Talk:History of geodesy

Public domain
This text is taken from chapter 1 of the public domain resource Geodesy for the Layman at http://www.ngs.noaa.gov/PUBS_LIB/Geodesy4Layman/TR80003A.HTM#ZZ0 – please Wikify as necessary.

Remarkable/non-remarkable? NPOV?
Riddle me this : how come Eratosthenes's estimation of the circumference of Earth, accurate to 0,3% in 200BC, is "non remarkable", while Aryabhata's estimate, accurate to 1% in 500 AD, seven hundred years later, is remarkable? 94.68.128.86 (talk) 19:42, 15 January 2011 (UTC)
 * I agree. The paragraph that begins "Had the experiment been carried out as described, it would not be remarkable if it agreed with actuality" perm is written with such arrogance. Not to mention that some of its oh-so-clever statements may well be wrong - such as claiming that Syene "is actually...22 miles north of the Tropic of Cancer"... Well, today, yes, but if the user who wrote it is so well-informed as his insolent tone suggests, maybe he'd have known that with nutation, Syene may well have been on the tropic 2200 years ago. It'd be good if someone rewrote the entire Eratosthenes section with a bit more neutrality. BigSteve (talk) 07:35, 1 March 2014 (UTC)
 * In fact, nutation in the north-south sense has an amplitude of only 20" and isn't cumulative. But there is indeed a slow secular change in the obliquity of the ecliptic and it is almost universally agreed that in Eratosthene's time the real obliquity was 23° 43’. That still puts Aswan 22' (that is 22 nautical miles) north of the tropic of Cancer, even in that time.

Eratosthenes Measurement
He did not need to know the direct distance between Syene and Alexandria, which would require him to travel across the desert. He only needed to know the distance in the N-S direction between them as long as he made his measurement of the angle of the Sun on the summer solstice, which he purportedly did. (128.138.64.39 (talk) 16:49, 9 August 2010 (UTC))

Stadia, not miles
The text judges the accuracy of Eratosthenes' estimate of the size of the Earth based on a number expressed in miles. This is nonsense because E. did not use miles, the type of mile is not specified, the number is rounded to the nearest 1000 miles, and the relation between E's original stadion and the mile that was used for the conversion is not correct. It says that 1 stadion is 1/10 of a statute mile. This is inaccurate by default, because the statute mile did not exist in E's time and it is totally unlikely that E's stadion was a neat fraction of the much later unit. So dump the miles and give E's results in stadia; then worry about the length of the stadion that he may have used. Tom Peters 22:57, 26 August 2006 (UTC)

Archimedes vandalism
Someone has a vandetta against Archimedes ? They inflated the values 1000%, the edit by BDPATTON2 was, ONLY, to make the quoted result 10 times too large,and leaving the text saying Archimedes was quite accurate, a little under the true size.

Although I wonder if its not a vandetta but an experiment to see how long it takes to get it fixed. — Preceding unsigned comment added by 115.69.16.149 (talk) 23:11, 22 December 2015 (UTC)

Figures for Geodesy for Layman
The figures in the NOAA and other versions of _Geodesy for the Layman_ are very bad in direct scans. While teaching at the Naval Postgraduate School I had these updated and placed on the Web there for my students. (With the permission of NIMA now NGA.) They are still there. A version of _Geodesy for the Layman_ with the DMA/NIMA/NGA text and new figures can be found at

www.oc.nps.edu/oc2902w/geodesy/geolay/gfl84b_t.htm

Jim Clynch (JimC728@gmail.com)

ps - More public domain geodesy items are also there. Some have been updated and expanded on my site clynchg3c.com —Preceding unsigned comment added by 69.154.11.157 (talk) 15:50, 11 November 2008 (UTC)

Medieval Period
This section contains the following: "It was not until the 15th century that his concept of the earth's size was revised. During that period the Flemish cartographer, Mercator, made successive reductions in the size of the Mediterranean Sea and all of Europe which had the effect of increasing the size of the earth." This must be either incorrect or worded poorly, as Mercator's lifespan was exclusively in the sixteenth century.

