CHAPTER I
THE PROBLEM AND ITS SCOPE

Introduction
Did you know who are the Maya? Maya are an American Indian people who developed a magnificent civilization in Central America. The civilization reached its period of greatest development about A.D250 and continued to flourish for more than 600 years.
The Maya produced remarkable architecture, paintings, pottery and sculpture. They made outstanding advancements in astronomy and mathematics and developed an accurate yearly calendar. Maya were also the first people in the Western Hemisphere to develop an advanced form of writing. Today, their territory is divided between Mexico and several Central American countries. Through this study, the researchers aim to have knowledge and ideas about them and to know their Mathematical system. In addition, researchers and students may find out that the research about Maya is an interesting topic because they will know how they predicted the past events in this world.

Statement of the Problem

This research aimed to know and share an idea that could enlighten the reader. Thus, this research sought to answer the following questions:
1. Who are the Maya?
2. What are their Calendar System and Numerals?
3. How did they predict the End of the World in 2012?
4. What are the other predictions predicted by the Maya?
5. How accurate was the predictions of the Maya in the past events?
Significance of the Study

Since we are now living in the modern age, some of the people who were born in this age might have not been aware about the past, and they might be curious about it.
Other schools nowadays, taught us about histories from the past, but not all. Others don`t know who are the Maya. They are more focused on the topic which they like or interested in it.

Scope and Limitations of the Study

This research is focused on the Maya Civilization, specifically on their Numeral and Calendar system, their predictions about the End of the World in 2012 and other past events.
Various people around the world researched this study. But nowadays, it is not anymore popular or well known in other countries like Philippines.

CHAPTER II
REVIEW OF RELATED LITERATURE This chapter contains the related literature, which provides insight about the research.
According to Hoffman (2005), the Mayawere settled in Mesoamerica after migrating from Siberia to Alaska over a land bridge that formed in the Bering Sea more than 11, 000 years ago. It is believed that they traveled along the west coast of North America and eventually established villages and then larger city states in Mexico and Central America. Maya civilization has been divided into 3 parts. The Preclassic period lasted from about 2000B.C to about A.D300; the Classic period lasted from about A.D300 to about A.D900; and then the Postclassic period lasted from about A.D900 to about A.D1500. The Maya expanded their writing system during the Classic period and devised a number system during the time. This was the success in building and construction, their creation of a calendar system, and their advance astronomical and scientific calculations.
The Maya Calculation Assistant published by Abreu (2004), is a calculator (matrix table) used by the Maya in their mathematical operation which also teaches the use of the Maya numerical system. As centuries ago, and today the Maya Calculation Assistant`s main purpose is to ease the executing of mathematical calculations such as multiplication, division and finding a square root. It is also similar to the abacus of other cultures that is a tool, used by the ancient Maya to simplify and speed up daily counting. (see figure 1, Appendix p.21)
Since the Maya numerical system is vegisimal. The Maya Calculation Assistant table contains 400 individual cells (20x20). Arabic numbers are also placed in each cell to simplify learning. To ease understanding, the table grows from top to bottom and from left to right, opposite to the Maya original.
When using the Maya Calculation table, please remember:
 The top row and the first column on the left are the main row and column, they are numbered 1-20.
 When Maya numbers are written vertically (in any vigesimal position) dots always go on top of bars.
 When there are dots, bars or a zero but separated by a wider space, they belong to a higher vigesimal position, so they have to be multiplied by the value of that position.
Learn THE MAYA MATHEMATICAL SYSTEM for these positional values.

