30. The number four

Footsteps of the number four


There are four options when people can get involved with the number four. The following theoretical (mental) positions can be distinguished:

————— 1. four as an unknown entity

————— 2. four in the mathematical world of numbers and figures

————— 3. four as a numerological unit

————— 4. four as a (division) base for a philosophy (known as quadralectics)

The absence of the number four (1) does not mean that this entity is absent in the context of a communication. The figure can be conceived as a philosophical subject, which has its place in a non-imaginary region of our mind. The intention to allow such a space to exist is of vital importance for the rest of the communication.

The mathematical approach (to numbers)(2) is often left behind after formal schooling, and its acquaintance is not always satisfactory.  The operations with numbers give pleasure to persons with a like for problem solving and can be useful for practical calculations.

The most favorable appearance of the number (four) is in day-to-day life (3) when four is a quantity, which presents itself to the observer in a visible reality. In it barest form the four is just a number, but the cipher can become involved in other meanings attributed to the number. The area of numerology covers all the actions of figures (including four) outside its character as a prime number. HARRISON (1976) defined numerology as an action: ‘to separate the numbers from their context and then to read more into the pattern of isolated numbers than is warranted.’ Such a formulation shows a cognitive technical process:

—————-   a. there are numbers in a context;

—————-   b. numbers can be released from their context;

—————-   c. numbers can be interpreted anew;

—————-   d. their can be read more in numbers than in their original context.

This procedure represents, whatever way one looks at it, a creative action. A mind performs an exercise on the ‘context’ of a number – based on the belief of its ‘right’ place – and can introduce deliberately changes to arrive at a different interpretation. These transformations are common in a dynamic understanding and part of every intellectual apparatus. In the end, it is the consensus of the context of a given fact, which determines its value in data exchange. However, this does not mean that this place or value should be fixed for the whole duration of a communication.

Finally, the number (four) points to the basic characteristic of every communication: division (4). This latter meaning has a philosophical undertone, but can easier be understood than the total absence of the first possibility. The ‘theory’ of quadralectics, for instance, is in fact an arithmetical exercise, which falls within the (second) field of mathematics and becomes operational in the fourth stage of a communication.

Numbers are dynamic parts, which are related to a specific type of division. They create their own reality (visibility) in the division. The development of their valuation depends on the treatment of the observer and the use by other representatives in a communication. This approach is, in my view, the way Pythagoras and his followers intended to use numbers.

Any observer plays an active role in the valuation-process (leading to a consensus). ‘Search and you will find’ is a golden rule in higher division-thinking (which Paul Kammerer extended to his ‘Law of Seriality’). If the multitude is accepted as an unlimited field of possibilities, then there will always be something to match (and make a bond of understanding). In the multitude is no limitation to create facts. And from these new facts, new worlds can be built.

No good reason exists to rate the new horizons lower (or higher) than the old ones. Numerology is just a name for an artificial frame of mind, constructed with numbers, which have lost their ‘neutrality’. Is a mind ruled by ‘figures out of context’ necessary inferior to a world guided by ‘figures in their right context’? Who decides what is ‘right’? The consensus, all right, but we all agree that the ideas of the multitude are floating and a valuation can only be a moment in time and space.

Kenneth HARRISON (1976) noted – in his book on the Anglo-Saxon history of England – that researchers (like LAPPENBERG, 1845) recognized a pattern in the data in the way old calendars were kept. ALCOCK (1971) concluded, that ‘a significant proportion of the annals followed a four-year cycle’ (p. 43). An argument could be that leap years – indicated by a B(issextilis) – were separated marked. The idea was that later data were added to existing notations rather than to a new field. For instance, remarks are written at the years 540, 544, 547, 552, 560, 565 en 568, which are all (minus 547 and 565) leap years.

Harrison proved convincingly that an extension of that period would result in a more or less random distribution. He pointed in this context to the so-called ‘Law of Titius-Bode‘ (1772), an ingenious rule, stating a fixed relation between the distances of the planets to the sun. The regularity is connected with the name of Johann Elert Bode (1747 – 1826)(fig. 192), but was actually discovered by J.D. Titius (1729 – 1796).

The numbers in the series                 0       3       6       12      24       48

                    are added by 4:                4       4       4         4        4         4

                                                            ————————————————— +

                                                                4       7      10       16      28       52

and their sum divided by 10:          0.4    0.7     1        1.6     2.8      5.2

These figures corresponded remarkably well with the distances of the planets to the sun expressed in the distance earth – sun: Mercurius (0.4), Venus (0.7), Earth (1.0), Mars (1.6), — (2.8), Jupiter (5.2). Saturn was next on the list with a distance of 10 (actually 9.5 astronomical units). The ‘Law’ gained even more status as a ‘Law of Nature’, when William Herschel discovered in 1781 a new planet (Uranus) at 19.2, where the ‘Law of Bode’ predicted a value of 19.6.

bodeFig. 192 – Johann Elert Bode (1747 – 1826) had his name linked to a ‘law of nature’, which no longer exists since additional observational evidence did not fit in its formula. In: van MAANEN  (1988/89).

Further supporting evidence for the ‘law’ came from the discoveries by the Italian astronomer-priest Giuseppe Piazzi (1746 – 1826), who filled the ‘gap’ between Mars and Jupiter. He discovered in the year 1801 a planet at 2.77 astronomical units (AU) of the sun. It turned out to be a member of a group of planetoids, now estimated to be numbering in their tens of thousands. They are the remnants of an exploded planet.

Unfortunately, the two newly found planets Neptune and Pluto did not fit into this rule. The ‘Law of Bode’ predicted a planet in the region of 38.8 astronomical units, but when he was found (in 1846) his distance was 30.05 AU. And Pluto, discovered in 1930, is 39.4 AU away from the sun.

The ‘Law of Bode’ had lost its precision and remained as ‘a curiosity, perhaps as a warning.’ This caution is valuable for all ‘laws of nature’. There is sometimes a thin line between ‘coincidence’ and rule, because it all depends on the ability of an observer to have the facts right and belief in them. A ‘law’ can turn into ‘coincidence’, if better observation provides more facts. In the same way, a ‘coincidence’ can become a law if the number of observation increases.

