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.
Reading 33
Reading 34
Reading 35
Reading 36
We will dissect this reading carefully so be ready to supply details to Archimedes' arguments.
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