THE ALEXANDRIAN PERIOD yy 



discoveries to those of his predecessors. He shows 

 how, in the first four volumes of his work, he has 

 generalized and extended the elements of the theory 

 already known. The third book enunciates new pro- 

 positions which make it possible to solve a problem 

 imperfectly treated by Euclid ; the fourth rectifies the 

 results of Conon relating to points of contact and inter- 

 section of conies. The rest of the work contains 

 further developments of the properties of conies and 

 their applications. 1 In fact, what is really new in 

 the work of Apollonius is his definition of conic 

 sections. Archimedes and Euclid defined these as 

 the sections taken perpendicularly to the sides of right 

 cones, i.e. cones whose axis is perpendicular to the 

 circle of the base, but of which the angle at the 

 apex may be a right, obtuse, or acute angle (Fig. 7). 

 Apollonius shows that the parabola, hyperbola and 

 ellipse can be obtained by sections taken on one and the 

 same oblique cone having a circular base. If through 

 the axis of this cone we take a plane perpendicular to 

 the circle of the base, we obtain the triangle formed by 

 the two sides of the cone and the diameter of the base. 

 If we now draw a plane perpendicular to the plane of 

 this triangle, the sides of this triangle will be cut at 

 two points, which will be the vertices of the curve. 

 A similar geometrical construction will enable us to 

 find a ratio indicating whether this curve or conic 

 section is an ellipse, hyperbola or parabola. The 

 geometrical constructions of lines and surfaces thus 

 play the same part as algebraical equations in analytical 

 geometry. But Apollonius not only expounds general 

 theories, he applies them to numerous and difficult 

 problems, carefully studying their conditions of possi- 

 bility. The collections of these problems were for a 

 long time in use in the school of Alexandria ; after- 

 wards they were lost, with the exception of those 

 1 15 Heiberg, Naturwiss., p. 56. 



