VIII] THE BISECTION OF A QUADRANT 583 



34° 3^1'. That is to say, a quadrant of a circle is bisected by a 

 circular arc cutting the side and the periphery of the quadrant 

 at right angles, when the arc is such as to include (90° — 34° 38'), 

 i.e. 55° 22' of the quadrantal arc. This determination of ours is 

 practically identical with that which Berthold arrived at by a 

 rough and ready method, without the use of mathematics. He 

 simply tried various w^ays of dividing a quadrant of paper by means 

 of a circular arc, and went on doing so till he got the weights of 

 his two pieces of paper approximately equal. The angle, as he 

 thus determined it, was 34-6°, or say 34° 36'. 



(2) The position of M on the side of the quadrant OA is given 

 by the equation OM = a cosec 6 — a cot 6 ; the value oi which 

 expression, for the angle which we have just discovered, is 0-3028. 

 That is to say, the radius (or side) of the quadrant will be divided 

 by the new partition into two parts, in the proportions, nearly, of 

 three to seven. 



(3) The length of the arc MP is equal to ad cot d; and the 

 value of this for the given angle is 0-8751. This is as much as to 

 say that the curved partition- wall which we ' are considering is 

 shorter than a radial partition in the proportion of 8| to 10, or 

 seven-eighths, almost exactly. 



But we must also compare the length of this curved antichnal 

 partition-wall (MP) with that of the concentric, or perichnal, one 

 {RS, Fig. 233) by which the quadrant might also 

 be bisected. The length of this partition is 

 obviously equal to the arc of the quadrant (i.e. 

 the peripheral wall of the cell) divided by V2; 

 or, in terms of the radius, = 77/2 a/2 = 1-111. 

 So that, not only is the anticlinal partition (such 

 as we actually find in nature) notably the best, 

 but the periclinal one, when it comes to dividing 

 an entire quadrant, is very considerably larger even than a radial 

 partition. 



The two cells into which our original quadrant is now divided 

 are equal in volume, but of very different shapes; the one is a 

 triangle (MAP) with circular arcs for two of its sides, and the other 

 is a four-sided figure (MOBP), which we may call approximately 

 oblong. How will they continue to divide? We cannot say as 



