The Koch house system is also called the Birthplace house system. Tables for this system first appeared in 1971 in an English translation of the work of Walter Koch. This sytem divides arcs and projects them onto the ecliptic as follows:
For cusps above the horizon, the semidiurnal arc (along a small circle) of the rising degree (ascendant) is trisected; then, altitude circles (small circles parallel to the horizon) are constructed through the points of trisection; finally, the cusps are determined by the intersections of these altitude circles with the ecliptic.
For cusps below the horizon, the semi-nocturnal arc is used. The MC and ascendant are the same, respectively, as the cusps of the 10th and 1st houses. This system is ill-defined at extreme latitudes.
Named for Porphyry (3rd century AD.This system uses great circles through the poles of the ecliptic to form the house cusps. The MC and ascendant are equal, respectively, to the 10th and 1st house cusps. The intervening house cusps are located by trisecting the ecliptic arc between the MC and ASC and between the ASC and IC. This system creates two pairs of three equally small houses in opposite quadrants and two pairs of large houses in the other quadrants. It is a variation of the Equal house system and attempts to divide the quadrants of the horoscope in a more equal manner. There are problems at extreme latitudes, but many astrologers like this system because it is relatively easy to calculate by hand.
Named after Placidus de Tito (1603-1668). The Placidian house system is often described as a "time-based" system (as opposed to a "space" system such as Campanus) because it is based on the division of an arc made by a moving point on the ecliptic. Although the MC and ascendant are respectively the 10th and 1st houses, the other cusps must be calculated according to division of their own semi-arcs, which is very laborious. This system is way off in extreme latitudes.
Named after Jean-Baptiste Morin (1583-1656), known as Morinus. The Morinus system uses great circles that pass through the poles of the ecliptic and through points that are spaced at 30-degree intervals along the Celestial Equator, beginning with the intersection of the Celestial Equator and the East Point. The cusps are determined by the intersections of these great circles and the ecliptic. The lunes of the houses are not of equal size. The MC is not the same as the 10th cusp. The ascendant is not the same as the 1st cusp. The great circle for the 1st house passes through the east point of the horizon. This system works in extreme latitudes but is not specific to birthplace.
Named after Regiomontanus (Johannes Muller), who devised the system 100 years after Campanus. The Regiomontanus system uses house circles which pass through the north and south points on the horizon. The MC and ascendant are, respectively, the cusps of the 10th and 1st houses. The arcs along the Celestial Equator between the MC and ASC and between the MC and IC are trisected, and house circles are then drawn perpendicular to the prime vertical through the trisection points. The points where the house circles intersect the ecliptic mark the cusps of the intermediate houses.
The Regiomontanus house system is a mathematically simpler version of the Placidian system. It gives slightly different cusp values from Placidus and does not use varying semi-arc ratios in determining the cusps. There are problems with this system in extreme latitudes.
Probably named after Johannes Campanus (? - 1297c). In the Campanus system, the houses are formed by great circles that pass through the north and south points of the horizon and a set of points spaced at 30? steps along the prime vertical, beginning at the zenith and proceeding eastward through the east point on the horizon. The cusps are located at the intersections of these house circles with the ecliptic. This system is ill-defined at extreme latitudes.
Where geocentric house systems are based on a projection of the birthplace to the center of the earth, the topocentric system keeps the birthplace on the center of the earth. It has a theoretical basis established through laborious empirical tests.
This is the personal favorite of the author.
The Solar house system takes the position of the Sun in a horoscope as the ascendant and assigns all other house cusps at equal 30-degree intervals from that position. The resultant chart is called a "Solar Chart." This method of house division is commonly used when the time of birth is unknown or unavailable. This system is not specific to birthplace.
The Meridian house system was proposed by the Australian astrologer Zariel (David Cope) in the early 1900s. It has never received much attention, but was investigated by American astrologers Bruce Lloyd and Garth Allen in the late 1950s. This system is also referred to as the "Axial Rotation System," a term which is somewhat misleading, as no rotation is involved, but derives from the use of the North and South Celestial poles for the construction of the great circles. Circles are drawn through the poles and twelve points that divide the equator equally, starting at the meridian. These points are then projected onto the ecliptic. The MC is identical to the 10th house cusp. The ASC is not equal to the 1st cusp, but is equivalent to the East Point or equatorial ascendant. This system works in extreme latitudes but is not specific to birthplace. The Meridian house system is most popular with Uranian astrologers.
The Equal house system is thought to have arisen about the time of Petosiris (1st century, BC). This system defines the house cusps by starting at the Ascendant and moving around the ecliptic in equal 30? segments. In the Equal system, the MC is not the 10th cusp, but the ascendant is the 1st cusp. Because of its simplicity, this system is popular.
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