the method of exhaustion that the area of the circle is equal to that of a right-angled triangle, in which the perpendicular is equal to the radius, and the base equal to the circumference, of the circle.
Prop. 3 is the famous proposition in which Archimedes finds by sheer calculation upper and lower arithmetical limits to the ratio of the circumference of a circle to its diameter, or what we call π; the result obtained is 31⁄7 > π > 310⁄71. Archimedes inscribes and circumscribes successive regular polygons, beginning with hexagons, and doubling the number of sides continually, until he arrives at inscribed and circumscribed regular polygons with 96 sides; seeing then that the length of the circumference of the circle is intermediate between the perimeters of the two polygons, he calculates the two perimeters in terms of the diameter of the circle.
His calculation is based on two close approximations (an upper and a lower) to the value of √3, that being the cotangent of the angle of 30°, from which he begins to work. He assumes as known that 265/153 < √3 < 1351/780. In the text, as we have it, only the results of the steps in the calculation are given, but they involve the finding of approximations to
the square roots of several large numbers: thus 11721⁄8 is given as the approximate value of √(137394333⁄64), 3013¾ as that of √(9082321) and 18389⁄11 as that of √(3380929).
In this way Archimedes arrives at 14688 / 4673½ as the ratio of the perimeter of the circumscribed polygon of 96 sides to the diameter of the circle; this is the figure which he rounds up into 31⁄7. The corresponding figure for the inscribed polygon is 6336 / 2017¼, which, he says, is > 310⁄71. This example shows how little the Greeks were embarrassed in arithmetical calculations by their alphabetical system of numerals.
On Conoids and Spheroids.