16
For an individual and his/her transits it wouldn't be such a problem but perhaps it would for comparing individuals as in synastry, because we don't know which fixed date to choose. But I don't know how precession correction users cope with this.
Hi Eddy,

An excellent point! I hadn't considered this difficulty. I get the feeling though that most of the astrologers advocating precession correction are event orientated and looking for what they assume is greater accuracy in timing events. I suspect that synastry wouldn't be high on many of these astrologers lists.

Mark
As thou conversest with the heavens, so instruct and inform thy minde according to the image of Divinity William Lilly

17
This old comment from Isaac Starkman in regards Andrew Marr seems relevant here:

Isaac Starkman wrote:
In all his books and articles Alexander Marr never used the Sidereal Zodiac like Cyril Fagan or Brigadier Firebrance, the editor of Spica. Indeed, he took some of their techniques and applied them into the Tropical Zodiac, like Fagan's PSSR system. I quote from the foreword of Marr's book Prediction I:


''The discoveries of the eminent astrologer Cyril Fagan split the astrological world into three main camps: Tropical, Sidereal, and what the late Brigadier Firebrance termed "Little Bastard Zodiac", where precession is expunged from charts whilst maintaining the tropical zodiac?I hope to persuade Siderealists that transits to the radix should be applied in the tropical manner, that is, disregarding precession. However, in regard to return charts, both Tropical and Sidereal Solar Returns are valid."
I thought referring to precession correction with the term the 'Frankenstein Monster' of predictive techniques was possibly skating a bit close to the ice. However, this is quite bland compared to Brigadier Firebrace!

Mark
As thou conversest with the heavens, so instruct and inform thy minde according to the image of Divinity William Lilly

18
Mark wrote:Fundamentally, I would suggest the tropical zodiac is internally self consistent. It doesn't need 'topping up' for precession correction by reference to a completely different astronomical framework.
It doesn't. What I am saying is that a precession-corrected tropical framework is sidereal, not tropical, so I am not suggesting anything about "topping up" the tropical zodiac.
Precession correction is only required if you wish to work from a sidereal perspective and incorporate the effects of precession on the tropical zodiac. From a tropicalist perspective I am left wondering though why bother?
You shouldn't bother. Precession correction is used when you want (or need) to work in a sidereal reference frame. The question is where do you establish the starting point of the sidereal zodiac you are using. In precession-corrected tropical, that starting point is the moment of birth.
If you think non-precessed charts create distortions
I don't think that. In fact, I don't care, I simply work sidereally because I think it gives better results.
doesn't it make much more sense to work sidereally in the first place?
If you read my quotes carefully, you will see at the end of part 6 at least 3 possible scenarios where working with either could make sense. If you are working with only one radix and are comparing positions from different times to it (including synastries), then it makes more sense to establish the time of the reference radix as the starting point. The zero point in time --the radix-- corresponds to zero Aries, and precession is removed from all positions from then on.

In practice, this would be the same as "fixing" the tropical zodiac of the radix in sidereal space. Once you fix it, it's no longer tropical, so it is simple prejudice and misunderstanding to refer to this zodiac as half-tropical-half sidereal, because it is 100% sidereal. This zodiac differs from any other sidereal zodiacs only in its starting point.

The starting point, needs not be seen as the tropical zodiac of birth, it can be seen simply as the time of birth, exactly as astronomers use the equinox and ecliptic of J2000, B1950, etc. as fundamental epochs and proceed to represent the positions sidereally from them. The astronomy and defintions involved are exactly the same (the details are discussed in my article quoted).
this seems like an ugly hybrid of two totally different systems to me.
I have given you the reasons why this view (as well as Astrodienst's) is incorrect, the result of a misrepresentation of the astronomical facts.
I will therefore call it the Frankenstein Monster of predictive astrology.
If you want to get theatrical, I must say that the amount of prejudice and mis-representation of what precession-correction stands for is the real monster. (I include here many siderealists that misrepresent it).

I agree with you that to suggest the use of precession correction for solar and lunar returns but not use it for transits is an absurdity.

Juan

19
Juan wrote:
What I am saying is that a precession-corrected tropical framework is sidereal, not tropical, so I am not suggesting anything about "topping up" the tropical zodiac.
Its stands to reason that precession correction requires a sidereal reference point. However, there are clearly astrologers working fully within tropical astrology who advocate precession correction. For example Marc Penfield on solar returns or Robert Hand (circa 1976) on transits. They dont tend to present the issue the way you do.

