Zheng He’s method of calculating latitude and longitude
This memo aims to summarise and combine the work of a number of distinguished professors. After it has been reviewed and errors corrected, it will be placed on the 1421 website. Layout is as follows
2 Outline of various methods and brief summary of Chinese methods
3 The principal method adopted – the slip between sidereal and solar time.
4 Establishing the prime Meridian and preparing ephemeris tables.
5 Establishing the local Meridian and calculating longitude on land.
6 Clocks and time measurement.
7 Calculating longitude at sea. Equation of time of the moon.
8 Latitude and Declination.
9 Beijing Ming Observatory.
1 – Acknowledgements
I am indebted to the following
(i) To Professor John Oliver and Marshall Payne, for encouraging me to discover how longitude could be calculated without clocks especially by eclipses of the moon.
(ii) Professor Robert Cribbs – for his pioneering work on calculating longitude by the slip between sidereal and solar time and, at sea, by using the equation of time of the moon coupled with the angular distance of stars.
(iii) Professor Helmer Aslaksen and Ng Say Tiong for their work on the Chinese Yuan astronomer Guo Shou Ping – notably on the gnomon and armillary sphere, the equation of time of the moon and (my suppositions) Guo Shou Ping’s influence on Cassini, Kepler, La Place and Newton.
(iv) Professor Rosa Mui, Paul Dong and Zhou Xinyan for their work on Jupiter’s satellites visible to and used by the Chinese to determine longitude.
(v) Ralph Mcgeehan and Dr Robert Massey for the transits of Venus.
(vi) Professor Joseph Needham for descriptions of the Chinese observatories at Beijing, Nanjing and Yang Cheng and for descriptions of Guo Shou Ping’s gnomons and armillary spheres.
(vii) Tai Peng Wang for his discovery of Zheng He’s records describing the actual stars used to calculate latitude and longitude at sea.
(viii) Professor William Penhallow for his experiments at Newport Rhode Island which show how longitude could be calculated there using the slip between solar and sidereal time.
(ix) Stan Lusby for discovering how latitude can be determined without clocks or sextants by the altitude of Alnilham.
(x) Professor Xiao Jun for kindly arranging a tour of the Ming dynasty observatory at Beijing.
(xi) Professor Yingsheng Liu for arranging a tour of the Yuan dynasty observatory at Nanjing.
(xii) Ross Prefect determining the lunar eclipses available during the 1405 – 1431 voyages of Zheng He.
(xiii) Dr Robert Massey and Louis Hissink for Venus rising and setting times and how to sharpen the image of Venus using natural features.
(xiv) Miss Hui for describing the clocks and time measuring devices available to Zheng He.
As will be seen neither I nor the 1421 Team has made any original contribution.
2 – A summary of the methods of determining longitude without clocks
(i) By eclipses of the moon. (Professor John Oliver and Marshall Payne) Observers on the prime and local meridian’s view this event at the same time. Each notices the star in line with Polaris at the same predetermined instant. The time of passage between star a and star b is the difference in longitude between prime and local meridians. This method works and is accurate but it depends on both observers seeing the eclipse and moreover the local meridian observer has to return to Beijing before longitude can be calculated. This method is described in ‘1421’.
(ii) By transit of Jupiter’s moons (Professors Rosa Mui, Paul Dong and Zhou Xinyan). This method is essentially the same as (i) save that it can be used on a daily basis. It was only possible when Jupiter’s moons could be seen with the naked eye which they cannot be today. (Zheng He did not have telescopes).
(iii) The slip between sidereal and solar time (Professor Robert Cribbs), this is described in para 3.
(iv) By using the equation of time of the moon coupled with its angular distance from a selected star (Professor Robert Cribbs). This is described in Para 7.
2(B) A brief description of the progress of Chinese Astronomy.
Summary of Chinese determination of longitude and latitude
(i) By the Tang dynasty latitude could be calculated.
(ii) By the Song longitude could be calculated at the Prime Meridian hence the map of China of 1137.
