Measuring the Non-Spherical Earth

 kw: book reviews, nonfiction, expeditions, science, geography

During the Age of Enlightenment theoretical and mathematical endeavors quickly outstripped the abilities of researchers ("natural philosophers", later called scientists) to apply them to the world around them. One such conundrum was the shape of the Earth. It was known that the Earth is "round" since the Earth's circumference was first measured in about 240 BC, and it was assumed for centuries to be a perfect sphere. One of the first to question this assumption from a scientific point of view was Isaac Newton. After discovering gravity, and considering that the Earth is rotating rapidly, he conjectured that the equatorial radius ought to be a little greater than the polar radius, based on an equilibrium between centripetal force and the self-gravity of the sphere. This describes an oblate ellipsoid.

In ensuing decades, others put forward reasons that the Earth might instead be prolate, that is, that the equatorial radius could be smaller than the polar radius. This may seem esoteric, but it has implications for navigation. Christopher Columbus, knowing along with the rest that the Earth is round, had proved himself wrong about the size of the Earth. "Everyone" knew that Earth was spherical in the 1490's, contrary to what was once taught in school, but Columbus thought the Indies would be "close", that the circumference was some 25,000 km (he didn't use km; this is using modern units) rather than 40,000 km. He thought India ought to be reachable by sailing only a few weeks west from Spain (as a famous poem relates, he was puzzled after 20 days of sailing and finding nothing, but "sailed on" for another 16 days). That 15,000 km error was enough to conceal two large continents, and his accidental discovery of the Americas helped trigger the Enlightenment. It also greatly increased the number of sailing expeditions across the open sea, and when you are at sea, it's essential to know where you are and the direction you need to sail to get where you are going.

If the Earth is not exactly spherical, the distance between latitudes will vary with latitude. The instruments in use prior to the 1900's were able to measure latitude with good accuracy, by sighting from the pole star, for example. Here is a quote from the Wikipedia article Earth's Circumference:

Measured around the Equator, it is 40,075.017 km (24,901.461 mi). Measured around the poles, the circumference is 40,007.863 km (24,859.734 mi).

The difference between the two is just over 67 km. Suppose a navigator calculates a rhumb line (a line to navigate by keeping a specific compass heading) to take his ship from Cadiz, Spain to Barbados. The uncertainties of navigating nearly 6,000 km might take the ship a few km. If the spherical-versus-spheroidal calculation adds another km or so of error, one might miss the island entirely.

As we read in Latitude: The True Story of the World's First Scientific Expedition by Nicholas Crane, the scientific societies of 18th Century Europe, particularly France, became convinced that it was necessarily to make measurements to determine with certainty whether the Earth is an oblate or prolate ellipsoid, and by how much. This illustration is a pictorial representation of the required calculation:

This exaggerated ellipse shows the difference between radii and lines normal (at right angles) to the surface. Latitude is measured by sighting the North Star. Its angle from the horizon is 0° at the equator and 90° (straight up) at the north pole. The green radius line shown is at a 15° angle, which would be 15° latitude on a sphere; the red radius is at 75°.

However, at the point where the green radius intersects the surface, the angle to the North Star is 47°, not 15°. Similarly, at the point where the red radius intersects the surface, the angle is about 85° rather than 75°. The other red line and green line show that to reach a position with the "right" latitude, as measured by the North Star, one must move away from the pole, unless one is at the pole or the equator already. And that means that a degree of latitude is longer on the surface of the Earth in the northerly regions than in the equatorial regions.

It was known that a degree of latitude had a certain length in Europe. However, any difference in the length of a degree (about 67 miles or 111 km in modern terms), measured in southern Europe compared to northern Europe, was too small for the academicians to clearly distinguish. The French Academy of Sciences decided to sponsor an expedition to the equatorial regions of South America, specifically to Ecuador, beginning at Quito, the city nearest the equator. At that time the area was part of the Viceroyalty of Peru, subject to Spain.

