Cycles, Mars, Moon and Maya Numbers. Abstract

Transcription

1 Cycles, Mars, Moon and Maya Numbers Laura Elena Morales Guerrero Centro de Investigación y Estudios Avanzados Cinvestav, México, D. F. lem@fis.cinvestav.mx lemgro@gmail.com Abstract We present here a numerical analysis of planetary and calendric cycles reinforcing the fact that the Maya had some knowledge about Greatest Common Divisor and least common multiples concepts as established somewhere else. Planetary cycles and time counting were of capital importance for them as shown in the Dresden Codex. We will learn about planets and Moon cycles and read several pages related to various astronomical facts like Mars behavior and solar-moon eclipses. To finish, we will decode some crucial pages on Maya numbers in mentioned original text. In addition we include an appendix to complement the information on Mars and show new exceedingly large numbers. Resumen Presentamos un análisis de los ciclos calendáricos y planetarios para reforzar lo dicho sobre el conocimiento que los mayas pudieron haber tenido de los conceptos máximo común divisor y mínimo común múltiplo, en otros textos. El códice Dresde muestra que los ciclos planetarios y la cuenta del tiempo fueron de importancia capital para ellos. Aprenderemos aquí sobre planetas, y ciclos lunares; leeremos páginas relacionadas a varios hechos astronómicos como el movimiento de marte, los eclipses solares-lunares además de decodificar páginas cruciales en el tema que aparecen en el texto original. Mostramos además un apéndice con información complementaria sobre Marte y números nuevos extremadamente grandes. Key words: Maya astronomy, Maya arithmetic, Mars, eclipses, lunations Introduction Cycles The Maya recognized several forms in which their astronomical observations of planets, Moon and Sun related. For instance, in the relation between the days of the ritual and agricultural calendar named Tzolkin the sacred almanac- of 260 days and the Haab-Uayeb year of 365 days they knew that each 52 years the calendars would coincide. This cycle is 47

2 known as the calendric round. The three more common cyclic computations performed by Mayas were the sacred almanac of 260 days (Tzolkin), the common year, or vague year, of 365 days and the calendric round of 52 years, calculated by Mayas. Using the calendar round, we can calculate the common cycle for the aforementioned calendars by looking at their prime factors: the numbers 2, 2, 5 y 13 for 260, and 5 and 73 for 365. We don t know what the Mayas did to find the 52 years. For us, the GCD of 260 and 365 is 5 and their lcm Those numbers are easily obtained from the factors: the least common multiple (lcm) of 260 and 365 is 2 2 x 5 x 13 x 73, that is, days. This is equivalent to 52 years (of 365 days per year) in common since = 52. Given the GCD is 5, then, 260 x 365=94,900, divided by 5= For all those and similar calculations they, alone or combined with their multiples and divisors, the Mayas distinguished among them the specific values of 4, 5, 9 and 13 represented in the Chichén Itzá pyramid. Planetary cycles When one considers the movement of a planet the cycle that governs its position in the sky is the synodic cycle, that is, the time it takes the considered planet in acquiring the same position relative to the Sun and the Earth. The Maya observed and recorded the movement of planets Venus and Mars, at least, as well as the Sun and the Moon to produce precise calendars. Omitting Venus, studied somewhere else, we mention that Mercury and Venus are inferior planets and make their appearance each one as evening and morning stars. Both can be seen mornings and evenings even though Mercury is much more difficult to see. 48

3 For Mars we can speak of the correlation between the 365 days of the terrestrial revolution and the 780 days of the Martian period. Results from multiplying 13x60, that is, 13x3 uinal, where 1 uinal has 20 days gives 780. For Earth and Mars cycles: = 284, 700. The GCD between the cycles is 5, then, = 284, 700 / 5 = 56,940 So, = 284,700. The lcm is 22 x 3x 5 x 13 x 73 = From this, 156 revolutions of the Earth are equivalent to 73 revolutions of Mars. Below, see Figure 1, is shown page 43 from the Dresden Codex (see References) in which appear numbers related to the visibility of Mars, solar and lunar eclipses according to experts in astronomy. In the first left column, below the date 3 (4 in original copy) Lamat and below what we call the Mars glyph (also visible twice in p. 44), one reads (from top to bottom): , 3 Lamat, equivalent to days, that is 780 x [The total of days was obtained from: (9 x = ) + (19 x 7 200= ) + (8 x 360=2 880) + (15 x 20=300) + (0 x 1=0).] Factorization gives: 2! A number like this is called a companion number by Satterthwaite (1964) and long reckonings by Thompson (1972) when its role in counting dates is analyzed. The previous date is followed by: Ahau, = (17x20) + (12x1) = =352 days, numbers related to Mars visibility. 49

