Constructions

Much of Euclidean geometry is based on two geometric constructions: the drawing of circles and the drawing of straight lines. To draw a circle with a compass, one needs to know the location of the center and some one point on the circle. To draw a line segment with a straightedge, one needs to know the location of its two end points. To extend a segment, one must know the location of it or a piece of it.

Three of the five postulates in Euclid's Elements say that these constructions are possible:

To draw a line from any point to any point.

To produce a finite straight line in a straight line.

To describe a circle with any center and distance.

Figure 1. Construction of an equilateral triangle. Illustration by Hans & Cassidy. Courtesy of Gale Group.

The constructions based on these postulates are called "straightedge and compass" constructions.

The Elements does not explain why these tools have been chosen, but one may guess that it was their utter simplicity which geometers found, and continue to find, appealing. These tools are certainly not the only ones which the Greeks employed, and they are not the only ones upon which modern draftsmen depend. They have triangles, french curves, ellipsographs, T-squares, scales, protractors, and other drawing aids which both speed the drawing and make it more precise.

These tools are not the only ones on which contemporary geometry courses are based. Such courses will often include a protractor postulate which allows one to measure angles and to draw angles of a given size. They may include a ruler-placement postulate which allows one to measure distances and to construct segments of any length. Such postulates turn problems which were once purely geometric into problems with an arithmetic component. Nevertheless, straightedge and compass constructions are still studied.

Euclid's first proposition is to show that, given a segment AB, one can construct an equilateral triangle ABC. (There has to be a segment. Without a segment, there will not be a triangle.) Using A as a center, he draws a circle through B. Using B as a center, he draws a circle through A. He calls either of the two points where the circles cross C. That gives him two points, so he can draw segment AC. He can draw BC. Then ABC is the required triangle (Figure 1).

Once Euclid has shown that an equilateral triangle can be constructed, the ability to do so is added to his tool bag. He now can draw circles, lines, and equilateral Figure 2. Illustration of Archimedes method for trisecting an arbitrary angle ABC. Illustration by Hans & Cassidy. Courtesy of Gale Group. triangles. He goes on to add the ability to draw perpendiculars, to bisect angles, to draw a line through a given point parallel to a given line, to draw equal circles, to transfer a line segment to a new location, to divide a line segment into a specified number of equal parts, and so on.

There are three constructions which, with the given tools, neither Euclid nor any of his successors were able to do. One was to trisect an arbitrary angle. Another was to draw a square whose area was equal to that of a given circle. A third was to draw the edge of a cube whose volume was double that of a given cube. "Squaring the circle," as the second construction is called, is equivalent to drawing a segment whose length is Π times that of a given segment. "Duplicating the cube" requires drawing a segment whose length is the cube root of 2 times that of the given segment.

In about 240 B.C., Archimedes devised a method of trisecting an arbitrary angle ABC. Figure 2 shows how he did it. Angle ABC is the given angle. ED is a movable line with ED = AB. It is placed so that E lies on BC extended; D lies on the circle; and the line passes through A. Then ED = DB = AB, so triangles EDB and ABD are isosceles. Because the base angles of an isosceles triangle are equal and because the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles, the sizes, in terms of x, of the various angles are as marked. Angle E is, therefore, one third the size of the given angle ABC; ABC has been trisected.

Why is this ingenious but simple construction not a solution to the problem of trisecting an angle? Line ED has to be movable. It requires a straightedge with marks on it. Simple as marking a straightedge might be, the Euclidean postulates don't make provision for doing so.

Archimedes' technique for trisecting an angle is by no means the only one which has been devised. Eves, in his History of Mathematics, describes several others, all ingenious. He also describes techniques for squaring the circle and duplicating the cube. All the constructions he describes, however, call for tools other than a compass and straightedge.

Actually doing these constructions is not just difficult with the tools allowed; it is impossible. This was proved using algebraic arguments in the nineteenth century. Nevertheless, because the goals of the constructions are so easily stated and understood, and because the tools are so simple, people continue to work at them, not knowing, or perhaps not really caring, that their task is a Sisyphean one.

The straightedge and compass are certainly simple tools, yet mathematicians have tried to get along with even simpler ones. In the tenth century Abul Wefa, a Persian mathematician, based his constructions on a straight-edge and a rusty compass—one that could not be adjusted. Nine centuries later it was proved by mathematicians Poncelet and Steiner that, except for drawing circles of a particular size, a straightedge and rusty compass could do everything a straightedge and ordinary compass could do. They went even further, replacing the rusty compass with one already-drawn circle and its center.

In 1797, the Italian mathematician Mascheroni published a book in which he showed that a compass alone could be used to do anything that one could do with a compass and straightedge together. He could not draw straight lines, of course, but he could locate the two points that would determine the undrawn line; he could find where two undrawn lines would intersect; he could locate the vertices of a pentagon; and so on. Later, his work was found to have been anticipated more than 100 years earlier by the Danish mathematician Mohr. Compass-only constructions are now known as Mohr-Mascheroni constructions.

Resources

Books

Birkhoff, George David, and Ralph Beatley. Basic Geometry. New York: Chelsea Publishing Co., 1959.

Euclid. Sir Thomas L. Heath, trans. Elements. New York: Dover Publishing Co., 1956.

Gardner, Martin. Mathematical Circus. New York: Alfred A. Knopf, 1979.

Gullberg, Jan, and Peter Hilton. Mathematics: From the Birth of Numbers. W.W. Norton & Company, 1997.

Hahn, Liang-shin. Complex Numbers and Geometry. 2nd ed. The Mathematical Association of America, 1996.

J. Paul Moulton

KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Compass

—A device for drawing circles having a given center and radius.

Construction

—A drawing of a geometrical figure made with the use of certain specified tools.

Straightedge

—An unmarked ruler of any length that can be used to draw lines through two given points.