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    We want to discuss certain creative cases in the history of
science within the theoretical framework we have delineated:

Case Ι:     Johannes Kepler.
Case Π:     Max Planck.

    Our double aim is to deepen our understanding of both
the theoretical framework and the structure and dynamic of
the history of science.

                           Case Ι
                      Johannes Kepler

Why Kepler?

What is so special about Kepler?
 Kepler happened to live in the era of the scientific
revolution; he was born after the death of Copernicus (1543)
and before the birth of Newton (1642). That era constituted an
unprecedented turning point and upheaval in our understanding
of nature and in world civilization. It gave birth to modern
science at the expense of pre-modern natural philosophy. To
understand Kepler is a key to understanding the scientific
revolution. After all, Kepler was a seminal figure, a giant, of
the scientific revolution, side by side with Copernicus, Gilbert,
Galileo, Descartes, Huygens, Leibniz and Newton.

  Kepler is the founder of modern astronomy (and optics).
   He radically changed our approach to celestial
   phenomena by destroying many of the old ways and

 Yet, he never made a clean break with per-modern science.
  He partook of both pre-modern and modern ways. He was
  more of a bridge between pre-modern natural philosophy
  and modern natural science. Accordingly, he could be
  considered an important key to the history of physical

 He combines in his work and career major elements of
  both pre-modern natural philosophy and modern science,
  which makes him an epitome of the totality of physical
  science – a basic node of scientific development?

  Let us discuss Kepler’s scientific career in relation to the
  model we have proposed for scientific production. We
  want to focus on Kepler’s epistemic heritage, his evolving
  research project, the tools he employed in his research
  work, and the way he arrived at his main astronomical
  discoveries. We shall also comment on the revolutionary
  features of Kepler’s work.

Kepler’s Epistemic Heritage:

  Kepler’s great contributions were not confined to
  astronomy, but covered also geometry and optics (he was
  the founder of scientific optics in Europe). However, we
  shall focus on his astronomy.

  The following broad stages         characterized   Kepler’s
astronomical epistemic heritage:

 1.   Aristotelian physics and cosmology.
 2.   Hellenistic Astronomy.
 3.   Arabic Medieval Astronomy.
 4.   Copernicus.
 5.   Tycho.
 6.   Pythagoras.
 7.   Materialism.
 8.   Gilbert.

1. Elements of Aristotelian Physics
 The Universe is uncreated.
 There is neither a starting point nor a terminating point to
 The Universe is finite in volume and variety.
 Space is the limit and boundary of matter. Thus, it is
   finite in extent. There is nothing beyond the boundary of
   the Universe – not even space and time.
 The Universe is spherical in shape, because the sphere is
   the perfect shape in solid geometry (a Platonic idea).
 Nature abhors a vacuum. There are no vacua in the
 Being finite, the Universe has a definite centre endowed
   with specific physical characteristics. It is the only centre
   of attraction in the Universe.
 The Earth is located at the centre of the Universe. Its
   centre coindes with the centre of the Universe.
 The Earth is spherical – a perfect sphere.

     [The circumference of the Earth in Graeco – Arabic
1. Aristotle’s Method (4th century BC):
   The ancients thought that all stars, apart from the planets,
   the Moon and Sun, are fixed relative to each other, and lie
   on the surface of a solid sphere rotating around the Earth.
   They noticed that the axis of rotation passed through the
   centre of the Earth and the polar star (almost). Thus, the
   latter remained fixed as the sphere of the stars revolved.

  Aristotle (and others) noticed that as one travels
  northwards, the angle made by the polar star with the
  horizon changes (increases).
  That was one piece of evidence for the sphericity of the
  Earth, and was used by Aristotle and others to measure
  the circumference of the Earth, as follows:
   α= angle polar star makes with horizon = < AOM

   C= Earth’s Circumference.

| AB | = θ g
  C      2π

 2. Eratosthenes` Method (3rd century BC):
    Eratosthenes noticed that, on 21st June at midday, a
    gnomon at Syene (modern Aswan) in the south of Egypt
    does not leave any shadow, but leaves a slight, but
    noticeable, shadow in Alexandria in the north of Egypt.
    Eratosthenes knew that the two cities lie almost exactly
    on the same longitude. To him, this slight difference in
    the shadow angle was a clear indication of the sphericity
    of the Earth. By accurately measuring the shadow angle
    at Alexandria and the distance between Syene and
    Alexandria, he was able to produce an accurate number
    for the circumference of the Earth, as follows:

S = Syene
A = Alexandria
α = Shadow Angle

   α   =   shadow angle = shadow length
                          gnomon length

  α = ISAI         Hence C.
  2π    C

3. Biruni’s Method (973-1051 AD):

The idea was to measure the horizon angle from the top of a
mountain next to a sea coast, and to measure this angle at
the foot of the mountain. From these measurements and
from measuring the height of the mountain, Biruni obtained
an accurate value for the circumference of the Earth, as

R= radius of Earth

      The Universe is fundamentally inhomogeneous. It
       consists of two qualitatively different realms: Earth
       and its immediate surroundings and the Heavens.
       They differ in the following respects:

             Earth                              Heavens
1. Obeys specific laws:            1. Obey different specific laws:
The laws of Aristotelian           the laws of Greek Astronomy;
physics; physics is the            astronomy is the science of the
science of Earth and its           heavens.
2. Subject to change,              2. Immutable and eternal; subject
corruption, birth and death.       to no change whatsoever.

