604 
Proceedings of the Royal Society 
reach to a thousand million of units in the twentieth place; that is 
to say, with five thousand million of terms we shall not he sure of 
the result in the tenth, place. 
Thus it is clear that M. Callet did not get his numbers by the 
summation of the series ; he must have got them elseways. These 
series, though interesting and valuable, are useless in calculation; 
they are not even established by the arguments adduced. The logic 
is as faulty as the logistics. 
When a polynome consists of a finite number of terms, each a 
multiple of an integer positive power of some variable x, and when 
the substitution of a definite value r for x renders that polynome 
zero, it is easy to show that the expression is divisible by the differ- 
ence X - r, the quotient being another polynome one degree lower in 
rank. 
The proof of this theorem rests essentially on the finitude of the 
expression; the very idea of divisibility implies a termination. 
Hence in extending this law to interminate progressions we destroy 
the foundation on which the argument rests. Granted the use of 
unending series, we may divide any finite polynome by any binome, 
the quotient being an endless progression ; there is now no indivisi- 
bility, and the adjective divisible ceases to have a meaning. It is 
absurd to say that the progression 
1 . . . 6 
+ &c. 
is divisible by 1 - ; the result of the division by any binome 
of the form 1 - ax'^ will be a progression arranged according to the 
ascending even powers of x. If we divide the quotient by another 
binome I - px‘^, the new quotient again by 1 - yx^^, and so on, we 
shall find no obstacle to the divisibility except the labour of the 
operation. Shall we thence argue that the original series is the 
continued product of — ax‘^){l - - yxf Truly not, for 
that would be to omit from consideration the essential item, the 
quotient resulting from the divisions. It is necessary to show that 
the successive divisions of the series for the cosine by 1 - 
