Using Pythagoras to Measure the Earth

© 2003 by Karl Hahn
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In Berkeley California, several circumstances of terrain and man-made structure coincide in such a way as to make it easy to measure the size of the earth using inexpensive and easily available (non-electronic) tools. More on this a little later. First...

Distance to the Horizon

Geometry of finding distance to the horizon. Notice that the observer sees the horizon depressed by a small angle below his horizontal view. In celestial navigation, that angle is called the "dip."

When you are at sea standing on the deck of a boat with your eye at some height, h, above the water, how far away is your horizon? Using the rule of Pythagoras, it's easy to see how you would compute this. The diagram to the right shows how you would go about it. The observer's view-line forms a right angle with the line segment that joins the horizon with the center of the earth. If R is Earth's radius and s is the distance from the observer's eye to his horizon, the Pythagorean rule requires that
   s2  +  R2  =  (h + R)2
Squaring out the right-hand side you find that
   s2  +  R2  =  h2 + 2hR + R2
Now cancel the R2 terms on both sides and factor an h out of the right-hand side:
   s2  =  h(h + 2R)
Even on the top spar of a very tall ship (or even on top of a mountain, for that matter), the observer's height, h, is tiny compared to the radius of the earth, R. Hence with almost no loss of accuracy we can say
   s2  ≈  2hR
or equivalently
   s  ≈  √2hR
So in what follows, we will take the above as the formula for distance to the horizon.

Claremont Canyon, the Golden Gate Bridge, and the Size of the Earth

In Berkeley California there is a hiking trail up the north rim of Claremont Canyon. As you follow it, it takes you higher and higher, and affords you a grand panoramic view of the Golden Gate Bridge, which is nearly due west of the trail. With favorable weather conditions, you can also see the Pacific Ocean through the Golden Gate out to a distant blue horizon. At a certain point on the trail, the tops of the towers of the Golden Gate Bridge line up exactly with the Pacific horizon. And it is this coincidence that allows you to measure the size of the earth.

When the tower-tops align with the horizon
Look carefully at the diagram to the right and think about the geometry of the situation. When the tower lines up with the horizon, the observer on the trail sees the same horizon point as another observer would see looking in the same direction from the top of the tower. Let Δs be the distance from the Berkeley observer to the Golden Gate Bridge. Let sB be the distance from the Berkeley observer to the horizon. Let sG be the distance from the top of the tower to the horizon. Then

   sB  =  Δs + sG
Applying the distance-to-the-horizon formula we have

   √2hBR  ≈

  Δs +
where hB is the height above sea level of the Berkeley observer, and hG is the height of the towers of the Golden Gate Bridge. Looking at the equation above, you can see that if you know hB, hG, and Δs, then you can solve for the radius, R, of the earth. Doing the algebra (which you should try for yourself) you find

   √R  ≈

  √2hB - √2hG

The height of the towers, hG, is known to be 746 ft above sea level. If I were to find, for example (these are not the actual measured numbers), that  Δs = 17 miles = 89760 ft,  and  hB = 1700 ft,  then I would find using the above formula that

R  ≈  4560 √ft

   R  ≈  20,800,000 ft  =  3939 miles
If you live within driving distance of Berkeley, you don't have to use my guesstimated numbers for Δs and hB. You can measure these for yourself. The equipment you'll need is a pneumatic altimeter and a homemade triangulation device (or you can use a sextant, if you can get your hands on one, instead of the homemade instrument. A sextant will give a more accurate measurement). Many high school and college physics labs have an altimeter. Or you might be able to borrow one from a general aviation facility if you explain what you want it for. For a homemade triangulation device, use a yard stick or meter stick. A steel one is best for this application. Take a short section of thin dowel and mark its midpoint. Now using colored pencils, mark off various distances from that midpoint in each direction, color-coding them as you go. So if you are using a yard stick, mark off one quarter inch from the midpoint, both right and left, in blue. Mark off one half inch from the midpoint, both right and left, in green. And so on. If you are using a meter stick, mark off centimeters. Now using a paperclip, fashion a way of attaching the dowel as a cross 'T' to the edge of the yard or meter stick. Do it so that you can slide the cross 'T' up and down the stick. The cross 'T' has to be at right angles to the yard or meter stick

You use this device to measure the distance from you to two distant objects that are near to each other. Do some practice sightings on trees or utility poles (sight from a point where the line of sight is nearly perpendicular to the line that joins the two objects). Hold the stick horizontally, sighting with your eye as close as possible to the zero inches or zero centimeters end of the stick. Now have your assistant slide the cross 'T' in or out until two marks on the cross 'T' of the same color line up with the two objects you are measuring. Read the number of inches or centimeters the cross 'T' was from your eye when it all lined up. Record that and the distance between the two colored marks that you lined up on. If you know the distance between the two objects you sighted, you can use similar triangles and your measurements to establish how far the two objects were from your eye. Practice this technique until you can do it easily before you go on your expedition to Claremont Canyon.

