The basic story is this. Rousseau, afraid of being imprisoned because of the religious controversy caused by his works, fled Paris in 1762, but found that he had nowhere to go; he ended up in a secluded place in Switzerland for a while. At some point, he developed the hope that Hume might help him receive asylum in England. This Hume did, and Rousseau was in England in January of 1766. Hume arranged a place for him to stay and began to take steps to get him a secret pension from the King. Hume regarded Rousseau with something like hero-worship; while regarding him as excessive, he repeatedly held him up as the exemplar of a genius. And as Hume put it to a friend, it would be a victory over the French to protect and encourage a genius they had persecuted; although later when writing his account of the dispute he put more emphasis on pity mixed with indignation at the thought of such an eminent man of genius reduced to indigence.
Things started happily enough. Rousseau was profuse with gratitude to Hume, and Hume waxed eloquent in letters about how sociable, warmhearted, amiable, and virtuous his new companion was. He did worry about Rousseau's impatience and 'extreme sensibility of temper', but it called forth mostly compassion; Rousseau, Hume noted, felt pain more deeply than pleasure, and was often unhappy due to a melancholic temper. Rousseau, however, seems to have begun to believe early on that Hume was often lying to him. Part of this was that Rousseau was slightly paranoid in the first place; but he also came across little examples of what seemed to him to be Hume's deceptions. For instance, Hume once made him believe that a carriage ride, which cost more than Rousseau could afford, had been obtained at a discount when in fact someone had paid for part of it. When Rousseau charged Hume with having been in on the scheme and Hume denied it, Rousseau became sullen for a long while. Suddenly, however, he embraced Hume, crying, and asking him to forgive him for being so ill-humored when Hume had done so much for him. Hume, surprised by the sudden change of mood, took it at the time as a sign of Rousseau's 'extreme sensibility and good heart.' According to Rousseau's later account of the event, Rousseau was not sullen; he found Hume staring at him in a way that spooked him. That Hume did stare at him in this way is extremely plausible; a dull stare is a characteristic noted by others who knew Hume. To Rousseau, however, the expression seemed mocking, and he could not reconcile the good man David was supposed to be with this mockery. In the end, however, out of remorse, he embraced Hume in the way described, and hoped it was enough to reconcile. On another occasion, Rousseau refused to receive mail, since he did not have the money to pay postage; Hume protested that if he did so, the mail would become entirely in the control of the clerks at the Post Office; Rousseau responded that he did not care. Hume then paid for the postage. This led Rousseau later to accuse Hume of reading his mail.
The major occasion for the rift, however, was the publication in the St. James Chronicle, and slightly later the London Chronicle, of a letter satirizing Rousseau as someone who had a persecution complex. The satire was anonymous; the author was Horace Walpole. Rousseau, however, was convinced that Hume had had a hand in it. He began to wonder if there might not be an ulterior motive in keeping the pension secret, so he wrote a letter declining the pension unless it were made public, and writing no more to Hume for a while. Hume was mystified by this behavior, but went about getting the secrecy condition removed; he assumed that Rousseau was not writing him because Rousseau was ashamed at having treated Hume poorly in this matter. He suffered a severe shock in this complacency, however, since Rousseau accused Hume of trying to dishonor him, saying to Hume that he had fooled the public but could not fool himself. Hume indignantly replied:
You say that I myself know that I have been false to you; but I say it loudly, and will say it to the whole world, that I know the contrary, that I know my friendship towards you has been unbounded and uninterrupted, and that though instances of it have been very generally remarked both in France and England, the smallest part of it only has as yet come to the knowledge of the public. I demand that you will produce me the man who will assert the contrary; and above all, I demand that he will mention any one particular in which I have been wanting to you. You owe this to me; you owe it to yourself; you owe it to truth and honour and justice, and to everything that can be deemed sacred among men.