3. The variety of bodies and       3. The heavens are made of a
material entities in it is         fifth, eternal substance, called the
constituted by four basic          ether, or, the quintessence. It is
elements: earth, water, air        neither heavy nor light. Thus, it
and fire. Earth is the heaviest    tends to move in uniform circular
element and naturally moves        motion; it neither falls nor rises
towards the centre of the          up. All the planets, stars and their
Universe, where it settles         carriers are made of it. It is
down. Water is heavy, but          heavenly stuff.
less so than earth. Air is
light,and,therefore,tends to
rise upwards. Fire is lighter
still, and tends to rise rapidly
towards the periphery of
Earth’s surrounding.
4. Bodies tend to move in          4. Celestial bodies move naturally
straight lines, whether            in uniform circular motion or a
downwards or upwards.              combination of such motions.
Natural motion is either           The stars move in a perfect
towards the centre of the         circular motion round the Earth.
universe, or away from it.        The Sun, Moon and five planets
External forces could cause       (Mercury, Venus, Mars, Jupiter,
bodies to deviate from their      and Saturn) move with complex
natural motions (forced           motions (varying speeds, sizes
motion).                          and brightnesses, and so-called
                                  retrograde motion). The
                                  challenge facing ancient
                                  astronomers was to describe this
                                  complexity with a combination of
                                  a limited number of uniform
                                  circular motions.
5. Force causes motion            5. There is a total absence of
(speed, not acceleration). The    gravitation in Aristotelian
speed of a falling body is        astronomy. Thus, the self-motion
directly proportional to its      of celestial objects is
weight and distance traveled      unimagineable. The
and inversely proportional to     sun,moon,planets and stars are
the density of the medium.        attached to solid ethereal spheres,
Thus, in the vacuum, speed        which rotate uniformly around
becomes infinite, which is        the Earth. The heavens are a
absurd. That enforces             system of concentric spheres (55
Aristotle’s contention that a     such spheres) rotating in various
vacuum is impossible. In          ways around the Earth, and
Aristotelian mechanics, the       carrying celestial bodies with
cornerstone is speed, and not     them. Each celestial body
acceleration, and the medium      partakes of one or more of such
is an essential factor of         motions. Hence, the complexity
motion. Galileo was to turn       of the motion of the sun, moon
all that upside down.             and planets. Aristotle’s Universe
                                  is like an onion, because it does
                                  not contain any vacua. If we
                                  move form the centre, we pass
                                  through the following regions
                              Mercury, Venus, Sun, Mars,
                              Jupiter, Saturn, Stars.
                               6. The sphere of stars is moved
                              by an unmoved mover (God?!),
                              and this motion is transmitted to
                              the rest of the Universe, including

2. Hellenistic Astronomy:
    Plato laid down the basis of the project of antique
   astronomy; how to explain the complexity of planetary
   motion (including the sun and moon) in terms of a limited
   number of uniform circular motions.
    Plato’s student, the mathematician, Eudoxos (4th century
   BC), formulated the first mathematical model embodying
   the Platonic project. However, it was quantitatively
    The Geek mathematician, Apollonius (3rd century BC),
   formulated the rudiments of an alternative, more accurate
   mathematical model.
    The Greek astronomer, Hipparchus (2nd century BC),
   inaugurated Greek astronomy as an exact science, by
   developing Apollonius’ model and wedding his model very
   closely to accurate observations. He was deeply influenced
   by Babylonian astronomy (500 BC- 1 AD).
    All these contributions were unified and completed in an
   integrated, comprehensive scientific theory and system by
   Claudius Ptolemy (2nd century AD), particularly in his
   great book, Almajest (the Majestic). Adding to
   Appolonius’ and Hipparchus’ innovations, he invented and
   employed three major mathematical devices to ―explain‖
   and describe the wealth of Babylonian and Greek
   observations of the sun, moon and planets. He succeeded
   in building a fairly accurate astronomical system of what
   we call today the solar system. These devices are:
(1). The Eccentric: The planet moves with uniform circular
motion about the centre of motion, but the centre is not
coincident with the centre of the Earth.

 (2). Epicycle and Deferent: The planet moves with uniform
circular motion about a centre, but the centre itself moves
with uniform circular motion about a fixed point. The circle
on which the planet moves is called the epicycle, whereas the
circle, which carries the epicycle, is called the deferent.
The result is a combination of two uniform circular motions.
A wide variety of motions could be accurately described with
this device by altering the parameters and directions,
including retrograde motion.

P = Planet
Ó = Epicycle centre
O = deferent centre

(3). The Equant: The planet is suuposed to be moving with
uniform circular motion, neither about the centre of the circle
nor about the Earth, but about a fictitious point at a distance
from both, called the equant.
3. Arabic Medieval Astronomy:

Arabic astronomy started at the end of the 8th century AD and
lasted till the 16th century. It passed through the following
   - The translation phase: whereby the major Greek,
      Sanskrit, Persian and Syriac astronomical texts were
      translated (several times) into Arabic.
   - The correction and updating phase whereby fresh
      calculations and observations were made, on the basis of
      which many Ptolemaic parameters were corrected and
   - The critique of Ptolemy phase: The great Arab physicist,
      Al-Hassan Ibn Al-Haitham (Al-Hazen), initiated a
      critical tradition in Arabic astronomy, called Al-Shukuk
      tradition, meaning casting doubts on the Ptolemaic
      system. This tradition subjected Ptolemy to a thorough
      theoretical critique. In particular, it emphasized the
      observation that the Ptolmaic devices deviated from the
      principles of Aristotelian physics, particularly the
      principle of uniform circular motion. Most critiques
      centred on the equant, which was generally considered
      an aberration. To Arabic scientists, this contradiction
      between mathematical astronomy and physics was
      intolerable. This accounts for their general
      dissatisfaction with Ptolemaic astronomy.
   - The new model construction phase: Having critiqued
      Ptolemy on the basis of Aristotle, Arabic astronomers
      embarked on constructing new models for celestial
      motions alternative to Ptolemaic models. Those
      astronomers strove to make their new models as
      accurate as Ptolemy’s models, without deviating from
   Aristotelian principles. That is these models were as
   close as possible to Aristotelian principles without
   compromising the accuracy of Ptolemy’s models. In
   particular, they all strove to eliminate the equant and
   replace it with a set of uniform circular motions. They
   invented new mathematical techniques for this purpose,
   such as the Urdi lemma and Tusi couple, and arrived at
   very elegant semi-Aristotelian models. Formost amongst
   those astronomers were: al-Urdi, Al-Tusi, Al- Shirazi
   and Ibn Al-Shatir. The main centres of this activity were
   Damascus and Maragha in North-Western Iran. There is
   mounting evidence that these new models were
   transmitted to Europe, and that Copernicus was well
   aware of some of them when he started developing his
   heliocentric model.

