Achieving closure: the first Bowditch adjusted traverse?

Origin of the Bowditch method. Published in 1802 Nathaniel Bowditch’s exposition on navigation is famous among surveyors for its description of the traverse adjustment method which bears his name. The method first appears for the computation of Traverse Sailing.  Bowditch correction was widely used until EDM replaced mechanical distance measurement and least squares took over. The title page says it all: ‘many thousand errors are corrected’. Thanks to Google books I took a look at an early* worked example and found it is more or less as taught to me at Vauxhall college, the method evolved to include proportional distribution of both angle and distance error and, such is the legendary status of his work, still bears the name ‘Bowditch’.


Among the many useful nautical wonders Nathaniel Bowditch offered in his 1802 ‘Epitome of Navigation’ is a method for the correction of survey travserses

Also_02which, until the computer age, was the standard method: the Bowditch Correction. Nathaniel describes, on p108,  the problem with alomst brutal clarity:

‘but it most frequently happens the numbers do not agree’ is an acceptance of ‘real world’  measurement with the advice ‘in which case the work must be carefully examined, and if no mistake be found, and the error be great, the place must be surveyed again…’  bourne out of experience. I don’t know if Bowditch himself carried out traverses but he certainly was familiar with travserse outcomes.

The first expression of the method is not quite what we know as Bowditch today. This is before the railway age. The method descibed is far from the backbone of civil engineering engineering practice which was some 20 years in the future, it is a version of a tried and tested dead reckoning correction for marine navigation. The mix of nautical and land survey terms reveals the navigational origin, for example Bowditch uses ‘Departure’ for the Easting axis and ‘Lattitude’ for the Northing and bearings are given relative to the 4 compass cardinals.

The first appearance of the method is a navigation example, here is the 1807 version (from the second edition in which the Surveying chapter makes no mention of traverse or adjustment) which appears as ‘Travserse Sailing’ the identification of the misclosure by tabulation is there:

The method appears in the Surveying chapter in or around 1840, by which time Mr Bowditch had passed on and the decimal system had been adopted for computation.

I’m not a surveyor of 1840s vintage so I found the worked example difficult to folow as:

  • The starting co-ordinates are not given
  • Neither whole circle bearings or internal angles are given
  • The direction of computation is not given
  • Co-ordinates are in mixed XY and YX axes
  • no angle sum check is used.

The figure, plotted by bearing and distance looks like this:

which shows the misclosure at A as:

Distance = 0.1264, Delta X = 0.0774, Delta Y = 0.1000

By the current method of included angle the figure looks like this:

The sum of the interior angles [ (N-2)*180] gives  540.  The angles given (A=155  B=15 C=215 D=85 E=70 ) sum to  540: there is no angular misclosure, he has used whole degree values. This is a theoretical example after all.

Bowditch gives the following as the corrections for each leg, rounded to 2 dp:

The misclosure in Northing (0.1) is divided by the number of legs, the misclosure in Easting is applied as an unequal fraction (42.10/.08) with the larger portion given to the longest legs. Bowditch tells us the Northing error is negative and the Easting is negative.

1st Description of proportional correction by length of leg. Bowditch then describes the correction thus:

Find the error in latitude, or the difference between the sums of southing and northing : also the sum of the boundary lines. Then say: As this sum is to the error in latitude, so is the length of any particular boundary to the correction of the corresponding difference of latitude, additive if in the column whose sum is the least, otherwise subtractive.

Which is an awkward way of saying:

  • Correction N= ΔN (Length of leg)/(Σ dist)
  • Correction E=ΔE (Length of leg)/(Σ dist)

The language is difficult but there it is: divide the sum of the distances by the difference in E and N for each leg to find the correction: I’m not certain this is as its described in the 1st 1802 edition and or how the later editions came about but the origin is clear enough: the ‘immacculate’ Bowdithch did indeed come up with a robust and simple method for adjusting traverses. Methods for determining the differences (partial co-ordinate method) may have come along later but the new idea is clear: proportional distribution of distance error.

Bowditch’s foldout frontispiece map shows his travserse across the Atlantic.


From March 25th to April 10th Bowditch’s path is plotted day by day.

*Google books has many digitised editions. The earliest I can find, is an 1809 version which has computation in imperial units, and the method is decribed in ‘Traverse Sailing.’ Later versions ( I have used an 1839 edition here) are decimal and include the method in a ‘Surveying’ chapter. Other than in the auction house pages the 1802 edition is an online rarity.



About billboyheritagesurvey

Heritage worker
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2 Responses to Achieving closure: the first Bowditch adjusted traverse?

  1. Roger J Clegg, LS says:

    If only the green surveyors would be admonished on the principle of “ the accepted practice of the day”; they wouldn’t be calling my father’s and grandfather’s monuments off. These guys take a gps stick out, get multipathing through the redwood canopy; then dump this dubious data through a least squares program.
    None of them know how to read a vernier on a transit or slope chain. One in a thousand could be classified a “retracement surveyor”, the other 999 “UFO surveyors” uninitiated fumbling operators.

    • Roger,
      This calls to mind the great quote from Digges in Pantometrioa 1571:

      “The geometer, how excellent be he, leaning only to discourse of reason, withut practice (yea and sundrie ways made) shall fall into manifole errors, or inextricable Laberinthes.”


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