Prime Vertical Parans

[Jim Eshelman]

I need a better name for this - but here's where I am at the moment:

I haven't asked for this yet because there is one value I can't tell you how to calculate even though I know what it is. (I'm decades past my trig prime when I could have just stopped and derived the formulae, and I haven't put the work into figuring it out. So I've held off mentioning this. - But this is the thread where I should introduce the request or idea - and we can sort out the fine points later.

The horizon (H), meridian (M), and prime vertical (PV) are always at right angles to each other. That means that a planet on any one of these is always conjunct, opposite, or square another planet on one of these. This theoretical construct has proven itself hands down - no doubt about it - in ingresses, and probably is equally true in returns and nativities (but, due to the rarity and the math difficulties, hasn't been actually studied by me and, probably, by anyone else).

In a sense, it's just another mundane aspect. If this were demonstrated and accepted, there is little reason not to lump it in with other in mundo aspects (since that's what it is: just a larger view of the landscape), but at the moment I'm using different language to avoid confusing everybody (since using PV aspects alone hasn't become normalized in everybody's thinking yet).

Here is what I know and what defines the aspects as explored thus far:

• The Vertex-Antivertex axis (either ecliptically or mundanely) is worse than useless in measuring an angularity effect in ingresses (and IMHO in return charts), though it seems to have some expression in nativities.
• Regardless of that fact, the mundane conjunctions, oppositions, and squares appear definitely powerful between a planet on the prime vertical and another planet that is on the horizon, meridian, or prime vertical. These come in three flavors.
• Functionally (while exploring them thus far) my working method has been to start with a planet within 3° of azimuth 90° or 270°, i.e., the Antivertex (due east) or Vertex (due west).
• FLAVOR 1: PV to PV. These are easy: These are conjunctions or oppositions in azimuth.
• FLAVOR 2: PV to Meridian. These also are easy: They are squares in azimuth where the PV planet fits the above criterion and the meridian planet is foreground.
• FLAVOR 3: PV to Horizon. This is the one where I don't have an easy direct way to tell you how to do the math. I've been using a kludge as an approximation (hopefully a close approximation). What we want is a measurement along the meridian in the same way that azimuth is a measurement along the horizon and PVL is a measurement along the PV.

The single principle is simple: For aspects between any two of Horizon, Meridian, and Prime Vertical, measure in the third circle. H-to-M aspects are measured along PV. PV-to-M aspects are measured along the horizon (in azimuth). PV-to-H aspects are measured along the meridian. - The problem is that measuring along the meridian (great circles through a planet and the EP and WP on the horizon, at right angles to the meridian) is so useless to astronomers that it doesn't even have a name. On the rare occasion I need a name, I've been calling it "meridian longitude," but there is probably a better name.

Here is how I've been faking it: I've the advantage that I'm only dealing with factors very close to the angle. When very close to the angle, several different ways of calculating will produce nearly identical results; e.g., for a planet within a few degrees of the horizon, both distance in PVL and in altitude will be similar and, often, identical. (My Moon is 3°15' off Dsc in PVL and 3°15' southern altitude.) Therefore, I use an alternative but similar pair of different measurements for PV-to-H squares: I take how far the first planet is past (plus) or before (minus) Vx/Av in azimuth, then how far the second planet is past (plus) or before (minus) the horizon in PV Amplitude (the latitude or declination analog to PVL) - a value I can squeeze out of Solar Fire. It's tedious and I probably miss some by inadvertence.

It would be better to figure out how to calculate this new value nobody else uses - meridian longitude - and make the measurement directly. I am prepared for the rude shock that the geometry of such a circle could be freakier than expected and produce surprise results but, if so, then it's because it's actually how things look.

[mikestar13]

I need some serious brushing up on my spherical trig before I tackle aspects in meridian longitude. But if I'm spared, it will be in 1.0 if not sooner.

[Jim Eshelman]

I need some serious brushing up on my spherical trig before I tackle aspects in meridian longitude. But if I'm spared, it will be in 1.0 if not sooner.

I keep thinking it should be a simple conversion of the formula for PVL, since the geometry is identical - but new variables need selecting. I couldn't make it work on a first pass (even if I ignored the usual quadrant misplacements).

[Jim Eshelman]

Reverse-engineering the spreadsheet:
TAN pvl = 1 / (COS phi / TAN ha + SIN phi * TAN dec /SIN ha)
where phi is geographic angle, ha is the planet's hour angle (RAMC - planet's RA), dec is planet's declination

Presuming I've reverse-engineered this correctly and thinking it through by visualization:

In the PVL problem,

• geographic latitude is distance from PV to the celestial equator
• declination is the planet's distance/direction from the celestial equator
• hour angle is the planet's distance from meridian [RAMC minus planet RA]

Let's orient ourselves as looking due east, such that the Meridian looks the same way for the Meridian Longitude problem that the Prime Vertical looks for the PV problem.

• Analog of hour angle is the tricky part, probably the key to the whole thing. In the PV problem, the equator's pole is on the meridian, but it isn't on the PV in the ML problem: This means that we can't use RA in adapting this formula. -- In the PV problem, the celestial equator and PV intersect at EP-WP. In the ML problem, the meridian intersects the horizon at SP-NP. Furthermore, in the PV problem, the middle divider (Meridian) is at right angles to the celestial equator; in the ML problem, the middle divider (PV) is only at right angles to the horizon (excluding the PV for other reasons).
• Therefore, we have to use the horizon for our starting values: azimuth and altitude in the ML problem are the substitutes for RA and Dec in the PV problem. Azimuth would be planet's azimuth minus 90° (reversing the equation because azimuth and RA are measured in opposite directions).
• Analog to geographic latitude: Distance from Meridian to the celestial equator OR the horizon (in this orientation) is always 90°. If this is correct, cosine 90° is 0 which wipes out the first part altogether, sine 90° is 1 which simplifies the next part.
• analog of declination is altitude, as deduced above

If I have all of this correct, the formula becomes:

TAN ML =1 / (COS 90° / TAN ha + SIN 90° * TAN alt /SIN ha)
where ha is horizon angle = planet azimuth -90°, alt is planet's altitude

This simplifies to: TAN ml = 1 / (TAN alt / SIN ha)
or probably: TAN ml = 1 / (TAN alt / COS azi)
Unbelievable! If this works, it's gross simplicity. I'll try to test it.

