machtundwissenschaft.files.wordpress.com€¦ · web viewstars). this, along with the observation...
TRANSCRIPT
Aylecia LattimerMath 4150
5/4/14
This Science, Above all Things, Could Make Men See[2]
Since the dawn of human history, scientists and philosophers alike have felt driven to
explain the mysterious motion of the cosmos. Today, the basic concept of this motion, that all
planets orbit the sun, is well known. But it was not always so. When Nicolaus Copernicus
published his theories of heliocentrism (that is, that the sun is the center of the solar system), he
began a revolution in the scientific community. Up until that point in history, scholars were
convinced that the earth must be at the center of the universe. That is, they favored geocentrism,
or the belief that the sun, moon, and planets, all orbit the earth. The most favored model of
geocentric planetary motion was the Ptolemaic model, named for and developed by second
century A.D. astronomer Claudius Ptolemaeus [1]. Despite Ptolemy’s now obvious error in using
geocentrism, the model he developed was used for nearly 1500 years, until it fell during the
Copernican Revolution. In his work the Almagest, Ptolemy records the details of the model,
including well-reasoned arguments to support his conclusions. Although incorrect, the Ptolemaic
model was surprisingly accurate when applied practically, and remained an important instrument
for those wishing to study the stars for many years.
I. Introduction and Background
In the Almagest, Ptolemy begins with several preliminary assumptions on which to base
his work: 1) “the heaven is spherical in shape” [2], 2) “in position [the earth] lies in the middle of
the heavens” [2], and 3) “has no motion from place to place” [2]. For the first point, Ptolemy
pointed out that the stars were observed to be “carried from east to west along circles which were
always parallel to each other” [2]. He also observed that the positions of the stars relative to each
other did not vary, rather staying fixed at their own point in the sky (in relation to the other
Aylecia LattimerMath 4150
5/4/14
stars). This, along with the observation of circumpolar stars (stars that are always visible above
the horizon) orbiting what seemed to be a fixed point, now known to be the poles, are the
arguments he presents that the heavens must be spherical. After all, he asked, if the stars moved
in a straight line “towards infinity” [2], how could they then be observed to begin their apparent
motion through the sky at the same point each day? This concept of the stars on a fixed sphere is
still used to today, in what is known as the celestial sphere. The celestial sphere is a visualization
tool used by astronomers, in which the stars are fixed to a clear sphere, at the center of which is
the earth. However, what is understood today as a merely conceptual tool was seen as fact for
Ptolemy, who held that the stars were all attached to a great sphere that rotated around the earth,
resulting in the motion of the stars as observed from earth.
Ptolemy also argued that the earth must be in the center of the heavens. A fact that he
claimed supported this hypothesis was that one hemisphere of the night sky was always visible.
He argued that if the earth were not at the center of the cosmos, “the plane of the horizon would
divide the heavens into a part above the earth and a part below the earth which are unequal and
always different” [2]. In other words, the visible portion of the night sky would either be more or
less than a full hemisphere, rather than exactly one hemisphere as is observed. As any of his
readers could see, this phenomenon was contrary to what is plainly observable in the night sky,
leading Ptolemy to state that the earth therefore must be in the middle of the heavens. Another
argument was that since, as Ptolemy held, “all bodies fall to the center of the universe” [1], the
earth must be fixed at the center, otherwise falling objects would not, as observed, “be seen to
drop toward the center of the Earth” [1]. Ptolemy also applied a similar argument to support his
hypothesis that the earth was stationary at the center of the universe.
Aylecia LattimerMath 4150
5/4/14
The third major point addressed by Ptolemy in the beginning of the Almagest is that the
earth is stationary, and has no motion. Building on his earlier statement that the earth must be at
the center of the universe, Ptolemy called attention to the path of motion of “all bodies
possessing weight” [2]. He stated that the path of motion for a body is “always and everywhere
at right angles to the rigid plane drawn tangent to the point of impact” [2]. He argued that if a
body were not halted by the earth, it would certainly fall to the center of the universe (as he had
already established that the earth occupied said the center). In an argument similar to the one
used to fix the earth at the center of the universe, Ptolemy stated that “if the Earth rotated once
every 24 hours, a body thrown vertically upward should not fall back to the same place” [1], but
rather fall behind as the earth rotated away from it. As before, this was so obviously contrary to
the observed motion of falling bodies that it seemed to instantly disprove the idea that the earth
could be moving.
