Since several mathematical issues I discussed might be too modern for the traditional forum, I moved part of my february 7 post to this new thread. I think it is instructive to discuss the mathematical side of astrology because astrology would be lost without maths. I'm not sure how to start this thread nor do I have much time at the moment to 'host' a thread, but anyway I plan to calculate planetary positions in (modern) mundo type of positions with the Kiev case chart as a basis. Let's consider this a thread under construction with a mainly mathematical colour.

Perhaps issues like lunar parallax, time-light abberation and (modern) primary directions would deserve their own thread to be discussed thoroughly, on the other side discussing these here as in an umbrella thread could also be useful. Often a longer thread goes a certain direction for a while and sometimes changing, so feel free to add your mathematical views/theories.

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the remainder of my moved post from the traditional board Primary Directions :
Thinking about reference frames is not rather about hierarchies but searching for a way to fixate planets (as in radix, natal charts). In astronomy its relevance is different from astrologers because in addition to astronomy astrologers apply times comparisons to the frames. This is what I see the core of the approach and from which differences in uses of zodiacs can be derived as Ed points out. The problem even gets more difficult when symbolic time is used as in cycle a = cycle b theories, as in progressions and directions. Since large differences can occur when different concepts of a time is used. If one unit of time is expressed as a return of a planet to its original position then surely a definition of original position has to be made. Problems increase when inbetween times/spaces are looked for (as in aspects to original positions). Sometimes this drives me insane when I try to implement this in primary directions, hence I need to ?escape? to traditional methods (Ptolemy, including his key). However I often soon find myself experimenting again.

The ecliptic by the way is not the oldest frame. Babylonian astrologers positioned the planets in equatorial paths using seasonal hours and even earlier the Sun?s positions throughout the seasons were determined according to the ?gates? along the horizon, so a kind of horizontal system. Although the ecliptic is a frame in which I see a kind of perfection for the use of slow movements (planets, transits) the Earth/equator is well served with frames bound on the rotation. The ecliptic can be defined in different ways, the Earth-Moon barycenter being the most used one. I?m not sure though whether I understand Ed?s remark that astrologers use Earth-Moon barycentric positions. I think that planets are expressed in geocentric positions whilst making use of a projection to the Earth?s center of the reference frame of the barycentric ecliptic. Hence we can see in astronomical ephemerides that the Sun usually has latitude (eg. in www.ephemeris.com). This latitude varies with the draconic months. It can easily be ignored in astrology however it?s interesting for mathematical astrology theorising. I for example mainly use at geocentric coordinates and rather than using the barycentric definition of the ecliptic I feel for a definition of the ecliptic in which the the equatorial rotation intersects with the direction of the precessional rotation. So rather than an Earth zodiac vs. Solar zodiac I feel for a ?fast? Earth zodiac (Equatorial rotation) vs. ?slow? Earth zodiac (precessional rotation). Note that the Earth?s equatorial rotation gives not an entire ?fixedness? for planets because of the continental shift and polar motion (not to be confused with precession or nutation).
For the definitions of meridian and horizon I also use mathematical constructs, like the geocentric horizon rather than geographic or geodetic. Note that the geodetic meridian can also slightly differ from the line from geocenter to location through meridian of the circle through the momentary poles of rotation. Although this are miniscule differences, and not always necessary to integrate in the calculations, these form different bases in our approach to astrology. And indeed the question of using atmospheric refraction and time-light correction is part of forming a theoretical basis. I understand that my view largely differs from the traditional view. My choice for the tropical zodiac is not mainly to be found in climatological foundation but rather in mathematical terms of circles and frames. My geometrical and geocentrical approach also has to face problems which I don?t manage to solve at once. Observing the native at the moment of birth means that you look from a geocentric position at your birth or body as if you would look at a planet to be placed in a chart, while at the same time many people can be born which you don?t place in the chart. This has always been a problem to me when I tried to find a causative model of astrological effects. I sometimes also think that the difficulty to place locality in the ecliptic frame would mean that in the ecliptic frame no MC and Asc should be placed at all.

And the question which one is correct is very difficult to answer. I usually find the effects of primary directions extremely intangible. In all different cycle comparisons and different keys I manage to find significant results. Like the Moon on the horizon or its ecliptical position on the horizon and much more. They all sometimes give results and sometimes not, so it?s quite tricky to base oneself upon the experience of its effects, there might be a preference for a certain method involved.
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In the meanwhile I'll do some calculations for several frames, from which parts of it are scattered in the traditional primary directions thread

