Earth/matriX
Science in Ancient Artwork

# The Integer (20) Calendar Reckoning and Astronomical Tables: Ancient Mexico

### byCharles William Johnson

Science in Ancient Artwork Series
Number One
New Orleans, Louisiana.

`D e d i c a t e d    t o`
Carl P. Munck
whose work on deciphering the
code of numbers found in ancient
cultures throughout the world has
had bearing on my own efforts.

ACKNOWLEDGEMENT
A word of thanks to the Middle-American Research Institute's library at
Tulane University in New Orleans, for the possibility of examining rare and
out-of-print works, as a public-access policy.
 Table of Contents The Integer (20) Calendar Reckoning and Astronomical Tables: Ancient Mexico Part I The 360 Day-Count Part II The 260 Day-Count Part III The Day-Counts of Venus, Mars, Jupiter, Saturn, and Mercury; the Earth's Moon The Table of Additional Days The Table of Calendar Rounds Observations

# THE INTEGER (20) CALENDAR RECKONING AND ASTRONOMICAL TABLES: ANCIENT MEXICO

### byCharles William Johnson

Part I

The 360 Day-Count

The numbering system of mayas consisted of three symbols: a shell-shaped figure that represented zero (); a dot or point for the number 1 ( ); and, a straight bar figure represented the number 5 ( ). These symbols were arranged vertically at distinct positional levels.

The positional levels determined specific values, whereby the given number at a particular level would be multiplied by that level's corresponding constant value. The following is generally cited as the basis of the system, although now it is known that there were more than five positional levels.

" " " " 23,040,000,000 days = alautun 1,152,000,000 = kinchiltun 57,600,000 = calbatun 2,880,000 = pictun 144,000 = baktun 7,200 = katun 360 = tun 20 = uinal 1 = kin

The maya system of the long count counted time-cycles in such a manner, that 20 kins (days) equaled 1 uinal (1 month); 20 uinals equal 1 tun (a year); and successively in this manner of multiples of twenty. We obviously have no commonday words in our language for an alautun, a time-cycle of 23,040,000,000 days, other than stating that that is a long time, over 63 million years.

For example, the number 1 (), in column one would have a distinct value depending upon its placement at each of the five levels. At positional level:

 V = 44,000 (or 400 years); IV = 7,200 (or 20 years); III = 360 (or 1 year); II = 20 (one month); I = 1 (or one day);

Such would be the translation of the constants into a specific number of years, given that 360 was the calendar count involved (360 days = 1 year, plus 5 extra days of the uayeb).

Similarly, regarding the number 5 (), the values would change as the number five would now be multiplied by the constants. Then, the number 5 would represent at positional level:

 V = 720,000 (or 5 times 144,000 days; 5 baktuns); IV = 36,000 (or 5 times 7,200 days; 5 katuns); III = 1,800 (or 5 times 360 days; 5 tuns); II = 100 (or 5 uinals; 5 times 20 days); and, I = 5 (5 times 1 day; 5 kins).

The maya long count, then, would be the cumulative sum of values generated in a specific column of figures. Let us examine a particular number example; the number 36,108 would be expressed in the following manner:

 = 5 times 7,200 = 36,000 = 0 times 360 = 0 = 5 times 20 = 100 = 8 times 1 = 8 36,108

The creative nature of the maya numbering system becomes obvious at once. In fact, a similar counting system was used by other peoples of ancient Mexico (the mexicas and aztecs), while some authors consider that the original counting system was developed by the olmec culture.

Without entering into a discussion of who-did-what-first, one is concerned with the procedure of such extensive calculations with numbers running into the millions on the fifth positional level and beyond. For even though the ancient peoples of Mexico might have been able to visually represent the numerical result of 18 times 144,000 (i.e., 2,592,000), it becomes intriguing to enquire how they actually effected the multiplications. With that concern in mind, we have attempted to understand how these calculations and reckonings may have been achieved without the need for multiplication.

The enquiry began with the numbers 1 through 20 and the corresponding values for each positional level. The search has the purpose of understanding why those particular constant values were chosen as cut-off points on those levels.

The integer 360 is obtained by multiplying 18 x 20, which in calendaric terms represents 18 months of 20 days. The older calendar reckoning was based on 13 months of 20 days and yielded a day-count of 260. But, we shall first examine the 360 count, since that pertains to the values cited in the tun figure.

