Sources page biographical material

6.O. PASSING A CUBE THROUGH AN EQUAL OR SMALLER CUBE --

PRINCE RUPERT'S PROBLEM
The projection of a unit cube along a space diagonal is a regular hexagon of side 2/3. The largest square inscribable in this hexagon has edge 6 - 2 = 1.03527618. By passing the larger cube on a slant to the space diagonal, one can get the larger cube having edge 32/4 = 1.06066172.

There are two early attributions of this. Wallis attributes it to Prince Rupert, but Hennessy says Philip Ronayne of Cork invented it. I have discovered a possible connection. Prince Rupert of the Rhine (1619-1682), nephew of Charles I, was a major military figure of his time, becoming commander-in-chief of Charles I's armies in the 1640s. In 1648-1649, he was admiral of the King's fleet and was blockaded with 16 ships in Kinsale Harbor for 20 months. Kinsale is about 20km south of Cork.

Ronayne wrote an Algebra, of which only a second edition of 1727 is in the BL. Schrek has investigated the family histories and says Ronayne lived in the early 18C. This would seem to make him too young to have met Rupert. Perhaps Rupert invented the problem while in Kinsale and this was conveyed to Ronayne some years later. Does anyone know the dates of Ronayne or of the 1st ed (Schrek only located the BL example of the 2nd ed)? I cannot find anything on him in Wallis, May, Poggendorff, DNB, but Google has turned up a reference to a 1917 history of the family which Schrek cites, but I have not yet tried to find this.

Hennessy's article says a little about Daniel Voster and details are in Wallis's . His father, Elias (1682   >1728) wrote an Arithmetic, of which Wallis lists 30 editions. The BL lists one as late as 1829. The son, Daniel (1705   >1760) was a schoolmaster and instrument maker who edited later versions of his father's arithmetic. The 1750 History of Cork quoted by Hennessy says the author had seen the cubes with Daniel. Hennessy conjectures that his example was made specially, perhaps under the direction of a mathematician. It seems likely that Daniel knew Ronayne and made this example for him.

John Wallis. Perforatio cubi, alterum ipsi aequalem recipiens. (De Algebra Tractatus; 1685; Chap. 109) = Opera Mathematica, vol. II, Oxford, 1693, pp. 470 471, ??NYS. Cites Rupert as the source of the equal cube version. (Latin and English in Schrek.) Scriba, below, found an errata slip in Wallis's copy of his Algebra in the Bodleian. This corrects the calculations, but was published in the Opera, p. 695.

Ozanam Montucla. 1778. Percer un cube d'une ouverture, par laquelle peut passer un autre cube égal au premier. Prob. 30 & fig. 53, plate 7, 1778: 319-320; 1803: 315-316; 1814: 268-269. Prob. 29, 1840: 137. Equal cubes with diagonal movement.

J. H. van Swinden. Grondbeginsels der Meetkunde. 2nd ed., Amsterdam, 1816, pp. 512 513, ??NYS. German edition by C. F. A. Jacobi, as: Elemente der Geometrie, Friedrich Frommann, Jena, 1834, pp. 394-395. Cites Rupert and Wallis and gives a simple construction, saying Nieuwland has found the largest cube which can pass through a cube.

Peter Nieuwland. (Finding of maximum cube which passes through another). In: van Swinden, op. cit., pp. 608 610; van Swinden Jacobi, op. cit. above, pp. 542-544, gives Nieuwland's proof.

Cundy and Rollett, p. 158, give references to Zacharias (see below) and to Cantor, but Cantor only cites Hennessy.

H. Hennessy. Ronayne's cubes. Phil. Mag. (5) 39 (Jan Jun 1895) 183 187. Quotes, from Gibson's 'History of Cork', a passage taken from Smith's 'History of Cork', 1st ed., 1750, vol. 1, p. 172, saying that Philip Ronayne had invented this and that a Daniel Voster had made an example, which may be the example owned by Hennessy. He gives no reference to Rupert. He finds the dimensions.

F. Koch & I. Reisacher. Die Aufgabe, einen Würfel durch einen andern durchzuschieben. Archiv Math. Physik (3) 10 (1906) 335 336. Brief solution of Nieuwland's problem.

M. Zacharias. Elementargeometrie und elementare nicht-Euklidische Geometrie in synthetischer Behandlung. Encyklopädie der Mathematischen Wissenschaften. Band III, Teil 1, 2te Hälfte. Teubner, Leipzig, 1914-1931. Abt. 28: Maxima und Minima. Die isoperimetrische Aufgabe. Pp. 1133-1134. Attributes it to Prince Rupert, following van Swinden. Cites Wallis & Ronayne, via Cantor, and Nieuwland, via van Swinden.

U. Graf. Die Durchbohrung eines Würfels mit einem Würfel. Zeitschrift math. naturwiss. Unterricht 72 (1941) 117. Nice photos of a model made at the Technische Hochschule Danzig. Larger and better versions of the same photos can be found in: W. Lietzmann & U. Graf; Mathematik in Erziehung und Unterricht; Quelle & Meyer, Leipzig, 1941, vol. 2, plate 3, opp. p. 168, but I can't find any associated text for it.

W. A. Bagley. Puzzle Pie. Op. cit. in 5.D.5. 1944. No. 12: Curios [sic] cubes, p. 14. First says it can be done with equal cubes and then a larger can pass through a smaller. Claims that the larger cube can be about 1.1, but this is due to an error -- he thinks the hexagon has the same diameter as the cube itself.

H. D. Grossman, proposer; C. S. Ogilvy & F. Bagemihl, solvers. Problem E888 -- Passing a cube through a cube of same size. AMM 56 (1949) 632 ??NYS & 57 (1950) 339. Only considers cubes of the same size, though Bagemihl's solution permits a slightly larger cube. No references.

D. J. E. Schrek. Prince Rupert's problem and its extension by Pieter Nieuwland. SM 16 (1950) 73 80 & 261 267. Historical survey, discussing Rupert, Wallis, Ronayne, van Swinden & Nieuwland. Says Ronayne is early 18C.

M. Gardner. SA (Nov 1966) = Carnival, pp. 41 54. The largest square inscribable in a cube is the cross section of the maximal hole through which another cube can pass.

Christoph J. Scriba. Das Problem des Prinzen Ruprecht von der Pfalz. Praxis der Mathematik 10 (1968) 241-246. ??NYS -- described by Scriba in an email to HM Mailing List, 20 Aug 1999. Describes the correction to Wallis's work and considers the problem for the tetrahedron and octahedron.