Date/Time: 2007.08.21 00:00:00 UTC (GMT - Delta T), JD = 2454333.500000
Sidereal Time: 21:55:45
Planet Longitude Latitude Right Asc. Declination
Sun 27 Leo 35'13" - 0?00'01" 09:59:07 12?18'41"
Moon 27 Sco 37'54" - 5?12'23" 15:36:16 -24?41'58"
Mercury 02 Vir 49'35" 1?39'44" 10:21:34 12?01'02"
Venus 23 Leo 05'27"R - 8?10'52" 09:30:58 6?04'46"
Mars 08 Gem 43'07" - 0?55'22" 04:28:36 20?50'43"
Jupiter 10 Sgr 13'38" 0?27'54" 16:34:42 -21?31'26"
Saturn 28 Leo 24'15" 1?16'45" 10:04:04 13?13'45"
Uranus 17 Psc 28'48"R - 0?48'36" 23:15:12 - 5?41'34"
Neptune 20 Aqr 26'10"R - 0?16'49" 21:31:43 -14?56'35"
Pluto 26 Sgr 22'55"R 6?48'33" 17:45:00 -16?35'04"
Moon's Node 07 Psc 07'45"R 0?00'00" 22:35:22 - 8?53'38"
Source: www.ephemeris.com

Kiev, Ukraine on Tuesday 21-8-2007 at 3:00 East European time (= 0h UT+2h timezone+1h daylight saving time)

Kiev, Ukraine : 50?26?N (90?-Lat=39?34?), 30?31?
0h GMT, 3h EET
Sidereal Time Greenwich: 21h55m45s
Sidereal Time Kiev: 21h55m45s + 2h2m4s = 23h57m49s, in degrees: 359?27?15??
Obliquity Ecliptic: 23?26?18??
Ascendant: 25?18? Cancer
MC: 29?24? Pisces

At the last moment I decided to put the lists of positions in a separate post. The more detailed descriptions are found below, and in the next post the positions according to house degrees are given. This gives a better overview.

First the easiest types of charts expressed in a position according to the reference system. Whole sign to start with. With ascendant in Cancer, this sign is house I. The planets in the houses keep the same number of degrees.

In an ecliptic based chart, 0?Aries is normally the basis point, but since the houses count from the Ascendant, the planetary positions could be counted from there. With the Ascendant at 25?18? Cancer set as the 0? point all positions are moved 115?18?. Here we get the planetary positions as seen from the Equal (Ascendant) houses perspective.


MC as 0? point of cusp X. With MC at 29?24? Pisces, cusp I is MC+90? which is 90?36? further and will serve as 0? point. An Equal (MC) houses perspective.

Apart from the Whole sign systems other house systems are based upon an integration of horizon and meridian. In the projection on the ecliptic this shows as an integration of Asc and MC. In two classical systems the ecliptical definition of the ascendant/horizon is essential for the houses, Porphyry and Alchabitius.

Porphyry calculates the intermediate cusps by dividing the distances between MC-Asc, Asc-IC, IC-Desc, and Desc-MC. In the example we get: 115?42? for X, XI, XII, 64?18? for I, II, III, 115?42? for IV, V, VI and 64?18? for VII, VIII, IX respectively. The sizes of the houses each is found by dividing these numbers by three. To express the planetary positions in these houses as ?house degrees?, we set the houses as if they were 30? each or the quadrants as if they were 90?. We have to ?squeeze? or ?stretch?, the chart till the angles are 90? apart. The proportions will be 90?/115?42? or 90?/64?18? depending on the planets? placements. With the distances of the planets to the angles multiplied by the proportions, we get the planetary positions expressed in ?house degrees?.

Alchabitius does something similar to Porphyry, except that the MC and Asc are converted into equatorial positions, then the intermediate cusps are derived from dividing the distance measured along the equator in three. The planetary positions in house degrees are found by converting the ecliptical positions of the planets to equatorial. However one does not use the latitude of the planets but set them on 0?. This is because of the definition of Alchabitius. The house division depends on the intersection point of ecliptic and horizon and not on the horizon itself as in the mundane quadrant systems. For example a planet in house I will be under the horizon when 0? latitude is used. In some cases the body of the planet might be above the horizon and one might expect the planet to be in house XII. Using Alchabitius though with latitude still will give the planet?s house position in house I, even when above the horizon. The positions lay close to those of Porphyry.

Koch houses are defined by the semi arc of MC. This is divided in 6 and the ascendants at the moment that MC is each 1/6th part of its semi arc further (and before)
are the next (and previous cusps). It can?t be used as mundane for a planet in mundo rising simultaneously with MC will not be on the meridian when MC is. The moments when the planets will rise/have rose/ will set/have set, in proportionate relation to the MC?s semi arc determine their ?house degree?.


The Meridian system is the first step outside the ecliptical framework. In astronomy this is the mainly used system to position planets. There it is expressed in hours and minutes according to a certain epoch. The vernal equinox is the 0?. Note that this list is not yet the house positions list.
Converted to degrees we get (measured from the equinox):
Sun 149?47?
Moon 234? 4?
Mercury 155?24?
Venus 142?45?
Mars 67? 9?
Jupiter 248?41?
Saturn 151? 1?
Uranus 348?48?
Neptune 322?56?
Pluto 266?15?
(Node 338?51?)
I placed the Node between brackets because it?s an intersection of lunar orbit and ecliptic, converting this ecliptical position to equatorial positions would fit less in the equatorial frame. There is an intersection of lunar orbit and equator and it oscillates back and forth along the equinox with a maximum distance of about 12?. As the equator obviously is always 0? declination, its intersection with the horizon is always exactly in the east. This as an ?Ascendant? is therefore always in 90? to the north-south meridian. It has a resemblance with the equal MC house system and the house degree positions of the planets in the meridian system lay close to those in the equal MC system. However, planets with a big latitude may have a quite different equatorial position. The Moon?s ecliptical position in relation to the vernal equinox is 3.5? further than equatorial. Cusp X of the meridian system is equal to the sidereal time expressed in degrees (multiply ST with 15). Cusp I is always 90? further.