Juan wrote:
In practice, this would be the same as "fixing" the tropical zodiac of the radix in sidereal space. Once you fix it, it's no longer tropical, so it is simple prejudice and misunderstanding to refer to this zodiac as half-tropical-half sidereal, because it is 100% sidereal. This zodiac differs from any other sidereal zodiacs only in its starting point.
A 'sidereal' chart that begins with a fiducial in the tropical zodiac? Am I the only one that sees an obvious contradiction there? Surely, if the chart was '100% sidereal' as you keep insisting it would have had a fiducial derived from a precession free zodiac in the first place!

Beyond this many tropical astrologers working with precession corrected charts wouldn't accept your idiosyncratic definition of terms. For example, if we examine precession corrected transits to a natal chart it can still be argued the radix is fully tropical not sidereal. The element that takes into account the sidereal framework is the calculation of the transits themselves. Nevertheless, the chart itself remains fully tropical.

That is why I and others see this approach as hybridized.

Mark
As thou conversest with the heavens, so instruct and inform thy minde according to the image of Divinity William Lilly

20
Mark wrote:A 'sidereal' chart that begins with a fiducial in the tropical zodiac? Am I the only one that sees an obvious contradiction there? Surely, if the chart was '100% sidereal' as you keep insisting it would have had a fiducial derived from a precession free zodiac in the first place!
The radix is tropical, defined by the position of the equinox at birth. There is no question on this. But the dynamical model used to calculate the passage of time away from this radix is 100% sidereal, including all return charts.

What I see here is the assumption (or prejudice) that a point in time -or the position of the equinox at this point in time-- cannot be a sidereal fiducial. This assumption is historically unwarranted. I suggest you study Raymond Mercier's hypothesis about the sidereal zero point of the Toledan Tables, which is also that of the "Tables of the Shah" used by Masha'Allah and Abu'Mashar, which --apparently-- was established this way. You find the reference in the documentation to the Swiss Ephemeris, and I wrote a summary of his explanation in a post dated April 16 2003 that you can find here:

http://www.expreso.co.cr/centaurs/posts ... ghdad.html

(please see the next post)

If a respected historian of astronomy suggests that the zero point of the Sasanian and Toledan astronomical canon was established by the position of the equinox at __the time__ of an important cultural/astronomical event in the 6th Century, and if astronomers all over the world today establish their standard sidereal zero point by means of the position of the equinox at a the time of the epoch, and if I see the mathematical and astrological logic and beauty of it, and if I see that those who oppose it are ignoring all of this...

... you know where I'm getting at.
Beyond this many tropical astrologers working with precession corrected charts wouldn't accept your idiosyncratic definition of terms.
It is not about "my" definitions but about astronomical facts. Please tell me where I am wrong astronomically, I will re-send your message to JPL and the Naval Observatory... or accept that I was wrong!
The element that takes into account the sidereal framework is the calculation of the transits themselves. Nevertheless, the chart itself remains fully tropical. That is why I and others see this approach as hybridised.
Precession-correction, as you are pointing out, is about dynamics, not about looking at a static zero point, and this dynamical model is fully sidereal, so...

... the fact that it is historically and astronomically consistent to define a sidereal zero point by means of a tropical marker puts in evidence how unrealistic and sterile (and boring and repetitive and dogmatic...) is this either/or separation between tropical and sidereal.

Juan
Last edited by Juan on Mon Feb 21, 2011 1:23 am, edited 2 times in total.

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Here is my summary. The reference and link are in the previous post.

Date: Wed, 16 Apr 2003 10:10:39 -0600

Mercier's article "Studies in the medieval concept of precession" is quite large (65 pages) and complex. I will try to summarize what is relevant and more interesting to this discussion, in an effort to understand it myself.

The analysis begins with the Toledan Tables (11th Century). Mercier points out that these tables are based on the sidereal year, unlike the tables of Ptolemy. The use of the sidereal year had its antecedents in the Kwarizmian tables (9th Century) and the Babylonian methods of the Indian astrological/astronomical tradition.

