400 



NATURE 



{Feb. 26, 1880 



We can therefore reduce a map to a single district by 

 successive operations of throwing two districts into one 

 by rubbing out the boundary or boundaries between two 

 districts of which one has less than six boundaries. 

 Conversely we can develop a map, starting from a single 

 district and adding boundaries, at each stage dividing a 

 district into two, one of which has less than six boundaries. 

 Suppose at any stage of its development by this process 

 a map can be coloured with four colours [red, blue, green, 

 and yellow]. Let these colours be indicated by coloured 

 wafers placed on the districts. Proceed to the next opera- 

 tion, this divides a wafered district into two districts. Shift 

 its wafer on to the district of these two which is not the 

 one which has less than six boundaries : if both have 

 less than six boundaries shift the wafer on to either. If 

 the district (W) which is left without a wafer is only 

 touched by three colours it can be coloured the fourth, 

 and a wafer may be put on it representing that colour. 

 But if it is touched by all four colours we must take 

 another step. This can only be necessary if W has four or 

 five adjacent districts. These may either all surround or 

 all be surrounded by or some surround and some be 

 surrounded by it. Take first the case in which four 

 districts are adjacent, all surrounding or surrounded by 

 W. Let abed be the districts, taking them in the 

 order in which they stand. Let a be red, b blue, c green, 

 and a yellow. If, starting from a, we can get to c, going 

 only through red and green districts, and not passing 

 through any points of concourse, we cannot, starting from 

 b, get to d, going similarly only through blue and yellow 

 districts, for otherwise two tracks which pass through 

 different districts would cross. Thus b forms one of a 

 group (G) of blue and yellow districts which are cut off 

 from the rest of the map by encircling red and green ones. 

 We can accordingly interchange the blue and yellow 

 wafers in G without changing any others ; this makes 6 

 yellow, and we can put a blue wafer on W. Similarly, if 

 we cannot pass from a to c, a belongs to an isolated 

 group of red and green districts. Interchanging the 

 wafers in these, a becomes green, and a red wafer can 

 be put on W. Precisely similar reasoning applies in the 

 ease of five surrounding or surrounded districts, viz., e 

 red, f blue, g green, h blue (two must of course be the 

 same colour), and k yellow. Here either e belongs to an 

 isolated group of red and green districts, or /* to one of 

 yellow and green districts, or f to one of blue and yellow, 

 and h to one of blue and red districts. In the first case, 

 interchanging wafers as before, e becomes green, and a 

 red wafer can be put on W; in the second k becomes 

 green, and a yellow wafer can be put on W ; in the third 

 / becomes yellow, and // red, and a blue wafer can be 

 put on W. In all cases before putting the wafer on W 

 we can interchange the colours of districts, e.g., we can 

 put red wafers in the place of all the green ones and vice 

 versd. Thus we can make the three colours adjacent to 

 W any three we please. If therefore the districts adjacent 

 to W belong to different groups of districts surrounding 

 and surrounded by W, and so detached from each other, 

 we can rearrange the wafers in each group so that only 

 three colours in all shall be adjacent to W, which can 

 therefore have a wafer of the fourth colour placed on it. 

 Thus in any case the district W can be wafered with a 

 wafer of one of the four colours. Thus if the map can be 

 coloured as developed at any stage it can be coloured at 

 the next. Hence since it can obviously be coloured at 

 the first stage when there is only one district, it can be 

 coloured at the last. 



Take then two copies, P and O, of the map we wish to 

 colour, one of which, Q, is on a slate or in pencil, so that 

 the boundaries can easily be obliterated. Pick out a dis- 

 trict with less than six boundaries. Rub out in Q the 

 boundary or boundaries (if there be more than one) be- 

 tween this and any other district which is adjacent to it. 

 K umber with a (1) the corresponding boundary or boun- 



daries in P. Repeat the operation, numbering the 

 boundary or boundaries in P this time (2). Continue the 

 process until a map is arrived at which can obviously be 

 coloured with four colours. This will generally happen 

 long before we reduce the number of districts to four. 

 Put wafers on the middle of the districts of Q, repre- 

 senting the colours. Proceeding as before shown with 

 the process of adding boundaries in the order indicated 

 by the numbering of P taken backwards, and of shifting 

 the wafers so as to be able to add a wafer to the W 

 of each stage, we finally arrive at a stage when Q is in 

 its original state. The map can then be coloured as 

 indicated by the colours of the wafers. 



This method applies equally to maps drawn on the 

 surface of globes, but fails in the case of surfaces which 

 are not necessarily divided into two parts by an endless 

 line, these in general requiring more than four colours. 



A. B. Kempe 



THE LIPARI ISLANDS 

 /"\N inquiring in Rome for the Stato Maggiore map of 

 v --' the Lipari Islands, I was told that it was out of 

 print, and when afterwards I succeeded in getting one in 

 Florence, I found that owing to the large scale, the islands 

 from Vulcano to Stromboli, in a north-easterly direction, 

 and from Vulcano to Alicudi, to the north-west, were given 

 on three separate sheets, too unwieldy to use for practical 

 purposes, except in their disconnected form. Our own 

 Admiralty chart (scale xinnrsii) constructed from the maps of 

 a French hydrographer, M. Darondeau, gives all the islands, 

 save Ustica, at one view, accompanied by soundings, 

 and a general diagrammatic view of the principal group. 

 The Comitato Geologico of Rome has not yet published a 

 geological map of the islands, and the only complete one 

 that exists, as far as I know, is that to be found in the 

 antiquated " Vulkanen-Atlas " of N. C. von Leonhard, which 

 is taken from the survey of Fr. Hoffmann. In this map 

 Alicudi, Felicudi, Salina, and the major part of Lipari 

 are represented as composed of tuff with porphyritic 

 lava. Panaria, with the surrounding islets Dattolo, 

 Lisia Bianca, &c, is stated to consist entirely of trachyte. 

 The greater part of Vulcano, and about half Stromboli 

 are given as old felspathic lava, while the craters of 

 Vulcano, Vulcanello, and Stromboli are described as 

 nochforldauernde viilk. Bildungen. Pumice and obsidian, 

 are shown in various parts of Lipari. 



Since Admiral Smyth visited the .Eolian Island in 

 1 Sr 5, numerous observers have followed in his footsteps. 

 He has devoted thirty pages of a quarto volume on 

 "Sicily and its Islands" to this subject, and two of the 

 three Admiralty charts which relate to these islands, 

 contain engravings executed from his drawings. 



In 1874 Prof. J. W. Judd visited the islands, and he 

 has embodied the results of his observations in some 

 valuable memoirs contributed to the Geological Magazine, 

 accompanied by reproductions of drawings made on the 

 spot. To him we are indebted for the accompanying 

 view of Vulcano and Vulcanello. 



We were surprised at the complete ignorance mani- 

 fested both by the Romans and the Neapolitans, in regard 

 to everything connected with Lipari and the members of 

 its group. Everybody said "You must tell us all about 

 them when you come back." In fact the islands are very 

 little visited ; communication with the mainland is at the 

 best only twice a week ; the boats are small and incon- 

 venient, and they start at midnight ; and worse than all, 

 the most important island of the group (Lipari) is cursed 

 by the presence of some 400 convicts, who are sent to this 

 penal settlement, much as we sent ill-disposed persons to 

 Botany Bay forty years ago. We left the harbour of 

 Messina at midnight, having on board ten of these 

 manacled manutengoli, guarded by carabinieri. At six 

 the next morning we were off Lipari, and soon afterwards 



