INTRODUCTION As many have read, astronomy has been reported as the birth-mother of mathematics. The oldest neolithic calendar has been reported in an Anatolian temple dated to 8,000 or 9,000 BCE. Its construction suggests the site defined an ancient lunar calendar within an arithmetic system that used only quotients. Like Mayan arithmetic, many lunisolar calendars did not use remainders.

What is important computationally, not considering remainders, translated into a 3 x 4 matrix, during Babylonian and Egyptian 135-moon eras, is that lunisolar calendars reduce to the movement of the solstices (SS) across 12-moons, adding 11 days each year and alternatively subtracting 29 or 30 days every time. This count gives the location of the SS and locations of the other seasonal moons. Making this count, as reported in Babylonian astronomy in an 135-moon context, used until 499 BCE, and replaced by Metonic 235-moon method, an equivalent count is 29+30+29 days = 8*11 days. Thus, the SS stays 8 solar years in the first three moons, the first column. After that the SS jumps to the second column. This give us the basic statement:

8 solar year = 8 lunar year + 3 moons = 99 moons,

which forms the Octaeteris, the well known lunisolar calendar of antiquity.

The basics of a Canary Island 12-moon and 270-moon lunar eclipse calendar context were likely created during the period 900 BCE to 400 AD based on an Egyptian or Libyan 135-moon lunar calendar. The Canary Island calendar is used today and is named acano. The acano calendar uses three colors and is parsed into a 3 x 4 matrix revealing several mathematical topics. Matrix math subjects include 135-moon, 270-moon, and 405-moon lunar eclipse calendars plus a solstice number system that are easy to explain face-to-face, with drawings, reports Jose Barrios Garcia, Professor of Mathematics at the University of Laguna, Canary Islands (Spain). Without drawings, a lengthy discussion is required to explain the modern acano method. An attempt to explain its details, outside of its three-colors, follows.

LUNAR CALENDARS Imagine a chessboard of 3 (horizontal) by 4 (vertical) files, equaling 12 squares alternatively painted in black and red. Each square represents one synodic moon, let us say, from new moon to new moon. So the acano represents 12 consecutive moons, alternatively colored red and black. The moons are ordered vertical, let us say, from left to right.

1 4 7 10

2 5 8 11

3 6 9 12

From this point of view, the acano is an (eventually eternal) temporal framework measuring the pass of time from the endless repetition of this exact group of 12 synodyc moons. There are not intercalary moons in this pattern. Time is measured by the eternal sequence of synodic moons packed in groups of exactly 12 moons. Red and black chessboards have been preserved in ancient caves on the Canary Islands.

Attending to a 19th century indirect source (La luna es la madre de los tiempos) the alternative coloring helps to track the duration of an actual moon. If we assume an average synodic month of 29.5 days, let red moons to be 29 days and black moons to be 30 days (or vice versa). On this way, two consecutive moons average 29.5 days. The acano accounts for a total of 6*29+6*30=354 days.

Let the Sun move across the acano, assume a solar year of 365 days, and suppose that we begin to count just from the coincidence of the New moon with the Summer Solstice in just the first day of the first month of the first acano.

First acano:

Since equinoxes and solstices are never less than three moon spaced, if Summer solstice (SS) occurs in the first day of the first moon of the first acano, the Autumn equinox (AE) occurs in the 4th moon, the Winter solstice (WS) in the 7th moon and the Spring Equinox (SE) in the 10th moon. So the equinoctial and solstice moons (from now on the seasonal moons) are aligned in the first file of the acano.

A Canarian priest working on this pattern would have readily located on the acano the very important (from astronomical, calendrical, social and agricultural perspective) the four seasonal moons. So they would prepare the social and economical activities for the year knowing exactly and anticipated in which moon everything occurs.

Second acano:

Now 12 moons have passed and we are in the first moon of the second acano. Since there is a 11 days difference between the acano and the Solar year, SS would occur in the 11th day of the first moon, so the seasonal moons continue to be aligned in the first file of the acano.

Third acano:

Another 12 moons have passed so the SS would occur in the 22th day of the first month and the seasonal moons continue to be aligned in the first file of the acano.