It must always be remembered that the ‘laws of nature’ are constructed in the minds of man. It can be concluded that the boundary between fact and fiction is a dynamic one, and it is better to realize this fact (or is it a fiction after all?) in order to understand the position of numerology.

Four is commonly used as a numerological reference, because the figure is aptly situated between the few and many. It has the advantages of the many, without the necessity for limitation. However, the number can also be counted on the fingers of one hand and is within reach of people’s sphere of imagination. Athanasius KIRCHNER (1665), in his ‘Arithmologia‘ (Pars VI, Cap. 3), referred to ‘De Mysterijs Tetradis seu quaternary‘ (fol. 259), and gave the following inventory:

Chaos est divisum in quatuor Elementa

Coelum in quatuor partes seu angulos

Aer in quatuor ventos

Zodiacus in quatuor triplicitates

sub Coelo quatuor temporum qualitates

sub qualitatibus quatuor Elementa

sub Elementis substantia, quantitas, qualitas, motum

sub substantia corporea: Entitatiuum,Vegetabile,

 Sensituum, Intelectiuum

Quantitas in punctum, longitudinem, latitudinem & profunditatem

qualitas in quatuor, siccum, frigidum, humidum, calidum

motus in ascendentem, descendentem, rectum, circularem

This list is only a general enumeration of the mirage of four-fold features. The great standard-work on numerology by Petrus Bungus of Bergamo (or Peter Bungi) tried to be more exhaustive. The book was called ‘Numerorum Mysteria ex abditis plurimarum disciplinarum fontibus hausta‘ and had a first print in 1585, but the main edition dated from 1618. The incredible rich material offered a wealth of numerological examples, surpassing the earlier popular manual, the ‘Liber numerorum qui in sanctis scripturis occurrunt‘ of Isidore of Seville (RATHOFER, 1970).

The well-known list of major four-partions was given in Bungi’s book in the chapter on ‘De Numero IV’ (p. 193): quatuor partibus anni, quatuor partibus diei naturalis, quatuor orbis terrae and also the Substantia, Quantitas, Qualitas and Locus. The names of God (ESSE), as a four-letter word used in different languages, were given as follows:

 DEUS                                                              FROE                                                        IDIO

AIVS                                                                 BEDY                                                        DIOS

ROMA                                                             ABDI                                                          DIEV

SYRE                                                                ESAR                                                        DIUS

ORSI                                                                GOTT                                                        DIEU

ADAD                                                              ITGA                                                         ALLA

UNUM                                                            BOVH                                                        ESGI

ODEN                                                              BVVH                                                        ABIR

THOR                                                              BOEG                                                        ADAM

AMUN                                                             IESU                                                          TEOS

A part of the previous inventory was used by Giordano Bruno (1548 – 1600) in his treatise ‘De Monade‘ (1591) and also by the prolific writer Luigi Novarini (1594 – 1650), a medical doctor from Verona, in his ‘Schediasmata sacrofana’ (Lib. I, no. 27, p. 7), published in Leiden (Lugduni) in 1635 (AUBER, 1884/1975). These publications proved that numerological thoughts were firmly established in the period between 1585 and 1635.

MICHEL (1962/1973) described, in an elucidate book on the cosmology of Giordano Bruno (1548 – 1600), the development of his ‘discursus‘. These cognitive steps indicated the course between ignorance and understanding (cognizance). It reflected, in effect, a history of division thinking within one person.

Bruno started with his thoughts in a multitude, when he followed the neo-Platonian eight division in his work ‘De umbris idearum’, published in 1582. ‘Bruno was haunted by the images and formulas of neo-Platonism’ said Michel. The exuberance of division thinking was soon diminished to five (mens, sensus, imaginatio, ratio, intellectus) in 1583. The symbolism of the pentad had an esoteric significance in relation to God, Soul, Matter, Form and Intelligence.

A year later, in ‘De ‘infinito’ (1584), he arrived at a four-division, when ‘sensus‘ and ‘imaginatio‘ were joined together. Further limitation took place in 1591, when he combined ‘intellectus‘ and ‘mens‘. Now the triad ‘sensusratiointellectus’ governed his thoughts. After these drastic restraints, God (or the invisible invisibility of ‘mens‘) was virtually lost out of sight and the ‘discursus‘ was fully a human affair.

Soon Bruno realized the poorness of such an approach and changed – in the same year 1591 – to a Pythagorean quadripartite division, which was recorded by Aristotle in his ‘De anima’. The final stage of Bruno’s discourse as it was reached in his work ‘Summa terminorum metaphysicorum’, written in Zürich in 1591 and published in 1595, is as follows:

 —————————————-             divine        ———————————————

   I.   Mens     (superior intuition: not accompanied by discourse)

  II.  Sensus  (accumulation: attention, imaginatio, phantasia)

 III.  Ratio     (organize: from plurality of images to unity of the idea)

  IV. Intellectus (unification: pure intuition by reason, derived from   argumentation)

 —————————————-            human     ———————————————-

Bruno envisaged at this (final) point of his ‘discursus‘-development even some sort of cyclic mechanism: a unifying process would ‘jump’ to the transcendent ‘mens‘ after reaching the stage of ‘intellectus‘, closing the human circle of understanding. He grasped the ultimate knowledge as put forward by Plotinus in his ‘Enneads‘ (VI, IX, 7): ‘He who learns who he is, will consequently know whence he comes’. This can only be said in a cyclic environment. The Unity (Oneness) is the center of the soul, the beginning and the end.

Numerological features became a favorite way of artistic expression in the sixteenth and seventeenth century of the European cultural history. Number-related metaphors provided the ‘humanistic’ material to replace the supremacy of religious images of the previous ages.

