7.3 Double Proportionate Measurement Methods

Principles of Double Proportionate Measurement

Under the BLM Manual of Surveying Instructions, which Rule 7.3 directs Mississippi surveyors to follow, a lost corner whose position was fixed by lines running in all four cardinal directions is restored by double proportionate measurement.

Double proportionate measurement is designed to restore a corner that controls four surrounding sections or four surrounding townships. The method guarantees that measurement discrepancies (excess or deficiency) in latitude (north-south) and departure (east-west) are resolved independently, without allowing chaining errors in one cardinal axis to corrupt the alignment or position of the other.

                         NORTH CONTROL (Cn)
                    Existent/Obliterated Section Corner
                                    |
                                    |  Record: 80.00 ch (N-S)
                                    |  Controls LATITUDE (Northing)
                                    |
         WEST CONTROL (Cw)          v          EAST CONTROL (Ce)
     Existent/Obliterated +------( CORNER )------+ Existent/Obliterated
        Section Corner     Record:   |   Record:    Section Corner
                           80.00 ch  |   80.00 ch
                                     |  Controls DEPARTURE (Easting)
                                     |
                                     v
                             SOUTH CONTROL (Cs)
                    Existent/Obliterated Section Corner

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Scope of Application: Corners Restored by Double Proportion

Under the Manual, double proportionate measurement applies to:

1. Lost Interior Section Corners: Section corners common to four interior sections (e.g., corner of Sections 9, 10, 15, and 16).
2. Lost Interior Township Corners: Township corners common to four townships on a continuous grid (e.g., corner of Townships 4 and 5 North, Ranges 2 and 3 East).
3. Other corners of four intersecting lines: in general, a lost corner whose position depended on lines run in all four directions. Corners established when a township exterior was run are otherwise restored along that exterior line (see the prohibited cases below).

Where Double Proportion is STRICTLY PROHIBITED

Applying double proportion where single proportion is legally required is a fatal surveying error. Double proportion must never be used for:

• Quarter-Section Corners: All quarter corners are restored by single proportion along the specific section line.
• Township or Section Corners on Standard Parallels: Corners on standard parallels were established on a senior east-west line; they are restored by single proportion along the parallel.
• Closing Corners: when a section line closed upon a township exterior or standard parallel, a lost closing corner is restored on the line closed upon, at its proportionate position between the nearest corners on that line, not by double proportion.

Quad-Directional Control: Identifying Controlling Corners

To restore a lost interior section corner by double proportion, the surveyor must establish quad-directional control by locating the nearest existent or obliterated corners in all four cardinal directions:

• North Control ($C_n$): Nearest verified section or township corner to the north.
• South Control ($C_s$): Nearest verified section or township corner to the south.
• East Control ($C_e$): Nearest verified section or township corner to the east.
• West Control ($C_w$): Nearest verified section or township corner to the west.

The "Missing Adjacent Corner" Rule

What happens if the immediately adjacent section corner 1 mile away is also lost?

Extended control: a surveyor cannot use an unrestored or assumed corner as control. If the immediate adjacent section corner is lost, the surveyor must extend the control survey in that direction—2 miles, 3 miles, or more—until reaching the nearest existent or obliterated corner. The proportionate calculation then incorporates the cumulative record distances across all intermediate sections.

The Mathematical Protocol: Coordinate Intersection

In modern geodetic and plane coordinate surveying (such as Mississippi State Plane Coordinates, NAD83 East or West Zone), double proportion is executed through independent single proportions on Northing (Latitude) and Easting (Departure):

                               [Cn] (N_n, E_n)
                                     |
                                     |  North-South Line:
                                     |  Determines Latitude
                                     |  (Northing Line: Y = Y_restored)
                                     |
   [Cw] (N_w, E_w) ------------------*------------------ [Ce] (N_e, E_e)
                                (N, E)
                                     |  East-West Line:
                                     |  Determines Departure
                                     |  (Easting Line: X = X_restored)
                                     |
                               [Cs] (N_s, E_s)

Step-by-Step Procedure

1. North-South Single Proportion (Latitude / Northing):

   • Calculate the total record distance along the north-south line between $C_s$ and $C_n$: $R_{\text{NS}} = R_{\text{south\_segment}} + R_{\text{north\_segment}}$.
   • Determine the modern measured difference in Northing between $C_s$ and $C_n$: $\Delta N_{\text{total}} = N_n - N_s$.
   • Compute the restored Northing coordinate: $$N_{\text{restored}} = N_s + \Delta N_{\text{total}} \times \left( \frac{R_{\text{south\_segment}}}{R_{\text{NS}}} \right)$$
   • Note: The Easting coordinate calculated on the north-south line is discarded.

2. East-West Single Proportion (Departure / Easting):

   • Calculate the total record distance along the east-west line between $C_w$ and $C_e$: $R_{\text{EW}} = R_{\text{west\_segment}} + R_{\text{east\_segment}}$.
   • Determine the modern measured difference in Easting between $C_w$ and $C_e$: $\Delta E_{\text{total}} = E_e - E_w$.
   • Compute the restored Easting coordinate: $$E_{\text{restored}} = E_w + \Delta E_{\text{total}} \times \left( \frac{R_{\text{west\_segment}}}{R_{\text{EW}}} \right)$$
   • Note: The Northing coordinate calculated on the east-west line is discarded.

