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ANALYTICAL GEOMETRY 47 

of space occupied by the end of a pen or even an " atom " of 
matter is merely a mark of position. We cannot, then, 
speak of a point moving ; the very essence of point is to be 
position. The motion of a point of space, as distinguished 
from a point of matter, is a fiction, and is the supposition that 
a given point can be now one point and now another. Motion, 
in the ordinary sense, is only possible to matter and not to 
space. Thus, when we speak of a " variable position," we are 
speaking absurdly if we wish our words to be taken literally. 
But we do not really wish so when we come to think about it ; 
what we are doing is this : we are using a picturesque phrase 
for the purpose of calling up an easily imagined thought which 
helps us to visualize roughly a mathematical proposition which 
can only be described accurately by a prolix process. The 
ancient Greeks allowed prolixity, and it was only objected to 
by the uninitiated. Modern mathematics up to about sixty 
years ago successfully warred against prolixity ; hence the 
obscurity of its fundamental notions and processes and its great 
conquests. The great conquests were made by sacrificing very 
much to analogy : thus, entities like the integer 2, the ratio 2/1, 
and the real number which is denoted by " 2 " were identified, 
as we have seen, because of certain close analogies that they 
have. This seems to have been the chief reason why the 
procedure of the mathematicians has been so often condemned 
by logicians and even by philosophers. In fact, when mathe- 
maticians began to try to find out the nature of mathematics, 
they had to examine their entities and the methods which they 
used to deal with them, with the minutest care, and hence to 
look out for the points when the analogies referred to break 
down, and distinguish between what mathematicians had 
usually failed to distinguish. Then the people who do not 
mind a bit what mathematics is, and are only interested in 
what it does, called these earnest inquirers "pedants" when 
they should have said "philosophers," and "logic-choppers" 
whatever they may be when they should have said " logicians." 
We have tried to show why ratios or fractions, and so on, are 
called " numbers," and apparently said to be something which 
they are not ; we must now try to get at the meaning of the 
words " constant " and " variable." 

By means of algebraic formulae, rules for the reconstruction 

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