Instruct: Given a question, retrieve relevant passages that can help answer the question.
Query: What is the circumference of the Earth?
Earth's circumference is the distance around the Earth, either around the equator (40,075.017km [24,901.461mi][1]) or around the poles (40,007.86km [24,859.73mi][2]).
Abu Rayhan Biruni (976–1048) first described a polar equi-azimuthal equidistant projection of the celestial sphere. He was regarded as the most skilled when it came to mapping cities and measuring the distances between them, which he did for many cities in the Middle East and the Indian subcontinent. He often combined astronomical readings and mathematical equations, in order to develop methods of pin-pointing locations by recording degrees of latitude and longitude. He also developed similar techniques when it came to measuring the heights of mountains, depths of the valleys, and expanse of the horizon. He also discussed human geography and the planetary habitability of the Earth. He also calculated the latitude of Kath, Khwarezm, using the maximum altitude of the Sun, and solved a complex geodesic equation in order to accurately compute the Earth's circumference, which was close to modern values of the Earth's circumference. His estimate of 6,339.9 km for the Earth radius was only 16.8 km less than the modern value of 6,356.7 km. 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.
The European Age of Discovery during the 16th and the 17th centuries, where many new lands were discovered and accounts by European explorers such as Christopher Columbus, Marco Polo, and James Cook revived a desire for both accurate geographic detail, and more solid theoretical foundations in Europe. The problem facing both explorers and geographers was finding the latitude and longitude of a geographic location. The problem of latitude was solved long ago but that of longitude remained; agreeing on what zero meridian should be was only part of the problem. It was left to John Harrison to solve it by inventing the chronometer H-4 in 1760, and later in 1884 for the International Meridian Conference to adopt by convention the Greenwich meridian as zero meridian.
Aryabhatiya also estimates the circumference of Earth, with an error of less than 1%. He gave the circumference of the earth as 4 967 yojanas and its diameter as 1 5811/24 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24 835 miles, which is an excellent approximation to the currently accepted value of 24 902 miles. Aryabhata gives the radii of the orbits of the planets in terms of the Earth-Sun distance as essentially their periods of rotation around the Sun. He also gave the correct explanation of lunar and solar eclipses and that the Moon shines by reflecting sunlight.
Later Greek scientists determined that the Earth was round, although there was some debate as to the veracity of this claim. During the conquests by Alexander the Great, Hellenistic knowledge of the world was expanded, resulting in an improved map of the world by Eratosthenes. He also calculated the circumference of the Earth, and published two treatises about geography, titled "Geographica" and "On the Measurement of the Earth". Contemporaries of Eratosthenes questioned the validity of his methods, although his calculation of the Earth's circumference was quite accurate.
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 almost exactly 40,000 km and 360 × 60 nautical miles long.[18]
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 ​71⁄2 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.[11] 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 24,000mi (39,000km), not much short of the actual circumference of 24,901mi (40,074km).[11] 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 18,000mi (29,000km).[12] 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.[13]
The first published reference to the Earth's size appeared around 350 BC, when Aristotle reported in his book "On the Heavens" that mathematicians had guessed the circumference of the Earth to be 400,000 stadia. Scholars have interpreted Aristotle's figure to be anywhere from highly accurate to almost double the true value. The first known scientific measurement and calculation of the circumference of the Earth was performed by Eratosthenes in about 240 BC. Estimates of the accuracy of Eratosthenes’s measurement range from 0.5% to 17%. For both Aristotle and Eratosthenes, uncertainty in the accuracy of their estimates is due to modern uncertainty over which stadion length they meant.
Syene had been used by Eratosthenes of Cyrene in c.220 BC as a point of reference to measure the circumference of the Earth (by observing the angle of a shadow cast at Alexandria on the day of the summer solstice—eighty-three degrees—and deducting that from the ninety-degree right angle of the sun over Syene on the same day, from that deducing the acute angle—seven degrees—at the apex of the segment of the Earth’s circumference represented by the known distance from Syene to Alexandria—504 stadia—and then multiplying that distance by the value of that angle and dividing it by the 360 degrees of the whole circumference of the Earth: 252,000 stadia, or 39,690 km, an error of less than one per cent.
Earth radius Spherical Earth Nautical mile
From these observations, measurements, and/or "known" facts, Eratosthenes concluded that since the angular deviation of the sun from the vertical direction at Alexandria was also the angle of the subtended arc (see illustration), the linear distance between Alexandria and Syene was 1/50 of the circumference of the Earth which thus must be 50×5000 = 250,000 stadia or probably 25,000 geographical miles. The circumference of the Earth is 24,902 mi (40,075.16 km). Over the poles it is more precisely 40,008 km or 24,860 mi. The actual unit of measure used by Eratosthenes was the stadion. No one knows for sure what his stadion equals in modern units, possibly it was the Hellenic 185 m stadion.
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.[17]
In 1615 Snellius, after the work of Eratosthenes in Ptolemaic Egypt in the 3rd century BC, probably was the first to try to do a large-scale experiment to measure the circumference of the earth using triangulation. He was helped in his measurements by two of his students, the Austrian barons Erasmus and Casparus Sterrenberg. In several cities he also received support of friends among the city leaders ("regenten"). In his work "The terrae Ambitus vera quantitate" (1617) under the author's name ("The Dutch Eratosthenes") Snellius describes the methods he used. He came up with an estimate of 28,500 Rhineland rods – in modern units 107.37 km for one degree of latitude. 360 times 107.37 then gives a circumference of the Earth of 38,653 km. The actual circumference is 40,075 kilometers, so Snellius underestimated the circumference of the earth by 3.5%.
