Principia of Polyzodiacal Astrology by Ptolomei Svarogich

( ptolomei@levante.org )

( Ph.D Thesis in Astrology. English editing by Ed Falis. The public defence of the thesis took place at the United Russian Astrological Congress 22 June 1996 11h48m UTC in Moscow. The related materials and software can be found at www.levante.org )

“…Astrologia, omni superstitione…eliminata,…” Tommaso Campanella

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Contents 1. Introduction. History of astrology and medieval primary sources.

Astrology as a knowledge system............................................................................................. 3 1.1. The age of modern astrological tradition............................................................................ 5 1.2. Development of the conceptual framework and

calculating algorithms of medieval astrology. .................................................................... 8 1.3. Astrology in 20th century. ................................................................................................... 9 1.4. The conceptual framework of astrology and natural science principles of thinking.. ...... 12

2. The Zodiac of a massive heavenly body ................................................................................ 17 2.1. Introduction. ..................................................................................................................... 17 2.2. Theoretical model of a zodiac for a born, situated in the vicinity of a heavenly body. .... 18 2.3. Choice of the reference point on the local equator (zodiacal circle). ............................... 25 2.4. Examples of zodiacs.. ....................................................................................................... 27

2.4.1. Terrestrial zodiac. ................................................................................................. 27 2.4.2. The Lunar zodiac.. ................................................................................................ 32

2.4.2.1. The Lunar nodes. .................................................................................... 33 2.4.3. Solar and planetary zodiacs................................................................................... 33

3. The mutual projection of two zodiacs. .................................................................................. 34 4. Symbolic times......................................................................................................................... 36

4.1. Symbolic mappings. The world line of a born ................................................................. 36 4.2. Traditional systems of symbolic time............................................................................... 38

4.2.1. Solar-terrestrial progressions. ............................................................................... 38 4.2.2. Solar-terrestrial directions..................................................................................... 39 4.2.3. Profections. ........................................................................................................... 40

5. Aspects, orbs and the technology of interpreting pinpoint accuracy.................................. 41 5.1. Aspects.. ........................................................................................................................... 41 5.2. Orbs.. ................................................................................................................................ 41 5.3. Experimental orb of "exact" aspects and the accuracy of astronomical and astrological

calculations....................................................................................................................... 43 5.4. Event-trigger points of the horoscope.. ............................................................................ 43

6. Astronomer’s notes. ................................................................................................................ 45 6.1. Coordinates of planets for the location of a born. The correction for parallax. ................ 45 6.2. Direct and retrograde planets. Stationary points............................................................... 46 6.3. Zodiacal conjunctions....................................................................................................... 47 6.4. The Natal horoscope as a multi-dimensional chart........................................................... 48

7. Precision rectification of the creation (birth) time and symbolic times.............................. 51 8. Conclusions.. ............................................................................................................................ 54 9. Thanks...................................................................................................................................... 56 10. Definitions. ............................................................................................................................... 57 11. References................................................................................................................................ 60

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1. Introduction. History of astrology and medieval primary sources. Astrology as a knowledge system. Man has long been interested in the order of the universe and in the place of human beings within that order. A longing for a knowledge, exact knowledge is perhaps the most distinctive characteristic of the human species. Namely cosmogonist hypotheses were among the first significant human descriptions of the surrounding world. Many such systems eventually froze in their development and became systems of doctrine, eventually transforming into religious systems. Such ideologies no doubt met certain human needs for a stable context in which to conduct societal activities. In spite of this tendency, certain communities of people, fulfilling a role similar to that of the modern scientific community, continued to study and develop conceptual frameworks for understanding the world.

In any sufficiently rich and formalized conceptual system it is always possible to find statement that is impossible to prove or disprove within its set of assumptions1. This characteristic of formal systems guarantees that no definitive and exhaustive description of the universal order can be attained. The Universe, Nature, God is always a developing and self-enriching essence. Not only does our knowledge of the Universe evolve, develop and become more profound, but the Universe itself does as well. We do well to consider the Universe as the development and realization of the divine idea. In the Gospel of John, God is identified with Logos (Greek λογος), a term which is clearly a conceptual framework for the Universe as it is used in the original of that text in ancient Greek.

Among the oldest known doctrines are the Gnostic writings, which are fairly criticized by both modern Christian historians and the ancient representatives of official theology [1]. The ancient Gnostic schemes known today can hardly cause other sensations than of dying or of impending doom. However alongside the dead Gnostic schemes, another form of living and developing knowledge of the Universal order (or God) existed from extreme antiquity – known to us under the name of astrology. The most prominent quality of astrology is its dialecticalness, as its main object of study is time and the qualitative changes of divine creations with time. The structure and essence of the time, the fundamental difference of one moment of time from another, is the main object of astrological study.

The core of astrology is not its calculations or techniques, but its symbolism, the laws and structure of which permeate the world and form the backbone of the Universe. This symbolism allows us to see intrinsic relationships among phenomena based upon their symbolic unity.

Astrology also highlights how the future crystallizes within the past. Our ability to forecast the short-term future that is the scope of a human life, its dynamics, its fullness of events of universal importance or of personal character illustrates of the existence of the future as a seed within the times at which the planets were just being formed. Astrological calculations based on the motions of the planets are in fact “counting sticks” by which we learn the divine idea of which our Universe is

1 In mathematics this statement is known as Gödel’s Theorem.

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woven. In assimilating the rhythms of this weaving, we form inside ourselves the spark of the divine consciousness, becoming like God.

The particular value and distinctiveness of astrology among systems of human knowledge is that the astrological "counting sticks" allow us to most profoundly understand this divine idea of our Universe. Perhaps this is the real reason astrology is the most ancient system of the human knowledge. As the author of this work supposes that we are all doomed (or blessed) to eternal life in this inconceivably old and young, beautiful and horrible world, rather than in some paradise or Hades, the best thing we can do is to penetrate as deeply as possible into God’s idea. We can become active parties in the creative work that is continually taking place in our sight. So, although forecasting and analysing events is not the primary task of astrology, it is one of the best ways on the human plane of evolution to recognize and to integrate the divine idea.

The 20th century is the age of the revival of astrology as a system of exact knowledge. Astrology has been reborn with a substantially changed appearance. It has gained a more humanistic, even humanitarian, nature. To a considerable extent it was revitalized as a conceptual basis of psychology. Some of the outstanding psychologists of 20th century (e.g. Carl Gustav Jung) have found that its notional system for describing human consciousness is very close to the structure of astrological symbolism. The majority of prominent astrologers of the 20th century (e.g. Dane Rudhyar) have been representatives of its psychological branch.

New names have even been invented for astrology such as astrosophy, astropsychology, cosmobiology and so on. Certainly, these names reflect the interests of people using the astrological notional structure and its methods of calculation in their activities. But the use of such euphemistic names in place of “astrology” shows not only the widespread intolerance of astrology, but also the sheer affront that the existence of such a system of knowledge offers to certain powerful communities in the modern world. The Christian fundamentalist communities are a prominent example.

This situation reflects the significant degree of dogmatism in religious doctrines and the drift of religious world-views as forms of knowledge toward becoming the empty shards of their formerly vital engagement with the universal order. The largest astrological organization in the USA carries the faint-hearted and ambiguous name “The National Council for Geocosmic Research”2.

It is quite possible, that the growing influence and authority of the Catholic Church in the 16th and 17th centuries was a contributor to the decline of medieval astrology. Nicholas Campion3 considers that the "denouncement" of astrology by Saint Augustine, one of the most authoritative church fathers, had a significant negative impact on the development of astrology and brought serious problems to the astrologers of Christian Europe [2]. Still, astrology had not completely disappeared; it continued to exist into the 18th and 19th centuries in a form similar to fortune telling. This period left an imprint of charlatanry on astrology, and led to a “cookbook” form of practice. Astrology still has not completely recovered from this heritage of its own "Dark Ages”.

2 Let us note here that the loss of a true name can be sufficient cause for the loss of the essence of an organization. 3 The president of the Astrological Association of England.

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In the 20th century, not only has astrological symbolism been rehabilitated: the methods of astrological calculations as an exact knowledge have also been restored. One of the outstanding figures in the astrology of the 20th century was Alfred Witte (2 March 1878 - 2 August 1941) [3]. While not all the techniques of his “Uranian” school are indisputable, his development of Kepler’s method of discrete symmetries (planetary pictures), his use of zodiacal harmonics, and his simultaneous use of multiple equal house divisions as facets of interpretation, have had a significant effect on the development of astrology, and will continue to do so for the foreseeable future. The second major event in the astrology of the 20th century was the development (or perhaps rediscovery) of the “Topocentric” method of domification as a natural successor to Placidus domification. The Topocentric method was made possible by enhancements to the accuracy and precision of astronomical calculations [4,5].

It is regrettable that astrology as an exact science is unknown not only to the mass of the educated public, but also to the majority of professional astrologers.

Unlike the majority of modern sciences, the history of astrology comprises many centuries. The full reclamation of the achievements and discoveries of medieval and ancient astrologers is impossible without the most attentive study of the history of astrology as it interacts with general history. Among existing disciplines only theology and history compete with astrology in age. But theology discredited itself with dogmatism not only in the view of 20th century intellectuals, but much earlier - in the 18th and 19th centuries. This fact was even reflected in some European languages such as French and Italian (e.g. compare the meaning of words teoretico and teorico in Italian).

At the same time, the discipline of historical study has too often been used as a tool for political domination. Authorities have never been interested in broad (and overdue) revisions to its methods and conclusions. Each generation of politicians has preferred to not touch what their predecessors have done, and has merely modified interpretations of historical periods to suit their own ideological needs. Furthermore the scholastic tradition in science that appeared during the Renaissance perniciously influenced many scientific disciplines for several centuries. Only the physical sciences have completely overcome its influence. Even biology has not completely freed itself from the scholastic tradition.

1.1. The age of modern astrological tradition. The traditional and generally accepted opinion is that the names of the zodiacal signs are approximately two thousand years old. This dating is calculated from the value of the precession of the equinox of 50" of ecliptic longitude per year and that the signs of the tropical zodiac, representing the 30º sectors of the ecliptic as measured from the vernal equinoctial point, have taken their names from the corresponding constellations superimposed on them at that time and have gone from them by 30°. We can assert with confidence that the existence of the 12-fold division of the ecliptic, together with the widely used names of the zodiacal signs, is a sufficient proof of the age of the fundamental notional system of astrology, despite the loss of this original system.

µαθηµατικη συνταξις (i.e. The Mathematical Systematic Treatise), known under the name of the Almagest and ascribed to the ancient astronomer Claudius Ptolemy

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from Alexandria, is considered a second proof of the ancient origin of astrology. The "Mathematical Treatise in Four Books", known as the Tetrabiblos (τετραβιβλος) [6], is also ascribed to Ptolemy. The generally accepted belief in the existence of a developed astrological knowledge system in antiquity is founded on the dating of the Tetrabiblos consistent with the author’s life and with the dating4 of the Almagest.

Almagest is a collection of 13 books of astronomical contents, including the results of astronomical observations. It is alleged that the period of the astronomical observations included in Almagest corresponds to 127-141 CE (or AD – anno domini in latin medieval tradition). These figures are the rationale for dating the writing of the Almagest in the 2nd century CE. However, a modern mathematical analysis of the astronomical observations of the Almagest, carried out by the well-known 20th century astronomer Robert Newton, caused him to conclude that the Almagest was composed in some other epoch. Robert Newton even published the book The Crime of Claudius Ptolemy, in which he attempted to prove that the data of the stellar catalogue was forged and that it was actually the result of recalculation [7].

The astronomical observations of Ptolemy were independently analysed by the Russian mathematician A.T. Fomenko5 and his co-authors [8]. Let us sketch their results. First, the astronomical stellar catalogue of Almagest is a copy of the catalogue of Hipparchus, the ancient Greek astronomer from Rhodes. The traditional chronology places the life of Hipparchus in the 2nd century BCE. Second, the stellar catalogue of Almagest could only have been created over the interval from the 7th through 13th centuries CE. Moreover the most probable time of its creation is the 10th or 11th century. This dating is consistent with the new version of the universal chronology, presented in the works of A.T. Fomenko and his co-workers [9]. From this standpoint Ptolemy was no fraud: he simply lived more than a thousand years later (13th or 14th century) than is accepted in the traditional chronology. But what are we to make of the displacement of the zodiacal signs relative to the corresponding constellations as given in Almagest? The history of the first printed editions of Almagest shed light on this question.

The first edition of Tetrabiblos was printed in Nürnberg in 1535 CE [6] in Greek, but it contained the Latin translation as well. The first edition of Almagest was printed 2 years later in 1537 CE in Cologne in Latin (“Nunc primum edita, Interprete Georgio Trapeuzuntio”) with stellar coordinates given for 16th century. The Albrecht Dürer engravings of constellation pictures used within the text to locate stars in the catalogue have original inscriptions dating them at 1515 CE. Therefore, the stellar catalogue description appeared after 1515 CE6. Some of the constellations — Ara and Pegasus - can be seen inverted from their normal appearance in the sky of the northern hemisphere. This absurdity seriously disturbed the medieval astronomers, including Coppernic (1473-1543). Therefore, Dürer must have considered his drawings of the constellations only as works of art; he did not even bother to verify how these constellations were seen in the sky. Succeeding editions, including the edition in Greek from Basel (1538), nowadays considered as "original", contain the stellar coordinates in which ecliptic longitudes

4 We mean dating based upon its astronomical catalogue. 5 Currently a member of the Russian Academy of Sciences. 6 The description of the stars’ positions in the sky, provided by the catalogue, uses small details of

the constellation pictures in Dürer’s engravings.

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correspond to those of the 2nd century CE. Perhaps the publisher of the Latin edition was alerted to the discrepancy between the stellar coordinates in the catalogue and the dating of the of Ptolemy’s life; he "corrected" the discrepancy in later editions by shifting the ecliptic longitudes of stars by 20º exactly7. Exactly thus located zodiacal signs (relative to the equinox of the 2nd century CE) could give matching names to the corresponding areas of the starry sky. These collected facts force us to question the approach of the so-called Age of Aquarius. Besides, from our standpoint the zodiac itself and the notion of the equinoctial point as a crossing point of the ecliptic and the celestial equator gives no basis in favour of this generally accepted opinion.

If we reject the legends about astrologers of antiquity of the same ilk as the story about Nectanebus, the astrologer of Philip of Macedonia, who had predicted the birth of Alexander the Great (and who could actually be the father of Alexander), we have then no other reliable material in hand beyond the treatises of the medieval astrologers (astronomers8), dated not earlier than the 14th century, and Almagest, whose author lived in 13th or 14th century. The dating of the life of Campanus (1239 CE–1296 CE), the author of one of the oldest house systems, is also questionable. It is possible that he lived even before the author of Almagest9. We had no opportunity to get acquainted with Ptolemy’s treatises using the first editions of the 15th century. Judging from the numerous works of later medieval astrologers, devoted to the interpretation and explanation of his methods of astrological calculations, we surmise an absence of clarity and completeness in the description of these techniques as they are presented in his works. In the opinion of one of the most famous French astrologers of 20th century, Alexandre Volguine, the astrological part of Ptolemy’s heritage (Tetrabiblos) is a compilation presenting to the reader three different approaches without stating the author’s position [10]. But this could be caused by the peculiarities of the Tetrabiblos languages, Greek and Latin, as synthetic archaic languages. In our opinion, these languages require not phrasal but block perception of the text. This peculiarity increases the probability of its incorrect interpretation. This may happen either as a consequence of the ancient languages’ evolutions, or due to mistakes in transcription and translation. Without a first-hand acquaintance with the primary sources we do not dare to make even a preliminary judgment on this.

Here it is necessary to note a very interesting characteristic of the stellar catalogue of Almagest [8]. The list of stars and constellations in the catalogue begins from the elevated pole of the celestial equator. This gives grounds to suppose that the measurements in the initial catalogue were carried out in equatorial coordinates, which are much more suitable from the standpoint of measurement technology. The conversion to ecliptic coordinates occurred after the catalogue’s creation. The catalogue begins with α Ursae Minoris, the North Star. But this star came to be

7Furthermore, in the Basel edition, the declinations of the stars were corrected. This is connected

with a more precise determination of the ecliptic position in the sky [8]. In other words, the publisher of the Almagest did not consider it as a relic of past, and it was updated to include the results of contemporary measurements.

8 We should note that before the 18th century, scientists were not divided into astronomers and astrologers; in Spanish dictionaries, it is even specified that in old texts under the word ‘astrologer’ it is necessary to understand an astronomer.

9 This is provided that Ptolemy really existed and was the author of the Almagest. As is well known, no biographic data on him exists.

