Broadly speaking a sextant is an instrument that measures the angle between two objects that are visible. Primarily, it is used to measure the angle between a celestial body and the horizon, a process normally known as sighting the object or taking a sight. The angle measured and the time at which it was measured is then used to identify the location of the user on the grid map of the world. Thus sextants are basically navigational tools and have been successfully used by seaman and even other travelers over the years. The most common process of this is to sight the sun at noon to find the latitude of one’s location.
Measuring the angle between two objects that are visible, which is the purpose of a sextant, has nothing to do with curvature. Every measurement is based on lines, line of siight, mirriors, angles, parallel, and vertical. In fact, the mirrors must be maintained or the measurement will be off. Mirrors within reflect directly, and cannot reflect curve. Notice that identification of location is charted on GRID MAP. A grid map is flat. That they can translate these lines and angles to a globe is just another step. But its an unnecessary one, unless one wants you to believe you are on a globe.