The Ancient Greek Cosmological Principle

 

by Konrad Rudnicki

 

 

2.01. Cosmology after the epoch of Ancient India

 

Several cultures arose, reaching their prime and then gradually declining, between the oldest known, the culture of ancient India, and the culture we want to refer now. One can mention here the cultures of ancient Iran, Egypt, Chaldea, and Babylon. All of them had quite definite views on celestial phenomena. Some nations, Chaldeans in particular, contributed much to astronomy but no account was left of their ideas about the Universe as a whole. Much is known about their views concerning the relationships between the Earth and the Moon, the Earth and the Sun, Venus or other particular celestial bodies, but neither contemporary records from those cultures nor documents written in later centuries but reflecting earlier views on the entire Cosmos (as was the case with the Bhagavad-Gita in respect to the outlook of ancient India) have yet been found by historians and archaeologists.

Taking into account the highly developed spirituality of ancient Egyptians, it seems very unlikely for them not to have any definite ideas about the Cosmos as such. The ancient Egyptian priest-astronomers developed the concepts of the Sidereal Zodiac (cf. Powell and Treagold 1979), which reveals that a certain view of the overall structure of the Universe must have existed in ancient Egyptian culture. This view, however, remains unknown to date. It was customary in those times that the highest wisdom should be preserved and passed solely to initiated persons orally or written in such a form that strangers should not understand it. Therefore it is very probable that any written records from that epoch concerning cosmological ideas, which would be of so much interest to us, never existed. Of course, it cannot be excluded that another collection of c1ay tablets or papyrus rolls will be discovered some day, delivering us information about Persian, Chaldean, Babylonian or Egyptian cosmological ideas. But as this has not happened yet, we have to skip those remarkable historical cultures and pass directly to ancient Greece. Here we have enough contemporary documents to reconstruct the general philosophical assumption, on which a number of mathematical models of the Universe were based, called the Ancient Greek Cosmological Principle or Cosmological Principle of the Ancients. This principle was reconstructed in 1972 by Michal Heller (Heller and Rudnicki 1972).

 

2.02. Greco-Roman culture

 

Classical western culture is not merely Greek culture. Its other major constituent is the Roman element. Therefore it is usually called, with reason, Greco-Roman culture. Yet I will be concerned with the Greek component only, because the Romans contributed mainly to the development of legal and political ideas, rather than to scientific or philosophical ones. In these latter areas they took over the achievements of Greeks.

Of course, that Greco-Roman culture was not detached from neighboring cultures and civilizations. Those connections as such, as well as their relation to cosmological ideas, may be of interest for the history of these times. For our aim, however, for the description of the evolution of cosmological principles, I will limit myself to a slightly simplified depiction of the Greek ideas.

When speaking about ideas of the ancient Greeks, it has to be said that it is impossible to find one consistent attitude towards the entire Universe; there were quite a variety of cosmological ideas instead. Traces of ancient Indian ideas are to be found there, as well as some original ideas, possibly cognate to some Chaldean or Egyptian views unknown to us. Some researchers presume that traces of anticipated elements of cosmological principles of our epoch could be found in ideas of some Greek philosophers. I will return to this problem in later chapters. Here I want to describe the main stream of Greek cosmology, which prevailed through at least two millennia and led to mathematical calculations of many historically and methodologically important and remarkable models of the Universe.

 

2.03. Logical Thinking - Philosophy

 

The particular intellectual atmosphere of Greek culture is strictly connected with the evolution of the human attitude towards cognition. For a priest in ancient Egypt, who was a scientist par excellence of that time, science was secret in the sense that its source was not research but revelation. The Egyptians knew a number of mathematical theorems, but they did not prove them logically. Rather, these theorems were revealed, in the sense that somebody first had the privilege to "perceive," to "see" a theorem in the realm of ideas, after which he could share his "revelation" with those worthy of it. Such a theorem was treated as something given from above to the chosen, to the initiated only. Nobody could arrive at such knowledge by an effort of his own. Revelation, not logical thinking could be of assistance in the cognitive process. Of course it was connected with thinking but with a thoroughly different manner of thinking than our contemporary, logical thinking.