Multiplication
Numbers from 2 to 20 of the main row can be multiplied by any other from 2 to 20 of the main row can be multiplied by any other from 2 to 20 on the main column, the result is easily found at the intersecting cell of the two numbers multiplied.
Squares
The first power of any number from 2 to 20 can be obtained in the same manner by multiplying any one of these main row numbers by the same number in the main column, the result is also in the intersecting cell of the two numbers multiplied.
Division
Any number from 4 to 400 located in the table`s cells (dividend), except in the main row and column, can be divided by a number in the main row (divisor). The result is found in the corresponding cell of the main column. This is also true if the divisor is in the main column instead, in which case the answer is found in the corresponding cell of the main row.
Square root
The square root of numbers from 4 to 400 located in cells of the diagonal row (that goes from 1 to 400) of the table, can be found in the corresponding cell of the main row or column.
According to Abreu (2004) that the decimal system widely used today around the world in schools and originated by counting with the fingers a person has. Counting with the fingers and toes started the Maya vigesimal system. The vigesimal system is based on groups of twenty units. The decimal system goes from 1, 10, 100, 1000, 10000, etc. and the Maya vigesimal system goes 1, 20, 400, 8000, 160000, etc. While in the decimal system there are ten possible digits for each placeholder [0-9], in the Maya vigesimal system each placeholder has possible twenty digits [0-19]. For example, in the decimal system 21=10*2+1 while in the vigesimal system 21=20+1.They added, Mayas were masters of education and they have also discovered and used the zero. Their zero is represented by an ovular shell .
The Characteristics of the Maya Mathematical System
a.) It is vegisimal, this means that it is based on 20 units [0-19] instead of the 10 units [0-9] of the decimal system. This table shows the first 20 numbers and their Arabic equivalents. (see figure 2, Appendix p.21 )

b.) It only uses three symbols, alone or combined, to write any number.
These are: the dot- worth 1 unit, the bar- worth 5 units and the zero symbolized by a shell.
c.) It also uses a vigesimal positioning system, in which numbers in higher places grow multiplied by 20`s instead of the 10`s of the decimal system, compare number 168,421 in both systems (see table 1)

Table 1. Compare number 168,421 in both system
Place Number
168,421 Place`s Decimal Value Equals & is written Place`s
Vegisimal
Value Equals & is written
6th 1 X 100,000= 100,000 3`200,000
5th 6 X 10,000= 60,000 160,000

4th 8 X 1,000= 8,000 8,000

3rd 4 X 100= 400 400

2nd 2 X 10= 20 20

1st 1 X 1= 1 1

TOTAL Arabic 168,421 Maya 168,421

d.) Numbers in the Maya system can be written vertically or horizontally. In vertical writing, the bars are placed horizontally and the dots go on top of them, in this case the vigesimal positions grow up from the base. When written horizontally, the bars are placed vertically and the dots go to their left and higher vigesimal positions grow to the left of the first entry.
The Maya would write 168, 421 (see figure 3, Appendix p.21)
Thus when writing vertically the vegisimal positioning system, to write 20 a zero is placed in the first position (base) with a dot on top of it, in the second position. The dot in this place means one unit of the second order which equals to 20. To write 21, the zero would change to a dot (1 unit) and for the subsequent numbers the original 19 number count will follow in the first position. As they in turn reach 19 again another unit (dot) is added to the second position. Any number higher than 19 units in the second position is written using units of the third position. A unit of the third position is worth 400 (20x20), so to write 401 a dot goes in the first position, a zero in the second and a dot in the third. Positions higher than the third also grow multiplied by twenties from the previous ones. Examples of the numbers mentioned (see figure 4 and figure 5, Appendices p.22 )
Maya Calendar was a catalog that registers all the days of a year, distributed in weeks and months, with astronomical data, such as time of sunrise and sunset, the moon phases, or with religious information such as patron saints and festivities. It also was a time division system, all of the world`s cultures have their calendars initially lunar and afterwards lunar-solar. The Chaldeans and Babylonians passed their calendric knowledge to the Egyptians, these in turn to the Greeks and these finally to the Romans who adopted it for their common use. Soul 2005)
Soul 2005) added, from the beginning of Maya civilization there has been a very close link between astrology and the development of the calendar. The importance of this connection is evident considering the need to determine the times for the most basic functions of early societies such as agriculture and the collaboration of religious events.
According to Douma (2008) The Maya calendar uses three different dating systems in parallel, the Long Count, the Tzolkin (divine calendar), and the Haab(civil calendar). Only the Haab has a direct relationship to the length of the year.
A typical Maya date looks like this: 12.8.16.2.6.3 Cimi 4 Zotz.
12.8.16.2.6is the Long Count date.
3 Cimiis the Tzolkindate.
4 Zotz is the Haabdate.

