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hand, a navigator in, say, the Indian Ocean must know only which day of the cycle it is, which he calculates by the number of sunsets that have occurred since he left Nanjing. If he left Nanjing on day 61 of the cycle and has noted eighty sunsets, then it is day 141. On the tables, he can see that Aldebaran is in line with Polaris on day 141 (to the Nanjing observer).
    However, in the Indian Ocean he observes another, unrecognized star in line with Polaris. He consults his star map and confirms from the tables that it is Betelgeuse. He can now make one of two calculations: he can note the difference in right ascension between Aldebaran and Betelgeuse, which will equal the difference in longitude between the observer in Beijing and himself; or he can note the time it takes for Aldebaran to come into line with Polaris. If this is, say, six hours (one quarter of twenty-four hours), then his longitude difference from Beijing is 90 degrees (one quarter of 360°).
    For the calculation to be accurate, both the observer in Nanjing and the navigator in the Indian Ocean must be looking due north at Polaris. If they wish to use the second method to calculate longitude, both must have precisely the same midnight. They do this as follows: First they use a vertical stick to measure the sun’s shadow. When the shadow is shortest, the sun is at its maximum height at midday and is due south. Both observers build a trench running due north-south, a trench that can be flooded to see the reflection of Polaris at night and emptied of water to measure the sun’s shadow at midday.
    The sun’s shadow when at its shortest can be measured on the trench. To get the precise second, the shadow is sharpened by employing a pinhole camera atop a pole called a gnomon (described on the website). By using identical gnomons and a standardized pinhole camera, the observers in Nanjing and the Indian Ocean can each determine the same due south/north and the same instant when the sun is at its highest—that is, midday. Our experiments described on the 1434 website have shown that they can calculate this to within two seconds. They can now use a standardized clock to calculate midnight, twelve hours after midday. The 1434 website explains how this Chinese clock worked and how, in Zheng’s era, refinements were built to compensatefor different temperatures and air pressures, which would otherwise have affected the number of drips coming out of the clock. Thus time was accurate to within two seconds.
    Using the water clock, the observer in Nanjing and the observer in the Indian Ocean establish the same midnight. After sunset the trench is flooded and two poles are placed on either side of the trench; a line is suspended horizontally between the poles. Another line is hung vertically so the observer can see the reflection of the vertical string in the water of the trench in line with Polaris. At the instant of midnight, the navigator in the Indian Ocean looks at the star in line with Polaris reflected in the water, which is in line with the string. (In our example, on day 141 this star is Betelgeuse.) His tables for day 141 say that in Nanjing the star is Aldebaran. From that, he can determine his longitude. According to Robert Cribbs, the method is accurate to within two seconds, which amounts to a maximum error of three degrees in longitude, negligible for mapping the world.
    This method requires the navigator to be on land. However, Professor Cribbs has also developed a method of determining longitude at sea by using the equation of time of the moon and the angular distance between the moon and a selected star. To deploy this method (see 1434 website) some calculus is required to establish the future position of the moon for the 1,461-day cycle. By 1280, Guo Shoujing had established a system very similar to calculus. The results appeared in his tables and calendar, which were adopted by the Ming in 1384. Consequently, they were available to