Construction by Compasses Alone

Compass Geometry

1. Euclidean constructions

In Book I of Euclid's Elements there are four postulates for constructions in plane geometry. Postulates 1 and 2 construct straight lines given two points. Postulate 3 constructs a circle given a center and radius Postulate 5 constructs points of intersections for nonparallel lines. Although Euclid did not state postulates for constructing intersection points for two circles, and intersection points for a circle and a line, such unstated postulates are used. For instance, an intersection point for two circles is needed in Proposition I.1, and an intersection point for a circle and a line is used in I.2.

The rest of the constructions in Euclidean geometry are built out of the constructions of the postulates. Such constructions include angle bisection (I.9), line bisection (I.10), perpendicular lines (I.11 and I.12), triangles given the three sides (I.22), parallel lines (I.31), and many others.

Our construction of a compass is precisely that of Euclid's Postulate 3. This doesn't correspond precisely to what a physical compass can do, since a physical compass can also transfer distances; given three points A, B, and C, a physical compass can be set to an opening of size AB, then moved to draw a circle with center C and radius equal to size AB. Nonetheless, Postulate 3 is enough, as Euclid shows in I.3 how to transfer distances. Three Euclidean constructions

We are interested in constructing the points of intersections of circles and lines, rather than the circles and lines themselves. Although we cannot construct a line with a compass that only draws circles, we will be able to find the intersection points of that line with other lines and with circles.

Essentially, there are three methods to construct points in Euclidean geometry, CC, LC, and LL.

CC: construct the intersections of two circles given their centers and a point on each circumference.: For example, in the diagram at the right, the circle with center A and radius AB intersects the circle with center C and radius CD at the two points E and F.
LC: construct the intersections of a line given by two points and a circle given its center and a point on its circumference.: For example, the intersections of the line GH with the circle with center C and radius CD are the two points K and L.
LL: construct the intersection of two lines, each given by two points.: For example, the line GH intersects the line MN at the point P.

Frequently, the circles and lines don't intersect, but it is only when they do intersect that the points are constructed, so the possibility of nonintersection is not relevant here.

Next part: 2. Reduce Euclidean constructions to circular constructions

April, 1998; March, 2002.
David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

Email: djoyce@clarku.edu
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