The Emergence of Science
The Copernican Revolution
On the morning of
February 19, 1600 an event that was a harbinger of things to come ushered in
the new century. Several hooded members of a group known as the Company of
Mercy and Pity took a young man from the Nona
Tower, a secular prison in Rome, separate from the "holy" one of Castel
Sant'Angelo, put him in a wooden wagon and hauled him over to the Campo dei
Fiori, the Square
of Flowers. Along the way, Jesuit and Dominican priests
mumbled their imprecations to the young man in an attempt to elicit a last
minute recantation of his beliefs, lest he forever be damned as a heretic. It would have been difficult for him to
respond, even if he desired to do so in the affirmative --- which he did not
--- since his jaw had been clamped shut with an iron gag, one long spike that
pierced his tongue and another that had been driven through his lower jaw and
upper palate. Perhaps the priests,
afraid of the impotence of their supplications to get the young man to recant,
were even more afraid that he would use his tongue to incite the crowd of
onlookers along the way
As the wagon passed, bystanders asked who the young man was. "A Lutheran", the priests replied,
in those days a synonym for anyone branded a heretic. The young man was no Lutheran; he was a
Dominican himself, a member of the group that would soon execute him for his
beliefs. He had been charged with eight
offenses. Upon several he had
wavered. On one, he stood firm. He believed in the infinity of the cosmos and
the plurality of habitable worlds. He
believed that God was infinite and so was his handiwork. He believed that the Earth traveled around
the Sun and that there were many systems like it that contained other living
creations of God.
The young man was stripped naked and a crucifix pressed to his
face. He turned away, in obvious
disgust., sending shrieks through the crowd.
Then, Giordano Bruno was burned, the priests chanting their litanies
while the crowd looked on.
3.1 Nicolaus Copernicus (1473 - 1543)
"This fool wishes to reverse the entire science of astronomy;
but sacred scripture tells us that Joshua commanded the sun to stand still, and
not the earth!"
Martin
Luther - Table Talk
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Insert Figure
of Copernicus
The edifice of modern science
rests on precise measurements. There is
a Latin phrase that describes the essence of the legal system that we have
adopted to govern ourselves --- Leges sine moribus sunt vanae, or, laws
without morals are vain. A similar
phrase could be coined to define the essence of science --- Ratio sine
observatione est vana, or, theory without observation is vain. Unless firmly founded on detailed
observation, any theory fabricated to explain some phenomenon is likely to be
an illusion at best and misleading at worst, particularly if an ill-founded and
incorrect theory is adopted and perpetuated as authoritative dogma.
How and when did modern
science begin? We can make a strong case
that it started with the act of taking precise measurement and using those
measurements to discriminate in an objective way between conflicting
theories. The protagonists of the
geocentric model of the universe could argue persuasively that the Earth was
immobile, bringing all their powers of rhetoric into the fray; they could
appeal to authority; they could bludgeon their opponents with clubs, but
rhetoric alone was doomed to fail as the final arbiter of nature’s truth. Once
humans began to make accurate observations of phenomena and precise
measurements of their characteristics, using unbiased instruments, it became
increasingly difficult to deny that the Earth was in motion around the Sun, for
to do so, forced the dissenter into the unenviable position of directly
demonstrating that a precise, unbiased measurement, made by a careful observer,
was wrong. Like the basis of our legal
system, the burden of proof came to rest squarely upon the shoulders of the
accuser, not the accused. Attack a
theory if you will, but be prepared to back up your attack with observation and
measurement to support your point of view --- not just rhetoric, or appeal to
authority, or, perish the thought, the strength of your fist or the power of
your military.
A strong argument can be made
that the modern scientific era began with the Copernican hypothesis,
published in his de Revolutionibus in 1543, and the subsequent attempts
to prove it true or false. Copernicus
stated that Sun was the center of the solar system and the Earth was not,
relegating it too just one of many other, similar worlds. If true, this
heretical proposition would topple the Earth from its exalted position as God’s chosen child, occupying the most special place among
all his subjects --- a single, isolated unique world, positioned at the center
of the universe, upon which a lone human species resided, paying homage to its
creator. Sir Isaac Newton demonstrated
144 years later in 1687 in the Principia, that this hypothesis was true
beyond all reasonable doubt. Newton’s theory describes completely the motion of any
macroscopic object regardless of its location in the universe, in terms of
several, simple, universal laws.
