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Engineering Calculus Notes 147

Engineering Calculus Notes 147 - show that the equation of...

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2.1. CONIC SECTIONS 135 6. The Quadratrix of Hippias: Pappus describes the construction of a curve he calls the quadratrix , which can be used for the quadrature of the circle as well as trisection of an angle. He ascribes it to Nicomedes ( ca. 280-210 BC ), but Proclus (411-485), a later commentator on Euclid and Greek geometry as important as Pappus, ascribes its invention to Hippias of Elis ( ca. 460-400 BC ), and Heath trusts him more than Pappus on this score (see [ 24 , vol. 1, pp. 225-226]). The construction is as follows [ 13 , p. 95]: the radius OX of a circle rotates through a quarter-turn (with constant angular speed) from position OC to position OA , while in the same time interval a line BD parallel to OA undergoes a parallel displacement (again with constant speed) from going through C to containing OA . The quadratrix is the locus of the intersection of the two during this motion (except for the final moment, when they coincide). (a) Assuming the circle has center at the origin and radius a and the final position of the radius OA is along the positive x -axis,
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Unformatted text preview: show that the equation of the quadratrix in polar coordinates is πr sin θ = 2 aθ. (b) Show that if P is on the arc of the circle in the Frst quadrant, then the angle ∠ POA can be trisected as follows: let F be the intersection of OP with the quadratrix, and let FH be the vertical line segment to the x-axis. If F ′ is one-third the way from H to F along this segment, and F ′ L is a horizontal segment with L on the quadratrix, then show that ∠ LOA = 1 3 ∠ POA . (c) i. Show that if the quadratrix intersects OA at G , then OG = 2 a π . (You can use calculus here: in the proof by Dinostratus ( ca. 390-320 BC ), it is done by contradiction, using only Euclidean geometry.) ii. Conclude from this that ⌢ CA : OA = OA : OG. iii. Show how, in light of this, we can construct a line segment equal in length to ⌢ CA . iv. Show that a rectangle with one side equal to twice this line segment and the other equal to a has the same area as the circle of radius a ....
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