Part 1 © James Taylor 2000 Dove Productions

A Little History The discovery of the conics is attributed to Menaechmus (350 - 330 BC). A pupil of both the philosopher Plato and the scientist Eudoxus of Cnidus, Menaechmus was led to the study of conics by his efforts to solve the problem of constructing a cube twice as large by volume as a given cube, the so-called Delian problem. It is impossible to say whether he regarded the curve as the locus of a point in a plane or as a section of a cone.

But there seems no doubt that Aristaeus, who lived shortly after Menaechmus, based his investigations on the cone. The plane of section was drawn at right angles to the generator of the cone, and different species of conics were obtained by altering the vertical angle of the cone: the acute angled cone giving the ellipse, the right -angled cone giving a parabola and the obtuse-angled cone giving a hyperbola.

The Method of Aristaeus Acute angle Right angle Obtuse angle Ellipse Parabola Hyperbola

A cone can be generated by fixing a central point on a straight line segment and moving the end of the line segment around a circle. Each half of the cone thus generated is called a nappe. Euclid wrote four books on the conic sections, but his work is completely lost. Archimedes succeeded in finding the area of the ellipse and of a sector of the parabola by means of a method closely akin to that of the integral calculus.

The pinnacle of Greek geometry, and perhaps of Greek mathematics in general, was reached by Apollonius of Perga (247-205 BC) in his eight books on conic sections., only the first seven of which have survived, but they contain the elementary theory of conics in a complete form. Apollonius was the first to show that all conics are sections of any circular cone, right or oblique.

In studying these curves Apollonius treated them as plane curves, however, without regard to their spatial origin. The terms ellipse, parabola and hyperbola were introduced by Apollonius. The terms refer to whether the slice is less than (ellipsis), equal to (para), or greater than (hyper) the angle of the cone.

The Method of Dandelin Spheres The focus-directrix property is mentioned by Pappus (300 AD) but little use was made of it until Newton (1642-1727) called attention to it in the Principia; the theory of the real foci was worked out by Keplar (1571-1630), to whom the term focus is due. It is not at all obvious that the figures obtained by the method of slicing the cone correspond to what we now manipulate algebraically and call by the same names: they certainly have the same general shape, but so do many curves.

The purpose of this presentation is to show that this correspondence is correct. The method which we are about to encounter (which uses focal spheres, or Dandelin spheres) is a recent discovery and is due to Pierre Dandelin (1794-1847), a professor of mechanics at Liège University, and Morton (1825). To begin, we need to agree on definitions for the figures. The method uses locus definitions, where P is a point on the locus and S, S' are given fixed points. In the case of the parabola, the straight line is also given.

we shall take our definition to be that P moves so that The Ellipse Given two fixed points we shall take our definition to be that P moves so that PS’ + PS = constant

Slice the cone with a plane that will not intersect the upper nappe, and which is not parallel to a generator or the base.

The spheres are the largest that can touch the wall of the cone, and the slanting plane. The points of contact with the plane are S and S’.

The green planes are the planes through the circles of contact. P is a point on the curve.

Tangents are drawn from P to the lower sphere. Tangent along the generator Tangent to S’

Similarly, tangents are drawn to the upper sphere.

But the distance AB along a generator is constant.

Hence PB + PA is constant. Hence PS’ + PS is constant. Hence the curve is an ellipse.

Note: the lines of intersection of the planes are the directrices of the ellipse.

we shall take our definition to be that P moves so that The Hyperbola Given two fixed points we shall take our definition to be that P moves so that PS’ - PS = constant

Cut the cone with a plane that slices through both nappes.

The spheres are the largest that can touch the wall of the cone, and the slanting plane. The points of contact with the plane are S and S’.

P is a point on the curve. Tangents are drawn from P to the lower sphere.

The tangent PB passes through the vertex of the cone.

The tangents PB and PS’ are equal.

Similarly the small tangents PS and PA are equal.

Hence : PS’ - PS = PB - PA

= AB.

But AB is constant. Hence PS’ - PS is constant, and the curve is a hyperbola.

Note: the lines of intersection of the planes are the directrices of the hyperbola.

The Parabola Given a fixed point S

PS = PM we shall take our definition to be that P moves such that and fixed line w PS = PM where PM is perpendicular to w

Cut the cone with a plane parallel to one of the generators.

The sphere is the largest that can touch the wall of the cone, and the slanting plane. The point of contact with the plane is S .

P is a point on the curve.

The generator is drawn through P.

Tangents are drawn from P to the sphere. Tangent along the generator Tangent to S

The line PM is drawn perpendicular to the line of intersection of the planes.

The line MA is drawn to meet the opposite point of contact of the sphere

ZA = ZD

and ZD is parallel to PM

By similar triangles, PM = PA

But PA = PS

Hence PM = PS, and the curve is a parabola.

Again, the line of intersection of the planes is the directrix of the curve.

This completes the presentation This completes the presentation. If you enjoyed the presentation or would like to make a comment, send me a note: taylj@ozemail.com.au Peace in every step