Since the longitude of the Sun in the Toledan tables is calculated sidereally, then it follows that its position will correspond to the tropical position of the Sun with respect to the equinox of one specific date, and of only that date. In all the tables in the Indian tradition to which the Toledan Tables apparently belong, this date is found somewhere between A.D. 450-650. The zero point of this tradition agrees with the end point of the lunar mansion called Batn al-hut in Arabic, Revati in Sanskrit, and K'uei in Chinese, determined by the junction stars: beta Andromedae, zeta Piscium, and eta Andromedae.

There is a record of one Guillaume de St Cloud, a leading figure in Astronomy by the end of the 13th Century, who proceeded to calculate the motion of the 8th sphere in order to compare astronomical tables known at his time. For this, he observed the exact time of the Spring equinox in the year 1290, and deduced that the 8th sphere had moved 10;13. There is no record of how he did all this. The time of the equinox he observed according to Mercier was 16h after noon of Sunday 12th, March 1290 (julian date 2192301.6 "ignoring the correction due to the difference in longitude between Greenwich and Paris").

Mercier proceeds to calculate the true longitude of the Sun at that date according to the Toledan Tables, and he obtains 19,47'59" Pisces, i.e., 10,12'01" from the true tropical position obtained by Guillaume. This "ayanamsa" (Mercier does not use this term), compared to Guillaume's figure of 10,13' shows that he was calculating his result using the Toledan Tables.

Using Tuckerman's tables, Mercier calculates the correct time of the equinox in 1290 as 17h50m after noon of 12th March. This would make the ayanamsa 10,09'. Rough calculation shows that the zero point of the Toledan Tables in somewhere in the 6th century. Mercier tries to make a more precise calculation by comparing the mean motion of the Sun according to the Toledan Tables (sidereal) and the modern tropical mean motion, obtaining a difference of 50.616" per year, and this allows him to put the zero point in A.D. 563/564.

Then he examines the Tables of al-Kwarizm, also based on the sidereal year. He calculates the "movement of the 8th sphere" according to the tables using the same procedure (i.e., calculating the deviation from zero of the true longitude of the Sun according to the tables at the times of the Spring equinox). He obtains the following:

A.D. Kwarizmian Toledan
560 -0,02'10" -0,02'41"
561 -0,01'15" -0,01'54"
562 -0,00'21" -0,01'07"
563 0,00'33" -0,00'19"
564 0,01'27" 0,00'28"

Mercier shows that this zero point is only 0,10' East of zeta Piscium, and notes that although this star was not necessarily the origin of longitudes in these tables, it was indeed used as such in the Indian tradition of the Surya Siddhanta, written around A.D. 600. He reasons (p.206) that "the authors of both the Kwarismian and Toledan tables must have had the date 562/3 explicitly in mind, for in no other way could it have been arranged that the two solar Tables [--which differ in everyone of the 4 solar parameters--] could agree."

The question is how did they arrive at that date?

He mentions a study made by Billard of the Indian material, where the result represents "not the exact date at which any particular deviation vanishes, but rather the date at which the sum of squares of selected deviations is minimum". In the case of the Brahmasphutasiddhanta (A.D.628), Billiard concludes that A.D. 562.8 "might be regarded as the date which represents overall the origin of the sidereal coordinates". This tends to confirm what has been suggested by several scholars that the Kwarizmian Tables descend in part from the Brahmasphutasiddhanta, if one assumes that this is the system known to the Arabs as the Sindhind.

But Mercier shows that an exact calculation using the Brahmasphutasiddhanta would put the zero point in A.D. 580, making it clear that the position of the Sun in the Kwarizmian Tables is not derived from it. Another explanation must be sought for the A.D.562/3 date, and this explanation is found in the Sassanian Zij al-Shah, known as the Tables of the Shah, associated with the reforms of Khusrau Anushirwan (reigned 431-579). A close examination of an account by al-Biruni in his "India" (Qanun'l-Mas'udi) of the reform of the Zij al-Shahriyaran in A.D. 556 ordered by Khusrau permits to reconstruct the rationale of this date, and the key is the role played by the planet Jupiter.

