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The problem of winding the greatest possible inductance with a given length of wire is solved for multilayered air-cored toroids with circular window sections. Calculations indicate significant improvements over single-layer designs, and are confirmed by measurements on devices using two different fabrication methods. The optimum designs are robust against fabrication errors: provided the wire length is correctly measured, accurate inductance is assured. To achieve the same inductance, multilayer devices use significantly less wire and are much more compact. A theory is developed for the optimum multilayer toroid, which is shown to scale similarly to the Brooks solenoid, but requires more wire for the same inductance.