by Jamie Longstreet
I wonder why people DON'T like Placidus. Other than for those born at extreme north latitudes (in which case an Alcabitius system can be substituted), it seems to me to be the perfect system. It assumes that each degree of the zodiac traces its own diurnal arc through our sky relative to our focus point on the earth. If we change latitudes, for instance if we head north in the summer, we see the sun for a longer period of time in the sky. This is because its diurnal arc is long relative to the viewer and its nocturnal arc is short relative to the viewer. In the Placidus system, the arc itself is measured (i.e., how long does it take the sun to rise on a given day at a given latitude) and divided into three equal parts.
But, in my mind, this is where the Placidus system gets interesting. I find it so because there is no easy way to measure/divide the arc in a linear fashion because the Sun's path does not move at a constant rate of speed. Therefore, it must be found by successive iteration, because at any given time, the speed of the Sun in that arc is different than it was at a previous moment in time. Each cusp must be calculated according to the relative solar motion over the earth for that moment. I believe this is what accounts for (IMHO) the cusp's exceptionally sensitive accuracy; the cusp's degree is based upon the Sun's motion at that moment.
A larger problem with different house systems lies in the complexity of the mathematical derivation. Knowing the difference between right ascension and the degree of the midheaven exists, while understanding why this discrepancy exists can be a challenging task.
The conic nature of an orbit is based upon its acceleration which is measured in three dimensions (x,y,z). The angle is represented by two separate focuses. One to the perigee, and the other to the object's current position. Thus, geometrically, you come up with a kind of odd-looking wedge off in the corner of a two-dimensional pie. One side of the wedge is the focus of the perigee, the other side is a moving object.
Essentially it is a problem similar to the one Einstein worked out in his theory of relativity which stated that the motion of an object is based upon the viewer's position relative to the action of that object.
Also, consider that right ascension is based upon the radius of a conic section; it changes its distance relative to its position along the conic axis. This is Kepler's law. The area of arc traced during any given place along the axis is equal to the area of arc traced along another similar section for the same duration. Simply explained, the body accelerates as it approaches a larger body and decelerates as it separates from a larger body but when the forces are taken as a whole they remain constant. The motion is based upon the constant "gravitational" forces between the two bodies.
As far as visualization goes, I always base this on a model of the Sun as it shines on the earth, whose tilt causes shadows to appear longer at certain times of the year. If you extrapolate the shadow, and somehow slice it into 12 equal parts based upon the speed that the shadow moves across the earth's surface, you should be able to visualize that these positions change from moment to moment and are completely dependent on this relative speed. Koch, Placidus, Meridian, Regiomontanus, etc... all have their own methods for sub-dividing this space. Equal House does it like the world map we looked at as kids where Greenland is huge in comparison to it's real size (Canada, Alaska for that matter), primarily because it must be stretched out to appear flat on a map.