Sorry, but that isn't correct either.

the error for 1 degree is 1/60:th of the distance travelled.

1 mile (US) = 5280 feet
5280/60= 88 feet for a 1 degree error.

I am glad you brought up the 1/60 rule Bork. A lot of people have never heard of it.
The 1/60 rule is a very useful approximation for course corrections, especially for pilots in small planes.
They usually count it over 60 miles and being miles off course.

The 1/60 rule simplifies the calculation, but the result you get is the distance off for slightly under one degree instead of a full degree.
(0.96 degrees)

If you use the circle calculation for the arc of a circle you get
2pi*5280/360 = 92.1533845
It is still an approximation because you are measuring the arc instead of the straight line(chord) but it is a bit closer.

If you use the sine of 1 degree you get 92.148706 feet
If you use the tangent of 1 degree you get 92.1627428 feet

The 1/60 rule is a lot easier to use though, and the results it gives are fine for most uses.
If we were able to get close enough with our hand held compasses that being 4 feet off in a mile mattered we should be very pleased with ourselves indeed.