Release the book from the Dwemer contraption | Main story mode - Path of Knowledge TES V: Skyrim - Dragonborn Guide

This is the last and easiest part of the mission. Return to the reading room (where the Black Book is) and press the button by which Neloth is standing. That way you will open the device in which the book has been locked - you will be finally able to use it. You will activate another mission - Gardener of Men.