What is the Long Count?
The Long Count is really a mixed base-20/base-18 representation of a number, representing the number of days since the start of the Maya era. It is thus a kin to the Julian Day Number.
The basic unit is the kin (day), which is the last component of the Long Count. Going from right to left the remaining components (see table 2).
Table 2. From right to left remaining components
Uinal( 1uinal= 20 kin = 20 days)
Tun (1 tun= 18 uiinal = 360 days =approx. 1 year)
Katun (1 katun= 20 tun = 7,200 days = approx. 20 years)
Baktun (1 baktun= 20 katun= 144,000 days= approx. 394 years)

The kin, tun, and katun are numbered from 0 to 19.
The uinal are numbered from 0 to 17.
The baktun are numbered from 1 to 13.
Douma (2008) also said that the first date in the Long Count should be 0.0.0.0.0, but as the baktun (the first component) are numbered from 1 to 13 rather than 0 to 12, the first date is actually written 13.0.0.0.0.
He also stated that there authorities disagree on what 13.0.0.0.0 corresponds to in our calendar. He had come across three possible equivalents. (See table 3).
Table 3. Three possible equivalents
13.0.0.0.0= 8 Sep 3114 BC (Julian) = 13 Aug 3114 BC (Gregorian)
13.0.0.0.0= 6 Sep 3114 BC (Julian)= 11 Aug 3114 BC (Gregorian)
13.0.0.0.0= 11 Nov 3374 (Julian) = 15 Oct 3374 BC (Gregorian)

Assume one of the first two equivalences, the Long count will again reach 13.0.0.0.0 on 21 or 23 December Ad 2012 – a not too distant future.
The date 13.0.0.0.0 may have been the Mayas` idea of the date of the creation of the world.

What is the Tzolkin? The Tzolkin date is a connection of two “week” lengths. While our calendar uses a single week of seven days, the Maya Calendar used two different lengths of week:
 A numbered week of 13 days, in which the days were numbered from 1 to 13.
 A named week of 20 days, in which the names of the days (see table 4).
Table 4. Names of the Tzolkin days
0.Ahau 1.Imix 2.Ik 3.Akbal 4.Kan
5.Chicchan 6.Cimi 7.Manik 8.Lamat 9.Muluc
10.Oc 11.Chuen 12.Eb13.Ben 14.Ix
15.Men 16.Cib 17.Caban 18.Etznab 19.Caunac

As the named week is 20 days and the smallest Long Count digit is 20 days, there is a contemporary between the two; for example, the last digit of today`s Long Count is 0, today must be Ahau; if it is 6, it must be Cimi. Since the numbered and the named week were both “weeks”, each of their name/number change daily; therefore, the day after 3 rolls around, 20 days later, it will be 10 Cimiinstead of 3 Cimi. The next 3 Cimi will not occur until 260 (or 13 x 20) days have passed. The 260-day cycle also had good luck or bad luck associations connected with each day, it became known as the “divinatory year.” The “years” of the Tzolkin calendar are not counted. The Tzolkin started when the Long Count 13.0.0.0.0 corresponds to 4Ahau.

What is the Haab? The Haab was the civil calendar of the Mayas. It consisted of 18 “months” of 20 days each, followed by 5 extra days, known as Uayeb. This gives a year length of 365 days.The names of the months (see table 5).
Table 5. Name of Haab months
1.Pop 7.Yaxkin 13.Mac
2.Uo 8.Mol 14.Kankin
3.Zip 9.Chen 15.Muan
4.Zotz 10.Yax 16.Pax
5.Tzec11.Zac 17.Kayab
6.Xul 12.Ceh 18.Cumku

