Hello Curtis,

Martin Gansten wrote:
Please note that what has commonly been called 'mundane aspects' or 'aspects in mundo' since the time of Placidus is something entirely different than zodiacal aspects with latitude.

see my
http://heavenastrolabe.net/primary-dire ... -software/

bottom page "aspects in mundo". A sextile will be distant 4 hours (2 houses), a square 6 hours (3 houses), a trine 8 hours (4 houses).

margherita

Martin Gansten wrote:

zoidsoft wrote:I'm wondering if anyone here knows what constitutes latitude of an aspect (such as the dexter trine of Venus)? I'm setting up the classes for the significator/promissor object in the Alchabitius method right now... Zodiacal directions are obvious, but is there an in mundo latitude to an aspect in Ptolemy?

Ptolemy doesn't mention latitude (nor, to my knowledge, do other Greek-language authors), but the Arabic authors do. You will find information on this on pp. 68-70 of my book (see also pp. 88-91). Briefly, aspects may be taken in a great circle passing through the body of the aspecting planet and inclined to the ecliptic by the extent of that planet's latitude. This is the most common method, though others have been proposed (Morin, for instance, had his own method).

Please note that what has commonly been called 'mundane aspects' or 'aspects in mundo' since the time of Placidus is something entirely different than zodiacal aspects with latitude.

I might assume that for an opposition for instance that if north latitude of +2 degrees, then the opposition would be -2 degrees below the ecliptic and of course proportional according sextile / Trine, and the square would be 0 here by my assumption - this is in keeping with my idea that aspect rays should be cast on the great circle from the planet casting the ray (this probably has a name by some author in history because I can't believe that I would have been the first to think of this issue).

No, you're not! This is the most common method, as mentioned above. It is usually named after Bianchini (Blanchinus).

Thanks. It's becoming clear now.

margherita wrote:Hello Curtis,

Martin Gansten wrote:
Please note that what has commonly been called 'mundane aspects' or 'aspects in mundo' since the time of Placidus is something entirely different than zodiacal aspects with latitude.

see my
http://heavenastrolabe.net/primary-dire ... -software/

bottom page "aspects in mundo". A sextile will be distant 4 hours (2 houses), a square 6 hours (3 houses), a trine 8 hours (4 houses).

margherita

Thanks.

I might assume that for an opposition for instance that if north latitude of +2 degrees, then the opposition would be -2 degrees below the ecliptic and of course proportional according sextile / Trine, and the square would be 0 here by my assumption - this is in keeping with my idea that aspect rays should be cast on the great circle from the planet casting the ray (this probably has a name by some author in history because I can't believe that I would have been the first to think of this issue).

No, you're not! This is the most common method, as mentioned above. It is usually named after Bianchini (Blanchinus).

This reminds me of an older thread http://skyscript.co.uk/forums/viewtopic ... ght=#37151 p. 1. Later I found that there's a mathematical error in Bianchini's reasoning. If I may quote here from that discussion,

Eddy wrote:Do you mean 'angular separation' here, measuring the distance between the two planets as points directly?

No. An aspect circle is a great circle passing through the body of a planet (i.e., its actual position, with latitude). In its simplest form, this circle is inclined to the ecliptic by the exact latitude of the planet, so that the planet represents one of the two points of maximum latitude in the aspect circle. The other such point is of course exactly opposite the planet. This aspect circle will always cut the ecliptic at a distance of 90? from the planet.

Take the noontime (CET) position of Venus today, around 14?45' of tropical Aries with a latitude of +7?48'. Drawing a straight line from Venus through the centre of the earth, its opposition point will be at 14?45' of tropical Libra with a latitude of -7?48'. The points halfway between Venus and its opposition (90? removed: square aspect) will intersect the ecliptic and therefore have 0? latitude. The sextile points (60? removed) will have half the latitude of Venus (+3?54'), while the trine points (120? removed) will have half the latitude of the opposition (-3?54').

Any of these points may be directed in the same fashion as a planet with latitude. When projected onto the ecliptic, the trines and sextiles will differ slightly from the corresponding aspects taken without latitude.