The numbers in brackets are the positions measured from meridian cusp I and the declination, which remained the same). These will be necessary for they are needed to calculate the mundane quadrant systems which will be discussed now. I therefore sometimes call the meridian system the mother of the mundane quadrant systems because it is via this mathematical step one can find the mundane quadrant systems. Possibly there are shorter more direct ways but imagining the systems via this procedure is a helpful path to the understanding of the following systems.

Campanus house system divides the prime vertical in equal parts of 30?. The prime vertical is the great circle from east to west via the zenith and the nadir under our feet. The great circles from north to south through these 30? divisions intersect the ecliptic which form the house positions as seen in the ecliptic. As the ecliptic usually is in oblique position with the prime vertical the houses appear as unequal, hence they are distorted. Yet this is only because of this projection. The planetary mundane positions in campanus are measured along the prime vertical, which is the base plane with the north and south points on the horizon as poles of the coordinate system and the line connecting these along the horizon as the 0? point of the coordinate system. The n-s line through the zenith determines the meridian. Note that this is the first system in this overview in which the horizon entirely coincides with the 0? point of cusp I. In other words any planet on the eastern horizon is on 0? I. Its calculation is easy, with the conversion formula for equator to ecliptic and vice versa ( http://en.wikipedia.org/wiki/Ecliptic_c ... quantities ) the meridian house positions are converted to campanus positions. The difference is that the ? value of the conversion is not the obliquity of the ecliptic but similar to the latitude of the birth place which for Kiev is 50?26? N.


The positions between brackets are useful for the following system,

the Horizon house system because the base plane has to be tilted 90?, prime vertical in campanus to horizon in horizon system, easier to convert. The attractiveness of the horizon system is that it is closest to what we actually see when we go outside. The cusps start in the east and we turn to the left to see the other cusps. The horizon as base plane is what we are standing on. Note that the place where the ecliptic intersects the horizon does not determine cusp I. The great circle through cusp I is now the prime vertical, and where it crosses the ecliptic would be cusp I when depicted in an ecliptical chart. This intersection point can be above or below the horizon. This chart is a kind of compass chart with planets direction west and south-east ect. Furthermore it shows height above and below the horizon so you can look to the south-west and 20? above the horizon to see a planet.


Since the campanus system (and therefore the horizon system too) are considered to be somewhat ?stiff? systems because nothing of the diurnal movement appears to be in these systems, astrologers tried to implement the equatorial movement in the house system. Alchabitius being an old example of before campanus,

Regiomontanus is a younger one after campanus with a closely related mathematical approach. Now the equator serves as base plane of the coordinate system, just like in the meridian system, however maintaining the north-south great circles as dividing lines and not the equatorial poles as in meridian system. This means that just like in campanus the circle which determines cusp I is also the horizon. However it?s base plane, the equator is not perpendicular to the poles (the line through north and south points on the horizon which are perpendicular to the prime vertical) but tilted according to the latitude, in this case 50?26?. To determine the planet?s house positions here, the planets? points on the great n-s circles are calculated where they intersect the equator. The places of these intersection points determine the house degrees and for their calculations we can use the results of the campanus house degrees.


Although there?s some dynamic element in regiomontanus, to others this may not be enough because the equator is still the dominant plane in that system and divisions are determined upon that plane. Every planet lays on a circle around Earth so one could also look for divisions based upon that particular circle.

The Placidus division does this. For each planet we therefore have to calculate the specific elements belonging to their own particular diurnal and nocturnal semi circles or semi arcs. As the planets usually aren?t on the equator the semi arcs vary and we have to calculate the ascensional difference with the points on the equator. With these differences and with the distance of the planet on the circle to these ascensional difference points we get oblique ascension. The proportion of the oblique ascension in relation with the whole diurnal or nocturnal semi arc leads to the house degree or mundane positions of the planets.

For the actual workouts of the houses, post follows soon.

edit: I just added Koch to the series.

Eddy,

Couldn't resist, eh?

I'm not ready to jump into this one at the moment. But if you add some thoughts, they may "get me started".

Nice summary of the various house systems considered on their own rather than for placement of planets in houses via the relationships of their ecliptic intercepts to the ecliptic intercepts of the house boundaries.


- Ed

Hi Ed,

About a month ago it was either quitting or changing views. The former would hurt a bit so I chose the latter. I?m thinking about reading a bit on ant colony algorithms etc. to get some inspirations of metaphors for modeling.

