Where is the relevance ? No detectable curve anywhere in plane surveying up to 100 square mile area. Where is this addressed . Adding spherical trig to plane survey does not change a plane into a curved surface. Be specific .

Where does it say, "no detectable curve..."? That's not what it says at all. It says:

"Plane surveying assumes the earth is flat. Curvature and spheroidal shape of the earth is neglected. In this type of surveying all triangles formed by joining survey lines are considered as plane triangles. It is employed for small survey works where errors due to the earth's shape are too small to matter.[14]

In geodetic surveying the curvature of the earth is taken into account while calculating reduced levels, angles, bearings and distances. This type of surveying is usually employed for large survey works. Survey works up to 100 square miles (260 square kilometers ) are treated as plane and beyond that are treated as geodetic.[15] In geodetic surveying necessary corrections are applied to reduced levels, bearings and other observations."

What it's saying is that for the most part the type of surveying performed is dependent on the size and scope of the job, so to speak.

Again, read the first few pages in the chapter on 'Leveling' in the book you cited. (Chapt. 5, page 60) There are 3 types of level surveying: Differential (Gravitational), Trigonometric, & Barometric. Differential (Gravitational) being the most accurate. It doesn't simply take a plane state map and apply math to it. It's an actual survey where you measure contours, slopes, altitudes as well as and in respect to measuring the curvature of the earth. If you can understand the tools used, you can understand how they directly measure curvature.