They are fighting around one of the great sites of our Western civilisation, from where, the Bible tells us, came the progenitor of the three great monotheistic religions, Abraham. Ur of the Chaldees dates back almost 5,000 years.
A little further up the road, on the left hand sweep towards Baghdad, is Uruk, probably the world's first city. It had a population of 50,000 almost 5,000 years ago.
Still further up that Baghdad road along the Euphrates is Kish, believed to be the first city rebuilt after the Flood.
As they get closer to Baghdad and before yesterday's sandstorms, American soldiers must have been able to see the golden dome of the mosque of Najaf. There lie the remains of one of the most revered figures in Islam, Ali ibn Abu Talib. Ali was the closest male relative of the prophet, Muhammad, and became the third caliph of Islam in 656 CE, 24 years after the death of the prophet.
His leadership was challenged and he was murdered. His followers (the Shias), in accordance with his declared wishes, tied his body to a camel and let it roam in the desert, stopping to rest at Najaf. There Ali was buried and the mosque built over his tomb. It is now one of the great places of pilgrimage for the Shias.
Kerbala, closer to Baghdad, has also been on the graphics of the television war generals, is another place of holiness. There the great massacre of the Islamic civil war occurred, when Hussein, son of Ali, died with a sword in one hand and the Koran in another, having proclaimed "death with dignity is better than a life of humiliation".
Right beside Kerbala is Al'Ukhairid, where there is an amazing desert palace dating from the 7th century. Nearby is Kufa, where Ali was assassinated. And then Babylon, the centre of a great empire in the 5th and 6th centuries BCE, where Israelites were taken into exile after the destruction of Jerusalem by Nebuchadnezzar in 586 BCE. Ashur, 70 miles south of Mosul, is one of the ancient capitals of the Assyrian empire. Mecca and Jerusalem are the sacred places of Islam, Judaism and Christianity, but Iraq is the sacred place of our shared civilisation.
They are dismantling that civilisation, not just its symbols and archaeological heritage, but the fabric of "civilised" coexistence that was falteringly woven from that history.
The Americans and British may have underestimated the effort of subduing Iraq. The number of ground troops deployed may be regarded as insufficient for a swift defeat of the Hussein regime. They seem to have underestimated the scale of the opposition they have faced. But they can hardly fail to prevail. The question now is at what cost?
Civilian casualties so far appear to be low and the flight of refugees has not happened. But who can tell from here of the humanitarian crisis enfolding in Basra, deprived of water and electricity? Who can foretell what is likely to happen in Baghdad once the assault on the city commences from the ground?
And for what?
Remember this war was supposedly to disarm Saddam. So far, there is no evidence of weapons of mass destruction. Maybe even before this column appears such evidence will have emerged, but it seems peculiar that Saddam has not used such weapons if he has them. (If he has them and is not prepared to use them, how was it he was such a threat?)
Joschka Fischer, the German Foreign Minister, in an interview with Der Speigel, published yesterday, said last September Paul Wolfowitz, the US Deputy Secretary for Defence, and the ideologue for this war, told him (Fischer) "the US had to liberate a whole string of countries from their terrorist rulers, if necessary by force".
Is this the first of such "liberations" to be undertaken in defiance of world opinion and of the United Nations and at a cost in lives, welfare and financial resources that is horrifying.
There has been "shock and awe" all right. Shock and awe over the declared cost of the war - $75 billion according to the Bush administration, between that and $500 billion according to William D. Nordhaus, a Yale economist, quoted in yesterday's New York Times. This at a time when the Bush administration is proposing a tax cut of $500 million for the richest 1 per cent of the American population, a benefit that (again according to yesterday's New York Times) will require cuts similar to that amount in health care, child care, education and environmental protection.
On Monday, the US State Department announced in some triumph that the US government was going to contribute $102 million to the United Nations and international organisations for humanitarian relief in Iraq. A little over 1 per cent of the cost of the war is to be given in humanitarian relief.
Postscript: Jean Hasset of Cork, the president of GROW (the self-help organisation for people suffering from mental illness) died suddenly. She was an extraordinary woman. Physically: tiny and frail. Personally: warm, kind, humorous, deceptively shrewd and determined. She did more for vulnerable people than all but a few I know of and she did it with a light touch and a mischievous smile.