The following Table of Numbers is then implied in the procedure of multiplying the numbers 1-20 with each one of the five positional level's constants:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

By carrying through this multiplication procedure with every combination cited, the following Table of Numbers obtains:

 144,000 288,000 432,000 576,000 720,000 7,200 14,400 21,600 28,800 36,000 360 720 1,080 1,440 1,800 20 40 60 80 100 1 2 3 4 5

 864,000 1,008,000 1,152,000 1,296,000 1,440,000 43,200 50,400 57,600 64,800 72,000 2,160 2,520 2,880 3,240 3,600 120 140 160 180 200 6 7 8 9 10

 1,584,000 1,728,000 1,872,000 2,016,000 2,160,000 79,200 86,400 93,600 100,800 108,000 3,960 4,320 4,680 5,040 5,400 220 240 260 280 300 11 12 13 14 15

 2,304,000 2,448,000 2,592,000 2,736,000 2,880,000 115,200 122,400 129,600 136,800 144,000 5,760 6,120 6,480 6,840 7,200 320 340 360 380 400 16 17 18 19 20

This Table of Numbers (360c) reveals many significant aspects which we shall now examine in detail. The positional level I involves the unit-numbers 1 through 20. We did carry out the positional level II multiplication to the 40th place, but proved to be unnecessary as numbers began to repeat themselves. Given that the number 20 is the significant integer of the system, carrying the calculation out to the 20th place is sufficient to comprehend the system.

The values of positional level I are 20 times less than those of level II; level II is 18 times less than level III; the values of level III are 20 times less than those of level IV; and, the values of level IV are 20 times less than those of positional level V. Hence, another way of expressing this is: level V is 144,000 times greater than level I, the values of level IV are 7,200 times greater than the values of level I; etc. In all cases, the different levels represent multiples of 20.

One should notice the placement of the zeros at each particular level, for this shall be meaningful in the remainder of the analysis regarding methods of calculations. For example, positional level V shows values with at least three zeros ending their terms; level IV with at least two zeros; and levels III and II with at least one zero at the end of each value.

The most significant aspect of the Table of Numbers is the cut-off point 360 at the positional level II, 18. This integer then becomes situated at positional level III, 1. The number 360, as mentioned, is the product of 18 x 20, which involves the relation months/days.

The relationship concerns the calendar count of 360 based on time-cycles.

Positional level III ends at the 20th integer with the value of 7,200, which begins also positional IV. It should be no surprise that positional level IV ends with the values of 144,000, the same constant that initiates positional level V.

The Table of Numbers, then, appears to arbitrarily be set at 360, since according to the logic of the integer 20, positional level III should begin with the number 400. However, the Table of Numbers reflects the logic of calendar reckoning, which in this case utilizes the earth's approximate time-cycle in seasonal years as a means for programming the entire set of values listed on the table. The positional levels III, IV, and V, then, become multiples of 360.

 V 144,000 ÷ 360 = 400 288,000 ÷ 360 = 800 432,000 ÷ 360 = 1,200 etc. IV 7,200 ÷ 360 = 20 14,400 ÷ 360 = 40 21,600 ÷ 360 = 60 etc. III 360 ÷ 360 = 1 720 ÷ 360 = 2 1,080 ÷ 360 = 3 etc.

Now, let us analyze the numbers appearing in the table. The numbers situated at positional level II may also be obtained by doubling the numbers corresponding to positional level I (1 - 20), and then simply adding a zero;

 18 + 18 = 36 + 0 = 360

In other words, multiplying by the integer 20 can also be achieved by simply doubling the number and adding a zero to the product. Observe the positional values at levels I and II on the table of numbers:

In each case, at each level and integer one can identify this possibility:

 1 + 1 = 2 + 0 = 20 2 + 2 = 2 + 0 = 40 3 + 3 = 6 + 0 = 60 etc.

Now, observe that this same relationship does not exist between the values listed on positional levels II and III, since the difference between these two levels is 18; not 20. It would appear as though the calendar count has become interrupted, no longer following the integer 20 logic. It is not possible to simply double the number on level II and add a zero to it order to obtain the number on level III. By selecting the positional level to begin with 360, instead of the number 400 (20 x 20; the value on space 20), then the apparent logic has been broken.

However, as we examine levels III, IV and V, we now notice that the relationship of 20 does exist among these three levels. Again, instead of multiplying, one may revert to the procedure of doubling the corresponding number on the lower level to obtain the next higher level, and simply add a zero.

