356 12. MODERN GEOMETRIEScubic curves generally intersect in nine points, so that some sets of nine points do
not determine the curve uniquely (see Problem 12.7). This fact was later (1748)
noted by Euler as well, and finally, by Gabriel Cramer (1704-1752), who also noted
Maclaurin's priority in the discovery that a curve of degree m and a curve of degree
ç meet generally in mn points. This interesting fact is called Cramer's paradox
after Cramer published it in a 1750 textbook on algebraic curves. Although he
correctly explained why more than one curve of degree ç can sometimes be made
to pass through n(n + 3)/2 points—because the equations for determining the co-
efficients from the coordinates of the points might not be independent—he noted
that in that case there were actually infinitely many such curves. That, he said,
was a real paradox. Incidentally, it was in connection with the determination of the
coefficients of an algebraic curve through given points that Cramer stated Cramer's
rule for solving a system of linear equations by determinants.^5
2. Projective and descriptive geometry
It is said that Euclid's geometry is tactile rather than visual, since the theorems
tell you what you can measure and feel with your hands, not what your eye sees.
It is a commonplace that a circle seen from any position except a point on the line
through its center perpendicular to its plane appears to be an ellipse. If figures did
not distort in this way when seen in perspective, we would have a very difficult time
navigating through the world. We are so accustomed to adjusting our judgments of
what we see that we usually recognize a circle automatically when we see it, even
from an angle. The distortion is an essential element of our perception of depth.
Artists, especially those of the Italian renaissance, used these principles to create
paintings that were astoundingly realistic. As Leonardo da Vinci (1452-1519) said,
"the primary task of a painter is to make a flat plane look like a body seen in
relief projecting out of it." Many records of the principles by which this effect was
achieved have survived, including treatises of Leonardo himself and a very famous
painter's manual of Albrecht Durer (1471 1528), first published in 1525. Over a
period of several centuries these principles gave rise to the subject now known as
projective geometry.
2.1. Projective properties. Projective geometry studies the mathematical rela-
tions among figures that remain constant in perspective. Among these things are
points and lines, the number of intersections of lines and circles, and consequently
also such things as parallelism and tangency, but not things that depend on shape,
such as angles or circles.
A less obvious property that is preserved is what is now called the cross-ratio
of four points on a line.^6 If A, B, C, and D are four points on a line, with  and
(^5) As mentioned in Chapter 8, the solution of linear equations by determinants had been known
to Seki Kowa and Leibniz. Thus, Cramer has two mathematical concepts named after him, and
in both cases he was the third person to make the discovery.
(^6) Although this ratio has been used for centuries, the name it now bears in English seems to go
back only to an 1869 treatise on dynamics by William Kingdon Clifford (1845 1879). Before that
it was called the anharmonic ratio, a phrase translated from an 1837 French treatise by Michel
Chasles (1809-1880). This information came from the website on the history of mathematical
terms maintained by Jeff Miller of Gulf High School in New Port Richey, Florida. The url of the
website is http://member8.aol.com/jeff570/mathword.html.