Eccentrics And Epicycles

Plato initiated the paradigm of uniform circular motion. Working within the paradigm, Eudoxus devised combinations of concentric spheres. The combined motions were intended to mimic observed planetary motions. Combinations of concentric spheres, however, cannot produce changing distances from the Earth. Next, schemes that could do so were developed, though not in Athens.

Not long after Plato and Eudoxus flourished in Athens, the center of scientific activity in the Greek intellectual world shifted to Alexandria. This port city was established by Alexander the Great in 332 b.c. on the western edge of Egypt's Nile River Delta.

When Alexander died in 323 b.c., his generals split his empire into three major kingdoms: Greece, Asia, and Egypt. With its fertile land along the Nile River, Egypt was wealthy and became even more so after the Egyptian ruler Ptolemy I gained possession of Alexander's body and made his tomb at Alexandria a profitable tourist attraction. In another economic coup, Ptolemy I stopped the export of grain until famine abroad brought higher prices.

Ptolemy I used money from these and other enterprises to begin construction of a lighthouse. Nearly 400 feet high when it was completed by Ptolemy II, it was the tallest building on Earth. Even more wondrous, its mirror reflected light (sunlight during the day and fire at night) that could be seen more than 35 miles offshore. The lighthouse was one of the Seven Wonders of the Ancient World—and the only wonder with a practical use. It may have collapsed during an earthquake in a.d. 1303. A French underwater archaeology project begun in 1994 has recovered much material from the harbor, including pieces of the lighthouse.

Ptolemy I also founded the Museum, around 290 b.c. It was home to a hundred scholars subsidized by the government. There were lectures, and specimens of plants and animals were collected for study.

Not to be outdone, Ptolemy II established the Library. Its famous collection of perhaps half a million books was obtained by purchasing private libraries, including possibly Aristotle's. Astronomical instruments were constructed for use at the Library, and the matching of theory with observation was undertaken on a systematic and sustained basis.

In 47 b.c. Julius Caesar arrived at Alexandria, chasing Pompey. Pompey had won fame in battles, including victory over the remnants of Spartacus's army of slaves and the successful siege of Jerusalem. He married Caesar's daughter Julia and joined Caesar in a ruling triumvirate. After Julia and the third member of the triumvirate died, Pompey gained temporary ascendancy in Rome. Caesar, defying orders of the Roman Senate, crossed the Rubicon River from Gaul with his army to battle Pompey. At Alexandria a traitor surprised the fleeing Pompey and delivered his head and signet ring to Caesar's ship.

Caesar tarried in Alexandria. He was a scholar, and the Museum and Library were major attractions for him. Cleopatra smuggled herself into Caesar's presence only after he went ashore.

Caesar selected thousands of books from the Library to take back to Rome, but they were lost in a fire that spread to the docks from the Alexandrian fleet set afire by Caesar. Later, Mark Anthony, who succeeded Caesar after he was assassinated in 44 b.c., may have given Cleopatra over 200,000 scrolls for the Library.

Both the Museum and the Library suffered in the fourth century a.d. Under the Roman emperor Constantine, Christianity triumphed and pagan institutions were destroyed. In a.d. 392 the last fellow of the Museum was murdered by a mob, and the Library was pillaged.

Whatever remained of the Library was further damaged following the Arab conquest of Alexandria in a.d. 640. Three hundred years later, a Christian bishop known for his critiques of Muslim atrocities asserted, without evidence, that the conquering Caliph reasoned that the books in the great library reputed to contain all the knowledge of the world either would contradict the Koran, in which case they were heresy, or would agree with the Koran, in which case they were superfluous. Supposedly it took six months to burn all the books as fuel for the bathhouses of the city.

In ancient Alexandria much prestige was attached to scholarship and scientific research, and the Ptolemies sought thus to enhance their reputations. Plato had lamented that inasmuch as no city held geometry in high regard, inquiries in the subject languished. This deplorable situation was no longer true, at least not in Alexandria.

The problem set by Plato and pursued by Eudoxus and Callippus in Athens, to account

A new library building rose in Alexandria in 2002, constructed over 12 years at a cost of $210 million by the Egyptian government and UNESCO (the United Nations Educational, Scientific and Cultural Organization). The slanting roof, made of aluminum and glass, looks like a computer microchip. No provision was made for books to fill the building, other than donations. Ironically, modern Alexandria and its new Library are surrounded by widespread illiteracy, Islamic fundamentalism, and cultural repression, including censorship of books.

for the observed motions of the planets, the Sun, and the Moon with a combination of uniform circular motions, now guided astronomers in Alexandria.

Unfortunately, if all too typically, little of the historical record has survived. Around a.d. 140 Claudius Ptolemy (not related to the rulers of Egypt) summed up previous astronomical work in his Mathematical Systematic Treatise. The Almagest, as Ptolemy's great work came to be called, was so comprehensive that its predecessors were rendered obsolete; they ceased to be copied and failed to survive.