4 Figure 1 Martian cycle multiples appear in the following 5 columns, they are: 2x780, 3x780, 5x780, 140x780, 168x780 and 194x780, as well as the numbers 3 380, , and , that are days concerned with Mars visibility in the sky. Reading those 5 columns we have: 2 nd column , 3 Lamat: (15x7 200)+(3x360)+(6x20)+(0x1)= = =780x , 3 Lamat: (1x )+(1x7 200)+(0x360)+(6x20)+(0x1)= = =780x rd column , 3 Lamat: (18x7 200)+(4x360)= = =780x , 3 Lamat: (10x360)+(15x20)= =3 900=780x5. 4 th column , 3 Lamat: (10x7 200)+(2x360)+(4x20)= = = 780 x , 3 Lamat: (9x360)+(7x20)= =3 380 = 780 x th column 50

5 , 3 Lamat: (9x7 200)+(13x360)+(6x20)= = Change to then = 780 x 80 (see below) , 3 Lamat: (6x360)+(9x20)= =2 340=780x3. 6 th column , 3 Lamat: (4x7 200)+(5x360)+(17x20)= = = 780 x , 3 Lamat: (4x360)+(6x20)= =1 560=780x2. The previous information together with the one given in page 44F (shown in appendix along with page 59) permits us to write the Martian sequence as: 78 x 1, 78 x 2 78 x 9, 10, 20, 30. Then follows: 780 x (= 3 380), 780 x 5 (= 3 900), 780 x (= ), 780 x 20 (= ), 780 x (= ), 780 x 80 (= ), 780 x (= ), 780 x 140 (= ), 780 x 168 (= ), 780 x 194 (= ). A correction can be made here in what appears to be a copyist s error, changing to ( to in the original Maya form). The last eight numbers of these: , , , , , , , , are good approximations to a solar eclipse interval, according to Smiley (1965). We know that information about planet Saturn was found in a Mayan classical inscription (Tablero I of Dumbarton Oaks Relieve) by James Fox and John Justeson according to Morley (1998: p. 550). For Saturn one has 378 days in its synodic cycle. This can be written as 3x7x18 = 378 and 29x13 + 1, or, 19x20-2 days, in terms of relevant Maya numbers. According to our calculations the product of Saturn and Earth cycles is = The GCD is 1 and its lcm is 2 x 33 x 5 x 7 x 73 = Hence 51

6 Earth makes 378 revolutions and Saturn 365 but we don t know yet if Mayas registered, somehow, this relation. The assertion is valid also for planet Mercury and Earth cycles. Cycles Combinations In the same way it happens with ciphers to reach the Mayan Age of Tun years (the 13 Baktún), the planets orbits also registered several combinations. Thus, 780 Martian revolutions multiplied by 9: 9x780=7 020, are the same number of days as 60 revolutions of Mercury: 117x60= This comes also from the multiplication of 12 Venus orbits of 585 days: 12x585= Other well known relations are: for 5 Venus cycles (5x584=2 920) there are 8 Haabs (365x8=2920) and nearly 99 Moon revolutions (99x29.5= ). Two Calendar Rounds (52x2) equal 65 Venusian cycles (104x365=37960= 65x584). These numbers are all shown in the Dresden Codex. Moon and Sun cycles, their relations and eclipses were as well of capital importance to the Maya and gave rise to profound studies to determine their sequences. Eclipse Table There is another table in pages of the Dresden Codex that contains predictions over lunar phases during about 33 Haab years of 360 days (equivalent to days), probably dates of new moons (Teeple 1931: p. 86). The Lunar month began with the appearance of the new moon, Landa reported in Relaciones. Evidence of lunar dates begins at (357 AD) in an inscription in Stela 18 from Uaxactún for a moon age of 25 days. Much of the Maya computations recorded in inscriptions are by moons (Teeple 1931: pgs. 48, 50, 64). 52

