As part of the calendar project, Guo presided over a nationwide programme of astronomical observations. He selected twenty-seven sites for astronomical observation throughout the country, which covered a wide area from latitude 15° N to 65° N and longitude 128° E to longitude 102° E. The items of observation included the length of the shadow of the gnomon, the angle of the North Star from the ground surface, and the beginning times of day and night on the vernal equinox and the autumnal equinox…. Guo also examined nearly nine hundred years of astronomical records from 462 to 1278 and selected six figures from the records for calculating the duration of the tropical year. Guo’s result was 365.2425 days, which was the same as that of the Gregorian calendar, the calendar now widely used across the world….
Guo Shou Jing and the other astronomers worked for four years and completed the calendar in 1280. They made numerous calculations converting the data of the ecliptic coordinate and the equatorial coordinate systems, and used twice interpolations to solve the variations in the speed of the sun’s movement, which affected the accuracy of the calendar. The calendar was unprecedented in accuracy. It adopted the winter solstice of the year 1280, the ninth year of the Yuan dynasty, as the epoch, the point of reference for the calendar, and established the duration of a tropical year of 365.2425 days and that of a lunar month 29.530593 days. The error between the duration of its tropical year and that of the revolution of the earth around the sun was only 26 seconds. The calendar was named the Shoushi, meaning “measuring time for the public.”
Issuing calendars was the prerogative of the emperor alone. Accuracy was necessary to enable astronomers to predict eclipses and comets—a sign that the emperor enjoyed heaven’s mandate. If predictions proved incorrect, the astronomer responsible was severely punished, often with death.
The Shoushi calendar produced by Guo Shoujing was officially adopted by the Ming Bureau of Astronomy in 1384. This is the calendar that Zhu Di and the Xuan De emperor would have ordered Zheng He to present to foreign heads of state (discussed in detail in later chapters).
The Shoushi calendar can be viewed in the Yuan shi-lu, the official history of the Yuan dynasty. However, copies also came into the possession of Europeans, notably the diarist Samuel Pepys and the famous scientists Robert Boyle and Robert Hooke. The Japanese and Koreans also copied the calendar, and translations from those languages can be viewed on our website.
The calendar contained the length of a solar day at the latitude of Beijing. This is the duration from the time when the sun is at its maximum height (altitude) in the sky from one day to the next. We tend to think of this as twenty-four hours. It is not. The earth rotates around its own axis every twenty-three hours and fifty-six minutes while also traveling round the sun. The combination of the two movements means that the earth’s position relative to the sun, compared with its position relative to the stars, varies by about four minutes each day. Moreover, the earth’s trajectory around the sun is not a circle but an ellipse. The sun is not at the center of this ellipse, so that as the earth nears the sun it accelerates. As the earth recedes from the sun, on the longer leg of the ellipse, it decelerates. Its rotation also speeds up approaching the sun and slows down receding from the sun.
Thus, the length of the solar day varies throughout the year. The difference of this length is called the equation of time of the sun. To predict the length of the year at 365.2425 days, which is accurate to within ten seconds a year, Guo Shoujing had to take into account four of these movements. In order to accomplish that, he must have known how the solar system worked, including the facts that the earth travels around the sun in an ellipse and is not at the center of the universe and that the earth is attracted to the sun’s much bigger mass.
A diagram showing how the earth travels in an ellipse around the sun.
Guo Shoujing’s calculations for the lunar month of 29.530593 days were even more impressive, requiring a more complex trigonometry. The moon travels around the earth as the earth is moving in an ellipse around the sun. This means that as the earth approaches the sun, the moon’s attraction to the sun’s mass increases, so the speed at which the moon travels around the earth accelerates. Then, as the earth recedes from the sun on its elliptical path, the moon decelerates. Hence, to make his extraordinarily accurate calculations, Guo had to be aware not only that the earth travels around the sun in an ellipse but also that the moon circles the earth. He had to have understood spherical trigonometry and to have employed calculus and have had an accurate idea of the respective masses of earth, sun, and moon.
However, there are further ramifications to Guo Shoujing’s achievements. The earth’s trajectory around the sun is not constant: it changes over the years. Guo knew of these changes, which he had gathered from Chinese observations stretching back eight hundred years. The great French astronomer Pierre-Simon Laplace credited Guo Shoujing with knowledge of what Laplace called the “diminution of the ecliptic”—essentially, the fact that the earth’s ecliptic path around the sun had grown flatter over the centuries.
Even further refinements were taken into account by Guo Shoujing. The earth is not a perfect sphere but an oblate spheroid with flattened poles. Its center of gravity is somewhat below the center of its volume. This means the earth has a slight wobble, which can be deduced by the apparent position of the stars—in particular by Polaris, the Pole Star, which apparently moves over a 26,000-year period. This movement had been compensated for by the Chinese before Guo Shoujing’s era. Templates had been made to adjust for the apparent movement of Polaris.
Finally, Guo Shoujing knew of the planets’ orbits around the sun, and even of Jupiter’s rotation and its circling moons. The American writer Rosa Mui and colleagues Paul Dong and Zhou Xin Yan have kindly informed me of the work of Professor Xi Zezong, a Chinese astronomer based in Beijing, who has found that Jupiter’s satellites or moons were first discovered two thousand years before Galileo by the Chinese astronomer Gan De.
Since A.D. 85, Chinese astronomers have made accurate observations of the period of planetary revolutions around the sun (synodic intervals). They are correct to within a few hours—Mercury 115 days, Venus 584 days, Mars 779 days, Jupiter 398 days, Saturn 378 days. (In later chapters, we provide evidence that Copernicus, Galileo, Kepler, Hooke, and Newton were aware of the Chinese astronomers’ work.)
