Before an audience of about 15,000 on Old Campus on Sunday afternoon, former President Bill Clinton LAW ’73 urged the members of the class of 2010 to work for the public good once they leave Yale as private citizens.

In a day marked by calls to public service, Clinton’s Class Day address focused on the importance of collaboration and open-mindedness in a partisan, 21st-century world. But Clinton’s mentions of themes such as poverty and global warming were some of the few somber moments of the afternoon’s festivities; students celebrated the occasion with old traditions — singing “Bright College Years” while waving embroidered handkerchiefs — and new, as the usual Class History speech was converted into a short film set to be posted on YouTube following the ceremony.

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And then there were the hats.

From flying pigs to birthday cakes, nearly every speaker at Class Day wore a spectacular hat — save Clinton.

“If it were a little bigger, I would turn it into a doo rag,” the former president said, holding up the white Class Day handkerchief given to him by University President Richard Levin.

Clinton appeared relaxed, especially given the earlier events of the day: A black Secret Service SUV carrying Clinton to Old Campus was hit from behind on the Merritt Parkway, state police said. Though Clinton arrived as planned shortly after 2 p.m., his SUV on Elm Street showed signs of damage on its back, right bumper and tail light. The Yale and New Haven Police departments also blocked off a section of High Street near Chapel Street earlier in the day after a suspicious package was found near the Skull and Bones tomb.

The former president took the stage with a sheaf of papers in hand, but he later showed the News a copy of his prepared remarks, which he had almost completely crossed out with a Sharpie before speaking. With a few exceptions, he said, his speech was impromptu.

During his address, Clinton wondered aloud why his wife, Secretary of State Hillary Clinton LAW ’73, who spoke at Class Day in 2001, had not warned him about Yale’s tradition of unorthodox headgear for the occasion. But his running commentary on the class of 2010’s “sartorial splendor” soon gave way to a more serious discussion of what the graduating seniors can look forward to once they leave Yale.

Much of Clinton’s 34-minute speech addressed major global issues he said the graduates will grapple with in their careers and private lives, from health care reform to environmental concerns. Though the country has seen many social barriers fall and technological advances have helped to close the gap between different cultures, inequality still exists, said Clinton, who is now the director of the charitable William J. Clinton Foundation.

Case in point, he said, was the class of 2010: Yale students are privileged to earn Yale degrees, he said, making it all the more important that they help to diminish inequities.

“You need to do things, because you got a good deal out of that one-tenth to [one-half] percent of genetic makeup that was different,” he said, referring to an earlier point about how all humans are genetically similar except for a small amount of variation. “You’ve got a lot of choices going forward. Some of those choices should be to do public good as private citizens.”

Clinton said one of the most important public questions of the next 20 years will be what he called the “how” question: How will people solve global problems?

He suggested that the best way to resolve issues is what game theorists refer to as a “nonzero-sum game,” in which it is possible for two parties both to emerge as winners, rather than a “zero-sum game,” in which one party must lose — an arrangement he compared to the annual Harvard-Yale game.

“Someone has to be a winner, and someone has to be a loser,” he said. “We play until someone drops.”

Public service was also a theme of Levin’s Baccalaureate address, given once on Saturday and twice more on Sunday morning, as well as of Bradford Williams’s ’10 class reflection speech, titled “Why We Serve.” Senior Insight speaker Celina Kirchner ’10 chose a different track, peppering her remarks with jokes tailored to the students in the audience. She ended the speech with lines from Journey’s “Don’t Stop Believin’,” the unofficial Toad’s Place anthem.

Clinton’s speech was accompanied by fair, partly cloudy weather, but as the ceremony continued, the wind picked up, threatening to topple some hats in the audience.

After Clinton left the stage, Yale College Dean Mary Miller stepped in to introduce the Yale College prizes traditionally given to outstanding seniors. Before attending to the awards, though, Miller gave the class of 2010 directions for the Monday’s Commencement ceremony and asked the seniors to “explode” with cheers and applause when she formally confers degrees upon them tomorrow.

Sixteen prizes were awarded to 14 seniors, starting with Tiffany Petrosino ’10 (a former copy and city editor for the News), Leslie Golden ’10 and Katherine Woodfield ’10, who received the Lewis Prize for intramural athletic participation from a rapping Timothy Dwight College Master Robert Farris Thompson. Deborah Gruen ’10, a world record-holding Paralympic swimmer who has had spina bifida since birth, won twice for athletic achievement and personal character, while Mihan Robin Lee ’10 — who has never earned a grade at Yale lower than an A- — took the prize for top academic achievement in both the humanities and social sciences.

The Sudler Prize for achievement in the arts went to vocalist Lucy Fitz Gibbon ’10. Susan Scanlon ’10, meanwhile, won the Chittenden Prize for the highest academic achievement in the natural sciences. Alice Baumgartner ’10 (a staff columnist for the News) and Eli Bildner ’10 took the Haas and Snow prizes, respectively, for general high scholarship, character and humanity.

One member of the audience, Anthony Snowden, said he had come to New Haven from Arizona to see his niece graduate — but Clinton’s speech was “definitely one of the reasons why we were trying to get here,” Snowden said.

“It was very inspiring,” he said. “They were good points to hit on at a graduation.”

In an interview after the ceremony, former provost Andrew Hamilton, for one, said he found it “exactly right.”

“Yale does Commencement just perfectly,” he said.