Cecil said that the straight line that surveyors measure by sighting between aligned landmarks, a line known as a “great circle”, isn’ t a straight line. Wrong. A great circle is universally agreed to be what a “straight line” means on the Earth. He justified his claim by saying that a geodesic can deviate from a parallel. Yes, and that’s because a parallel (other than the equator) is not a straight line. Drive along a parallel other than the equator, and you’re continually (if very gradually) turning your car right or left. Drive along any great circle, and you’re driving in a straight line.
When you start a thread in response to Cecil’s column, it’s helpful to other readers to provide a link. In this case, I presume, it’s How did the states establish long straight borders before GPS? - The Straight Dope
Of course. That was the point of the column.
I think the OP is splitting hairs. Cecil’s point was a “line-of-sight” straight line on the Earth isn’t the same as a straight line on a map (particularly, on a map where the lines of latitude are straight). And he’s right.
If you think Cecil should have said “a straight line doesn’t always look the way you might naively expect it to on a map” instead of “a straight line isn’t always a straight line”, well, I’d say that pithiness is part of why he’s employed as a newspaper columnist and you’re (presumably) not. Anyway, Cecil elaborated in such a way as to make the meaning of this phrase perfectly clear. What more do you want?
Let’s see if I remember my geometry.
Each line of latitude is about 60 miles apart. Technically, 100km or about 62.5mi.
If you sliced the earth along the 45deg latitude, you’d get a giant beanie with the rim being (cos 45)12,756.2km in diameter, or about 4510km radius.
For the 44 latitude slice, that radius is 4588km.
So the circumference difference is 78PI2=490.1km greater.If we had a 1km slice of land from the 44th to 45th parallel; but the sides went true north; that slice would be:
Circumference at 44lat = 4588PI2= 28827.3km
Circumference at 45lat = 4510PI*2 = 28337.2km
so we would have a sort of curved trapezoidal shape 1 km at the bottom, 100 km long, and (28337.2/28827.3)=0.983km across the top.
In 100km you lose 27m or about 28yards at 45 latitude.
Similarly, then, if you went west 1km, north 100km, and east 1 km, your 4th side would be off by about 28 yards in 100km. 880 yards in a half mile, so you’d need 880/28 or about 3100km (2000 miles) to be off a half mile.
Any serious errors on a short distance would probably be surveyor error…
Of course, between 89lat and the pole, you get a triangle; at the equator, you almost don’t notice the difference. It gets worse the farther north you go.
Or did I slip on the calculator buttons?
Uh, something fishy here…
I guess that’s meant to mean “borders that run east and west”, not “borders that separate east from west”. Gotcha. Carry on.
The result seems about right, but I think you’re answering a question that wasn’t asked. What you seem to be doing is determining how much a rectangle surveyed on a sphere would differ from a rectangle surveyed on a plane. As it happens, Cecil addressed essentially this question recently in his Straight Dope Chicago column - see:
http://chicago.straightdope.com/sdc20091008.php
The more pertinent question here, I think, is how much a geodesic would differ from a parallel of latitude. My trigonometry is too rusty to allow me to compute this exactly, but my understanding is that at the mid-latitudes a geodetic arc of 10 miles would differ from the parallel starting at the same point by about 50 feet. The rate of divergence increases with distance.
Folks:
Although an engineer with an extensive mathematics background, I try to tell people that they can derive almost any answer to any problem just by using what they already know.
What’s the circumference of the earth at the 45th parallel? Try this inexact, but close, method.
What is the distance between Portland, Oregon and Boston Mass? About 3000 miles, right? How many time zones are there between Portland and Boston? Four, right? Let’s do some simple math – 3000 miles divided by 4 (time zones) equale 750 miles per time zone, right? How many time zones are there on earth? (No one seems to get this one at first). Of course - there are 24 time zones. Now comes the answer – if there are 24 time zones on earth and the distance between time zones is about 750 miles at the 45th parallel then 750 miles x 24 time zones equals 18,000 miles.
Now, that’s pretty close and it shows many people that they can derive close answers using facts they already know. Not everyone knows algebra, trig or calculus, but everyone wants to be able to answer questions.
“Circumference of the Earth” having a very elastic meaning in your problem. 
I don’t think it’s exactly common knowledge that the distance between Portland and Boston is about 3000 miles.
Powers &8^]