Problem of Apollonius - History

History

A rich repertoire of geometrical and algebraic methods have been developed to solve Apollonius' problem, which has been called "the most famous of all" geometry problems. The original approach of Apollonius of Perga has been lost, but reconstructions have been offered by François Viète and others, based on the clues in the description by Pappus. The first new solution method was published in 1596 by Adriaan van Roomen, who identified the centers of the solution circles as the intersection points of two hyperbolas. Van Roomen's method was refined in 1687 by Isaac Newton in his Principia, and by John Casey in 1881.

Although successful in solving Apollonius' problem, van Roomen's method has a drawback. A prized property in classical Euclidean geometry is the ability to solve problems using only a compass and a straightedge. Many constructions are impossible using only these tools, such as dividing an angle in three equal parts. However, many such "impossible" problems can be solved by intersecting curves such as hyperbolas, ellipses and parabolas (conic sections). For example, doubling the cube (the problem of constructing a cube of twice the volume of a given cube) cannot be done using only a straightedge and compass, but Menaechmus showed that the problem can be solved by using the intersections of two parabolas. Therefore, van Roomen's solution—which uses the intersection of two hyperbolas—did not determine if the problem satisfied the straightedge-and-compass property.

Van Roomen's friend François Viète, who had urged van Roomen to work on Apollonius' problem in the first place, developed a method that used only compass and straightedge. Prior to Viète's solution, Regiomontanus doubted whether Apollonius' problem could be solved by straightedge and compass. Viète first solved some simple special cases of Apollonius' problem, such as finding a circle that passes through three given points which has only one solution if the points are distinct; he then built up to solving more complicated special cases, in some cases by shrinking or swelling the given circles. According to the 4th-century report of Pappus of Alexandria, Apollonius' own book on this problem—entitled Ἐπαφαί (Epaphaí, "Tangencies"; Latin: De tactionibus, De contactibus)—followed a similar progressive approach. Hence, Viète's solution is considered to be a plausible reconstruction of Apollonius' solution, although other reconstructions have been published independently by three different authors.

Several other geometrical solutions to Apollonius' problem were developed in the 19th century. The most notable solutions are those of Jean-Victor Poncelet (1811) and of Joseph Diaz Gergonne (1814). Whereas Poncelet's proof relies on homothetic centers of circles and the power of a point theorem, Gergonne's method exploits the conjugate relation between lines and their poles in a circle. Methods using circle inversion were pioneered by Julius Petersen in 1879; one example is the annular solution method of HSM Coxeter. Another approach uses Lie sphere geometry, which was developed by Sophus Lie.

Algebraic solutions to Apollonius' problem were pioneered in the 17th century by René Descartes and Princess Elisabeth of Bohemia, although their solutions were rather complex. Practical algebraic methods were developed in the late 18th and 19th centuries by several mathematicians, including Leonhard Euler, Nicolas Fuss, Carl Friedrich Gauss, Lazare Carnot, and Augustin Louis Cauchy.