regions on a unit SPHERE. A SOLID ANGLE of a region is
simply the measure of surface AREA of that region projected onto a unit sphere. Units of solid angle are called
steradians. The full surface area of the sphere is 4π
steradian.
See also BABYLONIAN MATHEMATICS; DIHEDRAL;
SLOPE; TRIANGLE; TRISECTING AN ANGLE.
angle trisection See TRISECTING AN ANGLE.
annulus (plural, annuli) An annulus is the region
between two CONCENTRIC circles in a plane. The
AREA of the annulus is the difference of the areas of
the two circles.
If a cyclist rides a perfect circle, the region between
the tracks made by the front and rear wheels is an
annulus. If the bicycle is r feet long (axel to axel), then,
surprisingly, the area of this annulus is πr
2 feet squared,
irrespective of the size of the circle the cyclist traces.
antidifferentiation (integration) The process of
finding a function with a given function as its DERIVATIVE is called antidifferentiation. For example, x
2 is an
antiderivative of 2x, since
. The MEAN
VALUE THEOREM shows that two antiderivatives of
the same function differ only by a constant. Thus all
the antiderivatives of 2x, for example, are functions
of the form x
2 + C.
The antiderivative of a function f(x) is denoted
∫f(x)dx and is called the indefinite integral of f. It is
defined up to a constant, and so we write, for example,
∫2x dx = x
2 + C. (The constant C is referred to as a
“constant of integration.”) The notation is deliberately
suggestive of a definite integral of a function, ∫
b
a f(x)dx,
for the area under the curve y = f(x) over the INTERVAL
[a,b]. The FUNDAMENTAL THEOREM OF CALCULUS
shows that the two notions are intimately connected.
See also INTEGRAL CALCULUS.
antilogarithm See LOGARITHM.
antipodal points (antipodes) Two points on a
SPHERE at the opposite ends of a diameter are said to be
antipodal. For example, the north and south poles are
antipodal points on the EARTH, as are any two points
EQUIDISTANT from the equator, with longitudes differing by 180°.
The famous Borsuk-Ulam theorem, first conjectured by Stanislaw Ulam and then proved by Karol
Borsuk in 1933, states:
Let f be a continuous function that assigns two
numerical values to each and every point on
the surface of a sphere. Then there must exist
two antipodal points which are assigned precisely the same pair of values.
An amusing interpretation reads:
At any instant there exist two antipodal points
on the Earth’s surface of precisely the same air
temperature and air pressure.
Although the proof of this theorem is difficult, a onedimensional version of the result follows as an easy
consequence of the INTERMEDIATE-VALUE THEOREM.
apex (plural, apices) The point at the top of a POLYGON or a POLYHEDRON, such as the vertex of a triangle
opposite its BASE or the vertex of a pyramid, is called
the apex of the figure. The distance from the base of
the figure to its apex is called the height of the figure.
Apollonius of Perga (ca. 262–190 B.C.E.) Greek Geometry Born in Perga, Greek Ionia, now Antalya,
Turkey, Apollonius worked during the Golden Age of
Greek mathematics and has been referred to throughout
history as the Great Geometer. His famous work, The
Conics, written in eight volumes, greatly influenced the
development of mathematics. (The names ELLIPSE,
PARABOLA, and HYPERBOLA for the three CONIC SECTIONS, for instance, are said to have been coined by
Apollonius.) Copies of the first four volumes of this
work, written in the original Greek, survive today. Arabic translations of the first seven volumes also exist.
Little is known of Apollonius’s life other than what
can be gleaned from incidental comments made in the
prefaces of his books. As a young man it is known that
he traveled to Alexandria to study with the followers of
EUCLID, who then introduced him to the topic of conics.
d
dx
x
x
( )
2
2
=
Apollonius of Perga 15
simply the measure of surface AREA of that region projected onto a unit sphere. Units of solid angle are called
steradians. The full surface area of the sphere is 4π
steradian.
See also BABYLONIAN MATHEMATICS; DIHEDRAL;
SLOPE; TRIANGLE; TRISECTING AN ANGLE.
angle trisection See TRISECTING AN ANGLE.
annulus (plural, annuli) An annulus is the region
between two CONCENTRIC circles in a plane. The
AREA of the annulus is the difference of the areas of
the two circles.
If a cyclist rides a perfect circle, the region between
the tracks made by the front and rear wheels is an
annulus. If the bicycle is r feet long (axel to axel), then,
surprisingly, the area of this annulus is πr
2 feet squared,
irrespective of the size of the circle the cyclist traces.
antidifferentiation (integration) The process of
finding a function with a given function as its DERIVATIVE is called antidifferentiation. For example, x
2 is an
antiderivative of 2x, since
. The MEAN
VALUE THEOREM shows that two antiderivatives of
the same function differ only by a constant. Thus all
the antiderivatives of 2x, for example, are functions
of the form x
2 + C.
The antiderivative of a function f(x) is denoted
∫f(x)dx and is called the indefinite integral of f. It is
defined up to a constant, and so we write, for example,
∫2x dx = x
2 + C. (The constant C is referred to as a
“constant of integration.”) The notation is deliberately
suggestive of a definite integral of a function, ∫
b
a f(x)dx,
for the area under the curve y = f(x) over the INTERVAL
[a,b]. The FUNDAMENTAL THEOREM OF CALCULUS
shows that the two notions are intimately connected.
See also INTEGRAL CALCULUS.
antilogarithm See LOGARITHM.
antipodal points (antipodes) Two points on a
SPHERE at the opposite ends of a diameter are said to be
antipodal. For example, the north and south poles are
antipodal points on the EARTH, as are any two points
EQUIDISTANT from the equator, with longitudes differing by 180°.
The famous Borsuk-Ulam theorem, first conjectured by Stanislaw Ulam and then proved by Karol
Borsuk in 1933, states:
Let f be a continuous function that assigns two
numerical values to each and every point on
the surface of a sphere. Then there must exist
two antipodal points which are assigned precisely the same pair of values.
An amusing interpretation reads:
At any instant there exist two antipodal points
on the Earth’s surface of precisely the same air
temperature and air pressure.
Although the proof of this theorem is difficult, a onedimensional version of the result follows as an easy
consequence of the INTERMEDIATE-VALUE THEOREM.
apex (plural, apices) The point at the top of a POLYGON or a POLYHEDRON, such as the vertex of a triangle
opposite its BASE or the vertex of a pyramid, is called
the apex of the figure. The distance from the base of
the figure to its apex is called the height of the figure.
Apollonius of Perga (ca. 262–190 B.C.E.) Greek Geometry Born in Perga, Greek Ionia, now Antalya,
Turkey, Apollonius worked during the Golden Age of
Greek mathematics and has been referred to throughout
history as the Great Geometer. His famous work, The
Conics, written in eight volumes, greatly influenced the
development of mathematics. (The names ELLIPSE,
PARABOLA, and HYPERBOLA for the three CONIC SECTIONS, for instance, are said to have been coined by
Apollonius.) Copies of the first four volumes of this
work, written in the original Greek, survive today. Arabic translations of the first seven volumes also exist.
Little is known of Apollonius’s life other than what
can be gleaned from incidental comments made in the
prefaces of his books. As a young man it is known that
he traveled to Alexandria to study with the followers of
EUCLID, who then introduced him to the topic of conics.
d
dx
x
x
( )
2
2
=
Apollonius of Perga 15