Beginning from about the sixth century before Christ, another approach toward cognition was born. That was philosophy, which consisted of clear, logical thinking. The knowledge of the world gradually lost its occult character. One did not need to be previously initiated. Whoever could think was able to get knowledge. Mathematical truths were no longer valid by revelation, but rather by logical proof, as was the case with the formulation of classic geometrical theorems by Euclid. Personal, individual thinking was discovered as a new human capacity. And as with everything new, here also the power of the new - of thinking - was overestimated. People hoped to understand and explain everything by means of mere thinking. At the beginning, however, there were only a few people who had really developed the thinking abilities to a sufficient degree. Therefore, with the epoch of philosophy an epoch of human authorities also began. There began a period of reverence for people who could think by themselves. Those individuals were now admired in a somewhat similar way that revealed truths as such had been admired before. In the late Middle Ages, when the period of mental enlivening was in decline, this reverence for authority grew exuberantly, like weeds. In certain academic circles, adducing an authority counted for more than any independent observation and independent thinking. But we are concerned now with the first, positive, ascending branch of that cultural and scholarly trend.

 

2.04. Respect for the sensory world

 

In the ancient Indian manner of thinking, only spirit was of any importance. The world accessible to our senses was taken as a gross illusion, as maya. In the realm of spirit everything was in its real shape, and this meant also that everything was good. Even something which in the material, sensual world seemed to be evil was understood as good in the spiritual world of true reality. Thus evil was maya - merely an illusion. In the ancient Iranian culture, the material world obtained the same level of reality as the spiritual one. To transform, to elaborate this material world according to norms of spirit, was considered to be the primary goal of humans. Evil became reality, and opposing it was the highest human obligation. The eternal struggle between the principles of Good and Evil must end eventually in the victory of the former, but presently both possess about equal power. In the same manner, matter and spirit were considered as staying in a kind of equilibrium.

The material world was considered even more seriously in the three cultures that developed on the adjoining territories of Asia and Africa: Egyptian, Chaldean, and Babylonian. To be sure, they considered the spiritual world as the foundation of all existence, and their attitude towards it was that of high esteem. However, except for some highly initiated individuals, exploring contacts with heavenly beings served them first of all for more conveniently arranging earthly life, which was considered the most important one for human beings. And so it was for the Egyptians; they preserved earthly human bodies after death as mummies. And so for the Chaldeans, who developed astrology to be able to interpret the intentions of deities and thus to carry out their worldly affairs in a more effective way. And so for the Babylonians who developed systems of magic to harness spiritual energies for mundane purposes. Some elements of preserving bodily remains of the dead, of reading the stars, and of magical ceremonies were known in almost all previous cultures as well. But only these three brought them to perfection in their efflorescence.

A still deeper stage of the process of "materialization" is to be seen in the period of Greek culture which began some centuries before Alexander the Great. Obviously, an ancient Greek was by no means an atheist. He worshipped gods and believed in his own life after death. The average member of the middle or upper class of that time, was convinced that the gods themselves are concerned mainly with earthly problems. A Greek foresaw his future spiritual life after death as a very miserable one indeed.1 One can put it this way: they believed in gods but did not believe in God; they believed in spirits but not in the Spirit.

 

2.05. Two ways to atheism

 

Some prominent personalities of that epoch did not share those prevailing beliefs. Just in this cultural environment some philosophers developed highly spiritualized notions like the logos of Heraclites of Ephesus (540-480 B.C.) or the nuos of Anaxagoras of Klazomenae (ca. 500-428 B.C.). They acknowledged that there were either some high spiritual principles of existence or just one highest Spiritual Principle and were not ready to accept the widespread concept of small gods concerned mainly with earthly affairs. So did, for example, Euhemer, who lived at the turn of the 4th century B.C. He considered all the deities worshipped by people to come from among people themselves but to be venerated and deified by the others.

