5.1. A cosmic comparison

The history of astronomic discoveries provides, seen as a whole, an insight in human observation. The early attention of mankind to the heavens was guided by a fear for the gods, who were supposed to live in the firmament. In general, it could be said that the initial stage of looking to the sun, the moon and the stars had a static character: the heavenly bodies and space represented a timeless existence, which was associated with the invisibility of the gods.

Certain dynamic ideas about the earth and the sun and their relation as physical bodies in the sky developed in an early age. The Babylonians in Mesopotamia (around 2000 BC) were imaginative in their interpretation of the stars and their role in an astronomical system. Their creation epic, the Enuma Elish (When Skies Above), was written on a clay tablet in cuneiform (i.e. wedge-shaped) symbols, started off with a description of the beginning:

 ——————–  When skies above were not yet named

———————  Nor earth below pronounced by name,

——————— Apsu, the First One, Their begetter

——————— And Mummu Tiamat, who bore them all,

——————— Had mixed their waters together,

——————— But had not formed pastures, nor discovered reed-bed,

——————— When yet no Gods were manifest,

——————— Nor Names pronounced, nor destinies decreed,

——————— Then Gods were born within Them.

The Babylonians were skilled astronomers. They developed the number system of the Sumerians (a culture which flourished before 3500 BC) to provide the mathematical background for the calculations of the heavenly bodies. Their sexagesimal system, with a base of 60, resulting in hours of 60 minutes and minutes of 60 seconds, is still in use today.

The Ancient Greeks were the first to ponder about a cosmological model of the heavenly bodies. Practical knowledge of the stars had been available for a long time. The poet Hesiod, living around 700 BC, described the farmers life in a poem called ‘Works and Days’. Demetrius Chalcondyles printed the first edition around 1493. The complete works were published by Aldus Manutius in Venice in 1495. Hesiod was aware of the rising and setting of constellations in order to start certain agricultural actions:

But when Orion and Sirius are come into mid-heaven,and the rosy-fingered Dawn sees Arcturus, then cut off all the grapes-clusters, Perses, and bring them home. Show them to the sun ten days and ten nights: then cover them over for five, and on the sixth day draw off into vessels the gifts of joyful Dionysus.

The physical properties of the stars (and their positions) were, in the Greek way of thinking, subordinate to an idea and their ultimate significance was found in an abstract configuration. The position of the stars was imagined as lying on a celestial sphere, which rotated about the spherical Earth every twenty-four hours. This idea provided a psychological framework, which consciously accepted the partly subjective nature of its observations (in order to reach a state of beauty).

The mathematician Eudoxus of Cnidus (Asia Minor), who lived from 408 to 355 BC, build an observatory in his birthplace and another one near Heliopolis (in Egypt), just to develop an ideal planetary theory. Two works on the results of the observations on the rising and setting of stars, the ‘Mirror’ and the ‘Phaenomena’, have now been lost. His most famous book ‘On velocities’, in which he developed a system of homocentric spheres to describe the movement of planets, is also not preserved. Aristotle handed down the particulars (of the complete system with twenty-seven spheres) in his book ‘Metaphysics’. A subsystem of three spheres was set into a fourth sphere to enable the description of the daily rotation of stars.

Eudoxus’ theory was influenced by the philosophy of the Pythagoreans through his teacher Archytas, who was a follower of Pythagoras. It is likely that Eudoxus regarded his ideas of the spheres in terms of an abstract geometrical model rather than a description of the physical world (like Aristotle did).

Eratosthenes of Cyrene (276 – 194 BC) was a librarian at the famous library of Alexandria. He came up with more solid results. He measured the distance to the sun as 804.000.000 stadia and the distance to the Moon as 780.000 stadia. The length of the circumference of the Earth was established as 250.000 stadia, using an inventive system of comparison between the shadow of a pole (obelisk) of a given length at Syene (now Aswan) and Alexandria. The accuracy of these values depends on the length of the stadium, but are still fairly near the mark if a value of 157.2 meter is used (as deduced from values given by Pliny).

At about the same time Archimedes (died 212 BC) discussed in his ‘Sand-Reckoner’ the way to represent very large numbers. He posed the problem of the counting of all the sand grains in the universe. To complicate the matter, he did not choose a geocentric setting, which was accepted at the time, but a heliocentric cosmos as proposed by Aristarchus of Samos (ca. 310 – 230 BC). The idea of an earth out of the center was therefore known long before Nicolas Copernicus’ (1473 – 1543) publication of ‘De Revolutionibus Orbium Coelestium’ in 1543 put the sun in the center again. It still took nearly a century (and a proper telescope) before Galileo’s ‘Dialogue Concerning the Two Chief World System’ (1632) had the scientific minds turned away from Aristotle’s fixed earth.

