When a corner is genuinely lost, meaning its position cannot be determined from any surviving evidence, it has to be restored by computation from the corners that still exist. Single proportionate measurement is the method used when the lost corner's position was controlled in one direction only: along a single line, between two identified corners on that line.
The idea is simple and deliberately impartial. The original surveyor's measurement between the controlling corners disagrees with yours by some amount. Single proportion assumes that disagreement accumulated uniformly along the line, and distributes it among the intervals in proportion to their record lengths. It does not favour either end, it does not depend on which direction you measured, and two competent surveyors applying it to the same evidence get the same answer.
When single proportion applies
- Single proportionate measurement
- The restoration of a lost corner by distributing the difference between the record and the measured length of a line, between two identified existing corners on that line, in proportion to the record distances of the intervals along it.
- Lost quarter-section corners on a line between two identified section corners.
- Lost corners on a base line, standard parallel, principal meridian or guide meridian, restored along that controlling line and never across it.
- Lost closing corners, restored on the governing line, since their position was fixed by the intersection rather than by distance from the corners beside them.
- Lost meander corners on a section or township line, restored along that line between identified corners.
- Lost corners on a township exterior where control exists in one direction only, such as on the boundary of a survey where nothing lies beyond.
- Lost quarter-quarter corners and other subdivisional corners on a line, once the controlling corners of that line have been established.
The unifying test is directional control. If the lost corner's position depended on measurement along one line and on nothing else, single proportion is the method. If its position was fixed by the intersection of two lines running in different directions, as an interior section corner is, then double proportionate measurement is required instead.
The computation
d′ = d × (M ÷ R)Two things about that expression are worth stating plainly. First, the ratio M ÷ R applies to every interval on the line, so the restored positions of several lost corners on one line are computed with the same factor and always sum correctly. Second, d is the record distance, not the nominal distance. Where the record itself is irregular, as in the north tier of a township, the proportion honours the irregular record rather than imposing a regular one.
Worked example: a lost quarter corner on a regular line
Two section corners on a north-south section line are recovered and identified as original. The record distance between them is 80.00 chains, with the quarter corner recorded at 40.00 chains from the south corner. The remeasured distance between the recovered corners is 79.60 chains.
| Quantity | Value |
|---|---|
| Record distance, south corner to north corner (R) | 80.00 ch |
| Record distance, south corner to quarter corner (d) | 40.00 ch |
| Measured distance, south corner to north corner (M) | 79.60 ch |
| Proportion factor M ÷ R | 0.9950 |
| Proportional distance d′ = 40.00 × 0.9950 | 39.80 ch |
| Remaining distance, quarter corner to north corner | 39.80 ch |
The deficiency of 0.40 chains is split evenly because the two record intervals were equal. Nothing about the result depends on which end you started from, which is the property that makes the method reproducible.
Worked example: a lost quarter corner on an irregular line
The second example matters more, because it is where people go wrong. Take a section in the north tier of a township, where the excess and deficiency of the township was placed. The record shows 40.00 chains from the south section corner to the quarter corner, and 39.14 chains from the quarter corner to the north section corner on the township boundary, a record total of 79.14 chains. The line remeasures 79.30 chains.
| Quantity | Value |
|---|---|
| Record distance, south corner to north corner (R) | 79.14 ch |
| Record distance, south corner to quarter corner (d) | 40.00 ch |
| Measured distance, south corner to north corner (M) | 79.30 ch |
| Proportion factor M ÷ R | 1.0020 |
| Proportional distance, south corner to quarter corner | 40.08 ch |
| Proportional distance, quarter corner to north corner | 39.22 ch |
The two proportional distances sum to 79.30 chains, matching the measurement. Notice that the quarter corner is not placed at the midpoint. The record put it at 40.00 chains from the south, deliberately, because the deficiency of the township belonged in the north half-mile. Single proportion preserves that intent. A surveyor who reflexively halves the measured distance would move the corner nearly half a chain and destroy the design of the original survey.
Alignment, not just distance
Distance is only half of the restoration. The restored corner must also lie on the correct line. For a lost quarter corner between two section corners, the corner is placed on the straight line joining the two recovered corners, at the proportional distance along it. For a lost corner on a standard parallel or other controlling line, the restored position is placed on that line, because the line's direction was fixed by the original survey independently of the corner.
Where the record shows the original line was not straight, or where a corner was originally set with a recorded offset, the restoration honours the record. Proportioning is a rule for distributing measurement discrepancy, not a licence to straighten a line the original surveyor did not run straight.
Practical cautions
- Confirm the corner is lost, not obliterated. Proportioning a corner whose evidence still exists produces a wrong answer that looks authoritative.
- Use the nearest identified existing corners on the line as control. Reaching further than necessary imports other people's error into your result.
- Verify that the corners used as control are original, or are restorations properly made from original evidence. Control built on a bad corner propagates the error.
- Take record distances from the original field notes and plat, and check whether any later dependent resurvey supersedes them.
- Never proportion across a standard parallel, a guide meridian closing, or the boundary of another survey system. Those lines interrupt control.
- Record the computation, the control used and the record distances in the report, so the restoration can be checked and repeated.
Single proportion is the workhorse of lost-corner restoration and it is arithmetically trivial. Its difficulty is entirely in the decisions that precede the arithmetic: whether the corner is really lost, which corners control, and what the record actually says. Those three questions are the job.