Despite the limitations of the data obtained by Ptolemy and his predecessors, for example
the fact that ancient astronomers were forced to rely upon naked-eye observations, these
arguments are well-reasoned and logical. However, it is easy now, in hindsight, to see where
Ptolemy’s reasoning failed him. For example, knowledge of gravity as it is understood today
would have negated the arguments both for the position of the earth at the universe’s center and
against its motion. Additionally, Ptolemy and his contemporaries greatly misjudged the vast
distances of the cosmos. This led to arguments such as that of the uneven hemispheres in support
of the position of the earth. As the above points were the basis of Ptolemy’s model of planetary
motion, these limitations greatly influenced his work and played a key role in the development of
his model.
Aylecia LattimerMath 4150
5/4/14
II. Mathematics of the Ptolemaic Model
a. General Description
Ptolemy’s concept of motion of the heavens was not necessarily unique. Built from the
works of Plato, Hipparchus, and Aristotle, Ptolemy’s works did provide more detail and
precision than those before him [3]. One idea central to the Ptolemaic model is that of uniform
motion, defined as a body traveling “uniformly around a point” [3]. However, despite his
arguments that the earth must be at the center of the universe, Ptolemy does not fix the center of
uniform motion as the earth. Referred to by medieval astronomers as the equant [3], this point
around which the celestial bodies rotate creates uniform motion, as proposed by Ptolemy’s
predecessors. Unlike those who came before him, however, this uniform motion is not
necessarily uniform as it is viewed from earth. The use of the equant in the Almagest is the
earliest known use of the model, though it has been argued that Indian astronomers may have
also used the concept of the equant [3].
Ptolemy used the equant to solve one problem set forth by the geocentric model: the
sometimes anomalous motion of the celestial bodies, such as the sun and moon. However, other
problems with the geocentric model arose, such as that of apparent retrograde motion. This
forced Ptolemy to add complications to his model, eventually resulting in a mathematically
complex model of planetary motion that was cumbersome, although surprisingly accurate, to use
[4].
i. Apparent Retrograde Motion
One problem that had plagued proponents of the geocentric model for years was that of
apparent retrograde motion. Easily explainable in the heliocentric model as when the earth
Aylecia LattimerMath 4150
5/4/14
passes, or ‘laps,’ an outer planet with a larger orbit, apparent retrograde motion is when a planet,
such as Mars, ceases to move in its customary direction of motion across the sky. For a time, the
planet appears to move backward, in the opposite direction of its regular motion. While the
heliocentric model of the universe provides a relatively simple explanation for this phenomenon,
geocentric models struggled to explain how the planets could suddenly reverse their motion.
Ptolemy’s predecessors did not see apparent retrograde motion as relative motion of the planets
to a moving earth. Rather, it was thought that the planets exhibiting apparent retrograde motion
physically reversed their orbits for periods of time. Ptolemy’s model, however, used a
reasonable, though complicated, method of accounting for retrograde motion.
ii. Epicycles and Deferents
Ptolemy’s model employed epicycles to explain the sudden reversal of a planet’s orbit.
An epicycle is a small circle upon which the planet travels. The center of this small epicycle then
orbits the earth along a larger circle, known as a deferent [4]. This “circle on circle motion” [4]
creates loops in the planet’s overall orbit around earth
(see Fig. 1). As viewed from earth, this pattern of
loops would account for apparent retrograde motion.
As stated above, to explain the sometimes
anomalous motion of celestial bodies, the center of
uniform motion was not always the earth, with some
of the deferents were centered on the equant. This,
coupled with the use of epicycles, resulted in a fairly
complicated model of the universe (see Fig. 1). An
Figure 1- A diagram of the epicycle system used to explain apparent retrograde motion. Note that the deferent is not centered on the earth (Terra). [5]
Figure 1- A diagram of the epicycle system used to explain apparent retrograde motion. Note that the deferent is not centered on the earth (Terra). [5]
Figure 1- A diagram of the epicycle system used to explain apparent retrograde motion. Note that the deferent is not centered on the earth (Terra). [5]
Figure 1- A diagram of the epicycle system used to explain apparent retrograde motion. Note that the deferent is not centered on the earth (Terra). [5]
Figure 1- A diagram of the epicycle system used to explain apparent retrograde motion. Note that the deferent is not centered on the earth (Terra). [5]
Aylecia LattimerMath 4150
5/4/14
example of this unnecessarily complicate model of motion can be found in the way that Ptolemy
modeled the three then-known outer planets- Mars, Jupiter, and Saturn. According to Ptolemy,
these planets orbited earth each on their own fixed deferent [5]. However, the east-west motion
of the each epicycle along the deferent was uniform
with respect to the equant, not the earth (see Fig. 2) [5].
