We will soon admire the accomplishments of Ptolemy, who collected accurate measurements of the sky and developed a model of the motions of the planets that was used for nearly 15 centuries. Maya astronomers also made remarkably accurate measurements. The accomplishments of these early astronomers seem incredible, but what they lacked in sophisticated technology and accurate instruments, they made up by patient observing over long periods of time. Here is a comparison of four basic periods of motion of celestial bodies, showing that both Ptolemy and the Mayas came incredibly close to the modern, accurate values. Ptolemy, in the second century A.D., used observations made by Hipparchus (2nd century B.C.) and Babylonian astronomers of the 5th and 6th centuries B.C. as well as his own to calculate average values of astronomical periods, combining some 800 years of observations. From the accuracy they achieved, it seems likely that the Maya astronomers had access to equivalently long records of observations.