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nearest to the elevated pole only in the 9th century CE. Before this time, the nearest star to the elevated pole was β Ursae Minoris. Thus, having begun from the North Star, the compiler of the Hipparchus-Ptolemy catalogue has signed out the epoch of his own observations - approximately the 9th century CE or later. This remark is not principal for the dating of Ptolemy’s catalogue. However it allows us to assume the time of creation of its more ancient prototype.

1.2. Development of the conceptual framework and calculating algorithms of medieval astrology.

"Since the moment that the manuscript of Tolomeo made its debut in the world of astrology, many were the people who tried to understand the mysterious directional technique of the grand maestro but who did not arrive at the true result." [11]. It is not an exaggeration to say that the attempt to understand the true techniques of Ptolemy was the leitmotif of most astrological treatises of the 15th through 17th centuries.

Prior to reviewing the history of the development of the ideas and calculation methods of medieval European astrology, we wish to beg the pardon of the reader for the inexactness, even the associability of what we write on this subject. We have restricted ourselves to our own highly immature version of the development of only European astrology (plus Egypt). This is more a glimpse than a history, and is given only to place the current work into the context of the development of astrological ideas. We have not encountered well-documented versions of the history of the eastern (Chinese and Indian) astrologies in European languages, though we are aware for instance that the Chinese Academy of Sciences has conducted such studies for many years. Even the references to the articles in Chinese are known. With the history of Arabic astrology the situation is much better: the medieval primary sources are known in sufficient detail and quantity. But the familiarity of the author with the history of astrology in the Arab countries is restricted for several reasons. First among these is his ignorance of Arabic language. The experience of working with the primary sources in Latin and Greek demonstrates the necessity of this condition for true understanding of the material. Any translation is fit only for a quick overview and orientation to the material. Second, many Arabic primary sources need to be re-dated. Third, many Arabic archives and documents are not open for use by non-Moslems. Without overcoming these obstacles it is difficult to construct a reasonable picture. Finally, since the purpose of this work is not to present a history of astrology, no attempt has been made to be comprehensive. The medieval methods are used mainly to validate the concept of pinpoint accuracy calculations according to the principle of correspondence.

We note with regret that, unlike in the conventional sciences, even serious researchers of astrological concepts, methods and history rarely make exact references to their sources. We have to take on trust a number of statements on the history of medieval astrology from the publications we cite here. This may have led to mistakes in the following review of the medieval history of European astrology. We hope that eventually we will have a chance using primary sources to verify all facts posited here. The situation with astrological information in our country, as well as throughout the world, arouses our pity and vexation. Many libraries do not subscribe to astrological journals and books. We hope that in the future the

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traditions of the British Museum’s library and the US Library of Congress — to subscribe to practically all that is published in the world, regardless of ideology — will be supported by the largest Russian libraries, including the Russian state library and the library of the Russian Academy of Sciences.

A small problem in identifying medieval authors is the transliteration of names from national languages to Latin, Greek and Arabic in citations. Most likely, this has not caused problems, but mistakes are possible.

The oldest known system of domification is that of Campanus (Campanus in Latin and Campano in Italian). It is generally agreed that he was a personal astrologer of the Pope and lived from 1239 to 1296 CE. The geometry of the Campanus system of houses gives grounds to believe that it preceded and was an influence on the development of the Regiomontanus system of houses.

The German astrologer Königsberger (Regiomontanus in Latin), known in the history of astronomy under his real name of Johann Muller (1436-1475 CE), developed the house system known by his pseudonym. It is believed [11] that Muller and his contemporaries were certain that this system of houses was used in the calculations described in Tetrabiblos. William Lilly used this system in horary astrology and for primary directions, although he did use other domification methods.

Francesco Giuntini (Junctinus in Latin), the famous Florentine astrologer of the 16th century, also used the Regiomontanus system. He was the author of the treatise "Speculum Astrologiae" [11]. He could not have known the house system carrying the name of Ptolemy (Placidus) since it was not published until 1604 CE by the mathematician from Padova, Giovanni Luciano Magini (Antonius Maginus in Latin10) under the title translated to English as The Second Measure of Time [12]. It is agreed that the method of Regiomontanus was spatial, but that the method of Ptolemy as interpreted by Maginus was temporal. William Lilly mentioned in his book [13] that a part of the work published by him was based on the treatises and utterances of the famous Tycho Brahe (1546–1601 CE).

Once Placidus de Titi (1606-1668 CE) had used the Ptolemaic system of houses to formulate his new, very exact system of equatorial directions, known in 20th century under the name of “mundane directions of Ptolemy-Placidus” (or “Placidean mundane primary directions”), the majority of astrologers moved to the use of the Ptolemaic domification in its Maginus variant. Now this domification is known as the Placidus domification. The Regiomontanus system of houses leads to significant mistakes in astrological calculations, as noted by Johannes Kepler (1571-1630 CE), the pupil of Tycho Brahe. It is believed that it was for this reason that he rejected the use of houses in astrological calculations. The Regiomontanus system of houses persisted as the most common system of domification up to the end of the 16th century, then almost immediately gave way to the much more exact system of Placidus.

1.3. Astrology in 20th century. This concludes our overview of the history of European medieval astrology. The Placidus domification and the equatorial directions of Ptolemy-Placidus remained

10 Note that we are not confident that this is the same person.

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the most exact methods of calculation in the astrology of events until the middle of the 20th century. The development of astronomical calculation methods, the augmentation of accuracy in ephemeris calculations following the discovery by Kepler of his three laws of planetary motion, and later by Newton (and/or by Hooke (1635-1703 CE)) of the laws of gravity has gone apace, but astrology came to a standstill in its evolution for three centuries11. It is necessary to note that there were purely astronomical reasons for this standstill. Formulation of an exact system of astronomical calculations was accomplished only at the beginning of the 20th century in the works of the outstanding American astronomer Newcomb (1835-1909 CE). This formulation permits further development of astrological computing methods and an increase in their accuracy.

"Charlatanry is not only a modern phenomenon,” noted Alexandre Volguine when speaking about texts on astrology and citing an ancient manuscript as an example [10]. We really are more familiar with the charlatanry of our own century. This phenomenon occurs not only in astrology, but in the natural sciences as well. The specific form of charlatanry in the astrology of the 20th century is the invention of a new house system. Altogether there are dozens of house systems. When attempting to understand the underlying geometry of various house systems, we usually find that they have no geometry at all, only an algorithm of obscure origin with an undesignated scope of applicability. Notice here that the house systems we’ve inherited from past centuries all have well-described and comprehensible geometries. Their formulae are secondary and can be derived by any competent mathematician from a description of the geometry. Among the house systems that have appeared in the 20th century, only three have a real geometry: Topocentric, Koch and that of local space.

In the early sixties Koch invented a system of houses [14] that his astrological colleagues have "perfected" by simplification of its calculations [15]. Note that some computer programs compute something different under the name of Koch house cusps. From here on, we will only discuss the Placidus and Topocentric systems.

As mentioned earlier in this work, astrologers tend to lump together distinct concepts under the notion of house system. There are at least two conceptually different constructs indicated by the term “houses”. The first is the collection of Equal House Systems, derived by equal division of the ecliptic. One can use several of these in the analysis of a single chart. These systems are most logically and systematically treated in the Uranian astrology of Witte [3]. It uses sectors of 30° on the ecliptic, starting from the Sun, Moon, MC, Ascendant and other points. Each set of divisions corresponds to a different facet in interpretation. Quadrant (or “unequal”) house systems are constructs of a fundamentally different nature. We also call them systems of domification. In these systems we assign to points on the celestial sphere a longitudinal coordinate, independent of time, in the first

11 This statement follows from the date of the publication of the Topocentric house system - the beginning of the 1960s. However, we cannot dismiss the feeling that the geometry of the Topocentric house system is much older and was elaborated 1-2 centuries before its publication by Vendel Polich and Nelson Page.

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equatorial system of astronomical coordinates [16]12. This mapping is performed in a manner such that the longitudinal coordinate for a point exactly on the celestial equator coincides with its usual uniform angular measure on the equator (i.e. the difference of its right ascension, expressed in degrees, from some predetermined starting point)13. There can only be one such mapping that is correct, just as 2⋅2 = 4, and not 3 or 5, depending on the result we seek in order to fit to our preconceptions. Once such a parameterization of the celestial sphere has been determined, we can vary the reference point in a manner similar to that used on the ecliptic in the Uranian techniques.

We did not check whether points such as the Vertex or Antivertex, which are held to be similar to the ascendant and descendant, are effective in astrological forecasting. We surmise that it is possible to introduce another 10 points, complementing the Vertex or Antivertex to a 12-fold division along the prime vertical. However it is not necessary to treat such a partitioning as a house system, since it is yet another completely different construct. The use of the same name to designate it can only increase the terminological mess.

We restrict the term domification to those systems providing a parameterisation of the celestial sphere, independent of time, in the first equatorial system of coordinates and connected with the celestial equator. This is the notion of houses that has its basis in medieval astrological tradition. If the systems of Koch and of local space as 12-fold divisions of the ecliptic are really effective, they are still not house systems in the traditional medieval sense.

With Newcomb’s formulation of the system of astronomical calculations at the beginning of the 20th century, it became possible to advance the accuracy of astrological forecasting techniques. In the early sixties a series of articles appeared in the journal Spica, describing the so-called Topocentric domification [4,5]. It is interesting that the authors understand their system as a house system in the traditional medieval (Ptolemaic) sense: they determine both the house cusps and each planet’s coordinate within the houses. The geometry (reconstructed by the author of the present work according to one of the formulae of the Topocentric system) of this system is uncommonly elegant. It is a natural development of the Regiomontanus domification. But the positions of the house cusps on the ecliptic, and of the planets within houses, are very close to the corresponding positions under the Placidus system. If we consider the positions of the Topocentric house cusps to be the best approximation, given present-day accuracy, to the true geometry that mediates the triggering of events in our world, then the calculation errors of the Placidus method do not exceed 20' even at the latitude of Moscow. In this system, the mathematical and geometric elegance of Regiomontanus has been harmoniously combined with the practical accuracy of Placidus.

Unfortunately the Spica articles describing the system are full of ambiguities and do not contain a description of its underlying geometry. The articles give only some

12 Throughout this work widely known astronomical notions are used without references (e.g. the first equatorial system of astronomical coordinates). The reader can find their definitions in any textbook on general astronomy. 13 Note that the oldest of the medieval systems — that of Campanus, satisfies the first requirement, but not second. The medieval astrologers after Campanus noticed that the house system must be naturally connected with the rotation of Earth around the Polar axis and consequently with the celestial equator. Campanus is a division of the prime vertical.

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practical formulae for the calculation of the positions of the house cusps on the ecliptic and of the positions of the planets in houses. The declared accuracy of calculations in this system was stated to be 3' – 5'. This is so, as the formula for the positions of the planets in houses is indeed approximate; if it is taken as exact, it actually contradicts the formula for calculation of the house cusps on the ecliptic. The error of the approximation compared with the exact formula (which is consistent with the algorithm for house cusp calculation) is about 5' at middle latitudes.

Furthermore, their algorithm for application of the system within the circumpolar regions is unfortunately absent from the articles. This algorithm, known to the authors but omitted in the articles, is most likely incorrect. There is no description of the algorithm in the articles, but there is mention of the characteristics of its results. The author of the present work has performed a mathematically accurate reconstruction of the geometry of this domification, based on the Topocentric cusp positions on the ecliptic. The exact formula for the calculation of a planet’s position in the houses has been obtained, and the analytical extension of the house system geometry into the circumpolar regions has been constructed. The characteristics of the reconstructed geometry within the Polar circle are radically different from those of the Topocentric domification in the Spica articles. Their confusion of an approximate formula for planetary positions, and its inconsistency with the formula for cusp determination, leads us to doubt the mathematical competence of the authors of these articles. However, as a first approximation adequate for practical calculation without a computer, this formula demonstrates a good understanding of the geometries of both the new house system and of the Placidus system. It also shows the high competence of a man who derived it. We suggest that the reader should make his own conclusion as to the possible reasons for this ambiguous situation.

This situation has compelled the author of the present work, in order to avoid misunderstanding, to use a different name for the house system reconstructed on the basis of the cusp formula of the Topocentric domification. We propose to name it the Zodiacal house system. One of the main tasks of the present work is to provide an accurate mathematical description of the characteristics of the Zodiacal house system, and to derive the formulae for practical calculations. The word Zodiacal has not been chosen lightly. From our standpoint the houses are the result of the accurate projection of one zodiac onto another. The geometry and formulae derived below are a consequence of this conceptual orientation to domification. In the perspective of this work, the generally accepted notion of houses is a projection of the terrestrial (earth) zodiac into the solar (conventional) zodiac.

1.4. The conceptual framework of astrology and natural science principles of thinking.

The main task we have put before ourselves is to reformulate both astrological calculations and the concepts upon which they are based in accordance with the notional systems of the physical sciences of the 20th century.

The phenomenologicalness (i.e. the absence of a physical basis for its rules) of modern astrology is its essential and as yet insurmountable defect. This defect cannot be considered grave: physics itself does not answer the question "why"; it only describes "how". When we find an answer to the question "why", physics

13

transforms into mathematics. It is agreed that we know why 2·2 = 4. In this sense, the general theory of relativity, correctly named geometrodynamics, is the mathematics describing the spacetime of the world. Newton’s law of gravitation, on the other hand, was physics. Since geometrodynamics is no longer physics, but the mathematics of spacetime, it is necessary for consistency to describe all phenomena occurring in space and time in accordance with its conceptual basis14. Here we have in mind not the relativistic laws of coordinate transformation, nor the curvature of space, nor Einstein’s equations, but the basic notional system underlying the language in which these concepts are expressed15.

In order to achieve this goal, we first need to articulate a fundamental principle, in accordance with which astrological laws manifest in the Universe. We need to clearly understand the nature of the phenomena represented by astrology. Holistic philosophy provides the best language in which to examine this question.

In holistic philosophy the world comprises different wholes interacting with each other, coming into existence and passing away in the course of their interaction. Furthermore a whole is an ideal object. In some sense, real objects cannot exist in time. The moving hand of a clock exists only as an abstraction, though it allows us "to measure time". For the purpose of measuring time we treat it as a certain kind of abstracted or ideal object, whose spatial attitude changes with respect to the face of the clock. Such a clock hand described to an accuracy of one atom does not exist as such: it is impossible to determine a discrete boundary between it and the surrounding world. However, the fact that we manage to perceive the clock hand as a whole, and to use its changing attitude relative to another whole to measure time, shows that this ideal object is real.

Since wholes are born, persist and die in time, it is possible to treat time as an internal measure of wholeness. It is interesting that while the clock hand dies on one level of abstraction and becomes a different whole when a piece breaks off, on another level it continues as a whole that can be used to measure time. In other words, several wholes manifest or live within the same clock hand, each of which has its own lifetime. It’s no large leap to say that it is precisely its moments of birth and death that define and designate the specific whole with which we are dealing.

While time can be used as an internal measure of wholeness, space can assume an analogous role as an external measure - a measure of simultaneity. For the measurement of distance between two wholes it is necessary to have a metric – yet another whole.

From the standpoint of astrology an event is the death of one whole and the birth of another. An event occurs at a particular place and time, thus complying with the notion of event in the theory of relativity as a collection of spatial and temporal

14 If our attempt here to describe the computing methods of astrology consistently with the notional system of the theory of general relativity has been successful, then astrology really has not "lagged behind" the other natural sciences. The International Astronomical Union only adopted the theory of general relativity as the conceptual basis of its measurements in 1992 (!). 15 Note that the accurate calculation of the space-like geodesic could be necessary in the situation of a conjunction of a planet with the Sun (superior conjunction for inferior planets), if the accuracy of calculations must be on the order of 1” or less. The accuracy of the system of calculations proposed in this work is not less than 10”. It is probably possible to improve the accuracy of these calculations by 1-2 orders of magnitude with no change to the conceptual basis. Current impediments to such improvement are discussed below.

14

coordinates. For astrology, time and space cannot be continuous - they are like the set of rational numbers. In astrology two moments of time are distinguished by the wholes that are born or die within them. To emphasize this point in this perspective, if from one moment of time to another nothing has been born or died, then there is no third temporal point between the two.16 Yet at the same time between them there exists a certain measure of continuous physical time. The same can be said about space. Note that one of the basic concepts for understanding space is the notion of simultaneity.

The closest scientific analogue to the astrological notion of event is the idea of catastrophe - a leap in a system’s state under continuous (smooth) variation of its parameters. In astrology, we understand an event as the same kind of discontinuous change: one whole at its death is transformed into another being birthed.

We are now ready to characterize astrology as a knowledge system: Astrology is a science connecting the events taking place in the world with the motion of heavenly bodies.

