Can You Really Derive Conic Formulae from a Cone? - Conics as Orthogonal Sections of Cones - Introduction

According to Pappus in his Collection from the fourth century, "Apollonius, who completed the four books of Euclid's Conics and added another four, gave us eight books of Conics. Aristaeus, who wrote the still extant five books of Solid Loci supplementary to the Conics, called the three conics [the] sections of an acute-angled, right-angled and obtuse-angled cone respectively." Pappus goes on to say that "Euclid regarded Aristaeus as deserving credit for his contributions to conics, and did not try to anticipate him or to overthrow his system .... Accordingly he wrote so much about the locus as was possible by means of the Conics of Aristaeus, but did not claim finality for his proofs." [14, p. 487]

Unfortunately, we know little about Aristaeus, and while his Solid Loci was extant in Pappus' time, it is lost now. Euclid's Conics is lost as well, but from Pappus' description the work was likely a book containing the "state of the art" of conics at the time, much like Euclid's Elements represented the culmination of the basics of geometry at the time. Aristaeus' time is placed in the late fourth century B. C., after Menaechmus, and before Euclid (ca. 300 B.C.). Fortunately, Archimedes (287 - 212 B.C.) writes a great deal about properties of conic sections, and he refers in his works to "the elements of conics," which we may infer as the works of either Aristaeus or Euclid, or both. In this way Archimedes provides some evidence as to what was in these works.

Recall that on Pappus' evidence we know that Aristaeus and Euclid considered sections of acute-angled, right-angled and obtuse-angled cones. We do in fact have Euclid's definition of a cone, in Elements Book XI, Definitions 18-20:

Def. 18. When a right triangle with one side of those about the right angle remains fixed is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a cone. And, if the straight line which remains fixed equals the remaining side about the right angle which is carried round, the cone will be right-angled; if less, obtuse-angled; and if greater, acute-angled.

Def. 19. The axis of the cone is the straight line which remains fixed and about which the triangle is turned.

Def. 20. And the base is the circle described by the straight line which is carried round [10].

At this time conics were intersections of a plane at right angles to the side that when rotated generates the cone. Let's see how this generates the three conic sections:

Above: The section of a right-angled cone (also known as an orthotome, now known as a parabola)
Right click and drag to change the viewing angle.

Above: The section of an acute-angled cone (also known as an oxytome, now known as an ellipse)
Right click and drag to change the viewing angle.

Above: The section of an obtuse-angled cone (also known as an amblytome, now known as an hyperbola)
Right click and drag to change the viewing angle.

While we know how the conics were generated, what we do not have is evidence of how the symptom of the conic is derived. The symptom is the relationship between the abscissa and ordinate of a point on a conic that determines which type of conic is present. (Today we would express this as something like y = x²). Using the geometry of the day, however, we can see how the symptom of a conic might have been derived. We will consider the section of a right-angled cone later, so first consider the oxytome, the section of an acute-angled cone.