There are two positive ways towards atheism. In the first, one does not accept the existence of the highest creator and ruler of the world, accepting only lower hierarchies of spirits (one "believes in saints and angels but not in God"). This way usually leads through superstitious belief in spirits of nature, and, in its eventual stage, brings one to accept the notion of inanimate laws of nature. The other way consists in accepting the existence of the highest creator or the highest principle of all being but denying the existence of lower spiritual hierarchies, especially those having any contact with the earth and individuals dwelling on it (one "believes in God but not in angels"). This leads through sublime but usually dry considerations and adoration of the Creator towards searching for a philosophical principle of the highest necessity. Euhemer is considered to be the precursor of atheism though he did accept the Supreme Being. In present times this other tendency reappears as an attempt at reducing all spiritual phenomena to intellectual ones, all intellectual to mental, all mental to biological, all biological to chemical, all chemical to physical, and all physical to the unified theory of all interactions. The entire content of the Universe thus is comprised in one set of mathematical equations - what a lofty goal! It makes men gods, knowing everything good and evil.

Both those opposite trends can be seen in the classical Greek culture. Each of them tended from a different side to the same point: atheism.

 

2.06. Halfway cosmological principle

 

As already stated, classical culture was not atheistic, but was on the way to atheism. At that intermediate stage, the attitude of Greeks toward the world was quite particular. On the one hand, they looked upon their physical, worldly environment in the same way as average people of our 20th century. To be sure, there could still be a nymph immersed in a stream; there could be a satyr hiding behind a tree, and one could meet a goddess when traveling. But any physical object was not a mask or external disguise, not maya, not just a symbol of some higher entity, but it was exactly as perceived by the human senses and a reality in itself. Something was beautiful or ugly exactly as much as it presented itself to one's senses. Greek and Roman sculptures and paintings represent the beauty and importance of prominent persons not by furnishing them with a nimbus or symbolic vestments but by realistically reproducing those features of one's inner constitution that manifest themselves externally. The Greeks were, in fact, masters of realistic plastic art. But on the other hand, the Greeks did not look upon celestial bodies as physical objects. They could not be either touched or smelled or even heard (not everybody can listen to the harmony of spheres). One could only see them, and that was not sufficient for considering them as consisting of physical matter. Therefore the attitude of Greeks towards celestial bodies much resembled the attitude of ancient Indians towards any things. Celestial bodies were for them but tokens of workings of higher worlds; they were a sort of maya. Even when a Greek actually did say that celestial bodies consisted of matter, he had in mind another, sublime kind of matter.

Only physical existence was regarded as truly important. The only place of physical existence was our Earth. Even the deities concentrated their activities on earthly affairs. Thus the Earth was the natural center of all. Every reasonable description of astronomical reality had to be geocentric. The Cosmological Principle of the Ancients reflecting the Greek common outlook can be put today in a following way:

 

Our Earth is the natural center of the Universe.

 

In other words: the structure of the Universe must reveal a symmetry or quasi-symmetry in respect to the Earth. This principle was never formulated exactly so, at least no such c1assical texts are available, but for about two thousand years all the known models of the Universe involved this assumption. The symmetry was understood in a geometric, not kinematic, sense and concerned positions of celestial circ1es or spheres. On the other hand, the presence of privileged axes of rotations, even a number of them, inc1ined to one another in various ways, were not considered to break that quasi-symmetry.