Of Ptolemy, himself, little is known. He reported observations made between the ninth year of Hadrian's regime (a.d. 125) and the fourth year of Antoninus Pius's reign (a.d. 141) "in the parallel of Alexandria." This could have been at Alexandria, itself, or at Canopus, 15 miles east of Alexandria. One scholar suggests that Ptolemy had his home in Canopus because it offered better possibilities for a quiet life of study than did the noisy capital of the Hellenistic world. Another scholar counters that Canopus was renowned in the ancient world for its dissolute and licentious lifestyle.

One of the very few of Ptolemy's predecessors known by name is Apollonius. He was born in the city of Perga, in what is now Turkey, sometime during the reign of an Egyptian king who ruled from 246 to 221 b.c. Apollonius moved to Alexandria, where he may have studied with pupils of Euclid, famous for his summary of Greek geometry.

Apollonius is famous for his book on conic sections (i.e., parabola, hyperbola, and ellipse: the curves cut from a right circular cone by a plane). The first four parts of his mathematical book have survived in the original Greek, and parts five through seven are preserved in Arabic translations. Part eight is lost, as is everything Apollonius wrote on astronomy.

Ptolemy wrote in his Almagest that a preliminary proposition regarding the retrograde motions of the planets was demonstrated by a number of mathematicians, notably Apollonius. Elsewhere in the Almagest, without attributing them to any particular individual, Ptolemy described what have become known as the eccentric and epicycle hypotheses:

[I]t is first necessary to assume in general that the motions of the planets . . . are all regular and circular by nature. . . . That is, the straight lines, conceived as revolving the planets or their circles, cut off in equal times on absolutely all circumferences equal angles at the centers of each, and their apparent irregularities result from the positions and arrangements of the circles . . .

But the cause of this irregular appearance can be accounted for by as many as two primary simple hypotheses. For if their movement is considered with respect to a circle . . . concentric with the cosmos so that our eye is the center, then it is necessary to suppose that they [the planets] make their regular movements either along circles not concentric with the cosmos [eccentric circles], or along concentric circles; not with these [concentric circles] simply, but with other circles carried upon them called epicycles. For according to either hypothesis it will appear possible for the planets seemingly to pass, in equal periods of time, through unequal arcs of the ecliptic circle which is concentric with the cosmos. (Almagest, III 3)

Figure 5.1: Eccentric and Epicycle Hypotheses

In the eccentric hypothesis (above, left), the planet moves around the circle ABCD centered on E with uniform (unchanging) velocity. The observer, however, is not at the center E, but at F, from which perspective apparently nonuniform planetary motion is observed.

Ptolemy wrote: "For if, in the case of the hypothesis of eccentricity, we conceive the eccentric circle ABCD on which the planet moves regularly, with E as center and with diameter AED, and the point F on it as your eye so that the point A becomes the apogee [point in the planet's orbit farthest from the Earth/observer at F] and the point D the perigee [point in the planet's orbit closest to the Earth/observer at F]; and if, cutting off equal arcs AB and DC, we join BE, BF, CE, and CF, then it will be evident that the planet moving through each of the arcs AB and CD in an equal period of time will seem to have passed through unequal arcs on the circle described around F as a center. For since angle BEA = angle CED, therefore angle BFA is less than either of them, and angle CFD greater" (Almagest, III 3).

In the epicycle hypothesis (above, right), the planet is carried around the small circle FGHK with uniform velocity, while that circle is simultaneously carried around on circle ABCD, also with uniform velocity. The combined motion of the planet as observed from E is not uniform.

Ptolemy wrote: "And if in the hypothesis of the epicycle we conceive the circle ABCD concentric with the ecliptic with center E and diameter AEC, and the epicycle FGHK carried on it on which the planet moves, with its center at A, then it will be immediately evident also that as the epicycle passes regularly along the circle ABCD, from A to B for example, and the planet along the epicycle, the planet will appear indifferently to be at A the center of the epicycle when it is at F or H; but when it is at other points, it will not. But having come to G, for instance, it will seem to have produced a movement greater than the regular movement by the arc AG; and having come to K, likewise less by the arc AK" (Almagest, III 3).

Figure 5.2: Eccentric Solar Orbit. The Sun moves in its orbit with constant speed. It traverses equal distances along the circumference of the circle in equal times. Viewed from the Earth rather than from the center of its circle, the Sun is seen to move through the 90-degree angle from 1 to 2 faster (in less time) than it moves through the 90-degree angle from 2 to 3. The journey from 3 to 4, cutting off another 90-degree angle, consumes even more time. Thus the Sun's constant angular speed relative to the center of its orbit appears irregular when viewed from any other reference point.

Figure 5.3: Change of Distance in the Epicycle Hypothesis. The large circle (the deferent) is rotating counterclockwise around the Earth at its center. This rotation carries the small circle (the epicycle) attached to the deferent from the left side of the drawing to the right side. Were the epicycle not rotating about its center, the planet carried on the epicycle would end up on the far right side of the drawing as the deferent rotated through an angle of 180 degrees (half a circle) and would always be the same distance from the Earth. If, instead, the epicycle is rotating about its center, through 180 degrees in the same time that the deferent rotates through 180 degrees, the planet will end up closer to the Earth, as shown in the above diagram.