7 It is seen that in each 2 Tzolkins, that is, in each 2x260=520 days, three eclipses occur, so, each days, one happens. This way, in about 33 Haab years of days, 69 eclipses take place. The next figure is page 56 of the Eclipses Table in the Dresden codex. Figure 2 Astronomer Charles Smiley (1965) presents evidence that the Maya were searching for a pattern for the dates of solar eclipses. It is impossible to determine from the Table s form whether it is used for solar or lunar eclipses since the Table could be employed for both with the addition or subtraction of fifteen days. This table is universally interpreted as a series of eclipse-intervals. It contains an upper register designated a and a lower register 53

8 designated b. This is the old pagination which is still used because of all the past studies which use this system. The Table can be divided into four sections: glyphic text; pictures; numbers (bars and dots); and dates. The dates divide the numbers into a bottom section of eclipse intervals and an upper section of totals. The totals, intervals, and dates make up the heart of the Eclipse Table. The observed quantity of 177 days is not a regular pattern since in the codex, after the date (p. 52ª) there is a period of 148 days. In page 57ª one also finds another of 178 days. The big question has been where to place the 148 and 178 (p.57a)-day periods that seem to be needed to adjust for a realistic value for the Moon period. Förstemann and others studied the table carefully and have shown it is an arrangement of 405 consecutive moons in 69 groups of 5 and 6 moons each. Of the 69 groups, 53 are of 6 moons=177 days, 7 groups are of 6 moons= 178 days, and 9 groups, each followed by a picture, are of 5 moons=148 days (Teeple 1931:p. 86). In the Maya tables for eclipses it is noteworthy to see there are written not only the visible eclipses in Yucatán but also the non visible ones that according to their calculations would occur by the interposition of the moon, earth and sun and would be visible somewhere else in the world. Lunations From Time and Reality by M. Leon-Portilla (1988: p.13) the so called lunation period is a cycle that corresponds to a complete revolution of the Moon around the Earth. In pages (Leon-Portilla, 1988) one can see what is known as the Eclipse Table used by the Maya to predict Sun and/or Moon eclipses as explained above. 54

9 The eclipse table is valid for predicting 69 possible eclipses in a period of nearly 33 years. The total number of days recorded (11,959) ( days = 46 cycles of 260 days) is only of a day less than the true time computed by the best modern method, certainly a remarkable achievement for the aboriginal mind. (Morley, An introduction to, Chap. II). By means of this table they were able to reduce the discrepancies among the new moons forecast by their computations and the new moon of astronomical actuality. The Maya may have elaborated their lunar corrections as follows. We may suppose they began with a lunar cycle of 30 days but soon they d realize that real new moon were present before that time. Then perhaps they d have assigned 29 days for a lunation only to discover that these lasted longer. Maybe then they tried successive lunations alternately that is, of 29 and 30 days. But this correction also failed. Two lunations registered that way give an average of 29.5 days per lunation but the real number is slightly higher. Modern astronomy gives us a value of days so that lunar calendar was ahead of time by of a day per each lunation in an accumulated error that would amount to a whole day after two years and two thirds (Morley, 1956, p. 548). One could conjecture that as time went by the Maya determined that days would correspond to 149 moons which would amount to days (4400/149). This value approximates quite well the real modern period of days. The 405 consecutive lunations (some 32 and ¾ of a year) are disposed in 69 groups, sixty of them are composed of six lunations each and the other nine by five each one. In the 60 groups of six lunations each one amounts to a total of 178 or 177 days, according to the use of 3 or 4 months of 30 days to give: =178 days or, 55

