INTRODUCTION Two modern math methods assists in decoding aspects of ancient Mayan math and astronomical texts. The first method, congruences, includes a Mayan version of the Chinese Remainder Theorem (CRT). The CRT may date to 1950 BCE when the known planets lined up like a string of pearls (confirmed by NASA) a few years ago. The 'string of pearls' legend initiated the Chinese calendar, a calendar that was kept in alignment by the CRT for almost 4,000 years.

Mayans before 600 BCE created lunar and solar calendars by observing solar cycles and planetary cycles, much as the Chinese and other advanced cultures had done with their mathematics. The Mesoamericans created several calendars that were kept in alignment by a great year of 52,000 years.

Ancient Near East lunar calendars were written in a form of the exact remainder arithmetic. Solar calendars used in China, India, and the Hellene world also used remainder arithmetic. Joseph Needham wrote of the Chinese version reaching the Hellene and medieval worlds via the Silk road. Fibonacci writing in the Liber Abaci, documented the indeterminate equations aspect of the CRT, as well as the CRT itself.

The second method includes Mesoamerican calendar that decodes Mayan arithmetic texts within a practical 3 x 4 (base 4 x base 5) abacus. A closely related Canary Island abacus/calendar dating method may have indirectly reached the Mesoamerican world by Mayans reading lunar cycles. The Canary Island is 3 x 4 acano "checker board" is best understood in full color, red and black for 29 and 30 day months, and white for recording other facts. The lunar eclipse calendar dating method has been used by several ancient cultures. Given that Mayans and Mesoamericans began with a nine (9) lunar month equivalent, 260 days, the smallest lunar eclipse calendar, and a 360 day calendar, the second 3 x4 lunar calendar was likely unknown to Mesoamericans, thereby limiting this discussion to Mayan topics.

PEDAGOGY Astronomy and lunar calendars were the birth-parents of mathematics. The observable cycles of our universe were parsed 11,000 years ago, possibly first in Anatolia. Writing, as a method of secondary thought, was built upon 6,000 years of 'star gazer' number systems. Unobservable lunisolar calendars emerged late, 499 BCE in the Babylonian Metonic 19 year cycle. Hence unobservable lunisolar calendars will not play a central role in this discussion.

LUNAR CALENDARS Mesoamerican calendars are closely related to an acano number system used in the Canary Islands. Jose Barrios Garcia reports the relationship this way, "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."

Dates were created in lunar calendars as Jose Barrios Garcia goes on to say, "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."

MAYAN LUNAR CALENDAR DATING SYSTEM Mayans and Mesoamericans altered the lunar eclipse calendar dating method as parsed by George I Sanchez, "Arithmetic in Maya" from Lowland Maya within an abacus.

AZTEC ARITHMETIC A second entry point to Mesoamerican number system story line is provided by an Aztec text reported in "Science" journal article. The article asks a mesoamerican arithmetic question phrased within a regional use of prime number divisors and Mayan mathematical astronomy methods. The 4/4/08 issue of ("Science") proposes that several algorithms may have calculated area in an Aztec manner. Two manuscripts - one found in a library in France and the other in Mexico - were written on European paper by Aztecs a couple of decades after the conquest, using the Aztec system. The article offers an optimist view that nearby Mayan cosmology and prime number arithmetic was involved. The paper did not include modern or ancient views of the fundamental theorem of arithmetic, as reported from the history of number theory, and a lunar calendar door to the past. The paper's oversight may be corrected by overlaying a Mayan abacus form of the Chinese Remainder theorem used to connect lunisolar 260 day calendars, lunisolar 405 moon calendars, great cycle calendars of 18980 days, and larger great years calendars without using fractions/remainders.

Arithmetic examples are needed to shed appropriate light on related Aztec and Mayan topics. The first defines a modular congruence. Floyd Lounsbury, writing on the number 1.5.5.0 of the Mayan Venus Table shows that a good history of number theory textbook allows observers to view the Mayan data outside our modern 10 decimal context, an academic mandate. Using only the history of number theory's use of prime numbers large chunks of mesoamerican mathematical astronomy and arithmetic 'jump out of' otherwise unreadable codices. Lounsbury chose Burton's number theory text. Sanchez selected Ore's history of number theory text.

A META-METHOD, MODULAR CONGRUENCES The modular congruence method follows Lounsbury and others who pointed out great years within Mayan lunar cycles connected to the 584 solar day Venus calendar context. The ancient Mesoamerican calendar info was likely processed by a Mesoamerican abacus, detailed by Sanchez in 1961. That is, an error in the 4/4/08 paper's use of a Western 'arrow of time' geometric proportion, reading the paper within modern base 10 decimals, is correctable by overlaying number theory congruences, and a Mayan abacus arithmetic detailed by Sanchez. The interesting 4/4/08 "Science" paper had inappropriately named a geometric congruence, rather than a number theory congruences, as a pre-1521 method that assisted Aztecs in determining taxes/tributes paid on the areas of properties owned/operated.

