Venus Pages in the Dresden Codex

This is a series of letters between me (Greg Reddick) and Lloyd Anderson that took place in February and March of 2000 discussing my analysis and discoveries in the Venus Pages of the Dresden Codex.

Edited notes not in the original email are supplied in Green Text.

From: Gregory Reddick
Sent: Wednesday, February 2, 2000 5:24 PM
To: Lloyd Anderson
Subject: The 1.5.14.4.0 number on page 24 of the Venus Pages

Lloyd-

Two days ago, I discovered how the 1.5.14.4.0 number on page 24 of Dresden works. Before I go raving about my discovery, I just want to check to make sure that you haven't heard of anyone else coming up with a plausible explanation of this number (and, yes, Thompson was wrong.) There is no question in my mind how this works: it's simple, elegant, and effective. It also conclusively proves that they were using the 584285 correlation constant in Dresden. Assuming that I'm not rediscovering the wheel, I'm writing a paper and I'll make a presentation at the Maya Meetings. I can explain in more detail in a follow-up.

Incidentally, having re-examined the 1.5.5.0 number, I don't think I can definitively prove one way or the other whether it was used the way that I explained last year. It's reasonable and works, but I can't say that's how the Maya used it. Unless I or someone else comes up with a better explanation of the number, it will have to stand as a possible way that the number is used.

Greg Reddick

From: Lloyd Anderson
Sent: Monday, February 14, 2000 10:03 PM
To: Gregory Reddick
Subject: Re: The 1.5.14.4.0 number on page 24 of the Venus Pages

Greg:

Glad to hear you are alive and well. Just got back yesterday from trip to Galapagos and highland Ecuador. Please send the more detailed explanation, so I have some content to go from, and I'll let you know whether it sounds familiar.

Lloyd

From: Gregory Reddick
Sent: Monday, February 14, 2000 11:49 PM
To: Lloyd Anderson
Subject: RE: The 1.5.14.4.0 number on page 24 of the Venus Pages

Lloyd-

If you take the long count 9.9.9.16.0 1 Ahaw 18 K'ayab explicitly arrived at from the ring number of page 24 and simply add 1.5.14.4.0, you arrive at 10.15.4.2.0 1 Ahaw 18 Wo, which just so happens to be a heliacal rising of Venus in the morning sky. It's the only 1 Ahaw 18 Wo that results in a helical rising of Venus in the morning between 8.5.0.0.0 and 12.12.0.0.0 (the range I ran the test for) using the 584285 constant. It explains why the 1 Ahaw 18 Wo so pointed comes right after the 9.9.9.16.0 long count date on page 24, when this obviously isn't the calendar round that goes with that long count. So then besides the 10.15.4.2.0 run of the tables, the tables are run twice forward and backwards from this date, using the Teeple correction of 9.11.7.0 minus 4.12.8.0 every time. So arrange in sequence, the five runs of the table are:

10.5.6.4.0 1 Ahaw 18 K'ayab + 9.11.7.0 - 4.12.8.0 =
10.10.5.3.0 1 Ahaw 8 Yax + 9.11.7.0 - 4.12.8.0 =
10.15.4.2.0 1 Ahaw 18 Wo + 9.11.7.0 - 4.12.8.0 =
11.0.3.1.0 1 Ahaw 13 Mak + 9.11.7.0 - 4.12.8.0 =
11.5.2.0.0 1 Ahaw 3 Xul

The tables only list the 1 Ahaw 18 K'ayab, 1 Ahaw 13 Mak, and 1 Ahaw 3 Xul runs of the table, leaving the other two implied.

All sorts of consequences fall from this conclusion for how the tables work. But Thompson didn't see this explanation in his 1972 commentary, and Schele didn't mention this in the 1997 workbook. It seems so obvious when I look at it that I'm finding it hard to believe that no one noticed it before. Most people (including Thompson) seem to be using the sequence:

1 Ahaw 18 K'ayab + 4.12.8.0 =
1 Ahaw 18 Wo + 9.11.7.0 - 4.12.8.0 =
1 Ahaw 13 Mak + 9.11.7.0 - 4.12.8.0 =
1 Ahaw 3 Xul

which doesn't work against the astronomical data in any pattern of long counts; you have to fit the 1 Ahaw 8 Yax run in.

I then tested all correlation constants that have been proposed that I'm aware of. The only one that matches the data is 584285. The 584283 only give a 3 degree elongation. No others are even close.

BTW, another interesting "coincidence": 10.10.5.3.0 1 Ahaw 8 Yax using the 584285 constant is the helical rising of Venus immediately after a transit of Venus five days before. Eight years later, on the second line of the table, it happened again. The astronomy programs show that the transits would have been clearly visible from the Maya region. Not proof, but I don't see how they would have missed the transits when they certainly would have been looking for Venus to start the next run of the tables.

Greg

From: Lloyd Anderson
Sent: Tuesday, February 15, 2000 6:19 AM
To: Gregory Reddick
Subject: Re: RE: The 1.5.14.4.0 number on page 24 of the Venus Pages

Greg:

From your discussion, and what you do not say, it seems you are not using Floyd Lounsbury's treatments of this? Better check them before you proceed.

Do you need me to send you a copy?

Floyd calculated exactly when one would need to make which adjustments, when Venus would get far enough out of line with the simple corrected table (foreshortened) that a different correction would be needed, and showed this with graphs of Venus actual vs. Venus predicted, a zig-zag line that gradually gets more out-of-whack until something is done about it.

That's a quick response, not even with the title of Floyd's article. Floyd did refer to 1 Ahau 18 Uo, I think in the same way you do. That doesn't mean that your searching for all occurrences within your time range that fit your criteria may not be a confirmation of his. I haven't looked at his article again to see whether he claimed to have done just that, or whether what you say and what I remember from him differ in any way. If you've done this yourself without knowing of his work, you deserve a prize. He believes he solved that one, but was stumped by the 1.5.5.0 number though he attempted to deal with the 1.5.5.0 again in a later article, I think in Aveni's edited "Sky in Mayan Literature" with the green slip cover.

Tell me if you need the Lounsbury article I was referring to.

Best wishes,
Lloyd

From: Gregory Reddick
Sent: Tuesday, February 15, 2000 6:47 AM
To: Lloyd Anderson
Subject: RE: RE: The 1.5.14.4.0 number on page 24 of the Venus Pages

Lloyd-

I have Floyd's 1.5.5.0 article (BTW, where was it published? I only have it from the Kinko's collection). I don't have the other one. If you need to fax it, my fax number is [Phone number suppressed]. Incidentally, if you have Teeple's original article on the corrections, I'd like to read that, too.

Thanks,

Greg

From: Gregory Reddick
Sent: Tuesday, February 15, 2000 6:56 AM
To: Lloyd Anderson
Subject: RE: RE: The 1.5.14.4.0 number on page 24 of the Venus Pages

Lloyd-

Paste this into Excel or something to see my analysis of how the Maya did their corrections. It seems they are within bounds for a very long time if they do a foreshortening every However, the tables do not have the extra foreshortening between any of five runs.

[data formatted into a table for the web]

Greg

From: Lloyd Anderson
Sent: Tuesday, February 15, 2000 9:36 AM
To: Gregory Reddick
Subject: Re: RE: RE: The 1.5.14.4.0 number on page 24 of the Venus Pages

Much too long to Fax, lounsbury is.
Give me your postal address, thx, lloyd

From: Gregory Reddick
Sent: Tuesday, February 15, 2000 6:23 PM
To: Lloyd Anderson
Subject: RE: RE: RE: The 1.5.14.4.0 number on page 24 of the Venus Pages

Lloyd-

Greg Reddick
[Address Suppressed]

Thanks,

Greg

From: Gregory Reddick
Sent: Tuesday, February 15, 2000 9:23 PM
To: Lloyd Anderson
Subject: Oops, I do have Lounsbury's article

Lloyd-

It struck me that I might already have it. I dug through a binder of things I had taken from the Kinko's archive years ago and found Lounsbury's "Maya Numeration, Computation, and Calendrical Astronomy." So thanks anyway. I'm giving it a read now and will tell you what I think.

Greg

From: Gregory Reddick
Sent: Tuesday, February 15, 2000 11:50 PM
To: Lloyd Anderson
Subject: The 1.5.14.4.0 number on page 24 of the Venus Pages

Lloyd-

I've read Lounsbury's articles. I think my paper (about 3/4th done) mostly affirms what Floyd said. However, I make two assertions that Floyd (and everyone else I've read) seems to have missed:

1) While he did point out in the paper that there is a connection between a 1 Ahaw 18 K'ayab and the 1 Ahaw 18 Wo through the 1.5.14.4.0 number, he did not point out the connection between the particular 9.9.9.16.0 1 Ahaw 18 K'ayab long count and the 1 Ahaw 18 Wo that reaches a helical rising. He says "The last of the numbers of this tier, 1.5.14.4.0, if applied to 1 Ahau 18 Kayab, leads again to a day 1 Ahau 18 Uo; but it is one that is eight calendar rounds later (1.1.1.14.0) that that attained by 4.12.8.0. The accumulation of error in eight calendar rounds, or four unabridged runs through the entire table, is 20.8 days. It is not likely, then, that two such days separated from each other by eight calendar rounds, were both intended to be designated as Venus epochs." (Maya Numeration, Computation, and Calendrical Astronomy, page 786)

This is entirely correct. However, he didn't reach the conclusion that I did from this. The 9.9.9.16.0 1 Ahaw 18 K'ayab is not a Venus epoch. But if you add the 1.5.14.4.0, it DOES reach a Venus epoch. It's all there, but as the paragraph cited above continues, it makes it obvious that he missed it.

