The Brooks inductor solves a problem, first studied by Maxwell and Gauss, of obtaining the greatest possible inductance with a given piece of wire. A simple procedure, presented graphically and algebraically, facilitates rapid design of air-cored solenoids to required inductance and current loadings. Calculations show that if the winding cross-section is not restricted to rectangles, as assumed hitherto, inductance values may be raised by about 1% above the Brooks limit.