PAPERS ON GEOLOGY AND GEOGRAPHY 195 



A GRAPHICAL METHOD OF DETERMINING THE 

 AVERAGE INCLINATION OF A LAND SUR- 

 FACE FROM A CONTOUR MAP 



John L. Rich, University of Illinois 



One of the most significant elements of geographic environ- 

 ment is the slope or inclination of the land surface. On it de- 

 pends, in a large measure, the rate of run-off of the precipita- 

 tion; the rate of erosion; indirectly, the character of the soil; 

 and finally, the possible utilization of the land for agriculture. 

 In connection with problems of water supply and Hood con- 

 trol, a determination of the average inclination of the land sur- 

 face is of vital importance to the engineer. 



In geographical descriptions of a region we seldom find any 

 quantitative mention of the inclination of the surface. The 

 more recent descriptions commonly state the stage in the ero- 

 sion cycle, and some of them give an indication of the average 

 relief, but these are not sufficient, in all cases, to convey a defi- 

 nite idea of the topography. The description of such a re- 

 gion would, obviously, be improved by adding to the discus- 

 sion of its stage in the erosion cycle and to the mention of its 

 average relief a statement of the average inclination of its 

 surface. 



By the average inclination of the surface of a region is 

 meant an average in which the inclination of each individual 

 portion is weighted in proportion to the ratio which its area 

 bears to that, of the whole. Stated mathematically, the for- 

 mula 1 is : 



B = Bigl + B & ■ • + B gn 



G G G 



where B is the average inclination of the region; B a , B 2 , etc., 

 the average inclination of the individual portions; g t , g 2 , the 

 areas of the individual portions ; and G the whole area. 



The principle heretofore used for obtaining such a weighted 

 average is that published in 1890 by Finsterwalder. 2 This 

 writer shows that the sum of the lengths of the contour lines 

 on a map multiplied by the contour interval, reduced to like 

 units, and divided by the area gives a result which is as near 



1 Penck, A., "Morphologic der Erdoberflaeche," I. Buch, s 47. Stuttgart, 1894. 



2 Finsterwalder, S., "I'ebcr den mittleren Boschungswinkel und das wahre Areal 

 einer topographischen Fiaeche." Sitzber. K. Ak. der Wiss., Math.-Phys. Kl., 

 XX, 1890, s 3S-S2. 



