How does a surveyor traipse into the woods and come out with accurate heights, positions, distances and property lines for artifacts in the middle of nowhere? It’s magic, of course, but it is mathematical magic that has been well understood for a good two-thousand years. In last week’s Dispatch I covered the core of that mathematics, which is the simple but incredible marriage between proportions and triangles.

I finished by presenting a fact little understood by the typical person: because of this mathematics *you can measure the distance to anything in the world by simply pointing to it*. No direct measurement is needed. This week in my continuing series on the magic of surveying I’m going to show how it is done, finishing with an Adirondack example of note.

Here’s where we are as of last week. We know that if two triangles have the same shape they are proportional – that is, the ratio (or fraction) of each triangle’s height to base is the same:

That means if we know the height and base of the smaller one and we can measure the base of the bigger one, then we can instantly calculate the height of the bigger one without measuring it. Go read last week’s Dispatch if you don’t follow this, but trust me this is a profoundly powerful thing.

(a quick digression: savvy readers like that Paul fellow may notice my examples are all right triangles – that is they have a right angle so the base is perfectly horizontal and the height is perfectly vertical – and in the real world most triangles would be odd shapes with no right angle. Fear not: I use right triangles because it makes the math easier to understand. The branch of mathematics known as trigonometry – literally “triangle” “measure” from the Greek – generalizes the stuff I am showing to all triangles regardless of shape. But if I said I was teaching trigonometry the *Almanack* would suffer a fatal loss in readership, so we’ll stick to right triangles)

Let’s see an example of how this works in the real world. Suppose you want to calculate the height of the Empire State building. Short of getting a really long rope and a helicopter, or hiring out King Kong, it is obvious that you would not be able to measure its height directly. In fact consider how difficult it would be to even get a reasonable estimate. But with our triangle power all you need to have in order to get a very good measurement is a small stick and a measuring tape. Oh, and a sunny day.

Like any other structure on a sunny day the empire State building casts a shadow. A diagram might look like this:

You may not be able to measure the height of the building, but if you wanted to you could measure the length of the shadow it casts because that’s on the ground, where you are. Let’s suppose you do that with your measuring tape and get a distance of 485 feet. Now you take your stick and stand it on end in any sunny spot. It casts a shadow too:

Obviously, because there is only one sun sending down rays in only one direction, the triangle made by the shadow of the stick *is the same shape* as the triangle made by the shadow of the Empire State building. Therefore they are proportional.

But your stick and the shadow it throws are under your control. You can measure both. Suppose the stick is 60 inches tall and the shadow it throws is 20 inches. This is the half of the proportion you **know**: h/b = 60/20 or 3 to 1. It is then child’s play to **project **that knowledge onto the Empire State building:

H = 3 x 485 or **1,455 feet**. Consider how powerful that is. You just obtained the height of the Empire State building *with a stick*.

At this point we are razor close to being able to put this all in the context of surveying in the wilderness. We need take but one more step.

What if you wanted to measure the height of the Empire State building on a cloudy day? Suppose there were no shadow? On a sunny day the sun makes triangles for you. On a cloudy day you have to make your own triangles. That’s easy to do: just point at the top of the building!

Here’s how that would work. Imagine holding a device that has two straight rods that are hinged. One rod has a powerful telescopic sight on it, just like one might have on a rifle. The device, which I will call a *triangle measuring device* or **TMD**, would look like this:

With the TMD you can hold the bottom rod level and rotate the top rod from the hinge to point at anything you want with the telescopic sight. So what you do is stand anywhere you want in relation to the Empire State building, but in a good spot and far enough away so that you can get a direct line of sight on the top of it. Then you measure the distance you are standing away from the Empire State building. Since you can stand anywhere you want you can make it a nice round distance, say 300 feet. This is what is known as establishing your **baseline**.

Then you hold up the TMD, keeping the bottom rod level and look through the telescopic sight at the building. You rotate the top rod upward until you see the exact top of the building through the sight.

A diagram would look like this:

But here’s the deal: as you open up the TMD to line-up with the top of the Empire State building you *make* a triangle with it. Since it is in your hot little hands you can easily measure the height and base (**h** and **b** in the diagram). Since the telescopic rod is in line with the line of sight to the top of the building, the triangles are the same, thus proportional, thus once again:

So suppose you make your triangle and measure the base at 5 inches and the height at 24.25 inches. Then in seconds:

H = (300 x 24.25) / 5 or **1,455** feet.

