1. Mick West

    Mick West Administrator Staff Member

    Given that some people claim the curvature of the Earth is not accounted for in surveying, I thought it would be interesting to document various historical instances of this, and of accounting for refraction.

    Curvature comes up in two ways. Firstly it arises when determining latitude and longitude by observation of sun and stars. The Earth here is assumed to be a sphere, and the stars essentially fixed in a "celestial sphere" infinitely far away.

    Secondly it comes up in the practice of levelling which is the art of determining how high things are above a nominal level surface, such as the surface of the sea. Interestingly this use of the word "level" is often misinterpreted as meaning "flat", when the books on surveying make quite clear this is not the case.

    Update: two additional ways in which curvature comes up in surveying, mentioned by @Jon Leighton

    Spherical Excess In general, the internal angles of any large surveyed triangle will sum to more than 180 degrees.

    Distance corrections including the chord to arc correction, and its inverse, the arc to chord correction.

    A treatise on surveying, containing the theory and practice: , John Gummere, 1853, page 239
    https://archive.org/stream/treatiseonsurve00gumm#page/238/mode/2up/
    20170623-140537-99xh8-1.

    A discussion of refraction adjustments, being 1/6 of the curvature adjustment
    https://archive.org/stream/treatiseonsurve00gumm#page/242/mode/2up/
    20170623-141102-8mjek.



    A definition of "levelling"

    The Theory and Practice of Surveying, Robert Gibson, 1814
    https://archive.org/stream/theoryandpracti00gibsgoog#page/n293/mode/2up
    20170623-133957-tw6f8.

    Actually from:
    A Treatise of Practical Surveying, Robert Gibson, 1777
    1808 version on archive.org:
    https://archive.org/stream/atreatisepracti01gibsgoog#page/n283/mode/2up
    20170623-134651-ljhpc.

    Unfortunately the plates were improperly scanned. What is being referred to is on the left here:

    20170623-134823-a3xcu.



    1789 Version does not contain the section on levelling, but does mention the globe and celestial poles
    https://archive.org/stream/treatiseofpracti00gibs#page/242/mode/2up/
    20170623-135706-4shqo.

    A similar figure (likely derived from the Gibson work) is found in:
    A treatise on surveying and navigation : uniting the theoretical, practical, and educational features of these subjects, by Robinson, Horatio N, 1858
    https://archive.org/stream/treatiseonsurvey00robirich#page/n157/mode/2up
    20170623-141513-idadg.
    (again, note the definition of a level surface)


    Slightly off topic, but in this older book, A treatise of practical geometry, published in 1745 (from a 1695 college book written in Latin), we have a discussion on the size of the earth:


    https://archive.org/stream/atreatisepracti00greggoog#page/n51/mode/2up
    20170623-142639-163yd.
    Unfortunately again the figures were not well scanned, and this is all that remains of Fig 22. 20170623-142944-4a1en.
     
    Last edited: Jul 11, 2017
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  2. Mick West

    Mick West Administrator Staff Member

    Back to levelling, the correction that is generally being referred to is a correction the value of what we call "drop" - i.e. the amount the curve of the earth falls below a plane tangent to it at the view point. Often we use "8 inches per mile squared", as that's the expected value without refraction. Not very useful for surveyors though, as the atmopshere gets in the way. So they use a different value, for example a correction of 1/6 the drop seen above, or here:

    The Principles and Practice of Surveying: Higher surveying, Charles Breed, 1908
    https://books.google.com/books?id=T2o7AAAAMAAJ&pg=PA212#v=onepage&q&f=false
    20170623-144634-ajvgq.
    20170623-144923-mgxi0.

    0.57 feet per mile squared squared is 6.84" instead of 8".
     
  3. Mick West

    Mick West Administrator Staff Member

    Some interesting info on the history of geodetic refraction:

    https://books.google.com/books?id=FdzrCAAAQBAJ&pg=PA2
    20170626-130708-nyd1b.

    Picard seems to crop up more in discussion of astronomy, but he was also a geodetic surveyor.
    https://books.google.com/books?id=eiQOqS-Q6EkC&pg=PA57&lpg=PA57
    20170626-131101-19ljb.

    And very interested in the daily variations of refraction
    https://books.google.com/books?id=Of5YAAAAYAAJ&pg=PA323&lpg=PA323
    20170626-131409-4d48j.

    The "Gaussian Refraction Coefficient" of 0.13 +/- 25% is about 1/7. So it seem like the 1/7 correction dates back to 1826.
     
  4. Mick West

    Mick West Administrator Staff Member

    I tried to buy a copy of this and serendipitous bought a different book: A Treatise on Surveying - Part I, by Middleton and Chadwick, 1899. It's an excellent book for describing how surveyors account for both curvature and refraction with all kinds of examples and discussions
    20170706-173438-321j3.
    20170706-173535-nqp9v.

    A slightly later (1904) version can be found here:
    https://archive.org/stream/atreatiseonsurv00boglgoog#page/n261/mode/2up
     
  5. Mick West

    Mick West Administrator Staff Member

    I also discovered why many of the diagrams in the scanned version of this book, and the other books above, have truncated diagrams. It seems common that surveying books of the era were normal hardback size, but with quite large fold out segments.

    20170706-174350-bs0iu.
    20170706-174405-dofwo.

    The semi-automated scanning process obviously did not include unfolding the maps, so you get messed up scans.
     
    Last edited: Jul 6, 2017
  6. Mick West

    Mick West Administrator Staff Member

    This section is particularly relevant to most of the surveying related Flat Earth objections.
    20170706-175839-ouqcv.
    20170706-175904-7gi8n.
     
  7. Mick West

    Mick West Administrator Staff Member

    And of course "history" extends up to the present day. I don't want people to get the idea that this is some archaic technique. The shape of the Earth has not significantly changed in the last 200 years, nor has atmospheric refraction. So the same numbers still apply.

    Here's a course in surveying leveling from Fresno State Lyles College of Engineering:

    Source: https://www.youtube.com/watch?v=kKcafQ-WxnY


    20170708-095154-3wwrm.

    The above slide shows the combined effects of curvature and refraction, at 0.574 feet per mile squared, or 6.89 inches per mile squared.

    That's the same correction as given in 1908
    The Principles and Practice of Surveying: Higher surveying, Charles Breed, 1908
    https://books.google.com/books?id=T2o7AAAAMAAJ&pg=PA212#v=onepage&q&f=false
    20170708-095453-9ga4q.
     
    Last edited: Jul 8, 2017
  8. Mick West

    Mick West Administrator Staff Member

    But back to history, in "A Text-book of Plane Surveying", Raymond, 1901, we have:
    https://books.google.com/books?id=R...epage&q=curvature refraction leveling&f=false
    20170708-100434-e862d.

    Again the 1/7th correction. And an odd way of measuring the effect of curvature. 0.001 feet in 220 feet. So essentially .001 feet per (miles*24)^2 (as 220 feet is 1/24th of a mile). That works out as .001*24^2 = 0.576 feet per mile squared. The same as the other references.
     
  9. Jon Leighton

    Jon Leighton New Member

    Mick,

    I think spherical excess is also worthy of a mention. In general, the internal angles of any large surveyed triangle will sum to more than 180 degrees.

    Then there's distance corrections including the chord to arc correction, and its inverse, the arc to chord correction.
     
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  10. Mick West

    Mick West Administrator Staff Member

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    Mick West Administrator Staff Member

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    Mick West Administrator Staff Member