through which the varying ellipse passes as it merges into

the hyperbola.

These three curves may be defined by a single law of motion of a point in a plane, and for purposes of study this is more convenient.

A point so moves that its distances from a fixed point, F, called the focus, and from a fixed line, DD', called the directrix, are in a given ratio, e, the eccentricity of the curve. Now the form of the curve and the class to which it belongs, ellipse, parabola or hyperbola, depends upon the value given to e. In the figure F is the fixed point, P

is the moving point on the curve and DD is the directrix or fixed line.

In Fig. I e is less than I and the curve is an ellipse. It is seen that it is symmetric to the line YY' and therefore must have another directrix, DD', on the right and also a second focus, F.

In Fig. II e = = 1 and the curve is the parabola. This curve constantly recedes from the line, yet ever curves to it. It may be thought of as the left half of an ellipse of which the right focus has been pulled out to the right an infinite distance; it is an open curve—that is, the two arms of the curve never join again.

In Fig. III is seen the third case, where e is greater than I; the hyperbola with two branches. In the generation of this curve the point starting at A' recedes indefinitely downward to the right. It next appears coming back on the upper half of the left branch, passing along that branch to an infinite distance and finally coming back

along the upper right of the right branch. It is convenient sometimes to think of the two ends of the curve being joined by a single infinite point and thus preserve continuity in the motion of the moving point. The two branches of the hyperbola constantly approach without

ever reaching the two intersecting lines OX' and OY' in the figure; that is, the curves are said to be asymptotic to these lines, which are called the asymptotes of the

curve.

In the full-page figure is seen the relation which exists between the foci and directrices of the plane figure and the cone itself. The plane AB cuts the ellipse from the cone. If a sphere be dropped in the cone so that it will be in the cone and just touch the plane the point of touching or tangency will be a focus. Two such spheres are possible, the small one above the plane and the large one below; the foci are F and F. These spheres touch the cone in circles. If planes be passed through these circles, as AC and BC, they will cut the original cutting plane AB in the lines AM and BN, which are the directrices. The futility of the argument that it is vain to cultivate truth for truth's sake is well seen in the case of the Conics of Apollonius. This monumental work lay dormant and did not reach fruition until seventeen centuries after, when Kepler found the paths of the planets to be ellipses and Newton subjected to law the wanderer of the celestial seas, the comet, whose path is an ellipse if it is a regular visitor of the solar system. If the path of the comiet is not an ellipse, it is a parabola, and it comes but once under the influence of the sun and then forever loses itself in the vastness of space.

Antiquity has left us three famous problems: Thequadrature of the circle, the duplication of the cube, called. the Delian problem, and the trisection of the angle, or more generally the problem of the inscription of the regular polygons in a circle.

The quadrature of the circle, popularly known as squaring the circle, is the problem of finding the side of a square which has the same area as a given circle. The philosopher Anaxagoras occupied himself with this problem in his prison. Hippocrates of Chios made one of the most