
Definitions 
Brief comments 
1 
Commensurable magnitudes are said to be those measured by the
same measure, 


and incommensurable those for which it is not possible for anything
to become a common measure of them. 

2 
Straightlines are commensurable in power whenever the squares
on them are measured by the same area, 


and incommensurable whenever it is not possible for any area
to become a common measure of the squares on them. 

3 
Given these suppositions it is proved that to a proposed line there
exist straightlines infinite in number which are commensurable and incommensurable,
some in length only, others in power. 
Note that what preceeds are called suppositions and that this states
a theorem. 

And so let the proposed straightline be called 'rational',
and those commensurable with this line, whether in length and in power
or in power alone be called 'rationals,' 


and let those which are incommensurable with this be called 'irrationals'. 

4. 
And let the square on the proposed line be called 'rational'
and those commensurable with this 'rationals', 


and those incommensurable with this 'irrationals', and let those
whose power they are be called 'irrational', if they are squares,
the sides themselves, and if they are some other rectilinear figure, those
which describe the squares equal to them. 
