The board of trustees of any road improvement district, created by sections 39-1601 to 39-1609, shall have the power to grade, establish grade, change grade, pave, repave, gravel, regravel, macadamize, remacadamize, widen or narrow roads, resurface or relay existing pavement, or otherwise improve any road or roads, or parts thereof, at public cost or by levy of special assessment on the property especially benefited thereby, proportionate to benefits; Provided, that none of the improvements hereinbefore named shall be ordered, except as provided in section 39-1611 or 39-1612.