the curve
(independently of tangent properties), discusses how many normals can be
drawn from particular points, finds their feet by construction, and
gives propositions determining the centre of curvature at any point and
leading at once to the Cartesian equation of the evolute of any conic.

The other treatises of Apollonius mentioned by Pappus are --1st, [Greek:
Logou apotomae], _Cutting off a Ratio_; 2nd, [Greek: Choriou apotomae],
_Cutting of an Area_; 3rd, [Greek: Dioris menae tomae], _Determinate
Section_; 4th, [Greek: Epaphai], _Tangencies_; 5th, [Greek: Neuseis],
_Inclinations_; 6th, [Greek: Topoi epipedoi], _Plane Loci_. Each of
these was divided into two books, and, with the _Data_, the _Porisms_
and _Surface-Loci_ of Euclid and the _Conics_ of Apollonius were,
according to Pappus, included in the body of the ancient analysis.

1st. _De Rationis Sectione_ had for its subject the resolution of the
following problem: Given two straight lines and a point in each, to draw
through a third given point a straight line cutting the two fixed lines,
so that the parts intercepted between the given points in them and the
points of intersection with this third line may have a given ratio.

2nd. _De Spatii Sectione_ discussed the similar problem which requires
the rectangle contained by the two intercepts to be equal to a given
rectangle.

An Arabic version of the first was found towards the end of the 17th
century in the Bodleian library by Dr Edward Bernard, who began a
translation of it; Halley finished it and published it along with a
restoration of the second treatise in 1706.

3rd. _De Sectione Determinata_ resolved the problem: Given two, three or
four points on a straight line, to find another point on it such that
its distances from the given points satisfy the condition that the
square on one or the rectangle contained by two has to the square on the
remaining one or the rectangle contained by the remaining two, or to the
rectangle contained by the remaining one and another given straight
line, a given ratio. Several restorations of the solution have been
attempted, one by W. Snellius (Leiden, 1698), another by Alex. Anderson
of Aberdeen, in the supplement to his _Apollonius Redivivus_ (Paris,
1612), but by far the best is by Robert Simson, _Opera quaedam reliqua_
(Glasgow, 1776).

4th. _De Tactionibus_ embraced the following general problem: Given
three things (points, straight lines or circles) in positi