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Boundary retracement

Record chains reconciled against a measured distance

Three record calls in chains between two existent monuments do not sum to the measured distance. Distribute the difference proportionately and report the restored positions in both feet and chains.

Recall· about 15 minutes by hand· 5 steps

Given

  • A retracement along a straight section line between two existent original corners, one mile apart by record.
  • Record calls from the original survey, running from the south corner: to a marked stone, 22.50 chains; thence to the center of a creek, 31.20 chains; thence to the north corner, 26.30 chains. The record total is 80.00 chains.
  • One chain is 66 ft, so the record total is 5,280.00 ft.
  • The measured distance between the two existent corners is 5,296.70 ft, in US survey feet, to 0.01 ft.
  • Neither the stone nor any evidence at the creek call was recovered, and the intervening points are to be restored by proportion. Distances are reported to 0.01 ft.

Required

  • The excess between the measured and the record distance, and its relative size.
  • The proportionate factor and the restored distance to each record point.
  • The restored distances expressed in chains.

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

Compare record with measured

Convert the record to feet before comparing anything. The excess is the measured distance less the record distance, and it should always be expressed as a relative figure as well as an absolute one, because a foot of excess in a chain is a blunder and a foot of excess in a mile is ordinary.

Here the excess is 16.70 ft in 5,280.00 ft, which is one part in 316. That is a plausible figure for a line originally chained with a Gunter's chain over uneven ground, and it gives no reason to suspect either existent corner. An excess of one part in twenty would be a different conversation entirely and would send the crew back to look for a different monument.

record total = 80.00 ch x 66 = 5,280.00 ft measured = 5,296.70 ft = 80.253 ch excess = 5,296.70 - 5,280.00 = +16.70 ft relative excess = 16.70 / 5,280.00 = 1 part in 316 excess per record chain = 16.70 / 80.00 = 0.2087 ft

The proportionate factor

Distributing the difference in proportion to the record lengths is the same operation as scaling every record call by a single factor: the measured distance divided by the record distance. Carrying the factor to eight figures is not excessive, because the longest course is over 2,000 ft and a factor good to only five figures would leave a hundredth of a foot of avoidable error in it.

This is the same reasoning that governs proportionate measurement between existent corners: no record call is favoured over another, and each absorbs the discrepancy in proportion to what it claimed.

k = 5,296.70 / 5,280.00 = 1.00316288

Distribute the excess

Multiply each record distance in feet by the factor. The correction to each course is that course's share of the 16.70 ft excess, and the corrections should be recorded alongside the results so that a reviewer can see the distribution rather than infer it.

1,485.00 x 1.00316288 = 1,489.6969 -> 1,489.70 ft 2,059.20 x 1.00316288 = 2,065.7130 -> 2,065.71 ft 1,735.80 x 1.00316288 = 1,741.2901 -> 1,741.29 ft
Record calls scaled to the measured distance
CourseRecord (ch)Record (ft)Correction (ft)Restored (ft)Restored (ch)
South corner to stone22.501,485.00+4.701,489.7022.571
Stone to creek31.202,059.20+6.512,065.7131.299
Creek to north corner26.301,735.80+5.491,741.2926.383
Totals80.005,280.00+16.705,296.7080.253

Restored distances from the south corner

Accumulating the restored courses gives the distance from the south existent corner to each restored point, which is what a crew actually needs to go and set them. Reporting the same figures in chains as well as feet keeps the retracement legible against the original notes, which were written in chains.

south corner to the stone position = 1,489.70 ft = 22.571 ch south corner to the creek position = 1,489.70 + 2,065.71 = 3,555.41 ft = 53.870 ch south corner to the north corner = 3,555.41 + 1,741.29 = 5,296.70 ft = 80.253 ch

What the distribution assumes, and when it is wrong

Proportioning assumes that the original chaining error accumulated uniformly along the line, which is a reasonable default and nothing more. It is the right treatment when nothing is recovered between the two existent corners, as here.

It is the wrong treatment the moment evidence turns up. If the marked stone were found, the stone controls its own position absolutely and the line is then treated as two separate problems: the record calls between the south corner and the stone are reconciled against the measured distance to the stone, and the calls beyond it against the remaining measured distance. A natural monument such as the center of a creek controls even more strongly, since a creek is where it is regardless of how the original chain behaved. Distributing a discrepancy across a recovered monument is one of the few ways to make a retracement worse than doing nothing.

Answer

  • Excess of measured over record = 16.70 ft in 5,280.00 ft, one part in 316
  • Proportionate factor k = 1.00316288
  • Restored courses: 1,489.70 ft (22.571 ch), 2,065.71 ft (31.299 ch), 1,741.29 ft (26.383 ch)
  • Restored positions from the south corner: stone at 1,489.70 ft, creek at 3,555.41 ft, north corner at 5,296.70 ft

Check

The three restored courses sum to 1,489.70 + 2,065.71 + 1,741.29 = 5,296.70 ft, exactly the measured distance, with no rounding residual to absorb.

The three corrections sum to 4.70 + 6.51 + 5.49 = 16.70 ft, the full excess.

Ratio check: 1,489.70 / 2,065.71 = 0.72115 and the record ratio 22.50 / 31.20 = 0.72115, so the record proportions survive the distribution unchanged.

Chains check: 80.253 measured chains x 66 = 5,296.70 ft, confirming the conversion back and forth.

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