A Paper Quadrant

A paper version of the classic surveying and navigation instrument.


Before GPS, radios, magnetic compasses, or detailed road maps, how did travelers know where they were in the world? If you were traveling to a distant city in a foreign land through wilderness, the landmarks would probably be unfamiliar and useless to you for most of the trip. If you were sailing across the open ocean, you would have no landmarks whatsoever to guide you. Without technology, you have to rely on nature's guidance system — the sky.

To use this navigation and time-keeping system, you need to be able to read and understand the map in the sky. To use the sky as a precise clock and compass, you need to measure the locations of things in the sky precisely. A quadrant is one way to do that.

How high is the sun? How high is that star? To name "how high" something is in the sky, we measure the vertical angle between the object and the horizon. More precisely, we measure the angle between two lines: a horizontal line, and our line of sight to the target object. This angle gives us the "altitude" angle of the thing in the sky, in degrees. Measuring this angle is the job of a quadrant. A quadrant is essentially a protractor pointed at the sky.

A Simple Traditional Quadrant

Most traditional hand-held quadrants were simple quarter-circles, with a weight on a string. One holds the quadrant upright, and sights along the upper edge, aligning that edge with one's line of sight. The weight is allowed to dangle freely, in which case the string marks the line of gravity (which of course is perpendicular to a horizontal line), and it falls along a scale graduated in degrees, thus indicating the angle between the upper edge and a horizontal line.

A Simple Paper Quadrant

The handout below contains a printable template for a rudimentary quadrant. To make it, print it onto thick cardstock and cut out the quarter-circle. (You could of course print it onto normal paper and then glue it onto a stiffer base material, like cardboard.) Then you simply press a pin or a paperclip through the mark at the center of the arc, and then dangle a weight on a string from the pin or the paperclip.

preview image of linked PDF file
A Simple Paper Quadrant

In practice, I find that the string was rather clumsy and tended to dangle the wrong way, and it would often become twisted or tangled. For this reason, the design also includes an optional swing-arm. It is less traditional and perhaps less accurate than a freely dangling string, but for classroom purposes I think it works better. To attach the swing arm, simply press a pin of some kind through the alignment marks in the corner of the quadrant and the hub of the swing arm. An earring post and back works well for this. You will also want to attach a weight to the bottom of the swing arm. A binder clip is quick and easy, or you could glue a coin or two to the tab at the bottom.

If you want a traditional-style quadrant, or if you want to try making a heavier quadrant by pasting the design onto cardboard or foamboard, or if you want to try laminating your quadrant, then this simple flat design is the design I would use. However, if you just want a cut-and-fold design that is a little sturdier and more usable than the floppy paper design, then I would recommend the following:

A Foldable Paper Quadrant

In addition to replacing the string with a swing-arm, I added a couple of folds to the edges of this design to help stiffen it, and to provide hand-holds. If you use a hole-punch to remove the two small circles, then you will also have two sighting holes to look through. You may find this easier on your eyes than trying to sight along the upper edge of the quadrant. (Note that your line of sight must be parallel to the upper edge of the quadrant, but it doesn't necessarily have to pass directly along the edge.)

preview image of linked PDF file
A Foldable Paper Quadrant

To make this version, print it onto cardstock, cut along the solid black lines, score and fold all dashed lines, and glue all shaded areas. You will also need to use a hole punch to remove the two sighting circles, and you will need to poke a hole for the axle pin, and you may find it more convenient to do these things before gluing the quadrant together. If you want to get really fancy, you could try to glue some threads across the sighting holes as "cross-hairs".

Assembling the Quadrant
An Assembled Quadrant

You can figure out a technique for using the quadrant that works best for you, but I suggest starting with the following: Hold the front flange in your left hand as a handle, and grasp the right face lightly with your fingertips. Allow the arm to swing freely, but be ready to pinch it gently with your fingertips to hold it in place. Now raise the quadrant, holding it vertically, with the "90" mark towards your face and the corner of the quadrant towards the thing you want to measure. Center the thing you want to look at in both sighting holes. When you are ready, gently press your finger against the swing arm to hold it in place while you lower the quadrant and turn it to look at the reading.

For practice, you might have fun trying to measure the altitudes of stars, and seeing if you can notice how they change hour by hour. You might especially want to try measuring the altitude of Polaris, a.k.a. the North Star. The altitude angle of Polaris should equal your latitude on Earth. Measuring that angle was an easy way for early ocean explorers to know their latitude.

You can also use the quadrant to measure the altitude of the sun, but I suggest aligning the quadrant by using shadows and a beam of sunshine, rather than sighting on the sun directly. (Under normal circumstances, a casual glance at the sun will not harm your eyes, especially if it is low in the sky, but you nevertheless want to avoid staring at it. Having to stare at the sun through navigation instruments may be one reason pirates wore eye patches.)

Measuring the Altitude of the Sun

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