The calculations are for illustrative purposes so we all can see what we actually are doing, and base a choice for a system and its attendant mathematical philosophy. I?m calculating primary directions for these systems according to their definition and some of the easier cusp calculating systems are difficult systems for primary directions and vice versa. Although I still have my preferences, systems like Porphyry and Alchabitius for example appear much less attractive than Placidus now. I?m still struggling with the definition of the latter (no polar stuff though). I?ll write down some reflections soon, also the directions of Venus to the sextile of Mercury (or the other way round in some cases).

(I added calculations for Koch, in the post before and in following list.)
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Placements of the planets from the viewpoint of houses as equal segments.

Whole sign
Sun 27?35? II
Moon 27?38? V
Mercury 2?50? III
Venus 23? 5? II
Mars 8?43? XII
Jupiter 10?14? VI
Saturn 28?24? II
Uranus 17?29? IX
Neptune 20?26? VIII
Pluto 26?23? VI
Node 7?08? IX

Equal Ascendant
Sun 2?17? II
Moon 2?20? V
Mercury 7?32? II
Venus 27?47? I
Mars 13?25? XI
Jupiter 14?56? V
Saturn 3?06? II
Uranus 22?11? VIII
Neptune 25?08? VII
Pluto 1?05? VI
Node 11?50? VIII

Equal MC
Sun 27?59? II
Moon 28?02? V
Mercury 3?14? III
Venus 23?29? II
Mars 9?07? XII
Jupiter 10?38? VI
Saturn 28?48? II
Uranus 17?53? IX
Neptune 20?50? VIII
Pluto 26?47? VI
Node 7?32? IX

Porphyry
Sun 15?11? II
Moon 15? 9? V
Mercury 22?32? II
Venus 8?53? II
Mars 23?46? XI
Jupiter 24?57? V
Saturn 16? 3? II
Uranus 13? 2? IX
Neptune 5?11? VIII
Pluto 7?30? VI
Node 28?33? VIII

Alchabitius
Sun 17? 4? II
Moon 12?43? V
Mercury 24?19? II
Venus 10?45? II
Mars 21?37? XI
Jupiter 22?50? V
Saturn 18?12? II
Uranus 14? 8? IX
Neptune 7? 1? VIII
Pluto 6?10? VI
Node 0?11? IX

Koch
Sun 15?10? II
Moon 0? 9? V
Mercury 22?35? II
Venus 8?46?
Mars 8?31? XI
Jupiter 9?45? V
Saturn 16?18? II
Uranus 13?15? IX
Neptune 5? 01 VIII
Pluto 24?57? V
Node 28?41? VIII

Meridian
Sun 0?20? III (60?20? / +12?19?)
Moon 24?37? V (144?37? / -24?42?)
Mercury 5?57? III (65?57?/ +12? 1?)
Venus 23?18? II (53?18? / +6? 5?)
Mars 7?42? XII (337?42? / +20?51?)
Jupiter 9?14? VI (159?14? / -21?31?)
Saturn 1?34? III (61?34? / +13?14?)
Uranus 19?21? IX (259?21? / -5?42?)
Neptune 23?29? VIII (233?29? / -14?57?)
Pluto 26?48? VI (176?48? / -16?35?)

Campanus
Sun 7?53? II (37?53? / +52?13?)
Moon 18?25? V (138?25? / +8? 1?)
Mercury 15?42? II (45?42? / +55?12?)
Venus 5?39? II (35?39? / +43? 0?)
Mars 29?57? XI (329?57? / -2?40?)
Jupiter 0?28? VI (150?28? / +1?11?)
Saturn 8?31? II (38?31? / +53?41?)
Uranus 11?24? IX (251?24? / -55?48?)
Neptune 27?13? VII (207?13? / -49?43?)
Pluto 15? 8? VI (165? 8? / -8? 5?)

Horizon
Sun 28?32? II / -22? 6?
Moon 19?20? VI / -41? 5?
Mercury 4? 6? III / -24? 6?
Venus 18?56? II / -25?14?
Mars 26?55? XII / +30? 1?
Jupiter 28?38? VI / -29?32?
Saturn 0? 6? III / -21?39?
Uranus 17?46? IX / +32?11?
Neptune 23? 0? VIII / +17? 8?
Pluto 8?22? VII / -14?43?

Regiomontanus
Sun 20?42? II
Moon 5?40? V
Mercury 28? 8? II
Venus 18?24? II
Mars 17?45? XI
Jupiter 18?21? V
Saturn 21?20? II
Uranus 17?54? IX
Neptune 8?55? VIII
Pluto 7?23? VI

Placidus
Sun 24?15? II
Moon 9?42? V
Mercury 1?10? III
Venus 20? 0? II
Mars 21?53? XI
Jupiter 22?35? V
Saturn 25?10? II
Uranus 18?28? IX
Neptune 13?48? VIII
Pluto 10?18? VI

Reflection upon the house systems / diurnal reference frames.

The result is that all planetary positions are given in equal sector positions. The concept of houses usually is that the relation between cusp X and I is one of three ?sectors? difference. Although not the same as signs the twelve fold division reflects its similar use in houses. Therefore in basis all houses are equal in expression. The question is whether some are more equal than others? Most of it will depend on the taste of the individual astrologer. 'It works for me? or ?papa smurf always says? are the usual arguments used to support a system, but this is not enough and not really satisfying because of its utter subjectiveness and inherent refusal to think about the problem. All experience is somewhat subject to subjectivism, but a more fundamental theory gives a basis to apply in a broader sense.

