Every closed traverse fails to close. The misclosure is small, it is real, and the question is never whether to remove it but how to spread it across the courses. Two rules answer that question, they give different answers, and the difference between them is a statement about which of your observations you trust.
The compass rule - the Bowditch rule in older texts - distributes the closure in proportion to course length. The transit rule distributes it in proportion to the latitude and departure of each course. Both drive the closure to zero exactly; neither improves the survey. What they do is decide where the error is assumed to have come from, and that assumption is the whole content of the choice.
What follows is the arithmetic of both rules on one traverse, the difference it makes to the adjusted coordinates, and a rule for choosing that survives contact with real field work.
What is being distributed
Every course of a traverse resolves into a latitude, its north-south component, and a departure, its east-west component. A closed loop returns to its starting point, so the latitudes must sum to zero and so must the departures. What they actually sum to are the closures in latitude and departure, and the resultant of those two is the linear misclosure.
latitude = D · cos α departure = D · sin α
e = √(ΣLat² + ΣDep²) precision = 1 : (ΣD ⁄ e)Nothing above depends on the adjustment rule. Closure, misclosure and precision are properties of the observations, so they are computed once, before any adjustment, and they are what tells you whether the traverse deserves adjusting at all.
A traverse to work on
Five courses, bearings observed to the nearest fifteen seconds, distances to a hundredth of a foot. This is an ordinary boundary loop of roughly two thousand feet of perimeter around a parcel of about six acres.
| Course | Bearing | Distance (ft) | Latitude (ft) | Departure (ft) |
|---|---|---|---|---|
| A-B | N 42°29′30″ E | 460.19 | +339.333 | +310.851 |
| B-C | S 61°18′30″ E | 380.58 | −182.715 | +333.851 |
| C-D | S 25°44′30″ W | 502.34 | −452.488 | −218.173 |
| D-E | S 81°21′00″ W | 307.84 | −46.299 | −304.338 |
| E-A | N 19°40′30″ W | 363.22 | +342.014 | −122.291 |
| Σ | - | 2,014.17 | −0.155 | −0.101 |
The closure in latitude is −0.155 ft and in departure −0.101 ft, so the linear misclosure is 0.185 ft over a perimeter of 2,014.17 ft - a precision of 1:10,897. That is a respectable closed traverse for boundary work and a poor one for control, which is the first thing the number tells you and the only thing it tells you before adjustment.
The compass rule
The compass rule assumes that angles and distances are of comparable precision, and therefore that error accumulates in proportion to how far you have walked. Each course absorbs a share of the closure equal to its share of the perimeter.
correction to latᵢ = −ΣLat · (Dᵢ ⁄ ΣD)
correction to depᵢ = −ΣDep · (Dᵢ ⁄ ΣD)On this traverse C-D is the longest course at 502.34 ft, which is 24.94 percent of the perimeter, so it absorbs 24.94 percent of both closures: +0.039 ft of latitude and +0.025 ft of departure. D-E, the shortest at 307.84 ft, absorbs 15.28 percent of each. Nothing about the direction of a course enters the calculation, which is exactly the point - the rule is indifferent to bearing because it assumes the bearings are as good as the distances.
The transit rule
The transit rule assumes the angles are markedly better than the distances. If the directions are effectively right, then the error lives in the distances, and a course contributes error to the northing budget in proportion to how much northing it carries.
correction to latᵢ = −ΣLat · (|latᵢ| ⁄ Σ|latᵢ|)
correction to depᵢ = −ΣDep · (|depᵢ| ⁄ Σ|depᵢ|)The latitude and departure budgets are now separate. On this traverse Σ|lat| is 1,362.85 ft and Σ|dep| is 1,289.50 ft, and a course such as D-E - nearly due west, carrying 304.34 ft of departure and only 46.30 ft of latitude - takes almost a quarter of the departure correction and almost none of the latitude correction. The compass rule would have given it 15 percent of both.
The two adjustments side by side
| Course | Compass Δlat | Transit Δlat | Compass Δdep | Transit Δdep |
|---|---|---|---|---|
| A-B | +0.035 | +0.039 | +0.023 | +0.024 |
| B-C | +0.029 | +0.021 | +0.019 | +0.026 |
| C-D | +0.039 | +0.051 | +0.025 | +0.017 |
| D-E | +0.024 | +0.005 | +0.015 | +0.024 |
| E-A | +0.028 | +0.039 | +0.018 | +0.010 |
| Σ | +0.155 | +0.155 | +0.101 | +0.101 |
Both columns sum to the same total, because both rules remove the whole closure; they simply allocate it differently. The largest disagreement is on D-E, where the two latitude corrections differ by nearly five times, and on C-D, where they differ by 0.012 ft. Everything else is within a hundredth.
| Station | Compass N | Compass E | Transit N | Transit E |
|---|---|---|---|---|
| A | 5,000.000 | 5,000.000 | 5,000.000 | 5,000.000 |
| B | 5,339.368 | 5,310.874 | 5,339.371 | 5,310.875 |
| C | 5,156.683 | 5,644.744 | 5,156.677 | 5,644.752 |
| D | 4,704.233 | 5,426.595 | 4,704.240 | 5,426.596 |
| E | 4,657.958 | 5,122.272 | 4,657.947 | 5,122.281 |
The largest coordinate difference between the two adjustments is 0.011 ft, at station E. The enclosed area comes out 6.1683 acres under either rule - identical to four decimal places, far finer than any area a plat reports.
Choosing between them
- Use the compass rule by default. It is the standard for a traverse run with a total station where angles and distances are observed with comparable care, and it is what most software, most agencies and most exam questions assume.
- Use the transit rule when the directions are genuinely stronger than the distances - a traverse on astronomic or GNSS-derived azimuths with taped or stadia distances is the classic case.
- Never choose on the basis of which gives the answer you want. Decide from the observation scheme before you compute, and record which rule you used on the computation sheet.
- Neither rule applies to a traverse that has not passed its angular closure check first. Adjust the angles, recompute the bearings, and only then distribute the linear misclosure.
- Do not adjust a traverse twice. Applying a second rule to already-adjusted latitudes and departures corrects a closure of zero and simply moves points for no reason.
What neither rule does
Both rules are arbitrary in the strict sense: they satisfy the closure condition and nothing else. A least-squares adjustment weights every observation by its own estimated precision, produces residuals you can test, and gives error ellipses at every station. The compass and transit rules produce coordinates and no statistics whatever.
Neither rule detects a blunder, either. A transposed digit in one distance produces a misclosure, and both rules will obligingly smear that blunder across every course, leaving five slightly wrong stations instead of one obviously wrong one. Before adjusting, check whether the closure azimuth points along one of the courses, which is the signature of an error in that distance, or perpendicular to a course, which points at an angle. Here the closure azimuth is 33°08′, which lines up with nothing - the mark of ordinary accumulated error rather than a mistake.
That is the honest summary of both rules. They are a convention for closing a figure, defensible because they are consistent and disclosed rather than because they are optimal. Say which one you used, show the unadjusted closure beside the adjusted coordinates, and the reader can judge the survey rather than the arithmetic.