[mikestar13]

I don't use a trig formula for PVL. Swiss Ephemeris has a function that calculates true house position, which for Campanus houses is PVL. No doubt there is a trig formula under the hood, but I don't know what it is.

[mikestar13]

If you get all the kinks out, the above is quite doable.

[Jim Eshelman]

After some false starts trying to modify the existing PVL spreadsheet, I started working directly from the formula. It confirmed the aspects I expected, although the orbs were far tighter than expected. Therefore, it still needs more confirmation. There were also quadrant issues which, however, don't matter for the practical use (and are probably resolvable with some tinkering - but I won't worry about them right now.)

Conway Theater Fire
So, using the straight-from-formula approach and not worrying about quadrants for now, here are some examples. A classic example of PVP aspects is the Canlunar for the Conway Theater fire. After citing the primary feature (Mars opposite Neptune across the horizon), I wrote:

Additionally, the proximity of Sun, Mercury, and Jupiter to the prime vertical leads to nine PVP aspects, all but one of which is partile. In order of orb, these are: Mercury-Neptune (0°01'), Sun-Mars (0°21'), Sun-Jupiter (0°32'), Mercury-Mars (0°33'), Mars-Jupiter, (0°53'), Sun-Mercury (0°54'), Sun-Neptune (0°55'), Mercury-Jupiter (1°26'), and, for the entertainment venue, Jupiter-Neptune (1°27'). Some of these fit very well, some adequately, and a couple seem misfits.

Leaving out the three that are azimuth-to-azimuth aspects leaves six others, PV to Horizon. Does the formula confirm these? Expecting Sun, Mercury, and Jupiter to aspect Mars and Neptune, I get these positions:

Pluto 28°39'
Sun 29°00'
Jupiter 29°30'
Mercury 29°51'
Neptune 0°02'
Mars 0°11'
Venus 0°40'

Yes - all those and a few more! (The orbs are coming out MUCH smaller than I thought they were. Going 3° on these may be excessive.) In addition to the expected Sun, Mercury, and Jupiter to Mars and Neptune, we get several more.

Rhythm Club Fire
Rhythm Club fire, Capsolar. I had identified a 1°14' Mars-Saturn PVP square. The formula shows it, but with a 0°39' orb. I'm suspicious that it folds other planets in, but they all were at least near the horizon or PV east-west axis. (My original proximity filtering may be necessary.) Here are the tightly clustered planets.

28°41' Moon
29°15' Neptune
29°23' Saturn
29°49' Jupiter
0°02' Mars

Hartford Circus Fire
Harford Circus Fire, Caplunar. I cited a Sun-Pluto PVP square 0°50'. The formula gives it 1°01' (similar), with Saturn just over 2° away. (Saturn is 5° off Antivertex in azimuth so I didn't catch that aspect and, perhaps, it shouldn't be considered. Maybe the proximity filtering still needs to be applied.)

Durunkah Fire
Durunkah Fire, Caplunar. I cited Pluto PVP square Uranus-Neptune. This one interests me because of larger celestial latitudes and possible distortion. It shows quite easily: Neptune 29°10', Uranus 29°36', Pluto 29°54'. (Rising Venus is nearby and could easily join except it is FAR from the PV in azimuth. Again, the filtering factor I've used probably still needs to be applied.)

Conclusion
The formula works! I suggest an approach of (1) filter first for planets within 3° of azimuth 90° or 270°, and then (2) see if they fit any of the three types of PVP aspects, either by azimuth-to-azimuth or (for PV-Horizon content) by using the formula above to calculate Meridian Longitude.

[Jim Eshelman]

Mike, I think this is now done enough to use for the intended purpose. I haven't solved the quadrant issue and probably won't try, at least in the near future (it doesn't affect usability). The formula for calculating Meridian Longitude is:

TAN ml = 1 / (TAN alt / SIN ha)

alt is altitude, ha (in this case) is "horizon angle" which is 90°00' minus azimuth (with 90°00' being the azimuth of due east, the eastern side of the prime vertical).

The problem with this is I have no other source to confirm the right answer. The basis for saying this is correct is that it finds the same aspects as my estimation method (in every test I've run on it).

It also grabs more aspects than I got by estimation because everything is tightened - other planets are "dragged in." Based on what I've seen and good astrological principle, I think these have to be prefiltered. The method would be:

• Find any planet within 3° of azimuth 90° or 270° (arbitrary value but one that has been demonstrated to produce great results)
• If two of these are conjunct or opposite in azimuth within a specified orb (say 3°), there is an aspect.
• If one of these squares another planet in azimuth (which means the other planet is on MC or IC), there is an aspect.
• If one of these squares another planet in meridian longitude (which means the other planet is on Asc or Dsc), there is an aspect.

Given the residual quadrant problem in the ML formula, the last item can simply be aspect on a 90° sort - other conditions specified here will automatically handle obscuritiess.

As usual, whatever aspect (of whatever type) has the closest orb will "win" if these are turned on.

