176 



UNICURSAL PROBLEMS 



[CH. IX 



The Konigsberg bridges lead to a network with four odd 

 nodes; hence, by Euler's fourth proposition, it cannot be 

 described unicursally in a single journey, though it can be 

 traversed completely in two separate routes. 



The first and second diagrams figured below contain only 

 even nodes, and therefore each of them can be described uni- 

 cursally. The first of these is a regular re-entrant pentagon ; 

 the second is the so-called sign-manual of Mohammed, said to 

 have been originally traced in the sand by the point of his 

 scimetar without taking it off the ground or retracing any part 

 of the figure — which, as it contains only even nodes, is possible. 

 The third diagram is taken from Tait's article: it contains 

 only two odd nodes, and therefore can be described unicursally 

 if we start from one of them, and finish at the other. 



The re-entrant pentagon, figured above, has some interest 

 from having been used by the Pythagoreans as a sign — known 

 as the triple triangle or pentagram star — by which they could 

 recognize one another. It was considered symbolical of health, 

 and probably the angles were denoted by the letters of the word 

 vyieia, the diphthong ei being replaced by a 6. Iamblichus, who 

 is our authority for this, tells us that a certain Pythagorean, when 

 travelling, fell ill at a roadside inn where he had put up for 

 the night ; he was poor and sick, but the landlord, who was a 

 kind-hearted fellow, nursed him carefully and spared no trouble 

 or expense to relieve his pains. However, in spite of all efforts, 

 the student got worse. Feeling that he was dying and unable 

 to make the landlord any pecuniary recompense, he asked for 

 a board on which he inscribed the pentagram star; this he 

 gave to his host, begging him to hang it up outside so that 

 all passers by might see it, and assuring him that the result 



