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*Preliminary draft for student use only. Not for citation or circulation without permission of editor.
I know that I am mortal, a creature of a day; but when I search into the multitudinous revolving spirals of the stars my feet no longer rest on the earth, but, standing by Zeus
himself, I take my fill of ambrosia, the food of the gods.
- Ptolemy, Cosmography1
Chapter 3) Ptolemaic Astronomy: Mathematical Theories of the Heavens
Introduction: From Geocentric to Heliocentric Planetary Theories
One of the great unifying concepts in the natural sciences of the Renaissance
was the nearly universal belief in a well-ordered, comprehensible ‘cosmos’. Properly
speaking, the science of the cosmos was cosmography, a term much broader than the
modern field of cosmology, since it included ‘Earth sciences’ like geography and ‘moral
sciences’ like harmonics. There was a powerful aesthetic element at work in the concept since the noun derives from the Greek verb kosmeo, which means “to set in order,
to marshal, or to arrange”. It was often used in connection with military commanders or
civic leaders. But it also carried an aesthetic connotation; the 'setting in order' should
be a beauty-enhancing order. Indeed, it was Pythagoras who coined the word kosmos
’to express his belief in a well-ordered, beautiful, harmonious world—a world knowable
1
Kepler used this quotation from Ptolemy’s
Cosmographicum (1596).
Geography [??] as the epigram to his
Mysterium
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through number. Combining this aesthetically- and mathematically-informed image of
the natural world with another Greek word, graphikös, which means “to write, draw, or
record in texts & images”, made cosmography fundamentally into a mapping and descriptive enterprise. Certainly for Renaissance cosmographers, the principal challenge
of the science was to locate themselves in time and space, with as much precision and
elaboration as they could muster.
Cosmography took as its domain the combined territories of cosmology; the general structure of the universe at large and the Earth's place in it; cosmogony, or the origins of the cosmos; and geography, the location of the Earth and the disposition of its
parts. Yet it also included stellar and planetary astronomy, especially theories of their
motions; astrology, or the study of the influence of celestial movements on the terrestrial
world; and harmonics, or music theory, as a means of unifying terrestrial and celestial
music. Since the seventeenth century the movement toward specialization has broken
cosmography into a number of smaller disciplines, ranging from cartography and navigational science to meteorology and celestial mechanics. But in its day cosmography
was the ‘central science’. It was not only a shared set of mathematical techniques and
a thoroughgoing commitment to systematic description of the large-scale structure of
the cosmos. It was also "world view" in the sense that cosmographers—whether working in the fields of geography or astronomy—understood their activities to be part of a
grand, unified conception of the cosmos as a coherent, intelligently structured, and orderly entity. As such cosmography excited the imagination of scholars from the High
Middle Ages to the end of the seventeenth century. Thus, the breakup of cosmography
and the transformations of its component into more or less independently functioning
scientific disciplines constitute some of the most important episodes in the history of
early modern science.
Like all Renaissance sciences, the technical content of cosmography had its origins in antiquity, and indeed, largely in the works of one man, Claudius Ptolemy.
Ptolemy is noted most for his work on planetary theory, the Syntaxis—better known un-
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der its Arabic title, Almagest, or ‘The Greatest’. But he in fact was the author of several
other works of perhaps even greater, or at least broader, influence. In addition to his
classic in planetary astronomy, Ptolemy also wrote the Tetrabiblios (or “four books”) on
astrology; the Harmonics which combines music theory and astronomy in a manner
broadly consistent with the Pythagorean program; and Geography (often also called
Cosmography) on positional geography, projections, and cartography.2 With the exception of the Harmonics which was recovered and translated only in the early seventeenth
century, Ptolemy’s literary corpus was readily available to Renaissance cosmographers
in printed translations from the late fifteenth century onward.3
Despite their common source in the works of Ptolemy and their shared investment mathematical and empirical methods, the branches of Ptolemaic cosmography
experienced very different fates in the sixteenth and seventeenth centuries. The literary
conversations with Ptolemy as spokesman for ancient cosmography was sometimes
cordial—Kepler speaks of him with highest respect even when he disagrees with
him—sometimes intense and intimate—Copernicus organized the chapters of his treatise so that there is nearly a one-to-one match up with those of Ptolemy. And sometimes the discussion dissolves into an argument as, for example, when Copernicus implies that the ‘system’ Ptolemy had created was no system at all but a “monster”. It
really was not until after the middle of the seventeenth century that Ptolemy was
dropped from the discussion across the range of cosmographical disciples, just as cosmography as a unified, central science fragmented into its modern specialties.
