INTRODUCTION:

A Mayan abacus was suggested by George I. Sanchez, "Arithmetic in Maya" in 1961. The abacus was parsed from Lowland Mayan in the 1950's while working with Morley and other Mayanists.

Sanchez's Mayan abacus was 3 x 4 on one level, accidently agrees with a Canary Island 3 x 4 red and black colored 3 x 4 12-moon acano calendar . The acano calendar predicts lunar eclipses within 270-moon cycles, and not Mayan 405-moon 260 day calendars.

Mayan calendars seem not to predict 46 lunar eclises within a Canary Island 270-moon calendar per TARA: A STUDY ON THE CANARIAN ASTRONOMICAL PICTURES Part II: The acano chessboard (Jose Barrios Garcia, Departamento de Analisis Matemetico Universidad de La Laguna 38271 La Laguna (Tenerife) Canary Islands. SPAIN. Jose Barrios Garcis reports:

"As Aaboe (1972) has shown, this ancient 135-moon eclipse count, most likely known in Babylon and Egypt and certainly known in China and Mesoamerica, can be easily derived with a simple arithmetical scheme from a good estimation of the eclipse year and the eclipse limits.

As a matter of fact, the calculus proposed by Aaboe (1972) can be performed on the acano in a much more simple and effective way. Indeed, once the eclipse year is measured and the date of a central eclipse is recorded, one can easily compute on the acano the dates of the successive solar passages by the nodes simply each time jumping 6 moons and subtracting 3 o 4 days. This calendar can be easily adjusted from time to time by actual observation of eclipses. Note that the date of a solar passage by a node locates on the acano the solar and lunar eclipses occurring at that node, provides a measure of their respective magnitudes and help to separate the eclipse limits of the sun from those of the moon. This calculus also provides a simple and graphic method for searching eclipse cycles. The 24-half moon acano is especially suited for the record and graphic calculus of eclipses. "

attachment:

Paper read at the IV SEAC International Conference on Archaeoastronomy and Ethnoastronomy Salamanca (Spain), 2-6 September 1996

TARA: A STUDY ON THE CANARIAN ASTRONOMICAL PICTURES Part II: the acano chessboard

IV SEAC International Conference on Archaeoastronomy and Ethnoastronomy Salamanca, Spain, 3-6 September 1996.

Publicado en: C. Jaschek & F. Atrio (eds.), Proceedings of the IVth SEAC Meeting "Astronomy and Culture" (Salamanca, Spain, 3-6 Sep. 1996).

Salamanca, Universidad, 1997, pp: 47-54.

Abstract

In the first part of this paper (SEAC Conference’95) I studied the archaeological, ethnohistorical and linguistic evidences that led me to propose that in 14th-15th centuries the Berber populations of Grand Canary Island systematically recorded numerical, astronomical and calendrical data by mean of certain geometrical figures named tara, painted in white, red and black on wood planks and on the walls of certain caves. One main conclusion was the use of a chessboard of 3 vertical x 4 horizontal squares (that I name acano) to represent 12 moons.

In this second part I study for first time the acano as a lunar calendar and show how to number its squares to force the solstitial, equinoctial and eclipse moons to move across the board with very simple and stable patterns. These patterns provide a safe and clear mnemonic guide for performing on the acano an easy calculus of seasonal and eclipse moons over extended periods of time, just using the difference in days of the lunar year with either the solar year or the eclipse year to perform an elementary saw function on the squares. This calculus establish the octaeteris, the metonic cycle and the 135-moon eclipse cycle as basic periods of the acano.

It is well known that the Canarians observed the summer solstice and had important festivals on the crescent moon that followed, so to complete the evidence I present two notices from ancient written sources supporting that they measured one and half eclipse year as 520 days.

The proposed calculus on the acano would reveal an unsuspected high level of Canarian mathematical astronomy and pose the question of the origin of this set of techniques.

Dedication

It is a happy coincidence that a number of important notices on Canarian astronomy come from the writings of the Canarian doctor Tom‡s Marin de Cubas (1643-1705), who study medicine and taught astrology at this Salamanca University. To him is dedicated this painted paper.

