LOC-Cribbs-Marco Polo

Introduction

For the past 300 years, navigators determined their longitude by using a chronometer, the equation of time, and measurements with a sextant—that is, before radio navigation and global positioning satellites. There are older maps that show coastlines and cities that are registered with somewhat accurate latitude and longitude. This paper shows how longitude can be measured by an explorer in one day without a chronometer, telescope, or radio or satellite navigation. This method or one based on similar principles could have been used by Zheng He.

Background

Ancient people understood longitude and described a method for measuring it. Rather than using a chronometer, the time synchronization was provided by an eclipse of the moon—no matter where you are on earth, the time of maximum eclipse is the same. This method is described in Gavin Menzies’ book “1421.” This method works but has practical limitations. Lunar eclipses are rare, the eclipse must be predicted, if it is cloudy at either the observer’s position or at the prime meridian, one must wait for the next eclipse, the eclipse can be seen from only half the earth… Furthermore, the longitude can only be determined when the observer returns to the prime meridian where his data is compared with that collected at the prime meridian on the same day and the calculations are made.

Technical Background

If you were to ask anyone what event describes a 24-hour day, you might get the reply that it is one revolution of the earth on its axis. That is the wrong answer. As Figure 1 illustrates, the earth must rotate once plus an extra four minutes for the sun to be over the same point on earth because the earth orbits the sun in an arc. One revolution is called a sidereal (star) day (23 hours, 56 minutes, 4 seconds.) In one sidereal day, a star moves from meridian to meridian. On the average, in one solar day (24 hours), the sun moves from meridian to meridian. The apparent period of the stars is exact (within a small fraction of a second), but the motion of the sun varies throughout the year. One day might be 16 minutes advanced and another 14 minutes late from the mean solar day. This is a result of the fact that the orbit of the earth is elliptical rather than circular. This irregularity repeats on a yearly basis.

Simplified Discussion of the Method

To simplify the discussion, assume that the difference between a solar and sidereal day were exactly 4 minutes. Then, at the prime meridian, a table could be made of the time interval between solar and a bright reference star (such as Regulus) passage through the meridian each day. This table would repeat after 4 years (there are 365-1/4 days in a year).

If a measurement were made at the anti-meridian (point B in Figure 1), the time difference would be displaced by 2 minutes in time because the earth moved halfway between the measurement points at the prime meridian. If the observer made a time interval measurement T at any location on one day and compared it with the table, he could obtain the difference in longitude between him and the prime meridian:

Longitude = .

How could the measurements be made? The minimum equipment required is a small pool of water, two poles, a string, and a clock that produces a uniform “tick” for 24 hours. The procedure is as follows:

Place the poles into the ground in a North-South orientation across the pool.
Pull the string taught between the poles.
Move one pole laterally so that it is simultaneously true that the reflection of the string and the North Star cannot be seen because the string is in the way as the eye scans from one end of the string to the other. This defines the local meridian. Once this is done, the equipment is calibrated, as long as it is not disturbed.
Establish a clock that is uniform for 24 hours.
Now position the eye so that the reflection of the string is hidden and note the time when the reflection of the reference star disappears behind the string.
With the eye in the same position, note the time when the edge of the sun is just visible crossing the string.
The next night, note the time of passage of the reference star.

The last step allows the clock to be calibrated because star-to-star passage is a uniform 23 hours, 56 minutes, 4 seconds, or 86164 seconds. The interval is then

T = 86164 seconds.

The table might look as follows, using our modern units of time:

Day	Hours	Minutes	Seconds	Seconds Increment
:	:	:	:	:
60	10	19	18	236
61	10	23	14	236
62	10	27	11	236
63	10	31	06	:
:	:	:	:	:

If the reading were 10 hours, 23 minutes, 57 seconds, then the observer knows it is between noon on day 61 and 62 at the prime meridian. His reading is 42 second after day 61, so he is at

Longitude = = 64.1^o

It might be daytime when the reference star is passing the meridian. The interval of star-to-star passage is constant for any pair of stars so a set of intervals for other stars allows them to be used. This also allows multiple measurements to be made each day for greater accuracy.