In astronomy, or more precisely planetary theory, Copernicus’ rejection of the
geocentric hypotheses of Ptolemy was the beginning of the ‘astronomical’ or ‘Coperni-
2
[Mention what else Ptolemy wrote: Optics, Spherical geometry, ‘Handy Tables, etc. ??]
See publication history under each of the selections from Ptolemy below. It is worth noting
here, however, that in the sixteenth and seventeenth centuries, Ptolemy’s Almagest went
through just four editions (Venice, 1515 & 1528; Basel, 1538; and Wittenberg, 1549)—and was
not to be printed again until the nineteenth century. His Geography, on the other hand, went
through more than 50 [??] editions during the same period, and the Tetrabiblos (first printed in
1484) went through about the same number of editions. [Check numbers??]
3
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can revolution’, which in turn is seen as the central episode in the Scientific Revolution
of the seventeenth century.4 The shift in consensus first among astronomers and
mathematicians and then among natural philosophers and the educated elite also led to
the abandonment of the finite, spherical, spatially closed, and geocentric cosmos of Aristotle in favor of the heliocentric universe of the infinite, or perhaps only ‘indefinitely
large’, universe of Descartes and Newton.
This process of cosmological consensus-building was neither continuous nor
smooth. The arguments for and against the Copernican and Ptolemaic systems were
technically complex and laden with theoretical presuppositions. There was no direct
empirical evidence that the sun revolved around the sun until a hundred years after Copernicus’ death and compelling proof waited until the second quarter of the eighteenth
century.5 And all sides confused the matter by making frequent appeals to philosophy
(e.g., Kepler was inordinately fond of animism, and Descartes of mechanism) and theology (e.g., Galileo’s biggest mistake may well have been his decision to instruct the
Catholic hierarchy on how best to re-interpret Scripture in light of his telescopic discoveries). Still, by the end of the seventeenth century, when Newton’s theory of gravitation
and laws of motion were beginning to be broadly assimilated, the grand edifice of Aristotelian geocentric cosmos had been discarded in favor of a very different image of the
cosmos. Much of the astronomical content of the transition from a geocentric to a heliocentric cosmos is carried in the works of Nicholas Copernicus, Tycho Brahe, Johannes
Kepler, and Isaac Newton.
Despite having been born in northern Poland and far from the centers of Renaissance culture, Copernicus (1473-1543) obtained an excellent humanist education in
4
(see ‘Introduction’, pp. 000 above and vol. 2, ‘Introduction’, pp. 000
See J. L. Heilbron, “Astronomia Reformata,” The Sun in the Church: Cathedrals as Solar O bservatories (Cambridge, Massachusetts: Harvard University Press, 1999), pp. 101-119 for a
discussion of the of the first observational evidence, by the Jesuit Giambattista Riccioli, in favor
of the Copernican theory, ca. 1650. See Anton Pannekoek, A History of Astronomy (New York:
Dover Publications, 1961), pp. 289-293 for a discussion of James Bradley’s discovery of stellar
aberration, which provided additional proof of the Earth’s motion around the sun.
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classics. Indeed, his first publication was a translation from Greek into Latin of the work
of a minor Greek poet. He also received excellent training in law, medicine, and of
course mathematics. In 1491 he matriculated at the University of Krakow, then considered the best university in central Europe. There were two chairs in astronomy, one in
mathematical astronomy and one in astrology, with a century-long tradition of critical.