The acano chessboard

In the first part of this paper I concluded that in 14th-15th centuries the Canarians (Berber populations of Grand Canary) systematically recorded numerical, astronomical and calendrical data by mean of certain geometrical figures named tara, painted in white, red and black on wood planks and on the walls of certain caves. One main conclusion was the use of a painted chessboard of 3 vertical x 4 horizontal squares to represent a count of 12 moons. Since written sources use to call acano or achano the Canarian lunar year, I am going to take the freedom of naming also acano this particular chessboard.

Figure 1. Two versions of a black-and-red acano

The Painted Cave of G‡ldar

The main archaeological evidence on the use of the acano chessboard as a lunar count comes from the decoration of the left and central panels of the Painted Cave of G‡ldar (Grand Canary Island). An artificial cave located very close to the ‘palace’ of the guanartemes or ‘kings’ of (northern half of) the island in the centuries preceding the Spanish conquest of the island, occurred on late 15th century.

Figure 2. Left panel (4.2 x 1.2 m)

The cave was casually found in 1873, declared National Archaeological Monument in 1949 and National Monument in 1972. It is known that there were other painted caves in the island with similar decorations but this is the only one preserved, so its great archaeological importance.

Figure 3. Central panel (5.0 x 1.3 m)

At present the site around the cave is being object of a deep archaeological study formerly directed by late Prof. C. Mart’n de Guzm‡n and now by Dr. J. Onrubia Pintado. Their researches have confirmed the extraordinary archaeological richness of the site, stratigraphical data using some 30 radiocarbon dates having proved a continuous occupation of the site during at least the whole nine centuries preceding the Spanish conquest, that is to say, from early 7th to late 15th century AD. The chronology of the Painted Cave itself remains unknown, most likely comprising at least the last two centuries preceding the Spanish conquest.

Lunar counts

Let an acano of 3 x 4 squares painted in red and black represents a count of 12 moons (synodic months). Certainly its pattern codes a cultural classification of the moons that elegantly synthesises the basic arithmetic of number 12.

By colour they are classified as red or black, the pattern suggesting that colour alternates from moon to moon, that is, odd moons are black and even moons are red or vice versa. By columns they are divided into 4 groups of 3 moons what immediately suggests a solar division according to equinoxes and solstices. Their division by files in 3 groups of 4 moons is less clear although there are examples in other African calendars.

In any case, the 12 moons must be counted on the acano in a certain order that needs to be investigated. Naturally, this order was culturally determined so, potentially, may vary from the most natural to the most unexpected one. As a matter of fact, basic combinatory establish that there are exactly 12! different ways of numbering the acano, that is to say, some 500 millions possibilities. In this paper I have try to deal only with the most simplest assumptions.

To illustrate the situation Figure 4 shows three different counts defined by the arrows. The first count is vertical, the second one is horizontal, and the third one is diagonal1. How to choose the correct one, if any?

1 Each numbered acano only represents one possible numeration compatible with the pattern defined by the set of arrows aside. Each pattern could be equally read from bottom to top and/or right to left.

1 4 7 10

2 5 8 11

3 6 9 12

1 2 3 4

5 6 7 8

9 10 11 12

1 2 4 7

3 5 8 10

6 9 11 12

Figure 4. Three patterns of counts

Certainly, there are several criteria we can try to reduce the search. For example:

1. 1. If colour alternates from moon to moon I would retain the first possibility and disregard the other two.

2. 2. The second criterion locates the solstitial and equinoctial moons assuming they are 3-month spaced. Certainly, I would expect the four stations of the sun nicely disposed on the pattern. Figure 4 shows on the right the seasonal moons generated by each count assuming that summer solstice occurs in the first moon. On this base I would retain again the vertical count and reject the other two.

I have tried with these criteria a number of patterns and all of them points to the vertical count as the best choice. Furthermore, it is the only one compatible with the most natural solar division of the acano by columns. On this base I am going to centre my study on this particular count2.

