  It is shown that the sum of class numbers of orders in totally complex
quartic fields with no real quadratic subfield obeys an asymptotic law similar
to the prime numbers, as the bound on the regulators tends to infinity. Here
only orders which are maximal at a given set of primes containing an even
number of elements are considered. The proof is accomplished by means of a
Prime Geodesic Theorem for symmetric spaces formed as compact quotients of
SL(4,R). This result extends work of Sarnak in the real quadratic case and of
Deitmar and Hoffmann in the complex cubic case.