[mikestar13]

So if neither planet is within 3 degrees of the prime vertical by azimuth there is no pvp. What is the significance if any of a planet within 3 degrees of the meridian by meridian longitude which is 90 degrees away from the ml of another planet? I'm having a hard time visualizing this. In any case we don't consider pvp unless within 3 degrees of an angle.

[Jim Eshelman]

First, for orientation: As we are viewing the celestial sphere facing east with Zenith at top, if we were to draw this as a chart the vertical axis would be the prime vertical, the horizontal line would be the horizon, and the surrounding circle of the wheel would be the meridian.

The left edge of the horizontal (where meridian circle intersects horizon) is the northpoint; the right end of the horizontal is the southpoint. Similarly, the top point of the vertical (where meridian intersects PV) is Zenith, the bottom point is the Zenith except for the way these are wrapping around) we can functionally say that the top point is the Antivertex, the bottom point the Vertex (analogize MC/IC in the standard chart).

What is the significance if any of a planet within 3 degrees of the meridian by meridian longitude which is 90 degrees away from the ml of another planet? I'm having a hard time visualizing this.

Meridian longitude will show aspects between the prime vertical (vertical line) and horizon (horizontal line). You're having a hard time visualizing the meridian because it isn't in a specific place in the chart - it's the whole circle. (The meridian has no ML because it has all MLs, just as Earth's equator has no longitude because it has all longitudes.)

Did I answer the right question?

In any case we don't consider pvp unless within 3 degrees of an angle.

This is how it seems: Like the celestial equator, no aspects as such except relationships that act like aspects when near "angles." The main purpose here is to show the right angle of horizon and PV (just as the horizon is the framework for showing the right angle between the meridian and PV, or the PV is the framework for showing the right angle between the meridian and horizon).

[Jim Eshelman]

Agh, this still might be wrong (though the most likely problem is related to quadrants). I ran your chart, Mike, expecting Mars to show as before Ascendant with Moon-Neptune after the Vertex axis, meaning Mars would be 4-5° from square Moon-Neptune. They came out aligned in a tight opposition that doesn't look right. FWIW, here are your position (the "house" numbers are just convenient 30° steps to make it more readable). Planet - "house" - degree - minute.

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Mon	3	28	38
Sun	3	18	41
Mer	3	26	57
Ven	3	16	6
Mar	9	28	45
Jup	3	4	36
Sat	3	25	43
Ura	8	26	3
Nep	3	29	45
Plu	3	0	25

[Jim Eshelman]

Mike, let's table this. No rush on it. I think what I've given is right but I keep wishing I had a way to validate it. Maybe solve the quadrant problem. Maybe draw planets on a celestial globe and sketch my own circles to see if the positions make any sense.

I'll break this sub-topic off to a new thread.

[Jim Eshelman]

Mike, I've got it. I know how to calculate these.

If possible, what I'd like - as a next step - is that at the earliest opportunity (that doesn't slow down other stuff) add ML to the data table (say, after the Alt? column). This will let us look at the coordinate for a while and make some decisions about aspect options. (I have thoughts, but that's a later step and more complicated implementation. Just getting the coordinate done should be pretty simple.)

I'll write it up in the next post.

One of the problems is that THIS LOOKS FRIGGIN' CRAZY. But it's actually that way. For example, in my chart it looks like in the lower right quadrant (which is north half of the sky below the horizon) the planets are going BACKWARDS. They aren't really - but since we are looking straight through from the eastern to western halves of the sky, it ends up collating them the opposite.)

Here's your chart with a big surprise (presuming these are legit aspects in natals): Noto only do you have Moon-Neptune exactly opposite in azimuth across the Vx-Av axis (PVP opposition), they are both in orb of square (PV to horizon) your rising Mars - giving you Mars square Moon-Neptune. I think we only want to look at the aspects of planets that are directly on one of the great circles, but here are all of yours:

26°03' 5H - Uranus
4°36' 6H - Jupiter
16°06' 12H - Venus
18°41' 12H - Sun
25°43' 12H - Saturn
26°57' 12H - Mercury
28°38' 12H - Moon
29°45' 6H - Mars
29°45' 6H - Neptune

[Jim Eshelman]

Orientation: Looking at the celestial sphere from dead-on the Eastpoint. Outer circle is meridian. Horizontal axis is horizon. Vertical axis is prime vertical. Southpoint at left ("9 o'clock"), Zenith at top ("12 o'clock"), Northpoint at right ("3 o'clock"), Nadir at bottom ("6 o'clock"). The center of the wheel is the EP on the near side, passing straight through the sphere to the WP on the far side.

To understand the circle and how the quadrants behave: On the resulting chart wheel the upper left quadrant is above the horizon, southern sky; upper right quadrant is above the horizon, northern sky; lower left quadrant is below the horizon, southern sky; and lower right quadrant is below horizon, northern sky. (You can't think of this in terms of houses, it will drive you batty; but we can label it as if it were houses for convenience in communicating about it.) The sequence of planets away from the horizon will be very nearly a sort by altitude (not always, but usually).

TAN ml = TAN alt / COS azi

ml = Meridian Longitude
azi = azimuth
alt = altitude

If the answer is negative, add 360.

This will always give a value from 270° to 90° via 360°/0° (what we might call 3rd house around east to 10th house - at least, that's how the numbers come out). Therefore, it is either exactly right or exactly 180° off. For our uses, it doesn't really matter which (same aspects) but, to be pedantic, there is one correction: If the value is 270°-360° but the planet (by altitude) is below the horizon, subtract 180°. If the value is 0°-90° but the planet (by altitude) is above the horizon, add 180°.

The result is Meridian Longitude.

(This has been nuts... it has taken me years to figure out how to do this! I have a sneaking suspicion that the weirdness is only now beginning.)