The Greek astronomer (Claudius) Ptolemy, living in Alexandria from around 85 – 165 AD, was perhaps the most influential in the history of astronomy. His book, called the ‘Almagest’ (corrupted from the Arabic, meaning ‘the greatest’), was written around 140 AD and is a treatise in thirteen books. Book I gives a broad outline of the geocentric plan of the solar system. Book II deals with the trigonometry of the stars. A star catalog (based on that of Hipparchus, 129 BC) is given in book VII and VIII.

The scientific text of the ‘Almagest’ was, together with Euclid’s ‘Elements’, historically the longest in use. Ptolemy’s representation of the world and the sky was a model using circles (and circles upon circles, so-called epicircles) around a fixed Earth. These universal attributes (a point and a circle) made his model acceptable to all types of division thinkers, and might explain the long period of its acceptance.

The general acceptance of Ptolemy’s geocentric model came to an end when the dual way of thinking reached an all-time high during the sixteenth and seventeenth century in Europe. The concentration on antagonism (of earth and cosmos) either led to nowhere (in the speculative thoughts about the infinity of worlds by Giordano Bruno) or to the acceptance of a new cosmic reality (by Copernicus and Galileo). The rigorous limitations in the number of initial divisions in a communication created a focus on identity (in the social realm) and facts (in the scientific domain). The socio-political implications were sorted out in a considerable number of revolutions and wars (ASTON, 1965).

The idealistic cosmology (of Ptolemy), partly grown on the fertile subjective soil of higher division thinking, was superceded by a heliocentric cosmos, based on objective facts. Copernicus still retained the circular motion. Johannes Kepler (1571 – 1630) assumed, in his book ‘Astronomia Novade Motibus Stellae Martis’ (Prague, 1609), an elliptical motion of the planets. The slight aberration in the orbit of Mars, as already meticulous recorded by Tycho Brahe (1546 – 1601), provided the vital evidence for the rejection of the cyclic motion of the planets.

Further conformation of an unstable universe came to light in November 1572, when the Italian mathematician Francesco Maurolyco (1495 – 1575) discovered a ‘new’ star in the configuration of Cassiopeia. This supernova was also seen by Tycho Brahe and later described in 1574 (fig. 57). It had the brilliance of Jupiter and could be seen in daylight for about two weeks. The ‘Stella Nova’ subsequently became to bear his name.


Fig. 57 – The position of the ‘Stella Nova’ (I) as seen in 1572 by the Danish astronomer Tycho Brahe. The appearance coincided with a new awareness of motion in the communication with the universe.

A cosmic phenomenon of special interest to the present study is the shift in brightness of a certain class of variable stars. The discovery of the variable stars started with David Fabricius (1564 – 1617) in the year 1596, when he detected a star of non-equal brightness. He observed the star Omicron Ceti, located in the constellation of Cetus (the Whale). The brightness varied from a magnitude of two to be virtually too dim to see. The period of variation was about eleven-month. Mira (‘wonderful’), as the star was subsequently called, is a red super giant, pulsing by inner actions.

This observation of new stars, like the ‘Stella Nova’ (supernova) of 1572 and 1604 (the latter named after Kepler), comets and stars with variable brightness did not fit into the scientific belief of the time. The universe was regarded to consist of fixed stars, which remained steady and unaltered. Due to this fact, it lasted more than seventy years before the Italian polymath Geminiano Montanari (1633 – 1687) discovered another variable star. He noted in 1670 that the second brightest star in the constellation of Perseus (beta Perseï) was changing in light intensity (fig. 58).


Fig. 58 – The graph of the light intensity of the eclipsing binary of Algol in the configuration of Perseus (beta Persei) is derived from the four stages in the position of a small star circling the large star.

The star Algol became the best-known example of the eclipsing binaries, in particular, after the astronomer John Goodricke, again some hundred years later, calculated the period of Algol to 68 hours and 50 minutes.

