Math (Несколько текстов для зачёта), страница 18

Exercise 4. By analogy with Assumption 2, consider a sphere as being composed of spherical shells centered at the center of the sphere, where each shell weighs the same as a circle of equal area. The justification follows exactly as in Proposition 2: Consider two pans suspended at equal distances from the fulcrum of a balance. On one pan, place a sphere of center A and radius AB and on the other a line CD of length equal to AB. For each E on AB there is a spherical shell passing through E, and consider a circle of area equal to this spherical shell with center at F lying on CD, where CF equals AE, and such that the circle is perpendicular to CD. The resulting figure is a cone with base the area of the sphere and height the radius of the sphere; since it balances the sphere, the claim is justified.

The similarity of this argument to the one of Proposition 1 suggests that Archimedes may have been implicitly aware of the ideas of this paper. Moreover, the reader may verify that the heuristic of this exercise and its justification directly generalize to higher dimensions (a different generalization is given in [19]):

Proposition 3. The volume of an n-dimensional ball is equal to the volume of a cone whose base has n - 1-dimensional volume equal to the (n - 1)-dimensional volume of the boundary of the ball and height equal to the radius of the ball.

Exercise 5. The procedure, when applied to the spiral, yields a section of a parabola. The general formula for such areas was computed by Archimedes in The Quadrature of the Parabola, and in this case it states that the resulting area is four-thirds the triangle with same base and height as the section of the parabola. Since the height and base are equal to the final radius and half the final radius, respectively, Proposition 2 follows.

Exercise 6. Further extensions of Archimedes's method could be a subject for investigation. As Archimedes wrote in The Method [6, Supplement, p.13],

I deem it necessary to expound the method partly because I have already spoken of it but equally because I am persuaded that it will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me.

Acknowledgment

I would like to thank Alain Herreman, Reviel Netz, and David Wilkins for helpful comments.

1 Hence I question the curriculum of St. John's College, which purports to educate its students by following an historical sequence of original sources. Its reading list also includes the ancient textbook [47].

2 Archimedes is addressing Eratosthenes of Cyrene (circa 284-194 B.C), director of the library of Alexandria, famous for his accurate measurement of the circumference of the earth [14] and his sieve to compute prime numbers [47].

3 Archimedes requested that a diagram of a sphere inscribed in a cylinder along with their proportion be placed on his grave, which Cicero reported finding in 75 B.C. when he was treasurer of Sicily [54, Vol. 2, p. 33].

4 In The Quadrature of the Parabola Archimedes gave what he considered to be a rigorous proof using the mechanical method of a result conjectured in a similar way in The Method, but using infinitesimals.

5 A similar method was used by Rabbi Abraham bar Hiyya (1070-1136), see V.J. Katz, review of "Force and Geometry in Newton's Principia" by F. de Gandt, American Math. Monthly 105 (1998), 386-392 and F. Sanchez-Faba, Abraham Bar Hiyya and his "Libro de Geometria, " (Spannish) Gac. Math., I 32 (1998), 101-115.