When Einstein created geometrodynamics, his guiding philosophy was Mach’s principle [17]. Mach’s principle is extremely important in the interpretation of the Universal order. It reveals the deepest conceptual basis of the natural-scientific world-view. Einstein aspired to develop a theory in which Mach’s principle would be physically expressed. He did not achieve this, but the theory he constructed with its inspiration turned out to be true. Mach held that space is created by matter. But the "ether drag" by ordinary matter (e.g. the Earth) within the framework of geometrodynamics is extremely small: for the Earth the ratio of its size to its gravitational radius is approximately equal to 10–9. Mach’s principle also posits that long-range action, which is formally denied by modern science, lies at the base of modern physical concepts. All observed physical phenomena are described within the framework of a maximum speed of interaction propagation; but the notion of simultaneity is used as the interpretative basis of modern physical theories.

For our work Mach’s principle is very important because it puts a question of the theoretical necessity for the existence of long-range phenomena. Let us briefly formulate here the relevant argument. Let an astronaut fly in a rocket a sufficient distance from any significant accumulation of matter. If he ignites the engine, he feels an acceleration that presses him into the chair. Observing carefully, he notices that he feels a pressure when he moves with acceleration relative to the remote matter (i.e. he feels the influence of this matter on himself). Moreover this influence is a long-range phenomenon, since he feels the pressure immediately upon starting the engine, as though he were in the immediate vicinity of the accumulation of matter. This effect indicates that matter can act in some long-range way currently unknown to us. Einstein’s geometrodynamics imply that the Earth’s part in the formation of the space-time at the distance, equal to its size, is about 10–

9.

In the 20th century many attempts have been made to explain astrological rules in terms of the known types of physical interactions. All of these attempts have failed.

16 Indeed, it is not a continuum, but a denumerable set. If we limit the denominator of the rational numbers we consider, as is done in astrology by a practical limitation of the harmonic numbers considered when using aspects, the situation of absence of a third point between two others divided by the finite interval becomes possible.

15

One can, of course, set one’s hopes on the discovery of an additional type of interaction. It seems to us, however, that the physical phenomenon lying at the base of astrology is not connected to either a particular type of interaction or to their joint effects. The responsible phenomenon could turn out to be long-range. It is interesting that the preliminary analysis of astrological observations performed on the basis of the concepts formulated in this work supports this opinion. That is to say: the planetary configurations causing one or another event are calculated for the time of a coming event without taking into account the effects of planetary and stellar aberration (i.e. without taking into account the finite velocity of interaction propagation — the velocity of light). This simultaneity is taken from the standpoint of the frame of reference of the native with whom the described event occurs.

A zodiacal circle, the main instrument in astrological calculation and interpretation, must be generated by a material body according to the above understanding of astrology. The conceptual object we call the Zodiac with a capital letter is the way to describe the astrological influences on surrounding wholes from the Sun. The relative dynamics of the Sun and a native, whose events we study, define its geometry.

The house cusps on the conventional zodiacal circle are projections of the zodiac generated by the Earth onto the solar zodiac. This sense of two “zodiacs” has been known from antiquity. Numerous attempts have been made to define the lunar zodiac (i.e. the zodiac generated by the Moon [18]). However no event calculations have been attempted with the resulting constructs.

We now understand how to describe the terrestrial and solar zodiacs in a unified way, using a common model and derivation despite the apparent differences in their geometries. Having developed this common model, we also know how to derive the zodiac of any massive heavenly body. It is clear a priori that the greater the distance of a body from the Earth, or the smaller its mass, the less effect it will have on the events of the native. From this standpoint, it seems necessary to derive and work with a properly constructed zodiac of the Moon, as it is the closest body to us after the Earth.

If we study the events taking place in a person’s life, we become inclined to think that the person’s spirit uses planetary configurations taken in the context of one or several zodiacs as potential turning points in his evolution. These opportunities are realized as events. When such events are frequently and obviously indicated in a given zodiac, we have more reason to use that zodiac in our calculations and interpretation.

From this perspective on time and events the astrologer’s stance toward each instant of time (epoch in astronomical terminology) becomes clear. Each instant of time differs from the next not quantitatively, but qualitatively, as characterized by the wholes that come into existence or pass away with it. Astrology is precisely the study of the qualitative description of each moment of time17.

The whole paradigm of astrology is built on the idea of the presence of the divine, ideal order in the material world. The Zodiac is a reification of the dialectical, evolutionary idea within the framework of the idealist Weltanschauung of the

17 This is most obvious in horary astrology. The practice of horary astrology is the best way to learn to conceive the character of time.

16

Universe. Even a materialist will not argue with the statement that the world is ordered by primary principles, fundamental laws of nature – what an idealist would name ideas, forms or eternal truths. Materialism only allows a certain degree of free will in its view of the world. Moreover, it admits free will only in the form of chaos, or as rebellion against the divine idea, rather than as a participation in the divine work. From the astrological perspective, order in the material world is no less harmonious and complete than in the ideal; the apparent chaos of the material world is merely an indicator of our own inability to grasp this order as it is manifested.

It is possible that the notion of causality can be understood more profoundly in the astrological perspective than within the framework of the modern natural-scientific Weltanschauung. The future continuously and inevitably crystallizes in the past; the closer the moment of an event, the less opportunity there is to change its character. The only way to exercise true free will, rather the illusion of it, is to foresee the possibilities of the future insofar as possible, and to become aware of the remote consequences of our actions.

It is important to note that one of the most difficult problems in all branches of the modern natural sciences is forecasting the evolution of long duration of a system. Attempts to solve this problem always seem to expose fundamental technical or conceptual difficulties. This comes as no surprise to the astrologer: permanent evolution is overwhelmingly determined by the laws of the spirit or Logos and is effected by means of material influences.

17

2. The Zodiac of a massive heavenly body

2.1. Introduction. Modern development of the quantum theory of gravitation already allows us to construct the phenomenological model of a zodiac of a massive heavenly body in concordance with the notions of modern non-classical physics. We have in view the Unruh effect — the excitation of a detector moving with acceleration relative to a vacuum. It is possible to distinguish the acceleration caused by gravitational attraction from that connected with the non-inertiality of the reference system. Here we apply only the fundamental possibility of distinguishing the force of gravitational attraction from inertial force. So, there are no grounds to consider that the basis of astrological rules is gravitational interaction. Furthermore, recent developments in quantum field theory and in theories of "Grand Unification" leave little hope of discovering a new interaction that accounts for astrological rules.

The basis of our conception is a zodiac model for a born that is in the vicinity of a massive body. For an observer on the Earth’s surface three bodies predominate — the Sun, Earth and Moon — and correspondingly, their three zodiacs. We cannot however neglect planetary influences18. A zodiac is a dynamic construct in that it depends on the motion of a given born relative to a designated heavenly massive body. By born we mean a wholeness or person for which we study events.

The motion of an observer stationary on the Earth’s surface relative to the Earth’s centre, or to the Sun, is the simplest case due to its closeness (with a certain accuracy) to circular motion. The deflections of the angular parameters of the solar and terrestrial zodiacs due to the deviation of the observer’s motion from ideal circle reach only several angular minutes. It is exactly the relative insignificance of these deviations that has given a static character to their geometries and thus allowed ancient astronomers to formulate the concept of a zodiac using only elementary geometric constructions. In traditional astrological language the solar zodiac is the ordinary Zodiac, and the projection of the terrestrial zodiac onto the solar is a house system.

The motion of a born relative to the Moon is much less circular. The taking into account the deviation of this motion from circular changes the angular parameters of the lunar zodiac by degrees. It is likely that this characteristic is exactly what prevented a comprehensive quantitative description of this zodiac in antiquity, though some steps were taken in this direction. Examples are the lunar zodiac of Chinese astrologers and the draconic astrology of the West. The difference of these models from the exact construction is so great that it is impossible to produce quantitative calculations. The lunar zodiac is a dynamic construction, in which the velocities of planets on the zodiacal circle experience significant variations not through changes of their observable velocities in the sky, but due to the complex nature of the motion of an observer around the Moon.

18 It may be necessary to mark the positions of the local planetary nodes on the circles of solar,

terrestrial and lunar zodiacs. By local planetary nodes we understand the longitudinal positions of the two crossing points of the equator of the corresponding planetary zodiac with the local equator of the zodiac under consideration.

18

In the model we elaborate an event as a qualitative change of a born occurs when certain group correlations (in the language of the group theory, or aspects in astrological language) arise among the elements of a zodiac. The concept of aspect - one of the central and very old notions of astrology - can be most precisely described by the language of algebraic group theory. The elements of a zodiac are completely determined by the motion of massive heavenly bodies and a born. Since the motion of heavenly bodies can be calculated with good accuracy for many years into the future, and that of a born can to a large extent be considered as fixed19, there results the possibility of forecasting events.

2.2. Theoretical model of a zodiac for a born, situated in the vicinity of a heavenly body.

We posit that a zodiac of a born is a way to describe the spatial anisotropy at the born’s location, which is generated by a massive body, relative to which the born moves20. Such an approach to describing spatial non-homogeneity is a construct exactly suitable for forecasting events. If a born is surrounded by several massive bodies, then it is possible to construct a corresponding number of zodiacs. When constructing the zodiac of one of these bodies, other massive bodies become its elements or planets (in ancient Greek sense of the word πλανης). The most interesting zodiacs are those of the closest and heaviest bodies. These zodiacs are the most stable due to the born’s dynamics governed by the gravitational fields of these bodies. We call such a massive body that generates a given zodiac its central body; we call the remaining bodies simply bodies or planets.

A zodiac as a dynamic structure is defined in general by three directions: 1) the vector of force acting on a born from the central body as a whole, 2) the velocity vector of the central body in the reference system of the born21 and 3) the axis of space anisotropy generated by the central body in the location of the born. The first two vectors determine the plane of a local equator (Fig. 1). When a born moves with acceleration relative to the central body, the vector of force of the central body and the axis of space anisotropy generated by it do not coincide. The angle between the plane of the local equator and the axis of anisotropy is called the dynamic angle. If the size of the central body is far less than the distance to it, then the direction to it in space can be used as its axis of anisotropy. That is, we use its position in space without correction for planetary and stellar aberration.

Thus, when studying the zodiac of a distant massive body, the angle between the central body’s vector of force and its axis of space anisotropy is the angle of the correction for aberration. This is due to the fact that such a distant body acts on a person for the most part only by gravitation. For a person on the Earth’s surface the correction for aberration for planets of the solar system is some dozens of angular seconds22. However, for the terrestrial zodiac the dynamic angle is much larger, being approximately equal to the latitude of the point on the Earth’s surface23.

19 Provided he does not travel to a nearby planet of the solar system. 20 The usual physical notions of length, time, simultaneity etc. are given in the reference system

accompanying born in accordance with the general theory of relativity. 21 The velocity vector of the central body (without aberration) in the non-rotating proper reference

system of the born. 22 To be fair it is necessary to note that there is no sufficient experimental basis for the introduction

of the notion of the axis of space anisotropy different from the direction of attraction by the gravitational field. However it seems to us that this follows by necessity from the conceptual basis of astrological rules

19

Fig.1 Geometry of the local zodiac of a born in the vicinity of a massive body. e — unit vector of anisotropy generated by the central body at the point of the born, A - unit east point vector (fundamental reference point) of the local zodiac of the born, P — polar vector — the direction to the north pole of the local equatorial system of coordinates, F — force acting on the born from the central body as a whole, V — velocity of the central body in the proper non-rotating reference system of the born, E — east point (fundamental reference point), W — west point, S — south point, N — north point, C — point of conjunction, T — point of opposition, M0 — projection of the central body on the celestial sphere, ϕd — dynamic angle.

and the impossibility to explain the mechanism of astrological calculations on the basis of some known interaction, including gravitation. The experimental aspect of the problem will be discussed below.

23 Instead of the astronomical horizon it is necessary to take the gravitational one, i.e. the plane perpendicular to the force of gravitational attraction (force of gravity minus inertial force).

20

Let us define the local equatorial system of astronomical coordinates of a born with respect to a given central body. The equatorial plane of this coordinate system has been introduced above in the definition of the dynamic angle — it is the plane of a local equator, whereon lie two vectors: the force vector F acting on the born on the part of the central body, and its velocity vector V in the proper non-rotating reference system without correction for aberration. The polar angle ν (also called the latitude) is measured from the equatorial plane in the positive direction toward the North Pole, and in the negative direction to the south. The vector product P= F×V defines a vector directed to the north pole of this system of astronomical coordinates.

The azimuth angle µ in the equatorial plane (also called the longitude) is measured in the positive direction (counter-clockwise if one looks from the north pole of the coordinate system) from the fundamental reference point on the local equator. It is fitting to label the fundamental reference point of the coordinate system as the east point of the central body. The vector A towards the east point of the central body is given by vector product A= e×P, where e is the vector of space anisotropy of the central body acting on the born, and P is the previously-defined polar vector of the coordinate system.

For a central body whose size is many times smaller than the distance to it, it is appropriate to designate its space anisotropy vector as a unit vector e tangent to a purely spatial geodesic, connecting its centre of mass and the born. The vector of anisotropy generated by the Earth is practically coincides with the vector of gravitational attraction at a given point on its surface.

It is necessary to distinguish the notions of local equator of a central body and its local equatorial coordinate system as introduced in this paper etc, from such traditional astronomical notions as the celestial equator, first and second system of equatorial coordinates etc. For the definitions of the traditional astronomical concepts see [16].

At this point, the concept of a zodiac is almost elaborated. Let us designate massive bodies other than the central one as zodiac elements. Then a zodiac is a collection of longitudes of massive bodies surrounding the born expressed in zodiacal coordinates. The zodiacal coordinate system in which we describe the “position” of the central body and other massive bodies is a curvilinear coordinate system distinct from the system of spherical coordinates on the local equator. It is not a coordinate system in the usual sense, as some points on the sphere are simultaneously associated with three values of zodiacal longitude24. The zodiacal longitudinal coordinates converge to the spherical longitudinal coordinates of the local equatorial coordinate system as the dynamic angle approaches zero.

For a complete introduction of zodiacal longitude it is necessary to consider several mathematical notions. Zodiacal coordinates can be considered a generalization of spherical coordinates. Let us denote the zodiacal longitude by τ. We may imagine it as a circumference of longitudes, called zodiacal circle in astrology. Let us move to the definition of the zodiacal longitudinal coordinate on the celestial sphere. For this it is necessary to construct a mapping of the sphere onto the circumference, and to introduce a distance function on the circumference as well as a reference point.

24 In topology, such a mapping is called a triple covering of the sphere.

21

Let us call this procedure of assigning a zodiacal longitudinal coordinate to a point on the celestial sphere a parameterisation, since it can be described by the movement of a great semicircle25 along the surface of the sphere.

To determine the zodiacal coordinate we restrict ourselves to a very narrow class of motions. We consider the celestial sphere as a sphere of unit radius with its centre in the location of the born. We identify the great circumference with distance function and reference point with the intersection of the celestial sphere and the local equatorial plane. The reason for this choice is that any point having zero spherical latitude has identical zodiacal and spherical longitudes.

We define a plane called the local horizon of anisotropy, or the local horizontal plane, as the plane perpendicular to the axis of anisotropy generated by the central body on the point of born. We define the local meridian plane as the plane perpendicular to the local horizontal and equatorial planes. The crossing points of the local horizontal and meridian great circles we call the north point and the south point. The north point is situated on the celestial sphere closer to the north pole of the local equatorial coordinate system of born.

The fundamental reference point on the local equator (zodiacal circle), or east point of the central body, is defined as that crossing of the local equator and the local horizontal plane the direction to which forms an obtuse central angle to the velocity vector of the central body in the non-rotating proper reference system of born. The opposite point it is called the west point. The crossing point of the line common to local meridian and equatorial planes with the celestial sphere, the direction to which from the centre of sphere forms an acute angle with the direction along the axis of anisotropy from the born to the central body, is called the point of conjunction, and its opposite point is called the point of opposition26.

At the moment τ = + 0 of virtual time27 the moving great semicircle (the parameterizing semicircle) lies in the horizontal plane and passes through the east point. The straight line passing through the edges of this great semicircle is rotated by the angle –β (i.e. by the angle β clockwise) from the straight line connecting the north and south points as viewed from the point of opposition, which we consider by definition to be above the plane of the horizon. The angle β is given by equation28

25 One half of a great circle. A great circle is a section of a sphere made by a plane going through its

center. 26 Projections of these points of the terrestrial zodiacal circle onto the solar zodiacal circle are the IC

and MC respectively. It is suitable to visualize for this construction that the born is on the Earth’s surface at a middle latitude in the center of the celestial sphere. The local meridian plane of the terrestrial zodiac will practically comply with the plane of celestial meridian. The local equatorial plane will coincide with the plane of the instantaneous equator. The local horizontal plane will coincide with the plane of the gravitational horizon. The difference between the true horizon of anisotropy and the gravitational horizon will be extremely small. Here one should not muddle the gravitational horizon with the plane perpendicular to the plumb line. The plumb gives a direction of the gravity force, which is the sum of the gravitational force and the inertial force. The latter force appears because of the rotation of the reference system motionless with respect to the Earth’s surface.

