History professors Tom Devine, Christopher Smout and Michael Lynch, and archaeologist Dr Anna Ritchie - all former trustees of the National Museums of Scotland - say they fear standards will slip without expert advice at a crucial time.
They have now written to the new Scottish Government to make their concerns clear about the make-up of the current museum leadership.
They claim the NMS board, chaired by banker Sir Angus Grossart, now has more financial, managerial and legal clout than academic expertise.
The Society of Antiquaries, which was gifted a statutory place on the board but not given a seat during the last round of appointments, has also joined the protest.
Retiring president Barbara Crawford, said the society's candidates were rejected at the last round of appointments and questioned whether skills in "management, industrial relations or administration" are now more important qualifications than archaeology for museum trustees.
Smout, of St Andrews University, the Historiographer Royal in Scotland, warned: "The danger is that scholarly standards begin to slip, and mistakes in the presentation of exhibits, or in the acquisition of things for display, are made."
Lynch, professor of history at Edinburgh University, said it was "almost unthinkable" that there was no-one on the board with an expertise in Scottish history or archaeology.
The row has emerged as NMS prepares for the reopening of the Royal Museum building in Edinburgh in July with 16 new galleries after a three-year, 46 million overhaul.
Lynch, who left the board last year, said he had raised the matter in meetings with the museum's director, Gordon Rintoul, Grossart, and the culture minister, Fiona Hyslop. The Scottish Government appoints trustees in consultation with the chairman, but Lynch said the response of the SNP administration so far, after an exchange of letters, has been to "fudge the issue".
Devine, the University of Edinburgh historian and award-winning author, stressed the NMS was an "absolutely excellent institution" and he was looking forward to a "stunning set of new galleries" at the Royal Museum.
But he noted that he and other past trustees had worked closely with the NMS on displays at the new Museum of Scotland building in the 1990s. "The basic issue is the expertise and advice," he said. "It's disappointing that the tradition of having an historian or an archaeologist has at least temporarily been broken."
All the academics involved in the dispute stress they are not criticising the museum's staff or current board members. But they felt compelled to write to Hyslop, reappointed last week as Scotland's culture supremo, expressing "deep concern".
"Certainly, there has, in the past 20 years, never been a board of trustees which has not had at least one scholar - and at times two - with expertise in the field of archaeology or history," the letter states.
"It is all the more extraordinary at a time when, it is often said, there has been a renaissance in the study of Scotland's past, led by a generation of scholars of archaeology and history... and it is all the more surprising that such a position has come about under a government of the Scottish National Party."
NMS runs five museums in Scotland, including the flagship Royal Museum on Chambers Street. There are two academics on the board at present, Professor Malcolm MacLeod, a leading ethnologist and a former director of the Hunterian Museum in Glasgow and Professor Stuart Monro, a geologist and scientific director of Our Dynamic Earth in Edinburgh. Both were appointed in 2005.
But the dispute flared privately when the three trustees named last year - as Lynch departed - included Miller McLean, a chartered banker and solicitor, James Troughton, an architect and real estate developer who has worked with several galleries, and Bruce Minto, who co-founded law firm Dickson Minto.
A Scottish Government spokeswoman said: "The Scottish Government and the National Museums Scotland regularly review the balance of skills required for the board of trustees of the National Museums of Scotland, and reflect those requirements in the criteria for recruitment rounds.
"All appointments are selected on merit against agreed criteria under public appointments' principles."
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