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86 THE NATURE OF MATHEMATICS 

in a thought-out form seems quite tenable ; in a crude form, 
it seems like excluding the number 2, as distinguished from 
classes of two things, from notice as a figment of the brain, 
because it is not visible or tangible like a poker or a bit of 
radium or a mutton-chop. 

In fact, a perfected geometry reduces to a series of deduc- 
tions holding not only for figures in space, but for any abstract 
things. Spatial figures give a striking illustration of some 
abstract things ; and that is the secret of the interest which 
analytical geometry has. But it is into algebra that we must 
now look to discover the nature of Mathematics. 

We have seen that Egyptian arithmetic was more general 
than Egyptian geometry : like algebra, by using letters to 
denote unknown numbers, it began to consider propositions 
about any numbers. In algebra and algebraical geometry this 
quickly grew, and then it became possible to treat branches of 
mathematics in a systematic way and make whole classes of 
problems subject to the uniform and almost mechanical working 
of one method. Here we must again recall the economical 
function of science. 

At the same time as this rapid growth of methods algebra 
and analytical geometry and the infinitesimal calculus grew 
up from the application of mathematics to natural science, the 
new conceptions which influenced the form which mathematics 
took in the seventeenth, eighteenth, and nineteenth centuries. 
The ideas of variable and function became more and more 
prominent. These ideas were brought in by the conception of 
motion, and, unaffected by the doubts of the few logicians in 
the ranks of the mathematicians, remained to fructify mathe- 
matics. When mathematicians woke up to the necessity of 
explaining mathematics logically and finding out what Mathe- 
matics is, they found that, in mathematics the striving for 
generality had led, from very early times, to the use of a 
method of deduction used but not recognised and distinguished 
from the method usually used by the Aristotelians. I will try 
to indicate the nature of these methods, and it will be seen 
how the ideas of variable and function, in a form which does 
not depend on that particular kind of variability known as 
motion, come in. 

A proposition in logic is the kind of thing which is denoted