4. Copernicus (1473-1543):

  Nicolaus Copernicus (a pole or German) initiated the
   scientific revolution, which culminated in Newtonian
   mechanics, the paradigm of the new physics.
  His age was the Renaissance age of discovery (America
   was discovered by the Europeans in 1492) — an age of
   unprecedented expansion at all levels.
  He studied astronomy and medicine in Renaissance Italy
   (Bologna), but lived most of his life as a clerical official
   (as the canon of Frauenberg) in Poland — a rather quiet
   and secluded life.
  Copernicus was famous for proposing a helio-centric
   model of the universe, reducing Earth to a mere moving
   planet. Here is a schematic figure showing the order of
   the planets.
 The idea of a moving Earth was indeed proposed in
   antiquity (some Babylonians, Philolaos the Pythagorean,
   Heraclides of Pontus, Aristarchus of Samos, Aryabhatta
   of India), but it was a peripheral idea. Copernicus not
   only revived the idea but also turned it into a detailed
   mathematical and astronomical scheme comparable in
   sophistication and explanatory and predictive power to,
   if not superior to, Ptolemy’s system. He spent almost
   forby years in formulating it.
 Aroud 1514, he produced the first written account of his
   helio-centric scheme, which was a 40-page manuscript
   outlining the principles of his system, the basis of his
   later elaboration. It was distributed on a very limited
   scale, and was known ever since as the Commentariolus.
   The Roman Catholic Church approved of it, but Martin
   Luther condermned it with vehemence.
 His major work, commonly called De Revolutionibus
   (On the Revolutions of the Celestial Spheres), was
   published in 1543, and it is said that he received the first
   copy of it when he was on his dying bed. It contained a
   detailed comprehensive elaboration of his helio-centric
   system, and is comparable, in scope and mathematical
   sophistication, to Ptolemy’s Al-Majest — a veritable
   revolutionary alternative.
 What innovations did Copernicus introduce in European
   and World astronomy? In fact, he introduced two
   innovative schemes in European astronomy, with
   worldwide ramifications:
(a) He was as opposed to Ptolemaic devices, particularly
      the equant, as Arabic astronomy had been before him.
      That is why he sought to introduce Arabic
      mathematical techniques into European astronomy,
      reminiscent of Urdi, Tusi and (particularly) Ibn Al-
      Shatir. Whether or not he reinvented these
      techniques, or plagiarized his Arabic predecessors, is
    still very much a controversial issue. In any case, he
    could be considered a continuation of Arabic
    astronomy, and that aspect of his work could be
    considered the conservative side of his work.
(b) Following his Pythagorean bend, he placed the sun
    near the centre of the Universe (of motion) and
    considered it fixed, and transformed the Earth into a
    mere planet revolving around itself and around the
    centre of the universe. That is, many of the apparent
    motions of celestial bodies were reduced and
    attributed to motions of the Earth. That was indeed
    the revolutionary side of Copernicus’s work, since it
    contradicted both common sense and the very essence
    of Aristotelian physics.
     What motivated Copernicus to introduce these
       innovations, and spend almost forty years of hard
       work to reformulate astromomy on seemingly
       crazy bases?
(a) The spirit of the age (the Rennaissance): It was an age
    of expansion and discovery. It was also an age of
    dissatisfaction with past and existing structures. It felt
    the need to challenge the old and to critique
    traditional wisdom and knowledge. It was impatient
    with all authority — a veritable revolutionary age.
(b) Copernicus adhered very strongly to the Platonic idea
    of uniform circular motion in the heavens, and he
    veheremtly apposed all Ptolemaic deviations from
    this principle. This motivated both his helio-centrism
    and revival of Arabic innovations.
(c) Ptolemy’s system was too complicated for
    Copernicus’s aesthetic taste. To him a viable theory
    should be simple, reflecting the simple beauty of
    Nature. Besides, he noticed that Ptolemy’s system
    increased in complexity with time; a good theory
    ought to move from complexity to simplicity, not the
    other way around.
(d) Related to that is Copernicus’s firm belief that a
    theory should not be a pile of ad hoc elements, but
    rather an integrated, coherent, structure. To
    Copernicus, Ptolemy’s theory lacked coherence and
    contained too many ad hoc assumptions. That is why
    he likened it to a monster constructed from incoherent
    limbs. His ideal was a whole governed by necessity.
(e) Copernicus’s Pythagoreanism no doubt drove him to
    favour the sun at the expense of the Earth. It was only
    natural for a Pythagorean like himself to consider the
    sun the centre and source of life and motion.

      What were the achievements and difficulties of
        Copernicus’s new theory?
      Achievements:
     (a) Copernicus’s system was a powerful
          comprehensive alternative to Ptolemy’s system.
          It rivaled the latter in sophistication and was
          superior to it in explanatory power.
     (b) It was more coherent, and more of an integrated
          coherent system, whereby the parts were
          systematically related to each other.
     (c) Its explanation of the retrograde motions of the
          planets was more physical and more natural
          than its Ptolemaic counterpart.
     (d) It provided natural explanations for the ad hoc
          assumptions in Ptolemy’s system.
     (e) It reduced the number of large epicycles via the
          Earth’s motion.
      Difficulties:

     (a) It did not achieve greater accuracy than
         Ptolemy’s system.
     (b) It could not remove all Ptolemaic devices; it had
         to reintroduce them to achieve a sufficient
      degree of accuracy, thus reintroducing in itself
      the complexity it sought to eradicate.
(c) Thus, it was a contradictory system with the
      new features uneasily coexisting with many of
      the old Ptolemaic features.
(d) Copernicus’s theory contradicted both common
      sense and a very well-established theory of
      matter and motion—namely, Aristothe’s. It did
      not offer (and could not possibly have offered)
      an alternative mechanics. Such a mechanics
      awaited the genius of Galileo Galilei, in the first
      third of the 17th century, to crystallize.
(e) Copernicus’ theory necessitated the observation
      of an annual parallax of the fixed stars due to
      Earth’s motion around the sun. No such parallax
      was observed. So either the theory was wrong or
      the stars were at an unimaginably large distance
      from us. Many could not bring themselves to
      accept the idea of such an inflated universe, and,
      thus, had to reject the theory. The Copernicans
      had to contend with such an absurdly large
      universe. The conclusive parallax measurement
      had to await the refinement of astronomical
      instruments, particularly the telescope, as late as
      the middle of the 19th century.
 To start with, the publication of De Revolutionibus
   did not cause a noticeable stir, as it was a very
   technical     and      difficult   work.      However
   Copernicus’s theory had enormous far-reaching
   implications, which could not be ignored for long.
   Both the scholarly and religious authorities were
   soon to realize these volcanic implications, and
   reacted strongly (and, sometimes, violently) to
   What were these implications? In the first place,
   we have the mind-boggling size of the Copernican
        universe, even implying the possibility of an
        infinite universe. And that was precisely the
        implication drawn by the fiery Italian cleric,
        Giordano Bruno, who speculated that the Universe
        was infinite, and populated with an infinite number
        of planetary systems similar to ours, realizing that
        the fixed stars were suns similar to our own sun.
        Secondly, having fixed stars no longer required
        that they be at an identical distance from us. On the
        contrary, it is more likely that they are distributed
        throughout three- dimensional space at varying
        distances from us, just as Bruno speculated.
        Thirdly, the Earth was demoted from a fixed solid
        centre of the universe to a mere moving planet.
        Thus, there are no qualitative differences between
        Heaven and Earth. They are of the same ilk. That
        cast a strong doubt on Aristotle’s cosmology and
        theory of matter and motion. Fourth, the fact that
        events on Earth occurred as though it were fixed
        pointed towards an inertial physics, which
        completely opposed Aristotle’s mechanics.
        These great implications were to be developed
        painstakingly within the next half century,
        primarily by Kepler and Galileo.