Solar counts

A first property of the vertical count is that solstitial and equinoctial moons are forced to be aligned on the acano. Once you know where is one, you know where are all. This is very good for tracking the movement of the seasonal moons across the lunar calendar.

Indeed, in round numbers, 12 moons are 11 days shorter than a solar year, so in a lunar calendar solar dates occurs 11 days later each year. For this reason seasonal moons jump each 2-3 years to the next square of the acano. Each time a seasonal moon jumps, the other three jump behind since they are never less than 3-moon spaced, so they are kept aligned.

If we follow the movement of the seasonal moons through a whole round of the summer solstice across the acano we get the basic calendar of Figure 5 adjusted to the Metonic cycle

2 It is worth noting that most of Lybic-Berber alphabetical inscriptions of the Canary Islands are written in vertical lines.

(19 solar years ≈ 19 lunar years + 7 moons). Note that the equivalence “8 solar years ≈ 8 lunar years + 3 moons”, base of the octaeteris, measures the pass of the sun through a column3.

8 solar years 8 solar years 9 solar years 8 solar years

99 moons 99 moons 111 moons 99 moons

Figure 5. Lunisolar calendar

It is true that this basic calendar must be adjusted from time to time since not always summer solstice jumps in first place and, on the long run, the whole pattern shifts. But using this basic scheme as a guide you know how things work and where are the critical points.

As a matter of fact, to record a date on the acano you only need to write a number from 1 to 30 on one of its squares. The selected square fixes the moon while the number fixes the day of the moon counted, let us say, from new to new. Accordingly, it is possible to record unambiguously on a single acano the 33 successive dates fixing a whole round of the summer solstice through the lunar year. What is of the utmost importance is that this can be accomplished either through the years by actual observation, either at any desired moment by performing an easy arithmetical exercise on the acano.

Indeed, once recorded on the acano the date of a particular summer solstice, we obtain the dates of the next summer solstices simply adding 11 days by year to the previous number. Each time the accumulated shift is greater than 29 or 30 days, we jump to the next square, reduce the shift by 29 or 30 days, write the new date on the square and continue the count. Actually, this exercise can be done even mentally for a number of years.

This is a very easy and natural way for computing on the acano a reliable ephemeris for the summer solstice for a number of years. Such ephemeris would provide the backbone of the lunisolar calendar. This basic calendar can be easily adjusted for the other three seasonal moons with an estimation of the number of days between equinoxes and solstices4. The same calculus provides ephemerides for the stars and even for the planets, provided their shifts are known. This working model can be easily adjusted from time to time by mean of actual

3 8 solar years ≈ 8 lunar years + 3 moons ≈ 5 Venus synodic revolutions.

4 Note that this calculus lies at the roots of Babylonian ephemerides (Neugebauer 1975 I).

observation on well located points of the acano, or just using longer and more accurate cycles.

On the other hand, it is well known that Canarians observed the summer solstice and had important festivals just on the moon that followed, so certainly they were in a very good position to measure the 11-day difference between lunar and solar years.

Eclipse counts

Now it is a pleasure to answer the insightful question that Dr. Arnold Lebeuf posed me after the Sibiu Conference, when he noted me that, since red is the colour of the eclipsed moon and black is the colour of the eclipsed sun, I should look for eclipses in the Cave’s decoration. What I promised him.

Indeed, as Prof. Neugebauer (1975 I: 525) stated, “it is probably one of the oldest empirical discoveries in astronomy that lunar eclipses are spaced regularly in 6-month intervals with an occasional 5-month gap between very small eclipses.”. So the second notable property I have discovered in the vertical count is that eclipse moons are forced to be paired and move backward on the acano with the concise pattern of Figure 6. Furthermore, this pattern is based on a 135-moon eclipse cycle decomposed by three 5-moon jumps, the eclipse moons traversing the acano in twice 135 moons. Figure 6 displays a standard count of the 270 moons fixing on the acano the whole 46 possible eclipse moons.