In comparing to the Tzolkin dates, the Haab month names changed every 20 days instead of daily; so the day after 4 Zotzwould be 5Zotz, followed by 6Zotz up to 19 Zotz, which followed by 0 Tzec. The days of the month were numbered from 0 to 19. The use of a 0th day of the month is a civil calendar is unique to the Maya system; it is believed that the Maya have discovered the number zero, and the uses to which it could be put, centuries before it was explored in Europe and Asia.
The Uayeb days acquired a very derogatory reputation for bad luck; known as “days without names” or “days without souls”, and were observed as days of prayer and mourning. Anyone born on those days was “doomed to a miserable life.” The years of the Haab calendar are not counted. The length of the Tzolkin year was 260 days and the length of the Haabyear was 365 days. The smallest number that can be divided evenly by 260 and 365 is 18, 980, or 365x52; this was known as the Calendar Round. If a day is, for example, “4 Ahau 8 Cumku” the next day falling on “4 Ahau 8 Cumku” would be 18,980 days or about 52 years later. Among the Aztec the end of a Calendar Round (see figure 6, Appendix, p. 23) was a time of public panic as it was thought the world might be coming to an end. When the Pleiades crossed the horizon on 4 Ahau 8 Cumku, they knew the world had been granted another 52-year extension. The Haab day started when the Long Count 13.0.0.0.0 corresponds to 8 Cumku.
Douma (2008) added that although there were only 365 days in the Haab year, the Mayas were aware that a year is slightly longer than 365 days. In fact, there many of the month-names were associated with the seasons; Yaxkin, for example, means “new or strong sun” and, at the beginning of the Long Count, 1 Yaxkin was the day after the winter solstice, when the sun starts to shine for a longer period of time and higher in the sky. When the Long Count was put into motion, it was started at 7.13.0.0.0, and 0 Yaxkin corresponded with Midwinter Day, as it did at 13.0.0.0 or 1,101,600 days. We can therefore derive a value for the Maya estimate of the year by dividing 1, 101, 600 by 365, subtracting 2, and taking that number and dividing, 1,101, 600 by the result, which gives us an answer of 365.242036 days, which is slightly more accurate than the 365.2425 days of the Gregorian calendar.
(This apparent accuracy could, however, be a simple coincidence. The Mayas estimated that a 365-day year precessed through all the seasons twice in 7.13.0.0.0 days. These numbers are only accurate to 2-3 digits. Suppose the 7.13.0.0.0 days had corresponded to 2.001 cycles rather than 2 cycles of the 365-day year).

In ancient times, the Mayans had a tradition of a 360-day yearly. But by the 4th century B.C.E. They took a different approach than either Europeans or Asians. They maintained three different calendars at the same time. In one of them, they divided a 365-day year into eighteen 20-day months followed by a five-day period that was part of no month. The five-day period was considered to be unlucky.
According to Johnson and Conger (2007) In the Calendar Round, the 260 days of the Tzolk'in calendar are paired with the 360 days and five nameless days of the Haab calendar. The two calendars are matched the same way the Tzolk'in day. This gives the Calendar Round 18,890 unique days, a time period of around 52 years.
At the time, the Calendar Round was the longest calendar in Mesoamerica. Contemporary historians, however, wished to record Mayan history for generations to come. To accomplish that, they needed a calendar that would take them through hundreds, even thousands, of years.

According to Soul 2005) the Maya predicted the End of the World, because it was the end of the cycle. Some 400 years before their current cycle ended, they erected the next cycle, which would end in 2012. According to the Popol Vuh, a sacred book of the Maya, they were living in the fourth world.The book describes the first three worlds that the gods failed in making and the creation of the successful fourth world where men were placed. Maya believed that the fourth world would end in catastrophe and the fifth and final world would be created that would signal the end of mankind. The last creation ended on a long count of 13.0.0.0.0. Another 13.0.0.0.0 will occur on December 21, 2012, and it has been discussed in many New Age articles that this will be the end of this creation. Maya abbreviated their long counts to just the last five vigesimal places. There were infinitely larger number of units that were usually not shown. It is expressed as 13.13.13.13.13.13.13.0.0.0, where larger units are evidently supposed to be 13s in all larger places.
The Classic period Maya did not believe that the end of this age would occur in 2012. There will be a Baktun ending in 2012, a significant event being the end of a 400 year period, but the end of the age.

According to Raaj (2003) The Maya predicted Japan`s 8.9 earthquake on the exact date it happened, and their calendar suggested a plate shift on western side of the world. Maya`s study of the stars and planets was far more advance than even the Hubble Telescope, that they calculated the distance of each planet from the sun to 99.45% accuracy when Mathematics wasn`t even invented or explored. Maya explored the sun`s spot cycle to 95% accuracy, as a sun`s solar cycle lasts 11.4 years, while Maya`s were off by 3 months to 11.1 years, but perhaps during their time, sun cycle only lasted 11.1 instead of 11.4 of today.

