'The Europeans' - from the Italian who gave us 'The Italians'; The Europeans, by Luigi Barzini. New York: Simon & Schuster. 267 pp. $14.95.

''On the eve of Sarajevo no passport was needed to go from one European nation to another.'' Why does Europe not unite today? Luigi Barzini sees Europe in terms of national differences and foibles that prove stronger than the original desire for continental unity. Barzini, to quote the chapter headings of his book, regards the Europeans as ''elusive,'' the British as ''imperturbable,'' the Germans ''mutable,'' the French ''quarrelsome, '' the Italians ''flexible,'' the Dutch ''careful,'' and the Americans ''baffling.''

European unity was conceived as a panacea for national ills and as an antidote to the superpowers. It has proved to be neither. Indeed, the European Common Market has exacerbated differences between member states. Witness the unseemly agricultural lamb ''war'' between France and England, Margaret Thatcher's strident demands for financial rebates and the awesome ''butter mountain,'' the constant bickering between member states, and the desperate longing of the British Labour Party to withdraw as soon as possible from the Common Market, almost echoing Winston Churchill's words: ''We are with the Europeans, but not of them.'' Are Europeans simply too cussed to unite?

When Barzini suggests that an Englishman is a European ''without pride'' and with ''resignation,'' then surely one should first define the term ''European.'' Barzini can't. He admits ''it is almost impossible to tell what part of Europe the young you meet are from; they appear identical, belonging to the same tribe.''

This reflects the state of European thought. Older generations tend to think nationally, even nationalistically. Younger generations - except the hooligan element at soccer matches - far less so.

Understandably Barzini is most enlightening about his own countrymen, ''the flexible Italians,'' about whom he has already written a best seller: ''The Italians.'' He quotes Alcide de Gasperi: ''I must have a united Europe to absorb in her vast bosom three problems we Italians alone will never be able to solve and on which our future depends'': the Church (''a state within the state''), unemployment, and communism. Some bosom!

Barzini is also quite good on the Germans. The descriptions of that people's metamorphosis from post-World War I laxity and debauchery to national discipline and ultimate disaster are the stuff of the journalism in which Barzini excels. He describes scenes, however harrowing, beautifully: ''White Russian refugees, nice girls from decayed good families . . . wept on the rumpled bed after making love when they accepted money.''

He actually met Hitler, who ''shook my hand, held it for a while between his two, looked steadily at me, and delivered a short statement for my benefit.'' Comically Barzini wondered afterward whether the following war could have been avoided if he had passed the message on to the right quarters. Unfortunately, he ''did not understand a word!''

Considering the Benelux countries, fraught as they are with internal divisions, he quotes a Dutch joke about the Belgians: ''The Amsterdam control tower to a Belgian pilot: 'What is your position exactly.' The Belgian pilot: 'I'm sitting in the cockpit!' ''

Barzini terms the Americans ''baffling,'' yet sees how their country reflects Europe's problems. He dwells on ''the American contradictions,'' particularly between ideals and behavior. But, recognizing their immense power, he concludes: ''The United States cannot afford to make mistakes today.''

Finally of European unity, he writes: ''We must admit that in a modest way the dream of European unity has been a success'' in avoiding war and establishing ''a shopkeepers' ideal continental market of sorts . . . , but the principal obstacle is the German problem.''

Surely this is the heart of the matter. It's not European cussedness, but the Iron Curtain and all that goes with it that prevents European unity. How can Europeans unite with Germany cut down the middle?

It's a book to be enjoyed for its author's distilled wisdom, literary skill, and sensitivity rather than as an example of profound political analysis; elegant but not substantial, yet delightful to read: a Roman sage's nostalgic and provocative view of Europe.