5. Tycho Brahe ( 1546-1601):

  Tycho was a great observer of the heavens. It could be
  said that he represented the apex of naked-eye
  observational astronomy. He had a crucial impact on
  Kepler’s work. It could even be said that without his
  discoveries, Kepler would, most likely, not have been
  able to arrive at his lasting astronomical discoveries.

  Tycho’s main contributions to astronomy were:

(a) He made systematic and regular measurements and
observations of celestial objects, particularly Mars. These
measurements were unprecedented in precision. They
were certainly the most accurate and precise to date. He
perfected naked-eye astronomical instruments to such a
pitch the he was able to improve celestial measurements
many times over. That precision was crucial to Kepler,
the great theoretician.
(b) In 1572, a supernova appeared in the sky and lasted
for several weeks. Tycho conducted extensive
measurements on it, and found out that it exhibited no
noticeable parallax. That showed not only that the
supernova occurred beyond the sphere of the Moon, but
that it also belonged to the distant sphere of the stars. This
was a clear refutation of Aristothe’s idea that the heavens
were unchangeable — eternally fixed and unchanging. It
cast a big doubt on Aristotle’s division of the universe
into a corrupt Earth and a pure unchanging Heaven.
(c) Tycho discovered that comets move beyond the
sphere of the moon, and that they cut through a number of
the support solid spheres as they orbit the sun. This
discovery dealt a heavy blow to the Aristotelian notion of
solid spheres and left celestial bodies moving with no
tangible support (almost hung in empty space) leaving the
door open for field inertial physics.
(d) Tycho proposed an alternative model of the Universe
to both Ptolemy’s and Copernicus’s models. His model
was a compromise between the two, which tried to retain
the ―attractive‖ features of both. It tried to adopt
Copernicus’s system after plucking the sting (Earth’s
motion) out of it. It fixed the Earth at the centre of the
Universe, and let the moon, sun and distant stars revolve
around it, with the planets revolving around the sun,
which carried them with it around the Earth.

    Physically speaking, it was a very unacceptable repulsive

 6. William Gilbert ( 1540-1603):

This English physician and physicist started the modern
science of magnetism with his important book, De Magnete
(1600). He showed that the Earth is a huge magnet, and that
all magnets are always bipolar. This book had a tremendous
impact on both Kepler and Galileo. In Kepler’s case, it
triggered the idea of action-at-a-distance or field interactions,
and thus the idea of field physics, which Kepler used very
effectively as a powerful tool in his astronomy.

6. Pythagoreanism:

Two peripheral currents (peripheral in Greek and Arabic
antiquity) were strongly revived in the 16th century in
Renaissance Europe: pythagoreanism and atomic materialism.
Originally, Pythagoreanism stipulated that the Universe, in
its inmost structure and reality, was made of numbers and
geometric figures, that its secret lay in mathematical
relations, and that it was governed by musical ratios, and,
thus, could be viewed as a huge musical instrument played by
the gods and producing the so-called ―music of the spheres‖.
Kepler’s Pythagoreanism, which played a significant role in
his scientific methodology, was of a Christian variety. He
believed that the Universe was a created entity, that God was
a grand mathematician, that He was basically a grand
mathematical reason or mind, that He created the Universe
according to mathematical principles — i.e., the model of
divine creation was mathematical and geometric — and that
He constructed it on musical terms. This Pythagorean trend
and conviction remained with Kepler throughout his life.

 7. Materialism:

Kepler was a true and passionate Pythagorean, but he was
definitely not a materialist. However, he could not escape
totally the impact of atomic materialism. In fact, this form of
materialism played a significant role in the emergence of
modern science. In a sense, it could be said that Bruno,
Galileo, Descartes, Gassendi, Boyle, Hooke, Huygens and
Newton were atomists of sorts. In his work, Kepler was
guided not only by Christian Pythagoreanism. but also by
materialism. The tension in his thought reflects the obvious
contradiction between the Pythagorean and materialist trends
in his methodology. His Pythagoreanism was constantly
checked and counterbalanced by his materialism. He viewed
celestial objects not as mere divine or Pythagorean symbols,
but also as material bodies interacting physically — i.e., he
thought about them in physical terms. Without this strong
physical intuition, he would never have arrived at his great
discoveries, and his work would have belonged wholly to
pre-modern times. The dimension of modernity in his thought
resided in his materialism and physicalism. Also, his strict
adherence to data and precise measurement — i.e., his strict
empiricism — was a clear sign of his materialism. His
creativity was fired by this intense clash between the antique
(pythagoreanism) and the modern (materialism).

   Kepler imbibed his epistemic heritage, with its
    variegated elements, at the University of Tübingen in
    southern Germany. The European Renaissance, with its
    cultural and commercial vitality, had animated European
    universities with the intellectual, artistic, scientific and
    philosophical gems of antiquity, both Greek, Roman and
    Arabic. Without this process of animating European
    university life, a Kepler would not have been possible.
   Kepler became a Copernican at an early stage, during
    his academic sojourn at Tübingen University, under the
   direct influence of his astronomy teacher, Michael
 Whilst still at university, Kepler formed his life-long
   research project out of his passionate involvement in his
   astronomical epistemic heritage. His passionate and
   deep commitment and devotion to this project were
   phenomernal. They sustained his work under very
   adverse conditinas and severe hardship. Only a
   passionately devoted man would work for hours on end
   working along a very uncertain path, doing laborious
   calculations, scrapping them several times, and redoing
   them several times as well.
 Kepler’s research project could be summarized as
(a) Defending the Copernican system and showing its
      superiority to the other systems.
(b) Defending it by resolving its contradictions and
      eradicating the left-overs of the Ptolemaic system.
(c) Explaining the number of the planets; why God
      created six, and only six, planets.
(d) Explaining the distances of the planets from the sun;
      why God created them with such distances.
(e) Finding the relationships between the distances of the
      planets from the sun and their periods (years) — i.e.,
      how the planets are related to each other and how
      they constitute an integrated system.
(f) Reading the mind of God;how and why God created
      the Universe in this specific manner, and on what
      principles. This was the ultimate justification of his
      endeavours, which would confer meaning on his life
      and on human life in general. It was the ultimate
      purpose of his scientific enterprise and of knowledge
      as such.

