Secretary of State Cyrus Vance's mission to the Middle East, which was completed Monday night, was the first step in a phased and methodical effort by the Carter administration to make major progress in 1977 toward an Arab-Israeli peace agreement.

On the surface, Vance's whirlwind trip provided little grounds for optimism about breaking a decades-long deadlock. Vance listened much and said little, while "the parties" (as he called them in his cool, lawyer-like way/ delivered unsurprising recitations of familiar and conflicting positions.

After hearing the views of six countries in seven days, he reported that the positions of Arabs and Jews are separated by a wide, deep gap. Nearly anybody could have told him so before he walked out the door of the State Department.

Skeptics also say Vance is only the latest in a long list of "fact-finders" who went to the Middle East to hear oppising views with the idea of forging a plan for peace - including John Foster Dulles, Gunner Jarring, George Ball, William Scranton, William Rogers, Henry Kissinger, and Kurt Waldheim, among others.

In this respect the learning process of world statesmen about the Middle East conflict is reminiscent of something a perceptive observer said of Indochina about 1968: "The United States hasn't been in Vietnam 14 years. It has been here one year, 14 times."

Despite all this, there appears to be a real chance - far less than a probability but greater than a remote prospect - that Secretary Vance and President Carter will be able to succeed where others have failed. This prospect arises from two underlying facts, one in the Middle East and the other in Washington. Both involve major shifts - and also major uncertainties.

First, there is little doubt that the 1973 Middle East war and its aftermath, including the diplomacy of former Secretary of State Henry A. Kissinger, have jarred fixed positions and led to a greater desire for a settlement among the Arab states. The leaders of Egypt, Syria and, above all, Saudi Arabia have far greater stake in regional stability - and far more to lose in another war - than ever before. Moreover, Israel - while fearful of the risks inherent in a settlement - faces unspeakable dangers including an erosion of U.S. support if the pattern of recurrent war remains unbroken.

A crucial question is whether the states in the region will remain stable long enough to make a settlement possible and sustainable.Egypt's President Anwar Sadat, particularly, is in highly precarious shape. Jordan's King Hussein may or may not be seriously hurt by last week's revelations of CIA payments. Israel is in the midst of selecting new leadership. And the Palestinian movement - in the past, a formidable barrier to peace - is in the midst of decision-making about its future relations with Israel.

As Vance is aware, the present favorable circumstances could become unglued very quickly. The Middle East is not known for happy surprises.

The second underlying fact that improves the prospect for the peace process is the changed role of the United States and the personal involvement of a new American President. Due to circumstances and Kissinger's efforts, the United States is now the vital outside power for nearly all states in the region. The role of the Soviet Union has faded, and the British, French and other European powers are far less important in the region than before.

Recognizing these faces, Vance and Carter decided before Inauguration Day to make a serious, high-priority effort to use U.S. influence in a push for peace. Vance's just-completed trip was conceived to give the U.S. effort high-level attention and visibility at the outset. Carter's scheduled meetings with Arab and Israeli leaders in March, April and may are to be the beginning of the bargaining process, with Carter starting to nudge the parties toward agreements. He is preparing for personal presidential involvement of unprecedented dimensions.

A return trip by Vance to the Middle East in June or thereafter is expected to exploit any tendencies to compromise. There is surprising confidence within Vance's entourage, given what is publicly known, that the United States will succeed in convening a new Geneva conference in the latter half of this year. Vance's persistent predictions to this effect constitute the next thing to a U.S. commitment to make it so.

The crucial question, which could not be answered aboard Vance's Air Force jet, is how much of Jimmy Carter's domestic political capital he will be willing to lay on the line for a Middle East settlement. Any effort to muscle Israel toward compromise will be met with massive political counter-pressure from Israel's friends. Yet without a very determined Carter stand for compromises on both sides - despite his numerous pro-Israel campaign statements - the peace process would be doomed.

In his methodical fashion, Vance has prepared a tabular comparison of the starting positions of the Middle East parties as he heard them in the past week. Such a way of summarizing things appeals to Carter's technocratic instincts. The future of Israel, the world's oil supply, superpower rivalries and much more are riding on future changes in the positions described in Vance's grid.