These had already been proved in earlier treatises, which Archimedes refers to as the “Elements of conics”.
Each of the other theorems about conics may be stated for cones of the second order.
Two conics which have four common tangents have always one and only one common polar-triangle.
Astronomy was also enriched by his investigations, and he was led to several remarkable theorems on conics which bear his name.
These conics meet at S1, and at some other point T where the line of intersection of α1 and β1 cuts the surface.
To do this we draw two planes α1 and β1 through S1, cutting the surface Φ in two conics which we also denote by α1 and β1.
Similarly, all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic.
The two surfaces Φ and Φ′ have therefore the points S and S1 and the conics α1 and β1 in common.
It will cut the two conics first at T, and therefore each at some other point which we call A and B respectively.
The corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point.