In truth, the tracking of celestial objects is as much shipcraft as river science. In 1609 (and indeed, for thousands of years before that and to this very day), celestial navigation was a crucial aspect of nautical life.
We use a traditional instrument called a quadrant to measure the angle of celestial objects above the horizon; these include the sun and various stars.
Captain Hudson himself would have used this instrument to determine his latitude (his position on the north-south axis of the globe). The quadrant is so named because it forms one quarter of a circle (a 90-degree angle). 0 degrees represents the equator, while 90 degrees represents the north pole. Gravity pulls the dangling plumb bob straight down toward the center of the Earth. A navigator in the northern hemisphere can look up along the quadrant's flat edge to observe Polaris, the north star; the resulting angle indicated by the plumb bob would then indicate the user's northern latitude (see the Daily Logs for our latitudes at each anchorage throughout the voyage).