Thus even as an undergraduate Copernicus had probably learned something about the
strengths and weaknesses of Ptolemaic planetary theory. It was also in Krakow that the
young Copernicus became a member of the secret learned society known as the
‘Brotherhood of the Vistula’, named after the river that runs through the city and dedicated in part to the revival of the Pythagorean philosophy. Copernicus continued his
education in northern Italy, first in law at the University of Bologna where he also studied with the Italian astronomer Domenico Maria da Novara, then medicine at the University of Padua, and finally canon law at the University of Ferrara. He returned permanently to Poland in 1503 and took up his duties as a canon (or church administrator)
attached to the cathedral in Formbork in Varmia, Poland where his maternal uncle had
recently been named bishop. His earliest account of the heliocentric system, the Commentariolis, or ‘Little Commentary”, was written and circulated among friends in manuscript form soon after 1512. By 1531 he had transformed the qualitative sketch of the
Commentariolis into a mathematical model capable of predictions. Yet he published his
masterpiece only in 1543, reviewing the proofs while lying on his deathbed.
Historians of astronomy have often noted that, in regard to principles, methods,
and goals, Copernicus was engaged in a program of ancient astronomy. Indeed, he
seems to have seen himself not as a revolutionary nor even as a reformer. Instead, like
the humanist he was, Copernicus saw himself to be a restorer of the lost knowledge of
true celestial motions. In keeping with Plato’s program of ancient Greek astronomy;
namely, to find “what are the uniform and ordered movements by the assumption of
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which the motions of the planets can be accounted for”,6 Copernicus sought to construct
a heliocentric theory using only uniform circular motion, strictly defined. The nonuniform motion Ptolemy had introduced into planetary theory, in the form of the equant
construction, had offended not only Copernicus’ aesthetic sensibilities but also those of
generations of Arabic astronomers before him.7 Principles of simplicity and harmony
also informed Copernicus’ criticism of the absence of systematic coherence in
Ptolemy’s planetary hypotheses. Such considerations—along with a host of technical
matters—drove Copernicus to prefer what he saw as the elegance of his heliocentric
system over the authority of received teachings.
The price of elegance, however, was implausibility. There were a number of
physical objections to placing the earth in motion, and Copernicus marshaled what evidence he could in is own defense. It was an uphill battle, because the most widelyaccepted theory of motion was the qualitative physics of Aristotle, which depended fundamentally on a fixed, non-rotating, central earth.8 Copernicus himself could offer only
vague notions of ‘local gravitation’ attached to each planet and not a thorough-going,
self-consistent alternative physics.
{TBA: paragraph on possibility of calculating the size of the solar system in the
Copernican theory . . . }
The challenge confronting the generation of astronomers who came of age after
Copernicus was the unenviable one of choosing between two highly technical and perfectly incompatible theories, one ancient and broadly consistent with a Christianized
Aristotelian cosmos and world view, the other tainted by novelty, in an age that still ven-
6
Heath, Greek Astronomy. p. xliv.
See vol. 2, 2.4 Swerdlow & Neugebauer, “Arabic Astronomy”, pp. 000. Copernicus was
certainly aware of the work of his Arabic predecessors and, indeed, was indebted to them for
solutions to technical problems raise by a heliocentric theory.
8
For a discussion of Aristotelian physics of motion, 4.2 Aristotle, “Physics”, pp. 000 below.
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erated tradition, and manifestly incompatible not only with common sense but with
scriptural teachings. Both were sophisticated pieces of astronomy—the two theories
surely ranked as the most technically difficult science of the day—and, alas, both were
imperfect with regard to predictions.
Faced with this impasse Tycho Brahe (1546-1601) turned to observation as the
only means to settle the debate. Unwilling to accept the physical implications of a rotating and revolving earth, Tycho was nonetheless fascinated by the simplicity and coherence the Copernican system seemed to offer. In a remarkable career of systematic
observation that spanned the last three decades of the sixteenth century, Tycho first
attempted to demonstrate the correctness of the heliocentric theory through a series of
observations aimed at establishing the relative distances between Mars and the Earth
and between the Earth and the Sun. The Copernican theory predicted that, at closest
approach, the Mars-Earth distance is much less than the Earth-Sun distance. Despite
the deployment of large naked-eye astronomical instruments expressly designed for the
purpose, Tycho failed to establish the relationship between the distances that would
have resolved the question in Copernicus’ favor.9
These ‘failed’ observations, along with his parallax measurements for the comet
of 1577, which showed that comets were celestial phenomena and not phenomena of
the Earth’s atmosphere as Aristotle had argued, encouraged Tycho to pursue a third
solution. Somewhere between a compromise and a synthesis, the Tychonic, or geoheliocentric, system had two centers of motion; Mercury, Venus, Mars, Jupiter, and Saturn orbit about the Sun as the Sun, accompanied by its entourage of planets, orbits
about a stationary and central Earth.