We have arrived at the startling assertion from the end of last week’s Dispatch:you can measure the distance to anything in the world **by simply pointing to it**! You only need two things for this magic: a direct line of sight to the thing you want to measure, as through a telescopic sight, let’s say; and a **baseline**, some straight line whose distance you know, from which you can measure and compare.

Now we’re finally ready for surveying. Let’s do an Adirondack example. Suppose you want to map out the mountains in the High Peaks region, starting from Lake Champlain and going west. First you need to have a baseline that is fully measured. You would preferably find two easy points that are specific, visible and whose latitude and longitude are known. Lighthouses are good for that: they are certainly specific, they are obviously visible and their positions are exactly known having long ago been surveyed and fixed by the Coast Guard. It so happens that there are two lighthouses on Lake Champlain which are due east of the High Peaks: the Crown Point Lighthouse and the Barber’s Point Lighthouse (click on the links for interesting historical information on each). These lighthouses are about eight miles apart and make a nice, clear baseline. Here’s a map:

In this map the red dot is the location of the Crown Point lighthouse and the orange dot is the Barber’s Point Lighthouse. The yellow line is therefore the baseline. Now you want to pick the first mountain to measure to. As you look due west one of the first peaks you come to is Bald Peak just east of Mineville. It has the advantage of being largely treeless at its summit, so it is easy to see to and from with a telescopic sight. On the map above I have marked Bald Peak with a yellow dot.

We want as precise a measurement as we can obtain, so we send someone to climb Bald Peak and stand at the true summit with a very bright signal mirror so that the reflected sunlight will make an unmistakable beacon (we don’t have to do this, but our result will be much less accurate since we’d then be ball-parking the true summit instead of zoning right in on it).

Then we stand at the Crown Point Lighthouse with our TMD and line up the bottom rod so it points to the Barber’s Point Lighthouse – in other words, it points along the baseline. Then we point the top rod with our telescopic sight directly at the beacon atop Bald Peak. Here’s the map of that:

In this map the red line is our line of sight through the TMD telescope. By pointing at our lighthouse and at bald peak we have made a triangle out of our TMD with height **h** and base **b** that is the same shape as this huge virtual triangle extending into the Adirondacks with a height **H**, which is the distance from the Barber’s Point Lighthouse to Bald Peak – the orange line – and base **B** which is our 8 mile yellow baseline. We measure **h** and **b**, from the TMD triangle which is literally in our hands. Since the triangles are the same shape, one more time:

So the calculation of **H** is quick work. Now we now know the distance from the Barber’s Point Lighthouse to Bald Peak, a straight-line route that would be devilishly difficult –if not impossible – to measure directly. Magic.

We already knew the length of the baseline, so at this point we know the lengths of two sides of our “Great Triangle.” Thanks to the famous theorem of the great Greek mathematician Pythagoras, if we know the lengths of two sides of a triangle, the length of the third side can be easily calculated. So in fact now we know the distance from the Crown Point Lighthouse to Bald Peak as well. We know *exactly* where our first Adirondack peak is… and it took merely pointing at it.

Here’s the beautiful part: the distances we computed in our Great Triangle have been added to our knowledge, so they themselves can now be used as baselines! For example the orange line can now be used as a baseline to calculate the distances to Hurricane Mountain next, as depicted in red on the following map. Then we can use one the shorter of those Hurricane red lines as a baseline to get the distance to Giant, purple on the map. And then maybe Dix after that. And on and on until we map the whole damned park:

Now you know how it’s done! Mostly. The more attentive among you might notice something key is missing yet. Try to think what it is if you like, but I promise I’ll get to it.

In the meantime, readers well versed in Adirondack history may already have recognized what I’m about to reveal. This Great Triangle to Bald Peak I made in this example was not a random choice. Allow me to offer two excerpts from a report issued nearly 140 years ago, when the interior of the park was still largely unknown:

*It was my intention to commence the work by the careful measurement of a great primary triangle, near Port Henry, on Lake Champlain, of which the base would be the distance between the center of the light-house on Crown Point, and the center of the light-house on Barber’s Point, the distance between those points having been determined to the decimal part of a metre by the United States Coast Survey…*

* …Having secured the services of two men for axe-work and pack carrying, we proceeded to the foot of Bald Peak; the weather being quite hot and this the first mountain ascent of the season. The summit being achieved, the instruments were placed, and the axe-men proceeded to level such of the trees as still obstructed the view, either of Barber’s Point light-house – which far below seemed like a speck at the water’s edge* *– or of the prospect northwestward…*

– Verplanck Colvin, *Annual Report on the Progress of the TopographicalSurvey of the Adirondack Region of New York*, 1874

Our example is in fact exactly how Verplanck Colvin began his great Adirondack Survey. His baseline was our baseline, his mountain was Bald Peak and he had a TMD too, a device to point and make triangles. Of course “TMD” is my acronym, meaningless in the real world. In Verplanck’s surveying world, his device was called a Theodolite . Examine the Wikipedia page on this link and you will clearly see it is a device for very accurate pointing, a more elegant version of our TMD. You may have seen pictures of a theodolite before but if you are like most non-surveyors it was a mystery to you; its magic was unintelligible. Now you know how it works.

