Eudoxus And Concentric Spheres

Plato encouraged a new approach to astronomy: to devise a combination of uniform circular motions to reproduce the observed motions in the heavens. Whether the Platonic paradigm would die in infancy or grow in strength depended, in part, on the support it received.

Greek society supported playwrights when the citizens of Athens paid to see the productions of Aeschylus, Sophocles, and Euripides; and prizes were awarded at festivals to poets and musicians. No city held geometry in high regard, however, and inquiries in this subject languished, so Plato lamented. Practicing physicians and architects charged fees for their services, and so could philosophers were they able to attract pupils. Alternatives included inherited wealth and the beneficence of a wealthy patron.

The first major attempt to develop the fledgling paradigm of uniform circular motion into a successful science was made by Eudoxus. He arrived in Athens a poor youth, about 23 years old, traveling as assistant to a physician. Diogenes Laertius, a Greek biographer of the third century a.d. , wrote in his Lives of Eminent Philosophers that when Eudoxus was about 23 years old and in straitened circumstances, he was attracted by the reputation of the Socratics and set sail for Athens with Theomedon the physician who provided for his wants. Diogenes also quoted other writers on Eudoxus, noting that Sotion in his Successions of the Philosophers said that Eudoxus was also a pupil of Plato and that Apollodorus stated that Eudoxus flourished about the 103rd Olympiad and died in his fifty-third year.

"Flourishing" often meant one's 40th year. Adding 40 to 368 (the beginning of the 103rd Olympiad) gives 408 b.c. for Eudoxus's birth. Subtracting 23 from 408 gives 385 b.c. as the year of his arrival in Athens. However, Eudoxus's flourishing could have been earlier, in his twenties when already he was famous or later in his life with the development of his planetary theory.

From Athens, Eudoxus traveled to Egypt, where he stayed perhaps for several years and became familiar with priests' astronomical observations. Later he established a school in the Greek city of Cyzicus, and later still, he moved with some of his pupils to Athens.

Unfortunately, nothing Eudoxus wrote has survived. His astronomical system was described briefly by Aristotle in his book Metaphysics and, much later (around a.d. 500), by Simplicius in his commentary on Aristotle's book On the Heavens. According to Simplicius, Callippus, who had studied Eudoxus's system in Cyzicus, traveled to Athens, where he stayed with Aristotle correcting and completing, with Aristotle's help, the discoveries of Eudoxus. By Simplicius's time, both Eudoxus's and Callippus's books about the planetary system were lost. Simplicius had his information from Sosigenes (around a.d. 250), who had relied on a history of astronomy by Eudemus (a pupil of Aristotle, around 350 b.c.).

In the nineteenth century a.d. , Giovanni Schiaparelli, an Italian astronomer more famous—or infamous—for his observations of canali on Mars, attempted to reconstruct Eudoxus's system. The reconstruction is compatible with what Aristotle and Simplicius wrote about Eudoxus, but it assumes that Eudoxus's system accounted for many of the astronomical phenomena known in Schiaparelli's time; thus it may attribute to Eudoxus a more advanced knowledge of astronomy than he actually possessed and, consequently, also a more detailed and accurate astronomical model.

A basic observational fact that any astronomical system must account for is the movement of the stars overhead each evening. Eudoxus placed all the stars on one sphere rotating with a uniform speed around the central Earth in 24 hours. This is equivalent to a rotating Earth and a fixed sphere of the stars; the observational consequences of the systems are identical. (Relative motion of a star with respect to another would not be detected for nearly two thousand years and thus did not trouble Eudoxus or complicate his model.)

The apparent motion of the Sun presents a more difficult problem. First, there had to be an outer sphere rotating with a period of 24 hours to produce the apparent daily movement of the Sun across the sky. Again, as with the stars, the outer sphere for the Sun produced the apparent motion now attributed to a rotating Earth.

Eudoxus next would have needed a second sphere rotating with a period of a year and its axis tilted relative to the axis of the outer sphere to move the Sun higher in the sky in summer and lower in winter as well as around the heavens with a period of a year. The axis of this inner sphere would have been fixed to the outer sphere and thus carried around with a 24-hour period.

Eudoxus ignored the changeable velocity of the Sun, already discovered by his time. This decision, whatever its justification, saved him much trouble. Eudoxus did add a third sphere, though, to account for a belief now known to have been mistaken. Continually, it would be a problem to decide which observations were accurate and should be incorporated into a model. Eudoxus

Solar Declination

Figure 4.1: The Sun in the Summer and Winter Skies. The Sun is seen high in the summer sky and low in the winter sky because the Earth's axis of rotation is not perpendicular to the plane of the Earth's orbit around the Sun, but is inclined at an angle of about 23.5 degrees. Thus the Sun's apparent annual motion carries it alternately above and below the plane of the Earth's equator.

