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Quadrature of the Parabola

Quadrature of the Parabola. What is QOP?. 24 Propositions 1-3: Euclid’s Elements of Conics 5 &6: Parabolic Properties 6-17: Mechanical Propositions 18-24: Geometric Propositions. The Main Theorem.

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Quadrature of the Parabola

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  1. Quadrature of the Parabola

  2. What is QOP? • 24 Propositions • 1-3: Euclid’s Elements of Conics • 5 &6: Parabolic Properties • 6-17: Mechanical Propositions • 18-24: Geometric Propositions

  3. The Main Theorem • Proposition 24: Every segment bounded by a parabola and a chord Qq is equal to four-thirds of the triangle which has the same base as the segment and equal height.

  4. Proof (Case 1) • Proposition 21: If Qq be the base, and P the vertex, of any parabolic segment, and if R be the vertex of the segment cut off by PQ, then ∆PQq = 8∆PRQ.

  5. Proof (Case 1) • Proposition 23: Given a series of areas A, B, C, D, … Z, of which A is the greatest, and each is equal to four times the next in order, thenA + B + C + D + … + Z + (1/3)Z = (4/3)A.

  6. Proof (Case 2) • Proposition 22: If there be a series of areas A, B, C, D, … each of which is four times the next in order, and if the largest, A, be equal to the triangle PQq inscribed in a parabolic segment PQq and having the same base with it and equal height, then (A + B + C + D + … ) < (area of segment PQq).

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