258 MISCELLANEOUS PROBLEMS [CH. XII 



square inclosing the small square within four segments equal 

 and similar to the four quarters of the larger of the given squares, 

 and consequently equal in area to the two together. 



Montucla's Dissection. Demonstrations of a few similar pro- 

 positions had long been current, but towards the close of the 

 eighteenth century attention was recalled to this kind of solution 

 by Montucla who proposed and solved the problem of dividing a 

 rectangle so that the parts could be put together in the form of 

 a square; he also solved the converse problem. Later, other 

 solutions of this problem were given by P. Busschop and 

 de Coatpont who made respectively eight-part and seven-part 

 dissections. The former also had constructions for making a 

 square from a five-part division of a regular hexagon, and from 

 a seven- part division of a regular pentagon * 



Polygonal Dissections. The more general problem of the 

 dissection of a given polygon of any number of sides, and the 

 rearrangement of the parts in the form of another polygon of 

 equal area, had been raised by Bolyai, and a method of solution 

 indicated by Gerwien. The question continued to attract occa- 

 sional attention. In particular a solution for the case of a poly- 

 gon and a triangle was given by Euzet in 1854, and the wider 

 problem of two polygons was discussed by E. Guitel in 1895, and 

 by E. Hoist in 1896, the results being obtained by triangular 

 reductions f. 



Minimum Dissections. The above mentioned writers aimed 

 only at finding a solution, and did not in general trouble them- 

 selves with considering the smallest number of pieces required. 

 In 1905, the special cases of a four-part dissection of a pair of 

 triangles, a four-part dissection of a triangle and parallelogram, 

 and dissections of a pair of parallelograms were given by H. M. 



* G. G. P. Montucla, Ozanam's R&criations MatMmatiques, 1803, English 

 edition, vol. i, pp. 292—298, 1840 edition, pp. 127—129; Paul Busschop, 

 Nouvelle Correspondence Mathlmatique, Brussels, 1875, vol. ii, p. 83 ; de Coat- 

 pont, Ibid. 1876, vol. iii, p. 116. 



+ P. Gerwien, Grelle's Journal, 1833, p. 228 ; M. Euzet, Nouvelles Annales de 

 Mathematiques, 1854, vol. xiii, pp. 114 — 115; E. Guitel, Association Francaisepour 

 Vavancement des sciences, 1895, pp. 264 — 267; E. Hoist, L'IntermSdiaire des 

 Mathematiciens, 1896, vol. iii, pp. 91 — 92. 



