A home-made sextant

Professional sextants

A sextant is an instrument for measuring angles between objects. They have been used in navigation for centuries, because latitude can be inferred from a measurement of the angle between the horizon and the noontime sun. For this reason, professional sextants have a way of sighting the sun angle without looking at the sun.

This sun-observation method is certainly NOT a possibility with the sextant described below, because it lacks a sun-blocking mechanism, and looking directly at the sun is extremely dangerous.

The purpose of the sextant provided here is to for use in finding angles (with respect to the horizontal), which allows a person to infer the heights of flagpoles, buildings, and even hills, by measuring a horizontal distance and one or two angles.

A homemade sextant

A crude sextant measuring angles with respect to the horizontal plane can be used to determine the heights of objects, using a method invented a thousand years ago by Al Biruni (see Peter Lynch’s blog posting “Al Biruni and the Size of the Earth.” ThatsMaths, June 10, 2021. https://thatsmaths.com/2021/06/10/al-biruni-and-the-size-of-the-earth/). Carrying out such measurements and performing the associated calculations can be a fun way to spend some time outside, and it’s also a good way to see that high-school trigonometry can be useful!

To make the sextant, follow these steps:

Print either https://github.com/dankelley/sextant/raw/main/man/figures/sextant1.pdf or https://github.com/dankelley/sextant/raw/main/man/figures/sextant2.pdf on a sheet of paper. (These files differ in how the angles are laid out. Pick whichever one you find easier to read.)
Paste or tape this PDF file onto a piece of cardboard or thin wood.
If you wish, cut along the gray outer semicircle, to eliminate corners.
Create a pivot at the point marked “P”. If the sextant is mounted on wood, a pivot can be made with half-hammered nail, or a thin dowel (perhaps a round toothpick) inserted into a pre-drilled hole. If it is mounted on a piece of corrugated cardboard, a round toothpick can inserted by hand. In either case, the pivot must be inserted at right angles to the plane of the sextant. To keep it so aligned, try using some glue.
Attach fishing line, sewing thread, or thin string, at the “P” pivot. The line should be long enough to extend past the bottom of the sextant. Attach something relatively small but heavy at the free end, which will weigh the line down, making it run vertically along the sextant, pointing in the downward direction. A bolt or washer is a good choice since you can attach it by tying a knot, but you could also use a heavy coin, attached by tape.
Insert a mail, dowel or toothpick at the spot marked “S” (the sighting spot).

Taking angle readings

Hold the sextant vertically, with the plumb line extending down to the round scale. Then position it so that you can sight an object of interest. Rotate it in the vertical plane until the “P” and “S” points are aligned with the object. You might need to wait a second or two for the plumb bob to stop moving. When it does, press your finger on the line, below the scale. Holding your finger there, turn the sextant so that you can read off the angle at the spot where the plumb line intersects the round scale. This is the angle along the sighting line, with respect to the horizontal plane.

Height of object at known distance

This method can be used to find the height of an object at a known distance, measured across a level plane. It requires making only a single angle measurement.

Figure 1. Definition sketch for the computation of the height of a object on level terrain. \(h\) is the height to be computed, while \(L\) is a measured distance along the ground, and \(\theta\) is the angle found by sighting from the marked point to the top of the object.

As illustrated in Figure 1 above, you can find the height \(h\) of a flagpole, tree, building, etc., on flat ground, by pacing off a horizontal distance \(L\) from it’s base, and then finding the angle \(\theta\) to the top. (You may determine \(L\) by counting how many paces it takes you to cover 10 sidewalk stones, and then using a ruler to measure one stone to get a conversion factor.) With \(L\) and \(\theta\) now known, you may calculate the object height above the sighting plan as

\[ h = L\tan\theta \]

so that the height of the object above the ground is given by

\[\begin{equation} H = h_e + L\tan\theta \end{equation}\]

where \(h_e\) is the height of your eye above the ground. Note that \(L\) may be measured in “pace” units, converted to metres (or feet) using a calibration factor developed by pacing out a number of sidewalk paving stones, then measuring the length of one of them with a ruler.

If you don’t have a calculator handy, you may use the table in the Appendix to get the \(\tan\theta\), and carry out the calculation by hand to a couple of digits. (As an exercise, repeat the pace-measure-calculate exercise a few times, to get an idea of the uncertainty of the method.)

Height of object at unknown distance

This method can be used to find the height of an object at a unknown distance, so long as distance can be measured along locally flat terrain. Two angles must be measured, one at the near end of this local terrain and the other at the far end.