Time Measurement

How might the clock be made? Many possibilities might be imagined. One way is to have two pendulums—one of long period and one short (a few seconds). The short period pendulum could be adjusted to produce an integer number of cycles in the period of the long period pendulum. A water clock could then be used to keep track of the number of cycles of the long period pendulum. As each pendulum ran down, a gentle push as the pendulum went through zero would keep it going. Finally, the observer could provide uniform verbal counts between “clicks” of the short period pendulum to provide sub-second intervals.

There are many possible variations of a 24-hour clock. The description provided above is based on ideas from several ancient civilizations. There is reference to a mechanical mechanism to supplement the water clock for astronomical purposes in the Chinese literature, but I have not yet found a detailed description.

The above procedure would provide accuracy to several degrees, enough for a crude idea of where the observer was and for a crude map of the earth. What follows are details that could lead to obtaining accuracies close to 1/10 degree. This is accompanied with a description of the Chinese mathematics that was likely used.

History and More Detailed Description

What the following arguments show is that measurements of the sun can be used to determine longitude to about 1^o. Measurements of the moon provide about 0.1^o (6 arc-minutes or 5 miles) accuracy at 40^o latitude. The longitude can be determined in the field if there are predictions of the sun-star or moon-star passage interval available in the field. Otherwise, the determination of longitude can be determined only after the observer returns to the prime meridian to compare the readings with readings made the same day at the prime meridian.

By 1421 the Chinese had the mathematics to:

Interpolate readings using quadratic or cubic curves, even if there were occasional cloudy days (irregular data);
Determine the required degree of the interpolation polynomial;
Provide equations that describe or predict astronomical events over a period of one year, like the equation of time.

I have found only indirect evidence that they could predict the meridian passage of the moon. That pattern repeats after 19 years. The Chinese calendar is lunar and the moon had been the most studied astronomical object. It seems likely to me that such predictions could have been made in 1421.

The Chinese use a lunar month for its calendar. Twelve lunar months are short of a year by about 11 days, so the Chinese add a “leap month” about every three years to keep the lunar calendar in line with the seasons. Predicting the first day of each month, the years that require a leap month and eclipses has been an age-old preoccupation of Chinese astronomers. At first they thought that the month was a consistent 29.5306 days. The Chinese astronomer Jia Kui (? – 92 AD) discovered that the motion of the moon is inconsistent.

The inconsistent motion of the sun was discovered by Zhang Zixin in the 6^th century. The actual motion was fully understood by the famous astronomer Yi Xing (683-727 AD).

The stars are consistent. The interval for a star to cross the meridian is 23 hours, 56 minutes, 4.091 seconds. The variation from day to day is a small fraction of a second. The interval for the sun varies throughout the year. Figure 2 shows the time when the sun crossed the meridian at Greenwich in 2004. There are only 4 days that are 24 hours long: 15 April, 14 June, 1 September, and 25 December. The time of sun passage varies from about plus 16 minutes to minus 14 minutes. Figure 3 shows the variation from day to day of the duration of the meridian crossing. This period can change up to 30 seconds per day. An accurate calculation of longitude using the method described in this paper requires cubic interpolation which was used in China by the famous astronomer Guo Shoujing (1231-1314).

The shape of the equation of time is consistent from year to year. However, there is an offset due to the fact that there are 365.25, or more precisely 365.2425 mean solar days in each year. With the equation of time for any year, one can find the equation for any other year by interpolating the curve knowing the time of the winter solstice at the prime meridian as shown in Figure 4. The pattern nearly repeats every 4 years when a leap year is skipped to maintain long-term accuracy. In two of every three century years a leap year is skipped to maintain long term accuracy. This was last done in the year 1900 and will be done again in 2100. A set of values as shown in figure 4 can provide the interpolation constant for future years.

The variation in the period of the moon is far more irregular and does not repeat from year to year. It approximately repeats every 19 years when there are 235 lunar months. If one knew the 19-year curve, a calibration constant could accurately predict the next 19 year cycle.