Therefore it would be useful to ponder on the mathematical sides, because ones choice affects or at least should affect ones symbolic approach. We saw that the house degrees of planets in ecliptic based systems can?t be expressed in mundane (planets with latitude positions) because these depend on the ascendant as intersection point rather than part of the horizon. This is important if you consider your symbolic preference rather in the horizon than in the intersection point of two circles. Saying something like Saturn is rising whilst using an ecliptic based house system doesn?t make much sense if the planet is still below the horizon. When the ecliptic based systems are used it is more proper to speak about the intersection point of horizon and ecliptic is conjuncting Saturn. (Even the horizon (and meridian) can be defined in several ways; I hope to elaborate more on this soon, because this is essential for the use of frames.) This second expression of ?rising? implies that the used reference frame of fixing planets is the ecliptic while the horizon/Ascendant is moving through it. The first (closer to real) expression of rising implies a reference frame related directly to the Earth?s body, to which the planets are fixed. This element with motion involved (which complicates the matter) will get more attention after my calculations of directions according to the several diurnal reference frames (which I try to finish soon).

Pondering on the easiest systems, Whole sign, Equal (Asc or MC) I personally already have some objections. The terminology of houses imply a ?beginning?, an initial fixed point through which planets pass. However with the ecliptic it?s rather this frame that is fixed with the planets but also the angles moving through. In my option using house numbers from the ecliptic confuses the matter. The only ?real? (or rather more realistic optional) beginning points on the ecliptic are those related to the various (astrological) definitions of the fixed ecliptic itself (equinoxes, solstice points, fixed star projections, historical event momentary ecliptics etc.). Using a moving object like the Ascendant or MC seems a bit arbitrary to me. Why not use Sun or Jupiter as a first house cusp from which the following houses are derived?

The domification methods formed by the integration of MC and Asc in one system are a faint shadow of the aim to express the quadrant formed by horizon and meridian which are always 90? apart. With the mutual MC and Asc as intersection points on the ecliptic as a basis for division the result gives usually a varying distance between the two. To equalize this one literally has to squeeze and stretch the houses. This is most apparent in Porphyry but also in Alchabitius which is similar to the former but with MC and Ascendant expressed in right ascension. Koch is another system depending on the intersection point of an angle, the MC. As its definition depends on the rising of the MC the curvature of the horizon is inherent to Koch?s MC. In other words a planet with latitude being conjunct MC wouldn?t be on the north-south meridian nor be conjunct with the MC from an ecliptical reference perspective (at least the MC system would have the virtue to do this latter). Therefore a planet with latitude on the n-s meridian can be in 29?IX Koch ?house degrees? and 1?X MC (ecliptical) house degrees. However Koch needs the ecliptic MC in the first place to define the whole system. Very confusing, you will cry if you understand the technique of this house system.

Campanus is a clear system entirely depending on the quadrant and in which the horizon ? meridian actually appears as a square. Most astrologers see as biggest drawback the fact that nothing in it is related to the Earth?s rotation. The Horizon system (which simply is Campanus tilted 90?) doesn?t do either. Regiomontanus is a reconciliation between Campanus? rigidness and the Earth?s rotation. A mention of the Ascendant Parallel Circle (APC) system of the Dutch astrologer Ram is interesting in this light. This method has first been proposed by Haly Abenragel but never actively used. It looks like a kind of transitional phase from Alchabitius to Regiomontanus. Where in the former the arc of Asc to MC is expressed in right ascension, Abenragel/APC expresses this arc and the whole semi arc of the Asc through the great circles /positioning circles through north to south. It?s an unusual system in which II and VIII (and the other opposing intermediate houses) usually are not 180? apart as in all other house systems. It?s complexity to calculate it probably explains its disuse. Regiomontanus also uses a parallel, namely the one of the equator and the position circles running through on the equal 30? divisions on the equator. When projected on the prime vertical one gets unequal results (except on the Earth?s equator). Closely related to Campanus it includes the dynamics of the equatorial rotation.

