--- title: "Gale's table for traverse ABCDA" subject: "Surveying II" chapter: 1 key: "semester_4/surveying_ii:gale-table-abcda-2072a" source: https://www.bctengineer.com/q/semester_4/surveying_ii/gale-table-abcda-2072a site: BCT Engineering --- # Gale's table for traverse ABCDA Surveying II · Chapter 1: Traversing - Asked in: 2072 Asoj, 10 marks - Video: [Balance the traverse by using Bowditch rule | Traverse adjustment | Latitude Departure Closing error](https://www.youtube.com/watch?v=Jkb8qO98QZA) — SBK Concept - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/gale-table-abcda-2072a - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-1 ## Question The following observations are made in a traverse ABCDA. | Traverse Leg | Horizontal Distance (m) | Traverse Station | Horizontal Angle | |---|---|---|---| | AB | 71.5 | A | 78°41'25" | | BC | 42.0 | B | 101°18'38" | | CD | 70.0 | C | 89°59'41" | | DA | 56.0 | D | 90°00'21" | Bearing of CD = 314°58'04". Coordinate of C (500 m N, 500 m E). Complete the Gale's Table with final adjusted length and bearing of each line. ## Short answer Angular error +5” distributed equally. Bowditch correction gives adjusted lengths and bearings: AB 71.466 m, 146°16'26”; BC 41.997 m, 224°59'20”; CD 70.033 m, 314°58'18”; DA 56.004 m, 44°56'08”. Closing error 1:2125. ## Answer Method: check and adjust the angles, find the bearings from the known bearing, compute latitudes and departures, balance them by Bowditch's rule and compute the coordinates from the given station. #### Step 1: Angular check and adjustment For a closed traverse of $n=4$ sides, the sum of interior angles must be $(2n-4)\times 90^\circ=360^\circ$. Observed sum = 360°00'05", so the angular error is 0°00'05" (excess). The correction is distributed equally: 1.25" (subtracted) per angle. | Station | Observed angle | Correction | Adjusted angle | |---|---|---|---| | A | 78°41'25" | -1.25" | 78°41'24" | | B | 101°18'38" | -1.25" | 101°18'37" | | C | 89°59'41" | -1.25" | 89°59'40" | | D | 90°00'21" | -1.25" | 90°00'20" | | **Sum** | 360°00'05" | | 360°00'00" | #### Step 2: Bearings The traverse is taken as clockwise (interior angles on the right), so $$ \text{Bearing of next line}=\text{Bearing of previous line}+180^\circ-\text{interior angle at the station} $$ Starting from CD = 314°58'04": | Line | Length (m) | Bearing | |---|---|---| | AB | 71.50 | 146°16'20" | | BC | 42.00 | 224°57'44" | | CD | 70.00 | 314°58'04" | | DA | 56.00 | 44°57'44" | Check: carrying the bearing round the traverse brings back CD = 314°58'04", as given. #### Step 3: Latitudes, departures and closing error $$ \text{Lat}=L\cos\theta,\qquad \text{Dep}=L\sin\theta $$ | Line | Length (m) | Bearing | Latitude | Departure | Corr. lat | Corr. dep | |---|---|---|---|---|---|---| | AB | 71.500 | 146°16'20" | -59.466 | +39.700 | +0.0269 | -0.0202 | | BC | 42.000 | 224°57'44" | -29.718 | -29.679 | +0.0158 | -0.0119 | | CD | 70.000 | 314°58'04" | +49.470 | -49.525 | +0.0263 | -0.0198 | | DA | 56.000 | 44°57'44" | +39.624 | +39.572 | +0.0210 | -0.0159 | | **Sum** | 239.500 | | -0.090 | +0.068 | +0.0900 | -0.0678 | Closing error: $e_N=-0.090$ m, $e_E=+0.068$ m, $$ e=\sqrt{e_N^2+e_E^2}=0.1127\ \text{m},\qquad \text{relative error}=\frac{e}{\Sigma L}=\frac{0.1127}{239.50}=\frac{1}{2125} $$ #### Step 4: Bowditch correction and adjusted lines Correction to a line $=-\text{(total error)}\times L/\Sigma L$. The corrected latitudes and departures give the final length $=\sqrt{\text{Lat}^2+\text{Dep}^2}$ and bearing $=\tan^{-1}(\text{Dep}/\text{Lat})$: | Line | Adj. lat | Adj. dep | Adj. length (m) | Adj. bearing | |---|---|---|---|---| | AB | -59.439 | +39.680 | 71.466 | 146°16'26" | | BC | -29.702 | -29.691 | 41.997 | 224°59'20" | | CD | +49.496 | -49.545 | 70.033 | 314°58'18" | | DA | +39.645 | +39.556 | 56.004 | 44°56'08" | #### Step 5: Coordinates (C = 500.000 N, 500.000 E) | Station | Northing (m) | Easting (m) | |---|---|---| | A | 589.141 | 490.011 | | B | 529.702 | 529.691 | | C | 500.000 | 500.000 | | D | 549.496 | 450.455 | **Answer:** the final adjusted lengths and bearings are as tabulated above, and the traverse closes with a relative error of 1 in 2125. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For the closed 4-sided traverse, what should the interior angles sum to, and how is the 5" excess handled? A. 360°; subtract 1.25" from each angle B. 180°; subtract 5" from the last angle C. 720°; add 1.25" to each angle D. 360°; ignore the excess Answer: A. (2n-4) x 90° = 360° for n = 4, and the observed excess of 5" is spread equally as -1.25" per angle. (Key point: Angle sum check) 2. For this clockwise traverse with interior angles, how is the bearing of the next line found? A. Previous bearing + interior angle B. Previous bearing + 180° - interior angle C. Previous bearing - 180° + interior angle only D. Previous bearing - interior angle Answer: B. Interior angles lie on the right in a clockwise traverse, so next bearing = previous bearing + 180° - interior angle (adjusting by 360°). (Key point: Bearing propagation) 3. How are the closing errors in latitude and departure (-0.090 and +0.068) distributed in Gale's table? A. Equally among the four lines B. In proportion to line length, correction = -error x L/ΣL, giving positive latitude corrections and negative departure corrections C. In proportion to the latitude of each line D. Entirely to the longest line Answer: B. Bowditch's rule: corrections are proportional to length and opposite in sign to the error, so +0.090 total in latitude and -0.068 in departure. (Key point: Bowditch correction) 4. How are the final adjusted length and bearing of each line obtained? A. Length = sqrt(adj lat^2 + adj dep^2) and bearing = tan^-1(adj dep/adj lat) B. Length is unchanged and only the bearing changes C. Length = adj lat + adj dep D. They are the observed values reduced by 1 in 2125 Answer: A. After Bowditch correction the adjusted latitudes and departures are converted back to polar form, giving the adjusted length and bearing. 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