Figure 4.1: The Sun in the Summer and Winter Skies. The Sun is seen high in the summer sky and low in the winter sky because the Earth's axis of rotation is not perpendicular to the plane of the Earth's orbit around the Sun, but is inclined at an angle of about 23.5 degrees. Thus the Sun's apparent annual motion carries it alternately above and below the plane of the Earth's equator.

may not necessarily have intended to account for all observations, however much modern scientists try to.

Eudoxus devised a similar system of spheres for the Moon. First, as with all celestial objects, there was the outer sphere rotating once every 24 hours, producing appearances now attributed to the daily rotation of the Earth.

The Moon circles the Earth approximately once a month. To produce this monthly motion, Eudoxus added a second sphere attached to the first and rotating west to east with a period of one lunar month.

There is also a small variation in the latitude of the Moon. It moves at times slightly above and at other times slightly below an imaginary plane containing the Earth and the Sun. (The modern explanation for this observed phenomenon is that the plane of the Moon's orbit around the Earth is inclined at an angle of approximately five degrees to the plane of the Earth's orbit around the Sun.) Eudoxus added a third sphere, presumably to produce variations in lunar latitude, although we cannot be certain that he knew of this phenomenon. Reversing the order of the middle and inner lunar spheres would bring Eudoxus's model into better agreement with modern observation, and historians should not ignore the possibility that an error occurred somewhere along the way in the transmission of the model. On the other hand, rewriting ancient reports to conform to modern knowledge would be a highly questionable way of doing history.

Eudoxus did not take into account variation in the Moon's speed. Perhaps he was unaware of it, though Callippus certainly knew of this motion about three decades later, around 325 b.c. Alternatively, Eudoxus might have been aware of the phenomenon but chose not to recognize it as requiring a place in his system.

Planetary motions presented a more difficult problem than did the motions of the Sun and the Moon. The planets display retrograde motions: sometimes they cease their motions relative to the stars, turn back temporarily, retrace small parts of their paths, and then change direction once more and resume their voyages around the heavens. Eudoxus's task was to devise a model consisting of uniform circular motions only, yet producing the apparent retrograde motions.

Eudoxus first gave each planet an outer sphere to carry it around the Earth with a period of 24 hours. Second spheres moved the planets around the heavens, with periods of a year for Venus and for Mercury, and longer periods for the outer planets. To produce observed motions in latitude, Eudoxus added third spheres for each planet. So far, the planetary solutions followed the solutions for the Sun and the Moon.

To produce the observed retrograde motions, Eudoxus added a fourth sphere. By a clever combination of inclinations and speeds of revolution of the third and fourth spheres, Eudoxus could have produced, in an approximate fashion, the observed retrograde motions. The diagram presented here, "Retrograde Motion from Concentric Spheres," is only a crude and imaginative suggestion of what might have been Eudoxus's system. The actual details of his system have been lost to time.

Retrograde motion can be produced from combinations of spheres rotating with constant velocities. Even four concentric spheres, however, cannot simultaneously produce with quantitative accuracy both the length of the retrograde motion westward and the length of the motion in latitude (north—south) for all the planets.

sphere of the stars

sphere of the stars

Esferas Celeste Eudoxo

Figure 4.2 : Modern Explanation for the Appearance of Retrograde Motion of a Planet. As seen from the Earth at times 1, 2, 3, and 4, the planet apparently moves against the sphere of the stars from 1 to 2, turns back to 3, and then resumes its forward motion to 4. The Earth, moving faster than the outer planet, overtakes and passes it.

Figure 4.2 : Modern Explanation for the Appearance of Retrograde Motion of a Planet. As seen from the Earth at times 1, 2, 3, and 4, the planet apparently moves against the sphere of the stars from 1 to 2, turns back to 3, and then resumes its forward motion to 4. The Earth, moving faster than the outer planet, overtakes and passes it.

Evidently Eudoxus's contemporaries detected flaws in his system, because there occurred a series of modifications after his death, modifications constituting what Thomas Kuhn might characterize as normal science. The first modification was made at Eudoxus's school in Cyzicus by his pupil Polemarchus. A second modification was made by Polemarchus's pupil Callippus. They continued their efforts after moving to Athens, where Callippus also worked with Aristotle. In his Metaphysics, Aristotle wrote: "Callippus made the position of the spheres the same as did Eudoxus and assigned the same number as did Eudoxus to Jupiter and to Saturn; but he held that two more spheres are to be added to the Sun as well as to the Moon, if one is to account for the phenomena, and one more to each of the other planets" (Metaphysics, I8: 1073b17-1074a15).

