The first volume of The Conics simply reviews elementary material about the topic and chiefly presents
results already known to Euclid. Volumes two and
three present original results regarding the ASYMPTOTEs
to hyperbolas and the construction of TANGENT lines to
conics. While Euclid demonstrated a means, for
instance, of constructing a circle passing through any
three given points, Apollonius demonstrated techniques
for constructing circles tangent to any three lines, or to
any three circles, or to any three objects be they a combination of points, lines, or circles. Volumes four, five,
six, and seven of his famous work are highly innovative
and contain original results exploring issues of curvature, the construction of normal lines, and the construction of companion curves to conics. Apollonius
also applied the theory of conics to solve practical
problems. He invented, for instance, a highly accurate
sundial, called a hemicyclium, with hour lines drawn
on the surface of a conic section.
Apollonius also played a fundamental role in the
development of Greek mathematical astronomy. He
proposed a complete mathematical analysis of epicyclic
motion (that is, the compound motion of circles rolling
along circles) as a means to help explain the observed
retrograde motion of the planets across the skies that
had confused scholars of his time.
Apollonius’s work was extraordinarily influential, and his text on the conics was deemed a standard reference piece for European scholars of the
Renaissance. JOHANNES KEPLER, RENÉ DESCARTES,
and SIR ISSAC NEWTON each made reference to The
Conics in their studies.
See also CIRCUMCIRCLE; CYCLOID.
Apollonius’s circle Let A and B be two points of the
plane and let k be a constant. Then the set of all points
P whose distance from A is k times its distance from B
is a CIRCLE. Any circle obtained this way is referred to
as one of Apollonius’s circles. Note that when k = 1 the
circle is “degenerate,” that is, the set of all points
EQUIDISTANT from A and B is a straight line. When k
becomes large, the Apollonius’s circle approaches a circle of radius 1.
To see that the locus of points described this way
is indeed a circle, set A to be the origin (0,0), B to be
the point (k + 1, 0) on the x-axis, and P to be a general point with coordinates (x,y). The DISTANCE FORMULA
then gives an equation of the form
. This is equivalent to
, which is indeed the equation of a circle, one of radius
. APOLLONIUS OF
PERGA used purely geometric techniques, however, to
establish his claim.
Apollonius’s theorem If a, b, and c are the sidelengths of a triangle and a median of length m divides
the third side into two equal lengths c/2 and c/2, then
the following relation holds:
This result is known as Apollonius’s theorem. It can be
proved using two applications of the LAW OF COSINES
as follows:
Let B be the ANGLE between the sides of length
a and c. Then m
2 = a
2 + (c/2)
2
–ac cos(B) and b
2
= a
2 + c
2 – 2ac cos(B). Solving for ac cos(B) in
the first equation and substituting into the second yields the result.
See also MEDIAN OF A TRIANGLE.
apothem (short radius) Any line segment from the
center of a regular POLYGON to the midpoint of any of
its sides is called an apothem. If the regular polygon
has n sides, each one unit in length, then an exercise in
TRIGONOMETRY shows that each apothem of the figure
has length
.
An analog of PI (π) for a regular polygon is the
RATIO of its PERIMETER to twice the length of its
apothem. For a regular n-sided polygon, this ratio has
value n tan(180/n). The SQUEEZE RULE shows that this
quantity approaches the value π as n becomes large.
See also LONG RADIUS.
r
n
=
1
2
180
tan
a b
c
m
2
2
2
2
2
2
+
=
+
k
k − 1
x
k
k
y
k
k
− −
+
=
−
2
2
2
2
1
1
x y k x k
y
2
2
2
2
1
+ =
− − +
(
)
16 Apollonius’s circle
results already known to Euclid. Volumes two and
three present original results regarding the ASYMPTOTEs
to hyperbolas and the construction of TANGENT lines to
conics. While Euclid demonstrated a means, for
instance, of constructing a circle passing through any
three given points, Apollonius demonstrated techniques
for constructing circles tangent to any three lines, or to
any three circles, or to any three objects be they a combination of points, lines, or circles. Volumes four, five,
six, and seven of his famous work are highly innovative
and contain original results exploring issues of curvature, the construction of normal lines, and the construction of companion curves to conics. Apollonius
also applied the theory of conics to solve practical
problems. He invented, for instance, a highly accurate
sundial, called a hemicyclium, with hour lines drawn
on the surface of a conic section.
Apollonius also played a fundamental role in the
development of Greek mathematical astronomy. He
proposed a complete mathematical analysis of epicyclic
motion (that is, the compound motion of circles rolling
along circles) as a means to help explain the observed
retrograde motion of the planets across the skies that
had confused scholars of his time.
Apollonius’s work was extraordinarily influential, and his text on the conics was deemed a standard reference piece for European scholars of the
Renaissance. JOHANNES KEPLER, RENÉ DESCARTES,
and SIR ISSAC NEWTON each made reference to The
Conics in their studies.
See also CIRCUMCIRCLE; CYCLOID.
Apollonius’s circle Let A and B be two points of the
plane and let k be a constant. Then the set of all points
P whose distance from A is k times its distance from B
is a CIRCLE. Any circle obtained this way is referred to
as one of Apollonius’s circles. Note that when k = 1 the
circle is “degenerate,” that is, the set of all points
EQUIDISTANT from A and B is a straight line. When k
becomes large, the Apollonius’s circle approaches a circle of radius 1.
To see that the locus of points described this way
is indeed a circle, set A to be the origin (0,0), B to be
the point (k + 1, 0) on the x-axis, and P to be a general point with coordinates (x,y). The DISTANCE FORMULA
then gives an equation of the form
. This is equivalent to
, which is indeed the equation of a circle, one of radius
. APOLLONIUS OF
PERGA used purely geometric techniques, however, to
establish his claim.
Apollonius’s theorem If a, b, and c are the sidelengths of a triangle and a median of length m divides
the third side into two equal lengths c/2 and c/2, then
the following relation holds:
This result is known as Apollonius’s theorem. It can be
proved using two applications of the LAW OF COSINES
as follows:
Let B be the ANGLE between the sides of length
a and c. Then m
2 = a
2 + (c/2)
2
–ac cos(B) and b
2
= a
2 + c
2 – 2ac cos(B). Solving for ac cos(B) in
the first equation and substituting into the second yields the result.
See also MEDIAN OF A TRIANGLE.
apothem (short radius) Any line segment from the
center of a regular POLYGON to the midpoint of any of
its sides is called an apothem. If the regular polygon
has n sides, each one unit in length, then an exercise in
TRIGONOMETRY shows that each apothem of the figure
has length
.
An analog of PI (π) for a regular polygon is the
RATIO of its PERIMETER to twice the length of its
apothem. For a regular n-sided polygon, this ratio has
value n tan(180/n). The SQUEEZE RULE shows that this
quantity approaches the value π as n becomes large.
See also LONG RADIUS.
r
n
=
1
2
180
tan
a b
c
m
2
2
2
2
2
2
+
=
+
k
k − 1
x
k
k
y
k
k
− −
+
=
−
2
2
2
2
1
1
x y k x k
y
2
2
2
2
1
+ =
− − +
(
)
16 Apollonius’s circle