60 GEOMETRICAL RECREATIONS [CH. Ill 



meet at a point, that point will be a crunode {i.e. a real double 

 point) on the contour-line through it, and such a point is 

 called a pass. The contour-line may enclose a region of de- 

 pression: if two such regions meet at a point, that point 

 will be a crunode on the contour-line through it, and such 

 a point is called a fork or bar. As the heights of the corre- 

 sponding level surfaces become greater, the areas of the regions 

 of elevation become smaller, and at last become reduced to 

 points: these points are the summits of the corresponding 

 mountains. Similarly as the level surface sinks the regions of 

 depression contract, and at last are reduced to points: these 

 points are the bottoms, or immits, of the corresponding valleys. 



Lines drawn so as to be everywhere at right angles to 

 the contour-lines are called lines of slope. If we go up a line 

 of slope generally we shall reach a summit, and if we go 

 down such a line generally we shall reach a bottom: we may 

 come however in particular cases either to a pass or to a fork. 

 Districts whose lines of slope run to the same summit are 

 hills. Those whose lines of slope ruu to the same bottom are 

 dales. A watershed is the line of slope from a summit to a 

 pass or a fork, and it separates two dales. A watercourse is 

 the line of slope from a pass or a fork to a bottom, and it 

 separates two hills. 



If n + 1 regions of elevation or of depression meet at a 

 point, the point is a multiple point on the contour-line drawn 

 through it; such a point is called a pass or a fork of the 

 nth order, and must be counted as n separate passes (or forks). 

 If one region of depression meets another in several places at 

 once, one of these must be taken as a fork and the rest as 

 passes. 



Having now a definite geographical terminology we can 

 apply geometrical propositions to the subject. Let h be the 

 number of hills on the earth (or an island), then there will be 

 also h summits ; let d be the number of dales, then there will 

 be also d bottoms ; let p be the whole number of passes, p x that 

 of single passes, p t of double passes, and so on ; let / be the 

 whole number of forks, f r that of single forks, /„ of double 



