Why does a mile have 5280 feet in it?

Since there is no surviving documentation of the choice of 5280 feet as the
definition of the mile, we can only speculate.  Such speculation is inherently
risky, but we proceed nonetheless.

Note that a mile is eight furlongs, where a furlong is 660 feet.  This is sometimes
explained as the distance that an ox can plow in a day, but this is a clear example
of post hoc justification.  So while the word "mile" probably derives from a
thousand paces, which is about eight furlongs, we cannot explain the length of the
mile by reference to the furlong.

5,280 factors to:

  2, 2, 2, 2, 2, 3, 5, and 11

This suggests that it was chosen at least in part because it is evenly
divisible by:

  2
  3
  5

  4 = 2 * 2
  6 = 2 * 3
  10 = 2 * 5
  15 = 3 * 5

  8 = 2 * 2 * 2
  12 = 2 * 2 * 3
  20 = 2 * 2 * 5
  30 = 2 * 3 * 5

  16 = 2 * 2 * 2 * 2
  24 = 2 * 2 * 2 * 3
  40 = 2 * 2 * 2 * 5
  60 = 2 * 2 * 3 * 5

and so on.

We have not divided by 11, and it seems like a curious factor.  Why is
it in there?

The key to this may be the approximation to pi of 22/7.

Suppose that the definers of the mile wished to define the unit so that
if a circle has a integral number of feet in its diameter, then it will
have an integral number of feet in its circumference.  This might be
desirable if building units (like bricks) come in standard lengths, like feet.
Since pi is irrational this cannot be done with absolute precision,
but nothing in nature is absolutely precise.  The definers only needed
to work within a certain accuracy.

Here is a table of the various integral fractions of a mile and the
diameter that corresponds to a circle with circumference equal to
that fraction, using 22/7 as the ratio of circumference to diameter.
Because of the factor of 11 in 5280, each of these diameters is a integer.

1       5280     1680
2       2640      840
3       1760      560
4       1320      420
5       1056      336
6        880      280
8        660      210
10       528      168
12       440      140
15       352      112
16       330      105
20       264       84
24       220       70
30       176       56
40       132       42
48       110       35
60        88       28
80        66       21
120       44       14
240       22        7

Because 22/7 is such a good approximation to pi, the relative error of
the integral diameter to the true diameter is 0.04%.  This corresponds to
less than one hundredth of an inch over one foot.  Even today it probably
is not feasible to mass produce bricks with that degree of accuracy.
It certainly was not possible at the time that the mile was defined.

Here is a C++ program showing that 5280 is the unique answer for all
numbers from 100 to 10,000:

//
// What number from 100 to 10000 has the property that it is evenly divisible
// by all the factors in array d, and if a circle is built with 
// (number / factor) units in it, then the diameter of that circle
// will be an integral number of units within better than one part in 1000?
// Answer: 5280
//

#include <cstdlib>
#include <iostream>

#using namespace std;

#define lengthof(x) (sizeof(x)/sizeof(*x))

long d[] = {
  2,
  3,
  4,
  5,
  6,
  8,
  10,
  12,
  15,
  16,
};

struct elem {
  double err;
  double rad[lengthof(d)];
  long num;
} a[10000];

int compare(const void *v1, const void *v2)
{
  double diff = ((elem *)v1)->err - ((elem *)v2)->err;
  return (diff > 0) ? 1 : ((diff < 0) ? -1 : 0);
}

int main()
{
  double pi = 3.141592653589793238462643383276;
  for (long i = 0; i < 10000; ++i) {
    long n = i + 100;
    a[i].err = 0;
    for (long j = 0; j < lengthof(d); ++j)
      if (n % d[j])
        a[i].err += 10;
      else {
        long frac = n / d[j];
        a[i].rad[j] = (frac - long(frac / pi) * pi) / frac;
        if (a[i].rad[j] > 0.001) ++a[i].err;
      }
    a[i].num = n;
  }
  qsort(a, lengthof(a), sizeof(*a), compare);
  for (long i = 0; i < 10; ++i)
    cout << a[i].num << "=" << a[i].err << " " << endl;
}