coefficients, Lindemann had demonstrated the impossibility 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 NONEUCLIDEAN GEOMETRY under the direction of CHRISTIAN FELIX KLEIN (1849–1925), and was awarded a
doctoral degree from Erlangen in 1873. He completed
an advanced habilitation degree in 1877 at the University 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 doctorate, French mathematician Charles Hermite published his proof that the number e is transcendental.
Lindemann traveled to Paris to meet Hermite and to
discuss the methods of his proof. Using the famous formula e
iπ = –1 of LEONHARD EULER (1707–83), Lindemann realized that Hermite’s methods could be
extended to also establish the transcendence of π. Lindemann published his proof in his 1882 paper Über die
Zahl (On numbers).
Lindemann was also interested in physics and contributed to the studies of electrons. He also worked to
translate and revise the work of the mathematician
JULES HENRI POINCARÉ (1854–1912).
In 1894 Lindemann was elected to the Bavarian
Academy of Sciences. He was also praised with an honorary 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 GEOMETRY, a line is usually understood to be straight, but it
is difficult to properly define what is meant by this.
The geometer EUCLID (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 construct has no breadth. Euclid, however, did state that
between any two points A and B in 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 AXIOMs
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, PYTHAGORAS’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 TRANSFORMATIONs between VECTOR SPACEs, 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) variables appearing in the equation are multiplied
together. For example, the equation 2x – 3y + z = 6 is
linear, but the equations 2x
3 – 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 a and
b. More generally, a function of several variables of
the form
f(x 1 ,x 2 ,…,x n ) = a 0 + a 1 x 1 + a 2 x 2 +…+ a n x n
for some constants a 0 , a 1 , a 2 , …, a n is called linear.
314 line
given circle using the classical tools of a straight-edge
and compass alone.
Lindemann wrote a thesis on the topic of NONEUCLIDEAN GEOMETRY under the direction of CHRISTIAN FELIX KLEIN (1849–1925), and was awarded a
doctoral degree from Erlangen in 1873. He completed
an advanced habilitation degree in 1877 at the University 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 doctorate, French mathematician Charles Hermite published his proof that the number e is transcendental.
Lindemann traveled to Paris to meet Hermite and to
discuss the methods of his proof. Using the famous formula e
iπ = –1 of LEONHARD EULER (1707–83), Lindemann realized that Hermite’s methods could be
extended to also establish the transcendence of π. Lindemann published his proof in his 1882 paper Über die
Zahl (On numbers).
Lindemann was also interested in physics and contributed to the studies of electrons. He also worked to
translate and revise the work of the mathematician
JULES HENRI POINCARÉ (1854–1912).
In 1894 Lindemann was elected to the Bavarian
Academy of Sciences. He was also praised with an honorary 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 GEOMETRY, a line is usually understood to be straight, but it
is difficult to properly define what is meant by this.
The geometer EUCLID (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 construct has no breadth. Euclid, however, did state that
between any two points A and B in 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 AXIOMs
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, PYTHAGORAS’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 TRANSFORMATIONs between VECTOR SPACEs, 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) variables appearing in the equation are multiplied
together. For example, the equation 2x – 3y + z = 6 is
linear, but the equations 2x
3 – 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 a and
b. More generally, a function of several variables of
the form
f(x 1 ,x 2 ,…,x n ) = a 0 + a 1 x 1 + a 2 x 2 +…+ a n x n
for some constants a 0 , a 1 , a 2 , …, a n is called linear.
314 line