Areas of government lots on a north township boundary

1Where the deficiency of the mile sits

When a township was subdivided, the work ran from the south and the east, so all the accumulated excess or deficiency was thrown into the last half-mile, the one adjoining the township boundary. On this north boundary that means the north quarter corner was set at the full record 40.00 chains west of the north-east section corner, and whatever was left of the mile fell between the quarter corner and the north-west section corner. That is why the sections along the north and west tiers of a township are lotted at all, and it is described in the chapter of the BLM Manual of Surveying Instructions (2009) on the subdivision of sections.

In this retracement all three monuments on the north boundary are existent, so each half-mile is simply what it now measures: 40.04 chains on the east and 39.68 chains on the west. The east half has run 0.04 chain long and the west half is 0.32 chain short, and the mile as a whole is 0.28 chain short of 80.00.

2Placing the lot lines

Each half-mile of the north boundary is bounded at both ends by an existent original corner, so the quarter-quarter corner controlling the lot line inside it is placed midway between those two corners. Applying that to both half-miles gives two equal lots in the east half and two equal lots in the west half, and the deficiency of the mile therefore shows up entirely in Lots 3 and 4.

It is worth stating plainly what this computation is and is not. The areas below are what the retracement measurement supports. The lot areas protracted on the original plat are the areas that were conveyed, and where the two differ it is the plat that describes the parcel; the retracement describes the ground.

3Lot areas

Each lot is a rectangle of its own width by the 20.00 chain depth of the tier. Working in square chains is the natural unit here, because ten square chains is exactly one acre and no conversion factor has to be remembered or looked up.

Lots 1 and 2 come out slightly over 40 acres and Lots 3 and 4 slightly under, and the pattern is the signature of the last-half-mile rule: on a plat of a fractional section the two easterly lots along the north boundary are habitually near 40.00 acres while the two westerly ones carry whatever the mile did not have.

4Check the total

The four lots together are the whole north tier, so their total must equal the measured length of the north boundary multiplied by the 20.00 chain depth. That single multiplication is an independent route to the total and catches any slip in the halving or in the individual areas.

5The deficiency read as area

The mile is 0.28 chain short, and the tier is 20.00 chains deep, so the missing area is 0.28 times 20.00, or 5.60 square chains, which is 0.56 acre. That figure agrees with the shortfall computed from the totals, and it makes a fast mental check that does not require redoing any of the lot arithmetic.

Note that the whole of the shortfall lies in Lots 3 and 4, at 0.32 acre each, while Lots 1 and 2 are each 0.04 acre over nominal. The excess and the deficiency did not cancel because they occurred in different half-miles, and there is no mechanism in the rectangular system that would make them cancel.