In the case of a sloping hill, it is difficult to measure the horizontal distance between the top of the hill and the observer. Pacing a distance off gives distance along the slope, which is not the same as distance in the horizontal direction. But we need the horizontal distance to use the formula from Example 1. What to do?

Trigonometry can come to the rescue, provided that there is a flat plain nearby. Just measure the angles at two spots on that plain, along with the distance between them. (Pacing works in this case because the ground is level.)

Figure 2. Definition sketch for the two-angle case. \(L_1\) is an unknown distance, and \(L_2\) is a known distance. Angles \(\theta\) and \(\phi\) are measured by sighting the top of the object from the two marked points.

Consider the figure above, in which the two angles are \(\theta\) and \(\phi\), the first being the value observed nearer the hill. The distance between the observation spots is \(L_2\). This may be measured by pacing off distance along the flat plain, as a horizontal distance was measured in Example 1.

In this case, then, the goal is to infer hill height from measurements of \(\theta\), \(\phi\), and \(L_2\).

If we knew \(L_1\), we could apply the method of Example 1 at both spots, giving two estimates of height. However, we cannot determine \(L_1\) without without tunneling through the hill, so we cannot use the formula for either spot.

Luckily, a little trigonometry can save us a lot of tunneling! At the nearby spot we have

\[ h = L_1\tan\theta \]

and at the farther-away spot we have

\[ h = (L_1\ +\ L_2)\tan\phi \]

but these must yield the same \(h\) value, so we can combine the two equations, yielding

\[ L_1 \tan\theta = (L_1\ +\ L_2)\tan\phi \]

which can be rearranged to

\[ L_1 (\tan\theta\ -\ \tan\phi) = L_2\ \tan\phi \]

and then to

\[ L_1 = L_2 \frac{\tan\phi}{\tan\theta-\tan\phi} \]

at which point we have a formula for the unknown distance \(L_1\), based on the easily-measured \(\theta\), \(\phi\) and \(L_2\). (This illustrates the magic of mathematics, for we found that answer just by moving symbols around, instead of visualizing calculations.)

Now, we can use this result our original formula for height at the nearby spot, to get

\[\begin{equation} H = h_e + L_2 \frac{\tan\theta\ \tan\phi}{\tan\theta-\tan\phi} \end{equation}\]

to get the height of the object above the ground. Here, again, \(h_e\) is the height of your eye above the ground.

A test case

The method was used to to estimate a downward distance, sighting from an apartment window to the base of a nearby building. For an independent value, I calculated the observation height as \(17.0\)m \(\pm 0.5\)m, based on my stature together with an estimate of the elevation of the first floor above ground-level, and taking into account the floor from which I made the observation.

A map from https://www.openstreetmap.org revealed the horizontal distance to be \(L=43 \pm 1\)m. The sextant gave \(\theta=20^\circ \pm 1^\circ\) as the downward angle. These values gave the height of my eye as \(17.2\pm 1.2\)m.

In this test case, the height estimates based on floor count and sextant angle are in agreement, to within the measurement uncertainty.

Appendix: trigonometry tables

This table shows the \(\tan\theta\) for \(\theta\) ranging from 0 to 89\(^\circ\). To find the value for a given angle, scan down to the row labelled with the first digit of \(\theta\) (after adding a \(0\) to the left of angles below \(10^\circ\)), and then scan across to the column labelled with the second digit. For example, the tangent of \(21^\circ\) is \(0.383864\), which, after rounding, matches the value in the third row and second column.

     x=0   x=1   x=2   x=3   x=4    x=5    x=6    x=7    x=8    x=9
0x 0.000 0.017 0.035 0.052 0.070  0.087  0.105  0.123  0.141  0.158
1x 0.176 0.194 0.213 0.231 0.249  0.268  0.287  0.306  0.325  0.344
2x 0.364 0.384 0.404 0.424 0.445  0.466  0.488  0.510  0.532  0.554
3x 0.577 0.601 0.625 0.649 0.675  0.700  0.727  0.754  0.781  0.810
4x 0.839 0.869 0.900 0.933 0.966  1.000  1.036  1.072  1.111  1.150
5x 1.192 1.235 1.280 1.327 1.376  1.428  1.483  1.540  1.600  1.664
6x 1.732 1.804 1.881 1.963 2.050  2.145  2.246  2.356  2.475  2.605
7x 2.747 2.904 3.078 3.271 3.487  3.732  4.011  4.331  4.705  5.145
8x 5.671 6.314 7.115 8.144 9.514 11.430 14.301 19.081 28.636 57.290

Dan Kelley (https://orcid.org/0000-0001-7808-5911)

2023-05-22