 115, A. 3 (588-589) on the interactions of heavenly bodies with terrestrial causes. Of the periodicity of celestial motion, Aquinas makes scant further comment, except in passing in Part 2, Q. 2, A. 3 (392-393). Of the divine cause of eternity of celestial motion, Aquinas is a devotee of the Aristotelian theory of causes, time, and aeveternity championed most fully in the Summa’s Part 1, Q. 10, A.  2, A. 4, and A. 5, (41-45) among other citations (945-946; 1017-1020). In dissent, Galileo’s Two World Systems posits the cosmos in terms of mathematical laws; both the periodicity of motion and the eternity of motion are rendered explicable in a geocentric cosmos (see Fourth Day 245), whereupon Copernicus hazards generalization in a comprehensively mathematized anti-Ptolemaic system (Revolutions of the Heavenly Spheres 507-508; 513-514; 628-629; 675-678; 740; 784-785). The Keplerian program admits of the repudiation of the perfect circularity of orbits, obviating the Nicomachean mathematics of Deferents and Equants present in Ptolemy’s Almagest (Epitome of Copernican Astronomy 846-848; 888-891; 892-893; 926-933; 968-979; 984-985, Harmonies of the World 1018). The Newtonian mathesis converges in the determination of fundamental forces with the common use of algebraic, geometric, and analytic methods. Of the periodicity of celestial motion, Newton advances Proposition 53 of Book 2 (266-267) and the scholium of of the Principia, “Granting the quadratures of curvilinear figures, it is required to find the forces with which bodies moving in given curve lines may always perform their oscillations in equal times.”, furnishing a generalization of Kepler’s Law of Areas. Of the eternity of celestial motion, the Principia advances Proposition 10 of Book 3 (on the general theory of forces) “that the motions of the planets in the heavens may subsist an exceedingly long time” – the proof not relying upon any premise of abstract philosophy, but by the impedance of motion furnished by atmospheric drag, of which Newton’s Second Law permits a full resolution. Book 3 of the Optics (540-541) addresses the same topic in passing. Of the form of celestial motion, Newton is more silent, tacitly according to Kepler the resolution of the Ptolemaic system; nonetheless, Book 1, Proposition 2 (42-43) of the Principia casts the Keplerian system in terms of centripetal forces and uniform rectilinear motion, and Proposition 17 furnishes a complete generalization: “Supposing the centripetal force to be reciprocally proportional to the squares of the distances of places from the centre, and that the absolute quantity of that force is known; it is required to determine the line which a body will describe that is let go from a given place with a given velocity in, the direction of a given right line” (47-48). It is the laws of celestial motion – their underlying mathematical logos – that the Newtonian mathesis fundamentally unifies; the highlights of the theory – proved using the theory of conic sections and mechanics, can be found perhaps most concisely in Proposition 4, Corollary 6, “If a body revolves in an ellipsis; it is required to find the law of the centripetal force tending to the focus of the ellipsis” (36). The apparent motions of the celestial sphere are subjected to a uniform, natural law. Further illustrative results of this type can be in Book 1, Propositions 1-3 and scholium (32-35); Propositions 11-13 (42-46); Proposition 15 (46-47); Proposition 17 (48-50); Book 2, Propositions 51-53, scholium (259-267); Book 3 (270-271); Phenomenon 1 – Proposition 7 (272-282); Proposition 13 (286); Proposition 35, scholium (320-324); Proposition 40 (337-338); and the general scholium (369, 371-372). In each case, observation is predicated upon mathematical law, and predictive empirical propositions are subjected to the primacy of mathematical reality.