[Jim Eshelman]

I decided I had to knuckle down and make this happen because I needed the real figure for the Saturn-Pluto PVP square in the new Arisolar - didn't want to get that one wrong. So now I've spent all day just working on this math problem instead of writing the DUE REALLY SOON NOW new monthly forecast. (Caplunar is Wednesday.)

One example of weirdness is my Pluto (on my 12th cusp normally flips to the other side of the chart because Pluto is (1) above the horizon and (2) just north of [shy of] the Antivertex/PV, so it goes on the right half of the chart just above the horizon. (It shows my "new" Moon-Pluto conjunction just over 1°.)

But a really weird one is Steve's chart. He has Mars, Saturn, and Pluto in the 8th and 9th houses, but these flip to the upper left quadrant because they are (1) above the horizon and (2) south of the PV. (Houses in brackets are ones where the hemisphere had to flip 180°.)

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	HOUSE	°	'
Mon	1	4	60
Sun	11	17	53
Mer	12	6	48
Ven	11	23	28
Mar	12	14	24
Jup	1	0	30
Sat	10	24	32
Ura	[7]	5	11
Nep	12	3	3
Plu	10	24	42

Arena is CRAZY! (Well, not Arena, but her chart! <g>). Almost everything flips back and forth across the PV line left and right:

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	HOUSE	°	'
Mon	[7]	5	36
Sun	[6]	25	51
Mer	[6]	21	12
Ven	[6]	26	34
Mar	[6]	26	16
Jup	1	0	41
Sat	[7]	0	44
Ura	[6]	29	40
Nep	[7]	1	48
Plu	[7]	2	30

The interesting aspects that fall out of this (there would be a lot more but I cut it off at 3° from an angle):

21°12' Mercury
25°51' Sun
26°16' Mars
26°34' Venus
29°40' Uranus
--------------------
0°41' Jupiter
0°44' Saturn
1°48' Neptune
2°30' Pluto
5°36' Moon

[SteveS]

Jim wrote:

But a really weird one is Steve's chart. He has Mars, Saturn, and Pluto in the 8th and 9th houses, but these flip to the upper left quadrant because they are (1) above the horizon and (2) south of the PV. (Houses in brackets are ones where the hemisphere had to flip 180°.)

Jim, I don't understand what the possible astrological meaning of this "flip" means for my chart. If it means that Mars, Saturn, & Pluto in my Natal would exert more powerful symbolic meanings than conventional stuff, then yes---it would make sense to me. Is this the case you are exploring?

[Jim Eshelman]

Steve, there surely is no meaningful astrological significance. I'm just describing what it looks like spatially (these are all weird). The only real relevance seems to be when a OVO is formed (No such aspects to your rising Jupiter). Too long to answer on my phone.

[SteveS]

Got it, thanks.

[mikestar13]

The following is the planet data from my chart as calculated by TMSA 1.0:

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 -------------------------------------------------------------------------
 Pl Longitude   Lat   Speed    RA    Decl    Azi     Alt     PVL    Ang FG
 Mo  2Ar18'11"  2N 4 +12°30'  23°46' 12N 8  92°48' +26°11' 333°47'   4%  b
 Su 17Pi31'46"  0N 0 +59'12"  10°44'  4N37 108°19' +32°30' 326° 8'   1%  b
 Me 29Pi40'55"  0N35 + 1°56'  21°50'  9N48  96° 8' +26°31' 333°21'   3%  b
 Ve 14Pi13' 7"  1S22 + 1°14'   8°13'  2N 4 112°33' +32°51' 325° 2'   2%  b
 Ma 15Ta 8'36"  1N14 +37'44"  67°24' 23N 4  59°52' - 2°29'   2°52'  98% A 
 Ju  0Vi50'53"  1N33 - 7'12" 176° 1'  3N25 307°49' -37°46' 135°33'  14%  b
 Sa 20Sc 5'56"  1N49 - 0'52" 253° 8' 20S42 238°15' + 8° 4' 189°28'  77% D 
 Ur  8Cn45' 1"  0N37 - 0'28" 125°20' 20N 7  13°42' -34°43'  71° 8'  30%   
 Ne  7Li37'12"  1N48 - 1'31" 210°14' 10S24 270°43' -19°53' 160° 7'  25%   
 Pl  4Le 6'55" 11N23 - 1'01" 154°40' 22N43 341°16' -30°56' 118°11'  44%   
 Er 14Pi43'47" 23S 6 + 0'42"  17°29' 17S40 121°59' +12°54' 344°53'  49%   
 Se  1Ar50'28" 10S30 + 0'36"  27°54'  0N14 100°56' +16° 8' 343°35'  43%   
 As 10Ta 7'54"                       21N 0           0° 0'   0° 0'
 Mc 20Cp41'10"               317°18' 16S23 180° 0' -50°22' 270° 0'
 Ep 25Ar36' 1"                47°18' 17N41 
 Vx  7Li 4'31"                       11S54 270° 0' -21°39'
 -------------------------------------------------------------------------

Notice that RA, Decl, Az, Alt, and PVL are given for certain angle where the values are well-defined for that angle.
1.Has anything been left out?
2. I am adding a ML column. Which angles should have ML calculated?

[Jim Eshelman]

Good-morning, Mike. This is exciting to see

Notice that RA, Decl, Az, Alt, and PVL are given for certain angle where the values are well-defined for that angle.
1.Has anything been left out?

To my eye, it's perfect.

