£20 On the Mensuration of Timber. 



For example : suppose a hewn tree of any length, and 

 that when callipered in the middle, on the angles or cor^ 

 pers, or in the directions Bb and Bl>, fig. 7, the same are 

 ghown by -the rule to be $6\ and 25^ inches ; add these to-* 

 gether and take the half thereof, and we have 26 inches for 



the 

 i a 



The mathematical reader will readily perceive the mode 

 of calculating the content of a tree, either by the round or 

 feirt method or by the square or calliper method, to be the 

 same, except in determining the area of the section at the 

 girting place ; and that the numbers in the second column 

 of the above table, expressing the content square measure 

 jn different cases when the round measure is unity, do ex- 

 press also the ratio of the areas of the girting or middle sec- 

 tions in each case: thus, in the first line of the table -/O -f 

 An . . ir , 3-14 15Q3 X 3-141593 



'70 5= -49 is the calliper area, and -— — — x 



r 4x4 



•6168503 is the girt area; whence, as -6168503 : 1 : : -49 : 

 '7943579 the number in column the second : and thu« the 

 numbers answering to *71, *72, :J3 9 &c. were determined. 

 In the second line of the table, as there is to be a decrease 

 of 1 5th or 2-10ths in the content, we have given -8 for 

 the number in the second column; whence, as 1 : "6168503 

 ; : -8 : -4934802 the calliper area, whose square root is 

 \ 70248 15 as in the first column ; and thus, when there is a 

 decrease of l-6th, l-7th, or an increase of 1-1 Oth, l-9th, 

 Sec. are the numbers determined. 



Let fig. 8 represent one quarter of the end of a hewn tree, 

 (as fig. 7 delineated the whole end,) draw the line CG, 

 making the angle ECG (= GCH) = 45°, and join FC; 

 then in the 20th line of the table, since the side is to be 

 " l -5th of the perimeter," half the side (or DF) is to be 

 equal to 4-5ths of the l-8th part of the perimeter (or DF 

 4- FG), and DF is equal to 4FG; whence we have re- 

 quired to divide an angle of 45° into two such parts, that 

 the sine of the greater part may be equal to four times the 

 arc of the lesser part. By the help of Dr. Button's or Cal- 

 let's tables, and the method of trial and error, the greater 

 angle will in this case be found = 36° 28' 59*42" = DCF, 

 whose natural cosine is -8040316, the number in the first 

 column ; and in like manner we proceed when the side is 

 to be l-6th, I -7th, &c. of the perimeter. The methods of 

 procedure in the remaining cases are sufficiently evident. 

 It may, however, be proper to state, that the prices being 

 reciprocally as the quantities, we have, in the second line, as 

 .8 : 1 : : 100 : 125 shillings per load, as in the last column. 



