A review to the introductory pages of the Table of Eclipses of the Dresden Codex is made. In particular, the section of three intercalated Maya numbers in columns A, B, C, of page 52a and the list of thirteen 13’s (two bars and three points) present in column D of the same page are studied and interpreted as a way to construct the sequence of 69 eclipse prediction numbers for a total period of 11.958 days. Those intercalated numbers (red & Black) in the codex represent big numbers that are multiples of the 11.960-day period. In combination with the draconic month (as studied by Chinese culture (27,2122)), the sequence of 177, 178, 148 days, intervals can be reproduced. The consequent conclusion is that these intercalated numbers could represent numerical fractions.

Keywords: maya numbers, fractional numbers, eclipse prediction, moon’s latitude

In this letter, I am presenting a possible interpretation of the intercalated numbers written in the section of aperture of the Table of Eclipses that are present in the page 52a of Dresden Codex. This new interpretation suggests that Maya Mathematics has made the use of fractions while generating the table for eclipse prediction inscribed in pages 53a to 58a and following from 51b through 58b. Also, a possible procedure to have the sequence of lunations will be discussed.

The Dresden Codex is one of the four original codices attributed to the Maya Culture before the arrival of Spaniards. The other three are: Paris Codex, Madrid Codex and Maya Codex, which was known as the Grolier Codex.¹ It is also, the codex having the greatest number of astronomical features.² For example, the Table of Venus, the Table of Mars and the lunar cycles, are some of the most important ones. The Table of Eclipses is the main issue analysed here.

The Dresden Codex is a folded book made of amate paper, covered by a thin layer of stucco in order to give a surface apt for writing. It has been accepted that the writing is in original Mayan hieroglyphic language, and it has 39 leaves, most of them are painted on both sides. The size of each leaf is 20,4 cm height by 9,0 cm wide; the total length of the codex is 3,50 m. Its name comes from the fact that it is conserved at the Saxony State Library in Dresden, Germany.²

In Figure 1A, it can be seen the page of interest for this proposal: 52a. There are several columns with hieroglyphic text at the top of it that can be read partially. The maya numbers present in the middle part of the first three columns are count of days as stated by Erik Velazquez in his interpretation. In Figure 1B, a schematic description of the page is given following the analysis made by Bricker. It is of highest interest to look at them and at the 13 13’s in the central column. The three intercalated pairs of numbers adjacent to it have been deciphered due to its colours: Red and Black.

As it can be seen in the figure, the six numbers of the three columns are related with the whole extent of the table: the number of days reach multiples of 11.960 (rounding total of 11.958 corresponding to 69 predicted eclipses). I will come back to these numbers ahead. For the moment, the series of 13’s has been interpreted as a way to recycle the table but without any mention a possible way to use it: Eric Velasquez says that “As Bricker and Bricker (2011) say, the rest of the sentence is a number of 1.828 days (5.1.8), which serves to update the table and be able to use it in the future, leading to new start dates, the same function that they attribute to the thirteen red numbers 13 located in the lower half of this column” (2016: 74). Other interpretations of this column of 13’s refers to a very big number (13 levels meaning an infinite) or as the thirteen levels of the world incorporated in the Maya cosmovision.

Figure 1 Contents of page 52a: a) Facsimile of the original codex;² b) Section of Table 1 of the schematic deciphered contents of the page.⁷

From page 53(a) to 58(b) there are a total of 79 columns (69 numeric plus 10 including figures), for my proposal, only the sections of the black number that represents the number of days accumulated from the beginning column of the table and the number corresponding to 177 or 148, which in Mayan notation are written (8.17) and (7.8), respectively, in the lower part of each column (see Figure 2A. In several places, the cumulative total reaches a number that assumes a 178 has been added but in the lower part a 177 is written: the sequence of lapses’ days for the occurrence of an eclipse could be, only, 177, 178 or 148³ as it can be seen in Figure 3B.

