When the council shall determine to open, lay out, alter, widen or extend any public highway, street, avenue, lane or alley, or shall determine to make any improvements, the cost of which or part thereof is to be paid by a special assessment, the council shall appoint two competent persons, who with the assessor shall constitute commissioners to make such assessments. The clerk shall issue to them his warrant commanding them to make the assessment ordered, within thirty days from the date thereof. The commissioners before entering upon the duties of their office shall be sworn by the clerk or some person authorized to administer oaths, faithfully and impartially to make such assessment to the best of their ability; and in making such assessment, shall determine the damages, if any, accruing to the owner or owners of any land in opening, laying out, altering, widening, or extending any such public highway, street, avenue, lane or alley, or in making any such improvements; and determine the benefits, if any, accruing to the owner or owners of land thereby, and also determine the benefits accruing thereby to the public, and in all things make such assessments according to law; and shall file their assessment roll with the warrant, and their certificate attached, with the clerk, within the time prescribed by the warrant. Provided no public highway, street, avenue, lane or alley shall be opened, laid out, altered, widened or extended, only upon the petition of two-thirds of the owners of lands through which the same may pass or upon the petition of the owners of one-half of the property fronting on the same, or upon the petition of one-half of the legal voters, or the petition of fifty resident freeholders, of said village.