Book VII

(mainly on conjugate diameters). Normals are treated, not in connexion with tangents, but as _minimum_ or _maximum_ straight lines drawn to the curves from different points or classes of points. Apollonius discusses such questions as the number of normals that can be drawn from one point (according to its position) and the construction of all such normals. Certain propositions of great difficulty enable us to deduce quite easily the Cartesian equations to the _evolutes_ of the three conics.

Several other works of Apollonius are described by Pappus as forming part of the 'Treasury of Analysis'. All are lost except the _Sectio Rationis_ in two Books, which survives in Arabic and was published in a Latin translation by Halley in 1706. It deals with all possible cases of the general problem 'given two straight lines either parallel or intersecting, and a fixed point on each, to draw through any given point a straight line which shall cut off intercepts from the two lines (measured from the fixed points) bearing a given ratio to one another'. The lost treatise _Sectio Spatii_ dealt similarly with the like problem in which the intercepts cut off have to contain a given rectangle.

The other treatises included in Pappus's account are (1) On _Determinate Section_; (2) _Contacts_ or _Tangencies_,