copyright © 2003 Tatiana Hamboyan

From Generation of Conic Sections by Blaise Pascal:

"Corollary I: It is therefore clear that if an infinite straight line is drawn from the vertex to an arbitrary point either on the circumference [of the base] or on the conic surface, this straight line is entirely on the conic surface, that is to say, it is a generatrix."

"Definition III: Two or more lines, however situated, are always said to converge, either at a finite distance if they meet in one and the same point, or at an infinite distance if they are parallel."

"Corollary: It is therefore clear that if the eye is at the vertex of a cone, and if what is presented to the view is the circumference of the circle which is the base of the cone, and if the surface on which it is represented is the plane which meets the surface of the cone on either side, then the conic section which is generated by that same plane in the conic surface -- be it a point, a straight line, an angle, an antobola [ellipse], a parabola, or an hyperbola -- will be an image of the circle."

"Corollary: With the same things given, if the plane of projection does not pass through the vertex and is not parallel to any generatrix, that is to say, to any ray, and consequently generates an antobola [ellipse], it is manifest that all the points of the circumference project their images on the section’s plane of projection at a finite distance."

"Scholium: This is the reason that the antobola [ellipse] returns on itself and encloses a space."

"Corollary: With the same things given, if the plane of projection is parallel to one of the generatrices only, that is to say to a single ray, and consequently generates a parabola, it is manifest that all the points of the circumference project their images on the section’s plane of projection at a finite distance, with the exception of one point which has no image, unless it be at an infinite distance."

"Scholium: "This is the reason that the parabola extends to infinity and generates an infinite space, although it is the image of the circumference of the circle which is finite and which encloses a finite space."

Since two lines converge at either infinity or a finite distance, all figures at infinity will be closed ones. For if the figure is closed already, as a triangle is, at infinity it will still be closed; and if the figure is open, as two lines that form an angle are, at infinity they will meet and thus close the figure. Furthermore, if the parabola is said to be an image of a circle with one missing point, the image of that missing point will occur at infinity and the parabola will appear to have the shape of an ellipse. What is meant by ellipse here is not that the parabola at infinity will have all the properties of an ellipse, but that the parabola will become a closed curved figure that resembles the shape of an ellipse more closely than it resembles the shape of its generating circle.

What does it mean to say that the parabola is an image of a circle with a missing point at finite distances and no missing points at infinity? Though Pascal speaks of the parabola and the ellipse as images of the circle, and therefore somehow generated by the circle, it is easier to understand the connection between the parabola and ellipse to the circle if we are shown both instead of trying to generate them from the circle. We will, therefore, create both the ellipse and the parabola using Apollonius’s methods instead of trying to generate the parabola and ellipse from the circle.

In Book I, Proposition 11, Apollonius generates the parabola by first finding an axial triangle. When the axial triangle is found, line DG is drawn parallel to AEC, one of the sides of the axial triangle. Line FGH is then drawn perpendicular to the diameter of the base of the cone. The parabola is created by connecting points D, F, and H on the surface of the cone. In Proposition 13, Apollonius generates the ellipse by drawing line DF that is neither parallel to any side of any axial triangle nor to the circumference of the base through the center of the cone. The ellipse is created by connecting points D and F on the surface of the cone. The circle that these conic sections are images of is created by drawing line DE through the cone parallel to the circumference of the base and connecting points D and E on the surface of the cone.

With both the circle and the conic sections drawn, the connection between the two can more easily be understood. First, we will look at the ellipse since it has no missing points of image from the circle and is, therefore, simpler. Each point on the circumference of the circle DE either can be connected to a point on the ellipse or is already a point on the ellipse, as point D is. Now, Euclid does state that a line can be found between any two points, but these points are being connected by a very particular line. They are being connected by what Pascal calls a "generatrix."

With the parabola, the generatrices that connect the points of the circle to the points on the parabola are not so easily seen. Part of the difficulty stems from the parabola being an open, not closed, figure. We cannot see the entire parabola, but only a section of it. Even with this difficulty, it is pretty clear that for each point of the circle but one, the generatrix that passes through that point will also pass through a point on the parabola. The one point that will not either be on the parabola, as point D is, or have a generatrix that connects it to the parabola at a finite distance will be point E. Recall that line DG was drawn parallel to AEC, and from that the parabola will also be parallel to line AEC. Since, as was already stated, all lines that are parallel will meet at infinity, the lines FDH (the points F and H are not the end points of the line, but merely the points used in the diagram to describe the parabola created by the line) and DG will meet the line AEC at infinity. Before this point, a point on the section of the parabola DH was connected to one point on the circle and an analogous point on the opposite section, DF, connected to another. When point E will be connected to the parabola, it will meet both sections, DF and DH, of the parabola. This means that, at infinity, the parabola will become a closed figure that appears to have the shape of an ellipse -- in the sense that it will be a closed curved figure that is clearly not a circle.