DPU (talk) 01:00, 18 January 2010 (UTC)
 * Agreed, and while I don't have access to suitable refs to improve the section, the notion that Mercator (or anybody else) changed the size of Europe, the Med and the Earth is bizarre; presumeably the passage means that he revised his estimates of these sizes. -- Timberframe (talk) 12:27, 18 January 2010 (UTC)

Biruni section
There appears to be no connection between the formulae quoted in this section and the adjacent "explanatory" diagram. What are "θ1" and "θ2" in relation to the angle "α" in the diargram. Is the Earth's radius "R" - or is it "r"? What is "d"? —Preceding unsigned comment added by 87.113.150.32 (talk) 18:43, 30 April 2011 (UTC)

Time synchronization
I found this article very interesting, but I am surprised that the author fails to mention, in the sources of error discussion, the difficulty of synchronizing clocks between the two locations. The article states "At the same time, he observed the sun was not directly overhead at Alexandria", as if all he had to do was look at his watch. I realize that it has to do with knowing that both cities were at the same longitude, but even so, the means for knowing the time accurately is not discussed. Were there mechanical timekeepers in his time? Did he use a sundial to know when it was noon? How accurately could he measure the time? How sensitive is his calculation of the earth's circumference to an error in the time? I don't mean to critisize the article, I just think a few words on the subject of time synchronizing would be helpful. — Preceding unsigned comment added by Rbarline (talk • contribs) 18:51, 16 June 2014 (UTC)

No the issue is the shadow toward the north or south, NOT the east /west component. .. You can tell the time from the shadows .. When he says there was no shadow, he could say that it was definitely the solstice at noon, as the place was on the tropic. So the shadow was TELLING the time, no chronograph needed.

The shadow a little further north would always have a northward component, but as long as there was no east or west component, then it was noon. The error introduced by being able to tell midday precisely, eg a sixth of an hour  in error,  does not then lead to an error worth calculating, when the measurement of the long distance between the two observation points has much more error.

115.69.16.149 (talk) 23:16, 22 December 2015 (UTC)

Desperately Seeking Stadion: A Tale of Two Cities
Subsection History of geodesy: Eratosthenes' work On the measurement of the Earth is now lost, but was quoted extensively by Strabo and others ([1], p. 359). This means that many details, including the measurement of the distance Alexandria/Syene, which was legendary measured by counting steps(!) must be taken with a pinch of salt.

From the wiki-article: "(...) to measure the distance [from Alexandria to Syene], which came out to be equal to 5000 stadia or (at the usual Hellenic 185 m per stadion) (...)". The stadion is not at all the 'usual' 185 meter; in fact its value is hotly debated: "It has been estimated that these varied in length from 149 m to 298 m, but agreement has not been reached on the specific length of the unit employed by ERATOSTHENES." ([1], p. 359). That is a range of 100% relative to 149 meter! The meter as length unit didn't exist in the time of Eratosthenes and came into use only after 1800 AD.

Then there is the figure of 250,000 for the circumference in stadia as determined by Eratosthenes. But the modern Arabic numerals we use to express numbers didn't exist in the time of Eratosthenes; they used Greek numerals with special notation for numbers greater than 9,999. This means that writing a number like 250,000 was no tidy affair; possibly it caused headaches for later authors who had to convert to Roman and later modern numerals. Summarizing, it is well-possible that '250,000' is an incorrect interpretation of the original text by a later scholar. The same problems affects the number 5000 (distance in stadia between Alexandria and Syene) and the fraction $1/undefined$.

It is clear then that ancient Greek numerals are a rather complicated affair, prone to ambiguity. Now from the time of Eratosthenes, thousands of transcripts and translations of his calculations must have been made, each used as a source for yet another chain of transcripts. Some copyists must have been insecure about the meaning of the numerals or calculations Eratosthenes used, thereby making mistakes, which propagated through the transcript-chain. Which means, that the quoted figure for the circumference of 250,000 stadia is unreliable.

The distance between Alexandria and Aswan/Syene. The |The distance 'as the crow flies' is 843 km/524 miles. If this was done by the legendary surveyor counting his steps, the surveyor would have travelled by night to escape the desert heat. If he travelled a max of 8 hours per day/5 km per hour/40 km per day, it would have taken him 843/40 days, i.e. more than 20 days. From the map, it can be seen that he would have to cross the Nile at least four times. Without a compass he couldn't have known the direction to Alexandria, unless he found a way to navigate with the help of the stars(?). He would have to negogiate the loose sand, possibly (human) predators, make sure to walk in a straight line, not to slide away with each step and to control his step size - and that for +/- 800,000 steps!! Impossible - the error in the measured distance Alexandria - Syene would have been gigantic.

