The integral symbol:
The notation was introduced by the German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings; it first appeared publicly in the article "De Geometria Recondita et analysi indivisibilium atque infinitorum" (On a hidden geometry and analysis of indivisibles and infinites), published in Acta Eruditorum in June 1686. The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite sum of infinitesimal summands.
Typography in Unicode and LaTeX
The original IBM PC code page 437 character set included a couple of characters ⌠ and ⌡ (codes 244 and 245 respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they still remain in Unicode (U+2320 and U+2321 respectively) for compatibility.
Extensions of the symbol
Meaning Unicode LaTeX Double integral ∬ U+222C
Triple integral ∭ U+222D
Quadruple integral ⨌ U+2A0C
Contour integral ∮ U+222E
Clockwise integral ∱ U+2231 Counterclockwise integral ⨑ U+2A11 Clockwise contour integral ∲ U+2232
Counterclockwise contour integral ∳ U+2233
Closed surface integral ∯ U+222F
Closed volume integral ∰ U+2230
Typography in other languages
In other languages, the shape of the integral symbol differs slightly from the shape commonly seen in English-language textbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe) is upright, and the Russian variant leans slightly to the left to occupy less horizontal space.
By contrast, in German and Russian texts, the limits are placed above and below the integral symbol, and, as a result, the notation requires larger line spacing, but is more compact horizontally, especially when longer expressions are used in the limits:
- Gottfried Wilhelm Leibniz, Sämtliche Schriften und Briefe, Reihe VII: Mathematische Schriften, vol. 5: Infinitesimalmathematik 1674–1676, Berlin: Akademie Verlag, 2008, pp. 288–295 Archived 2021-10-09 at the Wayback Machine ("Analyseos tetragonisticae pars secunda", October 29, 1675) and 321–331 Archived 2016-10-03 at the Wayback Machine ("Methodi tangentium inversae exempla", November 11, 1675).
- Aldrich, John. "Earliest Uses of Symbols of Calculus". Retrieved 20 April 2017.
- Swetz, Frank J., Mathematical Treasure: Leibniz's Papers on Calculus – Integral Calculus, Convergence, Mathematical Association of America, retrieved February 11, 2017
- Stillwell, John (1989). Mathematics and its History. Springer. p. 110.
- "Mathematical Operators – Unicode" (PDF). Retrieved 2013-04-26.
- "Supplemental Mathematical Operators – Unicode" (PDF). Retrieved 2013-05-05.
- "Russian Typographical Traditions in Mathematical Literature" (PDF). giftbot.toolforge.org. Archived from the original (PDF) on 28 September 2012. Retrieved 11 October 2021.
- Stewart, James (2003). "Integrals". Single Variable Calculus: Early Transcendentals (5th ed.). Belmont, CA: Brooks/Cole. p. 381. ISBN 0-534-39330-6.
- Zaitcev, V.; Janishewsky, A.; Berdnikov, A. (1999), "Russian Typographical Traditions in Mathematical Literature" (PDF), Russian Typographical Traditions in Mathematical Literature, EuroTeX'99 Proceedings