Discussion and Homework
Sixth Reading Assignment

Notes for the Reading

from On the Sphere and Cylinder I

Proposition 2: Given two unequal magnitudes, it is possible to find two unequal straight lines such that the greater straight line has to the less a ratio less than the greater magnitude has to the less.

Proposition 3: Given two unequal magnitudes and a circle, it is possible to inscribe a polygon in the circle and to describe another about it so that the side of the circumscribed polygon may have to the side of the inscribed polygon a ratio less than that of the greater magnitude to the less.

Proposition 23: The surface of the sphere is greater than the surface descibed by the revolution of the polygon inscribed in the great circle about the diameter of the great circle.

Proposition 25: [paraphrasing] The surface of a figure inscribed in a sphere is less than four times the greatest circle in the sphere.

Proposition 28: The surface of the figure circumscribed to the given sphere is greater than that of the sphere itself.

Proposition 30: The surface of a figure circumscribed as before about a sphere is greater than four times the great circle of the sphere.

Proposition 32: If a regular polygon with 4n sides be inscribed in a great circle of a sphere, as ab...a'...b'a, and a similar polygon AB...A'...B'A be described about the great circle, and if the polygons revolve with the great circle about the diameters aa' and AA' respectively, so that they describe the surfaces of solid figures inscribed in and circumscribed to the sphere respectively, then
(1) the surfaces of the circumscribed and inscribed figures are to one another in the duplicate ratio of their sides, and
(2) the figures themselves [i.e. their volumes] are in the triplicate ratio of their sides. [see figure on page 135, Calinger]

from Quadrature of the Parabola

Proposition 1: If from a point on a parabola a straight line be drawn which is either itself the axis or parallel to the axis, as PV, and if Qq be a chord parallel to the tangent to the parabola at P and meeting PV in V then QV = Vq. Conversely, if QV = Vq, the chord Qq will be parallel to the tangent at P. (See the figure on the top left of page 143, Calinger).

Proposition 2: If in a parabola Qq be a chord parallel to the tangent at P, and if a straight line be drawn through P which is either itself the axis or parallel to the axis, and which meets Qq in V and the tangent at Q to the parabola in T, then PV = PT.

Proposition 16: Suppose Qq to be the base of a parabolic segment, q not being more distant than Q from the vertex of the parabola. Draw through q the straight line qE parallel to the axis of the parabola to meet the tangent at Q in E. It is required to prove that (area of segment) = 1/3EqQ.

Questions for Discussion

Reading 33

  1. We have an expression for Assumption 1. What is it? Is it always true?
  2. Draw a picture representing Assumption 2.
  3. Where is the Postulate of Archimedes in Eudoxus (Reading 23)?

Reading 34

  1. What does Propostion 33 say in terms of Pi?
  2. Look again at Reading 23. What is Archimedes doing in his proof of Proposition 33?
  3. What does a fortiori mean?

Reading 35

  1. Without going into details, what is the basic idea behind how is Archimedes getting his approximation of the ratio of any circle to its diameter?

Reading 36

  1. Can you see the Mean Value Theorem in the explanation accompanying Proposition 18?
  2. In the explanation accompanying Proposition 19, fill in the details why PV = 4/3 RM follows from PV = 4 PW.
  3. Where have we seen Proposition 20 and its corollary before?
  4. In Propositions 22 and 23, think series. How would we probably state these two propositions?

We will dissect this reading carefully so be ready to supply details to Archimedes' arguments.

Homework Problems

  1. Use calculus to find the surface of revolution of the circle and of the inscribed polygon. Use the figure on page 135, that is, use an octagon. Let the radius of the circle be r, and you can center everything at the origin. Note that the octagon touches the circle at 45° intervals.
    Note that the area of a polygon is 1/2 hP where h is the apothem and P is the perimeter.
  2. Let A be the area of the circle, let K be the area of the triangle, let PI be the area of the inscribed polygon, and let PC be the area of the circumscribed polygon. Write out the two parts (I and II) of the proof of Proposition I using this notation to see if the proof is clearer.
  3. Look up a formula for area of a regular polygon. Find the area of both the inscribed and circumscibed polygon (using a circle of radius 1) for polygons of side n = 6, 12, 24, 48, and 96.

A pdf version of this page and the accompanying reading assignment.

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