This idea of spherical symmetry of the Universe is necessarily connected with the hypothesis (or discovery) that the Earth is a sphere. Presumably Pythagoras (ca. 572-497 B.C.) first stated that the Earth had a spherical shape. Some believe that it must have been Parmenides of Elea (ca. 540-470 B.C.) or even Hesiod (7th century B.C.) to arrive first at this conc1usion (cf. Ley 1963). Regardless, it was Eudoxos of Knidos who made the idea widely accepted. The conviction of the spherical shape of the Earth, born at about the same time as the new cosmological principle, was probably the departure point for the belief that the entire Universe had a spherical shape. But even if we suppose that the spherical shape of the sky 2 was the first and foremost argument for representing the Universe as a set of spheres, the spherical form of the Earth was an important fact in support of that view.

 

2.07. The sublunary and superlunary world; circular motions

 

Another idea was that the entire Universe was divided in two main parts: the sublunary and superlunary one. To the sublunary one belonged the solid Earth and atmosphere with clouds and all its phenomena. In contrast to superlunary planets, they were still considered physical.

Another important assumption was that the only movements admitted in the superlunary world were uniform circular motions. All known models of the Universe involving the Ancient Greek Cosmological Principle rely also on this assumption. But it is not as fundamental as the assumption of spherical symmetry; it does not constitute a part of the cosmological principle. The latter assumption is valid only for the observable parts of the Universe. Constructing models based on the Ancient Greek Cosmological Principle did not lead to the concept of a cosmological horizon in the contemporary sense. They provided, however, in their geometrically understood space, a concentric region reserved for the physica11y unobservable, purely spiritual world. This was the outermost part of the Universe. No systematic motions were possible there. Thus the assumption of exclusive1y circular, uniform motions, although very characteristic for those models, did not hold for the entire Universe. Different laws of motion governed in the innermost, sublunary sphere and in the outermost, invisible regions. Thus the circular uniform motion principle should be treated not as a cosmological principle or some part of such, but as a separate, additional assumption. The connection between the point symmetry and the circular motions around that point is obvious. It is remarkable that in fact also radial systematic motions do conform to the same symmetry, but such kind of motions was not regarded by the ancients as suitable for the Universe. Only in the 20th century did the Hubble law admit such kinds of movements in the Cosmos.

 

2.08. Systems of spheres

 

Two main kinds of mode1s involving the Ancient Greek Cosmological Principle, i.e. geocentric models, were developed. The first was constructed solely of concentric rotating spheres. This kind of model fulfills the condition of exact point symmetry. Models of the second kind, besides a sphere or system of spheres, also inc1uded hierarchical circ1es. The circ1es were not necessarily concentric and some minor ones (epicyc1es) were situated so that they did not encirc1e the center of the Universe, i.e. the Earth, at all. The point symmetry was nearly preserved, in the sense that the Earth remained the central body of the entire model. Planets, which inc1uded the Moon and the Sun, were considered in those models as basic constituents of the Universe. The stars were also taken into consideration, but only as a far-off background for the apparent planetary movements.

The construction of the first geocentric model of the first kind is attributed to Anaximander of Miletus, (611-546 B.C.). This model consisted of less than twenty spheres. The most famous model was constructed by Eudoxos from Knidos (ca. 408-355 B.C.). In Eudoxos' model, each sphere except for the outermost one which was at rest, rotated about its axis, which was fixed at the next, larger sphere. The axes had various inclinations. The first moving sphere here was the sphere of fixed stars. Its axis was the straight line connecting the celestial poles and was considered the most important axis in the Universe, 3 its rotation is the daily rotation of the celestial sphere. Several spheres, several axial inclinations, several periods of (uniform!) rotations had to be applied to account for the motion of a planet, to imitate its apparent wanderings over the sky, involving deviations from the ecliptic plane, deviations from a constant velocity, and loops. Another difficulty in that construction was that every sphere's movement included all the other spheres within it. Thus, spherical motions corresponding to individual planets were not independent from each other.

Luckily enough, the knowledge of planetary motions was not yet very accurate by Eudoxos' times; he succeeded in constructing a model of the Universe using no more than 27 spheres. Another model of this kind was described by Plato (ca. 427-347 B.C.) in the final chapter of his Republic. An even more widely known model was that of Aristotle (384-322 B.C.), consisting of 56 spheres. These kinds of models, although used in later times, as the accuracy of astronomical observations improved, became too intricate for practical construction and gradually gave way to developing models of the second type.

