What if I told you that the specifics of our American system of land measurement, with its miles and acres and such, was the direct result of a bunch of oxen standing tired in a field during a morning’s plowing more than a thousand years ago. Would you believe me? Read on.

If you peruse historical documents pertaining to the great Adirondack surveys you will encounter a variety of measurement units. Some, like feet and miles, will be common knowledge to you. Others, like acres, will be familiar terms though you may not know precisely what they are. But a few, like the chain, which seems to be the fundamental unit of surveying distance, may well be unknown. Every major land division in the Adirondacks was originally measured in chains using an actual metal chain called a *Gunter’s chain*.

I know my mathematics and I understand triangulation but I must confess that I was ignorant of the details of surveying measurements until two years ago when I became interested because of our acquisition of Lost Brook Tract. The sole surveying map relating to the property fixed the position and dimensions of the lot in terms of chains. I had heard that terminology before but had no idea what it actually meant. I had to look it up to learn that a chain was 66 feet. Like anyone would think I took that as a rather unusual number. I knew enough about the plethora of anachronistic units of measure that have been used in the past to chalk up the 66 foot distance to an accident of history.

Have you ever wondered why our British or Imperial units of measure are so odd? We are so used to them that we don’t often step back and think how strange it is that, for example, 5,280 feet make a mile. Why not 5,000 feet (as it was during Roman times)? Why not some other round number? The Imperial system is full of opportunities to ask questions like that. In the metric system the boiling point of water is 100 degrees centigrade and the freezing point is 0 degrees centigrade. There you have some nice round numbers that seem sensible. In the Imperial system those numbers are 212 degrees and 32 degrees Fahrenheit, respectively. Is that not goofy when you think about it?

Look at the dozens of ways we measure volume in the Imperial system and contrast that to the metric system where everything is decimal-based, in other words based upon multiples of ten. It’s an eye-opener that shows our system to be dumb in comparison. I would challenge any reader to successfully convert teaspoons to quarts, let’s say, without a happy and involved session of looking up units of measure and writing multiplication problems all over a piece of paper. In the metric system the closest analog to a teaspoon is the milliliter and the closest analog to the quart is the liter. A liter is 1,000 milliliters – three places of ten, baby. If you want to convert milliliters to liters you simply move your decimal point three places and you are done in one second flat. Because of that a decimal system such as the metric system is much easier to use. Of course the metric system has the advantage of having been designed from the ground up to be consistent and sensible, whereas the Imperial system is a collection of measurements cobbled together from every corner of the globe over the entire span of human history. That’s why it’s such a mess.

The chain is an Imperial unit of measure so I was not exactly shocked to learn it was a quirky number. I think it was a day or two after I looked it up that my casual interest in gave way to obsessive fascination. My penchant for messing around with numbers had me staring at 66 one afternoon and all of a sudden it hit me. I grabbed a piece of paper and wrote the number of feet in a mile, 5,280. Then I divided it by 66.

5,280 ÷ 66 = **80**.

Aha! A round number! So the length of a Gunter’s chain was no accident after all! Now I had to know the whole story.

When it comes to the incredible richness of the history of measurement I am pretty much a dilettante. Fortunately while nosing around on-line I found an extensive dictionary of units of measurement http://www.unc.edu/~rowlett/units/index.html that is concise yet descriptive enough to weave together a narrative of how we came to adopt certain units. I contacted its creator, Russ Rowlett, Professor of Mathematics Education at the University of North Carolina at Chapel Hill and he generously granted permission for me to use his work. A portion of what follows , especially a number of details on the old English measurements, comes from his dictionary.

Gunter’s chain was developed in 1620 by an English mathematician named Edmund Gunter. Gunter was a clergyman, having graduated from Christ Church in Oxford and entered the divinity. But his greatest love was mathematics. Gunter was my kind of guy, a master of the triangle magic I have been writing about. As a theoretician he made important extensions to the mathematics of trigonometry and logarithms. But he is more typically characterized as a pragmatic mathematician who applied his knowledge to solve practical problems of the day. His chain was a very clever solution to contemporary land surveying problems that were caused by the inelegant collision of two great measurement systems used in England at the time. Like all essential English conflicts across the great arc of human endeavor and identity, this one comes down to the Normans and the Saxons.