With the APC and the Regiomontanus systems in mind, one can imagine other parallel circle systems, like those of the Sun, Moon and all individual planets, or a combination of all these; here the Placidus system is born. Placidus? (mundane) positioning of planets can be seen as a working out of the principles of the working out of positioning of planets on their own parallels (semi arcs) with great positioning circles running through the north-south points on the horizon. This is a spatial definition of Placidus positioning with individual position circles. All these parallels mutually have points in common; on each parallel there?s a ?one third? of the semi arc. Connecting all these common points, results in curves instead of great circles. All this is fine but when used for circumpolar objects (sorry it has to be mentioned) this definition gets into trouble. The Otto Ludwig solution elaborated by Michael Wackford for positioning circumpolar objects in the semi arc division can?t be defined in those individual planet parallel?s great circles because these depend on the north and south points on the horizon. Using that solution with this certain definition in mind, the addition of it appears as a sudden change with a different reference method. An ascensional definition would address the problem more properly. All parallels are ascending and descending. Where they meet the eastern horizon the diurnal semi arc begins which ends at the western horizon. This relation with the horizon is expressed in oblique ascension. The obliquity is the biggest in the east. Objects near the equator rise fastest in relation to the horizon (In astronomy annuals with twilight duration one will see that the twilight in Spring and Autumn are shorter than those in Summer and Winter.) Further north or south the ascension more and more ?flattens? in relation to the horizon in order to skim the horizon exactly in the north and in the south. Further upwards (or downwards) the usage of the north to north pole line (or the south to south-pole line) by Wackford results in a right ascension approach of positioning. It resembles the meridian house division except for the division in 6 instead of 12 because the circle is treated as a semi arc. The great strength of this definition is the fact that the change in obliquity is a smooth one including the usage of circumpolar positioning while the former definition needs a sudden change of definition for that latter positioning.

When drawn on a sphere the Placidus division doesn?t show a traditional coordinate system like Campanus, Horizon, Meridian system and the Whole and Equal variants, nor great circles with a base plane which is squeezed and stretched, but curves with a ?kink? where the circumpolar region starts. While this can be viewed as a problem from ?aesthetical? geometrical view, it is acceptable if the focus isn?t geometry but proportionate division. For the 'aesthetics' reason I have strongly opposed to this system for a long time and perhaps I won?t get entirely used to it, but now I can at least appreciate it as a model in which its proportionate division is always equal in basis (something which more or less is absent in Porphyry and Alchabitius). However the integration of the horizon in this system still places me before problems. With the horizon as an essential part of the system the crossings of planets on the horizon can be seen as major significance. Circumpolar points which can?t cross the horizon can derive their significance from the closest approach to the horizon. In this view the element of height is introduced and is combined with the concept of proportionate semi arc division. At first sight the problem seems solved, highest point in the south meridian and lowest in the north meridian. Ascension and descension imply the upwards and downwards movements of planets and stars. This is fine for the slow planets, the stars and equinox but for a fast moving object as the Moon it can mean that the highest point doesn?t coincide with the meridian. At latitude of Kiev the Moon can be at highest point a few minutes (up to an extreme of 10 minutes) later or earlier than exactly in the south meridian. The difference can therefore be up to 2.5?. Although not eye catching it does change the definition which can make a big difference in primary directions, especially further north. In fact the whole reference frame alters because at the same moment of the Moon?s highest point the Sun can be at its highest point a degree further while retrograde Venus in its highest point is another degree further. They all are in their highest point but in a different positioning circle/curve. Moreover the semi arcs of the planets are ?spirals? rather than parallels. So how to solve this? One could use the horizon meridian quadrant as basis and divide it according to the discussed definitions. However the immediate connection with the ?time? systematics is somewhat lost. In this view Placidus too is a ?space? system with parallels crossing the horizon and where these don?t its parallels are closest to the horizon when near the meridian.

Since every model has its inconveniences the space Placidus approach solves the meridian/highest-lowest issue in exchange for a strict time division. The time element still is very near so this is a very close approximation of time basis. Moreover the definition of an entirely fixed Earth?s surface and related rotation might be so difficult that none of the systems may have no choice than being approximations of a perfect system.

Where my subjective aesthetical view makes me hesitant towards the complicated spherical expression of Placidus, the ?perfect? spherical systems of Campanus and Horizon lack the dynamics. For a long time, my personal set of preferences for a reference frame has consisted of firstly a coordinate system in the traditional sense with base plane equally divided and poles perpendicular to it and secondly the base plane as a rotating plane. The latter element excludes Campanus and Horizon and except ecliptic positioning leaves only the Meridian system. I find this a very attractive system however the horizon is not a real integrated part of it. One can use it as a system where the planets can rise and set in different houses without affecting the system but the aspects of the planets could be troublesome. A planet with a semi arc of 150? and on 3/5th of its path in the diurnal motion, will expressed in Placidus degrees be 108? past the eastern horizon and 72? before the western horizon, together 180?. The meridional position will be 90? from the eastern horizon and 60? from the western. While this introduces an interesting feature it still is somewhat alien to the meridian definition. However using the east point as ascendant/reference point any difference in latitude of location is irrelevant. This system therefore has as inconvenience the difficulty to integrate horizon. On the other side, I think that the representation of the concept of time reaches its closest approximation in the meridian system

Campanus and Horizon system lack the second element of the two preferences but use a coordinate system. While normally in house systems the emphasis is on longitude, right ascension, prime vertical degrees, azimuth or whatever you would call the division along the base plane (or the quasi plane), the concept of height in Horizon could open interesting perspectives. If the relations of height between planets is considered an extra dimension is given. The height problem in Placidus is much less a problem in this view. It?s worth to give this some theoretical examination.

