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. First, let's factor 5280. Its factors are
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 in the well-known 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 (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 //

  1. include <iostream>
  2. include <stdlib.h>
  3. 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);

}

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 / dj?; 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;

}

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