Application of these laws leads to the inescapable conclusion that the
Sun and all the planets could not orbit the Earth. After Newton,
no one could refute the notion that the Earth orbited the Sun in any rational
way. The recognition that the Earth was
not the center of the universe was perhaps the most profound discovery ever
made by the human species. Humankind no
longer occupied a special place in the cosmos and this realization opened up
the possibility of the existence of similar species inhabiting other, similar
worlds. The gradual awakening of this
idea threatened the Aristotelian underpinnings upon which the Church had
securely rested for centuries. It is no wonder that it so bitterly opposed this
newly emerging discipline whose embarrassing discoveries continued to erode its
foundations. As humankind entered the 16th
century, the Church regulated its behavior and dictated its views regarding all
things including the behavior of the natural world. By the end of the 17th century,
science, or natural philosophy as it was then known, had effectively eliminated
the Church as the final arbiter of questions concerning the way nature
worked. Its pronouncements on such
matters were now given no more credence than were the claims of mystics or
believers in the supernatural. The
natural world now fell under the domain of science.
It is astonishing how few
people are aware of the sequence of events that led to the emergence of the
science and technology upon which modern society is based. Even those who are supposedly educated have
more than likely never heard of Kepler's laws of planetary motion. The events leading up to their discovery and
proceeding from them had such a profound impact upon the development of modern
civilization and the emergence of the human species as an operationally intelligent
one, that awareness of them ought to be more widespread. We argued in the
previous chapter that many factors had to be established by a society in order
to set a proper stage for the beginnings of science. And even if the right factors are set in
place, modern science, as we now practice it, still might not emerge. Indeed, after almost starting up with the
Greeks of the 6th through the 2nd century B.C.E.,
continued development halted almost dead in its tracks and did not kick into
full gear again until the Renaissance. And it might not have happened then were
it not for the occurrence of many serendipitous events.
Nowhere is it written down
that the emergence of modern science among a potentially “intelligent species”
is inevitable, even though most people, including many, if not most,
scientists, think that it is. We ought
to examine this premise closely, for if it is not inevitable, then the
likelihood that other life forms that emerge in some other favored spot in the universe
will discover science and in doing so, become operationally intelligent, might
be much smaller than we think. The only
way we can even begin to assess this likelihood is to attempt to discover those
factors that led to the emergence of the modern scientific enterprise among our
own species. In order to make such an
assessment, we turn the clock back about three centuries to address a problem
that re-emerged after it had seemingly died with a few exceptional Greeks
almost two millennia earlier.
The problem centered on the
observed motion of the planets. Their
apparent motion is more complicated than meets the eye because the observations
are made from a moving frame of reference --- the Earth. Because of this complication, each planet
periodically executes retrograde motion --- it appears to reverse its
normal course of progression through the panoply of stars that serve as a
stationary backdrop. This problem had
plagued ancient astronomers for centuries, and to their great credit, they
invented incredibly ingenious geometrical schemes to deal with it. But from the moment Plato charged astronomers
with the mandate to only use uniform circular motion to “¼save the appearances” astronomers got stuck in the quagmire of epicycles and
deferents. Copernicus realized that
retrograde motion could be explained much more simply as an effect of the
relative motion between the planets, each one in orbit around a centrally
located Sun. The retrograde motion is an
apparent motion that occurs when one planet overtakes another. The effect is similar to how a driver of a
vehicle moving at 60 mph perceives another one traveling at 55 mph in the lane
to his right. As the faster moving vehicle
passes the slower moving vehicle, it appears to be moving backwards.
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Insert Figure
of retrograde motion from heliocentric point of view
And so Copernicus broke with
the past by advancing a heliocentric model of the solar system, but he
did not make a clean break. In fact,
Copernicus was a confirmed Platonist who, like Plato, firmly believed that the
planets executed perfectly uniform circular motion! It was Ptolemy, one of the staunchest
proponents of a geocentric model of the solar system, who actually
defied Plato by adopting the equant in his unwieldy system in order to force
the planets to travel at varying speeds along perfect circular paths. Thus,
Copernicus’ breakthrough was actually something of a throwback C he managed to re-institute Plato’s perfect, uniform circular motion requirement, albeit
in circles that ran around the Sun instead of the Earth. His scheme of motion, though in principle
simpler than that of Ptolemy’s, was still
riddled with the ferris-wheel-like structure of epicycle turning upon
cycle. He had no choice but to adopt
such a cumbersome structure if he was to obtain some degree of agreement
between the observed and predicted positions of the planets --- and insist that
the planets obey the Platonic dictate.
