Book v

., where he treats of normals as minimum and maximum straight lines drawn from given points to 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 position, 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 extension of the theory of irrationals expounded in Euclid,