Maya were the first civilization to learn the art of Black Magic, and the art of controlling the spirits to serve them. They knew the energies that existed during Full Moon eclipse and solar eclipse which enhanced their magic rituals; through which they saw the visions in the future and communicated with beings that may had not been in this planet. They also have medicinal knowledge, not all celestial knowledge.
These are the lists of the Maya predictions:
1. Gujarat India earthquake 26 Jan 2001
2. The Rhodes earthquake 26 June 1926
3. Taiwan earthquake 26 April 2010
4. Japan earthquake 26 Feb 2010
5. NE Japan earthquake 26 May 2003
6. North America earthquake 26 Jan 1700
7. Yugoslav earthquake 26 July 1963
All the earthquakes happened on `8’ as 2+6=8, and it is the most significant number in Maya astrology and numerology.
Other Historical disasters:
8.Krakatau volcanic eruption 26 Aug 1883 (36,000 dead)
9.China earthquake 26 July 1976 (255,000 dead)
10.Portugal earthquake 26 Jan 1951 (30,000 dead)
11.Kansu, China earthquake 26 Dec 1932 (70,000 dead)
12.Turkey earthquake 26 Dec 1939 (41,000 dead)
13. Sabah Tidal waves 26 Dec 1996 (1,000 dead)
14.Bam, Iran earthquake 26 Dec 2003 (60,000 dead)

CHAPTER III
RESULTS AND DISCUSSION
This chapter presents and interprets the data gathered in the Maya Civilization Numerals and Calendar System.
The Mayas were the first people who lived in the New world, they were the first people who introduced the counting numbers using the fingers and the feet. They were masters of education. During the Preclassic period lasted about 2000B.C to about A.D300, the Maya civilization began. During the Classic period lasted about A.D300 to about A.D900 it was one of the peaks of large-scale construction and Urbanism. Their population numbered in millions that they created multitude kingdoms, developed an elaborate hieroglyphic and devised a number system. During the Postclassic period lasted A.D900 to about A.D1500 the development in the northern centers persisted, and characterized by increasing diversity of external influence.
In their Mathematical System, they use the vigesimal system, which is based on groups of twenty units. The Maya vigesimal system goes 20, 400, 8000, 16000, etc. and each placeholder has possibly 20 digits [0-19]. For example, 32=20+12. They are also the first people who discovered and used the zero. The Maya Calculation Assistant table is similar to abacus of other cultures, and its main purpose is to execute mathematical calculations such as multiplying, dividing, square, and square roots.
The Maya numeral system shown in figure 2, I like the counting numbers we have been using nowadays, you just have to subtract and add, and also multiply. Numbers in the Maya system can be written vertically and horizontally; they call it the Vigesimal positioning system. For example, to write 20 a zero is placed in the first position which is the base with a dot on top of it, in the second position. (See figure 7, Appendix, p. 23). There are exceptions between the Mathematical count and the calendric count, they gave the third position a value of 360 instead of 400, the higher positions are also multiplied by 20. (Figure 4, and figure 5, Appendix p.22). Maya didn`t invent or discover the their calendars. They just used the calendars of the Mesoamerican people living in their community like the Aztecs. There are three corresponding calendars of the Maya. First, the Long Count, Second the Tzolkin and the third is Haab.
The Long Count is a mixed base-20, it represents the number of the days since the start of the Maya era. The basic unit of the Long count is Kin which means (day) and the last component of the Long Count. It is also an astronomical calendar which we used to track longer periods of time. The Long Count started 13.0.0.0.0.
The Tzolkin it is also known as the divine calendar/Sacred Round, it is a 260 day calendar, with 20 periods of 13 days used to determine the time of religious and ceremonial events. The Haab is the civil calendar, which consisted of 18 months of 20 days each, followed by 5 extra days, known as Uayeb and that gives the year a length of 365 days. The Uayeb days are derogatory reputation for bad luck; known as the days without names, or days without souls. Anyone who is born on that day was doomed to a miserable life.
The smallest number can be divided evenly by 260 and 365 is 18,980, or 365x52 is known as the Calendar Round. And because there were only 365 days in the Haab year, the Mayas were aware that a year is slightly longer than 365 days and true that there are many of the month-names are associated with the seasons, Yaxkin, for example, means “new or strong sun” and at the beginning of the Long Count, 1 Yaxkin was the day after winter solstice, when the sun starts to shine for a long period of time and higher in the sky. The Long Count started at 7.13.0.0.0 and 0 Yaxkin corresponded with Midwinter day, as it did at 13.0.0.0.0 back in 3114 BCE.
We can therefore derive a value for the Maya estimate if the year by dividing 1, 101, 600 by 365, subtracting 2, and taking that number and dividing 1, 101, 600 by the result which gives us an answer of 365.242036 days, which is slightly more accurate than the 365.2425 days of the Gregorian calendar. Suppose the 7.13.0.0.0 days had corresponded to 2.001 cycles rather than 2 cycles of the 365-day year. In ancient times, the Maya had a tradition of a 360-day yearly. They maintained the three different calendars at the same time, They divided a 365 day year into eighteen, 20-day months, followed by a five-day period that was part of no month. The Maya predicted the End of the World because it was the end of the cycle. Maya believed that the last creation ended on a long count of 13.0.0.0.0 and the other will occur on December 21, 2012, and discussed in many new age articles. When the cycle had ended, they erected the next cycle in 2012. According to the sacred book of the Maya, they were living in the fourth world and described the first three worlds that the god failed in making the creation of the successful fourth world where men will be placed. The Maya believed that the fourth and the fifth world will also end. The Maya shortens the Long Counts to just the last five vigesimal places. When larger unit were shown it is expressed as 13.13.13.13.13.13.13.0.0.0, where the larger units are evidently supposed to be 13s in all larger places. A few years ago, the Maya had predicted many past events. They have predicted the Japan`s 8.9 magnitude on the exact same date it happened, and their calendar suggested a plate shift on western sides of the world and also calculated the distances of the planets from the sun. The accuracy of it was 99.45% and discovered the sun`s spot cycle to 95%. Maya were the first civilization to learn the art of Black magic, and controlling the spirits to serve them. They have a knowledge in medicine, not only in the celestial knowledge. The only thing which Maya predicted but didn`t happen was the End of the world and the accuracy of that was 20% because of Classic period of the Maya didn`t agree and believe about the end of the age would occur in 2012 because will be a Baktun ending in 2012, a significant event being the end of the 400 year period, but not the end of the age.

