here's an early run (circa 1830) of a plane coordinate system for Massachusetts. It divides the state into five regions, develops its own ellipsoid from observations, measurements and mathematics, and provides fairly detailed instructions for its use. You get 20 gold stars in the great self-study grade book in the sky if you can relate the length of a degree in a meridian and the length of a degree perpendicular to a meridian to two of James Clynch's radii.
Go to the tables for Section I first, Tables 1 - 4, to get the gist of the effort. I used the Pythagorean Theorem to verify some of the grid distances. The published distances are "close," but the coordinates were calculated as averages, so calculated distances are going to be off. There are no scale or elevation factors, so everything stays as grid. The text is not for everybody, probably boring for most, but just skimming it can turn up some interesting tidbits.
A bit more background information, including elevations, is here: https://www.jstor.org/stable/1005341?seq=1#metadata_info_tab_contents ?ÿ , but the fact that a run at the equivalent of a State Plane Coordinate System was done so early is the main takeaway.
The south end of the base line has an NGS data sheet, pid = MZ1670. The north end mark is pid MZ1622, but it has apparently been destroyed.
WOW! Very cool.
I hope to have some time this winter to dig into this puppy.
Thanks,
Loyal
The Borden survey was very cool stuff. In 2002, a friend and I recovered one of its stations, which is also an NGS station. The data sheet says that the station was monumented by the Survey of the Coast, but I believe it actually was monumented by Borden or his crews. Borden did use X-scribed copper bolts as monuments.
It was ahead of its time. Here are a couple of quotes. The first is from the "Tables of Bearings, ..." and the second is from the "Account of a Trigonometrical..."
Generally, by awakening in some, and keeping alive in others, a
fondness for exactness in their field surveys and calculations, and thus
leading to that reform in the present method of land surveying, which the
rapidly increasing value of real estate in the greater part of the State,
loudly calls for.
The state of Massachusetts is so densely settled, that we could without much
inconvenience find lodgings with the inhabitants near almost all of our stations; so that
we did not provide ourselves with camp equipage.
Things haven't changed much, huh?
To: MathTeacher?ÿ?ÿ?ÿ your request (see your 3 line down) about James R. Clynch's radii is very Trivial.
For you it would be better to read Map Projection Equations?ÿ by ?ÿFrederick Pearson II. You will want to
read chapter 3 for sure and see equations ?ÿ(3.3.1) and (3.3.3) on page 75 and finish the chapter.
?ÿ
JOHN NOLTON
TO: MathTeacher, I forgot to put this in my post above. ?ÿJ.R. Clynch's 6 page paper (in my opinion)
is not that good. After you read Pearson's work you can write a small paper to post here and explain
all.
?ÿ
JOHN NOLTON
TO: MathTeacher:?ÿ I kept thinking I had another book by Frederick Pearson II but could not find it
on my bookshelf. So after dinner I came back and looked and found it "Out of place".
Map Projections: Theory and Applications?ÿ by Pearson
Chapter 3 will be the same chapter to read but the equations will be?ÿ numbered differently. Read the whole book!!
?ÿ
JOHN NOLTON
?ÿ
?ÿ
?ÿ
?ÿ
?ÿ
If you look carefully at pages 75 and 76 in the 1990 edition of Pearson's "Map Projections Theory and Applications," you will find that formulas 21 and 28 are the same as two of Clynch's formulas.
Now you should read Borden's "Tables..." paper and write down the length of a degree in the meridian and the length of a degree perpendicular to the meridian at the latitude of the State House found there. Then use the polar and equatorial axes published in Borden's paper, along with the two formulas from either Clynch or Pearson, to calculate those two lengths of a degree in perpendicular directions. This little exercise is good for two purposes: 1) to check Borden's work, and 2) to further our own understanding.
To get a another perspective, compare Borden's lat/lon for Alander Mountain to the modern-day values given for NGS PID MZ2084, Mount Lincoln to MZ1615, and Becket Station to MZ2000. The ellipsoids are different and there's no guarantee that everything has remained in place for almost 200 years, but....
I see how Borden used Paine's astronomical measurements and his own ground measurements to calculate the length of the meridional degree, but how he got the perpendicular value is beyond my understanding. That is a surveying question and I'm pretty bad at both comprehending and answering those.
It's amazing to see how geodesy was applied in 1830 and how much knowledge those guys left for us to use today.
In the years between 1830-1850 Geodesy was in its infancy. I think the only way to compare the old positions with the new
is to get from NGS as many points in Mass. on the U.S. Standard Datum and compare them with the work of John G. Palfrey
which you gave reference to. This comparison would also have to have another calculation done and that would be "change of ellipsoid"
because Bessel's ellipsoid was not used in this survey.
See: 1. Transformation of coordinates between geodetic systems, Survey Review #137, pp. 128-133 (1965) By T. Vencenty
2. Transformation of geodetic data between reference ellipsoids, Journal of Geophysical Research, 71(10), pp. 2619-2624 (1966)
by T. Vencenty
You can find this information in Geodesy by Bonford, 3rd edition, 1971 pp. 201-204
JOHN NOLTON
Note: since Bomford was from the UK he uses the word "Spheroid". So on page 201, under 2.54; "Change of spheroid via
Cartesian coordinates"
Geodesy being so young in 1830 is one of the things that makes this survey so remarkable. Another is its anticipation of what is now State Plane more than a hundred years in advance. Although maps were common, accurate ones were not so common. The concept of a full-sized flat map on the scale of the whole earth and then deriving coordinates from measurement instead of by formulas was one heckuva leap.
