"Cooler heads prevail" origin?

A cool head probably arose as the antithesis of a hothead, someone who is short tempered, impetuous and irascible. See https://www.etymonline.com/word/hothead

Antecedents of "cooler heads prevailed" go back at least to the early 1800s. For example, in a speech by the chancellor of the exchequer in connection with allegations of corruption involving the Duke of York, from "British and Foreign History for the Year 1809," chapter 2, in The New Annual Register, Or General Repository of History (July 1810):

He begged the house to recollect who it was they would endeavour to turn out,—almost the first subject in the kingdom,—and he hoped they would pause before they committed an act that would disgrace the house of commons. He was sorry to observe that the generous, open, candid, and manly feelings of the honorouble gentleman who brought forward the charges, had been led away, so as to be prevail'd on by the advice of cooler heads than his, and persons who meant more than he did, to fritter them down in the way that had been done by the motion of that night. He could not be supposed desirous of going out of his way to compliment the honourable gentleman; but as the present motion was so very different from what he had given reason to suppose he would bring forward, in consequence of the direct charges he had made, he felt it to be his duty to propose to the house, to say directly and distinctly, whether the duke of York had been guilty of the corruption or connivance of which he had been accused, and for that purpose would tender an amendment or resolution, on which the house might determine that question.

As a set phrase, however, "cooler heads prevailed" seems to have emerged in the United States during the last quarter of the 1800s. Here are nine instances from the period 1876–1889, drawn from the results of Elephind, Hathi Trust, and Google Books searches.

From an untitled item in the [Jefferson City, Missouri] Sate Journal (August 4, 1876):

Cincinnati is the scene of a great deal of discontent among the laboring classes. At the Esplanade a few days since there were demonstrations that were painful and serious. Large crowds of unemployed laborers met and talked over their grievances. It was no sham of suffering, an though nothing came of it, a strong effort was made to inaugurate a bread riot. "Bread or Blood," was the frequent cry and thoughtful men looked on with pain and apprehension. The counsels of cooler heads prevailed and no disturbances occurred.

From "Fight with a Band of Thieves," in the Richmond [Virginia] Daily Dispatch (November 9, 1883):

Clifford was lodged in jail to-night. He states that there are at least sixty of the escaped prisoners depredating in the vicinity of Welsh Mountain [near Lancaster, Pennsylvania]. Clifford, at the time of his arrest, made a narrow escape from being lynched, owing to the shooting of young Bard. Ropes were provided, but the advice of cooler heads prevailed. The mountain is surrounded to-night by a large posse, and it is believed to be impossible for any of the thieves to escape.

From an untitled item in the San Antonio [Texas] Daily Light (October 23, 1884):

San Antonio is a very peaceable city. It is also law abiding, and while many yesterday and the night before were clamorous for the life of the negro rapist, cooler heads prevailed and his miserable life was spared to take his chances with the slow proceedings of the courts. The penalty for rape should be a stout rope and short shrive.

From "Natural Gas: The Agent of Great Destruction and Death in Pittsburg, PA.: Three Terrible Explosions Occur in Almost Immediate Succession," in the Wichita [Kansas] Daily Eagle (February 1, 1885):

The threats of the people living in the vicinity of to-day's disaster to tear up the pipe were not carried out, the counsel of cooler heads prevailing. It is thought, however, that measures will be taken to prevent the piping of natural gas through the streets until safety pipes are adopted by the various companies. All the victims of the explosion were still living and resting easy at midnight.

From an untitled item in the Salt Lake [City, Utah] Evening Democrat (October 17, 1885):

Great excitement was created in front of the Democratic headquarters at Columbus, Ohio, at 3 o'clock yesterday afternoon, and a riot was narrowly avoided. Soon after noon some one about the headquarters caused to be strung across the street in large form a bloody shirt, with the inscription: "New North, new South : a new deal. The last of the bloody shirt." It is stated that a G. A. R. [Grand Army of the Republic] man called and gave the committee half an hour to have the shirt taken down, and by orders of a member of the committee, it was being done, when a man from the Republican headquarters grabbed the garment and after soaking it in kerosene oil returned and burned the shirt in front of the Democratic headquarters. A crowd of several hundred gathered and the greatest excitement prevailed. The counsel of cooler heads prevailed and after much loud talk and threats the crowd dispersed.

From "History of Davies County," in History of Knox and Daviess Counties, Indiana (1886):

Had the younger men of the county had their way, very few, if any [of those implicated in the murder of Captain Eli McCarty in 1864], would have been tried by a military commission or escaped, for their determination was to hang every one believed to have been guilty to the nearest tree, but the counsel of cooler heads prevailed.

From John McMaster, Benjamin Franklin as a Man of Letters (1887):

The nearness of the foe increased both curiosity and alarm. Some, who had never beheld a band of frontiersmen in their life, rode off to Germantown, and on their return described the Paxtons as a fine set of fellows, dressed in blanket coats and moccasins, and armed with knives, tomahawks, and guns. Others were for marching out, surrounding the men from Lancaster, and taking them prisoners. But cooler heads prevailed, and a committee, of which Franklin was one, met the malcontents on Tuesday morning, remonstrated with them, and received a written remonstrance in return.

From "A Fiend Incarnate," in the Coronado [California] Mercury (July 29, 1887):

A bench warrant was issued by the Superior Court this morning for the arrest of Michael Clancy. His bail has been fixed at $5,000, and up to the present he has been unable to raise that amount. Clancy is charged with committing a criminal assault upon Miss Louisa Hoenig, at Linda Vista. At the time of the perpetration of the dastardly deed, the worthy people of Linda Vista threatened to lynch him, but cooler heads prevailed, and he was handed over to the proper officials. Since his preliminary examination, at which he was held to answer in $2,000 bonds, he has industriously circulated slanderous reports regarding the poor girl's previous character, alledging a want of chastity.

From "The Faith Cure?" in the Los Angeles [California] Daily Herald (June 24, 1889):

The indignation of the citizens was aroused to such a pitch that a coroner's jury, after spending two days examining the case [of an Iowa woman's death], made a report condemning prayer and faith cure treatment in strong terms. James Clark, the fourteen-year-old son of a photographer of Salida, Cal., was treated lately by these fanatics for dropsy. They would not permit a physician in the house, and, as a result, the boy died. For three days after his death the deluded parents claimed that the boy was only sleeping, and would be brought back to life by prayer. Three weeks after his burial his sister was taken sick. Medicines were refuted by the heartless parents. Soon the circumstances spread, and the citizens organized a law and order society, and relieved the inhuman parents of the care of the child. It was almost impossible to restrain the people. Tar and feathers and rails were provided, but cooler heads prevailed.

The striking think about these U.S. instances of "cooler heads prevailed" is that virtually all of them involve situations where a riot or lynching seems imminent but is avoided when advocates of calm or the rule of law persuade an angry crowd not to become immediately violent. This is in sharp contrast to the 1810 instance from England, in which what prevails is not a nonviolent alternative to mob action, but a watered-down accusation of misconduct lodged in place of a more direct and forceful one.

The two earliest U.S. examples cited above—and five of the first six—use the form "the counsels [or advice or counsel] of cooler heads prevailed," which suggests that this was the original form of the expression. But the shorter form "cooler heads prevailed" appears in four of the nine instances, including one as early as 1884.

Today, "cooler heads prevailed" can apply to any situation where a more temperate action, response, or policy wins out over a less temperate one; but in its earliest manifestations in the United States, "cooler heads prevailed" seems to have been applied specifically to situations where mob violence was being incited by hotheads but was avoided through the intercession of calmer and wiser persons.