2) In the numeration article, he makes the first listed run of the table be 10.10.11.12.0 1 Ahaw 18 Kayab (page 787). This is not possible because the elongation of Venus will be over 15 degrees on that date. In the later 1.5.5.0 article, he correctly picks up that the 1 Ahaw 18 K'ayab run of the tables must match the long count 10.5.6.4.0, but then on page 5 goes on to say that the 4.12.8.0 was applied to this to reach the 1 Ahaw 18 Wo. This cannot be correct if the original long count is 10.5.6.4.0, because the long count reached (10.9.18.12.0) would have a Venus elongation of 6.3 degrees in the EVENING sky. This double foreshortening is too much, and a single foreshortening is called for. The single foreshortening reaches the date 10.10.5.3.0 1 Ahaw 8 Yax which works remarkably well (6.26 elongation in the MORNING sky). Then another single foreshortening reaches the aforementioned 10.15.4.2.0 1 Ahaw 18 Wo.

After applying these corrections to his articles, everything else he says fits. My interpretation locks the Venus tables into history, exactly matching the astronomy, and the 1.5.14.4.0 number is the smoking gun. In the 1997 Maya Workshop workbook, Linda also seems to be touching all the points, but misses the conclusion. On page 166, she actually lists exactly correct four of the five long counts that I claim are the epochs of the table (except that she lists them as Julian dates). However, for some reason she lists another 1 Ahaw 18 Kayab in the second entry, which points out that she missed the sequence here, too. She does not list the 10.10.5.3.0 1 Ahaw 8 Yax run of the table.

So it seems that everyone had all the pieces, but nobody seems to have fit them all together to finish the puzzle. The sequence that fulfills all the requirements is:

10.5.6.4.0 1 Ahaw 18 K'ayab (5.3 degree elongation in morning sky)
10.10.5.3.0 1 Ahaw 8 Yax (6.2 degree elongation in morning sky)
10.15.4.2.0 1 Ahaw 18 Wo (7.4 degree elongation in morning sky)
11.0.3.1.0 1 Ahaw 13 Mak (9.7 degree elongation in morning sky)
11.5.2.0.0 1 Ahaw 3 Xul (10.0 degree elongation in morning sky)

And the 10.15.4.2.0 1 Ahaw 18 Wo can be linked to the page 24 long count through the 1.5.14.4.0 distance number.

The one thing about the sequence above is that there are no double foreshortenings between the epochs. The double foreshortening would have come at the end of the 1 Ahaw 3 Xul run of the table. This means that we cannot 100% prove that the Maya realized that they had to do double foreshortenings. I think a preponderance of the evidence suggests that they used it, but not beyond a reasonable doubt.

And as another note, I point out that 10.10.5.3.0 1 Ahaw 8 Yax comes immediately after a Venus transit, and that another one happened exactly 8 years later.

[I went back and read a couple of the things I've sent you in the last couple of days. Sorry if I can't seem to finish sentences or get my verbs to agree with my subjects (which must be particularly annoying to a linguist)! I'm teaching computer programming for eight hours a day right now and the schedule is exhausting--for some reason my English suffers.]

Greg

From: Lloyd Anderson
Sent: Wednesday, February 16, 2000 8:44 AM
To: Gregory Reddick
Subject: 1.5.14.4.0 -- What Lounsbury understood

Greg:

Wow, that was fast reading. (By the way, I completely missed, perhaps as trivial, the fact that your teaching computer programming was leading to a new dialect of English. Not completely surprising, though.)

I'm glad you were *unaware* of Lounsbury's treatment until you did your own, because you *may* be more likely to discover other things he missed. Without reading all of this immediately (and not having it all fresh in my head) I cannot say. However, let me act as devil's advocate, responding now *only* to your words, not to re-reading Lounsbury. (By the way, have you also read his later article in Aveni editor "The Sky in Mayan Literature", the one which precedes the article on 1.5.5.0? That might clarify some things. I do have that one handy, and can send it.

In each of the following sections marked off by ***, I quote you in the first half, then after the single * my own comments.

***

You write:

"1) While he did point out in the paper that there is a connection between a 1 Ahaw 18 K'ayab and the 1 Ahaw 18 Wo through the 1.5.14.4.0 number, he did not point out the connection between the particular 9.9.9.16.0 1 Ahaw 18 K'ayab long count and the 1 Ahaw 18 Wo that reaches a helical rising. He says "The last of the numbers of this tier, 1.5.14.4.0, if applied to 1 Ahau 18 Kayab, leads again to a day 1 Ahau 18 Uo; but it is one that is eight calendar rounds later (1.1.1.14.0)
that that attained by 4.12.8.0. The accumulation of error in eight calendar rounds, or four unabridged runs through the entire table, is 20.8 days. It is not likely, then, that two such days separated from each other by eight calendar rounds, were both intended to be designated as Venus epochs." (Maya Numeration, Computation, and Calendrical Astronomy, page 786)

"This is entirely correct. However, he didn't reach the conclusion that I did from this. The 9.9.9.16.0 1 Ahaw 18 K'ayab is not a Venus epoch. But if you add the 1.5.14.4.0, it DOES reach a Venus epoch. It's all there, but as the paragraph cited above continues, it makes it obvious that he missed it."

*

I would believe the opposite. Please do not misunderstand me, but people always (myself included) want to believe we have found something new. I have many reasons to think Floyd saw what you say he did not, or at least part of it. He was very careful. My past reading of the paper (in which he gave the zig-zag line showing how much they were off at each stage, in his view, I think that is the paper you just read) led me to understand that of course some of the theoretical stations were not Venus stations in actuality, because the system had slipped too much. Floyd was very careful about that. I believe I always understood that the station *from which* such an adjustment number led was not likely to be good astronomically, that was one reason why an adjustment would be used. It might merely be some station which tradition had maintained, but which the Mayans knew had to be corrected, and they just had not undertaken a public calendar reform. Note the great resistance to the shift from Julian to Gregorian, and how long it took, and the fact that it did not occur everywhere at the same time. So I think Floyd *did* intend that the earlier date linked by that 1.5.14.4.0 DN was likely to *not* be good astronomy, and the later date reached by that DN *was* likely to be an astronomically good one. My vague memory, colored of course now by your hypothesis, so not reliable. You may find quotes to prove that Floyd really did miss this. But at the moment, using only your exact words so far, I doubt it.

***

"2) In the numeration article, he makes the first listed run of the table be 10.10.11.12.0 1 Ahaw 18 Kayab (page 787). This is not possible because the elongation of Venus will be over 15 degrees on that date. "

*

I don't think that follows merely from the logic of your sentence above. The first listed run does not have to be astronomically accurate, as explained above.

***

"In the later 1.5.5.0 article, he correctly picks up that the 1 Ahaw 18 K'ayab run of the tables must match the long count 10.5.6.4.0, ..."

*

Was he merely more explicit then, or did he change his mind?

***

"but then on page 5 goes on to say that the 4.12.8.0 was applied to this to reach the 1 Ahaw 18 Wo. This cannot be correct if the original long count is 10.5.6.4.0, because the long count reached (10.9.18.12.0) would have a Venus elongation of 6.3 degrees in the EVENING sky. This double foreshortening is too much, and a single foreshortening is called for. The single foreshortening reaches the date 10.10.5.3.0 1 Ahaw 8 Yax which works remarkably well (6.26 elongation in the MORNING sky). Then another single foreshortening reaches the aforementioned 10.15.4.2.0 1 Ahaw 18 Wo."

*

Most of the above sounds good, and may support your view as I understand it. It is using the same kind of careful reasoning
Floyd used. But you also have to consider the point at which Venus becomes visible, how many degrees (6.26 actually sounds not bad, does Floyd consider 7.45 degrees better? which you do not state in the paragraph just above but do list in the table you sent me).

Remember also that predictions slightly in advance are better than predictions that fall too late.

Since you mention the correlation constant 584285 not ...283, I will mention that there was a meeting in Albany at which Lounsbury, Aveni, Justeson and a student of J's (at least those three) discussed ...285 vs. ...283. The arguments are EXTREMELY difficult, because of the slippage. Justeson and student considered the lunar records, and slippages of fractions of a single day, to say which correlation constant gave overall better set of total results. No single record could resolve such a thing, I believe.

***

"After applying these corrections to his articles, everything else he says fits. My interpretation locks the Venus tables into history, exactly matching the astronomy, and the 1.5.14.4.0 number is the smoking gun. In the 1997 Maya Workshop workbook, Linda also seems to be touching all the points, but misses the conclusion. On page 166, she actually lists exactly correct four of the five long counts that I claim are the epochs of the table (except that she lists them as Julian dates).

However, for some reason she lists another 1 Ahaw 18 Kayab in the second entry, which points out that she missed the sequence here, too. She does not list the 10.10.5.3.0 1 Ahaw 8 Yax run of the table."

*

So it seems to me that the easiest way to present your alternative view is to set up two sets of epochs hypothesized for the table, and show that *AS A TOTAL SET*, you wish to propose a better alternative than that so far offered. The comparison which people will evaluate is set vs. set, not interpretation of one individual date record vs. another individual date record. My advice anyhow is to do that, and to accumulate every possible way of evaluating the two sets against each other, *including* any which do not support your conclusion, just as Floyd did that. Measure in fractions of days where you can. And so on.

***

"So it seems that everyone had all the pieces, but nobody seems to have fit them all together to finish the puzzle. The sequence that fulfills all the requirements is:

10.5.6.4.0 1 Ahaw 18 K'ayab (5.3 degree elongation in morning sky)
10.10.5.3.0 1 Ahaw 8 Yax (6.2 degree elongation in morning sky)
10.15.4.2.0 1 Ahaw 18 Wo (7.4 degree elongation in morning sky)
11.0.3.1.0 1 Ahaw 13 Mak (9.7 degree elongation in morning sky)
11.5.2.0.0 1 Ahaw 3 Xul (10.0 degree elongation in morning sky)

And the 10.15.4.2.0 1 Ahaw 18 Wo can be linked to the page 24 long count through the 1.5.14.4.0 distance number."