10 =177 days. In the other nine of five lunations one finds periods of 148 days. In those pages (51-58) one could read a solar eclipse table given that the last days of each of these groups are those in which, under certain conditions, a solar eclipse would be visible in some place over the earth. The additional lunar months of 30 days are so skillfully interpolated that in any moment during those 405 successive lunations it amounts to not even a day the difference between the calendar position and the real advent of new moons. This, comments Morley (1956, p. 289), is a truly colossal achievement for any chronological system, whether ancient or modern. According to modern calculations, the period of the lunar revolution is , or approximately 29½ days. For 405 revolutions the accumulated error would be =12.15 days. This error the Maya obviated by way of using values of days, the Maya understood the lunar cycle in numerical ways that are equivalent to 29.5 in some calculations and 29.6 in others, the latter offsetting the former. Thus the first 17 revolutions of the sequence are divided into three groups; the first 6 revolutions being computed at 29.5, each giving a total of 177 days; and the second 6 revolutions also being computed at 29.5 each, giving a total of another 177 days. The third group of 5 revolutions, however, was computed at 29.6 each, giving a total of 148 days. The total number of days in the first 17 revolutions was thus computed to be =502, which is very close to the time computed by modern calculations, (Morley,1956) In our study of the table in page 58b one reads: (2 nd col.) equivalent to days. But from (Teeple 1931, p. 86) the table numbers total days, however, apparently the length of the group for computation was intended to be days 56

11 ( ). We also see multiples 2, 4 and 5 of are found in the context. They are 2x5 980 = ; 4x2 990 = ; 5x2 392 = Registered planet orbits and the calculations for eclipses involve decimals. But Mayas did not write decimals. However as Smiley says, the Maya did not have decimal fractions but they seem to have had something almost as useful (Smiley 1965, p. 129). He based his conclusions in the following interesting situations. Let s take a look at pages of the Dresden Codex, there we find a list in order of the integral multiples of 65 up to 28 x 65= Then follows x 2, 3, 4; next comes the last multiple: (7 280 is x 4) x 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 15. In the latter sequence we note that three products are missing, namely, x 8, 11, 14, or, , and days. Charles Smiley finds that the factors missing in the above sequences, namely, x 8, 11, 14, or, , and days deviate from solar eclipse intervals by +5, -7, and +9 days, respectively. Examining the given x 10, 12, or and days the same way, he finds that these intervals deviate from solar eclipse intervals by +7 and +8 days. Following Smiley (1965: 129): Here we have five intervals that are close approximations to solar eclipse intervals, three of which were omitted on purpose and the other two, not greatly different, were included (Smiley, 1965, pgs. 128, 129). Let us assume that each interval represents an integral number of synodic lunar periods and determine the average length of the synodic period corresponding to each interval. We then have: = x

12 = x = x = x = x The modern value for the synodic period of the Moon is From the above we can conclude that Maya astronomers had a value quite close to the official one, that is, Smiley ends his paper by saying: Ultimately it may be possible to use Mayan astronomical observations to improve our modern tables of the motions of the planets. Serpent and other Numbers Thompson (1941, pgs. 61, 62) and 69 one finds the so called Serpent Numbers. On page 69 (Thompson, 1941) (not displayed) there is only one serpent. There are ten numbers in total, ten dates which are written in black and red color in the coils of five serpents. They are equivalent to about years. The extremely long distances placed in the coils of those serpents are not free of mistakes. This was recognized since early studies of Maya numbers by Cyrus Thomas and Hermann Beyer, as pointed by Thompson (1941, p.47). What would such lengthy distance numbers be used for? We still don t know. The least in pages 61 and 62 is somewhat higher than days. None appears to be related to the Mayan years. In page 62 we find an Initial Series given by: and another in page 64 as that are also given in page 31 (which in its lower part displays the 11 and 13 alternately in 8 columns) of the Dresden Codex. The next figures show pages 61, 62, 63 and 64 of Dresden Codex (taken from 58