The number four had a prominent profile in the display of numerological paraphernalia: from the four times of the day, the seasons, the directions of the winds, the parts of the world to the four cardinal virtues and the wheel of fortune. A good example of the use of numerological symbolism was provided by a world map of Willem Jansz. Blaeu and his son Joan in the atlas ‘Le theatre du monde’, published in 1635 (fig. 193). The map was engraved by Josua van den Ende and first published in 1606 on loose sheets. The four personifications of the four seasons (Quatuor Anni Tempestates) are visible to the right. The representations of the four elements (Quatuor Elementa) are depicted opposite to the left.


Fig. 193 – A world map of Willem Jansz. Blaeu and his son Joan as given in the atlas ‘Le theatre du monde’, 1635. The map was made at the height of numerological consciousness in Europe. The examples of elements, times of the year, planets and wonders of the world along the edges of the map are derived from Goltius and his school. The upper ribbon symbolized the planets, with the sun (Sol) on a quadriga. The lower ribbon exhibited the Seven Wonders of the World. They are, from left to right: the hanging gardens of Babylon, the Colossus of Alexandria, the pyramids of Egypt, the labyrinth (mausoleum), the temple of Diana, the statue of Jupiter and the pharos of Corinthe. Scheepvaartmuseum, Amsterdam. In: DONKERSLOOT-DE VRIJ (1992).

The roots and fascination of the numerological aspects of the tetrad were traced back to the first centuries AD. in the cultural melting pot of Alexandria (Egypt). The gnostic Logos, as the possible remains of the original Egyptian tetradic way of thinking, specified God with a secret name: the Tetragrammaton. In this approach, the division-thinking was seen as a holy process and God was equal to the principle of division.

KEYSERLING (1965, p. 396) talked of ‘dem vierfältigen Gott der Gnosis, der Selbst-erlösung: der tiefste menschliche Abgrund wird zu ihrem Ausgangspunt‘ (the tetradic God of the Gnosis, the self-redemption: the deepest human void becomes its point of departure). Furthermore, BAYLEY (1912/1968) gave – in his standard work on the ‘Lost Language of Symbolism’ – a summary of the name of God as a four-letter word and noted that ‘almost all peoples of antiquity possessed a name for the Deity composed of four letters’ (Tetragrammaton).

Four is in numerology the number of material order and associated with the earth (ENDRES & SCHIMMEL, 1984). Man travels through a chaotic world and invents divisions to assist in the orientation in time and place (fig. 194).


Fig. 194 – The eternal things on an Ojibway medicine lodge parchment. This parchment of the medicine man, collected in the early twentieth century and described as ‘ very old’, displayed the tour of the bear Mugwa through the four worlds. The following explanation of the figures was given by Olive P. DICKASON (1984): 1. Manido; 2. medicine man; 3. Medicine trees; 4. Snake manido watching the entrance; 5. Frog manido; 6. Shell with hands; 7. Lynx manido watching the road to the other world; 8. Wabeno wigwam; 9. Medicine man with cross; 10. Medicine man; 11. Chief manido of the third world; 12. Manido standing on a snake; 13. Medicine man, watching medicine wigwam; 14. Bear manido watching.

Division is of prime importance in the evocation of a cosmological picture, as given by the shamans of the people living in the Northern hemisphere (fig. 195). The position of the observer, as a micro cosmos within the universe, is a central theme.


Fig. 195 – The cosmology of the shamans, medicine-man of tribes living in the northern Hemisphere. Top left: A drum from the Altaier. In: ENDRES  (1984). Top right: A drum from Lycksele, Lapland. Painted on rendeer-hide. Statens Etnografiska Museum, Stockholm, Sweden. In: PALLOTTINO  (1966). Bottom left: World-tree on a shaman drum from Iceland. In: ELIOT  (1976). Bottom right: The shaman son of the earth. Jenissej-area. In: ENDRES (1984).

Witchdoctors and soothsayers have busied themselves, since way back when and in the most remote places of the world, which the design of a universe where the human soul could rest and take refuge of the oppressive powers of life and nature (fig. 196).


Fig. 196 – A drawing from a ‘bius pustaha‘ found in the area of Lake Toba, in northwestern Sumatra (Indonesia) as given by NIESSEN (1985). The ‘bius‘ is an agricultural ritual. A ‘pustaha‘ is an accordion-book made from bark with a wooden cover. Two snakes encircle the central ‘bindoe matoga’. They are the mystical ‘pane’ or ‘naga padoha’, figures from the underworld, making their annual passage through time.

The word ‘mandoedoe‘ indicated on the Indonesian isle of Sumatra a ritual, which was used to ward off the bad omen and gain happiness according to the prescriptions of the wizard. If these secret forces were evoked, it was necessary to take precautions: a magical figure – the ‘bindoe matoga’ – was painted in yellow, white and black flour close to the entrance of a house, where the ‘mandoedoe‘ was performed (fig. 197). The sides of the ‘bindoe matoga’ were approximately one meter long. In the middle laid an axe and a fowl’s egg. The right hand corner pointed to the east (Habinsaran) and the right side must be parallel to the house. An elevation or altar (langgatan) was erected in the left-hand corner. A red dog, a cock, a spade, a rake, a plough, a yoke, a comb and weaving tools are placed near the altar. Furthermore, some food is arranged in banana-leaves. The figure was washed away with the bodies of the sacrificial animals after the calling of the gods and the sacrifice of a dog and a cock (SCHNITGER, 1939 (p. 139)/1989).


Fig. 197 – The  ‘bindoe matoga‘ as given by SCHNITGER (1939; p. 139).

A complex world of four-fold symbolism can be found in Middle and South America. The cosmic representation of the Aztecs included a world view based on numerological division-thinking. The figures 4, 5, 7, 9 (4 + 4 + 1), 13 (4 + 4 + 4 + 1) and 20 (4 + 4 + 4 + 4 + 4) played an important role (RAYNAUD, 1901). The images on the so-called Sun or ‘calendar’-stones – flat rocks with circular inscriptions with the appearance of a mill-stone – symbolized a unity (fig. 198,199, 201).