There, a team of academicians and technicians and two Spanish officers (and a multitude of helpers) were to accurately measure at least one degree of latitude, from the equator south. They eventually decided to measure three degrees, to obtain a more accurate result. That was to mean traversing more than 200 miles of mountainous terrain with quadrants, telescopes, and other equipment, tons of it.

A team of ten was sent, as the Geodesic Mission to the Equator. Not all returned, and those that did returned nearly ten years later, having suffered privations and disasters beyond what any of them could have imagined. I find it hard to understand how any of them survived. Just measuring a selected star as it crossed the zenith was a torturous trial, with the added complications of dramatic temperature and humidity variations changing the shape of the building to which the telescope was affixed, occasional earthquakes knocking it out of alignment or stopping the pendulum of the "official" clock, and cloudy nights so frequent that taking a single measurement could take a week, or weeks, of trying. One team member was an instrument maker, a former clock maker, who was kept very busy.

The team spend nearly a year to reach Quito, at a time of year that any roads that existed were muddy morasses, and much of the route had no roads. They had started out in May, 1735, and it was the rainy (or "somewhat rainier than usual") season in 1736 when they reached Quito, not all at the same time. The "team", about as badly led as any team in history, split up at one point, and a few took a different route, which delayed them; the opposite of their intention. Almost everything they did went contrary to expectation.

As I read I remembered my sessions of Summer Field Camp. Living in a tent, first in a mountainous area of Nevada, and later in a wilderness basin among glaciers in the Sierras, used up all my tolerance for camping out. And that was just three months. Compared to ten years! I remember that one reason I picked the graduate school I went to, several years later at age 30, was that their field camp was not too far from the city. I wanted to avoid another season of tent living. Wimp! 

The book delineates their many privations, but it also illuminates the significant science they were able to produce in spite of them all. They succeeded in laying out and measuring a baseline in a 7-mile-long valley (that now hosts the Quito airport), and then laying out a succession of triangles, south through about 100 miles of a "corridor" between Andean ranges, to Riobamba, and another 100 miles to some distance beyond Cuenca, where the layout was much more challenging, there being no "corridor".

A couple of years into their expedition, the Mission learned that a second Mission had been commissioned to measure a degree of latitude in northern Europe near the Arctic Circle. Another year later, at which time they had initially thought their task would be complete, they learned that the measurement at the Arctic Circle was a success: the degree measured 0.66% longer than a degree measured near Paris, 57,437 toises versus 57,060. This was decades before the invention of the meter. A French toise is just over 1.949 meters (I had to look this up; the author doesn't tell us), so the two measurements were 111.946 km versus 111.211 km. This in itself proved that the Earth's shape is oblate. However, the Mission pressed on, not just to confirm the finding (which they most decidedly did), but for the sake of many other observations and measurements of natural history, historiography, and geography they performed along the way, including measuring the speed of sound at various elevations (using borrowed cannons).

Near the end of January 1743, after collecting the angular measurements of a couple of hundred triangles over the 200-mile stretch, and doing days and days of pen-and-paper calculations, they obtained their result: one degree of latitude at the equator is 56,573 toises, or 110.262 km. Modern geodesy shows this result to be low by only a quarter of a percent, and the accepted figure today is 110.567 km, or 305 m greater.

Calculating from these figures the ellipsoid for the earth yields an interesting result, that the equator is farther from the center of the Earth than the poles by more than 22 km, and the radius at 28°N, the latitude of Mount Everest, is about 8 km less than the equatorial radius. That means that sea level near the equator is almost as far from the center of the Earth as is the peak of Mt. Everest, which is 8,848 m. In terms of distance from the center of our planet, all the high peaks in the equatorial Andes are "higher" than Mt. Everest, and the highest is Chimborazo, a 6,263 m peak, as measured from local sea level.

The author relates in an endpiece that he sought to tell a story rather than produce biographies, or relate the science in detail. Numerous books do so already. While I might prefer a few more scientific details, it is indeed an enthralling story, a real page-turner. Very enjoyable, if at times horrific in the sufferings of the members of the Geodesic Mission.