I'm sorry if this is a bit confusing but this is incorrect because the function depends on trigonometry. It's easily illustrated with applying conversion formulae to big latitudes like 30?, or you can verify the thing with declination tables of the Sun (the declination of the Sun at 30? (as a sextile point from its maximum declination at 0?Cancer is not 23.44?/2=11.72? but 11.47? ). Although mathematically this is incorrect, in practical use with Venus having the biggest latitude this is not disturbing and there's barely any difference (difference less than 0?1').

Briefly, aspects may be taken in a great circle passing through the body of the aspecting planet and inclined to the ecliptic by the extent of that planet's latitude.

Seen from this perspective of the planet's great circle, the other planets and the Sun will have a latitude towards this plane. Also the aspects slightly change. An exact sextile of Sun to Venus (usually) will be a quarter of a degree bigger from the Venus great circle.

If you want to measure the distance between stars directly then a great circle is drawn through two points. This is the angular separation, the way how we would see the distance between planets in the sky without imagining their projection on the ecliptic or equator or imagining in semi-arc proportions.

The angular separation between two stars is given by the expression:

Cos (A) = sin(d1)*sin(d2) + cos (d1)*cos(d2)*cos (ra1-ra2)

where A is the angular separation between star 1 and star 2 (degrees)
d1 is the declination of star 1 (degrees)
d2 is the declination of star 2 (degrees)
ra1 is the right ascension of star 1 (degrees)
ra2 is the right ascension of star 2 (degrees)

source: http://www.bobatkins.com/photography/te ... ength.html To apply this to planets, for 'd' use latitude and for 'ra' use longitude.

Of course aspects might not just be cast from a frame of reference such as the plane of the ecliptic; one could use right ascension to cast aspects as well with very different results, but I do not know what these variations go by name wise by historical convention.

Having mainly used the ecliptic reference for theoretical and anecdotical evidential reasons and therefore rejecting other reference frames, I lately receded from this because it's interesting anyway. Years ago I calculated in different reference frames and although some might believed not to 'work' it's informative to ponder on them. I regained my joy in doing this.

Although angular separation may appear as the most logic according to what we see, what we perceive might be different. Last night when at a friend?s, he said to me get yourself a beer. Looking for a bottle, I couldn?t find it. He said there were tin cans, so I looked for it with a tin can in my mind. Still I couldn?t find it, until I saw a sixpack that hadn?t been opened yet. This didn?t correspond with the referential images of ?a beer? I had in my mind to search for the beer, hence my temporal inability to see it. After it we discussed how our minds can be tricked sometimes by this.

With a certain Aristotelian aesthetical preference for the circle and the sphere one could have ?conscientious? objections with the various divisions differing from these forms. The divisions are either variants of the usual coordinate systems with poles and base planes, or the combinations thereof, making use of great circles. Svarogich?s perception of placidean division also appears to make usage of great circles, but I don?t manage to get a grasp on his explanations.

Apart from the systems with ?perfect? spheres and circles, we could also imagine other shapes. When searching for the formula above I found subjects about non-euclidian spherical geometry. Apart from spheres, ellipsoids, paraboloids etc. can be used for measurements, making trines look like squares for example and straight lines like curves and vice versa, possibly leaving only the conjunctions somewhat safe (but probably not).

One could apply for example the ellipsoid (or rather the geoid), reflecting the Earth's shape in your mind but there's little difference since the Earth is close to spherical. One could imagine aspects based upon the ellipse shape of orbits. This illustrates that the reference you have in mind alters the things how you perceive them. Changing this perspective can be very difficult at times. Note how shocking the concept of elliptic orbits must have sounded in Kepler's days. On the other side, the perfectness of the sphere could also be restraining. I remember a painting of William Blake of a person escaping out of a circle.

I realize that this falls out of the scope of this thread and perhaps these views appear as complete madness. Perhaps I?d rather discuss these matters in another thread and forum section in a few months when I have more time.

So far I haven't run into examples of what to do with arcs greater than 180 degrees (or is it simply measured in the opposite direction?).

Been thinking it would be nice to expand the range of PD calculations several rotations into the future (10 rotations ~ 3600 years) for the sake of mundane astrology. Is it simply the addition of 360 degrees for the next pass on the same primary direction to the first incidence of it?

zoidsoft wrote:So far I haven't run into examples of what to do with arcs greater than 180 degrees (or is it simply measured in the opposite direction?).

