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The phrase back of an (or the) envelope is used figuratively to indicate that something (such as a calculation, a plan, etc.) has been formulated or devised hurriedly, roughly or carelessly (as though sketched or scribbled on the back of an envelope), and is lacking in detail or not fully formed.
—Synonyms: back of a napkin and back of a cigarette packet.
The actual practice of jotting down something on the back of an envelope was evoked, for example, in the following passage from Jalna, by the Canadian author Mazo de la Roche (1879-1961)—as published in The Atlantic Monthly: A Magazine of Literature, Science, Art, and Politics (Boston and New York: The Atlantic Monthly Company) of July 1927 [page 123, column 2]:
—Note: There is a dangling participle in the first sentence of this passage:
Sitting in a tearoom, the first lines of a new poem began to form in his mind. Pushing his plate of cinnamon toast to one side, he jotted them down on the back of an envelope. A quiver of nervous excitement ran through him. He believed they were good.
The earliest attributive uses of back of an (or the) envelope that I have found are as follows, in chronological order—this phrase seems to have originated in American English:
1-: From Losses, by Bruce Barton, published in several newspapers on Sunday 27th December 1931—for example in The Greenville News (Greenville, South Carolina, USA) [page 5, column 3]—the reference is to the Wall Street Crash, i.e., the dramatic collapse of share prices on the New York Stock Exchange in October 1929:
My friend uttered an important truth.
“The trouble with these fellows in Wall Street is that they have taken their losses fifteen times a day for two years,” he said. “Think of it, fifteen times seven hundred. What a loss that makes. Nobody can stand a loss like that. If they’d put away their lead pencils; if they’d quit figuring on the back of envelopes and the margins of newspapers, and forget the whole thing, they would be much better off. Taking your loss fifteen times a day doesn’t get you anywhere. It uses up brain cells and nervous energy that might be used for progress.”
[…]
[…] I am weary of the boys who tell me how much they would have had if they had sold everything in the summer of 1929—the back-of-the-envelope lads who take their losses fifteen times a day.
2-: From the column New York Day by Day, by ‘O. O.’ McIntyre (1884-1938), published in several newspapers on Tuesday 2nd May 1933—for example in The North Adams Transcript (North Adams, Massachusetts, USA) [page 4, column 6]:
New York—In clearing the desk today I came upon some back-of-an-envelope jottings made at the annual Dutch Treat dinner a few weeks ago. They were for no purpose—merely idle observations of artists and writers who dotted the various tables.
3-: From the column Today’s Inquest, by John Barry, published in the Boston Evening Globe (Boston, Massachusetts, USA) of Friday 6th October 1933 [page 32, column 7]—the reference is to Rockingham Park racecourse in Salem, New Hampshire:
Here you are, folks, the figgers from Rockingham for the benefit of all who are interested in what the other fellow is making. […]
But to go back to Rockingham for a moment—and this should interest Neal O’Hara, too, for his back-of-an-envelope figuring made the profit $750,000—let’s look at track figgers.
4-: From Dante Proves A Problem for Movie Experts, by H. H. Niemeyer, published in the St. Louis Post-Dispatch (St. Louis, Missouri, USA) of Saturday 24th November 1934 [page 2D, column 1]:
HOLLYWOOD, Nov. 23. An off hand, back-of-an-envelope check up would show that in Hollywood there are at least an even hundred thousand motion picture technical advisers—taking in the back seat cinema drivers and grandstand managers—who stand ready and waiting for the call to come from the studio to help make a big picture, one of those super epics the publicity departments love to write about.
5-: From the column The Prowler, by Dallas E. Wood, published in the Daily Palo Alto Times (Palo Alto, California, USA) of Thursday 20th December 1934 [page 1, column 1]:
WHEN IT COMES TO illegible handwriting, I vote myself the first prize without the formality of calling a competition. Not only is my chirography meaningless to others, but unless I transcribe my back-of-envelope pencil jottings into typewriting before my memory of their meaning has faded, the ideas will be forever lost upon me—and the world.
6-: From This Nerve-Racked World, by the Scottish journalist and author Ritchie Calder (1906-1982), published in the Daily Herald (London, England) of Thursday 5th December 1935 [page 12, column 5]:
To give a crude, back-of-an-envelope sketch of the nervous system with its countless millions of cells, fibres and nerve endings, one might compare it to a telephone system.
7-: From A Morning After A Night Of Paris Clubs, a short story by the U.S. author and socialite Mabel Herbert Urner (1881-1957), published in the Sunday Journal and Star (Lincoln, Nebraska, USA) of Sunday 13th June 1937 [Section C and D, page 6, column 5]:
The maid brought in the laundry. Waiting while Helen checked it over from a back-of-envelope list.
A large bundle. Traveling so fast, a fortnight’s accumulation.
8-: From the Buffalo Evening News (Buffalo, New York, USA) of Saturday 8th April 1939 [Saturday Magazine: page 5, column 3]—the reference is to the U.S. musician Walter Gross (1909-1967):
Scribbles Music On Envelope Backs
Sending his suits to the cleaners always is an exciting time for Walter Gross, radio conductor.
When he runs through his pockets before sending a suit out, he usually finds dozens of scraps of paper with bits of melody jotted down on them. Several of his original compositions have had such back-of-the-envelope beginnings including his “Cowbell Serenade” which Paul Whiteman played at Carnegie Hall.
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