610 
Froceedings of the Royal Society 
comes close up to A, and at the same time, e to O. Hence for a? = 1 , 
the area eOG/ becomes zero, and thus the ultimate area AOG . . . B , 
has for its value the constant C less the limiting value of the 
rectangle under the two logarithms ee and EC. 
How on halving the minute distance eO , we render ec rather less 
than half ; hut instead of doubling the line EC , when close to AB , 
we only augment its length by KN , the logarithm of 2. And thus 
the rectangle EC . ec is almost exactly halved; the halving being more 
and more nearly exact the more minute AE is made ; hence the limit- 
ing value of EC . ec is zero, and the constant C is nothing else than 
the representation of the area AOG . . . B , or of the sum of the 
series of inverse squares ; that is 
C = 1 -64493 40668 = {- + ^ ^ ^ + &c. 
How the square of the number tt is 9.86960 44011, the sixth part 
of which is exactly C, so that 
7t2^L L L L 
+ 52+02+ 
This process, although it bring out the agreement of the value of 
the sum of the series with that of the sixth part of the square of tt, 
is unsatisfactory, because it gives us a mere arithmetical coincidence, 
without showing why that coincidence should be. 