*

The table you give above certainly is extremely regular, though I think Lounsbury might argue that the last date was too far off, that is, the prediction was coming too late?

Among the tools to evaluate your proposal vs. earlier ones, create ALSO a zig-zag diagram like Lounsbury's, and compare the two zig-zag diagrams.

Perhaps the Mayans did use a table such as you display above, but were dissatisfied with its long-term slippage (much too much), so other intervals were used to get back closer, a double-foreshortening then?

If you adopt your table, and try to judge in Lounsbury's fashion just WHERE they would have undertaken a double-foreshortening, do you come back to Lounsbury's conclusion? Or to something different from both Lounsbury's and your present simpler system using only single foreshortenings?

Put in other words, at WHAT point would a doulbe foreshortening bring Venus back to an ideal predictive station, around 6 ??? degrees elongation in the morning sky?

***

Well, that is the best I can do at the moment, not looking at Lounsbury's article, not trying to reanalyze every step of your reasoning, just trying to read your comments from today somewhat carefully.

No matter what, congratulations on having worked so hard and having concluded so much on your own.

Now move on ? also to the two Lounsbury articles in Aveni's "The Sky in Mayan Literature"?

Best wishes,
Lloyd

From: Gregory Reddick
Sent: Thursday, February 17, 2000 4:40 AM
To: Lloyd Anderson
Subject: RE: 1.5.14.4.0 -- What Lounsbury understood

Lloyd-

I only read Floyd's commentary on the Venus pages, not the complete 60 page treatise, hence my quick read. I'm not sure that we are talking about the same paper, though. This one does not have a zigzag line in it, although I think I must have seen or read that one at some point a long time ago, because I vaguely remember that graph.

One thing to point out, Teeple in 1930 had figured out most of the Venus foreshortening stuff. I haven't read his paper (I'm trying to find a copy), but Thompson's commentary explains it pretty well--and I have gone over Thompson's commentary with a fine tooth comb. I can't take credit for reinventing that--I can only affirm that one double foreshortening for every four singles works in my analysis. Floyd lists Teeple's work, as well as Thompson's commentary, in his bibliography, so I don't think he could claim originality here either, although he doesn't cite Teeple in the body of his work (probably because he is trying to derive all the info empirically).

Before I continue, I should give one other caveat about my numbers: all elongations from Venus in my analyses are absolute elongations taken at 6 a.m. at GMT-6 (~Chichén Itzá) on the given day, unmodified for the bending of light in the Earth's atmosphere. That bending can be significant at the horizon, so my elongation numbers may be a little off for ground-based astronomy. The program I wrote to generate the elongations hooks into Excel and can generate hundreds of elongations very quickly, but does have that caveat. The calculations are based off the best NASA models (I stole the algorithm and modified it to my needs) for how the sky works. Skyglobe tends to agree within about 2 degrees on the elongations that I've checked, and it should be taking into account the bending of light. I've also tested several other astronomy programs, which agree within a small percentage of error. I am reasonably sure that my absolute numbers for elongation are accurate, at least to the point that NASA can predict where things were in the sky a thousand years ago. All date calculations are performed with my Maya Calendar engine, also hooked into Excel. I would never have been able to try as many things as I have without the tools that I built first.

|(By the way, have you also read his later article in Aveni editor
|"The Sky in Mayan Literature", the one which precedes the article on 1.5.5.0?
|That might clarify some things. I do have that one handy, and can send it.

I haven't and it would be useful. I can reimburse you for copying and postage at the Maya meetings. Send to [Address Suppressed]. If it's short, the fax number is [Phone number Suppressed].

|So I think Floyd *did* intend that the
|earlier date linked by that 1.5.14.4.0 DN was likely to *not* be good
|astronomy, and the later date reached by that DN *was* likely to be an
|astronomically good one.

Let me give the whole paragraph, because I think it makes it reasonable that, at least at the point that he wrote the paper that I'm quoting, that he didn't realize the link I'm talking about. I do have great respect for Floyd's calculations, and have used some of his formulae in my Maya Calendar program. Here's the complete quote:

"The last of the numbers of this tier, 1.5.14.4.0, if applied to 1 Ahau 18 Kayab, leads again to a day 1 Ahau 18 Uo; but it is one that is eight calendar rounds later (1.1.1.14.0) than that attained by 4.12.8.0. The accumulation of error in eight calendar rounds, or four unabridged runs through the entire table, is 20.8 days. It is not likely, then, that two such days separated from each other by eight calendar rounds, were both intended to be designated as Venus epochs. (The interval 1.5.14.4.0 is 17.36 days longer than 317 mean Venus periods of 583.92 days.) These two numbers, the second and fourth in this tier (counting from the right) were either to be applied to two different bases 1 Ahau 18 Kayab, eight calendar rounds apart, not both reckoned as heliacal risings of Venus, and were to lead to the same 1 Ahau 18 Uo; or else they were to be applied to the same 1 Ahau 18 Kayab and were to lead to two different days 1 Ahau 18 Uo, not both of which were to be counted as days of helical rising of Venus. The first alternative is the one generally accepted."

While I think his reasoning is manifestly true, I think it also points out that he missed the connection between the base date on page 24 and the date in the tables reaching a helical rising *in this paper*. If he had, I'm pretty certain that he would have pointed it out in this paragraph. He may have done just that in a paper that I haven't read--I wouldn't be surprised at all (as I've said, I find it surprising that no one that I've read has). I do think that he changed his mind between the two papers on the epoch of the first run of the table. He says in the 1.5.5.0 paper: "This is because 10.5.6.4.0 was the date of the only 1 Ahaw 18 Kayab ever, that coincided with a heliacal rising of Venus, four days after an inferior conjunction, in agreement with the record of the Maya table." My analysis agrees with his (at least for the 584283 or 584285 correlations), although in my sequence it does finally repeat in 6808 AD, which is as close to "ever" as you could want! In contrast, the Maya Numeration article (p 814) lists in a table: "1038: Epochal date for Venus table as appearing in Dresden Codex (10.10.11.12.0)." Neither paper gives the chronology that I laid out. I actually don't have the publication dates for either paper as they both came from the Kinko's collection without citing where they originally came from, so I'm assuming that the 1.5.5.0 paper comes later. I need to get the complete bibliographic information to cite Floyd's papers in my work.

|Since you mention the correlation constant 584285 not ...283,
|I will mention that there was a meeting in Albany at which Lounsbury,
|Aveni, Justeson and a student of J's (at least those three) discussed
|...285 vs. ...283. The arguments are EXTREMELY difficult,
|because of the slippage. Justeson and student considered the lunar
|records, and slippages of fractions of a single day, to say which
|correlation constant gave overall better set of total results.
|No single record could resolve such a thing, I believe.

I might agree with that, although I think it *might* be possible to conclude that a particular correlation constant was in use in Dresden, if not the entire Maya region. If it can be accepted that a particular run of a Venus or Lunar table was contemporary to when the tables were created, and not predictive, then you could conclusively prove that the Maya were using a particular constant. Also, you could argue that if a 584283 constant was being used, it would throw the entire numbers for the table off, because almost all the heliacal risings would appear too close to the Sun. I doubt, though, that anyone will ever be able to argue at better than a +/- 1 day resolution. Even an eclipse recording (as opposed to prediction) on a day may have been saying that "we record an eclipse that happened yesterday."

|So it seems to me that the easiest way to present your alternative view
|is to set up two sets of epochs hypothesized for the table,
|and show that *AS A TOTAL SET*,
|you wish to propose a better alternative than that so far offered.
|The comparison which people will evaluate is set vs. set,
|not interpretation of one individual date record vs. another individual
|date record.
|My advice anyhow is to do that, and to accumulate every
|possible way of evaluating the two sets against each other,
|*including* any which do not support your conclusion,
|just as Floyd did that. Measure in fractions of days where you can.
|and so on.

I don't have the paper with the zigzag, so I would have to recreate Floyd's data. But I can produce my data independently and then compare it to Floyd's when I get it. This might be better anyway, because I can't be influenced by his data. I have attached a gif of the graph of the elongations using my model of the date sequences and the 584285 correlation.

|The table you give above certainly is extremely regular,
|though I think Lounsbury might argue that the last date was too far off,
|that is, the prediction was coming too late?

I would argue that a double foreshortening should occur at the end of the 1 Ahaw 3 Xul run of the table. The table would then continue:

11.9.14.8.0 1 Ahaw 8 Ch'en (8.51 degree elongation in morning sky)
11.14.13.7.0 1 Ahaw 18 Pohp (11.92 degree elongation in morning sky)
11.19.12.6.0 1 Ahaw 13 Keh (9.53 degree elongation in morning sky)
12.4.11.5.0 1 Ahaw 3 Tzek (14.02 degree elongation in morning sky)
12.9.10.4.0 1 Ahaw 18 Muwan (10.44 degree elongation in morning sky)
12.14.2.12.0 1 Ahaw 3 Wayeb (5.24 degree elongation in morning sky)
12.19.1.11.0 1 Ahaw 13 Sak (8.77 degree elongation in morning sky)

|If you adopt your table, and try to judge in Lounsbury's fashion
|just WHERE they would have undertaken a double-foreshortening,
|do you come back to Lounsbury's conclusion? Or to something
|different from both Lounsbury's and your present simpler system
|using only single foreshortenings?

|Put in other words, at WHAT point would a doulbe foreshortening
|bring Venus back to an ideal predictive station, around 6 ??? degrees
|elongation in the morning sky?