In the work of Johannes Kepler (1571-1630), ‘assistant’ to Tycho in the last years
of his life and devout Copernican, we encounter the first thoroughgoing break with the
precepts of ancient astronomy. Whatever degree of originality we may grant to Coper-
9
Owen Gingerich and James R. Voelkel, “Tycho Brahe’s Copernican Campaign,” Journal for
the History of Astronomy, 1998, 29: 1-34.
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nicus, his models of planetary motion were, like those of Ptolemy, kinematic constructions that assumed, but did not explain the causes of, uniform circular motion. Kepler,
on the other hand, sought to understand the causes behind planetary motions, and his
insistence upon celestial dynamics led him to reject not only epicycle-on-deferent and
equant constructions but also the uniform circular motion that had been the fundamental
axiom in the two-thousand-year tradition of ancient Greek astronomy. The Keplerian
system, while ‘Copernican’ in the sense of being heliocentric,10 was really quite different
in its details. All planetary orbits were elliptical instead of circular, and their foci were
coincident with one another and located within the body of the sun which was the center
of action for planetary motions. Kepler guessed, correctly, that the sun rotates on its
axis and he postulated, incorrectly, that lines of anima motrix (or ‘motive force’) radiating
outward from the rotating sun swept the planets along in their orbits. Planetary motions
were non-uniform but law-like and therefore calculable. Finally, predictions of planetary
positions based on the Keplerian system far exceeded the accuracy of either the Ptolemaic or Copernican systems.
Even as an undergraduate at the Lutheran university in Tübingen (where he
studied under Michael Mästlin), Kepler was absolutely convinced of the physical truth of
the Copernican system. And the grounds of his conviction was, from a modern perspective, a seemingly eccentric mixture of neo-Pythagorean aestheticism (something he
shared with Copernicus), neo-Platonic animism (Kepler was the last great astronomer to
defend astrology as a science), and a stunningly original commitment to the mechanics
of celestial motions (a program he invented virtually by himself out of whole cloth). Kepler was also one of the great human calculating machines of the age, and many of the
patterns of celestial order he discerned in the chaos of planetary motions yielded themselves to him in large part under the sheer weight of his computational skills.
10
Technically speaking, the Copernican system was heliostatic (meaning the sun was at rest)
but not heliocentric since none of the centers of the planetary orbits were in the sun, nor was the
center of any one planetary orbit coincident with any other orbital center.
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In his first publication, written before he turned twenty-five, Kepler demonstrated
his understanding of the dilemma astronomers faced at the end of the sixteenth century.
And nowhere does the metaphor of a conversation between Ancients and Moderns
come more alive than in Kepler’s evaluation of the advantages of the Copernican system over the Ptolemaic. His enthusiasm for the former inspired him to venture his own
theory in response to his won question as to why there should be six and only six planets in the Copernican system. His answer, because there are five and only five regular
solids. By circumscribing spheres about the regular solids and selecting a particular
sequence of nested spheres-within-solids, Kepler believed he could geometrically reproduce the spacings of the orbits of the Copernican system. This was his first—and,
as he believed, his greatest—discovery, for here he had comprehended the geometrical
archetype that god has used in the construction of the cosmos.
Thirteen years, several books, and thousands of calculations later, Kepler published his Astronomia nova, or ‘New Astronomy’, based on a celestial dynamics.
