Ronald Reagan came to Europe to persuade people that he is not the shallow, nuclear cowboy of certain unkind assessments. Said White House spokesman David Gergen, on the eve of departure, "Some in Europe do not know or understand him."

But now that the president has been among them for over a week, Europeans may think they got him right the first time.

In Rome, he made a stab at identifying himself as a "pilgrim for peace." But by the time he got to London he had reverted to type as a cold warrior. And yesterday in Bonn, he reiterated his commitment to "peace through strength"--which is fancy talk for continuing the nuclear arms race.

If the two speeches defined him, they also showed him at his best. His timing, his delivery were perfect; his jokes floated up on wings of laughter. In Bonn he put down a heckler with a deft "Is there an echo in here?" This was after a second interruption from someone sitting somewhere near him in the big, modern brass and black chamber of the Bundestag of the Federal Republic of Germany.

His peers in the ornate gold and brocade setting of the Royal Gallery of the British Parliament were smitten by the high literary gloss of his speech and his polished technique. They were fair overcome by the technology of the teleprompting device he had brought with him.

It was only the day after that they pondered the substance--and the irrelevance--of what he had said. The Guardian called him a "wonderful old smoothie," and commentator Michael White suggested that the speech "could have been written by a CIA computer."

Other columnists thought it sounded like a commencement speech for a small Midwestern university, or a pre-presidential Rotary Club offering from the leader of the Western world.

The members of both houses of Parliament listened politely to his call to arms for a global campaign to consign the Soviets to the "ash heap of history." They are familiar with his views; many share them.

They probably agree that the men of Moscow "take subversion and conflict around the globe to further their barbarous assault on the human spirit." They can see for themselves that the Soviets cannot feed their own people or govern them with their consent.

They certainly hope that Reagan is right that the Soviets are in decay. They may, on the other hand, wonder why he drains his country's treasury to make weapons to counter a weak and doomed foe.

The Brits were waiting for just one thing, words of support for their action in the Falklands. When they came, the ancient chamber shook with applause. They clapped again when the president suggested a swap of air time between him and Brezhnev. They could see as he spoke who would win that particular conflict.

"Although I'm sure he meant every word of it," said one Liberal MP, "when you came right down to the nub, there was really only that one suggestion. I'm afraid I think it is just a gimmick."

In Bonn the next day, Reagan told Russia's neighbors that he knows better how to deal with the brutes. He even expects them to respond rationally provided they understand the strength and unity of the Western alliance.

The 519 members of the Bundestag hung on his every word. They stopped him with applause 18 times. Chancellor Helmut Schmidt sat in the front row clapping along with the rest.

Nuclear agitations have greatly complicated the chancellor's situation. While the members were cheering Reagan and hearing him say that parades and petitions are not the way to deal with the nuclear horror, thousands of Germans were massing outside for still another demonstration. Schmidt's future, like that of many of the politicians in the chamber, will be decided in part by Reagan's success in overcoming skepticism about his true philosophy.

The president went to inordinant lengths to ingratiate himself with his German hosts. He all but suggested they had invented democracy. He made reference to the American Revolution, in which, of course, the Hessians fought on the other side.

"The German people have given us so much," he gushed at one point. They gave us, among other things, World War I and World War II.

"The U.S. is proud of your democracy," he said in another lavish, but not quite accurate, compliment, "but we cannot take credit for it."

As a matter of fact, we can. Germany's present democratic government was imposed after World War II at gunpoint by a large occupying army which rooted the Nazis out of its national life.

The signs are that it worked. The soldiers who lined up to greet Ronald Reagan at the Bonn White House do not goosestep. They have long hair and much patience with inquiring reporters stumbling along in Reagan's train.

Inside the Bundestag where once Adolf Hitler ranted and screamed, the members were jovial and raucous and relaxed. There were a number of women among them, and several of them spoke. Germany has changed. Reagan may have a point. Maybe the Russians will change. Maybe he will.