TWO INGENIOUS PROCEDURES FOR measuring the radius of the earth have been sent to me independently by Joseph L. Gerver of Honolulu and Dennis Rawlins of San Diego. The procedures are elegant but simple enough for an amateur to be able to repeat easily. You might enjoy seeing how close you can come to the true size of the earth and how subtle variations in the methods alter the results.

Figure 1: Joseph L. Gerver's method of calculating the radius of the earth |

Mount the ruler upright about 15 feet from the pointed rock. measuring the distance with the string. Gerver labels the distance *b*. For the first measurement stand on the other side of the ruler from the rock and adjust your eye level until you can line up the ocean horizon. the top of the rock and a point on the ruler. Note where your line of sight intersects the ruler. Move to the rock and adjust your line of sight until the top of the rock is lined up with the ocean horizon in the opposite direction. Again note where your line of sight intersects the ruler. The distance between the two points of intersection on the ruler is labeled *a*.

The last measurement you need is your height above sea level. Place two stones on the beach so that they are equidistant from your observation point on the hill and a measured distance apart. Return to the observation point, hold the ruler at arm's length, sight on the stones and note their apparent separation on the ruler. Then, shifting the ruler to a vertical position by rotating your fist, measure the distance from the stones to the ocean horizon in units of the apparent separation. Your height above sea level (call it *h*) is the number you get multiplied by the true separation between the stones. Finally, calculate the radius of the earth as being approximately equal to 8hb^{2}/a^{2}.

Figure 2: The geometry of Gerver's method |

Why does Gerver's formula work? The geometry of the experiment, which is shown in Figures 2, 3, and 4, includes four important triangles. They are similar triangles, that is, they have equal angles. Since the triangles are similar, Gerver could employ them to devise several ratios involving the earth's radius *r* and the three measurements in the experiment: *a, h* and *b*. He was then able to solve for *r*.

I shall sketch out his proof here but leave the details to you. From triangles 1 and 2 you can write the ratio CO/*r* = *r*/ (*r* + *h*). From triangles 2 and 3 you have the ratio AC/BC = CO/AC. From triangles 2 and 4 the ratio to be used is *b*/(*a*/2) = CO/AC. Finally, BC is *r* + *h* - CO. By eliminating CO, AC and BC through appropriate substitutions in these ratios you can solve for *r*. At one point in the calculations Gerver makes the approximation that the altitude of the hill is insignificant compared with the diameter of the earth, an approximation that introduces into the results an error of less than .01 percent.

The height *h* of the experimental position above sea level was measured by sighting on the stones on the beach. Gerver makes the approximation that the ratio of *h* and the true separation between the stones is the same as the ratio of the apparent separation between the stones and the apparent distance from the stones to the nearest ocean horizon. In Gerver's scheme for measuring the distance you measure the distance to the ocean horizon in terms of the apparent separation of the stones, and so you automatically have the second of the two ratios. When you multiply by the true separation of the stones, you get the height above sea level.

Figure 3: Triangle BFG between the rock and the ruler |

Another factor to consider is the waves on the water; they too act to raise the horizon in your field of view. This effect can be taken into account by subtracting half of the mean wave height (half of the crest-to-trough distance reported by the Coast Guard and other services) from the observation height *h* determined by Gerver's method.

One source of random error in the experiment arises in judging the alignment of the tip of the rock, the ocean horizon and a mark on the ruler. It is not only the usual kind of error inherent in measuring the length of anything but also the error arising from the fact that these particular things do not cast sharp images on the retina.

Still another source of error comes in estimating the height of the observation site above sea level. The method of estimating the height on the basis of the distance between the stones on the beach assumes that a line to the ocean horizon is horizontal. It is actually angled downward because of the curvature of the earth. This angle, which is called the dip angle, is proportional to 3 the square root of the observer's height above sea level.

With proper care in your observations and with multiple observations Gerver's method will enable you to come close to the true radius of the earth. (What you are actually measuring is not, of course, the equatorial radius of the earth but the local radius of curvature of the ocean surface.)

Rawlins published his method for determining the size of the earth in the February issue of *American Journal of Physics*.

Figure 4: The similar triangles employed in Gerver's calculations |

To follow Rawlins' procedure you need a clear view of the sunset from a beach overlooking a relatively calm ocean or large lake. The eastern side of one of the Great Lakes may be good, and places around Great Salt Lake, which typically has low waves, may be ideal. A stopwatch is preferable for the timing, but a sweep second hand on a wristwatch will also serve. You should be very careful about how you view the sun. Even at sunset looking directly at the sun for any length of time can harm your eyes. The greatest danger is from the invisible radiation, which can dam age the eye without causing any immediate pain. It is best if you avoid gazing at the sun's disk until it is mostly below the horizon. You can keep track of the disk's motion by looking off to the side of it or by giving it an occasional quick glance.

