S² : S′ ² > V : V′ > S3/2 : 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 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.
The preface addressed to Dositheus shows, as we may also infer from internal evidence, that the whole of this book also was original. Archimedes first explains what his conoids and spheroids are, and then, after each description, states the main results which it is the aim of the treatise to prove. The conoids are two. The first is the right-angled conoid, a name adapted from the old name (“section of a right-angled cone”) for a parabola; this conoid is therefore a paraboloid of revolution. The second is the obtuse-angled conoid, which is a hyperboloid of revolution described by the revolution of a hyperbola (a “section of an obtuse-angled cone”) about its transverse axis. The spheroids are two, being the solids of revolution described by the revolution of an ellipse (a “section of an acute-angled cone”) about (1) its major axis and (2) its minor axis; the first is called the “oblong” (or oblate) spheroid, the second the “flat” (or prolate) spheroid. As the volumes of oblique segments of conoids and spheroids are afterwards found in terms of the volume of the conical figure with the base of the segment as base and the vertex of the segment as vertex, and as the said base is thus an elliptic section of an oblique circular cone, Archimedes calls the conical figure with an elliptic base a “segment of a cone” as distinct from a “cone”.
As usual, a series of preliminary propositions is required. Archimedes first sums, in geometrical form, certain series, including the arithmetical progression, a, 2a, 3a, ... na, and the series formed by the squares of these terms (in other words the series 1², 2², 3², ... n²); these summations are required for the final addition of an indefinite number of elements of each figure, which amounts to an integration. Next come two properties of conics (Prop. 3), then the determination by the method of exhaustion of the area of an ellipse (Prop. 4). Three propositions follow, the first two of which (Props. 7, 8) show that the conical figure above referred to is really a segment of an oblique circular cone; this is done by actually finding the circular sections. Prop. 9 gives a similar proof that each elliptic section of a conoid or spheroid is a section of a certain oblique circular cylinder (with axis parallel to the axis of the segment of the conoid or spheroid cut off by the said elliptic section). Props. 11-18 show the nature of the various sections which cut off segments of each conoid and spheroid and which are circles or ellipses according as the section is perpendicular or obliquely inclined to the axis of the solid; they include also certain properties of tangent planes, etc.
The real business of the treatise begins with Props. 19, 20; here it is shown how, by drawing many plane sections equidistant from one another and all parallel to the base of the segment of the solid, and describing cylinders (in general oblique) through each plane section with generators parallel to the axis of the segment and terminated by the contiguous sections on either side, we can make figures circumscribed and inscribed to the segment, made up of segments of cylinders with parallel faces and presenting the appearance of the steps of a staircase. Adding the elements of the inscribed and circumscribed figures respectively and using the method of exhaustion, Archimedes finds the volumes of the respective segments of the solids in the approved manner (Props. 21, 22 for the paraboloid, Props. 25, 26 for the hyperboloid, and Props. 27-30 for the spheroids). The results are stated in this form: (1) Any segment of a paraboloid of revolution is half as large again as the cone or segment of a cone which has the same base and axis; (2) Any segment of a hyperboloid of revolution or of a spheroid is to the cone or segment of a cone with the same base and axis in the ratio of AD + 3CA to AD + 2CA in the case of the hyperboloid, and of 3CA − AD to 2CA − AD in the case of the spheroid, where C is the centre, A the vertex of the segment, and AD the axis of the segment (supposed in the case of the spheroid to be not greater than half the spheroid).