13 59

14 Figure 3 Dresden Codex We now read directly the serpent numbers (Dresden Codex, pgs. 61, 62 and 69). In page 61 (Dresden Codex) left side serpent contains two dates. The one written in black numbers is read, from top to bottom, 4 Piktún, 6 Baktún, 14 Katún, 13 Tun, 15 Uinal, 1 Kin, that is, The 1 in the first place equals 1 (1 1); the 15 in the second place, 300 (15 20); the 13 in the third place, (13 360); the 14 in the fourth place, ( ); the 6 in the fifth place, ( ); and the 4 in the sixth place, ( ). The sum of these six products equals ( ). The column in red is read as: In right side serpent, one reads, in black: In red one has: Page 62 (Dresden Codex) shows, for the serpent in left side, in black: , and in red, The black , reduced to units of the first order reads: = = = = = = The red number in the first serpent is , which reduces to = = = = = = 2 60

15 In the second serpent in the right side we have, in black: , and in red, The black number in the second serpent, , reduces as follows: = = = = = = The red number in the second serpent reads and reduces as follows: = = = = = = In page 69 (Dresden Codex) there is only one serpent, its black numbers read: (8 or 9?). In red numbers one has: and are not shown neither reduced here. We ll also see interesting numbers and multiplication tables in pages 2, 32, 63 and 64 of the Dresden Codex. We show them next. -91 Days Tables In page 63 (Dresden Codex) one can read for instance, in the third column, from left to right, two dates, one in black and another in red. From top to bottom, in black: 61

16 , and, in red, , below them, apparently, 3 Chicchan 13 Pax. After those dates one finds the ring number: , 4 Ahau 8 Cumhú. Many years recorded indeed. In the same page 63 (Dresden Codex )one can also read a multiplication table beginning in the right half of this page. There we find the numbers 728, 819, 910, 1638, 1729 that are factorized by 91 that is, 91x8, 9, 10, 18, 19. Note that 7x13 is 91. The information on this page together with the total one in page 64 gives a name to the so called 91 Days Tables. See for instance, below the middle line of page 64 we find (from left to right and from top to bottom) the multiples of 91 as: 7x91, 6x91, 5x91, 4x91, 3x91, 2x91 and 1x91. That section then shows the multiples, from 1 to 7, of 91. Above the middle line in page 64 one reads also: 1 547, 1 456, 1 365, 1 274, 1 183, 1 092, All those columns differ from the consecutive one by 91 days. They obtained the multiples not by multiplication but by addition, it seems. On top, where numbers are worn out, we could read something: 320, 240, 160, 80 and 60. These columns differ from one another by 80 days, except for the sixth and fifth that differ by 20. Moreover there is another 91-day table in p.32 (Dresden Codex) that presents the multiples 91x1, 2, 3, 5, 6, 7, 9, 10 and11. Distinguished numbers there are 11 and 13 in the lower section of the page alternating in six columns. A missing factor in the sequence is 91x4=364 (one computing year) appears in the left side of the same page as: 364x1, 2, 4, 5, 10, 20, 40 and 60. Probably one could also read 364x80, 100, 200. The other missing factor 8x91=728 is placed at the start of the second half of the page and also found as 728x10 = in pages being related to a solar eclipse interval. The factor 9 x 91= 819 is a cycle related to the rain God. We find 91x100= in the Venus table of page 24 among 62

17 the so-called peculiar numbers (explained somewhere else). Besides, in page 2 (of Dresden s Kings version -p. 45) one finds the factors of the computing year: 364x 1, 2, 3, 4, 5, 10, 15, 20, 40, 60, 80. Finally, the total combined information of those pages presents the sequence: 91x1, 2, 3, 18, 19, 20. This last factor (1 820) is equivalent to 5 computing-years of 364. The following numbers then appear as multiples of 364 for a total of equivalent to a 400-computing-year period. The value 91 is a symbolic number for Mayas, not only in the 91 days tables do we find it: 91 are the days of each station in the Maya computing-year (91x4 = 364). There are 91 steps in Chichén-Itzá castle and 91 is nicely divided by 7 and 13, privileged numbers that become important in calendars as there is a number from 1-13 before the names of the 20 (13+7) days. Curiously in the middle of page 48 of the Dresden Codex with information related to Venus phases, we find: 148 days. This number of days we find mainly in the Eclipse table (Dresden Codex, pgs ). There it represents a period of 148 days for a five Moon period, one in a group of five lunations or revolutions. The 148 that arises in thatvenus table (p ) is the difference between middle cols. 3 and 4 = 148. Also, when subtracting columns 1 and 2 (in same page 48) we find the distinguished 91 days for which the Maya constructed the 91 Days Tables studied above. Interestingly enough, if we take the number of column 4 (in the middle of p. 48) to be an integral multiple of Venusian revolutions, for instance, 59 revolutions, we have that the Venusian period is That is = 59 x This value we now average with the 584 that one finds in the bottom section of same Venus pages (46-50) to 63