(iii) In 1267 the Arab astronomer Zamaruddin built the first wooden global sphere with latitudes and longitudes and correct ratio sea/land of 70/30.
(iv) By the Yuan dynasty Guo Shoujing had improved upon Jia Kui (Eastern Han) work on the equation of time of the moon and on Zhang Zixin (Sui) equation of time of the sun. Guo Shoujing’s method was to use a third degree formula of interpretation. Guo Shoujing also designed a new equatorial torquetum on which the positions of the sun, moon, ecliptic, Polaris and other key stars could be placed. (Professor Helmer Aslaksen)
(v) The effect of Guo Shoujing’s work was that the Chinese now had a torquetum which could be used to determine longitude by using the slip between sidereal and solar time or the slip between sidereal and lunar time (work of Professor Robert Cribbs).
(vi) Zheng He’s fleets used Guo Shoujing’s methods to determine latitude and longitude as can be noted from the Datong Li (Great Unified System of Calendrical Astronomy) officially adopted by the Ming Bureau of Astronomy in 1384 (research of Tai Peng Wang).
3 – Determining longitude by the slip between sidereal and solar time – Professor Robert Cribbs
Using this method, longitude can be determined on any clear day without waiting for a lunar eclipse and without sending messages back to the observer at the prime Meridian.
Each day the apparent time of passage of any particular star through the meridian changes about 4 minutes from the previous day. This apparent change is caused by the earth’s rotation coupled with the earth’s progression round the sun. Assume there were 365.25 days in a year and Polaris were at exactly true north (where it is twice each day), then at any location whether prime or local meridian a table could be made of the interval between solar passage and star passage for a period of 4 years after which it would repeat. This method does not require time synchronisation with the prime meridian. This is the procedure.
(i) Determine the prime meridian by using the sun or Polaris (details follow in para 4)
(ii) Note the star in line with Polaris when Polaris is due north for each day of the 1461 day cycle. Make an ephemeris table and issue this to the junk captains. (described in para 5)
(iii) Determine the local meridian by using the same method as in (i) – (described in para 6).
(iv) Note the star which is in line with Polaris on the local meridian for the day in question using the same method as (ii).
(v) Compare the time interval between the star at the prime meridian (ii) and the star at the local meridian (iv) for the appropriate day. For example of on day 61 the star at the prime meridian was Betelgeuse and at the local meridian was Aldeboran and the time difference between the two was six hours (quarter of 24 hours) then the longitude difference is 90 degrees (quarter of 360 degrees).
4 – Establishing the prime meridian
In Zheng He’s era the prime Meridian was the Yuan Observatory in Nanjing. It was shifted to the new observatory in Beijing c1442. Steps are as follows.
(i) Build a platform running precisely due north by aligning with either the sun when at its maximum altitude (due south) or with Polaris when due north.
(ii) This platform must be capable of measuring the length of the sun’s shadow at midday and also when flooded seeing the chosen star reflected in the water in line with Polaris (by using a slit or piece of string).
(iii) Using the sun’s shadow (Professor Aslaksen and Ng Say Thong). A rod of predetermined length (same length used on both prime and local meridian) is placed vertically at the south end of the platform (ii above). The length of the shadow will determine the solstice – maximum for winter, minimum for summer. However Chinese astronomers would have been faced with insignificant changes in the length of the noon shadow around the solstice and the indistinct shadow produced by the long gnomon. Guo Shoujing (Yuan dynasty) was able to solve both of the problems by making use of a cross bar fixed at the top of his 40 foot gnomon and a device called a shadow definer. This made use of the principle of the pin hole camera which was able to produce a distinct shadow of the crossbar within a bright spot of light. This distinct shadow was then visible for the length of the 40 foot gnomons noon shadow on a scale incised on the stone trench. This can be seen today in the Beijing observatory.
