### What is 'Straight'? How Can We Draw a Straight Line?

Two Cornell professors use history to develop students' understanding of geometry concepts, as in this example:

"When using a compass to draw a circle, we are not starting with a model of a circle; instead we are using a fundamental property of circles that the points on a circle are a fixed distance from a center. Or we can say we use Euclid's definition of a circle. So, now what about drawing a straight line: Is there a tool (serving the role of a compass) that will draw a straight line? One could say: We can use a straightedge for constructing a straight line. Well, how do you know that your straightedge is straight? How can you check that something is straight? What does 'straight' mean? . . .

"An exact straight-line linkage in a plane was not known until 1864-1871 when a French army officer, Charles Nicolas Peaucellier (1832-1913), and a Russian graduate student, Lipmann I. Lipkin (1851-1875), independently developed a linkage that draws an exact straight line. . . .The drawing in Figure 7 depicts the working parts of the Peaucellier-Lipkin linkage. The linkage works because the point *P* is inverted to the point *Q* through a circle with center at *C* and a radius squared equal to *s*^{2} - *d*^{2}. The point *P* is constrained to move on a circle that has center at *D* and that passes through *C* and thus *Q* must move along a straight line. If the distance between *C* and *D* is changed to *g* (not equal to *f*), then *Q* instead of moving along a straight line will move along an arc of a circle of radius (*s*^{2} - *d*^{2})*f*/(*g*^{2} - *f*^{2}). This allows one to draw an arc of a large circle without using its center. Note that the definition of a straight line used here is: 'A straight line is a circle of infinite radius.' For a learning module on these topics, see [http://kmoddl.library.cornell.edu/tutorials/04/]."

--from Daina Taimina and David W. Henderson, "How to Use History to Clarify Common Confusions in Geometry," in From Calculus to Computers: Using the Last 200 Years of Mathematics History in the Classroom, MAA Notes #68 (2005): 57-73. Link to MnCat Record