The fall of the unfortunate Michel Barnier can be explained by just one figure number — 47. That is the number of seats held by Monsieur Barnier’s classic conservative party, Les Républicains, out of the 577 seats in the current National Assembly.

In that context it’s something of a surprise that Mr. Barnier was able to hang on as prime minister for one hundred days and was dismissed only by an unlikely motion of no confidence on a budgetary technicality on December 5. He just lacked seats in the legislature

Without a sizable base, one cannot really play politics, whatever one’s talents or personal popularity. What should be expected, then, from the new premier, François Bayrou, who succeeded Mr. Barnier Friday? His own centrist party, Modem, has 36 seats — nine fewer than Les Républicains.

Mr. Bayrou is supposed to lead a “central bloc” coalition that would bring together everybody except the hard right, Marine Le Pen’s National Rally, and the far left flank, Jean-Luc Mélenchon’s La France Insoumise, or France Unbowed. That, though, was precisely Mr. Barnier’s task before Mr. Bayrou, and he failed.

Moreover, Mr. Barnier came with a flattering reputation, as a former European commissioner for more than 20 years, something that ranks above most domestic political positions in France. Whereas Mr. Bayrou, 73, has been seen usually as a loser.

He was born at Bordères, agricultural village of 600 in Basses-Pyrénées, now Pyrénées-Atlantiques, the westernmost département in the French Pyrenees. He still lives and breeds horses in the area — the ancien régime province of Bearn — and led his entire political career from there, as member of the National Assembly and chairman of the département’s council or, more recently, as mayor of Pau, the local capital.

A brilliant student, Mr. Bayrou won in a competitive examination the agrégation in classical studies (French, Latin and Greek), France’s highest and most prestigious academic degree. In 1979, he turned to politics as an assistant to a leading centrist politician, Pierre Méhaignerie.

The word “centrist” holds a particular meaning in French politics — an accurate translation would be “Christian Democratic.” In the late 1940s, as Europe lay in shambles and Communism was threatening, big Christian-Democratic parties emerged everywhere with a fourfold agenda: a strong military and economic partnership with America, European unity, a measure of a welfare state, and family values.

The German Christian-Democrats have stayed in power, or alternated with the Social-Democrats, until this very day. The Italian Christian Democracy party ruled as a hegemonic party until the 1980s. The French Republican Popular Movement, however, fared less well.

The French movement dominated, along with the Socialists, the short-lived Fourth Republic, which obtained between 1946 and 1958, but was subsequently obliterated by the Gaullist party, which was defiant of both American leadership and European federalism.

Yet a diminutive Christian Democratic party stayed the course, under a new “centrist” tag, with about 15 percent of the vote. In 1965, its candidate, Jean Lecanuet, prevented de Gaulle from being reelected president in the first round of voting.

Later, the centrists attempted to play the role of arbiter — or, as they claimed, of reconciler — in an increasingly fragmented French political landscape, either offering their support to the Gaullists in exchange for various concessions or aligning themselves with their conservative or socialist opponents.

This is the context in which François Bayrou’s career unfolded from the 1980s onward. His Béarnais origins and a certain megalomania led him to identify with King Henri IV, the leader born at Pau who ended the Wars of Religion at the end of the 16th century by reconciling — albeit temporarily — Catholics and Protestants.

Does this historical figure truly resonate with the French people of today, though? That is far from certain. What is certain, however, is that Mr. Bayrou has shown a great deal of pragmatism, if not outright versatility, with rather unconvincing results.

He has run for president of the Republic three times. His best result was in the 2007 election, when he garnered 18.57 percent of the vote in the first round but failed to qualify for the second. In the 2002 and 2012 presidential elections, he secured only 6.84 percent and 9.13 percent of the vote, respectively.

Similarly, the centrist parties or coalitions he leads typically receive between only 8 percent and 12 percent of the vote in legislative or European elections. Enough to secure an important ministerial position from time to time, but not to wield decisive influence over events.

In September 2016, as Emmanuel Macron’s presidential candidacy began to take shape, Mr. Bayrou was furious. The newcomer was claiming the centrist and conciliatory position in the political arena that had always been Mr. Bayrou’s up to that point.

In an interview with TV channel BFMTV, Mr. Bayrou bluntly accused Mr. Macron of being “the candidate of Big Money.” As Mr, Macron rose in the polls, though, Mr. Bayrou changed his tone and ultimately chose to ally himself with him.

Once elected, Mr. Macron declined to put Mr. Bayrou in charge of the government, as the would-be “Henri IV” had expected. He offered Mr. Bayrou a regal position as Minister of State and Garde des Sceaux, or Attorney General, instead — only to withdraw it from him when charges of misuse of public funds were issued against Mr. Bayrou and the MoDem party.

Mr. Macron went on like that with Mr. Bayrou for seven years, even after the latter was ultimately cleared by the court for lack of evidence. Last July, after catastrophic early legislative elections that deprived the president of any majority, the MoDem leader expected to be appointed prime minister at long last, but it was Mr. Barnier who was chosen instead.

Even after Mr. Barnier’s fall, Mr. Macron toyed with Mr. Bayrou’s nerves, meeting with him several times, having lunch with him, and ultimately pretending to turn to another candidate. The name of the new premier was to be disclosed before noon on December 13, and the red carpet was already in place at Hôtel de Matignon, the head of government’s residence in the aristocratic seventh district at Paris.

Still, nobody knew who was going to fill the job until the last moment. It appears that Mr. Bayrou issued a genuine ultimatum to Mr. Macron, who eventually gave in. So Mr. Bayrou is busy building a cabinet — at least for now. The test will be whether he manages to have both Les Républicains on his right, and the Socialists on his left, sitting with him.

As a conservative, Mr. Barnier failed to detach the latter from their alliance with Mr. Mélenchon. As a centrist, Mr. Bayrou stands a better chance. In the meantime, Moody’s is lowering a bit further France’s debt rating, servicing the national debt is turning into the budget’s top priority, and investment is declining.

The Macronists’ latest fantasy was that the National Rally would pay dearly for having joined Mr. Mélenchon’s far left in the vote of no-confidence that toppled Mr. Barnier. This is hardly going to happen.

The first presidential poll taken after the vote confirms Marine Le Pen’s ascendancy: she is expected to lead decisively in the first round with between 34 and 38 percent of the vote, some 10 to 15 points ahead of a Macronist candidate — Mr. Macron himself, who has been elected twice, cannot run.

And what if she is barred from the election? Prosecutors have charged her with the misuse of public funds in a case strikingly similar to Mr. Bayrou’s. Madame Le Pen has pleaded not guilty, but the case, in which a ruling is expected in the spring of 2025, could lead to prison and civic indignity. However, the poll shows that in such a case, her vote potential would simply be transferred to her designated heir, Jordan Bardella, who could garner a stunning 34 percent.