In all, his heavenly model required some forty-eight epicycles, eight
more than Ptolemy's!
Why didn't Copernicus get any
further than he did? His firm belief in
Platonic dogma certainly constrained his thought process. But, had he access to the large number of
precisely measured planetary positions and times when the planets were at those
positions that Kepler did, he might have freed himself of the Platonic
shackles. When forced to do so by
overwhelming experimental evidence, subsequent thinkers, Kepler among them,
managed to acknowledge the incorrectness of their pet notions. But Copernicus was never constrained by
precise measurements. Indeed, in his own
book, de Revolutionibus, he listed only twenty-seven observations of his
own that he made during the fifty-year course of his scientific lifetime. Most of the observations that he used in
putting together his revolutionary heliocentric proposition came from the Greek
astronomers Hipparchus, Ptolemy and others, made more than a thousand years
earlier! The degree of precision that
they had achieved was unknown. It was
taken for granted that they were as good as they could be. Accuracy was necessary if one wanted to
construct calendars and navigational charts.
But apart from these considerations, the necessity for precision, or an
assessment of the degree of precision, of a particular observation, was not
appreciated by anyone. This attitude,
incomprehensible to a modern mind trained in science, is due in no small part
to that great scientific authority, Aristotle, who emphasized qualitative
rather than quantitative measurement.
Given such an operational paradigm, only a scientific deviant would be
interested in precision for its own sake.
Besides, a geometry of the heavens, consisting only of cycles and
epicycles, made of perfect circles, did not require a great number of
observational data points for its construction.
A circle needs only three points on its circumference or a center and a
radius for it to be completely specified.
Thus, once Copernicus constructed an adequate theory of planetary motion
out of perfect circles based upon a few, fairly qualitative and inexactly
measured data points, no more needed to be done. It was good enough --- and he had
accomplished his goal of resurrecting Plato's dogma of perfect, uniform
circular motion.
The Copernican model was
presented in his treatise, de Revolutionibus (Concerning the Revolutions)
in 1543. The monk, Osiander, who had
been charged with publishing the book later bastardized the title to the more
palatable de Revolutionibus Orbium Coelestium (Concerning the Revolutions of
the Heavenly Spheres). Osiander was
most probably motivated towards this presumption, not to improve Copernicus'
choice of wording, but to appease church authorities, in hopes that they would
not block the publication of what was then a most heretical viewpoint. Copernicus never saw a copy of his book until
the day of his death in 1543. This
treatise, published in Latin, slowly began to percolate through a society, long
mentally dormant, just emerging from the deep, intellectual sleep of the dark
ages. The book was never a best seller;
it was imminently unreadable, but even so, its central premise would eventually
become known to a number of great thinkers. It would infuse them with a new
mode of thought that would dethrone the world and humanity along with in it.
3.2 Tycho Brahe (1546 - 1601)
"but for us, who, by divine kindness were given an accurate
observer such as Tycho de Brahe, for us it is fitting that we should
acknowledge this divine gift and put it to use . . . Henceforth, I shall lead
the way toward that goal according to my own ideas. For, if I had believed that
we could ignore those eight minutes, I would have patched up my hypothesis
accordingly. But since it was not
permissible to ignore them, those eight minutes point the road to a complete
reformation of astronomy."
Kepler
- The New Astronomy
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Insert Figure
of Tycho
Never was there a more
unlikely pair, Johannes Kepler, a pauper, from a family of misfits; Tycho
Brahe, a nobleman, from pure Danish stock.
Tycho's father, the Governor of Helsingborg Castle had promised his
childless brother, Tycho’s uncle Joergen, a country squire and vice-admiral,
that if he had a son, uncle Joergen could adopt him and raise him as his own.
The Governor reneged on his promise and uncle Joergen eventually stole the
first-born, Tycho. The Governor
eventually cooled down and gave his blessing to Joergen, knowing that Tycho
would eventually inherit much of Joergen's fortune. This came to pass somewhat sooner than
expected. Uncle Joergen had an accident
that turned out to be lethal, returning from battle with the Swedes in the
coterie of King Frederick II. The King
fell into the water while passing over a bridge connecting Copenhagen to the royal castle. Uncle Joergen jumped in and saved the King
but died of pneumonia soon after.
Being kidnapped and raised by
the irascible vice-admiral certainly helped turn Tycho into one of history's
most interesting eccentrics. This was
evidenced even in his physical appearance.