The passage of Biruni refers to the doctrine of the "Thousands":
In their opinion the seven planets and the two nodes follow one another with a constant number of years for each; we call this number fardariyat. They agree that in the 25th year of Anushirvan, 4 years of the Fardariyat of Jupiter had elapsed, and there remained 8. Then the fardariyat of Mercury, 13 years; then Saturn, 11; then the Tail, 2; then Mars, 7; Venus, 8 and the Sun for 10 and the Moon 9 and the Head 3, the again Jupiter around the Zodiac in relation to its exaltations, during the space of 75 years.
We mentioned the above date (25th year of Anushirvan) because at that time the astronomers of Persia met to correct the Zij al Shahriyaran known as al-Shah, and they recorded the measures of times and stations. They knew from the position of Jupiter at the end of the month Aban that 3 hazar had passed and of the 4th, 851 years. This if we discard 75-year cycles 51 times, there remain 26 years, which begins with the fardariyat of the Sun and ends with the first 4 years of the fardariyat of Jupiter. From this point to the beginning of the reign of Yazdagird there are 76 years: 23 of Anishirwan, 12 of Hormuz, 37 of Parviz, 4 of Shiruya. Thus at the beginning of the reign of Yazagird 5 years had elapsed of the fardariyat of Jupiter.
Mercier notes:

- 25th year of Anushirwan = A.D. 556
- the fardariyat of Jupiter ran from 552 to 564.4
- of all these fardariyat, only that of Jupiter has any astronomical significance, since we have then the sidereal period.
- end of Aban = early March
- 3851 = 51 x 75 + 26
- Khusrau Anushirwan = 531-579; Hormuz IV = 579; Khusrau Parviz = 591-628; Shiruya (and others) = 628-632; Yezdagird = 632-651

I quote Mercier now:
When the planet was observed in Aban 556 it was visible during much of the night, but at the termini of the fardariyat, in 552 and 564, it was in conjunction with the Sun almost exactly at the moment of the Spring Equinox. Nowhere else in the sequence of fardariyat of Jupiter or the other "stars" is there any clear astronomical phenomenon at the termini. (page 208)

The outstanding astronomical event at that time is the conjunction of Jupiter with the Sun at the time of the Spring Equinox of 564, and I had already argued that the event was crucial in fixing the parameters of the [Kwarizmian] Tables. This conjecture is now fully justified by the passage in the Qanun'l-Mas'udi, where we saw that observations of Jupiter were carried out under Khusrau Anushirwan in 556 by way of fixing a cycle of Jupiter for the interval 552-564...
Observations of Jupiter play a major role in ancient and medieval calendars, and seem to have served to link observations of the day and night sky, so important obviously in the phenomenon of precession...
Kugler in his study of Babylonian tablets of Jupiter's motion found that the planet was used as a "calibration marker" in a sense, regulating the lunar calendar in Babylonian astronomy... (page, 43 of part II)
There are many other details (references and ancient primary sources) that Mercier gives to strengthen his explanation based on the role of Jupiter, but I don't have the time to put them all. They all refer to the same idea

Juan

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Martin wrote:Regardless of one's personal stance on the underlying philosophy of astrology, it is a matter of historical fact that the earliest definitions of the zodiac were sidereal. In other words, the people who invented the 'ecliptic reference frame' (which obviously made sense to them) were not tropicalists. Indeed, the tropical zodiac doesn't seem to have caught on in earnest until several centuries after Ptolemy.
Hi Martin

This appears to disagree with my understanding of the history of the zodiac, but maybe I have misunderstood your point (?). Before the definition of the ecliptic there were various systems of time-related and star-related measurement, but these were not using the ecliptic (obviously, because this had not yet been defined), and so were not utilising celestial longitude as the primary means of planetary measurement as modern sidereal astrologers do.

For example, in early Mesopotamian astrology, the early lists of constellations which held the planetary movements were based upon those which formed the backdrop to the Moon?s movement, not the Sun?s (which can?t be observed against the stars and so has to be calculated mathematically). The Moon?s path is close to the Sun?s but it is not the same because it has a 5? inclination to the ecliptic (approx), so the technological breakthrough came when astrologers were able to define the ecliptic and the seasonal pivots precisely in the creation of the tropical zodiac, and then use that as a mathematical frame of reference instead of the earlier references which depended upon observed movement of the planets against the stars or the equatorial grid. Therefore, in its use of ecliptic-based longitude, a modern sidereal zodiac makes use of the system of measurement that the tropical zodiac provided and it developed as a variant of the tropical zodiac (not the other way around).