3. Intersection of Coordinates:

   • The restored corner is monumented at the exact coordinates: $$\mathbf{P}_{\text{restored}} = (N_{\text{restored}}, E_{\text{restored}})$$

Detailed Step-by-Step Worked Calculation with Numbers

A Mississippi surveyor is tasked with restoring the lost interior section corner common to Sections 15, 16, 21, and 22. Exhaustive field investigation confirms that all physical and collateral evidence at the corner is gone. The surveyor recovers the four nearest existent section corners and establishes their Mississippi State Plane Coordinates:

Recovered Control Corner Coordinates

• North Control ($C_n$) [Corner of Secs 9, 10, 15, 16]:
  • Coordinates: $N_n = 105,320.00\text{ ft}$, $E_n = 200,000.00\text{ ft}$
  • Original GLO Record: 80.00 chains (5,280.00 ft) North of lost corner
• South Control ($C_s$) [Corner of Secs 21, 22, 27, 28]:
  • Coordinates: $N_s = 94,700.00\text{ ft}$, $E_s = 200,080.00\text{ ft}$
  • Original GLO Record: 80.00 chains (5,280.00 ft) South of lost corner
• West Control ($C_w$) [Corner of Secs 16, 17, 20, 21]:
  • Coordinates: $N_w = 100,040.00\text{ ft}$, $E_w = 194,680.00\text{ ft}$
  • Original GLO Record: 80.00 chains (5,280.00 ft) West of lost corner
• East Control ($C_e$) [Corner of Secs 14, 15, 22, 23]:
  • Coordinates: $N_e = 99,980.00\text{ ft}$, $E_e = 205,340.00\text{ ft}$
  • Original GLO Record: 80.00 chains (5,280.00 ft) East of lost corner

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Step 1: North-South Proportion for Northing (Latitude)

1. Total GLO Record N-S Distance: $$R_{\text{NS}} = 80.00\text{ ch} + 80.00\text{ ch} = 160.00\text{ chains} = 10,560.00\text{ ft}$$
2. Modern Measured Northing Difference (between $C_s$ and $C_n$): $$\Delta N_{\text{total}} = N_n - N_s = 105,320.00 - 94,700.00 = 10,620.00\text{ ft}$$
3. Total N-S Excess: $$\Delta D_{\text{NS}} = 10,620.00\text{ ft} - 10,560.00\text{ ft} = +60.00\text{ ft}$$
4. Proportional Ratio for South Segment: $$\text{Ratio}_{\text{South}} = \frac{80.00\text{ ch}}{160.00\text{ ch}} = 0.500000$$
5. Compute Restored Northing Coordinate: $$N_{\text{restored}} = 94,700.00 + (10,620.00 \times 0.500000) = 94,700.00 + 5,310.00 = \mathbf{100,010.00\text{ ft}}$$

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Step 2: East-West Proportion for Easting (Departure)

1. Total GLO Record E-W Distance: $$R_{\text{EW}} = 80.00\text{ ch} + 80.00\text{ ch} = 160.00\text{ chains} = 10,560.00\text{ ft}$$
2. Modern Measured Easting Difference (between $C_w$ and $C_e$): $$\Delta E_{\text{total}} = E_e - E_w = 205,340.00 - 194,680.00 = 10,660.00\text{ ft}$$
3. Total E-W Excess: $$\Delta D_{\text{EW}} = 10,660.00\text{ ft} - 10,560.00\text{ ft} = +100.00\text{ ft}$$
4. Proportional Ratio for West Segment: $$\text{Ratio}_{\text{West}} = \frac{80.00\text{ ch}}{160.00\text{ ch}} = 0.500000$$
5. Compute Restored Easting Coordinate: $$E_{\text{restored}} = 194,680.00 + (10,660.00 \times 0.500000) = 194,680.00 + 5,330.00 = \mathbf{200,010.00\text{ ft}}$$

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Step 3: Monument the Restored Corner

The restored section corner is monumented at the exact coordinate intersection:

$$\mathbf{P}_{\text{restored}} = (N = 100,010.00\text{ ft}, E = 200,010.00\text{ ft})$$

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Why Coordinate Averaging is a Fatal Surveying Error

A common error made by unlicensed technicians is calculating the simple arithmetic average (centroid) of the four control corners:

• Average Northing: $(105,320 + 94,700 + 100,040 + 99,980) / 4 = 400,040 / 4 = 100,010.00\text{ ft}$
• Average Easting: $(200,000 + 200,080 + 194,680 + 205,340) / 4 = 800,100 / 4 = \mathbf{200,025.00\text{ ft}}$

Comparing the results:

• The true double proportionate Easting is 200,010.00 ft.
• The four-corner coordinate average yields 200,025.00 ft.
• Averaging coordinates introduces a 15.00-foot error in the position of the section corner! This occurs because the north and south control corners had a slight easterly drift ($E_n = 200,000$, $E_s = 200,080$) that corrupted the east-west coordinate when averaged. Under the Manual's double-proportion method, north-south measurements do not affect departure, and east-west measurements do not affect latitude.

Unequal Record Segments: Extended Multi-Mile Control

When one or more immediate section corners are also lost, the distance to the controlling corner will span multiple sections. Proportionate ratios must reflect the actual record lengths:

Example Scenario: Suppose the corner 1 mile south of the subject corner was also lost, requiring the surveyor to use the township exterior corner 2 miles to the south ($R_{\text{south}} = 160.00\text{ ch}$), while the north control is only 1 mile away ($R_{\text{north}} = 80.00\text{ ch}$).

1. Total Record N-S: $160.00 + 80.00 = 240.00\text{ chains} = 15,840.00\text{ ft}$
2. Proportion Ratio from South Control: $\frac{160.00}{240.00} = \frac{2}{3} \approx 0.666667$
3. Proportion Ratio from North Control: $\frac{80.00}{240.00} = \frac{1}{3} \approx 0.333333$

The modern measured difference in Northing between the south and north control corners is multiplied by $\frac{2}{3}$ to establish the Northing of the restored corner. This preserves the historical geometry of the entire multi-section block.

Major Exam Traps in Proportionate Measurement