Level III = 0; Level IV = 00; and Level V = 000

In order to make the translation from level II to level III, one would have to account for the two places (19 and 20) that were skipped, when the table was broken off at the value of 18 (360).

As a proposal for a method of calculation, let us run through a specific example.

For the sake of example, let us offer the case the number 15.

a) Take the number 15 and double it and add a zero to that sum:

 15 + 15 = 30 + 0 = 300 (= level II value)

b) Again double the number 15:
 15 + 15 = 30

c) Subtract the result of b from a:
 300 - 30 = 270

d) Double 270 and add a zero to that result:
 270 + 270 = 540 + 0 = 5400 (= level III value)

e) Double 5400 and add a zero to that result:
 5400 + 5400 = 10800 + 0 = 108000 (= level IV value)

f) Double 108000 and add a zero to that result:
 108000 + 108000 = 216000 + 0 = 2160000 (= level V value)

In this manner, one is able to obtain the corresponding numbers of each of the five positional levels on the Table of Numbers for the number 15:

 V 2,160,000 IV 108,000 III 5,400 II 300 I 15

The method of calculation, without multiplication, for obtaining the corresponding numbers on the Table of Numbers for the 360-count, for each positional level can be summarized as follows:

Choose a number 1 to 20:

Example:

 15 double it and add a zero 30 0 subtract 10% from that 27 0 double it and add a zero 540 0 double it and add a zero 1080 0 0 double it and add a zero 2160 0 0 0 double it and add a zero 4320 0 0 0 0 double it and add a zero 8640 0 0 0 0 0 double it and add a zero ad infinitum.

In this manner one can obtain any number series on the Table of Numbers, thereby reconstructing the entire table without actually having it written down, and without performing any long method of multiplication. Calculations into the millions or billions, and beyond, may be obtained with relative ease.

In the following part of this essay we shall now review the 360c reckoning with the old calendar 260, in order to perceive their possible relationship.

Part II

The 260 Day-Count

Over the millennia, much of ancient knowledge has been destroyed, both voluntarily and involuntarily through conquest. It is impossible to know exactly how much was destroyed. We may attempt, however, a recreation of the numerical tables that were understandably employed in calendar reckonings of ancient times. The Table of Numbers that we have discussed here might suggest the existence at one time of other tables of reckoning.

We have reviewed the counting system based on the integer twenty method, and the the multiplication of 18 x 20, which offers a pattern based on the cut-off point 360. The 360c has been related to the calendar round of 360 days, with the necessary remaining five days being added on (in maya the name for these five days is the uayeb; in nahuatl, it was nemonetemi).

But, there was an older calendar based on the 13 x 20 relation, producing the 260-day count (260c). This calendar of 13 months and twenty days existed within various cultures of Mexico. "We know that this 260-day cycle count of the days, called the tzolkin by the maya and tonalpohualli by the aztecs, lay at the core of all Mesoamerican calendars, at least since the sixth century B.C. This cycle rose to prominence, I think, because it approximated the length of several fundamental life-sustaining periods: It was a measure of the duration of the agricultural season, nine lunar months (266 days), as well as the Venus appearance interval ..., and it was equal to 13 x 20, both sacred numbers." (from Aveni, p.79).

We shall now consider what might have been the manner for reckoning the 260c in the light of the Table of Numbers (360c) presented in Part I.

We proceeded to follow the same logic of the integer 20 system as in the 18 x 20 (= 360c) calendar round, and devised the following 260c Table of Numbers:

 Mars Mars Mars 104000 208000 312000 416000 520000 624000 728000 832000 936000 1040000 5200 10400 15600 20800 26000 31200 36400 41600 46800 52000 260 520 780 1040 1300 1560 1820 2080 2340 2600 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 8 9 10

 Mars Mars Mars 1144000 1248000 1352000 1456000 1560000 1664000 1768000 1872000 1976000 2080000 57200 62400 67600 72800 78000 83200 88400 93600 98800 104000 2860 3120 3380 3640 3900 4160 4420 4680 4940 5200 220 240 260 280 300 320 340 360 380 400 11 12 13 14 15 16 17 18 19 20

In order to produce this table of number based on the 260c, there was again no need to effect a single multiplication. The procedure that we exemplified for calculating the numbers as shown earlier was employed.