18 find Clearly, when this value (583.90) is compared with the modern one (583.92), it is seen the Maya were off by only 2/100 of a day, only some thirty minutes of a day in 481 years (calculated somewhere else) when determining Venus synodic period. Indeed, Mayan astronomical achievements portray a magnificent intellectual achievement in an age in which Europe was subsumed in Middle Ages obscurity. Conclusion Even though we do not know how far the Maya went with their knowledge of arithmetic and the handling of arithmetic operations, because evidence was destroyed, from the analysis of what is left in the Dresden Codex, and the many Stelae there still exist to be read, one can infer an advanced mathematical development which includes working with systems of different numerical bases. We studied in this paper planetary, Moon and Sun cycles and relations among them considering the Maya could have known about mathematical concepts like GCD and lcm when observing their orbits. We found interesting information for the Martian sequence in pages of the Dresden Codex devoted to this planet. The Eclipse Table and the Lunations phenomena were carefully studied and the Maya Moon period was shown to be equivalent to , a value quite close to the modern one We noted that the error between real and tabulated times of the position of Venus would be off by about half an hour in 481 years. Pages showing multiplication tables, like the 91 Days pages, were studied. We finish showing how the Maya numbers relate to each other in several ways. The concept of cycles is inherent in all of Maya time and calendars contributions. The knowledge the Maya had of the apparent movement of celestial bodies (Sun, Moon and planets) was much higher than that Egyptians and Babylonians had. Early 64

19 Astronomy, a science never sufficiently well appraised in the culture of the lowest aborigines, can be used to give us an idea of the intellectual potential of minds working together as a group. An intelligence proof for tribes, nations and other groups can be inferred from the knowledge those social groups had of the celestial bodies movement. On this base, Mayas, Egyptians and Babylonians, groups of people who developed mathematics early on, would be, without doubt, catalogued as gifted, having creative minds. 65

20 APPENDIX The purpose of this appendix is to include information related to Mars that is both complementary but different to the one shown in the core paper. In addition information about extremely large numbers found in Mars page 59 (Dresden Codex) is given. An Interpretation of page 59 in the Dresden Codex This page is naturally divided into four sections and nine columns. The middle of the page is outlined by the two rows of diurnal dates 13 Muluc and 9 (different day signs). The other two lines are the rows identified by the 13 Muluc diurnal date only. We consider this page 59 as another Mars table only because of numbers in it. A reason we have to doubt is that the Mars glyph (or any other) is not present (as is in pages 43-44F) nor is the diurnal date, base of this table: 3 Lamat 1 Zotz. Instead this page is identified with the date 13 Muluc. Between them there is a distance of 101 days. Nevertheless, various multiples of Martian revolution periods occupy this page. Page 44F (Dresden Codex) is accepted as containing Martian information and only the numbers: and (from first line in page 44F) are not present in the legible portion of page 59 that contains another numbers and many more multiples of the Martian year than page 44F. Pages 44F and 59 can be understood as complementary to each other. The synodic period of Mars is 780 days. In this table we find multiples of this number and of 78 as well. Note also that 780= 3x260 which is three times a Tzolkin calendar and one can consider the multiples of Mars year as multiples also of the sacred calendar. In the upper part of the page numbers and signs are obliterated and some proposals are made here by the authors of this article to explain the correlations among numbers with the legible information left there. 66

