MEASUREMENT OF T I MB EX. 59 



equal length, when the contents of the thicker 

 piece, measured on this method, gives 9 X 4 25 = 



14^6- ft., or more than the whole- tree measured in 

 one piece. 



I observe that Hopper, in his introduction to his 

 Tables for Timber Measurement, gives a similar 

 instance as an illustration of the peculiarity of 

 timber; but the peculiarity is in the system of 

 measurement, which is based on the erroneous 

 assumption that the mean area of a pyramid 

 or cone is at the middle, as it is in the wedge 

 form B. 



The common mathematical rule for finding the 

 contents of a cone or pyramid is to multiply the 

 length by one-third the area of base. 



To get at the contents of the frustrum of a cone 

 the rule is : Add together the area of the greater 

 and lesser ends, and the mean proportional area, 

 and divide the sum by 3 for the mean area. 

 (The mean proportional is obtained by multiplying 

 the greater by the lesser diameter, and the product 

 by -7854.) For example, the lower half of the 

 cone E is 25 feet long, with 12 in. greater, and 6 in. 

 lesser diameter; then i* + + i^xex x -7864 



= 11 *45, contents of the log. 



By Hopper's method we have the square of the 

 quarter girth (7 2 ) ; then 4 =8*50, showing a 



