1 / 6

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.

hyman
Download Presentation

Quadrature of the Parabola

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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).

More Related