# Apollonius of Perga (c. 262–c. 190 BC)

Figure 1. Apollonius of Perga.

Figure 2. Apollonius circles.

Apollonius of Perga was a highly influential Greek mathematician and astronomer, born
in a region of what is now Turkey, who became known as the "Great Geometer."
In his famous eight-part work *On Conics*, he introduced such terms
as "ellipse," "parabola," and "hyperbola" – the conic
sections that, as we now know, describe the shapes of various types
of orbit. Euclid and others had written earlier about the basic properties of conic sections
but Apollonius added many new results, particularly to do with normals and
tangents to the various conic curves. In particular, he showed that the
conic curves can be obtained by taking plane sections at different angles
through a cone.

Apollonius also helped found Greek mathematical astronomy. Ptolemy says in his *Syntaxis* that Apollonius introduced the theory of epicycles to explain the apparent motion of the planets across the sky. Although this
isn't strictly true, since the theory of epicycles was mooted earlier, Apollonius
did make important contributions, including a study of the points where
a planet appears stationary. He also developed the *hemicyclium*, a
sundial with hour lines drawn on the surface of a conic section to give
greater accuracy. See also Greek astronomy.

One of the most famous questions he raised is known as the Apollonius
problem (see below). He also wrote widely on other subjects including science, medicine,
and philosophy. In *On the Burning Mirror* he showed that parallel
rays of light are not brought to a focus by a spherical mirror (as had been previously thought) and he discussed the focal properties of
a parabolic mirror. A few decades after his death, the Emperor Hadrian collected
his works and ensured their publication throughout his realm.

## Apollonius problem

The Apollonius problem is: given three objects in the plane,
each of which may be a circle *C*,
a point *P* (a degenerate circle),
or a line *L* (part of a circle with
infinite radius), find another circle that is tangent to (just touches) each of the three. This problem was first recorded in *Tangencies*, written around 200 BC by Apollonius
and so is named after him.

There are ten cases: *PPP*, *PPL*, *PLL*, *LLL*, *PPC*, *PLC*, *LLC*, *LCC*, *PCC*, *CCC* (Fig 2).
The two easiest involve three points or three straight lines and were first
solved by Euclid. Solutions to the eight
other cases, with the exception of the three-circle problem, appeared in *Tangencies*; however, this work was lost. The most difficult case,
to find a tangent circle to any three other circles, was first solved by
the French mathematician François Viète (1540–1603)
and involves the simultaneous solution of three quadratic equations, although, in principle, a solution could be found using just
a compass and a straightedge.

Any of the eight circles that is a solution to the general three-circle
problem is called an **Apollonius circle**. If the three circles
are mutually tangent then the eight solutions collapse to just two, which
are known as Soddy circles. If, having
started with three mutually tangent circles and having created a fourth
– the inner Soddy circle – that is nested between the original
three, the process is repeated to yield three more circles nested between
sets of three of these, and then repeated again indefinitely, fractal is produced. The points that are never inside a circle form a fractal set
called the **Apollonian gasket**, which has a fractional dimension
of about 1.30568.