I would argue that the double foreshortening would have to be immediately AFTER the 1 Ahaw 3 Xul run of the table and every fifth run after that. This is actually the same argument that Floyd makes, except for WHERE I place the double foreshortening in the sequence. I think that the graph comparison will show that my numbers are about as good overall as Floyd's, but my first five runs are better, and better fit the internal data that is in the Codex. Also, I don't think it necessarily follows that predicting early is better than late. Because the tables are approximations, the Maya priest is going to have to be watching the sky carefully in that period. The tables are predicting a "hot zone" to be on the lookout for Venus, plus or minus several days. One astronomy web site I hit pointed out that under ideal conditions, Venus theoretically could cross inferior conjunction in less than 24 hours and be seen in the evening and morning sky on the same night--the Earth would have to be at its closest point to the Sun, Venus at its closest point to Earth, be near the winter solstice in the Northern hemisphere (to give a long night), and have an extremely sharp-eyed observer. Under such conditions it would totally screw up the entire 8 day inferior conjunction thing for the Maya. The web site did not mention if that theory every played out in history.

|Now move on ? also to the two Lounsbury articles in Aveni's
|"The Sky in Mayan Literature"?

Yes, please send them, unless they are titled "A Solution for the Number 1.5.5.0 of the Mayan Venus Tables" or "Maya Numeration, Computation, and Calendrical Astronomy." These are the articles that I have.

BTW, I did have a really good theory about why the 236, 90, 250, 8 sequence, but after spending hours generating pictures of the sky, it didn't pan out. I'm still hoping to figure this out some day, because it is driving me nuts.

One other question that is somewhat unrelated to the above: What style guide is used in most papers on the Maya? It looks like most people are using the APA style, as opposed to the MLA or some other style guide.

Thanks,

Greg

From: Lloyd Anderson
Sent: Thursday, February 17, 2000 8:43 AM
To: Gregory Reddick
Subject: Reddick - Lounsbury parallels etc.

Greg:

The following which you quote from Lounsbury:

"These two numbers, the second and fourth in this tier (counting from the right) were either to be applied to two different bases 1 Ahau 18 Kayab, eight calendar rounds apart, not both reckoned as heliacal risings of Venus, and were to lead to the same 1 Ahau 18 Uo; or else they were to be applied to the same 1 Ahau 18 Kayab and were to lead to two different days 1 Ahau 18 Uo, not both of which were to be counted as days of helical rising of Venus. The first alternative is the one generally accepted."

is most consistent with what I thought I remembered him as meaning. That is, going from two different bases, either one of which may have slipped and come to be in error, leads to a single base 1 Ahau 18 Uo, which as a consequence of the use of the intervals as corrections should have had more of a claim to astronomical accuracy. That is, the date at the end of the interval 1.5.14.4.0 should be a Venus station, that at the beginning of that interval need not be, because it might rather be only the working out of a traditional scheme which no longer corresponded to actual Venus.

Lounsbury not only reasoned extremely carefully, he also wrote in a way which requires much more from his readers than is today commonly the practice. So the paragraph you quote from him (just above) does to me require that the *reader* conclude as I have stated.

You say:

"I think it also points out that he missed the connection between the base date on page 24 and the date in the tables reaching a helical rising *in this paper*. If he had, I'm pretty certain that he would have pointed it out in this paragraph. "

What I am saying is that, within the conventions of writing which he followed, he *did* point it out in that paragraph, just not in the way which you and I are most accustomed to. By saying both intervals reach a common end point, he is implying (quite directly, in his mind, I believe, requiring nothing but proper reading) that it is the common end point which must be an accurate Venus station, for the real Venus not the traditional purely calculated one.

*

"Also, you could argue that if a 584283 constant was being used, it would throw the entire numbers for the table off, because almost all the heliacal risings would appear too close to the Sun. "

That is the kind of reasoning Floyd used to argue for his choice of the correlation constant.

*

"I would argue that the double foreshortening would have to be immediately AFTER the 1 Ahaw 3 Xul run of the table and every fifth run after that. This is actually the same argument that Floyd makes, except for WHERE I place the double foreshortening in the sequence. I think that the graph comparison will show that my numbers are about as good overall as Floyd's, but my first five runs are better, and better fit the internal data that is in the Codex."

THAT is the kind of statement I am looking for.

However, in the real world, I think it is undermined by the following:

"Also, I don't think it necessarily follows that predicting early is better than late. "

because the entire field (?) operates with that assumption, or at least many do, has been my perception. I may have it wrong, but just a warning if you bet on something that requires people to change this belief, they may say you're simply wrong.

*

So I would say, put your tables together, line them up on one page, put Lounsbury's on another page laid out in exactly the same format, so easy to compare, put them on ***FACING*** pages in your paper, and send it off for comments to these three:

Tony Aveni
Richard Johnson
John Justeson

*

I am priority-mailing the other paper from the Aveni edited volume this morning. I'll try to get the paper with the zig-zag lines for you too, don't know where I have it.

Lloyd

From: Lloyd Anderson
Sent: Thursday, February 17, 2000 9:33 AM
To: Helen Alexander; Anthony Aveni; Richard Johnson; John Justeson; Gregory Reddick
Subject: Lounsbury paper?

Could any of you please tell me where was published that one of Lounsbury's papers on the Venus tables Dresden which contained the zig-zag line chart detailing his ideas on how the system would get slightly out-of-line, and when it might have been corrected.

I read this paper long ago, and because one of the yearly Austin meeting folks may be coming up with an alternative to Lounsbury's interpretation of where the base dates were located, it is important for them to read this paper. I may have misplaced my own copy because I was using if for something...

Thanks for any help,
Lloyd Anderson

From: Lloyd Anderson
Sent: Thursday, February 17, 2000 10:26 AM
To: Gregory Reddick
Subject: Re zigzag.gif

Greg:

Nice chart. Now can you add one for the long counts within the classical period, the ones where you choose differently than Lounsbury? Will have to be done by hand, since not constant equal intervals from one to the next.

Lloyd

From: Lloyd Anderson
Sent: Saturday, February 19, 2000 8:31 AM
To: Gregory Reddick
Subject: Lounsbury paper

Response from Aveni:

Try the Manchester ICA proceedings published by BAR in 1983.

***

And Helen Alexander called last night about scheduling our next Dresden SIG, delaying it,...wondered whether I might pass on to her and to Sharon Bowen any of the things you have sent me (table of dates, reasoning), of course only if you say it is OK.

She will not be in Austin for full week this year, at most only for the weekend.

Lloyd

From: Gregory Reddick
Sent: Sunday, February 20, 2000 5:05 AM
To: Lloyd Anderson
Subject: More Venus Pages

Lloyd-

You can pass on what I've sent to Helen and Sharon, with the provision that they do not further disseminate the information until I can finish my paper. If you'll supply their email addresses, I will CC them on further mail.

I will make an ad-hoc presentation at the Maya meetings on some evening on the weekend or early in the week. I've got use of a high-quality computer projector until about Tuesday to make the presentation with, as I intend to show Venus in the sky using astronomy programs. I'll be aiming the presentation at the Maya audience in general, not just those of us into Dresden. My theories will be only a part of the presentation; much will be just talking about the general theory of the Venus pages.

I just received John Teeple's 1926 paper that I ordered from the library, titled "Maya Inscriptions: The Venus Calendar and Another Correlation." This is the original paper that laid out the correction factors the Maya used. Nice work, especially for that early in Maya studies. He lays out the correction scheme in the paper, what Lounsbury calls foreshortenings. The correlation he proposes, though, works out to 492622 (they apparently weren't using correlation constants in 1926, so I had to derive it). He tries to sync the tables to mentions of Venus on inscriptions on Classic Period monuments using this correlation.

Also, I was wrong when I said in my last email that my elongations were at 6 a.m.--they are all calculated at midnight. That means that all the elongations listed throughout will be a tiny fraction more at sunrise.

I just did some further analysis. There are only three possible sequences of single and double foreshortenings that include all three runs shown in the Venus Tables within a reasonable span of time. They are:

Sequence A
1 Ahaw 18 K'ayab
1 Ahaw 18 Wo
1 Ahaw 13 Mak
1 Ahaw 3 Xul

Sequence B
1 Ahaw 18 K'ayab
1 Ahaw 8 Yax
1 Ahaw 13 Mak
1 Ahaw 3 Xul

Sequence C
1 Ahaw 18 K'ayab
1 Ahaw 8 Yax
1 Ahaw 18 Wo
1 Ahaw 13 Mak
1 Ahaw 3 Xul

The Sequence B has the drawback that it doesn't mention an 18 Wo, which is mentioned on Page 24, and is reached by the 1.5.14.4.0 distance number. Sequence A has the double foreshortening between the 18 K'ayab and 18 Wo. Sequence B has the double foreshortening between the 8 Yax and 13 Mak. Sequence C doesn't have any double foreshortenings, so presumably it would occur at the end of the 3 Xul run. I think most people have been arguing that the tables should be used with Sequence A, and I'm arguing that they should be used with Sequence C. All three sequences can use the one double foreshortenings to four single foreshortenings formula that will be an extremely accurate predictor of Venus. Each of the sequences could start at any 1 Ahaw 18 K'ayab. A graph of the sequences is attached showing various long counts for the 1 Ahaw 18 K'ayab. Using the 584285 constant, it displays 15 possibilities, with the initial start date listed for each one. Of these, only six have even a single data point between 0 and 10 degrees in the morning sky. I have attached these individual graphs as well. The graph shown in sequence4.gif is the one that I'm arguing is the one the Maya used. It is a sequence C starting on 10.5.6.4.0. I believe Lounsbury would be arguing for either sequence2.gif or sequence5.gif. Sequence 5 also allows reaching the 1 Ahaw 18 Wo by the 1.5.14.4.0 distance number, and would be the one alternative to sequence 4 that I consider even remotely possible, but it's initial data point is over 15 degrees above the horizon.

Do you have access to the paper Tony mentioned?

Greg

From: Lloyd Anderson
Sent: Sunday, February 20, 2000 11:34 AM
To: Gregory Reddick
Subject: Re: More Venus Pages

Greg:

Thanks, the format of this last message is the kind of comprehensive study of possibilities that I think many will find the most convincing.

