Free Gale's Table Generator
Closed Traverse Survey Calculator

A Gale's Table is the standard columnar computation sheet used in plane surveying to process a closed traverse. It organises all intermediate calculations - angular closure, bearing propagation, latitude and departure computation, Bowditch or Transit corrections, running coordinates, and adjusted bearings - in a logical sequence that allows each stage to be independently verified. It is named after its systematic, column-by-column layout and is widely taught in civil engineering, geomatics, and quantity surveying programmes across the UK, Commonwealth countries, and South Asia.

A closed traverse is a series of connected survey legs that returns to its starting point, forming a closed polygon. Because the start and end are the same physical point, the geometry demands that the sum of all interior angles equals (n-2)×180° and that the algebraic sum of all latitudes and departures each equals zero. Any deviation from these conditions represents measurement error, which can be quantified (as misclosure), assessed against permissible limits, and distributed using the Bowditch or Transit rule.

The tool sums all entered horizontal angles and compares with the theoretical sum (n-2)×180°. The difference is the angular misclosure. This is distributed using a weighted approach: the total correction is divided into whole arc-seconds, with the base correction applied to all angles and any remainder seconds assigned first to the largest observed angles. This ensures the corrections are integers in arc-seconds and their exact sum cancels the misclosure.

Using the corrected angles: WCB_next = (BackBearing(WCB_prev) + corrected_angle) mod 360°. BackBearing = (WCB + 180°) mod 360°. With one reference bearing and all corrected angles, all other bearings are computed cyclically around the traverse - both forward and backward from the reference.

In a closed traverse all bearings are mathematically linked through the corrected interior angles. Only ONE independent reference bearing is needed - all others follow by the bearing propagation formula. When you edit any bearing cell, that row becomes the new reference and every other bearing recalculates cyclically using the corrected angles. This mirrors what happens in the field: you measure one known bearing and derive all others from the angle observations.

The permissible angular error is the maximum acceptable angular misclosure for a traverse of n stations measured with an instrument of least count LC (in arc-seconds). The formula LC × √n comes from the theory of random errors: if each angle is measured to ±LC, and errors are random, the standard deviation of the sum of n angles is LC × √n. For example, a 20-arc-second theodolite on a 6-station traverse gives 20 × √6 ≈ 49 arc-seconds.

Both rules distribute the linear closing error across the legs, but they use different weights. Bowditch rule distributes corrections proportional to leg length: c_L = −δL × (l_i/ΣL). It is standard for modern EDM traverses. Transit rule distributes corrections proportional to the magnitude of the latitude or departure of each leg. It is used when angular observations are significantly more precise than distance measurements.

The Line column shows the leg that ARRIVES at the station. Station B shows line AB (the leg travelling from A to B), station C shows line BC, and so on. The starting station (e.g. A) shows '-' because no leg arrives there. The last data row shows the final leg arriving back at the starting station, completing the closed polygon.

The precision ratio 1:n expresses the quality of a traverse as the ratio of the total perimeter length to the linear closing error: Precision = ΣL / e, where e = √(δL² + δD²). Acceptable values depend on the survey purpose: rough/reconnaissance 1:500, ordinary engineering 1:1000–1:3000, good engineering 1:3000–1:10,000, first-order control above 1:25,000.

Starting from the known or assumed starting coordinates (E₀, N₀), each station's coordinates are computed by adding the corrected latitude and departure of the arriving leg: N_i = N_(i-1) + CorrLat_i and E_i = E_(i-1) + CorrDep_i. For a correctly balanced traverse, the final accumulated coordinate must exactly equal the starting coordinates (E₀, N₀).

After applying Bowditch corrections, each leg has slightly different latitude and departure from its measured values. The adjusted length is recomputed as √(CorrLat² + CorrDep²), and the adjusted bearing is derived from arctan(|CorrDep|/|CorrLat|) with the quadrant determined by the signs of CorrLat and CorrDep. These adjusted values represent the best estimate of the actual leg geometry after distributing the closing error.

This calculator works in any consistent length unit. Enter all leg lengths in the same unit and all output coordinates and distances will be in that unit. If your field measurements are in feet, chains, or other units, convert them to metres first using our free Length Unit Converter at engineersviews.com/tools/length-unit-converter/ before entering them into this tool.

The angle correction column only populates when ALL horizontal angles have been entered for every station. The correction requires computing the total misclosure across the full traverse, which is only possible once the complete sum of angles is known. Enter all n interior angles to see the individual corrections.

This tool automates the full sequence: angular misclosure computation and correction with weighted remainder distribution, cyclic bearing propagation from any reference bearing, latitude and departure calculation with correct quadrant signs, Bowditch or Transit corrections, running coordinate accumulation, adjusted bearing and length recomputation, and all verification checks. It also provides real-time updates, step-by-step explanations, an SVG traverse plot, and CSV/PNG export.