   How did kepler implement his project? He did that
    through four principal stages, each of which was
    associated with a major work.

    (a) The Mysterium Cosmographicum (1596)
In this (his first major) work, Kepler spelled out his research
project, offered a potent defense of the Copernican system
and proposed a Pythagorean theory of the Universe. It is a
clear embodiment of Kepler’s Christian pythagoreanism. In
it, Kepler looked for the geometric model on which God
based his construction of the universe. Also Kepler had the
peculiar notion that God projected part of his nature onto the
universe. Kepler found the holy trinity reflected in the basic
cosmic arrangement of the Copernican system, with the sun
representing the Father, the fixed stars representing the son
and the intervening space (the planets) representing the Holy

   What was the geometric model proposed by Kepler as the
model of divine creation? In fact, it was the five regular
Platonic solids: the cube, tetrahedron, dodecahedron,
icosahedron, and octahedron. It had been shown b Euclid that
whereas there is an infinite variety of two-dimensional
regular polygons, there can only be five regular solids.

   Kepler fitted these five solids between six spheres. The
cube, for example, would encase a sphere and would be
encased by another sphere; likewise with the other solids. The
five regular solids would, thus, give six spheres and, thus, six
planets only. Also Kepler was able to reproduce, with this
device, the Copernican distances of the planets
approximately. That was a moment of great triumph. With
one blow, he was able to account for the number and
distances of the planets. Kepler would remain a firm believer
in this fantastic Pythagorean scheme till the end of his life.
And in a sense, he was justified in adhering to it. After all, it
could be considered a scientific hypothesis of sorts. Its was a
precisely formulated scientific hypothesis, which could be

  (b) The Astronomia Nova (1609):

This is one of the most revolutionary and significant scientific
works in the history of astronomy. It was indeed the start of
an astronomia nova — a new astronomy. It contained the first
two Kepler laws of planetary motion — the first two correct
laws of physical astronomy. They are counted among the first
correct laws of modern natural science.
This work is a clear embodiment of the field physical
(materialist) element in Kepler’s thought. That does not mean
that Kepler renounced his Christian Pythagoreanism in this
work. Not in the least. It was just that for the moment, the
materialist element was ascendant.
How did Kepler arrive at his two famous laws in his magnum
(a) Kepler started by accepting Tycho’s conclusion that the
Aristotelian solid spheres do not exist. Instead, he introduced
the concept of definite orbits. That was a tremendous
revolutionary step, which introduced a basic element of
classical physics. It also led him to raise a new question.
What moved planets in their definite orbits?
(b) In this book, kepler introduced a new approach to
astronomy, which he termed physical astronomy. His new
approach consisted in viewing the solar system as a physical
system of interacting material bodies. Admittedly, his
conception of interaction was primitive and qualitative. Yet
he was the first astronomer in history to think in this novel
way of the Universe. In particular, he considered the sun to be
the main source of the field force that moved the planets and
animated the solar system. This idea, which could be

considered the starting point of classical field physics, guided
Kepler in this book in all his subsequent moves.
(c) He starting by transferring emphasis from the mean sun
(the centre of the planetary orbits) to the sun itself as a
material body that exerts field forces on the planets. He
refused to relate planetary motions to a mere mathematical
point. Instead, he related them and calculated all related
parameters in reference to the sun. That was a crucial step
(d) In analogy with light, on which Kepler was an expert, he
postulated that the field force of the sun diminishes as the
distance from the sun increases, and vice versa. Since the sun
did not lie at the centre of planetary orbits, this meant that a
planet would move quicker as it approached the sun, and
slower as it receded from the sun. With this notion, Kepler
destroyed the age-old idea that natural motion in the heavens
was uniform motion. According to Kepler, celestial motion
was of necessity non-uniform, because it depended on
distance from the sun. That step and overturning of tables
represented a tremendous upheaval in the history of science.
Kepler reversed a cherished principle, which had lasted for
two millennia.
(e) The next question that faced Kepler was: How does the
speed of the planet vary as it orbits the sun? To answer this
question, Kepler assumed (wrongly, of course) that the field

force of the sun obeyed an inverse law (         ) , rather than

an inverse square law (        2
                                ). Even though the assumption
was wrong, it led to a correct result — namely, Kepler’s
second law of planetary motion. This law states that the
planet sweeps equal areas in equal times. That is , the vector
emanating from the sun towards the planet sweeps equal
areas in equal times. Thus, Kepler’s second law was actually
discovered before the first law, and for circular orbits.

(f) Then, Kepler started enquiring about planetary orbits —
not yet about their shapes, but about their relationship to the
sun and the Ptolemaic devices. He used his physical intuition,
and his physical field model, to evaluate the Ptolemaic
devices. He decided that the epicycles. were unphysical; why
should a planet revolve in uniform circular motion about a
mere moving geometric point? Accordingly, he discarded
epicycles. Instead, he selected and revived Ptolemy’s equant.
That was a truly revolutionary move. After all, for a whole
millennium, Arabic astronomers and, later, Copernicus had
made their utmost to get rid of the equant. They had
considered it the weakest spot in Ptolemy’s system. They had
done that on the basis of Aristotelian physics. But, here we
find an astronomer (Kepler), who had the audacity to reverse
all that, just as he did with the assumption of uniform motion,
and assert the equant on the basis of an embryonic field
physics, a field materialism if you wish. He realized that
Ptolemy’s supposed aberration , his equant, held the key to
solving the problem of planetary motion. So, he constructed a
model of planetary motion consisting of a circular orbit and
an equant.
Kepler embarked on matching his model with Tycho’s data
for Mars, taking into account Earth’s motion. He tried to
specify his model and reconcile it with those data. However,
he was left with a discrepancy of 8'. He followed all sorts of
ways to remove this discrepancy, but with no avail. Of
course, such a tiny discrepancy would have been ignored in
the past, because it would have been contained within the
margin of error of past data. But, not with Tycho’s
unprecedented accurate data. With Tycho’s data, the
discrepancy meant some sort of a failure of Kepler’s circular
model. And tried as he could, no circular orbit would remove
the discrepancy and match Tycho’s data. Of course, Kepler
could have artificially removed it by resorting to epicycles.
But, he would not. He absolutely rejected epicycles on
physical grounds. And he was ready to sacrifice the principle
of circular motion rather than to violate his celestial physics.
He had to surge forward even if this meant violating a
hallowed Platonic principle that had lasted for two whole
millennia. Thus, he first attempted an oval orbit, which, of
course, did not work. However, from the discrepancy, he was
able to deduce a formula, which he later recognized as the
formula of an ellipse. The latter proved to be a perfect match
to Tycho’s data. Thus, Kepler arrived at his first law of
planetary motion, which states that a planet moves around the
sun in an elliptical orbit one of whose foci is occupied by the
sun. The other (empty) focus was actually the equant. Thus,
the circular model with an equant, with which he had started
his quest, was actually a first approximation of the elliptical
orbit. Ptolemy had intuited that when he invented the equant.
So did Kepler when he adopted it despite Arabic astronomers
and Copernicus. It was a truly revolutionary move.