27 It is convenient to represent the main parameter of this longitudinal parameterization of points on the celestial sphere as a virtual time, and the procedure of parameterization as a motion of the great semicircle with this virtual time.

28 When ϕd=0 the zodiacal coordinate system coincides with the spherical equatorial coordinate system. A nonzero value of ϕd skews the zodiacal coordinate system, eventually converting it into a triple covering of the celestial sphere. For now the characteristics of this distortion is known only for the case

22

tgsin dβ

ϕπ

=2

(1)

where ϕd – the dynamic angle, the analogue of latitude for the terrestrial zodiac – can differ from the geographic latitude by several angular minutes. The parameterizing semicircle, moving in a specified manner, returns to its initial position after 360 units of virtual time τ. Zodiacal longitude, like sidereal time, can be measured in hours or in degrees as is done here. Each point on the celestial sphere is marked by the moment at which the parameterizing semicircle passes through it. We consider each such marking moment as the point’s zodiacal longitudinal coordinate.

The description of the zodiacal longitudinal parameterisation is not yet complete. We describe the motion of the great semicircle by the angle θ (τ) between the polar axis connecting the north and south poles of the local equatorial coordinate system of a born and the parameterizing half-plane whose intersection with the celestial sphere yields the parameterizing semicircle. But this still does not give the exact orientation of the parameterizing semicircle. According to the second condition to complete the description given earlier, the crossing point of the parameterizing semicircle and the equator must have uniform motion along the equator29 (Fig. 2).

Therefore, when at the zodiacal positions 0°, 90°, 180° and 270° the parameterizing semicircle must correspondingly pass through the east point, the point of conjunction, the west point and the point of opposition. At the same time, the straight line that supports the parameterizing semicircle in the horizontal plane, and which is the instantaneous axis of rotation, moves in the positive direction (counter-clockwise as viewed from the north pole of the equatorial coordinate system) from the angle –β(τ = + 0), through the north-south axis, and up to the angle + β as the parameterizing semicircle approaches the west point (τ = 180°– 0).

Moving further the supporting straight line discontinuously changes its position, again forming the angle – β with the north-south axis (τ = 180°+ 0), and then moves again in the positive direction up to the angle + β (τ = 360°– 0). This motion ensures that the parameterizing half-planes in positions τ and 180°+ τ form a single plane. At the moment τ = + 0 the angle between the parameterizing half-planes to

when the plane of dynamic angle is perpendicular to the plane of local equator (owing to the known geometry of the terrestrial zodiac). For the solar and planetary zodiacs the aberration plane coinciding with the plane of the dynamic angle practically complies with the plane of equator and the aberration angle is on the order of tens of angular seconds - a noticeable value if we suppose that the distortion can linearly depend on the aberration angle for this orientation. It is hardly possible that the geometry of the distortion of the zodiacal coordinate system relative to the spherical coordinate system is determined only by the value of the angle between the plane of local equator and the vector of anisotropy, rather then the orientation of the aberration angle. As a result, when constructing the algorithm for calculation of the local solar zodiac for a computer program, we have preferred to put ϕd equal to zero (i.e. to use as the zodiacal coordinates the corresponding spherical coordinates) and to analyze the errors appearing in the process of practical calculations. For the moon zodiac we can consider the dynamic angle to be equal to zero, since the correction for planetary and stellar aberration is less than an angular second.

29 The angle between the polar axis and the parameterizing half-plane plus uniform motion of the crossing point of the parameterizing semicircle along the equator do not uniquely fix the motion. For each τ two different positions satisfying these conditions are possible. The choice of solution is defined by the continuity of the parameterising motion (except for the leaps of the supporting straight line in the plane of the horizon for zodiacal longitudes 0° and 180°) and by the character of motion described above for the supporting straight line in the plane of horizon.

23

the polar axis is equal to ϕd. For an arbitrary moment τ, the angle θ(τ) is defined by the following expression, for (– 180°< τ < + 180°):

tg tg dθτ

ϕ= −°

�

��

�

�� ⋅1

90 (2)

The sign of the angle θ in the formula is conventional, and the resulting tangent with the specified sign is substituted into formula (3).

It is interesting to note that the plane with which the parameterizing semicircle coincides in the position τ = + 90° passes through the south and north points and is identical to the plane passing through the points with spherical longitude µ = + 90° in the local equatorial coordinate system of the born. So, when converting from spherical coordinates to zodiacal coordinates the central body, which always has a spherical longitude µ = + 90° with respect to the fundamental reference point, does not change its longitude. The same is true for the majority of points with spherical longitude µ = + 90°. Only those points with spherical longitude µ = + 270° and a latitude ν lying in the range + 90°– ϕd < ν < + 90° in the local equatorial coordinate system of the born will have a zodiacal longitude τ = + 90°. Similarly, points with spherical longitude µ = + 90° and latitude ν in the range – 90°< ν < – 90°+ ϕd in the local equatorial coordinate system will have a zodiacal longitude τ = + 270°.

Fig. 2 Illustration of the parameterizing motion of the half-plane from the cusp of the 7th station to the end of the 12th station (the range of zodiacal longitude 180°+0 to 360°–0). The notation is that of Fig. 1. The angle β is given by (1). The great

24

semicircles limited by the supporting straight lines NiSi, lying in the plane of the local horizon of anisotropy, represent the two-dimensional cusps of the stations of the local zodiac of a massive body. The great semicircle NS lies in the plane of the local meridian and represents the cusp of the 10th station. The straight lines NiSi are the axes of instantaneous rotation of the parameterizing semicircle in positions 30°×(i–1) of zodiacal longitude. On the local equator ECWT, the spherical longitude counted from the east point E in usual angular measure is equal to the zodiacal longitude. Its (and the zodiacal longitude’s) values for the cusps of the stations are marked on the drawing at the crossing points of the two-dimensional cusps of the stations with the great circle of the local equator.

It is quite possible that unlike the other massive bodies there is no need to mark the central body on the zodiacal circle. Its presence is always implicitly indicated by the fundamental reference point, the point of conjunction, or possibly by all 4 cardinal stations of the east point: the east point itself, the point of conjunction, the west point and the point of opposition.

Unfortunately it is not clear how to define the latitudinal coordinate of the zodiacal coordinate system. We hope it is possible to derive the formula theoretically from the conditions of complex analyticity of the mapping of the pair of angular variables, longitude and latitude, ( µ, ν ) of the local equatorial coordinate system of a born onto the pair of angular variables ( τ, σ ) of the corresponding zodiacal coordinate system, as its inference from experimental data could require a great deal of time.

To conclude this section, let us write down the formula for converting the longitudinal coordinate of a point on the sphere from its spherical equatorial coordinates into zodiacal coordinates. The zodiacal and spherical longitudes of a point are different whenever the point has non-zero spherical equatorial latitude. The zodiacal longitude of a point (given the condition that the reference point is set to the corresponding east point or fundamental reference point) is obtained with domain and range (-180°< τ < + 180° and – 180°< µ < + 180°) by:

( )sin tg tgτ µ θ ν− + ⋅ = 0 (3) where ( µ , ν ) are the spherical longitude and latitude of the point in the local equatorial system; τ is the longitude of planet in the zodiacal system; tg θ is the tangent of the angle between the parameterizing half-plane in position τ and the polar axis of the local equatorial coordinate system of a born with the sign conventionally defined by formula (2).

Before solving this equation, it is necessary to test an additional condition to define whether the point is above or below the horizon (the positions "below the horizon" include the central body).

( )cos sin cos sin sind dϕ µ ν ϕ ν⋅ ⋅ − ⋅><��

��

0 . (4)

The upper sign of inequality is used for a position below the horizon, 0°< τ < + 180°; the lower for a position above the horizon, – 180°< τ < 0°.

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2.3. Choice of the reference point on the local equator (zodiacal circle).

A zodiacal circle or zodiac in astrology is a circumference of zodiacal longitudes with a designated reference point and sensitive points located on it. The sensitive points can be the points with the longitudes of massive bodies as well as some other points. Once the identification of a circumference of zodiacal longitudes with the local equator of a born has been made, the points of massive bodies on the zodiacal circle can be considered as the projections of these bodies onto this equator in the zodiacal coordinate system.

Once we have an equatorial circle with sensitive points on it, we can choose its reference point. We have already seen one variant: the fundamental reference point (east point) of the central body. The experience of Uranian astrology, founded by Alfred Witte, shows that the intended facet of interpretation determines the choice of reference point. In Uranian astrology six systems of equal houses are used. In the language of the present work this means that the reference point is chosen from six points in addition to the fundamental reference point. The principle becomes clear in the context of the technique of discrete symmetries on a zodiacal circle, which was used actively by Alfred Witte and his adherents. The forgotten originator of this technique was Johannes Kepler.

If a sensitive point is sufficiently strong, astrological experience shows that it reveals itself through a number of additional points, symmetrically located in the corners of a regular N-agon inscribed in the zodiacal circle30. Let us call these points the Nth order family of the point, and designate the point as the first point of the family. The points of the Nth order family of the fundamental reference point (located 90° clockwise from the central body) are the cusps of the stations of the central body of the Nth order. For N=12, we call them simply the cusps of the stations31.

By the term station we understand an interval on the zodiacal circle from a station cusp to the cusp of the next station, in accordance with astrological tradition. When we talk about zodiacal sign cusps of the Nth order we mean a family of a reference point that does not coincide with the fundamental reference point.

As the number N increases the subsidiary points of a family become weaker. Uranian astrology demonstrates that the points of the 4th order family work for any point representing a massive body. The points chosen in astrology as reference points are so strong that the subsidiary points of their 12th order families are also noticeable. The Sabian symbols for degrees demonstrate that very strong points can generate a family of effective points even at the 360th order [20]. This also testifies in favour of the special significance of certain positive integers in the zodiac.

30 In the language of group theory this can be formulated: the sensitive points of the Nth order family

of a given point can be generated from the given point by the action of the regular representation of the cyclic subgroup of the Nth order of the one-parametric group of rotations.

31 It is important for the number of stations to be divisible by 4, so that one of the station cusps coincides with the central body (point of conjunction). We introduced the notion of station to avoid confusion with the notion of house. By “house” (more exactly house cusp) we understand a projection of a station cusp onto another zodiac, as it is accepted in traditional medieval astrology when projecting the mondial (terrestrial) zodiac onto the solar one.

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In astrology it is not only aspects of sensitive points to the station or house cusps that are important; the areas from one cusp (station, sign or house) to another are meaningful as well. In traditional astrology these areas are called houses when projecting the terrestrial zodiac onto the solar one (the sections between 2 subsequent house cusps32), or signs of the Zodiac for the solar zodiac when the reference point is taken as the crossing of the celestial equator and the ecliptic. These sign sectors are considered to carry certain qualities that modify the expression of a sensitive point found within them.

We can note the dialectical character of this scheme, where symbolic motion in the positive direction along the zodiac circle describes a cycle of evolution resulting in an accumulation of characteristics that manifest as qualities when entering a new area. Traditional astrology makes the reference point absolute; nevertheless, its meaning is highly relative. Each time we choose a reference point we obtain a new zodiacal circle. The zodiacal circle is actually an ensemble of evolutionary cycles, an ensemble of ‘vernal equinoctial points’ and, in accordance with symmetry order, an ensemble of sets of phases of development.

By the notions of station, house and sign we imply one- and two-dimensional areas. By station, house or sign we mean the area between the border of a considered station (house, sign), called its cusp, and the cusp of the following station (house, sign). If this makes sense.

As a one-dimensional station (house, sign) we mean an arc on the zodiacal circle identified with the equator of a local zodiac. By definition, a station cusp is a point on a zodiacal circle whose longitude is equal to the part of the full circle 360 1°⋅ −( )n

N, where n is the number of cusp, and N is the order of a family of the

fundamental reference point.

We understand a two-dimensional station (house, sign) as an ensemble of points on the celestial sphere having the same longitudes on the considered zodiacal circle as the corresponding points of one-dimensional station, house or sign. The cusp of such a station (house, sign) is the great semicircle of points on the celestial sphere

with zodiacal longitude equal to 360 1°⋅ −( )n

N.

Astrology commonly uses both a fundamental reference point (the terrestrial zodiac), and a reference point fixed in the crossing point of a local equator with the local equator of the zodiac of another central body (the solar zodiac). The choice of a reference point in the crossing point of two equators testifies that this is a strongly manifesting point. The issue of choosing the reference point from the two crossing points is not clear. We can only state a preference for either point from natural science considerations. This leads us to the CPT-theorem of quantum field theory33 as a possible principle for resolution, though the specific choice remains unclear. The East point, or fundamental reference point, is a strong sensitive point in its own

32 Near the polar circle and within it the projections of the station cusps of the terrestrial zodiac onto

the solar zodiac (ecliptic) cease to follow one another in order. In this situation it is impossible to consider a house as an area from its cusp to the next one.

33 In a certain sense the CPT theorem can be proved within the framework of the special theory of relativity, i.e. remaining within the framework of classical description.

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right, since it is connected with the singularity that occurs when crossing the horizon – the leap in the motion of the parameterizing semicircle34 [21].

2.4. Examples of zodiacs.

2.4.1. Terrestrial zodiac.

What was written above about the structure and geometry of the zodiac of a massive body is very close to the structure and geometry of the terrestrial zodiac. The characteristic motion of a typical born — a person nearly always motionless with respect to the Earth’s surface (with the exclusion of a cosmonaut) — closely approaches an ideal rotation with respect to the terrestrial polar axis. That is, it is nearly a circular motion with a constant velocity. Therefore, although the geometry of the terrestrial zodiac is complicated due to its considerable dynamic angle, it is static. The longitudinal coordinates of the celestial sphere points, which are motionless with respect to an observer on the Earth surface, do not change with time35. This allowed a correct description of the geometry of the terrestrial zodiac, with sufficient accuracy, as early as the 17th century as the Placidus system of houses and the "mondial" (mundo) positions of planets in the houses [19] (by Ptolemy). It is easy to see that the instantaneous equator is a local equator of a born for the zodiac with the Earth as its central body. The plane of the dynamic angle is perpendicular to the plane of the equator, with high precision. This is a unique zodiac in that it has both a known geometry and a non-zero dynamic angle36.

For pinpoint accuracy calculations it is necessary to take into account three factors:

1. It is necessary to use the gravitational horizon rather than the usual gravity horizon (Fig. 3). Because the born is so close to the Earth the correction for aberration is many times less than for the Moon37. So for the vector of anisotropy it is possible to take the direction of the gravitational attraction by Earth. The usual horizon, defining the geographical or, taking into account the plumb deviation, the astronomical coordinates of a point on the Earth’s surface, is a plane perpendicular to the gravity force. This force is the sum of the gravitational attraction and of the inertial force generated by the rotation of a reference system motionless with respect to the Earth’s surface. To achieve a precision of several angular seconds it is sufficient to take into account the following correction to the geographic latitude:

∆ϕ( ) sinradR

g= −

ωϕ

2

22 , (5)

We add this correction to the geographical latitude of a point. Here ϕ is the geographical latitude of place; ω is the angular velocity of the Earth’s rotation expressed in radians per second; R is the radius of the Earth; g is free fall acceleration. The correction to the latitude calculated by this formula is expressed

34 Here it is worthwhile to note that the axis opposition-conjunction contains another singularity —

one family of parameterizing semicircles is changed for another. 35 The longitudinal zodiacal parameterization of the terrestrial zodiac is stationary in the first

equatorial and horizontal astronomical coordinate systems. 36 Therefore such constructions as solar and planetary zodiacs, for which the plane of the aberration

angle is close to the plane of equator and is on the order of tens of angular seconds, as concerns calculations of pinpoint accuracy (near 1"), should be subjected to study for the purpose of reconstruction of the geometry of zodiacs with a free orientation of the dynamic angle.

37 For the Moon it is a part of angular second.

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in radians. The use of the dynamic angle instead of the usual geographical latitude in the terrestrial zodiac will cause, for instance, a difference in the house cusp longitudes with respect to the usual determination of the terrestrial zodiac understood as a house system38.

Fig. 3. A section of the globe along the polar axis OP.

O —Earth’s centre. OQ — the plane of the terrestrial equator. N — the point of a born. R — the geocentric radius vector of the point N. ϕg — the geocentric latitude. ϕ — the usual astronomical latitude defined as the angle between the plumb and the plane of equator. The direction of the plumb is the direction of the gravity force mg acting on the born at the point N. It is the sum of the gravitational attraction force e, which in this case practically complies with the axis of anisotropy, and of the inertial force i, whose value is defined by formula i =ω2Rcosϕg. The surface of the terrestrial ellipsoid is perpendicular (with a certain accuracy) just to the plumb. The dynamic angle, or the gravitational latitude ϕd, can be obtained with a precision of several angular seconds by adding the correction given by formula (5) to the astronomical latitude ϕ.

