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70 THE NATURE OF MATHEMATICS 

exact results are due to a compensation of errors : the error 
resulting from the false supposition made, for example, by re- 
garding a curve as a polygon with an infinite number of sides 
each infinitely small and which when produced is a tangent 
of the curve, is corrected or compensated for by that which 
springs from the very processes of the calculus, according to 
which we retain in differentiation infinitely small quantities of 
the same order alone. In fact, after having introduced these 
quantities into the calculation to facilitate the expression of 
the conditions of the problem and after having regarded them 
as absolutely zero in comparison with the proposed quantities, 
with a view to simplify these equations, in order to banish the 
errors that they had occasioned and to obtain a result perfectly 
exact, there remains but to eliminate these same quantities 
from the equations where they may still be. 

But all this cannot be regarded as a strict proof. There 
are great difficulties in trying to determine what infinitesimals 
are : at one time they are treated like finite numbers and at 
another like zeros or as "ghosts of departed quantities," as 
Bishop Berkeley, the philosopher, called them. 

Another difficulty is given by differentials " of higher orders 
than the first." Let us take up again the considerations of 

ds 

the fourth chapter. We saw that v = , and found that s was 

at 

got by integration : s = (v . dt. This is now an immediate 

ds di} 

inference, since dt = ds. Now, let us substitute for v in . 
dt dt 

Here t is the independent variable, and all of the older mathe- 
maticians treated the elements dt as constant the interval of 
the independent variable was split up into atoms, so to speak, 
which themselves were regarded as known, and in terms of 
which other differentials, ds, dx, dy, were to be determined. 
Thus 



dt dt dt dt 2 ' 

" d z s " being written for " d(da) and " dt 2 " for " (cfe) 2 ". Thus 
the acceleration was expressed as " the second differential of 

d 2 8 
the space divided by the square of dt." If ^ were constant^