Fourth acano:

Another 12 moons have passed so the SS would occur in the 33th day of the acano, that is to say, in the 4th day of the second moon. Since the SS has jumped to the second file of the acano, the other three seasonal moons also jump, so they remain aligned!, now in the second file.

And so on for the next followings years. A report from the Canary Islands considers 520 days as one and one-half eclipse cycle, as discussed by Barrios. A suggestion offered by others is that Mayans could have adopted the 520 calendar as a 260 day calendar, the smallest one that predicts lunar eclipses. The Canary Island 520 day cycle a minor cycle of its 270-moon calendar, a calendar that also calculates 135-moon and 405-moon calendars possibly within a 260 day cycle.

Acano Number System

Quoting Barrios, "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."

DISCUSSIONS Aaboe argues for the knowledge of several classes lunar eclipse calendars in ancient cultures, with one being the acano. Several scholars argue the Egyptian and Babylonian 135-moon case, including red to denote life, and black to denote death, following an acano cycle. Jose Barrios Garcia argues the Canarian 270-moon case, including alternating red and black colors. F. Lounsbury and A. Lebeuf argue the Mayan math and 405-moon calendar case, including red to denote east (good), and black to denote west(almost bad), again following an acano cycle. Aztecs used black to denote north, and death. Acano lunisolar eclipse calendars were well known to several ancient cultures built from 9-moon (260 day) and 99-moon calendars.

Two recent acano-type uses aligned civil and scientific calendars connected to eclipses tables, and validated longitudes in land (map making) and ocean navigation. The far ranging Phoenicians circumnavigated Africa in 600 BCE, made later trips, as did Columbus in 1492 assisted by lunar eclipse tables and the "Armed Guards of Polaris", calibrated daily by ' 1/2 hour glasses' for longitude, and Polaris for latitude.

Acano-type lunar eclipse prediction tables, referenced to Spain port city longitudes, were carried by Columbus on four New World trips. Columbus, like the Phoenicians/Libyans had used the 135-moon Egyptian eclipse tables. Phoenicians/Libyans reached the Canary Islands in 900 BCE as deposed Egyptian pharaohs (denoted by mummification and calendars), and stayed there until 400 AD, as a stop over on long voyages. Columbus debarked from the Canary Islands and observed two eclipses on his voyages, determining longitudes more accurate than historians have credited him (by only reading Bishop de Landa's edited data). Columbus had based his longitude calculations on Ptolemy's Almagest definition of the earth's circumference. Bishop de Landa edited Columbus' 'armed guards of Polaris' data as political protection from the treaty of 1494 sanctions - in which Spain may have lost New World lands had Columbus' longitude hand written data been given to the Pope or to Portugal.

SUMMARY The best known acano cycle is marked by 33 solar years and 34 lunar years. The acano cycle uses 270 = 2*135 moons in which 46 lunar eclipses, and a complete number system. Lounsbury reports Mayan eclipse data, beginning with a 260 day calendar in the Dresden Codex as 405 moons = 3*135 moons. Related but different lunisolar calendars have been discovered around the world. Yet, several 135-moon, 270-moon and 405-moon lunar calendars show common features. One regional lunar calendar used a red and black acano 3 x 4 matrix placed in Canary Island caves next to mummies. The red and black 3 x 4 type matrix/abacus innovation may pop up elsewhere in paintings, suggesting an academic connection. As individual cultural achievements lunar eclipse tables were prepared by Egyptians, Libyans, Phoenicians, Mesoamericans, and other advanced cultures. Several ancient lunar eclipse tables assisted in aligning civil and religious calendars, map making, and ocean navigation.

Bibliography

1

Asger Aaboe, "Remarks on the theoretical treatment of eclipses in antiquity",Journal for the History of Astronomy (Cambridge), 1972.

2

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

3

A. Lebeuf, Astronomía en Xochicalco, in La Acrópolis de Xochicalco, México- Instituto de Cultura de Morelos, 1995.

4

Floyd Lounsbury, "Maya numeration computation and calendrical reckoning", Dictionary of Scientific Biography, Charles Scribner and Sons, 1978.