THE MEASUREMENT OF TIMBER. I3I 



Mathematical Formula for finding the Perimeter of a Circle 

 or Ellipse. 



1. Circle - D- 



, E,,ip S e = d + 2 (D-d)- (15 4^L 



If trees grew uniformly and retained the same shape or form 

 throughout their lifetime no difficulty would be experienced 

 in taking measurements for calculating their volume. As a 

 matter of fact not only do trees not grow uniformly, but different 

 portions of the same stem differ in form. 



Most trees show a very rapid taper for several feet above 

 the root, and this portion resembles most nearly the form of 

 a truncated neiloid (Fig. 5). A portion of the stem above 

 this may show practically no taper, and will, therefore, partake 

 of the form of a cylinder, while the taper from that point may 

 become gradual, and then more rapid, giving the third section 

 the form of a truncated paraboloid. The top portion of the 

 stem usually resembles in shape that of a cone. Thus by 

 measuring a tree in sections, which correspond in form to the 

 contour or outline of one or other of these geometrical solids, 

 a very accurate calculation can be made of the volume of 

 timber in the tree. 



This method, however, is not practical for ordinary work, as 

 it involves the taking of too many measurements. In the great 

 majority of cases, it will be found that the stem of a true 

 resembles most closely that of a truncated paraboloid. It 

 should be noted, however, that no matter what formula or 

 combination of formulas are adopted for ascertaining the volume 

 of a tree or log, the length and sectional area at one or more 

 points are essential. The measurement of the length is an easy 

 matter, but that of the sectional area is not so. The horizontal 

 cross-section of all the geometrical solids, described above, is 

 represented by a circle, but unfortunately an exceedingly small 

 percentage of trees will give a cross-section of a true circle. 

 Only in the case of a true circle can the area be calculated 

 either from the measurement of the circumference or girth, or 

 of the diameter. With a given perimeter the circle encloses 

 a greater area than any other figure, and this being so, whenever 

 the section of a log or tree deviates in form from that of a true 

 circle an erroneous result will be obtained by calculating the 

 area from the measurement of the circumference or girth. For 