Tycho was born with a silver spoon in his mouth and soon had a nose to
match. As a young man he fought a sword
duel with a fellow student to resolve an argument over who was the better
mathematician of the two. Tycho might
have won the verbal argument but lost the physical one when a good chunk of his
nose was sliced off by his agile opponent.
He replaced the lost piece with one fashioned from an alloy of silver
and gold. In portraits, the metal nose
stands out in sharp contrast with the fleshy features of Tycho's large,
baldhead, bulbous eyes, and curled handlebar mustache.
At the end of Tycho’s first year at the University of Copenhagen,
he saw something that changed the course of his whole life: a partial eclipse
of the Sun. It had been announced
beforehand and it struck the young Tycho as nothing short of miraculous that
men could predict so accurately the future occurrence of celestial events. He began to neglect his education in the
classics and instead, committed himself fully to the study of astronomy.
The predictability of
astronomy intrigued him. It stood in complete contrast with the
unpredictability of his life in the carnival that was the Brahe family. In
spite of their disapproval, he immersed himself in the design and construction
of massive astronomical instruments. One of these was a huge quadrant,
thirty-eight feet in diameter and turned by four handles.
$
Insert Figure
of Tycho’s quadrant
Tycho used this instrument
and others of his own design to mark the locations of all the planets, the
Moon, stars, comets and any other celestial body that could be seen by naked
eye but with an accuracy that far surpassed that obtained by any of his
predecessors. This painstaking work,
carried out over the duration of his life, established his place in history as
the father of modern observational astronomy.
His devotion to precision --- making measurements accurate to about a
minute of arc, the very limit of accuracy obtainable with the naked human eye
--- was totally original. This was his legacy.
His work demonstrated conclusively that the scientific method was only
of value when firmly based on precise and continual observation. No wonder Kepler called him the Phoenix of
Astronomy.
You might now have some
inkling regarding the intent of Kepler's comment in the quote at the beginning
of this section. Tycho was fifty-three
years old when Kepler, then twenty-nine, finally managed to meet him and gain
his patronage. By then Tycho had
achieved world renown and Kepler had been pining for years to gain access to
his precise data on planetary positions and distances. Had Tycho never made his measurements, Kepler
probably would have accepted his solution to the planetary orbits based on the
nesting of the five perfect Platonic solids that he had published in his book, Mysterium
Cosmographicum. Discrepancies
between his model and the meager data of Copernicus and Ptolemy never would
have fazed him. But this lackadaisical
attitude toward observation was on downward ebb at this time in history. Ocean-going navigators were beginning to
demand increased precision in compasses and clocks. Such a climate led to a
newfound respect for hard fact and exact measurement. The centuries old debate between Ptolemy and
Copernicus would no longer be settled by religious dogma or rhetoric alone; no
--- the argument would be decided in the new arena of precision experiment and
in that arena, Tycho reigned supreme.
Tycho's quest for precision was
partly stimulated by his desire to check the validity of the Copernican model
(in which he never believed). Mostly
though, achieving precision for its own sake was his passion. His catharsis had come with the solar eclipse
and the dumbfounding realization that the occurrence of astronomical events
could be predicted ahead of time. A
second occurrence of an opposite kind solidified the future course of his life;
the upsetting experience that an astronomical prediction could be in great
error. On 17-Aug-1563 he observed a
conjunction of Saturn and Jupiter in which they were so close together that
they appeared to be one. He discovered
that the Alphonsine tables (the currently accepted table of planetary positions
based on the Ptolemaic calendar) erred by a month in this prediction and the
more recently generated Copernican tables by several days. Tycho thought this situation
intolerable. His noble family
disapproved of lowly stargazers and this failure confirmed their opinion of
their worthlessness. Tycho resolved that
from this point on, this particular stargazer would show them how the job
should be done
A third heavenly event
established Tycho's fame as the leading astronomer of his time: on the night of
11-Nov-1572, Tycho gazed up at the evening sky and saw the most miraculous
wonder there that he or any other human being on Earth could ever be privileged
to see --- a new star --- brighter than Venus, in a place where no star
had been before, a little northwest of the constellation, Cassiopeia, the
familiar "W" in the sky near the Big Dipper. He could not believe his eyes. It was so bright that people with sharp eyes
could see it during the day! And there
it stayed, never changing its position relative to the other “fixed” stars, but
gradually dimming in the days to come until it finally faded from view some
eighteen months later. Other astronomers
saw it too. But no “western world”
observers had ever reported such a thing since the year 125 B.C.E. when the
great Greek astronomer, Hipparchus had seen a similar, new star in the sky.