Fig. 198 – These circular artifacts were found – and are still being found – in different areas in Mexico. The stones were used as calendars. The (Mid) American Indian cultures had a strong sense of division in time and place, which was reflected in their cosmological model. A four-fold element was nearly always present. Calender-stones from Mexico: 1. Peabody (Tenochtitlan); 2. Tizoc (Tenochtitlan); 3. Teocalli (Tenochtitlan); 4. Xochimilco (Distrito Federal); 5. Piedra del Sol (Tenochtitlan); 6. Humboldt (Zona del Golfo); 7. Xochicalco (Morelos); 8. Chichen-Itza (Zona de Yucatan); 9. Huaguechula (Puebla). In: TOMPKINS (1976).



Fig. 199 – Another sun wheel found during archeological excavations in Tonina (Chiapas, Mexico) (Photos: Marten Kuilman, 1988).


Fig. 200 – Explorer in Tonina (Chiapas, Mexico), 1988.

The history of the Indian people in Middle America is regarded as an ethnographic entity with a collective background. They are best known (to European observers) through the rule of the Aztecs, who lived in the post-classical period from around 900 to 1500 AD. (Tenochtitlan). The classical period from 200 to 900 AD. was dominated by the Maya in Yucatan (Tikal) and the Zapotecs (Monte Alban).


Fig. 201 – The central part of an Aztec calender-stone has a dominant four-fold division. The inner circle contains Tonatiuh, the Sun God, with a mask of fire, his attribute as King of the Planets. The sign ‘ome acatl‘ on his forehand points to the beginning of the year-count of ‘xiuhmolpilli’. The tongue hanging from the mouth, in the shape of an obsidian knife, urges his need for human blood and hearts. In the second circle contains the so-called ‘Ollin‘ symbol (of an earthquake or movement).

The four cardinal points are of primary importance. In the squares are depicted: 1. The jaguar (top right); 2. The crocodile head, god of the air (top left); 3. The rain and celestial fire (left below) and 4. The head of the water goddess Chalchiuhtlicue (right below). The squares depict the four world ages, which were earlier terminated with a catastrophe. Left from the upper triangle (pyramid): 1. The headdress of a fighter signifying a northern direction; to the right: 2. A knife of obsidian (tecpatl) indicates the east and the emergence of the sun. Below left: 3. The house of the rain god Tlalocan symboli-zing the west and right below: 4. A monkey represents the south. In: ANTON (1986) and: Explanation in the ‘Aztec Calender Easy Map‘; Garcia Valades Editores, S.A., Mexico 13, D.F.

The (Middle) American Indians envisaged the origin of (linear) time to lay in an event whereby the four Tetzcatlipocas strove to become a sun. The fifteenth century ‘Codex Borgia’, consisting of seventy-six pages on thirty-nine sheets of buckskin,  showed a picture with Omoteotl (or Tonacatecutli) in the middle – as the Father and Mother, Lord of Time – surrounded by the four Tetzcatlipocas. They represented, together with Omoteotl, the five periods of the world history (Von FRANZ, 1978). The Aztecs pictured themselves as living in a fifth and final age.

The four directions, colors and symbolic representations were given by SELER (1906) – in an older edition of the manuscript – as follows:

east                          –                  black                       –                  cipactli              –            crocodile

north                       –                  yellow                     –                  miquiztli            –           death

west                         –                  blue                         –                  ocomatli            –           monkey

south                       –                  red                           –                  cozcaquauhtli   –          vulture

The four-fold theme was further elaborated in plate 43 of the facsimile edition of the ‘Codex Borgia’ by Karl Anton NOWOTNY (1976). It depicts the ‘Underworld of the West’ as the final part of a series (Plates 29 – 46). SELER (1906) called this series ‘Die Höllenfahrt der Venus‘ (Venus’ ride to hell). The godhead is pictured in a ‘mamacouhticac‘-setting, which means a position with arms and legs spread out (fig. 202).


Fig. 202 – The ‘Road to Hell’ in the ‘Codex Borgia’ (Cholula-Tlaxcala area). In: ENDRES (1984).

RAYNAUD (1901) described – in an article on the sacred numbers and cruciform features of the Precolombian culture in Middle America – the ‘mamacouhticac‘ as ‘un dieu place dans un carrefour et de ses quatre membres indiquant les quatre chemins‘ (a god placed on a crossroad with its four members indicating the four roads). The symbolism is reminiscent of the cross and crucifixion in the Christian tradition.


Fig. 203 – A tetradic-cosmographical representation of the Aztecs from the ‘Codex Ferjervery-Mayer‘. The manuscript is kept in the Merseyside County Museum, Liverpool. In: NICHOLSON  (1967/1983).

The ‘Codex Ferjervery-Mayer’ gave another graphical expression of the four Tezcatlipocas standing at the four corners of creation (fig. 203). In the center is the god Tepeyollotl, the ‘Heart of the Mountain’, one of the Lords of the Night. He is depicted as a warrior. Rivers of blood flow towards the center-square and four holy trees grow from it towards the holy Tezcatlipocas, the four children of god with their own colors: black in north, red in the east, blue in the south and white towards the west. These colors differ, in this case, from the normally accepted compass colors: white (N), red (E), yellow (S) and black (W) and also from the above-given interpretation of the Tlaloc figures of SELER (1906). The difference may be due to a ‘night-time’ vision, rather than a ‘day-time’ vision.

The topmost cardinal point is identified as east, where the sun rises. A quetzal bird sits in a flowering tree, identifying the ‘Holy Land’ whence Quetzalcoatl arose. Two gods, one of the sharp cutting stone (Itzli) and another of the rising Sun, are in opposition, symbolizing dualistic powers.

On the right-hand side is the ‘Tree of the North’ with Cinteotl, the maize god, and Mictlantecuhtli, Lord of the Dead, facing each other. The theme of Life and Dead is obvious in the black land of the north.