Mathesis in cosmology also furnishes a battleground between Platonic and Aristotelian notions of the status of mathematics between pure codification of pure observation and purely predictive ontology. In Book VII of the Republic, Plato argues that the study of astronomy is necessary for guardians, and that the study of mathematics is necessary in turn for the study of astronomy (394-396), and in the Timaeus, he posits that mathematics provides a “likely account” for astronomical phenomena (451; 455). In Book 1, Chapter 13 of the Posterior Analytics (108); Book 2, Chapter 2 of the Physics (270); and Book 2, Chapter 14 of On the Heavens (388), Aristotle defends the primacy of observation in astronomy, relegating mathematics to an explanatory or discursive role. This trend is continued in Book 3, Chapter 2; Book 3, Chapter 8; and Book 13, Chapter 3 of the Metaphysics (516; 603-604; 609). It is also discussed incidentally in Book 1, Chapter 1 of the Parts of Animals (161). Book 5, Parts 3-6 (27-28) of the Confessions and Book 2, Chapter 29 (651) of the Christian Doctrine of Augustine adopt the Aristotelian cosmology and implicitly doubt the normativity of mathematical physics over metaphysics or observational physics. Aquinas’ Summa (175-178; 424-425; 779-780) also mirrors this view. In contrast, in his Arithmetic, Nicomachus argues for the indispensability of geometry to astronomical motions (813-814), the prediction of which constitute a foundation of the Ars Quadrivium. The Almagest employs Euclidean methods to systematize Aristotelian notions of the motion of terrestrial and celestial matter; the theories of Apollonius of Perga and Euclid employ an extensive theory of conics, which Ptolemy holds as indispensable to his scientific explorations (5-6; 14-24; 26-28). In the Revolutions of the Heavenly Spheres, Copernicus follows this Aristotelian reasoning and argues that mathematical techniques are descriptive of physical reality, but might not perfectly predict counterfacts to Aristotelian notions of the material uniformity of the physical universe (507-508; 510; 532-556). Book 5 of Kepler’s Epitome of Copernican Astronomy argues contrary to Aristotle and Copernicus that mathematical physics can be used to find the true, underlying structure of empirical reality (964-965; 968-986), a theme continued in Book 3 of Newton’s Principia (1-2; 269-372), in which the calculus of fluxions furnishes a full generalization of the Copernican-Keplerian astronomy, and in Descartes’ Meditations (76). Kant holds that the remarkable success of pure mathematics in the formulation of Newtonian mechanics vindicates Plato for supposing that mathematics, apart from practical experience, could be predictive of cosmological truths. In the Critique of Practical Reason (361) and Critique of Judgment (551-552), Kant holds that a dialectic between mathematics and astronomy is present, in which – as Plato holds – astronomical discoveries drive discoveries in mathematics; however, Kant also takes the reverse dialectic to be even more primary. In this sense, Kant can be conceived as uniting the thoroughgoing empiricism of Bacon’s Advancement of Learning (37; 46) and the Platonic perspective championed by Kepler The motions of the moon form a foundation for cosmological and related mathematical theories of nature, and are a prime subject of the coincidentia oppositorum between Platonistic and Aristotelian tendencies in mathematics. The Platonic thesis that the irregularities of the lunar procession are for the purpose of timekeeping is advanced in parallel in the Cratylus (98); the Apology (204-205); and, most paradigmatically, in the Timaeus (251). In the Cratylus, the dialogue between Socrates and Hermogenes is aimed at a deconstruction of Anaxagoras’ notion of lunar motion as Σελαναία (selenia), for σέλα νέον τε καὶ ἕνον (her gleam is always old and new) (98). This semiotic understanding of lunar motion is not asserted either in the Apology or the Timaeus, the former of which is interested in the exploitation of lunar motions for the keeping of time, and the latter of which is focused upon a theoretical excursus in fundamental cosmology. The Aristotelian counterpoise in Book 2, Chapters 11, 12, and 14 of the Heavens is concerned not with naming or calculation but with an explanatory taxonomy of lunar motions (see, for instance, 383; 284-385; 389). The logos is not a decision of eidetic causes, but of descriptive phenomenology. In Book 5 of De Rerum Natura (see especially 67-71), we find a resurgence of the Platonic project of a fundamental understanding of the logos of lunar motion amid a poetic vindication of Epicurean atheism: “We must suppose the moon and all the stars \ Which through the mighty and sidereal years \ Roll round in mighty orbits, may be sped \ By streams of air from regions alternate” (70). In Lives, Plutarch, like Plato, attributes the first theory of lunar motion to Anaxagoras, but attributes Socrates with the perfection of the theory in Nicias (435); the search for a logos of lunar motions is advanced in passing in Solon (74) Aemelius Paulus (220-221) and Dion (789-790). The Ptolemaic mathesis is presented in the Almagest, Books IV-VI using Nicomachean and Hipparchian geometry and Aristotelian cosmology. Books IV and V treats the motions, Parallaxes, and distances of the sun and moon from the earth; Book VI is a theory of Epicycles applied to the prediction of Eclipses (108-222). The mathematical structure of the discussion is predictive rather than explanatory, in keeping with the methodology of the Heavens and Physics. To the contrary, Plotinus’ neo-Pythagorean mysticism and platonic fascination with celestial causes is evident in Chapter 5, Treatise 3 of the Second Ennead: “The truth is that while the material emanations from the living beings of the heavenly system are of various degrees of warmth- planet differing from planet in this respect- no cold comes from them: the nature of the space in which they have their being is voucher for that” (43-44). In refutation, the Summa of Aquinas, uniting the epistemologies of Augustine and Aristotle, considers the lunar sphere and the relative and apparent magnitudes and sizes of the moon, sun, and stars in Q. 70, A. 1, Rep. 5, “On the Work of Adornment, As Regards the Fourth Day” (362-264). The counter-reaction of Bacon in Book 2 of Novum Organon, Aphorism 36, among diverse locations in the text wherein Newton (presumably via Propositions 43-45 and 66 of Book 1 and Book 3, Propositions 3-4 and 22-38 of the Principia that all bear explicitly upon the rectification of perceived irregularities in lunar motion; see 92-101; 118-128; 275-278; 294-329) is credited with explicating the underlying mathematical philosophy of diverse phenomena, including lunar motions; more than twenty arguments of the Organon are devoted to the triumph of inductive and empirical thinking (as opposed to mythological thinking) in lunar mechanics (167). In Light, Huygens follows a similar mathematical course, preferring mathematical generality as a philosophical mathesis (554-555).

Works Consulted.

Pagination has been updated to reflect references from the 1952 edition of the Great Books of the Western World, in the public domain thanks to the generosity of the Library of India. 

Adler, Mortimer Jerome. The great ideas: A syntopicon of great books of the western world. Vol. 1. Encyclopaedia Britannica, 1952.

Adler, Mortimer Jerome. The great ideas: A syntopicon of great books of the western world. Vol. 2. Encyclopaedia Britannica, 1952.

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