Whereas both the Ptolemaic and Copernican theories were essentially exercises in
planetary kinematics; i.e., geometrical descriptions of planetary motions, Kepler sought
to explain those motions in terms of the forces that actually caused their motions. Kepler begins his treatise by rehearsing the technical matters under debate, though now
the disputants include Tycho Brahe and Kepler himself as well as Ptolemy and Copernicus. Kepler moves with ease among the systems, comparing, contrasting, and striving
to preserve continuity even as he introduces radically new ideas into the discussion. In
response to the physical objections raised against heliocentric models, Kepler propounds an original, though still qualitative theory of gravitational attraction. He also addresses the objections brought against the Copernican theory based on a literalist
reading of selected passages from the Bible and presents a defense of the compatibility
between scripture and heliocentrism.11
11
See 8.3 Galileo & Inquisition, “Documents”, pp. 000.
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Despite the excitement caused by Galileo’s telescopic observations, which he
published in 1610, just a year after Kepler’s New Astronomy, and the commotion
caused by his campaign to have the Copernican—though, ironically, not the Keplerian—system accepted as the true description of the cosmos, there were few important
developments in planetary theory in the first half of the seventeenth century. The
planetary theories contending for consensus were the Ptolemaic, though seriously
weakened by Galileo’s observations, the Tychonic, and the Copernican/Keplerian and a
handful of minor variants of each.12 And despite improvements in the design of tel
e-
scopes, neither Galileo’s original round of observations nor those conducted over the
following decades provided compelling empirical evidence in favor of any one of these
theories. In the absence of decisive observation, there was much philosophizing—and
theologizing—about the implications the new theories might have for cosmography
(both sacred and profane) and natural philosophy in general.
By the middle of the century, the most astute astronomers understood that the
critical problem concerned the orbits of the planets. In the absence of the sort of rigid,
rotating crystalline spheres that Aristotle had postulated, how were planets held in their
orbits and their motions maintained? Kepler had certainly understood the problem and
framed a solution in his theory of anima motrix, but his speculations, however insightful,
convinced few.
Descartes’ qualitative model of planetary vortices was more popular
than Kepler’s scheme but scarcely any less speculative.13 By the 1660s and 70s Gi ovanni Borelli (1608-1679), Christiaan Huygens (16??-????), and Robert Hooke (16351702) had discussed or published partial solutions to the “problem of the planets” based
on mathematical approaches. Hooke appended an account of the problem—and his
hasty outline of the likely solution—to a short pamphlet he published in 1674 on his attempts to observe stellar parallax resulting from the motion of the Earth. In a bitter twist
of fate, Hooke’s observations lacked the precision to measure stellar parallax, which is
12
13
See Figure ?.?.>, p. 000 below [Kircher’s diagram of six systems, from Iter Istaticum??]
See 4.10 Descartes, “Principles of Philosophy”, pp. 000 below.
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in fact a necessary consequence of the Earth’s motion, and his sketch of an inversesquare law of universal gravitation lacked the precision necessary to achieve what he
correctly foresaw as “the true perfection of Astronomy.”
More bitterness was in store. Hooke had written a condescending letter to Newton in 1672 regarding the latter’s pioneering theory of light and color—a topic perilously
close to Hooke’s own work on optics, the Micrographia (1666?). The tone had provoked
Newton ire and ill-feeling characterized their first exchange of correspondence. In 1779
Hooke, now secretary of the Royal Society, again took up his pen and wrote to Newton
regarding what he thought would be a safe topic free from the taint of priority or rivalry.
He offered to Newton the intriguing idea that "it is possible that planetary motion could
be the result of a tangential inertial force and a central force to produce a closed orbit".
Newton, who had not yet published anything on gravitation, replied in such a way so as
not to reveal that he himself had worked intensively on the very same problem. In a
friendly gesture, Newton reciprocated by presenting his solution to a hypothetical
problem: namely, what is the shape of the path traced out by an object falling through a
‘transparent’ the Earth? His answer was a spiral centered on the Earth’s geometric
center.
Hooke had promised earlier not to publish anything of Newton's from their correspondence. Hooke, however, saw Newton's solution as a blunder and evidently could
not resist the temptation to expose it publicly. At a meeting of the Royal Society in London, he called attention to Newton's incorrect solution and presented his own; i.e., the
object goes into orbit around the Earth's center of mass. Newton, in Cambridge,
learned of the meeting and was outraged. He sent a letter to the Royal Society, correcting himself and offering a different solution, a three-cusped cycloid. Hooke responded by pointing out that that solution depended upon the assumption that the force
of attraction to be constant; when in fact, its strength decreases as 1/r2.
