346 THE PHILOSOPHY OF BIOLOGY 



and 



og' 



proportional to the sum of the tangents — and — , 



oe og 



divided by 2. 



THE NOTION OF THE LIMIT 



Suppose that we wish to find the rate of variation 

 of volume for a pressure change in the immediate 

 vicinity of the value bi, that is, the rate of variation as 

 the pressure changes from a little less than ^i to a little 

 more than bi. If we find the point b on the curve 

 corresponding to b}, and if we then draw a line ^,, 

 touching the curve at the point b, we shall obtain the 

 angle 0^1. It might appear now that the tangent of 



this angle, that is, the ratio -^, would give us a measure 



of the rate of variation of volume. 



But the reasoning would be faulty. The line ^i 

 only touches the curve, it does not coincide with an 

 element of the curve. Also at the point bi the pressure 

 has a certain definite value, and there is no change. At 

 the corresponding point b^ the volume also has a 

 certain definite value, and there is no change. There 

 can therefore be no rate of variation. The value of 

 the tangent does not give us a measure of the rate of 

 variation : it gives us the limit to the rate of varia- 

 tion, when the pressure is changing in the immediate 



vicinity of bi. 



We must stick to the notion of a pressure change 

 in the immediate vicinity of b^. What do we mean by 

 " immediate vicinity " ? We mean that we are think- 

 ing of a range of pressure- values in which the particular 

 pressure-value b^ is contained, but not as an end-point. 

 We mean also that we choose a definite standard of 

 approximation to the value b^, so that any pressure- 

 value within our interval differs from b^ by less than this 



