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 that is why other methods have been developed, which approach the

problem from different angles. In this thesis we extend a method of Schoof that was

designed for real cyclotomic fields of prime conductor to real cyclotomic fields of

conductor equal to the product of two distinct odd primes. Our method calculates

the index of a specific group of cyclotomic units in the full group of units of the field.

This index has h+ as a factor. We then remove from the index the extra factor that

does not come from h+ and so we have the order of h+. We apply our method to

real cyclotomic fields of conductor < 2000 and we test the divisibility of h+ by all

primes < 10000. Finally, we calculate the full order of the l-part of h+ for all odd

primes l < 10000.