8 Why was it important to Menaechmus solution of the problem

8. Why was it important to Menaechmus’ solution of the problem of two mean proportionals that the plane cutting the cone be at right angles to one of its generators?

Solution

Menaechmus is said to have made his discovery of conic sections while attempting to solve the problem of doubling the cube. Menaechmus\'s solution to finding two mean proportionals led to discovery of conic sections.

Suppose that we are given a, b and we want to find two mean proportionals x, y between them, i.e. a : x = x : y = y : b. Now

a/x = y/b so xy = ab,

x/y = y/b so y² = bx, and

a/x = x/y so x² = ay.

Menaechmus gave two solutions. The first comes from the rectangular hyperbola and the parabola . The values of x and y are found from the intersection of the parabola y² = bx and the rectangular hyperbola xy = ab.

Thus, Manaechmus uninentionally discoved Conic sections.

But now was the problem of practical visualization of the curves discovered.

This could have been done only by cutting cross sections of regular bodies at various angles and observe the profile of the cross sections.

what kind of cut could have given a profile which we now know as parabola or a hyperbola and which was discovered by Manaechmus ?

If a cone was cut at right angle to its axis then a circular profile was obtained.

But how will this circular profile appear if we project it in a plane which is at half of the cone angle to the horizontal plane?

This plane is nothing but a plane perpendicular to the axis of the generator of the cone.

In this plane the projection of the circular cross section would appear like a parabola or a hyperbola .

These sections of the cone obtained in a plane perpendicular to the generator defined the curves discovered by Menaechmus .

However, having said so, curves obtained in other planes inclined at different angles will also give conic sections, but that will not be a true representation of the conic section of the cone considered.

Only the plane perpendicular to the generator represents the angle of the cone and hence is unique .

This plane defies the conics of the cone taken.