It has just begun to sink in. President William Jefferson Clinton is on trial for his political life. It was not until Chief Justice William Rehnquist appeared in the Senate chamber to swear in the 100 senators that the reality of what was happening to the President came home to most people. He has been impeached for alleged "high crimes and misdemeanours" and is now being tried on charges of perjury and obstruction of justice.

We are in for a period of "split screen" presidency, where television pictures of the Senate trial and the accused going about his presidential duties will be beamed to a baffled nation. How has it come to this for a President at the peak of his popularity across a country enjoying the greatest spell of economic prosperity in its history? What happened to all the schemes for letting him off with a censure to stop wasting the time of Congress over the details of a tawdry sexual dalliance in the White House?

A lot of people are asking those questions as the Senate has run straight into a partisan wrangle over how to conduct the trial under rules drawn up for that of President Andrew Johnson 131 years ago. And can 100 politicians really do "impartial justice" to the President in their role as jurors? Were some of them committing perjury as they swore that oath, some commentators speculated.

Republican Senator Ted Stevens of Alaska put it well when he asked: "Can we get to the point where people can really think as jurors rather than as Republicans or Democrats?"

Everybody is playing around with the arithmetic. There are 55 Republican senators and 45 Democratic senators. Most of them will make up their minds, if they have not done so already, on political rather than forensic grounds such as DNA traces on a blue dress from The Gap.

The whole business hinges on the numbers. It will take 67 senators - a two-thirds majority - to convict Mr Clinton on one or both of the articles of impeachment. Assuming the 45 Democrats are determined not to convict, then the 55 Republicans are short the 12 votes needed to remove the President and allow Vice-President Al Gore take over.

The conventional wisdom - a rather battered concept in Washington these days - says there is no way a two-thirds majority can be found to convict. That is why last week's compromise plan to halt the trial and have a "test vote" to show the numbers are lacking seemed such a good idea.

Senate majority leader Trent Lott and his Democratic counterpart, Tom Daschle, were ready to accept this. It would have halted a trial which was going nowhere and at the same time given the Senate a chance to drum up a censure motion which the President would have gratefully accepted as the lesser of two evils.

But Senator Lott, who is a noted conservative himself, did not reckon with the anti-Clinton zealots from the House Judiciary Committee who are now the prosecutors or "managers" in the Senate trial.

Led by Henry Hyde, a courtly lawyer from Chicago, these Republicans are demanding a trial where witnesses like Monica Lewinsky and White House aides can be called. And it must be said that an unknown number of the Republican senators want this as well.

One of the prosecutors, Asa Hutchinson from Arkansas, says that when there is a conflict of testimony - as there is between Mr Clinton and Miss Lewinsky on the details of their trysts - "the only way to resolve it is to see the demeanour of the witnesses, evaluate their credibility, look into their eyes and determine where the truth is".

But for some Republican senators the prospect of Miss Lewinsky being questioned on the Senate floor about intimate details leaves them as well as the White House appalled. If there were just six who balked at that kind of a trial, they could join with the 45 Democrats and vote by simple majority to end it.

There are 19 Republican senators up for re-election in two years. Polls show that moderate Republican voters have no desire to see a drawn-out trial.

The White House is playing a canny game by offering to agree to a trial based on the thousands of pages of grand jury evidence in the report of Independent Counsel Kenneth Starr, but without any witnesses being called. If, on the other hand, the Republicans get their way and there are to be witnesses, then "all bets are off" according to White House spokesman Joe Lockhart.

The White House legal team will then call its own witnesses, file motions, seek new evidence and so "you're talking about a process that is open-ended and could go on for ever and ever," warns Mr Lockhart.

And who would get the blame? The Republicans, of course, for their vindictive attempts to lynch a popular President.

But is it that black and white? The Senate has the duty under the Constitution to try the President on the impeachment charges. Would it not be failing in its duty by aborting a proper trial through holding a "test vote"? Has it not a duty to hear witnesses and judge the evidence accumulated by Mr Starr for themselves?

The problem is that this is not really a trial as most people understand it. Even if two-thirds or even all the 100 senators end up believing the President committed perjury when asked about the details of his affair with Miss Lewinsky, they could still honestly conclude this does not rise to the level of "high crimes and misdemeanours" which the Constitution does not spell out but leaves to the Senate to decide for itself.

So Mr Clinton could be guilty of perjury and still stay President. This is why the voting, if it comes, will be essentially political - oath to do "impartial justice" or no oath.

The obstruction of justice charge would be harder to prove. The Republicans would have to show conclusively that Mr Clinton asked Miss Lewinsky to deny their affair under oath and coached his secretary, Betty Currie, and other aides to give false testimony.

If the polls are to be believed, most Americans believe the President lied but that this does not warrant dismissal from office. Most of the 100 senators almost certainly believe this as well. But now they are stuck with holding a trial few people want to come to the same conclusion.