 

2.09. Systems of circles

 

The first model of the second kind is attributed sometimes to Anaximander of Miletus (611-546 RC.), sometimes to Heraclites of Ephesus (540-480 B.C.) or even to Seleukus (a Babylonian, living probably in the 3rd century B.C.). The scheme is here less complicated. The sphere of fixed stars, rotating with the period of one sidereal day, is the same as that of Eudoxos, but, for reproducing planetary motions, flat circles surrounding the Earth are used instead of spheres. They are located within the sphere of stars according to inclinations of their orbits. The circles are independent of each other. The motion of one planet does not influence the others. Thus it is possible to obtain a good agreement from observations of each planet separately, without regarding the motions of others.

The motions along those circles had to have constant angular velocity. A circle surrounding the Earth, the center of the system, is a great circle projected on the celestial sphere. To account for all the departures from great circles and from constant velocities in planetary motions, several auxiliary circles were introduced by Hipparchus (190-125 RC.). The main, large circles corresponding to the basic position of a planetary orbit are called deferenses. A small circle, called epicycle, moves along each deferens. Planets were situated on epicyc1es and so were capable of performing quite complicated movements. It was presumably Hipparchus who noticed that it is better not to set the Earth in the very center of the main circ1e but a little off it. This disposes of one epicyc1e. Thus, it is quite plausible that Hipparchus displaced the Earth from its central position (but one cannot be quite sure whether it was he who invented the eccentric circ1e in cosmological mode1s); indeed, if it was not he, then it must have been Ptolemy. I am not going to describe such models in detail here. This can be found in any book on history of astronomy (e.g. Ley 1963).

As astronomy progressed, with the collection of more astronomical data, the planetary movements appeared more and more complicated. Accordingly, over time, the mode1s became more accurate, but they also became more complicated. The epicyc1es of higher order, moving along other epicyc1es were introduced. The number of epicyc1es with different inc1inations from one to another can be increased following accuracy requirements, reflecting the increasing precision of astronomical measurements. The planet itself was always located on the last epicyc1e.

Such a system of epicyc1es is equivalent to the Fourier analysis and every quasi periodical motion can be thus decomposed into a finite number of strict periodic motions within the required degree of accuracy. Any motion can be expressed in terms of a Fourier series and thus be reproduced by a complex system of epicyc1es. Nevertheless, with the purpose of reducing the number of epicyc1es, some other constructions were also introduced. Beginning with Ptolemy, the angular velocity along the deferenses was no longer assumed to be constant even though it seems to be so when viewed from the center of another circ1e, the aequant. This was a way to preserve the principle of constant velocity, albeit in a very abstract sense indeed.

 

2.10. Various geocentric models

 

Besides those two major types of models, there were some less typical ones. Such was, for one, the late mode1 of Jan of Glog�w (16th century) that involved no spheres and no circ1es but a sort of space tunnel instead. The sufficiently large diameters of these tunnels in respect to diameters of planets allowed for departures from the strictly uniform motions.

The number of known models based on the Cosmological Principle of Ancients is enormous. The best known ones have already been mentioned. The last model of considerable importance for history and astronomy based on the Ancient Greek Principle is the model of Tycho Brahe (1546-1601). It is remarkable in that it conforms to two cosmological principles at the same time. I will return to it in a later chapter.

By all means the most famous of all those mode1s is the model of Claudius Ptolemy constructed in the second century after Christ. This is described in Ptolemy's work entitled in Greek E Megale Syntaxis or E Megiste Syntaxis but more often going under its Latinized Arabic name Almagestum. Ptolemy's model, with its numerous later corrections (i.e. provision of additional epicycles for obtaining a better agreement with the even more accurate knowledge of actual movements of planets over the celestial sphere), was widely accepted by the scholarly community and was dominant for about thirteen centuries. For many people, a geocentric model, a geocentric system of the Universe, remains to date synonymous to Ptolemy's.

