Just registered to add to this thread, even if it's a few months old. Recently had the privilege
of sailing on part of Hokulea's worldwide voyage via her sister canoe Hikianalia. I wanted to second the recommendation of David Lewis' We, the Navigators, as well as Will Kyselka's An Ocean in Mind for background on ancient and now "modern" non instrument Polynesian Voyaging. (David Lewis also describes Micronesian methods.) Hokulea has a flashy new website documenting the voyage, but the older site Hawaiian Voyaging Traditions
is a wealth of knowledge on some of the ancient and recreated methods.
If you are math-inclined, you can get a lot of these methods yourself by using spherical trigonometry applied to the same navigator's triangle that the celestial (sextant) navigation method uses.
sin(D) = sin(L)*sin(H) + cos(L)*cos(H)*cos(Z). D is declination of celestial object, L is your latitude, H is height of object above the horizon, and Z is azimuth of object measured from north "clockwise looking-down" as in NESW.
For example, Hokulea is the Hawaiian name for Arcturus, whose declination of 19.5º is about the same as the latitude of the southern part of the Big Island of Hawaii. So once a sidereal day there (or at another place with the same latitude, like Mexico
, Saudi Arabia, etc.) Hokulea passes directly overhead. Mathwise, if H = 90º (overhead), that fat equation simplifies down to sin(D) = sin(L), so if a star or the sun is directly overhead, its declination is your latitude.
But you can get direction info also: if Hokulea (Arcturus) is on the horizon, H = 0º, then the equation becomes sin(D) = cos(L)*cos(Z). If you are close to the equator, and most of Polynesia is, then cos(L) ~ 1 so sin(D) ~ cos(Z), so Z = 90º - D or 270º + D. So for Hokulea, Z = 70.5º (when it's on the horizon, rising, which is a bit harder to anticipate), or Z = 289.5º (when it's on the horizon, setting).
So you can use lots of stars, not just Polaris, to get not only heading but also latitude information. The heading info, along with the speed of your vessel and time on course can enable you to mental dead reckon your position, and the latitude info can act as a double-check. And there are a bunch of other less obvious techniques, such as how certain pairs of stars rise together at the same time, or form a vertical line over due north or due south, etc.
Then there's a wealth of info to be had by looking at swell periods and directions (and less reliably, wind) when the sky is overcast. And finally, methods to detect islands, especially high volcanic islands, 100 miles or more away.
The open ocean navigation might not be very accurate, even compared to celestial (sextant) navigation, but the land finding helps narrow things down at the end. And anyway, having also had the privilege
of sailing aboard the USS San Francisco
aren't always entirely accurate anyway.