- PROJECTIVE AND DESCRIPTIVE GEOMETRY 361
One of Diirer's devices for producing an accurate painting. The
artist's assistant at the left holds a needle at a particular point on
the lute being painted, while the artist sticks a pair of crosshairs on
the frame to mark the exact point where the thread passes through
the window. The needle and thread are then to be removed, the
door holding the canvas closed, and the spot where the crosshairs
meet marked on the canvas. © Corbis Images (No. SF1906).The most remarkable properties of the sections of a scroll are com-
mon to all types, and the names Ellipse, Parabola, and Hyperbola
have been given them only on account of matters extraneous to
them and to their nature.Desargues was among the first to regard lines as infinitely long, in the modern
way. In fact, he opens his treatise by saying that he will consider both the infinitely
large and the infinitely small in his work, and he says firmly that "in this work every
straight line is, if necessary, taken to be produced to infinity in both directions."
He also had the important insight that a family of parallel lines and a family of
lines with a common point of intersection have similar properties. He said that
lines belonged to the same order 12 if either they all intersected at a common point
or were all mutually parallel. This term was introduced "[to] indicate that in the
one case as well as in the other, it is as if they all converged to the same place"
[emphasis added].
2 Now called a pencil or sheaf.