54 GEOMETRICAL RECREATIONS [CH. Ill 



These numbers are obtained by finding convergents to the con- 

 tinued fraction 



111 



+ 1 + 1 + 1 + •"■ 



Dissection Problems. The above paradoxes naturally sug- 

 gest the consideration of dissection problems. An excellent 

 typical example is to cut a square into 20 equal triangles, and 

 conversely to construct a square of 20 such triangles. 



There is an interesting historical example of such a problem. 

 Two late Latin writers, Victorinus and Fortunatianus, describe 

 an Archimedean toy composed of 14 ivory polygons which fitted 

 exactly into a square box, and they suggest that the puzzle was 

 to fit the pieces into the box. A recent discovery* has shown 

 that its association with the name of Archimedes is due to the 

 fact that he gave a construction for dividing a square into 14 

 such pieces (namely, 11 triangles, 2 scalene quadrilaterals, and 

 one pentagon) so that the area of each piece is a rational 

 fraction of the area of the square. His construction is as 

 follows : let ABGD be the square, and E, F, G, H, the mid- 

 points of the sides AB, BG, GD, DA. Draw HB, HF, HG, and 

 let J, K, L be the mid-points of these lines: draw AKG cutting 

 HB in M, and let N be the mid-point of AM, and P the mid- 

 point of BF. DrewBN. Draw AP cutting HB in Q. DrawPJ". 

 Draw BL, and produce it to cut DG in R. Draw FL cutting 

 AG in S. Draw LQ. Rub out the lines AQ and BL. The 

 remaining lines will give a division as required, each figure 

 being an integral multiple of l/48th of the square. Why 

 Archimedes propounded so peculiar a division it is impossible 

 to guess, but no doubt the problem has a history of which we 

 are ignorant. 



Colouring Maps. I proceed next to mention the geo- 

 metrical proposition that not more than four colours are neces- 

 sary in order to colour a map of a country (divided into districts) 



* H. Suter, Zeitsehrift fiir Mathematik und Physik, Abhandlungen zw 

 Gesch. der Math. 1899, vol. xuv, pp. 491—499. 



