Saturday, January 19, 2013

Lost Brook Dispatches: The Magic of Surveying

Surveying Tools, 1728Today I begin a series of Dispatches on surveying, one of the greatest and richest interactions between humans and their natural environment, rife with beauties,  drama and challenge.  And magic.

There are many perspectives from which to tell the story of the history of the Adirondacks.  Indeed the numerous Adirondack history books available to the curious reader feature a wide variety of approaches.  Some are essentially chronological in nature; some are cultural; some are political.  I especially enjoy the many historical writings about the region that are thematically organized around the personalities of the unequaled cast of characters whose fates were intertwined with the Adirondack Mountains.  From To Charles Herreshoff to John Brown to Ned Buntline to Thomas Clark Durant the variety of people and their various enterprises is remarkable.

There is one group of men (sadly no women I know of) who interest me the most, who as the byproduct their adventures and their toil tell the story of the exploration of the Adirondacks better than anyone else: the surveyors.  Verplanck Colvin is the most famous of them and arguably the single most important person in the history of the park, but he is only one of many, known and unknown, who did – and do – what few people understand.  Most of them were not principally surveyors.  But they knew how to do it and took the work either as one of several ways in which they made a living in the wild north or of the necessity to survey their own lands when there was no one else to do it.  It is their work that gives us our Adirondacks today, in so many respects, and it is from their work that we get so many early accounts of the interior of what was a completely unknown wilderness.

In the more public Adirondack surveying pantheon we have names you have heard: Colvin, Emmons and perhaps Gilliland (William Gilliland, founder of Willsboro and Essex, was an early land baron who among other things discovered AuSable Chasm).  In these Dispatches you will become acquainted with names you may very well not know: Campbell, Brodhead, Richards, Sanford and Davis among others.  But it will not be my purpose in this series of Dispatches to tell the story of the surveyors themselves.  There are many written accounts that are far more thorough than anything I could offer here, including their own writings.  Rather, my true focus will be to write something about the magic of surveying itself and to place this magic squarely in the heart of the forest preserve, including my own beloved Lost Brook Tract.

Focusing on the magic seems like an obvious approach to me: after all, what surveyors do is nothing short of astonishing by the standards of the typical Adirondack hiker.  Consider the following scenario.  I drop you off at any High Peaks trail head of your choice.  For the fun of it let’s say you begin at the Garden.  You are given the exact latitude and longitude of the trail head.  You are allowed to carry in any equipment you desire from a compass to a telescope to a measuring tape, even to a Ouija board if you like… save for two things: no map and no GPS (which for the remainder of my life shall represent nothing more than abject cheating and will never grace my presence in the woods except to be sneered at).  Then you are given a simple task:  hike to the summit of Gothics, then to Big Slide, then back to the Garden.  You may take as long as you like.  At the end you are to give me a written report of your journey.

Of course it is a little more complicated than that.  You are not allowed to use any existing trails.  You must bushwhack only.  If you cross a trail so be it, but you are not allowed to know where on the trail you crossed.  Your report at the conclusion of your hike must include the following basic information: the exact distance, to within a foot, of the summits of Gothics and Big Slide from the trail head and from each other, the bearings to and from each, the distances and bearings to the lesser peaks contained by the route you walked, the elevations of all these peaks and the total acreage contained within the loop made when you hiked.  Of course you couldn’t even come close to being able to do it.  Can you even imagine how you’d do it?  No.  Unless you were a surveyor.

Despite the obvious wow factor to what surveyors do there are not many accounts that focus on how they do it.  I think it is because of an assumption – or fear – that it will be too hard or too dry – in short, too mathematical – for the average reader.  As a math teacher I beg to differ.  Surveying is immensely complex and requires a great deal of training and discipline.   But the basic concepts are not that difficult, though they are truly magical.  I’m going to briefly lay out some of them them so that the world of surveying makes a little more sense.  I myself am not a professional surveyor and at first I would look like a fool on a survey team but I understand the principles perfectly because they are based upon the most beautiful and powerful part of mathematics and that’s my thing.

Oh yes, be warned: here comes a little math.

You’ve been warned.

Last chance.

Still here?  Good.  Let’s go.  It’s really quite easy.

Here’s the magic of surveying in a nutshell: fractions.

Yuck, right?  Stay with me: this is gorgeous stuff.

There is nothing more powerful in mathematics than two fractions that are equal.  For example:

Fraction One

It is obvious that these are equal, right?  For example, suppose you decide that you are going to do a goofy walk.  For every 2 steps you walk you are going to stop and wink 3 times.  Then if you walk 4 feet you will have done the winking twice, for a total of 6 winks.  There’s nothing to the math.

We call two equal fractions a proportion.  You may not find anything awe inspiring about two equal fractions.  But as soon as you put them in the real world and have them mean something… well then you’ve got a game changer.  You see, in the real world each of the two fractions plays a role.  One fraction stands for something you know and the other stands for something you don’t know but want to know.  Thus a proportion relates the known to the unknown.  This is the essence of math, not calculating, which is one boring step in a symphony of power.  Math is about relating things and using those relationship to know things we didn’t know before.  The calculations are of secondary importance at best.

Here is a really simple example.  You are working at a summer camp and you need to feed pancakes to a passel of hungry kids.  You have a recipe that requires 3 cups of flour and makes 20 pancakes.  In order to feed all the kids you need to make 100 pancakes.  If from that you can figure out how many cups of flour you need then you are doing proportions.  The math looks like this:

Fraction Two

You can always figure out x by just cross multiplying and dividing: x = (3)(100)/(20)= 15 cups flour.