Either eccentrics or epicycles, each carrying planets around with uniform circular motions, can produce seemingly irregular motion and thus save the planetary phenomena.

The changeable velocity of the Sun, ignored by Eudoxus, is easily accounted for qualitatively by the eccentric hypothesis. Uniform circular motion (as measured by constant angular velocity about the center of a circle and also as measured by constant velocity along the circumference of the circle) appears nonuniform to an observer not at the center. Thus the Sun appears to an observer not at the center of its circle to move sometimes faster and sometimes slower. Also, the Sun's distance from an eccentric observer changes.

The Sun's distance from an observer on the Earth also changes in the epicycle hypothesis. The large circle (the deferent) rotates around the Earth. Attached to the deferent circle and carried about by it is the center of the epicycle (the small circle). The planet is attached to and carried around by the epicycle rotating around its center. The epicycle thus moves the planet (or the Sun or the Moon) alternately closer to and farther from the Earth.

Gears from the Greeks

In 1900 a sponge diver discovered an ancient shipwreck at a depth of about 140 feet off the tiny Greek island of Antikythera. Along with marble and bronze statues of nude women, jewelry, pottery, and amphorae of wine were a few corroded lumps of bronze. Analysis of the pottery and amphorae suggested that they came from the island of Rhodes and that the ship, probably sailing to Rome with its cargo, sank around 65 b.c., plus or minus 15 years.

X-ray photographs of the corroded lumps of bronze later revealed about 30 separate metal plates and gear wheels. They would have fit into a wooden box about the size of a shoe box. An inscription on one of the metal plates is similar to an astronomical calendar written by someone thought to have lived on Rhodes about 77 b.c. This evidence pretty much ruled out the possibility that the clockwork mechanism had been dropped onto the wreck at a later date, or that it had been left behind by alien astronauts!

Clearly it was some sort of astronomical device, perhaps for navigation. Or possibly it was a small planetarium. The Roman Cicero had written in the first century b.c. about an instrument recently constructed by Poseidonius, which at each revolution reproduced the same motions of the Sun, the Moon, and the five planets. Archimedes was also said to have made a small planetarium, before 200 b.c. Maybe such devices actually existed.

Cleaning away the corrosion of centuries proceeded slowly. Not until the 1950s was a more detailed analysis begun. In the 1970s high-energy gamma rays were used to examine the interiors of the clumps of corroded bronze. Then similarities were revealed between the device, now in the Greek National Archaeological Museum in Athens, and a thirteenth-century-a.d. Islamic geared calendar-computer in the Museum of History of Science at Oxford, showing on dials various cycles of the Sun and the Moon. (The Arabs then had access to ancient Greek texts now lost to us.) Furthermore, both inscriptions and gear ratios from the ancient device were linked to astronomical and calendar ratios.

Still, most historians doubted the conclusion. Certainly the ancient Greeks had possessed the theoretical knowledge necessary to have built such a device, but this was the first physical evidence suggesting that they had attained such an advanced technology. Literary evidence in the form of ancient written accounts of Rhodes' military technology, including a machine gun catapult with gears powering its chain drive and feeding bolts into its firing slot, had been largely ignored or disbelieved.

In 1983 the Science Museum in London acquired a sixth-century-a.d. device that seemingly is one of the missing links between the Greek mechanism of the first century b.c. and the thirteenth-century-a.d. Islamic computer now at Oxford. More recently, in 2002, a new analysis of the Greek mechanism suggests that in addition to accounting for the motions of the Sun and the Moon, it may also have reproduced the motions of the planets using the epicycle model—just as Cicero wrote almost two thousand years ago about a mechanism constructed by Poseidonius.

Non Concentric Circle Motion

Figure 5.4: Retrograde Motion in the Epicycle Hypothesis. The large deferent circle rotates around the Earth. The center of the small epicycle circle is attached to the deferent and carried around by it. The epicycle rotates around its own center. The planet is attached to the epicycle and carried around by it. As seen from the Earth, the planet appears to move against the sphere of the stars from 1 to 2, back to 3, and then resumes its forward motion toward 4.

Figure 5.4: Retrograde Motion in the Epicycle Hypothesis. The large deferent circle rotates around the Earth. The center of the small epicycle circle is attached to the deferent and carried around by it. The epicycle rotates around its own center. The planet is attached to the epicycle and carried around by it. As seen from the Earth, the planet appears to move against the sphere of the stars from 1 to 2, back to 3, and then resumes its forward motion toward 4.

Retrograde motion is the appearance of a planet slowing down in its orbit, stopping, briefly reversing course, and then turning to resume its path once more around the heavens. Retrograde motion also can be saved, or reproduced with a combination of regular circular motions, with the epicycle hypothesis, at least qualitatively. A suitable combination of deferent and epicycle sizes and uniform angular velocities can produce the appearance of a planet moving irregularly against the sphere of the stars. In actual practice, quantitatively reproducing the observed widths and spacings of the retrograde arcs is far from simple. This difficulty, however, did not arise in the initial, qualitative period of Greek geometrical astronomy.

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