SteveS wrote:Mucho appreciated Jim! If I correctly understand your post, we should also allow for Paran combinations with any of the meridian/horizon angles with the auxiliary angles & the accented points of the Vertex axis?

If we take the basic definition of a paran as "two planets simultaneously on an angle," then that would be the conclusion, yes?

But I didn't address the entire range of "auxiliary angles" etc. (I don't know what yoy mean by "accented points" btw.) To finish this out:

**What about the Eastpoint-Westpoint axis?**
These are, primarily, squares to the Meridian as measured along the equator, so that EP or WP with the MC or IC are simply measured in RA. Easy enough.

But EP/WP are more complicated than that. The actual points mark the intersections of the Horizon and the Prime Vertical. (Such points are always, therefore, 90 degrees from the MC as measured either along the equator

*or* as measured along the circle of the Horizon

*or* as measured along the Prime Vertical. It's quite elegant when viewed spatially.)

The practical question then becomes: How would be best show parans between planets on EP or WP with planets (a) on the Asc/Dsc or (b) on the Vx/AVx? The answer is that I don't think there is a common framework for displaying this, and, at best, we'd have to overlap two wheels as I did with Horizon vs. PV. If a planet were literally ON the EP/WP - exactly in that exact point in space - then it would also be simultaneously ON the Horizon and the PV (since the EP/WP are the two points where the Horizon and PV intersect). But we're not requiring that a planet be in one exact spot to be "on" the EP or WP, just that it be somewhere along the great circle passing

*through* one of these points

*at right angles to the equator.*
This complex picture is worth elaborating: The EP is that point on the eastern half of the sky where the Horizon and PV great circles intersect. It is, therefore,

*always* exactly

*due east* on the horizon. This is also the exact point where the celestial equator crosses bot hthe Horizon and PV, because the celestial equator always rises due east and sets due west. Three important great circles thus intersect there. (The ecliptic, in contrast, only rarely rises due east. In all but two moments a day, it crosses the Horizon either north or south of due east.)

If we draw a circle through this point at right angles to the

*ecliptic* and passing through the ecliptic poles, we get the

*celestial longitude* of this point. That is, we get the celestial longitude that the Sun (or any planet with 0° celestial latitude) would have if it were exactly at this point 90° from the MC along the celestial equator. This is the longitude that we write onto a horoscope and call "the Eastpoint." We aren't interested in ecliptical conjunctions with it, though - we only use those to approximate the RA conjunction.

If we draw a circle through this point at right angles to the

*celestial equator,* and passing through the equatorial poles (which are always located on the great circle of the Meridian, at an altitude equal to the co-latitude of the location; i.e., on the Meridian, as far below the Zenith as one's geographic location is above the geographic equator), we get the first cusp of the Morinus house system; i.e., the ecliptical square to the Midheaven.

Those are interesting in their own right. And we already know how to measure parans between EP/WP planets and those on the MC/IC, by measuring purely in RA. But how about parans of these planets and the other angles?

I'm talking in circles in order to paint the bigger picture, but, as I said before, I don't think a single chart will display it. I have in front of me the chart of a woman with Mercury exactly setting mundanely (it looks to be 4° above the Dsc in the ecliptic, but has 0°48' altitude) and the Sun on the WP (it is 0°34' away on the ecliptic and 0°33' in RA - remember, the Sun has zero latitude, so exact ecliptic and equatorial conjunctions will always be exactly the same). But there is no single great circle that they have in common.

Aren't the both exactly on the Horizon circle? No, the Sun is nearly 7° below the Horizon and Mercury right on it. The PV? No, the distance is also 7° there. Azimuth? No, Mercury, though exactly setting, is doing so 23° south of due west. And of course they don't match on the ecliptic or equator.

All we can do is note that Mercury is 0°48'

*before* the Dsc, and the Sun is 0°33'

*after* the WP.

**However, here is an important distinction:** We can't really count this as an aspect - a spatial separation between the planets. We'd have to redefine "aspect" to mean a separation in

*time* rather than in

*space*. There is no aspect in space.)

**With Vertex vs. Asc/Dsc, there actually IS a measuring circle that they share in common - it's the Meridian. We just don't have a way, with existing and available software, to display this. But with the EP/WP vs. Horizon or PV, there's actually NO CIRCLE THEY HAVE IN COMMON. They don't actually ever make an ***aspect.* They are only *co-angularities.*
**This, therefore, comes down to terminology. Do we truly take the ***paran* definition of "two planets on an angle at the same time" to include *any* co-angularity? Or do we require that the planets' conjunction, opposition, or square be measurable in a common framework?
I don't have a firm opinion. On the one hand, I lean toward distinguishing parans that are measurable in a single space-based framework from those that are not, and calling the others "co-angularities." On the other hand, in practice it's awfully functional to think of any two planets simultaneously on any angles as parans (at least casually), and this gives us the practical information we need for astrological interpretation (the fact that two planets are simultaneously on the angles.)

So I leave that as a cultural problem to resolve, not a mathematical one.

**Considering the Northpoint/Southpoint.** These are the squares to the Vertex. I'm probably the first to ever call them this (which I did in

*Interpreting Solar Returns*). Here is the explanation.

The actual NP/SP are the two points where

*the Meridian intersects the Horizon.* That is, they are literally the two points that are

*due north* and

*due south* on the Horizon. Now, if you draw a great circle through these two points

*at right angles to the ecliptic,* they turn out to be... the ecliptical squares to the Vertex.

So the squares to the Vertex are the

*longitudes* of the NP and SP.

Do planets "on" the NP/SP form parans with planets on other angles? Well, they certainly form co-angularities. But is there any "mundane" aspect to them?

In Right Ascension, a planet on the NP/SP would be exactly on the MC/IC (at the point dividing the MC from the IC - no way to distinguish which half of the Meridian). In PV, the NP/SP are exactly conjunct each other and exactly on the Horizon. In fact, a planet or star exactly on the MC or IC in RA, with a declination equal to the co-latitude of one's location, would be

*simultaneously on the MC/IC and Asc/Dsc at the same moment* - actually on all four angles! And, it would be conjunct the Soutpoint (or, in southern latitudes, conjunct the Northpoint).

But almost never will a planet conjunct the NP or SP in longitude

*also* be exactly on the NP or SP in space. They likely will be quite a bit off the Horizon and also quite a bit off the Meridian, with no expectation that they even be near the Prime Vertical.

So there is (as far as I can tell) no way to measure these positions mundanely that's relevant to the current question.