In a series of papers Burgers introduced various mathematical models to illustrate some aspects of turbulence. The major paper initiating this endeavour seems to be forgotten and Burgers' name is only attached to the nonlinear equations v_t+2vv_y-v_yy=0, that model free turbulence. The original equations that Burgers proposed describe the phenomena of energy transfer from large scales to small scales. In this paper the author derives an estimate for the dimension of the attractor which is of the same order as the square root of a Reynolds number. This rigorous estimate improves the one obtained by Burgers without omitting any lower-order terms. He also shows that this estimate is optimal with respect to the exponent of Reynolds number.