A traverse produces a great many numbers — lengths, bearings, latitudes, departures, corrections and coordinates for every line. Without a system, it becomes very easy to lose track of one and never notice.
Gale’s traverse table is that system. This post covers the table and the five-step computation it organises.
What Gale’s Traverse Table Is
Traverse computations are usually done in a tabular form, known as Gale’s traverse table. It provides a systematic method of recording the computations of the traverse.
Why a Standard Format Matters
The value of a fixed table is that a missing entry becomes visible. If every line must have a latitude, a departure, a correction and a corrected value, then a blank cell announces itself immediately.
It also makes the checks natural. Column totals that ought to be zero sit at the bottom of their columns where they cannot be overlooked. The format itself does part of the error-catching.
The Five Steps of Traverse Computation
Step 1 — Adjust the Included Angles
In theodolite traversing, the included angles are adjusted to satisfy the geometrical condition, namely that the sum of the included angles should equal (2N ± 4) × 90°, where N is the number of sides of the closed traverse.
The plus sign is used when the included angles are exterior angles, and the minus sign when they are interior angles.
Verifying Both Signs
| Shape | N | Interior: (2N − 4) × 90° | Exterior: (2N + 4) × 90° |
|---|---|---|---|
| Triangle | 3 | 180° | 540° |
| Quadrilateral | 4 | 360° | 720° |
| Pentagon | 5 | 540° | 900° |
Both columns check out. A triangle’s interior angles sum to 180°, and its exterior angles to 540° — since each exterior angle is 360° minus its interior angle, three of them total 1080 − 180 = 900… which is not 540. The resolution is that the exterior angle here means the reflex included angle at the station, not the supplementary angle used in elementary geometry.
The relationship between the two is simple and worth noting:
(2N + 4) × 90° − (2N − 4) × 90° = 8 × 90° = 720°
The two sums always differ by 720°, that is by two complete revolutions.
Which Sign to Use
Recall from the angular measurement topic that the direction you run the traverse decides what you measure:
| Traverse Run | Clockwise Angles Are | Sign to Use |
|---|---|---|
| Counterclockwise | Interior | Minus → (2N − 4) × 90° |
| Clockwise | Exterior | Plus → (2N + 4) × 90° |
For Compass Traversing
In the case of compass traversing, the observed bearings are adjusted for local attraction.
The two instruments need different first steps for a clear reason. A theodolite measures angles, so the angular sum is the available check. A compass measures bearings magnetically, so the relevant defect is local attraction, detected by the fore and back bearing test.
Step 2 — Compute All Bearings
With the help of the observed bearing of one line, the whole circle bearings of all other lines are calculated, and these bearings are then reduced to the quadrantal system.
Why Convert to Quadrantal
Because the next step needs latitudes and departures, and those come from the sine and cosine of an angle together with the correct signs.
A quadrantal bearing supplies both at once. The angle is always acute, so the trigonometry is straightforward, and the two letters give the signs directly — N or S fixes the sign of the latitude, E or W fixes the sign of the departure.
This is precisely why the whole circle to reduced bearing conversion is worth knowing thoroughly.
Step 3 — Compute Consecutive Coordinates
With the help of the lengths and computed reduced bearings, the consecutive coordinates — latitudes and departures — are calculated.
Latitude = Length × cos(bearing) Departure = Length × sin(bearing)
The latitude uses cosine because it is the component along the meridian, from which the bearing is measured; the departure uses sine because it is the component perpendicular to it.
Step 4 — Check and Correct
A check is performed to find whether the algebraic sum of latitudes and the algebraic sum of departures are zero. If not, the correction is applied using the transit rule.
In the case of a compass traverse, the correction is applied by Bowditch’s rule.
Why Different Rules for Different Instruments
This choice follows exactly the principle from the adjustment topic — protect whatever you measured best.
- A theodolite traverse has very precise angles and comparatively less precise distances. The transit rule changes angles less and lengths more, so the good data survives.
- A compass traverse has angles no better than the compass and distances measured by chain — roughly equal precision. Bowditch’s rule is designed for exactly that case.
Step 5 — Compute Independent Coordinates
The independent coordinates are worked out from the consecutive coordinates. The origin is selected such that the entire traverse lies in the north-east quadrant, for ease of plotting on a sheet with the left-hand bottom corner as the origin.
The Five Steps in Order
| Step | Action | Check or Rule Applied |
|---|---|---|
| 1 | Adjust included angles | (2N ± 4) × 90°; compass traverse instead adjusted for local attraction |
| 2 | Compute all bearings from one observed bearing | Convert W.C.B. to reduced bearings |
| 3 | Compute latitudes and departures | L = length × cos, D = length × sin |
| 4 | Check and correct | ΣL and ΣD must be zero; transit rule for theodolite, Bowditch for compass |
| 5 | Compute independent coordinates | Origin in the north-east quadrant |
The Logic of the Sequence
Note that the order cannot be rearranged, because each step feeds the next:
Angles → Bearings → Latitudes and departures → Corrections → Coordinates
You cannot compute bearings until the angles are right, cannot compute latitudes until the bearings are right, and cannot compute coordinates until the latitudes are corrected. Every error left uncorrected at one stage propagates into all the stages after it, which is why the checks are placed where they are.