Here are Sharon's and Helen's email addresses. I have sent them your message from today with attachments, and also the attachments printed out in the post, in case they have any trouble with .mim attachments.

I think you would contribute a lot of joy to Richard Johnson if you would send your ideas to him too, even in preliminary form. It will help him to feel rewarded for shepherding the Dresden group for so many years.

I will try again to find that Lounsbury paper, it is the same one I was first looking for. If I can figure out the exact citation (it is in the bibliography of one of his latest papers?) I can also try to get it again at Library of Congress.

You should have a copy of his paper preceding the 1.5.5.0 paper very shortly, if not delivered already or yesterday, then on Tuesday for sure.

Lloyd

Sharon Bowen:
<[email address suppressed]>

Helen Alexander:
<[email address suppressed]>

Richard Johnson:
<[email address suppressed]>

From: Gregory Reddick
Sent: Sunday, February 20, 2000 6:47 PM
To: Lloyd Anderson
Cc: Sharon Bowen; Helen Alexander
Subject: Citation et.al.

Lloyd-

[Sharon and Helen: Welcome to the conversation. I will fill you in on previous email leading up to this one shortly so that this makes sense.]

From the back of the 1.5.5.0 paper:

Aveni, Anthony F., and Gordon Brotherston, eds.
1983 Calendars in Mesoamerica and Peru: Native American Computations of Time (Proceedings, 445th International Congress of Americanists, Manchester 1982; = BAR International Series, 174). Oxford.

----------

I just got Floyd's paper you sent from the P.O. Box. Very interesting! I pretty much reproduced everything he said in his paper almost totally independently. I find it very interesting, though, that he proposes the sequence:

10.5.6.4.0 1 Ahaw 18 K'ayab
10.10.11.12.0 1 Ahaw 18 K'ayab
10.15.4.2.0 1 Ahaw 18 Wo
11.0.3.1.0 1 Ahaw 13 Mak
11.5.2.0.0 1 Ahaw 3 Xul

I find it inexplicable to go forward two complete calendar rounds here, when an intervening 10.10.5.3.0 1 Ahaw 8 Pax works perfectly in between the original 10.5.6.4.0 1 Ahaw 18 K'ayab and the 10.15.4.2.0 1 Ahaw 18 Wo. The only argument that can be made in favor of the 10.10.11.12.0 date is the fact that a double foreshortening, followed by a single foreshortening sequence seems to be implied by the 4.12.8.0 and 9.11.7.0 corrections. But that sequence will explain a double foreshortening, followed by a single foreshortening, anywhere in a sequence and doesn't necessarily have to be off the initial date. To summarize where his sequence fits in my previous argument, Floyd is doing no foreshortening, followed by a double foreshortening to make up for the fact that no foreshortening was done (so it is a modified sequence 5 from my previous email, prepending an addition 1 Ahaw 18 K'ayab before the sequence). This will screw things up after the end of the table as well, as another double foreshortening is required immediately after the 1 Ahaw 3 Xul run, when his reasoning would delay it for another full run of the table if the normal 1:4 ratio of doubles to singles is maintained. It doesn't make sense.

In sequence 4 (from the previous email), it would be explained by the double foreshortening happening immediately before a Ahaw 18 K'ayab date, not after. The Venus epoch before sequence 4 would be 10.0.13.14.0 1 Ahaw 13 Kank'in. This argument is admittedly a little weak based off the internal evidence of the Codex, but I think it is far stronger than not having a foreshortening at all between the 10.5.6.4.0 1 Ahaw 18 K'ayab and the 10.15.4.2.0 1 Ahaw 18 Wo. I think it actually strengthens the remainder of the argument that Floyd made throughout the paper (which, of course, are my arguments as well).

So except for the 1 Ahaw 8 Pax (Yax) date, I concede that Floyd said *everything* I've said, and he said it first! I reconstructed his arguments almost entirely from scratch. The one additional thing I throw in, though, is the fact that the 10.10.5.3.0 1 Ahaw 8 Pax (Yax) is immediately after a transit of Venus.

Greg

From: Gregory Reddick
Sent: Sunday, February 20, 2000 8:58 PM
To: Sharon Bowen; Helen Alexander; Richard Johnson
Cc: Lloyd Anderson
Subject: Venus Pages Correspondence on the Web

Rather than forwarding everyone the email that Lloyd and I have exchanged over the last couple of weeks, I have posted the correspondence onto my web site. The address is http://www.mayainfo.org/works/venuspages/default.asp.

You might find it an interesting read. Basically I have independently confirmed (without knowing about them) some of Floyd Lounsbury's observations on the Venus Pages and added a couple of notes of my own.

Greg Reddick

From: Lloyd Anderson
Sent: Monday, February 21, 2000 8:46 AM
To: Gregory Reddick
Subject: Re: Venus

Greg:

I hope to re-read everything shortly, to make sure I am understanding what you have done.

I already copied the prior correspondence and posted it to Helen and Sharon, as well as emailing them copies, just for their convenience so they can access it more easily. Good you put it on your web site also.

Lloyd

From: Lloyd Anderson
Sent: Monday, February 21, 2000 8:59 AM
To: Gregory Reddick; Helen Alexander; Sharon Bowen
Subject: Good work!

Greg, don't be discouraged, Floyd is simply amazing. You probably now appreciate that better than most do, given how much work it takes to draw these kinds of conclusions!

You may have discovered a way to present this material which is more easily understood by others, and that is VERY IMPORTANT, since none of us succeed in holding it in our heads very well, it is so complex. Your presentation of alternatives A, B, C seems to hold that promise, and Floyd never did the many graphs you presented. All of that may be very useful.

And if you have discovered even one new date of relevance, that is something too.

Congratulations,
Lloyd

From: Lloyd Anderson
Sent: Monday, February 21, 2000 12:06 PM
To: Gregory Reddick
Subject: Venus table intervals

Greg:

(I have re-read the correspondence, but not tried to recheck any of the math myself.)

It seems from your latest message that you are suggesting one need not read the Venus table intervals to be reaching the date 1 Ahau 18 Uo by *two* intervals from *two* preceding dates?

That is, the interval 1.5.14.4.0 does lead to 1 Ahau 18 Uo from an earlier 1 Ahau 18 K'ayab,

but the interval 4.12.8.0, even though it would also lead to a 1 Ahau 18 Uo from a 1 Ahau 18 K'ayab, need not be read as intended to do that (Lounsbury did read it so).

Have I understood you?

If so,

That may bring in any other evidence there is for reading multiplication tables and prefaces to the astronomical almanacs, to support or refute such a suggestion.

Or was it merely Lounsbury's supposition that since the interval 4.12.8.0 *could* lead to a 1 Ahau 18 Uo from a 1 Ahau 18 K'ayab, that it was intended to be used in such a way? If merely an assumption, not grounded in the structure of this and other similar tables, then we are free to disregard it. Are you suggesting that?

***

Twice in the most recent message you referred to 1 Ahau 8 Pax, where I think you mean Yax 10.10.5.3.0 was the date.

***

Your graph of the many Venus elongations for dates through to the next Pictun is interesting. I think I see something in this graph that I have not seen in other similar graphs, though I haven't checked it with logic, and you may find it easy to do so. It may be almost purely random. Is it?

You group five heliacal risings (base points) for the table, with a pattern of single foreshortenings between them, and a double foreshortening between each group of five. That should presumably even out the five different places where Venus appears in the sky. I don't know how fast that five-pointed star rotates, or slips relative to these commensuration schemes in the foreshortened Venus tables, but wondered, because of the following recurring patterns in the graph you constructed.

The smooth increase in elongation for the first five bases in your graph, beginning with 10.5.6.4.0, is repeated more or less in the group of five beginning 1.0.0.12.0.0. Each time these are followed by five at relatively greater elongations, then a drop to a low point, the second base after 12.4.11.5.0 or after 1.1.19.17.1.0.

The grouping of a pair, then a pair at greater elongation, than a single at greater elongation, recurs in two places, in the five base points *following* (not including) 16.2.14.16.0, and in the five centered on 1.2.19.6.6.0, Each time these are followed by five at relatively large elongations. The same pattern also recurs once at higher elongations, starting one base point before 14.3.9.15.0.

***

All for today,
Lloyd

From: Gregory Reddick
Sent: Monday, February 21, 2000 5:44 PM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: FW: Venus table intervals

Lloyd-

| but the interval 4.12.8.0, even though it would also
| lead to a 1 Ahau 18 Uo from a 1 Ahau 18 K'ayab,
| need not be read as intended to do that
| (Lounsbury did read it so).

That is exactly what I mean.

| was it merely Lounsbury's supposition that since
| the interval 4.12.8.0 *could*
| lead to a 1 Ahau 18 Uo from a 1 Ahau 18 K'ayab,
| that it was intended to be used in such a way?
| If merely an assumption, not grounded in the structure
| of this and other similar tables, then we are free to
| disregard it. Are you suggesting that?

Yes, that's what I'm suggesting. I think the assumption has been (going all the way back to Teeple's original 1926 paper), that you must apply a double foreshortening, followed by a single foreshortening in a positive direction to the base 1 Ahaw 18 K'ayab. This does allow you to reach a 1 Ahaw 18 Wo, and 1 Ahaw 13 Mak from that date. However, I think it's a false argument. If the corrections can be applied to any Venus epoch, and there is already a pattern of applying single foreshortenings between dates that is not explicitly mentioned in the preface (1 Ahaw 13 Mak to 1 Ahaw 3 Xul), then why does the double foreshortening have to come after 1 Ahaw 18 K'ayab? If you instead applied 4.12.8.0 as a *subtraction* to the 1 Ahaw 18 K'ayab, reaching a 1 Ahaw 13 K'ank'in not mentioned in the Codex, then applied the 9.11.7.0 to the 1 Ahaw 13 K'ank'in, you wind up reaching my 1 Ahaw 8 Yax date. When reordered, the sequence becomes:

1 Ahaw 13 K'ank'in
1 Ahaw 18 K'ayab
1 Ahaw 8 Yax

This pattern of subtraction of the 4.12.8.0 followed by the addition of 9.11.7.0 is followed between the 1 Ahaw 13 Mak and 1 Ahaw 3 Xul, so is not unprecedented.

