Men of Mathematics (51 page)

The gist of quaternions is the algebra which does for rotations in space of three dimensions what the algebra of complex numbers does for rotations in a plane. But in quaternions (Gauss called them mutations) one of the fundamental rules of algebra breaks down: it is no longer true that
a
×
b = b
×
a,
and it is impossible to make an algebra
of rotations in three dimensions in which this rule
is
preserved. Hamilton, one of the great mathematical geniuses of the nineteenth century, records with Irish exuberance how he struggled for fifteen years to invent a consistent algebra to do what was required until a happy inspiration gave him the clue that
a
×
b
is not equal to
b
×
a
in the algebra he was seeking. Gauss does not state how long it took him to reach the goal; he merely records his success in a few pages of algebra that leave no mathematics to the imagination.

If Gauss was somewhat cool in his printed expressions of appreciation he was cordial enough in his correspondence and in his scientific relations with those who sought him out in a spirit of disinterested inquiry. One of his scientific friendships is of more than mathematical interest as it shows the liberality of Gauss' views regarding women scientific workers. His broadmindedness in this respect would have been remarkable for any man of his generation; for a German it was almost without precedent.

The lady in question was Mademoiselle Sophie Germain (1776–1831)—just a year older than Gauss. She and Gauss never met, and she died (in Paris) before the University of Göttingen could confer the honorary doctor's degree which Gauss recommended to the faculty. By a curious coincidence we shall see the most celebrated woman mathematician of the nineteenth century, another Sophie, getting her degree from the same liberal University many years later after Berlin had refused her on account of her sex. Sophie appears to be a lucky name in mathematics for women—provided they affiliate with broadminded teachers. The leading woman mathematician of our own times, Emmy Noether (1882-1935) also came from Göttingen.
^IV

Sophie Germain's scientific interests embraced acoustics, the mathematical theory of elasticity, and the higher arithmetic, in all of which she did notable work. One contribution in particular to the study of Fermat's Last Theorem led in 1908 to a considerable advance in this direction by the American mathematician Leonard Eugene Dickson (1874 ←).

Entranced by the
Disquisitiones Arithmeticae,
Sophie wrote to Gauss some of her own arithmetical observations. Fearing that Gauss might be prejudiced against a woman mathematician, she assumed a man's name. Gauss formed a high opinion of the talented correspondent whom he addressed in excellent French as “Mr. Leblanc.”

Leblanc dropped her—or his—disguise when she was forced to divulge her true name to Gauss on the occasion of her having done him a good turn with the French infesting Hanover. Writing on April 30, 1807, Gauss thanks his correspondent for her intervention on his behalf with the French General Pernety and deplores the war. Continuing, he pays her a high compliment and expresses something of his own love for the theory of numbers. As the latter is particularly of interest we shall quote from this letter which shows Gauss in one of his cordially human moods.

“But how describe to you my admiration and astonishment at seeing my esteemed correspondent Mr. Leblanc metamorphose himself into this illustrious personage [Sophie Germain] who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it; the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and a superior genius. Indeed nothing could prove to me in so flattering and less equivocal manner that the attractions of this science, which has enriched my life with so many joys, are not chimerical, as the predilection with which
you
have honored it.” He then goes on to discuss mathematics with her. A delightful touch is the date at the end of the letter: “Bronsvic ce 30 Avril 1807 jour de ma naissance—Brunswick, this 30th of April, 1807, my birthday.”

That Gauss was not merely being polite to a young woman admirer is shown by a letter of July 21, 1807 to his friend Olbers. “. . . Lagrange is warmly interested in astronomy and the higher arithmetic; the two test-theorems (for what primes 2 is a cubic or a biquadratic residue), which I also communicated to him some time ago, he
considers 'among the most beautiful things and the most difficult to prove/ But Sophie Germain has sent me the proofs of these; I have not yet been able to go through them, but I believe they are good; at least she had attacked the matter from the right side, only somewhat more diffusely than would be necessary. . . .” The theorems to which Gauss refers are those stating for what odd primes
p
each of the congruences
x
³
≡ 2 (mod
p
),
x
⁴
≡ 2 (mod
p
) is solvable.

*  *  *

It would take a long book (possibly a longer one than would be required for Newton) to describe all the outstanding contributions of Gauss to mathematics, both pure and applied. Here we can only refer to some of the more important works that have not already been mentioned, and we shall select those which have added new techniques to mathematics or which rounded off outstanding problems. As a rough but convenient table of dates (from that adopted by the editors of Gauss' works) we summarize the principal fields of Gauss' interests after 1800 as follows: 1800-1820, astronomy; 1820-1830, geodesy, the theories of surfaces, and conformal mapping; 1830-1840, mathematical physics, particularly electromagnetism, terrestrial magnetism, and the theory of attraction according to the Newtonian law; 18411855, analysis situs, and the geometry associated with functions of a complex variable.

