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Earth's circumference is thedistance aroundEarth. Measured around theequator, it is 40,075.017 km (24,901.461 mi). Measured passing through thepoles, the circumference is 40,007.863 km (24,859.734 mi).[1]
Treating the Earth as asphere, its circumference would be its single most important measurement.[2] The first known scientific measurement and calculation was done byEratosthenes, by comparing altitudes of the mid-day sun at two places a known north–south distance apart.[3] He achieved a great degree of precision in his computation.[4] The Earth's shape deviates from spherical byflattening, but by only about 0.3%.
Measurement of Earth's circumference has been important tonavigation since ancient times. In modern times, Earth's circumference has been used to define fundamental units of measurement of length: thenautical mile in the seventeenth century and themetre in the eighteenth. Earth's polar circumference is very near to 21,600 nautical miles because the nautical mile was intended to express one minute of latitude (seemeridian arc), which is 21,600 partitions of the polar circumference (that is 60 minutes × 360 degrees). The polar circumference is also close to 40,000 kilometres becausethe metre was originally defined to be one ten millionth (i.e., a kilometre is one ten thousandth) of the arc from pole to equator (quarter meridian). The accuracy of measuring the circumference has improved since then, but the physical length of each unit of measure had remained close to what it was determined to be at the time, so the Earth's circumference is no longer around number in metres or nautical miles.
The measure of Earth's circumference is the most famous among the results obtained byEratosthenes,[5] who estimated that the meridian has a length of 252,000stadia, with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres;[4] the exact value of the stadion remains a subject of debate to this day; seestadion).
Eratosthenes described his technique in a book entitledOn the measure of the Earth, which has not been preserved; what has been preserved is the simplified version described byCleomedes to popularise the discovery.[6] Cleomedes invites his reader to consider two Egyptian cities,Alexandria and Syene (modernAswan):
According toCleomedes'sOn the Circular Motions of the Celestial Bodies, around 240 BC, Eratosthenes calculated thecircumference of the Earth inPtolemaic Egypt.[8] Using a vertical rod known as agnomon and under the previous assumptions, he knew that at local noon on the summer solstice inSyene (modernAswan, Egypt), the Sun was directly overhead, as the gnomon cast no shadow. Additionally, the shadow of someone looking down a deep well at that time in Syene blocked the reflection of the Sun on the water. Eratosthenes then measured the Sun's angle of elevation at noon in Alexandria by measuring the length of another gnomon's shadow on the ground.[9] 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.[10] This angle was about 7°, or 1/50th the circumference of acircle; assuming the Earth to be perfectly spherical, he concluded that its circumference was 50 times the known distance from Alexandria to Syene (5,000 stadia, a figure that was checked yearly), i.e. 250,000 stadia.[11] 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%.[11] A value for the stadion of 157.7 metres has even been posited by L.V. Firsov, which would give an even better precision, but is plagued by calculation errors and false assumptions.[12] 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.[10]
Eratosthenes's method was actually more complicated, as stated by the same Cleomedes, whose purpose was to present a simplified version of the one described in Eratosthenes's book. Pliny, for example, has quoted a value of 252,000 stadia.[13]
The method was based on severalsurveying trips conducted by professionalbematists, whose job was to precisely measure the extent of the territory of Egypt for agricultural and taxation-related purposes.[4] Furthermore, the fact that Eratosthenes's measure corresponds precisely to 252,000 stadia (according to Pliny) might be intentional, since it is a number that can be divided by all natural numbers from 1 to 10: some historians believe that Eratosthenes changed from the 250,000 value written by Cleomedes to this new value to simplify calculations;[14] other historians of science, on the other side, believe that Eratosthenes introduced a new length unit based on the length of the meridian, as stated by Pliny, who writes about the stadion "according to Eratosthenes' ratio".[4][13]
