380 L2. MODERN GEOMETRIES
of the two principal curvatures derived by Euler would be the same for any two
mutually perpendicular planes cutting a surface. She therefore recommended this
average curvature as the best measure of curvature. Her approach does indeed
make sense in elasticity theory,^28 but turns out not to be so useful for pure geom-
etry.^29 Joseph Liouvillc (1809-1882), who founded the Journal de mathematiques
pures et appliquees in 1836 and edited it until 1874, proved that conformal maps
of three-dimensional regions are far less varied than those in two dimensions, being
necessarily either inversions or similarities or rigid motions. He published this result
in the fifth edition of Monge's book on the applications of analysis to geometry. In
contrast, a mapping (x, y) H-> (U, V) is conformal if and only if one of the functions
u{x, y) ± iv(x, y) is analytic. As a consequence, there is a rich supply of conformal
mappings of the plane.
After Newton differential geometry languished in Britain until the nineteenth
century, when William Rowan Hamilton (1805-1865) published papers on systems
of rays, building the foundation for the application of differential geometry to dif-
ferential equations. Another British mathematician, George Salmon (1819-1904),
made the entire subject more accessible with his famous textbooks Higher Plane
Curves (1852) and Analytic Geometry of Three Dimensions (1862).
3.7. Riemann. Once the idea of using parameters to describe a surface has been
grasped, the development of geometry can proceed algebraically, without reference
to what is possible in three-dimensional Euclidean space. This idea was understood
by Hermann Grassmann (1809-1877), a secondary-school teacher, who wrote a
philosophically inclined mathematical work published in 1844 under the title Die
lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Lineal
Extensions, a New Branch of Mathematics). This work, which developed ideas
Grassmann had conceived earlier in a work on the ebb and flow of tides, contained
much of what is now regarded as multilinear algebra. What we call the coefficients
in a linear combination of vectors Grassmann called the numbers by means of which
the quantity was derived from the other quantities. He introduced what we now call
the tensor product and the wedge product for what he called extensive quantities.
He referred to the tensor product simply as the product and the wedge product as
the combinatory product. The tensor product of two extensive quantities 53 Ï-ô^ô
and ^2Pses was
The combinatory product was obtained by applying to this product the rule that
[er,es] = — [es,er] (antisymmetrizing). The determinant is a special case of the
combinatory product. Grassmann remarked that when the factors are "numerically
related" (which we call linearly dependent), the combinatory product would be
zero. When the basic units eT and es were entirely distinct, Grassmann called the
combinatory product the outer product to distinguish it from the inner product,
which is still called by that name today and amounts to the ordinary dot product(^28) In particular, her concept of the average curvature plays a role in the Navier-Stokes equations
(http://www.navier-stokes.net/nsbest.htm).
(^29) However, the average curvature must be zero on a minimal surface.