Modern mapmakers use specialized equipment to take measurements and mathematics to figure out what those measurements mean. We have maps that cover nearly every part of the world, including under the oceans, across mountains, and over the polar ice caps. Perhaps the most important foundation upon which surveying and mapmaking stands is the mathematical discipline of trigonometry.

Measuring Long Distances by Using Triangles

The burghers of Leiden, Holland, must have wondered what that odd professor from the university was up to. Throughout the years 1614 and 1615, they saw Willebrord Snell, a young professor of mathematics, repeatedly climbing up and down the town’s church steeples and bell towers, lugging a huge quarter circle of iron. Then they saw him carefully rolling out and then rolling up a long metal chain, all the while carefully writing down notes in a notebook. What could this fellow have been up to?

It turns out he was busy making scientific history. Professor Snell was inventing the science of geodesy and laying the baselines for the future practice of surveying.

Specifically, Snell was attempting to figure out the exact size of the earth. The first attempt to do so was made almost two millennia earlier by the Greek scientist Eratosthenes of Cyrene, who used the different lengths of the sun’s shadows at noon as the basis of his calculations. The figure Eratosthenes came up with was pretty accurate, but Snell thought he could do better using a new technique. His idea was a mathematical process called triangulation that he had invented for the purpose of measuring long distances. Snell’s method was a game changer, and soon, people of science all over Europe saw what an incredibly valuable tool triangulation was.

Prior to Snell’s work, the only way to measure the distance between two towns or landmarks was to measure it directly. A common method was to make a couple of extremely long rulers and leapfrog them along the flattest and most direct path possible between the two points. That got old fast, so somebody came up with the slightly better idea of measuring the number of turns of a wagon wheel and then applying basic algebra to figure out how many feet or yards the wagon had traveled. Neither of these methods was very accurate, and if there was something in the way of the end points, say a mountain or a river, these methods didn’t work at all.

As a mathematics professor, Snell was well versed in the trigonometric principles formulated by early Greek and Arab mathematicians. From their work, he understood that every triangle is composed of three sides and three angles. Further, Snell was aware that if you know the values of two angles and one side, or two sides and one angle, you can accurately calculate all of the other sides and angles.

Specifically, there are two trigonometry formulas—the law of cosines and the law of sines—that Snell used to calculate distances. Using these meant that he and the thousands of land surveyors who followed him could make much more accurate and usable maps than the cartographers who relied solely on direct measurement methods.

Snell invented the practice of triangulation: the calculation of the distance between points by first making an accurately measured baseline and then measuring the angles made between distant points and the two ends of the baseline. With this technique and his trusty quadrant (the large iron quarter circle that mystified the burghers), he charted Holland. Soon after, surveyors and mapmakers the world over were redrawing the world, making maps far more exact and boundaries much more precise than they had ever been before.

Triangles: The Mapmaker’s Best Friend

The word trigonometry means “measurement of triangles” in Greek. It is an ancient mathematical discipline first explored by the Egyptians and Babylonians and then refined by the ancient Greeks.

Triangulation is just the use of trigonometry to measure distances. It isn’t a terribly difficult technique to understand. In this chapter, we’ll make a simple piece of measuring equipment that allows anyone to calculate the distances necessary for making good maps.

Suppose you need to determine the distance from point A to point C as shown in Figure 9-1. Because of the mountains, you can’t measure the distance directly, although you can see point C from both point A and point B.

The first thing you need to do is to make an accurately measured baseline, in this case from point A to point B since there are no obstructions (see Figure 9-2). The longer your baseline is, the more accurate your subsequent distance calculations will be. Use a reel-type tape measure or other direct measuring device for this.

Then you need to go to point A and measure the angle between point B and point C using an accurate angle-measuring device. You can use a sextant, a quadrant, or a circumferentor. Making your own circumferentor, which is described later in this chapter, is easy and provides reasonably accurate results. Next, go to point B and measure the angle between point A and point C (surveyors call this shooting an angle [see Figure 9-3]). You now know two angles. Use the fact that the three angles in a triangle must add up to 180 degrees to calculate the third angle.

It’s an easy job to calculate the length of segment AC, the distance between the church spire and the city hall in this diagram. You know the length of segment AB, and the angle values for angle ACB and angle ABC. Just look up the sine values (in degrees, not radians), set up the equivalent ratios using the Law of Sines, and solve for the length of segment AC (see Figure 9-4).

Building a Circumferentor

The key to accurate triangulation is measuring angles precisely. To make it possible for you to do that, we’ll make an instrument for measuring angles called a circumferentor.

How to Build Your Circumferentor

Before you begin, take a look at Figure 9-5 to see what you’re aiming for.

Follow these steps:

1. Drill a ¹³/₁₆-inch hole in the center face of the 1″×1″×½″ plastic block.
2. Cut screw threads in the hole using the ¼-inch UNC tap.

   The tool you use to cut a male threaded piece is called a die. When you make a female threaded piece, you use a tap. In this case, you are making threads in a hole, and that means you’re making a female threaded piece.

   

   1. Start the threading process by carefully positioning the main axis of the tap parallel to the hole.
   2. Turn the tap a half turn and then back out the tap a quarter turn to remove shavings so the tap doesn’t get clogged.
   3. Keep doing this until the hole is fully threaded.
3. Glue the block with the tapped hole to the angle finder (see Figure 9-6).

   

4. Glue the sighting nails to the knobs on the angle finder (see Figure 9-7).

   

5. Screw the tripod mounting screw into the tapped hole.

To find angles between distant objects, use the nails as sights, lining up the object with both the front and back nails. Note the angle separating the objects on the dial.

Now, all you need is to do the trigonometry and you can measure the distance to just about anything you can see!

Surveying the Modern World

Willebrord Snell’s quadrant was a state-of-the-art piece of surveying gear for his time, but not long after, it was replaced by smaller, lighter, and more accurate equipment. By the 18th century, the theodolite, a device that could measure both vertical and horizontal angles, came into widespread use. With theodolites, surveyors and cartographers could use Snell’s triangulation techniques to make maps and draw boundaries quickly and with great accuracy.

With the introduction of electronic instrumentation and satellites in the 20th century, there are few spots, if any, for which accurate maps do not exist.