Finally, the surveyor had to make a mark on say papyrus after each step (in the dark!) which forced him to pause him a second or two. 800,000 marks; that would have required tons of writing material which he would have to carry around together with water and food! So it had to be done in another way, or Eratosthenes made an educated guess. Let us assume an measurement error of 10 % in the distance between the two cities.

Note that the line between the two cities doesn't coincide with the meridian - more errors.

The angle of the overhead sun in Alexandria. "(...) the midday sun shone to the bottom of a well in the town of Syene (Aswan). At the same time, he observed the sun was not directly overhead at Alexandria (...)" "At the same time". That requires synchronized watches, modern electronic communication or a system of mirrors. He simply couldn't have known about any event in Syene at the same time. It is not clear what the experimental set up was in Alexandria.

One way to circumvent (well, sort of...) the problem of simultaneity is to wait for summer solstice i.e. wait for the moment that the shadow cast by the pole at noon is at it shortest; then one  knows that the sun is overhead in Syene. Problem: at solstice the height of the sun at noon varies very slowly, (gradually accelerating as it returns to  autumn equinox) and perceptible changes in the length of the shadow might take days or even weeks either side of solstice - by that the sun is no longer overhead in Syene...

Let us assume a relative error of 10 % the angle (incorporating the slow movement at solstice and the fact that the line connecting Alexandria with Syene is inclined about 3 degrees with meridian (10 % of 7 degrees is 0.7 degrees which is not overly pessimistic).

What possible values for the circumference of the Earth? Let θ be the angular deviation of the sun from the vertical direction at Alexandria, d the linear distance between Alexandria and Syene, R the radius of the earth and s the circumference of Earth. Then s = 2πR = 2π $d/θ$ which means that the relative error in s, is the sum of the relative errors in d and θ, i.e.  10 + 10 = 20% = $1/undefined$. The absolute error is in s is $1/undefined$ x 250,000= 50,000 stadia and 200,000 < s < 300,000 stadia.

The lowest value reported for the stadion in [1] is 149 m; $200,000 x 0.149 = 29,800$ km. The highest value is 1 stadion = 298 m; $300,000 x 0.298 = 89,400$ km. The calculated interval for the circumference $s$ is $29,800 < s < 89,400$ km or $s = 59,600$. The relative error is 50% (!) and there is only signifanct digit: $s = 60,000$ km. The modern value for $s$ is 40,030.2 km (NASA). With such a huge margin, figures become meaning less, one might as well throw darts.

Figures like 0.4 % precision  for the circumference, quoted in the article, are completely spurious:


 * The work by Eratosthenes is lost, he is only quoted by later commentators like Strabo and only the quotations survive. The scope of his work is much wider than just the measurement of the circumference of Earth. [2] (downloadable) is a thorough discussion of all extant fragments, as handed down to us by among others Strabo, Hipparchus and Plinius; section II B. Erdmessung., p. 99 - 142, is relevant to us. On p. 126 Berger notes that when studying and comparing the fragments it is clear that Eratosthenes' procedure is nowhere handed down to us in full and could not have relied on just one observation; "Verschiedene Berichtstatter berichten über verschiedene Punkte seiner Methode". On p. 136 he refers to the total silence of the ancient authors about the variety of stadion measures. After studying these pages, it is still not clear to me how Eratosthenes performed his measurements, in particular how he measured the linear distance between Alexandria and Syene.
 * The inclination of the line Alexandria/Syene with the meridian (about 3 degrees) is not taken into account, or rather, it is unknown if Eratosthenes was aware of it and if he knew how to correct for this; it tends to underestimate the measured angle of the sun with the vertical in Alexandria.
 * Syene/Aswan is not ON, but about 40 arcminutes distant from the Tropic of Cancer, which translates, if I calculate correctly, to about 73 km. Due to the nutation of the Earth the angular distance to the Tropic changes at most 9 arcseconds, 16% of the angular distance to the Tropic. So the minimum distance from Syene to the Tropic is about 60 km, somewhat more than 1% of 5000 stadia: we can neglect the error due to Syene not on the Tropic (See discussion Remarkable/non-remarkable? NPOV? above, on this Talk-page).
 * 2000 years have passed since the manuscripts by Eratosthenes/Strabo, his work has crossed several cultures, it was transcribed, errors and 'improvements' were made by the copyists. Modern Western Arabic/European numerals were introduced in the West in the 12-th century and the translation from the Greek numerals to modern digits probably wasn't a  trivial exercise; in particular large numbers might have caused confusion = error.
 * The meter was introduced around 1800. Because no one is sure about the version of the stadion commonly accepted in the time of Eratosthenes, this potentially introduces a huge systematic error in the value of the circumference in km.

