- PROJECTIVE AND DESCRIPTIVE GEOMETRY 357
A circle seen in perspective is an ellipse.C both between A and D and C between  and D, their cross-ratio is
^•"-^
It is not difficult to show, for example, that if the rays PA, PB, PC, and PD from
a point Ñ intersect a second line in points Á', B', C', and D', the cross-ratio of
these new points is the same as that of the original four points. Coolidge (1940, p.
88) speculated that Euclid may have known about the cross-ratio, and he asserted
that the early second-century mathematician Menelaus did know about it.
Some theorems that might appear difficult to prove from the standard Eu-
clidean techniques of proportion and congruence can be quite easy when looked at
"in perspective," so to speak. For example, it is easy to prove that if two tangents
to a circle from points A and C meet at a point P, then the line from Ñ to the
midpoint of the chord AC meets the circle in a point (namely the midpoint of the
arc AC) at which the tangent to the circle is parallel to the chord AC. To prove
that same theorem for an ellipse using analytic geometry is a very tedious compu-
tation. However, remembering that the ellipse was obtained as the intersection of a
cone with a plane oblique to its base, one has only to note that projection preserves
tangency, intersections, parallelism (usually), and midpoints. Then, projecting the
cone and all the lines into the base plane yields the result immediately, as shown in
Fig. 4. 7 Similarly, it could be shown that the bisector of a chord from the point of
intersection of the tangents at the endpoints of the chord passes through the center
of the ellipse.2.2. The Renaissance artists. The revival of interest in ancient culture in gen-
eral during the Renaissance naturally carried with it an interest in geometry. The
famous artist Piero della Francesca (14107-1492) was inspired by the writings of
Leonardo of Pisa and others to write treatises on arithmetic and the five regu-
lar solids. The scholar Luca Pacioli (1445-1517), who was influenced by Piero
della Francesca and was a friend of Leonardo da Vinci, published a comprehensive7 Of course, in the figure the "circle" in the base is really an ellipse because it has been projected
onto the page.