In their published paper entitled “Calendars, Interpolation, Gnomons and Armillary Spheres in the Works of Guo Shoujing (1231–1314),” Ng Say Tiong and Professor Helmer Aslaksen of the Department of Mathematics, National University of Singapore, note that the inconsistent motions of the moon and sun were discovered in the Eastern Han period (A.D. 25–200), and during the North and South dynasty (A.D. 386–589), respectively. The method of interpolation employed by A.D. 554–610 was the equal interval second difference method. (Please refer to our 1434 website for further explanation.) Guo Shoujing improved on this by using a third difference method of interpolation, which enabled him to determine the equation of time of the sun and moon and hence to predict their positions. Guo Shoujing had developed the forward distance method of interpolation subsequently further developed by Newton into calculus.
The Shoushi calendar, which Zheng He’s fleets presented to heads of state, based upon Guo Shoujing’s pioneering work, contained a mass of astronomical data running to thousands of observations. It enabled comets and eclipses to be predicted for years ahead, as well as times of sunrise and sunset, moonrise and moonset. The positions of the sun and moon relative to the stars and to each other were included, as were the positions of the planets relative to the stars, sun, and moon. Adjustments enabled sunrise and sunset, and moonrise and moonset, to be calculated for different places on earth for every day of the year. As described in detail in chapter 4, the calendar enabled longitude to be calculated by using the slip between solar and sidereal time, by eclipses of the moon, or by the angular distance between the moon and selected stars or planets. Please refer to the 1434 website and to t
he endnotes for further explanation.
Tai Peng Wang has found the specific stars by which Zheng He’s fleet navigated. We can set these up on the “Starry Night” computer program for the dates when Zheng He’s fleet was transiting the Indian Ocean en route for the Malabar Coast of India and Cairo. We can also compare these stars with those included in Zheng He’s navigational tables and the almanac for the year 1408, now in the Pepys Library at Cambridge. (The 1408 tables contain similar astronomical information as that contained in the Shoushi calendar.)
Thus Zheng He was able to provide Europeans with maps, navigational tools, and an astronomical calendar beyond anything they had yet been able to produce on their own. Supplied with this revolutionary knowledge, the barbarians would be able to make their way to the Middle Kingdom, appropriately “with deference.”
4
ZHENG HE’S NAVIGATORS’ CALCULATION OF LATITUDE AND LONGITUDE
There are no signposts in the open ocean. The only way a navigator can determine his position is by using the stars, planets, sun, and moon.
As a first step, a navigator must have a system of providing markers across the oceans. This system of markers, adopted by all seafaring civilizations for millennia, is latitude and longitude. It involves drawing imaginary horizontal and vertical lines over the globe. Horizontal lines are called latitude lines, and the vertical are longitude lines.
Latitude lines are parallel with the equator; each longitude line passes through both the North and South Poles. So a navigator’s precise position can be fixed on the globe using a common system.
In order to have produced an accurate map of the world by 1418, the Chinese fleets must have had such a system to determine their positions at sea. Without an accurate system, captains could not have known the true locations of newly discovered lands, and any map derived from their disparate calculations would have been an incoherent mess.
Unlike the Europeans, who followed Babylonian astronomers with 360 degrees of longitude, the Chinese employed 3651/4 degrees. The Chinese used latitude degrees below Polaris (at 90° elevation). The Europeans used latitude above the equator (Polaris 0° elevation). The results are the same for both systems.
Diagrams showing the lines of latitude and longitude around a globe.
After establishing a common system for the earth, the Chinese next had to establish a common map of the heavens. Each navigator would have had to use the same name for the same star as well as the same star map from which longitude would have been determined.
How the Chinese Fixed the Stars’ Positions in the Sky
In the thirteenth century, the astronomer Guo Shoujing fixed the positions of key stars relative to Polaris (the Pole Star). Polaris appears on an extension of the earth’s axis, billions of miles away above the North Pole. Because of the earth’s rotation, the heavens appear to rotate around Polaris. The farther north one goes, the more of the heavens one can see.
Diagram showing the positions of ships A and B on a globe. Ship A is at 20° N 20° W, Ship B is at 0° N 20° E.
Ships A and B discovering new lands at point C will have the same position for the new land: 10° N 0° E.
In 1964 I was navigator of HMS Narwhal, a submarine operating under the polar ice cap. Now and then we would find clear-water “lakes,” called polynyas, where we would surface in order to fix our position by the stars. The heavens appeared like a vast globe above us. As we approached the North Pole, we seemed to be inside a bowl looking at a hemisphere of stars spreading in an arc down to the horizon all around us.
At the North Pole, the Chinese could fix the position of every star in the Northern Hemisphere relative to Polaris. The stars are so far away that to an observer on earth they never change their positions relative to one another.
The Chinese divided the sky into twenty-eight segments or mansions. Picture an orange with its skin sliced; the cuts start where the orange was fixed to its tree and continue vertically downward. They called each mansion a hsiu. They fixed the position of stars at the top of each of the twenty-eight mansions relative to the Pole Star (ABC).
The Chinese fixed the position of stars at the top of each of the 28 lunar mansions relative to the Pole Star.
Then they fixed stars in the lower part (DEF) of each segment relative to those in the upper part (ABC). Because stars never change their position relative to one another, even if the Chinese were not near the North Pole and hence could not see the stars in the lower part of each segment (because these stars were below the horizon), they always knew the stars’ positions. So they could produce star maps.
They noted the vertical positions of each star below Polaris (none can be above Polaris) and the horizontal position of each hsiu relative to Nanjing (longitude). The Chinese called the 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 and longitude 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 minutes but 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 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°).
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