Space-Time and Astrological House Systems

Around 300 BCE the Greek mathematician Euclid wrote his famous treatise on plane geometry. In it, he presented a small number of axioms about the geometry of a plane surface, from which he logically derived the theorems of his system. Euclid’s geometry shaped the worldview of scholars until the 19th century when mathematicians realized that non-Euclidean geometries could exist and may even better describe the reality we inhabit. The most controversial of Euclid’s postulates was the parallel lines remain at a constant distance from each other and never intersect, even if extended to infinity. Scholars began to realize that if the space in which we live were curved, then parallel lines would either curve away from each other, or curve toward each other and intersect, with increasing distance.

Albert Einstein was one of the scholars who became fascinated by such non-Euclidean geometries. After a series of thought experiments, he came up with his theory of relativity in which there is no time without space, and our experience of time depends on our velocity through space as the observer. In other words, we do not live in a 3-dimensional Euclidean universe in which time is a separate dimension. Instead, we live in a 4-dimension non-Euclidean universe in which time and space are inseparably linked as “space-time.” My belief is that astrological house systems should be consistent with the prevailing scientific worldview of the epoch in which they are used.

Image from https://commons.wikimedia.org/wiki/File:GPB_circling_earth.jpg

John Wheeler summarized Einstein’s theory of relativity briefly in two sentences: “Space-time tells matter how to move. Matter tells space-time how to curve.”

Einstein’s theories have important implications for astrological house systems, which the textbooks tend to divide, someone erroneously, into “space-based” or “time-based” systems. I will not discuss here the system of whole sign “places” (topoi) which over time morphed into, and were replaced by, quadrant house systems, especially the Alcabitious system during the medieval period.

One of the earliest space-based quadrant systems was that attributed to Porphyry which divided the ecliptic circle into four quadrants bounded the the horizon and meridian. These quadrants were divided into thirds by taking the arc along the ecliptic between the horizon and meridian in each quadrant and dividing by 3.

Later astrologers who favored space-based systems felt that ideally the terrestrial houses should enclose equal volumes of space within the celestial sphere, which the Porphyry system failed to do by simply dividing each quadrant into thirds along the ecliptic circle.

To achieve houses which enclosed equal volumes of space, Campanus proposed equal divisions of the Prime Vertical, the great circle passing through the East and West Points of the horizon and through the Zenith and Nadir, at the location of birth. He then trisected each quadrant of the Prime Vertical, and projected these divisions onto the ecliptic using great circles connecting them to the North and South Points of the horizon.

Regiomontanus, a brilliant mathematician, felt that a better alternative to the Campanus system was to use the Equator rather than the Prime Vertical to generate house cusps. He favored using the Celestial Equator over the Prime Vertical because it was directly related to the diurnal rotation of the Earth on its axis and because it resembled the method of Claudius Ptolemy in his description of “releasing” and the use of primary directions to determine the length of life. Nonetheless, Regiomontanus was a completely spaced-based system which ignored the role of time in the structure of the universe, as if time had a separate existence.

Interestingly, Ptolemy in his description of primary directions appears to have had an inkling of Einstein’s notion of space-time. It is important to realize that Ptolemy was not describing a system of house division in this passage. In his description of primary directions to determine the length of life, Ptolemy writes (Ashmand translation, bold and italics mine):

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“Whenever the prorogative and preceding place is actually on the eastern horizon, we should take the times of ascension of the degrees up to the meeting place; for after this number of equinoctial periods the destructive planet comes to the place of the prorogator, that is, to the eastern horizon. But when it is actually at the mid-heaven, we should take the ascensions on the right sphere in which the segment in each case passes mid-heaven; and when it is on the western horizon, the number in which each of the degrees of the interval descends, that is, the number in which those directly opposite them ascend. But if the precedent place is not on these three limits but in the intervals between them, in that case the times of the aforesaid ascensions, descensions, or culminations will not carry the following places to the places of the preceding, but the periods will be different. For a place is similar and the same if it has the same position in the same direction with reference both to the horizon and to the meridian. This is most nearly true of those which lie upon one of those semicircles which are described through the sections of the meridian and the horizon, each of which at the same position makes nearly the same temporal hour. Even as, if the revolution is upon the aforesaid arcs, it reaches the same position with reference to both the meridian and horizon, but makes the periods of the passage of the zodiac unequal with respect to either, in the same way also at the positions of the other distances it makes their passages in times unequal to the former.”

“We shall therefore adopt one method only, as follows, whereby, whether the preceding place occupies the orient, the mid-heaven, the occident, or any other position, the proportionate number of equinoctial times that bring the following place to it will be apprehended. For after we have first determined the culminating degree of the zodiac and furthermore the degree of the precedent and that of the subsequent, in the first place we shall investigate the position of the precedent, how many ordinary hours it is removed from the meridian, counting the ascensions that properly intervene up to the very degree of
mid-heaven, whether over or under the earth, on the right sphere, and dividing them by the amount of the horary periods of the precedent degree, diurnal if it is above the earth and nocturnal if it is below.”

“But since the sections of the zodiac which are an equal number of ordinary hours removed from the meridian lie upon one and the same of the aforesaid semicircles, it will also be necessary to find after how many equinoctial periods the subsequent section will be removed from the same meridian by the same number of ordinary hours as the precedent. When we have determined these, we shall inquire how many equinoctial hours at its original position the degree of the subsequent was removed from the degree at mid-heaven, again by means of ascensions in the right sphere, and how many when it made the same number of ordinary hours as the precedent, multiplying these into the number of the horary periods of the degree of the subsequent; if again the comparison of the ordinary hours relates to the mid-heaven above the earth, multiplying into the number of diurnal hours, but if it re1ates to that below the earth, the number of nocturnal hours. And taking the results from the difference of the two distances, we shall have the number of years for which the inquiry was made.”

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While the above quote may sound rather technical, Ptolemy is saying that the measurement of the distance between points intermediate between the horizon and meridian has to take into account both space and time. A purely space-based or time-based system can only give approximate results.

For some reason Regiomontanus in the 15th century contented himself with a system that was a close approximation to Ptolemy’s thinking. Perhaps he could not find a workable mathematical solution to integrating both space and time into the generation of house cusps. In any case, the Regiomontanus system is rooted in pre-19th century Newtonian physics and Euclidean geometry.

Placidus in the 17th century, influenced by Kepler and his laws of planetary motion, especially the idea that the planets travel in elliptical orbits around the Sun in which they trace out “equal areas in equal amounts of time,” proposed and popularized what we now call the Placidus system of houses, a method was was described earlier by Ibn Ezra which was called the “hour circles” method. Alain G. Cablais comments in an article about Ibn Ezra and Placidus houses: “Ibn ‘Ezra then goes on to state the astronomical principle of this division of the twelve houses. This foundation is none other than the occurrence of a configuration between a celestial body and an angle by virtue of the perpetual motion of the sphere and according to the hourly pace of that celestial body.”

The Placidus system is consistent with Ptolemy’s description of “releasing” and the primary directions of points intermediate between the horizon and meridian, which takes into account both space and time, anticipating Einstein’s notions of space-time and the non-Euclidean geometries of the 19th century, implicit in Ptolemy’s original writings in the 2nd century CE. It is not surprising that Placidus fascinated British astrologers in the 19th century and became the predominant house system in the English-speaking world.

Image from http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html