# Earth's circumference

Earth's circumference is the distance around Earth. 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).[1]

Measurement of Earth's circumference has been important to navigation since ancient times. The first known scientific measurement and calculation was done by Eratosthenes, who achieved a great degree of precision in his computation.[2] Treating the Earth as a sphere, its circumference would be its single most important measurement.[3] Earth deviates from spherical by about 0.3%, as characterized by flattening.

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 one minute of latitude (see meridian 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 because the 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 a round number in metres or nautical miles.

## History

### Eratosthenes

The measure of Earth's circumference is the most famous among the results obtained by Eratosthenes,[4] 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).[2] Eratosthenes described his technique in a book entitled On the measure of the Earth, which has not been preserved.

Measure of Earth's circumference according to Cleomedes' simplified version, based on the approximation that Syene is on the Tropic of Cancer and on the same meridian as Alexandria

Eratosthenes' method to calculate Earth's circumference has been lost; what has been preserved is the simplified version described by Cleomedes to popularise the discovery.[5] Cleomedes invites his reader to consider two Egyptian cities, Alexandria and Syene, modern Assuan:

1. Cleomedes assumes that the distance between Syene and Alexandria was 5,000 stadia (a figure that was checked yearly by professional bematists, mensores regii).[6]
2. 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.
3. 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.

Under the previous assumptions, writes Cleomedes, you can measure the Sun's angle of elevation at noon of the summer solstice in Alexandria, by using a vertical rod (a gnomon) of known length and measuring the length of its shadow on the ground; it is then possible to calculate the angle of the Sun's rays, which he claims to be about 7°, or 1/50th the circumference of a circle. Taking the Earth as spherical, the Earth's circumference would be fifty times the distance between Alexandria and Syene, that is 250,000 stadia. Since 1 Egyptian stadion is equal to about 157.7 metres,[7] the result is roughly 39,425 km, which is 1.5% less than the real circumference, 40,008 km.[citation needed]

Eratosthenes' 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' book. 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.[2] Furthermore, the fact that Eratosthenes' measure corresponds precisely to 252,000 stadia 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;[8] 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".[2][9]

### Aryabhata

Around AD 525, the Indian mathematician and astronomer Aryabhata 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 14,200 kilometers, too large by 11%.[13] Another gives 15,360 km, too large by 20%.[14] Yet another gives 13,440 km, too large by 5%.[15]

### 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%.[16]

Diagram showing how al-Biruni was able to calculate the Earth's circumference by measuring the dip of the horizon from a point at a known height.

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.[16] From the top of the mountain, he sighted the dip angle which, along with the mountain's height (which he determined beforehand), he applied to the law of sines formula. This was the earliest known use of dip angle and the earliest practical use of the law of sines.[17] 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.[16]

### Columbus's error

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 by Toscanelli, 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 not Asia, but rather a New World.[18]

## Historical use in the definition of units of measurement

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 (1/60) of one degree of latitude. As one degree is 1/360 of a circle, one minute of arc is 1/21600 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.[19]

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, the French Academy of Sciences commissioned Jean Baptiste Joseph Delambre and Pierre Méchain to lead an expedition to attempt 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. The length of the first prototype 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 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.[20] This is why the polar circumference of the Earth is actually 40,008 kilometres, instead of 40,000.

## References

1. ^ Humerfelt, Sigurd (26 October 2010). "How WGS 84 defines Earth". Archived from the original on 24 April 2011. Retrieved 29 April 2011.
2. ^ a b c d Russo, Lucio (2004). The Forgotten Revolution. Berlin: Springer. p. 273–277.
3. ^ Shashi Shekhar; Hui Xiong (12 December 2007). Encyclopedia of GIS. Springer Science & Business Media. pp. 638–640. ISBN 978-0-387-30858-6.
4. ^ Russo, Lucio. The Forgotten Revolution. p. 68.
5. ^ Cleomedes, Caelestia, i.7.49–52.
6. ^ Martianus Capella, De nuptiis Philologiae et Mercurii, VI.598.
7. ^ Donald Engels (1985). The Length of Eratosthenes' Stade. American Journal of Philology 106 (3): 298–311. doi:10.2307/295030 (subscription required).
8. ^ Rawlins, Dennis (1983). "The Erathostenes-Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Erathostenes' Experiment?". Archive for History of Exact Sciences. 26 (3): 211–219. doi:10.1007/BF00348500. S2CID 118004246.
9. ^ Pliny, Naturalis Historia, XII §53.
10. ^ a b Posidonius, fragment 202
11. ^ Cleomedes (in Fragment 202) stated that if the distance is measured by some other number the result will be different, and using 3,750 instead of 5,000 produces this estimation: 3,750 x 48 = 180,000; see Fischer I., (1975), Another Look at Eratosthenes' and Posidonius' Determinations of the Earth's Circumference, Ql. J. of the Royal Astron. Soc., Vol. 16, p.152.
12. ^ John Freely, Before Galileo: The Birth of Modern Science in Medieval Europe (2012)
13. ^ Kak, Subhash (2010). "Aryabhata's Mathematics". arXiv:1002.3409 [cs.CR].
14. ^ "Journal of the Royal Asiatic Society of Great Britain and Ireland". 1907.
15. ^ "The_Aryabhatiya_of_Aryabhata_Clark_1930".
16. ^ a b c Mercier, Raymond (1992). "Geodesy". In Harley, J.B.; Woodward, David (eds.). The History of Cartography, Volume 2, Book 1. The University of Chicago Press. pp. 175–188. ISBN 9780226316352.
17. ^ Behnaz Savizi (2007), "Applicable Problems in History of Mathematics: Practical Examples for the Classroom", Teaching Mathematics and Its Applications, Oxford University Press, 26 (1): 45–50, doi:10.1093/teamat/hrl009
18. ^ Gow, Mary. "Measuring the Earth: Eratosthenes and His Celestial Geometry, p. 6 (Berkeley Heights, NJ: Enslow, 2010).
19. ^ Marine Insight, Why Nautical Mile and Knot Are The Units Used at Sea?
20. ^ Alder, Ken (October 2003). The Measure of All Things: The Seven-Year Odyssey and Hidden Error That Transformed the World. Simon and Schuster. ISBN 978-0-7432-1676-0.

## Bibliography

• Krebs, Robert E.; Krebs, Carolyn A. (2003). "Calculating the Earth's Circumference". Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Ancient World. Greenwood Publishing Group. p. 52. ISBN 978-0-313-31342-4.
• Nicastro, Nicholas (25 November 2008). Circumference: Eratosthenes and the Ancient Quest to Measure the Globe. St. Martin's Press. ISBN 978-1-4299-5819-6.
• Gow, Mary (1 July 2009). Measuring the Earth: Eratosthenes and His Celestial Geometry. Enslow Publishing, LLC. ISBN 978-0-7660-3120-3.
• Lowrie, William (20 September 2007). Fundamentals of Geophysics. Cambridge University Press. ISBN 978-1-139-46595-3.