This is because Ptolemy defined uniform motion as the
motion of a body that “travels uniformly around a
point” [3], but he did not require that point to be the
earth [3]. The motion of the inner planets, Mercury and
Venus was even more complicated than that of the outer
planets. To model the fact that the inner planets “remain
always within fixed distances from the sun” [5], each
planet moved about its own epicycle at a uniform rate
determined by the sun’s motion. The result was an exaggerated form of apparent retrograde
motion, with each planet moving toward and away from the sun. [5]. While Venus’s motion was
otherwise modeled like that of the outer planets, Mercury’s was not. Where the other planets
traveled along a fixed deferent, Mercury moved along a “moving eccentric” [5], a deferent with a
center that revolved about the line joining earth and the equant (see Fig. 3) [5]. This complicated
pattern of motion accounted for the fact that Mercury always
remains within a fixed distance from the sun [5].
Ptolemy, like those both before and after him, believed
that the heavens must be perfect and unchanging. This is the
reason that his model uses
Aylecia LattimerMath 4150
5/4/14
only perfect circles. By trying to account for the sometimes irregular movements of celestial
bodies, such as the fact that planets move faster at perihelion than at aphelion, using only circles,
Ptolemy imposed severe limits on himself and his model of planetary motion. Some 1500 years
later, Kepler would go on to derive his three laws of planetary motion, showing that celestial
bodies actually orbit in ellipses. This fact can account for much of the irregular motion that
presented problems in the development of the Ptolemaic model.
In the same way that Ptolemy believed the celestial sphere to be a physical object, he
proposed that the planets moved by a similar mechanism. He hypothesized “the physical
existence of crystalline spheres, to which the heavenly bodies were said to be attached” [5]. The
spheres to which the planets were attached were proposed to lie within the celestial sphere. The
theory of nesting spheres, of which he was a proponent, was also later used by European and
Islamic astronomers in the medieval period [3]. In this theory, each planet is attached to a sphere
so that a planet’s “greatest distance from the Earth is equal to the closet distance of the planet
above it” [3]. Ptolemy used the assumption inherent in this theory (that there is no empty space
in the universe), along with his estimate of the distances to the sun and moon to calculate the
distances to other planets. This idea of a “void free” [3] model of the universe agreed with the
ideas passed down to him by Aristotle and other philosophers, from which he drew great
inspiration [3]. However, this assumption also raised yet another problem for Ptolemy: when he
calculated the distances of Mercury and Venus, he was left with “a void of 81 Earth radii
between the outermost sphere of Venus and the innermost sphere of the Sun” [3].
Ptolemy held the belief that each celestial body was “driven by its own soul” [3], and that
this motion was also what moved the sphere to which it was attached. Furthermore, he disagreed
with the idea proposed by his predecessors that each sphere moved because it was attached to an
Aylecia LattimerMath 4150
5/4/14
axis [3], saying that each sphere was in its “natural place” [3] and “did not require a mechanism
to drive it” [3]. Ptolemy contended that to fill the sphere model of the universe, 34 spheres were
needed (see Fig. 4) [3].
Figure 4- Ptolemy's nesting sphere model. This is a model for a planet that is not Mercury. CD is the celestial equator; GF is the path of the planet along the ecliptic. The planet travels within the light grey sphere, which is the same width as the epicycle.[3]
b. Use of Chords
In order to explain the mathematics necessary to describe the motion present in his
model, Ptolemy introduced what would now be seen as trigonometric methods of calculation.
Many of these were based on the Crd function. This is related to the sine function by sin a = (Crd
2a)/120) [6]. From “using chords of a circle and an inscribed 360-gon” [6], Ptolemy
approximated a value of pi equal to 3.14166, and by using √3 = chord 60°, he found √3 =1.73205
[6]. In the Almagest, Ptolemy stated when introducing the geometry he would use that “it is first
necessary to explain the method of determining chords” [2]. He went on to use different
formulae for the Crd function, which were “analogous to our formulae for sin(a + b), sin(a - b)
Aylecia LattimerMath 4150
5/4/14
and sin a/2” [6], to formulate a table containing the values of the Crd function at .5 degree
intervals [6]. In the beginning of the Almagest, Ptolemy provides several examples of the use of
arcs and chords in determining celestial distances. He made use of spherical geometry and
trigonometry to calculate lengths of arc, as in the excerpt from Book I of the Almagest found
below. Ptolemy goes on to state that the method detailed in the proof can be used to “compute
the sizes of [the other] individual arcs” [2], meaning that the method could be used to calculate
what we would now call a star’s declination, the astronomical equivalent of latitude.
