In the Maya view, the heavens consist of 13 layers, each with its own god. The astronomers job was to keep an eye on them, and to learn their habits. Maya astronomers developed an (unknown) technique to spot cyclical motion in the sea of the stars. The details of their observations, the method of recording them, and the instruments they used are totally unknown. We do know that the calculations the mathematicians made, based on that data, are so accurate that we assume they managed to assemble an extremely detailed record of their observations over a very long period of time.
Their calculation of the solar cycle was so accurate that the calendar they based on it is still 1/10,000th of a day more accurate than our own. In the highlands of Guatemala Maya day-keepers still count the days in the ancient way. Their calendar has not lost a single day in 2500 years.
Their lunar calculations are off by only 33 seconds after 1500 years.
The complex orbit of Venus was predicted within 1 day in 6000 years. Their predictions of the orbit of Mars are equally accurate.
Their table predicting solar and lunar eclipses can still be used today.
Maya scribes wrote about "a dark rift in the center of the world that is the home of all evil and the doorway to the underworld" 3000 years before modern astronomers discovered the black hole in the center of the Milky Way. One Maya calendar charts the precession of the Earth. The cycle begins and ends on the ONLY day in 25,920 years when the rising sun precisely eclipses that black hole. The fact that black holes emit no visible light, and that only .0027% of the cycle can be observed in a human lifetime of 72 years, make the accuracy of that calendar mind-boggling.
While the best minds in Europe thought the world was 5000 years old, the Maya were writing accurate descriptions of celestial events that happened nearly 500 million years in the past.
They wrote about cycles in the heavens that are longer than the known age of the universe. We don't know how accurate they were.
All of these calculations were done by hand, based on observations made without telescope, sextant, protractor, compass, or any other known instrument. The idea that the Maya, essentially a stone age tribe, could produce calculations of such extreme accuracy that Europe could not duplicate them for thousands of years would seem totally incredible were it not so well documented.
The Maya astronomical accomplishments are clearly recorded on paper in the Dresden Codex in Germany. They are carved in stone in the ruins at Quirigua, Guatemala. You can see them expressed in the architecture of El Caracol and El Castillo in Chiten Itza, Mexico where the play of light and shadow depict the feathered serpent's decent into the earth and return to the sky, at sunrise and sunset, twice each year on the equinox days. Another example is in the Temple of the Sun at Palenque, Mexico where tiny darts of sunlight pierce the innermost corners of the building only at first light on 3 days each year. There may be more examples waiting in the jungle, as yet undiscovered. The ancient Maya were clearly masters of the sky.