The importance of this event
cannot be dismissed. Indeed, it might
have been a miracle, but though the new star resided in heaven, church
hierarchy would soon wish to consign it to hell. Its existence there contradicted all
religious and scientific dogma of the day --- Christian, Aristotelian and
Platonic. Supposedly, the eighth, or
heavenly, sphere contained only the fixed stars and it was perfect and
immutable from the day of divine creation until eternity. All change, generation, growth and decay were
confined to the innermost, sub-lunar sphere that contained the Earth. This “scientific” dogma originated with the
Greeks back in the third century B.C.E. and was advocated by Aristotle. Ultimately, Aristotle’s view was refined,
made consistent with the Christian religion and then adopted by it so that it
became a heresy to speak against it.
Tycho's reputation grew to
such an extent following his discovery of the new star of 1572 that King
Frederick II of Denmark,
whose life had been saved by the late Uncle Joergen and had already promised
Tycho a prominent position, began to fear that the brilliant young man might leave
to take up residence somewhere other than Denmark. King Frederick made Tycho an offer he
couldn't refuse --- his own island, Hveen, in the Sund between Copenhagen
and Elsinore castle, over two thousand acres
rising on magnificent white cliffs out of the sea. It was here that Tycho built his famed
observatory, Uraniborg, the Castle in the Heavens, at Denmark's
expense. He was given an annual stipend
and various grants that would make his income one of the highest in Denmark. Tycho deliberated for a week and then
accepted the island and the fortune that came with it.
Comparing Tycho's work at
Uraniborg to anything that came before is like comparing a series of still
shots in a slide show to the continuous image of a VCR display. His survey of the solar system was remarkable
for its precision and quantity. He measured planetary positions to an accuracy
of arc minutes, at thousands of orbital points.
He located the positions in the sky of a thousand stars. On the basis of his measurements, he built a
cosmology, a model of the solar system, that has been given little credence by
most historians of science, but in fact closely represents the motion of the
heavens as seen from the moving Earth!
Tycho believed, for good reason, that the Earth was motionless. He could
not detect a parallax for one single star out of the thousand whose positions
he had determined with unprecedented accuracy.
This implied that either the Earth was at rest or the stars were
unimaginably far away. His mind rebelled
at the latter conclusion even though it proved to be the correct one!
$
Insert Figure
and explanation of Tycho’s model
of the solar system
3.3 Johannes Kepler (1571 - 1630)
My
aim is to show that the heavenly machine is not a kind of divine, live being,
but a kind of clockwork ... insofar as nearly all the manifold motions are
caused by a most simple, magnetic, and material force, just as all motions of
the clock are caused by a simple weight.
And I also show how these causes are to be given numerical and geometrical
expression.
Kepler,
Letter to Herwert von Hohenberg, Catholic Chancellor of Bavaria, 10-Feb-1605
$
Insert Figure
of Kepler
The stage was set for
Kepler. The mystery of the way the solar
system worked lay hidden in Tycho's tables of planetary data, but Tycho lacked
both the desire and the ability to figure it out. He needed a Kepler --- and he knew it. Fate brought them together at precisely the
right moment.
King Frederick, Tycho's
patron, was fond of drink, a pleasure he engaged in so excessively that
ultimately it claimed his life in 1588.
Tycho treated the young successor, King Christian IV, with such
condescension and arrogance that Christian terminated Tycho's support and threw
him out in 1597. After wandering through
much of Europe, the Tychonic caravan touched down in Prague, having secured the position of Imperial
Mathematicus to Emperor Rudolph II. As part of the package deal, Rudolph
gave Tycho a substantial income and a castle about 22 miles northeast of Prague. Kepler would become his protégé six months
later.
This sequence of events
turned out to be one of the most fortuitous of the scientific renaissance. If the
vituperative Tycho hadn't been so overbearing and gotten himself thrown out of Denmark so unceremoniously --- if he hadn't
settled near Prague
just when he did, Kepler never would have been able to join Tycho. The expense of getting to Denmark was far too great for the poor Kepler,
but Prague was
not too distant from Gratz where Kepler was Mathematicus and it was just within
his reach. How fortunate that Kepler
made it there and that they were able to collaborate --- for the eighteen
months Tycho had left to live.
