For more than two decades, most Secretaries of State for Northern Ireland and prime ministers have worked on the theory of finding the middle ground, establishing a devolved administration there and patiently building up a consensus from the centre out.
In this theory, the extremes were to become gradually more and more isolated and would finally wither away. This approach had many drawbacks, not the least of which was that it never worked. In 1974, the only time that the first hurdle was overcome and a power-sharing executive created, the whole experiment fell apart under extremist pressure in less than six months.
One very obvious problem was the Rev Ian Paisley, whose anti-establishment instincts generally ensured he would attack any agreement among the Northern Ireland parties. Another problem lay in the fact that there are in fact two middle grounds, one Unionist and one nationalist. Even moderate nationalist politicians insist that their Irishness should be recognised in some form of Irish dimension and link with Dublin. But even moderate Unionist politicians were unsettled by this, worrying that it would weaken their link with Britain.
A further blow to the theory came with the emergence of Sinn Fein as a significant political presence in the early 1980s. Until then, part of the standard rhetoric of government ministers was to declare that only a tiny minority supported violence. Sinn Fein's achievement of winning the support of three out of every ten nationalist voters put an end to that line of argument.
Another problem was that as the years passed and the various attempts failed, fewer and fewer people maintained any belief that this course might lead to peace.
But lack of success with this approach did not stop Northern Ireland Office ministers and officials persevering with it. Sir Patrick Mayhew recently complained that people accused him of whistling in the dark in his efforts to restart inter-party talks, saying only a few weeks ago: 'While the talks retain so much potential, the priority must be to carry them forward, not cut across or duplicate them.'
Yesterday's joint declaration represents a dramatic departure from that traditional approach. The concept of building incrementally from the middle out is, for the moment at least, taking second place to the new idea of attempting to draw the extremists into the political processes.
The new approach received such widespread support in London and Dublin yesterday that it is easy to forget how controversial a move it seemed a short time ago.
When it emerged in the spring that John Hume, the SDLP leader, was engaged in dialogue with Gerry Adams of Sinn Fein, there were strong responses. The Independent reported at the time: 'Reaction has ranged from furious Unionist denunciation to warm endorsement of his actions, but between these two poles there is considerable bewilderment about the significance of it all.'
At that point the idea of talking to terrorists or those associated with them was one of the most delicate and dangerous issues in Irish politics. When Mr Hume announced that he and Mr Adams had reached a measure of agreement and said he was presenting this to the Irish government, some Dublin sources made it clear they were furious about having to handle anything bearing Mr Adams's thumbprints.
Dublin's decidedly cool reaction to the Hume-Adams initiative was followed by outright rejection from John Major, who declared in the Commons that the thought of talking to Mr Adams 'turned his stomach'. At that point the initiative seemed to be dead, but a wave of popular nationalist support, north and south, not only revived it but propelled it to centre-stage of the Anglo-Irish scene. Albert Reynolds was forced to reverse his original judgement and become the idea's chief proponent. His position was explained by an associate: 'The Hume-Adams idea is so big that you can't just leave it on the landscape and walk away and say, 'Very interesting idea but we have something else to try.' It's far too big to do that: you have to resolve it into success or failure.'
For some weeks it seemed the Hume-Adams initiative - sanitised, by this stage, into the Reynolds initiative - was going nowhere fast. An Anglo-Irish negotiation opened in which the Taoiseach sought to persuade John Major that the IRA could be induced to lay down its weapons and take a political path. Hume-Adams has never been published, but it is said to include a draft declaration on self-determination, assurances to Unionists that majority consent would be required for a united Ireland, emphasis on agreement, a reference to a new European context, and assurances that Sinn Fein can enter political life if the IRA abandons terrorism.
The Major-Reynolds declaration touches on all these points, though both governments are playing down its antecedence. For the moment the thumbprints of neither Hume nor Adams are unwelcome.
The Taoiseach's conversion to the new approach was sudden; the Prime Minister's change of tack came more gradually and was effected more elegantly. His problem was that he was anxious to retain the parliamentary support of James Molyneaux and his nine Ulster Unionist votes; and Mr Molyneaux clearly thought the new approach was not worth trying.
The Unionist leader has deep reservations about the declaration: his opinion seemed to harden during the day yesterday, culminating in his observation last night that it 'has the makings of a betrayal'. But he was briefed before publication and indicated to the Government that he would not denounce it.
Mr Major had therefore contributed to a remarkable political feat: the assembly of a document which stood some chance of being acceptable to the IRA, and which at the same time appeared to have the acquiescence of Northern Ireland's largest Unionist party.
While it has the potential to do both, there is no guarantee that it will do either. The considered reactions of the IRA, and the Unionist community, will take some time to emerge but it is quite possible that the coming days and weeks could bring heightened dangers - at worst, no IRA cessation, coupled with a rise in loyalist violence.
But even such a disastrous short-term outcome is unlikely to kill off completely the notion that addressing the terrorists directly could bring eventual peace. Even if the approach does not work immediately, there will be further efforts to bridge the gap between the IRA and the political system.
A fundamental change has been made to the political agenda, for the tantalising prospect that peace could be achieved by this path is simply too big to be ignored. There is more and more hope that it is an idea whose time has come.