When Equality walked into the dais to learn what career the Council of Vocations prescribed to him, this was an important moment because his career leads him to meet important people and discover many things. Equality knew he was a street sweeper when he walked into the dais and talked to the council, "...for the lips of the oldest did not move as they said: "Street Sweeper" (Rand 26).
Equality sees Liberty for the first time and it brings new feelings to him which is important because this is the start of Equality taking notice of the outside world. When Equality sees Liberty he takes notice of everything about her thinking, "Their body was straight and thin as a blade of iron. Their eyes were dark and hard and glowing, with no fear in them, no kindness and no guilt" (Rand 39).
"We must not keep our secret to ourselves, not buried under the ground" (Rand 60).
Equality discovers electricity and questions to himself everything he once knew about society. Equality writes "No single one can possess greater wisdom than the many Scholars who are elected by all men for their wisdom. Yet we can. Yet we do" (Rand 54).
"Where have you been?" (Rand 65).
Liberty went to gather Equality some water so he could drink the water out of Liberty's hand. When Liberty lifted her hands to Equality's mouth, Equality drank the water and continued holding his lips to her hands which is technically their first "kiss."
Equality was messing in the box of stuff from the Unmentionable Times and found glass and other pieces to use so that he could make a light bulb. He created the light bulb and wants to present it to the World Council of Scholars.
Equality was taken to the Palace of Corrective Detention for not telling the Council of the Home where he had been. Equality was beaten by two men until he passed out.