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VIEWS OF LIMITS AND NUMBERS 79 

The logical scrutiny to which, during the last century, the 
processes and conceptions of mathematics have been subjected, 
showed very plainly that it was a sheer assumption that such 
a process as 1*4142 . . ., though all its terms are less than 2, 
for example, has any limit at all. When we replace numbers 
by points on a straight line, we feel fairly sure that there is 
one point which behaves to the points representing the above 
sequence in the same sort of way as 2 to the sequence 1, 1 + , 
1 + & + i Now, if our system of numbers is to form a 
continuum as a line seems to our thoughts to be, so that we 
can affirm that our number system is adequate, when we in- 
troduce axes in the manner of analytical geometry, to the 
description of all the phenomena of change of position which 
take place in our space, 1 then we must have a number ^2 which 
is the limit of the sequence 1*4142 . . . if 2 is of the series 
1 + i + i + , for to every point of a line must correspond 
a number which is subject to the same rules of calculation as 
the ratios or integers. Thus we must, to justify from a logical 
point of view our procedure in the great mathematical methods, 
show what irrationals are, and define them before we can prove 
that they are limits. We cannot take a series, whose law is 
evident, which has no ratio for sum, and yet such that the 
terms of the corresponding sequence all remain less than some 

fixed number (such as l + 7+T^+T7o^+f7o^4+ i when 

all the terms of the corresponding sequence are less than 3, 
for example), and then say that it "defines a limit." All 
we can prove is that if such a series has a limit, then, if the 
terms of its corresponding sequence do not decrease as we read 
from left to right (as in the preceding example), it cannot have 
more than one limit. 

Some mathematicians have simply postulated the irrationals. 
At the beginning of their discussions they have, tacitly or not, 

1 The only kind of change dealt with in the science of mechanics 
is change of position, that is, motion. It docs not seem to me to 
be necessary to adopt the doctrine that the complete description of 
any physical event is of a mechanical event ; for it is possible to 
assign and calculate with numbers of our number-continuum to 
other varying characteristics (such as temperature) of the state of 
a body besides position. 

F