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40 THE NATURE OF MATHEMATICS 

called "the equation of the curve," and which contains, in 
germ as it were, all the properties of the curve considered. 
This constant relation between x and y is a relation like y z = 
4o#. We must distinguish carefully between a constant 
relation between variables and a relation between constants. 
We are always coming across the former kind of relation in 
mathematics ; we call such a relation a " function " of x and 
y the word was first used about fifty years after Descartes' 
Geometry was published, by Leibniz and write a function of 
x and y in general as "f(x, y)." In this notation, no hint is 
given as to any particular relation x and y may bear to each 
other, and, in such a particular function as y 1 4ax, we say 
that "the /OTTO of the function is constant," and this is only 
another way of saying that the relation between x and y is 
fixed. This may be also explained as follows. If x is fixed, 
there is fixed one or more values of y, and if y is fixed, there 
is fixed one or more values x. Thus the equation ax + by+c 
= gives one y for each x and one x for each y ; the equation 
y z 4aa; = gives two y's for each x and one x for each y. 1 

Consider the equation ax + by + c = 0, or, say, the more 
definite instance x + 2y 2 = 0. Draw axes and mark off 
points : having fixed on a unit of length, find the point je=l 
on the o;-axis, on the perpendicular to this axis measure where 
the corresponding y, got by substituting x = 1 in the above 
equation, brings us. We find y = . Take x = J, then y = % ; 
and so on. We find that all the points on the parallels to the 
7/-axis lie on one straight line. This straight line is determined 
by the equation x + 2y 2 = ; every point off that straight 
line is such that its x and y are not connected by the relation 
x + 2y 2 = 0, and every point of it is such that its x and y 
are connected by the relation x + 2^ 2 = 0. Similarly we 
can satisfy ourselves that every point on the circumference of 
a circle of radius c units of length, described round the point 
where the axes cross, is such that ic 2 + / 2 = c 2 , and every point 

1 We also denote a function of x by "f(x) " or " F(x) " or 
"0(0:)", fie. Here "/" is a sign for "function of," not for a 
number, just as later we shall find "sin" and "A" and "d" 
standing for functions and not numbers. This may be regarded as 
an extension of the language of early algebra. The equation 
y=f(x) is in a good form for graphical representation in the 
manner explained below.