A section is the working unit of the PLSS: nominally one mile square, 80 chains on a side, containing a nominal 640 acres. Everything smaller is described by cutting the section into halves and quarters, repeatedly, and naming the pieces from the smallest outward. Those pieces are the aliquot parts, and a description built from them is the most common form of land description in thirty states.
The grammar is simple once you see it. Read the description from right to left. Each term you meet is a subdivision of everything to its right. NE¼ SW¼ Sec 14 means: take section 14, take its southwest quarter, then take the northeast quarter of that. The result is a nominal 40 acres lying south and west of the center of the section.
The aliquot ladder
- Aliquot part
- A subdivision of a section formed by successive halving and quartering, so that it divides the section evenly. The half, the quarter, the half of a quarter and the quarter-quarter are aliquot; an arbitrary strip described by feet is not.
| Aliquot part | Fraction of a section | Nominal acres | Nominal dimensions |
|---|---|---|---|
| Section | 1 | 640 | 80 ch × 80 ch |
| Half section, N½ | 1/2 | 320 | 80 ch × 40 ch |
| Quarter section, NE¼ | 1/4 | 160 | 40 ch × 40 ch |
| Half of a quarter, N½ NE¼ | 1/8 | 80 | 40 ch × 20 ch |
| Quarter-quarter, NE¼ SW¼ | 1/16 | 40 | 20 ch × 20 ch |
| Half of a quarter-quarter, S½ NE¼ SW¼ | 1/32 | 20 | 20 ch × 10 ch |
| Quarter of a quarter-quarter, NE¼ NE¼ SW¼ | 1/64 | 10 | 10 ch × 10 ch |
A = 640 ÷ (d₁ × d₂ × … × dₙ)The arithmetic behind those figures is worth carrying in your head. One chain is 66 feet and eighty chains make a mile, so a section is 80 ch × 80 ch = 6,400 square chains. One acre is ten square chains, or 43,560 square feet, which makes the section 640 acres exactly. A quarter-quarter of 20 ch × 20 ch is 400 square chains, or 40 acres. Chains make PLSS arithmetic almost mental; feet do not.
Reading NE¼ SW¼ Sec 14, T3N, R2W
- Start at the right with the township and range: T3N, R2W of the named principal meridian identifies the six-mile block.
- Section 14 lies in the third tier of that township, second column in from the east side.
- SW¼ takes the southwest 160 acres of that section, a nominal 40 ch × 40 ch square.
- NE¼ of that quarter takes its northeast 40 acres, a nominal 20 ch × 20 ch square whose northeast corner is the center of section 14.
Notice what the final step implies. The northeast corner of the NE¼ SW¼ is the center of the section, a point that no original surveyor ever monumented. The center of section was not set in the original survey; it is established later by the local surveyor at the intersection of the lines joining opposite quarter corners. Aliquot descriptions therefore routinely depend on corners that have to be created rather than recovered.
What is not an aliquot part
Aliquot descriptions are convenient precisely because their boundaries are determined by rule from the section corners. Descriptions that look similar but are not aliquot do not enjoy that convenience, and confusing the two is a frequent source of trouble.
- The north 660 feet of the NE¼ is not an aliquot part. It is a strip measured from a boundary, and its south line depends on where you measure from and along what bearing.
- Government lots are not aliquot parts. A lot is an irregular parcel created by the original survey and identified by number, with an acreage returned on the plat.
- The N½ of a government lot is not an aliquot description of the section; it is a division of a lot and depends on how the lot itself is bounded.
- The east 40 acres of the NW¼ is an area-based description, not an aliquot one. Its boundary line is wherever it must be to enclose 40 acres, which requires a computation.
- The SE¼ of the SE¼ of the SE¼ of the SE¼ is aliquot but nominally only 2.5 acres, and at that scale the difference between the nominal and the actual becomes proportionally large.
How aliquot parts are actually located
The rule for subdividing a section is proportional, not nominal. The quarter corners on the section boundaries were monumented in the original survey, generally at the midpoint of each side. The center of the section is the intersection of the two straight lines joining opposite quarter corners. The quarter-quarter corners are then set at the midpoints of the lines joining the section corners, quarter corners and center.
This means that if a section measures 79.20 chains across instead of 80.00, its quarter-quarters are proportionally smaller and each contains slightly less than 40 acres. The deficiency is shared, not dumped on one corner, except in fractional sections along the north and west tiers of a township where the design deliberately puts the remainder in the outermost tier. Knowing which case you are in is the difference between a correct retracement and a wrong one.
For computing the actual area of an aliquot part once its corners are located, coordinate geometry is the practical tool: reduce the corners to coordinates and use the coordinate area method. The nominal acreage from the description then serves as a check on gross error, nothing more.