Tycho gave Kepler the task of
working out the orbit of Mars, a problem that had confounded all previous
attempts using circles or combinations thereof.
This choice of task for Kepler proved to be extremely fortunate, since
Mars alone held the key to a long-concealed truth about planetary orbits ---
they were elliptical, not circular --- and of all the planets visible to the
naked eye, the
elliptical orbit of Mars is the most eccentric. Furthermore, Mars was the
closest outer planet, so it made many revolutions about the Sun during a
person's lifetime. Thus, Tycho had
amassed an enormous amount of observational data on Mars. If Tycho had given
Kepler any other planet to study, Kepler might well have prematurely concluded
that circles could be made to work and the true laws of planetary motion might
have remained buried forever in Tycho's columns of numbers.
Tycho knew that he alone
could never disentangle the true shape of the Martian orbit from the erratic
wanderings that his own extensive and precise data had uncovered. By now he was too old. He lacked the
requisite imagination and mathematical tools.
He knew that Kepler was the only person who could work his way through
it and that probably nothing would get in his way from doing so. It galled him that it would be this common
upstart who would reap the fruit of his own lifelong work and so he gave Kepler
his data on Mars, grudgingly and slowly.
The frustrated Kepler could work on the problem only in brief snatches.
Another of those fortuitous
chance events that seemed to keep cropping up precipitated the end of their
tumultuous relationship. On
13-Oct-1601, Tycho was a dinner guest at the illustrious Baron Rosenberg's
castle in Prague. Although Tycho was used to drinking large
quantities of wine and could handle it well, during the course of the banquet
he held back his water beyond all demands of propriety. When he returned home, he could hardly
urinate at all. He developed a fever,
gradually becoming delirious, and still the critical passage remained
blocked. Even so, he continued to eat,
compounding his difficulty. He expired,
from uremia on 24-Oct-1601, his last words a refrain he repeated again and
again:
Let
me not seem to have lived in vain.
He had not.
On 6-Nov-1601, Kepler was appointed Tycho's successor to the post of
Imperial Mathematicus. He held it until
the death of Emperor Rudolph in 1612. By then he had worked out the laws of
motion of the heavenly objects that constituted the universe as it was then
known.
Another fortuitous event
occurred that further cleared the way for Kepler, this one initiated by Kepler
himself; he pilfered Tycho's data --- an act he calmly admitted in a
subsequent letter to Heyden, an English colleague and admirer:
…
I confess that when Tycho died, I quickly took advantage of the absence, or
lack of circumspection, of the heirs, by taking the observations under my care,
or perhaps usurping them...
Clearly, Kepler had been
driven to do such a thing by a long-standing lust for Tycho's treasured
data. It is a good thing that he did it
when he did, for soon thereafter, Junker Tengnagel, whose claim to Tycho's
estate had been staked by illicitly impregnating Tycho's daughter, Elisabeth,
whom he subsequently married, got hold of all Tycho's remaining possessions. He sold the observations and instruments to
the Emperor, but couldn't deliver the observations since Kepler now had them in
his possession. Subsequent litigation slowed down the publication of Kepler's
treatise, the New Astronomy, but it didn't prevent his work.
The critical point regarding
Kepler's struggle with Mars is that he let the data speak for itself. Somehow, he was able to rid himself of all
prejudice regarding the motion of the
planets. No one before him had been able
to do that. From the early Greeks up
through Copernicus, the Platonic dogma in one form or another held sway over
ideas of how the heavens worked.
Planetary orbits were manifestations of Platonic “true forms” that could only be unmasked by pure thought --- most
certainly not by observation. They moved
at uniform speed in perfect circles because Plato said circles were perfect
forms, whose true perfection could only be experienced with the mind's eye, not
the ones residing above one’s nose. Aristotle put the planets on rotating,
immutable spheres, whose perfection could never be attained by sub-lunar
residents. The Christian church readily
embraced the Aristotelian position as support for its own brand of dogma,
namely that the perfect, omnipotent one true god resided in the only perfect
place available, the outermost, heavenly, or eighth Aristotelian sphere. No
thinker of antiquity had ever been able to discard such confining thought and
pressure. Somehow, Kepler managed to do
it. He believed that the planets orbited
the Sun, dictated to do so --- not by a “prime mover” --- but somehow by
natural causes. Kepler believed that the Sun created a force on the planets
that caused them to move the way they did. Kepler never figured out a way to
describe that force in a mathematical sense nor did he understand how objects
responded to forces. That discovery awaited the brilliant Newton, but Kepler was way ahead of his time
in attributing natural causes to the behavior of physical systems. This is why he followed Tycho's data wherever
it took him. He knew that ultimately it
would tell him the truth, but he had to look at the problem with a mind freed
from the dogma of the past.
