The Marked Ruler

Abstract

A marked ruler is simply a straightedge with two marks on its edge. With an appropriate starter set, an immediate use of the marked ruler is to draw the line through two given points and mark off unit segments on the line. Since the two marks thus provide a scale in the sense of a rusty dividers, then the ruler and dividers theory can be applied to marked ruler constructions. Throughout this chapter, the two points determined by the two marks on the marked ruler in a particular position will be called R and S. We suppose the marks are one unit apart. So RS = 1, as in Figure 9.1a. For a second use of the marked ruler, we can set one mark on a given point R and rotate the marked ruler until the second mark falls on a given line s at a point S, whenever the unit circle with center R intersects the line s. See Figure 9.1b. The unit circle with center R can be taken as a poncelet circle. The Poncelet-Steiner Theorem now implies all the ruler and compass constructions are possible with the marked ruler alone. The characteristic use of the marked ruler is called verging or insertion. Given point V and two lines r and s, by verging through V with respect to r and s we determine two points R and S that are one unit apart and such that V$\overleftrightarrow {{\text{RS}}}$ is on , R is on r, and S on s. See Figure 9.1c. Thus, in verging through V with respect to r and s, the marked ruler is placed down to pass through V with one mark on r and the other on s. Using the marked ruler in this way to solve two of the three classical construction problems goes back to the Greeks who, according to Pappus,“moved a ruler about a fixed point until by trial the intercept was found to be equal to the given length.” A verging is sometimes called by its Greek name neusis. Although Apollonius’s book Neusis on the subject is lost, J. P. Hogendijk has reconstructed the text from Arabic traces of the work.

After the dark ages in Europe, the first important mathematical work was that of the Italian school. It contributed to arithmetic and produced the ultimate in classical algebra. Symbolic algebra per se first appeared at the very end of the sixteenth century, however, in the ideas of Vieta, who preceded by a short time interval the men whom we have called the forefathers of modern mathematics, namely, Descartes and Fermat.

Edna E. Kramer