The advantage of the moon is that it appears to move through the stars 12 times faster than the sun, so an error in the readings produces an error in longitude that is 1/12th the corresponding error using the sun. The disadvantage of the moon is that it is more difficult to predict its meridian passage—the pattern repeats every 19 years rather than each year like the sun. With only the equation of time, the observer can determine his position. A reading of the moon could also be taken to compare with readings taken on the same day at the prime meridian when the ship returns. This comparison which could be used for maps does not require prediction and is 12 times more accurate.

Issues Influencing Accuracy

How accurately can meridian passage be determined without a telescope? Let us first consider a star. The resolution of the unaided eye is considered to be ½ arc-minute; that is, two stars can be resolved if they are ½ of an arc-minute apart. The diffraction limit of the eye is about 1/3 arc-minute.

If a star were to go behind a string whose thickness was the diffraction limit of the eye, it would appear to go from full brightness to extinction to full brightness as it traveled 1/3 arc-minute. The time of extinction could be estimated to 1/6 arc-minute or 10 arc-seconds.

A star near the plane of the ecliptic (the path of the sun) moves 360^o (1296000 arc-seconds) in a sidereal day (86164 seconds) or 10 arc-seconds in 2/3 second of time. If the interval between star and sun passage changes 236 seconds per day (the average change), then a 2/3 second error corresponds to 1^o error in longitude using the slip of the sun. Using the moon, the error would be about 1^o/12 or 5 arc-minutes, or 5 statute miles at the 40^o latitude of Beijing.

The moon could be measured the same way as the sun. Today we define the position of the moon by its center, but most ancient civilizations define it by its edge, which is easier to measure accurately. Thus, the passage time is the instant when the first or last light is seen crossing the string depending on the phase of the moon.

A direct reading of the sun is dangerous to the eye without very dark glasses. What the Chinese did was to erect a structure about 50 feet high with a plate containing an 80-mil (1/12 inch) hole. This “camera obscura” casts an image of the sun. It has been used by many ancient civilizations to track the motion of the sun.

The sun is traveling at 360^o/24 hours, or 15 arc-seconds/second. At a 50-foot projection, the image of the sun is moving 44 mils/second (in 23 seconds it travels 1 inch) with a blur of 80 mils. The time of passage of the projection of the edge of the sun on a scribed meridian line can be estimated to about ½ second.

What does the error budget look like? The North Star was about 1^o off north in 1421 and is about 1^o off (in the opposite phase because of the procession of the earth) today. The star is true north twice a day as it circles the true north position. The Chinese had a pictograph of the position of the other stars at these two instances so that the error in the North Star position was much less than 1^o. This error turns out to be irrelevant in determining longitude. A 1^o error in the meridian produces essentially the same error in the sun passage and star passage if the star is near the plane of the ecliptic so the difference in time has no significant error due to a small meridian error.

Using a telescope, meridian passage can be estimated by amateurs no better than 1/3 second. Today this is normally done by amateurs by recording the ticking of the time standard radio station WWV on a tape recorder. At star passage, the observer says “T.” Analysis of the tape allows resolution to sub-second accuracy. The human mind is quite good at interpolating over short time intervals. Galileo discovered the law of falling bodies by adjusting the string used in musical instruments over a groove and allowing balls to roll through the groove. The positions of the strings were adjusted to produce equal intervals of time.

Perhaps the greatest unknown is the accuracy with which time was measured. From my understanding of water clocks, they do not have the necessary time resolution. The Chinese literature suggests that the water clocks that were used for astronomy were used in conjunction with a mechanical device. This combination could achieve accuracies to about 10 seconds per day. It is not accuracy, but uniformity that is important, and I have not found a discussion of this. The accuracy can be calibrated by the method described above. The ancient Chinese discussed temperature compensation, but I can only speculate how that was done.

