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The expression Bastille Day designates the fourteenth of July, the national holiday of the French republic, commemorating the storming of the Bastille * on 14th July 1789 and the Fête de la Fédération, which was held on 14th July 1790 to celebrate the French Revolution and the unity of the French people. The fourteenth of July became an official holiday in 1880.
* The Bastille was a fortress in Paris, built in the 14th century, and used as a state prison until its destruction in 1789, at the beginning of the French Revolution.
Note: There is no French expression similar to Bastille Day. The French expressions that designate the national holiday of the French republic are le quatorze juillet, literally the fourteenth of July, and la fête nationale.
—Cf. Quatorze Juillet.
It seems that the expression Bastille Day was coined by the Scottish historian and political philosopher Thomas Carlyle (1795-1881)—cf., below, quotation 1. Incidentally, the fourteenth of July was dear to him, it being the birthday of his wife, the Scottish author Jane Baillie Carlyle (née Welsh – 1801-1866)—cf., below, quotations 2 & 5.
These are, in chronological order, the earliest occurrences of the expression Bastille Day that I have found:
—Quotations 1 & 4 refer to the assassination, on 13th July 1793, of the revolutionary leader and journalist Jean-Paul Marat (1743-1793) by Charlotte Corday (1768-1793):
1-: From The French Revolution: A History. In Three Volumes (London: James Fraser, 1837), by Thomas Carlyle [Vol. 3: The Guillotine; Book 4: Terror; Chapter 1: Charlotte Corday, page 235]:
It is yellow July evening, we say, the thirteenth of the month; eve of the Bastille day,—when ‘M. Marat,’ four years ago, in the crowd of the Pont Neuf, shrewdly required of that Besenval Huzzar-party, which had such friendly dispositions, “to dismount, and give up their arms, then;” and became notable among Patriot men!
2-: From a letter from Thomas Carlyle to his wife, dated London, Monday 13th July 1846, published in Thomas Carlyle: A History of his Life in London 1834–1881 (New York: Charles Scribner’s Sons, 1884), by James Anthony Froude [Vol. 1, Chapter 14, page 327]:
I send thee a poor little card-case, a small memorial of Bastille day, and of another day also very important to me and thee. My poor little Jeannie! no heart ever wished another more truly ‘many happy returns.’
3-: From Mirabeau: A Life-History. In Four Books (London: Smith, Elder and Co., 1848), by John Stores Smith [Vol. 2: Triumph!; Book 4, Chapter 4, page 59]—however, here, the expression Bastille Day refers to 14th July 1789, not to the national holiday:
He rescued from an attacking mob the Baron de Besenval, who was flying out of France. This gentleman had commanded the troops in Paris on and before the Bastille-day, and had ordered his troops to disperse a peaceable crowd; in doing which several women and an old man had been slaughtered, and for which act a prosecution was instituted against him.
4-: From Stabbed Paganism Struggling to Be Avenged, published in The Methodist New Connexion Magazine, and Evangelical Repository (London, England) of April 1865 [page 211]:
It was the eve of the great Bastille-day, in which all Republicans and Patriots rejoiced; and since that day four years Marat had gone through scenes which made him really worth seeing. But Charlotte Corday—for this was her name—“was a Republican before” that day, and wanted to see Marat, who, as we have said, was sick from Revolution work, and other maladies not to be written down here.
5-: From Reminiscences (London: Longmans, Green, and Co., 1881), by Thomas Carlyle, edited by James Anthony Froude [Vol. 2; Jane Welsh Carlyle, page 117]:
Some time in autumn 1800 (I think) the young Haddington doctor married; my wife, his first and only child, was born July 14 (Bastille-day, as we often called it) 1801.
6-: From the title of an article published in the San Francisco Chronicle (San Francisco, California, USA) of Thursday 14th July 1881 [page 2, column 3]:
BASTILLE DAY.
Some Facts About the Old Fortress Prison.
ARRESTS UNDER LETTRES DE CACHET.
A Picture of Life in the Gloomy Castle—Voltaire’s Experience—Exercises in This City To-Day.
7-: From Brieflets, published in the Boston Evening Transcript (Boston, Massachusetts, USA) of Monday 14th July 1884 [page 8, column 4]:
Today is Bastille Day, the ninety-fifth anniversary. This evening the French people of Boston will celebrate the anniversary by a banquet. It was at first intended to hold a picnic and fête, but on account of the comparatively small number of French people in the city, this plan was abandoned.
8-: From The London and China Telegraph (London, England) of Saturday 29th August 1885 [page 776, column 1]—the celebrations took place in Shanghai:
Bastille day was celebrated July 14 on the French Concession with all the usual honours and rejoicings. Innumerable tricoloured banners waived from the heads of flagstaffs along the Bund and principal streets; and at night the whole Concession was ablaze with thousands of coloured lanterns and gas illuminations. The decorations were for the most part almost identical with those of last year. The celebration began at eight o’clock in the morning, when the members of La Lyre Sociétè [sic] Musicale assembled in the garden of the old French Consulate and saluted the hoisting of the tri-colour with the strains of the French National Anthem. At nine o’clock M. Collin de Plancy, the Consul, held a reception in the great hall of the French Municipal Buildings, at which almost every French resident was present, as well as many foreigners of other nationalities. The hall had been handsomely decorated for the occasion. At night, when the lanterns and gas jets were lighted up, the French Bund presented a most brilliant spectacle. So far as the Chinese were concerned, there was very little disturbance; but a few foreigners disgraced themselves by picking a quarrel with the French police.