Now, instead of using the number 360 as the cut-off point, the number 260 is used. It is significant to note that the numbers related to the 52 and 104 cycles, so important in ancient reckoning in Mexico, appear on the 260c Table of Numbers, although they do so in multiples. Remember that these numbers do not appear on the table of numbers of the 360c. The fact that the cycles of 52 (years) and 104 (years) may have come from the older calendar of the 260c reckoning would seem to be logical from an historical perspective. Nonetheless, those same cyclical periods remained relevant to the 360c.

Such a system of reckoning would then serve in making calculations of whole cycles between one calendar round and another. For, as some authors have pointed out, ancient cultures were very much concerned with identifying the relationships of the cosmos in terms of whole cycles: "The essence of this mathematically predictive astronomy consisted of the art of defining how many whole cycles of one set of phenomena accorded with how many of another set. Maya and late Babylonian astronomers alike were engaged in exploring and discovering commensurate numbers, embedded in nature, that described cycles of recurring phenomena: 99 moons and 5 Venus cycles, 151 Mars cycles and 284 years, the eclipse cycle of 223 moons and 19 seasonal years, and so on." (Aveni, p. 123).

A general belief existed that patterns existed in nature, and that these patterns could be and were expressed mathematically. In this sense, the Table of Numbers of the distinct calendar rounds (360c and 260c) reflect the systematic study of the solar system. The respective counting systems by their very logic of cut-off points are a product of that concern. Let us examine a specific relationship of equivalency between the 260c and 360c:

 1872000 ÷ 260 = 7200 rounds 93600 ÷ 260 = 360 " 4680 ÷ 260 = 18 " 260 ÷ 260 = 1 " 13 ÷ 260 = .05 " (= 1 day) 1872000 ÷ 360 = 5200 rounds 93600 ÷ 360 = 260 " 4680 ÷ 360 = 13 " 260 ÷ 360 = .72222 " 13 ÷ 360 = .03611 "
 Conversions: 7200/260c = 5200/360c 18/260c = 13/360c

Further considerations of an infinite number of time-cycle relationships could be as follows:

 13 rounds of 360c = 18 rounds of 260c = 8 rounds of Venus (less 8 days) = 6 rounds of Mars = 158.5 lunar cycles (less .5 a day) etc.

In other words, the relationships that exist in nature, as the planets revolve around the sun, in relation to one another, may be expressed numerically. This possibility seemed to convey a hidden meaning that could be revealed in the pattern of numbers themselves. Students of numerology might make it appear as though only the numbers are significant. But, let us remember that ultimately the relationships that are being measured and converted to numerical expression are what is important. But, the numbers, being the reflection of the relations, take on that same sacredness.

In order to better comprehend the numbers and the relations that they reflect, let us review the distinctions in the counting systems.

As was mentioned earlier, the cut-off point the Table of Numbers is the first place of the third positional level (III, 1). One might initially consider that the very distinction lies in the two system of multiplication; that of 13 x 20 and that of 18 x 20, which forms the distinguishing aspects of the two calendar counts. As we have explained, levels I and II are easily translated between themselves, and then, as a separate set, levels III, IV and V are translatable among themselves. And, note that this occurs equally on the 260c calendar, although the cut-off point is now distinct (260c; not 360c). Furthermore, now all of the levels III, IV and V are divisible by the number 260.

The Table of Numbers, then, as it must have been originally designed, can accommodate any number of an orbital cycle (or day-count), of any particular planet (or moon,s orbit), in order to create a set of numbers relevant to that astronomical body. In a very real sense, then, not just a numbering system or a counting system was created by the 13 x 20 or the 18 x 20 reckonings, but the basis for creating astronomical tables was developed through a method of procedure as we have attempted to show.

Hence, it is at the positional level of III, 1 that the time-cycle number of 360 or 260 may be substituted for that of the orbital time of any other planet. Now, since the Table of Numbers functions on the basis of the integer 20 (as one of its main multipliers), then it is necessary to maintain that the chosen orbital day-count number be an even number (divisible by 20), while the other main multiplier may be any even/odd integer.

Let us now substitute the 260c and the 360c numbers for those relating to other planets in the solar system that were considered to have been known then.