Some of the elements of meridian/horizon ? MC/Asc definition have already been touched upon but need more exploration. First I soon will give the calculations for primary directions of Venus to sextile Mercury when I have finished them. The observation of the time element, comparing initial time with actual time learns a lot about the construction of house systems. Moreover the calculations which I?ve done already which display the difficulty of the primary directions calculations of relatively easy systems and the relatively difficulty (or rather elaborateness) of easy concepts make me believe that the development of most house systems is mainly a research for the integration of semi arc positioning in ecliptical systems with as result the Placidus house system on the ecliptical frame. However problems rose when this ecliptical expression of Placidus met problems in circumpolar issues. I therefore think that notwithstanding our mathematical preferences for any which system, a mixture with the ecliptical reference frame should not be the aim.

Eddy wrote:...I therefore think that notwithstanding our mathematical preferences for any which system, a mixture with the ecliptical reference frame should not be the aim.

"main aim", anyway. Keep it up, Eddy.

Thanks Ed, I think many problems would be solved if more attention would be paid to the mathematical background. Some of the first astrology books (of the ?30s to ?50s) I read, contained for one third of technical introduction. Hardly a trace of this is to be found in today's astrology. I?m afraid that the development of computer programs may have introduced the deathblow to astrology, making the art easily accessible to the vulgar, expressed in sayings like:

someone in the caverns of cyberspace wrote: My personal experience is, that the higher an intellect is the more one should use Koch houses. For mundane astrology I would use Placidus houses.

But if you have a client/custumer that lives on a higher level of spirit, Koch houses are absolutly indicated!

sigh!

Eddy wrote:

someone in the caverns of cyberspace wrote: My personal experience is, that the higher an intellect is the more one should use Koch houses. For mundane astrology I would use Placidus houses.

But if you have a client/custumer that lives on a higher level of spirit, Koch houses are absolutly indicated!

sigh!

Yes, that one cracked me up too, and is an example of an all too common kind of witlessness. Sort of like the old Alice Bailey number with the esoteric rulers of the signs. All the astrologer had to do was somehow figure out whether the "client" was sufficiently evolved to use the esoteric rulers.

- Ed

Too sad that this slush still permeates astrology. I've done my part of moderns bashing but it's mainly because of this issue and not necessarily inherent to it. Probably such doctrine has been of all ages and not limited to today's astrology.

I just looked over the fence at the 5? rule. Most of the confusion around it seems futile, because the characteristics of house systems is ignored. Take for instance Porphyry for a certain moment at a certain time with Asc 29? Sagittarius and MC in 26?, the intermediate cusps XI and XII will be at 27? and 28?. The 5? rule places the beginning of the house I at 24? Sagittarius covering all houses from MC and onwards. Not up to much as a rule huh? In the houses I have discribed, 'translated' to equalness, the 5? rule would perfectly apply as 1/6th of the house.

Eddy wrote:Too sad that this slush still permeates astrology. I've done my part of moderns bashing but it's mainly because of this issue and not necessarily inherent to it. Probably such doctrine has been of all ages and not limited to today's astrology.

I just looked over the fence at the 5? rule. Most of the confusion around it seems futile, because the characteristics of house systems is ignored. Take for instance Porphyry for a certain moment at a certain time with Asc 29? Sagittarius and MC in 26?, the intermediate cusps XI and XII will be at 27? and 28?. The 5? rule places the beginning of the house I at 24? Sagittarius covering all houses from MC and onwards. Not up to much as a rule huh? In the houses I have discribed, 'translated' to equalness, the 5? rule would perfectly apply as 1/6th of the house.

Not up to much as a rule if confined to the ecliptic, and not much of a rule based on the reasons presented to support it, which turned out to be pretty much what I asked not be presented.

- Ed

In the following series a Venus Mercury or Mercury Venus direction is discussed. In cases where the zodiacal frame is an essential part of the definition of a house system, Mercury was directed to the sextile (measured forwards along the ecliptic) of Venus. In these reference systems the direction of Venus to the sextile of Mercury (measured backwards along the ecliptic) were calculated too. In classical astrology these are known as direct and converse directions respectively.

For the mundane variants (in the modern sense, without any ecliptical reference involved), only the motion of Venus to the mundane sextile of Mercury was calculated.

Whole sign is easy to calculate. Mercury in III, Venus in II, Ascendant/ cusp I in Cancer. When Asc enters Leo, Venus in Leo is also in I. House/sign I is in sextile by sign to house/sign III. In 6?22? ?times?/years Asc enters Leo and the sextile by sign will be active till Virgo rises, i.e. during 42? 3? times/years. If we prefer to use the pre Ptolomeian arithmetical method of calculating risings of signs, then the period will last 40?33? years.

Equal Ascendant moves Mercury to the sextile of Venus with the same speed as the Ascendant?s motion. Note that Mercury doesn?t really move with the Ascendant but is treated as being attached to it. Therefore it?s also called Ascendant-arc method, like the Solar-arc method. 70?20?

Equal MC has the same, MC-arc method. Result: 47?12?.

