Feb. 26, 18S0] 



NATURE 



399 



Manufactures, are well worthy of attention, but as they 

 deal with the artistic rather than with the scientific aspect 

 of those industries, we cannot dwell upon them. 



Amongst the instructions handed to each artisan re- 

 porter at the outset, were suggestions to ascertain the 

 prices and cost of production, the relative amount of 

 machinery employed in production, the hours of labour 

 and the manner of living of the French artisans. Much 

 useful information has been collected on most of these 

 heads. Almost all the reports agree that while cost of 

 living is perhaps a little cheaper in Paris than in London, 

 wages are on the whole much lower ; so that it is only by 

 working longer hours and by thrift and steadiness that the 

 French workman can live. The remark is almost uni- 

 versally made that drunkenness is extremely rare ; while 

 the absence of almost everything that constitutes home life 

 is equally conspicuous in the habits of the Parisian work- 

 man. 



In one or two points the volume before us is, from the 

 nature of things, strangely defective. Almost all the re- 

 porters who mention the subject at all, appear to have 

 misapprehended the nature and status of the Carles 

 cuvrieres or Corporations ouvriircs, which are the nearest 

 approach in France to the Trades Unions of this country, 

 and the comparisons drawn between the two are in con- 

 sequence often irrelevant or incorrect. These bodies in 

 France cannot legally extend beyond the limits of the 

 " commune " or parish ; they are usually semi-political or 

 socialistic in character, and while they concern them- 

 selves with the conditions of labour, are not exclusively 

 occupied in matters of wages and hours of work, and 

 do not, from the local restriction on their operations, 

 exercise an influence in any measure comparable to that 

 exercised by the English Unions over the price or 

 conditions of labour. Again it is impossible to derive 

 from the reports any ideas upon the relation between 

 skilled labour and the educational systems in operation 

 in Paris or in the provinces of France, for the simple 

 reason that not one of the reporters appears to have 

 been made acquainted with those educational systems as 

 a whole. A few of the more prominent technical schools, 

 the jzcole d Apprentis, the Horological Schools, and the 

 Typographic School of MM. Chaix and Co., are indeed 

 mentioned ; but beyond these exceptional institutions and 

 a chance reference to the free evening schools of drawing 

 and modelling which are to be found in every quarter of 

 Paris, there is no reference to the educational systems of 

 the country or to their influence on the artisan, the fore- 

 man, and the employer. Any account of the conditions 

 of the skilled industries in France which leaves these out 

 of consideration mast be regarded as imperfect in the 

 extreme. 



One result is however unmistakable. The artisans 

 who drew up these reports were fully alive to all the 

 advantages of which accrue to an industry from the 

 extension of labour-saving appliances, and from the dis- 

 semination of higher technical knowledge. They have 

 faithfully pointed out those departments of industry in 

 which we excel, and those in wheh we are excelled. 

 They have in most cases stated their opinions as to the 

 causes which have brought about these results. It will 

 be our own fault if we do not strengthen the weak points 

 and fill the gaps now revealed to us. The strides made 

 by some of our foreign competitors are so great as to 

 leave us no margin for indolence or wastefulness on our 

 part. The less favoured nation may more than make up 

 for the material disadvantage of having to import raw 

 products and fuel by the superior thrift and the better 

 training of its workmen. All these things point to 

 the need at home to lose no opportunity of pushing 

 forward the scientific and artistic culture of the workers 

 and of their employers, so that their training may at least 

 be not inferior to that of their Continental rivals. 



SlLVANUS P. THOMTSON 



HOir TO COLOUR A MAP WITH FOUR 

 COLOURS 

 CINCE the publication in the American Journal of 

 •^ Pure and Applied Mathematics, vol. ii. part 3, of 

 the solution of this problem obtained by me, and referred 

 to in Nature, vol. xx. p. 275, I have succeeded in 

 obtaining the following simple solution in which mathe- 

 matical formula? are conspicuous by their absence. It 

 may be premised that the problem is to show how the 

 districts of a map may be coloured with four colours, so 

 that no two districts which have a common boundary or 

 boundaries shall be of the same colour. The object of 

 this colouring being to make the division of the map into 

 districts clear without reference to boundary lines, which 

 may be confused with rivers, Sec., it is obvious that 

 nothing will be lost if districts which are remote from 

 each other, or touch only at detached points, are coloured 

 the same colour. 



The only parts of the map that it is necessary to con- 

 sider are the districts, boundaries, and points of concourse, 

 i.e., points at which boundaries terminate. Two districts 

 may have a single common boundary, or they may have 

 two or more such boundaries. Any two districts which 

 have more than one common boundary, inclose one or 

 more groups of districts ; in any one of these groups two 

 districts which have more than one common boundary 

 inclose one or more groups of districts, and so on. Pro- 

 ceeding in this way, we limit the area under consideration 

 more and more at each step, and must finally come either 

 to a group which has no pair of districts which have 

 more than one common boundary, or to a single district 

 having only two boundaries, one in common with each 

 of its two surrounding neighbours. Thus every map 

 must have at least one pair of adjacent districts which 

 have only one common boundary (ji). 



Every boundary is either continuous like a circle, or has 

 two ends which lie at the srme or at different points of 

 concourse. Every point of concourse may be called 

 triple, quadruple, &c, according to the number of lines 

 radiating from it. I expressly say lines and not boun- 

 daries, because if two ends of any boundary lie at the 

 same point of concourse two of the lines radiating from 

 the latter will belong to only one boundary. If a boundary 

 whose ends lie at two different points of concourse be 

 rubbed out, the number of lines radiating from each of 

 those points of concourse will be reduced by one, thus 

 if the two points were each triple points, they will become 

 double points, i.e., they will no longer be points of con- 

 course, the two remaining lines which radiate from each 

 becoming one boundary. The result is that rubbing out 

 a single boundary may reduce the number (B) of boun- 

 daries in the map by three. It can, however, never cause 

 a o-reater reduction, and may cause a smaller, e.g., rubbing 

 out a continuous boundary, or one which ends in two 

 quadruple points reduces the number of boundaries by 

 one only. 



Now the obliteration of the boundary /3 causes the 

 two districts it separates to become one. thus reducing 

 the number of districts (D) in the map by one, and the 

 map still remains a map, and has therefore a pair of 

 districts having only one common boundary. Obliterate 

 this common boundary, and so on. We finally get a blank 

 sheet, i.e. a single district and no boundary, and each reduc- 

 tion of D by one cannot involve a reduction of B by more 

 than three ; thus 3D must be greater than B, consequently 

 6O must be greater than 2B ; but 2B is the number which 

 would be arrived at if we counted both sides of every 

 boundary, i.e., the number which would be arrived at if 

 we counted the number of boundaries to each district 

 and added them all together ; thus the number arrived at 

 by the latter computation must be less than 6D, i.e., it is 

 impossible that every district can have as many as six 

 boundaries, i c, every map contains at least one district 

 with less than six boundaries. 