Things that are debatable (but I think you got right):
1. Asc RA? I think there is no practical use. Some people might want to experiment with this. I think they will get misled and either this shouldn't be here OR they need to speak up and explain why they need it first.
2. EP-a Azi? Is this line showing the astronomical EP or (for lack of a better term) a pseudo-point? The EP's azimuth is always 90° if it's the astronomical point. In a nit-picky Virgo way, I'm unsure about this EXCEPT that it doesn't add anything useful. (Maybe we talked this through in the past.) The point on the ecliptic that is EP-a has no universal or useful azimuth, the actual EP (intersection of prime vertical and horizon) is always 90.

2. I am adding a ML column. Which angles should have ML calculated?

My thoughts, angle by angle: two yeses, one maybe not, and one no.

ML of all points on the horizon is either 0 or 180. Asc ML could be either (depending on whether its declination is N or S): 0° ML is the SP, 180° is the NP, Asc/Dsc wanders left and right along the horizontal ("1st/7th" axis) throughout the day. Listing Asc ML is useful in the same way that listing Vertex's Azimuth is useful (to define the parameters for someone looking at the column), so I say: Asc yes.

ML of all points on the prime vertical is either 90 or 270. Vertex ML could be either (depending on whether its altitude is N or S): 90° ML is the nadir, 270° is the zenith, Vx/Av wanders up and down along the vertical ("4th/10th" axis) throughout the day. Listing Vx ML is useful much as listing Vertex's Azimuth is useful, so I say yes.

Listing Asc and Vx infers (if someone thinks it through) that the measurement is a way of seeing angles between the prime vertical and the horizon.

The meridian's ML is undefined (the meridian circle is the basis of the measurement). MC's ML is on the outer circle, though, where the ecliptic crosses the meridian. This is always as far above the SP ("1st cusp") as geographic co-latitude plus MC declination [but surely easier to calculate by the usual ML formula instead of futzing with all these positive and negative values]. I can fantasize that this might be interesting, but I have a hard time imagining it would be useful. Decision could go either way and I lean toward not including it.

If EP-a is the pseudo point, its ML is irrelevant. If EP-a is being treated as the astronomical EP, it's dead-center on the wheel and undefined (or: simultaneously all positions in the sense that the north pole is simultaneously all longitudes). So: No.

[Mike V]

TAN ml = TAN alt / COS azi

The trigonometric functions in Python accept parameters in radians, not degrees. I got a little help from AI, and it mostly makes sense, but I'm not sure if this is actually correct; can you double check me?

In pseudocode -

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tangent of ML = tan(altitude in radians) / cos(azimuth in radians)
ML in radians = arctan(tangent of ML)
ML in degrees = degrees(ML in radians)

And then we proceed with ML in degrees?

If the answer is negative, add 360.

This will always give a value from 270° to 90° via 360°/0° (what we might call 3rd house around east to 10th house - at least, that's how the numbers come out). Therefore, it is either exactly right or exactly 180° off. For our uses, it doesn't really matter which (same aspects) but, to be pedantic, there is one correction: If the value is 270°-360° but the planet (by altitude) is below the horizon, subtract 180°. If the value is 0°-90° but the planet (by altitude) is above the horizon, add 180°.

The result is Meridian Longitude.

I think I understand this part, but I could use a double check; does this normalization logic look right? (Actual code because I think it's as clear as pseudocode would be)

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if meridian_longitude < 0:
    meridian_longitude += 360
    
if meridian_longitude >= 270 and meridian_longitude <= 360 and altitude < 0:
    meridian_longitude -= 180
        
if meridian_longitude >= 0 and meridian_longitude <= 90 and altitude > 0:
    meridian_longitude += 180

[Jim Eshelman]

Some languages have a function for degree/radian conversion (even Excel has it built in) but, yes, the standard is to do the math in radians. Yes, you have it exactly right, convert altitude and azimuth to radians, perform the functions, convert the final answer back to degrees.

Here is your chart from my spreadsheet:

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	Azi	Alt	TAN alt	COS Azi	(ratio)	Atan	Final
Mon	73.26161698	-30.64381638	-1.687960205	0.28800211	-0.170621386	-9.682644553	350.3173554
Sun	304.3454118	-63.57902965	-0.496860585	0.564180626	-1.135490807	-48.63038774	311.3696123
Mer	280.131793	-48.7130175	-0.878118997	0.175912993	-0.200329333	-11.32807499	348.671925
Ven	272.5839509	-33.54392495	-1.508321531	0.045083165	-0.029889625	-1.712039654	358.2879603
Mar	14.73192248	-68.96119841	-0.384641242	0.967126221	-2.514359139	-68.31150961	291.6884904
Jup	114.1268854	58.01673632	0.624463267	-0.408758753	-0.654576138	-33.20780103	326.792199
Sat	286.6494849	-51.69144964	-0.78999475	0.286515939	-0.362680814	-19.93473678	340.0652632
Ura	295.4376951	-59.50516474	-0.588923604	0.429529348	-0.729346463	-36.10500905	323.894991
Nep	289.8788728	-53.64888715	-0.735947385	0.340032805	-0.462034124	-24.79854831	335.2014517
Plu	26.0345005	-48.32093998	-0.890312122	0.898529919	-1.009230243	-45.26321038	314.7367896

 
Or, in a "housey" format:

Code: Select all

Mon	12	20	19
Sun	11	11	22
Mer	12	18	40
Ven	12	28	17
Mar	10	21	41
Jup	11	26	48
Sat	12	10	4
Ura	11	23	54
Nep	12	5	12
Plu	11	14	44

In pseudocode -

Code: Select all

tangent of ML = tan(altitude in radians) / cos(azimuth in radians)
ML in radians = arctan(tangent of ML)
ML in degrees = degrees(ML in radians)

And then we proceed with ML in degrees?

Exactly.

