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36 THE NATURE OF MATHEMATICS 

any fixed values, x becomes fixed. The signs "a" and "6" 
denote ambiguously, not uniquely like "2" does; and "x" 
does not always denote ambiguously when a and 6 are fixed. 
Thus, in the above case, when a = 2, 6= 1,- "x" denotes 
the one negative number 1. What is meant is this: In 
each member of the class of problems got by giving a and 6 
fixed values independently of one another, there is an un- 
known x, which may or may not denote different numbers, 
which only becomes known when the equation is solved. Con- 
sider now the equation ax + by = c, where a, 6, andc are known 
quantities and x and y are unknown. We can find x in terms 
of a, 6, c, and y, or y in terms of a, b, c, and x ; but x is only 
fixed when y is fixed, or y when x is fixed. Here in each case 
of fixedness of a, 6, and c, x is undetermined and " variable," 
that is to say, it may take any of a whole class of values. 
Corresponding to each x, one y belongs ; and y also is a " vari- 
able " depending on the " independent variable " x. The idea 
of " variability " will be further illustrated in the next chapter ; 
here we will only point out how the notion of what is called 
by mathematicians the "functional dependence" of y on x 
comes in. The variable y is said to be a " function " of the 
variable x if to every value of x corresponds one or more values 
of y. This use has, to some extent, been adopted in ordinary 
language. We should be understood if we were to say that 
the amount of work performed by a horse is a function of the 
food that he eats. 

Descartes also adopted the custom if he did not arrive at 
it independently advocated by Harriot of transferring all the 
terms of an equation to the same side of the sign of equality. 
Thus, instead of "x=l," "ax + b = c," and "Sx^ + g^hx." 
we write respectively " x 1 = 0," " ax + (b c) = 0," and 
"3a; 2 hx j rg = 0." The point of this is that all equations of 
the same degree in the unknown we shall have to consider cases 
of more unknowns than one in the next chapter that is to say, 
equations in which the highest power of x (x or a; 2 or x 3 . . . .) 
is the same, are easily recognisable. Further, it is convenient 
to be able to speak of the expression which is equated to 0, 
as well as of the equation. The equations in which a; 2 , and no 
higher power of a;, appears are called "quadratic" equations 
the result of equating a " quadratic " function to ; those in