Points of the compass limit of the compass point. By the middle of the 18th century, the 32-point system had been further extended by using half- and quarter-points to give a total of 128 directions. These fractional points are named by appending, for example east, east, or east to the name of one of the 32 points. Each of the 96 fractional points can be named in two ways, depending on which of the two adjoining whole points is used, for example, NE is equivalent to NbEN. Either form is easily understood but alternative conventions as to correct usage developed in different countries and
Straightedge and compass construction these straightedge and compass constructions have been shown to be logically impossible to perform. (The problems themselves, however, are solvable, and the Greeks knew how to solve them, "without" the constraint of working only with straightedge and compass.) The most famous of these problems, squaring the circle, otherwise known as the quadrature of the circle, involves constructing a square with the same area as a given circle using only straightedge and compass. Squaring the circle has been proven impossible, as it involves generating a transcendental number, that is, . Only certain algebraic numbers can be constructed with ruler and compass