Modern number theory congruences also solve other planetary indeterminate equation problems and lunar eclipse counts. Mayans reduced eclipse findings to 260 day, 405 moon, 11960 day, and 11960 day calendars. A 408 solar moon cycle (33 solar years and 34 lunar years) aligned Islamic calenders and an acano method, close to a 405 moon cycle that assisted alignment of Mayan calendars. Alignment connected a 260 day lunar calendar, a 365 day calendar, and a LCM baed 18980 day calendar. Anthony Aveni summarized the connection this way, "Students of the codices need not be reminded that the eclipse table immediately follows and is attached to the Venus table. Moreover, these juxtaposed tables are related numerically: one great cycle (GC) = 37960 days - 3 x 11960 + 8 x 260 days... My attempts to seek solutions to this problem were stimulated by the work of Lounsbury (1983) who offered a plausible scheme for the placement of the Venus table in real time."

The Mayan great cycle calendar was not new. An ancient Chinese oral tradition reports the Chinese calendar was aligned by the visible planets, moon, and sun lining up as a 'string of pearls', within a sidereal lunar calendar. NASA 'proved' the Chinese tradition by modeling the 'string of pearls' planetary cycles to Feb 26, 1953 BCE through March 5, 1953 BCE. It may be important to note that a solar lunar calendar of 135 was commonly used in China, Babylon, Egypt, and Mesoamerica (Aaboe).

That is, the Dresden Codex and its lunar and planet related calendars were created by forms of the Chinese Remainder Theorem written into an abacus. The maze of calendars can be understood by beginning with a 405 cycle lunar calendar. One of the oldest lunar calendars was the 99 moon cycle Babylonian Calendar. Using lunar cycles "dated by solar passages (solstices) accuracy of solar lunar calendars" increased in the ancient Near East to 135 solar moons and 235 solar moons. The authors of the 4/4/08 "Science" paper may wish to update its use of the West's 'arrow of time' metaphor by considering modular lunar calendars written a Mayan version of the acano system, and Sanchez's "Arithmetic in Maya" (click to read the book by a slide show) parsed by Occams's Razor, the simplest method was the historical method.

A Canary Island lunar calendar method used a 3 x 4 'checkerboard'. The method shows the reliability of "Arithmetic in Maya" form of arithmetic used by Lowland Maya within a least common multiple abacus context, one that did not generally use fractions.

Mesoamerica should be considered a mathematical region for several reasons, one being an abacus likely used by adjoining cultures. It is well known that Mesoamerican cultures generally did not use fractions in astronomical calculations, hence Mesoamerican calendars were closely related. Yet, despite the need to find a common method, scholars have reported Mayan and Mesoamerican numerical themes in disconnected and confusing ways: a: Mayan paper, another Mayan paper, a Mesoamerican paper, and other Mesoamerican discussions; with the last link mentioning a sixth grade level "Voyage of the Mini" topic. Sixth graders can compute sidereal and solar time by viewing the Big Dipper (Ursa Major). The sidereal and solar time models lose an average 4 minutes per night based on the earth's orbit around the sun. Sidereal and solar time readings can be compared to increase poor 15 minute inaccuracies of each star clock readings for use in longitude estimations. The method allows latitude and longitude to be computed as Columbus and Portuguese navigators read the location of the North Star, and the nightly rotation of Ursa Major. Ursa Major's star clock is reading from the "Armed Guards" of Polaris pointer stars, terms used by Columbus. A research paper may connect this astronomical time clock method to decode geographic locations on Aztec 'surveying maps', since this method was also used by many ancient cultures.

SUMMARY Mayan 405-moon data (11980 days) and 584-day Venus calendars were likely dated on a Mayan abacus. A Mayan abacus and its calendar arithmetic was reported by Sanchez in 1961. The Mayan abacus did not employ remainders, a major feature of Mayan calendars. Yet, rational numbers may have been found in Aztec area texts. Spanish found areas to calculate 'tribute' owed in a manner that used rational number remainders, and an unknown Aztec, or Spanish round-off method. An Aztec remainder round-off method may be an exception to a standard Mesoamerican abacus.

Bibliography

1

Anthony F. Aveni, "The Moon and the Venus Table", THE SKY IN MAYAN LITERATURE,Oxford Press, 1992.

2

Jose Barrios Garica, TARA: A Study on the Canarian Astronomical Pictures, Part II: The acano chessboard, Universidad de Laguna (Spain), 1996.

3

David Burton, "Elementary Number Theory, Allyn and Bacon, 1976.

4

Floyd Lounsbury, "A Solution for the Number 1.5.5.0 of the Mayan Venus Table", THE SKY IN MAYAN LITERATURE, ed. A. Aveni, Oxford Press, 1992.

5

Oystein Ore, Number Theory and its History, McGraw-Hill, 1948.

6

George I. Sanchez, Arithmetic in Maya,Austin-Texas, 1961.