On the Domification Problem

by Bob Makransky

The basic problem of house division theory arises from the fact that a house system is an attempt to represent a three-dimensional situation in two dimensions. A house system isn't a pie, but rather a tangerine.  The divisions between wedges aren't lines, but planes. The problem of house division theory lies in the fact that certain information which makes sense from a three-dimensional point of view becomes highly distorted when squeezed into two dimensions.

Ideally, a house system should fulfill two conditions:

                        1)   It should model the earth's rotation. That is to say, the diurnal motion of a planet should be constant - it shouldn't take more time to pass through some houses than others.

                        2)   It should preserve the ASC, MC, DESC, and IC as house cusps. Another way of saying this is: both the horizon and meridian planes should delimit segments of the tangerine. We can only "see" the ASC as a cusp if we are "sighting down" the horizon plane (if our viewpoint lies on the plane of the horizon); and we can only see the MC as a cusp if we are sighting down the meridian plane (if our viewpoint lies on the plane of the meridian). Therefore, we can only see both the ASC and MC at once if both the horizon and meridian planes delimit segments of the tangerine.

The basic problem of house division theory lies in the fact that these two conditions contradict each other; and all the different house systems known to man represent different people's ideas of how to resolve that contradiction.

In fact, there is no way to resolve this contradiction (it is mathematically impossible to resolve it).  In practice what different house systems do is either ignore condition 1); ignore condition 2); ignore both conditions; or make a pretense of satisfying both conditions and end up satisfying neither of them.

Consider condition 1).   In order for a house system to model the earth's rotation, the axis of the tangerine must be the earth's axis, and our point of view must lie upon it. We squash the tangerine down onto the plane of the equator (or some plane parallel to it). We are looking down from the viewpoint of the north celestial pole, so our viewpoint is stationary, and everything else rotates around us (at a constant rate).  This is how the Meridian and Alcabitius systems are defined. The Meridian system is a perfect model of rotation.  In Alcabitius, a body's rotation is constant east of the meridian, but at the meridian it "jumps the tracks" and rotates at a different constant rate west of the meridian. As we shall see later on, there is a similar discontinuity in rotation in the Placidus and Koch systems, except at the horizon. That is, in Placidus and Koch, a planet passes through the houses above the horizon at a different speed than it passes through the houses below the horizon.

The Campanus, Regiomontanus, Sunshine, Horizontal, and Porphyry systems make no pretense of fulfilling condition 1), and therefore they are not good models of rotation. This is because the tangerine axes in these systems are not the earth's axis, and therefore as the world turns these models "wobble" (rotate at varying rates rather than smoothly at a constant rate).  In these systems our point of view is not stationary, but is itself rotating around the earth's axis.

The Placidus system doesn't wobble per se because of a rather elegant geometrical trick. Placidus is the only house system in which the segments of the tangerine are not delimited by planes, but rather by curves. The edges of the tangerine segments are not flat, but have a wave to them, like potato chips. Because our line of sight along these edges "bends", it is possible to maintain a fix on both the ASC and MC at the same time. However, Placidus has the same rotational flaw as Koch: rotation is constant above the horizon, and rotation is constant below the horizon, but as a body crosses the horizon it either hits the accelerator or slams on the brakes.

Now, just as the first condition (that rotation be smooth and constant) requires that our viewpoint be looking down from the north celestial pole, so too does the second condition (that the angles be cusps) require that our point of view be looking south from the north point on the horizon. In other words, condition 1) implies that the axis of the tangerine is the earth's axis; whereas condition 2) implies that the axis of the tangerine is the line formed by the intersection of the horizon and meridian planes (this line cuts through right where we are standing, and runs due north and south across the floor).  The two axes intersect at an angle equal to our latitude on the earth. This is the reason why conditions 1) and 2) contradict each other: each one requires a different point of reference.

The Equal House and Morinus systems resolve the problem by ignoring both of these conditions. In these systems the axis of the tangerine is the line joining the poles of the ecliptic (our point of view is the north ecliptic pole), so not only does the tangerine wobble, but also we can't sight down either the horizon or meridian planes in these systems because our point of view (the ecliptic pole), doesn't lie on either of these planes. The Equal House and Morinus systems, for these reasons, seem somewhat shameless in their pretensions to be considered house systems at all. 

