Five-Sided Boundary Traverse That Fails Both Close Checks
A junior surveyor closes a five-sided traverse ABCDE around a proposed site boundary using a 20-second theodolite and a 30 m steel tape. The five measured interior angles sum to 540°04′20″. The legs are AB = 310.25 m, BC = 402.10 m, CD = 285.40 m, DE = 368.75 m and EA = 483.50 m. The unadjusted latitudes and departures sum to ΣL = +0.35 m and ΣD = −0.23 m. The contract specification requires the angular misclosure to be within 20″√n and the relative precision of the closed traverse to be better than 1 in 5,000. As a cross-check, the surveyor compared his bearings with compass bearings taken on the same lines and found agreement within 3′. He proposes to distribute the closing error by the Bowditch rule and submit the plan, arguing that the error is 'only 42 cm in nearly 2 km' and that the compass check has already confirmed the angles.
1. Compute the angular misclosure and the relative precision, and give a verdict against the specification.
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Angular misclosure. For n = 5, theoretical sum = (2n − 4) × 90° = 540°00′00″. Measured 540°04′20″ → misclosure = 4′20″ = 260″.
Permissible = 20″ × √5 = 44.7″. The misclosure is 260/44.7 ≈ 5.8 times the limit → the angular work fails.
Linear precision. Perimeter = 310.25 + 402.10 + 285.40 + 368.75 + 483.50 = 1,850.00 m.
Closing error e = √(0.35² + 0.23²) = √(0.1225 + 0.0529) = √0.1754 = 0.419 m.
Relative precision = 1,850/0.419 = 1 in 4,416, worse than the specified 1 in 5,000.
Verdict. Both criteria fail, and the angular failure is gross rather than marginal. The traverse cannot be adjusted and submitted; the angles must be re-observed first.
2. The surveyor wants to apply the Bowditch rule and rely on the 3′ compass check. Assess both arguments.
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Bowditch is a distribution rule, not a repair. It spreads a misclosure in proportion to leg length, assuming random error roughly proportional to √distance - the classic chain/compass traverse error model. It is valid only when the misclosure is already inside tolerance.
Here the angular misclosure is 260″ against a limit of 45″. That is not random tape error; it points to a blunder - a misread angle, one face booked instead of two, a wrongly recorded minute, or the wrong station occupied. Spreading 260″ over five stations loads about 52″ per station, more than twice the theodolite's least count of 20″ and larger than the permitted total misclosure. The adjustment would produce a plan that looks closed and is silently wrong.
The compass check proves nothing. 3′ = 180″, four times the total permissible misclosure of 45″, and a magnetic compass is subject to local attraction of exactly that order in a built-up or rocky area. Agreement within 3′ is consistent with a 4′ angular error, not a refutation of it.
Correct action. Re-observe all five angles on both faces with horizon closure at each station, verify the booking, recompute bearings and coordinates, then re-check precision. Only a misclosure within 45″ and 1 in 5,000 permits Bowditch.
3. The executive engineer asks whether the plan can be accepted 'with a note' because the layout is urgent. Advise him, quantifying the positional consequence.
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Quantify before deciding. A 0.42 m loop misclosure, even after Bowditch adjustment, leaves the absolute accuracy of boundary points uncertain by roughly half the misclosure at the middle of the traverse - of the order of 0.2 m. That is four times the 0.05 m accuracy normally needed to set out pile positions, and at a site boundary it is a dispute waiting to happen.
Why 'accept with a note' is wrong here. The 1 in 5,000 and 20″√n limits are contract requirements, not targets. Endorsing a plan at 1 in 4,416 with the angles 5.8 times out transfers the risk to the engineer; any later boundary dispute will be decided against the record he signed.
Practical reply. Return the traverse for re-observation, requiring the field book, the two-face readings and the horizon closures to be submitted; check the tape against a standardised tape. If the programme is genuinely tight, ask the surveyor to re-observe first the two stations where the blunder is most likely (usually where the angle is far from the mean) - a single blunder, not random error, is the probable cause, and correcting it should bring the traverse inside tolerance within a day.