- THE EARLIEST GREEK GEOMETRY 273
bFIGURE 1. Left: turning a triangle into a rectangle. Right: turn-
ing a rectangle into a square (s^2 = ab).Á â Ó Â
FIGURE 2. Application with defect. Euclid, Book 6, Proposition 28.problem does not have a solution for all given lines and areas, since the largest
parallelogram that can be formed under these conditions is the one whose base is
half of the given line (Book 6, Proposition 26). Assuming that condition, let AB
be the given line, Ã the given polygonal region, and Ä the given parallelogram
shape. The dashed line from Β makes the same angle with AB that the diagonal
of the parallelogram Ä makes with its base. The line ΑÈ is drawn to make the
same angle as the corresponding sides of Ä. Then any parallelogram having its
sides along AB and ΑÈ and opposite corner on the dashed line will automatically
generate a "defect" that is similar to Ä. The remaining problem is to find the one
that has the same area as Ã. That is achieved by constructing the parallelogram
ÇÎÐÏ similar to Ä and equal to the difference between ΑΕÇÈ and Ã.
The Greek word for application is parabole. Proclus cites Eudemus in assert-
ing that the solution of the application problems was an ancient discovery of the
Pythagoreans, and that they gave them the names ellipse and hyperbola, names that
were later transferred to the conic curves by Apollonius. This version of events is
also confirmed by Pappus. We shall see the reason for the transfer below.
Although most of Euclid's theorems have obvious interest from the point of
view of anyone curious about the world, the application problems raise a small
mystery. Why were the Pythagoreans interested in them? Were they merely a
refinement of the transformation problems? Why would anyone be interested in
applying an area so as to have a defect or excess of a certain shape? Without
restriction on the shape of the defect or excess, the application problem does not