Below is the ‘Tree of the West’ showing Xochiquetzal, the good goddess of flowers, in opposition with the bad goddess of drunkenness and witchcraft Tlazotlteotl. A hummingbird perched in the tree.

Finally, the tree of the South (to the left) is split. The southern area could be a place of redemption, and the crack might point to a place of escape (NICOLSON, 1967/1983). The rain god Tlatoc faces an unknown god, probably the underworld. Van ZANTWIJK (1977, p. 63) drew attention to the resemblance of this scheme with the geographical lay-out of the capital Tenochtitlan at the time of foundation in four quarters, with four dams as boundaries.

The spatial (four-fold) division of the Aztec people, symbolized in the four Tezcatlipocas, had its parallel in the (later) Mayan culture. Now they were called the four Bacabs and had a similar relation to the points of the compass. The Nahuas enlarged the idea of a quadruple ‘compass-god’ to such an extend that it came close to the medieval conception of the ‘four humors’. In both cases, a ‘philosophy’ – or rather a deliberate use of an ‘a priori‘ conceptual system based on a tetradic division – could have provided the analogy of ideas.

There were two Aztec calendars, the so-called Tzolkin-cycle of 260 days (or ‘tonalpoalli‘, counting the days, resulting in the ‘tonalamatl‘-calender) and the ‘haäb‘-cycle of 360 + 5 days. Each day had therefore two names, one for every cycle.

The combination of Tzolkin– and Haäb-cycles produced a cyclic period of fifty-two (short) years (or 73 tzolkins): 52 x 365 = 18.980 = 73 x 260 (GILBERT & COTTERELL, 1996; p. 28). The fifty-two years (combination) cycle is divided in four parts of each thirteen-year (fig. 204). The four parts are associated with the directions of the wind, joining time and place together. The north-direction is to the left. A similar ‘calendario‘ in the work of Juan de Tovar has the ‘rabbit’ cycle orientated to the north (Afb. X.2 in VAN ZANTWIJK, 1977, p. 204).


Fig. 204 – The combination of the two Aztec calendars – the tzolkin– and haäb-cycle – resulted in a cyclic period of fifty-two years. The subdivision in four series (of thirteen years) was composed of the four primary day-signs: Canas (acatl, reed), Pe Dernales (tecpatl, flint knife), Casas (calli, house) and Conejos (tochtli, rabbit). In:  DURAN (1971).

The cyclic movement of time started in the east with the series of the ‘Canas‘ (reed) and turned anti-clockwise to the north of the ‘Pe Dernales‘ (the flint knife). The second series (of the knife) made the same movement to the ‘Casas‘ (house) in the west. The third series (of the houses) rotated anti-clockwise to the ‘Conejos‘ (rabbit) in the south. And this last series (of thirteen years) jointed again with the ‘Canas‘ to complete the cycle (of fifty-two years). When this happened the feast of Nexiuhilpiliztli (the Completion or Connection of the Perfect Circle of Years) took place (DURAN, 1971).

This short excursion in the worldwide occurrence of tetradic thinking will be cut short here. The preliminary conclusion can be reached that this type of thinking is a – generally accepted – cognitive stage to allow static and dynamic contrasts to interact in a balanced way. Its association with the earth is wholly justified, because everything, which occurs on earth – including the human world of thoughts – fits in a tetradic frame.

The tetradic structure included the four elements: fire, air, earth and water. The division covered every aspect of the material world and provided – by analogy – the base for dynamic interplay of thoughts. At present, it is often not realized that behind apparent ‘numerological’ features, a world of essential division thinking is hidden. The use of the main directions of the compass is, for example, a living evidence of a quaternion, dividing the unlimited space of the universe.

Robert Boyle (1627 – 1691) initiated – with his book ‘The Sceptical Chymist’ (1661) – the gradual decline of the time-honored picture of ‘undividable’ elements. He criticized the Aristotelian physics on the high tide of dual thinking in Europe. His contemporaries in the seventeenth century were completely forgotten that ‘undividability’ never was the absolute criterion for the existence of the four elements in the first place. In stead, as Aristotle might have envisaged, it were the elements that stood for the different worlds of visibility, characterized by their own particular way of ‘division’.

The (Western) view on a cyclic, tetradic world in the sixteenth and seventeenth century was strongly influenced by the conquest of Middle and South America. The Spaniards (and later the Portuguese, Dutch and English) learned to their temporary advantage, that higher division thinking (of the Indians) made them vulnerable and led to defeat.

The European dualistic outlook (around the year 1500 AD) denoted the tetradic world as fatalistic and deterministic. However, some five hundred years later, we are able to see that this opinion only reflected a time-bound frame of mind. Determinism and fatalism are, in itself, dualistic terms, born in a world of opposites. The outcome of events was, in a wider sense, not relevant. This conclusion might be tragic in the short run, but it provides a form of insight, which is the highest reward in life.

The boundary between original tetradic thinking and the use of the four-fold as a convenient way of ordering, is difficult to establish. The development of the latter kind was particularly manifest in the rationalistic era of the European cultural history, starting after 1500. The four-fold had in this period no association with tetradic thinking, although it frantically copied its outward appearances. A modern researcher, keen on quantity, could easily be misled in the belief that the relative abundance of four-fold imagery, was also a sign of the actual use of tetradic thinking. This was not the case.

A good example is an etching by an artist from the school of Hendrik Goltzius (1558 – 1616), depicting a physician as God, angel, human and devil (fig. 205). These appearances are related to the position of the medical doctor in the eyes of a patient: first – as a God who can decide over life or death; second – after recovery sets in, as a ministering angel; third – when everything is well again, doctor and patient are on equal terms, both humans; and fourth – when the bill is presented, which is obviously too high. Then the doctor is seen as a devil (HUISMAN, 1992).

The allegorical picture was published in 1587. Robert de Baudous in Amsterdam published a new series, based on Goltzius’ example and engraved by Johannes Galle, in 1609.


