Earliest Uses of Grouping Symbols

Last revision: June 24, 1999


Vinculum below. The first use of the vinculum was in 1484 by Nicolas Chuquet (1445?-1500?) in his Le Triparty en la Science des Nombres. The bar was placed under the parts affected (Cajori vol. 1, pages 101 and 385). Chuquet wrote:

The above expression in modern notation is sqrt (14 + sqrt 180). This use of a vinculum appears to be the earliest use of a grouping symbol of any kind mentioned by Cajori.

Vinculum above. According to Cajori, the first use of the vinculum above the parts affected was by Frans van Schooten (c. 1615-1660), who "in editing Vieta's collected works, discarded the parentheses and placed a horizontal bar above the parts affected." In Van Schooten's 1646 edition of Vieta, is used to represent B(D2 + BD).

Ball (page 242) says the vinculum was introduced by Francois Vieta (1540-1603) in 1591. This information may be incorrect.

Grouping expressed by letters. In the late fifteenth century and in the sixteenth century various writers used letters or words to indicate grouping. The earliest use of such a device mentioned by Cajori (vol. 1, page 385) is the use of the letter v for vniversale by Luca Paciolo (or Pacioli) (c. 1445 - prob. after 1509) in his Summa of 1494 and 1523.

Parentheses. Parentheses ( ) are "found in rare instances as early as the sixteenth century" (Cajori vol. 1, page 390). Apparently the earliest work Cajori names in which round parentheses are found is General trattato di numeri e misure by Nicolo Tartaglia (c. 1506-1557) in 1556. Round parentheses occur once in Ars magna by Cardan, as printed in Opera (1663) (Cajori vol. 1, page 392; Cajori does not indicate whether the parentheses occur in the original 1545 edition).

Cajori (vol. 1, page 391) says that Michael Stifel (1487 or 1486 - 1567) does not use parentheses as signs of aggregation in his printed works, but that they are found in one of his handwritten marginal notes. Cajori expresses the opinion that these parentheses are actually punctuation marks rather than mathematical symbols.

Kline says parentheses appear in 1544. He presumably refers to Arithmetica integra by Michael Stifel.

Brackets. Brackets [ ] are found in the manuscript edition of Algebra by Rafael Bombelli (1526-1573) from about 1550 (Cajori vol. 1, page 391).

Ball (page 242) and Lucas say brackets were introduced by Albert Girard (1595-1632) in 1629. This information appears to be inaccurate.

Kline says square brackets were introduced by Vieta (1540-1603). He presumably refers to the 1593 edition of Zetetica, which according to Cajori uses both braces and brackets.

Braces. Braces { } are found in the 1593 edition of Francois Vieta's Zetetica (Cajori vol. 1, page 391).

Grouping symbols in numeration. In the writing of large numbers, various methods have been used to separate numerals into groups, including dots, vertical bars, commas, arcs, colons, and accent marks.

In 1202, Leonardo of Pisa in Liber Abaci directs that the hundreds, hundred thousands, hundred millions, etc., be marked with an accent mark above, and that thousands, millions, thousands of millions, etc., be marked with an accent below (Cajori vol. 1, page 58).

The earliest example of the modern system of simply separating the numeral into groups of three with commas shown by Cajori is in 1795 in the article "Numeration" in Mathematical and Philosophical Dictionary by Charles Hutton.


Front Page | Operation | Grouping | Relation | Fractions and Decimals | Constants | Variables | Functions | Geometry | Trigonometry | Calculus | Matrices and Vectors | Set Theory and Logic | Number theory | Statistics | Sources