- DIFFERENTIAL GEOMETRY 381
when applied to vectors in physics. Grassmann remarked that parentheses have no
effect on the outer product -in our terms, it is an associative operation.^30
Working with these concepts, Grassmann defined the numerical value of an
extended quantity as the positive square root of its inner square, exactly what we
now call the absolute value of a vector in ç-dimensional space. He proved that
"the quantities of an orthogonal system are not related numerically," that is, an
orthogonal set of nonzero vectors is linearly independent.
Historians of mathematics seem to agree that, because of its philosophical tone
and unusual nomenclature, Ausdehnungslehre did not attract a great deal of notice
until Grassmann revised it and published a more systematic exposition in 1862. If
that verdict is correct, there is a small coincidence in Riemann's use of the term
"extended." which appears to mimic Grassmann's use of the word, and in his focus
on a general number of dimensions in his inaugural lecture at the University of
Gottingen. Riemann's most authoritative biographer Laugwitz (1999, p. 223) says
that Grassmann's work would have been of little use to Riemann, since for him
linear algebra was a trivial subject.^31 This lecture was read in 1854, with the
aged Gauss in the audience.^32 Although Riemann's lecture "Uber die Hypothesen
die der Geometrie zu Grunde liegen" ("On the hypotheses that form the basis of
geometry") occupies only 14 printed pages and contains almost no mathematical
symbolism—it was aimed at a largely nonmathematical audience;—it set forth ideas
that had profound consequences for the hiture of both mathematics and physics.
As Hermann Wcyl said:The same step was taken here that was taken by Faraday and
Maxwell in physics, the theory of electricity in particular, ... by
passing from the theory of action at a distance to the theory of
local action: the principle of understanding the world from its
behavior on the infinitesimal level. [Narasimhan, 1990, p. 740]In the first section Riemann began by developing the concept of an ç-fold ex-
tended quantity, asking the indulgence of his audience for delving into philosophy,
where he had limited experience. He cited only some philosophical work of Gauss
and of Johann Friedrich Herbart (1776 1841), a mathematically inclined philoso-
pher whose attempts to quantify sense impressions was an early form of mathemat-
ical psychology.^33 He began with the concept of quantity in general, which arises
when some general concept can be defined (measured or counted) in different ways.
Then, according as there is or is not a continuous transformation from one of the(^30) To avoid confusing the reader who knows that the cross product is not an associative product,
we note that the outer product applies only when each of the factors is orthogonal to the others.
In three dimensional space the cross product of three such vectors, however they are grouped, is
always zero.
(^31) One can't help wondering about the muiitlinear algebra that Grassmann was developing. The
recognition of this theory as an essential part of geometry is explicit in Felix Klein's 1908 work
on elementary geometry from a higher viewpoint, but Riemann apparently did not make the
connection.
(^112) At the time of the lecture Gauss had less than a year of life remaining. Yet his mind was still
active, and he was very favorably impressed by Riemann's performance.
(^33) Herbart's 1824 book Psychologie als Wissenschaft, neu gegriindet auf Erfahrung, Metaphysik,
und Mathematik (Psychology as Science on a New Foundation of Experiment, Metaphysics, and
Mathematics) is full of mathematical formulas involving the strength of sense impressions, ma-
nipulated by the rules of algebra and calculus.