The story of Eratosthenes' circumference and its uncanny accuracy is no more than a case of 'reverse engineering': one assumes the correct (based on what??) circumference to be 250,000 stadia (without a thorough discussion of potential error sources and in absence of his original work) and then searches for a value for the stadion in meter, such that the modern value rolls out: bingo!

Size of the Earth as determined by Posidonius. Similar story

[1] https://www.jstor.org/stable/41133819 Edward Gulbekian, The Origin and Value of the Stadion Unit used by Eratosthenes in the Third Century B. C., Archive for History of Exact Sciences Vol. 37, No. 4 (1987), pp. 359-363.

[2] HUGO BERGER, Die geographischen Fragmente des Eratosthenes (Leipzig, 1880).

--Gerard1453 (talk) 17:22, 31 August 2017 (UTC)


 * Ooops! Errors are of course statistical quantities and then the theory of 	error propagation applies. Recalculating the (absolute value) of the relative error in the value for the circumference s, as determinded by Eratosthenes, then yields |Δs/s| = √2 x 10% = 14 %, rather than 20 %. Slightly better it seems, but very much cosmetic: we do not have a clue regarding the true values of the uncertainties involved; I merely gave a numerical example to get  some idea. -- Gerard1453 (talk) 14:25, 11 October 2017 (UTC)

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Islamic world/al-Biruni - The Unbearable Lightness of Measuring
The figure '6339.6 km' quoted in the text for the radius of the earth, which al-Biruni arrived at, derives from the lemma al-Biruni in the Mac Tutor online archive, see ref. [1]. According to Mac Tutor ''“Important contributions to geodesy and geography were also made by al-Biruni. He introduced techniques to measure the earth and distances on it using triangulation. He found the radius of the earth to be 6339.6 km,...” followed by the compulsory, “...a value not obtained in the West until the 16th century (see [50])”''. Mac Tutor doesn't specify who this 16th century scholar is, how he arrived at this figure and where we can find more information.

The value “obtained in the West” strangely enough duplicates al-Biruni's precisely and so cannot be considered an improvement. But this talk of the relative merits of both measurements is as relevant as the (in)famous debate concerning the number of angels able to dance on the head of a pin: we do not know the old length measures employed and cannot translate in modern (kilo)meter let alone say anything about the accuracy and precision - the precision to 5 digits is spurious.

This reference '[50]' in the Mac Tutor lemma is an article by one K Norhudzaev, see ref. [3].

I cannot find a 'K Norhudzaev' on the internet let alone his academic credentials, neither is there a trace of his publication. I wrote to O'Connor and Robertson, editors of Mac Tutor, requesting a pdf of the article. Robertson supplied a link to http://ams.math.uni-bielefeld.de/, but unfortunately access is restricted (user name and password required), which makes it useless as a source.

MacTutor spends no more than one(!) line on the radius measuremnt by al-Biruni, just “He found the radius of the earth to be 6339.6 km [using triangulation] (...)” That's it, no discussion of the experimental set-up, no error sources, no mention of conmtemporary distance measures and the fact that the (kilo) meter was not known to al-Biruni. For details one is immediately referred to yet another publication (by said 'K Norhudzaev') which turns out to be a cul-de-sac.

In other words: The MacTutor lemma on al-Biruni is NOT a 'reliable source' .

