Basic Lemmata on Parabolas or OrthotomesReturn to Vignettes of Ancient Mathematics

by Henry Mendell (Cal. State U., L.A.)

Archimedes, Quadrature of the Parabola, introduction and contents

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*
1-5.

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: Theorem 0 B) 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: Theorem 0D 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: Theorem 0C |

And these have been proved in the *Conic Elements*.

4. Go to proof

Let there be a segment ABG enclosed by a straight line and an orthotome, and from the middle of AG let there be drawn BD parallel to the diameter or itself a diameter, and let straight line BG be joined and extended. If in fact some other line ZQ parallel to BD intersects the straight line through points B, G, ZQ will have the same ratio to QH which DA has to DZ.

5. Go to proof

Let there be a segment ABG enclosed by a straight line and an orthotome, and from A and parallel to the diameter let ZA be led, and from G and tangent to the orthotome at G let GZ be led. If in fact some line parallel to AZ be led in triangle ZAG, the line led will be cut by the orthotome and AG will be cut by the line led in the same ratio, but the segment of AG at A will be in the same part of the ratio as the segment of the led line at A.

The property Archimedes describes is: QK : LQ = AK : KG.