on, to
describe a circle passing through the given points, and touching the
given straight lines or circles. The most difficult case, and the most
interesting from its historical associations, is when the three given
things are circles. This problem, which is sometimes known as the
Apollonian Problem, was proposed by Vieta in the 16th century to
Adrianus Romanus, who gave a solution by means of a hyperbola. Vieta
thereupon proposed a simpler construction, and restored the whole
treatise of Apollonius in a small work, which he entitled _Apollonius
Gallus_ (Paris, 1600). A very full and interesting historical account of
the problem is given in the preface to a small work of J.W. Camerer,
entitled _Apollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in
hos Libras, cum Observationibus, &c_. (Gothae, 1795, 8vo).

5th. _De Inclinationibus_ had for its object to insert a straight line
of a given length, tending towards a given point, between two given
(straight or circular) lines. Restorations have been given by Marino
Ghetaldi, by Hugo d'Omerique (_Geometrical Analysis_, Cadiz, 1698), and
(the best) by Samuel Horsley (1770).

6th. _De Locis Planis_ is a collection of propositions relating to loci
which are either straight lines or circles. Pappus gives somewhat full
particulars of the propositions, and restorations were attempted by P.
Fermat (_Oeuvres_, i., 1891, pp. 3-51), F. Schooten (Leiden, 1656) and,
most successfully of all, by R. Simson (Glasgow, 1749).

Other works of Apollonius are referred to by ancient writers, viz. (1)
[Greek: Peri tou pyriou], _On the Burning-Glass_, where the focal
properties of the parabola probably found a place; (2) [Greek: Peri tou
kochliou], _On the Cylindrical Helix_ (mentioned by Proclus); (3) a
comparison of the dodecahedron and the icosahedron inscribed in the same
sphere; (4) [Greek: Hae katholou pragmateia], perhaps a work on the
general principles of mathematics in which were included Apollonius'
criticisms and suggestions for the improvement of Euclid's _Elements_;
(5) [Greek: Okutokion] (quick bringing-to-birth), in which, according to
Eutocius, he showed how to find closer limits for the value of [pi] than
the 3-1/7 and 3-10/71 of Archimedes; (6) an arithmetical work (as to
which see PAPPUS) on a system of expressing large numbers in language
closer to that of common life than that of Archimedes' _Sand-reckoner_,
and showing how to multiply such large numbers; (7) a great