Fig. 205 – The physician as God, angel, human and devil. An example of the use of four-fold division in a symbolic way in the late sixteenth century by an artist of the school of Hendrik Goltzius (1558 – 1616); Haarlem, 1587. Although four phases are used, it is obvious that the oppositional element (God versus devil) is more important.

The Museum Boerhave at Leiden acquired in 1992 four oil paintings of the Antwerp-born Jan Jozef Horemans the Younger (1714 – 1790) dated from 1752, with the same motif (REITSMA, 1992). It proofs the persistence of a rather unknown iconographical element in time (fig. 206).



Fig. 206 – The doctor as God (top left), angel (topright), human (bottom left) and devil (bottom right). Paintings  by Jan Jozef Horemans the Younger (1714 – 1790) dating from 1752. Boerhave Museum, Leiden.

HUISMAN (1992) pointed to the text of an early sixteenth century epigram on one of Horemans’ paintings, which was derived from the book ‘De medico monacho‘. Here the physician is portrayed as a ‘Aesculapius trifons’. The reference to a triple division is clear:

Tres medicus facies habet, unam quando rogatur

‘Angelicam’. Mox est cum juvat esse ‘Deus’

Post ubi curato poscit sua praemia morbo

Horridus apparet terribilisque ‘Sathan’

(Three faces has a physician, if he comes as desired, he is an angel. If his cure helps, he is a God. However, when he wants his pay, he looks like a terrible and horrifying devil)

The meaning of numerology – as the collective noun for all numerical approaches to reality – is in itself determined by the type of division thinking, which is used to validate it:

In a dual world of opposites, numerology is either mythical nonsense or bears the ultimate truth. There is no way in between. The decision (of its value), taken on the outset of the communication, determines the further handling of the matter. In a (dynamic) three-division an intermediate state can be envisaged, and nuances can express themselves. Finally, in a quadralectic world, the meaning of numerology can be ascribed to four cognitive positions:

I. Numerology in an undivided world. This situation is unimaginable and leads to a kind of ‘Gödel’s theorem‘;

II. Numerology as an idea (comparable with Vaihinger’s ‘hypothesis‘) provides a model, which is dependent on the number of divisions;

III. Numerology as the application of a fixed division in a communication;

IV. Numerology in a world of infinite division-possibilities (comparable with Vaihinger’s ‘Fiction‘).

These interpretations exist together and none of them can claim an exclusive right. Partners in any discussion should make their position clear. Once this is done, a wide field of understanding is possible, without being confused about intentions and interpretations.

The Biblical books of the evangelists Matthew, Marc, Luke and John in the Christian faith articulate, for instance, the same idea as the four steps of redemption in the Islam. The ‘schari’a’ (the law), the ‘tariqa‘ (the way), the ‘haqiqa (the truth) and the ‘marifa‘ (the gnosis) are associated with the transcendental stages of mankind: from ‘nasut’ (mankind), ‘malakut‘ (angels), ‘dschabarut‘ (power) to ‘lahut’ (god). It is a challenging thought that Christian and Islamic beliefs can find each other in a peaceful tetradic world view rather than fighting each other in dualism.

The figure four is also connected with an intellectual development in European history, known as alchemy or ‘black magic’. The major influence of these early branches of chemistry and psychology can be placed, with imprecise boundaries, between 1200 and 1800.

Masters of the alchemic-magical research were Albertus Magnus (1193 – 1280), Raymond Lull (1235 – 1312), Arnaldus de Villanova (1235 – 1313), Henry Cornelius Agrippa (1486 – 1535), Paracelsus (1493 – 1541), Jakob Boëhme (1575 – 1624), Michael Maier (1568 – 1622) and Robert Fludd (1574 – 1637). A number of these masters were depicted on the title page of Crollius’ ‘Basilica Chymica‘ (1612; first edition 1609) (fig. 207).


Fig. 207 – The title page from Crollius’ book ‘Basilica Chymica’ exhibited some of the great alchemists.  Oswald Croll (ca. 1560 – 1609) gave many details of alchemical processes and advocated the doctrine of signatures. The work became a major work of iatrochemistry, published in many editions. A symmetrical, three-fold division is most prominent in this picture.

There are different opinions about the origin of the word ‘alchemy’ (THOMPSON, 1932). It is possible that the Arabic ‘al‘ is placed before the Greek word ‘kimia‘. The Roman emperor Diocletian (who reigned between 284 and 305 AD) used this latter word in a decree against the Egyptian ‘kimia‘ or transmutation of gold and silver. Suidas, in the eleventh century, pointed to the ‘knowledge of Egyptian art, Chemi or Cham of the Black Country, which was the old name for Egypt.’ Other writers refer to Alexander of Aphrodisias, as the first to use the word ‘chymike‘.

The reference of alchemy and ‘black magic’ to Egypt and Alexandria suggests a connection with tetradic thinking, which was known and practiced in the old Egyptian culture. BURCKHARDT (1960) called the late-Egyptian city of Alexandria a melting pot of different cultures. Even so, the four-fold way was not dominant in the alchemical endeavors in Europe. Most of the above-mentioned writers had a notion of division thinking (fig. 208), but none of them saw it as a system of thought.


Fig. 208 – A healthy person attacked from four directions by diseases. This picture was given in Robert Fludd’s book ‘Integrum Morborum Mysterium’ (1631). In:  DEBUS  (1978).

Even today there remained a fascination for the mystical and alchemical achievements, despite the often-negative approach by modern scientists. A possible explanation can be derived from a viewpoint of power: there might be a certain envy in the world of higher division-thinking towards the easiness and sweet ignorance in which power can be obtained on lower levels of division-thinking.

Robert Fludd’s early geological thoughts about the mechanics of the earth do express some of the honesty in which creative-dualistic thinking can develop. GODWIN (1979) called him adequately a ‘Surveyor of Two Worlds‘. Brian VICKERS (1984) painted a picture of a ‘fight’ in the early seventeenth century (1619) between the new scientific sense of identity and the old magical thoughts of Fludd. This approach has since been challenged – and with good reasons – by William R. NEWMAN (2009).