A system of four concentric spheres can, in principle, give a satisfactory account of the actual motions (in longitude, in latitude, and retrograde) of Jupiter and Saturn, and of Mercury to some extent. For Venus, however, and even more so for Mars, combinations of four concentric spheres produce larger

Figure 4.3: Retrograde Motion from Concentric Spheres. The inner sphere is centered on the Earth. Its axis of rotation is horizontal and in the plane of the diagram. As the inner sphere rotates it carries the planet up and down and into and out of the plane of the diagram.

The outer sphere is not visible in the diagram. It, too, is centered on the Earth; the inner and outer spheres are concentric. The outer sphere's axis of rotation is vertical and in the plane of the diagram. As it rotates the outer sphere carries everything within it from right to left.

As the inner sphere moves the planet from 2 to 3 to 4 and the outer sphere moves everything also from right to left, the combined motion of the planet is rapid. But if the speed of the left to right motion from 4 to 1 to 2 imparted by the inner sphere is greater than the steady right to left speed imparted by the outer sphere, then the planet will appear to slow down and briefly move to the right during the passage from 4 to 1 to 2. Illustration after V. A. Mann, in Norriss S. Hetherington, Astronomy and Civilization (Tucson: Pachart Publishing House, 1987), p. 125.

deviations in latitude than are actually observed. Callippus was satisfied with Eudoxus's system for Jupiter and Saturn, but he added an additional sphere for each of the other planets.

Neither Aristotle nor Simplicius provides much detail about Eudoxus's system. They report planetary periods only in round numbers of years. It may be that Eudoxus's theory was not fully determined quantitatively. He might have been content solving the problem of retrograde motion qualitatively and geometrically. Determining the various curves traced by a point on the innermost sphere could have been so stupendous and absorbing that it provided its own justification. Lack of descriptive and predictive accuracy would not necessarily have weighed heavily against a theory intended primarily to give conceptual unity to the celestial motions (i.e., to show in a general way that a small number of principles could account for a large number of phenomena). It is tempting, however, to combine with ancient passages a modern analysis of the potential of a system of concentric spheres, then assume that ancient

Concentric Spheres Aristotle

astronomers were in possession of accurate observational parameters, and conclude, with much more certainty than is warranted, that we now understand what was taking place in the minds of astronomers more than two thousand years ago.

Aristotle adopted the ingenious and beautiful geometrical scheme of Eudoxus and Callippus, but he also thought of the spheres as material bodies. The problem for Aristotle was to connect all the spheres physically, yet prevent or compensate for transmission of the motions of outer planets to inner planets. To accomplish this objective he added more spheres to the system. In his Metaphysics, Aristotle wrote: "However, if all the spheres combined are to account for the phenomena, there must be for each of the planets other spheres . . . moving counter to these and bringing back to the same position the outermost sphere of the star [planet] . . . ; for thus alone can all the movements combine to produce the complex movement of the planets" (Metaphysics, I8: 1073b-1074a15).

Saturn was the outermost planet then known. It was acceptable in planetary models to transmit Saturn's daily rotation around the heavens due to its outermost sphere to Jupiter, but not to transmit inward the motions of Saturn's other three spheres. Aristotle inserted between Saturn and Jupiter three spheres, each rotating around the pole of one of Saturn's inner three spheres, with an equal and opposite velocity. Each of these added spheres neutralized the motion of its corresponding sphere among Saturn's. Thus none of Saturn's motion (except that of its outer sphere) was transmitted down to Jupiter's system of spheres.

Similarly, Aristotle added neutralizing spheres between each subsequent set of planetary spheres and also for the Sun. But none were added for the Moon, because there were no planetary spheres below it to protect from transmission of forces from above. To Callippus's 33 spheres (4 each for Saturn and Jupiter; 5 each for Mars, Mercury, Venus, the Sun, and the Moon) Aristotle added 22 neutralizing spheres (3 each for Saturn and Jupiter; 4 each for Mars, Mercury, Venus, and the Sun). This made a total of 55 (or 56, counting the outermost sphere of the stars), and Aristotle wrote that the total number of moving and countermoving spheres was 55.