Figure 2 Lapses of days for the occurrence of an eclipse in Dresden Codex: a) Part of the facsimile pages (53a, 54a, 55a),² with accumulation in Red; b) sequence of 69 intervals of days.³

It can be probed, as I myself do in another paper, that the dates corresponding to the sequence of 177, 178, and 148 days, lapses can be correlated with eclipses observed at somewhere on Earth, not necessarily in the zone of Maya influence.^3,4

I now ask the critical question about the Table of Eclipses of Dresden Codex: Is there any relation between the information in page 52a, mainly the list of 13’s and the multiples of 11.960, and the sequence of intervals of Figure 3B? Any of the many studies have addressed ideas towards this question.^5,6

In order to give a possible answer to this matter, I will discuss briefly the lunar observations made in ancient China. Christopher Cullen has published a very interesting book, “Heavenly Numbers” (2021), about the astronomy in the early imperial China. I am using the Table of Ying-Yang Sequence shown in Table 1. The contents of the table give us a very important information about what is called the Draconic month: a cycle of 27.212220 days taken by the Moon to make two passages through the same node (intersection of its orbit with the ecliptic).

The column 1 gives the number of days for the Moon being above or below the ecliptic, because the angle between the orbit of the Moon and the ecliptic is about 6 du, which means that the alignment of the three celestial bodies (Earth, Moon and Sun) and, in consequence, the propitious moment for an eclipse (lunar or solar) is of 13 days and a fraction. The corresponding fractions shown in last row of the table are, for a whole cycle 0,13363 and 0,64693. In the real world both fractions change due to several astronomical factors like precession or tidal forces. The reported ones are long term means.

The extension in space of the Moon and Sun makes possible three types of solar eclipses: partial, annular and total

The number thirteen (13 or two bars and three points) becomes relevant to the observation of the latitude of the Moon. Let me show how the 13’s and a fraction could work to obtain the sequence of 69 intervals of the table shown in Figure 2B.

The first thing to notice is that summing 13 13’s we arrive to 169 days. Then we can assume that column D of page 52a (see Figure 1B) make a base of 169 that need 8 days to arrive to 177 and 9 days to arrive to 178. In Table 2, I show the fractions needed to generate the three lunation cycles for an eclipse, assuming that the fractions are the same for all the periods.

The proper application of this table 3 would allow to reproduce the whole 69 intervals of the sequence; however, if one think about an alternative and nonhomogeneous way to arrive to it, it is possible to assign (observed?) fractions to the sequence. This is why I have look forward to the intercalated numbers in columns A, B, C, of page 52a (see Figure 1B). Taking the ratio between Black and Red numbers, we arrive to the following three fractions: 3/5; 2/3; and 0,28804.

My interpretation of these B&R numbers on the Dresden Codex because they are not exact, would imply that in constructing the table, some observational criteria could have been used as it is shown in an alternative reconstruction of the sequence of Table 4.

Table 4 An alternative way to build the intervals sequence

I can conclude with these words: In one of the introductory pages (52a) of the Table of Eclipses of the Dresden Codex, the list of 13 13’s in Column D, can be used coupled to the intercalated B&R numbers of columns A, B, C, representing fractions, to elaborate the sequence of lunations (intervals of 177, 178, 148 days) on the remaining pages of the table.

It follows to look forward for instructions to be made in order to have the correct sequence. The texts on the top of the columns must be interpreted accordingly.^7,8

I want to give thanks to the collaborators of the project of MAMH (Monumental Antikythera Mechanism for Hermosillo) for their useful comments: Ezequiel Rodríguez, Julio Saucedo, Alfredo Carmona, and J. Clemente Olvera. Also, my appreciation to Eduardo Rodas, Jesús Galindo and Stanislaw Iwaniszewski, for their critics and for being introduced in the field of Dresden Codex.

Author declares there are no conflicts of interests.

None.

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