On the next, I'm working off a 12" screen and don't have things so I can see all the pieces at once, so I might miss something... reacting just to what you wrote.... I think you have it right because it's pretty straightforward. (Also, the 180 flip isn't crucial - I'm willing to accept that we might stumble on this at first and it won't make a difference in practice because it only confuses a conjunction with an opposition.) Yes, add 360 to make it a positive number. Without working it out on paper again, I think the rest that you wrote is exactly right.

Cool

[Mike V]

I noticed Mike N did (altitude + 180) mod 360 right after it was calculated (by the Swiss Ephemeris), before it was displayed or went into any calculations. Does that sound right?
I guess it is, since it gives me the values for azimuth and altitude that you start with (and they match Solar Fire).

However, my output then deviates from yours; here's my Moon figures for example:

---------------Azi--------------Alt-------------TAN alt------- COS Azi---------(ratio)------------Atan----------Final
Yours**73.26161698**-30.64381638***-1.687960205**0.28800211**-0.170621386**-9.682644553**350.3173554
Mine***73.26235141**-30.64344070***-0.592422170**0.28798983**-2.057094030**-64.07456375**115.9254362

Sorry for the weird formatting, I'm trying my darnedest to make it readable.

Our tan(altitude) is different...

<Removed a bunch of tests that all said the same thing>

Omg! I think I figured it out. You must be taking the COtangent of altitude, not the tangent. Cotan(-30.6438 degrees) = -1.6879

Should the formula have cotangent? I guess it then needs to have arccotangent applied to the ratio, too.

Also: when I apply that (cotan and acotan), I think we have the same values right up until the normalization at the very end...
I get -9.682 for the "atan" value like you do, and after adding 360, I have 350.x like you do... but then I subtract 180 since the ML is > 270 and altitude is negative. Is that part right? Or should I only be subtracting 180 if the atan value is > 270?
My final final value using cotangent/arccotangent is 170.x, yours is 350.x, so I think we have the same calculations with that change.

[Jim Eshelman]

Apologies if I did make this mistake since it put you through extra work I may not be able to look at this until I get home (and the post is marked Read so I may not see it again. Nudge me Monday if I haven't confirmed.

I may be able to access the spreadsheet from here, in which case I'll be back in a few minutes. (We're heading up to Oklahoma today on a long drive - roughly like driving from LA to the Bay Area - so I won't have more than a few minutes at the computer.)

[Jim Eshelman]

OK, this will take reverse engineering. I hope I didn't screw up implementation, though I spent a lot of time on this originally. My spreadsheet uses: tangent (90-x), which would give the same value as the cotangent.

Now... why did I do that?

It's quite purposeful. I'd have to change it to see what answers I got. If you can see a reason I might have done that, then we're good, otherwise I have to figure out what I did. I think I'd have to go back to one of the mundane astrology cases that had such pronounced PVPs and see if removing the "90-" gives nonsense results, as I think it will.

[Jim Eshelman]

COT (x) is exactly the same result COT (90-x), so that can be substituted PRESUMING I have the essential formula right.

Spot checking a few entries confirms that using TAN (x) produces, in some cases, nonsense values. But none of these were "critical case" examples. To feel confident this is right I have to dig out those ingress examples in another thread somewhere that were really good PVP examples. If they produce nonsense with TAN and right answers with COT, we know the story. - Now, I'm off for the road.

[Mike V]

Safe travels! No rush on this stuff, I've got plenty to do in the meantime. (I screwed up chart rendering somewhere in this, so I have to figure that part out...)

90 - x is indeed quite purposeful. I wish I could help directly so you don't have to re-check examples, but I'm about 18 years removed from the last time I was doing trig in high school, and... well, I did fail geometry! (That was mostly because I refused to do homework and I slept in class, but still.)

By some miracle I can visualize the directionality of meridian longitude looking from due east, altitude, azimuth, etc, but I don't have any understanding of how the various quadrants behave within all of this math and the various addition/subtraction of 90* multiples.

[Mike V]

Nudge me Monday if I haven't confirmed.

To feel confident this is right I have to dig out those ingress examples in another thread somewhere that were really good PVP examples.

No rush at all - figuring out minor angle strength discrepancies is my current primary target before putting up a beta release - but poking you in case this got lost.

[Jim Eshelman]

Here is the prior post where I found important ingresses with clear PVPs and used as test cases. Let me apply the current formula to them an ensure that the answer comes out right. I'll calculate the ingresses from TM.

Conway Theater Fire (Canlunar)

December 5, 1876, 11:17 PM LMT, Brooklyn, NY
In addition to the obvious Mars opposite Neptune across the horizon, there were nine PVPs. Excluding those that were pure azimuth-to-azimuth, what else do we find? (I expect Sun, Mercury, and Jupiter aspects to Mars-Neptune). From the spreadsheet:

Jupiter 28°28'
Sun 28°58'
Mercury 29°50'
Neptune 0°02'
Mars 0°11'

Just what I expected.

Rhythm Club Fire (Capsolar)

April 29, 1940, 11:30 PM CST, 31N33'33" 91W23'51"
I expected a Mars-Saturn PVP. Mars is on Dsc, other planets on Antivertex (and some straight azimuth aspects may be happening - I seem to have missed the Jupiter Vx). For all planets meeting the 3° orb pre-screening, the spreadsheet gives:

29°23' Saturn
29°49' Jupiter
0°02' Mars

Correct.


I don't think I have to check the others in the same thread. These are all correct. Switching the trig function screws it up. The formula as stated (either using tan(90-x) or cot(x), since they're the same thing) works.

[Mike V]

Awesome, thank you. Now, about the logic to add/subtract 180...

