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MODERN MATHEMATICS ALGEBRA 31 

class of beasts of burden," the sign " ab " has been used to 
denote " the class of horned beasts of burden." We see that 
here ab =6a, just as in the multiplication of numbers, and the 
above operation has been called, partly for this reason, " logical 
multiplication," and denoted in the above way. Here we meet 
the practice of mathematicians and of all scientific men of 
using words in a wider sense for the sake of some analogy. 
This habit is all the more puzzling to many people because 
mathematicians are often not conscious that they do it, or even 
talk sometimes zs if they thought that they were generalising 
conceptions instead of words. But, when we talk of a "family 
tree," we do not indicate a widening of our conception of trees 
of the roadside. 

We shall not need to consider these modern algebras, but 
we shall be constantly meeting what are called the " generalisa- 
tions of number" and transference of methods to analogous 
cases. Indeed, it is hardly too much to say that in this lies 
the very spirit of discovery. An example of this is given by 
the extension of the word " numbers " to include the names of 
fractions as well. The occasion for this extension was given 
by the use of arithmetic to express such quantities as distances. 
This had been done by Archimedes and many others, and had 
become the usual method of procedure in the works of the 
mathematicians of the sixteenth century, and plays a great 
part in Descartes' work. 

Mathematicians, ever since they began to apply arithmetic 
to geometry, became alive to the fact that it was convenient 
to represent points on a straight line by numbers, and numbers 
by points on a straight line. What is meant by this may be 
described as follows. If we choose a unit of length, we can 
mark off points on a straight line corresponding to units 
which means that we select a point, called "the origin," to 
start from, 1 unit, 2 units, 3 units, and so on, so that " the 
point m," as we will call it for short, is at a distance of m 
units from the origin. Then we can divide up the line and 
mark points corresponding to the fractions , , f, y 1 ^, , 
or the point between 1 and 2 which is the same distance 
from 1 as f is from 0, and so on. Now, there is nothing 
here to distinguish fractions from numbers. Both are treated 
exactly in the same way ; the results of addition, subtraction, 

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