Part III

The Day-Counts of Venus, Mars, Jupiter, Saturn , and Mercury; the Earth's Moon

In this section we shall offer the different day counts that refer to the planets Venus, Mars, Jupiter, Saturn and Mercury, alongwith that of the Earth,s moon. After having produced the Table of Numbers for the 260c, a generic, non-orbital count, of astronomical significance, we consider that similar tables of numbers may be created for each of the planets' and the Moon. The day-counts of the different planets, orbits are in terms of the 365-day orbit of earth, expressed therefore in terms of their synodic revolutions; the time that it takes a planet to revolve around the sun on its orbit and return to its point of departure. (It is understood that the planet never returns to its exact position in space, but to the same orbital position relative to the sun/earth relationship).

In order to produce the Tables of Numbers of the different planets, we shall employ the following data and considerations:

 Planet/body Multipliers Calendar Reckoning Actual Orbit Days/round Earth Seasonal Year 18 x 20 360c 365.2421987 Earth Astronomical Year 13 x 20 260c (365.2421987) Venus 29 x 20 580c 584 (583.92) Mars 39 x 20 780c 780 Jupiter 20 x 20 400c 398.8846 Saturn 19 x 20 380c 378.0919 Mercury (5.75 x 20) 115c 115.8774 The Moon 1.5 x 20 30c 29.53058857

For the reckonings in the different tables, we have chosen the numbers closest to the actual orbital day-counts. The fractional differences will be treated in the tables of the additional day-count numbers (either through addition or subtraction).

In the frist six cases listed above, the number 20 is employed as a principal multiplier alongwith an even/odd integer. While in the case of Mercury and the Moon, the integer 20 is accompanied by a fractional number (5.75 and 1.5 respectively), given the fact that we have chosen a day-count number as close as possible to their orbital numbers. However, these fractions pose no special problems in the calculations, because as we stated earlier, only addition is involved in the procedure.

 260 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

 360 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

 580 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

 30 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

 115 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

 780 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

The multipliers, although definitely determinant for the time-cycles expressed in the values of the table, are implied aspects that remain somewhat outside of the picture. To make them more visible, we shall offer a Table of Calendar Rounds, which will offer the count for each specific positional level and place (i.e., I - V, and 1 - 20). For each Table of Numbers is based on the same number of calendar rounds (proportionately to each level/place), as we shall observe later.

Now, let us examine each Table of Numbers for the particular planets that we have selected for the analysis.

 232000 464000 696000 928000 1160000 1392000 1624000 1856000 2088000 2320000 11600 23200 34800 46400 58000 69600 81200 92800 104400 116000 580 1160 1740 2320 2900 3480 4060 4640 5220 5800 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 8 9 10

 2552000 2784000 3016000 3248000 3480000 3712000 3944000 4176000 4408000 4640000 127600 139200 150800 162400 174000 185600 197200 208800 220400 232000 6380 6960 7540 8120 8700 9280 9860 10440 11020 11600 220 240 260 280 300 320 340 360 380 400 11 12 13 14 15 16 17 18 19 20

Venus: 580c

For the planet Venus, we have chosen the 580c to produce the Table of Numbers, since its orbital rotation value is generally accepted as 584 days, although it is specifically 583.92 days. However, we shall attempt to work without the fractions, in order to be able to comprehend the whole cycle relationships better. Furthermore, this probably follows the general rule of thumb of ancient astronomy which conceived of the relationship between the orbits of Venus and Earth as constituting a 5:8 ratio.

Mars: 780c
For Mars', orbit, we have maintained the cited figure of 780 days, wich is generally offered, and which, as is readily obvious equals three times the 260c (3 x 260 = 780). Hence, there are an exceptional amount of inmediate associations of whole cycles that can be made between these two table of numbers. For convenience, we have indicated on this Table of Numbers: 780c, the corresponding columns to the 260c table.

 312000 624000 936000 1248000 1560000 1872000 2184000 2496000 2808000 3120000 15600 31200 46800 62400 78000 93600 109200 124800 140400 156000 780 1560 2340 3120 3900 4680 5460 6240 7020 7800 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 8 9 10

 342000 3744000 4056000 4368000 4680000 4992000 5304000 5616000 5928000 6240000 171600 187200 202800 218400 234000 249600 265200 280800 296400 312000 8580 93600 10140 10920 11700 12480 13260 14040 14820 15600 220 240 260 280 300 320 340 360 380 400 11 12 13 14 15 16 17 18 19 20

Saturn: 380c
Saturn's synodic cycle involves 378.0919 days, and therefore we have designated it as a 380c, thereby having to subtract 2.0919 days from the whole cycle calculations on the table.

