7.2 Single Proportionate Measurement Methods

Principles of Single Proportionate Measurement

When an exhaustive field search confirms that a PLSS corner is legally lost—meaning all physical marks, accessories, and reliable collateral evidence are gone beyond reasonable doubt—its position must be restored through mathematical proportionate measurement. Under the BLM Manual of Surveying Instructions, which Rule 7.3 directs Mississippi surveyors to follow, the restoration method depends on whether the corner is controlled along one line or in two directions.

Single proportionate measurement is applied to restore a lost corner whose position is constrained along a single line between two identified controlling corners (existent or obliterated corners). The fundamental principle of proportionate measurement is:

The Principle of Relative Proportion: The difference between the modern measured total distance and the original record total distance (excess or deficiency) is distributed among the intermediate segments in direct proportion to their original record lengths. Neither the original Gunter's chain nor modern electronic distance meters (EDM/GNSS) are assumed to be mathematically perfect; rather, every chain or foot of record distance is expanded or contracted by an identical scale factor.

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Scope of Application: When Single Proportion Governs

Under the Manual, single proportionate measurement restores these lost corners:

1. Quarter-Section Corners on Interior Section Lines: All lost quarter corners on interior north-south and east-west section lines.
2. Corners on Township Exterior Boundaries: All lost quarter-section corners and section corners on range lines and latitudinal township exterior lines.
3. Corners on Standard Parallels and Correction Lines: All lost standard section corners, standard quarter-section corners, and standard township corners along a Standard Parallel or Base Line.
4. Corners on Guide Meridians: All lost section and quarter-section corners along established Guide Meridians.
5. Corners on Fractional Section Lines: All lost quarter-section corners and sixteenth corners on fractional closing lines abutting the north and west boundaries of a township, or lines abutting meandered rivers and private land claims.
6. Closing Corners on Township Exteriors: A lost closing corner (CC) is re-established along the line closed upon (such as a standard parallel) by single proportion between the nearest identified corners on that line, constrained to the intersection of the closing meridional line.

+-------------------------------------------------------------------------+
|               SUMMARY OF SINGLE PROPORTION APPLICATIONS                |
+-------------------------------------------------------------------------+
| Line / Boundary Type             | Controlling Termini Corners          |
+----------------------------------+--------------------------------------+
| Interior East-West Section Line  | Existent/Obliterated Section Corners |
| (Quarter-Section Corner)         | to the East and West                 |
+----------------------------------+--------------------------------------+
| Interior North-South Section Line| Existent/Obliterated Section Corners |
| (Quarter-Section Corner)         | to the North and South               |
+----------------------------------+--------------------------------------+
| Township Range Line              | Existent/Obliterated Section or      |
| (Section or Quarter Corner)      | Township Corners on Range Line       |
+----------------------------------+--------------------------------------+
| Standard Parallel                | Existent/Obliterated Standard Corners|
| (Standard Section or 1/4 Corner) | along the Parallel (East and West)   |
+----------------------------------+--------------------------------------+
| Fractional Closing Tier Line     | Interior Section Corner and Township |
| (Section 6, 7, 18, 19, 30, 31)   | Exterior Closing Corner              |
+----------------------------------+--------------------------------------+

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The Mathematical Formula for Single Proportion

Let:

• $R_1, R_2, \dots, R_n$ = Original GLO record lengths of individual segments.
• $R_{\text{total}} = \sum R_i$ = Total GLO record distance between the two controlling corners.
• $M_{\text{total}}$ = Modern measured horizontal ground distance between the two controlling corners.
• $M_i$ = Modern proportionate distance to be laid off for segment $i$.

