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
Compass equivalence theorem points A, B, and C, construct a circle centered at A with radius the length of BC (that is, equivalent to the solid green circle, but centered at A). It is possible to prove compass equivalence without the use of the straightedge. This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) in proofs of the Mohr–Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone. Given points A, B, and C, construct a circle centered at A with the radius BC, using