The story begins back in history far before that, at the dawn of civilization. As human beings organized into social groups and communities they needed common understandings of all sorts of measurable things like sizes of fields, numbers of seeds or crops, various distances and so on. All these ancient measurement systems were based upon the human body in two important ways. Let’s look at both.

The first way in which the human body was important was in the development of standard units of measurement. Units of measurement for length were based upon lengths of forearm (the Egyptian cubit) , the palm, the finger and thumb (inches), the distance around a waist (a version of the yard) and a walking pace (the mile), among others.

By far the most important such measure was based upon the human foot. Here are portions of the entry from Rowlett’s dictionary for the foot:

a traditional unit of distance. Almost every culture has used the human foot as a unit of measurement. The natural foot (pes naturalis in Latin), an ancient unit based on the length of actual feet, is about 25 centimeters (9.8 inches). This unit was replaced in early civilizations of the Middle East by a longer foot, roughly 30 centimeters or the size of the modern unit, because this longer length was conveniently expressed in terms of other natural units:

1 foot = 3 hands = 4 palms = 12 inches (thumb widths) = 16 digits (finger widths)

This unit was used in both Greece and Rome… …the modern foot (1/3 yard or about 30.5 centimeters) did not appear until after the Norman conquest of 1066. It may be an innovation of Henry I, who reigned from 1100 to 1135. Later in the 1100s a foot of modern length, the “foot of St. Paul’s,” was inscribed on the base of a column of St. Paul’s Church in London, so that everyone could see the length of this new foot. From 1300, at least, to the present day there appears be little or no change in the length of the foot.

So the Normans brought a version of the Roman foot to Saxon England and the foot as we know it was standardized shortly thereafter.

The other way in which the human body affected measurement was in the development and organization of number systems themselves. The great majority of early number systems counted by tens: in other words they were base-10 or decimal systems. Ancient Egyptian, Chinese, Hindu, Greek and Roman number systems were all decimal systems in part or in whole. If you want to understand why this is and how it relates to the human body you need only look at your hands. As we are still wont to do from time to time, ancient peoples counted on their fingers (in a delightful side note I learned during my research that the Mayans had a base-20 system because they counted on both their fingers and toes!).

As described before, the advantage of a decimal system organized with place value in columns of tens is the ease with which various calculations can be done, since multiplying or dividing by ten simply requires moving the decimal point. This is the entire rationale behind the metric system. But while one version or another of the base-10 system was in common use for centuries, the development of a decimal point and its rapid calculating power was just coming into common use on the European continent in Edmund Gunter’s time. The Flemish mathematician Simon Stevin wrote a book called *The Tenth* in 1585 the purpose of which was to show people “how to perform with an ease unheard of, all computations necessary between men by integers without fractions .” Then the Scottish mathematician John Napier took up championing the decimal point, predicting that its calculating power would revolutionize mathematics. He brought it into common use in the early 16^{th} century and the decimal point as we know it became the standard in England in 1619, just one year before Gunter developed his chain. This timing was no coincidence, as we shall see.

Meanwhile in medieval England the Saxons were doing their own thing with measurement. Like other ancient systems of measurement Saxon units were derived from the need to measure distances in agriculture. In fact they came directly from Saxons’ experience plowing their fields with teams of oxen. The details are fascinating.

Teams of oxen were quite difficult to turn; consequently the Saxons would minimize the turns needed by making their fields long and narrow, plowing lengthy strips of furrows. The length of a furrow depended upon how far a team of oxen could plow before needing to stop for a rest; at that point they would be turned to plow back the other way. This distance – a furrow, long – became the *furlong*.

For shorter measurements the Saxons had the length of a pole. There is evidence that it was based upon the pole that was used to urge the oxen along. Whatever its origin it became known as a *rod* and it was the fundamental unit of measurement in their system. The length that oxen could plow before resting was about 40 pole-lengths, so the furlong was standardized as 40 rods.