CHAPTER IV
CONCLUSION

Maya civilization had contributed many things in the Mathematics society. They were the first people who introduced the counting numbers using the fingers and feet, they introduced the vigesimal (base 20) system; and they are the people who had many predictions and three corresponding calendars. I choose this topic, because It is interesting and I also want to seek knowledge and ideas about their brief history, their predictions from the past events and in the End of the World, their mathematical system and calendar system.
Therefore, I conclude that this research can be useful for those who want to seek information about the Maya civilization, their Numeral and Calendar system, their predictions from the past. I also encourage all the readers and researchers to read articles, and books so that you can have more ideas about the unknown histories.

Appendix

Figure 1. Maya Calculation Assistant Figure 2. The first 20 numbers and their Arabic equivalents

Figure 3. The Maya would write 168,421

Figure 4. These are the examples of the Mathematical count numbers
(Note: The Maya made one exception to this order, only their calendric calculations they gave the third position a value of 360 instead, instead of 400, the higher positions though, are also multiplied by 20.) Figure 5. The Calendric count

0 Xix im 10 lahun
1 hun 11 buluc 20 Hun kal 400 Hun bak
2 caa 12 lahca 40 Cakal 800 Cabak
3 ox 13 oxlahun 60 oxkal 1,200 Ox bak
4 can 14 canlahun 80 Can kal 1,600 Can bak
5 hoo 15 hoolahun 100 Hookal 2,000 Hoobak
6 uac 16 uaclahun 120 Uackal 8,000 pic
7 uuc 17 uuclahun 140 Uuckal 160,000 calab
8 uaxac 18 uaxaclahun 160 Kahookal 3`200,000 kinchill
9 bolon 19 bolonlahun 180 Ox hookal 64,000,000 alau
Maya Names for Numbers

Figure 6. Calendar Round

Figure 7. Face of the 20 (base)

CHAPTER VI
BIBLIOGRAPHY

Online Resources: http://mathsforeurope.digibel.be/Numerals.htm http://mathcentral.uregina.ca/RR/database/RR.09.00/hubbard1/ http://www.mayacalendar.com/mayamath.html http://www.abovetopsecret.com/forum/thread183760/pg1 http://www.krschannel.com/mayan.html http://www.webexhibits.org/calendars/calendar-mayan.html http://www.mayacalendar.com/calasist.html http://books.google.com.ph/books?id=sN9kOUelUAAC&printsec=frontcover&dq=the+history+of+the+maya+ancient+maya&hl=en&sa=X&ei=H5YVU5m5MI2ciAf_loGgBw#v=onepage&q=the%20history%20of%20the%20maya%20ancient%20maya&f=false