2. To achieve the maximum accuracy it is necessary to take into account the motion of the polar axis within the Earth’s body. The value of the latitude correction for this factor is about 1".

3. It is also necessary to take into account the plumb deviation as a correction to the latitude. The typical correction for flat country is about 1-2"; in the mountains it can reach several minutes of arc. However on the Earth’s surface, there are several

38 For Moscow the correction to the geographic latitude is approximately equal to 5'30".

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regions of "anomalous gravitation" in geodesic terminology, where flat land has a considerable plumb deviation, reaching 10" and more. One such region is the area of Moscow and its suburbs. Indeed, one of the largest known leaps in plumb deviation is in the Kremlin, adjacent to the belfry of Ivan the Great39.

No additional peculiarities in the geometry of the terrestrial zodiac appear when passing into the circumpolar regions. However, areas exist on the celestial sphere above the north point and beneath the south point where points have three zodiacal longitudes. This occurs even in equatorial regions of the globe. In these areas equation (3) has 3 solutions for a single spherical coordinate pair ( µ , ν ) (Fig. 4). When approaching the polar region the size of these areas grows. At the latitude of Archangelsk and Reykjavik these areas are already possible for retrograde Venus40 and for the Moon. That is, Venus can be represented by three points on the circle of the terrestrial zodiac. In circumpolar regions, the Sun and all planets other than Pluto sometimes pass through these areas. Because Pluto moves close to the plane of the equator, it does not enter the multiplicity area. It is obvious that a planet enters the multiplicity area when it becomes non-descending (i.e. when it passes over the horizon near the north point). But it also enters into this area somewhat earlier, while still crossing below the horizon in its daily motion.

Fig. 4. A part of the multiplicity area on the celestial sphere. This area is situated entirely above the horizon near the north point N. A similar area is situated beneath the horizon near the south point. MN is part of the great circle of the local meridian plane. ENW is the horizon great circle. For this example, suppose this is the terrestrial zodiac. Arcs R1R2 and S1S2 are the ascensional paths of planets for a born near the polar circle. R1R2 is the path of a non-descending planet; S1S2 is the path of a planet crossing the horizon but still falling into the multiplicity zone. P is a point on the path R1R2, whose zodiacal longitudinal arcs τ1, τ2 and τ3 are shown.

Let us describe the longitudinal dynamics of a star or planet on the terrestrial zodiacal circle for a single sidereal day, when it is at a point on the celestial sphere that goes under the horizon for a very short interval of time yet still enters into the multiplicity area (Fig. 5). As an example we may take the Sun for a point on the

39 So to achieve an accuracy of 1" it is necessary to use maps of the plumb deviation. 40 In retrograde motion the ecliptic latitude of Venus can reach 8°.

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globe near the polar circle and on a day not long before or after the summer solstice. For the simplicity we ignore any proper motion of the Sun in ecliptic longitude for the day under consideration.

Fig. 5. Dynamics during a sidereal day of the zodiacal longitude coordinate of a point that is motionless with respect to the 2nd system of equatorial coordinates.

Terrestrial zodiac station cusps of a born in the vicinity of the polar circle are marked with large Arabic numerals. The point we consider falls into the multiplicity zone, despite crossing the horizon in its daily path. The small Arabic numerals designate zodiacal longitudes for the following moments.

1 — At astronomical noon (assuming the point we consider to be the Sun).

2 — A point on the border of the multiplicity area on the celestial sphere, within the 7th station. When the point crosses this border on the terrestrial zodiacal circle, two additional solutions to formula (3) appear near the antiscia point of the first solution, within the 12th station.

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3 — The point crosses the horizon. Two solutions in the 7th and 12th stations disappear following their conjunctions with 1st and 7th cusps. The solution in 10th station disappears as well; a single solution 3" opposite it near the cusp of 4th station appears.

4 — The single solution 4' disappears near the cusp of 4th station when the point crosses the horizon, and three solutions 4" appear.

5 — The point leaves the multiplicity zone. The second and third solutions disappear after merging in the 7th station.

6 — A new astronomical noon.

We begin with an astronomical noon when Sun is on the celestial meridian conjunct the MC. In the terminology of this work, the Sun is conjunct the cusp of the10th terrestrial station. As it approaches the north point, the retrograde motion of the Sun through 7th station slows, and at a specific moment two additional solutions appear within the 12th terrestrial station. They move in opposite directions. The second solution moves quickly toward the 10th station. The third solution moves slowly toward the 1st station, reflecting the motion of the first solution near the cusp of the 7th station.

The Sun crosses the horizon before passing the north point. At the time of sunset the first and third solutions simultaneously conjunct the cusps of the 7th and 1st stations respectively and disappear. The second solution also disappears, replaced by a solution exactly opposite it. At the time of sunset the longitude of second solution is greater than 270°. If, for example, its longitude is 275°, then the single solution immediately after sunset is the longitude 95° = 90°+ ( 275°– 270°). The longitude of this solution decreases with time. At the moment when Sun passes under the north point the longitude of this solution is 90° - the Sun is conjunct the cusp of the 4th terrestrial station.

Continuing with this example, when the terrestrial zodiac longitude of the Sun becomes equal to 85° = 90°– ( 275°– 270°), the Sun again crosses the horizon line as it rises. The solution with terrestrial zodiac longitude 85° disappears and is replaced by the solution with the longitude 265° = 270°– ( 275°– 270°). We consider it a second solution as the Sun once again falls into the multiplicity zone after it crosses the horizon. We consider the solution near the cusp of 1st station as the first solution. The third solution is a point near the cusp of 7th station. As the Sun rises almost tangentially to the horizon in its movement away from the north point, the second and third solutions approach each other with increasing velocity and disappear as they coincide. At this point we have again the usual situation of the rising Sun that moves toward the cusp of 12th terrestrial station.

The central body of the terrestrial zodiac always has a zodiacal longitude τ= + 90°. Within the framework of traditional terminology it is possible to say that the Earth is always conjunct the cusp of 4th house or the IC. Since by tradition one projects the terrestrial zodiac with the fundamental reference point onto the solar zodiac, the Earth is not represented as a celestial body in the system of astrological calculations, but is instead implicitly included in the system through house interpretation. However the point of the Earth on the solar zodiac will not coincide with the cusp of 4th house. So its consideration on the solar zodiacal circle promises to be of interest.

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The point identified as the Ascendant in traditional astrology is the projection of the fundamental reference point, or east point, of the terrestrial zodiac onto the solar zodiac. The reference point coinciding with the crossing point of the terrestrial and solar equatorial circles (the equator and ecliptic in traditional terminology) is also used implicitly in traditional astrology. This reference point is required for a well-founded formulation of the methods of progressions and directions, as presented below. Here the words ‘directions’ and ‘progressions’ are used in the traditional sense. In this work the meaning of these words will be given by exact definitions. The newly corrected definitions of traditional symbolical methods will have names of the form solar-terrestrial progressions and so on.

2.4.2. The Lunar zodiac. It is likely that the lunar zodiac is the most interesting consequence of the theory of the dynamical origin of zodiac presented in this work. While ancient astrologers suspected the existence of the lunar zodiac, they had no mathematical means upon which to construct it quantitatively. It was then only possible to construct it on the basis of a static geometry, mimicking the geometry of the solar zodiac. While the traditional solar zodiac differs from its exact dynamic variant only by several angular minutes, the statically constructed lunar zodiac differs strikingly from its dynamic prototype. Because of this its static construction cannot have forecasting power, though it may work in a descriptive way. We mean, for example, the 28 stations of the Moon in Chinese astrology [18].

As in the case of the solar zodiac we can choose more than one reference point. In addition to the fundamental reference point it is possible to choose the crossing of the moon’s local equator with the local equator of either the solar or the terrestrial zodiac (i.e. set the reference point in the solar or terrestrial node).

Note an additional peculiarity of the lunar zodiac. The inclination of the moon’s orbit to the ecliptic is only about 6º. This means that the projections onto the solar zodiacal circle of the 12th order family of the reference point will remain practically 30º from each other. In astrological language these points can be called the cusps of the lunar houses. If we choose the reference point on the lunar zodiac to be the solar node, the projections of the cusps of the lunar signs onto the solar zodiac will be close to the cusps of the signs of the solar zodiac with the reference point in the moon’s node (draconic astrology). With this choice of reference points on the local solar and lunar zodiacs it will be difficult to differentiate the interpretations of the solar and lunar zodiacs.

The lunar zodiac is likely the best case for testing the concept of a zodiac of a massive body. The aberration angle of the Moon for a born located on the Earth’s surface does not much exceed 1 angular second. Therefore the geometry of the lunar zodiac is a common spherical geometry. But the local equatorial plane of the lunar zodiac constantly oscillates with considerable amplitude. The rotation of a born around the Earth’s polar axis strongly deforms his circular motion around the Moon. This is because the Moon’s linear orbital velocity around the Earth is only a little greater than a born’s linear rotational velocity around the Earth’s polar axis.

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2.4.2.1. The Lunar nodes.

When interpreting a natal chart considerable attention is given to the position of the Moon’s nodes in the signs and houses. In general, little or no attention is paid to the exact position of the node, for example, to an exact conjunction with a house cusp. If we consider the crossing point of the ecliptic with the equator (i.e. in the terminology of this work, the crossing of the equators of the terrestrial and solar zodiacs), the vernal and autumnal equinoctial points, as very important, then the crossing point of the ecliptic with the equator of the local lunar zodiac should also be of considerable importance. These two points of intersection (called here the local lunar nodes) are located at some distance from the traditional lunar nodes (whether the mean or the true nodes). Most likely it is these local lunar nodes that should be interpreted in the spirit of the north and south lunar nodes, as actively used in modern astrological consulting practice. When the local lunar nodes exactly conjoin a house cusp we would expect striking phenomena.

2.4.3. Solar and planetary zodiacs.

The solar zodiac was the first to be well understood. Its geometry is both simple and stationary — more precisely, nearly stationary. By this, we mean that a point on the surface of the earth follows a nearly circular path in its movement around the Sun. The plane formed by the velocity vector of a point on the Earth’s surface with respect to the Sun, and the vector pointing to the Sun with account for aberration41 (i.e. plane of the local solar equator), is oscillating with a period of one day and amplitude up to several arc minutes relative to the ecliptic. Remember that the ecliptic is a plane formed by the motion of the Earth-Moon barycentre42 around the Sun. At moderate latitudes this can result in a difference in the longitudinal coordinate of a planet on the solar zodiac of up to 1-2' compared to its ecliptic longitude. Here it is essential to apply the principle of locality: the events for a born with a given birthday and location on the Earth’s surface cannot be defined by a dynamic construction determined by the barycentre of the Earth-Moon system.

Planetary zodiacs have an additional important characteristic. At those moments when the central body changes its motion from direct to retrograde or vice versa, the plane of the local equator turns by 180°. The longitudinal zodiacal positions of other planets on the considered zodiac experience significant displacements as a result. The accuracy of the calculation of the local equatorial plane orientation falls by several orders of magnitude. We have not yet analysed the implications in detail.

41 In a strict sense it is necessary to take the vector tangent to the light geodesic, going from the

center of the Sun to the considered point on the Earth surface for this direction. 42 Center of mass.

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3. The mutual projection of two zodiacs. In astrology when analysing some past event or making a forecast one traditionally projects the cusps of the stations of the terrestrial zodiac onto the solar zodiac and calls them house cusps. The mutual projection of zodiacs becomes possible after the identification of two zodiacal circles with reference points and their corresponding equators, i.e. after establishing the correspondence between a zodiac as a dynamical structure and a geometrical construction on the celestial sphere. Any sensitive point on the first equator can be projected onto the second equator according a simple rule from traditional astrology: the projection (image) of a point on the first equator is a point on the second equator having the same zodiacal longitude on the first zodiacal circle as its prototype43.

Note a number of simple properties of this projection. If we project a point on the second equator that is the projection of a point on the first equator back onto the first equator, this second projection will not coincide with the initial prototype on the first equator. This property can give rise to apparent contradictions in the simultaneous analysis of several zodiacs. If the point on the zodiacal circle that is to be projected onto another zodiacal circle is the image of a massive body located beyond the first equator (i.e. that has latitude in the first zodiac), then it is necessary to use its direct image on the second equator rather than the projection of its image on the first zodiacal circle.

This remark is of considerable significance for interpretation in the framework of traditional astrological analysis of a natal chart. The interpretation of a planet in a house is in common practice the interpretation of a point obtained by projection on the terrestrial zodiac of a point that is already a projection of this planet on the solar zodiac. For planetary house positions, it is better to interpret directly in the terrestrial zodiacal circle.

Let us call the crossing points of two equators and their images on the corresponding zodiacal circles local nodes. By definition, the local node of the equator of the second zodiac on the equator of the first zodiac is the ascending node, if after a small displacement of the crossing point in the positive direction along the second zodiacal circle, the resulting point is closer to the north pole of the equatorial coordinate system of the first zodiac. The same crossing point of two equators can be identified in two ways, depending on the zodiacal circle from which it is considered. For instance, the vernal equinoctial point can be called an ascending solar node on the terrestrial equator or a descending terrestrial node on the solar equator (ecliptic).

The zodiacal longitude τ2 on the second zodiacal circle of the projection of a point of the first zodiacal circle with longitude τ1 is calculated using the following formula. We choose the fundamental point (east point) as the reference point on both zodiacal circles:

( ) ( )( )tg

sin

cos cos sin tgτ τ

τ τ

τ τ ε ε θ2 2

1 1

1 112 12

11

2

2

− =−

− ⋅ + ⋅�

�

�

(6)

where ε12 — angle between two vectors of infinitesimal displacements of the

43 This image is not a function, as some points can have three images.

35

crossing point of two equators in the positive direction along the considered equators. One can see from the determination of ε12 that this value is not simply the angle between the planes of the two equators, since it falls within the range 0° and 180º. tg θ 1 is defined by formula (2) for the first zodiacal circle. The positive choice of sign for the angle ε12 between the planes of the equators corresponds to the zodiacal longitude τ� 2

1 of the descending local node of the second zodiacal

circle on the first equator and to the zodiacal longitude τ�1

2 of the local ascending

node of the first zodiacal circle on the second equator.

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4. Symbolic times

4.1. Symbolic mappings. The world line of a born The use of symbolic times for forecasting is one of the most intriguing enigmas of astrology. Even if the main hypothesis of this work can be considered an adequate explanation of the nature of a zodiac, the correct determination of symbolic time mappings remains a mystery. Both the traditional linear determinations of progressions, directions and profections, and the collection of additional symbolic times invented by astrologers of the 20th century, give an impression of artificiality, of dependence on human convention.

Let us briefly review the generally accepted astrological definitions of these methods. Variants of the methods differ in details, so it is important to state the underlying idea for each category of methods. Progressions: 1 year of real time is equal to 1 day of progressed time. Directions: 1 year of real time is equal to 4 minutes of directed time, or 1º of zodiacal circle rotation. Profections: 1 year of real time is equal to 30º of zodiacal circle rotation. Symbolic times refer to the evolution of a specific object as a whole, born at a specific moment of time in a specific location. In the stated equalities symbolic and real times are counted out from this moment. With astrological methods, we study different kinds of wholes: persons, animals, nations, organizations, states etc. The symbolic time mappings stated above are not smooth due both to variations in the length of the solar day caused by the elliptical nature of the Earth’s orbit, and to the non-uniformity of the Earth’s rotation. If a smooth, uniform mapping is used, it is based on the average solar day, the result of human convention rather than astronomical fact. It is clear that a well-grounded determination of symbolic time mappings must be founded on a single conception, and on the real motions of massive bodies.

It was already understood in the later Middle Ages that a temporal mapping was the basis of “directions”. The definition of the technique of directions as presented within the framework of the Naibod method [12] or the method of Ptolemy-Placidus [11], implies a direct relationship between the interval of (transit) time lived by a born from his birth and a second interval also counted from the same birth44. There is also an opinion that the definition of progressions can be found in the Bible, a much older text than the medieval sources mentioned in the introduction to astrological methods. In fact, the temporal mapping45, the relating of two time intervals, is implied in the Bible.

Having determined a symbolic time corresponding to the transit time according to one of these mappings (progressed or directed), the medieval astrologer erected a chart for this (progressed or directed) moment of time. Those methods in which one shifts the house cusps or planets on the solar zodiacal circle by a uniform number of degrees per year or month etc. most likely appeared no earlier than the 19th or 20th centuries. We do not know of earlier sources for these methods. Most likely their appearance is connected with the period of decline in astrology, when it was largely transmitted by people having only an elementary mathematical education, insufficient for carrying out the sophisticated calculations characteristic

44 The question of what we understand by the moment of birth will be discussed below. 45 Day for year.

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of medieval astrology. It is necessary to note that there is some merit to parts of these methods (i.e. some calculations taken from them can give sufficiently good results). We think it exactly due to this phenomenon that such methods did not die off immediately.