Only from the point of view of the north point of the horizon (squashing the tangerine into the plane of the prime vertical) , where we can sight down both the horizon and meridian planes, can the ASC and IC be observed simultaneously. This is the viewpoint taken in the Campanus. Regiomontanus, and Sunshine House systems, and only these systems perfectly fulfill condition 2), while ignoring condition 1).

Some house systems ignore condition 2) altogether.  For example, the ASC is not a cusp in the Meridian and Horizontal systems, and the MC is not a cusp in Equals (the way it is usually defined).  Other house systems use some sort of trick or gizmo to pull both angles in as cusps. 

The Porphyry system has a bit more shame than Equal House, but not much. It wobbles just as badly, but at least lip service is paid to preserving both angles as cusps.  However the Porphyry system gives up on geometry, and solves the problem by waving a magic wand and pulling the MC out of a hat. 

The Koch system is the only house system in which the tangerine lacks a central axis. The planes which divide segments of the tangerine are tilted, so instead of intersecting in a line they intersect in a point at the center of the tangerine, forming a double cone (Figure 1).  The Koch house cusps are not the planes which intersect the double cone at the lines, but rather are the planes which are tangent to the double cone at these lines (i.e. they delimit the double cone). Our viewpoint in the Koch system is the center of the tangerine - the point where all planes (lines of sight) meet.

If left to itself, this double cone would rotate smoothly around the earth's axis (and hence be a perfect model of rotation, as the Meridian system is). Unfortunately what happens is that be-cause all other house systems take a viewpoint located on the surface of the tangerine (the north celestial pole, north ecliptic pole, or north point on the horizon), we can "look down on" the whole tangerine at once. But if our point of view is the center of the tangerine, then we have to be looking either one way or the other - either up the double cone (north) or down it (south).  We can't look both ways at once; so if we are looking at the ASC, we can never see the DESC. We can see the point P - the point on the double cone which lies directly across from the ASC, but this point is not the DESC (it's the ASC's antiscion - it has declination opposite to that of the DESC).

Nor can we ever see the MC or IC, since the meridian plane is not tangent to the double cone (it cuts through it, see Figure 2 ).  So the double cone had to be "split" at the meridian into two half-cones. Every time a body reaches the meridian, we have to shift our point of view from north to south (or the reverse) to keep it in sight. And at that precise instant, when we have to whirl around, we are able to steal a quick glance out to the side (down the meridian plane) and "see" the body transiting the meridian.

This is the basic problem with the Koch house system (apart from the logical contradictions it engenders) - it just doesn't make any sense. The Campanus, Regiomontanus, Sunshine, Meridian, Porphyry, Horizontal, and Morinus systems all make some kind of sense. There's a logic to them, flawed though it might be. Even the Placidus system makes sense at first glance (but falls apart under close scrutiny). In all of these house systems there's a fixed point of view. We don't have to be jumping about and waving our arms and looking this way and that. We can just sit there peacefully and watch the thing rotate.

One attempt to improve on the Koch concept of a double cone of rotation is the Topocentric house system, in which the cone is not a cone per se but rather a foil.  Its cross-section isn't a circle, but a spiral; our line of sight is an Archimedean spiral which curls into the meridian, so we can see the IC as well as the ASC.  The problem with the Topocentric system is that as the thing rotates our horizon keeps bobbing up and down, so we feel as though we are being tossed in a blanket (the angle between the the earth's axis, and the line through our feet north and south across the floor, keeps fluctuating between zero and our latitude).  

In view of all these problems, it is not surprising that some astrologers eschew the use of houses altogether.  Unfortunately for the theoretically-minded astrologer, the houses have a undeniable ability to work symbolically in the natal chart:  Bill Gates has Jupiter-Pluto conjunct Regulus in the 2^nd, and so on.  This is one of astrology's undoubted teasers.  It makes one wish transits and directions to intermediate house cusps worked as well as transits and directions to angles do.  However, for that a theory would be needed as to how to calculate these cusps.  Perhaps we're all just going about it completely wrong; but in over thirty years of thinking on this question I haven't found any new approach.  Maybe you will.