The tropical zodiac had its most important period of development between the mid 8th - mid 6th centuries BC (the mid-6th century being when it is first seen to have been put to full use in records that remain available to us). After this there was a lengthy period of overlap whilst some astronomers continued to record planetary positions by observable references to constellations, others adopted the mathematical reference of the tropical zodiac, and some used a combination of both. But ecliptic based astronomy provided a rapid development of knowledge of the planetary cycles, so this was increasingly used as the astronomical default. Ptolemy didn?t make any personal development of the system or set a new definition that the zodiac began its zero point at the spring equinox; he was only reporting what was already the well established principle of his day.

There may be a point of confusion over the use of the term zodiac. As I understand it, the term is specifically related to ecliptic-based division (which is itself intrinsically tropical); and would not, for example, relate to measurements by right ascension or constellation groupings, lunar mansions, etc. Maybe there is a reason to question that (I am researching this so I welcome other views); but generally the zodiac is historically defined as the belt of celestial space which is centred upon the ecliptic and extends 8-9? either side of it, therefore covering the area which holds the movement of all the visible planets. Am genuinely interested in early references that might show the use the term 'zodiac' for measurement which was not ecliptic-based, in case I've been making a definition which is too narrow to be precise.

Best wishes
Deb

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Hi Deb,

I have only got time for a brief reply just now, so I'll cut straight to the point:

Arguably, applying either label (tropical/sidereal) to a context where there was no knowledge of precession, and the positions of the constellations were therefore thought to be fixed in relation to the seasons, is anachronistic. But in the common understanding of the term 'tropical zodiac', the vernal equinoctial point defines 0? Aries. To the best of my knowledge, Ptolemy was the first astrological author to advocate this definition. It was certainly not present in Mesopotamia in the 5th century BCE (which, if memory serves, is the dating Rochberg gives for the earliest use of a zodiac consisting of 12 equally-sized signs).

(Later addition, just before I run out the door: There is nothing intrinsically tropical about the ecliptic, which is just the Sun's apparent path against the background of the stars. It is the combination of the equator and the ecliptic that gives us the tropical (equinoctial and solstitial) points.)

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Hi Martin
You wrote:? in the common understanding of the term 'tropical zodiac', the vernal equinoctial point defines 0? Aries. To the best of my knowledge, Ptolemy was the first astrological author to advocate this definition. It was certainly not present in Mesopotamia in the 5th century BCE (which, if memory serves, is the dating Rochberg gives for the earliest use of a zodiac consisting of 12 equally-sized signs).
I think astrologers are in a good position now to clarify that the zodiac was developed in order to provide ecliptic-based measurement that aligned to the Sun?s seasonal pivots. To keep this brief I will just summarise a few points:

? The earlier stage of planetary measurement in Mesopotamia (pre- 8th century BC) used the definition of 17 constellations which contained the Moon?s circuit ? this is usually referred to as the ?forerunner to the zodiac? although the logic of the arrangement is distinct from that of the zodiac which identifies the Sun?s path instead of the Moon?s path (the ecliptic being designed to move away from this).
- We know that in the mid 8th century BC, the astronomers of the Assyrian King Nabonassar were moving towards the development of the zodiac by keeping track of the Sun?s movement along its ?quarters? (using knowledge gleaned from eclipses/synodic conjunctions, etc).
- Also, that the earliest Babylonian associations (built into triplicity relationships, for example) associate zodiac signs with calendar months (showing a seasonal/calendrical basis to the earliest use of the zodiac signs)
-According to 3rd century reports, the Greek astronomer/philosopher Thales (650-545 BC) was able to teach ?knowledge of the Sun?s path from one end of the ecliptic to the other? (Diogenes Laertius, Lives and Opinions of Eminent Philosophers; I.I-IX.). It?s likely that Thales was promoting to the Greek world developments which were occurring in Babylonia. Tablets have been dated to around this time which show the zodiac in use as a means to establish planetary position.

Ptolemy wasn?t the first author to advocate the definition of the zodiac as commencing at the vernal point; he was following the convention of earlier Greek astronomers. We find a clear definition in the Phaenomena of Geminos, for example, the compilation of which we possess having been dated by the text translators (Evans and Berggren), to between 90-35 BC, so predating Ptolemy?s text by two centuries.

