coefficients, Lindemann had demonstrated the impossi-
bility of constructing a square of the same area of a
given circle using the classical tools of a straight-edge
and compass alone.
Lindemann wrote a thesis on the topic of
NON
-
EUCLIDEAN GEOMETRY
under the direction of C
HRIS
-
TIAN
F
ELIX
K
LEIN
(1849–1925), and was awarded a
doctoral degree from Erlangen in 1873. He completed
an advanced habilitation degree in 1877 at the Univer-
sity of Würzburg and was appointed a faculty position
at the University of Freiburg that same year. He later
transferred to the University of Königsberg, and then
eventually accepted a chair at the University of Munich
in 1893, where he remained for the rest of his career.
In 1873, the year Lindemann was awarded his doc-
torate, French mathematician Charles Hermite pub-
lished his proof that the number eis transcendental.
Lindemann traveled to Paris to meet Hermite and to
discuss the methods of his proof. Using the famous for-
mula eiπ= –1 of L
EONHARD
E
ULER
(1707–83), Linde-
mann realized that Hermite’s methods could be
extended to also establish the transcendence of π. Lin-
demann published his proof in his 1882 paper Über die
Zahl (On numbers).
Lindemann was also interested in physics and con-
tributed to the studies of electrons. He also worked to
translate and revise the work of the mathematician
J
ULES
H
ENRI
P
OINCARÉ
(1854–1912).
In 1894 Lindemann was elected to the Bavarian
Academy of Sciences. He was also praised with an hon-
orary degree from the University of St. Andrews in
1912. He died in Munich, Germany, on March 6, 1939,
and will always be remembered in history for bringing a
close to the classic problem of squaring the circle.
line A
CURVE
is sometimes called a line. In
GEOME
-
TRY
, a line is usually understood to be straight, but it
is difficult to properly define what is meant by this.
The geometer E
UCLID
(ca. 300–260
B
.
C
.
E
.) provided
the intuitive definition of a line as a “length with no
breadth,” but he never attempted to define what is
meant by a length or what it means to say that a con-
struct has no breadth. Euclid, however, did state that
between any two points Aand Bin the plane, there is
such a thing as a straight line that connects them.
Today mathematicians take this as the starting point of
geometry, leaving the terms line and point (and plane)
as undefined terms, but taking the properties we
expect them to possess (such as “between every two
points there is a line that connects them”) as
AXIOM
s
for the theory of geometry.
If one is working with a theory of geometry (or
of shape and space) in which there is a clear notion of
a distance between two points, then one could define
a straight line between two points to be the shortest
path between those points. For instance, P
YTHAGO
-
RAS
’
S THEOREM
, in some sense, establishes that
straight paths, as we intuitively think of them, are
indeed the shortest routes between two points. On
the surface of a
SPHERE
, the shortest paths between
points are arcs of great circles, and it is therefore
appropriate to deem these as the “straight” paths in
SPHERICAL GEOMETRY
.
See also
COLLINEAR
;
CONCURRENT
;
EQUATION OF A
LINE
;
LINEAR EQUATION
;
SLOPE
.
linear algebra The study of matrices and their
applications is called linear algebra. As matrices are
used to analyze and solve systems of
SIMULTANEOUS
LINEAR EQUATIONS
and to describe
LINEAR TRANSFOR
-
MATION
s between
VECTOR SPACE
s, this topic of study
unites geometric thinking with numerical analysis. As
the set of all invertible matrices of a given size form a
group, called the
GENERAL LINEAR GROUP
, techniques
of
ABSTRACT ALGEBRA
can also be incorporated into
this work.
See also
MATRIX
.
linear equation An equation is called linear if no
variable appearing in the equation is raised to a
power different from 1, and no two (or more) vari-
ables appearing in the equation are multiplied
together. For example, the equation 2x– 3y+ z= 6 is
linear, but the equations 2x3– 5y+ z–1 = 0 and 4xy +
5xz = 7 are not.
A function of one variable is said to be linear if it
is of the form f(x) = ax + b, for some constants aand
b. More generally, a function of several variables of
the form
f(x1,x2,…,xn) = a0+ a1x1+ a2x2+…+ anxn
for some constants a0, a1, a2, …, anis called linear.
314 line