The solution of this problem with a determination of the limits of possibility are given in a fragment by Archimedes, discovered and preserved for us by Eutocius in his commentary on the book; they are effected by means of the points of intersection of two conics, a parabola and a rectangular hyperbola. Three problems of construction follow, the first two of which are to construct a segment of a sphere similar to one given segment, and having (a) its volume, (b) its surface, equal to that of another given segment of a sphere. The last two propositions are interesting. Prop. 8 proves that, if V, V' be the volumes, and S, S' the surfaces, of two segments into which a sphere is divided by a plane, V and S belonging to the greater segment, then S^2 : S'^2 > V : V' > S^(3/2) : S'^(3/2). Prop. 9 proves that, of all segments of spheres which have equal surfaces, the hemisphere is the greatest in volume. _The Measurement of a Circle._ This treatise, in the form in which it has come down to us, contains only three propositions; the second, being an easy deduction from Props. 1 and 3, is out of place in so far as it uses the result of Prop. 3. In Prop. 1 Archimedes inscribes and circumscribes to a circle a series of successive regular polygons, beginning with a square, and continually doubling the number of sides; he then proves in the orthodox manner by 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 [pi]; the result obtained is 3-1/7> [pi] > 3-10/71. 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 [root]3, that being the cotangent of the angle of 30 deg., from which he begins to work. He assumes as known that 265/153 < [root]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 1172-1/8 is given as the approximate value of [root](1373943-33/64), 3013-3/4 as that of [root](9082321) and 1838-9/11 as that of [root](3380929). In this way Archimedes arrives at 14688/(4673-1/2) as the ratio of the perimeter of the circumscribed polygon of 96 sides to the diameter of the circle;