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P. Svarogich

(English editing by Ed Falis)

Principia of polyzodiacal astrology

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 t 2 on the second zodiacal circle of the projection of a point of the first zodiacal circle with longitude t 1 is calculated using the following formula. We choose the fundamental point (east point) as the reference point on both zodiacal circles:
where e12 — angle between two vectors of infinitesimal displacements of the crossing point of two equators in the positive direction along the considered equators. One can see from the determination of e12 that this value is not simply the angle between the planes of the two equators, since it falls within the range 0 and 180. tgq1 is defined by formula (2) for the first zodiacal circle. The positive choice of sign for the angle e12 between the planes of the equators corresponds to the zodiacal longitude of the descending local node of the second zodiacal circle on the first equator and to the zodiacal longitude of the local ascending node of the first zodiacal circle on the second equator.

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