The picture above is meant to represent the astronomer Claudius Ptolemy (an early champion of the geocentric model), who lived around 150 AD. It is an old picture, though not old enough for the artist to have actually known what Ptolemy looked like. But what is that gentleman holding?
It isn't a religious symbol --- the proportions are not right, and the marks on the stick do not seem appropriate. It is actually a cross staff (or “Jacob's staff"), a tool widely used by astronomers and navigators before the invention of the telescope, and for a while afterwards. It consists of a main staff with a perpendicular crosspiece, attached to the staff at its middle and able to slide up and down along it.
The device was likely invented by Rabbi Levi ben Gershon (1288-1344), a Jewish scholar who lived in Provence, in southern France, also referred to as “Gersonides." Claudius Ptolemy lived more than 1000 years earlier, so the drawing above indeed takes considerable artistic license.
Astronomers used the cross-staff for measuring the angle between the directions of two stars. Other, older instruments for this purpose existed, used by scholars such as Hipparchos and Ptolemy, but none was as portable, which made the cross-staff eminently suitable for navigation at sea. Ships' officers used it to measure the elevation angle of the noontime Sun above the horizon, which allowed them to estimate their latitude (see section on navigation). The problem of getting dazzled by the Sun later led to the invention of the 'backstaff, where the sunlight fell onto a target, not into the eye. Columbus may well have used one. This was greatly improved around 1594 by Captain John Davis, so perhaps the “Mayflower" used the upgraded design.
To measure the angle between two stars, an astronomer would place the staff just below one eye (see Figure above) and slide the cross-piece up and down. The cross-piece would have a pair of open sights sticking out perpendicular to the drawing at symmetric locations such as B and B' (often several pairs of sights, some spaced further apart than others). The astronomer would slide the cross-piece up and down, until sight B covered one of the stars and sight B' the other. For use at night, slits make convenient sights.
After that was achieved, the instrument would be lowered and the distance \(AC\) would be measured. Then if \(\alpha\) was the angle between the staff and the direction of one star, from the definition of the tangent:
The distance \(BC\) between the sight and the stick was already known to the astronomer --- so, using a table of tangents, the angle denoted by \(\alpha\) could be calculated. Since the instrument was symmetric, the angle between the directions of the stars was \(2\alpha\).