When using generally accepted definitions of the progressed or directed moment corresponding to a transit moment of time, it is necessary to decide which geographical coordinates to use for the erection of the natal and progressed or directed chart. In the literature we find a variety of opinions. This situation reflects the fact that it is necessary to take into account the displacement of a born during his life when determining his temporal mappings. Earlier we proposed to name as an event a combination of the location of a born with the moment of time when he was there. So instead of a purely temporal mapping it is necessary to consider an event mapping or a mapping of the world line of a born46. This also follows from the exact definitions given below, since the temporal mapping includes among its parameters the coordinates of the born at different moments in his life. We name the sequence of events in the life of a born, characterized as combinations of both coordinates and moments of time, the world line of the born. Using this definition, it is possible to say that we construct a mapping of the world line of a born into itself. The event of birth as a combination of birth time and coordinates of the birthplace is a fixed point that is mapped into itself.

To construct a smooth temporal mapping we consider two zodiacs. Each zodiacal circle has a fundamental reference point and a reference point that is a crossing point of the local equator of this zodiacal circle with the local equator of the other. Each fundamental reference point moves relative to the point of intersection. Let us identify the motion of 2 fundamental reference points relative to the point of intersection47. Specifically, we put a time interval T into correspondence with another interval t so that the displacement ∆τ∩2

1 of the local ascending node of the second zodiac on the first one with respect to the fundamental reference point of the first zodiac for the considered time interval T is equal to the displacement ∆τ∪1

2 of the local descending node of the first zodiac on the second one with respect to the fundamental reference point of the second zodiac for the time interval t. ∆τ ∆τ∩ ∪=

2 1

1 2( ) ( )T t . (7) Since the motion of each fundamental reference point relative to the local node takes place in real time, but with different velocities in units of zodiacal longitude, we have defined a temporal mapping. Let us call the moment of birth of a considered object as a whole a moment of creation.

Let a time T elapse from the moment of creation. We now choose as first the zodiacal circle on which the fundamental reference point moves with a smaller velocity expressed in longitudinal units relative to the crossing point of the two local equators48. Let during the specified time T the fundamental point has moved through the angle Φ (= ∆τ ∆τ∩ ∪=

2 1

1 2( ) ( )T t ) on the first zodiacal circle. On the second zodiacal circle the fundamental reference point is displaced by the same angle Φ in a smaller time t. We have constructed the temporal mapping. The point

46 According to the theory of relativity. 47 This mapping does not depend on the choice of one of two points of intersection. 48 This condition follows from nowhere and is not necessary.

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with the coordinate t of symbolic time relative to the time of creation corresponds to the point with the coordinate T of real-time relative to the same initial moment.

If we consider that real time is the same for all49 in a given location, symbolic time is like an internal time of a born, existing only while this born exists as whole, detached from real time at the moment of creation. Let us call a scale of symbolic time defined in this way a progressed time.

Such a mapping can be constructed for any pair of zodiacs. However this mapping can have no property of one-to-one correspondence when the crossing point of two zodiacs changes its direction of motion with respect to either of the fundamental reference points.

4.2. Traditional systems of symbolic time.

4.2.1. Solar-terrestrial progressions.

Let us now move to the construction of the conformal variants of familiar symbolic times. We start with the solar / terrestrial zodiac pairing. It is logical to call the corresponding mapping a solar-terrestrial progression. The exact solar-terrestrial progression constructed by means of local zodiacs is fully covered by the general definition.

Consider a determination of an approximate version of the solar-terrestrial progression, constructed on the basis of the usual (solar) Zodiac and the circle of houses considered as the terrestrial zodiac. It is easy to see that the angular displacement of the east point (fundamental reference point) of the terrestrial zodiac with respect to the vernal point will be the sidereal time interval expressed in angular units of 360º for 24 hours of sidereal time. The angular displacement of the fundamental reference point on the solar zodiac will be given by the change in solar ecliptic longitude. To be consistent, the Sun should not be marked on the solar zodiacal circle. However the displacement of the conjunction point (located exactly 90º counter clockwise from the fundamental reference point), with which the Sun coincides on the local solar zodiac, is measured with good accuracy (for the traditional astrology) by the change in the ecliptic longitude of Sun. In this definition one sidereal day is equal to one tropical year. However in intermediate points such a definition does not give a linear mapping, since the motion of the Sun along the solar zodiacal circle is non-uniform50. To get an interval of progressed time from the moment of creation, it is necessary to calculate the change in the ecliptic longitude of the Sun from the moment of creation up to the considered moment of transit time. Moreover, the value of the angular displacement is not limited to one revolution (360°). Having determined this displacement, we next find the moment of time at which the angular displacement of the vernal point (i.e. the interval of sidereal time) from the moment of creation is equal to this

49 In the theory of relativity the time goes differently for each reference frame (for each world line). 50 In winter the Earth is closer to the Sun than in summer. Therefore in winter the Earth moves more

quickly in its orbit. The distance from the autumnal point to the vernal point is traversed more quickly by a week, than from the vernal point to the autumnal point.

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displacement of the Sun along the ecliptic. The resulting moment of time is the progressed time corresponding to the given transit time51.

The progressed chart (in one of the zodiacal circles such as the solar zodiac) is calculated for this moment of time and for the location of the born at that moment. Just as the angular displacement of the Sun on the ecliptic depends on the location of a born at the chosen moment of transit time (in order to correct for parallax52), the determination of the corresponding moment of progressed time depends on the location of born at the progressed time. Therefore a temporal mapping depends on the location of the born over time, i.e. on its world line. It is most correct to say that the constructed mapping is the world line mapping into itself.

4.2.2. Solar-terrestrial directions.

A direction, or more exactly a solar-terrestrial direction, is obtained through the composition of two solar-terrestrial progressions (i.e. the repetition of the progressed solar-terrestrial mapping). The directed time is the progressed time of the progressed time. It can be written by formula in the following way. If the progressed mapping is denoted by the function t = P (T), the directed mapping can be written as t = D (T) = P ( P (T)). It is interesting to note that the variant of directions formulated by Placidus [11] differs from that defined in this work by the non-linearity of the latter.

In connection with this definition, it is important to know the location of a born throughout the first 3 months of his life. The world line of a born during the first three months of life almost completely defines the progressed mapping for the first 90 years of transit time. To obtain the directed time it is necessary to calculate the corresponding progressed time twice, for the second step substituting the progressed time obtained in the first mapping for the transit time argument of the mapping. The first 90 years of transit time correspond to approximately 6 hours of directed time. However, for the construction of this temporal mapping it is necessary to know the movements of a born not only during the first 6 hours of life, but through the first 3 months as well. This requirement is probably the most unusual aspect of the method of conformal directions as defined in the framework of the proposed concept. It is not difficult to subject this to experimental verification if one finds a born with known and significant movements (from city to city) during the first months of life.

To distinguish different types of progressions in the formulae, we enter indices for the mappings P and D: t = PST (T) and t = DST (T) (this example is for the solar-terrestrial progressions and directions). It is clear that from 3 zodiacs it is possible to choose 3 pairings and consequently to construct 3 kinds of progressions and directions: solar-terrestrial, solar-lunar and lunar-terrestrial.

51 For instance, for a born whose age is exactly 10 tropical years the Sun will be displaced along the

ecliptic by 3600°. The corresponding displacement of vernal point is equal to 240 sidereal hours or 10 sidereal days.

52 The parallax correction for the Sun (for the Moon it can reach 50') is small, but it is necessary to take it into account.

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4.2.3. Profections.

In addition to the composite functions used to formulate directions by composing two identical progressed mappings, it is also possible to construct mixed composite functions. A possible association to traditional methods is the profection, since one such mapping t = RSL–ST (T) ≡ PSL (PST (T))53 gives results close to those of the traditional profection (here, SL means solar-lunar, ST – solar-terrestrial).

Since the progressed mappings entering this definition are non-linear, there are two variants of profection for any two given pairings of zodiacs. They are distinguished by the order of application of the component mappings. We call a fast profection a composition of mappings in which we first apply the faster mapping to the transit time, followed by application of the slower mapping. The faster mapping is the one that yields a greater compression of time. For instance, the fastest mapping of the considered progressions is the solar-terrestrial. We call the other profection the slow profection. Here are the functional forms of the conformal profections that are closest to the traditional profection: t = RSL–ST (T) ≡ PSL (PST (T)) — the quick solar-lunar solar-terrestrial profection. t = RST–SL (T) ≡ PST (PSL (T)) — the slow solar-terrestrial solar- lunar profection.

All conformal mappings constructed according to these principles must be considered hypothetical, excepting the solar-terrestrial directions that have been confirmed by the methods of Naibod and Ptolemy-Placidus. A significant body of observational data is necessary to prove them relevant to the events of a born.

53 Sign ≡ means “equality by definition”.

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5. Aspects, orbs and the technology of interpreting pinpoint accuracy.

5.1. Aspects. The notion of an aspect on a given zodiac can be defined for two arbitrary points of a zodiac circle and accordingly for their prototypes on the celestial sphere. Two points on a zodiac circle are in an aspect of the Nth harmonic if one point belongs to the family of the Nth order of the second point. In the context of a given harmonic the specific aspects are distinguished by a number n that determines the angular

distance between two points on the zodiac circle to be 360°⋅n

N. An event as a

qualitative transition of the state of a born occurs in the presence of an aspect between zodiacal elements.

In particular we want to emphasize that an aspect between two points on the celestial sphere is not defined by the corresponding angular distance (great circle distance) between them. An aspect is the angular distance between the images of two points on a zodiacal circle. Therefore (and this is known since Ptolemy54 [19]) the number of aspects between two points on the celestial sphere can be as great as the number of zodiacs you consider.

5.2. Orbs. The proper use of orb in event calculations using astrological techniques and symbolism is a difficult problem. In astrology the term orb designates the deflection from exactness of the difference in longitude between two points, from some value considered to be an aspect. Aspects are generally understood as some rational part of the full circle (360°). Orb is also used to designate the maximum deflection from the exact value for which it is still possible to speak of an aspect existing in some sense between two points. Here we use the word ‘sense’ to emphasize that different maximum orbs apply, depending on the type of interpretation to be done. Each kind of interpretation has its own associated values of maximum orb.

When interpreting a natal chart the orb of some aspects can be several degrees. It is useful to call aspects with such a large orb psychological aspects, whose use is the determination of the nature of a born: his reactions, his relations with other people and society as a whole, his scope of interests, profession and so on.

However in event calculations when interpreting the nature of an event it is necessary to use an orb of approximately one degree. Let us briefly describe the technique for event interpretation that we learned from Markina N. Yu, which she in turn attributes to Vaisberg V.A.

The nature of an event and its approximate time of realization are connected with the formation of an ensemble of aspects related to it. We consider aspects of planets

54 According Ptolemy two planets could simultaneously be in two aspects: the usual (solar) and the

“mundo” (terrestrial).

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among themselves, and those between planets and house or station55 cusps. By house cusps we understand the projections of the stations of another zodiac onto the one we are considering. The nature of an event is described by the houses or stations that are activated by the aspects of the ensemble. A house or station is considered activated if its planet-ruler and planet-significator56 have an aspect between them, and both of these planets have an aspect to the cusp of the house or station. We mark on a zodiacal circle both natal and transit planets and cusps. In the calculations the aspects between two natal points, between natal and transit points and between two transit points are considered. Furthermore, instead of transit points we also use the progressed or directed points57 (Fig. 6).

Fig. 6. The structure of an ensemble of aspects for an event. The aspects determining the development of this event in time are marked with bold lines.

When carrying out the research on the framework of this work we found an additional kind of aspect interpretation. This is an aspect that directly induces the coming of an event. The ensemble of aspects we have discussed so far exists (within the orb of 1°) for a certain duration. How do we determine the exact time of the event? Some events comprise several sub-events that occur in sequence. For

55 Aspects of planets to station cusps in the angular measure of the terrestrial zodiac were used in the

method of directions in medieval astrological practice. Currently, almost no one uses them. 56 We question the existence of some strict basis for the notions of ruler, significator, exaltant and

their antipodes. More likely this is a simple way to note the power and intensity of a planet by sign or house (station). We have simply outlined the widespread variants of techniques for event calculations. Sometimes one happens to take an exaltant or significator for the analysis of some event, or simply one of the planets whose characteristics are combined with (or opposite to) the sign or house (station).

57 Most often the points of some symbolic chart are considered, since the cusps of transit houses move too quickly in real time to be used to consider events other than an accident, a splintering of dishes etc.

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example, when moving to a new apartment you first buy it and only later move to it. In the ensemble of aspects of an event there are long-lived and short-lived aspects. The short-lived aspects are the aspects to the transit (progressed, directed etc.) cusps of the radix and to the transit (progressed, directed etc.) planets. The moments of time when these aspects become exact define the exact time of an event, or of one of its sub-events, when there are several.

5.3. Experimental orb of "exact" aspects and the accuracy of astronomical and astrological calculations.

The experimental orb of "exact" aspects is about 30". For calculations in our research we used the astrological program CONCEPT. The precision of the calculations of planetary coordinates with this program is about 1"58. But the error in determining the orientation of the Earth can reach 15". The precision of the Earth orientation calculation will affect the precision of the house cusp determination, which can be several times less precise than that of the Earth orientation59. The CONCEPT program was written according to the algorithms presented in the paper by Sergei Tarassov, the author of the astrological program ALMAGEST.

Of the astrological programs with which we are familiar, ALMAGEST produces the most exact astronomical calculations for both planetary coordinates and the orientation of the Earth. Currently it is not clear whether the 30" orb is due solely to accumulated calculation errors, or whether it is fundamental; that is, whether 30" is an actual spread of astrological sensitivity. It is important to note that a large number of mistakes (if not superstitions) in astrology are due to confusing accumulated astronomical calculation errors with fundamental astrological orbs. We hope that the orb of 30" is of a purely computational nature, and that the increase in precision in calculating the orientation of the Earth will reduce the experimental orb. For now the precision of the Earth orientation calculation of 15” is sufficient, since its proper use requires geographical coordinates of a born at different moments of his life precise to 400 metres. Even a determination with a precision better than 300 metres (about 10") does not improve the situation, since the attainment of greater precision requires the use of gravimetric maps. Such levels of precision will hardly interest an astrologer, but are of great interest to a physicist, as they would allow a reliable proof of the unnecessity of the aberration correction in astronomical calculations used for astrological purposes60.

5.4. Event-trigger points of the horoscope. The question of what points on a zodiac are connected with the occurrence of an event is not as simple as it might seem at first glance. Not all points on the zodiac circle are connected with an event occurrence. The prime examples of such ineffective points are the cusps of the solar zodiac signs. The position of a planet in a solar zodiac sign displays strongly in the personality; the entrance into a new

58 For the 20th century. 59 The deterioration of accuracy in house cusp calculation is due to the fact that the house cusp is a

crossing point of the house cusp plane and the ecliptic plane. The smaller the angle between the two planes, the worse is the accuracy. This is particularly important at high latitudes. For instance, at the latitude of Moscow the accuracy can fall by one order of magnitude.

60 In the language of the general theory of relativity this means the use of a purely spatial geodesic rather than the light geodesic for the determination of the planets’ positions on zodiac circle.

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solar zodiac sign by a progressed planet occasionally causes obvious changes in the person’s life. But the actual events associated with such a change occur on one or more aspects of the considered planet to a house cusp or to another planet immediately before or after the entrance into the new sign61.

Event occurrence is carried by the planets, and by the house cusps as projections of the station cusps of terrestrial zodiac onto the solar one. This is known from medieval European astrological tradition (Ptolemy-Placidus directions): events occur on aspects of planets to station cusps in the terrestrial zodiac. Thus, we draw the conclusion that event occurrence is effected jointly by the planets, and the cusps of the zodiacal stations and of the houses as projections of zodiacal station cusps onto another zodiac. For the terrestrial zodiac this is proven by centuries of astrological practice.

61 Such inexactness can be also connected with the fact that when the local ecliptic (local equator of

solar zodiac) oscillates, the mutual distances of planets in the measure of the local solar zodiac change less than their longitudinal coordinates as counted from the local vernal point.

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6. Astronomer’s notes.

6.1. Coordinates of planets for the location of a born. The correction for parallax.

The positions of planets on the celestial sphere depend on the observer’s location. As an extreme but very convincing example it is very easy to see that the geocentric and heliocentric coordinates of the Moon or some planet differ considerably. But even displacement along the Earth’s surface results in displacements of the planets - or more exactly their images on the celestial sphere. The closer a planet to the Earth, the larger its angular offset when moving an observer along the Earth’s surface. For the planets nearest the Earth, this offset from a geocentric position, called parallax, is some tens of angular seconds. For the Moon the difference on the celestial sphere can reach almost 2° (Fig. 7).