 

2.11. Unobservable regions of the Universe

 

The Greeks' division of the Universe into two parts, the sublunary and the superlunary, was, as I mentioned above, equivalent to the division into the physically existing part of the Universe and the part with a sublime, superphysical existence. Only first regions of that second part were accessible to the human eye through manifestations as celestial phenomena. The rotating sphere of fixed stars was considered to be the ultimate constituent of the Cosmos still able to be perceived by humans. That sphere or the next one, enclosing it, was widely known under the Latin name given to it in much later, in the Middle Ages by Nicolas of Cusa (1401-1464): primum mobile. This was the border between the observable and unobservable parts of the Universe. No physical signal could be obtained from beyond it, where unmoving and invisible spheres were located. It was the world of the heavens, called in later centuries Empyrean, a region filled with heavenly fire and brilliance, which in the conception of ancient pre-Christian philosophers (e.g. Aristotle), and in the opinion of Christian thinkers was the abode of God himself, his angels and saints. From the general considerations (based on the cosmological principle adopted), it was known that it surrounds the Earth and the other celestial circles and spheres. Any other, more detailed knowledge about the Empyrean could be obtained only through supernatural revelation.

This was a specific kind of thinking indeed, establishing for spiritual, not physical, beings an appropriate location in space. It is paradoxical enough to provide room in a geometrically conceived model of the Universe for imponderable, sublime beings. This is, of course, related to a certain inconsistency of the ancient Greek world view which stopped halfway between the spiritual Indian philosophy and the modern materialistic one. The spiritual nature of the immediate human environment, of minerals, plants, and animals, was no longer accepted, at least not in daily life. But still, it was believed that physical matter is not the sole constituent of reality. Thus, for all that was not physical one had to assign a particular region, sufficiently removed from the human abode. For the ancient Indians, all that existed had two aspects, two sides. One spiritual, the fundamental one, and the other the physical, not quite real one, the maya. The dividing line did not lead through geometric space. In models based on the Ancient Greek Cosmological Principle, the physical world that could be observed through our senses was geometrically separated from the higher worlds.

It may be proper to add here that the rise of geocentric models based on the Ancient Greek Cosmological Principle took place in the last centuries before and the first centuries after Christ. Medieval European astronomy lived only with the echoes of ancient Greek and Arabian astronomies. The cosmological model prevailing at those times was a vulgarized model of Eudoxos. Thus the geometric, philosophic, and theological interpretation of that time concerning the observable and unobservable parts of the Universe did not have any reliable cosmological base. So it is not easy for the contemporary mode of thinking to grasp medieval consideration and compare them to the original ancient or later ones of the Renaissance. But for our purposes, exact delimiting of epochs is not necessary here.

In general, it can be stated that in all the known geocentric models the same outermost space was assigned for the part of the Universe not accessible to observation. Not too much could be said about these parts. Their sole property resulting directly from the accepted cosmological principle was that they have a certain (quasi)symmetry in respect to the Earth. One could say that the Cosmological Principle of the Ancients was a very "weak" one. It set little constraints on models in respect of unobservable regions of the Universe; nonetheless that is still enough to include it to the family of cosmological principles.

There is a subtle difference between the old Greek and our contemporary idea of the unobservable. At present the cosmologists believe that the regions of the Universe situated beyond the cosmological horizon are not connected with us in any causal sense. No event occurring in those distant regions can in any way influence us; no signal, no information can reach us from "out there." And vice versa, we cannot have any influence on anything that happens there. Such a strong separation did not exist in relation to the highest spheres in the ancient models of the Universe. The spiritual entities abiding there could come down even to the very Earth and human prayers were able to reach those highest invisible spheres. The separation was understood only in a physical sense, only for sensual perception. In that epoch the Universe was still considered both physical and spiritual. Its physical component was but one part of it. However, when those ancient models of the World are expressed using notions of contemporary cosmology, the difference disappears. The unobservable parts of the Universe were in a physical sense as little connected causally with us as is the case in the contemporary models with the regions beyond the cosmological horizon.