Al-Biruni is also mentioned in ref. [2] by physicist John Edward Huth. In chapter The Modern Legend of al-Biruni, p.215f., he discusses the manner in which al-Biruni measured the radius of the Earth. He first determined the height of a 'nearby' mountain by triangulation, then found the dip angle to the horizon from the top of the mountain “by long view across the dusty plains of the Punjab”. He reported the value of the dip angle (34 arc minute) and the height of the mountain in length units for which we do not have a precise modern equivalent. We do not know which mountain he chose and so cannot verify his value for the angle. Neither can we verify the height of this 'nearby' (?) mountain. Even less can we say anything of the measurent errors. If Huth is not familiar with the length unit used by al-Biruni, why then is he able to report the radius in kilometer?

Huth continues:

''“Al-Biruni's measurement, while clever, had at least one fatal flaw: he neglected to take into account the effect of Atmospheric refraction (...) There has been a modern mythology that has grown around al-Biruni, purporting that he made the most precise measurement of the Earth's radius well before modern times. In 1973 an engineer, Saiyid Samad Husain Rizvi, published an article reporting the discovery of a long lost book by al-Biruni, the Ghurrat-Uz-Zijat (...) This [the problem of the error due to refraction] has not deterred the numerous supporters of this measurement, who uncritically accepted it as highly precise and evidence of his genius (...)”''

The problems due to refraction are serious; as Huth explains (p. 217): “In reality it is quite possible that atmospheric layering could have given a flat [=infinite radius] or even concave Earth in the same way the layering [of the atmosphere] can give rise to strange mirages.” Given that the temperatures in the Punjab can reach 50 degrees in summer, this is a serious impediment to accurate measurements. “(...) without any knowledge of the refractive effects, it's impossible to assign any accuracy to this; it's more like winning a lottery.”

Unfortunately, there is an error source far more devastating than Huth's refraction: as stated above, it is the complete absence of any knowledge of the distance measure employed by al-Biruni.

Note that the the article by Saiyid Samad Husain Rizvi and the paper published by K. Norhudžaev (as cited by [1]) both have the same publication date -- 1973.

I then searched for Saiyid Samad Husain Rizvi and found several occurences:


 * Samad_Rizvi. Here Rizvi (july 23, 1924 – December 17, 2009) is considered to be an...astrologer, not the engineer Huth [2] considers him to be. His 'Research papers' include two contributions to astrology, the third is a publication Unique And Unknown Book Of Al - Beruni: Ghurrat - Uz - Zijat Or Karana Tilara, Islamic Culture, January 1965 - Number 153, which I cannot find on the internet. The publication year is 1965, not 1973 and the fact that it was published in a religious journal doesn't bode well for its reliability. In fact, it is about astrology, see below.


 * Samad Husain Rizvi (Pakistan), Astrological Poetry Of Ameer Khusro, SaptarishisAstrology (2009) ([5]). An article written by Rizvi and again it is about astrology. Rizvi is introduced as “a noted astronomer and astrologer of Pakistan (...) he has learned astrology from Pandits of Banaras. After earning BSc (Engineering) (...) he migrated to Pakistan (...) his milestone work is the revival of Al-Beruni's new moon sighting.” That's it! No work on measuring the Earth! I really do not understand where Huth [2] has his information from.

Among the “numerous supporters of this measurement, who uncritically accepted” the precise measurement of the Earth's radius by al-Biruni, Huth [2] counts the MacTutor online archives of mathematics [1], discussed earlier and the physicist Jim Al-Khalili, in his role as presenter of the BBC Four three-part series Science and Islam. Al-Biruni appears in part 2 The Empire of Reason, from minute 17:00 on, see ref. [4].

Huth [2] notes (p. 217) “In a recent BBC documentary, 'Science and Islam', which first aired on Janaury 12, 2009, the narrator describes the previous attempts at the measurement of the circumference of the Earth (...) The narrator goes on to descibe al-Biruni's dip angle method, even using a large astrolabe as a dramatic prop, and illustrates the technique, giving the dip angle formula.”

