The principle is almost forced upon us. If we think of Polaris as located at the hub of the sky, directly over the North Pole on earth, if I am standing at that point (Lat = 90º N), the star is directly overhead, with its height above the horizon equal to 90º.
If I then travel due south, I leave the point where the star is overhead, so it descends in the sky. If I travel 60 nmi south, then my new latitude will be 89º N, because 60 nautical miles is defined as the distance spanned by 1º of latitude. At this point, I can look up and confirm that the star is no longer over head; it is lower in the sky, and indeed by exactly the 1º I moved away from the pole. If I travel on down to 87º N, I have moved down the globe by 3º, and the star will then have descended by 3º, and its sextant height above the horizon will now be 87º. And this behavior continues as I head down into more likely cruising waters; at 30º N, the height of Polaris will be 30º above the horizon. The height of Polaris above the horizon observed from anywhere in the Northern Hemisphere is always equal to the latitude of the observer.
We could prove that mathematically with geometry and some discussion of parallel light rays from distant stars, but wouldn't this be our first guess without any math: I go 2º away; it goes 2º down. The sketch below summarizes the geometry.
Light rays from stars (millions of millions of miles away) arrive parallel, across the entire span of the earth's orbit (<200 million miles) about the sun. At the pole, the star is overhead; at latitude Lat, it is at angle H above the horizon. Angle sides in red and blue are mutually perpendicular, so the angles are equal, H = Lat.
The problem with applying this simple principle to navigation is Polaris is not in fact located at the hub of the sky. In modern times, it is located about 0º 40' off of the hub, so it, like all other stars, circles the true pole of the sky once a day. This one just happens to be on a very tiny radius of 40' so we cannot detect that motion with the human eye; we do not see it move throughout the night. In the days of Columbus, it was 3.5º off of the hub.
So the principle needs clarification: It is the height above the horizon of the true pole of the sky that matches our latitude, not the height of Polaris itself, which is moving around it.
Polaris circles the true pole at a radius of 0º 40' today; it was off by 3º 30' in the time of Columbus.
In the earliest days of Portuguese navigation, the height of Polaris was not used for a direct measure of latitude, which was not even marked on charts in those days, but rather to keep track of distance traveled from Lisbon, but this circular motion of the star (3.5º off the pole at the time) led to complex procedures. These procedures were later simplified by various sets of rules for making the correction, called regiments.
According to EGR Taylor in her "Navigating Manual of Columbus", the first regiments were too complex for the average marnier—she puts Columbus in that category—but then a breakthrough came by describing the location of Polaris relative to the pole in terms of the two stars on the tip of the cup of the Little Dipper, thought of then as a horn. These stars are called The Guards; one of them (Kochab) and Polaris define a clock hand, as the stars circle the pole. Indeed, that was the trick. The Guards had been used "for centuries" earlier on for time keeping, by both mariners and shepherds, relative to an imagined human figure in the sky with outstretched arms to specify the quadrants. See our own prescription for a modern star clock.
Example of Guards marking the effect of the rotation of Polaris around the Pole in late 1400s. The "line" is between Polaris and Kochab. The numbers mark the height of Polaris in degrees viewed from Lisbon with the Guards in those positions. Lisbon is at about 38.5º N, so Guards in the left shoulder meant Polaris was directly below the pole (see previous graphic), and Guards on the right hip meant Polaris directly over the Pole. A sailor farther south would then measure the height of polaris and note the location of the Guards to compare with this diagram (or printed Regiment) to learn how far south of Lisbon they were.
There is record of a Portuguese observation of Polaris in 1462 by Diego Gomez Cintra, who used a quadrant to measure the height of Polaris to determine latitude. He went on to "discover" Sierra Leone in 1465. It is not known what Regiment he used.
Eventually the reference to Lisbon was replaced with a more generic reference to Latitude based on the location of the Guards relative to this imaginary figure. The earliest such references were:
"Guards in the head, North Star 3° under the Pole,
Guards in the feet, Star 3° above the Pole"
The Guards stars (tips of the horn, above the word GVARDAS) are strangely not marked as the other stars in this original sketch, and it is not entirely clear what was meant to be the POLE—though we can piece this together since we know where these stars really are.
Cortes states that he uses 4.9º for the maximum offset, because that is what the astronomers told him, and they know the most about stars, but he acknowledges that mariners believe it is 3.5º, which is in fact what it was in the late 1400s, which probably reflects the actual time of its common usage. This was some 60 or 70 years earlier than the Cortes book. At the time of Cortes is was just about 3º.
This classic Spanish text was translated to English by Richard Eden in 1561. The full text is online.
I have not tracked down how this method finally faded away. We had good star almacs by early 1700s which would rule out such needs. It was used at least into the early 1600s. Here is an example from a 1595 voyage of Thomas Hariot who had made an improvement to the earlier Regiments, but it seems to have been lost. The process seems a bit complex (from David Water's book).
With modern almanacs and super tools like Stellarium, we can now investigate these early procedures and indeed make up our own rules, such as this Modern Regiment of the North Star. As celestial navigation itself fades into the horizon for land based navigation, these earlier, simpler tools may find a place again in the backup kit of prudent navigators.