Fludd (1574 – 1637) had a medical background, which provide him with the imagery to depict the movements inside of the earth as generated by two huge syringes under the influence of the sun. This depiction was, in its most primitive form, a forerunner of the modern theory of the convection currents and plate tectonics (fig. 209).


Fig. 209 – The geology of the earth as presented by Robert Fludd in his book ‘Philosophia Moysaica’ (1638). This unadorned illustration is a good example of oppositional thinking. It led, in this case, to a creative solution for what is presently explained by a theory of convection currents and plate tectonics.   In: FLUD,  Rob.,  al  de Fluctibus.  Doctore Oxoniensi  (1638).  Philosophia Moysaica.  Petrus Rammazenius,  Bibliopola, Gouda (Courtesy Biblioteca Hermetica, Amsterdam). Sectionis Primae, Lib. V., fol. 52/53. In: GODWIN (1979) and VICKERS  (1984).

Genuine tetradic thinking in a magical environment or in the field of astrology was often disguised by the emphasis on material proof. A frantic search for numerological characteristics, for instance, in the division of the zodiac and planets (fig. 210) or in the division of the (twelve) ‘houses‘, should give some sort of truth. However, the mechanism of thinking was lost out of sight. In fact, the cyclic displacement between the observer and the observed, and the new values created in this process, were far more important than the ‘proof’ which was found in an arbitrary quantity of numerological features.


Fig. 210 – Two examples of division thinking in the occult sciences. Left: a study of geometrical angles and aspects of the Zodiac and the planets depicted on a German woodcut from the sixteenth century. Astrology uses many forms of division, but the system as a whole – although dynamic – is not related to a specific form of division thinking (KENTON, 1974). Right: A horoscope of the Bohemian army commander and politician Albrecht von Wallenstein (1583 – 1634), created by Kepler around 1608. The German mathematician, astronomer and astrologer Kepler (1571 – 1630) recorded in his manuscripts the horoscopes of some eight hundred people (Kepler-Manuskripte (18.250v) in Pulkowo, Leningrad) (TEICHMANN, 1985).

At the end of this short survey of the number four, attention is drawn to the so-called magic squares. These squares are composed of rows of numbers, which provide, if added together, the same sum in either horizontal, vertical or diagonal direction (CAZALAS, 1934).

Athenaseus Kirchner (1601 – 1680) described in his erudite book about the ‘Arithmologia (sive De abditis Numerorum mysterijs)’ (1665, p. 233) ‘De Magicis Amuletis’ and assigned the different types of magic squares to planetary deities: the three-row version is named after Saturn (15), the four-row version after Jupiter (34), the five-row after Mars (65), the six-row after the Sun (111), the seven-row after Venus (175), the eight-row after Mercury (260) and the nine-row after the Moon (369). The figures between brackets give the total sum of a single row, added either in horizontal, vertical or diagonal direction. The magic squares were widely used in the seventeenth century as charms and amulets, with the intention to gain the magical power of the deity (MICHELL, 1969) (fig. 211).


Fig. 211 – A Jupiter-amulet. The sum of an individual row of numbers is 34, independent of the way the numbers are added either horizontal, vertical or diagonal. The total sum is 136. Richness, peace and unity were in store for the wearer of this silver amulet, if Jupiter was dominant when the amulet was engraved. The symbol of Jupiter is a stylized number four. The eagle is his symbol. In: ENDRES (1984).

The most famous magic square was depicted in Albrecht Dürer’s woodcut ‘Melancholia‘ (or ‘Melencolia‘ in his own spelling), dating from 1514. It was the year when his mother died (17.5.1514) and the influence of Saturn, with the two-fold face of fate, was immanent. The magical square on the wall behind a sitting angel was a ‘Jupiter’ diagram (and not a ‘Saturn’-diagram, like BUTLER (1970, p. 9) indicated wrongly).

CAZALAS (1934) reckoned the picture of the magic square in Dürer’s ‘Melancholia‘ to be the first of this kind published in Western Europe. He mentioned, just like PUSCHMANN (1983), Cornelius Agrippa and his book ‘De Occulta Philosophia’ (II, Ch. XXII) as a possible inspiration. The origin was Chinese (Square of Lo Shu, 650 BC) and carried forwards by the  Arabs, when they came in contact with the Indian culture and their mathematical combinatorics. Arabic manuscripts on this subject are known from the seventh and eighth century (Ibn Esra).  Magic squares of order 5 and 6 were given in the Rasa’il Ikhwan al-Safa, the ‘Encyclopedia of the Brethren of Purity’, dating from c. 983 AD. The Byzantine scholar Manuel Moschopoulos wrote a book called Erotemata grammatika’ (‘Grammatical Questions’, first printed in Milan, 1493), but he also busied himself with a work on magic squares (about 1315), while being in jail.

Durer’s woodcut of ‘Melancholia I’ was full of symbolical attributes. Melancholia, associated by Aristotle with creative talent, was long regarded as one of the four human characteristics with the least value. A reappraisal took place in the early sixteenth century. The association with deep insight and wisdom (leading to sadness) became dominant. The upgrading was, according to Von WINTERSTEIN (1929/1989) due to the best-selling work of Marsilio Ficino, titled ‘de Vita triplici’ (better: De vita libri tres, 1482-1489). Ficino described – in book 1 (De vita sana) – the characteristics of man born under the unlucky sign of the planet Saturn (KASKE & CLARK, 1989).

The polarity of the character of Saturn – pointing on one side to the god Chronos as the oldest, saddest and loneliest deity and on the other side to an overpowering deity (of time) – was understood in the dualistic setting of the European cultural history at that time. Saturn (and melancholia as a temperament) symbolized opposition: from the highest peaks of moral consciousness to the deepest abyss in mental being. In psychiatry this type of personality has been defined as ‘manic-depressive’.

