P. Svarogich

Principia of polyzodiacal astrology

4.2.1 Solar-terrestrial progressions.
4.2.2 Solar-terrestrial directions.
4.2.3 Profections.

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 20^th 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 1ş of zodiacal circle rotation. Profections: 1 year of real time is equal to 30ş 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 19^th or 20^th 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 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 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 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.
. (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 F  (=) on the first zodiacal circle. On the second zodiacal circle the fundamental reference point is displaced by the same angle F in a smaller time t. We have constructed the temporal mapping. The point 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 comformal 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 360ş 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 90ş counterclockwise 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 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.

4.2.2Solar-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 = P_ST (T) and t = D_ST (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.

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 = R_SL–ST (T) ş  P_SL (P_ST (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 = R_SL–ST (T) ş  P_SL (P_ST (T)) — the quick solar-lunar solar-terrestrial profection.
t = R_ST–SL (T) ş  P_ST (P_SL (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.