  The class numbers $h^{+}$ of the real cyclotomic fields are very hard to
compute. Methods based on discriminant bounds become useless as the conductor
of the field grows and methods employing Leopoldt's decomposition of the class
number become hard to use when the field extension is not cyclic of prime
power. This is why other methods have been developed which approach the problem
from different angles. In this paper we extend one of these methods that was
designed for real cyclotomic fields of prime conductor, and we make it
applicable to real cyclotomic fields of conductor equal to the product of two
distinct odd primes. The main advantage of this method is that it does not
exclude the primes dividing the order of the Galois group, in contrast to other
methods. We applied our algorithm to real cyclotomic fields of conductor $<$
2000 and we calculated the full order of the $l$-part of $h^{+}$ for all odd
primes $l$ $<$ 10000.