With, then, further single foreshortenings after the 1 Ahaw 8 Yax, the rest of the epoch dates are reached. This forms my full and complete sequence:

1 Ahaw 13 K'ank'in
1 Ahaw 18 K'ayab
1 Ahaw 8 Yax
1 Ahaw 18 Wo
1 Ahaw 13 Mak
1 Ahaw 3 Xul

I just realized that this may be evidence that the Maya understood the 1:4 ratio that has widely been mentioned. Lounsbury, in his 1.5.5.0 paper says: "Eric Thompson noted that the optimum mix of corrections is four singles to one double...Thompson noted this only to point out that the device invented by the Maya for this purpose was one of *potentially* great precision; but there was no knowing whether the Maya calendar specialists were aware of this ratio or had sufficient information to have deduced it. If the present line of reasoning [in the 1.5.5.0 paper] is correct, we can know now that they did not."

If my sequence is correct, then it is obvious that the Maya understood a 1:n ratio where n >= 4. To work out, the Maya must apply another double foreshortening at the end of the 1 Ahaw 3 Xul run, making n = 4. And they would have, too, because Venus would be rising late at that point, and later still if the double foreshortening wasn't done. Even with the double foreshortening, Venus gets excessively late for a while, before getting pulled back in line (see the zigzag graph in previous email). Lounsbury argues in the 1.5.5.0 article that the evidence shows that the Maya in actuality only used a 1:2 ratio. But this is based on his incorrect, in my thinking, placement of the double foreshortening in the sequence. If the double foreshortening is placed before the 1 Ahaw 18 K'ayab run, and at the end of the 1 Ahaw 3 Xul run, then there is some circumstantial evidence that the 1:4 ratio is in the Venus Tables. If so, this is a remarkable accomplishment, and means that the Maya must have been studying Venus for a *very* long time before those tables could be constructed.

| Twice in the most recent message you referred to
| 1 Ahau 8 Pax, where I think you mean Yax
| 10.10.5.3.0 was the date.

I meant Yax. I'm must have been really tired when a Pax and a Yax look the same.

| [Description of pattern]

I had noticed that pattern, too. If you'll also notice, it slowly slips downward, as the correction factors slowly overcompensate for the movement of Venus. But I think one day in a piktun is pretty good accuracy!

Greg

From: Lloyd Anderson
Sent: Monday, February 21, 2000 6:56 PM
To: Gregory Reddick
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: Hope you're right about 1:4 ratio

Greg:

I hope your reasoning holds up. Here's my attempt to point to a possible weakness, just in case. You write:

>If the corrections can be applied to any Venus epoch,
>and there is already a pattern of applying single foreshortenings
>between dates that is not explicitly mentioned in the preface (1 Ahaw 13
>Mak to 1 Ahaw 3 Xul), then why does the double foreshortening have to
>come after 1 Ahaw 18 K'ayab?

[so far OK]

>If you instead applied 4.12.8.0 as a
>*subtraction* to the 1 Ahaw 18 K'ayab, reaching a 1 Ahaw 13 K'ank'in not
>mentioned in the Codex, then applied the 9.11.7.0 to the 1 Ahaw 13
>K'ank'in, you wind up reaching my 1 Ahaw 8 Yax date. When reordered, the
>sequence becomes:
>
>1 Ahaw 13 K'ank'in
>1 Ahaw 18 K'ayab
>1 Ahaw 8 Yax

The reasoning here bothers me slightly, because if an interval in a multiplication table *were* intended as a subtraction, would that not be indicated in some way, rather than giving that interval in the same manner as other intervals which are intended as additions?

>This pattern of subtraction of the 4.12.8.0 followed by the addition of
>9.11.7.0 is followed between the 1 Ahaw 13 Mak and 1 Ahaw 3 Xul, so is
>not unprecedented."

I am not actually looking at the codex tonight, and without it I do not know why you believe the subtraction is warranted *as a subtraction* (which I gather is necessary for the parallel to hold).

Can you strengthen this part?

Lloyd

From: Gregory Reddick
Sent: Tuesday, February 22, 2000 12:19 AM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: Strengthening the argument

Lloyd-

See the attached diagram for a calculator to help figure out where you get from single and double foreshortenings. In the calculator, the box one line below and to the left of any box is the haab reached by a double foreshortening (reached by adding 4.12.8.0). The box one line below and to the right is the haab reached by a single foreshortening (reached by adding 9.11.7.0 minus 4.12.8.0 or 4.18.17.0). The box two lines down, immediately below a box is the haab reached by a double plus a single, or single plus a double (reached by adding 9.11.7.0). The boxes in gray are the epochs mentioned in the Codex. All of the tzolk'ins will be 1 Ahaw.

These are the three possible sequences that I laid out before:

Sequence A
1 Ahaw 18 K'ayab
1 Ahaw 18 Wo
1 Ahaw 13 Mak
1 Ahaw 3 Xul

Sequence B
1 Ahaw 18 K'ayab
1 Ahaw 8 Yax
1 Ahaw 13 Mak
1 Ahaw 3 Xul

Sequence C
1 Ahaw 18 K'ayab
1 Ahaw 8 Yax
1 Ahaw 18 Wo
1 Ahaw 13 Mak
1 Ahaw 3 Xul

I'll add a sequence D, which is the one that Lounsbury uses:

Sequence D
1 Ahaw 18 K'ayab
1 Ahaw 18 K'ayab
1 Ahaw 18 Wo
1 Ahaw 13 Mak
1 Ahaw 3 Xul

To use the calculator to add two calendar rounds, move one line down and two boxes to the right.

You can see from the calculator, A, B, & C are the only three possible sequences connected by single and double foreshortenings that include all three Venus epochs mentioned in the tables. Only in sequence B could you possibly have only double foreshortenings followed by single foreshortenings that touches all three epochs (you'd have to prefix a 1 Ahaw 13 K'ank'in to the sequence to make it work); but sequence B doesn't have an 18 Wo anywhere in it, and also the astronomy doesn't work (see the previous graphs), which makes it improbable. Sequence A, B, and D all require that you have a single foreshortening without a preceding double foreshortening to get from 1 Ahaw 13 Mak to 1 Ahaw 3 Xul. I show this as proof that the Maya used single foreshortenings individually without a preceding double foreshortening (unless the improbable sequence B is used).

Now there are three possible ways that you can calculate a single foreshortening without a preceding double foreshortening to arrive at a given Venus epoch:

1) You can subtract 4.12.8.0, then add 9.11.7.0.

2) You can add 9.11.7.0, then subtract 4.12.8.0.

3) You can subtract 4.12.8.0 from 9.11.7.0, producing the number 4.18.17.0, then adding the 4.18.17.0.

All three get you the same resulting date. All three require that you do subtraction of a 4.12.8.0. So no matter how it was done, the Maya performed subtraction of the 4.12.8.0. So if they performed subtraction of that number in one place, they could do it in another. So why not start by doing the subtraction to the initial date of sequence, which is 1 Ahaw 18 K'ayab?

There is no reason at all for the Maya to mention the number 9.11.7.0 on page 24 unless you have a double foreshortening followed by a single foreshortening somewhere in the sequence. Sequences A, B, and D do just that. But the sequence that I'm arguing best matches the astronomy and makes most sense, C, doesn't have a double foreshortening between *any* of the listed epochs. So the double foreshortening must come either before or after the sequence, or both. There is no other reason in sequence C to even list the 4.12.8.0 number or 9.11.7.0 number. Instead you would just list the 4.18.17.0 calculated number and be done with it. The only reason to list the 4.12.8.0 number is if double foreshortenings were performed, and the only reason to list the 9.11.7.0 is if a double foreshortening followed by a single foreshortening were performed. In sequence C, the double must be performed *before* the 1 Ahaw 18 K'ayab. So there is another implied mention of a Venus epoch in the tables if sequence C is to be accepted: 1 Ahaw 13 K'ank'in, which would come before the 1 Ahaw 18 K'ayab, which when combined with the other information presented in previous email, must be on the long count 10.0.13.14.0 1 Ahaw 13 K'ank'in. On that date, the Venus elongation is already at 14.3 degrees elongation in the morning sky, and would require a double foreshortening at the end of the run to pull it back in, because Venus would be a full 8 days early toward the end of the run.

So to summarize:

1) I showed that the Maya performed single foreshortenings without preceding double foreshortenings.

2) I showed that this required the Maya did use subtraction of 4.12.8.0 from a given epoch.

3) I showed that in sequence C the subtraction would be performed on the 1 Ahaw 18 K'ayab date, placing the double foreshortening immediately before this epoch.

Greg

From: Gregory Reddick
Sent: Tuesday, February 22, 2000 4:18 AM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: Analysis of Floyd's paper

Lloyd-

I just read Floyd's paper, "The Base of the Venus Table of the Dresden Codex, and it Significance for the Calendar-Correlation Problem," which I just found in some Kinko's collection photocopies that I missed on my first search through my collection. I want to start with a quote from the paper:

"After having completed the main part of this paper, in which I presented the results of my own inquiry, and turning then to a review of previous interpretations of the problem--and after finishing with those of Teeple and Thompson and going on to those with which I had been only slightly if at all acquainted--I found myself in the not unfamiliar circumstance of having discovered things that others had discovered long before; in this case the date A.D. 934 November 20, the uniqueness of the combination of calendrical and astronomical events of that date, and their signal importance. Under most such circumstances that would have left me without a paper; but not quite this time. Although crucial pieces of the puzzle had been located and identified earlier, their proper assembly has remained incomplete to the present time. So also has their full documentation."