Oxen were good to plow for a morning, not a full day. In a full morning of work they could plow furrows to a width of about 4 rods. Thus the dimensions of a Saxon plowed field was standardized as four rods by one furlong. The old English word for “field” was *acre*, the word and dimension we still use today. Considering that an acre had a length of 40 rods and a width of 4 rods, thus an area of 160 square rods, you can see that the Saxon system was based upon multiples of 4, not multiples of 10.

The Saxons had a version of a foot, from north Germany. The rod was about 15 of these feet. Therefore an acre was 600 feet by 60 feet. But when the Normans prevailed at the Battle of Hastings they brought with them decimal measurements and the Roman foot, which was a little bit shorter than the north German one. This led to some adjustments. From Rowlett’s dictionary:

…when the modern foot became established in the twelfth century, the royal government did not want to change the length of the rod, since that length was the basis of land measurement, land records, and taxes. Therefore the rod was redefined to equal 16 ½ feet, because with reasonable precision that happened to be its length in terms of the new foot.

It is said that Queen Elizabeth herself decreed that the mile be redefined from from it Roman definition of 5,000 feet to fit into the scheme of rods and furlongs. Since a furlong was now 660 feet under the modern foot (40 rods X 16 ½ feet per rod = 660 feet), the mile was rounded up to the next whole furlong, which was 8 furlongs: 8 furlongs X 660 feet per furlong = 5,280 feet! Now you now where that odd duck came from.

Land surveys in England continued to be performed with ropes and stakes using rods, furlongs and acres. But now an acre was no longer 600 feet by 60 feet under the old measurements. Under the new furlong it was 660 feet long. And its width? 4 rods X 16 ½ feet per rod = **66** feet. Does that number ring a bell?

Now we are finally ready for Edmund Gunter in 1620. He understood well the awkward calculations that were involved in seventeenth-century English surveying with its multiples of 4 and fractional parts of feet. Decimal calculations using a decimal point, a far superior method of calculation, had just been standardized. So he figured out a solution that married the two. He had a local blacksmith forge a chain of a hundred links, totaling 66 feet. A chain would be more durable and consistent than a rope. Now a furlong was ten chains and the the width of an acre was one chain, giving an area of ten square chains. Parts of a chain, measured in links, were hundredths. Voila! Furlongs, acres and portions thereof were now all measurable in decimals.

So well did Gunter’s chain work that it revolutionized surveying and carried the legacy of rods, furlongs and acres – carried figuratively on the backs of medieval English teams of oxen – right into the American colonies, the Adirondacks, the Totten and Crossfield Purchase and even Lost Brook Tract.

If you own some land in the Adirondacks that is measured in chains and acres, you now know what it means and where it came from. Next week we will take our new knowledge of how surveying distances are calculated and plunge back into the Adirondack wilderness.

*Photo One: A team of Saxon oxen, with rod, from the 11th Century. Photo courtesy of the The Foxearth and District Local History Society , Great Britain.*

*Photo Two: A Gunter’s chain, courtesy of Wikipedia*

This was a very informative article. Thanks for being obssesive in your research into this subject.

Amazing how this evolved over the course of human history.

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A fine article, full of things I’ve wanted to know! Thanks for doing the research.

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Hello Pete,

Interesting stuff, especially as I’ve always used bread crumbs as a navigational tool.

If you haven’t heard the song, look up Sailing to Philadelphia, by Mark Knopfler. It is a surveyor’s ballad, see the lyrics below.

Joe H.