He quickly threw out
combinations of circles. The idea that a
planet moved in epicycles made no physical sense to him. He tried to fit an eccentric circle to the
data. Hipparchus had made that work
fifteen centuries earlier but not with Tycho's data. Kepler could not make it work, but he came very
close. In fact, Kepler was able to get
an eccentric circle to fit ten different observations that Tycho had made of
Mars with no discrepancy between observed and calculated positions in excess of
2 arc minutes. Kepler was ecstatic but
to test his model, he picked out two more observations of Mars from Tycho's
storehouse of data and, to his great despair, discovered that they did not
fit. When he tried to adjust his
eccentric circle by including these observations, a single observation remained
steadfast in its disagreement by eight minutes of arc. Eight damned minutes of arc! What to do?
Kepler could not ignore it.
Probably Ptolemy and Copernicus would have thrown out the offending
observation. Not Kepler. He knew that Tycho’s observation was good to within ±1 arc minute.
Tycho couldn't have been off by eight minutes of arc. So Kepler threw out his hypothesis and
in doing so, launched a new era in the history of scientific investigation ---
no longer would good observational data in conflict with one’s pet hypothesis
play second fiddle to it! Hypothesis and
observation would become equal partners in the search for a true theory.
Still, Kepler had become
convinced that the shape of the orbit of Mars looked something like a flattened
circle but he didn't know what mathematical formula described that shape.
Eventually, Kepler stumbled upon a mathematical relationship between several
characteristics of the sought-after curve that defined an ellipse (see Focus
Box 1) and he was able to fit it to all of Tycho's data points for Mars to
within an accuracy of ±1 arc minute. This discovery became Kepler’s first law
of planetary motion.
------------------------------------------------------------------------------------------------------------------
Focus Box 1 - The Ellipse
An ellipse is the locus of
points such that the sum of their distances from each focus is a constant. We can use this property to draw an ellipse:
first, place two thumbtacks on a piece of paper. These are the two foci. Then link them with a loop of string whose total
length is longer than twice the distance between the two thumbtacks. Place a pen in the loop of string and pull
the loop taut. Move the pen around the
thumbtacks, keeping the string taut. The
pen will trace out an ellipse. You can
change the shape of the ellipse by changing the length of the string or the
distance between the two thumbtacks. The
major axis is the line connecting two points on the ellipse that passes through
the two thumbtacks, or foci. The
semi-major axis is half this distance. Since
planetary elliptical orbits are very nearly circles, the two foci for such
ellipses are extremely close together.
------------------------------------------------------------------------------------------------
Kepler actually discovered
and used what would become his second law of planetary motion (the planets
sweep out equal areas in equal times; see Focus Box 2) while in the process of
discovering the first law and he published these two laws in his magnum opus,
the Astronomia Nova, in 1609. Its
full title was:
A
New Astronomy Based on Causation
or A Physics of the Sky
derived from Investigations of the
Motions
of the Star Mars
Founded on Observations of The Noble
Tycho Brahe
------------------------------------------------------------------------------------------------------------------
Focus Box 2 - Kepler’s 2nd Law - Equal Areas
Note the two
pie-shaped segments in the ellipse in the figure. Each represents the area swept out by a
planet in a fixed period of time (say one month going from 1 ® 2
and also 1 month going from 3 ® 4) as it travels around the Sun (at one of the foci of
the ellipse). Kepler's second law states
that these two areas are equal. As the
planet moves around the Sun from position 1 to 2 towards perihelion, it speeds
up as it gets a little closer to the Sun, sweeping out the short, squat sector. As it moves around the Sun, from position 3
to 4 towards aphelion, it slows down as it moves further away, sweeping out the
long skinny sector. The speed must vary
in a precise, mathematical way if it is to preserve the equality of these
areas.
----------------------------------------------------------------------------------------------------
His final discovery, the
third law of planetary motion, took him a total of twenty-two years from the
time he first laid his hands on Tycho's data. It was first published in Harmonice Mundi,
or Harmony of the Worlds, in 1618.
