Noel Zinn, Math 5310, Summer 2017, Essay #2 Eratosthenes of Cyrene: Geodesist "Geodesy is from the Ancient Greek word γεωδαισία (geodaisia), literally meaning division of the Earth. A branch of applied mathematics and earth sciences, [geodesy] is the scientific discipline that deals with the measurement of the Earth, including its gravitational field." 1 Preface After a 40-year career as an oil industry geodesist, largely self taught though I studied geodesy, surveying, mathematics and geoscience at this university, I began an MS in applied mathematics in retirement to shore up my credentials. That on-campus MS quickly changed to an online MA after a semester of commuting, a time waster better left behind in my workaday life! In my career I occasionally taught introductory geodesy to my colleagues... always beginning with Eratosthenes. So, it is with pleasure that I now integrate Eratosthenes and geodesy into my mathematics education with this essay. Eratosthenes A remarkable polymath, Eratosthenes was born 276 BCE in the Greek town of Cyrene, which is now part of Libya. In Athens he studied Stoicism under that movement's founder, Zeno of Citium, then a more cynical philosophy under Ariston of Chios, and later Platonism at the Platonic academy. His first treatise was on the mathematical foundations of Plato's philosophies. He wrote on geography, mathematics, philosophy, chronology, literary criticism, grammar, and poetry. He rose to become the Chief Librarian of the library at Alexandria, where he spend the rest of his life. He died at age 82 (194 BCE) by self-imposed starvation because he had gone blind. 2 Determination of Earth Circumference Eratosthenes is most famous for being the first to credibly measure the circumference of Earth, and he did so within 15% of its correct value. That effort is the 1
topic of this essay. The following graphic 3 from a 40-year-old US Army student pamphlet (which calls Eratosthenes the "father of geodesy") illustrates the method. Two cities in Egypt are involved, Alexandria, where Eratosthenes resides, and Syene to the south, presumably along the same meridian, which means due south. Syene is currently known as Aswan. Eratosthenes knew that the sun was much larger than the Earth (another essay there), so he expected that the rays of light would strike the Earth in parallel, an important assumption, and he was correct. Eratosthenes also knew that the spin axis of the Earth was tilted with respect to the plane of the orbit of the Earth about the sun (another essay there). Because of this tilt (about 23.5 ), the rays of the sun strike the Earth at varying angles as the Earth orbits about the sun during the course of a year. The tilt defines the latitudes of the Tropics of Cancer (north) and Capricorn (south) and also of the Arctic and Antarctic circles. At the summer solstice (June 21st) the orientation of the Earth and sun is such that the rays of the sun strike the Earth vertically along the Tropic of Cancer at local noon (and nowhere else). 2
Next, Eratosthenes knew of a water well in Syene and of reports that at noon on the summer solstice a person's head looking down into the well would block the sun. He concluded that the well in Syene was on the Tropic, that is, that the rays of the sun were vertical at the moment of that astronomical event. Another fact known to Eratosthenes was that the distance from Alexandria to Syene was 5,000 stadia. There is some dispute about the metric length of the stade (singular of stadia) that he used, maybe 176.4m, maybe 184.8m, maybe something else 2. But worse, the total distance (about 900km) was estimated by the average number of days that camel caravans took to travel between the cities and an estimate of how fast a camel walked. As a humorous aside, the quality of geodetic land surveys have recently been categorized into first order (1:100,000, or a maximum of 1 meter of error per 100km), second order (1:20,000) and third order (1:5,000). Although the soon-to-be-described method of Eratosthenes was totally innovative, the 15% quality of his results forces us to conclude that his camels were only third order! This was his method. At noon on the summer solstice in Alexandria, just when the sun was directly above the well in Syene, Eratosthenes observed the shadow of the pointer of a sun dial in Alexandria. If Alexandria were on the Tropic, there would be no shadow, but Alexandria is not, and there was a shadow. Measuring the length of the shadow and the height of the pointer, Eratosthenes computed the angle of the rays of the sun ("a" in the graphic above). This angle was 1/50 of a full circle, or about 7.2 in today's measure. Applying a little geometry, angle "a" is also the angle at the center of the Earth between the two cities. That implies that the 5,000 stadia from Alexandria to Syene is 1/50 of the circumference of the Earth, which then would be 250,000 stadia or about 45,000,000 meters using an average stade. In fact, it's closer to 40,000,000 meters. 3
