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GROWTH OF MATHEMATICAL SCIENCE 21 

start with a line of any length, there are other lines whose 
lengths do not bear to the first length the ratio of one integer 
to another, no matter if we have all the integers to choose 
from. Of course, any two fractions have the ratio of two 
integers to one another. In the above case of the diagonal, if 
the diagonal were in length some number x of units, we should 
have a; 2 = 2, and it can be proved that no fraction, when 
" multiplied " in the sense to be given in the next chapter 
by itself gives 2 exactly, though there are fractions which give 
this result more and more approximately. 

On this account, the Greeks drew a sharp distinction be- 
tween "numbers," and "magnitudes'' or "quantities" or 
measures of lengths. This distinction was gradually blotted 
out as people saw more and more the advantages of identifying 
numbers with the measures of lengths. The invention of 
analytical geometry, described in the third chapter, did most 
of this blotting out. It is in comparatively modern times 
that mathematicians have adequately realised the importance 
of this logically valid distinction made by the Greeks. It is 
a curious fact that the abandonment of strictly logical thinking 
should have led to results which transgressed what was then 
known of logic, but which are now known to be readily in- 
terpretable in the terms of what we now know of Logic. This 
subject will occupy us again in the sixth chapter. 

The question of loci is connected with geometrical analysis, 
and is difficult to dissociate from a mental picture of a point 
in motion. Think of a point under certain restrictions, so that 
it cannot move in more than two directions at any instant, and 
so can only move in some curve. Thus, a point may move so 
that its distance from a fixed point is constant ; the peak of 
an angle may move so that the arms of the angle pass 
slipping through two fixed points, and the angle is always a 
right angle. In both cases the moving point keeps on the 
circumference of a certain circle. This curve is a " locus." It 
is evident how thinking of the locus a point can describe may 
help us to solve problems. 

We have seen that Thales discovered that a triangle is 
determined if its base and base angles are given. When we 
have to make a survey of either an earthly country or part of