THE ELEMENTS. BOOKS IX-X 
403 
of Pappus goes on to speak of the share of Euclid in the 
investigation: 
‘ As for Euclid, he set himself to give rigorous rules, which he 
established, relative to commensurability and incommensura 
bility in general; he made precise the definitions and the 
distinctions between rational and irrational magnitudes, he set 
out a great number of orders of irrational magnitudes, and 
finally he made clear their whole extent.’ 
As usual, Euclid begins with definitions. ‘ Commensurable ’ 
magnitudes can be measured by one and the same measure; 
‘ incommensurable ’ magnitudes cannot have any common 
measure (l). Straight lines are ‘commensurable in square’ 
when the squares on them can be measured by the same area, 
but ‘ incommensurable in square ’ when the squares on them 
have no common measure (2). Given an assigned straight 
line, which we agree to call ‘ rational ’, any straight line which 
is commensurable with it either in length or in square only is 
also called rational; but any straight line which is incommen 
surable with it (i.e. not commensurable with it either in length 
or in square) is ‘irrational’ (3). The square on the assigned 
straight line is ‘ rational ’, and any area commensurable with 
it is ‘ rational ’, but any area incommensurable with it is 
‘ irrational ’, as also is the side of the square equal to that 
area (4). As regards straight lines, then, Euclid here takes 
a wider view of ‘ rational ’ than we have met before. If a 
straight line p is assumed as rational, not only is— p also 
‘ rational ’ where m, n are integers and m/n in its lowest terms 
is not square, but any straight line is rational which is either 
commensurable in length or commensurable in square only 
with p; that is, p is rational according to Euclid. In 
the case of squares, p 2 is of course rational, and so is —p 2 ; but 
hi 
/— • p 2 is not rational, and of course the side of the latter 
v n 
square P is irrational, as are all straight lines commen 
surable neither in length nor in square with p, e.g. Va± Vh 
or (*>/k + V A). p. 
D d 2