Here is the view showing the Golden Gate Bridge from high up on the Claremont Canyon trail. You can see Alcatraz Island just in front of the left tower of the bridge. To the right is Marin County, and in front of that is Angel Island. The near shore shows the city of Berkeley, with the UC campus nearest the viewer. The foreground is the north rim of Claremont Canyon. Photo Credit: Qian Yu. Used by permission.
Here is an enhanced enlargement of the Golden Gate Bridge. You can see by comparing the tops of the towers to the horizon that Ms. Yu had already hiked past the critical point when she took this picture. Photo credit: Qian Yu. Used by permission.

The expedition to Claremont Canyon requires good clear weather, with little or no fog out to sea. Most people will find it to be a strenuous hike, so wear good shoes for it -- quality athletic shoes or hiking boots that still have a good tread on their soles. Bring at least a pint of drinking water per person. Optional items you might also bring include binoculars and a camera. Of course the altimeter and the triangulation device are required equipment, along with a notebook on which to record your observations, and don't forget a pencil.

You need to make your measurement in the morning hours while the sun will be behind you as you face the Golden Gate Bridge. Start at sea level, say at Jack London Square in Oakland, so that you can calibrate the altimeter. Then drive up Claremont Avenue until you are near the Claremont Hotel. Find a legal parking place in that neighborhood and proceed on foot. Walk up Claremont Avenue past Avalon and Tanglewood until you see a street on the left called Stonewall. Follow Stonewall up the slope until it makes a hairpin turn to the right. You will find the trail-head at that turn.

Hike up the trail. It will be uphill almost all of the way. At first you will be in the eucalyptus trees. But as soon as you get above the tree line, you will catch sight of the Golden Gate Bridge to the west. Keep hiking, but every hundred yards or so, check your view of the bridge to see where the Pacific horizon is in relationship to the tops of the towers. When you reach the point on the trail where the alignment of the tower-tops with the Pacific horizon occurs, read and record your altimeter. That number is your value for hB. Now point your triangulation device at the Golden Gate Bridge. Start with the cross 'T' at the far end of the stick and then slowly move it toward your eye until you get an alignment of two marks of the same color with the two towers of the bridge. It is known that the towers are 4200 ft apart. If, for example, you found that the two marks that are 1.5 inches apart on the cross 'T' align with the towers when the cross 'T' is 32.0625 inches from your eye, then by similar triangles you have

   4200 ft
  32.0625 in
    1.5 in

from which you can easily solve for Δs (multiplying through by 4200 ft gives you Δs in feet). And don't forget to record all your numbers.

Note, by the way, that from Claremont Canyon, you have nearly the perfect angle on the bridge to make this measurement -- that is the line joining the towers is nearly perpendicular to your line of sight. If this weren't the case, you'd have to do some further trigonometry to get the correct value of Δs.

Here is the view down the canyon itself, looking down at Oakland. You can see the Oakland Bay Bridge connecting the near shore to San Francisco on the far shore. Note that the mountains behind San Francisco are not necessarily higher than Ms. Yu's viewpoint, even though they block the sea horizon. At the altitude this photo was taken, the sea horizon is depressed downward from the horizontal by nearly three quarters of a degree, which is more than the diameter of a setting sun. Photo credit: Qian Yu. Used by permission.

If you do decide to make the trip to Claremont Canyon, be warned that the trail is very steep in places. So please be careful with your footing, especially on the way back down. And while you're up there, take some time to admire the natural beauty of the place and the spectacular view you get of Berkeley, Oakland, and San Francisco Bay. When you get back, please send me an email telling how it went.

If you use a sextant instead of the yard-stick instrument, you will have to calibrate the sextant standing on an ocean beach before you go to Claremont Canyon. There are books and webpages on how to do this. From the Claremont Canyon viewpoint, where the towers align with the horizon, turn the sextant on its side. Holding it in that position, look through the eye-piece and measure the angle from tower to tower by adjusting the sextant scale until you've aligned the top half of one tower with the bottom half of the other. Read the degrees and minutes and record those numbers. To calculate the distance, convert the degrees and minutes to radians, then divide the radian angle into 4200 ft. The result will be the distance to the bridge in feet.

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