So far so good. But Aristotle seemingly continued: "But if one does not add to the Sun and to the Moon the movements we have suggested, all the spheres will number only forty-seven. So much for the number of the spheres" (Metaphysics, I8: 1073b17-1074a15). Callippus and Aristotle had added 2 spheres each for the Sun and the Moon to Eudoxus's 3 each, which further required 2 neutralizing spheres for the Sun (but none for the Moon). Thus the total added for the Sun and the Moon was 6 spheres, not the 8 implied by reduction from 55 to 47. Might Aristotle have overlooked the fact that the Moon's motions need not be neutralized and that he had not added two neutralizing spheres for the Moon? Or might some scholar at a later date, not fully understanding what he was writing about and overlooking the fact that the Moon's motions need not be neutralized, have "corrected" what he mistakenly thought was an error by Aristotle. Or might a mistake in the long chain of copying and translation somewhere have changed 49 to 47? Or could the 47 be correct and our understanding in some way deficient?

Eudoxus's system was improved and brought into better agreement with observed planetary motions. There was one phenomenon, however, for which it could not account. The planets move at different times closer to and farther from the Earth. In the case of the Moon, apparent changes in size (due to changing distance from observers on the Earth) were observed directly by the Greeks. For the planets, apparent changes in size, and hence in actual distance, were inferred from changes in apparent brightness. (According to Aristotle, there was no change in the heavens; thus an apparent change in brightness was not real and was explained by a change in distance from the observer.) A system of concentric spheres cannot produce changes in distances of objects from the center of the spheres, where the Earth was assumed to reside.

Simplicius wrote in his commentary on Aristotle's On the Heavens:

Nevertheless the theories of Eudoxus and his followers fail to save the phenomena, and not only those which were first noticed at a later date, but even those which were before known and actually accepted by the authors themselves. . . . I refer to the fact that the planets appear at times to be near to us and at times to have receded. . . . The Moon also, even in the perception of our eye, is clearly not always at the same distance from us. . . . The same fact is moreover confirmed if we observe the Moon by means of an instrument; for it is at one time a disc of eleven fingerbreadths, and again at another time a disc of twelve fingerbreadths. . . . Polemarchus of Cyzicus appears to be aware of it [this inequality in the distances of each star (planet) at different times] but to minimize it as being imperceptible, because he preferred the theory which placed the spheres themselves about the very center in the universe. Aristotle, too, shows that he is conscious of it when, in the Physical Problems, he discusses objections to the hypotheses of astronomers arising from the fact that even the sizes of the planets do not appear to be the same always. In this respect Aristotle was not altogether satisfied with the revolving spheres, although the supposition that, being concentric with the universe, they move about its center attracted him. (Heath, Aristarchus of Samos, 221—23)

Greek astronomers after Eudoxus accepted the issue of planetary distances as a legitimate problem. Concentric spheres could not account for observed changes in distances, and eventually they were abandoned.

Nonetheless, Eudoxus's planetary model and its continuation by Callippus are impressive. Eudoxus went beyond mere philosophical speculation about the construction of the universe and attempted to account for planetary motions with a geometrical model. Callippus supplied observational facts necessary to test the theory, and he modified it, bringing it into better agreement with observation. This early example of continuity in science illustrates the cumulative advance possible when a problem receives continued attention. Astronomy was on its way to becoming an exact science and one of the most impressive achievements and legacies of Greek civilization.

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    When did eudoxus 55 concentric spheres?
    7 years ago
  • Shirley
    How did the followers of plato explain the retrograde motion of planets?
    2 years ago
  • cora goodchild
    What is the purpose of 27 concentric spheres of Eudoxus?
    2 years ago
  • Bernd Hueber
    Why Eudoxus did not assign any sphere for Earth ?
    2 years ago
  • nils rantanen
    How many spheres do we use by eudoxus to model the universe?
    1 year ago
  • roxy
    How many concentric spheres accommodate sun moon and stars in eudoxus model?
    1 year ago
  • jonas sommer
    How many sphere were used by eudoxus to model the universe?
    1 year ago
  • Estella
    How many concentric spheres accommodate the Sun, Moon, planets, and stars in Eudoxus model?
    1 year ago
  • Aisha Thomson
    How many spheres were used by eudoxos to model the universe?
    1 year ago
  • girmay sayid
    What are the three concentric sphere according to eudoxus?
    1 year ago
  • bruno
    What are the 27 concentri sphere?
    1 year ago
  • isacco
    Who invented the 56 concentric spheres?
    11 months ago
  • bertha
    How does eudoxus nested sphere model work?
    2 months ago

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