On the Mathematics of Circumambulations
© 2014, Jun 3 Curtis Manwaring
I have a convention in Delphic Oracle with the circumambulation listings (same as directing through the bounds by oblique ascension) of Dorotheus/Ptolemy where the bound lord or the aspect ray is indented to show which is primary and it can be switched either way. There are actually two sections in the program because oblique ascension is not astronomically correct motion for anything that is not on the ascendant, but because Valens and others used oblique ascensions for other bodies as well I included the older but "objectionable to Ptolemy" methods. The primary directions section though does directions through the bounds using the 3 types mentioned above (typically called Placidian semi-arc, in addition to the more modern circle of position method, Placidus under the pole and Polich-Page topocentric methods. These variations all have to do with what point on the celestial sphere is being directed to another point on the celestial sphere and what frames of reference you are using to measure the distance using spherical trigonometry. One of the principle problems with directing points not on the ascendant or midheaven is that these points almost never come into direct contact with each other but pass either above or below the other object. Therefore the reference frame becomes critical in measuring these arcs. There can be no argument about when some planet crosses the ascendant or midheaven because these are planes which all objects must pass through, but points can have any number of planes defined perpendicularly to them.
It depends upon who you talk to what the definition of a converse direction is. Martin Gansten uses the more traditional definition (which doesn't involve direction of motion at all because the motion is always from east to west above the horizon). Traditionally a converse direction is when a significator is moved toward a promissor. Significators are supposed to remain "fixed" so when you move them, it's converse. Those working with Morinus will tend to think of converse directions as being "against the diurnal rotation".
In your graphic above if the red lines represent the motion of the stars in the sky by diurnal rotation, they would not cross but be parallel to each other. In the proportional semi-arc method, in order for a promisor to reach a significator, one has to find the significators diurnal semi arc and nocturnal semi arc and find what proportion of that arc matches the significators arc (in its own arc). Assuming mid northern latitudes here, if the significator is 2/3rds of the way to the MC at roughly the cusp of the 11th house and the promisor happens to be on the ascendant, then one has to find the significators diurnal semi arc and calculate what would be 2/3rds of that arc and then do the same with the promissor in its own arc and when it reaches 2/3rds of it's own arc the direction is complete. That would be the Ptolemaic method. If the latitude of the significator is relatively low then it will be low in the sky and probably have a small DSA and relatively large NSA: for instance the DSA might be about 60 degrees and therefore the NSA will be 120 degrees but if the promissor passes above the significator then the DSA value will be larger and its NSA will be smaller. 2/3 of 60 would be 40, but if the value of the DSA for the promissor is about 90, then 2/3 is 60 in its own arc. Once we have the "arc of direction, then it is a matter of applying a time key to represent the passage of degrees. It's more complicated than I state here because one needs to find the latitude of birth, equinoctial distance of the sig and pro, etc before one can find DSA's and NSA's to derive the proportional point...
In the circle of position method one finds the "artificial horizon" for the significator or promissor and finds the oblique ascension...
In order to get converse directions in Morinus you have to swap significator and promissor because Morinus uses the modern definition of converse. I make a distinction in Delphic Oracle and call these modern converse directions "neo-converse" as per Martin's convention and no swapping is necessary (I'm a graduate of Martin Gansten's Primary Directions course and based the PD's in Delphic Oracle on this).
I should be more clear about neo-converse arcs in the above paragraph but it is late... suffice it to say that there are really 3 different possible directions (traditional converse, neo-converse and direct) and they are measured differently in different arcs. I wasn't able to confirm my calculations of PD's in Delphic Oracle on traditional converse directions using Morinus because Morinus uses the modern definition of converse, but I was able to confirm the neo-converse directions.
If you're going to use tables like this for primary directions at least make sure you are doing interpolation and stay away from extreme latitudes so that your results are at least within a year of being correct. Better yet use the formula:
oblique ascension = Right ascension - c
sin(c) = tan(geo.latitude) x tan(declination)
C is the "correction factor". If we remember the rules for sides of an equation in trig, the Sin(C) becomes ArcSin as below:
c = arcsin(tan(geo.latitude) x (tan(declination))
This works for directions of the ascendant, but Ptolemy said that it is not an accurate representation mathematically for planet to planet directions (unless one of them happens to be on the ascendant).
Planets to the MC simply subtract their right ascensions for the difference applied to the key in years which is the simplest kind of direction.
Planet to planet directions are another formula entirely (if you are following Placidus/Alchabitius method). The formula is much longer than these and involves many steps. Rumen Kolev did a nice mathematical simplification of Zoller's Alchabitius method in a paper back in 2003.
Beware that primary directions are not simply a matter of getting the oblique ascensions of planets in most cases.