- PROJECTIVE AND DESCRIPTIVE GEOMETRY 365
QFIGURE 9. Pascal's theorem (third lemma),the two directions can be described as
M " ((Ä-OsnV^)·
Newton noted that this kind of projection preserves the degree of an equation.
Hence a conic section will remain a conic section, a cubic curve will remain a cubic
curve, and so on, under such a mapping. In fact, if a polynomial equation p(x, y) = 0
is given whose highest-degree term is xmyn, then every term xpyq, when expressed in
terms of î and 77, will be a multiple of îñç' 1 /{Ä—î)ñ+'', so that if the entire equation
is converted to the new coordinates and then multiplied by (Ä - £)m+n, this term
will become îñçç(Á — ^)m+n_p^9! which will be of degree m + n. Thus the degree of
an equation does not change under Newton's mapping. These mappings are special
cases of the transformations known as fractional-linear or Mobius transformations,
after August Ferdinand Mobius (1790-1868), who developed them more fully. They
play a vital role in algebraic geometry and complex analysis, being the only one-
to-one analytic mappings of the extended complex plane onto itself. According to
Coolidge (1940, p. 269), it was Edward Waring (1736-1798) who first remarked, in
1762, that fractional-linear transformations were the most general degree-preserving
transformations.
2.6. Charles Brianchon. Pascal's work on the projective properties of conies was
extended by Charles Julien Brianchon (1785-1864), who was also only a teenager
when he proved what is now recognized as the dual of Pascal's theorem: The pairs