The eclipsing binaries are part of a class of variable stars (or ‘variables’) which have their variation in brightness in common. However, the causes of the variability might differ considerably (and the eclipsing type is only a small minority). About thirty thousand are now known, and many more are to be discovered. In fact, most stars, including our own Sun, vary in brightness if measured in great detail. Some display regular variations of light intensity, while others lack any particular pattern.

The most famous example is the star Altais or Delta Cephei, a pulsating variable in the constellation of Cepheus, discovered by John Goodricke. He calculated its period to 128 hours and 45 minutes. The super giant has nearly completed its life cycle (by using the majority of the hydrogen thermonuclear energy). The change in brightness is due to an interaction between gravity and radiation. Delta Cephei became the prototype of the so-called Cepheid variables. This kind of stars became later the major players in the quest for the boundaries and age of the universe.

The English astronomer John Goodricke (1764 – 1786) was born in Groningen (northern Netherlands) as a son of a British diplomat and a Dutch merchant daughter. He was born deaf-mute, but learned to read lips, to speak and use an early method of sign language. His parents send him to Edinburgh for further education. As a seventeen-year-old boy, while living in Yorkshire (England), he began watching the stars and in particular, the star Algol. He noticed during his observations of this star a regular loss of a full magnitude of brightness.

The young astronomer reported his findings in 1783 at the Royal Society of London. Two theories were brought forward as an explanation: either a dark body periodically occults a distant sun or the star itself had a darker region that faced the earth periodically. The first theory, which was the right one (but could only be proven in 1889), made him the first discoverer of occultating variable stars. Goodricke was admitted to the Royal Society in April 1786, but he could not recognize this honor anymore, because he died that same month in York by pneumonia.

Algol (or beta Perseï) is a multi-star system in the configuration of Perseus, some ninety-six light-years away with a massive, bright central star and an orbiting secondary star with a yellow-red color. The later one circles around the central star in an orbit, which was nearly edge-on to the Earth. Light from the main source is taken away when the orbiting star passes in front of the main source. The decrease in brightness is less if the orbiting star is at the backside of the central star. A third star (and possibly more) orbits the double star system of Algol, but its influence on the brightness is only slight, with a change in the spectrum over a period of 1862 years.

An eclipsing binary is – in general – a double star oriented so that the two stars alternately pass in front of each other. The variation in brightness is called a light curve and exhibits two depressions and a shallow one in the middle (fig. 59). The recline is caused by an eclipse of the bright star, while the shallow one is the result of an eclipse of the dim star. A full period is the distance between two deep depressions. The depth in the curve also depends on the angle of the orbital inclination (zero degrees is edge-on).


Fig. 59 – The graphs of the light intensity of various variable stars. The vertical axis gives the sum of the brightness, the horizontal axis the change in time. One period covers the full cycle of a Small Star circling around a Large Star, starting in a position when the Small Star is right in front of the Large Star.

The movement of the orbiting star (a period) can be broken down in four particular stages, which represent the ultimate positions in brightness:

1. The Small Doublestar is positioned right in front of the Large Doublestar. The Small Doublestar ‘blackened’ the Large Doublestar as a matter of speaking, resulting in a diminishing of the total light intensity for the earthly observer;

2. The Small Doublestar moves to the right and ‘dissolves’ from the Larger Doublestar. The collective light intensity of the two stars reaches a maximum for an observer on earth;

3. The Small Doublestar disappears – for the observer – behind the Great Doublestar. Again, some light gets ‘lost’, but since the Small Doublestar contributes a fraction to the total of light intensity, and the diminishing luminosity is only marginal;

4. The Small Doublestar reappears behind the large Doublestar and again, the maximum collective light intensity is reached. The situation is a mirror image of stage two.

The graph may differ in detail, depending on the form and behavior of the Small Star, looping around the Large Star, but the structural setting remains the same: a large depression, a saddle, a small depression, a saddle, and a large depression. A period is measured between the two largest depressions, i.e. as the Small Star is right in front of the Large Star and causes the maximum loss of light. These features bear a remarkable resemblance to the visibility period X in the CF-graph (as pictured in fig. 45 and in more detail in fig. 51).


The similarity between the graph of the light-intensity of the ecliptic double stars (like the star Algol) and a part (X) of the CF-graph is not a coincidence. In both cases, it concerns the movement of two units passing each other in a cyclic setting. This movement can be divided in four phases. The double star derives the quadruple division from the geometric positions, while the abstract four-division consists, by definition, of four quadrants. The graphic results of the change in light intensity (of certain binary stars) and the depiction of the shift between two four-divisions are similar because their major premises are the same: a Small and a Large Part occultating in a circular setting.