We see the presenter visiting one Dr. Sami Chalhoub (Institute for The History of Arabic Science, Aleppo University), to discuss the work of al-Biruni. Chalhoub is another completely unkown quantity in the world of academia; he hands over to Al-Khalilli a notebook of recent manufacture, with fresh white pages upon which are scribbled notes purported to be the original thoughts of al-Biruni. We only get a very quick glimpse of the handwriting and it is impossible to say anything sensible about its meaning. It is unclear where Chalhoub found the original work by al-Biruni, how he convinced himself of its authenticity and in which libary it is stored. There is no mention by him of K. Norhudzaev or Saiyid Samad Husain Rizvi (see above).

The calculations which also appear in the text of the wiki-lemma discussed here are made by Chaloub himself, on the spot! They are nowhere to be seen in the notebook. After staring for a second at the very unauthentic manuscript, Al-Khalili is utterly convinced and decides to try the experiment himself.

What now unfolds in the BBC documentary reminds me of a Monty Python-like sketch for two gentlemen walking thru a landscape: Al-Khalili and an assistant, who holds a huge flimsy, home made astrolabe, made of plywood. The unknown landscape is near a see (which one?) - no “dusty plains of the Punjab” (Huth) in sight. Now the measurements are made, on a not very horizontal plain, of the two angles on the base of a rather small base line d of  100 meter: the assistant is holding the 'astrolabe' in his hands, in a by definition, not very firm grip; it is not firmly supported and without a level. Al-Khalilli 'measures' the two angles and without much ado he declares the mountain to be 530 meter high (zero error of course).

Note that the mathematical symbols used in the documentary didn't exist in the time of al-Biruni - they are a Western development, just as the use of letters for algebraic variables (François Viète). Note e.g. the equals-sign, introduced around 1600. So al-Biruni had to tell the 'story' in words. He also had to have at hand trigonometric tables.

We now turn to the formula in the wiki-article for calculating the Earth's radius purported to have been developed by al-Biruni. It's source is the BBC Four televsion series Science and Islam - part 2, discussed above (see [4]). The formula is R = h x (cos θ)/(1 - cos θ). h is the (unknown) height of mountain, R radius Earth, θ the dip angle. Because cos θ ≈ 1, the numerator of the expression for R can be considered to be equal to h: the formula is fairly robust against errors in h. The angle θ is the real 'killer': with θ very small, the denominator (1 - cos θ) ≈ 0 and R becomes very sensitive to errors in the dip angle.

Numerical example:
 * with h = 0.530 km as suggested in the BBC documentary (Ref. [4]) and radius R = 6339.6 km, θ = 0.013 radian = 0.74° from which 1 - cos θ = 8.3 x 10^(-5).
 * If the dip angle is measured with precision, say, 20% (reasonable, given the imperfections of eye, imperfections of the instrument, hazy horizon, reading errors), then for dip angle θ : (0.80*0.74) < θ < (1.20 * 0.74) or 0.6 < θ < 0.9 degree, from which 4393 < R < 9996 km, or properly rounded (given the lack of precision) : 4000 < R < 10,000 km. (We ignore the systematic error due to refraction (see above), which might even lead to an infinite Earth!).

Ergo: what you win by using the much vaunted small baseline d in the triangulation formula for the determination of height h, you lose by the extreme sensitivity of R to the dip angle and hugh swings in the value of R  - nothing is gained.

Summarizing: In the section on al-Biruni, two main sources are used:
 * The MacTutor History of Mathematics archive [1] which dedicates just one line to the Earth measurement, immediately referring us to a 1973 publication by a certain K. Norhudzaev, who is unknown and has unknown credentials. His publication [3] is not available on-line, and:


 * the BBC Four three-part series Science and Islam - part 2 The Empire of Reason. I do not think a television documentary can ever serve as a reliable source and in this particular case the way in which the matter is presented comes close to being a farce. The narrator visits one Dr. Sami Chalhoub (Aleppo University), yet another unknown 'quantity', who hands over to the presenter Al-Khalilli a manuscript of highly dubious authenticity and reliability.


 * Another author, physicist John Edward Huth [2], claims that one Samad Husain Rizvi (Pakistan) in 1973 reported “the discovery of a long lost book by al-Biruni, the Ghurrat-Uz-Zijat”. The wiki-lemma Samad_Rizvi portrays him as an astrologer and mentions his article Unique And Unknown Book Of Al - Beruni: Ghurrat - Uz - Zijat Or Karana Tilara, Islamic Culture, January 1965 - Number 153, but the article dates from 1965 and is untraceable on the internet. I can find only one publication by Rizvi - about the Astrological Poetry Of Ameer Khusro . Not a trace of the circumference of the Earth here.


