ASSUME THE EARTH MODEL TO BE A SPHERE OF CIRCUMFERENCE 25,000 MILES.

QUESTION: WHAT IS THE LENGTH OF A DEGREE OF A MERIDIAN??

All meridians are halves of great circles. So, the length of one degree is 25,000/360 or 12,500/180 = 69.44 (approximately) miles. One degree measured along a meridian tells you how far north or south you are of the equator. So, this also is the length of one degree of latitude. Note that along a single meridian, every point will have different latitude (along a single parallel, every point will have the same latitude).

QUESTION: WHAT IS THE LENGTH OF A DEGREE OF A PARALLEL??

One parallel is a great circle. All other parallels are small circles. So, the length of a degree of a parallel measured along the equator is, as above, 69.44 miles. Along all other parallels, it depends on the size of the small circle. For the following sequence of steps, please refer to the figure below.

1. r=25,000/2p =3978.8769381111, the radius of the equator
2. Find s, the radius of a small circle. Given that r and q are known; q is the central angle in the sphere that measures latitude. Because s and r are parallel, q also appears with P as a vertex (alternate interior angles of parallel lines cut by a transversal are equal).
3. The NS polar axis is perpendicular to s (the way the system of parallels was set up).
4. So, looking at the right triangle with s as one side and r as another, it follows that cos q =s/r=s/3979 so that s = 3979 cos q .
5. So, for example, at 40 degrees north latitude, s=3979 cos 40 = 3048 which is the length in miles of the radius of the small circle at that latitude. The circumference of that small circle is 2pi times the radius or 19152 miles. Therefore, the length of one degree measured along the small circle at 40 degrees north is 19152/360=53.2 miles. This is also the length of one degree on longitude at 40 N. Every point on this small circle has different longitude. Every point on a single meridian has the same longitude.

A RELATED QUESTION:

AT WHAT LATITUDE IS THE LENGTH OF ONE DEGREE OF LONGITUDE EXACTLY HALF THAT OF ONE DEGREE OF LONGITUDE MEASURED AT THE EQUATOR?

At latitude 60 degrees…the radius s is half the length of the radius r; cos 60 =0.5.

Or, using the calculations above,

Half of 69.44 is 34.72. So, we need to have 2*pi*s/360=34.72; solving for s, s=1989.311144993; but also, s=3979 cos X. So, cos X = 1989.311144993/3979=0.499952537068; X=60.00.

This same strategy would then work not only for "half" but for any other fraction.