To summarise, Geminos tells us (1.3) that the word ?sign? is used two ways: the first refers to the 12th parts of the zodiac circle (these being mathematically derived, making the signs equal in size), the other referring to constellation images and position of the stars which are ?not equal in size; nor are they formed from equal number of stars; nor do they all exactly fill the spaces assigned to the twelfth-parts.? 1.4-5). He also tells us that it is the first definition, of the 12th parts, which is divided into degrees (1.6); and that this is defined by the period that the Sun completes its annual circuit (1.7) by which the first degree of Aries identifies the spring equinox (1.9) ...and so on the other signs and quarters.

Tropical zodiac division is clearly seen to be the norm in Geminos? text and the late Babylonian and early Greek astrological texts that are available to us, and this is why I felt the need to comment after your earlier remark ?Indeed, the tropical zodiac doesn't seem to have caught on in earnest until several centuries after Ptolemy?. I also felt it was misleading to suggest it is a historical fact that ?the earliest definitions of the zodiac were sidereal?. I?m not quite sure what you mean by that but it?s clearly open to misinterpretation since the earliest definitions of the zodiac were intended to be mathematical and based on identification of the unobservable solar path.

This is not to say that the zodiac cannot accomodate different perspectives very well; I'm just trying to clarify the historical position.

Regards
Deb
Last edited by Deb on Mon Feb 21, 2011 4:23 pm, edited 1 time in total.

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Martin Gansten wrote:There is nothing intrinsically tropical about the ecliptic, which is just the Sun's apparent path against the background of the stars. It is the combination of the equator and the ecliptic that gives us the tropical (equinoctial and solstitial) points.)
Right. All that is needed to accurately define or plot the ecliptic in the sky is the observation and recording of lunar eclipses, as the Babylonians actually did, so the ecliptic was defined by them on an observational basis with reference to the background stars.

Juan

26
A final comment, so as not to totally derail the thread (sorry Mark):

Martin in reference to your addition ?There is nothing intrinsically tropical about the ecliptic, which is just the Sun's apparent path against the background of the stars. It is the combination of the equator and the ecliptic that gives us the tropical (equinoctial and solistial) points?

I don?t disagree with your definition of how we obtain the seasonal points, but I would argue that knowledge of the ecliptic was purposefully sought in order to define the Sun?s (apparent) annual track, and to purposefully seek alignment with the equinoxes and solstices. Hence the historical information does show that the original intention was to define the tropical zodiac, although once defined, this can be used as a circle for other perspectives of measurements too. We just need to clarify that the tropical zodiac is not a misconceived add-on to a pre-existing tool that was designed for another purpose; but was the original motivation behind the development of ecliptic measurement which creates degrees of longitude.

It?s of relevance that the oldest references indicate that the original alignment between signs and constellations was closer to the middle of the constellations than their extremities; and that once the zodiac was determined the problem of precession was originally dealt with by periodic updates which moved its commencement point backwards through an initially fixed frame of reference which can be likened to the modern sidereal zodiac (so the updates periodically describe the *beginning* of the zodiac being at 15 Aries, then 12 Aries, then 8 Aries as described by Manilius).

These updates show attempts to keep the start of the zodiac aligned with the equinoxes, so that the 12th-parts moved backwards against the original plot of the ecliptic line. It?s obviously not practical to keep moving the start of what is supposed to be a standard frame of reference, especially once it reached a point where the sign symbolism would slip (sign-symbolism being so heavily weighted towards seasonal influences). This is why I believe (although I am still trying to research this and so can only speak from a personal view) that Hipparchus didn?t so much ?discover? precession as take a position on how to ?fix? the problem in Greek astronomy by establishing that the 12th-parts should begin with their own zero point wherever this fell against the zodiac as initially determined by the Babylonians. All of the references to a slipping commencement point show us that the zodiac being talked about was a tropical one.

27
A final comment, so as not to totally derail the thread (sorry Mark):
Hi Deb,

I think underlying this topic is the point I have been trying to make about the validity of the tropical zodiac as a self consistent system. So I do think your comments on the historical antecedents of the tropical zodiac are most useful to inform our discussion. I hadn't reallly expected that most of the people responding, here that work with precesssion corrected tropical returns and transits, would be essentially sidereal in outlook. :shock:

Mark
As thou conversest with the heavens, so instruct and inform thy minde according to the image of Divinity William Lilly