The transfer matrix method is used to study the quantum phase transitions of the uniform and periodic anisotropic XY quantum spin chain in a transverse field, which is defined by H=-½Εn[Σx/nσx/n+1+aσy/nσy/n+1]+hσz/n. In zero temperature, it is found that the quantum phase transition point corresponds to h/J=1+a, for uniform chain (Jn=J). For periodic chain, there is more than one phase transition point at some parameter region. In case the couplings take two alternating values, with ratio g, the number of phase transition points are dependent on the parameters (a and g) and the structure of the systems. These are different from that of quantum Ising chain in a transverse field. The critical points and the conditions of their existence are obtained analytically for period-two and three chains. The results are in good agreement with numerical results. The reasons of quantum phase transitions are discussed.