(c) The Harmonice Mundi or Harmony of the World (1619):

When Kepler arrived at his elliptical model on empirical
grounds, he did not consider it a refutation of his regular solid
Pythagorean model enunciated in his Mysterium
Cosmographicum. On the contrary, he felt the need for both
models. He was convinced of the truth of both. He was not
ready to sacrifice either. Thus, he set himself the task of
reconcling the Pythagorean model with the physical empirical
model. That task was fulfilled in his 1619 book entitled the
Harmonice Mundi (Harmony of the World).
     However, in this book, he also set himself another task —
namely, treating the Universe as an integrated physical
system with interlocked elements. He realized that his
treatment of planetary motion in the Astronomia Nova had
been too disconnected, too unsystemic. He had treated each
planet on its own as though it were not part of a solar system
or a universe. Thus, he set himself the task of relating

planetary motions and illuminating the unity of the Universe
and its systemic essence.
    He fulfilled the first task and resolved the contradiction
between the two models by considering the regular solid
model (geometric Pythagorean) correct, but insufficient. He
convinced himself that God, in creating the Universe, was
guided not only by geometry but by other principles as well.
In fact, Kepler convinced himself that God was not only a
super mathematician, but also a super musician. Had the
Universe been static, God would have confined himself to a
geometric model, the regular solid model. However, because
the Universe was dynamic and fall of motion, God did not
confine himself to geometry, but resorted also to the laws of
harmony —to music. Kepler convinced himself that he could
actually explain his elliptical model with a combination of
Pythagorean geometry and Pythagorean music (harmony).

      The second task led him to thoroughly investigate he
relationship between the periods (years) of planets and their
distances form the sun. He had already suggested a simple
relationship in the Mysterium, but he had realized right from
the start that it strongly contradicted the data. This time,
however, he was able to come up with a complex relationship
that matched the data perfectly. Thus, in the Harmonice
Mundi, he announced his third Law of Planetrary Motion, ten
years after announcing his first and second laws. The third
law states that the square of the perid (year) of a planet is
directly proportional to the cube of the mean distance of the
planet from the sun. Fifty years after this announcement,
Newton was to read the law of universal gravitation in this

(d) Epitome to Copernican Astronomy ( 1621):

In the Epitome, Kepler brought his great astronomical work
to a conclusion. He applied his discoveries and insights to all
the planets and to the whole solar system, corrected past
mistakes, removed errors and inconsistencies, tied up loose
ends, and combined and reconciled his various principles and
    Unfortunately, Kepler’s work was not properly received
and appreciated after his death in 1630. In a sense, it was too
anachronistic and too advanced for his age. It was too
Pythagorean, mystical and obscure on the one hand, and too
revolutionary and daring on the other. Only the genius of a
Newton could sift through the Keplerian Corpus, and capture
the Keplerian gem, with which he could construct his
mathematical mechanics.

                           Case II
                         Max Planck

   Three events or processes marked the age in which Max
      Planck was born and moulded:
  (i)      The belated German industrial revolution, which,
           within two decades, turned Germany into the
           foremost industrial power in the world.
  (ii)     German unification, which created a powerful
           imperial state with global ambitions
  (iii)    The emergence of the German university system
           — a novel rigorous system with powerful links to
           both government and industry.

