“I am here to draw my legitimacy from you,” Prime Minister Essam Sharaf told the raucous, flag-waving assembly. “You are the ones to whom legitimacy belongs.”  Even some protesters dismissed the speech as the savvy move of an ambitious politician in a time fraught with anxiety. Yet it was perhaps the symbolism itself that said the most about Friday’s moment when, just a day after his appointment, an Egyptian leader chose to make his first stop the square that helped topple his predecessor.  The burst of euphoria that greeted uprisings in Tunisia and Egypt has faded somewhat, with the bloodshed in Libya and the retrenchment of governments in Yemen and Bahrain.  But protesters said Mr. Sharaf’s appearance illustrated the new, if hesitant, calculus in the Arab world: the power of protests — or, simply, the expression of popular demands — to bring about change long left to a clique of officials around Arab strongmen.  The sentiment coursed through the protest, which rivaled some of the more modest days of the 18-day Egyptian uprising. In a celebratory atmosphere that was tinged with anger and resolve, the demonstrators seized the opportunity to demand that Mr. Sharaf undertake far deeper change than the largely cosmetic reforms Egypt’s military rulers have parceled out since taking power from President Hosni Mubarak on Feb. 11.  “This shows you that the power of our ideas — the idea of democracy, of people claiming their own legitimacy, of our right to choose — have come to reality,” said Mohammed Ali, a 42-year-old film director. “Freedom can do the impossible.”  Protesters had described Friday’s gathering as a “day of determination,” pressing their call for the resignation of Mr. Sharaf’s predecessor, Ahmed Shafiq, whom Mr. Mubarak had appointed to lead the cabinet before resigning. But the Supreme Council of the Armed Forces, with timing that appeared aimed at heading off bigger crowds, announced Mr. Shafiq’s resignation on Thursday with three terse lines on its Facebook page.  A former transportation minister, Mr. Sharaf served in government only briefly, until December 2005, then became one of the higher-profile politicians to occasionally join the demonstrations in Tahrir Square, endearing him to many there.  On Friday, protesters suggested that he had been their candidate to replace Mr. Shafiq, a former air force general like Mr. Mubarak, who came across as haughty at best, inept at worst.  His challenges, though, remain vast, not least in choosing new figures as foreign, interior and justice ministers. Other demands — from dismantling the more odious police forces to freeing thousands of political prisoners — may be beyond his purview in a landscape where the military, in almost uncontested fashion, makes the decisions.  “In a transition period, you could have a change of cabinet every few weeks,” said Samer Soliman, a political science professor at the American University in Cairo. “It’s a time of political instability.”  “They,” he added, referring to the military’s Supreme Council, “are probably just trying the guy out to see if he works with the population.”  Even Mr. Sharaf seemed to plead with the protesters to have patience. “I beg you, you did something great and together we will do more,” he said, as the crowd chanted “We are with you.” He added, “I have a heavy task and it will need patience.”  Mr. Sharaf entered the square a little before Friday Prayer services, drawing cries from the crowd and a few surprises, as he headed toward the Mugamma, a sprawling bureaucratic fortress redolent of Soviet-era design. “The prime minister has come!” one man shouted excitedly. To which another responded, “What’s his name again?”  If Mr. Sharaf meant to strike a different note on Friday, he did. He traveled with a few members of the military police in red berets and a few more men in dark sunglasses, their numbers paling before the phalanxes of security guards that usually accompany Egyptian officials.  In his remarks, there was none of the stentorian paternalism of Mr. Mubarak, who addressed Egyptians as his sons and daughters in his last speech. “If you would permit me,” Mr. Sharaf repeatedly asked the crowd. Dispensing with customary formal Arabic, he spoke in Egyptian slang, standing before them in a gray jacket and white shirt, with no tie.  In a way, the speech was a striking legacy of the uprising: the reimagining of power that once sought prestige though its very distance from those without it.  “Sharaf is the first official that everyone wants,” said Mohamed Mostafa, a young member of the Muslim Brotherhood from the Nile Delta. “He promises to implement all our demands, and we trust him because he was protesting in Tahrir with us.”  The utopian sense of Tahrir Square has long faded. Only a few people picked up trash on Friday and gone were the pharmacies, health clinics and the uprising’s equivalent of soup kitchens that cared for the tens of thousands who camped out here for days. Yet while some thought Mr. Mubarak’s resignation might dissipate the protests, Friday’s turnout represented a clear signal that the demonstrations had the vitality to continue.  In fact, Mr. Shafiq’s resignation was unlikely without the continuing protests.  “From here until the elections for a new parliament, the only way to voice our demands is protests,” said Ashraf Ismail, a 29-year-old engineer. “They have to go on.”  A banner put it differently: “We want the overthrow of the rest of the regime.”  The demands articulated Friday remain formidable — from forcing out the most loathed ministers to dismantling the nefarious State Security service. Others called for a trial of government figures, from Mr. Mubarak down, and the lifting of the Emergency Law, which has granted the government extensive powers to arrest and detain people for decades.  Mr. Ali, the film director, acknowledged the difficulty in realizing those demands. But, he added, “this is the accomplishment of the revolution. There’s a tomorrow.”  Sharon Otterman, Liam Stack and Mona El-Naggar contributed reporting.
Egyptâs new prime minister delivered a speech to a crowd of tens of thousands in Tahrir Square on Friday.