What is certain is that they had some clock starting with Jia Kui (?-92 A.D.) and had accurate clocks with high-resolution by the time of Guo Shoujing (1231-1314). The Chinese astronomers correctly concluded that it was both the sun and moon that were irregular. Most early observatories only measured the heavenly bodies as they passed the meridian. Elevation and time of meridian passage are the only measurements that could be made.

One could have concluded that the angle between the moon and some star follow a non-linear relationship by measuring the angles with a Torqueta. The issue would then be—was it the motion of the stars or the moon that was irregular? The correct conclusion that it was the moon could only be determined with a clock or a remarkably good guess. I have no suggestion of how the irregularity of the sun could be determined except with a high-resolution clock.

Was a table used to keep track of the equation of time? It probably was not used. Most ancient civilizations realized that any important function that depended on long tables would fail because of the lack of printing presses. Errors would creep in and multiply.

In the Congaing era (1102-1106) Songshi came up with a set of equations for the length of a shadow of an obelisk at 34◦ 48’ 45” latitude. The equations were, using modern notation:

where t is the number of days after the winter solstice. The instructions were as follows:

{

S₁ (t) if 0 ≤ t < 62.2

S₃ (182.62 – t) if 62.2 ≤ t < 91.31

S₂ (182.62 – t) if 91.31 ≤ t < 182.62

S₂ (t – 182.62) if 182.62 ≤ t < 273.93

S₃ (t – 182.62) if 273.93 ≤ t < 303.04

S₁ (365.24 – t) if 303.04 ≤ t < 365.24

This illustrates that the Chinese had sufficiently sophisticated mathematics in the 10^th century. This function is graphed in figure 5. I speculate that there was an equivalent algorithm for calculating the equation of time. Cubic interpolation is well documented so the constants in Figure 4 could be used to correct this equation to any year. I suggest that the Chinese had the mathematical skill and the good sense to avoid long tables.

The same skill at interpolation allowed the interval of sun and star passage to be estimated between the readings at the prime meridian, even when the period was changing rapidly or inflecting. This skill allows the cubic curve rather than straight line interpolation as I suggested in the simplified discussion at the beginning of this paper. Without going into details, the Chinese algorithm allows interpolation of either the dependant or independent variable, so longitude could be calculated directly knowing the equation of time, the coefficient for any particular year, and the star-sun interval.

So what accuracy might be expected? The biggest unknown is the uniformity and resolution of the clock that was used. A person of skill could probably make errors of 1 second in both star and sun passage by taking several days of measurements (including missing days due to clouds) and estimate the time to 1.4 seconds, excluding errors in the clock. This provides about 2^o error for the solar and 10 arc-minutes using the moon. This translates to 56 miles and 4.6 miles, respectively, at 40^o latitude.

References

I used material from an excellent history of the astronomy and mathematics of the Chinese: “Calendars, Interpolation, Gnomons, and Armillary Spheres in the work of Guo Shoujing (1231-1314)” by Ng Say Tiong under the direction of Helmer Aslaksen of the Department of Mathematics, National University of Singapore. This work contains the detailed equations for quadratic and cubic interpolation including interpolation with irregular data (cloudy days). A comprehensive work on the history of Chinese mathematics is in “A History of Chinese Mathematics” by Jean-Claude Martaloff, Springer, 1997. This work contains the equations for Figure 5. There are over 500 references in this work.

Acknowledgements

Gavin Menzies initiated my interest in this paper. He constantly finds new, ancient maps with various degrees of accuracy in longitude. I wrote him an e-mail suggesting that the method described in his book “1421” was used, but was impractical for the time scale in the book. He asked me to write a paper on this alternate method. Gavin put me in e-mail contact with Stan Lusby in New Zealand. Stan supplied many sanity checks on the method. He has agreed to cooperate with me in measuring the difference in longitude between California and New Zealand for a future paper on this subject. At that time we should have hard data on accuracy using this method.

List of Figures

Difference Between Solar and Siderial Day
Equation of Time
Variation of Length of Solar Day Each Day
Winter Solstice at Greenwich
Length of Shadow of Gnomon 8 Chi Long

Sun