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/ A discussion of refraction adjustments, being 1/6 of the curvature adjustment https://archive.org/stream/treatiseonsurve00gumm#page/242/mode/2up/ A definition of "levelling" The Theory and Practice of Surveying, Robert Gibson, 1814 https://archive.org/stream/theoryandpracti00gibsgoog#page/n293/mode/2up Actually from: A Treatise of Practical Surveying, Robert Gibson, 1777 1808 version on archive.org: https://archive.org/stream/atreatisepracti01gibsgoog#page/n283/mode/2up Unfortunately the plates were improperly scanned. What is being referred to is on the left here: 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/ 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 (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 Unfortunately again the figures were not well scanned, and this is all that remains of Fig 22.

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

Some interesting info on the history of geodetic refraction: https://books.google.com/books?id=FdzrCAAAQBAJ&pg=PA2 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 And very interested in the daily variations of refraction https://books.google.com/books?id=Of5YAAAAYAAJ&pg=PA323&lpg=PA323 The "Gaussian Refraction Coefficient" of 0.13 +/- 25% is about 1/7. So it seem like the 1/7 correction dates back to 1826.

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

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. The semi-automated scanning process obviously did not include unfolding the maps, so you get messed up scans.

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 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

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 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.

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.

Indeed: https://books.google.com/books?id=C...nepage&q="spherical excess" surveying&f=false A Treatise on Land Surveying and Levelling, Henry James Castle, 1845

Science and Industry, Volume 3, 1898 https://books.google.com/books?id=B...ge&q="true level" vs "apparent level"&f=false

Now here's a lovely book: https://books.google.com/books?id=S...s "apparent level"&pg=PP7#v=onepage&q&f=false The Complete Dictionary of Arts and Sciences. In which the Whole Circle of Human Learning is Explained, and the Difficulties Attending the Acquisition of Every Art, Whether Liberal Or Mechanical, are Removed, in the Most Easy and Familiar Manner - 1765 A discussion of the horizon, including accounting for refraction. And an extensive discussion of leveling https://books.google.com/books?id=SsEtm9CNCc0C&pg=PA6#v=onepage&q&f=false

The excellent site FlatEarth.ws has a small collection of books specifically on Railways, which the Flat Earth folk sometimes claim are built without "accounting" for the curve. Of course railway construction uses leveling even more so than road construction, because the exact grade is very important to trains, which dislike even the slightest hill. https://flatearth.ws/railroads Here's the two historical books: Field-book for Railroad Engineers – John B. Henck, 1869 A Treatise on the Principles and Practice of Levelling: Showing Its Application to Purposes of Railway Engineering and the Construction of Roads – Frederic Walter Simms — 1875 https://play.google.com/books/reader?id=beFMAAAAYAAJ&hl=en&pg=GBS.PA4 Note the use of 8 inches per mile squared as an approximation, here given in feet, since 8 inches = 8/12 = 2/3 of a foot.