Fig. 7 Diagram illustrating the origin of the parallax correction for the ecliptic coordinate of the Moon. The local (topocentric) ecliptic longitude of the Moon can differ from the corresponding geocentric longitude by up to 50'. This correction is at its maximum when the Moon is near the horizon, and at its minimum near the celestial meridian.

For medieval astrologers this was not a problem — they simply observed the sky. With the appearance of ephemeredes compiled for an observer at the Earth’s centre,

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many people have used them without understanding the nature of the coordinates listed in the tables. At that, if the error in the position of planets and Sun is small, for the Moon it can reach one degree. When the Moon is close to the MC, its true position62 (topocentric coordinates) is determined practically correctly by the geocentric ephemeris, but when it is close to the Ascendant or Descendant, its true position is differs from the geocentric one by almost a degree63.

6.2. Direct and retrograde planets. Stationary points. The comprehensive account of geometric and physical effects in the calculation of the visible motion of the planets not only enables us to improve the accuracy of astrological calculations; it also reveals a number of new phenomena valuable to the astrologer in an interpretive context.

Fig. 8. Diagram of the daily motion of a planet relative to the local vernal point: a) when the planet is direct (prograde) in its ecliptic coordinates; b) when the planet is retrograde in its ecliptic coordinates.

What we see is a change in the direction of a planet’s motion with respect to the local vernal point from direct to retrograde and vice-versa. The local vernal point oscillates with respect to the usual vernal point (true or mean) with a period of 1 day. This is caused by the daily oscillation of the local ecliptic plane with respect to the astronomical ecliptic. The amplitude of the fluctuation of the local vernal point is not large - only 10'-20'. But the velocity of this fluctuation is so high at times that all planets except the Moon become retrograde for part of the day. For instance, if Mercury is fast direct it becomes retrograde for several hours a day. If on the other hand it is retrograde it becomes direct for several hours a day.

62 For an observer on the Earth’s surface 63 The coordinates giving the true position of a planet for an observer on the Earth’s surface are

called topocentric in astronomy.

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This information can be a powerful instrument for the determination of the effectiveness of various kinds of action during a day in some location. Periods when the local vernal point moves quickly backwards, making even outer planets fast direct, are best for expansion, attack, the propagation of ideas - any activity influencing the surrounding world. Periods of general retrogradation are suited for internal work, contemplation, and heart-to-heart meetings – those activities where it is important to listen to one’s internal state and to become transparent to outer influences.

When the velocity of the local vernal point changes sign with respect to the usual vernal point, the planets change their direction at times that depend on their own velocities with respect to the stars. Nearly all the planets except the Moon and periodically the Sun, Mercury and Venus, change their directions relative to the local vernal point twice a day.

The most interesting aspect of this is the analysis of local stationary points in the context of symbolic temporal mappings – directions and progressions. In conform progressions and directions the local stationary points of planets play a greater role in horoscopes of people for whom planets change their direction in the first 2-3 hours of life. The presence of such points in Ptolemaic-Placidean directions should strongly manifest, since these points require a reorientation of the corresponding spheres of life from expansion to internal development or vice-versa. Such moments could manifest as serious life crises. Similarly, in progressed time analogous changes occur in each sphere of life twice a year.

Observation of such periods in a person’s life is of interest since it allows us to sense the difference in interpretation of progressed and directed planetary configurations and the events corresponding to them.

When a planet at a stationary point makes an aspect to a natal house cusp, this should be powerfully revealed by an event. If the stationary point, which can be determined to some minutes of real time using a computer with ephemeredes accurate to 1", does not show in some event, it will still be noticeable subjectively. Most relevant here are calculations not in transit time, but in progressed or directed time. Let us repeat again that for each point on the Earth’s surface the exact moment of stationarity of a planet is different. The accuracy in determination of the stationary moment of about 1-2 minutes of progressed time corresponds to about one day of transit time64.

6.3. Zodiacal conjunctions. The full use of the terrestrial zodiac reveals another unusual phenomenon. A planet can be in exact conjunction with a star twice a day throughout a period of several days or even weeks. This pertains to two slow planets as well. Consider the conjunctions of Uranus and Neptune that from a solar zodiacal perspective occurred three times in 1993 (February 2, August 19 and October 25). Since the planets had different ecliptic latitudes, they coincided only in ecliptic longitude. However, the conjunction can take place at other moments using a different definition of longitude. Terrestrial zodiacal longitude is an example. Because of the complex geometry of the terrestrial zodiac even motionless objects on the celestial sphere

64 It turns out to be rather curious to observe oneself on the days of the progressive planet turnabout

in one’s own horoscope.

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(e.g. stars) move along the it with varying relative velocities. A consequence of this non-uniformity of motion was that Uranus and Neptune were conjunct in the terrestrial zodiac twice a day throughout all of 199365. Our personal experience shows that these moments of exact conjunction can reveal themselves in events66.

If we consider the fundamental reference point on the solar zodiac, all planets other than Mercury and Venus are always retrograde, and the Moon is always direct. It is interesting to study the stationary points of Mercury and Venus from the standpoint of this fundamental reference point, i.e. those moments when the zodiacal velocities of Mercury and Venus are equal to that of the Sun.

6.4. The Natal horoscope as a multi-dimensional chart. The use of a zodiacal circle for both psychological study and event calculation illustrates the reduction of the interactions of sensitive points on the two-dimensional celestial sphere to a one-dimensional approach on a zodiacal circle. Marking house cusps (or station cusps or signs) on a zodiac does not make the approach two-dimensional. In our opinion, a two-dimensional approach comprises the simultaneous consideration of two zodiacs. Since planets have a non-zero latitudinal coordinate, i.e. planets are not moving exactly along the local equator of one or another zodiac, they can be in exact conjunction in one zodiac and in the aspect in another. For example, let us examine the horoscope of a person born when Pluto and Uranus were both on the plane of the horizon. For the terrestrial zodiac this means that planets are both in conjunction with the cusp of the 7th station. At the same time they were in a semi-square aspect in the ordinary solar zodiac. Horoscopes of this kind obviously call for special interpretation (Fig. 9).

In addition, as soon as we move to the joint analysis of two events, this consideration takes on an extra dimension. The first event is usually the moment of creation (birth). The second is some event on the world line of the same born. As the second event one can take the event itself, or another on the same world line related to it by a symbolic mapping (progression, direction, or profection).

Practically we try to analyse the second event as it is determined by the preceding event (most often event of birth). We place the elements of the transit or symbolic zodiacal configuration on the zodiacal circle of the natal configuration, thereby equating the two zodiac circles of one central body for the born, but calculated for different points on the born’s world line (fate). No new analytical possibilities appear in such a one-dimensional view - we merely consider mutual aspects of the points of both configurations.

A real two-dimensional consideration does yield new analytical possibilities. Let us consider an example that is easily understood by any astrologer. The intersection of a slowly transiting planet or progressed planet to a natal house cusp, particularly the Ascendant, often causes if not an event, a noticeable change in the structure of consciousness of a born (however, such a change still needs to be anchored by an external event). But the conjunction of a planet with the Ascendant does not mean the intersection of the planet with the plane of the horizon, which is the cusp of the first house treated as a two-dimensional station of the terrestrial zodiac. This is connected with the fact that the planets are not in general exactly on the ecliptic

65 For Moscow the last such conjunction took place 16 January 1994. 66 Here we mean the events of everyday life.

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(equator of solar zodiac). For example, the Moon or Venus can cross the horizon one hour later or earlier than the time of their conjunction with Ascendant (Fig. 10). The rising of the Moon is defined as the conjunction of the Moon with the cusp of 1st station of the terrestrial zodiac. As another example, transiting Pluto can conjunct the natal Ascendant seven years earlier than it will cross the natal horizon.

Fig.9 Example of the simultaneous consideration of two zodiacs for one birth event. Arabic numerals mark the stations of the terrestrial zodiac.

Fig.10 A view of the celestial sphere near the east point of the terrestrial zodiac. It is easy to see that even with the house cusps marked on the solar zodiacal circle, the one-dimensional view of the solar zodiac gives a deceptive picture of the positions of Pluto and Venus. If we treat this picture as the sky of the natal horoscope, and transit planets move only by changing their ecliptic coordinates, it is easy to see that the ecliptic point of Pluto (its point on the usual solar zodiac) crossed Ascendant some years ago and is now in the first house, whereas Pluto itself is above the horizon within the 12th station of the terrestrial zodiac.

For analysis of such situations it is useful to consider a special chart that we call the "transit in the natal sky". We set a one-to-one correspondence of the zodiacal coordinates of two zodiacs as it existed at the moment of creation. We place the elements (planets) of the event under consideration with latitudes on the first zodiac, and then project the elements of the considered event onto the second zodiac, as if the second zodiac were oriented with respect to the first as it was

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oriented at the moment of creation. If for the pair, we select the solar zodiac as the first element and the terrestrial as the second, we have the "transit in the natal sky" chart. Such a construction is possible for any pair of zodiacs. The single technical restriction is that it is impossible to take the terrestrial zodiac as the first element of the pair. This is because of the current uncertainty about the treatment of a latitude coordinate in the terrestrial zodiac.

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7. Precision rectification of the creation (birth) time and symbolic times. The attainment of reliable results when working with symbolic times is wholly dependent on the correct choice of the fixed point of the progressed and other symbolic mappings, i.e. by the choice of the moment of creation (birth). The accuracy of determination of this point on the timescale is determined by the astronomical precision of the calculation of the Earth’s orientation. Since the precision of the calculation of the Earth’s orientation using the CONCEPT program is about 15", it becomes necessary to know the moment of creation with an accuracy of about 1 second of time. This requirement is much more stringent than can be expected of the information taken from hospital archives. Even when we observe the birth process ourselves in order to time it, it is incomprehensible what moment in the birth process to take for the moment of creation.

The investigations conducted by the author of the present work with Alexandre Bochkour have shown that the true moment of creation is not connected with some particular action of an obstetrician, or of the woman in birth, or of the newborn. It is necessary to define the moment of creation by purely astrological means, although other approaches are possible67 — that is, the rectification procedure is necessary. We state our understanding of the procedure below. We will consider the case when we reliably know the medical registration time (the preferred situation), or when we have a first approximation to the birth time as a result of one of the effective procedures for rectification that give it with an accuracy of several minutes68.

In the section on timing events, we formulated a hypothesis whose effectiveness has shown itself repeatedly: at a given place and moment of time something occurs if and only there is an exact aspect in one of the local zodiacs.69 The accuracy of the aspect is crucial to a correct understanding of this statement. We mean a concrete accuracy of 0.5 angular minutes that could not be attained using the methods of medieval astrology. We do not pretend to more intelligence than the ancients. If we have discovered something new, astonishingly simple and practically very effective, it is only because they did not have our modern instrumental capabilities. Otherwise these techniques would have been known long ago. What they did not have was the accuracy of modern astronomical calculations. For any kind of aspect the orb could not be less than the errors in the precision of astronomical observations. If we take the orb as one degree between 10 (or 7) planets and 12 house cusps, it is always possible to find at least one aspect of some planet to some house. When working with an orb of 30", it is rare for even

67 On this matter please write directly to the author. 68 It is necessary to note that books on astrology are overcrowded with descriptions of various birth

time rectifications. For the most part, they are superstitions of their authors without presentation of the procedure used. Some methods are simply wrong. An example of a wrong procedure sanctified by centuries is the Trutine of Hermes. The only effective method of birth time rectification known to author is rectification using events once the ascending zodiacal sign has been determined. The determination of an unknown ascending sign according to the character, constitution and reactions of born is the most difficult and exciting art that a practicing astrologer can master.

69 An aspect of a planet in the solar zodiac to the cusp of a terrestrial house, or an aspect of a planet in the terrestrial zodiac to the cusp of a station.

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one such aspect to occur70. A moment when such an aspect exists is obviously distinguished relative to other moments.

Furthermore, an exact aspect at which an event occurs defines a certain profound symbolic meaning of the event for the human spirit. The first steps on the way of usage of exact methods of astrological calculation as proposed in this work, in the determination of an exact aspect in a directed or progressed chart for a particular event, have forced us to revise our astrological and philosophical understanding of events in the human life.

Each event is in correspondence with a fundamental triad: planet, aspect and station or house. From one perspective, the situation has been simplified: we no longer need to deal with the whole bundle of planets, aspects and houses simultaneously. On the other hand, such an analysis causes us to revise our understanding of the meaning of a considered event. In the long run, work on event analysis according the principle of the fundamental triad brings about a change to our interpretation of planets, aspects, and houses or stations.

The birth of a person is a significant event. It too must occur on an exact aspect. Furthermore, based on astrological symbolism, this must be an exact aspect of a planet to the Ascendant. In the language used in this work, we mean an aspect of a planet to the cusp of 1st terrestrial station in terrestrial zodiac, or to the Ascendant (as the projection of 1st terrestrial station onto the solar zodiac) in the solar zodiac. This statement is not a hypothesis following from the consistency of the notional construction given in this paper; it is an empirically verified fact.

In those birth events where such an aspect has been found, the most frequent aspect to the 1st station or Ascendant was from one of the slow planets. It is often easy to find such an aspect within several minutes of the time recorded in a hospital archive or that resulting from rectification. The check described below usually confirms a candidate moment whether it is alone or one of a cluster of triads close to each other in time71. In such cases it is often, but not always, one of the slower planets that both participates in a fundamental triad and is the ruler of the ascending sign72. It is interesting that it was often straightforward to choose the correct triad based on the psychological profile of the born; the planet fixing the moment of creation shows itself prominently in the character of the individual. By ‘check’ we mean event checking of the obtained time by means of conform progressions and directions. The check is considered positive if for considered events there is an exact aspect (triad) in one of the symbolic times.

In a large number of natal horoscopes (more than half) we did not find such a triad. We did not seek for an exact percentage of failures, as the sample we used was obviously non-representative. We have noticed that for bright creative natures it is not difficult to find such a triad. In these cases the aspect in the triad is most often both (a) major and (b) in the terrestrial zodiac. We posit two reasons for the cases when no triad is found:

70 Practically it is necessary to limit aspects to those of harmonics no greater than the twelfth. 71 Most often there are two, one an aspect in the terrestrial zodiac, the other in the solar. 72 It is exactly in the process of such investigations that we come to doubt that the notions of ruler,

exile, exaltation and fall have a conceptual basis, except as an approximate indication of the strength of a planet in a sign. A significant proportion of exceptions was observed for the formulated rule. Besides, for those ascending signs where the rulers are considered to be internal planets, we cannot show a single case of which we’re confident.

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1. It’s possible that the triads occur in zodiacs other than the solar or terrestrial, to which our work has been mostly confined. The leading candidate to consider is the lunar zodiac.

2. It may be necessary to look not only at aspects to the 1st station or its projections on the solar and other zodiacs, but to the 4th station and to the point where the central body is located and its projection onto another zodiac. Or perhaps it is necessary to consider the whole 4th order family of the conjunction point, i.e. the Descendant and the MC73. On the other hand, the increase by 4 of the harmonic number (denominator) of aspect as element of fundamental triad of birth could give us the possibility to restrict ourselves to the consideration of the Ascendant only.

The best way to solve this problem would be to scan the time interval surrounding the time of birth on a powerful computer in order to find that moment for which the events of the born’s life in symbolic time have corresponding fundamental triads with an exact aspect in one of the zodiacs. In the future we plan to implement this idea.

73 It seems to be sufficient to take into consideration the Ascendant and IC, since there is always a

simultaneous aspect to the opposite points whose harmonic (denominator of fraction of the full circle) is the same or 2 times higher or lower.

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8. Conclusions. Let us summarize the purpose, main conclusions and results of this work:

1. The main purpose of this work is to reformulate the computing algorithms of event forecasting astrology in accordance with modern physical and mathematical description of space and time. We took as a basis the concepts of geometrodynamics, more widely known as the general theory of relativity. The name of geometrodynamics, mainly used by experts, reflects the nature of this theory, and we prefer to consider it as a mathematics of space-time. If we separate the conception of geometrodynamics from Einstein’s equations, geometrodynamics is more mathematics than physics. Although we can encounter phenomena that it does not predict, it is still necessary to describe such phenomena using its language. In this sense geometrodynamics, just as mathematics as a whole, is not a theory but a language in which the strict notional systems existing in nature are described.