Oddly enough, that is *exactly* how I feel. I could have written that paragraph!

He repeats much from the paper you sent me (I think this one came first), but does include a nice survey of other's work--some of which I've seen, and many others that I haven't.

He seems obsessed with keeping the 10.10.11.12.0 1 Ahaw 18 K'ayab date. I don't think there is any reasonable justification for it. He proves conclusively (although it seems that Spinden beat him to it in 1928!) that the 10.5.6.4.0 1 Ahaw 18 K'ayab date is the only one in history that matches the astronomy. He shows that the 1.5.14.4.0 distance number must reach a 1 Ahaw 18 Wo used in the table. But why he is so attached to going a full, uncorrected run of the table between the 10.5.6.4.0 1 Ahaw 18 K'ayab and 10.10.11.12.0 1 Ahaw 18 K'ayab is inexplicable when the 1 Ahaw 8 Yax is a much better explanation that fits the astronomy better as well. My zigzag graph differs from his in that his vertical axis is the number of days from inferior conjunction, whereas mine shows the elongation of Venus in the morning sky, but otherwise they are equivalent. Since the 1 Ahaw 18 Wo run of the table is not listed, the 1 Ahaw 8 Yax can also be missing without undue alarm.

Another point he makes in the paper: There was a conjunction of Venus with Jupiter on the 1 Ahaw 18 Wo date. I had discovered the same thing, and had already noted it in my paper, but haven't mentioned it before. I also note that it is actually a triple conjunction as Mercury is nearby, too, but Floyd didn't mention it. I will attach a screen shot of the sunrise on the morning mentioned, and a QuickTime movie to the end of this message on the website (http://www.xoc.net/maya/venuspages.asp), but won't attach them as they are large. The movie is produced by the astronomy program Starry Night Backyard.

Greg

The above is sunrise on Dec 6, 1129 (Julian), which is 10.15.4.2.0 1 Ahaw 18 Wo using the 584285 correlation. Produced by SkyGlobe.

Click here to view a QuickTime movie of this sunrise, produced from Starry Night Backyard. It is slightly unrealistic in that you can still see Venus and Jupiter after sunrise, but the effect is awesome.

From: Lloyd Anderson
Sent: Tuesday, February 22, 2000 7:49 AM
To: Gregory Reddick; Helen Alexander; Sharon Bowen
Subject: Floyd's graph and yours

Greg:

>My zigzag graph differs from his in that his vertical
>axis is the number of days from inferior conjunction, whereas mine shows
>the elongation of Venus in the morning sky, but otherwise they are
>equivalent. Since the 1 Ahaw 18 Wo run of the table is not listed, the
>1 Ahaw 8 Yax can also be missing without undue alarm.

What I was suggesting was that if you want to present this, it would be useful to make a transparency of Floyd's zigzag graph, and then to make your own, using the dates YOU think are significant, so the two can be visually compared. Keep the long one you did which
goes into the 1st pictun, but produce also a simplified one so we can *see* the ways in which you believe your selection of dates, including the 1 Ahau 18 Uo and the 1 Ahau 8 Yax works the same as or better than Floyd's. Then give your OTHER reasons in favor (conjunction you just mentioned, transits, etc.). Then give arguments against, if you can contrive any (need to subtract when the table in the Dresden does not indicate any difference between addition and
subtraction in this case, if that is so).

Ooooh, nice "Haabpattern.gif".
A new form of visual communication, I'll have to study it.

I look forward to the other items.

Lloyd

From: Gregory Reddick
Sent: Tuesday, February 22, 2000 1:27 PM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: correction line not part of the multiplication table

Lloyd-

One thing I'd like to point out: The line that contains the corrections on page 24, the 1.5.14.4.0, 9.11.7.0, 4.12.8.0, and 1.5.5.0 numbers, is not part of the multiplication table. The numbers aren't used the same way as the other numbers shown. So the fact that a 4.12.8.0 is used for subtraction here is not unusual. In fact, I think the way that the 4.12.8.0 and 9.11.7.0 number is used is very similar to the way that a ring number is used. You subtract the 4.12.8.0 from the base date (the Venus epoch), then add the 9.11.7.0 to reach the new date. If you think of the 4.12.8.0 as a ring number working off a base that is not 4 Ahaw 8 Kumk'u (hence no ring), it makes more sense.

If you use the visual calculator (haabpattern.gif), to explain this, the actual movement pattern you make is not to go directly from one date to the one down and to the right to do the single foreshortening. Instead, you subtract 4.12.8.0 (move up one line and to the right), then add 9.11.7.0 (move two lines straight down). In no case, though, in sequence C did the Maya record the date reached by subtracting the 4.12.8.0. In one case, this would reach Floyd's insisted upon 1 Ahaw 18 K'ayab, but I don't think the Maya recorded that date.

Greg

From: Gregory Reddick
Sent: Sunday, February 27, 2000 8:40 AM
To: Lloyd Anderson
Subject: Floyd's paper

Lloyd-

I sent you a copy of Floyd's zigzag paper last Tuesday by priority mail. You should have it by now.

Greg

From: Lloyd Anderson
Sent: Tuesday, February 29, 2000 6:16 AM
To: Gregory Reddick
Subject: Re: Floyd's paper

Thanks, got it.
I still believe your presentation of a zig-zag diagram as closely analogous as you can to Floyd's would show your points clearly, in a manner in which people can make the two comparable.

Note that Thompson's also looks better than Floyd's, yet Floyd argues (perhaps with little foundation?) that his is better despite that. What are the reasons supporting Floyd's argument here? How would they apply or not apply to your proposal?

How would you compare your proposal with Thompson's? Are they not more similar than either is to Floyd's? (I have not finished reviewing, only started a deeper examination, so that is a real question, not a rhetorical question.)

Lloyd

From: Gregory Reddick
Sent: Wednesday, March 1, 2000 1:39 PM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: RE: Floyd's paper

Lloyd-

I have placed my zigzag graph on the web site, which you can quickly reach from here: http://www.xoc.net/maya/venuspages.asp#20000301. I will follow up later with more.

Greg

My data (in red) overlaid over Floyd Lounsbury's graph. I only add one additional point to his graph, point Y, at 10.10.5.3.0 1 Ahaw 8 Yax. This data point would be between zero and one depending on how Floyd did his calculations. There is no internal evidence in the Codex for points A2 (Th) and B (Th), but they would be extrapolated from the 1:4 ratio. (After F.G. Lounsbury, "The Base of the Venus Table of the Dresden Codex, and Its Significance for the Calendar-Correlation Problem." 44th International Congress of Americanists, Manchester, September 1982.)

Lounsbury's notes on the figure above:

Mean-error graph for predicted times of inferior conjunction of Venus:
(a) SOLID LINES, according to the prescriptions of the Dresden Codex tables;
(b) BROKEN LINES, Thompson's hypothetic early base sequence.
Abscissas represent time, marked at two-calendar-round intervals (104 years).
(Abscissas of the bases are tabulated inside of the frame of the graph.)
(Julian-Christian equivalents are by the 584285 correlation.)
Ordinates represent mean errors, in days, averaged for the five successive inferior conjunctions immediately prior to each of the specified bases.

From: Gregory Reddick
Sent: Thursday, March 2, 2000 6:26 AM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: Contrasts between Reddick, Lounsbury, and Thompson

Lloyd-

I feel honored placing my name in such company!

The discussion below refers to the graph found at http://www.xoc.net/maya/venuspages.asp#20000301.

| Note that Thompson's also looks better than Floyd's,
| yet Floyd argues (perhaps with little foundation?)
| that his is better despite that. What are the reasons
| supporting Floyd's argument here? How would they
| apply or not apply to your proposal?

Floyd's is better in one respect over Thompson's: it includes point D, which I think must be included in any sequence of Maya dates.

I will attempt to reverse engineer how Floyd came up with his sequence...

To create a sequence, one must somehow create a list of dates in the long count that include the three calendar round epochs listed in the tables (18 K'ayab, 13 Mak, and 3 Xul), and the long counts must fall on heliacal risings of Venus. [In the discussion below, let's call these the LC epochs.] Also, there must be some mechanism to connect the long count (9.9.9.16.0) given on page 24 with the derived long counts from your sequence. [In the discussion below, let's call this the correction DN.]

Many others (including Thompson) have proposed sequences that include points E, F, G, H. For points F, G, and H, I think they are correct, so that would include me, too. I think that Floyd also accepted point E without critically examining it. His other examinations showed that point D must also be in the sequence. So his LC epochs are points D, G, and H. To form his correction DN, he then figured that there must be a straight sequence of double calendar rounds without correction to connect points A through E. Then, in his reasoning, suddenly the Maya realized that things were getting out of whack and started implementing the corrections. The drawback to this scheme is that in the periods *before* the LC epochs, the points don't have any relation to what Venus is doing in the sky. How the Maya would form a sequence like this is beyond me. Also, between points E and F, the graph shows that the tables would be getting excessively out of sync with what Venus was doing in the sky. [Note: Because I have great respect for Floyd's reasoning, and the obvious flaws in this sequence, I actually *don't* think that Floyd believed in this sequence, regardless of the fact that it's what's in his paper. However, in none of his papers that talk about the Venus pages does he propose a sequence that I think he would believe in. It's too bad we can't ask him. Does he have any other later papers on Venus that I haven't read yet?]

Thompson, on the other hand, also included points F, G, and H in his sequence. He then used his contrived number of 1.5.5.0 + 13.0 (13.0 because he assumed that the Maya had a scribal error being off 260 days) to form the DN correction. Then by applying correction numbers, he could include the points F, G, and H. His LC epochs are points E, G and H. The drawbacks to this sequence is that: 1) The DN correction is entirely contrived by Thompson, and 2) It does not include point D which Lounsbury (and others, including me), have conclusively proved must be in the sequence if you are to use any correlation in the vicinity of 584285.

