Eratosthenes The measure of Earth's circumference is the most famous among the results obtained by
Eratosthenes, who estimated that the meridian has a length of 252,000
stadia, with an error on the real value between −2.4% and +0.8% (assuming a value for the stadion between 155 and 160 metres; Cleomedes invites his reader to consider two Egyptian cities,
Alexandria and Syene (modern
Aswan): • Cleomedes assumes that the distance between Syene and Alexandria was 5,000 stadia (a figure that was checked yearly by professional
bematists,
mensores regii). • He assumes the simplified (but inaccurate) hypothesis that Syene was precisely on the
Tropic of Cancer, saying that at
local noon on the summer
solstice the Sun was directly overhead. Syene was actually north of the tropic by something less than a degree. • He assumes the simplified (but inaccurate) hypothesis that Syene and Alexandria are on the same meridian. Syene was actually about 3 degrees of longitude east of Alexandria. According to
Cleomedes's
On the Circular Motions of the Celestial Bodies, around 240 BC, Eratosthenes calculated the
circumference of the Earth in
Ptolemaic Egypt. Using a vertical rod known as a
gnomon and under the previous assumptions, he knew that at local noon on the summer solstice in
Syene (modern
Aswan, 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. 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; 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. 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. The method was based on several
surveying trips conducted by professional
bematists, whose job was to precisely measure the extent of the territory of Egypt for agricultural and taxation-related purposes. 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; 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". It is generally thought that the stadion used by Posidonius was almost 1/10 of a modern statute mile. Thus Posidonius's measure of 240,000 stadia translates to , not much short of the actual circumference of .
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.
Aryabhata Around AD 525, the Indian mathematician and astronomer wrote
Aryabhatiya, in which he calculated the diameter of earth to be of 1,050
yojanas. The length of the
yojana intended by Aryabhata is in dispute. One careful reading gives an equivalent of , too large by 11%. Another gives , too large by 20%. Yet another gives , too large by 5%.
Islamic Golden Age Around AD 830,
Caliph Al-Ma'mun commissioned a group of
Muslim astronomers led by
Al-Khwarizmi to measure the distance from Tadmur (
Palmyra) to
Raqqa, in modern
Syria. 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%. A more convenient way to 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 locations,
al-Biruni developed a new method of using
trigonometric calculations, based on the angle between a
plain and
mountain top, which made it possible for it to be measured by a single person from a single location. 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 the
al-Ma'mun expedition. ==Historical use in the definition of units of measurement==