1 7

2 8

3 9

7x6+5 moons

6x6+5 moons

7x6+5 moons 6x6+5 moons

4 10

5 11

6 12

7x6+5 moons

7x6+5 moons

Figure 6. Eclipse calendar

As Aaboe (1972) has shown, this ancient 135-moon eclipse count, most likely known in Babylon and Egypt and certainly known in China and Mesoamerica, can be easily derived with a simple arithmetical scheme from a good estimation of the eclipse year and the eclipse limits.

As a matter of fact, the calculus proposed by Aaboe (1972) can be performed on the acano in a much more simple and effective way. Indeed, once the eclipse year is measured and the date of a central eclipse is recorded, one can easily compute on the acano the dates of the successive solar passages by the nodes simply each time jumping 6 moons and subtracting 3 o 4 days. This calendar can be easily adjusted from time to time by actual observation of eclipses. Note that the date of a solar passage by a node locates on the acano the solar and lunar eclipses occurring at that node, provides a measure of their respective magnitudes and help to separate the eclipse limits of the sun from those of the moon. This calculus also provides a simple and graphic method for searching eclipse cycles. The 24-half moon acano is especially suited for the record and graphic calculus of eclipses.

On this respect it is worth noting that, although astronomical counts with explicit numbers are certainly rare in Canarian written sources, at least two of them can be related with accurate measures of the eclipse year, what undoubtedly lends some support to the above proposed calculus.

The first one was recorded in 1592 by the Italian engineer Leonardo Torriani in his celebrated description of the Canaries. After him, Guanarteme ‘El Bueno’, one of the last guanartemes of G‡ldar, said to the Portuguese invader Diego de Silva [c. 1460 AD].

… que si nos bastara el haberos ahuyentado infinitas veces de nuestras costas y dado muerte, y muchas veces detenido como prisioneros (como de vuestro obispo Diego L—pez lo sabŽis, 520 esplendores de la luna que es nuestro cautivo), podr’amos hacer cuenta de que la ira de Dios se ha aplacado contra nosotros…

Torriani (1978 [1592]: 124)

Note that 520 days is an accurate count of one and half eclipse year. Since apparently the bishop Diego L—pez de Illescas never was retained prisoner in the island, at least for so long time, it seems that Torriani recorded a notable 520-day eclipse count related with a symbolic prison (eclipse) of the invader’s religious chief.

The second count was recorded twice by Tomas Marin de Cubas.

Contaban el a–o llamado acano por las lunas, de veinte i nuebe soles, ajust‡banlo por el st’o onde en la primera luna hac’an nuebe d’as de fiestas i regocijos a el recojer sus cementeras, pintaban en unas tablas de drago i en piedras, i en paredes de las cuebas, con almagra, i rayas, i otros caracteres llamados tara, i onde los pon’an tarja a modo de scudos de armas, dec’an que su origen era de la parte de elsur de çfrica i tambiŽn se–alaban a el oriente; y segœn dec’an era mui antigua la poblaci—n de yslas.

Marin (1986 [1687]: 77 v.)

Contaban su a–o llamado Acano por las lunaciones de veinte y nueve soles desde el d’a que aparec’a nueva empesaban por el st’o, quando el sol entra en Cancro a veinte y uno de junio en adelante la primera conjunci—n, y por nueve d’as continuos haz’an grandes vailes y convites, y casamientos, haviendo cojido sus sementeras haz’an raias en tablas, pared o piedras; llamaban tara, y tarja aquella memoria de lo que significaba.

Marin (1986 [1694]: 254)

Although the confuse redaction of Mar’n suggests a 29-day synodic month, note that a 29-day month is easily derived from the earlier 520-day eclipse count (520 Ö18 ≅ 29). So, for example, six 29-day months adds up to 174 days, a good and practical approximation of half eclipse year but a very crude estimation of half lunar year.

Acknowledgements

I want to thank Drs. Arnold Lebeuf, Elzbieta Siarkiewicz and Mariusz Ziolkowski for many enlightening conversations on technical and cultural aspects of time reckoning.

References

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