Book X

; (9) _On the burning-mirror_, dealing with spherical mirrors and probably with mirrors of parabolic section also; (10) ωκυτοκιον {ôkytokion} ('quick delivery'). In the last-named work Apollonius found an approximation to π {p} closer than that in Archimedes's _Measurement of a Circle_; and possibly the book also contained Apollonius's exposition of his notation for large numbers according to 'tetrads' (successive powers of the myriad).

In astronomy Apollonius is said to have made special researches regarding the moon, and to have been called ε {e} (Epsilon) because the form of that letter is associated with the moon. He was also a master of the theory of epicycles and eccentrics.

With Archimedes and Apollonius Greek geometry reached its culminating point; indeed, without some more elastic notation and machinery such as algebra provides, geometry was practically at the end of its resources. For some time, however, there were capable geometers who kept up the tradition, filling in details, devising alternative solutions of problems, or discovering new curves for use or investigation.

Nicomedes, probably intermediate in date between Eratosthenes and Apollonius, was the inventor of the _conchoid_ or _cochloid_, of which, according to Pappus, there were three varieties. Diocles (about the end of the second century B. C.) is known as the discoverer of the _cissoid_ which was used for duplicating the cube. He also wrote a book περι πυρειων {peri pyreiôn}, _On burning-mirrors_, which probably discussed, among other forms of mirror, surfaces of parabolic or elliptic section, and used the focal properties of the two conics; it was in this work that Diocles gave an independent and clever solution (by means of an ellipse and a rectangular hyperbola) of Archimedes's problem of cutting a sphere into two segments in a given ratio. Dionysodorus gave a solution by means of conics of the auxiliary cubic equation to which Archimedes reduced this problem; he also found the solid content of a _tore_ or anchor-ring.

Perseus is known as the discoverer and investigator of the _spiric sections_, i. e. certain sections of the σπειρα {speira}, one variety of which is the _tore_. The _spire_ is generated by the revolution of a circle about a straight line in its plane, which straight line may either be external to the circle (in which case the figure produced is the tore), or may cut or touch the circle.

Zenodorus was the author of a treatise on _Isometric figures_, the problem in which was to compare the content of different figures, plane or solid, having equal contours or surfaces respectively.

Hypsicles (second half of second century B. C.) wrote what became known as '