 * The experiment by al-Biruni was performed in the plains of Nandana in the Punjab (see Al-Biruni). It is not clear which  mountain was used nor what its height h is. Neither is there knowledge of the ancient length measure and its translation in modern units. It is unclear if al-Biruni was familiar with atmospheric refraction and if he knew how to correct for it. The region is very hot in summer possibly giving rise to unexpected atmospheric phenomena like mirages which uncontrollably ruin the observations.


 * The expression used for the calculation of the Earth radius R is useless from the point of experimental physics. R is fairly robust against measurement errors in the height h , but hopelessly sensitive to the dip angle θ: even small observational deviations (due to imperfections of eye, imperfections of the instrument, hazy horizon, reading errors) will send R to anything between zero and infinity: the precision to 5 digits in R, claimed by Mac Tutor is very misleading. Even if al-Biruni used an extremely robust and precise inclinometer with level, anchored to the mountain and used under very favorable razorsharp atmospheric conditions, we would still have to contend with the unfamiliarity of the ancient distance unit.

I suggest to scrap the section on al-Biruni's Earth measurement entirely; the sources are very dubious.

[1] Abu Arrayhan Muhammad ibn Ahmad al-Biruni (1999), in John J O'Connor& Edmund F Robertson (ed.), MacTutor History of Mathematics archive, School of Mathematics and Statistics, University of St Andrews, Scotland

[2] John Edward Huth, The Lost Art of Finding Our Way (2013), p.215f. (via google books)

[3] K Norhudzaev, al-Biruni and the science of geodesy (Uzbek), in Collection dedicated to the 1000th anniversary of the birth of al-Biruni (Tashkent, 1973), 145-158.

[4] BBC Four three-part series, Science and Islam - part 2 The Empire of Reason, from minute 17:00 (2009), narrated by physiscist Jim Al-Khalili.

[5] Samad Husain Rizvi (Pakistan), Astrological Poetry Of Ameer Khusro, SaptarishisAstrology (2009)

--- Gerard1453 (talk) 17:19, 6 October 2017 (UTC)


 * This refutation has gone uncontested for over half a year, I think I'll go ahead and scrap the section as suggested2600:1015:B101:7442:806F:E0D6:B0A3:1A86 (talk) 23:09, 24 April 2018 (UTC)


 * I consider Huth pretty conclusive on the subject. Al-Biruni's method seems to have been genuinely new and is a clever idea, however failing to allow for atmospheric refraction makes it more or less unusable as a real life calculation. There's a fascinating paper by Alberto Gomez Gomez which discusses the maths (and errors) in great detail. Too much detail for this article, and it wouldn't be considered a reliable source anyway, but very interesting. There are so many problems with the calculation that any accuracy would be a fluke, but I think one of the most interesting parts is where Al-Biruni measures a value between 34 and 35 minutes of degree and decides to use a value of 34. The whole point of the method was to provide a relatively simple way to validate the previous measurement (which involved walking across deserts); he already knew roughly the answer he wanted, and there's a strong suggestion that he fitted his measurement to match the result he was expecting. Merlinme (talk) 22:03, 21 April 2020 (UTC)

Early modern period/Europe
It says: In 1505 the cosmographer and explorer Duarte Pacheco Pereira calculated the value of the degree of the meridian arc with a margin of error of only 4%, when the current error at the time varied between 7 and 15%. I have a hard time corroborating the source, because Pereira's work wasn't published until 1892! He kept it secret, because of the papal division of the world in the 15th century. Citing Pereira's work like this implies we had substantially good maritime navigation from 1505 forward, but we didn't. Then we skip forward 165 years to Picard's survey in 1670. Huh? Nothing happened for 165 years? What about Snell, Grimaldi and Riccioli? There's also Mouton in ~1670. It was the golden age of cartography, and we skipped right over it. Sbalfour (talk) 18:00, 15 January 2018 (UTC)

Stadia math?
the math in hellenistic section doesn't make sense

It says...

"Eratosthenes concluded that... 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."