 152000 304000 456000 608000 760000 912000 1064000 1216000 1368000 1520000 7600 15200 22800 30400 38000 45600 53200 60800 68400 76000 380 760 1140 1520 1900 2280 2660 3040 3420 3800 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 8 9 10

 1672000 1824000 1876000 2128000 2280000 2432000 2584000 2736000 2888000 3040000 83600 91200 98800 106400 114000 121600 129200 136800 144400 152000 4180 4560 4940 5320 5700 6080 6460 6840 7220 7600 220 240 260 280 300 320 340 360 380 400 11 12 13 14 15 16 17 18 19 20

Mercury: 115c
The planet Mercury presents a rotational value ending in an odd number, but which is one-fourth of 20. Therefore the numbers on the table will be closely related to the 20 integer. However, for the first time we shall be observing fractions on the Table of Numbers for a planet due to this very same fact. The flexibility of the system of the tables of numbers would allow one to choose the orbital count of 120c, in order to approach some of the other whole cycle rounds, and then subtract the extra 5-plus days. But, that is not necessary for this essay.

 46000 92000 138000 184000 230000 276000 322000 368000 414000 460000 2300 4600 6900 9200 11500 13800 16100 18400 20700 23000 115 230 345 460 575 690 805 920 1035 1150 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 8 9 10

 506000 552000 598000 644000 690000 736000 782000 828000 874000 920000 25300 27600 29900 32200 34500 36800 39100 41400 43700 46000 1265 1380 1495 1610 1725 1840 1955 2070 2185 2300 220 240 260 280 300 320 340 360 380 400 11 12 13 14 15 16 17 18 19 20

The Moon: 30c
The Moon is considered to have a 29.5 day-count, although it is generally accepted as 29.53, and more exactly as 29.53058857 (which becomes for all purposes almost immeasurable). We shall offer a 30c tables of numbers for the Moon, and then work out the fractions on the table of additional numbers.

 12000 24000 36000 48000 60000 72000 84000 96000 108000 120000 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 30 60 90 120 150 180 210 240 270 300 20 40 60 80 100 120 140 160 180 200 1 2 3 4 5 6 7 8 9 10

 132000 144000 156000 168000 180000 192000 204000 216000 228000 240000 6600 7200 7800 8400 9000 9600 10200 10800 11400 12000 330 360 390 420 450 480 510 540 570 600 220 240 260 280 300 320 340 360 380 400 11 12 13 14 15 16 17 18 19 20

We shall now offer a table of numbers regarding the additional days that are required for calculating some of the planets' time-cycles. Again the same logic of the Table of Numbers is followed. However, we have listed this table according to the calendar rounds in absolute numbers of whole cycles of 1 - 8000, with respect to the additional numbers of days of 1 - 6. For example, 6 additional days would be required to reach the 266c from the 260c; 5 additional days for the 365c from the 360c, as well as the 6 days for leap year; 4 additional days are required for the 584c coming from the Venus 580c; etc.

One would simply have to know the whole cycle numbers of the main count, and then consult the same number of cycles and the respective number of additional days required.

 6 Days Rounds 4400 4800 5200 5600 6000 6400 6800 7200 7600 8000 26400 28800 31200 33600 36000 38400 40800 43200 45600 48000 5 22000 24000 26000 28000 30000 32000 34000 36000 38000 40000 4 17600 19200 20800 22400 24000 25600 27200 28800 30400 32000 3 13200 14400 15600 16800 18000 19200 20400 21600 22800 24010 2 8800 9600 10400 11200 12000 12800 17600 14400 15200 16000

 6 Days Rounds 400 800 1200 1600 2000 2400 2800 3200 3600 4000 2400 4800 7200 9600 12000 14400 16800 19200 21600 24000 5 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 4 1600 3200 4800 6400 8000 9600 11200 12800 14400 16000 3 1200 2400 3600 4800 6000 7200 8400 9600 10800 21000 2 800 1600 2400 3200 4000 4800 5600 6400 7200 8000

 6 Days Rounds 220 240 260 280 300 320 340 360 380 400 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 5 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 4 880 960 1040 1120 1200 1280 1360 1440 1520 1600 3 640 700 740 780 840 900 960 1020 1080 1140 2 440 480 520 560 600 640 680 720 760 800