Sailing to Philadelphia

I am Jeremiah Dixon

I am a Geordie boy

A glass of wine with you, sir

And the ladies I’ll enjoy

All Durham and Northumberland

Is measured up by my own hand

It was my fate from birth

To make my mark upon the earth…

He calls me Charlie Mason

A stargazer am I

It seems that I was born

To chart the evening sky

They’d cut me out for baking bread

But I had other dreams instead

This baker’s boy from the west country

Would join the Royal Society…

We are sailing to Philadelphia

A world away from the coaly Tyne

Sailing to Philadelphia

To draw the line

The Mason-Dixon line

Now you’re a good surveyor, Dixon

But I swear you’ll make me mad

The West will kill us both

You gullible Geordie lad

You talk of liberty

How can America be free

A Geordie and a baker’s boy

In the forest of the Iroquois…

Now hold your head up, Mason

See America lies there

The morning tide has raised

The capes of Delaware

Come up and feel the sun

A new morning is begun

Another day will make it clear

Why your stars should guide us here…

We are sailing to Philadelphia

A world away from the coaly Tyne

Sailing to Philadelphia

To draw the line

The Mason-Dixon line

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Fascinating – makes sense of so many things – has me wondering about a nautical mile

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I wish you had been one of my math teachers. I’ve learned (and understand) more by reading your articles than I learned during high school!

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Dee:

That is the best compliment a teacher can get and I thank you for it. Math is an art and an adventure. It deserves stories, not boring, rote calculations. I’m very glad that all this is followable and interesting, at least to some.

Joe:

Thanks for the lyrics! I will have to look up and see from whence this song came.

Alex:

I forget what a nautical mile is, but I know it has to do with degrees of arc so that it fits navigation by latitude and longitude.

Dan, Dan, Dan:

Oh wait, you haven’t commented here.

Thanks all, for the ongoing comments.

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I had no idea that “furrow” and “furlong” were related, and that the work ethic of oxen had anything to do with it. Bravo!

Something I tend to pedantically harp about: It is common to think that there is something special about “ten” and the decimal system. But replace this combination with, say, “twelve” and the duodecimal system, and you’ll find that you have the same mathematical conveniences. In fact, base-twelve would have been a better choice, since you can divide twelve evenly by three or four. That’s why we use “dozen” (10 in base twelve) and “gross” (100 in base twelve) for things that must be put in boxes or arranged in baking pans. Less directly, it’s why there are 360 degrees in a circle, and 24 hours in a day. Oh, if only we had twelve fingers!

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Curt:

I can hear the chiding now: “Curt Austin is a math geek.” And indeed you are, Curt! You are also absolutely correct. I would disagree in only one respect: this stuff isn’t pedantic, it’s gorgeous.

A number of you readers have let me know that you’re really into all this number stuff so far. That is of course a foolish thing to tell me because it encourages me and I could go on about this stuff all day. So look out, here we go.

The base-10 system is an accident of biology: ten fingers. The advantages of a decimal point for calculating are obvious but they are not unique to base-10, as Curt points out. A decimal point strategy would work for any number system with place value in columns, regardless of base.

But here’s the important part: is Curt right about base-12 being better? Yes, he is. Why? Here’s some food for thought.

I often tell my students that mathematics is about intuition, emotions and feelings, not Poindexterish, calculative, rule-following slavishness. At first they look at me like I’m crazy. But they get it after a couple of weeks.

Try this is you like. Write two numbers on a piece of paper: 19 and 24. Now circle the one you

likebetter. You may pick one or the other and argue for it, but what you cannot argue against is that these two numbersfeelvery different. You know they do.Try it with 35 and 36. Same thing, they feel very different even though they are only one apart, right?

Finally, write 25 on a piece of paper. Look away for a second, look back at it and write down the first thing you thought about it when you looked at it again.

Go do those things and then come back.

All right. If you are like the vast majority of people in the world you circled 24 and 36 and your first thought about 25 was that it is 5 x 5.

We like numbers we can split evenly. In mathematics we call that factoring. Not only do we like it, we are wired to do it, each of us. By and large the old saw that some people are math types and some aren’t is an unhelpful fiction. We all want to split, instinctively. We are practically compelled to do it. If you give a preschooler a pile of pebbles (and tell them not to throw any!) he or she might do a lot of things with them. But trying to make various even piles will most assuredly be one of those things.

We like numbers we can factor because we can do more with them. We like 24 better because it is 2 x 12, 3 x 8 or 4 x 6, whereas 19 is prime – it cannot be factored evenly. 35 can be factored by 5 or 7, but that ain’t nothing compared to 36 which can be factored by 2, 3, 4, 6, 9, 12 and 18. So we prefer 36. If you have $36 in singles you can divide it up a lot of ways.

