198 ILLINOIS ACADEMY OF SCIENCE 



its area. To a certain extent this may be done by judgment 

 alone, but some empirical method which, so far as possible, 

 eliminates the personal equation is preferable. A method 

 which gives good results and is here recommended, is to divide 

 the map into large or small squares of equal area, and, by the 

 ordinary method, run as many random lines across each square 

 as is necessary to yield a reliable average of the inclination of 

 its surface. Add all distances graphically along the paper edge 

 and obtain the sum of the ups and downs in the usual way. It 

 is essential that the same number of lines be run in each square, 

 otherwise those in which the most lines are run receive the 

 greatest weight in the average. This method should yield re- 

 sults with an error of 5 per cent or less, the accuracy de- 

 pending on the care with which the work is done and the size 

 of the squares used as units. The accuracy thus obtainable 

 compares favorably with that obtained by Finsterwalders' 

 method of measuring the lengths of the contours. 



In judging the character of the net of profiles necessary to 

 obtain the average slope of an area it should be borne in mind 

 that a single measurement of a truly conical hill, or of one in 

 which the slopes are uniform, gives a correct average, but that 

 on a hill or valley head of circular form which has a parabolic 

 profile, with the steeper slope at either top or bottom, the 

 results are in error on account of the inequality of the areas 

 having the various degrees of slope. 



On most examples of mature or old topography the ma- 

 jority of the slopes are parabolic, but since, in general, the 

 valley heads are the inverse of the spurs, the errors due to the 

 parabolic curve of the surface profiles tend to balance each 

 other where both valley heads and spurs are included in the 

 measurements. 



For certain determinations, however, such as that of the 

 average inclination of a conical mountain, e. g., a volcano, 

 having a parabolic profile, a special method of compensation 

 must be employed. This may be accomplished by drawing 

 concentric circles round the center of the mountain with radii 

 of 1, 2, 3, 4, 5, 6, 7, 8, etc., and increasing the number o<f 

 measurements in each succeeding belt in proportion to its 

 area, namely, (if the area of the first belt be taken as one) for 

 the first belt, one measurement ; for the second, three ; for the 



