The two civil servants looked at what might reasonably be achieved by reasonable men. Neither was an ideologue. TK Whitaker was 52 at the time of their meeting. Sir Arthur Snelling was two years his senior. One was in the throes of modernising the economy of a young Republic. The other was managing the decline of a once-vast empire.
It happened just over half-a-century ago last week Whitaker came over from Dublin to meet Snelling and other officials to discuss the state of the Republic's sterling reserves - this was long before the EU and the euro but only two years since the re-emergence of the UVF on the streets of Belfast. It was also - though the two men could not have known it - the last year of peace in the North for three decades.
They talked about sterling, about North-South economic cooperation, and the problems of agriculture. Snelling was Under Secretary at the British Foreign Office and had served as High Commissioner to Ghana around the time of independence. He had also dealt with white supremacists who led Southern Rhodesia into a unilateral declaration of independence. That was three years before his meeting with Whitaker.
Aware of his interlocutor's particular experience, Whitaker broached the old question. What was the current attitude of the British government towards Irish unity?
"He said," wrote Whitaker, "that he thought the British government, having been plagued with the Irish question for so long, wanted nothing more now than not to be disturbed by any problems relating to the unification of Ireland. This did not mean that they adopted a frigidly neutral attitude. Rather was their attitude one of benevolence towards any solution that might be agreed upon in Ireland between Irishmen. They would not, however, do anything to push Northern Ireland into a United Ireland."
Snelling's account largely matches that of his Irish counterpart and remembers how Whitaker wanted to know if the British would still subsidise Northern Ireland after unity.
"It was a new thought to me and I could not imagine an enthusiastic reception in any quarter in London," he minuted. As far as ending partition was concerned, that was "primarily the job of the man in the South to woo the girl in the North. If successful I thought we would be prepared to give the bride away and lead her to the altar but not prepared to put pressure on her".
Such was the language of the last months before reasonable men were shoved aside. The following October - after months of steadily growing tension - police baton-charged a Civil Rights march in Derry. The slide to catastrophe had begun.
Spool the haunted film forward to the same week four years later. Two British soldiers were killed in a landmine attack in Cullyhanna, Co Armagh. An IRA man was shot dead in a gun battle with the RUC. Six years later the same week: two Catholics - 16-year-old Thomas Donaghy and 18-year-old Margaret McErlean - were shot dead by loyalist paramilitaries as they arrived for work in Newtownabbey, Co Antrim.
Twelve years later, Constables Joseph Rose and Winston Howe were killed in an IRA landmine attack in Fermanagh. Eighteen years to the week after Snelling and Whitaker sat down, the IRA burst into a bar in Maguiresbridge, Fermanagh and shot dead Constable Derek Breen and the bartender, John McCabe. One was Protestant, the other Catholic. Twenty six years later, the INLA leader Dominic McGlinchey was shot dead in Drogheda.
Twenty-eight years on, the IRA ended its ceasefire by detonating a huge bomb at Canary Wharf, killing Inan Ul-Haq Bashir and John Jeffries. They were both victims of an 'inadequate' bomb warning from the IRA.
Thirty years later Robert Dougan was shot dead near Belfast. He was a leading loyalist and the IRA was blamed for the killing. We need to keep naming these names. What else can make a better claim for peace than these remembered names, dates, places?
Yet a few months after the killing of Robert Dougan - on Good Friday, 1998 - a peace deal was agreed. The parties in Northern Ireland decided they would share power in a devolved administration. The Republic gave up its constitutional claim on the six counties that constitute Northern Ireland. Irish unity would only come about by consent. Dublin now had an acknowledged say in the affairs of the North after years of unionist rejectionism.
Sinn Fein was in government - a government that operated within the constitutional framework of the British state. Sinn Fein MPs still refused to take their seats at Westminster but they had opted - for now - in favour of Michael Collins's politics of 'stepping stones'.
It was the right thing to do. Morally. Practically. Unity by consent - as TK Whitaker had espoused decades before - was the choice of reasonable men. As President Higgins put it after Whitaker's death in 2017: "The Good Friday Agreement would be unimaginable without TK Whitaker."
The question now is whether the Good Friday Agreement remains imaginable at all. It is too convenient to place all the blame on the distractions and tensions of Brexit. The agreement is also imperilled by an absence of will.
It is as if the further away we move from the terrible events of the Troubles, the less commitment there is to ensuring they never return. I want to say until I am hoarse: It is not over. The war can come back.
British cabinet ministers are leaking that Theresa May is worried a no-deal Brexit will lead to a border poll. The purpose of this leak is presumably to warn the DUP's supporters that they could be on a slippery slope to a United Ireland if their leaders fail to back whatever deal Mrs May eventually puts before parliament.
In Dublin, Leo Varadkar raises the spectre of resurgent Republican violence in the event of a hard border. All of this underlines how far we have drifted from the days of Blair and Ahern and a united approach to resolving the disaster of the Troubles. Certainly a hard border would give Republican ultras something to attack. And a no-deal Brexit could conceivably push nationalist (with a small 'n') Catholics and more moderate unionists to look favourably on a United Ireland within Europe.
But what then? Look beyond Brexit. No backstop and no border poll can cure the sectarianism which remains rife. It cannot disarm paramilitaries who still nauseatingly present themselves as the protectors of their communities, especially in loyalist areas.
It cannot create a deal between Sinn Fein and the DUP to end the political stalemate.
Agreements between London and Brussels can take the immediate tension out of the air. But they can't cure the old malady. That is up to ourselves... and we are running out of time.
Fergal Keane is a BBC Special Correspondent