If we use the value for a stadia (the only one )presented in the article, 185m, then 250,000 stadia equals 46,250 km or 28738 miles which is WAY out side the 0.4% error the article marvels at in the next paragraph. — Preceding unsigned comment added by 2602:301:777C:C750:59F9:A5AC:8DFA:3FFC (talk) 19:50, 17 June 2018 (UTC)

Proposal to merge Flat Earth

 * Opposed. There is too much material for a responsibly sized, single article. Flat Earth deserves a mention in History of geodesy, but the concept is mostly distinct; geodesy has little meaning on a flat earth model. It is just trigonometry at that point. Some of the duplicative material in History of geodesy should be deleted. Strebe (talk) 18:32, 13 July 2018 (UTC)
 * Strongly oppose. Strebe has nicely identified one of the problems. I would add that geodesy primarily concerns the modern mathematical description of the shape of earth, whose prehistory encounters speculations about the flat earth.  But the concept of the flat earth is a cosmological concept that includes non-mathematical — sometimes mythological — cosmologies, which don't fit well with discussions of geodesy. --SteveMcCluskey (talk) 22:30, 15 July 2018 (UTC)

Proposal to merge Spherical Earth

 * Opposed. There is too much material for a responsibly-sized, single article. Spherical Earth should be about the development of the concept of earth as a sphere, which it is. History of geodesy can include mention of that, but geodesy is much more about mensuration and refinement of the geoidal model than it is about the spherical earth. Strebe (talk) 20:29, 13 July 2018 (UTC)
 * Oppose. As Strebe has pointed out, the concept of Spherical Earth (like that of the Flat Earth) is too large to fit into this article and the concepts arising do not mesh well with the concepts of geodesy. My comments about the role of cosmologies in Flat Earth also apply here, although in the philosophical and religious senses of the term more than the mythological.
 * I would also note that this proposal was made by who subsequently made a series of major edits intended to shift the focus of the article on Spherical Earth.  Any such major change to Spherical Earth should be sorted out on that article's talk page before even considering moving the article into History of Geodesy. --SteveMcCluskey (talk) 22:44, 15 July 2018 (UTC)

The main problem I was trying to solve was that Flat Earth and Spherical Earth overlap too much, because "where and when people didn't believe the Earth is flat" is mostly the same subject as "where and when people believed the Earth is spherical". There is some parallelism because this article discusses the history of determinations of the size of the sphere, but you are both right that the other two articles don't overlap with this one enough to justify a merge. I'll see if I can find a better merge target title. Thanks for pondering this question! -- Beland (talk) 03:20, 16 July 2018 (UTC)

Proposal to split the section : 19th Century - Prime meridian and standard of length into another article titled Meridian arc of Delambre and Méchain

 * Opposed. The section is not about the meridian arc of Delambre and Méchain, which was measured in the 18th century, and is only briefly mentionned in this article in the section : Early modern period - Europe.

Moreover, in the section Prime meridian and standard of length several important works of 19th century geodesy are mentionned which justified the choice of the metre as international standard of length and that of Greenwich meridian as prime meridian. Splitting these informations prevents the reader to fully understand these choices. Charles Inigo (talk) 12:15, 28 August 2020 (UTC)

Eratosthenes' measure of Earth circumference
In this change, replaced the illustration File:Eratosthenes_measure_of_Earth_circumference.svg with File:Eratostene--Calcolo_Raggio_Terrestre.jpg.

Though I agree with the new caption noting that Syene and Alexandria are not on the same meridian, I believe that the original image is more accurate as the angle is to scale and being an SVG, it can be easily translated.

May I get a second opinion on this?

Thanks, cm&#610;&#671;ee&#9094;&#964;a&#671;&#954; 22:27, 26 August 2021 (UTC)
 * ✅ I've updated it as there has been no objection. cm&#610;&#671;ee&#9094;&#964;a&#671;&#954; 22:43, 19 April 2022 (UTC)

Archimedes & geographical mile
I was confused about the distances mentioned wrt Archimedes. Especially the “geographical mile”, which I had never heard of, as different from the “mile”. Why bring up this rare unit? Why not just use the “convert” template and 3,000,000 * 185 km = 555,000,000 meters? 555,000 km? TomS TDotO (talk) 03:49, 25 March 2024 (UTC)