So take Curt’s base-12. Even though 12 is a small number it can be factored by 2, 3, 4 and 6. That makes it very powerful for a little number and that is why Curt is right.

60 is even more powerful. It contains 12 with all its power, plus it is divisible by 5, which is a most useful addition. Look at its factor list: 2, 3, 4, 5, 6, 10, 12, 15, 20, 30! To a geek like me that’s mathematical red meat.

Here are some facts. The oldest number systems we know were base-60. There are 60 seconds in a minute. There are 60 minutes in an hour. Ancient civilizations tried their damnedest to make a year 360 days, 360 being 6 x 60. Even after observations of the Earth’s cycle around the sun showed that a year was actually a little more than that, 365.25 days, some cultures stubbornly hung on to 360 and even invented mythology to explain the inconvenient difference. We still use 360 degrees for an orbit, or a circle.

Between 12 and 60 you have our entire system of time and a good part of measurement (dozen eggs, 12 inches per foot, 360 degrees, etc.). Do you think any of these occurrences of 12 and 60 are coincidences?

Going the other way, certain of our basic numbers just cause trouble. They don’t split, they don’t fit into easy patterns, they wreak numeric havoc. The first two such troublemakers we encounter as we count are 7 and 13.

7 is either a lucky, unlucky or charmed number in nearly every major culture in the world. We have a seven day week; ancient monotheistic religious tradition tells us that six days were the hard work of creation and the seventh was

entirely different– rest – thus worthy of great prohibition and direction: thou shalt not on the sabbath and so forth. Meanwhile there are still buildings built to this day that lack a 13th floor and Friday the 13th is a stupid artifact of today’s culture. Do you think these facts are coincidences?There is always a reason for numbers being what they are in the world, and it almost always is connected with how we feel about them. These feelings we have are of great value; they make us much more powerful mathematicians.

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Which number sounds better? Hey, that’s a great bridge from the on-paper dryness of math and the real world of numerical relationships. Oh, I could get really carried away here – and I have: I gave a talk on wildflowers to the local garden club, and wandered into the subjects of parametric equations and fractal geometry (they bought some of my books anyway!)

And yesterday, I was deep into the actual sounds firmly attached to numbers, not just the treacherous overtone series of a French horn while on stage with the Lake George Chamber Orchestra, but desperately counting long rests during a complex Rachmaninoff piano piece. A rest usually spans one or more eight-bar phrases, labeled 8, 16, 24 etc. Should I count these out using all ten fingers in the decimal system? Or since I can catch errors by keying on the phrase breaks, should I convert to the octal system and count without my thumbs? I get lost while thinking about this.

Pete will no doubt write a book about the Adirondacks someday, and it will titled “Blue Line Calculus”. I’m looking forward to it.

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Well Curt, in my last comment I accused you of being a math geek. But after this latest comment from you I can go ahead and confidently refer to you as a crazed, desperate lunatic.

You play french horn. What could possibly possess a person to do that by any sort of sane free choice? That is the world’s most difficult instrument to play.

Seriously, I’d rather try to play the oboe. My credentials for this opinion? 35 years playing the trumpet, 20 of that professionally. You, sir, are a brave man. French horn is an instrument of sublimity in two ways: musical tone and deeply psychotic frustration.

Music and math.. a beautiful intertwining, proving that math is wound into the very fabric of existence.

Octal, once comfortable, would be superior. However, given the frequent use and rhythmic power of triplets in music one could argue for base-12 as better – or even base-24, if you were facile enough -subdivided as needed.

Anyhow, we just lost 17/24 of the readership…

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True. It’s all true. Except that beta-blockers have become an accepted bravery booster.

On the program for March 5: Septet for Piano, Trumpet and Strings by Saint-Saens, featuring Denise Foster of Warrensburg on trumpet – excellent!

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Pete:

Great article. When we used the Gunter’s chain in forest surveying, I knew it was an old English form of measurement, but I had no idea that it harkened back to 1620. No that’s a long life for any human technology! Especially these days. Very cool.

Dan

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