Does this still seem right?

meridian_longitude = cot(ratio)
if meridian_longitude < 0:
meridian_longitude += 360

if meridian_longitude >= 270 and altitude < 0:
meridian_longitude -= 180

if meridian_longitude <= 90 and altitude > 0:
meridian_longitude += 180

Does meridian longitude begin above the horizon? i.e. the first quadrant of 0-90* ML is in the top right of the circle, 91-180* is in the bottom right (below the horizon), etc?

If so...

meridian_longitude >= 270 and altitude < 0 - this means that we calculated the body to be in the top left quadrant, but since it's BELOW the horizon, it should really be in the BOTTOM left or right quadrant, and vice versa for the other case?

Do I understand it at least kind of?

[Jim Eshelman]

The only ways I can answer this are to rederive it from scratch OR to trust the original way I worked it out. I choose the latter (to save me a few hours; and, besides, it's almost certainly right; and, besides besides, if I'm wrong it doesn't matter in a practical way.)

Going back to here: https://www.solunars.com/viewtopic.php? ... 077#p53018

The orientation is looking at the sphere with Eastpoint at the center. This means south is at left, north is at right. Upper left quadrant is south-above, etc. - NOTE: You can tell I wasn't all that concerned with the quadrant, since this part is a mess.

This is very confusing. Rereading my own notes a couple of times gives me two different answers. The actual "where the circle starts" is entirely arbitrary, but we do want to be on the same page. Working it out...

Looking at my spread sheet with my own natal: Moon closest to the horizon (and below it) has ML 359.764, while Sun (next most people the horizon) has 357.158. Therefore, the "9 o'clock" position on the wheel (southern point of the horizon) is 0°/360°, increasing clockwise. Moon's azimuth is 274°+ so it is past 270° (belongs on right side). Sun's azimuth is 81°+ so it is before 90° (also belongs on the right side). They should have 180° subtracted from the calculated ML to show that.

Does that agree with the paragraph near the bottom of that page? ... All the answers will be 270° to 90° (via 360/0°), which (as stated on that page) means pseudo-3rd through pseudo-10th houses (bottom of the chart, around past left, up to top). I think I got the next part wrong then (it seems to say ML 270°-360° is above the horizon and isn't, so ignore that).

OK... confirming... output will be 270°-90°, which means on left side of chart, with 270°-360° being Nadir to SP (below the horizon) and 0°-90° being SP to Zenith (above the horizon). For quadrant adjustments we need to check for (1) above/below and (2) south/north.

if meridian_longitude < 0:
meridian_longitude += 360

Yes. Let's be positive.

if meridian_longitude >= 270 and altitude < 0:
meridian_longitude -= 180

I think this is backward. 270°-360° should be below horizon (Nadir to SP). It should flip 180° if altitude > 0. -- Example, my Jupiter and Uranus are above the horizon. Their calculated ML is 329°+ (both). That implies below the horizon, so subtract 180 to get it above the horizon (149°+). This wrongly puts them on the north side, but we'll come back to that.

if meridian_longitude <= 90 and altitude > 0:
meridian_longitude += 180

A value 0°-90° means it has crossed the horizon - in the upper left quadrant (above-south). Therefore, I had these two backwards and this one should be if ML < 90° [0 to 90] and alt < 0, then put it above the horizon (add 180°). - Like the one above, this might put it on the wrong left-right side and we can handle that next.




So, I left out the left-right correction entirely. Let's see how that would go. (I think it will work to do top-bottom correct first and then handle left-right.)

Above the horizon is ML 0°-180°. Below the horizon is 180°-360°.
Left (south) side is 270°-90°. Right (north) side is 90°-270°.
Southern azimuth is 90°-270°. Northern azimuth is 270°-90°.

We began with values 270° to 90° and some of them may have had 180° added. Therefore, we have values in any part of the circle. We start by assuming above-below horizon is correct since we've already checked for that: The correction process is different this time. Breaking it down... doing it in my head but easy to check on paper...

If above the horizon and ML 0-90° [south]: If azi = 180°-360° [north],
-- ML = 90+(90-ML) = 180-ML [same distance either side, still Above]
If above the horizon and ML 90-180° [north]: If azi = 0°-180° [south],
-- ML = 90-(ML-90) = 180-ML [same distance either side, still Above]
These are both the same! So, if Above and south-north mismatch, ML = 180-ML

If below the horizon and ML 270-360° [south]: If azi = 270°-90° [north],
-- ML = 270-(ML-270) = 540-ML = 180-ML [oh my: This just got really simple, I think!]
If below the horizon and ML 180-270° [north]: If azi = 90°-270° [south],
-- ML = 270+(270-ML) = 540-ML = 180-ML
270-(ML-270) = 540-ML = 180-ML [oh my!]

Please check my work. It seems, though, that in all cases the above-below fix and north-south fix (if needed) are both 180°-ML.

1. Test for above-below mismatch: If ML = 270-360° but alt > 0, or ML = 0°-90° and alt < 0, correct with 180 - ML.
2. Test for north-south mismatch: If ML = 270°-90° but azi 270°-90°, or ML = 90°-270° but azi = 90°-270°, correct with 180-ML.

(I think. I just did this sitting here at the typewriter and visualizing. It should totally make sense on paper.)

Does meridian longitude begin above the horizon? i.e. the first quadrant of 0-90* ML is in the top right of the circle, 91-180* is in the bottom right (below the horizon), etc?

0° is the left (9 o'clock) position, which is Southpoint. It increases clockwise, so 0°=90° is that would be 12th to 10th houses in a normal chart (SP rising along meridian to Zenith). 0-90° is upper left, 90-180° upper right, 180-270° lower right, 270-360° lower left. (This is arbitrary. I made it up.)

At least some of the confusion was my own screwing something up in the original. (That statement presumes I have it right now.)