Aylecia LattimerMath 4150
5/4/14
Figure 2- Excerpt detailing a proof showing the calculation of the length of segments of arc.[2]
Aylecia LattimerMath 4150
5/4/14
III. The Ptolemaic Model in History
Until Copernicus published his book De Revolutionibus Orbium Coelestium (“Concerning
the Revolutions of the Heavenly Spheres” [4]) in 1543, shortly before his death, the Ptolemaic
model was the dominant model of planetary motion. Over the years, the concept of earth as the
center of the universe had become “engrained in Christian theology, making it a doctrine of
religion as much as natural philosophy” [7]. This religious endorsement of the geocentric
universe lent itself to the proliferation of the Ptolemaic model. Despite what we now know to be
a more correct hypothesis of planetary motion, Copernicus’ model went largely unsupported for
many years. The main reason for this was the fact that the new heliocentric model was not any
more precise than Ptolemy’s model that had been in use for almost 1500 years [4].
The root of the inaccuracy in the Copernican model was one that was shared with the
Ptolemaic one. While Copernicus had readily decided to “overturn Earth’s central place in the
cosmos” [4], he had still founded his model on the deep-rooted belief that the celestial bodies,
being heavenly and perfect, must move in circles. Much like Ptolemy, Copernicus was forced to
add a system of circles upon circles to model the irregular motion of the planets as uniform
motion. [4]. Although Copernicus also made use of epicycles, his model needed far fewer than
the Ptolemaic model due to the new position of the sun at the center of the cosmos. However, the
continued use of epicycles resulted in a new system that was just as mathematically complicated
as the old one, and which worked with no more accuracy [4].
The church so vehemently defended the Ptolemaic model that to support the heliocentric
model of the universe was to run the risk of heresy. As a result, a heliocentric view of the
universe gained support slowly. It was not until Galileo began to observe celestial phenomena
Aylecia LattimerMath 4150
5/4/14
that went against the geocentric view that the Copernican model began to garner support. Several
such observations made by Galileo included: the observation of the asteroid-riddled lunar
surface, proving that celestial bodies were not perfect, and could be just as flawed as the earth,
and the observation of the Galilean moons of Jupiter, which proved that not every heavenly body
orbited the earth. Galileo also observed and mapped the phases of Venus, which “proved that the
planet orbits the Sun” [7].
It was these observations that set the stage for Johannes Kepler. Kepler was the first
astronomer to suggest that the orbits of the planets were not perfect circles. This led to Kepler’s
three laws of planetary motion. Derived empirically (that is, only from observation), Kepler’s
laws are also derivable from Newton’s later laws of motion. As such, Kepler’s model is now
thought to be the correct model of planetary motion. Despite its use of ellipses instead of circles,
Kepler’s model gained acceptance by the scientific community, likely for the fact that it could
“predict planetary positions with far greater accuracy” [4] than the Ptolemaic model.
IV. Accusations of Fraud
Interestingly, over the years Ptolemy’s work in the Almagest and the Planetary
Hypotheses has faced several accusations of merely being copied from his predecessors [6]. The
first to make this accusation was Tycho Brahe, who discovered that there was “a systematic error
of one degree in the longitudes of the stars” [6] in the star catalogues that Ptolemy had published.
Brahe claimed that this proved that the data in the catalogue was not actually obtained by
Ptolemy himself, but instead converted from “a catalogue due to Hipparchus corrected for
precession” [6]. The latest accusation of forgery against Ptolemy came in 1977, in R.R.
Newton’s book The Crime of Claudius Ptolemy. Newton accused Ptolemy of a “crime against
Aylecia LattimerMath 4150
5/4/14
science and scholarship” [6], saying that he “deliberately fabricated observations from the
theories so that he could claim that the observations prove the validity of his theories” [6].
Newton proposed that Ptolemy did this after discovering that “certain astronomical theories […]
were not consistent with observation” [6].
It is almost impossible to determine if Ptolemy committed the fraud of which he has been
accused. It is apparent that his works draw heavily upon those of his predecessors, such as Plato,
Aristotle, and Hipparchus. However, it is difficult to know if the calculations, theories, and
observations ascribed to Ptolemy were actually the work of someone else entirely. F. Boll, in
Studien über Claudius Ptolemäus (Leipzig, 1894), stated, perhaps correctly, that one should
“credit Ptolemy with giving an essentially richer picture of the Greek firmament after his
eminent predecessors” [6].