Posidonius calculated the Earth's circumference by reference to the position of the starCanopus. As explained byCleomedes, Posidonius observed Canopus on but never above the horizon atRhodes, while atAlexandria he saw it ascend as far as7+1⁄2 degrees above the horizon (themeridian arc between the latitude of the two locales is actually 5 degrees 14 minutes). Since he thought Rhodes was 5,000stadia 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.[15] It is generally thought[by whom?] that the stadion used by Posidonius was almost 1/10 of a modern statute mile.[citation needed] 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).[15]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).[16]Pliny the Elder mentions Posidonius among his sources and—without naming him—reported his method for estimating the Earth's circumference. He noted, however, thatHipparchus 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 hisGeography. This was the number used byChristopher Columbus in order to underestimate the distance to India as 70,000 stades.[17]
Around AD 525, the Indian mathematician and astronomerAryabhata wroteAryabhatiya, in which he calculated the diameter of earth to be of 1,050yojanas. The length of theyojana intended by Aryabhata is in dispute. One careful reading gives an equivalent of 14,200 kilometres (8,800 mi), too large by 11%.[18] Another gives 15,360 km (9,540 mi), too large by 20%.[19] Yet another gives 13,440 km (8,350 mi), too large by 5%.[20]
Around AD 830,CaliphAl-Ma'mun commissioned a group ofMuslim astronomers led byAl-Khwarizmi to measure the distance from Tadmur (Palmyra) toRaqqa, in modernSyria. They calculated the Earth's circumference to be within 15% of the modern value, and possibly much closer. How accurate it actually was is not known because of uncertainty in the conversion between the medieval Arabic units and modern units, but in any case, technical limitations of the methods and tools would not permit an accuracy better than about 5%.[21]
A more convenient way to estimate was provided inAl-Biruni'sCodex Masudicus (1037). In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two locations,al-Biruni developed a new method of usingtrigonometric calculations, based on the angle between aplain andmountain top, which made it possible for it to be measured by a single person from a single location.[21] From the top of the mountain, he sighted thedip angle which, along with the mountain's height (which he determined beforehand), he applied to thelaw of sines formula. This was the earliest known use of dip angle and the earliest practical use of the law of sines.[22] However, the method could not provide more accurate results than previous methods, due to technical limitations, and so al-Biruni accepted the value calculated the previous century by theal-Ma'mun expedition.[21]
1,700 years after Eratosthenes's death,Christopher Columbus studied what Eratosthenes had written about the size of the Earth. Nevertheless, based on a map byToscanelli, he chose to believe that the Earth's circumference was 25% smaller. If, instead, Columbus had accepted Eratosthenes's larger value, he would have known that the place where he made landfall was notAsia, but rather aNew World.[23]
In 1617 the Dutch scientistWillebrord Snellius assessed the circumference of the Earth at 24,630 Roman miles (24,024 statute miles). Around that time British mathematicianEdmund Gunter improved navigational tools including a newquadrant to determine latitude at sea. He reasoned that the lines of latitude could be used as the basis for a unit of measurement fordistance and proposed the nautical mile as one minute or one-sixtieth (1/60) of onedegree of latitude. As one degree is1/360 of a circle, one minute of arc is1/21600 of a circle – such that the polar circumference of the Earth would be exactly 21,600 miles. Gunter used Snellius's circumference to define a nautical mile as 6,080 feet, the length of one minute of arc at 48 degrees latitude.[24]
In 1793, France defined the metre so as to make the polar circumference of the Earth 40,000 kilometres. In order to measure this distance accurately, theFrench Academy of Sciences commissionedJean Baptiste Joseph Delambre andPierre Méchain to leadan expedition to attempt to accurately measure the distance between a belfry inDunkerque andMontjuïc castle inBarcelona to estimate the length of themeridian arc through Dunkerque. The length of the firstprototype metre bar was based on these measurements, but it was later determined that its length was short by about 0.2 millimetres because of miscalculation of theflattening 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.[25] This is why the polar circumference of the Earth is actually 40,008 kilometres, instead of 40,000.
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