It is obviously important that the gnomon is precisely vertical. This can be achieved by two trays of water at the top of the gnomon at right angles to each other. When water in each tray is level the gnomon is vertical. (This explains the slit in the sides of the sloping pyramid shape observatories – the slit enables the gnomon to be adjusted to vertical by an observer standing on the tower).
(iv) The precise determination of the sun’s minimum shadow length gives noon for a particular day and hence due south. The platform can be flooded at night to view Polaris and the chosen star. This dual requirement can be seen in the water troughs and drainage mechanism in the trench at the Ming observatory in Beijing.
(v) Preparing tables for 1461 day cycle (Professor Cribbs) For night viewing the trough is flooded and two poles are placed upright either side of the trough. A string is suspended horizontally between the poles with a vertical pendulum in the middle. One end of the string is adjusted so that it is simultaneously true that both the reflection of the string in the water and Polaris are observed by the string as the eye moves along the string. This establishes a vertical plane through the poles. Professor Cribbs estimates this system is accurate to about one second of time or 1.5 degrees (60 miles) at 40 degrees latitude. Determining the same precise interval between the meridian passage of the sun and the transit of the chosen star across Polaris for both local and prime meridian is of great importance. The accuracy achieved in determining this interval is discussed in para 6.
(vi) Ephemeris tables
The method summarised above would give a chosen star in line with Polaris for each day of the 1461 day cycle. Guo Shou Jing’s equatorial torquetum could list other stars which would be available for use at local meridians across the globe. Guo Shou Jing’s torquetum can be viewed at the Ming observatory in Beijing today. Here is Dr Gunnar Thompson’s photo and the diagram taken from Professor Needham ‘Science and Civilisation in China’.
The observer could view Polaris by looking up through the centre of the equatorial circle (j) which is six foot in diameter, then through the small circle (b) and along the arrow pointing to Polaris.
He could also look at a particular star through the central sighting tube (i) which is here shown pointing vertically upwards. The central sighting tube could be rotated within the declination ring (f) for the observation of various stars. Thus whatever star was bright in the sky for that particular day of the 1461 day cycle could have its position determined by virtue of the fact that Polaris was centrally sighted. The azimuth circle (m) and the altitude measurement circle (n) were, like other circles, graduated with degrees incised in their rims. So the declination and right ascension of bright stars could be written down in the ephemeris table together with the chosen star at the Prime Meridian for that particular day.
5 – Establishing the local meridian and calculating longitude in land
Exactly the same procedure would be employed as for the prime meridian. The observer would need to construct –
(i) A trench/platform pointing due north/south.
(ii) A gnomon exactly vertical of the same length as the prime meridians. Vertically achieved by deploying a slit in the side of the tower/pyramid to enable the observer to adjust the gnomon so both water trays at its top had level water).
(iii) Precisely the same clock as used at the prime meridian. The accuracy of the clock was less important than the requirement that the clocks were identical – i.e. if water clocks that both must use water of the same temperature and salinity and have identical systems for making adjustments for air pressure.
6 – Clocks and Time Measurements (Miss Hui Research)
A brief history of Chinese clocks
104 – 101 BC Han Wudi Luoxiayi – armillary sphere
130 AD Zhangheng – water transportation armillary sphere, a forerunner of the mechanical chronometer.
725 AD Zhang Sui and Liang Lingzan water transportation copper armillary sphere, a forerunner of the European mechanical clock.
1092 AD Su Song and Han Gonglian – Water transportation machine combining armillary sphere and mechanical calculagraph. This was the world’s first mechanical chronometer equipped with release mechanism. Su Song’s book on the construction of this chronometer has 60 pictures including those of its 150 separate parts.
1276 AD Guo Shoujing – Damingdian water clock.
By 1276 the Chinese had compensating mechanisms for reduced water flow and hence water pressure as time elapsed; for different air pressure; for different water temperature and salinity. I have been unable to find how accurate Gao Shoujing’s water clock was, but it seems to me it was invented in parallel with his shadow definer on the gnomon. In short the clock could be calibrated by the number of drips from one meridian passage of the sun to the next. Halving the number of drips between the two meridian passages of the sun would give the precise time for observers at the prime and local meridians to view the chosen star for the particular day in the 1461 day cycle. Differences between two meridian passages could probably have been calculated to about 1.5 seconds (comments would be appreciated).