 

2.12. The Geocentric system in contemporary astronomy

 

Geocentric observation of the Universe brought about major developments in the knowledge of the sky. Such words as "astronomy," "planet," "comet," all of Greek origin, testify to that fact and provide evidence that the geocentric view really had its point.

Not all astronomers, not to mention the general public, are aware that the Ancient Greek Cosmological Principle is still alive today. To be sure, it hardly underlies any mainstream cosmological research; nevertheless, it is used in many astronomical ephemeredes and tables in astronomical yearbooks and almanacs, since astronomical data quite often are given in the geocentric coordinate system. Although the belief that our Earth is the natural center of the Universe is no longer accepted, in our astrometry measurements the geocentric system is applied as the most comfortable one. It is more objective than the topocentric system in which our measurements are actually performed and much easier for reduction than the other systems used in today's astronomy. Besides, since we consider (like the ancient Indians did) that there is no favored point in the Universe, all its points being of equal importance, our Earth can be freely adopted as a conventional center of the Universe as can any other celestial body or individual point in the Universe.

 

2.13. The Generalized Ancient Greek Cosmological Principle

 

As was said, the Ancient Greek Cosmological Principle divided the Universe geometrically into two parts; one is our ordinary physical world, and the other is where the visible celestial bodies perform their movements and the beings and phenomena are neither physical nor purely spiritual. By implication, this view led to accepting the Earth as the center of the universe. But we can say laconically that the content of this principle is: the Universe does possess a distinguished center, the Earth. It is easy to generalize this statement by simply dropping its last words after the comma. We obtain then a simple proposition: the Universe does possess a distinguished center. This generalized assumption is fulfilled not only by all the ancient models with spheres or circ1es but also by the models of Copernicus and Kepler (with the Sun as the center).

In such a generalized form, that cosmological principle reappears time and again in contemporary cosmology. For example, Ellis, Maartens and Nel (1978) put forward a model of a spherically symmetrical Universe with our Galaxy in the center. It is considered to be a methodological exercise rather than an actual model. The intention of its authors was to show that even such an exotic model did not contradict observations.

Sometimes the relativistic point solution of Karl Schwarzschild (cf. 0.08 and 9.07) is considered a cosmological model. It also fulfills the Generalized Ancient Greek Cosmological Principle.

 

Notes

up1 Homer says in his Odyssey: "Better a most poor herdsman on the earth than a king in the kingdom of shadows" (realm of souls of the dead).

up2 In fact, the shape of the firmament, as it presents itself to our perception, is not strictly spherical. The "canopy of heaven" has rather an ovaloid shape, that of a two-axial ellipsoid. The actual shape is contingent on landscape, weather, time of day, placement and temper of the observer. The distance to its highest point (zenith) always seems to be smaller than the distance to the horizon.

up3 The term world axis taken from that model is still in use in many languages and sometimes even in English in modern spherical astronomy.

Bibliography

� Konrad Rudnicki 1995

From: Professor Rudnicki�s book: The Cosmological Principles

Previous chapter: The Cosmological Principle of Ancient India

Konrad Rudnicki is a professor at Jagiellonian University, Cracow, Poland. He is a member of the Free European Academy of Science (Holland), member of the Commission of Galaxies of the International Astronomical Union and member of the Mathematical-Astronomical Section at the Goetheanum, (Switzerland). Prof. Rudnicki has been Senior Research Fellow at the California Institute of Technology (1965-67), visiting professor at Rice University, USA (1988-89). His areas of interest are: extragalactic astronomy, cosmology, philosophy of science and methodology of science.