    Max Planck was a distinguished product of this vigorous
  situation. He was solidly grounded in his epistemic
  heritage, and endowed with the necessary philosophical,
  ethical and mathermatical backgrounds and constitutive
  structures. He was devoted to an ideal of rigor and integrity
  — to a vision of an ordered cosmos governed by necessity
  and universality. However, his ultimate loyalty and
  commitment was to the facts in their theoretical unity. He
   always tried to save a well-established theory or belief, but
   he always knew when to withdraw his support. He would
   vigorously defend a theoretical stance. However, his great
   expertise would ultimately drive him to realize that there
   was no way out but to violate it or abandon it. In this
   respect, he was a thorough empiricist, in the deep sense of
   the word.
   Planck was deeply steeped into the solid philosophical,
scientific and mathematical traditions of the German
academia. His rigor, skill and dedication were a witness to
   His original research project consisted in completing
classical physics, tidying it up, clarifying its concepts, and
establishing its deep internal interconnections.
Planck’s Epistemic Heritage
In contrast to Kepler’s epistemic heritage (antique astronomy
+ Copernicus + Tycho), Planck’s epistemic heritage was the
whole of classical physics as it existed in the second half of
the nineteenth century.
Basically, Classical physics consisted of four major
components or theoretical schemes: classical mechanics,
electromagnetic theory, thermodynamics, and statistical
mechanics (+ gas theory + atomic and molecular theory).
1. Classical Mechanics:
The core of classical mechanics is Newtonian mechanics. The
main steps in the formation of classical mechanics may be
summarized as follows:
(a)    Kepler, whose contributions we have already discussed
in detail.
(b) Galileo: His major contributions could be grouped into
three classes:
(i) Methodological:
(1) He initiated a project whereby natural science is a quest
for physical quantities, which are determined with
mathematics and precise measurement, and their inter-
(2) He initiated a new method of thinking of natural
processes, whereby one separates the essential from the
contingent or peripheral, builds models by abstracting the
essential from the peripheral (i.e., idealized models), and
creates experimental situations corresponding to these
models. This method is an essential feature of all scientific
(ii) Astronomical:
    Galileo was the first scientist to study the heavens
    systematically with a telescope. He used these studies
    effectively in devastating antique physics and astronomy,
    supporting the Copernican system, and establishing a new
    tradition of astronomical measurement and observation. He
    discovered with his telescope the materiality of the moon,
    the sun spots, the phases of Venus, the nature of the Milky
    way, four moons of Jupiter and other such astronomical
(iii) Mechanical:
    (1) He was the first to formulate a law of inertia. With
          that, he inaugurated inertial physics. However, his
          formulation was defective, in that he asserted that a
          body naturally maintained its uniform state of motion
          as long as it followed a circular path. He thought
          (wrongly of course) that circular motion entailed no
          acceleration. His concept of inertia lay in between the
          correct law of inertia and the Aristotelian notion of
          natural motion.
    (2) He was the first to formulate the principle of
          (Galilean) relativity, which states that, mechanically,
          all inertial frames of reference are equivalent. That is,
          we cannot detect absolute velocity, and distinguish
          between the state of rest and the state of uniform
          motion, by conducting mechanical experiments in our
          frames of reference.
    (3) Galileo discovered the correct law of free fall-
          namely, that the distance covered by a falling body is
        directly proportional to the square of the time(s=½gt2)
        assuming that air resistance is negligible. His law
        emphasized the contingency of the medium and the
        importance of acceleration, unlike Aristotle’s law of
        free fall, which emphasized the necessity of the
        medium and the importance of velocity.
  (4) Galileo discovered that a projectile on Earth follows a
        parabola, not a semi-circle, as had been thought
        before. This discovery was equivalent to discovering
        the method of analyzing motion into its components.
  (5) Galileo was the first to discover the principle of
        equivalence in its simplest from—namely, that
        inertial and gravitational masses are exactly equal to
        each other. Two bodies dropped in a vacuum would
        reach the ground simultaneously irrespective of their
  In fact, Galileo laid down the basis of the launching pad of
  Newtonian mechanics.

(c) René Descartes (1596-1650)

(1) Descartes discovered analytical geometry (how to
translate algebra into geometry and vice versa), which has
become an indispensable tool in physics.
(2) He was the first to draw a comprehensive materialist
program for explaining natural phenomena. He envisaged
nature as an infinite material system of colliding particles and
ethers. Strict mechanical laws controlled nature.
(3) Descartes envisaged the act of divine creation as a one
instant affair. God created matter and imparted a fixed and
conserved amount of motion (energy, power, force) to it.
Thus, after the instant of creation, the Universe would be self-
sufficient, and run on its own without divine intervention. In
a sense, Descartes marginalized God and made Him
redundant and irrelevant. The Universe is basically a self-
contained sustainable system of matter.

This type of theology led Descartes to a great idea—namely,
that the universe is governed by a system of conservation
laws. This idea has prevailed in physics ever since.
Conservation principles are the basic principles which
constrain and govern events.
However, Descartes’s proposal for the basic conservation law
of nature was erroneous. He erroneously believed that the
fundamental conserved quantity is the product of the volume
of a body with its speed (Vv). That sparked a controversy as
to whether that quantity involved the speed, its square or the
velocity. That controversy was resolved only in the second
half of the 18th century.

(4) Descartes was the first to formulate the law of inertia
correctly. He realized that it involved uniform motion in a
straight line rather than a circular path.

(d) Christian Huygens:
(1) Huygens was the first to formulate correctly the law of
conservation of linear momentum (mass times velocity) and
to explain collisions with it.
(2) He was the first to arrive at the correct mathematical
expression of the centripetal acceleration (v2/r). However, he
wrongly thought that the centripetal force was directed

(e) Isaac Newton:
(1) Newton started by identifying significant discoveries in
mechanics by his predecessors, and separating them from
hazy notions—namely, Galileo’s terrestrial laws of motion
and Kepler’s celestial laws of planetary motion.
(2) With the idea that uniform circular motion was basically a
process of free fall, Newton was able to synthesize Galileo’s
terrestrial laws with Kepler’s celestial laws in a universal
system of mechanics — a system of universal laws—
consisting of the laws of motion and the law of universal
(3) Newton was the first to tie gravitation with matter as such
(irrespective of location and duration( , rather than to Earth
only. With Newton, gravitation has become universal.
(4) Newton invented a new class of laws—universal
mathematized laws, which apply everywhere, at all times, and
under any conditions.
(5) With his great mathematical invention—the calculus—
Newton was able to specify and clarify the concepts of
(6) By formulating the laws of nature in terms of the calculus,
he created a powerful tool of knowledge production out of
them. He turned them into a powerful explanatory and
predictive tool.
(7) He could be considered the founder of theoretical physics.
His theory was the first of its kind. It was the first universal
mathematized scientific theory in history, and it became a
paradigm of subsequent scientific work.
(8) With this powerful tool, Newton was able to explain
quantitatively all of the phenomena of the solar system
hitherto known.

 (f) Leibniz and Maupertuis:
Leibniz highlighted the concept of energy (particularly
kinetic energy), which was absent from Newton’s mechanics.
Maupertuis introduced the notion of action (energy times
time, or momentum times distance) and was able to derive
Newton’s laws from an action principle.
(g) Lagrange and Hamilton: Lagrange and, later, Hamilton
were able to combine all that and generalize the result to a set
of very general partial differential equations and action
integrals. Their system has survived classical physics and
persisted in modern physics — relativity, quantum mechanics
and quantum field theory. In a sense, classical mechanics
reached its climax and final form in Lagrange and Hamilton.

  2. Classical Electrodynamics

   Between 1750 and 1850, most of the laws of electricity,
magnetism and optics were discovered (Coulumb, Oersted,
Ampere, Faraday, Young, Fresnel). In 1864, with the aid of
Faraday’s concept of field, Maxwell was able to express
those laws locally as a set of partial differential equations.
Using arguments of symmetry, he was able to complete these
equations and use them to deduce the existence of
electromagnetic waves and that light is an electromagnetic
wave. Later, Oliver Heaviside put these (Maxwell) equations
in the new familiar vector form.

  3. Thermodynamics

   Thermodynamics proper started in the 1840’s with the
discovery that heat is a form of mechanical motion, followed
by the discovery of the principle of conservation of energy
(Joule, Meyer, Helmholz). That important principle was
considered the first law of thermodynamics.