Note that all systems which depend on ecliptic positions move the planets with the same speed as the reference frame or the combination of MC and Asc. So they are all in fact similar to the Solar arcs. As a comparison Solar arcs progression of Me to sextile of Venus takes 50?34? periods. Solar arcs are often seen as a watered down version of primary directions, on the other side these ecliptic based directions are likewise a complicated version of solar/MC/Asc arcs. None of them are close to ?real? events. Moving only the Sun, MC or Asc is closer to reality, however what is ?real? is also debatable because it depends on our frame through which we observe these events.

Porphyry places us for quite a problem for finding the position of a directed planet depends on one known and two unknown points. For example the question when planet A will be on cusp XI gives us the problem to find out when planet A will be on 1/3d of the distance MC-Asc. We therefore have to make use of an equation formula approaching the position with a desired accuracy.

The formula to find the position for direct direction of Me ? Ve+60?. As Ve+60? is just past the IC use this house quadrant to calculate the proportion. Use the distance of Venus to the IC expressed in house degrees as a basis. Divide this by 90?(the translated to Porphyry quadrant size) for the proportion. Subtract Venus? Porphyry degree position of the Mercury position. Result is the ?new? IC. Then calculate the accompanying Descendant with this MC. The distance IC-Desc multiplied by the proportion gives the new position. Repeat the procedure till the results don?t change anymore. The motion passed by the MC in ST gives the result.
Direct 45?48?
Converse 64?35?

Alchabitius uses the same equation method, but first the positions of the angles are converted to right ascension, like in the house system itself.
Direct, 45?20?
Converse, 69? 2?.

Koch directions are not very difficult. We saw that the planets? positions first had to be calculated according the period before or after they rose, culminated, set. Calculate the directed planet?s position for the same moment as it will do the same as the planet(?s aspect) did. 70?29?. An odd problem occurs when you think about what happens when a planet would have to move more than 180? to reach a directed position. In this cases which we usually don?t think about because of the short length of life it becomes apparent that one sometimes need to use the opposition point. This is because Koch houses are calculated for the MC?s diurnal arc and added 180? for the nocturnal cusps. This is a bit inconsequent. If the nocturnal arc cusps had to be calculated with the nocturnal arc, then the cusps wouldn?t be 180? from the ?opposite? cusps, like the usual cases. In this it resembles the APC system, discussed a few posts ago. APC was more consequent, Koch simply ?mirrors? the diurnal arc through the horizon. I assume that I?m not evolved enough to appreciate this system.

Meridian is the easiest. The distance between Venus and ME-60 along the equator is the number of years. The equatorial rotation is equal to the distance.
47?21?

The Campanus, Horizon and the Regiomontanus systems were a joy to calculate. Not too complicated and purely geometrical. For Campanus first look up the distance from Venus to Me-60? according to the house degrees. Use the great circle through this point. This point, its Campanus projection on the prime vertical and the East point (where horizon and prime vertical coincide form the triangle to be solved by Napier formulae. The angle where equator crosses the great circle (which increases up till 90? (hence right-ascension) at the meridian / IV or X circle) forms a new ?ascension?. Then calculate the moment when Venus will ?rise? on this great circle with the same formulae for calculating risings and settings. For Horizon, the same as Campanus applies but here the zenith-nadir great circles form the lines of ?ascension? or passing through. For Regiomontanus the basis of the equatorial element has to be used, as in the house degree calculations. The following in mundo results are found.

Campanus: 68?15?

Horizon: 63?39?

Regiomontanus: 48?50?

Placidus (mundane in the modern sense) finally is relatively easy since it depends on the same calculations needed for the house degree positions. The method of calculation (as well as the traditional direct/converse ecliptic based directions, which I also calculated) are found in the top post in: http://skyscript.co.uk/forums/viewtopic ... &start=195

Modern
Placidus: 44?49?

Traditional
Direct: 54?14?
Converse: 63?19?

While the difference between direct and (traditional) converse troubled me a lot, the solution is quite simple. The problem lays in the terminology. For a moment I believed that converse was really back in time but disguised as positive post birth time. However this isn?t the case. The problem is simply that the medieval people didn?t like to direct an aspect. If I may quote from the discussion on primary directions in the traditional board:

How often is it the case that aspects of promissors are directed to aspects of significators? Was this common historical practice?

Never, to my knowledge. An aspect point cannot be a significator.

So if the aspect point is directed, then change the name to promissor and call the motion converse. However this is rather confusing because all motion is direct whether it is an aspect point or a planet itself. For example holding a hand blender by the part that is put in the pudding or whatever will make the handle move when switched on. It looks awkward but it?s the same motion. Moreover this terminology swap won?t work for a 360 year period. Take for instance Jupiter conjunct Saturn on cusp XI, ignoring secondary motion Jupiter and Saturn return to the cusp after 360 one-degree-periods. So directly moved Jupiter and Saturn move to the Jupiter and Saturn point as significators and conversely moved to the Jupiter and Saturn point they are called promissor. So two terms for the same planet. I don?t say this to annoy traditionalists, I simply just got this insight what really happens. It, like all the previous calculations, provides an illustration of what we are doing. Knowing what you are doing is essential for forming a basis of astrology.

Eddy,

Thanks for putting the time into this, and for sharing it.

- Ed