Andy Nold's post on March 2, Calc points on Latitude, brought forth several replies that included references to tables of offset lengths for given tangents and latitudes. Tables were the "black boxes" of their day and, like their modern electronic counterparts, there are interesting theory and formulas supporting them.
For tangents to latitude, table numbers are based on the geometry in this diagram:
Calculating the offset, d, turns out to be an application of the Pythagorean Theorem. The right angle is at the intersection of the radius and the tangent. Here are the formulas that achieve the result:
And here's a sample calculation. Sorry about the decimal degrees.
One of the interesting things is the radius of the parallel, R. It is the same thing as the mapping radius for a single-parallel Lambert conformal projection. Its formula is in the 2009 Manual of Surveying Instructions.
This is an easy application for a spreadsheet or a programmable calculator, so the tables can be updated to a modern "black box."
MathTeacher
Overall your answer is not too bad for the rounding you did.
Clark 1866 Ellipsoid, semi-major axis = 6378206.4 meters (held fix)
6378206.4 * 3.28083333333333333 = 20,925,832.164 feet
R= 24,973,380.9674 not 24,973,380.77
d= 20.09385918 or rounded to 20.0939 not 20.0903
Azimuth no change to the number of places you show.
e^2= 0.006768657997 as per TM 5-237 Surveying Computer's Manual, 1964
Respectfully
JOHN NOLTON
Tombstone, AZ.
You're right, that was a typo. My spreadsheet shows 20.09383506, which agrees with your calculation.
But there is one anomaly in the historical records. Table V in the 1894 Manual has lots of numbers that don't match the other sources. It's available as a free Google ebook. For example, if you look at a 6-mile tangent at 30 degrees latitude, that table shows 12.88 feet, while all the others and the formula approach give 12.83. Clarke did make an error in converting meters to feet and more than one set of parameters exist for his 1866 ellipsoid. Or perhaps that table is based on a different ellipsoid. I didn't pursue the source of the differences.
The numbers in the tables are calculated directly from the ellipsoid with no adjustment for elevation and, evidently, used at the surface for surveying good results.
Thanks for the discussion. I am a little overloaded with projects right now but will try to come back, absorb and comment on the posts when I can.