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"You mentioned taking the sine of very small numbers which the logarithmic tables didn't cover.  I note the log (sin (1'.74)) = 6.704.  I worked that in reverse asin (10^(6.704-10)) by calculator to verify.  Did you obtain this intermediate result by calculator or by a method available 100 years ago?"

This is actually easy, and the "method" has been known for centuries. The sine or tangent of a small angle is equal to the angle. Of course that angle has to be converted from minutes of arc to a pure number (also known as "radians") by dividing it by 3438 (or 60·180/pi for full accuracy). This small angle equivalence is accurate to about 1 part in 10,000 for angles up to 5°, and it's accurate to better than one part in a million for angles smaller than 0.5°. If you want to jump straight to the logarithm (+10) of the function, it's easy to show that you can calculate it from
  logsin(A) = log(A) + 6.464
when A is the angle in minutes of arc.

Just a reminder: you can calculate the angular size of any object in minutes of arc from
  angle(m.o.a.) = 3438 · (size across the line of sight) / (distance out, along the line of sight).

Frank Reed