Mathematical recreations and essays . hematics, London, 1890, vol. xxiv, pp. 332—338; and 1897,vol. XXXI, pp. 270—285. t Proceedings of the Royal Society of Edinburgh^ July 19, 1880, vol. x,p. 729; Philosophical Magazine, January, 1884, series 5, vol. xvii, p. 41; andCollected Scientific Papers^ Cambridge, vol. ii, 1890, p. 93. CH. Ill] GEOMETRICAL RECREATIONS 67 meets any other line except at one of the vertices, which is allthat we require for the map theorem; but it has not been this limitation it is not correct. For instance theaccompanying figure, representing a closed netw
Mathematical recreations and essays . hematics, London, 1890, vol. xxiv, pp. 332—338; and 1897,vol. XXXI, pp. 270—285. t Proceedings of the Royal Society of Edinburgh^ July 19, 1880, vol. x,p. 729; Philosophical Magazine, January, 1884, series 5, vol. xvii, p. 41; andCollected Scientific Papers^ Cambridge, vol. ii, 1890, p. 93. CH. Ill] GEOMETRICAL RECREATIONS 67 meets any other line except at one of the vertices, which is allthat we require for the map theorem; but it has not been this limitation it is not correct. For instance theaccompanying figure, representing a closed network in threedimensions of 15 lines formed by the sides of two pentagonsand the lines joining their corresponding angular points,cannot be coloured as described by Tait. If the figure is inthree dimensions, the lines intersect only at the ten verticesof the network. If it is regarded as being in two dimensions,only the ten angular points of the pentagons are treated asvertices of the network, and any other point of intersection of. the lines is not regarded as such a vertex. Expressed in tech-nical language the difficulty is this. Petersen* has shown thata graph (or network) of the 2nth. order and third degree andwithout offshoots {or fetiilles) can be resolved into three graphsof the 2nth order and each of the first degree, or into two graphsof the 2nth order one being of the first degree and one of thesecond degree. Tait assumed that the former resolution wasthe only one possible. The question is whether the limitationsmentioned above exclude the second resolution. Assuming that the theorem as thus limited can be estab-lished, Taits argument that four colours will suffice for a mapis divided into two parts and is as follows. * See J. Petersen of Copenhagen, VIntermediaire des Mathimaticiens, vol. v,1898, pp. 225—227; and vol. vi, 1899, pp. 36—38. Also Acta MatJiematica,Stockholm, vol. xv, 1891, pp. 193—220. 58 GEOMETRICAL RECREATIONS [CH. Ill First, suppose that the
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