As an historical aside, the meter was originally defined in 1793 to be 1/10,000,000 of the distance from the Equator to the pole (one quarter the circumference of the Earth). Problems With The Method Wikipedia 2 makes some insightful comments about the method. "He [Eratosthenes] made five important assumptions (none of which is perfectly accurate): 1. That the distance between Alexandria and Syene was 5000 stadia, 2. That Alexandria is due north of Syene 3. That Syene lies on the Tropic of Cancer 4. That the Earth is a perfect sphere. 5. That light rays emanating from the Sun are parallel." Point 1: I've already commented on Eratosthenes' third-order camels. Point 2: Alexandria is actually about 3 longitude west of Syene, or about 1650 stadia. A little Pythagorean computation reduces the skewed 5000 stadia hypotenuse distance between Alexandria and Syene to 4720 N/S stadia, a reduction in the circumference of the Earth of 6% in the right direction. Point 3: Syene is actually 0.65 latitude north of the Tropic. The anecdote of the head in the well blocking the sun turns out to be inadequate for the method. There will be a shadow in Syene at noon of the summer solstice in the same direction as the shadow in Alexandria. That will reduce the angle measured in Alexandria from 7.2 to 6.55 or 1/55 of a full circle. This effect alone will increase the circumference of the Earth by 10% in the wrong direction (yet larger). Point 4: The Earth is not a perfect sphere. Earth is better approximated by an oblate ellipsoid of revolution with a length reduction in the spin axis of 1:300 (compared to the Equatorial axis), though the difference is well within Eratosthenes' error budget. However, the gravitational level surface called the geoid is very irregular. See the next section on 4
Geodesy. Point 5: As mentioned, Eratosthenes got it right (despite Wikipedia's implication to the contrary). Problems aside, Eratosthenes' empirical approach to a basic geodetic question was clever and extraordinary for the times. A first in the history of geodesy. Geodesy Geodesy has been called "the forgotten geoscience". We all know about geology and geophysics, but few know about geodesy. Geodesy has two branches: physical and geometrical. Physical geodesy studies the irregular gravitational field of the Earth. Geometrical geodesy pertains to accurate positioning on the Earth by either terrestrial or satellite methods (e.g. GPS). The two are related, of course, since the gravitational field affects the alignment of survey instruments, the orbits of satellites and even the angle of rays of the sun at noon at summer solstice in Syene. The following graphic from my consultancy website 4 exhibits the geoid exaggerated 10,000 times. It turns out that the (unexaggerated) undulations of the geoid are small compared to the arc angle between Syene and the Tropic of Cancer previously discussed. So, 5
Eratosthenes could safely ignore physical geodesy in his creative exercise in geometrical geodesy in determining the circumference of the Earth. Another great mathematician later in history was also a practicing geodesist, Carl Friedrich Gauss. Gauss was the first to apply least-squares estimation to the computation of geodetic land surveys that he managed during the warm German summers and he developed the world's most used map projection, the Transverse Mercator, upon which to plot those surveys. Gauss's practical geodetic work led to remarkable advances in theoretical differential geometry later in his life 5. I look forward to that chapter of our History of Mathematics course. Notes and References (1) https://en.wikipedia.org/wiki/geodesy (paraphrased) (2) https://en.wikipedia.org/wiki/eratosthenes (paraphrased) (3) Defense Mapping School, DMS No. ST 203, "Basic Geodesy", 1977, page 5 (4) http://www.hydrometronics.com/ (4) W.K. Bühler, "Gauss, A Biographical Study", Springer, 1981, paraphrased 6