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Measurement of Earth's circumference has been important to navigation since ancient times. It was first calculated by Eratosthenes, which he did by comparing altitudes of the mid-day sun at two places a known north–south distance apart. In the Middle Ages, Al Biruni calculated a more accurate version, becoming the first person to perform the calculation based on data from a single location.
In modern times, Earth's circumference has been used to define fundamental units of measurement of length: the nautical mile in the seventeenth century and the metre in the eighteenth. Earth's polar circumference is very near to 21,600 nautical miles because the nautical mile was intended to express 1/TH of a degree of latitude (i.e. 60 × 360), which is 21,600 partitions of the polar circumference. The polar circumference is even closer to 40,000 kilometres because the metre was originally defined to be one 10-millionth the distance from pole to equator. The physical length of each unit of measure has remained close to what it was determined to be at the time, but the precision of measuring the circumference has improved since then.
According to Cleomedes' On the Circular Motions of the Celestial Bodies, around 240 BC, Eratosthenes, the librarian of the Library of Alexandria, calculated the circumference of the Earth in Ptolemaic Egypt. Using a scaphe, he knew that at local noon on the summer solstice in Syene (modern Aswan, Egypt), the Sun was directly overhead. (Syene is at latitude 24°05′ North, near to the Tropic of Cancer, which was 23°42′ North in 100 BC.) He knew this because the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. He then measured the Sun's angle of elevation at noon in Alexandria by using a vertical rod, known as a gnomon, and measuring the length of its shadow on the ground. Using the length of the rod, and the length of the shadow, as the legs of a triangle, he calculated the angle of the sun's rays. This angle was about 7°, or 1/50th the circumference of a circle; taking the Earth as perfectly spherical, he concluded that the Earth's circumference was 50 times the known distance from Alexandra to Syene (5,000 stadia, a figure that was checked yearly), i.e. 250,000 stadia. Depending on whether he used the "Olympic stade" (176.4 m) or the Italian stade (184.8 m), this would imply a circumference of 44,100 km (an error of 10%) or 46,100 km, an error of 15%. In 2012, Anthony Abreu Mora repeated Eratosthenes's calculation with more accurate data; the result was 40,074 km, which is 66 km different (0.16%) from the currently accepted polar circumference.
Posidonius calculated the Earth's circumference by reference to the position of the star Canopus. As explained by Cleomedes, Posidonius observed Canopus on but never above the horizon at Rhodes, while at Alexandria he saw it ascend as far as 7 1⁄2 degrees above the horizon (the meridian arc between the latitude of the two locales is actually 5 degrees 14 minutes). Since he thought Rhodes was 5,000 stadia due north of Alexandria, and the difference in the star's elevation indicated the distance between the two locales was 1/48 of the circle, he multiplied 5,000 by 48 to arrive at a figure of 240,000 stadia for the circumference of the earth. It is generally thought that the stadion used by Posidonius was almost exactly 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to 24,000 mi (39,000 km), not much short of the actual circumference of 24,901 mi (40,074 km). Strabo noted that the distance between Rhodes and Alexandria is 3,750 stadia, and reported Posidonius's estimate of the Earth's circumference to be 180,000 stadia or 18,000 mi (29,000 km). Pliny the Elder mentions Posidonius among his sources and without naming him reported his method for estimating the Earth's circumference. He noted, however, that Hipparchus had added some 26,000 stadia to Eratosthenes's estimate. The smaller value offered by Strabo and the different lengths of Greek and Roman stadia have created a persistent confusion around Posidonius's result. Ptolemy used Posidonius's lower value of 180,000 stades (about 33% too low) for the earth's circumference in his Geography. This was the number used by Christopher Columbus in order to underestimate the distance to India as 70,000 stades.
A more accurate estimate was provided in Al-Biruni's Codex Masudicus (1037). In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, al-Biruni developed a new method of using trigonometric calculations, based on the angle between a plain and mountain top, which yielded more accurate measurements of the Earth's circumference, and made it possible for it to be measured by a single person from a single location.
1,700 years after Eratosthenes's death, while Christopher Columbus studied what Eratosthenes had written about the size of the Earth, he chose to believe, based on a map by Toscanelli, that the Earth's circumference was one-third smaller. Had Columbus set sail knowing that Eratosthenes's larger circumference value was more accurate, he would have known that the place that he made landfall was not Asia, but rather the New World.
Both the metre and the nautical mile were originally defined as a subdivision of the Earth's circumference; today the circumference around the poles is very nearly 40,000 km and 360 × 60 nautical miles long.
In 1617 the Dutch scientist Willebrord Snellius assessed the circumference of the Earth at 24,630 Roman miles (24,024 statute miles). Around that time British mathematician Edmund Gunter improved navigational tools including a new quadrant to determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement for distance and proposed the nautical mile as one minute or one-sixtieth (1/) of one degree of latitude. As one degree is 1/ of a circle, one minute of arc is 1/ of a circle – such that the polar circumference of the Earth would be exactly 21,600 miles. Gunter used Snell's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude.
In 1791, the French Academy of Sciences selected the circumference definition over the alternative pendular definition because the force of gravity varies slightly over the surface of the Earth, which affects the period of a pendulum. To establish a universally accepted foundation for the definition of the metre, more accurate measurements of this meridian were needed. The French Academy of Sciences commissioned an expedition led by Jean Baptiste Joseph Delambre and Pierre Méchain, lasting from 1792 to 1799, which attempted to accurately measure the distance between a belfry in Dunkerque and Montjuïc castle in Barcelona to estimate the length of the meridian arc through Dunkerque. This portion of the meridian, assumed to be the same length as the Paris meridian, was to serve as the basis for the length of the half meridian connecting the North Pole with the Equator. The problem with this approach is that the exact shape of the Earth is not a simple mathematical shape, such as a sphere or oblate spheroid, at the level of precision required for defining a standard of length. The irregular and particular shape of the Earth smoothed to sea level is represented by a mathematical model called a geoid, which literally means "Earth-shaped". Despite these issues, in 1793 France adopted this definition of the metre as its official unit of length based on provisional results from this expedition. However, it was later determined that the first prototype metre bar was short by about 200 micrometres because of miscalculation of the flattening of the Earth, making the prototype about 0.02% shorter than the original proposed definition of the metre. Regardless, this length became the French standard and was progressively adopted by other countries in Europe.
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It was Biruni, not Avicenna, who found a way for a single man, at a single moment, to measure the earth's circumference, by trigonometric calculations based on angles measured from a mountaintop and the plain beneath it – thus improving on Eratosthenes' method of sighting the sun simultaneously from two different sites, applied in the ninth century by astronomers of the Khalif al-Ma'mun.