Al-Farghānī (Latinized as Alfraganus) was a Persian astronomer of the 9th century involved in measuring the diameter of the Earth, and commissioned by Al-Ma'mun. His estimate given above for a degree (56 Arabic miles) was much more accurate than the 60 Roman miles (89.7 km) given by Ptolemy. Christopher Columbus uncritically used Alfraganus's figure as if it were in Roman miles instead of in Arabic miles, in order to prove a smaller size of the Earth than that propounded by Ptolemy.Abu Rayhan Biruni (973–1048) used a new method to accurately compute the Earth's circumference, by which he arrived at a value that was close to modern values for the Earth's circumference. His estimate of 6,339.6 km for the Earth radius was only 31.4 km less than the modern mean value of 6,371.0 km. In contrast to his predecessors, who measured the Earth's circumference by sighting the Sun simultaneously from two different locations, Biruni developed a new method of using trigonometric calculations based on the angle between a plain and mountain top. This yielded more accurate measurements of the Earth's circumference and made it possible for a single person to measure it from a single location.
Biruni's method was intended to avoid "walking across hot, dusty deserts," and the idea came to him when he was on top of a tall mountain in India. From the top of the mountain, he sighted the angle to the horizon which, along with the mountain's height (which he calculated beforehand), allowed him to calculate the curvature of the Earth.
He also made use of algebra to formulate trigonometric equations and used the astrolabe to measure angles.
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.[6] 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[7]) 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.[8] 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.[9] 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.[10] Depending on whether he used the "Olympic stade" (176.4m) or the Italian stade (184.8m), this would imply a circumference of 44,100km (an error of 10%) or 46,100km, an error of 15%.[10] In 2012, Anthony Abreu Mora repeated Eratosthenes's calculation with more accurate data; the result was 40,074km, which is 66km different (0.16%) from the currently accepted polar circumference.[9]
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 . 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.
Eratosthenes, a Greek astronomer from Hellenistic Cyrenaica (276–194 BC), estimated Earth's circumference around 240 BC. He had heard that in Syene the Sun was directly overhead at the summer solstice whereas in Alexandria it still cast a shadow. Using the differing angles the shadows made as the basis of his trigonometric calculations he estimated a circumference of around 250,000 "stades". The length of a 'stade' is not precisely known, but Eratosthenes's figure only has an error of around five to fifteen percent. Eratosthenes used rough estimates and round numbers, but depending on the length of the stadion, his result is within a margin of between 2% and 20% of the actual meridional circumference, . Note that Eratosthenes could only measure the circumference of the Earth by assuming that the distance to the Sun is so great that the rays of sunlight are practically parallel.
Longditude lines, also known as meridian lines, follow the polar circumference, and are all exactly the same length.[4]
A more accurate estimate was provided in Al-Biruni's Codex Masudicus (1037).[14][15] 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.[16]
The metre was originally defined as of the meridian arc from the North pole to the equator passing through Dunkirk. The Earth's circumference is therefore approximately 40,000 km. The equatorial circumference is slightly longer than the polar circumference – the measurement based on this (40,075.017 km × × = ) is known as the geographical mile.
Pythagoras in the 6th century BC and Parmenides in the 5th century stated that the Earth is spherical, and this view spread rapidly in the Greek world. Around 330 BC, Aristotle maintained on the basis of physical theory and observational evidence that the Earth was spherical, and reported an estimate of its circumference. The Earth's circumference was first determined around 240 BC by Eratosthenes. By the second century AD, Ptolemy had derived his maps from a globe and developed the system of latitude, longitude, and climes. His "Almagest" was written in Greek and only translated into Latin in the 11th century from Arabic translations.
Earth's circumference lines are the foundation of the coordinate system of latitude and longitude, introduced in the mid-second century BC by Hipparchus.[5]
In the 15th century, Christopher Columbus used al-Farghani's estimate for the Earth's circumference as the basis for his voyages to America. However, Columbus mistook al-Farghani's 7091-foot Arabic mile to be a 4856-foot Roman mile, causing him to underestimate the Earth's circumference, believing he could take a shortcut to Asia.
The Earth's circumference is quite close to 40 million metres. Many globes are made with a circumference of one metre, so they are models of the Earth at a scale of 1:40 million. In imperial units, many globes are made with a diameter of one foot, yielding a circumference of 3.14 feet and a scale of 1:41,777,000. Globes are also made in many other sizes.
Posidonius (or Poseidonius) of Apameia (c. 135–51 BCE), was a Greek Stoic philosopher who traveled throughout the Roman world and beyond and was a celebrated polymath throughout the Greco-Roman world, like Aristotle and Eratosthenes. His work "about the ocean and the adjacent areas" was a general geographical discussion, showing how all the forces had an effect on each other and applied also to human life. He measured the Earth's circumference by reference to the position of the star Canopus. His measure of 240,000 stadia translates to , close to the actual circumference of .
He was informed in his approach by Eratosthenes, who a century earlier used the elevation of the Sun at different latitudes. Both men's figures for the Earth's circumference were uncannily accurate, aided in each case by mutually compensating errors in measurement. However, the version of Posidonius' calculation popularised by Strabo was revised by correcting the distance between Rhodes and Alexandria to 3,750 stadia, resulting in a circumference of 180,000 stadia, or . Ptolemy discussed and favored this revised figure of Posidonius over Eratosthenes in his "Geographia", and during the Middle Ages scholars divided into two camps regarding the circumference of the Earth, one side identifying with Eratosthenes' calculation and the other with Posidonius' 180,000 stadion measure.
Let "C"  be the Earth's circumference, "R"  be its radius, "c"  be the added string length and "r"  be the added radius. has a circumference of 2"R" ,