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.
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.
A diagram showing how the earth travels in an ellipse around the sun.
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 the 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.”
Notes Chapter 3
1. I am Indebted to the research of Tai Peng Wang, whose work has been the foundation for this chapter. See titles of papers in bibliography.
2. Needham Vol 27 p.145
3. Needham Vol 30 pt.2 p.83
For calendars, see Needham, vol. 3, pp. 49, 125, 378–381.
ZHENG HE’S NAVIGATORS’ CALCULATION OF LATITUDE AND LONGITUDE
There are no signposts in the open ocean. The only way a naviga-tor 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