## Why are there 63360 inches per mile?

#### January 14, 2010

Sometimes people complain about the British system of units, which now only the Americans use, the British having gone metric like everyone else. For example, some people don't like the fact that there are 63360 inches per mile. They say this is a silly number that's hard to remember.

These people must be nuts! The number

63360 = 27 × 32 × 5 × 11

is perfectly nice. I can't imagine anything simpler than this. Besides, the USA reached its technological preeminence on the basis of this system, and we're not going to change now!

Seriously:

There are 12 = 22 × 3 inches per foot,
3 feet per yard,
and 1760 = 25 × 5 × 11 yards per mile,

and multiplying these we get 27 × 32 × 5 × 11 = 63360.

All the above numbers are fairly nice round numbers except the number of yards per mile. Where did that nasty factor of 11 come from???

Delving into this mystery, one meets a unit of distance called the "chain". From the 1600s to the end of the 1800s, surveyors liked to measure distances using chains. They were less stretchy than the earlier "cords". A fellow named Edmund Gunter is associated with the chain system. A mile was defined to be 80 of Gunter's chains; each chain was 66 feet long.

So, the factor of 11 comes from having chains that were 66 feet long.

But where does the 66 come from?

In Gunter's system each chain was divided into 100 links, so a link is .66 feet long. That's almost 2/3 of a foot. At first I thought this is where the factor of 11 snuck in: from trying to approximate 2/3 with decimals. But this doesn't make too much sense, since 2/3 is closer to .67.

Eventually I found out that the factor of 11 comes from something completely different. It goes back to an earlier British system of measuring fields, which involved a unit of distance called a "rod" or "pole" or "perch". Four rods equal 66 feet. And, to make his system compatible with this older system, Gunter made his chains 4 rods long!

You can find more details in this book, which Gerry Myerson brought to my attention:

I quote:
Four perches measured 22 yards, a strange distance that makes sense only in the context of the traditional units used for measuring land. Like all units of land measurement, a perch, also known as a rod or a pole, originally varied according to the quality of ground: a perch of poor soil was longer than one of fertile soil, but in the course of the sixteenth century it became standardized at 16.5 feet. This inconvenient length was derived from the area of agricultural land that could be worked by one person in a day - hence the variability. The area was reckoned to be 2 perches by 2 perches (33 feet by 33 feet). Thus a daywork amounted to 4 square perches. Conveniently, there were 40 dayworks in an acre, the area that could be worked by a team of oxen in a day, and 640 acres in a square mile. It was significant that all of them were multiples of 4, a number that made it simpler to calculate the area of a four-sided field.

Gunter divided the chain into 100 links, marked off into groups of 10 by brass rings. On the face of it, the dimensions make no sense: each link is a fraction under 8 inches long; 10 links make slightly less than 6 feet, 8 inches; and the full length is 66 feet. In fact, he had made a brilliant synthesis of two otherwise incompatible systems, the traditional English land measurements, based on the number 4, and the newly introduced system of decimals based on the number 10.

The "long" units of distance fit together reasonably well in this system:

There are 4 rods per chain,
10 chains per furlong,
and 8 furlongs per mile.

So do the "short" units:

There are 12 inches per foot
and 3 feet per yard.

But the short units clash horribly with the long ones: there are 5.5 = 11/2 yards per rod, or 16.5 = 33/2 feet per rod. One has to wonder why people settled on this number and not some more convenient number like 16.

In fact, this decision was very old. Richard Tobin pointed out to me these remarks by Alan Chace:

An often repeated and published myth within the surveying profession concerns the origin of the measuring tool known as the rod. The surveyor's rod, also known as a pole or a perch, is a particularly important piece of material culture. Essentially a long wooden staff of 16½ English feet, this standardized length allowed early surveyors to measure distances accurately. More importantly, this length was the basis of a system of area calculation that also allowed the early surveyor to very accurately lay out a specified area of land without complicated calculation. This particular length of 16½ feet has had an enormous influence on surveying and land development, although the rod itself has not been in common use for over two centuries.

The classic text Boundary Control and Legal Principles by Brown, et al., notes that in the 16th century, the rod was established as the length of the left feet of the first 16 men out of church one Sunday morning. Brown cites "The Amazing Story of Measurements." This same story is often repeated in surveying texts and publications. Indeed, it is a part of the folklore of the surveying profession.

While the story that the rod was originally meant to represent 16 human feet is probably true, the reality is that the rod had been in common use many centuries earlier than Brown suggested. As early as 1270, Edward I of England issued a Royal ordinance titled Assize of Weights and Measures, which at this date defined into law the length of a perch. This indicates that the perch was in use prior to this time.

Usually, a larger measurement is defined in terms of being an equal number of a smaller measurement, much in the sense that one pound is considered to be 16 ounces. And yet as early as 1270, the perch is not considered as 16 feet but rather specifically defined as 16½ feet. This is a critical point to consider when attempting to date the origins of the rod itself. The parties who considered this issue in the 1270s were actually unwilling to define the pole in terms of an even number of feet. It is as if that even by this early date, a pole of a particular length, an inconvenient 16½ English feet, had been established for so long that it was not considered feasible to define it otherwise.

It is worth special note that the English rod or perch is within .03 feet of equaling 16 old Danish "fods", and is within a similar tolerance to equaling 16 old Prussian "Rheinfuss", both of which are lengths of measure slightly longer than, but no doubt corresponding in origin with, the English foot. We can therefore speculate, but not document, that the origin of the rod may go back as far as the Germanic Invasions of England around the 9th century, when the rod may well have equaled 16 fods. Several scholars have noted that the old North German land measure of 16 Rhineland feet, the "ruthen", is similar to the rod in length and may have been introduced into England by the Saxons. The use of the term "perch" for this tool is most likely derived from the Latin word "Pertica".

For more details, including references which I have omitted here, see:
Michael Press points out an interesting nuance: while Chace writes "the rod itself has not been in common use for over two centuries", canoe trippers still measure their portages in rods.

Summarizing, it seems that the number 63360 comes from these facts:

There are 12 inches per foot,
33/2 feet per rod,
4 rods per chain,
and 80 chains per mile.

Multiplying these numbers we get 12 × 33/2 × 4 × 80 = 63360.

On second thought, I agree: this system sucks!

Of course the SI units of length is really not much better except that all the units of length are related by powers of 10. I don't mean to downplay the tremendous advantage of working in consistently in a given base, like base 10. I'm not knocking metric: it's great. But is it really so scientific and sensible to define a meter as:

the distance that light emitted by a cesium 133 atom transitioning between the two hyperfine levels of its ground state will travel as it vibrates exactly 9,192,631,770 / 299,792,458 times?

A careful study of where these numbers came from would lead us down some very tangled paths, just like the question why there are 63360 inches in a mile.

Indeed, such puzzles are nice illustrations of what I half-jokingly call the fractal texture of history. Every historical fact or event becomes more and more complex as one studies it in greater and greater detail. We started by wondering why the number of inches in a mile was divisible by 11... and our quest for the answer led us to unsolved questions about the Germanic invasions of England!