After doing this, the leftmost button will unlock.
After doing this, the second button from the left will unlock.
Push the second button from the right four times, waiting for the machine to finish moving each time.
The solution to the puzzle is as follows: There is an inanimate skeleton named Drokt lying in the tower, with a journal on the ground next to it that provides a clue to the puzzle. This room houses an oculory that holds the Dragon Elder Scroll. The Elder Scroll needed to learn the Dragonrend shout is found inside the Tower's main rooms. If the Dragonborn is returning to Blackreach, the Great Lift at Raldbthar is the entry point closest to the Tower. The tower is located in the south-southwest corner of Blackreach. Blackreach can be found by traveling through Alftand, Mzinchaleft or Raldbthar of the major Dwemer ruins, but the quest marker will always direct the Dragonborn to Alftand. It is also possible to access it by forcing a plate against the wall and sprinting into it. The only way to do this is to have started the quest, "Discerning the Transmundane." Septimus Signus' Attunement Sphere is required to activate a Dwarven mechanism opening a door to Blackreach. To reach the other side, Blackreach must first be traversed.