Been thinking it would be nice to expand the range of PD calculations several rotations into the future (10 rotations ~ 3600 years) for the sake of mundane astrology. Is it simply the addition of 360 degrees for the next pass on the same primary direction to the first incidence of it?

Unless you take secondary motion into account, which the old guys apparently did not.

- Ed

Ed F wrote:

zoidsoft wrote:So far I haven't run into examples of what to do with arcs greater than 180 degrees (or is it simply measured in the opposite direction?).

Been thinking it would be nice to expand the range of PD calculations several rotations into the future (10 rotations ~ 3600 years) for the sake of mundane astrology. Is it simply the addition of 360 degrees for the next pass on the same primary direction to the first incidence of it?

Unless you take secondary motion into account, which the old guys apparently did not.

- Ed

I was also thinking of precession, but is it really that simple... just find a direction and the date of its first instance, then add 360 degrees in some key and recalculate the new date? Why wouldn't current programs I've seen calculate this?

Mine does:

Primary Directions for Transit Event

23 Jan 2011 13:28:54 Zone: 5:00W
Lat: 42 40' 45"N Long: 70 50' 30"W Alt: 20m

Key = NAIBOD


Date Promissor -> Significator

14 Feb 2511 mu ASC D -> 45 Ura (l = r)
5 Mar 2511 mu Ven (l = r) D -> 135 Plu (l = r)
1 Apr 2511 mu MC D -> sx MNo (l = r)
9 Sep 2511 mu Moo (l = r) D -> 135 Mer (l = r)
16 Sep 2511 mu Sun (l = r) D -> 135 Mar (l = r)
5 Nov 2511 mu Nep (l = r) D -> sx MNo (l = r)

Ed F wrote:Mine does:

Primary Directions for Transit Event

23 Jan 2011 13:28:54 Zone: 5:00W
Lat: 42 40' 45"N Long: 70 50' 30"W Alt: 20m

Key = NAIBOD


Date Promissor -> Significator

14 Feb 2511 mu ASC D -> 45 Ura (l = r)
5 Mar 2511 mu Ven (l = r) D -> 135 Plu (l = r)
1 Apr 2511 mu MC D -> sx MNo (l = r)
9 Sep 2511 mu Moo (l = r) D -> 135 Mer (l = r)
16 Sep 2511 mu Sun (l = r) D -> 135 Mar (l = r)
5 Nov 2511 mu Nep (l = r) D -> sx MNo (l = r)

Is there an algorithm that bypasses the quadrant issue? You have to get the Diurnal Semi Arc or Nocturnal Semi Arc and then find the proportion of the meridian distance in the quadrant of the significator that matches the proportion for the promissor. This means finding whether it is in quadrant 1, 2, 3 or 4 before you can know which to call. It seems cumbersome to me and I'm almost certain that there is a more mathematically elegant way to handle proportional semi arcs than the way I've seen it stated.

I use an algorithm to determine whether the point is above or below the horizon, then figure out the quadrant later. If you want, pm with an email address and I can send some Ada code, which should be pretty straightforward to follow.

- Ed

I was also thinking of precession,

Since 3600 years of the direction is in fact related with the first 10 days after the beginning chart, there would hardly be any change and precession could be neglected. Unless you want to mix the actual time with directed time.

I came up with this simple formula for Blanchinus using interpolation:

// Aspect is the number of degrees between 0 and 180
function TDirPoint.GetBianchiniLatitude(EcLatitude, Aspect: Real): Real;
var
Proportion: Real;
begin
Proportion := (90 - Aspect) / 180;
Result := EcLatitude * Proportion * 2;
end;

I noticed what was posted on the pymorinus site:

Bianchini assigned different latitudes(calculated from the latitude of the promissor) to the aspects(conjunctio:latitudeofpromissor, sextil:latitudeofpromissor/2, quadrat:0, trigon:-latitudeofpromissor/2, oppositio:-latitudeofpromissor) and he always used the latitude of the significator

seems incorrect for the sextile and trigon because of the /2. Take for example a latitude of -6 degrees, the opposition will be +6 degrees, the squares will be 0. If one takes interpolation between 90 and 180 for the trine it is 1/3 of that angle, not 1/2 (unless I'm missing something here).