Page:A History of Mathematics (1893).djvu/66

 hyperbola, respectively. To be sure, we find the words 'parabola' and 'ellipse' in the works of Archimedes, but they are probably only interpolations. The word 'ellipse' was applied because $$\scriptstyle{y^2px}$$.

The treatise of Apollonius rests on a unique property of conic sections, which is derived directly from the nature of the cone in which these sections are found. How this property forms the key to the system of the ancients is told in a masterly way by M. Chasles.[13] "Conceive," says he, "an oblique cone on a circular base; the straight line drawn from its summit to the centre of the circle forming its base is called the axis of the cone. The plane passing through the axis, perpendicular to its base, cuts the cone along two lines and determines in the circle a diameter; the triangle having this diameter for its base and the two lines for its sides, is called the triangle through the axis. In the formation of his conic sections, Apollonius supposed the cutting plane to be perpendicular to the plane of the triangle through the axis. The points in which this plane meets the two sides of this triangle are the vertices of the curve; and the straight line which joins these two points is a diameter of it. Apollonius called this diameter latus transversum. At one of the two vertices of the curve erect a perpendicular (latus rectum) to the plane of the triangle through the axis, of a certain length, to be determined as we shall specify later, and from the extremity of this perpendicular draw a straight line to the other vertex of the curve; now, through any point whatever of the diameter of the curve, draw at right angles an ordinate: the square of this ordinate, comprehended between the diameter and the curve, will be equal to the rectangle constructed on the portion of the ordinate comprised between the diameter and the straight