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vertical height of each star below Polaris “declination” and its position around the equator from Nanjing “right ascension.” So for the stars in the sky, the Chinese had the same system of measurement they used to determine latitude and longitude. This system was called the equatorial system—vastly simpler than the equinoctial system, used in medieval times before Guo Shoujing, which relied on the ecliptic or the horizon. After 1434, Europeans adopted the Chinese system, which remains in use today.
    Next, the Chinese needed precise instruments to measure each star’s position. Guo Shoujing provided the tools. A sighting tube was first positioned by pointing it at Polaris at precisely the angle of the observer’s latitude—that is, if the observer was at the North Pole, the sighting tube would be at 90° elevation. On this diagram, the instrument is aligned to Polaris at 39°49' N, the latitude of Beijing. Once positioned, the instrument was bolted down—because if the angle changed from the latitude of the observer, it became useless.
    The observer then selected a star, looking at it through another tube attached to a circle marked in degrees. The movement of the tube along the circle gave the number of degrees below Polaris of the selected star (the arc y-z), which is the star’s declination.
    The horizontal angle, the angle from Nanjing, was found by rotating the ring around the equatorial circle, which gave the horizontal angle of the star from Nanjing (its right ascension). The position of the star then was entered in the star tables. The Chinese entered 1,461 stars in their tables, a process that required many astronomers and hundreds of years.
    Tables were printed and, along with a star map, given to each navigator. Thus all navigators possessed a common system of latitude andlongitude to fix their positions on the globe, and an identical map of the heavens, which enabled them to recognize each star.

    A torquetum based on the equatorial system, as used by Zheng He’s navigators and pioneered by Guo Shoujing.
    How the Star Tables Allowed Longitude to Be Calculated
    For the following description, I am indebted to Professor Robert Cribbs, who has tested the method described to prove its efficacy. This method allows longitude to be determined on any clear day without waiting for a lunar eclipse and without sending messages back to the observer in Beijing. It is a much more advanced method than that described in my book 1421 (that method, kindly explained to me by Professor John Oliver and Marshall Payn, is dependent on eclipses of the moon, which do not happen all that frequently).
    Professor Cribbs’s method is based on the fact that the earth not only rotates on its own axis every twenty-three hours and fifty-six minutesbut also travels in an ellipse around the sun—something Guo Shoujing had worked out back in 1280. The combination of these two movements means there is a slip of four minutes each day between the time when the earth is in the same position relative to the sun (solar time, twenty-four hours) and the time when the earth is in the same position relative to the stars (sidereal time, twenty-three hours and fifty-six minutes). This slip between sidereal time and solar time amounts to one day every 1,461 days, or four years. The effect is that every midnight, twelve hours after the sun has hit its highest point in the sky, a different star will be in line with Polaris than the day before.

    This is a typical star map as used by Zheng He and his navigators.
    Astronomers in Nanjing observed the night sky for every day of the 1,461-day cycle and noted the star in line with Polaris at precisely midnight. They produced a table of 1,461 days, which was dispensed to navigators. The 1408 astronomical calendar covers 366 days of that cycle. A copy of a page of the 1408 astronomical tables is reproduced later in the color insert of this book.
    With the tables in