For the Scottish philosopher, Rousseau's action seemed a collapse into madness of someone who had already been on the border of it a very long time. Hume, who always had admitted that his ruling passion was his love of his literary reputation, began to be seriously afraid, from the fact that knowledge of the rift began to spread, and from remarks Rousseau had made, that the Swiss philosopher would publish something that would harm that name which he had spent his life trying to earn. To forestall any chance that Rousseau would damage his reputation, then, Hume preemptively published all their correspondence and his own account of the dispute. This poisoning of the wells was very calculated: if Rousseau published before Hume, Hume would have to convince people who had already read Rousseau's account -- which no doubt would be very well-written; worse, it might be published after Hume's death (in which case Hume could not respond); and even worse, it might be published after Rousseau's death (in which case Hume would be seen as attacking a dead man). Several of Hume's friends in France (Voltaire was an exception) had urged him to publish his own account; but his friends in Scotland and England had overwhelmingly urged him not to do so, insisting that Rousseau's charges needed no answer. Hume appeared to accept his British friends' advice in this matter; but then he later claimed that his French friends 'extorted' the publication from him, claiming that Rousseau had sent letters to all the courts of Europe. The British were taken aback by the move; Walpole's reaction was typical: "Good God! my dear Sir, could you pay any regard to such fustian?"
Hume was certainly generous in his aid to Rousseau, and Rousseau's excessive 'sensibility of temper', his manic-depressive behavior, is hard to sympathize with. Nonetheless, le bon David was not perfectly good in the proceedings. Hume in general tended to handle criticism very poorly; it shows in his correspondence with his friends, since he tends to make remarks about the character of people criticizing his arguments. He had always very wisely refrained from responding to criticism at great length or in print, however. In this case, however, he was pushed by fear and vanity into doing exactly what he had wisely avoided doing all this time. Rousseau he regarded as so dangerous that the mere hint that Rousseau might blacken his reputation by publishing against him leads him to blacken Rousseau's preemptively. And let us not have any illusions: the Concise and Genuine Account is not an account free of bias, but is a deliberate attempt to destroy any credibility Rousseau might have in the matter. Before the incident, Rousseau is a misunderstood genius, a little excessive and unhappy in society, but good-hearted and friendly, deserving of compassion for his impatience and melodrama; afterward he is suddenly the most blackhearted of rascals, a wicked liar whose compositions, while eloquent, are praised above their merit. This is, whether it is usually recognized or not, as wild and startling a swing as Rousseau's. And certain curiosities about Hume's account have been noticed. For instance, Hume's account of the satirical letter that played such a major role in the rift uses terms very similar to those used by the St. James Chronicle when it published it. It can be read two ways; either Hume imprudently plagiarized St. James Chronicle, or he had more of a hand in it than he lets on and has inadvertently let a clue to that effect slip. Much that is found in the account admits of such double reading; it raises as many questions as it answers to those who do not accept Hume's assumption of Hume's complete innocence going into it.
It is also interesting that defenses of Hume tend to be based on two assumptions, neither of which allow for objective view of the evidence. (1) That Hume was all amiability and sociability. There is no question that Hume was indeed amiable and sociable, good-natured and generous. That he also was able to pull a Rousseau on a smaller scale is clear from his letters to Strahan. There was more than one person at the time who noted that Hume and Rousseau were similar in vanity. (2) That Rousseau was mad. No one will say Rousseau was what we could call emotionally stable, given his obvious mood swings, and he certainly had difficulty telling imagined attempts to mock him from the many real attempts to mock him, but the accusations of madness are based on reading a great deal into the emotional excess of Rousseau's unguarded letter of accusation.
Seen in a charitable light, one can well believe that Hume was misunderstood by Rousseau; that he intended nothing but good to his philosophical hero; that he was shocked by finding out just how flawed Rousseau really was. But it also shows him to be a man so dominated by a love of reputation that all the pity he felt toward Rousseau could be extinguished if his reputation were threatened, even though that pity had been for exactly the same personality traits that now threatened it. It shows him willing to blacken another's reputation at the mere hinted threat of his own being blackened. It's a very human flaw. The jury is still out on how honest le bon David really was in his concise and genuine account; in the documents we have Rousseau, overwrought, takes no trouble to present himself in a good light, but Hume very clearly does, and a serious, careful, and unbiased evaluation is required before one can say how much Hume's account really accords with the evidence, how much of it is based on an accurate interpretation of Rousseau's actions, and how much of it stands when the merely self-justificatory is removed. But a first approximation shows the picture of a man, not a saint, faced with another man of very different personality and background, not a lunatic.