 6 Days Rounds 20 40 60 80 100 120 140 160 180 200 120 240 360 480 600 720 840 960 1080 1200 5 100 200 300 400 500 600 700 800 900 1000 4 80 160 240 320 400 480 560 640 720 800 3 60 120 180 240 300 360 420 480 540 600 2 40 80 120 160 200 240 280 320 360 400

 6 Days Rounds 11 12 13 14 15 16 17 18 19 20 66 72 78 84 90 96 102 108 114 120 5 55 60 65 70 75 80 85 90 95 100 4 44 48 52 56 60 64 68 72 76 80 3 33 36 39 42 45 48 51 54 57 60 2 22 24 26 28 30 32 34 36 38 40

 6 Days Rounds 1 2 3 4 5 6 7 8 9 10 6 12 18 24 30 36 42 48 54 60 5 5 10 15 20 25 30 35 40 45 50 4 4 8 12 16 20 24 28 32 36 40 3 3 6 9 12 15 18 21 24 27 30 2 2 4 6 8 10 12 14 16 18 20

THE TABLE OF CALENDAR ROUNDS

This table of numbers offers the whole cycle numbers that are implied in each one of the Tables of Numbers for the specific day-count that we have offered in this study. In each Table of Numbers, the corresponding cut-off day-count (30c, 115c, 260c, 360c, 584c, etc.) becomes a constant for that table's calculation, inasmuch as the positional levels of III, IV and V are divisible by that respective day-count number. Each specific level and place represent an absolute number of calendar rounds (or orbital rotations of a particular astronomical body, except the non-orbital amount of 260c). The number of days in the count may change, but the number of calendar rounds, then, remains constant for the table of numbers.

 Rounds: 4400 4800 5200 5600 6000 6400 6800 7200 7600 8000 400 800 1200 1600 2000 2400 2800 3200 3600 4000 220 240 260 280 300 320 340 360 380 400 20 40 60 80 100 120 140 160 180 200 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 Days: 220 240 260 280 300 320 340 360 380 400 20 40 60 80 100 120 140 160 180 200 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10

From the previous chart, it is evident that the set of constant numbers of rounds (or whole cycles) may reflect all numerical possibilities to infinity. For within the addition of the numbers listed from 1 to 8,000 calendar rounds, one could feasibly extend the table of numbers to any level beyond V. Furthermore, although the number 500 calendar rounds is not listed on the table of numbers and calendar rounds, one may simply add the value of 400 rounds to that of 100, and with that one would obtain the corresponding 500-rounds value. The possibilities are limitless.

It should now be obvious, that with the astronomical Tables of Numbers as presented, one coulds ompare almost any possible set of time-cycles and calendar reckonings with respect to the astronomical bodies treated herein. The whole-cycle patterns are readily visible through simple addition and subtraction.

Observations

Tables of numbers do exist within the maya culture in ancient Mexico. One has only to examine the Dresden Codex which has been interpreted as a registration of the movement of Venus and corresponding eclipses. Although we may never find exact example of the Tables of Numbers cited herein, the tables reflect the procedure implied in the 18 x 20 and 13 x 20 calendar reckonings. In fact, the 260c and the 360c calendar reckonings are themselves already the product of related calculations.

Multiplication is ultimately a form of addition. The Tables of Numbers, as illustrated in this essay, through their method of calculation, do not require a multiplication procedure; simple addition is sufficient to effect the calculations. The counting of time-cycles seems to take for granted a system of counting; something that is often overlooked. Furthermore, the ancient concept of time-cycles and the comparison of whole cycles, becomes more profound as we contemplate the relationship of one year to that of millions of days. Such a comparison, in itself, reflects a profound degree of human consciousness, that we still have difficulty in comprehending ourselves. In other words, if the ancient peoples represented numerically millions of days, then they consciously thought in terms of thousands of years.

The significant aspect, then, might not be that they abstracted time-cycles into seven or eight-digit numbers, but rather that they abstracted the progression of time through mathematics. The simplicity in calculation reflects a complexity of mental conceptualization that in turn implies a need for such extensive calculations. Such calculations were probably not only necessary for the sowing and harvesting of crops, but possibly dealt with a greater knowledge of the Cosmos; a knowledge that we have yet to discover completely.

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Earth/matriX
Science in Ancient Artwork Number One
The Integer (20) Calendar Reckoning and Astronomical Tables: Ancient Mexico
January 1995