Postulate 3

To describe a circle with any center and radius. This is the third assumed construction in the Elements. It corresponds to drawing a circle with a compass.

 Circles were defined in Def.I.15 and Def.I.16 as plane figures with the property that there is a certain point, called the center of the circle, such that all straight lines from the center to the boundary are equal. That is, all the radii are equal. The given data are (1) a point A to be the center of the circle, (2) another point B to be on the circumference of the circle, and (3) a plane in which the two points lie. In the first few books of the Elements, there is but one plane under consideration and needn't be mentioned, but in the last three books which develop solid geometry, the plane has to be specified.

Note that this postulate does not allow for the compass to be moved. The usual way that a compass is used is that is is opened to a given width, then the pivot is placed on the drawing surface, then a circle is drawn as the compass is rotated around the pivot. But this postulate does not allow for transferring distances. It is as if the compass collapses as soon as it's removed from the plane. Proposition I.3, however, gives a construction for transferring distances. Therefore, the same constructions that can be made with a regular compass can also be made with Euclid's collapsing compass.

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