The basic geometry involved in Rawlins' procedure is shown (for an observer on the Equator) in the illustration below. You observe the first sunset while you are lying at point M. As soon as the last visible arc of the sun's disk disappears below the horizon you jump up and wait for the same thing to happen from an elevation of *h*, the height of your eyes. The last ray in the first observation is tangent to the earth's surface at your location and therefore forms an angle of 90 degrees with the vertical, but the last ray in the second observation is not tangent. With your eyes above sea level the ocean horizon forms an angle (labeled 0 in the illustration) with the vertical; it is 90 degrees minus the "dip angle."

Figure 5: The geometry of Dennis Rawlins' sunset-watching method |

Rawlins recognized that the triangle QNO was a right triangle and thus that the sides were related by the Pythagorean theorem. One of the legs, QN, is approximately equal to the earth's radius multiplied by the angle through which the earth turned between the two sunsets. (The leg is therefore approximately equal to the arc between points, Q and M.) Putting this information together with the correspondence between the angle and the elapsed time, Rawlins devised an equation relating the time between the two sunsets, the earth's radius r and the eye-level height h of the observer. (He made one additional approximation: the eye-level height is negligible compared with the earth's radius.) The earth's radius in kilometers is equal to a proportionality constant of 3.78 X 10^{5} multiplied by the eye-level height (expressed in meters) and divided by the square of the time between sunsets (measured in seconds).

Figure 6: The role of refraction |

Suppose you make the first observation at normal eye level *h* while you are standing on the beach and then make the second observation after running up a flight of steps. The effective height difference for the calculation is the square of the difference of the square roots of the altitudes of your two observations. I shall give an example of this calculation below.

Although in the ideal situation you would make your observations on a calm ocean or lake, it is more likely that you will have to deal with waves. They complicate the observations in two ways. First, waves near the shore obscure the ocean horizon when you lie on the beach for your first observation. Hence the first observation must be made from a point higher than the height of the waves. Second, waves at the ocean horizon, which should be a few miles away, raise the horizon. The best you can do to account for them is to estimate their average height and subtract it from each of the heights from which you make your observations. Rawlins takes the average height for the distant waves to be .6 meter, which he bases on oceanographic observations.

A subtler correction has to do with the curvature of light rays by refraction in the earth's atmosphere. When a light ray passes from one medium to another medium with a different index of refraction, the path of the light is bent at the interface separating the two materials. The only exception comes when the ray is perpendicular to the interface; then the direction does not change. The extent of the bending in all other instances depends on the angle at which the ray meets the interface and on the difference between the refractive index of one medium and that of the other. The direction of the bending is governed by whether the refractive index increases or decreases across the interface. If it increases, the ray is closer to being perpendicular to the interface. If it decreases, the ray bends the other way.

Figure 7: The A and B factors employed by Rawlins |

If you are an amateur astronomer, you may have noticed the effect with stars. When a star is overhead, it is seen in its correct position because light from it is not refracted much. When a star is just above the horizon, however, it is seen slightly out of position (too high by about half a degree at the horizon) because the light rays the eye eventually receives enter the atmosphere at such a large angle with respect to the vertical that they are significantly refracted. The refraction makes the star seem to take longer to set because when it is actually a fraction of a degree below the horizon, the bending of its light rays extends the image over the horizon.

So it is with the sun. You can see it even after it should be out of sight below the horizon. Since the same thing happens at sunrise, the daylight lasts a little longer than it would without refraction. For example, since the sun sets perpendicularly to the horizon at the Equator, it can be seen for at least an additional 2.3 minutes because of refraction. Near the poles the additional time is much longer because the sun sets at a considerable angle to the vertical and therefore drops behind the horizon much more slowly.

Refraction makes trouble both far Gerver's method and for Rawlins'. A ray leaving the horizon travels not in a straight line to the observer but roughly in a circle that has a radius six times the radius of the earth. (A good discussion of this effect appears in a classic work by Simon Newcomb, *A Compendium of Spherical Astronomy*, which was first published in 1906. It can be found on pages 198-203 in the current Dover reprint.)

Figure 8: Geometry of the method attributed to Eratosthenes |

One cannot account for the effect of refraction in any consistent way because it depends on the temperature and pressure in the air layers through which the light rays travel. Those quantities change frequently, not only from place to place in the atmosphere but also at a given place. As an approximate measure of the refraction Rawlins multiplies his basic equation for the radius by 1.2. This correction rests on the assumption that the effects of refraction lower the computed radius by about 1.2. The correction effectively reduces the measured time between sunsets (more precisely the square of the time) to the value one would obtain in the absence of atmospheric refraction.