While it might be nice to transform the original coordinates to a modern ellipsoid and map projection, my goal has been to understand the original coordinate determination and get some feel for how accurate it was. My first step was to see how well the ellipsoid Borden defined fits his measurements.
Borden computed the ellipsoid's axes from his determination of the lengths of a meridional degree and a degree in the plane perpendicular to the meridian at the latitude of his cardinal point, 42° 21' 30" North. These lengths are:
Meridional degree = 364,356 feet
Perpendicular degree = 365,511 feet
From these lengths, his values for the equatorial and polar radii are:
Equatorial radius = a = 20,914,728 feet
Polar radius = b = 20,854,128 feet
To check the internal consistency of Borden's ellipsoid with his feet/degree determinations, we can use Clynch's RN (Pearson's RP) to calculate the perpendicular value and RM (same for Pearson and Clynch) to calculate the meridional value. First, though, we need e^2 for the ellipsoid, which is e^2 = 1 - b^2/a^2 = 0.005786564.
For the meridional degree, RM/(2pi) = 364,353.3 feet and for the perpendicular degree, RN/(2pi)= 365,511.3. The perpendicular degree agrees, but the meridional degree is different by 3 feet. There were some problems with the latitude of the State House, so it's not unexpected that there might be a discrepancy here.
Anyway, retracing this work through the papers of Borden and Paine is an engaging exercise. Since so much of it was based on measurements, I think that a good understanding of surveying would help a great deal, but that's something that I don't have.
Thanks for your valuable comments, John, and for the additional references.
A quick update. Borden revised his ellipsoid in the paper published in the American Philosophical Society's transactions in 1846. Apparently, Superintendent Hassler, the first superintendent of the Coast Survey, had several criticisms of the original work. One of those was his perceived lack of agreement between Borden's ellipsoid and Bessel's 1841 ellipsoid.?ÿ
Here is a comparison of Borden's published values and the same values calculated from the formulas for radii in the plane of the meridian and in the plane perpendicular to the meridian. These formulas would have been available to Borden, but he did not share the details of all of his calculations. Note how the distances per degree changed and how much better those values fit the revised ellipsoid.
For a little modern-day touch, here's a calculation from NGS' INVERSE that uses Borden's revised ellipsoid to calculate the distance between a point that is one degree south of the State House and another that is one degree north of the State House, both on the meridian of the State House. It covers two degrees along the meridian, so divide the distance by 2 and then convert it to survey feet. Compare the answer to the values for the length of a meridional degree in the table above.
Borden was subsequently inducted into the society. Hassler died in 1843 and Bessel died in 1846.
I would argue (pedantically) that Geodesy in the 1800s was hardly a ??young science.? As a science centered on the ??size and shape of the Earth? recollect the efforts of the ??father of geodesy? Eratosthenes (240 BCE), Snellius (aka Snell) early 1615, Jean-Felix Picard (1669), the Cassini expedition to modern-day Ecuador (1736), and others who sought to measure the size and shape of the Earth. ?ÿ
In fact the surveys of Picard and the Cassini contributed to Newton??s theory of gravity as well as answering whether the Earth??s shape was oblate or prolate . Lots of interesting history. ?ÿ
?ÿ
?ÿ
I am referring to geodesy in the USA (which I should have put in just for you). Here in the US it was not till Feb. 10, 1807 that the Coast Survey
came into being (on paper, because the Govt. moves slow). The President at the time Thomas Jefferson picked Ferdinand R. Hassler.
In 1811 Hassler went overseas to get the equipment he needed. He did not return till 1815. In 1816 Hassler began geodetic work in the vicinity
of New York. In 1818 by act of Congress The Survey of the Coast was transferred from the Treasury Dept. to the Navy. Hardly any Geodetic work was done after that till ""1832"" when Congress restored the Act of 1807, returning the Survey of the Coast to the Treasury Dept.. Hassler was named again to head the Survey. HISTORY!!
JOHN NOLTON
As I indicated, I was merely quibbling. That said, those interested in the history of geodetic surveys by the USC&GS should enjoy Joseph Dracup’s history of the period 1807 to 1940 available here: https://geodesy.noaa.gov/library/pdfs/geodetic-surveys-in-us-beginning-and-next-100-years.pdf
A good bit of the math was in place, for sure, though some of it lacked proof. Here's a timeline that might be used to judge the pace of discovery.
http://www.in-dubio-pro-geo.de/?file=register/mstones&english=1
Note that at year 150 they said latitude when they must have meant longitude.
At 1342, do they mean Jacob's staff was some form of measurement device, rather than being the support for an instrument?
I think there are a couple achievements that are credited twice.?ÿ For instance, see rotation of the earth by pendulum 1661 and 1851.
I think longitude is right. This one gives insight into the Jacob's staff. http://www.surveyhistory.org/jaco b's_staff1.htm
The 1851 reference to Foucault is correct, Viviani worked with pendulums, but I'm not sure about his connecting them to earth rotation.
Lists like this are best used as starting points. Thanks for pointing those things out.
I think longitude is right. This one gives insight into the Jacob's staff. http://www.surveyhistory.org/jaco b's_staff1.htm ?ÿ
You learn something every day (if you aren't careful).