The understanding of the quadralectic intensity (as expressed in the degree of commensurability in the CF-graph) may be enhanced by looking at the discoveries made in astronomy, with regard to a visibility-in-general and the conceptual position of ‘Observation-point Earth’ in the huge universe. The keyword in the search is an understanding of distance, not measured in a static comparison, but in a dynamic shift.

A special tribute must be given here to Henrietta Swan Leavitt (1868 -1921). She graduated from the Society for Collegiate Instruction of Women (later Radcliffe College) and joined the Harvard College Observatory as a volunteer in 1895. She was appointed to the permanent staff in 1902 and later became head of the photometry department (fig. 60).


Fig. 60 – The American astronomer Henrietta Leavitt (1868 – 1921). She discovered the Cepheid variable period-luminosity relationship. Photo: AAVSO/ American Association of Variable Star Observers.

The variable stars (of the delta Cephei type) of the Small Magellanic Cloud had her special interest. The discovery (in 1908) of the relation between the length of a cycle (period) and its average brightness (its luminosity as observed from earth) can be regarded as a major breakthrough. She found that the longer the period, the brighter the star. Large, bright Cepheids pulsate over a longer period, just as large bells resonate at a lower frequency (or graphically seen as a longer period). A circular (no. 173; March 3, 1912) of the Harvard College Observatory (under the direction of Edward C. Pickering) pointed towards the difficulties of obtaining the data:

‘A Catalogue of 1777 variable stars in the two Magellanic Clouds is given in H.A. 60, no. 4. The measurements and discussion of these objects present problems of unusual difficulty, on account of the large area covered by the two regions, the extremely crowded distribution of the stars contained in them, the faintness of the variable, and the shortness of their periods. As many of them never become brighter than the fifteenth magnitude, while very few exceed the thirteenth magnitude at maximum, long exposures are necessary, and the number of available photographs is small. The determination of absolute magnitudes for widely separated sequences of comparison stars of this degree of faintness may not be satisfactorily complete for some time to come. With the adoption of an absolute scale of magnitudes for stars in the North Polar Sequence, however, the way is open for such a determination.’

The distance of a Cepheid can be calculated from its period (the length of its cycle) and its average apparent brightness. Miss Leavitt published her findings in the above-mentioned circular with a chart of twenty-five Cepheid periods and their apparent brightness. Later, she calibrated the photographic magnitudes of forty-seven stars to which all other stars could be compared.

The observation of a Cepheid’s variation in luminosity over time will give an average apparent luminosity. The apparent luminosity decreases as the light travels longer. In theory, it falls off in proportion to the square of the distance to the object. The apparent luminosity can be compared with an absolute luminosity (that is, the apparent brightness the star would have if it were a standard distance of 10 parsecs away). The ratio of its absolute brightness to its apparent brightness will give an indication of its distance.

This method of cosmic distance-calculation has a strong resemblance, at least as a theoretical method, with the generation of CF-values in the quadralectic domain.

Once the relation between the period and luminosity was established these stars became distant markers for galaxies and offered the first hint for the distances within the universe. It became the ‘yardstick to the universe’, used by Edwin P. Hubble (1889 – 1953) and others to shape their view of the universe. However, there are some problems in the method. The dust between stars, for instance, can diminish the apparent luminosity and the chemical composition of the Cepheids might also influence the brightness. Therefore, some secondary distance indicators were developed, which use the Cepheid distance scale to calibrate. The study of a special category of supernovae, which show catastrophic explosions signaling the death of certain low mass stars, might give further answers. Furthermore, the measurement of the brightness and rotations of velocities of entire spiral galaxies (the Tully-Fisher relation) can hold a clue. The high luminosity galaxies (or the Large Part) have more mass than low-luminosity galaxies (the Small Part), and so the bright galaxies rotate slower than the dim galaxies. The relation of the velocities of the communication partners might give a theoretical shift-pattern, which can be used to measure distances.

The eclipsing binaries, or rather their physical setting, gained interest in the beginning of the twentieth century when their potential was recognized. The principle of a (slight) decrease in light intensity due to the passing of an object in front of a star, became the very condition for finding new planets orbiting other stars. The idea that the earth would be the only planet around a star (the sun) in a Milky Way galaxy consisting of hundred billion stars seems very unlikely.

