Rational and Measurable

Pi illustrates the distinction between 'rational' and 'commensurable' suggested by Eudoxus.  It is a Ratio--between the Circumference and the square of the Radius of a Circle.  Hence, it is not literally 'irrational' qua not expressible as Ratio'. But it does consist in an incommensurability between those two, often obscured by the glossing of it as 22 : 7.  Furthermore, the incommensurability is not a characteristic of one of the two terms, but of the relation between the two.  On the standard basis that it is a characteristic of a term, if R = 1, C = Pi, and, hence, C, alone is the presumably 'irrational' or the 'incommensurable' of the two terms.  But the Ratio of 2Pi : Pi is an example of Commensurability as a relation.  Now, 'commensurability' means 'co-measurable', so the shift in terminology shifts focus from some features of Reality to the measuring of those features.  Hence, Pi mediates the measurement of R and the measurement of C, not R and C themselves.  Now, a sharper illustration of the concept of Commensurability involves a contrast of R with a alternative type of measurement of C, in which the latter is, indeed, like the former, a Natural Number, yet, the two are plainly not commensurable.  This is when the span of C is given as 360--degrees.  In this case, Incommensurability is due to two absolutely independent systems of Measurement being involved.  So, the shift from Rational to Commensurable signifies a further, more radical, perhaps earliest critique of Rationalism--that of Protagoras, i. e. that it signifies not Reality but the measuring of Reality.