Without elaborate refinements of Gerver's and Rawlins' procedures the only way to get around the daily variation in refraction is to repeat the experiments over many days and average the calculated radii. The corrective factor of 1.2 might normally be off by a few percent. If the atmosphere is strongly stratified into layers with sharp differences in refractive index, the setting sun will even appear to be chopped into layers. The corrective factor of 1.2 will then probably be quite incorrect and your computation of the earth's radius will be far off the mark.

Rawlins makes two more alterations in his basic equation. They both have to do with the fact that in most places the sun does not set perpendicularly to the horizon. The ideal vertical setting can be seen only at the Equator at the time of the spring and fall equinoxes. For any other latitude or date sunsets are slower because the sun sets at an angle to the vertical. Rawlins handles this effect with two factors he calls A and B. The first factor is designed to take care of the date, the second to handle the latitude. A is the square of the cosine of the sun's declination on the day of the experiment. B is the square of the sine of the latitude. (Whether it is north or south latitude does not matter.)

The sun's apparent path through the sky for the year lies in the plane of the ecliptic, which is tilted with respect to the plane of the Equator by 23.44 degrees. The sun crosses the equatorial plane twice a year: on the vernal equinox, which is about March 20, and on the autumnal equinox, which is about September 23. (The precise times of the equinoxes shift backward every four years except when leap year is skipped.) The sun's declination is the angular distance north or south of the earth's equatorial plane. For example, at the equinoxes the declination is zero, at the winter solstice (December 21) it is about 23.4 degrees south, and at the summer solstice (June 21) it is about 23.4 degrees north. Precise tables of the sun's declination can be found in The American Ephemeris and Nautical Almanac, which is published yearly by the Government Printing Office. You can probably find your approximate latitude in an atlas. With this latitude and the date of your observations you can then take the associated factors A and B from the the table in Figure 7. To use these corrective factors subtract B from A and then divide the result into Rawlins' basic equation.

Figure 9: Gnomon method for determining the orientation of the sun's rays |

Rawlins next approximated the height added to the ocean horizon by the distant waves as being .6 meter. This number was subtracted from both of his observation heights, which then became 1.12 and 8.35 meters. To get the effective height difference needed for the calculation he computed the square root of both of these numbers, found the difference between them and squared the result. The answer was 3.35 meters.

Squaring the time, dividing the result into the proportionality constant of 3.78 X 10^{5} and then multiplying by the effective height difference of 3.35 meters gave Rawlins 3,300 kilometers (2,100 miles) as the radius of the earth. That was about 48 percent too low, but the factors involving the date, the latitude and the refraction were still to be taken into account.

Interpolating from the tables, Rawlins found that April 5 has an A factor of .988. The B factor for his latitude was approximately .295. A minus B was .693, which he divided into the previous result to get a radius of 4,800 kilometers (3,000 miles), a result about 25 percent too low. To take refraction into account he multiplied this result by 1.2 to get his final value for the radius: 5,700 kilometers (3,500 miles). This is about 10 percent less than the official value.

If you live near the ocean or a large lake, you might like to repeat either Gerver's procedure or Rawlins', gathering enough data to enable you to find an average value for the radius of the earth. Since atmospheric refraction affects their procedures in opposite directions, by following both procedures and averaging the results you could almost eliminate the effects of refraction. I find it impressive that one can come anywhere near the earth's true radius with rough measurements and without any equipment other than a ruler or a watch and even more impressive that values within 10 to 20 percent of the official ones can be achieved.

Long ago some people realized that the earth was round because of several clues. Stars were in noticeably different positions at places widely separated from north to south. An outbound ship disappeared below the horizon. The earth's shadow on the moon during a lunar eclipse was round. Nevertheless, the idea of a round earth was difficult for many people to accept.

Legend records that the Greek scholar Eratosthenes, working in Alexandria made the first measurement of the size of the earth about 2.200 years ago. He employed a simple (even crude) method but obtained surprisingly good results. At Syene (now Aswan) he either noted or was told that at noon on the summer solstice sunlight entered a well vertically. He reasoned that he could calculate the circumference of the earth if he knew the angle (from the vertical) of the sun's rays at noon on the same date in Alexandria, which was north of Syene and was situated on about the same meridian.

According to the story, Eratosthenes' measurement in Alexandria showed that the rays were slanted from the vertical by one-fiftieth of a circle (7.2 degrees). Since the rays at Alexandria were parallel to those at Syene (he must have realized that the distance to the sun greatly exceeded the radius of the earth), the difference in angle had to be the result of the curvature of the earth. The arc between the two cities along the earth's surface therefore occupied an angle of 7.2 degrees with respect to the earth's center.

