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The GPS Observer
James P. Reilly Ph.D.
State lines and how they were surveyed.
Figure 1. Astronomic latitude and longitude defined.
The original territory of the United States, as defined by the treaties with Great Britain of November 30, 1782, and September 3, 1783, was bounded on the north by Canada, on the south by the Spanish Colonies of East and West Florida, on the east by the Atlantic Ocean, and on the west by the Mississippi River. Additions to the original territory, the acquisition of Louisiana from France in 1803 and of Florida from Spain in 1819, the Texas annexation in 1845, the establishment of American title to Oregon territory in 1846, the Mexican cession of 1848 and the Gadsden Purchase from Mexico in 1853, brought the aggregate area of the conterminous United States within its present boundaries.
The 13 original states organized the Federal Union under the name of "The United States of America" by ratifying the Articles of Confederation and the Constitution. The boundaries of these states were not defined by the Acts of Ratification, but, in general, the states maintained their claim to their colonial boundaries, which had been established by royal decree or by agreement. Some overlapping territorial claims were not settled until many years later by Supreme Court decision. Other states were admitted into the Union by Acts of Congress, usually upon petition of the citizens residing in the territories in question. The boundaries of these states were defined by enabling acts. The boundaries of Missouri and Texas were changed by subsequent legislation.
Figure 2. The deflection of the vertical.
How are boundaries established? Three of the most common means of defining a boundary are:
By a water boundary, such as a stream, lake or other body of water;
By a divide between two drainage basins; or
By a meridian of longitude or a parallel of latitude.
Figure 3. Geodetic latitude and longitude defined.
Let's discuss the third method. When a parallel of latitude is specified, the numbering always begins with zero at the equator. Today, the numbering of the meridians ranges from zero to the east and west of Greenwich. This has not always been so; in many countries of the world the initial meridian is the one passing through the capital. In the United States, the Washington meridian was used as a point of departure in much of the 19th century.
The act of September 28, 1850, stipulated "That hereafter the meridian of the observatory at Washington shall be adopted and used for all astronomic purposes and…that the meridian of Greenwich shall be adopted for all nautical purposes." The act was repealed August 22, 1912 and the meridian of Greenwich was adopted for both astronomic and nautical purposes.
The Washington meridian passed through the center of the dome of the old Naval Observatory at 24th Street and Constitution Avenue, N.W. in Washington, D.C. It is 77° 03' 02.3" west of Greenwich. This longitude was derived by astronomic observations. With reference to the North American Datum of 1927, the geodetic latitude is 3.8 seconds of arc greater. Just to show how important this is, the meridian boundaries of the territories of Arizona, Colorado, Dakota, Idaho, Montana, Nevada and Wyoming, and the states of Kansas, Nebraska, New Mexico and Utah were referenced to the Washington meridian.
Figure 4. Go to the NGS website and click on DEFLEC99 from the Geodetic Tool Kit page.
I could continue on, but it's obvious that state lines that are parallels of latitude or meridians of longitude were established as astronomic parallels and meridians. Because of the lower order of accuracy in early astronomic instruments, many state lines are not on a true meridian or parallel. In surveying, until recent times, we used astronomic positioning and astronomic azimuth in geodetic triangulation to determine the deflection of the vertical. With GPS, this is not necessary unless you want to survey a state line.
What is the difference between an astronomic latitude and longitude, and a geodetic latitude and longitude for a point on the ground? Figures 1 through 3 show this relationship. Figure 1 shows how astronomic latitude, F, and longitude, L, are defined. No ellipsoid is involved; F and L are two of the natural coordinates of the Earth's gravity field. When we introduce the ellipsoid needed for geodetic coordinates, we have the situation described in Figure 2. The gravity vector is perpendicular to the geoid and the normal is perpendicular to the ellipsoid. The angle between the two is the deflection of the vertical, defined as d. Figure 3 shows the definition of geodetic latitude, j, and geodetic longitude, l; they are referenced to the ellipsoid.
Figure 5. Enter the data for DEFLEC99.
Given an astronomic latitude and longitude of a monument on a state boundary, how do you determine the geodetic latitude and longitude? For an existing monument, you can position a carrier-phase GPS receiver over the monument and use relative positioning from a nearby HARN or CORS station, or have the U.S. National Geodetic Survey determine the geodetic positioning using OPUS. If the monument is missing, but you know the astronomic position, you can calculate the geodetic position as follows:
In Figure 2, d is the deflection of the vertical. Rather than use d, we must have the two components of d, referred to as x (Xi) and h (Eta). The following equations express the components x and h of the deflection of the vertical in terms of geographic (astronomical and geodetic) coordinates.
(1) x = F – f
(2) h = (L – l) cos j
Now we can answer the question from earlier. Given an astronomic position, we can calculate the geodetic position if we know x and h by rearranging equations 1 and 2, as follows:
(3) f = F – x
(4) l = L – h
cos j
The two components of the deflection of the vertical can be calculated (computed) by going to the NGS website at www.ngs.noaa.gov, clicking on the Geodetic Tool Kit and opening the program DEFLEC99. The following example shows this calculation.
Figure 6. Enter the latitude and longitude for the example.
For this example, let's use the border between Maryland and Pennsylvania. A portion of this line is the famous Mason and Dixon line.
Mason and Dixon determined the latitude of this line to be 39° 43' 17.6" N. They began their work in 1763 but were stopped by Indians in 1767, after having run about 244 miles. For our example, let's calculate the geodetic position on the North American Datum of 1983 (NAD 83) for the following astronomic position.
F = N 39° 43' 17.6"
L = W 77° 00' 00"
The figures show the procedure. What is needed is x (Xi) and h (Eta) for that position.
The solution, as shown in Figure 7, is Xi = 8.30", Eta = –5.55"
Note: In practice, the argument in DEFLEC99 is geodetic latitude and longitude, not astronomic latitude and longitude.
We can now compute the geodetic position. Using equation 3,
j= F – x
j = (39° 43' 17.6") – 8.30"
j = N 39° 43' 09.3"
We can now use the computed value of j in equation 4,
l = L – h
&nbspcos j
l = (77° 00' 00") – (–7.21")
l = 77° 00' 7.21"
If we assume that 1 second of arc is 30 meters in the meridian, and you mistakenly entered the astronomic position in your GPS receiver, you would be about 1,000 feet away from the correct position. But if you use the geodetic position computed as we did above, you will be at the correct position within the tolerances of the original survey.
This article is not telling you how to survey a state line. Proper boundary surveying techniques must be used, and you survey from monument to monument.
Figure 7. The solution to the deflection of the vertical.
My reference material for the historical information came from a valuable reference book: "Boundaries of the United States and the Several States," U.S. Geological Survey, Geological Survey Professional Paper 909, 1976.
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James P. Reilly, PhD., is a past president of ACSM and retired department head of the Department of Surveying Engineering at New Mexico State University.
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