Each ordinance passed by the Council shall be published at least once within fifteen (15) days after its adoption in a newspaper published in Sanilac County. The publication of any ordinance in full after its final passage as a part of the published proceedings of the Council shall constitute publication of such ordinance as required herein. All ordinances and revisions, alterations, or amendments of ordinances shall be recorded by the Clerk in a book to be called "The Ordinance Book", and it shall be the duty of the Mayor and Clerk to authenticate such records by their official signatures thereon.