APPENDIX 347 



standard of approximation. It means further that, no 

 matter how small is the number representing this 

 standard of approximation, any pressure- value within 

 the interval will differ from hi by less than this number. 

 This is what we really mean when we say that the 

 interval we are thinking about is an " infinitely small 

 one." 



Now corresponding to this interval of pressure- 

 values in the immediate vicinity of h^, there will be an 

 interval of volume-values in the immediate vicinity of 

 6n, and, as before, any one of these volume- values will 

 differ from h^ by less than any number representing 

 a standard of approximation to b^. We then find the 

 point on the curve corresponding to both h^ and 6,1, 

 that is h, and v/e draw the line^i, and find the tangent 

 of the angle which this line makes with op. The value 

 of this tangent is the limit of the rate of variation of 

 the volume of the gas when the pressure undergoes a 

 change in the immediate vicinity of h-^. 



" Rate of variation " is a function of the argument 

 " pressure." This function has the limit I for a value 

 of its argument b^, when, as the argument varies in the 

 immediate vicinity of b^, the value of the function 

 approximates to / within any standard whatever of 

 approximation.^ 



We should not, of course, find the rate of variation 

 of volume of the gas by this means. We should calcu- 



late the value of the differential co-efficient ^- from 



dp 



the equation pv=k{i-\-at) : this would be - ^ ^ f' . 



P 

 But the reasoning involved in the methods of the 



calculus are those which we have attempted to outline. 



^ If the reader does not understand this, he should read Whitehead's 

 " Introduction to Mathematics." He should read this book in any case. 



