Barack Obama will have weighed every word carefully as he announced the successful operation to kill Osama Bin Laden. He will know that his statement in the White House will be played again and again on news bulletins around the world today. It was an announcement with one major fact in it – Bin Laden’s death – but with ramifications that would be felt around the world.

His tone was faultless; serious, sober, not a hint of triumphalism, a tribute to the forces who carried out the operation, with a warning that this was not the end of the story, and the important message that the West is not at war with Islam.

None of that could stop hundreds of Americans rushing to the White House to celebrate, some shouting ‘four more years’, seeing political mileage in the capture and killing of America’s Most Wanted. There may well be political benefits, but I suspect Obama would be the last to want that kind of reaction right now. 9/11, and Bin Laden’s role in it, are seared deep in the US psyche, across the admittedly bitter political divide. He got the tone right. Those shouting ‘four more years’ are not.

At this stage, unrurprisingly, the Western media are focusing on reaction here, but as the day develops the reactions of politicians and people in the Arab and Muslim world will have just as much significance. In the past, Al Qaida leaders, though none as senior as Bin Laden, have been taken out, and reprisals have followed. It is why David Cameron and shadow foreign secretary Douglas Alexander have been sensible to point out the continuing need for vigilance.

Bin Laden is dead. But the terrorist groups he led and inspired are not, which is why Obama rightly eschewed triumphalism, and got the tone of his announcement spot on.