Newton now felt both tricked and humiliated—and his theory of gravitation
threatened. His anger and rivalry with Hooke spurred him to take up again the problem
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of gravitation, which he had not worked on for several years. He reconsidered the
question Hooke had posed concerning orbits under an inverse square force and found
that the solution had to be one of the conic sections (i.e., a hyperbola, parabola, or ellipse). More critically, he was now able to derive an elliptical orbit from the inverse
square law. But again Newton did nothing with one of the most important derivations of
seventeenth-century astronomy. Finally in May of 1684, Edmund Halley, a close friend
and the foremost astronomer in England, asked Newton if he could solve the shape of
the curve resulting from a 1/r2 force. Newton told him yes, he knows the shape is an
ellipse but has forgotten where he put the papers with the proof. But promises to send
them when he finds them. When Halley finally saw the proof, he urged Newton to publish them quickly since he knew Hooke was nearing the same solution. Newton quickly
composed a short tract entitled 'De Motu' and had it read before the Royal Society. It
was favorably received and he is encouraged to continue. By 1687, Newton had finished the Principia, which was printed at Halley’s expense.14
Building on the work of Kepler, Galileo, Descartes, and of course Hooke, Newton was able to supply what Copernicus knew was missing from his astronomy; namely,
a physics compatible with a moving Earth. Newton’s three laws of motion and the inverse-square law of gravitational attraction were first and foremost universal laws: they
applied to all types of matter everywhere and at all times. Gone was the old Aristotelian
distinction between the celestial and terrestrial regions each with its own characteristic
type of matter and distinctive ‘natures’. Newton’s laws obtained on Earth, on and
among all the planets, and throughout the vast extend of the universe. And these laws
were thoroughly mathematical in regard to their interdependence as conceptual entities
and their dependence upon the measurable behavior of matter. Whereas Descartes’
vortices were qualitative models resistant both to mathematical treatment and empirical
confirmation and his laws of impact demonstrably wrong, Newton’s law of gravitational
14
Koyré, Newtonian Studies, pp. ?? [account of correspondence between Hooke and Newton]
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attraction was embodied in a system of internally consistent axioms and propositions
fashioned in direct imitation of the axiomatic-deductive structure of Euclidean geometry—Descartes’ dream realized. The ‘Newtonian synthesis’, then, was a triple synthesis; of terrestrial and celestial mechanics, of theory and experimental, and of mathematics and physics. Fittingly, the work in which he lays out his theory he called the
“Mathematical Principles of Natural Philosophy” (Philosophia naturalis principia mathematica).
Before presenting the mathematical principles of his new physics, Newton felt
obliged to engage in a bit of metaphysics. In the scholium, or explanatory note, following his first several definitions, Newton elaborates upon two ontological categories, absolute space and absolute time. Treating them almost like ‘common notions’ (in the
Euclidean sense) that can neither be reduced to other entities nor dispensed with,
Newton establishes them as the foundations of his mechanics. Absolute space bears a
strong resemblance to the fictitious imaginary space of Euclidean geometry, yet it functions as the universal arena in which the motions and collisions of matter take place.
Absolute time is the temporal equivalent to absolute space and a close relative of the
‘Eternity’ to which Plato referred when he spoke of “the instruments of time” (i.e., the
stars and planets) as “the moving image of eternity”.15 Interestingly, despite his deep
commitment to measurement and his hostility to entities that could not be derived from
phenomena,16 Newton knew he could measure neither motion with respect to absolute
space (the famous ‘bucket experiment’ accounts only for rotational motion and not
translational motion) nor duration with respect to absolute time. The inconsistencies
between Newtonian physics and metaphysics were what Ernst Mach and Albert Einstein challenged some two centuries later.
15
[Passage in Plato ??}
See his ‘Rules for the Study of Natural Philosophy’ in 3.7 Newton, “Principia”,
pp. 000 below.
16
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