Eight years ago, Herbert R. Collins, curator of political history at the Smithsonian, was watching the inaugural parade and saw a Secret Service agent take off his overcoat and hand it to President Nixon, who was coatless for the chilly ride down Pennsylvania Avenue. On the reviewing stand, the president returned the coat to its owner.

"I called up the next day and asked the Secret Service guard whether we might have the cost when he finally decided to discard it. Only last month his wife called and sent us the coat," Collins said yesterday with the satisfaction of a man who made a valuable acquisition.

The borrowed coat has become one of the odd artifacts in the collection of the Smithsonian. The Nixon coat won't be in the special exhibit opening today at the National Museum of American History. But one item will be the overcoat worn by President James Garfield while taking his oath of office in 1881.

This year's special inauguaral exhibit at the Smithsonian spans ceremonies from George Washington to the present. There are clothing buttons worn to celebrate George Washington's inaugural in 1789. And there is the daytime jacket suit worn by Nancy Reagan at the Republican National Convention.

"We made the mannequin to fit the dress. therefore, the measurements should be quite close," explained Margaret B. Lapthor, who, like Collins, is a Smithsonian curator of political history.

Klapthor's first memory of presidential inaugurals dates back to childhood. She remembers attending the second inaugural of Franklin Delano Roosevelt in 1937 and didn't miss being on the parade route until four years ago, when arthritis limited her outdoor attendance. She expects to be watching the Reagan inaugural activities on television this year, looking for such things as a borrowed overcoat that might becomes a historical artifact for a Smithsonian inaugural exhibit in the future.

The Smithsonian can put on a new show every four years because it has such a large collection of memorabilia on inaugural events.

"Then there are certain pieces that are so extraordinary that we do not hesitate to display them each time so that the public can see them," Klapthor said. "There is the Lincoln table [used by Lincoln when he took his oath of office in 1865]. And, of course, the quarter panel from the Washington coach [which took George Washington to his second inaugural in Philadelphia in 1793]."

This year's inaugural exhibit, which will run until March 15, includes a display of inaugural medals, brought up to date by the original plaster model used to strike the Reagan inaugural medals. Elvira Clain-Stefanelli, a Smithsonian curator, arranged the medals for this exhibit.

"I Do Solemnly Swear" is the title of the section that covers inaugural oath-taking, parades, balls and souvenirs.

"The only thing that is necessary is a simple oath of 35 words," Klapthor said. "Everthing else is window-dressing -- the ceremonies on the Capitol steps, the parade, the inaugural balls and festivities. It's now grown to a week of ceremonies."

The simple oath has been repeated by every president entering the office. But each has brought his own touch to the inaugural address. The shortest was the 135-word address by George Washington at his second inaugural in 1793. The longest (8,445 words) was delivered during a snowfall by William Henry Harrison. He paid for his prolixity. Harrison caught a cold and died of pneumonia a month after his inaugural.

"Just as Harrison was nearing the last paragraph, his speech was interrupted for the oath-taking. Then he returned to his address. I don't know whether they wanted to get the oath over before he died or though the speech had to be over," Collins observed.

The Harrisons were given to great length in speeches and parades. Benjamin Harrison's parade was so long in 1889 that darkness overtook the marchers, some of whom disbanded before passing the reviewing stand. Since the first inaugural, seven have been in snow and 10 in rain.

If Ronald Reagan has his "Thumbs Up" anthem for the 1981 inaugural, President Monroe also has his "march for the piano forte, to which is added a favorite waltz."

The National Museum of American History will be the scene of one of the Reagan inaugural balls.It is another tie to the history of presidential inaugurals. The Arts and Industries Building, then just completed, was used for the inaugural ball of 1881 for President Garfield.

Mrs. Reagan's jacket suit of oyster white was worn at both the Republican convention and on the campaign trail with her husband. It was designed by Adolfo, who gave it to the Smithsonian. The "Salute to the First Lady" section of the inaugural exhibit also includes photos of Mrs. Reagan from Women's Wear Daily to show her variety of clothes.