If a cone is cut by a plane through its axis, and also by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, and if the diameter of the section produced meets one side of the axial triangle beyond the vertex of the cone, and if any straight line is drawn from the section to its diameter such that this straight line is parallel to the common section of the cutting plane and of the cone's base, then this straight line to the diameter will equal in square some area which is applied to a straight line [the parameter] (to which the straight line which is added along the diameter of the section - such that this added straight line subtends the exterior angle of the [vertex of the axial] triangle - has the same ratio as the square on the straight line drawn - parallel to the section's diameter - from the cone's vertex to the triangle's base has to the rectangle contained by the sections of the base which this straight line from the vertex makes when drawn), such that that applied area (which has as breadth the straight line on the diameter from the section's vertex to where the diameter is cut off by the straight line drawn from the section to the diameter) projects beyond (hyperballon) by a figure (eidos), similar and similarly situated to the rectangle contained by the straight line subtending the exterior angle of the [vertex of the axial] triangle and by the parameter. And let such a section be called an hyperbola.🏁

If a cone is cut by a plane through its axis, and also by another plane cutting the base of the cone in a straight line perpendicular to the base of the axial triangle, and if the diameter of the section produced meets one side of the axial triangle beyond the vertex of the cone, and if any straight line is drawn from the section to its diameter such that this straight line is parallel to the common section of the cutting plane and of the cone's base, then this straight line to the diameter will equal in square some area which is applied to a straight line [the parameter] (to which the straight line which is added along the diameter of the section - such that this added straight line subtends the exterior angle of the [vertex of the axial] triangle - has the same ratio as the square on the straight line drawn - parallel to the section's diameter - from the cone's vertex to the triangle's base has to the rectangle contained by the sections of the base which this straight line from the vertex makes when drawn), such that that applied area (which has as breadth the straight line on the diameter from the section's vertex to where the diameter is cut off by the straight line drawn from the section to the diameter) projects beyond (hyperballon) by a figure (eidos), similar and similarly situated to the rectangle contained by the straight line subtending the exterior angle of the [vertex of the axial] triangle and by the parameter. And let such a section be called an hyperbola.🏁