[Jim Eshelman]

From the spreadsheet, my own chart: Azimuth, Altitude, ML [first calculation]. Azimuth starts at due north 0° so is southern at 90-270, northern at 270-90.

Code: Select all

274.1739799	-3.236059302	359.7642131
81.77609057	-19.13838505	357.1581665
75.22375326	-43.60530848	346.3461229
70.1038248	-57.97575397	331.4490257
294.7550123	-59.99602632	324.0518276
114.46004	54.59840314	329.7748141
69.95290926	-39.27808936	344.3389764
114.1995155	55.01708061	329.638597
76.82471712	-26.93803938	353.3930847
87.22241454	31.65823409	1.711495964

Working through one at a time, taking ML=0 as "1st cusp," 270° = "4th cusp"

1. Test for above-below mismatch: If ML = 270-360° but alt > 0, or ML = 0°-90° and alt < 0, correct with 180 - ML.
2. Test for north-south mismatch: If ML = 270°-90° but azi 270°-90°, or ML = 90°-270° but azi = 90°-270°, correct with 180-ML.

MOON: 359°should be south-below. Moon is north azimuth, below horizon, so 180-ML gives a -179°+, add 360° to make it positive, gives 180.2357869 (180°14').

SUN: 367°+ should be south-below. Sun is north azi and below. 180-ML = -177°+, add 360° to make it positive = 182.8418335 (182°51')

MERCURY: 346°+ is south-below. Mercury is north azi and below. 180-ML = -166°+, add 360° to make it positive = 193.6538771 (193°39')

VENUS: 331°+ is south-below. Venus is north azi and below. 180-ML = -151°+, add 360° to make positive = 208.5509743 (208°33')

MARS: 324° is south-below. Mars is north azi and below. 180-ML = -144°+, add 360° to make positive = 215.9481724 (215°57')

JUPITER: 329°+ is south-below. Jupiter is south azi and above. THIS GIVES THE WRONG ANSWER. It looks like it only needs one correction. 180°-ML = -149°+, add 360° to make positive = 210.2251859. However, this is exactly 180° off. I'm not sure how to generalize the solution. In this case the non-computer logic is: Oops, that put it below the horizon and we know that it's above the horizon, so add 180°. (Is this a general check? We'd already determined above-below was correct.) Add or subtract 180° puts it at 30.2251859 (30°14') which is correct.

URANUS: A repeat of Jupiter: 329°+ is south-below. Uranus is south azi and above horizon. The north-south correction is 180°-ML = -149°+, add 360° to make positive = 210.361403. As before, this is exactly 180° off. Seeing it no loner agrees on above-below, add or subtract 180° to get 30.361403 (30°22').

SATURN: 344°+ is south-below. Saturn is north azi and below. 180-ML = -164°+, add 360° to make positive = 195.6610236° (195°40')

NEPTUNE: 353°+ is south-below. Neptune is north azi and below. 180-ML = -173°+, add 360° to make it positive = 186.606153 (186°36')

PLUTO: 1°+ is south-above. Pluto is north azi and above. 180-ML = 178.288504036 (178°17')

Using only the spreadsheet, I got:

Code: Select all

Mon	12	29	46
Sun	12	27	9
Mer	12	16	21
Ven	12	1	27
Mar	11	24	3
Jup	11	29	46
Sat	12	14	20
Ura	11	29	38
Nep	12	23	24
Plu	1	1	43

From this step-through, I got:

Code: Select all

Mon	6	29	46
Sun	6	27	9
Mer	6	16	21
Ven	6	1	27
Mar	5	24	3
Jup	11	29	46
Sat	6	14	20
Ura	11	29	38
Nep	6	23	24
Plu	7	1	43

Gratifyingly, they were all correct EXCEPT eight of them were 180°. I knew this could happen, didn't correct for it, and didn't really care for my purposes. So I'm happy with that.

[Jim Eshelman]

I'm glad I did this planet by planet. Aside from the reminder to check and correct for negative numbers at every stage, there is a final check that needs to be. Surely there is an easier way that someone who REALLY knows trig could provide, but here is the brute force approach:

1. Check for above-below horizon. If mismatched, correct with: ML = 180-ML
2. Check for north-south hemisphere. If mismatched, correct with: ML = 180-ML
3. Because #2 may have made above-below incorrect, check for above-below. If mismatched, correct with: ML = ML + 180

[Mike V]

Thank you for writing all of this out in detail. I programmed this and verified it against your planets. Here's the logic of how this came out for me:

Code: Select all

ratio = cot(altitude) / cos(azimuth)
ML = acot(ratio)
if ML is negative, add 360

# If there are above/below or north/south errors
# fix using a flip about the prime vertical,
# i.e. mirror to other side of the meridian about the Y axis

if altitude > 0 and ML > 270   # above horizon but sub-horizon ML
OR
altitude < 0 and ML < 90   # below horizon but supra-horizon ML
OR
ML 270-90 and azimuth 270-90  # northern azimuth but southern ML
OR
90 < ML < 27 and 90 < azimuth < 270:  # southern azimuth but northern ML
ML = 180 - ML

if ML is negative, add 360

# Recheck above/below using a pure 180° correction, NOT mirroring
if altitude > 0 and ML > 180 
OR
altitude < 0 and ML < 180: 
ML -= 180

if ML is negative, add 360

I finally understand where I got caught up. We don't do the above/below check, THEN the south/north check, THEN another above/below check, which is what I originally thought was happening.

We make a series of above/below/south/north checks, resulting in a maximum of one correction, a mirroring about the Y axis - and THEN we do a pure above/below horizon check with a pure 180° correction if necessary. That gives me the same intermediate and final values you have.