axioms:
x>y or x<y or x~y

(here "<" is a binary relation)

by comparing x and copies of x to y you can deduce the interval of sizes w.r.t x in which y falls. call phi(x) the measure of x. demand phi(x) > phi(y) <=> x > y /\ phi(x) = phi(y) <=> x~y.

phi(x)/phi(y) = psi(x)/psi(y) for some other measure psi.

=> phi is a ratio scale

if you also have an arbitrary zero point => phi is an interval scale