368 THE FOKMS OF TISSUES [ch. 



narrower margin of its quadrilateral portion; and these incre- 

 ments will be in proportion to the angles of arc, viz. 55° 22' : 34° 38', 

 or as '96 : -60, i.e. as 8 : 5. And accordingly, if we may assume 

 (and the assumption is a very plausible one), that, just as the 

 quadrant itself divided into two halves after it got to a certain 

 size, so each of its two halves will reach the same size before 

 again dividing, it is obvious that the triangular portion will be 

 doubled in size, and therefore ready to divide, a considerable 

 time before the quadrilateral part. To work out the problem in 

 detail would lead us into troublesome mathematics ; but if 

 we simply assume that the increments are proportional to the 

 increasing radii of the circle, we have the following equations :- — 



Let us call the triangular cell T, and the quadrilateral, Q 

 (Fig. 151) ; let the radius, OA, of the original quadrantal cell 

 = a = 1 ; and let the increment which is required to add on a 

 portion equal to T (such as PP'A'A) be called x, and let that 

 required, similarly, for the doubling of Q be called x'. 



Then we see that the area of the original quadrant 



^T +Q = l7ra2 = .7854a^ 

 while the area of T ^Q= •S927a^. 



The area of the enlarged sector, p'OA', 



= {a + xY X (55^ 22') -^ 2 = -4831 (a + xf, 

 and the area OP A 



= a2 X (55° 22') -^ 2 = •4831a2. 



Therefore the area of the added portion, T', 



= -4831{(a + a;)2-a2}. 



And this, by hypothesis, 



= T = •3927a2. 



We get, accordingly, since a = 1, 



a:2 + 2a; - •3927/-4831 = -810, 

 and, solving, 



a; + 1 = VbSi = 1-345, or a; - 0-345. 



Working out x' in the same way, we arrive at the approximate 

 value, a;' + 1 = 1-517. 



