Least Squares for 1875 Retracement

I'm sure others will have discovered how useful Star*Net (and probably other LSA software without the limitations of the approach implemented in some programs such as SurvNet) can be for analyzing unusual problems, but here's an example. One of the things I did with it this weekend was to estimate the variation that Hays County Surveyor B.C. Hardin had set off in his compass in 1875 when he surveyed a certain 160-acre tract out of the public domain in what is now Comal County, Texas.

Mr. Hardin failed to note the variation to which he had adjusted his compass in running the lines of the 160 acres and it would be nice to know that value in advance since virtually all of the original 19th century surveys were made at declinations other than the the actual value at the time. Instead, those early surveys were run in relation to some "North" direction that differed significantly, but semi-systematically from true.

At the Northwest corner of the survey, Mr. Hardin reported that he took bearings to three distant topo features that modern topo maps show were between 5.6 miles and 10.5 miles distant from the reputed vicinity of the corner. The features are unmistakable and are still known by the same names that Mr. Hardin used: West Twin Sister, Lone Woman Mountain, and Lone Man Mountain.

From the Northwest corner, County Surveyor Hardin reported that he ran South, 950 varas (2640 ft.) to where he made the Southwest corner and took a bearing to what he called "Campbell Mountain" that almost certainly was the high tip of a hill about 2.7 miles away.

To make a preliminary estimate, I took positions of the four features from US Geological Survey 7.5-minute quadrangle maps and created an input file for Star*Net that duplicated Mr. Hardin's observations. The multiple bearings taken at the Northwest corner and the line running South on the same bearing basis were entered as a set of directions, DN and DM (for the line running South an approximately known distance).

Likewise, the bearing taken to Campbell Mountain at the Southwest corner was entered as a direction in relation to the line connecting the NW and SW corners of Mr. Hardin's survey.

I assigned standard errors of 0°20' (1200") to Mr. Hardin's bearings to topo features since he expressed them all to the nearest degree. A compass bearing taken to the nearest 1°, if subject mainly to rounding error, would be +/-0°30' and 68% of the values in that range would be +/-0°20'. So 0°20' was my estimate of the standard error of his compass bearings in relation to some yet undetermined North direction.

The least squares adjustment using those admittedly crude values gave an estimated bearing of S1°47'E (compared to Mr. Hardin's call of "South") and estimated distance of 2650 ft. (vs. his call for 950 varas = 2640 ft.) between the NW corner and the SW corner.

So what was most likely the variation he used in his work? The best estimate of the actual magnetic declination in the locality in 1875 was about 9°13'E. So:

[pre]

B.C. Hardin 1875

9°13’ True Az Mag North 1875
10°35’ Survey variation
-------
-1°22’ True Az Survey North
-0°25’ Convergence
-------
-1°47’ Grid Az Survey North
[/pre]

As far as I know, Mr. Hardin only ever used two variations, 10°30'E and 10°00'E, during the decade when he was County Surveyor. So, the value of 10°35' estimated from the above calculation means that Mr. Hardin most likely ran off the 160 acres with a variation of 10°30' set in his compass.