The universal single proportion equation is:

$$M_i = R_i \times \left( \frac{M_{\text{total}}}{R_{\text{total}}} \right)$$

Alternatively, defining the Proportion Factor (Scale Ratio) $k$:

$$k = \frac{M_{\text{total}}}{R_{\text{total}}}$$ $$M_i = R_i \times k$$

The total excess or deficiency $\Delta D = M_{\text{total}} - R_{\text{total}}$ is distributed across each segment:

$$\Delta d_i = \Delta D \times \left( \frac{R_i}{R_{\text{total}}} \right)$$ $$M_i = R_i + \Delta d_i$$

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Worked Calculation 1: Regular Interior Section Line

A Mississippi surveyor is retracing the east-west section line between Section 14 and Section 23. The quarter-section corner common to Sections 14 and 23 is determined to be lost.

Sec. Cor.                                   Lost 1/4 Cor.                           Sec. Cor.
(14, 15, 22, 23)                               (14 | 23)                         (13, 14, 23, 24)
      +-------------------------------------------O-------------------------------------+  
      | <-------- Record: 40.00 ch -------------> | <-------- Record: 40.00 ch --------> |
      |            (2,640.00 ft)                  |            (2,640.00 ft)            |
      |                                                                                 |
      | <====================== Measured: 5,312.40 ft ================================> |
      | <------ Modern: 2,656.20 ft ------------> | <------ Modern: 2,656.20 ft --------> |

Step 1: Compile Original GLO Field Note Records

• West Section Corner (14, 15, 22, 23) to Quarter Corner: $R_1 = 40.00\text{ chains} = 2,640.00\text{ ft}$
• Quarter Corner to East Section Corner (13, 14, 23, 24): $R_2 = 40.00\text{ chains} = 2,640.00\text{ ft}$
• Total Record Distance: $R_{\text{total}} = 80.00\text{ chains} = 5,280.00\text{ ft}$

Step 2: Measure Modern Distance Between Controlling Corners

• Controlling corners recovered: West Section Corner and East Section Corner are both existent.
• Modern measured horizontal distance: $M_{\text{total}} = 5,312.40\text{ ft}$

Step 3: Determine Excess or Deficiency

$$\Delta D = 5,312.40\text{ ft} - 5,280.00\text{ ft} = +32.40\text{ ft} \quad (\text{an excess of } 32.40\text{ ft})$$

Step 4: Calculate the Proportion Factor

$$k = \frac{M_{\text{total}}}{R_{\text{total}}} = \frac{5,312.40}{5,280.00} = 1.00613636...$$

Step 5: Compute Segment Lengths

• West Segment (West Section Corner to Quarter Corner): $$M_1 = 2,640.00 \times 1.00613636 = 2,656.20\text{ ft} \quad (40.2455\text{ chains})$$
• East Segment (Quarter Corner to East Section Corner): $$M_2 = 2,640.00 \times 1.00613636 = 2,656.20\text{ ft} \quad (40.2455\text{ chains})$$
• Check: $2,656.20\text{ ft} + 2,656.20\text{ ft} = 5,312.40\text{ ft}$. The math closes exactly.

Step 6: Field Alignment

The lost quarter-section corner is established on the straight line connecting the West and East section corners at a distance of 2,656.20 feet from each section corner. Because the record segments were equal (40.00 ch each), the 32.40-foot excess is divided equally (16.20 feet added to each half-mile).

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Worked Calculation 2: Fractional Closing Section Line

A critical exam scenario involves restoring a lost quarter corner along a fractional closing tier (e.g., Sections 1–6 along the north township boundary, or Sections 6, 7, 18, 19, 30, 31 along the west range line).

Consider the latitudinal line between Section 18 and Section 19, closing west upon the Range Line. The quarter corner between Section 18 and Section 19 is lost.

