“An expert is a man who avoids the small errors as he sweeps on to the grand fallacy” is a jocular statement showing that experts can be technically correct in small matters and yet wildly incorrect in larger matters. Laurence Stallings (1894-1968) wrote in the Introduction to The First World War: A Photographic History (1933):
 
” A military expert, to paraphrase, is one who carefully avoids all the small errors as he sweeps on to the grand fallacy.”
 
Benjamin Stolberg (1891-1951), a labor historian, was quoted in the Reader’s Digest in 1936, “An expert is a man who avoids the small errors as he sweeps on to the grand fallacy.” Stolberg is usually given credit for the line today, but it’s likely that he was simply quoting Stallings.
 
   
Wikipedia: Laurence Stallings
Laurence Tucker Stallings (November 25, 1894 - February 28, 1968) was an American playwright, screenwriter, lyricist, literary critic, journalist, novelist, and photographer. Best known for his collaboration with Maxwell Anderson on the 1924 play What Price Glory, Stallings also produced a groundbreaking autobiographical novel, Plumes about his service in World War I, and published an award-winning book of photographs, The First World War: A Photographic History.
 
OCLC WorldCat record

The first World War : a photographic history
Author: Laurence Stallings
Publisher: New York : Simon & Schuster, 1933.
Edition/Format:   Book : English
(From the Introduction—ed.)
In this anthology of pictures of the first world war there was no effort to satisfy any special interest or taste. A militarist will be disappointed in them for there are not enough pictures of guns and tactical groups. A pacifist will not find enough horror, nor enough of cadavers. And a student of war can hardly follow, from these pictorial representations, the methods of infantry combat slowly evolving from close-packed slaughter of the trenches to the loosely-held butcheries later on…The editor is conscious of his short-comings in the matter of captions. Many should be more expert, more military. A military expert, to paraphrase, is one who carefully avoids all the small errors as he sweeps on to the grand fallacy. This book, at least, avoids that fallacy. There is no conclusion to it. Man made this world in four years, and saw that it was good, if we are to believe Versailles. Well, here it is in the making, just as man made it, caught by many a camera eye. The pictures are placed more or less chronologically, but for the most part in a senseless fashion…If this picture book survives, doubtless it will get in time another preface, and one which will make sense out of chaos.
 
2 August 1933, Charleston (WV) Daily Mail,  “Books and Things” by Lewis Gannett, pg. 6, col. 6:
Mr. Stallings writes: “The editor is conscious of his shortcoming in the matter of captions. Many should be more expert, more military. A military expert, to paraphrase, is one who carefully avoids all the small errors as he sweeps on to the grand fallacy.”
 
1 February 1934, San Antonio (TX) Express, pg. 8, col. 5:
(An excerpt from Laurence Stalling’s book, The First World War: A Photographic History—ed.)
“A militarist will be disappointed in them, for there are not enough pictures of guns and tactical groups. A pacifist will not find enough horror…A military expert, to paraphrase, is one who carefully avoids all the small errors as he sweeps on to the grand fallacy. This series, at least, avoids that fallacy.”
 
Google Books
The Reader’s Digest
Volumes 28-29
1936
Pg. 92:
BENJAMIN STOLBERG:
AN EXPERT is a man who avoids the small errors as he sweeps on to the grand fallacy.
 
27 August 1936, The Robesonian (Lumberton, NC), pg. 5, col. 3:
Benjamin Stolberg: An expert is a man who avoids the small errors as he sweeps on to the grand fallacy.
 
Google Books
The Anti-Communist Manifestos: Four Books That Shaped the Cold War
By John V. Fleming
New York, NY: W. W. Norton & Company, Inc.
2009
Pg. 30:
Among the other members was Benjamin Stolberg, the labor historian and the aphorist who first defined “an expert” as “a person who avoids small error as he sweeps on to the grand fallacy.”