21 Table Illegible = = = = = = Illegible Illegible =180x780 =178x780 =172x780 =171x780 =152x780 =133x780 = 180 rev = 178 rev = 172 rev = 171 rev = 152 rev = 133 rev = 57 rev = 38 rev = 19 rev = 18 rev = 17 rev = 16 rev = 15 rev = 14 rev =13 rev X X X X X X X X X = 9x78 = 8x78 = 7x78 = 6x78 = 5x78 = 4x78 = 3x78 = 2x78 = 1x = = = =10 x 780 = 9 x 780 = 8 x 780 = 7 x 780 but = 5 x = 6x 780 Page 59, Dresden Codex 78/780 DAYS In the two upper rows and in the bottom one we find multiples of 780. In the top row we read the multiples, 133, 152, 171, 172, 178, 180. The lower half shows, in the first row, a nice set of the multiples from 1 to 9 of 78 that can be seen also in the smaller page 44F (see below page 44F (Dresden Codex). Underthe top line we find the multiples from 13 to 19 together with the 38 and 57 multiples of 780. In the bottom line we find the multiples from 5 to 10 of 780 and three very large numbers: ; and One of those three numbers ( ) is extremely large exceeding 12,489,781, a number found in page 61 and, according to Morley the highest number found in the 67

22 codices (Morley, 1915, p. 156). Seems Morley missed this one. These numbers would require further consideration since their association to Martian periods is only through a subtraction of two of them. They could be perhaps related to multiples of eclipse intervals. The equivalent in years of the largest is higher than 168, 000 years. When the smallest and the largest of the three large numbers are subtracted ( = = ) the resulting number is equivalent to Martian revolutions. The middle number is not explained and no effort is made here to do it. Neither had it intended to even suggest numbers in the illegible columns in the upper side, for the moment. As a final comment we mention that those pages (43, 44F, 59) were not for Thompson representations of Mars or of any other planet s revolutions (Thompson, Maya Hieroglyphic 1959: 229). He considers the pages 43-44F, for instance, only tables with multiples of 6x13, one of the several tables of multiples (4, 5, 6, 7) of the number thirteen in the Codex and, on another side, he associates the day Lamat with Venus and considers the glyphs (or Sky beasts) in those pages as rain animals... Nonetheless, scholars such as Willson (1924) take them as Mars tables. This is not definitive since Spinden (1942:43) finds the characteristic date 3 Lamat (Spinden, 1942) associated to Jupiter. In turn, Spinden (1942) relates page 59 to Saturn and an unknown writer in an unpublished paper associates pages 43-44F to Mercury (Thompson 1959, 1971, p. 258). We could conclude, nonetheless, that if The Maya observed Venus, an undeniable fact, they could as well have observed other planets as Mars and accept the tables (Spinden, pgs. 43, 44F, 59) as a consequence of that observation. That 78 is not an integral multiple of the Martian revolution (780) but a constant in the tables, its presence is understood as a multiple of 13 68

23 (6x13) naturally related to 780 because of their common factors with the exception of 2x5, a fact easily noticed by the Maya and equally handled in their vigesimal arithmetic. NOTE: In the bottom of the 4th column (left to right) 13 uinal must not appear there, seems a copyist mistake. Following the pattern we propose, = = 6x780. Explicit information of 780 multiples, used in main text and not shown there, in next table: Table 2 page 44, Dresden Codex, for comparison = = = 702 = 623 (but 624) = 546 = 468 = 20x780 = 17x = 9x78 = 8x78 =7x78 = 6x78 X X X X X X = 780 = 390 = 312 = 235 (but 234) = 156 = 78 = 1x780 = 5x78 = 4x78 = 3x78 = 2x78 = 1x78 We have made two small corrections of one day in the third column (l to r; upper and lower parts) to accomplish the shown pattern of multiples of

