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44 THE NATURE OF MATHEMATICS 

arily for the number of square units in a square whose sides are x 
units in length, but there is no necessity in this. We shall often use 
the latter kind of measurement in the fourth and fifth chapters. 
The equation of a straight line can be made to satisfy two 
given conditions. We can write the equation in the form 



and thus have two ratios, and -, that we can determine 

a a 

according to the conditions. The equation ax + by + c = has 
apparently three "arbitrary constants," as they are called, but 
we see that this greater generality is only apparent. Now we 
can so fix these constants that two conditions are fulfilled by 
the straight line in question. Thus, suppose that one of these 
conditions is that the straight line should pass through the 
origin the point (0, 0). This means simply that when x = 0, 
then 2/ = 0. Putting, then, x = Q and y = in the above 

M 

equation, we get - = 0, and thus one of the constants is 
a 

determined. The other is determined by a new condition that, 
say, the line also passes through the point (, 2). Substituting, 

then, in the above equation, we have, as - = 0, as we know 

a 

already, + = 0, whence - = . Hence the equation of 
a a 



the line passing through (0, 0) and (, 2) is x \y = 0, or y Qx. 
Instead of having to pass through a certain point, a condition 
may be, for example, that the perpendicular from the origin on 
the straight line should be of a certain length, or that the line 
should make a certain angle with the awixis, and so on. 

Similarly, the circle whose equation is written in the form 



is of radius c and centre (a, 6). It can be determined to pass 
through any three points, or, say, to have a determined length of 
radius and position of centre. Fixation of centre is equivalent 
to two conditions. Thus, suppose the radius is to be of unit 
length: the above equation is (a; a) z + (y 6) 2 = 1. Then, 
if the centre is to be the origin, both a and 6 are determined to 
be 0, and this may be also effected by determining that the circle 
is to pass through the points (, 0) and ( - , 0), for example.