During the period 1821-1848 Gauss was scientific adviser to the Hanoverian (Göttingen was then under the government of Hanover) and Danish governments in an extensive geodetic survey. Gauss threw himself into the work. His method of least squares and his skill in devising schemes for handling masses of numerical data had full scope but, more importantly, the problems arising in the precise survey of a portion of the earth's surface undoubtedly suggested deeper and more general problems connected with all curved surfaces. These researches were to beget the mathematics of relativity. The subject was not new: several of Gauss' predecessors, notably Euler, Lagrange, and Monge, had investigated geometry on certain types of curved surfaces, but it remained for Gauss to attack the problem in all its generality, and from his investigations the first great period of
differential geometry
developed.

Differential geometry may be roughly described as the study of properties of curves, surfaces, etc., in the immediate neighborhood of a point, so that higher powers than the second of distances can be
neglected. Inspired by this work, Riemann in 1854 produced his classic dissertation on the hypotheses which lie at the foundations of geometry, which, in its turn, began the second great period in differential geometry, that which is today of use in mathematical physics, particularly in the theory of general relativity.

Three of the problems which Gauss considered in his work on surfaces suggested general theories of mathematical and scientific importance: the measurement of
curvature,
the theory of
conformal representation
(or mapping), and the
applicability
of surfaces.

The unnecessarily mystical motion of a “curved” space-time, which is a purely mathematical extension of familiar, visualizable curvature to a “space” described by four coordinates instead of two, was a natural development of Gauss' work on curved surfaces. One of his definitions will illustrate the reasonableness of all. The problem is to devise some precise means for describing how the “curvature” of a surface varies from point to point of the surface; the description must satisfy our intuitive feeling for what “more curved” and “less curved” signify.

The total curvature of any part of a surface bounded by an unlooped closed curve
C
is defined as follows. The
normal
to a surface at a given point is that straight line passing through the point which is perpendicular to the plane which touches the surface at the given point. At each point of
C
there is a normal to the surface. Imagine all these normals drawn. Now, from the center of a sphere (which may be anywhere with reference to the surface being considered), whose radius
is equal to the unit length, imagine all the radii drawn which are parallel to the normals to
C.
These radii will cut out a curve, say C, on the sphere of unit radius. The
area
of that part of the spherical surface which is enclosed by
C
′ is defined to be the
total curvature
of the part of the given surface which is enclosed by
C.
A little visualization will show that this definition accords with common notions as required.

Another fundamental idea exploited by Gauss in his study of surfaces was that of
parametric representation.

It requires
two
coordinates to specify a particular point on a plane. Likewise on the surface of a sphere, or on a spheroid like the Earth: the coordinates in this case may be thought of as latitude and longitude. This illustrates what is meant by a
two-dimensional manifold.
Generally: if
precisely n
numbers are both necessary and sufficient to specify (individualize) each particular member of a class of things (points, sounds, colors, lines, etc.,) the class is said to be an
n-dimensional manifold.
In such specifications it is agreed that only certain characteristics of the members of the class shall be assigned numbers. Thus if we consider only the pitch of sounds, we have a one-dimensional manifold, because one number, the frequency of the vibration corresponding to the sound, suffices to determine the pitch; if we add loudness—measured on some convenient scale—sounds are now a two-dimensional manifold, and so on. If now we regard a
surface
as being made up of
points,
we see that it is a
two-dimensional manifold
(of points). Using the language of geometry we find it convenient to speak of
any
two-dimensional manifold as a “surface,” and to apply to the manifold the reasoning of geometry—in the hope of finding something interesting.

The foregoing considerations lead to the parametric representation of surfaces. In Descartes' geometry
one
equation between
three
coordinates represents a surface. Say the coordinates (Cartesian) are
x,
y,
z.
Instead of using a single equation connecting
x, y, z
to represent the surface, we now seek
three:

x = f(u, v), y = g(u, v), z = h(u, v),

where
f(u, v), g(u, v), h(u, v)
are such functions (expressions) of the new variables
u, v
that when these variables are eliminated (got rid of—“put over the threshold,” literally) there results between
x, y, z
the equation of the surface. The elimination is possible, because
two
of
the equations can be used to solve for the
two
unknowns
u, v;
the results can then be substituted in the third. For example, if

x = u + v, y = u
—
v, z = uv,

we get
u = ½(x + y), v = ½(x—y)
from the first two, and hence
4z
=
x
²
—y
²
from the third. Now as the variables
u, v
independently run through any prescribed set of numbers, the functions
f, g, h
will take on numerical values and
x, y, z
will move on the surface whose equations are the three written above. The variables
u, v
are called the
parameters
for the surface, and the three equations
x = f(u, v), y = g(u, v), z = h(u, v)
their parametric equations. This method of representing surfaces has great advantages over the Cartesian when applied to the study of curvature and other properties of surfaces which vary rapidly from point to point.