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Continued from Part 1 on medieval land area.
Surveying is a mechanized way of doing the waggle dance. It includes all three dimensions when measuring the height of a distant building or tree or mountain.
I tried a real experiment in my house with my kit-built astrolabe. Polistra and friends demonstrate inside the Homette, which closely resembles my house. We’re measuring the height of curtains.
First use parallax to find the distance to the curtain. We stand at two locations, as far apart as we can manage in the room. In this case the width is 4 feet.
Each astrolabe is first set to agree with the same standard, in this case the floor tiles. Surveyors use transits, which are vertical and horizontal astrolabes mounted on tripods. Early transits had alidades; after lenses were added, the centerline on the lens was still called the alidade. Surveyors use a calibrated compass (GPS in modern times) for the standard direction.
Each surveyor sights the left end of the curtains and checks the angle between the standard and the curtain, as seen through the alidade on the astrolabe.
Then we calculate the distance using tangents.
Here’s the triangle given what we know for sure. Neither surveyor is exactly perpendicular to the distance we want, and we don’t have a third perpendicular astrolabe between them to separate the two right triangles. So we have to imagine that the destination is centered, divide the known width into two equal triangles and use half of the width as the Opposite of one right triangle.
A tangent is Opposite/Adjacent. The width is known to be 4 feet, so each average Opposite is 2 feet. The sum of the two astrolabe angles is 15 degrees, so we use 7.5 as the angle of the average half-triangle. We then solve for the Adjacent, which gives us the perpendicular distance.
Opposite = 2.0, Angle = 7.5
Tangent = tan(7.5) = 0.13
Adjacent = Opposite / Tangent = 2.0 / 0.13 = 15 feet
Now that we have the distance, we can use the astrolabe vertically to find the height. In this case one surveyor sights through the alidade toward the bottom and top of the tree or curtain, finding the difference in angles between top and bottom.
Again we have the known triangle, and we imagine a symmetrical version.
In this case the angle is 21 degrees, so we use 11.5 for the average half-angle. The known Adjacent is 15 feet. Solving for Opposite:
Angle = 11.5
Tangent = tan(11.5) = 0.2
Opposite = Tangent x Adjacent = 0.2 x 15 = 3
This is the Opposite of the average half-triangle, so the height of the full curtain is 6 feet. My astrolabe angles checked pretty closely with the actual measurements in this easily accessible experimental setup.
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In the real world, the surveyors need to find the distance and height of a distant tree or mountain, which may be inaccessible due to property or geography. Polistra and friends are measuring a tree at the Mill. The tree is across the creek, so it’s hard to reach.
We use the railroad track for a baseline and set our astrolabes to match. Then we sight through the alidades and measure the angles.
After determining the distance, we get the top and bottom height of the tree and figure the actual height.
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Around 1880 surveyors developed the hypsometer, a vertical astrolabe with a built-in analog computer to figure the height all in one step.
You’d find the distance first, by parallax or direct measurement. Then set the slider to indicate the distance. Then tilt up the hypsometer, looking through the lens and alidade to center the top of the tree or mountain in the alidade’s reticule.
When you find the top, look into the mirror to read the height directly. The scale for the height is reversed and angled so it looks proper when seen in the mirror.
You could also get another person to read the scale directly, but the mirror makes it easy to find the top and read the height at the same time without moving anything. Modern hypsometers are digital of course, using digital cameras and screens and embedded microprocessors.
When the tree is accessible so the distance can be measured directly, surveyors use a simpler method without astrolabes or angles. Surveyors always have a long calibrated stick. Put the stick at a known distance, sight directly to the top and bottom of the tree, and mark the heights on the stick. Then use proportions to determine the height of the tree. If the distance to the stick is 10 feet and the distance to the tree is 30 feet, the tree must be 3 times as high as the difference between the marks on the stick.
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Personal note: This piece was especially satisfying. In the push to finish and check courseware, I had to put aside Tech History fun. Now that courseware is done I can unwind and get back to recreation. This item included some Real Science™ and some geometrical tricks that I couldn’t have done when I was young. I envied the Math Masters who had a repertoire of tricks to solve problems or prove theorems. After decades of experience with geometry in electronics and graphics, those tricks are available now.
Polistra's Mill 