The magical-astrological features on Dürer’s woodcut has been subject to divergent speculation (GIEHLOW, 1903/04). On the wall hangs the ‘mensula Jovis‘, a talisman. Jupiter, the son of Saturn-Cronos, trew his father in the river Tartarus and chained him. He conquered an evil power in the act. One can subsequently gain some of his powers by wearing his amulet and avoid the influence of Saturn.

Rosemarie PUSCHMANN (1983) studied the iatromathematical importance of the magical quadrants extensively by means of Thomas Mann’s book ‘Doktor Faustus’ (1947). She mentioned four ways (‘modi‘) to read the natural sequence of figures on the Jupiter amulet (p. 42: 6.4.3. Die Figuren der vier Modi im Magischen Quadrat). Pushman suggested that Thomas Mann (1875 – 1955) was drawn to these possibilities by Hans Peschick. She referred to a letter to him dated the 24th of November 1949 (fig. 212). The relation of Albrecht Dürer with Thoman Mann was also described in an article by Michael PALENCIA-ROTH (1980).


Fig. 212 – The four ways (‘modi’) of reading the natural sequence of figures in a magical quadrant.

The four ‘modi‘ are, in fact, eye-movements performed to read the sixteen figures in a natural sequence. The first row (1 – 4) was read from the bottom right (1), to the top (2, 3) and back again to the bottom left (4). The eye-movement made an arch. The next three rows resulted in the same graphical figures. The movements were called: R – UK – K en U:


——————– 1.  R   –  the ground row (‘die Grundgestalt oder Reihe‘: 1, 2, 3 4);

——————– 2.  UK –  the reversal of the retrograde row (5, 6, 7, 8);

——————– 3.  K    –  the retrograde row (‘der Krebs‘: 9, 10, 11, 12);

——————– 4.  U   –  the reversal of the ground row (13, 14, 15, 16).

The four ‘modi‘ or ways of reading the magical (Jupiter) quadrant are part of the theoretical foundation of the twelve-tone system in music, as developed by Arnold Schönberg (1874 – 1951) in the early decades of the twentieth century. Thomas Mann associated the four ‘modi‘ with sound figures and the ‘musikalisch-literarische Zwölftonreihe‘. The twelve-tone system, as it was developed in the early twentieth century, used a chromatic scale with four ‘modi‘: a row, a reversed row, a retrograde row and a reversed retrograde row (ADORNO, 1941/1975; SCHÖNBERG, 1976).

The modern twelve-tone system puts an emphasis on the tension between the horizontal (melody) and the vertical (chords). In the ‘old’ polyphony there is a priory for the melody (the succession of tones), while in the new sound of homophony, the accent is shifted to the chords (the simultaneity). Music is therefore subject to a ‘rationale Durch-organisation‘: the tones of a piece of music have to fit, like the numbers in the magic square, into a particular frame (in case of the ‘mensula Jovis‘: a sum of 34 (C = 34).

The beauty of music is no longer primary found in the succession of tones (the melody), but in the order of tones within a certain system (the chord). There is dialectic tension between melody and chord, in which the first view regards the second as ‘atonal’. Within a two-fold (dialectic) frame of mind these different approaches are oppositional and difficult to reconcile.

The continuity (of the melody) was comparable with the ‘constante rhythme‘ of the three-fold way of thinking, as it was highlighted by Bach (1685 – 1750) and  Händel (1685 – 1759). The sequence pointed to the priority of a ‘vertical’ scale. The hierarchical organization provided the Baroque musicians, and later the ‘classical’ composers, with a great freedom (HÖWELER, 1947, p. 173). This freedom ended in the development of the twelve-tone system.

ADORNO (1941/1949), stated in his book on the ‘Philosophie der neuen Musik’: there are no ‘free tones’ anymore. Composing music became, in the opinion of PUSCHMANN (1983, p. 54), a kind of ‘Geheimwissenschaft, verwandt dem Arkanum der Magie’ (secret science related to the arcanum (deep wisdom) of magic).

All observers agree, that Dürer’s portrayal of ‘Melencolia I’ was full of symbolic meaning, but the interpretations leave room for different opinions. Albrecht Dürer does not make himself particular clear, just like in much of his other work, in the use of symbolic or numerological elements. He was teasing his viewers from early on, without giving them a clue of the figurative meaning (if there was any).

Von WINTERSTEIN (1929/1989) emphasized, in a psycho-analytical study of Dürer’s illustration, the anal-sadistic element of melancholics. The female figure of ‘Melancholia’ was associated with his mother (who died in the same year as the woodcut was made (1514) and the putti with himself, as an angel-like boy. The sleeping dog was interpreted as a symbol of his deceased father.

‘The magical square underneath the clock gives away’, in Von Winterstein’s observation, ‘in its spatial setting an unconscious-psychic relation with the penis functions and the process of counting (neurotic urge to count).’ This interpretation appeared rather farfetched, but might satisfy the believers in Freudian psycho-analysis.

It can be noted that Anasthasius Kirchner in his ‘Arithmologia sive De abditis Numerorum mysterijs’ (1665), CAZALAS (1934) and MICHELL (1969) gave a different type of the Jupiter-quadrant, with the same sum (34), but with a different arrangement of the figures:

                                        4                       14                            15                               1

                                        9                         7                             6                               12

                                        5                        11                            10                              8

                                       16                        2                              3                              13

There are, in fact, 880 possibilities of a magic square of the fourth order (DUDENEY, 1917).


It would lead to far astray to explore the interesting subject of the magic square. The relation between this mathematical curiosity and the genesis of the quadralectic way of thinking is not clear cut, but the link with atonal music is interesting in its own right.

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WINTERSTEIN, von, Alfred (1929/1989). Dürers ‘Melancholie‘. Een psychoanalytische beschouwing (tr. Wilfred Oranje). Also in: Imago, 15 (1929). Boom, Meppel/Amsterdam. ISBN 90-6009-894-3

ZANTWIJK,  van,  Rudolf (1977).  Handel en wandel van de Azteken. De sociale geschiedenis van voor-Spaans Mexico.  Van Gorcum,  Assen/Amsterdam. ISBN 90 232 1509 5

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