Officials in Washington expect up to four million people to attend the presidential inauguration, most of them gathered on the National Mall.

Think about that.

Four million is nearly seven times the population of the city itself.

It's more than twice the size of the crowd that cheered the Phillies World Series parade.

It's the number of people who have attended the Eagles' regular-season home games - for the last seven years. Except at the inauguration they'd be at the same place at the same time.

Even if that four million estimate is off, and some believe it's way off, the volume of humanity that promises to press onto the Mall and its environs Jan. 20 will present unusual challenges:

Like, what happens when a crowd that size needs to go to the bathroom?

Mesa Waste Services, a national toilet-rental cooperative, estimates the need at 40,240 Port-a-Potties, minimum.

A four-million-person inauguration would rank among the biggest crowds to have gathered anywhere at any time. It would easily be the all-time-largest assembly in the United States.

"I haven't found anything that big," said Temple University instructor Ira Rosen, who studies crowds and who has put on big public events through his company, Entertainment on Location. "When they say unprecedented, they really do mean it."

In 2007, 70 million Hindus descended on the Indian city of Allahabad for Ardh Kumbh Mela, the world's largest religious festival. Three years earlier, about 30 million Hindus traveled to Ujjain.

But those events lasted weeks. The largest crowd for a single event?

Four million attended the closing Catholic Mass at World Youth Day in Manila in the Philippines in 1995. Pope John Paul II spoke to three million in Poland in 1979, and between two million and four million attended his funeral in Rome in 2005. A crowd of 750,000 to 1.5 million celebrated Earth Day in New York in 1990.

The largest gathering in Washington is said to be the 1965 inauguration of Lyndon B. Johnson, which drew 1.2 million, according to newspaper articles. But those accounts were almost surely in error because Johnson was inaugurated on the east side of the Capitol, facing a sea of buildings, not the expanse of the Mall. The next-largest crowds were one million for the 1976 Bicentennial Celebration and 800,000 for the first inauguration of Bill Clinton.

During the fall campaign, President-elect Barack Obama routinely drew enormous crowds: 75,000 for a rally in Portland, Ore.; 84,000 to hear him accept the Democratic nomination in Denver; 200,000 in Chicago on the night of his election.

But Clark McPhail says there is no way four million people will be standing on the Mall on Inauguration Day.

And he should know.

'Junior high math'

McPhail, a professor emeritus of sociology at the University of Illinois and author of

The Myth of the Madding Crowd

, has spent decades studying ways to produce reliable estimates. His laboratories have ranged from 1960s civil-rights rallies to more recent events such as the March for Life, which he has studied for 25 years. He has attended several inaugurations.

He says that if people fill every available space on the Mall and its immediate surroundings on Jan. 20, the crowd will total about two million.

Tops.

"It's a junior high school math problem," McPhail says.

Traditionally, the people responsible for estimating crowds have been elected officials, police, journalists and event organizers. The big problem for them is that when viewed from street level or slightly above, such as from a stage, crowds appear much more densely packed than they are. And because the view narrows over distance - down a railroad track, for instance - the crowd can appear uniformly thick.

Actually, crowds are invariably thickest at the front, near the stage, and thinner on the sides and at the rear.

McPhail credits Herb Jacobs, a professor at the University of California, Berkeley, with creating the gold standard for divining trustworthy estimates.

Jacobs photographed rallies from overhead so he could count the people. By calculating the square footage of the demonstration site, the density of the crowd, and the percentage of the site that was occupied, he developed a measure that could be widely applied.

At an event where everyone is standing, Jacobs determined, people shoulder to shoulder at the front cover 21/2 square feet each. That space increases through the middle of the crowd, reaching a relatively roomy 71/2 square feet per person at the back.

On average, a person at a typical event covers about five square feet - slightly larger than two newspaper pages. That figure was adopted by the U.S. Park Police in Washington.

The Mall runs from Third Street near the Capitol to 14th Street near the Washington Monument, 4,261 feet by 615 feet. That's 2,620,515 square feet.

So, McPhail calculates, if everyone stands close together and the crowd is uniformly dense so that people at the back take up the same 21/2 square feet as people at the front, the Mall could support 1,048,206 people.

That's a long way from four million.

But McPhail is willing to go further. If you include the grounds of the Washington Monument, 1,875,000 square feet, and the Ellipse, 696,960 square feet, you could fit an additional 1,028,784 people.

Include also the 240,000 who'll get free tickets for the area closest to the swearing-in at the Capitol. Grand total: 2,316,990 people.

Cold calculation

Inauguration planners say they'll place giant TV screens on the Mall so people can see and hear what they're there for. Security will be strict. People near the Capitol won't be allowed to carry so much as an umbrella, even if it's raining.

"It's going to be like a major military maneuver," says Temple University tourism professor Michael Jackson.

He and McPhail share a big worry: The potential for freezing weather.

"When you continue to hear this number, four million, popping up again and again, people say, 'Let's go,' " McPhail says. "I hope the excitement of being there and watching this on a Jumbotron will offset the frostbitten toes."

Contact staff writer Jeff Gammage at 610-313-8110 or jgammage@phillynews.com.