EXETER, N.H. — There was a man with a “Stop Puppy Mills” T-shirt and another whose shirt read “National Sarcasm Society.” There was a woman, dressed entirely in white, holding a banner reading “Lead Us to Clean Energy.” There was a man with an Apache haircut. There was even a little old lady in tennis shoes.

This could only be a Bernie Sanders rally.

And the lady in tennis shoes? She was here mainly out of curiosity. She voted for Mitt Romney in the last two New Hampshire primaries.

Then there was the candidate himself. He wore a dress shirt, open at the neck, and his speech started early and ended late. He used the word “billionaire” more than half a dozen times, and he sprinkled his talk with references to “Corporate America.” He spoke about big campaign contributions (he has none, wouldn’t take any) and the “grotesque level of income and wealth inequality in America” (he deplored it) and won his biggest applause when he said, “This is a rigged economy, an economy that is not sustainable, and that is not an American economy.”

But he wasn’t done yet. In the sweltering confines of the Exeter Town Hall — every seat filled, the back of the hall five deep with standees, the balcony jammed and every one of the seven granite steps outside occupied with the devout, the devoted and the determined, all drenched in heavy perspiration — he launched into his speech: full employment, the Citizens United decision, gay marriage, voter suppression, the Trans Pacific Partnership, student debt, climate change, acidification of the oceans, access to abortion, energy efficiency, the criminal justice system, prison reform, mental health and crumbling infrastructure. In one sentence he crammed in the words “racism,” “sexism” and “homophobia.”

But wait. We’re not nearly done yet. Elimination of tuition at all public colleges. Guaranteed single-payer health care. Assuring that police are no longer an “oppressor force.” Paid family leave. Paid vacations.

“This,” he said at one point, not remotely finished, “is some of what we have to do.”

And to think that a century ago Woodrow Wilson defeated Charles Evans Hughes on the basis of a mere six words: He kept us out of war.

Those six words from 1916 might actually suffice today, but Mr. Sanders was having none of it. This audience, whipped into a sticky delirium, was wailing with delight, and by the time he rolled into the topic of undocumented aliens — a crowd favorite even though New Hampshire is 94 percent white — the reaction was about the same as when the Rolling Stones finally get around to playing the opening strains of “(I Can't Get No) Satisfaction” at a muggy stadium concert.

Mr. Sanders — the most prominent socialist candidate for president since Eugene V. Debs easily edging out Norman Thomas, six-time contender — could have gone on for another sweaty hour. For what started out as a political meeting in a site where Abraham Lincoln spoke in the 1860 campaign had been transformed into a revival meeting, and the ones who should take notice aren’t so much the billionaires he excoriated as the Hillary Rodham Clinton strategists he should terrorize.

For this is where a good portion of the Democratic Party is today — proud of electing Barack Obama but disappointed he didn’t go far enough, skeptical of trade pacts, fired with revulsion about police excesses, convinced economic mobility is more a phrase from the American past than a touchstone of the American future, wary of half-steps on the economy and ritualistic bows to progressive issues. The New Math may be a discredited phrase from the 1960s, but for Democratic activists there is a new math, and it focuses on the portion of American wealth held by a fraction of the population.

So when Mr. Sanders is at full throat he says things like this: “This campaign is sending a message to the billionaire class and that message is that you cannot have it all.” In Democratic presidential primary campaigns at the end of the last century, Rep. Richard A. Gephardt, whose lobby clients now include Boeing, General Electric and Goldman Sachs, used far less incendiary language and was vilified for employing “class warfare.”

Mr. Gephardt was serving 3.2 beer and was accused of running a rogue saloon and corrupting minors. Mr. Sanders is serving straight 151 proof Bacardi rum — so incendiary that each bottle comes with a steel flame arrester — and it’s considered merely a chaser to Ms. Clinton. For while Mr. Sanders is chasing Ms. Clinton in national polls and the money derby, he has pulled within 10 percentage points of her in the NBC News/​Marist Poll completed only two and a half weeks ago. And that was before federal reports this month disclosed that fewer than five dozen individuals and groups already had pumped an astonishing $120 million into super PACs in a campaign that is barely underway.

“He wants to get big money out of politics and bring democracy back and give everyone a chance to get ahead in America,” Al Porsche, a retired veterans’ counselor, said outside Exeter’s town hall. “There’s an excess of wealth at the top that hasn’t left much room for people at the bottom.”

Two points need to be made in Ms. Clinton’s defense. One is that those two sentences from Mr. Porsche might easily have found their way into the text of a Clinton speech. The other is that Ms. Clinton has been battling for progressive causes for almost as long as Mr. Sanders.

But the problem for Ms. Clinton — and the reason why Mr. Sanders’ extremely unlikely challenge to her nomination has been transformed into Ms. Clinton’s political challenge — is simple to describe but difficult to combat: Mr. Sanders, while lacking the record and experience of his rival, seems to possess more audacity and authenticity.

The language Mr. Sanders uses on the stump is, to be sure, in her texts. But somehow it doesn’t seem to be in her heart. He’s fiery, she’s forced. He seems committed, she seems controlled.

The larger problem for the Democrats, perhaps the only issue gone unmentioned in Mr. Sanders’ appearance here: Their primary struggle is being fought on a field that may be alien territory for the voters who drifted away during the Reagan years and who were lured back in the Bill Clinton years.