In the northern hemisphere, we can find the south by aligning the hour hand with the angle bisector of 12 o'clock when finding the solar time. Similarly, in the southern hemisphere, we can find the north in a similar way. The principle behind this only involves local time, longitude, and the watch dial.
At noon, the sun is located in the due south, and the local time is 12 o'clock exactly. Let's take this as the starting point.
The Earth takes 24 hours to complete one rotation, so the angular velocity of the sun across the sky is a uniform 360°/24h = 15°/h.
The dial represents 12 hours, so the angular velocity of the hour hand across the dial is a uniform 360°/12h = 30°/h, which is twice the speed of the sun across the sky.
Let's imagine a virtual sun hand with the same angular velocity as the sun across the sky, starting from 12 o'clock and coinciding with the hour hand. Because the angular velocity is only half of the hour hand's, this virtual sun hand will always be on the angle bisector between the 12 o'clock mark and the angle formed by the hour hand.
When the virtual sun hand is aligned with the sun, the 12 o'clock mark will always point to the direction where the sun is at noon, which is due south. Similarly, when the hour hand points to the sun, the virtual sun hand points to the due south.
The position of the sun can be accurately indicated by a vertical rod, so the accuracy of due south depends on the accuracy of local time.
We know that the local time at the 0° meridian is Greenwich Mean Time (GMT), so as long as we know the precise longitude of the local area, we can accurately calculate the local time based on the angular velocity of the sun, with a difference of 1 hour for every 15°, and then accurately determine the position of due south.
In addition, in the opposite way, we can also calculate the longitude of a location based on the precise compass and local time, using the sun. By observing the angle between the noon sun and the horizon, we can calculate the precise latitude and then determine our own position.