Sources page biographical material

6.AE. 6" HOLE THROUGH SPHERE LEAVES CONSTANT VOLUME
Hamnet Holditch. Geometrical theorem. Quarterly J. of Pure and Applied Math. 2 (1858) ??NYS, described by Broman. If a chord of a closed curve, of constant length a+b, be divided into two parts of lengths a, b respectively, the difference between the areas of the closed curve, and of the locus of the dividing point as the chord moves around the curve, will be πab. When the closed curve is a circle and a = b, then this is the two dimensional version given by Jones, below. A letter from Broman says he has found Holditch's theorem cited in 1888, 1906, 1975 and 1976.

Richard Guy (letter of 27 Feb 1985) recalls this problem from his schooldays, which would be late 1920s-early 1930s, and thought it should occur in calculus texts of that time, but could not find it in Lamb or Caunt.

Samuel I. Jones. Mathematical Nuts. 1932. P. 86. ??NYS. Cited by Gardner, (SA, Nov 1957) = 1st Book, chap. 12, prob. 7. Gardner says Jones, p. 93, also gives the two dimensional version: If the longest line that can be drawn in an annulus is 6" long, what is the area of the annulus?

L. Lines. Solid Geometry. Macmillan, London, 1935; Dover, 1965. P. 101, Example 8W3: "A napkin ring is in the form of a sphere pierced by a cylindrical hole. Prove that its volume is the same as that of a sphere with diameter equal to the length of the hole." Solution is given, but there is no indication that it is new or recent.

L. A. Graham. Ingenious Mathematical Problems and Methods. Dover, 1959. Prob. 34: Hole in a sphere, pp. 23 & 145 147. [The material in this book appeared in Graham's company magazine from about 1940, but no dates are provided in the book. (??can date be found out.)]

M. H. Greenblatt. Mathematical Entertainments, op. cit. in 6.U.2, 1965. Volume of a modified bowling ball, pp. 104 105.

C. W. Trigg. Op. cit. in 5.Q. 1967. Quickie 217: Hole in sphere, pp. 59 & 178 179. Gives an argument based on surface tension to see that the ring surface remains spherical as the hole changes radius. Problem has a 10" hole.

Andrew Jarvis. Note 3235: A boring problem. MG 53 (No. 385) (Oct 1969) 298 299. He calls it "a standard problem" and says it is usually solved with a triple integral (??!!). He gives the standard proof using Cavalieri's principle.

Birtwistle. Math. Puzzles & Perplexities. 1971.

Tangential chord, pp. 71-73. 10" chord in an annulus. What is the area of the annulus? Does traditionally and then by letting inner radius be zero.

The hole in the sphere, pp. 87-88 & 177-178. Bore a hole through a sphere so the remaining piece has half the volume of the sphere. The radius of the hole is approx. .61 of the radius of the sphere.

Another hole, pp. 89, 178 & 192. 6" hole cut out of sphere. What is the volume of the remainder? Refers to the tangential chord problem.

Arne Broman. Holditch's theorem: An introductory problem. Lecture at ICM, Helsinki, Aug 1978. Broman then sent out copies of his lecture notes and a supplementary letter on 30 Aug 1978. He discusses Holditch's proof (see above) and more careful modern versions of it. His letter gives some other citations.