7 - Calculating Longitude at sea
(i) Before Zheng He (Tai Peng Wang research)
In 14th century Wang Da Yuan achieved a transoceanic passage from Mozambique to Sri Lanka using a combination of compass and star linked positions called Guoyang Qianxing to determine latitude and longitude.
(ii) Zheng He’s fleets (Tai Peng Wang)
According to Gong Zhen in “Xiyang Banguo Zhi” – Notes on Barbarian countries in the Western seas. Zheng He’s fleets were relying on sightings of the rising or setting sun and moon to help determine how far the ship had travelled east or west (longitude) and the height of the stars to determine the distance of the ship from the stars (lat and long).
The “Chart’s of Zheng He’s voyages” says “gauging the vertical positions of the given stars above the horizons in the east, west, north and south [he] reached Sri Lanka”. This book further states that on their voyages from Dandi Bandar (16ºN 73º 03’E) to Jabal Khamis (22º 25’N 59º 27’E) they used the declination of Zhini (The woman weaver) and of Nanmen Shuangxi (Sagittarius). For the westward route they used Gemini and Procyon to determine latitude and longitude. For further details please consult Xi Fei Long, Yangxi, Tang Xiren eds. “Zhongguo Kexue Jishi Shi, Jia Tong Quan” – The History of Chinese Science and Technology, volume on Transportation – Science Publisher Beijing 2004 at pages 395 – 397.
(iii) Chinese Astronomical Almanacs available to Zheng He.
The “Datong Li” (Great universal system of calculating Astronomy) based upon Guo Shou Jing’s system of calendrical astronomy was officially adapted by the Ming Bureau of Astronomy in 1384. This contained phases of the moon and predictions of lunar and solar eclipses.
(iv) Using the slip and sidereal and lunar time to determine longitude (Professor Robert Cribbs)
The advantage of this method is that it is more accurate by the number of lunar months in a year – about 12. The problem is that the motion of the moon needs to be predicted – it speeds up going to the sun and slows when retreating, both speeding and slowing being different at the earth’s apogee and perigee. Is there a lunar equivalent of the equation of time? (May 2005).
More recently Professor Cribs thinks the Chinese could predict the equation of time of the moon and hence if you know the orbit of the moon and local time and can determine (by a sextant) the position of the moon with respect to some star to one tenth of its diameter then you can determine longitude to about 1.5 degrees.
Ng Say Tiong and Professor Helmer Aslaksen state Jia Kui in the Eastern Han period (25 – 200 AD) discovered the insignificant motion of the moon and that Guo Shoujing used a third degree formula of interpolation to record the equation of time of the sun [which formula could also have been used to record the equation of time of the moon].
It would therefore seem entirely feasible that Gong Zhen’s statement that Zheng He’s fleet “were relying only on the sightings of the rising or setting sun or moon to help determine how far to the east or west the slip has gone” was correct.
8 – Chinese determination of latitude
By the time Zheng He set sail, Chinese astronomers had centuries old, proven, methods of determining latitude – in the northern hemisphere by the sun or Polaris, in the southern hemisphere by the sun.
Polaris was in the 1420’s near an extension of the earth’s axis, twice each day it was at due north. By then the Chinese carried circumpolar constellation templates which enabled them to determine when Polaris was due north. By measuring the altitude of Polaris when due north and when 12 hours away from due north and taking the average altitude, they could determine latitude – 90º N when Polaris was right above them and 0º N when Polaris was on the equator. Centuries earlier the Chinese had determined the earth to be round and so by sailing north between the equator and N pole they could make continuous estimates of the quarter circumference of the earth. Polaris is not visible in the S Hemisphere.
The Chinese had realised