Greek Science of the Hellenistic Era by Irby-Massie Georgia L. Keyser Paul T

Author:Irby-Massie, Georgia L.,Keyser, Paul T.
Language: eng
Format: epub, pdf
Publisher: Taylor & Francis (CAM)

Prop. 1 [construction of a parabolic mirror]

Let there be a parabola KBM, with a line cutting it in two AZ, and let half the parameter of the squares on the ordinates be line BH [a given length, which is the distance from the parabola's vertex to the apex of the right-angled cone generating it; this is Archimedes’ way of generating cones: compare Apollonios, Konika 1.11, Dijksterhuis [1956/1987] 59–63; the parabolic invariant is 2 · BH · BΓ = ΘΓ·ΘΓ]. Let BE on the axis be equal to BH, and let BE be bisected at point Δ. Let us draw a line tangent to the section at an arbitrary point, namely line ΘA, and draw line ΘΓ as ordinate to AZ [i.e., ΘΓ is perpendicular to AZ (Figure 7.10)]. Then we know that AB = BΓ [in Euclid's Conics, see Archimedes, Method 1 (Chapter 2.2), and Apollonios, Conics 1.35, 2.49] and that the line drawn from Θ perpendicular to ΘA meets AZ beyond E [BE = BH; so ΓZ = BE as below; the perpendicular from the tangent falls at Γ below B; so Z must be below E]. So let us draw ZΘ perpendicular to ΘA, and join ΘΔ.

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