This works, but there is one detail missing, and depending on your latitude the picture could be misleading. The sun moves approximately one finger width across the sky each 15 minutes, but the sun does not necessarily set vertically down onto the horizon. You have to use your hand to measure the distance the sun has to travel along it's path, not the distance above the horizon. At latitudes and seasons where the sun sets at an acute angle to the horizon, it would take a bit of practise to get it right.
How does this work
The earth spins at a constant speed, so the sun moves across the sky at a constant rate. That rate is 360 degrees per day, 15 degrees per hour, or 3.75 degrees in 15 minutes.
Angular size of a finger
A typical human has an arm about a meter long, and a pinky finger about 1.5cm wide (ref, and hopefully you have one you can measure). 1.5cm at one meter is a little over one degree. Three middle fingers is about 5 degrees, and across a closed fist is about 10 degrees. The image in the question suggests an arms length finger is 15 minutes or 3.7 degrees. This seems a bit high, but might work if the wrist is and elbow are bent, bringing the the finger closer to the eye than in the examples in the references. It will also make a difference if the arm is held out straight ahead, or held to the side and the head turned.
Sun path across the sky
Unless you are in the tropics at just the right time of year, the sun does not drop directly into the horizon. The sun sets when it hits the horizon (and it gets dark when it gets 6, 12 or 18 degrees lower, depending on definition). So if you want to measure the distance the sun will travel until it sets, you have to measure along the right path.
Putting it all together
At 15 degrees per hour the actual speed is more like three fingers in 20 minutes for a typical person, but you could be more accurate by "calibrating" your arm length and finger width. And you need to measure along the sun's path, not the height above the horizon. Estimating that path will be easy in the tropics where the path is nearly vertical, and harder in the arctic where it is at a small angle to the horizon, as a small mistake in estimating the angle makes a bigger difference in path length. But if you get that right, it does work.
- Everyone has different hands, fingers, and arms. So to be more accurate, everyone should measure the angular size of their hands and fingers.
- If there are mountains on the horizon, then this technique can be used to measure the time until the sun goes behind the mountain. Obviously, if you then drive round the other side of the mountain, the time will be different.
- If you're in the arctic at midsummer, the sun won't set. You can still measure how far along it's path it will go in a given time, but that path never touches the ground.