This book was the climax of his lifelong obsession. Kepler was a true Pythagorean. Harmonice Mundi was Kepler's attempt
to expose the ultimate secret of the universe via a complete synthesis of
geometry, music, astrology, astronomy and epistemology. Kepler completely regressed to Platonic dogma
in this endeavor. He was the last
scientist to do so. He was ecstatic
about this final discovery, the so-called “harmonic law,@ for he was convinced that it vindicated his firm
belief that the planets played musical notes as they danced around the
Sun. For example, when Saturn is at
aphelion (furthest from the Sun) it moves at the rate of 106 arc seconds per
day and when it is nearest at perihelion, at 135 arc seconds per day. Thus, it "plays notes" in the ratio
of 4:5 --- a major third in musical terms.
The ratio of Jupiter's slowest to its fastest motion is a minor third,
and so on for the rest of the planets.
If the human ear hadn't been overwhelmed by the cacophony that filled
the gross, sub-lunar sphere, we could hear the harmony of the outer spheres and
the laws of planetary motion would have been obvious to us since the beginning
of time. Shades of Pythagoras.
Kepler's third law of
planetary motion, which relates the distance of a planet from the Sun to
the time it takes the planet to go around it, lay buried in this book, a
beautiful pearl of wisdom hidden away in the midst of mystic mumblings. The first two laws had been discovered in
spite of his blundering; the third was the result of patient, tireless
pursuit. When he hit upon it, he
realized the dreams of his youth --- the universe was a harmonious place. Without such a law, the cosmos would have
made no sense to him. If the Sun,
somehow, caused the motion of the planets, then their speed around it had to
depend on their distance from it.
Indeed, it did. Though his search
was driven by mystic underpinnings, his unerring belief that motion was somehow
governed by physical causes ultimately led him to the truth.
3.3.1 Kepler's Three Laws of Planetary Motion
$
Kepler's first
law of planetary motion is known as the law of ellipses. It states that the planets travel around
the Sun in elliptical orbits, with the Sun at one of the foci. No Platonic circles! No Ptolemaic epicycles! No Aristotelian set of fifty-five mechanized
wheels grinding away on each other! Just
one simple, clean curve in space, traced out by each planet in its motion
around the Sun (Focus Box 1).
$
Kepler's second
law, known as the law of equal areas, states that a line (the radius
vector) connecting the Sun and a planet, sweeps out equal areas in equal times. This law describes the way the speed of a
planet varies as it travels around the Sun in its elliptical orbit, speeding up
as it approaches the Sun and slowing down as it recedes from it. (Focus Box 2).
$
Kepler's third
law states that the square of the period of revolution of a planet about the
Sun is proportional to the cube of the semi-major axis of its elliptical orbit.
The third law can be expressed mathematically in the following succinct way
Where K
is a constant of proportionality whose value depends on the units in which the
orbital period, T, and the semi-major axis, R, are
expressed. The law takes a particularly
simple form
… if time is expressed in
years and distance in astronomical units.
Using this law we can easily calculate the time it takes for a planet to
orbit the Sun if the mean radius of its orbit is known or vice-versa (Example
1).
------------------------------------------------------------------------------------------------------------------
Example 1
Suppose we know that it takes
the planet, Jupiter, 11.86 yrs to complete one orbit around the the Sun. Then using the above equation we have
One can make a quick mental estimate
of the solution to the equation above, without actually solving it, in the
following way: first, note that 11.86 yrs is about 12 yrs and that 122
= 144. The radius of the orbit has to be
a number whose cube is a little less than 144.
But 53 = 125 and therefore the radius has to be a little bit
more than 5 AU. The actual solution is
obtained by taking the cubed root of both sides of the equation.
------------------------------------------------------------------------------------------------------------------
Kepler's third law of
planetary motion works exceedingly well.
Take a look at Table 3.3.1 where we have listed the periods and mean
radii of the planets known to Kepler at the time of the discovery of his third
law. The fourth and fifth columns in the
table show the calculated values Pplanet2 and Rplanet3. The accuracy of the agreement is particularly
astonishing, given that the values listed for the periods and mean radii are
modern values --- values that Kepler did not have at his disposal 400 years
ago!
Table
3.3.1 Planetary Orbital Data
|
Planet
|
Tplanet (years)
|
Rplanet (AU)
|
Tplanet2
|
Rplanet3
|
|
Mercury
|
0.241
|
0.387
|
0.0581
|
0.0580
|
|
Venus
|
0.615
|
0.723
|
0.378
|
0.378
|
|
Earth
|