RANGE LINE                                                                   INTERIOR
Sec. Cor.                                    Lost 1/4 Cor.                   Sec. Cor.
(13, 18, 19, 24)                               (18 | 19)                  (17, 18, 19, 20)
      +-------------------------------------------O-------------------------------------+  
      | <-------- Record: 38.50 ch -------------> | <-------- Record: 40.00 ch --------> |
      |            (2,541.00 ft)                  |            (2,640.00 ft)            |
      |                                                                                 |
      | <====================== Measured: 5,210.00 ft ================================> |
      | <------ Modern: 2,555.22 ft ------------> | <------ Modern: 2,654.78 ft --------> |

Step 1: Compile Original GLO Field Note Records

Under original GLO instructions, the deputy surveyor ran west from the interior section corner, set the quarter-section corner at exactly 40.00 chains, and closed on the west range line, throwing all excess or deficiency into the western fractional half-mile:

• Interior Section Corner (17, 18, 19, 20) to Quarter Corner: $R_{\text{East}} = 40.00\text{ chains} = 2,640.00\text{ ft}$
• Quarter Corner to West Range Line Corner (13, 18, 19, 24): $R_{\text{West}} = 38.50\text{ chains} = 2,541.00\text{ ft}$
• Total Record Distance: $R_{\text{total}} = 40.00 + 38.50 = 78.50\text{ chains} = 5,181.00\text{ ft}$

Step 2: Measure Modern Distance Between Controlling Corners

• Controlling corners recovered: Interior Section Corner (existent) and Range Line Section Corner (existent).
• Modern measured horizontal distance: $M_{\text{total}} = 5,210.00\text{ ft}$

Step 3: Determine Excess or Deficiency

$$\Delta D = 5,210.00\text{ ft} - 5,181.00\text{ ft} = +29.00\text{ ft}$$

Step 4: Calculate the Proportion Factor

$$k = \frac{M_{\text{total}}}{R_{\text{total}}} = \frac{5,210.00}{5,181.00} \approx 1.005597375$$

Step 5: Compute Proportional Segment Lengths

• East Segment (Regular Half-Mile): $$M_{\text{East}} = 2,640.00 \times 1.005597375 = 2,654.78\text{ ft} \quad (40.2239\text{ chains})$$
  • Excess absorbed: $2,654.78 - 2,640.00 = +14.78\text{ ft}$
• West Segment (Fractional Half-Mile): $$M_{\text{West}} = 2,541.00 \times 1.005597375 = 2,555.22\text{ ft} \quad (38.7155\text{ chains})$$
  • Excess absorbed: $2,555.22 - 2,541.00 = +14.22\text{ ft}$
• Check Sum: $2,654.78\text{ ft} + 2,555.22\text{ ft} = 5,210.00\text{ ft}$.
• Check Excess: $14.78\text{ ft} + 14.22\text{ ft} = 29.00\text{ ft}$.

The "Midpoint Fallacy" on Fractional Lines

Notice the vital legal lesson:

• If a surveyor mistakenly placed the quarter corner at the arithmetic midpoint ($5,210.00 / 2 = 2,605.00\text{ ft}$), the East segment would be assigned 2,605.00 ft instead of 2,654.78 ft—an error of 49.78 feet!
• If a surveyor mistakenly held the original record 40.00 chains (2,640.00 ft) from the interior corner and threw all 29.00 feet of excess into the west half, the West segment would be assigned 2,570.00 ft instead of 2,555.22 ft—an error of 14.78 feet!
• Under the Manual, both segments are proportioned according to their record lengths, preserving the original ratio of 40.00 chains to 38.50 chains across the modern measured distance.

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Single Proportion on Curved Lines: Standard Parallels

When restoring a lost corner on a Standard Parallel (Correction Line) or Base Line, alignment is constrained by the true parallel of latitude, which is a curved line on the earth's surface rather than a straight geodetic chord:

1. Distance: Distributed by single proportion along the parallel between the nearest existent or obliterated standard corners.
2. Alignment: The corner must be placed on the true parallel. For long distances between controlling corners, the surveyor must calculate and apply latitude offsets from the secant or tangent line run in the field, ensuring the restored corner falls on the true curve of the parallel.