Boundary retracement
Lost corner restored by single proportionate measurement
A section corner on a straight range line is lost between two existent original corners. Restore it by single proportionate measurement, working in coordinates and proving the record ratio is preserved.
Given
- A north-south range line two miles long, with existent original corners at both ends and the intermediate section corner lost. All four corner positions are wanted in US survey feet on a local plane grid; coordinates and distances are carried to 0.01 ft.
- South existent corner: N 482,610.44, E 1,927,845.20.
- North existent corner: N 493,176.03, E 1,927,912.86.
- Original survey record along the line: south corner to the lost corner, 79.86 chains; lost corner to the north corner, 80.12 chains. One chain is 66 ft, so the record total is 159.98 chains = 10,558.68 ft.
- A thorough search has produced no monument, no accessories and no acceptable collateral evidence at the intermediate corner, and no other original line or corner controls its position.
Required
- The measured distance and direction between the two existent corners.
- The proportionate distance from the south corner to the restored position.
- The coordinates of the restored corner, and proof that the restoration honours the record.
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
Establish that single proportion is the right method
A corner is lost only when its position cannot be recovered from evidence, and a corner that can be recovered from bearing trees, a stone hole, fence evidence or reliable local testimony is obliterated rather than lost. The distinction matters because an obliterated corner is found, not computed, and no amount of proportioning may override evidence of the original position. That distinction is drawn in the chapter of the BLM Manual of Surveying Instructions (2009) on the restoration of lost or obliterated corners.
Once a corner is genuinely lost, single proportionate measurement applies where the corner lies on a line between two existent corners and its position on that line is controlled in one direction only. A corner on a straight range line, controlled by the corners a mile either side of it, is the standard case. Single proportion distributes the whole discrepancy between the record and the measured length along the line in proportion to the record distances, which is the only distribution that gives every record call the same relative respect.
Measure between the existent corners
Inverse between the two existent corner coordinates. The line is very nearly meridional, so the departure is small, but it is not zero and must be carried; discarding it would put the restored corner more than a foot off the line.
The measured length exceeds the record by 7.13 ft in two miles, which is one part in 1,481. That is an unremarkable excess for original chaining and gives no reason to doubt either existent corner.
dN = 493,176.03 - 482,610.44 = +10,565.59 ft
dE = 1,927,912.86 - 1,927,845.20 = +67.66 ft
measured D = sqrt(10,565.59^2 + 67.66^2) = 10,565.81 ft
azimuth = 0°22′01″, that is N 0°22′01″ E
record total = 159.98 ch x 66 = 10,558.68 ft
excess = 10,565.81 - 10,558.68 = 7.13 ft, or 1 part in 1,481Compute the proportionate factor
The proportionate factor is the record distance to the lost corner divided by the record total. It is a pure ratio, so it can be computed in chains or in feet with the same result, and it is what gets applied to the measured quantities.
Because the line between the two existent corners is straight, proportioning the latitude and the departure separately gives exactly the same point as proportioning the distance along the line and then computing the components. Working with the components is simpler by hand and avoids any trigonometry beyond the inverse already taken.
record to the lost corner = 79.86 ch x 66 = 5,270.76 ft
record beyond it = 80.12 ch x 66 = 5,287.92 ft
factor = 5,270.76 / 10,558.68 = 0.4991874
proportioned distance along the line = 10,565.81 x 0.4991874 = 5,274.32 ft
remaining distance = 10,565.81 - 5,274.32 = 5,291.49 ftCoordinates of the restored corner
Apply the factor to the latitude and to the departure and add both to the south existent corner. The restored corner falls on the straight line between the two existent corners by construction, which is what single proportion is meant to produce.
N = 482,610.44 + 0.4991874 x 10,565.59 = 482,610.44 + 5,274.21 = 487,884.65
E = 1,927,845.20 + 0.4991874 x 67.66 = 1,927,845.20 + 33.78 = 1,927,878.98| Quantity | Total (ft) | x factor 0.4991874 | Restored value (ft) |
|---|---|---|---|
| Latitude (northing) | +10,565.59 | +5,274.21 | 482,610.44 + 5,274.21 = 487,884.65 |
| Departure (easting) | +67.66 | +33.78 | 1,927,845.20 + 33.78 = 1,927,878.98 |
| Distance along the line | 10,565.81 | 5,274.32 | - |
Prove the record ratio survived
The whole point of proportionate measurement is that the ratio of the restored distances equals the ratio of the record distances. Inverse from the rounded restored coordinates to each existent corner and form both ratios; they must agree to as many figures as the rounding supports.
They agree to six figures. The restored distances also sum to the measured distance, which confirms that the restored point is genuinely on the line and not merely near it.
south corner to restored = 5,274.32 ft = 79.914 ch
restored to north corner = 5,291.49 ft = 80.174 ch
sum = 10,565.81 ft, equal to the measured distance
restored ratio = 5,274.32 / 5,291.49 = 0.996755
record ratio = 5,270.76 / 5,287.92 = 0.996755 (agrees)What to set and what to record
Set a durable monument at the computed position, establish accessories, and record the search that established the corner as lost, because the restoration is only defensible if the record shows that evidence was genuinely exhausted first. The computation itself is trivial to redo; the field search is not.
Note also that the restored corner is a computed position, not a recovered one. If evidence of the original corner later comes to light within a plausible distance, that evidence governs and the proportioned point yields to it. The two figures are not competing measurements of the same thing: one is a monument and the other is an estimate of where a monument used to be.
Answer
- Measured distance between the existent corners = 10,565.81 ft on N 0°22′01″ E; record total 10,558.68 ft, an excess of 7.13 ft
- Proportionate factor = 0.4991874
- Proportioned distance from the south corner = 5,274.32 ft (79.914 chains)
- Restored corner: N 487,884.65, E 1,927,878.98
Check
The two restored distances, 5,274.32 and 5,291.49 ft, sum to 10,565.81 ft, exactly the measured distance between the existent corners, so the restored point lies on the line.
Restored ratio 0.996755 equals the record ratio 5,270.76 / 5,287.92 = 0.996755 to six figures.
Working the proportion from the north end instead gives 10,565.81 x (5,287.92 / 10,558.68) = 5,291.49 ft back from the north corner, which places the same point, N 487,884.65 E 1,927,878.98.
More boundary retracement
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