The problem of measuring angles has been considered by mathematicians since ancient times. The ability to measure angles with some precision is important in fields such as astronomy, physics, engineering, and architecture, as well as in various practical supporting fields such as carpentry. It is difficult to imagine the ancient Egyptians trying to construct the pyramids without a precise system for measuring angles, particularly right angles.

The problem of measuring angles is compounded by the need for a satisfactory definition of the object to be measured. It was not until the concept of a set was developed in the late nineteenth century that the modern definitions of geometric objects such as sets of points were possible, and it was well into the twentieth century before such definitions began to appear in geometry textbooks. As late as the 1920s an angle was typically defined by...