Mr. Wiseman on Timber-measuring. 205 



" sliieJd of protection to designing knavery," and the practice 

 of "legerdemain," Jiave long since fallen plmiip "to the 

 ground." 



I would here just observe, that timber so tapering that one 

 diameter is above triple the other, is rarely found in practice. 

 A tree may occur now and then, having that disparity of dia- 

 meters ; but it usually happens that the tree naturally divides 

 Itself by sudden swells into two or more distinct parts, and 

 consequently must "be measured" dtiobiis vel pluribus frustis. 



I will supply the investigation of the principle, as relates to 

 round timber, to which the abovementioned rJidin^ rule is 

 adapted. 



Let s denote half the sum, and d half the difference of the 

 extreme diameters of any tapering piece of timber; / the 

 length, and p be = -7854. 



Then, the diameters themselves being s + d and 5— rf, the 



formula ^ pi x {s + df + (s-d)' + x/s+d'x s-d' = ^pl x 



2s^ + 2d' + s' -,d'=Xpi^ 3?+rf^ = pi s^-+^pl d' expresses (as 

 is well known) the true content. In which (s being the dia- 

 meter at the midlength) 2^1 s^ is the cylindrical content, and 

 i pi d- the difference between that and the true content. Now 

 since 13-5405 in inches, is the diameter of that cylinder whose 

 length and content in feet are always equal to each other, and 

 since the contents of cylinders of the same length are in the 

 duplicate ratio of their diameters; therefore 13-5405- (the 

 square of the diameter of the standard cylinder) : / (its content) 

 : : 52 (the square of the given diameter) : the cylindrical con- 

 tent required; and in the same way 13-5405" : ^ /:: rf^ • the 

 difference, which added to the cylindrical, will give the true 

 conical content. Now two logarithmic lines, sliding one 

 upon the other as in sliding rules, one being of a double 

 r:.<i us .V .li respect to the other, will, it is plain, solve these 

 analogies by an operation as short and easy as in the old 

 mode; that is, the length (/) on one line being set to 13-54 on 

 the other (usually called the girt) line, the middle diameter (s) 

 on this last line will stand against the cylindrical content on 

 the first line; and one-third of the length {^l) on the first line 

 being set to 13-54 on the other, the half difference of the two 

 extreme diameters [d] on the second line will stand a^rainst 

 the quantity on tlie first line, which added, makes the tnie 

 content. 



In a similar way the true content can be obtained by this 

 sliding rule, when, instead of the diameters, the corresponding 



quarter 