In this case you took something you knew (a pancake recipe) and used that knowledge – or projected that knowledge, if you will – in order to solve a problem you didn’t know (how to make enough pancakes for camp).   You may still not be heavy breathing over this but be patient: the power of this proportion stuff to project knowledge is incredible, as we shall see .  Note my use of the idea of projecting the knowledge: that word was used on purpose and we will return to it later.

A more dramatic example is this proportion:

Fraction Three

The fraction on the left is something we know, learned at the cost of years of medical research and testing.  It is the dosage rate of amoxicillin to give a sick child who has a bacterial infection.  The fraction on the right is what we want to know, namely how much to give that sick child.  Now to use this proportion all we need to do is weigh the child and put that number on the bottom.  Then the calculation of x to make the two fractions the same is automatic.  Therefore this proportion can project a dosage rate onto any child – meaning it can save literally any kid on earth, as long as you can weigh him or her.

The beautiful part is that this projecting power is unlimited.  That is, you can use proportions to project knowledge into realms that would seem impossible to investigate.  The exact same approach used for pancakes or medicine dosages  can accurately count fish in the ocean or tell the makeup and age of a distant star, things that are unimaginable to do directly.

But the power elevates to true magic when we add a discovery that goes back to the Sumerians in primitive form and following them the Greeks who perfected it: mathematical proportion is the same as physical shape.  This insight has changed the world; it has literally made the modern world possible.   Let’s see what this earth-shaking discovery is about.

Consider the following three triangles:

Triangles Unlabeled

If asked which two were most the same everyone would say the first and the third, regardless of whether they were preschoolers or PhD’s in mathematics.  This is worth thinking about because it’s a little bit remarkable.  We live in a size-obsessed culture if ever there was one.  If size were our most important criterion we would pick the first and second to be most the same as they are pretty close in size whereas the third one is a shrimp.  Yet we ignore size in favor of a much more powerful response, our response to shape.  The first and third have the same shape so we pick those because we are visually locked into a profound sense of shape from birth on.

Ladies and gentlemen, guess what: shape is a fraction and two shapes that are the same constitute a proportion.   Consider the triangles again, this time with labels for their heights and the length of their bases:

Triangles Labeled

The fact that one and three are the same shape is exactly because the ratio of each triangle’s height to base is the same:

Fraction Four

So in this example they are both 2 to 1 height to base.   There it is: “same shape” is a proportion.

Now remember that the power of proportions comes into play when we choose one thing to represent something known, which we then project to the other thing to determine something that was unknown.   In the context of that triangle example it means that if you know the shape of one triangle you can figure out the shape of the other, regardless of the other’s size.

For example, suppose you are given just those two triangles and told their shape is exactly the same:

Triangles Algebraic







Now if I give you the base of the larger one you can immediately calculate its height.  If I say the base is 12 inches then you can say the height is 24 inches.  If I say the base is 3 feet then you can say the height is six feet.  And so on.  Remember that we are interested in surveying for a moment: if I tell you the base of the larger one is 5.5 miles, you can still tell me the height, 11 miles, without measuring it directly.

What you are doing is projecting the shape of the smaller triangle onto the larger one, right?  But now you can grasp why I used the word “projecting.”  The marriage of shape and proportion is magical because it allows you to project knowledge into actual physical space.  If you have a triangle, if you know everything about even just a single one of them, you theoretically own all space.  You can determine any distance in the world without actually measuring it.

All you need is two things.  First, you need to “make” a larger triangle of the same shape as your smaller one, which you do by simply pointing the same direction as your small triangle points.  Second, you need to measure the base of the larger triangle.  In surveying, mathematics and colloquial English we refer to that as establishing a baseline.  The rest is a mere few seconds of calculating.

The bottom line of all this , the how-to’s of which we shall explore a little further next week, is an incredible fact little understood by the average person on the street : because of the mathematical magic that similar shapes are proportions you can measure the distance to anything in the world by simply pointing to it.

The next Dispatch will use an Adirondack example, start to finish, to show exactly how this pointing power works, but in the mean time let me leave you with a little food for thought.

Here is a map of the hiking area from our example as you might see it:

Gothics Big Slide Map Plain

Here is how a surveyor sees it:

Gothics Big Slide Map Surveyed














Triangles, triangles and triangles…

Top Photo: Surveying Tools, 1728.  Courtesy of Wikipedia.

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Pete Nelson is a teacher, writer, essayist and activist whose work has appeared in a variety of Adirondack publications, and regularly in the Adirondack Almanack since 2005. Pete is also a founder and current Coordinator of the Adirondack Diversity Advisory Council, which is working to make the Park more welcoming and inclusive.When not writing or teaching mathematics at North Country Community College, Pete can be found in the back country, making music or even walking on stilts, which he and his wife Amy have done professionally throughout the United States for nearly two decades.Pete is a proud resident of Keene, and along with Amy and his dog Henderson owns Lost Brook Tract, a forty-acre inholding deep in the High Peaks Wilderness.

4 Responses

  1. Bob H says:

    I get it. Looking forward to the next article!

  2. Awesome stuff. Can’t wait to read the next one.

  3. Sally says:

    Who else could have snagged us into a (fantastic!) math lesson?! Thanks!

  4. Big Burly says:

    Thanks Pete,
    Very helpful for my grandson — the practicals of why anybody needs to learn math aren’t taught today the same as when I was a kid.
    I’ve shared different examples with him — yours are real world too which he will get since he loves the outdoors.

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