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I would start by detemining the theoretical mid-length sag of a cable. The standard engineering mechanics equation for the sag of a catenary curve (i.e., a freely sagging cable) is:
sag = (T/W) *[cosh( (W*L)/(2*T) )- 1]
where
T= mid-span cable tension
W = weight per unit length of the cable
L = distance between the supports
A 250 lb monofilament fishing line has a diameter of 1.6mm. Assuming a density for nylon of 0.036 lbs/in^3 yields a weight/inch of 0.003116 lbs/in for the fishing line.
If the attachment points are 2 miles apart the midspan sag is about a half-mile! Here are the more precise calculations via Excel.
How should the sag be measured? From a straight line between the two end points or relative to a curve with a radius of Earth? In the real world example it would be relative to the arc of the Earth.
If you'd like to know more about the shape of the earth I recommend:
Measure of the Earth: The Enlightenment Expedition That Reshaped Our World
It's a surprisingly interesting story.