21 November 2007

Acres are strange.

Canada to Announce Vast New Park, from tomorrow's NYT.

The size of the park is referred to as "25.5 million acres", which seems kind of silly to me. The whole point of an acre is that it measures areas which are too large for, say, square feet and too small for square miles. I have no idea how large 25.5 million acres is, until I convert it to square miles -- 40,000 square miles. That's approximately the area of Pennsylvania, or five times the area of New Jersey. If the park were circular, it would have a radius of 112 miles -- that's kind of a useful way to picture it, since that says that if you were in the center of the park the borders would be over a hundred miles away. I have some idea what a hundred miles looks like.

But that relies on the fact that an area is the square of a distance. An acre is 43,560 square feet, or 1/640 of a square mile. What's the square root of an acre? 208.7 feet, which isn't any conventional length unit. The history is that an acre is a rectangle one furlong by one chain, or so Wikipedia says, so 208.7 feet is the geometric mean of a furlong (660 feet, or 1/8 mile) and a chain (66 feet, or 1/80 mile), or √10 chains. The weird definition is agricultural -- "[t]he acre was selected as approximately the amount of land tillable by one man behind an ox in one day", and it's easier to plow a long, narrow rectangle than a square. But haven't we moved past this?

Of course, in the end trying to rationalize the customary measurement system is silly. I suspect many of you are just saying "why don't you Americans use metric already?" In fact, one of my students, who is not American, actually wrote this once on a homework assignment, in his solution to a problem which applied calculus to physics where measurements were given in customary units.

11 comments:

Michael Cassidy said...

Oh wonderious amazing Isabel,

[you and John are my heros, though I've got a couple of others].

you can comprehend 40,000 square miles??!!??
I can't; except as a BIG area.

I think one of the problems of evolution astronomy and geology is the numbers.
Most people have no idea of what a million years is; in fact what 3,000 years means.

So when we all talk in terms of geological and astronomical times and distances we might as well be talking equations/math except geological and astronomical times and distances are not scary just incomprehensible.

Anyway Happy Thanksgiving and Jimmy Carter was right the US should have switched to metric.

.mau. said...

it's really fun, since in Piedmont (Italy) we have a similar measure for area, the giornata (literally, "day worth") which is exactly the amount of land tillable by one man behind an ox in one day.
Actually a giornata is 5% less than an acre. Who knows, maybe in Piedmont we use square fields, so plowing is more difficult :-)

John Armstrong said...

Here's how I'd come to grips with 25.5 million acres. Houses in many suburban developments are on quarter-acre lots, so I know what that looks like.

Now we're talking about a quarter of a hundred million acres here. That's enough to divide up the entire population of the USA into three-person families and give each of them a suburban house on its own quarter-acre lot in the region.

Mark Reid said...

With people like you questioning the usefulness of imperial units maybe there is still hope for the US converting eventually.

That would then just leave Liberia and Burma. :)

Paul Soldera said...

It seems to me that the original definition of an acre - been some combination of area and time, is a hell of a lot more intuitive than its definition in square feet. 25 million acres makes more sense as eons of ploughing - like, a really, really, large area!

Maybe we should go BACK to defining areas in the same way, just update the technology? 25 driving days to cross? 4 plane hours? Seems to make sense.

John Armstrong said...

Maybe we should go BACK to defining areas in the same way, just update the technology? 25 driving days to cross? 4 plane hours?

Except there's no standardization, and that breaks possibilities of communication. Is the car a Prius that doesn't have to stop for gas very often, or a Hummer that has to stop every 100 miles? Does the plane have a tailwind or not?

Every gain you make in intuition you lose in precision. And there's only so much memory one can be expected to devote to units. Look at how few people really have a good feel for both farenheit and celsius, just as an example. Since we need precision sometimes, it's better to stick with well-defined measurements and just try to build an intuition for them as we go.

Paul Soldera said...

just as an example. Since we need precision sometimes, it's better to stick with well-defined measurements and just try to build an intuition for them as we go.

I don't see why you can't have both. If you're not worried about a conveying things to the exact square mile, why use it? An acre started out as a commonly held convention about average work by an average man in an average day. Works for cars, planes just as well. Besides, we use comparisons all the time for area, height, weight etc. It's not like you need to learn 'units' for common sense comparisons, that's absurd.

John Armstrong said...

But that's exactly what you're proposing: units for common sense comparisons. And as for why not have both, it's exactly as I said. People are, on average, barely good at keeping one set of units straight at a given scale, let alone one for rough ideas and one for precision.

Paul Soldera said...

I'd argue no on that one I think. I think people are GREAT at keeping 'units' for precise and rough estimates. We do it all the time. In fact, I'd say it is the ONLY way we understand measurements outside of our daily scale. No one knows what a really small measurement is until you talk about it in terms of the 'width of a human hair', for instance.

Scott Carter said...

A foot is a convenient unit. It is divisible into 12 inches, and 12 can be divided by 1,2,3,4 and 6. I mentioned this to friend who honestly suggested using 30 cm units. Yes, 30 is a nice size, nearly a foot. But then you are moving out of that mysterious base 10 predjudice into some strange mixed metaphor.

The only advantage in metric units that I see is conversions from length to area to volume. On the other hand, units that are divisible by 12 are good for various countings. I never anticipate eggs being packaged in a 10 pack.

On a personal history note, my dad marketed eggs sold as a baker's dozen: a cartoon of 12 with a coupon for a 13th. The consumer could save twelve coupons and get dozen free.

Now what about these annoying conversions. One teaspoon is 5 ml. This makes a tablespoon 15 ml, and and ounce 30 ml. Thus one cup is 240 ml. And a 12 ounce can of carbonated beverage is 360 ml. But the side of the can states that it contains 354 ml --- round off error.

Meanwhile, I measure my gas mileage in miles per gallon. Converting to metric measures fuel efficiency in per square meter. Can any one descibe why that is a mesure of fuel efficiency?

The debate to convert to metric units is ill founded. If the aveage person has to convert units, that person might learn how to do arithmetic, or helpmake TI more rich.

cwitty said...

Converting to metric measures fuel efficiency in per square meter. Can any one descibe why that is a mesure of fuel efficiency?

Well, it makes more sense to look at the inverse of that number, and measure fuel efficiency in square meters.

Consider a car that doesn't have a gas tank. Instead, it can only drive along specially constructed roads that have a trough down the middle holding gasoline. The fuel efficiency as measured in square meters is the cross-sectional area of the trough at which the car uses exactly all the gasoline in the trough as it drives.

For a car that gets 50 miles per gallon, this cross-sectional area is about 0.047 mm^2; if the trough has a square cross-section, that's about (1/5 mm)^2.