VIII] THE BISECTION OF A QUADRANT 589 



matics; but if we simply assume that the increments are propor- 

 tional to the increasing radii of the circle, we have the following 

 equations : 



Call the triangular cell T, and the quadrilateral Q (Fig. 237); 

 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 doubhng of Q be called x' . 



Then we see that the area of the original quadrant 



= T + Q = Itto" = 0-7854a2, 



while the area of T =Q = 0-3927a2. 



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



= {a + x)^ X (55° 22') ^ 2 = 0-4831 (a + x)^, 



and the area OP A 



= a^x (55° 22') -^ 2 = 0-4831a2. 



Therefore the area of the added portion, T\ 



= 0-4831 {{a + x)^ - a^}. 



And this, by hypothesis, 



= T = 0-3927a2. 



We get, accordingly, since a = I, 



x^ + 2x= 0-3927/0-4831 = 0-810, 

 and, solving, 



x-{-l = Vrsi = 1-345, or x^ 0-345. 



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

 value, x' + 1 = 1-517. 



This is as much as to say that, supposing each cell tends to 

 divide into two halves when (and not before) its original size is 

 doubled, then, in our flattened disc, the triangular cell T will tend 

 to divide ,when the radius of the disc has increased by about a 

 third (from 1 to 1-345), but the quadrilateral cell, Q, will not tend 

 to divide until the Hnear dimensions of the disc have increased by 

 fully a half (from 1 to 1-517). 