2. In the opinion of the author, this work has achieved its purpose. But in order to realize this purpose we had to abandon the philosophy of short-range action alleged to be a basis of the modern natural-scientific picture of the Universe. The possibility of the existence in nature of long-range action phenomena does not contradict the basic concepts of this picture. To avoid contradictions, it is sufficient to abandon the attempt to consider one of the short-range interactions as a base of astrological laws. One of the main notions by which we realize the stipulated program is that of simultaneity. We constantly took the opportunity to construct purely spatial three-dimensional sections of four-dimensional spacetime in this work. It is well known that it is possible to formulate classical geometrodynamics in the invariant four-dimensional language. This supports the conjecture that the as yet unknown physical basis of astrology is not classical.

3. The need for a unified description of such two such outwardly different phenomena as the Zodiac and the house system requires the possibility of working only with pure spatial sections of four-dimensional spacetime. The possibility of such a description also requires that we consider the Sun as the body that generates the Zodiac, since its structure when emphasizing the principle of locality is defined by the mutual dynamics of a born and the Sun. In this approach a house system is a projection of another zodiac generated by the Earth onto the solar Zodiac.

4. Compared to traditional astrological calculations, the results obtained according to the new algorithms differ by less than the orbs used by the majority of astrologers in event calculations. Therefore, the proposed concept and resulting algorithms agree with the traditional astrological calculations and interpretations that we’ve inherited from the Moyen Age. Nevertheless we have proposed the technique of exact aspects, through which the accuracy of astrological event calculations increases by 2 orders of magnitude. This technique translates a theoretical difference between the proposed and traditional algorithms onto a purely practical plane.

5. The proposed technique for pinpoint accuracy event calculations allows us to attain temporal resolutions of 1-2 seconds of transit time, a half-hour of real-

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time for progressions and 3 days for directions. To achieve this accuracy the conception of symbolic progressed and directed time has been changed. The linear temporal mappings widely used in medieval European astrology have been replaced by conform (non-linear) mappings of the event (world) line of a born. These mappings allow us to attain pinpoint accuracy in event calculations, but do require detailed information about the movements of the born in the first three months of life. Additionally, a born’s location must be known to an accuracy of several hundred metres.

6. The unified conception of the derivation of the Zodiac as a dynamic construction generated by the Sun enables us to determine the structure of the lunar and planetary zodiacs with a high accuracy as well. The lunar zodiac is of considerable importance not only for event calculations but also for the development of astrological symbolism. Unlike the solar and terrestrial zodiacs it cannot be described by means of a static, elementary geometry. Because of this, we can understand the failure of ancient astrologers to define the lunar zodiac in a way suitable for event calculations and interpretation.

7. The unified description of the equinoctial points, local lunar and planetary nodes may serve to redefine the significance of their positions for the interpretation. The local lunar nodes are situated near the usual lunar nodes at a distance of 10-20º for most of the day, but at certain times each day they diverge 100º or more. In these cases it is particularly easy to find the interpretative value of the local lunar nodes in natal charts.

8. Within the framework of the idea of the fundamental reference point of a central massive body we proposed to use previously unknown sensitive points in event calculations. These points are the cusps of the stations of a zodiac and their projections onto other zodiacs (except house cusps and stations of the terrestrial zodiac and the so called cusps of equal solar houses on ecliptics), a generalization of the familiar house cusps (projections of the terrestrial station cusps) and the equal solar house cusps (station cusps of the solar zodiac) on the ecliptic.

9. Within the framework of the exact aspect technique we introduced the hypothesis of the central interpretative value of a "fundamental triad" (planet–aspect–cusp). We also formulated the “eventless” exact-aspect technique for precise rectification to 1-2 seconds of time. This technique has shown its efficacy in a considerable number of natal horoscopes.

10. At first we were well satisfied by the state of astrological symbolism used for interpretation, and did not plan any research in this area. But the system of ideas and methods for pinpoint calculations proposed in this work requires a fundamental revision of the conceptual basis for traditional astrological calculations. We think that this revision could eventually have a significant impact on the evolution of the symbolic structures in astrology.

To conclude we would like to note one more characteristic (and maybe in our opinion the main one) of the presented work. The contemporary situation in astrology, where it is covered over by mysticism and doubtful calculation techniques, as well as the broad proliferation of "astrological" charlatanry, can only frighten off people with a sound natural-scientific education and with research experience in these fields. In fact, the more precisely a man thinks the greater the

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likelihood that he will disqualify astrology on first exposure from any claim to being an exact knowledge system. We hope that this work has decreased the probability of such a dismissal of astrology at first sight.

9. Thanks. We would like express our gratitude to our astrology teachers — Natalie Markina and Michail Levin, whose aid was invaluable in our attempts to comprehend astrological symbolism— the only valuable knowledge in astrology. All the rest in astrology is no more than the skill to think clearly.

The Author also thanks Sergei Tarassov, developer and programmer of the astrological program ALMAGEST, for the realization in the CONCEPT program of the presented algorithms. Without practical testing of the proposed ideas, which could not have been realized without a computer, we could not have dared to present them for public discussion.

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10. Definitions. Astrology — the science linking the events occurring in the Universe with the motions of the massive

heavenly bodies.

Aspect — on a given zodiac an aspect is defined for two arbitrary points of the zodiacal circle and accordingly, for their prototypes on the celestial sphere. Two points on a zodiacal circle are in an aspect of the Nth harmonic, if one point belongs to the Nth order family of the second point. An aspect is also the corresponding angular distance between images of two given points on a zodiacal circle.

Dynamic angle — the angle between the plane of the local equator and the axis of anisotropy.

Zodiacal circle — a circle of zodiacal longitudes with a reference point and sensitive points on it. Conform progressed mapping (or conform progression) t = P(T) puts into correspondence to the

interval T of (transit) time another interval t (T > t) of (progressed) time such that the displacement

∆τ∩2

1 for the considered time interval T of the local ascending node of the second zodiac on the

first one with respect to the fundamental reference point of the first zodiac is equal to the

displacement ∆τ∪1

2 for the time interval t of the local descending node of the first zodiac on the

second one with respect to the fundamental reference point of the second zodiac.

∆τ ∆τ∩ ∪=2 1

1 2( ) ( )T t .

Conform directed mapping (or conform direction) is a composition of two identical conform progressions, i.e. the repetition of the conform progressed mapping: t = D (T) = P ( P (T)). The directed time is the progressed time of a progressed time. Conform mapping of a profection (or Conform profection) is a composition of two different conform progressions, i.e. of 2 progressions of 2 different pairs of zodiacs of a born. For example, the conform solar-lunar solar-terrestrial profection — t = RSL–ST (T) ≡ PSL ( PST (T)) (SL means solar-lunar, ST — solar-terrestrial) is close to the usual definition of a profection (30° per year). Fast conform profection — a composition of two different conform progressions, in which at first the fast mapping applies to transit time, and afterwards the slow. The Fast is identified with the mapping that "compresses" time by more times. Slow conform profection — a composition of two different conform progressions, in which the slow mapping applies firstly to a transit time, and afterwards the fast. By its full name it can be distinguished from the fast conform profection. The solar, terrestrial and lunar zodiacs generate 3 pairs of profections. Examples: Fast profection of the 1st kind (solar-lunar solar-terrestrial): t = RSL–ST (T) ≡ PSL ( PST (T)). “30° per year” Slow profection of the 1st kind (solar-terrestrial solar-lunar): t = RST–SL (T) ≡ PST ( PSL (T)). “30° per year” Fast profection of the 2nd kind (lunar-terrestrial solar-terrestrial): t = RLT–ST (T) ≡ PLT ( PST (T)). “12° per year” Slow profection of the 2nd kind (solar-terrestrial lunar-terrestrial): t = RST–LT (T) ≡ PST ( PLT (T)). “12° per year” Fast profection of the 3rd kind (lunar-terrestrial solar-lunar): t = RLT–SL (T) ≡ PLT ( PSL (T)). “day per year” Slow profection of the 3rd kind (solar-lunar lunar-terrestrial): t = RSL–LT (T) ≡ PSL ( PLT (T)). “day per year”

Cusp of a station of a central body of Nth order — a point of the family of Nth order of the fundamental reference point (locating at 90° clockwise from the central body), i.e. a point on a zodiacal circle

with longitude equal to the part of the full circle 360 1°⋅ −( )n

N, where n — number of cusp, N —

order of the fundamental reference point family. For N=12 — simply the cusp of station. Cusps of the zodiacal signs of Nth order — points of the Nth order family of some reference point

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different from the fundamental reference point. Cusp of a house — a projection of a station cusp onto another zodiac. Cusp of a two-dimensional station — a great semicircle of points on the celestial sphere with the

zodiacal longitude360 1°⋅ −( )n

N.

Local equator (plane) is determined by two vectors: the force vector F acting on a born on the part of the central body as a whole, and the velocity vector V of the central body without taking account of aberration in the non-rotating proper reference system of a born. North pole of the local equatorial coordinate system is given by the unit vector in the direction of the vector product P = F × V. Fundamental reference point on the local equator (zodiacal circle) or the east point of the central body is given by the crossing point of the local equator and the plane of the local anisotropy horizon, the direction to which from the centre of the celestial sphere forms an obtuse angle with the velocity vector V of the central body in the proper non-rotating reference system of a born. East point of the central body (fundamental reference point) is also given by the unit vector in the direction of the vector A= e × P, where e is the anisotropy vector of the central body. West point — the point on the celestial sphere opposite the east point. Anisotropy vector e — phenomenological vector characterizing the space anisotropy caused by a massive body at the point of a born. For distant massive bodies this is the direction to its centre ignoring aberration, i.e. the tangent vector to the purely spatial geodesic connecting the born and the distant massive body. For the Earth this is the vector toward the gravitational attraction on the born from the Earth. Local meridian plane — the plane perpendicular to the plane of the local horizon and the plane of the local equator. Plane of the local horizon (of space anisotropy) — the plane perpendicular to the vector of space anisotropy caused by the central body. Point of conjunction — crossing point of the straight line of intersection of the local meridian plane and the plane of the local equator with the celestial sphere, in the direction that forms an acute angle with the anisotropy vector e. Point of opposition — the point opposite the point of conjunction.

World line of a born — the sequence of events in the life of a born expressed as combinations of coordinates and moments of time.

Moment of creation — the moment of birth of a born as a wholeness. We introduce this term in order to avoid futile debates as to what constitutes the exact moment of birth, as each astrologer has his own particular opinion in this matter. Let us emphasize that this is an exact moment, rather than a time interval, chosen by following events of a born’s life in the context of the techniques of the exact aspect and of the fundamental triad.

Orb of aspect — the maximum deflection of the zodiacal longitudinal difference between 2 points from the exact angular value characterizing an aspect, for which it is possible to consider the aspect as holding between the points for a given purpose.

Parameterization — the procedure of assigning the zodiacal longitudinal coordinate to points on the celestial sphere.

Planet as a zodiacal element — the point on a zodiacal circle having the same zodiacal longitude as a point on the celestial sphere representing a planet. A point representing a planet on the celestial sphere is defined by the direction of the anisotropy vector caused by this planet in the location of a born.

Projection (image) of a point on a first equator onto a second equator has the same zodiacal longitude in the coordinate system of the first zodiac, as its prototype (the source point on the first equator).

Born — a wholeness, whose world line (fate as an ensemble of events) is of interest to us.

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Nth order family of points generated by the first point on a zodiacal circle — N points, symmetrically located in the corners of a regular N-agon inscribed within a zodiacal circle. More strictly: the points of the Nth order family of a given point can be generated from the first point by the action of a regular representation of the cyclic subgroup of the Nth order of the one-parametric group of rotations. First point of a family — the point generating the family.

Event — the death of one wholeness and the birth of another. Any event occurs at a specific time and place, i.e. in this sense the term complies with the notion of an event as a collection of spatial and temporal coordinates, as used in the theory of general relativity.

Station (house, sign) — an interval on a zodiacal circle from the cusp of a station (house, sign) up to the cusp of the following station (house, sign). One-dimensional station (house, sign) — an arc on a zodiacal circle identified with the equator of the local zodiac. Two-dimensional station (sign) —ensemble of points on the celestial sphere, having the same zodiacal longitudes on the considered zodiac, as the points of the one-dimensional station (house, sign) on the local equator (zodiacal circle) of the considered zodiac.

Exact aspect — in the present work by an exact aspect we mean an aspect with an orb of 30". It is possible that this is not a fundamental orb, but an error in astronomical and astrological calculations. When we increase the precision of calculations we expect a decrease in the orb of an exact aspect to at least 1" without modifying the conception presented here.

Triad (fundamental) of an event: a planet, aspect, and the cusp of station (house). The planet and cusp are in exact aspect. The moment this aspect becomes exact fixes the occurrence of the event.

Local nodes — the crossing points of two local equators of a born and of their images on both zodiacal circles. Ascending local node (of the equator of a second zodiac on the equator of a first zodiac) — the local node, for which after a small displacement from the considered crossing point in the positive direction along the second zodiacal circle, the obtained point is closer to the north pole of the equatorial coordinate system of first zodiac, than the crossing point. Descending local node — the second crossing point, opposite to the ascending local node. For example, it is possible to consider the vernal point as the ascending solar node on the terrestrial equator or as the descending terrestrial node on the solar equator (ecliptic).

Central body: a massive body generating the zodiac under consideration.

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11. References.

1. V.V. Bolotov. Lectures on the history of ancient church. V. 2. History of church at the period before Constantine the Great. (StPetersburg., 1907). (in Russian)

2. Nicholas Campion. St. Augustine on Astrology. Astrology Quarterly, vol. 64/3, Summer 1994. 3. Witte-Lefeldt. Rules for Planetary Pictures. Plantation (Florida, USA): Penelope Publications,

1990. 4. Wendel Polich and A.P. Nelson Page. The topocentric system of houses. Spica, vol. 3, N 3,

1964, p. 3-10. 5. Wendel Polich and A.P. Nelson Page. Answer to Cyril Fagan's objections. Spica, vol. 5, N 3,

1966, p. 38-44. 6. Claudius Ptolemaeus. Quadripartitum. Venetiis, Bonetus Locatellus, XIII Kal. Jan. (20 Dec.)

1498, 2°. 7. R. Newton. The crime of Claudius Ptolemy. Moscow: Nauka, 1985. (in Russian)74 8. A.T. Fomenko, V.V. Kalashnikov and G.V. Nosovskiy and. Geometrical and statistical methods

of analysis of star configurations. Dating of Ptolemy’s Almagest. Moscow, Factorial, 1995. (in Russian)75

9. G.V. Nosovskiy and A.T. Fomenko. New chronology and the concept of the ancient history of Russia, England and Rome. V. 1 and 2. Moscow, 1995. (in Russian)

10. Alexandre Volguine. L'astrologie en Grece. L'Astrologue n. 106, 2° trim. 1994. 11. Giovanni Zattini. Confronto pratico sulle direzioni mondane Tolomeo/Placido e l'elevazione

polare degli astri. Linguaggio Astrale. Anno VII n.4, N. 101, II Semestre 1995. P. 116-136. 12. Noel Tyl. Prediction in Astrology. St. Paul (Minnesota, USA): Llewellyn Publications, 1991, p.

59. 13. William Lilly. Christian Astrology. Houston, Texas: JustUs & Associates, 1986. 14. W. Koch und E. Schaeck. Gebürtsortes Hausetabellen. Saarbrucken (Deutschland): Schaeck

Verlag.76 15. Ralf William. Bases of determination of the house cusps77. 16. P.I. Bakoulin, E.V. Kononovich and V.I. Moroz. Course of general astronomy. Moscow:

Nauka, 1977. (in Russian).78 17. E. Mach. Die Mechanik in ihrer Entwickelung historisch-kritisch dargestellt, F.A.

Brockhaus,Leipzig, 1904, S. 236; E. Mach. In: “The Monist”, Vol. XIV, 1903. 18. A. Volguine. Astrologie lunaire. Paris: Dervy-Livres, 1977. 19. Giovanni Zattini. L'aspetto mondano di Claudio Tolomeo. Acts of the I Congresso

Internazionale del Centro Italiano di Astrologia. Venezia, November 25-27th, 1994, N 43. 20. Dane Rudhyar. Astrology of personality. Moscow: Antaris, 1991. (in Russian) 21. Ptolomei Svarogich. Is the modernization of astrology possible? Russian astrology. N2, 1993, p.

4. (in Russian). 22. Ptolomei Svarogich. Calcoli bidimensionali con la precisione di 30" nell’astrologia

previsionale. Acts of the I Congresso Internazionale del Centro Italiano di Astrologia. Venezia, November 25–27th, 1994, N 41.

74 It is possible to find an English edition. 75 It is possible to find an English edition. 76 Reference without year of publication is taken from the brochure: HP-67/HP97. Users' Library

Solutions. Astrology. Hewlett-Packard, 1978 (?). 77 We have no better reference to this book. We used a photocopy of illustrations from this book.

The book is in English. This is the best book on houses that we found. So we dare to give an inexact reference in the hope that the interested reader will be able to find it.

78 Or any other book on general astronomy.