“Now you’re repeating your hypothesis,” Petry said, leaning over him from the stage. “But how do you justify it?” He hesitated in confusion, and other protesters joined him. A teen-age girl began to speak from prepared notes, saying that the AfD denied climate change. “You have to hold the mike closer to your mouth,” Petry interrupted, and then rocked from foot to foot, marking the slow tempo of the girl’s speech. “Your party claims that CO₂ is not dangerous, but how do you explain all the people dying from air pollution in China?” the girl asked.

“I’m a chemist,” Petry said. “The problem is not CO₂—it’s the nitrogen and sulfur oxides that make the smog. So many people make this mistake.” She went on, “Let me ask you a question. If you dissolve CO₂ in water and the temperature rises, will you have more or less CO₂?” It was a trick question that Petry often uses.

“More,” the girl said, meaning CO₂ in the atmosphere.

“Exactly wrong,” Petry said, meaning in the water. She made a dismayed face to the audience. “There’s a huge amount of misinformation out there,” she said. “When you see what’s in their school textbooks, it’s no surprise they believe these things.”

Petry spent half an hour more raking through the protesters’ arguments, expressing concern that Germany’s youth could be led so badly astray and exasperating the students with her pedantry. Both the protesters and the audience were relieved when she finally began her speech.

For decades, the German far right has been a limited force, with easily recognizable supporters—nicotine-stained ex-Nazis in the sixties and seventies, leather-clad skinheads in the eighties and nineties. Petry is something different, a disarmingly wholesome figure—a former businesswoman with a Ph.D. in chemistry and four children from her marriage to a Lutheran pastor. During a month I spent with her this summer as she drove around Germany giving speeches, she drew connections between politics and laboratory science, sprinkled her speech with Latin phrases, and steered discussions about German culture toward the cantatas of Bach.

Petry is not a gifted orator. Her speeches tend to be dull, with ornate sentences and technocratic talking points, and she is more comfortable citing economic studies than discussing the lives of ordinary people. Her manner belies the extremism of the AfD’s views. At the start of this year, Petry said that, in the face of the recent influx of refugees (many of them fleeing the war in Syria), the police might have to shoot people crossing the border illegally. In April, the Party said that head scarves should be banned in schools and universities, and minarets prohibited. Party members called for a referendum on whether to leave the euro; for the expulsion of Allied troops, who have been stationed in Germany since 1945; and for school curriculums that focus more on “positive, identity-uplifting” episodes in German history and less on Nazi crimes. Most contentious of all was the declaration “Islam does not belong in Germany.”

By American standards, especially in the age of Donald Trump, contemporary German politics is decorous and understated. But although Petry’s crisp style is in many ways the opposite of Trump’s, her rise has similarities to his. She, too, has come late to politics and relishes her outsider status. Like him, she often works by insinuation, fanning right-wing conspiracy theories not merely to stir up grievances but to bind members together with a sense of shared beliefs. Like him, she has been accused of financial improprieties. Like him, she castigates the media for liberal bias but also thrives on media attention. Petry and her colleagues have mastered the art of dominating the news cycle, to the point where a visitor to Germany listening to the radio or reading the newspapers could be forgiven for thinking that the AfD is the party in power.

“Do I want the job? Huh, I never thought about it like that.”

Cartoon by Tom Chitty

Two years ago, the AfD won its first seats in regional parliaments. (Petry was elected to the parliament of Saxony, one of Germany’s sixteen federal states.) Earlier this year, support for the AfD reached fifteen per cent in national polls, three times more than for any previous right-wing party, and well beyond the five-per-cent threshold required to enter the Bundestag after next year’s national elections. In a recent election in Mecklenburg-West Pomerania, where Germany’s Chancellor, Angela Merkel, has her constituency, the AfD got more than twenty per cent of the vote, edging Merkel’s party—the center-right Christian Democratic Union—into third place. A week ago, the AfD won its first seats in the state parliament of Berlin, traditionally a social-democratic stronghold, in an election that brought the C.D.U.’s worst ever result in the city.