  Between 1851 and 1865, Rudolf Clausus formulated the
2 Law of thermodynamics in terms of a new concept — the

concept of entropy (change in heat divided by temperature).

   Together, these two fundamental laws constituted the core
of the science of thermodynamics. Clausius and others
succeeded in expressing the basic laws and concepts of
thermodynamics in terms of a set of partial differential
  4. Statistical Mechanics

In the 2nd half of the 19th century, physicists were faced with
three general theoretical systems: classical mechanics,
electromagnetic field theory and thermodynamics.
These systems were deeply related, but looked qualitatively
different. The challenge was to integrate them within a
unified theoretical framework— i.e., to unity them.
     Initially, the prevailing mood was mechanical. Mechanics
was considered the heart of physics, and physicists sought to
base thermodynamics and electromagnetism on mechanics. In
fact, they sought to reduce them to mechanics. The first such
attempt was atomic gas theory, which was revived by Rudolf
Clausius. That attempt consisted in accounting for
macroscopic gas properties in terms of molecular motion.

   Beginning with 1870, Maxwell and, principally,
Boltzmann introduced statistics into molecular and atomic
dynamics. Boltzmann introduced a highly controversial
statistical interpretation of entropy and the 2nd law of
thermodynamics. According to that interpretation, the 2nd law
was no longer an absolute law, but, rather, a probabilistic law.

Max Planck’s Road to the Quantum:

    Planck started with thermodynamics, focusing on the
problem of the 2nd law. In his Ph.D thesis, he dealt with
thermodynamics and based himself on Rudolf Clausius’s
work. He tried to clarify thermodynamics concepts by
reformulating Clausius’s thermodynamics and stating the 2nd
law in unambiguous terms. And, indeed, he came up with a
statement that has become the standard statement of the 2nd
law To him, the 2nd law meant that the entropy of a closed
system tends spontaneously to increase, and can never
     Next, he realized that the 2nd Law differed in kind
(qualitatively) from the 1st Law and the Laws of classical
mechanics. Basically, the difference lies in that the 2nd law is
irreversible, whereas the 1st law and the laws of classical
mechanics are reversible. That is the essence of the
difference. Thus his research project consisted in exploring
and understanding how the 2nd law is related to classical
mechanics— how irreversibility emanated from reversibility.
That is, he wanted to understand the 2nd law in mechanical
terms despite the apparent contradiction. In a sense, he was
looking for ways and means to resolve the contradiction
between thermodynamics and mechanics.

   In the last quarter of the 19th century, there were two
competing views of the 2nd law. The first was the energeticist
view, which denied the existence of atoms, and molecules,
stressed the continuity of matter, believed the 2nd law was an
absolute and universal law, and, thus, considered it totally
unrelated to mechanics.
The 2nd view was Boltzmann’s statistical mechanics, which
believed in atoms and molecules, interpreted entropy as a
measure of disorder of a molecular system, and considered
the 2nd law a probabilistic law rather that an absolute laws,
based on mechanics. Planck subscribed to neither view. He
considered the 2nd law absolute rather than probabilistic.
However, he believed it was deeply related to mechanics and
was ultimately based on it and derivable from it. However, he
rejected Boltzmann’s statistical atomistic approach, because
he simply rejected atomism. Boltzmann’s probabilistic
interpretation of the 2nd law was anathema to Planck. He
wanted to retain both its absolute character and its mechanical
base. He thought this could be done by basing
thermodynamics on the mechanics of continuous media,
rather than molecular systems. This consideration ultimately
led him to black-body radiation. In fact, he interpreted
Maxwell’s equations of electromagnetism as mechanical
equations of a continuous medium (the ether). Thus, he
considered electromagnetic systems perfect systems for
achieving his project of deriving irreversibility from
Black-Body Radiation

   Black-body radiation is the radiation emitted and absorbed
by a perfect black body. The latter absorbs all the radiation
that falls on its surface, and its surface is a perfect emitter of
   The best example of black-body radiation is the radiation
contained in a cavity under conditions of thermal equilibrium
(at constant temperature). The challenge that faced physicists
at the end of the 19th century was to discover and explain how
energy was distributed on the various colors comprising
black-body radiation—i.e., how radiation energy density was
related to frequency.

Planck’s Quantum of Action:

     To achieve his research project, Planck embarked on
working on cavity radiation in 1894. He applied his
thermodynamic definitions and Maxwell eqs. to cavity
radiation. He thought of the cavity system as a set of field
(not corpuscular) resonators interacting with the radiation
     Whilst conducting this theoretical work, an experimental
group in Berlin was conducting experimental work on cavity
radiation and obtaining important data.
The most promising empirical relation between energy
density and frequency at that time was Wien’s law. Planck set
out to derive it using his thermodynamic apparatus of
concepts and equations. In his endeavour to explain the 2nd
law and derive Wien’s law, he was to realize that his method
was leading him to a dead end, and that he had to introduce
Boltzmann’s statistical mechanics in his analysis. That
recognition of failure occurred around 1898.

   Eventually, Planck arrived at a relatively satisfactory
derivation of Wien’s law. However, fresh experimental
results coming from the Berlin group were soon to reveal the
limitations and inaccuracies of Wien’s law. Accordingly,
Planck used thermodynamic arguments to modify his basic
equations, and to surmise a new radiation law of which
Wien’s law was an approximation. This new law agreed
perfectly with all know experimental results, and has
withstood the test ever since.

   However, Planck was not wholly satisfied with his great
discovery. He aspired to ground it on firmer theoretical
grounds—i.e. to derive it from more fundamental principles.
To him, theoretical grounding was as important as
experimental corroboration.

   This time he openly resorted to Boltzmann’s so-called
combinatorial method. Following Boltzmann, he parceled the
radiation energy absorbed and emitted by oscillators into
discrete elements, and studied their statistical distribution on
oscillators. He found out that this method would lead to his
correct radiation equation if he assumed that these elements
were physically real (i.e., not mere mathematical ploys), and
that each is proportional to frequency:

    E = hf           E = energy
                     f = radiation frequency
                     h = so-called Planck Constant.

   That is, instead of letting the elements approach zero, and a
summation approach an integral, which would have led to an
absurd result, he retained the summation and kept the
elements finite and dependent on frequency only.

   That assumption marked the birth of quantum theory, and
was subsequently called the quantum hypothesis. It
eventually led to one of the greatest upheavals in the history
of knowledge—quantum mechanics and quantum field


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