Boundary retracement
Lost interior section corner by double proportionate measurement
An interior section corner is lost with existent corners a mile or so in each of the four cardinal directions. Restore it by double proportionate measurement and verify that the record latitude and departure ratios are preserved.
Given
- An interior section corner, lost, with the nearest existent original corners recovered on the section lines in all four cardinal directions. Coordinates are in US survey feet on a local plane grid, carried to 0.01 ft.
- West existent corner: N 100,271.18, E 200,118.62.
- East existent corner: N 100,313.44, E 210,680.35.
- South existent corner: N 95,008.20, E 205,390.06.
- North existent corner: N 105,535.90, E 205,431.18.
- Original record east-west: west corner to the lost corner 80.00 chains, lost corner to the east corner 79.72 chains.
- Original record north-south: south corner to the lost corner 80.00 chains, lost corner to the north corner 79.44 chains.
- One chain is 66 ft. The corner is genuinely lost; no accessories, no stone hole and no acceptable collateral evidence were found.
Required
- The measured distance and direction on each of the two controlling lines.
- The proportion point on each line.
- The coordinates of the restored corner and the cardinal offsets from each proportion point.
Work it through yourself before reading on — the solution below shows every step, so there is no way to skim it without giving the answer away.
Worked solution
Why the corner needs two proportions, not one
An interior section corner is controlled in both directions. Its east-west position was fixed by the chaining along the latitudinal section line, and its north-south position by the chaining along the meridional line. Neither line alone determines it. Double proportionate measurement, described in the chapter of the BLM Manual of Surveying Instructions (2009) on the restoration of lost or obliterated corners, applies each proportion in the direction it actually controls and takes the corner at the intersection.
In practice that means: proportion between the east and west existent corners and keep the resulting easting; proportion between the north and south existent corners and keep the resulting northing. The restored corner is then the intersection of a meridian through the first point and a parallel through the second. The two proportion points will not coincide, and the small cardinal offsets between them and the restored corner are a normal and expected product of the method.
Measure the two controlling lines
Inverse across each pair of existent corners. Neither line is exactly cardinal; the east-west line bears N 89°46′15″ E and the north-south line bears N 0°13′26″ E, which is typical of original work and of the local convergence.
Both measured lines exceed their record. The east-west line runs 20.29 ft long over two miles, one part in 520; the north-south line runs 4.74 ft long, one part in 2,220. Neither is large enough to cast doubt on an existent corner.
west to east: dN = +42.26, dE = +10,561.73, D = 10,561.81 ft, N 89°46′15″ E
record east-west total = (80.00 + 79.72) ch x 66 = 159.72 x 66 = 10,541.52 ft
south to north: dN = +10,527.70, dE = +41.12, D = 10,527.78 ft, N 0°13′26″ E
record north-south total = (80.00 + 79.44) ch x 66 = 159.44 x 66 = 10,523.04 ftThe two proportionate factors
Each factor is the record distance from one existent corner to the lost corner divided by the record total on that line. They differ, because the record excess and deficiency were not distributed alike on the two lines: the east-west record is 0.28 chain short of two miles and the north-south record is 0.56 chain short.
east-west factor = (80.00 x 66) / 10,541.52 = 5,280.00 / 10,541.52 = 0.5008765
north-south factor = (80.00 x 66) / 10,523.04 = 5,280.00 / 10,523.04 = 0.5017561The two proportion points
Apply each factor along its own line. Only one coordinate of each proportion point survives into the answer: the easting from the east-west proportion and the northing from the north-south proportion.
east-west point: 10,561.81 x 0.5008765 = 5,290.17 ft from the west corner
N = 100,271.18 + 0.5008765 x 42.26 = 100,292.35
E = 200,118.62 + 0.5008765 x 10,561.73 = 205,408.74 (kept)
north-south point: 10,527.78 x 0.5017561 = 5,282.38 ft from the south corner
N = 95,008.20 + 0.5017561 x 10,527.70 = 100,290.54 (kept)
E = 205,390.06 + 0.5017561 x 41.12 = 205,410.69| Line | Measured D (ft) | Factor | Proportioned distance (ft) | Point N | Point E |
|---|---|---|---|---|---|
| West to east | 10,561.81 | 0.5008765 | 5,290.17 | 100,292.35 | 205,408.74 |
| South to north | 10,527.78 | 0.5017561 | 5,282.38 | 100,290.54 | 205,410.69 |
| Restored corner | - | - | - | 100,290.54 | 205,408.74 |
The restored corner and its cardinal offsets
Taking the northing from one proportion point and the easting from the other gives the restored corner. The offsets from each proportion point to the restored corner are cardinal by construction, and they are the figures a set of field notes should record so that anyone re-checking the work can see the geometry rather than only the answer.
Both offsets here are under two feet, which is what should be expected when the four existent corners are well recovered and the original chaining was reasonably consistent. Offsets of tens of feet are a signal to go back and question whether one of the four corners is really an original monument.
restored corner: N 100,290.54, E 205,408.74
offset from the east-west proportion point: 1.81 ft south (cardinal)
offset from the north-south proportion point: 1.95 ft west (cardinal)Verify against the record ratios
The correct check on a double proportion is not on the straight-line distances, which the cardinal offsets have disturbed, but on the components. The departure from the west corner to the restored corner, divided by the departure onward to the east corner, must reproduce the record east-west ratio; the same must hold for the latitudes on the north-south line.
Both ratios reproduce their record to six figures, which confirms that the proportions were applied to the right components and in the right direction. Comparing the straight-line distances instead would show small discrepancies that mean nothing, and chasing them is a common way to talk oneself out of a perfectly correct restoration.
departure west to corner = 5,290.12 ft = 80.153 ch
departure corner to east = 5,271.61 ft = 79.873 ch
ratio = 1.003511 ; record 80.00 / 79.72 = 1.003512 (agrees)
latitude south to corner = 5,282.34 ft = 80.035 ch
latitude corner to north = 5,245.36 ft = 79.475 ch
ratio = 1.007050 ; record 80.00 / 79.44 = 1.007049 (agrees)Answer
- East-west proportion point: N 100,292.35, E 205,408.74
- North-south proportion point: N 100,290.54, E 205,410.69
- Restored lost section corner: N 100,290.54, E 205,408.74
- Cardinal offsets: 1.81 ft south of the east-west proportion point and 1.95 ft west of the north-south proportion point
Check
Departure ratio from the restored corner, 5,290.12 to 5,271.61 = 1.003511, reproduces the record east-west ratio 80.00 to 79.72 = 1.003512.
Latitude ratio, 5,282.34 to 5,245.36 = 1.007050, reproduces the record north-south ratio 80.00 to 79.44 = 1.007049.
Component sums close: 5,290.12 + 5,271.61 = 10,561.73 ft, the full measured departure west to east; 5,282.34 + 5,245.36 = 10,527.70 ft, the full measured latitude south to north.
Both cardinal offsets are under 2 ft, consistent with four well-recovered existent corners; a much larger offset would indicate that one of the four controlling monuments should be re-examined.
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