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28 THE NATURE OF MATHEMATICS 

nowadays quite sufficiently for our present purpose. Now, 
one of the rules of algebra is that any term can be taken from 
one side of the sign " = " to the other if only the " + " or " " 
belonging to it is changed into " " or " + ," as the case may 
l)e. Thus, in the present case, we have: "x = 33 + 8 = 41." 
This absurdly simple case is chosen intentionally. It is essential 
in mathematics to remember that even apparently insignificant 
economies of thought add up to make a long and complicated 
calculation readily performed. This is the case, for example, 
with the convention introduced by Descartes of using the last 
letters of the alphabet to denote unknown numbers, and the 
first letters to denote known ones. This convention is adopted, 
with a few exceptional cases, by algebraists to-day, and saves 
much trouble in explaining and in looking for unknown and 
known quantities in an equation. Then, again, the signs 
" +," " ," " = " have great merits, which those unused to 
long calculations cannot so readily understand. Even the 
saving of space made by writing " xy " for " x x y " (" x multi- 
plied by y ") is important, because we can obtain by it a shorter 
and more readily surveyed formula. Then, too, Descartes made 
a general practice of writing " powers " or " exponents " as we 
do now ; thus "x 5 " stands for "xxx" and "x 5 " for some less 
suggestive symbol representing the continued multiplication of 
five x's. 

One great advantage of this notation is that it makes the 
explanation of logarithms, which were the great and laborious 
discovery of John Napier (1550-1617), quite easy. We start 
from the equation "x m x n = x m+n ." Now, if yf y, and we cull 
p the " logarithm of y to the base x " ; in signs : "p = log z y " ; 
the equation from which we started gives, if we denote x m by " u " 
and x n by "v," so that ra = log x u and n = log x v, that log.,, (uv) = 
\og x u + log x v. Thus, if the logarithms of numbers to a given 
base (say x = 10) are tabulated ; calculations with large numbers 
are made less arduous, for addition replaces multiplication, 
when logarithms are found. Also subtraction of logarithms 
gives the logarithm of the quotient of two numbers. 

Let us now shortly consider the history of algebra from 
Diophantus to Descartes. 

The word "algebra" is the European corruption of an