24 References Aveni, Anthony F. (1981). Archæastronomy in the Maya region : A review of the past decade. Archaeoastronomy, no. 3, S1 - S16. Dresden Codex in Förstemann and Kingsborough versions. When Codex was found it was in a poor condition so Europeans assigned a page numbering that, later, was proved to be wrong. The numbering was conserved almost the same, however, the order of the pages inside the Codex was probably 1-24, y We find Dresden Codex in FAMSI page: Förstemann and Kingsborough versions differ slightly in page numbering. Pages Kingsborough 1, 2, are the Förstemann s 44, 45. Pages F1 and F2 are K44 and K45. From page 3 to 43, have the same numeration. From page 46 to the last page, 74, numbering coincide. The Dresden Codex, as it appears in the Lord Kingsborough was prepared with the cooperation and courtesy of the Jay Kislak Foundation, Miami Lakes, FL. Arthur Dunkelman, Director. Although the artist Agostino Aglio followed the original codex with great precision, there are subtle differences in the rendering of some of the glyphs and figures. Dresden Codex, see also: Förstemann, Ernst Wilhelm ( ) ErläuterungenzurMayahandschrift der Königlichen Öffentlichen Bibliothek zu Dresden Onñline book digitalized by Google from the original in Harvard University Library. It is also found in English: Försteman, Ernst Wilhelm (1989) Commentary on the Maya Manuscript in the Royal Public Library of Dresden, trans. Selma Wesselhoeft and A. M. Parker, Papers of the Peabody Museum of Archaeology and Ethnology. The Peabody Museum, Cambridge, 4, #2, unabridged reprint of the 1906 edition, Aegean Park Press, Laguna. Leon-Portilla, Miguel, (1988), Time and Reality in the thought of the Maya, 2 # ed. Enlarged, Norman: University of Oklahoma Press. Millbrath, S. (1999). Star Gods of the Maya, Texas University Press. Morley, Sylvanus Grisworld. (1956, 1938). The Ancient Maya, third edition, Stanford University Press. The first edition of this book was published by Fondo de Cultura Económica (FCE), México (1947). Since this book deals with the most authorized history on the great Maya Civilization, it has been reedited since,first by its author and, after his dead, by his disciples and collaborators. The fourth edition (in English) was prepared by Robert J. Sharer and corresponds to the second in Spanish: La Civilización Maya, FCE, It has been notably modified and enlarged. This is the most complete book on Maya Civilization. Morley is without doubt the recommended author; along with Sharer s (1994, 5th ed. Stanford Univ. Press.) Also by Sylvanus G. Morley: An Introduction to the Study of Maya Hieroglyphs (1915) Smithsonian Institution Bureau of American Ethnology. 70

25 Bulletin no. 57, Washington, D. C. (Also in Dover publications, New York, 1975); The Inscriptions at Copaci (1920), The Inscriptions of Peten (1938, 5 vols.). Satterthwaite, Linton (1962). Long count positions of Maya dates in the Dresden Codex, with notes on lunar positions and the correlation problem In Proceedings of the International Congress of Americanists (35 session, Mexico.). v. 2, pp , Mexico. Also Satterthwaite (1964) in: Dates in the new Maya hieroglyphic text as Katun-Baktun anniversaries. Estudios de Cultura Maya 4: ,Mexico. Smiley, C. H. (1965). Solar Eclipse Intervals in the Dresden Codex, Journal of the Royal Astronomical Society of Canada, Vol. 59, p.127 Online: Astrophysics Data System (ADS) Spinden, H. J. (1942) Time Scale for the New World. Proc. 8 th Am. Scient. Congr. Washington, 1940, 2, Teeple, John, D. (1931). Maya Astronomy, Carnegie Institution Publication no. 403, Washington, D. C. Edited in Spanish (1937), Museo Nacional de México. Thompson, John Eric Sidney (1972). A Commentary on the Dresden Codex; a Maya Hieroglyphic Book, Philadelphia, American Philosophical Society. Also, Maya Arithmetic (1941), Contributions to American Anthropology and History, Vol. II, no. 36. Thompson, John Eric Sidney, (1959), Maya Hieroglyphic Writing: An Introduction. Norman: University of Oklahoma Press: Norman. Reedited in Thompson, J. Eric. S. (1960). Maya Hieroglyphic Writing, 3rd Ed., University of Oklahoma Press. Willson, R. W. (1924) Astronomical Notes on the Maya Codices. Papers Peabody Mus. Harvard Univ. 6, no.3, Cambridge, Mass. 71