Basic Lemmata on Parabolas or OrthotomesReturn to Vignettes of Ancient Mathematics
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 four propositions about the principal orthotome, where the ordinates are at right angles to the abscissa:
- the construction of the orthotome;
- The ordinates with right orientation to the same point on the diameter are equal.
- The squares on ordinates have the same ratios as their respective abscissae.
- The tangent to a point on an orthotome is the line extended from diameter from the vertex equal to the abscissa.
Archimedes, Lemmata for Orthotomes from the Quadrature of the Parabola
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: Theorem 0 B)
Case where BD is parallel to the diameter
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: Theorem 0D
Case where BD is parallel to the diameter.
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.
And these have been proved in the Conic Elements.