To find the circumference of the earth all Eratosthenes needed to do was to find the ratio of this angle and 360 degrees and equate it with the ratio of the arc distance between the cities and the circumference of the earth. It was not easy, however, to find the distance between the cities. Eratosthenes resourcefully measured the distance in terms of the travel time of camels, since he knew that a caravan typically took 50 days to travel between the cities. Knowing how far a camel walked in a day, he was able to compute the distance between Alexandria and Syene and thereby calculate the circumference of the earth. His result (in modern units) was 46,250 kilometers, which is roughly 16 percent too high.

According to history, the next measurements of the circumference of the earth were made in about 85 B.C. by Posidonius of Apamea, who used the distance between Rhodes and Alexandria as Eratosthenes did the distance between Alexandria and Syene. Posidonius' means of measuring the distance was a boat that traveled between the two places. The angle was measured by sighting on the star Canopus: when it was on the horizon at Rhodes, it was 7.5 degrees above the horizon at Alexandria. By doing the same type of calculation that Eratosthenes is said to have done Posidonius computed the circumference of the earth. By modern reckoning his figure is low by about one part in six, or about 17 percent.

Still another measurement was reportedly made later by Abdullah al-Mamun, who obtained a value that was within 3.6 percent of the true figure. He had fairly accurate astronomical data for finding the needed angle. He also went to the trouble of measuring the distance of his arc by laying sticks down end to end.

Modern techniques are of course far more accurate, exploiting artificial satellites, lasers and reflectors on the moon. As a result the size of the earth is well known, as is the fact that the earth is not exactly a sphere. We have come a long way from the time of the early Greeks, who believed the earth was flat and was held up by four huge elephants standing on the back of a huge turtle. We also no longer have the philosophical problem of deciding what the turtle stands on.

You might like to measure the earth's radius by the technique attributed to Eratosthenes. Your two cities should be close to the same meridian, although this condition is not critical. Try to choose cities as far apart on the meridian as is practical. Ascertain the distance by means of a map or a table of distances.

To measure the orientation of the sun's rays in the two cities you will need a friend to do the experiment in one city on the same day you make your measurements in the other city. Each of you should erect a gnomon, a vertical rectangular post with one face perpendicular to the meridian. When the sun reaches its zenith, measure the length of the shadow cast by the gnomon. (It will be shortest then.) The tangent of the sun's angular distance from the zenith is equal to the length of the shadow divided by the height of the gnomon. Although you could then compute the angle from a table of trigonometric functions, I suggest you work with a pocket calculator to avoid the problem of interpolation between the values in a table. The angle between the vertical and the sun's rays is of course the same angle.

After making the measurements, subtract from them the angles you and your friend have obtained. Follow Eratosthenes' procedure by dividing the angular difference into 360 degrees and then multiplying by the arc distance between the two cities. The result is the circumference of the earth.

A problem with this method of measurement is that the far edge of the shadow is indistinct. The shadow fades from total brightness through an area of partial illumination (the penumbra) and into an area of maximum darkness (the umbra). The penumbra is partly illuminated because the plate blocks the rays originating from the lower section of the sun but does not block those from the upper section. Inside the umbra all the direct rays from the sun are blocked by the plate.

To find the angular distance between the zenith and the center of the sun you should measure the shadow to the center point of the penumbra. This measurement is difficult to make with precision It inevitably introduces error, perhaps as much as a quarter of a degree if you use either edge of the penumbra in your determination of the angle of the sun's rays. The penumbra is .5 degree wide because the sun subtends an angle of.5 degree in one's field of view. The error inherent in locating the center of the penumbra makes the gnomon an imprecise instrument for measuring the size of the earth.

One way to get around the problem is to replace the gnomon with another de. vice. Mount a horizontal pin at the center of a graduated circle placed so that its plane is vertical and aligned parallel to the meridian. The shadow cast by the pin has penumbras running the length of the shadow. Since they are symmetrical on the two sides of the shadow, you can easily find the center of the shadow and read its position on the graduated circle. When the circle is properly oriented, your reading will be the angular distance of the sun from the zenith and so will be the angle of the incident rays, without any systematic error introduced because of the penumbras.

REFRACTION IN A CLOUD-FREE ATMOSPHERE. Hans Neuberger in *Introduction to Physical Meteorology.* The Pennsylvania State University, 1957.

A COMPENDIUM OF SPHERICAL ASTRONOMY. Simon Newcomb. Dover Publications, Inc., 1960.

DOUBLING YOUR SUNSETS, OR HOW ANYONE CAN MEASURE THE EARTH'S SIZE WITH A WRISTWATCH AND METERSTICK. Dennis Rawlins in *American Journal of Physics*, Vol. 47, No. 2, pages 126-128; February, 1979.