An interior section corner was not established by measuring along one line. It was established where two lines crossed, one running north and south and one running east and west, and its position therefore depends on control in both directions at once. When such a corner is lost, restoring it by single proportion along either line alone would ignore half the evidence. Double proportionate measurement uses all of it.
The method computes two independent proportions, one along the north-south control and one along the east-west control, and then places the restored corner at the intersection: on the parallel of latitude through the north-south proportional point, and on the meridian through the east-west proportional point. The result is a single position that honours the record in both directions.
When double proportion applies
- Double proportionate measurement
- The restoration of a lost corner by taking the nearest identified existing corners north, south, east and west of it, computing a proportional position along each of the two lines, and locating the corner at the intersection of a latitudinal line through the one and a meridional line through the other.
- Lost interior section corners, common to four sections, whose position was fixed by the crossing of two subdivisional lines.
- Lost township corners common to four townships, where control exists in all four directions.
- Lost corners of the original survey whose establishment depended on measurement in both cardinal directions rather than along a single line.
- It does not apply to quarter-section corners, which are controlled along one line and are restored by single proportion.
- It does not apply to closing corners, which are restored on the governing line by single proportion.
- It does not apply where control is missing in one direction, such as on the boundary of a survey; there, single proportion or another special rule governs.
The procedure
- Confirm the corner is lost after exhausting every means of recovery, and identify the nearest existing corners north, south, east and west along the lines through it.
- Measure the total distance between the north and south controlling corners, and the total distance between the east and west controlling corners.
- Take from the original field notes the record distances from each controlling corner to the lost corner.
- Compute the north-south proportional distance from the south controlling corner, using the ratio of the measured to the record north-south total.
- Compute the east-west proportional distance from the west controlling corner, using the ratio of the measured to the record east-west total.
- Place a temporary point at the north-south proportional distance, and a second temporary point at the east-west proportional distance.
- The restored corner lies at the intersection of a line run east and west through the north-south point with a line run north and south through the east-west point.
- Monument the corner, and record the control, the record distances, the measurements and the computation.
d′ = d × (M ÷ R), computed independently for the north-south and the east-west controlWorked example with unequal control
An interior section corner is lost. The nearest identified existing corners are two miles to the west, one mile to the east, one mile to the south and two miles to the north. That asymmetry is typical, and the method accommodates it.
| Quantity | Value |
|---|---|
| Record distance, west controlling corner to lost corner | 160.00 ch |
| Record distance, lost corner to east controlling corner | 80.00 ch |
| Record total, west to east (R) | 240.00 ch |
| Measured total, west to east (M) | 239.28 ch |
| Proportion factor M ÷ R | 0.9970 |
| Proportional distance from the west corner | 159.52 ch |
| Remaining distance to the east corner | 79.76 ch |
| Quantity | Value |
|---|---|
| Record distance, south controlling corner to lost corner | 80.00 ch |
| Record distance, lost corner to north controlling corner | 160.00 ch |
| Record total, south to north (R) | 240.00 ch |
| Measured total, south to north (M) | 240.60 ch |
| Proportion factor M ÷ R | 1.0025 |
| Proportional distance from the south corner | 80.20 ch |
| Remaining distance to the north corner | 160.40 ch |
The east-west line is 0.72 chains short of its record and the north-south line is 0.60 chains long, and each discrepancy is distributed in proportion to the record intervals on its own line. The restored corner is then placed where a north-south line through the point 159.52 chains east of the west controlling corner meets an east-west line through the point 80.20 chains north of the south controlling corner.
Why the intersection, and not an average
It is tempting to compute a position along each line and average the two results, or to run a traverse and adjust it. Neither is correct. The two proportions are answering different questions: the east-west proportion determines how far east the corner is, and tells you nothing about how far north; the north-south proportion does the reverse. Each contributes one coordinate, and the corner is the point that satisfies both.
In modern practice the computation is done in coordinates. Reduce the four controlling corners to a plane coordinate system, compute the two proportional points, and take the departure from one and the latitude from the other. The result is the restored corner. Because the two proportions are independent, the restored position does not depend on the order of computation or on the direction of measurement, which is exactly the reproducibility the rule exists to provide.
Where double proportion goes wrong
- Applying it to a quarter-section corner. Quarter corners are controlled along one line and take single proportion.
- Applying it across a standard parallel, a guide meridian closing, or the boundary of another survey system. Control does not pass through those lines.
- Using control corners that are not original or not properly restored. Every error in the control passes straight into the result.
- Taking nominal record distances instead of the record distances shown in the field notes, which matters most where the section is fractional.
- Declaring a corner lost when accessories, traces or testimony would have recovered it. The most precise double proportion is worthless if the corner was never lost.
- Failing to record the control and the computation, so that the next surveyor cannot check the work and restores the corner somewhere else.
Like single proportion, double proportion is arithmetic that a spreadsheet handles in seconds. The professional content of the work is in establishing that the corner is genuinely lost, choosing the correct control, reading the record correctly, and documenting the result well enough that the restoration is stable for the next generation.