machtundwissenschaft.files.wordpress.com€¦ · web viewstars). this, along with the observation...

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


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.


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.


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


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]






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


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


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)


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.


5/4/14

Figure 2- Excerpt detailing a proof showing the calculation of the length of segments of arc.[2]


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


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

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hipparchus.html


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.


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,


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.


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

machtundwissenschaft.files.wordpress.com€¦ · web viewstars). this, along with the observation...

Documents