This week I return to my series on surveying. Two weeks ago we got as far as revealing the basic idea and magical power of triangulation. This wedding between shape and mathematical proportion transformed human knowledge and literally made all modern science, engineering, geography, architecture and cartography possible.
The power of triangulation is truly unlimited; just for fun, here is one last example. Contrary to the old saw that people thought the Earth was flat until the age of the great European sea explorations when sailors managed to not fall off of the edge, the ancient Greeks knew it was round. In fact the Greek mathematician Eratosthenes actually measured the circumference of the Earth using triangulation more than two hundred years before the birth of Christ. He used basically the same stick-and-shadow technique that I described in my previous surveying Dispatch for measuring the height of the Empire State building. No one knows for sure what unit of distance he employed but if we assume he used the unit considered most likely by scholars then his measurement was less than 2% off of the actual measurement, which is darned good for basically using a stick and a shadow.
Eratosthenes’ method was quite clever. He used two cities in ancient Egypt. First was Syene which was located almost exactly on the Tropic of Cancer, meaning that at noon on the summer solstice the sun was directly overhead and cast no shadow. Then he used his home city of Alexandria (where he was the head of the Great Library of Alexandria), 450 miles to north of Syene and thus partway around the Earth, where consequently the sun did cast a shadow. At Alexandria on the solstice he measured the angle made by the sun’s shadow and he had all he needed. For those of you who are actually getting into this math stuff a little bit, here’s a diagram of his method: notice all the triangles!
See if you can figure out what he did; if not, here’s a link to a good description.
For those of you who think all this stuff irrelevant in the age of GPS, think again. How does GPS work? You guessed it: triangulation. Here is an animated picture, courtesy of Wikipedia. The dashed green lines represent paths from multiple satellites to a GPS receiver, which is how it’s done: triangle city.
So we have triangulation as our most powerful surveying tool. But I left the previous entry in this series with a comment that something was missing, something that is quite important. What is it? Well, let me return to GPS for a moment. If you want to get an excellent description of exactly how it works, this article, from the Web site How Stuff Works, is very good. The fifth sentence in this article lauds the power of GPS with this statement: “As long as you have a GPS receiver and a clear view of the sky, you’ll never be lost again.”
Aha: a clear view of the sky. As I have twice written before, the magic of triangulation boils down to the astounding fact that we can measure the distance to anything by merely pointing to it. That sounds great but you do have to be able to point to it. Obviously that means you have to be able to see it. Whether measuring the distance to a mountain, a bluff, a light house or a GPS satellite you must have a line of sight.
Now there are a lot of things you can see in the Adirondacks so long as the weather is good, such as the tops of mountains, to take the most obvious example. The distance and location of these mountain summits can be therefore determined by triangulation. Even so it must be said that the accuracy of your measurement to a mountain summit will clearly depend upon how clearly and accurately you can see the actual top. In order to increase the accuracy of his Adirondack survey Verplanck Colvin cleared many mountain summits of their trees so that he could position a signal mirror exactly at the highest point and ensure that he had a clear and exact line of sight to his remote triangulating stations.
(By the way, there is a common misconception that most of our bare summits, the great majority of which are well below the tree line, are bare because Colvin cleared them as he worked on his survey. Generally this is not the case. Indeed he did clear summits to an extent, but almost of all that has grown back over time. The real culprit was the series of great forest fires near the beginning of the twentieth century which in many cases burned mountain tops to the soil, leaving them utterly bare and making regrowth extremely difficult.)
So here’s the problem. Suppose you stand on the summit of Blue Mountain, at the top of the fire tower. In the far distance to the south the conical peak of Snowy Mountain is visible. It sits to the left of Blue Ridge, over many miles of intervening and largely trail-less wilderness that is speckled with ponds and streams and crossed by numerous ridges and valleys. You can triangulate to Snowy Mountain and get its distance from Blue. But what about all that stuff in between? Suppose your job is to create a detailed map of the entire region between you and Snowy. You might be able to triangulate the distance from Blue to the shore of Lake Durant, visible below you. But suppose next you decide to get the distance from Lake Durant to Sprague Pond, deeper in the woods and not visible at all. You can hike down to the shore of Lake Durant. Now what? Can you remove the ridge shielding Sprague Pond? Are you going to cut down all the trees so you can see the very top of that ridge, along with all the other landmarks you need to see?
Any novice Adirondack hiker soon gets used to mostly not being able to see much of anything in the woods. Go fifty yards off trail in the Adirondack forest and you have a stellar opportunity to have absolutely no idea where you are, much less where anything else is. Just finding your way to safety has the potential to be a trying challenge, to say nothing about having the slightest notion how you could map it out. So the missing piece is pretty important: you have to be able to find where you are and where you are going even when you don’t have a line of sight.
This makes me think of my own surveying challenge, with which I will end my series some weeks hence. There is not a single landmark of note clearly visible from any point along the eastern border of Lost Brook Tract. Much of it is just about as dense as Adirondack forest gets, which is mighty dense indeed. Thus visual triangulation, magical as it is, would be useless in determining the line. Yet when that line was surveyed for the first time ever in 1948, long before the era of GPS, our stalwart surveyor and his team placed it within about an inch of the team that resurveyed line in 2011 using their modern stuff: laser levels, GPS and infrared total stations. That’s pretty good. I surveyed the line myself, before the 2011 team. I’ll share the comparatively pathetic result of that effort in a future Dispatch.
So how is it done? Well, it’s done like everyone knows it’s done. You have to begin from an established point and with an established direction to go. You can start from a spot given by previous maps or surveys and/or triangulate to a visible starting point, then off you go. But as you proceed you must be able to do two things very well: maintain your direction accurately and measure the distance you have traveled accurately. Once you reach a visible point you can triangulate to verify and correct your bearings, but in between the entire game depends upon you maintaining an accurate bearing and distance. This is far easier said than done.
These two things are entirely separate matters, each requiring entirely separate tools and techniques and each coming with a unique set of challenges. Finding a bearing relied primarily upon one tool for centuries: the magnetic compass, which only became obsolete for surveying in the last hundred and twenty years. On the other hand figuring distance has undergone many changes over the centuries. But both give us a chance to move away from framing our discussion as a “how to” and instead talk about them by telling a little Adirondack history. Next week we’ll plunge into the woods – and into the past – for a surveying journey.
Top Photo: Deep in the Woods of Lost Brook Tract