itself revolves at uniform speed about the origin as a fixed point.
Props. 1-11 are preliminary, the last two amounting to the summation of certain series required for the final addition of an indefinite number of element-areas, which again amounts to integration, in order to find the area of the figure cut off between any portion of the curve and the two radii vectores drawn to its extremities.
Props. 13-20 are interesting and difficult propositions establishing the properties of tangents to the spiral.
Props. 21-23 show how to inscribe and circumscribe to any portion of the spiral figures consisting of a multitude of elements which are narrow sectors of circles with the origin as centre; the area of the spiral is intermediate between the areas of the inscribed and circumscribed figures, and by the usual method of exhaustion Archimedes finds the areas required.
Prop. 24 gives the area of the first complete turn of the spiral (= 1⁄3π (2πa)², where the spiral is r