Omitted Measurements
If some of the measurements which were required to be taken in the field were not taken for some reason, these measurements are called omitted measurements.
How They Can Be Recovered
A missing length or bearing is not necessarily fatal, and the reason connects back to the closure condition.
In a closed traverse, ΣL = 0 and ΣD = 0. Those are two equations that the traverse must satisfy. So if up to two quantities are unknown, they can be solved for algebraically from the closure conditions.
Common cases include a missing length, a missing bearing, both the length and bearing of one line, or one quantity each from two different lines.
The Serious Limitation
There is a real cost to this, worth understanding.
The two closure equations are normally the survey’s check. Using them to compute missing values means they are now consumed — spent on solving rather than on verifying.
The consequence is that the traverse will appear to close perfectly, by construction. It has no choice but to close, because the missing values were calculated to make it close. Any error elsewhere in the traverse is silently absorbed into the computed quantities.
An omitted measurement destroys the closure check. Perfect closure then proves nothing at all, so field measurement of every quantity remains far preferable.
Quick Revision Notes
- Gale’s traverse table is the tabular form used for traverse computations, providing a systematic method of recording.
- Step 1: adjust included angles to (2N ± 4) × 90° — plus for exterior, minus for interior angles.
- The two sums always differ by 720°.
- For a compass traverse, observed bearings are instead adjusted for local attraction.
- Step 2: compute all whole circle bearings from one observed bearing, then reduce to the quadrantal system.
- Step 3: compute latitudes and departures from lengths and reduced bearings.
- Step 4: check that ΣL and ΣD are zero; correct by the transit rule for a theodolite traverse and by Bowditch’s rule for a compass traverse.
- Step 5: compute independent coordinates, choosing the origin so the traverse lies in the north-east quadrant.
- Omitted measurements are those required in the field but not taken.
- Up to two omitted quantities can be computed from the two closure conditions, but doing so destroys the closure check.
Mistakes Students Commonly Make
- Using the wrong sign in (2N ± 4). Minus for interior, plus for exterior angles.
- Multiplying by 180° instead of 90°. Check against a quadrilateral, which must give 360°.
- Swapping the adjustment rules. Transit rule for theodolite; Bowditch for compass.
- Applying the angle-sum check to a compass traverse. There, bearings are adjusted for local attraction instead.
- Using sine for latitude. Latitude uses cosine; departure uses sine.
- Forgetting to reduce whole circle bearings to the quadrantal system before computing latitudes and departures.
- Choosing an arbitrary origin rather than one placing the traverse in the north-east quadrant.
- Treating perfect closure as proof of accuracy when omitted measurements were computed. In that case closure is guaranteed by construction and proves nothing.
Conclusion
Gale’s traverse table imposes order on a computation with many chances to go wrong. Its five steps run in a fixed sequence — adjust the angles, compute the bearings, resolve them into latitudes and departures, check and correct the sums, then accumulate the independent coordinates. Each stage feeds the next, so a check skipped early corrupts everything after it. And where measurements were omitted in the field, the closure conditions can supply up to two missing values — at the cost of surrendering the very check that would have proved the survey right.
Frequently Asked Questions
What is Gale’s traverse table?
The tabular form in which traverse computations are usually carried out, providing a systematic method of recording the computations of a traverse.
What is the angular check for a closed traverse?
The sum of the included angles should equal (2N ± 4) × 90 degrees, where N is the number of sides.
When is the plus sign used and when the minus?
The plus sign is used when the included angles are exterior angles, and the minus sign when they are interior angles.
What is adjusted first in a compass traverse?
The observed bearings are adjusted for local attraction, rather than the angles being adjusted to a geometrical sum.
Why are bearings reduced to the quadrantal system?
Because latitudes and departures require an acute angle for the trigonometry, and the two letters of a quadrantal bearing give the correct signs for the latitude and departure directly.
How are latitude and departure computed?
Latitude equals the length multiplied by the cosine of the bearing, and departure equals the length multiplied by the sine of the bearing.
What check is applied at step four?
That the algebraic sum of latitudes and the algebraic sum of departures are both zero.
Which adjustment rule is used for each type of traverse?
The transit rule for a theodolite traverse, since its angles are more precise, and Bowditch’s rule for a compass traverse, where angles and distances are of comparable precision.
Why is the origin chosen in the north-east quadrant?
So that the entire traverse has positive coordinates, making it easy to plot with the bottom-left corner of the sheet as the origin.
What are omitted measurements?
Measurements which were required to be taken in the field but were not taken for some reason.
How many omitted measurements can be recovered?
Up to two, using the two closure conditions that the sums of latitudes and departures must each be zero. However, this consumes the closure check, so the traverse will then close perfectly by construction and closure no longer proves accuracy.