Basic Lemmata on Parabolas or Orthotomes
by Henry Mendell (Cal. State U., L.A.)

In the theory of conic sections before Apollonius, i.e. in Euclid (lost), Aristeas (lost), and Archimedes, a conic section is cut at right angles to the side of the cone.  The section is then determined by the type of cone, whose vertex is either right (parabola), acute (ellipse), or obtuse (hyperbola).  The terms, 'parabola', 'ellipse', and 'hyperbola' would appear to belong to the later theory.  Following Dijsterhuis, I shall call the section of the right angled cone the orthotome.

The following six theorems constitute the basis of the theory as presented in Archimedes, Quadrature of the Parabola.  Theorem 0 does not occur separately in Archimedes' list of lemmata, but is required for the other theorems.

Theorem  0:  This is actually three propositions:

1. the construction of the orthotome;
2. The ordinates with right orientation to the same point on the diameter are equal.
3. If there is an orthotome HFG with diameter FK of a cone with vertex at A, where AF is the line from the vertex of the cone to the vertex of the orthotome, then for any ordinate SN parallel to the tangent at F from the section to the diameter and abscissa FS from the vertext F to the ordinate:  the square on SN is twice the rectangle FS * AF.

1. If there is an orthotome, ABG, and BD is parallel to its diameter or itself a diameter, and AG is parallel to the tangent to the orthotome at B, then AD will be equal to DG. And if AD is equal to DG, then AG and the tangent to the orthotome at B will be parallels. Case where BD is the diameter Case where BD is parallel to the diameter

2. If there is an orthotome ABG, and BD is parallel to the diameter or is itself a diameter, and ADG is parallel to the tangent of the orthotome at B, and EG is tangent to the orthotome at G, then BD and BE will be equal. Case where BD is the diameter Case where BD is parallel to the diameter.

3. If there is an orthotome ABG, and BD is parallel to the diameter or is itself a diameter, and certain lines parallel to the tangent to the orthotome at B are led out, then as BD is to BZ so is AD in power to EZ. Case where BD is the diameter Case where BD is parallel to the diameter

And these have been proved in the Conic Elements.