V. Conclusions
Despite the perhaps dubious history of Ptolemy’s work, the model itself was widely used
throughout Europe in the middle ages up through the 17th century. It wasn’t until Isaac Newton
dealt the model its death-blow via his three laws of motion and theory of gravitation that it fell
out of use. Some of the ideas expressed in Ptolemy’s work, such as the hypothesis of nesting
spheres that make up the universe, made appearances in the works of Islamic scholars. Several
medieval scholars refer to theories of distances and sizes of planets that can be traced back to
Ptolemy’s work in the Planetary Hypotheses, a restating of his work in the Almagest [3]. It
would seem that these authors were unaware of the origin of the ideas they drew on, as they did
not credit Ptolemy for the calculations they used.
Aylecia LattimerMath 4150
5/4/14
Ptolemy wrote several other works besides the Almagest and the Planetary Hypotheses.
He wrote a book entitled Analemma, which explored the finding of angles necessary to construct
a sundial involving the “projection of points on the celestial sphere” [6]. In one of his major
works, Geographike Hyphegesis (Guide to Geography) [1], he attempted to map the world as it
was known at the time, giving the latitudes and longitudes of major cities [6]. Many of the maps
included in this book, however, were highly inaccurate, as Ptolemy was forced to rely only upon
the data he had readily available, which was of “very poor quality” [6] and could be “severely
distorted” [6] for anything that lay outside the Roman Empire. Furthermore, several other
mistakes can be found. For example, the earth’s equator is farther north than in reality. Also, the
given value of the circumference of the earth was nearly 30 percent less than an already
determined, more accurate value [1]. However, like the Almagest and the Ptolemaic model, the
Guide was widely used throughout history. In 1775, for example, it was still believed that “the
Indian Ocean was bounded by a southern continent, as Ptolemy had suggested” [1].
From studying all of his works, Ptolemy can be seen as not only an astronomer, but as a
mathematician who viewed calculation as “eternal and unchanging […] neither unclear nor
disorderly” [2]. As seen in his opening arguments in the Almagest, many of Ptolemy’s
hypotheses were logical and well-reasoned. It is, naturally, easy for modern astronomers to see
where Ptolemy’s reasoning led him astray from the physical reality of the cosmos. For example,
one might wonder how the Ptolemaic model might be different had he known of the force of
gravity. Had Ptolemy known of the vast distances between celestial bodies, and of the even more
distant, ever receding edge of the universe, his theories concerning the earth’s placement in the
heavens might have been changed dramatically. Forced to rely only on observations that he
could make with his own eyes, in a universe that is much more vast than the eye can perceive,
Aylecia LattimerMath 4150
5/4/14
Ptolemy was restricted by limitations over which he exerted no control, and of which he was
likely unaware. Like those who came before him, and those who would come after, Ptolemy
fought to reconcile the divine with the earthly, the faultless with the flawed, and the familiar with
a universe that is still largely unknown.
Aylecia LattimerMath 4150
5/4/14
VI. References
[1] "Ptolemy." Ptolemy. University of Oregon, n.d. Web. 08 Apr. 2014.
[2] Ptolemy. Ptolemy's Almagest. Trans. G. J. Toomer. New York: Springer-Verlag, 1984. Print..
[3] Hamm, Elizabeth Anne. Ptolemy's Planetary Theory: An English Translation of Book One,
Part A of the "Planetary Hypotheses" with Introduction and Commentary. Diss.
University of Toronto, 2011. Toronto: n.p., 2011. Web. 8 Apr. 2014.
[4] Bennett, Jeffrey, Megan Donahue, Nicholas Schneider, and Mark Voit. the essential cosmic
perspective. 6th ed. Boston, MA: Addison-Wesley, 2012. Print.
[5] "Ptolemaic Astronomy in the Middle Ages." Princeton University. N.p., n.d. Web. 08 Apr. 2014.
[6] "Claudius Ptolemy." Ptolemy Biography. University of St. Andrews, n.d. Web. 13 Apr. 2014.
[7] "Planetary Motion: The History of an Idea That Launched the Scientific Revolution: Feature Articles. “Planetary Motion: The History of an Idea That Launched the Scientific Revolution: Feature Articles. NASA, n.d. Web. 13 Apr. 2014