Harlow SHAPLEY (1958) summed up the arguments in his popular work ‘Of Stars and Men’. The discovery that the nebulae are actually galaxies of stars meant that there are ‘more than one hundred million million million sources of light and warmth for whatever planets accompany these radiant stars.’ Also the average density in the expanded universe must initially have been much greater and therefore, collisions of stars and gravitational disruptions happened more often: ‘at that time countless millions of other planetary systems must have developed, for our sun is of a very common stellar type’ (DICK, 1998).

John Gribbin’s conclusion that ‘it may, indeed be not only possible but likely that life exists on other planets circling other suns’ – as quoted here in the beginning of this book – offers a daring prospect. The key to their discovery lies in the geometrical characteristics similar to an eclipsing binary, which cause small fluctuations (perturbations) in the light intensity curve. Many stars may, in fact, have planets circling around them.

The search for new planets has been fairly successful recently, and the latest count of Sun-like stars with perturbations, pointing to a body orbiting them, is out of date the moment it is published. These structures reveal a staggering variety in size, distance, revolution time and shape of the orbits. Several extrasolar planets are at least three times as massive as Jupiter, which is the largest planet in our solar system. Known physical laws seems to break down when such large planets (‘hot Jupiters’) are formed in the proximity of their intense hot mother star. The migration theory suggests that they arrived from elsewhere and might be on a collision course.

In particular, any relative smaller planet, like the earth, is – as yet – difficult to detect around a far-away star. Michael Mayor and Didier Queloz of the Geneva Observatory in Switzerland found the first earth-like planet in 1995. They used the Doppler planet-detection technique to discover a ‘telltale wobble’ in the spectrum of 51 Pegasi in the constellation of Pegasus. The orbital period of the planet – our ‘year’ – was established at 4.2 days. Other research revealed six more planets of which Tau Bootis, 55 Cancri and Upsilon Andromedae are in the same class as 51 Peg. Their orbital periods are 3.3, 14.7 and 4.6 days respectively. It can be expected that more planets will be found soon.

The relation between ‘seeing’ a planet as a wobble in a light intensity curve and the ‘seeing’ of an object (or being) in a cosmic communication poses a great challenge. The ‘absolute brightness’ can be translated in an ‘absolute being’ and this quality is estimated from a period of light change (or intensity change in more general terms). The conclusion can be drawn that the ultimate existence (or being), which has been the philosopher’s grail for such a long time, is found in the reading of an intensity change.

The trail leads to the visibility boundaries of the CF-graph and the interpretation of the inflection points. This action is probably the most important event in the quadralectic philosophy (from an empiric point of view): the use of the CF-graph as a tool in the understanding of life. The transfer of the communication-model from the theoretical Second Quadrant into the practical Third Quadrant is a moment of truth: the ‘wobble’ takes on the life of a real planet.

The actual fact-creating act in this process comprises the definition of two points on the CF-graph. Such a demarcation or imprint is called the point of recognition (POR).

A point of recognition (POR) is a marker point in an exchange of information and is established by an observer in a process of comparison between the visibility in a particular situation and a similar representation on the universal communication graph.

It is a mathematical necessity within the application of quadralectic thinking to know at least two points of the communication graph to be able to project its full course. If two points of recognition (POR’s) are known, then the whole CF-graph is known and the nature of a communication can be known.

These primary choices carry a ‘subjective’ load, but that is typically for the higher stages of multiplicity thinking. The communication partner has to recognize a ‘something’ (a certain type of visibility – not necessarily a visible visibility) and compare the given impression – in a quadralectic analogy-process – with the characteristics of the known universal (CF) graph. These markers, in turn, are not fixed in an absolute sense, but are established in a process of maximum comparison (affinity) with other facts and situations within the known context.

The decision-making process towards the Points of Recognition constitutes the most essential act in a communication. It introduces an empiric structure in the continuous shifting pattern of visibility and provides the outlines for further progress in due course. The practical implications of this approach will be dealt with in the remainder of this book. From this day forwards, there are only calculated and subjective decisions, which will lead us to the boundaries of imagination.

It is worthwhile to learn, before we are carried away in this fascinating world, that certain phenomena in the human history might have had the same function as the CF-graph, i.e. as a tool in the search for a valuation of a particular displacement. The old remains of standing stones and stone circles, which are found over greater parts of Europe, could have been our earliest indicators of a ‘shift’. They will be our next point of interest.



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