It is the case that Colvin had a little more trouble than we did in our example because he discovered that some land jutting out from Vermont obscured his line of sight from Crown Point to Barber’s Point. But in surveying any obstacle can be overcome because anything on Earth, no matter how lofty, remote or obscured, can be connected by triangles to the rest of the surveyed world. So he made a small adjustment and established a Great Triangle to Bald Peak, which anchored his survey of everything else in the Adirondacks and thus became the basis of the mapped landscape we know today.

Now as to that key missing thing, which you may or may not have figured out… see you next week.

* Top Photo: Theodolite. Courtesy of Wikipedia*

Thanks for the great summary of how surveyors work, especially in the days before satellites and GPS. It should now be clear that establishing the horizontal position of each point is of primary importance for map making, while establishing elevations (the most important for hikers) is secondary. For the record, however, the Bald Peak that Colvin used first in his survey was a small, now wooded peak east of Mineville that due to lumbering/wood gathering was probably quite bald as of 1870.

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Tony:

Thank you for the correction. And here is the risk of secondary sources!

After your comment I went to a copy of Colvin’s original drawing and indeed you are right. That was sloppy of me and at the same time a good old Adirondack cautionary tale about the confusion of peaks and ponds with the same name.

I will correct the maps in the article as soon as I can. My apologies and thanks for the catch.

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The maps are corrected.

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If you like crawling around the Adirondacks sometimes on hands and knees and are interested in Colvin and the work he did plus surveying in general come join the Colvin Crew. It’s an informal group of surveyors and others interested in the above that does just that several times a year. For more information go to colvincrew.org to see just who we are and what we do. All are welcome and it’s cheap entertainment.

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I love the stan-helio in your logo. Sounds fun!

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Thanks Pete! I am lucky enough to have the actual 1874 book and reading it [the 1st part anyway] is reading an adventure book as well as a scientific book. Looking forward to that “mystery” missing thing next week.

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Your articles are most timely. The Town of Webb Historical Assoc. (315-369-3838) has just opened our 2013 exhibit on mapping. Your explanations on surveying are concise and most helpful to the uninitiated (me). Would it be permissible to copy these articles and place them in a notebook that visitors to the exhibit could read?

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Dear Teresa: you may do with these articles whatever you wish. I’m glad you could follow them!

You could put a triangle-shadow display in your exhibit, show how it’s done visually…

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Fine pedagogy, Pete!

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I am fascinated by surveying (a transit is on my diplomas from RPI, in the logo) and appreciate your attempt to explain how it is done, or was done. However, I think your Empire State Building example works only because there is a right triangle (one angle is 90 degrees), whereas your Bald Mountain problem has no right triangle. You need the angle from Barber’s Point (if no laser rangefinder is at hand). You’ll also need – gasp! – a table of tangents.

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Indeed, you are right. Hence the fifth paragraph in my Dispatch above which says exactly that. Trigonometry simply generalizes the power to any triangles. You need to point more than once depending upon what you already know, but the magic is essentially the same. It’s just easier to visualize with right triangles.

For those of you who want to believe me but don’t want to learn trig, draw a triangle of any shape you like. Now drop a line from the biggest angle so that it hits the opposite side at a 90 degree angle. Viola! Two triangles, both right triangles. All triangles, no matter the shape, can be made from two right triangles. If we own right triangles we own all of them. The rest is details, as they say.

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How is the elevation of the peak determined in the Bald Peak example?

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Tom: the notes Colvin wrote in his reports of establishing the great triangle do not specify how he determined Bald Peak’s elevation but typically he did so with a mercury barometer, first calibrated to a lower elevation.

With that said, note that pointing to a summit also makes a vertical triangle, thus taking two such bearings from connected points would allow for a trigonometric solution as well.

In any event Colvin needed the elevation of Bald Peak in order to adjust distances in his great triangle, as all land surveys measure on the horizontal plane and must be adjusted for any vertical rise.

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