In my sequence, I also include points F, G, and H. However, my DN correction is between A and F, using the 1.5.14.4.0 number from page 24. All of my LC epochs can then be created from using the 9.11.7.0 and 4.12.8.0 corrections that are adjacent to the 1.5.14.4.0 number. So my LC epochs are D, G, and H, the same as Floyd's. However, my sequence is more similar to Thompson's than Floyd's. I place my one double foreshortening in a different place in the sequence than he did, though, which causes me to hit point D, and thus also point Y. Point Y has huge advantages over point E that it replaces, in that things do not get excessively out of whack before a correction is done. You'll notice that my zigzag is amazingly regular. I don't see *any* obvious flaws with it.

One more thing: I think my sequence has more respect for the Maya. 1) I don't think the Maya made scribal errors of 13.0. 2) I think the Maya had been watching Venus a very, very long time, and would have already known that their whole number arithmetic could not describe the motion of Venus without correction by the time Dresden was composed, or even 9.9.9.16.0. 3) Neither Thompson's, nor Lounsbury's sequences allow for the Maya to have figured out the 1:4 double to single foreshortening ratio before Dresden was composed. My sequence doesn't positively confirm it, but it doesn't rule it out either.

Note that my sequence does leave some things in the Venus Pages unexplained: 1) The 1.5.5.0 number. 2) The 236, 90, 250, 8 periods. 3) The "zodiac" descriptions on pages 46-50. 4) the mentions of the moon on the texts on pages 46-50. I have looked at each of these, and have yet to form any firm conclusions on them, although I am working on some theories. I have no explanation for Mexican gods on these pages. I also have yet to decide whether I think the Maya were in the 18 Wo run or the 13 Mak run of the tables when Dresden was composed.

Greg

From: Lloyd Anderson
Sent: Thursday, March 2, 2000 10:32 PM
To: Gregory Reddick
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: Your zig-zag justifications

In a message dated 3/1/2000 5:40:49 PM, Gregory Reddick writes:

>http://www.xoc.net/maya/venuspages.asp#20000301

Very nice zig-zag chart.
I agree with your reasoning.
I especially liked numbers 2) and 3) in the following:

"One more thing: I think my sequence has more respect for the Maya. 1) I don't think the Maya made scribal errors of 13.0. 2) I think the Maya had been watching Venus a very, very long time, and would have already known that their whole number arithmetic could not describe the motion of Venus without correction by the time Dresden was composed, or even 9.9.9.16.0. 3) Neither Thompson's, nor Lounsbury's sequences allow for the Maya to have figured out the 1:4 double to single foreshortening ratio before Dresden was composed. My sequence doesn't positively confirm it, but it doesn't rule it out either."

On the other hand, your number (1) may not be an error at all. Check with Helen Alexander on the pattern of celebrating a day which just preceded an important event by precisely one Calendar Round. Such days, 13.0 in advance, are on Copan (and Quirigua?) stelas, before K'atun endings, thus on dates X.Y.19.5.0. I have a priori no reason to expect that any of them would reflect events similar to one intended by the 1.5.5.0 of the Dresden page 24. But is there any analog?

Lloyd

From: Gregory Reddick
Sent: Saturday, March 4, 2000 1:20 PM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: Dating Dresden

Lloyd-

Now that I think I've convinced you of my date sequence, I've started to move on to other things. I've started looking at how to date when Dresden was composed. These are the long counts that are talked about or derived from the Serpent Pages (pages 61-64, see http://www.xoc.net/maya/ring.asp) and the Venus pages (pages 26, 46-50). The implied dates in parens:

Notice that the dates overlap. From this, the earliest I would date Dresden's composition is 623AD and the latest is 1416AD. This should be a surprise to no one. The serpent numbers range from 779AD to 1052AD, forming a tighter range. I think it unlikely that the Venus pages were composed before 934AD and the latest I would think is 1324AD. The intersection of the Venus epochs and Serpent numbers is between 934AD (10.5.6.4.0) and 1052AD (10.11.5.14.5).

I don't make any strong conclusions from this, but do find it interesting. Notice, by the way that I have been careful to say "composed." Dresden may be a copy of an earlier work, so the actual painting of this copy may have come later. But the formulation of the content must have happened sometime in the wider range mentioned.

Greg

From: Gregory Reddick
Sent: Saturday, March 4, 2000 4:59 PM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: 1.5.5.0 number

Lloyd-

Okay, here's another speculation on how the 1.5.5.0 number might be used...I've already shown that the Maya could use the correction numbers on page 24 in a way similar to ring numbers, subtracting one then adding the other. What if you start at 10.5.6.4.0 1 Ahaw 18 K'ayab, subtract 1.5.5.0, then add 4.12.8.0 to reach 10.8.13.7.0 1 Ahaw 3 Sotz'. This reaches the last of morning sky recorded on the 9th line of page 47. Interestingly enough on this morning at sunrise, there was a conjunction of Venus with Jupiter. What's interesting about it is that the 1 Ahaw 18 Wo reached by the 1.5.14.4.0 also is a conjunction of Venus with Jupiter at sunrise.

I come to this conclusion because Floyd mentioned in passing in his 1.5.5.0 paper that the distance between the two 1 Ahaws on page 47 are exactly 9100 days. The second 1 Ahaw is the one a 4.12.8.0 correction reaches. So if you subtract a further 1.5.5.0 you reach the last of morning sky that is the earlier 1 Ahaw. Checking the astronomy on that morning produced the interesting result.

I don't know how common conjunctions of Venus with Jupiter are, but having them happen on days 1 Ahaw must be rare, and further happening on a day mentioned in the tables must be more rare.

Greg

From: Lloyd Anderson
Sent: Saturday, March 4, 2000 7:59 PM
To: Gregory Reddick
Subject: Re: 1.5.5.0 number

In a message dated 3/4/2000 9:03:33 PM, Gregory Reddick writes:

>I've already shown that the Maya could use the correction numbers
>on page 24 in a way similar to ring numbers, subtracting one then adding
>the other.

Well, I'm not sure you have shown that. Perhaps I missed it.

Your sentence above is ambiguous, "could use" maybe = "might have used" or "could use" maybe = "were able to use, as shown by the fact that they did use in X instance"

Perhaps most of the uses of those numbers on p.24 are as intervals to reach a fixed later date? That is the overwhelming structure in which two differnent intervals reach the same final date 1 Ahau 18 Wo (Or I tried that and got only two of them to work, the two already known?)

>What if you start at 10.5.6.4.0 1 Ahaw 18 K'ayab, subtract
>1.5.5.0, then add 4.12.8.0 to reach 10.8.13.7.0 1 Ahaw 3 Sotz'. This
>reaches the last of morning sky recorded on the 9th line of page 47.

Or it can be used to reach any other last of morning sky, as Floyd and others noted? (I have not phrased that exactly right, just pointing out the generality of it...)

>Interestingly enough on this morning at sunrise, there was a conjunction
>of Venus with Jupiter.

My memory when I checked astronomy for all of the dates you had included in previous messages was that I found a Venus-Jupiter conjunction or nearly on TWO of the dates. Don't remember at the moment which ones. Perhaps the same ones you are talking about. I would suspect there might be a reference in the text, but think not? Anyhow, I am inclined to

>What's interesting about it is that the 1 Ahaw 18
>Wo reached by the 1.5.14.4.0 also is a conjunction of Venus with Jupiter
>at sunrise.

That's the one I started from (your remark earlier?) and therefore noticed the Venus-Jupiter involvement at other dates too.

Lloyd

From: Gregory Reddick
Sent: Saturday, March 4, 2000 9:13 PM
To: Lloyd Anderson
Cc: Helen Alexander; Richard Johnson; Sharon Bowen
Subject: RE: 1.5.5.0 number

Lloyd-

| Well, I'm not sure you have shown that.
| Perhaps I missed it.

Well, I showed that there were three different ways you could get from one number to another. Refer to this message: http://www.xoc.net/maya/venuspages.asp#20000222. Add one, subtract the other. Subtract one, then add the other. Subtract one from the other and add the result. The more I think about it, I think it's the second one: subtract one, then add the other. I can't prove it, but there is precedent in the ring numbers. It doesn't really matter how they did it, because the result is the same; you reach the same date. I did show that they would use one number in a positive direction and the other in a negative direction.

| Or it can be used to reach any other last of morning sky,
| as Floyd and others noted? (I have not phrased that exactly right,
| just pointing out the generality of it...)

Yes, from any last of evening sky it will always reach a last of morning sky (or vice versa). However, this last of evening sky is actually special, because it is the one reached by the 4.12.8.0 number from a Venus epoch.

| That's the one I started from (your remark earlier?)
| and therefore noticed the Venus-Jupiter involvement at other
| dates too.

You're right. The 9.9.9.16.0 date has Venus and Jupiter very close to each other at sunset in the Western sky. [BTW, other notable astronomic events: 10.0.13.14.0 Venus comes very close to occulting Spica. 10.5.6.4.0 Venus in conjunction with Mars. 10.10.5.3.0 comes immediately after an transit of Venus. 10.15.4.2.0 Venus in conjunction with Jupiter.]

So if the Venus tables were composed shortly after 10.15.4.2.0 1 Ahaw 18 Wo, they might have been recording when previous conjunctions with Jupiter occurred, thus justifying both the 1.5.14.4.0 number and the 1.5.5.0 number. So they would be working *backwards* from 1 Ahaw 18 Wo, not forward. I will need to do a test to see how many times Venus is in conjunction with Jupiter on a 1 Ahaw in the date range. If it only shows these three times, I might feel justified in arguing that the 1.5.5.0 number is used that way. It would also give a reason *why* they picked 9.9.9.16.0 as the base on page 24. It sure would be nifty if one of those glyphs on the pages substituted for Jupiter.

Greg

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